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Communicating over outage-limited multiple-antenna and cooperative wireless channels
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Communicating over outage-limited multiple-antenna and cooperative wireless channels
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COMMUNICATING OVER OUTAGE-LIMITED MULTIPLE-ANTENNA AND COOPERATIVE WIRELESS CHANNELS by Raj Kumar Krishna Kumar A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2009 Copyright 2009 Raj Kumar Krishna Kumar Acknowledgements I would like to thank my advisor Prof. Giuseppe Caire, for his invaluable guidance and support over the past four years. I am particularly grateful for the generous amount of time he dedicated towards my research work, and for the enthusiasm and interest he showed in my work. His comments and ideas have shaped this thesis in many ways; it has been a pleasure working with him. Thanks are also due to Profs. Keith Chugg, Ramesh Govindan, Gerhard Kramer and Urbashi Mitra for having been on my dissertation committee, and to Profs. Todd Brun, Vijay Kumar, Susan Montgomery, Robert Scholtz and Bharath Sethuraman for their support and guidance. I am grateful to my colleagues at the Communication Sciences Institute (CSI), Daniel, Faruk, Hooman, Marco, Marjan, Ozgun, Petros, Rahul and Tung for their com- pany, and for all the good times both on and off-campus. Tim Boston, Diane Demetras, Milly Montenegro, Gerrielyn Ramos and Mayumi Thrasher helped make my stay at CSI pleasant, by helping me with many administrative issues and taking care of my assistantships and fellowships. Finally, Iwould like to dedicate this thesis tomyparents andgrandparents, for their love and affection, and their unwavering support over all these years. ii Table of Contents Acknowledgements ii Abstract ix Chapter 1: Introduction and Outline of the Thesis 1 Chapter 2: Background: The Diversity-Multiplexing Tradeoff 16 Chapter 3: Space-Time Codes From Structured Lattices 20 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.1.1 Lattice Space-Time (LaST) codes . . . . . . . . . . . . . . . . . . 22 3.1.2 ST Codes from CDA . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.3 Lattice Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 The Structured LaST Code Construction . . . . . . . . . . . . . . . . . 29 3.2.1 CDA ST Codes as Lattice Codes . . . . . . . . . . . . . . . . . . 30 3.2.2 The S-LaST Construction . . . . . . . . . . . . . . . . . . . . . . 31 3.2.3 Performanceunderlow-complexityMMSE-GDFELatticeDecoding 37 3.3 The S-LaST TCM Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.1 Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3.2 Decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.3 Construction of suitable lattice partition chains . . . . . . . . . . 48 3.3.4 Code construction examples . . . . . . . . . . . . . . . . . . . . . 51 3.4 Conclusions and Further Extensions . . . . . . . . . . . . . . . . . . . . 52 Chapter 4: Outage Analysis for Correlated MIMO Fading Channels 55 4.1 Correlated MIMO Channel Models . . . . . . . . . . . . . . . . . . . . . 56 4.2 DMT of Correlated MIMO Channels . . . . . . . . . . . . . . . . . . . 58 4.2.1 Joint Distribution of Eigenvalues . . . . . . . . . . . . . . . . . . 59 4.2.2 Correlated Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . 60 4.2.3 Correlated Rician Fading . . . . . . . . . . . . . . . . . . . . . . 61 4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 iii Chapter 5: AsymptoticPerformanceofLinearReceiversinMIMOFad- ing Channels 67 5.1 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2 System model and linear receivers . . . . . . . . . . . . . . . . . . . . . 71 5.3 Diversity-Multiplexing Tradeoff . . . . . . . . . . . . . . . . . . . . . . . 78 5.3.1 Discussion and numerical results . . . . . . . . . . . . . . . . . . 84 5.4 MMSE receiver with coding across antennas . . . . . . . . . . . . . . . . 88 5.5 Outage probability of linear receivers in the large antenna regime . . . . 92 5.5.1 Asymptotic Gaussianity of the mutual information . . . . . . . . 94 5.5.2 Joint cumulant moments of the SINRs of order 1 and 2 . . . . . 97 5.5.3 Gaussian approximation and outage probability . . . . . . . . . . 109 5.5.4 Simulations and comparisons . . . . . . . . . . . . . . . . . . . . 112 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Chapter 6: CodingandDecodingfortheDynamicDecodeandForward Relay Protocol 123 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2 Problem definition and background . . . . . . . . . . . . . . . . . . . . . 126 6.2.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.2.2 Diversity-Multiplexing TradeoffofCo-operativeDiversityProtocols 129 6.2.3 Existing DDF code designs . . . . . . . . . . . . . . . . . . . . . 132 6.3 DMT of the DDF Protocol with finite length . . . . . . . . . . . . . . . 133 6.3.1 Outage probability analysis . . . . . . . . . . . . . . . . . . . . . 134 6.3.2 Achievability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3.3 Computing the DMT and comparisons . . . . . . . . . . . . . . 158 6.4 DMT optimal codes for the single relay DDF channel. . . . . . . . . . . 160 6.4.1 Design tradeoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.4.2 Approximately universalX s . . . . . . . . . . . . . . . . . . . . . 163 6.4.3 Decoding decision function φ and Forney’s decision rule . . . . . 167 6.4.4 Low complexity MMSE-GDFE Lattice Decoding . . . . . . . . . 172 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Chapter 7: Channel State Feedback over the MIMO-MAC 185 7.1 Channel Model and Problem Statement . . . . . . . . . . . . . . . . . . 186 7.2 Upper bound on the distortion SNR exponent . . . . . . . . . . . . . . . 189 7.3 Separated source-channel coding for the MIMO-MAC . . . . . . . . . . 193 7.3.1 Diversity multiplexing tradeoff (DMT) of the MIMO-MAC . . . 193 7.3.2 Distortion exponent with separated source-channel coding . . . . 196 7.3.3 Performance of successive interference cancellation (SIC) receivers 199 7.4 Hybrid Digital-Analog Coding Scheme for the MIMO-MAC . . . . . . . 200 7.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 iv Chapter 8: Conclusions 209 Bibliography 214 Appendices 223 A DMT of Linear Receivers: Proof of P(A) =O(1) . . . . . . . . . . . . . 223 B Novikov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 C Fluctuations of eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 227 D Higher order cumulants are vanishing. . . . . . . . . . . . . . . . . . . . 229 E Separated Relay Activity Detection . . . . . . . . . . . . . . . . . . . . 235 E1 Infinite block-length . . . . . . . . . . . . . . . . . . . . . . . . . 237 E2 Finite block-length . . . . . . . . . . . . . . . . . . . . . . . . . . 239 F Publications Based on this Thesis . . . . . . . . . . . . . . . . . . . . . . 242 F1 Journal Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 F2 Conference Papers . . . . . . . . . . . . . . . . . . . . . . . . . . 243 v List of Figures 1.1 Three possible space-time architectures. π and π −1 in (b) denote inter- leaving and de-interleaving. . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The single-antenna single relay fading channel. . . . . . . . . . . . . . . 12 3.1 Illustrating the Sphere-Encoder: Hexagonal Lattice, Q = 16, linear map (left) and sphere-encoded map (right) . . . . . . . . . . . . . . . . . . . 35 3.2 Effectoffundamentalcodinggainonperformance: 2×2STcodesderived from CDA, 16 bpcu,N = 2, MMSE-GDFE lattice decoding . . . . . . . 39 3.3 Effect of shaping gain on performance: 2×2 ST code derived from CDA, 16 bpcu,N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Comparing the Golden Code with the Rotated Gosset Lattice ST Code, N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 Performance of the 2×2 Golden code, Golden-Gosset andGA+ S-LaST codes at R = 16 bpcu. The inset shows a portion of the plot zoomed for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 3×3 ST Codes under MMSE-GDFE lattice decoding, N = 3 . . . . . . 42 3.7 Increasing the Coding Length, M = N = 2, T = 2,4,6, R = 16 bpcu, MMSE-GDFE lattice Decoding . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 S-LaST TCM Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.9 Two level partition of the example constructions . . . . . . . . . . . . . 52 3.10 16-state trellis used for the example constructions . . . . . . . . . . . . . 53 3.11 Performance of theGolden-Gosset andGA+ S-LaST TCMschemes,R = 5 bpcu,T = 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.1 Outage Probabilities for CHANNELS 1 and 2 at 4 bpcu . . . . . . . . . 65 5.1 Three possible space-time architectures: (a) unrestricted space-time cod- ing scheme; (b) coding across the antennas, with linear spatial equaliza- tion; (c) purespatial multiplexing with linear spatial equalization. π and π −1 in (b) denote interleaving and de-interleaving. . . . . . . . . . . . . 74 5.2 Outage probabilities of ZF and MMSE receivers, 2× 2 i.i.d. Rayleigh channel, R= 1 and 5 bpcu. . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Diversity of the MMSE receiver with joint spatial encoding: solid lines represent the outage probability in (5.5) and the dash-dot lines represent the corresponding upper bounds (5.13). M =N =4, ratesR are in bpcu. 91 vi 5.4 ComparingtheoutageprobabilityofoptimalandMMSEreceivers,R = 3 bpcu.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.5 Mean of the MMSE mutual information per antenna (C mmse 1 /M) as a function of β, for M = 2,5,10 and 20. The solid lines are analytical re- sults, and thecorrespondingdash-dotlines areempirical results obtained from Monte Carlo simulation. Diamonds denote 3 dB, circles 10 dB and triangles 30 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.6 Variance of the MMSE mutual information (C mmse 2 ) as a function of β, for M = 2,5,10 and 20. The solid lines are analytical results, and the corresponding dash-dot lines are empirical results obtained from Monte Carlo simulation. Diamonds denote 3 dB, circles 10 dB and triangles 30 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.7 CDF of the mutual information (MI) for the MMSE and optimal re- ceivers, forM = 2,3, β = 0.5 andρ= 3,30 dB. The solid blue line is the analytical result for the MMSE, the dot-dash blue is MMSE empirical, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. . . . . . . . . . . . . . . . . . . . . . . . 118 5.8 CDF of the mutual information (MI) for the MMSE and optimal re- ceivers, for M = 5,10, β = 0.5 and ρ = 3,30 dB. The solid blue line is the analytical result for the MMSE, the dot-dash blue is MMSE empiri- cal, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. . . . . . . . . . . . . . . . . . . . . . . 119 5.9 CDFofthemutualinformation(MI)fortheZFandoptimalreceivers, for M = 3,10, β =0.5 andρ= 3,30 dB. The solid blue line is the analytical result for the ZF receiver, the dot-dash blue is ZF empirical, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.1 The single-antenna single relay fading channel. . . . . . . . . . . . . . . 127 6.2 Negativeρ-exponentoftheprobabilityof therelay decodingafter exactly m-subblocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.3 The regionB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.4 TheDMToftheDDFchannelwithfinitelymanydecodingdecisiontimes. 158 6.5 X s is arotated QAMcode,T = 1, M = 4, R = 4bpcu,relay implements φ 1 (·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.6 X s is arotated QAMcode,T = 1, M = 4, R = 4bpcu,relay implements φ 2 (·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.7 X s is arotated QAMcode,T = 1, M = 4, R = 4bpcu,relay implements φ 3 (·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.8 X s is arotated QAMcode,T = 1, M = 4, R = 4bpcu,relay implements φ F (·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 vii 6.9 X s is a permutation code, T = 1, M = 4, R = 4 bpcu, relay implements φ F (·). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.10 X s is arotated QAMcode,T = 1, M = 4, R = 4bpcu,relay implements Forney’s or modified Forney’s rule. . . . . . . . . . . . . . . . . . . . . 184 7.1 DMT of the MIMO-MAC . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.2 Hybrid digital-analog scheme for bandwidth expansion . . . . . . . . . . 201 7.3 Distortion SNR exponent for the MIMO-MAC, N t = 1, M =K . . . . . 205 7.4 Comparinganalog feedback with digital feedback usingseparated source- channel coding, K =M = 4, N t = 1 . . . . . . . . . . . . . . . . . . . . 206 7.5 Distortion obtained for users with very different SNRs, K = M = 4, N t = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A1 The unit hemisphere and a spherical cap. . . . . . . . . . . . . . . . . . 224 viii Abstract Thebroadobjective ofthis thesisis toexplorethefundamentallimits ofcommunication over outage-limited wireless fading channels (including point-to-point and some multi- user scenarios), and design coding schemes that perform close to these limits. Inthefirstpartofthisthesis,wedealwiththepoint-to-pointmultiple-inputmultiple- output (MIMO) channel. We present codes derived from lattices and cyclic division al- gebras that achieve the diversity-multiplexing tradeoff (DMT) of MIMO channels, and further have excellent performances in terms of codeword error probability at practical signal-to-noise ratios. Subsequently, we analyse the performance limits in terms of the DMT of such optimal codes over correlated Rayleigh and Rician MIMO channels, which model wireless channels observed in practice. We then shift our attention to receiver design, and characterize the performance of low-complexity linear MIMO receivers that are currently being considered as promising candidates for practical implementations. Having treated the three major components of a point-to-point MIMO communica- tion system viz., the encoder, channel and decoder, we shift our attention to multi-user problems. We first consider the single relay fading channel, and derive the DMT for finite block-length transmission under a dynamic decode and forward protocol. We ix also present practical codes that have excellent performance with reasonable decoding complexity for this channel. Subsequently, we consider the problem of channel state feedback in cellular systems, and model this problem as the transmission of analog sources over a MIMO-MAC (MIMO multiple access channel). We develop a framework for analyzing this problem in terms of minimizing the distortion, and come up with separated source-channel schemes and joint source-channel schemes that performbetter than traditional analog feedback. The results are validated through simulations using very simple channel codes. x Chapter 1 Introduction and Outline of the Thesis Shannon’s fundamental paper [81, 83], titled “A mathematical theory of communica- tion”, introducedthe subjectof information theory in 1948. Itwould notbean exagger- ation to assert this work lent a significant impetus to the entire digital communications revolution, from cell phones to the internet. Early telecommunication systems, including telegraphy and telephony, employed a physical wire to transmit information. These wireline channels are modeled accurately by the popular additive white Gaussian noise (AWGN) channel model, the complex discrete baseband equivalent of which takes on the form y =x+w, 1 where y is the output of the channel, x is the input symbol, and the additive noise w is assumed to be distributed as complex Gaussian with mean zero and unit variance CN(0,1). The input is subject to a power constraint E |x| 2 ≤ρ, whereE(·) denotes theexpectation operator andρtakes on themeaningof thetransmit signal to noise ratio (SNR). The capacity of the AWGN channel (which represents the number of bits of information that can be reliably transmitted per channel use) was solved by Shannon in his fundamental 1948 paper [81], and is given by C AWGN = log 2 (1+ρ). Wireless systems play a major role in modern telecommunication. The wireless channel, however, presents some unique features that make it more difficult to handle than wired channels. The signal transmitted through a wireless channel propagates through free space along multiple paths, bouncing off multiple reflectors. The signals at the destination may add constructively or destructively, resulting in a fading channel that is modeled as y =hx+w, where h is a channel fading coefficient that represents the attenuation of the channel. Assuming we work in a rich scattering environment (this is usually the case in most 2 practical scenarios of interest), an application of the central limit theorem would in- dicate that h is Gaussian. A typical assumption is that h is distributed as CN(0,1) (corresponding to Rayleigh fading), and further, that it remains constant for a block of T uses of the channel following which it changes independently. This corresponds to a quasi-staticchannelmodel,andisvalidinthescenariowherethechannelcoherencetime is much larger than the symbol time (again, this assumption is reasonable in practical wireless channels). Conventional wisdom treated fading as a detrimental phenomenon, but more recently, it has been realized that fading can actually be exploited to obtain performance improvements, both in terms of throughput and reliability. The key idea behind exploiting fading to obtain the above mentioned twin ben- efits is to employ multiple antennas at both the transmitter and receiver of wireless communication systems. Each receive antenna receives a faded superposition of signals from the transmit antennas, corrupted by AWGN. Such a system is termed a MIMO (multiple-input multiple-output) system, and is modeled as follows. The quasi-static, frequency-flatfading(complex)MIMOchannelwithM transmitandN receiveantennas and coding block-length T channel uses is described by Y =HX+W, (1.1) where X denotes the M ×T transmitted codeword matrix drawn from a space-time (ST) codeX, Y is the N×T received signal matrix, H is the N×M channel matrix andW is theN×T noise matrix. The entries of the channel matrixH are assumed to 3 be constant over a block length ofT channel uses and the entries ofW are independent and identically distributed complex Gaussian with zero mean and unit variance, i.e., i.i.d. CN(0,1). Some of our subsequent results will hold for arbitrary channel fading statistics, and some others will hold only for the i.i.d. Rayleigh fading model, in which casetheentries ofHarei.i.d. CN(0,1). Inthesequel,wewillassumethei.i.d. Rayleigh fading model, and will state explicitly if the result holds true for more general fading statistics. The input constraint EkXk 2 F ≤T ρ (1.2) is enforced, whereE(·) denotes the expectation operator andρ takes on the meaning of the transmit SNR (total transmit energy per channel use over the noise power spectral density). The channel matrix H is assumed to be known perfectly at the receiver but not at the transmitter. The use of ST codes over MIMO channels is known to provide two kinds of benefits: better reliability through diversity gain, and higher data rates in terms of multiplexing gain. The ergodic capacity of the MIMO channel (1.1), given by [92] C erg (ρ), max tr(S)≤ρ S≥ 0 E h logdet(I+HSH † ) i , was shown in [33] for high ρ to behave like C erg (ρ)≈ min{M,N}logρ+o(1). 4 Thus,thecapacityoftheMIMOchannelisapproximatelymin{M,N}timesthecapacity ofanAWGNchannelathighSNR.Further,itwasshown[90,35]thatthroughintelligent design of the ST codebook X, the pairwise error probability (PEP) between any two codewords X 1 and X 2 could be made to decay as fast as PEP(X 1 →X 2 )∝ρ −MN , i.e., a maximum diversity of MN was possible. However, the amount of diversity and multiplexing that could be simultaneously obtained was not clear from these early works. The diversity-multiplexing tradeoff (DMT) [109] captures in a succinct and elegant way the tradeoff between these two quantities in the high SNR regime. The DMT specifies the maximum possible diversity that can be obtained at each possible value of multiplexing gain, and has become a standard performance metric to evaluate ST schemes, and a tool to compare different ST schemes. We will review the DMT formulation in Chapter 2, since we will make use of it extensively in our work. Families of codes that achieve the DMT of MIMO fading channels have been pro- posed. Perhaps the most notable in terms of performance and generality are Lattice ST (LaST) codes and codes obtained from cyclic division algebras (CDA). An ensemble of randomly generated LaST codes was shown to be DMT optimal under minimum mean squared error generalized decision feedback equalizer (MMSE- GDFE) lattice decoding for T ≥ M +N − 1 [24]. In this case, DMT optimality is 5 shown in a random coding sense (i.e., with respect to error probability averaged over the random lattice ensemble) and for the Rayleigh i.i.d. fading statistics. Families of carefully constructed CDA codes enjoy the so-called non-vanishing de- terminant (NVD) property (to be defined subsequently), which in turns implies that these codes, under ML decoding, achieve the optimal DMT in a universal sense, i.e., over any channel fading statistics [28, 54]. Codes achieving the optimal DMT over any fading statistics are called “approximately universal” in [91]. Furthermore, these codes allow for minimum block length, i.e., there exist optimal codes for all T ≥M [28, 54]. A review of LaST and CDA ST codes are presented in Chapter 3. However, sincetheDMTisaperformancemetricvalidathighSNR,DMToptimality does not capture performance at finite SNRs. Following [24, 28], attention has shifted towards constructing ST codes that not only achieve the DMT, but also perform well at finite (practical) values of SNR. For example, generating codes at random from the ensemble of [24] yields typically performances that stay at 1 to 3 dB from outage probability (that can be regarded an effective “quasi-lower bound” on the performance of any code at meaningful SNR, i.e., for probability of block error not too large (say, ≤ 10 −1 )). In this perspective, our first result in Chapter 3 presents a construction of structuredLaST(S-LaST)codes 1 thatachieve theDMTandperformwellatfiniteSNR, for small to moderate block-lengths (i.e., T is equal to or slightly larger than M). In the second part of Chapter 3 we turn to the case of large block lengthsT ≫M. This is motivated by the fact that in practical wireless communication systems, information is 1 We use the term “structured” to distinguish these codes from the random lattice approach of [24]. 6 encoded and sent over the channel in packets, together with training symbols, protocol information, and guard intervals. Therefore, packets cannot be too small, for otherwise the overhead would be a large part of the overall capacity. We target the case where data packets span a number of channel uses T considerably larger than the number of transmit antennas M, but nevertheless smaller than a fading coherence interval. Then, the fading channel is constant over the whole codeword of duration T channel uses. Unfortunately, the LaST and CDA constructions do not generalize, in practice, to T ≫ M since the decoding complexity grows rapidly with T. Furthermore, with constructions such as those in [24, 28] it is not clear how to exploit the large block length to obtain codes with improved coding gain. Therefore, the challenge here is to design ST codes for large T that have good coding gain and low decoding complexity. In this regard, the authors in [46] have proposed a trellis coded modulation (TCM) scheme based on partitions of the Golden code [5]. For prior work on ST TCM, see [87, 48]. Building on these ideas, in the second half of Chapter 3, we propose a general techniquefortheconstruction ofST-TCMschemeswithgoodcodingandshapinggains. These codes can be decoded using the Viterbi Algorithm where the branch metrics are computed using a low complexity MMSE-GDFE lattice decoder. Simulation results reveal that both codes for the short and long block-length cases have performance that is the state of the art among codes with similar encoding/decoding complexity. Notice that our S-LaST construction for the case of short block-lengths is approxi- mately universal, as mentioned before. This means that these constructions achieve the DMT over any fading channel. The DMT was derived in closed form only for the case 7 of i.i.d. Rayleigh fading in [109]. Most practical wireless channels however have non- trivial correlations, and it is hence of interest to know the performance limits that we can achieve over correlated MIMO channels using approximately universal ST codes. This is the topic we address in Chapter 4, where we show using tools from random matrix theory that the DMT is unchanged for arbitrary correlations, as long as the correlation matrix is full rank. In Chapter 5, we continue focussing on the point to point MIMO channel, turning our attention to the important practical issue of receiver complexity. The topic is of particular relevance, since the forthcoming standards in wireless local area networks such as IEEE 802.11n make MIMO technology a central component of their physical (PHY) layer. We have presented “very good” ST codes in Chapter 3, that are both DMT optimal, and have excellent performance in terms of error probability at practical SNRs. Unfortunately, thereceiver complexity of manyof theseschemes is prohibitivein terms of implementation using current VLSI technology, given the high data rates that are being targeted by 802.11n. The current standardization trend in 802.11n focuses on simpler schemes, where a linear equalizer is used to separate the signals from the M transmit antennas, thus creating a set of M “virtual” parallel channels. Standard coding techniques, based on binary codes and QAM modulation, are then applied as an outer coding layer to the resulting parallel channels. Figure 1.1 depicts these linear receiver models: (a) shows the optimal (non-linear) receiver, with coding and decoding performed over all antennas; (b) represents a linear receiver architecture with encod- ing across antennas; and (c) shows a linear receiver architecture based on pure spatial 8 multiplexing. The DMT of this simple scheme has surprisingly not been fully charac- ST enc. ST dec. decoded info bits info bits π π −1 Linear Receiver (a) (b) (c) Linear Receiver info bits info bits info bits SISO enc. SISO enc. SISO enc. SISO dec. SISO dec. decoded info bits decoded info bits decoded info bits SISO dec. x x x H H H y y y Figure1.1: Threepossiblespace-timearchitectures. πandπ −1 in(b)denoteinterleaving and de-interleaving. terized before. The first part of Chapter 5 provides such a characterization and reports a negative result: we show that both linear Zero-Forcing (ZF) and linear Minimum Mean-Square Error (MMSE) linear receiver front-ends achieve the same DMT, which is equaltothatof*one*oftheparallelchannels, eventhoughoutercodinganddecodingis performedacross the antennas. Hence, linear receivers incur a dramatic loss of diversity gain with respect to other non-linear schemes. It turns out that while the behaviour of the ZF receiver is captured exactly by the DMT analysis, the MMSE receiver exhibits full-diversity at fixed low rates. In order to explain this, we provide in the latter part of Chapter 5 an approximate quantitative analysis of the different behavior of the MMSE and ZF receivers at finite rate (zero 9 multiplexing gain). It is shown that while the ZF receiver achieves poor diversity at any finite rate, the MMSE receiver error curve slope flattens out progressively, as the coding rate increases. InthesecondpartofChapter5wetakeacloser lookattheperformanceofthelinear MMSE and ZF receivers at finite SNR. Since this is very diffcult to capture in closed form, we explore a second type of asymptotic regime, where we fix the SNR and let the number of antennas become large. Using random matrix theory, we show that in this case the limiting distribution of the mutual information of the parallel channels induced by the linear receiver is Gaussian, with mean and variance that can be computed in closed form. The analysis provides accurate results even for a moderate number of antennas and allows to quantify how the performance loss in terms of diversity suffered bylinearreceiversmayberecoveredbyincreasingthenumberofantennas. Thisprompts to the conclusion that in order to achieve a desired target spectral efficiency and block- error rate, at given SNR and receiver complexity, increasing the number of antennas and using simple linear receiver processing may be, in fact, a good design option. Armedwithagoodunderstandingandintuitiongainedthroughtheabovementioned works on single user MIMO fading channels, we switch our attention at this point to multi-user channels. Quite a few practical applications of wireless communication in- volves dealing with multi-user channels: for example, the uplink of a cellular system involves a multi-access channel with several users communicating with a single base station, while the downlink is a broadcast channel with the base station transmitting to several users simultaneously. More general scenarios where several co-located users 10 communicate information amongst themselves (sensor networks, for example) may in- volve relay channels (where one or more nodes may facilitate the transmission between a source and destination) and interference channels (where multiple source-destination pairs talk to each other simultaneously). It is well known that optimal strategies for the single user case do not carry over to the multi-user case (i.e., strategies like time and frequency division between users are not optimal in general). Multi-user information theory is the discipline of information theory that deals with such channels. Although a large amount of work has been done in the area, quite a few fundamental problems remain open over several decades. For example, the capacity regions of even very simple channels like the relay channel and the interference chan- nel are unknown in general. Needless to say, the capacity of a general multi-terminal network is unknown, and may be thought of as a holy-grail of multi-user information theory. Loosely speaking, the main complicating factor in the analysis of such problems is the interference between signal streams that may arise from different terminals that are destined to one or many terminals. Ourinterestwillmostlybefocussedonmulti-userchannelsthatinvolve fadingchan- nels along the various links, as these relate to typical wireless environments. Further, we will mostly be concerned with outage limited settings, i.e., where each codeword experiences only a finite number of channel realizations owing to delay constraints. In Chapter 6, we focus on the fading relay channel, that involves a single relayR facilitat- ing communication between a sourceS and a destination, see Figure 1.2. In particular, we focus on a protocol known as the dynamic decode and forward (DDF) protocol, 11 Figure 1.2: The single-antenna single relay fading channel. introduced in [2, 62, 53]. In the DDF protocol, communication may be split into two phases. In the first phase, the source transmits information while both the relay and the destination listen. When the relay has listened for a sufficiently long time such that it is able to decode the message of the source, it switches to transmit mode and sends symbols to help the destination decode the source message (this will be made more precise in Chapter 6). The DDF protocol has been analyzed in terms of the DMT only in the infinite block length limit [2], this would merely constitute an upper bound on the DMT for finite length codes. We characterize the finite block length DMT and give new explicit code constructions. The finite block length analysis illuminates a few key aspects that have been neglected in the previous literature: 1. we show that one dominating cause of degradation with respect to the infinite block length regime is the event of decoding error at the relay; 12 2. we explicitly take into account the fact that the destination does not generally know a priori the relay decision time at which the relay switches from listening to transmit mode. Both the above problems can be tackled by a careful design of the decoding algorithm. In particular, we introduce a decision rejection criterion at the relay based on For- ney’s decision rule (a variant of the Neyman-Pearson rule), such that the relay triggers transmission only when its decision is reliable. Also, we show that a receiver based on the Generalized Likelihood Ratio Test rule that jointly decodes the relay decision time and the information message achieves the optimal DMT. Our results show that no cyclic redundancy check (CRC) for error detection or additional protocol overhead to communicate the decision time are needed for DDF. Finally, we investigate the use of minimum mean squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoding at both the relay and the destination, and show that it provides near optimal performance at moderate complexity. We then turn our attention to the problem of channel state information feedback for cellular systems. We consider an FDD (frequency division duplex) cellular system with sufficient frequency spacing between the uplink and downlink channels (in practi- cal systems, the uplink and downlink channels are placed far away from each other to minimize cross-talk between the two channels). This results in the uplink and downlink channels being independent of each other. In order for the base station (BS) to serve multipleuserterminals (UT),theBStypically implements usingsomeMIMObroadcast channel precoding technique, such as linear beamforming, Dirty-Paper Coding or some 13 low-complexity non-linear precoding approximation thereof. All these schemes require that the BS know the downlink channels to the various UTs. While the UT may es- timate their respective downlink channels through training, the BS can only learn the downlink channels through explicit feedback from the UTs. This feedback involves the UTs communicating to the BS over a MIMO-MAC channel, with both the BS and the UTs potentially employing multiple antennas. Very little attention has been devoted to properly design the CSIT feedback scheme exploiting the MIMO-MAC nature of the uplink channel: most previous works assume perfect feedback at fixed rate, or implic- itly assume that the feedback information is piggybacked “somehow” into the uplink transmissions. This may pose problems, since the CSIT feedback must have extremely low latency, therefore, its coding block length must be very short. In Chapter 7, we consider the problem of designing very low latency and low complexity CSIT feedback schemes for the uplink MIMO-MAC. It is shown in [52, 12] how the average distortion at which the base station recon- structs thedownlinkcoefficients determines thedownlinkrateachievable bythesystem. We hence use the average distortion as our performance metric, and in particular work with the negative SNRexponentof the average distortion, known as thedistortion SNR exponent. We investigate the distortion SNR exponent achievable by separated source- channel coding schemes and joint source-channel coding schemes, and show that these improve upon the distortion SNR exponent of analog feedback (i.e., when the channel coefficients are transmitted unquantized by the UTs and the BS employs linear esti- mation). Further, we present simulation results using very simple channel codes, and 14 demonstrate that the proposed “digital” feedback schemes outperform analog feedback at practical values of SNR. We provide some conclusions and a few pointers for future work in Chapter 8, the final chapter of this thesis. A list of publications based on this thesis may be found in Appendix F. 15 Chapter 2 Background: The Diversity-Multiplexing Tradeoff For the quasi-static MIMO channel defined in (1.1), where H is arbitrarily correlated and Gaussian, the Shannoncapacity is zero. Hence, no positive rate can be transmitted with arbitrarily small error probability, at any given SNR ρ. The relevant limiting performance metric is then the tradeoff between rate and error probability. This was captured in the high-SNR regime by the Diversity-Multiplexing Tradeoff (DMT) [109]. Define a space-time scheme {X(ρ)} as a family of space-time codes of block length T, indexed by their operating SNR. In order to achieve a non-vanishing fraction of the capacity, we consider schemes whose data rate R(ρ) in bits/channel use scales linearly with logρ. 16 The scheme{X(ρ)} is said to achieve spatial multiplexing gain r and diversity gain d if the data rate R and average codeword error probability P e satisfy 1 lim ρ→∞ R(ρ) logρ =r, lim ρ→∞ logP e (ρ) logρ =−d. This latter relation is written asP e . =ρ −d in the exponential equality notation of [109]. The optimal Diversity-Multiplexing Tradeoff (DMT) is the best possible error prob- ability exponent d ∗ (r) achievable by any space-time scheme at multiplexing gain r. Without further constraints on the code construction and block length, the standard theory of ǫ-capacity [101] readily yields that d ∗ (r) is the exponent of the information outage probability defined as P out (R,ρ) = inf S :S≥ 0 tr(S)≤ρ P[logdet(I +HSH H )≤R]. (2.1) In the regime of high SNR, [109] showed that P out (R,ρ) . = P h logdet(I +ρHH H )≤R i . ThustheinputcovariancedoesnotinfluencetheSNRexponentoftheoutageprobability at high SNR. This holds irrespective of the fading statistics. 1 More generally, the DMT could be defined for any communication channel characterized by a rate R and error probability Pe, and is not limited to MIMO point-to-point channels only. 17 Let R =rlogρ andλ 1 ≤λ 2 ≤...≤λ A be the non-zero eigenvalues ofH H H, where A, min{M,N}. We obtain P out (r) . =P " A Y i=1 (1+ρλ i )<ρ r # , where, with some abuse of notation, we let P out (r) , P out (rlogρ,ρ) for large SNR. Letting λ i . = ρ −α i with α 1 ≥ α 2 ≥··· ≥ α A , and considering the limit for high SNR, we eventually arrive at [109] P out (r) . =P " A X i=1 (1−α i ) + <r # , where (x) + denotes max{0,x}. The outage probability is a characteristic of the fading distribution of the channel. Given the pdf p(α 1 ,...,α A ), p(α) obtained from the joint pdf of the eigenvalues of H H H, we can compute the limiting outage probability P out (r) . = Z P A i=1 (1−α i ) + <r α 1 ≥α 2 ≥···≥α A p(α)dα. The exponent of the above expression for i.i.d. Rayleigh fading channel is computed in [109] by using a large-deviation technique. This results in the following remarkably succinct and elegant result: 18 Theorem 2.0.1. For the space-time channel in (1.1) with i.i.d. Rayleigh fading, the optimal DMT d ∗ (r) is given by the piecewise linear function interpolating the points r =k, d= (M−k)(N−k) for k = 0,1,...,min{M,N}, and is zero for r> min{M,N}. For coding schemes of fixed block length T, the optimal DMT may be less than d ∗ (r). It is shown in [109] and [24] respectively that a Gaussian random coding ensem- ble generated with i.i.d. entries and the class of lattice space-time codes achieve d ∗ (r) ifT ≥M +N−1. Further results in [28] showed non-random coding scheme construc- tions that achieve d ∗ (r) for all T ≥ M. In particular, these coding schemes have an error probability conditioned on the no-outage event (i.e., for any channel matrix H with log(I +ρHH H )>rlogρ) that decays exponentially with SNR, irrespective of the statistics of the fading. This property is known as approximate universality [91]. This results in Theorem2.0.2. [109, 28] The optimal DMT for codes with fixed block lengthT is equal to d ∗ (r) for T ≥M. While the DMT was formulated in the context of point to point MIMO channels in [109],itisclear thatonemaycomputetheDMTforothersingleandmulti-userchannels as well. Since the introduction of the DMT, it has been adopted by the community as a standard performance metric for various outage limited channels, following which we will make use of the DMT extensively in this thesis. 19 Chapter 3 Space-Time Codes From Structured Lattices We work with the ST channel model defined in (1.1). In this chapter, since we will be dealing with both the complex MIMO channel in (1.1) and its real equivalent (to be introduced subsequently), we will use the superscript c to explicitly denote complex quantities. Hence we will rewrite (1.1) for use in this chapter as Y c =H c X c +W c . (3.1) The results of this chapter will hold for arbitrary channel fading statistics, but we will use the standard i.i.d. Rayleigh fading model for our simulations, in which case the entries of H c are i.i.d. CN(0,1). As mentioned previously, the ensemble of LaST codes and ST codes derived from CDA with the NVD property were shown to be optimal in terms of DMT [24, 28]. In 20 somesense, thepresentworkmaybethoughtofasaconfluenceofthesetwoapproaches. Weconstructcodesthatretaindesirablepropertiesfrombothfamilies: notonlyarethey are non-random explicit constructions from CDAs, but they also employ the nested lattice construction that enables shaping gains and the reduced complexity MMSE- GDFE lattice decoding akin to the LaST codes. The main target is to construct codes that are not only DMT optimal, but also perform well at finite SNRs. Wewilltreatthecaseofshortblock-lengthsfirst(i.e.,T isequaltoorslightlygreater than M). We present the Structured-LaST construction, that carves ST constellations from dense lattices that have good shaping properties. Decoding is done through a re- ducedcomplexity MMSE-GDFE lattice decoder,andsomeoftheexampleconstructions presented out-perform the perfect ST code construction, that was the best previously knownconstruction intermsoferrorprobabilityperformance[70,29]. Wethenturnour attention to the case of long block-lengths (T ≫M), and present a family of ST TCM schemes thathave good codingandshapingproperties. We show construction examples based on the Gosset lattice E 8 and lattices drawn from the Golden+ algebra [45] that yield, to the best of the authors’ knowledge, the current state-of-the art performance among codes with similar encoding/decoding complexity. In Section 3.1 we review LaST codes and ST codes from CDAs, as these form the two main ingredients for our construction. We also review some concepts relating to lattice packings that will be used subsequently. Code design for the short block-length case is presented in Section 3.2, and Section 3.3 deals with the construction of TCM 21 schemes. Simulations results are provided alongside each construction, and illustrate the effectiveness of the constructions. 3.1 Background 3.1.1 Lattice Space-Time (LaST) codes An n-dimensional real lattice Λ is a discrete additive subgroup of R n defined as Λ = {Gu : u ∈ Z n }, where G is the n×n (full-rank) real generator matrix of Λ. The fundamental Voronoi cell of Λ, denoted as V(Λ), is the set of points x∈ R n closer to zero than to any other pointλ∈ Λ. The fundamental volume of Λ is V f (Λ),V(V(Λ)) = Z V(Λ) dx = q det(G T G). An n-dimensional lattice code C(Λ,u 0 ,R) is the finite subset of the lattice translate Λ +u 0 inside the shaping region R, i.e., C = {Λ +u 0 }∩R, where R is a bounded measurable region ofR n . LaST codes are more easily illustrated by considering the real vectorized channel model equivalent to (3.1), y =Hx+w, (3.2) wherex∈R 2MT andy,w∈R 2NT denote respectively the vector equivalents ofX c ,Y c and W c obtained by separating real and imaginary part and by stacking columns, and 22 where H = I T ⊗ Re(H c ) −Im(H c ) Im(H c ) Re(H c ) , according to the well-known construction as in [24]. We say that an M ×T space-time coding scheme X is a full-dimensional LaST code if it’s vectorized (real) codebook (corresponding to the channel model in (3.2)) is a lattice codeC(Λ,u 0 ,R), for some n-dimensional lattice Λ, translation vector u 0 , and shaping region R, where n = 2MT. Given the equivalence of the real vector and the complex matrix representation of X, we shall not distinguish between them explicitly and write simplyX =C(Λ,u 0 ,R). Any linear-dispersion ST code, including the constructions of [28], can be represented as a LaST code, for a suitable shaping region. For later use, we define the lattice quantization function as Q Λ (y), argmin λ∈Λ |y−λ| and the modulo-lattice function [y] mod Λ =y−Q Λ (y). We also define the notion of a non-vanishing determinant (NVD) for an infinite LaST code (i.e., disregarding the shaping region R) as follows. A LaST code has the NVD 23 property if and only if the minimum determinant corresponding to its infinite lattice Λ is bounded away from zero by a constant independent of SNR, i.e., 1 min ΔX c =X c i −X c j , x i 6=x j , x i ,x j ∈Λ+u 0 det h ΔX c (ΔX c ) H i ˙ ≥ SNR 0 . Notice that since Λ is a lattice, this is equivalent to min x∈Λ+u 0 det h X c (X c ) H i ˙ ≥ SNR 0 . 3.1.2 ST Codes from CDA For a detailed exposition of ST codes from CDA, we refer the reader to [80, 28, 54] and references therein. We provide a very brief review in the sequel. LetQ denote the field of rational numbers and ı, √ −1. Set F =Q(ı). The construction of a CDA calls for the construction of an n-degree cyclic Galois extension L/F with generator σ. Then a CDA D(L/F,σ,γ) with center F, maximal subfield L and index n is the set of all elements of the form P n−1 i=0 z i ℓ i , wherez is an indeterminate satisfyingℓz =zσ(ℓ)∀ℓ∈ L and z n =γ. The elementγ needs to be a properly chosen non-norm element in order to ensure thatD is a division algebra, see [80, 28, 54] for details. Every element in the 1 We make use of the exponential equality notation from [109], defined as a . =ρ −b ⇔b =− lim ρ→∞ loga logρ . The notations ˙ ≥ and ˙ ≤ are defined similarly. 24 CDA can be associated with an n×n matrix through the left regular representation, which is of the form ℓ 0 γσ(ℓ n−1 ) γσ 2 (ℓ n−2 ) ... γσ n−1 (ℓ 1 ) ℓ 1 σ(ℓ 0 ) γσ 2 (ℓ n−1 ) ... γσ n−1 (ℓ 2 ) . . . . . . . . . . . . . . . ℓ n−1 σ(ℓ n−2 ) σ 2 (ℓ n−3 ) ... σ n−1 (ℓ 0 ) , (3.3) where ℓ i ∈L. The trace and determinant of the above matrix are respectively defined to be the reduced trace tr r (·) and reduced norm N r (·) of the element it represents. The ST code with M =T =n is a finite collection of matrices of the above form, scaled to satisfy the power constraint in (1.2). Choosingγ∈Z[ı] and restricting theℓ i to belong to the ring of integers O L of L bestows the NVD property on the ST code. One such choice for the ℓ i corresponds to choosing ℓ i = n X k=1 e i,k β k , e i,k ∈A QAM , (3.4) withA QAM ={a+ıb | −Q+1≤a,b≤Q−1, a,b odd},andwhereβ k ,k = 1,2,...,n isanintegralbasis(i.e.,abasisasamodule)forO L /O F . Moregenerally, wecouldchoose {β k } n k=1 to constitute anO F -basis for any idealI⊆O L . In this case,|X| =Q 2n 2 . The results of [28, 91] show that codes derived from CDA with NVD are approximately universal. 25 In the recent work [45], ST codes are obtained from maximal orders in CDAs. For thesake oflater use,abriefreviewfollows. AZ[ı]−order inanF−algebraD isasubring O ofD, having the same identity element as D, and such thatO is a finitely generated module overZ[ı] and generates D as a linear space over F. An orderO is called maximal if it is not properlycontained in any otherZ[ı]−order. ThediscriminantofaZ[ı]−orderO iscomputedasd(O/R) = det([tr r (b i b j )] m i,j=1 ),where {b 1 ,...,b m } is anyZ[ı]−basis of O. AllmaximalordersofaCDAsharethesamevalueofthediscriminant,andalsohave the smallest possible discriminant among all orders within a given CDA. An important property of elements of an order of a CDAD(L/F,σ,γ) is that their reduced norm (i.e., the determinant of their matrix representation) is an element of the ring of integers O F = Z[ı] of the center F. This property ensures that ST codes carved out of orders in suitably constructed CDAs are endowed with the NVD property. The choice of a subset of elements of D corresponding to (3.4) amounts to choosing a particular order O known as the natural order. It is established in [45] that the discriminant of an order in a CDA is directly proportional to the fundamental volume of the ensuing lattice (they are in fact equal for the case when the center of the CDA isF =Q(ı)). Therefore, in order to maximize the energy efficiency of the code, a sensible design guideline is to use the maximal order of the CDA to derive ST codes, owing to them having the minimum possible discriminant. All previous constructions of ST codes from CDAs, including the ones in 26 [80, 28, 70, 5, 29] have used the natural order, which is not guaranteed to be maximal in general. As an illustration of the technique, the authors in [45] construct a 2×2 ST code derived from the maximal order of a CDA named the Golden+ Algebra (GA+), whose minimum determinant improves upon that of previously known constructions. We will revisit this construction subsequently in Section. 3.2, and use it to construct some of our examples. 3.1.3 Lattice Packings The classical sphere packing problem is to find how densely a large number of identical spheres can be packed together in n-dimensional space. A packing is called a lattice packing if it has the property that the set of centres of the spheres forms a lattice in n-dimensional space. An excellent reference for this area is the book by Conway and Sloane [16]. The density Δ of a lattice packing is given by Δ , Proportion of space that is occupied by the spheres = volume of one sphere V f (Λ) . A related quantity is the center density δ, given by δ = Δ V n , 27 where V n is the volume of an n-dimensional sphere of radius 1, given by V n = π n/2 (n/2)! = 2 n π (n−1)/2 ((n−1)/2)! n! (the second form avoids the use of (n/2)! when n is odd). A related parameter is the fundamental coding gain γ c (Λ), defined as: γ c (Λ),4δ 2/n = d 2 min (Λ) V(Λ) 2/n , (3.5) where d min (Λ) denotes the minimum distance of the lattice Λ. It is evident from the definition that the fundamental coding gain is a normalized measure of the density of the lattice. Further, thefundamental codinggain also possesses the desirable properties of being dimensionless, and invariant to scaling and any orthogonal transformation (rotation) [31]. For the cubic lattice, γ c (Z n )= 1. The problem of finding dense packings (i.e., those with high values of γ c (Λ)) in n- dimensional space has a long and interesting history. In two dimensions, Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fe- jes T´ oth proved that the hexagonal lattice is indeed the densest of all possible plane packings. In 1611, the German astronomer Johannes Kepler stated that no packing in three dimensions can be denser than that of the face-centered cubic (f.c.c.) lattice arrangement which fills about 0.7405 of the available space. It took mathematicians some 400 years to prove him right, with Thomas Hales proving the conjecture in 1998 (Gauss showed in 1821 that the f.c.c. lattice is the densest possible lattice packing in 28 three dimensions). The densest possible lattice packings are known for all dimensions n≤ 8. The checkerboard latticesD 4 andD 5 are the densest possible lattice packings in 4 and 5-dimensions respectively while Gosset’s root lattices E 6 ,E 7 and E 8 are optimal among lattice packings in 6,7 and 8-dimensions. It is also known that the densest lat- tice packings in dimensions 1 to 8 are unique. Although not proven, it seems likely that Coxeter-Todd lattice K 12 , the Barnes-Wall lattice Λ 16 ∼ = BW 16 and the Leech lattice Λ 24 are the densest lattices in dimensions 12,16 and 24 respectively [16]. Tables of the best known lattice packings in n-dimensions are available in the literature [16] and in the online catalogue of lattices [68]. For later use, we define a lattice Λ with generator matrixG to be an integral lattice if the Gram matrix A,G T G has integer entries. It turns out that many of the best known lattices in terms of packing belong to this class, when suitably scaled. 3.2 The Structured LaST Code Construction This section deals with code design for the case of short block-lengths, i.e., T is equal to or slightly larger than M. Before we present the construction, we first explore the LaST formulation of space-time codes derived from CDA. 29 3.2.1 CDA ST Codes as Lattice Codes We will illustrate the equivalent lattice structure with an example of a 2×2 ST code derived from CDA. From (3.3), any codeword matrix is of the form X c = ℓ 0 γσ(ℓ 1 ) ℓ 1 σ(ℓ 0 ) . The real vector corresponding to X c in the equivalent channel model of (3.2) is given by x= h Re(x c ) T Im(x c ) T i T , where x c = [ℓ 0 ℓ 1 γσ(ℓ 1 ) σ(ℓ 0 )] T ∈C 4 . Let{β 1 ,β 2 } denote an integral basis over Z[i] for some ideal I ⊆O L . Then, in accor- dance with (3.4),X c represents a point in the (complex) lattice whose generator matrix is given by G c = β 1 β 2 0 0 0 0 β 1 β 2 0 0 γσ(β 1 ) γσ(β 2 ) σ(β 1 ) σ(β 2 ) 0 0 , (3.6) i.e., x c =G c [a 1 a 2 a 3 a 4 ] T , {a i } 4 i=1 ∈Z(ı). 30 The corresponding real lattice generator matrix is given by G= Re(G c ) −Im(G c ) Im(G c ) Re(G c ) . It is now evident that the choice of parameters γ and {β 1 ,β 2 } completely determines the lattice structure of the ST code (assuming a particular generator σ for the group of automorphisms). Furthermore, the choice of these parameters in conjunction with (3.4) amounts to the choice of a particular subsetL ofO L to be the signaling alphabet. The key to ensuring good constellation shaping lies in an intelligent choice of the non-norm element and the integral basis. In [70], these parameters arechosen to ensurethat theresultant lattice generated by G is a rotated version of the cubic lattice Z 2MT , i.e., that G is a unitary matrix. The cubic shaping is in fact the best possibleshaping that we can obtain by a linear encoder over thereals(linear-dispersioncode). Noshapinggaincanbeachieved byalinearmap: at most, the encoder does not increase the transmit energy. This is indeed obtained by G unitary, that is an isometry ofR 2MT . The authors in [70] provide such constructions for2×2, 3×3, 4×4and6×6(square)STcodeswithNVDandhavetermedtheresultant ST codes as perfect codes. More recently, [29] presented perfect ST code constructions for arbitrary number of transmit antennas and also for the rectangular case (T ≥M). 3.2.2 The S-LaST Construction We wish to obtain LaST codes with the following properties: 31 1. the NVD property; 2. the underlying lattice Λ c (referred to as the coding lattice in the following) has large fundamental coding gain γ c (Λ c ) (see (3.5)); 3. the shaping regionR is as close as possible to a sphere. We term the resulting codes as Structured-LaST (S-LaST) codes. The third property yields good shaping gain γ s , defined as the ratio of the normalized second moment of an n-dimensional hypercube to that of the shaping region R. If the shaping region is an n-dimensional hypercube, as in the case of perfect codes, then γ s = 1. Choosing a bettershapingregionRdoesnotchangethegeometricarrangementofthelatticepoints, but the average transmitted energy is decreased thanks to shaping. The above three requirement are simultaneously achieved using a nested lattice (Voronoi) construction and a non-linear modulo-lattice encoder nicknamed sphere encoder. 2 Let G p denote the generator matrix of a perfect code (unitary), and let G Λ denote the generator matrix of a good 2MT-dimensional integral lattice Λ, that is, a lattice with large fundamental coding gain (such lattices are available in the literature [16]). Define Λ c to be the lattice with generator matrixG Λc =G p G Λ and let Λ s (referred to as the shaping lattice) be a sublattice of Λ c such that Λ s has good shaping gain. Let [Λ c |Λ s ] denote the nesting ratio, that is, the cardinality of the quotient group Λ c /Λ s . 2 Tree-searchalgorithms toperform theClosest LatticePointSearch(CLPS),basedonPohstenumer- ation [104] and generalized in [18, 67] are generally nicknamed “sphere decoders” if used for minimum distance lattice decoding or “sphere encoders” if used for modulo-lattice precoding, in the current communication and coding theoretic literature. The reason of the nickname follows from the bounded- distance enumerative decoding of the Pohst lattice point enumeration and variants thereof. 32 Then, we construct a structured LaST code X as the set of all distinct points x given by x= [λ+u 0 ] modΛ s as λ varies in Λ c , and u 0 is a translation vector used to symmetrize the code. Although not necessary, in all cases considered in this chapter we let Λ s = QΛ c , Q∈Z + for simplicity, i.e., we use a self-similar shaping lattice. The rationale behind this choice is that it is well-known that for moderate dimensions, the best lattices with respecttocodinggainarealsogoodquantizers,i.e., havegoodshapinggain. Thecoding rateisgivenbyR = 1 T log[Λ c |Λ s ]= 2M logQ. Noticealsothatbecauseofthe“rotation” matrix G p and the fact that Λ is an integral lattice, the set of pointsX represented as complex matrices has the NVD property. Theorem 3.2.1. The space-time code X derived from the lattice G Λc = G p G Λ using a nested-lattice structure corresponds to a space-time code derived from CDA with non- vanishing determinant and hence achieves the optimal DMT over any fading channel statistics. Proof. Recall that G p corresponds to a ST code with NVD, i.e., the set of all non-zero lattice vectors z ∈ G p Z 2MT , represented as complex matrices Z c , have det Z c (Z c ) H boundedawayfromzerobysomeconstanttermSNR 0 (uptoorderofexponentofSNR). Since Λ is an integral lattice, there exists ak∈R such thatkG Λ generates a sublattice 33 of Z 2MT . It follows that the LaST code kX generated by kG p G Λ is a sublattice of G p Z 2MT and therefore satisfies min X∈X: X6=0 det(XX H ) ˙ ≥ k −2M SNR 0 . = SNR 0 . The proof of DMT optimality now follows from [28, 91]. The modulo-Λ s “sphere-encoder” is easily implemented by some CLPS, using some “sphere decoding” algorithm [18, 67]. The shaping effect of sphere-encoding is best illustrated using a 2-dimensional example. Suppose that Λ c is the hexagonal lattice in two dimensions. Set Q = 16. The constellations corresponding to the linear map (centred at the origin) and the sphere-encoder are shown in Fig. 3.1. As the value ofQ increases, the sphere-encoded constellation fills the fundamental Voronoi region of the hexagonal lattice uniformly. Although both constellations correspondto signalling from the hexagonal lattice, the energy saving of the sphere-encoder is evident. Example 1. (The Golden-Gosset S-LaST code) WhenM = 2, we chooseG p to be the lattice generator matrix of the Golden code [5] and G Λ to be the generator matrix of the Gosset lattice E 8 , which are respectively given by G p = 1 √ 5 Re(G c p ) −Im(G c p ) Im(G c p ) Re(G c p ) , 34 −20 0 20 −15 −10 −5 0 5 10 15 −10 0 10 −15 −10 −5 0 5 10 15 Figure 3.1: Illustrating the Sphere-Encoder: Hexagonal Lattice, Q = 16, linear map (left) and sphere-encoded map (right) where G c p = η θη 0 0 0 0 η θη 0 0 γσ(η) γσ(θ)σ(η) γσ(η) γσ(θ)σ(η) 0 0 , 35 θ = 1+ √ 5 2 , σ(θ) =1−θ, η = 1+ı−ıθ, σ(η) = 1+ı−ıσ(θ), γ =ı, and G Λ = 2 −1 0 0 0 0 0 0.5 0 1 −1 0 0 0 0 0.5 0 0 1 −1 0 0 0 0.5 0 0 0 1 −1 0 0 0.5 0 0 0 0 1 −1 0 0.5 0 0 0 0 0 1 −1 0.5 0 0 0 0 0 0 1 0.5 0 0 0 0 0 0 0 0.5 . Example 2. (The Golden+ Algebra (GA+) S-LaST code) Our second example is based on a 2×2 ST code derived from a maximal order of a CDA [45]. The Golden+ algebra [45] is defined to be GA+ = (Q(δ)/Q(ı),σ,ı), where δ is the first quadrant square root of 2+ı and the automorphismσ is determined byσ(δ) =−δ. The maximal order O ofGA+ is generated by the following orderedZ(ı)−basis: 1 0 0 1 , 0 1 ı 0 , 1 2 ı+ıδ ı−δ −1+ıδ ı−ıδ , 1 2 −1−ıδ ı+ıδ −1+δ −1+ıδ . (3.7) 36 The Golden+ code [45] corresponds to the left ideal of the maximal order generated by M = (1−δ) 3 0 0 (1+δ) 3 . (3.8) In this case, we choose G Λ to be the lattice generator matrix corresponding to this left ideal of the maximal order and G p = I (trivial rotation). Notice that this choice does not maximize the fundamental coding gain (the Golden-Gosset S-LaST code has a higher density), but the minimum determinant of the Golden+ S-LaST code is better than that of the Golden-Gosset code. It is a priori not clear which effect will dominate the performance in terms of error probability; this will be answered in the simulation results to follow. 3.2.3 Performance under low-complexity MMSE-GDFE Lattice De- coding Unfortunately, due to the usage of a non-linear encoding to achieve shaping gain, ML decoding of the resulting code is very complicated, requiring essentially the exhaustive enumeration of the whole codebook. Notice that a similar problem arises in the case of theGA+ code in [45], where linear encoding would result in very bad shaping. The au- thorsin [45]have obtained shapingbyenumeratingtheminimumenergycodewords and perform exhaustive decoding, both these are feasible only for low spectral efficiencies. Hence, we resort to suboptimal MMSE-GDFE lattice decoding (see [24, 18] for details). It has been proven that this decoder achieves the optimal DMT in the random 37 coding sense, for a specific ensemble of random lattices. Here, we use it with our deterministic non-random constructions. We do not claim that the resulting schemes achievetheoptimalDMTunderlatticedecoding. Nevertheless,theperformanceofthese codes is outstanding. In our simulations, we make use of a random translation vector u 0 , uniformly distributed over a very large hypercube with volume much larger than the volume of the shaping region. This random “dithering” is known to the receiver, and is subtracted before decoding, as explained in [24]. With this “trick”, we ensure that the transmitted points have energy exactly equal to the second moment of Λ s and have exactly zero mean. Furthermore, ditheringsymmetrizes the scheme and makes the error probability independent of the transmitted codeword. Fig. 3.2 compares the performance of two 2×2 ST codes derived from CDA with R = 16 bpcu and N = 2. The two ST codes chosen in this case have γ c (Λ c ) equal to 0.8365 and1.4142 respectively. SphereencodingandMMSE-GDFE lattice decodingare used in both cases. We notice about one dB of gain due to better fundamental coding gain of the lattice. In order to illustrate the benefit of constellation shaping, we plot in Fig. 3.3 the performance of a (2×2) ST code derived from CDA first using linear encoding of the information symbols and ML decoding and then using sphere encoding and MMSE- GDFE decoding (R = 16 bpcu, N = 2). The particular ST code chosen has γ c (Λ c ) = 0.8365. Quite a significant gain of about 3.5 dB results from codebook shaping in this particular case. 38 24 26 28 30 32 34 36 38 40 42 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. Fundamental Coding Gain = 0.8365 Fundamental Coding Gain = 1.4142 Outage Probability, 16 bpcu Figure 3.2: Effect of fundamental coding gain on performance: 2×2 ST codes derived from CDA, 16 bpcu,N = 2, MMSE-GDFE lattice decoding 24 26 28 30 32 34 36 38 40 42 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. (2 × 2) CDA code, Linear map, ML Decoding (2 × 2) CDA code, Sphere Encoded map, MMSE−GDFE Lattice Decoding Outage Probability, 16 bpcu Figure 3.3: Effect of shaping gain on performance: 2×2 ST code derived from CDA, 16 bpcu,N = 2 39 5 10 15 20 25 30 35 40 45 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. Golden Code, 4 bpcu, MMSE−GDFE Gosset ST Code, 4 bpcu, MMSE−GDFE Golden Code, 4 bpcu, ML Decoding Golden Code, 16 bpcu, MMSE−GDFE Gosset ST Code, 16 bpcu, MMSE−GDFE Golden Code, 16 bpcu, ML Decoding Outage, 16 bpcu Outage, 4 bpcu Figure 3.4: Comparing the Golden Code with the Rotated Gosset Lattice ST Code, N = 2 For thecase ofM = 2, wecomparetheperformanceof theGolden Code[5], whichis aperfect2×2STcode(withγ c (Λ c )= 1),withtheGolden-Gosset2×2S-LaSTcodefrom Example 1, (γ c (E 8 ) = 2). Fig. 3.4 shows plots of the Golden code under ML decoding andMMSE-GDFE lattice decodingincomparison with theGolden-Gosset S-LaSTcode with MMSE-GDFE lattice decoding at rates of 4 and 16 bpcu. At 4 bpcu, the (real) information symbol constellation corresponds to BPSK signaling on each dimension (Q = 2). In this case, the signal points of the Golden code in 8-dimensional space lie on the surface of a sphere (they are vertices of the rotated hypercube). Therefore, the 2×2 perfect code construction is optimal for 4 bpcu also in terms of shaping. This intuition is verified by the plots corresponding to 4 bpcu in Fig. 3.4. However, when the number of bits per channel use increases, the effect of the coding gain of the lattice and the shaping gain begin to show up. At 16 bpcu, the Golden-Gosset S-LaST code 40 withMMSE-GDFElatticedecoding(marginally)outperformstheGoldencodewithML decoding (see Fig. 3.4). These plots also serve to illustrate that MMSE-GDFE lattice decoding is near-ML in performance, while offering significant reductions in complexity. In Fig. 3.5, we present comparisons of the Golden code with ML decoding, the Golden-Gosset S-LaST code (see Example 1) and the GA+ S-LaST code (see Exam- ple 2), at 16 bpcu. While the fundamental coding gain of the lattice corresponding to the GA+ code is less than the coding gain of E 8 , the loss in density is compensated for by an increase in the minimum determinant. Both the Golden-Gosset and theGA+ S-LaST codes with MMSE-GDFE lattice decoding outperform the Golden code with ML decoding. Figure 3.5: Performance of the 2×2 Golden code, Golden-Gosset and GA+ S-LaST codes at R = 16 bpcu. The inset shows a portion of the plot zoomed for clarity. 41 5 10 15 20 25 30 35 40 45 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Codeword Error Prob. 3*3 Perfect Code (Elia et al.), 6 bpcu 3*3 Perfect Code (Oggier et al.), 6 bpcu 3*3 Rotated Λ 18 Lattice, 6 bpcu 3*3 Perfect Code (Elia et. al.), 24 bpcu 3*3 Perfect Code (Oggier et. al.), 24 bpcu 3*3 Rotated Λ 18 Lattice, 24 bpcu 3*3 MIMO Outage, 6 bpcu 3*3 MIMO Outage, 24 bpcu Figure 3.6: 3×3 ST Codes under MMSE-GDFE lattice decoding, N = 3 For the 3×3 case, we compare the performance of two perfect codes from [29] and [70] (with base alphabets QAM and HEX respectively) with an S-LaST code based on a rotated version of the Λ 18 lattice, which is the best known lattice packing in 18- dimensions [16]. MMSE-GDFE lattice decoding is used for all cases. The results shown in Fig. 3.6 show a significant gain for both 6 and 24 bpcu resulting from the increased lattice coding gain and shaping. In Fig. 3.7 we compare the performance of the 2×2 Golden-Gosset S-LaST code (T = 2) with rectangular 2×4and 2×6S-LaSTcodes constructed usingthehorizontal- stacking construction [28]in conjunction with theBarnes-Wall (Λ 16 )(γ c (Λ 16 )= 2.8284) and Leech (Λ 24 ) (γ c (Λ 24 ) = 4) lattices respectively. The length-24 cyclic codeG 24 (Z 4 ) constructedin[10]wasusedtoconstructanisomorphicversionoftheLeechlatticeusing 42 33 34 35 36 37 38 39 40 41 42 10 −3 10 −2 SNR (dB) Codeword Error Prob. Rotated Gosset Lattice, T = 2 Rotated BW−16 Lattice, T = 4 Outage (16 bpcu) Rotated G 24 (Z 4 ) lattice, T = 6 Figure 3.7: Increasing the Coding Length, M = N = 2, T = 2,4,6, R = 16 bpcu, MMSE-GDFE lattice Decoding construction-A[16]. MMSE-GDFElatticedecodingisusedforallthreeSTcodes. Inac- cordance with intuition, the performance approaches outage probability asT increases, owing to better values of γ c (Λ c ). 3.3 The S-LaST TCM Scheme Motivated by the fact that in practical wireless communications M is limited by trans- mitter complexity tobeasmall integer (typically 2or4, incurrentIEEE802.11n MIMO extension of wireless local area networks) while T may be of the order of 100 channel uses,ourobjective inthissection istoconstructM×T STcodesforthecaseofT ≫M. For ease of exposition and without loss of fundamental generality, we will focus on the 43 Figure 3.8: S-LaST TCM Encoder case where T =LM, for some integer L. TCM has the nice feature that a single trel- lis code can generate any desired block length, with decoding complexity linear in L, using a Viterbi decoder. Furthermore, the construction of TCM schemes is rather well understood and a rich literature exists for the Gaussian channel (see [98, 32, 77] and references therein), the scalar fading channel (see [49] and references therein) and for the MIMO fading channel [90, 87, 48]. 3.3.1 Encoder Consider a three level partition Λ t ⊃ Λ m ⊃ Λ b (where the subscripts indicate ‘top’, ‘middle’ and ‘bottom’) of lattices in R n , with n = 2M 2 . Let [Λ t |Λ m ] =M and let the cosets of Λ m in Λ t be indicated by C i , {v i +Λ m }, for i = 1,...,M, where each v i is a coset representative of C i . From each coset C i , we carve a finite set of N points, denoted by{v i +c j :c j ∈ Λ m ,j = 1,...,N}. These points are chosen via a modulo-Λ b 44 sphere encoder, that will be described in the following. Also, we choose Λ b such that N = [Λ m |Λ b ]. In all the examples presented here, we use Λ b =QΛ m , for some Q∈Z + (i.e., we use again a self-similar shaping lattice). In this case,N =Q 2M 2 . We make use of Forney’s general “coset coding” framework [31]. A block diagram of the encoder is shown in Fig. 3.8. During each block k = 1,...,L comprising of M channel uses each, a block of (logM)/r + logN information bits enters the encoder. The top (logM)/r information bits are input to a convolutional encoder of (binary) rater, that outputs logM coded bits, which select the indexi k ∈{1,...,M} of a coset in Λ t /Λ m . The remaining logN information bits select the index j k of a point in the finite constellation carved from the selected cosetC i k . The transmitted vector at time k is given by x k = [c j k +v i k +u k ] mod Λ b (3.9) where u k is an optional random dithering signal known to the receiver, that serves to symmetrize the overall TCMcodeand to inducethe uniformerror property. Thevector x k is then mapped into anM×M complex matrix and transmitted inM channel uses across the MIMO channel. The rate of the S-LaST TCM scheme is given by R = (logM)/r+logN M bits/channel use. 45 It should be noticed that x k =c j k +v i k +u k −λ k for some λ k ∈Λ b that is a function of c j k ,v i k ,u k . Further, x k ∈ V(Λ b ). Since [Λ m |Λ b ] = N, the mapping between the uncoded bits and the constellation points in each coset is one-to-one. 3.3.2 Decoder The (real equivalent) received point at each block k is given by y k =Hx k +w k , for k = 1,...,L. In general, the trellis of the S-LaST TCM scheme has N parallel transitions per trellis branch, corresponding to the N points in the intersection C i ∩ V(Λ b ), on each branch labeled by the coset C i . Consider time k, and a branch labeled by coset C i . The corresponding branch metric for a ML trellis decoder (implemented via the Viterbi algorithm) is given by B i,k = min c∈Λm∩V(Λ b ) |y k −H(v i +c+u k )| 2 . (3.10) Computing this branch metric amounts to exhaustive enumeration of all points of Λ m in the Voronoi regionV(Λ b ) of the shaping lattice. Since exhaustive enumeration is usually too complex, we resort once again to a suboptimalMMSE-GDFE lattice decoder along the lines of [24], in orderto compute an approximateMLbranchmetricfortheViterbidecoder. First,werelaxtheminimization 46 in (5.17) to take into account all points of Λ m (Lattice decoding), i.e., we consider the suboptimal branch metric B i,k = min c∈Λm |y k −H(v i +c+u k )| 2 . (3.11) This amount to solving a CLPS problem for the channel-modified lattice HΛ m , with respect to the point y k −H(v i +u k ), where u k is a known dithering vector and v i depends on the label of the branch for which we compute the metric. The surviving path among the parallel paths corresponds to the argument c that minimizes (3.11). Then,wefurthermodifythesuboptimalmetricfollowingtheMMSE-GDFEparadigm (see [24] for the details). Let F and B denote the forward and backward filters of the MMSE-GDFE as defined in [24]. At each time k, the receiver obtains the following set of modified channel observations y ′ i,k =Fy k −B(v i +u k ), 1≤i≤M. Using the properties of the matrices F and B, these can be written as y ′ i,k = F[H(c j k +v i k +u k −λ k )+w k ]−B[u k +v i ] = B(c j k +v j k −λ k −v i )−[B−FH](c j k +v i k −λ k +u k )+Fw k = B(c j k +v j k −λ k −v i )−[B−FH]x k +Fw k , B(c j k +v j k −λ k −v ℓ )+e ′ k . 47 Notice thatx k is uniformly distributed overV(Λ b ) and is hence independent ofc j k and v j k [24]. Itcan beshownthatthenoiseplusself-noisevectore ′ k hasthesamecovariance matrixoftheoriginalnoisew k ,althoughitisgenerallynon-Gaussian. Also,v i k −v i = 0 (i.e., it belongs to Λ m ) ifi k =i, while it belongs to some coset of Λ m in Λ t not equal to Λ m if i k 6=i. For each branch labeled by coset C i , the low-complexity Viterbi decoder computes branch metric B i,k = min z∈Z 2M 2 y ′ i,k −BG Λm z 2 where G Λm denotes a generator matrix for Λ m . This can be obtained by a sphere decoder applied to the channel-modified latticeBΛ m . It is clear that the branch metric for the correct coset (i.e., for i = i k ) will be smaller than the branch metric for an incorrect coset, with high probability. 3.3.3 Construction of suitable lattice partition chains In order to ensure good performance, we choose the componentM×M code of the S- LaSTTCMschemetobeapproximatelyuniversal. WewillthereforechooseΛ t tobethe lattice correspondingto an ST codederived from CDA with NVD. In orderto construct Λ m and Λ b , we will first discuss the important special case when Λ t corresponds to a perfect code, and then treat the more general case. 48 Partitions of perfect codes Let Λ t be the lattice corresponding to a perfect code [70, 29], with generator matrix G p . Then, Λ t is a rotated version of the cubic lattice Z 2M 2 . Following what was done before for the case of short block codes, we choose Λ m to be the best known integral lattice packing in 2M 2 −dimensionalspace, rotated byG p . Also, we set Λ b =QΛ m . For example, when M = 2, we choose Λ m to be the Golden Gosset lattice. The resulting code shall be named the Golden-Gosset S-LaST TCM scheme. S-LaST TCM from maximal orders in CDAs We choose Λ t to be the lattice corresponding to the maximal order of a given CDA. An example for the case when M = 2 would be the lattice corresponding to theGA+ code that we made use of for the short block-length case in Example 2. Similar to the approach used in [14, 46] for the cubic lattice case, we will use idealsβO of the maximal order for the sublattice Λ m . The element β yielding a good sublattice is obtained through a computer search, that makes use of the following lemma. Lemma 3.3.1. Let D(L/Q(ı),σ,γ) be a cyclic division algebra of index n, and let O denote an order of D. If β is an element of the order, then [O|βO] =|N r (β) n | 2 . 49 Proof. Although this lemma is well known to the mathematics community, we provide a sketch of the proof for completeness. Consider any β ∈ O. Then β induces a trans- formation on O with image βO. These are finitely generated free modules over Z, and so the index of partition is just the determinant of β in this action. We may compute thedeterminantover thecorrespondingfield. D has rank2n 2 over Q. First viewingD as a (right) vector space of dimensionn 2 overQ(ı), we see that the determinant of multiplication byβ isN r (β) n . We then apply the norm fromQ(ı) toQ to obtain the determinant. The computer search performs the following: 1. Fix a desired index of partitionM = [Λ t |Λ m ], and a sufficiently large integer ν. 2. Let O ν denote the integral closure of {−ν,−ν +1,...,ν−1,ν}⊂Z in O. More specifically, if γ 1 ,γ 2 ,...,γ 2M 2 constitutes a basis for O over Z, then O ν , 2M 2 X i=1 g i γ i −ν≤g i ≤ν, g i ∈Z∀i . Notice that such a basis always exists, since every algebraic number field has at least one integral basis [75]. 3. For eachβ∈O ν that generates a partition with required indexM, i.e., satisfying N r (β) M 2 =M, compute the fundamental coding gain of the lattice correspond- ing to βO, and let β max denote a maximizer. 4. Set Λ m to be the lattice corresponding to β max O. 50 Finally, as before, we use the self-similar shaping lattice Λ b =QΛ m , for some Q∈Z + . 3.3.4 Code construction examples Inthissection,wepresenttwoconstructionexamplesofS-LaSTTCM,theperformances of which are compared by simulation. • The Golden-Gosset S-LaST TCM construction (see Example 1): here Λ t =G p Z 8 , Λ m =G p E 8 and Λ b =QΛ m , Q∈Z + . • TheGA+S-LaSTTCMconstruction: wechooseΛ t tobethelatticecorresponding totheGA+S-LaSTcodeinExample2. Λ m isobtained usingthecomputersearch given above, and corresponds to the left ideal of β 2 O generated by M (given in (3.8)), whereO is the maximal order of theGA+ algebra (see Example 2) and the coordinates of β in terms of the ordered basis in (3.7) are (−1,−1,1−ı,−1−ı). We then set Λ b =QΛ m , Q∈Z + . Both these codes correspond to a 16−ary partition Λ t /Λ m , as shown in Fig. 3.9. The minimumdeterminantincreases as onegoes downthepartition chain. We usethetrellis showninFig.3.10thatisdesignedsuchthatthetransitionsleaving/merging intoastate have maximum possible minimum determinant. In our simulations, we have used block length T = 260 channel uses, corresponding to 1300 information bits per packet, atR = 5 bpcu. Fig. 3.11 shows the performance in terms of packet error probability of the above two S-LaST TCM schemes in comparison with the Golden ST TCM (GST-TCM) scheme [46] at 5 bpcu. Also shown is the 51 Figure 3.9: Two level partition of the example constructions performanceof the “uncoded Golden code” construction [46], which consists of stacking 130 Golden code matrices next to each other (coding is performed only over 2 time- slots). The proposed S-LaST TCM construction is seen to gain around 1 dB over the GST-TCM scheme. 3.4 Conclusions and Further Extensions In this chapter, we have advocated the use of structured lattices that are endowed with good packing and shaping properties in the design of space-time codes with both short and long block-lengths. The constructions presented have reasonable decoding complexity, and exhibit excellent performance in terms of error probability. Quiteafewresearchtopicsoccurnaturallyaspotentialfollow-upworks. Whilecodes with shortblock-length have performances that are very close to the outage probability, there is still quite a significant gap from outage for the case of long block-lengths. 52 Figure 3.10: 16-state trellis used for the example constructions Designing better codes for this scenario remains a challenging open problem. It would also be interesting to explore if there exist better algebraic frameworks that allow us to choose sublattices with good packing and shaping properties. 53 4 6 8 10 12 14 16 18 20 22 10 −3 10 −2 10 −1 10 0 E b /N 0 (dB) P e Golden ST−TCM, 16 states Gosset S−LaST TCM, 16 states Uncoded Golden Code Golden+ S−LaST TCM Outage, 5 bpcu Figure 3.11: Performance of theGolden-Gosset andGA+ S-LaST TCMschemes,R = 5 bpcu,T = 260 54 Chapter 4 Outage Analysis for Correlated MIMO Fading Channels In a rich scattering wireless environment, the assumption of an i.i.d. Rayleigh fading model for the channel matrixH of a wireless communication channel is quite accurate. However, in a typical wireless cellular environment where the users are not stationary, thechannelvarieswithtime,albeitslowly. Thereareseveralfactorsthatmightmakethe i.i.d. Rayleighassumptioninaccurate, owingtocorrelationsamongfadecoefficients, and the presence of a strong line of sight component in the received signal. Characterizing the fundamental limits of such channels is the topic of interest of this chapter, and is especiallyrelevantsinceapproximatelyuniversalcodesareknowntoexistfortheMIMO channel. We will work with the MIMO channel model in (1.1). 55 4.1 Correlated MIMO Channel Models The central limit theorem, when applied to typical wireless channels characterized by multi-path fading, allows us to accurately model the fade coefficients between antenna pairs as complex Gaussian random variables. Most initial research in the area of MIMO systems (see for example [90, 35, 92, 109] and references therein) assumed the entries of the channel matrixH to be i.i.d. CN(0,1). This i.i.d. Rayleigh fading assumption is valid when the transmitter and receiver are located in a “rich-scattering” environment and there is sufficient spacing among the antenna elements. Most MIMO channels encounteredinpracticehowever havenon-trivialcorrelations between thevariousfading coefficients. In general, correlations between the MN fading coefficients constituting the channel matrix H = [h ij ] are characterized by a correlation matrix Q = [q ij ] ∈ C MN×MN , defined as follows. Let h ∈ C MN denote the vectorized equivalent of H, obtained by stacking the columns of H one below the other. Then the entries q ij are given by q ij =E h (h i −E(h i ))(h j −E(h j )) H i for i,j = 1,...,MN, with h i denoting the i th element of h. Thus the entries of Q represent the correlations 1 between each pair of fade coefficients. If the mean of the (Gaussian) fading coefficients is non-zero, then the channel is said to be Rician. 1 More precisely, they represent the covariance between fading coefficients. A more appropriate nomenclature forQ would be covariance matrix, but we will use the term correlation matrix in confor- mity with the literature in this area. 56 Several papers have addressed the issue of modeling this correlation. In [84], an abstract “one-ring” scattering model was considered, and it was shown that this sce- nario is well modeled by what is now known as the semi-correlated Kronecker channel. The prefix semi-correlated refers to the fact that correlations are present either at the transmitter or the receiver only. More generally, a “fully-correlated” Kronecker (or sep- arable) correlation model has been widely used (see for example [84, 72] and references therein) to model a significant variety of practical channels, owing to ease of analysis. The correlation of H = [h ij ] is said to be Kronecker or separable if the correlation between h ij and h i ′ j ′ is given by E h (h ij −E(h ij ))(h i ′ j ′−E(h i ′ j ′)) H i =R ii ′T jj ′, where R and T are (N ×N) and (M×M) receive and transmit correlation matrices respectively. The model is said to be semi-correlated if either R or T is identity. The Kronecker model, although simple, has it’s own deficiencies, see [71] for details. Our main result will hold for the case of the most general full-rank correlation scenario (i.e., Q is an arbitrary (MN×MN) full-rank matrix), of which Kronecker correlations are a particular case. The main result of this paper, a characterization of the DMT for general Raleigh andRicianchannelswithfull-rankcorrelations, ispresentedinSection4.2. Weconclude by providing some simulation results in Section 7.5. 57 4.2 DMT of Correlated MIMO Channels Following the characterization of the DMT of the i.i.d. Rayleigh channel in [109], there have been attempts made to characterize the DMT for more general fading channel models. In a recent result [108], the authors compute bounds on the DMT for cer- tain correlation scenarios including Kronecker correlated Rayleigh and Rician channels, Nakagami and Weibull fading. The bounds computed are tight for long block-lengths. The DMT of Kronecker correlated channels is shown to be unchanged from that of the i.i.d. Rayleigh case. In this section, we present the main result of this paper, which establishes that the DMT of Rayleigh and Rician channels with arbitrary full-rank correlations is identical to that of the i.i.d. Rayleigh channel. Notice that the analysis is not restricted to the popular Kronecker correlation model, and holds for any (full-rank) correlation model. The analysis is based upon the fact that the problem of computing the joint pdf of eigenvalues of H H H can be recast in terms of computing integrals over unitary groups [86]. Such integrals may be evaluated using methods from the field of representation of Lie groups [15, 47]. We begin with a review of this approach. 58 4.2.1 Joint Distribution of Eigenvalues Define the Vandermonde determinant of a vector β as Δ(β 1 ,...,β n ) = Δ(β), det 1 1 ··· 1 β 1 β 2 ··· β n β 2 1 β 2 2 ··· β 2 n . . . . . . . . . β n−1 1 β n−1 2 ··· β n−1 n . Further, it can be shown that Δ(β)= Q 1≤i<j≤n (β j −β i ). Let A, min{M,N} as before and B , max{M,N}. Consider the following coor- dinate transformation (SVD) H =U diag(μ)V H , where U,V are unitary and μ is the vector of singular values of H, i.e., μ i = √ λ i . Including the Jacobian of this transfor- mation results in dH =C B,A Δ(λ) 2 A Y i=1 λ B−A i dλ dU dV, where C B,A is a constant, dλ , dλ 1 dλ 2 ···dλ A , and dU, dV are the standard Haar integration measures for the corresponding unitary matrices. Therefore, we can express the joint pdfP(λ) as P(λ) . = Δ(λ) 2 A Y i=1 λ B−A i · Z V Z U p(H =U diag(μ)V H )dU dV, (4.1) 59 wherep(H) is the probability distribution of the matrixH. For the special case of i.i.d. Rayleigh fading, thepdfofeigenvalues can becomputedinclosed form[65], andisgiven by p(λ) . = Δ(λ) 2 " A Y i=1 λ B−A i # exp " − A X i=1 λ i # . (4.2) 4.2.2 Correlated Rayleigh Fading We are now ready to compute the outage probability for correlated Rayleigh channels. Theorem 4.2.1. The DMT for Rayleigh fading channels with arbitrary full-rank cor- relations is unchanged from the case of i.i.d. Rayleigh fading. Proof. Let{λ i } A i=1 denote the non-zero eigenvalues of H H H, whereH is Rayleigh with arbitrary full-rank correlations characterized by theMN×MN matrixQ. Comparing (4.1) and (4.2), it is enough we show that Z V Z U p(H =U diag(μ)V H )dU dV . = exp " − A X i=1 λ i # . (4.3) Let h denote the vectorized equivalent of H, obtained by stacking the columns of H one below the other. By assumption, h is a zero-mean circularly symmetric complex Gaussian random vector with correlation matrix Q. Therefore, p(H =U diag(μ)V H ) = det(πQ) −1 exp h −h H Q −1 h i . = exp h −h H Q −1 h i . (4.4) 60 The Rayleigh quotient for any matrix G is given by [57] R G (h) = h H Gh |h| 2 . It is shown in [57] thatλ min (G)≤R G (h)≤λ max (G), whereλ min (G) andλ max (G) are the minimum and maximum eigenvalues of G. Setting G=Q −1 , we obtain λ min (Q −1 )|h| 2 ≤h H Q −1 h≤λ max (Q −1 )|h| 2 . SinceQ −1 is full-rank and positive definite with entries independent of SNR, we obtain h H Q −1 h . = |h| 2 = kHk 2 F = P A i=1 λ i . We use the above relation in conjunction with (4.4) to evaluate the left hand side of (4.3). The integrals over the unitary group are now trivial integrals over the Stiefel manifold (space of unitary matrices), and evaluate to constants. The claim follows. 4.2.3 Correlated Rician Fading In this section, we generalize the results for the correlated Rayleigh channel to the correlated Rician case. Theorem 4.2.2. The DMT for Rician fading channels with arbitrary full-rank corre- lations is unchanged from the case of i.i.d. Rayleigh fading. 61 Proof. LetH beRician, with full-rankcorrelation matrixQ and (non-zero) mean (M× N) matrix H 0 . Further, let{γ i } A 0 i=1 be the non-zero eigenvalues of H H 0 H 0 , where A 0 is the rank of H 0 , and let h 0 denote the vectorized equivalent of H 0 . The pdf p(H) = det(πQ) −1 exp h −(h−h 0 ) H ·Q −1 (h−h 0 ) i . = exp h −(h−h 0 ) H Q −1 (h−h 0 ) i . (4.5) Using the Rayleigh quotient as before, we arrive at p(H =U diag(μ)V H ) . = e −|(h−h 0 )| 2 =e −kH−H 0 k 2 F . Notice thatatthispoint,wehavereducedtheproblemtothatofanuncorrelatedRician channel. Continuing, p(H) = exp[−tr{(H−H 0 ) H (H−H 0 )}] = e − P A i=1 λ i − P A 0 i=1 γ i exp h tr{H H 0 Udiag(μ)V H } +tr{Vdiag(μ) H U H H 0 } i . (4.6) 62 Using (4.6) in (4.1), we obtain P({λ i }) . = Δ(λ) 2 A Y i=1 h λ B−A i e −λ i i Z V Z U dU dV· exp h tr{H H 0 Udiag(μ)V H +Vdiag(μ) H U H H 0 } i . = Δ(λ) 2 A Y i=1 h λ B−A i e −λ i i Z V Z U dU dV· exp h 2Re tr{H H 0 Udiag(μ)V H } i . Notice that the elements of diag(μ) are the singular values ofH, i.e.,μ i = √ λ i . Setting λ i =ρ −α i , P({α i }) . = Δ(λ) 2 A Y i=1 h λ B−A+1 i i λ i =ρ −α i · A Y i=1 exp −ρ −α i Z V Z U dU dV· exp h 2Re tr{H H 0 Udiag( p ρ −α )V H } i . Since for α i < 0, lim ρ→∞ exp h −ρ −α i +c p ρ −α i i → 0 irrespective of the constant c, and further, for α i ≥ 0, lim ρ→∞ exp h −ρ −α i +c p ρ −α i i ˙ → ρ 0 , 63 wemay restrictattention onlytotheregionα i ≥ 0∀iinsofar as determiningtheoutage probability is concerned. Underthis restriction, the integrals over the unitarygroup are once again trivial and evaluate to a constant. That the outage probability is exactly equal (asymptotically) to that of the uncorrelated Rayleigh case follows immediately from [109]. 4.3 Simulation Results We will use the following two Kronecker correlated channel scenarios as illustrative examples in this section: • (CHANNEL 1) Rayleigh channel with zero mean, uncorrelated receiver, transmit cross correlation= 0.8, i.e., T = 1 0.8 0.8 1 ,R = 1 0 0 1 . • (CHANNEL 2) Rayleigh channel with zero mean, transmit and receive cross correlations= 0.8, i.e., T =R = 1 0.8 0.8 1 . In[85],theauthorscomputetheoutageoptimalinputcovariance matrixS o forthesemi- correlated MIMO Kronecker channel (correlated transmitter, uncorrelated receiver), focusing primarily on the case of M = 2. We will use these results in our simulations. 64 Outage probabilities at a rate of 4 bpcu obtained using Monte-Carlo simulation are plotted in Fig. 4.1. The two curves for CHANNEL 1 correspond to using an identity input covariance and the covariance S o that minimizes the outage probability in (5.3) respectively 2 . As predicted by theory, it is seen that the identity covariance is optimal athighSNRs. Significantgainsofaround1dBthroughusingS o isobservedonlyatvery lowSNRs. ThecorrespondingcurvesfortheCHANNEL2caseusingtheoutageoptimal 6 8 10 12 14 16 18 20 22 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) P out CHANNEL 1, Outage Optimal Covariance CHANNEL 1, Identity Covariance CHANNEL 1, Ergodic Optimal Covariance IID Channel CHANNEL 2, Outage Optimal Covariance CHANNEL 2, Identity Covariance CHANNEL 2, Ergodic Optimal Covariance Figure 4.1: Outage Probabilities for CHANNELS 1 and 2 at 4 bpcu covariances obtained for CHANNEL 1 are also shown in Fig. 4.1. Notice however that this covariance may not be optimal for the CHANNEL 2 case. A trend similar to the CHANNEL 1 case is observed. 2 The notion of outage probability using an identity covariance is an abuse of notation; by definition, the outage probability is obtained by maximizing over all possible input covariances. 65 Veryloosely speaking,theeffect offull-rankcorrelations on MIMOsystems amounts to a penalty in the “coding gain” alone with no loss of “diversity”. The outage proba- bilities of the example channels presented above will attain the same asymptotic slope, albeit at SNRs higher than that required for the i.i.d. channel. 66 Chapter 5 Asymptotic Performance of Linear Receivers in MIMO Fading Channels The next generation of wireless communication systems is expected to capitalize on the large gains in spectral efficiency and reliability promised by MIMO multi-antenna com- munications [92, 109,8,72] andincludeMIMO technology as afundamental component of their physical layer [110]. The information theoretic analysis and the efficient design ofspace-time(ST)codesfortransmissionovertheseMIMOsystemshavebeenactive ar- eas of research over the past decade. Also, suboptimal low-complexity receiver schemes have been widely proposed and investigated as a low-complexity alternative to the op- timal Maximum-Likelihood (ML) or ML-like receivers [18, 67]. These schemes range from the iterative interference (soft) cancellation (e.g., [25]), to successive interference 67 (hard) cancellation (e.g., [107, 11]), to the even lower complexity “separated” architec- ture,basedonlinearspatialequalization followed bystandardsingle-inputsingle-output (SISO) decoding. 1 Inthischapter, wepresenttwo typesof asymptotic performanceanalysis ofthis low- complexity MIMO architecture. First, we consider the Diversity-Multiplexing Tradeoff (DMT) [109] (see Chapter 2), which captures the performance tradeoff between rate and block-error probability in the high-SNR, high spectral efficiency regime. We deter- mine the DMT achieved by low-complexity MIMO architectures that use Zero-Forcing (ZF) or Minimum Mean-Square Error (MMSE) linear receivers and apply conventional SISO outer coding before the MIMO transmitter and conventional SISO decoding to the output of the linear receiver. The DMT analysis reveals that both ZF and MMSE linear receivers are very suboptimal in terms of their achievable diversity. Furthermore, we observe that while the DMT analysis accurately predicts the behavior of the ZF receiver at all finite rates, the performance of the MMSE receiver is in stark contrast to that predicted by the DMT analysis at low rates. In fact, we observe that for suffi- ciently low rates the MMSE receiver exhibits an ML-like performance. On the contrary, when working at higher rates (and correspondingly higher SNR) the MMSE receiver approaches the ZF performance. We provide an approximate analysis that explains this behavior both qualitatively and quantitatively. 1 It should be noticed that the current MIMO WLAN standard [110] is based on MIMO-OFDM, therefore, linear equalization is performed in the space and in the frequency domains. For simplicity, in this chapter we restrict ourselves to the standard frequency-flat case where equalization is purely spatial. 68 In the second part of this chapter we take a closer look at the performance of the linear MMSE and ZF receivers at finite SNR. Since this is very difficult to capture in closed form, we explore a second type of asymptotic regime, where we fix SNR and let thenumberofantennasbecomelarge. Usingrandommatrixtheory,weshowthatinthis case the limiting distribution of the mutual information of the parallel channels induced by the linear receiver is Gaussian, with mean and variance that can be computed in closed form. The analysis provides accurate results even for a moderate number of antennas and allows to quantify how the performance loss in terms of diversity suffered bylinearreceiversmayberecoveredbyincreasingthenumberofantennas. Thisprompts to the conclusion that in order to achieve a desired target spectral efficiency and block- error rate, at given SNR and receiver complexity, increasing the number of antennas and using simple linear receiver processing may be, in fact, a good design option. The chapter is organized as follows. First, in Section 5.1, we briefly comment on concurrent existing literature. In Section 5.2, we define the system model and recall the main facts the ZF and MMSE linear receivers considered in this work. Section 5.3 presents the DMT analysis and some illustrative numerical examples. Section 5.4 is devoted to thefixed-rateanalysis of theMMSE receiver performancewithcodingacross theantennas and provides anapproximate quantitative analysis of theslopeof theerror probability versus SNR. Section 5.5 deals with the limiting distribution of the mutual informationfortheMMSEandZFreceivers foralargenumberofantennasandprovides some illustrative numerical examples on the validity and limitations of this analysis. 69 Conclusions are pointed out in Section 5.6 and some technical details of the proofs are deferred to the Appendices. 5.1 Related literature Sinceits introduction inthe seminalwork[109], the DMT hasbecomea standard tool in the characterization of the performance of slowly-varying fading channels in the high- SNR, large spectral efficiency regime. Space-time coding schemes have been character- ized in terms of their achievable DMT in a series of works, including lattice coding and decoding[24]andZForMMSEdecision feedbackreceivers(seeforexample[50,51]). The multipath diversity achievable by linear equalizers in frequency-selective SISO channels has also attracted some attention and was recently solved in [89]. The spatial diversity achievable by MIMO linear receivers and separated detection and decoding was inves- tigated in [42, 41, 56]. In this respect, it is worthwhile to stress the differences between our work and [41]: 1) we investigate the full DMT curve, while [41] focuses only on the fixed-rate case (corresponding to zero multiplexing gain); 2) [41] develops only lower bounds to the diversity order, based on upper bounds on the outage probability, while we have both lower and upper bounds and show that they are tight; 3) the analysis on the diversity order of the ZF receiver in [41] is based on the conjecture that “the diversity arising from coding across channels with independent power levels is no worse than coding across channels whose power levels are random and correlated.” Our anal- ysis, instead, is based on the proper correlation structure of the SINR’s and does not rely on their assumed independence. We notice, in passing, that if the SINR’s were 70 independent, then the diversity order would be different, as explained in a comment at the end of Section 5.3.1. With respecttothelarge-system analysis of linear receivers presentedinSection 5.5, we notice that asymptotic Gaussianity was shown for the MIMO channel mutual infor- mation given by the “log-det” formula, whose cumulative distribution function (cdf) yields the block-error rate achievable under optimal decoding. This was shown in var- ious works, such as [43, 64, 88, 38]. At the same time the marginal asymptotic Gaus- sianity of the SINR of a single MMSE and ZF receiver channel was derived in [96, 59], without looking at the joint Gaussianity of all SINRs for all these channels. While the marginal Gaussianity is useful in the case of pure spatial multiplexing, where each antenna (or “spatial stream”) is independently encoded and decoded, we would like to remark here that the joint Gaussianity is crucial in the analysis of the most relevant case where outer coding is applied across antennas. In Section 5.5 we characterize the limiting joint Gaussian distribution of theSINRs andobtain thestatistics of the mutual information oflinear MMSEandZFreceivers forthecase ofcodingacross theantennas. Our approach is novel and does not follow as a simple extension of the analysis of the marginal statistics as done previously. 5.2 System model and linear receivers Fig. 5.1shows threetypes of MIMOarchitectures, employingM transmitandN receive antennas. Since the focus of this chapter is on linear receivers, we shall assumeN ≥M throughout this chapter. Scheme (a) puts no restriction on the choice of the space-time 71 coding and decoding scheme: the M channel inputs are jointly encoded, and the N channel outputs are jointly and possibly optimally decoded. Scheme (b) is based on interleaving and demultiplexing over the M inputs the codewords of a SISO code. A linear spatial equalizer (referred briefly as “linear receiver” in the following) processes each N-dimensional channel output vector (purely spatial processing) and creates M virtual approximately parallel channels (details are given later on). The output of these virtual channels are thendemultiplexed and deinterleaved, and eventually fed to aSISO decoder that treats them as scalar observations, thus disregarding the possible depen- dencies introduced by the underlying MIMO channel. Notice that in scheme (b) coding is applied across the antennas. Finally, scheme (c) is based solely on “spatial multiplex- ing”, that is, M independently encoded streams drive the M transmit antennas and are approximately separated by the linear receiver, the outputs of which are fed to M independent decoders. The output of the underlying frequency-flat slowly-varying MIMO channel is given by (this channel model is equivalent to the one in (1.1)) y t =Hx t +w t , t = 1,...,T, (5.1) where x t ∈ C M denotes the channel input vector at channel use t, w t ∼ CN(0,N 0 I) is the additive spatially and temporally white Gaussian noise and H ∈ C N×M is the channelmatrix. InthisworkwemakethestandardassumptionthattheentriesofHare i.i.d. ∼CN(0,1), andthatH is randombutconstant over thedurationT of acodeword 72 (quasi-static Rayleigh i.i.d. fading [92, 109]). The input is subject to the total power constraint 1 MT E kXk 2 F ≤E s , (5.2) where X = [x 1 ,...,x T ] denotes a space-time codeword, uniformly distributed over the space-time codebookX, andk·k F denotes the Frobenius norm. Furthermore, following thestandardliteratureofMIMOchannelsandspace-timecoding,wedefinethetransmit SNRρ as the total transmit energy per time-slot over the noise power spectral density, i.e., ρ=ME s /N 0 . We assume no Channel State Information (CSI) at the transmitter. In this work we consider the case of very large block length (and consequently of very slowly-varying fading). Under the quasi-static assumption, it is well-known that the capacity and the outagecapacity(orǫ-capacity)areindependentoftheassumptiononCSIatthereceiver [7]. Hence, assuming perfect CSI at the receiver incurs no loss of generality. We focus on the MIMO detector/decoder blocks in Fig. 5.1. Under the fully uncon- strained ST architecture (a), the optimum receiver for the MIMO channel in (5.1) is the maximum likelihood (ML) decoder, with minimum distance decision rule given by ˆ X= argmin X∈X kY−HXk 2 F . This entails joint processing of the symbols across all antennas at the receiver, over the whole block length T, and is typically implemented using algorithms like Sphere Decoding (see [18] and reference therein) and their tree search sequential decoding 73 Figure 5.1: Three possible space-time architectures: (a) unrestricted space-time coding scheme; (b)codingacross theantennas, with linear spatial equalization; (c) purespatial multiplexing with linear spatial equalization. π and π −1 in (b) denote interleaving and de-interleaving. 74 generalization [67], possibly coupled with ML Viterbi algorithm if the underlying code has a trellis structure (e.g., [55, 14]). The performance of this decoder is characterized by the information outage probability given by P out (R,ρ) = inf S:S 0 tr(S)≤ 1 P logdet(I +ρHSH H )≤R . (5.3) where the optimization is over the Hermitian symmetric non-negative definite matrix S subject to a trace constraint, reflecting the channel input power constraint (5.2). Several lower complexity suboptimal decoders have been proposed in the literature. In particular, architectures (b) and (c) in Fig. 5.1 involve a linear memoryless receiver defined by the matrix G, such that the output of the linear receiver is y ′ t = Gy t . Classical choices for G are the ZF or the MMSE spatial filters, or any diagonal scaling thereof. Under the assumption of Gaussian inputs, very large block length T and ideal interleaving, the linear receiver createsM “virtual” parallel channels that, without loss of generality, can be described by y ′ k,t = √ γ k x k,t +w ′ k,t , k = 1,...,M, (5.4) 75 where we normalize the input and output such that E[|x k,t | 2 ] = E[|w ′ k,t | 2 ] = 1, and where γ k denotes the Signal to Interference plus Noise Ratio (SINR) at the k-th linear receiver output. 2 Under the above assumptions, the performance of such schemes is characterized by the following two outage probabilities. With coding across antennas (scheme (b)), the outage probability of interest is given by P lin out (R,ρ),P M X k=1 log(1+γ k )≤R ! ; (5.5) Under pure spatial multiplexing (scheme (c)), the relevant outage probability is given by P sp mult out (R,ρ),P M [ k=1 log(1+γ k )≤ R M ! . (5.6) where we used the fact that, by symmetry, without CSI at the transmitter the optimal performance of spatial multiplexing with linear receivers is achieved by allocating the same rate R/M to each stream. For completeness and for later use, we recall here the expressions of the SINRs for the ZF and the MMSE linear receivers. Here we follow closely the approach presented in [41]. 2 In order to avoid any misunderstanding, it should be noticed here that “interference” is uniquely caused by the generally non-perfect separation of the transmitted symbols inxt by the linear receiver G. We consider a strictly single-user setting, with no multiuser interference. 76 ZFreceiver. Inthiscase, thematrixGischosen asG=DH + , whereDis asuitable diagonal scaling matrix andH + is the Moore-Penrose pseudo-inverse of H [100]. Since H has rank M with probability 1, this takes on the form H + = (H H H) −1 H H . In the absence of transmitter CSI, the signal power is allocated uniformly across the transmitter antennas. It is immediate to show that the SINRs on the resulting M parallel channels are given by γ k = ρ/M [(H H H) −1 ] kk , (5.7) where the notation [A] kk indicates the k th diagonal entry of a matrix A. MMSEreceiver. Inthis case, thematrixGis chosen inorder tomaximize theSINR γ k for eachk, over all linear receivers. It is well-known that this is achieved by choosing G = DH mmse , where D is a suitable diagonal scaling matrix and H mmse is the linear MMSEfilter[100]thatminimizestheMSEE[kx t −H mmse y t k 2 ]. Usingtheorthogonality principle, we find H mmse = ρ M H H h I+ ρ M HH H i −1 = H H H+ M ρ I −1 H H . (5.8) 77 A standard calculation [100] yields the SINRs γ k of the resulting set of virtual parallel channels in the form γ k = ρ M h H k h I+ ρ M H k H H k i −1 h k = 1 h I+ ρ M H H H −1 i kk −1, (5.9) where H k denotes the N×(M−1) matrix obtained by removing the k th column, h k , from H. 5.3 Diversity-Multiplexing Tradeoff A compact and convenient characterization of the tradeoff between rate and block-error probabilityofMIMOquasi-static fadingchannels inthehigh-SNRregimeisprovidedby the DMT introduced by [109]. Recall from Chapter 2 thatd ∗ (r) is equal to the negative ρ-exponent of the information outage probability (5.3). Recall also that for the space- time channel in (5.1), d ∗ (r) is given by the piecewise linear function interpolating the points (r,d) with coordinates r =k, d= (M−k)(N−k) for k = 0,1,...,min{M,N}, and is zero for r> min{M,N} [109]. While d ∗ (r) is achievable under the optimal receiver (a) in Fig.5.1, the following result characterizes the DMT of the MIMO channel in (5.1) under schemes (b) and (c), when the linear receiver is either the ZF or the MMSE receiver defined above: 78 Theorem 5.3.1. The DMT of the M-transmit, N-receive i.i.d. Rayleigh MIMO chan- nel with N ≥ M, constrained to use Gaussian codes under either MMSE or ZF linear receivers is given by 3 d ∗ lin (r)= (N−M +1) 1− r M + , (5.10) for both the cases of coding across antennas or pure spatial multiplexing. Proof. The theorem is proved by developing upper and lower bounds onP lin out (R,ρ) for the MMSE receiver in the configuration (b) of the block diagram of Fig. 5.1. A simple upper bound on the outage probability for the ZF receiver extends immediately theresulttothiscase. For configuration(c)theresultfollows asanimmediatecorollary. Lower bound on the outage exponent. Let λ min (A) and λ max (A) denote the minimum and maximum eigenvalues of a Hermitian symmetric matrix A, and λ 1 ≤ λ 2 ≤ ··· ≤ λ M denote the ordered eigenvalues of the M ×M Wishart matrix H H H, with joint pdf given by [92] p(λ) =K M,N M Y i=1 λ N−M i · Y i<j (λ i −λ j ) 2 exp − M X i=1 λ i ! , (5.11) where K M,N is a normalization constant and we have assumed M ≤N. 3 Note: (x) + Δ = max{x,0}. 79 Using (5.9), we can write the mutual information with Gaussian coding across the antennas and the MMSE receiver as I mmse (H) = − M X k=1 log I+ ρ M H H H −1 kk . (5.12) Since the function−log(·) is convex, using Jensen’s inequality we have I mmse (H) ≥ −Mlog 1 M M X k=1 I+ ρ M H H H −1 kk ! = −Mlog 1 M Tr I+ ρ M H H H −1 = −Mlog 1 M M X k=1 1 1+ ρ M λ k ! . Using this bound in (5.5) we obtain P mmse out (R,ρ) ≤ P log 1 M M X k=1 1 1+ ρ M λ k ! ≥− R M ! = P 1 M M X k=1 1 1+ ρ M λ k ≥ρ − r M ! , (5.13) where in the last line we let R = rlogρ. Finally, we can use the trivial asymptotic upper bound P 1 M M X k=1 1 1+ ρ M λ k ≥ρ − r M ! ˙ ≤P 1 ρλ 1 ≥ρ − r M (5.14) 80 First, we notice that the asymptotic outage probability upper bound in the RHS of (5.14) vanishes only if r/M < 1. Hence, the outage exponent lower bound is zero for r/M ≥ 1. When r/M < 1, we can write P λ 1 ≤ρ r M −1 = Z ρ r M −1 0 dλ 1 M Y i=2 Z ∞ λ 1 dλ i p(λ) ≤ Z ρ r M −1 0 dλ 1 M Y i=2 Z ∞ 0 dλ i p(λ) = Z ρ r M −1 0 p 1 (λ 1 )dλ 1 = κ 1 ρ (N−M+1)(r/M−1) , (5.15) whereκ 1 isaconstantandwherewehaveusedthewell-knownfact[92]thatthemarginal pdf of λ 1 = λ min (H H H), denoted by p 1 (λ) in (5.15), satisfies p 1 (λ)∝ λ N−M for small argument λ≪1. The resulting outage exponent lower bound is d ∗ mmse (r)≥ (N−M +1) 1− r M + . (5.16) The same result can be obtained by following the by-now standard technique of [109] based on the change of variable λ i =ρ −α i , integrating the resulting pdf of α 1 ,...,α M over the outage region and applying Varadhan’s lemma [109]. 81 Upper bound on the outage exponent. Using the concavity and the mono- tonicity of the log(·) function, we obtain from (5.12) and Jensen’s inequality that I mmse (H)≤Mlog 1 M M X k=1 1 h (I+ρH H H) −1 i kk . (5.17) Consider the decomposition H H H = U H ΛU, where U is unitary and Λ is a diagonal matrix with the eigenvalues ofH H H on the diagonal. Definingu k to be thek th column of U and e k to be the column vector that has a one in the k th component and zeros elsewhere, we have that h (I+ρH H H) −1 i kk = e H k U H (I+ρΛ) −1 Ue k = u H k (I+ρΛ) −1 u k = M X ℓ=1 |u ℓk | 2 1+ρλ ℓ . Hence, the term inside the logarithm in (5.17) can be upperbounded as 1 M M X k=1 1 h (I+ρH H H) −1 i kk = 1 M M X k=1 1 P M ℓ=1 |u ℓk | 2 1+ρλ ℓ = 1 M M X k=1 1 |u 1k | 2 1+ρλ 1 h 1+ P M ℓ=2 |u ℓk | 2 |u 1k | 2 1+ρλ 1 1+ρλ ℓ i ≤ (1+ρλ 1 ) 1 M M X k=1 1 |u 1k | 2 (5.18) 82 LetA denote the event n 1 M P M k=1 1 |u 1k | 2 ≤c o , wherec is some constant (independent of ρ). We have that P mmse out (R,ρ) ≥ P (A)P log (1+ρλ 1 ) 1 M M X k=1 1 |u 1k | 2 ! ≤ R M A ! ≥ P (A)P log((1+ρλ 1 )c)≤ R M . = P log(1+ρλ 1 )≤ r M logρ (5.19) where the last exponential equality holds if P (A) is a O(1) non-zero term, i.e., it is a constant with respect to ρ bounded away from zero. This is indeed the case, as shown rigorously in Appendix A. It is immediate to check that the last line of (5.19) is asymptotically equivalent to (5.14). Therefore, applying the same argument as in (5.15) we find that the upper bound on the outage probability exponent coincides with the previously found lower bound. The proof of Theorem 5.3.1 is completed by observing that in the case of the ZF receiver a lower bound on the SINR γ k is readily obtained from the inequality h (H H H) −1 i kk ≤λ max [(H H H) −1 ] = 1 λ min (H H H) = 1 λ 1 , 83 that holds for allk = 1,...,M. Using this in the mutual information expression for the ZF receiver with coding across the antennas we obtain P zf out (R,ρ) ≤ P log(1+ρλ 1 )≤ r M logρ (5.20) Noticing that (5.20) coincides with the asymptotic lower bound (5.19) for the MMSE receiver, and that the MMSE receiver maximizes the mutual information over all linear receivers, under Gaussian inputs and the system assumptions made here, we immedi- ately obtain that the ZF also achieves the outage exponentd ∗ lin (r) given in (5.10). Finally, asfarasspatialmultiplexingisconcerned(nocodingacrosstheantennas),it is clear from (5.5) and (5.6) that, for any linear receiverG, P lin out (R,ρ)≤P sp mult out (R,ρ). On the other hand, it is immediate to show that spatial multiplexing achieves the same DMT (5.10). Details are trivial, and then are omitted. 5.3.1 Discussion and numerical results Theorem 5.3.1 shows that, in terms of DMT, there is no advantage in using interleaving and coding across the antennas when a linear receiver is used in order to spatially separate the transmitted symbols. In order words, the linear receiver front-end kills the transmit diversity gain offered by the MIMO channel. In fact, the DMT (N −M + 1) 1− r M + of Theorem 5.3.1 has the following intuitive interpretation: this coincides with the DMT of a SIMO (Single-Input, Multiple-Output) channel (receiver diversity only) with N−M +1 receive antennas, used at a rate R/M. 84 This fact shows also that the channel gains of the virtual parallel channels are strongly statistically dependent. For example, it is well-known that the ZF receiver applied to a M ×N channel with N ≥ M and i.i.d. Rayleigh fading yields channel gains γ k = 1 [(H H H) −1 ] kk that are marginally distributed as central Chi-squared random variables with 2(N −M + 1) degrees of freedom [106]. If the gains γ 1 ,...,γ M were statistically independent, by coding across the antennas we would obtain the DMT of the parallel independent channels, given by [94] d ∗ parallel,i.i.d. (r)≥ (N−M +1)(M−r) + , which is much larger than the DMT given by Theorem 5.3.1. In contrast, the channel gains in the regime of high SNR are essentially dominated by the minimum eigenvalue of the matrix H H H and therefore are strongly correlated: if one subchannel is in deep fade, they are all in deep fade with high probability. This is the reason why coding across the transmit antennas does not buy any improvement in terms of DMT with respect to simple spatial multiplexing. Having said so, we should also remark that the picture about linear receivers is not totally grim as it may appear from the high-SNR DMT analysis. Indeed, coding across antennasyieldsaverysignificantperformanceadvantage withthelinearMMSEreceiver at fixed and not too large rate (notice that fix rate R corresponds to the case of zero multiplexing gain, r = 0.) In order to illustrate these claims, we provide simulations results for the following outage probabilities under i.i.d. Rayleigh fading: 85 • MIMO outage probability (5.3) with input covariance (ρ/M)I (scheme (a) in Fig.5.1); • outageprobability(5.5)withZFandMMSEreceivers withcodingacrossantennas ((scheme (b) in Fig.5.1)); • outage probability (5.6) with ZF and MMSE receivers under pure spatial multi- plexing, i.e., without coding across antennas (scheme (c) in Fig.5.1). Fig. 5.2 shows the correspondingplots at ratesR = 1 and5 bits per channel use(bpcu). Several interesting observations can be drawn from this figure. We observe that while at high rates the MMSE with coding across antennas behaves as predicted by the DMT analysis, the behavior at low rates is in stark contrast to the asymptotic result (this fact was also noticed in [41]). In fact, the MMSE exhibits an apparent “full diversity”behavioratsmallrate(e.g.,R = 1bpcuinFig.5.2). Incontrast, thebehavior of the ZF receiver is accurately predicted by the asymptotic analysis at all rates. This remarkablebehavioroftheMMSEreceiverisexplainedthroughanapproximateanalysis in Section 5.4. FromFig.5.2weobservealsothatcodingacross antennasdoesachieve anadvantage overspatialmultiplexing. FortheMMSEreceiveroperatingatsmallratestheadvantage is very significant, and corresponds to the diversity advantage discussed above. At high rates the advantage is moderate and consists only of a horizontal shift (dB gain) of the error curve, not in a steeper slope. 86 −15 −10 −5 0 5 10 15 20 25 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage probability 2 × 2 i.i.d. Rayleigh Channel, R = 1,5 bpcu MIMO Outage ZF, Coding across antennas MMSE, Coding across antennas ZF, spatial multiplexing MMSE, spatial multiplexing Figure 5.2: Outage probabilities of ZF and MMSE receivers, 2× 2 i.i.d. Rayleigh channel, R = 1 and 5 bpcu. 87 5.4 MMSE receiver with coding across antennas The difference between the performances of the ZF and MMSE receivers is best ex- plained by comparing their corresponding upperbounds on outage probability in (5.20) and (5.13). While only the minimum eigenvalue appears in the ZF case in (5.20), all eigenvalues play a role in the case of the MMSE receiver in (5.13). Although at asymptotically high SNR and high coding rates the minimum eigenvalue dominates (and therefore determines the corresponding DMT), the other eigenvalues appear to be relevant at lower rates and provide higher effective diversity for the MMSE receiver. In order to substantiate this intuition, we compare in Fig. 5.3 the outage probability of the MMSE receiver with coding across antennas for the case M = N = 4 with the corresponding upper bound in (5.13). The upper bound is found to be very accurate across a wide range of rates and SNRs. The particular choice of rates for this plot will be made clear in the sequel, where we analyze the high SNR behavior of the outage probability upper bound (5.13). Define T k , 1 1+ ρ M λ k and T , M2 − R M . We use a change of variables λ k = ρ −α k , whereα k denotes the level of singularity of the correspondingeigenvalue [109]. For ease of analysis we make the assumption that the channel eigenvalues fall into one of the following two categories: • α k < 1, i.e., λ k is “much larger” than the inverse SNR 1/ρ: in this case, T k → 0 as ρ→∞. • α k > 1, i.e., λ k is “much smaller” than 1/ρ: in this case, T k → 1 as ρ→∞. 88 Recall that the{α i } are ordered according to α 1 ≥···≥α M . Suppose that the rate R is such that m−1<T≤m, for some integer m = 1,2,...,M, i.e., M log M m ≤R<M log M m−1 . (5.21) For all i = 1,...,M define the event E i ={α 1 ,...,α i > 1}∩{α i+1 ,...,α M < 1}. (5.22) Then, for large ρ, the following approximation holds ( M X k=1 T k ≥T ) ≈ M [ i=m {α 1 ,...,α i > 1}∩{α i+1 ,...,α M < 1} = E m ∪E m+1 ∪···∪E M . (5.23) In the above approximation we are neglecting the cases where the eigenvalues take on values that are comparable with 1/ρ, and therefore contribute to the sum P M k=1 T k in (5.13)byaquantitybetween 0and1. Itcanbeexpectedthatasρ→∞,theprobability of such intermediate values decreases, and our approximation becomes tight. Using the union bound, we find an approximate upper bound on (5.13) given by P M X k=1 T k ≥T ! . M X i=m P(E i ). (5.24) 89 DefiningP(E i ) . =ρ − ˜ d i (R) , i = 1,...,M, using the joint pdf of theα k ’s, given by [109] p(α) = K M,N [log(ρ)] M M Y i=1 ρ −(N−M+1)α i Y i<j ρ −α i −ρ −α j 2 exp − M X i=1 ρ −α i ! . = " M Y i=1 ρ −(2i−1+N−M)α i # exp − M X i=1 ρ −α i ! , and applying Varadhan’s lemma as in [109], we obtain ˜ d i (R) = inf α j > 1∀ j≤i α j < 1∀ j >i α j ≥ 0∀j M X j=1 (2j−1+M−N)α j = i X j=1 (2j−1+M−N)×1+ M X j=i+1 (2j−1+M−N)×0 = i(i+N−M). (5.25) From (5.24) and (5.25), we eventually conclude that P M X k=1 T k ≥T ! ˙ . P(E m ). This yields the diversity of the MMSE receiver with spatial encoding at a finite rate R as d mmse (R) ≈ m(m+N−M). (5.26) In particular, whenM =N, d mmse (R) ≈ m 2 where m and R are related by (5.21). 90 To illustrate the effectiveness of the above approximation, consider the plots in Fig. 5.3 for the caseM =N = 4. The coding rates areR = 0.7706, 2.7123, 5.6601 and 12bpcu,correspondingtoT = 3.5, 2.5, 1.5and0.5respectively. Thediversities16, 9, 4 and1predicted bytheanalysis in(5.26)well approximate themeasuredslopes(for high SNR) of the outage curves, that are 15.15, 10.69, 5.55 and 1.3 in the logP mmse out (R,ρ) vs. logρ chart observed in Fig. 5.3. −15 −10 −5 0 5 10 15 20 25 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage Probability R = 0.7706 R = 2.7123 R = 5.6601 R = 12 Figure 5.3: Diversity of the MMSE receiver with joint spatial encoding: solid lines rep- resenttheoutageprobabilityin(5.5)andthedash-dotlines representthecorresponding upper bounds (5.13). M =N = 4, rates R are in bpcu. 91 5.5 Outage probability of linear receivers in the large an- tenna regime In order to motivate this section, consider the following system design issue: for a given target spectral efficiency, block-error rate, operating SNR, and receiver computational complexity (including power consumption, VLSI chip area etc.) how many antennas do we need at the transmitter and receiver? Consider the outage probability curves of Fig. 5.4 and suppose that we wish to achieve a rate of R = 3 bpcu with block-error rate of 10 −3 at SNR not larger than 15 dB. With M = N = 2 antennas this target performance is achieved by an optimal receiver, but is not achieved by the MMSE receiver. However, with M = 2,N = 4 or M = N = 3 the target performance is achieved also by the MMSE receiver. It turns out that, in some cases, adding antennas may be more convenient than insisting on high-complexity receiver processing. It is therefore interesting to analyze the outage probability of a linear receiver with coding across the antennas in the regime of fixed SNR ρ and rate R. This analysis is difficultdueto the fact that, forfiniteM,N, thejoint distribution of thechannel SINRs {γ k } in (5.4) escapes a closed-form expression. This problem can be overcome by con- sideringthesystem inthe limit of a large numberof antennas. Specifically, we willshow that the mutual information for the linear MMSE and ZF receivers becomes asymptot- ically Gaussian. Therefore, the outage probability for large but finite dimensions and fixed SNR can be accurately approximated by a Gaussian cdf with appropriate mean and variance, that we shall give in closed form. 92 Figure 5.4: Comparing the outage probability of optimal and MMSE receivers, R = 3 bpcu. 93 In the next subsection we will discuss the methodology used to show the asymptotic Gaussianity of the mutual information. The method is general and applies to both MMSE and ZF linear receivers. Subsequently, in Section 5.5.2 we will calculate the first and second cumulant moments of the SINR for the MMSE and ZF receivers, which suffice to characterize the mutual information limiting distribution. 5.5.1 Asymptotic Gaussianity of the mutual information The mutual information at the output of a linear receiver withM transmit andN ≥M receive antennas and coding across the antennas is given by I N , M X k=1 log(1+γ k ) (5.27) withγ k given by(5.9)for theMMSE case andby(5.7)for theZFcase. Inthefollowing, we fix the ratioβ =M/N ≤ 1 and consider the limit for largeN and the “fluctuations” around this limit. In order to prove the asymptotic Gaussianity of these fluctuations, we will need to analyze the characteristic function of the mutual information, given by Φ N (ω),E e jωI N . (5.28) We start by considering the cumulant generating function [61], defined as φ N (ω), log(Φ N (ω)) = ∞ X n=1 (jω) n n! C n , (5.29) 94 where the coefficient C n is the n-th cumulant moment of the mutual information. In general, the joint cumulant of m random variables X 1 ,...,X m is defined as E c (X 1 ;...;X m ), X π (|π|−1)!(−1) |π|−1 Y B∈π E " Y i∈B X i # , where π runs through all partitions of {1,...,m}, |π| denotes the number of blocks in π and B runs through the list of all blocks of π. We will call the above moment irreducible, with respect to the random variables X 1 ,...,X m , when in each argument of the cumulant moment only one random variableX i appears. By contrast a reducible cumulant moment with respect to the same random variables has arguments containing mixed products of these random variables. In general, an n-order reducible cumulant moment can bewritten in terms of a sum of products of irreducible cumulant moments, with each term in the sum having moments with order summing up to n. The n th cumulant momentC n of a random variable X is defined to be C n ,E c (X;...;X | {z } n times ). For example, the first few cumulant moments of X are C 1 = E[X] mean (5.30) C 2 = Var[X] variance C 3 = Sk[X] skewness 95 The probability density of I N can be expressed in terms of (5.28) and (5.29) as follows p(y) = 1 2π Z ∞ −∞ e −jωy Φ N (ω)dω = 1 2π Z ∞ −∞ exp −jω(y−C 1 )− ω 2 2 C 2 + X n>2 (jω) n n! C n ! dω. (5.31) In Section 5.5.2, we will show that in the limit of large N and M =βN with β≤ 1, C 1 = m 1 +o(1) (5.32) C 2 = σ 2 +o(1) (5.33) where m 1 =Mc 10 +c 11 , and where c 10 , c 11 and σ 2 are constants independent of N for which wegive closed-form expressions forboth MMSE andZFcases. In AppendixDwe will also show that all higher-order cumulants of the mutual information asymptotically vanish for largeN. Therefore, φ N (ω) is a quadratic function ofω with corrections that vanish as N →∞. As a result the mutual information is asymptotically Gaussian, i.e. I N −m 1 σ d → N(0,1). (5.34) This follows directly from (5.31) by setting y =z+m 1 and taking the large N limit p(z) = lim N→∞ 1 2π Z ∞ −∞ exp −jωz− ω 2 2 σ 2 +o(1) dω (5.35) = 1 √ 2πσ 2 e − z 2 2σ 2 . 96 Before moving on to the proofs, we would like to comment on the nature of this result. This states that the probabilityP(|I N −m 1 |>z) approaches a Gaussian probability for sufficiently largeN and fixed distance z of the mutual information from its mean. This is quite different from stating that for fixedN the mutual information distribution falls off like a Gaussian random variable for anyz andρ. As a matter of fact, for fixedN and largeenoughSNRthisGaussianapproximation isnolongervalid, sincethehigher-order cumulants will no longer be small. It is also worth pointing out here that the variance of I N is O(1) (a finite constant) for large N. This another manifestation of the fact that the SINRs of the parallel channels{γ k } are strongly correlated, in agreement with the outage analysis of previous sections. In contrast, if they were independent, or nearly independent, the variance would be roughly linear in N, as the central limit theorem would suggest. This fact is in line with the well-known behavior of the mutual information logdet(I+ ρ M HH H ) under the optimal receiver [43, 64, 88, 38], where again the variance is O(1) for large N, indicating the strong correlation among the eigenvalues of H H H. 5.5.2 Joint cumulant moments of the SINRs of order 1 and 2 Our goal is to calculate the cumulant moments of I N . Since I N consists of a sum of mutual informations of the virtual channels (see (5.27)), the n th cumulant moment of I N can be written as C n = M X k 1 ,...kn=1 E c [log(1+γ k 1 );...;log(1+γ kn )]. (5.36) 97 The building blocks of the above cumulant moments are the joint cumulant moments of the SINRs{γ k }, i.e. E c [γ k 1 ;γ k 2 ;...;γ kn ]. (5.37) In fact, by expanding the logarithms in (5.36) in Taylor series, we can express (5.36) in terms of (5.37). Even calculating these joint cumulants amounts generally to a formidable task. However, with the help of Theorem B.1 (Novikov’s theorem) given in Appendix B and due to simplifications that occur in the large N limit, we will show that this computation is possible. To obtain a feel for the computation, we will first calculate the first two joint cumulants of{γ k } and defer the proof that the higher-order cumulants vanish sufficiently fast with N to Appendix D. Cumulant moments for the MMSE receiver Starting with the case of the MMSE receiver, we recall from (5.9) that the SINR of the k-th virtual channel induced by the MMSE receiver can be written as γ k =αh H k h I+αH k H H k i −1 h k (5.38) whereH k is theN×(M−1) matrix obtained by eliminating thek-th columnh k from thechannelmatrixH, andcontains i.i.d. Gaussianelements∼CN(0,1/N) andwehave defined for convenience α=ρN/M =ρ/β. 98 Theasymptotic mean ofγ k in the limit of largeN andM =βN has been calculated in [102] in the context of large-system analysis of CDMA with random spreading, and successively rederived in various ways (e.g., [93, 96, 21]). Due to symmetry, the result does not depend on the indexk. Hence, without loss of generality we can choosek = 1. We have E[γ 1 ] = αE 1 N Tr h I+αH 1 H H 1 i −1 . (5.39) The leading order in N of the above trace can be evaluated as g mmse 1 (α,β) = lim N→∞ αE 1 N Tr h I+αH 1 H H 1 i −1 = α 1+ αβ 1+g mmse 1 (α,β) . (5.40) Solving for g mmse 1 (α,β) in (5.40), we obtain g mmse 1 (α,β) = 1 2 h α(1−β)−1+ p (α(1−β)−1) 2 +4α i . (5.41) To be able to calculate the O(1) correction to the mean mutual information, we need to evaluate the next to leading (O(1/N)) correction toE[γ 1 ]. The correction follows by noticing that the term β in (5.40) should be replaced by the aspect ratio of the matrix H 1 . For large but finite N, this is equal to (M − 1)/N = β− 1/N. Therefore, the 99 correction can beevaluated byreplacingβ byβ−1/N in (5.41). Using the Taylor series expansion, this amounts to computing E[γ 1 ] = g mmse 1 α,β− 1 N (5.42) = g mmse 1 (α,β)− 1 N ∂ ∂β g mmse 1 (α,β)+O N −2 , (5.43) where ∂ ∂β g mmse 1 (α,β) = − α 2 " 1+ α(1−β)−1 p (α(1−β)−1) 2 +4α # . (5.44) For later use, we define also the following asymptotic moments g mmse m (α,β) , lim N→∞ α m E 1 N Tr h I+αH 1 H H 1 i −m , (5.45) which can be obtained by repeatedly differentiatingg mmse 1 (α,β) with respect toα using the recursive relation g mmse m+1 (α,β) = α 2 m ∂ ∂α g mmse m (α,β), m≥ 1. (5.46) Thus we have g mmse 2 (α,β) = α 2 ∂ ∂α g mmse 1 (α,β), (5.47) g mmse 3 (α,β) = α 3 2 α ∂ 2 ∂α 2 g mmse 1 (α,β)+2 ∂ ∂α g mmse 1 (α,β) . (5.48) 100 For large SNR, i.e. α = ρ/β ≫ 1 and β < 1, g mmse 1 (α,β) is approximately α(1−β). This result indicates that only the ≈ N(1−β) zero eigenvalues of the matrix H 1 H H 1 contribute to the SINR for large α. Similarly, g mmse m (α,β) ≈ (1−β)α m for large ρ and β < 1, while for β = 1, g mmse m (α,β) ≈ k m ρ m−1/2 , where the constant k m satisfies k m+1 = Q m j=1 (1−1/2j). Next we calculate the matrix of the joint cumulants of order 2 with elements Σ mmse i,j =E c [γ i ;γ j ]≡E[γ i γ j ]−E[γ i ]E[γ j ]. Given the symmetry, all diagonal elements (i =j) are equal, and so are all off-diagonal ones (i6=j). Therefore, it is sufficient to compute Σ mmse 1,1 and Σ mmse 1,2 . We start withE c [γ 1 ;γ 1 ]. For convenience, we defineB 1 , I+αH 1 H H 1 −1 , and let (B 1 ) ij denote the (i,j) th element of B 1 and h 1i denote the i th element of h 1 . Then, a direct application of (5.38) and (B4) yields Σ mmse 1,1 = E c [γ 1 ;γ 1 ] = α 2 X a,b,c,d E (B 1 ) ab (B 1 ) cd h ∗ 1a h 1b h ∗ 1c h 1d − δ a,b N δ c,d N = α 2 X a,b,c,d E[(B 1 ) ab (B 1 ) cd ]E c [h ∗ 1a h 1b ;h ∗ 1c h 1d ] = α 2 E 1 N 2 Tr(B 2 1 ) → g mmse 2 α,β− 1 N N = v mmse d M +O(1/N 2 ) (5.49) 101 wherev mmse d =βg mmse 2 α,β− 1 N . We see that the leading correction in the autocorre- lation is non-vanishing only due to the random character of the vector h 1 [96]. We now turn to the more complicated computation of Σ mmse 1,2 to leading order in N. To simplify notation, we define the matrices B i = I+αH i H H i −1 , for i = 1,2, as before, and B 12 = I+αH 12 H H 12 −1 where H 12 is obtained by striking out from H both columnsh 1 andh 2 . Therefore, γ 1 = αh H 1 B 1 h 1 γ 2 = αh H 2 B 2 h 2 (5.50) Using the same notation as before, we rewrite the cumulant moment of γ 1 , γ 2 as E c [γ 1 ;γ 2 ]=α 2 X abcd E c [h ∗ 1a (B 1 ) ab h 1b ;h ∗ 2c (B 2 ) cd h 2d ] (5.51) In the following we will make extensive use of the following matrix identities, obtained by applying the Sherman-Morrison matrix inversion lemma, B 2 = B 12 −B 12 h 1 h H 1 B 12 α 1+αh H 1 B 12 h 1 B 1 = B 12 −B 12 h 2 h H 2 B 12 α 1+αh H 2 B 12 h 2 (5.52) 102 WewillnowuseNovikov’stheorem(TheoremB.1inAppendixB)tosuccessivelyaverage over the variables h 1 and h 2 . For example, considering the general term for indices (a,b,c,d) in (5.51) we write E c [h ∗ 1a (B 1 ) ab h 1b ;h ∗ 2c (B 2 ) cd h 2d ] = E[h ∗ 1a (B 1 ) ab h 1b h ∗ 2c (B 2 ) cd h 2d ]−E[h ∗ 1a (B 1 ) ab h 1b ]E[h ∗ 2c (B 2 ) cd h 2d ] = 1 N E ∂ ∂h 1a ((B 1 ) ab (B 2 ) cd h 1b h ∗ 2c h 2d ) − 1 N E ∂ ∂h 1a ((B 1 ) ab h 1b ) E[h ∗ 2c (B 2 ) cd h 2d ] (5.53) = 1 N E ∂h 1b ∂h 1a (B 1 ) ab (B 2 ) cd h ∗ 2c h 2d + 1 N E ∂(B 2 ) cd ∂h 1a (B 1 ) ab h 1b h ∗ 2c h 2d − 1 N E ∂h 1b ∂h 1a (B 1 ) ab E[h ∗ 2c (B 2 ) cd h 2d ] (5.54) where in (5.53) we have applied Novikov’s theorem formally replacing h ∗ 1a with 1 N ∂ ∂h 1a inside the expectations. We remind the reader that, as explained in Section B, in the above manipulations we treat the complex variables h ka and h ∗ nb as distinct and independent for all k,n,a,b, such that partial derivatives are performed individually with respect to these variables. In order to compute ∂(B 2 ) cd ∂h 1a we use the matrix inversion lemma (5.52) for B 2 . After some algebra, we obtain E c [γ 1 ;γ 2 ] = α 2 N E c h Tr(B 1 );h H 2 B 2 h 2 i (5.55) − α 3 N E h H 2 B 12 B 1 h 1 h H 1 B 12 h 2 1+αh H 1 B 12 h 1 (5.56) + α 4 N E " h H 1 B 12 B 1 h 1 h H 2 B 12 h 1 h H 1 B 12 h 2 1+αh H 1 B 12 h 1 2 # . (5.57) 103 The term in (5.55) results by summing over all indices the first two terms in (5.54), and the terms in (5.56) and (5.57) result by summing the last term in (5.54) after applying the partial derivative with respect to the elements of h 1 appearing in the numerator and the denominator of the matrix inversion lemma expansion of B 2 . It is important to notice that the order of magnitude of the first term isO(1), while the last two terms areO(1/N). The reason is that the last two terms are the result of applying the partial derivative inh 1a toB 2 , wherethe term that dependsonh 1 is scaled byafactorO(1/N) compared to the remaining matrix. WeproceednowbyapplyingNovikov’s theorem totherandomvariablesh ∗ 2a appear- ing in thenumerator of (5.56) and (5.57) and exchanging the correspondingexpectation with a derivative ∂/∂h 2a . However, with some hindsight we only apply the derivative toh 2 and not toB 1 , which would give a subleading term in 1/N. Therefore, to leading order in 1/N, we have (5.56)≈− α 3 N 2 E h H 1 B 2 12 B 1 h 1 1+αh H 1 B 12 h 1 ≈− α 3 N 2 Tr(B 3 12 ) N 1+ α N Tr(B 12 ) ≈− 1 N 2 g mmse 3 α,β− 2 N 1+g mmse 1 α,β− 2 N (5.58) (5.57)≈ α 4 N 2 E h H 1 B 12 B 1 h 1 h H 1 B 2 12 h 1 (1+αh H 1 B 12 h 1 ) 2 ≈ α 4 N 2 Tr(B 2 12 ) N 2 1+ α N Tr(B 12 ) 2 ≈ 1 N 2 g mmse 2 α,β− 2 N 2 1+g mmse 1 α,β− 2 N 2 (5.59) 104 where the approximation sign≈ means to leading order in 1/N. The second expression in each line occurred by averaging over h 1 to leading order, i.e. only on the numerator. In the last equation in each line we used the fact that 1 N Tr(B 12 )≈g mmse 1 α,β− 2 N . Next we may go back to (5.55) and expandB 2 using (5.52). After applying exactly the same methods as above we arrive at the following expression (5.55)≈ α 2 N 2 E c [Tr(B 12 );Tr(B 12 )]+ 2 N 2 g mmse 2 α,β− 2 N 2 1+g mmse 1 α,β− 2 N 2 − 1 N 2 g mmse 3 α,β− 2 N 1+g mmse 1 α,β− 2 N . (5.60) We collect all terms and use (C5) to reach the final result, Σ mmse 1,2 = E c [γ 1 ;γ 2 ] ≈ 1 N 2 β− 2 N α 4 1+2α 1+β− 2 N +α 2 1−β+ 2 N 2 2 (5.61) + 1 N 2 3g mmse 2 α,β− 2 N 2 1+g mmse 1 α,β− 2 N 2 − 2g mmse 3 α,β− 2 N 1+g mmse 1 α,β− 2 N ! = v mmse od M 2 +O(1/N 3 ), where we let v mmse od = β 2 3g mmse 2 α,β− 2 N 2 1+g mmse 1 α,β− 2 N 2 − 2g mmse 3 α,β− 2 N 1+g mmse 1 α,β− 2 N + β− 2 N α 4 1+2α 1+β− 2 N +α 2 1−β + 2 N 2 2 . (5.62) 105 For large α, v mmse od ≈ρ 2 whenβ< 1, and v mmse od ≈ρ 2 /16 when β = 1. We collect the results of (5.61) and (5.49) by writing the correlation matrix for the SINRs{γ k } to leading order as: Σ mmse i,j =δ i,j v mmse d M +(1−δ i,j ) v mmse od M 2 (5.63) It is worth pointing out that despite the fact that the off-diagonal elements are much smaller compared to the diagonal ones, they all contribute to the eigenvalues of Σ. In fact, these can be computed in closed form and are given by λ 1 (Σ mmse ) = v mmse d M +(M−1) v mmse od M 2 ≈ v mmse d +v mmse od M and λ k (Σ mmse ) = v mmse d M − v mmse od M 2 ≈ v mmse d M for all k = 2,...,M. Cumulant moments for the ZF receiver The corresponding results for the ZF receiver can be derived directly from the previous section by observing that the SINR for the k-th channel of the ZF receiver, given by 106 (5.7), can be deduced from the corresponding expression (5.9) for the MMSE receiver in the limit of infiniteα, i.e. γ zf k = α h (H H H) −1 i kk = α lim α 0 →∞ γ mmse k (α 0 ) α 0 (5.64) = α lim α 0 →∞ α 0 I+α 0 H H H −1 kk −1 A subtle point needs to be stressed here: the results for the ZF receiver cannot be obtained simply as the “limit for high SNR” of the results for the MMSE receiver. Rather, we have to distinguish between the channel SNR (contained in the parameter α) and the SNR parameter in the linear receiver matrix expression (indicated by α 0 above) that we let to infinity in order to obtain the ZF results. It can be shown that for β < 1 the analysis of the previous section involving the matrices B 1 ,B 2 and B 12 can be carried out in this limiting case. In addition, as seen in Appendix C, the condition for the validity of the manipulation of the first term of (5.61) is thatf(x)= (1+αx) −1 is a smooth function of x in the region of support of the eigenvalue spectrum. This is not true in the vicinity of x = 0 for arbitrarily large α, specifically when α = O(N). Thus when β = 1, in which case the asymptotic eigenvalue spectrum includes x = 0, the above approximation is not valid. As a result, this method breaks down at β = 1. 107 From (5.42) we get the mean SINR for the ZF receiver 4 E[γ zf 1 ]= α(1−β+1/N) β < 1 0 β = 1 (5.65) in agreement with [96]. Similarly, the second order moments can be obtained from E c h γ zf i ;γ zf j i =α 2 lim α 0 →∞ E c h γ mmse i (α 0 );γ mmse j (α 0 ) i α 2 0 . (5.66) Thus we get Σ zf 11 = v zf d M = α 2 β(1−β+1/N) M β< 1 0 β = 1 (5.67) and Σ zf 12 = v zf od M 2 = α 2 β 2 M 2 β < 1 ρ 2 16M 2 β = 1 . (5.68) While in the case ofβ < 1 the covariance matrixΣ is well-defined and positive-definite with eigenvalues λ 1 (Σ zf ) = v zf d M +(M−1) v zf od M 2 ≈ α 2 β 2 βM and λ k (Σ zf ) = v zf d M − v zf od M 2 ≈ α 2 β 2 (1−β) βM 4 For simplicity we neglect the subleading terms in the following equalities, i.e., we omit O(1/N 2 ) in (5.65) and O(1/N 3 ) in (5.67), (5.68), respectively. 108 for all k = 2,...,M, the case β = 1 is problematic. Specifically, it results in (narrowly) negative eigenvalues forΣ, therebyinvalidating the Gaussian approximation fortheγ i ’s and as a result the further treatment of the mutual information as a Gaussian variable. The case β = 1 for the ZF receiver is therefore excluded in the subsequent Gaussian approximation of the mutual information. 5.5.3 Gaussian approximation and outage probability In this section we use the previous results together with the asymptotic Gaussianity that follows from the fact that higher-order moments are vanishing (see Appendix D) to give an explicit Gaussian approximation for the outage probability of linear MMSE and ZF receivers with coding across the antennas in the regime of fixed SNR and large number of antennas. We start with the meanE[I N ]. Due to the symmetry with respect to the terms γ k , we have C 1 =E[I N ] = ME[log(1+γ 1 )] =ME[log(1+E[γ 1 ]+γ 1 −E[γ 1 ])] (5.69) = Mlog(1+E[γ 1 ])−M ∞ X n=1 (−1) n n E[(γ 1 −E[γ 1 ]) n ] (1+E[γ 1 ]) n 109 In the above expansion, all termsn> 2 involve cumulants ofγ 1 higher than 2, thus can be neglected. For the MMSE receiver, this yields C mmse 1 = E[I N ] = Mlog 1+g mmse 1 α,β− 1 N − v mmse d 2(1+g mmse 1 α,β− 1 N ) 2 +o(1) ≈ Mlog 1+g mmse 1 (α,β)− 1 N ∂ ∂β g mmse 1 (α,β) − v mmse d 2(1+g mmse 1 (α,β)) 2 ≈ Mlog(1+g mmse 1 (α,β))−β ∂ ∂β g mmse 1 (α,β) 1+g mmse 1 (α,β) − v mmse d 2(1+g mmse 1 (α,β)) 2 , (5.70) where g mmse 1 (α,β) and v mmse d are given by (5.41) and (5.49) respectively. For the ZF receiver the mean is given by C zf 1 =Mlog(1+α(1−β))+ αβ 1+ α(1−β) 2 (1+α(1−β)) 2 +o(1). (5.71) Similarly, we can calculate the variance of the mutual information as follows C 2 = ME c [log(1+γ 1 );log(1+γ 1 )]+M(M−1)E c [log(1+γ 1 );log(1+γ 2 )] = M ∞ X m,n=1 (−1) m+n mn E c [(γ 1 −E[γ 1 ]) m ;(γ 1 −E[γ 1 ]) n ] (1+E[γ 1 ]) n+m + M(M−1) ∞ X m,n=1 (−1) m+n mn E c [(γ 1 −E[γ 1 ]) m ;(γ 2 −E[γ 2 ]) n ] (1+E[γ 1 ]) n+m . (5.72) 110 As we see above, the first terms in both the above summations give the leading order of the variance of the mutual information, which we denoted in (5.33) byσ 2 . The variance of the MMSE mutual information is given by C mmse 2 = v mmse d +v mmse od 1+g mmse 1 α,β− 1 N 2 +o(1), (5.73) where v mmse od is given by (5.62). The corresponding variance for the ZF receiver is C zf 2 = βα 2 (1+1/N) (1+α(1−β+1/N)) 2 +o(1) (5.74) forβ< 1. As mentioned above, the caseβ =1 does not result in a well-behaved jointly Gaussian behavior of theγ k ’s, and therefore the Gaussian approximation of the mutual information cannot be derived with this approach. As anticipated at the beginning of this section, from (5.70) and (5.71) we see that the mean mutual information is expressed in the form m 1 = Mc 10 +c 11 , where the coefficient c 10 was found in previous works (e.g., [102]), considering the limit N →∞ of the normalized mutual information per transmit antenna, and the term c 11 is a correction term that captures the correlation between the SINRsγ k . Under this Gaussian approximation, we can easily evaluate the outage probability for fixed SNR,β and number of antennas M as follows: P lin out (R,ρ)≈Q R−C 1 √ C 2 (5.75) 111 where Q(x) = R ∞ x 1 √ 2π e −t 2 /2 dt is the Gaussian tail function. We conclude this section with a discussion on the range of validity of the Gaussian approximation. For the MMSE receiver, in the largeρ limit we have thatC mmse 2 =O(ρ) for β = 1, while C mmse 2 = O(1) for β < 1. This fast increase of the variance of the distribution for β = 1 and ρ≫ 1 is a spurious result in this approximation, due to the neglected terms which are negligible for fixed ρ and increasingly large N, but become important for fixedN and large ρ. The behavior of the ZF receiver for β = 1, when the jointly Gaussian behavior of the γ k ’s breaks down, exacerbates the above large ρ behavior of the MMSE receiver. In fact, as was discussed before, the ZF case is in some sense the “infinite ρ limit” of the MMSE case. Therefore the problematic situation appearing in the ZF receiver for β = 1 has the same roots as the problems faced in the large (but finite)ρ limit whenN is also finite but not large enough. 5.5.4 Simulations and comparisons In this section we first validate the asymptotic analysis by comparing the asymptotic approximation for C 1 and C 2 with the exact moments obtained by finite-dimensional Monte Carlo simulation. We then compare the Gaussian approximation to the outage probability with finite-dimensional Monte Carlo simulation. For the sake of comparison, we also consider the outage probability of the optimal receiver, given by the log-det cdf P(I opt N ≤ R), where I opt N = logdet(I+ ρ M HH H ). As said in the Introduction, the asymptotic Gaussianity of the log-det mutual information 112 is well-known and holds under very general models of channel correlation across the antennas (not considered in this work). For completeness, we recall its expressions under the i.i.d. channel coefficient assumptions and in the notation of this chapter. We have I opt N −Mμ ν d →N(0,1) (5.76) where μ = log(1+g mmse 1 (α,β))+ 1 β log(1−α(1−β)+g mmse 1 (α,β))+ g mmse 1 (α,β) αβ − 1 β , (5.77) and ν 2 =−log 1− 1 β 1− g mmse 1 (α,β) α 2 ! . (5.78) Thefirsttermin(5.77)coincideswiththecoefficientofM inthefirsttermof (5.70),i.e., it is the asymptotic capacity per antenna of the linear MMSE receiver. The additional terms in (5.77) represent the so-called “non-linear” gain of the optimal versus linear MMSE receiver, as discovered in [103]. It is worth pointing out that for large SNR, the asymptotic mean capacity is given by E[I opt N ]≈M logρ (5.79) 113 while the variance has the following behavior ν 2 ≈ −log(1−β) β< 1 1 2 logρ β = 1 (5.80) Using the Gaussian approximation P out (R,ρ)≈Q R−Mμ ν ≈ exp − (R−Mμ) 2 2ν 2 with R =rlogρ, in the large ρ limit we find − logPout(R,ρ) logρ ≈ (M−r) 2 logρ 2|log(1−β)| β< 1 − logPout(R,ρ) logρ ≈(M−r) 2 β = 1 (5.81) For β = 1, the Gaussian approximation yields an outage probability exponent equal to (r−M) 2 for r ∈ [0,M], that closely approximates the exact exponent d ∗ (r) [109]. However, forβ< 1theGaussianapproximationyieldsacompletelyinaccuratebehavior. In fact, in this case the exponent obtained through (5.81) would be infinite, while we know from [109] that d ∗ (r) ≤ MN. The reason for this spectacular failure of the Gaussian approximation is that, in the large-N approximation, it is implicitly assumed that forβ6= 1 the eigenvalue distribution at very small eigenvalues is zero, as described by the Marcenko-Pastur law [97]. Instead, for largeρ the eigenvalues that dominate the outage probability are exactly the very small ones, of the order of 1/ρ, i.e., exactly the ones that the Gaussian approximation neglects. 114 Weconcludethissectionbypresentingsomenumericalresults. Fig.5.5comparesthe analytical mean of the MMSE mutual information per antennaC mmse 1 /M with the cor- responding empirical mean obtained from Monte Carlo simulation. The corresponding comparison between the analytical variance C mmse 2 and the empirical variance is pre- sented in Fig. 5.6. Using the results for the mean and the variance, we plot the CDF of the (Gaussian) mutual information for the MMSE and optimal receiver in Figs. 5.7,5.8. Both analytical and empirical results are plotted, for a wide range of M,N and SNRs. For brevity, we have only reported plots of the CDF for the mutual information for the ZFcase, seeFig. 5.9. Theplots are forM = 3,10,β = 0.5 andρ= 3,30 dB.Theresults are similar in flavor to the MMSE case. We notice that the analytical and empirical results match closely, for even moderate number of antennas and not too large SNRs, in line with the comments made earlier regarding the validity of the analysis. It is also worthwhile noticing that the accuracy of the Gaussian approximation for linear receivers appears to be slightly inferior to that of the Gaussian approximation for the optimal receiver case, especially for very small N and large SNR. 5.6 Conclusions Novel wireless communication systems are targeting very large spectral efficiencies and will operate at high SNR thanks to hot-spots and pico-cell arrangements. For example, a system with bandwidth of 20 MHz and operating at 100 Mb/s requires a spectral effi- ciencyof5bit/s/Hz, correspondingtocodingrateR = 5bpcuinnotation adoptedhere, 115 Figure 5.5: Mean of the MMSE mutual information per antenna (C mmse 1 /M) as a func- tion of β, for M = 2,5,10 and 20. The solid lines are analytical results, and the cor- responding dash-dot lines are empirical results obtained from Monte Carlo simulation. Diamonds denote 3 dB, circles 10 dB and triangles 30 dB. 116 Figure 5.6: Variance of the MMSE mutual information (C mmse 2 ) as a function of β, for M = 2,5,10 and 20. The solid lines are analytical results, and the corresponding dash- dot lines are empirical results obtained from Monte Carlo simulation. Diamonds denote 3 dB, circles 10 dB and triangles 30 dB. 117 Figure 5.7: CDF of the mutual information (MI) for the MMSE and optimal receivers, for M = 2,3, β = 0.5 and ρ = 3,30 dB. The solid blue line is the analytical result for the MMSE, the dot-dash blue is MMSE empirical, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. 118 Figure 5.8: CDF of the mutual information (MI) for the MMSE and optimal receivers, for M = 5,10, β = 0.5 and ρ = 3,30 dB. The solid blue line is the analytical result for the MMSE, the dot-dash blue is MMSE empirical, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. 119 Figure 5.9: CDF of the mutual information (MI) for the ZF and optimal receivers, for M = 3,10, β = 0.5 and ρ = 3,30 dB. The solid blue line is the analytical result for the ZF receiver, the dot-dash blue is ZF empirical, the dashed black is Optimal receiver analytical and the dotted black is the optimal receiver empirical. 120 if one neglects non-ideal effects such as pilot symbols, guard band and guard intervals, cyclic prefix redundancy for OFDM, etc. For such systems, the use of low-complexity linear receivers in a separated detection and decoding architecture as those examined in this chapter may be mandatory because of complexity and power consumption. In this chapter we investigated the asymptotic performance of such separated linear detection and decoding architectures in two relevant asymptotic regimes. In the regime of fixed number of antennas and increasing SNR and coding rate, we showed that linear detection may be very suboptimal. Furthermore, due to the strong correlation between the SINRs of the parallel channels induced by the linear receiver, coding across the an- tennasdoesnothelpintermsoftheachievableDiversity-Multiplexing Tradeoff. Wealso illuminated the very peculiar behavior of the linear MMSE receiver with coding across the antennas, that exhibits a diversity order(slope of theoutage probability curve) that changes depending on the rate. Then, we analyzed the asymptotic behavior of the lin- ear MMSE and the ZF receivers with coding across the antennas in the regime of fixed SNR and large (but finite) number of antennas. We showed that the correspondingmu- tual information has statistical fluctuations that converge in distribution to a Gaussian random variable, and we computed its mean and variance in closed form. This yields a simple Gaussian approximation of the outage probability in this asymptotic regime, within the limitations that have been thoroughly discussed. Based on the analysis carried out in this work, we may summarize some consid- erations on system design. In order to achieve a required target spectral efficiency at given block-error rate and SNR operating point, an attractive design option may 121 consists of increasing the number of antennas (especially at the receiver) and using a low-complexity linear receiver. However, pure spatial multiplexing (independent coded streams directly fed into the transmit antennas) and/or linear ZF receivers should be avoided. In contrast, coding across antennas and a linear MMSE receiver can achieve a very good tradeoff between performance and complexity in a wide range of system operating points. 122 Chapter 6 Coding and Decoding for the Dynamic Decode and Forward Relay Protocol 6.1 Introduction Employingmultipleantennasatthetransmitterandthereceiver ofwirelesscommunica- tions is known to provide significant benefits in terms of both throughput (multiplexing gain) and reliability (diversity gain) (see [109] and references therein). When physical constraints limitthenumberofantennasthatcanbeinstalled onasinglewirelessdevice (e.g., small sensors insensor networks), theusage of cooperative wireless relay protocols is a promising alternative strategy. In these protocols, two or more terminals cooperate in order to mimic a super-user with multiple antennas. The relay channel was introduced by van der Meulen [99] and was studied in detail byCover andElGamal [17], whocharacterized thecapacity for thediscrete memoryless as well as for the Gaussian degraded cases. The relay channel with fading was examined 123 by Sendonaris et al., [79], where an achievable rate region was provided. In the case of slow fading, the outage behavior of half-duplex wireless relay channels was studied by Lanemanetal.,[58],andsimplecooperative diversityprotocolsforsignallingacrossthese channels (such as amplify and forward and decode and forward) were introduced. In [2], Azarian et al. used the diversity-multiplexing tradeoff (DMT) formulation of [109] to study the outage behavior of slowly-fading relay channels in the high-SNR regime, and also introduced new classes of protocols such as the non-orthogonal amplify and forward (NAF) and the dynamic decode and forward (DDF). An improved DDF protocol based on code superposition was later proposed in [76]. The DDF protocol for the single relay case was subsequently studied in [66], where simplified variants of the protocol were introduced and some code design issues were addressed. Code design for the DDF protocol is also addressed in the recent contribution [27]. ThepresentchapteralsofocusesontheDDFprotocolforthehalf-duplex,singlerelay single-antenna case. With respect to [66] and [27], we analyze explicitly the achievable DMTofpracticalcodeswithfiniteblocklengthandproposeasimpleDMToptimalcode construction that makes use of approximately universal codes for the parallel channel and of the Alamouti code. Approximately universal codes for the parallel channel may be obtained either from using a QAM base alphabet and a suitable unitary precoding matrix (lattice codes) or from permutation codes derived from universally decodable matrices (UDM) [91, 34]. We treat both cases and give construction examples and comparisons. Remarkably, our codes perform very close to the outage probability and have generally lower decoding complexity than those previously proposed. 124 Furthermore,wediscusstwooftenneglectedissues: 1)theeffectofdecodingerrorsat therelay, andhowtomitigateit;2)thefactthatthedestinationdoesnotgenerallyknow a priori the relay decision time. In order to tackle 1), we introduce a decision rejection criterion at the relay, such that the relay triggers transmission only when its decision is reliable. We show that the Forney’s decision rule (a variant of Neyman-Pearson rule) yields almost optimal performance with practical finite length codes, while previously proposed options suffer from significant degradation. In order to tackle 2), we treat the channel “seen at destination” as a compound channel, where each compound member corresponds to a different relay decision time. We prove that a receiver based on the Generalized Likelihood Ratio Test (GLRT) rule, that jointly decodes the relay decision time and the information message, achieves the optimal DMT. We also show that a simpler scheme that performs separate detection of the relay decision time, by ignoring the structure of the coded signal and treating it as random, is generally suboptimal and it becomes optimal only in the limit of infinite block length. As an aside, our results show that no side information channel or additional protocol overhead is needed in order to inform the destination about the relay decision time. This may yield to much simplified actual protocol design for the DDF scheme, at the cost of an augmented decoder at the destination. With the lattice codes advocated in this chapter, the decoder at the relay has to solve a closest lattice point problem with a rank deficient lattice matrix. It is well- known that standard sphere decoding [104, 40] yields exponential complexity in this case. In order to address this problem (again, often neglected in the current literature) 125 we advocate the use of the minimum mean squared error generalized decision feedback equalizer (MMSE-GDFE) lattice decoder of [67, 20]. Via simulation of the performance of our explicitly constructed codes, we demonstrate that this lattice decoder is able to provide near optimal performance at moderate complexity. In Section 6.2, we introduce the system model we work with and review relevant previous results. Section 6.3 presents the main result of this chapter, a characterization oftheDMToftheDDFprotocolforfiniteblocklength. Explicitcodeconstructionsthat achieve this DMT are provided in Section 6.4, and methods to enable error detection at the relay and low complexity decoding of these codes are also dealt with. 6.2 Problem definition and background 6.2.1 System model We consider the single relay channel shown in Fig. 6.1, where S, R and D denote the source, relay and destination, and h, g 1 and g 2 denote the fading coefficients between the source-relay, source-destination and relay-destination terminals, respectively. The channel fading coefficients are i.i.d. CN(0,1) random variables, corresponding to i.i.d. Rayleigh fading. Following the standard outage setting [58, 2, 109], we assume that the channel coherence time is considerably larger than the allowed decoding delay. Invoking a time-scale decomposition argument (see for example [94]) this setting is modeled by the so-called quasi-static fading channel, where the channel coefficients are random butremain constant over the wholeduration of a codeword, although the latter 126 Figure 6.1: The single-antenna single relay fading channel. can be very large. We consider slotted transmission where a source codeword spansM slots of length T symbols each, resulting in a total block length of MT. The relay operates in half-duplex mode. In decode and forward protocols, the block of lengthMT symbols is split into two phases. In the firstphase the relay is in listening mode and receives the signal from the source. At a certain instant, referred to as the decision time in the following, the relay tries to decode the source information message. In the second phase, from the decision time to the end of the block, the relay switches to transmit modeand sends symbols to help the destination decode the source message. The DDF protocol is characterized by the fact that the decision time is not fixed a priori. On the contrary, the relay decides when to decode and switch to transmit mode depending on the channel coefficient h and the received signal. Therefore, the decision time is a random variableM. Without loss of generality, we restrict the decision time to coincide with the end of a slot 1 , i.e., M takes on values in the set {1,2,...,M}, 1 Notice that T is a design parameter. Letting T = 1 provides an unrestricted decision time. In this way, there is no loss of generality in this assumption. 127 whereM =M corresponds to the case where the relay does not help the destination. During phase 1 (listening phase) the signal received by the relay is y r,k =hx s,k +v k , k = 1,2,...,MT, (6.1) and the signal received by the destination is y k =g 1 x s,k +w k , k = 1,2,...,MT. (6.2) During phase 2 (relay transmit phase), the signal received by the destination is y k =g 1 x s,k +g 2 x r,k +w k , k =MT +1,MT +2,...,MT. (6.3) Here,x s = [x s,1 ···x s,MT ] T denotes the source codeword, drawn from a codeX s ⊂C MT of rateR bits per symbol. Without loss of generality, we may assume that the symbols x r,k transmitted by the relay are from an auxiliary code X r ⊂ C MT with rate R and block length MT, but only the last (M−M)T symbols of a codeword are effectively transmitted in phase 2, while in phase 1 the relay transmitter is idle because of the half-duplex constraint. 128 The noise at the relay and destination, denoted by v k ∼ CN(0,σ 2 v ) and w k ∼ CN(0,σ 2 w ), form two white mutually independent sequences. We impose the same per- symbol average power constraint for both the source and the relay, given by E |x s,k | 2 , E |x r,k | 2 ≤E, where E denotes the symbol energy, and define the SNRs of the S-D and the S-R links to beρ=E/σ 2 w andρ ′ =E/σ 2 v , respectively. For later use, we introduce the following notation: let y j i , y j r,i , x j s,i and x j r,i , each ∈ C (j−i)T , denote respectively the received signals at the destination and at the relay from symbol time iT +1 tojT, the source transmit signal from time iT +1 to jT and the relay transmit signal from timeiT +1 tojT, where the latter is assumed to be zero for all times k≤MT. The quantities w j i andv j i are defined similarly. 6.2.2 Diversity-Multiplexing Tradeoff of Co-operative Diversity Pro- tocols We will use the DMT (see Chapter 2) as our performance metric when we analyze cooperative diversity protocols. It is clear that the DMT of the MIMO channel with one receive and two transmit antennas provides an upper bound to the performance of any relay protocol for the channel of Fig. 6.1. This bound, known as the transmit diversity bound [58], is given by d tx.div.bd. (r) = 2(1−r). 129 The DMT of the DDF protocol, proposed and analyzed in [2], is given by d ∗ (r) = 2(1−r), 0≤r≤ 1 2 (1−r)/r, 1 2 ≤r≤ 1 . (6.4) This result is obtained by analyzing the information outage probability with Gaussian inputs, and it is achievable (e.g., by using a Gaussian random coding argument) in the limit of bothM →∞ and T →∞. The relay decision time is given by M= min M, MR log(1+|h| 2 ρ ′ ) , (6.5) i.e., M is set to the minimum m = 1,2,...,M −1 such that the mutual information between x m s,0 and y m r,0 for fixed and known h, given by mT log(1+|h| 2 ρ ′ ), exceeds the number of information bits per message MTR. If such an m exists, the relay triggers the decoding of the whole information message and switches to the transmission mode. Ifnosuchmexists,thenM=M andtherelayremainssilent. BoththelimitoflargeM andT are necessary to achieve the DDF DMT in (6.4). In fact, the normalized decision time M/M must converge to a continuous random variable distributed in [0,1] and, for every decision timeM=m, the number of symbolsmT received by the relay must be arbitrarily large, such that the decoding error event coincides with the information 130 outage error event. In this way, the corresponding probability of decoding error is arbitrarily close to the information outage probability P log(1+|h| 2 ρ ′ )≤ MR m , and the probability of undetected error (i.e., the relay accepts a wrong decision) is arbitrarily small. In brief,T →∞ is necessary in order to fix the optimal decision time based only on the channel strength|h| 2 and be sure (with arbitrarily high probability) that the decoded message is the correct one. We should also notice that, in the limit of T →∞, the outage probability does not depend on the knowledge of h at the relay decoder and of (g 1 ,g 2 ) at the destination decoder (see for example [7]). On the other hand, a common assumption made in previous works is that the destination knows exactly the relay decision time M. In practice, this assumption requires some form of protocol to provide side information to the destination. In the DMT analysis, one should pay great care to ensure that the error probability of such side information protocol does not dominate the decoding error probability, i.e., in designing any side information protocol we must ensure that its probability of error decreases not slower than ρ −d ∗ (r) . Practical code design for the DDF protocol considers finite, possibly very short, M andT. In the following, we will make an explicit assumption of perfect receiver channel state information (CSIR), that is relatively easy to acquire using pilot symbols and is a common assumption in the DMT analysis of even finite-length codes (see [109] and 131 [94]). Onthecontrary, we explicitly addressthefact that the destination doesnot know a priori the relay decision timeM and tackle this problem by analyzing an augmented decoder based on the GLRT rule. 6.2.3 Existing DDF code designs In [66], a variant of the DDF protocol is proposed where the relay codeX r is such that the signal received at the destination reduces to an Alamouti constellation [1]. We will refer to this scheme as the “Alamouti-DDF” scheme, and review it briefly in the sequel since we make use of the same approach. With the Alamouti-DDF, assuming that the relay decodes correctly at the decision timeM=m, the signal transmitted by the relay at time k is given by [66] x r,k = x ∗ s,k+1 , k =mT +1,mT +3,... −x ∗ s,k−1 , k =mT +2,mT +4,... , (6.6) which reduces the signal seen by the destination formT +1≤k≤MT to an Alamouti constellation. Through linear processing of the received signal y M 0 , the destination obtains the sufficient statistics for decoding, given by ˜ y k = g 1 x s,k +w k , k = 1,...,mT p |g 1 | 2 +|g 2 | 2 x s,k + ˜ w k , k =mT +1,...,MT , (6.7) 132 where the statistics of ˜ w k are identical to those ofw k . In this case, it is easy to see that the mutual information per symbol at the destination, forM =m and i.i.d. Gaussian inputs, is given by m M log 1+|g 1 | 2 ρ + M−m M log 1+(|g 1 | 2 +|g 2 | 2 )ρ (6.8) andcoincides withthat of theoriginal DDF schemedefinedby(6.2)and(6.3), whenthe codebooks X s and X r are also drawn independently from an i.i.d. Gaussian ensemble. Hence, theAlamouti-DDF modification entails noloss inDMT comparedtotheoriginal DDF protocol [66]. 6.3 DMT of the DDF Protocol with finite length In this section, we characterize the achievable DMT of the DDF protocol with finiteM and T. First, we find an upper bound on the DMT by letting T →∞, assuming that the destination has perfect knowledge of the relay decision time M, and using outage probability. Then, we shall analyze the performance of Gaussian random codes with finite length, with the assumption that the destination has no knowledge of M, and find a lower bound that matches the upper bound. Since for i.i.d. Gaussian inputs the Alamouti-DDF yields the same mutual infor- mation as DDF, as far as outage probability is concerned we can refer to the channel defined in (6.7). This is a set of parallel channels for m = 1,...,M, with dependent channel gains. In particular, there are two types of sub-channels: one representing the 133 S-D link, and another set representing the composite (S,R)-D link (except for the case when m = M, which corresponds to when the relay remains inactive for the whole block; in this case, only the S-D link appears). The switching point between the two channels is controlled by the random variable M. We will refer to this channel as a random switch channel (RSC). Given a particular switching instant M = m, we will call the ensuing channel as a m-switch channel (m-SC). The RSC belongs to the class of “mixed channels” (see [39]), that is, a compound channel with an a priori probability distribution on the compound members. In this case, the probability distribution on the channel members (the m-SCs in (6.7)) is induced by the triple (M,g 1 ,g 2 ). 6.3.1 Outage probability analysis We compute the DMT of the RSC defined above for arbitrarily large T under the assumption that the destination receiver has perfect knowledge of M, and hence find an upper bound on the DMT exponentd ∗ M (r) for the finite-length DDF protocol. This is established by the following theorem. Theorem 6.3.1. The DMT of the single relay DDF scheme with decision times m = 1,2,...,M and finite slot length T ≥ 1 is upper bounded by d ∗ M (r)≤d out (r) = min 1≤m≤M d m (r)+d m (r) , 134 where d m (r)= 1− Mr m−1 , 0≤r≤ m−1 M 0, m−1 M <r≤ m M ∞, m M <r≤ 1 , (6.9) d m (r) = 2−2r, m< M 2 M(1−r) m , m≥ M 2 (6.10) for r≥ 1 2 , and d m (r)= 2−2r, m< M 2 2− rM M−m , M 2 ≤m<M(1−r) M(1−r) m , m≥M(1−r) (6.11) for r< 1 2 . Proof. LetM denote the random decision time as defined in (6.5) and P out (r) denote the outage probability of the corrsponding RSC. Also, let P m−SC out (r) denote the outage probability of the m-SC for given m. Then, the law of total probability yields P out (r)= M X m=1 P(M =m)P m−SC out (r). (6.12) 135 Since in the regime of very high SNR that characterizes the DMT, scaling SNR by a constant does not change the DMT, we allow both ρ,ρ ′ →∞ and the DMT shall not depend on the (constant) ratio ρ ′ /ρ=σ 2 w /σ 2 v . Define P out (r) . = ρ −dout(r) , P m−SC out (r) . = ρ −dm(r) , 1≤m≤M, P(M =m) . = ρ −dm(r) , 1≤m≤M. Then, it is clear from (6.12) that d out (r) = min 1≤m≤M d m (r)+d m (r) . Furthermore, fromstandardarguments basedon Fano inequality [109]andbecausehere we are assuming that the destination receiver is enhanced by the side information on M, it is also immediate to conclude that d ∗ M (r)≤d out (r). It remains to prove (6.9) and (6.10), (6.11). Notice that d m (r) is solely a function of the S-R link and d m (r) is a function of the R-D and S-D links. We analyze these quantities separately as follows. 136 Analysis of ρ −dm(r) Let’s consider first the case m < M. Set R = rlogρ. The probability that the relay decodes after m sub-blocks P(M =m), 1≤m≤M−1, corresponds to the event mT log(1+|h| 2 ρ ′ )>MRT > (m−1)T log(1+|h| 2 ρ ′ ) ⇔ n Mr m logρ< log(1+|h| 2 ρ ′ )< Mr m−1 logρ o ⇔ ρ Mr m −1 ρ ′ <|h| 2 < ρ Mr m−1 −1 ρ ′ . (6.13) Since|h| 2 is exponentially distributed andρ ′ . =ρ, we compute P(M =m) . = Z ρ Mr m−1 −1 ρ Mr m −1 e −z dz = e −ρ Mr m −1 −e −ρ Mr m−1 −1 . According to the value of the multiplexing gain, we analyze the above quantity for each 1≤m<M as follows. • r> m M : This corresponds to Mr m −1, Mr m−1 −1> 0. In this case 2 P(M =m) . =ρ −∞ . 2 The notation P . = ρ −∞ indicates that P decreases faster than any polynomial function of ρ. 137 • m−1 M <r≤ m M : This corresponds to Mr m −1≤ 0, Mr m−1 −1> 0. In this case, P(M =m) . =ρ 0 . • r≤ m−1 M : This corresponds to Mr m −1≤ 0, Mr m−1 −1≤ 0. In this case, using a power series expansion, P(M =m) = " 1−ρ Mr m −1 + ρ 2( Mr m −1) 2! +··· # − " 1−ρ Mr m−1 −1 + ρ 2( Mr m−1 −1) 2! +··· # . = ρ Mr m−1 −1 . A similar analysis for P(M =M) results in P{M =M} . = ρ Mr M−1 −1 , 0≤r≤ M−1 M ρ 0 , M−1 M <r≤1 . Therefore, the result for all 1≤m≤M can be compactly expressed by (6.9), shown in Fig. 6.2. 138 Figure 6.2: Negative ρ-exponent of the probability of the relay decoding after exactly m-subblocks. Analysis of d m (r) From (6.8), the outage probability of the m-SC is given by P m−SC out (r) =P I m−SC ≤MTR , whereI m−SC =mT log(1+|g 1 | 2 ρ)+(M−m)T log[1+(|g 1 | 2 +|g 2 | 2 )ρ]. Defining|g 1 | 2 = ρ −α 1 and|g 2 | 2 =ρ −α 2 and applying standard approximations in the regime of large ρ, we eventually obtain P m−SC out (r) . = P ((M−m)max{[1−α 1 ] + ,[1−α 2 ] + }+m[1−α 1 ] + ≤rM), 139 where [x] + , max{0,x}. Since |g 1 | 2 and |g 2 | 2 are independent exponential random variables, the joint pdf of (α 1 ,α 2 ) is given by f(α 1 ,α 2 ) . =e −ρ −α 1−ρ −α 2 ρ −α 1 −α 2 . Therefore, P m−SC out (r) . = Z B ρ −α 1 −α 2 dα 1 dα 2 , where B is the two-dimensional region defined by the inequalities (M −m)max{[1− α 1 ] + ,[1−α 2 ] + }+m[1−α 1 ] + ≤rM and α i ≥ 0∀ i. Using Varadhan’s lemma [22], we obtain d m (r)= inf B {α 1 +α 2 }. (6.14) Define β = m M . The regionB is equivalently defined by (1−β)max{[1−α 1 ] + ,[1−α 2 ] + }+β[1−α 1 ] + ≤r, α i ≥ 0∀ i. It is obvious that we may restrict attention to α i ≤ 1 ∀ i insofar as computing the infimum in (6.14) is concerned. We analyzeB according to the following cases: 140 Figure 6.3: The regionB. • α 1 ≥α 2 : We have β(1−α 1 )+(1−β)(1−α 2 )≤r ⇔βα 1 +(1−β)α 2 ≥ 1−r. This line has intercepts 1−r β and 1−r 1−β on theα 1 and α 2 axes respectively. • α 1 <α 2 : We have β(1−α 1 )+(1−β)(1−α 1 )≤r ⇔α 1 ≥ 1−r. 141 The region B is depicted in Fig. 6.3. The solution to the problem in (6.3) corresponds to choosing the least non-negative k such that the line α 1 +α 2 = k touches B. The analysis should be done according to whether 1−r β ≷ 2− 2r ⇔ β ≶ 1 2 and whether 2−2r ≷ 1 ⇔ r ≶ 1 2 . It is immediate from Fig. 6.3 that the solution to (6.14) when r≥ 1 2 is at (α ∗ 1 ,α ∗ 2 )= 1−r β ,0 forβ≥0.5, and (α ∗ 1 ,α ∗ 2 ) = (1−r,1−r) forβ < 0.5. For the case whenr< 1 2 , the solution to (6.14) is at (α ∗ 1 ,α ∗ 2 )= (1−r,1−r) forβ< 0.5, at (α ∗ 1 ,α ∗ 2 )= 1,1− r 1−β forβ≥ 0.5 and 1−r β > 1, and at (α ∗ 1 ,α ∗ 2 ) = 1−r β ,0 forβ≥ 0.5 and 1−r β ≤ 1. The final solution is compactly expressed by (6.10), (6.11). This concludes the proof of Theorem 6.3.1. 6.3.2 Achievability We consider finite lengthT and no a priori knowledge ofM at the destination decoder. We have the following result: Theorem 6.3.2. The upper bound of Theorem 6.3.1 is achievable. Therefore, d ∗ M (r)= d out (r). Proof. We consider the original DDF protocol (not the Alamouti variant) defined by (6.1), (6.2) and (6.3). For this channel we construct a particular coding scheme and analyze its performance. Codebook generation: For givenM,T andR,wegenerateX s ⊂C MT andX r ⊂C MT of cardinality ρ rMT independently, with i.i.d. components ∼ CN(0,E). We let x s (ω) 142 and x r (ω) denote the codewords in X s and in X r , respectively, corresponding to the information message ω∈{1,...,ρ rMT }. Relay decoding: We define the relay outage event at slot m as O m = ( h∈C:|h| 2 ≤ ρ rM m −1 ρ ′ ) (6.15) Differently from the case of arbitrarily large T, the relay may decode in error at time m even though h / ∈ O m . In the presence of such undetected error the relay would switchtotransmitmodeandsendacodewordcorrespondingtoanincorrectinformation message, thusjammingthedestination receiver. Inordertoavoid this eventweconsider a bounded distance relay decoding decision functionψ δ defined as follows (see [23]): for m = 1,...,M −1, define the regions S m (ω) of all points y ∈ C mT for which ω is the unique message that is contained in a sphere of squared radius mT(1+δ)σ 2 v centered at y, i.e., |y−hx m s,0 (ω)| 2 ≤mT(1+δ)σ 2 v . Then, let ψ δ (y m r,0 ,h) =b ω∈{1,...,ρ rMT } if both the following conditions are satisfied: 1. h / ∈O m ; 2. y m r,0 ∈S m (b ω); (the relay has perfect knowledge of its own channel coefficient h, by the perfect CSIR assumption). If these conditions are satisfied, then M = m and the relay switches to transmit mode, sending the signal x M r,m (b ω) for the remaining part of the block. Other- wise, it refrains from making a decision and waits for the next slot. 143 It should be noticed that the condition 2) above is a test on the typicality of the estimated channel noise. In fact, if ω is the transmitted message, we have that |y m r,0 −hx m s,0 (ω)| 2 =|v m 0 | 2 is a central chi-squared random variable with 2mT degrees of freedom and meanmTσ 2 v , that provides an empirical estimate of the noise variance. Destination decoding: The destination is not aware of the relay decision time M. Hence, it makes use of an augmented decoder that simultaneously detects the decision time and the information message according to the GLRT rule: {b ω,b m} = argmax ω,m p y M 0 |ω,m,g 1 ,g 2 . (6.16) where p(y M 0 |ω,m,g 1 ,g 2 ) is the decoder likelihood function, i.e., the pdf of the signal received by the destination over the whole block length, under the hypothesis that the source transmitted the information message ω, that the relay decision time is m, and given the channel coefficients g 1 ,g 2 (recall that we assume perfect CSIR). 144 Error probability analysis: LetE denote the decoding error event at the destination andE r denote the decoding error event at the relay. 3 We can write P(E) = M X m=1 P(M =m)P(E|M =m) = M X m=1 P(M =m) P(E,E r |M =m)+P(E,E r |M =m) ≤ M X m=1 P(M =m) P(E r |M=m)+P(E|E r ,M=m)P(E r |M =m) ≤ M X m=1 P(M =m) P(E r |M=m)+P(E|E r ,M=m) . (6.17) First, we bound the effect of the undetected decision error at the relay. Our analysis follows closely the analysis of the MIMO-ARQ scheme in [23]. In fact, the relay applies a scheme very similar to ARQ: when it is sure about its decision it stops receiving and starts transmitting, while if it is not sure about its decision it waits for the next slot. We have P(E r |M =m) = ρ −rMT ρ rMT X ω=1 P [ b ω6=ω y m r,0 ∈S m (b ω) ω ≤ P |v m 0 | 2 >mT(1+δ)σ 2 v ≤ (1+δ) mT e −mTδ , (6.18) 3 The complement of an eventA is denoted byA. 145 where the last line follows from the Chernoff bound on the tail of the chi-squared distribution. Letting δ =μlogρ, we find P(E r |M=m) ˙ ≤ρ −mTμ . Notice thatP(E|E r ,M =m) ˙ ≥ρ −dm(r) whered m (r) is the exponent of the information outage probability of the m-SC channel given in (6.10), (6.11) and is not larger than 2. Hence, it is sufficient to choose μT > 2 in order to make the terms P(E r |M = m) exponentially irrelevant in (6.17). Next, let us examine the probabilities P(M =m). LetU m = S ρ rMT ω=1 S m (ω) denote thesubsetoftherelaychanneloutputspaceC mT suchthatify m r,0 ∈U m thenthereexists a unique codeword within the bounded distance decoder’s decoding sphere centered at y m r,0 . For m = 1, we have P(M = 1) = P {h / ∈O 1 },{y 1 r,0 ∈U 1 } ≤ P(h / ∈O 1 ) . = ρ −d 1 (r) . (6.19) For brevity we letD m ={h / ∈O m }∩{y m r,0 ∈U m }. Then, for 1<m<M, we have P(M =m) = P D 1 ,...,D m−1 ,D m ≤ P D m−1 ,D m . (6.20) 146 For 1<m<M, from (6.20) we can write P(M=m) ≤ P n {h∈O m−1 }∪{y m−1 r,0 / ∈U m−1 } o ,{h / ∈O m−1 }, y m r,0 / ∈U m ≤ P ({h∈O m−1 },{h / ∈O m })+P {h / ∈O m−1 },{y m−1 r,0 / ∈U m−1 } , (6.21) where the second inequality follows from the fact that for events A,B,C and D, we have using the distributive law and the union bound that P ({A∪B}∩{C∩D}) = P {A∪(B∩A)}∩{C∩D} ≤ P(A∩C)+P(B∩A). Finally, for m =M, we have P(M =M) = P D 1 ,...,D M−1 ≤ P D M−1 = P {h / ∈O M−1 },{y M−1 r,0 / ∈U M−1 } +P (h∈O M−1 ). (6.22) We notice that the event {h ∈ O m−1 }∩{h / ∈ O m } coincides with (6.13) and there- fore the first term in (6.21) decreases as ρ −dm(r) . It is also immediate to see that P (h∈O M−1 ) . = ρ −d M (r) . Hence, we are left with the analysis of the probability P {h / ∈O m },{y m r,0 / ∈U m } (6.23) 147 for all m = 1,...,M−1. Averaging with respect to the random coding ensemble, we may choose without loss of generality ω = 1 as the reference transmitted message. We have U m ⊆ |v m 0 | 2 >mT(1+δ)σ 2 v ∪R m (1), whereR m (1) are the points y m r,0 such that|y m r,0 −hx m s,0 (1)| 2 ≤mT(1+δ)σ 2 v , and there exists some ω6= 1 for which also|y m r,0 −hx m s,0 (ω)| 2 ≤mT(1+δ)σ 2 v . Letting for brevity Δx(ω) =x m s,0 (ω)−x m s,0 (1), we can write R m (1) = [ ω6=1 n |v m 0 −hΔx(ω)| 2 ≤mT(1+δ)σ 2 v , |v m 0 | 2 ≤mT(1+δ)σ 2 v o . Using the union bound and the Chernoff bound we have P {h / ∈O m },{y m r,0 / ∈U m } ≤ P |v m 0 | 2 ≥mT(1+δ)σ 2 v +P ({h / ∈O m },R m (1)) ≤ (1+δ) mT e −mTδ + X ω6=1 P {h / ∈O m }, n |v m 0 | 2 ≤mT(1+δ)σ 2 v o , n |v m 0 −hΔx(ω)| 2 ≤mT(1+δ)σ 2 v o (6.24) Let us consider one term in the sum in the last line of (6.24) for a given message ω and given channel h, averaged over the random coding ensemble. Noticing that for vectors a and b and Γ> 0 we have {|a+b| 2 ≤ Γ,|b| 2 ≤ Γ}⊆{|a| 2 ≤ 4Γ}, 148 we can bound this probability as P n |v m 0 −hΔx(ω)| 2 ≤mT(1+δ)σ 2 v o , n |v m 0 | 2 ≤mT(1+δ)σ 2 v o h (6.25) ≤P |hΔx(ω)| 2 ≤ 4mT(1+δ)σ 2 v h (a) = P ρ ′ |h| 2 χ≤ 2mT(1+δ) h ˙ ≤ρ −mT[1−ν] + , (6.26) where (a) follows from the fact that for the randomly generated codewords, χ = |Δx(ω)| 2 /E is a central chi-squared random variable with mean 2mT and 2mT de- grees of freedom, and the last line follows by letting|h| 2 =ρ −ν and from the fact that the chi-squared cdf satisfies P(χ≤u) =O(u mT ) for small u and P(χ≤u) =O(1) for large u. Summing over the ρ rMT −1 messages ω 6= 1 and integrating with respect to the pdf of|h| 2 over the setO m , we obtain P {h / ∈O m },{y m r,0 / ∈U m } ˙ ≤ Z {ν≥0,[1−ν] + ≥ Mr m } ρ −ν ρ −mT[1−ν] + +rMT dν . = ρ − e dm(r) , (6.27) where, from a standard application of Varadhan’s lemma, we have e d m (r)= inf ν≥0,[1−ν] + ≥ Mr m {ν+mT[1−ν] + −rMT}. (6.28) The domain of ν over which the infimum is calculated is non-empty only for r ≤ m M . This means that the set of channels for which the probability in (6.23) has a polynomial 149 decrease is empty for r > m M and therefore e d m (r) = ∞ for r > m M . For r ≤ m M it is not hard to see that for all T ≥ 1 we have e d m (r) = 1− Mr m . Comparing e d m (r) with d m (r) we see that the former dominates the latter for all r ∈ [0,1]. It follows that for our relay bounded distance decoder and the Gaussian random coding ensemble P(M =m) ˙ ≤ρ −dm(r) . So far we have shown that in the upper bound (6.17) the terms P(E r |M =m) are asymptotically negligible and the terms P(M = m) are upper bounded by the same exponent of the outage probability based, infiniteT, case. It remains to show that the terms P(E|E r ,M =m) have exponent d m (r) given in (6.10), (6.11), and the proof will be complete. We consider the GLRT decoder at the destination. This decoder ignores the knowl- edge of the a priori distribution ofM and treats it as a deterministic unknown param- eter. Hence, we are in the presence of a compound channel formed by the family of m-SC component channels, without any a priori knowledge ofM. Again, without loss of generality we assume message 1 is transmitted. While for the sake of notational simplicity, we omit the explicit conditioning with respect to E r , it is understood that the relay has perfect knowledge of the transmitted information message. We omit also the explicit conditioning with respect to CSIR and denote y M s,0 simply byy since no ambiguity is possible at this point. Hence, the likelihood function 150 p y M 0 |ω,m,g 1 ,g 2 shallbedenoted simplybyp(y|ω,m). Thepairwiseerrorprobability for some ω6= 1 can be upper bounded as follows: P(1→ω|M =m) = P max m ′ p(y|1,m ′ )≤ max m ′ p(y|ω,m ′ ) M=m ≤ P p(y|1,m)≤ max m ′ p(y|ω,m ′ ) M =m = P M [ m ′ =1 p(y|1,m)≤p(y|ω,m ′ ) M =m ! ≤ M X m ′ =1 P p(y|1,m)≤p(y|ω,m ′ ) M=m . (6.29) We shall analyze separately the terms inside the above sum, averaged over the random coding ensemble. Define the event E 1 = p(y|ω,m ′ ) p(y|1,m) ≥ 1 . We first analyze the probability of the event E 1 , which we then use to compute P(E). AssumingM=m, the actual received signal is y m 0 = g 1 x m s,0 (1)+w m 0 y M m = g 1 x M s,m (1)+g 2 x M r,m (1)+w M m . (6.30) We consider the case m ′ ≥ m and leave the case m ′ ≤ m to the reader, since it follows in an almost identical manner. Define the partial codeword differences Δx m s,0 = 151 x m s,0 (1)−x m s,0 (ω),Δx m ′ s,m =x m ′ s,m (1)−x m ′ s,m (ω),Δx M s,m ′ =x M s,m ′ (1)−x M s,m ′ (ω),andΔx M r,m ′ = x M r,m ′ (1)−x M r,m ′ (ω). The error eventE 1 can be written as E 1 = |g 1 | 2 Δx m s,0 2 +2Re n g 1 (w m 0 ) H Δx m s,0 o + g 1 Δx m ′ s,m +g 2 x m ′ r,m (1) 2 + +2Re n (w m ′ m ) H h g 1 Δx m ′ s,m +g 2 x m ′ r,m (1) io + g 1 Δx M s,m ′ +g 2 Δx M r,m ′ 2 + +2Re n (w M m ′) H g 1 Δx M s,m ′ +g 2 Δx M r,m ′ o ≤ 0 o . (6.31) After a little algebra, we obtain the compact expression E 1 = n 2Re{z H w}≤−|z| 2 o , wherez is defined as z, g 1 Δx m s,0 g 1 Δx m ′ s,m +g 2 x m ′ r,m g 1 Δ M s,m ′ +g 2 Δx M r,m ′ For given codebooksX s ,X r , the variance of 2Re{z H w} is equal to 2|z| 2 σ 2 v , which leads to P(E 1 |X s ,X r ,M =m,g 1 ,g 2 )≤Q |z| p 2σ 2 v ! ≤e −|z| 2 /(4σ 2 v ) . Define the following notation, ξ i = [x s,i (1) x s,i (ω) x r,i (1) x r,i (ω)] T , 1≤i≤MT, 152 and ξ, [ξ T 1 ξ T 2 ···ξ T MT ] T ∈C 4MT×1 . It can be verified that|z| 2 =ξ H Mξ, for a block diagonal M of the form M = M 1 . . . M MT , where for 1≤k≤mT, M k =|g 1 | 2 1 −1 0 0 −1 1 0 0 0 0 0 0 0 0 0 0 , for mT +1≤k≤m ′ T, M k = |g 1 | 2 −|g 1 | 2 g 2 g ∗ 1 0 −|g 1 | 2 |g 1 | 2 −g 2 g ∗ 1 0 g 1 g ∗ 2 −g 1 g ∗ 2 |g 2 | 2 0 0 0 0 0 , 153 and for m ′ T +1≤k≤MT, M k = |g 1 | 2 −|g 1 | 2 g 2 g ∗ 1 −g 2 g ∗ 1 −|g 1 | 2 |g 1 | 2 −g 2 g ∗ 1 g 2 g ∗ 1 g 1 g ∗ 2 −g 1 g ∗ 2 |g 2 | 2 −|g 2 | 2 −g 1 g ∗ 2 g 1 g ∗ 2 −|g 2 | 2 |g 2 | 2 . It turns out that the matricesM k have rank 1, for all 1≤k≤MT. It follows that the eigenvalues of each M k are tr(M k ),0,0,0. We now average P(E 1 |X s ,X r ,m,g 1 ,g 2 ) over the ensemble of random Gaussian codebooks. In order to do so, we use the following well-known resultonthecharacteristic function of Hermitian quadratic formof complex Gaussian random variables (briefly, HQF-GRV). Lemma 6.3.3. [78, Appendix 4] The characteristic function of the HQF-GRV Δ = z H Fz, where z∼CN(z,R) is given by Φ Δ (s) =E[exp(−sΔ)]= exp(−sz H F(I+sRF) −1 z) det(I+sRF) . Therefore, P(E 1 |m,g 1 ,g 2 ) ≤ E Xs,Xr h e −|z| 2 /(4σ 2 v ) i = Φ |z| 2 1 4σ 2 v = 1 det(I+ ρ 4 M) . 154 Explicitly, we have 1 det(I+ ρ 4 M) = 1 1+ ρ 2 |g 1 | 2 mT · 1 1+ ρ 4 (2|g 1 | 2 +|g 2 | 2 ) (m ′ −m)T · 1 1+ ρ 2 (|g 1 | 2 +|g 2 | 2 ) (M−m ′ )T . = 1 [1+ρ|g 1 | 2 ] mT · 1 [1+ρ(|g 1 | 2 +|g 2 | 2 )] (M−m)T . (6.32) We notice that (6.32) does not depend onm ′ , at least in the exponential equality sense. Summing over all m ′ = 1,...,M and over all messagesω6= 1, we eventually can bound the average probability of error of the GLRT decoder conditioned on M = m and on g 1 ,g 2 as P(E|E r ,M =m,g 1 ,g 2 ) ≤ X ω6=1 P(1→ω|M =m,g 1 ,g 2 ) ≤ X ω6=1 M X m ′ =1 P(E 1 |m,g 1 ,g 2 ) ˙ ≤ Mρ rMT [1+ρ|g 1 | 2 ] mT · 1 [1+ρ(|g 1 | 2 +|g 2 | 2 )] (M−m)T . (6.33) Next,weshallevaluatethediversityexponentofP(E|E r ,M =m)=E g 1 ,g 2 [P(E|E r ,M = m,g 1 ,g 2 )]. In order to do so, we separate the outage event from the no-outage event. Define the outage event of the m-SC as A m = (M−m)max{[1−α 1 ] + ,[1−α 2 ] + }+m[1−α 1 ] + −rM ≤ 0 . (6.34) 155 Then, P(E|E r ,M =m) = P(E,A m |E r ,M =m)+P(E,A m |E r ,M =m) ≤ P(A m )+P(E,A m |E r ,M =m). (6.35) Recall that P(A m ) =P m−SC out (r) . =ρ −dm(r) , where d m (r) is evaluated in (6.10), (6.11). In order to evaluate P(E,A m |E r ,M = m), we use (6.33) and write the exponential inequality P(E|E r ,M =m,g 1 ,g 2 ) ˙ ≤ρ −Tgm(α 1 ,α 2 ,r) , where g m (α 1 ,α 2 ,r) = (M−m)max{[1−α 1 ] + ,[1−α 2 ] + }+m[1−α 1 ] + −rM. (6.36) Therefore, using again Varadhan’s lemma, we obtain P(E,A m |E r ,M =m) . = ρ −d G,m (r) , 156 where d G,m (r)= inf g m (α 1 ,α 2 ,r)> 0 α 1 ,α 2 ≥ 0 {α 1 +α 2 +Tg m (α 1 ,α 2 ,r)}. (6.37) The above infimum is achieved wheng m (α 1 ,α 2 ,r)↓ 0, yielding d G,m (r)=d m (r). This concludes the proof of Theorem 6.3.2. Remark. The proof of Theorem 6.3.2 is not only conceptually appealing, but also reveals a few very important and often neglected features that should be taken into account in the design of a DDF scheme. First, the proof sheds light on the fact that the relay must make its decision based not only on the outage condition, but also on the reliability of the decoding decision. Then, it shows also that despite the fact that the destination does not know the relay decision time, there is no need for an explicit protocol that provides this side information. In Appendix E, we analyze a simpler decoder, nicknamed relay activity detector, based on separated detection of the relay decision time by treating the codewords as random Gaussian signals (i.e., ignoring the structure of the code). We show that such a simple “energy detector” is optimal if we let T →∞ first, and then consider the high SNR performance, but it is dramatically suboptimal if we do the limits in the reverse order. In fact, for any finite T, the relay activity detector yields a constant error probability, that does not vanish with SNR. 157 6.3.3 Computing the DMT and comparisons Obtainingaclosed-formsolution totheDMT expressioninTheorem6.3.1 appearstobe intractable. We plot in Fig. 6.4 values of d ∗ M (r) for M = 2,5,10 and 20 in comparison with the optimal DMT of the DDF protocol (corresponding to M =∞). 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 r d * M (r) M = 2 M = 5 M = 10 M = 20 Optimal DDF (M = ∞) Figure 6.4: The DMT of the DDF channel with finitely many decoding decision times. With increasingM,d ∗ M (r)is seen toapproach theoptimal tradeoff veryrapidly. For practical code design, even a relatively small value of M is therefore expected to have close to optimal performance in terms of diversity. 158 Remark. The authors in [66] consider a related problem, where T → ∞ and the relay is restricted to a finite number of decision times (say N). These time instants coincide with the end of blocks {M j } N j=1 , with 1≤ M 1 <··· < M N < M ∀ j (notice: with this notation, in our case we would have N = M and M j = j). Further, define M 0 , 0, M N+1 ,M, and a set of “waiting fractions”{f j } N+1 j=0 by f j , M j M . Thus f 0 = 0<f 1 <···<f N <f N+1 = 1. In [66], it is proved that for any fixedN no set of waiting fractions yields a DMT curve that dominates all others. Then, a particular set of waiting fractions are chosen that yield for any fixed N a DMT curve that is not uniformly dominated 4 by any other protocol with the same number of decision timesN. The resulting DMT is derived and it is summarized by the following lemma from [66]. Lemma 6.3.4. [66] For the DDF protocol with a given numberN of decision times, let f p 1 = 1 2 and f p j = 1−f p j−1 2− 1+ 1 f p N f p j−1 , for 1<j≤N, then no set of fractions uniformly dominates{f p j } N j=1 . Further, the DMT corresponding to the set of fractions {f p j } N j=1 is given by d p (r) = 1−r+ 1− r f p N + . (6.38) 4 According to the definition in [66], protocol A uniformly dominates protocol B if, for any multiplex- ing gain r, dA(r)≥dB(r). A protocol that is not uniformly dominated by any other protocol is said to be Pareto-optimal. 159 A few interesting observations can be made about this result. As it is remarked in [66], the DMT obtained through {f p j } is not asymptotically optimal, i.e., it does not converge to the optimal DMT of the DDF protocol as N → ∞. Indeed, it is evident from (6.38) thatd p (r) consists of two straight line segments, sayL 1 for 0≤r≤f p N and L 2 forf p N ≤r≤ 1. AsN →∞,L 1 andL 2 can at best be tangential to the curved part oftheDMToftheDDFprotocol(i.e., the0.5≤r≤ 1region)in(6.4). Inparticular, the optimal value of the DMT of the DDF protocold ∗ (0.5) = 1 is never approached even in thelimitbyd p (r). Incontrast, theDMTd ∗ M (r)derivedinthischapter isasymptotically optimal. As the number of decoding points increases, d ∗ M (r) dominates over d p (r) for almost all values of r, and is strictly less for only an exceedingly small range of values ofr. Asymptotically, it is clear that the only set of points whered p (r) dominatesd ∗ M (r) is a very small set of points around the point whered p (r) is tangent to the curved part of the DMT of the DDF protocol. 6.4 DMT optimal codes for the single relay DDF channel The authors in [2] used the ensemble of random Gaussian codes of asymptotically large block-lengths to show the achievability of the DMT of the DDF protocol. Subsequently, aconstructionofcodesderivedfromcyclicdivisionalgebras(CDA)wasshowntoachieve the DMT of the DDF channel for arbitrary number of relays [26]; i.e., they achieve the corresponding tradeoff for a particular number of decoding instants. As we increase the block-length and the number of decoding instants, the DMT of these codes tends towards the optimal DMT of the DDF protocol given in (6.4). In a recent submission, 160 [27], the authors present a division-algebraic construction based on the Alamouti code that is similar in flavor to the construction to be presented in this chapter. However, for the codes in [27], the parameterT is fixed to 2; on the contrary, we will see that our code construction is valid for arbitrary values of T including the special case of T = 1, and is hence a minimum delay construction. Decoding these codes involves sphere or sequential decoding [18, 67] over a large dimensional lattice. It is hence of interest to construct codes that achieve the DMT of the DDF protocol and permit low complexity decoding. Since our construction is of minimum delay, the dimensionality of the lattice to be sphere decoded is half that of the corresponding case in [27]. In order to completely specify a signalling scheme (X s ,φ,X r ) for the DDF channel, we need to define the following: 1. A codeX s that is used by the source. 2. A causal decoding decision function φ(·,·) : (C,C MT )→{1,2,...,M}, that dic- tates when the relay attempts to decode the source’s transmission based on the S-R channel gain h and the signal y r received at the relay. In particular, if φ(h,y r )=M, the relay will not attempt to decode the transmission of the source. Ifφ(h,y r )=m, 1≤m<M, then the relay attempts to decode the transmission of the source upon completion of the m th block. Because of the causality con- straint, we assume that the output of φ at time m depends only on h (CSIR of the relay) and on y m r,0 . 161 3. A codeX r used by the relay. In the following, we will only consider the case that the relay implements the Alamouti-DDF scheme [66] given in (6.6); hence X r is the same asX s upto coordinate permutations, sign change and conjugation. 6.4.1 Design tradeoffs Despite the importance of the decoder at the destination, as evidenced in the proof of Theorem2, inthissection wetake ashortcutandwedonottreatthetheGLRTdecoder explicitly. For the sake of simplicity, our simulations assume a genie-aided destination, ideally informed of the relay decision time. The main focus of this section is on the design of the codebook X s and on the efficient implementation of the relay decoding decision function, in order to trigger the relay transmission only when decisions are reliable. Choosing a good relay decoding decision decision function φ is critical to ensure good performance: a conservative φ that makes the relay wait for too long before decoding results in low relay error probabilityP(E r ), butincreases the destination error probability P(E,E r ) since the relay has less time to help the destination. Vice-versa, a φ that is too aggressive and makes the relay decode too early yields low P(E,E r ) but results in a large P(E r ), since the blocklength of the signal observed at the relay is too shortto copewithatypical noise. Weshallalso see throughsimulations that undetected decoding errors at the relay have a huge impact on performance, since the relay ends up jamming the destination with high probability. We will present our choices of X s and φ in the following two subsections. 162 6.4.2 Approximately universal X s The equivalent channel resulting from the use of the Alamouti scheme for the relay code is a parallel channel (6.7) with statistically dependent fading coefficients. We will choose X s to be a code of length MT that is approximately universal over the parallel fading channel. A code that is approximately universal over the parallel channel (a notion introduced in [91]) meets the DMT over any parallel channel. Such a code has an error probability that decays exponentially withρ for all parallel channel gains such that the corresponding mutual information is larger than the coding rate, i.e., for all channel gains in the no-outage region. Therefore, such an approximately universal code X s meets the DMT of the relay DDF channel for any M. This means that, for any fixed rateR and sufficiently large SNR, the decay of error probability with SNR of our code (with finite T) exhibits the same slope of outage probability. However, the “gap from outage” (i.e., the horizontal distance in dB between the outage probability and the actual probability of error) is not captured by the DMT optimality and in practice it may be very large, thus making a DMT-optimal scheme totally useless for practical purposes. We shall discuss ways to close this gap in the next section, by an appropriate choice of the relay decision function φ. We may obtain approximately universal X s from either suitable algebraic lattices [3, 105] or from permutation codes through UDMs [91, 34]. In the following we briefly review these constructions. 163 Rotated QAM codes from algebraic lattices LetL be anMT-dimensional extension ofQ(ı) and let the Galois groupGal(L|Q(ı)) = {σ 1 ,...,σ MT }. Denote the ring of integers ofL asO L and letI be an ideal ofO L . Let N L|Q(ı) (·) denote the algebraic norm fromL toQ(ı). We define the codeX s as follows: X s = σ 1 (ℓ) σ 2 (ℓ) . . . σ MT (ℓ) ℓ∈S , whereS is some finite subset ofI. X s has the desirable property of a “non-vanishing” product distance, since we have for each x∈X s that MT Y j=1 |x j | = MT Y j=1 σ j (ℓ) = N L|Q(ı) (ℓ) ≥ 1, since the norm N L|Q(ı) (·) of an algebraic integer in L is an element of Z[ı]. This non- vanishingproductdistancepropertyensuresthatX s is approximately universalover the parallel channel [91, 27]. It can be verified thatX s can equivalently be rewritten as a lattice code, i.e., X s ={Gb|b∈B}, (6.39) 164 for suitable G ∈ C MT×MT and B ⊂ Z[ı] MT . A particular choice of G and B that is good in terms of shaping consists of constructing G to be unitary and B to be a set of points in Z[ı] MT contained in a hypercube that is centered around the origin 5 . For the algebraic details regarding the construction of such unitary G, see [3, 105]. Notice also that choosing the information setB to be a bounded subset ofZ[ı] MT corresponds, in practice, to choosing information symbols from a QAM alphabet, which is appealing for practical implementation. The rate ofX s in this case is R = log|B| MT bpcu. Parameters for simulations: In the simulations to follow in Sec. 7.5, we construct the matrix G using the cyclotomic construction given in [3]. For M = 4 and T = 1, G is a complex 4×4 matrix, or equivalently a real 8×8 matrix. We chooseB to be a cartesian product of Q 2 -QAM alphabets, B ={a+ıb|−Q+1≤a,b≤Q−1, a,b odd} MT , for some even integer Q. Thus |B| = Q 2MT . For example, by choosing Q = 4 with M = 4 and T = 1 we obtain a rate of R = 4 bpcu. 5 Notice howeverthatchoosingGunitaryisoptimal onlywhenweareconstrained tousea linearmap to encode the information vector onto the code symbols. An alternate approach is to use a constellation carved out of a dense lattice inR n and employ a non-linear sphere encoder and a mod-Λ MMSE-GDFE lattice decoder; as we saw in Chapter 3, this approach yields significant performance improvements over unitary shaping. However, for simplicity of exposition, we will restrict our attention tothe case of linear encoding in this chapter. 165 Permutation codes from UDM Approximately universal code construction from UDM were introduced in [91] and a generalalgebraicconstructionvalidforanynumberofsub-channelswasprovidedin[34]. Definition 1. [34] Let n and L be some positive integers and let q be a prime power. The L matrices A 0 ,...,A L−1 over F q of size n×n are (L,n,q)-UDMs if for every (k 0 ,...,k L−1 ) such that 0≤k ℓ ≤n ∀ ℓ, P L−1 ℓ=0 k ℓ ≥n, the ( P L−1 ℓ=0 k ℓ )×n matrix com- posed of the firstk 0 rows ofA 0 , the firstk 1 rows ofA 1 , ..., the firstk L−1 rows ofA L−1 has full rank. ♦ The authors in [34] provide an algebraic construction of such (L,n,q)-UDMs for any L≤ q +1. It is shown in [91] that an approximately universal permutation code for the parallel channel with L-branches can be obtained from (L,n,q)-UDMs, in the following manner. Assume that we have to transmit 2n information symbols from F q . We encode independently n-symbols each onto the I and Q sub-channels. Let u∈F n q denote the firstn input information symbols. Map the sequence ofF n q symbols {A 1 u,A 2 u,...,A L u} componentwise onto a L-length vector of q n -PAM symbols, and transmit the components on the I sub-channel. The next n information symbols are 166 similarlyencodedandtransmittedontheQsub-channel. Therateofsuchapermutation code is R = 2nlogq L bpcu. In our case, we set L=MT to obtain codes for the DDF channel. Parameters for simulations: ThesimulationsinvolvingpermutationcodesinSec.7.5 for M = 4, T = 1 are derived from (4,4,4)-UDMs, leading to R = 4 bpcu. In order to completely specify the code, we need to provide the mapping to PAM symbols that was used. We construct the Galois fieldF 4 using the primitive polynomialX 2 +X+1. Thus any element inF 4 may be associated with a polynomialb 1 X+b 0 , where theb i are either 0 or 1, and X is a primitive element. Hence we may also associate each element inF 4 with the binary string b 1 b 0 . In order to map an F 4 4 vector v (which is one of the A j u considered previously) to the PAM alphabet, first concatenate the binary strings correspondingtov i ∈F 4 ,i = 1,2,3,4 toobtainan8-lengthbinaryvectorb. Thisbinary vector is mapped to the centered 256-PAM alphabet by computing 2 P 7 i=0 b i 2 i −255. 6.4.3 Decoding decision function φ and Forney’s decision rule A first choice for φ, which we shall denoteφ 1 , would be to allow the relay to decode as soon as the mutual information between the source and the relay exceeds MTR, i.e., φ 1 (h) = min M, MR log(1+|h| 2 ρ ′ ) , 167 where ρ ′ is the SNR of the source-relay link. This rule is asymptotically optimal for largeT, infactitcoincides withtheruleintheoriginal formulation oftheDDFprotocol (6.5). For finite T, φ 1 is suboptimal since it ignores the actual signal received by the relay, i.e., the atypical behavior of the noise may dominate the error probability for short block lengths. As an illustration of the inefficacy of this decision function at finite block-length, consider the simulation results in Fig. 6.5. In the simulations to follow, we choose X s to be a rotated QAM code. We will subsequently compare these results with those obtained by choosingX s to be a permutation code, and observe very similar trends. We consider ML decoding at both the relay and destination for all the simulations in this sub-section. Further, we assume that the source-relay link SNR ρ ′ is 3 dB above the SNR ρ of all other links in all our simulations (the X-axis on all our plots is the SNR ρ in dBs). The simulations in Fig. 6.5 are for the case when X s is a rotated QAM code, T = 1, M = 4 and R = 4 bits per channel use (bpcu). The seemingly strange non-monotonic behavior of the error probability can be understood by the following intuitive explanation. At low SNRs, the relay hardly ever triggers before m = 4, re- sulting in P(E) being dominated by the error probability at the destination P(E,E r ), and hence P(E) is large and decreasing. Then, there is an intermediate region of SNR where the relay attempts to decode, but it decodes incorrectly with high probability and causes significant interference at the destination. Thus P(E) is dominated by the relay error probability P(E r ), and increases in this region. For sufficiently large SNR, the relay decodes correctly with high probability and therefore helps the destination, 168 thus providing the required cooperative diversity (slope of the overall error curve at high SNR). However, this happens at very large gap from the outage probability, that can be regarded as a de-facto optimal performance also for finite-length codes and not asymptotically high SNR. This simulation reveals a phenomenon that has been scantily treated in previous works: the effect of decoding errors at the relay clearly dominates the overall performance. This fact has often been neglected since it is neither captured by the T →∞ case, where the atypicality of the noise has no effect and triggering the relay based on the outage event is exact, nor by the DMT formulation, that does not capture the gap from outage, but just the asymptotic error probability curve slope. One immediate remedy consists of adopting a conservative relay decoding decision function, which we will denote as φ 2 , defined as φ 2 = min M, MR log(1+|h| 2 ρ ′ ) +1 . Simulation results using this strategy are shown in Fig. 6.6, once again for the case of X s being a rotated QAM code, T = 1, M = 4 andR = 4 bpcu. In this case, the relay error probability is so low that no errors were recorded in our Monte Carlo simulation (no such curve is shown in Fig. 6.6). The downside of this strategy however is that since the relay is over-conservative, it helps the transmitter too late, and the overall error probabilityP(E,E r ) suffers from significant degradation with respect to outage probability. 169 In [66], the authors prove that there is no loss in DMT for the DDF protocol using the Alamouti type relay by using the following relay decoder function φ 3 : φ 3 = min M,max M 2 , MR log(1+|h| 2 ρ ′ ) , i.e., the relay is allowed to decode and transmit only after the codeword from the source is at least half-way through. Simulation results of this protocol shown in Fig. 6.7 reveal that this scheme also suffers from a significant penalty at high SNRs due to the P(E r ) term dominating the overall error probability. In [66], the authors use an extra layer of cyclic-redundancy check (CRC) coding to enable the relay to perform error detection (and wait for another round if incorrect decoding is detected) - while this strategy is effective in reducing P(E r ), there is an inherent loss of rate. In fact, it is shown in [23] for the MIMO-ARQ channel that CRC is suboptimal in terms of DMT since the undetected error probability must decrease with SNR at least with the same exponent of error probability itself, and this requires a number of CRC bits that grow linearly with logSNR. The same consideration applies here. Hence, we wish to avoid the use of CRC in order to detect if the relay decodes in error. We present a novel strategy to enable error detection at the relay without further layers of coding at the transmitter. We make use of a criterion introduced by Forney in [31] in the context of retransmission (ARQ) protocols to decide whether the decoder is in error or accept the decoding outcome. Here, we apply this criterion to the relay 170 decoder, that we refer to as Forney’s decision rule. Interestingly, Forney’s decision rule is similar in essence to the bounded distance decoder that we have considered in the proofofTheorem6.3.2. However, whiletheboundeddistancedecoderiseasyto analyze but only asymptotically optimal, Forney’s decision rule has the remarkable property of striking an optimal balance between the probability of undetected error at the relay and the probability of rejecting the decision and waiting for the next slot (probability of decision “erasure”, in the language of [31]). To the best of our knowledge, this decoding decisionrulewasnotproposedbeforeinthecontextofrelaycooperativecommunication. We define the decoding decision function φ F (h,y r ) using Forney’s decision rule as follows: 1. If φ 1 (h) =M, don’t decode and setφ F (h,y r ) =M (worthless trying to decode if we are in outage). 2. If φ 1 (h) =m<M, decode after the m th block and apply the following threshold test. Let b ω denote the outcome of the relay decoder. Accept the decision and trigger the transmission mode if p(y m r,0 |b ω,h) P ω6=b ω p(y m r,0 |ω,h) ≥τ, (6.40) where τ a suitable threshold set empirically for each SNR. If the threshold is not exceeded, wait for the next block and repeat this step until either the threshold is exceeded or m =M. 171 φ F is found to be extremely effective in suppressing the error probability at the relay without being too conservative and refraining from helping the destination when pos- sible. Simulation results for the case whenX s is a rotated QAM code, T = 1, M = 4 and R = 4 bpcu are shown in Fig. 6.7. The error probability is within 1 dB from the corresponding outage probability. Fig. 6.9 shows the results when we choose X s to be a permutation code, with T = 1, M = 4 and R = 4 bpcu. In this figure we considered two cases: the case of a genie aided relay, where a genie provides the relay with the source message as soon as the mutual information exceeds MTR, and the case where the relay performs minimum distance decoding in conjunction with Forney’s rule. The results are similar in flavour to the case where X s is a rotated QAM code, with the permutation code losing 1 dB with respect to the rotated QAM code. Notice that despite the DMT optimality, these codes may performdifferently dependingon their shapingandcoding gain. Inthis case, it is apparent that the rotated QAM code outperforms the permutation code, although they achieve the same diversity. 6.4.4 Low complexity MMSE-GDFE Lattice Decoding As we saw in (6.39), the choice of rotated QAM codes makes X s a lattice code. Let Λ be the 2MT-dimensional lattice corresponding to the generator matrix G in (6.39). MMSE-GDFE lattice decoding has been shown to be DMT optimal for the class of lattice space-time (LaST) codes over MIMO channels [24], and has also been shown to perform well for deterministic structured LaST (S-LaST) codes (see Chapter 3). Let 172 V(Λ) denote the fundamental Voronoi cell of an n-dimensional lattice Λ (See [16] for definitions relating to lattice theory). The lattice quantization function is defined by Q Λ (y), argmin λ∈Λ |y−λ| and the modulo-lattice function is given by y mod Λ,y−Q Λ (y). Inthesequel,wewillworkwiththerealchannelmodelwhichisequivalentto(6.1),(6.2) and(6.3),obtainedbywritingsignalsexplicitlyintermsoftheirrealandimaginaryparts (see for example [92] for details regarding the equivalence between real and complex channel models). By slight abuse of notation, we will refer to the real equivalent of the complex vectors and matrices x s ,y,y r ,Λ,G using the same notation. In order to use reduced complexity MMSE-GDFE lattice decoding [24], information needs to be encoded onto cosets of a sublattice Λ s of Λ, as follows. Choose Λ s = QΛ, where Q ∈ Z + . Thus |Λ/Λ s | = Q 2MT . Let C denote the set of points {Gz | z ∈ Z 2MT Q }, where Z Q , {0,1,...,Q−1}. The transmitter selects a codeword c∈C, generates a pseudo-random dither signal u with uniform distribution over V(Λ s ), and obtains the transmitted codeword x s = [c−u] mod Λ s . 173 Thus information is encoded onto the cosets of the partition Λ/Λ s : x s is a coset rep- resentative of the coset onto which the information is encoded, and belongs to the fundamental Voronoi region of Λ s . Let C ω denote the coset of Λ s in Λ onto which the information corresponding to message ω is encoded. From (6.1), (6.2) and (6.3) the (real equivalent) received signalsy r =y m r,0 forM=m at the relay andy s =y M s,0 at the destination may be written as y r =H r x s +v (6.41) and y s =Hx s +w, (6.42) where H r ∈ C 2mT×2MT and H∈ C 2MT×2MT denote the (real) equivalent channels at the relay and destination, and v and w denote the (real equivalent) noise at the relay and destination respectively. Notice from (6.41) that decoding at the relay corresponds to solving an under-determined system of linear equations. We follow the approach of [20] in this case, where it was shown how MMSE-GDFE lattice decoding may be used to efficiently solve under-determined systems of linear equations. We focus on decoding at the relay in the sequel, the decoder at the destination is identical upon replacing the relevant signals and parameters at the relay with those at the destination. LetF andB denote the forward and backward filters of the MMSE-GDFE (see for example [24] for the definition of these matrices in terms of H r and the relay SNR). The relay produces the modified observation y ′ r ,Fy r +Bu, 174 and computes b z= arg min z∈Z 2MT |y ′ r −BGz| 2 . The relay then decides in favor of the cosetC b ω that contains the point [Gb z] mod Λ s . In order to work with the lattice coding and decoding scheme, Forney’s decision rule (6.40)needstobemodifiedtotakeintoaccountthefactthatinformationisencodedonto cosets as against points in the lattice. Encoding information into cosets is equivalent to consider a modulo-Λ s channel with output y ′ r modulo BΛ s . Hence, the relevant likelihood function is given by e p(y ′ r |ω,h) = X λs∈Λs p e w (y ′ r −B(c ω +λ s )|h) with domainy ′ r ∈V(BΛ s ), wherec ω is a coset representative ofC ω and wherep e w (w|h) denotes the pdf of the noise induced by the modulo-Λ s channel with the dithering, that is, e w =y ′ r −Bx s wherex s is the transmitted signal. Unfortunately,p e w is difficult if not impossible to determine in closed form. However, a good practical choice that works well for good shaping lattices is to let p e w be a Gaussian pdf with i.i.d. components ∼N(0,σ 2 v /2) (see [24] for a theoretical asymptotic justification of Gaussianity in this 175 context). Then,theproposedmodificationof Forney’sdecision rule(6.40)forthelattice MMSE-GDFE decoder is given by: accept b ω at time m if P λs∈Λs p e w (y ′ r −B(c b ω +λ s )|h) P ω6=b ω P λs∈Λs p e w (y ′ r −B(c ω +λ s )|h) ≥τ, (6.43) where, againτ is a suitable threshold set empirically for each SNR. The infinitesums at numeration anddenominator can besafely truncatedbyrestricting toanumberof most likely lattice points, which may be done as follows. Generate a list ofN ={Bλ i :λ i ∈ Λ} N i=1 of lattice points of the lattice generated byBG that are closest toy ′ r . Such a list may be generated, for example, by using a standard lattice decoder with a sufficiently large search radius. For any given message ω, check whether λ i belongs to the coset c e ω +Λ s . If yes, then this point makes a contribution towards the numerator of (6.43), elsetowards thedenominator. Ifthereexistsb ω forwhichthecorrespondingratio crosses the thresholdτ then accept the decision, otherwise reject and wait for the next slot. The modified Forney’s rule in (6.43) is seen to be quite effective for the case when MMSE-GDFE lattice decoding is performed at both the relay and the destination. The simulations in Fig. 6.10 compares the performance of rotated QAM codes with T = 1, M = 4, R = 4 bpcu that use Forney’s rule (6.40) with ML decoding and modified Forney’s rule (6.43) with MMSE-GDFE lattice decoding. The low complexity lattice decoder tracks the ML performance within 1 dB. 176 6.5 Conclusion Wepresentedacharacterization oftheachievableDMTofthesingle-relayDDFprotocol for finite block length. Our achievability proof yields insight on the design of actual coding schemes. In particular, we stressed the importance of a relay decoding function that checks the reliability of its decision, in order not to jam the destination. Also, we showed thatthedestination neednotbeaware oftherelay decision time, sincea GLRT- based decoder achieves optimal DMT performance. This may have some impact on the design of practical DDF protocols, since it essentially shows that no complicated side- informationchannelneedstobeimplementedinordertoexplicitlynotifythedestination about when the relay starts transmitting. Inourproofs,weconsideredaboundeddistancedecoderattherelayandanensemble of random Gaussian codes. Then, we constructed practical and very simple codes based on lattices (rotated QAM constellations) and permutation codes. We demonstrated via simulation that the impact of undetected decoding errors at the relay may be huge. In order to tackle this problem, we have proposed the use of Forney’s decision rejection rule, that proves to be very effective. Finally, we have investigated the use of a reduced complexity MMSE-GDFE lattice decoderandmodulo-Λlatticecodes,thatyieldsthewell-knownlow-complexitydecoding even at the relay. It should be remarked that the relay invariably has to decode an undetermined linear system, therefore standard sphere decoding algorithms fail. A few comments relating to future work are in order here. We have not exploited the low-complexity quantization-based decoding approach for permutation codes [91], 177 owing to the fact that it is not completely clear as to how we can apply a good decision rejection rule at the relay in this case. Another interesting problem relates to code design for finite but large T; in this case, neither the constructions presented in this chapter, nor those in [26, 27] are fully controllable in terms of coding gain and both entail very high decoding complexity. Concatenation of short codes based on rotated QAM constellations or permutation codes with some form of outer coding (along the lines of [36]) may prove to be appropriate for this scenario. 178 0 10 20 30 40 50 60 70 80 90 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage Pr(Destination decodes in error,Relay decodes correct) Pr(Relay decodes in error,Relay attempts to decode) Total error probability Figure 6.5: X s is a rotated QAM code, T = 1, M = 4, R = 4 bpcu, relay implements φ 1 (·). 179 0 5 10 15 20 25 30 35 40 45 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage Pr(Destination decodes in error,Relay decodes correct) Pr(Relay decodes in error,Relay attempts to decode) Total error probability Figure 6.6: X s is a rotated QAM code, T = 1, M = 4, R = 4 bpcu, relay implements φ 2 (·). 180 0 10 20 30 40 50 60 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage Pr(Destination decodes in error,Relay decodes correct) Pr(Relay decodes in error,Relay attempts to decode) Total error probability Figure 6.7: X s is a rotated QAM code, T = 1, M = 4, R = 4 bpcu, relay implements φ 3 (·). 181 0 5 10 15 20 25 30 35 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage P(Destination decodes in error,Relay decodes correct) P(Relay decodes in error,Relay attempts to decode) Total error probability Figure 6.8: X s is a rotated QAM code, T = 1, M = 4, R = 4 bpcu, relay implements φ F (·). 182 0 5 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage Total error probability, Genie aided relay Total error probability, relay implements Forneys test Figure 6.9: X s is a permutation code, T = 1, M = 4, R = 4 bpcu, relay implements φ F (·). 183 0 5 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Outage ML relay and destination MMSE−GDFE lattice decoder relay and destination Figure 6.10: X s is a rotated QAM code, T = 1, M = 4, R = 4 bpcu, relay implements Forney’s or modified Forney’s rule. 184 Chapter 7 Channel State Feedback over the MIMO-MAC Consider a frequency division duplex (FDD) cellular system with sufficient frequency spacing between the uplink and downlink channels, such that the uplink and down- link channel coefficients are independent. A base station (BS) withM antennas wishes to serve K user terminals (UTs), with N t antennas each, using some MIMO broad- cast channel precoding technique, such as linear beamforming, Dirty-Paper Coding or some low-complexity non-linear precoding approximation thereof. Essential to all these techniques is the availability of accurate channel state information at the transmitter (CSIT), that is, the BS must know the user downlink channels. We assume that the UTs have perfect knowledge of their downlink channels, e.g., obtained through down- link training. Then, at each time slot, they send this information to the BS on the uplink. We can model this CSIT feedback as signaling over a MIMO multiple-access 185 channel (MAC). It has been shown in [52, 12] that the relevant performance measure that dominates the downlink rate gap 1 is the mean-square distortion at which the BS is able to represent the UTs channel coefficients (this will be made more precise in the sequel). It follows that the CSIT feedback problem consists of lossy transmission under an end-to-end mean-square distortion constraint of a set of K Gaussian sources (i.e., the channel coefficients) over a MIMO-MAC channel affected by block fading. It should also be noticed that, apart from some “analog” feedback schemes where the channel coefficients are sent unquantized and in parallel by all users (see [52, 12] and references therein), very little attention has been devoted to properly design the CSIT feedback scheme exploiting the MIMO-MAC nature of the uplink channel: most works assume perfect feedback at fixed rate, or implicitly assumethat the feedback information is pig- gybacked “somehow” into the uplink transmissions. This may pose problems, since the CSITfeedback musthave extremely low latency, therefore, its coding block length must be very short. In this chapter we consider the problem of designing very low latency and low complexity CSIT feedback schemes for the uplink MIMO-MAC. 7.1 Channel Model and Problem Statement Thecomplex-basebandequivalentofT channelusesofsuchachannelcanberepresented as Y = √ ρ K X i=1 H i X i +W, (7.1) 1 The gap between the rate achievable with imperfect CSIT from the optimal rate achievable with perfect CSIT and the optimal MIMO broadcast channel coding strategy. 186 whereYdenotestheM×T signalreceivedattheBS,X i istheN t ×T signaltransmitted bythei th UT,H i isthe(uplink)channelbetweenthei th UTandtheBSthatisassumed to remain constant over T channel uses, and W is the additive noise. We assume that the entries ofH i andW are distributed as i.i.d. complex Gaussian with zero mean and unit variance CN(0,1). Further, we assume that the channels remain constant for a block of T channel uses. We impose an average per-user power constraint E h kX i k 2 F i ≤T ∀ i, wherek·k F denotes the Frobenius norm. Hence ρ takes on the meaning of the uplink transmit SNR. We consider that each UT needs to transmitS samples of a complex i.i.d. Gaussian source 2 overT uses of the MIMO-MAC. We defineb,T/S as the bandwidthefficiency. Associated with a particular b, we consider a family of coding schemesC(ρ) indexed by the SNRρ. Correspondingto the coding schemeC(ρ), we defineD(ρ) to be the average (squared error) distortion involved in reproducing the source, averaged over the source, channel and the noise. The distortion SNR exponent of the family is given by [13, 37, 6] δ(b) =− lim ρ→∞ logD(ρ) logρ . Thedistortion SNRexponentof the channelδ ∗ (b) is given bythesupremumofδ(b) over all possible coding schemes. 2 The source length S corresponds to the overall number of channel coefficients that need to be fed back at any time slot. 187 ThekeyquantitythatdominatestherategapintheMIMOdownlinkwithimperfect CSIT is the product between the downlink transmit power and the channel estimation distortion. Up to constants, this is given by [52, 12] ρ·D(ρ), provided that the uplink and the downlink SNRs differ by a constant factor. If the above product vanishes with ρ → ∞, then the asymptotic high-SNR rate gap goes to zero and the imperfect CSIT scheme yields close-to-optimal performance. If the above product converges to a constant, then a constant rate gap is achieved. If the above product grows as a power ρ μ , then the downlink sum-rate scales only as min{M,KN t }(1−μ)logρ, i.e., a loss of degrees of freedom by the factor 1−μ occurs. Finally, if the above product grows as ρ (i.e., D(ρ) =O(1) as ρ→∞), then the user rate saturates to a constant term and the system becomes interference limited (unless only a single user is served at each slot, of course). It is therefore clear that the MIMO downlink performance depends critically on the distortion exponent δ(b): we wish to achieve at least δ(b) = 1. This is easily achieved with the analog feedback scheme, in the case of b > 1. Our goal is to study more efficient techniques that achieve δ> 1 at some b≥ 1. WefirstpresentanupperboundonthedistortionSNRexponentinSec.7.2andthen analyse the performance of a separated source-channel coding scheme in Sec. 7.3. The sub-optimality of the separated scheme prompts us to investigate joint source-channel coding schemes for the MIMO-MAC, following which we presenta hybrid digital-analog coding scheme in Sec. 7.4. Simulation results that highlight the superiority of digital feedbackover conventional analogfeedback, andestablishtherelevanceofthedistortion SNR exponent in practical scenarios are presented in Sec. 7.5. 188 7.2 Upper bound on the distortion SNR exponent Ourupperboundisinspiredbytheonepresentedforthesingle-userMIMOblockfading channel in [13, 37]. First, notice that we consider the simple case of UTs with symmetric statistics, and since the UTs have no uplink channel state information, they all send the same coding rate, by symmetry. DefineH = [H 1 H 2 ···H K ]. Consider an augmented channel where all UTs are provided with the knowledge of the coding rate R(H) that can be transmitted such that the rate point R(H) = (R(H),...,R(H)) is inside the capacity region of the MIMO-MAC with given channel vectors and transmit SNR ρ, denoted byC MAC (H;ρ). Using this knowledge, each UT can employ a separated source-channel coding scheme with a source coding rate R s = bR(H) nats per complex sample, and achieve an end-to-end instantaneous distortion given by D(H) = exp(−bR(H)). This would result in an average distortion D(ρ)=E[exp(−bR(H))]. From the expression of the MIMO-MAC capacity regionC(H;ρ) we have the follow- ing conditions on the rate R(H). R(H)≤ 1 |K| logdet I+ρ X k∈K H k H H k ! (7.2) 189 for all subsetsK⊆{1,...,K}, where|K| denotes the cardinality of the setK. Denote H K to be the equivalent M×|K|N t channel comprising of the channels of the users in K stacked next to each other. We can rewrite (7.2) as R(H)≤ 1 |K| logdet I+ρH K H H K . Let m K , min{M,|K|N t } and n K , max{M,|K|N t }. We define λ 1 ≤···≤λ m K to be the m K ordered non-zero eigenvalues of H K H H K , and rewrite the above as R(H)≤ 1 |K| log " m K Y i=1 (1+ρλ i ) # . Shannon’ssource-channel separation theorem holds for a single-user system and any fixed channel. Therefore, as far as each of the above constraints are concerned, we have a situation completely equivalent to a single-user MIMO channel where the |K| users act as a single cooperative transmitter with a source of length |K|S, operating over a MIMO channel with block length T and channel coding rate |K|R(H). It follows that the corresponding distortion obtained by this genie-aided scheme is a lower bound on any achievable distortion for the actual MIMO-MAC. This is given by D LB (ρ) = E[exp(−bR(H))] ≥ E " m K Y i=1 (1+ρλ i ) − b |K| # . (7.3) 190 We set 3 λ i . = ρ −α i . The joint pdf of the random vector α = (α 1 ,...,α m K ) is given as [109] p(α) . = ρ − P m K i=1 (2i−1+n K −m K )α i , α 1 ≥···≥α m K ≥ 0 ρ −∞ , otherwise . (7.4) Using (7.4) to compute (7.3) and an application of Varadhan’s lemma as in [109] results in an upper-bound on the distortion SNR exponent δ ∗ (b)≤ inf α 1 ≥···≥αm K ≥0 m K X i=1 b |K| (1−α i ) + +(2i−1+n K −m K )α i ], where (x) + , max{0,x}. It can be verified that the above infimum is achieved by α ∗ i = 0, b |K| ≤ 2i−1+n K −m K 1, b |K| > 2i−1+n K −m K . Hence δ ∗ (b)≤ m K X i=1 min b |K| ,2i−1+n K −m K , ∀K⊆{1,...,K}. For a fixed|K|, define δ |K| (b) = m K X i=1 min b |K| ,2i−1+n K −m K . (7.5) 3 We use the exponential equality notation of [109], x . =y⇔ lim ρ→∞ logx logρ = lim ρ→∞ logy logρ 191 We hence have that δ ∗ (b)≤ min{δ 1 (b),δ 2 (b),...,δ K (b)}. We term the above upper-bound as the informed transmitter bound. We now investigate the informed transmitter bound for the important special case of N t = 1 and M =K. Notice that for b≤K, we have that δ 1 (b) =δ K (b) =b. Let us now examine how the δ |K| in (7.5) behave, for 1<|K| <K in the range of b≤ K. In particular, we claim that min b |K| ,2i−1+K−|K| = b |K| , i= 1,2,...,|K|. This is true for b≤K since for some ǫ≥ 0, 2i−1+K−|K|< K |K| −ǫ ⇔K < |K| |K|−1 (|K|−2i+1−ǫ) ⇔K <|K|−|K| 2i−2+ǫ |K|−1 , for i = 1,2,...,|K|. Setting i= 1, this requires that K <|K|−ǫ |K| |K|−1 ⇒K <|K|, 192 which is not true. This implies that δ |K| (b) = b ∀ b ≤ K and ∀ 1 ≤ |K| ≤ K. In particular,δ |K| (K) =K. Now consider the caseb>K. Notice thatδ 1 (b) =K ∀b>K. Sinceδ |K| (b) is an increasing function of b, we have that δ |K| (b) ≥ δ |K| (K)∀ b≥K = K. This implies that δ |K| (b)≥δ 1 (b)∀ b. The informed transmitter bound hence reduces to δ ∗ (b)≤δ 1 (b) = min{b,K}. 7.3 Separated source-channel codingfor the MIMO-MAC In this section, we compute the achievable distortion SNR exponent under separated source-channel coding. 7.3.1 Diversity multiplexing tradeoff (DMT) of the MIMO-MAC We first present a very brief overview of the DMT [109] of the MIMO-MAC [95], which is a metric of performance for channel coding. According to the DMT formulation, we 193 scale the rate of transmission of the coding schemeC(ρ) as rlogρ, wherer denotes the multiplexing gain. We define the diversity gain of the system to be d(r) =− lim ρ→∞ logP e (ρ) logρ , whereP e (ρ)denotesthecodeworderrorprobability. TheDMTd ∗ (r)isdefinedtobethe supremumofallachievablediversitygains. Weconsideracommondiversityrequirement of d(r) for the transmission of all UTs, and equal rate transmission. For the general K user MAC withN t antennas at each UT andM antennas at the BS, the DMT is given by [95] d ∗ MAC (r) = d ∗ Nt,M (r), r≤ min n N t , M K+1 o d ∗ KNt,M (Kr), r≥ min n N t , M K+1 o , whered ∗ nt,nr (r) is the DMT for the single user MIMO channel with n t transmit andn r receive antennas [109], given by the piecewise linear function interpolating the points (r,(n t −r)(n r −r))forintegralr = 0,1,...,min{n t ,n r }(seeChapter2). TheDMTcurve of the MIMO-MAC is illustrated in Fig. 7.1. The tradeoff performance can hence be dividedintotworegimes,thelightlyloadedregimecorrespondingtor≤min n N t , M K+1 o , and the heavily loaded regime corresponding to r ≥ min n N t , M K+1 o . In the lightly loaded regime, the DMT of the MAC is as though there was only one user in the system, i.e., single-user performance is achieved. In the heavily loaded regime, the DMT of the MAC is as though all the users pooled their antennas together into a single “super-user”, and transmit at K times the single-user rate. The authors in [95] were 194 Figure 7.1: DMT of the MIMO-MAC 195 also abletoshow thatthedominanterror event inthelightly loaded regimecorresponds to a single user decoding in error, while the dominant error event in the heavily loaded regime is that of all users decoding in error. 7.3.2 Distortion exponent with separated source-channel coding We now turn to the analysis of the distortion SNR exponent of a separated source- channel coding scheme for the MIMO-MAC. A separated source-channel coding scheme consists of concatenating a quantizer of rateR s bits/source sample with a channel code of rate R c bpcu, with R s = bR c . It was shown in [44] that the end-to-end distortion achievable by a separated scheme is upper-bounded by D sep (ρ)≤D Q (R s )+κP e (ρ), where D Q (R s ) denotes the quantizer distortion-rate function, P e (ρ) is the error proba- bility of the channel code, andκ is a constant independent of ρ. Let R s andR c denote the rates of the quantizer and the channel encoder respectively, with corresponding multiplexing gains r s and r c . Thus, D sep (ρ) ˙ ≤ ρ −brc +κρ −d ∗ MAC (rc) . Notice thatthereexistverysimplescalar quantizers withrate-distortion optimalscaling ρ −brc [13], and DMT optimal codes with finite blocklengthKN t +M−1 [95]. The best 196 possibledistortionSNRexponentisobtainedbychoosingr c suchthatthetwoexponents of the above expression are balanced, i.e., such that br c =d ∗ MAC (r c ). Theorem 7.3.1. If N t ≤ M K+1 , or if N t > M K+1 and b > K+1 M d ∗ Nt,M M K+1 , then the distortion SNR exponent of the separated source channel coding scheme δ sep (b)≥b jd ∗ Nt,M (j−1)−(j−1)d ∗ Nt,M (j) b+d ∗ Nt,M (j−1)−d ∗ Nt,M (j) , b∈ d ∗ Nt,M (j) j , d ∗ Nt,M (j−1) j−1 , for j = 1,...,min{N t ,M}. Else, the exponent is given by δ sep (b)≥b jd ∗ KNt,M (j−1)−(j−1)d ∗ KNt,M (j) b−K 2 [d ∗ KNt,M (j)−d ∗ KNt,M (j−1)] , for b between Kd ∗ KN t ,M (j) j and Kd ∗ KN t ,M (j−1) j−1 , for j = 1,...,min M, Nt K . Proof. For N t ≤ M K+1 , the MIMO-MAC DMT coincides with the single-user DMT for allr. Theoptimal distortion exponentin this case is thesameas that forthe single-user case, obtained in [13]. For the case when N t > M K+1 , the transition between the lightly loaded and the heavily loaded regimes occurs atr c = M K+1 , at which point the diversity gain can be shown to be d ∗ Nt,M M K +1 = M K +1 − M K +1 2 M K +1 −(N t +M−1) + N t − M K +1 M− M K +1 . 197 Theslopeofthelineconnectingtheoriginwith M K+1 ,d ∗ Nt,M M K+1 is K+1 M d ∗ Nt,M M K+1 . Hence ifb> K+1 M d ∗ Nt,M M K+1 , then once again the single-user solution from [13] holds. For the case whenb< K+1 M d ∗ Nt,M M K+1 , we need to solve br c =d ∗ KNt,M (Kr c ). (7.6) For r c = j K , j = 0,1,...,min M, Nt K , we have b = Kd ∗ KNt,M (j) j Thenδ sep (b)interpolatesthepoints Kd ∗ KN t ,M (j) j ,d ∗ KNt,M (j) forj = 0,1,...,min M, Nt K . Forr c ∈ h j−1 K , j K i ,thefunctiond ∗ KNt,M (Kr c )linearlyinterpolatesthepoints j−1 K ,d ∗ KNt,M (j−1) and j K ,d ∗ KNt,M (j) . This gives us that d ∗ KNt,M (Kr c )=K d ∗ KNt,M (j)−d ∗ KNt,M (j−1) Kr c − j−1 K +d ∗ KNt,M (j−1), r c ∈ j−1 K , j K . Solving (7.6), we hence obtain the exponent br c =b jd ∗ KNt,M (j−1)−(j−1)d ∗ KNt,M (j) b−K 2 h d ∗ KNt,M (j)−d ∗ KNt,M (j−1) i, for b between K d ∗ KN t ,M (j) j and K d ∗ KN t ,M (j−1) j−1 . 198 For the special case whenN t =1 andM =K,d ∗ MAC (r c )=K(1−r c ), leading us to choose r c = K K +b ⇒ δ sep (b)≥ bK K +b . (7.7) 7.3.3 Performance of successive interference cancellation (SIC) re- ceivers We now consider the performance of low complexity MMSE and ZF SIC receivers (i.e., MMSE and ZF V-BLAST receivers [107]) in terms of distortion SNR exponents. We show that these low complexity receivers are very suboptimal (irrespective of the order in which users are decoded), which suggests that joint decoding techniques might be inevitable in this scenario. The suboptimality in terms of distortion exponent is a direct consequence of the suboptimality of both MMSE and ZF SIC receivers in terms of DMT. For simplicity of exposition we restrict ourself to the case of N t = 1 and M =K. It was shown in [95] that these receivers with no user ordering achieve a DMT ofd ∗ SIC (r)= 1−r c . It was later shown in [51] that noordering, includingthe V-BLAST optimal ordering [107] can improve upon this DMT. In the separated case, this would result in a very suboptimal distortion-SNR exponent of δ sep,SIC (b) = b 1+b . 199 7.4 HybridDigital-AnalogCodingSchemefortheMIMO- MAC While the separated scheme is close to optimal for very low and very high bandwidth efficiencies, it is in fact very suboptimal for a wide range of bandwidth efficiencies, including the ones that are most relevant for practical CSIT feedback scheme, such as b between 2 and 6. One therefore needs to consider joint source channel coding to improve upon the performance of the separated source channel coding scheme. This approach has been employed with good success in the case of the single-user MIMO channel in the series of works in [13, 37, 6]. In the following subsection, we consider a generalization of the hybrid digital-analog scheme of [13] for the MIMO-MAC, and evaluate the distortion SNR exponent for this case. We restrict our attention to the case of bandwidth expansion, i.e.,b≥ 1 (the case of bandwidth compression b< 1 is not relevant for our purpose since it would result in a loss of downlinkmultiplexing gain [12]). Along the lines of [13], we consider the encoder shown in Fig. 7.2 for each UT. We assume that M ≥ K in this section. Define N ′ t to be the maximum integer less than or equal to N t such that KN ′ t ≤ M. The block of S input symbols corresponding to the i th UT is first fed to a separated source-channel encoder that produces “digital” encoded data X d i of size N t ×T d . The quantization error e is scaled by 1/ p N ′ t D Q (R s ) and reformatted to produce “analog” data X a i of size N ′ t ×S/N ′ t . The analog data is to be transmitted over a pre-selected set of N ′ t 200 Reconstruct MUX C Q Userk X a k X d k 1 √ D Q (R s ) × − X k Figure 7.2: Hybrid digital-analog scheme for bandwidth expansion antennas. The digital and analog data are multiplexed in time to producethe signalX i transmitted by the i th UT, over T =T d +S/N ′ t time-slots. We have that r c = S T d r s and b= T d +S/N ′ t S , which result in r c = N ′ t r s N ′ t b−1 . The BS receives a faded superposition of the signals transmitted by the various UT in AWGN, according to (7.1). The BS firsttries to decode the digital data, byusing the first T d columns of Y, denoted as Y d . Up to SNR exponents, the probability of error can be replaced with the probability of outage for the MAC. The outage event is given byO = {H i } K i=1 :(R c ,...,R c ) / ∈C MAC (H;ρ) . 201 If the decoding is successful, the BS obtains a linear MMSE estimate of the esti- mation error e from the last S/N ′ t columns of Y (denoted as Y a ), and adds it to the decoded digital information to produce a reconstruction. Let H a k denote the M×N ′ t channel from thek th UT to the BS corresponding to theN ′ t antennas chosen for analog transmission and H a , [H a 1 ··· H a K ]. Notice that Y a = √ ρ K X k=1 H a k X k +W a = √ ρ[H a 1 H a 2 ···H a K ] | {z } ,H a X a 1 X a 2 . . . X a K | {z } ,X a +W a . StackingthecolumnsofY a andX a onebelowtheothertoobtainy a andx a respectively, we obtain the equivalent vector model y a = √ ρ H a H a . . . H a | {z } ,Heq=I S/N ′ t ⊗H a x a +w a 202 Notice that we may obtain an LMMSE estimate ofe by first linearly estimating x a fromy a , and then byrearranging components and scaling. In order toestimatex a from y a , we compute the covariance matrices K x a x a∗ = I K x a y a∗ = √ ρH H eq K y a y a∗ = I+ρH eq H H eq . From the orthogonality principle, the minimum mean squared error mmse(x a |H) . = tr I−ρH H eq (I+ρH eq H H eq ) −1 H eq = tr h (I+ρH H eq H eq ) −1 i = m ′ X k=1 S/N ′ t 1+ρλ k , wherem ′ , min{M,KN ′ t } andλ 1 ≤···≤λ m ′ denote the ordered eigenvalues ofH H H. Thus mmse(e|H) = D Q (R s ) m ′ X k=1 S/N ′ t 1+ρλ k . It can be shown that the distortion for the hybrid scheme is given by D hybrid (ρ) ˙ ≤ Z O c mmse(e|H)P(H)dH+κP(O) ≤ Z mmse(e|H)P(H)dH+κP(O) (7.8) 203 where κ is a constant independent of ρ. Let λ i . =ρ −α i . Using the joint pdfp(α) given in (7.4), we compute Z mmse(e|H)P(H)dH . = ρ −rs Z α i ≥0 ρ −(1−α 1 ) + − P m ′ i=1 (2i−1+|M−KN ′ t |)α i dα . = ρ −rs ρ − inf α i ≥0 ∀ i (1−α 1 ) + + P m ′ i=1 (2i−1+|M−KN ′ t |)α i . = ρ −(1+rs) . From (7.8), the distortion SNR exponent for the hybrid scheme is obtained by equating 1+ rc N ′ t (N ′ t b−1) =d ∗ MAC (r c ) This equation may be solved in a manner similar to the proof of Theorem 7.3.1. For brevity, we only present the solution whenN t = 1,M =K, in which case the distortion SNR exponent is given by δ hybrid (b)≥ 1+ (b−1)(K−1) K +b−1 . Similar to the separated source channel coding case, it can be shown that successive interference cancellation is very suboptimal in this case also. Fig. 7.3 shows plots of the distortion exponent for K = 4 obtained from separated source channel coding and hybrid digital-analog coding in comparison to the informed transmitter upper bound and the analog scheme. 204 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 b Distortion Exponent Upper bound Separated scheme Hybrid scheme Analog feedback Figure 7.3: Distortion SNR exponent for the MIMO-MAC, N t = 1, M =K 7.5 Simulation results In this section, we present simulation results comparing the case of digital and analog CSIT feedback over the MIMO-MAC. We focus on the case of N t = 1 andK =M = 4 and restrict attention to separated source channel coding for the digital feedback case. We make use of uncoded QAM as our channel code, owing to the following reason. The optimal DMT for theMAC underconsideration isd ∗ MAC (r) =K(1−r)[95]. Ithas been shown in [109] that for the K×K MIMO channel, the DMT achievable by space-only coding (i.e., one is allowed to code across antennas but not across time) is given by d s−o (r) =K(1−r/K). We conjecture that this DMT is achievable by uncoded QAM transmission and ML decoding (our simulation results support this conjecture). Such a transmission is a valid strategy for the MAC, and the corresponding MAC DMT, obtained by replacing the rater/M withr ind ∗ s−o (r), would beK(1−r)(subject to the 205 8 10 12 14 16 18 20 22 24 26 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) D Digital, b = 2 Digital, Zero−error MAC code, b = 2 Analog, b = 2 Digital, b = 4 Digital, Zero−error MAC code, b = 4 Analog, b = 4 Figure 7.4: Comparing analog feedback with digital feedback using separated source- channel coding, K =M = 4, N t = 1 conjecture). Thisisexactly equaltotheoptimalDMTd ∗ MAC (r). Inthesequel, uncoded QAM will be seen to achieve the right distortion SNR exponent through simulations. Fig. 7.4 compares the per-user squared error distortion using analog and digital feedback for the case whenb= 2 andb= 4. For the analog feedback case, we use time- division with 2 users transmitting un-quantized Gaussian sources simultaneously while satisfying the average power constraint (such a time-division strategy has been proven to be the optimal strategy for analog feedback [12]). Notice that we fix T = 4 for the digitalcase, andvarythenumberofsamplesofthetransmittedGaussiansourceS = 2,1 to achieve b = 2,4 respectively. We vary the codebook size with SNR according to the optimal multiplexing gainsr c = 2/3 andr c = 1/2 correspondingtob= 2,4 respectively, 206 computed using (7.7). For example, at b = 4, this would correspond to 4−QAM and 16−QAM constellations at SNRs of 12.0412 dB and 24.0824 dB respectively (in turn correspondingtoratesR c = 2,4bpcuandR s = 8,16bits/complexsamplerespectively). We make use of scalar uniform quantizers from [13] and use gray maps to assign bits to the QAM alphabets. While the slope of analog feedback can never exceed 1, we see that the slopes of the digital feedback in Fig 7.4 are close to their theoretical values of δ sep (2) ≥ 4 3 and δ sep (4) ≥ 2. While analog feedback is better at b = 2, we observe that the digital feedback significantly outperforms analog feedback at b = 4. In order to illustrate that the channel code that we have picked is near optimal, we also plot the distortion achieved by an ideal zero-error MAC code (i.e., the only distortion is due to the quantizer distortion). We observe that the distortion performance of the chosen MAC code is almost identical to the zero-error case. Inapracticalcellularsystem,theSNRexperiencedbyUTsclosetotheBSandthose at the cell fringes may differ by several 10s of dBs, owing to significantly different path- loss. Aswesawbefore,thedownlinkrate-gapisdeterminedbyρD(ρ),whichnecessitates thatthedistortion decaywithanexponentstrictlylarger thanonetoensureavanishing rate gap. Analog feedback is insufficient for this purpose, leading at best to a rate-gap that does not decay with SNR (the presence of sub-logarithmic terms results in the exponent being slightly less than one in practice). Digital feedback is hence a necessity in this practical scenario. Corresponding to this scenario, we present simulations of a K = 4usersystem, wheretwo high-SNRusersoperateat24dB,andtwolow-SNRusers operate at 12 dB. We fix b = 4, and choose the rates of the users in accordance with 207 12 14 16 18 20 22 24 26 10 −5 10 −4 10 −3 10 −2 10 −1 SNR (dB) D Digital feedback Digital F/B, Zero−error MAC code Analog feedback Users 1 and 2 operate at 12.0412 dB (base alphabet 4−QAM) Users 3 and 4 operate at 24.0824 dB (base alphabet 16−QAM) Figure 7.5: Distortion obtained for users with very different SNRs,K =M = 4,N t = 1 the distortion-optimal multiplexing gain of r c = 1/2 (hence low SNR users signal using 4-QAM and high SNR users with 16-QAM). We plot the per-user distortions obtained for the low and high SNR users against their respective SNRs in Fig. 7.5, and observe that the distortion decays with a slope that is exactly equal to the slope for the case of b = 4 in Fig. 7.4. This provides a very strong motivation for the use of intelligently designed digital feedback schemes in practice, and is also testament to the practical relevance of the theoretical distortion exponents derived in this chapter. 208 Chapter 8 Conclusions Inthis thesis, we have explored thefundamentalperformancelimits of certain canonical single and multi-user wireless channels, and constructed codes approaching these limits. Our focus has been on the outage-limited regime, that is typical of many practical in- door and outdoor cellular and wireless local area network channels. While constructing codes, we have strived to come up with practical schemes that are both information theoretically optimal with respect to asymptotic performance metrics, and in addition work well over practical wireless channels with finite SNR. We have also constructed codes that involve reasonable encoding and decoding complexity, and are hence promis- ing candidates for future wireless standards. In the sequel, we wrap up this thesis by recalling the salient features of our main contributions, and briefly providing pointers for future research. Our initial attention was on the single-user MIMO channel, for which we proposed a general technique for the construction of space-time codes of both short and long 209 blocklengths termed as S-LaST codes. The resulting codes are endowed with excel- lent packing and shaping properties, and are currently among the best known in terms of error-probability performance. In particular, our constructions improve upon the Golden code [5], which is a 2× 2 perfect code that has been adopted as part of the IEEE 802.16 WiMax standard. While the community had relied upon hypercube (uni- tary) shaping prior to our work to obtain good performance at finite SNR, the S-LaST methodologyservestoillustratethatonecanindeedimproveuponunitaryshaping. Fur- ther, this methodology allows us to employ dense ST codes with non-unitary shaping derived from maximal orders of division algebras [45], which would involve considerable encoding and decoding complexity without our approach. The improvements obtained from the S-LaST approach become increasingly significant when one employs a large number of antennas, and we expect that these codes will assume increased significance in future, when VLSI technology permits implementing sphere decoders working over large-dimensions. We also sketched a few open problems in this area, involving the algebraic construction of suitable lattice partition chains. Following this, we investigated the performance of such approximately universal codes over arbitrary correlated Rayleigh and Rician fading channels, and showed that the DMT is unchanged from that of i.i.d. Rayleigh fading. This shows that one does not lose diversity while employing an approximately universal ST code over a practical wireless channel involving dynamic correlation environments. We then shifted our attention to the issue of receiver design for MIMO systems. The issue of complexity of the optimal MIMO receiver is currently among the biggest 210 bottlenecks towards the widespread implementation of MIMO systems. We analyzed the performance of linear MMSE and ZF receivers, that are currently being considered as promising low-complexity (albeit suboptimal) alternatives to the optimal MIMO re- ceiver. We presented two asymptotic analysis of these receivers, the first one assuming large SNR and the other assuming large number of antennas. While our first analy- sis revealed the suboptimality of these receivers in terms of the DMT, further analysis revealed that under suitable operating conditions the MMSE receiver in fact exhibits full-diversity. An interesting design conclusion was to employ a low-complexity linear receiver with a few additional antennas at the base station rather than an optimal receiver, while maintaining identical performance. Further research in this direction pertains to investigating the performance of other sub-optimal receivers to determine if similar full-diversity behaviour is obtained for certain operating regimes. Some promis- ingalternatives toconsiderwouldbedecision feedbackreceivers andotherreceivers that bridge the gap between low-complexity linear receivers and optimum sphere decoders in terms of complexity. Subsequently, we dealt with two multi-user wireless channels, the fading relay chan- nel and the cellular MIMO-MAC. We characterized the DMT of the DDF protocol for thefadingrelaychannelforfiniteblock-lengths, andpresentedanalgebraicconstruction of DMT optimal codes. Our results also took into account two important practical con- cerns that had been scantily treated in the previous literature: the problem of incorrect decoding at the relay, and the need to inform the destination of when the relay starts 211 transmitting. Our codes constructed have very reasonable decoding complexity, and operate within a dB from outage probability. Several interesting question relating to fading relay channels remain unanswered. A quantize-and-map scheme has been recently shown to achieve the optimal DMT of the single relay channel, and certain other network topologies [73]. This work makes use of the recently proposed deterministic channel model for wireless networks. The problem of explicit code design for this protocol remains open, and we believe that lattice and algebraic codes are promising strategies to explore. In a broader sense, more light needs to be shed on the fundamental limits of half-duplex wireless networks, and the deterministic model looks to be a powerful tool towards this end. Ourfinalproblem related toCSIfeedback for cellular systems, an issueof significant practical relevance. Code design for CSI feedback has not received sufficient attention in the prior literature, with most works assuming the presence of a finite-rate noiseless feedbacklink,orthatthefeedbackinformationissomehowpiggy-backed ontotheuplink data. We present a framework towards solving this problem in terms of minimizing the distortion, and present separated source-channel and joint source-channel coding schemes that improve significantly on the performance of traditional analog feedback. Coming up with better achievability schemes that close the gap to the upper-bound on the optimal distortion-SNR exponent is an open problem, as is the problem of dealing with imperfect CSI at the user terminals. 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[110] Draft Standardization Document: IEEE P802.11n/D2.00, February 2007. 222 Appendices A DMT of Linear Receivers: Proof of P(A) =O(1) Our aim is to provide a lower-bound to the quantity P(A) =P 1 M M X k=1 1 |u 1k | 2 ≤c ! . It is well known that U and Λ are independent random matrices and that U is Haar distributed (i.e., is an isotropically random unitary matrix distributed uniformly over the Stiefel manifold). Therefore, the vector u 1 , (u 1,1 ,u 1,2 ,...,u 1,M ) correspond- ing to the first row of U is uniformly distributed on the unit M-dimensional hyper- sphere (or M-sphere) O M (0,1), 1 and satisfies |u 1 | 2 = P M i=1 |u 1i | 2 = 1. The point p = 1 √ M , 1 √ M ,..., 1 √ M is a point on the unit M-sphere. Consider the spherical cap C of the unit sphere that is cut out by the sphereO M (p,ǫ), where ǫ is a small positive number (See Fig. A1). The coordinates of any point u in this spherical capC is lower bounded by u j ≥ 1 √ M −ǫ, ∀ j = 1,...,M. Therefore, 1 M M X j=1 1 |u j | 2 ≤ 1 1 √ M −ǫ 2 . 1 We will use the notationOM(c,δ) to denote an M-sphere centred atc with radius δ. 223 1 h a 1−h φ ε Figure A1: The unit hemisphere and a spherical cap. Defining the constant c= 1 √ M −ǫ −2 , we have that P(A) ≥ P (u 1 ∈C) = Surface area ofC Surface area ofO M (0,1) , where the latter equality holds since u 1 is isotropic. In order to compute the above surface areas, consider Fig. A1. The surface area ofC, denoted by Ω(φ) is given by [82] Ω(φ) = (M−1)π (M−1)/2 Γ M+1 2 Z φ 0 (sinθ) M−2 dθ, 224 and the surface area of an unit M-sphere is [82] S M (1) = Mπ M/2 /Γ(M/2 + 1). All that remains is to compute the angle φ, which is accomplished by solving the following equations obtained from the two right-angled triangles in Fig. A1: h 2 +a 2 = ǫ 2 a 2 +(1−h) 2 = 1. Solving for a,h, we obtain h = ǫ 2 2 , a=ǫ r 1− ǫ 2 4 . Therefore, φ = tan −1 ǫ q 1− ǫ 2 4 1− ǫ 2 2 , and P(A)≥ Ω(φ) S M (1) > 0, as desired. B Novikov’s Theorem We introduce here a usefultrick which makes the connection between Gaussian integra- tion and differentiation over Gaussian random variables. Theorem B.1. [69] (Novikov) Let H be an N×M matrix with i.i.d. elements drawn from CN(0,1/N). Let f(H,H H ) be a scalar function of the matrix elements of H and 225 their complex conjugates, which does not grow faster than the inverse of the probability density of H, i.e., such that lim |h ij |→∞ p(H)f(H,H H )= 0 (B1) for all matrix elementsh ij . Then, for any set of indicesi,j, the following relation holds E h h ij f(H,H H ) i = 1 N E " ∂f(H,H H ) ∂h ∗ ij # , (B2) where h ij and h ∗ ij are to be treated as individual variables in the differentiation. 2 Sketch of the Proof. Even though the general proof is involved, we present here a simple proof for a single real Gaussian variableN(0,1) by integrating by parts: E[zf(z)] = Z ∞ −∞ dz √ 2π zf(z)e − z 2 2 (B3) = − e − z 2 2 f(z) √ 2π ∞ −∞ + Z ∞ −∞ dz √ 2π f ′ (z)e − z 2 2 = E f ′ (z) 2 This means that ∂h ∗ i,j ∂h i,j = 0 when computing the partial derivative in (B2). 226 Example. A useful and illustrative example of the application of Novikov Theorem is to evaluate the fourth moment of a i.i.d. complex Gaussian vector with elements drawn fromCN(0,1/N). We have E[h ∗ i h j h ∗ k h l ] = 1 N E ∂ ∂h i (h j h ∗ k h l ) = δ ij N E[h ∗ k h l ]+ δ il N E[h ∗ k h j ] (B4) = δ ij δ kl N 2 + δ il δ kj N 2 C Fluctuations of eigenvalues Let B = I+αHH H −1 (C1) where H∈C N×M has i.i.d. elements CN(0,1/N), and let β =M/N. The normalized trace of B is given by 1 N Tr(B) = 1 N N X k=1 1 1+αλ k (HH H ) =η (N) (α) where η (N) (α) is the η-transform [97] of the empirical eigenvalue distribution of HH H . For largeN, the variance of Tr(B) =Nη (N) (α) is of order unity and can be calculated in closed form [4, 74], E c [Tr(B);Tr(B)] = P Z λmax λ min dλ Z λmax λ min dμK 2 (λ,μ) 1 (1+αλ)(1+αμ) , (C2) 227 whereP denotes the principal part of the integral,λ min,max denote the extremal values of the support of the Marcenko-Pastur law [97], given by λ min,max = 1± p β 2 , (C3) and K 2 (λ,μ) is an integral kernel representing the deviation of the joint eigenvalue distribution of the eigenvaluesλ,μ from theproductof their marginal distributions and is given asymptotically by K 2 (λ,μ) = 1 2π 2 1 p (λ−λ min )(λ max −λ) ∂ ∂μ " p (μ−λ min )(λ max −μ) λ−μ # . (C4) It can be verified that the above function is symmetric inλ andμ. After integration by parts, we get E c [Tr(B);Tr(B)] = 1 2π 2 P Z λmax λ min dλ Z λmax λ min dμ (μ−λ min )(λ max −μ) (λ−λ min )(λ max −λ) 1/2 · α (μ−λ)(1+αλ)(1+αμ) 2 = α 2 β (1+2α(1+β)+α 2 (1−β) 2 ) 2 =O(1). (C5) The result (C5) is used in the calculation of the correlations of γ 1 , γ 2 in (5.61). 228 Furthermore, in [4, 74] two important and more general results are shown. In par- ticular, for any functions f 1 (x), f 2 (x) the following result is true in the limit of large N E c [Tr(f 1 (B));Tr(f 2 (B))] = P Z λmax λ min dλ Z λmax λ min dμK 2 (λ,μ)f 1 (λ)f 2 (μ) = O(1) (C6) as long as these functions are bounded and smooth enough within the support of the asymptotic eigenvalue spectrum (for example f(x) = [x] + is not smooth, while f(x) = α/(1+αx) 2 is smooth). Also, in [4, 74] it is shown that all higher order cumulants of such smooth functions vanish in the large N limit, i.e., for n> 2 E c h Tr f 1 (HH H ) ;...;Tr f n (HH H ) i =Q n (N) =o(1) (C7) where we have denoted the (arbitrary for our purposes) scaling of the above cumulant moment with N asQ n (N) for future reference. We will use this result in Appendix D to prove that all cumulant moments ofI N of order higher than two vanish for largeN. D Higher order cumulants are vanishing After the calculation of the mean (5.70) and the variance (5.72) of the mutual informa- tion in Section 5.5 we now need to show that the higher order cumulants of the mutual 229 information vanish in the limit of large N. This will conclude the proof of the asymp- totic Gaussianity of the mutual information, as discussed in the beginning of Section 5.5. We need to show that the cumulant moments defined in (5.36) as C n = M X k 1 ,...kn=1 E c [log(1+γ k 1 );...;log(1+γ kn )]. vanish for n> 2 when N →∞. Despite the fact thatC n is a sum of O(N n ) terms, we shall show that it is in fact of order o(1). We discuss in some detail the case of the MMSE receiver. At the end of this ap- pendix, a short argument is given in order to reach the same conclusions for the ZF case. While a formal proof would be lengthy and tedious and would not add much value to the paper, we shall provide a sketch the basic steps of the proof leaving out several technicalities. We start by recalling that each γ k i in the above sum is defined as γ k i = αh H k i B k i h k i ,whereB k i = I+αHH H −αh k i h H k i −1 . Usingthematrixinversionlemma, we see that the denominator in the expression of B k i includes all columns of H other than h k i . For every n-tuple{k 1 ,k 2 ,...,k n } we define the matrix B {k i } , such that B {k i } = I+αH {k i } H H {k i } −1 , 230 where H {k i } is obtained by removing columns h k 1 ,h k 2 ,...,h kn from H. If for some i 6= j we have k i = k j , then we only remove column k i once. For any finite n, the following approximation holds B k i =B {k i } +O 1 N , (D1) in the sense that the elements of the matrix N(B k i −B {k i } ) are almost surely finite in thelimit oflargeN. Roughlyspeaking, sincetheelements ofH arezero-mean Gaussian with variance 1/N, the difference B −1 k i −B −1 {k i } =α P k j 6=k i h k j h H k j adds a term of order O(1/N) to each element of B, which can be neglected for largeN. For example, of the aboveapproximation can beproved inaniterative fashion,byshowingthatB k 1 −B k 1 ,k 2 is O(1/N), then adding to writing B k 1 = B k 1 −B k 1 ,k 2 +B k 1 ,k 2 −B k 1 ,k 2 ,k 3 and then showing that B k 1 ,k 2 −B k 1 ,k 2 ,k 3 is also O(1/N) etc. Each such difference can be shown to bealmost surelyO(1/N) by applyingthe matrix inversion lemma and observingthat the elements of h k i areCN(0,1/N). As a result, to leading order in 1/N, we have C n ≈ X {k i } E c [log(1+αh H k 1 B {k i } h k 1 );...;log(1+αh H kn B {k i } h kn )] where P {k i } is a sum over all possible 1≤k 1 ,...,k n ≤M. 231 Since now the random vectorsh k 1 ,...,h kn do not appear inB {k i } , we can explicitly average over them in the above expression. At this point it is convenient to expand the logarithms in Taylor series, such that each term in the sum above becomes ∞ X l 1 ,...,ln=1 (−α) l 1 +...+ln l 1 l 2 ...l n X {k i } E c h H k 1 B {k i } h k 1 l 1 ;...; h H kn B {k i } h kn ln . (D2) We may now apply Novikov Theorem to average over the h k i ’s. This is in general a formidable exercise in combinatorics. Instead, we only need to find how the leading terms scale withN. Specifically, in the following we will fix then-tuplel 1 ,l 2 ...,l n and show that the corresponding term of the type P {k i } E c [·] is o(1) as N →∞. We first notice that since there areL {l i } =l 1 +...+l n pairs ofh H ’s andh’s, we will have an overall factor of N −L {l i } after averaging over all h’s. Also, we can decompose the sum over {k i } into subsets or “shells”, where each shell has the same number of distinct indicesk i ’s. For example, there are M n =O(N n ) terms containing all distinct indices, andnM!/((M−n+1)!(n−1)!) =O(N n−1 ) terms havingn−1 distinct indices and one repeated index. In general, the shell with q distinct indices contains O(N q ) terms. If the k i ’s are all distinct, then the resulting cumulant moment will be of order n. Possible terms that may appear include, for example, E c [Tr B {k i } l 1 ;...;Tr B {k i } ln ] (D3) E c [Tr B {k i } l 1 −2 Tr h B 2 {k i } i ;...;Tr B {k i } ln ] 232 whereaterm as in thesecond linecan onlyappearforl 1 ≥ 2. Therefore, after averaging over the h k i ’s we are left with order-n moments, having as arguments products of the randomvariablesTrB m j {k i } ,for1≥m j ≥l j ,withj = 1,...,n. Letx j bethetotalnumber of traces appearing in the j-th argument of a particular term, such that x j ≤ l j . For example, in the first line of (D3), we have x 1 = 1, while in the second line we have x 1 = 2. Since generally x i ≥ 1, such cumulant moments can be reduced to sums of products of irreducible moments with respect to these random variables. To estimate the leading scaling inN of these reducible moments, we recall that the first moment of the trace is E c [TrB m {k i } ] = O(N), the second cumulant moment is O(1) (C6), while all higher cumulant moments are o(1) (C7). Let the leading term in the expansion of the reducible moments into irreducible ones have d 1 cumulants of order one, d 2 cumulants of order 2, d 3 cumulants of order 3 etc. The only constraint we need to impose is that d 1 ≤d 1max = n X i=1 [x i −1] + ≤L {l i } −n, whichisvalidduetotheshift-invarianceofirreduciblecumulantmomentsoforderhigher thanone. Inthecasethatd 1 =d 1max ,weneedtohaved n = 1andd k = 0forallk6= 1,n. Collecting all powers ofN this term will be of orderN n−L {l i } +d 1max Q n (N)≤Q n (N) = o(1), where we recall that the factor N n is due to the O(N n ) possible combinations of the distinctk i ’s that appear in the corresponding sum in (D2), while the factorN −L {l i } comes from the averages over the h k i . Otherwise, when d 1 < d 1max , the leading term 233 will have, if that is at all possible, d 2 = ( P i x i −d 1 )/2 and d k = 0 for k > 2. In this case, the scaling of this term with N is N n−L {l i } +d 1 ≤N −1 =o(1). The above argument can be extended to the case when there areq<n distinctk i ’s. The difference is that now additional terms may appear, since Novikov’s formula may give derivatives across different arguments of the cumulant moment, thus reducing the order of the cumulant, e.g. if k 1 =k 2 we will get the term E c [Tr B {k i } l 1 −1 Tr B {k i } l 2 −1 Tr h B 2 {k i } i ;Tr B {k i } l 3 ...;Tr B {k i } ln ]. (D4) In general, for the shell with q < n distinct indices the resulting terms will include cumulant moments with orders m, such that q ≤ m ≤ n. When m = n, we can directly apply the argument used when q = n, only replacing the number of possible combinations of the distinct combinations from N n to N q . In the case that m < n, when one expresses the reducible cumulant moments in terms of sums over products of irreducible ones, the maximum number of order one cumulants that may appear is now bounded by d 1 ≤d ′ 1max = m X i=1 [x i −1] + <L {l i } −q. The crucial difference is that d ′ 1max < L {l i } −q, which is due to the fact that, as seen in (D4), in order to reduce the order of the cumulant moment to m<n, one needs to produce traces that span different arguments of the original cumulant moments. In this case, the leading term will be of order N q−L {l i } +d 1 ≤N q−L {l i } +d ′ 1max =o(1). 234 Followingtheaboveargument,wecanshowthatallcumulantmomentsofthemutual information with order n> 2 are negligible in the limit N →∞. As far as the ZF receiver is concerned, recall that (see Section 5.5.2) we can obtain theSINRofthevirtualchannelsoftheZFreceiver bysettingtheparameterαinsidethe correspondingSINRsexpressionoftheMMSEreceiverto∞. Specifically,expressingthe relation in terms of the matrices B k , where we have explicitly denoted the dependence on α, we have γ zf k = α lim α 0 →∞ h H k B k (α 0 )h k (D5) = α lim α 0 →∞ h H k h I+α 0 H k H H k i −1 h k As mentioned in Section 5.5.2, the above analysis involving the matricesB {k i } etc. can be carried out in the case of zero-forcing if β < 1. In addition, as seen in Appendix C, for β < 1 all n-th order cumulant moments of traces of products of B {k i } are given by (C6) and (C7). As a result, all finite n> 2 order cumulant moments of the mutual information areo(1) for the ZF receiver too. E Separated Relay Activity Detection In this Appendix we treat a side problem. An intuitive low-complexity scheme for detecting the relay decision time consists of treating M as a random parameter, and use ML detection by disregarding the structure of the channel codes. Intuitively, the destination should be able to detect a transition in the received power, between the 235 listening phase and the transmission phase of the relay. This approach is referred to as separated Relay Activity Detection (RAD), since the decision time and the source codeword are separately decoded, in contrast with the GLRT decoder analyzed in the proof of Theorem 6.3.2. We shall show that separated RAD yields no performance loss when we consider the limit of T →∞. On the contrary, it is suboptimal and actually may perform very poorly when limits are taken in the reverse order, that is, for each finiteT we consider the performance as SNR gets large. We assume that the source uses an i.i.d. random Gaussian code and the relay implements the Alamouti-DDF scheme. As before, letM denote the decision time. An ML decision time detector that is ignorant of the codebooks treats the channel input as a random Gaussian signal. The detection rule is given by c M= argmax m p(y|M =m,g 1 ,g 2 ). where p(y|M = m,g 1 ,g 2 ) shall be denoted in the following simply by p(y|m) for sim- plicity, and it is given by p(y|m) = 1 [π(|g 1 | 2 ρ+1)] mT exp − |y m 0 | 2 |g 1 | 2 ρ+1 (E1) · 1 {π[(|g 1 | 2 +|g 2 | 2 )ρ+1]} (M−m)T exp −|y M m | 2 (|g 1 | 2 +|g 2 | 2 )ρ+1 . (E2) SupposeM=m, we define the pairwise error event {m→m ′ }, p(y|m ′ ) p(y|m) ≥ 1 . 236 The detector error probability is lower bounded by P(M6= c M)≥ max m6=m ′ P(m→m ′ ), and is upper bounded by the union bound P(M6= c M)≤ (M−1) max m6=m ′ P(m→m ′ ). Hence, we shall study the diversity exponent ofP(m→m ′ ) for general m6=m ′ . If this does not depend on m,m ′ we have determined the diversity exponent of the separated RAD. E1 Infinite block-length IfM=m and T →∞, the law of large numbers yields the almost sure convergence of the limits: 1 T |y n n−1 | 2 → |g 1 | 2 ρ+1, 1≤n≤m and 1 T |y n n−1 | 2 → (|g 1 | 2 +|g 2 | 2 )ρ+1, m+1≤n≤M. 237 Thus, for large T we have p(y|m)≈ exp −MT −mT log π(|g 1 | 2 ρ+1) −(M−m)T log π((|g 1 | 2 +|g 2 | 2 )ρ+1) . Consider the case m ′ >m (the other case follows in the same way and it is omitted for brevity). We have p(y|m ′ ) ≈ exp −mT−(M−m ′ )T −(m ′ −m)T (|g 1 | 2 +|g 2 | 2 )ρ+1 |g 1 | 2 ρ+1 − −m ′ T log π(|g 1 | 2 ρ+1) −(M−m ′ )T log π (|g 1 | 2 +|g 2 | 2 )ρ+1 . After some simplifications, the pairwise error probability for T →∞ is given by P(m→m ′ |g 1 ,g 2 )=P (1−X 1 +logX 1 ≥ 0), (E3) where we let X 1 = (|g 1 | 2 +|g 2 | 2 )ρ+1 |g 1 | 2 ρ+1 . Since logx≤x−1 ∀ x≥ 0, we see that {m→ m ′ } can occur only if |g 2 | 2 = 0, which is an event of measure 0. Therefore, we conclude that P(m→m ′ )↓ 0 for any fixed ρ, as T →∞. This shows that for the infinite T case, even a very simple separated RAD scheme at the destination yields perfect knowledge of the relay decision time without any need of a side information channel that involves some protocol overhead. 238 E2 Finite block-length We now fixT to bean arbitraryfinitevalueandstudythe diversity exponentofP(m→ m ′ ) as ρ→∞. Again, we consider only the case m ′ >m. The likelihood function for the hypothesis m ′ whenM =m is given by p(y|m ′ ) = exp −m ′ T log π(|g 1 | 2 ρ+1) −(M−m ′ )T logπ (|g 1 | 2 +|g 2 | 2 )ρ+1 − |y m 0 | 2 +|y m ′ m | 2 |g 1 | 2 ρ+1 − |y M m ′ | 2 (|g 1 | 2 +|g 2 | 2 )ρ+1 ! . After some algebra, we find that P(m→m ′ |g 1 ,g 2 ) =P χ≤ (m ′ −m)T X 2 log(1+X 2 ) , (E4) where χ= y m ′ m 2 1+(|g 1 | 2 +|g 2 | 2 )ρ is a central chi-squared random variable with 2T(m ′ −m) degrees of freedom and mean T(m ′ −m), and we define X 2 = |g 2 | 2 ρ |g 1 | 2 ρ+1 . As an aside, notice that 1 x log(1+x) is a decreasing function ofx that is less than 1 for allx> 0, and approaches 1forx↓ 0. Therefore, theterm (m ′ −m)T X 2 log(1+X 2 )in(E4)is always strictly less thanE[χ] = (m ′ −m)T for all|g 2 |> 0. Therefore, as an application of the the large deviation theorem [22], we find thatP(m→m ′ )↓ 0 exponentially with 239 T for all finite ρ and |g 2 | > 0. Thus, we recover in a more rigorous way the result obtained before by letting T →∞ directly in the detector decision metric. Returning to the case of finite T, we have using well-known properties of the chi- squared distribution that P(X ≤u) = 1 ((m ′ −m)T)! u (m ′ −m)T +O(u (m ′ −m)T+1 ) for small u, and obviously P(X ≤u)=O(1) whenu =β(m ′ −m)T for some constantβ >0. Fix an arbitrary0<β < 1. From what was said before, there exists an x 2 > 0 such that 1 x 2 log(1+x 2 ) = β. Hence, consider the event E(ρ,β) = {X 2 ≤x 2 } = |g 2 | 2 ρ≤x 2 (1+|g 1 | 2 ρ) . (E5) It is clear that for all (g 1 ,g 2 )∈E(ρ,β), the pairwise error probabilityP(m→m ′ |g 1 ,g 2 ) in (E4) is exponentially equivalent to a constant as ρ→∞, i.e., P(m→m ′ |g 1 ,g 2 ) . = ρ 0 , (g 1 ,g 2 )∈E(ρ,β). 240 Averaging with respect to g 1 ,g 2 , and using the standard variable substitution |g 1 | 2 = ρ −α 1 ,|g 2 | 2 =ρ −α 2 , we find P(m→m ′ ) ˙ ≥ Z E ρ 0 e −ρ −α 1−ρ −α 2 ρ −α 1 −α 2 dα 1 dα 2 . = Z E ′ ρ −α 1 −α 2 dα 1 dα 2 , where, from (E5), E ′ = (α 1 ,α 2 )∈R 2 + : 1−α 2 ≤ [1−α 1 ] + . Using Varadhan’s lemma, we find that the diversity exponent of the pairwise error probability is given by Δ = inf (α 1 ,α 2 )∈E ′ {α 1 +α 2 }= 0, since the point α 1 =0,α 2 = 0 belongs to the boundary of the regionE ′ . This shows that for any finiteT, a separated RAD scheme based on optimal (Max- imum Likelihood) detection of the relay decision time M that ignores the codebook structure and treats the transmitted signals as random processes is very suboptimal. In fact, the probability of error of such a scheme is constant with SNR and eventually will dominate the performance of the whole destination decoder. In some way, this result shows that the joint detection of the relay decision time and of the information message is necessary in order to achieve the optimal (infiniteT) DDF DMT. 241 F Publications Based on this Thesis F1 Journal Papers 1. K. Raj Kumar and Giuseppe Caire, “Coding and Decoding for the Dynamic De- code and Forward Relay Protocol,” IEEE Trans. Inform. Theory, Vol. 55, No. 7, pp. 3186 - 3205, Jul. 2009. 2. K. Raj Kumar,GiuseppeCaire and Aris L.Moustakas, “Asymptotic Performance of Linear Receivers in MIMO Fading Channels,” To appear in IEEE Trans. In- form. Theory. Available at http://arxiv.org/abs/0810.0883. 3. K.RajKumarandGiuseppeCaire,“Space-TimeCodesfromStructuredLattices,” IEEE Trans. Inform. Theory, Vol. 55, No. 2, pp. 547 - 556, Feb. 2009. 4. GiuseppeCaireand K.Raj Kumar,“Information Theoretic Foundations of Adap- tive Coded Modulation”, Invited paper, Proceedings of the IEEE (Special issue on Adaptive Modulation/Transmission), Vol. 95, No. 12, Dec. 2007. 5. GiuseppeCaire,PetrosEliaandK.RajKumar,“Space-TimeCoding: AnOverview”, Invited paper, Journal of Communications Software and Systems (JCOMSS), Vol. 2, No. 3. 242 F2 Conference Papers 1. K. Raj Kumar and Giuseppe Caire, “Channel State Feedback over the MIMO- MAC,” Proc. IEEE Int. Symp. Inform. Theory (ISIT - 09), Seoul, Jul. 2009. 2. K. Raj Kumar and Giuseppe Caire, “Coding and Decoding for the Dynamic De- code and Forward Relay Protocol,” Proc. IEEE Int. Symp. Inform. Theory (ISIT - 08), Toronto, Jul. 2008. 3. K. Raj Kumar and Giuseppe Caire, “Code Design for the Dynamic Decode and Forward Relay Protocol,” Proc. of the 45th Allerton Conf. on Communications, Control and Computing, Illinois, Sept. 2007. 4. K.RajKumar,GiuseppeCaireandArisL.Moustakas,“TheDiversity-Multiplexing Tradeoff of Linear MIMO Receivers,” Proc. IEEE Inform. Theory Workshop (ITW ’07), Lake Tahoe, Sep. 2007. 5. K.RajKumarandGiuseppeCaire, “StructuredLatticeSpace-TimeTrellisCoded Modulation”, Proc. IEEE Int. Symp. Inform. Theory (ISIT - 2007), Nice, France, Jun. 2007. 6. K. Raj Kumar and Giuseppe Caire, “Outage Analysis and Code Design for Cor- related MIMO Fading Channels”, Proc. IEEE Int. Symp. Inform. Theory (ISIT - 2007), Nice, France, Jun. 2007. 7. K. Raj Kumar and Giuseppe Caire, “Construction of Structured LaST Codes”, Proc. IEEE Int. Symp. Inform. Theory (ISIT - 2006), Seattle, July 2006. 243
Abstract (if available)
Abstract
The broad objective of this thesis is to explore the fundamental limits of communication over outage-limited wireless fading channels (including point-to-point and some multiuser scenarios), and design coding schemes that perform close to these limits.
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Asset Metadata
Creator
Krishna Kumar, Raj Kumar
(author)
Core Title
Communicating over outage-limited multiple-antenna and cooperative wireless channels
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
08/07/2009
Defense Date
06/01/2009
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
cooperative communication,MIMO receivers,MIMO systems,OAI-PMH Harvest,relay channel,space-time codes,wireless communication
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Caire, Giuseppe (
committee chair
), Chugg, Keith M. (
committee member
), Govindan, Ramesh (
committee member
), Kramer, Gerhard (
committee member
), Mitra, Urbashi (
committee member
)
Creator Email
k.raj55@gmail.com,rkkrishn@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m2545
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UC1496609
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etd-Kumar-3036 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-178213 (legacy record id),usctheses-m2545 (legacy record id)
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etd-Kumar-3036.pdf
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Dissertation
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Krishna Kumar, Raj Kumar
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texts
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University of Southern California Dissertations and Theses
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Repository Location
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Repository Email
cisadmin@lib.usc.edu
Tags
cooperative communication
MIMO receivers
MIMO systems
relay channel
space-time codes
wireless communication