Cooperative Communication for Multi-User Cognitive Radio Networks

Cooperative Communication for Multi-User Cognitive Radio Networks

MAKSYM GIRNYK

Licentiate Thesis in Telecommunications Stockholm, Sweden 2012

TRITA-EE 2012:024 ISSN 1653-5146 ISBN 978-91-7501-399-2

KTH, School of Electrical Engineering Communication Theory Laboratory SE-100 44 Stockholm SWEDEN

Akademisk avhandling som med tillst˚ and av Kungl Tekniska h¨ ogskolan framl¨agges till offentlig granskning f¨or avl¨ aggande av teknologie licentiatexamen fredagen den 8 juni 2012 klockan 13.15 i h¨orsal Q2, Kungl Tekniska h¨ ogskolan, Osquldas v¨ag 10, Stockholm. c Maksym Girnyk, June 2012  Tryck: Universitetsservice US AB

Abstract In recent years, the main trend in wireless communications has been shifted from voice transmission to data-centric communication. This shift has caused an increase in the data rate requirements for future wireless communication systems. These requirements result in need for large bandwidth. Being a limited and thus expensive resource, wireless spectrum needs to be used efficiently. For higher spectral efficiency, new transmission techniques as well as new dynamic spectrum-allocation policies are needed. Cognitive radio is a promising approach for increasing spectral efficiency of wireless systems. By exploiting advanced signal processing techniques and sophisticated transmission schemes, cognitive radio devices allow to serve new wireless users within the existing crowded spectrum. Typically, a cognitive radio network is installed in parallel to an existing primary network, a legacy owner of the spectrum. The cognitive radio network adapts to its electro-magnetic environment in order to limit or even avoid the disturbance to the primary network. This thesis focuses on the underlay cognitive radio paradigm, which assumes that both the primary network and the ad hoc cognitive radio network operate within the same time and frequency band, as well as at the same geographic location. The cognitive network is able to estimate the interference caused to the primary network by means of channel training and possible feedback. This knowledge is then used to adjust the cognitive network’s transmissions in such a way that the disturbance to the primary network is below some acceptable threshold. In the first part of the thesis, we discuss the multi-hop line cognitive networks, in which the information content before reaching its destination passes through several hops from node to node within the cognitive network. In this way, transmission power at the source terminal may be decreased, thus producing less interference to the primary network. Moreover, the powers at each terminal within the cognitive network may be optimally allocated so that the interference constraint at the primary network is satisfied. This power allocation can be realized in both centralized and decentralized ways, depending on the available information about the channel state. We discuss both of these allocations subject to different interference constraints employed at the primary network. In the second part of the thesis, we discuss the reliability of transmission within the line cognitive ad hoc networks in terms of outage probability and diversity. iii

iv We also illustrate the benefit of network coding for such networks and provide a heuristic algorithm for optimal scheduling. In the final part of the thesis, we study the uplink relay-assisted cellular cognitive radio scenario. Both, the cognitive network and the primary network, contain a set of multi-antenna users that communicate with a corresponding base station. The users create mutual interference and hence limit each other’s performance. Using certain mathematical tools originally developed within the field of statistical physics, we are able derive a closed-form expression for the ergodic mutual information for arbitrary channels inputs, which enables characterization of the achievable rate region of such scenario.

Sammanfattning Under de senaste ˚ aren har den st¨orsta trenden inom tr˚ adl¨os kommunikation flyttats fr˚ an r¨ ost˚ atergivning till data-centrerad kommunikation. Denna f¨ or¨andring har medf¨ort en o¨kning av datahastigheten f¨or framtida tr˚ adl¨osa kommunikationssystem, vilket resulterar i krav p˚ a stor bandbredd. Eftersom tr˚ adl¨ost spektrum ar en begr¨ansad och dyr resurs, beh¨over det anv¨ ¨ andas effektivt. F¨or h¨ogre spektraleffektivitet kr¨ avs nya s¨andningstekniker samt nya principer f¨or dynamisk spektrumtilldelning. Kognitiv radio ¨ar en lovande metod f¨ or att o¨ka spektraleffektiviteten hos tr˚ adl¨osa system. Genom att utnyttja avancerad signalbehandlingsteknik och sofistikerade s¨ andningsmetoder, s˚ a m¨ojligg¨or kognitiv radioenheter att nya tr˚ adl¨osa anv¨ andare kan betj¨anas inom existerande begr¨ ansade frekvenser. Typiskt ¨ar ett kognitivt radion¨ at installerad parallellt med en befintligt prim¨arn¨at, en legal a¨gare av spektrumet. Det kognitiva radion¨ atet anpassar sig till den elektromagnetiska milj¨ on i syfte att begr¨ansa eller undvika st¨orningen till den prim¨ara n¨ atverket. Denna avhandling fokuserar p˚ a underlay-paradigmen f¨or kognitiv radio, vilket f¨oruts¨atter att b˚ ade prim¨ar-n¨atverket och kognitiva radion¨ atet arbetar inom samma tid och frekvensband, och p˚ a samma geografiska plats. Det kognitiva radion¨atet kan skatta vilken st¨orning som orsakas till det prim¨ ara n¨atverket med hj¨ alp av uts¨anda tr¨ aningssekvenser och eventuell feedback. Denna kunskap anv¨ands sedan f¨or att anpassa kognitiva radion¨ atets s¨andning p˚ a ett s˚ adant s¨att att de samverkar under en acceptabel gr¨ ans. Vi unders¨ oker tandem multi-hop kognitiva radion¨at, d¨ar inar flera steg, fr˚ an nod till nod, inom kognitiva n¨ atverket. P˚ a formationsinneh˚ allet g˚ detta s¨att kan s¨andningseffekten vid k¨ allterminalen minskas, vilket skapar mindre interferens f¨ or det prim¨ara n¨ atverket. Dessutom kan s¨ andeffekten vid varje terminal inom kognitiva radion¨ atet anpassas optimalt s˚ a att st¨orningsniv˚ aerna inom det prim¨ara n¨ atverket h˚ alls p˚ a en acceptabel niv˚ a. Denna effektf¨ordelning kan ˚ astadkommas b˚ ade centraliserat och decentraliserat, beroende p˚ a tillg¨anglig information om kanaltillst˚ andet. I den f¨orsta delen av avhandlingen diskuterar vi b˚ ada dessa principer, f¨ or ett antal olika formuleringar av st¨orningsvillkoren f¨or det prim¨ ara n¨ atet. I den andra delen av avhandlingen diskuterar vi tillf¨ orlitligheten f¨or ¨overf¨ oring inom kognitiv radion¨ at i termer avbrottssannolikhet och diversitet. Vi illustrerar aven f¨ ¨ ordelen av s˚ adana n¨ at och tillhandah˚ aller en heuristisk algoritm f¨ or optimal v

vi schemal¨ aggning. I den sista delen av avhandlingen studerar vi ett uppl¨anksscenario d¨ar rel¨ aer anv¨ ands f¨or att st¨odja cellul¨ar kognitiv radio. B˚ ade det kognitiva n¨atverket och det prim¨ ara n¨ atverket har ett antal anv¨ andare, utrustade med multipla antenner, som kommunicerar med en motsvarande basstation. B˚ ada p˚ averkar varandra o¨msesidigt och kan d¨ armed begr¨ ansa varandras prestanda. Med hj¨ alp av matematiska verktyg fr˚ an statistisk fysik, h¨ arleder vi en slutet uttryck f¨or ergodiska o¨msesidiga informationen f¨or s˚ adana kanaler, vilket g¨ or det m¨ojligt att karakt¨arisera omr˚ adet av uppn˚ aeliga datatakter.

Acknowledgments Foremost, I would like to express my gratitude to my supervisor Prof. Lars K. Rasmussen for his support and guidance throughout my research. His encouragement and positive attitude helped me to overcome difficulties during the research process. I am thankful to Asst. Prof. Ming Xiao for his help and collaboration. Ming always had many inspiring ideas and valuable comments, whenever I came to him. I am also grateful to Dr. Mikko Vehkaper¨a. I enjoyed very much working with him on some topics covered by this thesis. In particular I am indebted to Mikko for his guidance through the minefield of statistical physics. I also would like to thank Dr. Vishwambhar Rathi for many valuable discussions on the related research topics. I gratefully acknowledge the joint work with my colleagues Fr´ed´eric Gabry, Nan Li and Nicolas Schrammar within the QUASAR-ACROPOLIS project cooperation. In addition, my appreciation goes to Prof. Mikael Skoglund for giving me the opportunity to join the Communication Theory lab and to experience its great scientific atmosphere. I am grateful to Mikko, Dr. Chao Wang, Hamed Farhadi, Iqbal Hussain, Dennis Sundman and Ali Zaidi for proofreading some parts of this thesis. I also thank Prof. Mats Bengtsson for the quality review and valuable comments. I would like to express my thanks to Dr. Fredrik Rusek for taking time to act as opponent for this thesis. It was a privilege to share the office with Amirpasha Shirazinia all these years. I am also thankful to all my current and former colleagues at “plan 4” for a creative and friendly working environment. Special thanks goes to Annika Augustsson, Tetiana Viekhova, Ir´ene Kindblom and Raine Tiivel for taking care of the administrative issues. L˚ angholmen Football Club is acknowledged for giving me the possibility to maintain my physical shape during these years. I am thankful to my family and friends for supporting and encouraging me. Finally, more than anyone else, I would like to thank Karina for her love and care, for all the happiness she brings to my life. This thesis is dedicated to her. Maksym Girnyk Stockholm, May 2012

vii

Contents Abstract

iii

Sammanfattning

v

Acknowledgments

vii

Contents

ix

List of Acronyms

xiii

Notation

xv

List of Figures

xxi

1 Introduction 1 1.1 Evolution of Wireless Communication Systems . . . . . . . . . . . . 1 1.2 Cooperative Communication . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Network Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Cognitive Radio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Contribution and Related Work . . . . . . . . . . . . . . . . . . . . . 7 1.5.1 Optimal Resource Allocation in Multi-Hop CRNs . . . . . . . 7 1.5.2 Network-Coded Cooperation in Multi-User Multi-Hop CRNs 8 1.5.3 Asymptotic Sum-Rate of Relay-Assisted Cellular MIMO CRNs 9 1.6 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Contributions Outside the Scope of the Thesis . . . . . . . . . . . . 12 2 Fundamentals 2.1 Elements of a Digital Communication System 2.2 Entropy and Mutual Information . . . . . . . 2.3 The AWGN Channel Model . . . . . . . . . . 2.4 The Wireless Channel . . . . . . . . . . . . . 2.4.1 Slow Fading Scenario . . . . . . . . . . 2.4.2 Fast Fading Scenario . . . . . . . . . . ix

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The Relay Channel . . . . . . . . . . . . . . Basics of Network Coding . . . . . . . . . . Cognitive Radio Scenario . . . . . . . . . . Multi-Antenna Transmission . . . . . . . . . Statistical Physics and The Replica Method 2.9.1 Replica Trick . . . . . . . . . . . . . 2.9.2 Replica Symmetry . . . . . . . . . . 2.10 Convexity and Convex Optimization . . . . 2.10.1 Lagrange Duality . . . . . . . . . . . 2.10.2 The KKT Conditions . . . . . . . .

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3 Optimal Power Allocation 3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Aggregate-SI-Constrained Throughput Maximization . . . . . . . . 3.2.1 Power Allocation in the Absence of Intra-CRN Interference 3.2.2 Power Allocation in the Presence of Intra-CRN Interference 3.2.3 High- and Low-SNR Approximate Power Solutions . . . . . 3.2.4 Distributed Power Allocation . . . . . . . . . . . . . . . . . 3.2.5 Limited-Feedback Solution . . . . . . . . . . . . . . . . . . 3.3 Individual-SI-Constrained Throughput Maximization . . . . . . . . 3.3.1 Centralized Solution . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Distributed and Limited-Feedback Solutions . . . . . . . . . 3.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Aggregate SI Constraint . . . . . . . . . . . . . . . . . . . . 3.4.2 Individual SI Constraints . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Diversity Network-Coded Cooperation 4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . 4.2 Evaluation of Transmission Strategies . . . . . . . . . 4.2.1 Conventional Regenerative TDD Transmission 4.2.2 Binary Network-Coded Transmission . . . . . . 4.2.3 Diversity Network-Coded Transmission . . . . 4.3 Three-User Network Example . . . . . . . . . . . . . . 4.3.1 Conventional TDD Transmission . . . . . . . . 4.3.2 BNC-Based Transmission . . . . . . . . . . . . 4.3.3 DNC-Based Transmission . . . . . . . . . . . . 4.3.4 Simulation Results . . . . . . . . . . . . . . . . 4.4 Optimal Scheduling . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Asymptotic Sum-Rate Analysis 83 5.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Asymptotic Sum-Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 86

xi 5.3 5.4

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Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Gaussian Channel Inputs . . . . . . . . . . . . 5.4.2 Quadrature Phase-Shift Keying Channel Inputs Numerical Illustration . . . . . . . . . . . . . . . . . . 5.5.1 Amplify-and-Forward Relay Channel . . . . . . 5.5.2 Communication in the Presence of Interference 5.5.3 Relay-Assisted Interference MIMO channel . . Summary . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Conclusions and Future Work 111 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 A Omitted Steps from the Proof of Theorem 5.1 A.1 Derivation of Covariance Matrices Q0 , Q1j and Q2 . . A.2 Evaluation of G(u) (Q0 , Q2 ) . . . . . . . . . . . . . . . (u) A.3 Evaluation of G1 (Q1j ) . . . . . . . . . . . . . . . . . A.4 Derivation of the Third Term of (5.41) . . . . . . . . . A.5 Derivation of the Eighth Term of (5.41) . . . . . . . . A.6 Saddle-Point Conditions . . . . . . . . . . . . . . . . . A.7 Evaluation of the Free Energy . . . . . . . . . . . . . . A.8 Evaluation of Second Term of the Mutual Information Bibliography

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115 115 118 119 119 121 123 131 134 141

List of Acronyms

1G 2G 3G 4G AF AMPS AWGN BNC BPSK bpcu CF CR CRN CSI CDMA DF DMT DNC DSA FCC FDMA GEV GSM i.i.d. KKT LF LICQ LSL LTE MGF MIMO MMSE

First generation cellular mobile communication system. Second generation cellular mobile communication system. Third generation cellular mobile communication system. Fourth generation cellular mobile communication system. Amplify-and-forward. Advanced mobile phone system. Additive white Gaussian noise. Binary network coding. Binary phase-shift keying. bits per channel use. Compress-and-forward. Cognitive radio. Cognitive radio network. Channel side information. Code division multiple access. Decode-and-forward. Diversity-multiplexing trade-off. Diversity network coding. Dynamic spectrum allocation. Federal communication commission of the U.S. Frequency division multiple access. Global encoding vector. Global system for mobile communication. Independent and identically distributed. Karush-Kuhn-Tucker. Limited feedback. Linearly independence constraint qualification. Large system limit. Long-term evolution. Moment-generating function. Multiple-input and multiple-output. Minimum mean square error. xiii

xiv MSE NMT OFDM Ofcom pdf pmf PTS PU QAM QoS QPSK RT r.v. SF SI SU SNR TDD TDMA UMTS UWB WLAN XOR ZMCSCG

Mean square error. Nordic mobile telephony. Orthogonal frequency-division multiplexing. Office of communications in the U.K. Probability density function. Probability mass function. Swedish post and telecom authority. Primary user. Quadrature amplitude modulation. Quality of service. Quadrature phase-shift keying. Relay terminal. Random variable. Store-and-forward. Secondary interference. Secondary user. Signal-to-noise ratio. Time division duplex. Time division multiple access. Universal mobile communication system. Ultra-wideband. Wireless local area network. Exclusive or. Zero mean circular symmetric complex Gaussian.

Notation CN ×M RN ×M ai = [a]i aij = [A]i AT AH a∗ a A−1 Re{a} Im{a} j |a| a diag(a1 , . . . , aN ) tr{A} IN 0N ×m 1N E{X} |A| A\B A∪B A∩B CN (m, C) O(·) arg{·} s.t. ∀ ∇ δ (x) GF(q)

The set of complex-valued N × M matrices. The set of real-valued N × M matrices. ith entry of a vector a. ith entry of a matrix A. The transpose of a matrix A. Hermitian transpose of a matrix A. The complex conjugate of a scalar a. Optimal solution of an optimization problem. Inverse of a square matrix A. Real part of a complex scalar a. Imaginary part of a complex scalar a. √ The imaginary unit, −1. absolute value of a scalar a. L2 norm of a vector a. The diagonal matrix with a1 , . . . , aN on the main diagonal. The trace of a square matrix A. The N × N identity matrix. The N × M matrix of zeros. The N -vector of ones. Expectation of a random variable X. Cardinality of a set A. Set-theoretic difference of A and B. Union of A and B. Intersection of A and B. The circular symmetric complex Gaussian distribution with mean vector m and covariance matrix C. Order of a function / computational complexity. Argument / phase of a complex variable. Means “subject to”. The universal quantifier. The gradient operator. The Dirac function. The Galois field of size q. xv

xvi

Notation ⊕

Exclusive or operation, i.e., addition in binary field GF(2).

√α A

The pathloss exponent. Diagonal element of the channel matrix of a fixed MIMO √ channel z = AIx + w. The inverse temperature. Network codeword for user i. Capacity of a point-to-point channel. Outage capacity of a point-to-point channel. ith constraint of an optimization problem. The domain of a function. Matrix of gradients for all constraints ci of an optimization problem. Diversity order for user i. Diversity order for user i using conventional TDD transmission. Diversity order for user i using conventional BNC transmission. Diversity order for user i using conventional DNC transmission. Distance between nodes i and j in a network. The error event for a message m. Energy. The (normalized) free energy. Forwarding matrix of jth relay terminal. Complementary cumulative density function of |h|2 . Interference threshold. Outage threshold. Global encoding vector of user i. The dual function. Channel power gain of the link between nodes i and j. Channel power gain of the link between secondary node i and primary node j. Instantaneous channel side information at the receiver. MIMO channel matrix. Effective MIMO channel matrix consisting of scaled channel matrices of MIMO links from all primary users to the primary base station through the direct path and through the relay terminals. Effective MIMO channel matrix consisting of scaled channel matrices of MIMO links from all secondary users to the primary base station through the direct path and through the relay terminals.

β Ci C Cε ci D Dc Di DiC DiB DiD dij Em E(·) F Fj Fc (·) γ G Gi g(·) gij g˜ij H H Hp

Hs

Notation ˆ H Hp01k Hp1jk H21j Hs01k Hs1jk H H(X) h hij h(X) h(X, Y ) h(Y |X)

Ii I(·) I(Y ; X) I(Y ; X|Z) λ Lij L(·) M· μ(·) n n01 n1j n21 n ω

xvii Effective MIMO channel matrix consisting of scaled channel matrices of MIMO links from all relay to the primary base station. Channel matrix of the MIMO link between primary user k and the primary base station. Channel matrix of the MIMO link between primary user k and relay terminal j. Channel matrix of the MIMO link between relay terminal j and the primary base station. Channel matrix of the MIMO link between secondary user k and the primary base station. Channel matrix of the MIMO link between secondary user k and relay terminal j. Hamiltonian. The entropy of a discrete random variable X. Channel gain. Channel gain of the link between nodes i and j. The differential entropy of a continuous random variable X. The joint differential entropy of continuous random variables X and Y . The conditional differential entropy of a continuous random variable Y given the knowledge about the random variable X. Information packet of user i. The rate function. The mutual information between random variables Y and X. The mutual information between random variables Y and X, given the knowledge about the random variable Z. Dual variable. Link between nodes i and j. The Lagrangian. Moment generating function. Probability measure. AWGN vector of a MIMO channel. AWGN vector added at the primary base station during direct transmission from primary and secondary users. AWGN vector added at relay terminal j during transmission from primary and secondary users. AWGN vector added at the primary base station during transmission from relay terminals. AWGN in a channel. Dual variable.

xviii

Notation Oij P max Pi P p (n) pmax pm po po (Ii ) p(x) p˜ Q ˜ Q Q ˜ Q q q˜ ρ ρp01k ρp1jk ρ21j ρs01k ρs1jk R σ2 Sx Tcoh T t ti tC i tB i

Outage event for link Lij . Power constraint. Transmit power at user i’s terminal. Vector of powers of all users in a network. Diagonal element of a symmetric matrix Q. Maximal probability of error for a transmission. Conditional probability of error for a transmitted message m. Link outage probability. Outage probability for a packet Ii . of a disProbability mass/density function1 crete/continuous random variable X. ˜ Diagonal element of a symmetric matrix Q. Covariance matrix of a vector V. Auxiliary matrix of the same size as Q. Set of all Q’s. ˜ Set of all Q’s. Off-diagonal element of a symmetric matrix Q. ˜ Off-diagonal element of a symmetric matrix Q. Signal-to-noise ratio. Signal-to-noise ratio of the direct MIMO link between primary user k and the primary base station. Signal-to-noise ratio of the MIMO link between primary user k and relay terminal j. Signal-to-noise ratio of the MIMO link between relay terminal j and the primary base station. Signal-to-noise ratio of the direct MIMO link between secondary user k and the primary base station. Signal-to-noise ratio of the MIMO link between secondary user k and relay terminal j. Rate. Noise variance. Support set for x. Coherence time interval. Total number of time instances/slots. Time instant. Number of time-slots allocated to user i. Number of time-slots allocated to user i using conventional TDD transmission. Number of time-slots allocated to user i using conventional BNC transmission.

1 For notational convenience, we do not distinguish between the random variables and their realizations.

Notation tD i w X x x xpk xsk

x Y y y01 y1j y21 Y y yj Z(·) z

xix Number of time-slots allocated to user i using conventional DNC transmission. √ AWGN vector of a fixed MIMO channel z = AIx + w. Set of all possible discrete channel inputs. Channel input. Input vector to a MIMO channel. Signal vector transmitted by primary user k. Signal vector transmitted by secondary user k. The MMSE estimate of x. Set of all possible discrete channel outputs. Output vector of a MIMO channel. Noisy signal vector received at the primary base station directly from all primary and secondary users. Noisy signal vector received at the relay terminal from all primary and secondary users. Noisy signal vector received at the primary base station all relay terminals. Support set for the channel outputs. Channel output. Signal received at user kth terminal. The partition function. √ Output vector of a fixed MIMO channel z = AIx + w.

List of Figures 2.1 2.2 2.3 2.4 2.5 2.6

Typical digital communication system. . . . . . . . . . . . The relay channel. . . . . . . . . . . . . . . . . . . . . . . Store-and-forward relaying on the butterfly network. . . . Network Coding on the butterfly network. . . . . . . . . . Network Coding in the uplink cooperative relay scenario. Interference channel. . . . . . . . . . . . . . . . . . . . . .

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Multi-hop cognitive radio network. . . . . . . . . . . . . . . . . . . . . . Multi-hop transmission with overhearing upstream and downstream. . . Example of a two-hop CRN. . . . . . . . . . . . . . . . . . . . . . . . . . Optimal power allocation for a two-hop CRN. . . . . . . . . . . . . . . . Thee-hop transmission with overhearing upstream and downstream. . . End-to-end throughput as function of the aggregate SI threshold of the primary network for different power allocation strategies. . . . . . . . . End-to-end throughput as a function of the SI threshold of PU 1 for different power allocation strategies. . . . . . . . . . . . . . . . . . . . .

36 37 42 43 52

Multi-user multi-hop CRN. . . . . . . . . . . . . . . . . . . . . . . . . . Schedule for the conventional TDD regenerative transmission. . . . . . . Schedule for the BNC transmission. . . . . . . . . . . . . . . . . . . . . Schedule for the DNC transmission. . . . . . . . . . . . . . . . . . . . . Three-user CRN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conventional TDD transmission. . . . . . . . . . . . . . . . . . . . . . . BNC transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DNC transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outage probability of SU 3 when using conventional TDD DF, BNC and DNC strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Outage probability of SU 2 when using conventional TDD DF, BNC and DNC strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Outage probability of SU 1 when using conventional TDD DF, BNC and DNC strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 64 65 67 69 70 71 73

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4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

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. . . . . .

. . . . . .

. . . . . .

56 58

75 76 77

xxii 4.12 Outage probability for different users the DNC to the equal time-slot assignment. . . . . . . . 4.13 Outage probability for different users the DNC to the optimized schedule. . . . . . . . . . . . .

Notation transmission according . . . . . . . . . . . . . . transmission according . . . . . . . . . . . . . .

Uplink of a cellular non-regenerative relay-assisted cognitive MIMO network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 AF relay MIMO channel. . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Average mutual information per dimension versus SNR of the first-hop link ρ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Communication in the presence of interference. . . . . . . . . . . . . . . 5.5 Average mutual information per dimension versus SNRs ρp1 and ρs1 . . . . 5.6 Average mutual information per transmit antenna versus SNRs ρp1 and ρs1 for different combinations of PU’s and SU’s signalings. . . . . . . . . 5.7 Relay-assisted interference MIMO channel. . . . . . . . . . . . . . . . . 5.8 Average mutual information per dimension versus SNRs ρp1 and ρs1 . . . . 5.9 Mobile-relay-assisted interference MIMO channel with shadowing. . . . 5.10 Achievable rate region for the mobile-relay-assisted multi-access interference MIMO channel. . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 80

5.1

84 104 105 106 106 107 108 108 109 110

Chapter 1

Introduction In this chapter we start with a brief review the history of communication systems, and then in Sections 1.2, 1.3 and 1.4, we give a short introduction to cooperative communications, as well as cognitive radio. Furthermore, we go through the previous work related to the topic in Section 1.5, and provide an outline of the thesis in Section 1.6. The final section of this chapter informs the reader about the work that is not covered by the present thesis.

1.1

Evolution of Wireless Communication Systems

Wireless communication is a century old field of industry, which remains one of the most successful and fast growing fields at present. Being one of the fundamental needs of a human, social interaction through communication stimulates continuous development for connecting people all over the world. Recent progress in technology enables production of tiny devices able to realize very complex signal processing tasks consuming limited power that allows implementation of more and more sophisticated communication devices. In recent years, we have witnessed a great success of cellular mobile telephony, which has become an important part of people’s everyday life in all developed countries. As a consequence, the demand of new audio, video and data services has significantly increased over the past decade and continues growing from year to year. In this perspective, new techniques and tools for fast, efficient and reliable communication over the wireless channel are needed. The first digital communication system (telegraph network), developed by Samuel Morse, was demonstrated to the public in 1838. Later on, after the discovery of electro-magnetic waves by James Maxwell in 1864, and the series of experiments by Heinrich Hertz in 1887, during which the radio waves were physically transmitted in the free space, Alexandr Popov demonstrated to the public the world’s first shortrange transmission of continuous radio signal in 1895. Already in 1901, Guglielmo Marconi established the first Trans-Atlantic reception. Since then, radio transmis1

2

Chapter 1. Introduction

sion methods and services advanced rapidly enabling more reliable transmission, using smaller and cheaper devices, and thereby leading to appearance of new applications, such as radio and TV broadcast, mobile communication, navigation, remote control, etc. The first generation (1G) cellular mobile systems appeared in 1980s when the advances in technology allowed to implement the ideas of mobile communication. Although these systems were not the first mobile radio networks, they were the first to consider the cellular structure of coverage. This approach allowed to significantly increase the capacity of the network and provided support for mobility. One of the first 1G systems implemented in Europe was Scandinavian NMT (Nordisk Mobiltelefoni). The system was designed for wireless speech service; it used analog signaling and was based on frequency division multiple access (FDMA). Two years later, Advanced Mobile Phone Service (AMPS) was launched in the U.S. [Mol11]. The development of integrated circuits together with new digital signal processing algorithms led to the replacement of the analog 1G systems by new digital second generation (2G) cellular mobile systems. In 1992, the first such system – Global System for Mobile Communication (GSM) – was implemented in Europe. Allowing better speech quality and increased capacity, GSM made an important contribution to the development of the wireless communication systems. Namely, it provided users with new services, such as roaming, handover and SMS-messaging [Red98]. The latter shifted the emphasis of the communication system design from voice calls to data transmission. This shift established a new era in wireless communications, the era of data-centric services, such as web-browsing, file downloading, video streaming, gaming, on-line banking, etc. All these applications increased the data rate demands on the networks. Even though being already a successful mobile communication system, GSM was not able to provide high enough data rates to satisfy these demands. Thus, current – third generation (3G) – systems appeared to be designed indeed to meet these high rate demands. European Universal Mobile Telecommunication System (UMTS) is based on a wideband code division multiple access (CDMA) standard enabling high data rates up to 2 Mbps [HT04]. Further evolution follows by the forth generation (4G) communications systems, which is currently being under investigation by the 3GPP, a group of telecommunication associations responsible for the finalization of the standard. Long-Term Evolution (LTE) Advanced standard, which is finalized as one of the candidates, includes the target of supporting 1 Gbps data rates, advanced MIMO options, coordinated multiple point transmission and reception, relaying and autonomous component carrier selection for uncoordinated femto-cell deployment [MK09]. At the same time, another class of wireless communication systems – wireless local area networks (WLANs) – developed into a large field of short-range communication systems. WLANs are destined for communication at high data rates within a small region between objects that are stationary or moving at pedestrian speeds. The systems operate within unlicensed bands, and in order to mitigate interference from one WLAN to others, the constraint on the transmit power is

1.2. Cooperative Communication

3

imposed to all user terminals within the given area and the given band. Often used protocols for WLANs are IEEE 802.11 family in the U.S. and HIPERLAN/2 in Europe [DAB+ 02]. There are also small-scale WLAN standards, such as Bluetooth [Haa00], and ultra-wideband (UWB) [IOH04] standards used for substitution of cabling. WLANs typically use either centralized or ad hoc topology [CCL03]. While for centralized WLANs there is a central node (access point) which manages the connections within its range, in ad hoc WLANs the central node is absent and the network organizes itself into a set of links between pairs of nodes using relaying of information and routing algorithms. Further development of the communication systems has to deal with growing number of applications and services that demand higher quality of service (QoS) for users, in terms of data rates, reliability, security, fairness, etc. In addition, the fact that communication becomes more data-centric puts new constraints on latency. New services may exploit all range of latency requirements, from bursty applications to on-line streaming.

1.2

Cooperative Communication

One of the main challenges in the design of a communication system is the timevarying nature of the wireless channel due to multi-path radio wave propagation, distance attenuation and possible shadowing by obstacles. This phenomenon, also known as fading, may cause poor performance of a system even when high transmit powers are used. The reason of this poor performance is that the aforementioned fading effects significantly increase the probability for the channel to be in deep fade. When the channel is in deep fade, the signal experiences severe degradation while the noise power remains the same. Hence, the communication system will most likely suffer from errors. A natural approach to combat fading is to incorporate diversity, i.e., to ensure that multiple signal copies pass through different paths experiencing statistically independent changes. The probability that all the paths turn to deep fade decreases dramatically, and reliable communication is possible as long as at least one path is strong enough to carry the signal. The diversity may be obtained via different diversity techniques: • Temporal diversity may be obtained by repetition of a signal over time. • Frequency diversity is achieved when the signal is sent over several sub-carriers of the orthogonal frequency division multiplexing (OFDM) system. • Spatial diversity is acquired by using multiple antennas at the transmitter and/or the receiver. The easiest way of obtaining diversity is the so-called repetition coding, i.e., the very same signal is transmitted over several paths separated in time, frequency or space. Although repetition coding achieves full diversity available in the channel, it

4

Chapter 1. Introduction

is not optimal in the sense of the degrees of freedom available in the channel [TV05]. In other words, repetition coding does not use all the potential of the channel. For instance, more sophisticated coding schemes, additionally to diversity gain, may provide coding gain, which is also useful for increasing reliability of transmission. As an example, interleaving of coded symbols in time, used in GSM [Red98], may illustrate an efficient diversity acquisition technique. Another way of obtaining diversity gains is to use of multiple-input multipleoutput (MIMO) transmission [FG98], [Tel99]. In this scenario, both the transmitter and the receiver are equipped with multiple properly spaced transmit/receive antennas. In this way, the transmitter may send the same signal from each of its antennas, so that the receiver obtains several copies of the signal. On the other hand, when transmitting different signals from different antennas the transmitter may produce several data streams, which may be received at the receiver simultaneously, thereby increasing the data rate. The latter advantage of MIMO is often referred to as multiplexing gain. These two gains of MIMO are shown to form the diversity-multiplexing trade-off (DMT) which is the fundamental trade-off for any communication system [ZT03]. The advantages described above are coupled with the availability of multiple antennas at the terminals. In practice, antennas have to be separated in space, which may be a problem for the mobile phones. Fortunately, single-antenna terminals may obtain some of the benefits of MIMO systems by implementing cooperative communication techniques. In this way, a virtual MIMO system may be created from a set of mobiles sharing their antennas for the transmission. The terminals may assist each other by transmitting the partner’s signal. Hence, the signals arriving at the receiver traverse several independent paths allowing for diversity gains. Another important benefit of cooperative communication is coverage extension. If the direct communication between the transmitter and the receiver is not possible (e.g., due to the far geographical location or some obstacles on the way), the partner terminal may still deliver the signal to the destination through multi-hop link. The benefits of the cooperative communication come at cost of sharing the terminal’s transmission power and computation resources with others. However, this loss of own power may be counteracted when helping terminal sends its own signal, which can be then relayed by others. Thus, although the benefits depend of users’ willingness to cooperate, the cooperation may potentially lead to significant resource savings for the whole network. The fundamental block of the cooperative communication, the so-called relay channel, was firstly introduced by van der Meulen in 1968 [vdM71]. Further, Cover and El Gamal analyzed the relay channel from the information theoretic point of view and developed several fundamental relaying strategies [EC79]. The main idea of the relay channel is that a relay terminal can overhear the signal from the transmitter and retransmit it towards the receiver. In this way, the relay channel then can be regarded as superposition of a broadcast channel [Cov72], [KM77] and a multiple access channel [Ahl71], [CES80] well investigated before. Cover and El Gamal provoked high interest to the cooperative communication, which remains a

1.3. Network Coding

5

hot topic within the area of communication theory. Interested reader is referred to [KMY06] for an excellent overview of the topic. There are many relaying strategies proposed in the literature. We may roughly divide those into two classes: non-regenerative and regenerative. A typical representative of non-regenerative strategies is the so-called amplify-and-forward (AF) strategy. The basic underlying principle of the strategy is the amplification of the noisy received signal at the relay terminal and its retransmission towards the destination. This is quite an old technique used by radio engineers to increase the coverage of the microwave transmission almost sixty years ago, e.g., [RSF51], [PT58]. Within the context of cooperative communication, the AF strategy was firstly introduced and investigated by Lanemann et al. in [LTW04]. The most frequently used regenerative relaying strategy is decode-and-forward (DF), originally suggested in [EC79]. The key idea of the DF strategy is that the received signal is first decoded at the relay, then re-encoded and retransmitted to the destination. Another representative of the class of regenerative strategies is compress-and-forward (CF), also initially suggested in [EC79]. The idea here is that the relay quantizes the received signal and encodes the samples into a new message which is forwarded to the destination serving as additional redundancy for the signal received directly from the source.

1.3

Network Coding

The regenerative relaying strategies discussed above are implemented on the packet level, so that an information message (or packet) received at the relay terminal is processed, stored and retransmitted towards the destination. For instance, when two packets are received by a relay node at the same time, and assuming that the capacity of the relay-destination link is one packet per channel use, only one of these two packets may be transmitted at a time. The other packet has to be stored and transmitted afterwards. Therefore, this class of strategies is referred to as storeand-forward (SF). This approach is shown to be sub-optimal in terms of network throughput for networks with one information source and many information sinks, i.e., so-called multicast scenario. Instead, a new ground-breaking technique, named network coding is proposed by Ahlswede et al. in [ACLY00]. With network coding, each intermediate node in the network is allowed to mix the incoming packets in a certain way, which may provide the highest possible throughput for multicast networks. A large amount of literature has appeared after the discovery of network coding. In [LYC03], it was shown that linear network coding suffices for achieving the min-cut capacity for multicast sessions. In other words, all the intermediate nodes combine the incoming packets via linear combinations with coefficients picked from a finite field and assigned to each node. In this way, each sink knows which coefficients are used by which node. In [HMK+ 06], a practical suggestion was proposed to choose the coefficients randomly and then send the information

6

Chapter 1. Introduction

about the coefficients en route with a packet as an overhead. Knowing all the random coefficients, a sink can easily recover the packets. This approach is referred to as random network coding. In turn, K¨ otter and M´edard, in [KM03], proposed to track the network coding process via transfer matrices, thereby establishing a convenient algebraic framework for design and analysis of network codes. For instance, the problem of finding the feasible coding scheme is turned into a search for a set of non-singular transfer matrices corresponding to all source-destination pairs. For more details on the topic the reader is referred to the textbooks on network coding [Yeu08], [FS06], [FS07] and [HL08].

1.4

Cognitive Radio

The development of new wireless applications and services requires increased data rates which is coupled with higher demands for wireless spectrum. The electromagnetic spectrum is a limited natural resource, access to which is regulated by governmental agencies. For example, in the U.S., the usage of spectrum is regulated by the Federal Communications Commision (FCC), in the U.K., it is done by the Office of Communications (Ofcom), or in Sweden, by Post- och Telestyrelsen (PTS). Spectrum is assigned to network operators within some geographic regions on a long-term basis. This policy is proved inefficient by recent measurements [RHLM06, VMB+ 10] since large parts of spectrum remain unutilized or partially utilized during certain periods of time, whereas the other frequency bands may be heavily exploited. Furthermore, wireless spectrum is a very expensive resource, and therefore it must be utilized efficiently, so that the resource is allocated only as long as it is needed. Hence, there is a clear need for new techniques for increasing the effectiveness of the present spectrum policies. A promising approach of the dynamic spectrum allocation (DSA) allows to overcome this problem by exploiting advanced digital communication and signal processing techniques at the communication terminals. The DSA policy is based on the technology of cognitive radio (CR), firstly introduced by Mitola and Maguire [MM99] and expanded further in Mitola’s Ph.D. thesis presented at KTH - Royal Institute of Technology, Sweden, in May 2000 [Mit00]. The CR is defined as an intelligent wireless communication system that is aware of its environment and is able to change its transmitter parameters based on interactions with its environment with objectives of highly reliable communication and efficient spectrum utilization [FCC02]. For some good survey material on the topic the reader is referred to [Hay05], [ALVM08] and [ALVM06]. CR technology allows for coexistence of an adaptive cognitive radio network (CRN), also called a secondary network, together with a primary network, the legacy owner of the spectrum. The CRN is capable to sense and analyze its surrounding environment as well as reconfigure its operation in accordance with this radio environment. In this way, based on the available channel side information (CSI), the CRN may dynamically access the spectrum of the primary network

1.5. Contribution and Related Work

7

whenever it does not harm the latter. Examples of potential secondary networks may include the aforementioned WLANs working under IEEE 802.11, Bluetooth or WiMAX standards and exploiting unlicensed bands. CR systems can be classified into three groups according to their operation principle [GJMS09]: • Underlay: CRs are allowed to operate in parallel to the primary network under the constraint that the secondary interference (SI) from the CRN towards the primary users (PUs) does not degrade performance of the primary network. • Overlay: CRs are allowed to operate simultaneously with the primary network provided that they enhance its performance by having access to PUs’ codebooks and applying advanced precoding and interference cancelation techniques. • Interweave: CRs are allowed to opportunistically access the underutilized parts of the spectrum without interfering the primary network. In this thesis we focus on the underlay CR paradigm. Apart from cellular CRNs, we also discuss cognitive ad hoc networks [ABZ09], [ALC09], which use the unlicensed spectrum for peer-to-peer content delivery. By transmitting the information in multi-hop fashion the CRN can extend its coverage without increasing the amount of power used. Vice versa, by reducing transmit power driven by splitting the direct path into multiple links, the CRN may decrease the SI towards the primary network while keeping the same coverage.

1.5

Contribution and Related Work

In this thesis we look at different aspects of a multi-hop CR system. Firstly, we consider the problem of throughput optimization of multi-hop underlay CRNs under the constraint on the SI towards the primary network. Secondly, we take a look at the problem of reliable network-coded cooperative communication in multi-user multi-hop CRNs. Finally, we touch upon the primary network in cellular setting by looking at the relay-assisted interference MIMO channel. In the present section we provide the references directly related to the topics of the thesis.

1.5.1

Optimal Resource Allocation in Multi-Hop CRNs

Multi-hop CRNs allow the same infrastructure to be reconfigured in different ways in order to deliver the information from source to destination. For example, as already mentioned, by splitting the whole distance between the source and the destination into set of smaller links, the CRN may reduce the power used at each node, thus reducing the interference to the primary network. On the other hand, there can be blind zones due to shadowing from surrounding buildings or in tunnels.

8

Chapter 1. Introduction

The presence of wireless nodes around such zones can help in relaying information in a multi-hop fashion and avoid deep fades. The interference to a primary receiver created by the CRN is determined by the aggregate power of secondary transmitters weighted with the channel gains of the paths to the primary receiver. In the corresponding chapter we address the issue of optimal power allocation in a multi-hop CRN such that the end-to-end data throughput is maximized under a constraint on the secondary interference power at a set of nearest primary receivers. Optimal power allocation for multi-hop relayed transmission over Rayleigh fading channels, within a non-CR network, is considered in [HA04]. Outage probability of the weakest link is used as the optimization criterion. The optimization problem is shown to be convex and is efficiently solved via Lagrangian duality. In [DGA04], bandwidth and power allocation for ergodic end-to-end throughput maximization was studied for multi-hop FDMA-based systems with line topology. The logarithmic link capacity expression was approximated by the square root of its argument, which allowed to simplify the non-linear problem, thereby obtaining a closed-form solution. Upon this result, the authors of [SZQY05] proposed three sub-optimal power and bandwidth allocation strategies for multi-hop OFDMAbased line networks. The question of fundamental power allocation for cooperative relay networks is investigated in [KC09]. A multi-hop OFDM-based network is generalized to arbitrary number of hops and different paths towards destinations. The objective is to minimize network throughput subject to a power constraint. The optimal power allocation is shown to be unique and found by the famous water-filling solution. Moreover the optimal water-levels are shown to be the same for all paths. Finally, the problem of transmit power allocation in dual-hop CRNs is solved in [ML09]. The optimization problem considered here is the minimization of the sum transmit power of the relays subject to a set of constraints: constraint on the output SNR of the maximum ratio combiners at the destination, constraint on the SI towards the primary network and the set of individual power constraints of each relay node. The problem is shown to be a linear optimization problem, and hence an efficient centralized solution is established. Moreover, fully and partially decentralized solutions are provided as well. Yet, none of the preceding references considered interference within the network. Therefore, in our work we study the optimal power allocation for a multihop CRN of line topology, in which the nodes operate within the same frequency band and hence interfere with each other. The optimization criterion is the end-toend throughput of the line network. We constrain the interference from the CRN towards the primary network in two different ways; namely, via the aggregate SI constraint at the primary network or via the individual SI constraint at each receiver. Furthermore, we provide fully decentralized power allocation as well as the limited-feedback (LF) solution.

1.5. Contribution and Related Work

1.5.2

9

Network-Coded Cooperation in Multi-User Multi-Hop CRNs

In linear network-coded systems, relays perform a linear combination of the incoming packets over a finite field. For example, in [XFKC07] the authors proposed a two-user scheme where each user linearly combines its own message with that of the other user over a binary field GF(2). The scheme is referred to as binary network coding (BNC) and provides substantial coding gains. Meanwhile, the BNC scheme is shown to be suboptimal for multi-user multirelay networks in terms of achievable diversity [XS09b]. For such networks, the authors propose a special kind of linear network codes that allow to achieve full diversity, and hence protection against channel fading. The authors also show the existence of such codes. We refer to this coding strategy as diversity network coding (DNC). In the parallel paper [XS09a], the authors show that full diversity in the network is achieved when a network code has linearly independent global encoding vectors for all possible source-relay channel outage situations. Both contributions are summarized and extended in [XS10]. Some simplified network code constructions are provided as well. In [RUFLV12], the aforementioned results are further generalized. The design of the network codes maximizing diversity order is stated as equivalent to the design of linear block codes over a non-binary finite field GF(q) under the Hamming metric. The authors show that the generalized network codes achieve better tradeoff between rate and diversity than the result of their precedents. Finally, in [WX11] the DMT analysis of network-coded wireless relay networks is carried out and a new cluster-based transmission protocol for optimizing the diversity order of the transmission is proposed. In the corresponding chapter of this thesis we apply the aforementioned idea of the DNC to multi-user ad hoc CRNs with line topology. We show that the DNC strategy with linearly independent global encoding vectors outperforms the BNC strategy and the benchmark TDD-based strategy in terms of diversity gain. The analysis of the diversity order of the proposed scheme is carried out through outage probabilities.In addition, we explore the problem of optimal transmission scheduling within a given number of available time-slots and derive an effective heuristic algorithm maximizing the minimum diversity.

1.5.3

Asymptotic Sum-Rate of Relay-Assisted Cellular MIMO CRNs

In the last part of the thesis, we look at the topology of the cellular relay-assisted CR scenario. In addition to the CRN, in this case a set of relay terminal (RTs) is present in the field, assisting both networks in communication to their corresponding receivers. For instance, these relays can be shared or belong to the CRN offering diversity and/or multiplexing gains to the primary network, thus being able to compensate the loss in performance due to the SI of the CRN.

10

Chapter 1. Introduction

A suitable model for the analysis of the discussed topology is the so-called relayassisted interference channel [SVJS08], [MDG09], [SSE09]. Within this model, two source-destination pairs transmit simultaneously, thereby creating interference to each other. An (infrastructure) RT assists both transmissions by performing different relaying and/or coding techniques. Despite of recent attention to this scheme, the capacity of such scenario is still an open problem. Within the MIMO setting, a relevant scenario was addressed in [SRS10]. The relay-assisted cognitive interference multi-access channels were analyzed in terms achievable rates and outage probability under the assumption of the DF relaying in combination with precoding/precancelation techniques. In this thesis we focus on the multi-user cooperative MIMO relay transmission within the CR scenario aiming at analyzing the achievable ergodic rates of such scheme. In our model, the set of relays have no access to the messages and each relay terminal, employing only the AF strategy. This scenario has not yet been discussed in the literature, in spite of being simple for implementation and hence of practical importance. Due to symmetry of the scenario, we, mainly, study the asymptotic sum-rate of the primary network in the presence of secondary interference. The sum-rate of the secondary network may be found in the same way. The key technique for our analysis is the so called replica method, invented by Edwards and Anderson [EA75] within the field of statistical physics. It is used for the analysis of the macroscopic behavior of the system consisting of the large number of microscopic bodies. The method per se is not rigorous, since it takes a sequence of steps that have not been proven correct yet. For instance, rigorous justification of such assumptions as replica symmetry, replica continuity and selfaveraging is the weak point of the replica method, and it is still an open problem in mathematical physics. However, the method works as it is, providing good approximations for many cases where systematic approaches fail, e.g., computing mutual information for the MIMO systems with discrete signal constellations. Tanaka was the first one to introduce this approach to the field of communication theory (vide [Tan01], [Tan02]); he derived the asymptotic mutual information of a CDMA system with antipodal inputs. His study was generalized to arbitrary inputs by the authors of [GV05]. Later, in [Mul03] and [MS03], the replica method was applied to evaluation of the capacity of a MIMO system. Finally, in [WRW08] and [WW10], the method was applied to cooperative communications. The authors of both references, by means of the replica method, analyzed the asymptotic achievable ergodic rates of the AF MIMO relay channels.

1.6

Outline of the Thesis

Apart from increasing the efficiency of the spectral usage, a CRN should provide high QoS for secondary users (SUs) in terms of data rate and reliability of communication, while keeping the QoS of the PUs high. In this thesis we consider

1.6. Outline of the Thesis

11

multi-hop CRNs, and analyze those from the point of view of achievable network throughput and reliability. The remainder of the thesis is organized as follows. • Chapter 2 introduces a few fundamentals needed to proceed into the topic, describes the methods and concepts used in the thesis. • Chapter 3 considers the problem of optimal power allocation within a multihop regenerative CRNs under different SI constraints towards the primary network. We derive and compare both centralized and distributed power allocations, as well as high- and low-SNR approximations. The chapter is based on the paper [GXR11]

M. A. Girnyk, M. Xiao, and L. K. Rasmussen, “Optimal power allocation in multi-hop cognitive radio networks,” in Proceedings of International Symposium on Personal, Indoor and Mobile Communications (PIMRC), Toronto, Canada, 2011, pp. 472–476.

• Chapter 4 addresses the problem of reliability of network-coded transmission within the multi-user multi-hop regenerative CRNs. We discuss several relaying strategies and analyze those in terms of diversity and outage probability. A heuristic algorithm for finding the optimal scheduling is derived as well. The chapter is based on the paper [GXR12]

M. A. Girnyk, M. Xiao, and L. K. Rasmussen, “Cooperative communication in multi-source line networks,” in Proceedings of Wireless Communication and Networking Conference (WCNC), Paris, France, 2012, pp. 2406–2410.

• Chapter 5 turns the reader’s attention to the primary network. Here we analyze the achievable sum-rate of the relay-assisted non-regenerative cognitive interference MIMO channel. We derive the closed-form expression of the mutual information for the primary multi-access channel under the assumption that the signals from the SUs are treated as noise. The chapter is based on the papers

12

Chapter 1. Introduction

[GVR12a]

M. A. Girnyk, M. Vehkaper¨ a, and L. K. Rasmussen, “On the asymptotic sum-rate of the relay-assisted amplify-and-forward cognitive mimo channel,” in Proceedings of International Symposium on Personal, Indoor and Mobile Communications (PIMRC), Sydney, Australia, 2012, submitted.

[GVR12b]

M. A. Girnyk, M. Vehkaper¨ a, and L. K. Rasmussen, “On the asymptotic sum-rate of uplink MIMO cellular systems in the presence of non-gaussian inter-cell interference,” in Proceedings of IEEE GLOBECOM, Anaheim, U.S.A., 2012, submitted.

• Chapter 6 provides the reader with the conclusions and possible directions of the future work. Technical details of some proofs are left to appendices.

1.7

Contributions Outside the Scope of the Thesis

Some results obtained during the studies were not included in the present thesis, mainly because lack of coherence with the topic covered here. The corresponding papers are listed below in order of appearance. [GR11]

M. A. Girnyk and L. K. Rasmussen, “Myopic multi-hop transmission strategies in layered wireless networks,” in Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Communications (PIMRC), Toronto, Canada, 2011, pp. 1773– 1777.

[GLS+ 12]

F. Gabry, N. Li, N. Schrammar, M. Girnyk, E. Karipidis, R. Thobaben, L. K. Rasmussen, E. G. Larsson, and M. Skoglund, “Secure broadcasting in cooperative cognitive radio networks,” in Proceedings of Future Networking and MobileSummit (FNMS), Berlin, Germany, 2012, to appear.

[GSG+ 12]

F. Gabry, N. Schrammar, M. Girnyk, N. Li, R. Thobaben, and L. K. Rasmussen, “Cooperation for secure broadcasting in cognitive radio networks,” in IEEE International Conference on Communications (ICC), Ottawa, Canada, 2012, to appear.

In [GR11], a general framework was developed for the analysis of a layered multihop network with overhearing. In particular, the network is analyzed in terms of bit

1.7. Contributions Outside the Scope of the Thesis

13

error probability for zero-forcing and linear minimum mean square error detectors at the destination. The scheme is also analyzed in terms of available diversity, coming from possible overhearing between layers. The collaboration with fellow doctoral students resulted in two following papers. In [GLS+ 12], we considered the CR scenario, where the CRN is a potential eavesdropper of the message of primary transmitter. Hence, there is a trade-off between cooperation and secrecy in the CR system. We have derived the achievable rate regions from an information theoretic prospective. The scenario was further generalized to multiple secondary receivers. In [GSG+ 12], the precedent study was twisted to the cases, where the CRN is aware and unaware of the primary message. The achievable rate regions for both cases are investigated and compared. Finally, cooperation was shown to be beneficial in spite of the secrecy constraint of the primary system.

Chapter 2

Fundamentals In this chapter we introduce a few fundamentals necessary for understanding the material in the following chapters. We begin with the discussion of the mathematical model of a general communication system. Further, we introduce entropy and mutual information as information measures and define the notion of the capacity for the Gaussian channel. In Section 2.4 we discuss the wireless channel and its inherent features, followed by Section 2.5, where the relay channel is discussed. In Section 2.6 we describe the minimal basics of network coding required for understanding the material from Chapter 4. Section 2.7 is devoted to the cognitive radio scenario. Section 2.8 briefly discusses the multi-antenna transmission. In particular, we motivate studies of systems with large antenna arrays. The discussion is followed by a brief description of some basic concepts of statistical physics necessary for the analysis of very large systems. Finally, in Section 2.10 we cover some necessary parts of convex optimization theory. The chapter is targeted the general audience, and hence, the following should note be considered as a general introduction to the topics. More detailed and comprehensive information can be found in the references within each section of the present chapter.

2.1

Elements of a Digital Communication System

We begin with a description of a general digital communication system and its elements. The goal of any communication system is to transfer information from a source to an information consumer. Figure 2.1 illustrates the block-scheme of a typical digital communication system. An information source produces an output (analog or digital) which is converted into a sequence of bits by the source encoder. At this stage, the data is aimed to be compressed to as few bits as possible in order to remove the inherent redundancy and increase the speed of communication, taking into account the end user’s requirements. Then, the channel encoder introduces some controlled redundancy to the sequence of bits in order to increase the reliability of the transmission against distortions introduced to the signal on its way. 15

16

Chapter 2. Fundamentals

Source

Source encoder

Channel encoder

Modulator

Channel

Consumer

Source decoder

Channel decoder

Demodulator

Figure 2.1: Typical digital communication system.

The encoded binary sequence then reaches the modulator, which translates the discrete symbols into electrical signals (waveforms) which can be transmitted over a physical medium, such as wires, air, water, etc. This medium is referred to as a communication channel, and is used to deliver the signal from the transmitter to the receiver. The physical characteristics of the channel may vary widely, which makes mathematical channel models essential for the design of efficient communication systems. Communication channels typically can distort the signal in a random manner, due to different mechanisms, e.g., thermal noise, interference created by other systems, etc. After traversing the channel, the physical signal reaches the receiving end, where the demodulator translates it into a binary sequence representing the estimates of the transmitted symbols. This sequence is then fed to the channel decoder that attempts to reconstruct the sequence of compressed data, which was the input of the channel encoder at the transmitting end. The harmful effects of the channel are removed with help of the controlled redundancy injected at the transmitter. Finally, the source decoder converts the compressed data sequence into a format suitable for the information consumer.

2.2

Entropy and Mutual Information

In order to mathematically describe the process of information transmission, we first introduce two basic information measures, viz., entropy and mutual information. The entropy represents a measure of uncertainty about a random variable (r.v.) and hence characterizes its information content. Mutual information, in turn, measures the amount of information, which one r.v. contains about another. This section closely follows Chapters 2 and 8 of [CT91] and Chapter 10 of [Yeu08].

2.2. Entropy and Mutual Information

17

The entropy is a measure of uncertainty contained in a r.v., representing the average number of bits needed to describe the r.v. Let x ∈ X be a discrete r.v. with probability mass function (pmf) p(x). The entropy of x is then defined as1 

H(x)  −

p(x) logb p(x),

(2.1)

x∈X

where units depend on the base b of the logarithm. If b = 2, then the entropy is measured in bits. If b = e, then the entropy is measured in nats. For the rest of the chapter we adopt bits as the unit of information. In this thesis we consider discrete-time continuous memoryless channels. Therefore, we need to define a suitable information measure for such channels, viz., differential entropy. Let x be a continuous r.v. with probability density function (pdf) p(x). Then differential entropy2 is defined as  h(x)  − p(x) log2 p(x)dx, (2.2) S§

where Sx is the support set of a r.v. x. We define the joint differential entropy of two discrete r.v.s with joint pdf p(x, y) as  h(x, y)  − p(x, y) log2 p(x, y)dxdy, (2.3) Sx,y

where Sx,y is the support of a vector [x, y]T . The conditional differential entropy of r.v. y given the knowledge of the r.v. y is defined as  h(X|Y )  − p(x)p(x, y) log2 p(x|y)dxdy. (2.4) Sx,y

According to these definition, we may formulate the chain rule of entropy as follows. h(x, y) = h(x) + h(y|x) = h(y) + h(x|y).

(2.5) (2.6)

We further define the mutual information between continuous r.v.s x and y  p(x, y) dxdy (2.7) I(x; y)  p(x, y) log2 p(x)p(y) = h(x) − h(x|y) (2.8) = h(y) − h(y|x), (2.9) 1 Note

that for convenience, we abuse the notation and do not distinguish between the random variables and their realizations. 2 In every definition involving integral and/or density, we assume that those exist.

18

Chapter 2. Fundamentals

which measures the reduction of uncertainty in x due to observation of y, or vice versa, the reduction of uncertainty in y due to knowledge of x. The conditional mutual information between r.v.s x and y given some variable z is defined as I(x; y|z)  h(y|z) − h(y|x, z) = h(x|z) − h(x|y, z).

2.3

(2.10) (2.11)

The AWGN Channel Model

In this section, we present the additive white Gaussian noise (AWGN) channel model. This model very important for discrete-time continuous alphabet communication channels due to its adequacy and simplicity. Since the additive disturbance in a channel is caused by a large number of different effects, by central limit theorem, it is approximately Gaussian. In the following, after the discussion of the continuous and AWGN channels, we arrive at the definition of the channel capacity, which is the central concept of the field of information theory. Namely, the capacity describes the maximal amount of information that can potentially be transmitted through a channel. This section follows the definitions from Chapter 9 of [CT91] and Chapter 11 of [Yeu08]. A continuous channel is a system with input random variable x and output random variable y related through the conditional pdf p(y|x). A continuous channel is said to be memoryless if its output yt at time instant t depends only on channel inputs xt at time t and conditionally independent of all previous inputs and outputs, i.e., (2.12) p(yt |{x}t1 , {y}t−1 1 ) = p(yt |xt ). The AWGN channel is a discrete-time continuous channel, for which the output yt at time t is the sum of the input xt and the noise term nt , so that yt = xt + nt ,

(2.13)

where nt is referred to as AWGN and is drawn from a zero-mean circular symmetric complex Gaussian (ZMCSCG)3 distribution with variance σ 2 , i.e., nt ∼ CN (0, σ 2 ). Moreover, assume all nt ’s are independent and identically distributed (i.i.d.) and also independent from xt . This is a common model for wired communication channels. Due to practical hardware constraints, a limitation should be put on the maximal transmission power. A common assumption is the average power constraint, i.e., for every codeword xi transmitted over the channel, it holds that M 1  |xi |2 = E{|x|2 } ≤ Ps . M i=1

(2.14)

3 For a detailed discussion on complex circular symmetric Gaussian r.v.s, the reader is referred to Appendix A in [TV05].

19

2.3. The AWGN Channel Model

This constraint limits the amount of information that could potentially be transmitted over the channel. An (M, n) code for the Gaussian channel with power constraint Ps consists of a message set {1, 2, . . . , M }, an encoding function f : {1, 2, . . . , M } → X n

(2.15)

mapping a message m from the message set to a block {x}n1 (m) of channel inputs (codewords), provided that power constraint (2.14) holds, and a decoding function g : Y n → {1, 2, . . . , M }

(2.16)

mapping a block of channel outputs {y}n1 to a message estimate m. ˆ The rate in bits per channel use (bpcu) of an (M, n) code is defined as log2 M . (2.17) n If estimated message m ˆ differs from m, which was originally sent, an error event occurs ˆ = m|m sent}. (2.18) Em  { m R

The conditional probability of error is then defined as pm  p (Em ) .

(2.19)

The maximal probability of error is defined as p(n) max 

max

m∈{1,2,...,M}

pm .

(2.20)

A rate R is said to be achievable for an   AWGN channel with power constraint Ps if there exists a sequence of 2nR , n codes satisfying the power constraint, (n) for which the maximal error probability pmax → 0 as block length n → ∞. The capacity of the channel is defined as the supremum of all its achievable information rates. Shannon’s noisy-channel coding theorem [Sha48] states that if the information rate is equal to or less than the channel capacity (i.e., R ≤ C) then there exists a coding technique which enables transmission over the noisy channel with no errors. And vice versa, if R > C, then the probability of error is close to 1 for every symbol. Thus, the meaning of the channel capacity is the maximum rate of reliable (error-free) information transmission through the channel. Shannon also derived the capacity for the AWGN channel, which for the case of ZMCSCG noise can be expressed as follows C=

max

p(x):E{|x|2 }≤Ps

= log2 (1 + ρ) ,

I(x; y)

(2.21a) (2.21b)

where x is the input and y is the output of a channel, and ρ  Ps /σ 2 is the signal-to-noise ratio (SNR).

20

Chapter 2. Fundamentals

2.4

The Wireless Channel

The communication channel provides the connection between the transmitter and the receiver. It may represent different physical media, from aqueous molecular medium to optical fiber. In this thesis we focus on wireless channels, which have several distinguishing properties: • Pathloss: When propagating, the radio waves that carry the signal are spread in all directions. Therefore, only a limited portion of the radiated power reaches the receiver. • Shadowing: This effect is created by large objects in the surrounding (e.g., buildings, tunnels, hills) leading to worsening of the channel conditions even when the transmitter and the receiver are close to each other. • Multipath propagation: When propagated in all directions, radio waves experience reflection, scattering and diffraction from various objects on their way. Therefore, they arrive at the receiver via multiple paths with different delay and phase rotation in each path, thereby interfering each other. This causes small-scale fluctuations of the received signal power. These channel effects are summarized as signal fading. This phenomenon is often modeled as block fading; that is, the channel gain is assumed to be constant during a block of several consecutive discrete time instants (t ∈ {0, . . . , Tcoh }), and the channel gains of different blocks are assumed to be independent and identically distributed (i.i.d.)4 . The length of such block, Tcoh , is called the coherence interval. The discrete-time complex baseband representation of a fading channel (vide Section 2.2.3 in [TV05]) is given by yt = ht xt + nt ,

(2.22)

where yt denotes the signal received at the receiver and xt is denotes the signal transmitted by a transmitter at time instant t. At the transmitter, power constraint E{|xt |2 } ≤ Ps is used. Here, nt represents the additive white Gaussian noise (AWGN). This randomness is introduced in additive manner mainly at the internal components of the receiver, and is sometimes called thermal noise. Other sources of noise are external to the system and may include atmosphere noise, vehicle ignition noise, relict radio emission, etc. Moreover, transmitters of other systems may produce additional interference, which can be modeled as an additive noise. The noise per se is modeled as a zero mean circularly symmetric complex Gaussian (ZMCSCG) r.v., that is, nt ∼ CN (0, σ 2 ). ht represents the fading process, which is widely modeled as a ZMCSC r.v., hence with its magnitude having Rayleigh distribution (vide Section 2.4.2 in [TV05]). Thus, the model is referred to as Rayleigh fading. 4 In general, there may also be correlation between different blocks, but we do not consider it in the thesis.

21

2.4. The Wireless Channel

For additional information on modeling of wireless channels, the reader is referred to standard books on wireless communications, e.g., [TV05], [Gol05], [Rap02]. In the wireless communication literature, the effect of fading is broadly divided into the following two phenomena, depending on the interrelation between the symbol duration and the coherence interval of the channel. • Slow fading: The variations of the channel gain are random, but slow in comparison to symbol rate, i.e., coherence interval of the channel is larger than several symbol durations. This situation is often modeled by the quasi static scenario, where the channel gains are random, but are assumed to be constant over the transmission duration. • Fast fading: Channel variations are fast, so that a codeword length spans a large number of coherence intervals. As this number increases, the fading process becomes ergodic; that is, averaging over time becomes equivalent to averaging over ensemble of fading realizations. Hence, the randomness of the channel may be averaged out over time allowing for constant long-term transmission rates. In this thesis, we assume that the channel coefficients are i.i.d. and can be tracked at the receiver. This knowledge of the channel coefficients is referred to as channel side information (CSI) and is an important assumption for the analysis of the achievable rates and reliability.

2.4.1

Slow Fading Scenario

Let us first consider the case, where the channel gain is random, but stays constant during the whole transmission. The received signal y at time t consists of the transmitted signal xt scaled with the channel coefficient h and AWGN n. Hence, the channel has the following discrete-time complex baseband representation yt = hxt + nt ,

(2.23)

where power constraint E{|xt |2 } ≤ Ps is used, and nt ∼ CN (0, σ 2 ). For ease of notation, we drop the index t from now on. Since channel gain h does not change over transmission duration and is known at the receiver, its effect can be removed. Therefore, the only randomness left is the AWGN in the channel. The capacity of this point-to-point AWGN channel can be directly found from (2.21a), that is, C = log2 (1 + |h|2 ρ),

(2.24)

where ρ is the SNR, and the value |h|2 ρ is referred to as the received SNR. Since the received SNR depends on channel gain h, it is a random variable. If the transmitter encodes the data at rate R bpcu, the channel conditions may be too bad to support this rate, and the channel is said to be in deep fade. In

22

Chapter 2. Fundamentals

this case, the decoding error cannot be made arbitrary small via coding at the transmitter. Hence, the communication link is said to be in outage, and we define the corresponding event as O  {R > log2 (1 + |h|2 ρ)} = {|h| < G}, 2

(2.25a) (2.25b)

where G = (2R − 1)/ρ is the outage threshold. For Rayleigh fading channels, h ∼ CN (0, 1) and, according to [LTW04], the outage probability can be evaluated as (2.26) po = p(O) = 1 − e−G . Deep fade becomes the typical error event and it cannot be removed with coding, and since the probability of deep fade for fading channel is non-zero, communication with an arbitrary small error probability is not possible. Thus, the capacity of the slow fading channel is zero. An alternative performance measure for such channels is known as outage capacity, defined as the largest possible transmission rate R such that the outage probability is bounded from above by some small ε, i.e., po ≤ ε. Thus,   (2.27) Cε = log2 1 + Fc−1 (1 − ε) , where Fc (x) = p(|h|2 > x) is the complementary cumulative distribution function of |h|2 .

2.4.2

Fast Fading Scenario

Consider a discrete-time complex baseband representation for fast block fading y = hx + n,

(2.28)

where h remains constant over bth coherence interval Tcoh and is i.i.d. across different coherence intervals b = 1, . . . , B accommodated within a single symbol duration. For finite B there is no meaningful definition for the capacity, in sense of maximum rate of communication with vanishing error probability, due to the fact that the instantaneous capacity is random and can drop below any target rate R with a non-zero probability. However, when B grows large, the fast fading channel becomes ergodic, and the randomness of the channel state can be averaged out. By coding over many independent coherence intervals, the rate   (2.29) C = Eh log2 1 + |h|2 ρ can be achieved with arbitrary small error probability. Conversely, for all rates R > C, error probability is bounded away from zero. Therefore, such C is called the ergodic capacity, and it is achieved when Gaussian channel inputs are used.

23

2.5. The Relay Channel

Chan.

Source

Relay

Chan.

Chan.

Destination

Figure 2.2: The relay channel.

2.5

The Relay Channel

In the previous section we highlighted main features of the wireless channel and introduced the notion of fading. In the present section we will narrow down the discussion to the relay channel, described in the introductory chapter. The relay channel is a fundamental building block of cooperative communications. It will be useful for understanding the results presented in the following chapters. A relay channel depicted in Figure 2.2 consists of three terminals: a source, a destination and a relay. The relay terminal (RT) is used for aiding the transmission from the source to the destination. Practically, the relay terminal cannot transmit and receive at the same time due to cross-interference between the transmitted and the received signals at the relay. This limitation is referred to as half-duplex constraint, and it requires that the transmission session is split into two time-slots (or orthogonal frequency carriers)5 . In the first time-slot, both the relay and the destination receive the signal transmitted by the source. The relay performs some processing on the received signal, and sends it to the destination during the second time-slot. Then, the destination observes signals from the source and the relay, and combines both observations to jointly decode those. Note that the efficiency of the relay channel may be increased by allowing source to transmit during the second time-slot. However, in this thesis we do not consider this scenario. When using the assisting RT, the source-destination pair obtains following benefits: • Reliability is increased through diversity. The signal from the source reaches destination through two statistically independent paths (one direct path and one path via the relay), thereby providing spatial diversity. Acting individually, single-antenna terminals cannot generate spatial diversity. 5 Full-duplex regime assumes that the half-duplex constraint is overcome, i.e., the node can transmit and receive at the same time. Full-duplex transmission may be used as an idealistic benchmark to realistic transmission schemes.

24

Chapter 2. Fundamentals • Throughput maybe increased if the direct source-destination link is very poor. Then, it may be beneficial to split it into two better quality channels, so that the resulting rate increases albeit two time-slots used. • Coverage extension is again achieved by splitting the direct link into two shorter links, so that less power has to be spent by the source in order to reach the destination. • Interference reduction is useful if another system within the same area experiences interference from the source. The signal is transmitted in two hops, therefore the power is reduced, so is the interference level.

In this thesis we will consider the following well investigated strategies available for the relay: • Amplify-and-Forward (AF): Upon the reception of the signal from the source after the first time-slot, the relay terminal amplifies the noisy signal and retransmits it towards the destination without any special processing. • Decode-and-Forward (DF): The RT tries to decode the signal received from the source and get the message intended to the destination. Then, it re-encodes the message and transmit it to the destination during second time-slot. The AF strategy works better when the relay is situated closer to the destination, i.e., when the relay-destination link is good and the relay does not need to use much power. Otherwise, if the relay is closer to the source, more amplification is needed at the relay in order to reach the destination. And since the relay forwards the noisy received signal, the noise is amplified and its effect becomes more severe. On the contrary, the DF strategy works better if the relay terminal is close to the source, since then it is more likely to decode the source’s message. If the relay moves farther from the source, it becomes harder for it to decode the message, and the relay becomes less useful.

2.6

Basics of Network Coding

Having introduced the basic model for cooperative communications (the relay channel), we now shift the reader’s attention again to the concept of network coding (cf. Section 1.3), which is principal for the design of many band-efficient transmission schemes for wireless networks. The classical illustration of the benefits of network coding is based on the so-called butterfly network example [Yeu08], depicted in Figure 2.3. We consider the communication at the packet level, where the source node wants to communicate packets m1 and m2 to a set of two sinks (multicast scenario). There are four intermediate relays in the network and the specific network topology is shown in the figure. The capacity of each link is assumed to be one packet per channel use, meaning that we cannot transmit more than one packet through a link

25

2.6. Basics of Network Coding

Source m1

m2

m1

Relay 1

Relay 2

m2 Relay 3 m1

m1

m2

Bottleneck! m1

Relay 4

m1

Sink 1

m1

Sink 2 m2

m1

m2

Figure 2.3: Store-and-forward relaying on the butterfly network.

at a time. Let us consider the store-and-forward (SF) relaying first. On the way to the sinks both packets meet relay node 3, where they both have to pass through a single link. Since the capacity is one packet, we cannot transmit both m1 and m2 . We transmit only one of them, the other one has to be stored until the link is free for transmission. This link becomes the bottleneck of the network in sense that it limits the network throughput. An additional time-slot is needed for both packets to pass through the bottleneck. Thus, the whole multicast transmission takes 5 time-slots. On the other hand, when the network coding approach is used (vide Figure 2.4), an “exclusive or” (XOR) operation of both packets is passed through the bottleneck and then to both sinks. In this case, both sinks receive m1 ⊕ m2 and can recover the missing packet from the received combination. For instance, sink 1 receives m1 directly from relay 1 and m1 ⊕ m2 from relay 4. The it can perform the following operation m1 ⊕ (m1 ⊕ m2 ) = m2 . (2.30) Similarly, sink 2 recovers m1 by performing m2 ⊕ (m1 ⊕ m2 ) = m1 .

(2.31)

26

Chapter 2. Fundamentals

Source m1

m2

m1

Relay 1

Relay 2

m2 Relay 3

m1

m2

m1 ⊕ m2

m1 ⊕ m2 Relay 4 m1 ⊕ m2 Sink 2

Sink 1

m1

m2

m1

m2

Figure 2.4: Network Coding on the butterfly network.

Thus, both packets are recovered at both sinks after 4 time-slots and we conclude that usage of network coding, in comparison with SF relaying, reduces the number of transmissions needed for communication over a multicast network, thereby increasing the throughput of a network. Another important benefit of using network coding is the increased reliability via diversity gain. To illustrate this fact, we consider the uplink cooperative relay scenario depicted in Figure 2.5. Two mobile users communicate their corresponding messages m1 and m2 to the BS. A cooperative RT overhears their transmissions during the first time-slot, decodes both packets and forwards their XOR combination to the base station (BS), which can recover those. Clearly, the system is protected against one link outage, no matter at which link it appears. Again link RT – BS is the bottleneck, and if network coding was not used, one of the users would have been less protected against outages.

2.7

Cognitive Radio Scenario

In this section we discuss the models of the cognitive radio channel, which will be used in further chapters. A comprehensive review of the main paradigms of the

27

2.7. Cognitive Radio Scenario

m1 User 1 m1

m2

m1 ⊕ m2

Relay m2 Base station

User 2 Figure 2.5: Network Coding in the uplink cooperative relay scenario.

Prim. Tx

Sec. Tx

Prim. Rx

Sec. Rx

Figure 2.6: Interference channel.

28

Chapter 2. Fundamentals

present section can be found in [GJMS09] and references therein. As we mentioned in the previous chapter, within the underlay CR paradigm, no sophisticated precoding and cancelation techniques are available. The CRN is simply allowed to access the spectrum owned by the primary network under the condition that the performance of the latter does not degrade. Namely, the specific quality of service (QoS) requirements of the primary users (PUs) (in terms of achievable rates, reliability, etc.) have to be fulfilled. Therefore, secondary interference (SI) generated by the CRN to the primary receivers should be below a certain acceptable interference threshold. This constraint can be translated into a constraint on transmit power, using knowledge of the CSI at secondary transmitters. Moreover, each secondary user (SU) has a maximum power constraint When both the primary network and the CRN consist of a single transmitterreceiver pair, and the system is modeled as the classical interference channel [Sat77], [Car78]. For instance, the uplink communication within the primary and the secondary network with mutual interference is shown in Figure 2.6. By transmitting a signal to the secondary receiver, the secondary transmitter interferes with the primary receiver. Vice versa, the primary transmitter interferes with the secondary receiver. This interference reduces the rate at which the opposite side can operate. Therefore, a trade-off in performance of both networks may be described by the achievable rate region, which captures all possible rate-pairs achievable by both networks. Knowing the achievable rate region and minimum QoS requirement of the PU being fixed, we may directly find out what is the maximum achievable rate for the CRN. The largest possible region of achievable rates is referred to as the capacity region and remains an open problem even for a simple case of only two users. A CRN is generally assumed to know the locations of the primary receivers and the primary transmitters, as well as signal parameters and pilot information. It is also widely assumed that the CRN may cooperate with the primary receiver and/or primary transmitters so that it is aware of the CSI of both the secondary channels and the primary channels. In this way, the underlay CRN may efficiently manage the SI towards primary users by realizing corresponding power allocation. The CSI may be obtained via approximation via reciprocity if a secondary transmitter can overhear a transmission from the secondary receiver’s location. Alternatively, the primary and the secondary receivers have to send some feedback with CSI, as discussed. Having no CSI at all at the secondary transmitters will result in poor performance of the CRN. If the CRN is centralized, the secondary receiver may collect the CSI of all the channels, compute the transmitters’ powers and communicate those to corresponding SUs. On the other hand, if the CRN is ad hoc and there is no central node. Either one of the nodes must take a central place of a base station, or the power allocation has to be done in a distributed fashion. That is, each terminal must compute its own power in such a way that the CRN is guaranteed to operate without harming the primary network. Hence, we conclude that new techniques for the SI management and power allocation within CRNs are required. In order to increase performance, an infrastructure RT may be incorporated

2.8. Multi-Antenna Transmission

29

between the CRN and the primary network. The model has been recently introduced in [SE07] and is called the relay-assisted interference channel. Within this model, two source-destination pairs transmit simultaneously creating interference to each other. An RT assists both transmissions by performing different relaying and/or coding techniques. Despite significant attention to this set-up (vide [SVJS08], [SSE09], [MDG08]), the capacity of such channel is still an open problem. For the MIMO setting, achievable rates and outage probability performance of relay-assisted interference channels were analyzed in [SRS10], where DF relaying was assumed in combination with precoding techniques.

2.8

Multi-Antenna Transmission

Multi-antenna communication is becoming a widely used technology. It has been incorporated into emerging wireless broadband standards. For example, the LTE standard allows for up to 8 antenna ports at the base station [DPSB08]. With multiple antennas, the performance of a communication channel can be dramatically improved both in terms of data rate and link reliability. Conceptually, there are two important gains the MIMO systems offer: • Diversity gain: When properly spaced, the transmitter-receiver antenna pairs experience different fading. Therefore, by transmitting the same information through all antennas, the reliability of the communication will be determined by the strongest path between antennas. Clearly, this enhances the reliability of the communication as compared to the single-antenna systems. • Multiplexing gain: By having several paths connecting the transmitter and the receiver, the former may transmit different data blocks via different antennas. In this way, there are several data streams carrying different information that are transmitted and received in parallel. This leads to increased data rate of the communication. These two advantages of MIMO transmission form the aforementioned DMT, which illustrated the fundamental limits of communication. Consider a communication channel with M antennas installed at the transmitter and N antennas at the receiver. The input-output relationship for such channel is given by y = Hx + n, (2.32) where y ∈ CN is the received signal vector, x ∈ CM is the transmitted vector, H ∈ CN ×M is the matrix of i.i.d. complex channel coefficients, often assumed to be CN (0, 1), and n ∈ CN is the additive ZMCSCG noise, n ∼ CN (0N ×1 , IN ). The ergodic capacity of the MIMO channel (2.32) is given by   C = EH log2 det IN + ρHHH . (2.33)

30

Chapter 2. Fundamentals

This capacity is achieved when Gaussian channel inputs are used at the transmitter. However, a practical approach would restrict the signal constellations to be discrete, e.g., BPSK, QPSK, 16-QAM, etc. A natural question is what would be the highest possible data rate when the discrete channel inputs are used? In order to compute (2.33) with discrete inputs, one has to evaluate the mutual information I(x; y|H) = h(y|H) − h(y|x, H) = −Ey log2 p(y|H) + Ey,x log2 p(y|x, H)     2 2 = −Ey log2 Ex e−y−Hx + Ey,x log2 e−y−Hx .

(2.34) (2.35) (2.36)

which is, however, not easy even via Monte-Carlo simulations, especially when number of antennas grows large. In order to address those questions, and motivated by the emerging interest in massive MIMO [RPLL12], we consider the MIMO systems with very large numbers of antennas, i.e., N, M → ∞. When the number of antennas grows large, the MIMO systems which behave randomly become somewhat deterministic. For example, the distribution of the singular values of the channel matrix converges to a deterministic function [MP67]. As a consequence, the effect of small-scale fading can be averaged out. Moreover, this opens a door to the methods of random matrix theory [TV04] and statistical physics [Mer10], which allow to analyze some generally intractable cases. The design and analysis of such MIMO systems is a fairly new subject that is attracting substantial interest[NLM11], [HtBD11], [DM05]. Such systems, destined to operate at 60 GHz, have already been successfully implemented [STS08]. In the corresponding spectrum region, each individual antenna can have a small physical size, and can be built from inexpensive hardware. The size of an antenna array is then reduced to reasonable dimensions. Another important advantage of large MIMO systems is that they allow for the reduction of the transmission power. In the uplink, reducing the transmit power of the terminals will exhaust their batteries slower. In the downlink, reducing the emitted RF power would help in cutting the electricity consumption of the BS. In the following section we present the replica method, a mathematical tool borrowed from the field of statistical physics and used for the analysis of the performance of the large systems. In contrast to the random matrix theory, the replica method allows to analyze the performance of the MIMO systems using arbitrary channel inputs (not only Gaussian). The downside is that the method itself is not proven rigorous. It invokes several assumptions that could be criticized by the mathematicians, e.g., replica continuity, replica symmetry, interchangeability and existence of limits. However, the assumptions are quite common in both physics and information theory, and the results obtained via the replica method meet the simulations.

2.9. Statistical Physics and The Replica Method

2.9

31

Statistical Physics and The Replica Method

Random matrix theory is a well investigated field. One of its main results states that the dependence of the performance of a system on a particular realization of parameters vanishes as the system’s size grows large without bound. Namely, expectation over the realizations and/or time may be described by the matrix of the states per se, provided that its size grows large. This property is often referred to as self-averaging [M¨ 04]. On the other hand, in the context of physics, it is nothing but the description of the fundamental law that the fluctuation of macroscopic behavior of a system consisting of microscopic particles disappears as the number of particles becomes large. Such systems are studied within the field of statistical physics via probabilistic models for the particle interactions. Let us consider a system consisting of a large number of bodies. The microscopic state of the system is described by the configuration of K variables grouped into a vector x = [x1 , . . . , xK ]T . The energy associated with each state is called the Hamiltonian, and is denoted as E(x). Let p(x) be the probability that the system is in configuration x. At thermal equilibrium, the entropy of the system  H=− p(x) ln p(x) (2.37) x

is maximized given the average energy 

E = p(x)E(x)

(2.38)

x

is preserved. The entropy maximizing distribution is known to be the Boltzmann distribution e−βE(x) p(x) =  −βE(x) , (2.39) xe where β = 1/T > 0 is the inverse temperature. Namely, the most probable configuration is the one that minimizes the Hamiltonian E(x). The normalization factor  e−βE(x) (2.40) Z x

is referred to as the partition function, which describes the statistical properties of the system in the thermodynamic equilibrium. The partition function characterizes all the important macroscopic quantities of a system. However, a more convenient macroscopic quantity is the (normalized) free energy

1 H F=

E − (2.41) K β 1 ln Z. (2.42) =− βK

32

Chapter 2. Fundamentals

At thermal equilibrium, the entropy of the system is maximized, whereas the free energy is minimized. Unfortunately, except for some special cases, the explicit computation of the free energy is very complicated due to large number of particles in the system. A spin glass [Dot01], a magnetic material consisting of a large number of directional spins, is an example of the case where direct computation of the free energy is infeasible. Basically, (2.41) depends on the random interactions between spins. These interactions reflect the realization of the spin glass and do not change over time. Therefore, in physics literature they are referred to as quenched disorder in the system. In statistical physics, it is often postulated that in the large many-body systems the normalized free energy (2.41) has self-averaging property, i.e., − lim

K→∞

1 1 ln Z = − lim E ln Z, K→∞ βK βK

(2.43)

where the expectation is taken with respect to all quenched randomness in the system. For example, consider a gas in a container; its properties (e.g., pressure) are certainly determined by the interactions between the individual molecules. However, in the thermodynamic limit, i.e., when K → ∞, the explicit dependence on these individual interactions vanishes. This opens the door to evaluation of the free energy, which is the macroscopic characteristic of the system. For wider discussion on the topic, we refer reader to textbooks [Nis01], [Tal10] and [Dot01], as well as overview articles [CCFM05] and [WTV11].

2.9.1

Replica Trick

Unfortunately, direct evaluation of (2.43) is prohibitive due to the logarithmic term. For instance, the partition function (2.40) is a sum, and hence computation of E ln Z, is difficult in practice. A useful identity is6 E ln Z = lim

u→0

∂ ln E {Z u } . ∂u

(2.44)

The problem reduces to computation of the uth moment EZ u of the function of the random vector x, where the evaluation has to be done for real-valued u in order to perform limits. To proceed, we need to assume the generalization of EZ u to real-valued u. This assumption is referred to as replica continuity. The conditions under which this assumption hold are unknown. The mathematical rigor of this step (known as the replica trick ) is still an open problem within the field of mathematical physics. 6 This

is true for u ∈ R

33

2.10. Convexity and Convex Optimization Assuming that the replica trick is valid, let Z = 

u Z(v)dv Zu = =

n 



Z(x)dx7 , then

Z(v (u) )dv (u) .

(2.45) (2.46)

u=1

Hence, instead of calculating Z u , we create u replicas of v, such that replicated variables v (u) may be assigned some desired properties.

2.9.2

Replica Symmetry

Consider a computation of the uth moment of Z of the following kind  1 EZ u = ln eKf (v1 ,v2 ) dv1 dv2 . K

(2.47)

According to Cram´er’s theorem [Ell06], in the limit of K → ∞, the integral will be dominated by those values of v1 and v2 that maximize f (v1 , v2 ). In other words,  1 lim ln eKf (v1 ,v2 ) dv1 dv2 = maxf (v1 , v2 ). (2.48) v1 ,v2 K→∞ K If the function in the exponent is multivariate, the task of finding its extremum may be prohibitive. One common assumption made at this point is the so called replica symmetry. For example, the value of the function may be independent of the exact order of the variables, such that f (v1 , v2 ) = f (v2 , v1 ) = f (v).

(2.49)

This step significantly reduces the dimension of the optimization problem maxf (v1 , v2 ) = maxf (v). v1 ,v2

v

(2.50)

Unfortunately, the replica symmetry does not always hold. For instance, in many practical cases, the replica-symmetric solution is not correct. This type of problems is referred to as replica symmetry breaking [ZMdMM08].

2.10

Convexity and Convex Optimization

In this section we introduce some basics of convex optimization. We start with the definition of a convex optimization problem. Further, we introduce the concept of Lagrange duality, the dual problem and the constraint qualification conditions. Finally, we end with the famous Karush-Kuhn-Tucker optimality conditions. For more information about the topic, the reader is referred to standard textbooks [BV04] [BNO03] and [Nes04]. 7 The integrals hereafter are assumed to be taken over the entire support unless specified otherwise. For notational simplicity, integral limits are omitted whenever they are clear from context

34

Chapter 2. Fundamentals An optimization problem of the following kind min

f0 (x)

s.t.

fi (x) ≤ 0, i ∈ {1, ..., N }

x

(2.51)

is said to be a convex optimization problem if the objective function f0 (x), as well as constraints fi (x) are convex functions. Function f (x) is said to be convex if it satisfies the following inequality f (θx1 + (1 − θ)x2 ) ≤ θf (x1 ) + (1 − θ)f (x2 ),

(2.52)

for all θ ∈ (0, 1). In other words, the chord between any two points in the domain of the function lies above its graph. Alternatively, function f (x) is concave if −f (X) is convex. For convex optimization problems, a local optimum is guaranteed to be a global optimum and it is unique provided that the objective function is strictly convex. The analysis of such problems is a well developed field and although, in general, it is hard to find closed-form solution, very efficient practical algorithms for solving such problems have been developed. For instance, interior-point methods allow to solve convex optimization problems within several dozens of iterations [BV04].

2.10.1

Lagrange Duality

One of the methods for solving (2.51) is the so called Lagrange relaxation. The key idea is to incorporate the constraints into the objective function and penalize their violation. In this way, the constraints are relaxed and the problem is turned into a non-constrained problem. For problem (2.51) the Lagrangian is defined as L(x, λ) = f0 (x) +

N 

λi fi (x),

(2.53)

i=1

where λi ∈ R+ , ∀i are referred to as the Lagrange multipliers associated with corresponding constraints. The new optimization problem reads as min L(x, λ) = f0 (x) + x

N 

λi fi (x).

(2.54)

i=1

Further, we define the dual function as the minimum of the Lagrangian over x g(λ) = minL(x, λ). x

(2.55)

The dual function is always concave, no matter whether problem (2.51) is convex or not. Moreover, it provides a lower bound on the optimal value f0 (x ) of the

2.10. Convexity and Convex Optimization

35

problem. Suppose x is the optimal solution of (2.51). Then, it can be verified that for each λi ≥ 0, ∀i the following holds g(λ) ≤ f0 (x ).

(2.56)

By maximizing the dual function over the values of λi , we may obtain the best possible lower bound for the optimal value. This is done via the dual problem, which is stated as follows max g(λ) λ

s.t.

λi ≥ 0, i ∈ {1, ..., N }.

(2.57)

Since the dual problem always has a concave objective function and a convex set of constraints, it is guaranteed to be a convex optimization problem. Let λ be the optimal solution of (2.57), then (2.56) implies that f0 (x ) ≥ g(λ ),

(2.58)

which is referred to as weak duality property. The duality gap is defined as the difference between these optimal values, Δ  f0 (x ) − g(λ ),

(2.59)

and it is always non-negative. The dual problem (2.57) may be used to evaluate the optimal value of the primal problem (2.51) since it is always a convex problem and is easy to solve. The difficulty, though, may lie in the duality gap which can be arbitrary large in general. If the duality gap is zero, i.e., f0 (x ) = g(λ ),

(2.60)

then we say that strong duality holds, meaning that the best lower bound obtained via solving (2.57) gives the exact optimal value to the primal problem. The conditions under which strong duality holds are called constraint qualifications. Several different constraint qualifications have been formulated and studied in the literature [ref]. In this thesis we will use only two of those, namely • Slater’s constraint qualification [BV04]: For a convex problem, there exists a strictly feasible point, i.e., such x that fi (x) < 0, i ∈ {1, . . . , N }.

(2.61)

• Linear independence constraint qualification (LICQ) [Hen92]: For a general non-linear optimization problem, the gradients of the active constraints are linearly independent at the optimal point x .

36

Chapter 2. Fundamentals

2.10.2

The KKT Conditions

If a solution x is optimum, it has to satisfy a certain set of conditions, namely, Karush-Kuhn-Tucker (KKT) conditions are given by • Optimality conditions: ∇f0 (x ) +

N 

∇fi (x ) = 0

(2.62a)

fi (x ) ≤ 0, i ∈ {1, . . . , N }.

(2.62b)

i=1

• Primal feasibility conditions:

• Dual feasibility conditions: λi ≥ 0, i ∈ {1, . . . , N }.

(2.62c)

λi fi (x ) = 0, i ∈ {1, . . . , N }.

(2.62d)

• Complementary slackness:

Hence, for any optimization problem, the KKT conditions are necessary conditions for a solution x to be optimal, provided that the strong duality holds for the given problem (i.e., one of the aforementioned constraint qualification conditions is satisfied).

Chapter 3

Optimal Power Allocation In this chapter, we study the problem of optimal power allocation for multi-hop ad hoc CRNs. The network configuration of interest is depicted in Figure 3.1, where a CRN operates simultaneously with a primary network which owns the given frequency band. The operation at the same area and the same band with the primary network is allowed for the CRN under the condition that the disturbance created by the CRN at the receivers of the primary network is below a certain acceptable threshold. Obviously, this restriction on the secondary interference (SI) level limits the achievable throughput of the secondary network. Within the CRN, a source node tries to deliver certain content to a destination. The transmission is assisted by a set of intermediate nodes within the CRN which relay the information in a multi-hop regenerative fashion. However, the secondary user (SU) terminals may create interference both to the primary network and to each other. The main question discussed throughout this chapter is how to optimally choose the powers at each secondary transmitter in such a way that the throughput of the CRN is maximized while the constraint on the interference to the primary network is satisfied.

3.1

System Model

Consider a K-hop CRN illustrated in Figure 3.1 operating in the presence of a primary network consisting of M primary users (PUs). An SU 1 wants to deliver certain information content to the destination K +1 with help of intermediate nodes that can relay the content via multiple hops. The network topology illustrated in Figure 3.2 is referred to as line (or tandem) topology since the node connections of the network visually form a straight line. We begin with stating several assumptions that we use throughout this chapter. First, we assume that an appropriate transmission protocol is employed at a higher level so that a route between the source and the destination is selected and the relaying nodes along the route form a line network. We also assume that the 37

38

Chapter 3. Optimal Power Allocation

g˜11

PU 3

PU 2

PU 1 g˜12

g˜13

g13

g2,K+1 g3,K+1

PU M g˜1M

g1,K+1

SU 1 g12 SU 2

SU K+1

g23 SU 3

Figure 3.1: Multi-hop cognitive radio network.

relaying nodes within the CRN operate in full-duplex mode. Namely, nodes can receive and transmit information at the same time. In this way, the transmission is realized simultaneously by all nodes causing interference to each other. The transmit power used by the SU i is denoted as Pi , and it should not exceed its own transmit power constraint P max . The links of the primary and the secondary networks are assumed to be AWGN channels subject to slow fading such that the channel gains are random but constant during a transmission from the source to the destination. The corresponding inputoutput relation of the link Li,j between nodes i and j within the CRN (cf. (2.28)) is given by √ yj = gi,j xi + nj . (3.1) The channel power gain gi,j of link Li,j absorbs both Rayleigh fading and pathloss effects, so that 2 gi,j = d−α (3.2) i,j |hi,j | , where di,j is the distance between SUs i and j, α is the pathloss exponent, and channel gains are modeled as zero mean circular symmetric Gaussian (ZMCSCG) random entries, so that hi,j ∼ CN (0, 1). The channel power gain g˜i,j between the SU i and the PU j isdefined  likewise. Link Li,j is subject to ZMCSCG noise with variance σi2 , i.e., CN 0, σi2 . Throughout this chapter we measure the influence of the CRN on the primary network according to the following criteria.

39

3.1. System Model

SU1

SU i−B

SU i−1

SU i

SU i+1

SU i+2

SU i+F+1

SU K+1

Figure 3.2: Multi-hop transmission with overhearing upstream and downstream.

• Aggregate SI constraint represents the tolerable level of the accumulated suminterference produced by the CRN towards the primary network. Mathematically, the corresponding constraint can be written as follows K  M 

g˜i,j Pi ≤ γ,

(3.3)

i=1 j=1

where γ is the acceptable threshold on the aggregate sum SI produced by the CRN towards all the receivers of the primary network. • Individual SI constraint represents the maximum tolerable sum interference power at a given receiver j of the primary network, K 

g˜i,j Pi ≤ γj , ∀j ∈ {1, . . . , M },

(3.4)

i=1

where γj is the acceptable threshold on the sum SI produced by the CRN towards the PU j. The CRN is assumed to know the type of the interference constraint of the primary network as well as the corresponding thresholds. Each node of the CRN is also expected to have the knowledge of the channel gains between itself and every primary receiver, g˜i,j , ∀i, j. It is further assumed that all the CRN nodes, via standard training sequence methods, are able to get information about the channel gains of their incoming links in order to implement regenerative relaying. In this chapter, we will consider both centralized and distributed resource allocation. For the former case we assume that all channel gains are available at the source before the transmission starts. In this way, the source can determine the optimal power allocation and forward the allocation information upstream as overhead with the message. In the latter case, we assume that the source has the same knowledge as the rest of nodes, i.e., channel gains to the primary network and its interference thresholds. We will also discuss the case where at each SU the information about the channel gains of the outgoing links is available. We refer to this case as limited feedback (LF) solution. Transmissions from the SU i to the SU i + 1 are assumed to be overheard by other terminals within the tandem. As we will see in the next chapter, in certain scenarios, overhearing may be helpful leading to increased diversity. However, the

40

Chapter 3. Optimal Power Allocation

receivers would need to have the CSI of the overhearing links. Since it was assumed that SU i + 1 knows only its incoming channel from SU i, the former node can only treat the overheard signals as noise. As shown in Figure 3.2, the transmitted signal from the SU i may be overheard upstream by F and downstream by B SUs. These numbers describe the network topology and, in general, may differ for different SUs depending on their geographical positions and their own power constraints. However, for simplicity, we assume that F and B are the same for all SUs. Another important assumption is that the interference from node i to receiver j within the CRN may be regarded as the AWGN with zero mean and variance gi,j Pi . Therefore, according to the network model, the capacity1 of the link between node i and node i + 1 can be written as ⎛ ⎞ ⎜ ⎜ Ci = log2 ⎜1 + ⎝

⎟ gi,i+1 Pi ⎟ ⎟. F B   ⎠ 2 σi + gi−j,i+1 Pi−j + gi+1+j,i+1 Pi+1+j j=1

3.2

(3.5)

j=1

Aggregate-SI-Constrained Throughput Maximization

From [EFS56] it is known that the maximum possible end-to-end throughput of a network with a single source and a single sink is determined by the sum of the capacities of the channels belonging to the minimum cut that separates the source and the sink within the network. Since in our model the channel gains of the overhearing links are not known at the SUs and overheard signals are treated as interference, the end-to-end throughput of the line network is determined by the capacity of its weakest link, i.e., C = min {C1 , C2 , . . . , CK }. Denoting ϕi  following form

M

˜i,j j=1 g

(3.6)

we may rewrite the aggregate SI constraint in the K 

ϕi Pi ≤ γ.

(3.7)

i=1

Hence, the end-to-end throughput maximization problem for a given channel realization may be written as max min{C1 , C2 , . . . , CK } Pi

s.t.

K 

(3.8)

ϕi Pi ≤ γ

i=1

0 ≤ Pi ≤ P max , 1 Here,

∀i.

the link capacity is regarded as the capacity of a single point-to-point channel in the absence of the other links of the network.

3.2. Aggregate-SI-Constrained Throughput Maximization

41

In the following sections, we discuss the influence of interference among the SUs (intra-CRN interference) on the end-to-end throughput of a line CRN. We further provide centralized, distributed and LF solutions to the optimization problem above.

3.2.1

Power Allocation in the Absence of Intra-CRN Interference

We begin with the simple case where interference among nodes within the CRN is absent. This assumption may be interpreted as follows. Each secondary node has adjusted its power in such a way that its transmission reaches only the following node upstream, i.e., F = 0. Interference backwards may be, for instance, avoided by using the directional antennas with main lobes directed upstream [Wan02], so that B = 0. In this scenario, the expressions of the link capacities are reduced to the following

gi,i+1 Pi Ci = log2 1 + , ∀i ∈ {1, . . . , K}. (3.9) σi2 Since logarithm is a concave function and (3.7) is a linear constraint, (3.8) is a convex optimization problem. Therefore, provided that the strong duality holds, the optimal solution of (3.7) is unique and must satisfy the KKT conditions. Lemma 3.1. For convex optimization problem (3.8) without intra-CRN interference strong duality holds. Proof. We show that Slater’s constraint qualification condition is satisfied, which yields in strong duality [BV04]. By choosing transmit powers according to the following rule   ε  max ,P Pi = min , ∀i ∈ {1, . . . , K}, (3.10) ϕi γ , we show that there exists a strictly feasible interior point P = for some 0 < ε < K   T (P1 , . . . , PK ) , and hence, Slater’s constraint qualification condition is satisfied.

Since problem (3.8) is a convex optimization problem, the KKT conditions are necessary and sufficient conditions for a solution to be optimal. Therefore, based on the KKT conditions we formulate the optimal power allocation conditions summarized in the following theorem. Theorem 3.1. For a multi-hop tandem CRN without intra-CRN interference and with a constraint on the aggregate SI the end-to-end throughput is maximized if and only if the capacities of all channels of the network are equated, i.e., C1 = C2 = · · · = CK

(3.11)

42

Chapter 3. Optimal Power Allocation

and the aggregate SI constraint is met with equality, i.e., K 

ϕi Pi = γ.

(3.12)

i=1

Proof. We rewrite problem (3.8) in the epigraph form2 [BV04] and omit the individual power constraints of transmitters Pi ≤ P max . min − t s.t. t − C1 ≤ 0 .. . t − CK ≤ 0 K 

(3.13)

ϕi Pi − γ ≤ 0.

i=1

The constraints will be incorporated later, once (3.13) is solved. It can be shown that by so we do not change the final result. T T Define two vectors P = (P1 , . . . , PK ) and ϕ = (ϕ1 , . . . , ϕK ) . The Lagrangian for (3.13) over domain D = {Pi : Pi ≥ 0, ∀i} can be written as L (t, P, λ, ω) = −t +

K 

  λi (t − Ci ) + ω ϕT P − γ ,

(3.14)

i=1 T

where λ = (λ1 , . . . , λK ) and ω are the dual variables. From (3.14) we can write down the KKT conditions ∂L (t, P, λ, ω) = 0, ∀i ∈ {1, . . . , K}, ∂Pi ∂L (t, P, λ, ω) = 0, ∂t t − Ci ≤ 0, ∀i ∈ {1, . . . , K}, 2 For

(3.15a) (3.15b) (3.15c)

the optimization problem min

f0 (x)

s.t.

fi (x) ≤ 0,

min

t

i ∈ {1, ..., N }

the epigraph form is written as

s.t.

f0 (x) − t ≤ 0 fi (x) ≤ 0,

i ∈ {1, ..., N }.

The latter problem is equivalent to the original problem, viz., both problems have the same optimal solutions.

3.2. Aggregate-SI-Constrained Throughput Maximization ϕT P − γ ≤ 0, λi ≥ 0, ∀i ∈ {1, . . . , K},

43 (3.15d) (3.15e)

ω ≥ 0, λi (t − Ci ) = 0, i ∈ {1, . . . , K},  T  ω ϕ P − γ = 0.

(3.15f) (3.15g) (3.15h)

Taking partial derivatives of (3.14) we get the optimality conditions (3.15a) and (3.15b) of form ∂L (t, P, λ, ω) = ϕi ω − ⎛ ∂Pi ⎜ 2 ⎝σi +

gi,i+1 λi j+B  k=j−F k =i+1

⎞

= 0,

(3.16a)

⎟ gk,i+1 Pk ⎠ ln2

K  ∂L (t, P, λ, ω) = −1 + λi = 0, ∂t i=1

(3.16b)

. From (3.16a) we observe that any λi = 0 directly leads to ω = 0 which, in turn, yields λj = 0, ∀j = i, so that (3.16b) cannot be satisfied. Therefore, the only possible case for dual variables is where λi > 0, ∀i ∈ {1, . . . , K} and ω > 0. From complementary slackness (3.15g) and (3.15h) we get Ci = t, ∀i ∈ {1, . . . , K},

(3.17)

which completes the result of the theorem, C1 = C2 = · · · = CK , T

ϕ P = γ.

(3.18) (3.19)

In order to solve (3.8), the source node has to have complete CSI as discussed in the beginning of the chapter. Solution of the K equations (3.18) provides the  T source with the optimal power allocation vector P = (P1 , . . . , PK ) which is then transmitted to the rest of nodes. However, since each secondary terminal i has its own constraint on the available transmit power P max , the constraint may be violated by some of the powers Pi . In this case, the transmit powers have to be adjusted in such a way that max Pi = P max , (3.20)  while keeping C1 (P1 ) = C2 (P2 ) = · · · = CK (PK ). Thus, both the total SI constraint and the SUs’ individual power constraints are guaranteed to be satisfied. Figure 3.3 shows a simple example of the network topology corresponding to the case discussed in the present section. A CRN consisting of K + 1 = 3 nodes

44

Chapter 3. Optimal Power Allocation

PU 2

PU 3

PU 1 g˜12

PU 4

g˜13

g˜11

g˜14

g12

g23 C2

C1 SU 1

SU 2

SU 3

Figure 3.3: Example of a two-hop CRN operating in parallel to a primary network with four receivers.

operates simultaneously to a primary network with M = 4 receivers. Intra-CRN interference is absent, and the SI created towards the primary network is limited by the aggregate SI constraint. Figure 3.4 illustrates the result of Theorem 3.1 applied to this simple two-hop case. We note that the solution of (3.8) lies on the curve formed by the intersection of two logarithmic surfaces corresponding to link capacities, i.e., C1 = C2 . Further, an aggregate SI constraint together with the individual power constraints of each secondary node limit the throughput of the CRN. Thus, the optimal power allocation (P1 , P2 )T is situated at the intersection of the aforementioned curve with the boundaries of the region formed by the constraints, i.e., ϕT P = γ and Pi ≤ P max , ∀i.

3.2.2

Power Allocation in the Presence of Intra-CRN Interference

In this section we consider the more general case, where CRN nodes may interfere with each other. This effect limits the achievable end-to-end throughput of the line network via corresponding interference terms in the denominators of (3.9). The

3.2. Aggregate-SI-Constrained Throughput Maximization

45

Figure 3.4: Optimal power allocation at two transmitters corresponds to the intersection of the equal-capacity curve (cyan line) with the region of power constraints (limited by green planes).

expression for the capacity of link i becomes ⎛ ⎜ ⎜ Ci = log2 ⎜1 + ⎝

⎞

⎟ gi,i+1 Pi ⎟ ⎟, F B   ⎠ σi2 + gi−j,i+1 Pi−j + gi+1+j,i+1 Pi+1+j j=1

(3.21)

j=1

where F and B are corresponding maximum numbers of hops that create interference from nodes downstream and upstream, discussed previously. Under the aggregate SI constraint the optimization problem stays the same, max Pi

s.t.

min{C1 , C2 , . . . , CK } K 

ϕi Pi ≤ γ

(3.22)

i=1

0 ≤ Pi ≤ P max , ∀i ∈ {1, . . . , K}. a However, taking into account (3.21), since a function of type f (x) = log2 (1 + b+cx ) with a, b and c constants is not concave, problem (3.22) is not a convex optimization

46

Chapter 3. Optimal Power Allocation

problem any longer. Therefore, convex optimization tools may not be applied. Yet, for this problem we can show that the previous result holds. Theorem 3.2. For a multi-hop line CRN with intra-network interference and with a constraint on the aggregate SI the end-to-end throughput is maximized if and only if the capacities of all channels of the network are equated, i.e., C1 = C2 = · · · = CK

(3.23)

and the aggregate SI constraint is met with equality, i.e., K 

ϕi Pi = γ.

(3.24)

i=1

In order to prove Theorem 3.2 we first prove the following lemma. Lemma 3.2. For optimization problem (3.22) strong duality holds. Proof. Again, we rewrite the optimization problem in the epigraph form min − t s.t. c1 = t − C1 ≤ 0 .. .

(3.25)

cK = t − CK ≤ 0 cK+1 = ϕT P − γ ≤ 0. Next, we show that (3.25) satisfies the LICQ conditions [Hen92] and hence, the duality gap of this problem is zero. Define a vector of primal variables v  (P1 , . . . , PK , t)T . Constraints cj , j ∈ {1, . . . , K + 1} of problem (3.25) are said to satisfy the LICQ conditions if a matrix Dc = [∇v c1 , . . . , ∇v cK+1 ]

(3.26)

has full column rank, viz., the gradients of the active constrains are linearly inde  T ,t ) . pendent at the optimal point v = (P1 , . . . , PK The matrix of gradients has the following form ⎡ −A

··· 11 B12 0 −A22 B23 B31 0 −A33

⎢ ⎢ Dc = ⎢ . ⎣ ..

0 1

B1,B ··· B34

0 0 ··· B2,1+B 0 ··· ··· B3,2+B ···

.. ··· 1

0 1

BK,K−1−F 1

··· 1

.

0 0 0

ϕ1 ϕ2 ϕ3

⎤

⎥ ⎥ , .. ⎥ . ⎦

BK,K−2 0 −AK,K ϕK 1 ··· 1 0

(3.27)

3.2. Aggregate-SI-Constrained Throughput Maximization

47

where Ai,i = ⎛ ⎜ 2 ⎝σi + Bi,j =

gi,i+1 j+B  k=j−F k =i+1

⎞

,

(3.28a)

⎟ gk,i+1 Pk ⎠ ln2

Pj gi,j+1 gj,j+1 ⎞ ⎛ . ⎞ ⎟ ⎜ j+B j+B   ⎜ 2 ⎟⎜ ⎟ gk,j+1Pk ⎠ ⎜σ 2 + gk,j+1Pk ⎟ ln2 ⎝σi + ⎠ ⎝ k=j−F k=j−F ⎛

k =j+1

(3.28b)

k =j k =j+1

Since gi,j , ∀i, j are positive, non-zero and i.i.d. with probability one, σi2 and Pi , ∀i are strictly positive, then Ai,j and Bi,j are also strictly positive and indepen T dent with probability one. Therefore, for a set of optimal powers (P1 , . . . , PK ) matrix Dc in (3.27) has full rank with probability one and the LICQ conditions are satisfied. Thus, for (3.25) strong duality holds. Proof of Theorem 3.2. From Lemma 3.2 we know that the KKT conditions are necessary conditions for a solution of (3.25) to be optimal. We now show that the KKT conditions provide a unique feasible solution corresponding to the situation where all link capacities are equated and then maximized, provided that the aggregate SI constraint (3.24) is met with equality. The Lagrangian over D = {Pi : Pi ≥ 0, ∀i} for (3.25) can be written as L (t, P, λ, ω) = −t +

K 

λi (t − Ci ) + ω(ϕT P − γ).

(3.29)

i=1

Taking partial derivatives from (3.29) we get the optimality conditions (3.15a) and (3.15b) ∂L (t, P, λ, ω) = ϕi ω − Ai,i λi + ∂Pi

i+B 

Bi,j λj = 0, ∀i ∈ {1, . . . , K},

(3.30a)

j=i−F −1

K  ∂L (t, P, λ, ω) = −1 + λi = 0, ∂t i=1

(3.30b)

where Ai,i and Bi,j are the same as in (3.28). There is only one negative component in each of the K equations (3.30a) and hence, all λi ’s have to be strictly positive in order to satisfy (3.30a). Otherwise, any single λi = 0 will immediately lead to λj = 0, ∀j = i and hence will violate (3.30b) will not be satisfied.

48

Chapter 3. Optimal Power Allocation Finally, if ω = 0, from (3.30a) we will get a linear system of equations ⎤⎡ ⎤ ⎡ −A B ··· B1,B 0 0 ··· 0 11 12 λ1 0 −A22 B23 ··· B2,1+B 0 ··· 0 λ2 ⎥ ⎢ B31 0 −A33 ⎢ B ··· B ··· 0 34 3,2+F ⎥ ⎣ λ3 ⎥ ⎢ = 0. ⎣ . .. ⎦ .. ⎦ .. .. . . . 0

···

0

BK,K−1−F

···

BK,K−2 0 −AK,K

(3.31)

λK

Since Ai,j and Bi,j are strictly positive and independent with probability one, the matrix above has full column rank with probability one, the system (3.31) has a unique trivial solution λi = 0, ∀i which leads to violation of (3.30b) and hence is not feasible. Therefore, the KKT conditions have a unique optimal solution corresponding to C1 = C2 = · · · = CK and K i=1 ϕi Pi = γ. In order to find the optimal power allocation that achieves the maximum throughput the source node has to solve a system of K − 1 non-linear equations (3.23) together with the aggregate SI constraint (3.24). If the optimal solution of (3.25) contains power levels Pi violating the individual power constraints P max at the SUs, the transmit powers of all SUs have to be readjusted such that max Pi = P max ,

(3.32)

 ). In this way, all the constraints while keeping C1 (P1 ) = C2 (P2 ) = · · · = CK (PK are satisfied.

3.2.3

High- and Low-SNR Approximate Power Solutions

For a large number of hops within the line network it is difficult to find a closed-form solution for (3.23) and (3.24). Yet, the problem can be solved numerically. Furthermore, in order to simplify the solution we may use a high-SNR approximation based on the fact that in the high-SNR regime network performance becomes interferencelimited and the influence of noise vanishes. Thus, if mini,j {gi,j Pi }  maxi σi2 the latter can be neglected and the link capacities become ⎛ ⎞ ⎜ ⎜ Ci = log2 ⎜1 + F  ⎝ j=1

⎟ gi,i+1 Pi ⎟ ⎟. B  ⎠ gi−j,i+1 Pi−j + gi+1+j,i+1 Pi+1+j

(3.33)

j=1

According to Theorem 3.2, when equating all capacities in (3.33) we get K − 1 equalities with K unknowns, while one more equation is formed by the aggregate SI constraint (3.24). Therefore, optimal Pi can be computed and, in general, this equation system is easier to solve than (3.23).

3.2. Aggregate-SI-Constrained Throughput Maximization

49

Similarly, in the low-SNR regime the network performance becomes noise-limited and hence, we can neglect the interference. The resulting link capacity expressions correspond to those of (3.9), so that

gi,i+1 Pi . (3.34) Ci = log2 1 + σi2 The system of K equations formed by (3.34) and (3.24) is much easier to solve than the one of (3.23) and (3.24).

3.2.4

Distributed Power Allocation

In order to solve (3.23) and (3.24), as well as for obtaining the approximate solutions, the source node has to know all the channel gains and potentially power constraints of all the nodes. If such information is not available, implementation of the optimal solution is no longer possible. However, each node may perform the power allocation in a distributed fashion setting its power according to the following rule.   γ max Pi = min P , M . (3.35) K j=1 g˜i,j Namely, the power is divided between the SUs in such a way that they all contribute equal fractions of the amount of interference restricted by the aggregate SI constraint. If the power level is higher than the individual SU’s power constraint, the former is adjusted to meet the letter. Although this solution is sub-optimal since it may not satisfy (3.23), it guarantees that the aggregate SI constraint is met and the CRN is allowed to operate at the given frequency band.

3.2.5

Limited-Feedback Solution

The performance of the distributed allocation may be improved by allowing for some feedback from the nodes upstream to the nodes downstream within the CRN. Namely, we may assume that each SU i, in addition, receives information about the channel power gain of its outgoing link towards the SU i + 1. Knowledge of this channel gain allows the SU to compute the capacity of the link Li,i+1 according to (3.9). The node is able to compute only the low-SNR approximation of the capacity since the overhearing links are not known. On the other hand, since the channel gain of the incoming link is known, SU i is able to estimate the transmit power of the SU i − 1, and hence compute the low-SNR approximation of the capacity of the link Li+1,i+2 . By doing so, the SU is able to adjust its transmit power such that the capacity of its incoming link is equal to the capacity of its outgoing link, and hence condition (3.23) holds in the approximation of low SNR. Thereby, the source node may start with the distributed choice of the transmit power (3.35), and each of the upstream nodes within the CRN chooses its power

50

Chapter 3. Optimal Power Allocation

according to

 Pi = min P

max

gi−1,i Pi−1 γ , , M gi,i+1 K j=1 g˜i,j

 ,

(3.36)

so that its own transmit power constraint together with the aggregate SI constraint are guaranteed to be met.

3.3

Individual-SI-Constrained Throughput Maximization

We now consider the case where each primary receiver has an individual threshold on the tolerable interference. In the same way as before, it is assumed that the interference thresholds are known at all nodes of the CRN. Under this assumption we formulate the end-to-end throughput maximization problem subject to the set of individual SI constraints max min{C1 , ..., CK } Pi

K 

s.t.

(3.37)

g˜i,j Pi ≤ γj , ∀j

i=1

0 ≤ Pi ≤ P max ,

3.3.1

∀i.

Centralized Solution

When full channel state information is available at the source node, problem (3.37) may be solved in a centralized way. The solution of this problem is along the line with the result of the previous section. We summarize the solution in the following theorem. Theorem 3.3. For a multi-hop line CRN with the SI constraints at the primary receivers the end-to-end throughput is maximized if and only if the capacities of all channels of the CRN are equated, i.e., C1 = C2 = · · · = CK

(3.38)

and the SI constraint of the most disturbed primary receiver is met with equality, i.e., K  g˜i,j  Pi = γj  , (3.39) i=1

where

 

j = arg max j

K ˜i,j Pi i=1 g

γj

 .

(3.40)

51

3.3. Individual-SI-Constrained Throughput Maximization

Since the maximizer in (3.40) corresponds to the PU, which either experiences the most severe interference or has the lowest interference threshold among the PUs, we call such user the most disturbed PU. To proceed with the proof of Theorem 3.3, we need to prove the following lemma. Lemma 3.3. For optimization problem (3.37) strong duality holds. Proof. The proof directly follows from Lemma 3.2 with slight modification. Likewise, we rewrite optimization problem (3.37) in the epigraph form min s.t.

−t c1 = t − C1 ≤ 0 .. . cK = t − CK ≤ 0 cK+1 =

K 

(3.41)

g˜i,1 Pi − γ1 ≤ 0

i=1

.. . cK+M =

K 

g˜i,M Pi − γM ≤ 0,

i=1

and show that (3.41) satisfies the LICQ conditions leading to strong duality. For (3.41)the matrix of gradients has the following form ⎤ ⎡ −A B ··· B 0 0 ··· 0 g ˜ ··· g ˜ 11

0

⎢ B31 ⎢ Dc = ⎢ . ⎣ ..

12

1,B

−A22 B23 0 −A33

··· B34

B2,1+B 0 ··· ··· B3,2+B ···

.. ··· 1

0 1

0 1

BK,K−1−F 1

··· 1

.

11

0 0

1M

g ˜21 ··· g ˜2M g ˜31 ··· g ˜3M

.. .

BK,K−2 0 −AK,K g ˜K1 ··· g ˜KM 1 ··· 1 0 ··· 0

⎥ ⎥ ⎥, ⎦

(3.42)

where Ai,i = ⎛ ⎜ 2 ⎝σi + Bi,j =

gi,i+1 j+B  k=j−F k =i+1

⎞

,

(3.43a)

⎟ gk,i+1 Pk ⎠ ln2

Pj gi,j+1 gj,j+1 ⎞ ⎛ . ⎞ ⎟ ⎜ j+B j+B   ⎟ ⎜ 2 ⎟⎜ gk,j+1Pk ⎠ ⎜σ 2 + gk,j+1Pk ⎟ ln2 ⎝σi + ⎠ ⎝ k=j−F k=j−F ⎛

k =j+1

k =j k =j+1

(3.43b)

52

Chapter 3. Optimal Power Allocation

Since Ai,j and Bi,j are strictly positive and independent with probability one,  T for a set of optimal powers (P1 , ..., PK ) matrix Dc in (3.42) has row rank of K + 1 with probability one. Thus, for (3.41) the LICQ conditions are satisfied and strong duality holds. Proof of Theorem 3.3. We show that the KKT conditions provide a unique feasible solution corresponding to the situation where all link capacities are equal and then maximized, such that the SI constraint (3.39) of the most disturbed PU is met with equality. The Lagrangian over D = {Pi : Pi ≥ 0, ∀i} for (3.41) can be written as K  K M    λi (t − Ci ) + ωj g˜i,j Pi − γj . (3.44) L (t, P, λ, ω) = −t + i=1

j=1

i=1

The KKT conditions hence give ∂L (t, P, λ, ω) = 0, ∀i ∈ {1, . . . , K}, ∂Pi ∂L (t, P, λ, ω) = 0, ∂t t − Cj ≤ 0, ∀j ∈ {1, . . . , K + 1}, K 

(3.45a) (3.45b) (3.45c)

g˜i,j Pi − γj ≤ 0, ∀j ∈ {1, . . . , M },

(3.45d)

λi ≥ 0, ∀i ∈ {1, . . . , K}, ωj ≥ 0, ∀j ∈ {1, . . . , M },

(3.45e) (3.45f)

i=1

ωj

K 

λi (t − Ci ) = 0, i ∈ {1, . . . , K}  g˜i,j Pi − γj

= 0, ∀j ∈ {1, . . . , M }.

(3.45g) (3.45h)

i=1

To get the optimality conditions (3.46a) and (3.46b) we take partial derivatives of (3.44) M

K

∂L (t, P, λ, ω)   = ωj g˜i,j − Ai,i λi + ∂Pi j=1 i=1 K  ∂L (t, P, λ, ω) = −1 + λi = 0, ∂t i=1

i+B 

Bi,j λj = 0,

(3.46a)

j=i−F −1

(3.46b)

where Ai,i and Bi,j are the same as in (3.43). There is only one negative component in each of the K equations (3.46a) and hence, all λi ’s have to be strictly positive in order to satisfy (3.46a). Otherwise, any

3.3. Individual-SI-Constrained Throughput Maximization

53

single λi = 0 would immediately lead to λj = 0, ∀j = i and hence (3.46b) will not be satisfied. From complementary slackness (3.45g) we have C1 = C2 = · · · = CK , which gives K − 1 equations. To complete the equation system one more equation is needed. The equation is selected in such a way that maximum powers are used to increase link capacities, whereas all the constraints are satisfied. After some manipulations of (3.45) we may rewrite the constraints as follows K  g˜i,j Pi i=1

γj

≤ 1, ∀j ∈ {1, . . . , M }.

(3.47)

In fact, only one of these constraints may be satisfied with equality and this can be described as K  g˜i,j Pi = 1. (3.48) max j γj i=1 Thus, two conditions of maximizing the throughput are C1 = C2 = · · · = CK , K 

(3.49)

g˜i,j  Pi = γj  ,

(3.50)

i=1

where

 

j = arg max j

K ˜i,j Pi i=1 g

γj

 .

(3.51)

Theorem 3.3 states that the optimal solution of the optimization problem (3.37) is achieved when all link capacities are equated and the SI constraint of the PU, which is the most disturbed by the secondary transmitters, for a given channel realization, is satisfied with equality. In order to find the optimal power allocation the source node has to solve a system of K − 1 non-linear equations (3.38) together with an equation formed by the SI constraint (3.39), where the PU j  , whose SI constraint has to be satisfied with equality. The problem is non-trivial and needs numerical solution. Similarly to the case, where the aggregate SI constraint was employed at the primary network, in order to solve the system of corresponding equations we may use the approximate solutions. The source node may solve (3.33) or (3.34) together with the SI constraint (3.4) and thus compute Pi . Then the transmit powers are to be adjusted according to (3.32) to meet the individual power constraints of the secondary terminals. In the following section we will discuss a decentralized power allocation for the problem (3.37) as well as the LF solution.

54

Chapter 3. Optimal Power Allocation

g13 g12 SU 1

g24 g34

g23

C1

C3

C2 SU 2

SU 4

SU 3 g14

Figure 3.5: Thee-hop transmission with overhearing upstream and downstream.

3.3.2

Distributed and Limited-Feedback Solutions

In order to solve (3.38) and (3.39) in a distributed way, each SU may independently set its power according to the following rule    γ max . (3.52) Pi = min P , min M j K j=1 g˜i,j In this way, each SU contributes equally to the interference experienced by the most disturbed PU. Even though being sub-optimal, the solution guarantees that the individual SI constraints of the primary receivers are satisfied. In (3.52), the transmit power constraints at each SU are also taken into account. If the CSI from the SUs upstream is available to the SUs downstream, each SU i may adjust its transmit power such that the capacity of its incoming link is equal to the capacity of its outgoing link in the low-SNR approximation. Again, the source node may start with the distributed solution for the transmit power (3.52). Then, each of its upstream neighbors chooses its power in accordance with    gi−1,i Pi−1 γ , min , (3.53) Pi = min  j gi,i+1 K M ˜i,j j=1 g so that the SI constraints of all the PUs are guaranteed to be satisfied.

3.4

Numerical Illustration

In this section we give an illustrative example to the results above. Consider a 3-hop network with possible one hop overhearing both upstream and downstream, i.e., F = 1 and B = 1, as shown in Figure 3.5. We consider constraints both on aggregate SI in the primary network and on SI at each primary receiver. By means

55

3.4. Numerical Illustration

of simulations we evaluate and compare different power allocation policies, namely, centralized and distributed allocations, high- and low-SNR approximate allocation and the LF solution. In the following section we discuss the resource allocation subject to the aggregate SI constraint at the primary network.

3.4.1

Aggregate SI Constraint

For the setting described in the figure, the link capacities are given by

g12 P1 , C1 = log2 1 + 2 σ1 + g32 P3

g23 P2 C2 = log2 1 + 2 , σ2 + g13 P1

g34 P3 . C3 = log2 1 + 2 σ3 + g24 P2

(3.54a) (3.54b) (3.54c)

From Theorem 3.2 we know that in order to obtain optimal P = (P1 , P2 , P3 )T we have to solve the system of equations above, together with (3.24). Even for this simple case, it is hard to find a closed-form solution. Yet, it can be done numerically. As discussed previously, when the transmission power is large, the network performance becomes interference-limited, and the noise term becomes negligible. The link capacities become C1 = log2 1 + C2 = log2 1 + C3 = log2 1 + Let r =

g12 g32 ,

s=

g23 g13

and q =

g34 g24 .

g12 P1 , g32 P3

g23 P2 , g13 P1

g34 P3 . g24 P2

i=1

(3.55b) (3.55c)

Thus, we can find that  P1 sq = 3 2, P3 r  rq P2 = 3 2, P1 s

3 

(3.55a)

ϕi Pi = γ.

(3.56a) (3.56b) (3.56c)

56

Chapter 3. Optimal Power Allocation

After solving the system of equations (3.56a) we obtain the optimal power allocation for the high-SNR regime P1 =

P2 = P3 =

γ

ϕ1

3

sq r2

ϕ1

3

sq r2

ϕ1

3

sq r2

3

sq r2

! , 2 + ϕ2 3 qrs + ϕ3 ! 2 γ 3 qrs ! , 2 + ϕ2 3 qrs + ϕ3 γ ! . 2 + ϕ2 3 qrs + ϕ3

(3.57a)

(3.57b) (3.57c)

In the low-SNR regime, the network performance is noise-limited and hence, we can neglect the interference from overhearing. The capacity expressions are simplified to

g12 P1 C1 = log2 1 + , (3.58a) σ12

g23 P2 C2 = log2 1 + , (3.58b) σ22

g34 P3 . (3.58c) C3 = log2 1 + σ32 By equating capacities in (3.58) we obtain a system of equations P1 g23 = , P2 g12 P2 g34 = , P3 g23 3 

ϕi Pi = γ.

(3.59a) (3.59b) (3.59c)

i=1

Solving (3.59a) we get the optimal transmit powers at low SNR γg23 g34 , ϕ1 g23 g34 + ϕ2 g34 g12 + ϕ3 g12 g23 γg34 g12 P2 = , ϕ1 g23 g34 + ϕ2 g34 g12 + ϕ3 g12 g23 γg12 g23 . P3 = ϕ1 g23 g34 + ϕ2 g34 g12 + ϕ3 g12 g23

P1 =

(3.60a) (3.60b) (3.60c)

Taking into account individual transmit power constraints we have to adjust the values found above according to max Pi = P max ,

(3.61)

57

3.4. Numerical Illustration

and C1 (P1 ) = C2 (P2 ) = C3 (P3 ). Here P  is the set of optimal power levels that violate the power constraints P max . The distributed power allocation is found as follows   γ max P1 = min P , M , (3.62a) 3 j=1 g˜1,j   γ P2 = min P max , M , (3.62b) 3 j=1 g˜2,j   γ max P3 = min P , M . (3.62c) 3 j=1 g˜3,j Finally, the LF solution is given by  P1 = min P

max

 P2 = min P

max

 P3 = min P

max

γ

, M 3 j=1 g˜1,j

 ,

g1,2 P1 γ , ,  g2,3 3 M ˜2,j j=1 g g2,3 P2 γ , , M g3,4 3 j=1 g˜3,j

(3.63a)  ,

(3.63b)

.

(3.63c)



Next, we plot the results for the concrete network with line topology illustrated in Figure 3.2. The network has K + 1 = 4 SUs and M = 4 primary receivers that experience SI from the CRN. The primary network is assumed to have similar line topology with distances between SUs, PUs and between both line networks normalized to 1. For simplicity, the noise variances are chosen to be the same at each link i, so that σi2 = σ 2 = 1 mW. Pathloss exponent is set to α = 4. Channel gain of the link L32 is modeled as g32 = G g23 , where antenna backward attenuation is set to G = 0.2. We note that the tolerable aggregate SI threshold γ is linearly related to the transmission powers in (3.3). Therefore, we use γ as x-label for plots. For instance, large values of γ clearly lead to high transmit powers for all nodes, and hence yield the high-SNR regime. The same holds for the low-SNR region. Figure 3.6 shows the performance of the optimal power allocation found by solving (3.23) versus interference threshold γ. High- and low-SNR approximate solutions (3.33) and (3.34), together with the distributed power allocation (3.35) and the LF solution (3.36) are presented as well. We observe that the low-SNR approximation is suitable for low γ values, while the high-SNR approximation tends to the optimal solution for large γ values. We also note that low-SNR, LF and distributed solutions are sub-optimal in the highSNR region due to neglecting interference terms in (3.21), which becomes significant

58

Chapter 3. Optimal Power Allocation

4

3.5

End-to-end trhoughput, bpcu

3

2.5

2

1.5

1 Optimal power allocation High−SNR approximation Low−SNR approximation Limited Feedback solution Distributed power allocation

0.5

0 −3 10

−2

10

−1

10

0

10 γ, Watt

1

10

2

10

3

10

Figure 3.6: End-to-end throughput as function of the aggregate SI threshold of the primary network for different power allocation strategies.

as SNR increases. However, both low-SNR approximation and the LF solution outperform the distributed solution in this region. This is because the equal-capacity argument is taken into account and the reason for performance degradation of these allocations lies in neglecting of interference terms in the link capacity expressions. The distributed solution, in turn, completely omits the equal-capacity condition aiming at satisfying the aggregate SI constraint. Although the distributed allocation performs worse than the rest of solutions, it is completely decentralized and does not need any extra feedback information.

3.4.2

Individual SI Constraints

In this section we power allocation subject to the SI constraints employed at each primary receiver. The network configuration stays the same as in the in the previous section. Again, we consider a 3-hop CRN with possible one-hop overhearing both upstream and downstream. The link capacity expressions are given by (3.54). According to Theorem 3.3, to find the optimal P = (P1 , P2 , P3 )T the source node has to solve the corresponding system of equations, where the aggregate SI constraint of the primary network is replaced with the individual SI constraint (3.39)

59

3.4. Numerical Illustration

of user j  found by (3.40) and referred to as the most disturbed PU. Then the optimal powers are adjusted according to max Pi = P max ,

(3.64)

and C1 (P1 ) = C2 (P2 ) = C3 (P3 ). Same holds for both high- and low-SNR approximate solutions. The distributed power allocation is found as follows    γ max P1 = min P , min , (3.65a) M j 3 j=1 g˜1,j    γ max P2 = min P , min , (3.65b) M j 3 j=1 g˜2,j    γ max P3 = min P , min . (3.65c) M j 3 j=1 g˜3,j Further, the LF solution is given by  P1 = min P

max



, min j

 P2 = min P



max



γ

3

M

j=1

g1,2 P1 , , min j g2,3

g2,3 P2 P3 = min P max , , min j g3,4

 

,

g˜1,j γ

3



M

˜2,j j=1 g γ

3

(3.66a)

M

˜3,j j=1 g

,

(3.66b)

.

(3.66c)



Figure 3.7 illustrates the results of the numerical simulation of the CRN under the SI constrains at the primary nodes. For the same set of parameters as in Section 3.4.1 we fix the SI threshold values for the PUs to be γ2 = γ3 = γ4 = γ = 1 W, and vary γ1 . The figure depicts all five power allocation strategies discussed throughout the chapter. We observe that the curve corresponding to the distributed allocation is no longer monotone and reaches its maximum near the point where γj = 1, ∀j. To explain this fact we first look at the case where γ1  γ. The network configuration illustrated in Figure 3.5 under the pathloss channel model guarantees that on average for secondary node i the channel power gain g˜ii is the strongest among all g˜ij , ∀j. Hence, condition γ1  γ leads to P1 < Pi , i = 1 which, in turn, guarantees that on average the end-to-end throughput of the CRN will have a bottleneck given by C1 , the minimum link capacity among all links. In case where γ1 ≈ γ, the throughput will be determined by any of the link capacities, depending on the channel realization. There is no pre-determined bottleneck and therefore, at this point the end-to-end throughput, on average, is higher than in previous case. Finally,

60

Chapter 3. Optimal Power Allocation

3.5

3

End-to-end trhoughput, bpcu

2.5

2

1.5

1

Optimal power allocation High−SNR approximation Low−SNR approximation Limited Feedback solution Distributed power allocation

0.5

0 −3 10

−2

10

−1

10

0

10 γ1 , Watt

1

10

2

10

3

10

Figure 3.7: End-to-end throughput as a function of the SI threshold of PU 1 for different power allocation strategies. The SI thresholds of the other PUs remain fixed.

γ1  γ leads to increased P1 , which yields in higher C1 , on average. However, since the throughput is determined by the minimum link capacity, C1 does not play any role. Furthermore, increased P1 increases interference to link 2 making it the bottleneck for the tandem. Increased interference for this case also makes the throughput smaller than that of the case where γ1 ≈ γ. Thus, the maximum of the curve for the distributed power allocation as function of γ1 is achieved at the point, where γ1 ≈ γ. Another observation is that LF solution outperforms the fully decentralized allocation for medium values of γ1 . However, in the high-SNR regime the difference vanishes. Clearly, for low γ1 values the LF solution takes benefit of its intrinsic lowSNR approximate solution. Whereas, for large values of γ1 the transmit power of secondary node 1 increases making C1 high, so that node 2 cannot equate capacities of links 1 and 2. Hence, it drops to the distributed power allocation. Node 3 then is able to equate capacities of links 2 and 3, though only in the approximation of low-SNR, which does not hold because of strong interference to link 2 from the increased P1 . Thus, the efficiency of the LF solution for large values of γ1 drops to that of distributed solution.

3.5. Summary

61

Finally, we note that the high-SNR approximation does not tend to the optimal solution for large γ1 . This is due to the fact that high γ1 in this case simply does not yield in the high-SNR regime. When γ1 increases, it forces P1 to increase which leads to higher C1 . Hence, the network throughput is determined by the capacities of other links and C1 plays no role. Since γ is fixed to be 1 W, which is not high enough for satisfying mini,j {gi,j Pi }  maxi σi2 , the high-SNR approximation does not provide good precision.

3.5

Summary

In this chapter we studied multi-hop cognitive radio networks with arbitrary overhearing interference among nodes at each hop. We considered two ways of limiting the interference from the secondary network towards the primary network. When the aggregate secondary interference at the primary network is limited, transmit powers of the secondary nodes can be optimally adjusted in order to maximize the end-to-end throughput. We have shown that in this case the maximum end-to-end throughput of a secondary network with line topology is achieved when all intermediate link capacities are equated and then maximized, so that the accumulated interference constraint is satisfied with equality. If this allocation violates the individual power constraints at the SUs, their powers have to adjusted such that the largest of those power satisfies the power constraints. We have also shown that when each primary user has an individual constraint on the received secondary interference, similar result holds. Namely, in order to maximize the throughput of the line network secondary nodes should aim to transmit at such power levels that link capacities are equal and the constraint of the most sensitive primary user is met with equality. We have also presented high- and low-SNR approximation solutions that provide good precision in corresponding SNR regions. For the case where channel states are not known at the source node we have presented a fully decentralized power allocation. In addition, a limited-feedback decentralized solution is proposed which uses the advantage of the low-SNR approximation at cost of the information about channel state of the outgoing link at each node of the tandem. Finally, we have presented numerical results of a concrete network utilizing different power allocation strategies. Simulation results illustrate the efficiency of the proposed solutions.

Chapter 4

Diversity Network-Coded Cooperation In this chapter we discuss the usage of the network coding for multi-source ad hoc CRNs. In contrast to the previous chapter, here there are several users willing to transmit their messages to the destination during a certain number of time-slots. Given a fixed total transmission duration each source may repeat its transmission, hence obtaining temporal diversity. Moreover, if the destination overhears the transmission of a source several times, it may use this information for combating fading by exploiting spatial diversity. Both diversity types lead to increased system performance in terms of robustness of transmission. Thus, by designing the transmission strategy one may significantly increase the gains coming from the diversity.

4.1

System Model

Consider a line CRN shown in Figure 4.1, where nodes 1, . . . , K represent secondary users (SUs) having corresponding messages I1 , . . . , IK to deliver to the secondary destination node K + 1. Each SU i is assigned a certain number of time-slots ti during which his packet Ii is transmitted. The total duration of the transmission is limited to T time-slots, so that K 

ti = T.

(4.1)

i=1

During each hop, the transmission may be overheard by upstream nodes which may relay the received packets further towards the destination. The system operates in the half-duplex regime, i.e., nodes receive and transmit messages during separate time-slots. Thereby, there is no interference among nodes during the transmission. We assume that nodes apply regenerative decode-andforward relaying; namely, each message is first decoded and then forwarded to 63

64

Chapter 4. Diversity Network-Coded Cooperation

PU1

PUM

PU2

I1

I3

I2

SU 1

SU K + 1

SU 2

SU 3

Figure 4.1: Multi-user multi-hop secondary network.

upstream nodes after some processing. We also assume usage of perfect channel codes, and hence any message at a relay is either decoded perfectly and forwarded to the nodes upstream, or discarded. Links of the network are assumed to be quasi-static Rayleigh fading channels with i.i.d. fading coefficients known at corresponding receivers during one transmission round. We assume block-fading channel model, so that channel coefficients remain fixed during a transmission round, and then are replaced with new realization in the following round. Hence, the signal received at node j via link Lij is modeled as (4.2) yj = hij xi + nj , where nj is complex AWGN, so that nj ∼ CN (0, σ 2 ), xi is the signal transmitted with power Ps by node i, and hij is the complex channel gain between nodes i and j, hij ∼ CN (0, 1). Throughout this chapter we assume that the power allocation is already implemented. Due to half-duplex restriction, in each time instance only a single user is transmitting. Therefore, each user sets its transmit power so that the ISI constraint of the most sensitive primary user (PU) is satisfied with equality, i.e., Ps = min i,j

γj ˜ i,j |2 |h

,

(4.3)

where ˜ hi,j is the channel gain between SU i and PU j, similarly to the previous chapter. Since we assumed that the channel codes are perfect, xi can be fully recovered at node j provided that the data rate of the transmission between i and j is less than or equal to the mutual information between yj and xi . Otherwise, a link outage event occurs, defined as (4.4) Oij  {Rij > I(yj ; xi )}.

4.2. Evaluation of Transmission Strategies

65

Since I(yj ; xi ) = log2 (1 + |hij |2 ρ), where ρ = Ps /σ 2 is the signal-to-noise ratio (SNR) which is assumed to be the same at all links, and assuming that rates of all links are the same (Rij = R, ∀i, j), we can rewrite the condition of the outage event as follows Oij = {|hij |2 < G}, (4.5) where G = (2R − 1)/ρ is an outage threshold. For Rayleigh fading channels the outage probability can be evaluated accordingly to [LTW04] po = p(Oij ) = 1 − e−G .

(4.6)

Throughout this chapter po is assumed to be the same for all the links within the CRN. For a packet Ii = [Ii1 , . . . , Iin ]T , we define a packet outage event as a situation, where the packet is not successfully delivered to the destination node while propagating through the network. By having multiple copies of the original message received at the destination through overhearing over independent links, users may obtain diversity gains which can be useful for combating fading. We define the diversity gain for SU i given a transmission strategy as Di  lim

ρ→∞

− log2 pout (Ii ) , log2 ρ

(4.7)

where pout (Ii ) is the probability of outage for packet Ii .

4.2

Evaluation of Transmission Strategies

In this section we discuss the transmission strategies that might be used for the multi-source multi-hop transmission within the network. We start with the simplest possible strategy, where each SU transmits his own packet during the timeslots assigned to him. Then we move to the network-coded transmission and show its advantages. Later on, we propose the transmission scheme that achieves full diversity for the line network.

4.2.1

Conventional Regenerative TDD Transmission

When conventional time-division duplex (TDD) transmission is realized, the users transmit sequentially one after another. The current user starts its transmission only after the message of the previous user downstream is delivered. The nodes upstream assist the current user by relaying the content to the destination. The corresponding transmission schedule for a given network is illustrated in Figure 4.2. At a time-slot assigned to the SU i, packet Ii is transmitted either by the SU i or by other SUs that are relaying SU i’s message. This transmission strategy is used as a benchmark throughout this chapter.

66

Chapter 4. Diversity Network-Coded Cooperation

I1 1 SU 1

SU 2

SU 3

SU K

SU K + 1

SU 2

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

I1 t1 SU 1

I2 t1 + 1 SU 1

SU 2

I2 t1 + t2 SU 1

K−1  i=1

SU 2

IK ti + 1 SU 1

SU 2

SU 3

SU K + 1

SU K IK

T SU 1

SU 2

SU 3

SU K

SU K + 1

time-slots

Figure 4.2: Schedule for the conventional TDD regenerative transmission.

Assume each SU transmits (or his message is relayed by other SUs) during K C time-slots in total, so that i=1 ti = T , where the superscript is used to signalize the transmission strategy used. Hence, temporal diversity is available and the destination may use it for combatting link outages. Unless all tC i copies of packet Ii are corrupted, the destination can decode the message from the SU i. Henceforth, the outage occurs only if all tC i transmissions fail. Thus, we may write the corresponding diversity gain distribution among users as tC i

DiC = tC i ,

i ∈ {1, . . . , K}.

(4.8)

67

4.2. Evaluation of Transmission Strategies

I1 1 SU 1

SU 2

SU 3

SU K

SU K + 1

SU 2

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

I1 t1 SU 1

I1 ⊕ I2 t1 + 1 SU 1

SU 2

I1 ⊕ I2 t1 + t2 SU 1

K−1  i=1

SU 2

I1 ⊕ I2 ⊕ ... ⊕ IK ti + 1 SU 1

SU 2

SU 3

SU K

SU K + 1

I1 ⊕ I2 ⊕ ... ⊕ IK T SU 1

SU 2

SU 3

SU K

SU K + 1

time-slots

Figure 4.3: Schedule for the BNC transmission.

4.2.2

Binary Network-Coded Transmission

Repetition is shown to be a poor approach for increasing diversity. Network coding is a more efficient way of exploiting available degrees of freedom [TV05] in the network. Instead of retransmitting only the packet of another user, SU i could add his own packet to it over the binary field GF(2). We refer to this strategy as binary network coding (BNC). In this way, the packets are mixed with each other and the destination recovers all the packets by solving the corresponding equation system. Therefore, for the same transmission duration as in the conventional TDD case, the BNC transmission contains packet Ii in a virtually larger number of packets as a summand. Thus, the strategy leads to increased diversity, provided that the equation system can be resolved.

68

Chapter 4. Diversity Network-Coded Cooperation

A possible transmission schedule for the BNC strategy is illustrated in Figure 4.3. For a given number of time-slots T the diversity gain for SU i is determined by the minimal number of link outages that make it impossible to recover packet Ii from the combined packet received at the destination. This leads to the result summarized in the following theorem Theorem 4.1. The diversity gain distribution among users for the BNC transmission strategy is (4.9) DiB = min tB j , i ∈ {1, . . . , K}, j∈{1,...,i}

where tB i is the number of transmissions of SU i. Proof. When using the BNC transmission, an outage situation occurs when the system of equations formed by mixed packets containing Ij , j ∈ {1, . . . , i} is rank deficient. Packets of users upstream do not contribute to recovery of Ii . Hence, the outage event occurs when all packets of SU j (where j ≤ i), who is allocated the smallest number of transmissions tB j , are in outage and packet Ij cannot be recovered at the destination. Consequently, Ii can neither be decoded, since the corresponding equation system will remain rank-deficient without Ij . Thus, SU j becomes the bottleneck of the tandem, and we have DiB =

min

j∈{1,...,i}

tB j ,

(4.10)

which completes the proof.

4.2.3

Diversity Network-Coded Transmission

Restriction to the binary field may be sub-optimal in terms of available diversity [XS10]. Instead, we will show that carefully designed non-binary network codes can achieve full diversity available in the network. Hence, we refer to such approach as diversity network coding (DNC). Figure 4.4 highlights a possible schedule for the DNC transmission. When this strategy is implemented, each node linearly encodes incoming messages and its own message over a finite field GF(q). The summation is realized in such way that the global encoding vectors (GEVs) are linearly independent. A GEV, Gi , denotes T T T the linear relation between the vector I = [IT of source messages 1 , I2 , . . . , IK ] Ii , i ∈ {1, . . . , K}, and node i’s corresponding output packet (network codeword) Ci , i.e., Ci = Gi I. (4.11) If the field size q is sufficiently large, one can always find GEVs that are linearly independent. In this way the system of equations at the destination is guaranteed to be solvable. By coding the transmitted packets in the way described above, each user may increase transmission robustness to link outages. Such coding decreases the chance

69

4.2. Evaluation of Transmission Strategies

I1 1 SU 1

SU 2

SU 3

SU K

SU K + 1

SU 2

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

SU 3

SU K

SU K + 1

I1 t1 SU 1

I1 + I2 t1 + 1 SU 1

SU 2

I1 + 2I2 t1 + t2 SU 1

K−1  i=1

SU 2

I1 + I2 + ... + IK ti + 1 SU 1

SU 2

SU 3

SU K

SU K + 1

I1 + 3I2 + ... + 2IK T SU 1

SU 2

SU 3

SU K

SU K + 1

time-slots

Figure 4.4: Schedule for the DNC transmission.

of the outage of each packet caused by a link outage. Moreover, when propagating through the line network, packet Ii becomes present in the increasingly large number of other packets, thus increasing its protection against outages. At some point the number of packets containing Ii becomes larger than the number of outgoing links from SU i’s terminal. In this case an unlikely event of failure of all the outgoing links will limit the available total diversity gain, since none of other users would be able to receive Ii . Thus, we conclude that the diversity order is limited by the diversity gain coming from network coding and the gain coming from space diversity and repeated transmissions. We summarize this statement in the theorem below. Theorem 4.2. The diversity gain distribution among users for the DNC transmis-

70

Chapter 4. Diversity Network-Coded Cooperation

sion strategy is given by

DiD = min{Di1 , Di2 },

(4.12)

Di1 = T − K + 1, i = 1, ⎧ ⎫ i−1 ⎨ ⎬ D Di1 = T − max tD , 2 ≤ i ≤ K, j , K + t1 − 2 ⎩ ⎭

(4.13a)

with

(4.13b)

j=1

Di2 = tD i (K − i + 1), 1 ≤ i ≤ K. where

tD i

(4.13c)

is the number of transmissions of SU i.

Proof. We denote a set of T packets received by the destination by C  {Ci,ki }, where i ∈ {1, . . . , K} are indices corresponding to users and ki ∈ {1, . . . , tD i } are indices corresponding to transmissions of SU i. Note that C1,k1 = I1 , ∀k1 . Only one of those tD 1 packets can contribute to the system of equations at the destination if received correctly. We denote a set containing the packets I1 received at the destination as I. Since for the DNC scheme GEVs are chosen to be linearly independent of each other, all received packets Ci,ki (except aforementioned I1 ’s) are linearly independent and can be used for construction of the equation system at the destination. Let K denote the set of linearly independent received packets. Obviously, when cardinality of K is less than K, the destination cannot recover all K messages. Therefore, in order to determine the diversity order for SU i we search for the minimal number of link outages Di1 that causes event Di  {|K| < K, for SU i}. On the other hand, the diversity order is limited by the number of links outgoing from SU i’s terminal, we denote this number as Di2 . Thus, we can write the diversity order as Di = min{Di1 , Di2 }, where Di1 is interpreted as the diversity gain that comes from coding, and Di2 is the gain that comes from space-time diversity. First, we look at the term Di1 . For SU i let Si be a set of all received packets that contain only combinations of packets I1 , . . . , Ii−1 , viz., packets transmitted by all users downstream prior to SU i. We can write Si  {Cj,ki , ∀ki , j ∈ {1, . . . , i − 1}}. We start with SU K who transmits packet IK . Set SK contains T − tD K packets, of which only K − 1 are linearly independent. Once a single packet from the set of CK,kK , ∀kK is received correctly at the destination, set K becomes completed (i.e., |K| = K) and all K packets can be recovered. Thus, the minimum number of link 1 needed for Di to occur is tD outages DK K. Considering SU K − 1, in a similar way we can show that |SK−1 | = T − (tD K + tD ), where K − 2 packets are linearly independent. Hence, in order to create an K−1 D + t packets must be in outage via corresponding link outage for SU K − 1 all tD K K−1 1 D failures and DK−1 = tD K + tK−1 . Repeating this analysis for the remaining users i−1 one can check that the following argumentation holds, |Si | = T − j=1 tD j and Di1 = T −

i−1  j=1

tD j

(4.14)

71

4.3. Three-User Network Example Time-slot 1

Time-slot 3

Time-slot 2 L14 L24

L13 L12

L34

L23 SU 3

SU 2

SU 1

SU 4

Figure 4.5: Three-user CRN.  D for every SU i up to some number i , for which |Si | ≤ |K∪I|, i.e., T − i−1 j=1 tj ≤ K−  1 D 1 + tD 1 − 1. Thus, for SUs 2, . . . , i , Di = T − K − t1 + 2, and we can rewrite it for general case as ⎧ ⎫ i−1 ⎨ ⎬ D tD , K + t − 2 . (4.15) Di1 = T − max j 1 ⎩ ⎭ j=1

Finally, for SU 1, all the packets are useful for recovery and hence, only K − 1 links are allowed to be non-outage. Thus, D11 = T − K + 1.

(4.16)

The number of outgoing links of node i equals K −i+1, and, taking into account the fact that SU i transmits tD i times, we can write that Di2 = tD i (K − i + 1),

(4.17)

which completes the proof.

4.3

Three-User Network Example

In the previous section we determined the diversity gains by using different transmission strategies. Here we will apply the analysis to the concrete network topology. Consider a three-hop line network as illustrated in Figure 4.5. SUs 1, 2 and 3 are all sources with corresponding independent messages I1 , I2 and I3 to be sent to the destination SU 4. These messages traverse different numbers of hops before reaching the destination. The figure highlights the possible schedule of the multi-hop transmission. During each hop, transmission may be overheard by upstream nodes that assist by relaying the received packets further.

4.3.1

Conventional TDD Transmission

Assume that the conventional TDD transmission is used, and hence the users transmit sequentially, as shown in Figure 4.3.1. The current user starts its transmission

72

Chapter 4. Diversity Network-Coded Cooperation I1 I1 I1

1 SU 1

SU 3 I1

SU 2

SU 4

I1

2 SU 1

SU 3

SU 2

SU 4 I1

3 SU 1

SU 3 I2

SU 2

SU 4

I2

4 SU 1

SU 3

SU 2

SU 4 I2

5 SU 1

SU 3

SU 2

SU 4 I3

6 SU 1

SU 3

SU 2

SU 4

time − slots

Figure 4.6: Conventional TDD transmission.

only after the message of the previous user downstream is delivered. The nodes upstream then assist this user by relaying the content to the destination. As shown in the figure, transmissions from SUs 1, 2 and 3 to the destination take three, two and one time-slots, respectively. The whole transmission finishes when messages from all sources have arrived to the destination, namely, when all phases are completed. Thus, for the given network the entire transmission cycle takes in total T = 6 time-slots. From the perspective of User 3, the destination receives I3 only once via link L34 and hence, the probability that the message of SU 3 is lost is pC o (I3 ) = po . Therefore, only one outage is needed for transmission to fail, and User 3 gets diversity of order D3C = 1. For SU 2, packet I2 is lost if direct link L24 is in outage, and either L23 , or L34 is in outage. Therefore, the probability that I2 cannot be successfully decoded is pC o (I2 ) = p(O24 ∩ [O23 ∪ O34 ]) =

2p2o

−

p3o

≈

2p2o .

(4.18a) (4.18b)

73

4.3. Three-User Network Example I1 I1 I1

1 SU 1

SU 2

SU 3 I1 ⊕ I2

SU 4

SU 3

SU 4

I1 ⊕ I2

2 SU 1

SU 2

I1 ⊕ I2 ⊕ I3 3 SU 1

SU 3 I1

SU 2

SU 4

I1 I1

4 SU 1

SU 2

SU 3 I1 ⊕ I2

SU 4

SU 3

SU 4

I1 ⊕ I2

5 SU 1

SU 2

I1 ⊕ I2 ⊕ I3

6 SU 1

SU 2

SU 3

SU 4

time − slots

Figure 4.7: BNC transmission.

Thus, SU 2 gets diversity of order D2C = 2. Finally, the probability that the packet of User 1 is in outage is pC o (I1 ) = p(O14 ∩ [O12 ∩ {O13 ∪ O34 } ∪ O34 ∩ {O12 ∪ O24 }]) =

3p3o

−

2p4o

≈

3p3o .

(4.19a) (4.19b)

Hence, SU 1 gets diversity of order D1C = 3 and thus has the best protection against fading.

4.3.2

BNC-Based Transmission

Consider the same three-hop network as in the previous section. The corresponding transmission schedule for the BNC strategy is shown in Figure 4.7. Each user decodes a received signal, performs binary network coding through “exclusive or” (XOR) operation on the the decoded message and his own message, and forwards the combination to the nodes upstream. When the destination has received com-

74

Chapter 4. Diversity Network-Coded Cooperation

binations of I1 , I2 and I3 , it can recover all three messages by solving the corresponding equation system. It takes a round of three time-slots to complete the BNC transmissions for all nodes. Since the objective is to maximize diversity as compared to the conventional scheme we can repeat the process for a second round within the six time-slot cycle. Since each user in this network transmits during two time-slots, from Theorem 4.1 we conclude that the diversity gain distribution among users for this transmission strategy is D1B = 2,

(4.20a)

D2B D3B

= 2,

(4.20b)

= 2.

(4.20c)

We shall verify this result by the following analysis. From the figure we observe that the destination receives the coded combinations I1 , I1 ⊕ I2 and I1 ⊕ I2 ⊕ I3 during two independent transmissions. In order to lose any of the messages, at least two of these combinations must be corrupted, i.e., link outage must occur at least two times. For SU 3, if outage occurs during both rounds in links L14 , L24 or L34 , I3 can never be recovered. This happens with probability 1 2 1 2 1 2 pB o (I3 ) = p([O14 ∩ O14 ] ∪ [O24 ∩ O24 ] ∪ [O34 ∩ O34 ])

= 3p2o − p6o ≈ 3p2o ,

(4.21)

z where, the superscript in Oij denotes round z ∈ {1, 2}. This outage probability corresponds to the diversity order of D3B = 2. In fact, this outage event is the worst-case outage situation for SU 3. Any other outage occurrence will require at least three link failures. In turn, SU 2 will not be able to recover I2 if link L14 or L24 are in outage during both rounds. Note that link L34 will only play a minor role in this case. If it is in outage both times, the destination receiver can just ignore it and recover I2 from the rest of received packets. Thus, the outage probability for I2 is 1 2 1 2 pB o (I2 ) = p([O14 ∩ O14 ] ∪ [O24 ∩ O24 ])

= 2p2o − p4o ≈ 2p2o ,

(4.22)

which corresponds to the diversity order D2B = 2 for SU 2. Finally, for SU 1 only link L14 is important. If it is in outage twice, I1 cannot be recovered. However, if all the other links are in outage, I1 is still extractable. Thus, 1 2 2 pB o (I1 ) = p(O14 ∩ O14 ) = po ,

and SU 1 also has diversity gain D1B = 2.

(4.23)

75

4.3. Three-User Network Example I1 I1 I1

1 SU 1

SU 2

SU 3 I1 + I2

SU 4

SU 3

SU 4

I1 + I2

2 SU 1

SU 2

I1 + I2 + I3 3 SU 1

SU 3 I1

SU 2

SU 4

I1 I1

4 SU 1

SU 3 I1 + 2I2

SU 2

SU 4

I1 + 2I2

5 SU 1

SU 3

SU 2

SU 4 I1 + 2I2 + 2I3

6 SU 1

SU 3

SU 2

SU 4

time − slots

Figure 4.8: DNC transmission.

4.3.3

DNC-Based Transmission

When utilizing the DNC strategy, each user linearly combines the incoming packets with his own packet over a finite field GF(q) in such a way that the GEVs of all users are linearly independent. For the considered three-user network, the field GF(4) is sufficient. In this case the destination receives I1 two times via direct transmission as well as I1 + I2 , I1 + I2 + I3 , I1 + 2I2 , I1 + 2I2 + 2I3 via relaying, each of which experiences independent fading (vide Figure 4.8). It is easy to check that the corresponding the GEVs are linearly independent. Theorem 4.2 provides us with the diversity gain distribution among users. D1D = 4,

(4.24a)

D2D D3D

= 3,

(4.24b)

= 2.

(4.24c)

Let us verify this diversity distribution through the analysis of the outage probability of each SU. Thus, for packet I3 , at high SNR the dominating outage event is

76

Chapter 4. Diversity Network-Coded Cooperation

when both transmissions of I1 are successful, both I1 + I2 and I1 + 2I2 are successfully received and both I1 + I2 + I3 and I1 + 2I2 + 2I3 are in outage. The probability of this event is 4 2 2 (4.25) pD o (I3 ) = (1 − po ) po ≈ po . Therefore, the diversity order for SU 3 is D3D = 2. Packet I2 can be in outage even when both transmissions of I1 are successful; however, link L14 must be in outage twice and any of the two relayed network-coded packets must be lost. The probability of this event is 2 2 2 2 3 pD o (I2 ) = [2po (1 − po ) + po ](1 − po ) po ≈ 3po .

(4.26)

The corresponding diversity gain for SU 2 is D2D = 3. For SU 1, in order for packet I1 to be lost, both direct transmissions have to be in outage and two (or more) out of four relayed packets must be lost. The probability of this event is

(

) 4 3 4 2 2 2 4 4 (I ) = p (1 − p ) + p (1 − p ) + p (4.27) p pD 1 o o o o o ≈ 6po , 2 o 2 o and the diversity gain for SU 1 is thus D1D = 4.

4.3.4

Simulation Results

Figure 4.9 shows the simulation results for the outage probability of SU 3 using all the three transmission strategies. We use the aforementioned three-user network topology depicted in Figure 4.5, and set R = 5. Dashed lines illustrate the analytic results obtained above, while single markers indicate the simulated values of the outage probabilities averaged over 109 Monte-Carlo iterations. We note that the analytic results match with simulation quite well, especially in the region of interest (high-SNR). From the slopes of the curves at the high-SNR region we may obtain the corresponding diversity gains for SU 3 for different strategies. We see that when conventional TDD relaying is used, SU 3 attains only diversity of order 1, while both the BNC and the DNC strategies provide SU 3 with diversity of order 2. Besides, we observe that the DNC has the lowest probabilities among the rest of strategies, and hence provides the best protection against link outages. Figure 4.10 shows the simulated outage probabilities of SU 2 for all the three strategies. From the figure we observe that both the conventional TDD and the BNC strategies allow for the diversity gain of 2. On the other hand, the DNC strategy provides SU 2 with the diversity gain of 3, again showing the best performance in the outage probability among the three strategies. From SU 1’s perspective (vide Figure 4.11) the DNC is again the most beneficial strategy allowing to achieve diversity of order 4, whereas the conventional TDD transmission provides with D1C = 3 and the BNC only with D1B = 2.

77

4.4. Optimal Scheduling 0

10

−1

Outage probability

10

−2

10

−3

10

−4

10

15

TDD, User 3 BNC, User 3 DNC, User 3 20

25 ρ, dB

30

35

Figure 4.9: Outage probability of SU 3 when using conventional TDD DF, BNC and DNC strategies. Dashed curves are obtained theoretically, markers indicate corresponding simulated values. The corresponding diversity orders can be observed by the slopes of the curves.

Therefore, we conclude that for the given multi-hop transmission scenario the DNC relaying performs best among other proposed strategies in terms of achievable robustness of transmission against fading. On the other hand, we note that the usage of the BNC transmission leads to a more fair diversity distribution among users.

4.4

Optimal Scheduling

In the previous sections we discussed three transmission strategies used in a multiuser line CRN. Given the total transmission duration of T in (4.1), one may readily determine the available diversity gains provided by the conventional transmission strategy, the BNC and the DNC strategies, according to (4.8), (4.9) and (4.12), respectively. We have also shown that the DNC strategy outperforms the other benchmark transmission strategies in terms of the diversity gain. In fact, under the assumption that a single packet from each user is intended for the destination and the users do not know each others packets in advance, we cannot do any better. The DNC

78

Chapter 4. Diversity Network-Coded Cooperation 0

10

−1

Outage probability

10

−2

10

−3

10

−4

10

−5

10

−6

10

15

TDD, User 2 BNC, User 2 DNC, User 2 20

25 ρ, dB

30

35

Figure 4.10: Outage probability of SU 2 when using conventional TDD DF, BNC and DNC strategies. Dashed curves are obtained theoretically, markers indicate corresponding simulated values.

strategy exploits full diversity available in the network; however, it is still not clear what is the optimal schedule of the DNC transmissions that achieves the highest possible diversity. Moreover, we have seen that the BNC strategy allows for fair failure protection for all users in the tandem, whereas the diversity gains of the DNC strategy are unequally distributed among users. This fact raises the question of fairness of a transmission strategy, and in the present section we address this question and develop an efficient algorithm for schedule optimization. Following the transmission schedule of the previous section, one may conclude that adding more hops to the tandem results in the higher diversity gain of SU 1. Certainly, for SU 1 this approach is most beneficial, however the last user in the tandem will not get much benefit out of it. On the other hand, the ideally fair diversity distribution would be uniform among the users. Therefore, we might use the optimization criteria that force the diversity distribution to be uniform. For instance, one could search for a time-slot allocation that maximizes the minimum diversity among the users.

79

4.4. Optimal Scheduling 0

10

−1

Outage probability

10

−2

10

−3

10

−4

10

−5

10

−6

10

15

TDD, User 1 BNC, User 1 DNC, User 1 20

25 ρ, dB

30

35

Figure 4.11: Outage probability of SU 1 when using conventional TDD DF, BNC and DNC strategies. Dashed curves are obtained theoretically, markers indicate corresponding simulated values.

For this purpose, we formulate an optimization problem as follows max min{Di1 , Di2 } ti

s.t.

i

M 

ti = T

(4.28)

i=1

ti ≥ 0, ∀i. The problem itself is an integer programming problem and in general it requires exhaustive search over all possible allocations of ti . The computational complexity of such search is of order O(T K ), which is difficult to implement even for a tandem with moderate number of users. In this section we propose a sub-optimal heuristic scheduling algorithm that solves the given problem efficiently with less complexity. Algorithm 1 is initialized with the equal time-slot distribution among users, and then iteratively aligns the available time-slots to equalize the diversity gains, thus maximizing the minimum diversity. The complexity of Algorithm 1 is only of order O(K 2 ) which is reasonable for practical use.

80

Chapter 4. Diversity Network-Coded Cooperation

Algorithm 1 Proposed Scheduling Algorithm Initialization: Distribute T time-slots equally among K users. If K is even, let users K/2 + 1, . . . , K have one extra time-slot. Compute the initial diversity distribution according to (4.12). 2 > K do while D12 − DK t1 = t1 − 1, ti = ti + 1, where i = argmin{Di2 }. i

Compute the diversity distribution according to (4.12). for j = 1 → K − 1 do 2 then if Dj2 > Dj−1 tj = tj − 1, ti = ti + 1, where i = argmin{Di2 }. i

Compute the diversity distribution according to (4.12). end if end for end while

Table 4.1: Optimal scheduling for diversity network-coded transmission found by equal time-slot assignment, exhaustive search and proposed algorithm. {t1 , . . . , tK }

{D1 , . . . , DK }

minDi

Equal Exhaustive Proposed

{2, 2, 3, 3} {2, 2, 2, 4} {1, 2, 2, 5}

{7,6,6,3} {7,6,4,4} {4,6,4,5}

3 4 4

5

Equal Exhaustive Proposed

{3, 3, 3, 3, 3} {2, 2, 2, 3, 6} {2, 2, 2, 3, 6}

{11,9,9,6,3} {10,8,6,6,6} {10,8,6,6,6}

3 6 6

6

Equal Exhaustive Proposed

{3, 3, 3, 4, 4, 4} {2, 2, 2, 3, 4, 8} {2, 2, 2, 3, 4, 8}

{16,14,12,12,8,4} {12,10,8,9,8,8} {12,10,8,9,8,8}

4 8 8

M

Method

4

i

81

4.4. Optimal Scheduling 0

10

−1

10

p0

−2

Outage probability

10

p2o

−3

10

p3o −4

10

p4o −5

10

p5o −6

10

−7

10

−8

10

User User User User 16

p6o

1 2 3 4

p7o

18

20

22

24

26

ρ, dB

Figure 4.12: Outage probability for different users the DNC transmission according to the equal time-slot assignment.

The output of the algorithm is the set of numbers of time-slots tD i assigned to every SU i. It is further assumed that the transmission goes sequentially, i.e., the leftmost user transmits first, then the second from the left, and so on. If the leftmost SU has no more time-slots left for transmission, the second SU from the left takes his role in starting transmission in the line. We do not consider the exact scheduling between SUs, but rather its simplified version. Table 4.1 presents the scheduling found by the proposed algorithm, the exhaustive search and the equal time-slot assignment among SUs, applied to a network of line topology. We relate the given number of time-slots T to the number of information sources through K2 + K . (4.29) T = 2 Here (4.29) represents the number of time-slots needed to complete the transmission with conventional TDD strategy used. From the table we can observe that both exhaustive search and Algorithm 1 outperform equal time-slot allocation in terms of minimum diversity provided for a user. One can also note that the outcome of the proposed algorithm either coincides with the result of the exhaustive search or provides a solution that performs equally well.

82

Chapter 4. Diversity Network-Coded Cooperation 0

10

−1

10

p0

−2

Outage probability

10

p2o

−3

10

p3o −4

10

p4o −5

10

p5o −6

10

−7

10

−8

10

p6o

User User User User 16

1 2 3 4

p7o

18

20

22

24

26

ρ, dB

Figure 4.13: Outage probability for different users the DNC transmission according to the optimized schedule.

Figure 4.12 illustrates the results of numerical simulations of the outage probabilities for all users in a four-user network when equal time-slot assignment among SUs. Solid lines indicate the simulated values of the outage probabilities, whereas dashed lines indicate curves of different slopes, corresponding to pio , where i ∈ {1, . . . , 7}. In other words, dashed curves indicate corresponding diversity orders of 1, . . . , 7. Hence, we can compare the diversity gains observed from the figure to those reported in Table 4.1. From the Figure 4.12 we see that for SU 1 the slope of the outage probability at high-SNR is parallel to that of p7o , which means that the diversity gain for SU 1 is 7. Similarly, for SU 2 and SU 3, diversity gain of 6 is read from the figure, since their slopes are parallel to that of p6o . The diversity gain for SU 4 is observed to be 3. These observations coincide with the diversity gains summarized in the table. Likewise, Figure 4.13 illustrates the simulation results for the outage probabilities for all users in a four-user network when optimized schedule is employed, so that the diversity distribution is close to uniform. Again, we can check that the results reported in Table 4.1 hold.

4.5. Summary

4.5

83

Summary

In this chapter we discussed joint scheduling and network coding for cooperative multi-source line CRNs with line topology. Several regenerative transmission strategies were applied to the network with the aim of achieving maximum diversity for the secondary users. A non-binary network coding strategy with linearly independent global encoding vectors was shown to outperform the conventional TDD regenerative forwarding as well as binary network-coded transmission in terms of robust transmission. Moreover, this scheme allows to extract maximum inherent diversity of a line network. The performance may be further improved in terms of fairness among users when an optimal scheduling is applied. The proposed strategy was analyzed in terms of diversity for networks with arbitrary number of sources. Furthermore, a computationally efficient scheduling algorithm for fair maximization of the user diversity was proposed. The results are shown to coincide with those obtained via exhaustive search and to be confirmed by numerical simulations.

Chapter 5

Asymptotic Sum-Rate Analysis In previous chapters we analyzed the performance of multi-hop ad hoc CRNs in terms of transmission rates and reliability. However, we did not take into account that the primary network can potentially cause interference towards the CRN. In the present chapter we analyze the achievable sum-rates of both the primary network and the CRN, accordingly to the relay-assisted multiple-access interference channel model, previously discussed in Section 2.7.

5.1

System Description

The set-up of interest is depicted in Figure 5.1. A primary network and a CRN contain K users each. The users’ terminals transmit simultaneously and hence interfere each other. Increasing the transmit power in one network results in increased interference at the other side. For higher achievable rates, the terminals of both networks are equipped with multiple antennas. Namely, primary users (PUs) and secondary users (SUs) have M transmit antennas each, and both base stations (BSs) have N receive antennas each. Moreover, within the same geographic area a set of infrastructure relay terminals (RTs) is deployed to assist both transmissions. The RTs have L antennas each and perform cooperative AF relaying of the received signals towards the secondary destinations. We consider the uplink channels, so that a set of multi-antenna PUs and SUs transmit their signals xpk and xsk , respectively, to their corresponding multi-antenna BSs. A set of multi-antenna RTs assists both transmissions by relaying the received superposition of the signals. We denote the matrix representing the MIMO channel between the PU k and the BS i by Hp0ik ∈ CN ×M , i ∈ {1, 2}, the matrix of the channel between the SU k and BS i by Hs0ik ∈ CN ×M , i ∈ {1, 2}, the matrix of the channel between the PU k and the RT j by Hp1jk ∈ CL×M , and the matrix of the channel between the SU k and the RT j Hs1jk ∈ CL×M . All channels are subject to fast Rayleigh flat fading, so that the channel matrices are assumed to have zero mean circular symmetric complex Gaussian (ZMCSCG) entries of unit 85

86

Chapter 5. Asymptotic Sum-Rate Analysis F1 y11

H211

Hp111 RT1

Hp011

H221 P U1 Hp021

Hp1J1 Hs011

Hs111

P UK

BS1 H21J

Hs021 SU1 Hs1J1

H22J

SUK

FJ y1J

BS2

RTJ

Figure 5.1: Uplink of a cellular non-regenerative relay-assisted cognitive MIMO network.

variance. These entries are assumed to be constant during one transmission and change independently from one interval to another. We assume that terminals operate in the half-duplex regime, so that the terminals may not transmit and receive at the same time. Hence the whole transmission takes two time-slots. During the first time-slot both BSs and RT j receive the following signals, respectively

y02

y1j



K ρp01k p p  ρs01k s H01k xk + H xs + n01 , M M 01k k k=1 k=1  p K K s  ρ  ρ02k s p p 02k H x + H xs + n02 , = M 02k k M 02k k k=1 k=1 *  s p K K  ρ1jk p p  ρ1jk s s H x + H x + n1j , = M 1jk k M 1jk k

y01 =

K 

k=1

k=1

(5.1a)

(5.1b)

(5.1c)

87

5.1. System Description

where xpk ∈ CM is the signal vector transmitted by the kth PU, xsk ∈ CM is the signal vector transmitted by the kth SU. Signal vectors of PUs and SUs are assumed H H to have have covariance matrices E{xpk xpk } = IM and E{xsk xsk } = IM , respectively. Here, ρp0ik and ρs0ik denote the SNRs of the links PU k - BS i and SU k – BS i, respectively; ρp1jk and ρs1jk denote SNRs of links PU k – RT j and SU k – RT j, respectively. Both noise terms n1j ∈ CL , i.e., the noise added at the RT j, and n0i ∈ CN , i.e., noise added at the BS i, i ∈ {1, 2} are assumed to be ZMCSCG unit variance vectors with identity covariance matrices. During the second time-slot, relay j amplifies the received signals with a forwarding matrix Fj = diag(f1j , . . . , fLj ) and forwards it to both BSs. The diagonal elements of the forwarding matrix are chosen to fulfil the average power constraint of the relay, given by ,, + + H H ≤ L. (5.2) Fj tr E Fj y1j y1j The signals received at the BSs after the second time-slot are J   ρ21j y21 = H21j Fj y1j + n21 , L j=1 J   ρ22j y22 = H22j Fj y1j + n22 , L j=1

(5.3a)

(5.3b)

where H2ij ∈ CN ×L is the matrix of the MIMO channel between RT j and BS i, ρ21j is the corresponding SNR, and n2i ∈ CN is ZMCSCG noise vector added at BS i ∈ {1, 2}. Since the system is symmetric in terms of expressions of received signals, from now on we focus only on the primary network. For the ease of notation we denote T T , y21 ]. We the vector received at the primary BS during both time-slots as y  [y01 further define the effective channel matrices as follows ⎤ ⎡   ⎢ Hp  ⎣ ⎡ Hs  ⎣ -

J 



j=1

J 

p ρ 011 M

p ρ 21j L

H21j Fj

 

j=1

ρs 21j L

ρs 011 M

...

p ρ 1j1 M

Hp 1j1

Hs011 

H21j Fj

ˆ  √ ρ211 0 H H211 F1 L

Hp 011 

... ...

...

J  j=1

...

ρs 1j1 M

Hs1j1 ...

√ ρ21J 0 L

H21J FJ

J  j=1

.



p ρ 01K M

p ρ 21j L

H21j Fj

 

ρs 21j L

Hp 01K 

ρs 01K M

p ρ 1jK M

Hp 1jK

⎥ ⎦

Hs01K 

H21j Fj

,

ρs 1jK M

Hs1jK

(5.4a)

⎤ ⎦

(5.4b) (5.4c)

where Hp ∈ C2N ×KM and Hs ∈ C2N ×KM are the effective channels that convey ˆ ∈ C2N ×JL represents the effective channel for all PUs and SUs, respectively, and H the noise amplified at the RTs. Hence, we may write the end-to-end input-output relation of the channel as ˆ 1 + n2 , y = Hp xp + Hs xs + Hn

(5.5)

88

Chapter 5. Asymptotic Sum-Rate Analysis T

T

where xp  [xp1 , . . . , xpK ]T ∈ CKM is the vector of signals transmitted by all T T PU’s, xs  [xs1 , . . . , xsK ]T ∈ CKM is the vector of signals transmitted by all SU’s, T T T n1  [n11 , . . . , n1J ] ∈ CJL is the vector of noises added at the RTs and n2  T T 2N [nT is the vector of noises added at the primary BS. The channel for 01 , n21 ] ∈ C the SUs may be described likewise. ˆ Now, For convenience, we define the instantaneous CSI as H  {Hp , Hs , H}. assuming that the channels are ergodic and the CSI is perfectly known at the corresponding receivers, the average mutual information in nats between xp and y is given by 1 I(y; xp ) = (EH {h(y|H) − h(y|xp , H)}) 2 2 1 ˆ = − Ey,H ln Exp ,xs ,n1 {e−y−Hp xp −Hs xs −Hn1  } 2 2 1 ˆ + Ey,xp,H ln Exs ,n1{e−y−Hpxp −Hs xs −Hn1  }, 2

(5.6a)

(5.6b)

where the multiplier 1/2 is due to half-duplex regime.

5.2

Asymptotic Sum-Rate

We define the large-system limit (LSL) as the limit of M , L, and N going to infinity at fixed ratios M/N , L/N , and M/L. Here we present the asymptotic closed-form expression for the average mutual information in (5.6) in the LSL. The result is summarized in the following theorem. Theorem 5.1. In the LSL, the asymptotic average per-dimension sum-rate1 of the AF relay-assisted primary multi-access MIMO channel in the presence of secondary interference is given by

K K 1 /! p 1  1  0 s s // I(y; xp ) p p/ = I zk ; xk Ak + I zk ; xk Ask M 2M 2M k=1

k=1

N N 1 L ln (1 + ε0 ) + ln (1 + ε2 ) − ξ0 ε0 − ξ2 ε2 + 2M 2M 2 2M J J L 1 1  − ξ1j ε1j + ln (1 + ξ2 t j (ε1j + 1)) 2 j=1 2M j=1

=1

−

1 2M

K  k=1

0 / s I z k ; xsk /

1 N N ln (1 + ε0 ) − ln (1 + ε2 ) A sk − 2M 2M J

L   1   1 ξ2 ε2 + ξ ε + ξ0 ε0 + 2 2M 2 j=1 1j 1j 1 We operate with per-dimension mutual information since the actual mutual information tends to infinity in the LSL.

89

5.2. Asymptotic Sum-Rate

−

J L   1   ln 1 + ξ2 t j ε1j + 1 , 2M j=1

(5.7)

=1

    where Tj = ρ21j FH j Fj = diag(t1j , . . . , tLj ), and ξ0 , ξ0 , ε0 , ε0 , ξ1,j , ξ1,j , ε1,j , ε1,j ,   2 ∀j, ξ2 , ξ2 , ε2 and ε2 satisfy the following set of fixed-point equations

ξ0 =

ξ1j =

M

N 0 K

L 

=1

1,

(5.8a)

k=1 ε0 + 1

ξ2 t j , M (1 + ξ2 t j (ε1j + 1))

(5.8b)

N , L (ε2 + 1) N , ξ0 = M (ε0 + 1) ξ2 =

 ξ1j =

L 

=1

ξ2 = ε0 =

ξ2 t j    ,  M 1 + ξ2 t j ε1j + 1

=

K + ,  + , ρs01k Ezpk ,xpk xpk − xpk 2 + Ezsk ,xsk xsk − xsk 2 , M M

K + ,  ρs1jk + , Ezpk ,xpk xpk − xpk 2 + Ezsk ,xsk xsk − xsk 2 , M M

ε2

=

(5.8h)

k=1

J  L 

t j (ε1j + 1) , L (1 + ξ2 t j (ε1j + 1))

K  ρs

(5.8j)

+ , s Ezsk ,xsk xsk − x k 2 ,

(5.8k)

  t j ε1j + 1    . L 1 + ξ2 t j ε1j + 1

(5.8l)

M

K  ρs1jk

M

J  L  j=1 =1

(5.8i)

+ , s Ezsk ,xsk xsk − x k 2 ,

01k

k=1

(5.8g)

k=1

K  ρp1jk

k=1

ε1j =

(5.8f)

01k

j=1 =1

ε0

(5.8e)

K  ρp

k=1

ε2 =

(5.8d)

N , L (ε2 + 1)

k=1

ε1j =

(5.8c)

2 The fixed-point equations (5.8) are solved iteratively. The initial values are set to be either very small positive or very large positive. The fixed-point equations may have multiple solutions. Among those, one should choose those minimizing both entropy terms in (5.7). In physics literature, this phenomenon is referred to as phase transition [Nis01].

90

Chapter 5. Asymptotic Sum-Rate Analysis s

In (5.8), I(zpk ; xpk | Apk ), I(zsk ; xsk | Ask ) and I(z k ; xsk | A sk ) represent the mutual information between the corresponding inputs and the corresponding outputs of the following fixed MIMO channels ! (5.9a) zpk = Apk IM xpk + wkp , zsk = s z k

=

Ask IM xsk + wks , A sk IM xsk +

s w k ,

(5.9b) (5.9c)

with wkp , wks , w sk ∈ CM being random vectors with ZMCSCG unit variance entries. The corresponding channel conditional distributions are given by ! √ p p 2 p 1 p(zpk |xpk , Apk ) = M ezk − Ak xk  , (5.10a) π √ s s s 2 1 (5.10b) p(zsk |xsk , Ask ) = M ezk − Ak xk  , π ! √ p 1 p p 2 p p(z k |xpk , A pk ) = M ez k − A k xk  , (5.10c) π where Apk  ρp01k q˜0k + Ask  ρs01k q˜0k +

J  j=1 J 

ρp1jk q˜1jk ,

(5.11a)

ρs1jk q˜1jk ,

(5.11b)

 ρs1jk q˜1jk .

(5.11c)

j=1 s

 A k  ρs01k q˜0k +

J  j=1

s

The terms xpk , xsk  and x k  denote the minimum mean square error (MMSE) estimates3 [Ver98] of xpk and xsk associated with the fixed MIMO channels (5.9a), (5.9b) and (5.9c), respectively, and are given by   / !

xpk   E xpk /zpk , Apk , (5.13a)   / (5.13b)

xsk   E xsk /zsk , Ask ,   / s s (5.13c)

x k   E xsk /z k , A sk , 3 The minimum mean square error (MMSE) estimate of a r.v. x on the basis of observation z is a function of z, denoted by x, that minimizes the mean square error (MSE):   Ez,x x − x2 . (5.12)

91

5.3. Proof of Theorem 5.1

with the expectations taken over the posterior channel distributions p(xpk |zpk , Apk ), s p(xsk |zsk , Ask ) and p(xsk |z k , A sk ), respectively. The posterior distributions can be obtained from the prior distributions p(xpk ), p(xsk ) and the channel conditional distributions (5.10) through Bayes’ formula. Although the expression (5.7) looks cumbersome, it has a meaningful interpretation. The first eight terms represent the sum-rate contributed by the signal vectors xp and xs , as well as the noise vector n1 . The rest of the terms represent the entropy removed due to noise and secondary interference.

5.3

Proof of Theorem 5.1

In this section we prove the result from Theorem 5.1 via the large system analysis. The proof follows the procedure described in [WWN11] with some modifications. We start with the first term of (5.6) and define the partition function (cf. (2.40)) ˆ

2

Z(y, H)  Exp ,xs ,n1 {e−y−Hp xp −Hs xs −Hn1  },

(5.14)

and the free energy (cf. (2.41)) F  −Ey,H ln Z(y, H).

(5.15)

The main difficulty in evaluating the free energy (5.15) lies in averaging of the logarithm over y and H. In order to proceed we employ the replica method, widely used in the field of statistical physics for analysis of the macroscopic behavior of large systems consisting of microscopic particles. We proceed with rewriting (5.15) as follows ∂ ln Ey,H {Z u (y, H)} . (5.16) F = − lim u→0 ∂u Unfortunately, evaluation of Ey,H {Z u (y, H)} for real-valued u is prohibitive. In order to proceed, we have to invoke a common assumption in statistical physics that the replica trick is valid and we can evaluate Ey,H {Z u (y, H)} for integer-valued u and then generalize it to real values when u tends to zero. Although mathematical rigor of the replica trick is still an open problem, the have been evidences that some solutions obtained via replica analysis match the ones derived via systematic approaches [Nis01]. Therefore, we believe that the replica analysis is valid as a mathematical tool. To evaluate the inner expectation of (5.16) we replicate the corresponding variables, i.e., rewrite the under-log term in (5.16) as follows   u (a) 2 (a) (a) 1 ˆ e−y−Hp xp −Hs xs −Hn1  dy , Ey,H {Z u (y, H)} = EXp ,Xs ,H,N1 π 2N a=0 (5.17) (a) (a) where xp and xs are the ath replica vectors of signals of the PUs and the SUs (a) (a) (a) with corresponding distributions p(xp ) and p(xs ). Likewise, n1 is defined as

92

Chapter 5. Asymptotic Sum-Rate Analysis (0)

(0)

the ath replica vector of noise added at the relays. In this way, xp , xs and (0) n1 represent the true signal and noise vectors. All replicas here are assumed to be statistically independent. We group those and define the following vectors containing the corresponding true and replicated vectors , . . . , xp(u)T ]T , Xp  [x(0)T p

(5.18a)

[x(0)T , . . . , xs(u)T ]T , s (0)T (u)T [n1 , . . . , n1 ]T ,

(5.18b)

Xs  N1 

(5.18c)

so that Xp ∈ CKM(u+1) ,Xs ∈ CKM(u+1) and N1 ∈ CJL(u+1) . We further define the following sets of random vectors  (a) v0,k (a) v1jk



*

 

(a) v2,j



ρp01k p p(a) H x + M 01k k ρp1jk M

(a) Hp1jk xpk

ρ21j H21j Fj L



 

k=1

ρs1jk

(a)

Hs xs , M 1jk k 

+ K 

(a) ρs01k s H xs , M 01k k

(a)

(a)

v1jk + n1j

,

(5.19a) (5.19b) (5.19c)

where v0,k ∈ CN , v1jk ∈ CL and v2,j ∈ CN ; define also (a)

(a)

(a)

(a)

v0



K  k=1

(a)

v1j 

K  k=1

(a)

v2



J  j=1

(a)

v0,k ,

(5.20a)

(a)

(a)

v1jk + n1j ,

(5.20b)

(a)

v2,j ,

(5.20c)

and group them into vectors (0)T

V0  [v0 V1j 

(u)T T

] ∈ CN (u+1) ,

, . . . , v0

(0)T (u)T [v1j , . . . , v1j ]T (0)

V2  [v2

T

(u)

, . . . , v2

T

∈C

L(u+1)

(5.21a)

,

(5.21b)

]T ∈ CN (u+1) .

(5.21c)

By the central limit theorem applied with respect to random channel matrices, in the LSL, V0 , V1j and V2 become approximately Gaussian zero mean random

93

5.3. Proof of Theorem 5.1 vectors with covariance matrices Q0 =

K 

(Q0k ⊗ IN ) ∈ CN (u+1)×N (u+1) ,

(5.22a)

(Q1jk ⊗ IL ) + IL(u+1) ∈ CL(u+1)×L(u+1) ,

(5.22b)

(Q2j ⊗ IN ) ∈ CN (u+1)×N (u+1) ,

(5.22c)

k=1

Q1j =

K  k=1

Q2 =

J  j=1

having corresponding entries ρp01k p(b)H p(a) ρs01k s(b)H s(a) x x xk + xk , M k M k p ρ1jk p(b)H p(a) ρs1jk s(b)H s(a) x x = xk + xk , M k M k ρ21j (b)H H (a) v = Fj Fj v1j , L 1j

Qab 0k = Qab 1jk Qab 2j

(5.23a) (5.23b) (5.23c)

for a, b ∈ {0, 1, . . . , u}. The details of the derivations can be found in Appendix A.1. We denote Q0  {Q0k }∀k , Q1  {Q1jk }∀j,k , and Q2  {Q2j }∀j for sets of covariance matrices above. With the Gaussian approximations (5.22), the expectation over the replicated vectors may then be rewritten as  (u)  (5.24) Ey,H {Z u (y, H)} = EQ0 ,Q2 eG (Q0 ,Q2 ) + O (1) , where the expression in the exponent is given by  (u)

G

(Q0 , Q2 )  ln

1

π

E { 2N V0 ,V2

u

(a) 2

e−y01 −v0



(a) 2

e−y21 −v2



}dy,

(5.25)

a=0

and O (1) is a vanishing constant in the LSL. The integral in (5.25) can be computed by means of the Gaussian integral 4 , resulting in G(u) (Q0 , Q2 ) = − 2N ln(u + 1) 2 3 − ln det IN (u+1) + Q0 (Σ ⊗ IN ) 2 3 − ln det IN (u+1) + Q2 (Σ ⊗ IN ) ,

(5.26)

4 Let x ∈ CM be a complex random vector with positive definite covariance matrix C, the Gaussian integral is  H πM e−x Cx dx = . det C

94

Chapter 5. Asymptotic Sum-Rate Analysis

1 (u+1)×(u+1) where Σ  Iu+1 − u+1 1u+1 1T . For the details the reader is u+1 ∈ R referred to Appendix A.2. The expectation in (5.24) may then be rewritten as an integral over the probability measure of Q0 and Q2 , i.e.,



Ey,H {Z u (y, H)} ≈

eG

(u)

(Q0 ,Q2 )

dμ(u) (Q0 , Q2 ),

(5.27)

where μ(u) (Q0 , Q2 ) is given by ⎧ K u ⎨ 1 0 (b)H (a) (b)H (a) δ ρp01k xpk xpk + ρs01k xsk xsk − M Qab μ(u) (Q0 , Q2 ) = EXp ,Xs ,H,N1 0k ⎩ a,b=0 k=1 ⎞⎫ ⎛  H K  J K ⎬   (a) (b) (a) (a) ⎠ , × δ ⎝ρ21j v1jk + n1j FH v1jk + n1j − LQab j Fj 2j ⎭ j=1

k=1

k=1

(5.28) with δ(·) being the Dirac function. In contrast to [Tan02] and [GV05], where the measure was dependent only on the input symbols, (5.28) depends on channels as well, which complicates the calculations and we need to proceed with further steps. Being sums of independent random variables, Q0 and Q2 can be shown to satisfy the large deviations property [Ell06]. Therefore, Cram´er’s theorem [Ell06] states that in the LSL μ(u) (Q0 , Q2 ) is dominated by the exponent of the rate function I (u) (Q0 , Q2 ). In order to obtain I (u) (Q0 , Q2 ), we first define the following momentgenerating function5 (MGF) for Q0 and Q2  M

(u)

˜0 , Q ˜ 2 ) =EXp ,Xs ,H,N1 (Q ⎛ × exp ⎝

 exp

K 

k=1 K 

 H ˜ 0k ρp01k Xpk (Q

⊗

H

˜ 0k ⊗ IM )Xs + ρs01k Xsk (Q k

IM )Xpk

J  j=1

k=1

H ˜ V1j (Q2j

⎞⎫ ⎬ ⊗ Tj )V1j ⎠ , ⎭ (5.30)

(0)T

(u)T

where Xpk  [xpk , . . . , xpk ]T ∈ CM(u+1) , Xsk  [xsk , . . . , xsk ]T ∈ CM(u+1) L×L ˜ 0  {Q ˜ 0k }∀k , Q ˜ 1  {Q ˜ 1jk }∀j,k , and and Tj  ρ21j FH . We denote Q j Fj ∈ C ˜ 2  {Q ˜ 2j }∀j for sets of symmetric auxiliary matrices. Q 5 The

(0)T

(u)T

moment-generating function of a random variable x is   Mx (t) = E etx ,

wherever the expectation exists.

t ∈ R,

(5.29)

95

5.3. Proof of Theorem 5.1

˜0, Q ˜ 2 ), The rate function is given by the Legendre-Fenchel transform6 [Ell06] of ln M (u) (Q namely, ⎧ K J ⎨   ˜ 0k Q0k } + L ˜ 2j Q2j } tr{Q tr{Q I (u) (Q0 , Q2 ) = max M ˜ 0 ,Q ˜2 ⎩ Q j=1 k=1  K   p pH ˜ p − ln EX ,X ,H,N exp ρ X (Q0k ⊗ IM )X p

 × exp

s

1

k=1 K 

01k

k

k



H

˜ 0k ⊗ IM )Xs ρs01k Xsk (Q k

⎞⎫⎫ ⎛ J ⎬⎬  H ˜ V1j (Q2j ⊗ Tj )V1j ⎠ exp ⎝ ⎭⎭ j=1

k=1

(5.31) Then, by Varadhan’s theorem [Ell06], in the LSL ln Ey,H {Z u (y, H)} → max

Q0 ,Q2

  G(u) (Q0 , Q2 ) − I (u) (Q0 , Q2 ) .

(5.32)

Although the expression for G(u) (Q0 , Q2 ) is given in (5.26), the rate function I (u) (Q0 , Q2 ) still depends on the channel through vector V1j . Therefore, we need to compute the third term of (5.31) in a similar way as in (5.27), ˜0, Q ˜2) ln M (u) (Q





= ln EXpk ,Xsk

exp

⎧ ⎨

⎛

× EV1j  = ln

⎩

exp ⎝

K 

k=1 J 

(5.33a)  H

˜ 0k ⊗ IM )Xp + ρs XsH (Q ˜ 0k ⊗ IM )Xs ρp01k Xpk (Q 01k k k k

H ˜ V1j (Q2j

j=1

⎞⎫⎫ ⎬⎬ ⊗ Tj )V1j ⎠ ⎭⎭

0  (u) 1 (u) exp ln EV1j eG1 (Q1j ) dμ1 (Q1j ),

(5.33b) (5.33c)

where (u)

G1 (Q1j ) 

J 

H ˜ V1j (Q2j ⊗ Tj )V1j ,

j=1 6 Suppose

μ is the probability measure such that a function defined as  H c(y) = ln ex y dμ(x)

is finite for all y. The Legendre-Fenchel transform of c(t) is given by   I(z) = max zH y − c(y) . y

(5.34)

96

Chapter 5. Asymptotic Sum-Rate Analysis

and the probability measure is given by  (u) μ1 (Q1 )

=E

s Xp k ,Xk

 exp

K 

k=1

 H ˜ 0k ρp01k Xpk (Q

⊗

IM )Xpk

+

H ˜ 0k ρs01k Xsk (Q

⊗

IM )Xsk

⎫ J u K 1⎬ 0 (b)H (a) (b)H (a) × . δ ρp1jk xpk xpk + ρs1jk xsk xsk − M Qab 1jk ⎭ a,b=0 j=1 k=1

(5.35) The exponent expression (5.34) is computed through the Gaussian integral, similarly to (5.26). (u) G1 (Q1j )

=−

J 

4

0

˜ 2j ⊗ Tj ln det IL(u+1) − Q

j=1

K 1 

5 (Q1jk ⊗ IL ) + IL(u+1)

.

k=1

(5.36) The details of the derivation can be found in Appendix A.3. Next, in order to compute the expectation in (5.34) using large deviation property, we write the MGF for Q1 ,  (u) ˜ M1 (Q 1)

=EXpk ,Xsk

 exp

K 

k=1

 H ˜ 0k ρp01k Xpk (Q

⊗

IM )Xpk

+

H ˜ 0k ρs01k Xsk (Q

⊗

IM )Xsk

⎛ ⎞⎫ J  K ⎬  H ˜ 1jk ⊗ IM )Xp + ρs XsH (Q ˜ 1jk ⊗ IM )Xs ⎠ , × exp ⎝ ρp1jk Xpk (Q 1jk k k k ⎭ j=1 k=1

(5.37) as well as the rate function for ⎧ ⎨

(u)

I1 (Q1 ) = max ˜1 Q

⎩

M

(u) μ1 (Q1j ),

J  K  j=1 k=1

⎫ ⎬ ˜ 1jk Q1jk } − ln M (u) (Q ˜1 ) . tr{Q 1 ⎭

(5.38)

Again, by Varadhan’s theorem we have that in the LSL  ln

(u)

eG1

(Q1j )

  (u) (u) (u) dμ1 (Q1j ) → max G1 (Q1j ) − I1 (Q1j ) . Q1j

(5.39)

Next, we plug all the terms calculated above to (5.16) and define Q  {Q0 , Q1 , Q2 } ˜  {Q ˜ 0, Q ˜1, Q ˜ 2 }. The free energy can be then written as and Q F = lim

u→0

∂ ˜ min max T (u) (Q, Q), ∂u Q Q˜

(5.40)

97

5.3. Proof of Theorem 5.1 ˜ is given by where T (u) (Q, Q) 4 T

(u)

˜ =2N ln(u + 1) + ln det IN (u+1) + (Q, Q)

K 

5 Q0k Σ ⊗ IN

k=1

⎡ + ln det ⎣IN (u+1) +

J 

⎤

Q2j Σ ⊗ IN ⎦ + M

j=1

+L

J 

˜ 2j Q2j } + M tr{Q

j=1

+

J 

0

˜ 1jk Q1jk } tr{Q

˜ 2j ⊗ Tj ln det IL(u+1) − Q

j=1



− ln EXp ,Xs

 exp

˜ 0k Q0k } tr{Q

k=1 J  K  j=1 k=1

4

K 

K 1 

5 (Q1jk ⊗ IL ) + IL(u+1)

k=1 K  k=1

H ˜ 0k ρp01k Xpk (Q

⊗

IM )Xpk

 +

H ˜ 0k ρs01k Xsk (Q

⊗

IM )Xsk

⎛ ⎞⎫ J  K ⎬  H ˜ 1jk ⊗ IM )Xp + ρs XsH (Q ˜ 1jk ⊗ IM )Xs⎠ . × exp ⎝ ρp1jk Xpk (Q 1jk k k k ⎭ j=1 k=1

(5.41) The first two terms in the above expression constitute G(u) (Q0 , Q2 ) in (5.26), the fourth and the fifth term come from (5.31), the sixth term comes from (5.38), the (u) seventh term is the expression for G1 (Q1j ) in (5.36), and the last term is the (u) ˜ logarithm of the MGF M1 (Q1 ) in (5.37). In this way, using large deviations, the averaging over the MIMO channels in (5.6) is turned into optimization problem (5.40). It is difficult to solve (5.40) directly, however we may proceed with a common assumption of replica symmetry (2.49). We rewrite covariance matrices together with the auxiliary matrices as follows Q0k = q0k 1u+1 1T u+1 + (p0k − q0k )Iu+1 , ˜ Q0k = q˜0k 1u+1 1T p0k − q˜0k )Iu+1 , u+1 + (˜ Q2j = q2j 1u+1 1T u+1 + (p2j − q2j )Iu+1 , ˜ 2j = q˜2j 1u+1 1T Q p2j − q˜2j )Iu+1 , u+1 + (˜

(5.42)

Q1jk = q1jk 1u+1 1T u+1 + (p1jk − q1jk )Iu+1 , ˜ 1jk = q˜1jk 1u+1 1T + (˜ Q p1jk − q˜1jk )Iu+1 . u+1 Here we have assumed that permutations of the replica indices do not affect the physics of the system being artificially introduced for convenience of computing the expectation. Although this step lacks rigorous justification and there might be

98

Chapter 5. Asymptotic Sum-Rate Analysis

cases where replica symmetry breaks, it is worth emphasizing that this assumption is quite common in statistical physics [Dot01] and information theory [Tan02]. Some results obtained under the replica symmetry assumption are shown to match those obtained via systematic approaches [Nis01]. Moreover, numerical simulations also show that the results derived in this chapter are accurate and reliable. With the replica symmetry assumption, the last term of (5.41) is simplified to   u √ u u √  K p(a)H  p(b) p p(a)H p(a)  Ap Ap − (Ap xk k xk k xk k −Bk )xk (u) ˜ a=0 a=0 b=0 ln EXp ,Xs e ln M (Q1 ) = − 1

k=1 u 

×e

√

(a)H

Ask xsk

a=0

u  b=0

√

(b)

Ask xsk

−

u 

(a)H

(Ask −Bks )xsk

a=0

(a)

xsk

 ,

(5.43) J J where Apk  ρp01k q˜0k + j=1 ρp1jk q˜1jk , Ask  ρs01k q˜0k + j=1 ρs1jk q˜1jk , Bkp  ρp01k p˜0k +  J p ˜1jk and Bks  ρs01k p˜0k + Jj=1 ρs1jk p˜1jk . Next, we introduce two auxilj=1 ρ1jk p iary random variables zpk , zsk ∈ CM and perform the Hubbard-Stratonovich transform7 [Hub59] to (5.43) in order to decouple the quadratic terms, so that we obtain (u)





H   ! p p 1 p p p 2 xp k Bk xk p − A x  E exp −z e xk k k k πM k=1 (  !

)u

! H H H × Expk exp zpk Apk xpk + Apk xpk zpk − (Apk − Bkp )xpk xpk dzpk

˜1) = − ln M1 (Q

K 

ln



   √ H 1 −zsk − Ask xsk 2 xsk Bks xsk s e E e xk πM k=1 (  H√ )u

√ s sH s s s s sH s s s × Exsk ezk Ak xk + Ak xk zk −(Ak −Bk )xk xk dzsk . −

K 

ln

Under the replica symmetry, the third term of (5.41) may be simplified to 4 5   K K   ln det IN (u+1) + Q0k Σ ⊗ IN = u ln det IN + (p0k − q0k )IN , k=1

7 For

(5.45)

k=1

and, similarly, the forth term of (5.41) is simplified to ⎡ ⎤ ⎞ ⎛ J J   ln det ⎣IN (u+1) + Q2j Σ ⊗ IN ⎦ = u ln det ⎝IN + (p2j − q2j )IN ⎠ , j=1

(5.44)

(5.46)

j=1

a complex random vector x ∈ CM , the Hubbard-Stratonovich transform is given by  H H H H 1 ex x = M e−(a a−a x−x a) da, π

where a is an auxiliary vector of the same size as x.

99

5.3. Proof of Theorem 5.1

as shown in Appendix A.4. The fifth, sixth and seventh terms of (5.41) are easily shown to be K 

M

˜ 0k Q0k } = M tr{Q

k=1

L

J 

J  K 

(u + 1)(˜ p0k p0k + u˜ q0k q0k ),

(5.47a)

k=1

˜ 2j Q2j } = L tr{Q

j=1

M

K 

J 

(u + 1)(˜ p2j p2j + u˜ q2j q2j ),

(5.47b)

j=1

˜ 1jk Q1jk } = M tr{Q

j=1 k=1

J  K 

(u + 1)(˜ p1jk p1jk + u˜ q1jk q1jk ).

(5.47c)

j=1 k=1

Finally, the eighth term of (5.41) can be evaluated as follows (vide Appendix A.5). 4 5 K J 0 1   ˜ 2j ⊗ Tj ln det IL(u+1) − Q (Q1jk ⊗ IL ) + IL(u+1) j=1

=u

J 

4

ln det IL − (˜ p2j − q˜2j )Tj

j=1

+

k=1



J 

K 

5 (p1jk − q1jk )IL + IL

k=1





ln det IL − (˜ p2j + u˜ q2j )Tj

j=1

K 

 (p1jk + uq1jk )IL + IL

,

(5.48)

k=1

and hence, with the replica symmetry assumption, (5.41) is simplified to ˜ , q, q ˜) T (u) (p, p

4

=2N ln(u + 1) + u ln det IN +

K 

5 (p0k − q0k )IN

k=1

⎡ + u ln det ⎣IN +

J 

⎤

(p2j − q2j )IN ⎦ + M

j=1

+L

J 

+u

J 

4

+

J 

−

k=1

4



ln det IL − (˜ p2j + u˜ q2j )Tj

j=1 K 



ln det IL − (˜ p2j − q˜2j )Tj

j=1

ln

1 πM



(u + 1)(˜ p0k p0k + u˜ q0k q0k )

k=1

(u + 1)(˜ p2j p2j + u˜ q2j q2j ) + M

j=1

K 

J  K 

(u + 1)(˜ p1jk p1jk + u˜ q1jk q1jk )

j=1 k=1 K 

5

(p1jk − q1jk )IL + IL

k=1 K  k=1

(p1jk + uq1jk )IL + IL

 √ p p 2 pH p p  p −z − Ak xk  xk Bk xk e Expk e k

5

100

Chapter 5. Asymptotic Sum-Rate Analysis (  H√ )u

√ H p H p p zp Ap xp + Ap xp zk −(Ap −Bkp )xp xk k k k k k k k p × Exk e dzk

   √ H 1 −zsk − Ask xsk 2 xsk Bks xsk s − ln e Exk e πM k=1 (  H√ )u

√ H H zsk Ask xsk + Ask xsk zsk −(Ask −Bks )xsk xsk s s × Exk e dzk . K 

(5.49)

In order to evaluate (5.40), we have to find the optimal values for the parameters in (5.49), and then take the partial derivative with respect to u in the limit of u → 0. We find the saddle point by setting all partial derivatives of (5.40) to zero (vide Appendix A.6). From the derivations, we find that p˜0k = p˜1jk = p˜2j = 0, ∀k, j, and hence Bkp = Bks = 0, ∀k. Furthermore, we define the following set of parameters ξ0k  q˜0k ,

(5.50a)

ξ1jk  q˜1jk ,

(5.50b)

ξ2j  q˜2j ,

(5.50c)

ε0k  p0k − q0k ,

(5.50d)

ε1jk  p1jk − q1jk ,

(5.50e)

ε2j  p2j − q2j ,

(5.50f) (5.50g)

and show that at the saddle point these parameters must satisfy a set of fixed-point equations (

ξ0k = M

N K  k=1

ξ1jk =

L 

),

(5.51a)

(ε0k ) + 1

ξ2j t j ) , (K  M 1 + ξ2j t j (ε1jk ) + 1

=1

(5.51b)

k=1

(

ξ2j = L

N K 

k=1  ξ0k =

( M

 = ξ1jk

=1

(5.51c)

(ε2j ) + 1 N

K  k=1

L 

),

),

(5.51d)

(ε0k ) + 1

 ξ2j t j ) , (K    M 1 + ξ2j t j (ε1jk ) + 1 k=1

(5.51e)

101

5.3. Proof of Theorem 5.1  ξ2j =

( L

N K 

k=1

),

(ε2j )

(5.51f)

+1

+ , ρs + , ρp01k Ezpk ,xpk xpk − xpk 2 + 01k Ezsk ,xsk xsk − xsk 2 , M M s ρp1jk + p , ρ + , 1jk Ezpk ,xpk xk − xpk 2 + Ezsk ,xsk xsk − xsk 2 , = M M ) (K  t (ε ) + 1 L

j 1jk  k=1 = (K ) , 

=1 L 1 + ξ t (ε ) + 1 2j j 1jk

ε0k = ε1jk

ε2j

(5.51g) (5.51h)

(5.51i)

k=1

ε0k ε1jk ε2j

+ , ρs = 01k Ezsk ,xsk xsk − xsk 2 , M + , ρs1jk Ezsk ,xsk xsk − xsk 2 , = M ) (K   t j (ε1jk ) + 1 L  k=1 = ) , (K  

=1 L 1 + ξ  t (ε1jk ) + 1 2j j

(5.51j) (5.51k)

(5.51l)

k=1

xpk ,

xpk 

xpk 

and are the MMSE estimates associated with the fixed MIMO where channels (5.9). The detailed derivation can be found in Appendix A.6. Finally, by letting u → 0, we evaluate the free energy

K !  p p p s s s F= I(zk ; xk | Ak ) + I(zk ; xk | Ak ) + 2N k=1



+ N ln 1 +

K 

⎛

 ε0k

+ N ln ⎝1 +

−M

ξ0k ε0k − L

k=1

+

J L  

=1 j=1

4

⎞ ε2j ⎠

j=1

k=1 K 

J 

J  j=1

ln 1 + ξ2j t j

ξ2j ε2j − M 

J  K 

ξ1jk ε1jk

j=1 k=1 K 

5

ε1jk + 1

,

(5.52)

k=1

where I(zpk ; xpk | Apk ) is the mutual information between zpk and xpk for some fixed MIMO channel ! (5.53) zpk = Apk IM xpk + wkp , with wkp ∈ CM being a random vector with ZMCSCG unit variance entries. Upon the observation of the channel output zpk , the estimate minimizing mean squared

102

Chapter 5. Asymptotic Sum-Rate Analysis

error (MSE) is found as

xpk 

 !  / p/ p = E xk yk , Apk ,

(5.54)

Where the expectation is taken over the posteriori distribution of the channel (5.53), viz., p(xpk |ykp , Apk ), which can be found from the channel conditional distribution, given by ! √ p p 2 p 1 p(zpk |xpk , Apk ) = M ezk − Ak xk  , (5.55) π and prior distribution p(xpk ) through the Bayes’ formula. Mutual information I(zsk ; xsk | Ask ) is defined likewise for the corresponding fixed channel (5.9b). Now, in order to compute the second term of the mutual information expression (5.6) we have to repeat the replica analysis, in contrast to [WWN11], where the corresponding term was easier to evaluate due to the absence of the interfering SUs (cf. equations (15) and (19) therein). We follow the same procedure as before and define the new partition function 2

Z  (y, xp , H)  Exs {e−y−Hs xp −Hs xp  },

(5.56)

and the new free energy F  = −Ey,xp ,H ln Z  (y, xp , H) ∂ ln Ey,xp ,H {Z u (y, xp , H)}. = − lim u→0 ∂u

(5.57a) (5.57b)

Since we are not averaging the inner part over xp any longer, the vector does not need to be replicated and, alike the previous case, here we do not need to define the vector Xp for the computation of the under-log term in (5.57). Hence, Ey,xp ,H {Z u (y, xp , H)}   u 2 1 −y−Hp xp −Hs x(a) s  dy e . = Exp ,Xs ,H π N a=0

(5.58)

Afterwards, we proceed with exactly the same steps as before up to the point where replica symmetry is introduced. Since xp is not replicated, we pass on the Hubbard-Stratonovich transform for the first term of (5.43), which is no longer required due to the absence of the quadratic term corresponding to xp . As shown in Appendix A.7, the free energy in (5.57) is evaluated as F =

K  k=1

s

I(z k ; xsk | 

+ N ln 1 +

A sk ) + 2N K  k=1

 ε0k

⎛ + N ln ⎝1 +

J  j=1

⎞ ε2j ⎠

103

5.3. Proof of Theorem 5.1

−M

K 

 ξ0k ε0k − L

k=1

+

J L  

J  j=1



J  K 

  ξ2j ε2j − M



 ln 1 + ξ2j t j

=1 j=1

 ξ1jk ε1jk

j=1 k=1 K 



ε1jk + 1

,

(5.59)

k=1 s

   where ξ0k , ξ1jk , ξ2j , ε0k , ε1jk , ε2j and A k are defined similarly to those in (5.52), s s s and I(z k ; xk | A k ) is the mutual information between the input and the output of a fixed MIMO channel s

s

A sk IM xpk + w k ,

z k =

(5.60)

Finally, the mutual information (5.6) is evaluated as I(y; xp ) F − F = M 2M

K ! 1  = I(zpk ; xpk | Apk ) + I(zsk ; xsk | Ask ) 2M k=1 ⎛ ⎞   K J   N N + ln 1 + ln ⎝1 + ε0k + ε2j ⎠ 2M 2M j=1

(5.61a)

k=1

J J K L  1  ξ2j ε2j − ξ1jk ε1jk 2M 2 j=1 j=1 k=1 k=1 4 5 K J L  1  + ln 1 + ξ2j t j ε1jk + 1 2M j=1

−

1 2

K 

ξ0k ε0k −

=1

−

1 2M

K 

k=1

s

k=1



−

s

I(z k ; x k | A sk )

N ln 1 + 2M

K  k=1

⎛

 ε0k

−

J 

⎞

N ln ⎝1 + ε2j ⎠ 2M j=1

K J J K L    1    1   ξ0k ε0k + ξ2j ε2j + ξ1jk ε1jk 2 2M j=1 2 j=1 k=1 k=1 K   L J  1    − ln 1 + ξ2j t j ε1jk + 1 , 2M j=1

+

=1

which results in (5.7).

k=1

(5.61b)

104

5.4

Chapter 5. Asymptotic Sum-Rate Analysis

Applications

In this section we apply the result from Theorem 5.1 to two particular communication scenarios. Namely, we discuss the case of Gaussian channel inputs in Section 5.4.1, and the case of QPSK signal constellations in Section 5.4.2. Both cases are later examined with Monte-Carlo simulations in Section 5.5. Since we consider an uncorrelated MIMO system, each of the fixed MIMO channels [cf., (5.9)] reduces to a set of scalar channels of the form zkp =

! Apk xpk + wkp ,

(5.62)

J where Apk = ρp01k ξ0 + j=1 ρp1jk ξ1j and wkp ∼ CN (0, 1). Channels (5.9b) and (5.9c) can be treated in the same way.

5.4.1

Gaussian Channel Inputs

For the case of Gaussian channel inputs xsk , the per-dimension mutual information terms can be directly computed as 0 / I zpk ; xpk /

Apk

1 = ln (1 + Apk ) ,

M

(5.63)

Moreover, as shown in Appendix A.6, the expressions for the MMSE estimates can be reduced to

xpk  =

Apk zpk . 1 + Apk

(5.64)

and the mean-squared error terms are then computed as

ε0 =

K K   ρp01k ρs01k + , p 1 + Ak 1 + Ask

k=1

ε1j =

K  ρp1jk k=1

ε0 =

K  k=1

ε1j =

K  k=1

(5.65a)

k=1

1 + Apk

+

1 + A sk

(5.65b)

k=1

ρs01k , 1 + A sk ρs1jk

K  ρs1jk , 1 + Ask

.

(5.65c)

(5.65d)

5.5. Numerical Illustration

5.4.2

105

Quadrature Phase-Shift Keying Channel Inputs

For the QPSK constellation, the prior distribution is

j 1 1 p p p (xk ) = δ xk = ± √ ± √ . 4 2 2

(5.66)

Therefore, the output of the MMSE detector is given by the following expression (vide [LT06]) ! !



1 j p p p p p 2Ak Re{zk } + √ tanh 2Ak Im{zk } , (5.67)

xk  = √ tanh 2 2 and the MSE terms are then given by



 K !  2 1 p p p − z2 tanh Ak + Ak z e ρ01k 1 − √ dz ε0 = 2π k=1  K 1 z2

0  1 + ρs01k 1 − √ tanh Ask + Ask z e− 2 dz , 2π k=1



 K !  1 z2 ε1j = tanh Apk + Apk z e− 2 dz ρp1jk 1 − √ 2π k=1  K 0 1 z2

 1 tanh Ask + Ask z e− 2 dz , ρs1jk 1 − √ + 2π k=1  K 0 1 z2

 1 s  s s  tanh A k + A k z e− 2 dz , ρ01k 1 − √ ε0 = 2π k=1  K 0 1 z2

 1 s  s s  tanh A k + A k z e− 2 dz , ρ1jk 1 − √ ε1j = 2π k=1

(5.68a)

(5.68b)

(5.68c)

(5.68d)

where the integration with respect to z ∈ R is done numerically from −∞ to ∞. It is worth emphasizing that when solving the system of fixed-point equations, one should pick only those solutions that minimize the free energies (5.15) and (5.57). Using the connection between the mutual information and the MMSE established by Guo et al. in [GSV05], we can numerically compute the per-dimension mutual information terms in (5.7), according to 1 0

 ! 

I zpk ; xpk | Apk  z2 1 = 2 Apk − √ ln cosh Apk + Apk e− 2 dz . (5.69) M 2π

5.5

Numerical Illustration

In this section we verify the result stated in Theorem 5.1 via Monte-Carlo simulation. We consider several popular scenarios and employ the results obtained

106

Chapter 5. Asymptotic Sum-Rate Analysis ρp0

PU ρp1

ρ2 BS

RT

Figure 5.2: Simulation set-up: single user, single relay and single base station. The user is moving around the BS along the circle trajectory.

above. We start from the AF relay MIMO channel, where there are no secondary users. Then we proceed with the communication scenario in the presence of interference, where relay terminal is switched off and the secondary user interfers the primary transmission. Finally, we consider the relay-assisted interference channel and compute the achievable rate region for this channel.

5.5.1

Amplify-and-Forward Relay Channel

First, we note that our results degenerate to those reported in [WWN11] for the case of uncorrelated MIMO channel with Gaussian inputs and in the absence of the SUs. Therefore, we simulate the corresponding setup by switching off the SUs and setting K = J = 1, and M = L = N . The resulting topology is the AF relay MIMO channel, as depicted in Figure 5.2. We fix SNRs ρ0 = 0 dB and ρ2 = 20 dB and vary ρ1 . This corresponds to the case, where the PU is moving around the primary BS along the circle trajectory, so that ρ1 is changing, whereas the RT remains still. Since we consider uncorrelated case, to fulfil the power constraint (5.2) the 1 IL . forwarding matrix is set to G = √1+ρ 1 Figure 5.3 illustrates the dependence of the average mutual information per transmit antenna on SNR ρ1 . From the figure we see that the analytic approximation matches well the numerical results for both Gaussian and QPSK inputs. The curve obtained from (5.7) for QPSK inputs coincides with the one reported in [WWN11]. Moreover, we observe the expected deviation of the mutual information curve for the Gaussian case from the one reflecting the QPSK-modulated transmission for large ρ1 .

107

5.5. Numerical Illustration

2

1.8

Analytic approximation, Gaussian inputs Simulation, M = N = L = 4, Gaussian inputs Analytic approximation, QPSK inputs Simulation, M = N = L = 4, QPSK inputs Result from [WWN11], QPSK inputs

I(y; xp )/M , bpcu

1.6

1.4

1.2

1

0.8

0.6

0.4 −10

−5

0

5

10

15

ρ1 , dB

Figure 5.3: Average mutual information per transmit antenna versus SNR of the first-hop link ρ1 . Solid curves denote analytic results, markers denote simulated values.

5.5.2

Communication in the Presence of Interference

For the present scenario, the simulation set-up is depicted in Figure 5.4. The PU communicates with the primary BS, while the SU is interfering to this communication. The SNRs of PU - BS and SU - BS links are denoted by ρp1 and ρs1 , respectively. Figure 5.5 illustrates the simulation results of this set-up. We observe that the simulations corroborate the validity of the analytical results from Theorem 5.1. The gap between the theoretical curve and the simulations for the QPSK constellation at high SNR is due to the fact that in that region LSL is more critical; that is, one needs more antennas to approach the asymptotic result. In addition, we verify that the result for the Gaussian signaling coincides with that presented in [MS03]. We also note that the QPSK constellation at both the PU and the SU leads to better performance in terms of mutual information. This is due to the fact that the most severe interference is indeed Gaussian. The corresponding result was first reported by Cahn in [Cah71]. Figure 5.6 illustrates this behavior for the case of Gaussian and QPSK signaling. We directly see that for the PU it is more beneficial, in terms of mutual information, to switch to the Gaussian signaling. On the other hand, Gaussian signaling when used by the SU creates more severe

108

Chapter 5. Asymptotic Sum-Rate Analysis

ρp1

ρp2

PU

BS1

RT

Figure 5.4: Simulation set-up: single primary user, single secondary user and single base primary station. Both the primary and the secondary user are approaching the primary BS.

2

Analytic approximation, both QPSK Simulation, M = N = 4, both QPSK Analytic approximation, both Gaussian Simulation, M = N = 4, both Gaussian Result from [MS03], both Gaussian

I(y; xs )/M s , bits/cu

1.5

1

0.5

0 −10

−5

0

5

10

15

ρs1 , ρi1 , dB

Figure 5.5: Average mutual information per transmit antenna versus SNRs ρp1 and ρs1 . Solid curves denote analytic results, markers denote simulated values.

109

5.5. Numerical Illustration

4.5 4 3.5

PU : Gauss − SU : Gauss PU : QPSK − SU : Gauss PU : Gauss − SU : QPSK PU : QPSK − SU : QPSK PU : Gauss − SU : Off PU : QPSK − SU : Off

I(y; xp )/M , bits/cu

3 2.5 2 1.5 1 0.5 0 −10

−5

0

ρp1 , ρs1 , dB

5

10

15

Figure 5.6: Average mutual information per transmit antenna versus SNRs ρp1 and ρs1 for different combinations of PU’s and SU’s signalings.

interference. Hence, the primary system might successfully handle several SUs if they use discrete constellations, whereas, when they use Gaussian signals, the system might be down even from a single SU.

5.5.3

Relay-Assisted Interference MIMO channel

Next, we add an RT performing AF relaying and assisting both transmissions, as shown in Figure 5.7. We vary ρp1 and ρs1 at the same time, whereas the rest of SNRs remain fixed. This corresponds to the situation, where the PU and the SU are moving around the primary BS along the circle at the same speed and the RT remains still. The simulation results for this set-up with Gaussian channel inputs are depicted in Figure 5.8. We see that the analytic results match the numerical values averaged over 108 Monte-Carlo iterations. When increasing numbers of antennas, numerical results approach closer to the LSL approximation. Slight deviation in the low-SNR region can be removed be further increasing the number of Monte-Carlo iterations. Being computationally simple (fixed-point equations are solved iteratively within 20-30 iterations), the presented solution allows us to characterize the achievable rate region of this specific interference channel. For instance, we consider both networks having a single transmitter-receiver pair, so that we may find out what is the max-

110

Chapter 5. Asymptotic Sum-Rate Analysis

ρp0 ρp1

ρ2

PU RT ρs1

BS1

ρs0

BS2 SU

Figure 5.7: Simulation set-up: single primary user, single secondary user, single relay and single base station. Both PU and SU are moving along a circle centered at corresponding BS.

1 Analytic approximation Simulation, M = L = N = 4 Simulation, M = L = N = 2 0.9

0.8

0.7

0.6

0.5

0.4

−10

−5

0

ρp1 , ρs1 dB

5

10

15

Figure 5.8: Average mutual information per transmit antenna versus SNRs ρp1 and ρs1 . Solid curve denotes analytic result, markers denote simulated values.

111

5.6. Summary

Shadowing

PU

BS1

0.5

RT

Shadowing

SU

BS2 1

Figure 5.9: Simulation set-up: single primary user, single secondary user, single relay and single base station. The RT is moving along the line from the users towards the BSs.

imum achievable rate for the SU under the condition that the PU’s data rate is limited from below by a certain threshold. Here, the PU, the SU and the corresponding BSs are on the fixed positions, while the mobile RT moves along the line in the middle, starting from the transmitters towards the receivers. The distances between PU, SU and their corresponding BSs are set to 1, whereas the distance between PU and SU is set to 0.5. Powers of all terminals are constrained to 10 dB and the signaling is limited to Gaussian. We employ the path-loss model with path-loss exponent α = 3. The direct links are attenuated by 5 dB to achieve some shadowing effect. By taking the convex hull of the individual achievable rate pairs for different link SNRs we may compute the achievable rate region as shown in Figure 5.10. The degenerate case, where the RT is absent may be directly obtained via the result derived in [MS03]. From the figure we see that as the RT moves closer to the BSs, AF relaying becomes more beneficial in terms of achievable rates. When the RT comes close to the BSs, the AF strategy outperforms the scheme with no relay at all, despite the fact that AF requires two time-slots, whereas the latter does not.

5.6

Summary

In this chapter we analyzed the asymptotic sum-rate of the primary network within the amplify-and-forward relay-assisted MIMO cognitive scenario. By means of the replica method we obtained a closed-form expression for the sum-rate as a func-

112

Chapter 5. Asymptotic Sum-Rate Analysis

1

R2 /M , bpcu

0.8

0.6

0.4

0.2

0 0

dP U,RT = 0.25 dP U,RT = 0.45 dP U,RT = 0.65 dP U,RT = 0.85 No relay 0.2

0.4

0.6 R1 /M , bpcu

0.8

1

Figure 5.10: Achievable rate region for the relay-assisted multi-access interference MIMO channel. Red curves with different markers show the rate regions for different positions of the relay. Black curve shows the rate region obtained in absence of the relay, so that only one time-slot is needed for the transmission.

tion of parameters satisfying a set of fixed-point equations. The solution is easy to implement and despite the fact that it is obtained for the large system limit, the solution approximates well the mutual information of systems with moderate numbers of antennas. The result holds for both Gaussian and discrete channel inputs, which was verified through Monte-Carlo simulations. Moreover, with help of the obtained closed-form expression, we attested that the worst-case interference is Gaussian. Finally, the simplicity of the obtained solution allowed us to characterize the achievable rate region of the relay-assisted interference channel.

Chapter 6

Conclusions and Future Work In this chapter we summarize the work done within the present thesis and draw some directions of the future work.

6.1

Conclusions

In this thesis we have investigated the cognitive radio networks from different points of view. Motivated by cognitive ad hoc networks, we considered multi-hop relaying within the secondary network, which is known to be helpful in terms of network throughput, transmission reliability and coverage extension. A useful example of a multi-hop network is the line network, which assumes that there has been formed a multi-hop route from the source to the destination and the signal is transmitted from one node to another within the route, such that the signal’s trajectory has a line topology. Within the cognitive radio scenario, the secondary network is allowed to access the spectrum owned by the primary network under condition that it does not harm the primary’s performance. One way of limiting the effect of interference from the secondary network is to incorporate an interference threshold at the primary network. In Chapter 3 we considered a single-source multi-hop cognitive radio network of the line topology. Previously reported results in [DGA04] and [KC09] were extended to the presence of the interference, which makes the optimization problem non-convex and hence difficult to deal with. We have shown that the optimal throughput of the secondary network is achieved when the powers are allocated in such way that the link capacities of each hop are equal and the constraint of the least tolerant primary user is satisfied with equality. We also proposed highand low-SNR approximated solutions of less complexity as well as distributed and limited-feedback solutions for the case where centralized coordination is not possible. The solutions are shown to perform well at the corresponding SNR regions. In Chapter 4 we investigated retransmission strategies for a multi-source multihop line secondary network in terms of reliability of users’ transmissions. Our 113

114

Chapter 6. Conclusions and Future Work

analysis was based on that reported in [XS10] and [RUFLV12] and extended to the case of multi-hop networks. As a benchmark we used a time division duplex transmission, where each user has an individual time-slot for his transmission. The two relaying strategies we compared to the benchmark are binary network-coded relaying and diversity network-coded relaying. Namely, each user mixes his own packet with the packet received from the user downstream in the corresponding finite field. We have shown that the DNC strategy has the best performance in terms of the achievable diversity gains for users provided that the global encoding vectors of each node are linearly independent. We further developed a heuristic algorithm for scheduling of the DNC-based transmission within the line network. The algorithm is shown to provide the optimal time-slot allocation such that the minimum diversity gain among the users is maximized; the allocation is shown to coincide with that obtained by the exhaustive search. Finally, in Chapter 5 we considered the relay-assisted multiple access cognitive radio scenario, where a set of primary users as well as a set of secondary users communicate to their corresponding base stations with help of a set of non-regenerative relay terminals. By means of the replica method we derived the asymptotic sumrate of the primary network in the large system limit, i.e., where all the terminals in the network are equipped with multiple antennas, the number of antennas at each terminal grows without bound and the ratios of receive and transmit antennas of each MIMO channel remains constant. The previously reported results in [WW10] and [WWN11] were extended to the presence of the secondary interference. The obtained result allows to characterize the achievable rate-region for the resulting relay-assisted interference channel. By the specific example we showed that the AF-based relaying may outperform the direct transmission when the relay terminal is situated close to the destination.

6.2

Future Work

Here we list some future research directions related to the work carried out in this thesis. • Power allocation for the multi-hop secondary networks of arbitrary topology. In Chapter 3 we considered line topology of the secondary network. A natural extension would be to investigate the optimal power allocation for general multi-hop topology, similarly to the problem in [KC09]. • Diversity-multiplexing trade-off analysis of the multi-user multi-hop cognitive radio network. Except gains in the transmission reliability, the DNC relaying scheme from Chapter 4 may offer the gains in throughput. To investigate the interplay of these two performance metrics within the DNC scheme, we can carry out the DMT analysis, similarly to [WX11]. • Asymptotic sum-rate for relay-assisted multiple-access interference channel with correlation. In Chapter 5 we investigated the sum-rate of the primary

6.2. Future Work

115

network in presence of secondary interference for the case of the Gaussian and QPSK signalling. We considered uncorrelated signals for both users and interferers. An extension of this problem may include the presence of correlation between antennas, similarly to the problem investigated in [WW10] and [MS03].

Appendix A

Omitted Steps from the Proof of Theorem 5.1 A.1

Derivation of Covariance Matrices Q0 , Q1j and Q2

The covariance matrix of vector V0 is given by Q0 = EHp0 ,Hs0 {V0 V0H } (a) (b)H

p = [Qab 0 ] = [EH0 ,Hs0 {v0 v0

}].

(A.1)

N ×N of the covariance matrix Q0 can be written as The (a, b) sub-matrix Qab 0 ∈C p Qab 0 =EH0 ,Hs0 {v0 v0

(a) (b)H

=EHp0 ,Hs0 {

K  k=1

ρp01k M

}

(A.2a) (a)

Hp01k xpk

(b)H

xpk

H

Hp01k +

K  k=1

(a) (a)H H ρs01k s H01k xsk xsk Hs01k } M

(A.2b) =

K  k=1

+

ρp01k M

K  k=1

=

EHp0 {

ρs01k M

M  m=1

EHs0 {

(a)

(b)∗

H

hp01km xpkm xpkm hp01km }

M 

(a)

(a)∗

H

hs01km xskm xskm hs01km }

(A.2c)

m=1

K M  H ρp01k  p(b)∗ p(a) xkm xkm EHp0 {hp01km hp01km } M m=1 78 9 6 k=1

+

K  k=1

=IN

ρs01k M

M 

(b)∗

H

(a)

xskm xskm EHs0 {hs01km hp01km } 6 78 9 m=1 =IN

117

(A.2d)

118

Appendix A. Omitted Steps from the Proof of Theorem 5.1

=

K K   ρp01k p(b)H p(a) ρs01k s(b)H s(a) xk xk IN + x xk IN . M M k k=1

(A.2e)

k=1

Defining a matrix Q0k ∈ C(u+1)×(u+1) with elements Qab 0k 

ρp01k p(b)H p(a) ρs01k s(b)H s(a) x x xk + xk M k M k

(A.3)

we can conveniently write the covariance matrix Q0 ∈ CN (u+1)×N (u+1) as Q0 =

K 

(Q0k ⊗ IN )

(A.4)

k=1

The covariance matrix of vector V1j is computed in a similar way. H Q1j = EHp1 ,Hs1 ,n1j {V1j V1j } (a) (b)H

p = [Qab 1j ] = [EH1 ,Hs1 ,n1j {v1j v1j }], ∀a, b ∈ {0, . . . , u}.

(A.5)

L×L of the covariance matrix Q1j can be written as The (a, b) sub-matrix Qab 1j ∈ C p Qab 1j =EH1 ,Hs1 ,n1j {v1j v1j }

(a) (b)H

=EHp1 ,Hs1 ,n1j { +

=

K  ρs1jk

k=1

+

=

M

+

k=1

M  m=1

EHp1 ,Hs1 {

M 

H

Hp1jk

H

Hs1jk + n1j nH 1j } (a)

(b)∗

(A.6b)

H

hp1jkm xpkm xpkm hp1jkm }

M 

(a)

(a)∗

H

hs1jkm xskm xskm hs1jkm } + En1j {n1j nH 1j }

(A.6c)

m=1

(b)∗

(a)

H

=IL

K M  ρs1jk 

K 

(a)H

(b)H

xpk

xp xp E p s {hp1jkm hp1jkm } M m=1 km km 6 H1 ,H1 78 9

k=1

=

EHp1 ,Hs1 {

M

k=1

M

(a)

K  ρs1jk

k=1 K  ρp1jk

k=1

(a)

Hp1jk xpk

Hs1jk xsk xsk

M

k=1 K  ρp1jk

K  ρp1jk

(A.6a)

ρp1jk M

(b)∗

H

(a)

xs xs E p s {hs1jkm hp1jkm } +IL M m=1 km km 6 H1 ,H1 78 9

(A.6d)

=IL (b)H

xpk

(a)

xpk

IL +

K  k=1

ρs1jk M

(b)H

xsk

(a)

xsk IL + 1(a = b)IL , ∀a, b ∈ {0, . . . , u}, (A.6e)

A.1. Derivation of Covariance Matrices Q0 , Q1j and Q2

119

with 1()˙ being the indicator function. Defining a matrix Q1jk with elements Qab 1jk 

ρp1jk M

(b)H

xpk

(a)

xpk

+

ρs1jk s(b)H s(a) x xk M k

(A.7)

we can write covariance matrix for V1j as Q1j =

K 

(Q1jk ⊗ IL ) + IL(u+1) ,

(A.8)

k=1

where Q1j ∈ CL(u+1)×L(u+1) . Similarly, the covariance matrix for V2 is given by Q2 = EH2 {V2 V2H } (a) (b)H

= [Qab 2 ] = [EH2 {v2 v2

}].

(A.9)

We compute Q2 element-wise for all a, b ∈ {0, . . . , u} as follows. (a) (b)H

} Qab 2 = EH2 {v2 v2 ⎧⎛ ⎞⎛ ⎞H ⎫  ⎪ ⎪ J  J ⎨  ⎬  ρ21j ρ21j (a) ⎠ ⎝ (b) ⎠ ⎝ H21j Fj v1j H21j Fj v1j = EH2 ⎪ ⎪ L L ⎩ j=1 ⎭ j=1 ⎧ ⎫ J  J ⎨ ⎬ ρ21j (a) (b)H H = EH2 H21j Fj v1j v1j  Fj  HH 21j  ⎩ ⎭ L j=1 j  =1 ⎧ ⎫ J ⎨ ⎬ ρ21j (a) (b)H H = EH2 H21j Fj v1j v1j FH j H21j ⎩ ⎭ L j=1 ⎧ ⎫ J L L ⎨ ⎬   ρ21j (a) (b)∗ ∗ H = EH2 h21jl fjl v1j

v1j fj

h21jl ⎩ ⎭ L j=1

=

J  j=1

(A.10b)

(A.10c)

(A.10d)

(A.10e)

l=1

J L  ρ21j  j=1

=

=1

(A.10a)

+ , (b)∗ ∗ (a) v1j fj

fj v1j EH2 h21jl hH 21jl L 6 78 9 l=1

(A.10f)

=IN

ρ21j (b)H H (a) v Fj Fj v1j IN , L 1j

(A.10g) (A.10h)

N ×N where Qab . 2 ∈C Again, defining a matrix Q2j with elements

Qab 2j 

ρ21j (b)H H (a) v Fj Fj v1j . L 1j

(A.11)

120

Appendix A. Omitted Steps from the Proof of Theorem 5.1

then we can write the covariance matrix Q2 ∈ CN (u+1)×N (u+1) for the vector V2 as J  (Q2j ⊗ IN ). (A.12) Q2 = j=1

A.2

Evaluation of G(u) (Q0 , Q2 )

Using the property of the Frobenius norm, we rewrite G(u) (Q0 , Q2 ) as follows.  (u)

G

(Q0 , Q2 ) = − 2N ln π + ln

u

(a) 2

e−y01 −v0



}dy01

a=0



u

EV2 {

+ ln

EV0 {

(a) 2

e−y21 −v2



}dy21

(A.13a)

a=0

= − 2N ln π + ln EV0 {e + ln EV2 {e

−

−

u  a=0 u  a=0

(a)

v0 2

(a)

v2 2

 e  e

H −(u+1)y01 y01 +2Re{(

H −(u+1)y21 y21 +2Re{(

u  a=0 u  a=0

(a)

v0 )H y01 }

dy01 }

(a)

v2 )H y21 }

dy21 } (A.13b) (A.13c)

Then, we apply the Gaussian integral with linear term1 and get G(u) (Q0 , Q2 )



= −2N ln π + ln EV0 



 exp −

u  a=0

u 



 (a)H (a) v0 v0

πN exp (u + 1)N u  u 



(A.14a)

 1 (a)H (b) ( v0 v0 ) u + 1 a=0 b=0  u  u 

1 v2 v2 ) πN u+1 ( a=0 b=0 e (A.14b) N (u + 1) a=0 ⎧ ⎛ ⎞⎫ ⎪ ⎪ u u  u ⎨ ⎬   u 1 ⎜ (a)H (a) (a)H (b) ⎟ = −2N ln(u + 1) + ln EV0 exp ⎝− v0 v0 + v0 v0 ⎠ ⎪ ⎪ u + 1 a=0 u + 1 a=0 ⎩ ⎭

+ ln EV2

exp −

(a)H

(b)

(a)H (a) v2 v2

b=0 b =a

1 For a complex random vector x ∈ CM with positive definite covariance matrix C the Gaussian integral with linear term reads as  H H π M bH C−1 b . e e−x Cx+2 Re{b x} dx = det C

(u)

A.3. Evaluation of G1 (Q1j ) ⎧ ⎪ ⎨

121

⎞⎫ ⎪ ⎬ 1 u ⎜ (a)H (a) (a)H (b) ⎟ + ln EV2 exp ⎝− v2 v2 + v2 v2 ⎠ ⎪ ⎪ u + 1 a=0 u + 1 a=0 ⎩ ⎭ b=0 ⎛

u 

u  u 

(A.14c)

b =a

    H H = −2N ln(u + 1) + EV0 e−V0 (Σ⊗IN )V0 + EV2 e−V2 (Σ⊗IN )V2  H H −1 1 = −2N ln(u + 1) + e−V0 (Σ⊗IN )V0 N (u+1) e−V0 Q0 V0 dV0 π det Q0  H H −1 1 + e−V2 (Σ⊗IN )V2 N (u+1) e−V2 Q2 V2 dV2 π det Q2

(A.14d)

(A.14e) (A.14f)

Finally, by applying the Gaussian integral once again we get 2 3 G(u) (Q0 , Q2 ) = − 2N ln(u + 1) − ln det IN (u+1) + Q0 (Σ ⊗ IN ) 2 3 − ln det IN (u+1) + Q2 (Σ ⊗ IN ) , where Σ  Iu+1 −

A.3

1 T u+1 1u+1 1u+1

(A.15a) (A.15b)

∈ R(u+1)×(u+1) .

(u)

Evaluation of G1 (Q1j ) (u)

We compute G1 (Q1j ) via the Gaussian integral as follows. (u)

G1 (Q1j ) = =

J 

 H  ˜ ln EV1j eV1j (Q2j ⊗Tj )V1j

(A.16a)

j=1 J 

 ln

j=1 J 

H

˜

eV1j (Q2j ⊗Tj )V1j

−1 H 1 e−V1j Q1j V1j dV1j (A.16b) π L(u+1) det Q1j



−1 H 1 ˜ e−V1j (Q1j −Q2j ⊗Tj )V1j dV1j (A.16c) L(u+1) det Q π 1j j=1 K 4 5 J 1  0  ˜ 2j ⊗ Tj =− ln det IL(u+1) − Q (Q1jk ⊗ IL ) + IL(u+1) .

=

ln

j=1

k=1

(A.16d)

A.4

Derivation of the Third Term of (5.41)

The third term of (A.65a) can be rewritten in the following way. 4 5 K  ln det IN (u+1) + Q0k Σ ⊗ IN k=1

(A.17a)

122

Appendix A. Omitted Steps from the Proof of Theorem 5.1 ⎡ = ln det ⎣IN (u+1) +

k=1

⎛ = ln det ⎝IN (u+1) + ⎡ = ln det ⎣

 p0k ··· q0k  ⎡ .. . . .. ⎣ . . .

K 

K 

⎡ ⎣

1 − u+1

u u+1 (p0k −q0k )

.. .

K

k=1 (p0k −q0k )IN

K

.. .

..

k=1 (p0k −q0k )IN

We define the following matrix. ⎡ K u U⎣

1+ u+1

k=1 (p0k −q0k )

..

1 − u+1

− u+1

⎤

⎦ ⊗ IN ⎦

u+1

⎤

u u+1 (p0k −q0k )

K

k=1 (p0k −q0k )IN

..

u IN + u+1

···

···

.

1 − u+1

u k=1 (p0k −q0k ) ··· 1+ u+1

(A.17b) ⎞

⎦ ⊗ IN ⎠ (A.17c)

.. .

.

1 − u+1

.

..

K .

.. . u

··· ···

⎤

1 ··· − u+1 (p0k −q0k )

.. ···

1 ··· − u+1

.. .1

1 − u+1 (p0k −q0k ) ···

k=1 u IN + u+1

q0k ··· p0k

u u+1

K .

⎤ ⎦.

(A.17d)

k=1 (p0k −q0k )IN

⎤

K

k=1 (p0k −q0k )

⎦.

..

K.

(A.18)

k=1 (p0k −q0k )

When subtracting the first line of U from all other lines, we have ⎡ u u+1

K 

1 − u+1

K 

(p0k − q0k ) (p0k − q0k ) ⎢1 + k=1 k=1 ⎢ ⎢ K K ⎢   ⎢ 1+ (p0k − q0k ) −1 − (p0k − q0k ) ⎢ k=1 k=1 U=⎢ ⎢ .. .. ⎢ ⎢ . . ⎢ K ⎣  −1 − (p0k − q0k ) 0 k=1

···

1 − u+1

··· ..

.

···

K 

⎤

(p0k − q0k ) ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ .. ⎥ ⎥ . ⎥ K ⎦  1+ (p0k − q0k ) k=1

k=1

(A.19) so that the desired term becomes 4 5 K  ln det IN (u+1) + Q0k Σ ⊗ IN = ln det (U ⊗ IN ) .

(A.20)

k=1

Then, according to the matrix determinant lemma [Har08], the log-det term can be rewritten as follows 5  4 K  (p0k − q0k )IN ⊗ Iu ln det IN + k=1

4  5 K K u  1  T (p0k − q0k )IN − − (p0k − q0k )IN ⊗ 1u + ln det IN + u+1 u+1 k=1 k=1 ⎡4 ⎤ 4 5−1  5⎞ K K   × ⎣ IN + −IN − (p0k − q0k )IN ⊗ Iu ⎦ (p0k − q0k )IN ⊗ 1u ⎠ 

k=1

k=1

123

A.5. Derivation of the Eighth Term of (5.41) 4 = ln det

IN +

K 

5



(p0k − q0k )IN ⊗ Iu

k=1



K

u  (p0k − q0k )IN u+1 k=1 54 5−1 K K   (p0k − q0k )IN IN + (p0k − q0k )IN

+ ln det IN + 4 −u − 4

1 u+1

k=1

K 

× −IN −

(p0k − q0k )IN

k=1

4

IN +

= ln det

k=1

5

K 

5 (p0k − q0k )IN ⊗ Iu

k=1

 = u ln det IN +



K 

(A.21a)

 (p0k − q0k )IN

.

(A.21b)

k=1

A.5

Derivation of the Eighth Term of (5.41)

For the forth line of (A.65a) we use similar approach J 

4

˜ 2j ⊗ Tj ln det IL(u+1) − Q

j=1

=

J  j=1

=

0

J  j=1

K 1 



4 p˜2j Tj ··· .. . . ln det IL(u+1) − . .

5 (Q1jk ⊗ IL ) + IL(u+1)

k=1 q˜2j Tj

5

.. .

q˜2j Tj ··· p˜2j Tj

4 W ··· 11 .. . . ln det . .

W12

.. .

K  k=1

4 p1jk IL ··· .. . . . .

(A.22a)

q1jk IL

.. .

5

 + IL

q1jk IL ··· p1jk IL

(A.22b)

5 ,

(A.22c)

W12 ··· W11

where  W11 = IL −

4 p˜2j Tj

 W12 = − q˜2j Tj

4

K 

k=1 K  k=1

5 p1jk + 1 + u˜ q2j Tj 5

K 

 q1jk

IL ,

(A.23a)

k=1

p1jk + 1 − (u − 1)˜ q2j Tj

K  k=1

q1jk + p˜2j Tj

K 

 q1jk

IL .

k=1

(A.23b)

124

Appendix A. Omitted Steps from the Proof of Theorem 5.1

Again we subtract the first row from the other rows of the matrix and have ⎤ ⎡ W11 W12 ··· W12 J ⎥ ⎢ W12 − W11 W11 − W12 · · · 0L  ⎥ ⎢ (A.24) ln det ⎢ ⎥, . . . . .. .. .. .. ⎦ ⎣ j=1 0L · · · W11 − W12 W12 − W11 where

4 W11 − W12 = IL − (˜ p2j − q˜2j )Tj

K 

5 (p1jk − q1jk )IL + IL .

(A.25)

k=1

By the matrix determinant lemma the log-det term can be simplified as follows. 4 K 5  J   ln det IL − (˜ p2j − q˜2j )Tj (p1jk − q1jk )IL + IL ⊗ Iu j=1

+

J 





ln det IL −

j=1

4 

4

− − q˜2j Tj

K 

p˜2j Tj 5

+ p˜2j Tj

k=1 K 

5

p1jk + 1 + u˜ q2j Tj

k=1



q1jk IL

 q1jk

IL

K 

q1jk IL

k=1

5 ⊗

K  k=1

p1jk + 1 IL + (u − 1)˜ q2j Tj

k=1 K 

4

1T u

k=1

⎡ ⎤ 4K 5−1  × ⎣ IL − (˜ p2j − q˜2j )Tj (p1jk − q1jk )IL + IL ⊗ Iu ⎦ k=1

4 

4

× − IL + (˜ p2j − q˜2j )Tj =u

J 

5 (p1jk − q1jk )IL + IL

k=1



4

ln det IL − (˜ p2j − q˜2j )Tj

j=1

+

K 

J 





ln det IL −

j=1



−u q˜2j Tj

4K 

4 p˜2j Tj 5

K  k=1

K 

J  j=1

5

ln det IL − (˜ p2j − q˜2j )Tj

K 

 q1jk

IL

k=1

p1jk + 1 IL + (u − 1)˜ q2j Tj

K 

q1jk IL + p˜2j Tj

k=1

4

(A.26a)

(p1jk − q1jk )IL + IL

p1jk + 1 + u˜ q2j Tj

k=1

=u

⊗ 1u 5

k=1



5

K  k=1

 q1jk IL

k=1

5 (p1jk − q1jk )IL + IL

K 

(A.26b)

125

A.6. Saddle-Point Conditions

+

J 



4

ln det IL − (˜ p2j + u˜ p2j )Tj

j=1

− u(˜ p2j + u˜ q2j )Tj =u

J 



J 

 q1jk IL

(A.26c)

k=1

4

K 

5 (p1jk − q1jk )IL + IL

k=1





ln det IL − (˜ p2j + u˜ q2j )Tj

j=1

A.6

p1jk + 1 IL

k=1

ln det IL − (˜ p2j − q˜2j )Tj

j=1

+

K 

5

K 

K 

 (p1jk + uq1jk )IL + IL

.

(A.26d)

k=1

Saddle-Point Conditions

In order to find the saddle point we take all the partial derivatives of (5.49), set those to zero and let u → 0. First, we take the derivative with respect to p2j ⎛ ⎡ ⎤ J  ∂ ∂ ⎝u ln det ⎣IN + ˜ , q, q ˜) = T (u) (p, p (p2j − q2j )IN ⎦ ∂p2j ∂p2j j=1 ⎞ J  +L (u + 1)(˜ p2j p2j + u˜ q2j q2j )⎠ (A.27a) j=1

=

1+

J

uN

j=1 (p2j

− q2j )

+ L(u + 1)˜ p2j .

(A.27b)

By setting the derivative to zero we get p˜2j = − which for u → 0 implies

uN ., K L(u + 1) 1 + k=1 (p2j − q2j ) p˜2j = 0, ∀j.

(A.28)

(A.29)

Next, we take the derivative with respect to q2j and set it to zero. ∂ uN ˜ , q, q ˜) = − T (u) (p, p + L(u + 1)u˜ q2j = 0. J ∂q2j 1 + j=1 (p2j − q2j )

(A.30)

Thereby, we obtain q˜2j =

N .. K L(u + 1) 1 + k=1 (p2j − q2j )

(A.31)

126

Appendix A. Omitted Steps from the Proof of Theorem 5.1

By setting u to zero, we get q˜2j =

L

N

- J

j=1 (p2j − q2j ) + 1

..

(A.32)

Similarly, by taking partial derivatives with respect to p0k , q0k , p1jk , q1jk , p˜2j and q˜2j we have p˜0k = 0, ∀k, p˜1jk = 0, ∀j, k, M

N

) , ∀k,

(A.33c)

q˜2j t j ) , ∀k, (K  M 1 + q˜2j t j (p1jk − q1jk ) + 1

(A.33d)

(

q˜0k =

K  k=1

q˜1jk =

L 

=1

(p0k − q0k ) + 1

4

p2j

q2j

(A.33a) (A.33b)

k=1

5 K tr{Tj }  = p1jk + 1 , L k=1 . ⎡ -K L t j  k=1 (p1jk + uq1jk ) + 1 ⎣ = L

=1 - . ⎤ K t j (p − q ) + 1 1jk 1jk k=1 .1 ⎦ . - − 0 K L 1 + q˜2j t j (p − q ) + 1 1jk 1jk k=1

(A.33e)

(A.33f)

Now, what is left to do is to compute the partial derivatives with respect to p˜0k , q˜0k , p˜0k and q˜0k , which are present in the last two terms of (5.49). First we take the derivative with respect to p˜0k . ∂ ˜ , q, q ˜ ) = M (u + 1)p0k T (u) (p, p ∂ p˜0k −

∂ ∂ p˜0k EXp {· · ·}   √  H √ p p(a) u p p(a) H u p p p(a) p(a) u EXp e( a=0 Ak xk ) a=0 Ak xk − a=0 (Ak −Bk )xk xk

−

∂ ∂ p˜0k EXs {· · ·}   √  . (A.34) u √ s s(a) u (a)H s(a) s(a) )H ( u Ask xk Ak xk − a=0 (Ask −Bks )xsk xk a=0 a=0 EXs e

Taking into account that (A.33a) and (A.33b) imply Bkp = Bks = 0, ∀k, we may

A.6. Saddle-Point Conditions

127

simplify the derivative as follows. ∂ ˜ , q, q ˜ ) = M (u + 1)p0k T (u) (p, p ∂ p˜0k   √ p p 2 ( p H √ p p √ p p H p p H p p )u  H p Expk ρp01k xpk xpk π1M e−zk − Ak xk  ezk Ak xk + Ak xk zk −xk Ak xk dzpk −   pH √ p p √ p pH p pH p p .u √ p p 2  p 1 Expk e−zk − Ak xk  dzpk Expk ezk Ak xk + Ak xk zk −xk Ak xk πM  ( )   −zs −√As xs 2 zsH √As xs +√As xsH zs −xsH As xs u s s sH s 1 k k k k k k k k k k k k s dzk Exk ρ01k xk xk πM e e −    . √ √ √ u  s s s 2 sH s s s sH s sH s s 1 Exsk e−zk − Ak xk  dzsk Exsk ezk Ak xk + Ak xk zk −xk Ak xk πM (A.35) Then, we set u → 0 and notice that p˜0k = 0, thereby having   √ p p 2 p pH p  1 −zp p Ak x k  k− p ρ E x x e dz xk 01k k k k πM ∂   √ p p 2 ˜ , q, q ˜ ) = M p0k − T (u) (p, p p p −z − A x  1 ∂ p˜0k k k k Expk dzk πM e   √  s s s 2 H Exsk ρs01k xsk xsk π1M e−zk − Ak xk  dzsk   √ − (A.36a) 1 −zsk − Ask xsk 2 s Exsk e dz M k π + , + , = M p0k − Expk ρp01k xpk 2 − Exsk ρs01k xsk 2 . (A.36b) 

Since

by setting

1 −zp −√Ap xp 2 p k k e k dzk = 1, πM

∂ (u) ˜ , q, q ˜) (p, p ∂ p˜0k T

p0k =

(A.37)

= 0 we get

+ , ρs + , ρp01k Expk xpk 2 + 01k Exsk xsk 2 . M M

(A.38)

Now we take the derivative with respect to q˜0k ∂ ˜ , q, q ˜ ) = M u(u + 1)q0k T (u) (p, p ∂ q˜0k ∂ {· · ·} ∂ q˜0k Exk p   √  − √ (a)H (a) u (a)H p(a)  u u Ap xp Ap xp − a=0 (Ap −Bkp )xp xk a=0 a=0 k k k k k k Exkp e −

∂ {· · ·} ∂ q˜0k Exk s   √ . √ H  (a)H (a)  u u s(a) xs(a) Ask xsk Ask xsk − u (Ask −Bks )xk a=0 a=0 a=0 k Exks e

(A.39a) (A.39b)

(A.39c)

128

Appendix A. Omitted Steps from the Proof of Theorem 5.1

We first take the derivative from second term in the expression above, noticing that p˜0k = 0 and Bkp = Bks = 0   u  u u  p(a)H   (a) (a)H (a) ∂ p p p p E k {· · ·} = Expk ρ01k xk xk − xk xk ∂ q˜0k xp a=0 a=0 a=0  u √ u u √   p(a)H  p(a) p(a)H p(a) Ap Ap − Ap xk k xk k xk k xk a=0 a=0 (A.40a) × ea=0

=Expk

⎧ ⎪ ⎪ ⎨ ρp

01k πM

u 

(b)H

xpk

(a)

xpk

⎪ ⎪ a,b=0 ⎩ b =a ( H√ )   √ √ H p H p u p −zp − Ap xp 2 zp Ap xp + Ap xp zk −Ap xp xk k k k k k k k k k k × e dzk e ⎧ ⎪ ⎪ u ⎨ ρp  pH p u  √ p p(b) 2 p(b)H −zp zk zk Ak x k  01k k− p x e e =Exk k M ⎪ π ⎪ a,b=0 ⎩ b =a ⎫ ⎪ u ⎬ √ √ (a) (c) p(a) −zp p − Ap xp 2 −zp − Ap xp 2 k k k k k k × xk e e dzk . ⎪ ⎭ c=0

(A.40b)

(A.40c)

c =a =b

Assuming that replicas are identical, we proceed ρp ∂ EXp {· · ·} = 01k ∂ q˜0k πM ×π



pH p zk

euzk

(u−1)M

(  ! )2 / H u(u + 1)π 2M Expk xpk p zpk /xpk , Apk

(  ! )u−1 / p/ p p dzpk Exk p zk xk , Apk

(A.41a)

 H 0 / 1 ⎤2 Expk xpk p zpk /xpk , Apk ⎣  0 / 1 ⎦ =ρp01k π uM u(u + 1) e p/ p p p Exk p zk xk , Ak ( 

)u+1 ! / × Expk p zpk /xpk , Apk dzpk . (A.41b) 

⎡

H p uzp k zk

Using the fact that the minimum MSE estimator xpk  is given by  H 0 1 Expk xpk p zpk |xpk , Apk  0 1 ,

xpk  = Expk p zpk |xpk , Apk

(A.42)

129

A.6. Saddle-Point Conditions we rewrite the expression of the partial derivative as follows.

∂ EX {· · ·} =ρp01k π uM u(u + 1) ∂ q˜0k p (   ! )u+1 / H pH p dzpk . (A.43) × euzk zk xpk 2 Expk xpk p zpk /xpk , Apk Similarly, we obtain the derivative for the term EXs {· · ·} ∂ EX {· · ·} =ρs01k π uM u(u + 1) ∂ q˜0k s   H 0 / sH s × euzk zk xsk 2 Exsk xsk p zsk /xsk ,

Ask

1.u+1

dzsk .

(A.44)

Then, the whole derivative becomes ∂ ˜ , q, q ˜ ) = M u(u + 1)q0k T (u) (p, p ∂ q˜0k  0 1.u+1  pH p ρp01k π uM u(u + 1) euzk zk xpk 2 Expk p zpk |xpk , Apk dzpk −   pH √ p p √ p pH p p pH p .u √ p p 2  −zp Ak x k  1 k− p p e E ezk Ak xk + Ak xk zk −Ak xk xk E dzpk M x x π k k 2 +  ,3u+1 s  sH s dzk ρs01k π uM u(u + 1) euzk zk xsk 2 Exsk p zsk |xsk , Ask − .   sH √ s s √ s sH s s sH s .u √ s s 2  s −zk − Ak xk  zk Ak xk + Ak xk zk −Ak xk xk 1 s s s e E e dz E x x M k π k k (A.45) Setting the derivative to zero gives

∂ ˜ , q, q ˜ ) =M q0k − T (u) (p, p ∂ q˜0k

 96 ! 7 Expk p zpk |xpk , Apk dzpk   1 −zp −√Ap xp 2 p k k k p Exk e dzk πM 6 78 9

ρp01k



8

 √ p Ak =p zp k|

xpk 2

=1

√ =p(zsk | Ask ) 8  0 96 17  uzsH zs s 2 s s s uM s dzsk ρ01k π e k k xk  Exsk p zk |xk , Ak    − , 1 −zsk −√Ask xsk 2 s Exsk e dz k πM 6 78 9 =1

(A.46)

130

Appendix A. Omitted Steps from the Proof of Theorem 5.1

so that at u → 0 we have + , + , ∂ ˜ , q, q ˜ ) = M q0k − ρp01k Ezpk xpk 2 − ρs01k Ezsk xsk 2 = 0, T (u) (p, p ∂ q˜0k

(A.47)

And hence we obtain q0k =

+ , ρs + , ρp01k Ezpk xpk 2 + 01k Ezsk xsk 2 . M M

(A.48)

Next, since we have that  Ezpk ,xpk

(xpk

−

2

xpk )



  p =Ezp Exp {xp k } xk  k

k

8 96 7 + , + , =Expk xpk 2 − 2 Ezpk ,xpk {xpk xpk } +Ezpk xpk 2 + , + , =Expk xpk 2 − Ezpk xpk 2 ,

(A.49a) (A.49b)

the MSE can be expressed as follows. + , + ,1 ρs  + , + , ρp 0 p0k − q0k = 01k Expk xpk 2 − Ezpk xpk 2 + 01k Exsk xsk 2 − Ezsk xsk 2 M M (A.50a) p s + , + , ρ ρ = 01k ρp01k Ezpk ,xpk xpk − xpk 2 + 01k Ezsk ,xsk xsk − xsk 2 , M M (A.50b) With exactly the same steps we compute the other MSE term p1jk − q1jk =

ρp1jk

+ , ρs1jk + , Ezpk ,xpk xpk − xpk 2 + Ezsk ,xsk xsk − xsk 2 . M M

(A.51)

Finally, the parameters have to satisfy the following system of fixed-point equations p˜0k = p˜1jk = p˜2j = 0, ∀j, k, N - ., q˜0k = K M k=1 (p0k − q0k ) + 1 q˜1jk =

L 

=1

q˜2j = p0k − q0k =

L

q˜ t 0 .1 , - 2j j K M 1 + q˜2j t j k=1 (p1jk − q1jk ) + 1

- J

N

j=1 (p2j − q2j ) + 1

.,

(A.52a) (A.52b)

(A.52c)

(A.52d)

+ , ρs + , ρp01k p ρ01k Ezpk ,xpk xpk − xpk 2 + 01k Ezsk ,xsk xsk − xsk 2 , M M (A.52e)

131

A.6. Saddle-Point Conditions

p2j − q2j

ρp1jk

+ , ρs1jk + , Ezpk ,xpk xpk − xpk 2 + Ezsk ,xsk xsk − xsk 2 , (A.52f) M M . - K L t j  k=1 (p1jk − q1jk ) + 1 - 0 .1 . = (A.52g) K ˜2j t j

=1 L 1 + q k=1 (p1jk − q1jk ) + 1

p1jk − q1jk =

which directly leads to (5.51). Note, though, that p0k , p˜0k , q0k , q˜0k , p1jk , p˜1jk , q1jk , q˜1jk do not depend on k, as well as p2j , p˜2j , q2j , q˜2j do not depend on j. For ease of notation, we redefine the parameters ξ0 = q˜0k , ∀k, ξ1j = q˜1jk , ∀k,

(A.53a) (A.53b)

ξ2 = q˜2j , ∀j,

(A.53c)

ε0 =

K 

(p0k − q0k ) ,

(A.53d)

(p1jk − q1jk ) ,

(A.53e)

(p2j − q2j ) ,

(A.53f)

k=1

ε1j =

K  k=1

ε2 =

J  j=1

and for the case of Gaussian channel inputs xpk and xsk we find the MMSE estimators as follows.

xpk 

 =  =  =  =  =

!

p p xk |zk , Apk dxpk 1 0 p zpk |xpk , Apk p (xpk ) 1 0 xpk dxpk p p p zk | Ak 1 0 p zpk |xpk , Apk p (xpk ) dxpk xpk  p(zpk |xpk , Apk )p (xpk ) dxpk √ p p 2 H p Ak x k  1 x p 1 −zp k− k xk p πM e πM e xk  dxpk √ H p p −zp − Ap xp 2 1 x p xk k k k k e dxk πM e

H p H√ p p p p pH p −zk zk +2Re zk Ak xk −(Ap xp k +1)xk xk k e 2M p π    1 −zpkH zpk +2Re zpkH √Apk xpk −(Apk +1)xpkH xpk p dxk . dxk π 2M e

xpk p

(A.54a)

(A.54b)

(A.54c)

(A.54d)

(A.54e)

132

Appendix A. Omitted Steps from the Proof of Theorem 5.1

Using the Gaussian integral with linear term, we get

√ p p pH p Ak zk Ap k zk zk exp p p Ak +1 Ak +1

xpk  = H p p Ap k zk zk exp Ap +1

(A.55a)

k

=

Apk zpk . Apk + 1

(A.55b)

Similarly, we can find the MMSE estimator for xsk

xsk  =

Ask zsk . Ask + 1

(A.56a)

Then, the MSE term corresponding to the PUs in (A.53) become + , Ezpk ,xpk xpk − xpk 2   Apk zpk 2 p = Ezpk ,xpk xk − p  Ak + 1 

 Apk zpk ! p p p p = Expk ,wkp xk − p Ak xk + wk 2 Ak + 1   Apk xpk + Apk wkp 2 p = Expk ,wkp xk −  Apk + 1   xpk − Apk wkp 2  = Expk ,wkp  Apk + 1  H  H Expk ,wkp xpk xpk + Apk wkp wkp = 2 (Apk + 1)

(A.57a) (A.57b) (A.57c)

(A.57d) (A.57e)

(A.57f)

=IM

8 96 7   H H tr{Expk xpk xpk } + tr{Apk Ewkp wkp wkp } 6 78 9 =IM

= =

(Apk

2

+ 1)

M , Apk + 1

(A.57g)

and, similarly, for the SUs the MSE terms become + , Ezsk ,xsk xsk − xsk 2 =

M . Ask + 1

(A.58)

133

A.7. Evaluation of the Free Energy

A.7

Evaluation of the Free Energy

Now according to (5.40) we can compute the free energy by taking the derivative of (5.49) with respect to u and setting u → 0. We also note that p˜0k = 0, ∀k, as well as Bkp = Bks = 0, ∀k. Thus, we get ∂ (u) ˜ , q, q ˜) T (p, p ∂u ⎡ ⎤ 4 5 K J   2N = + N ln 1 + (p0k − q0k ) + N ln ⎣1 + (p2j − q2j )⎦ u+1 j=1 k=1 4K 5 K   +M (˜ p0k p0k + u˜ q0k q0k ) + (u + 1) q˜0k q0k k=1

k=1

⎡ ⎤ J J   + L ⎣ (˜ p2j p2j + u˜ q2j q2j ) + (u + 1) q˜2j q2j ⎦ j=1

j=1

⎤ ⎡ J  J  K K   (˜ p1jk p1jk + u˜ q1jk q1jk ) + q˜1jk q1jk ⎦ +M⎣ j=1 k=1

+

J  L 

4



ln 1 − (˜ p2j − q˜2j )t j

j=1 =1

j=1 k=1 K 

5

(p1jk − q1jk ) + 1

k=1

0 - 1 . K K J  L t q ˜ (p + uq ) + 1 + (˜ p + u˜ q ) q 

j 2j 1jk 2j 2j k=1 1jk k=1 1jk . - − K 1 − (˜ p2j + u˜ q2j )t j j=1 =1 k=1 (p1jk + uq1jk ) + 1  √    √ H p √ p pH p H p  uE p {··· } −zp − Ap xp 2 Ap zp xk + Ak xk zk −Ap xp xk k k k k k k k p p ln Exk e Exk e dzpk e xk K  −   √ p pH p √ p pH p p pH p .u √ p p 2  p Expk e Ak zk xk + Ak xk zk −Ak xk xk Expk e−zk − Ak xk  dzpk k=1  √   √ s s 2 √ H H H  uE s {··· } −zp Ak x k  Ask zsk xsk + Ask xsk zsk −Ask xsk xsk s− s s e ln E dzsk E e e xk K x x k k  − .   √ s sH s √ s sH s sH s s .u √ s s 2  s Exsk e Ak zk xk + Ak xk zk −xk Ak xk dzsk Exsk e−zk − Ak xk  k=1 (A.59) Then, by letting u → 0 and noticing that all p˜0k = p˜1jk = p˜2j = 0, ∀j, k we obtain the free energy ∂ (u) ˜ , q, q ˜) T (p, p ∂u 4 =2N + N ln 1 +

K  k=1

⎤ J  (p0k − q0k ) + N ln ⎣1 + (p2j − q2j )⎦ 5

⎡

j=1

134

Appendix A. Omitted Steps from the Proof of Theorem 5.1

+M

K 

q˜0k q0k + L

k=1

+

J  L 

−

j=1

4

q˜2j q2j + M 

ln 1 + q˜2j t j

j=1 =1 J  L 

J 

4 t j q˜2j

j=1 =1

q˜1jk q1jk

j=1 k=1 K 

5

(p1jk − q1jk ) + 1

k=1 K 

J  K 

5

p1jk + 1

k=1

 √    √ H p √ p pH p H p −zp − Ap xp 2 Ap zp xk + Ak xk zk −Ap xp xk k k k k k k k p e ln Exk e E dzpk K   −  √ p p 2 1 −zp Ak x k  k− p k=1 e dzpk πM E π M xk 6 78 9 =1  √    √ √ H H H  −zp − Ask xsk 2 Ask zsk xsk + Ask xsk zsk −Ask xsk xsk s s s ln Exk e Exk e dzsk K   − .   √ 1 −zsk − Ask xsk 2 s s k=1 πM E e dz x k πM k 6 78 9 

xp k

=1

(A.60) The last term of (A.60) is further computed as −

K   k=1

 √ p p 2 p 1 Expk e−zk − Ak xk  ln Expk M 78 9 6 6π √ p p =p(zk |

−

K   k=1

=−

k=1

−

K   k=1

=π M

  √ 1 −zsk − Ask xsk 2 s e E ln Exsk x k M 78 9 6 6π √ s s =p(zk |

K  

Ak )

Ak )

 √ √ p pH p p pH s  p pH p e Ak zk xk + Ak xk zk −Ak xk xk dzpk 78 9  √ p p 2 pH p p 1 πM

Exp

k

−z − k

e

A x  k k

ezk

z k

 √  √ H H H Ask zsk xsk + Ask xsk zsk −(Ask −Bks )xsk xsk dzsk e 78 9  √ s s 2 H s =π M

1 πM

Exs

k

−z − k

e

A x  k k

s zs k

ezk

!

! H p / / p/ p p/ p zp zk k M ln π + ln p zk Ak + ln e dzpk p zk Ak

1 1 0 / 0 / sH s p zsk / Ask M ln π + ln p zsk / Ask + ln ezk zk dzsk

⎛

(A.61a)

⎞ !

!

  !

⎜ ⎟ / / / H ⎜h zp / Ap − M ln π p zp / Ap dzp − zp zp p zp / Ap dzp ⎟ = k k k k k k k k k k⎠ ⎝ k=1 6 78 9 K 

=1

135

A.7. Evaluation of the Free Energy ⎛ +

K  k=1

1 Ask − M ln π

⎜ 0 s/ ⎜h zk / ⎝



0 / 1 p zsk / Ask dzsk − 6 78 9

⎞



0 / H zsk zsk p zsk /

1 ⎟ Ask dzsk ⎟ ⎠

=1

⎛

=

!

K  / ⎜ ⎜h zp / Ap − k k ⎝ k=1

⎛ +

K  ⎜ s ⎜h(zk | As ) − k ⎝ k=1

=

⎞ ⎤ J  ⎟ −M ρp01k ⎣q˜0k + q˜1jk ⎦⎟ ⎠

(A.61b)

⎡

M ln π − M 78 9 6  / √ p p/ p =−h zk xk , Ak

j=1

⎡

⎤

⎞

J  ⎟ M ln π − M −M ρs01k ⎣q˜0k + q˜1jk ⎦⎟ ⎠ 78 9 6  / j=1 √  =−h zsk /xsk , Ask

K 0  /! / I zpk ; xpk / Apk + I zsk ; xsk /

Ask

(A.61c)

1

.

(A.61d)

k=1

Since, in the LSL the following three terms converge to the diagonal elements of matrices Q0k , Q1jk and Q2j 4 t j

K 

5 p1jk + 1 → p2j ,

(A.62a)

k=1 ρp01k + ρs01k → p0k , ρp1jk + ρs1jk → p1jk ,

(A.62b) (A.62c)

we may write down the final expression for the free energy F=



 K K 1 0  /! / I zpk ; xpk / Apk + I zsk ; xsk / Ask + 2N k=1

k=1

4

+ N ln 1 +

K 

5

(p0k − q0k ) + N ln ⎣1 +

K 

(p0k − q0k ) q˜0k − L

k=1

+

J  L  j=1 =1

4 ln 1 + q˜2j t j

J 

⎤ (p2j − q2j )⎦

j=1

k=1

−M

⎡



J 

(p2j − q2j ) q˜2j − M

j=1 K  k=1

which results directly in (5.52).

(p1jk − q1jk ) + 1

J  K 

(p1jk − q1jk ) q˜1jk

j=1 k=1

5 ,

(A.63)

136

Appendix A. Omitted Steps from the Proof of Theorem 5.1

A.8

Evaluation of Second Term of the Mutual Information

We start from the point where the Hubbard-Stratonovich transform is applied to s the corresponding term, and an auxiliary random variable z k ∈ CM is introduced. All the previous steps are exactly the same as before with one exemption. Since the inner part of (5.56)is not averaged over xp , we do not replicate the vector. We p s notice that B  k = B  k = 0 and simplify the given term as follows.  K  pH p 2  p pH p u p p ln EXp ,Xs e(u+1) A k xk xk − a=0 (A k −B k )xk xk − k=1  √  s s(a) H u √ s s(a) u  (a)H s(a) ( u A k xk ) A k xk − a=0 (Ask −B  sk )xsk xk a=0 a=0

×e

=−

K 

ln E

xp k

k=1

(A.64a)

  H p H p (u+1)2 Ap xp xk −(u+1)(A p −B p )xp xk k k k k k e

  √  √  s s(a) u s(a) H u s(a)H s(a) ( u Ask xk ) A k xk − a=0 Ask xk xk a=0 a=0 × EXs e =−

K  k=1



× EXs × e− =−

ln Expk 1 πM

u

K 

   p pH p eu(u+1)A k xk xk



H  sH  s z k +z sk

e−z k (a)H

a=0

Ask xsk

 ln

(a)

s xk

s

dz k pH

p

eu(u+1)A k xk

k=1

1 × M π



H

e

(A.64b)

−zsk z sk

u a=0

E

(a)

s Ask xk

a=0

+

u

√

a=0

(a)H

Ask xsk

)z sk

 (A.64c) xp k

 xsk

√

u

1 −xpH xp p e k k dxk πM H

e

zsk

√

Ask xsk +

√

H

H

Ask xsk z sk −xsk Ask xsk



s

dz k

   √ 1 −z sk − Ask xsk 2 s e E xk π M det(IM − u(u + 1)A pk IM ) k=1 (  H√ √ s sH  s s sH s )u s s s s dz k × Exsk ez k A k xk + A k xk z k −A k xk xk ⎛ ⎞ K J   p  =M ln ⎝1 − u(u + 1)ρ01k (˜ q0k + q˜1jk )⎠ =−

K 

(A.64d)

ln

k=1

(A.64e)

j=1

  √ s s 2  s 1 × M Exsk e−z k − A k xk  π (  √ )u √ H H H s Ask zsk xsk + Ask xsk z sk −Ask xsk xsk × Exsk e dz k .

(A.64f)

137

A.8. Evaluation of Second Term of the Mutual Information Then, similarly to the previous case, we get ˜  , q , q ˜) T (u) (p , p

4

=2N ln(u + 1) + u ln det IN +

K 

5  (p0k − q0k )IN

k=1

⎡ + u ln det ⎣IN +

J 

⎤

 (p2j − q2j )IN ⎦ + M

j=1

+L

J 

+u

J 

+

J 



ln det IL −

k=1

−

 q˜2j )Tj



ln det IL −

K 

(˜ p2j

4

j=1

+M

k=1

4

j=1

  (u + 1)(˜ p0k p0k + u˜ q0k q0k )

J  K 

  (u + 1)(˜ p2j p2j + u˜ q2j q2j ) + M

j=1

K 

(˜ p2j

+

 u˜ q2j )Tj

j=1 k=1 K 

(p1jk k=1 K 

(p1jk

 ln ⎝1 − u(u + 1)ρp01k (˜ q0k +

J 

5

−

 q1jk )IL

+

 uq1jk )IL

+ IL 5

k=1

⎛

  (u + 1)(˜ p1jk p1jk + u˜ q1jk q1jk )

+ IL

⎞  q˜1jk )⎠

j=1



  √ 1 −zsk − Ask xsk 2 s e E x k πM k=1 (  √ )u √ H H H s Ask z sk xsk + Ask xsk z sk −A sk xsk xsk s dz k , × Exk e

−

K 

ln

(A.65a)

which can be further simplified to ˜  , q , q ˜) T (u) (p , p



=2N ln(u + 1) + uN ⎡ + ln ⎣1 +

J 

4 ln 1 + ⎤⎞

K 

5 (p0k

−

k=1

 ⎦⎠ (p2j − q2j ) + M (u + 1)

j=1

+ L(u + 1)

+u

j=1 =1

K 

  (˜ p0k p0k + u˜ q0k q0k )

k=1

J 

(˜ p2j p2j

+

  u˜ q2j q2j )

j=1 J  L 

 q0k )

4

 ln 1 − (˜ p2j − q˜2j )t j

+ M (u + 1) 

J  K 

  (˜ p1jk p1jk + u˜ q1jk q1jk )

j=1 k=1 K 

k=1

 (p1jk − q1jk )+1

5

138

Appendix A. Omitted Steps from the Proof of Theorem 5.1

+

J  L 

4 ln 1 −

j=1 =1

+M

K  k=1

−

K 

 (˜ p2j

+

 u˜ q2j )t j

K 

⎛

ln

k=1

J 

 uq1jk )

+

k=1

 ln ⎝1 − u(u + 1)ρp01k (˜ q0k +



5 (p1jk

+1

⎞  q˜1jk )⎠

j=1

 √ s s 2  s Exsk e−z k − A k xk 

 √ )u ( √ H H H s Ask z sk xsk + Ask xsk z sk −Ask xsk xsk s dz k . × Exk e

(A.66a)

˜  , q , q ˜  ) with respect to u at u → 0, Then, we take the derivative of T (u) (p , p    noticing that p˜ok = p˜1jk = p˜2j = 0, ∀j, k ∂ (u)     ˜ ,q ,q ˜) T (p , p ∂u ⎡ ⎤ 4 5 K J     ⎦ =2N + N ln 1 + (p0k − q0k ) + N ln ⎣1 + (p2j − q2j ) j=1

k=1

+M

K 

  q˜0k q0k +L

k=1

+

J  L 

−

J  L  j=1 =1

−M

K  k=1

j=1

4 ln 1 +

j=1 =1  t j q˜2j

J 

  q˜2j q2j + M



 q˜2j t j

4K 

J  K 

  q˜1jk q1jk

j=1 k=1 K 

(p1jk k=1

−

 q1jk )

5

+1

5

p1jk

+1

k=1

 ρp01k (˜ q0k +

J 

 q˜1jk )

j=1

 H√   √ s s 2  √ H  s −zp A k xk  zsk Ask xsk +( A sk xsk )H zsk −(Ask −B sk )xsk xsk s− s s ln Exk e Exk e dz k K   − .   √ 1 −zsk − Ask xsk 2 s s k=1 e dz πM E x k πM k 6 78 9 =1

(A.67) Having defined the following vector channel s

z k =

s

A sk IM xsk + w k ,

(A.68)

A.8. Evaluation of Second Term of the Mutual Information

139

where wks ∼ CN (0, IM ), and hence, 1 √ s s 2 s 1 A sk = M ez k − A k xk  , π

0 / s p z k /xsk ,

(A.69)

we can rewrite K  

  √ 1 −zsk − Ask xsk 2 s e E x k M 78 9 6π j=1 k=1 k=1 /√   / s s   =p z k Ak  √  √ H H H s s s s s s s s s s × ln Exsk e A k z k xk + A k xk z k −A k xk xk dz k 6 78 9  √ s s 2 H s −M

K 

 ρp01k (˜ q0k

=π M

= −M

K  k=1

−

K  

1 πM

Exs

k

+

J 

−z

e

 ρp01k (˜ q0k +

 q˜1jk )

k

−

J 

−

A

x  k k

(A.70a)

 s z s k k

ez

 q˜1jk )

j=1



 sH  s s s s p(z k | A sk ) M ln π + ln p(z k | A sk ) + ln ez k z k dz k

(A.70b)

k=1

=

  K ! ! !  H p p p p p p h(z k | A pk ) − M ln π p(z k | A pk )dz k − zpk z k p(z k | A pk )dz k k=1 6 78 9 =1

  K  s s s sH s s + h(z k | A sk ) − M ln π p(z k | A sk )dz k − z k zsk p(z k | A sk )dz k k=1 6 78 9 =1

= −M

K  k=1

⎡  ρp01k ⎣q˜0k +

K  s + h(z k | A sk ) − k=1

=

K  k=1

⎛ ⎝I(z sk ; xsk |

J 

(A.70c)

⎤  ⎦ q˜1jk

j=1

⎡

J 

⎤

  ⎦ M ln π − M −M ρs01k ⎣q˜0k + q˜1jk 78 9 6 √ s j=1 s s

=−h(z

k |xk ,

A k)

⎡

 A sk ) − M (ρp01k + ρs01k ) ⎣q˜0k +

J  j=1

(A.70d)

⎤⎞

 ⎦⎠ . q˜1jk

(A.70e)

140

Appendix A. Omitted Steps from the Proof of Theorem 5.1

Thus, the free energy is expressed as F =

1 Ask + 2N

K 0  / s I z k ; xsk / k=1

4

K 

+ N ln 1 +

⎡

5

J 

 (p0k − q0k ) + N ln ⎣1 +

K 

  q˜0k q0k +L

k=1

+

J  L 

−

J  L 

4  q˜2j t j

−M



K 

p1jk

J  K 

(p1jk

−

 q1jk )

78

9

K 

J  K    (ρp01k + ρs01k )q˜0k −M (ρp1jk + ρs1jk )q˜1jk 6 78 9 78 9 j=1 k=1 6 k=1



+ N ln 1 +

→p1jk

1 Ask + 2N

K 

⎛

 ε0k

+ N ln ⎝1 +

K 

 ξ0k ε0k − L

k=1

+

J  L 

J  j=1

 ln 1 +

j=1 =1

J 

⎞ ε2j ⎠

j=1

k=1

−M

+1

+1

→p2j

K 0  / s I z k ; xsk / k=1

5

5

→p0k

=

  q˜1jk q1jk

j=1 k=1 K 

k=1

k=1

6

  q˜2j q2j + M

 q˜2j t j

ln 1 +

j=1 =1

J  j=1

4

j=1 =1

 ⎦ (p2j − q2j )

j=1

k=1

+M

⎤

 ξ2j t j

  ξ2j ε2j − M



J  K 

 ξ1jk ε1jk

j=1 k=1 K 



ε1jk IL

+ IL

.

(A.71)

k=1

Similarly to previous case, we get the following system of equations p˜0k = p˜1jk = p˜2j = 0, ∀j, k, N  - ., q˜0k = K  − q ) + 1 M (p 0k k=1 0k  = q˜1jk

L 

=1

q˜ t j - 2j .1 , K  t   M 1 + q˜2j

j k=1 (p1jk − q1jk ) + 1 0

(A.72a) (A.72b)

(A.72c)

A.8. Evaluation of Second Term of the Mutual Information  q˜2j =

 p2j − q2j

N

 k=1 (p2j

 )+1 − q2j

.,

+ , ρs01k s Ez sk ,xsk xsk − x k 2 , M + , ρs1jk s Ez sk ,xsk xsk − x k 2 , = M - . K   L t j (p − q ) + 1  1jk k=1 1jk - 0 .1 . = K    )+1 ˜2j t j (p − q

=1 L 1 + q 1jk k=1 1jk

 = p0k − q0k  p1jk − q1jk

L

- K

141

(A.72d)

(A.72e) (A.72f) (A.72g)

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