Distributed Space-Time Systems

EURASIP Journal on Advances in Signal Processing

Distributed Space-Time Systems Guest Editors: Ranjan K. Mallik, Arogyaswami J. Paulraj, Mrityunjoy Chakraborty, Keith Q. T. Zhang, and George K. Karagiannidis

Distributed Space-Time Systems


Distributed Space-Time Systems Guest Editors: Ranjan K. Mallik, Arogyaswami J. Paulraj, Mrityunjoy Chakraborty, Keith Q. T. Zhang, and George K. Karagiannidis

Copyright © 2008 Hindawi Publishing Corporation. All rights reserved. This is a special issue published in volume 2008 of “EURASIP Journal on Advances in Signal Processing.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Editor-in-Chief Phillip Regalia, Institut National des Télécommunications, France

Associate Editors Kenneth E. Barner, USA Richard J. Barton, USA Kostas Berberidis, Greece J. C. M. Bermudez, Brazil A. Enis Çetin, Turkey Jonathon Chambers, UK Liang-Gee Chen, Taiwan Huaiyu Dai, USA Satya Dharanipragada, USA Florent Dupont, France Frank Ehlers, Italy S. Gannot, Israel Fulvio Gini, Italy M. Greco, Italy Irene Y. H. Gu, Sweden Fredrik Gustafsson, Sweden Ulrich Heute, Germany Arden Huang, USA Jiri Jan, Czech Republic Magnus Jansson, Sweden

Sudharman K. Jayaweera, USA Søren Holdt Jensen, Denmark Mark Kahrs, USA Moon Gi Kang, South Korea W. Kellermann, Germany Joerg Kliewer, USA Lisimachos P. Kondi, Greece Alex Chichung Kot, Singapore C.-C. Jay Kuo, USA Tan Lee, China Geert Leus, The Netherlands T.-H. Li, USA Mark Liao, Taiwan Y.-P. Lin, Taiwan S. Makino, Japan Stephen Marshall, UK C. F. Mecklenbräuker, Austria Gloria Menegaz, Italy Ricardo Merched, Brazil Marc Moonen, Belgium

Vitor Heloiz Nascimento, Brazil Sven Erik Nordholm, Australia D. O’Shaughnessy, Canada Bjorn Ottersten, Sweden Wilfried Philips, Belgium Aggelos Pikrakis, Greece Ioannis Psaromiligkos, Canada Markus Rupp, Austria William Allan Sandham, UK B. Sankur, Turkey Dirk Slock, France Y.-P. Tan, Singapore George S. Tombras, Greece Dimitrios Tzovaras, Greece Jacques G. Verly, Belgium Bernhard Wess, Austria Jar-Ferr Kevin Yang, Taiwan Azzedine Zerguine, Saudi Arabia A. M. Zoubir, Australia

Contents Distributed Space-Time Systems, Ranjan K. Mallik, Arogyaswami J. Paulraj, Mrityunjoy Chakraborty, Keith Q. T. Zhang, and George K. Karagiannidis Volume 2008, Article ID 891036, 3 pages Distributed Space-Time Block Coded Transmission with Imperfect Channel Estimation: Achievable Rate and Power Allocation, Leila Musavian and Sonia A¨ıssa Volume 2008, Article ID 471327, 9 pages Cooperative Multibeamforming in Ad Hoc Networks, Chuxiang Li and Xiaodong Wang Volume 2008, Article ID 310247, 11 pages NAF, OAF, or Noncooperation: Which Protocol to Choose?, Ahmed Saadani and Olivier Traoré Volume 2008, Article ID 546470, 7 pages How to Solve the Problem of Bad Performance of Cooperative Protocols at Low SNR, Charlotte Hucher, Ghaya Rekaya-Ben Othman, and Jean-Claude Belfiore Volume 2008, Article ID 243153, 7 pages Censored Distributed Space-Time Coding for Wireless Sensor Networks, S. Yiu and R. Schober Volume 2008, Article ID 127689, 9 pages Code Design for Multihop Wireless Relay Networks, Frédérique Oggier and Babak Hassibi Volume 2008, Article ID 457307, 12 pages Link-Adaptive Distributed Coding for Multisource Cooperation, Alfonso Cano, Tairan Wang, Alejandro Ribeiro, and Georgios B. Giannakis Volume 2008, Article ID 352796, 12 pages Diversity Analysis of Distributed Space-Time Codes in Relay Networks with Multiple Transmit/Receive Antennas, Yindi Jing and Babak Hassibi Volume 2008, Article ID 254573, 17 pages On the Duality between MIMO Systems with Distributed Antennas and MIMO Systems with Colocated Antennas, Jan Mietzner and Peter A. Hoeher Volume 2008, Article ID 360490, 9 pages Power-Efficient Relay Selection in Cooperative Networks Using Decentralized Distributed Space-Time Block Coding, Lu Zhang and Leonard J. Cimini Jr. Volume 2008, Article ID 362809, 10 pages Low-Complexity Distributed Multibase Transmission and Scheduling, Hilde Skjevling, David Gesbert, and Are Hjørungnes Volume 2008, Article ID 741593, 9 pages

Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2008, Article ID 891036, 3 pages doi:10.1155/2008/891036

Editorial Distributed Space-Time Systems Ranjan K. Mallik,1 Arogyaswami J. Paulraj,2 Mrityunjoy Chakraborty,3 Keith Q. T. Zhang,4 and George K. Karagiannidis5 1 Department

of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India of Electrical Engineering, Stanford University, 232 Packard, 350 Serra Mall, Stanford, CA 94305, USA 3 Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, India 4 Department of Electronic Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong 5 Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54 124 Thessaloniki, Greece 2 Department

Correspondence should be addressed to Ranjan K. Mallik, [email protected] Received 8 January 2008; Accepted 8 January 2008 Copyright © 2008 Ranjan K. Mallik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Space-time or multiple-input multiple-output (MIMO) wireless communication, which exploits multiple antennas at the transmitter and the receiver nodes, has been a rich and exciting research area. This technology has now matured and is now seeing widespread applications such as in WLANs (IEEE 802.11n) and WMANs/mobile internet (IEEE 802.16e, 3GPP LTE, and UMB). MIMO offers significantly higher throughputs and increased link reliability. Millions of MIMO-enabled wireless devices are now being shipped annually and this number can reach hundreds of millions in a few years. For a variety of reasons including size and power limitations, wireless nodes (both user terminals and base stations) can only support a limited number of antennas. Therefore, the ability to use the antennas of other nodes in the vicinity can be very attractive. The large geographic antenna separation implied in using distributed nodes can often also significantly improve spatial multiplexing performance (due to better channel conditioning) and improve diversity performance (due to stronger fade decorrelation). Cooperative or distributed space-time techniques seek to realize the MIMO leverage by exploiting antennas spread across nodes. Cooperation may be used in transmission or in reception, or at both ends. Since such cooperating nodes need to communicate wirelessly (unlike in simple MIMO), the problem becomes significantly more complex and intellectually richer. Cooperative or distributed space-time techniques can be used in relay networks where multiple relay nodes (and optionally, source or sink nodes) can form a virtual antenna

array for MIMO operation. Similar opportunities can also arise in sensor networks. Another rich area is cellular systems where multiple user terminals can cooperate for up or down link MIMO operation. Likewise, multiple base stations can also cooperate. In many of these applications, the geographic separation of user terminals or base stations is the key advantage being exploited. In other applications, the increase in the number of effective antennas can boost the MIMO advantage. Cooperative MIMO is already supported in WIMAX 802.16e standard and has been field tested with marked success. Distributed or cooperative space-time methods raise a number of important questions: what is the network capacity? How does this capacity scale with different numbers of nodes and antennas? What are good space-time codes for distributed antennas? What are the channel models for this problem? What are good strategies for channel estimation? How do we do precoding? Can we use cross-layer information for scheduling? Does distributed space-time coding (DSTC) work well with OFDMA as it does with standard MIMO systems? How can we develop cooperative communication schemes to suit the low power needs of sensor networks? Cooperative space-time communications research has begun to attract significant attention. Many opportunities now exist to include this technology in the next generation broadband wireless systems. This special issue aims at capturing the state of the art in this emerging area. We received a total of fifteen submissions, and after a rigorous review

2 process, a total of eleven papers have been selected. They cover topics ranging from systems design, DSTC, applications to sensor networks, and routing. We hope this collection will be a significant contribution to the growth of this important field. The first four papers focus on system design and optimization. The paper “Distributed space-time block coded transmission with imperfect channel estimation: achievable rate and power allocation” by L. Musavian and S. Aissa investigates the effects of channel estimation error at the receiver on the achievable rate of distributed space-time block coding. The authors present efficient bounds on the mutual information of distributed space-time block codes (DSTBCs) when the channel gains and channel estimation error variances pertaining to different transmitter/receiver links are unequal. Furthermore, they propose an optimum power transmission strategy to achieve the outage capacity lower bound of DSTBCs under arbitrary number of transmit and receive antennas and provide closed-form expressions for this capacity metric. The next paper “Cooperative multibeamforming in ad hoc networks” by C. Li and X. Wang considers a scenario with multiple source nodes cooperatively forming multiple data-carrying beams toward multiple destination nodes. An iterative transmit power allocation algorithm is proposed under fixed beamformers subject to the maximal transmit power constraint as well as the minimal receive signal-to-interference-plus-noise ratio and receive power constraints. In addition, a joint optimization algorithm is developed to iteratively optimize the powers and the beamformers. Further, since channel state information (CSI) is required by the sources to perform this optimization, a cooperative scheme is proposed to implement a simple CSI estimation and feedback mechanism. The next paper “NAF, OAF, or noncooperation: which protocol to choose?” by A. Saadani and O. Traore addresses the problem of choosing between communicating without cooperation or by using one of the two well-known amplify and forward (AF) cooperative protocols, orthogonal or nonorthogonal. This problem is translated to a power sharing problem on the cooperation frame between source and relays, aiming to maximize the short-term channel capacity. The obtained solution shows that the cooperative protocol choice depends only on the available power at the relays. Furthermore, an efficient power allocation scheme is proposed, where relay selection improves the outage probability compared to the selective orthogonal and nonorthogonal protocols, with a significant capacity gain. The paper “How to solve the problem of bad performance of cooperative protocols at low SNR?” by C. Hucher et al. proposes some new adaptive AF and decode and forward (DF) protocols using a selection criterion which is a function of the instantaneous capacities of all possible transmission schemes (with or without cooperation). Results show that the adaptive cooperation protocols compensate for the performance degradation of cooperation protocols at low signal-to-noise ratio (SNR). The next three papers are on coding for distributed systems. The paper “Censored distributed space-time coding for wireless sensor networks” by S. Yiu and R. Schober deals with sensors using a common noncoherent DSTBC to for-

EURASIP Journal on Advances in Signal Processing ward their local decisions to a fusion center (FC) which makes the final decision. To overcome the problem of error propagation, censored DSTC, where only reliable decisions are forwarded to the FC, is proposed. Based on the performance analysis of a low complexity suboptimal decision rule, a gradient algorithm for optimization of the local decision/censoring threshold is derived. The next paper “Code design for multihop wireless relay networks” by F. Oggier and B. Hassibi considers an elaborate version of AF, where the relay nodes multiply their received signal with a unitary matrix, such that the receiver senses a space-time code. A full diversity condition is obtained for such codes using the rank criterion. A systematic way of constructing such codebooks with full diversity is also presented. The paper “Linkadaptive distributed coding for multisource cooperation” by A. Cano et al. presents a new protocol capable of achieving a diversity order up to the number of cooperating users and large coding gains. The diversity order is expressed as a function of the rank properties of the distributed coding strategy employed: a result analogous to the diversity properties of colocated multiantenna systems. The particular case of distributed complex field coding emerges as an attractive choice because of its high rate, full spatial diversity, and relaxed synchronization requirements. The next two papers deal with performance analysis. The paper “Diversity analysis of distributed space-time codes in relay networks with multiple transmit/receive antennas” by Y. Jing and B. Hassibi extends the concept of DSTC to wireless relay networks with multiple-antenna nodes and analyzes the pairwise error probability at high SNR. The paper “On the duality between MIMO systems with distributed antennas and MIMO systems with colocated antennas” by J. Mietzner and P. A. Hoeher investigates the loss in the capacity and error performance in wireless systems with individual transmit or receive antennas spatially distributed on a large scale. It is shown that owing to a strong duality between MIMO systems with colocated antennas (and spatially correlated links) and MIMO systems with distributed antennas (and unequal average link SNRs), the two systems can be treated in a single, unifying framework. The final two papers are on routing and transmission. The paper “Power-efficient relay-selection in cooperative networks using decentralized distributed space-time block coding” by L. Zhang and L. J. Cimini, Jr. presents a powerefficient relay selection strategy for decentralized distributed space-time block coding in a selective DF cooperative network, and by applying this idea to each relaying hop in a multihop network, a power-efficient hop-by-hop routing strategy is proposed. The paper “Low complexity distributed multibase transmission and scheduling” by H. Skjevling et al. addresses the problem of base station coordination and cooperation in multicell wireless networks. A distributed approach to downlink multibase beamforming, which allows for the multiplexing of multiple user terminals randomly located in a network with multiple base stations, is presented. It is shown that this scheme yields significant gains, when compared to schemes that do not allow cooperation between cells, without the extensive signaling overhead required in previously known multicell MIMO processing.

Ranjan K. Mallik et al.

3

ACKNOWLEDGMENTS Guest editors are grateful to all the authors who contributed their high-quality papers to this special issue. They thank all the reviewers who took time and consideration to assess the submitted manuscripts. They also acknowledge the Editorin-Chief and the Editorial Board members of the journal on Advances in Signal Processing for their support of this special issue. Ranjan K. Mallik Arogyaswami J. Paulraj Mrityunjoy Chakraborty Keith Q. T. Zhang George K. Karagiannidis


Research Article Distributed Space-Time Block Coded Transmission with Imperfect Channel Estimation: Achievable Rate and Power Allocation Leila Musavian and Sonia A¨ıssa INRS-EMT, University of Quebec, Montreal, QC, Canada Correspondence should be addressed to Leila Musavian, [email protected] Received 2 May 2007; Accepted 27 August 2007 Recommended by R. K. Mallik This paper investigates the effects of channel estimation error at the receiver on the achievable rate of distributed space-time block coded transmission. We consider that multiple transmitters cooperate to send the signal to the receiver and derive lower and upper bounds on the mutual information of distributed space-time block codes (D-STBCs) when the channel gains and channel estimation error variances pertaining to different transmitter-receiver links are unequal. Then, assessing the gap between these two bounds, we provide a limiting value that upper bounds the latter at any input transmit powers, and also show that the gap is minimum if the receiver can estimate the channels of different transmitters with the same accuracy. We further investigate positioning the receiving node such that the mutual information bounds of D-STBCs and their robustness to the variations of the subchannel gains are maximum, as long as the summation of these gains is constant. Furthermore, we derive the optimum power transmission strategy to achieve the outage capacity lower bound of D-STBCs under arbitrary numbers of transmit and receive antennas, and provide closed-form expressions for this capacity metric. Numerical simulations are conducted to corroborate our analysis and quantify the effects of imperfect channel estimation. Copyright © 2008 L. Musavian and S. A¨ıssa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

An effective way of approaching the promised capacity of multiple-input multiple-output (MIMO) systems is proved to be through space-time coding, which is a powerful technique for achieving both diversity and coding gains over MIMO fading channels [1]. Orthogonal space-time block codes (O-STBCs) that can extract the spatial diversity gains are specially attractive since they drastically simplify maximum likelihood (ML) decoding by decoupling the vector detection problem into simpler scalar detection problems [2, 3], thus yielding a process that can be viewed as an orthogonalization of the MIMO channel [4, 5]. The use of the MIMO technology along with STBCs is becoming increasingly popular in different wireless systems and networks. Specifically, in sensor and ad hoc networks where nodes are generally limited in terms of the number of antenna elements that can be implemented at the equipment, benefiting from the MIMO technology calls for cooperation

between nodes so as to form MIMO antenna arrays in a distributed fashion, and yield the sought for gains of MIMO under space-time block coding. Recently, there has been increasing interest in distributed space-time coded transmissions which employ STBCs in a cooperative fashion. Indeed, space-time coded cooperative diversity provides an effective way for relaying signals to the end user by multiple disjoint wireless terminals [6]. Cooperative transmit diversity is of particular advantage in sensor networks, where multiple transmit nodes collect information of the same kind and individually transmit the corresponding signals to a given destination, for example, multiple thermal sensors can measure temperature and transmit this information to a device that controls the desired temperature in the space where it operates. These nodes can be deployed to employ distributed STBCs (D-STBCs) in order to cooperatively achieve transmit diversity gains. This is particularly attractive when the links between the transmitting nodes and the receiver (referred to here as

2 subchannels) are of different quality, for instance, when a subset of transmitters are required to be positioned at specific locations, for example, sensors measuring the humidity of the soil in a dense environment, wherein not all transmitting nodes can have line-of-sight (LOS) with the receiver. Performance of D-STBCs with unequal subchannel gains has been investigated in [7] in terms of the outage probability. On the other hand, a memoryless precoder for D-STBCs in MIMO channels with joint transmit-receive correlation is provided in [8]. However, the analyses in [7, 8] rely on the availability of perfect state knowledge of all subchannels; an assumption which is hard to obtain in practice, whether the multiple-antenna configuration provides a MIMO link or is created in a distributed way. In addressing the effect of imperfect channel knowledge in single-input single-output (SISO) and MIMO configurations, recent information-theoretical studies assume different channel state information (CSI) uncertainties at the receiver. For instance, lower and upper bounds on the capacity of SISO channels under imperfect CSI at the receiver, with and without feedback to the transmitter, are provided in [9]. In [10], the capacity in the presence of channel estimation error at the receiver is evaluated when a fixed modified nearest neighbor decoding rule is employed. The same approach has been taken in [11, 12] for MIMO systems with independent and identically distributed (i.i.d.) Rayleigh fading channels. In particular, it has been proven that spatio-temporal water-filling is the optimal power allocation strategy that achieves the capacity lower bound [11]. In addition, the performance of space-time coding in the presence of channel estimation error is studied in [13– 15]. In particular, closed-form expressions for the pairwise error probability (PEP) of space-time codes in Rayleigh flatfading channels have been obtained in [15]. In this paper, we address the effects of channel estimation error at the receiver on the performance of D-STBCs. In particular, we derive lower and upper bounds on the mutual information for Gaussian input signals, and present a limiting value that upper bounds the gap between these bounds at any input transmit powers. We further show that the gap between the mutual information bounds increases as the disparity between the subchannel estimation error variances increases. In addition, assuming that the summation of the subchannel gains remains constant, we provide the information for positioning the receiving node so as to maximize the mutual information bounds of D-STBCs. Furthermore, we provide the power allocation scheme that achieves the outage capacity lower bound of D-STBCs, and derive closed-form expressions for this capacity metric and its associated power allocation. In detailing these contributions, the remainder of this paper is organized as follows. In Section 2, the system and channel models are introduced. Lower and upper bounds on the mutual information under channel estimation error for D-STBCs in Rayleigh fading channels are derived in Section 3. The tightness of these bounds is also analyzed in Section 3. Section 4 investigates the location of the receiver that maximizes the mutual information bounds, when the summation of the channel gains is constant. In Section 5,

EURASIP Journal on Advances in Signal Processing closed-form expressions for the lower bound on the outage capacity of D-STBCs are derived. Finally, sample numerical results are presented in Section 6 followed by the paper’s conclusion. 2.

SYSTEM AND CHANNEL MODELS

Throughout this paper, we use the upper-case boldface letters for matrices and lower-case boldface letters for vectors. AT , AH , |A|, and A2F indicate the transpose, Hermitian transpose, determinant, and Frobenius norm of matrix A, respectively. In stands for an n × n identity matrix, and the matrix (pseudo) inverse is denoted by [ · ]−1 . E [x] denotes the expectation of the random variable x, abs(x) indicates the absolute value of x, and x∗ its conjugate value. We consider a wireless communication system employing nT transmitters, each equipped with a single antenna, and a receiver equipped with nR receive antennas in a flat-fading environment. A linear model relates the nR ×1 received vector y to the signals sent from the nT transmitting nodes, that is, xi for i = 1, . . . , nT , via y=

nT

hi xi + n,

(1)

i=1

where the entries of n represent the zero-mean complex Gaussian noise with independent real and imaginary parts of equal power, and hi , i = 1, . . . , nT , indicate the channel transfer vector between the ith transmitter and the receiver. The elements of the nR × 1 channel transfer vectors, hi , i = 1, . . . , nT , are assumed to be independent zero-mean circularly symmetric complex Gaussian (ZMCSCG) random variables with variances γi , . . . ,γnT , referred to as channel gains. Furthermore, we assume that the receiver performs minimum mean square error (MMSE) estimation of hi , i = i + ei , where by the property of 1, . . . , nT , such that hi = h i and ei are uncorrelated. The elements of MMSE estimation h ei , i = 1, . . . , nT , are independent ZMCSCG random variables with variance σi2 . Finally, the average transmit power from each transmitter is constrained to P, and it is assumed that the transmitters cooperate to provide a distributed spacetime block encoder, and that the channel coefficients remain constant during the transmission of a space-time codeword. 3.

MUTUAL INFORMATION BOUNDS

We start by deriving lower and upper bounds on the mutual information of the distributed system employing Alamouti codes [3], when the receiver is equipped with two antennas. Generalization to a system with nT > 2 and nR > 2 follows. We assume that the signals at the input of the subchannels are independent Gaussian distributed, which is not necessarily the capacity achieving distribution when CSI at the receiver is not perfect [9]. The Alamouti scheme transmits symbols x1 and x2 from the first and second transmitters, respectively, during the first symbol period, while symbols −x2∗ and x1∗ are transmitted from the first and second transmitters during the second

L. Musavian and S. A¨ıssa

3

symbol period, respectively. The channels between the distributed transmitters and the receiver remain unchanged during these two symbol periods. Let us define vectors y1 and y2 as the received vectors at the first and second time periods. The receiver forms a rearranged signal vector y as y = [y1 y2∗ ]T that can be expressed as

(6), can be seen as the variance of an additive white Gaussian noise (AWGN). Furthermore, by following similar steps as in (2) to (7), one can find the mutual information lower and upper bounds of D-STBCs with arbitrary numbers of transmit and receive antennas such that

y = H eff x + Eeff x + n,

(2)

Clower = RE log2

T

where n = [ n1 n2 n∗3 n∗4 ] is the vector of noise samples, T x = [ x1 x2 ] , and the effective channel estimation and error matrices are given by ⎛

h11

H eff

h12

⎞

⎜ ⎟ ⎜h 22 ⎟ ⎜ 21 h ⎟ ⎟, =⎜ ∗ ⎜h ⎟ 12 −h ∗ 11 ⎠ ⎝ ∗ ∗ h22 −h21

⎛

z=

H + H eff

(3)

Eeff x + n ,

H

Clower

2

H −1 1 eff Eeff x 2F InR+cov H 2F σn2 H , = E log2 InR+P H nR Cupper 2

H 1 eff Eeff x 2F InR + cov H 2F + σn2 H = E log2 P H nR

H eff Eeff x|x −1 2F InR +cov H , × σn2 H

(5) H

where cov(H eff Eeff x) indicates the covariance matrix of the H H random vector H eff Eeff x, and cov(H eff Eeff x|x) denotes the H covariance matrix of the random vector H eff Eeff x given x. Then, inserting Eeff and H eff (3) into (5), one can derive the mutual information bounds and express them according to

Clower = E log2 1 + P

Cupper = E log2

2 H

2F

σn2 + P σ1 + σ22

Cupper = RE log2

, nT

2F + R σn2 + P σ2 1 P H i=1 i nT R σn2 + P σi2 X2i

,

i=1

(8)

where R denotes the communication rate of the STBC. We now investigate the tightness of the obtained lower and upper bounds on the mutual information to justify that they represent a good estimate of the true Gaussian mutual information. Define Δ as the gap between the mutual information bounds:

2F + σn2 + P σ12 + σ22 P H

σn2 + P σ12 X21 + σ22 X22

,

(6)

,

(7)

where X2i , i ∈ {1,2}, is a chi-squared random variable with two degrees of freedom and E [X2i ] = 1. Note that the term P(σ12 +σ22 ), appearing in the mutual information lower bound

Δ = RE log2

(4)

where the vector n = H eff n is zero-mean with covariance H matrix E [n n ] = σn2 H2F I2 . The lower and upper bounds on the mutual information can now be derived by adopting a similar approach as used in [11] yielding

2 P H 1 1+ nFT R σn2 + P σ2 i=1 i

⎞

e11 e12 ⎜ ⎟ ⎜e21 e22 ⎟ ⎟ Eeff = ⎜ ⎜e∗ −e∗ ⎟ . ⎝ 12 11 ⎠ ∗ ∗ e22 −e21

Note that the effective channel estimation is an orthogonal matrix. Then, the receiver multiplies the received vector y with the Hermitian transpose of H eff to obtain 2F I2 x H

σn2 + P σn2 + P

nT

nTi=1

σi2

σ 2 X2i i=1 i

,

(9)

then an upper bound on Δ at high transmit powers can be derived by adopting similar approach to that in [16] as follows:

lim Δ ≤ R·min

P →∞ nT →∞

σ2 1 ε , + log2 max 2 ln 2 2nT ln 2 σmin

,

(10)

2 2 where σmin and σmax are the minimum and maximum values 2 of σi for i = 1, . . . , nT , respectively, and ε = 0.577216 is the Euler-Mascheroni constant [17]. Furthermore, the gap between the mutual information bounds is shown to increase monotonically as a function of the input transmit power [18]; hence, Δ does not exceed the right-hand side of (10), or equivalently, the mutual information bounds are fairly close at any input transmit powers. We now assume that the receiver can estimate the channels pertaining to different transmitters with the same accuracy, that is, σ12 = · · · = σn2T σe2 . In this case, the gap between the mutual information bounds can be shown to be upper bounded by limP→∞, nT →∞ Δ ≤ R/(2nT ln 2), which shows that the gap between the mutual information bounds decreases as the number of transmitters increases. Proceeding with our investigation about the gap between the mutual information bounds, we now provide the following lemma.

Lemma 1. The gap between the bounds on the mutual information of distributed Alamouti codes with unequal channel error variances increases monotonically as the disparity between the error variances increases. Proof. Consider that the channel error variances σ12 and 2 2 − αe and σsum + αe . The σ22 are respectively given by σsum

4


gap between the mutual information bounds, Δ, can be simplified to

Δ = RE log2

2 σn2 +Pσsum 2 2 − α X + σ 2 +α X2 σn2 +P σsum e e 1 2 sum

. (11)

We now find the first partial derivative of Δ with respect to αe and prove that Δ is an increasing function of αe . We proceed as follows:

P X21 − X22 R ∂Δ

. = E 2

2 2 +α X2 ∂αe ln 2 σn +P σsum − αe X21 + σsum e 2 (12) Then, by using the fact that X21 and X22 are i.i.d. random variables, we can show that ∂Δ/∂αe |αe =0 = 0. Furthermore, one can now derive the second partial derivative of Δ with respect to αe which leads to ∂2 Δ/∂α2e ≥ 0. This implies that ∂Δ/∂αe is an increasing function of αe , hence, ∂Δ/∂αe ≥ 0 for 0 ≤ αe ≤ 1, which concludes the proof. 4.

define 0 ≤ α2 ≤ 1 such that γ2 = β(1 − α1 )(1 + α2 ) and γ3 = β(1 − α1 )(1 − α2 ). We can then simplify the mutual information lower bound (8) as follows:

Clower = RE log2

P γ 1 w 1 + γ 2 w 2 + γ 3 w 3 − σe 1+ R σn2 + 3Pσe2

2

3 i=1

wi

= RE log2 1 + Q ,

(13) where Q = a((1 + 2α1 )w1 + (1 − α1 )(1 + α2 )w2 + (1 − α1 )(1 − α2 )w3 − σe2 /β 3i=1 wi ), σe2 represents the channel estimation error variance, wi , i = 1, . . . , 3, are i.i.d. random variables according to Rayleigh distribution with unit variances, and a = Pβ/(R(σn2 + 3Pσe2 )). We need to prove that Clower is at its maximum when α1 = 0 and α2 = 0. We start by deriving the first and second partial derivatives of Clower with respect to α2 :

a 1 − α1 w2 − w3 R ∂Clower = E ∂α2 ln 2 1+Q

a 1 − α1 w2 − w3 ∂2 Clower R =− E ∂α2 2 ln 2 1+Q

,

(14)

2

.

(15)

OPTIMUM POSITIONING

In the communication system under consideration, we now assume that the transmitters are fixed in their position and that the receiver can estimate the channel gains pertaining to different transmitters with the same accuracy, and investigate the best position for the receiving node. Our transmitters can be sensor nodes placed, for example, at the corners of a room, and we investigate the best location of the receiver collecting data from these nodes, where we assume that nodes cooperate to provide a distributed space-time block coded transmission. In particular, we assume that when the channel gains pertaining to a subset of transmitter-receiver links improve, the gains of the rest of the subchannels degrade such that the summation of all gains remains constant, and provide the following lemma. Lemma 2. The mutual information bounds of D-STBCs are maximum when the channel gains pertaining to different transmitter-receiver links are equal, as long as the summation of these gains remains constant. Proof. The proof for this lemma can be obtained by adopting a similar approach as proposed in [19]. For completeness, we provide here the proof for a system with nT = 3 transmitting nodes and a single receive antenna. We refer to the channel gains by γ1 , γ2 , and γ3 , and define the constant 3β as the summation of these variances, that is, 3i=1 γi = 3β. Since the channel gains are real positive numbers, then at least one of them is bigger than or equal to β. Without loss of generality, we assume that γ1 ≥ β and define 0 ≤ α1 ≤ 1 such that γ1 = β(1 + 2α1 ). Hence, summation of the two remaining channel gains, γ2 and γ3 , can be found as γ2 + γ3 = 2β(1 − α1 ). Furthermore, we

Observe that the second derivative of Clower with respect to α2 (15) is nonpositive, therefore, the maximum on ∂Clower /∂α2 (14) occurs at α2 = 0, irrespective of α1 . Furthermore, by adopting similar steps as in [16], one can show that ∂Clower /∂α2 |α2 =0 = 0, which proves that the maximum on the mutual information lower bound occurs at α2 = 0 for any value of α1 . Note that since abs(∂Clower /∂α2 ) increases monotonically as a function of α2 , then not only the mutual information lower bound is at its maximum when α2 = 0, but, its robustness to the variations of α2 is also maximum at this point. We now prove that the maximum of Clower |α2 =0 occurs at α1 = 0. For this purpose, we define the function f (α1 ) = Clower |α2 =0 given by

f α1

= RE log2 1 + Q ,

(16)

where 3Q = a((1 + 2α1 )w1 + (1 − α1 )w2 + (1 − α1 )w3 − 2 σe /β i=1 wi ). Then, by obtaining the first and second derivatives of f (α1 ) with respect to α1 , one can show that ∂2 f (α1 )/∂α21 ≤ 0 and ∂ f (α1 )/∂α1 |α1 =0 = 0, hence, the maximum of f (α1 ) occurs at α1 = 0. Therefore, the maximum of Clower occurs at α1 = 0 and α2 = 0. In addition, since the gap between the mutual information bounds, Δ, does not depend on the variations of channel gains, then the mutual information upper bound is also maximum at α1 = 0 and α2 = 0; which concludes the proof.

According to the above analysis, one can conclude that the best position for the receiving node is the one that provides the condition of having equal subchannel gains. For instance, when the distributed transmit antennas are located


5

in the corners of a room, the best position for the receiving node is the center of the room, under the condition that the summation of the subchannel gains remains constant. 5.

OUTAGE CAPACITY

In the following, we assume that the transmitters, considered to cooperate to provide a distributed STBCed transmission, can adaptively change their input power according to the channel variations. The transmitting nodes use the same input power level, which can be calculated at the receiver that has access to the state information of each subchannel. The receiver then broadcasts the information about the required transmit power level, and the transmitters adapt their input power according to this information. Here, we investigate the adaptive power allocation scheme that achieves the outage capacity lower bound of the channel. Outage capacity is the maximum constant-rate that can be achieved with an outage probability less than a certain threshold [20, 21]. In this case, the transmitters invert the channel fading so as to maintain a constant power at the receiver. Using channel inversion, the capacity of fading channels and its closed-form expressions have previously been derived in [22, 23], respectively. This metric corresponds to the capacity that can be achieved in all fading states while meeting the power constraint. However, in extreme fading cases, for example, Rayleigh fading, this capacity is zero as the transmitter has to spend a huge amount of power for channel states in deep fade to achieve a constant rate. To alleviate this problem, an adaptive transmission technique, referred to as truncated channel inversion with fixed rate (tifr), which can achieve nonzero constant rates, was introduced in [22]. This technique maintains a constant received-power for channel fades above a given cutoff depth. Recalling that channel inversion technique provides a constant received power at the receiver such that 2F /(σn2 + PΣni=T1 σi2 )) = α, we can find the power (1/R)(P H allocation for the system with D-STBCs and imperfect channel estimation at the receiver according to

P=

αRσn2

+

nT 2F − αR H σ2 i=1 i

,

fλ (λ) =

g ui

(18)

(19)

1 Ki, j = u − j ui − j ! − γi + σi2 i ∂ui − j × ui − j ∂s

g

1−

γk − σk2

−uk

s

k=1,k= /i

. s=1/(γi −σi2 )

(20) We can then obtain ∞ a closed-form solution for the probability Pr{λ ≥ λ0 } = λ0 fλ (λ)dλ as follows:

λ j −1 −λ/(γi −σi2 ) , e 2 j Γ( j) γi − σi

where Γ(·) is the Gamma function [24], and the coefficients Ki, j are given by

Pr λ ≥ λ0 =

2F ≥ λ0 , Pr H

Ki, j

i=1 j =1

(17)

where the constant value for α is found such that the transmit-power constraint is satisfied and [x]+ denotes max{0, x}. We assume that the transmission is suspended for channel gains below a cutoff threshold λ0 such that the outage probability Pout is satisfied. Note that, at the same time, the transmission is suspended for channel gains smaller than H 2F ≤ αR ni=T1 σi2 ; hence, the acceptable value for α is limited to α ≤ λ0 /(R ni=T1 σi2 ). Therefore, the lower bound on the outage capacity can be obtained as λ0 Cout = R log2 1 + min α, nT 2 R i=1 σi

where Pr{H 2F ≥ λ0 } = 1 − Pout indicates the probability that the inequality H 2F ≥ λ0 holds true. It is worth noting that the expression derived in (18) does not represent the true channel outage capacity. However, one can guarantee that by using the power allocation scheme in (17), at least a minimum constant-rate according to (18) can be achieved by D-STBCs with imperfect CSI at the receiver. Also, recalling that the mutual information bounds (8) are proved to be tight at any input transmit powers, we conclude that (18) represents a good estimate for the true channel outage capacity. Hereafter, we use the parameter λ = H 2F for the ease of notation. To obtain a closed-form expression for the outage capacity, we start by deriving a closed-form expression for Pr{λ ≥ λ0 }. We proceed by defining ui as the number of transmitters with equal γi − σi2 and choose g such that g i=1 ui = nT . Without loss of generality, we assume that 2 γl − σl2 = / γk − σk for l = 1, . . . , g and k = 1, . . . , g, having k= / l. The probability density function (PDF) of λ, fλ (λ), can now be found by following similar steps as in [7] according to

g ui j −1 Ki, j e−λ0 /(γi −σi2 ) λk0 .

2 k Γ(k + 1) i=1 j =1 k=0 γi − σi

(21)

On the other hand, given that the transmission is suspended for the channel gains below the cutoff threshold, λ0 , we can find a closed-form expression for α by expanding the input power constraint as

P=

∞ λ0

αRσn2

nT

λ − αR

σ2 i=1 i

fλ (λ)dλ

g ui αRσn2 Ki, j ∞ λ j −1 = e−λ/mi dλ, j (λ − n) λ 0 i=1 j =1 Γ( j)mi

(22)

6


g ui Rσn2 Ki, j

P=α

j i=1 j =1 Γ( j)mi

×

j −2 k=0

j −2−k

mi nk e−λ0 /mi

l=0

− n j −1 e−n/mi Ei

Γ( j − k − 1) j −k−l−2 ml λ Γ( j − k − l − 1) i 0

n − λ0 mi

5 Mutual information bound (nats/s/Hz)

where mi = γi − σi2 and n = αR ni=T1 σi2 . The integration in (22) can be expanded by using the equalities λ j −1 − n j −1 = j −2 j (λ − n) k=0 nk λ j −2−k , x j e−x/mi dx = −mi e−x/mi l=0 ( j!/( j − l)!)mli x j −l , and (n j −1 /(x − n)) e−x/mi dx = n j −1 e−n/mi Ei((n − x)/mi ) such that

4.5 4 3.5 3 2.5 2 1.5 1 0.5

,

0

5

10

15

20

SNR (dB)

(23)

σ12 = σ22 = 0 σ12 = σ22 = 0.01

which leads to a closed-form expression for α.

σ12 = σ22 = 0.05 σ12 = σ22 = 0.1

σ12 = σ22 = 0.02

NUMERICAL RESULTS

In this section, we provide some numerical results in order to illustrate our theoretical analysis. For our simulations, we consider distributed Alamouti codes in Rayleigh fading channels and assume that SNR = P/σn2 and σn2 = 1; hence, a high SNR implies a high transmit power in the presented results. We start by comparing the mutual information bounds, Clower and Cupper , of D-STBCs with the same subchannel estimation error variances, σ12 = σ22 = σe2 , and with a single receive antenna for different values of σe2 . In Figure 1, the channel gains from the two transmitters are assumed to be γ1 = 1.5 and γ2 = 0.5. The steady and dashed lines correspond to the mutual information lower and upper bounds, respectively. Figure 1 shows that not only are the bounds fairly close at high SNRs, but also that the gap between the two bounds is small for low SNRs. We observe that at low SNRs, the capacity increases logarithmically as a function of SNR, but with smaller slope as compared to a system with perfect CSI at the receiver, that is, when σe2 = 0. Figure 1 also indicates that at high SNRs, the mutual information bounds saturate and do not increase logarithmically as a function of SNR. The gap between the mutual information bounds, Δ, for D-STBCs with two receive antennas and a constant measure for σ12 + σ22 , namely, σ12 + σ22 = 0.2, are plotted versus SNR in Figure 2. The plots show that when the SNR increases, Δ increases monotonically. The figure also illustrates that the gap between the mutual information bounds increases when the ratio between the subchannel estimation error variances, that is, σ12 /σ22 , increases. In Figure 3, we further plot the gap between the mutual information bounds for D-STBCs, SIMO subchannels, and distributed MIMO channel with two receive antennas, versus the channel estimation error variance of the first subchannel, that is, σ12 , at SNR = 20 dB. The channel estimation error variance of the second subchannel relates to σ12 through σ12 + σ22 = 0.1. The figure shows that the gap between the mutual information bounds of D-STBCs is relatively small compared

Figure 1: Mutual information lower and upper bounds for DSTBCs with single receive antenna; σ12 = σ22 . 0.7 0.6 Gap, Δ (nats/s/Hz)

6.

0.5 0.4 0.3 0.2 0.1 0 0

5

10

15

20 25 SNR (dB)

30

35

σ12 = σ22 = 0.1

σ12 : σ22 = 4 : 1

σ12

σ12 = 0.2, σ22 = 0

:

σ22 =

2:1

40

Figure 2: Gap between the mutual information bounds for DSTBCs with two receive antennas; σ12 + σ22 = 0.2.

to that of the SIMO and distributed MIMO channels. We also observe that the gap for D-STBCs changes slowly as the subchannel estimation error variances change, while Δ in SIMO subchannels increases significantly when the channel estimation error variance increases. The mutual information lower bound of D-STBCs with a single receive antenna and with γ1 = 1 + αγ and γ2 = 1 − αγ is plotted in Figure 4 for SNR = 15 dB. Variations of the bounds as a function of αγ are illustrated for various channel estimation error variances showing that the mutual information lower bound is at its maximum when αγ = 0, or equivalently, when γ1 = γ2 ; hence confirming the results of


7 3

1.4

σ12 = σ22 = 0.01 2.5

1 Cout (nats/s/Hz)

Gap, Δ (nats/s/Hz)

1.2

0.8 0.6 0.4

1.5

σ12 = σ22 = 0.1

1 0.5

0.2 0

2

0

0.02

0.06

0.04

0.08

0

0.1

0

5

αe D-STBC, σ12 = αe , σ22 = 0.1 − αe D-MIMO, σ12 = αe , σ22 = 0.1 − αe SIMO, σ12 = αe SIMO, σ22 = 0.1 − αe

15

Pout = 0.1 Pout = 0.05 Pout = 0.01

Figure 3: Gap between the mutual information bounds for DSTBCs with two receive antennas, and for its SIMO subchannels at SNR = 20 dB: variations as a function of σ12 = αe given σ12 +σ22 = 0.1.

Figure 5: Lower bounds on the outage capacity of D-STBCs with single receive antenna.

2.4 3.4

2.2 2

3.2 Cout (nats/s/Hz)

Mutual information lower bound (nats/s/Hz)

10 SNR (dB)

3 2.8

1.8 1.6 1.4 1.2 1

2.6

0.8 0.6

2.4 0

0.2

0.4

0.6

0.8

1

0.4

0

0.1

0.2

0.3

σ12 = σ22 = 0.01 σ12 = σ22 = 0.02 σ12 = σ22 = 0.05

0.4

0.5

0.6

0.7

0.8

0.9

αγ

αγ σ12 = σ22 = 0.01 σ12 = σ22 = 0.02 σ12 = σ22 = 0.03

Figure 4: Mutual information lower bounds for D-STBCs with single receive antenna at SNR = 15 dB, given γ1 = 1 + αγ and γ2 = 1 − αγ .

Figure 6: Lower bound on the outage capacity of D-STBCs with single receive antenna versus the channel gains variations at SNR = 15 dB.

Section 4. The figure also illustrates that the variations of the mutual information lower bound as a function of αγ is small around αγ = 0. In Figure 5, the outage capacity lower bound of D-STBCs with γ1 = 1.5 and γ2 = 0.5 and with a single receive antenna is plotted versus SNR for different values of Pout . The plots show that the outage capacity suffers a significant loss as a result of estimation errors at the receiver. Indeed, it can be

seen that the outage capacity of D-STBCs with σ12 = σ22 = 0.1 starts to saturate at SNR values as small as 5 dB. Finally in Figure 6, the lower bound on the outage capacity of D-STBCs with outage probability Pout = 1% and with subchannel gains γ1 = 1 + αγ and γ2 = 1 − αγ is plotted versus αγ at SNR = 15 dB for various channel estimation error variances. The figure shows that a capacity gain of 0.9 nats/s/Hz can be achieved by positioning the receiver such

8


that it provides γ1 = γ2 . Furthermore, comparing Figures 4 and 6 reveals that by optimum positioning, the increase in the capacity of a system with channel inversion technique is higher than that of a system with constant input power transmission. 7.

CONCLUSION

We have addressed the effect of channel knowledge uncertainty at the receiver on the mutual information of distributed space-time block coded transmission in Rayleigh fading channels. Specifically, we studied upper and lower bounds on the mutual information of the system when knowledge of the variance of the channel estimation error is available at the receiver and the transmitters. We provided a limiting value that upper bounds the gap between the mutual information bounds at any input transmit powers so as to justify that they represent a good estimate of the true channel mutual information for Gaussian input signals. We also showed that the tightness between the bounds increases when the number of transmitters increases as long as the receiver can estimate the channels pertaining to different transmitters with the same accuracy. In addition, we showed that when the disparity between the estimation error variances increases, the gap between the bounds increases. Also, assuming that the summation of the channel gains is constant, we determined the receiver’s position at which the mutual information lower and upper bounds of D-STBCs and their robustness to the variations of the subchannel gains are maximum. We further determined a lower bound for the outage capacity of D-STBCs with arbitrary numbers of transmit and receive antennas, and also obtained closedform expressions for this capacity metric and its associated power allocation scheme. Numerical results showed that the capacity increase, achieved by optimum positioning of the receiver, is higher in systems with channel inversion transmission technique as compared to constant input power transmission, and that the outage capacity suffers significant loss as a result of channel estimation errors at the receiver. ACKNOWLEDGMENTS This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Part of this work was presented at IEEE WCNC’07. REFERENCES [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744–765, 1998. [2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, 1999. [3] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451–1458, 1998.

[4] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications, Cambridge University Press, New York, NY, USA, 2003. [5] A. Maaref and S. A¨ıssa, “Performance analysis of orthogonal space-time block codes in spatially correlated MIMO Nakagami fading channels,” IEEE Transactions on Wireless Communications, vol. 5, no. 4, pp. 807–817, 2006. [6] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. [7] M. Dohler and H. Aghvami, “Information outage probability of distributed STBCs over Nakagami fading channels,” IEEE Communications Letters, vol. 8, no. 7, pp. 437–439, 2004. [8] A. Hjørungnes and D. Gesbert, “Precoding of orthogonal space-time block codes in arbitrarily correlated MIMO channels: iterative and closed-form solutions,” IEEE Transactions on Wireless Communications, vol. 6, no. 3, pp. 1072–1082, 2007. [9] M. Médard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Transactions on Information Theory, vol. 46, no. 3, pp. 933–946, 2000. [10] A. Lapidoth and S. Shamai, “Fading channels: how perfect need “perfect side information” be?” IEEE Transactions on Information Theory, vol. 48, no. 5, pp. 1118–1134, 2002. [11] T. Yoo and A. Goldsmith, “Capacity and power allocation for fading MIMO channels with channel estimation error,” IEEE Transactions on Information Theory, vol. 52, no. 5, pp. 2203– 2214, 2006. [12] A. Sabharwal, E. Erikp, and B. Aazhang, “On channel state information in multiple antenna block fading channels,” in Proceedings of International Symposium on Information Theory and Its Applications, pp. 116–119, Honolulu, Hawaii, USA, November 2000. [13] V. Tarokh, A. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications, vol. 47, no. 2, pp. 199–207, 1999. [14] W. Hoteit, Y. R. Shayan, and A. K. Elhakeem, “Effects of imperfect channel estimation on space-time coding performance,” IEE Proceedings-Communications, vol. 152, no. 3, pp. 277–281, 2005. [15] P. Garg, R. K. Mallik, and H. M. Gupta, “Performance analysis of space-time coding with imperfect channel estimation,” IEEE Transactions on Wireless Communications, vol. 4, no. 1, pp. 257–265, 2005. [16] L. Musavian, M. R. Nakhai, M. Dohler, and A. H. Aghvami, “Effect of channel uncertainty on the mutual information of MIMO fading channels,” IEEE Transactions on Vehicular Technology, vol. 56, no. 5, part 1, pp. 2798–2806, 2007. [17] R. M. Young, “Euler’s constant,” The Mathematical Gazette, vol. 75, no. 472, pp. 187–190, 1991. [18] L. Musavian and S. A¨ıssa, “Performance analysis of distributed space-time coded transmission with channel estimation error,” in Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC ’07), pp. 1275–1280, Hong Kong, China, March 2007. [19] L. Musavian and S. A¨ıssa, “On the achievable sum-rate of correlated MIMO multiple access channel with imperfect channel estimation,” to appear in IEEE Transactions on Wireless Communications.

L. Musavian and S. A¨ıssa [20] E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: information-theoretic and communications aspects,” IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2619– 2692, 1998. [21] G. Caire and S. Shamai, “On the capacity of some channels with channel state information,” IEEE Transactions on Information Theory, vol. 45, no. 6, pp. 2007–2019, 1999. [22] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Transactions on Information Theory, vol. 43, no. 6, pp. 1986–1992, 1997. [23] M.-S. Alouini and A. J. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity-combining techniques,” IEEE Transactions on Vehicular Technology, vol. 48, no. 4, pp. 1165–1181, 1999. [24] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1965.

9


Research Article Cooperative Multibeamforming in Ad Hoc Networks Chuxiang Li1 and Xiaodong Wang2 1 Marvell

Semiconductor, Inc., Santa Clara, CA 95054, USA of Electrical Engineering, Columbia University, New York, NY 10027, USA

2 Department

Correspondence should be addressed to Xiaodong Wang, [email protected] Received 24 April 2007; Revised 6 August 2007; Accepted 8 October 2007 Recommended by G. K. Karagiannidis We treat the problem of cooperative multiple beamforming in wireless ad hoc networks. The basic scenario is that a cluster of source nodes cooperatively forms multiple data-carrying beams toward multiple destination nodes. To resolve the hidden node problem, we impose a link constraint on the receive power at each unintended destination node. Then the problem becomes to optimize the transmit powers and beam weights at the source cluster subject to the maximal transmit power constraint, the minimal receive signal-to-interference-plus-noise ratio (SINR) constraints at the destination nodes, and the minimal receive power constraints at the unintended destination nodes. We first propose an iterative transmit power allocation algorithm under fixed beamformers subject to the maximal transmit power constraint, as well as the minimal receive SINR and receive power constraints. We then develop a joint optimization algorithm to iteratively optimize the powers and the beamformers based on the duality analysis. Since channel state information (CSI) is required by the sources to perform the above optimization, we further propose a cooperative scheme to implement a simple CSI estimation and feedback mechanism based on the subspace tracking principle. Simulation results are provided to demonstrate the performance of the proposed algorithms. Copyright © 2008 C. Li and X. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Recently, a new approach of achieving spatial diversity gain in relay networks, namely, cooperative diversity or user cooperation diversity, has received considerable interests [1– 5]. Cooperative diversity comes from the fact that multiple nodes in an ad hoc network can cooperatively form a virtual antenna array providing the potential of realizing spatial diversity. As an effective technique of exploiting spatial diversity in multiple-antenna systems, space-timing coding has been widely studied for cooperative ad hoc networks (e.g., see [6–9]). Beamforming is another important diversity technique in multiple-antenna systems, and several beamforming-based schemes have been developed in current literature for cooperative ad hoc networks. Specifically, distributed receive beamforming is treated in [10, 11]. The effects of phase noises in distributed beamforming schemes are analyzed in [12]. A probabilistic transmit beamforming scheme, namely, collaborative beamforming, is proposed in [13, 14]. In [15], the power optimization issue and also the beamforming at the relay side have been addressed in ad hoc wireless networks. The cooperative beamforming con-

cept and power efficiency issues in fading channels have been treated in [16]. In existing work, one key assumption is that the neighboring nodes which form one cluster can share the data information a priori. From the viewpoint of power consumption, this assumption is reasonable in the sense that the overhead requested by intracluster information sharing is relatively small due to the short distances among intracluster nodes. Another key issue is the synchronization among multiple cooperative nodes [12], for example, carrier frequency, phase, and timing synchronization. It is worth noting that one major problem brought by beamforming applications in wireless networks is the so-called “hidden node” problem. In particular, carrier-sense-multiple-access (CSMA) mechanism is employed in 802.11 standards, where each node attempts to access the network and transmits only when it detects no energy from other nodes. Such a CSMA mechanism brings the problem of potential collisions among different transmissions in the case that multiple nodes cannot sense one another’s transmission. The problem of potential collision is, namely, the hidden node problem [17, 18]. In the wireless networks employing beamforming schemes,

2


the hidden node problem becomes more severe due to the fact that a directional beam inevitably reduces the energy delivered to some unintended destination nodes in the network, and consequently, collisions happen more frequently and result in more retransmission, delay, and packet loss. In this paper, instead of considering the beamforming problem that a cluster of nodes cooperatively forms one beam toward one destination node (e.g., [13, 14, 18]), we treat the problem of simultaneously forming multiple beams for multiple concurrent data transmissions in wireless ad hoc networks. Figure 1 shows an example of multiple beamforming. This problem resembles the multiuser beamforming problem in MIMO systems which has been studied in [19]. Moreover, different from the probabilistic approach (e.g., see [18]) to resolve the hidden node problem, we propose a deterministic approach which imposes a link constraint on the minimum receive power at each unintended destination node. Therefore, the cooperative multiple beamforming problem can be formulated as a multiuser beamforming problem with extra receive power constraints for unintended destination nodes. To solve this problem, we first propose an iterative power allocation algorithm to maximize the balanced SINR ratio under fixed beamformers. Then we develop a joint optimization algorithm to iteratively optimize the powers and the beamformers. Note that channel state information (CSI) is required for the source nodes to perform the above optimizations, and thus, some CSI estimation and feedback mechanism are necessary. We then present a scheme for the source and destination clusters to cooperatively implement a simple CSI tracking mechanism. The remainder of this paper is organized as follows. In Section 2, the system model is described and the cooperative multiple beamforming problem is formulated. In Section 3, an iterative power allocation strategy is proposed under fixed beamformers. In Section 4, the joint power and beamforming optimization algorithm is developed. In Section 5, the subspace tracking based CSI feedback scheme is presented. Section 6 contains the conclusions. 2.

SYSTEM MODEL AND PROBLEM FORMULATION

The basic concept of cooperative multiple beamfomring is to simultaneously transmit several data-bearing signal beams toward some destination nodes and non-data-bearing signal beams toward unintended destination nodes. As shown in Figure 1, there are K nodes in the source cluster where M ones, namely, source nodes, have information to transmit; there are totally K nodes in the destination cluster, where M of them are the destination nodes, namely, destination nodes, and the other K-M ones are the unintended destination nodes. 2.1. Cooperative multiple beamforming Cooperative beamforming consists of two stages, local broadcasting and cooperative transmission. In particular, in local broadcasting, each source node broadcasts its databearing signal to the other ones in the source cluster; then in cooperative transmission, each node in the source cluster

Unintended destination nodes

Destination cluster Destination node 1

Destination node 2

Beam-1

Beam-2

Source node 1

Source node 2 Source cluster

Figure 1: Cooperative multiple beamforming in wireless ad hoc networks: two concurrent beams are formed; K = 10 nodes in the source/destination cluster; M = 2 source/destination nodes; K − M = 8 unintended destination nodes.

acts as a relay for the others, and the source cluster cooperatively forms multiple concurrent beams. Note that perfect synchronization is assumed in this paper. 2.1.1. Local broadcasting In the first stage, the received signal at node j in the source cluster from source node i is

yi, j = Pi,0 hi, j si + n j ,

1 ≤ i ≤ M, 1 ≤ j ≤ K, i = j,

(1)

where si is the data-bearing signal from source node i and E{|si |2 } = 1; Pi,0 is the transmit power of source node i; n j ∼CN (0, η) denotes the AWGN at node j; hi, j ∼CN (0, 1) is the channel response between the nodes i and j. The amplifyand-forward scheme is employed in the source cluster, that is, each node does not attempt to decode but directly forwards the received signal. Specifically, yi, j at node j is first normal ized by αi, j := E{| yi, j |2 }, that is,

si, j =

Pi,0 hi, j yi, j 1 = 2 si + 2 nj, αi, j Pi,0 hi, j + η Pi,0 hi, j + η 1 ≤ i ≤ M,

1 ≤ j ≤ K,

(2)

j =i.

Define the cooperative data-bearing signal vector toward each destination node Di as si := [si,1 , si,2 , . . . , si,K ]T , where si,i = si , 1 ≤ i ≤ M, and the non-data-bearing signal vector toward each unintended destination node D j as s j := [s j , s j , . . . , s j ]T , M + 1 ≤ j ≤ K.

C. Li and X. Wang ∗ K = 5; M = 2 ∼ 4; SINR∗ i = 6 dB and γi = 0.8, 1 ≤ i ≤ M.

2

Power sequences in the iterative power optimization: K = 5; M = 3; γi∗ = 0.8, 1 ≤ i ≤ M; PT /η = 10.

1.8 1.2

1.6

Feasible region: C(PT ) > 1

1.4

1

A (for Fig.3)

|| p||1 = PT

1.2 Power ratio

Maximum achievable SINR ratio C(PT )

3

1 0.8 0.6

0.6 PD4 /PT = 0.1

0.4

0.4 Infeasible region: C(PT ) ≤ 1

0.2 0

0.8

0.2 2 P1 4

0

6 P2 8

10 PT /η

12

14

16

18

20

0

1

In the second stage, each node j (1 ≤ j ≤ K) in the source cluster transmits the signal x j = Ki=1 Pi ui, j si, j , where ui, j is the beam weight at node j for the transmission toward destination node Di . Denote ui := [ui,1 , ui,2 , . . . , ui,K ]T and hH i := [h1,Di , h2,Di , . . . , hK,Di ], 1 ≤ i ≤ K, as the beamformer and the channel vector for the reception of si at Di , respectively. Then the received data-bearing signal si at destination node Di is given by

=

Pi hH i Λi ui si

(3)

+

Pi hH i Ξi ui ,

1 ≤ i ≤ M,

Pi,0 hi, j / Pi,0 |hi, j |2 + η, and Ξi := diag{ξ i,1 , . . . , ξ i,i−1 , 0,

ξ i,i+1 , . . . , ξ i,K } with ξ i, j := n j / Pi,0 |hi, j |2 + η, 1 ≤ j ≤ K and =i. Moreover, the received data-bearing signal s j at Di ( j =i) j is given by ID i =

M

P j hH i Λjujsj +

j =1, j =i

+

6

K

signal yDi = sDi + IDi + nDi at each destination node Di can be written as y Di =

M

P j hH i Λjujsj +

j =1

P j hH i Ξjuj

j =1 K

+

M

Pl hH i ul sl

+ nD i ,

(5) 1 ≤ i ≤ M.

l=M+1

2.1.3. Receive SINR and power

h j,Di ui, j si, j

j =1

where Λi := diag{βi,1 , . . . , βi,i−1 , 1, βi,i+1 , . . . , βi,K } with βi, j :=

5

Figure 3: Power distribution in the iterative power optimization algorithm (Algorithm 1): K = 5; M = 3; γ∗i = 0.8, 1 ≤ i ≤ M; PT /η = 10.

2.1.2. Cooperative transmission

4

The sequence of total power of all nodes The sequence of total power of active nodes The sequence of total power of silent nodes The sequence of received power at one silent node

Figure 2: Feasible region of problem (B): K = 5; M = 2 ∼ 4; SINR∗i = 6 dB and γ∗i = 0.8, 1 ≤ i ≤ M.

K

3 Iteration number

M=2 M=3 M=4

sDi = Pi

2

M j =1, j =i

P j hH i Ξjuj (4)

Pl hH i ul sl ,

l=M+1

where the first two terms come from the data-bearing signal =i), and the last term is from the non-datas j (1 ≤ j ≤ M, j bearing signal sl (M + 1 ≤ l ≤ K). Then the overall received

H Define Ωi := hi hH i and Ωi := E{Λi Ωi Λi }, 1 ≤ i ≤ K. For a given {h1 , h2 , . . . , hK }, the receive SINR at each destination node Di can be expressed as

SINRi Pi uH i Ωi ui

= M

P uH Δ u + j =1 j j i j

K

P uH Ωi u j − Pi uH i Ωi ui + η j =M+1 j j

,

1 ≤ i ≤ M, (6) where Δi := E{(Λ j + Ξ j )H Ωi (Λ j + Ξ j )} = E{ΛHj Ωi Λ j + ΞHj Ωi Ξ j } and Δi = diag{Ωi } for 1 ≤ j ≤ M. Further define γi as an increasing function of SINRi in (6) γ i :=

SINRi 1 + SINRi Pi uH i Ωi ui , K H H j =1 P j u j Δi u j + j =M+1 P j u j Ωi u j + η

= M

(7)

4


which is essentially equivalent to SINRi . It should be notified that the SINRi based analysis and optimization are quite involved in cooperative ad hoc networks, and the metric γi can help to make the analysis and optimization much more tractable. The optimization based on γi can be viewed as an approximation of the optimization based on SINRi . Note that we will adopt γi as the performance metric throughout this paper. For convenience, hereafter, we call γi the receive SINR at Di , though the receive SINR is in fact SINRi given by (6). The receive power at each unintended destination node D j is given by ⎡

⎤⎡

⎤

⎡

⎤

H uH P1 PDM+1 1 ΔM+1 u1 · · · uK ΩM+1 uK ⎢ ⎥⎢ . ⎥ ⎢ . ⎥ .. .. .. ⎢ ⎥⎢ . ⎥ = ⎢ . ⎥. . ⎣ ⎦⎣ . ⎦ ⎣ . ⎦ . . PK PDK uH · · · uH 1 ΔK u1 K ΩK uK

p

Θ

(8)

3.

OPTIMAL POWER ALLOCATION STRATEGY

3.1.

K

P j hH i u j s j + nD i ,

(9)

Pi uH i Ωi ui . H P j =1, j =i j u j Ωi u j + η

(10)

j =1

i = SINR K

Assuming that each relay receives broadcasting signals withi = out noises, we have Λi = IK , Ξi = OK , and Δi = Ω Ωi . Then (5) and (6) reduce to (9) and (10), respectively. Moreover, (9) and (10) also hold for the decode-and-forward scheme in relay networks assuming perfect decoding at relays. Hence, the assumption of perfect a priori sharing among source nodes is a special case of the general relay scenarios (5) and (6), and the existing distributed beamforming approaches still fall in the cooperative relay framework treated in this paper.

For a given beamforming matrix U, problem (A) reduces to the power allocation problem

p

p,U 1≤i≤M

γi (p, U) , γ∗i

⎧ K ⎪ ⎪ ⎪ ⎪ p = Pi ≤ PT , ⎪ 1 ⎪ ⎨

(11)

i=1

subject to ⎪ ⎪ ⎪C(p, U) ≥ 1,

⎪ ⎪ ⎪ ⎩P (p, U) ≥ P min , Dj j

1≤i≤M

γi (p) , γ∗i

⎧ K ⎪ ⎪ ⎪ ⎪ p = Pi ≤ PT , 1 ⎪ ⎨

M + 1 ≤ j ≤ K,

where U := [u1 , u2 , . . . , uK ]; PT is the maximal transmit power; γ∗i is the minimal SINR for destination node Di ; P min is the minimal receive power for unintended destinaj tion node D j .

(12)

i=1 subject to ⎪ ⎪ ⎪C(p) ≥ 1,

⎪ ⎪ ⎩

PD j (p) ≥ P min , j

M + 1 ≤ j ≤ K.

Note that a similar problem but without the receive power constraints has been treated in [19, 20], where a specific structure is exploited to calculate p∗ . Such a structure, however, does not exist for problem (B) due to the extra constraints on receive powers PD j (p). To solve problem (B), we further treat the following total power minimization problem:

) ρ p∗ = min (B p

K

Pi ,

i=1

⎧ ⎨γi (p) ≥ γ∗ i ,

subject to ⎩ PD j (p) ≥ P min , j

The cooperative beamforming problem is to find the optimal power and beamforming matrix to maximize the minimal receive SINR of destination nodes under the maximal transmit power constraint and the minimal receive power constraints for unintended destination nodes,

(B) C p∗ = max min

2.2. Problem formulation

(A) C p∗ , U∗ = max min

Optimal power allocation problem

pD

Remark 1. One key assumption in the existing literature on distributed beamforming is that one cluster can share information a priori. Under this assumption, the received signal and the SINR at each Di are given, respectively, by yDi =

Remark 2. In problem (A), an assumption of Θ in (8) is that H for each j (M + 1 ≤ j ≤ K), uH i Δ j ui < uk Ω j uk , 1 ≤ i ≤ M, M + 1 ≤ k ≤ K. This assumption is reasonable and necessary due to the hidden node problem. In particular, the hidden node problem exists when the receive powers at the unin H tended destination nodes are small, that is, M i=1 Pi ui Ω j ui in (8). Thus it is necessary to form the extra non-data-bearing beams to ensure certain receive powers. On the other hand, if H uH i Δ j ui ≥ uk Ω j uk , the minimum receive power constraints can be guaranteed by only allocating power to those databearing beams (i.e., let Pi = 0, 1 + M ≤ i ≤ K), and thus the hidden node problem becomes trivial [18].

1 ≤ i ≤ M,

(13)

M + 1 ≤ j ≤ K,

∗

which is to find p for a given U so as to minimize the total transmit power under the minimum constraints on receive ) and (B) are powers and SINRs. Note that the problems (B closely related [19] in the sense that without the minimum receive power constraints, they are equivalent and have the same solution if and only if ρ(p∗ ) = PT . Then it can be solved by an iterative approach where in each iteration, p∗ of prob ) is calculated under a given target SINR set {γ∗ } , lem (B i i and then increase {γ∗i }i if p∗ 1 is less than PT . As p∗ 1 approximates PT , C(p∗ ) will reach the maximal achievable value. With the minimum receive power constraints, however, it is difficult to find the optimal solution, and thus we propose to find an approximation of p∗ as follows. 3.2.

Iterative power optimization algorithm

Denote pM = [P1 , . . . , PM ]T and pK −M = [PM+1 , . . . , PK ]T . First, consider the optimal pM under a given pK −M . Since

C. Li and X. Wang

5

each γi in (7) is monotonically increasing with respect to Pi (1 ≤ i ≤ M) and monotonically decreasing with respect to =i) under a given pK −M , the optimal P j (1 ≤ j ≤ K and j pM of problem (B) only with the minimum receive SINR constraints can be achieved when γi (pM , pK −M , U) = γ∗i , 1 ≤ i ≤ M. Using (7), these M linear equations can be written into the matrix representation: ⎡ ⎡ H ⎢ ⎢ u1 Δ1 u1 ⎢ ⎢ ⎢ −1 .. ⎢ ⎢Γ Ψ − ⎣ . ⎢ ⎢ uH ⎣ 1 ΔM u1

⎤ ⎤⎥ · · · uH K Ω1 uK ⎥ ⎥⎥ .. .. ⎥⎥ p = η·1M , . ⎦⎥ . ⎥ ⎥ H · · · uK ΩM uK ⎦

(14)

Φ

where Γ := diag{γ∗1 , . . . , γ∗M }; 1M := [1, . . . , 1]T has a dimension of M; Ψ := [Ψ1 , OM ×(K −M) ], where Ψ1 := H diag{uH 1 Ω1 u1 , . . . , uM ΩM uM }. Next consider the optimal pK −M under a given pM . Using (8) with a given pM , the op ) with only the minimum receive timal pK −M of problem (B power constraints is achieved when Θp = pmin .

(15)

Iteratively optimizing pM and pK −M using (14) and (15) under increasing target SINRs, p1 will approximate PT . The iterative power allocation is summarized in Algorithm 1. Denote p∗ = [p∗M T , p∗K −M T ]T as the optimal solution ). In step (1), pK −M (1)1 = 0 ≤ p∗ of problem (B K −M 1 and pM (1)1 ≤ p∗M 1 , and thus p(1) ≤ p∗ 1 . In step (2), pK −M (2)1 ≥ p∗K −M 1 and pM (2)1 ≥ p∗M 1 , p(1) ≥ p(1)1 . In step and thus p(2)1 ≥ p∗ 1 , p(1)1 , (3), pK −M (3)1 ≤ p∗K −M 1 , and thus p(1)1 ≤ p∗ 1 , p(2)1 ≤ p(2)1 . In steps (4)–(6), we have pM (n+ 1)1 ≥ pM (n)1 due to γ∗i (n + 1) ≥ γ∗i (n) in (14), that is, pM (n)1 is increasing with respect to the iteration index n. Consequently, (15) further implies that pK −M (n + 1)1 ≤ pK −M (n)1 , that is, pK −M (n)1 is decreasing. Then the convergence of Algorithm 1 depends on whether p(n)1 = pM (n)1 + pK −M (n)1 is increasing with respect to n. Remember that the assumption of Θ stated in Remark 2 enH sures that for each M + 1 ≤ j ≤ K, uH i Ω j ui < uk Ω j uk , M M i ≤ M < k. Hence, we have i=1 Pi (n + 1) − i=1 Pi (n) ≥ K K k=M+1 Pk (n) − k=M+1 Pk (n + 1), that is, M K p(n + 1) = Pi (n + 1) + Pk (n + 1) 1 i=1

≥

M i=1

k=M+1

Pi (n) +

K

Pk (n) = p(n)1 .

3.3.

Figure 2 shows the achievable region of SINR ratios for problem (B) under a fixed beamforming matrix U. The results are the averaged performances over 1000 channel realizations. For each channel realization, ui in the fixed U is the optimal beamforming vector for node i’s single transmission, that is, the eigenvector corresponding to the largest eigenvalue of Ωi . The simulation conditions in Figure 2 are as follows: K = 5; M = 2∼4; the minimum receive SINR is γ∗i = 0.8 (i.e., SINR∗i = 6 dB), 1 ≤ i ≤ M; the minimum receive power is pmin = [1, . . . , 1]T . In Figure 2, the maximum achievable SINR ratio for problem (B) C(PT ) := C(p∗ ) depends on both PT and {γ∗i }i , and is monotonically increasing with respect to the total transmit power PT . The feasible region corresponds to the region C(PT ) > 1 in Figure 2, and depends on {γ∗i }i . It is seen from Figure 2 that P1 and P2 (P1 < P2 ) are the minimum total transmit powers to guarantee feasible solutions, respectively, for the cases of M = 2 and M = 3. For the case of M = 4, however, there exists no possible solution in the feasible region, that is, no feasible solution exists for problem (B) when M = 4. Hence, we conclude from Figure 2 that on the one hand, the more concurrent transmissions the system simultaneously supports, the higher the total transmit power required to guarantee feasible solutions is; on the other hand, under some cases, there exists no feasible solution even if PT →∞, and this has also been pointed out in [19] for multiuser beamforming scenarios. In the latter case, beamforming optimization will play an important role which will be demonstrated later. Figure 3 shows the sequences of total transmit power {p(n)1 } generated in Algorithm 1 under the same conditions as those in Figure 2, where M = 3 and PT /η = 10. Note that the maximum achievable SINR ratio in Figure 3 corresponds to the point A in Figure 2 (C(PT ) = 1.2). It is observed that p(n)1 is increasing and reaches PT (i.e., p(n)1 /PT →1) as n increases. Moreover, it is seen from Figure 3 that the total transmit power sequence for data-bearing transmissions { M i=1 Pi (n)} is also an increasing one; in contrast, the total transmit power sequence for non-data-bearing transmissions { Ki=M+1 Pi (n)} is a decreasing one. Figure 3 also shows that the receive power sequence for the unintended destination node PD4 (n) = P4min ∼ = 1 is approximately fixed as the minimum value. This implies that the power consumption to guarantee the receive power constraints on the unintended destination nodes is minimized. 4.

(16)

k=M+1

This guarantees the convergence of Algorithm 1, which is summarized as follows. Theorem 1. The sequence {p(n)1 } obtained in Algorithm 1 is a monotonically increasing sequence. The optimal solution to problem (B) is achieved when p(n)1 reaches PT .

Simulation results

4.1.

JOINT POWER AND BEAMFORMER OPTIMIZATION Optimal beamforming and duality property

Under a given power set p, problem (A) is then reduced to the beamforming problem (C1)

C ∗ = C U∗ = max min

U 1≤i≤M

γi (U) . γ∗i

(17)

It is observed from (7) that each γi is coupled with the entire beamforming matrix U = [u1 , u2 , . . . , uK ], and thus problem (C1) is hard to solve. Note that it has been proven in [19]

6


1: Given pK −M (1) = 0K −M = [0, 0, . . . , 0]T , calculate pM (1) using (14). If pD (p(1)) ≥ pmin , then stop the iteration and let p∗ = p(1), where p(1) = [pM (1)T , pK −M (1)T ]T . 2: Given pM (1), calculate pK −M (2) using (15), and then given pK −M (2), calculate pM (2) T T T T T T (1) = [pM (1) , pK −M (2) ] , and p(2) = [pM (2) , pK −M (2) ] . using (14). Then p T T T 3: Given pM (2), calculate pK −M (3) using (15). Then p (2) = [pM (2) , pK −M (3) ] . Let the target SINR be γ∗i (2) = γ∗i ; n ⇐ 3. (n − 1))/γ∗ 4: γ∗i (n) = C(n − 1)γ∗i (n − 1), 1 ≤ i ≤ M, where C(n − 1) = max 1≤i≤M (γi (p i (n − 1)). 5: Given pK −M (n), calculate pM (n) using (14), and then given pK (n), calculate pK −M (n + 1) T T T (n) = [pK (n) , pK −M (n + 1) ] . using (15). Then p(n) = [pM (n)T , pK −M (n)T ]T and p p(n)1 < PT , thenn ⇐ n + 1, and go to step (4); otherwise, stop andp∗ ← p(n − 1). 6: If Algorithm 1: Iterative power allocation algorithm.

that the downlink multiuser beamforming problem can be solved by alternatively treating the dual uplink problem due to the uplink-downlink duality for multiuser beamforming scenarios without receive power constraints. Then an interesting question is whether the duality still holds under the extra receive power constraints in the problem considered in this paper. Remark 3. In Section 3, we only assume that uH i Ω j ui < Ω u , i ≤ M < k and M + 1 ≤ j ≤ K. Hereafter, we uH j k k further assume that the channels of the unintended destination nodes fall in the orthogonal space spanned by the channels of the destination nodes, that is, uH i Ω j ui = 0 for 1 ≤ i ≤ M and M + 1 ≤ j ≤ K. In such a case, the extra nondata-bearing transmission (e.g., complementary beamforming [18]) is a must. Furthermore, under this assumption, p∗ for ) can be obtained by simultaneously solving (14) problem (B and (15), that is,

η1M Γ−1 Ψ − Φ p = min . p Θ

(18)

Now consider a virtual scenario with the same PT , pmin , Γ, and U as those in problem (B). Define the receive SINR for each destination node i in this virtual scenario as

uH i

Pi uH i Ωi ui

PΔ j =1 j j

+

K

k=M+1

(20)

where Υ in (18) is replaced by ΥT . The following lemma indicates the duality between problem (B) and the above virtual power optimization problem under the extra constraints on receive powers. Let C be the maximum achievable SINR ratio of this virtual problem. Lemma 1. For the same U, PT , and pmin , problem (B) and the above virtual power optimization problem have the same PT ). achievable SINR regions, that is, C(U, PT ) = C(U, Proof. To guarantee the minimum receive power constraints in problem (B), the transmit powers p should satisfy Θp = pmin . Based on the assumption stated in Remark 3, Θp = pmin can then be rewritten into the following one: Θ1 pK −M = pmin ,

Then problem (B) can be solved via the simplified version of ) is obtained from (18) Algorithm 1, where p∗ of problem (B for given {γ∗i }i , and then {γ∗i }i are increased if p∗ 1 < PT .

M

ΥT p = η ,

η

Υ

γi =

assumption stated in Remark 3, the optimal power vector for the virtual total power minimization problem can be obtained by solving a similar equation as (18) for solving prob) lem (B

Ωk + ηI ui

,

where Θ1 is the (K − M) × (K − M) bottom-right submatrix in Θ. It is observed from (21) that the receive powers for the unintended destination nodes only depend on the extra powers of non-data-bearing transmissions pK −M . Similarly, we have the same conclusion for the transmit powers T p = [P1 , . . . , PK ] in the virtual problem, that is, ΘT1 p K −M = pmin ,

1 ≤ i ≤ M. (19)

) by γ Replacing γi in problem (B) and problem (B i in (19), the power optimization problem and the total power minimization problem can then be formulated for the virtual scenario (19). The virtual power optimization problem can be solved by a similar approach as Algorithm 1, that is, iteratively solving the virtual total power minimization problem under the increasing target SINRs. In particular, under the

(21)

(22)

T K −M = [PM+1 , . . . , PK ] . Using (21) and (22), we have where p K i=M+1

Pi = 1T Θ1−1 pmin = 1T ΘT1

−1

pmin =

K

Pi .

(23)

i=M+1

That is, the total transmit powers for the non-data-bearing transmissions in the two problems are the same. Hence, given the same total transmit power PT , the total transmit powers for the data-bearing transmissions are also the same in the

C. Li and X. Wang

7 As revealed by Algorithm 1, the balanced SINR ratio C(n) := C(U(n), p(n)) for given U(n)

two problems, that is, M

K

P j = PT −

j =1

Pi = PT −

i=M+1

K

Pi =

i=M+1

M

Pj .

(24)

j =1

p

Given the same total power of data-bearing transmissions, it has been proven in [19] that the two problems have the same achievable SINR region. A direct consequence of Lemma 1 is that problem (A) can be solved by iteratively optimizing the powers and the beamformers using the dual problems. In particular, replacing γi in problem (C1) by γi in (19), we have the virtual beamformer optimization problem

(C2) u∗i = arg max γi ui = arg max ui

ui

uH i Ri ui , H ui Qi ui

1 ≤ i ≤ M,

(C3)

u∗j = arg max PD j uj

= arg max uHj P j Ω j u j , uj

M + 1 ≤ j ≤ K.

(28)

Using (27) and (28), we then have

γi ui (n + 1), p(n) ≥ C(n). 1≤i≤M γ∗i min

(29)

Similarly, for the given U(n+1), C(n+1) := C(U(n+1), p(n+ 1)) satisfies

γi ui (n + 1), p(n + 1) γ∗i γ ui (n + 1), p(n) ≥ min i . 1≤i≤M γ∗i

C(n + 1) =

K i and Qi := M i := Pi Ω where R j =1 P j Δ j + k=M+1 Pk Ωk +ηI. In problem (C2), each γi only depends on its own beamformer ui , and thus it is relatively easy to solve. The optimal beamformer u∗i to problem (C2) is given by the dominant gen i , Qi }, 1 ≤ i ≤ M eralized eigenvector of the matrix pair {R [19]. Moreover, for the non-data-bearing transmissions, the beamformer optimization problem is formulated as the receive power maximization:

γ ui (n), p(n) γ ui (n), p(n) = min i = i . ∗ 1≤i≤M γi γ∗i

(25)

γi ui (n), p 1≤i≤M γ∗i

C(n) = max min

(30)

It is shown from (29) and (30) that C(n + 1) ≥ C(n), that is, the sequence {C(U(n), p(n))} is a monotonically increasing one. Since the optimal solution to problem (A) is nonnegative and bounded, the monotonicity property implies the existence of a limited value as the global optimum lim n→∞ C(n), that is, {C(n)} approximates the global optimal solution.

(26) Then the optimal solution to problem (C3) is given by the eigenvector corresponding to the largest eigenvalue of the matrix {P j Ω j }. 4.2. Joint power and beamformer optimization algorithm In Sections 3.2 and 4.1, the power optimization algorithm under a given U and the beamformer optimization algorithm under a given p are developed, respectively. Then the algorithm for solving problem (A) (see Algorithm 2) is to iteratively optimize p using Algorithm 1 and optimize U using the algorithm in Section 4.1 until reaching convergence. Furthermore, the convergence of Algorithm 2 is revealed in the following theorem. Theorem 2. The sequence {C(U(n), p(n))} generated in Algorithm 2 is a monotonically increasing one, if only the optimum has not been reached. It approximates the global optimal solution of problem (A). Proof. From (25), ui (n + 1) = arg max ui γi (ui , p(n)) for given p(n), 1 ≤ i ≤ M, then

γi ui (n + 1), p(n) γ ui n), p(n) ≥ min i . 1≤i≤M 1≤i≤M γ∗i γ∗i min

(27)

4.3.

Simulation results

Figure 4 shows the achievable region of SINR ratios for problem (A). Note that different from Figure 2 where only power optimization is considered, we treat joint power and beamformer optimization in Figure 4. The simulation conditions are the same as those in Figure 2 with M = 4. It is also worth noting that the definition of C(PT , U) in Figure 4 is the same as that in Figure 2, that is, C(PT , U) := C(p∗ , U) = max p min i (γi (p, U)/γ∗i ). The quantities with index n denotes those in the nth iteration in the joint power and beamformer optimization algorithm (Algorithm 2), for example, U(n) denotes the optimal beamforming matrix in the nth iteration. It is seen from Figure 4 that C(PT , U(n)) is increasing as n increases. In particular, it is seen that the lowest curve (C(PT , U(1))) corresponds to the case of M = 4 in Figure 2, which always falls in the infeasible region. Moreover, when PT /η ≥ PT,0 /η = 10, as n increases, C(PT , U(n)) is successively increasing such that the following points C(PT , U(2)) and C(PT , U(3)) fall in the feasible region. This demonstrates that the optimization of beamformers can significantly improve the system performance. Figure 5 shows the convergence of Algorithm 2. The simulation conditions are the same as those in Figure 4. In particular, C(n) denotes the balanced SINR ratio after both power and beamformer optimization in the nth

8


1: n ⇐ 0; p(n) = [0, . . . , 0]T = 0K ; do the following iterative steps. i , Q(p(n − 1))}, 1 ≤ i ≤ M; u j (n) ⇐ vmax {P j (n − 1)Ω j }, 2: n ⇐ n + 1; ui (n) ⇐ vmax {R M + 1 ≤ j ≤ K; ui (n) ⇐ ui (n)/ ui (n)2 , 1 ≤ i ≤ K. 3: Calculate p(n) for the given U(n) using Algorithm 1, where (18) is replaced by (20). 4: If C(p(n), U(n)) − C(p(n − 1), U(n − 1)) < , then stop; otherwise, go back to step (2). Algorithm 2: Joint power and beamforming optimization algorithm.

K = 5; M = 4; γi∗ = 0.8, 1 ≤ i ≤ M

1.3

1.4

Feasible region of {γi∗ }i : C(PT , U) > 1

1.2

1.2

1.1 1

1

C(PT,0 , U(3)) C(PT,0 , U(2)) Operating points C(PT,0 , U(1))

0.9 0.8

SINR ratios

Achievable SINR ratio C(PT , U)

Convergence behavior: K = 5; M = 4; L = 1; γi∗ = 0.8, 1 ≤ i ≤ M; PT /η = 10.

0.7 0.6

0.4

PT,0 /η 0

2

4

6

8

0.6 0.4

Infeasible region of {γi∗ }i : C(PT , U) ≤ 1

0.5

0.8

0.2 10 PT /η

12

14

16

18

20

0

1

2

3

4

Iteration number U(1) U(2) U(3)

Figure 4: Feasible region of problem (A): K = 5; M = 4; γ∗i = 0.8, 1 ≤ i ≤ M; PT /η = 10.

iteration, that is, C(n) := C(PT , U(n)) = C(p(n), U(n)) = max p min i (γi (p, U(n))/γ∗i ); the SINR ratios after beamformer optimization and before power optimization in the nth iteration are denoted as {γi (p(n − 1), U(n))/γ∗i }i . Note that without power optimization in each iteration, {γi (p(n − 1), U(n))/γ∗ i }i are not necessarily balanced. Then min i (γi (p(n − 1), U(n))/γ∗i ) ≤ C(n) ≤ max i (γi (p(n − 1), U(n))/γ∗i ) in each iteration n. It is seen from Figure 5 that the convergence is achieved until the SINR ratios of all transmissions are balanced, that is, min i γi (p(n − 1), U(n)) = max i γi (p(n − 1), U(n)). Moreover, it is seen from Figure 5 that the convergence can be quickly achieved within only a few iterations. 5.

SUBSPACE TRACKING FOR COOPERATIVE BEAMFORMING

In Sections 3 and 4, we assume perfect CSI when optimizing the powers and the beamformers. In practical systems, however, only estimated CSI is available. In particular, in FDD systems, CSI has to be estimated at the destination cluster, and then fed back to the source cluster, namely, forward estimation and feedback. In TDD systems, CSI can be estimated

mini {γi (p(n − 1), U(n))/γi∗ } C(n) maxi {γi (p(n − 1), U(n))/γi∗ }

Figure 5: The convergence performance of the iterative joint power and beamformer algorithm (Algorithm 2): K = 5; M = 4; γ∗i = 0.8, 1 ≤ i ≤ M; PT /η = 10. Feedback Weight adjust Wodd /Weven Pilot Tx array W Data

Rx array Binary decision

Figure 6: Subspace tracking scheme with binary feedback in multiple-antenna systems.

either at the source cluster or at the destination cluster, and in the latter case, CSI estimates have to be further fed back to the source cluster, namely, backward estimation. Moreover, the data rate of the feedback channel is typically very low in practical systems. Hence, in this section, we propose to employ a simple subspace tracking scheme with only binary feedback to track channel variations [21, 22]. Note that we assume perfect feedback channels, which is reasonable because only binary feedback is required.

C. Li and X. Wang

9

Perfect CSI versus tracked CSI: K = 5; M = 4; γi∗ = 0.8, 1 ≤ i ≤ M; PT /η = 10.

Direct transmission vs. cooperative beamforming 20 18 Total achievable rate (bits/s/Hz)

0.75 0.7 SINR ratio (γi /γi∗ )

0.65 0.6 0.55 0.5 0.45 0.4

16 14 12 10 8 6

0.35 0.3

4 0

10

20

30

40

50

60

70

0

5

10

80

15

20

25

30

Transmit SNR (dB)

Iteration number Cooperative beamforming Direct transmission

SINR ratio using tracked CSI SINR ratio using perfect CSI

Figure 7: The performance of the subspace tracking based approach (Algorithm 3): the perfect CSI case versus the tracked CSI case; K = 5; M = 4; γ∗i = 0.8, 1 ≤ i ≤ M; PT /η = 10.

tions 2 and 3.1, cooperative multiple beamforming resembles the multiuser beamforming in multiple-antenna systems. Therefore, in Figure 6, we adopt the multiple-antenna system diagram as a simplified illustration to show the subspace tracking-based scheme for the cooperative multiple beamforming system. In particular, the transmitter modulates the signals with two different but related weights (ui,e and ui,o ) in two consecutive time slots, even and odd time slots, respectively. Then the receiver side evaluates the two different transmit weights, and generates a binary feedback sign(Ti ) which indicates the preferred transmit weight. For problem (C2) in (25), Ti is defined as the metric to maximize the receive SINR γi (ui ) under a given power set

Iterative optimization of power and beamforming: perfect CSI vs. tracked CSI 1.6 1.4

SINR ratios

1.2 1 0.8 0.6

0.4

Ti := γi ui,o − γi ui,e

0.2 0

Figure 9: The comparison between cooperative multiple beamforming and direct transmission: K = 4 and M = 2.

= 1

2

3

4

Iteration number Perfect CSI: mini γi (p(n − 1), U(n))/γi∗ Tracked CSI: mini γi (p(n − 1), U(n))/γi∗ Perfect CSI: maxi γi (p(n − 1), U(n))/γi∗ Tracked CSI: maxi γi (p(n − 1), U(n))/γi∗

Figure 8: The maximum achievable SINR ratios: the perfect CSI case versus the tracked CSI case; K = 5; M = 4; γ∗i = 0.8, 1 ≤ i ≤ M; PT /η = 10.

5.1. Beamformer optimization via subspace tracking Figure 6 shows the diagram of the subspace tracking scheme with binary feedback for multiple-antenna systems [21, 22]. Note that the source nodes in Figure 1 cooperatively form a virtual antenna array, and also, as we addressed in Sec-

uH uH i,o Ri ui,o i,e Ri ui,e − , H uH Q u u Q i,o i i,o i,e i ui,e

1 ≤ i ≤ M.

(31)

Similarly, for problem (C3) in (26), T j is defined to maximize the receive power PD j (u j ):

T j := uHj,even P j Ω j u j,even − uHj,odd P j Ω j u j,odd , M + 1 ≤ j ≤ K.

(32)

With the aid of such a binary feedback sign(Ti ), the transmitter can iteratively adjust the transmit weights to make the transmissions more adaptive to the channels [21, 22]. Such a subspace tracking-based approach is summarized in Algorithm 3. To compute Ti at the estimation end, pilot signals and certain cooperations are necessary. For instance, in the forward estimation and feedback scheme, the pilot signals (si ) of different nodes at the source cluster are successively transmitted. That is, only s1 is transmitted during the first time slot,

10


1: Given the adaptation rate β, the test perturbation vector μ, and the initial base weight ui,b , 1 ≤ i ≤ M, do the following iterative steps. 2: ui,e = ui,b + βui,b μ and ui,o = ui,b − βui,b μ, 1 ≤ i ≤ M. 3: Calculate Ti using (31) and (32). 4: If sign(Ti ) = 1, ui,b ⇐ ui,o ; otherwise, ui,b ⇐ ui,e , 1 ≤ i ≤ M. 5: Perform Gram-Schmidt orthogonalization on ui,b , 1 ≤ i ≤ M. Algorithm 3: Subspace tracking algorithm for beamformer optimization.

and then, only s2 is transmitted during the second time slot, and so on. Correspondingly, at the destination cluster, the receive powers at D j (1 ≤ j ≤ K) are simply measured during the successive time slots. After some local information sharing within the destination cluster, each node can then calculate its T j using (31), and ui,base in Algorithm 3 will converge i , Qi } for problem (C2) [21, 22]. to the optimal u∗i = vmax {R Similar pilot signals and cooperations can be employed in the backward estimation scheme, and T j in (32) can also be calculated using the local measurements in the source cluster. Remark 4. In the above implementation of the subspace tracking-based algorithm (Algorithm 3), we assume that the local measurements can be perfectly shared at the estimation end, for example, the destination cluster in the forward estimation and feedback scheme and the source cluster in the backward estimation scheme. 5.2. Power optimization scheme As mentioned in Sections 3 and 4, the optimal power vector p(n) for a given U(n) can be obtained by solving (18) or (20). According to the definition of Υ in (18), it is necessary to know hi, j := hTi u j (1 ≤ i ≤ K and 1 ≤ j ≤ K) to calculate p(n) in step (4) of Algorithm 2. It has been pointed out by [21, 22] that the equivalent channel estimates hi, j in the system shown by Figure 6 can be simply obtained by the mean of the even and the odd time slot channel estimates, that is, T ui,b = (h T u j,e + h T u j,o )/2. In the forward estimahi, j = h i i i tion and feedback scheme, hi, j (1 ≤ i, j ≤ K) are obtained at the destination cluster, and the optimal power vector p(n) can be calculated using (20) at the destination cluster. Then p(n) will be fed back to the source cluster. In the backward estimation scheme, both hi, j (1 ≤ i, j ≤ K) and p(n) can be directly extracted at the source cluster. Similarly as the forward estimation and feedback scheme, p(n) will also be sent to the destination cluster. 5.3. Simulation results Figure 7 shows the performance of the subspace tracking based approach (Algorithm 3). The simulation conditions are the same as those in Figure 5. In particular, Figure 7 demonstrates the achievable SINR of one destination node within one iteration of Algorithm 2. That is, for the given

p(n), γi (p(n), ui (n + 1)) = max ui γi (p(n), ui ). It is seen from Figure 7 that γi (p(n), ui (n + 1))/γ∗i where ui (n + 1) is tracked using Algorithm 3 can asymptotically approximate the optimal SINR where ui (n + 1) is calculated assuming perfectly CSI. Furthermore, Figure 8 shows the performance comparison between the joint power and beamformer optimization (Algorithm 2) based on the tracked CSI and that based on perfect CSI. Also, the conditions here are the same as those in Figure 5. It is seen from Figure 8 that when solving problem (A) using Algorithm 2, the achievable SINR ratio obtained using the tracked CSI can approximate those calculated assuming the perfect CSI. Therefore, we conclude from Figures 7 and 8 that Algorithm 3 is an efficient scheme to realize the cooperative beamforming in practice. Figure 9 shows the comparison between the proposed cooperative multiple beamforming scheme and the conventional direct transmission scheme. In Figure 9, K = 4; M = 2; K = 4; pmin = [1, . . . , 1]T . The direct transmission is achieved by simultaneously transmitting M independent links between the source and the destination clusters. Here, we compare the total throughput of the system. Note that the transmit power and the bandwidth are both normalized to guarantee a fair comparison. In particular, given p1 = PT , the rate of each cooperative transmission si is given by ri = log (1 + SINRi (p, U)); in contrast, the rate of each direct transmit link is given by ri = (M + 1) log (1 + SINRi (2p)). Note that the gains M + 1 and 2 in the direction transmission come from the bandwidth loss in the cooperative transmission due to the local broadcasting in the source cluster and the extra local broadcasting power required in the cooperative transmission, respectively. Also note that we here assume equal transmit power for each link in the direction transmission scheme. It is seen from Figure 8 that in the low SNR region, the direct transmission outperforms the proposed cooperative multiple beamforming scheme; in contrast, in the high SNR region which it is interference-dominant, the proposed cooperative multiple beamforming scheme evidently outperforms the direct transmission scheme, because the interferences among multiple concurrent transmissions can be effectively suppressed at the receivers. 6.

CONCLUSIONS

In this paper, we have analyzed the problem of cooperative multiple beamforming in wireless ad hoc networks. We have proposed the iterative power allocation algorithm for given

C. Li and X. Wang beamformers, and studied its convergence. Then we have developed the iterative joint power and beamformer optimization algorithm to solve the problem based on the duality analysis. Moreover, we have proposed to employ the simple subspace tracking-based algorithm with only binary feedback to practically track the channel variation in the system where only bandwidth limited feedback channels are available. We further presented the cooperative scheme to implement such a subspace tracking algorithm. Simulation results have been demonstrated to verify the performances of the proposed algorithms.

11

[14]

[15]

[16]

REFERENCES [17] [1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. [2] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. [3] R. Pabst, B. H. Walke, D. C. Schultz, et al., “Relay-based deployment concepts for wireless and mobile broadband radio,” IEEE Communications Magazine, vol. 42, no. 9, pp. 80–89, 2004. [4] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part I: system description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, 2003. [5] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part II: implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, 2003. [6] M. Dohler and H. Aghvami, “Information outage probability of distributed STBCs over Nakagami fading channels,” IEEE Communications Letters, vol. 8, no. 7, pp. 437–439, 2004. [7] M. Dohler, A. Gkelias, and H. Aghvami, “A resource allocation strategy for distributed MIMO multi-hop communication systems,” IEEE Communications Letters, vol. 8, no. 2, pp. 99–101, 2004. [8] M. Dohler, B. Rassool, and H. Aghvami, “Performance evaluation of STTCs for virtual antenna arrays,” in Proceedings of the 57th IEEE Semiannual Vehicular Technology Conference (VTC ’03), vol. 1, pp. 57–60, Jeju, Korea, April 2003. [9] R. U. Nabar, H. Bölcskei, and F. W. Kneubühler, “Fading relay channels: performance limits and space-time signal design,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 1099–1109, 2004. [10] J. C. Chen, L. Yip, J. Elson, et al., “Coherent acoustic array processing and localization on wireless sensor networks,” Proceedings of the IEEE, vol. 91, no. 8, pp. 1154–1162, 2003. [11] K. Yao, R. E. Hudson, C. W. Reed, D. Chen, and F. Lorenzelli, “Blind beamforming on a randomly distributed sensor array system,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1555–1567, 1998. [12] G. Barriac, R. Mudumbai, and U. Madhow, “Distributed beamforming for information transfer in sensor networks,” in Proceedings of the 3rd International Symposium on Information Processing in Sensor Networks (IPSN ’04), pp. 81–88, Berkeley, Calif, USA, April 2004. [13] H. Ochiai, P. Mitran, H. V. Poor, and V. Tarokh, “Collaborative beamforming for distributed wireless ad hoc sensor net-

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works,” IEEE Transactions on Signal Processing, vol. 53, no. 11, pp. 4110–4124, 2005. H. Ochiai, P. Mitran, H. V. Poor, and V. Tarokh, “Collaborative beamforming in ad hoc networks,” in Proceedings of the IEEE Information Theory Workshop (ITW ’04), pp. 396–401, San Antonio, Tex, USA, October 2004. A. F. Dana and B. Hassibi, “On the power efficiency of sensory and ad hoc wireless networks,” IEEE Transactions on Information Theory, vol. 52, no. 7, pp. 2890–2914, 2006. R. Madan, N. B. Mehta, A. F. Molisch, and J. Zhang, “Energyefficient cooperative relaying over fading channels with simple relay selection,” in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM ’06), pp. 1–6, San Francisco, Calif, USA, November 2006. P. Mehrotra, B. Jose, J. Brennan, and E. Casas, “Performance impact of smart antennas on 802.11 MAC layer,” in Proceedings of the 58th IEEE Semiannual Vehicular Technology Conference (VTC ’03), Orlando, Fla, USA, October 2003. Y.-S. Choi, S. M. Alamouti, and V. Tarokh, “Complementary beamforming: new approaches,” IEEE Transactions on Communications, vol. 54, no. 1, pp. 41–50, 2006. M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Transactions on Vehicular Technology, vol. 53, no. 1, pp. 18–28, 2004. W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP ’98), vol. 6, pp. 3337–3340, Seattle, Wash, USA, May 1998. B. C. Banister and J. R. Zeidler, “Tracking performance of a stochastic gradient algorithm for transmit antenna weight adaptation with feedback,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP ’01), vol. 5, pp. 2965–2968, Salt Lake City, Utah, USA, May 2001. B. C. Banister and J. R. Zeidler, “A simple gradient sign algorithm for transmit antenna weight adaptation with feedback,” IEEE Transactions on Signal Processing, vol. 51, no. 5, pp. 1156– 1171, 2003.


Research Article NAF, OAF, or Noncooperation: Which Protocol to Choose? Ahmed Saadani and Olivier Traore´ France Telecom Division of Research and Development, 38-40 Rue du Général Leclerc, 92794 Issy les Moulineaux Cedex 9, France Correspondence should be addressed to Ahmed Saadani, [email protected] Received 1 June 2007; Revised 20 September 2007; Accepted 1 November 2007 Recommended by G. K. Karagiannidis The two main Amplify and Forward cooperative protocols are the orthogonal (OAF) and the nonorthogonal one (NAF). In this paper, we consider a given source, N relays, a destination, and a channel realization and we try to resolve the following problem: what is the best way to communicate: without cooperation or using one of the two cooperative protocols? This is equivalent to a power-sharing problem on the cooperation frame between source and relays aiming to the short-term channel capacity maximization. The obtained solution shows that cooperative protocol choice depends only on the available power at the relays. However the decision to cooperate depends on the channel conditions. We show that our power allocation scheme with relay selection improves the outage probability compared to the selective OAF and the NAF protocols and has a significant capacity gain. Copyright © 2008 A. Saadani and O. Traoré. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Communications on wireless channels are limited by multiple impairment sources (multipath fading, shadowing, and path loss). Many diversity techniques have been developed to fight the fast fading such as multiple antennas for the spatial diversity, coding for the time diversity. Recently, cooperative diversity technique has attracted much attention because it is able to combat not only the fast fading but also the shadowing and the path loss [1, 2]. It considers a source, a destination, and several relay nodes distributed throughout the network. The relay set forms a virtual antenna array and by using cooperation protocols they can exploit the diversity as a multiple-in multiple-out (MIMO) system [3]. One can distinguish three main classes of cooperative strategies [2]: amplify-and-forward (AF), decode-and-forward (DF) and compress-and-forward (CF). A cooperation protocol is in general composed of two phases. In the first one, the source transmits the information to the relays and to the destination. In the second, when only the relays are authorized to transmit, the protocol is considered as orthogonal. In this case, the receiver processing is simple. However, when the source continues to transmit leading to a throughput increasing [4] the protocol is not orthogonal. The AF protocols have been more studied than others because of their simplicity. Indeed, the relay stations

have to only amplify and forward to the destination the signal received from the source by respecting a power constraint. A way to prolong the different network nodes lifetime and to optimize the system performance is to make a power allocation. The adaptive power allocation for wireless networks has been mainly addressed for orthogonal protocols. In [5–9], the ODF protocol ergodic capacity or the outage capacity was optimized. In [5, 10–12], the OAF protocol power allocation was optimized by considering the signal-tonoise ratio or the outage probability. They respect in general the source and relay maximum power constraints and a per frame power budget. Solutions are optimal power allocations to the source at the first cooperation slot and to the relay at the second one. In [13], only one relay is considered and the NAF protocol power allocation was obtained for downlink using iterative procedure considering separately the source and relay maximum power constraints. For some channel conditions, zero power was allocated to the relay leading to a direct transmission. Hence, selective cooperative protocols are obtained. Previous works studied separately these NAF and OAF protocols but the problem of the best protocol choice was rarely addressed. In our paper, we fix the sum power per slot over all transmitters and we consider a general problem of power sharing between the source and the relays under maximum power constraints at the relays. The power repartition

2


per slot is chosen to make fair the comparison with no cooperation case and to limit the interference level in the network. The considered criterion to be optimized is the instantaneous mutual information between the source and the destination. When the individual power constraint at the relay surpasses the transmitting one, the optimal solution is that the source and the relay should not share the power in the second slot: either the source or the relay should transmit and the choice is dictated by the channel conditions. However, when the cooperation is chosen and the relay has not sufficient power to achieve the allowed transmission level per slot, the remaining power is reallocated to the source to transmit in second slot. This is equivalent to the selective NAF protocol use. This paper is organized as follows. In Section 2, we describe the system model. The problem formulation is addressed in Section 3. In Section 4, we point out the best protocol to use with its optimal power allocation respecting the considered constraints. Section 5 gives simulation results that compare the outage behavior and the capacity of the proposed solutions compared to the selective OAF [2] and the NAF [4] protocols. In Section 6, we give conclusions. 2.

SYSTEM MODEL

We consider a network with N + 2 nodes uniformly distributed. It consists of a source (s), a destination (d) and the remaining N nodes can serve as potential relay nodes (ri ). The cooperation frame for the N relays is shown in Figure 1 and is composed of N subframes. Each one is divided into two slots. In the sequel h, fi , and gi denote respectively the instantaneous channel gains between source and destination, source and node i, and node i and destination. wi and nik denote, respectively, the additive noises at the ith relay node and at the destination during the ith cooperation subframe and the kth time slot. The channel gains are assumed to be independent, zero-mean complex gaussian distributed random variables with variances σ h , σ fi , and σ gi . The additive noises at the relay nodes and at the destination are assumed to be independent, zero-mean gaussian distributed random variables with variance N0 . We consider the NAF protocol proposed in [4] which is a general cooperative protocol representation since the OAF one and the direct transmission correspond to particular power allocations per slot. The source (s) transmits during the ith cooperation subframe duration to the destination (d), the relay (ri ) retransmits to the destination (d) by amplifying what it has received from the source (s) during the first time slot. The system can be characterized as follows:

yi1d = h P1 xi1 + ni1 ,

yi2d = h P2 xi2 + gi βi yir + ni2 , yir

(1)

= fi P1 xi1 + wi ,

where xi1 and xi2 are, respectively, the first and the second symbols transmitted by the source during the ith cooperation subframe. yi1d and yi2d are the first and the second symbols received at the destination during the ith cooperation sub-

s

α1

1

1

αN

1 − α1

r1

1 − αN

rN

Transmit Receive

d

Figure 1: General cooperative frame for N relays.

frame. yir is the symbol received by the ith relay node from the source, and βi is the scale factor of the ith relay node with

βi ≤

Pri , P1 | fi |2 + N0

(2)

where Pri is the relay i transmitting power that should satisfy the constrain Pri ≤ Prmax i

(3)

and P1 and P2 are, respectively, the transmitting power of the source at the first and the second slots. After vectorization, the received frame can be written as ⎛

H1 0 . . .

yd

⎜ ⎜ ⎜ =⎜ ⎜ ⎝

⎞

0 .. ⎟ . ⎟ ⎟

0 H2 0 ⎟ x + w, .. ⎟ . . 0 .. 0 ⎠ 0 . . . 0 HN

(4)

where yd = [y1d , . . . , yNd ]t with yid = [yi1d , yi2d ], x = [x1 , . . . , xN ]t with xi = [xi1 , xi2 ],

⎛ ⎜ ⎜ ⎜ Hi = ⎜ ⎜ ⎜ ⎝

h

P1 N0

⎞

0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ fi gi P1 βi P2 ⎠ h

2 2 2 2 N 1 + β g 0 i N0 1 + βi gi i

(5)

is the normalized channel matrix and w is the noise vector with w∼N (0, I). 3.

PROBLEM FORMULATION

We propose to determine the best protocol to use for a given channel realization h, gi , fi , a fixed sum over all transmitters power budget P1 per slot and relays power constraints . For this purpose, we consider the NAF protocol with Prmax i the general power allocation presented in Figure 1. At each second slot per subframe i, the power P1 is divided into a part P2 = αi P1 allocated to the source and Pri = (1 − αi )P1 to the relay with 0 ≤ αi ≤ 1. The power allocation is chosen to maximize the mutual information between the source and the destination

{α1 , . . . , αN } = arg max I x, y d with 1 − αi P1 ≤ Prmax . i

(6)

A. Saadani and O. Traoré

3 opt

Using (4) the mutual information is

I x, yd = log 2 det I2N + HHH N

(7)

I xi , yid ≤ N max i I xi , yid ,

=

i=1

where

|βi |2 | fi |2 |gi |2 P1 + |h|2 P2

N0 N0 1 + |βi |2 |gi |2 |h|4 P1 P2 + 2 2 . N0 1 + |βi |2 gi

I xi , yid = log 2 1 +

|h|2 P1

+

(8) Replacing βi by its maximum value (2), we obtain

I xi , yid

| fi |2 P1 |gi |2 Pri + |h|2 P2 N0 + | fi |2 P1

1+ + N0 N0 N0 + | fi |2 P1 + |gi |2 Pri

4 |h| P1 P2 N0 + | fi |2 P1

. + 2 N0 N0 + | fi |2 P1 + |gi |2 Pri

= log 2

In the sequel, the optimal power allocation per subframe i respecting the power constraints on the relay i is detailed. Without loss of generality, we distinguish two cases depending on the available power for all relays: if it is higher than the source one or not.

|h|2 P1

(9) Now, let a0 = |h|2 P1 /N0 , ai = | fi |2 P1 /N0 and bi = |gi |2 P1 /N0 . By replacing P2 and Pri by their values it is easy to obtain that

= (ii) when function (11) is decreasing, we have αi max (1 − Prmax /P1 , 0). Hence, the relay i can cooperate, i (iii) for each subframe, we should allocated all power to opt the relay (i.e., αi = 0) leading to an OAF protocol choice. However, when this power exceeds the individual power constrain, it is reallocated to the source (i.e., /P1 ) which means that the NAF protocol αi = 1 − Prmax i is selected; opt (iv) the optimal αi is expected to have infinite possibilities depending on h, fi , and gi , however there is only two possible values which make the feedback very simple (only one bit per relay is needed).

d

I xi , yi = log 2

ai bi 1 − αi + a0 αi 1 + ai

1 + a0 + 1 + ai + bi 1 − αi

a20 αi 1 + ai

+ . 1 + ai + bi 1 − αi

4.1.

In this case, the constraint (1 − αi )P1 ≤ Prmax is always met i meaning that the relay i could transmit with power P1 . The power allocation scheme used to maximize the system capacity is given by

opt αi

(10)

Hence, resolving problem (6) is equivalent to find for every i the αi that maximizes (10) under the constraint that (1 − opt αi )P1 ≤ Prmax . Once the optimal values αi are obtained, the i upper bound (7) is achieved by communicating with only one relay i0 that satisfies I(xi0 , yid0 ) = max i {I(xi , yid )}. This selection leads to a short-term cooperation protocol choice which could be NAF, OAF, or direct transmission.

≥ P1 Case Prmax i

1 if Ai Di − Bi Ci ≥ 0, 0 if Ai Di − Bi Ci < 0,

=

where Ai Di − Bi Ci is the term determining the derivation sign of (11) (see the Appendix). We recommend hence either the opt OAF protocol (i.e., αi = 1) presented in [2] or not to coopopt erate with the relay i (i.e., αi = 0). A relay station i will serve during the subframe i if the global channel capacity when relaying the source’s signal to the destination is enhanced. If there are more than one relay station and all of them have the same power constraint, one can select for the global cooperative frame the one that maximizes the following expression: γi =

4.

PROTOCOL SELECTION

I

xi , yid

= log 2

ai bi . 1 + ai + bi

(13)

opt

Resolving the formulated problem allows to find a method that selects the best protocol based on the power available at the relay nodes and the channel realizations. Let Ai = a0 (1 + a0 )(1 + ai ) − bi (1 + a0 + ai ), and Bi = ai bi + (1 + a0 )(1 + ai + bi ) and Ci = −bi and Di = 1 + ai + bi . Equation (10) is equivalent to

(12)

Ai αi + Bi . Ci α i + D i

(11)

The behavior of (11) is reflected by its first derivative sign. We show in the Appendix that for a fixed channels realization h, fi , and gi the ∂I(xi , yid )/∂αi has a constant sign and hence I(xi , yid ) is a monotonous function of αi ∈ [0, 1]. This is an important result, indeed opt

(i) when function (11) is increasing, we have αi = 1 which means that the relay i should not cooperate;

In fact, since αi = 0, it is easy to show that (13) maximizes (7) and the upper bound is achieved. The cooperative frame will be reduced to only one subframe. Our best relay selection leads to a selective OAF protocol that we call OAFPA (OAF with power allocation) protocol. We remind that selective OAF (S-OAF) protocol was addressed in [2] where the selection is based on the outage probability: The cooperation is used only when the direct link is in outage. However in our protocol, the cooperation can be used even if the direct link is not in outage since we select the transmission method that maximizes the instantaneous capacity. Performance comparison between the two protocols is done and discussed in Section 5. 4.2.

≤ P1 Case Prmax i

Here, the NAF protocol should be used, since the constraint is met if and only if αi ≥ (1 − Prmax /P1 ). (1 − αi )P1 ≤ Prmax i i

4


The power allocation scheme used to maximize the system’s capacity is given by =

Prmax ⎪ ⎩1 − i P1

if Ai Di − Bi Ci ≥ 0, if Ai Di − Bi Ci < 0.

(14)

As previously stated, a relay station will only serve if it performs better than the direct transmission. Unlike [4], there is a relay selection depending on the channel realizations. If there are more than one relay and all of them have the same power constraint, using (7) and (11) one can easily / 0), it suffices show that to achieve the upper bound of (αi = to select only the relay that maximizes γi =

opt Ai αi + Bi , opt Ci αi + Di

Outage probability

opt

αi

⎧ ⎪ ⎨1

Outage probability vs SNR in a symmetric one relay network

100

R=6

10−1

R = 10

10−2

R=2

R=4 10−3

(15)

0

opt

since αi = / 0. SIMULATION RESULTS

We consider a symmetric network with equal channel variances σ h = σ fi = σ gi = 1. The relay number N is fixed to one or three. The analyzed performance is the outage probability and the capacity. The proposed protocols based on the optimal power allocation with relay selection are compared with the following. (i) The S-OAF protocol proposed in [2] and reminded in Section 4.1. Our proposed protocol is OAFPA (OAF with power allocation). (ii) The NAF protocol proposed in [4] since there is any selective NAF yet known. We remind that the power is equally divided between the source and the relay at the second slot for every subframe [4]. Our proposed protocol is called NAFPA. For simplicity, we assume that all the relays have the same . As previously, we distinguish hence maximum power Prmax i the following two cases. 5.1.

Case Prmax i

20

(1) Outage probability Figure 2 compares the outage probability for N = 1 and different transmitting rates R (bits per channel use). The proposed solution and the S-OAF protocol have the same performance because they have the same outage criterion. However, for N = 3 the OAFPA protocol gives the best performance as shown in Figure 3. Indeed, it selects from the three relays the one that maximizes the system capacity corresponding to the upper bound of (7) when it is higher than the noncooperation ones. On the other hand, S-OAF tests first if the direct link is in outage. If it is the case, it uses the three relays to evaluate the capacity which is in general lower than the upper bound of (7).

50

60

Figure 2: Outage probability comparison for orthogonal protocols, N = 1.

100

Outage probability vs SNR in a symmetric three relays network

R=6

10−1

R = 10

10−2 R=2 R=4

10−3

10−4

0

10

20

≥ P1

In order to satisfy the power constraint at the relay, the Prmax i is fixed to P1 .

30 40 SNR (dB)

No cooperation S-OAF OAFPA

Outage probability

5.

10

30 40 SNR (dB)

50

60

70


Figure 3: Outage probability comparison for orthogonal protocols, N = 3.

(2) Ergodic capacity Unlike the outage probability, Figure 4 shows that even with one relay the OAFPA protocol capacity outperforms the SOAF ones. The ergodic capacity depends on R because this parameter is used for the relay selection criterion in the SOAF protocol. The fact that it decides not to cooperate when the direct link is not in outage, without considering if the capacity when relaying is better, degrades the performance. This is amplified for high spectral efficiencies R. The OAFPA

A. Saadani and O. Traoré

5

Capacity vs SNR in a symmetric one relay network

100

Outage probability vs SNR in a symmetric one relay network

101 Outage probability

Capacity (bit/s/Hz)

10−1

R = 10 R=6 R=4

100

R=6

10−2

R = 10 R=2

10−3

R=4

R=2

10

20

30 SNR (dB)

40

50

10−4

60

0

(1) Outage probability The outage probabilities with different spectral efficiencies R are presented for N = 1 and N = 3, respectively in Figures 5 and 6. The NAFPA protocol outperforms the NAF protocol for all cases thanks to the power allocation and the optimal relay selection. The NAF protocol performance suffers from the selection absence at low SNR. (2) Ergodic capacity In Figure 7, the NAFPA protocol capacity outperforms the NAF one for all SNR for N = 1. Indeed, this is due to the selection of the best way to communicate that maximizes the capacity. Both protocols have the same instantaneous capacity only when the cooperation is decided. 6.

CONCLUSION

In this work, we have proposed to find the best way to communicate under power constraints per slot at the relays. The NAF, OAF, or noncooperation protocols choice is equivalent

40

50

60

70

Figure 5: Outage probability comparison for nonorthogonal protocols, N = 1.

100

Outage probability vs SNR in a symmetric three relays network

10−1 Outage probability

We assume that Prmax = 3P1 /4 and hence P1 /4 is used by the i source in the second slot.

30

No cooperation NAF NAFPA

protocol capacity is very close to the noncooperation ones since the cooperation is not frequently decided for a symmetric network. Finally at very high SNR, the three protocols have the same capacity since the direct transmission is always selected. < P1 5.2. Case Prmax i

20

SNR (dB)


Figure 4: Ergodic capacity comparison for orthogonal protocols, N = 1 and minimum transmitting rate R.

10

R=6 R = 10

10−2 R=2 R=4

10−3

10−4

0

10

20

30

40

50

60

70

SNR (dB) No cooperation NAF NAFPA

Figure 6: Outage probability comparison for nonorthogonal protocols, N = 3.

to a power allocation problem to maximize the system capacity. The solution showed that the cooperation mode per subframe (OAF or NAF) depends only on the power constraints at the relays. We gave simple conditions needed to decide to cooperate or not. The obtained optimization leads to new proposed cooperation protocols that combines power allocation with relay selection (OAFPA and NAFPA protocols) respecting the per slot constraints.

6

EURASIP Journal on Advances in Signal Processing Case B (Ai < 0)

Capacity vs SNR in a symmetric one relay network

Now, to obtain the sign of (A.1) two subcases need to be considered: Ai Di − Bi Ci ≥ 0 and Ai Di − Bi Ci < 0.

101 Capacity (bit/s/Hz)

(1) Case Ai Di − Bi Ci ≥ 0 The mutual information derivative’s numerator is assumed to be positive. First, we rewrite this numerator as

Ai Di − Bi Ci = Ai Ci 100

D i Bi − . Ci Ai

(A.5)

Since Ai < 0 and knowing that Ci is always negative, (A.5) is positive if and only if (Di /Ci − Bi /Ai ) > 0. That means that 0

10

20

30 SNR (dB)

40

50

60

Ci

No cooperation NAF NAFPA

Figure 7: Ergodic capacity comparison for one relay and Prmax = i 3P1 /4.

xi , yid

∂αi

I(xi , yid )

is

Bi > 0 Di > 0 Ci < 0 −Di 1 + ai + bi = >1 Ci bi

Ci αi + Di ≥ 0,

(A.1)

(A.2)

.

(A.6)

(A.7)

(A.3)

Case A (Ai ≥ 0) The numerator of (A.1) is hence positive and the sign depends only on the denominator one. This latter is the product of two linear functions of αi with αi ∈ [0, 1]. The sign of each one has to be determined and to make a product afterwards. Since Ai ≥ 0, the ratio −Bi /Ai ≤ 0 and the function (Ai αi + Bi ) are positive for all αi ∈ [0, 1]. The positiveness of the denominator lies on the function (Ci αi + Di ) one which is the case as shown in (A.3). We then deduce that

∂I xi , yid ≥0 ∂αi

(A.4) opt

= 1.

=

(2) Case Ai Di − Bi Ci < 0 Similarly to the previous subcase, since Ai < 0 and knowing that Ci ≤ 0, the expression (A.5) is negative if and only if (Di /Ci − Bi /Ai ) < 0. That means that Ci

∀αi ∈ [0, 1].

for αi ∈ [0, 1] and the optimal choice is αi

for αi ∈ [0, 1] and as previously the optimal choice is αi 1.

−Di

The derivative sign analysis lies on the sign of Ai . For this purpose, two cases are distinguished.

Ai

opt

Ai D i − B i C i 1

. = ln2 Ai αi + Bi Ci αi + Di

From the expressions of Ai , Bi , Ci , and Di given previously, the following signs determination is obvious

−Bi

The derivative sign depends only on the denominator (Ai αi + Bi )(Ci αi + Di ) ones. But using (A.3), it only depends on the function Ai αi +Bi sign. It is easy to see that this latter is always positive for all αi ≤ − Bi /Ai . On the other hand, from (A.2) and (A.6) we have −Bi /Ai > 1 and consequently,

The first derivative of

and hence

−Bi

Ai

.

(A.8)

Unlike the previous subcase, (A.8) does not show if the denominator zero is greater than 1. Anyway, it is easy to see that the denominator is positive for all αi ≤ − Bi /Ai . Knowing that the numerator is negative, the derivative is negative for all αi ≤ − Bi /Ai . Moreover, before saying that the derivative is negative for αi ∈ [0, 1], we ensure that −Bi /Ai ≥ 1 which is equivalent to Ai + Bi ≥ 0. By using

Ai + Bi = 1 + a0 + ai + a0 ai + a0 1 + a0 1 + ai ,

(A.9)

we have that Ai + Bi is a sum of positive quantities and the sum is always positive. We can now write

∂I xi , yid ≤0 ∂αi

(A.10)

for αi ∈ [0, 1], however the power constraints at the relay opt /P1 ). impose that the optimal choice is αi = max (0, 1 − Prmax i Using (A.4), (A.7), and (A.10) it is shown that (11) is a monotonous function.

A. Saadani and O. Traoré ACKNOWLEDGMENT The authors thank Mrs Ghaya Rekaya from Ecole Nationale des Telecommunications de Paris for her precious arguments and help in this work. REFERENCES [1] A. Sendonaris, E. Erkip, and B. Aashang, “User coorperation diveristy—part 1: system description,” IEEE Transactions on Communications, vol. 51, no. 11, 2003. [2] J. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behaviour,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. [3] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1073–1096, 2003. [4] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4152–4172, 2005. [5] Y. Li, B. Vucetic, Z. Zhou, and M. Dohler, “Ditributed adapative power allocation for wireless relay networks,” IEEE Transactions on Wireless Communications, vol. 6, no. 3, pp. 948–958, 2007. [6] A. H. Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 2020–2040, 2005. [7] J. Luo, R. S. Blum, L. Cimini, L. Greenstein, and A. Haimovich, “Power allocation in a transmit diversity system with mean channel gain information,” IEEE Communications Letters, vol. 9, no. 7, pp. 616–618, 2005. [8] D. Gündüz and E. Erkip, “Opportunistic cooperation by dynamic resource allocation,” IEEE Transactions on Wireless Communications, vol. 6, no. 4, pp. 1446–1454, 2007. [9] E. Beres and R. S. Adve, “On selection cooperation in distributed networks,” in Conference on Information Sciences and Systems (CISS 2006), pp. 1056–1061, Princeton, NJ, March 2006. [10] X. Deng and A. M. Haimovich, “Power allocation for cooperative relaying in wireless networks,” IEEE Communications Letters, vol. 9, no. 11, pp. 994–996, 2005. [11] M. O. Hasna and M.-S. Alouini, “Optimal power allocation for relayed transmissions over rayleigh-fading channels,” IEEE Transactions on Wireless Communications, vol. 3, no. 6, pp. 1999–2004, 2004. [12] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-andforward relay networks: optimal power allocation versus selection,” in Wireless Communications, IEEE Transactions on Wireless Communications, vol. 6, no. 8, pp. 3114–3123, August 2007. [13] M. Pischella and J.-C. Belfiore, “Optimal power allocation for downlink cooperative cellular networks,” in Proceeding of the IEEE Vehicular Technology Conference (VTC ’07), pp. 2864– 2868, 2007.

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Research Article How to Solve the Problem of Bad Performance of Cooperative Protocols at Low SNR Charlotte Hucher, Ghaya Rekaya-Ben Othman, and Jean-Claude Belfiore Ecole Nationale Superieure des Telecommunications, 46 rue Barrault, 75013 Paris Cedex 13, France Correspondence should be addressed to Charlotte Hucher, [email protected] Received 1 June 2007; Accepted 27 August 2007 Recommended by Ranjan K. Mallik We propose some new adaptive amplify-and-forward (AF) and decode-and-forward (DF) protocols using a selection. The new selection criterion is a function of the instantaneous capacities of all possible transmission schemes (with or without cooperation). Outage probabilities and simulation results show that the adaptive cooperation protocols solve the problem of bad performance of cooperation protocols at low SNR. Moreover, they improve the asymptotic performance of their corresponding AF and DF protocols. Copyright © 2008 Charlotte Hucher et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Diversity techniques have been developed in order to combat fading on wireless channels and improve the reliability of the received message. Recently, cooperation has been proposed as a new mean to obtain “space-time” or “cooperative” diversity [1, 2]. Different nodes in the network cooperate in order to form a virtual MIMO system and exploit space-time diversity even if their hardware constraints do not allow them to support several antennas. Many cooperative protocols have been proposed [3–6] which can be classified in three main families: amplify-and-forward (AF), decodeand-forward (DF), and compress-and-forward (CF). In this paper we are interested in the two first families, which are the more natural ones. AF protocols have been studied the most due to their simplicity. Indeed, the relays just amplify the received signals and forward them. DF protocols require a bit more processing: this strategy consists in decoding the received signals at the relays and then forwarding them. They have interesting performance, however, and are even essential for multihop systems. Asymptotically, both protocols bring diversity and give better performance than SISO which only uses the direct link. However, it does not match noncooperation at low SNR. We propose here a new strategy named adaptive cooperation which can be applied either to AF or to DF protocols.

This new strategy consists in choosing the best transmission scheme, based on a new selection criterion, a function of the instantaneous capacities of all these possible transmission schemes. Selection between cooperation and noncooperation has already been proposed in literature for DF protocols [5, 7], as well as relay selection [8], but never adapted to AF protocols. Moreover, the usual selection criterion of DF protocol is based only on the source-relay outage probability, while the proposed selection takes all the channel links into account. Outage probability calculations and simulation results prove that the new adaptive AF and DF protocols perform asymptotically better than their corresponding AF and DF protocols, and more interesting, solve the problem of poor performance of cooperation protocols at low SNR. 2.

SYSTEM MODEL

We consider N + 1 terminals who want to transmit messages to the same destination D. The channel is shared in a TDMA manner, so each terminal is allocated a different time slot and the system can be reduced to a relay channel with one source, N relays, and one destination (Figure 1). The N + 1 terminals play the role of the source in succession while the others are used as relays. In the next sections, we will use the notation given in Figure 1. The channel coefficient of the link between source

2

EURASIP Journal on Advances in Signal Processing R1

R2 h1

g1 g2

h2 g0

S hn

Rn

D

gn

Figure 1: System model : a relay channel with one source, N relays, and one destination.

(b) SISO case: only direct link is used, symbols are sent over the source-destination link in a noncoded manner, at a rate of 1 symb. pcu. (c) NLOS case: only nonline-of-sight (NLOS) link is used, in a first phase symbols are sent over the source-relay link in a noncoded manner and forwarded by the relay in a second phase. The rate is then 1/2 symb. pcu. Therefore in order to have the same spectral efficiency of 1 symb. pcu case as in the other cases, we need to use a larger constellation. For example, if the other protocols use a 16-QAM constellation, the NLOS scheme must use a 256-QAM. The principle of this new adaptive AF strategy is to evaluate the qualities of the three schemes (SISO, AF, and NLOS) and to select the best of them. Generalization to the N-relay case

S and destination D is g0 , the one between source S and relay Ri is hi , and the one between relay Ri and destination D is gi . We consider a half-duplex channel; each terminal, and in particular the relays, cannot receive and transmit at the same time. The channel links are Rayleigh, slow fading, so we can consider their coefficients as constant during the transmission of at least one frame. We suppose that all terminals are equipped with only one antenna; the MIMO case is not considered in this work. We focus here on the protocol. So, for simplicity, we assume a uniform energy distribution between source and relays, with the total power kept constant. We will see in the following (see Sections 3.3 and 6.2) that channel state information needs to be known only at destination. 3.

NEW SELECTION FOR AF PROTOCOLS

AF protocols proposed in literature [3, 4, 6] bring diversity at high SNR, but their performance at low SNR is poorer than that of the noncooperative scheme. To solve this issue, we introduce the adaptive AF strategy where the choice of a transmission scheme is based on the channel links quality. 3.1. Presentation of the adaptive AF The idea leading to the definition of the adaptive AF strategy is to consider all possible transmission schemes and decide which one to select. In order to better understand this strategy, the one-relay case is detailed, before the generalization to the N-relay case. One-relay case There are only three possible transmission schemes as follows (Figure 2). (a) AF case: full cooperation scheme is used, symbols are sent using the AF protocol. In case of a full rate protocol (NAF [9]), the symbol rate is 1 symbol per channel use (1 symb. pcu).

This selection can be generalized quite easily to a higher number of relays. For example, for 2 relays R1 and R2 , the number of possible schemes is 7: (1) full cooperation: symbols are sent using the AF protocol for 2 relays. With a full rate protocol, the symbol rate is 1 symb. pcu; (2) cooperation with only relay R1 : symbols are sent using the AF protocol for only 1 relay. With a full rate protocol, the symbol rate is still 1 symb. pcu; (3) cooperation with only relay R2 ; (4) noncooperation: symbols are sent in a noncoded manner over the direct link: the symbol rate is 1 symb. pcu again; (5) NLOS link using only relay R1 : symbols are sent in a noncoded manner and the symbol rate is 1/2 symb. pcu; (6) NLOS link using only relay R2 ; (7) both NLOS links: the symbol rate is 1/2 symb. pcu. The number of cases grows with the number of relays. In the N-relay case, there are Nk=0 Nk = 2N different cooperation cases from the noncooperative one (no relay, k = 0) to the full cooperation one (N relays, k = N). If K > 2 terminals are transmitting simultaneously, the signal power has to be divided by K, which makes the signals too difficult to decode. That is why we consider only theNLOS cases with one or two relays, which corresponds to N1 + N2 + N(N + 1)/2 cases. We can remark as well that, in cooperation schemes, even if several relays are used, at each time slot, only two terminals transmit simultaneously. So finally, there are 2N +N(N +1)/2 different transmission schemes to consider. However, this high number of cases does not increase complexity that much. Indeed, only a simple test is necessary to determine the best one. As some schemes are identical except for exchanging coefficients (e.g., NLOS with relay R1 or relay R2 ), the decoding complexity reduces to only (N + 1) + 2 = N + 3 different algorithms. So the complexity of this new selection protocol increases linearly with the number of relays, which is quite reasonable.

Charlotte Hucher et al.

3 R

R

R

g1

h

S

g1

h

D

g0

S

(a) AF scheme

D

g0

S

D

(b) SISO scheme

(c) NLOS scheme

Figure 2: Three possible schemes in the 1-relay case.

Moreover, we will show in the example of Section 4 that depending on the chosen AF scheme, some cases can be omitted, which reduces the complexity even more. 3.2. Selection criterion In the previous subsection we have listed all the 2N + N(N + 1)/2 possible transmission cases. The question is now which criterion to use to select the best one. We propose to study all these schemes and to select the one which has the largest instantaneous capacity. Let us number the possible transmission schemes from 1 to NS = 2N + N(N + 1)/2 and note Ci (H) the instantaneous capacity of the ith scheme. The selected transmission scheme is the one offering the maximum instantaneous capacity arg max

i{1,...,Ns }

Ci (H)

(1)

with

Ci (H) = log 2 1 + SNRH H H .

(2)

3.3. Implementation constraints To implement the new adaptive AF strategy, a node in the network has to decide which transmission scheme to use. We suppose that this node is the destination. So it has to estimate the channel coefficient g0 of the direct link and the product channels gi βi hi for each relay Ri , calculate the instantaneous capacity of each possible transmission scheme, and determine the one to be used. Then it broadcasts no more than log 2 (2N + N(N + 1)/2) = N + 1 bits at both source and relays in order to inform them about its decision. As we consider a slow fading channel, an estimation is made for several frames and so the transmission strategy remains the same. When a new estimation is made and if the strategy has to change, it is effective after a delay of one frame during which the strategy is not optimal. 4.

Table 1: NAF protocol. S

x11

x12

R1

yr1

β1 yr1

R2 . . . RN D

4.1.

x21

x22

yr2

β2 yr2

···

.. y11

y12

y21

y22

xN1

xN2

yrN

βN yrN

yN1

yN2

.

···

NAF protocol

We consider the nonorthogonal AF (NAF) protocol proposed in [4] for the one-relay case and generalized in [5] to N > 1 relays. This protocol is schematized in Table 1 where xi1 , xi2 are the signals to be transmitted, yri is the received signal at the ith relay, yi1 ,yi2 are the received signals at destination, and βi is the scale factor of the ith relay. The source keeps transmitting: x11 during the first time slot and x12 during the second one, and so on. During the first time slot, the first relay listens yr1 , and, during the second time slot, retransmits a scale version of the signal βi yr1 . The optimum value of each scale factor βi has been calculated in [9]: 1 βi = √ 2 , 1 + SNRhi

(3)

where SNR stands for the signal-to-noise ratio. An equivalent model of the form Y = HX +W can be calculated for any number of relays. After vectorization and separation of real and imaginary parts of complex expressions, we obtain a lattice representation of the system. So decoding can be performed by using ML lattice decoders, such as the sphere decoder or the Schnorr-Euchner algorithm. It has been proven in [9] that this protocol is optimal when used with the distributed Golden code [10] for the onerelay case, or a distributed 2N × 2N perfect code [11] for the N-relay case.

EXAMPLE OF THE ADAPTIVE NAF PROTOCOL 4.2.

In order to better understand this new selection strategy and its possible simplifications, we develop in this section the example of the adaptive NAF protocol.

Adaptive NAF protocol

As can be seen immediately in Table 1, the NAF scheme is a parallel protocol. Indeed, the N relays of the NAF scheme

4


5.

100

Outage probability

play exactly the same role and are never used in the same time. By studying the instantaneous capacities of the cooperation schemes using NAF protocol with different number of relays, we can see easily that the greatest instantaneous capacity will be associated to a one-relay case. So we can avoid to study all the NAF strategies with several relays, which reduces considerably the complexity. Indeed, the adaptive NAF protocol is then the result of the selection of the best transmission scheme between the SISO scheme, the NAF schemes using only one relay, and the NLOS schemes using either 1 or 2 relays. Finally, we have only 1 + N + N(N + 1)/2 = (1 + N)(1 + N/2) possible transmission cases to study and 4 corresponding decoding algorithms; and so, we can remark that the decoding complexity does not increase with the number of relays.

10−1

10−2

10−3

10−4

0

20 30 SNR (dB)

40

SISO NAF Adaptive NAF

PERFORMANCE OF THE ADAPTIVE AF STRATEGY

Figure 3: 1-relay scheme: comparison of the outage probabilities of the noncooperative case, the NAF protocol, and the adaptive NAF for 4 bits pcu.

5.1. Outage probability The outage probability can be expressed as a function of the instantaneous capacity. For each scheme numbered from 1 to NS = 2N + N(N + 1)/2 as in Section 3.2:

(4)

where R is the spectral efficiency in bits per channel use (bits pcu). The principle of the adaptive AF protocol is to choose the transmission scheme that maximizes the instantaneous capacity Ci (H) over i. So the instantaneous capacity of the new adaptive AF protocol is larger than each Ci (H) for a fixedchannel realization H. Thus, the selection scheme is in outage if and only if the NS possible transmission schemes are all in outage. So we get

i ∈ 1, . . . , NS .

100

Outage probability

(i) Pout = P Ci (H) < R ,

(AAF) (i) ≤ Pout Pout

10

10−1

10−2

10−3

10−4 0

10

20 30 SNR (dB)

40


(5)

We can calculate and plot the outage probabilities of these different protocols as functions of the SNR thanks to Monte Carlo simulations. In Figure 3, we plot the outage probabilities of the SISO, NAF, and adaptive NAF protocols for a one-relay scheme and a spectral efficiency of 4 bits pcu. We can note that the adaptive NAF performs better than the NAF protocol. Indeed, we obtain a 4 dB asymptotic gain. Even more interesting is the fact that the adaptive NAF always performs better than SISO, even at low SNR, which was the main weakness of the NAF protocol without selection. In Figure 4, we plot the outage probabilities of the SISO, NAF and adaptive NAF protocols for a two-relay scheme and a spectral efficiency of 4 bits pcu. Here again, the enhancement of the adaptive NAF over the NAF protocol is verified, as we obtain a 5 dB asymptotic gain and solve the problem of bad performance at low SNR.

Figure 4: 2-relay scheme: comparison of the outage probabilities of the noncooperative case, the NAF protocol, and the adaptive NAF for 4 bits pcu.

5.2.

Simulation results

In Figures 5 and 6, we plot the frame error rate of the SISO, NAF, and adaptive NAF protocols as functions of the SNR for a spectral efficiency of 4 bits pcu. In Figure 5, the curves for a one-relay scheme are represented. The NAF protocol is implemented with a distributed Golden code [10] and a Schnorr-Euchner decoding. Simulation results confirm theoretical ones obtained by outage probability calculations. We can observe that the a3daptive NAF performs better asymptotically than the NAF protocol, with a 5 dB gain. Moreover, we can check that it solves the problem of bad performance at low SNR.

5

100

100

10−1

10−1

Outage probability

Frame error rate

Charlotte Hucher et al.

10−2

10−3

10−4

10−2

10−3

10−4 0

10

20 30 SNR (dB)

0

40

20 30 SNR (dB)

40

SISO Alamouti DF Adaptive Alamouti DF


Figure 5: 1-relay scheme: comparison of the performance of the noncooperative case, the NAF protocol, and the adaptive NAF for 4 bits pcu.

Figure 7: 1-relay scheme: comparison of the outage probabilities of the noncooperative case, the Alamouti DF protocol, and the adaptive Alamouti DF for 4 bits pcu.

100

100

10−1

10−1 Frame error rate

Frame error rate

10

10−2

10−3

10−2

10−3

10−4

10−4 0

10

20 30 SNR (dB)

40

0

10

20 30 SNR (dB)

40

SISO Alamouti DF Adaptive Alamouti DF


Figure 6: 2-relay scheme: comparison of the performance of the noncooperative case, the NAF protocol, and the adaptive NAF for 4 bits pcu.

Figure 8: 1-relay scheme: comparison of the performance of the noncooperative case, the Alamouti DF protocol, and the adaptive Alamouti DF for 4 bits pcu.

In Figure 6, the curves for the two-relay scheme are represented. The NAF protocol is implemented with a distributed 4 × 4 perfect code [11] and a Schnorr-Euchner decoding. The improved performances of the adaptive NAF are here again confirmed with a 7 dB gain over the NAF protocol. Besides, the problem of bad performance of the NAF at low SNR is solved with two relays too, since the adaptive NAF curve is always under the SISO curve.

problem as the AF protocols: poorer performance at low SNR than SISO.

6.

NEW SELECTION FOR DF PROTOCOLS

This new selection working quite efficiently on AF protocols, we propose to adapt it to DF protocols, which have the same

6.1.

Presentation of the adaptive DF

The adaptive DF strategy is based on the same principle than the adaptive AF strategy. However, relays do not amplify the signals but decode them for both DF and NLOS protocols. So there is one more criterion to take into account. Indeed, a DF or NLOS protocol is efficient only if signals are correctly decoded at relays. According to Shannon theorem, if a source-relay link is in outage, signals cannot be decoded without error at this

6


relay. On the contrary, if a source-relay link is not in outage, detection without error is possible and we can use either a DF or a NLOS protocol using this relay by assuming that no error occurs during detection. So a first selection step has to be added to the protocol. In the N-relay case, the strategy of an adaptive DF protocol is as follows: (1) select only the K relays whose source-relay link is not in outage, (2) select the best transmission scheme in the 2K + K(K + 1)/2 possible ones in term of instantaneous capacity.

Table 2: Alamouti DF protocol.

7.2.

7.

EXAMPLE OF THE ADAPTIVE ALAMOUTI DF PROTOCOL

7.1. Alamouti DF protocol The Alamouti DF protocol is a DF protocol designed for a 1-relay channel and based on the Alamouti space-time code [12]. It requires 4 channel uses to send 2 symbols: the symbol rate is 1/2 symb. pcu. As schematized in Table 2, in the first phase, the source sends the first line of the Alamouti codeword: x1 and x2 , while the relay listens. In the second phase, the relay sends a decoded version of the first line of the codeword, while the source sends the second line of the Alamouti codeword: −x2∗ and x1∗ . The destination keeps listening during the whole transmission. the reAssuming x1 and x2 have been correctly decoded, √ ceived signals can be written in the form Y = SNRHX + W with an equivalent channel matrix H being orthogonal. So, linear decoding can be performed as for the original Alamouti ST code. However, the Alamouti DF protocol can be used only if the signals are correctly decoded at the relay, which, according to Shannon’s theorem, is possible only if the source-relay link is not in outage. In the other case, we can not use the relay, so signals are sent in a noncoded manner over the direct link.

x1

x2

−x2∗

x1∗

R

yr1

yr2

x1

x2

D

y1

y2

y3

y4

Adaptive Alamouti DF protocol

The first test is on the source-relay link. Two cases can occur: (1) either it is in outage, then signals cannot be decoded without error at relay, so we only use the direct link; (2) or it is not in outage, then three different transmission schemes can be considered:

6.2. Implementation constraints As in the adaptive AF strategy, it is the destination who has to select the best transmission scheme. However, before considering the possible transmission schemes, it has to know which relays are usable, that is, which source-relay links are not in outage. We propose that each relay estimates its own source-relay link and transmits a single bit to the destination indicating whether it is in outage or not. Then, the steps are the same as for the adaptive AF: the destination estimates the direct link g0 and the relaydestination links gi for all K relays which are not in outage. Estimations of the source-relay links are not necessary as the relays decode the signals. Thanks to these estimations, it can calculate the instantaneous capacities of all possible transmission schemes and determine the best one. N + 1 bits are then necessary to broadcast the information on the chosen scheme to the source and relays.

S

(a) SISO scheme; (b) Alamouti DF scheme; (c) NLOS scheme. According to the same selection criterion as in Sec ure ??, we choose the one with the greatest instantaneous capacity. 8. 8.1.

PERFORMANCE OF THE ADAPTIVE DF STRATEGY Outage probability

The outage probability of the adaptive DF protocols can be proven to be lower than the outage probability of the corresponding DF protocols in the same manner than for the adaptive AF protocols. It comes directly from the selection criterion which minimizes the instantaneous capacity. In Figure 7, we plot the outage probabilities of the SISO, Alamouti DF, and adaptive Alamouti DF protocols as functions of the SNR, obtained through Monte Carlo simulations. We can see that for the DF protocols too, the new selection criterion brings a great improvement in asymptotic performance with a 4 dB gain, and solves the problem of bad performance at low SNR. 8.2.

Simulation results

We plot the performance simulations of the SISO, Alamouti DF, and adaptive Alamouti DF protocols as functions of the SNR in Figure 8. The improvements due to the new selection criterion are here again confirmed with a 3 dB asymptotic gain, and better or same performance as SISO for low SNR. 9.

CONCLUSION

We proposed adaptive amplify-and-forward (AF) and decode-and-forward (DF) protocols based on a new selection criterion derived from the calculations of the instantaneous capacities of all possible transmission schemes (SISO, cooperative schemes, NLOS schemes). For the adaptive DF protocol, an additional selection on the source-relay links is necessary to ensure an efficient decoding at relays. Both outage probability and performance from simulation results prove

Charlotte Hucher et al. that the adaptive cooperation enhances the performance of the initial cooperation schemes at high SNR, and solves the problem of poor performance at low SNR. REFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part I: system description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, 2003. [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part II: implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, 2003. [3] J. Laneman and G. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. [4] R. Nabar, H. Bölcskei, and F. Kneubühler, “Fading relay channels: performance limits and space-time signal design,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 1099–1109, 2004. [5] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in half-duplex cooperative channels,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4152–4172, 2005. [6] S. Yang and J.-C. Belfiore, “Towards the optimal amplify-andforward cooperative diversity scheme,” IEEE Transactions on Information Theory, vol. 53, no. 9, pp. 3114–3126, 2007. [7] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. [8] A. S. Ibrahim, A. Sadek, W. Su, and K. J. R. Liu, “SPC125: relay selection in multi-node cooperative communications: when to cooperate and whom to cooperate with?” in IEEE Global Telecommunications Conference (GLOBECOM ’06), pp. 1–5, San Francisco, Calif, USA, November-December 2006. [9] S. Yang and J.-C. Belfiore, “Optimal space-time codes for the MIMO amplify-and-forward cooperative channel,” in IEEE International Zurich Seminar on Digital Communications, pp. 122–125, Zurich, Switzerland, February 2006. [10] J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2 × 2 full-rate space-time code with non-vanishing determinants,” IEEE Transactions on Information Theory, vol. 51, no. 4, pp. 1432–1436, 2005. [11] F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, “Perfect space-time block codes,” IEEE Transactions on Information Theory, vol. 52, no. 9, pp. 3885–3902, 2006. [12] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451–1458, 1998.

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Research Article Censored Distributed Space-Time Coding for Wireless Sensor Networks S. Yiu and R. Schober Department of Electrical and Computer Engineering, The University of British Columbia, 2356 Main Mall, Vancouver, BC, Canada V6T 1Z4 Correspondence should be addressed to S. Yiu, [email protected] Received 22 April 2007; Accepted 3 August 2007 Recommended by George K. Karagiannidis We consider the application of distributed space-time coding in wireless sensor networks (WSNs). In particular, sensors use a common noncoherent distributed space-time block code (DSTBC) to forward their local decisions to the fusion center (FC) which makes the final decision. We show that the performance of distributed space-time coding is negatively affected by erroneous sensor decisions caused by observation noise. To overcome this problem of error propagation, we introduce censored distributed space-time coding where only reliable decisions are forwarded to the FC. The optimum noncoherent maximum-likelihood and a low-complexity, suboptimum generalized likelihood ratio test (GLRT) FC decision rules are derived and the performance of the GLRT decision rule is analyzed. Based on this performance analysis we derive a gradient algorithm for optimization of the local decision/censoring threshold. Numerical and simulation results show the effectiveness of the proposed censoring scheme making distributed space-time coding a prime candidate for signaling in WSNs. Copyright © 2008 S. Yiu and R. Schober. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

In recent years, wireless sensor networks (WSNs) have been gaining popularity in a wide range of military and civilian applications such as environmental monitoring, health care, and control. A typical WSN consists of a number of geographically distributed sensors and a fusion center (FC). The low-cost and low-power sensors make local observations of the hypotheses under test and communicate with the FC. Centralized detection schemes require the sensors to transmit their real-valued observations to the FC. However, this automatically translates into the unrealistic assumption of an infinite-bandwidth communication channel. In reality, the WSN has to work in a bandlimited environment. Moreover, as communication is a key energy consumer in a WSN, it is desirable to process the observation data as much as possible at the local sensors to reduce the number of bits that have to be transmitted over the communication channel. Therefore, the sensors typically make local decisions which are then transmitted to the FC where the final decision is made [1–5].

The resulting decentralized detection problem has a long and rich history. The decentralized optimum hypothesis testing problem was first formulated in [1] to provide a theoretical framework for detection with distributed sensors. Traditionally, the local decisions are assumed to be transmitted to the FC through perfect, error-free channels [1–6]. Realistically, the sensors typically work in harsh environments and therefore, fading and noise should be taken into account. The problem of fusing sensor decisions over noisy and fading channels was considered in [7, 8]. The fusion rules developed in [7] require instantaneous channel-state information (CSI). While the fusion rules in [8] do not require amplitude CSI, they still assume perfect phase estimation/synchronization. However, obtaining any form of CSI may not be feasible in large-scale WSNs and cheap sensors make phase synchronization challenging. To avoid these problems, simple ON/OFF keying and corresponding fusion rules were considered in [9]. Furthermore, power efficiency is improved in [9] by employing a simple form of censoring [10], where the sensors transmit only reliable decisions to the FC. The schemes in [7–9] assume orthogonal channels

2


between the sensors and the FC, which entail a large required bandwidth especially in dense WSNs with a large number of sensors. To overcome the bandwidth limitations of orthogonal transmission in WSNs, the application of coherent distributed space-time coding was proposed in [11]. In particular, in [11] each sensor is randomly assigned a column of Alamouti’s space-time block code (STBC) [12] and it is assumed that only two sensors are active randomly at any time. The quantized observations are encoded by the sensors using the respective preassigned columns of the STBC and transmitted to the FC via a common, noorthogonal channel. Since there are typically more sensors than STBC columns, the same column has to be assigned to more than one sensor resulting in a diversity order of 1. The performance degradation due to the diversity loss and the observation noise is analyzed in [11]. We point out that distributed space-time coding is usually employed in relay networks where a cyclic redundancy check (CRC) code can be used to avoid the retransmission of incorrect decisions by the relays [13–15]. In this context, selection relaying first introduced in [16] has some similarities to censoring in sensor networks [9, 10]. However, while in selection relaying the decision whether a relay retransmits a packet or not depends on the instantaneous CSI of the source-relay channel, the censoring decision depends on the observation noise at the sensor. Furthermore, relaying decisions in selection relaying are made on a packet-by-packet basis enabling coherent detection at the destination node but censor decisions are performed on a symbol-by-symbol basis making coherent data fusion at the FC practically impossible. In this paper, we consider noncoherent distributed spacetime block coding for transmission of censored sensor decisions in WSNs. In particular, we make the following contributions. (i) We show that the noncoherent distributed STBCs (DSTBCs) introduced in [14] eliminate the various restrictions and drawbacks of the coherent scheme in [11]. (ii) Moreover, it is shown that censoring of local decisions is essential for the efficient application of distributed space-time coding in WSNs. (iii) We derive the optimum maximum-likelihood (ML) and a suboptimum generalized likelihood ratio test (GLRT) noncoherent FC decision rules for the proposed signaling scheme. (iv) The bit-error rate (BER) at the FC for the GLRT decision rule is characterized analytically. (v) Based on the analytical expression for the BER, we devise a gradient algorithm for calculation of the optimum local decision/censoring threshold. (vi) Our numerical and simulation results show the effectiveness of the proposed transmission scheme and the ability of the noncoherent DSTBC to achieve a diversity gain in WSNs. This paper is organized as follows. In Section 2, we present the system model and introduce the proposed transmission scheme for WSNs. In Section 3, we derive the

H0 /H1 x1 Sensor 1

h1

xK

x2 ···

Sensor 2

u1

u2

DSTBC

DSTBC

s1

s2

Sensor K uK

···

DSTBC sK

···

h2

hK

n

r Fusion center u0

Figure 1: Parallel fusion model with K sensors and one FC. A censored DSTBC is used for transmission from the sensors to the FC.

ML and GLRT noncoherent FC decision rules and analyze the performance of the GLRT decision rule. A gradient algorithm for optimization of the local decision/censoring threshold is provided in Section 4. Simulation and numerical results are given in Section 5, while some conclusions are drawn in Section 6. Notation. In this paper, bold upper case and lower case letters denote matrices and vectors, respectively. [·]T , [·]H , ε{·}, ||·||2 , |·|, and ∪ denote transposition, Hermitian transposition, statistical expectation, the L2 -norm of a vector, the cardinality of a set, and theunion of two sets, respec√ 2 ∞ tively. In addition, Q(x) 1/ 2π x e−t /2 dt, IX , 0X ×Y , and √ j −1 denote the Gaussian Q-function, the X × X identity matrix, the X × Y all zeros matrix, and the imaginary unit, respectively. 2.

SYSTEM MODEL

The binary hypothesis testing problem under consideration is illustrated in Figure 1, where a set K {1, 2, . . . , K } of K distributed sensors tries to determine the true state of nature H as being H0 (the null hypothesis) or H1 (target-present hypothesis). Typical applications for binary hypothesis testing include seismic detection, forest fire detection, and environmental monitoring. The a priori probabilities of the two hypotheses H0 and H1 are denoted as P(H0 ) and P(H1 ), respectively. We assume that P(H0 ) = P(H1 ) = 0.5 throughout this paper. The details of the system model will be discussed in the following subsections. 2.1.

Local sensor decisions

We assume that the sensor observations are described by H0 :xk = −1 + nk , k ∈ K, H1 :xk = 1 + nk , k ∈ K,

(1)

S. Yiu and R. Schober

3

where the local observation noise samples nk , k ∈ K, are independent and identically distributed (i.i.d.). For convenience and similar to [8, 9, 11], we assume identical sensors in this paper and model nk as real-valued additive white Gaussian noise (AWGN) with zero mean and variance σ 2 ε{n2k }, k ∈ K. We note, however, that the generalization of our results to nonidentical sensors (e.g., sensors with different noise variances) is also possible. Upon receiving its own observation, each sensor makes a ternary local decision: ⎧ ⎪ ⎪ ⎨−1

uk = ⎪1 ⎪ ⎩0

if xk < −d, if xk > d, otherwise,

k ∈ K,

d−1 , σ d+1 Pw = Q , σ Pc = Q

(3)

d−1 d+1 . −Q σ σ

(4)

2.2. Noncoherent distributed space-time coding The general concept of DSTBC was originally proposed in [13] to achieve a diversity gain in cooperative networks with decode-and-forward relaying. The DSTBC scheme in [14] is particularly attractive for application in networks with a large number of nodes since its decoding complexity is independent of the total number of nodes. This scheme consists of a code C and a set of signature vectors G. The active relay nodes1 encode the (correctly decoded) source information using a T × N code matrix Φ ∈ C. Each active relay transmits a linear combination of the columns of the informationcarrying matrix Φ. The linear combination coefficients for each node are unique and are collected in a signature vector gk ∈ G, gk 22 = 1, k ∈ K, of length N. In this work, we consider the application of the DSTBC scheme in [14] in WSNs. In particular, sensors encode their local decisions using a noncoherent DSTBC. Since we consider here a binary hypothesis testing problem, C = {Φ0 , Φ1 } 1

sk = ⎪ EΦ1 gk ⎪ ⎩0 T ×1

if k ∈ H0 , if k ∈ H1 , if k ∈ S,

(5)

where E denotes the transmitted energy of sensor k per codeword. We note that sensor k transmits the T elements of sk in T consecutive symbol intervals. The total average transmitted energy per information bit is given by Eb = EK(Pw + Pc ). 2.3.

Channel model

We assume that the sensors transmit time synchronously and that the sensor-FC channels are frequency-nonselective and time-invariant for at least T symbol intervals.2 Therefore, using the equivalent complex baseband representation of bandpass signals, the signal samples received at the FC in T consecutive symbol intervals can be expressed as r =

√

√

sk hk + n = EΦ0 GH0 hH0 + EΦ1 GH1 hH1 + n,

k∈H0 ∪H1

respectively. The probability that a decision is censored is given by Ps = 1 − Pc − Pw = 1 − Q

⎧√ ⎪ ⎪ ⎨√EΦ0 gk

(2)

where d is the nonnegative decision/censoring threshold. While uk = −1 and uk = 1 correspond to hypotheses H0 and H1 , respectively, uk = 0 corresponds to a decision that is deemed unreliable by the sensor and thus censored. For future reference, we denote the sets of sensors with uk = 0, uk = −1, and uk = 1 by S, H0 , and H1 , respectively. Note that K = S ∪ H0 ∪ H1 . It is not difficult to show that the probabilities of correct and wrong sensor decision are given by

has only two elements. To optimize performance under noncoherent detection, we choose Φ0 and Φ1 to be orthogoH nal, that is, ΦH 0 Φ1 = 0N ×N and Φν Φν = IN , ν ∈ {0, 1} (cf. [17]). Each sensor is assigned a unique signature vector gk ∈ G, gk 22 = 1, k ∈ K, of length N. For the design of deterministic and random signature vector sets G, we refer to [14, 15] , respectively. The transmitted signal of sensor k is given by

The relays which fail to decode the source packet correctly remain silent.

(6) where hk and n denote the fading gain of sensor k and a complex AWGN vector, respectively. The columns of the N ×|H0 | matrix GH0 and N × |H1 | matrix GH1 contain the signature vectors of the sensors in H0 and H1 , respectively. The corresponding fading gains are collected in column vectors hH0 and hH1 which have lengths |H0 | and |H1 |, respectively. We model the channel gains hk , k ∈ K, as i.i.d. zero-mean complex Gaussian random variables (Rayleigh fading) with variance σ 2h = ε{|hk |2 } = 1.3 The elements of the noise vector n have variance σ 2n = N0 , where N0 denotes the power spectral density of the underlying continuous-time passband noise process. Equation (6) clearly shows the importance of censoring when applying DSTBCs in WSNs, since incorrect sensor decisions lead to interference. For example, for H = H0 , ideally the term involving Φ1 in (6) would be absent. However,√incorrect decisions√may cause some sensors to transmit EΦ1 gk instead of EΦ0 gk . The considered censoring 2

Time synchronous transmission can be accomplished if the relative delays between the relay nodes are much smaller than the symbol duration. This is usually a reasonable assumption for low-rate WSN applications. We refer the interested reader to [18] for a more detailed discussion on time synchronism in the context of WSNs. 3 This model is justified if the distance between any pair of sensors is much smaller than the distances between the sensors and the FC. The effect of unequal channel variances is considered in Section 5 (cf. Figure 7).

4


scheme reduces the number of incorrect decisions (by choosing d > 0) at the expense of reducing the number of sensors that make a correct decision. However, this disadvantage is outweighed by the reduction of interference as long as d is not too large (cf. Section 5). We note that censoring was not considered in any of the related publications, for example, [11, 13–15]. For example, in [13–15], DSTBCs were mainly applied for relay purposes, where a CRC code can be used to avoid the retransmission of incorrect decisions.

We note that the sums in the numerator and denominator of (7) both have 3K terms, that is, the complexity of the ML decision rule is of orde O(3K ) and grows exponentially with K. In addition, (8) reveals that for the ML decision rule the FC requires knowledge of the signature vectors of all sensors. These two assumptions make the implementation of the ML decision rule difficult, if not impossible in practice. Therefore, we will provide a low-complexity suboptimum FC decision rule in the next subsection.

2.4. Processing at fusion center (FC)

3.2.

The FC makes a decision based on the received vector r and outputs u0 = 1 if it decides in favor of H1 , and u0 = −1 otherwise. Different decision rules may be applied at the FC differing in performance and complexity. In this context, we note that coherent detection is not feasible in large-scale WSNs since the FC would have to estimate and track the channel gains of all (6) suggests that only the effective √ sensors. While√ channels EGH0 hH0 and EGH1 hH1 have to be estimated if distributed space-time coding is applied, this is also not feasible since the sets H0 and H1 typically change after T symbol intervals (i.e., for every new sensor decision). Therefore, only noncoherent decision rules will be considered in the next section.

The received vector can be expressed as

3.

FC DECISION RULES AND PERFORMANCE ANALYSIS

In this section, we present the optimum ML and the generalized-likelihood ratio test (GLRT) noncoherent decision rules. In addition, we provide a performance analysis for the GLRT decision rule.

GLRT decision rule

r = Φheff + neff ,

Φ ∈ Φ0 , Φ1 .

(10)

√

If H0 is the true hypothesis Φ = Φ0 , heff EGH0 hH0 , and √ n EΦ1 GH1 hH1 + √ n, while if H1 is true Φ = Φ1 , heff √eff EGH1 hH1 , and neff EΦ0 GH0 hH0 + n. Equation (10) suggests a two-step GLRT approach for the estimation of the transmitted codewor Φ. In the first step, heff is estimated assuming Φ is known, and in the second step the eff is used to detect Φ. Since the correlation channel estimate h matrix of the effective noise neff depends on GH1 or GH0 , the ML estimate for heff and thus the resulting GLRT decision rule depend on the signature vectors. Therefore, the complexity of this GLRT decision rule is still exponential in K. To avoid this problem we resort to the simpler least-squares (LS) approach to channel estimation. The LS channel estimate is given by

eff arg min r − Φheff 2 = ΦH r. h 2

(11)

heff

Now, the GLRT decision rule can be expressed as 3.1. Optimum maximum-likelihood (ML) decision rule

(7)

where P(H0 , H1 | H0 ) = Pc|H0 | Ps|S | Pw|H1 | and P(H0 , H1 | H1 ) |H | = Pc|H1 | Ps|S | Pw 0 denote the probabilities that the sets H0 , H1 occur for H0 and H1 , respectively. Since r conditioned on H0 , H1 is a Gaussian vector, the conditional probability density function (pdf) f (r | H0 , H1 ) is given by

f r | H0 , H1

exp − rH Br = , π T det(B)

(8)

where the T × T correlation matrix B is defined as B H H H 2 ε{rrH | H0 , H1 } = E(Φ0 GH0 GH H0 Φ0 +Φ1 GH1 GH1 Φ1 )+σ n IT . Now we can express the ML decision rule at the FC as

u0 =

1 −1

if Λo (r) ≥ 1, if Λo (r) < 1.

(9)

Φ∈{Φ0 ,Φ1 }

Φ∈{Φ0 ,Φ1 }

We first provide the optimum ML decision rule. For this purpose, we introduce the likelihood ratio (LR): f r | H1 Λo (r)

f r | H0

H0 ,H1 f r | H0 , H1 P H0 , H1 | H1

, = H0 ,H1 f r | H0 , H1 P H0 , H1 | H0

2

2 H eff = arg max Φ r2 , = arg min r − Φh Φ 2

(12) where all irrelevant terms have been dropped. The FC output = Φ0 , and u0 = 1 if Φ = Φ1 . Clearly, the GLRT u0 = −1 if Φ decision rule does not require CSI and the FC does not have to know the signature vectors of the sensors. 3.3.

Performance analysis for GLRT decision rule

For the optimum ML decision rule, a closed-form performance analysis does not seem to be feasible. However, fortunately such an analysis is possible for the more practical GLRT decision rule. In particular, the BER can be expressed as

Pe = P u0 = 1 | H0 P H0 + P u0 = −1 | H1 P H1 . (13) Since the considered signaling scheme is symmetric in H0 and H1 , (13) can be simplified to Pe = P(u0 = 1|H0 ). Expanding now P(u0 = 1|H0 ) leads to Pe =

H0 ,H1

P u0 = 1 | H0 , H1 P H0 , H1 | H0 ,

(14)


5

where P(u0 = 1 | H0 , H1 ) denotes the probability that u0 = 1 is detected assuming that uk = −1 for k ∈ H0 and uk = 1 for k ∈ H1 , and P(H0 , H1 | H0 ) is given in Section 3.1. Exploiting the orthogonality of Φ0 and Φ1 and using (6) and (12), P(u0 = 1 | H0 , H1 ) can be expressed as

P u0 = 1 | H0 , H1 = P Δ < 0 | H0 , H1 ,

where we have used the fact that P(u0 = 1 | H0 , H1 ) is independent of d and the remaining partial derivative is given by

(15)

∂Pc ∂d |S | |H0 | |H1 |−1 ∂Pw + |H1 |Ps Pc Pw . ∂d

+ |H0 |Ps|S | Pc|H0 |−1 Pw|H1 |

where Δ x22 − y22 , √

x EGH0 hH0 + ΦH 0 n, y

Using (3), (4) and the fundamental theorem of calculus [23], the derivatives in (23) can be expressed as

EGH1 hH1 + ΦH 1 n.

Since Δ is a quadratic form of Gaussian random variables, the Laplace transform ΦΔ (s) of the pdf of Δ can be obtained as 1 ΦΔ (s) = N

N

, i=1 1 + sλxi i=1 1 − sλ yi

2 Dx ε{xxH } = EGH0 GH H 0 + σ n IN ,

(18)

2 D y ε{yyH } = EGH1 GH H 1 + σ n IN ,

respectively. Thus, P(u0 = 1 | H0 , H1 ) can be calculated from [19] P u0 = 1 | H0 , H1

1 = 2π j

c+j ∞ c− j ∞

ΦΔ (s) ds, s

(19)

where c is a small positive constant in the region of convergence of the integral. The integral in(19) can be either computed numerically using Gauss-Chebyshev quadrature rules [19] or exactly using [20, 21]

P u0 = 1 | H0 , H1 = −

Residue

RHS poles

1 −(d+1)2 /2σ 2 ∂Pw = −√ , e ∂d 2πσ ∂Pc 1 −(d−1)2 /2σ 2 = −√ , e ∂d 2πσ 2 ∂Ps 1 −(d+1)2 /2σ 2 2 = √ e + e−(d−1) /2σ . ∂d 2πσ

(17)

where λxi and λ yi denote the eigenvalues of the N × N matrices

ΦΔ (s) , s

(20)

where RHS stands for the right-hand side of the complex plane. The BER at the FC for the GLRT decision rule can be readily obtained by combining (14) and (19). OPTIMIZATION OF CENSORING THRESHOLD d

Since a closed-form calculation of the optimum decision/ censoring threshold d which minimizes Pe does not seem to be possible, we derive here a gradient algorithm for recursive optimization of d. This algorithm is given by [22] d[i + 1] = d[i] + δ

∂Pe , ∂d[i]

(21)

where i is the discrete iteration index and δ is the adaptation step size. Using (14) the gradient in (21) can be expressed as

(24)

For d = 0, we have |S | = 0 and since ∂Pw /∂d < 0 and ∂Pc /∂d < 0 we obtain ∂Pe /∂d < 0. On the other hand, for d→∞, we get |H0 |→0 and |H1 |→0 which results in ∂Pe /∂d > 0.4 Therefore, by the mean value theorem, ∂Pe /∂d = 0 is valid for at least one value of 0 ≤ d < ∞ corresponding to at least one local minimum of Pe [23]. Although numerical evidence shows that there is exactly one local minimum (which therefore is also the global minimum), we cannot formally prove this due to the complexity of the involved expressions. Nevertheless, the above considerations suggest that we initialize the gradient algorithm with d[0] = 0 corresponding to the case of no censoring. The solution found by the algorithm is then guaranteed to yield a performance not worse than that of the no censoring case. Numerical examples will be given in the next section. We note that d will typically be calculated at the FC and the value of d has to be conveyed to the sensors over a feedback channel. However, this feedback channel can be very low rate assuming that the statistical properties of the forward channel and the sensors vary only slowly with time. 5.

4.

(23)

(16)

√

∂P H0 , H1 | | H0 ∂Ps = |S |Ps|S |−1 Pc|H0 | Pw|H1 | ∂d ∂d

SIMULATION RESULTS

In this section, we provide some numerical and simulation results for the proposed censored DSTBCs and the system model introduced in Section 2. We assume that T = 8 symbol intervals are available for transmission of one information bit, that is, orthogonal matrices Φ0 and Φ1 can be found for N ≤ 4. Here, we consider N = 1, N = 2, and N = 4, and generate Φ0 and Φ1 from the 8 × 8 Hadamard matrix H8 , where the orthogonal columns of H8 are normalized to unit length. For example, for N = 2Φ0 consists of the first two columns of H8 , whereas Φ1 consists of the third and fourth

∂P H0 , H1 | H 0 ∂Pe = P u0 = 1 | H0 , H1 , ∂d ∂d H0 ,H1

(22)

4

In fact, it can be shown that ∂Pe /∂d approaches zero from above if d →∞ corresponding to the maximum BER of Pe = 0.5.

6

EURASIP Journal on Advances in Signal Processing 10−1

1.4

10−1

N =1

1.2 1

10−2

N =2

Pe

d 0.6

Pe

0.8 10−2

N =4

10−3

0.4 0.2 0

10−4 0

2

4

6 i

N = 1, δ = 3 N = 2, δ = 1 N = 4, δ = 1 (a)

8

10 ×102

10−3

6 0

2

4

6 i

8

10 ×102

N = 1, δ = 3 N = 2, δ = 1 N = 4, δ = 1 (b)

Figure 2: d and Pe versus iteration number i for a WSN with K = 30 sensors using DSTBCs with N = 1, 2, and 4. 10 log 10 (Eb /N0 ) = 15 dB, σ 2 = 1/4.

column of H8 . For the set of signature vectors G, we adopted the gradient sets described in [14]. Unless stated otherwise, the sensors have a local noise variance of σ 2 = 1/4 corresponding to a signal-to-noise ratio (SNR) of 6 dB and we assume the suboptimum GLRT decision rule and Pe at the FC are obtained using the analytical results presented in Section 3.3.5 d and Pe versus i. First, we investigate the behavior of the adaptive algorithm described in Section 4 for optimization of d. Figure 2 shows d and the corresponding BER Pe at the FC as a function of the iteration number i for N = 1, 2, and 4, respectively. The considered WSN had K = 30 sensors and the channel SNR was 10 log 10 (Eb /N0 ) = 15 dB. d[i] was initialized with 0 and the step size parameter was chosen to achieve a fast convergence while avoiding instabilities. As can be observed from Figure 2 the adaptive algorithm significantly improves the BER over the iterations. While d itself requires more than 600 iterations to converge to the final optimum value, Pe does practically not change after more than 180 iterations for all considered cases. It is interesting to note that the optimum value for d decreases with increasing N, that is, for larger N less censoring should be applied. The reason for this behavior is that the maximum achievable diversity order of a DSTBC is N (cf. [14]) and therefore, the performance of the DSTBC improves notably with increasing number of transmitting sensors only until N sensors transmit. If more than N sensors transmit, the diversity order does not further improve and only a small additional coding gain can be real-

We note that we confirmed the analytical BER results for the GLRT decision rule presented in Section 3.3 by simulations. However, we do not show the simulation results here for conciseness.

10

12 14 16 10 log10 (Eb /N0 ) (dB)

18

20

σ 2 = 1/4, d = 0 σ 2 = 1/4, d = dopt σ 2 = 0, d = 0

Figure 3: Pe versus 10 log 10 (Eb /N0 ) for a WSN with K = 30 sensors using DSTBCs with N = 1, 2, and 4. Considered cases: error-free local sensor decisions (σ 2 = 0, d = 0), noisy sensor decisions without censoring (σ 2 = 1/4, d = 0), and noisy sensor decisions with optimum censoring (σ 2 = 1/4, d = dopt ).

ized. On the other hand, less censoring means that more erroneous decisions are forwarded to the FC which may negate the additional coding gain. Pe versus 10 log 10 (Eb /N0 ). In Figure 3, we consider the BER achievable with the proposed censored DSTBCs at the FC of a WSN with K = 30 sensors as a function of the channel SNR 10 log 10 (Eb /N0 ). For each considered N, we compare the BER for error-free local sensor decisions (σ 2 = 0, d = 0), noisy sensor decisions without censoring (σ 2 = 1/4, d = 0), and noisy sensor decisions with censoring (σ 2 = 1/4, d = dopt ), where dopt denotes the optimum decision/censoring threshold found with the gradient algorithm. Figure 3 clearly shows that DSTBCs suffer from a significant performance degradation due to erroneous decisions if censoring is not applied. Fortunately, with censoring this performance degradation can be avoided and a performance close to that of error-free local decisions can be achieved. Figure 3 also nicely illustrates the diversity gain that can be realized with censored DSTBCs. Pe versus K. In Figure 4, we investigate the dependence of the BER on the total number of sensors in the network for 10 log 10 (Eb /N0 ) = 15 dB. In particular, we show in Figure 4 the BER for error-free local sensor decisions and the GLRT decision rule at the FC (σ 2 = 0, d = 0), noisy sensor decisions with censoring and the GLRT decision rule at the FC (σ 2 = 1/4, d = dopt ), and noisy sensor decisions with censoring and the ML decision rule at the FC (σ 2 = 1/4, d = dopt ).6 6

5

8

We note that we use for the ML decision rule also the decision/censoring threshold dopt found by the proposed gradient algorithm which is based on the GLRT decision rule. Therefore, this threshold is not strictly optimum for the ML decision rule.


7 1.4

0.06

1.2

0.05

1 0.8

0.04

N =2

d

Pe

Pe

N =1

0.07

10−2

0.03

0.6

0.02

0.4

0.01

0.2

N =4

5

10

15

20

25

30

0

1

2

Figure 4: Pe versus total number of sensors K for aWSN using DSTBCs with N = 1, 2, and 4. 10 log 10 (Eb /N0 ) = 15 dB. Numerical results for error-free local sensor decisions and GLRT decision rule (σ 2 = 0, d = 0), numerical results for noisy sensor decisions with censoring and GLRT decision rule (σ 2 = 1/4, d = dopt ), and simulation results for noisy sensor decisions with censoring and ML decision rule (σ 2 = 1/4, d = dopt ).

The results for the GLRT decision rule were obtained numerically based on the analytical results in Section 3.3, whereas Monte Carlo simulation was used to obtain the results for the ML decision rule. For complexity reasons, for the latter case, we only show the results for K ≤ 5. For error-free local sensor decisions, BER is constant for K > N since the diversity order is limited to N and the DSTBC achieves the same performance as the related STBC C for colocated antennas if all K > N sensors transmit. The censored DSTBC with noisy sensor decisions approaches the performance of the DSTBC with error-free sensor decisions as the number of sensors increases. This is due to the fact that as K increases the decision/censoring threshold dopt increases making the transmission of erroneous sensor decisions less likely. Figure 4 also shows that the GLRT decision rule is almost optimum and only small additional gains are possible if the significantly more complex ML decision rule is used. Pe and d versus N. Assuming the GLRT decision rule and 10 log 10 (Eb /N0 ) = 15 dB at the FC, Figure 5 shows Pe and the corresponding optimum decision threshold d as a function of N for K = 1, 2, 4, 10 and 30. Similar to the observation we made in Figure 2, d decreases for increasing signature vector length N for all K. As we have mentioned before, the maximum achievable diversity order for DSTBC is N. For a given K, a smaller d allows more sensors to be active and thus exploits the the extra diversity benefit provided by the longer signature vectors. This figure also shows that d increases for increasing K. This can be also explained easily. For a given d and N, increasing K allows more sensors to transmit. However, our scheme only requires a certain number of sensors to be active to exploit the full diversity benefit and

4

0

1

2

K =1 K =2 K =4 (a)

3

4

N

N

K σ 2 = 1/4, d = dopt , GLRT σ 2 = 1/4, d = dopt , ML σ 2 = 0, d = 0

3

K = 10 K = 30

K =1 K =2 K =4

K = 10 K = 30

(b)

Figure 5: Pe and d versus N for aWSN with K sensors. σ 2 = 1/4 and 10 log 10 (Eb /N0 ) = 15 dB. GLRT fusion rule is shown for all K (solid curves) and ML fusion rule is shown for K = 1 and 2 (dashed curve).

achieve a certain target BER. On the other hand, increasing d decreases the chance of having erroneous decisions being transmitted to the FC. This suggests that our scheme tries to maximize the performance by only allowing the minimum number of sensors (with quality decisions) to transmit. Finally, it is interesting to see that the Pe performance actually deteriorates for N > K for the GLRT fusion rule. This is because for N > K the GLRT fusion rule implicitly estimates eff in a noisy environment the N × 1 effective channel vector h (cf. (11)) whereas the underlying channel vectors, hH0 and hH1 , have a smaller dimensionality K. The increased dimensionality causes a larger channel estimation error while no diversity benefit is achieved because the maximum diversity order is limited to K [14]. In light of this degradation for the GLRT fusion rule, we also simulated the ML fusion rule for K = 1 and K = 2 (dashed curves) and clearly, as expected, the ML decision rule does not suffer from the same degradation. We note that in the practically more relevant case of N < K ML and GLRT decision rules have similar performances (cf. Figure 4). Pe and d versus SNR of local sensors. We investigate the effect of local sensor observation noise on the Pe performance in Figure 6. In particular, we plot Pe versus the SNR of local sensors 10log 10 (1/σ 2 ) for different K and N. We assume the GLRT fusion rule at the FC and the corresponding optimum decision threshold d is also depicted. Furthermore, the channel SNR is fixed to 10 log10 (Eb /N0 ) = 15 dB for all cases. As expected, the network with K = 30 sensors performs better than the network with K = 10 sensors for any N regardless of the sensor observation noise. However, this gain is minimal for large sensor SNR. This is because as the sensor SNR

8

EURASIP Journal on Advances in Signal Processing 0.16

3.5

0.14

3

0.12

K = 10

2.5 K = 30

Pe

0.1

2

0.08

d

K = 30

1.5

0.06

1 K = 10

0.04

0.5

0.02 0

−5

0 5 10 15 10 log10 (1/σ 2 ) (dB)

0

−5

0

5

10

15

10 log10 (1/σ 2 ) (dB)

N =1 N =2 N =4

N =1 N =2 N =4

(a)

(b)

Figure 6: Pe and d versus 10 log 10 (1/σ 2 ) for a WSN with K = 10, and 30 sensors and DSTBC with N = 1, 2, and 4. 10 log10 (Eb /N0 ) = 15 dB.

I.n.d. Rayleigh fading. Until now, we have been considering i.i.d. Rayleigh fading channels. In our last example, we consider independent and nonidentically distributed (i.n.d.) fading channels. In particular, we consider a network with K = 30 sensors and the sensor nodes are uniformity distributed in a circle with radius r and the distance from the center of the circle to the FC is d. We assume i.n.d. Rayleigh fading between the sensors and the FC and the received power decreases as dk−α , where dk is the distance measured from sensor k to the FC and α = 3 is the path loss exponent. Figure 7 depicts the simulated Pe versus 10 log 10 (Eb /N0 ) for different r/d ratios. For a given N, the decision threshold d was optimized for r/d = 0 (corresponding to i.i.d. fading) and it was then used also for r/d > 0. It can be seen from the figure that, as expected, Pe increases with increasing r/d. It is also interesting to note that the performance degradation is larger for larger N. This can be explained as follows. For a given network size K, as we have seen in Figures 4 and 5, d decreases for increasing N. Since a smaller censoring threshold d corresponds to a larger number of active sensors, more sensors are negatively affected by the i.n.d. channels resulting in the greater performance degradation for larger N. 6.

10−1

N =1

10−2 Pe

N =2

10−3 N =4

10−4

6

8

r/d = 0.6 r/d = 0.4

10

12 14 16 10 log10 (Eb /N0 ) (dB)

18

20

r/d = 0.2 r/d = 0

Figure 7: Pe versus 10 log 10 (Eb /N0 ) for a WSN with K = 30 sensors using DSTBCs with N = 1, 2, and 4. σ 2 = 1/4 and i.n.d. Rayleigh fading channels.

increases, most of the sensor decisions will be correct and less censoring is required. This phenomenon is clearly supported by the corresponding d versus 10 log 10 (1/σ 2 ) figure where the optimum decision threshold d approaches zero for increasing sensor SNR. In addition, as more sensors transmit, the maximum achievable diversity order N and the channel SNR will be the ultimate factors which determine Pe and therefore, for a given N, the BER curves for K = 10 and K = 30 converge to the same value for large local sensor SNR.

CONCLUSION

In this paper, we have considered the application of noncoherent DSTBCs in WSNs. We have introduced censoring as an efficient method to overcome the negative effects of erroneous local sensor decisions on the performance of the noncoherent DSTBC. Furthermore, we have derived optimum ML and suboptimum GLRT FC decision rules, and we have analyzed the performance of the latter decision rule. Based on this analysis, we have devised a gradient algorithm for recursive optimization of the decision/censoring threshold. Numerical and simulation results have shown the effectiveness of censoring which eliminates the effect of local decision errors for practically relevant BERs if the number of sensors in the network K is greater than the length of the signature vectors N or in other words, if there are enough sensors to exploit the diversity benefit provided by the DSTBC. Finally, our results have shown that the suboptimum GLRT fusion rule performs very close to the optimum ML fusion rule while having a very low complexity and allowing noncoherent detection at the FC. ACKNOWLEDGMENTS This paper was presented in part at the IEEE Wireless Communications & Networking Conference, Hong Kong, China, March 2007. REFERENCES [1] R. R. Tenney and N. R. Sandell Jr., “Detection with distributed sensors,” IEEE Transactions on Aerospace and Electronic Systems, vol. 17, no. 4, pp. 501–510, 1981. [2] J. N. Tsitsiklis, “Decentralized detection,” in Advances in Statistical Signal Processsing, vol. 2, pp. 297–344, JAI Press, Greenwich, Conn, USA, 1993.

S. Yiu and R. Schober [3] R. Viswanathan and P. K. Varshney, “Distributed detection with multiple sensors—part I: fundamentals,” Proceedings of the IEEE, vol. 85, no. 1, pp. 54–63, 1997. [4] R. S. Blum, S. A. Kassam, and H. V. Poor, “Distributed detection with multiple sensors—part II: advanced topics,” Proceedings of the IEEE, vol. 85, no. 1, pp. 64–79, 1997. [5] P. K. Varshney, Distributed Detection with Multiple Sensors, Springer, Berlin, Germany, 1997. [6] J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth-constrained sensor network,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2617–2624, 2005. [7] B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, “Channel aware decision fusion in wireless sensor networks,” IEEE Transactions on Signal Processing, vol. 52, no. 12, pp. 3454– 3458, 2004. [8] R. Niu, B. Chen, and P. K. Varshney, “Fusion of decisions transmitted over Rayleigh fading channels in wireless sensor networks,” IEEE Transactions on Signal Processing, vol. 54, no. 3, pp. 1018–1027, 2006. [9] R. Jiang and B. Chen, “Fusion of censored decisions in wireless sensor networks,” IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2668–2673, 2005. [10] C. Rago, P. Willett, and Y. Bar-Shalom, “Censoring sensors: a low-communication-rate scheme for distributed detection,” IEEE Transactions on Aerospace and Electronic Systems, vol. 32, no. 2, pp. 554–568, 1996. [11] T. Ohtsuki, “Performance analysis of statistical STBC cooperative diversity using binary sensors with observation noise,” in Proceedings of the 62nd IEEE Vehicular Technology Conference (VTC ’05), vol. 3, pp. 2030–2033, Dallas, Tex, USA, September 2005. [12] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451–1458, 1998. [13] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. [14] S. Yiu, R. Schober, and L. Lampe, “Distributed space-time block coding,” IEEE Transactions on Communications, vol. 54, no. 7, pp. 1195–1206, 2006. [15] B. S. Mergen and A. Scaglione, “Randomized space-time coding for distributed cooperative communication,” in Proceedings of IEEE International Conference on Communications (ICC ’06), vol. 10, pp. 4501–4506, Istanbul, Turkey, June 2006. [16] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. [17] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 543–564, 2000. [18] X. Li, M. Chen, and W. Liu, “Application of STBC-encoded cooperative transmissions in wireless sensor networks,” IEEE Signal Processing Letters, vol. 12, no. 2, pp. 134–137, 2005. [19] E. Biglieri, G. Caire, G. Taricco, and J. Ventura-Traveset, “Computing error probabilities over fading channels: a unified approach,” European Transactions on Telecommunications, vol. 9, no. 1, pp. 15–25, 1998. [20] J. K. Cavers and P. Ho, “Analysis of the error performance of trellis-coded modulations in Rayleigh-fading channels,” IEEE Transactions on Communications, vol. 40, no. 1, pp. 74–83, 1992.

9 [21] E. B. Saff and A. D. Snider, Fundamentals of Complex Analysis for Mathematics, Science and Engineering, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, 1993. [22] J. Nocedal and S. J. Wright, Numerical Optimization, Springer, New York, NY, USA, 1999. [23] R. A. Adams, Single Variable Calculus, Addison-Wesley, Reading, Mass, USA, 1995.


Research Article Code Design for Multihop Wireless Relay Networks ´ erique ´ Fred Oggier and Babak Hassibi Department of Electrical Engineering, California Institute of Technology, Pasadena CA 91125, USA Correspondence should be addressed to F. Oggier, [email protected] Received 2 June 2007; Revised 21 October 2007; Accepted 25 November 2007 Recommended by Keith Q. T. Zhang We consider a wireless relay network, where a transmitter node communicates with a receiver node with the help of relay nodes. Most coding strategies considered so far assume that the relay nodes are used for one hop. We address the problem of code design when relay nodes may be used for more than one hop. We consider as a protocol a more elaborated version of amplify-andforward, called distributed space-time coding, where the relay nodes multiply their received signal with a unitary matrix, in such a way that the receiver senses a space-time code. We first show that in this scenario, as expected, the so-called full-diversity condition holds, namely, the codebook of distributed space-time codewords has to be designed such that the difference of any two distinct codewords is full rank. We then compute the diversity of the channel, and show that it is given by the minimum number of relay nodes among the hops. We finally give a systematic way of building fully diverse codebooks and provide simulation results for their performance. Copyright © 2008 F. Oggier and B. Hassibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Cooperative diversity is a popular coding technique for wireless relay networks [1]. When a transmitter node wants to communicate with a receiver node, it uses its neighbor nodes as relays, in order to get the diversity known to be achieved by MIMO systems. Intuitively, one can think of the relay nodes playing the role of multiple antennas. What the relays perform on their received signal depends on the chosen protocol, generally categorized between amplify-and-forward (AF) and decode-and-forward (DF). In order to evaluate their proposed cooperative schemes (for either strategy), several authors have adopted the diversitymultiplexing gain tradeoff proposed originally by Zheng and Tse for the MIMO channel, for single or multiple antenna nodes [2–5]. As specified by its name, AF protocols ask the relay nodes to just forward their received signal, possibly scaled by a power factor. Distributed space-time coding [6] can be seen as a sophisticated AF protocol, where the relays perform on their received vector signal a matrix multiplication instead of a scalar multiplication. The receiver thus senses a space-time code, which has been “encoded” by both the transmitter and the relay nodes with their matrix multiplication.

Extensive work has been done on distributed space-time coding since its introduction. Different code designs have been proposed, aiming at improving either the coding gain, the decoding, or the implementation of the scheme [7–10]. Scenarios where different antennas are available have been considered in [11, 12]. Recently, distributed space-time coding has been combined with differential modulation to allow communication over relay channels with no channel information [13–15]. Schemes are also available for multiple antennas [16]. Finally, distributed space-time codes have been considered for asynchronous communication [17]. In this paper, we are interested in considering distributed space-time coding in a multihop setting. The idea is to iterate the original two-step protocol: in a first step, the transmitter broadcasts the signal to the relay nodes. The relays receive the signal, multiply it by a unitary matrix, and send it to a new set of relays, which do the same, and forward the signal to the final receiver. Some multihop protocols have been recently proposed in [18, 19], for the amplify-and-forward protocol. Though we will give in detail most steps with a two-hop protocol for the sake of clarity, we will also emphasize how each step is generalized to more hops.

2


The paper is organized as follows. In Section 2, we present the channel model, for a two-hop channel. We then derive a Chernoff bound on the pairwise probability of error (Section 3), which allows us to derive the full-diversity condition as a code design criterion. We further compute the diversity of the channel, and show that if we have a two-hop network, with R1 relay nodes at the first hop, and R2 relay nodes at the second hop, then the diversity of the network is min(R1 , R2 ). Section 4 is dedicated to the code construction itself, and some examples of proposed codes are simulated in Section 5.

where gi j denotes the fading from the ith relay in the first hop to the jth relay in the second hop. The normalization factor c2 guarantees that the total energy used at the first hop relays is P2 T (see Lemma 1). The noise at the jth relay is denoted by w j . (4) At the receiver, we have y = c3

R2

h j B j x j + z ∈ CT

j =1

= c3 c2 c1

2.

j =1

A TWO-HOP RELAY NETWORK MODEL

Let us start by describing precisely the three-step transmission protocol, already sketched above, that allows communication for a two-hop wireless relay network. It is based on the two step protocol of [6]. We assume that the power available in the network is, respectively, P1 T, P2 T, and P3 T at the transmitter, at the first hop relays, and at the second hop relays for T-time transmission. We denote by Ai ∈ CT ×T , i = 1, . . . , R1 , the unitary matrices that the first hop relays will use to process their received signal, and by B j ∈ CT ×T , j = 1, . . . , R2 , those at the second hop relays. Note that the matrices Ai , i = 1, . . . , R1 , B j , j = 1, . . . , R2 , are computed beforehand, and given to the relays prior to the beginning of transmission. They are then used for all the transmission time. Remark 1 (the unitary condition). Note that the assumption that the matrices have to be unitary has been introduced in [6] to ensure equal power among the relays, and to keep the forwarded noise white. It has been relaxed in [4].

+ c3

⎤

⎡ R2

R2

f 1 g1 j .. ⎥ ⎥ . ⎦

⎢ h j B j A1 s, . . . , AR1 s ⎢ ⎣

h j B j c2

j =1

fR1 gR1 j

R1

gi j Ai vi + w j + z

i=1

= c3 c2 c1 B1 A1 s, . . . , B1 AR1 s, . . . , BR2 A1 s, . . . , BR2 AR1 s ⎡

f1 g11 h1

⎢ .. ⎢ ⎢ . ⎢ ⎢ f R gR 1 h 1 ⎢ 1 1 ⎢ .. ×⎢ ⎢ . ⎢ ⎢ f1 g1R2 hR2 ⎢ ⎢ .. ⎢ . ⎣

fR1 gR1 R2 hR2

S∈CT ×R1 R2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4)

H ∈CR1 R2 ×1

+ c3 c 2

R2 R1

h j gi j B j Ai vi + c3

i=1 j =1

R2

h j B j w j + z,

j =1

W ∈CT ×1

The protocol is as follows. (1) The transmitter sends its signal s ∈ CT such that E[s∗ s] = 1.

(1)

(2) The ith relay during the first hop receives

ri = P1 T fi s + vi ∈ CT ,

i = 1, . . . , R1 ,

(2)

c1

where fi denotes the fading from the transmitter to the ith relay, and vi the noise at the ith relay. (3) The jth relay during the second hop receives x j = c2

R1

gi j Ai c1 fi s + vi + w j ∈ CT ,

i=1

⎡

⎢ = c1 c2 A1 s, . . . , AR1 s ⎢ ⎣

fR1 gR1 j + c2

R1 i=1

gi j Ai vi + w j ,

Lemma 1. The normalization factors c2 and c3 are, respectively, given by

⎤

f 1 g1 j .. ⎥ ⎥ . ⎦ j = 1, . . . , R2 ,

where h j denotes the fading from the jth relay to the receiver. The normalization factor c3 (see Lemma 1) guarantees that the total energy used at the first hop relays is P3 T. The noise at the receiver is denoted by z. In the above protocol, all fadings and noises are assumed to be complex Gaussian random variables, with zero mean and unit variance. Though relays and transmitters have no knowledge of the channel, we do assume that the channel is known at the receiver. This makes sense when the channel stays roughly the same long enough so that communication starts with a training sequence, which consists of a known code. Thus, instead of decoding the data, the receiver gets knowledge of the channel H, since it does not need to know every fading independently.

(3) c2 =

c3 =

P2 , P1 + 1 P3 . P2 R1 + 1

(5)

F. Oggier and B. Hassibi

3

Proof. (1) Since E[r∗i ri ] = (P1 + 1)T, we have that

E c2 Ai ri ∗Ai ri = P2 T ⇐⇒ c22 P1 + 1 T = P2 T

If gi j and h j are known, then W is Gaussian with zero mean. Thus knowing fi , gi j , h j , H and s, we know that y is Gaussian.

2

⇐⇒ c2 =

P2 . P1 + 1

(6)

E[y] = c1 c2 c3 SH.

(2) We proceed similarly to compute the power at the second hop. We have

∗

E xj xj = E

R1

c22

gi j Ai ri

i=1

= c22

∗ R1

+ E w∗j w j

gk j Ak rk

(1) The expectation of y given s and H is

(2) Thevariance of y given gi j and h j is

E y − E[y] y − E[y]

E r∗i ri + T = P2 R1 + 1 T,

i=1

i=1 j =1

(7) so that

E c32 B j x j

+ c32 E ∗

⇐⇒ c3 =

P3 . P2 R1 + 1

=

(8)

hjBjwj

c32 c22

R1 R2 i=1

+ c32

Note that from (4), the channel can be summarized as y = c1 c2 c3 SH + W,

R2

j =1

B j x j = P3 T ⇐⇒ c32 P2 R1 + 1 T = P3 T

∗

= E WW ∗ R1 R2 R2 R1 ∗ 2 2 h j gi j B j Ai vi hl gkl Bl Ak vk = c3 c2 E

k=1

R1

(11)

k=1 l=1 R2

h l Bl w l

∗

l=1

gi j h j B j

R2

j =1

+ E zz∗

∗ ∗

∗

gil hl Bl

l=1

R2 2 h j IT + IT =: Ry , j =1

(12)

(9)

which has the form of a MIMO channel. This explains the terminology distributed space-time coding, since the codeword S has been encoded in a distributed manner among the transmitter and the relays.

where

Remark 2 (generalization to more hops). Note furthermore the shape of the channel matrix H. Each row describes a path from the transmitter to the receiver. More precisely, each row is of the form fi gi j h j , which gives the path from the transmitter to the ith relay in the first hop, then from the ith relay to the jth relay in the second hop, and finally from the jth relay to the receiver. Thus, though we have given the model for a two-hop network, the generalization to more hops is straightforward.

Summarizing the above computation, we obtain the obvious following proposition.

3.

Thus the maximum likelihood (ML) decoder of the system is given by

PAIRWISE ERROR PROBABILITY

In this section, we compute a Chernoff bound on the pairwise probability of error of transmitting a signal s, and decoding a wrong signal. The goal is to derive the so-called diversity property as code-design criterion (Section 3.1). We then further elaborate the upper bound given by the Chernoff bound, and prove that the diversity of a two-hop relay network is actually min(R1 , R2 ), where R1 and R2 are the number of relay nodes at the first and second hops, respectively, (Section 3.2). In the following, the matrix I denotes the identity matrix. 3.1. Chernoff bound on the pairwise error probability In order to determine the maximum likelihood decoder, we first need to compute

P y | s, fi , gi j , h j .

(10)

P2 P3 . c22 c32 = P1 + 1 P2 R1 + 1

(13)

Proposition 1.

P y | s, fi , gi j , h j ∗ 1 exp − y − c1 c2 c3 SH × Ry−1 y − c1 c2 c3 SH . = T π det Ry (14)

2

arg max P y | s, fi , gi j , h j = arg min y − c1 c2 c3 SH . s

s

(15) From the ML decoding rule, we can compute the pairwise error probability (PEP). Lemma 2 (Chernoff bound on the PEP). The PEP of sending a signal sk and decoding another signal sl has the following Chernoff bound:

P sk −→ sl

1 ∗ ≤ E fi ,gi j ,h j exp − c12 c22 c32 H ∗ × Sk − Sl Ry−1 Sk − Sl H .

4

(16)

4

EURASIP Journal on Advances in Signal Processing Proof. We first rewrite the channel matrix H as H = H f, with

Proof. By definition,

P sk −→ sl | fi , gi j , h j

⎡

= P P(y | sl , fi , gi j , h j > P y | sk , fi , gi j , h j = P ln P(y | sl , fi , gi j , h j − ln P y | sk , fi , gi j , h j > 0

≤ EW expλ ln P y | sl , fi , gi j , h j , − ln P y | sk , fi , gi j , h j

(17)

⎡

− ln P y | sk , fi , gi j , h j

−1 ∗ ∗ = −λ c12 c22 c32 H ∗ S∗ K − Sl Ry Sk − Sl H + c1 c2 c3 H ∗ −1 ∗ −1 × SK − S∗ l R y W +c1 c2 c3 W Ry Sk − Sl H ∗ = − λc1 c2 c3 Sk − Sl H + W × Ry−1 λc1 c2 c3 Sk − Sl H + W ∗ + λ2 − λ c12 c22 c32 H ∗ Sk − Sl Ry−1 Sk − Sl H

+ W ∗ Ry−1 W, (18)

EW expλ ln P(y | sl , fi , gi j , h j

=

−1

− ln P y | sk , fi , gi j , h j

g11 h1

⎤ ⎥ ⎥ ⎥ ⎥ gR1 1 h1 ⎥ ⎥ ⎥ ⎥ ∈ CR1 R2 ×R1 . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(22)

Thus we have, since f is Gaussian with 0 mean and variance IR1 , 1 ∗ E fi exp − c12 c22 c32 H ∗ Sk − Sl Ry−1 Sk − Sl H 4 ∗ exp − f ∗ f 1 exp − c12 c22 c32 f ∗ H ∗ Sk − Sl = R 1 π 4

=

× Ry−1 Sk − Sl H f df

∗ 1 1 exp − f ∗ IR1 + c12 c22 c32 H ∗ Sk − Sl π R1 4 × Ry−1 Sk − Sl H f df

exp − W RW W −1 expλ ln P y | sl , fi , gi j , h j T π det RW dW − ln P y | sk , fi , gi j , h j ∗

fR1

gR1 R2 hR2

and thus

⎤

⎢ .. ⎢ ⎢ . ⎢ ⎢ ⎢ ⎢ .. H =⎢ ⎢ . ⎢ ⎢g1R2 hR2 ⎢ ⎢ .. ⎢ . ⎣

where the last inequality is obtained by applying the Chernoff bound, and λ > 0. Using Proposition 1, we have λ ln P y | sl , fi , gi j , h j

f1

⎢ . ⎥ R1 ⎥ f =⎢ ⎣ .. ⎦ ∈ C ,

∗ −1 1 = det IR1 + c12 c22 c32 H ∗ Sk − Sl × Ry−1 Sk − Sl H .

2 ∗ = exp λ − λ c12 c22 c32 H ∗ Sk − Sl Ry−1 Sk − Sl H

4

(23)

(19) Similarly to the standard MIMO case, and to the previous work on distributed space-time coding [6], the full-diversity condition can be deduced from (21). In order to see it, we first need to determine the dominant term as a function of P, the power used for the whole network.

since Rw = Ry and

∗ 1 −1 exp − λc1 c2 c3 Sk − Sl H + W T π det RW × Ry−1 λc1 c2 c3 Sk − Sl H + W × dW = 1.

(20) To conclude, we choose λ = 1/2, which maximizes λ2 − λ, and thus minimizes −(λ − λ2 ). We now compute the expectation over fi . Note that one has to be careful since the coefficients fi are repeated in the matrix H, due to the second hop. Lemma 3 (bound by integrating over f). The following upper bound holds on the PEP:

P sk −→ sl

∗ −1 1 ≤ Egi j ,h j det IR1 + c12 c22 c32 H ∗ Sk − Sl Ry−1 Sk − Sl H

4

where H is given in (22).

(21)

Remark 3 (power allocation). In this paper, we assume that the power P is shared equally among the transmitter and the three hops, namely, P1 =

P , 3

P2 =

P , 3R1

P3 =

P . 3R2

(24)

It is not clear that this strategy is the best, however, it is a priori the most natural one to try. Under this assumption, we have that P , R2 (P + 3) 2 P , c22 c32 = R1 R2 (P + 3)2 3 P T . c12 c22 c32 = 3R1 R2 (P + 3)2 c32 =

Thus, when P grows, c12 c22 c32 grows like P.

(25)


5

Remark 4 (full diversity). It is now easy to see from (21) that if Sl − Sk drops rank, then the exponent of P increases, so that the diversity decreases. In order to minimize the Chernoff bound, one should then design distributed space-time codes =0 (property well known as such that det (Sk − Sl )∗ (Sk − Sl ) full diversity). Note that the term Ry−1 between Sk − Sl and its conjugate does not interfere with this reasoning, since Ry can be upper bounded by tr(Ry )I (see also Proposition 2 for more details). Finally, the whole computation that yields to the full-diversity criterion does not depend on H being the channel matrix of a two-hop protocol, since the decomposition of H used in the proof of Lemma 3 could be done similarly if there were three hops or more.

so that

P sk −→ sl

−1

λ2min c12 c22 c32 ≤ Egi j ,h j det IR1 + 2 H ∗ H ! 4 c32 c22 α+T c32 Rj =2 1 h j +1

, (28)

⎛ 2 2 ⎞ h g1 j ⎜ j ⎟ ⎜ ⎟ .. ⎜ ⎟ H ∗H = . ⎝ ⎠ j =1 h j 2 gR j 2 1 ⎛R ⎞ 2 2 2 ⎜ h j g1 j ⎟ ⎜ ⎟ ⎜ j =1 ⎟ ⎜ ⎟ ⎜ ⎟ . . ⎟, =⎜ . ⎜ ⎟ ⎜ ⎟ R2 ⎜ 2 2 ⎟ ⎝ h j gR j ⎠ R2

Proposition 2. Assuming that the code is fully diverse, it holds that the PEP can be upper bounded as follows:

R1

≤ Egi j ,h j

(29)

1

j =1

i=1

× 1+

λ2min c12 c22 c32 4T

which yields

! R2 −1 2 gi j 2 j =1 h j × ! 2 2 ! ! c32 c22 Rk=1 1 Rj =2 1 h j gk j +c32 Rj =2 1 h j +1

λ2min c12 c22 c32 det IR1 + 2 2 2 H ∗ H ! 4 c3 c2 α+T c32 Rj =2 1 h j +1

R1

=

R1

≤ Egi j ,h j × 1+

i=1

i=1 λ2min c12 c22 c32

2 2 2 λ2min c12 c22 c32 1+ 2 2 2 ! R2 2 h j gi j 4 c3 c2 α+T c3 j =1 h j +1 j =1 R

−1

,

where

4T

2 j =1 |h j gi j | ! ! ! c32 c22 (2R2−1) Rk=1 1 Rj =2 1 |h j gk j |2+c32 Rj =2 1 |h j |2+1

−1

.

(26)

α≤|α|

R1 R2 R2 ∗ ∗ ∗ g h B g h B = tr kj j j kl l l k=1

R1

Ry ≤ tr Ry IT

c32 c22

j =1

l=1

R2 R2 ∗ ∗ ∗ g h B g h B ≤ tr k j j j kl l l j =1 k=1 l=1 ( ) R2 R1 ) R2 ) gk j gk∗j h j h∗j B j B∗j tr gkl gkl∗ hl h∗l Bl Bl∗ , ≤ *tr

Proof. (1) Note first that

=

−1

(30) ! R2

×

c12 c22 c32

2 ! 4 c32 c22 α + T c32 Rj =2 1 h j + 1 −1 ∗ ∗ Sk − Sl H × H Sk − Sl

The goal is now to show that the upper bound given in (21) behaves like P min(R1 ,R2 ) when we let P grows. To do so, let us start by further bounding the pairwise error probability.

≤ Egi j ,h j det IR1 +

where λ2min denote the smallest eigenvalue of (Sk − Sl )∗ (Sk − Sl ), which is strictly positive under the assumption that the codebook is fully diverse. Furthermore, we have that

3.2. Diversity analysis

P sk −→ sl

R1 R2

tr

i=1

gi j h j B j

j =1

α

R2 l=1

j, j =1

l,l =1

(31)

gil∗ h∗l Bl∗

R2 2 + T c32 h j + 1 IT , j =1

k=1

(27)

where the last inequality uses Cauchy-Schwartz inequality. Now recall that B j , j = 1, . . . , R2 , are unitary, thus B j B∗j and Bl Bl∗ are unitary matrices, and

tr Bk Bk∗ ≤T

∀k, k .

(32)

6


Thus

so that

α≤T

( ) ) *

R1 ) R2 k=1

=T

=T

j, j =1

( ) ) *

R1 ) R2 k=1

j =1

R1 R2

k=1 j =1

gk j gk∗j h j h∗j

R2

R2 2 R2 R2 R2 2 h j gk j 2 + hl gkl 2 h j gk j ≤ h j gk j +

gkl gkl∗ hl h∗l

j =1

j =1

j =1 l=1,l =j

l,l =1

2 R2 2 hl gkl h j gk j

= R2

R2 R2 R2 h j gk j 2 + hl gkl 2 j =1

(33)

j =1 l=1,l =j R2

= 2R2 − 1

l=1

2 h j gk j .

h j gk j 2 ,

j =1

(38) which concludes the proof.

We can now rewrite

We now set xi :=

P(sk −→ sl ) R1

≤ Egi j ,h j i=1

2,

so that the bound

Egi j ,h j

+1

× 1+

i=1 λ2min c12 c22 c32

4T

γ1

! R2

−1 2 j =1 |h j gi j | × 2 2 ! ! ! c2 c3 2R2 − 1 Rk=1 1 Rj =2 1 |h j gk j |2 +c32 Rj =2 1 |h j |2 +1

R1

γ2

i=1

(39)

λ2min c12 c22 c32

× 1+

! 2 ! 2 ! 4 c32 c22 T Rk=1 1 Rj =2 1 h j gk j +T c32 c32 Rj =2 1 h j +1 −1 R2 2 2 h j gi j × , j =1

(34) which proves the first bound. (2) To get the second bound, we need to prove that R2 2 R2 h j gk j 2 . h j gk j ≤ 2R2 − 1 j =1

can be rewritten as R1

Egi j ,h j i=1

1 + γ1

(35)

j =1

j =1

R2 R2 R2 h j gk j 2 + h j gk j hl gkl . = j =1

(36)

l=1,l =j

Using the inequality of arithmetic and geometric means, we get h j gk j hl gkl =

+

2 ! R2 2 +1 j =1 h j k=1 xk + c3

.

(40)

λ2min P 3 T λ2min P 3 , 2 = 4T3R1 R2 (P + 3) 12R1 R2 (P + 3)2 2 2R2 − 1 P , γ2 = R1 R2 (P + 3)2 P c32 = . R2 (P + 3)

(41)

In order to compute the diversity of the channel, we will consider the asymptotic regime in which P →∞. We will thus use the notation y x . = lim . (42) x = y ⇐⇒ lim P →∞ log P P →∞ log P

R2 2 R2 2 h j gk j ≤ h j gk j

j =1

γ2

−1

xi

! R1

Note here that by choice of power allocation (see Remark 3), γ1 =

By the triangle inequality, we have that

j =1

j =1 |h j gi j |

R1

λ2min c12 c22 c32 1+ 2 2 2 ! 4 c3 c2 α + T c32 Rj =2 1 h j −1 R2 2 2 × hj gi j j =1

≤ Egi j ,h j

! R2

With this notation, we have that . . γ1 = P, γ2 = P 0 = 1,

. c32 = P 0 = 1.

(43)

In other words, the coefficients γ2 and c3 are constants and can be neglected, while γ1 grows with P. Theorem 1. It holds that

h j gk j 2 hl gkl 2 ≤h j gk j 2 + hl gkl 2 ,

R1

Egi j ,h j i=1

(37)

.

=P

1 + P ! R2

k=1 xk

−min{R1 ,R2 }

,

xi

+

! R2 2 + 1 j =1 h j

−1

(44)


7

!

where xi := Rj =2 1 |h j gi j |2 . In other words, the diversity of the two-hop wireless relay network is min(R1 , R2 ).

and that

2 gi j = P −βi j ,

i = 1, . . . , R1 , j = 1, . . . , R2 . (45)

We thus have that xi =

R2 R2 h j gi j 2 = P −(α j +βi j ) j =1

=P

(46)

j =1

max j {−(α j +βi j )}

=P

−min j {α j +βi j }

xk +

.

=P

.

=P

maxk (−min j (α j +βk j ))

meaning that in a product of exponentials, if at least one of the variables is negative, then the whole product tends to zero. Thus, only the integral where all the variables are positive does not tend to zero exponentially, and we are left with integrating over the range for which α j ≥0, βi j ≥0, i = 1, . . . , R1 , j = 1, . . . , R2 . This implies in particular that . . . P −min(min jk (α j +βk j ),min j α j ) + 1 = P −c + 1 = P max(−c,0) = 1

+ P max j (−α j ) + 1

max(−min jk (α j +βk j ),−min j α j )

since c > 0. This means that the denominator does not contribute in P. Note also that the (log P) factors do not contribute to the exponential order. Hence R1

j =1

k=1

i=1

.

=

+ 1.

.

=

The above change of variable implies that 2

and recalling that |h j |2 and |gi2j | are independent, exponentially distributed, random variables with mean 1, we get R1

Egi j ,h j i=1

=

∞

R1

0 i=1

−1

xi

! R2 2 + 1 j =1 h j k=1 xk +

1 + P ! R2

xi

1 + P ! R2

k=1 xk +

R1 R2

×

∞

−1

0 i=1 ∞ R1 0 i=1 ∞ R1

=

0 i=1

×

=

∞

R1

−∞ i=1

1+P

2 2 exp − gi j dgi j

.

× i=1 j =1 R2

×

+

i=1 j =1 R1 R2

P

R2

dβi j

P −βi j dβi j

P−α j dα j j =1 R2

P −α j dα j

j =1 R2

−βi j

dβi j

P −α j dα j , j =1

! R2

k=1 xk + ∞ R1 R2 − f (α j ,βi j ) 0

P

−inf f (α j ,βi j )

−1

xi

1 + P ! R2

=P

(49)

−1

.

P −min(min jk (α j +βk j ),min j α j ) + 1

i=1 j =1 R1 R2

i=1 j =1

=

−1

+

−(1−min j {α j +βi j })

i=1

P −min j {α j +βi j }

R1 R2

P

Egi j ,h j

j =1

j =1

P (1−min j {α j+βi j })

R1

2 2 exp − h j dh j

P

where (·)+ denotes max{·, 0} and the second equality is obtained by writing 1 = P 0 . By Laplace’s method [20, page 50], [21], this expectation is equal in order to the dominant exponent of the integrand

! R2 h j 2 + 1

+

(53)

i=1 j =1 R2

! R2

2 j =1 |h j | + 1 R1 R2 1−min j {α j+βi j } −1 −βi j k=1 xk

R1

−1

xi

1 + P ! R2 1+P

2

dgi j = (log P)P −βi j dβi j , (48)

Egi j ,h j

(47)

dh j = (log P)P −α j dα j ,

(51)

(52)

R2 R2 R2 2 . −min j {α j +βk j } −α j h j + 1 = P + P +1 j =1

k=1

,

. where the third equality comes from the fact that P a + P b = max {a,b} P . Similarly (and using the same fact), we have that R2

. = exp − P −a + P −b = exp − P −min(a,b)

Proof. Since we are interested in the asymptotic regime in which P →∞, we define the random variables α j , βi j , so that 2 h j = P −α j ,

exp − P −a exp − P −b

i=1 j =1

j =1 |h j | R2

dβi j

2

+1 (54)

dα j j =1

,

where

f α j , βi j

=

R1 ! i=1

,

1 − min α j + βi j

- +

j

+

R1 ! R2 ! i=1 j =1

βi j +

R2 ! j =1

(55)

exp − P −βi j (log P)P −βi j dβi j

exp − P

−α j

(log P) P

αj.

In order to conclude the proof, we are left to show that −α j

dα j .

j =1

Note that to lighten the notation by a single integral, we mean that this integral applies to all the variables. Now recall that . . exp − P −a = 1, a > 0, exp − P −a = 0, a < 0, (50)

,

-

inf f α j , βi j = min R1 , R2 .

α j ,βi j

(56)

(i) First note that if R1 < R2 , R1 is achieved when α j = 0, βi j = 0 and if R1 > R2 , R2 is achieved when α j = 1, βi j = 0. (ii) We now look at optimizing over βi j . Note that one cannot optimize the terms of the sum separately. Indeed, if

8

EURASIP Journal on Advances in Signal Processing !

!

βi j are reduced to make Ri=1 1 Rj =2 1 βi j smaller, then the first term increases, and vice versa. One can actually see that we may set all βi j = 0 since increasing any βi j from zero does not decrease the sum. (iii) Then the optimization becomes one over the α j : inf

α j ≥0

R1

, -+

1 − min α j j

i=1

+

R2

αj.

(57)

j =1

Using a similar argument as above, note that if α j are taken greater than 1, then the first term cancels, but then the second term grows. Thus the minimum is given by considering α j ∈ [0, 1] which means that we can rewrite the optimization problem as inf

R1

α j ∈[0,1]

,

1 − min α j

-+

j

i=1

+

R2

αj.

(58)

j =1

R1

, -

1 − min α j j

i=1

+

R2

, -

= R1 1 − min α j j

⎛

+

, -

j

R2

αj

j =1

(59) , -

+ R2 min α j , -

j

(iv) This final expression is minimized when α j = 0, j = 1, . . . , R2 for R1 < R2 and α j = 1, j = 1, . . . , R2 for R1 > R2 , since if R2 −R1 < 0, one will try to remove as much as possible from R1 . Since α j ≤ 1, the optimal is to take α j = 1. Thus if R1 < R2 , the minimum is given by R1 , while it is given by R1 + R2 − R1 = R2 if R2 < R1 , which yields min{R1 , R2 }. Hence infα j ,βi j f (α j , βi j ) = min{R1 , R2 } and we conclude that 1+P ! R2

k=1 xk +

−1

xi

! R2

j =1 |h j |

2

+1

Let us now comment the interpretation of this result. Since the diversity is also interpreted as the number of independent paths from transmitter to receiver, one intuitively expects the diversity to behave as the minimum between R1 and R2 , since the bottleneck in determining the number of independent paths is clearly min(R1 , R2 ).

⎞

− p0 − p1 − p2

.. 0 . 1 − pn−1

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

s = s0 + s1 θ + · · · + sn−1 θ n−1 ∈ K

(64)

Ms = s, C(p)s, . . . , C(p)n−1 s ,

(65)

where s = [s0 , s1 , . . . , sn−1 ]T and C(p) is the companion matrix of p(x). (2) Furthermore,

det Ms = 0 ⇐⇒ s = 0.

(66)

Proof. (1) By definition, Ms satisfies

1, θ, . . . , θ n−1 Ms = s 1, θ, . . . , θ n−1 .

1, θ, . . . , θ n−1 s = s.

We now discuss the design of the distributed space-time code S = B1 A1 s, . . . , B1 AR1 s, . . . , BR2 A1 s, . . . , BR2 AR1 s ∈ CT ×R1 R2 . (61) For the code design purpose, we assume that T = R1 R2 .

(67)

Thus the first column of Ms is clearly s, since

(63)

is of the form

CODING STRATEGY

0 0 0

Proposition 3. Let p(x) be a monic irreducible polynomial of degree n in Q(i)[x], and denote by θ one of its roots. Consider the vector space K of degree n over Q(i) with basis n−1 {1, θ, . . . , θ }. (1) The matrix Ms of multiplication by

.

= P −min{R1 ,R2 } .

(60)

4.

(62)

Set Q(i) := {a + ib, a, b ∈ Q}, which is a subfield of the complex numbers.

j

= R1 + (R2 − R1 )min α j .

0 ··· 0 1 .. .. . . 0 0

0 ⎜1 ⎜ ⎜0 C(p) = ⎜ ⎜. ⎜. ⎝.

αj

≥ R1 1 − min α j

i=1

p(x) = p0 + p1 x + · · · + pn−1 xn−1 + xn ∈ C[x],

j =1

Egi j ,h j

The coding problem consists of designing unitary matrices Ai , i = 1, . . . , R1 , B j , j = 1, . . . , R2 , such that S as given in (61) is full rank, as explained in the previous section (see Remark 4). We will show in this section how such matrices can be obtained algebraically. Recall that given a monic polynomial

its companion matrix is defined by

Now we have that

R1

Remark 5. There is no loss in generality in assuming that the distributed space-time code is square. Indeed, if one needs a rectangular space-time code, one can always pick some columns (or rows) of a square code. If the codebook satisfies that (Sk − Sl )∗ (Sk − Sl ) is fully diverse, then the codebook obtained by removing columns will be fully diverse too (see, e.g., [12] where this phenomenon has been considered in the context of node failures). This will be further illustrated in Section 5.

(68)

Now, we have that sθ = s0 θ + s1 θ 2 + · · · + sn−2 θ n−1 + sn−1 θ n = − p0 sn−1 + θ s0 − p1 sn−1 + · · · + θ n−1 sn−2 − pn−1 sn−1

(69)


9

since θ n = − p0 − p1 θ − · · · − pn−1 θ n−1 . Thus the second column of Ms is clearly ⎛

− p0 sn−1 ⎜ s −p s 1 n−1 ⎜ 0 ⎜ .. ⎜ ⎝ . sn−2 − pn−1 sn−1

⎞

⎛

0 ··· 0 1 .. .. . . 0 0

0 ⎜ ⎟ ⎜1 ⎟ ⎜0 ⎟=⎜ ⎟ ⎜. ⎠ ⎜. ⎝.

0 0 0

− p0 − p1 − p2

.. 0 . 1 − pn−1

⎞

⎛

⎟ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

s0 s1 .. .

(2) Define Ai = C(p)i−1 ,

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

sn−1

B j = C(p)

2

2

sθ = (sθ)θ = C(p)sθ = C(p) s,

j = 1, . . . , n.

1, θ, . . . , θ

n−1

Ms = s 1, θ, . . . , θ

n−1

,

(73)

that s is an eigenvalue of Ms associated to the eigenvector (1, θ, . . . , θ n−1 ). By applying σ j to the above equation, we have, by Q(i)-linearity, that

p(x) = x4 −

⎛ ⎜0 ⎜ ⎜1 ⎜ ⎝0 0

p(x) = x9 −

⎜0 ⎜ ⎜1 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

(75)

which concludes the proof.

Lemma 4. One has that C(p) is unitary if and only if 2 p0 = 1.

(76)

The family of codes proposed in [10] is a particular case, when p0 is a root of unity. The distributed space-time code design can be summarized as follow. (1) Choose p(x) such that | p0 |2 = 1 and p(x) is irreducible over Q(i).

(80)

i+2 , i−2

(81)

is irreducible over Q(i), with companion matrix

j =1

p1 = · · · = pn−1 = 0,

i+2⎞ 0 i − 2⎟ ⎟ 0 0 ⎟ ⎟. 0 0 ⎠ 1 0

0 0 1 0

⎛

The matrix Ms , as described in the above proposition, is a natural candidate to design a distributed space-time code, since it has the right structure, and is proven to be fully diverse. However, in this setting, C(p) and its powers correspond to products of Ai B j , which are unitary. Thus, C(p) has to be unitary. A straightforward computation shows the following.

(79)

Example 6 (R1 = R2 = 3). We need now a monic polynomial of degree 9. For example,

n

σ j (s),

i+2 , i−2

The matrices A1 , A2 , B1 , B2 are given explicitly in next section.

Thus σ j (s) is an eigenvalue of Ms , j = 1, . . . , n, and det Ms =

(78)

which are irreducible over Q(i). Its companion matrix is given by

1, σ j (θ), . . . , σ j θ n−1 Ms = σ j (s) 1, σ j (θ), . . . , σ j θ n−1 . (74)

(77)

For example, one can take

(72)

Now, it is clear, by definition of Ms , namely,

2 p0 = 1.

p(x) = x4 − p0 ,

(71)

and thus sθ j = C(p) j s is the j+1 column of Ms , j = 1, . . . , n− 1. (2) Denote by θ 1 , . . . , θ n the n roots of p. Let θ be any of them. Denote by σ j the following Q(i)-linear map: σ j (θ) = θ j ,

j = 1, . . . , R2 .

,

Example 5 (R1 = R2 = 2). We need a monic polynomial of degree 4 of the form

(70) We have thus shown that for any s ∈ K, sθ = C(p)s. By iterating this processing, we have that

i = 1, . . . , R1 ,

R1 ( j −1)

5.

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

i+2⎞ 0 i − 2⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ . 0 0 ⎟ ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎟ ⎟ 0 0 ⎠ 1 0

(82)

SIMULATION RESULTS

In this section, we present simulation results for different scenarios. For all plots, the x-axis represents the power (in dB) of the whole network, and the y-axis gives the block error rate (BLER). Diversity discussion In order to evaluate the simulation results, we refer to Theorem 1. Since the diversity is interpreted both as the slope of the error probability in log-log scale as well as the exponent of P in the upper bound on the pairwise error probability, one intuitively expects the slope to behave as the minimum between R1 and R2 .

10


Tx

Rx A2

100

A1

B1 Tx

B2

Rx A2

10−1

Figure 1: On the left, a two-hop network with two nodes at each hop. On the right, a one-hop network with two nodes.

BLER

A1

10−2

We first consider a simple network with two hops and two nodes at each hop, as shown in the left of Figure 1. The coding strategy (see Example 5) is given by

10−3

16

18

20

22

24

26

28

30

P (dB)

⎛

⎞

i+2 i − 2⎟ ⎟ ⎟ 0 0 ⎟, ⎟ 0 0 ⎠ 1 0 ⎞ i+2 0 ⎟ i−2 ⎟ i+2⎟ ⎟ 0 . i − 2⎟ ⎟ ⎟ 0 0 ⎠ 0 0

⎜0 0 0

A1 = I4 ,

⎜ ⎜ 1 0 A2 = ⎜ ⎜ ⎝0 1 ⎛

B1 = I 4 ,

0 0

⎜0 0 ⎜ ⎜ ⎜0 0 B2 = ⎜ ⎜ ⎜ ⎝1 0

0 1

(83)

2 nodes

2-2 (no) nodes

2-2 nodes

2 (no)-2 nodes

Figure 2: Comparison between a one-hop network with two relay nodes and a two-hop network with two relay nodes at each hop, “(no)” means that no coding has been done either at the first or second hop.

Tx

We have simulated the BLER of the transmitter sending a signal to the receiver through the two hops. The results are shown in Figure 2, given by the dashed curve. Following the above discussion, we expect a diversity of two. In order to have a comparison, we also plot the BLER of sending a message through a one-hop network with also two relay nodes, as shown on the right of Figure 1. This plot comes from [10], where it has been shown that with one hop and two relays, the diversity is two. The two slopes are clearly parallel, showing that the two-hop network with two relay nodes at each hop has indeed diversity of two. There is no interpretation in the coding gain here, since in the one-hop relay case, the power allocated at the relays is more important (half of the total power, while one third only in the two-hop case), and the noise forwarded is much bigger in the two-hop case. Furthermore, the coding strategies are different. We also emphasize the importance of performing coding at the relays. Still on Figure 1, we show the performance of doing coding either only at the first hop, or only at the second hop. It is clear that this yields no diversity. We now consider more in details a two-hop network with three relay nodes at each hop, as show in Figure 3. Transmitter and receiver for a two-hop communication are indicated and are plotted as boxes, while the second hop also contains a box, indicating that this relay is also able to be a transmitter/receiver. We will thus consider both cases, when it is either a relay node or a receiver node. Nodes that serve as relays are all endowed with a unitary matrix, denoted by either Ai at the first hop, or B j for the second hop, as explained in Section 4.

A1

B1

A2

B2

A3

B3

Rx

Figure 3: A two-hop network with three nodes at each hop. Nodes able to be transmitter/receiver are shown as boxes.

For the upcoming simulations, we have used the following coding strategy (see Example 6). Set ⎛ ⎜0 0 0 0 0 0 0 0

⎜ ⎜ ⎜1 ⎜ ⎜0 ⎜ ⎜0 Γ=⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

i+2⎞ i − 2⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0 ⎠ 0

A1 = I9 ,

A2 = Γ,

A3 = Γ2 ,

B1 = I9 ,

3

B3 = Γ6 .

B2 = Γ ,

(84)

In Figure 4, the BLER of communicating through the twohop network is shown. The diversity is expected to be three. In order to get a comparison, we reproduce here the performance of the two-hop network with two relay nodes already shown in the previous figure. There is a clear gain in diversity


11

100

BLER

10−1

10−2

10−3

16

18

20

2-2 nodes 2-3 nodes

22 24 P (dB)

26

28

30

3-2 nodes 3-3 nodes

Figure 4: Comparison among different uses of either two or three nodes at, respectively, the first and second hops.

Decoding issues All the simulations presented in this paper have been done using a standard sphere decoder algorithm [22, 23].

100

10−1

6.

BLER

10−2

10−3

10−4

10−5 16

Finally, we would like to mention that the scheme proposed does not restrict to the case where communication requires exactly two hops. In order to do so, we assume that one node among those at the second hop can actually be a receiver itself (see Figure 3). We keep the coding strategy described above and simulate a one-hop communication between the transmitter and this new receiver. The performance is shown in Figure 5, where it is compared with a onehop network (as in [10]). Both strategies have now noise forwarded from only one hop. However, the difference of coding gain is easily explained by the fact that we did not change the power allocation, and thus the best curve corresponds to having half of the power at the first hop relays, while the second curve corresponds to a use of only one third of the power. Diversity is of course similar. The main point here is to notice that the coding strategy does not need to change. Thus the unitary matrices can be allotted before the start of communication, and used for either one or two hops communication.

18

20

22 24 P (dB)

26

28

30

4 nodes 1 hop 4 nodes 2 hop

Figure 5: One hop in a one-hop network versus one hop in a twohop network.

obtained by increasing the number of relay nodes. We now illustrate that the diversity actually depends on min{R1 , R2 }, that is, the minimum number relays between the first and the second hops. We assume now that one node in the first hop is not communicating (it may be down, or too far away). We keep the same coding strategy, and thus simulate communication with a first hop that has two relay nodes, and a second hop that has three relay nodes. We see that the diversity immediately drops to the one of a network with two nodes at each hop. There is no gain in having a third relay participating in the second hop. This is true vice versa, if the first hop uses three relays while the second hop uses only two. Though the performance is better, the diversity is two.

CONCLUSION

In this paper, we considered a wireless relay network with multihops. We first showed that when considering distributed space-time coding, the diversity of such channels is determined by the hop whose number of relays is minimal. We then provided a technique to design systematically distributed space-time codes that are fully diverse for that scenario. Simulation results confirmed the use of doing coding at the relays, in order to get cooperative diversity. Further work now involves studying the power allocation. In order to get diversity results, power is considered in an asymptotic regime. In doing distributed space-time coding for multihop, one drawback is that noise is forwarded from one hop to the other. This will not influence the diversity behavior since the power can grow to infinity. However, for more realistic scenarios where the power is limited, it does matter. In this case, one may need a more elaborated power allocation than just sharing equally the power among the transmitter and relays at all hops. ACKNOWLEDGMENTS The first author would like to thank Dr. Chaitanya Rao for his help in discussing and understanding the diversity result. This work was supported in part by NSF Grant CCR0133818, by The Lee Center for Advanced Networking at Caltech, and by a grant from the David and Lucille Packard Foundation. REFERENCES [1] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless

12

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

EURASIP Journal on Advances in Signal Processing networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversity-multiplexing tradeoff in halfduplex cooperative channels,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4152–4172, 2005. P. Elia, K. Vinodh, M. Anand, and P. V. Kumar, “D-MG tradeoff and optimal codes for a class of AF and DF cooperative communication protocols,” to appear in IEEE Transactions on Information Theory. G. Susinder Rajan and B. Sundar Rajan, “A non-orthogonal distributed space-time coded protocol part I: signal model and design criteria ,” in Proceedings of the IEEE Information Theory Workshop (ITW ’06), pp. 385–389, Chengdu, China, October 2008. S. Yang and J.-C. Belfiore, “Optimal space-time codes for the amplify-and-forward cooperative channel,” IEEE Transactions on Information Theory, vol. 53, no. 2, pp. 647–663, 2007. Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Transactions on Wireless Communications, vol. 5, no. 12, pp. 3524–3536, 2006. P. Dayal and M. K. Varanasi, “Distributed QAM-based spacetime block codes for efficient cooperative multiple-access communication,” to appear in IEEE Transactions on Information Theory. Y. Jing and H. Jafarkhani, “CTH17-1: using orthogonal and quasi-orthogonal designs in wireless relay networks,” in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM ’07), pp. 1–5, San Francisco, Calif, USA, November 2007. T. Kiran and B. S. Rajan, “Distributed space-time codes with reduced decoding complexity,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’06), pp. 542–546, Seattle, Wash, USA, September 2006. F. Oggier and B. Hassibi, “An algebraic family of distributed space-time codes for wireless relay networks,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’06), pp. 538–541, Seattle, Wash, USA, July 2006. Y. Jing and B. Hassibi, “Cooperative diversity in wireless relay networks with multiple-antenna nodes,” to appear in IEEE Transactions on Signal Processing. F. Oggier and B. Hassibi, “An algebraic coding scheme for wireless relay networks with multiple-antenna nodes,” to appear in IEEE Transactions on Signal Processing. Y. Jing and H. Jafarkhani, “Distributed differential space-time coding for wireless relay networks,” to appear in IEEE Transactions on Communications. T. Kiran and B. S. Rajan, “Partially-coherent distributed spacetime codes with differential encoder and decoder,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’06), pp. 547–551, Seattle, Wash, USA, September 2006. F. Oggier and B. Hassibi, “A coding strategy for wireless networks with no channel information,” in Proceedings of 44th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Ill, USA, September 2006. F. Oggier and B. Hassibi, “A coding scheme for wireless networks with multiple antenna nodes and no channel information,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP ’07), vol. 3, pp. 413–416, Honolulu, Hawaii, USA, April 2007. X. Guo and X.-G. Xia, “A distributed space-time coding in asynchronous wireless relay networks,” to appear in IEEE Transactions on Wireless Communications.

[18] S. Yang and J.-C. Belfiore, “Distributed space-time codes for the multi-hop channel,” in Proceedings of International Workshop on Wireless Networks: Communication, Cooperation and Competition (WNC3 ’07), Limassol, Cyprus, April 2007. [19] S. Yang and J.-C. Belfiore, “Diversity of MIMO multihop relay channels-part I: amplify-and-forward,” to appear in IEEE Transactions on Information Theory. [20] C. Rao, “Asymptotics analysis of wireless systems with rayleigh fading,” Ph.D. Thesis, 2007. [21] D. Zwillinger, Handbook of Integration, Jones and Bartlett, Boston, Mass, USA, 1992. [22] B. Hassibi and H. Vikalo, “On the sphere-decoding algorithm I. Expected complexity,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2806–2818, 2005. [23] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1639–1642, 1999.


Research Article Link-Adaptive Distributed Coding for Multisource Cooperation Alfonso Cano, Tairan Wang, Alejandro Ribeiro, and Georgios B. Giannakis Department of Electrical and Computer Engineering, University of Minnesota, 200 Union Street, Minneapolis, MN 55455, USA Correspondence should be addressed to Georgios B. Giannakis, [email protected] Received 14 May 2007; Accepted 7 September 2007 Recommended by Keith Q. T. Zhang Combining multisource cooperation and link-adaptive regenerative techniques, a novel protocol is developed capable of achieving diversity order up to the number of cooperating users and large coding gains. The approach relies on a two-phase protocol. In Phase 1, cooperating sources exchange information-bearing blocks, while in Phase 2, they transmit reencoded versions of the original blocks. Different from existing approaches, participation in the second phase does not require correct decoding of Phase 1 packets. This allows relaying of soft information to the destination, thus increasing coding gains while retaining diversity properties. For any reencoding function the diversity order is expressed as a function of the rank properties of the distributed coding strategy employed. This result is analogous to the diversity properties of colocated multi-antenna systems. Particular cases include repetition coding, distributed complex field coding (DCFC), distributed space-time coding, and distributed error-control coding. Rate, diversity, complexity, and synchronization issues are elaborated. DCFC emerges as an attractive choice because it offers high-rate, full spatial diversity, and relaxed synchronization requirements. Simulations confirm analytically established assessments. Copyright © 2008 Alfonso Cano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

In distributed virtual antenna arrays (VAA) enabled by user cooperation, there is a distinction as to how users decide to become part of the VAA for a given transmitted packet. Most relaying techniques can be classified either as analog forwarding (AF), decode-and-forward (DF), and selective forwarding (SF) [1–3]. In SF, prospective cooperators decode each source packet and, if correctly decoded, they cooperate by relaying a possibly reencoded signal. In AF, cooperating terminals amplify the received (transmitted signal plus noise) waveform. Both strategies achieve full diversity equal to the number of users forming the VAA, and in some sense their advantages and drawbacks are complementary. One of the major limitations of AF is that it requires storage of the analog-amplitude received waveform, which strains resources at relaying terminals, whereas SF implementation is definitely simpler. However, relaying information in SF is necessarily done on a per-packet basis eventually leading to the dismissal of an entire packet because of a small number of erroneously decoded symbols. This drawback is sometimes obscured in analyses because it does not affect the diversity gain of the VAA. It does affect the coding gain, though, and in many situations, SF does not improve performance of

noncooperative transmissions because the diversity advantage requires too high signal-to-noise ratios (SNR) to “kickin” in practice [4]. Simple implementation with high diversity and coding gains is possible with the DF-based link-adaptive regenerative (LAR) cooperation, whereby cooperators repeat packets based on the instantaneous SNR of the received signal [4]. In LAR cooperation, relays retransmit soft estimates of received symbols with power proportional to the instantaneous SNR in the source-to-relay link—available through, for example, training—but never exceeding a given function of the average SNR in the relay-to-destination link which is available through, for example, low-rate feedback. With LAR-based cooperation, it suffices to perform maximum-ratio combining (MRC) at the destination to achieve full diversity equal to the number of cooperators [4]. Finally, link-adaptive cooperation was also considered for power-allocation purposes, as in [5, 6], and references therein. In the present paper, we extend LAR cooperation to general distributed coding strategies operating over either orthogonal or nonorthogonal channels. For that matter, we consider a multisource cooperation (MSC) setup, whereby a group of users collaborate in conveying information symbols to a common destination [7, 8]. In Phase 1, terminals

2 sequentially transmit their information bearing signals. Due to the broadcast nature of wireless transmissions, signals are overheard by other terminals that use these received waveforms to estimate the information sent by other sources. In Phase 2, sources (re)encode the aggregate information packet that they then transmit to the destination. Combining the signals received during both phases, the destination estimates the sources data. The goal of this paper is to analyze the diversity of LAR-MSC protocols in terms of general properties of the reencoding function used during Phase 2. Particular cases of reencoding functions include (LAR based) (i) repetition coding, (ii) distributed complex field coding (DCFC), (iii) distributed space-time (ST) coding, and (iv) distributed error control coding (DECC). The use of coding techniques (i)–(iv) in SF relaying has been considered in, for example, [8–12], where different diversity properties are reported. The use of repetition coding as in (i) with average SNR source-relay knowledge at the receivers was tackled in [9] using a piecewise linear (PL) detector that established diversity bounds. Alamouti codes [13] were considered as in (iii) with regenerative relays in [10, 11]. In particular, full diversity was demonstrated in [11] if the per-fading error probability of the relay can become available at the destination. DCFC and DECC cooperation in a multiple access channel, using a 2-phase protocol similar to the one proposed here in (ii) and (iv), was advocated by [8, 12], respectively. Assuming that to participate in the Phase 2, sources have to correctly decode the packets of all other peers, diversity order as high as the number of cooperating terminals was established. In general terms, the present work differs from existing alternatives in that LAR cooperation is employed to enable high error performance (in coding gain and diversity) even if packets are not correctly decoded and realistic channel knowledge is available at terminals and destination. Our main result is to show that the diversity order of LAR-MSC coincides with that of a real antenna array using the same encoding function used by the VAA created by MSC. In particular, this establishes that for a network with N users, the diversity orders are (i) 2 for repetition coding, (ii) N for DCFC, (iii) at least the same diversity order afforded by the ST code in a conventional antenna array when we use distributed ST coding, and (iv) for DECC, the same diversity achieved by the ECC over an N-lag block fading channel. Through simulations we also corroborate that, having the same diversity gain, LAR transmissions enable higher coding gains than those afforded by SF-based transmissions. The rest of the paper is organized as follows. In Section 2, we introduce the 2-phase LAR-MSC protocol. We define a generic encoding function and specialize it to repetition coding and DCFC. We then move on to Section 3 where we present the main result of the paper characterizing the diversity gain in terms of the properties of the distributed coder. We discuss the application of our result to repetition coding, DCFC, distributed ST coding, and DECC in Section 3.2. In this section, we also compare these four different strategies in terms of diversity, decoding complexity, synchronization, and bandwidth efficiency. Section 3.1 is devoted to prove the

EURASIP Journal on Advances in Signal Processing main result introduced in Section 3. We present corroborating simulations in Section 4. Notation 1. Upper (lower) bold face letters will be used for matrices (column vectors); [·]i, j ([·]i ) for the i, jth (ith) entry of a matrix (vector); [·]i,: ([·]:, j ) for the ith ( jth) row (column) of a matrix; [·]i: j will denote a vector formed extracting elements from i to j; IN the N × N identity matrix; 1N the N × 1 all-one vector; ⊗ the Kroneker product; · the Frobenius norm; R ∪ S (R ∩ S) the union (intersection) of sets R and S; |S | the cardinality of a set S; ∅ the empty set; and CN (μ, σ 2 ) will stand for a complex Gaussian distribution with mean μ and variance σ 2 . 2.

LINK-ADAPTIVE REGENERATIVE MULTI-SOURCE COOPERATION

Consider a set of sources {Sn }Nn=1 communicating with a common access point or destination SN+1 . Information bits of Sn are modulated and parsed into K × 1 blocks of symbols xn := [xn1 , . . . , xnK ]T with xnk drawn from a finite alphabet (constellation) As . Terminals cooperate in accordance to a two-phase protocol. In Phase 1, sources {Sn }Nn=1 transmit their symbols to the destination SN+1 in nonoverlapping time slots. Thanks to broadcast propagation, symbols transmitted by Sn are also received by the other N − 1 sources (m) {Sm }N represent the K × 1 m=1,m= / n ; see also Figure 1. Let yn block received at Sm , m ∈ [1, N + 1], m= / n from Sn , n ∈ [1, N]. The Sn → Sm link is modeled as a block Rayleigh fad(m) 2 ing channel with coefficients h(m) n ∼ CN (0, (σ n ) γ). Defining normalized additive white Gaussian noise (AWGN) terms wn(m) ∼ CN (0, IK ) for the Sn → Sm link, we can write the Phase-1 input-output relations as (m) yn(m) = h(m) n xn + wn ,

m ∈ [1, N + 1], n ∈ [1, N], n= / m, (1)

where we recall m = N + 1 denotes the signal received at the common destination SN+1 . For reference, we also define 2 := |h(m) the instantaneous output SNR of each link γ(m) n | n (m) 2 (m) and the corresponding average SNR as γn = (σ n ) γ [cf. (m) 2 h(m) n ∼ CN (0, (σ n ) γ) ]. After Phase 1, each source has available an estimate of the other sources blocks. Let xn(m) denote the estimate of the source block xn formed at source Sm , m ∈ [1, N], m= / n. Due to communication errors, xn(m) will generally differ from the original block xn and from estimates xn(l) at different sources Sl = / Sm . In Phase 2, each source transmits to the destination a block that contains coded information from other sources’ blocks. To be precise, consider the set of Phase-1 transmitted blocks, := {xn }Nn=1 . If x were perfectly known at Sm , it would have been possible to form a reencoded block vm of size L × 1 obtained from x through a mapping Mm , that is, vm = Mm (x).

(2)

Note, however, that x is not necessarily known at Sm . In fact, source Sm collects all estimates {xn(m) }Nn=1,n=/ m plus its own

Alfonso Cano et al.

S1

3

x1 (N+1)

h1

(m) γ1

√

x1

α 1 v1

√

α 2 v2

x2

(N+1)

h2 (N+1)

. . .

Sm

xm

γm

vm

αm γm

..

SN+1 (N+1)

(N+1)

(m) γN

SN xN

Figure 1: Transmitted and received signals at Sm during Phase 1 and Phase 2.

information xm in the set x (m) := {xm , {xn(m) }Nn=1,n=/ m }. The L × 1 vector vm built by Sm in Phase 2 is thus obtained from x (m) as

(3)

Comparing (2) with (3) we see that different from the MSC strategies in [7, 14], we are encoding based on a set of errorcorrupted blocks x (m) . To make this explicit, we denoted the mapped block as vm [cf. (3)] to emphasize that it may be different from the desired vm [cf. (2)]. Propagation of decoding errors can have a detrimental effect on error performance at the destination. To mitigate this problem, our approach is to have source Sm adjust its Phase2 transmitted power using a channel-adaptive scaling coefficient αm . The block transmitted from Sm in Phase 2 is thus √ αm vm . The signal y(N+1,2) received at the destination SN+1 is the superposition of the N source signals; see Figure 2. Upon := [v defining a matrix of transmitted blocks V 1 , . . . , v N ] = [M1 (x (1) ), . . . , MN (x (N) )], a diagonal matrix containing the √ √ αm coefficients Dα := diag([ α1 , . . . , αN ]) and the aggregate channel h(N+1) := [h(N+1) , . . . , h(N+1) ]T containing the 1 N coefficients from all sources to the destination, we can write y

(N+1,2)

αh = VD

(N+1)

+w

(N+1,2)

.

x = arg minKN x∈As

2 (N+1,2) (N+1) + y − VDα h ,

Example 1 (repetition coding). A simple cooperation strategy for Phase 2 is that each source retransmits the packet of one neighbor; that is, if we build a mapping

T T Mm : vm = 0T(m −1)P , xm

, 0(N −m)P

T

(6)

with m

= mod[m, N]+1, the mth terminal repeats the m−1)st signal’s estimate for m= / 1 and the first terminal repeats the Nth signal’s estimate. Note that the all-zero vectors appended T before and after xm

are to separate transmissions in time during Phase 2. With this definition, it can be seen that the optimum receiver in (5) simplifies for each entry k to

(N+1,1) [xm(N+1) ]k = arg min ym x∈As

k

+ y(N+1,2)

2

− h(N+1) x m

(m+1)K+k

√ (N+1) 2 − αm hm

x .

(7)

N 2 (N+1,1) yn − diag xn h(N+1)

n=1

The goal of this paper is to characterize the diversity of the 2-phase MSC protocol with input/output relations (1) and (4) and detection rule (5) in terms of suitable properties of the mappings Mm . In particular, we will show that for given mappings Mm , an appropiate selection of the coefficients Dα enables diversity order equal to an equivalent multi-antenna system with N colocated transmit antennas; that is, when no inter-source error occurs. Purposefully general, to illustrate notation, let us describe two examples for Mm yielding different MSC protocols.

(4)

The destination estimates the set of transmitted blocks x using the N blocks of K symbols yn(N+1,1) received during Phase 1 and the L symbols y(N+1,2) received during Phase 2. Assuming knowledge of the product Dα h(N+1) (through, e.g., a training phase), demodulation at the destination relies on the detection rule

Phase 2

Figure 2: Time-division scheduling for N sources during Phase 1 and simultaneous transmissions during Phase 2.

Phase-1 Phase-2

α N vN

Phase 1

min{minm=/ n {γn }, γ(N+1) } m γ(N+1) m

vm = Mm x (m) .

√

xN

hN (m)

αm : =

. ..

.

Example 2 (distributed complex-field coding). Define the (m) (m) T N ×1 column vector p(m) k := [[x1 ]k , . . . , [xm ]k , . . . , [xN ]k ] T and linearly code it using a row 1 × N vector θ m , taken as the mth row of a complex-field coding matrix Θ [15]. Repeating this process for all k, the mapping Mm now becomes

(m) , 0T(N −m)P Mm : vm = 0T(m−1)P , θ Tm p(m) 1 , . . . , pK

(5)

where V := [v1 , . . . , vN ] = [M1 (x), . . . , MN (x)]. The search in (5) is performed over the set of constellation codewords x with size |As |KN . Note that this is a general detector for performance analysis purposes but its complexity does not necessarily depend on the size of the set x; see also Section 4.

T

. (8)

In this case, the destination SN+1 can decode p(m) using the k following detection rule: 2

(N+1,1) (N+1) p k = arg min qk − diag(pk )h(N+1) N pk ∈As

2

+ qk(N+1,2) − diag(Θpk )Dα h(N+1) , (9)

4

EURASIP Journal on Advances in Signal Processing T

where qk(N+1,1) := [[y1(N+1,1) ]k , . . . , [yN(N+1,1) ]k ] and qk(N+1,2) := T [[y(N+1,2) ](k−1)N+1 , . . . , [yN(N+1,2) ]kN ] .

3.

ERROR-PROBABILITY ANALYSIS

The purpose of this section is to determine the high-SNR diversity order of MSC protocols in terms of suitable properties }N of the mapping Mm . For given channels H (d) := {h(N+1) n n=1 (m) N (s) from sources to destination and H := {hn }m,n=1,m=/ n between sources, we define the so-called conditional (or instantaneous) pairwise error probability (PEP) Pr {x= /x | H (s) , H (d) } as the probability of decoding x when x was transmitted and denoted as Pr {x → x | H (s) , H (d) }. The diversity order of the MSC protocol is defined as the slope of the logarithm of the average PEP as the SNR goes to infinity, that is,

log E Pr x −→ x | H (s) , H (d) γ→∞ log γ

d = min − lim x,x = /x

. (10)

For MIMO block-fading channels, the diversity order d depends on the rank distance between constellation codewords [16]. This will turn out to be generalizable to the VAA created in LAR-MSC systems. For that matter, define the set

X := n | xn − x n = /0

(11)

containing the indices of the sources transmitting different packets. For the same x and x consider the corresponding

We are interested in a subset of phase-2 blocks V and V.

columns of (V − V) that form a basis of the span of its columns. This can be formally defined as

V := n | span vn − v n

n∈V

, = span(V − V)

(12)

The coefficient αm in (13) is formed based on the instanN taneous SNR of the links through which blocks {yn(m) }n=1,n=/ m arrived (available, e.g., by appending a training sequence) and the average SNR of its link to the destination, which is assumed to slowly fade at long scale, and thus is feasible to be fed back. These same conventions have also been adopted in the context of DF protocols in [4, 9]. In [9], the average channel is assumed to be known for decoding at the destination, whereas in [4] average knowledge is assumed to be known at the relays; the latter has been proved to be fulldiversity achieving, while the former cannot achieve fulldiversity, which in our set-up amounts to N, the number of sources transmitting to the destination. As detailed in the next subsection, the diversity order can be assessed by establishing proper bounds on the PEP as in, for example, [9] or [4]. However, for systems with the same diversity order, comparing relative performance typically relies on their respective coding gains [17, Chapter 2]. Unfortunately, analytical assessment of coding gains is rarely possible in closed form especially for the DF-based cooperative systems even for simple constellations using repetition coding; see also [9] for similar comments. For this reason, we will resort to simulated tests in order to assess coding-gain performance in Section 4. 3.1.

Proof of Theorem 1

The difficulty in proving Theorem 1 is the possibility of having decoding errors between cooperating terminals, that is, x (m) = / x. Thus, define the set of sources’ indices that estimate x erroneously,

(m) . E := m | x= /x

(15)

where span(·) denotes the span of a set of vectors or columns = V, of a matrix. With reference to Figure 2, if we assume V the equivalent system can be seen as a MISO block-faded transmission and the achievable diversity order is related to

= |V | over all possible pairs, where rank(·) rank (V − V) denotes the rank of a matrix. We are now challenged to es= tablish similar diversity claims when V / V along with the contribution to diversity of X after Phase 1. The pertinent result is summarized in the following theorem we prove in Section 3.1.

By definition E ’s complement E contains the indices of the sources that decoded x correctly. For a given set E of correct decoders, one expects that sources {Sm }m∈E help to increase the detection probability, whereas sources {Sm }m∈E tend to decrease it. In terms of diversity, not all of the elements of E contribute to increasing its order. In fact, for Sm to contribute to the diversity order it also has to belong to the set (X ∪ V). Thus, we define the set

Theorem 1. Consider two distinct blocks x, x and the pairwiseerror indicator sets X and V defined in (11) and (12), respectively. Let the Phase-2 power-weighting coefficients {αn }Nn=1 be

The cardinality of C can be bounded as |C | ≥ |X ∪ V | − |E |. Note also that C ∩ E = ∅. Using these definitions, we can condition on the set of correct decoders E and bound (i) the probability Pr {x → N x (m) }m=1 | H (s) } of erroneous detection at the sources af{ x (m) }Nm=1 → x | ter Phase 1; and (ii) the probability Pr {x, { H (s) , H (d) } of incorrect detection at the destination after Phase 1 and Phase 2. The result is stated in the following lemma.

min min m=/ n γ(m) , γm(N+1) n , αm := (N+1) γm

(13)

(N+1) is the instantaneous (average) SNR of link where γ(m) n γm Sn − Sm (Sm − SN+1 ), m ∈ [1, N]. The average diversity order as defined in (10) of the MSC protocol defined in (1)–(5) is

log Pr{x −→ x } = min |X ∪ V | . γ→∞ log γ x,x = /x

d = min − lim x,x = /x

(14)

C := (X ∪ V) ∩ E .

(16)

Lemma 1. Consider a transmitted block x, a set of estimated N x (m) }m=1 at terminals {Sm }Nm=1 , and an estimated block blocks { x at the destination. Let (i) E and C denote the sets defined in

Alfonso Cano et al.

5

(m) (N+1) 2 (15) and (16); (ii) let γ(m) = |hn |2 and γ(N+1) = |h n | n n be the instantaneous SNRs in the Sn → Sm and Sn → SN+1 links; and (iii) let αm denote the power scaling constant in (13). Conditioned on fading realizations, N x (m) }m=1 at (a) the conditional probability of decoding { {Sm }N m=1 given that x was transmitted in Phase-1 can be bounded as

Pr x −→ x (m)

N m=1

| H (s) ≤ κ1 exp

− κ2

n∈E

min γ(m) n

m= /n

(17) for some finite constants κ1 , κ2 ; (b) the conditional probability of detecting x given that N x (m) }m=1 in Phase-2 is x was transmitted in Phase-1 and { bounded as

Pr x, x (m)

N

m=1

⎛

−→ x | H (s) , H (d)

⎞

As is well known [18, Chpater 14], the expected value of the right-hand side of (21) can be bounded as

E Q

κ3

(18)

Proof. See Appendices A and B. Using results (a) and (b) of Lemma 1, we can bound the PEP in (10) to obtain

≤

− κ2

κ1 exp

N ∀{ x (m) }m=1

⎛

n∈E

min γ(s) m,n m= /n

⎞

To interpret the bound in (19) let us note that the factors in (17) and (18) are reminiscent of similar expressions that appear in error-probability analysis of fading channels [18, Chapter 14]. Taking expected values over the complex Gaussian distribution of the channels in H (s) and H (d) allows us to bound the right-hand side of (17) as

− κ2

E exp

(m)

min γn

/n n∈E m=

≤ k1 γ

−|E |

(20)

for some constant k1 . With respect to (18), we expect decoding errors at Sn when some of the fading coefficients {γ(m) are small. In turn, since αn ≤ min m=/ n {γ(m) n }m = n } we / n expect n∈E αn γ(N+1) to be small since n ∈ E when decoding n errors are present at Sn . Thus, the right-hand side of (18) can be approximated as ⎛

for some constant k2 . Combining (22), (20), and (19), we could deduce that Pr {x → x } := E[Pr {x → x | H (s) , H (d) }] ≤ (k1 k2 γ)−|C |−|E | . Since |C | ≥ |X ∪ V | − |E |, we have |C | + |E | ≥ |X ∪ V |, which establishes Theorem 1. This argument is not a proof however, since (22) is a bound on the approximation (21). Furthermore, the factors in the products of (19) are depen(N+1) }/ γm(N+1) [cf. dent through αm := min {min m=/ n {γ(m) n }, γ m (13)] and cannot be factored into a product of expectations. The next lemma helps us to overcome these technical difficulties.

Pe γc , γe , ηc , ηe

≈Q

κ3

n∈C

αn γ(N+1) . n

κ3 γ c η c − κ4 γ e η e ≤ κ1 exp − κ2 γe Q κ3 γ c η c + κ4 γ e η e (23)

for some finite constants κ1 , κ2 , κ3 , κ4 , and γc ∼ Gamma (|C |, 1/γ), γe ∼ Gamma (|E |, 1/γ); γc , ηc , γe , and ηe are nonnegative and independent of each other, if the probability density functions p(ηc ) and p(ηe ) do not depend on γ, the expectation over γc , γe , ηc , and ηe is bounded as

Proof. See [19]. Combining Lemmas 1 and 2, we can prove Theorem 1 as we show next. Proof of Theorem 1. Using the definition of αm in (13), one can derive the following bounds on the probability expressed in (19): n∈E

αn γ(N+1) n

=

min min m γ(m) , γ n

γ

n∈E

≤

min

n∈E

n∈C

αn γ(N+1) = n

n∈C

× ≥

m

γ(m) n

γ(N+1) n

(N+1) n∈E γ n

γ

n∈C

,

γ(N+1) n

min min m γ(m) , γ n

n∈C

(21)

(24)

with k := E[k(ηc , ηe )] a constant not dependent on γ.

⎞

κ3 n∈C αn γ(N+1) − κ4 n∈E αn γ(N+1) n n ⎠ Q ⎝ (N+1) κ3 n∈C αn γn + κ4 n∈E αn γ(N+1) n

(22)

Pe ≤ (kγ)−|C |−|E |

κ3 n∈C αn γ(N+1) − κ4 n∈E αn γ(N+1) n n ⎠. × Q ⎝ (N+1) κ3 n∈C αn γn + κ4 n∈E αn γ(N+1) n (19)

−|C | ≤ k2 γ

n∈C

for some finite constants κ3 , κ4 .

αn γ(N+1) n

Lemma 2. For some error probability Pe {γc , γe , ηc , ηe } satisfying

κ3 n∈C αn γ(N+1) − κ4 n∈E αn γ(N+1) n n ⎠ ≤ Q ⎝ κ3 n∈C αn γ(N+1) + κ4 n∈E αn γ(N+1) n n

Pr x −→ x | H (s) , H (d)

γ(N+1) n

γ

γ(N+1) n

(N+1) n∈C γ n

min min ∀m,n∈C γ(m) ,γ n , γ (25)

where we set all instantaneous SNRs to have the same average γ; that is, one can pick the maximum average SNR among all

6


links of our setup and bound the performance of this system by another one with the same average SNR γ in all links, as demonstrated in [19]. If one defines γe := n∈E minn {γn(m) }, γc := n∈C γn(N+1) , ηe := min{min∀m,n∈C {γn(m) }, γ}/γ and ηc := n∈E γn(N+1) /γ, then we obtain the upper bound

Pr x −→ x | H (s) , H (d)

≤

κ1 exp − κ2 γe Q N

∀{ x (m) }m=1

κ3 γ c η c − κ4 γ e η e , κ3 γ c η c + κ4 γ e η e (26)

∼ Gamma (|C |, 1/γ), γ(s) where γ(d) c e ∼ Gamma (|E |, (N − (d) 1)/γ); γc , ηc , γ(s) , and η are nonnegative and independent e e of each other; that is, ηe(|E |−1) exp − ηe , |E | − 1 !

p ηc = ηc = 1

!

with pr

"

∀m,n∈C

min γ(m) n ≥ γ = exp − |C |(N − 1) ,

p ηc = |C |(N − 1) exp − |C |(N − 1)ηe !

with pr

"

(27)

Proof. If x and x differ in at least two subblocks, we have |X| ≥ 2. In the worst-case event in which x differs from x

in one unique sub-block, say the nth, we find X = {n}. If we use repetition coding and permute symbols in one position n} with n = as in (6), then V = {

/ n. Hence, the union of X and V has at least two elements and the detector in (7) achieves diversity min x,x =/ x {|X ∪ V |} = 2. Because information is forwarded without modification, this scheme can be interpreted as a relay scenario such as the one in [4]. Thus, Theorem 1 demonstrates diversity for classical relay schemes based on repetition coding. This result was already established in [4]. Repetition coding was the first reported cooperation strategy [3]. It features low-complexity detection and does not require symbol synchronization, because each source transmits frames over separate time slots. As demonstrated here, it can achieve diversity 2. With each source transmitting a frame of K symbols, and assuming that Phase 1 and Phase 2 have identical duration KN, the per-source bandwidth efficiency of repetition coding is η = K/(KN + KN) = 1/(2N). Example 4 (complex-field coding). In Section 2 we described the use of CFC to code blocks of symbols. In view of Theorem 1, we can now establish the following corollary.

min γ(m) n < γ = 1 − exp − |C |(N − 1) .

∀m,n∈C

Corollary 2. For the distributed CFC strategy in (8), if θ m is

k )= designed to guarantee that θ Tm (pk − p / pk and / 0 for any pk =

for any m, the detector in (9) achieves diversity d = N.

Finally, because |C | ≥ |X ∪ V | − |E |, we have

Pr {x −→ x } ≤

kγ

−|C |−|E |

N

∀{ x (m) }m=1

≤

kγ

−|X∪V |+|E |−|E |

−|X∪V | = k γ

N

∀{ x (m) }m=1

(28) N for some constant k that absorbs the sum over all { x (m) }m=1 ,

because the terms in the sum no longer depend on N { x (m) }m=1 ; that is, the bound is independent of the errors after Phase 1, and so is its diversity order. 3.2. Corollaries

Theorem 1 not only quantifies error performance bounds for our system, but also provides insight on how to design diversity-enabling mappings Mm for each Sm . The following examples illustrate these facts and establish desirable tradeoffs (summed up in Table 1) accounting for performance, complexity, spectral efficiency, and synchronism requirements. Example 3 (repetition coding). In Section 2 we described a specific example in which each source transmits information of neighboring sources in separate time slots [cf. (6)]. Now, in view of Theorem 1, we can establish the following corollary. Corollary 1. Repetition coding defined by the encoding strategy in (6) and the detector in (7) achieves diversity d = 2.

has

k )= Proof. If θ Tm (pk − p / 0 for any m, k, the matrix (V − V) (full) rank N; that is, |V | = N. Thus, min x,x =/ x {|X ∪ V |} = |V | = N.

The condition θ Tm (pk − p

k )= / 0 is the so-called maximumseparability criterion. Designs for θ m are available in [15] in the context of MIMO systems with either systematic or numerically optimized constructions. Interestingly, a matrix Θ enforcing maximum separability exists for any size N [15]. As with repetition coding, distributed CFC does not require synchronism at the symbol level. Likewise, the persource bandwidth efficiency is also η = K/(KN + KN) = 1/(2N), but higher diversity gains are possible. Example 5 (distributed ST coding). Theorem 1 also allows us to analyze the performance of the distributed ST coding. Among the several options one may consider, we here analyze the performance of any generic ST code designed for a MISO system in which the number of transmit antennas equals the number of sources in our setup (N). Its implementation follows these steps. Suppose source Sm builds an N × 1 vector p(m) := [[x1(m) ]k , . . . , [xm ]k , . . . , [xN(m) ]k ]T after Phase 1 k and maps it to a generic-size T × N matrix T(m) (with rows k denoting time and columns denoting space) using a generic ST mapping MST . Source Sm builds vm as follows: Mm : vm =

#

T(m) 1

$T :,m

,...,

#

T(m) K

$ T T :,m

,

(29)

Alfonso Cano et al.

7 Table 1: Comparison between distributed coding strategies. Diversity order (d)

BW efficiency η

Synchr. at symb. level

from dST to min {N, dST + 1}

1 N +T

Needed

Repetition

2

1 2N

Not needed

DCFC

N

1 2N

Not needed

1 N +P

Not needed

DSTC N × T

% !

min dmin , 1 + N 1 −

DECC KP parity bits

K

(K + P)log 2 As

that is, Sm concatenates the mth column of the K ST mapped (m) matrices T(m) 1 , . . . , TK . By construction, vm has size KP × 1. Now, the following corollary assesses its performance when applied to our VAA setup. Corollary 3. Given a generic ST mapping MST that enables diversity dST in a MIMO system, its distributed implementation as in (29) can achieve diversity at least d = dST and at most d = min {N, dST + 1}. Proof. If MST enables diversity dST , it means that for any k,

k ) = dST . Now, adding up contributions from all rank(Tk − T sources, matrix V has size KP × N and is built as [cf. (29)]

T T

V = TT1 , . . . , TK

.

(30)

= Equation (30) implies that for any k rank rank (V − V)

k ) = dST . If, by construction, MST is such that rank(Tk − T for some x , a worst-case event X ∈ V is possible, then d = min x,x =/ x {|X ∪ V |} = |V | = rank(V) = dST . If, instead, MST is such that for all x = / x, X ∩ V = / ∅, then d = min x,x =/ x {|X ∪ V |} = min {N, |X| + |V |} = min {N, dST + 1}.

Corollary 3 connects the diversity criteria for MIMO ST codes specified in, for example, [20], with its distributed error-prone implementation in multisource scenarios. It indeed demonstrates that a judicious distributed implementation of this ST code may increase its diversity by 1. Compared to repetition coding or distributed CFC, the distributed ST codes described here require symbol-level synchronism to operate and their performance may degrade if sources are not perfectly synchronized [21, 22]. Simulations in Section 4 will consider this effect. For a general timespan of the code T, the bandwidth efficiency of this strategy is η = 1/(N + T). Example 6 (distributed error-correcting codes). Suppose that each source transmits

T , 0T(N −m)P Mm : vm = 0T(m−1)P , vm

T

,

(31)

is a P × 1 vector comprising parity check bits of and vm the block x(m) . Such mapping implements the distributed channel-coding strategy in [7, 14]. As depicted in Figure 3,

"&

(N+1)

h1

√

α1 v1

x1

(N+1) h2

x2 ..

(N+1)

√

α2 v2 ..

.

.

xN

hN

Phase 1

√

α N vN

Phase 2

Figure 3: Time-division scheduling for N sources during Phase 1 and Phase 2.

the aggregate block sequence sent to the destination is [x1 , . . . , xN , v1 , . . . , vN ] and has size N(K + P). The first NK symbols sent during Phase 1 then correspond to the systematic symbols and the NP symbols sent during Phase 2 comprise the parity-check portion of a generic ECC scheme. The following corollary states its performance. Corollary 4. The distributed implementation of ECC codes as described in (31) achieves diversity order d = dC , where

% !

dC ≤ min dmin , 1 + N 1 −

K

(K + P)log 2 As

"&

(32) and dmin is the minimum Hamming (free) distance of the ECC. Proof. It is sufficient to observe that a sequence of systematicplus parity-bits [x1T , . . . , xNT , (v )T1 , . . . , (v )TN ], if transmitted over a point-to-point block-faded channel, achieves diversity dC . This is indeed demonstrated in [16]. Notice that (32) is the Singleton bound. As shown in (32), the code rate and the constellation-employed affect the maximum achievable diversity order of coded transmissions over fading channels [16]. We further remark that in order to achieve diversity dC , one has to judiciously design interleavers provided that systematic and parity bits are sent as shown in Figure 3. Finally, note that DECC features low synchronism requirements (frame-level as in repetition and distributed CFC) and per-source bandwidth efficiency η = 1/(N + P).

8 100

SIMULATIONS AND COMPARISONS

We present numerical simulations to test error performance of the proposed cooperative protocols. We employ binary phase-shift keying (BPSK). We suppose that all inter-source and source-destination links have the same average output SNR; that is, γ(m) n = γ, for all n ∈ [1, N], m ∈ [1, N + 1].

10−1 10−2 BER

4.


10−3

4.1. Distributed coding strategies 10−4

We will compare the diversity order achieved by the encoding schemes in Examples 1, 2, 3, and 4 for different numbers of cooperating sources N.

10−5 10−6

4.1.1. Distributed orthogonal ST codes

0

⎢

(1) k p

k = ⎢ # T ⎣ −

1 $∗ (1) p k 2

#

(2) k p

2 ⎥ $∗ ⎥ ⎦. (2) k p 1

(33)

#

(1) k p

$∗ # 2

,

(2) k p

$∗ 1

4.1.2. Repetition coding and distributed complex-field coding Figure 5 shows the BER when employing DCFC, repetition coding, and the PL detector in [9] for N = 1, 2, 3 cooperating sources. The CFC matrix Θ is chosen from [15]. For CFC and repetition-based transmissions, we employ the detectors in (9) and (7). For reference, we again depict the BER when sources are not cooperating. We can verify that, as established in Theorem 1, the slope of the BER varies with N when employing CFC and remains fixed to 2 when employing repetition coding with the same bandwidth efficiency. As a byproduct, we also outline the advantages of repetition-based link adaptation compared to [9]; whereas the former achieves diversity 2 in any scenario, the latter loses diversity when

LAR-DSTBC1 LAR-DSTBC2

10−1 10−2

(34)

with per-source bandwidth efficiency η = 1/3. In (33) we have |V | = 2 whereas in (34) |V | = 1 but still |X ∪ V | = 2. This same strategy can be generalized to N > 2 which corresponds to the ST orthogonal codes in [20]. Figure 4 depicts bit-error-rate (BER) as a function of the average SNRγ for different relay locations and schemes. Specifically, we compare (33) (LAR-DSTBC1), (34) (LARDSTBC2), and also [10] (DSTBC with no adaptation) and [11] (DSTBC with full channel knowledge at the destination). For reference, we also depict the BER when sources are not cooperating. Designs using LAR achieve diversity with different coding gains. Designs which do not exploit adaptivity suffer diversity loss. The performance is fairly close to that obtained with instantaneous channel knowledge [11].

25

100

BER

20

Figure 4: BER of DSTBC for N = 2, 3 sources and no cooperation.

The per-source bandwidth efficiency of this choice is η = 1/4. One can improve the rate without diversity loss by just sending k = − T

15

No cooperation DSTBC with no adaptation DSTC with full chl. knowl.

⎤

10 SNR

Here we rely on the ST codes proposed in [23]. If N = 2, then (m) p k has size 2 × 1 and we map it to ⎡

5

10−3 10−4 10−5 10−6

0

5

10 SNR

No cooperation PL detector N = 2 Repetition N = 2

15

20

Repetition N = 3 DCFC N = 2 DCFC N = 3

Figure 5: BER of DCFC versus repetition and PL relaying strategies for N = 2, 3 sources.

sources are sufficiently separated. As already mentioned, repetition coding is manifested in the well-known DF-relaying. This motivates us to also include comparisons with the coherent piecewise-linear (PL) detector of [9], which assumes that the average inter-source SNR is known at D. 4.1.3. Distributed convolutional codes Figure 6 illustrates BER performance when employing the distributed convolutional codes (DCC) of [14] for N = 2 and N = 4 users. We employ blocks of size K = 52 bits encoded through a rate 1/2 convolutional code (K = L) with

Alfonso Cano et al.

9 100

10−2

10−1

10−3

10−2

BER

BER

10−1

10−3

10−4

10−4

10−5

10−6 10

10−5 15

20

25

0

5

10

SNR

Figure 6: BER of DCC for N = 2, 4 sources.

generator in octal form [5, 7] with Hamming (free) distance dfree = 5. According to (32), the achievable diversity orders are dC = 2 for N = 2 and dC = 3 for N = 4. Figure 6 confirms that diversity orders are achieved as predicted by Theorem 1. From the same figure we can also observe that coding gain is reduced. This can be due to the fact that highly corrupted blocks processed by the Viterbi decoder severely degrade its optimality at low SNRs. 4.2. Effect of synchronization In the context of distributed setups, a fair comparison between distributed Alamouti, DCFC, and repetition coding should also account for synchronization issues. We fix the same bandwidth efficiency to be η = 1/4 and set the variation of timing offset to be uniformly distributed as U(−Ts , Ts ) around the optimum sampling instant. We assume raised cosine pulses with roll-off factor β = 0.22. Figure 7 confirms the severe degradation that simultaneous transmissions suffer when accounting for mistiming across sources. Moreover, performance degradation increases with the number of users, which clearly offsets the potential diversity gains. We also show the performance of nonsimultaneous transmissions such as CFC, which do not experience this degradation. 4.3. CRC-aided retransmissions versus adaptive techniques The advantages of MSC using SF as in [8] hinge upon the assumption of either error-free links between sources or, as is the case in practice, on correct error-detection decoding per frame. In this practical case, frames with errors are discarded and no signal is retransmitted. This strategy, however, can be inefficient at low SNR and/or when the CRC block size is large, because a single erroneous bit leads one to discard the

20

25

DSTBC N = 3 asynchr. DCFC N = 3 synchr.

Figure 7: BER of DCFC versus DSTBC for N = 3 and asynchronous transmissions.

100 10−1 10−2 BER

DCC N = 2 DCC N = 4 No cooperation

15 SNR

10−3 10−4 10−5 10−6

0

5

SF MSC N = 2 SF MSC N = 3

10 SNR

15

20

LAR-SF N = 2 LAR-SF N = 3

Figure 8: BER of adaptive versus selective retransmissions for packet length K = 200 bits and DCFC.

entire block. To delineate this assessment, we set both strategies to use the same error-correction strategy. For the LAR, we set αn = 1 if no error is detected at user n; otherwise, the block is transmitted with αn as in (13). This slight modification of our protocol, which we name LAR-SF, although not analytically proven here, can be reasonably expected to achieve full diversity. On the other hand, and for the sake of a fair comparison, we increase the average power of SF to match that of adaptive LAR-SF transmissions. Figures 8 and 9 compare the BER of these strategies for block sizes of K = 200 and K = 1024 bits, respectively. As expected, both systems achieve full diversity. Moreover,

10

EURASIP Journal on Advances in Signal Processing in E is

100

N

Pr x −→ x (m) | H (s) ≤

10−1

n=1,n= /m

(A.2)

BER

10−2

K

(m) Letting δn(m) := n ]k |2 denote the k=1 |[xn ]k − [x squared Euclidean distance between xn and xn(m) , and using the fact that function Q(·) is monotonically decreas(m) (m) ing, n ) = . can be bounded as hn (xn − x . its inner term

10−3 10−4

2

2

γn(m) (δn(m) ) ≥

10−5 10−6

# # $$ Q h(m) xn − xn(m) . n

N

0

5

10 SNR

SF MSC N = 2 SF MSC N = 3

15

20

minm=/ n {γn(m) } minm=/ n {(δn(m) ) }. And thus,

# # $$ Q h(m) xn − xn(m) n

m=1,m= /n

# 1 # $2 $ min δ (m) , ≤ (N − 1) exp − min γ(m) n n

LAR-SF N = 2 LAR-SF N = 3

2

m= /n

m= /n

(A.3)

Figure 9: BER of adaptive versus selective retransmissions for packet length K = 1024 bits and DCFC.

link-adaptive transmissions exhibit larger coding gain, which corroborates the fact that discarding large packets renders SF strategies inefficient.

where we also used the fact that Q(z) ≤ exp (−z2 /2). The probability that a set of sources E participates in error detection can be then readily bounded as

| H (s) m=1 ! " / Pr x −→ x (m) | H (s) ≤ κ1 exp − κ2 min γ(m) ≤ n

Pr x −→ x (m)

N

m∈E

5.

m∈E

m= /n

(A.4)

CONCLUSIONS

We have developed a link-adaptive relay protocol for use in multisource cooperative scenarios. General diversity performance was analyzed as a function of the rank properties of the distributed coding strategy. We included repetition coding, distributed CFC, distributed ST coding, and distributed ECC as particular cases of this general diversity analysis, concluding that the attainable diversity order is (i) 2 for repetition coding; (ii) N for DCFC; (iii) at least the same diversity order afforded by the ST codes in conventional antenna arrays when we use distributed ST coding; and (iv) for DECC the same diversity achieved by the ECC over an N-lag block fading channel. Simulations suggested that synchronization tasks are relevant to be included as part of the design of a VAA. In this context, we found that DCFC offers high-rate, full-diversity, and relaxed synchronization requirements.

for some finite constants κ1 and κ2 . B.

PROOF OF LEMMA 1(b) (N+1)

:= diag (h(N+1) ) ⊗ IK , For compactness, we define Dh (N+1,1) T (N+1,1) Dα : = Dα ⊗ I K , y := [(y1 ) , . . . , (yN(N+1,1) )T ]T N (N+1,1) T and x := [x1 , . . . , xN ] and rewrite − n=1 yn (N+1) (N+1) 2 (N+1,1) 2 diag (xn )h = y − Dh x . With these definitions, the probability of detection error in (5) is

−→ x | H (s) , H (d) m=1 2 (N+1) 2 (N+1,1) = Pr y − Dh x + y(N+1,2) − VDα h(N+1) 2 (N+1) 2

α h(N+1) , > y(N+1,1) − Dh x + y(N+1,2) − VD

Pr x, x (m)

N

(B.1) APPENDICES A.

where x := [x 1T , . . . , x NT ]T . This probability of error can be written as Pr{X > 0}, where

PROOF OF LEMMA 1(a)

#

The probability that Sm fails to detect the block xn for all n= /m sent from Sn can be bounded as

#

$

n(m) ) , Pr xn −→ xn(m) | H (s) ≤ Q h(m) n (xn − x √

-∞

(A.1)

where Q(z) := 1/ 2π z exp (−t 2 /2)dt. Considering that these are independent processes, the probability that Sm is

X := −2 Re

y(N+1,1)

$H

(N+1)

Dh

(x − x )

# $H

α h(N+1) − 2 Re y (N+1,2) (V − V)D

(N+1)

+ Dh

2

(N+1)

x − Dh

2

2

x + VDα h(N+1)

2

(N+1) . − VD αh

(B.2)

Alfonso Cano et al.

11

(N+1)

α h(N+1) + Using y (N+1,1) = Dh x+w(N+1,1) and y(N+1,2) = VD (N+1,1) , it follows that X in (B.2) is a Gaussian random variw able. Thus, √ the error probability is quantified by Pr{X > 0} = Q(−μ/ σ 2 ), where μ is its mean and σ 2 is its variance and is given by

Pr x, x (m)

N m=1

−→ x | H (s) , H (d)

(N+1) 2 D (x − x ) h = Q 2 D(N+1) (x − x

α h(N+1) 2

) + (V − V)D h (V − V)D − V)Dα h(N+1) 2

α h(N+1) 2 − (V + . 2 D(N+1) (x − x

α h(N+1) 2

) + (V − V)D h

(B.3) The second term in the denominator of (B.3) can be ex − V

)Dα h(N+1) 2 ≤ ( V

)Dα h(N+1) 2 + panded as (V − V 2 (N+1) . Defining the Euclidean distance δ :=

n (V − V)Dα h K

(N+1)

n ]k |2 , we can write Dh (x − x )2 k=1 |[xn ]k − [x 2 (N+1) δ , where we have used the definition of the n∈X γ n n

X in (12). Furthermore, one can bound this sum as n∈X

γ(N+1) δ 2n ≥ n

n∈X∩E

γ(N+1) δ 2n . n

=

set

(B.4)

Ma − V) and (V − V). Now we turn our attention to (V trix (V − V) has at most |E | linearly independent columns indexed by E ; one can thus compute its singular value de − V) = AΣB and choose B such that composition (V √ (N+1) ]n = n αn h(N+1) for n ∈ E and some nonzero [BDα h n constant n , and bound (V − V)Dα h(N+1) 2 ≥ αn γ(N+1) 2n λn , n n∈E

(B.5)

where λn is the associated singular value λn := [Σ]n,n . Like − V)

has at least |V ∩ E | linearly independent wise, (V columns indexed by V ∩ E and following the same reasoning as before, we can bound 2

(V − V)Dα h(N+1) ≤

n∈V ∩E

αn γ(N+1) n2 λn n

(B.6)

for some nonzero n2 and λn . Inequalities (B.4) and (B.5) are lower bounds, whereas (B.6) is an upper bound. √ √ Using the fact that Q((a − b)/ a + b) ≤ Q((c − d)/ c − d) if a ≥ c and b ≤ d, we can rewrite (B.3) as

Pr x, x (m)

N m=1

! " B − B −→ x | H (s) , H (d) ≤ Q √ , B − B

(B.7)

δ 2n + n∈X∩E αn γ(N+1) 2n λn , where B denotes n∈X∩E γ(N+1) n n n2 λn . Finally, noticing that sums B denotes n∈E αn γ(N+1) n over indexes n ∈ X ∩ E and n ∈ V ∩ E can be merged into a single sum with index n ∈ (X ∪ V) ∩ E , and bounding with appropriate nonzero constants, one can readily arrive to (18).

ACKNOWLEDGMENTS This work was supported through collaborative participation in the Communications and Networks Consortium sponsored by the US Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement no. DAAD19-01-2-0011. The US Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The work of the first author was supported by the Spanish Government Grant no. TEC2005-06766-C0301/TCM. REFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part I: system description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, 2003. [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part II: implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, 2003. [3] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003. [4] T. Wang, R. Wang, and G. B. Giannakis, “Smart regenerative relays for link-adaptive cooperation,” to appear in IEEE Transactions on Communications. [5] X. Deng and A. M. Haimovich, “Power allocation for cooperative relaying in wireless networks,” IEEE Communications Letters, vol. 9, no. 11, pp. 994–996, 2005. [6] M. O. Hasna and M.-S. Alouini, “Optimal power allocation for relayed transmissions over Rayleigh-fading channels,” IEEE Transactions on Wireless Communications, vol. 3, no. 6, pp. 1999–2004, 2004. [7] O. Shalvi, “Multiple source cooperation diversity,” IEEE Communications Letters, vol. 8, no. 12, pp. 712–714, 2004. [8] A. Ribeiro, R. Wang, and G. B. Giannakis, “Multi-source cooperation with full-diversity spectral-efficiency and controllablecomplexity,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp. 415–425, 2007. [9] D. Chen and J. N. Laneman, “Modulation and demodulation for cooperative diversity in wireless systems,” IEEE Transactions on Wireless Communications, vol. 5, no. 7, pp. 1785–1794, 2006. [10] G. Scutari, S. Barbarossa, and D. Ludovici, “Cooperation diversity in multihop wireless networks using opportunistic driven multiple access,” in Proceedings of the IEEE International Workshop on Signal Processing Advances for Wireless Communications, pp. 170–174, Rome, Italy, June 2003. [11] P. A. Anghel, G. Leus, and M. Kaveh, “Distributed space-time cooperative systems with regenerative relays,” IEEE Transactions on Wireless Communications, vol. 5, no. 11, pp. 3130– 3141, 2006. [12] T. E. Hunter and A. Nosratinia, “Diversity through coded cooperation,” IEEE Transactions on Wireless Communications, vol. 5, no. 2, pp. 283–289, 2006. [13] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451–1458, 1998.

12 [14] R. Wang, W. Zhao, and G. B. Giannakis, “Multi-source cooperative networks with distributed convolutional coding,” in Proceedings of the 39th Asilomar Conference on Signals, Systems and Computers, pp. 1056–1060, Pacific Grove, Calif, USA, October 2005. [15] Y. Xin, Z. Wang, and G. B. Giannakis, “Space-time diversity systems based on linear constellation precoding,” IEEE Transactions on Wireless Communications, vol. 2, no. 2, pp. 294–309, 2003. [16] R. Knopp and P. A. Humblet, “On coding for block fading channels,” IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 189–205, 2000. [17] G. B. Giannakis, Z. Liu, X. Ma, and S. Zhou, Space-Time Coding for Broadband Wireless Communications, John Wiley & Sons, New York, NY, USA, 2006. [18] J. G. Proakis, Digital Communications, McGraw-Hill Higher Education, New York, NY, USA, 4th edition, 2001. [19] T. Wang and G. B. Giannakis, “High-throughput cooperative communications with complex field network coding,” in Proceedings of the 41st Annual Conference on Information Sciences and Systems (CISS ’07), pp. 253–258, Baltimore, MD, USA, March 2007. [20] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744–765, 1998. [21] S. Jagannathan, H. Aghajan, and A. Goldsmith, “The effect of time synchronization errors on the performance of cooperative MISO systems,” in Proceedings of IEEE Global Telecommunications Conference Workshops (GLOBECOM ’04), pp. 102– 107, Dallas, Tex, USA, November 2004. [22] X. Li, F. Ng, J.-T. Hwu, and M. Chen, “Channel equalization for STBC-encoded cooperative transmissions with asynchronous transmitters,” in Proceedings of the 39th Asilomar Conference on Signals, Systems and Computers, pp. 457–461, Pacific Grove, Calif, USA, October-November 2005. [23] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, 1999.



Research Article Diversity Analysis of Distributed Space-Time Codes in Relay Networks with Multiple Transmit/Receive Antennas Yindi Jing1 and Babak Hassibi2 1 Department 2 Department

of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697, USA of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

Correspondence should be addressed to Yindi Jing, [email protected] Received 1 May 2007; Revised 13 September 2007; Accepted 28 November 2007 Recommended by M. Chakraborty The idea of space-time coding devised for multiple-antenna systems is applied to the problem of communication over a wireless relay network, a strategy called distributed space-time coding, to achieve the cooperative diversity provided by antennas of the relay nodes. In this paper, we extend the idea of distributed space-time coding to wireless relay networks with multiple-antenna nodes and fading channels. We show that for a wireless relay network with M antennas at the transmit node, N antennas at the receive node, and a total of R antennas at all the relay nodes, provided that the coherence interval is long enough, the high SNR pairwise MR 1/M if M = N, where P is the total power consumed error probability (PEP) behaves as (1/P)min {M,N }R if M = / N and (log P/P) by the network. Therefore, for the case of M = / N, distributed space-time coding achieves the maximal diversity. For the case of M = N, the penalty is a factor of log 1/M P which, compared to P, becomes negligible when P is very high. Copyright © 2008 Y. Jing and B. Hassibi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

It is known that multiple antennas can greatly increase the capacity and reliability of a wireless communication link in a fading environment using space-time coding [1–4]. Recently, with the increasing interestin ad hoc networks, researchers have been looking for methods to exploit spatial diversity using the antennas of different users in the network [5– 24]. Many cooperative strategies are proposed, for example, amplify-and-forward (AF) [11, 13, 14, 16, 21, 23], decodeand-forward (DF) [9, 10, 14, 16, 22], and coded cooperation [15]. In [7], the authors proposed the use of space-time codes based on Hurwitz-Radon matrices in wireless relay networks. This work follows the strategy of [5], where the idea of space-time coding devised for multiple-antennasystems is applied to the problem of communication over a wireless relay network. (Though having the same name, the distributed space-time coding idea in [5] is different from that in [14]. Similar ideas for networks with one and two relays have appeared in [6, 11].) In [5], the authors consider wireless relay networks in which every node has a single antenna and the channels are fading, and use a cooperative strategy called distributed space-time coding by applying a

linear dispersion space-time code [25] among the relays. It is proved that without any channel knowledge at the relays, a diversity of R(1 − log log P/ log P) can be achieved, where R is the number of relays and P is the total power consumed in the whole network. This result is based on the assumption that the receiver has full knowledge of the fading channels. Therefore, when the total transmit power P is high enough, the wireless relay network achieves the diversity of a multiple-antenna system with R transmit antennas and one receive antenna, asymptotically. That is, antennas of the relays work as antennas of the transmitter although they cannot fully cooperate and do not have full knowledge of the transmit signal. After the appearance of [5], code designs for distributed space-time coding have been proposed in [26– 31] and the differential use of distributed space-time coding has been introduced in [32–35]. The references [36, 37] analyze the diversity-multiplexing tradeoff of distributed spacetime coding. Distributed space-time coding in asynchronous networks is discussed in [38–43]. Other related papers can be found in [44–46]. This paper has two main contributions. First, we extend the idea of distributed space-time coding to wireless relay networks whose nodes have multiple antennas. Second and

2


more importantly, based on the pairwise error probability (PEP) analysis, we prove lower bounds on the diversity of this scheme. We use the same two-step transmission method in [5], where in one step the transmitter sends signals to the relays and in the other the relays encode their received signals into a linear dispersion space-time code and transmit to the receiver. For a wireless relay network with M antennas at the transmitter, N antennas at the receiver, and a total of R antennas at all the relay nodes, our work shows that when the coherence interval is long enough, a diversity of min{M, N }R if M = / N and MR(1 − (1/M)(log log P/ log P)) if M = N can be achieved, where P is the total power used in the network. With this two-step protocol, it is easy to see that the errorprobability is determined by the worse of the two steps: the transmission from the transmitter to the relays and the transmission from the relays to the receiver. Therefore, when M = / N, distributed space-time coding is optimal since the diversity of the first stage cannot be larger than MR, the diversity of a multiple-antenna system with M transmit antennas and R receive antennas, and the diversity of the second stage cannot be larger than NR. When M = N, the penalty on the diversity, because the relays cannot fully cooperate and do not have full knowledge of the signal, is R(log log P/ log P). When P is very high, it is negligible. Therefore, with distributed space-time coding, wireless relay networks achieve the same diversity of multiple-antenna systems, asymptotically. The paper is organized as follows. In the following section, the network model and the generalized distributed space-time coding are explained in detail. A training scheme is also proposed. The PEP is first analyzed in Section 3. In Section 4, the diversity for the network with an infinite number of relays is discussed. Then, the diversity for the general case is obtained in Section 5. Section 6 contains the conclusion. Proofs of some of the technical theorems are given in Appendices A–D. In Appendix E, we discuss heterogeneous networks. 2.

WIRELESS RELAY NETWORK

2.1. Network model and distributed space-time coding We first introduce some notation. For a complex matrix A, A, At , and A∗ denote the conjugate, the transpose, and the Hermitian of A, respectively. det A, rank A, and tr A indicate denotes the determinant, rank, and trace of A, respectively. A the vectorization of A formed by stacking the columns of X into a single column vector. In denotes the n × n identity matrix and 0m,n is the m × n matrix with all zero entries. We often omit the subscripts when there is no confusion. log indicates the natural logarithm. · indicates the Frobenius norm. P and E indicate the probability and the expected value. g(x) = O( f (x)) means that limx→∞ (g(x)/ f (x)) is a constant. h(x) = o( f (x)) means that limx→∞ (h(x)/ f (x)) = 0. a is the minimal integer that is not less than a. Consider a wireless network with R + 2 nodes which are placed randomly and independently according to some distribution. As shown in Figure 1, there are one transmit node and one receive node. All the other R nodes work

Relays

f11 Transmitter . . .

g11 g1N .. .

r1 t1

f1R fM1

. ..

fMR

. ..

gR1 gRN

rR tR

Step 1: time 1 to T

Receiver .. .

Step 2: time T + 1 to 2T

Figure 1: Wireless relay network with multiple-antenna nodes.

as relays. The transmitter has M transmit antennas, the receiver has N receive antennas, and the ith relay has Ri antennas. Since the transmit and received signals at different antennas of the same relay can be processed and designed independently, the network can be transformed to a network with R = Ri=1 Ri single-antenna relays by designing the transmit signal at every antenna of every relay according to the received signal at that antenna only. This is one possible scheme. In general, the signal sent by one antenna of a relay can be designed using received signals at all antennas of the relay. However, as will be seen later, this simpler scheme achieves the optimal diversity asymptotically although a general design may improve the coding gain of the network. Therefore, to highlight the diversity results by simplifying notation and formulas, in the following, we assume that every relay has a single antenna. Denote the channel vector from the M antennas of the transmitter to the ith relay as t fi = [ f1i · · · fMi ] , and the channels from the ith relay to the N antennas at the receiver as gi = [gi1 · · · giN ]. We use the block-fading model [2] by assuming a coherence interval T. From the two-step protocol that will be discussed in the following, we can see that we only need fi to keep constant for the first step of the transmission and gi to keep constant for the second step. It is thus good enough to choose T as the minimum of the coherence intervals of fi and gi . Also, perfect symbol-level synchronization is assumed in this network model. For asynchronized networks, please refer to [38–43]. The information bits are encoded into T × M matrices s, whose mth column is the signal sent by the mth transmit antenna. For the power analysis, s is normalized as E tr s∗ s = M.

(1)

To send s to the receiver, the same two-step strategy in [5] is used, as shown in Figure 1. In step one, the transmitter sends P1 T/Ms. The average total power used at the transmitter for the T transmissions is P1 T. The received signal vector and the noise vector at the ith relay are denoted as ri and vi . In step two, the ith relay sends ti . The received signal and noise at the receiver are denoted as X and w. The noises are assumed to be i.i.d. CN (0, 1). Clearly,

ri = P1 T/Msfi + vi ,

X = t1 · · · tR G + w, t

where G = [g1t · · · gRt ] .

(2)

Y. Jing and B. Hassibi

3

We use distributed space-time coding proposed in [5] by designing the transmit signal at relay i as a linear function of its received signal:

ti =

P2 Ai ri , P1 + 1

(3)

where Ai is a predetermined T × T unitary matrix known to both the ith relay and the receiver. It is fixed during training and data transmissions. For various methods on how to design the Ai , see [26–31]. P2 can be proved to be the average transmit power for one transmission at every relay. After some calculation, the system equation can be written as

X=

P1 P2 T SH + W, M P1 + 1

(4)

where

S = A1 s · · · AR s ,

W=

⎡

H=

R

P2 ⎣ gi1 Ai vi · · · P1 + 1 i=1

f1 g1 R

t

···

fR gR

⎤

giN Ai vi ⎦ + w.

t t

,

(5) (6)

Yp =

P2 G∗ G. 1 + P1

(7)

The covariance matrix of the equivalent noise matrix can be proved to be RW . The diversity analysis in this paper is much more difficult than that in [5] because in networks with single-antenna nodes, the covariance matrix of the equivalent noise is a multiple of the identity matrix. Here, for the diversity result, we need to analyze the eigenvalues of RW or find bounds on them. 2.2. Assumptions and training In this paper, we assume that fmi and gin have independent Rayleigh distributions; that is, fmi and gin are independent circulant complex Gaussian random variables with zero mean. For simplicity, we also assume that fmi and gin have the same variance, which is 1. The heterogeneous case, in which every channel has a different variance, is discussed in Appendix E. The same diversity results can be obtained in heterogeneous networks. We make the practical assumption that the relays have no channel information. However, we do assume that the receiver has enough channel information to do coherent detection. Thus, a training process is needed. For coherence ML decoding at the receiver, the receiver needs to know H and RW , or equivalently, H and G. We propose a training process that contains two steps and takes M p + 2N p symbol periods (other training methods can also be envisioned, and the one proposed here is one possibility).

QpMp UpG + w p, R

(8)

where Q p is the power used at every relay and w p is the M p × N noise matrix. Since there are RN unknowns (corresponding to the components of G) and min{M p , R}N independent equations, we need M p ≥ R. We could estimate G from U p using ML, MMSE, or other criteria. Then, we estimate H using distributed space-time coding discussed in Section 2.1. This takes 2N p symbol periods. The transmitter sends a full-rank N p × M pilot signal matrix s p and the relays perform distributed space-time coding. From (4), the received signal can be written as P1,p P2,p N p Xp = SpH + Wp,

(9)

M(P1,p + 1)

i=1

The received signal matrix, X, is T × N. S, which is T × MR, is the linear distributed space-time code. Since fi is M × 1 and gi is 1 × N, the equivalent channel matrix H is RM × N. W, which is T × N, is the equivalent noise matrix. Define RW = I +

Each step mimics the training process of a multiple-antenna system [47] as its system equation has the same structure. First, we estimate G, which takes M p symbol periods. Let U p be a predesigned full-rank M p × R pilot matrix. The ith relay sends the ith column of U p simultaneously. The receiver gets

where P1,p and P2,p are the powers used at the transmitter and every relay and

S p = A1 s p · · · AR s p

(10)

is the carefully designed N p × MR pilot space-time codeword. Now, let us discuss the number of training symbols needed in this step. Note that G is known from the first training step. Define

t

f = f1t · · · fRt .

(11)

By stacking the columns of X into one single column vector, we can rewrite (9) as P1,p P2,p N p Xp =

⎡

⎢ ⎢ ⎢ M P1,p + 1 ⎢ ⎣

S p diag g11 IM , . . . , gR1 IM

⎤

.. .

S p diag g1N IM , . . . , gRN IM P1,p P2,p N p =

⎡

⎢ ⎢ ⎢ M P1,p + 1 ⎢ ⎣

P1,p P2,p N p =

⎡

⎢ ⎢ ⎢ M P1,p + 1 ⎢ ⎣

g11 A1 s p · · · gR1 AR s p .. .

..

.. .

.

g1N A1 s p · · · gRN AR s p g11 IN p · · · gR1 IN p .. .

..

.

.. .

g1N IN p · · · gRN IN p

⎥ ⎥ ⎥f + W p ⎥ ⎦

⎤ ⎥ ⎥ ⎥f + W p ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

p. × diag A1 s p , . . . , AR s p f + W

(12)

4


Denote ⎡ ⎢ Hp = ⎢ ⎣

Theorem 2 (ML decoding and the PEP Chernoff bound). The ML decoding of the relay network is

⎤

g11 IN p · · · gR1 IN p .. .. ⎥ .. ⎥ . . . ⎦ diag A1 s p , . . . , AR s p . g1N IN p · · · gRN IN p

arg min tr X − sk

The number of independent equations in (9) equals the rank of H p , which is min{N p N, N p R, MR}. Since there are MR unknowns (corresponding to the components of f), we need min{N p N, N p R, MR} ≥ MR, which is equivalent to

(14)

While this condition is satisfied, we could estimate f from X p using ML, MMSE, or other criteria. The overall training process takes at least R+2 max{MR/N , M } symbol periods. The optimal designs of U p , Q p , S p (or s p ), and P1,p , P2,p are interesting issues. However, they are beyond the scope of this paper. 3.

PAIRWISE ERROR PROBABILITY AND OPTIMAL POWER ALLOCATION

To analyze the PEP, we have to determine the maximumlikelihood (ML) decoding rule. This requires the conditional probability density function (PDF) P(X | s k ), where s k ∈ S and S is the set of all possible transmit signal matrices. Theorem 1. Given that s k is transmitted, define

Sk = A1 s k A2 s k · · · AR s k .

(15)

Then conditioned on s k , the rows of X are independently Gaussian distributed with the same variance RW . The tth row of X has mean P1 P2 T/M(P1 + 1)[Sk ]t H with [Sk ]t being the tth row of Sk . Also,

P X | sk

−T = π N det RW

×e

−1 −tr(X − P1 P2 T/M P1 +1 Sk H)RW (X − P1 P2 T/M P1 +1 Sk H)

∗

. (16)

P1 P2 T Sk H M P1 + 1

(13)

N p ≥ max MR/N , M .

−1 X− × RW

P1 P2 T Sk H M P1 + 1

(17)

∗

.

With this decoding, the PEP of mistaking s k by s l , averaged over the channel realization, has the following upper bound:

P s k −→ s l ≤ E e−(P1 P2 T/4M(1+P1 )) tr (Sk −Sl ) fmi ,gin

∗

−1 ∗ (Sk −Sl )HRW H

.

(18) Proof. The proof is omitted since it is the same as the proof of Theorem 1 in [5]. As both H and RW are known at the receiver, sphere decoding can be used to perform the ML decoding in (17). The main purpose of this work is to analyze how the PEP decays with the total transmit power. The total power used in the whole network is P = P1 + RP2 . One natural question is how to allocate power between the transmitter and the relays if P is fixed. Notice that when R → ∞, according to the law of large numbers, the off-diagonal entries of (1/R)G∗ G go to zero while the diagonal entries approach 1 with probability 1. It is thus reasonable to assume (1/R)G∗ G ≈ IN for large R. With this approximation, minimizing the PEP is now equivalent to maximizing P1 P2 T/4M(1 + P1 + RP2 ). This is exactly the same power allocation problem in [5]. Therefore, we can conclude that the optimum solution is to set P1 =

P , 2

P2 =

P . 2R

(19)

That is, the optimum power allocation is such that the transmitter uses half the total power and the relays share the other half. As discussed in Section 2.1, for the general network where the ith relay has Ri antennas, the antennas are treated as Ri different relays. Therefore, in general, the optimum power allocation is such that the transmitter uses half the total power as before, but every relay uses a power that is proportional to its number of antennas, that is, P1 = P/2 and the power used at the ith relay is Ri P/2R.

Proof. See Appendix A.

4.

In view of Theorem 1, we should emphasize that for a wireless relay network with multiple antennas at the receiver, the columns of X are not independent although the rows of X are. (The covariance matrix of each row RW is not diagonal in general.) That is, the received signals at different antennas are not independent, whereas the received signals at different times are. This is the main reason that the PEP analysis in the new model is much more difficult than that of the network in [5], where X had only a single column. With P(X | s k ) in hand, we can obtain the ML decoding and thereby analyze the PEP. The result follows.

4.1.

DIVERSITY ANALYSIS FOR R → ∞ Basic results

As mentioned earlier, to obtain the diversity, we have to compute the expectations over fmi and gin in (18). We will do this rigorously in Section 5. However, since the calculation is detailed and gives little insight, in this section, we give a simple asymptotic derivation for the case where the number of relay nodes approaches infinity, that is, R → ∞. As discussed in the previous section, when R is large, we can make the approximation RW ≈ (1 + P2 R/(P1 + 1))IN . Denote the nth column of H as hn . From (5), hn = Gn f, where we


5

have defined Gn = diag{g1n IM , . . . , gRn IM }. Therefore, from (18) and using the optimal power allocation in (19),

P s k −→ s l E e−(PT/16MR)trH

∗

N

∗

n=1 hn

Sk −Sl

fmi ,gin

= E e

−(PT/16MR)f ∗ [

∗

×⎣

∗ n=1 Gn Sk −Sl

∗

.

(20) Since f is white Gaussian with mean zero and variance IRM , ⎡

E det

−1

gin

⎤

PT ∗ G (Sk − Sl )∗ (Sk − Sl )Gn ⎦ . 16MR n=1 n (21) N

⎣IRM +

Similar to the multiple-antenna case [4, 48] and the case of wireless relay networks with single-antenna nodes [5], to achieve full diversity, Sk − Sl must be full rank. Since the distributed space-time codes Sk and Sl are T × MR, in the following, we will assume T ≥ MR and the code is fully diverse. Denote the minimum singular value of (Sk − Sl )∗ (Sk − 2 2 Sl ) by σmin . From the full diversity of the code, σmin > 0. Therefore, the right side of (21) can be further upper bounded as

⎡

gin

=E

gin

R

⎛ ⎝1 +

i=1

16MR

2 gin ⎠

.

⎡

1 (N − 1)!R

⎣

∞ 0

2 PTσmin 1+ x 16MR

⎤R

−M

x

e dx⎦ .

N −1 −x

(23) 2 By defining y = 1 + (PTσmin /16MR)x, we have

P s k −→ s l

1 (N − 1)!

×

∞

1

R

16MR 2 PTσmin

NR 2

2

e16MR /PTσmin

(y − 1)N −1 −(16MR/PTσmin 2 )y e dy yM

R

P s k −→ s l

1 (N − 1)!R

16MR 2 Tσmin

min{M,N }R

⎧ R ⎪ ⎪ 2N −1 ⎪ ⎪ P −NR ⎪ ⎪ ⎪ ⎪ M−N ⎪ ⎪ ⎨ MR × log1/M P ⎪ ⎪ ⎪ ⎪ P ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩(N − M − 1)!R P −MR

if M > N, if M = N, if M < N. (25)

Therefore, the diversity of the wireless relay network is ⎧ ⎪ min{M, N }R ⎪ ⎪ ⎪ ⎨ d=⎪ ⎪ ⎪MR 1 − 1 log log P ⎪ ⎩

M log P

if M = / N, (26) if M = N.

Proof. See Appendix B.

n=1

(22)

⎞−M

Since gin are i.i.d. CN (0, 1), Nn=1 |gin |2 are i.i.d. gamma distributed with PDF (1/(N − 1)!)giN −1 e−gi . Therefore, P s k −→ s l

⎤R N − 1 ∞ l−M −(16MR/PTσ 2 )y min d y ⎦ . y e l 1

Theorem 3 (diversity for R → ∞). Assume that R → ∞, T ≥ MR, and the distributed space-time code is full diverse. For large total transmit power P, by looking at only the highestorder term of P, the PEP of mistaking s k by s l has the following upper bound:

N 2 PTσmin G∗ Gn ⎦ 16MR n=1 n

N 2 PTσmin

NR

The following theorem can be obtained by calculating the integral.

⎤

P s k −→ s l E det−1 ⎣IRM +

16MR 2 PTσmin

(24) (Sk −Sl )Gn ]f

fmi ,gin

P(s k −→ s l )

N −1 l=0

(Sk −Sl )hn

N

(N − 1)!R ⎡

(Sk −Sl )∗ (Sk −Sl )H

fmi ,gin

= E e−(PT/16MR)

1

4.2.

Discussion

With the two-step protocol, it is easy to see that regardless of the cooperative strategy used at the relay nodes, the error probability is determined by the worse of the two transmission stages: the transmission from the transmitter to the relays and the transmission from the relays to the receiver. The PEP of the first stage cannot be better than the PEP of a multiple-antenna system with M transmit antennas and R receive antennas, whose optimal diversity is MR, while the PEP of the second stage can have diversity not larger than NR. Therefore, when M = / N, according to the decay rate of the PEP, distributed space-time coding is optimal. For the case of M = N, the penalty on the decay rate is just R(log log P/ log P), which is negligible when P is high. If we can use the diversity definition in [49], since limP→∞ (log log P/ log P) = 0, diversity min{M, N }R can be obtained. The results in Theorem 6 are obtained by considering only the highest-order term of P in the PEP formula. In brief, we call the rth highest-order term of P in the PEP formula

6


the rth term. When analyzing the diversity, not only is the first term important, but also how dominant it is. Therefore, we should analyze the contributions of the second and also other terms of P compared to those of the first one. This is equivalent to analyzing how large the total transmit power P should be for the terms in (25) to dominate. The following remarks are on this issue. They can be observed from the proof of Theorem 3 in Appendix B. Remark 1. (1) If |M − N | > 1, from (B.13) and (B.22), the second term behaves as P − min{M,N }R+1 . The difference between the first and second terms is a P factor. Therefore, the first term is dominant when P 1. In other words, contributions of the second and other terms are negligible when P 1. (2) If M = N, from (B.16), the second term is

2M −1 R 16MR 2 (M − 1)!R Tσmin

MR

log P , P MR

5.1. A simple derivation The diversity analysis in the previous section is based on the assumption that the number of relays is very large. In this section, analysis on the PEP and diversity for networks with any number of relays is given. As discussed in Section 3, the main difficulty of the PEP analysis lies in the fact that the noise covariance matrix RW is not diagonal. From (18), we can see that one way of upper bounding the PEP is to upper bound RW . Since RW ≥ 0, RW ≤ tr RW

⎛

⎞

P2 gin 2 ⎠ IN . IN = ⎝N + P1 + 1 n=1 i=1

N

R

(28)

Therefore, from (18) and using the power allocation given in (19),

P s k −→ s l

E e−(PT/8MNR(1+(1/NR)

N

n=1

R

i=1

P s k −→ s l E

gin

2 gin ))tr H ∗ (Sk −Sl )∗ (Sk −Sl )H

fmi ,gin

(29)

R i=1

2 gi PTσmin 1+ 8MNR 1 + (1/NR) Ri=1 gi

−M

, (30)

2 is the minimum singular value of where, as before, σmin ∗ (Sk − Sl ) (Sk − Sl ) and gi = Nn=1 |gin |2 . Calculating this integral, the following theorem can be obtained.

Theorem 4 (diversity for wireless relay network). Assume that T ≥ MR and the distributed space-time code is full diverse. For large total transmit power P, by looking at the highest-order terms of P, the PEP of mistaking s k by s l satisfies

(27)

DIVERSITY ANALYSIS FOR THE GENERAL CASE

P(s k −→ s l )

R−1

which has one less log P than the first one. Therefore, the MR MR 2 first term, (1/(M − 1)!R )(16MR/Tσmin ) (log1/M P/P) , is dominant if and only if log P 1, which is a much stronger condition than P 1. When P is not very large, contributions of the second and even other terms are not negligible. (3) If |M − N | = 1, from (B.11) and (B.24), the second term behaves as P − min{M,N }R (log P/P). The difference between the first and second terms is log P/P factor. Therefore, the first term given in (25) is dominant if and only if P log P. This condition is weaker than the condition log P 1 in the previous case; however, it is still stronger than the normally used condition P 1. 5.

when P 1. If the space-time code is fully diverse, using similar argument in the previous section,

1

(N − 1)!R

×

8MNR 2 Tσmin

⎧ ⎪ ⎪ M ⎪ ⎪ ⎪ ⎪ ⎪ N(M − N) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨% &R

1 1+ N

min{M,N }R

R

P −NR

log1/M P P

if M > N,

MR

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' (R ⎪ ⎪ ⎪ 1 ⎪ ⎩ + (N − M − 1)! P −MR

N

if M = N, if M < N. (31)

Therefore, the same diversity as in (26) is obtained. Proof. See Appendix C. Although the same diversity is obtained as in the R → ∞ case, there is a factor of N in (31), which does not appear in (25). This is because we upper bound RW by (trRW )IN , whose expectation is N times the expectation of RW , while in the previous subsection we approximate RW by its expectation. This factor of N can be avoided by tighter upper bounds of RW . In the following subsection, we analyze the maximum eigenvalue of RW . Then in Section 5.3, a PEP upper bound using the maximum eigenvalue of RW is obtained. 5.2.

The maximum eigenvalue of Wishart matrix

Denote the maximum eigenvalue of (1/R)G∗ G as λmax . Since G is a random matrix, λmax is a random variable. We first analyze the PDF and the cumulative distribution function (CDF) of λmax . If entries of G are independent Gaussian distributed with mean zero and variance one, or equivalently, both the real and imaginary parts of every entry in G are Gaussian with mean zero and variance 1/2, (1/R)G∗ G is known as the Wishart matrix. While there exists explicit formula for the distribution of the minimum eigenvalue of a Wishart matrix, we could not find nonasymptotic formula for the maximum eigenvalue. Therefore, we calculate the PDF and CDF of λmax from the joint distribution of all the eigenvalues of (1/R)G∗ G in this section. The following theorem has been proved.


7

PDF of λmax of Wishart matrix

3 2.5

0.8

2 Pr (λmax < λ)

Pr (λmax = λ)

CDF of λmax of Wishart matrix

1

1.5

0.6

0.4

1 0.2

0.5 0

0

2

1

3

4

0

5

0

1

2

λ R = 10 N = 2 R = 10 N = 3 R = 10 N = 4

3

4

5

λ R = 40 N = 2 R = 40 N = 3 R = 40 N = 4

R = 10 N = 2 R = 10 N = 3 R = 10 N = 4

R = 40 N = 2 R = 40 N = 3 R = 40 N = 4

Figure 2: PDF of the maximum eigenvalue of (1/R)G∗ G.

Figure 3: CDF of the maximum eigenvalue of (1/R)G∗ G.

Theorem 5. Assume that R ≥ N and G is an R × N matrix whose entries are i.i.d. CN (0, 1).

Corollary 1. When R ≥ N, the PDF of the maximum eigenvalue of (1/R)G∗ G can be upper bounded as

(1) The PDF of the maximum eigenvalue of (1/R)G∗ G is RRN λR−N e−Rλ det F, n=1 Γ(R − n + 1)Γ(n)

pλmax (λ) = )N

(32)

RRN det F , Γ(R − n + 1)Γ(n) n=1

(34)

where 1 Γ(R − n + 1)Γ(n) n=1

C1 = )N

where F is an (N − 1) × (N − 1) Hankel matrix whose *λ (i, j)th entry equals fi j = 0 (λ − t)2 t R−N+i+ j −2 e−Rt dt. (2) The CDF of the maximum eigenvalue of (1/R)G∗ G is P λmax ≤ λ = )N

pλmax (λ) ≤ C1 λRN −1 e−Rλ ,

(33)

where F is an N × N Hankel matrix whose (i, j)th entry *λ equals fi j = 0 t R−N+i+ j −2 e−Rt dt. Proof. See Appendix D. A theoretical analysis of the PDF and CDF from (32) and (33) appears to be quite difficult. To understand λmax , we plot the two functions in Figures 2 and 3 for different R and N. Figure 2 shows that the PDF has a peak at a value a bit larger than 1. As R increases, the peak becomes sharper. An increase in N shifts the peak right. However, the effect is smaller for larger R. From Figure 3, the CDF of λmax grows rapidly around λ = 1 and becomes very close to 1 soon after. The larger R is, the faster the CDF grows. Similar to the PDF, an increase in N results in a right shift of the CDF. However, as R grows, the effect diminishes. This verifies the validity of the approximation G∗ G ≈ RIN in Section 4 for large R. In the following corollary, we give an upper bound on the PDF. This result is used to derive the diversity result for general R in the next subsection.

2N −1 RRN n=1 (R − N + 2n − 1)(R − N + 2n)(R − N + 2n + 1) (35)

× )N −1

is a constant that depends only on R and N. Proof. From the proof of Theorem 5, F is a positive semidef) inite matrix. Therefore, det F ≤ nN=−11 fnn . From (32), fnn can be upper bounded as fnn ≤ =

λ 0

(λ − t)2 t R−N+2n−2 dt

2 (R − N + 2n − 1)(R − N + 2n)(R − N + 2n + 1) × λR−N+2n+1 ,

(36) then we have det F 2N −1 n=1 (R − N + 2n − 1)(R − N + 2n)(R − N + 2n + 1)

≤ )N −1

× λRN −R+N −1 .

(37) Thus, (34) is obtained.

8


5.3. Bound on PEP from bound on eigenvalues

1 Γ(N − r + 1)Γ(r) r =1

C2 = )R

If the maximum eigenvalue of (1/R)G∗ G is λmax , the maximum eigenvalue of RW is 1+(P2 R/(P1 +1))λmax , and therefore RW ≤ (1 + (P2 R/(P1 + 1))λmax )IN . From (20) and using the power allocation given in (19), we have

P s k −→ s l | λmax = c ≤

E e−(P1 P2 T/4M(1+P1 +P2 Rλmax ))tr(Sk −Sl )

∗

Therefore, the same diversity as in (26) is obtained.

(Sk −Sl )HH ∗

fmr ,grn

∗

E e−(PT/8(1+λmax )MR)tr(Sk −Sl )

(Sk −Sl )HH ∗

fmr ,grn

Proof. When R ≥ N, . (38)

The only difference of the above formula with formula (20) is that the coefficient in the constant in the denominator of the exponent is 8(1 + λmax ) now instead of 16. This makes sense since c → 1 as R → ∞. Therefore, using an argument similar to the proof of Theorem 3, at high total transmit power, by looking at the highest-order terms of P,

2R−1 N RN . r =1 (N − R + 2r − 1)(N − R + 2r)(N − R + 2r + 1) (41)

× )R−1

P s k −→ s l | λmax = c

1 (N − 1)!

R

8(1+c)MR 2 Tσmin

⎧ R ⎪ 2N −1 ⎪ ⎪ ⎪ P −NR ⎪ ⎪ ⎪ ⎪ M−N ⎪ ⎨ MR 1/M × ⎪ log P ⎪ ⎪ ⎪ P ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (N − M − 1)!R P −MR

min{M,N }R

P s k −→ s l = ≤

∞ 0

∞ 0

8MR C1 2 (N − 1)!R Tσmin ×

if M > N,

P s k −→ s l

C+

8MR 2 (N − 1)!R Tσmin

⎧ R ⎪ ⎪ 2N −1 ⎪ ⎪ P −NR ⎪ ⎪ ⎪ M − N ⎪ ⎪ ⎪ ⎨ MR 1/M × ⎪ log P ⎪ ⎪ ⎪ P ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩(N − M − 1)!R P −MR

∞ 0

min{M,N }R

cRN −1 e−Rc (1 + c)min{M,N }R dc

⎧ R ⎪ ⎪ 2N −1 ⎪ ⎪ ⎪ P −NR ⎪ ⎪ ⎪ M−N ⎪ ⎪ ⎨ % & × log P R ⎪ ⎪ ⎪ ⎪ PM ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩(N − M − 1)!R P −MR

if M = N, if M < N. (39)

min{M,N }R

C1 cRN −1 e−Rc P s k −→ s l | λmax = c dc

using (34) in Corollary 1. From (39),

The following theorem can thus be obtained.

(42)

P s i −→ s i

Theorem 6 (diversity for wireless relay network). Assume that T ≥ MR and the distributed space-time code is full diverse. For large total transmit power P, by looking at the highest-order terms of P, the PEP of mistaking s k by s l can be upper bounded as

P sk −→ sl | λmax = c pλmax (c)dc

if M > N, if M = N, if M < N. (43)

Since ∞ 0

cRN −1 e−Rc (1 + c)min{M,N }R dc =

min{ M,N }R i=0

min{M, N }R (RN + i − 1)! , RRN+i i

(44)

if M > N, if M = N, if M < N, (40)

∞

where

⎧ ⎛ ⎞ min{ M,N }R min{M, N }R ⎪ ⎪ ⎪ ⎪ ⎝ ⎠ (RN + i − 1)! ⎪ C1 ⎪ ⎪ ⎪ RRN+i ⎪ i i=0 ⎨

0

%

cRN −1 e−Nc 1 +

if R ≥ N, =

+ C= ⎪

⎛ ⎞ ⎪ ⎪ min{ M,N }R min{M, N }R ⎪ ⎪ ⎪ ⎝ ⎠ (RN + i − 1)! ⎪ C ⎪ ⎪ ⎩ 2 Ri N RN i i=0

(40) is obtained. For the case of R < N, G∗ is an N × R (N > R) matrix whose entries are i.i.d. CN (0, 1). Denote the maximal eigenvalue of (1/N)GG∗ as λ max . Its PDF and CDF are given in Theorem 5 with R and N being switched. Using the facts that the maximal eigenvalue of (1/R)G∗ G is (N/R)λ max and &min{M,N }R

dc

min{ M,N }R min{M, N }R i=0

if R 0, Rμ > 0, n = 0, 1, 2, . . . ,

,

(N − M − 1)! (N − 1)!

(R %

16MR 2 Tσmin

&MR

1 P MR

%

+o

(B.2) n e−μx n+1 μ Ei(−μu) dx = ( − 1) xn+1 n!

+ ∞

n−1 e−μu (−1)k μk uk , un k=0 n · · · (n − k)

μ > 0, n = 1, 2, . . . , (B.3)

e

−μx

x

u

dx = −Ei(−μu),

Rμ > 0, u ≥ 0,

(B.4)

I=

χ

et dt, −∞ t

χ < 0,

(B.5)

=

is the exponential integral function [51]. To calculate I, we discuss the following cases separately. Case 1 (M < N). In this case, I=

N −1 l=M

⎛

+⎝

N −1 l

N −1

M−1

+

M −2

⎞ ⎠

∞

1

∞

e−

2 16MR/PTσmin

dy

(B.6)

l

N −1

16MR + M −Ei − 2 PTσmin

+

MR

RM 16MR 2 M!R Tσmin

1 P MR

MR+1

1

N −1 + log P + M−1

−(l−M+1)

M −2 N −1

l

l=0

%

(B.7)

1 M−l−1

−(N −M)

&

I = (N − M − 1)!

16MR 2 PTσmin

&−(N −M)

+ (N − 1)(N − M − 2)!

% & . / 16MR −(N −M −1) × + o P N −M −1 2

PTσmin

By only looking at the highest-order term of P, which is in the first term with l = N − 1, we have 16MR I = (N − M − 1)! 2 PTσmin

%

log P log P + o MR+1 . P MR+1 P (B.11)

The second highest-order term of P in the PEP behaves as log P/P MR+1 = P −(MR+1−log log P/ log P) . If N > M + 1,

+ lower-order terms of P.

+ O(1)

+ M log P + O(1).

16MR N −1 (l − M)! 2 l PTσmin

−1

1 16MR 2 M!R Tσmin

⎞

2 ), Using equalities (B.2)–(B.4) with u = 1, μ = (16MR/PTσmin and n = l − M or n = M − l − 1,

l=M

P s k −→ s l

N − 1 ∞ −(M −l) −(16MR/PTσ 2 )y ⎝ ⎠ min d y. y e

l=0

I=

16MR 2 PTσmin

−1

(B.10)

y

y

1

⎛

16MR 2 PTσmin

Therefore,

2

y l−M e−(16MR/PTσmin )y d y

. P MR (B.9)

While analyzing the performance of the system at high transmit power P, not only is the highest-order term of P important, but also how fast other terms decay with respect to it. Therefore, we should also look at the second highestorder term of P. To do this, we have to consider two different cases. If N = M + 1,

where Ei (χ) =

&

1

.

% =

16MR 2 PTσmin

&−(N −M) '

(N − M − 1)! + (N − 1) % &(

/

+ o P −(N −M) .

× (N − M − 2)!

(B.8)

1 16MR +o . 2 P PTσmin (B.12)

12


Therefore,

Thus,

(N − M − 1)! (N − 1)!R

P s k −→ s l

R%

16MR 2 Tσmin

&MR

1 P MR

N −1 N −1

(N − 1)(N − M − 2)(N − M − 1)! (N − 1)!R

% & 16MR MR+1 × 2

Tσmin

%

I=

∞

e−

&

y

1

+

N −2 N −1 ∞

l

l=0

1

(N − 1)!

dy (B.14) 2

y −(M −l) e−(16MR/PTσmin )y d y.

1 M−l−1

P s k −→ s l

(M − 1)!R

16MR 2 Tσmin

(B.15)

%

&MR

16MR 2N −1 R 2 R (M − 1)! Tσmin

+

%

' ≤

2N −1 (M − N)(N − 1)!

R

(R %

16MR 2 Tσmin

&NR

P −NR .

As discussed before, we also want to see how dominant the highest-order term of P given in the above formula is. If M > N + 1, M − l − 2 > N + 1 − (N − 1) − 2 = 0. From (B.3), % &

logR−1 P P MR

16MR 2N −1 1 2N −1 − +o . 2 M − N (M − N)(M − N − 1) PTσmin P (B.21)

Therefore,

P s k −→ s l

'

2N −1 (M − N)(N − 1)! '

(B.16)

×

&

P NR

+o

(R %

16MR 2 Tσmin

%

1

%

logR−1 P . P MR

+o

Tσmin

I
N). In this case, I=

M −2 l=0

N −1 l

∞

1

2

y −(M −l) e−(16MR/PTσmin )y d y.

Therefore, (B.17)

2 , and n = M −l −1, Using (B.3) with u = 1, μ = 16MR/PTσmin

I=

N −1 l=0

The second highest-order term in the PEP behaves as 1/(P NR+1 ). If M = N + 1, % & 1 16MR log P I< 2N −1 + . (B.23) + O 2 P Tσmin P

1 N −1 + lower-order terms of P. l M−l−1 (B.18)

P s k −→ s l

%

2R(N −1) 16MR 2 (N − 1)!R Tσmin %

+

&NR

2(R−1)(N −1) R 16MR 2 (N − 1)!R Tσmin %

+o

&

1 P NR &NR+1

log P P NR+1

log P , P NR+1 (B.24)


13

which indicates that the second highest-order term in the PEP behaves as log P/P NR+1 = R−(NR+1−log log P/ log P) . C. PROOF OF THEOREM 4 Proof. Since gi have PDF p(gi ) = (1/(N − 1)!)giN −1 e−gi ,

P s k −→ s l ≤

R

1

Ti1 ,...,ir ,

r =0 1≤i1 N, if M = N, if M < N. (C.14)

We should choose a negative α such that the exponent of the highest-order term of P in the above formula is minimized. In other words, if we denote the exponent of the rth term as f (r), choose an α < 0 such that maxr f (r) is minimized. If M > N, f (r) = −rM + αN(R − r) − rα(M − N) = αNR − rM(1 + α). If α ≤ −1, f (r) is an increasing function

⎢ ⎢ ⎣

Vg (Λ) = ⎢

g0 λ1 .. .

gN −1 λ1 ⎡

h0 λ1 ⎢ .. ⎢ Vh (Λ) = ⎢ . ⎣

hN −1 λ1

Fgh =

⎡ ⎢ f (t) ⎢ ⎣

⎤

···

..

.

g0 λN .. .

⎤

· · · gN −1 λN ···

..

.

h0 λN .. .

⎥ ⎥ ⎥, ⎦

⎤

· · · hN −1 λN

⎥ ⎥ ⎥, ⎦

g0 (t) .. ⎥ ⎥ . ⎦ h0 (t) · · · hN −1 (t) dt. gN −1 (t)

(D.1)


15

Define G as a complex Gaussian matrix whose entries’ real and imaginary parts have mean zero and variance one. Denote the ordered eigenvalues of G G ∗ as λ 1 ≥ λ 2 · · · ≥ λ N . It is well known that the eigenvalues have the following joint distribution [52]:

P λ 1 , . . . , λ N = C

N

λi R−N e−λi /2

i=1

2

λ i − λ j ,

1≤i< j ≤N

)

P λ1 , . . . , λN

dλ 1 · · · dλ N = P λ 1 , . . . , λ N dλ1 · · · dλN 4 5 = det diag{2R, . . . , 2R} 2Rλ1 , . . . , 2RλN N

= (2R) C

N

2Rλi

R−N

e−Rλi

i=1

= C(2R)

N RN

4

2R λi − λ j

52

1≤i< j ≤N

λi − λ j .

E.

In Section 2.2, it is assumed that fmi and gin have the same variance. Physically, this means that the distances between the transmitter/receiver and all relays are about the same, which may not be a practical assumption for networks with scattered nodes. In this appendix, we extend our diversity analysis to heterogeneous networks whose channels have different variances. We assume that the distributions of fmi and gin are CN (0, σ 2fmi ) and CN (0, σg2in ), respectively. By following the derivation in Section 4, compared with (21), the PEP for the heterogeneous case can be upper bounded by

= P λ1 = λ P λ, λ2 , . . . , λN dλ2 · · · dλN =

gin

=

=

λ

(N − 1)!

0

···

λ 0

C(2R)RN R−N −Rλ e λ (N − 1)! ×

gin

0

λ N 0 i=2

2

λ − λi λRi −N e−Rλi

2

P s k −→ s l

2≤i< j ≤N

λ 0

···

λ N 0 i=2

f (λi )

σf2i

i=1

−M

2 2 σg2 PTσmin N gin + i n=1 σ 2 16MR gin

.

Since |gin |2 /σg2in has the exponential distribution with mean 1 and gin ’s are independent, Nn=1 |gin |2 /σg2in has the i.i.d. gamma distribution (1/(N − 1)!)giN −1 e−gi . Thus, following the derivations in Section 4 and Appendix B, we can show that the PEP of heterogeneous networks has the following upper bound:

λi − λ j dλ2 · · · dλN

C(2R)RN R−N −Rλ = e λ (N − 1)!

R

(E.2)

···

P s k −→ s l E

P λ, λ2 , . . . , λN dλ2 · · · dλN

λ

PT ∗ ∗ G Sk − Sl ) Sk − Sl Gn , 16MR n=1 n (E.1) N

Σf +

2 , . . . , σ 2f mR } is the where Σf = diag{σ 2f 11 , . . . , σ 2f m1 , . . . , σ1R M covariance matrix of f. Denote σf2i = minm=1 {σ 2fmi } and σg2i = 2 minM m=1 {σgin }. We have from (E.1) that

λ≥λ2 ≥···λN

1

E det

(D.3)

−1

P λmax = λ

(D.5)

DISCUSSION ON HETEROGENEOUS NETWORKS

1≤i< j ≤N

To get the PDF of λ1 , we have to do the integral over λ2 , . . . , λN . Define f (x) = (λ − x)2 xR−N e−Rx and gi (x) = hi (x) = xi−1 . Thus,

t N −2

⎥ ⎥ ⎥ 1 t · · · t N −2 dt, ⎥ ⎦ *λ

P s k −→ s l

2

⎤

1 t .. .

whose (i, j)th entry is fi j = 0 (λ − t)2 t R−N+i+ j −2 e−Rt dt. The CDF of λ1 can be obtained similarly.

λRi −N e−Rλi

i=1

⎡ ⎢ ⎢ F = g(t) ⎢ ⎢ 0 ⎣ λ

(D.2)

where C = 2−RN / Nn=1 Γ(R − n + 1)Γ(n) is a constant. Denote the ordered eigenvalues of (1/R)GG∗ as λ1 ≥ λ2 · · · ≥ λN . Therefore, λ i = 2Rλi . The joint distribution of λ1 , . . . , λN is therefore

of λi , we only need to divide the new value by (N − 1)!. From Theorem 7,

× det Vg λ2 , . . . , λN det Vh λ2 , . . . , λN λ1 · · · λN

C(2R)RN R−N −Rλ = e (N − 1)! det F, λ (N − 1)! (D.4) where in the second equality we have changed the integral space from ordered λi to unordered one. From the symmetry

1 (N − 1)!R

R i=1

σf2i

− min{M,N } %

16MR 2 Tσmin

⎧ % &−(N −M) % N −1 &R R ⎪ ⎪ 2 ⎪ 2 ⎪ ⎪ σg i P −NR ⎪ ⎪ M −N ⎪ ⎪ i = 1 ⎪ ⎨ × % log1/M P &MR ⎪ ⎪ ⎪ ⎪ ⎪ P ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (N − M − 1)!R P −MR

&min{M,N }R

if M > N, if M = N, if M < N. (E.3)

16 Thus, the same diversity results as in (26) can be obtained. Similarly, the rigorous analysis in Section 5 also applies to this heterogeneous case. ACKNOWLEDGMENTS This work is supported in part by the National Science Foundation under Grants nos. CCR-0133818 and CCR0326554, by the David and Lucille Packard Foundation, and by Caltech’s Lee Center for Advanced Networking. A preliminary version of this paper and the related results first appeared in the Proceeding of the 2005 IEEE International Symposium on Information Theory [53]. REFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585–595, 1999. [2] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-antenna communication link in rayleigh flat fading,” IEEE Transactions on Information Theory, vol. 45, no. 1, pp. 139–157, 1999. [3] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multielement antennas,” Bell Labs Technical Journal, vol. 1, no. 2, pp. 41–59, 1996. [4] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Transactions on Information Theory, vol. 44, no. 2, pp. 744–765, 1998. [5] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Transactions on Wireless Communications, vol. 5, no. 12, pp. 3524–3536, 2006. [6] Y. Chang and Y. Hua, “Application of space-time linear block codes to parallel wireless relays in mobile ad hoc networks,” in Proceedings of the 36th Asilomar Conference on Signals, Systems and Computers (ACSSC ’02), vol. 1, pp. 1002–1006, Pacific Grove, Calif, USA, November 2003. [7] Y. Hua, Y. Mei, and Y. Chang, “Wireless antennas-making wireless communications perform like wireline communications,” in Proceedings of IEEE Topical Conference on Wireless Communication Technology, Honolulu, Hawaii, USA, October 2003. [8] Y. Tang and M. C. Valenti, “Coded transmit macrodiversity: block space-time codes over distributed antennas,” in Proceedings of IEEE Vehicular Technology Conference (VTC ’01), vol. 2, pp. 1435–1438, Atlantic City, NJ, USA, May 2001. [9] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity—part I: system description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, 2003. [10] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity-part II: implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, 2003. [11] R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: performance limits and space-time signal design,” IEEE Journal on Selected Areas in Communications, vol. 22, no. 6, pp. 1099–1109, 2004. [12] H. Bolcskei, R. U. Nabar, O. Oyman, and A. J. Paulraj, “Capacity scaling laws in MIMO relay networks,” IEEE Transactions on Wireless Communications, vol. 5, no. 6, pp. 1433–1444, 2006.

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Y. Jing and B. Hassibi [30] G. S. Rajan and B. S. Rajan, “Algebraic distributed spacetime codes with low ML decoding complexity,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’07), Nice, France, June 2007. [31] G. S. Rajan, A. Tandon, and B. S. Rajan, “On four-group ML decodable distributed space time codes for cooperative communication,” submitted to IEEE Transactions on Information Theory, http://arxiv.org/abs/cs.IT/0701067v1. [32] Y. Jing and H. Jafarkhani, “Distributed differential spacetime coding in wireless relay networks,” to appear in IEEE Transactions on Communications. [33] F. Oggier and B. Hassibi, “Cyclic distributed space-time codes for wireless networks with no channel information,” Submitted. [34] T. Kiran and B. S. Rajan, “Partially-coherent distributed space-time codes with differential encoder and decoder,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp. 426–433, 2007. [35] G. S. Rajan and B. S. Rajan, “Noncoherent low-decodingcomplexity space-time codes for wireless relay networks,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’07), Nice, France, June 2007. [36] P. V. Kumar, K. Vinodh, M. Anand, and P. Elia, “Diversitymultiplexing gain tradeoff and DMT-optimal distributed space-time codes for certain cooperative communication protocols: overview and recent results,” in Proceedings of Information Theory and Applications Workshop, San Diego, Calif, USA, January 2007. [37] P. Elia and P. V. Kumar, “Approximately universal schemes for cooperative diversity in wireless networks,” submitted to IEEE Transactions on Information Theory. [38] X. Guo and X.-G Xia, “A distributed space-time coding in asynchronous wireless relay networks,” to appear in IEEE Transactions on Wireless Communications. [39] Y. Li, W. Zhang, and X.-G Xia, “Distributive high-rate spacefrequency codes achieving full cooperative and multipath diversities for asynchronous cooperative communications,” to appear in IEEE Transactions on Vehicular Technology. [40] Z. Li and X.-G Xia, “A simple Alamouti space-time transmission scheme for asynchronous cooperative systems,” IEEE Signal Processing Letters, vol. 14, pp. 804–807, November 2007. [41] Y. Li and X.-G. Xia, “A family of distributed space-time trellis codes with asynchronous cooperative diversity,” IEEE Transactions on Communications, vol. 55, no. 4, pp. 790–800, 2007. [42] P. Elia, S. Kittipiyakul, and T. Javidi, “Cooperative diversity schemes for asynchronous wireless networks,” Wireless Personal Communications, vol. 43, no. 1, pp. 3–12, 2007. [43] P. Elia and P. V. Kumar, “Constructions of cooperative diversity schemes for asynchronous wireless networks,” in Proceedings of the IEEE International Symposium on Information Theory (ISIT ’06), pp. 2724–2728, Seattle, Wash, USA, July 2006. [44] G. S. Rajan and B. S. Rajan, “Distributed space-time codes for cooperative networks with partial CSI,” submitted to IEEE Transactions on Information Theory, http://arxiv.org/abs/ cs/0701068v1. [45] G. S. Rajan and B. S. Rajan, “A non-orthogonal distributed space-time coded protocol part I: signal model and design criteria,” submitted to IEEE Transactions on Information Theory, http://arxiv.org/abs/cs/0610161v1. [46] G. S. Rajan and B. S. Rajan, “A non-orthogonal distributed space-time coded protocol part II: code construction and DMG tradeoff,” submitted to IEEE Transactions on Information Theory, http://arxiv.org/abs/cs/0610160.

17 [47] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?” IEEE Transactions on Information Theory, vol. 49, no. 4, pp. 951–963, 2003. [48] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 543–564, 2000. [49] L. Zheng and D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1073– 1096, 2003. [50] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Transactions on Information Theory, vol. 45, no. 5, pp. 1456–1467, 1999. [51] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, NY, USA, 6th edition, 2000. [52] A. Edelman, Eigenvalues and Condition Numbers of Random Matrices, Ph.D. thesis, Massachusetts Institute of Technology (MIT), Department of Mathematics, Cambridge, Mass, USA, 1989. [53] Y. Jing and B. Hassibi, “Cooperative diversity in wireless relay networks with multiple-antenna nodes,” in Proceedings of the International Symposium on Information Theory (ISIT ’05), pp. 815–819, Adelaide, South Australia, September 2005.


Research Article On the Duality between MIMO Systems with Distributed Antennas and MIMO Systems with Colocated Antennas Jan Mietzner1 and Peter A. Hoeher2 1 Communication

Theory Group, Department of Electrical and Computer Engineering, The University of British Columbia, 2332 Main Mall, Vancouver, BC, Canada V6T 1Z4 2 Information and Coding Theory Lab, Faculty of Engineering, University of Kiel, Kaiserstrasse 2, 24143 Kiel, Germany Correspondence should be addressed to Jan Mietzner, [email protected] Received 1 May 2007; Revised 16 August 2007; Accepted 28 October 2007 Recommended by M. Chakraborty Multiple-input multiple-output (MIMO) systems are known to offer huge advantages over single-antenna systems, both with regard to capacity and error performance. Usually, quite restrictive assumptions are made in the literature on MIMO systems concerning the spacing of the individual antenna elements. On the one hand, it is typically assumed that the antenna elements at transmitter and receiver are colocated, that is, they belong to some sort of antenna array. On the other hand, it is often assumed that the antenna spacings are sufficiently large, so as to justify the assumption of uncorrelated fading on the individual transmission links. From numerous publications it is known that spatially correlated links caused by insufficient antenna spacings lead to a loss in capacity and error performance. We show that this is also the case when the individual transmit or receive antennas are spatially distributed on a large scale, which is caused by unequal average signal-to-noise ratios (SNRs) on the individual transmission links. Possible applications include simulcast networks as well as future mobile radio systems with joint transmission or reception strategies. Specifically, it is shown that there is a strong duality between MIMO systems with colocated antennas (and spatially correlated links) and MIMO system with distributed antennas (and unequal average link SNRs). As a result, MIMO systems with distributed and colocated antennas can be treated in a single, unifying framework. An important implication of this finding is that optimal transmit power allocation strategies developed for MIMO systems with colocated antennas may be reused for MIMO systems with distributed antennas, and vice versa. Copyright © 2008 J. Mietzner and P. A. Hoeher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Multiple-input multiple-output (MIMO) systems have gained much attention during the last decade, because they offer huge advantages over conventional single-antenna systems. On the one hand, it was shown in [1–3] that the capacity of a MIMO system with M transmit (Tx) antennas and N receive (Rx) antennas grows linearly with min {M, N }. Correspondingly, multiple antennas provide an excellent means to increase the spectral efficiency of a system. On the other hand, it was shown in [4–6] that multiple antennas can also be utilized, in order to provide a spatial diversity gain and thus to improve the error performance of a system. The results in [1–6] are based on quite restrictive assumptions with regard to the antenna spacings at transmitter and receiver. On the one hand, it is assumed that the

individual antenna elements are colocated, that is, they are part of some antenna array (cf. Figure 1(a)). On the other hand, the antenna spacings are assumed to be sufficiently large, so as to justify the assumption of independent fading on the individual transmission links. In numerous publications, it was shown that spatial fading correlations, caused by insufficient antenna spacings (cf. Figure 1(b)), can lead to significant degradations in capacity and error performance, for example, [7–9]. In this paper, we show that this is also the case when the individual transmit and/or receive antennas are distributed on a large scale (cf. Figure 1(c)), since the individual transmission links are typically characterized by unequal average signal-to-noise ratios (SNRs) caused by unequal link lengths and shadowing effects. Application examples include simulcast networks for broadcasting or paging applications, where multiple distributed transmitting nodes

2

EURASIP Journal on Advances in Signal Processing Tx1 Tx2 TxM Tx1

Tx2

Rx1

TxM

Rx1

RxN

(a)

RxN (b)

Tx1 Distributed transmitting nodes

Tx2

TxM

Receiving node Rx1 RxN (c)

Figure 1: MIMO systems with different antenna spacings, for the example of M = 3 transmit and N = 2 receive antennas: (a) MIMO system with colocated antennas (statistically independent links); (b) MIMO system with colocated antennas and insufficient antenna spacings at the transmitter side (spatially correlated links); (c) MIMO system with distributed antennas at the transmitter side (and unequal average link SNRs).

free exchange of messages between distributed antennas are possible. (In the application examples mentioned above, the exchange of messages can, for example, be performed via some fixed backbone network, possibly by employing some error-detecting channel code.) Thus, our simplified framework yields ultimate performance limits for MIMO systems with distributed antennas. Still, the major finding of this paper, namely that MIMO systems with distributed antennas and unequal average link SNRs behave in a very similar way as MIMO systems with correlated antennas (with regard to various performance measures) will also be valid if the assumptions of perfect synchronization and error-free message exchange between distributed antennas are dropped. The duality results presented here are based on two unitary matrix transforms. The first transform associates a given MIMO system with colocated antennas with a corresponding MIMO system with distributed antennas. This transform is related to the well-known Karhunen-Loève transform [12, Chapter 8.5], which is often used in the literature, in order to analyze correlated systems. Moreover, we introduce a second transform which associates a given MIMO system with distributed antennas with a corresponding MIMO system with colocated antennas. Although the performance of MIMO systems with distributed antennas has already been considered in various publications, mainly with focus on cooperative relaying systems, for example, [13–19], the impact of unequal average link SNRs, and particularly its close relation to spatial correlation effects, has not yet been clearly formulated in the literature. In fact, some papers on cooperative relaying neglect the impact of unequal average link SNRs completely; for example, see [13]. 1.1.

(typically base stations) serve a common geographical area by performing a joint transmission strategy [10], as well as future mobile radio systems, where joint transmission and reception strategies among distributed wireless access points are envisioned [11]. In particular, we show that there is a strong duality between MIMO systems with colocated antennas (and spatially correlated links) and MIMO systems with distributed antennas (and unequal average link SNRs). To this end, we will consider resulting capacity distributions as well as the error probabilities of space-time codes. An important implication of the above duality is that optimal transmit power allocation strategies developed for MIMO systems with colocated antennas can be reused for MIMO systems with distributed antennas, and vice versa. In practice, there are several important differences between MIMO systems with colocated antennas and MIMO systems with distributed antennas. For example, synchronization issues are typically more crucial when antennas are spatially distributed. Furthermore, in a scenario with distributed antennas, the exchange of transmitted/received messages between the individual antenna elements may entail error-propagation effects. In order to establish a strict duality between colocated and distributed antennas, we will employ a somewhat simplified common framework here. In particular, we will assume that perfect synchronization and an error-

Remark on spatially correlated MIMO systems

When referring to spatial fading correlations, the notion of “insufficient antenna spacings” is somewhat relative, because spatial correlation effects are not only governed by the geometry of the antenna array and the employed carrier frequency, but also by the richness of scattering from the physical environment and the angular power distribution of the transmitted/received signals [7–9]. In cellular radio systems with a typical urban environment, for example, antenna correlations are thus observed both at the base stations (since the transmitted/received signals are typically confined to comparatively small angular regions) and at the mobile terminals (since antenna spacings are typically rather small). Even in rich-scattering (e.g., indoor) environments, where spatial correlation functions typically decay quite fast with growing antenna spacings, there are usually pronounced side lobes within the spatial correlation functions, so that unfavorable antenna spacings can still entail notable spatial correlations. 1.2.

Paper organization

The paper is organized as follows. First, the system and correlation model used throughout this paper are introduced in Section 2. Then, the duality between MIMO systems with distributed antennas and MIMO systems with colocated antennas is established in Section 3, with regard to the resulting

J. Mietzner and P. A. Hoeher

3

capacity distribution (Section 3.1), the pairwise error probability of a general space-time code (Section 3.2), and the symbol error probability of an orthogonal space-time block code (OSTBC) [5, 6] (Section 3.3). The most important results are summarized in Theorems 1–3. Finally, optimal transmit power allocation strategies for MIMO systems with colocated and distributed antennas are briefly discussed in Section 4, and some conclusions are offered in Section 5.

Possibly, they are correlated due to some underlying spacetime code. We assume an overall transmit power constraint of P, that is, i σ 2xi ≤ P. For the time being, we focus on the case of equal power allocation among the individual transmit antennas, that is, σ 2xi = P/M for all indices i. Finally, the entries of n[k] are assumed to be zero-mean, spatially and temporally white complex Gaussian random variables with variance σ 2n /2 per real dimension, that is, n j [k]∼CN {0, σ 2n } for all indices j and E{n[k]nH [k ]} = σ 2n ·δ[k − k ]·IN .

1.3. Mathematical notation Matrices and vectors are written in upper case and lower case bold face, respectively. If not stated otherwise, all vectors are column vectors. The complex conjugate of a complex number a is marked as a∗ and the Hermitian transposed of a matrix A as AH . The (i, j)th element of A is denoted as ai j or [A]i, j . The trace and the determinant of A are denoted as tr(A) and det(A), respectively. Moreover, AF = tr(AAH ) denotes the Frobenius norm of A, diag(a) a diagonal matrix with diagonal elements given by the vector a, and vec(A) a vector which results from stacking the columns of matrix A in a single vector. Finally, In denotes the (n × n)-identity matrix, E{·} denotes statistical expectation, and δ[k − k0 ] denotes a discrete Dirac impulse at k = k0 . 2.

SYSTEM AND CORRELATION MODEL

Throughout this paper, complex baseband notation is used. We consider a point-to-point MIMO communication link with M transmit and N receive antennas. The antennas are either colocated or distributed and are assumed to have fixed positions. The discrete-time channel model for quasi-static frequency-flat fading is given by y[k] = Hx[k] + n[k],

(1)

where k denotes the discrete time index, y[k] the kth received vector, H the (N × M)-channel matrix, x[k] the kth transmitted vector, and n[k] the kth additive noise vector. It is assumed that H, x[k], and n[k] are statistically independent. The channel matrix H is assumed to be constant over an entire data block of length Nb , and changes randomly from one data block to the next. Correspondingly, we will sometimes use the following block transmission model: Y = HX + N,

(2)

where Y := [y[0], . . . , y[Nb − 1]], X := [x[0], . . . , x[Nb − 1]], and N := [n[0], . . . , n[Nb − 1]]. The entries h ji of H (i = 1, . . . , M, j = 1, . . . , N) are assumed to be zero-mean, circularly symmetric complex Gaussian random variables with variance σ 2ji /2 per real dimension, that is, h ji ∼CN {0, σ 2ji } (Rayleigh fading). The instantaneous realizations of the channel matrix H are assumed to be perfectly known at the receiver. The covariance between two channel coefficients h ji and h j i is denoted as σ 2i j,i j := E{h ji h∗j i } and the corresponding spatial correlation as ρi j,i j := σ 2i j,i j /(σ ji σ j i ). The entries xi [k] (i = 1, . . . , M) of the transmitted vector x[k] are treated as zero-mean random variables with variance σ 2xi .

2.1.

MIMO systems with colocated antennas

In the case of colocated antennas (both at the transmitter and the receiver side), all links experience on average similar propagation conditions. It is therefore reasonable to assume that the variance of the channel coefficients h ji is the same for all transmission links. Correspondingly, we define σ 2ji := σ 2 for all indices i, j. (A generalization to unequal variances is straightforward.) Moreover, we define

E HH H RTx := 2 , Nσ

E HHH RRx := 2 , Mσ

(3)

where RTx denotes the transmitter correlation matrix and RRx the receiver correlation matrix (with tr(RTx ) := M and tr(RRx ) := N). Throughout this paper, the so-called Kronecker-correlation model1 [7] is employed, that is, the overall spatial correlation matrix R := E{vec(H)vec(H)H }/σ 2 can be written as the Kronecker product

RTx := ρTx,ii

R = RTx ⊗ RRx , , RRx := ρRx, j j j, j =1,...,N ,

(4)

i,i =1,...,M

and the channel matrix H can be written as 1/2 1/2 GRTx , H := RRx

(5)

where G denotes an (N × M)-matrix with independent and identically distributed (i.i.d.) entries g ji ∼CN {0, σ 2 }. The eigenvalue decompositions of RTx and RRx are in the sequel denoted as RTx := UTx ΛTx UH Tx ,

RRx := URx ΛRx UH Rx ,

(6)

where UTx , URx are unitary matrices and ΛTx , ΛRx are diagonal matrices containing the eigenvalues λTx,i and λRx, j of RTx and RRx , respectively. 2.2.

MIMO systems with distributed antennas

Consider first a MIMO system with distributed antennas at the transmitter side. As a generalization to Figure 1(c), the individual transmitting nodes may in the sequel be equipped 1

Although the Kronecker-correlation model is not the most general correlation model, it was shown to be quite accurate as long as a moderate number of transmit and receive antennas are used [20].

4


with multiple antennas. To this end, let T denote the number of transmitting nodes, Mt the number of antennas employed at the tth transmitting node (1 ≤ t ≤ T), and let M again denote the overall number of transmit antennas, that is, t Mt =: M. As earlier, let N denote the number of receive antennas used. For simplicity, we assume that all transmit antennas are uncorrelated. (For antennas belonging to different transmitting nodes, this assumption is surely met.) A generalization to the case of correlated transmit antennas is, however, straightforward. Similar to Section 2.1, it is again reasonable to assume that all channel coefficients associated with the same transmitting node t have the same variance σ 2t . Correspondingly, we obtain

E

HH H N

= diag σ 21 , . . . , σ 2t , . . . , σ 2T =: ΣTx ,

(7)

σ 2t

occurs Mt times. Following the where each variance Kronecker-correlation model, we may thus write H :=

1/2 RRx GΣ1/2 Tx ,

E HHH = diag σ 21 , . . . , σ 2r , . . . , σ 2R =: ΣRx , M

(9)

1/2 that is, H := Σ1/2 Rx GRTx .

Duality with regard to capacity distribution

For the time being, we assume that no channel state information is available at the transmitter side. In this case, the (instantaneous) capacity of the MIMO system (1) is given by the well-known expression [2]

where C(H) =: r is a random variable with probability density function (PDF) denoted as p(r). 3.1.1. Capacity distribution for MIMO systems with colocated antennas The characteristic function of the instantaneous capacity, √ cfr (jω) := E{ejωr } (j = −1, ω ∈ R), was evaluated in [22]. The result is of form Kϕ(jω) det cfr (jω) = ψ(RA , RB )

tr E vec(H)vec(H)H

:= MN.

(10)

For MIMO systems with colocated antennas this means we set σ 2 := 1. For MIMO systems with distributed transmit or receive antennas, it means we set tr(ΣTx ) := M or tr(ΣRx ) := N. 3.

DUALITY BETWEEN DISTRIBUTED AND COLOCATED ANTENNAS

In the following, we will show that, based on the above framework, for any MIMO system with colocated antennas, which follows the Kronecker-correlation model (5), an equivalent MIMO system with distributed antennas (and unequal average link SNRs) can be found, and vice versa, in the sense that both systems are characterized by identical capacity distributions.

V(RB ) M(RA , RB , jω)

(12)

(see [22] for further details2 ), where

RA , RB :=

RTx , RRx RRx , RTx

if M < N, else.

(13)

Interestingly, the scalar term ψ(RA , RB ) as well as the Vandermonde matrix V(RB ) and the matrix M(RA , RB , jω) depend solely on the eigenvalues of RA and RB , but not on specific entries of RA or RB . Moreover, the terms K and ϕ(jω) are independent of RA and RB . The characteristic function cfr (jω) contains the complete information about the statistical properties of r = C(H). Specifically, the PDF of r can be calculated as [23, Chapter 1.1]

2.3. Normalization In order to treat MIMO systems with colocated antennas and MIMO systems with distributed antennas in a single, unifying framework, we employ the following normalization:

P HHH bit/channel use, (11) Mσ 2n

C(H) = log 2 det IN +

(8)

where the i.i.d. entries of G have variance one. Due to different link lengths (and, possibly, additional shadowing effects), the variances σ 2t will typically vary significantly from one transmitting node to another, since the received power decays at least with the square of the link length [21, Chapter 1.2]. Similarly, in the case of colocated transmit antennas and distributed receive antennas, where R denotes the number of receiving nodes, we obtain

3.1.

p(r) =

1 2π

+∞ −∞

cfr (jω)e−jωr dω.

(14)

3.1.2. Capacity distribution for MIMO systems with distributed antennas Since the characteristic function cfr (jω) according to (12) depends solely on the eigenvalues of RA and RB , any MIMO system having an overall spatial covariance matrix

H E vec(H)vec(H)H = UM RTx UH M ⊗ UN RRx UN ⊗ RRx , =: RTx

(15)

where UM is an arbitrary unitary (M × M)-matrix and UN an arbitrary unitary (N × N)-matrix, will exhibit exactly the same capacity distribution (14) as the above MIMO system 2

For simplicity, it was assumed in [22] that both matrices RA and RB have full rank and distinct eigenvalues. If the eigenvalues of RA or RB are not distinct, the characteristic function of r = C(H) can be obtained as a limiting case of (12).

J. Mietzner and P. A. Hoeher

5

with colocated antennas (because the eigenvalues of RTx and RTx and of RRx and RRx are identical). Specifically, we may H choose UM := UH Tx and/or UN := URx , in order to find an equivalent MIMO system with distributed transmit and/or distributed receive antennas:

UH Tx RTx UTx = ΛTx =: ΣTx ,

UH Rx RRx URx = ΛRx =: ΣRx . (16)

By this means, for any MIMO system with colocated antennas, which follows the Kronecker-correlation model, an equivalent MIMO system with distributed antennas can be found. Vice versa, given a MIMO system with distributed transmit and/or distributed receive antennas, the diagonal elements of the matrix ΣTx (ΣRx ), normalized according to Section 2.3, may be interpreted as the eigenvalues of a corresponding correlation matrix RTx (RRx ). In fact, for any num M (U N) ber of transmit (receive) antennas, a unitary matrix U can be found such that the transform M ΣTx U H U M =: RTx ,

N ΣRx U H U N =: RRx

To this end, consider the block transmission model (2). We assume that a space-time encoder with memory length ν (e.g., a space-time trellis encoder [4]) is used at the transmitter side—possibly employing distributed antennas. The space-time encoder maps a sequence of (Nb − ν) information symbols (followed by ν known tailing symbols) onto an (M × Nb ) space-time code matrix X (Nb > M). Assuming that the channel matrix H is perfectly known at the receiver, the metric for maximum-likelihood sequence estimation (MLSE) reads 2

:= Y − HX F , μ(Y, X)

denotes a hypothesis for code matrix X. The (avwhere X erage) PEP P(X→E), that is, the probability that the MLSE decoder decides in favor of an erroneous code matrix E = / X, although matrix X was transmitted, is given by [24]

P(X −→ E) = Pr μ(Y, E) ≤ μ(Y, X)

(17)

yields a correlation matrix RTx (RRx ) with diagonal entries equal to one and nondiagonal entries with magnitudes ≤ 1. Suitable unitary matrices are, for example, the (n×n)-Fourier √ matrix with entries ui j = ej2π(i−1)( j −1)/n / n (which exists for any number n), or the normalized (n × n)-Hadamard ma/ IM (ΣRx = / IN ), at least some nonditrix. Note that for ΣTx = agonal entries of RTx (RRx ) in the equivalent MIMO system with colocated antennas will have magnitudes greater than zero, that is, some of the transmission links will be mutually correlated. If only a single diagonal element of ΣTx (ΣRx ) is unequal to zero, one obtains an equivalent MIMO system with fully correlated transmit (receive) antennas. Finally, note that within our simplified framework there is no difference between distributed and colocated antennas, as soon as ΣTx = RTx = IM (ΣRx = RRx = IN ). The above findings are summarized in the following theorem. Theorem 1. Based on the presented framework, for any MIMO system with colocated transmit and receive antennas, which is subject to frequency-flat Rayleigh fading obeying the Kronecker correlation model, an equivalent MIMO system with distributed transmit and/or distributed receive antennas (and unequal average link SNRs) can be found, and vice versa, such that both systems are characterized by identical capacity distributions. 3.2. Duality with regard to the pairwise error probability (PEP) of space-time codes The results in Section 3.1 were very general and are relevant for coded MIMO systems with colocated or distributed antennas. In the following, we focus on the important class of space-time coded MIMO systems. Specifically, we will show that based on the above framework space-time coded MIMO systems with correlated antennas and space-time coded MIMO systems with distributed antennas (and unequal average link SNRs) are characterized by (asymptotically) identical pairwise error probabilities (PEPs).

(18)

=E Q

P H(X − E)F 2Mσ 2n

,

(19)

where Q(x) denotes the Gaussian Q-function. 3.2.1. PEP for space-time coded MIMO systems with colocated antennas In the sequel, we assume that the employed space-time code achieves a diversity order of MN (full spatial diversity). In [24], it was shown that the PEP (19) can be expressed in the form of a single finite-range integral, according to P(X −→ E) =

1 π

π/2 M N

1+

0

i=1 j =1

P ξ Tx,i λRx, j 4Mσ 2n sin2 θ

−1

dθ, (20)

where ξ Tx,1 , . . . , ξ Tx,M denote the eigenvalues of the matrix (X − E)(X − E)H RTx =: ΨX,E RTx and λRx,1 , . . . , λRx,N the eigenvalues of RRx , as earlier. 3.2.2. PEP for space-time coded MIMO systems with distributed antennas Based on the same arguments as in Section 3.1, by evaluating (16) we can always find a MIMO system with distributed receive antennas and overall spatial covariance matrix

E vec(H)vec(H)H = RTx ⊗ ΣRx

(21)

which leads to exactly the same PEP (20) as the above MIMO system with colocated antennas. Vice versa, given a MIMO system with distributed receive antennas, we can find an equivalent MIMO system with colocated antennas by evaluating (17). As opposed to this, a MIMO system with distributed transmit antennas and overall spatial covariance matrix

E vec(H)vec(H)H = ΣTx ⊗ RRx

(22)

6


(ΣTx := UH Tx RTx UTx ) will not lead to the same PEP (20), because the eigenvalues of the matrices ΨX,E RTx and ΨX,E ΣTx are, in general, different. (Note that we obtain a PEP expression for space-time coded MIMO systems with distributed transmit antennas, by replacing the eigenvalues ξ Tx,i in (20) by the eigenvalues of the matrix ΨX,E ΣTx .) Asymptotically, that is, for large SNR values, the PEP (20) is well approximated by [25]

P(X −→ E) ≤

P 4Mσ 2n

−MN

det ΨX,E RTx

−N

det RRx

−M

,

(23) where we have assumed that RTx and RRx have full rank. Since also ΨX,E has full rank (due to the assumption that the employed space-time code achieves full spatial diversity), we obtain

det ΨX,E RTx = det ΨX,E det RTx = det ΨX,E det ΣTx

can be evaluated based on an equivalent maximum-ratiocombining (MRC) system [26] z[k] = h2 a[k] + η[k]

with one transmit antenna and MN receive antennas, where h := vec(H), cf. (1), and η[k]∼CN {0, σ 2n }. The transmitted data symbols a[k] are i.i.d. random variables with zero mean and variance σ 2a = P/(MRt ), where Rt ≤ 1 denotes the temporal rate of the OSTBC under consideration. Using Craig’s alternative representation of the Gaussian Q-function [27], one can find closed-form expressions for the resulting average SEP, which are in the form of finite-range integrals over elementary functions [28]. For example, in the case of a Qary phase-shift keying (PSK) signal constellation, the average SEP can be calculated as

Ps =

= det ΨX,E ΣTx ,

(24) that is, the expression (23) does not change if RTx is replaced by ΣTx . Therefore, asymptotically the PEP expressions for MIMO systems with distributed transmit antennas and MIMO systems with colocated transmit antennas again become the same. The above findings are summarized in the following theorem. Theorem 2. Based on the presented framework, for any MIMO system with colocated transmit and receive antennas, which is subject to frequency-flat Rayleigh fading obeying the Kronecker correlation model and which employs a space-time coding scheme designed to achieve full spatial diversity, an equivalent space-time coded MIMO system with distributed transmit and/or distributed receive antennas (and unequal average link SNRs) can be found, and vice versa, such that asymptotically both systems are characterized by identical average PEPs.

3.3.1. Average SEP for OSTBC systems with colocated antennas Consider again the system model (1). In the case of uncorrelated antennas, the average SEP resulting for an OSTBC system with M transmit and N receive antennas (employing the associated widely linear detection steps at the receiver side)

1 π

(Q−1)π/Q MN 0

ν=1

sin2 (φ) dφ. sin (φ) + sin2 (π/Q)γν 2

(26)

Here γν = γ := P/(MRt σ 2n ) denotes the average link SNR in the equivalent MRC system (25), where we have again employed the normalization according to Section 2.3 (i.e., σ 2 := 1). If the antennas in the OSTBC system are correlated, we have E{hhH } = R, cf. (4). In this case, the resulting average SEP can still be calculated based on (26), while replacing the average link SNRs γν by transformed link SNRs [29] γν :=

Pλν , MRt σ 2n

(27)

where λν (ν = 1, . . . , MN) denote the eigenvalues of R. Moreover, assuming again that the OSTBC system follows the Kronecker-correlation model, the eigenvalues λν are given by the pairwise products λTx,i λRx, j (i = 1, . . . , M, j = 1, . . . , N) of the eigenvalues of RTx and RRx [30, Chapter 12.2]. Altogether, we can thus rewrite (26) according to3

3.3. Duality with regard to the symbol error probability (SEP) of OSTBCs In the sequel, we further specialize the above results and focus on MIMO systems that employ an orthogonal space-time block code (OSTBC) [5, 6] at the transmitter side. Based on the presented framework, it will be seen that in this case identical average symbol error probabilities (SEPs) result in MIMO systems with colocated antennas and MIMO systems with distributed antennas (for any SNR value, not only asymptotically).

(25)

Ps =

1 π

(Q−1)π/Q N M 0

i=1 j =1

sin2 (φ)MRt σ 2n × dφ. sin2 (φ)MRt σ 2n + sin2 (π/Q)PλTx,i λRx, j

(28)

3.3.2. Average SEP for OSTBC systems with distributed antennas Obviously, the complete SEP analysis for OSTBC systems with colocated antennas depends solely on the eigenvalues of RTx and RRx . Correspondingly, it is clear that P s will stay exactly the same, if we replace RTx by ΣTx := UH Tx RTx UTx and/or R U . By this means, we have found RRx by ΣRx := UH Rx Rx Rx an equivalent OSTBC system with M distributed transmit 3

Similar expressions can also be derived for quadrature-amplitudemodulation (QAM) and amplitude-shift keying (ASK) constellations.

J. Mietzner and P. A. Hoeher antennas and/or N distributed receive antennas. Vice versa, given a distributed OSTBC system, we can again find an equivalent OSTBC system with colocated antennas by evaluating (17). The above findings are summarized in the following theorem. Theorem 3. Based on the presented framework, for any MIMO system with colocated transmit and receive antennas, which is subject to frequency-flat Rayleigh fading obeying the Kronecker correlation model and which employs an OSTBC in conjunction with the corresponding widely linear detection at the receiver, an equivalent OSTBC system with distributed transmit and/or distributed receive antennas (and unequal average link SNRs) can be found, and vice versa, such that both systems are characterized by identical average SEPs. 3.4. Discussion The previous sections have shown that MIMO systems with distributed antennas and unequal average link SNRs behave in a very similar way as MIMO systems with colocated antennas and spatially correlated links (with regard to various performance measures). In other words, both effects entail very similar performance degradations. For example, spatial fading correlations (unequal average link SNRs) can lead to significantly reduced ergodic or outage capacities [7]. With regard to space-time coding, the presence of receive antenna correlations (distributed receive antennas) always degrades the resulting PEP, particularly for high SNRs. As opposed to this, the impact of transmit antenna correlations (distributed transmit antennas) depends on the employed spacetime code and the SNR regime under consideration [24]. Concerning the average SEP of OSTBCs, correlated antennas (unequal average link SNRs) always entail a performance loss [31, Chapter 3.2.5]. Note that although the assumptions within the presented framework are rather restrictive, the major finding that MIMO systems with distributed antennas and MIMO systems with colocated antennas behave in a very similar fashion will also hold, when more general scenarios are considered. For example, if error-propagation effects or nonperfect synchronization between distributed antennas come into play, distributed antennas will still behave like spatially correlated antennas. In particular, the performance degradations caused by error-propagation or non-perfect synchronization effects will simply come on the top of those caused by unequal average link SNRs, since the effects are independent of each other. Possible generalizations of the above results to frequency-selective fading channels and more general fading scenarios (e.g., Rician and Nakagami-m fading) were discussed in [31, Chapter 3.3]. 4.

OPTIMAL TRANSMIT POWER ALLOCATION SCHEMES

An important implication of the above duality is that optimal transmit power allocation strategies developed for MIMO systems with colocated antennas (see, e.g., [32] for an

7 overview) may be reused for MIMO systems with distributed antennas, and vice versa. As an example, we will focus on the use of statistical channel knowledge at the transmitter side, in terms of the transmitter correlation matrix RTx , the receiver correlation matrix RRx , and the noise variance σ 2n . Statistical channel knowledge can easily be gained in practical systems, for example offline through field measurements, ray-tracing simulations or based on physical channel models, or online based on long-term averaging of the channel coefficients [33]. Optimal statistical transmit power allocation schemes for spatially correlated MIMO systems were, for example, derived in [33–35] with regard to different optimization criteria: minimum average SEP of OSTBCs [33], minimum PEP of space-time codes [34], and maximum ergodic capacity [35]. Based on the presented framework, these optimal power allocation strategies can directly be transferred to MIMO systems with distributed antennas. Here, we consider the optimal transmit power allocation scheme for maximizing ergodic capacity [35]. Consider again a MIMO system with colocated antennas and an overall spatial covariance matrix R = RTx ⊗ RRx . In order to maximize the ergodic capacity of the system, it was shown in [35] that the optimal strategy is to transmit in the directions of the eigenvectors of the transmitter correlation matrix RTx . To this end, the transmitted vector in (1) is premultiplied with the unitary matrix UTx from the eigenvalue decomposition of RTx . Moreover, a diagonal weighting matrix W1/2 := diag

√

√

w1 , . . . , wM , tr(W) := M,

(29)

is used in order to perform the transmit power weighting among the eigenvectors of RTx . Altogether, the transmitted vector can thus be expressed as x[k] := UTx W1/2 x [k],

(30)

where the entries of x [k] have variance σ 2x i = P/M for all i = 1, . . . , M. Under these premises, the instantaneous capacity (11) becomes

C(H, Qx ) = log 2 det IN +

1 HQx HH bit/channel use, σ 2n (31)

where Qx := E{x[k]xH [k]} = P/M ·UTx WUH Tx denotes the covariance matrix of x[k]. Unfortunately, a closed-form solution for the optimal weighting matrix Wopt maximizing the ergodic capacity C(Qx ) := E{C(H, Qx )} is not known. The optimal power weighting results from solving the optimization problem [35]

maximize g(W) := E log 2 det IN + subject to tr(W) := M , wi ≥ 0

M wi λTx,i zi zH

i

i=1

σ 2n

∀ i,

(32) where the vectors zi are i.i.d. complex Gaussian random vectors with zero mean and covariance matrix ΛRx . Note that the optimum power weighting depends both on the eigenvalues

8


C Qx ≤ log 2 1 +

N min m=1

×

P Mσ 2n

m

m!

25

20 Ergodic capacity

of RTx and on the eigenvalues of RRx . Based on the same arguments as in Section 3, the resulting transmit power weighting will also be optimal for a MIMO system with distributed antennas and an overall covariance matrix R = ΣTx ⊗ RRx or R = RTx ⊗ ΣRx with ΣTx , ΣRx given by (16). In the case of distributed transmit antennas, the prefiltering matrix UTx reduces to the identity matrix. The expression (32) is, in general, difficult to evaluate. In the following, we will therefore employ a tight upper bound on C(Qx ), which is based on Jensen’s inequality and which greatly simplifies the optimization of W (see [36], where the case of equal power allocation is studied). One obtains

0

wi1 λTx,i1 · · · wim λTx,im

−5

where Nmin := min{M, N }. Furthermore, Im and Jm denote index sets defined as

Im := i := i1 , . . . , im | 1 ≤ i1 < i2 < · · · < im ≤ M

5

10

15

20

10 log10 P/(Mσn2 ) (dB) Uncorrelated (4 × 3)-MIMO system Correlated (4 × 3)-MIMO system (equal power allocation) Correlated (4 × 3)-MIMO system (optimal power allocation)

(33)

0

i∈Im

λRx, j1 · · · λRx, jm ,

10

5

j∈Jm

15

Jm := j := j 1 , . . . , j m | 1 ≤ j 1 < j 2 < · · · < j m ≤ N (34) (m ∈ Z, 1 ≤ m ≤ Nmin ). For a fixed SNR value P/(Mσ 2n ), the right-hand side of (33) can now be maximized numerically in order to find the optimum power weighting matrix Wopt . As an example, we consider a MIMO system with four colocated transmit antennas and three colocated receive antennas. (Equivalently, we could again consider a corresponding MIMO system with distributed transmit and/or distributed receive antennas.) For the correlation matrices RTx and RRx , the single-parameter correlation matrix proposed in [37] for uniform linear antenna arrays has been taken, with correlation parameters ρTx := 0.8 and ρRx := 0.7. Figure 2 displays the ergodic capacity as a function of the SNR P/(Mσ 2n ) in dB which results in different transmit power allocation strategies. Simulative results are represented by solid lines and the corresponding analytical upper bounds are represented by dashed lines. As can be seen, compared to the case of uncorrelated antennas (dark curve, marked with “x”) the ergodic capacity in the case of correlated antennas and equal power allocation (dark curve, no markers) is reduced significantly, especially for large SNR values. For the light-colored curve, the transmit power weights w1 , . . . , wM were optimized numerically, based on (33). As can be seen, compared to equal power allocation the ergodic capacity is notably improved. For SNR values smaller than −2 dB, the achieved ergodic capacity is even larger than in the uncorrelated case. Further numerical results not displayed in Figure 2 indicate that the knowledge of RRx at the transmitter side is of rather little benefit. When assuming RRx := IN at the transmitter side, the resulting power allocation is still very close to the optimum.

Figure 2: Ergodic capacity as a function of the SNR P/(Mσ 2n ) in dB, for different MIMO systems with M = 4 transmit and N = 3 receive antennas. Solid lines: simulative results obtained by means of Monte Carlo simulations over 105 independent channel realizations. Dashed lines: corresponding analytical upper bounds based on (33).

5.

CONCLUSIONS

In this paper, it was shown that MIMO systems with distributed antennas (and unequal average link SNRs) behave in a very similar way as MIMO systems with correlated antennas. In particular, a simple common framework for MIMO systems with colocated antennas and MIMO systems with distributed antennas was presented. Based on the common framework, it was shown that for any MIMO system with colocated antennas, which follows the Kronecker correlation model, an equivalent MIMO system with distributed antennas can be found, and vice versa, while various performance criteria were taken into account. An important implication of this finding is that optimal transmit power allocation strategies developed for MIMO systems with colocated antennas can be reused for MIMO systems with distributed antennas, and vice versa. As an example, an optimal transmit power allocation scheme based on statistical channel knowledge at the transmitter side was considered. REFERENCES [1] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multielement antennas,” Bell Labs Technical Journal, vol. 1, no. 2, pp. 41–59, 1996. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, 1998. [3] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585–595, 1999.

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9 [21] R. Steele, Ed., Mobile Radio Communications, IEEE Press, New York, NY, USA, 1994. [22] S. Park, H. Shin, and J. H. Lee, “Capacity statistics and scheduling gain for MIMO systems in correlated Rayleigh fading,” in Proceedings of the 60th IEEE Vehicular Technology Conference (VTC ’04), vol. 2, pp. 1508–1512, Los Angeles, Calif, USA, September 2004. [23] J. G. Proakis, Digital Communications, McGraw-Hill, New York, NY, USA, 4th edition, 2001. [24] J. Wang, M. K. Simon, M. P. Fitz, and K. Yao, “On the performance of space-time codes over spatially correlated Rayleigh fading channels,” IEEE Transactions on Communications, vol. 52, no. 6, pp. 877–881, 2004. [25] H. Bolcskei and A. J. Paulraj, “Performance of space-time codes in the presence of spatial fading correlation,” in Proceedings of the 34th Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 687–693, Pacific Grove, Calif, USA, October-November 2000. [26] H. Shin and J. H. Lee, “Exact symbol error probability of orthogonal space-time block codes,” in Proceedings of IEEE Global Telecommunications Conference (Globecom ’02), vol. 2, pp. 1197–1201, Taipei, China, November 2002. [27] J. W. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proceedings of IEEE Military Communications Conference (MILCOM ’91), vol. 2, pp. 571–575, McLean, Va, USA, November 1991. [28] M.-S. Alouini and A. J. Goldsmith, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” IEEE Transactions on Communications, vol. 47, no. 9, pp. 1324–1334, 1999. [29] X. Dong and N. C. Beaulieu, “Optimal maximal ratio combining with correlated diversity branches,” IEEE Communications Letters, vol. 6, no. 1, pp. 22–24, 2002. [30] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, New York, NY, USA, 2nd edition, 1985. [31] J. Mietzner, “Spatial diversity in MIMO communication systems with distributed or co-located antennas,” Ph.D. dissertation, Shaker, Aachen, Germany, 2007. [32] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 5, pp. 684–702, 2003. [33] S. Zhou and G. B. Giannakis, “Optimal transmitter eigenbeamforming and space-time block coding based on channel correlations,” IEEE Transactions on Information Theory, vol. 49, no. 7, pp. 1673–1690, 2003. [34] H. Sampath and A. Paulraj, “Linear precoding for space-time coded systems with known fading correlations,” IEEE Communications Letters, vol. 6, no. 6, pp. 239–241, 2002. [35] E. A. Jorswieck and H. Boche, “Channel capacity and capacityrange of beamforming in MIMO wireless systems under correlated fading with covariance feedback,” IEEE Transactions on Wireless Communications, vol. 3, no. 5, pp. 1543–1553, 2004. [36] H. Shin and J. H. Lee, “Capacity of multiple-antenna fading channels: spatial fading correlation, double scattering, and keyhole,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2636–2647, 2003. [37] A. van Zelst and J. S. Hammerschmidt, “A single coefficient spatial correlation model for multiple-input multipleoutput (MIMO) radio channels,” in Proceedings of the 27th General Assembly of the International Union of Radio Science (URSI ’02), pp. 1–4, Maastricht, The Netherlands, August 2002.


Research Article Power-Efficient Relay Selection in Cooperative Networks Using Decentralized Distributed Space-Time Block Coding Lu Zhang and Leonard J. Cimini Jr. Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716, USA Correspondence should be addressed to Lu Zhang, [email protected] Received 1 May 2007; Accepted 8 September 2007 Recommended by R. K. Mallik Distributed space-time block coding (Dis-STBC) achieves diversity through cooperative transmission among geographically dispersed nodes. In this paper, we present a power-efficient relay-selection strategy for decentralized Dis-STBC in a selective decodeand-forward cooperative network. In particular, for a two-stage network, each decoded node broadcasts a small amount of information with limited power. This node then utilizes its own and its neighbors’ information to decide whether or not to act as a relay for the source information. In this way, only part of the decoded set will act as relays. Further, by applying the idea of this relay-selection strategy to each relaying hop in a multihop network, a power-efficient hop-by-hop routing strategy is formulated. The outage analyses and simulations are presented to illustrate the advantage of these strategies. Copyright © 2008 L. Zhang and L. J. Cimini Jr.. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

Cooperative diversity is a set of techniques that exploits the spatial diversity available among a collection of distributed single-antenna terminals (e.g., see [1]). In most proposed cooperative systems, a two-stage relaying strategy is used. In the first stage, a source transmits and all the other nodes listen; in the second stage, the relays cooperate to retransmit the source message to the destination. Several relay management strategies can be employed. In selective decode-andforward relaying, a node is called a decoded node if it can correctly decode the source message; then some subset of the decoded nodes is selected to act as the relay set. In [2], a distributed space-time block code (Dis-STBC) was proposed in which each relay transmits one unique column of the underlying STBC matrix. So that each relay knows which column to transmit, most of the proposed Dis-STBC schemes [2–8] require a central control unit or full internode negotiations. Several decentralized Dis-STBC schemes have been proposed to implement code assignment at the relays without control signaling (e.g., see [9, 10]). In decentralized DisSTBC, one possible relay-selection strategy is for all the nodes in the decoded set to act as relays for the source information; we call this the all-select approach. For decentralized

Dis-STBC, it has been observed [11] that, when the number of relays is much greater than the number of columns in the underlying STBC matrix, any further increase in the number of relays, although consuming more power, will not result in additional diversity benefit. Obviously, then, such a strategy, where all the nodes in the decoded set retransmit the source message, might be wasteful of power, especially when the number of decoded nodes is large. In this paper, we propose a more power-efficient relayselection strategy for decentralized Dis-STBC. In the proposed relay-selection strategy, each decoded node broadcasts a small amount of information with limited power, then utilizes its own and its local neighbors’ information to decide whether to act as a relay or not. Based on this information, only a subset of the nodes in the decoded set will act as relays. In order to incur the least overhead while also being robust when a deep fade occurs over some internode channels, we do not require that each decoded node can correctly receive the information from all of the other decoded nodes. In particular, as an example, we focus on m-group Dis-STBC [9], in which each relay randomly and independently chooses one column from the underlying STBC matrix. The paper is organized as follows. The system model for a two-stage network is described in Section 2. In Section 3,

2


using m-group Dis-STBC as an example, we propose a power-efficient relay-selection strategy for decentralized DisSTBC. In Section 4, an asymptotic upper bound on the outage for the m-group Dis-STBC is derived, and the powerefficiency advantage of the proposed relay-selection strategy is illustrated. Simulation results in a two-stage network are given in Section 5. In Section 6, by extending the idea of the proposed relay-selection strategy and also using mgroup Dis-STBC as an example, a power-efficient hop-byhop routing strategy is formulated for a multihop network that uses decentralized Dis-STBC at each relaying hop. Finally, Section 7 provides concluding remarks. 2.

Destination

Source

Decoded set

Figure 1: A two-stage selective decode-and-forward cooperative network.

SYSTEM MODEL

We assume a two-stage protocol that uses a selective decodeand-forward relaying strategy, as illustrated in Figure 1. In particular, we consider a network with M single-antenna nodes. When one source-destination pair, (s, d), is active, all the remaining M − 2 nodes can serve as potential relays. Define the decoded set as the set of N (N ≤ M − 2) nodes that can correctly decode the transmitted signal from the source. Note that the decoded set is random, varying with the instantaneous channel gains. K (K ≤ N) decoded nodes are then selected to relay the source message. We assume that nodes cannot transmit and receive simultaneously. In addition, we assume perfect synchronization and a quasi-static propagation environment. When using the all-select relay-selection strategy, K = N; however, this decentralized relay-selection strategy might be inefficient. Each relay must also determine which column of the code matrix to transmit. Generally, the code assignment among the relays requires a central control unit or full internode negotiations. Several decentralized Dis-STBC schemes have been proposed to implement code assignment without control signaling (e.g., see [9, 10]). In this paper, we will focus on m-group Dis-STBC [9] which is described next. In m-group Dis-STBC, each relay independently chooses one column at random out of the L columns in the underlying STBC matrix. Specifically, denote S as the underlying Lcolumn STBC matrix, where the row of S indicates the time index and the column indicates the transmit antenna index. When S has m columns (i.e., L = m), it is equivalent to dividing the relays into m groups, where the relays within a certain group choose the same column. However, since some groups might be empty, this scheme does not ensure the maximum possible diversity order, L. In particular, the number of distinct columns randomly selected by the K (K ≤ N) relays is denoted as V (1 ≤ V ≤ L). Then, denote Bv (v = 1, . . . , V ) as the vth subset of the set of K relays, and Kv as the number of relays in Bv . The relays within Bv will transmit the vth column out of the V randomly selected distinct columns. This scheme is a special case of the randomized DisSTBC proposed in [10]. Let R = [r1 , r2 , . . . , rK ] represent the randomized matrix with L rows and K columns, where r j = [r1, j , . . . , rL, j ]T is the randomized coefficient vector independently generated by relay j ( j = 1, . . . , K). In the mgroup scheme, the random coefficients ri, j are drawn from the discrete-element set {0, 1}. In this case, r j belongs to

{di , i = 1, . . . , L}, where di is the vector of all zeros except for the ith position which is 1. Let the instantaneous channel coefficients αi, j capture the effects of path loss and flat Rayleigh fading between node i and node j. Denot α = [α1,d , . . . , αK,d ]T as the channel coefficient vector for transmissions from the K relays to the destination, and Z as additive white Gaussian noise with the variance N0 1 per complex dimension. Further, let βi,d = ri,1 α1,d + · · · + ri,K αK,d (i = 1, . . . , L) represent the equivalent instantaneous channel coefficients and β = [β1,d , . . . , βL,d ]T . Then, the received signal Y at the destination is

Y = SRα + Z = Sβ + Z.

(1)

Based on (1), by estimating the equivalent channel coefficients β, the conventional coherent detection algorithm of STBC can still be used for randomized Dis-STBC. 3.

POWER-EFFICIENT RELAY-SELECTION STRATEGY

For decentralized Dis-STBC, the all-select relay-selection strategy might result in a substantial waste of power. If only a subset of the decoded nodes is selected to act as relays, good performance might be achieved with much less power consumption in the second stage. In the m-group scheme, the number of randomly selected distinct columns V (1 ≤ V ≤ L) determines the achieved diversity benefit. If the random column selection is performed after the relay selection, only selecting part of the N decoded nodes as relays might result in a decrease in V (i.e., a decrease in the diversity gain). Thus, the challenge is to effectively select a subset of the decoded nodes as relays to try to maintain the same diversity gain as the all-select strategy, while introducing sufficiently small overhead. In what follows, by using power-limited one-way control signals, a power-efficient relay-selection strategy is presented for the m-group Dis-STBC. We call this strategy local-k-best relay selection, and it works as follows. (i) Each decoded node randomly chooses a column so that V (1 ≤ V ≤ L) becomes the number of distinct columns randomly selected by the N decoded nodes. 1

Without loss of generality, N0 is normalized to 1.

L. Zhang and L. J. Cimini Jr. (ii) Each decoded node broadcasts the local mean power gain of the channel from itself to the destination and the index of its randomly selected column, by using a low transmit powe Pbc . The broadcast transmissions by all the decoded nodes use a multichannel CSMA MAC protocol [12], which provides “soft” channel reservation by combining CSMA with CDMA, for example. By using this MAC protocol, the control signaling is one-way traffic (no response is required for broadcasting). The effect of the hidden-node problem could also be reduced well. (iii) At each decoded node, if it finds that there exist at least k (k ≥ 1) neighbors which have the same selected column as itself and which have larger local mean power gain than itself, then this decoded node will not act as a relay. In this strategy, for any particular random selection of columns by the N decoded nodes (i.e., for any given value of V ), each of the V randomly selected distinct columns will be transmitted by at least one relay such that V ≤ K ≤ N. Thus, when only a subset of all the decoded nodes is selected as relays, the achieved diversity gain is the same as using the all-select strategy. In addition, since the K selected relays have relatively larger local mean power gains, better performance will be achieved. The power overhead of the local-k-best strategy is NPbc . In order to incur the least possible overhead and also to be robust when a deep fade occurs over some internode channels, we do not require each decoded node to correctly receive the information from all of the other decoded nodes. With a low transmit power for the broadcast signal, the neighbors of each decoded node will only be a subset of the decoded set. Since the amount of the local information required to be broadcasted by each decoded node is quite small, by further using a multichannel CSMA MAC protocol [12] for broadcastings by all the decoded nodes, the resulting time overhead should be negligible when compared with the time used for the transmission of data packets. 4.

PERFORMANCE ANALYSIS

In this section, we illustrate the power-efficiency advantage of the local-k-best strategy by deriving an asymptotic upper bound on the outage probability for the m-group Dis-STBC. 4.1. Asymptotic upper bound on the outage for m-group Assume that the all-select or local-k-best strategy is used such that K (V ≤ K ≤ N) decoded nodes are selected as relays. A two-stage transmission is in outage if the receive SNR at the destination is below a given SNR threshold ηt . The outage probability at the destination is denoted as pout,d . Denote Ps as the transmit power of the source node and Pr as the transmit power of each relay. (When coding is used and the code rate is not equal to one, Ps and Pr represent the power per information symbol.) Denote the mean values of the channel

3 power gains |αs,d |2 and |α j,d |2 as μs,d and μ j,d ( j = 1, . . . , K), respectively. Further, denote μmin,v as the minimum value among μ j,d , j ∈ Bv (v = 1, . . . , V ). Next, an asymptotic upper bound on pout,d is obtained. Theorem 1. For any given decoded set and particular random column selection by all of the decoded nodes, an asymptotic upper bound on pout,d for the m-group Dis-STBC is given by pout,d ≤

Ps μs,d × PrV

ηVt +1 /(V + 1)! . × (μmin ,1 × · · · × μmin ,V )

(2)

This asymptotic upper bound is tight when Ps μs,d and Pr μmin ,v are sufficiently large for all v ∈ {1, . . . , V }. Proof. For any particular random column selection by the N decoded nodes, the nonzero equivalent channel coefficients βv,d (v = 1, . . . , V ) can be expressed as βv,d =

α j,d

(v = 1, . . . , V ).

(3)

j ∈Bv

Based on (3), it can be seen that the β1,d , . . . , βV ,d are independent complex Gaussian variables since there is no intersection among Bv (v = 1, . . . , V ). Thus, the power gains of the nonzero equivalent channels |βv,d |2 (v = 1, . . . , V ) are independent exponential random variables with means j ∈Bv μ j,d . By applying the conventional coherent detection algorithm for STBC [13] and combining the received signals from the two stages, the outage probability at the destination pout,d is

pout,d = Pr

2

Ps αs,d +

V v=1

2

Pr βv,d

≤ ηt .

(4)

It can be seen that, for a particular random column selection by any given N decoded nodes, the m-group scheme is equivalent to formulating V (1 ≤ V ≤ L) “virtual relays,” each of which transmits one distinct column out of the V selected columns. Here, the equivalent channel coefficients between the virtual relays v and the destination are βv,d (v = 1, . . . , V ), which are independent complex Gaussian variables. Thus, it can be viewed as applying the centralized Dis-STBC [2] with a V -column code matrix to the V “virtual relays.” By exploiting the results in [14] for the outage analysis of centralized Dis-STBC, for any given decoded set and particular random column selection by all the decoded nodes, we can express the upper bound on pout,d for the m-group Dis-STBC as pout,d ≤

Ps μs,d × Pr

ηVt +1 /(V + 1)! . j ∈B1 μ j,d × · · · × Pr j ∈BV μ j,d (5)

This upper bound is tight when Ps μs,d and Pr j ∈Bv μ j,d are sufficiently large for all v. Clearly, μmin,v ≤ j ∈Bv μ j,d (v = 1, . . . , V ). Thus, based on (5), we obtain the asymptotic upper bound as given in (2).

4


4.2. Advantage of local-k-best With the given total power consumption in a two-stage transmission, the asymptotic upper bound on pout,d can be optimized by using the local-k-best strategy when the values of Pbc and k (k ≥ 1) are properly chosen such that K < N. This is shown in the following theorem. Theorem 2. With a given source power Ps and a given power consumption in the second stage P2 , for the m-group Dis-STBC, the asymptotic upper bound on pout,d when using the local-kbest strategy is smaller than or equal to that when using the all-select strategy. Proof. With a given power consumption P2 in the second stage, we have Pr = P2 /N for the all-select strategy and Pr = P2 /K (V ≤ K ≤ N) for the local-k-best strategy. For any given decoded set and particular random column selection by the N decoded nodes, denote Dv (v = 1, . . . , V ) as the vth subset of the decoded set. The randomly selected column by the decoded nodes in Dv is the vth column out of the V randomly selected distinct columns. Obviously, Bv ⊆ Dv . Further, denote εmin,v as the minimum value among μ j,d , j ∈ Dv (v = 1, . . . , V ). Clearly, when the all-select strategy is used, K = N and Bv = Dv so that μmin,v = εmin,v . Since (2) is obtained for general K (V ≤ K ≤ N) and Bv (Bv ⊆ Dv ), according to (2), we get all-select: pout,d ≤

ηVt +1 /(V + 1)! , Ps μs,d P2V (1/N)V (εmin,1 × · · · × εmin,V )

(6)

local-k-best: pout,d ≤

ηVt +1 /(V + 1)! . Ps μs,d P2V (1/K)V (μmin,1 × · · · × μmin,V )

(7)

When the local-k-best strategy is used but some inappropriate values are set up for Pbc and k (k ≥ 1) such that K = N, we have Bv = Dv so that μmin,v = εmin,v for all v. In this case, the local-k-best strategy is equivalent to the all-select strategy; in particular, this might result when Pbc is very small or when k is large. The values of Pbc and k (k ≥ 1) could be properly chosen such that K < N (the optimal values of Pbc and k will be investigated by simulations). In this case, Bv ⊂ Dv for at least one v ∈ {1, . . . , V }. As we know, the local-k-best strategy is designed to select Kv decoded nodes from Dv to act as relays (v = 1, . . . , V ), and it also tries to choose the Kv relays that have larger local mean power gains when compared with the other decoded nodes in Dv . For any v with Bv ⊂ Dv , in the worst case, the Kv selected relays include the “poorest” decoded node in Dv (i.e., the node having the smallest local mean power gain among all the decoded nodes in Dv ) so that μmin,v = εmin,v . This situation might happen when the “poorest” decoded node in Dv has no neighbors or all of its neighboring decoded nodes choose different columns from itself. In the other situations, clearly, μmin,v > εmin,v . Thus, when K < N, we have μmin,v ≥ εmin,v (v = 1, . . . , V ). According to (6) and (7), when K < N, the asymptotic upper

bound on pout,d with the local-k-best strategy is smaller than that with the all-select strategy. 4.3.

Key parameters in the local-k-best strategy

Based on the discussion in the previous subsection, Pbc and k are the two key parameters in the local-k-best strategy. Pbc is the power used by each decoded node to broadcast its local information. If one decoded node finds that there exist at least k (k ≥ 1) neighbors which are better relay candidates than itself, it will not act as a relay. The value of Pbc will affect the number of neighbors for each decoded node and, subsequently, affect the number of relays K (V ≤ K ≤ N). If Pbc is large, the power overhead might be too large. On the other hand, if Pbc is too small, the number of neighbors of each decoded node might be zero so that K = N. In the next section, we will use simulations to investigate the effect on performance for different values of Pbc to obtain an appropriate range of values. With an increase in k (k ≥ 1), at each decoded node the possibility that there exist at least k neighbors which are better relay candidates decreases; then, the number of relays K increases. This will result in an increase in power consumption in the second stage. However, for the m-group scheme with the local-k-best strategy, whatever the value of k is, all V (1 ≤ V ≤ L) distinct columns which are randomly selected by all N decoded nodes will be transmitted by K (V ≤ K ≤ N) relays. That is to say, an increase in k will not provide more diversity benefit. Intuitively, when k is smaller, the power efficiency is better. In the next section, we will use simulations to show the effect on performance when varying k. 5.

SIMULATION RESULTS

In this section, under realistic propagation conditions, including the effects of path loss and flat Rayleigh fading, the outage performance of the m-group Dis-STBC is evaluated with different relay-selection strategies, including the localk-best and all-select strategies. 5.1.

Simulation environment

We consider a square coverage area with diagonal dimension dmax and M uniformly distributed single-antenna halfduplex nodes. To implement power allocation in a decentralized way, it is assumed that constant transmit power Pt is used for each node, that is, Ps = Pr = Pt .2 Thus, for the all-select strategy, the total power to transmit one message is P = Ps + NPr = (1 + N)Pt ; for the local-k-best (k ≥ 1) strategy, the power overhead resulting from broadcasting local information is included in the performance evaluation such that the total power to transmit one message is P = Ps + KPr + NPbc = (1 + K)Pt + NPbc . Here, the time overhead 2

Two ad hoc, yet more efficient, power allocation strategies are suggested in [15] for decentralized Dis-STBC.

L. Zhang and L. J. Cimini Jr. 100

Outage probability

resulting from broadcasting local information is not considered since it could be negligible when compared with the time used for transmitting data packets. The outage probability of the farthest (s, d) pair is evaluated. To determine the SNR threshold ηt , we follow a similar argument as in [16]; that is, ηt is determined as b × (22r − 1) for two-stage cooperative transmission. The parameter r (bps/Hz) is the achieved spectral efficiency of the noncooperative direct transmission. The parameter b ranges from 1 to about 6.4, depending on the degree of used coding [17]. To evaluate the performance in a more realistic environment, the wireless channels include the effects of path loss and flat Rayleigh fading. In addition, the geographic distributions of the potential relays are random. The outage probability is obtained by averaging over node locations and Rayleigh fadings. As in [16], the powers are normalized by Pmax which is the transmit power required, for the maximal possible separation of source and destination dmax , to achieve a given spectral efficiency r in direct transmission without shadow fading and Rayleigh fading. The outage curves are plotted as a function of the normalized average power Pav , which is the average consumed power per two-stage transmission.

5

10−2

10−3

0

0.1

0.2

0.3

0.4

0.5

Pbc /Pmax Pav = 3 dB Pav = 6 dB

Figure 2: Outage probability as a function of Pbc /Pmax for the twogroup scheme with the local-one-best strategy with M = 16, L = 2 (r = 2 bps/Hz).

100

Outage probability

5.2. Outage probability Here, we use the parameter ξ = Pbc /Pmax to investigate the effect on outage performance when varying Pbc . This is shown in Figure 2 for the two-group scheme with local-one-best strategy when there are M = 16 nodes in the network and L = 2 columns in the STBC matrix. In particular, we use an Alamouti code [18]. It can be observed that a ratio ξ in the interval [0.09, 0.11] achieves the optimal performance. Similarly, when M = 16 and L = 2, the optimal ξ, ξ opt , is around 0.1 for the two-group scheme with local-two-best strategy. In addition, when M = 32 and L = 2, the ξ opt is around 0.05 for the two-group scheme with local-k-best (k = 1, 2) strategy. As an empirical result, ξ opt is approximately equal to 1/(M − 2). Recall that M − 2 is the number of all potential relays in the network; thus, M − 2 is also the maximum possible number of the decoded nodes. With the empirically optimal value for Pbc /Pmax , the outage performance of the m-group scheme with local-k-best strategy is investigated when varying k. Simulation results are shown in Figure 3 for the two-group scheme with the localk-best (k = 1, 2) strategy and the all-select strategy, when M = 16 nodes and L = 2 using an Alamouti code. Clearly, it can be seen that, even with the overhead included, the localone-best strategy is much more power-efficient than the allselect strategy. In particular, a 2 dB advantage can be observed at an outage probability of 10−2 . When Pav is large, the local-two-best strategy is also more power-efficient than the all-select strategy. Obviously, the advantage of the localk-best strategy decreases with an increase in k. This is because an increase in k will not provide additional diversity benefit but will result in an increased power consumption in the second stage. Results are shown in Figure 4 when M = 32. Clearly, it can be seen that, as the number of nodes M increases,

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Pav (dB) All-select Local-one-best, Pbc /Pmax = 0.1 Local-two-best, Pbc /Pmax = 0.1

Figure 3: Outage probability as a function of the total transmission power of the two stages, Pav , for the two-group scheme with M = 16, L = 2 (r = 2 bps/Hz).

the performance gap between the all-select strategy and the local-k-best (k = 1, 2) strategy becomes larger. In this case, the local-one-best strategy is almost 3 dB better than the allselect strategy at an outage probability of 10−2 . With a given transmit power for the source, on average, the number of decoded nodes will increase with an increase in the number of total nodes, M. Thus, when M increases, the all-select strategy will waste more power in the second stage to achieve the required performance at the destination.

6

EURASIP Journal on Advances in Signal Processing outage probability at hop n (n = 0, . . . , J − 1) and pout,d as the outage probability at the destination, then we have

Outage probability

100

pout,d = 1 −

10−1

n=0

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2

4

6

8

10

Pav (dB) All-select Local-one-best, Pbc /Pmax = 0.05 Local-two-best, Pbc /Pmax = 0.05

Figure 4: Outage probability as a function of the total transmission power of the two stages, Pav , for the two-group scheme with M = 32, L = 2 (r = 2 bps/Hz).

6.

J −1

EXTENSION TO MULTIHOP NETWORK

There has been growing interest in applying Dis-STBC to a multihop wireless network to achieve cooperative diversity by using a virtual antenna array at each relaying hop [19–23]. In these works, it has been shown that this type of ST-coded cooperative routing has much better performance than traditional node-by-node single-relay routing. However, just as for a two-stage network, in most of these works, for a multihop network, the implementation of Dis-STBC at each relaying hop requires a central control unit or full internode negotiations so that every selected relay knows which column of the underlying STBC matrix to transmit. Obviously, this could require significant overhead. In this section, we will investigate applying decentralized Dis-STBC to a multihop network. In particular, by extending the idea of the power-efficient relay-selection strategy in a two-stage network and also using m-group Dis-STBC as an example, a power-efficient routing strategy will be proposed for a multihop network that uses decentralized Dis-STBC at each relaying hop. In the multihop case, since each relay might have multiple local mean power gains to the multiple receiving nodes, some modification must be done when utilizing the local mean power gain information at relays. In a decode-and-forward multihop network, since a successful end-to-end transmission requires the source message to be correctly decoded by some node(s) at each hop, the destination will be in outage if any one certain hop is in outage. Thus, the end-to-end outage performance is determined by the outage performance of each single hop. In particular, we consider a J-hop (J > 2) network. If we denote pout,n as the

1 − pout,n ≈

J −1

pout,n .

(8)

n=0

In the decentralized scenario, it is difficult to obtain global channel information. Thus, it is desirable to design a hopby-hop routing strategy which optimize pout,d by optimizing pout,n for every n ∈ {0, . . . , J − 1}. When designing a hop-by-hop routing strategy for a multihop network that uses selective decode-and-forward relaying, the relay selection at each relaying hop is the key to the design. Since we could optimize the performance independently for each single hop, the power-efficient relay-selection strategy in a two-stage network can be naturally applied to each relaying hop with appropriate modification. Then, the power efficiency of the routing can be improved. Note that, in this paper, a routing strategy just means a path-selection strategy; it is not a real routing protocol. 6.1.

Multihop network system model

We consider a J-hop (J > 2) network that uses a selective decode-and-forward relaying strategy, as illustrated in Figure 5. The J − 1 node sets S1 , . . . , SJ −1 are located from the source to the destination. The source is denoted as S0 and the destination is denoted as SJ . The Wn nodes in Sn (n = 1, . . . , J − 1) are potential forwarding relays at relaying hop n. Here, as an example, it is assumed that the node sets Sn (n = 1, . . . , J − 1) are formulated through a destinationinitiated power-limited flooding. As in a two-stage network, we assume that the instantaneous channel between any two single-antenna half-duplex nodes captures the effects of path loss and flat Rayleigh fading. In addition, we assume perfect synchronization and a quasi-static environment. Finally, we assume that the receiving nodes at each hop can only utilize the transmission in the current hop to make a decision. At relaying hop n (n = 1, . . . , J − 1), the transmitting node set is Sn ; the decoded set within Sn is defined as the set of Nn (Nn ≤ Wn ) decoded nodes that can correctly decode the transmission from hop n − 1. Note that the decoded sets are random, varying with the instantaneous channel gains. At relaying hop n (n = 1, . . . , J − 1), Kn (Kn ≤ Nn ) decoded nodes are selected to relay the source message. In particular, when m-group Dis-STBC is used at each relaying hop, the number of distinct columns randomly selected by the Nn decoded nodes in Sn is denoted as Vn (1 ≤ Vn ≤ L). Then, denote Bn,v (v = 1, . . . , Vn ) as the vth subset of the set of Kn (Kn ≤ Nn ) selected relays, and Kn,v as the number of relays in Bn,v . The relays within Bn,v will transmit the vth column out of the Vn randomly selected distinct columns. 6.2.

Power-efficient hop-by-hop routing strategy

When m-group Dis-STBC is used, the all-select relayselection strategy can be used at each relaying hop. In this case, at relaying hop n (n = 1, . . . , J − 1), all Nn decoded nodes in the transmitting node set Sn forward the source

L. Zhang and L. J. Cimini Jr.

7

Hop 0

Hop n

Hop 1

Hop J − 1

Hop n + 1

Destination

Source ···

···

Decoded set

Decoded set

Decoded set

Decoded set

S1

Sn

Sn+1

SJ −1

Figure 5: A J-hop selective decode-and-forward cooperative network.

message. We call this all-select routing. This routing strategy might result in a substantial waste of power, similar to the all-select relay-selection strategy in a two-stage network. If the local-k-best (k ≥ 1) relay-selection strategy is used at each relaying hop, a power-efficient hop-by-hop routing strategy might be formulated. However, in the multihop case, each relay in Sn (n = 1, . . . , J − 1) might have multiple local mean power gains to the Wn+1 (Wn+1 ≥ 1) receiving nodes in Sn+1 . Thus, we cannot directly use the local-k-best (k ≥ 1) relay selection. Intuitively, a good measurement of the channel power gain at each relay in Sn is an average over its local mean power gains to the Wn+1 receiving nodes in Sn+1 . Here, we choose the geometric average and denote this as the locally averaged mean power gain to the next node set. Then, the local-k-best relay-selection strategy for a two-stage network can be simply modified by letting each decoded node in Sn (n = 1, . . . , J − 1) broadcast its locally averaged mean power gain to Sn+1 , instead of broadcasting its local mean power gain to the destination. By applying the modified local-k-best (k ≥ 1) relay-selection strategy to each relaying hop, a power-efficient hop-by-hop local-k-best routing is formulated. When using the local-k-best routing strategy, the achieved diversity gain at each relaying hop is the same as using the all-select routing strategy; however, less power is used to relay the source message. In addition, at relaying hop n (n = 1, . . . , J − 1), since the Kn (Vn ≤ Kn ≤ Nn ) selected relays have relatively larger locally averaged mean power gains to the receiving node set Sn+1 , better performance will be achieved. 6.3. Performance analysis Based on (8), the end-to-end outage performance pout,d is determined by the outage probability at each hop. In this section, we illustrate the power-efficiency advantage of the local-k-best routing strategy by deriving an asymptotic upper bound on the outage probability at relaying hop n (n = 1, . . . , J − 1). “Relaying hop n is in outage” means that all nodes within Sn+1 cannot correctly decode the source message forwarded by the Kn selected relays within Sn . Denote Pt as the transmit power of each node. At relaying hop n (n = 1, . . . , J − 1), denote μi, j as the mean value of the channel power gain from

the selected relay i in Sn to node j in Sn+1 (i = 1, . . . , Kn , j = 1, . . . , W n+1 ). Further, denote gmin,v as the minimum value among j ∈Sn+1 μi, j , i ∈ Bn,v (v = 1, . . . , Vn ). Next, an asymptotic upper bound on pout,n (n = 1, . . . , J − 1) is obtained. Theorem 3. When m-group Dis-STBC is used at relaying hop n (n = 1, . . . , J − 1), for any given decoded set in Sn and particular random column selection by the Nn decoded nodes, an asymptotic upper bound on pout,n is given by pout,n ≤

ηVt n Wn+1 /(Vn !)Wn+1 . PtVn Wn+1 × gmin ,1 × · · · × gmin ,Vn

(9)

This asymptotic upper bound is tight when PtWn+1 gmin ,v is sufficiently large for all v ∈ {1, . . . , Vn }. The proof of Theorem 3 can be done through the quite similar way used in the proof of Theorem 1; thus, it is omitted for the sake of brevity. With a given power consumption at relaying hop n, the asymptotic upper bound on pout,n can be optimized by using the local-k-best routing strategy when the values of Pbc and k (k ≥ 1) are properly chosen such that Kn < Nn . This is shown in the following theorem. Theorem 4. With a given power consumption Pn for the transmission at relaying hop n (n = 1, . . . , J − 1), when m-group Dis-STBC is used at relaying hop n, the asymptotic upper bound on pout,n when using the local-k-best routing strategy is smaller than or equal to that when using the all-select routing strategy. The proof of Theorem 4 can be done through the quite similar way used in the proof of Theorem 2; thus, it is omitted for the sake of brevity. Since pout,n for each n ∈ {1, . . . , J − 1} can be improved by using the local-k-best routing strategy, based on (8), the end-to-end outage performance pout,d can be improved. 6.4.

Simulation results

In this subsection, under realistic propagation conditions, including the effects of path loss and flat Rayleigh fading, the end-to-end outage performance is evaluated with different

8


SJ = {destination},

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Pbc /Pmax

Pav = 4 dB Pav = 7 dB

SJ −2 = i | dist{i, j } ≤ dinthop , ... ...

j ∈ SJ −1 , i ∈ SJ ∪ SJ −1 ,

Sn−1 = i | dist{i, j } ≤ dinthop , ... ....

j ∈ Sn , i ∈ SJ∪ SJ −1∪· · ·∪ Sn ,

10−1

SJ −1 = i | dist{i, destination} ≤ dinthop ,

100

Outage probability

routing strategies, including the local-k-best routing and allselect routing strategies. As for a two-stage network, we consider a square coverage area with diagonal dimension dmax and M uniformly distributed single-antenna half-duplex nodes. The x- and ycoordinates of all nodes are i.i.d. uniformly distributed ran√ dom variables on the interval [0, dmax / 2]. Denote dist{i, j } as the distance between node i and node j. In simulations, when using destination-initiated power-limited flooding to form the node sets Sn (n = 1, . . . , J − 1), we simply use dinthop to represent the reliable coverage range resulting from a limited flooding power. For every particular geographic distribution of the M nodes, the node sets for a given (s, d) pair in a J-hop (J > 2) network are formulated as

Figure 6: Outage probability as a function of Pbc /Pmax for the local-one-best routing using the two-group scheme with M = 100, dinthop /dmax = 1/6, Jav ≈ 5.11, L = 2 (r = 2 bps/Hz).

(10) The processing stops when the source is found such that S0 = {source}. In simulations, for a given (s, d) pair, the geographic distributions of all the other M − 2 potential relays are randomly generated and a large number of realizations are considered. Thus, J is a dynamic value for a given (s, d) pair and particular dinthop . Define Jav as the average hop number where the average is taken over all considered realizations of random geographic distributions. In particular, we evaluate the end-to-end outage per√ formance for the (s, d) pair√with source√= (0, 0.5dmax / 2) and destination = (dmax / 2, 0.5dmax / 2). It is assumed that the constant transmit power Pt is used for each node. Thus, thetotal power to transmit one message over J hops is P = n=0∼J −1 Pn , where Pn is the power consumption at hop n (n = 0, . . . , J − 1). For both routing strategies, P0 = Pt . For the all-select routing strategy, Pn = Nn Pt (n = 1, . . . , J − 1); for the local-k-best routing strategy, the power overhead resulting from broadcasting local information is included in the performance evaluation such that Pn = Kn Pt + Nn Pbc (n = 1, . . . , J − 1). Similar to Section 5.1, in a J-hop (J > 2) network, the SNR threshold ηt is determined as b × (2Jr − 1) since the J-hop cooperative transmission has a 1 : J bandwidth penalty compared to the direct transmission. The powers are normalized by Pmax . The definitions of r, b, and Pmax are the same as in Section 5.1. The outage curves are plotted as a function of the normalized average power Pav , which is the average consumed power per J-hop transmission. As in a two-stage network, here, we also use the parameter ξ = Pbc /Pmax to investigate the effect of varying the broadcast power Pbc on the end-to-end outage performance. This is shown in Figure 6 for the local-one-best routing when there are M = 100 nodes in the network, dinthop /dmax = 1/6, and L = 2 columns in the underlying STBC matrix of the m-group Dis-STBC. It can be observed that a ratio ξ in the

interval [0.05, 0.1] achieves the optimal performance. Similarly, ξ opt is in the interval [0.05, 0.1] for the local-two-best routing when M = 100, dinthop /dmax = 1/6, and L = 2. In these simulations, Jav ≈ 5.11. Then, on average, the number of the relaying node sets is Jav − 1. Thus, on average, the maximum possible number of the decoded nodes per relaying hop is (M − 2)/(Jav − 1). As an empirical result, ξ opt is approximately equal to 1/[(M − 2)/(Jav − 1)]. According to the obtained range of values for ξ opt , with choosing Pbc /Pmax = 0.08 and using the m-group Dis-STBC, the end-to-end outage performance of the local-k-best routing is investigated when varying k. Simulation results are shown in Figure 7 for the local-k-best (k = 1, 2) routing and all-select routing when M = 100 nodes, dinthop /dmax = 1/6, and L = 2 using an Alamouti code. Clearly, it can be seen that, even with the overhead included, the local-onebest routing is much more power-efficient than the all-select routing. In particular, a 2.5 dB advantage can be observed at an outage probability of 10−2 . As in a two-stage network using local-k-best relay selection, the advantage of the local-kbest routing decreases with an increase in k. 7.

CONCLUSIONS AND FUTURE WORKS

In this paper, for a two-stage network that uses selective decode-and-forward relaying, we presented a power-efficient relay-selection strategy for a particular decentralized DisSTBC scheme (m-group). The power-efficiency advantage of the proposed local-k-best (k ≥ 1) relay-selection strategy was illustrated through the outage analysis. Under realistic propagation conditions, including the effects of path loss and flat Rayleigh fading, we evaluated the outage performance of the m-group scheme with different relay-selection strategies. It was found that, when compared with the allselect relay-selection strategy, the local-k-best relay-selection

L. Zhang and L. J. Cimini Jr.

9

Outage probability

100

investigated. Furthermore, in the multihop case, the distribution for the local mean power gains of all interhop channels might be another implementation issue worthy of concern. It is advisable to combine this information distribution into the process of forming and maintaining relay clusters (i.e., node sets defined in this paper). Besides the destinationinitiated power-limited flooding scheme used for simulations in this paper, so many other proposed (distributed) clustering schemes could be explored in further research. In the future, by paying more attention to these implementation issues, we will try to implement local-k-best strategies in practical communication protocols for wireless cooperative networks.

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10

ACKNOWLEDGMENTS

Pav (dB) All-select routing Local-one-best routing, Pbc /Pmax = 0.08 Local-two-best routing, Pbc /Pmax = 0.08

Figure 7: Outage probability as a function of the total transmission power of the J hops, Pav , for the two-group scheme with M = 100, dinthop /dmax = 1/6, Jav ≈ 5.11, L = 2 (r = 2 bps/Hz).

strategy is much more power-efficient even with the additional power overhead included. In addition, by using the modified local-k-best relay-selection strategy at each relaying hop, a power-efficient hop-by-hop routing strategy was proposed for a multihop, selective, decode-and-forward network that uses the m-group Dis-STBC at each relaying hop. Under realistic propagation conditions, the end-to-end outage performance was evaluated for different routing strategies. It was found that, when compared with the all-select routing strategy, the local-k-best routing strategy is much more power-efficient even with the additional power overhead included. Although, in this paper, the local-k-best (k ≥ 1) relay-selection/routing strategies were presented by using the m-group Dis-STBC as an example, these strategies can be naturally extended to other decentralized Dis-STBC schemes (such as the continuous randomized scheme [10]). To implement the local-k-best strategies, the implementation for all decoded nodes to broadcast local information is important. In a cooperative network with half-duplex limitation of nodes, since we try to implement the broadcastings of decoded nodes with incurring small overhead, the oneway control traffic is preferred. In addition, the local-k-best strategies would like to let the broadcasting by each decoded node reach all neighboring decoded nodes. Based on the considerations described above, when using a random access protocol to implement the broadcastings by all the decoded nodes, we would not like to utilize CSMA combined with the RTS/CTS mechanism. Instead, we currently consider using a multichannel CSMA MAC protocol [12]. This protocol combines CSMA with CDMA, for example; it reduces the effect of the hidden-node problem elegantly and is quite suitable for the scenario where the broadcastings are intended to reach all neighbors. Of course, other approaches will also be

This material is based on research sponsored by the Air Force Research Laboratory, under Agreement no. FA9550-06-10077. The US government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. REFERENCES [1] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. [2] J. N. Laneman and G. W. Wornell, “Distributed space-timecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2525, 2003. [3] S. Barbarossa and G. Scutari, “Distributed space-time coding for multihop networks,” in Proceedings of the IEEE International Conference on Communications (ICC ’04), vol. 2, pp. 916–920, Paris, France, June 2004. [4] S. Barbarossa and G. Scutari, “Distributed space-time coding for regenerative relay networks,” IEEE Transactions on Wireless Communications, vol. 4, no. 5, pp. 2387–2399, 2005. [5] H. T. Cheng, H. Mheidat, M. Uysal, and T. M. Lok, “Distributed space-time block coding with imperfect channel estimation,” in Proceedings of the IEEE International Conference on Communications (ICC ’05), vol. 1, pp. 583–587, Seoul, Korea, May 2005. [6] A. Stefanov and E. Erkip, “Cooperative space-time coding for wireless networks,” IEEE Transactions on Communications, vol. 53, no. 11, pp. 1804–1809, 2005. [7] P. A. Anghel, G. Leus, and M. Kaveh, “Distributed space-time cooperative systems with regenerative relays,” IEEE Transactions on Wireless Communications, vol. 5, no. 11, pp. 3130– 3141, 2006. [8] J. Mietzner and P. A. Hoeher, “Improving the performance of mobile broadcasting systems using multiple base stations and distributed space-time codes,” IET Communications, vol. 1, no. 3, pp. 348–353, 2007. [9] J. Luo, R. S. Blum, L. J. Greenstein, L. J. Cimini Jr., and A. M. Haimovich, “Link-failure probabilities for practical cooperative relay networks,” in Proceedings of the 61st IEEE Vehicular Technology Conference (VTC ’05), vol. 3, pp. 1489–1493, Stockholm, Sweden, May-June 2005.

10 [10] B. Sirkeci-Mergen and A. Scaglione, “Randomized space-time coding for distributed cooperative communication,” IEEE Transactions on Signal Processing, vol. 55, no. 10, pp. 5003– 5017, 2007. [11] L. Zhang, L. J. Cimini Jr., L. Dai, and X.-G. Xia, “Relaying strategies for cooperative networks with minimal node communications,” in Proceedings of the Military Communications Conference (MILCOM ’06), pp. 1–7, Washington, DC, USA, October 2006. [12] A. Nasipuri, J. Zhuang, and S. R. Das, “A multichannel CSMA MAC protocol for multihop wireless networks,” in Proceedings of Wireless Communications and Networking Conference (WCNC ’99), vol. 3, pp. 1402–1406, New Orleans, La, USA, September 1999. [13] E. G. Larsson and P. Stoica, Space-Time Block Coding for Wireless Communications, Cambridge University Press, Cambridge, UK, 2003. [14] J. Luo, R. S. Blum, L. J. Cimini Jr., L. J. Greenstein, and A. Haimovich, “Power allocation in a transmit diversity system with mean channel gain information,” IEEE Communications Letters, vol. 9, no. 7, pp. 616–618, 2005. [15] L. Zhang and L. J. Cimini Jr., “Efficient power allocation for decentralized distributed space-time block coding,” submitted to IEEE Transactions on Wireless Communications, 2007. [16] J. Luo, R. S. Blum, L. J. Cimini Jr., L. J. Greenstein, and A. M. Haimovich, “New approaches for cooperative use of multiple antennas in ad hoc wireless networks,” in Proceedings of the 60th IEEE Vehicular Technology Conference (VTC ’04), vol. 4, pp. 2769–2773, Los Angeles, Calif, USA, September 2004. [17] S. Catreux, P. F. Driessen, and L. J. Greenstein, “Data throughputs using multiple-input multiple-output (MIMO) techniques in a noise-limited cellular environment,” IEEE Transactions on Wireless Communications, vol. 1, no. 2, pp. 226–235, 2002. [18] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 8, pp. 1451–1458, 1998. [19] L. Liu and H. Ge, “Space-time coding for wireless sensor networks with cooperative routing diversity,” in Proceedings of the 38th Asilomar Conference on Signals, Systems and Computers (ACSSC ’04), vol. 1, pp. 1271–1275, Pacific Grove, Calif, USA, November 2004. [20] T. Miyano, H. Murata, and K. Araki, “Cooperative relaying scheme with space time code for multihop communications among single antenna terminals,” in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM ’04), vol. 6, pp. 3763–3767, Dallas, Tex, USA, November-December 2004. [21] T. Miyano, H. Murata, and K. Araki, “Space time coded cooperative relaying technique for multihop communications,” in Proceedings of the 60th IEEE Vehicular Technology Conference (VTC ’04), vol. 7, pp. 5140–5144, Los Angeles, Calif, USA, September 2004. [22] T. Koike, M. Tanaka, and S. Yoshida, “System-level evaluation of STC cooperative relaying scheme in multihop IVC networks,” in Proceedings of the 2nd IEEE VTS Asia Pacific Wireless Communications Symposium (APWCS ’05), pp. 60– 64, Hokkaido University, Japan, August 2005. [23] A. D. Coso, S. Savazzi, U. Spagnolini, and C. Ibars, “Virtual MIMO channels in cooperative multi-hop wireless sensor networks,” in Proceedings of the 40th Annual Conference on Information Sciences and Systems (CISS ’06), pp. 75–80, Princeton, NJ, USA, March 2006.



Research Article Low-Complexity Distributed Multibase Transmission and Scheduling Hilde Skjevling,1 David Gesbert,2 and Are Hjørungnes3 1 Department

of Informatics, University of Oslo, P. O. Box 1080, Blindern, 0316 Oslo, Norway Communications Department, Eurécom Institute, 2229 Route des Crêtes, BP 193, 06904 Sophia Antipolis Cédex, France 3 UNIK-University Graduate Center, University of Oslo, Instituttveien 25, P. O. Box 70, 2027 Kjeller, Norway 2 Mobile

Correspondence should be addressed to Hilde Skjevling, [email protected] Received 1 May 2007; Revised 16 October 2007; Accepted 25 November 2007 Recommended by M. Chakraborty This paper addresses the problem of base station coordination and cooperation in wireless networks with multiple base stations. We present a distributed approach to downlink multibase beamforming, which allows for the multiplexing of M user terminals, randomly located in a network with N base stations. In particular, we detail a low-complexity scheduling algorithm, which can be employed with different objective functions, exemplified here by two approaches: (1) maximizing the sum rate of the network; and (2) maximizing the number of users served, given a statistical constraint on the received rate per user. The optimizations are based on locally available information at each base station. Results show that our approaches yield significant gains, when compared to schemes that do not allow cooperation between cells. These gains are obtained without the extensive signaling overhead required in previously known multicell MIMO processing. Copyright © 2008 Hilde Skjevling et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1.

INTRODUCTION

The scarcity of spectral resources in cellular networks motivates aggressive frequency reuse, an approach that has shown promise of significant capacity gains. In many cases, however, this potential is severely limited by intercell interference [1]. The interference problem may be alleviated in different ways, for example, by exploiting the multiuser diversity [2]. Also, the employment of a system-wide resource distribution is beneficial, through power-allocation and scheduling of the users in the different cells [3]. With many of the existing joint resource allocation and scheduling schemes, the user terminals are still communicating with their preferred base station or access point. However, as a result of the coordination of concurrent transmissions in neighboring cells, the terminals will benefit from reduced interference. A limited form of network multiple input multiple output (MIMO) inspired coordination is presented in [4], where groups of co-located or distributed antennas transmit to a set of users, in a coherent and coordinated manner, with the aim of mitigating intercell interference.

Allowing all the antennas at the network’s base stations to act together as distributed antennas of a large-scale multipleantenna array, yet subject to per-base power constraints, is discussed in many recent papers. Such network coordination may improve the spectral efficiency of the communication, and reduce the interference from neighboring cells [5]. This form of cell coordination exploits a common signal-processing-based effort, and known multiuser MIMO transmission techniques, such as minimum mean square error, zero-forcing, or dirty paper coding, can be reused over the multibase antenna array [6–8]. In [9], the focus is on joint power control and optimal beamforming, allowing each mobile user to receive cooperative transmissions from all base stations in an active set. Alternatively, the subject of [10] spatial multiplexing over cooperating base stations, with limited, local channel state information. Theoretical analysis of scheduling or cooperative base station transmission schemes for downlink communication is complicated, in many cases prohibitively so. Still, advances have been made, deriving the capacity-maximizing power allocation for a two-cell system [11], and finding closedform expressions for (1) the per-cell sum rate of a distributed

2


multicell zero-forcing beamformer [12], and (2) the sum rates using different precoders [13], in both cases for nonfading channels. In [14], the authors consider solutions to optimal transmit beamforming for multiuser downlink with per-antenna power constraints at the base station, so extensions to having distributed antennas are conceivable. The optimum use of the distributed base antennas leads to a promising research direction. However, two major issues need to be addressed before such techniques can be considered in practical settings. First, the complexity of implementing multiuser MIMO solutions for a large number of cells and users is prohibitive. Second, the optimum antenna combining requires a large signaling overhead between the base stations of the network, which must exchange information on all the users’ channel responses. This is especially problematic in the downlink. Centralized approaches yield good performance, but remain of interest only for the optimization of very small networks or when dividing the network into clusters of cells. One handicap of clustering, however, lies in the edge effects it creates for users who sit in the neighborhood of two or more clusters, although this can be addressed by dynamic clustering [15]. To avoid the above-presented problems of high complexity and large overhead, in the case of large-scale networks, it is of great interest to derive multibase-aided cooperation techniques, which can be realized in a distributed manner and have a reasonable complexity. This is the main topic of this paper, and we explore approaches to distributed pro-cessing, using limited channel state information, for downlink communication in a multiuser, multibase, wireless network. We investigate some consequences and advantages of such solutions. The key ideas presented here can be summarized as (i) distributed beamforming and (ii) greedy scheduling. A first part of this work is presented in [16]. The proposed distributed beamforming framework exploits the base antennas so that each scheduled mobile station will receive coherently added versions of the desired signal, possibly from several bases. The scheduling technique attempts to assign users to base stations, one user being served by one or more base stations, and receiving interference from others. More specifically, we present the following contributions.

(i) the network sum capacity, adding up the rates of all the receiving users; and (ii) the number of users scheduled, with a statistical per-user rate constraint.

(1) The first contribution is a practical setup for distributed beamforming, where each base station only needs hybrid channel state information (CSI). By hybrid CSI, we consider instantaneous CSI on locally measured channels and long-term, statistical CSI on nonlocally measured channels. This latter information may be exchanged via a central unit, using a lowrate dedicated channel. (2) Next, we present low-complexity algorithms for multibase scheduling, where the base stations jointly select users, so as to optimize a chosen objective function, for which we will present the following two variations:

The organization of the rest of this paper is as follows. In Section 2, we present the system model and the distributed beamforming setup. Next, in Section 3, the two different optimization objectives are presented. In Section 4, we detail the user scheduling problem for the centralized case, while Section 5 presents the distributed approaches. Results from numerical simulations are presented in Section 6, and the concluding remarks are contained in Section 7. Use of notation: in this paper, A, a, and a denotes a matrix, a vector, and a scalar, respectively. For a realvalued function f with domain S, arg max x∈S f (x) is the set of elements in S that achieve the global maximum in S. Finally, we define the following three sets of indeces: N = {0, 1, . . . , N − 1}, M = {0, 1, . . . , M − 1}, and Nt = {0, 1, . . . , NTx − 1}. 2.

SYSTEM MODEL

We assume a setting with N base stations (BS) and M users or mobile stations (MS), the whole system being engaged in downlink communication. The base stations have Tx transmit antennas each, while, for ease of exposition, the MSs are equipped with a single antenna, Rx = 1. Each base station holds all or part of the same M-length symbol vector, s = [s0 , s1 , . . . , sM −1 ]T , where sm is intended for MSm , m ∈ M. The symbols are seen as uncorrelated, E[sm s∗ / k. k ] = 0, for m = The base stations schedule users and apply precoding in the form of transmit-side matched filtering. To this end, a base station BSn , n ∈ N , is required to have perfect, instantaneous CSI on the channels from itself to the M users. This can be done by a preamble using training sequences, enabling the base stations to measure and track the local channels. Note that this assumes a form of symbol-level synchronization between the bases, realizable if the relative distances between the neighboring bases are not too large. Synchronization between widely separated bases is not a requirement, because the larger path loss will in any case limit the need for cooperation between them. For the nonlocal channels between the N − 1 base stations BSl , l ∈ N \ n, and the M users, we assume that BSn has only long-term, statistical knowledge. Statistical knowledge is equivalent to knowledge of slow-varying macroscopic parameters of the channels, such as distance-based path loss and shadowing effects. See Figure 1 for an illustration of the network, and note that the coefficient Wl denotes the precoding at BSl , to be defined. For the user scheduling, we define a scheduling graph, represented by the N × M-sized matrix G:

T

G = g0 g1 · · · gN −1 ,

(1)

with gn being the scheduling vector of size M × 1 at BSn :

T

gn = gn0 gn1 · · · gn(M −1) ,

(2)

Hilde Skjevling et al.

3 precoding matrix W are (W)nt m = wnt m , where nt ∈ Nt and m ∈ M, such that

BSl

···

Wl

···

···

h∗mnt

wnt m = gnm Pt

hmn .

···

(Hl )i,: MSi (Hk )i,:

(H j )i,: ···

(6)

t

Wj

···

BS j

Wk

···

···

BSk

Figure 1: System model, showing the base stations as squares in a multicell network, while the mobile stations or users are depicted as circles. Arrows from BSk to MSi imply that the MS is scheduled by the base stations, so that BSk transmits (Wk )i si to MSi , over the channel (Hk )i,: . The interference is not shown.

Here, hmnt represents the channel from transmit antenna nt ∈ Nt , at BSn , to the receiving antenna at MSm , and n is related to nt as n = nt /Tx , where · denotes the floor function. Note that the matched filtering naturally lends itself to distributed implementation. From the definition of G in (1), it is evident that only a single row in each Wn contains nonzero elements. The transmit power per base station is limited as Wn 2F = Pt (in Watts), where ·F is the Frobenius norm. Now, BSn transmits xn = WTn s from its Tx antennas. The paths from BSn to the M receiving MSs are represented by the M × Tx -sized matrix Hn . The total channel matrix H includes all paths, is of size M × NTx , and is given as

H = H0 H1 · · · HN −1 . where each coefficient gnm is interpreted as ⎧ ⎨1,

gnm = ⎩ 0,

if BSn transmits to MSm ,

(3)

otherwise.

We schedule one user MSm , m ∈ M, per base station BSn , n ∈ N , at full power, at any given time. More generally, we assume that one user is assigned to each spectral resource slot available per cell (time, frequency, code, etc.). Any MSm is served by zero, one, or more base stations. For a given BSn , the optimization is thus limited to choosing the best MS, according to a chosen performance criterion. Thus, this is a pure scheduling problem. Among the possible objective functions, we will present two: (1) the network sum capacity, and (2) a fairness-oriented approach of maximizing the number of users served, with statistical rate constraints. For fairness, we may also rely on user mobility and time-variant channel conditions. The set of all feasible graphs, under the scheduling constraints above, is denoted by SG , and includes all G for which all the vectors gn , n ∈ N , contain a single nonzero element:

SG = G = g0 g1 · · · gN −1

T

: gn ∈ EM .

T

W = W0 W1 · · · WN −1 ,

The coefficient (H)mnt = hmnt gives the complex channel gain from transmit antenna nt ∈ Nt , at BSn , n = nt /Tx to MSm , and includes both fast (multipath) fading and more slowly changing effects. The M × 1 received vector at all the mobile stations is y = HWs + v,

(5)

where each Wn , of size M × Tx , is the scheduling and precoding matrix of BSn . The coefficients of the global

(8)

where the M × 1-sized vector v contains random noise coefficients, following a Gaussian, white distribution, vm ∼ CN (0, σv ). Each MSm receives both desired symbols, interfering symbols, and noise: d i + ym . ym = (H)m,: Ws + vm = ym

(9)

d is the desired part of the signal, Here, ym d = ym

Pt

NT x −1

gnm hmnt sm ,

(10)

nt =0

i contains the interference and noise, while ym

i = Pt ym

(4)

Here, the set EM = {e1 , e2 , . . . , eM } defines the standard basis for the real-vector space RM , so that em is an M × 1sized vector with 1 at the mth coordinate, and 0 elsewhere. The cardinality of SG is |SG | = M N , which mounts to a substantial size as the networks grow. We combine the user selection with matched filter precoding in the NTx × M-sized matrix

(7)

3.

NT x −1 nt =0

hmnt

h∗knt

gnk

hkn sk + vm . t k=0

M −1

(11)

k= /m

SYSTEM OPTIMIZATION

In the following, we present two possible objective functions for use with the distributed beamforming setup. First, in Section 3.1, we focus on the network sum capacity. Section 3.2 presents an alternative; counting the number of mobile stations that are served satisfying a statistical constraint on the received rates. 3.1.

The network sum capacity

There is no cooperation or coherent combining between the MSs, so the instantaneous sum capacity of the whole system

4


is simply the sum of the data rates of the M noncooperating MISO receive branches, under ideal single-user decoding assumption [17]: C(G, H) =

M −1

Cm (G, H) =

m=0

M −1 m=0

log2 1 + SINRm (G, H) . (12)

Here, Cm (G, H) is the data rate at MSm , and the signal-to-interference-plus-noise ratio (SINR) of user m is SINRm (G, H), and depends both on the channel H and the scheduling graph G. Using the assumptions that E[|sm |2 ] = ∗ σs2 , E[sm s∗k ] = 0 for m = ] = 0 for all / k, and that E[sk vm possible k and m, we develop the SINRm (G, H) as SINRm (G, H) d 2

Es ym 2 = Es,v y i

m

2 NT −1 Es Pt nt =x0 gnm hmnt sm

= M −1

2

NT −1

Es,v Pt nt =x0 hmnt k=0,k = m gnk h∗ knt / hknt sk+vm =

/

2 2

σ nt =0 gnm hmnt s .

NTx −1

2 ∗

M −1

Pt

σ 2 +σ 2

s v k=0,k = nt =0 hmnt gnk hknt / hknt /m

Pt

NTx −1

(13)

scheduling will be one where each base station transmits to a separate MS, as in the conventional, singlebase approach. Therefore, the choice of rate constraints is a crucial one. Although this scheme does not use power allocation in an attempt to minimize the total power used for transmission, the resulting power per served MS is naturally limited by the wish to serve, in a satisfactory manner, as many MS as possible. In Section 6, we study and compare simulation data resulting from use of the two different optimizations approaches described in this and the previous sections. 4.

USER SCHEDULING PROBLEM

We seek the scheduling graph G that optimizes our chosen measures of performance as described in Sections 3.1 and 3.2. The assumption on the scheduling of a single user at each base station is maintained for both approaches. Given the above presented constraints and assumptions, the optimization problem is expressed as finding the best scheduling graph, such that either (1) the sum capacity C(G, H), or (2) the number of served users with statistical rate constraint is maximized. The latter objective is expected to introduce an element of fairness among the MSs. The scheduling problem can be approached in different ways, first we present a centralized scheduler in Section 4.1, useful for comparison. In Section 5, we propose low-complexity, distributed schedulers.

From this, we get C(G, H) =

M −1 m=0

where V = Pt

log2 1 +

NTx −1 nt =0

NTx −1

2

gnm hmnt σs2 , 2 2 2 k=0,k = / m | V | σ s + σv (14) Pt

hmnt gnk (h∗knt )/(|hknt |).

3.2. Number of served users, under statistical rate constraints The sum capacity is not the only useful quantitative measure on the performance of a wireless network. A different view could be gained from counting the number of simultaneously served users, given a certain per-user minimum-rate constraint CR . This can be seen as a quality-of-service (QoS) guarantee for the scheduled users, one way to do QoS-based scheduling is described in [18]. In our case, with access only to hybrid CSI and distributed processing, the final rates are not guaranteed and we refer to the constraints as statistical. Given a certain scheduling matrix G, a channel realization H, and the rate constraint CR , we define the set

4.1.

Centralized scheduler

nt =0

QG,H,CR = MSm | m ∈ M and Cm (G, H) ≥ CR .

(15)

The cardinality of this set, denoted |QG,H,CR |, is the number of scheduled users whose rates satisfy the constraints. For a given channel realization, there are |SG |, possibly different, sets QG,H,CR . Obviously, it holds that 0 ≤ |QG,H,CR | ≤ M. Only the scheduled mobile stations MSm , for which N −1 n=0 gnm ≥ 0 can possibly contribute to |QG,H,CR |, and they only will if their received rates satisfy the constraints. Note that if the rate constraint is too modest, the resulting best

The centralized scheduling approach is governed by a central unit, which is required to have full, instantaneous CSI on the whole channel H. The optimization takes the form of an exhaustive search, where the central unit searches the entire M N -sized set of feasible graphs SG , and picks the one that maximizes the chosen objective function. For the case of maximum sum capacity, we denote this best scheduling graph by G∗SC , and write the optimization problem as G∗SC = arg max C(G, H). G∈SG

(16)

For the case when the objective is to maximize the number of users served with an acceptable rate, we find the best graph G∗MS as

G∗MS = arg max QG,H,CR . G∈SG

(17)

If |QG,H,CR | = |QG ,H,CR |, for G = / G , the chosen graph will be the one that gives the highest sum rate C(G, H). As mentioned, the cardinality of feasible graph set is |SG | = M N , so for a large network, the centralized scheduler is prohibitively complex and time-consuming. Furthermore, this implies a very large amount of feedback information between the MSs and the base stations to be centrally collected by the network, which is not practical for large networks in mobility settings. Theoretical analysis of these problems are highly nontrivial, but we give two very simple examples to illustrate the case of maximizing the sum capacity.


5

(1) Interference-limited case When the noise power is very small compared to the received interference, we neglect it and consider an interferencelimited scenario: lim C(G, H) =

σv →0+

M −1 m=0

Pt

log2 1 +

NTx −1

nt =0 gnm hmnt

M −1

2 k=0,k = / m V|

2

(18) From this expression, we observe that, with no interference, the sum capacity can theoretically be infinite. That is the case if all the base stations in the network schedule any single mobile station, towards which there is at least one nonzero channel. (2) Static channels Now, assume that all the channels are static and equal to unity, hmn = 1, {m, n} ∈ {M, N }. For the ease of exposition, we assume that the BSs are single-antenna Tx = 1, and define δ = σv2 /(Pt σs2 ). Now, (14) simplifies to C(G, H1 ) =

M −1 m=0

2 n=0 gnm . N −1 2 M −1 k=0,k = n=0 gnk +δ /m N −1

log2 1 +

(19) We give three example cases, assuming that there are as many MSs as BSs in the network M = N: when (1) all the BSs schedule a single MS, (2) all MSs are scheduled by a separate BS, and (3) half of the BSs schedule one MS and the rest schedule a second MS. The corresponding sum rates are

C1 = log2 1 +

N2 , δ

C2 = Mlog2 1 +

C3 = 2log2 1 +

1 , N −1+δ

(20)

(N/2)2 . (N/2)2 + δ

Using N ≥ 2 and a sensible range 0 < δ ≤ 1, it is obvious from these examples that the best rate is achieved in C1 , where all BSs schedule a single MS. In fact, from a sum-rate point of view in this case, scheduling any single MS is the best choice, which makes intuitive sense given the slope of the log function. Even when the channels have different, distance-based path loss, which is closer to reality and should diminish the interference problem between far-away nodes, the network very quickly becomes interference-limited, and scheduling more than a few users is suboptimal.

then each base station only needs to be told which MS to schedule. However, the exponential complexity increases and the need for full, centralized, instantaneous CSI motivates the search for low-complexity solutions with acceptable performance. In the following, we give some distributed user scheduling approaches. One approach to derive distributed algorithms is to break channel information into two sets, characterized as being local or nonlocal information. These sets of information are treated differently and dubbed together as hybrid CSI. Here, the term is used to describe the fact that BSn , n ∈ N , has full, instantaneous CSI on its local channels, defined as the Tx M channels linking BSn to all the M users, and represented by Hn . On the remaining M(N − 1)Tx channels, BSn has only long-term, statistical CSI, by which, for this scenario, we specifically refer to the path loss and the shadow fading. In Section 5.1, we describe a spatially distributed multibase scheduler of relatively low complexity and where only hybrid CSI is needed. For comparison, we also give a fully distributed scheduler, as well as a conventional singlebase scheduler, in Sections 5.2 and 5.3, respectively. Note that these comparisons are tailored neither to maximize the capacity nor the number of scheduled MSs, they simply illustrate alternative scheduling approaches. 5.1.

Iterative, distributed scheduling

We present an iterative scheme, in which the base stations successively make greedy scheduling decisions and update the common scheduling graph G. They all optimize the same objective function, thus benefiting from intercell cooperation, but have access only to hybrid CSI. This approach demands that statistical channel state information is distributed to all the base stations prior to optimization, and that the updated scheduling graph is always known to the base stations. In comparison with the centralized scheme, the feedback load is significantly reduced. The system starts from an initial graph G0 , known to all the base stations. Next, in a predetermined, nonoptimized order, all the BSn , n ∈ N , are allowed to update the scheduling graph once, including its own best scheduling vector gn∗ in Gn to form Gn+1 . The distributed scheduling is performed based on the choice of objective function and with access to hybrid CSI. We summarize the scheduling procedure for both choices of optimization functions, the maximum sum rate and the maximum number of served users, the latter with a statistical constraint on the user rate. For ease of exposition, we define the matrix T

n = g0 · · · gn−1 g n gn+1 · · · gN −1 , G

5.

(21)

DISTRIBUTED SOLUTIONS

The concept of the centralized scheduler is simple, as the scheduling graph G is constructed in a central unit, and

T

where Gn = g0 · · · gn−1 gn gn+1 · · · gN −1 , in other words, gn in row n of Gn is exchanged with gn .

6

EURASIP Journal on Advances in Signal Processing (1) Maximum network downlink sum capacity

where SNR(gn , Hn ) is defined as

Initilize G0 .

SNR gn , Hn =

for n = 0 : N − 1

Pt gn H n 2 σ 2 s

σv2

∗ n, H ; gn,SC = arg max EH n C G

gn ∈EM

T

(22)

∗ gn+1 · · · gN −1 ; Gn+1 = g0 · · · gn−1 gn,SC

end G = GN . (2) Maximum number of users served Initialize G0 . for n = 0 : N − 1

GMS = arg max EH n QG n ,H,CR ;

,

(26)

where Hn denotes a matrix with entries (Hn )mn = |(Hn )mn |. This represents the receive SNR in MSm , the single MS scheduled by BSn , for which gnm = 1. From a network point of view, one mobile station may be selected by multiple base stations, in which case it receives a coherently added sum of the desired signal, beamformed from all the antennas of these base stations. This method has low complexity and only local information is used, while statistical external information is not needed. One disadvantage is the limited amount of cell cooperation; the base stations are not aware of each other, and this will in turn limit network performance.

gn ∈EM

∗ n, H ; = arg max EH n C G gn,MS

(23)

gn ∈GMS

T

∗ gn+1 · · · gN −1 ; Gn+1 = g0 · · · gn−1 gn,MS

end G = GN . Here, the double use of arg max signifies that when several gn give the same |QG n ,H,CR |, we select the one yielding the maximum sum rate, based on the available CSI. In both approaches, Gn+1 is found in the same way as n , where gl , l = G / n, are taken from Gn , which contains all previously updated scheduling choices. Also, EH n denotes taking the expected value with respect to all channels in

n = H0 , H1 , · · · Hn−1 , Hn+1 , · · · HN −1 . H

(24)

This matrix contains all the channel coefficients of the fullchannel H, except Hn . As Hn , the local channel matrix from BSn to all MSs, is instantaneously known at BSn , there is no need to average over it, while BSn only has long-term n. statistical information on the rest of the channel; H In the above iterative procedure, for both objective functions, the scheduling graph is updated once for each of the N base stations. After traversing all the base stations, the last version of G is the final scheduling matrix. This calls for a central unit to hold and distribute the intermediate Gn , but the exchange of information to and from the users is moderate. 5.2. Fully distributed user scheduling This comparison is fully distributed and noncooperative, so no central unit is required for coordination. Each BSn schedules the MSm with the maximum receive signal-tonoise ratio (SNR), with no regard for the interference. In other words, BSn finds its own best scheduling vector gn∗ , such that

gn∗ = arg max SNR gn , Hn , gn ∈EM

(25)

5.3.

Conventional single base station assignment

Finally, we formalize a conventional singlebase approach for this scenario, in the sense that a receiving MS can only be scheduled by a single base station. A central unit goes through the N available base stations, and allows each base station to choose a previously unscheduled MS, if there are any left. The central unit holds and updates the scheduling graph, ensuring that one MS is scheduled by one base station only. For BSn , the user is selected by maximizing the receive SNR:

gn∗ = arg max SNR gn , Hn , gn ∈Se

(27)

where Se is a subset of the full RM standard basis {e1 , e2 , . . . , eM }, representing those users not already scheduled by a base station. Each BS only needs local CSI. The central unit exploits the available information by optimizing the scheduling order, at all times coupling the BS-MS pair that has maximum expected SNR, among those remaining. When there are no more base stations or users left to connect, the scheduling graph is finished. Note that the last two scheduling approaches, in Sections 5.2 and 5.3, are not linked to the two objective functions used in this paper, as presented in Section 3. 6.

NUMERICAL RESULTS

Next, we present some results of Mont Carlo simulations for the above-described schedulers, for both optimization objectives, as described in Sections 3.1 and 3.2. The focus is on how the low-complexity, iterative, and distributed scheduling approach in Section 5.1 performs when compared to the centralized, the fully distributed, and the conventional schemes; see Sections 4.1, 5.2, and 5.3, respectively. The base stations are placed in a grid, as seen in Figure 1, with a minimum distance d between neighbors. The positions of the mobile users are quasistatic, generated following a random and uniform spatial distribution over the entire network area.


7

Table 1: Simulation parameters.

4

Value 0 10 dB 1 Watt 16 dB 6 dB {30, 1} m 1800 MHz 0.5 km {1, 1} 50 200 4 b/s/Hz for all SNR

3 2.5 2 1.5 1 0.5

0

10

20

30 40 SNR (dB)

50

60

70

Centr., capacity-maximizing scheduling, full CSI Iter., capacity-maximizing scheduling, hybrid CSI Centr., max. number of users served scheduling, full CSI Iter., max. number of users served scheduling, hybrid CSI Non cooperative, fully distributed user scheduling Conventional single-base assignment

8 7 Capacity (bits/s/Hz/cell)

3.5 Number of users served

Parameter Shadow fading mean μχ Shadow fading standard dev. σχ Transmit power P Transmit antenna gain Gt Receive antenna gain Gr Antenna heights {hb , hr } Carrier frequency fc Smallest distance d between BSs Number of antennas {Tx , Rx } Random MS locations NMS Channel realizations Nchan Rate constraint, Cm (G, H)

6

Figure 3: Number of MS served versus edge-of-cell SNR for N = M = 4. Note that the iterative, and the centralized |QG,H,CR |maximizing approaches both schedule a relatively constant number of users, while the centralized and iterative capacity-maximizing approaches schedule fewer users as the SNR increases. The conventional approach schedules N users, regardless of the conditions. The statistical rate constraint was CR = 4 b/s/Hz.

5 4 3 2 1

0

10

20

30 40 SNR (dB)

50

60

70

Centr., capacity-maximizing scheduling, full CSI Iter., capacity-maximizing scheduling, hybrid CSI Non cooperative, fully distributed user scheduling Conventional single-base assignment Centr., max. number of users served scheduling with full CSI Iter., max. number of users served scheduling with hybrid CSI

Figure 2: Sum capacity per cell versus edge-of-cell SNR for N = M = 4. Note that the iterative, capacity-maximizing scheduling approaches lie between that of the centralized schemes and the interference-limited performance of the fully distributed and the conventional schedulers. Note also that the attempt to maximize the number of scheduled users with acceptable rate limits the sum capacity. The statistical rate constraint was CR = 4 b/s/Hz.

The channel from antenna nt ∈ Nt , located at BSn , n = nt /Tx , to MSm is hmnt = γmnt h mnt , where h mnt represents the complex random, Rayleigh distributed fast fading, h mnt ∼ CN (0, 1). The constant and slow-varying transmission effects are contained in γmnt . In dB scale, we write γmnt ,dB = Gt,dB − ρmnt ,dB + χmnt ,dB + Gr,dB ,

(28)

where Gt,dB and Gr,dB are the transmit and receive antenna gains, and ρmn,dB is the path loss, generated using the COST 231 model [19]. The distributed, long-term (shadow) fading χmn,dB is modeled as random, log-normal χmnt ,dB ∼ N (μχ , σχ ). Useful parameters are detailed in Table 1. All the simulations were run by averaging the resulting sum capacity over a total of NMS random MS locations and Nchan realizations of the instantaneously known channel coefficients. The expectation operator EH n , of (22) and (23), implies further averaging for each of the Nchan channel realizations. Simulations have been run for different scenarios, where performance is measured by both the network sum capacity of (14) per cell, with unit bits/second/Herz/cell, and by the number of mobile stations served, in different figures. First, we simulated a rather small network, with only 4 transmitting base stations and 4 receiving, mobile users, N = M = 4. For simplicity, the base stations are assumed equipped with a single antenna, as are the receiving users, Tx = Rx = 1. In Figure 2, the curves show how the network sum capacity develops with an increasing edge-ofcell SNR (reference value for single-user at distance dref ). The centralized scheduler of Section 4.1 and the iterative scheduler of Section 5.1 are both represented with two curves, one for each objective function, as shown in the figure legend. The remaining two curves are obtained by using the

8

EURASIP Journal on Advances in Signal Processing 7.5 7

Capacity (bits/s/Hz/cell)

6.5 6 5.5 5 4.5 4 3.5 3 2.5

4

6 8 10 12 14 Number of receiving MS (4 transmitting BS)

16

Iterative, capacity-maximizing scheduling with hybrid CSI Non cooperative, fully distributed user scheduling Conventional single-base assignment

Figure 4: Sum capacity per cell versus number of receiving MSs, for edge-of-cell SNR of 20 dB and N = 4 base stations. Note that the iterative, capacity-maximizing scheduling outperforms both the fully distributed and the conventional scheduling approaches.

3.8 3.6 Capacity (bits/s/Hz/cell)

7.

CONCLUSIONS

In this paper, we have presented approaches for base station coordination and cooperation in multibase, multiuser wireless networks. First, a framework for distributed, downlink beamforming was given, where each participating base station only needs access to hybrid channel state information, including instantaneous CSI on locally measured channels. Next, we have detailed some scheduling schemes to use with this framework, which may be tailored to different optimization needs; such as the maximization of the network sum capacity or the maximization of the number of MSs that can be scheduled while enjoying a certain rate. For both cases, the low-complexity approach of distributed, iterative, scheduling represents a middle course between the interference-limited fully distributed and conventional schemes, and the prohibitively complex centralized algorithm.

4

3.4 3.2 3 2.8 2.6 2.4 2.2 2

will suffer. This also applies for the corresponding case of maximizing the sum capacity, in which case the number of MS served will decrease with increased SNR. Focusing purely on the sum capacity of the network, we also fix the SNR to 20 dB and explore the network sum capacity when increasing the number of receiving users M = {4, 8, 12, 16}, while keeping a constant N = 4 base stations. The results are shown in Figure 4. In this case, as the M increases beyond N, note that only N of these users will be served at any given time. No simulation results for maximizing |QG,H,CR | were included here. Finally, in Figure 5, we present the simulation results when increasing the number of receiving users and base stations, M = N = {4, 8, 12, 16}. We observe that the sum capacity per cell is decreasing when increasing M and N together, and imagine one explanation for this being the increased levels of interference resulting from more base stations transmitting. In Figures 4 and 5, only three curves are plotted, as the centralized scheme of Section 4.1 is very time-consuming for larger networks. No simulation results for maximizing |QG,H,CR | were included here.

4

6 8 10 12 14 Number of receiving MS (and transmitting BS)

16

Iterative, capacity-maximizing scheduling with hybrid CSI Non cooperative, fully distributed user scheduling Conventional single-base assignment

Figure 5: Sum capacity per cell versus number of receiving MSs and base stations (N = M), for edge-of-cell SNR of 20 dB. Note that the iterative, capacity-maximizing scheduling outperforms both the fully distributed and the conventional scheduling approaches.

schemes described in Sections 5.2 and 5.3, in downward order. Next, in Figure 3, we show the number of users scheduled, for the same schemes as in the previous figure. When comparing the results of Figures 2 and 3, we observe that the choice of objective function indeed has an impact; when attempting to maximize |QG,H,CR |, the network sum capacity

ACKNOWLEDGMENT This work was supported by the Research Council of Norway through the Projects 160637/V30 “Advanced Signaling for Multiple Input Multiple Output (MIMO) Wireless Application to High Speed Data Access Networks” and VERDIKT 176773/S10 “OptiMO.” REFERENCES [1] S. Catreux, P. F. Driessen, and L. Greenstein, “Simulation results for an interference-limited multiple-input multipleoutput cellular system,” IEEE Communications Letters, vol. 4, no. 11, pp. 334–336, 2000. [2] R. Knopp and P. A. Humblet, “Information capacity and power control in signel-cell multiuser communication,” in Proceedings of IEEE International Conference on Communications (ICC ’95), vol. 1, pp. 331–335, Seattle, Wash, USA, June 1995.

Hilde Skjevling et al. [3] D. Gesbert, S. Kiani, A. Gjendemsjø, and G. Øien, “Adaption, coordination and distributed resource allocation in interference-limited wireless networks,” to appear in Proceedings of IEEE. [4] F. Boccardi and H. Huang, “Limited downlink network coordination in cellular networks,” in Proceedings of the 18th Annual IEEE International Symposium on Personal Indoor and Mobile Radio Communications (PIMRC ’07), Athens, Greece, September 2007. [5] M. K. Karakayali, G. J. Foschini, and R. Valenzuela, “Network coordination for spectrally efficient communications in cellular systems,” IEEE Wireless Communications, vol. 13, no. 4, pp. 56–61, 2006. [6] D. Gesbert, M. Kountouris, R. Heath, C.-B. Chae, and T. Salzer, “From single user to multiuser communications: shifting the MIMO paradigm,” to appear in IEEE Signal Processing Magazine. [7] S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via co-processing at the transmitting end,” in Proceeding of VTS 53rd IEEE Vehicular Technology Conference (VTC ’01), vol. 3, pp. 1745–1749, Rhodes, Greece, May 2001. [8] H. Zhang and H. Dai, “Cochannel interference mitigation and cooperative processing in downlink multicell multiuser MIMO networks,” EURASIP Journal on Wireless Communications and Networking, vol. 2004, no. 2, pp. 222–235, 2004. [9] C. Botella, G. Piñero, M. de Diego, A. González, and O. Lázaro, “An efficient joint power control and beamforming algorithm for distributed base stations,” in Proceedings of the 1st International Symposium on Wireless Communication Systems (ISWCS ’04), pp. 135–139, Port Louis, Mauritius, September 2004. [10] H. Skjevling, D. Gesbert, and A. Hjørungnes, “Receiverenhanced cooperative spatial multiplexing with hybrid channel knowledge,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP ’06), vol. 4, pp. 65–68, Toulouse, France, May 2006. [11] A. Gjendemsjø, D. Gesbert, G. E. Øien, and S. G. Kiani, “Optimal power allocation and scheduling for two-cell capacity maximization,” in Proceedings of the 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt ’06), pp. 1–6, Boston, Mass, USA, April 2006. [12] O. Somekh, O. Simeone, Y. Bar-Ness, and A. M. Haimovich, “CTH11-2: distributed multi-cell zero-forcing beamforming in cellular downlink channels,” in Proceedings of IEEE Global Telecommunications Conference (GLOBECOM ’06), pp. 1–6, San Francisco, Calif, USA, November 2006. [13] S. Jing, D. N. C. Tse, J. Hou, J. B. Soriaga, J. E. Smee, and R. Padovani, “Multi-cell downlink capacity with coordinated processing,” in Proceedings of the Information Theory and Applications Workshop (ITA ’07), San Diego, Calif, USA, January 2007. [14] W. Yu and T. Lan, “Transmitter optimization for the multiantenna downlink with per-antenna power constraints,” IEEE Transactions on Signal Processing, vol. 55, no. 6, part 1, pp. 2646–2660, 2007. [15] A. Papadogiannis, D. Gesbert, and E. Hardoin, “Dynamic clustering approach in wireless networks with multi-cell cooperative processing,” in Proceedings of IEEE International Conference on Communications, Beijing, China, May 2008. [16] H. Skjevling, D. Gesbert, and A. Hjørungnes, “A low complexity distributed multibase transmission scheme for improving

9 the sum capacity of wireless networks,” in Proceedings of the 8th IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC ’07), Helsinki, Finland, June 2007. [17] T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, New York, NY, USA, 1991. [18] M. Kountouris, A. Pandharipande, H. Kim, and D. Gesbert, “Qos-base user scheduling for multiuser MIMO systems,” in Proceedings of the 61st IEEE Vehicular Technology Conference (VTC ’05), vol. 1, pp. 211–215, Stockholm, Sweden, May-June 2005. [19] COST Action 231, “Digital mobile radio towards future generation systems, final report,” Tech. Rep. EUR 18957, European Commission, Brussels, Belgium, 1999.

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