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Distributed space-time coding for cooperative networks Yiu, Simon Tik-Kong 2007

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DISTRIBUTED SPACE-TIME CODING FOR COOPERATIVE NETWORKS by SIMON TIK-KONG YIU B.A.Sc, The University of British Columbia, 2002 M.A.Sc, The University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA October 2007 © Simon Tik-Kong Yiu, 2007 Abstract Cooperative networks exploit the broadcast nature of wireless channels to gain spatial diversity. This dissertation develops two distributed space-time cod-ing schemes for frequency-nonselective channels, and extends and optimizes the previously proposed distributed space-time filtering scheme for frequency-selective channels. Unlike many other distributed space-time coding schemes in the literature, the decentralized schemes proposed in this thesis are designed specifically for wireless networks with a large set of N decode-and-forward relay nodes. At any given time, an a priori unknown subset of nodes acts as relays to cooperatively assist the communication between the source and destination node. To facilitate the cooperation, each physically distributed single-antenna node in the network is assigned a signature vector (signature matrix for multiple-antenna nodes) or signature filter vector. We show that the proposed schemes guarantee a certain diversity gain regardless of which relay nodes in the network cooperate. In addition, the decoding complexity of the various schemes is independent of N and the (random) number of ac-tive nodes, and receiver designs advocated originally for traditional co-located antenna systems can be applied. The first scheme is called distributed space-time block coding. Depending on the chosen design, it allows for low-complexity coherent, differential, and noncoherent detection. The second scheme is referred to as distributed space-ii time trellis coding. We show that distributed space-time trellis coding inherits the coding gain of space-time trellis codes over space-time block codes even in the cooperative setting. The two aforementioned schemes are designed for frequency-nonselective fading channels. Finally, we optimize and extend dis-tributed space-time filtering, proposed originally by El Gamal and Aktas, to frequency-selective fading channels. For each scheme, the design criteria are derived. Then, using mathematical tools and classical optimization techniques, efficient methods for the design of the set of signature vectors, signature matrices, and signature filter vectors are provided. Furthermore, the achievable diversity gain, as well as the loss entailed by the distributed implementation of each scheme, are characterized and verified via simulations. Finally, we apply noncoherent distributed space-time block coding to a practical wireless sensor network and show that the proposed scheme is a promising solution for cooperative communication in future sensor, ad hoc, and wireless networks. iii Contents Abstract ii Contents iv List of Tables ix List of Figures x List of Abbreviations and Symbols xviii Acknowledgments xxii 1 Introduction 1 1.1 Cooperative Diversity 3 1.2 Distributed Space-Time Coding 5 1.3 Thesis Contributions 8 1.4 Thesis Outline 10 1.5 Related Publications 11 2 Preliminaries 14 2.1 Space-Time Block Coding 16 2.1.1 Coherent Space-Time Block Coding 16 2.1.2 Noncoherent Space-Time Block Coding 23 2.1.3 Differential Space-Time Block Coding 26 iv 2.2 Space-Time Trellis Coding 29 2.3 Space-Time Filtering . 31 2.4 System Model and Assumptions 35 2.5 Conclusions 38 3 Distributed Space-Time Block Coding 40 3.1 Introduction 40 3.2 Network Model and DSTBCs 42 3.3 Optimization and Receiver Design 44 3.3.1 Coherent DSTBCs 45 3.3.2 Differential DSTBCs 46 3.3.3 Noncoherent DSTBCs 48 3.3.4 Choice oi Nc 50 3.4 Design of the Node Signature Vector Set 51 3.4.1 Distribution Loss 51 3.4.2 Optimum Sets ^ o p t • 52 3.4.3 Influence of the Design Parameter Na 53 3.4.4 Gradient Sets Ggva,d 54 3.4.5 Grassmannian Sets Q&as 57 3.4.6 Harmonic Sets Ghaxm 58 3.4.7 Random Sets £rand 6 0 3.4.8 Some Specific Examples 60 3.5 Simulation Results 62 3.5.1 Ideal Channel Conditions 63 3.5.2 Practical Considerations 69 3.6 Conclusions 72 4 Distributed Space-Time Trellis Coding 74 v 4.1 Introduction 74 4.2 Network Model and DSTTCs 77 4.3 Decoding and Optimization of DSTTCs 79 4.3.1 Receiver Design 80 4.3.2 DSTTC Design .' 80 4.4 Design of Node Signature Matrix Set Q 87 4.4.1 Gradient Sets £ g r a d 87 4.4.2 Harmonic Sets £harm 87 4.4.3 Random Sets £ r a n d 8 8 4.4.4 Distribution Loss 89 4.4.5 Comparison of Different Designs 90 4.5 Simulation Results 94 4.5.1 Ideal Channel Conditions 95 4.5.2 Practical Considerations •'. 99 4.6 Conclusions • • • 104 5 Distributed Space-Time Filtering 106 5.1 Introduction 106 5.2 Network Model and DSTF 108 5.2.1 Distributed Space-Time Filtering 109 5.2.2 Channel Estimation and Detection . . . . I l l 5.3 Optimization of Distributed Space-Time Filtering 112 5.3.1 Approximate PEP Analysis of DSTF 112 5.3.2 Optimum Sets ^ o p t . 114 5.3.3 Gradient Sets QSTad 115 5.3.4 Harmonic Sets £harm 116 5.3.5 Random Sets Qiand 117 5.3.6 Influence of the Design Parameter Na 117 vi 5.3.7 Distribution Loss of DSTF 118 5.4 SFV Design for Frequency-Nonselective Channels 119 5.4.1 Relation to the Design of Signature Sequences for CDMA 120 5.4.2 Relation to the Design of Grassmannian Frames . . . . . 121 5.5 Simulation Results . 125 5.5.1 Gradient Algorithm 125 5.5.2 Distribution Loss 127 5.5.3 BER Performance under Ideal Conditions .129 5.5.4 Influence of Practical Issues 132 5.6 Conclusions 139 6 Applicat ion to Wireless Sensor Networks 141 6.1 Introduction 141 6.2 System Model •. 144 6.2.1 Local Sensor Decisions 145 6.2.2 Noncoherent Distributed Space—Time Coding 146 6.2.3 Channel Model 148 6.2.4 Processing at Fusion Center (FC) 149 6.3 FC Decision Rules and Performance Analysis 149 6.3.1 Optimum M L Decision Rule 150 6.3.2 GLRT Decision Rule 151 6.3.3 Performance Analysis for GLRT Decision Rule 152 6.4 Optimization of Censoring Threshold d . 154 6.5 Simulation Results 155 6.6 Conclusions 163 7 Conclusions and Future Work 165 vii 7.1 Research Contributions 165 7.2 Future Work 170 A Upper Bound on P E P of Noncoherent S T B C s 173 B Elements of a Gradient Set for N = 30, Na = Nc = 2 175 Bibliography 176 viii List of Tables 7.1 Comparisons of DSTBC, DSTTC, and DSTF ix List of Figures 1.1 A three-terminal relay network consists of a source (S), a relay (R), and a destination (D) node 4 1.2 A network with N = 8 DaF relay nodes and Ns — 2 active relay nodes. S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows 6 2.1 STBC encoder with Nc antennas 17 2.2 STTC encoder with iV c antennas 29 2.3 Generalized DD encoder with Nc antennas 32 2.4 A network with N = 8 DaF relay nodes and Ns = 2 active relay nodes (S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows, h: Relay nodes are uniformly distributed in a circle with radius R. l\: Distance from the source node to the center of the circle. l2- Distance from the center of the circle to the destination node 36 3.1 A network with N = 8 DaF relay nodes and Ns = 2 active relay nodes (S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. l\\ Relay nodes are uniformly distributed in a circle with radius R. l\\ Distance from the source node to the center of the circle, fa Distance from the center of the circle to the destination node. hn: Fading gain from relay node n to the destination node 43 3.2 Learning curve of gradient algorithm for N = 30 and Na = Nc = 2. Adaptation step size: /z[i] = 10~2 for 0 < i < 1999, H\i] = 10~3 for 2000 < i < 3999, fi[i] = 10~4 for 4000 < i < 5999, and = 10'5 for 6000 < i < 7999 56 3.3 Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = Nc = 2. Gradient sets Ggrad and harmonic sets £ h a r m are considered. The lower bound (3.26) is also shown 62 3.4 Maximum and average distribution losses as functions of the total number of nodes TY for a gradient set optimized for Na = Nc = 2 and different numbers of active nodes Ns 63 3.5 Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = Nc = 3. Gradient sets Ggrad and harmonic sets (/harm are considered 64 3.6 Average BER of a coherent DSTBC vs. 101og10(£6/7Vo) for a network with TY = 30 nodes and different numbers of active nodes Ns. B: Alamouti's STBC (T = Nc = 2) with QPSK modulation. The results for Alamouti's code with Nc = 2 co-located antennas and DSTF [59] are also shown 65 xi 3.7 Average BER of a coherent DSTBC vs. 101og1 0(£:6/iVo) for a network with N — 50 nodes and different numbers of active nodes Ns- Solid lines: Rate 3/4 orthogonal STBC (T = NC = 4) [78] with 16QAM modulation as code B. Dash-dotted lines: Single-antenna 8PSK transmission for Ns = 1 and Alamouti's STBC with 8PSK modulation as code B for Ns = 2. Dashed line: Rate 3/4 orthogonal STBC from [78] with 16QAM modu-lation and Nc = 4 co-located antennas 3.8 Average BER of a differential DSTBC vs. lOlog^f i / iVo) for a network with N = 30 nodes and different numbers of ac-tive nodes Ns- B: Diagonal differential STBC from [91] with T = Nc = 3 and rate R = 1 bit/(channel use) (solid lines). The results for single-antenna DBPSK transmission and the differ-ential STBC with Nc = 3 co-located antennas are also shown. 3.9 Average BER of a noncoherent DSTBC vs. 10 log10(JEVJV0) for a network with N = 50 nodes and different numbers of active nodes Ns. B: Noncoherent STBC from [90, Section V] with T = 8, Nc = 2, and rate R — 1 bit/(channel use) (solid lines). The results for the single-antenna code also given in [90, Section V] and the noncoherent STBC with Nc = 2 co-located antennas are also shown 3.10 Average BER of DSTBC vs. 10log10(Eb/N0) for a network with N — 30 nodes with i.n.d. fading gains and Ns = 2 active nodes. B: Alamouti's STBC (T = Nc = 2) with QPSK modulation (solid lines). Results for QPSK transmission without DSTBC are also shown (dashed lines) 3.11 Average BER of DSTBC vs. 101og1 0(£b/7vo) f o r a network with N = 30 nodes and two-hop transmission. In the first hop, each relay node listens with probability p. The received energy per bit in the first hop is E-i. B: Alamouti's STBC (T = Nc = 2) with QPSK modulation. Results for Alamouti's code with Nc = 2 co-located antennas and QPSK transmission without DSTBC are also shown (dash-dotted lines) 71 4.1 Processing performed at active relay node n £ S for Nc = 4 and NT = 2 78 4.2 Average distribution losses Lav(Ns) as functions of the total number of nodes N for a gradient set £/grad optimized for Na = NC/NT — 4/2 = 2 and different numbers of active nodes Ns-Both correlated (solid curves) and uncorrelated (dashed curves) antennas are considered. A correlation coefficient p = 0.8 is assumed for the correlated antenna cases 92 4.3 Average distribution loss Lav(Ns) as function of the total num-ber of nodes N for Na = Ns = NC/NT = 4/2 = 2, and p = 0.8. Gradient sets <7grad, harmonic sets (/harm> a n d random sets £ r a n d are considered 93 4.4 Pr{Ls > Lth} as functions of threshold L t h for random sets £rand with Na = Ns = NC/NT = 4/2 = 2, p = 0.8, and different randomization methods. LMAX(2) for harmonic and gradient sets designed for = 30 nodes is also shown. 94 4.5 Average FER of DSTTC vs. 10Tog10(£?6/iVb) (solid lines). For Ns = 4 results for DSTF I [59], DSTF II (Chapter 5), and a DSTBC (Chapter 3) are also shown (dashed lines). N = 30, Nc = 4, and NT = 1. Signature matrices: Gradient set (?grad- • • 97 xiii 4.6 Average FER of DSTTC vs. 101og10(Eb/A^0). For Ns = 2, results for a DSTBC (Chapter 3) are also shown. N = 30 relay nodes with NT = 2 antennas, iV~c = 4, and p = 0.8. A gradient set £grad is used for the signature matrices 98 4.7 Average FER of DSTTC vs. 101og10(£6/Wo). N = 30, Nc = 4, NT = Ns = 2, and p — 0.8. Gradient sets GSrad> harmonic sets ^harmj and random sets Grand are considered 99 4.8 FER of DSTTC vs. 101og10(.E|,/iVo) for R/l2 = 0.6 and l2 > R. N = 30, Nc = 4, NT = Ns = 2, p = 0.8, and a = 3. Gradient sets ^grad are optimized for both R/l2 = 0.6 and l2^> R. . . . . 100 4.9 Average FER of decentralized (solid lines) and centralized (dashed lines) DSTTC vs. 101og10(E6/iVo). N = 30, NT = 1 and NT — 2 (p = 0.8), Nc = 4, and Ei/E^ — 10. Signature matrices: Gradi-ent Set £ g r a d 101 4.10 Average FER of DSTBC (Chapter 3), DSTTC, and DSTF (Chap-ter 5) vs. 101og 1 0(£6/W 0). N = 30 relay nodes with 7VC = 4, NT = 1, and ExjEb = 10. p= 1/3 (dashed lines). p= 1/10 (solid lines). A gradient set ^ g r a d is used for the signature vectors. 103 5.1 A network with N = 8 DaF relay nodes and NS = 2 active relay nodes (S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. l\\ Relay nodes are uniformly distributed in a circle with radius it!. l\\ Distance from the source node to the center of the circle. l2: Distance from the center of the circle to the destination node. hn[l\. CIR from relay node n to the destination node 109 xiv 5.2 Learning curve of gradient algorithm for BPSK, N = 30, iV a = 2, Lg = 2, and 101og10(£?b/Aro) = 20 dB. Adaptation step size: H[i] = 10- 3 for 0 < i < 1999, fi[i] = 10~4 for 2000 < i < 3999, ti[i] = 10"5 for 4000 < i < 5999, and = 10"6 for 6000 < i < 7999 126 5.3 Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na — 2 and Lh — 1 127 5.4 Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = 2 and Lh = 2 129 5.5 Maximum and average distribution losses as functions of the SFV length Lg for Ns = Na = 2, Lh = 1, and N = 30 130 5.6 Average BER of DSTF with different SFV sets vs. 10 log 1 0(£ f e/7V 0) for TV = 30 nodes and Lh = l. Ns = Na = 2 and MLSE is em-ployed in all cases. The BERs of a DSTBC (cf. Chapter 3) and optimized DD for two co-located transmit antennas (cf. Section 2.3) are also shown 131 5.7 Average BER of DSTF with different SFV sets vs. 10 \ogw(Eb/NQ) for N = 30 nodes and Lh = 2. Ns = Na = 2 and MLSE is employed in all cases. The BER of optimized DD for two co-located transmit antennas (cf. Section 2.3) is also shown. . . 132 5.8 Average BER of DSTF vs. 101og1 0(£6//Vo) for Lh = 1, Lg = 5, N = 30, and MLSE. Gradient sets optimized for Na — 2, 5 are considered for Ns = 2, 5 active nodes 133 xv 5.9 Average BER of DSTF with different SFV sets vs. 10 \og10(Eb/N0) for Lh = 1, N = 30, and Ns = Na = 2. For L9 = 5 M M S E -DFE, DFSE, and MLSE are employed at the receiver. The BER of optimized DD for two co-located transmit antennas (cf. Sec-5.10 Required 101og10(i?h/A/o) to achieve an average BER of 10 4 vs. training sequence length NTS for DSTF with different SFV 5.11 Average BER of DSTF with perfect and imperfect time synchro-nization vs. 101og10(£b/7Vo) for flat Rayleigh fading, N = 30, and Ns = Na = 2. In the case of imperfect time synchro-nization, the timing offset of all nodes is independent and uni-formly distributed in the interval [-3T/4,3T/4]. For L9 = 2 and Lg = 5 gradient sets are employed and DFSE with two states is adopted in all cases 137 5.12 Average BER of DSTF in two-hop transmission vs. 10 \ogw(Eb/N0). Gradient set, Lg = 5, Lh = 1, N = 30, Na = 2, DFSE with two states. Each node listens with probability p. E\. Received en-ergy per bit in first hop. Eb: Received energy per bit in second hop 138 6.1 Parallel fusion model with K sensors and one FC. A censored DSTBC is used for transmission from the sensors to the FC. . . 145 6.2 d and Pe vs. iteration number i for a WSN with K = 30 sensors using DSTBCs with Nc = 1, 2, and 4. 101og10(J56/JVo) = 15 dB, tion 2.3) is also shown 134 sets. Lh = 2, N = 30, Ns = Na = 2, and MLSE 135 a2 = 1/4, 157 xvi 6.3 Pe vs. 101og10(£?6/JV0) for a WSN with K = 30 sensors using DSTBCs with Nc = 1, 2, and 4. Considered cases: Error-free local sensor decisions (a2 — 0, d — 0), noisy sensor decisions without censoring (a 2 = 1/4, d = 0), and noisy sensor decisions 6.4 Pe vs. total number of sensors K for a WSN using DSTBCs with 7VC = 1, 2, and 4. 101og10(£6/iVo) = 15 dB. Numerical results for error-free local sensor decisions and GLRT decision rule ( c r 2 = 0, d = 0), numerical results for noisy sensor decisions with censoring and GLRT decision rule (a2 = 1/4, d = dopi), and simulation results for noisy sensor decisions with censoring and M L decision rule (cr2 = 1/4, d = dopt) 159 6.5 Pe and d vs. 7VC for a WSN with K sensors, a2 = 1/4 and 101og10(£fe/iVo) = 15 dB. GLRT fusion rule is shown for all K (solid curves) and M L fusion rule is shown for K = 1, and 2 (dashed curve) 161 6.6 Pe and d vs. 101og10(l/cr2) for a WSN with K = 10, and 30 sensors and DSTBC with 7VC = 1, 2, and 4. 101og10(JE?6/JV0) = 15 6.7 Pe vs. 101og10(£;b/iVo) for a WSN with K = 30 sensors using DSTBCs with Nc = 1, 2, and 4. a2 - 1/4 and i.n.d. Rayleigh with optimum censoring (a2 — 1/4, d — dopt) 158 dB 162 fading channels 163 xvii List of Abbreviations and Symbols A c r o n y m s AaF Amplify-and-forward AWGN Additive white Gaussian noise BER Bit error rate BPSK Binary phase shift keying CDMA Code-division multiple access CIR Channel impulse response CRC Cyclic redundancy check CSI Channel state information DaF Decode-and-forward DBPSK Differential binary phase shift keying DD Delay diversity DFE Decision-feedback equalization DFSE Decision feedback sequence estimation DFT Discrete Fourier transform DPSK Differential phase shift keying DS-CDMA Direct sequence code—division multiple access xviii DST Distributed space-time DSTBC Distributed space-time block codes DSTF Distributed space-time filtering DSTTC Distributed space-time trellis codes EGC Equal-gain combining FC Fusion center F E R Frame error rate GLRT Generalized likelihood ratio test GSIC Generalized successive interference cancellation I.i.d Independent and identically distributed I.n.d Independent and non-identically distributed ISI Intersymbol interference ISM Industrial/scientific / medical L E Linear equalization LR Likelihood ratio LS Least-squares LS-CE Least-squares channel estimation MIMO Multiple—input and multiple—output M L Maximum-likelihood MLSE Maximum-likelihood sequence estimation MMSE Minimum-mean squared error M P S K M-ary phase shift keying M Q A M M-ary quadrature amplitude modulation MRC Maximal-ratio combining OFDM Orthogonal frequency division multiplexing P A M Pulse-amplitude modulation Pdf Probability density function PEP Pairwise error probability PSK Phase shift keying Q A M Quadrature amplitude modulation QPSK Quaternary phase shift keying SC Selection combining SFV Signature filter vector SISO Single-input single-output SNR Signal-to-noise ratio STBC Space-time block code STC Space-time code STC-OFDM Space-time coded-orthogonal frequency division multiplexing STTC Space-time trellis code T C M Trellis-coded modulation TR-STBC Time reversal STBC TS Training sequence USTM Unitary space-time modulation UWB Ultrawide bandwidth WSNs Wireless sensor networks Notations and Operators Throughout this work, bold upper case and lower case letters denote matrices and vectors, respectively. The remaining notations and operators used in this thesis are listed as follows: |-| Absolute value of a complex number or cardinality of a set || • ||2 Z/2-norm of a vector xx • U • Union of two sets • * • Discrete-time convolution • ® • Kronecker product S[] Dirac delta function Q(x) Gaussian Q-function, fx°° e - ' 2 / 2 dt £ {•} Statistical expectation det {•} Determinant of a matrix tr {•} Trace of a matrix vec {•} Vectorization of a matrix r (•) Rank of a matrix diag {x} Diagonal matrix with elements of vector x in its main diagonal [•]* Complex conjugate [•]T Matrix or vector transposition [•}H Matrix or vector hermitian transposition O x x Y 1 x 7 matrix with all zero elements Ix Identity matrix with dimension X x X j = y/—l Imaginary unit P {•} Probability of an event A j ( X ) Eigenvalues of the X x X matrix X \i(X)>\i+1(X),l<i<X-l xxi Acknowledgments Most special thanks go to my advisor, Professor Robert Schober, for his in-valuable support and guidance over the last five years. I am truly privileged to have the most dedicated and understanding advisor a graduate student could ever imagine having. I am especially grateful that he is always available when-ever I need his advice, even during weekends and holidays. Whenever I have an idea on my research, I can drop by his office, and every time emerge with a much clearer picture. Robert is a great mentor, a generous and patient aca-demic provider, and a constant source of inspiration. Ever since I became his student, I have benefited tremendously from his unique blend of energy, vision, creativity, and technical insight. Robert will continue to be my role model for my future career. I am also particularly appreciative of the many stimulating and helpful discussions with Professor Lutz Lampe on some of the work we have done together. I would also like to thank the other members of my doctoral committee, Vikram Krishnamurthy, Vijay Bhargava, and Arnaud Doucet, for their time and effort in evaluating my work and providing their valuable feedback and suggestions. It is also my great honor to have Professor Anna Scaglione from Cornell Uni-versity as my external committee member. I would like to express additional thanks to the members of the Communica-tions Theory Group and the faculty and staff of the Electrical and Computer xxii Engineering Department for providing an extraordinarily pleasant and collab-orative environment during my years at UBC. Last, but certainly not least, I wish to thank my family, my parents, my brother, my sister, my auntie and Kelly for their constant love, support, and encouragement throughout this process. The financial support of NSERC through a Canadian Graduate Scholarship (CGS) is also gratefully acknowledged. xxiii Chapter 1 Introduction Tremendous research effort has been directed towards multiple-input and multiple-output (MIMO) systems over the last ten years due to their ability to enable a spatial multiplexing gain and/or diversity gain. In fact, the potential benefit offered by space-time coding and MIMO technology has proven to be so promising that it has already been incorporated into standard such as the IEEE 802.16e-2005 [1]. Currently, IEEE 802.11n [2], which builds upon the previous IEEE 802.11 standards [3] by adding multiple transmit and receive an-tennas, is being finalized. In addition, draft IEEE 802.16j, which concentrates on the multihop relay specification, is being developed. In the future, MIMO technology is expected to be employed in many other wireless communication systems [4]. A MIMO communication system [5-8] employs multiple antennas at both the transmitter and receiver sides. Typically, transmission schemes over MIMO channels can be classified as spatial multiplexing schemes and space-time cod-ing techniques. Spatial multiplexing schemes maximize the channel through-put by sending independent data streams over the transmit antennas, and the number of parallel spatial channels available is limited by the minimum num-ber of transmit and receive antennas [6]. A well-known example of a spatial 1 multiplexing scheme is the Vertical Bell-labs layered space-time (V-BLAST) architecture [9]. The objective of space-time coding techniques [10-13], as opposed to spatial multiplexing, is to improve error performance by exploiting spatial diversity [14-16] in a MIMO system. Compared with other commonly considered diver-sity techniques in the literature such as time diversity and frequency diversity, spatial diversity is particularly attractive because, unlike the other two meth-ods, no bandwidth expansion is required to achieve a diversity gain. Spatial diversity can be further broken down into transmit diversity and receive diver-sity. Traditionally, multiple antennas have been employed at the receiver to achieve receive diversity [15]. In particular, the signals received at the antenna branches are combined using classical combining techniques such as maximal-ratio com-bining (MRC), equal-gain combining (EGC), or selection combining (SC) [17]. Later, Wittneben [18, 19] and Seshadri and Winters [20, 21] showed that di-versity can also be achieved at the transmitter side by employing some clever signal processing techniques. However, the development of transmit diversity did not really take off until the idea of space-time coding [10-13] was conceived. Since then, interest in the topic has grown rapidly. There are two general classes of space-time codes (STCs), namely, space-time block codes (STBCs) [10, 12, 13] and space-time trellis codes (STTCs) [11]. If NT and NR antennas are used at the transmitter and receiver, respectively, both schemes can achieve a diversity gain of d = NTNU. In addition, depending on the complexity of the trellis, STTCs provide a coding gain at the expense of increased receiver complexity. We will provide more detail on STCs in Chapter 2. The aforementioned attractive features of space-time coding systems can be 2 i fully realized only if the antenna arrays have sufficient spacing between them resulting in independent and identically distributed (i.i.d.) channels. Usually, the physical antenna spacing is selected to be greater than one-half the wave-length of the transmitted signal, to ensure low correlation between the MIMO channels. For example, for wireless sensor nodes operating in the 916 MHz [22] or 2.4 GHz [23] industrial/scientific/medical (ISM) band, antenna spacing of 16.38 cm and 6.25 cm, respectively is required. This requirement fundamen-tally limits the possibility of having multiple antennas on small communication devices. A question therefore arises: Given an area limitation on wireless devices, is it possible to obtain a diversity gain similar to that in MIMO systems without deploying multiple antennas? Recently, it has been shown that a virtual MIMO system can be created by node cooperation [24, 25]. The term cooperative diversity was introduced by Laneman et al. [24] to describe the diversity gain obtained by such a system. 1 .1 C o o p e r a t i v e D i v e r s i t y In applications such as sensor networks, ad-hoc networks, and uplink transmis-sion in mobile communication systems where the transmitter cannot accom-modate multiple antennas due to size, energy, and/or economical constraints, cooperative diversity [24, 25] allows single-antenna wireless devices to provide a diversity gain. The idea is to allow a collection of mobile users to share their antennas to emulate a virtual antenna array and exploit spatial diversity in wireless fading channels. Generally, cooperative diversity is the result of coop-erative communication, which refers to situations where mobile users interact to jointly transmit information. Early work on cooperative communication appeared in the information theory 3 Figure 1.1: A three-terminal relay network consists of a source (S), a relay (R), and a destination (D) node. literature, where the classical three-terminal relay model was examined [26, 27]. The considered relay network, which is depicted in Fig. 1.1, consists of a source (S), a destination (D), and a relay (R) node. Assuming that all nodes operate in the same frequency band and exploiting the broadcast nature of the wireless medium, the source node transmits the source information to the destination node with the assistance of the relay node. In [28], the capacity of this three-node relay network in an additive white Gaussian noise (AWGN) channel was extensively analyzed. The recent development of cooperative communication [29-37] was sparked by Sendonaris et al. [38, 39] and Laneman et al. [24, 25, 40, 41]. Sendonaris et al. introduced the concept of user cooperation diversity in [38, 39], where two mobile users with their own information to transmit share their antennas and resources to obtain a diversity gain through distributed transmission. In [25] and [40], Lanemen et al. proposed various cooperative transmission protocols and analyzed their performance in terms of outage behavior. It was shown that most of the protocols achieve a full diversity order equal to the number of cooperative nodes. 4 The terms decode-and-forward (DaF) and amplify-and-forward (AaF) relay-ing were introduced in [25, 40, 42], where the cooperating nodes simply act as relays for the source node and do not have their own information to transmit. The cooperation process can be broken down into two phases. In the first phase, the source node broadcasts the information to the relay nodes. The relay nodes then either AaF or DaF the received signal to the destination. In DaF [43-48], the relays decode the source information and forward the decoded signals to the destination node. In AaF [49-56], the relays simply amplify the received signals and retransmit them to the destination node. At the destina-tion node, clever signal processing techniques are used to combine the received signals to obtain a diversity gain. Diversity combining is facilitated either by assigning all nodes to orthogonal channels [42] or by application of a suitable distributed space-time (DST) coding scheme [25, 57]. In order to make effi-cient use of the available bandwidth and power resources, it is advantageous to employ a DST code in the second phase of transmission [25, 57]. 1.2 Distributed Space-Time Coding In a cooperative network with a large number of relay nodes, the relay nodes can be either active or non-active, depending on whether they participate in the cooperation or not. Furthermore, depending on the level of cooperation between the relay nodes, two different approaches to DST coding may be dis-tinguished. In centralized DST coding, the cooperating (active) nodes know their partners, and existing STCs designed for co-located antennas can be applied straightforwardly, cf. e.g. [58]. However, in many situations, decentral-ized DST coding schemes [25, 59, 60], where the cooperating nodes are not required to know their partners, may be more practical. For example, decen-tralized DST coding may be useful in DaF relaying with cyclic redundancy 5 Figure 1.2: A network with N = 8 DaF relay nodes and Ns = 2 active relay nodes. S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. check (CRC) coding in the first phase of transmission. In this case, only an a priori unknown subset of relay nodes that can decode the received message correctly forwards the message to the destination node in the second phase, cf. Fig. 1.2. Decentralized DST coding is also favorably in wireless sensor net-works (WSNs) [61-63] where sensors decide independently whether or not they forward their measurement to the fusion center (FC), cf. e.g. [64]. Since decentralized networks are more practical, in this work we only consider decentralized DST coding where neither the cooperating nodes nor the desti-nation node know which relay nodes are active. The literature on DST coding for node cooperation with unknown partners is surprisingly sparse. Most of the work in the literature belongs either to the class of centralized DST coding or is only applicable to networks with a small number of relay nodes. For ex-ample, Laneman and Wornell advocated in [25] the use of orthogonal STBCs for DST coding, where each node transmits a different column of a STBC ma-trix. For networks with N < Nc nodes, it was shown in [65] that orthogonal STBCs designed for Nc > 2 co-located transmit antennas achieve a diversity order of d — Ns if Ns < Nc nodes are active. However, if N > Nc, the same 6 column of the STBC matrix has to be assigned to multiple relay nodes, which results in a diversity order of d = 1 [59]. Since the design of full-rate com-plex orthogonal STBCs for Nc > 2 antennas is not possible and a rate loss is unavoidable for large Nc (cf. Section 2.1), the straightforward application of orthogonal STBCs is limited to networks with a small total number of nodes N. Replacing the orthogonal STBCs with full-rate non-orthogonal STBCs, which are available for any Nc, cf. e.g. [66], is also not a satisfactory solution, as the decoding complexity of these codes increases rapidly with Nc. The lim-itation of Laneman's scheme can be avoided if it is applied to a cluster-based cooperative environment [67] where a cluster head is used to select Ns = Nc out of the TV nodes to cooperate. Of course, any scheme that employs a cluster head requires a centralized network. There are also many studies in the literature that analyze the performance of orthogonal STBCs when they are applied in a distributed setting. For exam-ple, the performance of Alamouti's scheme in a 2-relay cooperative network with different propagation delays and path losses is considered in [68], and the asymptotic error performance of orthogonal STBCs over flat Nakagami-m fading channels with both identical and non-identical m-distributions is characterized in [69]. Recently, Hu and Beaulieu have presented a closed-form expression for the outage and error probabilities of DaF relaying with orthog-onal STBCs in dissimilar Rayleigh fading channels [70]. Tao and Kam have studied the performance of differential orthogonal STBCs over semi-identically distributed block Rayleigh fading channels [71]. Another popular protocol for cooperative relaying is opportunistic relaying [72-74]. In these schemes, as with traditional selection diversity [75], after the data broadcasting phase, only the relay with the strongest channel to the destination transmits. However, these schemes require either partial or full channel state 7 information (CSI) at the relays, which makes them less attractive. Distributed space-time filtering (DSTF) is El Gamal and Aktas' solution to decentralized DST coding [59]. If properly designed, this scheme can guarantee a diversity order of d if the number of active nodes Ns is not less than d. Unfortunately, the decoding complexity for DSTF increases exponentially with d. Both DSTF and the orthogonal STBCs in [25, 65] are coherent techniques that require channel estimation at the receiver. Tarasak et al. [76] have recently proposed a differential DST coding scheme that avoids channel estimation. However, if the cooperating nodes are not known a priori, this technique only guarantees full diversity in networks with N = 2 nodes. 1 .3 T h e s i s C o n t r i b u t i o n s In this thesis, we develop three decentralized DST coding schemes, which elim-inate the various drawbacks and limitations of the existing schemes while at the same time achieving a cooperative diversity gain. The proposed schemes are designed for networks with a potentially large set of DaF relay nodes M where at any given time only an a priori unknown subset of nodes <S C M is active and cooperates. We note that relay nodes may be active or inactive for various reasons. For example, for the two-hop scenario with DaF relaying and CRC considered in this work (cf. Fig. 1.2), only those relay nodes that can successfully detect the information received in the first hop correctly can be active in the second hop. In addition, nodes may be temporally inactive be-cause of an energy-saving policy of the network, and nodes may even become permanently inactive if the life-time of their batteries has expired. The considered network model is depicted in Fig. 1.2, which shows a decen-tralized network with N = \M\ = 8 DaF relay nodes located between the source and the destination node. The first and second phase of transmission 8 are denoted by the solid and dashed arrows, respectively. In this example, only relay nodes 2 and 8 are active during the second phase. This information is a priori unknown to both the relay nodes and the destination node. With-out requiring any CSI at the relay nodes, we show that the proposed schemes can guarantee a certain diversity order, regardless of which relay nodes in the network cooperate. The first scheme that we propose is distributed space-time block coding. De-pending on the chosen design, distributed space-time block codes (DSTBCs) allow for coherent, differential, and noncoherent detection, respectively. We refer to the second scheme as distributed space-time trellis coding. We show that distributed space—time trellis codes (DSTTCs) achieve not only coopera-tive diversity, but also a coding gain at the expense of increased receiver com-plexity, compared with DSTBCs. Both single-antenna and multiple-antenna relay nodes are considered. For both schemes, each single-antenna (multiple-antenna) node in the network is assigned a signature vector (matrix). The transmitted signal of an active node is the product of an information-carrying code matrix, which contains the correctly decoded source symbols and is there-fore identical for all nodes in S, and the signature vector (matrix) of that node. We show that if the relay nodes and destination node are equipped with NT and NR antennas, respectively, both DSTBCs and DSTTCs achieve a diversity order of min{NTNsNR, NCNR} , if Ns — \S\ nodes are active. The two aforementioned schemes are designed for frequency-nonselective fad-ing channels. We also provide a DST coding scheme designed for frequency-selective channels. Specifically, we optimize and extend DSTF [59] to frequency-selective fading channels. In DSTF, each node in the network is assigned a different signature filter vector. The transmitted signal of an active node is the convolution of the source symbol sequence, which is again identical to all 9 nodes in S, and the signature filter vector of that node. Here, we also con-sider several practical problems, such as suboptimum equalization, imperfect channel estimation, and imperfect timing synchronization. We show that the optimized scheme achieves a diversity order of min{L s + Lh. — 1, LhNs), where Lg and Lh denote the length of the signature filter vectors and the number of taps of the channel impulse response (CIR), respectively. To illustrate the practicality of the proposed schemes, we consider the appli-cation of the proposed DSTBCs in a WSN. In particular, sensor nodes encode their local decisions using noncoherent DSTBCs, which are then transmitted to the FC through fading and noisy channels for detection. Our numerical and simulation results show the effectiveness of the proposed transmission scheme in WSNs and the ability of the noncoherent DSTBC to achieve a diversity gain in these networks. 1 .4 T h e s i s O u t l i n e The rest of the thesis is organized as follows. Chapter 2 presents some relevant background material. Specifically, we introduce the basics of space-time coding and space-time filtering. The major assumptions that we make in Chapters 3-6 will be also outlined. Equipped with background knowledge on space-time coding and space-time filtering, we introduce DSTBCs, DSTTCs, and DSTF in Chapters 3, 4, and 5, respectively. For each proposed scheme, we outline the design criteria and receiver structure. We also quantify the achievable diversity gain and loss due to the distributed implementation of each scheme. Efficient methods for the optimization of the signature vectors, signature matrices, and signature filter vectors will also be presented. In Chapter 6, we apply noncoherent DSTBCs to a practical WSN. In particular, we introduce the signaling scheme and the 10 corresponding optimum and suboptimum decision rules, and demonstrate that noncoherent DSTBCs are attractive candidates for signaling also in WSNs. Finally, Chapter 7 summarizes the contributions of this thesis and identifies areas of future research. 1 .5 R e l a t e d P u b l i c a t i o n s The following is a list of journal and conference papers that are based on the research conducted for this thesis. Journal Papers [Yiul] S. Yiu, R. Schober, and L. Lampe. Decentralized distributed space—time trellis coding. Accepted for publication in the IEEE Transactions on Wireless Communications, 2007. [Yiu2] S. Yiu and R. Schober. Optimized distributed space-time filter-ing. IEEE Transactions on Wireless Communications, 6(3):982-992, March 2007. [Yiu3] S. Yiu, R. Schober, and L. Lampe. Distributed space-time block coding. IEEE Transactions on Communications, 54(7): 1195-1206, July 2006. [Yiu4] L. Lampe, R. Schober, and S. Yiu. Distributed space-time coding for multihop transmission in power line communications networks. IEEE Journal on Selected Area in Communications, 24(7): 1389-1400, July 2006. [Yiu5] S. Yiu and R. Schober. Censored distributed space-time coding for wireless sensor networks. Submitted to EURASIP Journal on Advances in Signal Processing, April 2007. 11 Conference Papers [Yiu6] S. Yiu and R. Schober. Censored noncoherent distributed space-time coding for wireless sensor networks. In Proceedings of IEEE Wireless Communications and Networking Conference (WCNC), Hong Kong, China, March 2007. [Yiu7] S. Yiu, R. Schober, and L. Lampe. Uncoordinated distributed space-time trellis coding. In Proceedings of the IEEE Interna-tional Conference on Communications (ICC), Istanbul,. Turkey, pages 2957-2962, June 2006. [Yiu8] S. Yiu and R. Schober. A new class of distributed space-time trellis codes. In Proceedings of the IEEE International Waveform Diversity and Design Conference, Hawaii, USA, January 2006. [Yiu9] S. Yiu, R. Schober, and L. Lampe. Distributed space-time block coding for cooperative networks with multiple-antenna nodes. In Proceedings of the IEEE Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Puerto Val-larta, Mexico, pages 52-55, December 2005. [YiulO] S. Yiu, R. Schober, and L. Lampe. Distributed space-time block coding. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), St. Louis, USA, pages 1592-1597, November-December 2005. [Yiull] S. Yiu, R. Schober, and L. Lampe. On distributed space-time filtering. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM), St. Louis, USA, pages 3324-3329, November-December 2005. 12 [Yiu 12] S. Yiu, R. Schober, and L. Lampe. Noncoherent distributed space-time block coding. In Proceedings of the IEEE Vehicu-lar Technology Conference (VTC), Dallas, USA, pages 947-951, September 2005. [Yiu 13] S. Yiu, R. Schober, and L. Lampe. Optimization of distributed space—time filtering. In Proceedings of the IEEE Vehicular Tech-nology Conference (VTC), Dallas, USA, pages 1829-1833, Septem-ber 2005. [Yiul4] S. Yiu, R. Schober, and L. Lampe. Differential distributed space-time block coding. In Proceedings of the IEEE Pacific Rim (PACRIM) Conference, Victoria, Canada, pages 53-56, August 2005. [Yiu 15] L. Lampe, R. Schober, and S. Yiu. Multihop transmission in power line communication networks: analysis and distributed space-time coding. In Proceedings of the Sixth IEEE Inter-national Workshop on Signal Processing Advances in Wireless Communications (SPAWC), New York, pages 1006-1012, June 2005. 13 Chapter 2 Preliminaries It is well known that bandwidth limitation and channel fading are two major obstacles to achieve reliable high data rate wireless communication. Recent interest in multiple-input and multiple-output (MIMO) systems [5] has been largely motivated by information theoretic results showing the substantial ca-pacity gain achieved by such systems [6, 7, 9, 77]. MIMO systems also can be used for diversity purpose. The various schemes that realize the poten-tial diversity gain of multiple-antenna systems are referred to as space-time coding schemes [10-13, 78-84]. In these schemes, a number of coded symbols is generated and transmitted simultaneously by each antenna. The receiver decodes the received signal using signal processing techniques to arrive at a diversity gain. The history of space-time codes (STCs) can be traced back to the early works on delay diversity (DD) proposed independently by Wittneben [19] and Se-shadri and Winters [20]. It is worth mentioning that DD is a special case of space-time filtering that refers to the class of transmission schemes employing space-time filters at the transmit antennas. The fundamental development of space—time coding first appeared in [11, 80], where the performance criteria for STCs were derived. In [11], several space—time trellis codes (STTCs) for two 14 to four transmit antennas, which perform close to the outage capacity outlined in [6] and [7], were also proposed. However, for space—time trellis coding, a Viterbi decoder [85, 86] is required at the receiver; therefore, the maximum likelihood (ML) decoding complexity of STTCs is high. Another class of STC is space-time block code (STBC), where the information data stream is encoded in blocks. STBC designs in the literature are mostly motivated by one or more of the following attractive properties: full diversity, full data rate, and simple linear M L detection. Orthogonal STBCs always achieve full diversity, and, unlike STTCs which require maximum-likelihood sequence estimation (MLSE) decoding, simple linear M L detection is possible at the receiver. They were first proposed by Alamouti for a two-transmit antenna system [10]. Later, Tarokh et al. applied the theory of orthogonal designs to create orthogonal STBCs for more than two transmit antennas [13]. The authors of [13] also showed that the Alamouti's STBC is the only complex orthogonal STBC that achieves full rate. A class of full-rate quasi-orthogonal STBCs for systems with four transmit antennas was proposed in [87, 88]. However, these quasi-orthogonal codes achieve only partial diversity. The aforementioned STBCs [10, 13, 87, 88] are coherent schemes which require channel state information (CSI) at the receiver. In situations where the fading channel changes quickly, the channel estimation overhead may make coherent detection unattractive. In light of this limitation, noncoherent [89, 90] and differential [91, 92] space-time coding methods have also been developed in order to eliminate the need for training symbols. The purpose of this chapter is to familiarize readers with some background materials and important results on space-time coding and space-time filtering that are relevant to our development of distributed space-time coding and dis-tributed space—time filtering schemes in Chapters 3-5. In particular, through 15 the analysis of the pairwise error probability (PEP), we characterize the di-versity gain and coding gain achieved by each scheme. The receiver structure for each scheme will be also presented. The rest of this chapter is organized as follows. In Section 2.1, we introduce STBCs. Specifically, we consider coher-ent STBCs, noncoherent STBCs, and differential STBCs, for their simplicity and practicality. In Section 2.2, we shift our attention to STTCs, and Section 2.3 is devoted to space-time filtering. Finally, in Section 2.4 we present the system model and outline the major assumptions we make for the subsequent chapters. To convey and emphasize the concept of transmit diversity provided by the various schemes, we assume a single receive antenna throughout this chapter. We introduce coherent space-time block coding, noncoherent space-time block coding, and differential space-time block coding in order in the following three subsections. 2.1.1 Coherent Space-Time Block Coding We first consider general coherent STBCs and then focus on orthogonal de-signs. In coherent space-time block coding, the information data stream is divided into groups of L source symbols {XQ, X \ , . . . , } and these symbols are encoded in a STBC matrix expressed as follows: 2.1 Space—Time Block Coding B = (2.1) T - l 16 XL-h • • • , X2, Xh X0 STBC Encoder bT_1} . . . , 6j, 6j &T-1> • • • j &?> &o r uNc hNc hNc ^ Figure 2.1: STBC encoder with iV c antennas. where 6™c, t = 0, . . . , T — 1, nc = 1,..., iV c , is the encoded symbol transmitted by antenna nc at time t. The STBC encoder is shown in Fig. 2.1. The encoded symbols 6"° are normalized to Uc=l J (2.2) to ensure that the total energy transmitted by the iV c antennas in any symbol interval t is equal to 1. £{•} in Eq. (2.2) denotes the statistical expectation of the argument. Since L symbols are transmitted in T symbol intervals, the code rate R is defined as R = L/T. Assuming a single receive antenna, the received signals in T symbol intervals can be collected in a column vector r = [ro ... rT-i]T given by r = Bh + n, (2.3) where h = [hi ... /ijvc]T and n = [no ... nT-i]T- hnc is the fading gain between antenna nc and the receive antenna, and is modeled as an independent and identically distributed (i.i.d.) zero-mean complex Gaussian random variable with unit variance (Rayleigh fading) [15]. It is customary to assume that h 17 is constant over T consecutive symbol intervals (quasi-static channels). The additive white Gaussian noise (AWGN) samples collected in the noise vector n are modeled as i.i.d. zero-mean complex Gaussian random variables with variance a\ = N0/Es, where Es and No denote the average received energy per (scalar) symbol interval and the one-sided power spectral density of the underlying continuous-time noise process, respectively. Assuming coherent detection and perfect channel estimation, the M L estimate B for the information-carrying vector B is given by where B is the set of all possible codewords and || • H2 denotes the Z/2-norm of a vector. We refer the readers to [93] for details on channel estimation techniques. In general, the channels can be estimated by transmitting training symbols, provided that the channels change slowly compared with the symbol rate. Given that B0 is the transmitted codeword, the PEP PE is the probability of receiving an incorrect codeword B\, {BQ,BI} G B,B0 ^ B\. For coherent STBCs, the PE is upper bounded by [11] at high signal-to-noise ratios (SNRs). Matrix A is defined as A = E E, E = B0 — B\, and Xi(A) and r(A) denote the ith eigenvalue of A and the rank of A, respectively. We define the coding gain and diversity gain as B = argmin {\\r - Bh\\22} (2.4) (2.5) (2.6) 18 respectively. Thus, both the diversity gain and the coding gain should be maximized to achieve the optimum performance. At high SNRs, the upper bound in Eq. (2.5) is dominated by the diversity gain d, which determines the asymptotic slope of the bit error rate (BER) curve in log scale. On the other hand, the coding gain specifies the offset of the BER curve from that of an un-coded system. Coherent Orthogonal Space—Time Block Coding The M L detection strategy described in Eq. (2.4) requires minimizing the decision metric over all possible combinations {XQ, XI, ..., XL-I} resulting in a decoding complexity that is exponential in L. In 1998, Alamouti [10] excited the communications community with the invention of a full-transmit diversity scheme for Nc = 2 transmit antennas. This scheme allows for simple linear M L decoding at the receiver. The source symbols are first divided into groups of two symbols {XQ,X\} and then fed to the space-time encoder. At any given symbol period, two encoded symbols are transmitted simultaneously from the two transmit antennas. In the first time slot, symbols XQ and x\ are transmitted by antenna 1 and 2, respectively. In the next time slot, antenna 1 transmits — x\ and antenna 2 transmits XQ. We note that L = 2 symbols are transmitted in T — 2 symbol intervals. Thus, the rate of this STBC is R = 1 and hence, there is no bandwidth expansion resulting from this coding scheme. The (orthogonal) STBC matrix can be written as x0 Xi B= * , (2.7) — Xi XQ and the received signals in the two symbol intervals can be collected in the received vector r given by Eq. (2.3). In particular, the received signals in the 19 two time slots can be expressed as ro = hiXo + h2Xi + no and r i = — h\xl + h2x*a + n\, (2.8) respectively. We define a new received vector y = [r0 rl]T such that Eq. (2.8) can now be written as y = Hb + n, (2.9) where b = [aj0 xi]T, n = [no n\}T, and H ± hi h2 hi -K (2.10) Assuming that the encoded symbols are uncorrelated, the source vector b is normalized to £{bbH} = ±-INc, (2.11) in order to satisfy the energy constraint presented in Eq. (2.2). Ix in Eq. (2.11) denotes an identity matrix with dimension X x X. Assuming coherent detec-tion and perfect channel estimation, the ML estimate b for the information carrying vector b is given by b= argmin {\\y - Hbf2} . b€B (2.12) At first sight, it appears that the M L detection strategy described in Eq. (2.12) requires minimizing the decision metric over all possible combinations {xn, xi}, resulting in a decoding complexity that is again exponential in L, as in Eq. (2.4). However, closer inspection reveals that the columns of the channel matrix H 20 are orthogonal, regardless of the channel coefficients, i.e., H H — calNci (2.13) where a = \hL\2 + \h2 Making use of Eq. (2.13), we obtain y A HHy = ab + h, (2.14) where h = HHn. It can be easily verified that the elements of h are still i.i.d.. Therefore, the decision rule in Eq. (2.12) becomes Remarkably, by using simple linear combining, the joint M L decoding for sym-bols {x0, xi} in Eq. (2.12) is decoupled into two individual decoding problems. Thus, the decoding complexity increases only linearly, as opposed to exponen-tially, with the code size L. We note that the linear combining technique presented above in Eq. (2.14) can be applied to any STBC that satisfies the following orthogonality property (cf. Eqs. (2.7) and (2.20)): Thus, orthogonal STBCs are very attractive due to their linear decoding com-plexity. Using Eqs. (2.9), (2.11), and (2.13), the SNR corresponding to each symbol interval of any orthogonal STBC can be calculated easily, and is given by (2.15) £{BHB} = -INc. (2-16) SNR = a Es _ aEb\og2M N~'N~o~ NCN0 (2.17) 21 where a is now defined as a = \hi •i\2 + • • • + \hNc\2 and Es = Eblog2M, (2.18) where Eb and M denote the received energy per bit and the signal constellation size of xi, 0 < I < L, respectively. For Alamouti's scheme, which clearly shows that Alamouti's scheme provides a diversity order of d — 2. Eq. (2.19) also suggests that there is a 3 dB penalty to Alamouti's scheme, com-pared with a system with 1 transmit and 2 receive antennas with maximal-ratio combining (MRC), due to the splitting of the total transmit power between the two transmit antennas. The general orthogonal STBCs that achieve full diversity and allow for linear processing of more than Nc — 2 transmit antennas have been studied exten-sively in [12, 13, 87, 94]. In general, there are two classes of STBCs from orthogonal designs. One class consists of those from real orthogonal designs for real constellations, such as pulse-amplitude modulation (PAM). For real constellations and a square matrix, i.e., Nc = T, it was proven in [13] that rate one (R = 1) codes can be achieved only for Nc = 2, 4, or 8 antennas. By allowing T > Nc, it was shown that STBCs with transmission rate R = 1 can be constructed for any number of transmit antennas [13] . The other class of STBCs consists of those from complex orthogonal designs for complex constellations, such as M-ary quadrature amplitude modulation (MQAM) and M-ary phase shift keying (MPSK). However, for complex or-thogonal designs, the situation is not as encouraging. For a square matrix (Nc = T), it was shown that a complex orthogonal design with R = 1 only SNR=(|/n|2 + |/i2|2) NCNQ' (2.19) 22 exists for Nc = T = 2 [13]. For Nc = T = 3 and Nc = T = 4, STBCs exist with R = 3/4 [13, 78]. For example, for 7VC = T = 4, XQ XI X2 X-[ XQ 0 —x\ 0 XQ 0 x5 — x 0 - Z 2 XQ (2.20) Furthermore, it was proven by Tarokh et al. that STBCs with complex con-stellations exist for any rate R < 1/2 for any number of transmit antennas [13]. Other efforts in designing higher-rate complex codes can be found in [95, 96]. Recently, it has been shown that orthogonal STBCs may be identified with packings on the Grassmann manifold [83]. STBCs from orthogonal designs always achieve full rank, i.e., r(A) = Nc (cf. Eq. (2.6)) and, therefore, guarantee maximum diversity. In this case, the coding gain defined in Eq. (2.6) can be written as G = det{EHE}1/N<, (2.21) where det{-} denotes the determinant of a matrix. Therefore, the code should ensure that the minimum value of det{EHE} is as large as possible. However, it is well known that orthogonal STBCs provide zero coding gain. Nevertheless, the special structure of orthogonal STBCs is still very appealing, because it achieves full diversity while also allowing simple (single-symbol) M L decoding. 2.1.2 Noncoherent Space-Time Block Coding The coherent STBCs presented in the last section require CSI at the receiver, cf. Eq. (2.12). In situations where the fading channel changes quickly from 23 burst to burst, training symbols are required for every burst, therefore coher-ent STBCs may not be suitable. Unitary space-time modulation (USTM), proposed in [89], does not require CSI at the receiver for detection. It was also shown in [89] that at high SNRs and T > Nc capacity can be achieved with unitary matrices. In [90], a systematic approach to the design of con-stellations of unitary space—time signals was presented. Mathematically, the transmission model can be also described by Eq. (2.3). However, here the information-carrying matrix B S B is a T x Nc unitary matrix, where T > Nc, i.e., BHB — J^INC, and h is piecewise constant. Other design approaches to unitary space—time signals can be found in [97-99]. Since the number of unitary space—time signals I = 2RT required to transmit R bits/(channel use) grows exponentially with both R and T, generating and storing this many T x Nc complex matrices is cumbersome if the signals do not have any additional structure [90]. Hochwald et al. proposed the following structure for the unitary matrices B = {3>i,..., [90], $ i = e i _ 1 * i , l<i<I, (2.22) where the columns of the T x Nc matrix $ i are drawn from a (normalized) T xT discrete Fourier transform (DFT) matrix and 0 ej(27r//)ui _ _ Q eJ(2n/I)uT (2.23) In Eq. (2.23) u\,...,ur £ {0, . . . ,1 — 1} are optimized for maximum BER performance, cf. e.g. [90]. It was shown in [89] that the noncoherent M L decision rule for unitary space-24 time signals is given by B = argmax {rHBBHr} . (2.24) BeB Clearly, Eq. (2.24) indicates that CSI is not required for M L decoding. In addition, the noncoherent decision rule presented in the above equation can be also viewed as a generalized likelihood ratio test (GLRT) [90], as follows: h ± argmin {||r - Bhf2} = ^BHr, (2.25) h 1 B = argmin | | | r - Bh\\l}. (2.26) BeB In Eq. (2.25), h is first estimated assuming B is known, and in Eq. (2.26), the channel estimate h is used to detect B. Combining Eqs. (2.25) and (2.26) and dropping irrelevant terms, the GLRT decision rule is identical to the optimum noncoherent decision rule shown in Eq. (2.24). We show in Appendix A that for high SNRs the PEP for noncoherent USTM can be upper bounded by 1 r { A ) 1 / P \ - r < A ) ^ n ^ ( « o - ) , (2.27) where ao = —1/\T(B0BQ — B\B^) > 0, and matrix A is defined as A = B0B$(B0B$ - B i B f ) , {B0, Bx] e B, B0 ^ Bv Obviously, the diversity 1 Note that, in general, Eq. (2.27) is not the Chernoff bound but a looser upper bound. Nevertheless, Eq. (2.27) is still useful for our development of D S T B C , to be presented in Chapter 3. 25 gain of USTM is d — r(A). We define the coding gain as l/r(A) G ± JJ \i(BQE<Q (BQBQ - BiBf)) (2.28) Furthermore, for full-rank noncoherent STBCs (r (A) = Nc), the coding gain can be simplified to 2 2.1.3 Differential Space-Time Block Coding We finish Section 2.1 on space-time block coding with a discussion of differen-tial STBCs. Differential space-time block coding [91, 92, 100-105] is attractive in time-varying fading channels where fading coefficients change continuously. In this case, it is reasonable to assume that CSI is not available at either the transmitter or the receiver. Based on the unitary space-time signals originally proposed for noncoherent STBC, differential USTM was developed in [91]. A similar idea was proposed independently in [92], where Hughes considered group codes. Tarokh et al. developed a differential STBC scheme [101, 102] where the differential modulation was combined with orthogonal space-time block coding [13]. The aforementioned differential STBCs can only be used in conjunction with constant modulus constellations, e.g., M-ary PSK. Since non-constant modulus constellations, e.g., M-ary Q A M , have a larger min-imum distance compared with M-ary PSK, differential STBCs using non-constant modulus constellations have also been considered [103]. We will show later in this section that, similar to differential PSK (DPSK) in single-antenna 2 This simplification is useful for comparison with the noncoherent DSTBCs to be devel-oped in Section 3.3.3. l/Nc G = det INC — BQ B\BX B0 (2.29) 26 systems, there is also a 3 dB SNR penalty to differential STBCs when differen-tial detection is used at the receiver3. It is shown in [104] that the gap can be narrowed down to 2 dB by introducing redundancy in the differential encoder. Furthermore, Schober and Lampe show that the power efficiency of differential STBCs can be improved by increasing the observation window [106]. In the following, we follow closely Hochwald and Sweldens' work on differential USTM [91]. For differential STBC, T = Nc holds and the STC matrices are differentially encoded, using B[k] = V[k]B[k-l], (2.30) where V[k] € V is a unitary matrix (i.e., VH[/c]V[&] = I;vc) containing the source information. Since the code matrices are encoded differentially, we introduce the discrete block index k G Z in Eq. (2.30). Assuming that the fading coefficients are constant over 2T symbol intervals, similar to Eq. (2.3), the received vectors in two consecutive blocks can be expressed as r[k - 1} = B[k - l}h + n[k - 1} and r[k] = B[k]h + n[k], (2.31) respectively. Using Eqs. (2.30) and (2.31), the received signal in the kth block can be written as r[k] = V[k]r[k - 1] + n[k] - V[k}n[k - 1]. (2.32) Realizing that the noise vectors are independent and statistically invariant to multiplication by unitary matrices, we may write Eq. (2.32) as r[k] = V[k]r[k - 1] + V2n'[k], (2.33) 3 This 3 dB SNR penalty can be recovered if coherent detection is used at the receiver. 27 where the elements of n'[k] are zero mean Gaussian random variables with unit variance. Comparing Eq. (2.33) with Eq. (2.3), we note that the signal V[k] appears to be transmitted through a channel with fading response r[k — 1], which is known to the receiver, and corrupted by noise with twice the variance resulting in the infamous 3 dB penalty. Considering Eq. (2.33), the ML estimate of V[k] for V[k] is [91] V[k] = argmin {\\r[k] - V[k]r[k - l]\\l} ; (2.34) v\k]ev remarkably, no knowledge about the channel vector h is required. In differential USTM, the information bits are encoded in a square matrix V = {<3>i,..., <£/}, where i — is a special case of the unitary space-time matrix presented in the last section (cf. Eqs. (2.22) and (2.23)) with T = Nc and 3>i = IT- Furthermore, by constructing the code matrix using the group constellations [91], the transmitter does not need to perform matrix multiplication to compute the transmitted matrix B\k). Eq. (2.33) suggests that we can consider the differential scheme as the coherent case (cf. Eq. (2.3)) with twice the noise variance. Therefore, the PEP for differential STBC is bounded by r { A ) 1 / F \ - r ( A ) where A is now defined as A = EHE, E = V0-Vu {V0, VL} € V, V0 ^ Vv The coding gain and diversity gain of differential STBC are also defined the same way as in the coherent case, cf. Eqs. (2.6) and (2.21). 28 2.2 Space-Time Trellis Coding Building upon the framework of DD [18, 19], Tarokh et al. established the fundamental description of STTC in [11]. In this section, a brief introduction to STTCs will be given. x[2], x[l),x[0) STTC Encoder M I Y - I ] , 6i[o] b2[K-l], . . . ,6 2 [1] ,& 2 [0] ' L J ' L J 1 L J bNc[K - 1], ..., bNc[l], bNc[0] Figure 2.2: STTC encoder with NC antennas. In STTCs, the space—time trellis encoder maps the source information symbols x[k] into iV c symbols 6nc[/c], 1 < n c < iV c, to be transmitted simultaneously by NC antennas at time k, 0 < k < K. The STTC encoding process is depicted in Fig. 2.2. The received signal at time k is given by r[k] = b[k]h + n[k], (2.36) where b[k] = [bi[k] b2[k] ... bNc[k]], and the fading gains corresponding to the NC transmit antennas are collected in h = [hi ... hNc]T. We do not include the time index in the channel vector h in Eq. (2.36) because it is customary to assume that the channel gains are constant over the frame length K and change independently from one frame to another. n[k) is the discrete AWGN sample at time k, modeled as i.i.d. zero-mean complex Gaussian random variables with variance a\ = NQ/ES. Furthermore, it is assumed that the elements of 29 b[k] are uncorrelated and normalized, such that £{b [k]b[k]} = -^-INC-The received signals r = [r[0] r[l] . . . r[K — 1]]T in the whole frame can be written as r = Bh + n, (2.37) where the K xNc matrix B = [bT[0] bT[l] ... bT[K- l ] ] r is the STTC matrix. Mathematically, Eq. (2.37) has exactly the same form as the channel equation for STBC (cf. Eq. (2.3)). However, it should be noted that the codeword matrices B for the two cases have completely different structures and are generated very differently. For STBC, groups of symbols are encoded in the STBC matrix all at once, whereas for STTC, the space-time trellis encoder maps one input symbol to produce iV c symbols at any time instance k. Since the encoder has memory, these symbols produced at different time instances are dependent on time. Decoding is performed via MLSE using a Viterbi decoder [85, 86] to compute the path with the lowest accumulated metric. In particular, assuming perfect channel estimation, the branch metric for STTC decoding at time k is given by [11] 2 (2.38) Since Eq. (2.37) has exactly the same form as Eq. (2.3), the PEP of STTC is also bounded by Eq. (2.5) and the coding gain and diversity gain are defined exactly the same way as in Eq. (2.6). Therefore, it is clear that in designing a STTC, the rank for the error matrix r(A) and G (cf. Eq. (2.6)) should be maximized, thereby maximizing both the diversity gain and coding gain. Similar to traditional trellis-coded modulation (TCM) in single-antenna sys-tems, STTCs provide a coding gain. Since they can also achieve full diversity, their main advantage over STBCs is the provision of a coding gain. However, NC n c =l 30 this advantage comes at the expense of increased receiver complexity. Several hand-crafted codes that achieve full diversity gain are provided in [11]. Since the introduction of the original STTC in [11], an extensive research effort has aimed at improving the performance of the original designs [81, 107-113]. However, only marginal gains were obtained in most cases. The performance of STTCs transmitted over independent and non-identically distributed (i.n.d.) rapid Rayleigh fading channels is considered in [112], and practical considera-tions such as channel estimation errors, mobility, multiple paths, and antenna correlation can be found in [114-116]. 2 . 3 S p a c e — T i m e F i l t e r i n g Space—time filtering is a generic term that is used to describe DD [19, 20], generalized DD [117], and optimized DD [118], since space-time filters are em-ployed in all of these schemes. DD was first proposed in [19, 20] for frequency-nonselective channels to effectively provide transmit. diversity. The idea is to introduce deliberate resolvable multipath distortion by transmitting the source symbols x[k] and their, delayed versions using multiple antennas and space-time filters. Specifically, the signal bnc [k] transmitted by antenna n c , 1 < nc < Nc, is given by bne[k] = x[k]*5[k-(nc-l)], (2.39) where * denotes the discrete-time convolution and 5[-} denotes the Dirac delta function. Let hnc be the fading gain between transmit antenna nc and the receive antenna. The received signal at time k is then given by r[k] = h^c • bnAk] + n[k] = x[k] * heq[k] + n[k], (2.40) n c = l 31 , x[2], x[l], x[0] 5[k] S[k - Lh] S[k - 2Lh] 6[k - (Nc - l)Lh] h[k] b2[k] h[k] bNc[k] Figure 2.3: Generalized DD encoder with NC antennas. where the equivalent overall channel impulse responses (CIRs) heq[k] of length L e q = NC is defined as heq[k] - ^ K c - S[k - (n c - 1)], (2.41) n c = l and n[k] is the AWGN noise sample taken at time k. Both n[k] and hnc are modeled as i.i.d. zero-mean complex Gaussian random variables and hnc has unit variance, whereas n[k] has variance a2 = NQ/ES. From Eq. (2.40), it is clear that DD transforms the multiple-input single-output (MISO) frequency-nonselective channels into a single-input single-output (SISO) frequency-selective channel. At the receiver, MLSE is used to completely resolve the multipath distortion, and full diversity order d = NC can be achieved. DD is generalized to frequency-selective channels with Lh taps in [117]. In 32 the resulting generalized DD scheme, the transmit antenna n c , 1 < nc < Nc, delays the transmitted data stream by (n c — l)Lh symbols4. Therefore, the signal bnc[k] transmitted by antenna n c , 1 < nc < Nc, in generalized DD is given by (cf. Fig. 2.3) Assuming a single receive antenna, the received signal at time k is given by Nc r[k] = J2hne[k]*bnc[k]+n[k], (2.43) n c = l where hnc[k], 0 < k < Lh — 1 is the discrete-time frequency-selective CIRs between transmit antenna nc and the receive antenna. According to Eq. (2.42), the received signal r[k] given in Eq. (2.43) can be written as where the equivalent CIRs heq[k] (of length L e q = NcLh) is defined as No heq[k] 4 £ hnc[k] * S[k - (n c - l)Lh]. (2.45) n c = l Obviously, full transmit diversity ci = NcLh can be achieved if optimum MLSE is employed at the receiver. However, the computational complexity of MLSE increases exponentially in MLhNc~l where M is the size of the signal constella-tions. Therefore, other known suboptimum SISO equalization techniques, such as decision-feedback sequence estimation (DFSE) [119] and minimum-mean squared error (MMSE) decision-feedback equalization (DFE) [120, 121], may 4Note that the L/, = 1 special case corresponds to conventional DD for frequency-nonselective channels. bnc[k] = x[k] * 5[k - (n c - l)Lh]. (2.42) r[k] = x[k] * heq[k] + n[k], (2.44) 33 be desirable. In general, the delays of the transmitted data stream can be realized using discrete-time filters with Lg taps. Specifically, the filter coefficients of antenna nc can be collected in a filter vector gHc == [gnc [0] gnc [1] . . . gnc [Lg — 1]]T, where 9nc[{nc — l)Lh] — 1 and gnc[l] = 0 V / ^ (n c — l)L/j. For a channel with Lh taps, it can be easily verified that Lg = Lh(Nc — 1) + 1 is a necessary condition to exploit the maximum available diversity [118]. However, Lg < Lh(Nc — 1) + 1 might be desirable in practice for complexity reasons. In [118], optimized DD was introduced, and it was shown that filters with complex coefficients and shorter filter length Lg < Lh(Nc — 1) + 1 outperform generalized DD significantly for SNRs of practical interest. As indicated in [118], a natural optimization criterion for the set of filter vectors is the minimization of the maximum BER of the transmitted data sequence. Unfortunately, Eq. (2.44) shows that the received signal is impaired by inter-symbol interference (ISI), and it seems not possible to derive a meaningful BER expression for continuous transmission that could be exploited for optimiza-tion of gn , V n c . On the other hand, it is well known that the matched-filter bound accurately predicts the BER of optimum MLSE for random fading chan-nels, cf. e.g. [122]. Therefore, the filter vectors were optimized for the PEP of single-symbol transmission in [118]. In particular, it was shown that the PEP of the resulting optimized DD scheme with optimum M L detection and arbitrary filter length Lg can be bounded by [118] Pe < : t t — , (2.46) e ~ det{J L e q + 7 G * G H } ' V J where L e q = Lg + Lh — 1 can be considered as the effective channel length. The L eq x LhNc matrix G is defined as G = [Gi G2 . . . GjvJ, where Gnc, 1 < nc < Nc is a L e q x Lh column-circulant matrix with vector [g^c 0 0 . . . 0] T 34 (length: Leq) as the first column. Furthermore, the LhNc x LhNc channel correlation matrix is defined as $ = £{hhH} with h = [h[ h\ ... hJfc]T, hnc = [hnc[0] hnc[l] ... hnc[Lh - 1]]T and the effective SNR 7 is given by 7 = ci2ninjBs/(4A^o), where d m j n denotes the minimum Euclidean distance of the signal constellation. At high SNRs, Eq. (2.46) can be written as r(A) p - U r n - { 2 A 1 ) where A == G&GH. According to Eq. (2.47), the diversity order d of the optimized DD scheme is given by the rank of G<bGH, i.e., d = r(G$GH). Assuming that G and $ both have full rank, the maximum diversity order is given by d = min{L e q , LhNc}. Finally, the optimum filter vectors gnc> 1 < nc < Nc were obtained by using a gradient algorithm to minimize the upper bound in Eq. (2.46) [118]. Extension of optimized DD to suboptimum equalizations, including DFE [123] and linear equalization (LE) [124], can be found in [125]. 2.4 System Model and Assumptions In the previous three sections, we have reviewed several conventional space-time coding and space-time filtering schemes advocated for co-located anten-nas. In Chapters 3-5, we consider a cooperative network with a large set of DaF relay nodes M, but in which at any given time only an a priori unknown subset of nodes S C J\f can be active and participate in the second phase of transmission. We assume that the N = \M\ relay nodes are uniformly dis-tributed in a circle with radius R. The distance from the source node to the center of the circle and the distance from the center of the circle to the des-tination node is denoted by l\ and 1%, respectively, cf. Fig. 2.4. Therefore, in 35 Figure 2.4: A network with N = 8 DaF relay nodes and Ns = 2 active relay nodes (S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. l\\ Relay nodes are uniformly distributed in a circle with radius R. l\\ Distance from the source node to the center of the circle. l2: Distance from the center of the circle to the destination node. contrast to Sections 2.1-2.3, where the transmit antennas are co-located, the antennas of the N relay nodes are physically distributed for the network model under consideration. In our simulation, we detect erroneous packets by simply comparing the trans-mitted packet with the packets detected by the relay nodes. In practice, a cyclic redundancy check (CRC) code may be used to enable the relay nodes to detect erroneous packets. To facilitate a fair comparison of the various pro-posed schemes with different numbers of active nodes, the average total energy transmitted by all of the active nodes <S per (scalar) symbol interval is nor-malized such that it is independent of the number of active nodes Ns — \S\. In a practical network, each node may transmit with a fixed power, and the total transmitted power may increase with the number of active nodes. We assume frequency-nonselective channels for our development of DSTBCs and DSTTCs in Chapters 3 and 4. Furthermore, following the related lit-36 erature, e.g. [25, 57, 59], we assume that all relay nodes transmit time syn-chronously. Time synchronous transmission is easily accomplished if the rela-tive delays between the relay nodes are much smaller than the symbol duration Ts. Considering the scenario shown in Fig. 2.4, and assuming that all relay nodes are inside a circle of radius R, the maximum possible delay between two nodes is r m a x = 2R/c, where c is the speed of light. For example, for R — 100 m we obtain r m a x = 0.67 fis, and Ts 3> r m a x is a realistic assump-tion for sensor-type applications, which are typically low rate. If the relative delays between relay nodes are not negligible, individual relay nodes have to advance or delay their transmit signals in order to achieve synchronism at the receiving node. In [126, 127], the asynchronous phenomenon in cooperative networks was analyzed, and space-time coded orthogonal frequency division multiplexing (STC-OFDM) was proposed to achieve full asynchronous cooper-ative diversity. It was shown in [128] that distributed diversity can be achieved in asynchronous wireless sensor networks (WSNs) by employing a ultrawide bandwidth (UWB) transmitted reference signaling scheme. Recently, Veronesi and Goeckel have proposed a multiple frequency offset compensation scheme to counteract the effect of multiple frequency offsets in a distributed MISO system [129], whereas litis et al. have suggested a generalized successive inter-ference cancellation (GSIC) scheme to estimate the sensor carrier offset and the virtual MIMO channels in a joint fashion [130]. We refer the readers to [131] for a more detailed discussion of time synchronism in the context of wireless sensor networks. For ease of exposition, we first assume single-antenna relay and destination nodes for our discussion in Chapter 3, and consider multiple-antenna relay and destination nodes in Chapter 4. For practical systems, knowledge about the location of the relay nodes is often not available. Therefore, for the design 37 of DSTBC in Chapter 3, we model the channels between the relay nodes and the destination node as i.i.d. random variables, which corresponds to the spe-cial case of l2 S> R in Fig. 2.4. The case of non-identical channel variances is also considered in Chapter 4 in the context of DSTTC. Frequency-selective channels are assumed for the development of distributed space-time filtering (DSTF) in Chapter 5. Unlike Chapters 3 and 4, we relax the timing synchro-nism restriction, because the effect of imperfect timing synchronization simply causes additional ISL In Chapter 6, we consider a binary hypothesis testing problem where a set JC = {1, 2 , . . . , K} of K distributed sensors tries to determine the true state of nature H as being Ho of H\. Unlike the aforementioned relay networks, where error detection can be facilitated by employing a CRC code, error detection is not possible at the sensor nodes. In contrast to the traditional assumptions that the local decisions are transmitted to the fusion center (FC) through perfect and error-free channels [132-137], we assume realistically that the lo-cal decisions are transmitted through noisy and frequency-nonselective fading channels. Single-antenna sensor nodes and i.i.d. channels between the sensors and the FC are also assumed. 2 . 5 C o n c l u s i o n s In this chapter, we have introduced the concept of space-time coding and space-time filtering for multiple co-located transmit antenna systems. In par-ticular, we have provided a concise presentation of the general theory of space-time block coding, space—time trellis coding, and various space-time filtering schemes. The upper bound of the PEP for each scheme has been characterized. Through the PEP analysis, we have illustrated the coding gain and diversity gain achieved by each scheme. In addition, we have also presented the receiver 38 structure of each scheme. Finally, the various assumptions we make in the following chapters have been outlined. 39 Chapter 3 Distributed Space-Time Block Coding 3 . 1 I n t r o d u c t i o n In this chapter, a new class of distributed space—time block codes (DSTBCs) is introduced. These DSTBCs are designed for wireless networks that have a large set of single—antenna DaF relay nodes W, but in which at any given time an a priori unknown subset of nodes S C J\f can be active, cf. Fig. 3.1. In the proposed scheme, the signal transmitted by an active relay node is the product of an information-carrying code matrix, which contains the correctly decoded source information and is therefore identical to all nodes, and a unique node signature vector of length iV c. Recall from Section 2.1 that Nc corresponds to the number of columns of a space-time block code (STBC) (Eq. (2.1)). This approach allows us to conveniently exploit existing coherent [10, 13], differ-ential [91, 101], and noncoherent [89, 90] STBCs originally designed for Nc co-located antennas (cf. Section 2.1). In particular, it is shown that exist-ing STBCs designed for Nc> 2 co-located antennas are favorable choices for 40 the code matrix, guaranteeing a diversity order of d = min{iV5,7V"C} if Ns nodes are active. For the most interesting case Ns > Nc, the performance loss entailed by the distributed implementation is analytically characterized. Fur-thermore, efficient methods for the optimization of the set of signature vectors are provided. Depending on the chosen design, the proposed DSTBCs allow for low-complexity coherent, differential, and noncoherent detection, respectively. Possible applications include ad hoc and sensor networks employing DaF re-laying. In summary, we make the following contributions in this chapter: • We provide design guidelines for the proposed DSTBCs. Unlike Lane-man's scheme [25], where the number of nodes in the network has to be less than the space-dimension of the STBC, the proposed scheme is applicable to networks with an arbitrary number of nodes. • We show that with a full-rank STBC designed originally for Nc co-located antennas, a diversity order of d = min{Ns, Nc} can be guaran-teed if Ns nodes are active. • We propose three efficient optimization techniques for signature vector design. • The performance loss entailed by the distributed implementation is an-alytically characterized for the most interesting case where Ns > Nc. • Our simulation results show that the proposed DSTBCs yield a signif-icant performance gain over single-hop transmission while allowing for low-complexity coherent, differential, and noncoherent detection. Organization: This chapter is organized as follows. In Section 3.2, the consid-ered communication network model is defined and the proposed DSTBCs are introduced. Design criteria for coherent, differential, and noncoherent DST-BCs are given in Section 3.3, while the design of the node signature vectors is 41 discussed in Section 3.4. In Section 3.5, simulation results are provided, and Section 3.6 concludes the chapter. 3.2 Network Model and DSTBCs The considered communications network is depicted in Fig. 3.1. For ease of exposition and practical relevance, we assume in this chapter that each node is equipped with a single antenna. However, the results presented in this chapter can be easily extended to the case where relay nodes are equipped with multiple antennas1. Each relay node is assigned a number n € AT, M = {1, 2, . . . , N}, and a unique, unit-norm signature vector gn S Q, \\gn\\l = 1, n S J\f, of length 7VC, where Q denotes the set of node signature vectors. As we have already mentioned, Nc coincides with the space—dimension of a STBC (Eq. (2.1)), and its role will become clear in Section 3.3. The Ns active nodes transmit in T > Nc consecutive symbol intervals the elements of the vector sn[k} = ^/PsB[k}gn, neS, (3.1) where k G Z is the discrete time index, Ps — Nc/Ns is a normalization constant, and B[k] € B is a T x Nc STBC matrix (cf. Section 2.1), which carries the information to be transmitted and is identical for all nodes. For convenience, the code B is normalized as £{BH[k]B[k]} = J^INC- There-fore, the average transmitted energy per node and (scalar) symbol interval is £{s"[k]sn[k]}/T = 1/Ns, i.e., the total transmitted energy per symbol inter-val is independent of the number of active nodes Ns. Assuming a single receive antenna at the destination node and perfect syn-xThe case of multiple-antenna relay and destination nodes will be considered in Chapter 4 in the context of distributed space-time trellis code (DSTTC). 42 Figure 3.1: A network with N = 8 DaF relay nodes and Ns — 2 active relay nodes {S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. l\: Relay nodes are uniformly distributed in a circle with radius R. l\\ Distance from the source node to the center of the circle. l2: Distance from the center of the circle to the destination node. hn: Fading gain from relay node n to the destination node. chronization2, the vector of signal samples received in T consecutive symbol intervals is modeled as r\k] = JFs J2Blkl 9nhn + n[k] = JFsB[k] Gs hs + n[k], (3.2) where n[k] contains the additive white Gaussian noise (AWGN) samples, and hn, n £ S, are the channel gains of the active nodes, cf. Fig. 3.1. The columns of the Nc x Ns matrix Gs are given by the node vectors gn, n £ S, and the elements of the channel column vector hs are the corresponding channel gains hn, n £ S. The noise vector n[k] has variance a\ — N0/Es. Since we do not assume any a priori knowledge about the location of the nodes, we model the channels between the relay nodes and the destination 2 A s mentioned in Section 2.4, time synchronous transmission can be accomplished if the relative delays between the relay nodes are much smaller than the symbol duration. However, it is worth mentioning that timing mismatch could be devastating in a distributed system because it destroys the space-time signal structure. 43 node hn, n G M as independent and identically distributed (i.i.d.) zero-mean Gaussian random variables (block Rayleigh fading) with variance o\ = \. For the scenario considered in Fig. 3.1, this channel model is valid for l2 S> R. In Section 3.5.2, we will investigate the effect of non-i.i.d. channel gains. For our discussion in Section 3.3, two different interpretations of Eq. (3.2) will be useful. First, Eq. (3.2) may be rewritten as r[k] = y/ps'Cs[k]hs + n[k], (3.3) where the effective DSTBC matrix is defined as Cs[A;] = B[k] Gs- This formu-lation shows that for performance optimization, the set C of DSTBC matrices Csf/c] has to be properly designed. On the other hand, since the node matrix Gs is unknown at the receiver, for receiver design it is preferable to express Eq. (3.2) as r[k] = B[k]fs + n[k], (3.4) where fs — VPsGshs denotes the effective channel vector. Note that the channel model described by Eq. (3.4) is identical to that for conventional STBCs with co-located antennas (cf. Eq. (2.3)). Both DSTBC optimization and receiver design will be addressed in the next section. 3.3 Optimization and Receiver Design In this section, we present coherent, differential, and noncoherent DSTBCs, provide criteria for optimization of these codes, and discuss suitable detection techniques. We also provide some guidelines for the choice of /V c. 44 3.3.1 Coherent DSTBCs We first consider the coherent receiver where channel state information (CSI) is required at the destination node. We note that for coherent DSTBCs, it is not required to estimate the individual channel gains of the channel vector hs, and that only knowledge of the effective channel vector fs has to be avail-able at the receiver, cf. Eqs. (3.4) and (2.3). For estimation of fs, known pilot matrices B[k] = Bp[k] may be inserted at the relay nodes, and channel estima-tion techniques originally developed for co-located transmit antennas can be applied at the receiver, cf. e.g. [93]. Assuming perfect channel estimation, the maximum-likelihood (ML) estimate B[k] for the information-carrying matrix B[k] is given by As usual in space-time coding [11], the Chernoff bound on the pairwise error probability (PEP) will serve as an optimization criterion for the proposed DSTBCs. Considering the similarity of Eqs. (2.3) and (3.3), at high signal-to-noise ratio (SNRs) the PEP is, therefore, bounded by Eq. (2.5) [11], which is also shown below for convenience of description in the sequel: The matrix A in Eq. (3.6) is now defined as A = Ps EH EGs, E = B0 - Bi, {B0, Bi} e B, B0 ^ Bv Eq. (3.6) shows that the diversity order d of the DSTBC is maximized by maximizing the rank, r(A). This procedure is achieved by ensuring that both E and Gs have full rank [138]. In this diversity order of d = r(A) = mm{Ns,Nc} is obtained [138, p. 13]. B[k] = argmin {||r[fc] - JB[fc]/5||!} • (3.5) 45 If Ns < Nc, the diversity order is limited by the number of active nodes. In this case, DSTBC design is complicated, since the code B and the set of node signature vectors Q have to be jointly optimized. On the other hand, our simulations in Section 3.5 show that DSTBCs designed for Ns > Nc also work well if Ns < Nc; therefore, Ns > Nc is assumed for code design. Note also that proper placement of a sufficiently large number of nodes can ensure that Ns > Nc is valid with high probability. The PEP expression given in Eq. (3.6) suggests that we define G = (Yll^ xi(A)YMA) a s t h e codin9 9ain i n analogy to Eq. (2.6) [11]. If E and Gs each have full rank, and Ns > Nc, G can be expressed as [138] (Nc \W° G = PS I HkiEGsG^E") j = Ps{det{EHE}det{GsG»})l/Nc. (3.7) This equation shows that for optimization of the DSTBC C, i.e., for mini-mization of the bound on the maximum PEP in Eq. (3.6), we may separately optimize the code B and the set of node signature vectors Q. The design of Q will be discussed in Section 3.4. The code B should ensure that E always has full rank and that the minimum value of det{EHE} is large. Since these are exactly the criteria for the design of coherent STBCs for 7YC co-located antennas (cf. Eq. (2.21) in Section 2.1.1), we can use existing STBC designs for B. For example, we may use the orthogonal designs proposed in [10, 13] (see also Section 2.1.1). In this case, the complexity of the M L decision rule in Eq. (3.5) can be reduced considerably as usual [13] (cf. Eq. (2.15)). 3.3.2 Differential DSTBCs Similar to differential STBCs for co-located antennas, for differential dis-tributed space-time block coding each node applies differential encoding before 46 transmission, as follows: B[k] = V[k] B[k - 1} (3.8) where T = Nc and V[k] G V is a unitary matrix [91] (cf. Section 2.1.3). Considering Eq. (3.4), and assuming an observation window of 2T scalar-which is identical to the M L decision rule for conventional differential STBCs with co-located antennas, cf. Eq. (2.34). Therefore, no knowledge about the channel vector hs or the node signature matrix Gs is required, i.e., channel estimation at the receiver is avoided. As in the co-located antenna case, the power efficiency of differential DSTBCs can also be improved by increasing the observation window [106]. However, this has no influence on the DSTBC design, which is the primary focus of this work. Similar to differential STBCs for co-located antennas (cf. Section 2.1.3), it is straightforward to show that the PEP for differential DSTBCs is bounded by where matrix A is now defined as A = PsG%BH[k - l]EHEB[k - 1]GS, E = VQ - V i , {V0, Vi} G V, V0 + V\- As in the coherent case, if both E and Gs have full rank, a diversity order of d.= r(A) = mm{Ns,Nc} is achieved. Using the same definition for the coding gain G as in the coherent symbol intervals, the M L estimate V[k] for V[k] is V[k) = argmin {\\r[k] - V[k]r[k - l}\\22} , (3.9) v[k}ev (3.10) 47 case, and assuming again Ns > Nc, we obtain G = Ps{det{EB[k-l]GsG^BH[k-l}EH}y/Nc = Ps {det{EHE} det {GSG^})1/Nc, (3.11) where the fact that B[k — 1] is unitary if the differential encoding in Eq. (3.8) is initialized with a unitary matrix has been used. Eq. (3.11) reveals that, similar to the coherent case, the code V and the set of node signature vectors Q can be optimized separately. As mentioned before, the design of Q will be addressed in Section 3.4. The resulting rank and determinant design criteria for V are the same as those for the differential STBCs designed for Nc co-located antennas given in Section 2.1.3. As a consequence, we may employ efficient known group [91, 100] or non-group [139] designs for V. 3.3.3 N o n c o h e r e n t D S T B C s For noncoherent DSTBCs, we assume that the information-carrying code ma-trix B[k] G B is a (scaled) T x Nc unitary matrix3, where T > iV c, i.e., BH[k]B[k) = jfclNc, c f - t 8 9] a n d Section 2.1.2. For M L detection, the prob-ability density function (pdf) of the effective channel vector fs has to be characterized. Assuming the nodes are active at random, Gs is a random ma-trix, which is drawn from a finite set with a certain probability. Therefore, in general, the pdf of fs is a finite sum of Gaussian pdfs with covariance matrices GsGs • Clearly, such a pdf will result in a very complex M L decision rule and code optimization will be complicated. Therefore, we resort to the generalized 3 T h e extension of our results to noncoherent DSTBCs employing non-unitary matrices (cf. e.g. [140]) is possible, of course, but beyond the scope of this work. 48 likelihood ratio test (GLRT) decision rule [90], as follows: B[k] = argmin {| |r[A;]-B[A;]/ 5 | | 2 2 }, (3.12) B[fc]ee h = YB"[k]r[k]- ( 3 - 1 3 ) Here, fs is the M L estimate for the effective channel vector fs, assuming that B[k] was transmitted. Combining Eqs. (3.12) and (3.13) results in the GLRT decision rule B[k] = argmax {rH[k]B[k]BH[k}r[k]} . (3.14) B[k]eB Once again, Eq. (3.14) is identical to the M L decision rule derived in Eq. (2.24) for noncoherent STBCs with co-located antennas [89]. Similar to the differ-ential case, the decision rule in Eq. (3.14) does not require knowledge about the channel vector hs or the node signature matrix Gs- Note, however, that for noncoherent DSTBCs, hs and the set S of active nodes may change after every code word B[k], whereas for differential STBCs hs should change only very slowly, and an error occurs with high probability if S changes. Therefore, noncoherent DSTBCs are particularly attractive for applications where indi-vidual nodes can reliably receive and relay information for only a very short period of time. The application of noncoherent DSTBCs in practical wireless sensor networks (WSNs) will be considered in Chapter 6. At high SNRs, the PEP for noncoherent DSTBC can be bounded by (cf. Ap-pendix A and Eq. (2.27)) r { A ) 1 / F \-rW ^ T m r w • ( 3 - 1 5 ) where ao = —1/XT(BQBQ — B\B±) > 0, and matrix A is now defined as 49 A 4 PSB0GSG%B$- (B0B$ - B i - B f ) , {B0, B1} G B, B0 ? Bx. If both Gs and B0BQ — B\Bf have full rank, the proposed noncoherent DSTBC achieves a diversity order of d = r(A) = min{Ns, Nc}. Assuming again Ns > Nc, the coding gain can be expressed as G = \^\[PSBoGsG^^—B^-B^B^ = P 5 ^ d e t { G 5 G f } d e t | ^ I N C - B^B.Bf B 0 | j (3.16) where we exploit the fact that BQBQ = JJ-INC- Eq. (3.16). shows that similar to coherent and differential DSTBCs, the code B and the set Q of node signature vectors can also be optimized separately for noncoherent DSTBCs. For Q, the same design criterion as in the coherent and differential cases results. On the other hand, I N C - ( ^ ) B^BiB^B0 should have maximum rank, and the minimum value of its determinant over all possible pairs {.Bo, B\} £ B, BQ ^ Bi, should be maximized. These are the same design criteria as those for noncoherent STBCs optimized for Nc co-located antennas, cf. [89] and Eq. (2.29) in Section 2.1.2. Therefore, we can use existing noncoherent STBCs [90] for B. 3.3.4 C h o i c e o f 7VC From the preceding discussion, it is clear that Nc is a design parameter of the proposed DSTBCs. If decoding complexity is not critical, Nc should be chosen to be equal to the expected number of active nodes. If A^c is smaller than the number of active nodes, diversity is wasted. On the other hand, if Nc is larger than the number of active nodes, reducing Nc to iV c — Ns usually improves performance, as existing STBCs B are generally optimized for full 50 diversity. If decoding complexity is a concern, then a small TVC is preferable. This is especially true for DSTBCs employing non-orthogonal codes B, since in that case decoding complexity increases rapidly with 7VC for a given data rate, e.g. [90, 91]. 3.4 Design of the Node Signature Vector Set Our analysis in Section 3.3 reveals that for high SNRs coherent, differential, and noncoherent DSTBCs are all affected by the node vectors in the same way. In particular, the node gain Gs — det{GsG$ } is relevant for optimization of Q. However, before we embark on the optimization of G, we introduce the so-called distribution loss. 3.4.1 D i s t r i b u t i o n L o s s The analysis in Section 3.3 also applies to a system with TV = TVC co-located antennas, where TV5 = TV = TVC and gn = en, 1 < n < TV (e„: column vector whose elements are all zero except for the element in position n, which is equal to one). Obviously, for co-located antennas Gs = 1 results. The coding gains in Eqs. (3.7), (3.11), and (3.16) show that for high SNRs and a given set of active nodes S, Ns > TVC, the distributed implementation entails a performance loss of 1 1 L S ~ PsW1 = Pstet{GsG$}W ( 3 ' 1 ? ) i.e., the DSTBC with TVs > Nc active nodes requires a 101og1 0(L5) dB higher SNR than the corresponding STBC for TVC co-located antennas. Therefore, Ls will be referred to as the distribution loss in the following. Note that Ls > 1 holds, since d e t { G 5 G ^ } 1 ^ < J - t r { G 5 G ^ } = l/Ps [138, p. 477] (tr{-} 51 denotes the trace of a matrix). The performance loss of DSTBCs compared with STBCs for co-located antennas is due to the fact that it is impossible to find N > Nc orthogonal node signature vectors. For future reference, we define the maximum and the average distribution loss for Ns active nodes as Lmax(Ns) = max {Ls} (3.18) \S\=NS and L&V(NS) =4 i - Ls, (3.19) \S\=NS respectively. In Eq. (3.19), KQ = ) is the number of different subsets S C M with Ns active nodes, and it is assumed that all nodes are active with equal probability. The maximum distribution loss Lmax(Ns) and the average distribution loss La,v(Ns) will be used in Section 3.4.8 for evaluation of different sets Q. 3.4.2 Optimum Sets QOPT The adopted optimization criterion is the minimization of the maximum PEP, as usual. For the optimization of Q, we assume that iV a > Nc nodes are active.4 The design problem for the optimum set Qopt of node signature vectors can be mathematically formulated as Gopt = argmax < min {Gs} ? , (3.20) S I |S |=JV a J subject to HflUl2 = l . l<n<N, (3.21) 4Note that iV a is a design parameter which may differ from the actual number of active nodes Ns, cf. Sections 3.4.3, 3.4.8, and 3.5. 52 i-e., Gopt maximizes the minimum node gain Gs (or equivalently minimizes the maximum distribution loss Lma,x(Na)) over all (£j ) possible sets <S. Note that Gopt is not unique, as Gopt and U G0ptVH yield the same distribution loss Ls for all S if U and V are arbitrary Nc x Nc and Na x Na unitary matrices, respectively. Also, while the constraint in Eq. (3.21) is equivalent to the con-vex constraint H0JI2 < 1, the cost function Gs in Eq. (3.20) is not convex in Gs- Therefore, the above optimization problem may have local maxima. Nevertheless, when applying the gradient algorithm given in the next subsec-tion, we never observed any local maxima, and the algorithm always produced the same minimum value for Gs regardless of the initialization chosen. This result is in accordance with those reported by Hassibi and Hochwald in [141], where a similar optimization problem was considered in the context of linear dispersion code design. 3.4.3 I n f l u e n c e o f t h e D e s i g n P a r a m e t e r iV a Ideally, we would need to maximize the minimum node gain Gs in Eq. (3.20) over all 2~ZJVS=I (JVQ P o s s ib le sets of active nodes. However, such an opti-mization would be computationally very demanding, especially for large N. To avoid this problem, we advocate here a pragmatic approach and optimize G assuming that exactly Na out of N nodes are active, where iV a is a design parameter that has to be carefully chosen to guarantee good performance for any Ns- Therefore, an important practical question to ask is how the distri-bution loss Ls is affected if G is optimized for |«S| = iV a but Ns > Na nodes are active. In this context, we first note that Ls may increase or decrease if an additional node n S becomes active, i.e., it seems to be impossible to make a general statement. Nevertheless, we observed in our simulations that for "good" sets G (e.g. for those found by the gradient algorithm discussed in 53 the next subsection) the average distribution loss Lav(Ns) always decreased if more nodes became active, i.e., La v(./Vs + 1) < Lav(Ns) (cf. Figs. 3.4, 3.6-3.9, 3.11, and the discussions in Sections 3.4.8 and 3.5). This result suggests that it is advisable to always design Q for the minimum required number of active nodes, Na = Nc, since (average) performance seems to improve if more nodes can cooperate. 3.4.4 G r a d i e n t S e t s QgTa,d In this section, we provide a gradient-based algorithm for node signature vec-tor optimization, and refer to the corresponding sets as gradient sets Ggr&d-Assuming that GsG^ is not singular, the gradient of Gs is given by [142] = de t{G 5 Gf} (GSG$r1 Gs. (3.22) Therefore, we propose the following algorithm for solving the optimization problem defined by Eqs. (3.20) and (3.21). 1. Initialization: Set iteration number i to i = 0, and generate a random initial set of complex unit norm vectors gn[0], 1 < n < N. 2. F ind worst set <Smin[i]: «5min[i] = argmin {Gs[i\} (3.23) s \S\=Na 3. Adaptation: X 5 m i n [ i ] [ i ] 4 G 5 m i n [ i][i]Gf n i i n [ i l[z] + piNc Gsmin[i}[i + 1] = G 5 m l n [ i ] [ i ] + det{X 5 m i n [ i][i]} (X 5 m i n [ i ][z])- 1 G S m l n [ i ] [ i ] (3.24) 54 4. Normalization: Replace gn[i +1] by gn[i + l]/\\gn[i + l}\\2, n e <Smin[z]. 5. Termination: If |G 5 m l n [ i ] [z +1] - G 5 m l n [i_i][i] | / |Gs m l a [ i][i + 1]| < e goto 6, otherwise increment i by one and goto 2. 6. End: gn[i], 1 < n < N, is the desired set <5grad-Now, some definitions and explanations are in order. In Step 3, 3 is a small positive constant (e.g. 8 — 10 - 5) that ensures that -Xsmin[i][i] is not singular. This regularization of -X^5min[i][i] is useful for the first iterations, and 3 = 0 may be used for the last iterations. In Eq. (3.24), denotes the adaptation step size in iteration i. For generation of the gradient sets QgTad used in this chapter, we adopted fj.[i] = 10 - 2 in the first iterations and decreased the value gradually to fi[i] = 10~5 in the final iterations. Step 4 ensures that the updated node signature vectors have unit norm. Finally, Step 5 terminates the algorithm if the (normalized) improvement from iteration i to iteration i + 1 is smaller than the termination constant e. e = 10 - 5 was used for the results shown in this chapter. As mentioned in the previous subsection, the global convergence of the algo-rithm cannot be proved. However, the algorithm always produced the same value for the minimum Gs for different random initializations. Fig. 3.2 i l -lustrates the convergence behavior of the proposed gradient algorithm. In particular, the maximum distribution loss Lmax(2) of a set of N = 30 signa-ture vectors is depicted as a function of the iteration index i. The set was optimized for Na = Nc = 2 using the method presented above. The adap-tation step size of the gradient algorithm was fi[i] = 10 - 2 for 0 < i < 1999, fi[i] = 10~3 for 2000 < i < 3999, fi[i] = 10~4 for 4000 < i < 5999, and fi[i] = 10 - 5 for 6000 < i < 7999, whereas the termination constant e was set to zero in this experiment to avoid premature termination. For initialization, a 55 91 1 1 1 1 r 8.5 - \ : 8 • : • 7.5 - : • 0 1000 2000 3000 4000 5000 6000 7000 i Figure 3.2: Learning curve of gradient algorithm for N = 30 and Na = Nc — 2. Adaptation step size: fi[i] = IO - 2 for 0 < i < 1999, p[i) = 10~3 for 2000 < i < 3999, = IO" 4 for 4000 < i < 5999, and /x[t] = 10~5 for 6000 < i < 7999. random harmonic set was chosen (cf. Eq. 3.30). As can be seen from Fig. 3.2, the gradient algorithm significantly decreases L m a x (2 ) , i.e., the minimum value of Gs is significantly increased. The corresponding set of signature vectors at i = 0 and i = 7999 are tabulated in Appendix B. Finally, we note that since the speed of convergence of the gradient algorithm is not a critical issue no attempt was made to optimize in this respect. The performance of the gra-dient set will be compared with the other sets to be proposed in the following three subsections in Section 3.4.8. 56 3.4.5 Grassmannian Sets ^ g r a s In this section, we focus on the case of Ns = 2 active nodes. Let us define the correlation between gn and gm as ps — gn9m> <5 = {ni m}- Taking into account the constraint Hf/JI2. = 1, 1 < n < iV, a Grassmannian frame is defined by [143] Ggras = argmin < max{\ps\} > • (3.25) S {isf=2 J For the sake of consistency, we will refer to Ggvas as a Grassmannian set. For the special case of Na — Nc = 2, we obtain Gs = 1 — | A S | 2 , i.e., Eq. (3.20) and Eq. (3.25) are equivalent. This means that for Na — Nc = 2, Grassmannian sets are optimum, which allows us to find a lower bound on the distribution loss. In particular, the maximum correlation | p m a x | — argmax {\ps\} is bounded S,|<S|=2 by |pmax| > max{ x/(A r - 2)/(2JV - 2), 1 - 2A''-1} [144, Theorem 2], where equality can be achieved only if N < N* = 4 [143]. Using this bound, for Na = Nc = 2 we can establish a lower bound on the maximum distribution loss j 2 - % , 2 < N < 5 Lm3X(2)>{ V ~ ~ , (3.26) A^>6 1 / JV 2 V 1-1/iV' i.e., for a large total number of nodes, the maximum distribution loss is propor-tional to y/N/2. Furthermore, since the bound on \pmax\ can be achieved with equality for N E {3, 4}, in these cases \ps\ = pmax = y/(N - 2)/(2N - 2), V<S, |<S| = 2, for Grassmannian (optimum) sets, cf. [143]. As a result, for Ns = Na = Nc — 2 and N € {3, 4} the minimum average distribution loss is given by Lav(2) = J 2 - ^ . (3.27) 57 This result means that the distributed implementation entails a minimum aver-age performance loss of 0.62 dB and 0.88 dB for N = 3 and N = 4, respectively. The findings of this section suggest that we adopt known construction methods for Grassmannian frames for the design of the set of node signature vectors. For example, assuming Na = Nc = 2 and using the results of [143, Section 2.1.2] and [145, Lemma 5], it is easy to find the Grassmannian (and optimum) sets for N = 3, as follows, 'gras - Govt — { —?= V2[l\ T 1 i 1 i 1 ' 7 1 eJ27r/3 ' 7 1 eJ47r/3 } (3.28) and N = 4: Ggtas — Gt opt "-0.95" 0.30 J -0.08 - jO.32 -0.25 +jO.91 -0.23 +jO.60 -0.76 - jO.02 -0.16 -.70.73' -0.51 - j'0.41 (3.29) On the other hand, for Na = 2 and N > N2 few results are available in the literature, but the proposed gradient algorithm produced excellent signature vector sets performing close to the lower bound (3.26), cf. Section 3.4.8. Finally, we emphasize that for the general case Nc> 2 and/or Na > 2, Grass-mannian sets £/ g r a s are not optimum in the sense of Section 3.4.2, cf. Eq. (3.20). 3.4.6 H a r m o n i c Se t s ^harm In [90], harmonic frames were proposed for construction of noncoherent STBCs (see also Section 2.1.2). Here, we apply harmonic frames for the design of node signature vectors and refer to the resulting sets as harmonic sets, £harm- In 58 particular, the elements of Ghmm are defined as 9n± 1 g j^u i (n - l ) e J ^ u 2 ( n - l ) _ _ _ g3^Uiv c (n-l) 1 < n < JV, (3.30) where the coefficients 0 < u\ < < ... < UNc < N — 1 are optimized for maximization of the minimum node gain Gs- For the special case Ns = Na = iV c = 2, it is easy to show that the minimum node gain is maximized for u\ = 0 and U2 = 1. The resulting maximum distribution loss is <(2) = . . N >2. (3.31) v y sin(7r/A r) v y This result means that for large iV, the maximum distribution loss of harmonic sets grows as N/ir, i.e., at a considerably faster rate than the related lower bound (3.26). For the general case Nc > 2 and/or Na > 2, the coefficients Ui, 1 < i < Nc, were found through a random search method, similar to that reported in [90]. Due to the additional structural constraints in Eq. (3.30), harmonic sets in gen-eral are not Grassmannian sets and are not optimum in the sense of Eq. (3.20). However, the special case o£ Nc = Na = 2 and N = 3 is an exception, cf. [143], and the Grassmannian set in Eq. (3.28) is also a harmonic set, i.e., in this special case we find £ o p t = Ggias = Gharm- The optimality of the harmonic set in this special case can also be verified by comparing Eqs. (3.26) and (3.31), which both yield the same result for N — 3. It is interesting to combine harmonic sets (5harm with the diagonal differential STBCs proposed in [91]. For diagonal differential STBCs, the code V is a set of diagonal matrices with constant modulus diagonal elements. As a consequence, if the differential encoding process in Eq. (3.8) is also initialized with a diagonal matrix, B[k] is a diagonal matrix with constant modulus entries bi[k], \bi[k] \ = 59 1, 1 < z < Nc, on its main diagonal. Therefore, the ith element sn,i[^] of transmit vector sn[k] is given by snti[k] = - L ^ ' ^ " - 1 ) bi[k], l<i<Nc, (3.32) v Ns which has also a constant modulus |sn,i[^]| = W-Ws. Hence, the combina-tion of harmonic sets (7harm with diagonal differential STBCs is interesting for practical implementation, since the transmit symbols have constant mod-ulus, which leads to a low peak-to-average power ratio of the transmitted continuous-time signal. 3.4 .7 R a n d o m S e t s £ r a n d Gradient sets Ggr&d and harmonic sets Gh&rm are deterministic sets and have to be pre-designed for a network of a given size N. If new nodes join an existing network, the signature matrices for all nodes have to be re-designed. For applications where new nodes frequently enter the network, this may not be practical. To circumvent this problem, random signature vectors have been recently proposed in [146] and we denote the corresponding set of signature vectors as Grand (cf. [146] for more detail). 3.4.8 S o m e S p e c i f i c E x a m p l e s In this section, we compare the maximum distribution losses Lmax(Ns) and the average distribution losses L&v(Ns) of some specific gradient sets Ggr&d and harmonic sets Gh&rm designed for different total numbers of nodes N using the techniques proposed in Sections 3.4.4 and 3.4.6, respectively. In Fig. 3.3, the distribution losses of sets optimized for Na = Nc = 2 are shown for Ns = 2 active nodes. The lower bound (3.26) for the maximum 60 distribution loss is also shown. Al l distribution losses shown in Fig. 3.3 in-crease monotonically with N. For N = 2, the distribution loss is zero, since orthogonal signature vectors exist. For N = 3, the gradient algorithm found an optimum Grassmannian set for which L m a x (2 ) and L av(2) are identical to the lower bound, i.e., GSrad — <7gras = Gopt yields the same performance as the optimum harmonic set (3.28). For N — 4, the gradient algorithm also found an optimum Grassmannian set, where again Lmax(2) and Lav{2) are identical to the lower bound, cf. Section 3.4.5. For most N > 4, the maximum dis-tribution loss of Ggrad is only less than 0.5 dB larger than the lower bound, which indicates that the proposed gradient algorithm works well. In contrast, as expected Lmax(2) of Ghmm grows much faster with N than the lower bound. Furthermore, the average distribution loss Lav(2) of GgTa.d is moderate, even for large N. For example, for a network with N = 30 and N = 50 nodes, Gsra.d entails a distribution loss of 2.2 dB and 2.4 dB, respectively. In Fig. 3.4, we compare the distribution losses of the gradient set considered already in Fig. 3.3 for different numbers of active nodes. Since this set was optimized for Na = 2 active nodes, it is not surprising that the maximum dis-tribution loss Lmax(Ns) may increase if > 2 nodes become active. However, for a given N, it is interesting to note that the average distribution loss de-creases as more nodes become active (Ns increases). For example, for N = 30, the average distribution loss reduces to 1.3 dB and 0.5 dB for Ns = 3 and Ns = 5, respectively. This result suggests that the performance of the pro-posed DSTBCs is fairly robust with respect to the number of active nodes, which is important in practice where the number of active nodes may not be known. Fig. 3.5 shows the distribution loss of sets optimized for Na — Nc = 3 for Ns = 3 active nodes. Whereas the maximum and the average distribution 61 0 5 10 15 20 25 30 35 40 45 50 N •• • Figure 3.3: Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = Nc = 2. Gradient sets GSTad and harmonic sets £ h a r m are considered. The lower bound (3.26) is also shown. losses of ^g rad are monotonic functions of N, this is not true for (?harm> which can be explained by the special structure of the signature vectors, cf. Eq. (3.30). In practice, expurgated harmonic sets could be employed to improve performance. For example, Fig. 3.5 suggests using in a network with N = 9 nodes the 9 "best" harmonic signature vectors obtained for N = 10 instead of the harmonic signature vectors obtained for N = 9. Furthermore, although (/ g r a d always results in a smaller L m a x (3 ) , in most cases Gharm has a similar L a v(3) as the gradient set. 3.5 Simulation Results In this section, we present simulation results for some specific DSTBC designs. As a performance measure, we consider the average bit error rate (BER) of the 62 10r PQ M A L Figure 3.4: Maximum and average distribution losses as functions of the total number of nodes iV for a gradient set optimized for Na = Nc = 2 and different numbers of active nodes Ns-network. Unless specified otherwise, we assume that all nodes are active with equal probability, and the BER is averaged with respect to both the active nodes and the fading gains. First, in Section 3.5.1, we consider the idealized channel model specified in Section 3.2 and assumed for code optimization. Then, we address the impact of practical issues such as non-i.i.d. fading gains and a variable number of active relay nodes in Section 3.5.2. 3.5.1 Ideal Channel Conditions First, we consider the performance of a coherent DSTBC in a network with AT = 30 nodes. Alamouti's STBC (T = Nc = 2, cf. Eq. (2.7)) [10] with quaternary phase-shift keying (QPSK) modulation is used for the code B, i.e., 63 Figure 3.5: Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = Nc — 3. Gradient sets G&&& and harmonic sets Gharm are considered. the data rate is R = 2 bits/(channel use). For the node signature vectors, a gradient set <?gra(j optimized for TY = 30 and Na = Nc = 2 is adopted. Fig. 3.6 shows the average BER vs. 101og10(£?b//Yo) (£(,: received energy per information bit) for this DSTBC for different numbers of active nodes Ns (solid lines). For comparison, the performances of Alamouti's STBC for Nc = 2 co-located antennas and distributed space-time filtering (DSTF) [59]5 for two active nodes are also shown in Fig. 3.6. As expected, for Ns, > 2 the DSTBC achieves a diversity order of d = 2. Furthermore, for Ns = 2 active nodes, the proposed coherent DSTBC outperforms DSTF by 2.5 dB at BER = 10"4. Note that low-complexity M L detection can be applied for the DSTBC, whereas 5 We consider here the original DSTF design proposed by E l Gamal and Aktas and not the optimized design to be presented in Chapter 5. 64 Figure 3.6: Average BER of a coherent DSTBC vs. 101og1 0(£ , 6/A ro) for a network with N = 30 nodes and different numbers of active nodes Ns- B: Alamouti's STBC (T = Nc = 2) with QPSK modulation. The results for Alamouti's code with Nc = 2 co-located antennas and DSTF [59] are also shown. a Viterbi algorithm with two states is necessary for M L detection of DSTF [59]. Fig. 3.6 also shows that at high SNRs the performance of the DSTBC with Ns = 2, Ns = 3, and Ns - 5 is approximately 2.3 dB, 1.4 dB, and 0.6 dB, respectively, inferior to that of the STBC with co-located antennas. These results are in good agreement with the respective average distribution losses reported in Section 3.4.8. If only Ns = 1 node is active, the BER of conventional QPSK transmission over a Rayleigh fading channel is achieved, cf. [147], i.e., the application of the DSTBC does not entail any additional performance penalty in this case. 65 Figure 3.7: Average BER of a coherent DSTBC vs. 101og 1 0(£ b/Ar 0) f o r a network with N = 50 nodes and different numbers of active nodes Ns- Solid lines: Rate 3/4 orthogonal STBC (T = NC = 4) [78] with 16QAM modulation as code B. Dash-dotted lines: Single-antenna 8PSK transmission for Ns = 1 and Alamouti's STBC with 8PSK modulation as code B for Ns = 2. Dashed line: Rate 3/4 orthogonal STBC from [78] with 16QAM modulation and Nc = 4 co-located antennas. In Fig. 3.7, results for a network with a data rate of R = 3 bits/(channel use) and N = 50 nodes are shown. The rate 3/4 orthogonal STBC (T = Nc = 4) proposed in [78, Eq. (22)] (also given in Eq. (2.20)) with 16-ary quadrature amplitude modulation (16QAM) is used for B (solid lines). A gradient set GgraA optimized for N = 50 and Na = Nc = A is employed for the node signature vectors. For comparison, we also show the performance of single-antenna 8-ary phase shift keying (8PSK) transmission and a DSTBC based on Alamouti's STBC with 8PSK modulation and a gradient set optimized for 66 5 10 15 101og1 0(£6/JVo) [dB] 20 25 Figure 3.8: Average BER of a differential DSTBC vs. 101og10(.E6/A'o) for a network with N = 30 nodes and different numbers of active nodes Ns- B: Diagonal differential STBC from [91] with T = Nc = 3 and rate R = 1 bit/(channel use) (solid lines). The results for single-antenna DBPSK trans-mission and the differential STBC with Nc — 3 co-located antennas are also shown. N = 50 and Na = Nc = 2 (dash-dotted lines). In addition, the performance of the rate 3/4 orthogonal STBC with 16QAM for Nc = 4 co-located antennas is also included (dashed line). As expected from our analysis in Section 3.3, the DSTBCs achieve diversity orders of d = 1, d = 2, and d = 4 if Ns = 1, Ns = 2, and Ns > 4 nodes are active, respectively. For Ns = 1 and Ns = 2 the 8PSK-based schemes benefit from the larger minimum Euclidean distance of 8PSK compared with 16QAM, i.e., for Ns < 2, Nc = 2 is preferable over JVC = 4 (cf. Section 3.3.4). Fig. 3.8 shows the performance of a differential DSTBC (T = Nc = 3) employ-67. ing a harmonic set (/harm of node signature vectors in a network with N = 30 nodes. For the code B, we adopt the diagonal differential STBC given in [91] for transmission with rate R = 1 bit/(channel use) and Nc = 3 co-located transmit antennas. In order to fully highlight the capabilities of differential DSTBCs, we assume here a time-varying fading channel with a temporal corre-lation that follows Jake's fading model [147] with normalized fading bandwidth / D T S = 0.005, where fn and Ts denote the Doppler bandwidth and the symbol duration, respectively. As expected, for Ns = 1, Ns = 2, and Ns > 3 active nodes, diversity orders of d = 1, d = 2, and d = 3 are achieved, respectively. The DSTBC suffers from a certain loss in performance compared with conven-tional differential binary PSK (DBPSK) if only Ns = 1 node is active. As more nodes become active, the gap to the co-located antenna case becomes smaller. Fig. 3.8 shows that for high SNRs the slope of all BER curves changes, which is typical for differential detection receivers in time-variant fading channels (cf. e.g. [91, 106]). Fig. 3.9 shows the performance of a noncoherent DSTBC (T = 8, Nc = 2) with R = 1 bit /(channel use) for a network with N — 50 nodes (solid lines). A gradient set £/grad is adopted for the node signature vectors. For the code B, the noncoherent STBC given in [90, Section V] was used. For comparison, we also show the results for single-antenna transmission (using the corresponding code given in [90, Section V]) and noncoherent space-time coding with Nc = 2 co-located antennas. The fading channel is time-invariant during the transmission of one code word (i.e., for T = 8 scalar symbol intervals) and changes to an independent realization for transmission of the next code word. Similarly, the set of active nodes changes after every code word. Fig. 3.9 confirms the excellent performance of the proposed DSTBCs, also in the noncoherent case. For Ns > 2, a diversity order of d = 2 is achieved, and the corresponding 68 1 0 - " l — -1 1 ' : .. 5 10 15 20 2 5 101og 1 0(£ h/JV 0) [dB] • Figure 3.9: Average BER of a noncoherent DSTBC vs. 101og 1 0(£ 6/./v 0) for a network with N = 50 nodes and different numbers of active nodes Ns. B: Noncoherent STBC from [90, Section V] with T - 8, Nc = 2, and rate R = l bit/(channel use) (solid lines). The results for the single-antenna code also given in [90, Section V] and the noncoherent STBC with Nc — 2 co-located antennas are also shown. performance losses compared with the co-located antenna case agree very well with the average distribution losses predicted from Fig. 3.4. 3.5.2 Practical Considerations In this subsection, we evaluate the performance of the proposed DSTBCs for non-ideal channel conditions. For this purpose, we consider a network with N = 30 nodes and the coherent DSTBC (T = Nc = 2, B: Alamouti's STBC, QPSK, Sgrad) from Fig. 3.6. 69 0 2 4 6 8 10 12 14 16 18 20 101og10(£6/ivo) [dB] -Figure 3.10: Average BER of DSTBC vs. 101og10(f?6//Y0) for a network with N = 30 nodes with i.n.d. fading gains and Ns = 2 active nodes. B: Alamouti's STBC (T = TVC = 2) with QPSK modulation (solid lines). Results for QPSK transmission without DSTBC are also shown (dashed lines). I.n.d. fading channels: First, we consider the network shown in Fig. 3.1 and assume that the nodes are uniformly distributed within a circle of radius R. We model the fading gains hn as independent and non-identically distributed (i.n.d.) complex Gaussian random variables. In particular, the variance of hn is proportional to d~3, where dn denotes the distance between relay node n G Af and the destination node. Fig. 3.10 shows the performance of the DSTBC assuming Ns = 2 active nodes, as well as that of QPSK with Ns — 1, for different ratios R/l2. For R/l2 = 0, i.i.d. fading gains result and the corresponding BER curves are identical to those shown in Fig. 3.6. As R/l2 increases, the differences in the variances of the fading gains of different nodes 70 T 1— 1 1 1 1 1 r 101og10(J576/iVo) [dB] • Figure 3.11: Average BER of DSTBC vs. 101og1 0(£6/iVo) for a network with N — 30 nodes and two-hop transmission. In the first hop, each relay node listens with probability p. The received energy per bit in the first hop is E\. B: Alamouti's STBC (T = Nc = 2) with QPSK modulation. Results for Alamouti's code with Nc = 2 co-located antennas and QPSK transmission without DSTBC are also shown (dash-dotted lines). become larger, which results in an SNR loss for both the DSTBC and QPSK. However, the diversity order is not affected by the i.n.d. fading and the DSTBC yields substantial performance gains compared with QPSK for all considered values of R/l2. Two-hop Transmission: In our second practical example, we consider a two-hop transmission where, for simplicity, we assume i.i.d. Rayleigh fading chan-nels in both the first and the second hop, i.e., !i » and l2 ^> R for the scenario in Fig. 3.1. In the first hop, a source node broadcasts data packets containing 260 bits to the N = 30 relay nodes, using QPSK modulation. To save energy the relay nodes listen only to the source node with probability 71 p. The nodes decide independently whether they listen or not. If they do not listen, they are inactive and cannot participate in the second hop. The average received energy per bit at each listening relay node is denoted by E\. Those listening nodes that can detect the transmitted data packet correctly form the set of active nodes S and use the DSTBC to forward the data to the destina-tion node in the second hop. In contrast to the previously considered scenarios, the number of active nodes N$ is now random. If S is empty after the first hop, the source node re-transmits the same data packet again for a new re-alization of the fading channel. Fig. 3.11 shows the BER of this second hop for different values of p and E\/Eb. For ExjEb = 1 and 101og10(i?b//Vo) < 2 dB, most of the time only one relay node is active and the DSTBC achieves a similar performance as QPSK. For higher values of Eb/N0 (and consequently higher values of E\/NQ) most of the time more than one node is active and the achieved diversity order is close to two for all values of p. If the first hop enjoys a higher SNR than the second hop (Ei/Eb = 10), the DSTBC outper-forms QPSK for all considered Eb/N0. For both values of Ex/Eh the DSTBC approaches the performance of the co-located antenna case for p = 1/3 and high SNRs, since the average number of listening nodes £ {Ns} = pN = 10 is relatively high in this case. For p = 1/5 and p = 1/7, fewer nodes are listening on average, which results in a certain performance degradation. 3.6 Conclusions In this chapter, we have proposed a new class of DSTBCs that enables efficient node cooperation in networks with large numbers of relay nodes. These DST-BCs are based on the combination of existing STBCs B originally designed for Nc co-located transmit antennas and a set of node signature vectors. If Ns nodes are active, the proposed DSTBCs guarantee a diversity order of 72 d = mm{Ns, Nc}. For Ns > Nc and high SNRs, the performance loss entailed by the distributed implementation depends only on the set of node signature vectors Q but is independent of the adopted STBC B. We have provided effi-cient methods for optimization of Q that yield a moderate average distribution loss even for networks with up to 50 nodes. The considered DSTBCs are highly attractive for applications such as ad hoc and sensor networks, since the coop-erating nodes do not need to know which other nodes are active. Furthermore, the proposed design method is highly flexible, allowing DSTBCs suitable for low-complexity coherent, differential, and noncoherent detection to be easily obtained. 73 Chapter 4 Dis t r ibuted Space-T ime Trell is Cod ing 4.1 Introduction Distributed space-time block codes (DSTBCs) were introduced for decentral-ized distributed space-time (DST) coding in the last chapter. In particular, we showed that for single-antenna decode-and-forward (DaF) relay nodes, DSTBCs consisting of a signature vector and a conventional space—time block code (STBC) [13] originally designed for i V c co-located antennas can achieve a diversity order of d = mm{Ns, Nc}, where N$ denotes the number of ac-tive nodes and there is no limitation on the total number of nodes N in the network. We note that although there has been increasing attention on cooperative diversity in the literature, most of the current research is built upon the as-sumption that user nodes are equipped with a single antenna, and results on multiple antenna nodes [148-150] are minimal. Therefore, in this chapter we consider the general case where each relay node is equipped with NT antennas 74 and the destination node is equipped with NR antennas. From space-time coding theory for co-located antennas it is well known that space-time trellis codes (STTCs) [11] can potentially offer a larger coding gain than STBCs. In principle, conventional STTCs can also be straightforwardly applied in a distributed setting. If any Ns out of the N multiple-antenna relay nodes are active, full-rank STTCs designed for iVc = NTN co-located antennas achieve full diversity order d = NTNSNR [58]. However, the number of states required to obtain full-rank STTCs increases with Nc [109]. Therefore, it is not prac-tical to apply STTCs designed for co-located antennas in large distributed networks, as the decoding complexity increases with the number of network nodes [59, 109]. Similarly, the family of asynchronous distributed space-time trellis codes (DSTTCs) proposed in [58] is also only suitable for small networks. This problem is avoided in distributed space-time filtering (DSTF) with fixed [59] or random filters [151], which may be viewed as a DSTTC over the field of complex numbers. However, unfortunately, a coding gain cannot be realized with DSTF [59]. In this chapter, we introduce a new class of DSTTCs that achieves a coding gain and whose decoding complexity is independent of the total number of nodes in the network. As a natural extension to DSTBC and to gain more insight into the problem, we consider multiple-antenna nodes for the develop-ment of DSTTC in this chapter. In the proposed scheme, each relay node is assigned a unique signature matrix but all active nodes encode the informa-tion sequence using the same STTC. We show that existing full-rank STTCs designed for Nc > 2 co-located antennas are a favorable choice for the code matrix. Furthermore, we make the following additional contributions in this chapter: • We provide design guidelines for the proposed DSTTCs and show that 75 with a full-rank STTC designed for Nc co-located antennas, a diver-sity order of d = mm{NcNR, NTNSNR} can be guaranteed if the Ns active relay nodes and the destination node have NT and NR antennas, respectively. • In contrast to Chapter 3, we do not assume that all relay-destination channels have identical variances. However, our results suggest that sig-nature matrices optimized for identical relay-destination channel vari-ances are also close to optimum for the case of non-identical variances. • We propose efficient techniques for signature matrix design. In partic-ular, we extend the optimized and random designs proposed for NT = NR = 1 in [146], cf. Chapter 3, (in the context of DSTBCs) to NT > 1 relay node antennas and NR > 1 destination node antennas. A compar-ison of the optimized and random signature matrix designs shows that while random designs perform well on average, a large percentage of ran-dom signature matrices performs significantly worse than the worst-case optimized signature matrices. • While the existing decentralized DST coding literature, e.g. [59, 109, 146], has focused on single-antenna relay nodes, we show that increasing the number of relay antennas can substantially improve performance, even if these antennas are strongly correlated. • Our simulations show that even for single-antenna relay nodes the pro-posed DSTTCs yield a significant performance improvement compared with DSTF [59] and the DSTBCs introduced in Chapter 3. Furthermore, equipping relay nodes with a second antenna can be highly beneficial, especially if only a few nodes can be active at any given time. Organization: This chapter is organized as follows. In Section 4.2, the proposed 76 DSTTCs are introduced. Design criteria for DSTTCs are derived in Section 4.3, while the design of the node signature matrices is discussed in Section 4.4. In Section 4.5, simulation results are provided, and Section 4.6 concludes this chapter. 4.2 Network Model and DSTTCs We again consider a communication network consisting of N potentially co-operating relay nodes and each node is assigned a number n G Af, M = {1, 2, N}, cf. Fig. 3.1. However, unlike Chapter 3, in this chapter we assume that each relay node is equipped with NT antennas and is assigned a unique Nc x NT signature matrix Gn, n G Af. As will be shown in Section 4.3, properly designed signature matrices ensure that the DSTTC inherits the full diversity gain of the underlying STTC. The set of signature matrices is denoted by Q and the normalization t r { G „ G f } = l , l < n < i V , (4.1) is used. It is assumed that for the considered transmission period only an a priori unknown subset of nodes S = {n i ,n2, . . . ,n^s} C J\f of size Ns — \S\ is active and cooperates to transmit the information sequence received in the first transmission phase, cf. Fig. 3.1. The active nodes encode the (same) information sequence with an STTC B [11], producing Nc sequences frnj/c], 0 < k < K, 1 < nc < Nc, where K is the codeword length. At time k, the active node n forms the vector sn[k] = [s^k] s^k] ... s^T[k]}, where 8n[k] = y/ps'b[k]Gn, neS, (4.2) 77 STTC Encoder Figure 4.1: Processing performed at active relay node n G S for Nc = 4 and NT = 2. and the n tth transmit antenna of node n transmits the symbol s™' [fc], cf. Fig. 4.1. In Eq. (4.2), b[k] = [bi[k] 62[fc] ... bNc[k]] and Ps = Nc/Ns is a normalization constant. We assume that the elements of b[k] are uncorrelated and normalized such that £{bH[k]b[k]} = JTINC- Thus, the average transmitted energy per node is £{sn[k}s^[k}} — l/Ns, i.e., the total transmitted energy per symbol interval is independent of the number of active nodes Ns- For convenience, we rewrite Eq. (4.2) as Sn = y/FsBGn, neS, (4.3) where Sn = [sT[0] sT[l] ... sT[K - 1]]T and the K x Nc STTC matrix B contains the vector b[k] in row fc. We again assume a frequency-nonselective block-fading channel and perfect time synchronization [25, 59]. Furthermore, we assume the general case of NR > 1 receive antennas at the destination node. The signal samples rnr[k], 0 < k < K, 1 < nr < NR, received by the NR receive antennas in K consecutive symbol intervals, are collected in the 78 K x NR matrix R = J2 SnHn + JV = v ^ B 5 ] GnHn + N (4.4) where the channel gains / i " n between antenna nt of active relay node n and receive antenna nr of the destination node are collected in the NT X NR channel matrix Hn. We assume that the fading gains of different nodes are mutually uncorrelated with variance o\ = ^{|^ "(„r|2} = ^ E l i O " 1 ] where dn and a denote the distance between relay node n and the destination node and the path-loss exponent of the channel, respectively. Because of the generally small nodes and the resulting insufficient antenna spacing, the elements of Hn may be correlated and are modeled as correlated zero-mean Gaussian random variables (Rayleigh fading). The elements of the K x NR noise matrix N are independent and identically distributed (i.i.d.) zero-mean Gaussian random variables with variance o\ = NQ/ES. Using the definitions Gs — [Gni Gn2 ... GnNs ] and Hs — [H^ H„„ H^N }T, Eq. (4.4) can be rewritten as where the Nc x NR matrix He^ is the effective channel matrix given by J f | f f = ^fPs~GsHs• Therefore, Eq. (4.5) has the same structure as Eq. (2.37), except that here multiple receive antennas are considered. 4.3 Decoding and Optimization of DSTTCs In this section, we discuss the decoding of DSTTCs and provide design guide-lines for the STTC B and the set qf signature matrices Q, considering both R=yJPsB GSHS + N = BHf + N, (4.5) 79 high and low signal-to-noise ratio (SNR) scenarios. 4.3.1 Receiver Design Let h^cTlr, 1 < nc < NC, 1 < nr < NR, be the elements of the iV c x NR effective channel matrix Hess. Since the signal model given in Eq. (4.5) is identical to that of space—time trellis coding with NC co-located antennas (cf. Section 2.2), the branch metric for DSTTC decoding at time k is given by [11] and the Viterbi algorithm [85, 86] can be used to compute the path with the lowest accumulated metric. Here, we assume that H"|ff is known to the re-ceiver. Note that Hss can be estimated directly if all active nodes transmit simultaneously identical training symbols. Therefore, both the decoding com-plexity and the channel estimation complexity of the proposed DSTTCs are identical to those of the underlying STTC B (cf. Section 2.2). Moreover, in contrast to other DSTTC designs [58, 59], both complexities are independent of the number of active relay nodes Ns and the total number of relay nodes As usual in space-time coding [11], the Chernoff bound on the average pairwise error probability (PEP) will serve as the optimization criterion for the proposed DSTTCs. If we define the difference sequence matrix E = Bo — Bx, where BQ and B\ are two distinct allowed STTC codewords, then according to [11] and our discussion of STTC for co-located antennas in Section 2.2, the PEP 2 (4.6) 4.3.2 DSTTC Design 80 is bounded by r ( A ) 1 / F \ ~ r ( y l ) at high SNRs where A = PS(INR®EGS)$S(INR®EGS)H and * 5 is the chan-nel correlation matrix given by $s = £ {vec{H s}vec{H s}H} • <g> and vec{-} denote the Kronecker product and the vectorization of a matrix, respectively. In the following, we assume that $ 5 has full rank r(&s) = NSNTNR. Using the properties of the Kronecker product [142], A can be expressed as A = (INR®E)RS(INR®E)H, (4.8) where RS = PS{INR ® GS)&S(INR ® GS)H. Following [11] and Eq. (2.6), we define the diversity order as d = r(A) (4.9) and the coding gain as G = I J[Xi(A)\ (4.10) of the DSTTC. Of course, similar to the design of STTC for co-located an-tennas, the goal of DSTTC design is to minimize the maximum PEP bound in Eq. (4.7), which is equivalent to maximizing d and G, respectively. In the following, we derive design guidelines for the STTC B and the signature matrix set Q. 81 Rank of EHE Exploiting the relations r(A) = T ( ( I N R (g> E ) H ( I N R <g> E)RS) and r(RS) = NRr(GsG%), the diversity order d of the DSTTC can be bounded as [142] [r(GsG$) + r(EHE) - Nc] • NR < d < min{r (G 5 Gf) , r(EHE)} • NR(4.U) In the following, we distinguish two cases: 1. NTNS > Nc If NTNS > Nc nodes are active, r(GsGg) = Nc can be achieved for all sets <S if Q is properly designed, cf. Subsection 4.3.2 and Section 4.4. In that case, Eq. (4.11) shows that the DSTTC achieves a diversity order of d = dcNR, where dc = r(EHE) denotes the diversity order of the underlying STTC B assuming a single receive antenna. 2. NTNS < Nc Assuming that Gs has full rank r(Gs) = NTNS, Eq. (4.11) simplifies to [NTNS + dc-Nc}-NR<d< mm{NTNs, dc} • NR. (4.12) If the STTC B has full rank, i.e., dc = Nc, the DSTTC has a diversity order of d = NTNSNR. On the other hand, if the STTC does not have full rank, the diversity order of the DSTTC may be substantially smaller than dcNR or NTNSNR. In practice, the number of active nodes is not known and depends on the first phase of transmission from the source to the relay nodes, cf. Fig. 3.1. Therefore, in order to achieve the best possible performance, only full-rank STTCs B should be used for DSTTCs! That is, STTCs whose design is based on the Euclidean distance criterion (cf. e.g. [109, 152]), are not well suited 82 for distributed implementation even, if NR > 1 receive antennas are available. For example, for N$ = 2 active relay nodes with NT = 1 antenna and a destination node with NR antennas, Eq. (4.12) shows that an STTC B with Nc = 4 and dc = 2 yields a diversity order of 0 < d < 2NR, which may lead to unacceptable performance. Note that this behavior is completely different from the co-located antenna case, where a diversity order of dcNR can be guaranteed. In contrast, for the distributed implementation only a full-rank STTC B with Nc = 4 and dc = 4 guarantees a diversity order of d = 2NR. In the following, we will only consider full-rank STTCs B. Therefore, the diversity order is ultimately limited by NCNR. In order to make optimum use of the available transmit antennas, it is advisable to choose Nc such that NC/NT is an integer (i.e., NTNS = Nc is possible), since otherwise antenna resources would be wasted. For example, for NT = 2 and Ns = 2, the choice Nc = 3 would limit the diversity order of the DSTTC to d = 3NR and would not make full use of the available four transmit antennas. Therefore, in the following, we will assume that NC/NT is an integer. Assumed Number of Active Nodes TVa Similar to the signature vector design of DSTBC in Chapter 3, ideally, we would have to minimize the maximum PEP in Eq. (4.7) over all 5ZJVS=I (N ) possible sets of active nodes. However, as we have discussed in Section 3.4.3, such an optimization would be computationally very demanding, especially for large TV. To avoid this problem we optimize the DSTTC, assuming that exactly Na out of N nodes are active, where Na is a design parameter that has to be carefully chosen to guarantee good performance for any Ns-The considerations in Subsection 4.3.2 show that matrix GsGg should have full rank for maximization of the diversity order d. Therefore, the chosen Na 83 should guarantee full rank of Gs for any Ns- This can be achieved if Q is optimized for Na = NC/NT- In this case, Gs has NTNa = Nc independent columns for any S with Ns = Na. Therefore, for Ns > Na any matrix Gs with NTNS > Nc columns will also have maximum rank r{Gs) = Nc. On the other hand, for Ns < Na, matrix Gs will have NTNs independent columns and the maximum possible rank r(Gs) = NTNS-Based on the findings in this subsection, in the following we will assume i V a = NC/NT for optimization of the DSTTC. Determinant of EHE For Ns = Na = NC/NT, full-rank Gs, and full-rank EHE, the coding gain can be simplified to Since det(Hs) is independent of B, the minimum value of det(EHE) should be maximized over all STTC words for maximization of the coding gain. There-fore, the design criteria for B are identical to those reported for STTCs (cf. [11] and Section 2.2) and are independent of the set of signature matrices Q, the number of active nodes, and the total number of network nodes. Therefore, it is not necessary to design new STTCs for distributed implementation, but existing full-rank codes optimized for Nc co-located antennas can be adopted (4.13) (cf. e.g. [11, 109]). 84 Optimization of Q Eq. (4.13) shows that the determinant of matrix Rs is relevant for optimization of the DSTTC. For Ns = Na = NC/NT, we obtain det{i*5} = P ^ N R (de t{G 5 Gf }) N * det{* 5}. (4.14) Therefore, for maximization of the coding gain G, the signature matrix set G should be optimized for maximization of the minimum value of Gs — det{GsGs } (det{$>s}y^NR- We define the optimum signature matrix set Qopt formally as £ o P t = argmax I min {Gs} > , (4.15) S {. |S|=JV 0 J subjectto t r{G„G^} = l , 1 < n < N. (4.16) Note that Qopt is not unique as Gopt and U GoptVH yield the same Gs for all S if U and V are arbitrary ./V,. x Nc unitary matrices. Interestingly, for the special case of identical correlation matrices for all considered subsets of active nodes, i.e., $ 5 = <&, V<S with Ns = Na, the optimum set Gopt does not depend on For example, for the scenario considered in Fig. 3.1 this special case holds if the antennas of all relay nodes have identical correlations and all relay-destination channel variances are identical (i.e., l2 » R in Fig. 3.1). The design of signature matrices will be discussed in more detail in Section 4.4. Low S N R Case The DSTTC design guidelines derived up to this point are based on Eq. (4.7) which was derived assuming high SNR. However, if multiple transmit and receive antennas are available, it has been shown that STTCs optimized for low 85 SNR may perform better for practically relevant SNRs than STTCs optimized for high SNR, cf. e.g. [109, 152]. This motivates us to briefly consider DSTTC optimization for low SNRs. For low SNRs, the PEP is bounded by [152, 153] P e ~ l + (Ea/4N0)tr{AY ( 4 ' 1 7 ) From Eq. (4.17), it is obvious that the minimum value of tr{A} should be maximized. While it is difficult to extract simple design rules for B and Q, some qualitative conclusions can be drawn from a lower bound on tr{A}. Using the identity tr{A} = t r { ( I N R <g> (EH E))Rs}, we can establish the inequality [142] NCNR (Rs)) < tv{A} »= i < NRNctr{EHE}tv{$s}, (4.18) where Aj(-) > A i + i ( - ) - Eq. (4.18) shows that in order to achieve a large lower bound on tr{A}, both EHE and Rs should have only non-zero eigenvalues. Therefore, both EHE and Gs should have full rank. As far as the lower bound is concerned, it is advantageous if all eigenvalues of INR®(EH E) (and therefore all eigenvalues of EHE) and Rs are approximately equal. This design goal is similar to the (high SNR) determinant criterion for EHE and Rs in Eq. (4.13), since equal eigenvalues maximize the determinant of a matrix under a trace constraint [142]. The qualitative considerations above show that DSTTCs optimized for high SNRs should also perform well for low SNRs. 86 4.4 Design of Node Signature Matrix Set Q For signature matrix design for DSTTCs, similar techniques as for signature vector design for DSTBCs proposed in the last chapter can be used. However, some changes are required, since in contrast to DSTBC we consider here the general case of (a) non-identical correlation matrices <&s and (b) multiple antenna nodes, i.e., NT > 1, NR > 1. In this section, we will generalize the deterministic designs proposed for DSTBCs in the last chapter and also briefly explain how the random signature vectors in [146] can be extended to the multiple-antenna case. 4.4.1 Gradient Sets Qgra,d Assuming that Gs is an iVc x NC full-rank matrix, the gradient of Gs with respect to Gs is [142] f | | = G 5 ( G f ) - 1 . (4.19) Using this result, the gradient algorithm given in Section 3.4.4 can be extended to the general case of non-identical correlation matrices $ 5 and NT > 1. We refer to the resulting sets as gradient sets, GSreA-4.4.2 Harmonic Sets (/harm Besides gradient sets, we also consider harmonic signature matrices whose elements are complex exponentials. Harmonic sets have been proposed in [90] for noncoherent space-time coding (cf. Section 2.1.2). They are attractive since only NC parameters need to be optimized. For harmonic sets, the signature matrices are constructed as [90] Gn = 0™"1 G i , 1 < n < N, (4.20) 87 where the NC x NT matrix G\ contains NT distinct columns of the x NC discrete Fourier transform (DFT) matrix, and 0 is given by © = d iag l^ ' 2 ™ 1 ^ , J2™*'", ej2™N</N}, (4.21) where 0 < un < N, 1 < n < NC (cf. Eqs. (2.22) and (2.23)) and diag{-} denotes a diagonal matrix. Good values for un, 1 < n < NC, can be found by a random search where the maximization of the minimum value of Gs over all possible sets <S is the optimality criterion. For this search, we assume again that NS = NA = NC/NT. 4.4.3 Random Sets ^ r a n d As mentioned in Section 3.4.7, gradient sets Ggr&d and harmonic sets <?harm are deterministic sets and have to be pre-designed for a network of a given size N. Therefore, the signature matrices for all nodes need to be re-designed if new nodes join an existing network. For applications where new nodes frequently enter the network, this may not be practical. It has been recently shown in [146] that the pre-designed signature vectors can be replaced by random signature vectors, which may be beneficial for applications where relay nodes frequently leave/enter the network. These random signature vectors can also be extended to random signature matrices for NT > 1, and we denote the corresponding signature matrix set by Grand- Following [146], we consider three different distributions for the design of signature matrices and briefly summarize them below for completeness: 1. Gaussian Distribution: In this case, the elements g%cTH, 1 < n c < NC, 1 < n < N, of the signature matrix Gn, n G S, are zero-mean, independent, and complex Gaussian 88 random variables with variance 1/(NCNT). Therefore, the signature ma-trices satisfy the average power constraint tr{£{GnG« }} = 1, n G S. (4.22) but not the instantaneous power constraint in Eq. (4.1). 2. Uniform Phase Distribution: The elements <7™ of the signature matrix Gn, n G <S, are drawn from eje, where 6 ~ f/(0, 27r). Each element is then multiplied by l/^/NcNT to ensure that the energy constraint in Eq. (4.1) is satisfied. 3. Uniform Distribution on a Complex Hypersphere: This distribution is similar to the Gaussian distribution except that the coefficients are normalized such that the instantaneous power constraint in Eq. (4.1) is satisfied. 4 . 4 . 4 Distribution Loss In order to facilitate the comparison of different signature matrix designs, we also define here the distribution loss for multiple-antenna nodes. For practical importance and to simplify our derivations, we assume that all subsets of active nodes have equal correlation matrices, i.e., $ 5 = 3? holds for a given Ns. Assuming a system with N = Nc uncorrected co-located antennas, we obtain det{J? 5} = 1 from Eq. (4.13). Therefore, for high SNR and a given set S of active nodes, which satisfy Ns > NC/NT, the distributed implementation entails a performance loss of 1 1 det{J^}V(iVcJv«) Psdet{(INR ® Gs)3>(INR <8> Gs)HV^N^' (4.23) 89 i.e., the DSTTC with Ns > NC/NT active nodes having NT possibly correlated antennas requires a 101og10(Ls) dB higher SNR to achieve the same perfor-mance as the corresponding STTC with 7VC uncorrelated co-located antennas. Consequently, we refer to Ls as the distribution loss in the following. Note that Ls > 1 holds, since [138, p. 477] 0 < Psdet{(INR®Gs)*(lNR®Gs)H}1KN°N*'> < Psdet{GsG^}1^<^-tv{GsG^} = l. (4.24) The maximum and average distribution losses for Ns active nodes are given by Eqs. (3.18) and (3.19) and are reproduced below for convenience: Lmax(Ns) = max {Ls}, (4.25) Lav(Ns) = ^ - . £ L * (4-26) u s w h e r e ^ ^ ( £ ) . 4 . 4 . 5 Comparison of Different Designs In this section, we compare the average distribution losses Lav(Ns) and maxi-mum distribution losses LmaK(Ns) of some specific gradient sets £ g r a d, harmonic sets (?harm, and random sets GTa.no\ designed for different total numbers of nodes N using the techniques proposed in Sections 4.4.1-4.4.3. Note that we have assumed $s = * fe>r all S of size Ns for derivation of the distribution loss. In Fig. 4.2, the Lav(Ns) (cf. Eq. (4.26)) of optimized for 7VQ = 2 is shown for Ns = 2, 3, and 4 active nodes. The relay nodes are equipped with NT — 2 antennas and Nc = 4. The parameters 7Va, NT, and 7VC are chosen according 90 to our discussion in Section 4.3.2. We consider both the uncorrelated antenna cases (dashed lines) and the correlated antenna cases (solid lines) where a correlation coefficient of p = 0.8 is assumed [154]. An important practical question is how the distribution loss is affected if Q is optimized for Na but Ns > Na nodes are active. In this context, we first note that the distribution loss may increase or decrease if an additional node n £ S becomes active and it seems impossible to generalize. Nevertheless, we observed in our simulations that for "good" sets Q (e.g., those found by the gradient algorithm discussed in Sections 4.4.1) Lav(Ns) always decreases if more nodes became active, i.e., Le,v{Ns + 1) < Lav(Ns). This behavior can be seen in Fig. 4.2. We also note that while Lav(Ns) = 0 when N = Ns for the uncorrelated antenna cases, Lav(Ns) for Ns = 2, 3, and 4 active nodes is 2.22 dB, 1.38 dB, and 0.97 dB, respectively, for the correlated antenna cases. These losses are due to the antenna correlation. For example, when Ns = 2, Eq. (4.23) can be simplified to det{Rs}l^N^ P l S det{G 5 Gf} 1 / iVcdet{$} 1 / ( iv c iv H ) = det{*}VW=*«) = L 6 ' ( 4 2 ? ) where $ is a block diagonal matrix with [1.0 0.8; 0.8 1.0] in its diagonal entry. Therefore, the distribution loss 101og10L,s = 2.22 dB is purely due to the an-tenna correlation for Ns = N = 2. Unfortunately, the above decomposition is only valid for Nc — NSNT, and therefore cannot be applied to Ns = 3 and 4. We would also like to point out that L a v(4) is lower than 2.22 dB for all Ns considered. In other words, the performance of Ns = 4 active nodes with NT — 2 correlated antennas (i.e., total of NsNT = 8 antennas) is always better 91 4 .5 4 h 3 .5 —B— N s = 2 , p=0.8 N s = 3 , p=0.8 - e - N s = 4 , p=0.8 -o- N s = 2 , p=0.0 H>- N g = 3 , p=0.0 - o - N s = 4 , p=0.0 Figure 4.2: Average distribution losses Lav(Ns) as functions of the total num-ber of nodes N for a gradient set ( ? g r a d optimized for 7Y0 = NC/NT = 4/2 = 2 and different numbers of active nodes Ns- Both correlated (solid curves) and uncorrelated (dashed curves) antennas are considered. A correlation coefficient p = 0.8 is assumed for the correlated antenna cases. than that of the correlated co-located antenna case. Fig. 4.3 depicts the average distribution loss Lav(Ns) for ( ? g r a d , (?harm, and £ r a n d as a function of the total number of active nodes TY. Accordingly, Ns = A^a = NC/NT = 4/2 = 2 and the node antenna correlation coefficient is p = 0.8. For (/rand the distribution loss does not depend on the total number of nodes TY, as each node generates its own signature matrix randomly whenever it becomes active. The average distribution losses shown in Fig. 4.3 for QraTl0\ were averaged over 1,000,000 realizations of Gs- We observe that for 7YT = 2, uniform spherical randomization yields the best performance, followed by 92 Figure 4.3: Average distribution loss Lav(Ns) as function of the total number of nodes N for Na = Ns = NC/NT = 4/2 = 2, and p = 0.8. Gradient sets C/grad, harmonic sets Gharm, and random sets Qrand a r e considered. uniform phase randomization and Gaussian randomization. These results are in accordance with those reported in [146] for NT = 1. We also note that as N increases, the performance gap between G&ad and Grand decreases. However, it should be noted that although La.v(Ns) for Grand is relatively close to that of GSrad and £harm for large N, there is no limit on Lmax(Ns) for £ r a n d -To gain more insight into this problem, Fig. 4.4 shows Pr{Ls > Ltn} vs. 101og10(Lth) for the 1,000,000 realizations of ^ r a n d considered in Fig. 4.3, where Pr{L,s > Zah} denotes the probability that the distribution loss Ls exceeds threshold Ltn. The vertical lines indicate the maximum distribution losses achievable with gradient and harmonic sets optimized for a network of N = 30 93 Figure 4.4: Pr{Ls > L th} as functions of threshold Ltn for random sets Grand with Na = Ns = NC/NT = 4/2 = 2, p = 0.8, and different randomization methods. L m a x (2 ) for harmonic and gradient sets designed for N = 30 nodes is also shown. relay nodes. Fig. 4.4 clearly shows that the flexibility gained by random sig-nature matrices comes at the price of a large performance degradation for individual transmissions. For example, 75.5%, 67.0%, and 66.0% of the matri-ces Gs generated by Gaussian randomization, uniform phase randomization, and uniform spherical randomization, respectively, have a higher distribution loss than the worst-case distribution loss of the gradient set. 4.5 Simulation Results In this section, simulation results for relay nodes with NT = 1 and NT = 2 antennas are provided. Since for most applications relay nodes must be 94 small, we assume a relatively high correlation coefficient of p = 0.8 for the NT = 2 case. For the destination node, NR = 1 is assumed. In addition, the frame error rate (FER) of the network is considered as a performance measure. For all presented results, a network with N = 30 nodes and a frame size of 260 bits is assumed. For code B, we selected the full-rank quaternary phase-shift keying (QPSK) STTC with 64 states given in [109, Table I] and designed for Nc = 4 co-located antennas. In the following subsection, we first consider the performance of DSTTC under the assumptions that all relay-destination channel variances are identical, which corresponds, for example, to the scenario considered in Fig. 3.1 to l2 3> R- The impact of practical issues such as independent and non-identically distributed (i.n.d.) fading gains and a variable number of active relay nodes is addressed in Section 4.5.2. 4.5.1 I d e a l C h a n n e l C o n d i t i o n s In this section, we assume that all relay-destination channel variances are identical, which corresponds to l2 ~> R in Fig. 3.1. In Fig. 4.5, the FER vs. 101og10(£?6/AT0) (Eb: energy per bit) is depicted for the proposed DSTTC (solid lines), assuming Ns active relay nodes with NT = 1 antenna. For the signature matrices, a gradient set Gsia,d was designed for Na = NC/NT = 4, as described in Section 4.4. As expected, the diversity order of the DSTTC is d — min{A^, 4}. For comparison, the performance of the STTC B (dash-dotted line) for Nc = A co-located uncorrelated antennas1 and the perfor-mances of DSTF I [59], DSTF II (cf. Chapter 5), and DSTBC (introduced in Chapter 3) are also shown (dashed lines)2. As expected from the discussion in Section 4.4.4, the DSTTC suffers from some performance loss compared with 1We note that a centralized D S T T C (also based on B) with Ns = 4 active nodes would achieve the same performance as the S T T C with co-located antennas. 2For the underlying STBC B, we employed the rate 3 / 4 orthogonal STBC (T = Nc = 4) given in Eq. (2.20) with QPSK modulation. 95 the STTC, even for Ns > 4. However, this loss decreases with increasing Ns-On the other hand, for a fixed number of active nodes, the DSTTC outper-forms both DSTF and the DSTBC, which were also optimized for iV c = 4. For example, for Ns = 4 and FER = 10 - 2 the DSTTC achieves performance gains of 2.5 dB, 2.2 dB, and 1.7 dB compared with the DSTBC (Chapter 3), DSTF I [59], and DSTF II (Chapter 5), respectively. We note that the DSTF schemes require the same decoding complexity as the DSTTC, whereas the DSTBC has a much lower decoding complexity. However, because of the non-availability of full-rate orthogonal STBCs for Nc = 4, the DSTBC has a rate of only 1.5 bits/(channel use), compared with the 2 bits/(channel use) of the DSTTC and DSTF 3 . The FER for a DSTTC optimized for relay nodes with NT = 2 correlated antennas is depicted in Fig. 4.6 for various Ns- A gradient set Ggrad is con-sidered. As expected from our analysis, the diversity order of the DSTTC is d — min{2A^5, 4}. For comparison, the performance of the STTCs B (dash-dotted line) for Nc = 4 uncorrelated and correlated co-located antennas, and DSTBC with Ns = 2, are also shown. We note that DSTTC with Ns = 6 per-forms better than STTC with correlated co-located antennas, which is in accor-dance with our discussion in Section 4.4.5 and with Fig. 4.2. At FER = 10 - 4 , DSTTC with Ns = 2 and 3 is about 4.1 dB and 2.2 dB inferior, respectively, to STTC with uncorrelated co-located antennas. These numbers are very close to our distribution loss analysis depicted in Fig. 4.2. A comparison of Figs. 4.5 and 4.6 shows that despite the relatively large correlation, a second antenna is highly beneficial as long as Ns < Nc. As Ns increases beyond Nc, the differ-ence between NT = 1 and NT = 2 diminishes. For example, for FER = 10 - 2 the performance gain achievable with NT = 2 and the gradient set is 6.6 dB, 3 We have not included the DSTTCs from [58] in our comparison as it seems to be difficult to design these codes for N = 30. For the results shown in [58] N < 4 was adopted. 96 101og1 0(£6/7V0) [dB] • Figure 4.5: Average FER of DSTTC vs. 101og 1 0(£ 6//V 0) (solid lines). For 7V5 = 4 results for DSTF I [59], DSTF II (Chapter 5), and a DSTBC (Chapter 3) are also shown (dashed lines). TV = 30, Nc = 4, and NT = 1. Signature matrices: Gradient set ( ? g r a d-2.0 dB, and 0.1 dB for one, two, and six active nodes, respectively. This is not unexpected, because Gs is rank deficient when Ns < Nc for NT = 1; therefore, in this case, DSTTC does not achieve the full diversity order Nc. Fig. 4.7 compares the performances of £ g r a d , (?harm, and QTand for Nc — 4 and N = 30 when Ns = 2 two-antenna relay nodes (p = 0.8) are active. For com-parison, the performance of the STTC B (dashed line) for 7VC = 4 uncorrelated co-located antennas is also shown. As expected, the optimized sets ( ? g r a d and £ h a r m perform better than the random sets ( / r a nd- At FER = 10 - 3 , the DSTTC with NSNT = 4 correlated antennas suffers from a loss of 3.4 dB, 3.7 dB, 4.0 97 Figure 4.6: Average FER of DSTTC vs. l01ogW(EB/N0). For NS = 2, results for a DSTBC. (Chapter 3) are also shown. N = 30 relay nodes with NT = 2 antennas, NC = A, and p = 0.8. A gradient set Q&aA is used for the signature matrices. dB, 4.1 dB, and 4.4 dB for £ g r a d, £harm> and Gr&nA with uniform spherical randomization, uniform phase randomization, and Gaussian randomization, respectively, compared with the STTC with NC = 4 uncorrelated co-located antennas. We note that these numbers are somewhat different from the cor-responding average distribution losses shown in Fig. 4.3. This difference can be attributed to the fact that SNR —> oo was assumed for the derivation of the distribution loss in Eq. (4.23), whereas Fig. 4.7 shows the FER for finite SNRs. 98 0 2 4 6 8 10 12 14 16 18 20 Mogw(Eb/N0) [dB] • Figure 4.7: Average FER of DSTTC vs. 101og10(Eb//Vo). N = 30, Nc = 4, NT = Ns = 2, and p = 0.8. Gradient sets Qgrad, harmonic sets Ohmm, and random sets £ r a n d are considered. 4.5.2 Practical Considerations In this subsection, we evaluate the performance of the proposed DSTTCs for non-ideal channel conditions. I.n.d. fading channels: In order to investigate the effect of non-identical relay-destination channel variances, we consider in Fig. 4.8 the FER in the second hop of the network shown in Fig. 3.1 for R/l2 = 0.6 and path-loss exponent a = 3. The following parameters are adopted: Nc = 4, TV = 30, Ns = 2, NT = 2, and p = 0.8. Fig. 4.8 shows the average FER (averaged over all possi-ble active nodes) and the FER of the subset >S with the worst performance for, respectively, a gradient set optimized for the considered channel (i.e., taking 99 5 10 101og10(£;6M)) [dB] Figure 4.8: FER of DSTTC vs. Wlogw(Eb/N0) for R/l2 = 0.6 and l2 » R. N = 30, Nc = 4, NT = Ns = 2, p = 0.8, and a = 3. Gradient sets Qgrad are optimized for both R/l2 — 0.6 and l2 >^ i?. into account the non-identical channel variances) and a mismatched gradient set optimized for identical relay-destination channel variances (i.e., assuming l2 R). Since the goal of the presented DSTTC optimization is to make the worst-case performance as good as possible, it is not surprising that the optimized gradient set yields a better performance than the mismatched gra-dient set for the worst subset of nodes. However, the achievable gain is quite small. In addition, as far as average FER is concerned, both signature matrix sets yield practically the same performance. This is an encouraging result, since in practice the relay-destination channel variances may not be known and optimizing the signature matrices assuming equal variances may be the only option. For comparison, Fig. 4.8 also includes the average FER for the 100 Figure 4.9: Average FER of decentralized (solid lines) and centralized (dashed lines) DSTTC vs. 10\og1Q(EB/NQ). N = 30, NT = 1 and NT = 2 (p = 0.8), NC = 4, and E\/Eb — 10. Signature matrices: Gradient set case where all nodes have identical relay-destination channel variances (i.e, k » R). Two-hop Transmission: In Fig. 4.9, we study the performance of the two-hop system depicted in Fig. 3.1, assuming l\ » R and l2 ~> R, i.e., all channels in the first and the second hop are modeled as i.i.d. flat Rayleigh fading. In the first hop, QPSK symbols are transmitted in packets of 130 symbols. To save energy, each of the iV = 30 relay nodes listens only to the transmission with a probability of p < 1. Those nodes that can detect the entire packet correctly forward the packet to the destination in the second hop, using the proposed decentralized DSTTC or a centralized DSTTC. We note that, in 101 contrast to Figs. 4.5-4.8, Ns here is random. For the centralized DSTTC, the columns of the STTC matrix B were assigned to the active nodes in an optimum manner in order to minimize FER. The received energy per bit in the first and the second hop is denoted by E\ and Ebl respectively. If no relay node can successfully decode the transmitted packet, a new packet is transmitted, assuming a new realization of the fading gains. For both the centralized and the decentralized DSTTC, Nc = 4, N = 30, and Ex/Eb = 10 are assumed. We also considered NT = 1 antenna and 7VT = 2 correlated antennas (p = 0.8). In the case of NT = 2, the relay nodes exploit the second antenna, also in the first phase of transmission, by applying maximal-ratio combining (MRC). Fig. 4.9 shows that the decentralized DSTTC achieves a similar performance as the centralized DSTTC. Note that finding the optimum assignment of the columns of B to the active nodes is a tedious task and must be repeated if the channel changes, e.g., due to movement of the nodes. For p = 1/10 and high SNR, £{Ns} = 3 nodes are active on average. In this case, having a second antenna is highly beneficial for distributed space—time block coding, despite the relatively large correlation coefficient. For example, for the decentralized scheme a 3 dB gain is achieved by having a second antenna at FER = 10~2. On the other hand, for p = 1/3 and high SNR on average £{Ns} = 10 relay nodes are active, already providing sufficient diversity for NT = 1; little can be gained by having a second antenna. Finally, we compare the performances of DSTBC, DSTTC, and DSTF in a two-hop system, as shown in Fig. 3.1. For this purpose, we assume that all the channels in the first and the second hop are modeled as i.i.d. Rayleigh flat-fading channels. In the first hop, QPSK symbols are transmitted in packets of 130 symbols. Similar to Fig. 4.9, each of the N — 30 relay nodes listens to the transmission only with a probability of p < 1. Those nodes, equipped 102 11 I I i i i I I i : I I 0 2 4 6 8 10 12 14 16 18 20 101og10(£i/M>) [dB] Figure 4.10: Average FER of DSTBC (Chapter 3), DSTTC, and DSTF (Chap-ter 5) vs. 101og10(Eb/7V0). N = 30 relay nodes with Nc = 4, NT = 1, and Ei/Eb = 10. p = 1/3 (dashed lines), p = 1/10 (solid lines). A gradient set (?grad is used for the signature vectors. with NT = 1 antenna, which can detect the entire packet correctly, forward the source packet to the destination in the second hop. If no relay node can suc-cessfully decode the transmitted packet, a new packet is transmitted, assuming a new realization of the fading gains. E\ and Eb again denote the received en-ergy per bit in the first and the second hop, respectively. Fig. 4.10 shows the FER of the second hop for DSTBC (proposed in Chapter 3), DSTTC, and DSTF (to be introduced in Chapter 5) with Qgiad designed for N = 30 and Na = 4. The other system parameters used for the various schemes are listed below: • DSTBC: B: Rate 3/4 orthogonal STBC (T = NC = 4) given in Eq. (2.20) 103 with QPSK modulation. • DSTTC: B: Full-rank QPSK STTC with 64 states (given in [109, Table I]) designed for NC = A. co-located antennas. • DSTF: Lg = 4 with QPSK modulation. In addition, E\/Eb = 10 is assumed for all simulations. When p = 1/3 and p = 1/10, £{NS) = 3 and £(NS) = 10 result, respectively. Therefore, a better FER performance is expected for p = 1/3, which can be seen from Fig. 4.10. For p = 1/10, DSTTC is about 2.2 dB better than both DSTF and DSTBC at high SNRs. For p = 1/3 and at FER = 10 - 3 , DSTTC is about 2.3 dB better than DSTBC and 4.0 dB better than DSTF. We note that the curves are bent at high SNRs for p = 1/3. This behavior can be attributed to the adopted (decentralized) energy-saving policy, which with a certain probability results in only one active relay node regardless of E\ and hence, a diversity loss. Finally, we note that DSTF and DSTTC have the same decoding complexity, whereas DSTBC has a much lower decoding complexity. However, because full-rate orthogonal STBC for NC = 4 antennas is not available, the DSTBC has a rate of only 1.5 bits/(channel use) compared with the 2 bits/(channel use) of the DSTTC and DSTF. 4.6 Conclusions In this chapter, we have proposed a new class of DSTTCs for wireless networks with a large set of DaF relay nodes each having NT antennas. The DSTTC consists of a set of i V c x NT signature matrices Q and a STTC B. While each node is assigned a unique signature matrix from Q, all nodes employ the same STTC B. We have shown that off-the-shelf full-rank STTCs designed for Nc 104 co-located antennas are optimum for B, and we have characterized the opti-mum signature matrix set Gopt- Furthermore, we have proposed two practical design methods for signature matrix sets and have compared the resulting sets with random signature matrices. The proposed DSTTCs guarantee a diversity order of d = min{NcNR, NTNsNR} if Ns nodes are active and the destination node has NR antennas. In contrast to previously proposed DSTTCs, the de-coding complexity of the proposed codes is independent of the total number of relay nodes and of the number of active nodes. The presented simulation results confirm the superior performance of the proposed DSTTCs compared with DSTBCs and DSTF. 105 Chapter 5 Dist r ibuted Space—Time Fi l ter ing 5.1 Introduction We have proposed distributed space-time block codes (DSTBCs) and dis-tributed space-time trellis codes (DSTTCs) for decentralized DST coding in the last two chapters. El Gamal and Aktas proposed distributed space-time filtering (DSTF) for decentralized networks; DSTF also does not require node coordination [59]. In DSTF, each node in the network is assigned a different signature filter vector (SFV) of length Lg. The transmitted signal of the ac-tive nodes is the convolution of the correctly decoded source symbol sequence, which is identical to all nodes in <S, and their respective SFVs. It was shown in [59] that for active nodes, full diversity order d = Ns can be achieved with Lg = Ns, and a simple, heuristic design for the set of SFVs Q was provided. We note that the aforementioned schemes are all designed for frequency-nonselective channels. However, wireless communication channels are usually frequency-selective, as is the case when partial response modulation is used, 106 or if the channel experiences a non-negligible delay spread. In [151, 155-157], intentional delays are introduced in different relay nodes, and at the destina-tion node, a minimum-mean squared error (MMSE) estimator is used to ex-ploit the intentional induced frequency selectivity. Li et al. extend their own work on asynchronous cooperative communications for frequency-nonselective channels [156] to frequency-selective channels in [158]. In particular, using the idea of orthogonal frequency division multiplexing (OFDM), high rate dis-tributed space-frequency codes are designed to achieve both full asynchronous cooperative diversity and multipath diversity. The performance of Laneman's distributed space-time block coding scheme in underwater acoustic communi-cations is considered in [159]. In particular, the underwater communication channel is modeled as frequency-selective. To cope with the intersymbol inter-ference (ISI), the time reversal space-time block code (TR-STBC) proposed in [160] for co-located antennas is extended to the distributed scenario. Similar to Laneman's scheme [25], the major drawback of [159] is that it cannot be applied to networks with a large number of relay nodes, cf. Section 1.2. In this chapter, we optimize and extend DSTF to frequency-selective channels. In particular, we make the following contributions in this chapter: • We derive a new criterion for optimization of the set of SFVs Q for both frequency-nonselective and frequency-selective channels. • Based on the proposed optimization criterion, we provide two practical and general methods for the design of optimum or close-to-optimum sets Q. • We compare DSTF with optimized delay diversity (DD) (cf. [118] and Section 2.3) for co-located antennas and provide an analytical expres-sion for the performance loss entailed by the distributed implementation, 107 which we refer to as the distribution loss. • For the special case of frequency-nonselective channels, we show that the SFV optimization problem is closely related to the design of signature sequences for code—division multiple access (CDMA) [161, 162] and the design of Grassmannian frames [143]. Exploiting this relationship, we derive a lower bound for the distribution loss and propose three optimum and close-to-optimum construction methods for SFV sets. • We show that the performance of DSTF can be significantly improved at the expense of higher receiver complexity if Lg is increased beyond Lg = Ns. • We study several practical problems such as suboptimum equalization, imperfect channel estimation, and imperfect timing synchronization. Organization: This chapter is organized as follows. In Section 5.2, the system model is presented. In Section 5.3, the design of the SFVs for general fading channels is discussed and the distribution loss is introduced. The SFV design for the special case of frequency-nonselective channels is studied in more de-tail in Section 5.4. In Section 5.5, simulation results are provided and some practical issues are discussed. Section 5.6 concludes the chapter. 5 .2 N e t w o r k M o d e l a n d D S T F The considered network model with N potentially cooperating nodes is shown in Fig. 5.1. Each node is assigned a number n, n 6 A/", A/" — {1, 2, . . . , N} and a signature filter with impulse response gn[l], 0 < I < Lg — 1, of length Lg. For ease of exposition, we assume that each node is equipped with a single antenna; however, generalization of the results of this chapter to multiple antennas is 108 Figure 5.1: A network with N = 8 DaF relay nodes and Ns = 2 active relay nodes (<S = {2, 8}). S: Source node. D: Destination node. Phase 1: Solid arrows. Phase 2: Dashed arrows. l\: Relay nodes are uniformly distributed in a circle with radius R. l\\ Distance from the source node to the center of the circle. l2- Distance from the center of the circle to the destination node. hn[l]: CIR from relay node n to the destination node. possible. For convenience, the complex impulse response coefficients of the signature filters are collected in a SFV, gn = [gn[0] flvjl] -.- • 9n[Lg — 1]]T, and the SFVs are normalized as H0JI2 = 1, 1 < n < TV. The set of all SFVs is denoted by Q. Out of the N nodes, only an a priori unknown subset S C M of nodes is active and transmits. where the independent and identically distributed (i.i.d.) data symbols x[k] are taken from an M-ary symbol alphabet A, such as M-ary phase shift keying (MPSK) or M-ary quadrature amplitude modulation (MQAM). The energy 5.2.1 Distributed Space-Time Filtering The signal transmitted by the active nodes is [59] n G S, (5.1) 109 of the data symbols is normalized to £{|a;[fc]|2} = 1. The normalization factor Ps — 1/Ns guarantees that the total transmit power J2nes £{\bn[k}\2} is equal to Es regardless of the number of active nodes Ns — \S\. Similar to the last two chapters, this normalization is necessary for a fair comparison of DSTF with different numbers of active nodes. Taking into account Ts-spaced sampling (Ta: symbol duration) at the destina-tion node, the channel between relay node n <G S and the destination node can be characterized by its discrete-time impulse response hn[l], 0 < I < Lh — 1 (cf. Fig. 5.1). The length Lh is chosen to be large enough to fully capture the ISI for all active nodes. The discrete-time CIRs hn[l] include the effects of transmit pulse shaping, wireless fading channel, receive filtering, and imper-fect timing synchronization. Note that even if the wireless channel is frequency nonselective, Lh > 1 results if the timing of the active nodes is not perfectly synchronized. We assume a quasi-static block Rayleigh fading channel, i.e., the CIRs are constant for the transmission of one packet but change indepen-dently from packet to packet. The (normalized) discrete-time received signal can be modeled as RW = -JW S h n [ k ] *K[k] + n[h] = ^ £ hs[l]x[k ~l]+n[k]> (5-2) V h s n€S 1=0 where the equivalent overall CIR of length Leq — Lh + L9 — 1 is defined as M*] = XXW*0»M' (5-3) neS and n[k] denotes an additive white Gaussian noise (AWGN) process with va,ri-ancea2n±£{\n[k}\2} = N0/Es. 110 5.2.2 Channel Estimation and Detection Eq. (5.2) shows that the overall transmission system can be modeled as a single-input single-output (SISO) system with scalar M-ary modulation over a SISO ISI channel. The destination node can directly estimate the equivalent overall channel impulse response (CIR) coefficients hs[l], 0 < I < Leq — 1. The individual CIRs hn[l], n € <S, and the set of active nodes S do not need to be known at the destination node. The estimation of hs[l] can be facilitated by transmitting a suitable training sequence (TS) of known symbols x[k] = Xrs[k], 0 < k < NTs — 1 at the beginning of each packet. If the length NTs of the TS is not smaller than 2 L e q — 1, the equivalent overall CIR can be estimated using the least-squares technique proposed in [163]. Similar to space-time filtering (cf. Section 2.3), any known optimum or subop-timum SISO equalization technique can be applied for detection of the trans-mitted data sequence. In particular, we will consider optimum maximum-likelihood sequence estimation (MLSE), decision-feedback sequence estima-tion (DFSE) [119], and MMSE decision-feedback equalization (MMSE-DFE) [120, 121]. According to Eq. (5.2), the number of states required for MLSE is given by M z , Q q ~ 1 . Therefore, the complexity of the optimum receiver in-creases exponentially with the SFV length Lg. On the other hand, we will show that the performance of DSTF improves significantly with increasing Lg. Hence, reduced-complexity equalization techniques,-which achieve a close-to-optimum performance, are attractive for application in networks with DSTF. I l l 5.3 Optimization of Distributed Space-Time Filtering In this section, we first provide an approximate upper bound on the pair-wise error probability (PEP) of DSTF. Based on this bound, we derive an optimization criterion and devise two practical construction methods for SFV sets, Q. We also compare DSTF with DD transmission for co-located transmit antennas (cf. Section 2.3) and introduce the distribution loss for DSTF. 5.3.1 Approximate PEP Analysis of DSTF As in space-time filtering (cf. Section 2.3), a natural optimization criterion for the set of SFVs Q is the minimization of the maximum bit error rate (BER) of the transmitted data sequence. Unfortunately, Eq. (5.2) shows that the received signal is impaired by ISI, and it seems to be not possible to derive a meaningful BER expression for continuous transmission that could be exploited for optimization of Q. For a given set of active nodes S, the received signal r[k] in Eq. (5.2) is identical to that of DD transmission with Ns co-located transmit antennas. Therefore, we can design the set of SFV Q using the same optimization criterion as in DD [118] (see also Section 2.3). In particular, we optimize Q for an upper bound on the PEP. The Chernoff bound on the PEP of DSTF, assuming single-symbol transmission and optimum M L detection is defined similarly as in Eq. (2.46) [118]: e^(<S) < 1 <5-4) det{J L e q + jPsGs^G^} where the matrix Gs is defined as Gs — [Gni Gn2 ... GnNs ], and Gn is an L e q x Lh column-circulant matrix with vector [g^ 0 0 • • • 0]T (length: L e q ) 112 as the first column. The channel correlation matrix of the active nodes S = {ni, n2, ... nNs} is defined as $ = £{ / i 5 / i f } with hs = [h^ hT2 ... hlNJT, hn — [hn[0] hn[l] ... hn[Lh — 1]]T. For the design of the set Q we assume that the nodes are perfectly synchronized, all CIRs are spatially uncorrelated, tem-porally correlated (because of transmit and receive filtering), and identically distributed, and that <fr has full rank1. Therefore, $ is a full-rank block diago-nal matrix and is independent of <S. Furthermore, the effective signal-to-noise ratio (SNR) 7 in Eq. (5.4) is given by 7 = d 2 n i n £ , s / (4A 7 0 ) , where d m i n denotes the minimum Euclidean distance of the signal constellation A. The assump-tion that all CIRs are identically distributed is justified, for example, for the two-hop system in Fig. 5.1 if l2 S> R and 2R/c <C T s , where c denotes the speed of light. As discussed in Section 2.4, even in cases where this assump-tion is not justified and only little information about the channel statistics is available, assuming identically distributed CIRs may still be the only viable option for SFV design. To gain further insight, we consider the high SNR case and rewrite Eq. (5.4) as (cf. Eq. (2.47)) P , ( 5 ) < 7 - ^ > n ^ , ( 5 . 5 ) where As = PsGs&Gs. Eq. (5.5) shows that the diversity order of the considered DSTF scheme is given by d = r(As). Assuming that Gs and $ both have full rank, the maximum diversity order is d = min{L e q , LhNs} = mm{Lh + Lg - 1, LhNs} (cf. Section 2.3). Therefore, for Lh = 1 an SFV length of Lg = Ns achieves the maximum diversity order, as pointed out in [59]. However, for Lh > 1, Lg = L m i n = Lh(Ns — 1) +1 is a necessary condition to exploit the maximum available diversity. 1 # could be rank deficient in an asynchronous system where the relative delays of the cooperative nodes are greater than the symbol duration. 113 5.3.2 Optimum Sets Gopt For optimization of the SFV set Q, we again assume that |<S| = Na nodes are active and introduce the cost function ds — det{iz,e q + ^PsGs&Gg}. Of course, the actual number of active nodes Ns in a network may be different from the design parameter Na (cf. discussion in Section 5.3.6). Considering Eq. (5.4), we define the optimum2 set of signature vectors as Gopt = argmax < mjn {ds} > , (5.6) subjectto |ff„H2 = 1» l<n<N, (5.7) i.e., Gopt minimizes the maximum value of the upper bound on the PEP in Eq. (5.4) over all (^ ) possible sets S. The optimum SFV set is not unique, as ds does not change if Gopt is left-multiplied by an arbitrary L e q x L e q unitary matrix. The optimization problem defined by Eqs. (5.6) and (5.7) is highly non-linear, and ds is a non-convex function of Gs-Since ds depends on 7 and <&, in general the optimum set Gopt will depend on the SNR and $ as well. However, assuming that both Gs and <& have full rank, for Lg > Lmin and 7 —* 00 we obtain NaLh ds = lNaLh A<(As) = hPs)NaLh det{*} • d e t { G £ G s } - (5.8) i=l Therefore, in this case, we may replace ds in Eq. (5.6) by det{Crf Gs}- Hence, for high SNRs and filter lengths that exploit the full available diversity, the optimum set Gopt is independent from the underlying channel correlation ma-trix and only depends on iV a , Lh, and N. This is a very important result, since in practice $ may not be known. 2Optimum means here optimum with respect to the adopted cost function ds- We cannot claim optimality with respect to the B E R for continuous transmission, of course. 114 5.3.3 G r a d i e n t S e t s Qgrad In this section, we devise an adaptive gradient algorithm to solve the optimiza-tion problem defined in the previous subsection. The resulting SFV sets will be referred to as gradient sets, Ggiad- For convenience, we collect the SFVs of all active nodes in a hyper-vector, gs = [g^ g„2 • • • g^N ]T- The gradient of ds with respect to gs is given by [118] dds A ( dds dds dds \ H , . dg*s " V%u[0] dgni[l] •" dgnNa[Lg-l}J = >,Psdsti{(ILwi + 'yPsGs*GsI})-1En.l*G»}, (5.10) where Enj denotes an L e q x LhNa matrix, the elements of which are equal to 1 at those positions where Gs has entries gna[l] and equal to zero other-wise. We are now ready to formulate the following gradient algorithm for the optimization problem defined by Eqs. (5.6) and (5.7): 1. Initialization: Set iteration number i to i = 0, and generate a random initial set of complex unit norm vectors gn[i], 1 < n < N. 2. F ind worst set «Smin[i]: «Smi„[i] = argmin {ds[i}} (5.11) 5 |S|=iV„ 3. Adaptation: Q ^ i +1] = + f ^ | j (5-12) 4. Normalization: Replace gn[i + 1] by gn[i + l]/\\gn[i + 1]||2, n G Smin[i}. 5. Termination: If \dsmin[i\[i + 1] - d5 m i n [ i - i ]NI/ |^ m i n [ i ] [^ + 1]| < e goto 6, 115 otherwise increment i by one and goto 2. 6. End: gn[i], 1 < n < N, is the desired set £/grad-The adaptation constant fi[i] may be optimized to maximize the speed of convergence of the algorithm, and e in Step 5 denotes a termination constant (typical value: e = 10 - 5). As mentioned before, the underlying optimization problem is not convex, and we therefore cannot prove global convergence of the algorithm. Nevertheless, the algorithm did converge in all of our experiments and it did produce the same minimum value for ds for different random initializations, i.e., we did not observe any local maxima. Similar findings were reported by Hochwald and Hassibi in [141], where they solve a similar non-convex optimization problem by a gradient algorithm. 5.3.4 H a r m o n i c Se t s £/harm The concept of harmonic sets, (/harm, was proposed in Chapter 3 for optimiza-tion of the signature vectors in the context of DSTBC (cf. Section 3.4.6). By inspection, it can be easily seen that (/harm can be also used here for our SFV design. In particular, we consider the SFVs n * 1 9 n = L9 1 < n < N, (5'.13) where the coefficients 0 < u\ < u2 < ... < u^g < N — 1 are optimized for maximization of the minimum ds, cf. Eq. (5.6). For large values of Lg and N, an exhaustive search over all possible coefficients may not be possible. In that case, we perform a random search over a large number of different vectors and choose those coefficients that yield the maximum value for the minimum ds-We note that the SFVs proposed by E l Gamal and Aktas in [59] result as a 116 special case of Eq. (5.13) if we set Lg = Na and Ui = i — 1, 1 < i < Lg. We refer to the SFV sets in [59] as GGA-5.3.5 R a n d o m Se ts £ r a n d In the last two chapters (cf. Sections 3.4.7 and 4.4.3), we advocated random sets, (?rand, originally proposed in [146], for the design of signature vectors and matrices for distributed space-time block coding and distributed space-time trellis coding. Clearly, Grand can be also applied to the design of SFVs. Again, the advantage of such designs is that the SFVs are generated randomly whenever a node becomes active, and therefore can be applied to dynamically changing networks where new relay nodes enter frequently the network. 5.3.6 I n f l u e n c e o f t h e D e s i g n P a r a m e t e r Na As mentioned before, in a network scenario, the actual number of active nodes Ns may be different from the design parameter 7Ya. It is interesting to inves-tigate the effect of such a mismatch. At high SNRs, the SFV design will be dominated by the diversity gain, i.e., optimum sets Gopt designed for Na >2 active nodes will achieve full diversity order d = mm{Lh + Lg — 1, LhNa}. This result is only possible if the matrices Gs have full rank r(Gs) = rain{Lh + Lg - 1, LhNa} for any subset <S of Na SFVs. However, considering the special form of Gs, this condition is only possible if any subset of min{iVa, Lg} SFVs is linearly independent. On the other hand, if any subset of minlA^, Lg} SFVs is linearly independent, any subset of min{iVs, Lg}, Ns < Na, SFVs will also be linearly independent. Therefore, a set of SFVs that achieves full diversity for Na active nodes also achieves full diversity for Ns < Na active nodes. If Ns > Na, full diversity cannot be guaranteed. Nevertheless, in our experience optimum or close-to-117 optimum SFV sets also achieve full diversity if Ns > Na. In general, we found that the choice of Na is not very critical. Excessive simulations have shown that a SFV set Q optimized for Na nodes, also performs close-to-optimum if Ns ^ Na, cf. Fig. 5.8, for example. On the other hand, for the relevant case iV 0 < N/2 the number of different subsets and the design complexity of Q increases with increasing Na. Therefore, in practice, the choice Na = 2 may be preferable. 5.3.7 Distribution Loss of DSTF It is interesting to compare DSTF with Ns active nodes and optimized DD [118] with Ns co-located antennas (cf. Section 2.3). The optimum DD fil-ter vectors are also defined by Eqs. (5.6) and (5.7) if we set N = Ns = Na, cf. Section 2.3. We denote the corresponding optimum DD filter matrix by GDD and the corresponding maximum value of the cost function by duD — det{i£ e q + R ) A D D ) , A D D = PsGDD$>G%D. If Gs has full rank for all possible subsets 5, DSTF achieves the same diversity order as DD, i.e., the distributed implementation does not entail any loss in diversity. However, the distributed implementation does cause an SNR penalty. Therefore, we define the distri-bution loss of DSTF as 7^ oo \ ds J 7-*oc V, det{iL e q + 7As} / fr{As) \ l / r ^ i.e., if the nodes in S are active, DSTF requires a lOTog^Ls) dB higher SNR to achieve the same BER as optimized DD. Since it is obvious that dun cannot be smaller than any ds, Ls > 1 holds for all subsets S. 118 If we assume Lg > Lm[n, r(A$) = NsLh results and we can infer from [118] that doo = (ryPs)NsLh&&{&}- Thus, using Eq. (5.8), the distribution loss can be rewritten as L s = det{G$Gsy/(W ( 5 ' 1 5 ) Interestingly, for Lg > Lm[n (and full rank <I>) the distribution loss is inde-pendent of the underlying channel, whereas it does depend on the channel correlation matrix for Lg < Lmm. Similar to Chapters 3 and 4, we define the maximum and the average distri-bution losses as Lmax(Ns) = max {Ls} and Lav(Ns) = ^ - £ Ls, (5.16) \S\=NS 0 5 \S\=NS respectively, where K0 = )• Lmax(Ns) is the maximum SNR loss that any subset of Ns nodes suffers compared with optimized DD, while Lav(Ns) gives the average SNR loss of DSTF with TV's active nodes, assuming all nodes are active with equal probability. 5.4 SFV Design for Frequency-Nonselective Channels In this section, we will show that for the special case of frequency-nonselective channels with Lh = 1, the SFV optimization problem is related to the design of signature sequences for CDMA and Grassmannian frames. We note that for Lh — 1 the channel correlation matrix is given by $ = INS and the cost function simplifies to ds = det{Jx,g + }, where the Ns columns of Gs are now simply the SFVs gn of the active nodes n £ S. 119 5.4.1 Relation to the Design of Signature Sequences for Consider a synchronous direct sequence code-division multiple access (DS-CDMA) system with N equal power users, where the nth user is assigned a unit-norm signature sequence gn, 1 < n < N, of length Lg. Assuming an AWGN channel with bandwidth B and an SNR per user of P57, the 7Ya-out-oi-N symmetric capacity is given by [161, 162] In [162], the design of signature sequences that maximize Csym is considered. Since the logarithm in Eq. (5.17) is a monotonic function, this design problem is closely related to the SFV optimization problem considered in this chapter. The main difference is that we assume in Eq. (5.6) that exactly Na nodes are active, whereas in Eq. (5.17) it is assumed that not more than Na users are active. In [162], a lower bound on C s y m is developed which can be used to find a lower bound on ds and an upper bound on the distribution loss L 5 . However, this bound is unfortunately quite loose for Na <C N, which is the interesting case for DSTF. For example, for N > LgNa the lower bound in [162] simplifies to Csym > B/(LgNa)log(l + NaPsj). Combining this result with Eq. (5.8) and Eq. (5.15) results in Ls < l i m ^ ^ P 5 7 / ( l + 7 ) 1 / ( W o L " ) = 00 for Na > 2, which is a trivial bound. In [162] a design method for signature sequence sets that achieve the lower bound with equality is also developed. This method can also be used to design SFV sets G, of course. Unfortunately, the SFV sets found by this method con-sistently performed worse than the sets found by the gradient search proposed CDMA (5.17) 120 in this chapter. Nevertheless, the relationship between the signature sequence design for maximization of the symmetric capacity C s y m and SFV optimization is interesting, since any future progress with respect to the former problem can be exploited for SFV optimization. 5.4.2 Relation to the Design of Grassmannian Frames In this subsection, we consider the design of SFVs for Na = 2 active nodes. We show that, in this special case, we can bound the distribution loss and construct optimum SFV sets by exploiting the rich theory of Grassmannian frames [143, 144]. Grassmannian Sets Qgias Recall from Eq. (3.25) in Section 3.4.5 that a Grassmannian frame is defined by [143] £ g r a s = argmin < max{\ps\} > , (5.18) Q {\S\=2 J subjectto | |0„||2 = 1 . l<n<N, (5.19) where ps — 9n19n2 1 S the correlation between two SFVs in S = { n i , n2}. We refer here again to QgTas as Grassmannian sets. Since we can rewrite the cost function as ds = (1 + 7-Ps)2 - 7 2-Pj|/0s| 2, Eqs. (5.6) and (5.18) are equivalent and Grassmannian sets are optimum, i.e., Qgras- = Gopt for iV a = 2. 121 Lower Bound on the Distribution Loss For N > Lg & lower bound on the maximum correlation between two SFVs is given by [144] p m a x = max{|p5|} > max j JL^N 1 - 2Y ^ i , (5.20) where equality can hold only if N < L2, and if and only if equality holds I As I = Pmax for all possible subsets <S, i.e., all vectors in Qgias are equi-correlated [143]. Assuming N > Lg and Lg > Na = 2, we can combine Eqs. (5.15) and (5.16) and obtain a lower bound for the maximum distribution loss W 2 ) > h 1 2 (5-21) V 1 - /'max with p m a x from Eq. (5.20). We note that for a given Lg and N S> Lg, the lower bound grows as N1^2(-L9~1^/2, i.e., the bound grows slower for larger Lg. If we assume N > Lg and N < L2 [143], the lower bound can be achieved and we obtain W 2 ) = i " ( 2 ) = | ^ l j ' ( 5 - 2 2 ) i.e., the maximum and the average distribution losses are identical. If both TY and Lg are large, both losses approach 0 dB, i.e., the distributed implemen-tation does not incur any performance penalty. For example, if we assume Lg = 5 and N = 25, the distribution loss is only 0.4 dB. By comparison, the SFVs proposed in [59] achieve L m a x (2 ) = 1/ sin(ir/N), i.e., for TY = 25 a maximum distribution loss of 9.0 dB results. 122 Construction of Grassmannian Sets Elegant methods for construction of Grassmannian sets that achieve the lower bound on the maximum distribution loss either exactly or approximately can be found in [143, Section 2.1], [144, Section III], and references therein. We note that SFV sets that achieve the lower bound with equality do not exist for all combinations of Lg and N, even if N < l?g. In the following, we briefly present three methods that we found useful for the construction of sets of SFVs. For conciseness, we do not explain the theory behind these construction methods, and instead refer the interested reader to [143] and [144]. a) Method I [143]: N = 2Lg, Lg = 2 > £ {1, 2, 3, ...}. Consider the matrix R = 1^ + ^ ^ C / v , where C V is calculated from the recursion '2m (5.23) with C2 = [j "Q1]. Furthermore, let vn = [vn[l] vn[2] ... vn[N]]T, 1 < n < N be the eigenvectors of R. Then a set of Grassmannian SFVs that satisfy Eq. (5.22) is given by [143] gn = V2[Vl[n] v2[n] ... vLg[n)}T, l<n<N. (5.24) This method allows Grassmannian sets to be easily constructed, but is restricted to N = 2Lg, where Lg must be a power of two. b) Method II [144]: Coefficients uu 1 < I < Lg, of £harm form a difference set. The coefficients ui, 1 < / < Lg, of a harmonic set £harm form a (N, Lg, K) 123 difference set if the Lg(Lg — 1) differences (uk — u{) mod N, k ^ I, take all possible values 1, 2, . . . , N — 1, with each value, exactly K times [144]. It has recently been shown in [144] that a harmonic set is a Grassmannian set if and only if its coefficients ui, 1 < I < Lg, form a difference set, i.e., in this case £ h a r m = £gras = Qopt holds. Exploiting known difference sets [144] presents several design methods for such Grassmannian harmonic sets for various N < L2 and Lg. Specific designs are tabulated in [144, Table I]. c) M e t h o d III [143]: N = L2g, Lg > 5, Lg prime number. In this case, good sets can be constructed as follows. Define the elements of vector vlk = ^ jM 1 ] Vikfi] • • • vik[Lg]}T, I, k 6 {1, 2, .. ' . , Lg}, as , i f c [ n ] ^ e x P ( J 2 7 r ( n - f e ) 3 + ^ - 1 ) ( n - 1 ) ) , 1 < n < L , (5.25) Then, the N = L 2 vectors vik, I, k € {1, 2, Lg}, form a set of SFVs G that almost achieves the bound (5.21). Since this construction method is related to Gabor frames [143], we refer to the corresponding sets as Gabor sets £ g a b o r - To be more precise, the correlation of Gabor sets is given by |ps| e {0,1/^/Z~g} [143]. Therefore, the distribution loss is Ls e {0, y/Lg/(Lg - 1)}, i.e., L m a x (2 ) = ^/Lg/(Lg - 1), whereas £max(2) > \/(Lg + 1)/Lg is obtained from the lower bound (5.21). For example, for Lg = 5 and N = 25, the Gabor set achieves a maximum distribution loss of 0.48 dB, which is only slightly higher than the lower bound of 0.4 dB. . We emphasize that the design methods proposed in this subsection are limited to specific values of Lg and iV and Na = 2. Most notably, these methods all assume N < L2.. In practice, however, we may be forced to design sets of SFVs 124 for N > L2g because of complexity constraints at the receiver. In this case, we need to rely on the gradient sets and harmonic sets introduced in Section 5.3. It is worth mentioning that for Na = 2, the modified Lloyd search algorithm introduced in [144] presents an alternative to the gradient algorithm proposed in this chapter. In fact, both algorithms achieve similar maximum distribution losses, and their complexity is also comparable. 5.5 Simulation Results In this section, we present numerical results for the distribution loss and sim-ulation results for the average BER of DSTF. Unless stated otherwise, binary phase shift keying (BPSK) modulation is considered. Moreover, Ns is fixed and we select the set <S of active nodes randomly from the N relay nodes of the network. The BER is then averaged over a large number of different sets S. We assume that the CIRs of different nodes are mutually independent but identically distributed. We consider frequency-nonselective Rayleigh fading (Lh = 1) and frequency-selective Rayleigh fading with two i.i.d. taps, which are separated by one symbol duration (Lh = 2). For optimization of the gradi-ent and harmonic SFV sets used to generate the results shown in this section, we assumed an SNR of 101og10(JE?&/iVo) = 15 dB unless stated otherwise average received energy per bit). However, before we turn our attention to the performance of DSTF, we consider the convergence behavior of the gradient algorithm proposed in Section 5.3.3. 5.5.1 Gradient Algorithm Fig. 5.2 shows the inverse minimum cost function I/dsmin\i\[i] vs. iteration num-ber i for the gradient algorithm proposed in Section 5.3.3 for construction of 125 0 1000 2000 3000 4000 5000 6000 7000 8000 i Figure 5.2: Learning curve of gradient algorithm for BPSK, TY = 30, Na = 2, Lg = 2, and 101og10(£ !6//Yo) = 20 dB. Adaptation step size: fi[i] = 10~3 for 0 < i < 1999, = 10~4 for 2000 < i < 3999, = 10"5 for 4000 < i < 5999, and fi[i] = 10~6 for 6000 < i < 7999. g g r a d. Here, BPSK, N = 30, Na = 2, Lg = 2, Lh = 1, and 101og 1 0 (£yY 0 ) = 20 dB are assumed. We consider 1 /dsmin\i][i] instead of the cost function d,smin[i][i] itself, since l/dsm l n[t][*] is directly related to the worst case PEP, cf. Eq. (5.4). The adaptation step size of the gradient algorithm was fi[i] = 10 - 3 for 0 < i < 1999, fj.[i] = 10 - 4 for 2000 < i < 3999, = 10~5 for 4000 < i < 5999, and = 10 - 6 for 6000 < i < 7999, whereas the termination constant e was set to zero in this experiment to avoid premature termination. For initialization, a random set of vectors gn[0], 1 < n < N, was chosen, as described in Section 5.3.3, and the curve in Fig. 5.2 was averaged over three different random initial-izations. As can be observed from Fig. 5.2, the gradient algorithm significantly decreases 1/ min,s{ds}, i.e., the minimum value of ds is significantly increased. 126 Figure 5.3: Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na = 2 and Lh = 1. Gradually decreasing the step size from = 10 - 3 to fi[i] = 10 - 6 further improves 1/ mins{ds}. By comparison QGA yields 1/mins-Jds} = 7.8 • IO - 3 . We note that since the speed of convergence of the gradient algorithm is not a critical issue no attempt was made to optimize in this respect. 5.5.2 Distribution Loss Fig. 5.3 shows the maximum distribution loss L m a x (2 ) (Eq. (5.16)) and the av-erage distribution loss Lm(2) (Eq. (5.16)) of gradient sets Ggiaxi and harmonic sets £ h a r m designed for Lh = 1 and Na = 2 as functions of the total number of nodes N. Ns = Na is valid, and SFVs with lengths Lg = 2 and Lg = 5 are considered. For Lg = 2, £ h a r m is identical to QGA proposed by El Gamal and Aktas in [59]. For comparison, the lower bound (5.21) on Lmax(2) is also in-127 eluded. For Lg = 2, the gradient set achieves considerably lower maximum and average distribution losses than QQA [59]. For Lg = 5, in general the gradient set also outperforms the harmonic set. For the harmonic set, L m a x (2 ) is not a monotonic function of N, which can be explained by the special structure of harmonic sets. As expected from [144, Table I], for N = 11 and N = 21, har-monic sets that achieve the lower bound can be found, cf. Method II in Section 5.4.2. For N < L2 both L m a x (2 ) and Lav(2) of the gradient sets are practi-cally identical to the lower bound, which suggest that the gradient algorithm proposed in Section 3.3 works well. Even for N > L2, where the lower bound is not achievable, the gradient set entails only a very moderate performance degradation of less than 0.5 dB compared with the bound. Furthermore, for Lg = 5, the average distribution losses of the gradient and the harmonic sets are practically identical. Fig. 5.4 shows Lmax(2) and Lav(2) of gradient and harmonic sets for Lh = 2 and Na = N$ = 2. SFV lengths of Lg = 3 and Lg = 5 are considered. Again the distribution loss of the harmonic sets is not a monotonic function of N. Interestingly, for N < 50 the average distribution loss of the gradient sets is limited to about 1 dB. Similar to the Lh = 1 case, increasing the SFV length to Lg = 5 > L m j n = 3 leads to a significant reduction of the distribution loss. The dependence of the distribution loss on Lg is depicted in detail in Fig. 5.5 for N = 30, Na = Ns = 2, and Lh = 1. For the gradient set, L m a x (2 ) and Lav(2) decrease monotonically with increasing Lg. For L2 > N = 30, both I/m a x(2) and Lav(2) for the gradient set are practically identical to the lower bound (5.21). The bound itself also decreases gradually as Lg increases. Although the maximum distribution loss of the harmonic set is considerably larger than that of the gradient set, the respective average distribution losses are practically identical for Lg > 4. 128 Figure 5.4: Maximum and average distribution losses as functions of the total number of nodes N for Ns = Na — 2 and Lh = 2. 5.5.3 BER Performance under Ideal Conditions In this subsection, we provide BER simulation results assuming that the SFV sets operate under the conditions they were designed for. In particular, we assume optimum MLSE at the receiver, perfect channel estimation, perfect timing synchronization, and Ns = Na = 2 active nodes. Figs. 5.6 and 5.7 show the corresponding BER results for Lh = 1 and Lh = 2, respectively, for N = 30 nodes. Fig. 5.6 shows the performance of DSTF for gradient sets and QGA [59], re-spectively. The curve for Lg = 1 corresponds to the case where both nodes transmit with BPSK without DSTF, i.e., the spatial diversity of the chan-nel is not exploited. For comparison, we also include the BER of a DSTBC (cf. Chapter 3) whose design is based on Alamouti's STBC [10]. The DSTBC 129 Figure 5.5: Maximum and average distribution losses as functions of the SFV length L G for Ns = Na = 2, L H = 1, and N = 30. has the same rate and was optimized under the same assumptions as the con-sidered DSTF scheme (see Chapter 3 for details). We observe that DSTF with gradient sets approaches the performance of DD optimized for Na = 2 co-located transmit antennas as L G increases from L G = 2 to L G = 7. We note that this performance improvement comes at the expense of an increase in detection complexity, cf. Section 5.2.2. However, even for L G = 2 the gradient sets perform significantly better than QGA- Assuming a BER of 4 • 10 - 4 , the performance losses of QGA and the gradient sets compared to optimized DD are 3.6 dB and 2.2 dB (LG = 2), 1.1 dB (LG = 3), 0.6 dB (LG = 5), and 0.3 dB ( L G = 7), respectively. These values are in excellent agreement with the respective average distribution losses shown in Figs. 5.3 and 5.5 for N = 30 nodes. This good agreement shows the relevance of the distribution loss as 130 101og 1 0 (£ 6 /W 0 ) [dB] • Figure 5.6: Average BER of DSTF with different SFV sets vs. 101og1 0(£6/iVo) for N = 30 nodes and Lh = 1. Ns = Na = 2 and MLSE is employed in all cases. The BERs of a DSTBC (cf. Chapter 3) and optimized DD for two co-located transmit antennas (cf. Section 2.3) are also shown. a performance measure. It also validates the adopted optimization criterion for SFV design, since Fig. 5.6 shows the BER for continuous transmission, whereas the distribution loss was derived based on the PEP for single-symbol transmission. Fig. 5.6 also shows that DSTF with gradient sets with Lg > 3 outperforms the DSTBC. However, we note that the decoding complexity of the DSTBC is significantly lower than that of DSTF. Similar observations as in Fig. 5.6 for Lh = 1 can be made in Fig. 5.7 for the frequency-selective channel (Lh = 2). Again, the performance of the gradient sets improves significantly as Lg increases and the performance of optimized DD is approached. For comparison, we have also included a BER curve for a 131 2 4 6 8 10 12 14 16 101og 1 0(^/iVo)[dB] • Figure 5.7: Average BER of DSTF with different SFV sets vs. 101og10(£ , b//v"0) for N = 30 nodes and Lh = 2. Ns = Na = 2 and MLSE is employed in all cases. The BER of optimized DD for two co-located transmit antennas (cf. Section 2.3) is also shown. (suboptimum) harmonic set with Ui = i — 1, 1 < i < Lg = Lmm = 3. Since this harmonic set was not optimized for the frequency-selective channel, a significant performance penalty compared with Ggx&d with Lg = 3 results. For a BER of 3 • 10 - 5 the performance loss of DSTF compared with optimized DD is 2.4 dB and 1.0 dB for gradient sets with Lg — 3 and Lg = 5, respectively. Again, these values are in excellent agreement with the respective average distribution losses in Fig. 5.4 for N = 30. 5 . 5 . 4 Influence of Practical Issues In this subsection, we study the performance of DSTF when the actual oper-ating environment differs from the environment the SFVs were designed for. 132 2 4 6 8 10 12 14 16 18 20 10log1 0(E6/7V0) [dB] • Figure 5.8: Average BER of DSTF vs. lO\ogw{Eb/N0) for Lh = 1, Lg = 5, N = 30, and MLSE. Gradient sets optimized for Na = 2, 5 are considered for Ns = 2, 5 active nodes. Mismatch between Na and Ns: As already mentioned in Section 3.5, in a practical network the actual number of active nodes Ns may differ from the design parameter Na. The impact of such a mismatch is studied in Fig. 5.8 for Lh = 1, Lg — 5, and iV = 30. In particular, we optimized two sets of gradient sets for Na = 2 and Na = 5, respectively, and simulated the resulting BER in each case for Ns = 2 and Ns = 5 active nodes. Fig. 5.8 shows that for a given Ns, the SFV sets achieve almost the same performance for both values of Na, which can also be deduced from the respective maximum distribution losses. For Ns = 2, the gradient sets achieve L m a x (2 ) = 0.7 dB and L m a x (2 ) = 0.9 dB for Afa = 2 and Na = 5, respectively, whereas for Ns — 5 we obtained L m a x (5 ) = 13.1 dB and L m a x (5 ) = 12.8 dB for Na = 2 and A^a = 5, 133 Figure 5.9: Average BER of DSTF with different SFV sets vs. 101og10(£6/7Vo) for Lh = 1, N = 30, and Ns = Na = 2. For Lg = 5 MMSE-DFE, DFSE, and MLSE are employed at the receiver. The BER of optimized DD for two co-located transmit antennas (cf. Section 2.3) is also shown. respectively. We observe similar results for Lh > 1 and different values of TY, Na, and Nsi suggesting that-the SFV design is fairly robust with respect to the adopted design parameter 7Ya, i.e., SFV sets optimized for Na active nodes are also close-to-optimum for Ns ^ Na. Therefore, it is generally advisable to design the SFVs for 7Ya = 2, since in this case the design complexity is lowest, cf. Section 5.3. Suboptimum Equalization: Figs. 5.6 and 5.7 illustrate that the achievable performance improves as Lg is increased. Unfortunately, increasing Lg also leads to an increase in the complexity of MLSE, cf. Section 2.2, and reduced-complexity suboptimum equalizers may have to be adopted. In Fig. 5.9, we 134 known CIR _ _ estimated CIR „ El Gamal & Aktas, L = 3 V 8 0 gradient set, L f l = 3 A gradient set, L=5 —.,. V -~ ~ ~ V - _ ~ - v - V - , 7 - v V ^ ^  v Si. *». V V V ^ 7 V S \ ~~ ~ - e - _ ~ - e - - _ - e e 9 O 5 O o o o A A— - - - A - -- - - A —& A ^ 10 35 N TS Figure 5.10: Required 101og10(i?6/./Vo) to achieve an average BER of 10 4 vs. training sequence length NTS for DSTF with different SFV sets. Lh = 2, TV" = 30, NS = NA = 2, and MLSE. show the performance of DSTF for a gradient set with Lg = 5 for M M S E -DFE, DFSE with different numbers of states [119], and MLSE receivers. NA = NS = 2, Lh = 1, and N = 30 are valid. For comparison, we also show the MLSE performance of DSTF with QGA [59] and a gradient set with Lg = 2. As expected, the performance for Lg = 5 deteriorates as the number of DFSE states decreases. However, even for M M S E - D F E (corresponding to DFSE with just one state) the gradient set with Lg = 5 outperforms the gradient set with Lg = 2 and MLSE reception at high SNRs. If we compare all schemes for two states and a BER of 4 • 10 - 4 , the gradient set with Lg = 5 outperforms the gradient set with Lg = 2 and QGA by 0.7 dB and 2.0 dB, respectively. Therefore, increasing Lg may be beneficial even if low-complexity equalization needs to be used at the receiver. 135 Imperfect Channel Estimation: Fig. 5.10 shows the SNR required to achieve a BER of 10 - 4 for estimated and known equivalent CIRs for DSTF with Lg = 3 and Lg = 5. Gradient sets with Lg = {3, 5} and a (suboptimum) harmonic set with Ui = i — 1, 1 < i < Lg = 3 are considered. Lh = 2, iV = 30, Ns = Na = 2, and MLSE are assumed. Least-squares channel estimation (LS-CE) with perfect training sequences (TSs) of length NTs is considered [163]. For long TSs, LS-CE always approaches the performance of the idealized case of known equivalent CIR. For short TSs the performance gain of Lg = 5 compared with Lg = 3 diminishes. In addition, the minimum required TS length N^ = 2 L e q - 1 [163] increases with Lg. Therefore, if only a small training overhead is desirable, relatively small Lg are preferable. Imperfect Timing Synchronization: So far we have assumed perfect timing synchronization among all the nodes. However, in practice it may be very challenging to perfectly synchronize all nodes. Therefore, in Fig. 5.11, the performance of DSTF with gradient sets with Lg = 2 and Lg = 5 is depicted, under the assumption that the timing offsets of different nodes are independent and uniformly distributed in the interval [—3Ts/4, 3T s/4]. The performance of BPSK transmission without DSTF (Lg = 1) is also shown. We assume a frequency-nonselective Rayleigh fading channel, Na — Ns = 2, and N = 30. The transmit filters of the nodes and the receive filter are square-root-raised cosine filters with a roll-off factor of 0.3. The gradient sets were optimized for zero timing error. Any timing error causes ISI, and for a fair comparison of different values of Lg, we use DFSE with two states in all cases for detection3. Fig. 5.11 shows that Lg = 1 benefits from timing errors, since frequency di-versity is introduced that can be exploited at the receiver. Nevertheless, for sufficiently high SNRs, the gradient sets with Lg = 2 and Lg = 5 outperform 3 In the absence of a timing error, symbol-by-symbol detection is already optimal for Lg = 1, and DFSE is identical to M L S E for Lg = 2. 136 Figure 5.11: Average BER of DSTF with perfect and imperfect time synchro-nization vs. 101og10(£?6/A7'o) for flat Rayleigh fading, N = 30, and Ns = Na— 2. In the case of imperfect time synchronization, the timing offset of all nodes is independent and uniformly distributed in the interval [—3T/4,3774]. For Lg = 2 and Lg = 5 gradient sets are employed and DFSE with two states is adopted in all cases. Lg = 1 by more than 2 dB also in the presence of the timing error. For Lg — 2, the timing error has a negligible effect on performance, whereas a loss of about 0.6 dB results in case of Lg = 5. The larger loss for Lg = 5 is due to the fact that for a given number of states the performance of DFSE deteriorates if the channel length is increased. Therefore, if timing errors are unavoidable and the receiver complexity is limited, moderate values of Lg may be preferable. Two-hop Transmission: We consider the two-hop system depicted in Fig. 5.1 and assume flat Rayleigh fading, l\ » R, l2^> R, and 2R/c < Ts. Therefore, all channels in the first and the second hop can be modeled as i.i.d. Rayleigh 137 lulog 1 0(£ h/;Vo) [dB] • Figure 5.12: Average BER of DSTF in two-hop transmission vs. 10\og10(Eb/N0). Gradient set, Lg = 5, Lh = 1, N = 30, Na = 2, DFSE with two states. Each node listens with probability p. Ex: Received energy per bit in first hop. Eb: Received energy per bit in second hop. fading channels with Lh = \. In the first hop, BPSK symbols are transmitted in packets of 100 symbols. To save energy, each of the N = 30 relay nodes listens only to the transmission with a probability of p < 1. These listening nodes, which can detect the entire packet correctly, use DSTF with a gradient set designed for Lh = 1, N = 30, Lg = 5, and Na = 2 in the second hop. We note that, in contrast to Figs. 5.6-5.11, here Ns is random. The received energy per bit in the first and the second hop is E\ and Eb, respectively. If no relay node can successfully decode the transmitted packet, a new packet is transmitted assuming a new realization of the fading gains. Fig. 5.12 shows the BER of the second hop for DFSE with two states. For comparison, we also show the BER of BPSK transmission with one active relay node and that 138 of DFSE with Ns = 2, 3, 5, 10 active relay nodes (dash-dotted lines). For E\/Eb = 1 and 101og10(.E&//Vo) < 4 dB, most of the time only one relay node is able to successfully detect the entire packet. Therefore, the resulting BER in. the second hop is similar to that of BPSK transmission. For higher SNRs, more nodes are able to successfully detect the packet, and node cooperation via DSTF results in a significant performance gain, compared with BPSK transmission. If the SNR in the first hop is significantly higher than in the second hop (i.e., E\/Eb = 10), node cooperation is beneficial in the entire considered Eb/No range. For high Eb/No, the two-hop BER curves become parallel to the BPSK curve. The reason for this behavior is the adopted (decentralized) energy-saving policy, which with a certain probability results in only one active relay node, regardless of E\. 5.6 Conclusions In this chapter, we have derived a novel cost function for the optimization of the SFVs of DSTF for general frequency-selective fading channels. Based on this cost function, two efficient algorithms for SFV design have been devel-oped. We have quantified the performance penalty entailed by the distributed implementation in our analysis of distribution loss. For the special case of frequency-nonselective fading channels, we have shown that the SFV opti-mization is related to the design of Grassmannian frames. This finding has facilitated the construction of optimum SFV sets and enabled the derivation of a lower bound on the maximum distribution loss. We have shown that increasing the length of the SFVs beyond the number of active nodes signifi-cantly improves performance, and that assuming two active nodes for the SFV design is also sufficient to achieve close-to-optimum performance if more nodes are active. Finally, we have studied the impact of suboptimum equalization, 139 imperfect channel estimation, and imperfect timing synchronization. Even if these effects are taken into account, the proposed SFV designs still achieve sig-nificant performance gains over SFV sets previously proposed in the literature and transmission without DSTF, respectively. 140 Chapter 6 Application to Wireless Sensor Networks 6.1 Introduction In recent years, wireless sensor networks (WSNs) have been gaining popularity in a wide range of military and civilian applications, including environmental monitoring, health care, and control. A typical WSN consists of a number of geographically distributed sensors and a fusion center (FC). The low-cost and low-power sensors make local observations of the hypotheses under test. A centralized detection scheme requires the sensors to transmit their real-valued observations to the FC. However, this procedure automatically translates into the unrealistic assumption of an infinite—bandwidth communication channel. In reality, the WSN has to work in a bandlimited environment. Moreover, as communication is a key energy consumer in a WSN, it is desirable to process the observation data as much as possible at the local sensors to reduce the number of bits that needs to be transmitted over the communication channel. Therefore, the sensors typically make local decisions that are then transmitted 141 to the FC where the final decision is made [132-136]. The resulting decentralized detection problem has a long and rich history. The decentralized optimum hypothesis testing problem was first formulated in [132] to provide a theoretical framework for detection with distributed sensors. Traditionally, the local decisions are assumed to be transmitted to the FC through perfect and error-free channels [132-137]. Realistically, the sensors typically work in harsh environments. Therefore, fading and noise should be taken into account. The problem of fusing sensor decisions over noisy and fading channels was con-sidered in [164, 165]. The fusion rules developed in [164] require instantaneous channel state information (CSI). While the fusion rules in [165] do not require amplitude CSI, they still assume perfect phase estimation/synchronization. However, obtaining any form of CSI may not be feasible in large-scale WSNs, and cheap sensors make phase synchronization challenging. To avoid these problems, simple ON/OFF keying and the corresponding fusion rules were considered in [166]. Furthermore, power efficiency was improved in [166] by employing a simple form of censoring [167] where the sensors transmit only reli-able decisions to the FC. The schemes in [164-166] assume orthogonal channels between the sensors and the FC. These schemes require a large bandwidth, es-pecially in dense WSNs with a large number of sensors. To overcome the bandwidth limitations of orthogonal transmission in WSNs, the application of coherent distributed space—time (DST) coding was proposed in [64]. In particular, in [64] each sensor is randomly assigned a column of Alamouti's space-time block code (STBC) [10], and it is assumed that only two sensors are active randomly at any time. The quantized observations are encoded by the sensors using the respective pre-assigned columns of the STBC and are transmitted to the FC via a common, non-orthogonal channel. Since 142 there are typically more sensors than STBC columns, the same column must be assigned to more than one sensor. However, as mentioned in Section 1.2, this kind of assignment results in a diversity order of 1. The performance degradation due to the diversity loss and the observation noise is analyzed in [64]. In this chapter, we consider noncoherent distributed space—time block coding for transmission of censored sensor decisions in WSNs. In particular, sensors use a common noncoherent distributed space-time block code (DSTBC) to forward their local decisions to the fusion center (FC), which makes the fi-nal decision. We consider specifically here noncoherent DSTBC and not the other schemes proposed in this thesis, because noncoherent DSTBC allows for low-complexity noncoherent detection which is particularly attractive in ap-plications of WSNs (cf. Sections 3.3.3 and 6.2.4). The contributions made in this chapter can be summarized as follows: • We show that the noncoherent DSTBCs introduced in Chapter 3 (cf. Sec-tion 3.3.3) eliminate the various restrictions and drawbacks of the coher-ent scheme in [64]. • Without any error-detection ability at the sensors, we show that the performance of distributed space-time coding is negatively affected by erroneous sensor decisions caused by observation noise. • We show that censoring of local decisions is essential for the efficient application of distributed space-time coding in WSNs. • We derive the optimum maximum-likelihood (ML) and a suboptimum generalized likelihood ratio test (GLRT) noncoherent FC decision rules for the proposed signaling scheme. 143 • The bit error rate (BER) at the FC for the GLRT decision rule is char-acterized analytically. • Based on the analytical expression for the BER, we devise a gradient al-gorithm for calculation of the optimum local decision/censoring thresh-old. • Our numerical and simulation results show the effectiveness of the pro-posed transmission scheme and the ability of the noncoherent DSTBC to achieve a diversity gain in WSNs. Organization: The rest of this chapter is organized as follows. In Section 6.2, we present the assumed system model and introduce the proposed transmission scheme for WSNs. In Section 6.3, we derive the M L and GLRT noncoherent FC decision rules and analyze the performance of the GLRT decision rule. A gradient algorithm for optimization of the local decision/censoring threshold is provided in Section 6.4. Simulation and numerical results are given in Section 6.5, and conclusions are drawn in Section 6.6. 6.2 System Model The binary hypothesis testing problem under consideration is illustrated in Fig. 6.1 where a set /C = {1,2, . . . , K} of K distributed sensors tries to de-termine the true state of nature H as being HQ (the null hypothesis) or H\ (target-present hypothesis). The a priori probabilities of the two hypotheses H0 and Hi are denoted as P(H0) and P(Hi), respectively. We assume that P(H0) = P(Hi) = 0.5 throughout this chapter. The details of the system model will be discussed in detail in the following subsections. 144 Sensor 1 Sensor 2 Ul DSTBC DSTBC 1 Sensor K uK Figure 6.1: Parallel fusion model with K sensors and one FC. A censored DSTBC is used for transmission from the sensors to the FC. 6.2.1 Local Sensor Decisions We assume that the sensor observations are described by H0 : xk = -l + nk, k e K., Hi : Xk — 1 + nfc, k G /C, (6.1) where the local observation noise samples nk, k G /C, are independent and identically distributed (i.i.d.) random variables. In particular, we model nk as real-valued additive white Gaussian noise (AWGN) with zero mean and 145 variance cr2 = £{n\}, k E K.. Upon receiving its own observation, each sensor makes a ternary local decision Uk = < — 1 if Xk < —d 1 iixk>d , kEJC, (6.2) 0 otherwise where d is the non-negative decision/censoring threshold. While uk = — 1 and uk = 1 correspond to hypotheses #0 and Hi, respectively, uk = 0 corresponds to a decision that is deemed unreliable by the sensor, and is thus censored. For future reference, we denote the sets of sensors with uk = 0, uk = —1, and Uk = 1 by S, Ho, and Tii, respectively. Note that IC = 5 U Tio U "Hi, where U is the union operator. The probabilities of correct and incorrect sensor decisions are given by Pc = Q{{d-l)/a) (6.3) and Pw = Q((d+l)/a), (6.4) respectively, where Q(x) = J x°° e - * 2 / 2 d£ denotes the Gaussian Q-function. The probability that a decision is censored is given by Ps = l-Pc-Pw = l- Q{{d - I) I a) - Q((d + l)/a). (6.5) 6.2.2 Noncoherent Distributed Space—Time Coding We introduced DSTBC for cooperative networks with decode-and-forward (DaF) relaying in Chapter 3. As mentioned before, DSTBC is also highly attractive for ad hoc networks and WSNs because it allows for low-complexity 146 detection at the receiver while achieving a diversity gain. Recall that a DSTBC consists of a code B and a set of signature vectors Q. The active relays encode the (correctly decoded) source information using a T x Nc code matrix 3> S B1. Each active node transmits a linear combination of the columns of the information-carrying matrix 3>. The linear combination coefficients for each node are unique and are collected in a signature vector gk E Q, \\gk\\2 — 1> k E JC, of length Nc. In this chapter, we consider the application of DSTBC in WSNs. In particular, sensors encode their local decisions using a noncoherent DSTBC. Since we consider here a binary hypothesis testing problem, B = {<&o, ^1} has only two elements. To optimize performance under noncoherent detection, we choose 3>o and <fri to be orthogonal, i.e., ^jf^i = ONCXNC {®XXY represents a, X xY matrix with all zero elements) and Q^$>V = INC, V E {0, 1}, cf. [89]. Similar to Chapter 3, each sensor is assigned a unique signature vector gk E Q, \\gk\\2 = 1, k E JC, of length Nc. For the design of deterministic and random signature vector sets Q we refer to Section 3.4 and [146], respectively. The transmitted signal of sensor k is given by VE*0gk iikEHo sk={ VE$xgk \ikEHi , ' (6.6) OTXI if k E S where E denotes the transmitted energy of sensor k per codeword. We note that sensor k transmits the T elements of sk in T consecutive symbol inter-vals. The total average transmitted energy per information bit is given by Eb = EK(PW + Pc). It should be noted that error detection is not possible at the sensors and therefore, wrong decisions may be forwarded to the FC 1 The relays that fail to decode the source packet correctly remain silent, cf. Chapter 3. 147 (cf. Eq. (6.7)). 6.2.3 Channel Model We assume that the sensor-FC channels are frequency-nonselective and time-invariant for at least T symbol intervals. Therefore, using the equivalent com-plex baseband representation of bandpass signals, the signal samples received at the FC in T consecutive symbol intervals can be expressed as r = £ skhk + n keHoUHi = VE$0GHohHo + VE^Gn.hn, + n , (6.7) where hk and n denote the fading gain of sensor k and a complex AWGN vector, respectively. The columns of the Nc x \rio\ matrix G-H0 and Nc x \H\\ matrix Gjix contain the signature vectors of the sensors in 7Yn and 7i\, respectively. The corresponding fading gains are collected in column vectors hn0 and , which have lengths \rio\ and \H\\, respectively. We model the channel gains hk, k € /C, as i.i.d. zero-mean complex Gaussian random variables (Rayleigh fading) with variance a\ = £{\hk\2} = l 2 . The elements of the noise vector n have variance a2 = No, where NQ denotes the power spectral density of the underlying continuous-time passband noise process. Eq. (6.7) clearly shows the importance of censoring when applying DSTBCs in WSNs, since incorrect sensor decisions lead to interference. For example, for H = Ho ideally the term involving <&i in Eq. (6.7) would be absent. However, incorrect decisions may cause some sensors to transmit y/~E$igk instead of VE&ogk. The considered censoring scheme reduces the number of incorrect 2 Again, this model is justified if the distance between any pair of sensors is much smaller than the distances between the sensors and the FC. The effect of unequal channel variances will be considered in Section 6.5 (cf. Fig. 6.7). 148 decisions (by choosing d > 0) at the expense of reducing the number of sensors that make a correct decision. However, this disadvantage is outweighed by the reduction of interference as long as d is not too large, cf. Section 6.5. We note that censoring was not considered in Chapters 3 and any of the related publications, e.g. [64, 146]. For example, in Chapter 3 and [146] DSTBCs were mainly applied for relay purposes, where a cyclic redundancy check (CRC) code could be used to avoid the re-transmission of incorrect decisions. 6.2.4 Processing at Fusion Center (FC) The FC makes a decision based on the received vector r and outputs UQ = 1 if it decides in favor of Hi and UQ = — 1 otherwise. Decision rules differing in performance and complexity may be applied at the FC. In this context, we note that coherent detection is not feasible in large-scale WSNs, since the FC would need to estimate and track the channel gains of all sensors. While Eq. (6.7) suggests that only the effective channels yfEG-n0h-n0 and VEG^xh^ need to be estimated if distributed space-time coding is applied, this is also not feasible since the sets Ho and Hi typically change after T symbol intervals (i.e., for every new sensor decision). Therefore, only noncoherent decision rules are considered in this chapter. 6.3 FC Decision Rules and Performance Anal-ysis In this section, we present the optimum ML and the GLRT noncoherent de-cision rules. In addition, we provide a performance analysis for the GLRT decision rule. 149 6.3.1 Optimum ML Decision Rule We first provide the optimum M L decision rule. For this purpose, we introduce the likelihood ratio (LR) A M A f(r\Hr) Ewo.wx /(rlMo ,tti)P(Mo ,tti |gi) °[ ] f(r\H0) E W o ,w 1 . /(»' |Wo,'Wi)P(Wo > W 1 |^o) ' l ' j where P(Ho,Hx\H0) = P^P^P]wl] and P(Ha,Hl\Hl) = P™P?\p™ denote the probabilities that the sets Ho, Hi occur for Ho and H\, respec-tively. Since r conditioned on Ho, H\ is a Gaussian vector, the conditional pdf f{r\Ho,Hi) is given by , , , exp (— rHBr)-/ ( r | « . , W , ) = liiet{B)', <«•») where the T xT correlation matrix B is defined as B = £{rrH\Ho,Hi} — E($OGH0G%0$O + ^IGHIGH^I) + a n J T - We can express the M L decision rule at the FC as UQ 1 if A 0 (r) > 1 W _ . (6.10) -1 i f A 0 ( r ) < l We note that the sums in the numerator and denominator of Eq. (6.8) both have 3K terms, i.e., the complexity of the ML decision rule is of order 0(3^) and grows exponentially with K. In addition, Eq. (6.9) reveals that for the M L decision rule the FC requires knowledge of the signature vectors of all sensors. These two assumptions make the implementation of the M L decision rule difficult, if not impossible in practice. Therefore, we will provide a low-complexity suboptimum FC decision rule in the next subsection. 150 6.3.2 GLRT Decision Rule The received vector can be expressed as r = $heff + n e f f , * e { $ 0 , * i } - (6.11) heft = yfEG^h^ and nefr — VE^iGuih-Hi + n if H0 is the true hy-pothesis, <fr = <fr0, while if Hx is true, $ = <&i, /i eff — ^PEGu-J^nx and n e f f = VE^0Gu0hHo + n. Eq. (6.11) suggests a two-step GLRT approach for estimating the transmitted codeword In the first step, heg is estimated assuming $ is known, and in the second step the channel estimate hes is used to detect Since the correlation matrix of the effective noise nes depends on Gnx or G ^ 0 , the ML estimate for Aieff and thus the resulting GLRT decision rule depend on the signature vectors. Therefore, the complexity of this GLRT decision rule is still exponential in K. To avoid this problem, we resort to the simpler least-squares (LS) approach to channel estimation. The LS channel estimate is given by heff = argmin {\\r - $hefl-\\22} = * " r . (6.12) The GLRT decision rule can be expressed as $ = argmin | | | r - ^^efrlll} * € { * o . * l } = argmax {\\$Hr\\l}, (6.13) * 6 { * o , * i } where all irrelevant terms have been dropped. The FC outputs UQ = — 1 if 3? = 3>0 and uo = 1 if <I> = 3>i. The resulting GLRT decision rule is similar to the previously derived decision rules for noncoherent STBCs and DSTBCs 151 (cf. Eqs. (2.24) and (3.14)). Clearly, the decision rule does not require CSI and the FC does not need to know the signature vectors of the sensors. 6.3.3 Performance Analysis for GLRT Decision Rule For the optimum M L decision rule, a closed-form performance analysis does not seem to be feasible. However, such an analysis is fortunately possible for the more practical GLRT decision rule. In particular, the BER can be expressed as PE = P(u0 = 1\H0)P(H0) + P(u0 = -l\Hi)P(Hi). (6.14) Since the considered signaling scheme is symmetric in Ho and Hi, Eq. (6.14) can be simplified to PE = P(UQ = 1\HQ). Expanding now P(uo = l\Ho) leads to PE = £ P(u0 = l\H0,Hi)P(H0,Hi\H0), (6.15) "Ho,Hi where P(UQ = 1\HQ,HI) denotes the probability that UQ = 1 is detected as-suming that Uk = —1 for k G HQ and uk = 1 for k G Hi, and P(Ho,Hi\H0) is as given in Section 6.3.1. Exploiting the orthogonality of $o and $ i and using Eqs. (6.7) and (6.13), P(UQ = l\Ho,H\) can be expressed as P(u0 = l\H0, Hi) = P ( A < 0\H0, Hi), (6.16) where A ^ \\x\\22 - \\y\\l (6.17) x = VEGnohHo + ^n (6.18) y = VEGmhm+Qin. (6.19) 152 Since A is a quadratic form of Gaussian random variables, the Laplace trans-form $A(S ) of the pdf of A can be obtained as where XXi and XVi denote the eigenvalues of the Nc x iV c matrices Dx±£{xxH} = EGnoG»0 + allNc (6.21) and Dy±£{yyH} = EGHlG%1+o*INe, (6.22) respectively. Thus, P(UQ = l\TCo,7ii) can be calculated from [168] c+joo p(Uo = i \ n 0 , n 1 ) = ^- f ^ - d s , (6.23) ZTTJ J S C—JOO where c is a small positive constant in the region of convergence of the integral and j = yf—1 denotes the imaginary unit. The integral in Eq. (6.23) can be computed either numerically using Gauss-Chebyshev quadrature rules [168], or exactly, using [169, 170]: P(uo = l\7io, TLi) = — ^ ] Residue RHS poles $ A ( S ) (6.24) where "RHS" stands for the right-hand side of the complex plane. The BER at the FC for the GLRT decision rule can be readily obtained by combining Eqs. (6.15) and (6.23). 153 6.4 Optimization of Censoring Threshold d Since a closed-form calculation of the optimum decision/censoring threshold d that minimizes PE does not seem to be possible, we derive here a gradient algorithm for recursive optimization of d. This algorithm is given by [171] BP d [ i + l ] = d W + ( j _ i > (6.25) where i is the discrete iteration index and 6 is the adaptation step size. Using Eq. (6.15) the gradient in Eq. (6.25) can be expressed as ~QJ = Z_, P(Uo = ^ O i W i ) ^ , (6-26) 'Hot'Hi where we have used the fact that P(UQ = l |Ho,7ii) is independent of d and the remaining partial derivative is given by 3P(Ho,Hi\Ho) _ | « j | p | s | - i p | W o | p | W i | ^ [ j . , | 7 / n i p | 5 | p | W o l - i p | W i | ^ 3d 1 1 3 c w 3d 11 011 ' c w 3d + i K a l P f l p l ^ l p ^ l - 1 ^ . (6.27) Using Eqs. (6.3)-(6.5) and the Fundamental Theorem of Calculus [172], the derivatives in Eq. (6.27) can be expressed as 3PW 1 3d 3PC 1 3d V2ncr 3PS 1 3d \2 e~^- (6.28) e-^f- (6.29) (6.30) e 20 -f e 2(7 For d = 0 we have |<S| = 0. Since 3Pw/3d < 0 and 3Pc/3d < 0, we obtain 3Pe/3d < 0. On the other hand, for d —• oo, we get \HQ\ 0 and \Hi\ —> 154 0, which results in dPe/dd > 0 3. Therefore, by the Mean Value Theorem, dPe/dd = 0 is valid for at least one value of 0 < d < oo corresponding to at least one local minimum of Pe [172]. Although numerical evidence shows that there is exactly one local minimum (which therefore is also the global minimum), we cannot formally prove this due to the complexity of the involved expressions. Nevertheless, the above considerations suggest that we initialize the gradient algorithm with d[0] = 0 corresponding to the case of no censoring. The solution found by the algorithm is then guaranteed to yield a performance no worse than that of the no-censoring case. Numerical examples will be given in the next section. 6.5 Simulation Results In this section, we provide some numerical and simulation results for the pro-posed censored DSTBCs. We assume that T = 8 symbol intervals are available for transmission of one information bit, i.e., orthogonal matrices 3>o and <&i can be found for Nc < 4. Here, we consider Nc = 1, Nc = 2, and Nc = 4, and generate 3>o and 3?i from the 8 x 8 Hadamard matrix Hs, where the orthog-onal columns of H$ are normalized to unit length. For example, for Nc = 2, 3>o consists of the first two columns of H8, whereas <fri consists of the third and fourth columns of Hs. For the set of signature vectors Q, we adopted the gradient sets described in cf. Chapter 3. Unless stated otherwise, the sensors have a local noise variance of a2 = 1/4, corresponding to a local SNR of 6 dB. We assume the suboptimum GLRT decision rule, and Pe at the FC is obtained using the analytical results presented in Section 6.3.34. 3 In fact, it can be shown that dPe/dd approaches zero from above if d —> oo corresponding to the maximum B E R of Pe = 0.5. 4 We note that we confirmed the analytical B E R results for the G L R T decision rule presented in Section 6.3.3 by simulations. However, for conciseness, we do not show the simulation results here. 155 d and Pe vs. i: First, we investigate the behavior of the adaptive algorithm described in Section 6.4 for optimization of d. Fig. 6.2 shows d and the corre-sponding BER Pe at the FC as a function of the iteration number i for Nc = 1, 2, and 4. The considered WSN had K = 30 sensors and the channel SNR was 101og10(£'b/A'o) = 15 dB. d[i] was initialized with 0 and the step size param-eter was chosen to achieve a fast convergence while avoiding instabilities. As can be observed from Fig. 6.2, the adaptive algorithm significantly improves the BER over the iterations. While d itself requires more than 600 iterations to converge to the final optimum value, Pe does not practically change after more than 180 iterations for all considered cases. It is interesting to note that the optimum value for d decreases with increasing Nc, i.e., for larger Nc less censoring should be applied. The reason for this behavior is that the maximum achievable diversity order of a DSTBC is Nc (cf. Chapter 3) and therefore, the performance of the DSTBC improves notably with an increasing number of transmitting sensors, until only Nc sensors transmit. If more than Nc sensors transmit, the diversity order does not further improve, and only a small addi-tional coding gain can be realized. On the other hand, less censoring means that more erroneous decisions are forwarded to the FC, which may negate the additional coding gain. Pe vs. 10logw(Eb/N0): In Fig. 6.3, we consider the BER achievable with the proposed censored DSTBCs at the FC of a WSN with K = 30 sensors as a func-tion of the channel SNR 101og10(£;h/A'o). For each considered Nc, we compare the BER for error-free local sensor decisions (a2 = 0, d = 0), noisy sensor decisions without censoring (a2 = 1/4, d = 0), and noisy sensor decisions with censoring (a2 = 1/4, d = dopt), where dopt denotes the optimum deci-sion/censoring threshold found with the gradient algorithm. Fig. 6.3 clearly shows that DSTBCs suffer from a significant performance degradation due to 156 1.2 0.8 0.6 0.4 0.21~ I / ' / / : / : ; / : / / /• 1 / : . / . / _ NC=1,5=3 - - N^=2,5=1 . _ . N°=4,5=1 c 1 / ' / l • . / L! i 10 10 ' 0 200 400 600 800 1000 i 10 1 1 1 1 — - N=I,5=3 - - N^=2,6=1 . _ . N°=4,5=1 c \ __ _ _ \ \ i 0 200 400 600 800 1000 i Figure 6.2: d and Pe vs. iteration number i for a WSN with K = 30 sensors using DSTBCs with Nc = 1, 2, and 4. 101og1 0(£6/A^0) = 15 dB, a 2 = 1/4. erroneous decisions if censoring is not applied. Fortunately, with censoring this performance degradation can be avoided and a performance close to that of error-free local decisions can be achieved. Fig. 6.3 also nicely illustrates the diversity gain that can be realized with censored DSTBCs. Pe vs. K: In Fig. 6.4, we investigate the dependence of the BER on the total number of sensors in the network for 10log1Q(Eb/N0) = 15 dB. In particular, we show in Fig. 6.4 the BER for error-free local sensor decisions and the GLRT decision rule at the FC (a2 = 0, d = 0), noisy sensor decisions with censoring and the GLRT decision rule at the FC (a 2 = 1/4, d = dopt), and noisy sensor decisions with censoring and the M L decision rule at the FC (a2 = 1/4, d = 157 6 8 10 12 14 16 18 20 101og10(£;6/Ar0) [dB] -Figure 6.3: Pe vs. 101ogio(£b/iV"o) f o r a WSN with K = 30 sensors using DSTBCs with Nc = 1, 2, and 4. Considered cases: Error-free local sensor decisions (a2 — 0, d = 0), noisy sensor decisions without censoring (cr2 = 1/4, d = 0), and noisy sensor decisions with optimum censoring (cr2 = 1/4, d = dQr,t)-d o p t ) - 5 The results for the GLRT decision rule were obtained numerically based on the analytical results in Section 6.3.3, whereas Monte-Carlo simulation was used to obtain the results for the M L decision rule. For complexity reasons, for the latter case we show only the results for K < 5. For error-free local sensor decisions BER is constant for K > Nc since the diversity order is limited to Nc and the DSTBC achieves the same performance as the related STBC B for co-located antennas if all K > Nc sensors transmit. The censored DSTBC with noisy sensor decisions approaches the performance of the DSTBC with error-5 W e note that for the M L decision rule, we also use the decision/censoring threshold d o p t found by the proposed gradient algorithm, which is based on the G L R T decision rule. Therefore, this threshold is not strictly opt imum for the M L decision rule. 158 I I I I I I I 5 10 15 20 25 30 K »~ Figure 6.4: Pe vs. total number of sensors K for a WSN using DSTBCs with Nc = 1, 2, and 4. 101og10(.E&//Vo) = 15 dB. Numerical results for error-free local sensor decisions and GLRT decision rule (a2 = 0, d = 0), numerical results for noisy sensor decisions with censoring and GLRT decision rule (cr2 = 1/4, d = dopt)) and simulation results for noisy sensor decisions with censoring and M L decision rule (cr2 = 1/4, d = dopt). free sensor decisions as the number of sensors increases. This is due to the fact that as K increases, the decision/censoring threshold d0pt increases, making the transmission of erroneous sensor decisions less likely. Fig. 6.4 also shows that the GLRT decision rule is almost optimum and only small additional gains are possible if the significantly more complex M L decision rule is used. Pe and d vs. Nc: Assuming the GLRT decision rule and 101og10(2£&/iVo) = 15 dB at the FC, Fig. 6.5 shows Pe and the corresponding optimum decision threshold d as a function of Nc for K = 1, 2, 4, 10, and 30. Similar to the observation we made in Fig. 6.2, d decreases for increasing signature vector 159 length Nc for all K. As we have mentioned before, the maximum achievable diversity order for DSTBC is Nc. For a given K, a smaller d allows more sensors to be active, and thus exploits the extra diversity benefit provided by the longer signature vectors: Fig. 6.5 also shows that d increases for increasing K, which can be explained easily. For a given d and Nc, increasing K allows more sensors to transmit. However, our scheme only requires a certain number of sensors to be active to exploit the full diversity benefit and achieve a certain target BER. On the other hand, increasing d decreases the chance of erroneous decisions being transmitted to the FC, which suggests that our scheme tries to maximize the performance by only allowing the minimum number of sensors (with quality decisions) to transmit. Finally, it is interesting to see that the Pe performance actually deteriorates for Nc> K for the GLRT fusion rule. This is because for iV c > K the GLRT fusion rule implicitly estimates the Nc x 1 effective channel vector /ieff in a noisy environment (cf. Eq. (6.12)), whereas the the underlying channel vectors, h^0 and h^, have a smaller dimensionality, K. The increased dimensionality causes a larger channel estimation error while no diversity benefit is achieved because the maximum diversity order is limited to K (cf. Chapter 3). In light of this degradation for the GLRT fusion rule, we also simulated the M L fusion rule for K = 1 and K = 2 (dashed curves); as expected, the ML decision rule does not suffer from the same degradation. We note that in the practically more relevant case of Nc < K, M L and GLRT decision rules have similar performance, cf. Fig. 6.4. Pe and d vs. SNR of local sensors: We investigate the effect of local sensor observation noise on the Pe performance in Fig. 6.6. In particular, we plot Pe vs. the SNR of local sensors 101og10(l/cr2) for different K and Nc. We assume the GLRT fusion rule at the FC, and the corresponding optimum de-cision threshold d is also depicted. Furthermore, the channel SNR is fixed to 160 0.07 Figure 6.5: Pe and d vs. Nc for a WSN with K sensors, cr2 = 1/4 and 101og10(£?6/iVo) = 15 dB. GLRT fusion rule is shown for all K (solid curves) and M L fusion rule is shown for K = 1, and 2 (dashed curve). 101og10(Et/A^o) = 15 dB for all cases. As expected, the network with K = 30 sensors performs better than the network with K = 10 sensors for any Nc, regardless of the sensor observation noise. However, this gain is minimal for large-sensor SNR. This is because as the sensor SNR increases, most of the sen-sor decisions will be correct and less censoring is required. This phenomenon is clearly supported by the corresponding d vs. 101og10(l/cr2) figure, where the optimum decision threshold d approaches zero for increasing sensor SNR. In addition, as more sensors transmit, the maximum achievable diversity order Nc and the channel SNRs will be the ultimate factors that determine Pe. There-fore, for a given Nc, the BER curves for K = 10 and K = 30 converge to the same value for large local sensor SNR. 161 o.ie 0.14 0.12 0.1 0.08 0.06 0.04 0.02 K = W o N =1 c V N =2 c • N =4 c ~K = 30 o N =1 c V N =2 c • N =4 c (a. s—• e O V y y &- - p -9 0 5 101og 1 0(l/a 2) [dB]-10 15 101og10(l/<72) [dB] • Figure 6.6: Pe and d vs. 101og10(l/cr2) for a WSN with i f = 10, and 30 sensors and DSTBC with Nc = 1, 2, and 4. 101og10(£(,/iVo) = 15 dB. I.n.d. Rayleigh fading: Until now, we have been considering i.i.d. Rayleigh fading channels. In our final example, we consider independent and non-identically distributed (i.n.d.) fading channels. In particular, we consider a network with K = 30 sensors, where the sensor nodes are uniformly distributed in a circle with radius R, and the distance from the center of the circle to the FC is d. We assume i.n.d. Rayleigh fading between the sensors and the FC, and that the received power decreases as d^a, where dk is the distance measured from sensor k to the FC and a = 3 is the path loss exponent. Fig. 6.7 depicts the simulated Pe vs. 101og10(jE&/./Vo) for different R/d ratios. For a given A^c, the optimum decision threshold d was optimized for R/d = 0 (corresponding to i.i.d. fading) and was then also used for R/d > 0. As expected, it can be seen from the figure that Pe increases with increasing R/d. It is also interesting 162 6 8 10 12 14 16 18 20 101og10(£6/AW [dB] -Figure 6.7: Pe vs. 101og10(E6/iVo) for a WSN with K = 30 sensors using DSTBCs with JVC = 1, 2, and 4. cr2 = 1/4 and i.n.d. Rayleigh fading channels. to note that the performance degradation is larger for larger /V c, which can be explained as follows. For a given network size K, d decreases for increasing Nc, as we have seen in Figs. 6.4 and 6.5., Since a smaller censoring threshold d corresponds to a larger number of active sensors, more sensors are negatively affected by the i.n.d. channels, resulting in greater performance degradation for larger Nc. 6.6 Conclusions In this chapter, we have considered the application of noncoherent DSTBCs in WSNs. We have introduced censoring as an efficient method to overcome the negative effects of erroneous local sensor decisions on the performance of 163 the noncoherent DSTBC. Furthermore, we have derived the optimum M L and a suboptimum GLRT FC decision rules, and have analyzed the performance of the latter decision rule. Based on this analysis, we have devised a gradient algorithm for recursive optimization of the decision/censoring threshold. Nu-merical and simulation results have shown the effectiveness of censoring, which eliminates the effect of local decision errors for practically relevant BERs if the number of sensors in the network K is greater than the length of the signature vectors Nc; in other words, if there are enough sensors to exploit the diversity benefit provided by the DSTBC. This makes our scheme particularly attractive in dense WSNs. Finally, our results have shown that the suboptimum GLRT fusion rule performs very close to the optimum M L fusion rule while having a very low complexity and allowing noncoherent detection at the FC. 164 Chapter 7 Conclusions and Future Work In this final chapter, we summarize our results and highlight the contributions of this dissertation. We also suggest topics and open problems for further research. 7.1 Research Contributions In this dissertation, we have proposed several novel distributed space-time (DST) coding schemes. Specifically, the three proposed schemes, namely dis-tributed space-time block coding, distributed space-time trellis coding, and distributed space-time filtering, belong to the class of decentralized DST coding schemes, which require neither channel state information (CSI) nor coordina-tion at the relay nodes. These schemes are designed particularly for wireless networks with an arbitrary number of decode-and-forward (DaF) relay nodes, where at any given time only an a priori unknown subset of nodes is active. In the following, we summarize our contributions and the main findings of this dissertation in further detail Since the proposed DST coding schemes are built upon the framework of space-time block coding, space—time trellis coding, and space-time filtering advo-165 cated originally for co-located antenna multiple-input and multiple-output (MIMO) systems, this dissertation commences in Chapter 2 with a concise discussion of these traditional space-time coding techniques. Specifically, the diversity gain and coding gain of each scheme are characterized through anal-ysis of the pairwise error probability (PEP). Chapters 3-5 are devoted to the development of the three DST coding schemes, respectively, and Chapter 6 considers the application of noncoherent DSTBCs in a practical wireless sen-sor network (WSN). In Chapter 3, we have provided design guidelines for the proposed distributed space-time block codes (DSTBCs). We have considered single-antenna relay and destination nodes for ease of exposition. The signal transmitted by an active relay node is the product of an information-carrying space-time block code (STBC) matrix and a unique node signature vector. We have shown that with full-rank STBCs designed originally for iV c co-located antennas, DSTBCs achieve a diversity order of d = mm{Ns, Nc} if Ns nodes are active. Interestingly, if Ns < Nc , the diversity order is limited by the number of active nodes; in this case, DSTBC design is complicated, since the information-carrying code matrix B and the set of signature vectors Q must be jointly optimized. On the other hand, for Ns > iV c, we have shown that we may separately optimize the code B and the set of node signature vectors Q. We have also shown that existing STBCs designed for co-located antennas are them optimal choice for B. We have proposed three efficient techniques for the optimization of Q. Furthermore, we have compared the performance of DSTBC with traditional STBC designed for MIMO systems and, in particular, we have characterized the distribution loss entailed by the distributed implementation. In conclusion, the proposed design method is highly flexible, and DSTBCs suitable for low-complexity coherent, differential, and noncoherent detection 166 can be easily obtained. In Chapter 4, we have proposed and analyzed the performance of distributed space-time trellis codes (DSTTCs). As a natural extension to Chapter 3 and to gain more insight into the problem, we have considered the general case where each relay node is equipped with NT antennas and the destination node is equipped with NR antennas. In this case, the transmitted signal of an ac-tive relay node is the product of an information-carrying space-time trellis code (STTC) matrix and a unique node signature matrix. We have shown that increasing the number of relay antennas can substantially improve the performance, even if these antennas are strongly correlated. To further gen-eralize the network model considered in Chapter 3, we have assumed inde-pendent and non-identically distributed (i.n.d.) relay-destination channels in Chapter 4. Our results suggest that signature matrices optimized for iden-tical relay-destination channel variances are also close to optimum for the case of non-identical variances. We have demonstrated that existing full-rank STTCs designed for Nc > 2 co-located antennas are a favorable choice for the information-carrying code matrix. In this diversity order of d = min{NcNR, NTNSNR} can be guaranteed if Ns nodes are active and the signature matrices are properly optimized using the various techniques pro-posed in chapter 4. In particular, we have extended the optimized design proposed in Chapter 3 (in the context of DSTBC) and the random design pro-posed in [146] to multiple-antenna relay nodes. Similar to Chapter 3, we have characterized the distribution loss entailed by the distributed implementation and have shown that DSTTCs inherit the coding gain of STTCs over STBCs, even in the distributed setting. Both DSTBCs and DSTTCs have been designed for frequency-nonselective channels. In Chapter 5, we have extended and optimized the original dis-167 tributed space—time filtering (DSTF) design proposed in [59] for frequency-selective channels. In DSTF, each node in the network is assigned a different signature filter vector (SFV) of length Lg, and the transmitted signal of the ac-tive nodes is the convolution of the source symbol sequence and their respective SFVs. We have derived a new optimization criterion for the design of the set of SFVs Q for both frequency-nonselective and frequency-selective channels. In particular, we have shown that a diversity order of d = min{Lh+Lg — 1, LhNs} can be achieved if Ns nodes are active and Lh is the number of channel taps. We have provided two practical and general methods for the design of opti-mum and close-to-optimum sets Q based on the proposed optimization crite-rion. For the special case of frequency-nonselective channels, we have pointed out that the SFV optimization problem is closely related to the design of sig-nature sequences for code—division multiple access (CDMA) and the design of Grassmannian frames. We have compared DSTF with optimized delay di-versity (DD) for co-located antennas and characterized the performance loss due to the distributed implementation. Several practical problems, such as suboptimum equalization, imperfect channel estimation, and imperfect timing synchronization, have also been studied. Table 7.1 summarizes the characteristics of the three proposed schemes. As we have argued in Chapter 4, DSTTCs achieve the best performance in terms of frame error rate (FER) due to their ability to provide a coding gain. However, the superior performance of DSTTCs comes at the expense of increased receiver complexity, as decoding for DSTTCs is performed via maximum-likelihood sequence estimation (MLSE) using a Viterbi decoder. On the other hand, DSTBCs have the lowest decoding complexity. DSTBC is also appealing for applications such as WSNs, as it is the only scheme that allows noncoherent detection. DSTF is attractive in the sense that it can be applied to frequency-168 Table 7.1: Comparisons of DSTBC, DSTTC, and DSTF DSTBC DSTTC DSTF Coding gain No Yes No Decoding complexity Low High Moderate Applicable to ISI channels No No Yes Allows timing synchronization errors No No Yes Noncoherent detection Yes No No selective channels and is robust to channel estimation and synchronization errors. Finally, it is worth mentioning that all of these schemes do not affect the receiver design, and traditional detection rules advocated originally for co-located antenna systems can be applied. In Chapter 6, we have specifically considered the application of noncoherent DSTBCs to WSNs. In particular, sensor nodes use a common noncoherent DSTBC to forward their local observations to the fusion center (FC), which makes the final decision. Unlike the relay model considered in Chapters 3-5, where a cyclic redundancy check (CRC) code can be use for error detection, the lack of any error-detection mechanism at the sensor nodes results in error propagation to the FC. We have shown that the performance of noncoherent DSTBCs is negatively affected by the propagation of erroneous sensor deci-sions resulting from observation noise. To overcome this problem, we have introduced censoring at the sensors, so that only reliable decisions are for-warded to the FC. We have proposed the optimum noncoherent maximum-likelihood (ML) and a low-complexity, suboptimum generalized likelihood ra-tio test (GLRT) FC decision rules. Furthermore, the performance of the GLRT 169 decision rule has been characterized analytically. Based on our performance analysis, we have derived a gradient algorithm for optimization of the local decision/censoring threshold. Numerical and simulation results show the ef-fectiveness of the proposed censoring scheme, making noncoherent DSTBC a prime candidate for signaling in WSNs. 7.2 Future Work While several enlightening results for decentralized DST coding schemes have been presented in this dissertation and in the literature, many issues remain to be addressed, and many possible directions for future research exist. We have concentrated on Rayleigh fading channels throughout our develop-ment in Chapters 3-5. It would be interesting to extend our results to other fading distributions, e.g., Ricean and Nakagami fading. Also, as we have seen before, DSTTCs achieve the best performance while DSTBCs have the low-est decoding complexity. The major advantage of DSTF over DSTBCs and DSTTCs is its ability to deal with intersymbol interference (ISI). The inability of DSTBCs and DSTTCs to combat ISI makes them less attractive. It would be desirable to extend the DSTBCs and DSTTCs proposed in this work to frequency-selective channels., To that end, traditional STBCs [160, 173] and STTCs [174, 175] designed specifically for frequency-selective MIMO chan-nels, or in combination with the orthogonal frequency division multiplexing (OFDM) approach [176], are particularly useful. Time synchronization is a critical issue for any distributed system, and is especially challenging for cooperative networks, as it is difficult to provide a precise clock reference for all the signals coming from many geographically distributed relay nodes. Existing timing synchronization methods designed for ad hoc networks, e.g., IEEE 802.11, are usually energy inefficient and therefore, 170 unattractive for the energy efficient WSNs. Therefore, DST coding schemes that are robust to timing synchronization errors would be a rich area for future research. The existing protocols for cooperative networks usually consist of two trans-mission phases of equal length. The source node broadcasts its message in the first phase and the relay nodes forward the message in the second phase to the destination node. A potential future research area would be to investigate and optimize protocols where the two transmission phases have different du-rations. This can be accomplished, for example, by using different types of modulation and coding in the two phases. A somewhat related area would be to develop power control schemes that maximize the error performance at the destination node. The power allocation issue is particularly important for networks where the relay-destination channel variances are different due to the locations of the physically distributed relay nodes. Most previous work on power allocation in cooperative networks considers the centralized approach, which requires instantaneous CSI at the transmitters [177, 178]. The liter-ature on power allocation with minimum overhead is limited. Recently, the power-allocation problem for DaF relaying has been studied under the as-sumption that only the mean channel gains are available at the transmitters [179]. Therefore, another interesting area for future research would be the de-sign of limited-feedback or adaptive-power allocation schemes for cooperative networks. As we have illustrated in Chapter 6, the proposed DST coding schemes are highly attractive in WSNs. In fact, the applications envisioned for WSNs vary from disaster-area monitoring and battle-field surveillance to space ex-ploration. In Chapter 6, by assuming statistical knowledge of the indepen-dent and identically distributed (i.i.d.) sensor noise, we have proposed a low-171 complexity non-orthogonal and noncoherent signaling scheme for WSN that takes into account the noisy and fading channels between the sensors and the FC. However, in situations where the sensors are located at different distances from the unknown target being monitored, the sensor noise would not be iden-tically distributed. In addition, knowledge of the noise distribution may not even be available in a hostile environment. Universal decentralized estima-tion techniques have been proposed for identical and nonidentical unknown sensor noise in [180] and [137], respectively. However, perfect communica-tion channels between the sensors and the FC were assumed in [137, 180]. Therefore, another rich area for future research would be the development of low-complexity DST coding schemes and the corresponding decision rules for WSNs with non-identical and unknown sensor noise. 172 Appendix A Upper Bound on PEP of Noncoherent STBCs Considering the decision rule in Eq. (2.24) and assuming transmission of B = B 0 E B, the probability that J3i € B, B 0 ^ Bi, is detected is Pe = Pr{A < 0}, where Pr{-} denotes the probability of the event in brackets, and A is defined as k ± r H { B 0 B $ - B l B \ H ) r . ( A . l ) Since r is a Gaussian random vector with zero mean and covariance £{rrH} = PSB0B$ + o*IT, Ps = T/Ne, the Laplace transform of the pdf of A is given by $(s) = l / d e t { J T + s<&} [181], where $ = {PsBQBg + ollT) (BQBQ - BXB»). (A.2) The latter matrix can be rewritten as $.= S + cr^M, where M = B0BQ — B\BX and S = PSB0B$M. To arrive at a meaningful bound for Pe, we need to characterize the minimum positive singularity s 0 of $(s) [168]. For this, we first note that matrix S has 173 Nc positive eigenvalues Xi(S) > 0, 1 < i < Nc, and T — Nc zero eigenvalues Xi(S) = 0, Nc + 1 < i < T. Therefore, taking the condition T > Nc into account, the inequality Ar(3>) > XT(S) + O^XT(M) = a2XT(M) results for the minimum eigenvalue of matrix cf. [138, p. 181]. On the other hand, since tr{J3^B 0} = tr{J3f Bx} = T, we get t r{M} = J2f=i x i ( M ) = 0> i-e-> XT(M) < 0. Therefore, we obtain for the minimum positive singularity of S 0 = — T T ^ T ^ ~ \ ~\—7T7\ > 0- ( A- 3) A T ($) (?l A T ( M ) According to [168] Pe is upper bounded by P e < $(a) for any value of a € (0, s0). Taking Eq. (A.3) into account it is easy to see that the PEP is also upper bounded by Pe < $(a0/al), where a0 = —1/XT(M). Therefore, the PEP can be upper bounded by P e ~ de t{ / r + {a0/ol) (PSB0B^ + alIT) (B0B» - BXB»)}' (A"4) In general, the tightness of this bound depends on the particular value of QJO-However, for the case a2 —> 0, the optimum value is ao = —l/Ar(Af) , and with al = N0/Es, Eq. (A.4) simplifies to Eq. (2.27). 174 Appendix B Elements of a Gradient Set for N = 30, N a = N c = 2 i = 0 i = 7999 9i -0.6091 - 0.7065J 0.3399 + 0.1200J -0.9808 - 0.0261J -0.0617 - 0.1830J 92 0.3382 - 0.4910J 0.7812 + 0.1849J 0.2944 - 0.1814J 0.4624 - 0.8165J 93 0.1846 - 0.4104J -0.7616 - 0.4663J 0.2232 - 0.5673J -0.3751 - 0.6984J 9A -0.2011 + 0.3655J -0.5988 + 0.6837J -0.6855 + 0.1211J 0.4805 + 0.5334J 9s -0.1572 - 0.7449J 0.1317 - 0.6348J 0.6893 + 0.4027J -0.5844 - 0.1454J 96 0.0642 + 0.5742J 0.1970 - 0.7921J -0.0055 + 0.1191J -0.7383 + 0.6639J 97 -0.3429 - 0.5387J 0.6725 - 0.3740J -0.3924 - 0.2372J 0.3224 + 0.8281J 98 -0.3454 + 0.6602J -0.6165 + 0.2543J -0.1636 - 0.3994J 0.0288 - 0.9016J 99 0.3857 + 0.3379J -0.6189 - 0.5949J 0.6332 - 0.6911J 0.2308 - 0.261 lj 910 0.2121 - 0.0494J 0.7495 + 0.6252J -0.6278 - 0.7088J 0.3084 + 0.0915J 9n 0.3931 + 0.2751J -0.3800 - 0.7908J -0.1098 - 0.4415J 0.8369 + 0.3044J 9l2 -0.5665 + 0.4146J 0.4822 + 0.5241J -0.6852 - 0.0216J 0.0065 - 0.7280J 9l3 0.0350 + 0.3214J -0.4634 + 0.8250J 0.3901 - 0.1649J 0.5600 + 0.7120J 9u 0.5222 + 0.6061J -0.2866 + 0.5271J -0.3985 + 0.9054J 0.1413 - 0.0394J 9l5 -0.4190 + 0.1922J 0.8824 - 0.0946J -0.0584 + 0.5338J 0.8387 + 0.0910J 9l6 0.9401 - 0.3152J -0.0638 + 0.1132J -0.3753 + 0.6786J 0.4766 + 0.4141J 9l7 0.4280 + 0.7326J -0.3286 - 0.4149J 0.7051 - 0.5416J -0.3924 - 0.2355J 9l8 0.7271 + 0.1216J -0.0260 - 0.6752J -0.1224 - 0.1961J -0.7415 + 0.6298J 919 0.2646 + 0.5737J 0.5211 - 0.5738J 0.5127 - 0.5104J 0.0967 + 0.6836J 920 0.8015 + 0.2897J -0.1678 + 0.4956J -0.1154 - 0.2221J 0.5667 + 0.7850J 921 -0.0252 + 0.0645J -0.9214 + 0.3823J 0.8790 + 0.0507J -0.1154 + 0.4598J 922 0.3362 - 0.6818J 0.3390 + 0.5542J -0.6808 + 0.4426J -0.2624 + 0.5213J 923 -0.9644 - 0.1533J -0.2014 + 0.0769J -0.8512 - 0.1635J -0.2032 - 0.4554J 924. 0.1162 - 0.0263J -0.9403 - 0.3188J 0.3929 + 0.6804J 0.1371 + 0.6032J 925 0.5064 - 0.3747J -0.2403 - 0.7385J -0.4803 + 0.4460J -0.7492 - 0.0954J 926 -0.4815 + 0.1495J -0.6414 - 0.5783J -0.7163 - 0.5974J -0.3352 + 0.1329J 927 -0.5546 + 0.6331J 0.2550 - 0.4760J -0.2138 - 0.6677J -0.0131 + 0.7129J 928 -0.5689 - 0.3917J 0.7071 + 0.1515J 0.4615 - 0.7694J -0.3426 + 0.2787J 929 -0.3373 - 0.7484J 0.4964 - 0.2823J 0.6094 - 0.0727J 0.7567 - 0.2251J 930 -0.3337 + 0.0594J 0.8385 - 0.4266J 0.2827 - 0.4768J -0.2375 + 0.7977J 175 Bibl iography [1] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems Amendment for Physical and Medium Access Control Layers for Com-bined Fixed and Mobile Operation in Licensed Bands, IEEE Std 802.16e-2005 and IEEE Std 802.16-2004/Cor 1-2005 (Amendment and Corrigen-dum to IEEE Std 802.16-2004), pages 0.1-822, 2006. 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