A Survey on MIMO Transmission with Discrete Input Signals

A Survey on MIMO Transmission with Discrete Input Signals: Technical Challenges, Advances, and

arXiv:1704.07611v1 [cs.IT] 25 Apr 2017

Future Trends Yongpeng Wu, Senior Member, IEEE, Chengshan Xiao, Fellow, IEEE, Zhi Ding, Fellow, IEEE, Xiqi Gao, Fellow, IEEE, and Shi Jin, Member, IEEE

Abstract

Multiple antennas have been exploited for spatial multiplexing and diversity transmission in a wide range of communication applications. However, most of the advances in the design of high speed wireless multipleinput multiple output (MIMO) systems are based on information-theoretic principles that demonstrate how to efficiently transmit signals conforming to Gaussian distribution. Although the Gaussian signal is capacity-achieving, signals conforming to discrete constellations are transmitted in practical communication systems. The capacityachieving design based on the Gaussian input signal can be quite suboptimal for practical MIMO systems with discrete constellation input signals. As a result, this paper is motivated to provide a comprehensive overview on MIMO transmission design with discrete input signals. We first summarize the existing fundamental results for MIMO systems with discrete input signals. Then, focusing on the basic point-to-point MIMO systems, we examine transmission schemes based on three most important criteria for communication systems: the mutual information driven designs, the mean square error driven designs, and the diversity driven designs. Particularly, Y. Wu is with Institute for Communications Engineering, Technical University of Munich, Theresienstrasse 90, D-80333 Munich, Germany (Email:[email protected]). C. Xiao is with Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO 65409, USA. (Email: [email protected]). Z. Ding is with Department of Electrical and Computer Engineering, University of California, Davis, California 95616, USA. (Email: [email protected]). X. Gao and S. Jin are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, P. R. China. (Emails: [email protected]; [email protected]).

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a unified framework which designs low complexity transmission schemes applicable to massive MIMO systems in upcoming 5G wireless networks is provided in the first time. Moreover, adaptive transmission designs which switch among these criteria based on the channel conditions to formulate the best transmission strategy are discussed. Then, we provide a survey of the transmission designs with discrete input signals for multiuser MIMO scenarios, including MIMO uplink transmission, MIMO downlink transmission, MIMO interference channel, and MIMO wiretap channel. Additionally, we discuss the transmission designs with discrete input signals for other systems using MIMO technology. Finally, technical challenges which remain unresolved at the time of writing are summarized and the future trends of transmission designs with discrete input signals are addressed.

G LOSSARY 3GPP

3rd Generation Partnership Project

5G

fifth Generation

ADC

analog-to-digital converter

AMC

adaptive modulation and coding

AF

amplify-and-forward

BER

bit error rate

BICM

bit-interleaved coded modulation

BOSTBC

block-orthogonal property of space time block coding

BC

broadcast channel

CQI

Channel Quality Indicator

CSI

channel state information

CSIT

channel state information at the transmitter

CDMA

code-division multiple access systems

DFE

decision-feedback equalization

DF

decode-and-forward

EPI

entropy power inequality

FER

frame error rate

2

GSVD

generalized singular value decomposition

H-ARQ

hybrid automatic repeat request

IPCSIR

imperfect channel state information at the receiver

IPCSIT

imperfect channel state information at the transmitter

KKT

Karush-Kuhn-Tucker

LCFE

linear complex-field encoder

LOS

line-of-sight

LTE

long term evolution

LDPC

low density parity check codes

ML

maximum likelihood

MRC

maximum ratio combining

MRT

maximum ratio transmission

MSE

mean squared error

MMSE

minimum mean squared error

MB

multichannel-beamforming

MAC

multiple access channel

MIDO

multiple-input double-output

MIMO

multiple-input multiple-output

MIMOME

multiple input multiple output multiple-antenna eavesdropper

MISO

multiple input single output

MISOSE

multiple input single output single-antenna eavesdropper

NVD

non-vanishing determinants

OFDM

orthogonal frequency division multiplexing

OFDMA

orthogonal frequency division multiplexing access

OSTBC

orthogonal space-time block code

PER

packet error rate

3

PEP

pairwise error probability

PGP

Per-Group Precoding

PCSIT

perfect channel state information at the transmitter

PCSIR

perfect channel state information at the receiver

PSK

phase-shift keying

pdf

probability distribution function

PAM

pulse amplitude modulation

QAM

quadrature amplitude modulation

QOSTBC

quasi-orthogonal space-time block code

SNR

signal-to-noise ratio

SISO

single-input single-output

SVD

singular value decomposition

STBC

space-time block code

ST-LCFE

space time linear complex-field encoder

ST-LCP

space time linear constellation precoding

STTC

space time trellis codes

SDMA

spatial division multiple access

SCSI

statistical channel state information

SCSIT

statistical channel state information at the transmitter

SCSIR

statistical channel state information at the receiver

SER

symbol error rate

TAST

threaded algebraic space time

THP

Tomlinson-Harashima precoding

V-BLAST

vertical bell laboratories layer space-time

WSR

weighted sum-rate

ZF

zero-forcing

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I. I NTRODUCTION The emergence of numerous smart mobile devices such as smart phones and wireless modems has led to an exponentially increasing demand for wireless data services. Accordingly, a large amount of effort has been invested on improving the spectral efficiency and data throughput of wireless communication systems. In particular, multiple-input multiple-output (MIMO) technology has been shown to be a promising means to provide multiplexing gains and/or diversity gains, leading to improved performance. By increasing the number of antennas at the base stations and mobile terminals, communication systems can achieve better performance in terms of both system capacity and/or link reliability [1]–[3]. So far, MIMO technology has been adopted by third generation partnership project (3GPP) long-term evolution (LTE), IEEE 802.16e, and other wireless communication standards.

A. Future Wireless Communication Networks The unprecedented growth in the number of mobile data and connected machines ever-fast approaches limits of 4G technologies to address this enormous data demand. For example, the mobile data traffic is expected to grow to 24.3 Exabytes per month by 2019 [4], while the number of connected Internet of Things (IoT) is estimated to reach 50 Billion by 2020 [5]. Also, emerging new services such as UltraHigh-Definition multimedia streaming and cloud computing, storage, and retrieval require the higher cell capacity/end-user data rate and extremely low latency, respectively. It will become evident soon that current 4G technologies will be stretched to near breaking points and still can not meet the quality of experience necessary for these emerging demands. Therefore, the development of the fifth generation (5G) wireless communication technologies is a priority issue currently for deploying future wireless communication networks [6], [7]. As illustrated in Figure 1, the fundamental requirements for building 5G wireless networks are clear. From system performance point of view, 5G wireless networks must support massive capacity (tens of Gbps peak data rate) and massive connectivity (1 million /Km2 connection density and tens of Tbps/Km2 traffic volume density), extremely diverging requirements for different user services (user experienced data rate from 0.1 to 1 Gbps), high mobility (user mobility up to 500 Km/h) and low latency communication

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(1 ms level end to end latency). From transmission efficiency point of view, 5G wireless networks must simultaneously realize spectrum efficiency, energy efficiency, and cost efficiency. The evolution towards 5G wireless communication will be a cornerstone for realizing the future human-centric and connected machine-centric wireless networks, which achieve near-instantaneous, zero distance connectivity for people and connected machines. To achieve these requirements, the 5G air interface needs to incorporate three basic technologies: massive MIMO, millimeter wave, and small cell [8]. For millimeter wave transmission, the millimeter wave beamforming formulated by large antenna arrays, including both analog beamforming and digital beamforming, is required to combat the higher propagation losses [9]. For small cell, cell shrinking also needs the deployment of large and conformal antenna arrays whose cost are inversely with the infrastructure density [10]. Therefore, MIMO technology will continue to play an important role in upcoming 5G and

5G Capabilities: The 5G Flower future Key wireless networks.

Performance Requirements

Efficiency Requirements

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14 IMT-2020(5G) PROMOTION GROUP. ALL RIGHTS RESERVED.

Fig. 1:

Fundamental requirements of 5G: The 5G flower in [11].

B. MIMO Transmission Design While the benefits of MIMO can be realized when only the receiver has the communication channel knowledge, the performance gains can be further improved if the transmitter also obtains the channel

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knowledge and performs certain processing operations on the signal before transmission. For example, MIMO transmission design can achieve double capacity at -5dB signal-to-noise ratio (SNR) and 1.5 b/s/Hz additional capacity at 5 dB SNR for a four-input two-output system over independent identically distributed (i.i.d.) Rayleigh flat-fading channels [12]. These are normal SNR ranges for practical applications such as LTE, WiFi, and WiMax. The gains of MIMO transmission design in correlated fading channels are even more obvious [13], [14]. For the massive MIMO technology in future wireless communication networks, it is shown in [15] that with perfect channel knowledge, linear precoding at the transmitters and simple signal detection at the receivers can approach the optimum performance when the number of the antennas at the base station tends to infinite. A two step joint spatial division and multiplexing transmission scheme is further proposed to achieve massive MIMO gains with reduced channel state information at the transmitter (CSIT) feedback [16]. Therefore, based on some forms of CSIT, MIMO transmission design has a great practical interest in wireless communication systems. The typical MIMO transmission designs can be mainly classified into three categories [17]: (i) designs based on mutual information; (ii) designs based on mean squared error (MSE); (iii) diversity driven designs. The first category adopts mutual information criterions from information theory such as ergodic or outage capacity for the transmission optimization. The second category performs the transmission design by optimizing the MSE related objective functions subject to various system constraints; The third category maximizes the diversity of the communication system based on pairwise error probability (PEP) analysis.

C. MIMO Transmission Design with Discrete Input Signals From information theory point of view, the Gaussian distributed transmit signal achieves the fundamental limit of MIMO communication [1]. However, the Gaussian transmit signal is rarely used in practical communication systems. This is mainly because of two reasons: (i) The amplitude of Gaussian transmit signal is unbounded, which may result in a substantial high power transmit signal at the transmitter; (ii) The probability distribution function (pdf) of Gaussian transmit signal is a continuous function, which will significantly increase the detection complexity at the receiver. Therefore, the practical transmit signals

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are non-Gaussian input signals taken from finite discrete constellations, such as phase-shift keying (PSK), pulse amplitude modulation (PAM), and/or quadrature amplitude modulation (QAM). Figure 2 compares the pdf of the standard Gaussian distributed signal and the probability mass function (pmf) of the QPSK signal. From Figure 2, we observe that the QPSK signal is significantly different from the Gaussian signal. Therefore, the communication based on discrete input signals will normally be quite different from that of the Gaussian signal and needs specific transmission designs [17]–[21]. The main difficulties of MIMO transmission design with discrete input signals are threefold: (i) The mutual information and the minimum mean squared error (MMSE) of MIMO channels with finite discrete constellation signals usually lack explicit expressions. As a result, the elegant water-filling type solutions for MIMO channels with Gaussian input signal does not exist anymore. Sophisticated numerical algorithms should be designed to search for both the power allocation matrix and the eigenvector matrix. (ii) The complexity of MIMO transmission design with finite discrete constellation signals normally grows exponentially with the number of antenna dimension and the number of the users, which often makes the implementation of the design prohibitive. (iii) Depending on different channel conditions, adaptive transmission is needed in practical systems which determine the best transmission policy among various criterions such as mutual information, MSE, diversity, etc. This process is challenging since it involves a selection of a large number of parameters.

D. The Contributions of This Paper In this paper, a comprehensive review of MIMO transmission design with discrete input signals is presented and summarized in the first time for various design criterions. We first introduce the fundamental research results for discrete input signals, e.g., the mutual information and the MMSE relationships [22]– [24], the properties of the MMSE function [25], [26], the definition of MMSE dimension [27], etc. Then, focusing on the basic point-to-point MIMO systems, we provide in details the transmission designs with discrete input signals based on the criterions of mutual information (e.g., [17], [28]–[31]), MSE (e.g., [32]– [34]), and diversity (e.g., [35]–[40]), respectively. In particular, a unified framework for the low complexity designs which can be implemented in large-scale MIMO antenna arrays for 5G wireless networks is

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Fig. 2:

The pdf of the Gaussian signal and the pmf of the QPSK signal.

provided in the first time. In light of this, the practical adaptive transmission schemes which switch among these criterions corresponding to the channel variations are summarized for both uncoded systems (e.g., [41]–[44]) and coded systems (e.g., [45]–[48]), respectively. The specific adaptive transmission schemes defined in 3GPP LTE, IEEE 802.16e, and IEEE 802.11n standards are further introduced. Moving towards multiuser MIMO systems, we introduce the designs based on the above-mentioned criterions for multiuser MIMO uplink transmission (e.g., [20], [49]–[51]), multiuser MIMO downlink transmission (e.g., [21], [52]–[54]), MIMO interference channel (e.g., [55]–[58]), and MIMO wiretap channel (e.g., [59]–[61]). Finally, we overview transmission designs with discrete input signals for other systems employing MIMO technology such as MIMO cognitive radio systems (e.g, [62]–[64]), green MIMO communication systems (e.g., [65], [66]), and MIMO relay systems (e.g, [67]–[70]). The overall outline of this paper is shown in Figure 3.

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Background Introduction (Sec. I)

Key Model MIMO systems

The importance The chanellege Core Objective First comprehensive review for MIMO transmission with discrete input signals

Design Criterion Mutual information MSE Diversity

Fundamental Results (Sec. II)

Basic relationships

Mutual information driven design

MSE driven Design Point-to-point MIMO (Sec. III) Diversity driven design

Adaptive transmission

Uplink transmission Organization

Downlink transmission Multiuser MIMO (Sec. IV) MIMO interference channel

MIMO wiretap channel

MIMO cognitive radio

Green MIMO communication Other MIMO Systems (Sec. V) MIMO relay

Others Conclusion (Sec. VI)

Fig. 3:

The overall outline of this paper

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II. F UNDAMENTAL R ESULTS FOR D ISCRETE I NPUT S IGNALS In this section, we briefly review the existing fundamental results for discrete input signals. To design transmission schemes for MIMO systems with discrete input signals has to rely on these fundamental results.

A. Fundamental Link between Information Theory and Estimation Theory The relationship between information theory and estimation theory is a classic topic, which dates back to 50 years ago for a relationship between the differential entropy of the Fisher information and the mean square score function in [71]. This relationship is called De Bruijn’s identity. The representation of mutual information as a function of causal filtering error was given in [72] as Duncan’s theorem. Other early connections between fundamental quantities in information theory and estimation theory can be found in [73]–[77] and reference therein. 1) Gaussian channel model: Recently, D. Guo et al. establish a fundamental relationship between the mutual information and the MMSE [22], which applies for discrete-time and continuous-time, scalar and vector channels with additive Gaussian noise. In particular, for the real-value linear vector Gaussian √ channel y = snrHx+n, D. Guo et al. find the following relationship regardless of the input distribution x 1 √ d I x; snrHx + n = mmse(snr) dsnr 2

(1)

where the mutual information is in nats and h √ 2 i mmse(snr) = E Hx − E Hx snrHx + n .

(2)

The right-hand side of Eq. (2) denotes the expectation of squared Euclidean norm of the error corresponding to the best estimation of Hx upon the observation y for a given SNR. Eq. (1) is proved based on an incremental-channel method in [22]. The key idea of this method is to model the change in the mutual information as the input-output mutual information of an equivalent Gaussian channel with vanishing SNR. The mutual information of this equivalent Gaussian channel is

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essentially linear with the estimation error. Therefore, it is possible to establish a relationship between the incremental of the mutual information and the MMSE. For the continuous-time real-value scalar Gaussian channel Rt =

√ snrXt + Nt , t ∈ [0, T ], where Xt

is the input process and Nt is a white Gaussian noise with a flat double-sided power spectrum density of unit height, D. Guo et al. establish another fundamental relationship between the causal MMSE and the noncausal MMSE with any input process 1 cmmse (snr) = snr

Z

snr

mmse (γ) dγ

(3)

0

where T

1 cmmse (snr) = T

Z

cmmse (t, snr) = E

h

t 2 i Xt − E Xt Y0 ; snr

1 T

Z

T

mmse (t, snr) = E

h

mmse (snr) =

cmmse (t, snr) dt

(4)

0

mmse (t, snr) dt

(5) (6)

0

2 i . Xt − E Xt Y0T ; snr

(7)

Eq. (3) is proved in [22] by comparing Eq. (1) to Duncan’s Theorem. The fundamental relationships in Eq. (1) and (3) and their generalizations in [22], are referred to the I-MMSE relationships. These I-MMSE relationships can be applied to provide a simple proof of Shannon’s entropy power inequality (EPI) [78]. The main idea is to use [22, Eq. (182)] to transform the EPI into the equivalent MMSE inequality, which can be proved much more easily. Similar approaches have been applied to prove the Costa’s EPI and the generalized EPI in [79] and a new inequality for independent random variables in [80]. I-MMSE relationships in [22] can be generalized. For the complex-value linear vector Gaussian channel with arbitrary signaling, D. P. Palomar et al. derive the gradient of the mutual information with respect to the channel matrix and other arbitrary parameters of the system through the chain rule [23]. It is shown in [23] that the partial derivative of the mutual information with respect to the channel matrix is equal to the product of the channel matrix and the error covariance matrix of the optimal estimation of the input given the output. Moreover, the Hessian of the mutual information and the entropy with respect to various parameters of the system are derived in [24]. In a special case where the left singular vectors of

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the precoder matrix coincide with the eigenvectors of the channel correlation matrix, it is proved in [24] that the mutual information is a concave function of the squared singular values of the precoder matrix. Another generalization direction is to extend the I-MMSE relationships to the abstract Wiener space for the scalar Gaussian channel in [81] and the vector Gaussian channel in [82]. D. Guo et al. investigate the properties of the MMSE as a function of the SNR and the input distribution for a real-value scalar Gaussian channel in [25]. It is shown in [25] that the MMSE is concave in the input distribution for a given SNR and infinitely differentiable for all SNRs for a given input distribution. An important conclusion in [25] is that the MMSE curve of a Gaussian input and the MMSE curve of a non-Gaussian input intersect at most once throughout the entire SNR regime. The MMSE properties are extended to a real-value diagonal MIMO Gaussian channel in [83], where each eigenvalue of the MMSE matrix with a Gaussian input is proved to have at most a single crossing point with that of the MMSE matrix with a non-Gaussian input for all SNR values. More results for the mutual information and the MMSE properties for the scalar Gaussian channel are obtained. Based on the Lipschitz continuity of the MMSE and I-MMSE relationship, Y. Wu et al. prove that the mutual information with input cardinality constraints converges to the usual Gaussian channel capacity as the constellation cardinality tends to infinity [84]. Y. Wu et al. further provide a family of input constellation generated from the roots of the Hermite polynomials, which achieves this convergence exponentially [85]. N. Merhav et al. introduce techniques in statistical physics to evaluate the mutual information and the MMSE in [86]. N. Merhav et al. establish several important relationships between the mutual infomration, the MMSE, the differential entropy, and the statistical-mechanical variables. Some application examples in [86] shows how to use the statistical physics tools to provide useful analysis for the MMSE. K. Venkat et al. investigate the statistic distribution of the difference between the input-output information density and half the causal estimation error [87]. G. Han et al. propose a new proof of the I-MMSE relationship by choosing a probability space independent of SNR to evaluate the mutual information [88]. Based on this approach, G. Han et al. extend the I-MMSE relationship to both discrete and continuous-time Gaussian channels with feedback in [88].

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A random variable with distribution P is at the receiver. The receiver performs the MSE estimation of this random variable under the assumption that its distribution is Q. This is referred to the mismatch estimation. S. Verdú initializes a new connection between the relative entropy and the mismatched estimation for a real-value scalar Gaussian channel [89]. The key relationship revealed in [89] is as follows: 1 D (P k Q) = 2

Z

∞

mseQ (γ) − mseP (γ) dγ

(8)

0

where D (P k Q) is the relative entropy. mseQ (γ) and mseP (γ) are MMSEs obtained by an estimator which assumes the input signal is distributed according to Q and P for a real-value scalar Gaussian channel, respectively. Eq. (8) is a generalization of Eq. (1) for the scalar Gaussian channel with a mismatch estimation. Similarly, using the mutual information and the relative entropy as a bridge, T. Weissman generalizes Eq. (3) into the case of mismatch estimation in [90]. The statistic properties of the difference between the mismatched and matched estimation losses are studied in [87]. The relationship between relative entropy and mismatched estimation is extended to the real-value vector Gaussian channel in [91]. For a more complicated case of mismatched estimation where the channel distribution is mismatched at the estimator instead of the input signal distribution, W. Huleihel et al. derive the mismatched MSE for the vector Gaussian channel using the statistical physics tools [92]. A brief summation of recent research results of the I-MMSE relationships for Gaussian channel is given in Table II. 2) Other Channel Models: The I-MMSE relationships are investigated in other channel models. For the scalar Poisson channel, D. Guo et al. prove that the derivative of the input-output mutual information with respect to the intensity of the additive dark current equals to the expected error between the logarithm of the actual input and the logarithmic of its conditional mean estimate (noncausal in case of continuoustime) [93]. The similar relationship holds for the derivative of the mutual information with respect to input-scaling, by replacing the logarithmic function with x log x [93]. Moreover, R. Atar et al. prove that the I-MMSE relationships hold for Gaussian channel [22], [89], [90], including discrete-time and continuous-time channel, matched and mismatched filters, also hold for scalar Poisson channel by replacing the squared error loss with the corresponding loss function [94]. L. Wang et al. derive the gradient of

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TABLE II: Recent research results of the I-MMSE relationships for Gaussian channel Paper

Model

Main Contribution

Real-value MIMO

Reveal a fundamental relationship between mutual information and MMSE

Discrete-time, Continuous-time

Reveal a fundamental relationship between causal and noncausal MMSEs

Complex-value MIMO

Derive the gradient of the mutual information

Discrete-time

with arbitrary parameters of the systems

Complex-value MIMO

Derive the Hessian of the mutual inforamtion and entropy

Discrete-time

with arbitrary parameters of the systems

M. Zakai [81]

Real-value scalar

Extend I-MMSE relationship in [22] to the abstract Wiener space (scalar version)

E. M.-Wolf et al. [82]

Real-value MIMO

Extend I-MMSE relationship in [22] to the abstract Wiener space (vector version)

Real-value MIMO

Obtain various properties of MMSE

Discrete-time

with respect to SNR and input distribution

D. Guo et al. [22]

D. P. Palomar et al. [23]

M. Payaró et al. [24]

D. Guo et al. [25] Real-value diagonal MIMO R. Bustin et al. [83]

Obtain properties of the MMSE matrix Discrete-time Real-value scalar

Prove the mutual information with input cardinality constraints converges to

Discrete-time

the Gaussian channel capacity as the constellation cardinality tends to infinity

Y. Wu et al. [84] Real-value scalar Y. Wu et al. [85]

Prove the convergence speed in [84] is exponential Discrete-time Real-value scalar

Introduce techniques in statistical physics to

Discrete-time

evaluate the mutual information and the MMSE

N. Merhav et al. [86] Real-value scalar, mismatched case Derive the statistical properties of difference between the K. Venkat et al. [87]

with/without feedback input-output information density and half the causal estimation error Continuous-time, Discrete-time Real-value scalar, feedback

G. Han et al. [88]

Extend the I-MMSE relationship to channels with feedback Discrete-time, Continuous-time Real-value scalar

Initializes a new connection between the relative entropy

Discrete-time, mismatched case

and the mismatched estimation

Real-value scalar

Establish a relationship between causal MMSE

Continuous-time, mismatched case

and non-causal MMSE in mismatched case

S. Verdú et al. [89]

T. Weissman [90] Real-value MIMO M. Chen et al. [91]

Extend the relationship in [89] to MIMO Discrete-time, mismatched case Real-value MIMO

Derive the MSE when the mismatched estimation

Discrete-time, mismatched case

occurs for the channel distribution

W. Huleihel et al. [92]

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mutual information with respect to the channel matrix for vector Poisson channel [95]. In addition, L. Wang et al. define a Bregman matrix based on the Bergman divergence in [95] to formulate a unified framework for the gradient of mutual information for both vector Gaussian and Poisson channels. The Bergman divergence is also used to characterize the derivative of the relative entropy and the mutual information for the scalar binomial and negative binomial channels in [96]. The I-MMSE relationship for a scalar channel with exponential-distributed additive noise is established in [97]. For continuous-time scalar channel with additive Gaussian/Possion noise in the presence of feedback, the relationships between the directed information and the causal estimation error are established in [98]. Some work consider more general channel models. A scalar channel with arbitrary additive noise is studied in [99]. It is found that the increase in the mutual information due to the improvement in the channel quality equals to the correlation of two conditional mean estimates associated with the input and the noise respectively. D. P. Palomar et al. represent the derivative of mutual information in terms of the conditional input estimation given the output for general channels (not necessarily additive noise) [100]. The general results in [100] embrace most of popular channel models including the binary symmetric channel, the binary erasure channel, the discrete memoryless channel, the scalar/vector Gaussian channel, an arbitrary additive-noise channel, and the Poisson channel. For a scalar channel with arbitrary additive noise, D. Guo obtain the derivative of relative entropy with respect to the energy of perturbation, which can be expressed as a mean squared difference of the score functions of the two distributions [101]. Furthermore, Y. Wu et al. analyze the properties of the MMSE as a function of the input-output joint distribution [26]. It is proved in [26] that the MMSE is a concave function of the input-output joint distribution. N. Merhav analyze the MMSE for both cases of matched and mismatched estimations by using the statistical-mechanical techniques [102]. The main advantage of the results in [102] is that they apply for very general vector channels with arbitrary joint input-output probability distributions. By exploiting the statistical-mechanical techniques, N. Merhav also establish relationships between rate-distortion function and the MMSE for a general scalar channel [103]. Then, simple lower and upper bounds on the rate-distortion function can be obtained based on the bounds of the MMSE. A brief summation of recent research results of the I-MMSE

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TABLE III: Recent research results of the I-MMSE relationships for non-Gaussian channels Paper

Model

Main Contribution

Scalar Poisson

Obtain the derivative of the mutual information with respect to the additive dark current

Discrete-time, Continuous-time

Obtain the derivative of the mutual information with respect to the input-scaling

D. Guo et al. [93] Scalar Poisson, mismatched case R. Atar et al. [94]

Obtain I-MMSE relationship Discrete-time, Continuous-time MIMO Poisson

Obtain gradient of mutual information with

Discrete-time

respect to the channel matrix

Scalar binomial and negative binomial

Obtain derivative of the relative entropy and the mutual information

Discrete-time

with respect to scaling factor

L. Wang et al. [95]

C. G. Taborda et al. [96] Scalar, exponential noise M. Raginsky et al. [97]

Obtain the I-MMSE relationship Discrete-time Scalar Gaussian and Poisson

Obtain the relationships between the directed information

Continuous-time with feedback

and the causal estimation error channels

T. Weissman et al. [98] Scalar, arbitrary additive noise D. Guo et al. [99]

Obtain the I-MMSE relationship Discrete-time General channel

Obtain the derivative of mutual information in terms of

Discrete-time

the conditional input estimation given the output

Scalar, arbitrary additive noise

Obtain derivative of relative entropy with respect

Discrete-time

to the energy of perturbation the output

Scalar, arbitrary additive noise

Analyze the properties of the MMSE as a function

Discrete-time

of the input-output joint distribution


D. Guo et al. [101]

Y. Wu et al. [26] MIMO, general channel N. Merhavet al. [102]

Analyze the MMSE Discrete-time, mismatched case Scalar, general channel

Establish relationships between rate-distortion

Discrete-time

function and the MMSE

N. Merhavet al. [103]

relationships for non-Gaussian channels is given in Table III.

B. Other Analytical Results with Discrete Input Signals 1) Asymptotic Results in Low and High SNR Regime: Shannon’s pioneering work [104] have shown that the binary antipodal inputs are as good as the Gaussian input in low SNR regime. Recently, S. Verdú investigate the spectral efficiency in low SNR regime for a general class of signal inputs and channels [105]. In [105], S. Verdú provides two important criterions to characterize the spectral efficiency in low SNR regime: the minimum energy per information bit required for reliable communication and the wideband slope of the spectral efficiency. S. Verdú further define that any input signals which achieve the

17

same minimum energy per information bit achieved by the Gaussian input are first-order optimal. If the perfect channel knowledge is available at the receiver, both BPSK and QPSK are first-order optimal for a general MIMO channel. Also, S. Verdú define that any first-order optimal input signals which achieves the wideband slope achieved by Gaussian input are second-order optimal. For a MIMO Gaussian channel with perfect channel knowledge at the receiver, it is proved that equal-power QPSK is second-order optimal. Moreover, V. V. Prelov et al. provide the second order expansion of the mutual information in low SNR regime for very general classes of inputs and channel distributions [106]. For a parallel Gaussian channels with m-ary constellation inputs, A. Lozano et al. provide both lower and upper bounds of the MMSE in high SNR regime [18]. For an arbitrary MIMO Gaussian channel, F. Pérez-Cruz et al. further derive high SNR lower and upper bounds of the MMSE and the mutual information [19]. For a scalar channel with arbitrary inputs and additive noise, Y. Wu et al. define the MMSE dimension when SNR tends to be infinity in [27]. The MMSE dimension represents the gain of the non-linear estimation error over the linear estimation error in high SNR regime. For MIMO Gaussian block Rayleigh fading channels with m-ary constellation inputs, it is proved that the constellation with a higher minimum Euclidean distance achieves a higher mutual information in high SNR regime [107, Appendix E]. For a scalar Gaussian channel with arbitrary input distributions independent of SNR, the exact high SNR expansion of the mutual information is obtained in [108] in the first time. The obtained asymptotic expression validates the optimality of the Gray code for bit-interleaved coded modulation (BICM) in high SNR regime. By using Mellin transform, A. C. P. Ramos et al. derive low and high SNR asymptotic expansions of the ergodic mutual information and the average MMSE for a scalar Gaussian channels with arbitrary discrete inputs over Rician and Nakagami fading [109]. High SNR expansions are derived for a MIMO Gaussian channel over Kronecker fading with the line-of-sight (LOS) [110]. A brief summation of the recent analytic results for discrete input signals in asymptotic SNR regime is given in Table IV. 2) Asymptotic Results in Large System Limits: Some work provide the analytic results for discrete input signals in large system limit. A technique which has been widely used in statistical physics, referred to

18

TABLE IV: Recent analytic results for discrete input signals in asymptotic SNR regime Paper

Model

Main results in asymptotic SNR regime

MIMO, low SNR

Analyze the minimum energy per information bit

A general class of channels

Analyze the wideband slope

S. Verdú [105] MIMO, low SNR V. V. Prelov et al. [106]

Obtain the second order expansion of the mutual information A general class of channels Parallel MIMO Gaussian channels

A. Lozano et al. [18]

Provide lower and upper bounds of the MMSE high SNR MIMO Gaussian channels

Provide lower and upper bounds of

high SNR

the MMSE and the mutual information

F. P.-Cruz et al. [19] Scalar, arbitrary additive noise Y. Wu et al. [27]

Define the MMSE dimension high SNR MIMO Gaussian channels

Prove the mutual information is a monotonously

block Rayleigh fading, high SNR

increasing function of the minimum Euclidean distance

Scalar Gaussian channels

Obtain the exact high SNR expansion

high SNR

of the mutual information

Scalar Gaussian channels, Rician fading

Obtain asymptotic expansions of the

Nakagami fading, low SNR, high SNR

ergodic mutual information and the average MMSE

MIMO Gaussian channels, high SNR

Obtain asymptotic expansions of

Kronecker fading with LOS

ergodic mutual information and the average MMSE

D. Duyck et al. [107]

A. Alvarado et al. [108]

A. C. P. Ramos et al. [109]

M. R. D. Rodrigues [110]

replica method, is initially exploited to analyze the asymptotic performance of various detection schemes of code-division multiple access systems (CDMA) in large dimension [111]–[114]. By using this replica method, R. R. Müller derive an asymptotic mutual information expression for a spatially correlated MIMO channel with BPSK inputs in the first time [115]. This expression is extended to arbitrary input at the transmitter in [116]. C.-K. Wen et al. obtain the asymptotic sum rate expression for the Kronecker fading channels for the centralized MIMO multiple access channel (MAC) and the distributed MIMO MAC in [117] and [118], respectively. The asymptotic sum-rate expression including the LOS effect for the centralized MIMO MAC is derived in [119]. C.-K. Wen et al. further obtain the asymptotic sum-rate expression for the Kronecker fading channels for the centralized MIMO MAC with amplify-and-forward (AF) relay [120]. Also, M. A. Girnyk et al. derive an asymptotic mutual information expression for i.i.d.

19

Rayleigh fading for K-hop AF relay MIMO channels [121]. For other performance criterions, R. R. Müller et al. obtain an asymptotic expression for the average minimum transmit power for i.i.d. Rayleigh fading MIMO channels with a non-linear vector precoding [122]. Similar problem is also considered for MIMO broadcast channel (BC) in [123]. A more general analysis, referred to one step replica symmetry breaking analysis, is used in [123] to reduce the asymptotic approximation errors in [122] for non-convex alphabet sets. The effect of the transmit-side noise caused by hardware impairments is investigated in [124] for point-to-point MIMO systems with i.i.d. discrete input signals and i.i.d. MIMO Rayleigh fading channels. It is assumed that the receiver has perfect channel state information (CSI) but the transmitter does not know the CSI. Also, the transmit-side noise is generally unknown at both transmitter and the receiver. As a result, the receiver employs a mismatched decoding without taking the consideration of the transmit-side noise. The general mutual information in [125] is adopted as the metric to evaluate the performance. An asymptotic expression of the general mutual information is derived when both the numbers of transmit and receive antennas go to infinity with a fixed ratio. Numerical results indicate that the derived asymptotic expression provides an accurate approximation for the exact general mutual information and the effects of transmit-side noise and mismatched decoding become significant only at high modulation orders. A brief summation of recent research results of the analytic results (transmitter side) for discrete input signals in large system limit is given in Table V. 3) Other Results: There are some other analytical research results for discrete input signals. For a scalar Gaussian channel with a finite size input signal constraint, N. Varnica et al. propose a modified Arimoto-Blahut algorithm to search for the optimal input distribution which maximizes the information rate [126]. The corresponding maximum information rate is referred to constellation constrained capacity. J. Bellorado et al. study the constellation constrained capacity of MIMO Rayleigh fading channel. It is shown in [127] that for a given information rate, the SNR gap between the uniformly distributed signal and the constellation constrained capacity when the constellation size tends to infinity is 1.53 dB. This is the same with the QAM asymptotic shaping gap for the scalar Gaussian channel in [128]. Also, it is conjectured in [127] that when the elements of input signal vector are chosen from a Cartesian product of one dimension

20

TABLE V: Analytic results (transmitter side) for discrete input signals in large system limit Paper

Channel

Criterion

R. R. Müller [115]

Point-to-point MIMO, spatially correlated Rayleigh

Asymptotic mutual information (BPSK)

C.-K. Wen et al. [116]

Point-to-point MIMO, Kronecker fading

Asymptotic mutual information


Distributed MIMO MAC, Kronecker fading

Asymptotic sum-rate


Centralized MIMO MAC, Kronecker fading

Asymptotic sum-rate


Centralized MIMO MAC, Kronecker fading with LOS

Asymptotic sum-rate


Centralized MIMO MAC with AF relay, Kronecker fading

Asymptotic sum-rate

M. A. Girnyk et al. [121]

K-hop AF relay MIMO, i.i.d. Rayleigh fading

Asymptotic mutual information

R. R. Müller et al. [122]

Point-to-point MIMO, i.i.d. Rayleigh fading, vector precoding

Asymptotic average minimum transmit power

B. M. Zaidel et al. [123]

MIMO BC, i.i.d. Rayleigh fading, vector precoding (1RSB ansatz)

Asymptotic average minimum transmit power

Point-to-point MIMO, i.i.d. Rayleigh fading, i.i.d. input signals M. Vehkaperä et al. [124]

General mutual information Transmit-side noise and mismatched decoding

PAM constellations, the constellation constrained capacity can be achieved by optimizing the distribution of each element of the signal vector independently. This reduces the computational complexity significantly. This conjecture is proved in [129]. M. P. Yankov et al. derive a low-computational complexity low bound of the constellation constrained capacity for MIMO Gaussian channel based on QR decomposition [130]. For one dimension channel with memory and finite input state, D. M. Arnold et al. propose a practical algorithm to compute the information rate [131]. For two dimension channel, tight low and upper bounds of the information rate are derived in [132]. O. Shental et al. propose a more accurate computation method for the information rate of two dimension channel with memory and finite input state based on generalized belief propagation [133]. III. P OINT- TO -P OINT MIMO S YSTEM In this section, we focus on the transmission designs for point-to-point MIMO systems with discrete input signals. We classify these designs into four categories: i) design based on mutual information; ii) design based on MSE; iii) diversity driven design; iv) adaptive transmission. A. Design Based on Mutual Information The mutual information maximization yields the maximal achievable rate of the communication link with the implementation of reliable error control codes [134]. For point-to-point MIMO system, the

21

maximization of the mutual information has been obtained in [1]. It is proved in [1] that the mutual information is maximized by a Gaussian distributed signal with a covariance structure satisfying two conditions. First, the signal must be transmitted along the right eigenvectors of the channel, which decomposes the original channel into a set of parallel subchannels. Second, the power allocated for each subchannel is based on a water-filling strategy, which depends on the strength of each subchannel and the SNR. Although the Gaussian signal is capacity-achieving, it is too idealistic to be implemented in practical communication systems. The discrete input signals are taken from a finite discrete constellation set, such as PSK, PAM, or QAM. Therefore, some work begin to investigate MIMO transmission designs based on the discrete constellation input mutual information. 1) Mutual Information Maximization: By exploiting the I-MMSE relationship in [22], A. Lozano et al. propose a mercury water-filling power allocation policy which maximizes the input-output mutual information for diagonal MIMO Gaussian channels with arbitrary inputs [18]. The essential difference between the mercury water-filling policy and the conventional water-filling policy is that it will pour mercury onto each of the subchannel to a certain height before filling the water (allocated power) to the given threshold. Because of the poured mercury, the classical conclusion for the Gaussian input signal that, the stronger subchannel receives more filling water (allocated power) does not hold any more. The asymptotic expansions of the MMSE function and the mutual information in low and high SNR regime are obtained in [18]. Based on these expansions, it is proved that allocating power only to the strongest channel is asymptotically optimal in low SNR regime, which is the same with the conventional water-filling allocation. For the optimal power allocation policy in high SNR regime, in contrast, the less power should be allocated to the stronger subchannel for equal constellation inputs. This is significant different from the conventional water-filling allocation. Also, it is revealed that the more power should be allocated to the richer constellation inputs for equal channel strength at high SNR. For the non-diagonal MIMO Gaussian channel with arbitrary inputs, an iterative numerical to search for the optimal linear precoder based on a gradient decent update of the precoder matrix is proposed in [23]. Moreover, necessary conditions

22

for the optimal linear precoding matrix maximizing the mutual information of the non-diagonal MIMO Gaussian channel with finite alphabet inputs are presented in [135], from which a numerical algorithm is formulated to search for the optimal precoder. Simulations indicate that the obtained non-diagonal and non-unitary precoder in [135] outperforms the mercury water-filling design in non-diagonal MIMO Gaussian channel. The design in [135] has been extended to the case where only statistical CSI (SCSI) is available at the transmitter [136]. In the mean time, F. Pérez-Cruz et al. also establish necessary conditions that the optimal precoder should satisfy for the non-diagonal MIMO Gaussian channel with finite alphabet inputs through Karush-Kuhn-Tucker (KKT) analysis [19]. The MMSE function and the mutual information of the non-diagonal MIMO Gaussian channel with finite alphabet inputs are expanded in asymptotic low and high SNR regime. It is proved that in low SNR regime, the optimal precoder design for finite alphabet inputs is exactly the same with the optimal design for Gaussian input. In high SNR regime, low and upper bounds of the mutual information are derived. Base on the bounds, it is shown that the precoder that maximizes the mutual information is equivalent to the precoder that maximizes the minimum Euclidean distance of the receive signal points in high SNR regime. This result indicates that the three important criterions in MIMO transmission design: maximizing the mutual information, minimizing the symbol error rate (SER), and minimizing the MSE are equivalent in high SNR regime. The optimal power allocation scheme for the general MIMO Gaussian channel is also obtained in [19]. For the real-valued non-diagonal MIMO Gaussian channels, M. Payaró et al. decompose the design of the optimal precoder into three components: the left singular vectors, the singular values, and the right singular vectors [137]. The optimal left singular vectors are proved to be the right singular vectors of the channel matrix. Necessary and sufficient conditions of the optimal singular values are established, from which the optimal singular values can be found based on the standard numerical algorithm such as Newton method. The optimal solution of the right singular vectors is a difficult problem. Based on standard numerical algorithm, a local optimum design can be found. Moreover, the optimal designs in low and high SNR regime are analyzed in [137]. For the complex-valued non-diagonal MIMO Gaussian channels, M. Lamarca proves that the optimal left singular vectors of the precoder is still the right singular

23

vectors of the channel matrix [138]. More importantly, when the entries of the channel matrix and the input signals are all real-valued, M. Lamarca proves that the mutual information is a concave function of a quadratic function of the precoder matrix in [138]. Then, an iterative algorithm is proposed to maximize the mutual information by updating this quadratic function along the gradient decent direction. For the complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs, C. Xiao et al. prove that the mutual information is still a concave function of a quadratic function of the precoder matrix [17]. Then, a parameterized iterative algorithm is designed to find the optimal precoder which achieves the global maximum of the mutual information. Moreover, the coded bit error rate (BER) performance of the obtained precoder is simulated. The BER gains of the obtained precoder comparing to the conventional designs coincides with that of the mutual information gains, which indicate that maximizing the mutual information is also an effective criterion for optimizing the coded BER in practical systems. X. Yuan et al. consider a joint linear precoding and linear MMSE detection scheme for finite alphabet input signals with perfect CSIT (PCSIT) and perfect CSI at the receiver (PCSIR) [139]. The linear precoder is set to be a product of three matrices: the right singular vector matrix of the channel, the diagonal power allocation matrix, and the normalized discrete Fourier transform matrix. When the detector and the decoder satisfy the curve-matching principle [139, Eq. (23)], an asymptotic achievable rate of the linear precoding and linear MMSE detection scheme as the dimension of the transmit signal goes towards infinity is derived. The derived achievable rate is a concave function of the power allocation matrix. As a result, the optimal power allocation scheme for maximizing the achievable rate can be obtained based on the standard convex optimization method. Very recently, K. Cao et al. propose a precoding design in high SNR regime by transforming the maximization of the minimum Euclidean distance into a semi-definite programming optimization problem [140]. Moreover, for a large number of antennas, the implementation complexity of digital beamforming may be significantly high. To reduce the complexity, the MIMO beamforming can be implemented in both the analog and digital domains, which is referred as hybrid beamforming [141]. R. Rajashekar et al. investigate the hybrid beamforming designs to maximize the mutual information for millimeter wave MIMO systems with finite alphabet inputs [142].

24

TABLE VI: MIMO transmissions with finite alphabet inputs based on mutual information maximization Paper

Model

Main Contribution

Diagonal MIMO

Obtain the optimal power allocation

PCSIT, PCSIR

Obtain the optimal power allocation in low and high SNR regime

A. Lozano et al. [18] Non-diagonal MIMO D. P. Palomar et al. [23]

Propose a gradient descent algorithm to search for the optimal precoder PCSIT, PCSIR Non-diagonal MIMO

Establish necessary conditions for the optimal precoder

PCSIT, PCSIR

Propose a numerical algorithm to search for the optimal precoder

Non-diagonal MIMO


SCSIT, PCSIR

Propose a numerical algorithm to search for the optimal precoder

Non-diagonal MIMO


PCSIT, PCSIR

Analyze the optimal precoder in low and high SNR regime

Real-valued non-diagonal MIMO

Obtain the left singular vectors and the singular values of the optimal precoder

PCSIT, PCSIR

Analyze the optimal precoder in low and high SNR regime

C. Xiao et al. [135]

C. Xiao et al. [136]

F. Pérez-Cruz et al. [19]

M. Payaró et al. [137] Obtain the left singular vectors of the optimal precoder in complex-valued channels Non-diagonal MIMO Prove that the mutual information is a concave function of a quadratic function M. Lamarca [138] of the precoder matrix for real-valued non-diagonal MIMO channels PCSIT, PCSIR Propose an iterative algorithm to maximize the mutual information Prove that the mutual information is still a concave function of a quadratic function Non-diagonal MIMO of the precoder matrix for complex-valued non-diagonal MIMO channels C. Xiao [17] Propose a global optimal iterative algorithm to maximize the mutual information PCSIT, PCSIR Examine the coded BER performance of the obtained precoder Non-diagonal MIMO

Design the linear precoder by a product of three matrices

PCSIT, PCSIR

Obtain an asymptotic achievable rate of this linear precoder and linear MMSE detection

Non-diagonal MIMO

Transform the maximization of the minimum Euclidean distance in high SNR regime

PCSIT, PCSIR

into a semi-definite programming optimization problem

Non-diagonal MIMO

Propose hybrid beamforming designs to maximize

PCSIT, PCSIR

mutual information for millimeter wave MIMO systems

X. Yuan et al. [139]

C. Kao et al. [140]

R. Rajashekar et al. [142]

2) Low Complexity Design: Although the proposed algorithm in [17] finds the global optimal precoder, capable of achieving the maximum of the mutual information for the complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs, its implementation complexity is usually difficult to afford in practice systems. There are two main reasons for the high computational complexity. First, both the mutual information and the MMSE function lack closed-form expression. Therefore, the Monte Carlo method has to be used to evaluation both terms, which requires an average over enormous noise realizations in order to maintain the accuracy. More importantly, the evaluation of the mutual information [135, Eq. (5)]

25

and the MMSE function [135, Eq. (8)] involves additions over the modulation signal space which goes exponentially with the number of transmit antennas. This results in a prohibitive computational complexity when the dimension of the transmit antennas is large. By employing the Jensen’s inequality and the intergral over the exponential function, a lower bound of the single-input single-output (SISO) mutual information with finite alphabet inputs is derived in [143, Eq. (9)]. The evaluation of this lower bound does not require to calculate the expectation over the noise. As a result, W. Zeng et al. extend this lower bound to the case of MIMO systems to solve the first factor of the high computational complexity due to the Monte Carlo average [144]. An algorithm is proposed by maximizing this lower bound. Numerical results indicate that the proposed design in [144] reduces the computational complexity without performance loss. W. Zeng et al. further investigate the linear precoder design with SCSI at the transmitter (SCSIT) over the Kronecker fading model [30]. A lower bound of the ergodic mutual information which avoids the expectations over both the channel and the noise is derived. Simulations over various fading channels show that with a constant shift, this lower bound provides an accurate approximation of the ergodic mutual information. Moreover, it is proved that maximizing this lower bound is asymptotically optimal in low and high SNR regime. An iterative algorithm is further developed by maximizing this low bound. Numerical results indicate that the proposed design in [30] achieves a near-optimal mutual information and BER performance with reduced complexity. The lower bound in [30] is utilized to design the linear precoder for practical MIMO-orthogonal frequency division multiplexing (OFDM) systems in [145]. When perfect instantaneous CSI is available at the transmitter, the MIMO channel can be decomposed into a set of parallel subchannels. For the second factor of the high computational complexity, S. K. Mohammed et al. propose X-Codes to pair two subchannels into a group [29]. X-Codes are fully characterized by the paring and a 2 × 2 single angle parameterized rotation matrix for each pair. Based on X-Codes structure, the mutual information can be expressed as a sum of the mutual information of all the pairs. In this case, the calculation of the mutual information only involves additions over the modulation signal space which goes polynomially with the number of transmit antennas. This significantly

26

reduces the computational complexity of evaluating the mutual information. Designs of the rotation angle and power allocation within each pair and the optimization pairing and power allocation among pairs are further proposed to increase the mutual information performance. T. Ketseoglou et al. extend this idea to pair multiple subchannels based on a Per-Group Precoding (PGP) technique in [31]. The PGP technique decomposes the design of the power allocation matrix and the right singular vector matrix into the design of several decoupled matrices with a much small dimension. An iterative algorithm is proposed to find the optimal solution of these decoupled matrices. Numerical results indicate that the proposed algorithm achieves almost the same performance as the design in [17] but with a significant reduction of the computational complexity. Moreover, T. Ketseoglou et al. propose a novel and efficient method to evaluate the mutual information of the complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs based on the Gauss-Hermite quadrature rule [146]. This approximation method can be combined with the PGP technique to further reduce the design complexity. When only SCSI is available at the transmitter, a low-complexity iterative algorithm to find the optimal design over the Kronecker fading model is proposed in [147]. The proposed algorithm relies on an asymptotic approximation of the ergodic mutual information in large system limit, which avoids the Monte Carlo average over the channel. Moreover, based on this approximation, the design of the power allocation matrix and the right singular vector matrix of the precoder can also be decoupled into the design of small dimension matrices within different groups. Simulations indicate that the proposed algorithm in [147] radically reduces the computational complexity of the design in [30] by orders of magnitude with only minimal losses in performance. Y. Wu et al. propose a unified low-complexity precoder design for the complex-valued non-diagonal MIMO Gaussian channel with finite alphabet inputs [148]. An asymptotic expression of the ergodic mutual information over the jointly-correlated Rician fading model is derived in large system limit. Combined with the approximation given in [30], the obtained asymptotic expression can provide an accurate approximation of the ergodic mutual information without the Monte Carlo average over the channel and the noise. The left singular matrix of the optimal precoder for this general model is obtained. Structures of the power allocation matrix and the right singular matrix of the precoder are

27

further established. The proposed structures decouple the independent data streams over parallel equivalent subchannels, which avoid a complete-search of the entire signal space for the precoder design. Based on these, a novel low computational complexity iterative algorithm is proposed to search for the optimal precoder with the general CSI availability at the transmitter, which includes the cases of PCSIT, imperfect CSIT (IPCSIT), and SCSIT. Here we provide Example 1 and Example 2 to analytically and numerically compare the complexity of the low-complexity algorithm in [148] and the complete-search design in [30], respectively. As observed from Table VII-X, the computational complexity of the complete-search design in [30] grows exponential and quickly becomes unwieldy. Simulations indicate that the proposed algorithm in [148] drastically reduces the implementation complexity for finite alphabet precoder design, but in such a way that the loss in performance—established based on the 3GPP spatial channel model [149]—is minimal. Moreover, some novel insights for the precoder design with SCSIT are revealed in [148]. A brief summation of the above-mentioned low-complexity MIMO transmissions with finite alphabet inputs is given in Table XI. Example 1: Assume Nt and Ns denote the number of transmit antenna and the number of streams in each subgroup, respectively. Assume Ns = 2 and QPSK modulation. The numbers of additions required for calculating the mutual information and the MSE matrix in the complete-search design in [30] and the algorithm in [148] are listed in Table VII for different numbers of transmit antenna. TABLE VII: Number of additions required for calculating the mutual information and the MSE matrix. Nt

4

8

16

32

Complete-search design in [30]

65536

4.29 e+009

1.84 e+019

3.4 e+038

Algorithm in [148]

512

1024

2048

4096

Example 2: Let us evaluate the complexity of the algorithm in [148] for different values of Ns . Matlab is used on an Intel Core i7-4510U 2.6GHz processor. Tables VIII–X provide the running time per iteration, for various numbers of antennas and constellations, with × indicating that the time exceeds one hour.

28

TABLE VIII: Running time (sec.) per iteration with BPSK. Nt

Ns = 2

Ns = 4

Ns = Nt

4

0.0051

0.0190

0.0190

8

0.0112

0.0473

11.6209

16

0.0210

0.1939

×

32

0.0570

0.4111

×

TABLE IX: Running time (sec.) per iteration with QPSK. Nt

Ns = 2

Ns = 4

Ns = Nt

4

0.1149

21.5350

21.5350

8

0.2029

23.3442

×

16

0.3001

48.1725

×

32

0.7094

98.7853

×

TABLE X: Running time (sec.) per iteration with 16-QAM. Nt

Ns = 2

Ns = Nt

4

28.0744

×

8

58.3433

×

16

106.6022

×

32

233.2293

×

B. Design Based on MSE 1) Linear Transmission Design: The MSE is an important criterion to measure the quality of the communication link with discrete input signals. With PCSIT and PCSIR, an optimal linear transceiver design to minimize the weighted sum MSE of all channel substreams under total average transmit power constraint is proposed in [150], developing upon early work in [151]–[154]. The minimization of the sum MSE of all channel substreams under maximum eigenvalue constraint is investigated in [155]. A unified framework to optimize the MSE function of all channel substreams under total average transmit power constraint for a multi-carrier MIMO system and linear processing at both ends of the link is established in [32]. It is assumed in [32] that the signal constellation and coding rate for all the subchannels have been determined before the transmission and the carrier-noncooperative scheme is adopted. The system model in [32] is illustrated in Figure 4. It was proved in [32] that if the MSE function is Schur-concave, the optimal

29

TABLE XI: Low-complexity MIMO transmissions with finite alphabet inputs Paper

Model

Main Contribution

MIMO W. Zeng et al. [144]

Design based on a lower bound without Monte Carlo average over the noise PCSIT, PCSIR MIMO

Design based on a lower bound without Monte Carlo average

SCSIT, PCSIR

over the channel and the noise

MIMO-OFDM

Evaluate MIMO transmissions with finite alphabet inputs

practical channel estimation

in practical MIMO test-bed

MIMO

Decouple the mutual information into a sum of

PCSIT, PCSIR

the mutual information of small dimension matrices

MIMO

Decompose the design of the power allocation matrix and the right singular vector matrix

PCSIT, PCSIR

into the design of several decoupled matrices with a much small dimension

MIMO

Propose a novel and efficient method to evaluate the mutual information

PCSIT, PCSIR

based on the Gauss-Hermite quadrature rule

MIMO

Propose a design which reduces the computational complexity of the design in [30]

SCSIT, PCSIR

by orders of magnitude with only minimal losses in performance

MIMO

Propose a unified low-complexity precoder design which applies

general CSIT, PCSIR

for the general CSI availability at the transmitter

W. Zeng et al. [30]

Y. R. Zheng et al. [145]

S. K. Mohammed et al. [29]

T. Ketseoglou et al. [31]

T. Ketseoglou et al. [146]

Y. Wu et al. [147]

Y. Wu et al. [148]

solution is to diagonalize the channel in each subcarrier directly, and if the MSE function is Schur-convex, the optimal solution is to pre-rotate the transmit symbol and then diagonalize the channel in each subcarrier. The above two families of objective functions embrace most of popular criteria used in communication systems, including the MSE, the SINR, and also the uncoded BER with identical constellations. The design in [32] is extended to a general shaping precoder constraint in [156]. Moreover, D. P. Palomar employs a primal decomposition approach to decompose the original optimization problem over multiple (e.g.,multicarrier) MIMO channels in [157] into several subproblems, each of which can be independently solved over a single MIMO channel. These subproblems are coordinated by a simple master problem. As a result, the optimization problem in [32] can be solved in a simple and efficient way. In the mean time, the transmission design which satisfies the specific MSE constraints with minimum transmit power is investigated in [158]. The optimal solution is to pre-rotate the transmit signal by a particular unitary matrix and then diagonalize the channel. Similar problem is also considered in [159] under the uncoded BER constraints instead of the MSE constraints. It is assumed that the constellation used for each substream is different. Then, the uncoded BER is neither Schur-convex nor Schur-concave function of the MSE. This

30

optimization problem is solved by the primal decomposition approach. Moreover, L. G. Ordón˜ ez et al. propose an adaptive number of substreams transmission scheme to minimize the uncoded BER under fix transmission rate constraint and total average transmit power constraint [160]. A widely linear MMSE transceiver which processes the in-phase and quadrature components of the input signals separately is designed in [161]. The widely linear processing is implemented by using a real-value representation of the input signals. Recently, a linear precoder design to minimize the sum MSE of all channel substreams under the joint total average transmit power constraint and maximum eigenvalue constraint is proposed in [162].

. . .

Input 1

Output 1

. . .

. . . noise Input k

Linear Precoder

MIMO Channel

Fig. 4:

Output k

. . .

. . . Input L

Linear Equalizer

. . .

Output L

Multi-carrier MIMO system with linear processing and carrier-noncooperative scheme.

However, perfect CSI is only reasonable for the slow fading scenario where the channel remains deterministic for a long period. Otherwise, the CSI estimated at the transmitter and the receiver may have mismatches with the actual channel. Therefore, S. Serbetli et al. begin to investigate the optimum transceiver structure in sense of minimizing the sum MSE of transmit data streams via imperfect CSI at both ends [33]. A MIMO system with correlated receive antennas is considered and the effect of channel

31

estimation is investigated in [33]. X. Zhang et al. investigate the statistically robust design of linear MIMO transceivers based on minimizing the MSE function under total average transmit power constraint [163]. The statistically robust design exploits the channel mean and channel covariance matrix (equivalently, the channel estimate and the estimation error) to design linear MIMO transceivers. Two cases are considered in [163]: i) IPCSIT and imperfect channel state information at the receiver (IPCSIR) with no transmit antenna correlation; ii) IPCSIT and PCSIR. For the former case, the optimal solution has the same structure as the perfect CSI case by replacing the instantaneous channel realization matrix and the noise covariance matrix with the channel mean matrix and equivalent covariance matrix (e.g., channel covariance matrix plus noise covariance matrix), respectively. For the latter case, a lower bound of the average MSE of all substreams is derived and the linear precoder minimizing this lower bound is obtained. The statistically robust design for the IPCSIT and IPCSIR case with transmit antenna correlation is investigated in [164]. By finding the optimal solution which satisfies the KKT conditions, the structures of the optimal precoder and equalizer minimizing the sum MSE of transmit data streams under total average transmit power constraint are obtained in [164]. In the mean time, J. Wang et al. investigate a deterministic robust design for MIMO system under total average transmit power constraint [165]. The deterministic robust design assumes the estimated imperfect CSI is within a neighborhood of actual channel. The case of IPCSIT and IPCSIR is considered in [165]. Given a pre-fixed linear receiver structure, the optimal transmission direction and the optimal power allocation for the linear precoder matrix minimizing the MSE of transmit data streams are found in [165]. For the same scenario, the optimal linear precoder structure given a pre-fixed linear receiver and the optimal linear equalizer structure given a pre-fixed linear precoder are obtained in [166]. Therefore, the precoder and the equalizer can be designed alternatively by an iterative algorithm. For the scenario considered in [165], the optimal linear MIMO transceiver design is found in [167]. On the other hand, to reduce the cost of analog-to-digital converter (ADC) chains at the receiver, MIMO analog beamformers are designed in [168] to minimize both the output MSE of the digital beamformer and the power consumption of the ADC by exploiting the statistical knowledge of the channel. A brief summation of linear transmission designs based on MSE is given in Table XII, where the notations LP and LE denote

32

abbreviations for linear precoder and linear equalizer, respectively. 2) Non-linear Transmission Design: Above-mentioned work are based on the linear processing at both ends of the link. However, research have shown that non-linear processing schemes can offer additional advantages over linear processing schemes [34], [169]–[173]. Two typical technologies are TomlinsonHarashima precoding (THP) at the transmitter and decision-feedback equalization (DFE) at the receiver. A unified transceiver structure of a MIMO system with THP and DFE is illustrated in Figure 5. Assuming PCSIT and PCSIR, a THP precoding with linear equalization which minimizes the sum MSE of transmit data streams is proposed in [34]. The work in [34] focuses on the design of the feedback precoder in Figure 5. The permutation matrix and the precoder matrix in Figure 5 are set to be identical matrix. The THP design with linear equalization is extended to MIMO-OFDM system in [169], where the input signal is pre-ordered by a permutation matrix and an additional precoder matrix is designed to improve the overall MSE performance. Two linear precoding designs with DFE at the receiver are proposed in [170] and [171], which optimize the sum MSE and the MSE function of the transmit data streams, respectively. A unified framework which is applicable for both THP with linear equalization and linear precoding with DFE is established in [172]. The proposed designs in [172] optimize the MSE function of the transmit data streams. It is proved that if the MSE function is Schur-convex, the optimal design forces the MSE of each transmit data stream to be equal, and if the MSE function is Schur-concave, the optimal THP leads to linear precoding and the optimal DFE leads to linear equalization. When only statistical CSI is available at the transmitter, O. Simenone et al. propose a THP with linear equalization design based on minimizing a lower bound of the sum MSE of the transmit data streams [173]. A brief summation of non-linear transmission designs based on MSE is given in Table XIII, where the notations NLP and NLE denote abbreviations for non-linear precoder and non-linear equalizer, respectively. 3) Performance Analysis: Since the publication of [32], some work begin to analyze the performance of the uncoded MIMO systems within the linear processing framework (also refereed to MIMO multichannelbeamforming (MB) systems [174]) in [32]. In [175], the first-order polynomial expansions for the marginal pdfs of all of the individual ordered eigenvalues of uncorrelated central Wishart-distributed matrix are

33

TABLE XII: Linear transmission designs with discrete input signals based on the MSE Paper

Systems

H. Sampath et al. [150]

MIMO

Criterion

Transceiver

CSI

Weighted

LP

PCSIT

sum MSE

LE

PCSIR

LP

PCSIT

LE

PCSIR

LP

PCSIT

LE

PCSIR

LP

PCSIT

Covariance matrix

LE

PCSIR

shaping

Sum power

Multi-carrier A. Scaglione et al. [155]

Sum MSE MIMO Multi-carrier


Maximum eigenvalue

MSE function MIMO

D. P. Palomar [156]

MIMO

Sum power

MSE function

Multi-carrier

MSE function

LP

PCSIT

MIMO

efficient solution

LE

PCSIR

LP

PCSIT

MIMO

Transmit power LE

PCSIR

D. P. Palomar [157]


Sum power

MSE

Multi-carrier

Uncoded BER

LP

PCSIT

MIMO

different constellations

LE

PCSIR

Uncoded BER

LP

PCSIT

Sum power

adaptive streams

LE

PCSIR

fixed rate

Widely LP

PCSIT

Widely LE

PCSIR

LP

PCSIT

Sum power

LE

PCSIR

maximum eigenvalue

Average sum MSE

LP

IPCSIT and IPCSIR

channel estimation effect

LE

no transmit correlation

LP

IPCSIT and IPCSIR

LE

no transmit correlation

Average MSE function

LP

IPCSIT

lower bound

LE

PCSIR

LP

IPCSIT and IPCSIR

LE

with transmit correlation

LP

IPCSIT

fixed LE

IPCSIR

Sum MSE

LP

IPCSIT

not optimal

LE

IPCSIR

Sum MSE

LP

IPCSIT

optimal

LE

IPCSIR

MSE, ADC power

LP

SCSIT

optimal

LE

SCSIR


L. G. Ordón˜ ez et al. [160]

F. Sterle [161]

J. Dai et al. [162]

S. Serbetli et al. [33]

X. Zhang et al. [163]

X. Zhang et al. [163]

M. Ding et al. [164]

J. Wang et al. [165]

J. Wang et al. [166]

H. Tang et al. [167]

V. Venkateswaran et al. [168]

Constraint

Sum power

MIMO

MIMO

MIMO

Sum MSE

Sum power

Sum MSE

MIMO

MIMO

Sum power

Average MSE function

Sum power

MIMO

MIMO

MIMO

Sum power

Average sum MSE

Sum power

Sum MSE

Sum power

MIMO

Sum power

MIMO

Sum power

MIMO

No

34

s

Permutation Matrix

a



Precoder matrix

b

Modulo

c

Channel

Feedback Precoder

nosie Quantizer

z

Equalizer Matrix

y

Feedback Equalizer

Fig. 5:

MIMO transceiver with THP and DFE.

TABLE XIII: Non-linear transmission designs with discrete input signals based on the MSE Paper

Systems

Criterion

Transceiver

CSI

R. F. H. Fischer et al. [34]

MIMO

Sum MSE

NLP

PCSIT

LE

PCSIR

NLP, LP

PCSIT

LE

PCSIR

LP

PCSIT

LE, NLE

PCSIR

LP

PCSIT

LE, NLE

PCSIR

NLP, LP, LE

PCSIT

LP, LE, NLE

PCSIR

Average sum MSE

NLP, LP, LE

SCSIT

lower bound

LP, LE, and NLE

PCSIR

MIMO A. A. D. Amico et al. [169]

F. Xu et al. [170]

Y. Jiang et al. [171]

M. B. Shenouda et al. [172]

O. Simeone et al. [173]

Sum power

Sum MSE OFDM MIMO

MIMO

MIMO

Constraint

Sum power

Sum MSE

Sum power

MSE function

Sum power

Sum MSE

Sum power

MIMO

Sum power

derived. Based on these results, the closed form expressions for the outage probability and the average uncoded BER performance for each subchannel of MIMO MB systems with fixed power allocation over an uncorrelated Rayleigh flat-fading channel in high SNR regime are obtained [175]. The obtained expressions reveal both the diversity gain and the array gain in each subchannel. Also, tight lower and upper bounds

35

for the performance with nonfixed power allocation such as waterfilling power allocation are obtained. For the correlated central Wishart-distributed matrix, the exact pdfs expressions for the ordered eigenvalues are derived in [176]. These expressions allow us to analyze the MIMO MB systems over a semicorrelated Rayleigh flat-fading channel under various performance criterions, including the outage probability and the average uncoded BER. Simple closed-form first-order polynomial expansions for these marginal pdfs are derived in [160], from which the diversity gain and the array gain of the MIMO MB systems are indicated. In the mean time, the exact pdfs expressions for the uncorrelated noncentral Wishart-distributed matrix are independently derived in [174] and [176], respectively. The first order polynomial expansions for these marginal pdfs are independently obtained in [160] and [174], respectively. Performance of MIMO MB systems over uncorrelated Rician fading channel can be analyzed similarly based on these expressions. For MIMO MB over a more general Rayleigh-product fading channel, H. Zhang et al. derive the first order polynomial expansions of the ordered eigenvalues of the channel Gram matrix [177]. The diversity gain and the array gain for MIMO MB systems with fixed power allocation over the Rayleighproduct fading channel are revealed in [177]. Later, the analysis for MIMO MB systems is extended to scenarios in the presence of co-channel interference. With fixed power allocation, L. Sun et al. obtain exact and asymptotic (low outage probability) expressions of outage probability of MIMO MB systems in the presence of unequal power interferers [178], where both the desired user and the interferes experience semicorrelated Rayleigh flat-fading. Also, S. Jin et al. obtain exact and asymptotic expressions of outage probability of MIMO MB systems in the interference-limited (without noise) scenarios with equal power interferers. It is assumed in [179] that the desired user experiences uncorrelated Rician fading and the interferers experience uncorrelated Rayleigh fading. Y. Wu et al. obtain exact and asymptotic expressions of outage probability of MIMO MB systems with arbitrary power interferers, where the desired user experiences Rayleigh-product fading and the interferers experience uncorrelated Rayleigh fading [180]. A brief summation of performance analysis of MIMO MB systems is given in Table XIV.

36

TABLE XIV: Performance analysis of MIMO MB systems Paper

user channel

interferer channel

interferer power

outage probability

L. G. Ordón˜ ez et al. [175]

Uncorrelated Rayleigh fading

No

No

Asymptotic

No

No

Exact

No

No

Asymptotic

Semicorrelated Rayleigh fading L. G. Ordón˜ ez et al. [176] Uncorrelated Rician fading Semicorrelated Rayleigh fading L. G. Ordón˜ ez et al. [160] Uncorrelated Rician fading S. Jin et al. [174]

Uncorrelated Rician fading

No

No

Exact, asymptotic

H. Zhang et al. [177]

Rayleigh-product fading

No

No

Asymptotic

L. Sun et al. [178]

Semicorrelated Rayleigh fading

Semicorrelated Rayleigh fading

Unequal

Exact

S. Jin et al. [179]

Uncorrelated Rician fading


Equal

Exact, asymptotic Interference-limited Y. Wu et al. [180]

Rayleigh-product fading


Arbitrary

Exact, asymptotic

C. Diversity Driven Designs 1) Early research for Open-loop Systems: For the open-loop MIMO system, it is usually assumed that the CSI is available at the receiver but not at the transmitter. To acquire receive diversity, exploiting diversity combining schemes at the receiver, such as equal gain combining, selection combining, and maximum ratio combining (MRC), can be tracked back to the 60 years ago with references in the paper [181]. Without CSIT, space-time techniques are designed to combat the deleterious effects of small scale fading. These space-time techniques can acquire additional transmit diversity and thereby improve the reliability of the link with discrete input signals. A simple and well known space-time technique is the Alamouti orthogonal space-time block code (OSTBC) for a two transmit antennas and two receive antennas system [35], as illustrated in Figure 6. For OSTBC in Figure 6, in time slot 1, the symbols s1 and s2 are transmitted in antenna 1 and 2, respectively. In time slot 2, the symbol −s∗2 and −s∗1 are transmitted in antenna 1 and 2, respectively. After passing the channel and the combiner in Figure 6, the signal s1,c and s2,c sent to the maximum likelihood (ML) detector are given by s1,c = |h11 |2 + |h12 |2 + |h21 |2 + |h22 |2 s0 + h∗11 n11 + h12 n∗12 + h∗22 n22 + h21 n21

(9)

s2,c = |h11 |2 + |h12 |2 + |h21 |2 + |h22 |2 s1 − h11 n∗11 + h∗12 n12 − h22 n∗22 + h∗21 n21 .

(10)

37

(9) and (10) are equivalent to that of four-branch MRC. Therefore, it achieves a diversity of four for the ML detector. Additional transmit diversity is obtained by utilizing OSTBC. For both real and complex OSTBC, the construction of OSTBC for any number of transmit antennas and receive antennas is proposed in [36]. The proposed design achieves maximum possible spatial diversity order with a low complexity linear decoding. The space-time block code (STBC) is extended to single-carrier MIMO system in [37] and MIMO-OFDM system in [182]. The proposed designs achieve the maximum diversity but with the increase of the decoding complexity since the detector needs to jointly decode all subchannels. To solve this problem, a subchannel grouping method with the maximum diversity performance and a reduced decoding complexity is proposed in [182]. The transmission rate of above OSTBC design is less than 1 symbol per channel use (pcu) for more than two transmit antennas. Therefore, a quasi-orthogonal spacetime block code (QOSTBC), which provides 1 symbol pcu transmission rate and half of the maximum possible diversity for four transmit antennas, is proposed in [183]. The QOSTBC divides the transmission matrix columns into different groups, and makes the columns in different groups orthogonal to each other. This quasi-orthogonal structure increases the transmission rate. Also, based on this structure, the ML decoding at the receiver can be done by searching symbols in each group individually instead of searching all transmit signals jointly. By rotating half of the transmit symbol constellations, QOSTBC can achieve 1 symbol pcu rate and maximum diversity for four transmit antennas [184] and higher rate than OSTBC and maximum diversity for MIMO systems [185]. Based on a different rotation method, C. Yuen et al. construct a class of QOSTBCs for MIMO systems, whose ML decoding can be reduced to a joint detection of two real symbols [186]. On the other hand, the space time trellis codes (STTC), which achieves both maximum spatial diversity and large coding gains, is designed in [187]. However, the decoding complexity of STTC grows exponentially with transmit rate and becomes significantly high for high order modulation. A space time linear constellation precoding (ST-LCP) is proposed in [188] as an alternative scheme to obtain transmit diversity. The baseband equivalent framework of ST-LCP is illustrated in Figure 7. The key idea of ST-LCP is to perform a pre-combination of the transmit signal by a linear precoder

38

s1 -s2*

n11,n12 s2 -s1*

h12 h21

h11,h12 Combiner

n21,n22

Fig. 6:

s1,c,s2,c

h21,h22 Channel Estimator

h22

h11,h12 Maximum liklihood detector

Channel Estimator

h11

s1,e,s2,e

h21,h22

The OSTBC for a 2 × 2 system.

matrix as in Figure 7. Then, the pre-processed data is mapped to the transmit antenna by a diagonal matrix and a unitary matrix. Based on the average PEP expression, linear precoders which achieve the maximum diversity and perform close to the upper bound of the coding gain are designed in closed-form. Near optimal sphere decoding algorithms are employed to reduce the decoding complexity of ST-LCP. Simulations indicate that ST-LCP achieve a better BER performance than classic OSTBC. On the other hand, Liu et al. study ST-LCP for MIMO-OFDM system [189]. The authors in [189] combine ST-LCP with a subcarrier grouping method to realize both the maximum diversity and the low decoding complexity.

noise

input

Linear Precoder

Fig. 7:

ST-LCP Mapper

MIMO Channel

Detection

output

The baseband equivalent framework of ST-LCP.

Although the above-mentioned work achieves good BER performance, their transmit rate can not exceed 1 symbol pcu. This limits the transmission efficiency of MIMO system. B. A. Sethuraman et al. construct

39

a high rate full diversity STBC using division algebras with high decoding complexity [190]. Also, many work try to construct full rate (Nt symbol pcu) full diversity space time codes. One approach is to extend the ST-LCP to the full rate full diversity design [191]. The key idea is to replace the linear precoder in Figure 7 with a linear complex-field encoder (LCFE). The LCFE divides the Nt2 transmit symbols into Nt layers, where Nt is the number of transmit antenna. In each layer, a closed form linear precoder matrix is used to combine the Nt symbols. Then, the ST-LCP mapper in Figure 7 is replaced with space time LCFE (ST-LCFE) mapper, where the pre-precessed Nt2 data is mapped into a Nt ×Nt matrix. Finally, the mapped data is transmitted by Nt antennas in Nt time slot. In this case, Nt symbol pcu transmission rate is achieved. It is proved that the ST-LCFE design also obtains the maximum spatial diversity of MIMO system. The price of the ST-LCFE is the exponential increase of the decoding complexity. Therefore, both diversitycomplexity tradeoff and modulation-complexity tradeoff schemes are proposed in [191]. Moreover, the ST-LCFE design is extended to frequency-selective MIMO-OFDM and time-selective fading channels in [191]. Another approach is to employ threaded algebraic space time (TAST) code [192]. When using TAST code, the size of the information symbols alphabet is increased since the degree of the algebraic number field increases. Therefore, full rate Nt symbol pcu and the maximum spatial diversity can be achieved simultaneously. A golden STBC with non-vanishing determinants (NVD) property when the signal constellation size approaches infinity is first constructed for 2 × 2, 3 × 3, and 6 × 6 antenna systems in [193], [194], and extended to arbitrary number of antennas MIMO systems in [195]. The NVD property guarantees that the STBC performs well irrespectively of the transmit signal alphabet size. The abovementioned work are some important initial space-time techniques with discrete input signals and PCSIR. These space-time techniques focus on improving the transmit diversity, coding gain, and transmission rate of the link. A brief comparison of these work is given in Table XV. For the noncoherent scenario where the CSI is neither available at the transmitter nor at the receiver, V. Tarokh et al. combine differential modulation with OSTBC to formulate transmission schemes, which can still obtain transmit diversity with PSK signals [196], [197]. Based on unitary group codes, differential STBCs to achieve the maximum transmit diversity of MIMO systems are proposed in [198], [199], where

40

TABLE XV: Some important initial space-time techniques with discrete input signals and perfect CSIR Paper

System

Coding gain

Rate

Decoding complexity

S. M. Alamouti [35]

2×2

Not optimal

1 symbol pcu

Low

V. Tarokh et al. [36]

MIMO

Not optimal

< 1 symbol pcu

Low

S. Zhou et al. [37]

Single-carrier MIMO

Not optimal

< 1 symbol pcu

High

Z. Liu et al. [182]

MIMO-OFDM

Large

< 1 symbol pcu

Lower than [37]

H. Jafarkhani [183]

4×4

Not optimal (half diversity)

1 symbol pcu

Reduced

N. Sharma et al. [184]

4×4

Not optimal

1 symbol pcu

Reduced

W. Su et al. [185]

MIMO

Not optimal

≤ 1 symbol pcu

Reduced

C. Yuen et al. [186]

MIMO

Not optimal

≤ 1 symbol pcu

Lower than [185]

V. Tarokh et al. [187]

MIMO

Large

< 1 symbol pcu

High

Y. Xin et al. [188]

MIMO

Near-optimal

1 symbol pcu

High

Z. Liu et al. [189]

MIMO-OFDM

Not optimal

1 symbol pcu

Lower than [37]

B. A. Sethuraman et al. [190]

MIMO

Near-optimal

> 1 symbol pcu

High

Not optimal

Nt symbol pcu

High

MIMO, MIMO-OFDM X. Ma et al. [191] MIMO time selective channels H. E. Gammal et al. [192]

MIMO

Not optimal

Nt symbol pcu

High

J.-C. Belfiore et al. [193]

2×2

Large (NVD)

2 symbol pcu

High

F. Oggier et al. [194]

2 × 2, 3 × 3, 6 × 6

Large (NVD)

Nt symbol pcu

High

P. Elia et al. [195]

MIMO

Large (NVD)

Nt symbol pcu

High

the optimal modulation scheme for two transmit antennas is obtained in [199]. A double differential STBC is designed for time selective fading channels in [200]. In addition, a differential STBC using QAM signals is constructed in [201]. For the partially coherent scenario where the CSI is not available at the transmitter and only imperfect CSI is available at the receiver, robust STBCs are designed in [202]–[204]. The effects of mapping on the error performance of the coded STBC have been studied in [205], [206]. A rate-diversity tradeoff for STBC under finite alphabet input constraints is obtained in [207] and the corresponding optimal signal constructions for BPSK and QPSK constellation are provided. The space-time techniques have also been extended to MIMO CDMA systems [208]–[210]. 2) Recent Advance for Open-loop Systems: Recent space-time techniques with discrete input signals focus on designing high rate full diversity STBCs with NVD and low decoding complexity. For a multiple input single output (MISO) system equipped with a linear detector, a general criterion for designing full

41

diversity STBC is established in [39]. The STBC with Toeplitz structure satisfies this criterion. Also, the rate of this Toeplitz STBC approaches 1 symbol pcu when the number of channel use is sufficiently large. For MISO system equipped with ML detector, this Toeplitz STBC achieves the maximum coding gain. A new 2 × 2 full rate full diversity STBC is proposed by reconstructing the generation matrix of OSTBC [211]. The proposed STBC reduces the complexity of ML decoding to the order of the standard sphere decoding. Another 2×2 full rate full diversity STBC is designed in [212]. During the ML decoding exhaustive search process for this STBC, the Euclidean distance can be divided into groups and each group is minimized independently. Henceforth, this STBC leads to an obvious decoding complexity reduction comparing to the Golden code in [193] with a slight performance loss. A unified design framework for full rate full diversity 2 × 2 fast-decodable STBC is provided in [213]. The ML decoding complexity of the designed code is reduced to the complexity of standard sphere decoding. For a 4 × 2 system, two QSTBCs are combined to formulate a new 2 pcu symbol rate and half diversity STBC with simplified ML decoding complexity [213]. K. P. Srinath et al. construct a 2 × 2 STBC which has the same performance as the Golden code in [193] and a lower ML decoding complexity for non-rectangular QAM inputs [214]. K. P. Srinath et al. further construct a 2 pcu symbol rate and full diversity 4 × 2 STBC which offers a large coding gain with a reduced ML decoding complexity [214]. H. Wang et al. construct a QOSTBC which experiences the same low decoding complexity as the design in [186], but has a better coding gain for rectangular QAM inputs [215]. K. R. Kumar et al. design a STTC for MIMO systems based on lattice coset coding [216]. The constructed code has a good coding gain and a reduced decoding complexity for large block length with a DFE and lattice decoding. In [217], a general framework of multigroup ML decodable STBC is introduced to reduce the decoding complexity of MIMO STBC. Multigroup ML decodable STBC is a full diversity STBC. It allows ML decoding to be implemented for the symbols within each individual group. As a result, multigroup ML decodable STBC provides a tradeoff between rate and ML decoding complexity. A specific multigroup ML decodable STBC based on extended Clifford algebras, named Clifford unitary weight, is constructed to meet the optimal tradeoff between the rate and the ML decoding complexity in [218]. Moreover, the

42

structure of fast decodable code is integrated into the multigroup code to formulate a fast-group-decodable STBC [219]. A new class of full rate full diversity STBC which exploits the block-orthogonal property of STBC (BOSTBC) is proposed in [220]. Simulation shows that this BOSTBC achieves a good BER performance with a reduced decoding complexity when QR decomposition decoder with M paths (QRDM decoder) is employed at the receiver. Also, this block-orthogonal property can be utilized [221] to reduce the decoding complexity of the Golden code in [193] with ML decoder. Rigorous theoretical analyses for the block-orthogonal structure of various STBCs are given in [222]. Some fast-decodable asymmetric STBCs are designed in [223]–[225]. In particular, R. Vehkalahti et al. establish a family of fast-decodable STBCs from division algebras for multiple-input double-output (MIDO) systems [223]. These codes are full diversity with rate no more than Nt /2 symbol pcu. By properly constructing the geometric structures of these codes, the ML decoding complexity can by reduced by at least 37.5%. In addition, explicit constructions for 4 × 2, 6 × 2, and 6 × 3 codes are given in [223]. N. Markin et al. propose an iterative STBC design method for Nt ×Nt /2 MIMO systems [224]. The iterative design constructs two Nt /2×Nt /2 original codewords with algebraic codewords based on cyclic algebra to create a new Nt × Nt codewords. The resulting code achieves Nt /2 symbol pcu rate and full diversity with fast decoding complexity. This iterative design is used to construct new 4 × 2 and 6 × 3 STBCs in [224] based on some well known STBCs such as Golden code in [193]. Moreover, a generalization framework of this iterative design is used in [225] to construct 2 symbol pcu rate and full diversity STBC with large coding gain and fast decoding complexity for MIDO systems. Based on this framework, explicit 4 × 2, 6 × 2, 8 × 2, and 12 × 2 STBCs are constructed in [225]. A brief comparison of above low decoding complexity STBC designs is given in Table XVI. Besides reducing the decoding complexity, there are some other space-time techniques recently, e.g., a full rate full diversity improved perfect STBC is designed in [226], which has larger coding gain than the perfect STBC in [194]. A full rate STBC under the BICM structure is designed in [226] to optimize the entire components of the transmitter jointly, including an error-correcting code, an interleaver, and a symbol mapper. A new spatial modulation technology which only simultaneously activates a few number

43

TABLE XVI: Recent low decoding complexity space-time techniques with discrete input signals Paper

System

Coding gain

Rate

Decoding complexity

J. Liu et al. [39]

MISO

NVD

Approach 1 symbol pcu

Linear

J. M. Paredes et al. [211]

2×2

NVD

2 symbol pcu

Reduced ML

S. Sezginer et al. [212]

2×2

Not optimal

2 symbol pcu

Reduced ML

2×2

Not optimal

2 symbol pcu

Reduced ML

4×2

Not optimal (half diversity)

2 symbol pcu

Reduced ML

2×2

NVD

2 symbol pcu

Lower than Golden code [193]

4×2

NVD

2 symbol pcu

Reduced

H. Wang et al. [215]

MIMO

Larger than [186]

1 ≤ symbol pcu

Same as [186]

K. R. Kumar et al. [216]

MIMO

Good

High

Reduced ML

S. Karmakar et al. [217]

MIMO

Not optimal

Flexible

Trade off with rate

G. S. Rajan et al. [218]

MIMO

Not optimal

Flexible

Trade off with rate

T. P. Ren et al. [219]

MIMO

Not optimal

Flexible

Lower than [213]

T. P. Ren et al. [220]

MIMO

Not optimal

Nt symbol pcu

Reduced QRDM

M. O. Sinnokrot et al. [221]

2 × 2 Golden code [193]

NVD

2 symbol pcu

Reduced ML

G. R. Jithamithra et al. [222]

MIMO

Classic STBCs

Flexible

Reduced SD

R. Vehkalahti et al. [223]

MIDO

NVD

Nt /2 symbol pcu

Reduced ML

N. Markin et al. [224]

Nt × Nt /2

NVD

Nt /2 symbol pcu

Reduced ML

4 × 2, 6 × 2

NVD, large 2 symbol pcu

Reduced ML

8 × 2, 12 × 2

NVD, large

E. Biglieri et al. [213]

K. P. Srinath et al. [214]


of antennas among the entire antenna arrays is described in [227]–[229]. Since the publication of [35], space-time techniques with discrete input signals have been studied extensively around the world [38], [230], [231]. 3) Close-loop Systems: The space-time technologies achieve the transmit diversity without CSI at the transmitter, which are applicable for open-loop systems. Alternatively, for close-loop MIMO systems in Figure 8, the CSI is available at the transmitter either by a feedback link from the receiver (frequency division duplex systems) or a channel estimation from the pilot signal of the receiver (time division duplex systems). In this case, proper transmission designs can be formulated to exploit the close loop transmit diversity of MIMO systems with discrete input signals. For example, a bit interleaved coded multiple beamforming technology is introduced in [232] to achieve the full diversity and full spatial multiplexing for MIMO and MIMO-OFDM systems over i.i.d. Rayleigh fading channels. The decoding complexity of

44

the bit interleaved coded multiple beamforming design is further reduced in [233]. An important criterion to enhance the transmit diversity for the close loop MIMO systems is to maximize the minimum Euclidean distance of the receive signal points. This criterion is directly related to the SER of the MIMO system if the ML detector is used and is proved to achieve the optimal mutual information performance in high SNR regime [19]. For a 2×2 system with BPSK and QPSK inputs, an optimal solution for a non-diagonal precoder, which maximizes the minimum Euclidean distance of the receive signal points (max-dmin ), is proposed in [234]. It is shown in [234] that for BPSK inputs, the optimal transmission is to focus the power on the strongest subchannel, which is the beamforming design. For QPSK inputs, the beamforming design is optimal when the channel angle is less than a threshold. When the channel angle is larger than the threshold, the optimal design can be found by numerically searching the angle of the power allocation matrix. In [40], a suboptimal precoder design for MIMO systems with an even number of transmit antenna and 4-QAM inputs is proposed. The authors transform the MIMO channels into parallel subchannels by the singular value decomposition (SVD). Each two subchannels are paired into one group to formulate a set of 2 × 2 subsystems. Then, 2 × 2 sub-precoders are designed similar as in [234]. The design in [40] is denoted as the X-structure design. It is noted this X-structure design enjoys both reduced precoding and decoding complexities. For M -QAM inputs, finding the optimal max-dmin MIMO precoder is rather complicated and the optimal solution varies for different modulations. Therefore, Ngo et al. propose a suboptimal design by selecting 2 × 2 sub-precoders between two deterministic structures [235]. These sub-precoders are then combined together to formulate a MIMO precoder based on the X-structure. The suboptimal design in [235] has good uncoded BER performance with a reduced implementation complexity. An exact expression for the pdf of the minimum Euclidean distance under this suboptimal precoder design is derived in [236]. Also, a max-dmin MIMO precoder design for a three parallel datastream scheme and M -QAM inputs is proposed in [237]. This work allows us to decompose arbitrary MIMO channels into 2×2 and 3×3 subchannels and then design the max-dmin MIMO precoder efficiently. On the other hand, a suboptimal real-value max-dmin MIMO precoder for M -QAM inputs is proposed in [238]. The proposed design has a order of magnitude lower ML decoding complexity than the precoders

45

in [40], [235]. Moreover, it is proved in [238] that the max-dmin precoder achieves full diversity of MIMO systems. In the mean time, two low decoding complexity precoders derived from a rotation matrix and the X-structure, called X-precoder and Y-Precoder, are proposed in [29]. X-precoder has a closed-form expression for 4-QAM inputs. Y-precoder has an explicit expression for arbitrary M -QAM inputs but with a worse error performance than X-precoder. Overall, both X-precoder and Y-Precoder are suboptimal in error performance and loses out in word error probability when comparing to max-dmin precoder [40], [235]. Two suboptimal max-dmin MIMO precoder designs for M -QAM inputs by optimizing the diagonal elements of the “SNR-like” matrix and the lower bound of the minimum Euclidean distance are proposed in [239] and [240], respectively.

noise

input

Linear Precoder

MIMO Channel

Receiver

output

Feedback CSI from Receiver

Fig. 8:

The baseband equivalent framework of close-loop system.

An alternative approach of exploiting the close loop transmit diversity is to design the precoder based on the lattice structure. A lattice-based linear MIMO precoder design to minimize the transmit power under fixed block error rate and fixed total data rate is proposed in [241]. The precoder design is decomposed into the design of four matrices. The optimal rotation matrix is given in a closed form structure, the bit load matrix and the basis reduction matrix are optimized alternatingly via an iterative algorithm, and the lattice base matrix is selected based on a lower bound of the transmit power. Linear precoders which achieve full diversity of Nt × Nt MIMO systems and apply for integer-forcing receiver are designed in [242] based on full-diversity lattice generator matrices. Some work also use lattice theory to design maxdmin precoder. For a 2×2 system with a square QAM constellation inputs, the optimal real-value max-dmin

46

TABLE XVII: Techniques for the transmit diversity of close loop MIMO systems with discrete input signals Paper

System

Modulation

minimum Euclidean distance

Design complexity

Decoding complexity

M -QAM

Not optimal

Low

High

MIMO E. Akay et al. [232] MIMO-OFDM B. Li et al. [233]

MIMO-OFDM

M -QAM

Not optimal

Low

Reduced

L. Collin et al. [234]

2×2

BPSK, QPSK

Optimal

Low

Low

B. Vrigneau et al. [40]

MIMO

QPSK

Near-optimal

Reduced

Reduced

Q.-T. Ngo et al. [235]

MIMO

M -QAM

Suboptimal

Reduced

Reduced


MIMO

M -QAM

Suboptimal

Reduced

Reduced


MIMO

M -QAM

Smaller than [40]

Lower than [235]

Lower than [235]

S. K. Mohammed et al. [29]

MIMO

M -QAM

Not optimal

Lower than [235]

Lower than [235]


MIMO

M -QAM

Larger than [235]

Reduced

High

C.-T. Lin et al. [240]

MIMO

M -QAM

Close to [29]

Lower than [29]

High

S. Bergman et al. [241]

MIMO

M -QAM

Not optimal

Reduced

High

A. Sakzad et al. [242]

Nt × Nt

M -QAM

Not optimal

Reduced

Reduced

X. Xu et al. [243]

2×2

Square M -QAM

Optimal

Low

Low

X. Xu et al. [244]

MIMO

QPSK

Larger than [235]

Reduced

High

D. Kapetanović et al. [245]

Nt × 2

Infinite Constellation

Asymptotic optimal

Low

Low

D. Kapetanović et al. [246]

MIMO

Infinite Constellation

Asymptotic optimal

Reduced

High

precoder is found by expanding an ellipse within a 2-dimensional lattice to maximum extent [243]. For MIMO systems with 4-QAM inputs, a real-value max-dmin precoder is directly constructed based on wellknown dense packing lattices [244]. The obtained precoder offers better minimum Euclidean distance and BER performance than the precoder based on the X-structure [40]. For Nt × 2 systems with infinite signal constellation, an explicit structure of the max-dmin precoder is obtained from the design of the lattice generator matrix [245]. Numerical results indicate that the obtained precoder achieves good mutual information performance with large QAM inputs. This precoder design is extended to MIMO systems in [246]. A brief summation of transmission designs with discrete input signals to optimize the transmit diversity for the close loop MIMO systems is given in Table XVII. Some work combine the STBC with the linear processing step in close loop system to further improve the performance of MIMO systems. The structure of a close loop STBC systems is illustrated in Figure

47

9. It is important to note that the main difference between the linear processing in Figure 7 and the linear precoder in Figure 9 is the usage of CSI. The linear precoder in Figure 7 does not need any knowledge of CSI and the linear processing in Figure 9 requires some forms of the CSI. With PCSIT, H. J. Park et al. exploit the constellation precoding scheme in [188] to design a full-diversity multiple beamforming scheme for uncoded systems [247]. For the cases of both PCSTT and SCSIT, D. A. Gore et al. investigate antenna selection techniques to minimize the SER for MIMO systems employing OSTBC [248]. Jongren et al. combine the beamforming design and the OSTBC with partial knowledge of the channel over frequency-nonselective fading channel [249]. The linear precoder design for a STBC system over a transmit correlated Rayleigh fading channel is proposed in [250]. The transmitter utilizes the transmit correlation matrix to design a linear precoding matrix, which minimizes an upper bound of the average PEP of the STBC system. This design is extended to transmit correlated Rician fading channel and arbitrary correlated Rician fading channel in [251] and in [252], respectively. Linear precoding for a limited feedback STBC systems is proposed in [253]. For the noncoherent scenario where the CSI is not available at the receiver, linear precoder designs for differential STBC systems over Kronecker fading channel and arbitrary correlated Rayleigh fading channels are provided in [254] and [255], respectively. For the partially coherent scenario where the imperfect channel estimation is performed at the receiver and the estimation error covariance matrix is perfectly known at the transmitter via feedback, STBC designs are proposed in [256] and [257] for the i.i.d. Rayleigh fading channels based on criterions of KullbackLeibler distance and the cutoff rate, respectively. The design in [258] is extended to the correlated Rayleigh fading channels where the transmit and receive correlation matrices are perfectly known at the transmitter. Moreover, a novel and efficient bit mapping scheme is proposed in [258]. A linear precoder is further designed in [259] to adapt to the degradation caused by the imperfect CSIR.

D. Adaptive MIMO Transmission The above-mentioned multiplexing, hybrid, and diversity transmission schemes can be employed adaptively according to the channel condition. Adaptive MIMO transmission is an effective approach to increase

48

noise

input

ST Code

Linear Processing

MIMO Channel

Receiver

output

Feedback CSI from Receiver

Fig. 9:

A close loop STBC system

the data rate and decrease the error rate for wireless communication systems [260]–[262]. Adaptive MIMO transmission techniques are investigated for both uncoded and coded systems. 1) Uncoded Systems: The capacity-achieving coding usually has long (infinitely) codewords and brings delay to the system. Therefore, for the delay-sensitive applications, the modulation part of physical layer is usually designed separately before the outer error-correcting coding is applied, as shown in Figure 10. For the uncoded SISO system subject to the average power and the BER (instantaneous or average) constraints, the maximization of the spectral efficiency is obtained by jointly optimizing the transmit rate and power over Rayleigh fading channels in [263], [264] and Nakagmi fading channels in [265], developing upon the previous adaptive transmission techniques [266]–[276]. Z. Zhou et al. investigate adaptive transmission schemes for i.i.d. Rayleigh flat fading MIMO channels with perfect or imperfect CSIT and CSIR [41]. With perfect CSIT and CSIR, the MIMO channels are decomposed into a set of parallel SISO subchannels by SVD. Then, the transmit rate and power allocation of each subchannel need to be optimally designed to maximize the average spectral efficiency under the average transmit power and instantaneous subchannel BER constraints. It is proved that this optimization problem can be transformed into several subproblems, each of which only involves one subchannel. Then, the adaptive transmission designs for SISO channels in [263], [264] can be exploited to find the optimal solution. Moreover, the effect of the discrete rate constraint and the imperfect CSIT on system performance are evaluated in [41]. When only partial CSI, i.e., the channel mean feeback, is available at the transmitter,

49

S. Zhou et al. provide an adaptive two-dimensional beamformer to exploit the transmit diversity by incorporating the Alamouti coded [42]. Based on this beamformer, the beamvectors, the power allocation policy, and the modulation schemes are designed for MIMO and MIMO-OFDM systems in [42] and [277], respectively, which maximize the average discrete spectral efficiency under the constant power and the average BER constraints. To further study the impact of the CSI error on adaptive MIMO transmission, S. Zhou et al. consider the case where the transmitter performs a pilot signal assisted channel prediction and examine the effect of the prediction error on the BER performance for a one-dimensional adaptive beamformer [278]. Numerical results show that when the normalized estimation error is below a critical threshold, the channel prediction error has a minor impact on the BER performance. For correlated MISO Rayleigh fading channels with a limited number of feedback bits at the transmitter, P. Xia et al. develop a strategy to adaptively select the transmission mode to maximize the average spectral efficiency under the average power and the average BER constraints [279]. For correlated MIMO fading channels where the correlation matrices are available at the transmitter and equal power is allocated to each transmit antenna, a strategy to determine the number of active transmit antenna and the corresponding transmit constellations is developed by maximizing a low bound of the minimum SNR margin with a fixed data rate [280]. R. W. Heath Jr et al. propose a simple MIMO adaptive transmission scheme by switching between spatial multiplexing and transmit diversity [281]. With perfect CSIR, both the minimum Euclidean distances of spatial multiplexing scheme and STBC scheme which achieve a fixed transmit rate are computed at the receiver. The scheme with larger minimum Euclidean distances is chosen and the decision is feedback to the transmitter via a low-rate feedback channel. For double-input multiple-output (DIMO) systems with both PCSIT and SCSIT, an adaptive scheme switching between OSTBC and spatial multiplexing with zero-forcing (ZF) receiver based on the average spectral efficiency is proposed in [282]. For an adaptive MIMO-OFDM system employing maximum ratio transmission (MRT) and MRC transceiver structure, the statistic properties of the number of bits transmitted per OFDM block and the number of outage per OFDM block over i.i.d. Rayleigh fading channels are derived in [283]. A brief comparison of above adaptive MIMO transmitter designs is given in Table XVIII.

50

TABLE XVIII: Adaptive MIMO transmitter designs for uncoded systems Paper

System

Criterion

CSI

Constraint

MIMO

Maximize average

PCSIT, IPCSIT

Instantaneous BER

i.i.d. Rayleigh fading

spectral efficiency

PCSIR

Average transmit power

MIMO

Maximize average

Channel mean feedback

Average BER

i.i.d. Rician fading

spectral efficiency

PCSIR

Constant transmit power

MIMO-OFDM

Maximize average

Channel mean feedback

Average BER

i.i.d. Rician fading

spectral efficiency

PCSIR


MIMO

Maximize average

Z. Zhou et al. [41]

S. Zhou et al. [42]

P. Xia et al. [277]

S. Zhou et al. [278]

Instantaneous BER Channel prediction error


spectral efficiency


MISO

Maximize average

Limited feedback

Average BER


spectral efficiency

PCSIR


MIMO

Maximize minimal

SCSIT

Fixed data rate

correlated Rayleigh fading

SNR margin

PCSIR

Equal power allocation

Spatial Multiplexing

Feedback decision

OSTBC switch

PCSIR

DIMO

Spatial Multiplexing

PCSIT, SCSIT


OSTBC switch

PCSIR

P. Xia et al. [279]

R. Narasimhan [280]

R. W. Heath Jr [281]

MIMO

Fixed data rate

J. Huang [282]

ZF receiver MIMO-OFDM

K. P. Kongara [283]

PCSIT

MRT

PCSIR

MRC

Performance Analysis i.i.d. Rayleigh fading

noise

input

Modulation

Precoder

MIMO Channel

Receiver

output

Adaptive Design

Feedback from Receiver

Fig. 10:

Adaptive MIMO transmission for uncoded systems

Some work study the adaptive MIMO transceiver designs. For perfect CSIT and CSIR, D. P. Palomar et al. investigate the transmit constellation selection and the linear transceiver design to minimize the

51

TABLE XIX: Adaptive MIMO transceiver designs for uncoded systems Paper

System


MIMO

A. Yasotharan et al. [285]

S. Bergman et al. [286]

C.-C. Li et al. [287]

Criterion

CSI

Constraint

Minimize

PCSIT

Instantaneous BER

transmit power

PCSIR

Fixed transmit rate

Minimize

PCSIT

ZF transceiver, constant transmit power

average BER

PCSIR

Fixed transmit rate

Minimize

PCSIT

DFE receiver, constant transmit power

weighted MSE

PCSIR

Fixed transmit rate

Maximize

PCSIT

Instantaneous BER

transmit rate

PCSIR


MIMO

MIMO

MIMO

transmit power with fixed transmit rate, where each subchannel is under the same BER constraint [284]. Assuming the symbol error probability of each subchannel is the same, the gap approximation method is used to select the transmit constellations. Then, necessary and sufficient conditions are established to examine the optimality of the subchannel diagonalization design. Moreover, an upper bound of the transmit power loss incurred by using the subchannel diagonalization design is presented. On the other hand, an optimal bit loading scheme among subchannels is developed in [285]. The proposed scheme minimizes a lower bound of the average BER of the system subject to the fixed transmit data, constant transmit power, and ZF transceiver structure constraints. S. Bergman et al. further obtain the optimal transceiver design and the bit loading scheme among subchannels with DFE receiver [286]. The p-norms of weighed MSEs is minimized under the fixed transmit data and the constant transmit power constraints. The subchannel diagonalization design is proved to be always optimal for the system model considered in [286]. C.-C. Li et al. establish a duality relationship between the joint linear transceiver and bit loading design of maximizing the transmit rate and that of minimizing the transmit power in [287]. Then, an algorithm can be developed to find the optimal design for the transmit rate maximization by using the solution of transmit power minimization problem. A brief comparison of above adaptive MIMO transceiver designs is given in Table XX. With perfect CSIR, Y. Ko et al. propose a rate adaptive modulation scheme combined with OSTBC for i.i.d. Rayleigh fading MIMO channels [288]. Close-form expressions for the average spectral efficiency

52

and a tight upper bound of the average BER in presence of feedback delay are derived. Then, optimal SNR switching thresholds for adaptive modulation, which maximize the average spectral efficiency under the average BER, the outage BER, and the constant transmit power constraints, are designed at the receiver. The constellation chosen at the receiver is feeded back to the transmitter and the impact of feedback delay on the average BER, throughout, and outage probability are analyzed. The design in [288] is extended to spatially correlated Rayleigh fading channels in [289], where a low complexity method to determine a set of near optimal SNR switching thresholds is provided. On the other hand, the design in [288] is also extended to the case with the channel estimation error and the average transmit power constraint in [290]. For MIMO-OSTBC systems, the channel estimation process of the rate adaptive modulation scheme is investigated [291]. The transmit constellation and the allocation of the time and power between the pilot and data symbols are optimally designed to maximize the average spectral efficiency under the instantaneous BER constraint. The adaptive MIMO transmission scheme for the energy efficiency maximization of MIMO-OSTBC systems with the channel estimation error subject to an instantaneous BER constraint is proposed in [43]. By utilizing the imperfect CSIT, Q. Kuang et al. incorporate the design of spatial power allocation between different transmit antennas into the adaptive modulation scheme in [290] to further increase the average spectral efficiency [44]. A brief comparison of above adaptive transmission designs for MIMO-OSTBC systems is given in Table XX. Other adaptive MIMO transmission schemes with outdated CSIT and imperfect CSIT are investigated in [292] and [293], [294], respectively. X. Zhang et al. and T. Delamotte et al. consider the adaptive power allocation for each subchannel with the instantaneous BER constraint based on the modified waterfilling and mercury water-filling algorithms in [295] and [296], respectively. T. Haustein et al. develop an adaptive power allocation strategy for each subchannel by minimizing the MSEs at the receiver under the maximum average BER constraint [297]. Some adaptive MIMO transmission schemes by providing the reconfigurable operation modes of the antenna arrays are discussed in [298], [299]. 2) Coded Systems: The capacity expression in [1] has provided a fundamental transmission limit for MIMO system. Since then, adaptive modulation and coding (AMC) schemes have been extensively

53

TABLE XX: Adaptive transmission designs for MIMO-OSTBC uncoded systems Paper

System

Criterion

CSI

Constraint

MIMO

Maximize average

Feedback decision with delay

Average BER, outage BER


spectral efficiency

PCSIR


MIMO

Maximize average

Perfect channel estimation

Instantaneous BER, average BER


spectral efficiency

Imperfect channel estimation


MIMO

Maximize average

Y. Ko et al. [288]

J. Huang et al. [289]

X. Yu et al. [290]

Average BER Imperfect channel estimation


spectral efficiency

MIMO

Maximize average

D. V. Duong et al. [291]

Average transmit power Instantaneous BER Channel estimation optimization


spectral efficiency


MIMO

Maximize

IPCSIT

Instantaneous BER


energy efficiency

PCSIR


MIMO

Maximize average

IPCSIT

Average BER


spectral efficiency

PCSIR


L. Chen et al. [43]

Q. Kuang et al. [44]

investigated to approach this performance limit. M. R. McKay et al. consider the AMC for the correlated Rayleigh fading MIMO channels [45]. The BICM is used for the coding. With SCSIT, two simple transmission schemes are switched: statistical beamforming or spatial multiplexing with a zero-forcing receiver. Then, the transmitter selects the best combination of the code rate, the modulation formate, and the transmission scheme to maximize the data rate subject to a general BER union bound. Adaptive DIMO switching between the spatial multiplexing and STBC based on the spectral efficiency maximization for a non-selective channel with practical constraints, impairments, and receiver designs is investigated in [46]. For spatial multiplexing or STBC, the spectral efficiency thresholds to select the best coding rate and the modulation format is determined based on the packet error rate (PER) vs. instantaneous capacity simulation curves (convolutional code). The selection is performed at the receiver with perfect CSIR and then feeds back to the transmitter. Also, for a time- and frequency-selective channel, a new physical layer abstraction and switching algorithm is proposed in [46], where a weighed sum of channel qualities is minimized to reduce the variance of the channel qualities. P. H. Tan et al. investigate the AMC for MIMO-OFDM systems in presence of channel estimation errors in slow fading channels [47]. The convolutional or low density parity check codes (LDPC) coded BER at the output of the detector is evaluated via curving fitting method. The packet error rate (PER) for each modulation and coding

54

scheme is accurately approximated based on the obtained coded BER expression. Then, the code rate, the modulation format, and the number of transmit streams are selected by maximizing the system throughout under a fixed PER constraint. A supervised learning algorithm, called k-nearest neighbor, is used to select the AMC parameters by maximizing the data rate under frame error rate (FER) constraints for MIMOOFDM systems with convolutional codes in frequency- and spatial-selective channels [300]. Assuming perfect CSIT and CSIR, M. D. Dorrance et al. consider the AMC for a vertical bell laboratories layer space-time (V-BLAST) MIMO system with rate-compatible LDPC codes and an incremental redundancy hybrid automatic repeat request (H-ARQ) scheme [301]. A constrained water-filling algorithm for finite alphabet input signals is provided to allocate power for each subchannel. Then, the modulation format used for each subchannel is determined based on a look-up table. Z. Zhou et al. incorporate the BICM into the adaptive transmission scheme in [41] to improve the BER robustness against the CSI feedback delay in [48]. Moreover, BICM technology is modified by using a multilevel puncturing and interleaving technique and combined with rate-compatible punctured code/turbo code to achieve a near full multiplexing gain. An AMC strategy for BICM MIMO-OFDM systems with soft Viterbi decoding is proposed in [302]. A system performance metric called the expect goodput is defined in [302], which is a function of the code rate, the modulation format, the subchannel power allocation, and the space-time-frequency precoder. The transmitter determines these AMC parameters by maximizing the expect goodput via the feedback of the perfect CSI and the effective SNR from the receiver. Spectral efficiencies of the adaptive MIMO transmission schemes in real-time practical test-bed are shown in [303]. A brief comparison of above AMC schemes for MIMO systems is given in Table XXI. The current wireless standards such as 3GPP LTE, IEEE 802.16e, and IEEE 802.11n define concrete implementation procedures for adaptive MIMO transmission in practice. We take the downlink transmission in LTE standard as an example. The code rate and modulation format are selected based on the received Channel Quality Indicator (CQI) user feedback. For example, the 4-bit CQI indices and their interpretations in LTE [304] are given in Table XXII. Then, the modulated transmit symbols are mapped into one or several layers. This antenna mapping depends on the Rank Indicator user feedback,

55

TABLE XXI: AMC schemes for MIMO systems Paper

System

Criterion

CSI

MIMO

Statistical beamforming

SCSIT


spatial multiplexing switch

PCSIR

DIMO, non-selective

Spatial multiplexing

Feedback decision

time- and frequency-selective

STBC switch

PCSIR

MIMO-OFDM

Maximize

slow fading

system throughout

MIMO-OFDM

Maximize average

PCSIT

frequency- and spatial-selective

spectral efficiency

PCSIR

V-BLAST MIMO

Maximize average

PCSIT

Rate-constrained

H-ARQ

spectral efficiency

PCSIR

power, bit allocation

MIMO

Maximize average

IPCSIT

Instantaneous BER


spectral efficiency

PCSIR


Maximize

PCSIT

expect goodput

PCSIR

Practical MIMO

Maximize average

Channel estimation

test bed

spectral efficiency

in practical systems

M. R. McKay et al. [45]

Constraint BER union bound

Y.-S. Choi et al. [46]

Practical PER

P. H. Tan et al. [47]

Channel estimation error

R. C. Daniels et al. [300]

Fixed PER

Fixed FER

M. D. Dorranceet al. [301]

Z. Zhou et al. [48]

I. Stupia et al. [302]

MIMO-OFDM

Soft Viterbi decoding

T. Haustein et al. [303]

Average BER

which includes a single antenna port, transmit diversity, and spital multiplexing [304, Table 7.2.3-0]. The precoding matrices for these transmission modes depend on the Precoding Matrix Indicator user feedback, which are defined in details in [305, Sec. 6.3.4]. The adaptive MIMO transmission schemes in IEEE 802.16e and IEEE 802.11n are similar. In IEEE 802.16e, 52 combinations of codes, code rate, and modulation format are defined as in [306, Table 8.4]. The transmission mapping matrices includes [306, Eq. (8.7)]: Matrix A, exploiting only diversity; Matrix B, combining diversity and spatial multiplexing; Matrix C, employing spatial multiplexing. Then, a transmission mode is selected based on the link quality evaluation. In IEEE 802.11n, 16 mandatory modulation and coding schemes are defined [307, Table 1]. A spatial multiplexing matrix is applied to convert the transmitted data streams into the transmit antennas and an additional cyclic deay can be applied per transmitter to provide transmit cyclic delay diversity [307, Figure 1]. Some practical AMC simulation results in these standards are shown in [262], [300], [307]–[313].

56

TABLE XXII: 4 bit CQI table in LTE standard CQI index

Modulation

0

Code rate × 1024

Efficiency

out of range

1

QPSK

78

0.1523

2

QPSK

120

0.2344

3

QPSK

193

0.3770

4

QPSK

308

0.6016

5

QPSK

449

0.8770

6

QPSK

602

1.1758

7

16QAM

378

1.4766

8

16QAM

490

1.9141

9

16QAM

616

2.4063

10

64QAM

466

2.7305

11

64QAM

567

3.3223

12

64QAM

666

3.9023

13

64QAM

772

4.5234

14

64QAM

873

5.1152

15

64QAM

948

5.5547

noise

input

Encoder

Modulation

Precoder

MIMO Channel

Receiver

output

Adaptive Design

Feedback from Receiver

Fig. 11:

Adaptive MIMO transmission for coded systems IV. M ULTI - USER MIMO S YSTEMS

In this section, we introduce the transmission designs with discrete input signals for multiuser MIMO systems. We discuss the following four scenarios: i) MIMO uplink transmission; ii) MIMO downlink transmission; iii) MIMO interference channel; iv) MIMO wiretap channel.

57

A. MIMO Uplink Transmission 1) Mutual Information Design: For the multiple access channel (MAC), the maximum mutual information region with uniformly distributed finite alphabet inputs is defined as the constellation-constrained capacity region [314], [315]. For the MIMO MAC with PCSIT, M. Wang et al. derive the constellationconstrained capacity region for an arbitrary number of users and arbitrary antennas configurations [20]. The boundary of this region can be achieved by solving the weighted sum-rate (WSR) maximization problem with finite alphabet and individual power constraints. Necessary conditions of the optimal precoders of all users are established and an iterative algorithm is proposed to find these optimal precoders. Moreover, a LDPC coded system with iterative detection and decoding is provided to examine the BER performance of the obtained precoders. Numerical results indicate the obtained precoders achieve considerably better performance than the non-precoding design and the Gaussian input design in terms of both mutual information and coded BER. For the MIMO MAC with SCSIT, asymptotic expressions for the mutual information of the MIMO MAC with arbitrary inputs over Kronecker fading channels are derived in large system limits [118]. The linear precoder design for the MIMO MAC is further developed based on the sum-rate maximization [316]. The proposed design embraces the case where non-Gaussian interferers are present. For a more general Weichelberger’s fading model in [317], W. Yu et al. investigate the linear precoder design based on the WSR maximization [318]. The asymptotic (in large system limits) optimal left singular matrix of each user’s optimal precoder is proved to be the eigenmatrix of the transmit correlation matrix of the user. This results facilitates the derivation of an efficient iterative algorithm for computing the optimal precoder for each user. A brief summation of above mutual information based precoder designs for the MIMO uplink transmission with discrete input signals is given in Table XXIII. 2) MSE Design: For the MIMO MAC with PCSIT and PCSIR, E. A. Jorswieck et al. first design the transmit covariance matrices of the users to minimize the sum MSE with the multiuser MMSE receiver and individual user power constraints [50]. A power allocation scheme among users is developed under a sum power constraint to further reduce the sum MSE. Based on the KKT conditions, the optimal transmit

58

TABLE XXIII: Mutual information based precoder designs for the MIMO uplink transmission with discrete input signals Paper

Model

Main Contribution

MIMO MAC

Propose an iterative algorithm to find

PCSIT, PCSIR

the optimal precoders which maximize the WSR

MIMO MAC, Kronecker fading

Derive asymptotic expressions for the mutual information

SCSIT, PCSIR

of the MIMO MAC with arbitrary inputs

M. Wang et al. [20]

C.-K. Wen et al. [118] MIMO MAC, Kronecker fading Propose an iterative algorithm to find M. A. Girnyk et al. [316]

SCSIT, PCSIR the optimal precoders which maximize the sum-rate non-Gaussian interferers MIMO MAC, Weichelberger’s fading

Find the asymptotic optimal left singular matrix of each user’s optimal precoder

SCSIT, PCSIR

Propose an efficient iterative algorithm to maximize WSR

Y. Wu et al. [318]

covariance matrices and the power allocation scheme can be found by an iterative algorithm. In addition, the achievable MSE region is studied in [50]. S. Serbetli et al. investigate how to determine the number of transmit symbols for each user [319]. Z.-Q. Luo et al. extend the MMSE transceiver design to MIMO MAC OFDM systems [320]. I.-T. Lu et al. further study the impact of the practical per-antenna power and peak power constraints [321]. The MMSE transceiver design for MIMO MAC with channel mean or covariance SCSIT feedback and PCSIR subject to individual user power constraints is investigated in [322], [323]. For both sum and individual user power constraints, X. Zhang et al. study the average sum MSE minimization problem for two user MIMO MAC with correlated estimation error IPCSIT and IPCSIR [324]. P. Layec et al. further consider the K user MIMO MAC case with additional quantization noise and individual user power constraints [325]. A near optimal closed-form transceiver structure, which minimizes the average sum MSE for MIMO MAC with i.i.d estimation error IPCSIT and IPCSIR subject to a sum power constraint, is provided in [326]. A brief summation of above MMSE precoder designs for the MIMO uplink transmission with discrete input signals is given in Table XXIV. 3) Diversity Design: For open-loop systems, based on the framework for analyzing the dominant error event regions in [327], M. E. Gärtner et al. derive space-time/frequency code design criterions for twouser fading MIMO MAC [328]. Moreover, an explicit two-user 2 × 2 coding scheme is constructed by concatenating two Alamouti schemes with a column swapping for one user’s codeword to achieve a

59

TABLE XXIV: MMSE precoder designs for the MIMO uplink transmission with discrete input signals Paper

Model

Main Contribution

MIMO MAC E. A. Jorswieck et al. [50]

PCSIT, PCSIR

Propose an iterative algorithm to minimize the sum MSE

Individual/sum power constraints MIMO MAC S. Serbetli et al. [319]

PCSIT, PCSIR

Investigate how to determine the number of transmit symbols

Individual power constraints MIMO MAC OFDM Z.-Q. Luo et al. [320]

PCSIT, PCSIR

Extend the MMSE transceiver design to OFDM systems

Individual power constraints MIMO MAC I.-T. Lu et al. [321]

PCSIT, PCSIR

Propose iterative algorithms to minimize the sum MSE

Per-antenna/peak power constraints MIMO MAC [322], [323]

SCSIT, PCSIR

Propose iterative algorithms to minimize the average sum MSE

Individual power constraints Two-user MIMO MAC X. Zhang et al. [324]

IPCSIT, IPCSIR


Individual/sum power constraints MIMO MAC P. Layec et al. [325]

IPCSIT, IPCSIR, Quantization noise


Individual power constraints MIMO MAC J. W. Huang et al. [326]

IPCSIT, IPCSIR

Provide closed-form near optimal MMSE precoder designs

A sum power constraint

minimum rank of three. A algebraic construction of STBC based on diversity and multiplexing tradeoff for MIMO MAC is proposed in [329]. Similar STBC designs are also provided for the asynchronous single-input multiple-output (SIMO) MAC in [330], non-uniform user transmit power MIMO MAC in [331], and the two-user MIMO multiple-access AF relay channel in [332]. A space-frequency code for MIMO-OFDM MAC is designed in [49], which achieves full diversity for every user. However, the codes in [49] suffer from high large peak-to-average power ratio since some elements in the codeword matrices are zero. Another group of STBC for MIMO MAC is proposed in [333] by minimizing a truncated unionbound approximation. Simulations show that the codes in [333] achieve better error probability than the

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TABLE XXV: Space-time techniques for the MIMO uplink transmission with discrete input signals Paper

Model

Main Contribution

MIMO MAC

Derive space-time/frequency code design criterions

Two-user

Construct an explicit coding scheme for two-user 2 × 2

MIMO MAC, asynchronous SIMO MAC, unbalanced MIMO MAC

Construct STBCs based on

Two-user MIMO multiple-access AF relay

diversity and multiplexing tradeoff

M. E. Gärtner et al. [328]

M. Badr et al. [329]–[332] Design a space-frequency code W. Zhang et al. [49]

MIMO-OFDM MAC which achieves full diversity for every user Design STBCs based on

Y. Hong et al. [333]

MIMO MAC truncated union-bound approximation MIMO MAC

[334], [335]

Design sphere-decodable STBCs Two-user MIMO MAC

[336], [337]

Design differential STBCs Two-user MIMO MAC

Investigate the linear precoding design

Outdated CSIT

subject to individual user’s transmit power constraint

J. W. Huang et al. [338] DIMO MAC Y.-J. Kim et al. [339]

Design a STBC with a linear detection Two-user, feedback Two-user MIMO MAC, K-user MIMO MAC

F. Li et al. [340], [341]

Propose full-diversity precoder designs PCSIT

codes in [49], [328]. H.-F. Lu et al. propose sphere-decodable STBCs for two-user MIMO MAC and provide an explicit construction for 2 × 2 case [334]. STBCs for two-user MIMO MAC with reduced average sphere decoding complexity are further constructed in [335]. Moreover, A differential STBC for two-user MIMO MAC is proposed in [336]. Another differential STBC for two-user MIMO MAC with low complexity non-coherent decoders is further designed in [337]. For close-loop systems, by exploiting the outdated CSI, J. W. Huang et al. investigate the linear precoding design for MIMO MAC with OSTBC to minimize PEP subject to individual transmit power constraint of each user [338]. Also, Y.-J. Kim et al. propose a STBC for two-user DIMO MAC with a linear detection by utilizing the phase feedback transmitted to one user [339]. By exploiting PCSIT, F. Li et al. propose full-diversity precoder designs for two-user and K-user MIMO MAC in [340] and [341], respectively. A brief comparison of above work is given in Table XXV. 4) Adaptive Transmission: For MIMO-OFDM MAC with PCSIT and PCSIR, Y. J. Zhang et al. propose an adaptive resource allocation scheme when each user’s signal is transmitted along the maximum singular

61

TABLE XXVI: Adaptive transmissions for the MIMO uplink transmission with discrete input signals Paper

Model

Main Contribution

MIMO-OFDM MAC, perfect CSI

Propose an adaptive resource allocation

Each user’s uncoded BER and data rate constraints

to minimize the total transmit power


Propose a low complexity adaptive resource

Each user’s uncoded BER and data rate constraints

allocation to minimize the total transmit power

MIMO-OFDM MAC, SCSIT, PCSIR

Propose an adaptive resource allocation

Each user’s uncoded BER and total power constraints

to maximize the sum data rate


Propose an adaptive resource allocation considering the

Each user’s uncoded BER and total power constraints

users’ fairness and maximizing the sum data rate

Y. J. Zhang et al. [51]

Y. J. Zhang et al. [342]

R. Chemaly et al. [343]

M. S. Maw et al. [344]

of its own channel [51]. The proposed scheme selects the subcarrier allocation, the power allocation, and modulation modes to minimize the total transmit power subject to each user’s uncoded BER and data rate constraints. It is assumed that these uplink transmission parameters are determined at the base station and then sent to the users via the error-free channel. Simulation results indicate that within a reasonable region of Doppler spread, the proposed scheme is also robust to the channel time variation. A low-complexity design with a neighborhood search for the optimal resource allocation solution is further proposed in [342]. For MIMO-OFDM MAC where each user employs the transmission design in [279] with its own SCSI, R. Chemaly et al. propose an adaptive resource allocation scheme to maximize the sum data rate subject to each user’s uncoded BER and the total power constraints [343]. By taking an additional fairness consideration among different users, another adaptive resource allocation scheme is given in [344] to maximize the sum data rate subject to each user’s uncoded BER and the total power constraints. A brief comparison of above work is given in Table XXVI.

B. MIMO Downlink Transmission 1) Mutual Information Design: For the 2-user MISO BC with PCSIT, PCSIR, and finite alphabet inputs, the precoder designs to maximize the sum rate and the corresponding simplified receiver structure are studied in [345]–[347]. Y. Wu et al. further investigate the linear precoder design to maximize the WSR of the general MIMO BC [21]. Explicit expressions for the achievable rate region are derived, which is applicable to an arbitrary number of users with generic antenna configurations. Then, it is revealed that the

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TABLE XXVII: Mutual information based precoder designs for the MIMO downlink transmission with discrete input signals Paper

Model

Main Contribution

MISO BC, 2-user

Propose the precoder designs and the corresponding

PCSIT, PCSIR

simplified receiver structure to maximize the sum-rate

MIMO BC

Propose an iterative algorithm to find

PCSIT, PCSIR

the optimal precoders which maximize the WSR

Cooperative multi-cell MIMO downlink

Propose two low-complexity algorithms to find

PCSIT, PCSIR

the optimal precoders which maximize the sum-rate

MISO downlink multicasting

Propose two transmit schemes

PCSIT, PCSIR

to maximize the multicast rate

R. Ghaffar et al. [345]–[347]

Y. Wu et al. [21]

W. Wu et al. [349]

S. X. Wu et al. [350]

sum rate loss will occur in high SNR regime for the improper precoder design because of the non-uniquely decodable transmit signals. An iterative algorithm is proposed to optimize precoding matrices for all users. A downlink multiuser system with LDPC code and iterative detection and decoding is further developed. Simulations illustrate that the proposed precoding design provides substantial WSR and coded BER gains over the best rotation design for SISO BC [348], the non-precoding design, and the Gaussian input design. W. Wu et al. investigate the linear precoder design for cooperative multi-cell MIMO downlink systems with PCSIT and finite alphabet inputs [349]. To reduce the complexity of evaluating the non-Gaussian interferers, W. Wu et al. use a Gaussian interferer to approximate the sum of non-Gaussian interferers at each user’s receive side and propose two iterative algorithms to maximize the approximated sum-rate under per base station power constraints. Comparing to the algorithm in [21], the proposed algorithms in [349] reduce the implementation complexity but with negligible performance losses. S. X. Wu et al. propose two transmit schemes to maximize the multicast rate of MISO downlink multicasting systems with PCSIT and finite alphabet inputs [350]. A brief summation of above mutual information based precoder designs for the MIMO downlink transmission with discrete input signals is given in Table XXVII. 2) MSE Design: For the linear processing, early research in [351], [352] consider to extend the MMSE transceiver design in [32] to MIMO BC. For MIMO BC with PCSIT, PCSIR, and a sum power constraint, S. Shi et al. establish a uplink-downlink duality framework between the downlink and uplink MSE feasible

63

regions [53]. Based on this framework, the downlink sum-MSE minimization can be transformed into an equivalent uplink problem. Then, two globally optimum algorithms are proposed to find the optimal MMSE transceiver design. S. Shi et al. further propose near-optimal low complexity iterative algorithms to minimize the maximum ratio of MSE and given requirement under a sum power constraint and to minimize the transmit power under various MSE requirements [353]. Moreover, R. Hunger et al. propose a low complexity approach to evaluate the MSE duality [354], which is able to support the switched off data streams and the passive users correctly. The MSE duality in [53] is extended to the imperfect CSI case for MISO BC [355]. D. S.-Murga et al. provide a transceiver design framework for MIMO-OFDM BC based on the channel Gram matrices feedback, where the CSI quantization error is also taken into consideration [356]. As an example, a robust precoder design with fixed decoder which minimizes the sum MSE subject to a sum power constraint is given in [356]. M. Ding et al. propose a robust transceiver design for network MIMO systems with imperfect backhaul links [357], which minimizes the maximum substream MSE subject to per base station power constraints. For the robust design where the actual channel is considered to be within an uncertain region around the channel estimate, a robust power allocation scheme among users is proposed for MISO BC to minimize the transmit power subject to each user’s MSE requirement [358]. N. Viˇcić et al. extend this to robust transceiver design for MISO BC and MIMO BC in [359] and [360], respectively. X. He et al. further investigate this robust transceiver design subject to the probabilistic MSE requirement of each user [361]. A robust resource allocation design to optimize the function of worst case MSE subject to quadratic power constraints for multicell MISO downlink transmission is proposed in [362]. Some work consider the MMSE precoder design with the fixed decoder. A. D. Dabbagh et al. obtain an MMSE based precoding technique for MISO BC with IPCSIT and a sum power constraint [363]. The limited feedback system model is also investigated in [363]. H. Sung et al. develop a generalized MMSE precoder for MIMO BC with perfect and imperfect CSI [364]. The designs for the minimization of the sum MSE subject to a sum power constraint and the minimization of the interference-plus-noise power at the receiver subject to individual user’s power constraints are obtained. P. Xiao et al. propose an improved

64

MIMO BC MMSE precoder with perfect and imperfect CSI subject to a sum power constraint for improper signal constellations such as amplitude shift-keying, offset quadrature phase shift keying, etc [365]. A brief summation of above linear MMSE precoder designs for the MIMO downlink transmission with discrete input signals is given in Table XXVIII. There are also some work studying the non-linear MMSE precoder designs for the MIMO downlink transmission with discrete input signals [53], [366]–[371]. 3) Diversity Design: In the downlink transmission, mitigating the multiuser interference is essential for the reliable communication and this normally needs the availability of some forms of CSI. Therefore, there are not much work for the open loop downlink systems. Q. Ma et al. propose a trace-orthogonal space-time coding, which is suitable for MIMO BC [372]. The proposed space-time coding reduce the complexity of ML with little performance loss. Some differential STBC designs for the data encoded using layered source coding are proposed for MIMO BC [373], [374]. For the close loop systems, the space-time techniques are usually combined with the linear precoding design to eliminate the cochannel interference and improve the diversity performance. R. Chen et al. propose a unitary downlink precoder design for multiuser MIMO STBC systems with PCSIT [375]. Using enough antenna dimension, the proposed precoder can effectively cancel the co-channel interference at each user’s side. Then, the additional antenna dimension can be used to select the precoder matrix for the diversity gain. R. Chen et al. further provide a more general transmission design for MIMO BC [52], which combines the unitary downlink precoder for co-channel interference mitigation with the space-time precoder design and antenna selection technique for SER minimization. Simulations indicate that the proposed design achieves significant diversity gains in terms of SER. D. Wang et al. combine nonlinear THP with two diversity techniques: dominant eigenmode transmission and OSTBC to improve the BER performance of MIMO BC [376]. B. Clerckx et al. investigate the transmission strategy for two-user MISO BC with outdated CSIT and obtain the analytical expressions for the corresponding average PEP over correlated Rayleigh fading channel [377]. By exploiting the further SCSIT, signal constellations are optimized to improve the average PEP performance. D. Lee et al. analyze the outage probability of the average effective SNR for MIMO BC employing OSTBC [378]. The analytical results indicate that the opportunistic scheduling

65

TABLE XXVIII: Linear MMSE precoder designs for the MIMO downlink transmission with discrete input signals Paper

Model

Main Contribution

MIMO BC S. Shi et al. [53]

PCSIT, PCSIR

Establish a uplink-downlink MSE duality framework

A sum power constraint MIMO BC

Minimize the maximum ratio of MSE and given requirement

PCSIT, PCSIR

Minimize the transmit power under various MSE requirements

S. Shi et al. [353] MIMO BC Establish a MSE duality, which is able to support R. Hunger et al. [354]

PCSIT, PCSIR the switched off data streams and the passive users correctly A sum power constraint MISO BC


IPCSIT, IPCSIR

Extend the MSE duality in [53] to the imperfect CSI case

A sum power constraint MIMO-OFDM BC Provide a transceiver design framework D. S.-Murga et al. [356]

Channel Gram matrix based on the channel Gram matrices feedback A sum power constraint Network MIMO systems


Imperfect backhaul links

Minimize the maximum substream MSE

Per base station power constraints MISO BC Propose a robust power allocation scheme among users M. Payaró et al. [358]

IPCSIT, IPCSIR to minimize the transmit power Each user’s MSE requirement MISO BC Propose a robust transceiver design

N. Viˇcić et al. [359]

IPCSIT, IPCSIR to minimize the transmit power Each user’s MSE requirement MIMO BC

Propose robust transceiver designs

IPCSIT, IPCSIR

for several MSE-optimization problem

N. Viˇcić et al. [360] MIMO BC Propose a robust transceiver design X. He et al. [361]

IPCSIT, IPCSIR to minimize the transmit power Each user’s probabilistic MSE requirement MISO BC Propose a robust resource allocation design

E. Björnson et al. [362]

IPCSIT, IPCSIR to optimize the function of worst case MSE Quadratic power constraints MISO BC, fixed decoder

A. D. Dabbagh et al. [363]

IPCSIT, PCSIR

Obtain an MMSE based precoding technique

A sum power constraint MIMO BC, fixed decoder

Minimize the sum MSE subject to a sum power constraint

Perfect and imperfect CSI

Minimize the interference-plus-noise power subject to individual user’s power constraints

H. Sung et al. [364] MIMO BC, fixed decoder Propose an improved MIMO BC MMSE precoder P. Xiao et al. [365]

Perfect and imperfect CSI for improper signal constellations A sum power constraint

66

TABLE XXIX: Space-time techniques for the MIMO downlink transmission with discrete input signals Paper

Model

Main Contribution

Q. Ma et al. [372]

MIMO BC

Propose a trace-orthogonal space-time coding

[373], [374]

MIMO BC

Propose differential STBC designs for the data encoded using layered source coding MIMO BC

Combine a unitary downlink precoder

PCSIT, PCSIR

with STBC

MIMO BC

Combine a unitary downlink precoder

PCSIT, PCSIR

with STBC and antenna selection

MIMO BC

Combine THP with dominant eigenmode

PCSIT, PCSIR

transmission and OSTBC

MISO BC

Optimize signal constellation

two-user, outdated CSIT, PCSIR

to improve the average PEP

MIMO BC

Analyze opportunistic scheduling and diagonalization-precoding

PCSIT, PCSIR

for MIMO BC with OSTBC

MIDO BC

Combine a interference cancellation scheme

PCSIT, no CSIR

with the Alamouti code

MIMO BC

Extend X-Structure precoder

PCSIT, PCSIR

to MIMO BC

R. Chen et al. [375]

R. Chen et al. [52]

D. Wang et al. [376]

B. Clerckx et al. [377]

D. Lee et al. [378]

L. Li et al. [379]

C.-T. Lin et al. [380]

scheme provide the effective SNR and the diversity gains compared to the block diagonalization-precoding scheme. With PCSIT and no CSIR, L. Li et al. propose a interference cancellation scheme for MIDO BC, where the transmitter sends the precoded Alamouti code to every user simultaneously [379]. Numerical results indicate that the proposed design in [379] has better diversity gains than the block diagonalization designs in [375], [378]. By using a regularized block diagonalization precoder to suppress the multiuser interference in the first step, C.-T. Lin et al. extend the X-Structure precoder in [40] to MIMO BC in [380]. A brief comparison of above work is given in Table XXIX. 4) Adaptive Transmission: For uncoded systems, most adaptive transmission literatures focus on the adaptive resource allocation for multiuser MIMO-OFDM systems. Typically, with perfect CSIT and CSIR, C.-F. Tsai et al. devise an adaptive resource allocation algorithm for multiuser downlink MISO orthogonal frequency division multiplexing access (OFDMA)/spatial division multiple access (SDMA) systems with multimedia traffic [54]. A dynamical priority scheduling scheme is developed to provide service to urgent

67

users based on time to expiration. Then, the resource for power, subchannel, and bit allocation is iteratively allocated to maximize the spectrum efficiency subject to various quality of service constraints. C.-M. Yen et al. extend this adaptive resource allocation design to multiuser downlink MIMO OFDMA systems [381], where a utility function is defined to distinguish the real time service and the nonreal time service more clearly. Other resource allocation schemes for practical multiuser MIMO-OFDM systems can be found in [382]–[384]. For coded systems, Y. Hara et al. design an efficient downlink scheduling algorithm for multiuser downlink MIMO systems by optimizing an equivalent uplink scheduling problem [385]. With perfect knowledge of the channel, the proposed algorithm selects the user for each transmit beam, the transmit weight for the user, and the corresponding modulation and coding scheme to maximize the overall system throughout subject to PER constraints. Moreover, M. Fsslaoui et al. propose a two-step adaptive transmission scheme for multiuser downlink MISO-OFDM systems [386]. In the first step, the users that should be simultaneously transmitted are selected based on the degree of orthogonality criterion. In the second step, the modulation and coding scheme for each user is determined adaptively by maximizing the throughput while satisfying the effective SNR constraint for each user. The proposed design is simulated for the IEEE 802.11ac. It is worth mentioning that as shown in [386, Fig. 4], even the adaptive transmission technology is employed, there are still obvious gaps between the practical schemes and the theoretical capacity achieved by Gaussian input. The design in [386] is extended to multiuser downlink MIMOOFDM systems and the fairness among users is taken into consideration additionally [387]. J. Wang et al. propose another transmission scheme for multiuser downlink MIMO-OFDMA systems, where the power allocation, the resource allocation policy, and the modulation and coding scheme for each user are adaptively determined to maximize the system throughout [388]. The proposed design is simulated for IEEE 802.16e and achieves good performance. The adaptive transmission for multiuser downlink MIMO-OFDM systems with limited feedback is provided in [389], where only a single spatial stream is transmitted for each user. A. R.-Alvari¯no et al. propose an adaptive transmission scheme for multiuser downlink MIMO-OFDM systems with limited feedback, which allows multiple spatial streams for each

68

TABLE XXX: Adaptive transmissions for the MIMO downlink transmission with discrete input signals Paper

Model

Main Contribution

Uncoded MISO-OFDMA/SDMA, perfect CSI

Devise an adaptive resource allocation algorithm

Various quality of service constraints

to maximize the spectrum efficiency

Uncoded MIMO-OFDMA, perfect CSI

Define a utility function to distinguish the

Various quality of service constraints

real time service and the nonreal time service

Coded MIMO, perfect CSI

Design an efficient downlink scheduling algorithm

PER constraints

to maximize the overall system throughout

Coded MISO-OFDM, perfect CSI

Propose a two-step adaptive transmission scheme

Effective SNR constraints

to maximize the spectral efficiency

Coded MIMO-OFDM, perfect CSI

Fairness among users is taken


into consideration of system designs

Coded MIMO-OFDMA, perfect CSI

Propose an adaptive scheme for the power allocation,


the resource allocation policy, and the modulation and coding scheme

C.-F. Tsai et al. [54]

C.-M. Yen et al. [381]

Y. Hara et al. [385]

M. Fsslaoui et al. [386]

M. Fsslaoui et al. [387]

J. Wang et al. [388] Coded MIMO-OFDM, feedback CSI Z. Chen et al. [389]

Propose an adaptive scheme to maximize the spectrum efficiency Effective SNR constraints Coded MIMO-OFDM, feedback CSI

Propose an adaptive scheme to maximize the overall system

FER constraints

throughout, which allows multiple spatial streams for each user

A. R.-Alvari¯no et al. [390]

user [390]. A greedy algorithm is used to select the users and the spatial modes. A machine learning classifier is used to select the modulation and coding schemes for users by maximizing the overall system throughout subject to FER constraints. A brief comparison of above work is given in Table XXX.

C. MIMO Interference Channel 1) Mutual Information Design: By directly extending the classical interference alignment scheme used for Gaussian input signal, B. Hari Ram et al. and Y. Fadlallah et al. investigate the precoder designs for K-user MIMO interference channel by maximizing the mutual information of the finite alphabet signal sets in [391] and [392], respectively. Following the interference alignment scheme, the precoders in [391], [392] only use half of the transmit antenna dimension to send the desired signals. H. Huang et al. consider to relax this dimension constraint by allowing the interfering signals to overlap with the desired signal and propose a partial interference alignment and interference detection scheme for K-user MIMO interference channel with finite alphabet inputs [393]. For the interfering signals which can not be aligned at the each receiver, the receiver detects and cancels the residual interference based on the constellation map. By

69

TABLE XXXI: Mutual information based precoder designs for the MIMO interference channel with discrete input signals Paper

Model

Main Contribution

MIMO interference channel

Extend interference alignment scheme

Global PCSIT, PCSIR

to precoder design with finite alphabet inputs


Propose a partial interference alignment

Global PCSIT, PCSIR

and interference detection scheme


Propose gradient decent based iterative algorithms

Global PCSIT, PCSIR

to find optimal precoders of all transmitters


Propose a novel low-complexity

Global PCSIT, PCSIR

fractional interference alignment scheme

[391], [392]

H. Huang et al. [393]

[55], [394]

[395], [396]

treating the interference as noise, Y. Wu et al. and A. Ganesan et al. independently derive the mutual information expressions for K-user MIMO interference channel with finite alphabet inputs in [55] and [394], respectively. Then, gradient decent based iterative algorithms are proposed in [55], [394] to find the optimal precoders of all transmitters. Instead of degree of freedom used for Gaussian input signal, B. Hari Ram et al. define a new performance metric to analyze the transmission efficiency for K-user MIMO interference channel with finite alphabet inputs: number of symbols transmitted per transmit antenna per channel user (SpAC) [395]. It is revealed in [55], [394] that simple transmission schemes can achieve the maximum 1 SpAC in high SNR regime. However, the joint detection of all the transmitter signals including the desired signals and the interfering signals at each receiver is needed in [55], [394] to achieve 1 SpAC. B. H. Ram et al. further propose a fractional interference alignment scheme, which does not decode any of the interfering signals [396]. The fractional interference alignment can achieve any value of SpAC in range [0, 1] and a good error rate performance for both uncoded and coded BER. A brief summation of above mutual information based precoder designs for the MIMO interference channel with discrete input signals is given in Table XXXI. 2) MSE Design: For MIMO interference channel with global perfect CSI, H. Shen et al. investigate linear transceiver designs under a given and feasible degree of freedom [56]. Two iterative algorithms are proposed to minimize both the sum MSE and the maximum per-user MSE. The transceiver design

70

TABLE XXXII: MMSE transceiver designs for MIMO interference channel with discrete input signals Paper

Model

Main Contribution

MIMO interference channel H. Shen et al. [56]

Minimize both the sum MSE and the maximum per-user MSE Global perfect/Imperfect CSI MIMO interference channel

C.-E. Chen et al. [397]

Minimize the maximum per-stream MSE of all users Global perfect CSI MIMO interference channel

F. Sun et al. [398]

Minimize the MSE of the signal and interference leakage Global perfect CSI MIMO interference channel

F.-S. Tseng et al. [399]

Minimize both the sum MSE and the maximum per-user MSE Random vector quantization feedback

for MIMO interference channel with channel estimation error is also provided in [56]. With perfect CSI, C.-E. Chen et al. extend the design in [397] to minimize the maximum per-stream MSE of all users. Also, F. Sun et al. propose a low complexity transceiver design based on minimizing the MSE of the signal and interference leakage at each transmitter [398]. The robust transceiver design for MISO interference channel with random vector quantization feedback mechanism is proposed in [399], where the impact of quantization error at the transmitter is considered. The optimal transmit beamforming vectors and the receiver decoding scalars for both the sum MSE and the maximum per-user MSE minimization are found by iterative algorithms. A brief summation of above work is given in Table XXXII. 3) Diversity Design: For the open loop system, L. Shi et al. design full diversity STBCs for two-user MIMO interference channel with the group ZF receiver [57]. Y. Lu et al. combine the blind interference alignment scheme with three STBCs for K-user MISO interference channel employing reconfigurable antenna [400]. These combinations provide tradeoff between diversity, rate, and decoding complexity, where threaded algebraic space-time codes have full diversity and high rate with high decoding complexity, OSTBCs have full diversity and low rate with the linear decoding complexity, and Alamouti codes have high rate and low linear decoding complexity but with full diversity loss. For the close loop system, when the global CSI is available at all transmitters and receivers, A. Sezgin et al. combine the STBC with interference alignment to achieve high diversity gains for MIMO interference channel [401]. However, the feasibility condition investigation for MIMO interference channel reveals that the separation of spacetime coding and precoding design may not be optimal in general [402]. Moreover, the diversity gains of

71

TABLE XXXIII: Space-time techniques with discrete input signals for MIMO interference channel Paper

Model

Main Contribution

Two-user MIMO interference channel L. Shi et al. [57]

Design full diversity STBCs No PCSIT, PCSIR MISO interference, reconfigurable antenna

Combine the blind interference alignment scheme

No PCSIT, PCSIR

with three STBCs


Combine STBC

Global PCSIT, PCSIR

with the interference alignment design


Reveals that the separation of space-time coding

Global PCSIT, PCSIR

and precoding design may not be optimal

3-user 2 × 2 interference channel

Combine the antenna selection technique

Global PCSIT, PCSIR

with the interference alignment design

Y. Lu et al. [400]

A. Sezgin et al. [401]

H. Ning et al. [402]

L. Li et al. [403]

various interference alignment schemes for MIMO interference channels are analyzed in [402]. L. Li et al. combine the antenna selection technique with the interference alignment design to improve the diversity gain of the worst case user for 3-user 2 × 2 interference channel [403]. A brief comparison of above work is given in Table XXXIII. 4) Adaptive Transmission: B. Xie et al. investigate the adaptive transmission for uncoded MIMO interference channels with imperfect CSIT [58]. Interference alignment based bit-loading algorithms are proposed to minimize the average BER subject to a fixed sum-rate constraint. Based on this, an adaptive transmission scheme switches among two interference alignment algorithms and the basic time-division multiple access scheme to combat the CSI uncertainty is further provided. M. Taki et al. investigate the adaptive transmission for coded MIMO interference channels with imperfect CSIT [404]. An interference alignment based adaptive transmission scheme is proposed, which adaptively selects the coding, modulation, and power allocation across users to maximize the weighted sum rate subject to a sum power constraint and BER constraints for every transmit stream.

D. MIMO Wiretap Channel 1) Mutual Information Design: For the wiretap channel, the finite-alphabet inputs may allow both the receiver and the eavesdropper to decode a transmitted message accurately if the transmit power is highenough. Therefore, the secrecy rate may not be high at a high SNR. This phenomenon has been observed

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in SISO single-antenna eavesdropper wiretap channels via simulations [405]. Therefore, S. Bashar et al. suggest to find an optimum power control policy at the transmitter for maximizing the secrecy rate [406]. For MISO single-antenna eavesdropper (MISOSE) wiretap channel, S. Bashar et al. investigate the power control optimization issue at the transmitter based on the conventional beamforming transmission for the Gaussian input case and develop a numerical algorithm to find the optimal transmission power [406]. This idea is extended to MIMO multiple-antenna eavesdropper (MIMOME) wiretap channel by exploiting a generalized SVD (GSVD) precoding structure to decompose the MIMOME channel into a bank of parallel subchannels [407]. Then, numerical algorithms are proposed to obtain the adequate power allocated to each subchannel. Although the precoding design in [407] achieves significant performance gains over the conventional design that relies on the Gaussian input assumption, it is still suboptimal. The reasons are twofold. First, part of the transmission symbols is lost by both the desired user and the eavesdropper according to the GSVD structure. This condition may result in a constant performance loss. Second, the power control policy compels the transmitter to utilize only a fraction of power to transmit in the high SNR regime, which may impede the further improvement of the performance in some scenarios (see [407, Fig. 3] as an example). As a result, Y. Wu et al. further investigate the linear precoder design for MIMOME wiretap channel when the instantaneous CSI of the eavesdropper is available at the transmitter [59]. An iterative algorithm for secrecy rate maximization is developed through a gradient method. For systems where the number of transmit antennas is less than or equal to the number of the eavesdropper antennas, only partial transmission power is necessary for maximizing the secrecy rate at a high SNR. Accordingly, the excess transmission power is used to construct an artificial jamming signal to further improve the secrecy rate. By invoking the lower bound of the average mutual information in MIMO Kronecker fading channels with the finite alphabet inputs [30], Y. Wu et al. also study the scenarios where the transmitter only has the statistical CSI of the eavesdropper [59]. W. Zeng et al. study the precoder design for MIMO secure cognitive ratio systems with finite alphabet inputs [408]. With the SCSIT of the eavesdropper, an iterative algorithm is proposed to maximize the secure rate of the secondary user under the control of power leakage to the primary users. The transmission design for MIMO decode-and-forward (DF)

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TABLE XXXIV: Mutual information based precoder designs for the MIMO wiretap channel with discrete input signals Paper

Model

Main Contribution

MISOME

Find an optimum power control policy

SCSIT of the eavesdropper

at the transmitter for maximizing the secrecy rate

MIMOME

Propose a GSVD precoder structure to

PCSIT of the eavesdropper

increase the secure rate performance

MIMOME

Propose iterative algorithms for secrecy rate maximization

PCSIT and SCSIT of the eavesdropper

Construct the jamming signal with the excess power

MIMO cognitive radio secure

Propose an iterative algorithm to maximize


the secure rate of the secondary user

MIMO DF relay secure

Design the optimal source power, signal beamforming

IPCSIT of all links

and jamming signal matrix for worst case secrecy rate maximization

MIMOME with a helper

Design a secure transmission scheme


with a help of source node

S. Bashar et al. [406]

S. Bashar et al. [407]

Y. Wu et al. [59]

W. Zeng et al. [408]

S. Vishwakarma et al. [409]

K. Cao et al. [410]

relay systems with the IPCST and finite alphabet inputs in presence of an user and J non-colluding eavesdroppers is investigated in [409]. Very recently, s secure transmission scheme with a help of source node is proposed in [410]. A brief summation of above mutual information based precoder designs for the MIMO wiretap channel with discrete input signals is given in Table XXXIV. 2) MSE Design: For MIMOME where the perfect instantaneous CSI is available at all parties, H. Reboredo et al. investigate the transceiver design minimizing the receiver’s MSE while guaranteeing the eavesdropper’s MSE above a fixed threshold [61]. A linear zero-forcing precoder is used at the transmitter and either a zero-forcing decoder or an optimal linear Wiener decoder is designed at the receiver. Moreover, the design is generalized to the scenarios where only the statistical CSI is available. Simulations show that maintaining the eavesdropper’s MSE above a certain level also limits eavesdropper’s error probability. This design is further extended to MIMO interference channel with a multiple antenna eavesdropper [411]. M. Pei et al. investigate the masked beamforming design for MIMO BC in presence of a multiple antenna eavesdropper [412]. It is assumed that the transmitter obtains the perfect/imperfect CSI of the desired receivers but no CSI of the eavesdropper. The transmit power used to generate the artificial noise is maximized subject to the MSE requirements of the desired receivers. Based on the same criterion, L.

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TABLE XXXV: MMSE transceiver designs for MIMO wiretap channel with discrete input signals Paper

Model

Main Contribution

MIMOME

Minimizing the receiver’s MSE while guaranteeing

Instantaneous/statistical CSI

the eavesdropper’s MSE above a fixed threshold

MIMO interference channel secure

Minimizing the receivers’ sum MSE while guaranteeing

Perfect CSI

the eavesdropper’s MSE above a fixed threshold

MIMO BC secure, Perfect/imperfect CSI of receivers

Maximize the transmit power to generate the artificial noise

No CSI of the eavesdropper

subject to the MSE requirements of the desired receivers

MIMOME, nonlinear THP design

Maximize the transmit power to generate the artificial noise

Perfect CSI of the receiver, no CSI of the eavesdropper

subject to the MSE requirements of the receiver

H. Reboredo et al. [61]

Z. Kong et al. [411]

M. Pei et al. [412]

L. Zhang et al. [413]

Zhang et al. investigate the nonlinear THP design for MIMOME [413]. A brief summation of above work is given in Table XXXV. 3) Diversity Design: For two transmit antennas, two receive antennas, multiple antennas eavesdropper wiretap channel, S. A. A. Fakoorian et al. design a secure STBC by exploiting the perfect CSI of the receiver but no CSI of the eavesdropper [414]. By combining QOSTBC in [183] with the artificial noise generation, the proposed design in [414] effectively deteriorates the uncoded BER performance of the eavesdropper while maintains the good uncoded BER performance of the receiver. For the same STBC system, H. Wen et al. propose a cross-layer scheme to further ensure the secure communication [415]. T. Allen et al. investigate two/four transmit antennas, one receive antenna, and one antenna eavesdropper wiretap channels employing STBCs, where no CSI of the receiver and the eavesdropper are available at transmitter [416]. By randomly rotating the transmit symbols at each transmit antenna based on the receive signal strength indicator, the proposed scheme provides full diversity of the receiver and enables the eavesdropper’s diversity to be zero. For MISOSE wiretap channel with the same CSI assumption, S. L. Perreau designs the transmit antennas weights based on STBC, which formulates beamforming along the receiver’s direction while disables the eavesdropper’s ability to discriminate the transmit signal [417]. With an assumption that the transmitter has the perfect CSI of the receiver based on the channel reciprocity while both the receiver and the eavesdropper have no channel knowledge, X. Li et al. investigate the secure communication for MISOME wiretap channel by using M-PSK modulation and OSTBC [60]. A phase shifting precoder is designed to align the transmit signals at the receiver side so that the receiver

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TABLE XXXVI: Space-time techniques with discrete input signals for MIMO wiretap channel Paper

S. A. A. Fakoorian et al. [414]

Model

Main Contribution

Two transmit antennas and receive antennas

Design a secure STBC

Multiple antennas eavesdropper

with low uncoded BER of the receiver


and high uncoded BER of the eavesdropper Propose a cross-layer scheme to

H. Wen et al. [415]

Same as [414] further ensure the secure communication

T. Allen et al. [416]

Two/four transmit antennas, one receive antennas

Design a secure STBC

one antennas eavesdropper

providing full diversity of the receiver

No CSI of the receiver and eavesdropper

and zero diversity of the eavesdropper

MISOSE

Design the transmit antennas weights based on STBC


to make the eavesdropper unable to decode the signal

MISOME

Design the secure communication scheme


by using M-PSK modulation and OSTBC

MISOME

Propose transmit antennas weights design method


to deliberately randomize the eavesdropper’s signal

MIMOME

Establish a lattice wiretap code design criterion minimizing


the eavesdropper’s correctly decoding probability

S. L. Perreau [417]

X. Li et al. [60]

X. Li et al. [418]

J.-C. Belfiore et al. [419] MISOME T. V. Nguyen et al. [420]

Analyze the SER of confidential information Perfect CSIT of the receiver, SCSIT of the eavesdropper

can perform a non-coherent detection for the transmit signals. At the same time, the information rate of the eavesdropper can be reduced to zero by the proposed design. For the same channel model, another transmit antennas weights design method to deliberately randomize the eavesdropper’s signal but not the receiver’s signal is proposed in [418]. For MIMOME wiretap channel without the eavesdropper’s CSI, J.-C. Belfiore et al. establish a lattice wiretap code design criterion, which minimizes the probability of correctly decoding at the eavesdropper’s side [419]. T. V. Nguyen et al. analyze the performance of the MISOME wiretap channel with perfect CSIT of the receiver and SCSIT of the eavesdropper [420]. The SER of confidential information is derived and the corresponding secure diversity is obtained. A brief comparison of above work is given in Table XXXVI.

V. OTHER S YSTEMS E MPLOYING MIMO T ECHNOLOGY In this section, we overview the transmission design with discrete input signals for other systems employing MIMO technology. We consider the following three typical scenarios: i) MIMO Cognitive

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Radio Systems; ii) Green MIMO communications; iii) MIMO relay systems.

A. MIMO Cognitive Radio Systems 1) Mutual Information Design: W. Zeng et al. investigate the spectrum sharing problem in MIMO cognitive radio systems with finite alphabet inputs [62]. With the global PCSIT, the linear precoder is designed to maximize the spectral efficiency between the secondary-user transmitter and the secondaryuser receiver with the control of the interference power to primary-user receivers. This design is extended to the secure communication of MIMO cognitive radio systems in [408]. However, these algorithms only apply for the equip-probable finite alphabet inputs. By exploiting the I-MMSE relationship in [23], R. Zhu et al. propose a gradient based iterative algorithm to maximize the secondary users’ spectrum efficiency for arbitrary inputs [421]. 2) MSE Design: B. Seo et al. investigate the transceiver design to minimize the MSE of the primary user subject to the secondary user’s interference power constraint [422]. For MIMO cognitive networks, X. Gong et al. jointly design the precoder and decoder to minimize the MSE of the secondary network subject to both transmit and interference power constraints [423]. The transceiver designs for ad hoc MIMO cognitive radio networks based on total MSE minimization are proposed in [64], [424]. For fullduplex MIMO cognitive radio networks, A. C. Cirik et al. minimize both the total MSE and the maximum per secondary users’ MSE [425]. 3) Diversity Design: For cognitive radio networks, by considering multiple cognitive radios as a virtual antenna array, W. Zhang et al. design space-time codes and space-frequency codes for cooperative spectrum sensing over flat-fading and frequency-selective fading channels, respectively [63]. X. Liang et al. design a flexible STBC scheme for cooperative spectrum sensing by dynamically selecting users to form the clusters [426]. A robust STBC scheme based on dynamical clustering is further proposed in [427]. The differential STBCs for cooperative spectrum sensing without CSI knowledge are also proposed in [428], [429]. J.-B. Kim et al. analyze the outage probability and the diversity order of various spectrum sensing methods for cognitive radio networks [430]. K. B. Letaief et al. design a space-time-frequency block

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code to exploit spatial diversity for cognitive relay networks with cooperative spectrum sharing [431]. A distributed space-time-frequency block code is further proposed to reduce the detection complexity [432].

B. Green MIMO Communications 1) Mutual Information Design: M. Gregori et al. study a point-to-point MIMO systems with finite alphabet inputs where the transmitter is equipped with energy harvesters [65]. A total number of N channels are used and the non-causal knowledge of the channel state and harvested energy are available at the transmitter. By obtaining the optimal left singular matrix and setting the right singular matrix to be identity matrix, M. Gregori et al. propose an optimal offline power allocation solution to maximize the spectral efficiency along N channel uses [65]. This design is extended to the SCSIT case in [433], where an iterative algorithm to find the entire optimal precoder is proposed. The further exploitation of partial instantaneous CSI along with the SCSI is studied in [434]. An resource allocation algorithm for the downlink transmission of multiuser MIMO systems with finite alphabet inputs and energy harvest process at the transmitter is proposed [435]. With the SCSIT and the statistical energy arrival knowledge, the precoders, the MCR selection, the subchannel allocation, and the energy consumption are jointly optimized by decomposing the original problem into an equivalent three-layer optimization problem and solving each layer separately. 2) Diversity Design: Y. Tang et al. propose a STBC for geographically separated base stations, which improves the energy efficiency at remote locations [436]. Y. Zhu et al. design a low energy consumption turbo-based STBC [437]. A cooperative balanced STBC is proposed for dual-hop AF wireless sensor networks to reduce the energy consumption [438]. M. Tomio et al. compare the energy efficiency of transmit beamforming scheme and transmit antenna selection scheme [439]. It is revealed that transmit antenna selection scheme is more efficient for most transmit distances since only a single radio-frequency chain is used at the transmitter. Novel non-coherent energy-efficient collaborative Alamouti codes are constructed by uniquely factorizing a pair of energy-efficient cross QAM constellations and carefully designing an energy scale [66], [440].

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C. MIMO Relay Systems 1) Mutual Information Design: W. Zeng et al. investigate the linear precoder design for dual hop amplify-and-forward (AF) MIMO relay systems with finite alphabet inputs [67]. By exploiting the optimal precoder structure similar as the point-to-point MIMO case, a two-step algorithm is proposed to maximize the spectral efficiency. The obtained precoder in [67] also achieves a good BER performance. Since the publication of [67], the transmission design for the spectral efficiency maximization under finite alphabet constraints has been investigated for various MIMO relays systems including: two-hop nonregenerative three-node MIMO relay systems [441], non-orthogonal AF half-duplex single relay systems [442], MIMO two-way relay systems [443], and MIMO DF relay secure systems [409]. 2) MSE Design: Linear transceiver designs via MMSE criterion for MIMO AF relay systems are proposed in [69], [444]–[450]. In addition, non-linear THP/DFE transceiver structures based on the optimization of MSE function are designed for MIMO AF relay systems in [451]–[455]. The average error probability of AF relay networks employing MMSE based precoding schemes is derived in [456]. At the same time, the MMSE based transceiver designs for non-regenerative MIMO relay networks are provided in [457]–[461]. The MMSE transceiver designs are further investigated in multiuser MIMO relay networks [462]–[466]. Two MMSE transceiver designs for filter-and-forward MIMO relay systems and single-carrier frequency-domain equalization based MIMO relay systems are given in [467] and [468], respectively. 3) Diversity Design: For AF relay systems with multiple antennas and finite alphabet inputs, various transmission schemes are provided to optimize the diversity gain of the system in [68], [469]–[472]. Moreover, the diversity order and array gains for AF relay MIMO systems employing OSTBCs are derived analytically in [473], [474]. For DF relay systems with multiple antennas and finite alphabet inputs, various transmission schemes are proposed based on minimizing the error probability of the system in [470], [475], [476]. Meanwhile, the error performance of DF relay MIMO systems are analyzed in [477], [478]. Further transmission schemes to increase the reliability of MIMO relay networks with finite alphabet inputs are designed and analyzed in [479]–[486]. The diversity designs for the non-coherent MIMO relay networks

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are investigated in [487]–[489] 4) Adaptive Transmission: I. Y. Abualhaol et al. propose an adaptive transmission scheme for MIMOOFDM systems with a cooperative DF relay [70]. The bit and power are adaptively allocated among different subchannels to maximize the receive signal power under the constraint that each subchannel is assigned either to the direct link or the relay link. D. Munoz et al. propose an adaptive transmission scheme for MIMO-OFDMA systems with multiple relay nodes [490]. Based on the channel conditions, the transmission mode adaptively switches between joint relay and antenna selection and space-frequency block coding to minimize the sum BER of the direct link and the relay link subject to a total power constraint.

D. Others An optimal precoding technique for coded MIMO inter-symbol interference MAC with joint linear MMSE detection and forward-error-correction codes is proposed in [491]. The geometric mean method, which obtains an equal SINR for all transmit streams, is applied to single user and multiuser MIMO systems [492]–[495]. There are some precoder designs for multiuser MIMO systems based on the receiver side’s SINR [496]–[503]. There are some transmission designs for MIMO X channel based on the diversity performance [504]–[509]. The diversity designs for cooperative systems, Ad Hoc networks, and distributed antenna systems are given in [510], [511], and [512], respectively.

VI. C ONCLUSIONS This paper has provided a first comprehensive summary for the research on MIMO transmission design with practical input signals. We introduced the existing fundamental understanding for discrete input signals. Then, to facilitate a better clarification of practical MIMO transmission techniques, we provided a detailed discussion of the transmission designs based on the criterions of mutual information, MSE, diversity, and the adaptive transmission switching among these criterions. These transmission schemes were designed for point-to-point MIMO systems, multiuser MIMO systems, MIMO cognitive radio systems,

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green MIMO communication systems, MIMO relay systems, etc. Valuable insights and lessons were extracted from these existing designs. For the forthcoming massive MIMO systems in 5G wireless networks, the transmission designs with discrete input signals will become more challenging. A unified framework for the mutual information maximization design for point-to-point massive MIMO systems with discrete input signals was introduced in this paper. However, there are a variety of open research issues currently. The tractable mutual information maximization design for multiuser massive MIMO systems with discrete input signals is still unknown. The robust MSE design for large scale antenna arrays with channel uncertainty due to pilot contamination and other channel variation effects is less studied. The high rate low detection complexity STBCs for large MIMO systems need to be designed. Most importantly, an integrity and efficient adaptive transmission scheme for practical massive MIMO systems is still missing yet. This problem is extremely challenging since it needs to determine the optimal resource allocation (time/space/frequency), the user scheduling, the transmission mode selection, etc, in a significant large dimension. The characteristics of massive MIMO channels should be exploited and heuristic methods from other research areas such as machine learning may be helpful.

R EFERENCES [1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Commun., vol. 10, pp. 585–595, Nov/Dec 1999. [2] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, Mar. 1998. [3] L. Zheng and D. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 1073–1096, May 2003. [4] Cisco, “Cisco visual networking index: Global mobile data traffic forecast update: 2013-2018,” Cisco, Feb. 2014. [5] UMTS, “Mobile traffic forecasts 2010-2020 report,” UMTS Forum, Jan. 2011. [6] Huawei, “5G: A technology vision,” White Paper, Nov. 2013. [7] Ericsson, “5G radio access,” White Paper, Apr. 2016. [8] J. G. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be,” IEEE J. Sel. Areas Commun., vol. 32, pp. 1065–1082, Jun. 2014. [9] Rohde & Schwarz, “Millimeter-wave beamforming: Antenna array design choices & characterization,” While Paper, Oct. 2016.

81

[10] A. Puglielli, A. Townley, G. Lacaille, V. Milovanović, P. Lu, K. Trotskovsky, A. Whitcombe, N. Narevsky, G. Wright, T. Courtade, E. Alon, B. Nikolić, and A. M. Niknejad, “Design of energy- and cost-efficient massive MIMO array,” Proceeding of IEEE, vol. 104, pp. 586–606, Mar. 2016. [11] IMT-2020 (5G) Promotion Group, “5G vision of IMT-2020 (5G) promotion group,” Sep. 2014. [12] M. Vu and A. Paulraj, “MIMO wireless linear precoding,” IEEE Signal Process. Mag., vol. 24, pp. 86–105, Sep. 2007. [13] Y. Wu, S. Jin, X. Gao, M. R. McKay, and C. Xiao, “Transmit designs for the MIMO broadcast channel with statistical CSI,” IEEE Trans. Signal Process., vol. 62, pp. 4451–4446, Sep. 2014. [14] Y. Wu, R. Schober, D. W. K. Ng, C. Xiao, and G. Caire, “Secure massive MIMO transmission with an active eavesdropper,” IEEE Trans. Inform. Theory, vol. 62, pp. 3880–3900, Jul. 2016. [15] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Trans. Wireless Commun., vol. 9, pp. 3590–3600, Nov. 2010. [16] A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatial division and multiplexing: The large-scale array regime,” IEEE Trans. Inform. Theory, vol. 59, pp. 6441–6463, Aug. 2013. [17] C. Xiao, Y. R. Zheng, and Z. Ding, “Globally optimal linear precoders for finite alphabet signals over complex vector Gaussian channels,” IEEE Trans. Signal Process., vol. 59, pp. 3301–3314, Jul. 2011. [18] A. Lozano, A. M. Tulino, and S. Verdú, “Optimum power allocation for parallel Gaussian channels with arbitrary input distributions,” IEEE Trans. Inform. Theory, vol. 52, pp. 3033–3051, Jul. 2006. [19] F. Pérez-Cruz, M. R. D. Rodrigues, and S. Verdú, “MIMO Gaussian channels with arbitrary inputs: Optimal precoding and power allocation,” IEEE Trans. Inform. Theory, vol. 56, pp. 1070–1084, Mar. 2010. [20] M. Wang, W. Zeng, and C. Xiao, “Linear precodinng for MIMO multiple access channels with fintie discrete inputs,” IEEE Trans. Wireless Commun., vol. 10, pp. 3934–3942, Nov. 2011. [21] Y. Wu, M. Wang, C. Xiao, Z. Ding, and X. Gao, “Linear precoding for the MIMO broadcast channels with fintie alphabet constraints,” IEEE Trans. Wireless Commun., vol. 11, pp. 2906–2920, Aug. 2012. [22] D. Guo, S. Shamai, and S. Verdú, “Mutual information and minimum mean-square error in Gaussian channel,” IEEE Trans. Inform. Theory, vol. 51, pp. 1261–1282, Apr. 2005. [23] D. P. Palomar and S. Verdú, “Gradient of mutual information in linear vector Gaussian channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 141–154, Jan. 2006. [24] M. Payaró and D. P. Palomar, “Hessian and concavity of mutual information, differential entropy, and entropy power in linear vector Gaussian channels,” IEEE Trans. Inform. Theory, vol. 55, pp. 3613–3628, Jan. 2009. [25] D. Guo, Y. Wu, S. Shamai, and S. Verdú, “Estimation in Gaussian noise: Properties of the minimum mean-square error,” IEEE Trans. Inform. Theory, vol. 57, pp. 2371–2385, Apr. 2011. [26] Y. Wu and S. Verdú, “Functional properties of minimum mean-square error and mutual information,” IEEE Trans. Inform. Theory, vol. 58, pp. 1289–1301, Mar. 2012. [27] ——, “MMSE dimension,” IEEE Trans. Inform. Theory, vol. 57, pp. 4857–4879, Aug. 2011.

82

[28] A. Lozano, A. M. Tulino, and S. Verdú, “Optimum power allocation for multiuser OFDM with arbitrary signal constellation,” IEEE Trans. Commun., vol. 56, pp. 828–837, May 2008. [29] S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “Precoding by pairing subchannels to increase MIMO capacity with discrete input alphabets,” IEEE Trans. Inform. Theory, vol. 57, pp. 4156–4169, Jul. 2011. [30] W. Zeng, C. Xiao, M. Wang, and J. Lu, “Linear precoding for finite-alphabet inputs over MIMO fading channels with statistical CSI,” IEEE Trans. Signal Process., vol. 60, pp. 3134–3148, Jun. 2012. [31] T. Ketseoglou and E. Ayanoglu, “Linear precoding for MIMO with LDPC coding and reduced complexity,” IEEE Trans. Wireless Commun., vol. 14, pp. 2192–2204, Apr. 2015. [32] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization,” IEEE Trans. Signal Process., vol. 51, pp. 2381–2401, Sep. 2003. [33] S. Serbetli and A. Yener, “MMSE transmitter design for correlated MIMO systems with imperfect channel estimates: Power allocation trade-offs,” IEEE Trans. Wireless Commun., vol. 5, pp. 2295–2304, Aug. 2006. [34] R. F. H. Fischer, C. Windpassinger, A. Lampe, and J. H. Huber, “Space time transmission using Tomlinson-Harashima precoding,” in Proc. 4th ITC Conf. on Source and Channel Coding, Berlin, Germany, Jan. 2002, pp. 139–147. [35] S. M. Alamouti, “A simple transmit diversity technique for wireless communication,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [36] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [37] S. Zhou and G. B. Giannakis, “Single-carrier spact-time block-coded transmission over frequency-selective fading channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 164–179, Jan. 2003. [38] H. Jafarkhani, Space-time coding: Theory and practice.

Cambridge: Cambridge University Press, 2005.

[39] J. Liu, J.-K. Zhang, and K. M. Wong, “Full diversity codes for MISO systems equipped with linear or ML detectors,” IEEE Trans. Inform. Theory, vol. 54, pp. 4511–4527, Oct. 2008. [40] B. Vrigneau, J. Letessier, P. Rostaing, L. Collin, and G. Burel, “Extension of the MIMO precoder based on the minimum Euclidean distance: A cross-form matrix,” IEEE J. Sel. Topic Signal Process., vol. 2, pp. 135–146, Feb. 2008. [41] Z. Zhou, B. Vucetic, M. Dohler, and Y. Li, “MIMO systems with adaptive modulation,” IEEE Trans. Veh. Technol., vol. 54, pp. 1828–1842, Sep. 2005. [42] S. Zhou and G. B. Giannakis, “Adaptive modulation for multiantenna transmissions with channel mean feedback,” IEEE Trans. Wireless Commun., vol. 1, pp. 1626–1636, Sep. 2004. [43] L. Chen, Y. Yang, X. Chen, and G. Wei, “Energy-efficient link adaptation on Rayleigh fading channel for OSTBC MIMO system with imperfect CSIT,” IEEE Trans. Veh. Technol., vol. 62, pp. 1577–1585, May 2013. [44] Q. Kuang, S.-H. Leung, and X. Yu, “Adaptive modulation and joint temporal spatial power allocation for OSTBC MIMO systems with imperfect CSI,” IEEE Trans. Commun., vol. 60, pp. 1914–1924, Jul. 2012. [45] M. R. McKay, I. B. Collings, A. Forenza, and R. W. Heath Jr, “Multiplexing/beamforming switching for coded MIMO in spatially correlated channels based on closed-form BER approximations,” IEEE Trans. Veh. Technol., vol. 56, pp. 2555–2567, Sep. 2007.

83

[46] Y.-S. Choi and S. M. Alamouti, “A pragmatic PHY abstraction technique for link adaptation and MIMO switching,” IEEE J. Select. Area Commun., vol. 26, pp. 960–971, Aug. 2008. [47] P. H. Tan, Y. Wu, and S. Sun, “Link adaptation based on adaptive modulation and coding for multiple-antenna OFDM system,” IEEE J. Select. Area Commun., vol. 26, pp. 1599–1606, Oct. 2008. [48] Z. Zhou and B. Vucetic, “Adaptive coded MIMO systems with near full multiplexing gain using outdated CSI,” IEEE Trans. Wireless Commun., vol. 10, pp. 294–302, Jan. 2011. [49] W. Zhang and K. B. Letaief, “A systematic design of full diversity multiuser space-frequency codes,” IEEE Trans. Signal Process., vol. 58, pp. 1732–1740, Mar. 2010. [50] E. A. Jorswieck and H. Boche, “Transmission strategies for the MIMO MAC with MMSE receiver: Average MSE optimization and achievable individual MSE region,” IEEE Trans. Signal Process., vol. 11, pp. 2872–2881, Nov. 2003. [51] Y. J. Zhang and K. B. Letaief, “An efficient resource-allocation scheme for spatial multiuser access in MIMO/OFDM systems,” IEEE Trans. Commun., vol. 53, pp. 107–116, Jan. 2005. [52] R. Chen, R. W. Heath Jr, and J. G. Andrews, “Transmit selection diversity for unitrary precoded multiuser spatial multiplexing systems with linear receivers,” IEEE Trans. Signal Process., vol. 55, pp. 1159–1171, Mar. 2007. [53] S. Shi, M. Schubert, and H. Boche, “Downlink MMSE transceiver optimization for multiuser MIMO systems: Duality and sum-MSE minimization,” IEEE Trans. Signal Process., vol. 55, pp. 5436–5446, Nov. 2007. [54] C.-F. Tsai, C.-J. Chang, F.-C. Ren, and C.-M. Yen, “Adaptive radio resource allocation for downlink OFDMA/SDMA with multimedia traffic,” IEEE Trans. Wireless Commun., vol. 7, pp. 1734–1743, May 2008. [55] Y. Wu, C. Xiao, X. Gao, J. Matyjas, and Z. Ding, “Linear precoder design for MIMO interference channels with finite-alphabet signaling,” IEEE Trans. Commun., vol. 61, pp. 3766–3780, Sep. 2013. [56] H. Shen, B. Li, M. Tao, and X. Wang, “MSE-based transceiver designs for the MIMO interference channel,” IEEE Trans. Wireless Commun., vol. 11, pp. 3480–3489, Nov. 2010. [57] L. Shi, W. Zhang, and X.-G. Xia, “On designs of full diversity space-time block codes for two user MIMO interference channels,” IEEE Trans. Wireless Commun., vol. 11, pp. 4184–4191, Nov. 2012. [58] B. Xie, Y. Li, H. Minn, and A. Nosratinia, “Adaptive interference alignment with CSI uncertainty,” IEEE Trans. Commun., vol. 61, pp. 792–801, Feb. 2013. [59] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “Linear precoding for finite-alphabet signaling over MIMOME wiretap channels,” IEEE Trans. Veh. Technol., vol. 61, pp. 2599–2612, Jul. 2012. [60] X. Li, R. Fan, X. Ma, J. An, and T. Jiang, “Secure space-time communications over Rayleigh flat fading channels,” IEEE Trans. Wireless Commun., vol. 15, pp. 1491–1504, Feb. 2016. [61] H. Reboredo, J. Xavier, and M. R. D. Rodrigues, “Filter design with secrecy constrains: The MIMO Gaussian wiretap channel,” IEEE Trans. Signal Process., vol. 61, pp. 3799–3814, Aug. 2013. [62] W. Zeng, C. Xiao, J. Lu, and K. B. Letaief, “Globally optimal precoder design with fintie-alphabet inputs for cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 30, pp. 1861–1874, Nov. 2012.

84

[63] W. Zhang and K. B. Letaief, “Cooperative spectrum sensing with transmit and relay diversity in cognitive radio networks,” IEEE Trans. Wireless Commun., vol. 7, pp. 4761–4766, Dec. 2008. [64] E. A. Gharavol, Y.-C. Liang, and K. Mouthaan, “Robust linear transceiver design in MIMO ad hoc cognitive radio networks with imperfect channel state information,” IEEE Trans. Wireless Commun., vol. 10, pp. 1448–1457, May 2011. [65] M. Gregori and M. Payaró, “On the precoder design of a wireless energy harvesting node in linear vector Gaussian channels with arbitrary input distribution,” IEEE Trans. Commun., vol. 61, pp. 1868–1879, May 2013. [66] D. Xia, J.-K. Zhang, and S. Dumitrescu, “Enery-efficient full diversity collaborative unitray space-time block code designs via unqiue factorization of signals,” IEEE Trans. Inform. Theory, vol. 59, pp. 1678–1703, Mar. 2013. [67] W. Zeng, Y. R. Zheng, M. Wang, and J. Lu, “Linear precoding for relay networks: A perspective on finite-alphabet inputs,” IEEE Trans. Wireless Commun., vol. 11, pp. 1146–1157, Mar. 2012. [68] Y. Ding, J.-K. Zhang, and K. M. Wong, “The amplify-and-forward half-duplex cooperative system: Pairwise error probability and precoder design,” IEEE Trans. Signal Process., vol. 55, pp. 605–617, Feb. 2007. [69] R. Mo and Y. H. Chew, “MMSE-based joint source and relay precoding design for amplify-and-forward MIMO relay networks,” IEEE Trans. Wireless Commun., vol. 8, pp. 4668–4676, Sep. 2009. [70] I. Y. Abualhaol and M. M. Matalgah, “Subchannel-division adaptive resource allocation technique for cooperative relay-based MIMO/OFDM wireless communication systems,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2008), Las Vegas, USA, Mar. 2008, pp. 1002–1007. [71] A. J. Stam, “Some inequalities satisfied by the quantities of information of Fisher and Shannon,” Inform. Contr., vol. 2, pp. 101–112, Jun. 1959. [72] T. E. Duncan, “On the calculation of mutual information,” SIAM J. Applied Math., vol. 19, pp. 215–220, Jul. 1970. [73] J. Ziv and M. Zakai, “Some lower bounds on signal parameter estimation,” IEEE Trans. Inform. Theory, vol. 15, pp. 386–391, May 1969. [74] T. T. Kadota, M. Zakai, and J. Ziv, “Mutual information of the white Gaussian channel with and without feedback,” IEEE Trans. Inform. Theory, vol. 17, pp. 368–371, Jul. 1971. [75] J. Seidler, “Bounds on the mean-square error and the quality of domain decisions based on mutual information,” IEEE Trans. Inform. Theory, vol. 17, pp. 655–665, Nov. 1971. [76] R. S. Bucy, “Information and filtering,” Inform. Sci., vol. 18, pp. 179–187, Jul. 1979. [77] A. R. Barron, “Entropy and the central limit theorem,” Ann. Probab., vol. 14, pp. 336–342, 1986. [78] S. Verdú and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inform. Theory, vol. 52, pp. 2165–2166, May 2006. [79] D. Guo, S. Shamai, and S. Verdú, “Proof of entropy power inequalities via MMSE,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2006), Seattle, USA, Jul. 2006, pp. 1011–1015. [80] A. M. Tulino and S. Verdú, “Monotonic decrease of the non-Gaussianess of the sum of independent random variables: A simple proof,” IEEE Trans. Inform. Theory, vol. 52, pp. 4295–4297, Sep. 2006.

85

[81] M. Zakai, “On mutual information, likelihood ratios, and estimation error for the additive Gaussian channel,” IEEE Trans. Inform. Theory, vol. 51, pp. 3017–3024, Sep. 2005. [82] E. M.-Wolf and M. Zakai, “Some relations between mutual information and estimation error in wiener space,” Ann. of Appl. Prob., vol. 17, pp. 1102–1116, Jun. 2007. [83] R. Bustin, M. Payaró, D. P. Palomar, and S. Shamai, “On MMSE properties and I-MMSE implications in parallel MIMO Gaussian channels,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2010), Austin, USA, Jun. 2010, pp. 535–539. [84] Y. Wu and S. Verdú, “Functional properties of MMSE,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2010), Austin, USA, Jun. 2010, pp. 1453–1457. [85] ——, “The impact of constellation cardinality on Gaussian channel capacity,” in Proc. Ann. Aller. Conf., Monticello, USA, Oct. 2010, pp. 620–628. [86] N. Merhav, D. Guo, and S. Shamai, “Statistical physics of signal estimation in Gaussian noise: Theory and examples of phase transitions,” IEEE Trans. Inform. Theory, vol. 56, pp. 1400–1416, Mar. 2010. [87] K. Venkat and T. Weissman, “Pointwise relations between information and estimation in Gaussian noise,” IEEE Trans. Inform. Theory, vol. 58, pp. 6264–6281, Oct. 2012. [88] G. Han and J. Song, “Extensions of the I-MMSE relation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2014), Honolulu, USA, Jun. 2014, pp. 2202–2206. [89] S. Verdú, “Mismatched estimation and relative entropy,” IEEE Trans. Inform. Theory, vol. 56, pp. 3712–3720, Aug. 2010. [90] T. Weissman, “The relationship between causal and noncausal mismatched estimation in continuous-time AWGN channels,” IEEE Trans. Inform. Theory, vol. 56, pp. 4256–4273, Sep. 2010. [91] M. Chen and J. Lafferty, “Mismatched esimation and relative entropy in vector Gaussian channels,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2013), Istanbul, Turkey, Jun. 2013, pp. 2845–2849. [92] W. Huleihel and N. Merhav, “Analysis of mismatched esitmation errors using gradient of partition functions,” IEEE Trans. Inform. Theory, vol. 60, pp. 2190–2215, Apr. 2014. [93] D. Guo, S. Shamai, and S. Verdú, “Mutual information and conditional mean estimation in Poisson channels,” IEEE Trans. Inform. Theory, vol. 54, pp. 1837–1849, May 2008. [94] R. Atar and T. Weissman, “Mutual information, relative entropy, and estimation in the Poisson channel,” IEEE Trans. Inform. Theory, vol. 58, pp. 1302–1318, Mar. 2012. [95] L. Wang, D. E. Carlson, M. R. D. Rodrigues, R. Calderbank, and L. Carin, “A Bregman matrix and the gradient of mutual information for vector Poisson and Gaussian channels,” IEEE Trans. Inform. Theory, vol. 60, pp. 2611–2629, May 2014. [96] C. G. Taborda, D. Guo, and F. P.-Cruz, “Information-estimation relationships over binomial and negative binomial models,” IEEE Trans. Inform. Theory, vol. 60, pp. 2630–2646, May 2014. [97] M. Raginsky and T. P. Coleman, “Mutual information and posterior estimates in channels of exponetial family type,” in Proc. IEEE Inform. Theory Workshop (ITW’2009), Volos, Greece, Jun. 2009, pp. 399–403. [98] T. Weissman, Y.-H. Kim, and H. H. Permuter, “Directed information, causal estimation, and communication in continuous time,” IEEE Trans. Inform. Theory, vol. 59, pp. 1271–1287, Mar. 2013.

86

[99] D. Guo, S. Shamai, and S. Verdú, “Additive non-gaussian noise channnels: Mutual information and conditional mean estimation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2005), Adelaide, Australia, Sep. 2005, pp. 719–723. [100] D. P. Palomar and S. Verdú, “Representation of mututal information via input estimates,” IEEE Trans. Inform. Theory, vol. 53, pp. 453–470, Feb. 2007. [101] D. Guo, “Relative entropy and score functions: New information-estimation relationships through arbitrary additive perturbation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2009), Seoul, Korea, Jun. 2009, pp. 814–818. [102] N. Merhav, “Optimum estimation via gradients of partition functions and information measures: A statistical-mechanical perspective,” IEEE Trans. Inform. Theory, vol. 57, pp. 3887–3898, Jun. 2011. [103] ——, “Rate-distortion function via minimum mean square error estimation,” IEEE Trans. Inform. Theory, vol. 57, pp. 3196–3206, Jun. 2011. [104] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423,623–656, Jul.-Oct. 1948. [105] S. Verdú, “Spectral efficiency in the wideband regime,” IEEE Trans. Inform. Theory, vol. 48, pp. 1319–1343, Jun. 2002. [106] V. V. Prelov and S. Verdú, “Second-order asymptotic of mutual information,” IEEE Trans. Inform. Theory, vol. 50, pp. 1567–1580, Aug. 2004. [107] D. Duyck, J. J. Boutros, and M. Moeneclaey, “Precoding for outage probability minimization on block fading channels,” IEEE Trans. Inform. Theory, vol. 59, pp. 8250–8266, Dec. 2013. [108] A. Alvarado, F. Brännströrn, E. Agrell, and T. Koch, “High-SNR asymptotics of mutual information for discrete constellations with application to BICM,” IEEE Trans. Inform. Theory, vol. 60, pp. 1061–1076, Feb. 2014. [109] A. C. P. Ramos and M. R. D. Rodrigues, “Fading channels with arbitrary inputs: Asymptotics of the constrained capacity and information and estimation measures,” IEEE Trans. Inform. Theory, vol. 60, pp. 5653–5672, Sep. 2014. [110] M. R. D. Rodrigues, “Multiple-antenna fading channels with arbitrary inputs: Characterization and optimization of the information rate,” IEEE Trans. Inform. Theory, vol. 60, pp. 569–585, Jan. 2014. [111] T. Tanaka, “A statistical-mechanics approach to large-system analysis of CDMA multiuser detections,” IEEE Trans. Inform. Theory, vol. 48, pp. 2888–2910, Nov. 2002. [112] D. Guo and S. Verdú, Communication, Information and Network Security.

New York: Kluwer Academic, pp. 229-277, 2002.

[113] R. R. Müller and W. Gerstacker, “On the capacity loss due to separation of detection and decoding,” IEEE Trans. Inform. Theory, vol. 50, pp. 1769–1778, Aug. 2004. [114] D. Guo and S. Verdú, “Randomly spread CDMA: Asymptotics via statistical physics,” IEEE Trans. Inform. Theory, vol. 50, pp. 1982–2010, Jun. 2005. [115] R. R. Müller, “Channel capacity and minimum probability of error in large dual antenna array systems with binary modulation,” IEEE Trans. Signal Process., vol. 51, pp. 2821–2828, Nov. 2003. [116] C.-K. Wen, P. Ting, and J.-T. Chen, “Asymptotic analysis of MIMO wireless systems with spatial correlation at the receiver,” IEEE Trans. Commun., vol. 42, pp. 349–362, Feb. 2006. [117] C.-K. Wen, K.-K. Wong, and J.-T. Chen, “Spatially correlated MIMO multiple-access systems with macrodiversity: Asymptotic analysis via statistical physics,” IEEE Trans. Commun., vol. 55, pp. 477–488, Mar. 2007.

87

[118] C.-K. Wen and K.-K. Wong, “Asymptotic analysis of spatially correlated MIMO multiple-access channels with arbitrary signalling inputs for joint and separate decoding,” IEEE Trans. Inform. Theory, vol. 53, pp. 252–268, Jan. 2007. [119] C.-K. Wen, K.-K. Wong, and J.-T. Chen, “Asymptotic mutual information for Rician MIMO-MA channel with arbitrary inputs: A replica analysis,” IEEE Trans. Commun., vol. 58, pp. 2782–2788, Oct. 2010. [120] C.-K. Wen and K.-K. Wong, “On the sum-rate of uplink MIMO celluar systems with amplify-and-forward relaying and collaborative base stations,” IEEE J. Sel. Areas Commun., vol. 28, pp. 1409–1424, Dec. 2010. [121] M. A. Girnyk, M. Vehkaperä, and L. K. Rasmussen, “Asymptotic performance analysis of a K-hop amplify-and-forward relay MIMO channel,” IEEE Trans. Inform. Theory, vol. 62, May 2016. [122] R. R. Müller, D. Guo, and A. L. Moustakas, “Vector precoding for wireless MIMO systems and its replica analysis,” IEEE J. Sel. Areas Commun., vol. 26, pp. 1–11, May 2008. [123] B. M. Zaidel, R. R. Müller, A. L. Moustakas, and R. de Miguel, “Vector precoding for Gaussian MIMO broadcast channels: Impact of replica symmetry breaking,” IEEE Trans. Inform. Theory, vol. 58, pp. 1413–1440, Mar. 2012. [124] M. Vehkaperä, T. Riihonen, M. Girnyk, E. Björnson, M. Debbah, L. K. Rasmussen, and R. Wichman, “Asymptotic analysis of SU-MIMO channels with transmitter noise and mismatched joint decoding,” IEEE Trans. Commun., vol. 63, pp. 749–765, Mar. 2015. [125] A. Ganti, A. Lapidoth, and I. Telatar, “Mismatched decoding revisited: General alphabets, channels with memory and the wide-band limit,” IEEE Trans. Inform. Theory, vol. 46, pp. 1953–1967, Nov. 2000. [126] N. Varnica, X. Ma, and A. Kav˘cić, “Capacity of power constrained memoryless AWGN channels with fixed input constellation,” in Proc. Global Commun. Conf. (Globecom 2002), Taipei, Taiwan, Jun. 2002, pp. 1339–1343. [127] J. Bellorado, S. Ghassemzadeh, and A. Kav˘cić, “Approaching the capacity of the MIMO Rayleigh flat-fading channel with QAM constellations, independent across antennas and dimensions,” IEEE Trans. Wireless Commun., vol. 6, pp. 1322–1332, Jun. 2006. [128] G. D. Forney, “Multidimensional constellations - part i: Introduction, figure of metrit, and generalized cross constellation,” IEEE J. Select. Areas Commun., vol. 7, pp. 877–886, Aug. 1989. [129] M. P. Yankov, S. Forchhammer, K. J. Larsen, and L. P. B. Christensen, “Factorization properties of the optimal signaling distribution of multi-dimensional QAM constellation,” in IEEE Int. Sym. Commun. Control, and Signal Process., Athens, Greece, May 2014, pp. 384–387. [130] ——, “Approximating the constellation constrained capacity of the MIMO channel with discrete input,” in Proc. Int. Conf. Commun. (ICC’2015), London, UK, Jun. 2015, pp. 4018–4023. [131] D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kav˘cić, and W. Zeng, “Simulation-based computation of information rates for channel with memory,” IEEE Trans. Inform. Theory, vol. 52, pp. 3498–3508, Aug. 2006. [132] J. Chen and P. H. Siegel, “On the symmetric infomration rate of two-dimensional finite-state ISI channels,” IEEE Trans. Inform. Theory, vol. 52, pp. 227–236, Jan. 2006. [133] O. Shental, N. Shental, S. Shamai, I. Kanter, A. J. Weiss, and Y. Weiss, “Discrete-input two-dimensional Gaussian channels with memory: Estimation and information rates via graphical models and statistical mechanics,” IEEE Trans. Inform. Theory, vol. 54, pp. 1500–1513, Apr. 2008. [134] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed.

New York: Wiely, 2006.

88

[135] C. Xiao and Y. R. Zheng, “On the mutual information and power allocation for vector Gaussian channels with finite discrete inputs,” in Proc. IEEE Global. Telecommun. Conf. (GLOBECOM’2008), New Orleans, USA, Dec. 2008, pp. 1–5. [136] ——, “Transmit precoding for MIMO systems with partial CSI and discrete-constellation inputs,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2009), Dresden, Germany, Jun. 2009, pp. 1–5. [137] M. Payaró and D. P. Palomar, “On optimal precoding in linear vector Gaussian channels with arbitrary inputs distribution,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2009), Seoul, Korea, Jun. 2009, pp. 1085–1089. [138] M. Lamarca, “Linear precoding for mutual information maximization in MIMO systems,” in Proc. Int. Symp. Wireless Commun. Sys. (ISWCS’2009), Siena, Italy, 2009, pp. 1–5. [139] X. Yuan, L. Ping, C. Xu, and A. Kavcic, “Achievable rates of MIMO systems with linear precoding and iterative LMMSE detection,” IEEE Trans. Inform. Theory, vol. 60, pp. 7073–7089, Nov. 2014. [140] K. Cao, Y. Wu, W. Yang, and Y. Cai, “Low-complexity precoding for MIMO channels with finite alphabet input,” Electron. Lett., vol. 53, pp. 160–162, Feb. 2017. [141] S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G,” IEEE Commun. Mag., vol. 53, pp. 186–194, Jan. 2015. [142] R. Rajashekar and L. Hanzo, “Hybrid beamforming in mm-Wave MIMO systems having a finite input alphabet,” IEEE Trans. Commun., vol. 64, Aug. 2016. [143] G. Abhinav and B. S. Rajan, “Two-user Gaussian interference channel with finite constellation input and FDMA,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2011), Quintana-Roo, Mexico, Mar. 2011, pp. 25–30. [144] W. Zeng, C. Xiao, and J. Lu, “A low complexity design of linear precoding for MIMO channel with finite alphabet inputs,” IEEE Wireless Commun. Lett., vol. 1, pp. 38–42, Feb. 2012. [145] Y. R. Zheng, M. Wang, W. Zeng, and C. Xiao, “Practical linear precoder design for finite alphabet multiple-input multiple-output orthogonal freqquency division multiplexing with experiment validation,” IET Commun., vol. 7, pp. 836–847, Jun. 2013. [146] T. Ketseoglou and E. Ayanoglu, “Linear precoding gain for large MIMO configurations with QAM and reduced complexity,” IEEE Trans. Commun., vol. 64, pp. 4196–4208, Oct. 2016. [147] Y. Wu, D. W. K. Ng, C. W. Wen, R. Schober, and A. Lozano, “Low-complexity MIMO precoding with discrete signals and statistical CSI,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2016), Kuala Lumpur, Malaysia, May 2016, pp. 1–6. [148] ——, “Low-complexity MIMO precoding for finite alphabet signals,” Online available: http://arxiv.org/abs/1606.03380. [149] J. Salo, G. Del Galdo, J. Salmi, P. Ky¨ osti, M. Milojevic, D. Laselva, and C. Schneider, (2005, Jan.) MATLAB implementation of the 3GPP Spatial Channel Model (3GPP TR 25.996), [Online]. Available: http://www.tkk.fi/Units/Radio/scm/. [150] H. Sampath, P. Stocia, and A. Paulraj, “Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion,” IEEE Trans. Commun., vol. 49, pp. 2198–2206, Dec. 2001. [151] K. H. Lee and D. P. Peterson, “Optimal linear coding for vector channels,” IEEE Trans. Commun., vol. 24, pp. 1283–1290, Dec. 1976. [152] J. Salz, “Digital transmission over cross-coupled linear channels,” At&T Tech. J., vol. 64, pp. 1147–1159, Jul.-Aug. 1985.

89

[153] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input-multiple-output (MIMO) transmission systems,” IEEE Trans. Commun., vol. 42, pp. 3221–3231, Dec. 1994. [154] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbank precoders and equalizer part I: Unification and optimal designs,” IEEE Trans. Signal Process., vol. 47, pp. 1988–2006, Jul. 1999. [155] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, and H. Sampath, “Optimal designs for space-time linear precoders and decoders,” IEEE Trans. Signal Process., vol. 50, pp. 1051–1064, May 2002. [156] D. P. Palomar, “Unified framework for linear MIMO transceivers with shaping contraints,” IEEE Commun. Lett., vol. 8, pp. 697–699, Dec. 2004. [157] ——, “Convex primal decomposition for multicarrier linear MIMO transceiver,” IEEE Trans. Signal Process., vol. 53, pp. 4661–4674, Dec. 2005. [158] D. P. Palomar, M. A. Lagunas, and J. M. Cioffi, “Optimum linear joint transmit-receive processing for MIMO channels with QoS constraints,” IEEE Trans. Signal Process., vol. 52, pp. 1179–1196, May 2004. [159] D. P. Palomar, M. Bengtsson, and B. Ottersten, “Minimum BER linear transceiver for MIMO channels via primal decomposition,” IEEE Trans. Signal Process., vol. 53, pp. 2866–2881, Aug. 2005. [160] L. G. Ordón˜ ez, D. P. Palomar, A. P.-Zamora, and J. R. Fonollosa, “Minimum BER linear MIMO transceivers with adaptive number of substreams,” IEEE Trans. Signal Process., vol. 57, pp. 2336–2353, Jun. 2009. [161] F. Sterle, “Widely linear MMSE transceivers for MIMO channels,” IEEE Trans. Signal Process., vol. 55, pp. 4258–4270, Aug. 2007. [162] J. Dai, C. Chang, W. Xu, and Z. Ye, “Linear precoder optimization for MIMO systems with joint power constraints,” IEEE Trans. Commun., vol. 60, pp. 2240–2254, Aug. 2012. [163] X. Zhang, D. P. Palomar, and B. Otterstein, “Statistically robust design of linear MIMO transceiver,” IEEE Trans. Signal Process., vol. 56, pp. 3678–3689, Aug. 2008. [164] M. Ding and S. D. Blostein, “MIMO minimum total MSE transceiver design with imperfect CSI at both ends,” IEEE Trans. Signal Process., vol. 57, pp. 1141–1150, Mar. 2009. [165] J. Wang and D. P. Palomar, “Robust MMSE precoding in MIMO channels with pre-fixed receiver,” IEEE Trans. Signal Process., vol. 58, pp. 5802–5817, Nov. 2010. [166] J. Wang and M. Bengtsson, “Joint optimization of the worst-case robust MMSE MIMO transceiver,” IEEE Signal Process. Lett., vol. 18, pp. 295–298, May 2011. [167] H. Tang, W. Chen, and J. Li, “Achieving optimality in robust joint optimization of linear transceiver design,” IEEE Veh. Tech., vol. 65, pp. 1814–1819, Mar. 2015. [168] V. Venkateswaran and A.-J. Veen, “Analog beamforming in MIMO communications with phase shift networks and online channel estimation,” IEEE Trans. Signal Process., vol. 58, pp. 4131–4143, Aug. 2010. [169] A. A. D. Amico and M. Morelli, “Joint Tx-Rx MMSE design for MIMO multicarrier systems with Tomlison-Harashima precoding,” IEEE Trans. Wireless Commun., vol. 8, pp. 3118–3127, Aug. 2008. [170] F. Xu, T. N. Davidson, J.-.K. Zhang, and K. M. Wong, “Design of block transceivers with decision feedback detection,” IEEE Trans. Signal Process., vol. 3, pp. 965–978, Mar. 2006.

90

[171] Y. Jiang, D. P. Palomar, and M. K. Varanasi, “Precoder optimization for nonlinear MIMO transceiver based on arbitrary cost function,” in Proc. Conf. Inform. Sci. Syst., Baltimore, USA, Mar. 2007, pp. 119–124. [172] M. B. Shenouda and T. N. Davidson, “A framework for designing MIMO systems with decision feedback equalization or TomlisonHarashima precoding,” IEEE J. Sel. Topic Signal Process., vol. 2, pp. 401–410, Feb. 2008. [173] O. Simeone, Y. B.-Ness, and U. Spagnolini, “Linear and nonlinear preequlization/equalization for MIMO systems with long-term channel state information at the transmitter,” IEEE Trans. Wireless Commun., vol. 3, pp. 373–378, Mar. 2004. [174] S. Jin, M. R. McKay, X. Gao, and I. B. Collings, “MIMO mutichannel beamforming: SER and outage using new eigenvalue distributions of complex wishart matrices,” IEEE Trans. Commun., vol. 56, pp. 424–434, Mar. 2008. [175] L. G. Ordón˜ ez, D. P. Palomar, A. P.-Zamora, and J. R. Fonollosa, “High-SNR analytical performance of spatial multiplexing MIMO systems with CSI,” IEEE Trans. Signal Process., vol. 55, pp. 5447–5462, Nov. 2007. [176] L. G. Ordón˜ ez, D. P. Palomar, and A. P.-Zamora, “Ordered eigenvalues of a general class of Hermitian random matrices with application to the performance analysis of MIMO systems,” IEEE Trans. Signal Process., vol. 57, pp. 672–689, Feb. 2009. [177] H. Zhang, S. Jin, M. R. McKay, X. Zhang, and D. Yang, “High-SNR performance of MIMO multi-channel beamforming in doublescattering channels,” IEEE Trans. Commun., vol. 59, pp. 1621–1631, Jun. 2011. [178] L. Sun, M. R. McKay, and S. Jin, “Analytical performance of MIMO multichannel beamforming in the presence of unequal power cochannel interference and noise,” IEEE Trans. Signal Process., vol. 57, no. 7, pp. 2721–2735, Jul. 2009. [179] S. Jin, M. R. McKay, K. K. Wong, and X. Gao, “MIMO multichannel beamforming in interference-limited Ricean fading channels,” in Proc. IEEE Global. Telecommun. Conf., New Orleans, USA, Dec. 2008, pp. 1–5. [180] Y. Wu, S. Jin, X. Gao, C. Xiao, and M. R. McKay, “MIMO multichannel beamforming in Rayleigh-product channels with arbitrarypower co-channel interference and noise,” IEEE Trans. Wireless Commun., vol. 3, pp. 3677–3691, Oct. 2012. [181] D. G. Brennan, “Linear diversity combining techniques,” in Proc. IEEE, vol. 91, Feb. 2003, pp. 331–356. [182] Z. Liu, Y. Xin, and G. B. Giannakis, “Space-time-frequency coded OFDM over frequency-selective fading channels,” IEEE Trans. Signal Process., vol. 50, pp. 2465–2476, Oct. 2002. [183] H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Inform. Theory, vol. 49, pp. 1–4, Jan. 2001. [184] N. Sharma and C. B. Papadias, “Improved quasi-orthogonal codes through constellation rotation,” IEEE Trans. Commun., vol. 51, pp. 332–335, Mar. 2003. [185] W. Su and X.-G. Xia, “Signal constellation for quasi-orthogonal space-time block codes with full diversity,” IEEE Trans. Inform. Theory, vol. 50, pp. 2331–2347, Oct. 2004. [186] C. Yuen, Y. L. Guan, and T. T. Tjhung, “Quasi-orthogonal STBC with minimum decoding complexity,” IEEE Trans. Wireless Commun., vol. 4, pp. 2089–2094, Sep. 2005. [187] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [188] Y. Xin, Z. Wang, and G. B. Giannakis, “Space-time diversity systems based on linear constellation precoding,” IEEE Trans. Wireless Commun., vol. 2, pp. 294–309, Mar. 2003.

91

[189] Z. Liu, Y. Xin, and G. B. Giannakis, “Linear constellation precoding for OFDM with maximum multipath diversity and coding gains,” IEEE Trans. Commun., vol. 51, pp. 416–427, Mar. 2003. [190] B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full diversity, high-rate space-time block codes from division algebras,” IEEE Trans. Inform. Theory, vol. 49, pp. 2596–2616, Oct. 2003. [191] X. Ma and G. B. Giannakis, “Full-diversity and full-rate complex-field space time coding,” IEEE Trans. Signal Process., vol. 51, pp. 2917–2930, Nov. 2003. [192] H. E. Gammal and M. O. Damen, “Universal space-time coding,” IEEE Trans. Inform. Theory, vol. 49, pp. 1097–1119, May 2003. [193] J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: A 2 × 2 full rate space-time code with nonvanishing determinants,” IEEE Trans. Inform. Theory, vol. 51, pp. 1432–1436, Apr. 2005. [194] F. Oggier, G. Rekaya, J.-C. Belfiore, and E Viterbo, “Perfect space-time block codes,” IEEE Trans. Inform. Theory, vol. 52, pp. 3885–3902, Sep. 2006. [195] P. Elia, B. A. Sethuraman, and P. V. Kumar, “Perfect space-time codes for any number of antennas,” IEEE Trans. Inform. Theory, vol. 53, pp. 3853–3868, Nov. 2007. [196] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity,” IEEE J. Select. Areas Commun., vol. 18, pp. 1169–1174, Jul. 2000. [197] H. Jafarkhani and V. Tarokh, “Multiple transmit antenna differential detection from generalized orthogonal designs,” IEEE Trans. Inform. Theory, vol. 47, pp. 2626–2631, Sep. 2001. [198] B. M. Hochwald and W. Sweldens, “Differential unitrary space time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041–2052, Dec. 2000. [199] B. L. Hughes, “Differetinal space time modulation,” IEEE Trans. Inform. Theory, vol. 46, pp. 2567–2578, Nov. 2000. [200] Z. Liu, G. B. Giannakis, and B. L. Hughes, “Double differential space-time block coding for time-selective fading channels,” IEEE Trans. Commun., vol. 49, pp. 1529–1539, Sep. 2001. [201] C.-S. Hwang, S. H. Nam, J. Chung, and V. Tarokh, “Differential space time block codes using nonconstant modulus constellations,” IEEE Trans. Signal Process., vol. 51, pp. 2955–2964, Nov. 2003. [202] S. Siwamogsatham and M. P. Fitz, “Robust space-time codes for correlated rayleigh fading channels,” IEEE Trans. Signal Process., vol. 50, pp. 2408–2416, Oct. 2002. [203] E. Baccarelli and M. Biagi, “Performance and optimized design of space-time codes for MIMO wireless systems with imperfect channel estimates,” IEEE Trans. Signal Process., vol. 52, pp. 2911–2923, Oct. 2004. [204] J. Giese and M. Skoglund, “Space-time constellation design for partial CSI at the receiver,” IEEE Trans. Inform. Theory, vol. 53, pp. 2715–2731, Aug. 2007. [205] Y. Li and X.-G. Xia, “Constellation mapping for space-time matrix space-time modulation,” IEEE Trans. Commun., vol. 53, pp. 764–768, May 2005. [206] N. Tran, H. Nguyen, and T. L.-Ngoc, “Coded unitrary space-time modulation with iterative decoding: Error performance and mapping design,” IEEE Trans. Commun., vol. 55, pp. 703–716, Apr. 2007.

92

[207] H.-F. Lu and P. V. Kumar, “Rate-diversity tradeoff of space-time codes with fixed alphabet and optimal constructions for PSK modulation,” IEEE Trans. Inform. Theory, vol. 49, pp. 2747–2751, Oct. 2003. [208] B. M. Hochwald, T. L. Marzetta, and C. B. Papadias, “A transmitter diversity scheme for wideband CDMA systems based on space-time spreading,” IEEE J. Select. Area Commun., vol. 19, pp. 48–60, Jan. 2001. [209] R. T. Derryberry, S. D. Gray, D. Mihai lonescu, G. Mandyam, and B. Raghothaman, “Transmit diversity in 3G CDMA systems,” IEEE Commun. Mag., vol. 40, pp. 68–75, Apr. 2002. [210] L. Hanzo, L. L. Yang, E. L. Kuan, and K. Yen, Single- and Multi-Carrier CDMA Multi-User Dectection, Space-Time Spreading Synchronisation and Standards.

New Jersey: Wiley-IEEE Press, 2003.

[211] J. M. Paredes, A. B. Gershman, and M. G.-Alkhansari, “A new full-rate full-diversity space-time block code with nonvanishing determinants and simplified maximum-likelihood decoding,” IEEE Trans. Signal Process., vol. 56, pp. 2461–2469, Jun. 2008. [212] S. Sezginer, H. Sari, and E. Biglieri, “On high-rate full-diversity 2 × 2 space-time codes with low-complexity optimum detection,” IEEE Trans. Commun., vol. 57, pp. 1532–1541, May 2009. [213] E. Biglieri, Y. Hong, and E. Viterbo, “On fast-decodable space-time block codes,” IEEE Trans. Inform. Theory, vol. 55, pp. 524–530, Feb. 2009. [214] K. P. Srinath and B. S. Rajan, “Low ML-decoding complexity, large coding gain full-rate, full diversity STBCs for 2 × 2 and 4 × 2 MIMO systems,” IEEE J. Sel. Topic Signal Process., vol. 3, pp. 916–927, Dec. 2009. [215] H. Wang, D. Wang, and X.-G. Xia, “On optimal quasi-orthogonal space-time block codes with minimum decoding complexity,” IEEE Trans. Inform. Theory, vol. 55, pp. 1104–1129, Mar. 2009. [216] K. R. Kumar and G. Caire, “Space-time codes from structured lattices,” IEEE Trans. Inform. Theory, vol. 55, pp. 547–556, Feb. 2009. [217] S. Karmakar and B. S. Rajan, “Multigroup-decodable STBCs from clifford algebras,” IEEE Trans. Inform. Theory, vol. 55, pp. 223–231, Jan. 2009. [218] G. S. Rajan and B. S. Rajan, “Multigroup ML decodable collocated and distributed space-time block codes,” IEEE Trans. Inform. Theory, vol. 56, pp. 3221–3247, Jul. 2010. [219] T. P. Ren, Y. L. Guan, C. Yuen, and R. J. Ren, “Fast-group-decodable space-time block code,” in Inform. Theory Workshop (ITW 10), Cairo, Egypt, Jan. 2010, pp. 1–5. [220] T. P. Ren, Y. L. Guan, C. Yuen, and E. Y. Zhang, “Block-orthogonal space-time code structure and its impact on QRDM decoding complexity reduction,” IEEE J. Sel. Topic Signal Process., vol. 5, pp. 1438–1450, Dec. 2011. [221] M. O. Sinnokrot and J. Barry, “Fast maximum-likelihood decoding of the golden code,” IEEE Trans. Wireless Commun., vol. 9, pp. 26–31, Jan. 2010. [222] G. R. Jithamithra and B. S. Rajan, “Construction of block orthogonal STBCs and reducing their sphere decoding complexity,” IEEE Trans. Wireless Commun., vol. 5, pp. 2906–2919, May 2014. [223] R. Vehkalahti, C. Hollanti, and F. Oggier, “Fast-decodable asymmetric space-time codes from division algebras,” IEEE Trans. Inform. Theory, vol. 58, pp. 2362–2385, Apr. 2012. [224] N. Markin and F. Oggier, “Iterated space-time code constructions from cyclic algebras,” IEEE Trans. Inform. Theory, vol. 59, pp. 5966–5979, Sep. 2013.

93

[225] K. P. Srinath and B. S. Rajan, “Fast-decodable MIDO codes with large coding gain,” IEEE Trans. Inform. Theory, vol. 60, pp. 992–1007, Feb. 2014. [226] ——, “Improved perfect space-time block codes,” IEEE Trans. Inform. Theory, vol. 59, pp. 7927–7935, Dec. 2013. [227] M. D. Renzo, H. Haas, and P. M. Grant, “Spatial modulation for multiple-antenna wireless systems: A survey,” IEEE Commun. Mag., vol. 49, pp. 182–191, Dec. 2011. [228] M. D. Renzo and H. Haas, “On transmit diversity for spatial modulation MIMO: Impact of spatial constellation diagram and shaping filters at the transmitter,” IEEE Trans. Veh. Tech., vol. 62, pp. 2507–2531, Jul. 2013. [229] M. I. Kadir, S. Sugiura, S. Chen, and L. Hanzo, “Unified MIMO-multicarrier designs: A space-time shift keying approach,” IEEE Commun. Surveys Tuts., vol. 17, pp. 550–579, Second Quarter 2015. [230] G. B. Giannakis, Z. Liu, X. Ma, and S. Zhou, Space-time coding for broadband wireless communications. New Jersey: John Wiley and Son, 2007. [231] B. Ahmed and M. A. Matin, Coding for MIMO-OFDM in Future Wireless Systems.

Springer, 2015.

[232] E. Akay, E. Sengul, and E. Ayanoglu, “Bit interleaved coded multiple beamforming,” IEEE Trans. Commun., vol. 9, pp. 1802–1811, Sep. 2007. [233] B. Li and E. Ayanoglu, “Full-diversity precoding design of bit-interleaved coded multiple beamforming with orthogonal frequency division multiplexing,” IEEE Trans. Commun., vol. 61, pp. 2432–2445, Jun. 2013. [234] L. Collin, O. Berder, P. Rostaing, and G. Burel, “Optimal minimum distance-based precoder for MIMO spatial multiplexing systems,” IEEE Trans. Signal Process., vol. 3, pp. 617–627, 2004. [235] Q.-T. Ngo, O. Berder, and P. Scalart, “Minimum Euclidean distance based precoders for MIMO systems using rectangular QAM modulations,” IEEE Trans. Signal Process., vol. 3, pp. 1527–1533, 2012. [236] O. J. Oyedapo, B. Vrigenau, R. Vauzelle, and H. Boeglen, “Performance analysis of closed-loop MIMO precoder based on the probability of minimum distance,” IEEE Trans. Wireless Commun., vol. 14, pp. 1849–1857, Apr. 2015. [237] Q.-T. Ngo, O. Berder, and P. Scalart, “Minimum Euclidean distance-based precoding for three-dimensional multiple input multiple output spatial multiplexing systems,” IEEE Trans. Wireless Commun., vol. 11, pp. 2486–2495, Jul. 2012. [238] K. P. Srinath and B. S. Rajan, “A low ML-decoding complexity, full-diversity full-rate MIMO precoder,” IEEE Trans. Signal Process., vol. 11, pp. 5485–5498, Nov. 2011. [239] Q.-T Ngo, O. Berder, and P. Scalart, “General minimum Euclidean distance-based precoder for MIMO wireless systems,” EURASIP J. Adv. Signal Process., vol. 39, pp. 1–12, Mar. 2013. [240] C.-T. Lin and W.-R. Wu, “X-structured precoder designs for spatial-multiplexing MIMO and MIMO relay systems,” IEEE Trans. Veh. Tech., vol. 62, pp. 4430–4443, Nov. 2013. [241] S. Bergman and B. Ottersten, “Lattice-based linear precoding for MIMO channels with transmitter CSI,” IEEE Trans. Signal Process., vol. 56, pp. 2902–2914, Jul. 2008. [242] A. Sakzad and E. Viterbo, “Full diversity unitrary precoded integer-forcing,” IEEE Trans. Wireless Commun., vol. 8, pp. 4316–4327, Aug. 2015.

94

[243] X. Xu, Q. Gu, Y. Jiang, and X. Dai, “Ellipse expanding method for minimum distance-based MIMO precoder,” IEEE Commun. Lett., vol. 16, pp. 490–493, Apr. 2012. [244] X. Xu and Z. Chen, “Recursive construction of minimum Euclidean distance-based precoder for arbitrary-dimensional MIMO systems,” IEEE Trans. Commun., vol. 62, pp. 1258–1271, Apr. 2014. [245] D. Kapetanović, H. V. Cheng, W. H. Mow, and F. Rusek, “Optimal two-dimensional lattices for precoding of linear channels,” IEEE Trans. Wireless Commun., vol. 12, pp. 2104–2113, May 2013. [246] ——, “Lattice structures of precoders maximizing the minimum distance in linear channels,” IEEE Trans. Inform. Theory, vol. 61, pp. 908–916, Feb. 2015. [247] H. J. Park, B. Li, and E. Ayanoglu, “Constellation precoded multiple beamforming,” IEEE Trans. Commun., vol. 59, pp. 1275–1286, May 2011. [248] D. A. Gore and A. J. Paulraj, “MIMO antenna subset selection for space-time coding,” IEEE Trans. Signal Process., vol. 50, pp. 2580–2588, Oct. 2002. [249] G. Jongren, M. Skoglund, and B. Ottersten, “Combining beamforming and orthogonal space-time block coding,” IEEE Trans. Inform. Theory, vol. 48, pp. 611–627, Mar. 2002. [250] H. Sampath and A. Paulraj, “Linear precoding for space-time coded systems with known fading correlations,” IEEE Commun. Lett., vol. 6, pp. 239–241, Jun. 2002. [251] M. Vu and A. Paulraj, “Optimal linear precoders for MIMO wireless correlated channels with nonzero mean in space-time coded systems,” IEEE Trans. Signal Process., vol. 54, pp. 2318–2332, Jun. 2006. [252] M. R. Bhatnagar and A. Hjørungnes, “Linear precoding of STBC over correlated Rician MIMO channels,” IEEE Trans. Wireless Commun., vol. 9, pp. 1832–1836, Jun. 2010. [253] H. Wang, Y. Li, and X.-G. Xia, “Unitrary and non-unitrary precoders for a limited feedback precoded OSTBC system,” IEEE Trans. Veh. Tech., vol. 62, pp. 1646–1654, May 2013. [254] X. Cai and G. B. Giannakis, “Differential space-time modulation with eigen-beamforming for correlated MIMO fading channels,” IEEE Trans. Wireless Commun., vol. 5, pp. 1832–1836, Apr. 2006. [255] M. R. Bhatnagar, A. Hjørungnes, and L. Song, “Differential coding for non-Orthogonal space-time block codes with non-unitrary constellations over arbitrarily correlated Rayleigh channels,” IEEE Trans. Wireless Commun., vol. 8, pp. 1832–1836, Aug. 2009. [256] M. J. Borran, A. Sabharwal, and B. Aazhang, “Design criterion and construction methods for partially coherent multiple antenna constellation,” IEEE Trans. Wireless Commun., vol. 8, pp. 4122–4133, Aug. 2009. [257] A. Yadav, M. Juntti, and J. Lilleberg, “Cutoff rate optimized space-time signal design for partially coherent channel,” IEEE Trans. Commun., vol. 60, pp. 1563–1574, Jun. 2012. [258] ——, “Partially coherent constellation design and bit-mapping with coding for correlated fading channels,” IEEE Trans. Commun., vol. 61, pp. 4243–4255, Oct. 2013. [259] ——, “Linear precoder design for doubly correlated partially coherent fading MIMO channels,” IEEE Trans. Wireless Commun., vol. 13, pp. 3621–3635, Jul. 2014.

95

[260] S. Catreux, P. F. Driessen, and L. J. Greenstein, “Data throughtputs using multiple-input multiple-output (MIMO) techniques in a noise-limited cellular environment,” IEEE Trans. Wireless Commun., vol. 1, pp. 226–235, Apr. 2002. [261] S. Catreux, V. Erceg, D. Gesbert, and R. W. Heath, “Adaptive modulation and MIMO coding for broadband wireless data networks,” IEEE Commun. Mag., vol. 40, pp. 108–115, Jun. 2002. [262] C.-B. Chae, A. F. Rearden, R. W. Heath Jr, M. R. Mckay, and I. B. Collings, “Adaptive MIMO transmission technieques for broadband wireless communication systems,” IEEE Commun. Mag., vol. 48, pp. 112–118, May 2010. [263] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218–1230, Oct. 1997. [264] S. T. Chung and A. J. Goldsmith, “Degrees of freedom in adaptive modulation: A unified view,” IEEE Trans. Commun., vol. 49, pp. 1561–1571, Sep. 2001. [265] M. S. Alouini and A. J. Goldsmith, “Adaptive modulation over Nakagami fading channels,” Kluwer J. Wireless Commun., vol. 13, pp. 119–143, May 2000. [266] J. F. Hayes, “Adaptive feedback communications,” IEEE Trans. Commun. Technol., vol. COM-20, pp. 29–34, Feb. 1968. [267] J. K. Cavers, “Variable-rate transmission for Rayleigh fading channels,” IEEE Trans. Commun., vol. COM-20, pp. 15–22, Feb. 1972. [268] S. Otsuki, S. Sampei, and N. Morinaga, “Square-QAM adaptive modulation/TDMA/TDD systems using modulation level estimation with Walsh function,” Electron. Lett., vol. 31, pp. 169–171, Feb. 1995. [269] W. T. Webb and R. Steele, “Variable rate QAM for mobile radio,” IEEE Trans. Commun., vol. 43, pp. 2223–2230, Jul. 1995. [270] Y. Kamio, S. Sampei, H. Sasaoka, and N. Morinaga, “Performance of modulation-level-controlled adaptive-modulation under limited transmission delay time for land mobile communications,” in Proc. IEEE Veh. Tech. Conf. (VTC’1995), Chicago, USA, Jul. 1995, pp. 221–225. [271] B. Vucetic, “An adaptive coding scheme for time-varying channels,” IEEE Trans. Commun., vol. 39, pp. 653–663, May 1991. [272] S. M. Alamouti and S. Kallel, “Adaptive trellis-coded multiple-phased-shift keying for Rayleigh fading channels,” IEEE Trans. Commun., vol. 42, pp. 2305–2314, Jun. 1994. [273] T. Ue, S. Sampei, and N. Morinaga, “Symbol rate and modulation level controlled adaptive modulation/TDMA/TDD for personal communication systems,” in Proc. IEEE Veh. Tech. Conf. (VTC’1995), Chicago, USA, Jul. 1995, pp. 306–310. [274] D. L. Goeckel, “Adaptive coding for time-varying channels using outdated fading estimates,” IEEE Trans. Commun., vol. 47, pp. 844–855, Jun. 1999. [275] K. J. Hole, H. Holm, and G. E. Oien, “Adaptive multidimensional coded modulation over flat fading channels,” IEEE J. Select. Area Commun., vol. 18, pp. 1153–1158, Jul. 2000. [276] S. Hu and A. Duel-Hallen, “Combined adaptive modulation and transmitter diversity using long range prediction for flat fading mobile radio channels,” in Proc. Global Telecommun. Conf. (Globecom’2001), San Antonio, USA, Nov. 2001, pp. 25–29. [277] P. Xia, S. Zhou, and G. B. Giannakis, “Adaptive MIMO-OFDM based on partial channel state information,” IEEE Trans. Signal Process., vol. 52, pp. 202–213, Jan. 2004. [278] S. Zhou and G. B. Giannakis, “How accurate channel prediction needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO channels,” IEEE Trans. Wireless Commun., vol. 3, pp. 1285–1294, Jul. 2004.

96

[279] P. Xia, S. Zhou, and G. B. Giannakis, “Multiantenna adaptive modulation with beamforming based on bandwidth-constrained feedback,” IEEE Trans. Commun., vol. 53, pp. 526–536, Mar. 2005. [280] R. Narasimhan, “Spatial multiplexing with transmit antenna and constellation selection for correlated MIMO fading channels,” IEEE Trans. Signal Process., vol. 51, pp. 2829–2838, Nov. 2003. [281] R. W. Heath Jr and A. J. Paulraj, “Switching between diversity and multiplexing in MIMO systems,” IEEE Trans. Commun., vol. 53, pp. 962–968, Jun. 2005. [282] J. Huang and S. Signell, “On spectral efficiency of low-complexity adaptive MIMO systems in Rayleigh fading channel,” IEEE Trans. Wireless Commun., vol. 8, pp. 4369–4374, Sep. 2009. [283] K. P. Kongara, P.-H. Kuo, P. J. Smith, L. M. Garth, and A. Clark, “Performance analysis of adaptive MIMO OFDM beamforming systems,” in Proc. IEEE Int. Commun. Conf. (ICC’2008), Beijing, China, May 2008, pp. 4359–4365. [284] D. P. Palomar and S. Barbarossa, “Designing MIMO communication systems: Constellation choice and linear transceiver design,” IEEE Trans. Signal Process., vol. 53, pp. 3804–3818, Oct. 2005. [285] A. Yasotharan, “Lower bound on the BER of a pre-and postequalized vector communication system,” IEEE Trans. Commun., vol. 53, pp. 453–462, Mar. 2005. [286] S. Bergman, D. P. Palomar, and B. Ottersten, “Joint bit allocation and precoding for MIMO systems with decision feedback detection,” IEEE Trans. Signal Process., vol. 57, pp. 4509–4521, Nov. 2009. [287] C.-C. Li and Y.-P. Lin, “On the duality of MIMO transceiver designs with bit allocation,” IEEE Trans. Signal Process., vol. 59, pp. 3775–3787, Aug. 2011. [288] Y. Ko and C. Tepedelenlioˇglu, “Orthogonal space-time block coded rate-adaptive modulation with outdated feedback,” IEEE Trans. Wireless Commun., vol. 8, pp. 290–295, Feb. 2006. [289] J. Huang and S. Signell, “On performance of adaptive modulation in MIMO systems using orthogonal space-time block codes,” IEEE Trans. Veh. Technol., vol. 58, pp. 4238–4247, Oct. 2009. [290] X. Yu, S.-H. Leung, W. H. Mow, and W.-K. Wong, “Performance of variable-power adaptive modulation with space-time coding and imperfect CSI in MIMO systems,” IEEE Trans. Veh. Technol., vol. 58, pp. 2115–2120, May 2009. [291] D. V. Duong and G. E. Øien, “Optimal pilot spacing and power in rate-adaptive MIMO diversity systems with imperfect CSI,” IEEE Trans. Wireless Commun., vol. 6, pp. 845–851, Mar. 2007. [292] Z. Zhou, B. Vucetic, Z. Chen, and Y. Li, “Performance analysis of adaptive MIMO OFDM beamforming systems,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2005), Berlin, Germany, Sep. 2005, pp. 1101–1105. [293] J. F. Paris and A. J. Goldsmith, “Adaptive modulation for MIMO beamforming under average BER contraint and imperfect CSI,” in Proc. Int. Commun. Conf. (ICC’2006), Istanbul, Turkey, Jun. 2006, pp. 1312–1317. [294] Z. Wang, C. He, and A. He, “Robust AM-MIMO based on minimized transmission power,” IEEE Commun. Lett., vol. 60, pp. 432–434, Jun. 2006. [295] X. Zhang and B. Ottersten, “Power allocation and bit loading for spatial multiplexing in MIMO systems,” in Proc. Int. Conf. on Acoustics, Speech and Signal Process. (ICASSP’2003), Hong Kong, China, Apr. 2003, pp. 53–56.

97

[296] T. Delamotte, V. Dantona, G. Bauch, and B. Lankl, “Power allocation and bit loading for fixed wireless MIMO channels,” in Proc. ITG Conf. on System, Communi. and Coding (SCC’2013), Munich, Germany, Jan. 2013, pp. 1–6. [297] M. I. Rahman, S. S. Das, Y. Wang, F. B. Frederiksen, and R. Prasad, “Bit and power loading approach for broadband multi-antenna OFDM system,” in Proc. Veh. Tech. Conf. (VTC’2007), Baltimore, USA, Sep. 2007, pp. 1689–1693. [298] B. A. Cetiner, H. Jafarkhani, J.-Y. Qian, H. J. Yoo, A. Grau, and F. D. Flaviis, “Multifunctional reconfigurable MEMS integrated antennas for adaptive MIMO systems,” IEEE Commun. Mag., vol. 42, pp. 62–70, Dec. 2004. [299] B. A. Cetiner, E. Akay, E. Sengul, and E. Ayanoglu, “A MIMO system with multifunctional reconfigurable antennas,” IEEE Antenna and Wireless Propagation Lett., vol. 5, pp. 463–466, Jan. 2006. [300] R. C. Daniels, C. M. Caramanis, and R. W. Heath Jr, “Adaptation in convolutionally coded MIMO-OFDM wireless systems through supervised learning and SNR ordering,” IEEE Trans. Veh. Technol., vol. 59, pp. 114–126, Jan. 2010. [301] M. D. Dorrance and I. D. Marsland, “Adaptive discrete-rate MIMO communications with rate-compatible LDPC codes,” IEEE Trans. Commun., vol. 58, pp. 3115–3125, Nov. 2010. [302] I. Stupia, V. Lottici, F. Giannetti, and L. Vandendorpe, “Link resource adaptation for multiantenna bit-interleaved coded multicarrier systems,” IEEE Trans. Signal Process., vol. 60, pp. 3644–3656, Jul. 2012. [303] T. Haustein, A. Forck, H. Gäbler, V. Jungnickel, and S. Schiffermüller, “Real-time signal processing for multiantenna systems: Algorithms, optimization, and implementation on an experimental test-bed,” EURASIP Journal on Appiled Signal Process., vol. 2006, pp. 1–21, Apr. 2006. [304] 3GPP, TS36.213, “Physical layer procedures (Release 8),” V8.8.0, Sep. 2009. [305] 3GPP, TS36.814, “Further advancements for E-UTRA physical layer aspects (Release 9),” V9.0.0, Mar. 2010. [306] J. G. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of WiMAX: Understanding Broadband Wireless Networks. New Jersey: Prentice Hall, 2007. [307] R. V. Nee, V. K. Jones, G. Awater, A. V. Zelst, J. Gardner, and G. Steele, “The 802.11n MIMO-OFDM standard for wireless LAN and beyond,” Wireless Personal Commun., vol. 37, pp. 445–453, 2006. [308] F. Peng, J. Zhang, and W. E. Ryan, “Adaptive modulation and coding for IEEE 802.11n,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2007), Hong Kong, China, Mar. 2007, pp. 656–661. [309] P. Mogensen, W. Na, I. Z. Kovács, F. Frederiksen, A. Pokhariyal, K. Pedersen, T. Kolding, K. Hugl, and M. Kuusela, “LTE capacity compared to the Shannon bound,” in Proc. Veh. Tech. Conf. (VTC’2007), Dublin, Ireland, Apr. 2007, pp. 1234–1238. [310] D. Marabiss, D. Tarchi, R. Fantacci, and F. Balleri, “Efficient adaptive modulation and coding techniques for WiMAX systems,” in Proc. Int. Commun. Conf. (ICC’2008), Beijing, China, May 2008, pp. 3383–3387. [311] C. Mehlführer, S. Caban, and M. Rupp, “Experimental evaluation of adaptive modulation and coding in MIMO WiMAX with limited feedback,” EURASIP Journal on Appiled Signal Process., vol. 2008, pp. 1–12, Jan. 2008. [312] C. Mehlführer, M. Wrulich, J. C. Ikuno, D. Bosanska, and M. Rupp, “Simulating the long term evolution physical layer,” in Proc. Euro. Signal Process. Conf. (EURASIP’2009), Glasgow, Scotland, Aug. 2009, pp. 1471–1478. [313] W. Guo, S. Wang, and X. Chu, “Capacity expression and power allocation for arbitrary modulation and coding rates,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2013), Shanghai, China, Apr. 2013, pp. 3294–3299.

98

[314] J. Harshan and B. S. Rajan, “On two-user Gaussian multiple access channels with finite input constellations,” IEEE Trans. Inform. Theory, vol. 57, pp. 1299–1327, Mar. 2011. [315] ——, “A novel power allocation scheme for two-user GMAC with finite input constellations,” IEEE Trans. Wireless Commun., vol. 12, pp. 818–827, Mar. 2013. [316] M. A. Girnyk, M. Vehkaperä, and L. K. Rasmussen, “Large-system analysis of correlated MIMO multiple access channels with arbitrary signalling in the presence of interference,” IEEE Trans. Wireless Commun., vol. 13, pp. 2060–2073, Apr. 2014. [317] W. Weichselberger, M. Herdin, H. Ozcelik, and E. Bonek, “A stochastic MIMO channel model with joint correlation of both link ends,” IEEE Trans. Wireless Commun., vol. 5, pp. 90–100, Jan. 2006. [318] Y. Wu, C.-K. Wen, C. Xiao, X. Gao, and R. Schober, “Linear precoding for the MIMO multiple access channel with finite alphabet inputs and statistical CSI,” IEEE Trans. Wireless Commun., vol. 14, pp. 983–997, Feb. 2015. [319] S. Serbetli and A. Yener, “Transceiver optimization for multiuser MIMO systems,” IEEE Trans. Signal Process., vol. 52, pp. 214–226, Jan. 2004. [320] Z.-Q. Luo, T. N. Davidson, G. B. Giannakis, and K. M. Wong, “Transceiver optimization for block-based multiple access through ISI channels,” IEEE Trans. Signal Process., vol. 52, pp. 1037–1052, Apr. 2004. [321] I.-T. Lu, “Joint MMSE precoder and decoder design subject to arbitrary power constraints for uplink multiuser MIMO systems,” in Sarnoff Symposium (SARNOFF’2009), Princeton, USA, Mar.-Apr. 2009, pp. 1–5. [322] E. A. Jorswieck, A. Sezgin, H. Boche, and E. Costa, “Multiuser MIMO MAC with statistical CSI and MMSE receiver: Feedback strategies and transmitter optimization,” in Proc. of Int. Conf. on Wireless Commun. and Mobile Computing (IWCMC’2006), New York, USA, Jul. 2006, pp. 455–460. [323] X. Zhang, E. A. Jorswieck, B. Ottersten, and A. Paulraj, “MSE based optimization of multiuser MIMO MAC with partical CSI,” in Proc. Asilomar Conference on Signals Systems and Computers (Asilomar’2006), Pacific Grove, USA, Oct.-Nov. 2006, pp. 374–378. [324] X. Zhang, D. P. Palomar, and B. Ottersten, “Robust MAC MIMO transceiver design with partial CSIT and CSIR,” in Proc. Asilomar Conference on Signals Systems and Computers (Asilomar’2007), Pacific Grove, USA, Nov. 2007, pp. 324–328. [325] P. Layec, P. Piantanida, R. Visoz, and A. O. Berthet, “Transceiver design for sum-MSE optimization in MIMO-MAC with imperfect channel estimation,” in Proc. Asilomar Conference on Signals Systems and Computers (Asilomar’2008), Pacific Grove, USA, Oct. 2008, pp. 321–325. [326] J. W. Huang, E. K. S. Au, and V. K. N. Lau, “Linear precoder and equalizer design for uplink multiuser MIMO systems with imperfect channel state information,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2007), Hong Kong, China, Mar. 2007, pp. 1296–1301. [327] R. G. Gallager, “A perspective on multiaccess channels,” IEEE Trans. Inform. Theory, vol. 31, pp. 124–142, Mar. 1985. [328] M. E. Gätner and H. Bölcskei, “Multiuser space-time/frequency code design,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2006), Seattle, USA, Jul. 2006, pp. 2819–2823. [329] M. Badr and J.-C. Belfiore, “Distributed space-time block codes for the MIMO multiple access channel,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2008), Toronto, Canada, Jul. 2008, pp. 2553–2557.

99

[330] M. Badr, M.-O. Damen, and J.-C. Belfiore, “Delay-tolerant distributed space-time block codes for the asynchronous multiple-access channel,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’09), Tokyo, Japan, Sep. 2009, pp. 521–525. [331] ——, “Unbalanced space-time block codes for non uniform energy distribution multiple access channels,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2009), Tokyo, Japan, Sep. 2009, pp. 315–319. [332] M. Badr and J.-C. Belfiore, “Distributed space-time block codes for the amplify-and-forward multiple access relay channel,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2008), Toronto, Canada, Jul. 2008, pp. 2543–2547. [333] Y. Hong and E. Viterbo, “Algebraic multiuser space-time block codes for a 2 × 2 MIMO,” IEEE Trans. Veh. Technol., vol. 58, pp. 3062–3066, Jul. 2009. [334] H.-F. Lu, R. Vehkalahti, C. Hollanti, J. Lahtonen, Y. Hong, and E. Viterbo, “New space-time code constructions for two-user multiple access channels,” IEEE J. Sel. Topic Signal Process., vol. 3, pp. 939–957, Dec. 2009. [335] J. Harshan and B. S. Rajan, “STBCs with reduced sphere decoding complexity for two-user MIMO MAC,” in Proc. IEEE Global. Telecommun. Conf. (GLOBECOM’2009), Honolulu, USA, Nov.-Dec. 2009, pp. 1–6. [336] M. R. Bhatnagar and A. Hjørungnes, “Differetial coding for MAC based two-user MIMO communication,” IEEE Trans. Wireless Commun., vol. 11, pp. 9–14, Jan. 2012. [337] S. Poorkasmaei and H. Jafarkhani, “Orthogonal differential modulation for MIMO multiple access channels with two users,” IEEE Trans. Commun., vol. 61, pp. 2374–2384, Jun. 2013. [338] J. W. Huang, E. K. S. Au, and V. K. N. Lau, “Precoding of space-time block codes in multiuser MIMO channels with outdated channel state information,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2007), Nice, France, Jun. 2007, pp. 2031–2035. [339] Y.-J. Kim, C.-H. Choi, and G.-H. Im, “Space-time block coded transmission with phase feedback for two-user MIMO-MAC,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2011), Kyoto, Japan, Jun. 2011, pp. 1–5. [340] F. Li and H. Jafarkhani, “Multiple-antenna interference cancellation and detection for two users using precoders,” IEEE J. Sel. Topics Signal Process., vol. 3, pp. 1066–1078, Dec. 2009. [341] ——, “Interference cancellation and detection for more than two users,” IEEE Trans. Commun., vol. 59, pp. 901–910, Mar. 2011. [342] Y. J. Zhang and K. B. Letaief, “Adaptive resource allocation for multiaccess MIMO/OFDM systems with matched filtering,” IEEE Trans. Commun., vol. 53, pp. 1810–1813, Nov. 2005. [343] R. Chemaly, K. B. Letaief, and D. Zeghlache, “Adaptive resource allocation for multiuser MIMO/OFDM networks based on partial channel state information,” in Proc. Global Commun. Conf. (Globecom’2005), St. Louis, USA, Nov.-Dec. 2005, pp. 3922–3926. [344] M. S. Maw and I. Sasase, “Efficient resource allocation for multiuser MIMO-OFDM uplink system to guarantee the proportional data rate fairness among users in a system,” in Int. Sym. Acc. Spaces (ISAS’2011), Keio University Yokohama, Japan, Jun. 2011, pp. 132–137. [345] R. Ghaffar and R. Knopp, “Channel capacity for linearly precoded multiuser MIMO for discrete constellations,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’09), Tokyo, Japan, Sep. 2009, pp. 1–5. [346] ——, “Near optimal linear precoder for multiuser MIMO for discrete alphabets,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2010), Capt Town, South Africa, May 2010, pp. 1–5.

100

[347] ——, “Linear precoders for multiuser MIMO for finite constellations and a simplified receiver structure under controlled interference,” in Proc. Asilomar Conference on Signals Systems and Computers (Asilomar’2009), Asilomar, USA, Nov. 2009, pp. 1–5. [348] N. Deshpande and B. S. Rajan, “Constellation constrained capacity of two-user broadcast channels,” in Proc. IEEE Global. Telecommun. Conf. (GLOBECOM’2009), Honolulu, HI, USA, Nov.-Dec. 2009, pp. 1–5. [349] W. Wu, K. Wang, W. Zeng, Z. Ding, and C. Xiao, “Cooperative multi-cell MIMO downlink precoding with finite alphabet inputs,” IEEE Trans. Commun., vol. 63, pp. 766–779, Mar. 2015. [350] S. X. Wu, Q. Li, A. M.-Cho, and W.-K. Ma, “Rank-two beamforming and stochastic beamforming for MISO physical-layer multicasting with finite alphabet inputs,” IEEE Signal. Process. Lett., vol. 22, pp. 1614–1618, Oct. 2015. [351] A. J. Tenenbaum and R. S. Adve, “Joint multiuser transmit-receive optimization using linear processing,” in Proc. Int. Conf. Commun. (ICC’2004), Paris, France, Jun. 2004, pp. 588–592. [352] J. Zhang, Y. Wu, S. Zhou, and J. Wang, “Joint linear transmitter and receiver design for the downlink of multiuser MIMO systems,” IEEE Commun. Lett., vol. 9, pp. 991–993, Nov. 2005. [353] S. Shi, M. Schubert, and H. Boche, “Downlink MMSE transceiver optimization for multiuser MIMO systems: MMSE balancing,” IEEE Trans. Signal Process., vol. 56, pp. 3702–3712, Aug. 2008. [354] R. Hunger, M. Joham, and W. Utschick, “On the MSE-duality of the broadcast channel and the multiple access channel,” IEEE Trans. Signal Process., vol. 57, pp. 698–713, Feb. 2009. [355] M. Ding and S. D. Blostein, “Uplink-downlink duality in normalized MSE or SINR under imperfect channel knowledge,” in Proc. Global Commun. Conf. (Globecom 2007), Washington DC, USA, Nov. 2007, pp. 3786–3790. [356] D. S.-Murga, M. Payaró, and A. P.-Iserte, “Transceiver design framework for multiuser MIMO-OFDM broadcast systems with channel gram matrix feedback,” IEEE Trans. Wireless Commun., vol. 11, pp. 1174–1786, May 2012. [357] M. Ding, J. Zou, Z. Yang, H. Luo, and W. Chen, “Sequential and incremental precoder design for joint transmission network MIMO systems with imperfect backhaul,” IEEE Trans. Veh. Technol., vol. 61, pp. 2490–2503, Jul. 2012. [358] M. Payaró, A. P.-Iserte, and M. A. Lagunas, “Robust power allocation designs for multiuser and multiantenna downlink communication systems through convex optimization,” IEEE J. Select. Area Commun., vol. 25, pp. 1390–1401, Sep. 2007. [359] N. Viˇcić and H. Boche, “Robust QoS-constrained optimization of downlink multiuser MISO systems,” IEEE Trans. Signal Process., vol. 57, pp. 714–725, Feb. 2009. [360] N. Viˇcić, H. Boche, and S. Shi, “Robust transceiver optimization in downlink multiuser MIMO systems,” IEEE Trans. Signal Process., vol. 57, pp. 3576–3587, Sep. 2009. [361] X. He and Y.-C. Wu, “Probabilistic QoS constrained robust downlink multiuser MIMO transceiver design with arbitrarily distributed channel uncertainy,” IEEE Trans. Wireless Commun., vol. 12, pp. 6292–6302, Dec. 2013. [362] E. Björnson, G. Zheng, M. Bengtsson, and B. Ottersten, “Robust monotonic optimization framework for multicell MISO systems,” IEEE Trans. Signal Process., vol. 60, pp. 2508–2522, May 2012. [363] A. D. Dabbagh and D. J. Love, “Multiple antenna MMSE based downlink precoding with quantized feedback or channel mismatch,” IEEE Trans. Commun., vol. 56, pp. 1859–1868, Nov. 2008.

101

[364] H. Sung, S.-R. Lee, and I. Lee, “Generalized channel inversion methods for multiuser MIMO systems,” IEEE Trans. Commun., vol. 57, pp. 3489–3499, Nov. 2009. [365] P. Xiao and M. Sellathurai, “Improved linear transmit processing for single-user and multi-user MIMO communications systems,” IEEE Trans. Signal Process., vol. 58, pp. 1768–1779, Mar. 2010. [366] R. F. H. Fischer, C. Stierstorfer, and J. B. Huber, “Precoding for point-to-multipoint transmission over MIMO ISI channels,” in Int. Zurich Seminar Commun., Zurich, Switzerland, Mar. 2004, pp. 208–211. [367] W. Hardjawana, B. Vucetic, and Y. Li, “Multi-user cooperative base station systems with joint precoding and beamforming,” IEEE J. Sel. Topic Signal Process., vol. 3, pp. 1079–1093, Dec. 2009. [368] C. Masouros, M. Sellathurai, and T. Ratnarajah, “Interference optimization for transmit power reduction in Tomlinson-Harashima precoded MIMO downlinks,” IEEE Trans. Signal Process., vol. 60, pp. 2470–2481, May 2012. [369] C.-H. Liu and P. P. Vaidyanathan, “MIMO broadcast DFE transceivers with QoS constraints: Min-power and max-rate solutions,” IEEE Trans. Signal Process., vol. 61, pp. 5550–5562, Nov. 2013. [370] K. Zu, R. C. de Lamare, and M. Haardt, “Multi-branch Tomlinson-Harashima precoding design for MU-MIMO systems: Theory and algorithms,” IEEE Trans. Commun., vol. 57, pp. 939–951, Mar. 2014. [371] L. Sun, J. Wang, and V. C. M. Leung, “Joint transceiver, data streams, and user ordering optimization for nonlinear multiuser MIMO systems,” IEEE Trans. Commun., vol. 63, pp. 3686–3701, Oct. 2015. [372] Q. Ma, J. Liu, and L. Yang, “A trace-orthogonal multi-user space-time coding,” in Proc. Wireless Commun. Networking and Mobile Computing (WiCom’2007), Shanghai, China, Sep. 2007, pp. 310–313. [373] E. G. Larsson, “Unitary nonuniform space-time constellations for the broadcast channel,” IEEE Commun. Lett., vol. 7, pp. 21–23, Jan. 2003. [374] C.-H. Kuo and C.-C. J. Kuo, “An embedded space-time coding (STC) scheme for broadcasting,” IEEE Trans. Broadcasting, vol. 53, pp. 48–58, Mar. 2007. [375] R. Chen, J. G. Andrews, and R. W. Heath Jr, “Multiuser space-time block coded MIMO system with downlink precoding,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2004), Paris, France, Jun. 2004, pp. 2689–2693. [376] D. Wang, E. A. Jorswieck, A. Sezgin, and E. Costa, “Joint tomlinson-harashima precoding with diversity techniques for multiuser MIMO systems,” in Proc. Veh. Tech. Conf. (VTC’2005), Stockholm, Swenden, Jun. 2005, pp. 1017–1021. [377] B. Clerckx and D. Gesbert, “Space-time encoded MISO broadcast channel with outdated CSIT: An error rate and diversity performance analysis,” IEEE Trans. Commun., vol. 63, pp. 1661–1675, May 2015. [378] D. Lee and K. Kim, “Reliability comparison of opportunistic scheduling and BD-precoding in downlink MIMO systems with multiple users,” IEEE Trans. Wireless Commun., vol. 9, pp. 964–969, Mar. 2010. [379] L. Li and H. Jafarkhani, “Multi-user downlink system design with bright transmitters and blind receivers,” IEEE Trans. Wireless Commun., vol. 11, pp. 4074–4084, Nov. 2012. [380] C.-T. Lin, W.-R. Wu, and W.-C. Lo, “X-structured precoder design for multiuser MIMO communications,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2013), London, UK, Sep. 2013, pp. 1366–1370.

102

[381] C.-M. Yen, C.-J. Chang, and L.-C. Wang, “A utility-based TMCR scheduling scheme for downlink multiuser MIMO-OFDMA systems,” IEEE Trans. Veh. Technol., vol. 65, pp. 4105–4115, Oct. 2010. [382] Z. Hu, G. Zhu, Y. Xia, and G. Liu, “Multiuser subcarrier and bit allocation for MIMO-OFDM perfect and partial channel information,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2004), New Orleans, USA, Mar. 2004, pp. 1188–1193. [383] W. W. L. Ho and Y.-C. Liang, “Optimal resource allocation for multiuser MIMO-OFDM systems with user rate constraints,” IEEE Trans. Veh. Technol., vol. 58, pp. 1190–1203, Mar. 2009. [384] S. Sadr, A. Anpalagan, and K. Raahemifar, “Radio resource allocation algorithms for the downlink of multiuser OFDM communications systems,” IEEE Commun. Surveys Tuts., vol. 11, pp. 92–106, Third Quarter 2009. [385] Y. Hara, L. Brunel, and K. Oshima, “Spatial scheduling with interference cancellation in multiuser MIMO systems,” IEEE Trans. Veh. Technol., vol. 57, pp. 893–905, Mar. 2008. [386] M. Esslaoui, F. Riera-Palou, and G. Femenias, “Fast link adaptation for opportunistic multiuser MIMO-OFDM wireless networks,” in Proc. Int. Sym. Wireless Commun. Sys. (ISWCS’2011), Aachen, Germany, Nov. 2011, pp. 372–376. [387] ——, “A fair MU-MIMO scheme for IEEE 802.11ac,” in Proc. Int. Sym. Wireless Commun. Sys. (ISWCS’2012), Paris, France, Aug. 2012, pp. 1049–1053. [388] J. Wang, Z. Lan, R. Funada, and H. Harada, “On scheduling and power allocation over multiuser MIMO-OFDMA: Fundamental design and performance evaluation in WiMAX systems,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2009), Tokyo, Japan, Sep. 2009, pp. 2752–2756. [389] Z. Chen, W. Wang, M. Peng, and F. Gao, “Limited feedback scheme based on zero-forcing precoding for multiuser MIMO-OFDM downlink systems,” in Proc. Int. Wireless Int. Conf. (WICON’2010), Singapore, Mar. 2010, pp. 1–5. [390] A. R.-Alvari¯no and R. W. Heath, “Learning-based adaptive transmission for limited feedback multiuser MIMO-OFDM,” IEEE Trans. Wireless Commun., vol. 13, pp. 3806–3820, Jul. 2014. [391] B. Hari Ram, W. Li, A. Ayyar, J. Lilleberg, and K. Giridhar, “Precoder design for K-user interference channels with finite alphabet signals,” IEEE Commun. Lett., vol. 17, pp. 681–684, Apr. 2013. [392] Y. Fadlallah, A. Khandani, K. Amis, A. A¨ıssa-El-Bey, and R. Pyndiah, “Precoding and decoding in the MIMO interference channel for discrete constellation,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2013), London, UK, Sep. 2013, pp. 1152–1156. [393] H. Huang and V. Lau, “Partial interference alignment for K-user interference channels,” IEEE Trans. Signal Process., vol. 60, pp. 4900–4908, Oct. 2011. [394] A. Ganesan and B. S. Rajan, “On precoding for constant K-user MIMO Gaussian interference channel with finite constellation inputs,” IEEE Trans. Wireless Commun., vol. 13, pp. 4104–4118, Aug. 2014. [395] B. Hari Ram and K. Giridhar, “Precoder design for fractional interference alignment,” in Proc. Asilomar Conference on Signals Systems and Computers (Asilomar’2013), Asilomar, USA, Nov. 2013, pp. 1–5. [396] ——, “Fractional interference alignment: An interference alignment scheme for finite alphabet signals,” Online Available: http://arxiv.org/abs/1310.4993. Feb. 2014.

103

[397] C.-E. Chen and W.-H. Chung, “An iterative minmax per-stream MSE transceiver design for MIMO interference channel,” IEEE Wireless Commun. Lett., vol. 1, pp. 229–232, Jun. 2012. [398] F. Sun and E. de Carvalho, “A leakage-based MMSE beamforming design for a MIMO interference channel,” IEEE Signal Process. Lett., vol. 19, pp. 368–371, Jun. 2012. [399] F.-S. Tseng and J.-F. Gu, “Robust beamforming design in MISO interference channels with RVQ limited feedback,” IEEE Trans. Veh. Technol., vol. 64, pp. 580–592, Feb. 2015. [400] Y. Lu, W. Zhang, and K. B. Letaief, “Blind interference alignment with diversity in K-user interference channels,” IEEE Trans. Commun., vol. 62, pp. 2850–2859, Aug. 2014. [401] A. Sezgin, S. A. Jafar, and H. Jafarkhani, “Optimal use of antennas in interference networks: A tradeoff between rate, diversity and interference alignment,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2009), Honolulu, USA, Nov.-Dec. 2009, pp. 1–6. [402] H. Ning, C. Ling, and K. K. Leung, “Feasibility condition for interference alignment with diversity,” IEEE Trans. Inform. Theory, vol. 57, pp. 2902–2912, May 2011. [403] L. Li, H. Jafarkhani, and S. A. Jafar, “Selection diversity for interference alignment systems,” in Proc. Global Commun. Conf. (Globecom’2013), Atlanta, USA, Dec. 2013, pp. 1773–1778. [404] M. Taki, M. Rezaee, and M. Guillaud, “Adaptive modualtion and coding for interference alignment with imperfect CSI,” IEEE Trans. Wireless Commun., vol. 13, pp. 5264–5273, Sep. 2014. [405] M. R. D. Rodrigues, A. S.-Baruch, and M. Bloch, “On Gaussian wiretap channels with M-PAM inputs,” in Proc. Eur. Wireless Conf., Lucca, Italy, Apr. 2010, pp. 774–781. [406] S. Bashar, Z. Ding, and C. Xiao, “On the secrecy rate of multi-antenna wiretap channel under finite alphabet input,” IEEE Commun. Lett., vol. 15, pp. 527–529, May 2011. [407] ——, “On secrecy rate analysis of MIMO wiretap channels driven by finite-alphabet input,” IEEE Trans. Commun., vol. 60, pp. 3816–3825, Dec. 2012. [408] W. Zeng, Y. R. Zheng, and C. Xiao, “Multiantenna secure cognitive radio networks with finite-alphabet inputs: A global optimization approach for precoder design,” IEEE Trans. Wireless Commun., vol. 15, pp. 3044–3057, Apr. 2016. [409] S. Vishwakarma and A. Chockalingam, “MIMO DF relay beamforming for secrecy with artificial noise, imperfect CSI, and finite alphabet input,” Online Available: https://arxiv.org/abs/1505.00573. May 2015. [410] K. Cao, Y. Wu, Y. Cai, and W. Yang, “Secure transmission with aid of a helper for MIMOME network having finite alphabet inputs,” IEEE Access, vol. 5, pp. 3698–3708, Feb. 2017. [411] Z. Kong, S. Yang, F. Wu, S. Peng, L. Zhong, and L. Hanzo, “Iterative distributed minimum total MSE approach for secure communication in MIMO interference channel,” IEEE Trans. Inform. Forensics Security, vol. 11, pp. 594–608, Mar. 2016. [412] M. Pei, J. Wei, K.-K. Wong, and X. Wang, “Masked beamforming for multiuser MIMO wiretap channels with imperfect CSI,” IEEE Trans. Wireless Commun., vol. 11, pp. 544–549, Feb. 2012. [413] L. Zhang, Y. Cai, B. Champagne, and M. Zhao, “Tomlinson-Harashima precoding design in MIMO wiretap channels based on the MMSE criterion,” in Proc. Int. Conf. Commun. (ICC’2015), London, UK, Jun. 2015, pp. 470–474.

104

[414] S. A. A. Fakoorian, H. Jafarkhani, and A. L. Swindlehurst, “Secure space-time block coding via artificial noise alignment,” in Proc. Conf. Rec. 45th ASILOMAR (ASILOMAR’2011), Pacific Grove, USA, Nov. 2011, pp. 651–655. [415] H. Wen, G. Gong, S.-C. Lv, and P.-H. Ho, “Framework for MIMO cross-layer secure communication based on STBC,” Telecommun. Syst., vol. 52, pp. 2177–2185, Aug. 2013. [416] T. Allen, J. Cheng, and N. A.-Dhahir, “Secure space-time block coding without transmitter CSI,” IEEE Wireless Commun. Lett., vol. 52, pp. 573–576, Dec. 2014. [417] S. L. Perreau, “A new space time coding scheme for the multiple input single output wiretap channel,” in Int. Conf. Signal Process and Commun. Syst. (ICSPCS’2014), Gold Coast, Australia, Dec. 2014, pp. 1–5. [418] X. Li, J. Hwu, and E. P. Ratazzi, “Using antenna array redundancy and channel diversity for secure wireless transmissions,” J. Commun., vol. 2, pp. 24–32, May 2007. [419] J.-C. Belfiore and F. Oggier, “An error probability approach to MIMO wiretap channels,” IEEE Trans. Commun., vol. 61, pp. 3396– 3403, Aug. 2013. [420] T. V. Nguyen, T. Q. S. Quek, Y. H. Kim, and H. Shin, “Secrecy diversity in MISOME wiretap channels,” in Proc. Global Commun. Conf. (Globecom’2012), Aneheim, USA, Dec. 2012, pp. 4840–4845. [421] R. Zhu, Y. Zhao, Y. Li, J. Wang, and H. Hong, “Optimal linear precoding for opportunistic spectrum sharing under arbitrary input distributions assumption,” EURASIP Journal on Advances in Signal Processing, vol. 59, pp. 1–11, Mar. 2013. [422] B. Seo, “Joint design of precoder and receiver in cognitive radio networks using an MSE criterion,” Signal Processing, vol. 91, pp. 2623–2629, Nov. 2011. [423] X. Gong, A. Ishaque, G. Dartmann, and G. Ascheid, “MSE-based stochastic transceiver optimization in downlink cogntive radio networks,” in Proc. Wireless Commun. and Networking Conf. (WCNC’2011), Quintana-Roo, Mexico, Mar. 2011, pp. 1426–1431. [424] Y. Zhang, E. Dall’Anese, and G. B. Giannakis, “Distributed optimal beamformers for cogntive radios robust to channel uncertainties,” IEEE Trans. Signal Process., vol. 60, pp. 6495–6508, Dec. 2012. [425] A. C. Cirik, R. Wang, Y. Rong, and Y. Hua, “MSE-based transceiver designs for full-duplex MIMO cognitive radios,” IEEE Trans. Commun., vol. 63, pp. 2056–2070, Jun. 2015. [426] X. Liang, Z. Bai, L. Wang, and K. Kwak, “Dynamically clustering based cooperative spectrum sensing with STBC scheme,” in International Conference and Ubliquitous and Future Networks (ICUFN’2012), Phuket, Thailand, Jul. 2012, pp. 227–231. [427] B. Wang, Z. Bai, Z. Xu, P. Gong, and K. Kwak, “A robust STBC based dynamical clustering cooperative spectrum sensing scheme in CR systems,” in International Conference and Ubliquitous and Future Networks (ICUFN’2013), Da Nang, Vietnam, Jul. 2013, pp. 553–557. [428] Z. Chen and W. Yen, “Differential space-time block coding based cooperative spectrum sensing over fading environments in cognitive radio sensor networks,” Journal of Information and Computational Science, vol. 9, pp. 4599–4606, Nov. 2012. [429] Y. Abdulkadir, O. Simpson, N. Nwanekezie, and Y. Sun, “A differential space-time coding scheme for cooperative spectrum sensing in cognitive radio networks,” in Proc. Int. Sym. on Personal, Indoor and Moblie Radio Commun. (PIMRC’2015), Hong Kong, China, Aug.-Sep. 2015, pp. 1386–1391.

105

[430] J.-B. Kim and D. Kim, “Outage probability and achievable diversity order of opportunistic relaying in cognitive secondary radio networks,” IEEE Trans. Commun., vol. 9, pp. 2456–2466, Sep. 2012. [431] K. B. Letaief and W. Zhang, “Cooperative communications for cognitive radio networks,” Proceeding of the IEEE, vol. 97, pp. 878–893, May 2009. [432] Q.-T. Vien, B. G. Stewart, H. X. Nguyen, and O. Gemikonakli, “Distributed space-time-frequency block code for cognitive wireless relay networks,” IET Commun., vol. 8, pp. 754–766, May 2014. [433] W. Zeng, Y. R. Zheng, and C. Xiao, “Online precoding for energy harvesting transmitter with finite-alphabet inputs and statistical CSI,” IEEE Trans. Veh. Tech., vol. 65, pp. 5287–5302, Jul. 2016. [434] X. Zhu, W. Zeng, and C. Xiao, “Throughput optimisation for energy harvesting transmitter with partial instantaneous channel state information and finite-alphabet inputs,” IET Commun., vol. 10, pp. 435–442, Apr. 2016. [435] W. Zeng, Y. R. Zheng, and R. Schober, “Online resource allocation for energy harvesting downlink multiuser systems: Precoding with modulation, coding rate, and subchannel selection,” IEEE Trans. Wireless Commun., vol. 10, pp. 5780–5794, Oct. 2015. [436] Y. Tang and M. C. Valenti, “Coded transmit macrodiversity: Block space-time codes over distributed antennas,” in Proc. IEEE Veh. Tech. Conf. (VTC’2001), Atlantic City, USA, May 2001, pp. 1435–1438. [437] Y. Zhu, L. Li, and C. Chakrabarti, “Energy efficient turbo based space-time coder,” in IEEE Workshop on Signal Processing Systems (SIPS’2003), Seoul, Korea, Aug. 2003, pp. 35–40. [438] A. Eksim and M. E. Celebi, “Extended cooperative balanced space-time block coding for increased efficiency in wireless sensor networks,” Lecture Notes in Computuer Science, vol. 5550, pp. 456–467, May 2009. [439] M. T. Kakitani, G. Brante, R. D. Souza, and M. A. Imran, “Energy efficiency of transmit diversity systems under a realistic power consumption model,” IEEE Commun. Lett., vol. 17, pp. 119–122, Jan. 2013. [440] F.-K. Gong, J.-K. Zhang, Y.-J. Zhu, and J.-H. Ge, “Energy-efficient collaborative Alamouti codes,” IEEE Wireless Commun. Lett., vol. 1, pp. 512–515, Oct. 2012. [441] X. Liang, Z. Ding, and C. Xiao, “On linear precoding of non-regenerative MIMO relays for QAM inputs,” in Proc. IEEE Int. Telecommun. Conf. (ICC’2012), Ottawa, Canada, Jun. 2012, pp. 2439–2444. [442] T. M. Syed, N. H. Tran, T. X. Tran, and Z. Han, “Precoder design for a single-relay non-orthogonal AF systems based on mutual information,” in Proc. Int. Wireless Commun. and Moblie Comput. (IWCMC’2013), Sardinia, Italy, Jun. 2013, pp. 567–572. [443] H. Feng, Z. Chen, and H. Liu, “Joint power allocation and mapping strategy design for MIMO two-way relay channels with finitealphabet inputs,” in Proc. IEEE Global. Telecommun. Conf. (GLOBECOM’2014), Austin, USA, Dec. 2014, pp. 3332–3336. [444] C. Xing, S. Ma, and Y.-C. Wu, “Robust joint design of linear relay precoder and destination for dual-hop amplify-and-forward MIMO relay systems,” IEEE Trans. Signal Process., vol. 58, pp. 2273–2283, Apr. 2010. [445] F.-S. Tseng and W.-R. Wu, “Linear MMSE transceiver design in amplify-and-forward MIMO relay systems,” IEEE Trans. Veh. Technol., vol. 59, pp. 754–765, Feb. 2010. [446] K.-J. Lee, H. Sung, E. Park, and I. Lee, “Joint optimization for one and two-way MIMO AF multiple-relay systems,” IEEE Trans. Wireless Commun., vol. 9, pp. 3671–3681, Dec. 2010.

106

[447] Y. Fu, L. Yang, W.-P. Zhu, and C. Liu, “Optimum linear design of two-hop MIMO relay networks with QoS requirements,” IEEE Trans. Signal Process., vol. 59, pp. 2257–2269, May 2011. [448] R. Wang and M. Tao, “Joint source and relay precoding designs for MIMO two-way relaying based on MSE criterion,” IEEE Trans. Signal Process., vol. 60, pp. 1352–1365, Mar. 2012. [449] C. Xing, S. Ma, Z. Fei, Y.-C. Wu, and H. V. Poor, “A general robust linear transceiver design for multi-hop amplify-and-forward MIMO relaying systems,” IEEE Trans. Signal Process., vol. 61, pp. 1196–1209, Mar. 2013. [450] H. Shen, J. Wang, B. C. Levy, and C. Zhao, “Robust optimization for amplify-and-forward MIMO relaying from a worst-case perspective,” IEEE Trans. Signal Process., vol. 61, pp. 5458–5471, Nov. 2013. [451] F.-S. Tseng, M.-Y. Chang, and W.-R. Wu, “Joint Tomlinson-Harashima source and linear relay precoder design in amplify-and-forward MIMO relay systems via MMSE criterion,” IEEE Trans. Veh. Technol., vol. 60, pp. 1687–1698, May 2011. [452] R. Wang, M. Tao, and Z. Xiang, “Nonlinear precoding design for MIMO amplify-and-forward two-way relay systems,” IEEE Trans. Veh. Technol., vol. 61, pp. 3984–3995, Nov. 2012. [453] C. Xing, M. Xia, F. Gao, and Y.-C. Wu, “Robust transceiver with Tomlinson-Harashima precoding for amplify-and-forward MIMO relay systems,” IEEE J. Select. Area Commun., vol. 30, pp. 1370–1382, Sep. 2012. [454] L. Zhang, Y. Cai, R. C. de Lamare, and M. Zhao, “Robust multibranch Tomlinson-Harashima precoding design in amplify-and-forward MIMO relay systems,” IEEE Trans. Commun., vol. 62, pp. 3476–3490, Oct. 2014. [455] M. Ahn, H.-B. Kong, T. Kim, C. Song, and I. Lee, “Precoding techniques for MIMO AF relaying systems with decision feedback receiver,” IEEE Trans. Wireless Commun., vol. 14, pp. 446–455, Jan. 2015. [456] C. Song, K.-J. Lee, and I. Lee, “Performance analysis of MMSE-based amplify and forward spatial multiplexing MIMO relaying systems,” IEEE Trans. Commun., vol. 59, pp. 3452–3462, Dec. 2011. [457] Y. Rong, X. Tang, and Y. Hua, “A unified framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems,” IEEE Trans. Signal Process., vol. 61, pp. 4837–4851, Dec. 2009. [458] Y. Rong, “Optimal linear non-regenerative multi-hop MIMO relays with MMSE-DFE receiver at the destination,” IEEE Trans. Wireless Commun., vol. 9, pp. 2269–2279, Jul. 2010. [459] C. Song, K.-J. Lee, and I. Lee, “MMSE based transceiver designs in closed-loop non-regenerative MIMO relaying systems,” IEEE Trans. Wireless Commun., vol. 9, pp. 2310–2319, Jul. 2010. [460] C. Xing, S. Ma, Y.-C. Wu, and T.-S. Ng, “Transceiver design for dual-hop nonregenerative MIMO-OFDM relay systems under channel uncertainties,” IEEE Trans. Signal Process., vol. 58, pp. 6325–6339, Dec. 2010. [461] Y. Rong, “Joint source and relay optimization for two-way linear non-regenerative MIMO relay communications,” IEEE Trans. Signal Process., vol. 60, pp. 6533–6546, Dec. 2012. [462] S. Jang, J. Yang, and D. K. Kim, “Minimum MSE design for multiuser MIMO relay,” IEEE Commun. Lett., vol. 14, pp. 812–814, Sep. 2010. [463] R. Wang, M. Tao, and Y. Huang, “Linear precoding designs for amplify-and-forward multiuser two-way relay systems,” IEEE Trans. Wireless Commun., vol. 11, pp. 4457–4469, Dec. 2012.

107

[464] K. X. Nguyen, Y. Rong, and S. Nordholm, “MMSE-based transceiver design algorithms for interference MIMO relay systems,” IEEE Trans. Wireless Commun., vol. 14, pp. 6414–6424, Nov. 2015. [465] J. Yang, B. Champagne, Y. Zou, and L. Hanzo, “Joint optimization of transceiver matrices for MIMO-aided multiuser AF relay networks: Improving the QoS in the presence of CSI errors,” IEEE Trans. Veh. Technol., vol. 65, pp. 1434–1451, Mar. 2016. [466] K. X. Nguyen, Y. Rong, and S. Nordholm, “Simplified MMSE precoding design in interference two-way MIMO relay systems,” IEEE Signal Process. Lett., vol. 23, pp. 262–266, Feb. 2016. [467] Y.-W. Liang, A. Ikhlef, W. Gerstacker, and R. Schober, “Two-way filter-and-forward beamforming for frequency-selective channels,” IEEE Trans. Wireless Commun., vol. 10, pp. 4172–4183, Dec. 2011. [468] P. Wu, R. Schober, and V. K. Bhargava, “Transceiver design for SC-FDE based MIMO relay systems,” IEEE Trans. Wireless Commun., vol. 12, pp. 4376–4391, Sep. 2013. [469] M. Safari and M. Uysal, “Cooperative diversity over log-normal fading channels: Performance analysis and optimization,” IEEE Trans. Wireless Commun., vol. 7, pp. 1963–1972, May 2008. [470] T. Cui, F. Gao, T. Ho, and A. Nallanathan, “Distributed space-time coding for two-way wireless relay networks,” IEEE Trans. Signal Process., vol. 57, pp. 658–671, Feb. 2009. [471] Y. Ding and M. Uysal, “Amplify-and-forward cooperative OFDM with multiple-relays: Performance analysis and relay selection methods,” IEEE Trans. Wireless Commun., vol. 8, pp. 4963–4968, Oct. 2009. [472] B. Maham, A. Hjørungnes, and G. Abreu, “Distributed GABBA space-time codes in amplify-and-forward relay networks,” IEEE Trans. Wireless Commun., vol. 8, pp. 2036–2045, Apr. 2009. [473] P. Dharmawansa, M. R. McKay, and R. K. Mallik, “Analytical performance of amplify-and-forward MIMO relaying with orthogonal space-time block codes,” IEEE Trans. Commun., vol. 58, pp. 2147–2158, Jul. 2010. [474] S. Chen, W. Wang, X. Zhang, and Z. Sun, “Performance analysis of OSTBC transmission in amplify-and-forward cooperative relay networks,” IEEE Trans. Veh. Tech., vol. 59, pp. 105–113, Jan. 2010. [475] Q. F. Zhou, Y. Li, F. C. M. Lau, and B. Vucetic, “Decode-and-forward two-way relaying with network coding and opportunistic relay selection,” IEEE Trans. Commun., vol. 58, pp. 3070–3076, Nov. 2010. [476] Q. You, Z. Chen, and Y. Li, “A multihop transmission scheme with detect-and-forward protocol and network coding in two-way relay fading channels,” IEEE Trans. Veh. Tech., vol. 61, pp. 433–438, Jan. 2012. [477] Q. Shi and Y. Karasawa, “Error probability of opportunistic decode-and-forward relaying in Nakagami-m fading channels with arbitrary m,” IEEE Wireless Commun. Lett., vol. 2, pp. 86–89, Feb. 2013. [478] C. Chen, L. Bai, Y. Yang, and J. Choi, “Error performance of physical-layer network coding in multiple-antenna two-way relay systems with outdated CSI,” IEEE Trans. Commun., vol. 63, pp. 3744–3753, Oct. 2015. [479] G. Scutari and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Trans. Wireless Commun., vol. 4, pp. 2387–2399, Sep. 2005. [480] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, pp. 3524–3526, Dec. 2006.

108

[481] Y. Shang and X.-G. Xia, “Shift-full-rank matrices and applications in space-time trellis codes for relay networks with asynchronous cooperative diversity,” IEEE Trans. Inform. Theory, vol. 52, pp. 3153–3167, Jul. 2006. [482] Y. Jing and H. Jafarkhani, “Using orthogonal and quasi-orthogonal designs in wireless relay networks,” IEEE Trans. Inform. Theory, vol. 53, pp. 4106–4118, Nov. 2007. [483] E. Koyuncu, Y. Jing, and H. Jafarkhani, “Distributed beamforming in wireless relay networks with quantized feedback,” IEEE J. Select. Area Commun., vol. 26, pp. 1429–1439, Oct. 2008. [484] X. Guo and X.-G. Xia, “A distributed space-time coding in asynchronous wireless relay networks,” IEEE Trans. Wireless Commun., vol. 7, pp. 1812–1816, May 2008. [485] Y. Jing and H. Jafarkhani, “Single and multiple relay selection schemes and their achievable diversity orders,” IEEE Trans. Wireless Commun., vol. 8, pp. 1414–1423, Mar. 2009. [486] H. Yan, H. H. Nguyen, and J. Su, “Distributed precoding for OFDM in two-way relaying communications,” IEEE Trans. Veh. Tech., vol. 64, pp. 1930–1941, May 2015. [487] D. H. N. Nguyen, H. H. Nguyen, and H. D. Tuan, “Power allocation and error performance of distributed unitrary space-time modulation in wireless relay networks,” IEEE Trans. Veh. Tech., vol. 58, pp. 3333–3346, Sep. 2009. [488] W.-M. Lai, Y.-M. Chen, and Y.-L. Ueng, “Raptor-coded noncoherent cooperative schemes based on distributed unitrary space-time modulation,” IEEE Trans. Commun., vol. 63, pp. 2873–2884, Aug. 2015. [489] M. R. Avendi and H. Jafarkhani, “Differential distributed space-time coding with imperfect synchronization in frequency-selective channels,” IEEE Trans. Wireless Commun., vol. 4, pp. 1811–1822, Apr. 2015. [490] D. Munoz, B. Xie, and H. Minn, “An adaptive MIMO-OFDMA relay system,” IEEE Wireless Commun. Lett., vol. 1, pp. 496–499, Oct. 2012. [491] X. Yuan, C. Xu, L. Ping, and X. Lin, “Precoder design for multiuser MIMO ISI channels based on iterative LMMSE detection,” IEEE J. Sel. Topic Signal Process., vol. 3, pp. 1118–1128, Dec. 2009. [492] Y. Jiang, J. Li, and W. W. Hager, “Joint transceiver design for MIMO communications using geometric mean decomposition,” IEEE Trans. Signal Process., vol. 53, pp. 3791–3803, Oct. 2005. [493] J. Liu and W. A. Krzymieñ, “A novel nonlinear joint transmitter-receiver processing algorithm for the downlink of multiuser MIMO systems,” IEEE Trans. Veh. Technol., vol. 57, pp. 2189–2204, Jul. 2008. [494] S. Lin, W. W. L. Ho, and Y. C. Liang, “Block diagonal geometric mean decomposition (BD-GMD) for MIMO broadcast channels,” IEEE Trans. Wireless Commun., vol. 7, pp. 2778–2789, Jul. 2008. [495] F. Liu, L. Jiang, and C. He, “Advanced joint transceiver design for block-diagonal geometric-mean-decomposition-based multiuser MIMO systems,” IEEE Trans. Veh. Technol., vol. 59, pp. 692–701, Feb. 2010. [496] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR contraints,” IEEE Trans. Veh. Technol., vol. 53, pp. 18–28, Jan. 2004. [497] A. Wiesel, Y. C. Eldar, and S. S. Shitz, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process., vol. 54, pp. 161–176, Jan. 2006.

109

[498] Z.-Q. Luo and S. Zhang, “Dynamic spectrum management: Complexity and duality,” IEEE J. Sel. Topic Signal Process., vol. 2, pp. 57–73, Feb. 2008. [499] E. Chiu, V. K. N. Lau, H. Huang, T. Wu, and S. Liu, “Robust transceiver design for K-pairs quasi-static MIMO interference channel via semi-definite relaxation,” IEEE Trans. Wireless Commun., vol. 9, pp. 3762–3769, Dec. 2010. [500] Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, “Coordinated beamforming for MISO interference channel: Complexity analysis and efficient algorithms,” IEEE Trans. Signal Process., vol. 59, pp. 1142–1157, Mar. 2011. [501] A. Mukherjee and A. L. Swindlehurst, “Robust beamforming for security in MIMO wiretap channels with imperfect CSI,” IEEE Trans. Signal Process., vol. 59, pp. 351–361, Jan. 2011. [502] W. C. Liao, T.-T. Chang, W. K. Ma, and C.-Y. Chi, “QoS-based transmit beamforming in the presence of eavesdroppers: An optimized artificial noise aided approach,” IEEE Trans. Signal Process., vol. 59, pp. 1202–1216, Mar. 2011. [503] K.-Y. Wang, Z. Ding, and C.-Y. Chi, “Robust MISO transmit optimization under outage-based QoS constraints in two-tier heteogeneous networks,” IEEE Trans. Wireless Commun., vol. 12, pp. 1883–1897, Apr. 2013. [504] L. Li, H. Jafarkhani, and S. A. Jafar, “When Alamouti codes meet interference alignment: Transmission schemes for two-user X channel,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2011), Saint-Petersburg, Russian, Jul.-Aug. 2011, pp. 2577–2581. [505] L. Shi, W. Zhang, and X.-G. Xia, “Space-time block code designs for two-user MIMO X channels,” IEEE Trans. Commun., vol. 62, pp. 3806–3815, Sep. 2013. [506] L. Li and H. Jafarkhani, “Maximum-rate transmission with improved diversity gain for interference networks,” IEEE Trans. Inform. Theory, vol. 59, pp. 5313–5330, Sep. 2013. [507] A. Ganesan and B. S. Rajan, “Interference alignment with diversity for the 2 × 2 X-network with four antennas,” IEEE Trans. Inform. Theory, vol. 60, pp. 3576–3592, Jun. 2014. [508] ——, “Interference alignment with diversity for the 2 × 2 X-network with three antennas,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2014), Honolulu, USA, Jun.-Jul. 2014, pp. 1216–1220. [509] A. Ganesan and K. P. Srinath, “Interference aligned space-time transmission with diversity for the 2 × 2 X-network,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT’2015), Hong Kong, China, Jun. 2015, pp. 621–625. [510] Z. Li and X.-G. Xia, “A simple alamouti space-time transmission scheme for asynchronous cooperative systems,” IEEE Signal Process. Lett., vol. 14, pp. 804–807, Nov. 2007. [511] W. Zhang, Y. Li, X.-G. Xia, P. C. Ching, and K. B. Letaif, “Distributed space-frequency coding for cooperative diversity in broadband wireless Ad Hoc networks,” IEEE Trans. Wireless Commun., vol. 7, pp. 995–1003, Mar. 2008. [512] Y. Liu, W. Zhang, S. C. Liew, and P.-C. Ching, “Bounded delay-tolerant space-time codes for distributed antenna systems,” IEEE Trans. Wireless Commun., vol. 13, pp. 4644–4655, Aug. 2014.

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