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UBC Theses and Dissertations

Advanced receivers for space-time block-coded single-carrier transmissions over frequency-selective fading… Wavegedara, Kapila Chandika B. 2008

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ADVANCED RECEIVERS FOR SPACE-TIME BLOCK-CODED SINGLE-CARRIER TRANSMISSIONS OVER FREQUENCY-SELECTIVE FADING CHANNELS by KAPILA CHANDIKA B. WAVEGEDARA BSc.Eng, The University of Peradeniya, 1998 M.Eng., Asian Institute of Technology, 2001 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) March 2008 c© Kapila Chandika B. Wavegedara, 2008 Abstract In recent years, space-time block coding (STBC) has emerged as an effective transmit- diversity technique to combat the detrimental effects of channel fading. In addition to STBC, high-order modulation schemes will be used in future wireless communica- tion systems aiming to provide ubiquitous-broadband wireless access. Hence, advanced receiver schemes are necessary to achieve high performance. In this thesis, advanced and computationally-efficient receiver schemes are investigated and developed for single- carrier space-time (ST) block-coded transmissions over frequency-selective fading (FSF) channels. First, we develop an MMSE-based turbo equalization scheme for Alamouti ST block- coded systems. A semi-analytical method to estimate the bit error rate (BER) is devised. Our results show that the proposed turbo equalization scheme offers significant perfor- mance improvements over one-pass equalization. Second, we analyze the convergence behavior of the proposed turbo equalization scheme for Alamouti ST block-coded sys- tems using the extrinsic information transfer (EXIT)-band chart technique. Third, burst-wise (BW)-STBC is applied for uplink transmission over FSF chan- nels in block-spread-CDMA systems with multiuser interference-free reception. The performances of different decision feedback sequence estimation (DFSE) schemes are investigated. A new scheme combining frequency-domain (FD) linear equalization and modified unwhitened-DFSE is proposed. The proposed scheme is very promising as the error-floor behavior observed in the existing unwhitened DFSE schemes is eliminated. Fourth, we develop a FD-MMSE-based turbo equalization scheme for the downlink of ST block-coded CDMA systems. We adopt BW-STBC instead of Alamouti symbol-wise (SW)-STBC considered for WCDMA systems and demonstrate its superior performance in FSF channels. Block spreading is shown to be more desirable than conventional spreading to improve performance using turbo equalization. We also devise approx- imate implementations (AprxImpls) that offer better trade-offs between performance and complexity. Semi-analytical upper bounds on the BER are derived. Fifth, turbo multicode detection is investigated for ST block-coded downlink trans- mission in DS-CDMA systems. We propose symbol-by-symbol and chip-by-chip FD- MMSE-based multicode detectors. An iterative channel estimation scheme is also pro- posed. The proposed turbo multicode detection scheme offers significant performance improvements compared with non-iterative multicode detection. Finally, the impact of channel estimation errors on the performance of MMSE-based turbo equalization in ST block-coded CDMA systems is investigated. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Space-Time Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 STBC for Broadband Communications . . . . . . . . . . . . . . . 7 1.2.3 Transmit-Diversity Techniques for DS-CDMA Systems . . . . . . 10 1.2.4 Decision Feedback Sequence Estimation (DFSE) for ST Coded Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.5 Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.6 Turbo (Iterative) Receivers for ST Coded DS-CDMA Systems . . 15 1.2.7 Channel Shortening for ST Coded Systems . . . . . . . . . . . . . 16 1.3 Theme and Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . 17 1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2 Turbo Equalization for Alamouti Space-Time Block-Coded Transmis- sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 MMSE-based Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1 SoftISoftO Linear-MMSE Equalizer . . . . . . . . . . . . . . . . . 41 2.3.2 SoftISoftO MMSE-Decision Feedback Equalizer . . . . . . . . . . 49 2.3.3 SoftISoftO Symbol Demapper . . . . . . . . . . . . . . . . . . . . 51 iii Table of Contents 2.4 Semi-Analytical BER Estimation . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 54 2.5.1 Performance of the Exact Implementation . . . . . . . . . . . . . 55 2.5.2 Performance of Approximate Implementations . . . . . . . . . . . 64 2.5.3 Comparison of semi-analytical and the simulated BER perfor- mances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3 Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 73 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 MMSE-based Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Matched Filter-based SoftISoftO Detector . . . . . . . . . . . . . 86 3.4 EXIT-band Chart Analysis for Turbo Equalization . . . . . . . . . . . . . 86 3.5 Obtaining Estimates of the Code BER (CBER) . . . . . . . . . . . . . . 93 3.6 Application to EDGE/EGPRS Systems . . . . . . . . . . . . . . . . . . . 94 3.6.1 Extrinsic Transfer Characteristics . . . . . . . . . . . . . . . . . . 96 3.6.2 EXIT-band Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4 Space-Time Coded CDMA Uplink Transmission. . . . . . . . . . . . . . 125 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.3 Low-Complexity Linear Equalization . . . . . . . . . . . . . . . . . . . . 134 4.4 Decision Feedback Sequence Estimation (DFSE) . . . . . . . . . . . . . . 136 4.4.1 Unwhitened DFSE (UDFSE) . . . . . . . . . . . . . . . . . . . . . 137 4.4.2 Proposed Combined Linear Equalization-MUDFSE (Comb. LE- MUDFSE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.4.3 Whitened DFSE (WDFSE) . . . . . . . . . . . . . . . . . . . . . . 138 4.5 Training-based Channel Estimation . . . . . . . . . . . . . . . . . . . . . 140 4.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 141 4.6.1 Case A: Comparison of the Performance of the proposed ST coded uplink system with the System without ST Coding . . . . . . . . 142 4.6.2 Case B: Performance Investigation of Different DFSE Schemes with Relatively Short Channels . . . . . . . . . . . . . . . . . . . 143 4.6.3 Case C: Performance Investigation of Different DFSE Schemes with Relatively Long Channels . . . . . . . . . . . . . . . . . . . . 146 4.6.4 Case D: Performance with training-based Channel Estimation . . 148 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 iv Table of Contents 4.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5 MMSE-based Turbo Equalization for Space-Time Block-Coded. . . . 154 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.2.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 5.2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.2.4 ST Block Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.3 MMSE-based Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . 165 5.3.1 Symbol-by-Symbol Estimation wise SoftISoftO Equalizer . . . . . 166 5.3.2 Chip-by-Chip Estimation wise SoftISoftO Equalizer . . . . . . . . 174 5.3.3 SoftISoftO Symbol Demapping . . . . . . . . . . . . . . . . . . . . 178 5.3.4 Comparison of Computational Complexities . . . . . . . . . . . . 179 5.4 Semi-Analytical Upper Bounds on the BER Performance . . . . . . . . . 182 5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.5.1 Performance obtained using the (symbol-by-symbol) ExactImpl with SBIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.5.2 Performance obtained using the ExactImpls for BIS . . . . . . . . 195 5.5.3 Performance obtained for time-variant channels . . . . . . . . . . 199 5.5.4 Performance obtained using approximate implementations . . . . 200 5.5.5 Comparison of the semi-analytical upper bounds with the simula- tion results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6 TurboMulticode Detection for Space-Time Block-Coded CDMADown- link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.2.2 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 6.3 MMSE-based Turbo Multicode Detection . . . . . . . . . . . . . . . . . . 224 6.3.1 Symbol-by-Symbol Estimation wise MMSE-based SoftISoftO Mul- ticode Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.3.2 chip-by-chip Estimation wise Multicode Estimator . . . . . . . . . 236 6.3.3 SoftISoftO Symbol Demapping . . . . . . . . . . . . . . . . . . . . 240 6.3.4 Computational-Complexity Comparison . . . . . . . . . . . . . . . 241 6.4 Iterative Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 6.5.1 Performance obtained with perfect channel information for QPSK modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 v Table of Contents 6.5.2 Performance obtained with perfect channel information for 16- QAM modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.5.3 Performance obtained with iterative channel estimation . . . . . . 256 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 6.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7 Effects of Channel Estimation Errors on Turbo Equalization. . . . . . 263 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 7.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 7.3 ST Block Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.4 MMSE-based Turbo Equalization . . . . . . . . . . . . . . . . . . . . . . 270 7.4.1 Symbol-by-Symbol SoftISoftO Equalizer . . . . . . . . . . . . . . 271 7.4.2 Chip-by-Chip Estimation wise SoftISoftO Equalizer . . . . . . . . 276 7.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 7.7 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8 Conclusions and Suggestions for Further Research . . . . . . . . . . . . 288 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 8.2 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . . . . 294 8.3 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Appendices A Space-Time Transmit Diversity Scheme Used in WCDMA Systems . 301 A.1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 vi List of Tables 5.1 Number of mathematical operations required per block of M symbols per iteration for different MMSE-based equalization algorithms. P =MG+L, Q =MG, spreading gain G, maximum channel order L, number of receive antennas NR, SBS: symbol-by-symbol, CBC: chip-by-chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.2 Number of mathematical operations required per block ofM = 8 symbols per iteration for different MMSE-based equalization algorithms. P = 74, Q = 64, spreading gain G = 8, maximum channel order L = 10, number of receive antennas NR = 1, SBS: symbol-by symbol, CBC: chip-by-chip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.1 Numbers of mathematical operations required per block of MD symbols per iteration in different MMSE-based SoftISoftO multicode detection algorithms: Spreading Gain GD, Maximum Channel Oder L, Number of multicodes assigned to the desired user J , and Q =MDGD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.2 Number of mathematical operations required per block of MD = 8 sym- bols per iteration in different MMSE-based SoftISoftO multicode detec- tion algorithms for GD = 16, L = 9, J = 5, and Q = 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 vii List of Figures 2.1 Equivalent Baseband System Model: Transmitter Section . . . . . . . . . 36 2.2 Equivalent Baseband System Model: Receiver Section . . . . . . . . . . . 36 2.3 Power delay profiles of the equivalent channel models for Typical Urban Areas (TUA), Typical Rural Areas (TRA) and Typical Hilly Terrains (THT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4 Average BER of the proposed turbo equalization scheme for MCS-5 and TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.5 Average BER of the proposed turbo equalization scheme for MCS-5 and TRA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.6 Average BER of the proposed turbo equalization scheme for MCS-5 and THT channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7 Average BER of the proposed turbo equalization scheme for MCS-7 and TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.8 Average BER of the proposed turbo equalization scheme for MCS-8 and TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.9 Average BLER of the proposed turbo equalization scheme for MCS-5 and TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.10 Average BLER of the proposed turbo equalization scheme for MCS-8 and TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.11 Comparison of the BER performance of AprxImpl-1 with the ExactImpl: MCS-5, TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.12 Comparison of the BER performance of AprxImpl-2 with the ExactImpl: MCS-5, TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.13 Comparison of the BER performance of the MF with the ExactImpl: MCS-5, TUA channel model . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.14 Comparison of the BER performance estimated using the semi-analytical method with that obtained directly from simulation: MCS-5, TUA chan- nel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.1 Equivalent Baseband System Model . . . . . . . . . . . . . . . . . . . . . 79 3.2 Model for EXIT-band chart analysis of turbo equalization . . . . . . . . . 89 3.3 Distribution of the mutual information IeE computed at the output of the equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4 Distribution of the mutual information IeD computed at the output of the decoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 viii List of Figures 3.5 EXIT characteristics of MAP decoder for the convolutional code with generators (133, 171, 145)8 for different code rates . . . . . . . . . . . . . . 96 3.6 EXIT characteristics of MMSE-based equalization (ExactImpl) for differ- ent Eb/N0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.7 EXIT characteristics of MMSE-based equalization (ExactImpl) for differ- ent Eb/N0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.8 EXIT characteristics of different equalization schemes at Eb/N0 = 5dB . . 100 3.9 EXIT characteristics of different equalization schemes at Eb/N0 = 7dB . . 101 3.10 EXIT-band chart for turbo equalization using the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.11 EXIT-band chart for turbo equalization using MMSE AprxImpl-1 and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.12 EXIT-band chart for turbo equalization using MMSE AprxImpl-2 and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.13 EXIT-band chart for turbo equalization using MF and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. . . . . . 107 3.14 EXIT-band chart with simulated trajectories of turbo equalization us- ing the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 3 dB: MCS-5, TUA channel model. . . . . . . . 110 3.15 EXIT-band chart with simulated trajectories of turbo equalization us- ing the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 5 dB: MCS-5, TUA channel model. . . . . . . . 111 3.16 EXIT-band chart with simulated trajectories of turbo equalization us- ing the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 7 dB: MCS-5, TUA channel model. . . . . . . . 112 3.17 EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl and the MAP decoder at Eb/N0 = 7 dB: MCS-5, equalization-test channel model. . . . . . . . . . . . . . . . . . . . . . . . 113 3.18 EXIT-band chart with average trajectories of turbo equalization using MMSE AprxImpl-1 and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.19 EXIT-band chart with average trajectories of turbo equalization using MMSE AprxImpl-2 and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.20 EXIT-band chart with average trajectories of turbo equalization using the MF and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model.117 3.21 EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 14 dB: MCS-8, TUA channel model. . . . 118 3.22 EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 18 dB: MCS-8, TUA channel model. . . . 119 ix List of Figures 3.23 EXIT/POIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 14 dB: MCS-8, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.24 EXIT/POIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 18 dB: MCS-8, TUA channel model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.1 Discrete-time baseband system model of the proposed ST coded CIBS- CDMA scheme for uplink transmissions, Part I: Transmitter Section . . . 129 4.2 Discrete-time baseband system model of the proposed ST coded CIBS- CDMA scheme for uplink transmissions, Part II: Receiver Section . . . . . 129 4.3 BER performance comparison between the proposed ST coded uplink system and the CIBS system without ST coding in [9] . . . . . . . . . . . 144 4.4 Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively short channels (µ = 1 in DFSE schemes) . . . . . . . . . . . . . . . . . . . . . . 145 4.5 Performance investigation of different DFSE schemes for the proposed ST coded system with two receiver antennas (NR = 2) for relatively short channels (µ = 1 in DFSE schemes) . . . . . . . . . . . . . . . . . . . . . . 146 4.6 Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively long channels (µ = 1 in DFSE schemes) . . . . . . . . . . . . . . . . . . . . . . 148 4.7 Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively long channels (µ = 2 in DFSE schemes) . . . . . . . . . . . . . . . . . . . . . . 149 4.8 Comparison between the BER performance obtained using training-based channel estimation and that obtained using perfect channel state informa- tion for the proposed ST coded system without receiver diversity (µ = 1 in all DFSE schemes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1 Transmitter section of the equivalent discrete-time ST block coded CDMA downlink system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.2 Receiver section of the equivalent discrete-time ST block coded CDMA downlink system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.3 Power-delay profiles of the equivalent channel models . . . . . . . . . . . . 188 5.4 Comparison of the BER performance of CS and BS obtained using the symbol-by-symbol ExactImpl for K = 8 users . . . . . . . . . . . . . . . . 190 5.5 BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users . . . . . . . . . . . . . . . . . . . . . 191 5.6 BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 4 users . . . . . . . . . . . . . . . . . . . . . 192 5.7 BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users without complex scrambling . . . . . 193 x List of Figures 5.8 FER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users . . . . . . . . . . . . . . . . . . . . . 194 5.9 BER Performance obtained using the exact implementations of the symbol- by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users196 5.10 FER Performance obtained using the ExactImpls of the symbol-by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users . . . . . 197 5.11 Comparison of the BER performance of BS and CS obtained using the chip-by-chip SoftISoftO equalizer . . . . . . . . . . . . . . . . . . . . . . . 198 5.12 Comparison of the BER performance of BS and CS for BIS with different number of users at Eb/N0 = 4 dB . . . . . . . . . . . . . . . . . . . . . . . 199 5.13 BER performance obtained using the AprxImpl of the symbol-by-symbol SoftISoftO equalizer with SBIS for K = 4 users . . . . . . . . . . . . . . . 201 5.14 BER performance obtained using the AprxImpl of the symbol-by-symbol SoftISoftO equalizer with SBIS for K = 8 users . . . . . . . . . . . . . . . 202 5.15 BER performance obtained using the approximate implementations of the symbol-by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.16 Comparison of the semi-analytical upper bounds with the simulated BER performance for K = 8 users . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5.17 Comparison of the semi-analytical upper bounds with the simulated BER performance for K = 4 users . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.1 Transmitter section of the equivalent discrete-time system model . . . . . 217 6.2 Receiver section of the equivalent discrete-time system model . . . . . . . 218 6.3 Power delay profile of the equivalent channel model for Typical Urban (TU) areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.4 BER obtained using the symbol-by-symbol AprxImpl-1 with QPSK mod- ulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . . . . 250 6.5 BLER obtained using the symbol-by-symbol AprxImpl-1 with QPSKmod- ulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . . . . 251 6.6 Comparison of the BLER performances obtained using the approximate implementations of the symbol-by-symbol and chip-by-chip SoftISoftO multicode detection schemes with QPSK modulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.7 BLER obtained using the symbol-by-symbol AprxImpl-1 with QPSKmod- ulation for Rec = 3/4 and J = 5 . . . . . . . . . . . . . . . . . . . . . . . . 253 6.8 Variation of the BLER obtained using the symbol-by-symbol AprxImpl-1 with PDI for QPSK modulation, Rec = 1/2 and J = 5 . . . . . . . . . . . 254 6.9 BER obtained using the symbol-by-symbol AprxImpl-1 with 16-QAM modulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . 255 6.10 BLER obtained using the symbol-by-symbol AprxImpl-1 with 16-QAM modulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . 256 xi List of Figures 6.11 Comparison of the BLER performances obtained using the approximate implementations of the symbol-by-symbol and chip-by-chip SoftISoftO multicode detection schemes with 16-QAM modulation for Rec = 1/2 and J = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.12 BLER obtained using turbo multicode detection with iterative channel estimation for QPSK modulation, Rec = 1/2, J = 5, and P P = 20% . . . 258 7.1 BER performance of MMSE-based turbo equalization for the case of SBIS at CESNR = 10 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 7.2 BER performance of MMSE-based turbo equalization for the case of BIS at CESNR = 10 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 7.3 BER performance of MMSE-based turbo equalization scheme versus CESNR for the case of SBIS at Eb/N0 = 5 dB . . . . . . . . . . . . . . . . . . . . 283 7.4 BER performance of MMSE-based turbo equalization scheme versus CESNR for the case of BIS at Eb/N0 = 5 dB . . . . . . . . . . . . . . . . . . . . . 284 A.1 Comparison of the BER performances of BW-STBC and STTD schemes in the downlink of WCDMA systems with BS . . . . . . . . . . . . . . . . 305 A.2 Comparison of the BER performances of BW-STBC and STTD schemes in the downlink of WCDMA systems with CS . . . . . . . . . . . . . . . . 307 A.3 Comparison of the BER performances of BW-STBC and STTD schemes in the downlink of WCDMA systems without scrambling . . . . . . . . . . 308 A.4 Comparison of the BER performances of BW-STBC and STTD schemes in the downlink of WCDMA systems with Channel Model-A for OI&P test environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 xii Acronyms 3G Third Generation AprxImpl Approximate Implementation ARQ Automatic Repeat Request AWGN Additive White Gaussian Noise BCJR Bahl-Cocke-Jelinek-Raviv BER Bit Error Rate BICM Bit-Interleaved Coded-Modulation BIS Blind Interference Suppression BLAST Bell Labs Layered Space-Time BLER Block Error Rate BMUIS Blind Multiuser Interference Suppression BPEF Backward Prediction Error Filter BPSK Binary Phase Shift Keying BS Block Spreading BW Burst-Wise CAI Co-Antenna Interference CBC Chip-by-Chip CBER Code Bit Error Rate CDMA Code Division Multiple Access CESNR Channel Estimation Signal-to-Noise Ratio CIBS Chip-Interleaved Block Spread CMA Constant-Modulus Algorithm CP Cyclic Prefix CPICH Common Pilot Input Channel CRC Cyclic Redundancy Check CS Conventional Spreading CSI Channel State Information D-BLAST Diagonal-BLAST xiii Acronyms DD Delay Diversity DDFSE Delayed Decision Feedback Sequence Estimation DFE Decision Feedback Equalization/Equalizer DFSE Decision Feedback Sequence Estimation DFT Discrete Fourier Transform DMT Discrete Multi-Tone DS Direct Spread EDGE Enhanced Data Rates for Global Evolution EGPRS Enhanced General Packet Radio Service EI Extrinsic Information erfc Complementary Error Function ExactImpl Exact Implementation EXIT Extrinsic Information Transfer FD Frequency Domain FEP Front-End Prefilter FER Frame Error Rate FFF Feed-Forward Filter FFT Fast Fourier Transform FIR Finite Impulse Response GDD Generalized Delay Diversity GMC Generalized Multi-Carrier GMSK Gaussian Minimum Shift Keying GOFDM Generalized Orthogonal Frequency-Division Multiplexing GSNR Geometric Signal-to-Noise Ratio HARQ Hybrid Automatic Repeat Request HSDPA High Speed Downlink Packet Access IBI Inter-Block Interference ICI Inter-Chip Interference IFFT Inverse Fast Fourier Transform IPI Inter-Path Interference ISI Inter-Symbol Interference LC Low Complexity LE Linear Equalization LLR Log Likelihood Ratio xiv Acronyms LP Linear Prediction MAI Multiple Access Interference MAP Maximum A Posteriori MCI Multicarrier Interference MCS Modulation Coding Scheme MF Matched Filter MFB Matched-Filter Bound MIMO Multiple-Input Multiple-Output ML Maximum Likelihood MLSE Maximum Likelihood Sequence Estimation MMSE Minimum Mean Square Error MMUDFSE Multistage-Modified Unwhitened DFSE M-PSK M-ary Phase Shift Keying MRC Maximum Ratio Combining MSE Mean Square Error MSSNR Maximum Shortening Signal-to-Noise Ratio MUDFSE Modified-Unwhitened DFSE MUI Multiuser Interference NSCC Nonsystematic Convolutional Code OFDM Orthogonal Frequency-Division Multiplexing OI&PTE Outdoor-to-Indoor and Pedestrian Test Environments QAM Quadrature Amplitude Modulation QPSK Quadrature Phase Shift Keying PAR Peak-to-Average Ratio PCC Parallel Concatenated Code PCCC Parallel Concatenated Convolutional Code PCI Perfect Channel Information PDF Probability Density Function PEF Prediction Error Filter PEP Pairwise Error Probability POIT A Posteriori Information Transfer PSK Phase Shift Keying QAM Quadrature Amplitude Modulation QS Quasi-Synchronous xv Acronyms SBIS Semi-Blind Interference Suppression SBMUIS Semi-Blind Multiuser Interference Suppression SBS Symbol-by-Symbol SC Single Carrier SCC Systematic Convolutional Code SD Sphere Decoding SDMA Space-Division Multiple Access SIRF Shortened Impulse Response Function SISO Single-Input Single-Output SoftISoftO Soft-In/Soft-Out SOVA Soft-Output Viterbi Algorithm SOVE Soft-Output Viterbi Equalizer SNR Signal-to-Noise Ratio SRC Square-root Raised Cosine SS Spreading Sequence SSNR Shortening Signal-to-Noise Ratio ST Space-Time STBC Space-Time Block Coding STF Space-Time Frequency STS Space-Time Spreading STTD STBC-based Transmit Diversity SW Symbol-Wise TDMA Time Division Multiple Access TEQ Time Domain Equalizer THT Typical Hilly Terrains TIR Target Impulse Response TR Time Reversal TRA Typical Rural Areas TU Typical Urban TUA Typical Urban Areas UDFSE Unwhitened DFSE UMTS Universal Mobile Telecommunications System VTE Vehicular Test Environments WCDMA Wideband-Code Division multiple Access WL Widely Linear ZF Zero Forcing ZP Zero Padding xvi Acknowledgements I would like to express my profound gratitude and sincere appreciation to my graduate supervisor, Prof. Vijay K. Bhargava, for his excellent guidance, invaluable advices and continual encouragement throughout my studies at the UBC. Without his guidance and support, this work would not have been possible. My sincere gratitude goes to Dr. Robert Schober and Dr. Lutz Lampe, the supervi- sory committee members, for their valuable suggestions and motivation for this research. I express my thanks to all of my former and present colleagues at the Signal and Systems Lab for their friendship. I owe special thanks to Pom, Dejan, Zeljko, Jahangir and Mamun for their help and support in many ways. My heartfelt thanks go to Ranil and Lalinda, as my Sri Lankan friends at UBC, for their encouragements, moral support and helping me in numerous ways during my stay at UBC. Finally, I would like to take this opportunity to express my deepest gratitude and thanks to my parents who made many sacrifices to bring me up to this level, parents-in- law, other family members, and my beloved wife Lasantha for their love, moral support, patience and encouragement. This piece of work is dedicated to them. xvii Statement of Co-Authorship Dr. D. Djonin has co-authored the manuscript (Chapter 4) entitled “Space-Time Coded CDMA Uplink Transmission with MUI-free Reception”. I identified and formulated the research problem in consultation with my graduate supervisor Prof. Vijay K. Bhargava. I, myself, performed all mathematical derivations. Dr. Djonin checked the mathematical derivations. Computer simulations and analyzes of the results were totally carried out by me. Dr. Djonin helped me in organizing and writing the manuscript. xviii Chapter 1 Introduction 1.1 Background and Motivation During the last decade or so, the ever-increasing demand for broadband wireless services has been the main driving force behind research aimed at developing new transceiver technologies to improve the signal quality and spectral efficiency of wireless systems. Two major research themes have been dominating the development of new transceiver technologies for broadband wireless communication systems: (1) multiple-input multiple- output (MIMO) communication, and (2) turbo (iterative) detection and decoding tech- niques. A wireless communication system where both the transmitter and receiver use multiple antennas, is referred to as a MIMO system. As pocket mobile phones are being replaced with Internet access devices, such as laptops, it has become feasible to employ two or more antennas at the mobile receiver unit. Transmission techniques developed for MIMO systems can be classified into two broad categories [1]: • Category 1 maximizes the data rate or the capacity over MIMO channels, and • Category 2 maximizes the diversity gain in order to improve the system perfor- mance. In the first category, generally referred to as spatial multiplexing, different data streams are simultaneously transmitted from each transmit antenna in parallel and sep- arated at the receiver. Usually, the number of receive antennas must be greater than or equal to the number of transmit antennas. It is a well known fact that given the number 1 Chapter 1. Introduction of transmit antennas NT and number of receive antennas NR, the capacity of a MIMO system increases linearly with the minimum of NT and NR. Foschini [2] proposed a diagonally-layered space-time architecture known as Diagonal-Bell-Labs Layered Space- Time (D-BLAST) for spatial multiplexing. D-BLAST uses multi-element antenna arrays at both the transmitter and receiver and an efficient diagonally-layered coding structure in which code blocks are dispersed across diagonals in space-time. A simplified version of the BLAST detection algorithm known as Vertical-BLAST (V-BLAST) was intro- duced in [3]. Falling in Category 2, transmit diversity techniques, such as space-time (ST) codes, are used to exploit the diversity advantage offered by the multiple transmit antennas. Recently, hybrid approaches comprising spatial multiplexing and ST coding have gained increasing interest. The time-varying fading nature of wireless channels, which basically limits the chan- nel capacity, is one of the main obstacles to increase the data rate supported by a wireless channel. Wireless channels also usually suffer from interference from other users. These factors make it difficult for the receiver to determine the transmitted signal accurately. Diversity techniques can be effectively used to suppress the detrimental effect of channel fading [4, 5]. Different diversity techniques, such as time (or temporal), frequency, and antenna (or spatial) diversity techniques, are used in various applications of wireless communications [6]. However, in most applications it may not be possible to utilize all types of diversity techniques. Among the diversity techniques, spatial diversity is more appealing, as it can be used without loss of spectral efficiency [7]. Over the last few decades, receiver-spatial diversity (referred to as receiver diversity) techniques have been thoroughly investigated and widely employed in practice. Nevertheless, transmit- spatial diversity (referred to as transmit diversity) techniques have recently become more appealing than receiver diversity techniques, in particular for cellular mobile communi- cation systems, since it is easier to implement multiple antennas at the base station transmitter than at hand-held mobile receivers. 2 Chapter 1. Introduction Different transmit-diversity techniques, including delay diversity (DD) and ST cod- ing, have been intensively investigated and developed. The concept of DD was intro- duced by Wittneben [8] for frequency-flat fading channels. In DD-based schemes, spatial diversity is converted into multipath (or frequency) diversity by transmitting the signal stream from the first transmit antenna and the delayed replicas of the symbol stream from the other transmit antennas. In systems using DD, the equivalent channel (as seen by the receiver) can be modeled as a single-output single-output (SISO) frequency- selective channel. It was shown in [9] that the standard DD scheme [10] is unable to attain the full spatial-frequency diversity order in frequency-selective fading channels. The generalized delay diversity (GDD) scheme [9], in which the delays of transmitted symbol streams from different antennas are the integer multiple of the channel delay spread, was proposed for frequency-selective fading channels. The GDD scheme exploits the full diversity. Nevertheless, the resulting equivalent SISO channel may be very long leading to high equalization complexity [11]. In the last decade, ST coding has emerged as an powerful transmit-diversity technique to combat the detrimental effects of the fast-fading nature of cellular-mobile channels. There are mainly two types of ST coding schemes: ST trellis codes [5] and ST block codes [12]. ST block codes are more suitable for use in practical wireless systems due to linear maximum likelihood (ML)-decoding complexity. Therefore, in this thesis, we are particularly interested in ST block-coded transmission. Broadband radio channels usually experience frequency-selective fading. This is es- pecially true in urban areas where radio channels become highly time-dispersive due to severe multipath propagations. High time-dispersion results in severe inter-symbol inter- ference (ISI) and inter-chip interference (ICI)1 in time division multiple access (TDMA) and code division multiple access (CDMA) systems, respectively. Various transceiver technologies used to combat the detrimental effects of multipath propagation experi- 1also called inter-path interference (IPI). 3 Chapter 1. Introduction enced in typical broadband wireless communications can be broadly categorized into three main groups [13]: • signal-carrier modulation with broadband equalization in the time-domain • multicarrier modulation with orthogonal frequency-division multiplexing (OFDM) • signal-carrier modulation with broadband equalization in the frequency-domain. In OFDM-based systems, the frequency-selective channel is converted into a set of frequency-flat (i.e., ISI or ICI-free) subchannels through the inverse fast Fourier trans- form (IFFT) and CP (cyclic prefix) insertion operations at the transmitter and the fast Fourier transform (FFT) and CP removal operations at the receiver [14, 15]. As a consequence, in OFDM-based systems, low-complexity single-tap equalization for each subcarrier channel is sufficient. However, OFDM-based systems suffer from the follow- ing well-known issues (see, e.g., [14, 16, 17]): (1) non-linear distortion, which is caused by the high peak-to-average ratio (PAR) of the transmitted signal power; (2) high sen- sitivity to subcarrier frequency offsets, which causes intercarrier interference; and (3) incapability of achieving inherent frequency diversity2 in uncoded OFDM systems. Equalization techniques used in systems that adopt single-carrier modulation can be broadly divided into two categories: linear and non-linear equalization schemes. Linear-equalization schemes, including zero-forcing (ZF) and minimum mean square er- ror (MMSE) equalization schemes, suffer from a noise enhancement problem on channels having spectral nulls. Since mobile channels often exhibit spectral nulls and thus, linear equalizers are not suitable for most wireless systems. Decision feedback equalization (DFE) is most commonly used non-linear equalization technique in mobile communi- cations systems. Although DFE outperforms linear equalization, it suffers from error propagation at low signal-to-noise ratio (SNR) values, due to erroneous feedback. In order to minimize the probability of symbol detection errors, the optimal solution is the 2also called multipath diversity 4 Chapter 1. Introduction maximum likelihood sequence estimation (MLSE) [18, 19]. Unfortunately, MLSE’s high computational-complexity makes it difficult to be used in practice, particularly with large channel order and/or high-order modulation. Nevertheless, DFSE [20, 21] has emerged as a promising suboptimal alternative to MLSE. A better trade-off between performance and complexity can be achieved through decision feedback sequence estimation (DFSE). Frequency-domain equalization has emerged as a promising alternative technique to handle frequency selectivity in single-carrier broadband transmissions. Frequency- domain equalization is more attractive than time-domain equalization because of its simpler computational complexity. Because discrete-Fourier transform (DFT) is used at the receiver to reduce the signal processing complexity, the computational complexity of frequency-domain equalization is comparable to that of OFDM. Another advantage of frequency-domain equalization over time-domain equalization is that noise enhancement is less severe, since frequency-domain equalization can handle deep spectral nulls better. Unlike OFDM, single-carrier modulation with frequency-domain equalization does not suffer from high sensitivity to non-linear distortions and subcarrier offsets [13, 14]. In recent publications, turbo (iterative) equalization has received intensive atten- tion as a powerful technique to improve the performance of single-carrier transmissions. Most wireless systems use channel coding to suppress the adverse effects of channel impairments, such as noise, interference and channel fading. In channel-coded sys- tems, additional performance improvements over traditional one-pass (i.e., non-iterative) equalization can be achieved using turbo equalization that iteratively exchanges the soft information of code bits between the soft-in/soft-out (SoftISoftO) equalizer and channel decoder. Although turbo equalization has been extensively investigated for single-input single-output (SISO) systems, only limited research has been conducted on developing efficient turbo equalization schemes for ST block-coded transmissions, particularly in multiuser CDMA systems. 5 Chapter 1. Introduction Since radio channels usually exhibit frequency-selective fading in broadband com- munications, inherent multipath (or frequency) diversity can be exploited by using a powerful equalization technique to improve the received signal quality. Nevertheless, under deep fading conditions, it is necessary to have an additional means of diversity in order to attain high performance and, eventually, to increase the system through- put. Therefore, in our view, ST coding will be an essential technique to be employed in wireless communication systems. STBC-based transmit diversity (STTD) has been considered in the third generation (3G) Universal Mobile Telecommunications System (UMTS) for downlink transmission [22]. It is also highly likely that high-order mod- ulation schemes will be used in future mobile communication systems to increase user data rates. For example, the UMTS with High Speed Downlink Packet Access (HSDPA) uses 16-quadrature amplitude modulation (16-QAM), in addition to quadrature phase shift keying (QPSK) modulation [23]. The evolution of Enhanced Data Rates for Global Evolution (EDGE) considers adopting 16-QAM and 32-QAM modulation schemes in ad- dition to Gaussian-minimum shift keying (GMSK) and 8-PSK [24]. When a high-order modulation scheme is used, we need to have very precise estimates of the phase and/or the amplitude of the transmitted signal to achieve high performance. As a consequence, advanced receiver schemes, such as turbo equalization, are highly desirable for future wireless communications systems in achieving high performance. Therefore, the devel- opment of advanced and computationally-efficient receiver schemes for ST block-coded single-carrier transmissions becomes very important. 1.2 Literature Review In this section, the literature pertinent to the research areas of this thesis is reviewed. The topics covered include space-time coding, ST coding for frequency-selective fading channels, transmit-diversity techniques for DS-CDMA systems, channel shortening for ST coded systems, and turbo (iterative) receivers for ST coded DS-CDMA systems. 6 Chapter 1. Introduction 1.2.1 Space-Time Coding ST codes were originally investigated and designed for frequency-flat fading channels. ST trellis codes were first proposed in [5]. ST trellis codes require a multidimensional Viterbi algorithm at the receiver for decoding. Therefore, the obvious disadvantage of using ST trellis codes in MIMO systems is high decoding complexity. The concept of ST block coding (STBC) was introduced for two transmit antenna systems by Alamouti [25]. Later, in [12], ST block codes were extended to an arbitrary number of transmit antennas using the theory of orthogonal designs. The main characteristic of the ST block codes proposed in [12] is the orthogonality of the codes and hence, those codes are referred to as orthogonal-ST block codes. Maximum likelihood (ML) decoding of ST block codes is possible only using linear processing at the receiver [12]. Lower decoding complexity of ST block codes makes them more attractive compared with ST trellis codes. The performance of ST block codes was investigated in [4]. In [12] it was shown that a complex orthogonal design and the corresponding ST block code, which provides full transmission rate and full diversity, does not exist for more than two transmit antennas. In [12], a design strategy was also proposed to provide the maximum possible diversity, while maintaining the orthogonality between the codes. Jafarkhani [26] proposed and investigated the performance of quasi-orthogonal block codes, which provide full rate, but only a half of the maximum possible diversity. 1.2.2 STBC for Broadband Communications Initially ST block codes were extensively investigated for frequency-flat fading channels. However, the frequency-flat fading assumption is no longer valid for broadband commu- nications, as wireless channels typically experience frequency-selective fading. During the past few years, there had been extensive research on STBC for OFDM-based multi- carrier systems. As mentioned before, in OFDM systems the frequency-selective channel is converted into a set of frequency-flat fading subchannels through the IFFT operation 7 Chapter 1. Introduction at the transmitter. Therefore, ST block codes designed for flat fading channels can be simply adopted with OFDM for frequency-selective fading channels. For example, in [27] the Alamouti SW-ST code was adopted in generalized OFDM (GOFDM) systems with constant-modulus algorithm (CMA)-based blind channel estimation. In [28] STBC was applied for generalized multi-carrier CDMA (GMC-CDMA) systems. However, it is not possible to achieve full potential diversity in frequency-selective channels3 using ST block codes, designed for flat fading channels, over the tones in OFDM [7]. In [29] ST- frequency (STF) codes, which are capable of achieving the maximum possible diversity gain, were developed for OFDM systems. In single-carrier broadband communications, inter symbol interference (ISI) becomes a major performance limiting factor. Hence, channel equalization is indispensable to mit- igate the detrimental effects of the ISI on the performance. Since in MIMO systems the subchannel between each transmit and receive antenna pair has to be treated separately, the complexity of channel equalization in MIMO systems is significantly higher than that in SISO systems. There are two main approaches proposed in the literature for the application of STBC in single-carrier broadband systems, which experience frequency- selective fading: the first is the direct application of SW-ST block codes, which were originally designed for frequency-flat fading channels, and the second approach is to use burst-wise (BW)-STBC schemes preferably with frequency-domain equalization. Direct Application of the Alamouti SW-ST Block Code in Broadband Systems Direct application of the Alamouti SW-ST block code for transmissions over frequency- selective fading channels was considered in e.g., [30], [31], and [32]. In these schemes channel equalization is carried out using a computationally expensive MIMO equalizer 3The maximum achievable diversity order in frequency-selective fading channels is given by the prod- uct of the number of transmit antennas, receive antennas, and the number of realizable multipath components. 8 Chapter 1. Introduction in time domain. ST block decoding is performed separately after MIMO channel equal- ization. In these schemes, it is necessary to have multiple receive antennas in order to obtain good performance. Recently, Gerstacker et al. [33] proposed novel equalization techniques for the Alamouti SW-ST block-coded systems. In [33], widely linear (WL) processing is used to exploit the rotational variance of the ST block-coded transmit signal. Interestingly, unlike schemes proposed in e.g., [32], it is not necessary to have multiple receiver antennas, when WL processing is used. In [33], it was shown that the Alamouti SW-STBC scheme outperforms the time-reversal (TR)-based BW-STBC scheme [34] under moderate-to-fast time-varying channel conditions. Nevertheless, the performances of one-pass (i.e., non-iterative) linear and DFB equalization schemes with WL processing are basically limited by intersymbol interference (ISI) and co-antenna interference (CAI). BW-STBC for Frequency-Selective Fading Channels The inherent problems experienced by OFDM-based multicarrier systems, such as non- linear distortion and high sensitivity to carrier frequency offsets, motivated investigation of burst-wise (BW)-STBC schemes for single-carrier systems experiencing frequency- selective fading channels [35]. The time-reversal (TR)-based BW-STBC scheme was first proposed in [34] for transmission over frequency-selective fading channels and later, generalized to multiple transmit and receive antennas in [36]. In the TR-based BW- STBC scheme [34], ST block decoding and equalization operations are fully performed in the time domain. The TR-based BW-STBC scheme suffers from edge effects due to interburst interference (IBI) [37]. Al-Dhahir [35] proposed another BW-STBC scheme, in which each ST encoded sym- bol block is appended with a cyclic prefix (CP) of length equal to the channel memory in order to prevent the IBI. The use of the CP makes the subchannel matrix between each transmit and receive antenna pair circulant. A circulant matrix can be diagonalized 9 Chapter 1. Introduction using the fast Fourier transform operation [38] leading to low-complexity frequency- domain equalization. Nevertheless, in [37], it was shown that the BW-STBC scheme in [35] does not exploit the inherent multipath (or frequency) diversity and suggested that either redundant or non-redundant linear precoding can be used to exploit multi- path diversity. Zhou and Giannakis [39] developed another BW-STBC scheme, which is also TR-based, for single-carrier transmissions over frequency-selective channels. In the BW-STBC scheme of [39] zero padding (ZP) is used to avoid IBI and hence, referred to as the ZP-only BW-STBC scheme. Interestingly, the ZP-only BW-STBC scheme is capable of achieving the maximum possible diversity gain without precoding [37]. In [39], a frequency-domain ST block decoding scheme, which facilitates low-complexity frequency-domain equalization was proposed for the ZP-only BW-STBC scheme. In [37], a generalized framework was developed for analyzing single-carrier ST block-coded transmissions over frequency-selective channels and BW-STBC schemes proposed in, e.g., [34, 35, 39], were subsumed as special cases. Although BW-STBC schemes are able to yield the full potential diversity order4, their extension to fast time-varying channels is yet to be addressed. In [33] a high error floor in the BER performance of the TR-based BW-STBC scheme was observed for fast time-varying channels. 1.2.3 Transmit-Diversity Techniques for DS-CDMA Systems Receiver schemes were developed for WCDMA systems using Alamouti ST block coded- based STTD with cell-specific scrambling [40, 41] and without cell-specific scrambling [42]. In [43], a transmit diversity scheme based on space-time spreading (STS), which is a combination of single-transmit antenna DS-CDMA and Almouti STBC scheme [25], was proposed for the downlink transmission over flat fading channels in CDMA sys- tems. Since STS-based scheme was designed considering frequency-flat fading chan- nels, it does not exploit the maximum possible diversity in frequency-selective channels. 4Note that the BW-STBC scheme [35] requires precoding to achieve full potential diversity in frequency-selective channels. 10 Chapter 1. Introduction In [44] the ZP-only BW-STBC scheme [39] was applied for the downlink transmission over frequency-selective fading channels in DS-CDMA systems and ST chip equalization scheme was proposed. In [44], it was shown that much improved performance can be obtained using ST block-coded transmission based on the ZP-only BW-STBC scheme [39] over the STS-based scheme [43] in frequency-selective channels. In [45] the ZP-only BW-STBC scheme [39] was combined with block-spread (BS)-CDMA systems for the downlink transmission over frequency-selective channels. A per-tone based frequency- domain equalization scheme, which is computational-efficient, was also proposed for the ST block-coded BS-CDMA systems. In [46], a novel transmit diversity scheme, which is the combination of spreading sequences and algebraic constellations was proposed for up- link transmission over multipath channels in CDMA systems. However, in this scheme, it is assumed that a distinct spreading code is assigned to each transmit antenna. 1.2.4 Decision Feedback Sequence Estimation (DFSE) for ST Coded Transmissions Delayed decision feedback sequence estimation (DDFSE) was originally proposed in [20] as a promising suboptimal alternative for Forney-type MLSE (i.e., based on whitened channels). In DDFSE, the channel is split into two parts: the leading and tail parts. Intersymbol interference (ISI) due to the leading part of the channel is handled by MLSE and ISI due to the tail of the channel is canceled by decision feedback means. Hence, DDFSE can be viewed as a combination of the MLSE and DFE schemes [47, 48]. As a consequence of truncating the channel to its first few multipath components, the perfor- mance of DDFSE becomes highly sensitive to the channel phase. The best performance can be achieved when the underline multipath channel is minimum phase as in this case error propagation is minimized [21]. In [21], DFSE is developed for unwhitened channels based on the Ungerböeck approach of MLSE. Unlike in the whitened case, the performance of unwhitened DFSE (UDFSE) does not depend on the channel phase and 11 Chapter 1. Introduction therefore, it is not required to convert the channel into its minimum phase equivalent. However, the fact that the output of unwhitened channels (i.e., the matched-filter out- put) depends on both past and future input symbols, degrades the performance of the UDFSE (see [21] for more details). In [21], a modified version of the UDFSE (MUDFSE) scheme was proposed to cancel the effect of anti-causal transmitted symbols on the selec- tion of survivor paths using conditional decisions. It was also shown that the performance can be further improved using the multistage MUDFSE (MMUDFSE) scheme in which decisions are made based on the previous stage at the expense of increased complexity. In order to avoid the effect of anticausal symbols, unwhitened channel can be con- verted into whitened one by prefiltering and then apply the whitened DFSE to the whitened output. In [20], noise whitening prefilter was used to convert the channel into a minimum-phase whitening channel. In [49], it was proposed to use a feedforward filter (FFF) of MMSE-FBE as the prefilter. However, in [50] it was shown that the approach of using the FFF of MMSE-FBE is highly sensitive to parameters. In [50], a low-complexity linear prediction (LP)-based algorithm was proposed to estimate a finite impulse response (FIR) prefilter. In [51], an MMSE-based prefilter was used instead of a noise whitening filter. It was shown that improved performance can be achieved using MMSE-DFSE compared with W-DFSE. MMSE-DFSE is preferable for severely distorted channels with spectral nulls. In [48] DFSE was considered for delay diversity- based and ST trellis coded systems and the performance was investigated for the EDGE systems. Moreover, an approach based on the maximizing shortening signal-to-noise ratio (MSSNR) criterion was proposed for designing a front-end prefilter (FEP), which is used to shorten the multipath channels. More recently, Schober et al. [52] DFSE was considered for TR-based BW-ST coded transmission. 12 Chapter 1. Introduction 1.2.5 Turbo Equalization The turbo principle was introduced with iterative decoding of parallel concatenated convolutional codes (so called ”Turbo Codes”) in [53]. Shortly after, in [54] the concept of turbo equalization (or turbo detection) was introduced for convolutionally coded information transmission over frequency-selective channels. Then, in [55] the concept of turbo equalization was reviewed. In principle, turbo equalization can be considered as to be equivalent to the serial concatenated convolutional coded system. In turbo equalization, we replace the inner convolutional code with the ISI channel (i.e., the channel with a memory) [55]. The BER performance can be improved significantly by exchanging the soft information of code bits between the equalizer and the channel decoder over iterations compared with one-pass (i.e., non-iterative) equalization and decoding. Hence, turbo equalization has recently received a considerable attention in the literature (see e.g., [56] for an overview on turbo equalization). The original turbo equalization scheme proposed in [54] used the soft output Viterbi equalizer (SOVE). In [57], the performance of turbo equalization was studied using the MAP-based SoftISoftO equalizer. Moreover, in [55], the performance of turbo equal- ization using different algorithms (namely, Log-MAP, Max-Log-MAP and SOVE/SOVA algorithms) were compared and investigated with perfect channel information and mis- matched channel estimation. Even though the MAP equalizer gives optimal perfor- mance, it is highly computationally expensive. Unfortunately, similarly to the MAP equalizer, its suboptimal variants also use the full-state trellis. Therefore, these schemes are also highly computationally-expensive for long channels and/or for large signal constellations. There are several reduced-complexity techniques proposed in the lit- erature. Those techniques can be mainly categorized into two groups: reduced-state trellis-based schemes and MMSE-based schemes. In [58], a DDFSE-based SoftISoftO equalizer was proposed for coded ISI transmission. As mentioned previously, the per- formance of DDFSE can be improved if the channel is minimum phase and the channel 13 Chapter 1. Introduction can be converted into its minimum phase equivalent through prefiltering. In [59], two reduced-complexity algorithms based on the Lee algorithm [60] were proposed. The first algorithm uses hard-decision feedback while the second uses soft-decision feedback. Furthermore, an algorithm, which is a combination of reduced state BCJR and the Lee algorithm, was also proposed. Due to low computational-complexity, MMSE-based turbo equalization is very at- tractive for practical mobile receivers compared with trellis-based equalizers and hence, in recent years, MMSE-based turbo equalization received significant attention. The con- cept of MMSE-based turbo equalization was introduced in [61]. Later, in e.g., [62, 63], MMSE-based turbo equalization was reexpressed with the complete a priori informa- tion (i.e., the mean and variance of symbols). In MMSE-based turbo equalization, the trellis-based SoftISoftO equalizer (e.g., MAP-based equalizer, SOVE) found in the origi- nal schemes is replaced with a combination of the soft ISI canceler and the MMSE filter. In [64], linear MMSE-based turbo equalization was extended to high-order modulation schemes (i.e., for QPSK and 8-PSK). It is noteworthy that in MMSE-based turbo equal- ization the output of the MMSE filter (i.e., soft estimates of symbols) is approximated by a Gaussian distribution. This approach facilitates the computation of the extrinsic log likelihood ratio values of code bits, which are transferred to the channel decoder as the a priori information. In [65], turbo equalization was applied for ST trellis coded transmission over MIMO frequency-selective channels. In the proposed scheme, channel shortening is used to reduce the complexity of MAP equalizer. In ST trellis coded systems, the equivalent channel model depends on the transmitted symbols and therefore, all-pass filter can not be used to convert the channel into its minimum-phase equivalent, which is necessary in reduced-state equalization [20] to attain high performance. Hence, alternatively, in [65] a set of prefilters was designed based on the mean square error (MSE) criterion to concentrate the energy of the channel to a few number of taps (in other words, to 14 Chapter 1. Introduction shorten the equivalent channel). 1.2.6 Turbo (Iterative) Receivers for ST Coded DS-CDMA Systems In [66], an MMSE-based turbo interference cancelation scheme was proposed for single- input single-output (SISO) DS-CDMA systems. In the turbo receiver proposed in [66], soft interference cancelation was performed using the a priori information of code bits fed back from the SoftISoftO channel decoder from the previous iteration. Soft inter- ference cancelation is followed by MMSE filtering to minimize residual MAI and ISI. The extrinsic information of each code bit is computed by modeling the output of the linear-MMSE filter using a Gaussian distribution. After deinterleaving, the extrinsic information of code bits is passed to the bank of MAP-channel decoders as the a priori information. Following a similar approach, in [67, 68], turbo receiver schemes were pro- posed for multiuser systems with multiple receiver antennas. However, in the schemes presented in [67, 68], only receiver diversity is considered, but not ST coding. The principle of turbo soft interference cancelation [66] was adapted in space-division multiple-access (SDMA) systems with transmit diversity for transmission over flat-fading channels in [69]. Nevertheless, this scheme needs multiple receiver antennas for proper receiver operation (the number of receiver antennas needed will depend on the product of the number of active users and the number of transmit antennas). Jayaweera et al. [70] proposed a MMSE-based turbo multiuser receiver for the uplink transmission over multipath channels in ST block-coded CDMA systems. A RAKE-based iterative receiver, in which MMSE filtering and ST block decoding are carried out separately for each multipath components, was also proposed to lower the computational complexity. Nevertheless, in [70], it was assumed that after spreading each transmit symbol is zero- padded to prevent ISI, which leads to low spectral efficiency, specially for channels with large delay spreads. Therefore, the scheme proposed in [70] may not be suitable when the channel is highly dispersive. 15 Chapter 1. Introduction 1.2.7 Channel Shortening for ST Coded Systems Channel shortening was originally considered discrete multitone (DMT) systems. In DMT transceivers, each input symbol block of size B is appended with a cyclic prefix (CP), which is the last ν symbols of the input block, in order to avoid interblock inter- ference (IBI) between successive transmitted blocks [71, 72]. The length of the CP is determined by the length of the channel impulse response L (i.e., ν ≥ L− 1). Inclusion of a CP in each transmitted block leads to reduce the data rate or spectral efficiency by a factor of ν/(B + ν). It is possible to reduce the loss in the data rate by increasing the block length B. However, it is not desirable to increase the symbol block length, as it will cause to increase the processing delay, computational complexity, and memory requirements [72]. In particular, for highly time-dispersive channels, the reduction in data rate will be significant, if we select ν ≥ L− 1 to prevent IBI. A channel shortening prefilter (also known as a time domain equalizer (TEQ)) can be used at the receiver to shorten the effective length of the channel to avoid using a long CP. There are different criteria developed in the literature to design TEQs (see for literature review e.g., [73]). In [72], a channel shortening impulse response filter was proposed based on the maximum shortening signal to noise ratio (MSSNR) criterion, which minimizes the energy outside the target window. Note that SSNR is defined as the ratio of the energy of the components in the target window to that of the components outside the target window. This algorithm requires the shortened impulse response function (SIRF) to be less than or equal to the length of the target impulse response (TIR). In [74], the MSSNR algorithm proposed in [72] is extended so that the SIRF can have any length. In [75], the coefficients of the optimum TEQ was obtained by maximizing the geometric signal to noise ratio (GSNR). In [71], a TEQ was designed by minimizing the mean square error (MSE) between the output obtained by equalizing with the TEQ and that obtained using the target impulse response without adding noise. A different approach was presented in [73] to obtain the optimal TEQ by maximizing 16 Chapter 1. Introduction the channel capacity. However, this scheme is not suitable for practical implementation due to high computational complexity. A near-optimal scheme, which minimizes the weighted sum of the ISI power (called min-ISI method), was also proposed in [73]. In [76], the MMSE impulse response shortening algorithm was reformulated and it was shown that MSSNR algorithm in [72] and the min-TEQ algorithm in [73] as a special case of MMSE-based algorithm. Al-Dhahir [77] has extended the MSE criterion for designing TEQs to MIMO chan- nels. In [65], an MSE-based channel shortening scheme was proposed for ST trellis coded systems in order to reduce the complexity of turbo equalization (i.e., by reducing the number of trellis states in the MAP equalizer). In the proposed channel shortening scheme, the TIR is determined by maximizing the sum of the shortened channel output signal-to-noise ratios (SNRs). It was shown that in this scheme [65] time diversity is transformed into receiver antenna diversity. Hence, the performance loss due to chan- nel shortening (as a result of reduction in the maximum achievable diversity gain) is partially obtained. In [78], a channel shortening prefilter was designed based on the MSSNR criterion for ST Trellis coded systems. Furthermore, in [78] a modification was presented to overcome the problems encountered in the modified-BCJR algorithm-based MAP equalizer in [79] over the original BCJR algorithm-based MAP equalizer. 1.3 Theme and Objectives of the Thesis During the last decade, STBC has emerged as an effective transmit-diversity technique to combat the detrimental effects of channel fading, and has been extensively investigated for frequency-flat fading channels. However, the frequency-flat fading assumption is no longer valid in frequency-selective broadband communications. Research on ST block- coded transmissions over frequency-selective channels, particularly in multiuser CDMA systems, has received relatively limited attention. In our view, future wireless commu- nication systems will not only include STBC, but also high-order modulation schemes 17 Chapter 1. Introduction to provide ubiquitous broadband wireless access. Therefore, it is necessary to use ad- vanced receiver schemes to achieve high system performance, especially in very hostile wireless environments. For example, turbo receiver processing techniques can be used to obtain improved performance over non-iterative receiver counterparts. The overall goal of this thesis is to investigate and develop advanced and computationally-efficient receiver schemes for single-carrier ST block-coded transmissions over frequency-selective fading channels, with possible applications in future generation mobile communication systems. The specific objectives of the thesis include: 1. To develop an MMSE-based turbo equalization scheme for Alamouti SW-ST block- coded systems, to devise a semi-analytical method to estimate the BER perfor- mance obtained using turbo equalization, and to investigate the performance of turbo equalization in Alamouti SW-ST block-coded EDGE/EGPRS systems. 2. To analyze the convergence behavior of turbo equalization in Alamouti SW-ST block-coded MIMO systems with finite block lengths, using the extrinsic informa- tion transfer characteristics (EXIT)-band chart technique which considers bit-wise mutual information transfer characteristics 3. To apply BW-STBC for uplink transmission over frequency-selective fading chan- nels in block-spread-CDMA systems with multiuser interference-free reception, and to investigate and propose advanced equalization schemes to achieve high system performance in ST block-coded CDMA uplink transmission. 4. To propose frequency-domain MMSE-based turbo equalization schemes for BW-ST block-coded transmission in the downlink of DS-CDMA systems, and to develop a semi-analytical approach to estimate the bit error rate (BER) performance ob- tained using turbo equalization. 5. To develop frequency-domain turbo multicode detection schemes for BW-ST block- coded transmission over frequency-selective fading channels in the downlink of DS- 18 Chapter 1. Introduction CDMA systems, and to investigate the performance of turbo multicode detection in a ST block-coded UMTS system with HSDPA. 6. To develop a frequency-domain MMSE-based turbo equalization scheme in the presence of channel estimation errors for BW-ST block-coded downlink transmis- sion in CDMA systems and to investigate the impact of channel estimation errors on the BER performance of turbo equalization. 1.4 Outline of the Thesis The remainder of the thesis is organized as follows: • In Chapter 2, we develop an MMSE-based turbo equalization scheme for Alam- outi SW-ST block-coded MIMO systems. We use widely linear processing at the receiver level to exploit the rotational variance of ST block-coded transmitted sig- nals. Two approximate implementations are proposed to reduce the computational complexity involved in MMSE filtering. We also devise a semi-analytical method to estimate the BER performance obtained using the proposed turbo equalization scheme after an arbitrary number of iterations. We investigate the performance of the proposed turbo equalization scheme for EDGE/EGPRS systems. • In Chapter 3, the probabilistic convergence behavior of MMSE-based turbo equal- ization in Alamouti ST block-coded MIMO systems with finite block lengths is investigated. The extrinsic information transfer characteristics (EXIT)-band chart technique, which was originally proposed for analyzing turbo decoding of parallel concatenated codes with finite block lengths, is applied. The impact on the con- vergence behavior using STBC is analyzed through the EXIT-band characteristics of the SoftISoftO equalizer. The convergence nature of the exact and approximate implementations of the MMSE-based SoftISoftO equalization scheme are compared 19 Chapter 1. Introduction through the EXIT-band chart analysis. We propose the hybrid EXIT/POIT5-band chart to analyze the convergence behavior of turbo equalization, when the a poste- riori information delivered by the channel decoder is transferred to the equalizer as the a priori information. The estimates on the BER performance are obtained from the EXIT-band chart using a semi-analytical approach after an arbitrary number of iterations. • In Chapter 4, we consider single-carrier ST block-coded uplink transmissions in chip-interleaved block-spread (CIBS)-CDMA systems. A system architecture is developed for ST coded uplink transmission over frequency-selective fading chan- nels with multiuser interference (MUI)-free reception. DFSE is applied for ST block-coded uplink transmission in CIBS-CDMA systems. The performances of different DFSE schemes are investigated and compared. A new UDFSE-based scheme is proposed to avoid the high error floor behavior experienced by the ex- isting UDFSE schemes. • In Chapter 5, we investigate frequency-domain minimummean square error (MMSE)- based turbo equalization for the downlink of ST block-coded CDMA systems. A novel system architecture, that considers both conventional (i.e., symbol-wise) and block spreading, is developed. We adapt burst-wise (BW)-STBC instead of Alamouti SW-STBC considered for STTD in WCDMA systems, and demon- strate its superior performance in frequency-selective fading channels. We develop frequency-domain symbol-by-symbol and chip-by-chip MMSE-based turbo equal- ization schemes. Approximate implementations are also devised for both symbol- by-symbol and chip-by-chip equalization schemes. The computational complexities of different proposed algorithms are evaluated and compared. We derive semi- analytical upper bounds on the bit error rate (BER) obtained using the proposed turbo equalization schemes. 5posteriori information transfer 20 Chapter 1. Introduction • In Chapter 6, frequency-domain turbo multicode detection is considered for the ST block-coded CDMA downlink. We propose frequency-domain symbol-by-symbol and chip-by-chip MMSE-based SoftISoftO multicode detectors. Approximate im- plementations are developed for both symbol-by-symbol and chip-by-chip SoftI- SoftO multicode detectors in order to reduce their computational complexities. The computational complexities of the exact and approximate implementations are evaluated and compared. We also propose an iterative channel estimation scheme. The performance of the proposed turbo multicode detection schemes is investigated for a WCDMA system with HSDPA. • In Chapter 7, we investigate the effects of channel estimation errors on MMSE- based turbo equalization in the downlink of ST block-coded CDMA systems. 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Anderson, “Concatenated decoding with a reduced-search BCJR algorithm,” IEEE Journal on Selected Areas in Communications, vol. 16, pp. 186– 195, Feb. 1998. 31 Chapter 2 Turbo Equalization for Alamouti Space-Time Block-Coded Transmission1 2.1 Introduction Turbo equalization has recently drawn considerable attention due to its improved per- formance over conventional one-pass (i.e., non-iterative) hard-decision based equaliza- tion, specially in severe intersymbol interference (ISI) channels. The concept of turbo equalization was introduced in [1]. In turbo equalization, the extrinsic information of code bits is exchanged between the soft-input soft-output (SoftISoftO) equalizer and the SoftISoftO channel decoder in an iterative manner. Initially, turbo equalization was considered using full-state trellis-based SoftISoftO equalizers (e.g., the soft-output Viterbi equalizer [1], maximum a posteriori (MAP) algorithm-based equalizer and its sub-optimal variants [2]). Unfortunately, trellis based soft equalizers are highly compu- tationally expensive for channels with a long delay spread and/or for large signal constel- lations. Specifically, the Enhanced Data Rates for Global Evolution (EDGE) systems use 8-ary phase shift-keying (8-PSK) high-order modulation to increase the achievable data rate and spectral efficiency. Therefore, trellis-based soft equalizers may not be 1A version of this chapter has been submitted for publication. Wavegedara, K. C. B. and Bhar- gava, V. K. Turbo Equalization for Alamouti Space-Time Block Coded Transmission. 32 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission suitable for EDGE systems. On the other hand, minimum mean square error (MMSE)- based turbo equalization schemes were proposed for single-input single-output (SISO) channels, e.g., in [3]. MMSE-based turbo equalization is very appealing for practical implementation due to its low computational complexity compared with trellis-based SoftISoftO equalizers. Space-time block coding (STBC) can be effectively used to alleviate the detrimental effects of channel fading. The concept of STBC was introduced for frequency-flat fading channels by Alamouti in [4]. Later, burst wise-STBC (BW-STBC) schemes were pro- posed for frequency-selective fading channels, e.g., [5, 6]. BW-STBC schemes assume that subchannels are invariant over two consecutive symbol blocks in order to satisfy the orthogonality condition. Although these schemes are able to yield the maximum possible diversity gain, their extension to fast fading channels is yet to be addressed. In [7], a high error floor in the bit error rate (BER) performance of the BW-STBC was observed for fast time-varying channels. Direct application of the Alamouti symbol-wise (SW) ST block code for transmission over frequency-selective channels was considered in [8]. However, in these schemes, it is necessary to have multiple receive antennas in order to achieve good performance. ST block decoding is performed separately af- ter conventional multiple-input multiple-output (MIMO) equalization. Recently, in [7], Gerstacker et al. proposed novel equalization techniques for the Alamouti SW-STBC scheme. In [7], widely linear (WL) processing is used to exploit the rotational variance of the ST block-coded transmit signal. Interestingly, in contrast to schemes such as [8], equalization techniques based on WL processing can achieve high performance without receiver diversity. In [7], it was also shown that the Alamouti SW-STBC scheme outper- forms BW-STBC under moderate-to-fast fading channel conditions. Nevertheless, the performances of the one-pass (hard-decision based) MMSE equalization schemes even with WL processing are highly limited by intersymbol interference (ISI) and co-antenna interference (CAI). 33 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission In [9] an MMSE-based turbo equalization scheme was proposed for MIMO systems, however, STBC was not considered. In [10], turbo equalization was considered for BW-ST block-coded EDGE systems. In this chapter, we therefore investigate turbo equalization for Alamouti SW-ST block-coded MIMO systems with NT = 2 transmit antennas and NR ≥ 1 receive antennas. The contributions and novelties of this chapter are summarized as follows: • The main contribution is the development of an MMSE-based turbo equaliza- tion scheme for Alamouti ST block-coded MIMO systems. In the proposed turbo equalization scheme, WL processing is used to improve the performance of ST block-coded transmission, as previously shown for one-pass hard-decision based equalization in [7]. Both linear and decision feedback-MMSE SoftISoftO equaliz- ers are derived. At the beginning of each iteration, soft ISI and CAI cancelations are carried out. Then, MMSE filtering is performed separately for each transmit- ted symbol.2 It is noteworthy that, unlike existing turbo equalization schemes (e.g., [3, 9]), since WL processing is considered in the proposed scheme, in addi- tion to the variance, we also require the pseudo variance of transmitted symbols for MMSE filtering. • In order to reduce the computational complexity involved in MMSE filtering, we propose two approximate implementations, that offer better trade-offs between performance and complexity. • We devise a semi-analytical method to obtain the BER estimates of the proposed turbo equalization scheme after an arbitrary number of iterations. • As a potential application for the proposed turbo receiver scheme, we consider Enhanced General Packet Radio Service (EGPRS), which is the packed-switched- 2Since we derive the MMSE filters for the equivalent signal model obtained after WL processing, MMSE filtering can be considered to be functionally equivalent to joint equalization and ST block decoding operations. 34 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission based mode of the EDGE technology. The performance of the proposed turbo receiver is thoroughly investigated for EGPRS systems using computer simulation. Notation: Bold lower case letters represent vectors while bold upper case letters de- note matrices; (.)T , (.)∗, and (.)H denote the operations transpose, complex conjugate (component-wise), and Hermitian transpose, respectively; IM and 0M×N denote the M ×M identity matrix and an all-zero matrix of size M × N ; diag{s} represents a diagonal matrix, where vector s is on the diagonal. <{x} and ={x} denote the real and imaginary parts of a complex random variable x. E{x} and Var{x} , E{|x − E{x}|2} represent the expected value and variance of x, respectively; Cov{x1, x2} , E{(x1 − E{x1})(x2 − E{x2})∗} stand for the covariance between random variables x1 and x2. d.e represents the ceiling function. 2.2 System Description We consider Alamouti SW-ST block-coded transmission over MIMO frequency-selective fading channels. The equivalent complex baseband transmitter and receiver models are shown in Figure 2.1 and Figure 2.2, respectively. The transmitter is equipped with NT = 2 antennas, while the receiver has either a single or multiple receive antennas (NR ≥ 1). A block of Ni information bits {b(k)}Ni−1k=0 , b(k) ∈ {0, 1} is fed into a binary convolutional encoder of constraint length ν and code rate Rc. The resulting block of code bits {c(m)}Nc−1m=0 is then interleaved by a random block interleaver, where Nc = Ni/Rc. The interleaved bit sequence is denoted by {c(m′)}Nc−1m′=0, where m′ = Π(m) and Π(.) denotes the interleaver function. Let Q represent the modulation order (e.g., Q = 1 for BPSK and Q = 3 for 8-PSK). The interleaved bit sequence is grouped into sets of Q bits, cn = {cn(0), cn(1), · · · , cn(Q− 1)}, n = 0, 1, · · · ,Ns − 1, where Ns = Nc/Q. Each set of Q modulating bits, cn is mapped into MPSK symbols s(n) ∈ {α0, α1, · · · , αM−1}, where αν = e j2piν/M , ∀ν, ν = 0, 1, · · · ,M − 1, and M = 2Q is the signal constellation size. Note that the signal constellation has zero mean Es , 1/M ∑M−1 ν=0 αν = 0 and 35 C h a p ter 2 . T u rb o E q u a liza tio n fo r A la m o u ti S p a ce-T im e B lo ck -C o d ed T ra n sm issio n Figure 2.1: Equivalent Baseband System Model: Transmitter Section Figure 2.2: Equivalent Baseband System Model: Receiver Section 36 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission unit variance σ2s , 1/M ∑M−1 ν=0 |αν |2 = 1. We assume that the symbols {s(n)}Ns−1n=0 are statistically independent. After constellation mapping, two consecutive symbols s(2n) and s(2n + 1) are fed into the ST block encoder. The ST block encoder outputs sequences s1(i) and s2(i) according to the following rule [4]: s1(i) =   s(2n); i = 2n −s∗(2n + 1); i = 2n + 1 (2.1) s2(i) =   s(2n+ 1); i = 2n s∗(2n); i = 2n+ 1 (2.2) ST block encoded sequences s1(i) and s2(i) are first pulse-shaping filtered and then sent to the first and second transmit antennas, respectively. The (continuous-time) received signal at each receive antenna is perturbed by an additive white Gaussian noise (AWGN) process ηnr(t), nr = 1, 2 · · · ,NR, with two- sided power spectral density N0/2. Let us define the discrete-time overall impulse response of the transmit pulse-shaping filter, the multipath subchannel between the ntth transmit antenna and the nrth receive antenna, and the receiver input filter as hnt,nr , [hnt,nr(0), hnt ,nr(1), · · · , hnt,nr(L)]T , nt = 1, 2, and nr = 1, 2, · · · ,NR, NR ≥ 1. Note that without loss of generality, we assume that the channel order L is the same for all subchannels. The discrete-time received signal in the nth symbol interval at the nrth receive antenna, which is obtained by sampling the output of the receive input filter at the symbol-rate, can be given as rnr(n) = L∑ l=0 [h1,nr(l)s1(n − l) + h2,nr(l)s2(n − l)] + ηnr(n). (2.3) ηnr(n) represents the (discrete-time) complex-valued zero-mean AWGN with variance σ2η , E{|ηnr (n)|2} = N0/Ts, where Ts is the symbol duration. We assume that the 37 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission multipath subchannels are time-invariant over a transmitted symbol burst and that the channel fading between symbol bursts is independent. Similar to its use in hard decision-based equalization schemes in [7], WL processing is used in the proposed turbo equalization scheme to exploit the rotational variance of the ST block-coded transmit signal. In a WL-based receiver, the received signal and its complex conjugate are processed together. Let us define a vector comprising two consecutive samples of the received signal at the nrth antenna and the complex conjugates of the polyphase components as [7] r̃nr(n) , [rnr(2n) rnr (2n+ 1) r ∗ nr (2n) r ∗ nr(2n + 1)] T , (2.4) for nr = 1, 2, · · · , NR. We can express r̃nr(n) as r̃nr(n) = L̃∑ l=0 H̃nr(l)s̃(n− l) + η̃nr(n), (2.5) where L̃ = dL/2e and s̃(n) , [s(2n) s∗(2n) s(2n+ 1) s∗(2n + 1)]T , η̃nr(n) , [ηnr(2n) ηnr(2n + 1) η ∗ nr(2n) η ∗ nr(2n + 1)] T , H̃nr(l) ,   h1,nr(2l) h2,nr(2l − 1) h2,nr(2l) −h1,nr(2l − 1) h1,nr(2l + 1) h2,nr(2l) h2,nr(2l + 1) −h1,nr(2l) h∗2,nr(2l − 1) h∗1,nr(2l) −h∗1,nr(2l − 1) h∗2,nr(2l) h∗2,nr(2l) h ∗ 1,nr(2l + 1) −h∗1,nr(2l) h∗2,nr(2l + 1)   . Note that, unlike [7], for convenience we change the order of the terms when s̃(n) is de- fined, and H̃nr(l) is accordingly modified. Then, we define a vector r̃(n) , [r̃ T 1 (n) r̃ T 2 (n) · · · r̃TNR(n)]T by stacking signal vectors of NR receive antennas. r̃(n) can be expressed 38 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission as r̃(n) = L̃∑ l=0 H̃(l)s̃(n− l) + η̃(n), (2.6) where H̃(l) = [H̃T1 (l) H̃ T 2 (l) · · · H̃TNR(l)]T , and η̃(n) = [η̃T1 (n) η̃T2 (n) · · · η̃TNR(n)]T . Let us now introduce a sliding-window model in the nth time interval as follows: r̃n = H̃s̃n + η̃n, (2.7) with r̃n , [r̃T (n+N2) · · · r̃T (n) · · · r̃T (n−N1)]T , s̃n , [s̃T (n+N2) · · · s̃T (n) · · · s̃T (n− N1 − L̃)]T , and η̃n , [η̃T (n +N2) · · · η̃T (n) · · · η̃T (n −N1)]T . The equivalent channel matrix of size 4NR(Ñf + 1)× 4(Ñf + L̃+ 1) is given as H̃ =   H̃(0) · · · H̃(L̃) 0 · · · · · · 0 0 H̃(0) · · · H̃(L) 0 · · · 0 ... . . . . . . ... 0 · · · · · · 0 H̃(0) · · · H̃(L̃)   , (2.8) where Ñf = N1 + N2. We input the vector r̃n of length 4(Ñf + 1) to the iterative equalizer in the nth symbol interval. In the next section, we describe the proposed MMSE-based turbo equalizer for Alamouti ST block-coded transmissions over a MIMO frequency-selective fading channel. 2.3 MMSE-based Turbo Equalization Different from the existing MMSE-based turbo equalization schemes (e.g., SISO sys- tems [11, 12] and MIMO systems [9]), our proposed scheme performs both equaliza- tion and ST block decoding jointly at each iteration. At the beginning of each itera- tion, the SoftISoftO channel decoder delivers the a priori log likelihood ratios (LLRs) LaE(c(m ′)) , ln P [c(m ′)=1] P [c(m′)=0] , m ′ = 0, 1, · · · ,Nc − 1, computed in the previous iteration, to 39 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission the SoftISoftO symbol mapper. The symbol a priori probability can be obtained using the a priori LLRs LaE(c(m ′)) as P (s(n)=αν) = Q−1∏ q=0 P (cn(q)=bν(q)) = Q−1∏ q=0 1 2 (1 + (2bν(q)− 1).tanh(LaE(cn(q))/2)) . (2.9) As will be seen, statistics including the mean s̄(n), variance υ2(n), and pseudo variance υ2p(n) of every symbol {s(n)}Ns−1n=0 are required in SoftISoftO MMSE-based equalization.3 The SoftISoftO symbol mapper computes these statistics using the symbol a priori probability as follows: s̄(n) , E{s(n)} = M−1∑ ν=0 ανP (s(n)=αν), υ2(n) , Cov{s(n), s(n)} = M−1∑ ν=0 |αν |2P (s(n)=αν)− |s̄(n)|2, and υ2p(n) , Cov{s(n), s∗(n)} = M−1∑ ν=0 (<{αν}2 −={αν}2)P (s(n)=αν) − ( M−1∑ ν=0 <{αν}P (s(n)=αν) )2 + ( M−1∑ ν=0 ={αν}P (s(n)=αν) )2 . We can easily show that for BPSK signals (i.e., Q = 1), s̄(n) = σstanh(L a E(cn(1))/2), and υ2(n) = υ2p(n) = σ 2 s − s̄(n)2. Note that since in the first iteration no a priori information of code bits is delivered to the SoftISoftO equalizer, we assume that the code bits {c(m′)} are equiprobable and i.i.d. corresponding to LaE(c(m′)) = 0, ∀m′. Hence, in the first iteration we can see that s̄(n) = 0, υ2(n) = σ2s , and υ 2 p(n) = ζσ 2 s , where ζ = 1 and ζ = 0 for BPSK and MPSK, M 6= 2, respectively. 3Note that since WL processing is not used in MMSE-based turbo equalization schemes, e.g., [9, 11, 12], the pseudo variance υ2p(n) is not required. 40 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 2.3.1 SoftISoftO Linear-MMSE Equalizer Let us first consider linear MMSE SoftISoftO equalizer. The soft estimate ŝ(2n+ j) for symbol s(2n+ j) obtained at the output of the linear MMSE equalizer can be expressed as [13]: ŝ(2n+ j) = E{s(2n + j)}+wHn,j[r̃n − E{r̃n}], (2.10) for j = 0, 1. The linear MMSE equalizer wn,j , [fn,j(N2), fn,j(N2 − 1), · · · , fn,j(0), · · · , fn,j(−N1)]H of length 4NR(Ñf + 1) is designed so that the Bayesian mean square error (MSE) between ŝ(2n+j) and s(2n+j) is minimized, i.e., wn,j = argminwn,j E {|ŝ(2n + j) − s(2n + j)|2 } . Using the Wiener filter theorem, we can show that wn,j is given by wn,j = Cov{r̃n, r̃n}−1Cov{r̃n, s(2n + j)}. (2.11) In accordance with the turbo principle, when the symbol s(2n + j) is estimated, the a priori information of the same symbol is not used. Therefore, we set s̄(2n+j) , E{s(2n+ j)} = 0 and υ2(2n + j) = σ2s , which yields in (3.10) E{s̃n,j} = 0 and E{r̃n} = H̃s̄n,j, where s̄n,j , [s̄T (n+N2), · · · , s̄T (n+ 1), d̄j(n), s̄T (n− 1), · · · s̄T (n−N1 − L̃)]T , j = 0, 1, with s̄(ñ) = [s̄(2ñ), s̄∗(2ñ), s̄(2ñ + 1), s̄∗(2ñ + 1)]T , ∀ñ, ñ 6= n, ñ = n − N1 − L̃, n−N1 − L̃+ 1, · · · , n+N2, and d̄0(n) = [0 0 s̄(2n + 1) s̄∗(2n + 1)]T , d̄1(n) = [s̄(2n) s̄∗(2n) 0 0]T . Let us now consider the input signal vector r̃n−E{r̃n} = H̃s̃n+ η̃− H̃s̄n,j to the linear MMSE filter, given in (3.10). Subtraction of H̃s̄n,j from the received signal vector H̃s̃n is functionally equivalent to soft ISI and CAI cancelations. Note that in the schemes [11, 14], only soft ISI cancelation is performed, as those schemes were designed for single-transmit antenna systems. Let us now define matrices Φ(ñ) , Cov{s̃(ñ), s̃(ñ)}, ∀ñ, ñ = n − N1 − L̃, n + 1 − 41 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission N1 − L̃, · · · , n+N2. We can show that Φ(ñ) =   υ2(2ñ) υ2p(2ñ) 0 0 υ2p(2ñ) υ 2(2ñ) 0 0 0 0 υ2(2ñ+ 1) υ2p(2ñ + 1) 0 0 υ2p(2ñ+ 1) υ 2(2ñ + 1)   . We can also show that when the symbol s(2n + j) is estimated, the covariance matrix of r̃n, Ψr̃r̃(n, j) , Cov{r̃n, r̃n} in (2.11) can be expressed as Cov{r̃n, r̃n} = H̃Rss,j(n)H̃H + σ2nI4NR(Ñf+1), (2.12) where Rss,j(n) , Cov{s̃n, s̃n} is a block diagonal matrix of size 4(Ñf + L̃+1)× 4(Ñf + L̃+ 1). Furthermore, it can be shown that Rss,j(n) = diag{Φ(n+N2), · · · ,Φ(n+1),Φ(n)+ Φ̃j(n),Φ(n− 1), · · · ,Φ(n−N1− L̃)}, (2.13) j = 0, 1, where Φ̃0(n) , Cov{d0(n),d0(n)} =   σ2s − υ2(2ñ) σ2sζ − υ2p(2ñ) 0 0 σ2sζ − υ2p(2ñ) σ2s − υ2(2ñ) 0 0 0 0 0 0 0 0 0 0   , and Φ̃1(n) , Cov{d1(n),d1(n)} =   0 0 0 0 0 0 0 0 0 0 σ2s − υ2(2ñ + 1) σ2sζ − υ2p(2ñ+ 1) 0 0 σ2sζ − υ2p(2ñ + 1) σ2s − υ2(2ñ + 1)   . 42 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission We can also show that Ψr̃s(j) , Cov{r̃n, s(2n + j)} in (2.11) is given by Ψr̃s(j) = [ 01×4NR(N2−L̃) ϑ T j H̃ T (L̃) ϑTj H̃ T (L̃− 1) · · · ϑTj H̃T (0) 01×4N1NR ]T , (2.14) j = 0, 1, where ϑ0 , Cov{s̃(n), s(2n)} = σ2s ·[1 ζ 0 0]T , and ϑ1 , Cov{s̃(n), s(2n+1)} = σ2s · [0 0 1 ζ]T . Substituting (7.15) and (6.18) in (2.11), the MMSE filter wn,j can be expressed as wn,j = [ H̃Rss,j(n)H̃ H + σ2nI4NR(Ñf+1) ]−1 [ 01×4NR(N2−L̃) ϑTj H̃ T (L̃) ϑTj H̃ T (L̃− 1) · · · ϑTj H̃T (0) 01×4N1NR ]T , (2.15) j = 0, 1. We refer to (3.13) as the exact implementation (ExactImpl). Note that since there is no a priori information delivered to the SoftISoftO equalizer in the first iteration, Rss,j(n) is the same for every symbol s(2n+ j), j = 0, 1, n = 0, 1, · · · ,Ns/2. Hence, it is sufficient to compute the inverse of Ψr̃r̃(n, j) only once per symbol burst in order to obtain the MMSE filter. Using the matrix inversion lemma,Ψ−1r̃r̃ (n, j) can be reexpressed as Ψ−1r̃r̃ (n, j) = 1/σ 2 n [ I4NR(Ñf+1) − H̃ ( σ2nR −1 ss,j(n) + H̃ HH̃ )−1 H̃H ] . (2.16) It is clear that in order to get (Ψr̃r̃(n, j)) −1 using (2.16), we need to compute the inverse of a 4(Ñf + L̃ + 1) × 4(Ñf + L̃ + 1) matrix. Hence, when Ñf  L̃ and NR ≥ 2, it is computationally efficient to obtain (Ψr̃r̃(n, j)) −1 using (2.16). The estimate ŝ(2n + j) for the symbol s(2n, j), obtained using the linear-MMSE equalizer, can be expressed as ŝ(2n+ j) = wHn,j[r̃n − H̃s̄n,j], j = 0, 1. In order to facilitate easy computation of the extrinsic LLR values of code bits (see Section III.C), we consider that the estimate ŝ(2n+ j) can be approximated using a 43 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission conditional Gaussian distribution, i.e., ŝ(2n+j) ∼ N (µ̂(2n+j)s(2n+j), σ̂2(2n+j)). Note that a similar approximation was previously adopted for turbo interference cancelation in CDMA systems, e.g., [15] and for turbo equalization, e.g., [11, 14]. Therefore, ŝ(2n+ j) can be represented as the output of the equivalent additive white Gaussian noise (AWGN) system model as follows: ŝ(2n + j) = µ̂(2n + j)s(2n + j) + η̂2n+j, (2.17) where µ̂(2n+j) and η̂2n+j denote the amplitude and zero-mean complex white Gaussian noise with variance σ̂2(2n+j) of the equivalent AWGN system model, respectively. After some manipulation, we can show that µ̂(2n+ j) , 1 σ2s · E{ŝ(2n+ j)s∗(2n + j)} = 1 σ2s wHn,jΨr̃s(j) and σ̂2(2n+ j) , E{|η̂2n+j |2} = σ2s ( µ̂(2n + j)− µ̂2(2n + j)) , for j = 0, 1. Approximate Implementations As was seen previously, in linear MMSE equalization using the ExactImpl, we need to compute the inverse of the covariance matrixΨr̃r̃(n, j) of size 4(Ñf+L̃+1)×4(Ñf+L̃+1) for each symbol separately (except in the first iteration). Therefore, the computational complexity of the proposed linear MMSE SoftISoftO equalizer is clearly dominated by the computation ofΨ−1r̃r̃ (n, j). In the following subsections, we propose two approximate implementations to reduce the complexity involved in MMSE filtering. For comparison purposes, the MF-based SoftISoftO detector is also considered. Approximate Implementation-1 In Approximate Implementation-1 (AprxImpl-1), the MMSE filter is computed using the time average of the covariance matrix, i.e., 44 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 1/L̃bs ∑L̃bs−1 n=0 Ψr̃r̃(n, j). Here, L̃bs , dLbs/2e, where Lbs denotes the burst size. Note that in [3], a similar approximation was considered in MMSE-based equalization for SISO systems. We can see that 1/L̃bs L̃bs−1∑ n=0 Ψr̃r̃(n, j) = 1/L̃bs L̃bs−1∑ n=0 ( H̃Rss,j(n)H̃ H + σ2nI4NR(Ñf+1) ) = H̃R̄ss,jH̃ H + σ2ηI4NR(Ñf+1), j = 0, 1, where R̄ss,j , 1/L̃bs ∑L̃bs−1 n=0 Rss,j(n). R̄ss,j can be computed as follows: R̄ss,j = diag { [Φ̄(N2), · · · , Φ̄(1), Φ̄(0) + ¯̃Φj , Φ̄(−1), · · · , Φ̄(−N1 − L̃)] } , (2.18) where Φ̄(ñ) , 1/L̃bs ∑L̃bs−1 n=0 Φ(n + ñ), for ñ = − N1 − L̃,−N1 − L̃ + 1, · · · , 0, · · · ,N2 and ¯̃Φj , 1/L̃bs ∑L̃bs−1 n=0 Φ̃j(n). For large burst lengths Lbs, we can simplify R̄ss,j by using the approximation Φ̄(ñ) ≈ Φ̄(n), ∀ñ, ñ ∈ {−N1 − L̃,N2}, and Φ̄(n) = 1/L̃bs L̃bs−1∑ n=0 Φ(n) =   ῡ2(0) ῡ2p(0) 0 0 ῡ2p(0) ῡ 2(0) 0 0 0 0 ῡ2(1) ῡ2p(1) 0 0 ῡ2p(1) ῡ 2(1)   , where ῡ2(j) = 1/L̃bs ∑L̃bs−1 n=0 υ 2(2n + j) and ῡ2p(j) = 1/L̃bs ∑L̃bs−1 n=0 υ̃ 2(2n + j). We observed that the use of the approximation Φ̄(ñ) ≈ Φ̄(n) to obtain R̄ss,j does not degrade the performance. In AprxImpl-1, the MMSE filter can be expressed as wAI1j = [ H̃R̄ss,jH̃ H + σ2ηI4NR(Ñf+1) ]−1 Ψr̃s(j). (2.19) It is clear from (2.19) that in AprxImpl-1 we need to compute the MMSE filter coefficients only twice (i.e., for j = 0, 1) per symbol burst provided that the MIMO channel is time- 45 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission invariant within a symbol burst (or channel variation is negligible). Similar to the ExactImpl, we represent the symbol estimate ŝ(2n+ j) obtained using AprxImpl-I as the output of an equivalent Gaussian system model. The amplitude µ̂AI1(2n+ j) and variance σ̂ 2 AI1(2n+ j) of the equivalent Gaussian model can be shown to be µ̂AI1(2n + j) = µ̂AI1j , 1 σ2s (wAI1j ) HΨr̃s(j) and( σ̂AI1(2n+ j) )2 = (wAI1j ) H [ H̃Rss,jH̃ H + σ2ηI4NR(Ñf+1) ] wAI1j − σ2s ( µ̂AI1j )2 , for j = 0, 1, and n = 0, 1, · · · ,Ns/2. Since computation of the variance ( σ̂AI1(2n+ j) )2 separately for each symbol is computationally very expensive, the same approximation (i.e., the time average of the covariance matrix of s̃n, R̄ss,j) can also be used for com- puting ( σ̂AI1(2n + j) )2 . We can then easily show that ( σ̂AI1(2n+ j) )2 ≈ (σ̂AI1j )2 , σ2s µ̂ AI1 j ( 1− µ̂AI1j ) , ∀j, n. Approximate Implementation-2 In Approximate Implementation-2 (AprxImpl-2), when the MMSE filter coefficients are computed, we consider that |LaE(c(m′))| → ∞, i.e., we assume that perfect a priori information is available and hence, υ2p(n ′) = 0,∀n′, n′ 6= n, n′ = 0, 1, · · · , Ns − 1. Note that a similar approximation was previously considered in MMSE-based turbo equalization for SISO systems [11]. In AprxImpl-2, the MMSE filter can be written as wAI2n,j , [ H̃Rss,j(n)H̃ H + σ2ηI4NR(Ñf+1) ]−1 Ψr̃s(j)||LaE(c(m′))|→∞ = %σ2s [ Ψr̃s(j)Ψ H r̃s(j) + σ 2 ηI4NR(Ñf+1) ]−1 Ψr̃s(j), (2.20) 46 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission where % = 2 and % = 1 for BPSK and MPSK, respectively. Using the matrix-inversion lemma, wAI2j can be reexpressed as wAI2j = %σ2s σ2η { I4NR(Ñf+1) − 1( σ2η +Ψ H r̃s(j)Ψr̃s(j) )Ψr̃s(j)ΨHr̃s(j) } Ψr̃s(j), j = 0, 1. After some manipulations, we can show that the MMSE equalizer can be written in the following form: wAI2j = %σ2s( σ2η +Ψ H r̃s(j)Ψr̃s(j) )Ψr̃s(j). (2.21) Note that in (3.15) ( σ2η +Ψ H r̃s(j)Ψr̃s(j) ) is a scalar. Hence, it is clear from (3.15) that using AprxImpl-2, computation of the inverse of 4NR(Ñf + 1) × 4NR(Ñf + 1) input covariance matrix required in the ExactImpl can be avoided. Furthermore, similar to AprxImpl-1, it is only necessary to compute the MMSE filter coefficients twice per sym- bol burst. Thus, computational complexity involved in MMSE filtering can be reduced to a great extent by using AprxImpl-2. Nevertheless, although the assumption that υ2(n′) = 0 and υ2p(n′) = 0,∀n′, n′ 6= n, n′ = 0, 1, · · · ,Ns − 1, might be approximately valid after a few iterations, in the first iteration, it does not hold since there is no a priori information provided to the SoftISoftO equalizer. Hence, intuitively, we can use the ExactImpl (or AprxImpl-1) in the first predetermined number of iterations and then, use AprxImpl-2 in the subsequent iterations.4 We denote the hybrid scheme between the ExactImpl and AprxImpl-2 as Hyb Exact/AprxImpl-2. We can show that the amplitude and variance of the equivalent Gaussian model, which is used to represent the estimate ŝ(2n + j) obtained using AprxImpl-2, can be 4Note that extrinsic information transfer (EXIT) chart-based analysis can be used to determine the number of iterations required using the ExactImpl (or AprxImpl-1), before switching to AprxImpl-2 as discussed in [11]. 47 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission computed as µ̂AI2(2n + j) = µ̂AI2j , 1 σ2s (wAI2j ) HΨr̃s(j) and( σ̂AI2(2n + j) )2 = (wAI2j ) H [ H̃Rss,jH̃ H + σ2ηI4NR(Ñf+1) ] wAI2j − σ2s(µ̂AI2(2n+ j))2, for n = 0, 1, · · · , Ns−1, and j = 0, 1. Since computation of the variance ( σ̂AI2(2n+ j) )2 separately for each symbol is computationally very expensive, we can use the same approximation (i.e., υ2(n′) = 0 and υ2p(n′) = 0,∀n′, n = 0, 1, · · · ,Ns − 1, n′ 6= n) in computing ( σ̂AI2(2n + j) )2 as well. Similar to AprxImpl-1, we can easily show that( σ̂AI2(2n + j) )2 ≈ (σ̂AI2j )2 , σ2s µ̂AI2j (1− µ̂AI2j ) , ∀j, n. Matched Filter-based SoftISoftO Detector Instead of the proposed AprxImpls for the MMSE equalizer, the matched filter (MF), which is matched to the equivalent channel H̃, can be used to further reduce the com- putational complexity of SoftISoftO equalization. Before matched filtering, soft ISI and CAI cancelations are performed in a similar manner as described above for the MMSE equalizer. The soft estimate ŝ(2n + j) for symbol s(2n + j) can be obtained using the MF-based detector as follows: ŝ(2n + j) = ẽjH̃ H [r̃n − H̃s̄n,j ], (2.22) j = 0, 1, where ẽj is the all zero vector of length 4(Ñf+1) except for a one at the (4N2+ 2j + 1)th position. The soft estimate ŝ(2n + j) obtained using the MF-based detector is also modeled using a conditional Gaussian distribution with the mean µ̂MF (2n + j)s∗(2n + j) and variance (σ̂MF (2n + j))2. It can be shown that µ̂MF (2n+ j) = µ̂MFj , 1 σ2s ẽjH̃ HΨr̃s(j) and (σ̂MF (2n+ j))2 = ẽjH̃ H [ H̃Rss,jH̃ H + σ2ηI4NR(Ñf+1) ] H̃ẽTj − σ2s ( µ̂MFj )2 . 48 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission Following an approach similar to that used for AprxImpl-2, in order to achieve high performance using the MF-based detector, one may use the MMSE SoftISoftO equalizer (either the ExactImpl or AprxImpl-1) in the first predetermined number of iterations, and then use the MF-based SoftISoftO detector in the subsequent iterations. We denote the hybrid scheme between the MMSE equalizer and MF-based detector as the Hyb MMSE/MF scheme. 2.3.2 SoftISoftO MMSE-Decision Feedback Equalizer In this subsection, we discuss the SoftISoftO MMSE-decision feedback equalizer (DFE) with WL processing for Alamouti ST block-coded MIMO systems. A SoftISoftO MMSE- DFE was previously developed for SISO systems in [3]. The MMSE-DFE consists of a 4NR(Ñf + 1) × 1 feedforward filter with coefficients fn,j(κ) of length 4NR, κ = − N1,−N1 − 1, · · · , 0, · · · , N2, and feedback filter with coefficients bn,j(κ) of length 4NR, κ = 1, 2, · · · , Nb, where Nb = N1 + L̃ is the order of the feedback filter. Note that since N1 > 0, the feedback filter is sufficiently long for complete cancelation of postcursor ISI and CAI. Following an approach similar to that used in [3], the soft estimate ŝ(2n + j) obtained using the SoftISoftO MMSE-DFE can be expressed as ŝ(2n+ j) = E{s(2n + j)} + N2∑ κ=−N1 fn,j(κ) (r̃(n+ κ)− E{r̃(n+ κ)}) − Nb∑ κ=1 bn,j(κ) ( ˆ̃sd(n− κ)− E{ˆ̃sd(n− κ)} ) , (2.23) where ˆ̃sd(n) , [ŝd(2n) (ŝd(2n))∗ ŝd(2n + 1) (ŝd(2n + 1))∗]T . ŝd(2n + n′) represents the previously detected symbol obtained using the proper decision rule applied on ŝ(n′), n′ = − 2Nb,−2Nb+1, · · · ,−1. We assume that previously detected symbols in the feedback filter are correct (i.e., ŝd(2n + n′) = s(2n+ n′), ∀n′). Similar to SISO systems [11, 16], we can show that the optimum feedback-filter coef- ficients can be given in terms of the channel coefficients and feedforward filter coefficients 49 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission as bn,j(κ) = L̃∑ l=0 H(l)fn,j(l − κ), κ = 1, 2, · · · ,Nb. (2.24) Hence, it can be shown that the soft estimate ŝ(2n+ j) obtained using the MMSE-DFE can be expressed as ŝ(2n+ j) = E{s(2n + j)} +wHn,j(r̃n − H̃s̄dn,j), (2.25) where s̄dn,j , [s̄ T (n+N2), · · · , s̄T (n+1), d̄Tj (n), (ˆ̃sd(n− 1))T , · · · (ˆ̃sd(n−Nb))T ]T . Ap- plying wn,j = Cov{r̃n, r̃n}−1Cov{r̃n, s(2n + j)}, the optimum MMSE feedforward filter wn,j can be shown to be as wn,j = [ ȞŘss,j(n)Ȟ H + σ2nI4NR(Ñf+1) ]−1 Ψr̃s(j), (2.26) where the equivalent channel matrix Ȟ is now a 4NR(Ñf + 1) × 4(N2 + 1) matrix [cf. (7.32)] defined as Ȟ ,   H̃(0) H̃(1) · · · H̃(L̃) 04×4 · · · · · · 04×4 04×4 H̃(0) H̃(1) · · · H̃(L) 04×4 · · · 04×4 ... . . . . . . ... 04×4 · · · · · · · · · 04×4 H̃(0) ... . . . . . . ... 04×4 · · · · · · · · · 04×4   (2.27) and Řss,j(n) is a 4(N2+1)×4(N2+1) block diagonal covariance matrix given as follows: Řss,j(n) = diag{Φ(n+N2),Φ(n +N2 − 1), · · · ,Φ(n+ 1),Φ(n) + Φ̃j(n)}, (2.28) 50 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission j = 0, 1. Note that when the MMSE-feedforward filter wn,j is derived, we assume that the previously detected symbols, ŝd(2n + n′), n′ = − 2Nb,−2Nb + 1, · · · ,−1, are known and hence, the variance υ2(2n + n′) = 0 and pseudo variance υ2p(2n + n′) = 0, ∀n′. Note also that similar to linear MMSE equalization, when computing ŝ(2n + j), we set E{s(2n + j)} = 0, υ2(2n + j) = σ2s , and υ2p(2n + j) = σ2sζ. Now, we can reexpress the symbol estimate obtained using the MMSE-DFE as ŝ(2n + j) = ΨHr̃s(j) [ ȞŘss,j(n)Ȟ H + σ2nI4NR(Ñf+1) ]−1 (r̃n − H̃s̄dn,j). Similar to linear MMSE equal- ization, we use the Gaussian approximation to model the estimates obtained using the MMSE-DFE. We can show that the amplitude µ̂(2n + j) and variance σ̂2(2n + j) of the equivalent Gaussian model can be computed as µ̂(2n + j) = 1/σ2sw H n,jΨr̃,s(j) and σ̂2(2n+j) = µ̂(2n+j)σ2s (1− µ̂(2n+j)), respectively. Note that following the approaches similar to that used for deriving the AprxImpls of the linear MMSE equalizer, we can derive the corresponding AprxImpls for the MMSE-DFE as well. 2.3.3 SoftISoftO Symbol Demapper The SoftISoftO demapper computes the extrinsic LLR values LeE(cn(q)) of code bits cn(q), q = 0, 1, · · · , Q− 1, and n = 0, 1, · · · ,Ns − 1, using the soft estimates of symbols ŝ(n) and the a priori information of code bits. We can show that LeE(cn(q)) can be computed as follows (see, e.g., [3, 12] for detailed derivations): LeE(cn(q)) = ln ∑ ∀s(n):cn(q)=1 P (ŝ(n)/s(n)) ∏ ∀q′,q′ 6=q P (cn(q ′))∑ ∀s(n):cn(q)=0 P (ŝ(n)/s(n)) ∏ ∀q′,q′ 6=q P (cn(q′)) = ln ∑ ∀s(n):cn(q)=1 exp { − |ŝ(n)−µ̂(n)s(n)|2 σ̂2(n) + ∑ ∀q′,q′ 6=q bv(q ′)LaE(cn(q ′)) } ∑ ∀s(n):cn(q)=0 exp { − |ŝ(n)−µ̂(n)s(n)|2 σ̂2(n) + ∑ ∀q′,q′ 6=q bv(q′)L a E(cn(q ′)) } , (2.29) 51 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission where ∀s(n) : cn(q) = {1, 0} denotes the subset of symbols having the qth bit, cn(q) = {1, 0}. For BPSK (i.e., Q = 1), we can easily show that LeE(cn(q)) , log P [ŝ(n)/s(n)= + 1] P [ŝ(n)/s(n) = −1] = 4Re{ŝ(n)µ̂(n)} σ̂2(n) . (2.30) After deinterleaving, the extrinsic information {LeE(c(m′))}Nc−1m′=0 is passed to the SoftI- SoftO channel decoder as the a priori information. 2.4 Semi-Analytical BER Estimation In this section, we develop a semi-analytical method for estimating the BER performance of the proposed MMSE-based turbo equalization scheme. In [11], a different approach has been proposed to obtain the BER estimates of MMSE-based turbo equalization, with long code blocks as a side-product of the extrinsic information transfer (EXIT) chart-based analysis. In the method proposed in [11], the relationship between the mu- tual information5 and the BER of the information bits is first obtained using open-loop simulations. Then, the mutual information values obtained from the iterative trajectory of turbo equalization are mapped to the BER values. A similar approach can be adapted for systems with a short block length, perhaps using the average output mutual infor- mation and the average iterative trajectories, which are obtained by taking the averages over several blocks. However, we would need to estimate the probability density function (pdf) of the extrinsic LLR values of code bits and then compute the mutual information using the estimated pdf. Hence, in this chapter, we propose a approach that does not require computing the mutual information (and is not based on the EXIT chart), to estimate the BER of the information bits in systems with a short block length. First, we consider systems with a systematic convolutional code (SCC). Without loss of generality, we assume that all of one information sequence is transmitted. Let 5i.e., the mutual information between the extrinsic output of the channel decoder and the code bits. 52 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission Lp[nb](b(k)) denote the a posteriori LLR values of information bits obtained from the channel decoder after a given number of iterations (in the nbth block). L p[nb](b(k)) can be written as the sum of the extrinsic output of the MMSE-based equalizer (i.e., the a priori input to the channel decoder) L e[nb] E (b(k)) and the extrinsic output of the channel decoder L e[nb] D (b(k)) as Lp[nb](b(k)) = L e[nb] E (b(k)) + L e[nb] D (b(k)), (2.31) for k = 0, 1, . . . , Ni. Extrinsic outputs of the MMSE-based equalizer and the channel decoder, L e[nb] E (b(k)) and L e[nb] D (b(k)), are considered as the samples of random vari- ables L e[nb] E and L a[nb] D , respectively. Note that L e[nb] E and L e[nb] D are weakly and pos- itively correlated. For simplicity in obtaining an expression for the BER of informa- tion bits, we assume that L e[nb] E and L e[nb] D are conditionally Gaussian distributed as L e[nb] E ∼ N ( (2b(k) − 1)µ[nb]E , (σ[nb]E )2 ) and L e[nb] D (b(k)) ∼ N ( (2b(k) − 1)µ[nb]D , (σ[nb]D )2 ) , where mean values µ [nb] E = (σ [nb] E ) 2/2 and µ [nb] D = (σ [nb] E ) 2/2, respectively.6 Hence, the a posteriori output LLRs are also Gaussian distributed with a mean value µ[nb] = µ [nb] E + µ [nb] D and variance (σ [nb])2 ≤ (σ[nb]E )2 + (σ[nb]D )2. A lower bound on the BER P [nb] e in the nbth block after a given number of iterations can be given as P [nb] e = 1 2erfc ( σ[nb] 2 √ 2 ) ≥ 12erfc ( 1 2 √ 2 . √ (σ [nb] E ) 2 + (σ [nb] D ) 2 ) , where erfc denotes the complementary error function. The mean values µ [nb] E and µ [nb] D (and hence, the variances (σ [nb] E ) 2 = 2µ [nb] E and (σ [nb] D ) 2 = 2µ [nb] D ) are obtained through simulations. The lower bound on the average BER P̄e can be obtained by taking the average over a number of blocks NB (i.e., taking the average over many channel realizations and over many different interleavers) and 6It is a known fact that the extrinsic output LLR values of a constituent MAP decoder of a turbo decoder can be well modeled using a consistent Gaussian distribution, i.e., one with a mean, which is one-half of its variance (see, e.g., [17] and references therein). 53 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission can be expressed as P̄e = 1/NB NB∑ nb=1 P [nb]e ≥ 1/NB NB∑ nb=1 1 2 erfc ( 1 2 √ 2 . √ (σ [nb] E ) 2 + (σ [nb] D ) 2 ) . (2.32) Let us now consider systems with a nonsystematic convolutional code (NSCC). The a posteriori LLR values of the information bits obtained from the channel decoder in the nbth block L p[nb] D (b(k)), for k = 0, 1, . . . ,Ni, are regarded as the observation sequence of the random variable L p[nb] D . Similar to in systems with a SCC, we assume that L p[nb] D is conditionally Gaussian distributed with variance (σ p[nb] D ) 2 and mean value µp[nb] = (σ p[nb] D ) 2/2. Hence, the BER P [nb] e in the nbth block can be given as P [nb] e ≈ 12erfc ( σp[nb] 2 √ 2 ) . The mean value µp[nb] (and hence, the variance (σp[nb])2 = 2µp[nb]) is obtained through simulation. The average BER P̄e can be obtained by taking the average over NB blocks and can be expressed as P̄e = 1/NB NB∑ nb=1 P [nb]e ≈ 1/NB NB∑ nb=1 1 2 erfc ( σp[nb] 2 √ 2 ) . (2.33) It is noteworthy that although in these proposed semi-analytical methods we need to perform closed-loop simulations to get the mean value µp[nb], the number of blocks needed to obtain a reliable estimate for the BER using the semi-analytical method is much less than that needed to the get an accurate BER merely using simulations. 2.5 Numerical Results and Discussion We investigate the performance of the proposed MMSE-based turbo equalizer for Alam- outi ST block coded transmission via simulations. In the simulations, we closely follow the radio downlink specifications of EDGE-EGPRS systems as given in [18, 19, 20]. We choose the following modulation and coding schemes (MCSs): MCS-5, MCS-7, and MCS- 8, which use 8−PSK modulation. Block lengthsNb = 448, Nb = 2×448 andNb = 2×544 54 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission binary data bits are used in the MCS-5, MCS-7 and MCS-8 schemes, respectively. A rate-1/3 nonsystematic convolutional code with generators (133, 171, 145)8 is used in all MCSs considered. Puncturing is used to achieve the desired code rate Rec = 0.37 and Rec = 0.92 in MCS-5 and MCS-8, respectively. For simplicity, pseudo-random block interleavers are used instead of the interleavers specified in [20]. After interleaving, the punctured code block is combined with a block of coded header and flag bits. The re- sulting block of 1392 bits is mapped into 4 bursts of 348 bits each. Each burst is Gray mapped into 8−PSK symbols. Each burst of MCS-5 (MCS-8) is composed of 2 × 52 (2× 51) data symbols, 2× 6 (2× 7) header symbols and flag symbols, 26 training sym- bols in the middle of the burst and three guard symbols at both ends. We use channel models for typical urban areas (TUA), typical hilly terrains (THT) and typical rural areas (TRA), as specified in [21]. A linearized Gaussian minimum shift-keying (GMSK) pulse-shaping filter [19] and a square-root raised cosine (SRC) filter are employed as the transmitter and receiver filters, respectively. For simplicity in the simulations, we assume the same power delay profile for all subchannels. We also assume that every subchannel is time-invariant within each burst and varies independently from burst to burst. This assumption is well justified for low-to-moderate mobile speeds based on the EDGE-EGPRS system specifications with ideal frequency hopping. Furthermore, it is assumed that perfect channel information is available to the receiver. The total average channel power gain is normalized to one. For channel decoding, we use the BCJR-MAP algorithm with the modifications found in [15]. Eb/N0 denotes the received bit energy-to-noise ratio. 2.5.1 Performance of the Exact Implementation In Figure 2.4, we show the average BER of MCS-5 obtained using the proposed MMSE- based turbo equalization scheme with the TUA channel model in Figure 2.4. Note that it was found that after the first iteration the SoftISoftO linear-MMSE equalizer 55 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative Delay (in µs) Av er ag e Po we r G ai n   TUA TRA THT Figure 2.3: Power delay profiles of the equivalent channel models for Typical Urban Areas (TUA), Typical Rural Areas (TRA) and Typical Hilly Terrains (THT) remarkably outperforms the SoftISoftO MMSE-DFE. Therefore, here we only show the performance obtained using the SoftISoftO linear-MMSE equalizer. In Figure 2.4, also shown is the BER of bit-interleaved coded modulation (BICM) in a flat fading channel having the squared channel amplitude, which is chi-square distributed with NTNR(L+1) degrees of freedom. This performance corresponds to the coded matched filter bound (MFB) for a NTNR(L + 1)th order diversity channel, and hence, is regarded as the reference performance (denoted by “RefPer”). Moreover, for comparison purposes, we also include the performance of SISO transmission obtained using the MMSE-based turbo equalization. Note that since after 5 iterations we cannot get considerable further 56 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission improvements, the BER performance is shown up to the 5th iteration. We observe that at a BER of 10−3 a performance improvement of ≈ 3 dB can be obtained using the proposed turbo receiver after five iterations, compared with the single-pass hard- decision based equalization. It can also be seen that Alamouti SW-STBC transmission clearly outperforms SISO transmission. In particular, we can see that after 5 iterations there is roughly a 2 dB improvement in the performance STBC over SISO transmission at BER = 5 × 10−3. Hence, these results reconfirm that we can obtain substantial performance improvements using the Alamouti ST block code along with WL processing at the receiver compared with SISO transmission in multipath fading channels [7]. 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer SISO, It#1 SISO, It#2 SISO, It#3 SISO, It#5 Figure 2.4: Average BER of the proposed turbo equalization scheme for MCS-5 and TUA channel model 57 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission The BER performances of MCS-5 for the TRA and THT channel models are shown in Figure 2.5 and Figure 2.6, respectively. It is clear that for TRA and THT channel models, we can make similar observations as for the TUA channel model. Furthermore, it can be seen that we obtain the best performance for the TRA and THT channel models after the first and the fifth iterations, respectively, which can be explained as follows. Among the three channel models considered, the THT model has the longest delay spread (i.e., the highest number of multipath components), whereas the TRA model has the shortest delay spread (i.e., the lowest number of multipath components). Intuitively, we get the best performance for the TRA channel model in the first iteration because the ISI and ICI are lowest. Nevertheless, with iterations the impact of ISI and CAI diminishes through soft ISI and CAI cancelations. Hence, we obtain the best performance for the THT channel model in the fifth iteration when the multipath diversity is highest. Therefore, it is clear from these simulation results that our proposed turbo receiver is capable of exploiting the multipath diversity in addition to the space (or antenna) diversity. In Figure 2.7 and Figure 2.8, we show the BER of MCS-7 and MCS-8, respectively, for the TUA channel model. It can be observed that the BER of MCS-8 degrades by about 10 dB compared with that of MCS-5 due to the fact that MCS-8 has much higher coding rates than MCS-5. More important, it can be seen that at a BER of 10−2 we can obtain a performance improvement of ≈ 5 dB for MCS-8. It is clear that the higher the coding rate the larger the performance improvement that can be obtained using the proposed turbo equalizer. Hence, the proposed turbo receiver is more useful in achieving improved performance at low Eb/N0 when a high coding rate is used. Figure 2.9 shows the average block error rate (BLER) performance of MCS-5. It is clear from the figure that we can get substantial improvements in the BLER performance using the Alamouti SW-STBC compared with SISO transmission. For example, after five iterations, at BLER = 10−2 we can get about 2 dB gain using the Alamouti SW- STBC compared with SISO transmission. Interestingly, we can also see that a significant 58 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 4 5 6 7 8 9 10 10−5 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer SISO, It#1 SISO, It#2 SISO, It#3 SISO, It#5 Figure 2.5: Average BER of the proposed turbo equalization scheme for MCS-5 and TRA channel model 59 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer SISO, It#1 SISO, It#2 SISO, It#3 SISO, It#5 Figure 2.6: Average BER of the proposed turbo equalization scheme for MCS-5 and THT channel model 60 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 9 10 11 12 13 14 15 16 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer Figure 2.7: Average BER of the proposed turbo equalization scheme for MCS-7 and TUA channel model 61 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 14 15 16 17 18 19 20 21 22 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer Figure 2.8: Average BER of the proposed turbo equalization scheme for MCS-8 and TUA channel model 62 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission improvement in the BLER performance is possible using iterative equalization compared with non-iterative equalization and decoding. Specifically, at BLER = 10−1 we can obtain about 3 dB improvement using the proposed MMSE-based iterative equalizer. We show the BLER performance of MCS-8 in Figure 2.10. The figure shows that even higher improvements in the BLER performance can be obtained for MCS-8 compared with MCS-5. Hence, these results demonstrate that our proposed turbo equalizer can be used to considerably reduce the number of erroneous block retransmissions. 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BL ER   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer SISO, It#1 SISO, It#2 SISO, It#3 SISO, It#5 Figure 2.9: Average BLER of the proposed turbo equalization scheme for MCS-5 and TUA channel model 63 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 14 15 16 17 18 19 20 21 22 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BL ER   STBC, It#1 STBC, It#2 STBC, It#3 STBC, It#5 RefPer Figure 2.10: Average BLER of the proposed turbo equalization scheme for MCS-8 and TUA channel model 2.5.2 Performance of Approximate Implementations In Figure 2.11, we compare the BER performances of AprxImpl-1 and the ExactImpl for MCS-5. Note that in the first iteration both implementations are identical. It is observed that there is no considerable performance degradation due to the approx- imation used in the AprxImpl-1 (i.e., the use of the average input covariance matrix, 1/L̃bs ∑L̃bs−1 n=0 Ψr̃r̃(n, j)). We show the performance of AprxImpl-2 in Figure 2.12. We see that at high Eb/N0 the performance of AprxImpl-2 obtained after 8 iterations (i.e., after the convergence) is even worse than that of the ExactImpl in the first iteration. Hence, it is clear that we cannot achieve improved performance using AprxImpl-2 alone since 64 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission the assumption (i.e., υ2p(n ′) = 0,∀n′, n = 0, 1, . . . ,Ns − 1, n′ 6= n) used in AprxImpl-2 does not hold in the first iteration. Nevertheless, it can be seen that we can obtain almost similar performance as AprxImpl-1 using the Hyb Exact/AprxImpl-2 scheme, which uses the ExactImpl in the first iteration.7 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   AprxImpl−1, It#2 AprxImpl−1, It#5 ExactImpl, It#1 ExactImpl, It#2 ExactImpl, It#5 RefPer Figure 2.11: Comparison of the BER performance of AprxImpl-1 with the ExactImpl: MCS-5, TUA channel model In Figure 2.13, we show the performances of the MF and the Hyb Exact/MF schemes. In the Hyb Exact/MF scheme, the ExactImpl is used in the first iteration. It is observed that at BER = 10−3, after 8 and 5 iterations (i.e., after convergence), the MF and the 7We have found using EXIT-band chart analysis that in this particular case of MCS-5 with the TUA channel model, it is sufficient to use the ExactImpl in the first iteration in order to achieve improved performance. 65 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   AprxImpl−2, It#1 AprxImpl−2, It#2 AprxImpl−2, It#5 AprxImpl−2, It#8 Hyb. Exact/AprxImpl−2, It#2 Hyb. Exact/AprxImpl−2, It#5 ExactImpl, It#1 ExactImpl, It#2 ExactImpl, It#5 RefPer Figure 2.12: Comparison of the BER performance of AprxImpl-2 with the ExactImpl: MCS-5, TUA channel model Hyb Exact/MF scheme yield improvements of about 1 dB and 1.5 dB, respectively, compared with the performance of the ExactImpl obtained after the first iteration (i.e., without using iterations). However, the performance obtained using either the MF or the Hyb Exact/MF scheme is even much worse than that of the ExactImpl obtained after two iterations. Furthermore, we have found that in the Hyb Exact/MF scheme, although we can use either the ExactImpl or AprxImpl-1 more than in the first iteration, we can not get substantial further improvements by using the MF in the following iterations. This behavior has been verified using EXIT-band chart-based analysis. 66 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   MF, It#1 MF, It#2 MF, It#5 MF, It#8 Hyb. Exact/MF, It#2 Hyb. Exact/MF, It#5 ExactImpl, It#1 ExactImpl, It#2 ExactImpl, It#5 RefPer Figure 2.13: Comparison of the BER performance of the MF with the ExactImpl: MCS- 5, TUA channel model 2.5.3 Comparison of semi-analytical and the simulated BER performances As in EDGE-EGPRS systems where a non-systematic convolutional channel code is employed, we use (2.33) to obtain estimates of the BER. In Figure 2.14, we compare the estimated BER of MCS-5 with the BER obtained directly from simulation. It is clear from the results shown in Figure 2.14 that we can use the proposed semi-analytical technique to obtain reliable BER estimates down to a BER of 10−3. Hence, although the proposed semi-analytical technique is not very useful at high Eb/N0 values, it can be used 67 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission to obtain accurate BER estimates in the low-to-moderate Eb/N0 region, which is of more interest in practice for system design purposes. The discrepancy is due to the consistent Gaussian approximation used in the semi-analytical method. In this case of MCS-5, we have found that it is sufficient to take the average over about NB = 100 to 200 blocks in order to obtain reliable BER estimates. On the other hand, at moderate Eb/N0 we need NB = 1000 to 5000 blocks to obtain the BER directly from simulation. Therefore, the proposed semi-analytical approach can be effectively used to avoid lengthy simulations in obtaining the approximate BER performance of the proposed turbo equalization scheme in the low-to-moderate Eb/N0 region. 3 4 5 6 7 8 9 10 10−4 10−3 10−2 10−1 100 Eb/N0 dB Av er ag e BE R   Sim, It#1 Sim, It#2 Sim, It#5 SemiAna, It#1 SemiAna, It#2 SemiAna, It#5 Figure 2.14: Comparison of the BER performance estimated using the semi-analytical method with that obtained directly from simulation: MCS-5, TUA channel model 68 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 2.6 Conclusion In this chapter, we have developed an MMSE-based turbo equalization scheme for Alam- outi ST block coded transmissions over MIMO frequency-selective fading channels. In the proposed receiver, WL processing is used to exploit the rotational variance of ST block coded signals. The equalization and ST block decoding functions are performed jointly at each iteration. We have proposed two approximate implementations that pro- vide better trade-offs between performance and complexity. A semi-analytical method has been developed to estimate the BER performance of the proposed tubo equalization scheme obtained after any given number of iterations. We have investigated the perfor- mance of the proposed turbo equalizer for EDGE-EGPRS systems through simulations. Simulation results have shown that high performance improvements can be obtained using the proposed turbo equalization scheme compared with one-pass equalization and decoding. The number of erroneous block retransmissions can be significantly reduced by using our proposed turbo receiver along with Alamouti STBC, and thus, considerably increasing system throughput. Further research in this area should include analysis of the convergence behavior of the proposed turbo equalization scheme using the EXIT- band chart based technique in EDGE-EGPRS systems. 69 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission 2.7 Bibliography [1] C. Douillard et al., “Iterative correction of intersymbol interference: Turbo equal- ization,” European Transactions on Telecommunications, vol. 6, pp. 507–511, Sept. 1995. [2] G. Bauch and V. Franz, “A comparison of soft-in/soft-out algorithms for ’Turbo- detection’,” in Proc. Int. Conf. on Telecomm. 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[18] Digital cellular telecommunications system (Phase 2+); Physical Layer on the Radio Path (General Description), ETSI Technical Specification 3GPP TS 05.01 version 8.9.0, Nov. 2004. [Online]. Available: http://www.etsi.org 71 Chapter 2. Turbo Equalization for Alamouti Space-Time Block-Coded Transmission [19] Digital cellular telecommunications system (Phase 2+); Modulation, ETSI Technical Specification 3GPP TS 05.04 version 8.4.0, Nov. 2001. [Online]. Available: http://www.etsi.org [20] Digital cellular telecommunications system (Phase 2+); Channel coding, ETSI Technical Specification 3GPP TS 45.003 version 6.8.0, Oct. 2005. [Online]. Available: http://www.etsi.org [21] Digital cellular telecommunications system (Phase 2+); Radio transmission and reception, ETSI Technical Specification 3GPP TS 05.05 version 8.20.0, Nov. 2005. [Online]. Available: http://www.etsi.org 72 Chapter 3 Convergence Behavior of Turbo Equalization in ST Block-Coded MIMO Systems with Finite Block Lengths1 3.1 Introduction Recently, turbo equalization has drawn tremendous attention in the literature as a promising technique to achieve improved performance in broadband communication systems over conventional one-pass (non-iterative) equalization. Initially turbo equal- ization was considered using trellis-based soft-input soft-output (SoftISoftO) equalizers such as the maximum a posteriori (MAP) equalizer. Later, minimum mean square er- ror (MMSE)-based turbo equalization was considered in e.g., [1] for single-input single- output (SISO) systems and [2] for multiple-output multiple-input (MIMO) systems with spatial multiplexing. MMSE-based turbo equalization is more appealing than trellis- based turbo equalization schemes due its low-complexity. In wireless communication systems, space-time (ST) block coding can be effectively used to suppress the detrimental effects of channel fading. High order modulation 1A version of this chapter has been submitted for publication. Wavegedara, K. C. B. and Bhargava, V. K. Convergence Behavior of Turbo Equalization in ST block-coded MIMO Systems with Finite Block Lengths. 73 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . schemes will be used in future wireless communication systems (e.g., EDGE Evolution). Therefore, although in broadband communications inherent multipath (frequency) di- versity can be exploited through a powerful equalization technique, additional means of diversity is required to obtain good system performance especially under deep fading conditions. Thus, we are in the opinion that STBC will be an essential technique to be employed in future wireless communication systems. In [3] (see Chapter 5), we proposed an MMSE-based turbo equalization scheme for Alamouti ST block-coded transmission. In the proposed turbo equalization scheme, widely linear processing is used to exploit the inherent rotational variance of ST block-coded signals as shown in [4]. ST block decoding and equalization are performed jointly. The knowledge of the probabilistic convergence behavior of turbo equalization is very useful for system design purposes. In turbo equalization, the extrinsic information (EI) is exchanged between the SoftISoftO equalizer and the SoftISoftO channel decoder. Therefore, we can analyze transfer characteristics of EI between the equalizer and the decoder in order to analyze the probabilistic convergence behavior of turbo equalization. In [5] a technique, which uses signal-to-noise ratio (SNR) transfer characteristics of the extrinsic information, was developed to analyze the iterative behavior of turbo decod- ing. Ten Brink [6] introduced the concept of the extrinsic information transfer (EXIT) chart considering the bitwise mutual information transfer characteristics to analyze the convergence behavior of iterative decoding in bit interleaved coded modulation (BICM) and later, applied to turbo decoding of parallel concatenated codes in [7]. For the EXIT chart analysis the constituent receiver components (e.g., constituent decoders in turbo decoding) are modeled as devices mapping the a priori information and/or channel observations into the EI. In the Ten Brink approach, the extrinsic trans- fer characteristics of the constituent receiver components are obtained by assuming the input log likelihood ratios (LLRs) of code bits are Gaussian distributed. This is based on the observation that the extrinsic output of the MAP channel decoder is well ap- 74 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . proximated using a single-parameter Gassing distribution. The EXIT chart analysis was applied for turbo equalization in [8] and for iterative multiuser detection in [9]. Later, Lee and Blahut [10, 11] extended the EXIT chart analysis to iterative decoding of parallel concatenated codes with finite (i.e., short) block lengths in AWGN channels. They introduced the concept of EXIT-band chart considering mutual information trans- fer characteristics. In [9], a similar approach was adopted along with the consistence Gaussian assumption on the distribution of the output EI to analyze the convergence behavior of iterative multiuser detection with finite block lengths in AWGN channels. In [8], the EXIT chart analysis was applied for turbo equalization for single-input single-output (SISO) systems with very large block lengths (i.e., the asymptotic con- vergence behavior). However, in wireless systems, finite (i.e., short) block lengths are generally considered for practical reasons such as latency and high block error rates. For example, in Enhanced General Packet Radio Service (EGPRS)/Enhanced Data Rates for Global Evolution (EDGE) systems, the maximum block size considered is 898 infor- mation bits, as specified for modulation coding scheme (MCS)-9. In [8] it was shown that although the extrinsic output of the equalizer cannot be well approximated using a consistent Gaussian distribution, the EXIT chart technique can still be used to analyze the convergence behavior of turbo equalization with very large block lengths. However, the applicability of the EXIT-band chart analysis for turbo equalization with finite (i.e., short) block lengths is still an open and imperative research issue. Therefore, in this chapter, the convergence behavior of turbo equalization in Alamouti ST block-coded MIMO systems with finite block lengths is studied. For this purpose, we adapt the EXIT-band chart technique considering bitwise mutual information transfer character- istics, originally proposed for analyzing iterative decoding of PCCs in [10]. We consider the MMSE-based turbo equalization scheme proposed in [12] (see Chapter 2). In sim- ulations, we closely follow the specifications of EGPRS/EDGE systems. The specific contributions of this chapter can be summarized as follows: 75 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . • We successfully accomplish the goal of the application of the EXIT-band chart technique [10] for turbo equalization in ST block-coded MIMO systems with finite block lengths. The transfer of EI between the SoftISoftO equalizer and the decoder is visualized as the iterative trajectory. Our results demonstrate that EXIT-band chart can be used as an effective engineering tool for analyzing the convergence behavior of turbo equalization with finite block lengths. Furthermore, we show that the insights obtained using the EXIT-band chart analysis on the convergence behavior can be used for system design purposes (e.g., to determine a suitable code rate). • The EXIT-band characteristics of different channel coding schemes used in EDGE systems are studied. The impact on the convergence behavior (hence, on the performance) of turbo equalization due to the use of STBC is analyzed through the EXIT-band characteristics. We also study the effects of having different number of receive antennas (NR) on the convergence behavior. • We compare the performance of the MMSE and matched filter (MF)-based equal- ization schemes using the EXIT-band chart analysis. Our results show that even at high a priori input values the performance of the MF-based scheme is considerably worse than that of the MMSE-based equalizer. • We also compare the convergence behaviors of the proposed exact implementa- tion (ExactImpl) and approximate implementations (AprxImpls) of the MMSE equalizer through the EXIT-chart band analysis. Our results show that a better trade-off between performance and complexity can be achieved using a hybrid ap- proach between the ExactImpl (or AprxImpl-1) and AprxImpl-2. Furthermore, the results show that the EXIT-band chart analysis can be used to determine the number of iterations performed using the ExactImpl (or using AprxImpl-1) in the hybrid approach. 76 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . • It is revealed through the EXIT-band chart analysis that when employing a convo- lutional code with a very high code rate 2 is employed, we cannot achieve consider- able performance improvements using turbo equalization. Nevertheless, it is shown that we can achieve remarkable performance improvements through turbo equal- ization by feeding a posteriori information of code bits as the a priori information to the SoftISoftO equalizer instead of the EI of code bits computed by the channel decoder. When the a posteriori information is fed as the a priori information to the equalizer, we propose the hybrid EXIT and POIT3-band chart (referred to as Hyb EXIT/POIT-band chart) for analyzing the convergence behavior of turbo equalization. • We obtain a lower bound on the average bit error rate (BER) from the EXIT-band chart using a semi-analytical approach. We show comparing with the simulated BER that the lower bound obtained from the EXIT-band chart is sufficiently tight (down to the BER of about 10−3) for practical purposes. Notation Bold lower case letters represent vectors while bold upper case letters de- note matrices; (.)T , (.)∗, and (.)H denote the operations transpose, complex conjugate (component-wise), and Hermitian transpose, respectively; IM and 0M×N denote the M ×M identity matrix and an all-zero matrix of size M × N ; diag{s} represents a diagonal matrix, where vector s is on the diagonal. <{x} and ={x} denote the real and imaginary parts of a complex random variable x. E{x} and Var{x} , E{|x − E{x}|2} represent the expected value and variance of x, respectively; Cov{x1, x2} , E{(x1 − E{x1})(x2 − E{x2})∗} stand for the covariance between random variables x1 and x2. d.e represents the ceiling function. 2which is obtained using severe puncturing 3Posteriori Information Transfer 77 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 3.2 System Description The equivalent complex-baseband system model is depicted in Figure 3.1. The transmit- ter is equipped with NT = 2 antennas, while the mobile receiver has either a single an- tenna or multiple antennas (NR ≥ 1). A block of Ni information bits {b(k)}Ni−1k=0 , b(k) ∈ {0, 1} is fed into a binary convolutional encoder of constraint length ν and code rate Rc. The resulting block of code bits {c(m)}Nc−1m=0 is then interleaved by a random block in- terleaver, where Nc = Ni/Rc. Interleaved code bits {c(m′)}Nc−1m′=0, are then grouped into sets of Q bits, cn = {cn(1), cn(2), · · · , cn(Q)}, n = 0, 1, · · · ,Ns − 1, where Ns = Nc/Q and Q is the modulation order. Each set of Q modulating bits, cn is mapped into MPSK symbols s(n) ∈ {α0, α1, · · · , αM−1}, where αν = ej2piν/M , ∀ν, ν = 0, 1, · · · ,M − 1, and M = 2Q is the size of the signal constellation. Note that the signal constellation has zero mean Es , 1/M ∑M−1 ν=0 αν = 0 and unit variance σ 2 s , 1/M ∑M−1 ν=0 |αν |2 = 1. We assume that symbols {s(n)}Ns−1n=0 are statistically independent. After constellation mapping, two consecutive symbols s(2n) and s(2n + 1) are fed into the ST block encoder. The ST block encoder outputs sequences s1(i) and s2(i) according to the following rule [13]: s1(i) =   s(2n); i = 2n −s∗(2n + 1); i = 2n + 1 (3.1) s2(i) =   s(2n+ 1); i = 2n s∗(2n); i = 2n+ 1 (3.2) ST block encoded sequences s1(i) and s2(i) are first pulse-shaping filtered and then sent to the first and second transmit antennas, respectively. 78 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . F ig u re 3. 1: E q u iv al en t B as eb an d S y st em M o d el 79 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . Let us define the discrete-time overall impulse response of the transmit pulse-shaping filter, the multipath subchannel between the ntth transmit antenna and the nrth receive antenna, and the receiver input filter as hnt,nr , [hnt,nr(0), hnt,nr(1), · · · , hnt,nr(L)]T , nt = 1, 2, and nr = 1, 2, · · · ,NR, NR ≥ 1. Note that without loss of generality, we assume that the channel order L is the same for all subchannels. The discrete-time received signal in the nth symbol interval at the nrth receive antenna, which is obtained by sampling the output of the receive input filter at the symbol-rate, can be expressed as rnr(n) = L∑ l=0 [h1,nr(l)s1(n − l) + h2,nr(l)s2(n − l)] + ηnr(n). (3.3) ηnr(n) represents the (discrete-time) complex-valued zero-mean additive white Gaussian noise (AWGN) with variance σ2η. We assume that the multipath subchannels are time- invariant over a transmitted symbol burst and that the channel fading between symbol bursts is independent. In [4] it was shown that WL processing, in which the received signal and its complex conjugate are processed together, can be used to exploit the rotational variance of Alam- outi ST block-coded signals. Therefore, WL processing was used in turbo equalization scheme developed for Almaouti ST block-coded systems in [12]. Let us now define a vector comprising two consecutive samples of the received signal at the nrth antenna and the complex conjugates of the polyphase components as [4]: r̃nr(n) , [rnr(2n) rnr(2n+ 1) r ∗ nr(2n) r ∗ nr(2n+ 1)] T , (3.4) for nr = 1, 2, · · · , NR. We can express r̃nr(n) as r̃nr(n) = L̃∑ l=0 H̃nr(l)s̃(n− l) + η̃nr(n), (3.5) 80 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . where L̃ = dL/2e and s̃(n) , [s(2n) s∗(2n) s(2n+ 1) s∗(2n + 1)]T , η̃nr(n) , [ηnr(2n) ηnr(2n + 1) η ∗ nr(2n) η ∗ nr(2n + 1)] T , H̃nr(l) ,   h1,nr(2l) h2,nr(2l − 1) h2,nr(2l) −h1,nr(2l − 1) h1,nr(2l + 1) h2,nr(2l) h2,nr(2l + 1) −h1,nr(2l) h∗2,nr(2l − 1) h∗1,nr(2l) −h∗1,nr(2l − 1) h∗2,nr(2l) h∗2,nr(2l) h ∗ 1,nr(2l + 1) −h∗1,nr(2l) h∗2,nr(2l + 1)   . Note that, unlike [4], for convenience we change the order of the terms when s̃(n) is de- fined, and H̃nr(l) is accordingly modified. Then, we define a vector r̃(n) , [r̃ T 1 (n) r̃ T 2 (n) · · · r̃TNR(n)]T by stacking signal vectors of NR receive antennas. r̃(n) can be expressed as r̃(n) = L̃∑ l=0 H̃(l)s̃(n− l) + η̃(n), (3.6) where H̃(l) = [H̃T1 (l) H̃ T 2 (l) · · · H̃TNR(l)]T , and η̃(n) = [η̃T1 (n) η̃T2 (n) · · · η̃TNR(n)]T . Let us now introduce a sliding-window model in the nth time interval as follows: r̃n = H̃s̃n + η̃n, (3.7) with r̃n , [r̃T (n+N2) · · · r̃T (n) · · · r̃T (n−N1)]T , s̃n , [s̃T (n+N2) · · · s̃T (n) · · · s̃T (n− N1 − L̃)]T , and η̃n , [η̃T (n +N2) · · · η̃T (n) · · · η̃T (n −N1)]T . The equivalent channel matrix of size 4NR(Ñf + 1)× 4(Ñf + L̃+ 1) is given as H̃ =   H̃(0) · · · H̃(L̃) 0 · · · · · · 0 0 H̃(0) · · · H̃(L) 0 · · · 0 ... . . . . . . ... 0 · · · · · · 0 H̃(0) · · · H̃(L̃)   , (3.8) 81 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . where Ñf = N1 + N2. We input the vector r̃n of length 4(Ñf + 1) to the iterative equalizer in the nth symbol interval. 3.3 MMSE-based Turbo Equalization In this section, we briefly review the MMSE-based turbo equalizer developed for Alam- outi ST block-coded transmission in [12]. At the beginning of each iteration, the Soft- ISoftO channel decoder delivers the a priori log likelihood ratios (LLRs) LaE(c(m ′)) , ln P [c(m ′)=1] P [c(m′)=0] , m ′ = 0, 1, · · · , Nc − 1, computed in the previous iteration, to the SoftI- SoftO symbol mapper. The SoftISofO symbol mapper first computes symbol a priori probabilities using the a priori LLRs LaE(c(m ′)) as P (s(n)=αν) = Q−1∏ q=0 P (cn(q)=bν(q)) = Q−1∏ q=0 1 2 (1 + (2bν(q)− 1).tanh(LaE(cn(q))/2)) , n = 0, 1, · · · , Ns − 1. Then, it computes statistics including the mean s̄(n), variance υ2(n), and pseudo variance υ2p(n) of every transmitted symbol {s(n)}Ns−1n=0 using the symbol a priori probabilities. Let us define a vector s̄n,j , [s̄T (n+N2), · · · , s̄T (n+ 1), d̄j(n), s̄T (n− 1), · · · s̄T (n−N1 − L̃)]T , (3.9) for j = 0, 1, with s̄(ñ) = [s̄(2ñ), s̄∗(2ñ), s̄(2ñ + 1), s̄∗(2ñ + 1)]T , ∀ñ, ñ 6= n, ñ = n − N1 − L̃, n − N1 − L̃ + 1, · · · , n + N2, and d̄0(n) = [0 0 s̄(2n + 1) s̄∗(2n + 1)]T , d̄1(n) = [s̄(2n) s̄ ∗(2n) 0 0]T . The linear-MMSE equalizer wn,j of size (4NR(Ñf +1)× 1) is designed so that the Bayesian mean square error (MSE) between ŝ(2n+j) and s(2n+j) is minimized, i.e., wn,j = argminwn,j E { |ŝ(2n + j)− s(2n+ j)|2 } . The soft estimate ŝ(2n+ j) of the symbol s(2n+ j) obtained at the output of the linear-MMSE equalizer 82 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . can be expressed as ŝ(2n + j) = E{s(2n+ j)} +wHn,j[r̃n − H̃s̄n,j}], (3.10) for j = 0, 1. Let us now define a matrix of 4(Ñf + L̃+1)×4(Ñf + L̃+1), Rss,j(n) , Cov{s̃n, s̃n}. It can be shown that Rss,j(n) is block diagonal and is given by Rss,j(n) = diag{Φ(n+N2), · · · ,Φ(n+1),Φ(n)+ Φ̃j(n),Φ(n− 1), · · · ,Φ(n−N1− L̃)}, (3.11) j = 0, 1, where Φ(ñ) , Cov{s̃(ñ), s̃(ñ)} =   υ2(2ñ) υ2p(2ñ) 0 0 υ2p(2ñ) υ 2(2ñ) 0 0 0 0 υ2(2ñ + 1) υ2p(2ñ + 1) 0 0 υ2p(2ñ + 1) υ 2(2ñ + 1)   , ∀ñ, ñ = n−N1 − L̃, n+ 1−N1 − L̃, · · · , n+N2, and Φ̃0(n) , Cov{d0(n),d0(n)} =   σ2s − υ2(2ñ) σ2sζ − υ2p(2ñ) 0 0 σ2sζ − υ2p(2ñ) σ2s − υ2(2ñ) 0 0 0 0 0 0 0 0 0 0   , Φ̃1(n) , Cov{d1(n),d1(n)} =   0 0 0 0 0 0 0 0 0 0 σ2s − υ2(2ñ + 1) σ2sζ − υ2p(2ñ+ 1) 0 0 σ2sζ − υ2p(2ñ + 1) σ2s − υ2(2ñ + 1)   . 83 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . We also define vectors Ψr̃s(j) , Cov{r̃n, s(2n+ j)}, j = 0, 1, which are given by Ψr̃s(j) = [ 01×4NR(N2−L̃) ϑ T j H̃ T (L̃) ϑTj H̃ T (L̃− 1) · · · ϑTj H̃T (0) 01×4N1NR ]T , (3.12) with ϑ0 , Cov{s̃(n), s(2n)} = σ2s · [1 ζ 0 0]T , and ϑ1 , Cov{s̃(n), s(2n + 1)} = σ2s · [0 0 1 ζ]T . The MMSE filter wn,j can be expressed as [12]: wn,j = [ H̃Rss,j(n)H̃ H + σ2nI4NR(Ñf+1) ]−1 Ψr̃s(j), (3.13) j = 0, 1. We refer to (3.13) as the exact implementation (ExactImpl). The soft estimate ŝ(2n+j) is modeled using a conditional Gaussian distribution with η̂2n+j with the mean σ̂ 2(2n+ j)s(2n+ j) and variance σ̂2(2n+ j) (see [12] and references therein). We can show that µ̂(2n+ j) , 1 σ2s · E{ŝ(2n + j)s∗(2n + j)} = 1 σ2s wHn,jΨr̃s(j), and σ̂2(2n+ j) , E{|η̂2n+j |2} = σ2s ( µ̂(2n + j)− µ̂2(2n+ j)) , j = 0, 1. The SoftISoftO demapper computes the extrinsic LLR values LeE(cn(q)) of code bits cn(q), q = 0, 1, · · · , Q− 1, and n = 0, 1, · · · ,Ns− 1, using the soft estimates of symbols ŝ(n) and the a priori information of code bits. We can show that LeE(cn(q)) = ln ∑ ∀s(n):cn(q)=1 exp { − |ŝ(n)−µ̂(n)s(n)|2σ̂2(n) + ∑ ∀q′,q′ 6=q bv(q ′)LaE(cn(q ′)) } ∑ ∀s(n):cn(q)=0 exp { − |ŝ(n)−µ̂(n)s(n)|2 σ̂2(n) + ∑ ∀q′,q′ 6=q bv(q′)L a E(cn(q ′)) } , where ∀s(n) : cn(q) = {1, 0} denotes the subset of symbols having the qth bit, cn(q) = {1, 0}. After deinterleaving, the extrinsic information {LeE(c(m′))}Nc−1m′=0 is passed to the SoftISoftO channel decoder as the a priori information. 84 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . Approximate Implementations The computational complexity of the linear-MMSE SoftISoftO equalizer is dominated by the computation of [ H̃Rss,j(n)H̃ H + σ2nI4NR(Ñf+1) ]−1 separately for each symbol. Therefore, in [12], we proposed two approximate implementations to reduce the com- plexity involved in MMSE filtering. For comparison purposes, the MF-based SoftISoftO detector is also considered. Approximate Implementation-1 In Approximate Implementation-1 (AprxImpl-1), the time average R̄ss,j , 1/L̃bs ∑L̃bs−1 n=0 Rss,j(n) of the covariance matrix Rss,j(n) is used in the MMSE filter wn,j, ∀n. R̄ss,j is computed as follows: R̄ss,j = diag { [Φ̄(N2), · · · , Φ̄(1), Φ̄(0) + ¯̃Φj , Φ̄(−1), · · · , Φ̄(−N1 − L̃)] } , (3.14) where Φ̄(ñ) , 1/L̃bs ∑L̃bs−1 n=0 Φ(n + ñ), for ñ = − N1 − L̃,−N1 − L̃ + 1, · · · , 0, · · · ,N2 and ¯̃Φj , 1/L̃bs ∑L̃bs−1 n=0 Φ̃j(n). Approximate Implementation-2 In Approximate Implementation-2 (AprxImpl-2), when the MMSE filter coefficients are computed, we consider that |LaE(c(m′))| → ∞ (i.e., perfect a priori information is as- sumed) and hence, υ2p(n ′) = 0,∀n′, n′ 6= n, n′ = 0, 1, · · · ,Ns − 1. With this approxima- tion, it can be shown that the MMSE filter is expressed as wAI2j = %σ2s( σ2η +Ψ H r̃s(j)Ψr̃s(j) )Ψr̃s(j). (3.15) Note that in (3.15) ( σ2η +Ψ H r̃s(j)Ψr̃s(j) ) is a scalar. 85 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 3.3.1 Matched Filter-based SoftISoftO Detector Instead of the AprxImpls, the matched filter (MF), which is matched to the equivalent channel H̃, may be used to further reduce the computational complexity of SoftISoftO equalization. Before matched filtering, soft ISI and CAI cancelations are performed in a similar manner for the MMSE equalizer. The soft estimate ŝ(2n+j) for symbol s(2n+j) can be obtained using the MF-based detector as follows: ŝ(2n + j) = ẽjH̃ H [r̃n − H̃s̄n,j ], (3.16) j = 0, 1, where ẽj is the all zero vector of length 4(Ñf + 1) except for a one at the (4N2 + 2j + 1)th position. 3.4 EXIT-band Chart Analysis for Turbo Equalization Similar to [8], for the EXIT-band chart analysis we model the SoftISoftO MMSE-based equalizer with WL processing and SoftISoftO channel decoder as the devices, which maps the received signal sequence and/or the a priori information of code bits into output extrinsic information. As shown in Figure 3.2, MMSE filter with WL process- ing, SoftISoftO symbol mapper and demapper, and the deinterleaver are included in the SoftISoftO equalizer as one unit, whereas the MAP channel decoder and the in- terleaver included in the SoftISoftO channel decoder as one unit. Let us denote the extrinsic information of code bits obtained from the equalizer and the channel decoder in the nbth code block (i.e., for a certain channel output sequence and interleaver) as {Le[nb]E (c(m′))}Nc−1m′=0 and {Le[nb]D (c(m))}Nc−1m=0 , respectively. Similarly, we can express the a priori information fed into the equalizer and channel decoder in the nbth code block as {La[nb]E (c(m′))}Nc−1m′=0 and {La[nb]D (c(m))}Nc−1m=0 , respectively. For a given channel output sequence and interleaver, we regard {Le[nb]E (c(m′))}Nc−1m′=0 and {Le[nb]D (c(m))}Nc−1m=0 as the samples of random variables L e[nb] E and L e[nb] D , respectively. Similarly, we define random 86 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . variables L e[nb] E and L e[nb] D to represent the input a priori sequences of the equalizer and channel decoder, respectively. In order to measure the information content in the a pri- ori information, we compute the input mutual information between the random variable La[nb] representing the distribution of the a priori information of either the equalizer or the channel decoder and code bit C as follows [7]: I(La[nb];C) = 1 2 ∑ c∈{1,0} ∫ +∞ −∞ pLa[nb](l|C = c) log2 2pLa[nb](l|C = c) pLa[nb](l|C = 1) + pLa[nb](l|C = 0) dl, (3.17) where I(La[nb];C) ∈ [0, 1] and pLa[nb](l|C = c) denotes the probability density function (PDF) of the input LLR values (i.e., a priori information) conditioned on code bit c ∈ {1, 0}. Note that in (3.17) we omit subscript e or d to commonly represent the equalizer and the decoder. Adopting an approach similar to that proposed in e.g., [7, 11] for the problem at hand, in computing the input mutual information I(L a[[nb]] E ;C) we assume that L a[nb] is Gaussian distributed with the PDF pLa[nb](l|C = c) , 1√ 2σa[nb] exp { −(l − µa[nb](2c− 1))2 2(σa[nb])2 } , (3.18) where the mean value µa[nb] = (σa[nb])2/2. Note that I(La[nb];C) is a function of single parameter σ a[nb] E . Let us now define an abbreviation J(σa) , I(σa). The two extreme values can be expressed as limσa→0 J(σa) = 0 and limσa→∞ J(σa) = 1, which correspond to the cases with no a priori information and perfect a priori information, respectively. The samples obtained from L a[nb] E [or L a[nb] D ] are fed into the equalizer [or the channel decoder] as the a priori information. Besides the samples obtained from L a[nb] E , after WL processing, we also input the channel output sequence {rnr(n)}, nr = 0, 1, . . . ,NR, to the SoftISoftO equalizer. At the output of the SoftISoftO equalizer [or the Soft- ISoftO decoder], we compute the output mutual information I(L e[nb] E ;C) ∈ [0, 1] [or 87 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . I(L e[nb] D ;C) ∈ [0, 1]] between random variable Le[nb]E [or Le[nb]D ] and code bits C using (3.17) with the pdf p L e[nb] E (l|C = c) [or p L e[nb] D (l|C = c)]. The PDF pLeE (l|C = c) [or pLeD(l|C = c)] of LeE [or LeD] is obtained using the histogram measurements of extrinsic LLR values {Le[nb](c(m′))}Nc−1m′=0 [or {Le[nb]D(c(m))}Nc−1m=0 ]. With a finite (i.e., short) code block length, we obtain different histograms of extrinsic LLR values using the equalizer for the same distribution (i.e., for the same value of (σaE) 2) of the a priori information using different channel output sequences {rnr(n)}, nr = 1, 2, . . . ,NR, and for different channel realizations at the same value of Eb/N0 (in dB, Eb: average received bit energy per receive antenna, N0: noise power spectral density).Unlike for systems with infinite (i.e., long) block lengths, where the EXIT characteristics is a single curve, with a finite block length, for a given Eb/N0 value the EXIT characteristics of the SoftISoftO equal- izer is a band of curves of I(LeE ;C) with respect to I(L a E ;C). The band of EXIT curves for the equalizer is denoted as the e-band. Similarly, with a finite block length, for dif- ferent sequences of a priori information {La[nb]D (c(m′))}Nc−1m′=0 with the same distribution (i.e., with the same value of (σaD) 2) of input a priori information, we attain different histograms of extrinsic LLR values using the channel decoder as well. Therefore, also for the channel decoder with a finite block length we obtain a band of curves I(LeD;C) with respect to I(LaD;C) as the EXIT characteristics. The band of EXIT curves of the decoder is referred to as the d-band. 88 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . F ig u re 3. 2: M o d el fo r E X IT -b an d ch ar t an al y si s of tu rb o eq u al iz at io n 89 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . In order to obtain the EXIT characteristics of the SoftISoftO equalizer and decoder, we perform open loop simulation for several code blocks with different information se- quences, different realizations of the MIMO multipath channel and interleavers. As shown in Figure 3.2, the extrinsic output of the equalizer represented by I(LeE ;C) be- comes the a priori input to the decoder represented by I(LaD;C) (i.e., I(L a D;C) = I(LeE ;C)) and the extrinsic output of the decoder represented by I(L e D;C) becomes the a priori input to the equalizer represented by I(LaE ;C) (i.e., I(L a E ;C) = I(L e D;C)). For notational brevity, from this point onwards we denote I(LaD;C), I(L a E ;C), I(L e D;C), and I(LeE ;C) using I a D, I a E, I e D, and I e E, respectively. We obtain the EXIT-band chart for turbo equalization by plotting the e-band and d-band together in a single diagram. In Figure 3.3 and Figure 3.4 we show the distribution of the mutual information IeE computed at the output of the equalizer for IaE = 0.5604 (i.e., σ a E = 5) and Eb/N0 = 5dB, and the distribution of the mutual information IeD computed at the output of the decoder for IaD = 0.5604, respectively. It is clear from Figure 3.3 and Figure 3.4 that I e E and IeD can be approximately modeled using (truncated) Gaussian distributions with means µIeE , mean{IeE} and µIeD , mean{IeD}, and variances σ2IeE and σ 2 IeD , respectively. Similar to the analysis of the convergence behavior of turbo decoding in systems with finite block lengths [11], the e-band is represented using µIeE and µI e E ± σIeE and the d-band represented using µIeD and µI e D ± σIeD . It is noteworthy that the consistent Gaussian assumption used for obtaining the EXIT characteristics is based on the fact that the extrinsic LLR values of the con- stituent receiver components: the SoftISoftO equalizer and channel decoder, can be modeled using a consistent Gaussian distribution. It is already known that the extrinsic information (i.e., LLRs) of code bits obtained using a MAP channel decoder can be well approximated with a consistent Gaussian distributed. Let us now consider the distri- bution of the extrinsic information of code bits obtained using the SoftISoftO equalizer. Recall that in order to compute the extrinsic LLR values of code bits, the soft estimates 90 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 Mutual Information IeE Pr ob ab ilit y De ns ity   Simulated PDF Standard truncated Gaussian PDF Figure 3.3: Distribution of the mutual information IeE computed at the output of the equalizer of symbols obtained using the MMSE-based equalizer is approximated with a conditional Gaussian distribution with the mean µ̂(n)s(n) and variance σ̂2(n). Therefore, we can easily show that the mean and variance of the extrinsic information of code bits at the output of the MMSE-based equalizer for BPSK modulated signals can be written as [8]: E{LeE(c(m))|c(m)=c} = 2µ̂(m) σ̂2(n) .c Cov{LeE(c(m)), LeE(c(m))|c(m)=c} = 4µ̂2(m) σ̂4(n) . It is clear that Cov{LeE(c(m)), LeE(c(m))|c(m)=c} = 2E{LeE(c(m))|c(m)=c} and thus, 91 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 Input Mutual Information IaD Pr ob ab ilit y De ns ity   Simulated PDF Standard truncated Gaussian PDF Figure 3.4: Distribution of the mutual information IeD computed at the output of the decoder the distribution of the extrinsic LLR values obtained using the MMSE-based equalizer has a similar property as the consistent Gaussian distribution, in which the covari- ance is twice of the mean value. We can easily show that the same property holds for QPSK modulated signals as well. Nevertheless, distinct from the consistent Gaussian distribution given in (3.18), the mean and variance of the distribution of LeE obtained using the SoftISoftO equalizer is not constant in m even for a certain realization of the MIMO multipath channel. Even though for systems using 8PSK modulation it is intractable to derive simplified expressions for the mean E{LeE(c(m))|c(m)=c} and vari- ance Cov{LeE(c(m)), LeE(c(m))|c(m)=c}, we can easily see from Chapter 2 that the mean and variance of the extrinsic LLR values are not constant in time index m. We observed 92 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . from simulations that, in general, the distribution of LeE cannot be well modeled using a single-parameter Gaussian distribution. Therefore, the use of consistent Gaussian dis- tributions to obtain the EXIT characteristic of the SoftISoftO channel decoder in turbo equalization is an approximation [8]. 3.5 Obtaining Estimates of the Code BER (CBER) In [7] it was shown that an estimate of the BER can be obtained from the EXIT chart after an arbitrary number of iterations in turbo decoding of parallel concatenated codes. In turbo decoding, the extrinsic information of the information bits (i.e., systematic bits) is exchanged between the constituent decoders. Distinct from turbo decoding, in turbo equalization, the extrinsic information of code bits is exchanged between the SoftISoftO equalizer and the SoftISoftO decoder. As a consequence, it is not possible to obtain estimates on the BER directly from the EXIT-band chart for turbo equalization. Nevertheless, instead of the BER of the information bits, following an approach similar to that used in [7], we can obtain a lower bound on the code-bit error rate (CBER) after an arbitrary number of iterations from the EXIT-band chart of turbo equalization. In addition to better understanding of the convergence behavior of turbo equalization that can be obtained using the CBER, we can also use the CBER to get a rough idea about the BER performance. Let us assume that all one-code bit sequence is transmitted without loss of generality. After a given number of iterations the a posteriori LLRs {L[nb](c(m))}Nc−1m=0 of code bits obtained at the output of the SoftISoftO channel decoder can be written as the sum of the a priori information (i.e., the extrinsic information of the MMSE-based equalizer) and the extrinsic information of the channel decoder (in the nbth code block) as L p[nb] D (c(m)) = L e[nb] E (c(m)) + L e[nb] D (c(m)), (3.19) 93 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . for m = 0, 1, . . . , Nc − 1. We consider that the extrinsic output of the equalizer {Le[nb]E (c(m))}Nc−1m=0 , the extrinsic output {Le[nb]D (c(m))}Nc−1m=0 of the channel decoder, and the a posteriori LLR values {Lp[nb]D (c(m))}Nc−1m=0 are the samples of random variables Le[nb]E , L a[nb] D , and L p[nb] D , respectively. For simplicity in deriving an expression for the CBER, similarly in [7], we assume that L e[nb] E and L a[nb] D are conditionally Gaussian distributed as L e[nb] E ∼ N ( (2c(m) − 1)µ[nb]E , (σ[nb]E )2 ) and L d[nb] D (c(m)) ∼ N ( (2c(m) − 1)µ[nb]E , (σ[nb]D )2 ) where mean values µ [nb] E = (σ [nb])2E/2 and µ [nb] D = (σ [nb])2D/2. Therefore, L p[nb] D is also Gaussian distributed with the mean µ[nb] = µ [nb] E + µ [nb] D and the variance (σ [nb])2 ≤ (σ [nb] E ) 2+(σ [nb] D ) 2. Note that we consider that L e[nb] E and L a[nb] D are weakly and positively correlated. We can obtain variances (σ [nb] E ) 2 and (σ [nb] D ) 2 as (σ [nb] E ) 2 ≈ J−1(Ia[nb]D ) and (σ [nb] D ) 2 ≈ J−1(Ie[nb]D ). The lower bound on the CBER p[nb]CBER obtained after an arbitrary number of iterations in the nbth radio block can be given as p [nb] CBER = 1 2 erfc ( σ[nb] 2 √ 2 ) ≥ 1 2 erfc   √ (σ [nb] E ) 2 + (σ [nb] D ) 2 2 √ 2   ≈ 1 2 erfc   √ J−1(Ia[nb]D ) + J−1(I e[nb] D ) 2 √ 2   , (3.20) where erfc denotes the complementary error function. Hence, using (3.20), we can obtain an approximate lower bound on the CBER at any given point (IaD, I e D) (or (I e E , I a E)) on the EXIT-band chart. The lower bound on the average CBER p̄CBER can be obtained by taking average over several code blocks NB (i.e., taking average over many different information sequences, channel realizations and random interleavers) and expressed as p̄CBER , 1/NB ∑NB nb=1 p [nb] BER ' 1 2NB ∑NB nb=1 erfc (q J−1(Ia[nb]D )+J−1(I e[nb] D ) 2 √ 2 ) . 3.6 Application to EDGE/EGPRS Systems In this section, we analyze the convergence behavior of MMSE-based turbo equalization using the EXIT-band chart technique for ST block-coded MIMO EDGE/EGPRS sys- 94 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . tems. We closely follow the radio downlink specifications as given in [14, 15, 16] with few exceptions. The following modulation and coding schemes (MCSs): MCS-5, MCS- 7, and MCS-8, which use 8PSK modulation, are chosen. Block lengths of Nb = 448, Nb = 2×448 andNb = 2×544 binary information bits are used in the MCS-5, MCS-7 and MCS-8 schemes, respectively. A rate-1/3 nonsystematic convolutional code with code polynomials (133, 171, 145) in octal notation is used in all MCSs considered. Puncturing is used to achieve the desired code rate Rec = 0.37, Rec = 0.76 and Rec = 0.92 in MCS-5, MCS-7, and MCS-8, respectively. For simplicity, instead of the interleavers specified in [16], pseudo-random block interleavers are used. After interleaving, the punctured code block is combined with a block of coded header and flag bits. The resulting block of 1392 bits is mapped into 4 bursts of 348 bits each. Each burst is Gray mapped into 8-PSK symbols. We use the channel model for typical urban areas (TUA) as specified in [17]. A linearized Gaussian minimum shift-keying (GMSK) pulse-shaping filter [15] and a square-root raised cosine (SRC) filter are employed as the transmitter and receiver filters, respectively. For simplicity in simulation, we assume the same power-delay profile for all subchannels. We also assume that every subchannel is time-invariant within each burst and varies independently from burst to burst. This assumption is well justified for low-to-moderate mobile speeds based on the EDGE/EGPRS system specifications (with ideal frequency hopping). Furthermore, it is assumed that perfect channel information is available to the receiver. The total average channel power gain is normalized to one. The noise variance σ2η is determined according to the SNR σ2η = σ2s ∑2 nt=1 ∑L l=0 α 2 nt,nr(l) QRec 10− (Eb/N0) 10 , (3.21) where α2nt,nr(l) is the average power gain of the lth multipath component. Note that since we assume the same power-delay profile for all subchannels, the noise variance σ2η is the same at all receiver antenna for a given Eb/N0 (dB) value. 95 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 3.6.1 Extrinsic Transfer Characteristics Figure 3.5 shows the EXIT characteristics of the MAP decoder for the MCS-5, MCS-7 and MCS-8 schemes with different code rates and block lengths. We use the input mutual information IaD and the output mutual information I e D on the abscissa and ordinate, respectively. It can be clearly observed from Figure 3.5 that the mutual information provided by the MAP decoder decreases as the effective coding rate Rec increases. We see that the output mutual information IeD of the MAP decoder is negligible (i.e., nearly zero) till the input mutual information about IaD = 0.6 and I a D = 0.8 for MCS-7 (with Rec = 0.76) and MCS-8 (with Rec = 0.92), respectively. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mutual information IaD at input of decoder M ut ua l i nf or m at io n Ie D  a t i np ut  o f d ec od er   MCS−5 MCS−7 MCS−8 σIe D d−band µIe D Figure 3.5: EXIT characteristics of MAP decoder for the convolutional code with gen- erators (133, 171, 145)8 for different code rates 96 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . In Figure 3.6 we present the EXIT characteristics of the SoftISoftO MMSE equalizer with WL processing using the ExactImpl for different Eb/N0 values. We use the input mutual information IaE and the output mutual information I e E on the abscissa and the ordinate, respectively. Here, we closely followed the specifications of the MCS-5 scheme. In particular, we used 8-PSK modulation and a block length of Nb = 1248 code bits (corresponding to 416 symbols). We can see that the output mutual information IeE of the MMSE equalizer increases with increase of Eb/N0 for a given value of I a E. In Figure 3.7 we show the EXIT characteristics of the SoftISoftOMMSE equalizer using the ExactImpl with different number of receive antennas NR. For comparison purposes, we also show the EXIT characteristics of MMSE equalization obtained for SISO transmission (i.e., without STBC). We can observe that as the number of receive antennas NR is increased, since the maximum possible diversity order NTNR(L+ 1) increases, the output mutual information IeE also increases. As NR is increased, the effect of channel variations on the performance of SoftISoftO equalization decreases, which causes low fluctuations in IeE . As a consequence, we observe that the width of the e-band (and also σIeE reduces as NR is increased.4 Furthermore, it can be seen that there is no much difference in IeE obtained for STBC (with NR = 1) and SISO, especially at the high I a E region. Nevertheless, we see that the width of the e-band (and also σIeE obtained for STBC (with NR = 1) is considerably lower than that obtained for SISO, which is due to the fact that spatial diversity obtained by using STBC reduces the effect of channel variations. Hence, it is clear from the EXIT characteristics that we can achieve improved performance using the Alamouti symbol-wise STBC scheme (with NR = 1) compared with SISO transmission in frequency-selective channels. In Figure 3.8 and Figure 3.9, we show the EXIT characteristics of different proposed equalization algorithms at Eb = 5dB and Eb = 5dB, respectively. As the reference per- formance, also included is the EXIT characteristics of 8-PSK (Gray-) demapper obtained 4We observed that when the MIMO channel is time-invariant (static), we get significantly a smaller e-band. 97 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mutual information IaE at input of Equalizer M ut ua l i nf or m at io n Ie E  a t o ut pu t o f E qu al ize r   Eb/N0=2 dB Eb/N0=5 dB Eb/N0=8 dB σIe E µIe E Figure 3.6: EXIT characteristics of MMSE-based equalization (ExactImpl) for different Eb/N0 using the maximum ratio combining (MRC) receiver for an equivalent channel model with a set of NTNR(L+1) parallel and independent flat fading subchannels. Hence, it is clear that µIeE obtained for 8-PSK demapper is the average matched-filter bound (MFB) for the ST block-coded system considered here. As expected, we see that the ExactImpl and the MF-based receiver provide the best and worst mutual information µIeE . We see that the difference in µIeE obtained using the ExactImpl and AprxIpml-1 is not signifi- cant. Even though µIeE obtained using either ExactImpl-2 or MF is nearly the same in the low IaE region, since ExactImpl-2 has a larger slope, µIeE provided by ExactImpl-2 is considerably larger than that provided by the MF-based receiver at large values of IaE . 98 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mutual information IaE at input of equalizer M ut ua l i nf or m at io n Ie E  a t o ut pu t o f e qu al ize r   µIe E  − STBC, NR=1 µIe E  − STBC, NR=2 µIe E  − STBC, NR=4 µIe E  − SISO Figure 3.7: EXIT characteristics of MMSE-based equalization (ExactImpl) for different Eb/N0 This result is due to the fact that in the MF-based receiver, the performance is highly affected by self-symbol interference (SSI), which is due to STBC, whereas in ExactImpl- 2 SSI is suppressed through MMSE filtering. Note that at IaE = 1 (i.e., with perfect a priori information, |LaE | → ∞) it can be observed that the ExactImpl, AprxImpl-1, and AprxImpl-2 schemes provide the same µIeE . With perfect a priori information the ExactImpl, AprxImpl-1, and AprxImpl-2 schemes can be shown to be identical. With perfect a priori (i.e., IaE = 1) since we know s(n ′)∀n′, n′ 6= n intersymbol interference (ISI) and co-antenna interference (CAI) can be completely removed from the received signal through soft ISI and CAI cancelations. Nevertheless, due to the presence of SSI, 99 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . µIeE provided by either ExactImpl, AprxImpl-1 or AprxImpl-2 is lower than that provided by the 8-PSK demapper at IaE = 1. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Mutual information IaE at input of equalizer M ut ua l i nf or m at io n Ie E  o u tp ut  o f e qu al ize r   avg{IeE}, ExactImpl avg{IeE}, AprxImpl−1 avg{IeE}, AprxImpl−2 avg{IeE}, MF avg{IeE}, 8−PSK Demapper, MFB Figure 3.8: EXIT characteristics of different equalization schemes at Eb/N0 = 5dB 3.6.2 EXIT-band Chart Intuitively, we can obtain significant improvements in the performance using turbo equal- ization when the iterative process converges successfully (i.e., converges at high IaE and IaD values). When there is an open tunnel (i.e., gap) between the e-band and the d-band in a given block the iterative process can converges successfully with a high probabil- ity. Hence, similarly in turbo decoding [18], we can study the intersection behavior of 100 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mutual information IaE at input of equalizer M ut ua l i nf or m at io n Ie E  a t o ut pu t o f e qu al ize r   avg{IeE}, ExactImpl avg{IeE}, AprxImpl−1 avg{IeE}, AprxImpl−2 avg{IeE}, MF avg{IeE}, 8−PSK Dempapper (MFB) Figure 3.9: EXIT characteristics of different equalization schemes at Eb/N0 = 7dB e-band and d-band to investigate probabilistic convergence behavior of MMSE-based turbo equalization. Figure 3.10 depicts the EXIT-band charts of turbo equalization using the ExactImpl of the MMSE-based equalizer and the MAP decoder for the MCS-5 scheme at different values of Eb/N0. We see that e-band and d-band intersect in the low I a E and I a E region at low Eb/N0 values. Therefore, the iterative process converges at low I a E and I a D with high probability. As Eb/N0 increases, the e-band gradually moves upwards eventually leading to an clear open tunnel between the e-band and the d-band. Particularly, at Eb/Nb = 6 although there is an open tunnel between the e-band and the d-band at low-to-moderate 101 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . IaD values, the e-band and the d-band intersect before reaching the CBER contour of 10−2. Therefore, we can expect that some blocks converge successfully while others fail do so. Hence, on average, we obtain moderate BER performance. Nevertheless, at Eb/N0 = 8 it can be observed that there is a large tunnel between the e-band and the d- band. Therefore, the iterative process converges successfully with very high probability leading to low BER performance. In Figure 3.11, Figure 3.12, and Figure 3.13, the EXIT-band charts of turbo equaliza- tion using AprxImpl-1, AprxImpl-2, and the MF-based receiver, respectively, are shown for the MCS-5 scheme at Eb/N0 = 6dB and Eb/N0 = 8dB. For comparison purposes we also show the EXIT characteristics of the ExactImpl. From Figure 3.11 it is clear that AprxImpl-1 possess nearly the same EXIT characteristics as the ExactImpl, especially for low Eb/N0 values. Therefore, we can attain almost the same performance using AprxImpl-1 as the ExactImpl. It can be observed from Figure 3.12 that for low-to-moderate Eb/N0 values the e- band of AprxImpl-2 and the d-band intersect at low IaD and I a E values and hence, the iterative process is unable to converge successfully. Therefore, it is not possible to achieve considerable performance improvements using AprxImpl-2 for low-to-moderate Eb/N0 values. We see from Figure 3.12 that at small values of IaE and I a D, the ExactImpl provides much higher IeE compared with AprxImpl-2 and as a consequence, there is an opening between the e-band of the ExactImpl and the d-band. It is clear from the EXIT-band chart that we can use a hybrid approach, in which we use the ExactImpl in the first predetermined number of iterations to exploit the good starting behavior of the ExactImpl and then, use AprxImpl-2 for the subsequent iterations. It is noteworthy that in the hybrid scheme one can use AprxImpl-1 instead of the ExactImpl as the AprxImpl- 1 has almost similar EXIT characteristics as the ExactImpl. The EXIT-band chart can effectively be used to determine the number of iterations in which the ExactImpl should be used to achieve high performance improvements using the hybrid scheme. For the 102 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . system considered, we can determine from the EXIT-band chart that the ExactImpl should be used in the first two iterations at low-to-moderate Eb/N0 values and in the first iteration at high Eb/N0 values. From the EXIT-band chart shown in Figure 3.13 it is clear that we can not attain substantial performance improvements using the MF-based receiver for Eb/N0 < 8 dB since the e-band and the d-band overlap. We observe that, on average, IeE provided by the MF-based receiver is much less than that provided by the ExactImpl for all values of IaE . As a result, there is no clear opening between the e-band of the MF-based receiver and the d-band at Eb/N0 = 8dB. It follows that even we use a hybrid approach, in which the ExactImpl (or AprxImpl-1) is used in the first predetermined number of iterations, the performance obtained using the Hyb ExactImpl/MF scheme is considerably less than that obtained using the ExactImpl. Furthermore, at high Eb/N0 values, it is clear from the EXIT-band charts that we can achieve nearly the same performance using the MF- based receiver as the Hyb ExactImpl/MF scheme. Therefore, the Hyb ExactImpl/MF scheme is not very useful. 103 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 2 dB   µIe D , MAP Decoder µIe E ,  MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 IeE = I a D I e E  =  I a D (b) Eb/N0 = 4 dB   µIe D  , MAP Decoder µIe E ,  MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 IeD = I a E I e E  =  I a D (c) Eb/N0 = 6 dB   µIe D , MAP Decoder µIe E  , MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 IeD = I a E I e E  =  I a D (d) Eb/N0 = 8 dB   µIe D ,  MAP Decoder µIe E ,  MMSE Equalizer d−band e−band 0.2 0.1 0.05 0.02 0.01 0.005 0.001 0.2 0.1 0.05 0.02 0.02 0.05 0.1 0.2 0.2 0.1 0.05 0.02 0.01 0.01 0.01 0.005 0.005 0.005 0.001 0.0010.001 Figure 3.10: EXIT-band chart for turbo equalization using the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. 104 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 6 dB   avg{IeD}, MAP Decoder avg{IeE}, MMSE Equ.−AprxImpl1 avg{IeE}, MMSE Euq.−ExactImpl 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (b) Eb/N0 = 8 dB   avg{IeD}, MAP Decoder avg{IeE}, MMSE Equ.−AprxImpl1 avg{IeE}, MMSE Equ.−ExactImpl 0.0050.005 0.01 0.02 0.001 0.001 0.01 0.02 0.05 0.1 0.05 0.1 0.2 0.2 Figure 3.11: EXIT-band chart for turbo equalization using MMSE AprxImpl-1 and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. 105 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 6 dB   avg{IeD}, MAP Decoder avg{IeE}, MMSE Equ.−AprxImpl2 avg{IeE}, MMSE Equ.−ExactImpl 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (b) Eb/N0 = 8 dB   avg{IeD}, MAP Decoder avg{IeE}, MMSE Equ.−AprxImpl2 avg{IeE}, MMSE Equ.−ExactImpl 0.01 0.2 0.2 0.10.1 0.050.05 0.020.02 0.01 0.005 0.005 0.0010.001 Figure 3.12: EXIT-band chart for turbo equalization using MMSE AprxImpl-2 and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. 106 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 6 dB   avg{IeD}, MAP Decoder avg{IeE}, MF avg{IeE}, MMSE Equ.−ExactImpl 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (b) Eb/N0 = 8 dB   avg{IeD}, MAP Decoder avg{IeE}, MF avg{IeE}, MMSE Equ.−ExactImpl 0.05 0.2 0.1 0.05 0.1 0.2 0.02 0.01 0.005 0.0010.001 0.005 0.01 0.02 Figure 3.13: EXIT-band chart for turbo equalization using MF and the MAP decoder with CBER scaling as contour plot: MCS-5, TUA channel model. 107 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . Trajectories of Iterative Equalization The iterative process starts with IaE = 0 (i.e., with no a priori information, L a E(c(m ′)) = 0, ∀m′) at the SoftISoftO equalizer. The output LLRs of the equalizer {LeE(c(m′))}Ncm′=0 (represented by IeE in the EXIT-band chart) are then fed into the channel decoder as the a priori information (i.e., IeE becomes I a D). The extrinsic output of the chan- nel decoder {Lad(c(m))}Ncm=0 (represented by IeD in the EXIT-band chart) is then fed back to the SoftISoftO equalizer as the a priori information (i.e., IaE becomes I e D) in the next iteration and so forth. Therefore, the iterative process is the trace between the EXIT characteristics bands of the equalizer and decoder. We can use the trace of iterative equalization, which is plotted in the EXIT-band chart, to investigate the validity of the EXIT-band chart analysis and also to gain more insight into the itera- tive process of turbo equalization. For the snapshot trajectory of the nbth block, I e E [or IeD] is computed using the PDF p [nb] LeE (l|C = ±1) [or p[nb]LeD (l|C = ±1)], which is ob- tained using the histogram measurements of extrinsic LLR values {Le[nb](c(m′))}Nc−1m′=0 [or {Le[nb]D(c(m))}Nc−1m=0 ] of the given block. For the averaged trajectory, IeE [or IeD] is computed using the average PDF p̄LeD(l|C = ±1) = 1/NB ∑NB nb=1 p [nb] LeE (l|C = ±1) [or p̄LeD(l|C = ±1) = 1/NB ∑NB nb=1 p [nb] LeD (l|C = ±1)] obtained by taking average over NB blocks. In Figures 3.14, 3.15, and 3.16 we show a set of 10 snapshot trajectories in 10 different blocks and the averaged trajectory obtained from simulation at Eb/N0 = 3 dB, Eb/N0 = 5 dB, and Eb/N0 = 7 dB, respectively. We see from Figures 3.14-3.16 that snapshot trajectories of turbo equalization obtained from simulation fit into the EXIT- bands fairly well. It can be observed from Figure 3.14 that as expected, at Eb/N0 = 3dB since the EXIT bands completely overlap from the beginning, most of blocks are unable to converge successfully. We see from Figure 3.15 that compared with Eb/N0 = 3dB at Eb/N0 = 5dB more number of frames are able to converge successfully since the EXIT- bands now intersect at a point between the CBER contours of 0.1 and 0.05. Nevertheless, 108 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . still a considerable number of blocks fails to converge successfully and consequently, on average, we can not achieve large performance improvements. It can be observed from Figure 3.16 that at Eb/N0 = 7dB, since there is a large open tunnel between the EXIT-bands of the MMSE equalizer and the MAP decoder, most of blocks converges successfully. We found that for Eb/N0 > 7 dB a block rarely fails to convergence and thus, the probability of convergence failure is low, which leads to high performance improvements. It can be seen Figures 3.14-3.16 that the averaged trajectories of MMSE-based turbo equalization approximately follow the mean EXIT curves µIeD and µI e E within the first few iterations. For comparison purposes, in Figure 3.17 the averaged trajectories obtained using the ExactImpl for the equalization test channel are shown at Eb/N0 = 3 dB and 5 dB. We notice that the trace of the iterative process slightly deviates from the mean EXIT curve µIeD of the decoder even in the first iteration (or in the first few iterations). Particularly, for the first iteration (or for the first few iterations) it can be observed that the averaged extrinsic output IeD of the MAP decoder traced by the trajectory is bigger and smaller than expected, for low and high IaD values, respectively. This discrepancy could be accounted for the fact that in obtaining the EXIT characteristics of the MAP decoder we consider that the a priori LLR values {LaD(c(m))}Nc−1m=0 are samples of a Gaussian distributed random variable, but we showed that the extrinsic LLR values {LeE(c(m′))}Nc−1m′=0 can not be well modeled using a single-parameter Gaussian distribution. 109 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (b) Average trajectory   µIe D , MAP Decoder µIe D , MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) A set of 10 snapshot trajectoreis   µIe E , MAP Decoder µIe E , MMSE Equalizer Average Trajectory 0.2 0.2 0.10.1 0.05 0.05 0.01 0.020.02 0.01 0.0050.005 0.001 0.001 Figure 3.14: EXIT-band chart with simulated trajectories of turbo equalization using the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 3 dB: MCS-5, TUA channel model. 110 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (b) Average trajectory   µIe D , MAP Decoder µIe E , MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) A set of 10 snapshot trajectoreis   µIe D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory 0.2 0.1 0.2 0.1 0.050.05 0.020.02 0.010.01 0.005 0.005 0.0010.001 Figure 3.15: EXIT-band chart with simulated trajectories of turbo equalization using the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 5 dB: MCS-5, TUA channel model. 111 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (b) Average trajectory   µIe D , MAP Decoder µIe E , MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) A set of 10 snapshot trajectoreis   µIe D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory 0.2 0.1 0.05 0.05 0.1 0.2 0.001 0.001 0.005 0.005 0.010.01 0.02 0.02 Figure 3.16: EXIT-band chart with simulated trajectories of turbo equalization using the MMSE ExactImpl and the MAP decoder with CBER scaling as contour plot at Eb/N0 = 7 dB: MCS-5, TUA channel model. 112 C h a p ter 3 . C o n v erg en ce B eh a v io r o f T u rb o E q u a liza tio n in S T B lo ck -C o d ed ... 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 3 dB   µIe D , MAP Decoder µIe E , MMSE Equalizer 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E I e E  =  I a D (a) Eb/N0 = 5 dB   µIe D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory Average Trajectory 0.02 0.01 0.0050.005 0.0010.001 0.05 0.1 0.20.2 0.1 0.05 0.02 0.01 Figure 3.17: EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl and the MAP decoder at Eb/N0 = 7 dB: MCS-5, equalization-test channel model. 113 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . For short block lengthsNi (i.e., for finite interleaving depths), the correlation between the extrinsic information of the SoftISoftO equalizer and the MAP decoder increases with iterations. As a result, we see (more prominently at Eb/N0 = 5 dB from Figure 3.15) that with iteration the gains in IeE and I e D gradually decrease and therefore, eventually, the averaged trajectory of turbo equalization ceases at a point very close to the mean EXIT curve muIeE of the SoftISoftO equalizer. It is noteworthy that even after a few iterations the average value of IeE for a given value of I a D is nearly the same as the expected value based on the mean EXIT characteristics muIeE of the equalizer. Hence, the trace of the iterative process closely follow the mean EXIT curve µIeE of the equalizer and the averaged trajectory is shifted towards the curve of µIeE . Figure 3.18 shows the averaged trajectory of turbo equalization obtained using the AprxImpl-1 at Eb/N0 = 7 dB. It can be observed that the average trajectories of both ExactIpml and AprxImpl-1 trace approximately similar paths and rest at very close points. We observed a similar behavior for different Eb/N0 values. Hence, as expected, it is clear that the nearly the same performance can be obtained using AprxImpl-1 as using the ExactImpl. In Figure 3.19 we show the averaged trajectories of turbo equalization obtained using AprxImpl-2 and Hyb ExactImpl/AprxImpl-2 schemes at Eb/N0 = 7 dB. In the Hyb ExactImpl/AprxImpl-2 scheme we use the ExactImpl for the first iteration. We see that after a few iterations the averaged trajectory obtained using AprxImpl-2 rests in a point even before reaching the CBER contour of 0.05 and hence, we are unable to attain improved performance using AprxImpl-2 alone. This results is due to the fact that some blocks fail to converge successfully as the EXIT-bands of the MAP decoder and AprxImpl-2 slightly overlap at low IaD and I a E values. Nevertheless, it can be seen that the averaged trajectory obtained using the Hyb ExactImpl/AprxImpl-2 scheme reaches very close to the CBER contour of 0.005. Therefore, the averaged trajectories confirm that we can obtain considerably higher performance improvements using the Hyb ExactImpl/AprxImpl-2 scheme than using the AprxImpl-2 scheme alone. 114 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E Ie E  =  Ia D   µIe D , MAP Decoder µIe E , MMSE Equ.−AprxImpl−1 µIe E , MMSE Equ.−ExactImpl Av. Trajectory, MMSE Equ.−AprxImpl−1 Av. Trajectory, MMSE Equ.−ExactImpl 0.2 0.1 0.05 0.02 0.01 0.005 0.001 Figure 3.18: EXIT-band chart with average trajectories of turbo equalization using MMSE AprxImpl-1 and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model. In Figure 3.20, we show the averaged trajectories of turbo equalization obtained using the MF-based receiver and the Hyb ExactImpl/MF scheme at Eb/N0 = 7 dB. Note that in the Hyb ExactImpl/MF scheme we use the ExactImpl for the first iter- ation. It can be observed that the averaged trajectory of the MF-based receiver dies out before passing the CBER contour of 0.5. Hence, it is clear that the performance of the MF-based receiver is even less than that of the MMSE-ExactImpl obtained after the first iteration. It is clear by inspecting the average trajectory that as expected, in the Hyb ExactImpl/MF scheme, after the first iteration we can not achieve considerable 115 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E Ie E  =  Ia D   µIe D , MAP Decoder µIe E , MMSE Equ.−AprxImpl−2 µIe E , MMSE Equ.−ExactImpl Av. Trajectory, MMSE Equ.−AprxImpl−2 Av. Trajectory, Hyb. ExactImpl/AprxImpl−2 Av. Trajectory, MMSE Equ.−ExactImpl 0.2 0.1 0.05 0.02 0.01 0.005 0.001 Figure 3.19: EXIT-band chart with average trajectories of turbo equalization using MMSE AprxImpl-2 and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model. improvements using the MF-based receiver. This results is due to the fact that the EXIT-bands of the MF-based receiver and the MAP decoder partially overlap even at high IaD values. As confirmed by the averaged trajectories, although slightly improved performance can be attained using the Hyb ExactImpl/MF scheme compared with the MF-based receiver, the performance of the Hyb ExactImpl/MF scheme is much worse than that of the ExactImpl. Figure 3.21 and Figure 3.22 show the EXIT-band cart and the averaged trajectory of turbo equalization obtained using the ExactImpl for the MCS-8 scheme with the 116 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E Ie E  =  Ia D   µIe D , MAP Decoder µIe E , MF µIe D , MMSE Equ. − ExactImpl Av. Trajectory − MF Av. Trajectory. − Hyb. ExactImpl/MF Av. Trajectory − MMSE Equ. ExactImpl 0.2 0.1 0.05 0.02 0.01 0.005 0.001 Figure 3.20: EXIT-band chart with average trajectories of turbo equalization using the MF and the MAP decoder at Eb/N0 = 7 dB: MCS-5, TUA channel model. code rate Rec = 0.92 at Eb/N0 = 14dB and Eb/N0 = 18dB, respectively. From the EXIT-band characteristics of the MAP decoder for MCS-8, it can be observed that the output mutual information IeD is negligible for the input mutual information even up to IeD = 0.8 due to severe puncturing. Therefore, it can be seen from Figure 3.22 that when a channel code having a very high code rate is used, even at high Eb/N0 values there is no clear opening between the EXIT-bands of the equalizer and MAP decoder. As a result, even at high Eb/N0 values, we can not obtain considerable performance improvements using turbo equalization as a high number of blocks are unable to converge successfully. Now we investigate the effect of transferring the a posteriori information (i.e., full 117 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E Ie E  =  Ia D   µIe D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory Figure 3.21: EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 14 dB: MCS-8, TUA channel model. information) of code bits obtained from the channel decoder instead of the extrinsic information of code bits to the equalizer. To analyze the iterative nature of turbo equal- ization, when a posteriori information is used as the a priori information to the equalizer, similar to the EXIT-band chart, we plot the EXIT-band characteristics of the equalizer and the a posteriori information transfer (POIT)-band characteristics of the channel de- coder in a single diagram. This diagram is referred to as Hyb EXIT/POIT-band chart. Figure 3.23 and Figure 3.24 depict the Hyb EXIT/POIT-band chart and the averaged trajectory of turbo equalization obtained using the ExactImpl for the MCS-8 scheme at Eb/N0 = 14dB and Eb/N0 = 18dB, respectively. It is clear from Figure 3.24 that 118 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IeD = I a E Ie E  =  Ia D   µIe D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory Figure 3.22: EXIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 18 dB: MCS-8, TUA channel model. when we provide the SoftISoftO equalizer with the a posteriori information of code bits obtained from the MAP decoder as the a priori information, there is a large opening between the EXIT-band characteristics of the equalizer and the POIT-band character- istics of the MAP decoder letting most blocks converge successfully. Hence, the Hyb EXIT/POIT-band chart demonstrates that significantly higher performance improve- ments can be achieved by transferring the a posteriori information of code bits obtained from the SoftISoftO channel decoder to the SoftISoftO equalizer than transferring the extrinsic information of code bits. The averaged trajectories of turbo equalization shown in Figure 3.23 and Figure 3.24 clearly confirm this fact. It is noteworthy that when the 119 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . a posteriori information of code bits is used as the a priori information to the equal- izer, the correlation between the extrinsic information of the code bits obtained from the equalizer and the a priori information is high. As a consequence, in this case, we observe that after the first iteration, the averaged trajectory does not closely follow the mean EXIT curve µIeE of the SoftISoftO equalizer. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IpD = I a E Ie E  =  Ia D   µIp D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory Figure 3.23: EXIT/POIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 14 dB: MCS-8, TUA channel model. 120 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 IpD = I a E Ie E  =  Ia D   µIp D , MAP Decoder µIe E , MMSE Equalizer Average Trajectory Figure 3.24: EXIT/POIT-band chart with average trajectories of turbo equalization using the MMSE ExactImpl at Eb/N0 = 18 dB: MCS-8, TUA channel model. 3.7 Conclusion In this chapter, we studied the convergence behavior of turbo equalization for Alamouti ST block-coded MIMO systems with finite block lengths. We successfully applied the EXIT-band chart technique, which was originally proposed for analyzing turbo decoding of PCCS with finite block lengths, for turbo equalization. The transfer of extrinsic information between the SoftISoftO equalizer and decoder was visualized as the iterative trajectory. We closely follow the subclassifications of EDGE/EGPRS systems. Our results showed that EXIT-band chart can be used as an effective engineering tool for 121 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . analyzing the convergence behavior of turbo equalization with finite block lengths. The insights obtained using the EXIT-band chart analysis on the convergence behavior can be used system design purposes (e.g., to determine an appropriate code rate). The impact on the convergence behavior and the performance due to use of STBC was analyzed through the EXIT-band characteristics. Although STBC and SISO possess nearly the same average extrinsic characteristics, STBC has a narrower e-band compared with SISO, which leads to improved performance using STBC over SISO. Results con- firmed that the performance of the MMSE-base equalizer is significantly better than that of the MF-based receiver. We compared the convergence behavior of the ExactImpl and AprxImpls of the MMSE-based equalizer through the EXIT-chart band analysis. Results showed that using a hybrid approach of the ExactImpl (or AprxImpl-1) and AprxImpl-2, a better trade-off between performance and complexity can be achieved. The EXIT-band chart analysis can be used to determine the number of iterations performed using the ExactImpl (or AprxImpl-1) in the hybrid approach. When a convolutional code with a very high code rate, that is obtained using severe puncturing, is employed, we cannot achieve considerable improvements using turbo equalization. Nevertheless, we showed through the EXIT-band chart analysis that instead of the extrinsic information of code bits delivered by the channel decoder, significant performance improvements can be ob- tained using a posteriori information of code bits as the a priori information to the SoftIsoftO equalizer. We proposed the Hyb EXIT/POIT-band chart for analyzing the convergence behavior, when a posteriori information is used as the a priori information to the equalizer. The estimates on the average bit error rate (BER) after an arbitrary number of iterations were obtained from the EXIT-band chart using a semi-analytical approach. Simulation results verified that the BER estimates obtained from the EXIT- band chart is substantially accurate for practical purposes. 122 Chapter 3. Convergence Behavior of Turbo Equalization in ST Block-Coded. . . 3.8 Bibliography [1] M. Tuchler, A. C. Singer, and R. Koetter, “Minimum mean squared error equal- ization using a priori information,” IEEE Trans. Commun., vol. 50, pp. 673–682, Mar. 2002. [2] X. Wautelet, A. Dejonghe, and L. Vandendorpe, “MMSE-based fractional turbo re- ceiver for space-time BICM over frequency-selective MIMO fading channels,” IEEE Trans. Signal Processing, vol. 52, pp. 1804–1809, June 2004. [3] K. C. B. Wavegedara and V. K. Bhargava, “MMSE-based turbo equalization for space-time block coded CDMA downlink,” IEEE Trans. Veh. Technol., submitted for publication. [4] W. 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[Online]. Available: http://www.etsi.org [16] Digital cellular telecommunications system (Phase 2+); Channel coding, ETSI Technical Specification 3GPP TS 45.003 version 6.8.0, Oct. 2005. [Online]. Available: http://www.etsi.org [17] Digital cellular telecommunications system (Phase 2+); Radio transmission and reception, ETSI Technical Specification 3GPP TS 05.05 version 8.20.0, Nov. 2005. [Online]. Available: http://www.etsi.org [18] H. Lee and V. Gulati, “Iterative equalization/decoding of LDPC code transmit- ted over MIMO fading ISI channels,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’02), Sept. 2002, pp. 1330–1336. 124 Chapter 4 Space-Time Coded CDMA Uplink Transmission with MUI-Free Reception1 4.1 Introduction Space-time (ST) coding has recently evolved as an effective transmit diversity technique for achieving high data rates over multiple-transmit antenna wireless systems. ST codes were originally designed and investigated for frequency-flat fading channels. In [1], a single-carrier time-reversal zero-padding (SC-TR-ZP) based block transmission scheme is developed for frequency-selective fading channels. It is shown that this scheme can yield the total diversity gain defined as the multiplication of the number of transmit, re- ceive antennas and the number of multipath components. However, maximum likelihood (ML) decoding is required to acquire the full diversity gain [2]. Unfortunately maximum likelihood sequence estimation (MLSE) can not be exploited in practice because of its high complexity (measured in number of trellis states), particularly with high channel order and/or high level modulation. Even though linear equalization and decision feed- back equalization (DFE) can be exploited as reduced complexity alternatives to MLSE, they suffer from noise enhancement and error propagation, respectively [3]. 1A version of this chapter has been published. Wavegedara, K. C. B., Djonin, D. V., and Bhargava, V. K. (2005) Space-time coded uplink transmission with MUI-free reception, IEEE Transactions on Wireless Communications, vol.4, pp. 3095-3105, Nov. 2005. 125 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . On the other hand, delayed decision feedback sequence estimation (DDFSE) is orig- inally proposed in [4] as a promising suboptimal alternative for Forney-type MLSE (i.e., based on whitened channels). In DDFSE, the channel is split into two parts: the leading and tail parts. Intersymbol interference (ISI) due to the leading part of the channel is handled by MLSE and ISI due to the tail of the channel is canceled by decision feedback means. Hence, DDFSE can be viewed as a combination of MLSE and DFE schemes [3]. As a consequence of truncating the channel to its first few multipath components, the performance of DDFSE becomes highly sensitive to the channel phase. The best perfor- mance can be achieved when the underline multipath channel is minimum phase as in this case error propagation is minimized [5]. In [5], DFSE is developed for unwhitened channels based on the Ungerböeck approach of MLSE. Unlike in the whitened case, the performance of unwhitened DFSE (UDFSE) does not depend on the channel phase and therefore, it is not required to convert the channel into minimum phase equivalent. However, the fact that the output of unwhitened channels (i.e., the matched filter out- put) depends on both past and future input symbols, degrades the performance of the UDFSE (see [5] for more details). In [3] DFSE is considered for two transmit diversity schemes (i.e., for delay diversity and ST trellis coding schemes) and the performance is investigated for the EDGE system. More recently, in [6] DFSE estimation is also considered for TR-based ST block coding [7]. Even though ST coding has been extensively studied for single user systems, so far, research on ST coding in multiuser systems has received relatively limited atten- tion, especially over frequency selective channels. In a K-user system where each user is equipped with NT transmit antennas, multi user interference (MUI) is caused by (K − 1)NT interfering signals instead of K − 1 interfering signals as in the system with users having single transmit antenna. Hence, utilizing ST coding in multiuser systems is challenging due to the presence of severe MUI [8]. In CDMA systems, MUI is consid- ered as a main performance limiting factor. Multiuser detectors can be used to suppress 126 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . detrimental effect of MUI to the system performance. However, in addition to high com- plexity, most multiuser detectors alleviate MUI statistically [9]. A promising MUI-free scheme known as chip-interleaved block-spread (CIBS)-CDMA based on block-spreading and zero-padding operations is proposed in [9]. The main feature of CIBS-CDMA is that the mutual orthogonality among users is preserved regardless of the underline mulitpath propagation. Hence, users can be separated deterministically using low-complexity code- matched filtering. Thanks to MUI free reception, the performance of CIBS-CDMA is not affected by the number of active users in the system. The main contribution of this chapter is development of a scheme, which enables ST block coded transmission over multipath fading channels in the uplink of CDMA systems. We propose a novel system architecture, which combines SC-TR-ZP based ST block coding and CIBS-CDMA for the uplink communication. In the proposed system, each user is equipped with NT = 2 transmit antennas and the receiver base station may be equipped with NR ≥ 1 receive antennas. The base station receiver may easily support more than one transmit antennas. However, we limit the number of transmit antennas to NT = 2 because of two reasons: the first is that we can not define full-rate ST block codes from complex constellations for more than two transmit antennas [10] and the second reason is due to the possible practical limitations of having more than two antennas at the mobile station. Recently, in [8] a scheme is proposed for downlink following a similar approach (i.e., combination of SC-TR-ZP based ST block coding and CIBS-CDMA). However, such a scheme does not seem to exist for uplink transmission. Second, low-complexity linear equalizers are derived based on the structure of the ST decoded signal. Third, we show how to use DFSE in the proposed scheme and extensively investigate the performance of different DFSE schemes. Initially, whitened DFSE is considered based on the unwhitened system model. Next, the unwhitened system model is converted into its minimum phase equivalent by using a finite impulse response (FIR) whitening prefilter and whitened DFSE (WDFSE) is then applied to 127 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . the output of the whitened system model. Herein, we adapt the linear prediction (LP)- based FIR whitening prefilter design approach. Furthermore, it can be observed that UDFSE schemes show high error-floor behavior, which is more visible for long channels. We propose a novel scheme, which is the combination of modified UDFSE (MUDFSE) and linear MMSE equalizer, to avoid the error-floor behavior. Finally, we appropriately modify the training-based channel estimation algorithm (originally proposed in [11]) for the ST coded CDMA system developed in this chapter. Notation: Bold lower case letters represent vectors while bold upper letters denote ma- trices; (.)T , (.)∗, and (.)H denote the operations transpose, complex conjugate (component- wise), and Hermitian transpose, respectively; ⊗ represents Kronecker product; IM and 0M×N denote the M × M identity matrix and an all-zero matrix of size M × N ; FM stands for a FFT matrix of size M × M of which (i, j)th element is given by (1/ √ M)e−j2pi(i−1)(j−1)/N ,∀i, j ∈ [1,M ]; PnJ is the J × J permutation matrix carrying the reverse operation followed by a right cyclic shift of over n positions of a given vector of length J [2]; diag(s) represents a diagonal matrix with s on the diagonal. 4.2 System Model We consider information transmission in the uplink of a ST coded CIBS-CDMA system. Each mobile user is equipped with two transmit antennas (NT = 2), whereas the base station receiver includes one or multiple antennas (NR ≥ 1). The transmitter and receiver section of the discrete-time base band system model in the uplink operation are depicted in Figure 4.1 and Figure 4.2, respectively. Note that only the transmitter of the kth user (out of maximum K users) is shown in Figure 4.1 and the receiver processing is shown for the same user in Figure 4.2. 128 C h a p ter 4 . S p a ce-T im e C o d ed C D M A U p lin k T ra n sm issio n ... Figure 4.1: Discrete-time baseband system model of the proposed ST coded CIBS-CDMA scheme for uplink transmissions, Part I: Transmitter Section Figure 4.2: Discrete-time baseband system model of the proposed ST coded CIBS-CDMA scheme for uplink transmissions, Part II: Receiver Section129 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 4.2.1 Transmitter Let us consider the transmitter of the kth user. The data symbol stream sk(m) is serial- to-parallel converted into blocks of length B, sk(i) = [sk(iB), . . . , sk(iB+B−1)]T . Then two consecutive symbol blocks sk(2i), sk(2i+ 1) are fed into the ST block encoder. The output of the ST block encoder can be given as [1],   sk1(2i) sk1(2i + 1) sk2(2i) s k 2(2i + 1)   =   sk(2i) −P0Bsk(2i+ 1)∗ sk(2i+ 1) P0Bs k(2i)∗   . (4.1) In this proposed system, symbol blocks of each user are separately ST encoded at symbol- block level before block-spreading. This differs from the downlink system in [8], where composite user signal is ST encoded at chip-block level. Each user is assigned with a distinct short signature sequence (e.g., Walsh-Hadamard code) of length G, ck = [ck(0), . . . , ck(G− 1)]T . Following a similar approach as given in [9], the block-spreading matrix of dimension G(B + L) × B for the kth user is constructed as Ck = ck ⊗ Tzp, where Tzp = [IB,0B×L]T is the zero-padding matrix, and L denotes the maximum channel order. As will be seen, the usage of zero-padding matrix overcomes inter block interference (IBI) arising due to multipath propagation. The output of ST block encoder, sknt(i), for nt = 1, 2 is multiplied by the same block-spreading matrix, C k. The chip blocks of length P = G(B + L), uk1(i) and u k 2(i) are transmitted through the first and second transmit antennas, respectively, at block time interval i. 4.2.2 Channel Model We consider a quasi-synchronous (QS) system in which asynchronism among mobile users is limited to only a few chips. Hence, user asynchronism can be incorporated in the channel model as a delay factor [9]. The multipath frequency-selective channel between the ntth transmit antenna of the kth user and the nrth receive antenna is modeled as a finite impulse response (FIR) filter hknt,nr = [h k nt,nr(0), . . . , h k nt,nr(L k nt,nr)] T . The channel 130 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . order is defined as Lknt,nr , L̃ k nt,nr + τ k, where L̃knt,nr denotes the actual channel order or the number of multipath components, and τk represents the relative delay of the kth user. The composite received signal from all active users at the nrth receiver antenna in the nth chip interval can be expressed as rnr(n) = K−1∑ k=0 [ Lk1,nr∑ l=0 hk1,nr(l)u k 1(n− l) + Lk2,nr∑ l=0 hk2,nr(l)u k 2(n− l) ] + ηnr(n), (4.2) where ηnr(n) denotes the zero mean-additive white Gaussian noise (AWGN) with vari- ance σ2η . Similar to the CIBS-CDMA system (without ST coding) considered in [9], the only channel knowledge required at each transmitter is the maximum channel order, L , maxk,nt,nr L k nt,nr , ∀k, nt, nr. 4.2.3 Receiver The composite received signal at each receiver antenna is sampled at the chip rate and then serial-to-parallel converted into blocks rnr(i) = [rnr(iP ), . . . , rnr(iP+P−1)]T . The received signal block at the nrth receiver antenna for ith block interval can be expressed as matrix-vector form rnr(i) = K−1∑ k=0 [ 2∑ nt=1 Hknt,nrC ksknt(i) + 2∑ nt=1 Ĥknt,nrC ksknt(i− 1) ] + ηnr(i), (4.3) where Hknt,nr , ∀k, nt, nr is the P ×P lower triangular Toeplitz matrix having [hknt,nr(0), . . . , hknt,nr(L), 01×(P−L−1)] T as the first column, Ĥknt,nr , ∀k, nt, nr is the P × P upper triangular Toeplitz matrix having [01×(P−L), hknt,nr(L), . . . , h k nt,nr(1)] as the first row, and ηnr(i) = [ηnr(iP ), · · · , ηnr(iP +P −1)]T denotes the AWGN vector with covariance matrix σ2ηIP . In (7.2), the term ∑2 nt=1 Ĥknt C ksknt,nr(i−1) represents IBI in the received signal from kth user. However, we see that the last L rows of matrix Ck are zero, because of zero-padding at the transmitter. Therefore, it is always guaranteed that 131 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . Ĥknt,nrC k = 0B×K ,∀k, nt, nr. Thus, IBI is completely avoided. Multiuser Separation Block-spreading and zero-padding operations at the transmitter preserve the orthogo- nality among the users even after transmitting through multipath fading channels. The received user signals can be separated deterministically using the block despreading ma- trix defined as [9], Dk = Ck ⊗ IB+L. After despreading, the received signal vector at receive antenna nr for kth user, y k nr(i) = (D k)Hrnr(i) of length M = B+L can be given as yknr(i) = H̄ k 1,nr(i)Tzps k 1(i) + H̄ k 2,nr(i)Tzps k 2(i) + η̄nr(i), (4.4) where H̄knt,nr(i), ∀k, nt, nr areM×M lower triangular Toeplitz matrices having [hknt,nr(0), . . . , hknt,nr(L),01×(B−1)] T as the first column and η̄nr(i) = (D k)Hηnr(i). It is clear from (4.4) that the original multiuser detection problem has been converted into a set of single user detection problems, even with multiple transmit antennas using only simple processing at the receiver. See [9] for the detailed derivation of the case without ST coding. ST Decoding Since the original multiuser detection problem has been converted into a set of single user detection problems, herein we adopt a similar ST block decoding approach developed for single user systems in [1]. Note that in the proposed system, ST decoding is performed at the symbol-block level after multiuser separation rather than at the chip-block level as in the downlink system developed in [8]. Performing ST decoding at the symbol-block level is obviously more computationally efficient than performing it at the chip-block level. First, we see that H̃knt,nr(i)Tzp = H̄ k nt,nr(i)Tzp, nt = 1, 2, where H̃ k nt,nr(i) is the corresponding circulant matrix of Toeplitz channel matrix H̄knt,nr(i). We replace the Toeplitz channel matrices in the received sample blocks given in (4.4) with the 132 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . corresponding circulant matrices (see [1] for details), yknr(i) = H̃ k 1,nr(i)Tzps k 1(i) + H̃ k 2,nr(i)Tzps k 2(i) + η̄nr(i). (4.5) We assume that multipath channels are invariant over two consecutive symbol blocks 2i and 2i+ 1 (i.e., channels are quasi-static). Hence, for notational brevity we can express H̃knt,nr(2i) = H̃ k nt,nr(2i + 1) = H̃ k nt,nr , nt = 1, 2. From the ST encoding rule given in (7.1), we can easily see the following relationships: Tzps k 1(2i + 1) = − PBMTzpsk2(2i)∗, and Tzps k 2(2i+1) = P B MTzps k 1(2i) ∗. Then we can express two consecutive sample blocks (after multiuser separation) for the kth user at the nrth receiver antenna as yknr(2i) = H̃ k 1,nrTzps k 1(2i) + H̃ k 2,nrTzps k 2(2i) + η̄nr(2i), (4.6) yknr(2i + 1) = −H̃k1,nrTzpPBMsk1(2i) + H̃k2,nrTzpPBMsk2(2i) + η̄nr(2i+ 1). (4.7) Before pursuing frequency-domain (FD) processing, we perform conjugate and per- mutation operations on block yknr(2i + 1), so that ST decoding is possible with lin- ear processing. Next, we transform yknr(2i), P B My k nr(2i + 1) ∗ into FD by defining ỹknr(2i) = FMy k nr(2i), ỹ k nr(2i+1) = FMy k nr(2i+1) and similarly, η̃nr(2i) = FMηnr(2i), η̃nr(2i+ 1) = FMP B Mηnr(2i) ∗. We can easily obtain the following FD relationship   ỹknr(2i) ỹknr(2i+ 1)   ︸ ︷︷ ︸ ,ȳknr (i) =   Dk1,nr Dk2,nr (Dk2,nr) ∗ −(Dk1,nr)∗   ︸ ︷︷ ︸ ,D̃ k +   η̃nr(2i) η̃nr(2i+ 1)   , (4.8) where Dknt,nr = diag{h̃knt ,nr}, h̃knt,nr is the discrete Fourier transform (DFT) of the channel impulse response hknt,nr . Let us define unitary matrix U k nr , D̃ k (I2⊗ (D̄knr)−1), where D̄ k nr , [(D k 1,nr) ∗Dk1,nr+(D k 2,nr) ∗Dk2,nr ] 1/2 is a real-valuedM×M diagonal matrix. The output of the ST decoder of the nrth receiver antenna, [z̄ k nr(2i) T , z̄knr(2i + 1) T ]T = 133 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . (Uknr) H ȳknr(i) can be given as   z̄knr(2i) z̄knr(2i + 1)   =   D̄knrFMTzpsk(2i) D̄ k nrFMTzps k(2i + 1)  +   wnr(2i) wnr(2i+ 1)   , (4.9) where resulting noise vectors [wTnr(2i),w T nr (2i+1)] T = (Uk)H [FM η̄ T nr(2i), FMP B M η̄ H nr(2i +1)]T . It is clear from (6.9) that the ST decoded block corresponding to transmitted symbol block sk(i) can be separated and expressed as z̄knr(i) = D̄ k nrFMTzps k(i) +wnr(i). (4.10) Next, following a similar approach as in [2] we combine all ST decoded blocks from each receive antenna, z̄knr(i)∀nr, in such a way that receiver diversity is obtained. We construct matrix Vk , Bk(B̄k)−1, where Bk = [D̄k1 , . . . , D̄ k nr ] T , B̄ k = [∑NR nr=1 ∑2 nt=1 (Dknt,nr) ∗Dknt,nr ]1/2 . Let us stack all ST decoded blocks as z̃k(i) = [z̄k1(i) T , . . . , z̄knr (i) T ]T . Finally, we obtain zk(i) = (Vk)H z̃k(i) as zk(i) = B̄ k FMTzps k(i) +w(i). (4.11) Note that noise vector w(i) = (Vk)H [w1(i), . . . ,wNR(i)] T is still white. The transmitted symbol block, sk(i),∀k, is detected from (4.11) after channel equalization. 4.3 Low-Complexity Linear Equalization In this section we derive low-complexity linear equalizers, namely, the zero-forcing (ZF) and MMSE equalizers, to mitigate the intersymbol interference (ISI). Note that unlike in [8], in the proposed ST coded uplink system, equalization is carried out at the symbol- block level rather than at the chip-block level. Input symbol vectors sk(i) are assumed to be uncorrelated with the covariance matrix, Rs , E [ sk(i)sk(i)H ] = σ2sIB. Using 134 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . well-known Wiener solution, linear block-MMSE equalizer based on the signal model given in (4.11) can be obtained as GkMMSE = [F H zpB̄ k FzpB̄ k + σ2η/σ 2 sIB ] −1FHzpB̄ k , (4.12) where Fzp = FMTzp and note that B̄ k is a real-valued diagonal matrix. We can obtain the ZF equalizer, GkZF by setting σ 2 η = 0. It is clear from (4.12) that exact implementation of linear equalization includes the B ×B matrix inversion and IFFT operations. Inversion of a general B ×B matrix has the computational-complexity of order O(B3). Nevertheless, [FHzpB̄kFzpB̄k + σ2η/σ2sIB] is a Hermitian Toeplitz matrix and the complexity incurred in the inversion of such a matrix reduces to order O(B2) [12]. However, for large block size B, the computational- complexity incurred in the matrix inversion may be still a burden to the receiver. We can further reduce the complexity of the linear equalizers by exploiting the fact that the equivalent channel matrix B̄ k is diagonal. The low-complexity (LC)-MMSE equalizer is derived in two steps following a similar approach as in [13]. First, let us define the MMSE equalizer for xk = Fzps k as Ḡk = σ2sFzpF H zpB̄ k [ σ2ηIP + σ 2 sB̄ k FzpF H zpB̄ k ]−1 . (4.13) Using approximation FzpF H zp ≈ (B/M)IM , we can simplify (4.13) as Ḡk = B̄k [(M/B) σ2η/σ 2 sIM + (B̄ k )2 ]−1 . Then we can find an estimate of the transmitted symbol block sk(i) as, ŝk(i) = [FHzpFzp] −1FHzpḠkzk. Now the complete LC-MMSE block equalizer can be given as follows GkLC−MMSE = F H zpB̄ k [ (M/B)σ2η/σ 2 sIM + (B̄ k )2 ]−1 . (4.14) The LC-ZF equalizer, GkLC−ZF can be obtained by taking the noise variance σ 2 η = 0. 135 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . Since low-complexity implementation of the linear equalizers involves the IFFT and diagonal matrix inversion operations, it has low-complexity of order O(M log2M) per symbol block. Although both LC-ZF and LC-MMSE equalizers are computationally efficient, generally they are not the minimum norm solution to (4.11). 4.4 Decision Feedback Sequence Estimation (DFSE) Following we show how to adopt DFSE to detect the transmitted symbol sequence from the ST decoded block given in (4.11). Note that we omit block index i to reduce the notational complexity. Let us define the vector, qk , FHM B̄ k zk. It can be shown that qk = Hkeb k + w̄, (4.15) where Hke , F H M (B̄ k )2FM , b k = Tzps k and correlated noise vector w̄ = FHP B̄ k w. The effective channel order for the kth user, Lke is defined as L k e , maxnt,nr L k nt,nr , ∀nt, nr. It can be shown that Hke is the M ×M Hermitian-circulant matrix with [αk0 , (αk1)∗, . . ., (αk Lke )∗, 0, 0, . . . , 0 , αk Lke , . . . , αk1 ] as the first row, where αka = NR∑ nr=1 NT=2∑ nt=1 L∑ l=0 hknt,nr(l − a)∗hknt,nr(l), ∀a, a{0, 1, . . . , Lke}. (4.16) Intuitively, qk can be viewed as the output of a conventional matched-filtered single- input single-output (SISO) system with the FIR, hke = [(α k Lke )∗, . . . , (αk1) ∗, αk0 , α k 1 , . . . , αk Lke ]T (note that hke(l) is defined ∀ l, −Lke ≤ l ≤ Lke). The output sequence of the equivalent SISO system can be expressed as follows qk(m) = hke(m) ∗ bk(m) + w̄(m), (4.17) 136 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . where ∗ denotes the convolution. It is clear that Forney-type MLSE, which is based on whitened system models, can not be directly applied to the output of the equivalent SISO system, {qk(m)}. Following a similar approach as in [2] for single user systems, we can derive the branch metric for UngerBöeck-type MLSE based on (4.11) for M > 2Lke . The metrics evaluated for the MLSE can be recursively computed in the Viterbi algorithm (for m = 0, 1, · · · , B + Lke − 1) as M(νm+1) =M(νm) +W(m), (4.18) whereM(νm) is the branch metric and state vector ν(m) = [bk(m−1), bk(m−2), . . . , bk(m −Lke)]T . It can be shown that the branch metric is given by W(m) = < { bk(m)∗ [ 2qk(m)− αk0bk(m)− 2 Lke∑ l=1 αkl b k(m− l) ]} . (4.19) Note that because of zero-padding, the initial and final state in the Viterbi algorithm (i.e., ν0 = νB+Lke−1 = 0Lke×1) are known. Hence, this implementation of the Viterbi algorithm is exact and it does not suffer from edge effects. The number of trellis states involved in MLSE is given byMLke , whereM is the size of signal constellation. Thus, MLSE is not feasible for channels with high memory order and/or with large signal constellations. On the other hand, DFSE can be viewed as a reduced-complexity suboptimal alternative to MLSE. Hence, we investigate the performance of DFSE for the proposed ST coded CDMA system. 4.4.1 Unwhitened DFSE (UDFSE) In [5], DFSE is studied for unwhitened ISI channel based on Ungerböeck-type MLSE. As opposed to whitened channel models, the output of unwhitened channel models depends on both past and future input symbols. Nevertheless, when tentative decisions are made 137 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . in UDFSE, the effect of ISI due to future symbols is simply ignored. This deteriorates the performance. In the modified UDFSE (MUDFSE) scheme, a modified decision rule is used to include the effect of anti-causal input symbols. In multistage-MUDFSE (MMUDFSE), the performance is further improved by performing MUDFSE iteratively at the expense of increased complexity. Moreover, in MMUDFSE scheme, the decisions made on the future inputs in the previous stage are used compute the bias term in the current stage. Although near ML performance can be achieved through large number of iterations, the computational-complexity and delay associated with the multistage configuration increase with the number of iterations (see [5] for details). 4.4.2 Proposed Combined Linear Equalization-MUDFSE (Comb. LE-MUDFSE) The first stage of MMUDFSE scheme is identical to MUDFSE. However, the MUDFSE scheme usually shows high error-floor behavior. As a consequence, the performance of MMUDFSE (with two stages) deteriorates at high signal levels. As will be seen in the simulation results, this would be more prominent for long channels. Hence, in this novel scheme we propose to use linear equalization for the first stage, instead of MUDFSE. In other words, the anti-causal symbols, which are used to compute the bias term in MUDFSE as the second stage, are determined using linear equalization (e.g., MMSE- based equalization) in the first stage. In order to reduce the computational-complexity, one can even adopt the low-complexity implementation of the MMSE-based equalizer given in Section III. 4.4.3 Whitened DFSE (WDFSE) The detrimental effect of anti-causal symbols, which causes to degrade the performance in UDFSE, can be avoided by converting the unwhitened channel into a whitened equiv- alent using a noise-whitening prefilter. Since it is crucial to have a minimum-phase 138 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . overall transfer function to achieve high performance in WDFSE, the prefilter has to convert the channel into its minimum phase equivalent [5]. In this chapter, we follow a linear prediction (LP)-based approach, which is originally proposed for SISO systems in [14], to compute a FIR prefilter. A similar approach is used in [6] for TR-based ST block coded transmission as well. Suppose Hke (z) denotes the (two-sided) z-transform of the equivalent discrete-time channel response (i.e., Hke (z) , ∑Lke l=−Lke h k e(l)z −k). Since hke(l) = (hke(−l))∗, the equiv- alent system transfer function, Hke (z) can be factorized and expressed as H k e (z) = Xmin(z)X ∗ min(1/z ∗), where Xmin(z) is the minimum phase polynomial. It is clear that the channel can be transformed into its minimum phase equivalent by filtering with all pole transfer function (i.e., the noise-whitening prefilter) defined as Fw(z) , 1/X∗min(1/z ∗). (4.20) The resulting overall channel transfer function can be expressed as Hkmin(z) = Fw(z) Hke (z) = Xmin(z). The all-pole transfer function of the noise-whitening prefilter, Fw(z) can be approximated by FIR filter F̃w(z) as F̃w(z) ≈ C 1 X∗min(1/z∗) , (4.21) where C is an arbitrarily constant. F̃w(z) can be obtained by defining a backward prediction error filter (BPEF), which can be computed using Yule-Walker equations (see [14] for details). The z-transform of the whitened sequence {qkw(m)} can be defined as Qkw(z) = F̃w(z)Q k(z), where Qk(z) is the z-transform of {qk(m)}. The coefficients of the minimum phase channel model, {h̃kmin} are obtained approximately as H̃kmin(z) = F̃w(z)H k e (z). After prefiltering, WDFSE can be applied to the output of the whitened 139 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . system model with the branch metric [4], W(m) = − ∣∣∣∣qkw(m)− µ∑ l=0 h̃kmin(l)b k(m− l)− Lke∑ l=µ+1 h̃kmin(l)b̂ k(m− l) ∣∣∣∣2. (4.22) Note that if we set µ = 0, then WDFSE becomes equal to the decision feedback equal- ization. 4.5 Training-based Channel Estimation So far, we have assumed that perfect channel state information (CSI) is available at the receiver. However, in practice, user channels have to be estimated before ST decoding and channel equalization. A training-based channel estimation scheme is proposed in [15] for ST block coded systems with two transmit antennas. However, in that scheme, it is considered that two consecutive symbol blocks are completely allocated for training sequences. Nevertheless, it is not possible to allocate complete symbol blocks for the training sequences in quasi-static channels, as channels have to be estimated in each two- block time interval separately. Hence, in this chapter training sequences are embedded as integral parts of transmitted symbol blocks. Following, we show how to adapt the original scheme [15] in the proposed ST coded uplink system. In the proposed system, channels of each user are estimated separately after multiuser separation (i.e., after despreading). Let us define the training sequences of length Nts for the kth user as bknt = [b k nt(0), . . . , b k nt(Nts − 1)]T , nt = 1, 2. We replace (before ST block encoding) the first Nts symbols in two consecutive symbol blocks s k(2i), sk(2i+1) with the training sequences, bk1 and b k 2, respectively. Then two symbol blocks are ST encoded according to the encoding rule in (7.1). Note that the operation of the transmitter remains the same as for symbol blocks with only data symbols. We assume that relative user delays τk due to asynchronism among users are known at the receiver and provided by delay estimation. Let us now consider estimation of the channels 140 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . between two transmit antennas of the kth user and receiver antenna nr. Without loss of generality, we assume that the channels between two transmit antennas of a mobile user and the nrth receiver antenna have the same actual channel order denoted by L̃knr . After multiuser separation, we form two vectors of length Nts − Lt + 1 from two consecutive received blocks as vknr(2i) = [y k nr(2i;Lt + τ k − 1), . . . , yknr(2i;Nts + τk − 1)]T and vknr(2i + 1) = [y k nr(2i + 1;B − Nts + Lt + τk − 1), . . . , yknr(2i + 1;B + τk − 1)]T , where Lt = L̃ k nr + 1 is the number of taps in the kth user channels. Following a similar approach as in [15], we define (Nts −Lt +1)×Lt Toeplitz matrices Uknt , nt = 1, 2, with [bknt(Lt − 1), bknt(Lt − 2), . . . , bknt(0)] and [bknt(Lt − 1), bknt(Lt), . . . , bknt(Nts − 1)]T as the first row and column, respectively. Let us also define matrix Ū as Ū =   Uk1 Uk2 (Ũk2) ∗ −(Ũk1)∗   , where Ũknt , nt = 1, 2 is the time reversal matrix of U k nt . We can define the linear least square channel estimation for quasi-static channels as ĥknr = [ (ĥk1,nr) T , (ĥk2,nr) T ]T =( ŪHŪ )−1 ŪH v̄knr , where v̄ k nr = [v k nr(2i) T ,vknr(2i + 1) T ]T . In order to achieve MMSE in channel estimation, the training sequences must have impulse like autocorrelation and zero cross-correlation (i.e., ŪHŪ = 2(Nts − Lt + 1)I2Lt) [15]. Furthermore, as the orthogonality among users is preserved in the proposed ST coded system, we can use the same pair of training sequences for all users. 4.6 Numerical Results and Discussion We consider the uplink of a ST block coded CIBS-CDMA system with NT = 2 transmit antennas at each mobile user and either NR = 1 or 2 transmit antenna/s at the receiver base station. Following we present the details of the simulation environment. • Each user’s QPSK modulated data symbol sequence is serial-to-parallel converted 141 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . into blocks of length B = 64. A distinct Walsh-Hadamard spreading code of length G = 16 is assigned to each mobile user. The maximum relative delay, τmax = maxk τ k ∀k, due to asynchronism among the users is considered as 7 chip intervals. • For simplicity in the simulations, we assume that all user channels have the same actual channel order. Unless otherwise mentioned, the effective channel order of Le = 3 (i.e., 4 taps) is assumed for all user channels. • In order to satisfy the postfix condition L ≥ Le + τmax, unless otherwise stated, the maximum channel order is set to L = 10 and correspondingly the transmitted chip block length M = B + L = 74. • It is assumed that Rayleigh distributed channel taps have equal variance 1NT (Le+1) . In this way, we normalize the transmitted power. We also assume quasi-static fading (i.e., channels remain constant within the duration of two blocks and then change independently). • In MMUDFSE, we restrict only to two stages (i.e., two iterations) as the complexity and delay increase with the number of stages. Moreover, in WDFSE the BPEF of order 15 is used. Note that as a result of MUI free reception, the user performance is not affected by the number of active users in the system. Unless otherwise stated, we assume that perfect channel state information (CSI) is available at the receiver. Eb/N0 stands for the average received bit energy to noise ratio at the receiver antenna. 4.6.1 Case A: Comparison of the Performance of the proposed ST coded uplink system with the System without ST Coding In Figure 4.3, the BER performance of the proposed ST coded system with NR = 1 and 2 receiver antennas for uplink transmission is compared with that of the CIBS- 142 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . CDMA system without ST coding in [9]. The results clearly indicate that a significant performance improvement is achieved through ST coding. These results reconfirm the effectiveness of ST coding as a transmit diversity technique in achieving high data rates. Moreover, we can clearly see the improvement in the performance with the number of receiver antennas. The maximum possible diversity order (defined as NTNRL̃ k nt,nr) is equal to 8 and 16 for the system with NR = 1 and 2, respectively. In this chapter we have assumed that there is no correlation between antennas. However, in practice, as a result of correlation among antennas, it may be impossible to achieve the maximum pos- sible diversity order defined above. Moreover, as expected, the MMSE linear equalizer shows improved performance compared to the ZF equalizer. Nevertheless, it is clearly visible that there is a significant performance gap between the ML performance and that obtained using the linear equalizers. Although the performance of the low com- plexity linear equalizers is not reported here, the LC-MMSE equalizer show practically no degradation in the performance compared to the exact MMSE equalizer. It can be also observed that the performance difference between the linear equalizers and MLSE decreases with the diversity order. This is due to the fact that as the diversity order increases, the effect of noise enhancement becomes less significant. 4.6.2 Case B: Performance Investigation of Different DFSE Schemes with Relatively Short Channels In this section, the performance of different DFSE schemes on the proposed system is studied for relatively short channels (with 4 multipath components). The performance obtained with µ = 1 (i.e., 4 trellis states) is presented in Figure 4.4. We see that for µ = 1, only Comb. LE-MUDFSE and WDFSE schemes yield a substantial perfor- mance improvement over MMSE linear equalization (MMSE-LE). We also observe that the Comb. LE-MUDFSE scheme slightly outperforms WDFSE. However, it should be noted that the performance of WDFSE also depends on the order of the PEF used. 143 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 2 4 6 8 10 12 14 16 10−7 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  (dB) BE R   ZF−LE (without ST coding [15]) MMSE−LE (without ST coding [15] ) MLSE (without ST coding [15]) ZF−LE  (prop. system with NR = 1) MMSE−LE (prop. system with NR = 1) MLSE  (prop. system with NR = 1) ZF−LE  (prop. system with NR = 2) MMSE−LE (prop. system with NR = 2) MLSE  (prop. system with NR = 2) Figure 4.3: BER performance comparison between the proposed ST coded uplink system and the CIBS system without ST coding in [9] 144 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . behavior of the first stage of MMUDFSE, which is identical to MUDFSE. 2 4 6 8 10 12 14 16 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  (dB) BE R   MLSE MMSE−LE UDFSE MUDFSE MMUDFSE Comb. LE−MUDFSE WDFSE DFE (µ=0) Figure 4.4: Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively short channels (µ = 1 in DFSE schemes) In Figure 4.5, we present the BER performance of the proposed ST coded system with NR = 2 receiver antennas. The performance obtained using the DFSE schemes with µ = 1 (i.e., with 4 trellis states) is compared with that of the MMSE-LE and MLSE. We can clearly see the improvement in the performance acquired through the addition of receiver diversity compared to the case without receiver diversity reported in Figure 4.4. Furthermore, similar observations can be made on the performance of the DFSE schemes as in the case without receiver diversity. However, it is worthy to note that in this case, 145 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . the proposed Comb. LE-MUDFSE scheme shows very close performance to MLSE. 2 3 4 5 6 7 8 9 10 11 12 10−6 10−5 10−4 10−3 10−2 Eb/N0  (dB) BE R   MLSE MMSE−LE UDFSE MUDFSE MMUDFSE Comb. LE−MUDFSE WDFSE DFE (µ = 0) Figure 4.5: Performance investigation of different DFSE schemes for the proposed ST coded system with two receiver antennas (NR = 2) for relatively short channels (µ = 1 in DFSE schemes) 4.6.3 Case C: Performance Investigation of Different DFSE Schemes with Relatively Long Channels In this section we study the performance of the DFSE schemes for relatively long channels (i.e., with 6 multipath components). Note that in this case the maximum channel order is set to L = 12 to satisfy the postfix condition. The performance of DFSE schemes with µ = 1 (i.e., with 4 trellis states) and with µ = 2 (i.e., with 16 trellis states) is 146 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . reported in Figure 4.6 and Figure 4.7, respectively. Ideally, the performance should be improved as the diversity order increases with the number of multipath components (in this case the diversity order is equal to 12). This can be confirmed by comparing the performance of MMSE-LE with that of the case for short channels in Figure 4.4 (This improvement may not be clearly visible on the performance of DFSE schemes, as we consider much smaller than full number of trellis states which leads to performance degradation). The performance of MLSE is not shown as computational-complexity of MLSE is prohibitively high for long channels. Instead, we show the performance of the MUDFSE scheme with perfect transmitted symbol knowledge used for computing the bias term for comparison purposes. We refer to this scheme as Per-MUDFSE scheme and expect that the performance of Per-MUDFSE is very close to ML performance. Simulation results indicate that UDFSE and MUDFSE schemes become useless for long channels. Although the MMUDFSE scheme with µ = 1 slightly outperforms MMSE-LE at low signal levels, the performance of MMUDFSE is worse at high sig- nal levels. Moreover, with µ = 2 (i.e., with 16-trellis states), the MMUDFSE scheme shows an improved performance compared to MMSE-LE for low-to-moderate Eb/N0 val- ues. Nevertheless, it shows error-floor behavior for high signal levels even with 16 trellis states. As expected, in this case of long channels, error-floor behavior is more prominent compared to the case of relatively short channels studied in the previous section. The proposed Comb. LE-MUDFSE scheme shows a slightly better performance compared to the WDFSE scheme. In the case of µ = 1, the deviation in the performance of the WDFSE scheme compared to the Comb. LE-MUDFSE scheme becomes more visible for high signal levels. Furthermore, it can be clearly observed that the error-floor behavior is completely eliminated in the Comb. LE-MUDFSE scheme, even with the minimum possible number of trellis states (in this case 4 states). 147 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 2 4 6 8 10 12 14 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  (dB) BE R   MMSE−LE UDFSE MUDFSE MMUDFSE Comb. LE−MUDFSE Per−MUDFSE WDFSE DFE (µ =0) Figure 4.6: Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively long channels (µ = 1 in DFSE schemes) 4.6.4 Case D: Performance with training-based Channel Estimation The training sequences are chosen from the L-perfect sequences of length Nt = 15, which are constructed from the QPSK signal constellation (see [15] for details). Figure 4.8 compares the BER performance obtained using training-based channel estimation with that obtained using perfect CSI at the receiver. We see that the use of training-based channel estimation degrades the performance of MMSE-LE and Comb. LE-MUDFSE schemes by ≈ 1.5 dB and the performance of WDFSE scheme by ≈ 2 dB at moderate signal levels. Even though it is possible to improve the performance by increasing the 148 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 2 4 6 8 10 12 14 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0   (dB) BE R   MMSE−LE UDFSE MUDFSE MMUDFSE Comb. LE−MUDFSE Per−MUDFSE WDFSE DFE (µ = 0) Figure 4.7: Performance investigation of different DFSE schemes for the proposed ST coded system without receiver diversity (i.e., NR = 1) for relatively long channels (µ = 2 in DFSE schemes) length of training sequences, in practice, it may not be desirable to use long training sequences in each symbol block, as it leads to low bandwidth efficiency. 149 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 2 4 6 8 10 12 14 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  (dB) BE R   MMSE−LE (Per. CSI) MMSE−LE (Tr. ch. est.) MUDFSE (Per. CSI) MUDFSE (Tr. ch. est.) MMUDFSE (Per. CSI) MMUDFSE (Tr. ch. est.) Comb. LE−MUDFSE (Per. CSI) Comb. LE−MUDFSE (Tr. ch. est.) WDFSE (Per. CSI) WDFSE (Tr. ch. est.) Figure 4.8: Comparison between the BER performance obtained using training-based channel estimation and that obtained using perfect channel state information for the proposed ST coded system without receiver diversity (µ = 1 in all DFSE schemes) 4.7 Conclusion In this chapter, we have shown how to combine SC-TR-ZP based ST block coding with CIBS-CDMA for multipath uplink transmissions. In the proposed system, ST decoding and channel equalization operations are performed at the symbol-block level. The simulation results have confirmed that a substantial performance improvement can be achieved through ST coding in the uplink of CIBS-CDMA systems. The effect of receiver diversity on the performance has also been studied. We have also shown how to use DFSE in the proposed ST-coded uplink system. The performance of different DFSE 150 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . schemes has been extensively investigated. In the case of WDFSE, a LP-based approach has been adopted for designing whitening prefilter, which is used to transform the system model into its minimum phase equivalent. The UDFSE and MUDFSE schemes have shown high error-floor behavior, which is more prominent for long channels. Simulation results have also indicated that even the MMUDFSE scheme is not suitable for long channels. A novel scheme known as Comb. LE-MUDFSE, which is the combination of linear equalization and MUDFSE, has been proposed. Interestingly, the simulation results have shown that the error-floor behavior is avoided in the proposed Comb. LE- MUDFSE scheme. Moreover, in order to reduce the computational complexity, the LC- MMSE equalizer can be employed as the first stage of the proposed Comb. LE-MUDFSE scheme. Finally, it can be concluded that either the proposed Comb. LE-MUDFSE or WDFSE scheme can be used as a promising reduced-complexity suboptimum alternatives to MLSE in the proposed ST coded CDMA system. One could consider channel coding with Turbo (iterative) equalization as an extension to this work. 151 Chapter 4. Space-Time Coded CDMA Uplink Transmission. . . 4.8 Bibliography [1] S. Zhou and G. B. Giannakis, “Space-time coding with maximum diversity gains over frequency-selective fading channels,” IEEE Signal Processing Lett., vol. 8, pp. 269–272, Oct. 2001. [2] ——, “Single-carrier space time block-coded transmissions over frequency-selective fading channels,” IEEE Trans. Inform. Theory, vol. 49, pp. 164–179, Jan. 2003. [3] W. Younis and N. Al-Dhahir, “Joint prefiltering and MLSE equalization of space- time-coded transmissions over frequency-selective channels,” IEEE Trans. Veh. Technol., vol. 51, pp. 144–154, Jan. 2002. [4] A. Duel-Hallen and C. Heegard, “Delayed decision-feedback sequence estimation,” IEEE Trans. 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Fragouli, N. Al-Dhahir, and W. Turin, “Training-based channel estimation for multiple-antenna broadband transmission,” IEEE Trans. Wireless Commun., vol. 2, pp. 384–391, Mar. 2003. 153 Chapter 5 MMSE-based Turbo Equalization for Space-Time Block-Coded CDMA Downlink1 5.1 Introduction In recent years, space-time block coding (STBC) has drawn tremendous attention as an effective transmit diversity technique for achieving high data rates over hostile wireless channels. Space-time (ST) block codes were originally designed for frequency-flat fad- ing channels [1, 2]. Later, burst- (or block-) wise (BW)-STBC schemes were proposed for frequency-selective channels, e.g., the time-reversal (TR) BW-STBC scheme [3] and the zero padding (ZP)-only BW-STBC scheme [4]. In [5], it is shown that BW-STBC schemes can yield the maximum diversity order, defined as the multiple of the number of transmit antennas, the number of receive antennas, and the number of multipath com- ponents. The symbol-wise Alamouti ST block code has been adopted for the downlink of third generation wideband code division multiple access (WCDMA) systems [6]. Hence, it becomes important to develop advanced receivers (still of moderate computational complexity) for downlink transmission in ST block coded CDMA systems. Previously, receiver structures (and/or equalization schemes) were developed for 1A version of this chapter has been submitted for publication. Wavegedara, K. C. B. and Bhargava, V. K. MMSE-based Turbo Equalization for Space-Time Block-Coded CDMA Downlink. 154 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . STTD-WCDMA systems both with and without cell-specific scrambling [7, 8] and [9]. In [10], an minimum mean square error (MMSE)-based (infinite length) chip-equalization scheme was proposed for TR-ST block-coded CDMA systems. The ZP-only BW-STBC scheme was adapted for downlink transmission in CDMA systems in [11]. In [12], the ZP-only BW-STBC scheme was also applied for downlink transmission of block-spread (BS)-CDMA systems with frequency-domain receiver precessing. All of these schemes, however, are limited to one-pass (hard decision-based) equalization. On the other hand, in channel-coded broadband wireless systems, performance can be improved can be im- proved through turbo equalization. In [13], an MMSE-based turbo interference cancelation scheme was proposed for single-input single-output (SISO) DS-CDMA systems. In [14], an MMSE-based turbo equalizer/inter-chip interference canceler was proposed for multicode transmission in the downlink of SISO DS-CDMA systems. In [15], an MMSE-based turbo multiuser receiver was proposed for the uplink of symbol-wise (SW) Alamouti ST block coded CDMA systems. In [15], each symbol after spreading is zero-padded in order to prevent ISI. This procedure may lead to low spectral efficiencies especially for channels with a long delay spread and/or for short spreading sequences. In this chapter, we develop an MMSE-based turbo equalizer for the downlink of ST block coded CDMA systems. Our proposed scheme is significantly different from the existing schemes (e.g., [13, 14]), mainly due to the fact that in the proposed downlink scheme we consider BW-STBC and frequency-domain receiver processing. It is possible to yield inherent temporal diversity of the wireless channel using channel coding and interleaving. However, when the channel is under deep fading condition, some additional means of diversity besides the inherent temporal diversity is necessary. Hence, we believe STBC is an essential technique to be used in future wireless communications systems to achieve high data rate transmission over hostile wireless channels. In contrast to the uplink scheme in [15], in our proposed downlink scheme, we consider BW-STBC instead 155 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . of the Alamouti SW-STBC and frequency-domain receiver processing. The specific contributions of this chapter can be summarized as follows: • We propose a novel downlink CDMA system architecture that facilitates both ef- ficient application of the (ZP-only) BW-STBC scheme [4] and MMSE-based turbo equalization. ST block encoding and decoding are performed at the chip-block level. We show through simulations that superior performance can be obtained using the (ZP-only) BW-STBC scheme, compared with the STTD scheme pro- posed for WCDMA systems. • Based on the equivalent SISO system model obtained after ST block decoding, we propose symbol-estimation wise and chip-estimation wise MMSE-based soft-in soft-out (SoftISoftO) equalization schemes in the frequency domain. • In the proposed unified CDMA system architecture, both conventional spreading (CS) and block spreading (BS) are considered. We show that BS is more desirable for achieving improved performance using turbo equalization. • The computational complexities of both symbol and chip-estimation wise MMSE- based SoftISoftO equalization schemes are dominated by the matrix inversion re- quired in the computation of the MMSE filter. To circumvent this problem, two computationally-efficient implementations are developed for the proposed equal- ization schemes. • We evaluate and compare the computational complexities of the different SoftI- SoftO equalization schemes proposed. • In high signal-to-noise ratio (SNR) regions, lengthy simulations are required to evaluate the performance of the proposed turbo equalization schemes. Instead, we can use convenient semi-analytical techniques to obtain the system performance. 156 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . Hence, we derive semi-analytical upper bounds on the bit error rate (BER) per- formance, based on the union bound technique. The rest of the paper is organized as follows. In the next section, we describe the pro- posed CDMA downlink system model. Then, in Section III, the proposed MMSE-based turbo equalization schemes are described. Next, in Section IV, we derive semi-analytical upper bounds on the BER performance of the proposed (symbol-estimation wise) turbo equalization scheme. Simulation results for the proposed schemes are presented in Sec- tion V. Finally conclusions are provided. Notation: Bold lower-case letters represent vectors, while bold upper-case letters de- note matrices; E{.}, Var{.}, Cov{.} stand for the expected value, variance and covari- ance operators, respectively; (.)T , (.)∗, and (.)H denote the transpose, complex con- jugate (component-wise), and Hermitian transpose operations, respectively; ∗ denotes the convolution operation and ⊗ represents the Kronecker product; IM and 0M×N de- note the M ×M identity matrix and an all-zero matrix of size M × N , respectively; FM stands for a FFT matrix of size M × M , where its (i, j)th element is given by (1/ √ M)e−j2pi(i−1)(j−1)/M ,∀i, j ∈ [1,M ]; PnJ is the J × J permutation matrix carrying the reverse operation followed by a right cyclic shift of over n positions on a given vector of length J ; diag{s} represents a diagonal matrix with vector s on its diagonal. 5.2 System Model We consider synchronous downlink transmission in a ST block coded CDMA system with K active users (K ≤ G, where G is the processing gain). The transmitter and receiver sections of the discrete-time baseband system model are shown in Fig.5.1 and Figure 5.2, respectively. Note that the transmitter processing is explicitly shown only for the kth user out of K users and that, likewise, the receiver is only shown for the desired user k0. In this system configuration, the base station transmitter is equipped with 157 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . NT = 2 transmit antennas, while the receiver of the desired user has a single antenna (NR = 1). No channel information is assumed at the transmitter and the perfect channel information is assumed at the receiver unless otherwise mentioned. 5.2.1 Transmitter The frame of binary information bits {dk(j)}Nu−10 for user k, k = 1, 2, · · · ,K is channel encoded using a binary convolutional encoder, where Nu denotes the length of a data frame. For simplicity, we assume that the same convolutional code with constraint length ν and code rate Rc is used for every user. The channel encoded frame of length Nc = Nu/Rc is then interleaved by a random block interleaver and the interleaved sequence is expressed as {ck(n′)}Nc−1n′=0 , where n′ = Πk(n), n′ = 0, 1, · · · ,Nc and Πk denotes the interleaver function for the kth user. After BPSK mapping, the coded symbol sequence is serial-to-parallel (S/P), converted into blocks of length M , bk(i) = [bk(iM), bk(iM + 1), · · · , bk(iM +M − 1)]T , for i=0, · · · , B − 1, where bk ∈ {+1,−1} and B = Nc/M denotes the number of blocks 2. 2Note that the frame length Nu should be chosen so that Nc is an integer multiple of block length M . 158 C h a p ter 5 . M M S E -b a sed T u rb o E q u a liza tio n fo r S p a ce-T im e B lo ck -C o d ed ... Figure 5.1: Transmitter section of the equivalent discrete-time ST block coded CDMA downlink system model 159 C h a p ter 5 . M M S E -b a sed T u rb o E q u a liza tio n fo r S p a ce-T im e B lo ck -C o d ed ... Figure 5.2: Receiver section of the equivalent discrete-time ST block coded CDMA downlink system model 160 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . A distinct channelization code of length G, sck = [s c k(0), s c k(1), · · · , sck(G − 1)]T is assigned to each user. In the proposed system model, we consider both CS and BS. BS is functionally equivalent to CS followed by block interleaving. Note that BS has also been considered in, for example, [16] and [12]. Let {ss(ι)} be the base-station-specific long scrambling code sequence. The spreading matrix of the kth user in the ith subblock interval, Sk(i) for k = 1, 2, · · · ,K, is constructed as Sk(i) =   φ(i) (IM ⊗ sck) ; Conventional Spreading (CS) (φ(i)sck) ⊗ IM ; Block Spreading (BS) , (5.1) where φ(i) = diag(ss(i)) is a diagonal matrix and ss(i) denotes the scrambling code vector in the ith block interval defined as ss(i) = [ss(i, 0), ss(i, 1), · · · , ss(i,MG − 1)]T and ss(i) = [ss(i, 0), ss(i, 1), · · · , ss(i,G − 1)]T for CS and BS, respectively. When the base-station-specific long scrambling code is absent, φ(i) = IMG. For both CS and BS, SHk (i)Sk(i) = IMG. It is worth noting that in BS, scrambling is performed in a block-by-block fashion (i.e., the same scrambling code vector is used for every symbol in a block) compared with symbol-by-symbol fashion scrambling in CS. Block-by-block fashion scrambling has previously been used for BS-based CDMA systems in [12, 17]. Although in the BS-based CDMA schemes developed in [12, 16] it is possible to separate user signals deterministically at the receiver, the bandwidth efficiency is less.3 The chip blocks of all users are then added together and the multiuser chip block is given by u(i) = ∑K k=1 αkuk(i) = ∑K k=1 αkSk(i)bk(i), where αk denotes the amplitude of the kth user signal. Then, two consecutive multiuser chip blocks u(2i),u(2i + 1) are fed into the ST block encoder, which performs ST encoding at the chip-block level. The 3The bandwidth efficiency of our proposed system can be given as ξ = M (M+L/G) . In our notations, the bandwidth efficiency of the BS systems considered in [4] and [12] is expressed as ξ = M (M+L) . Thus, the bandwidth efficiency of the systems in [4, 12] can be considerably low (for a given moderate value of M) for transmission over high-delay spread channels. 161 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . output of the ST block encoder can be given as [4],   ū1(2i) ū1(2i+ 1) ū2(2i) ū2(2i+ 1)   =   u(2i) −P0MGu∗(2i+ 1) u(2i+ 1) P0MGu ∗(2i)   . (5.2) Following ST block encoding, multiuser chip blocks are zero padded as ũnt(i) = TZP ūnt(i), nt = 1, 2, where the zero padding matrix of size P×MG, TZP = [IMG , 0MG×L]T and P = MG+ L. Here, L , maxk L̃k, ∀k, k = 1, 2, · · · ,K denotes the maximum channel order, where L̃k is the order of the equivalent discrete-time finite impulse response (FIR) subchannel between a base-station antenna and the kth user mobile receiver. After par- allel to serial (P/S) conversion of ũnt(i) = [ũnt,i(0), ũnt,i(1), · · · , ũnt,i(P − 1)], multiuser chip sequences {ũnt,i(n)} for n = 0, 1, · · · , P −1 and i = 0, 1, · · · , B−1 are filtered using the transmit chip-pulse shaping filter ϕT (t). The resulting continuous time baseband signal through antenna nt, nt = 1, 2, can be given as unt(t) = B∑ i=0 P∑ n=0 unt,i(n)ϕT (t− (iP + n)Tc), (5.3) where Tc is the chip period. Continuous time signal signals unt(t) are then sent to the transmit antennas. 5.2.2 Channel Model In this chapter, we consider a wideband frequency-selective Rayleigh fading channel model for each subchannel between the ntth transmit antenna and the kth user receive antenna with Lk,nt resolvable multipath components. The low-pass equivalent complex impulse response of the multipath subchannel between the ntth transmit antenna and the kth user receive antenna is given as gk,nt(t) = Lk,nt∑ l=0 αk,nt(l)δ(t − τk,nt(l)), (5.4) 162 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . where αk,nt(l) and τk,nt(l) denote the complex-valued fading coefficient and the delay of the lth resolvable path, respectively, and δ(t) denotes the Dirac-delta function. αk,nt(l) and αk,n′t(l), nt 6= n′t and/or l 6= l′ are assumed to be independent random variables. Each Rayleigh distributed channel coefficient may have different average gains. We also assume that multipath subchannels are invariant over two consecutive symbol blocks of length M and that the channel fading between units of two consecutive blocks is inde- pendent (i.e., quasi-static fading). For numerical results, we choose channel model-A and channel model-B for Outdoor-to-Indoor and Pedestrian Test Environments (OI&PTE) and channel model-A for Vehicular Test Environments (VTE), as specified in [18]. 5.2.3 Receiver We assume that the receive filter ϕR(t) is matched to the chip pulse shaping transmit filter (i.e., ϕR(t) = ϕ ∗ T (−t). The continuous-time receive signal r(t) = ∑2 nt=1 gk0,nt(t) ∗ unt(t)+η(t) at the receive antenna of the desired user k0 is first chip-matched filtered and then sampled at the chip rate, where η(t) denotes the complex additive white Gaussian noise (AWGN) process. Let hk0,nt,i(l) , ϕT (t)∗gk,nt(t)∗ϕR(t)|t=lTc for l = 0, 1, · · · , Lk0 be the chip-rate sampled discrete-time equivalent FIR channel between the ntth transmit antenna and the k0th user receive antenna in the ith chip block interval. Without loss of generality, we assume that both subchannels of user k0 have the same channel order Lk0 . It should be noted that hk0,nt,2i(l) = hk0,nt,2i+1(l) for l = 0, 1, · · · , Lk0 as we assume that subchannels are time invariant over two consecutive chip blocks. For notational simplicity, from this point onwards, we drop the block index i from channel coefficients. The resulting discrete-time received signal can be expressed as xk0,i(n) , ∫ ∞ −∞ r(τ)ϕ∗T (τ − (iP + n)Tc)dτ = Lk∑ l=0 [hk0,1(l)ũ1(n− l) + hk0,2(l)ũ2(n− l)] + η(n), (5.5) 163 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . for n = 0, 1, · · · , P − 1 and i = 0, 1, · · · , B − 1, where η(n) , ϕ(t) ∗ η(t)|t=nTc is the discrete-time zero mean AWGN process with variance σ2η/2 per dimension. Next, xk0,i(n) is serial-to-parallel (S/P) converted into blocks of length P . The received signal block in the ith block interval, xk0(i) = [xk0,i(0), xk0,i(1) · · · , xk0,i(P − 1)]T can be expressed as xk0(i) = Hk0,1(i)ũ1(i) +Hk0,2(i)ũ2(i) + η(i), (5.6) for i = 0, · · · , B − 1, where Hk0,nt(i), for nt = 1, 2 are the P × P lower triangular Toeplitz matrices with [hk0,nt(0), hk0,nt(1), · · · , hk0,nt(Lk0), 01×(P−Lk0−1)] T as the first column and η(i) = [η(iP ), η(iP + 1), · · · , η(iP + P − 1)]T denotes the AWGN vector with covariance matrix E{η(i)ηH(i)} = σ2ηIP . 5.2.4 ST Block Decoding In this chapter, we adopt a similar frequency-domain ST block decoding approach as in [4]. We can easily see that Hk0,ntTzp = H̃k0,nt(i)Tzp, nt = 1, 2, where H̃k0,nt(i) is the P ×P circulant matrix having [hk0,nt(0),01×(MG−1), hk0,nt(L), · · · , hk0,nt(1)]T as the first row. We can replace Topelitz matrices in (7.2) with the corresponding circulant matrices, and xk0(i) can be re-expressed as xk0(i) = H̃k0,1(i)ũ1(i) + H̃k0,2(i)ũ2(i) + η(i), (5.7) We take two consecutive chip blocks xk0(2i) and xk0(2i+1) as given in (6.7) and trans- form them into the frequency domain using the DFT transform matrix FP as yk0(2i) = FPxk0(2i) and yk0(2i+1) = FPP MG P x ∗ k0 (2i+1). Note that before transforming into the frequency domain, the complex conjugate of the second chip block is reverse-cyclic shifted byMG positions using the permutation matrix PMGP . Similarly, we define frequency do- main noise vectors as η̃(2i) = FPη(2i) and η̃(2i+1) = FPP MG P η ∗(2i+1). We know that the circulant matrices can be diagonalized using the DFT operation. Hence, the circulant 164 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . subchannel matrices H̃k0,nt(i), nt = 1, 2, are diagonalized using the DFT operation as H̃k0,nt(i) = F H P Dk0,nt(i)FP , where Dk0,nt(i) = diag{h̃k0 ,nt}. Here h̃k0,nt denotes the P - point DFT of the channel impulse response hk0,nt = [hk0,nt(0), hk0,nt(1), · · · , hk0,nt(L)]. After simple manipulations, we can show that [4],   yk0(2i) yk0(2i + 1)   ︸ ︷︷ ︸ ,y̌k0(i) =   Dk0,1(i) Dk0,2(i) D ∗ k0,2(i) −D∗k0,1(i)   ︸ ︷︷ ︸ ,Ďk0   ũ1(2i) ũ1(2i+ 1)  +   η̃(2i) η̃∗(2i + 1)   ︸ ︷︷ ︸ ,η̌(i) . (5.8) We define a real-valued P × P diagonal matrix D̄k0(i) , [|Dk0,1(i)|2 + |Dk0,2(i)|2]1/2. Let us also define a unitary matrix Ūk0(i) , Ďk0(i)(I2 ⊗ (D̄k0(i))−1). After some manipulations, the output of the ST decoder [rTk0(2i), r T k0 (2i + 1)]T = ŪHk0(i)y̌k0(i) can be shown as [4],   rk0(2i) rk0(2i + 1)   =   D̄k0(i)FPTzpu(2i) D̄k0(i)FPTzpu(2i+ 1)  +   η̄(2i) η̄(2i + 1)   , (5.9) where the resulting noise vectors are defined as [η̄T (2i), η̄T (2i + 1)]T = ŪHk0 η̌(i). It is clear from (6.9) that the ST block decoded signal vector in the frequency domain corre- sponding to the transmitted chip-block in the ith block interval, u(i) can be separated and given as rk0(i) = D̄k0(i)uf (i) + η̃(i), (5.10) where the frequency domain multiuser (zero-padded) chip block uf , FPTzpu(i). 5.3 MMSE-based Turbo Equalization In this section, we describe the proposed MMSE-based turbo equalization schemes. It is noteworthy that one can use optimal trellis-based SoftISoftO equalization schemes such as SOVE and MAP-based schemes. Nevertheless, the complexity of trellis-based 165 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . SoftISoftO schemes grows exponentially with the channel order. Hence, such trellis- based equalization schemes may not be feasible for most practical applications. There- fore, in this chapter, we consider low-complexity MMSE-based equalization in the fre- quency domain. The structure of the proposed turbo receiver is depicted in Figure 5.2. At the beginning of each iteration (except in the first iteration) SoftISoftO-symbol mapper computes the mean b̄k0(n ′) and variance υ2k0(n ′) of every symbol bk0(n ′) for n′ = 0, 1, · · · , Nc − 1, of the desired user k0 and input them to the SoftISoftO MMSE- based equalizer. These statistics are obtained as follows [19]: b̄k0(n ′) = tanh ( LpD(ck0(n ′)) 2 ) and υ2k0(n ′) = 1− |c̄k0(n′)|2, (5.11) where LpD(ck0(n ′)) , ln P [ck0(n ′)=1] P [ck0(n ′)=0] is the a priori log likelihood ratio (LLR) value of code bit ck0(n ′) delivered by the channel decoder in the previous iteration. Since no a priori code-bit information is available to the SoftISoftO equalizer at the beginning of the first iteration, we assume that {bk0(n′)}Nc−1n′=0 are equiprobable and i.i.d.; and hence, b̄k0(n ′) = 0 and υ2k0(n ′) = 1, ∀n′. 5.3.1 Symbol-by-Symbol Estimation wise SoftISoftO Equalizer First, we describe the proposed symbol-by-symbol estimation wise (from here onwards referred to as symbol-by-symbol) MMSE-based SoftISoftO equalizer. Let us consider soft estimation of the m′th symbol in the ith symbol block bk0,i(m ′) for m′ = 0, 1, · · · ,M−1. Before MMSE filtering, soft cancelation of ISI is performed in the frequency domain using the soft estimates of the other symbols in the block b̄k0,i(m ′), ∀m′, m′ 6= m. We define the following vector b̄k0(i) = [ b̄k0,i(0), b̄k0,i(1), · · · , b̄k0,i(M − 1) ]T . (5.12) 166 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . For notational brevity, let us also define matrixHf (i) , D̄k0FPTzp and vector h̃i(m ′) , Hf (i)Sk0(i)em′ , m = 0, 1, · · · ,M−1, where em is an all-zero column vector except a one at the mth position. After soft ISI cancelation on the ST block decoded signal vector rk0(i) with respect to symbol bk0,i(m ′), we get r̃k0,i(m ′) = rk0(i) − αk0Hf (i)Sk0(i) ( b̄k0(i) − b̄k0,i(m′)em′ ) , = Hf (i)  αk0Sk0(i)(bk0(i)− b̄k0(i)) + K∑ k=1,k 6=k0 αkuk(i)   + αk0 b̄k0,i(m ′)h̃i(m′) + η̃(i), (5.13) Note that in (6.13) term b̄k0,i(m ′)em′ is due to the fact that in accordance with the turbo principle the extrinsic information LeE(ck0(n)) computed at the output of SoftISoftO equalizer should be independent of the a priori information LpE(ck0(n)). Therefore, we need that b̂k0,i(m ′) does not depend on the a priori information LpE(ck0(B(i−1)+m′)) of the corresponding code bit (thus, we set b̄k0,i(m ′) = 0). Since in the first iteration there is no a priori information delivered to the SoftISoftO equalizer (i.e., b̄k0,m′(i) = 0M ), the soft ISI cancelation step can be simply skipped. It is noteworthy that the power delay profile of a typical wireless channel is usually confined to one symbol interval (i.e., strong multipath components are usually located within one symbol interval). Hence, with CS, interpath interference (IPI) caused by typical multipath propagation mainly consists of self-symbol interference (SSI) rather than ISI. However, when BS is used, the IPI will mostly be consists of ISI, since BS is functionally equivalent to CS followed by chip interleaving. Thus, when BS is used, soft ISI cancelation becomes more effective, leading to high performance improvements with iterations. Furthermore, it is clear from (5.1) that BS can be implemented with the same complexity as CS. It should also be noted that in the schemes proposed for the uplink of CDMA systems e.g., [13], both soft MAI and ISI cancelation are performed prior to MMSE-based filtering. In contrast, in the downlink operation, soft estimates of 167 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . interfering user symbols are not available since mobile receivers are generally not aware of the SSs of interfering users. Even if the SSs of interfering users are known to mobile receivers, in practice, it might be highly computationally expensive for a mobile receiver to obtain soft estimates of interfering users’ symbols. Hence, in the proposed symbol- by-symbol turbo equalization scheme for the CDMA downlink, only soft ISI cancelation is performed, using the soft estimates of its own symbols. Soft ISI cancelation is followed by linear MMSE filtering in order to suppress the residual ISI and MAI. The linear MMSE filter for symbol bk0,i(m ′) is designed to min- imize the mean square error between the filter output and the m′th symbol in the ith block bk0,i(m ′) as wk0,i(m ′) = arg min wk0,i(m ′) E {∣∣wHk0,i(m′)r̃k0,i(m′)− bk0,i(m′)∣∣2} , (5.14) for m′ = 0, 1, · · · ,M − 1 and i = 0, 1, · · · , B − 1. Using the well-known Wiener solution based on the orthogonal principle, the MMSE-based filter can be given by wk0,i(m ′) =[ E { r̃k0,i(m ′)r̃Hk0,i(m ′) }]−1 E {r̃k0,i(m′)bk0,i(m′)}. It can be shown that E { r̃k0,i(m ′)r̃Hk0,i(m ′) } = Hf (i) ( α2k0Sk0(i)Vk0(i)S H k0(i) +RMAI ) HHf (i) + α2k0(1− υk0,i(m′))h̃i(m′)h̃Hi (m′) + σ2ηIP , (5.15) where the covariance matrix of bk0(i), Vk0(i) , Cov {bk0(i)} = diag[υk0,i(0), υk0,i(1), · · · , υk0,i(M − 1)] and RMAI is the covariance matrix of the chip blocks of (K − 1) interfering users RMAI , E     K∑ k=1,k 6=k0 αkuk(i)     K∑ k=1,k 6=k0 αkuk(i)  H   . (5.16) Note that in (7.15), term α2k0(1 − υk0,i(m′))h̃i(m′)h̃Hi (m′) results from the fact that we set the variance of the m′ symbol υk0,i(m ′) = 1 based on the turbo principle. It can also 168 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . be shown that E { r̃k0,i(m ′)bk0,i(m ′) } = αk0h̃i(m ′). (5.17) Now using (7.15) and (7.16), we can express the MMSE equalizer as wk0,i(m ′) = αk0 [ Ψ(i) + α2k0(1− υk0,i(m′))h̃i(m′)h̃Hi (m′) ]−1 h̃i(m ′), (5.18) where we define a P × P matrix as Ψ(i) = Hf (i) ( α2k0Sk0(i)Vk0(i)S H k0(i) +RMAI ) HHf (i) + σ 2 ηIP . (5.19) It is clear from (7.17) that in order to obtain the MMSE-based equalizer, except in the first iteration, we need to compute the inverse of the P × P covariance matrix E { r̃k0,i(m ′)r̃Hk0,i(m ′) } separately for each symbol. This can be a burden for a mobile receiver, especially for large P . Hence, in the following, we derive a computationally effi- cient implementation. Applying the matrix inversion lemma, [ E { r̃k0,i(m ′)r̃Hk0,i(m ′) }]−1 can be re-expressed as [ E { r̃k0,i(m ′)r̃Hk0,i(m ′) }]−1 = Ψ(i)−1 − α 2 k0 (1− υk0,i(m′)) γi(m′) Ψ(i)−1h̃i(m′)h̃Hi (m ′)Ψ(i)−1, where γi(m ′) = 1 + α2k0(1 − υk0,i(m′))h̃Hi (m′)Ψ−1(i)h̃i(m′) is a scalar. After simple manipulations, we can show that the MMSE filter can be given by wk0,i(m ′) = αk0 γi(m′) Ψ−1(i)h̃i(m′). (5.20) Note that the matrix Ψ(i) is the same for all symbols in a given block. Hence, we need to compute Ψ−1(i) only once per block, meaning that using (7.19) the computational complexity of MMSE filtering can be significantly reduced. It is also noteworthy that in the first iteration since no a priori information is delivered to the SoftISoftO equalizer 169 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . by the channel decoder, υi(m ′) = 1, ∀m′, and hence, γi(m′) = 1, ∀m′, i. In the following subsections, we show how to obtain the covariance matrix RMAI of interfering user chip blocks under two scenarios: semi-blind and blind interference suppressions. Semi-blind Interference Suppression In this case, we assume that the SSs of interfering users are known to the mobile receiver of interest and hence, knowledge of SSs of interfering users is used in MMSE-based filtering. Nonetheless, soft MAI cancelation is not performed using the soft estimates of interfering users’ symbols.4 Hence, we refer to this case as semi-blind interference suppression (SBIS). In this case of SBIS it can be shown that RMAI = Hf (i)   K∑ k=1,k 6=k0 α2kSk(i)Vk(i)S H k (i)  HHf (i), (5.21) where Vk(i) = IM , ∀k, k = 1, 2, · · · ,K, k 6= k0, since the soft information of interfering users’ symbols are not available at the mobile receiver of interest. Substituting (7.21) into (7.18), we obtain Ψ(i) for systems using either CS or BS as Ψ(i) =Hf (i) ( K∑ k=1 α2kSk(i)Vk(i)S H k (i) ) HHf (i) + σ 2 ηIP . (5.22) Note that from here onwards, we refer to (7.19) along with Ψ(i) given in (7.22) as the exact implementation (ExactImpl) for the case of SBIS. Blind Interference Suppression In this case, we assume that knowledge of the interfering users’ SSs is not available at the mobile receiver of interest. Since neither soft estimates of interfering users’ sym- bols nor knowledge of the interfering users’ SSs is used in MMSE-based SoftISoftO 4Note that in this proposed scheme, we consider that at the mobile receiver of interest only the soft estimates of its own symbols are computed. 170 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . equalization, this case is referred to as blind interference suppression (BIS). In this case of BIS, we can show that for systems using either CS or BS RMAI = βIMG, where β , 1/G ∑K k=1,k 6=k0 α 2 k. Substituting RMAI = βIMG in (7.18), we obtain matrix Ψ(i) for this case of BIS as Ψ(i) = Hf (i) ( α2k0Sk0(i)Vk0(i)S H k0(i) + βIMG ) HHf (i) + σ 2 ηIP . (5.23) From here onwards, we refer to (7.19) along with Ψ(i) given in (7.23) as the exact implementation (ExactImpl) for the case of BIS. In this case of BIS, the computational complexity involved in MMSE filtering can be further reduced by exploiting the diagonal structure of the equivalent frequency- domain channel matrix D̄k0(i), which is obtained after ST block decoding. Using the approximation TzpT T zp ≈ (MG/P )IP and the fact that FPFHP = IP , we can express Hf (i)H H f (i) = D̄k0(i)FPTzpT T zpF H P D̄ H k0 ≈ (MG/P )D̄ 2 k0. Then, Ψ(i) can be given ap- proximately as Ψ(i) ≈ α2k0Hf (i)Sk0(i)Vk0(i)SHk0(i)HHf (i) +De(i), where De , (MG/P )βD̄ 2 k0 + σ 2 nIP is a real-valued diagonal matrix of size P × P . Applying the matrix inversion lemma, Ψ−1(i) can be re-expressed as Ψ−1(i) ≈ D−1e (i) −D−1e (i)Hf (i)Sk0(i) [ (α2k0Vk0(i)) −1 + SHk0(i)H H f (i)D −1 e (i)Hf (i)Sk0(i) ]−1 SHk0(i)H H f (i)D −1 e (i). (5.24) Note that in (7.25) De is a diagonal matrix. It is clear that, instead of taking the direct inversion of Ψ(2i) as given in (7.23), using (7.25), we need to compute only the inverse of a M×M matrix to obtain Ψ−1(2i).5 5Note that due to the numerical limitations of finite precision calculations, it might be difficult to get the inverse of Vk0,i(m ′) as vk0,i(m ′) approaches zero after a few iterations. However, in practice this 171 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . As will be seen later, computation of the extrinsic LLR values of code bits will be uncomplicated, if we model the soft estimates b̂k0,i(m ′) obtained using the MMSE- based equalizer as the output of an additive white Gaussian noise (AWGN) channel model with input bk0,i(m ′). In [20], it was shown that in CDMA systems after linear MMSE-based filtering, the residual MAI plus the receiver noise can be well modeled using a Gaussian distribution. Later, the Gaussian approximation was used to model the output of the MMSE-based filter in turbo soft interference cancelation [13] and in turbo equalization [19]. We assume that the Gaussian approximation also holds for the problem at hand. Hence, the soft estimates b̂k0,i(m ′) = wHk0,i(m ′)r̃k0,i(m ′) obtained using the linear MMSE-based equalizer is represented using an AWGN model as follows b̂k0,i(m ′) = µk0,i(m ′)bk0,i(m ′) + ηk0,i(m ′), (5.25) where µk0,i(m ′) , E { b̂k0,i(m ′)bk0,i(m ′) } is the amplitude of the equivalent AWGN model and ηk0,i(m ′) , Var{b̂k0,i(m′)} represents a zero-mean complex Gaussian noise with variance σ2k0,i(m ′). We can show that µk0,i(m ′) = αk0w H k0,i(m ′)h̃i(m′) (5.26) σ2k0,i(m ′) = µk0,i( ′m)− µ2k0,i(m′). (5.27) Approximate Implementation The complexity of MMSE-based turbo equalization is dominated by the computation of the inverse covariance matrix E { r̃k0,2i(m ′)r̃Hk0,2i(m ′) } . Although in the ExactImpl for the case of BIS we can avoid the direct inversion of the matrix Ψ using (7.25), it still involves a few complex matrix multiplications. Hence, the computational complexity of the MMSE-based turbo equalization might still be high for practical mobile receivers, especially with a large block length M and/or a large processing gain G. Therefore, problem can be easily avoided by setting a minimum allowable value for the variance of symbols. 172 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . we propose an approximate implementation (AprxImpl) to reduce the complexity of MMSE-based filtering. In the AprxImpl, we assume that υk0,i(m) = 0, ∀m, m ∈ {0, 1, · · · ,M − 1}, m 6= m′ and thus, Vk0(i) = diag{em′}. This is the case when perfect a priori code-bit information is available at the SoftISoftO equalizer. Note that a similar approximation was considered for MMSE-based turbo equalization in single-user SISO systems [19]. Using this assumption, we can express the covariance matrix as E { r̃k0,i(m ′)r̃Hk0,i(m ′) } = α2k0h̃i(m ′)h̃Hi (m ′) + Hf (i)RMAIHHf (i) + σ 2 ηIP . Note that the term α 2 k0 h̃i(m ′)h̃Hi (m ′) results from the fact that we set the variance of them′th symbol to υk0,i(m) = 1. For the AprxImpl, we consider only the case of BIS, and hence, have RMAI = βIMG. Using the approximation Hf (i)H H f (i) ≈ (MG/P )D̄2k0(i), we can re-express E { r̃k0,i(m ′)r̃Hk0,i(m ′) } as E { r̃k0,i(m ′)r̃Hk0,i(m ′) } ≈ α2k0h̃i(m′)h̃Hi (m′) + β̃D̄2k0 + σ2ηIP , (5.28) where β̃ = (MG/P )β. Using (5.28) and (7.16) the MMSE filter can be given as wAIk0,i(m ′) = [ α2k0h̃i(m ′)h̃Hi (m ′) +De(i) ]−1 h̃i(m ′), where De(i) = β̃D̄ 2 k0(i) + σ 2 nIP . Applying the matrix inversion lemma, after some manipulations, we can show that the MSSE filter can be written as wAIk0,i(m ′) = αk0 ζi(m′) D −1 e (i)h̃i(m ′), (5.29) where ζi(m ′) , 1 + α2k0h̃ H i (m ′)D−1e (i)h̃i(m′) is a scalar. Note that D −1 e (i) is a diagonal matrix. Hence, it is clear from (7.26) that the computational complexity can be reduced to a great extent using the AprxImpl. It should be noted that although the assumption of perfect a priori information is approximately valid after a few iterations, in the first iteration, since no a priori information is provided to the SoftISoftO equalizer, this assumption is not valid in the first iteration at all. Hence, intuitively, we can adopt a hybrid approach, in which the ExactImpl is used in the first or first predetermined 173 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . number of iterations, and the AprxImpl is then used in the subsequent iterations. Similarly, for the ExactImpl, we assume that the estimates obtained using the Aprx- Impl b̂AIk0,i(m ′) = (wAIk0,i(m ′))H r̃k0,m′(i) can be modeled using a (conditional) Gaussian distribution with mean µAIk0,i(m ′)bk0,i(m ′) and variance (σAIi (m)) 2, where µAI1i (m) is the amplitude of the equivalent Gaussian model. We can show that the parameters of the equivalent AWGN model for the AprxImpl can be expressed as µAIk0,i(m ′) = ( wAIk0,i(m ′) )H h̃i(m ′) = 1/ζh̃Hi (m ′)D−1e h̃i(m ′), (σAIk0,i(m ′))2 = ( wAIk0,i(m ′) )H [ E { r̃k0,m′(i)r̃ H k0,m′(i) }] wAIk0,i(m ′)− (µAIk0,i(m′))2. We need to compute the variance (σAIk0,i(m ′))2 explicitly for every symbol to obtain the extrinsic LLR values of code bits, which is computationally expensive. Hence, when (σAIk0,i(m ′))2 is computed, we use the same approximations as are used for de- riving the MMSE filter wAIk0,i(m ′). Then, we can easily show that (σAIk0,i(m ′))2 ≈ (1 − µAIk0,i(m ′))/µAIk0,i(m ′), ∀m′. 5.3.2 Chip-by-Chip Estimation wise SoftISoftO Equalizer In this subsection, we derive a chip-by-chip estimation wise (from here onwards referred to as chip-by-chip) SoftISoftO MMSE-based equalizer for ST block transmission in the downlink of CDMA systems. As mentioned before, chip-by-chip MMSE-based turbo equalization/multicode detection (in the time-domain) was originally proposed in [14]. However, in [14] only an approximate implementation was considered. Let us consider estimation of the q′th chip in the ith chip block of the desired user, ui,k0(q ′), for q′ = 0, 1, · · · , Q−1. First, soft ICI cancelation is performed using the regenerated soft values of chips in the frequency-domain. Let us define a vector ūk0(i) using the mean values of the chips as ūk0(i) , Sk0(i)b̄k0(i) = [ūk0,i(0), ūk0,i(1), · · · ūk0,i(Q− 1)]T . (5.30) 174 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . For notational brevity, we define a vector h̃i(q ′) = Hf (i)eq′ , where eq′ is the Q long all-zero column vector except for a one at the q′th position. After soft ICI cancelation, we have r̃ck0,i(q ′) = r(i)− αk0Hf (i) ( ūk0(i) − ūk0,i(q′)eq′ ) , = Hf (i) [ αk0 (uk0(i)− ūk0(i)) + K∑ k=1 αkuk(i) ] + αk0ūk0,i(q ′)h̃i(q′) + η̃(i). (5.31) Note that in the first iteration, since no a priori code-bit information is fed to the SoftISoftO equalizer, we have ūk0(i) = 0Q. The MMSE-filter is designed to minimize the mean square error between the filtered output and chip uk0,i(q ′) as w̃k0,i(q ′) = arg min w̃k0,i(q ′) E {∣∣w̃Hk0,i(q′)r̃k0,i(q′)− uk0,i(q′)∣∣2} , (5.32) ∀q′, i, q′ = 0, · · · , Q − 1, and i = 0, · · · , B − 1. Using the well-known Wiener solution based on the orthogonal principle, the optimum MMSE filter can be given by w̃k0,i(q ′) = [ E { r̃k0,i(q ′)r̃Hk0,i(q ′) }]−1 E { r̃k0,i(q ′)(uk0,i(q ′))∗ } . (5.33) Let us now define the Q×Q covariance matrix of uk0(i), Ṽk0(i) , Cov{uk0(i)uHk0(i)} = diag{υ̃k0,i(0)υ̃k0,i(1), · · · , υ̃k0,i(Q− 1)}, where υ̃k0,i(q) , Var{uk0,i(q)} = 1/Gυk0,i(q̃) is the variance of the qth chip, where q̃ = bq/Gc. Note that for chip-by-chip equalization, we consider only the case of BIS and hence, we have RMAI = βIQ. We can show that E { r̃k0,i(q ′)r̃Hk0,i(q ′) } = Hf (i) [ α2k0Ṽk0(i) + βIQ ] HHf (i) + α2k0(1/G − υ̃k0,i(q′))h̃i(q′)h̃Hi (q′) + σ2ηIP . (5.34) Note that we assume that the chip sequence uk0(i) is uncorrelated such that Cov{uk0,i(q) 175 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . u∗k0,i(q ′)} = 0, for q 6= q′, due to the use of a pseudo-random scrambling code. We can also show that E { r̃k0,i(q ′)(uk0,i(q ′))∗ } = αk0 G h̃i(q ′). (5.35) Let us define a P × P matrix Ψ̃(i) = Hf (i) [ α2k0Ṽk0(i) + βIQ ] HHf (i) + σ 2 ηIP . Using (5.34) and (6.38) the MMSE filter can be given as w̃k0,i(q ′) = αk0 G Hf (i) [ α2k0Ṽk0(i) + βIQ ] HHf (i) + α2k0(1/G − υ̃k0,i(q′))h̃i(q′)h̃Hi (q′) + σ2ηIP h̃i(q′). (5.36) After applying the matrix inversion lemma and some manipulations we can show that the optimum MMSE filter can be given as w̃k0,i(q ′) = 1 γ̃i(q′) Ψ̃ −1 (i)h̃i(q), (5.37) where γ̃i(q ′) = 1/αk0 [ G+ α2k0(1−Gυ̃k0,i(q′))h̃Hi (q)Ψ̃ −1 (i)h̃i(q) ] is a scalar. An unbiased estimate for chip uk0,i(q ′) can be obtained as ûk0,i(q ′) = 1/µ̃k0,i(q ′) w̃Hk0,i(q ′)r̃k0,i(q ′), where the bias value µ̃k0,i(q ′) is given by µ̃k0,i(q ′) , E{ûk0,i(q′)u∗k0,i(q′)} = αk0w̃Hk0,i(q′)h̃i(q). (5.38) Similar to symbol-by-symbol equalization, we assume that the unbiased chip estimates ûk0,i(q ′) can be approximately modeled using a (conditional) Gaussian distribution with the mean uk0,i(q ′) and variance σ̃2k0,i(q ′) as ûk0,i(q ′) = uk0,i(q ′) + η̃i(q′), (5.39) where η̃i(q ′) represents the zero-mean complex Gaussian process with the variance σ̃2k0,i(q ′) , Cov{η̃i(q′)}. We can show that σ̃2k0,i(q′) = 1/G(1 − µ̃k0,i(q′))/µ̃k0,i(q′). We stack 176 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . the soft estimates of the chips in block i into a vector ûk′0(i) = [ûk0,i(0), ûk0,i(1), · · · , ûk0,i(Q − 1)]T . The soft estimates of symbols can be obtained by despreading the soft estimates of the chips as b̂k0(i) = [b̂k0,i(0), b̂k0,i(1), · · · , b̂k0,i(M − 1)]T = SHk0(i)ûk0(i). We can express the soft estimate of symbol bk0,i(m) as b̂k0,i(m) = s ∗ k0,i(0)ûk0,i(0) + s ∗ k0,i(1)ûk0,i(1) + · · · + s∗k0,i(Q− 1)ûk0,i(Q− 1) =  Q−1∑ q=0 |s∗k0,i(q)|2   bk0,i(m) + Q−1∑ q=0 s∗k0(q)η̃i(q), (5.40) for m = 0, 1, · · · ,M − 1. Since {ûk0,i(q)}Q−1q=0 are Gaussian distributed, it is clear from (6.43) that b̂k0,i(m) is also Gaussian distributed with mean µk0,i(m) = 1 and σ 2 k0,i (m) = 1/G ∑Q−1 q=0 σ̃ 2 k0,i (q) = 1/G2 ∑Q−1 q=0 (1− µ̃k0,i(q))/µ̃k0,i(q). Approximate Implementation The computational complexity of the chip-by-chip SoftISoftO MMSE-based equalizer is very high since it is necessary to compute the MMSE filter for each chip separately. Therefore, here, we derive an approximate implementation (AprxImpl) in order to reduce the complexity of the chip-by-chip estimation. In the proposed the AprxImpl, instead of using the exact variance values of chips, we use the approximation υ̃k0,i(q) ≈ ¯̃υk0,i , 1 G − 1Q ∑Q−1 q=0 |ūk0,i(q)|2, ∀q, which are obtained using the variances of symbols. Note that a similar approach was previously adapted in [14]. Using this approximation, the covariance matrix E { r̃k0,q′(i)r̃ H k0,q′(i) } can be re-expressed as E { r̃k0,q′(i)r̃ H k0,q′(i) } ≈ ζiHf (i)HHf (i) + α2k0(1/G − ¯̃υi)h̃i(q′)h̃Hi (q′) + σ2ηIP , (5.41) 177 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . where ζi = α 2 k0 ¯̃υi+β. Following a similar approach as in the ExactImpl, it can be shown that the MMSE filter is given by w̃AIk0,i(q ′) = αk0 γ̃i(q′) Ψ̃ −1 (i)h̃i(q ′), (5.42) where Ψ̃(i) = ζiHf (i)H H f (i)+σ 2 ηIP and γ̃i(q ′) = [ G+ α2k0(1−G¯̃υi)h̃Hi (q′)Ψ̃ −1 (i)h̃i(q ′) ] . We can show that h̃Hi (q ′)Ψ̃ −1 (i)h̃i(q ′) = κ, ∀q′, q′ = 0, 1, · · · , Q−1, where κ is a constant and hence, we have γ̃i(q ′) = γ̃i = [ G+ κα2k0(1−G¯̃υi) ] , ∀q′. Therefore, we need to com- pute γ̃i only once per symbol block. Using the approximationHf (i)H H f (i) ≈ Q/P D̄2k0(i), we can show that Ψ̃(i) ≈ β̃D̄2k0(i) + σ2ηIP , which is a real-valued diagonal matrix of size P × P . Hence, it is clear that using the proposed AprxImpl, the computational com- plexity of the chip-by-chip SoftISoftO equalization scheme can be reduced to a great extent. 5.3.3 SoftISoftO Symbol Demapping The symbol demapper computes the extrinsic LLR value LeE(ck0,i(m)) of code bit ck0,i(m) using the soft estimate b̂k0,i(m) as [19], LeE(bk0,i(m)) , ln P ( bk0,i(m) = + 1|b̂k0,i(m) ) P ( bk0,i(m) = − 1|b̂k0,i(m) ) − ln P (bk0,i(m) = + 1) P (bk0,i(m) = − 1) = ln P ( b̂k0,i(m)|bk0,i(m) = + 1 ) P ( b̂k0,i(m)|bk0,i(m) = − 1 ) . (5.43) Using the Gaussian approximation to model the soft estimate b̂k0,i(m), we can easily show that LeE(bk0,i(m)) = 4Re{b̂k0,i(m)µ∗k0,i(m)} σ2k0,i(m) , (5.44) for m = 0, 1, · · · ,M − 1 and i = 0, 1, · · · , B − 1. The sequence of extrinsic LLR values {LeE(ck0(n′))}Nc−1n′=0 obtained from the symbol demapper is first deinterleaved and then, 178 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . passed to the channel decoder as the a priori information {LpD(ck0(n))}Nc−1n=0 . It is noteworthy that in this chapter, for simplicity, we consider BPSK modula- tion. However, our proposed turbo equalization schemes can be easily extended for higher modulation schemes such as QPSK, 16-QAM. Basically the same SoftISoftO MMSE-based equalizers can be employed even when higher modulation schemes are used. Nonetheless, according to the modulation scheme used, we will have to change the SoftISoftO-symbol mapper, which computes the mean and variance of symbols using the a priori LLR values of code bits, and SoftISoftO-symbol demapper, which computes the extrinsic LLR values of code bits using the soft estimates of symbols obtained from the MMSE-based equalizer (see e.g.,[21] for details). Note that when QAM modulation is used, in the symbol demapper we will have to use unbiased estimates of symbols to compute the extrinsic LLR values of code bits. 5.3.4 Comparison of Computational Complexities In Table 5.1, the number of mathematical operations (i.e., real additions and real mul- tiplications) required per symbol block per iteration in each SoftISoftO MMSE-based equalization algorithm is given. The number of real divisions required is not given, as it is negligible compared with the number of real additions or real multiplications. Furthermore, the lower-order terms are omitted for simplicity. In doing so, we assume that the maximum channel order L is comparable to the block length M (in symbols). For comparison purposes, the number of operations required for MMSE-based iterative equalization is also shown. As was seen previously, we need to compute the inverse of P × P complex covariance matrix Ψ(i) once per each code block in the exact im- plementations of both chip-by-chip and symbol-by-symbol equalizers. The number of operations required to compute Ψ−1(i) shown in Table 5.1 is obtained by assuming that the exchange method-based matrix inversion algorithm [22] is used.6 6Note that when the number of operations required for computing Ψ(i) is calculated, the fact that Ψ(i) is a hermitian matrix is not accounted for. We consider the exchange method-based algorithm for 179 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . We see from Table 5.1 that the computational complexity of the AprxImpl is ap- proximately twice higher than that of the ExactImpl for the case of BIS. Note that we assume that knowledge of the interfering users SSs is available at the receiver for the computation of the variance of the equivalent AWGN model in the symbol-by-symbol AprxImpl. As expected, it is apparent that among the symbol-by-symbol implementa- tions the AprxImpl with use of the approximation in computing the variance has the lowest complexity. It is apparent from Table 5.1 that the complexity of the chip-by-chip ExactImpl is comparable to that of the symbol-by-symbol ExactImpl. Among all pro- posed chip-by-chip and symbol-by-symbol algorithms, the AprxImpl of the chip-by-chip equalization scheme has the lowest computational complexity. We also observe that the computational complexity of the MMSE-based iterative equalization scheme is approxi- mately M -times greater than that of the proposed ExactImpl of the (symbol-by-symbol) equalization scheme for the case of SBIS. However, it should be noted that unlike in turbo equalization, in iterative equalization, soft channel decoding is not performed in each it- eration. Therefore, for more fair complexity comparison, the complexity involved in soft channel decoding should also be accounted for. It is also apparent from Table 5.1 that the (non-iterative) Petre et al. scheme [12], which uses per-tone based frequency-domain equalization, has much lower complexity compared with the other schemes. In Table 5.2, we show the number of mathematical operations required using different equalization algorithms for M = 8, G = 8, and L = 10, which will be used for obtaining the numerical results given in Section V. For the given set of parameters, it can be seen from Table 5.2 that the complexities of the ExactImpl and AprxImpl (with use of the approximation in computing the variance) of the symbol-by-symbol equalizer for the case of BIS are approximately 3 and 10 times greater than that of the AprxImpl of the chip-by-chip equalizer, respectively. computing the inverse of a general non-singular matrix. 180 C h a p ter 5 . M M S E -b a sed T u rb o E q u a liza tio n fo r S p a ce-T im e B lo ck -C o d ed ... Table 5.1: Number of mathematical operations required per block of M symbols per iteration for different MMSE-based equalization algorithms. P = MG + L, Q = MG, spreading gain G, maximum channel order L, number of receive antennas NR, SBS: symbol-by-symbol, CBC: chip-by-chip. Scheme Real Additions Real Multiplications SBS ExactImpl-SBIS (2P +Q+ 3M)2P 2 + (Q+M)4QP (2P +Q+ 3M)2P 2 + (Q+M)4QP SBS ExactImpl-BIS (6M − 1)P 2 + (2Q+ 3M + 8)2MP 6MP 2 + (2MQ+Q+ 3M2 + 12M)2P SBS AprxImpl 12MP 2 + (2Q+ 2M + 7)2MP 14MP 2 + (2M + 1)2QP + (2M + 13)2MP SBS AprxImpl (with the approx. (2Q+ 7)2MP (2MQ+Q+ 9M)2P in computing the variance) CBC ExactImpl (4P + 6Q+ 3)P 2 + 14QP (2P + 3Q)2P 2 + 26QP CBC AprxImpl (8Q+ 3)P + (3M + 1)2Q (7Q+ 5)2P + (M + 1)6Q Iterative Equalization (2P +Q+M)2MP 2 + 4MQ2P (2P +Q+M)2MP 2 + 4MQ2P Petre et al. Scheme [12] 2(2GNR + 21NR + 6M + 3)(M + L) 2(2GNR + 24NR + 6M + 9)(M + L) 181 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . Table 5.2: Number of mathematical operations required per block of M = 8 symbols per iteration for different MMSE-based equalization algorithms. P = 74, Q = 64, spreading gain G = 8, maximum channel order L = 10, number of receive antennas NR = 1, SBS: symbol-by symbol, CBC: chip-by-chip. Scheme Real Additions Real Multiplications SBS ExactImpl-SBIS 3.935 × 106 3.980 × 106 SBS ExactImpl-BIS 4.487 × 105 4.669 × 105 SBS AprxImpl 7.045 × 105 8.089 × 105 SBS AprxImpl (with the approx. 1.599 × 105 1.719 × 105 in computing the variance) CBC ExactImpl 3.810 × 106 3.851 × 106 CBC AprxImpl 4.130 × 104 7.051 × 104 Iterative Equalization 2.908 × 107 2.950 × 107 Petre et al. Scheme [12] 3.136 × 103 3.492 × 103 5.4 Semi-Analytical Upper Bounds on the BER Performance In this section, we derive semi-analytical upper bounds on the BER obtained using the proposed symbol-by-symbol turbo equalization scheme. Previously, a theoretical upper bound based on the union bound technique was proposed on the BER performance of parallel concatenated convolutional codes (PCCCs) in [23], and serial concatenated block and convolutional codes in [24]. An average upper bound was derived for PCCCs over correlated Rayleigh fading channels in [25]. On the other hand, the extrinsic information transfer (EXIT) chart technique, has received a considerable attention as a tool for understanding the convergence behavior of turbo receivers. The concept of the EXIT chart analysis was introduced in [26] to analyze the performance of iterative decoding of PCCCs and later, was applied for turbo equalization [19] and for iterative multiuser detection in [27]. The EXIT chart can also be used to estimate the BER performance at arbitrary number iterations at low-to-moderate SNR values. In e.g., [19, 26], the EXIT 182 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . chart analysis was considered for systems with infinite block length (i.e., asymptotic system behavior was considered). Nevertheless, in practical wireless systems, finite block lengths are used for practical reasons. The EXIT chart analysis for finite block lengths is yet to be thoroughly addressed. Especially, the reliability of BER estimates obtained based on the EXIT chart analysis for finite block lengths should be further investigated for turbo equalization. In this chapter, our goal is to devise a method to evaluate the performance of the proposed turbo equalization schemes at the high SNR region where it is intractable or even impossible to get the performance through simulations. Hence, the union bounding technique is chosen to analyze the performance of the proposed turbo receiver schemes at high SNR levels. In particular, we derive upper bounds on the BER performance obtained using the ExactImpl of the proposed symbol-by-symbol turbo equalization scheme for the two extreme cases: with no a priori information and with the perfect a priori information. It should be noted that following a similar approach upper bounds can easily be derived for the performance obtained using the chip-by-chip equalization scheme and the proposed approximate implementations as well. Also note that for the convenience of theoretical analysis, we assume that a linear convolutional encoder with κ number of input bits is employed at transmitter. As mentioned before, in the first iteration since there is no a priori code-bit informa- tion delivered to the SoftISoftO equalizer, we have b̄k0,i(m) = bk0,i(m) and υk0,i(m) = 1, ∀i,m. On the other hand, after a sufficient number of iterations to achieve conver- gence, it is reasonable to assume that the channel decoder feedbacks perfect a pri- ori code-bit information. When perfect a priori information is available, we have b̃k0,i(m) = bk0,i(m) and υk0,i(m) = 0, ∀i,m and therefore, ISI is completely removed through soft ISI cancelation. Hence, it is clear that with perfect a priori code-bit infor- mation the best possible performance can be obtained at the output of the SoftISoftO equalizer. Note that henceforth the desired user index k0 is omitted for notational sim- 183 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . plicity. The optimum MMSE filters for the cases with no a priori and perfect a priori code-bit information can be expressed as follows: wi(m) =   αk0 [ Hf (i) (∑K k=1 α 2 kSk(i)S H k (i) ) HHf (i) + σ2ηIP ]−1 h̃i(m); No a priori ; αk0 [ α2k0h̃i(m)h̃ H i (m) +RMAI + σ 2 ηIP ]−1 h̃i(m); Perfect a priori. As described before, the output of the MMSE-based equalizer b̂i(m) = w H i (m)r̃i(m), ∀i,m, i = 0, 1, · · · , B − 1 and m = 0, 1, · · · ,M − 1, is assumed Gaussian distributed. After interleaving, the output of the MMSE-based equalizer can be reexpressed as b̂(n) = µ(n)b(n) + η(n), (5.45) for n = 0, 1, · · · , Nc, where η(n) ∼ Nc ( 0, σ2(n) ) . As shown before, the amplitude and variance of the equivalent Gaussian system model can be computed as µ(n) = wH(n)h̃(n) and σ2(n) = µ(n)−µ2(n), respectively. Here, we assume that {µ(n)}Nc0 are statistically independent.7 In order to simplify the analysis, we divide b̂(n) by a factor of √ µ(n)− µ2(n) so that the variance of the equivalent model is normalized to 1. The normalized AWGN signal model can be given as ˜̂ b(n) = b̂(n)/ √ µ(n)− µ2(n) = µ̃(n)b(n) + η̃(n), (5.46) where µ̃(n) = µ(n)/ √ µ(n)− µ2(n) and η̃(n) = η(n)√µ(n)− µ2(n) ∼ Nc (0, σ̃2(n) = 1). In this analysis, we assume that soft maximum likelihood (ML) channel decoding is employed and that the perfect channel state information (CSI) is available at the receiver. In [28], the BER performance for the linear convolution codes over AWGN 7It was observed through simulations that {µk0,i(n)} M−1 j=0 for a given fading block i are almost the same. However, for the simplicity of the analyzes, we assume that after interleaving µ(n)Nc−1m=0 are statistically independent. This assumption is more accurate for a large number of fading blocks per frame. 184 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . channels is analyzed assuming soft ML decoding at the receiver. Following a similar approach for the problem at the hand, we can show that the BER performance for binary modulation at the output of the turbo equalization scheme (i.e., at the output of the soft channel decoder) is upper bounded by Pb < 1 κ Nc∑ d=dfree βdPPEP (d), (5.47) where k is the number of input bits to the convolutional encoder, βd = Adid, Ad is the number of paths with d distance to all-zero code sequence and id is the number of 1’s in the information sequence of the paths with d distance from all-zero code sequence. dfree denotes the minimum free distance of the code and the pairwise error probability (PEP), PPEP (d) is defined as the probability that the ML soft decoder chooses in favor of the codeword, which differs in d positions to the all-zero codeword transmitted. Let PPEP (d|µ) denote the PEP conditioned on the given pattern of distributing d non-zero bits throughout the frame µ = (n1, n2, · · · , nd), where nl is the bit position index. Using the central limit theorem it can be shown that PPEP (d|µ) = E  Q   √√√√2 d∑ l=1 µ̃2(nl)     = ∫ µ(n1) ∫ µ(n2) · · · ∫ µ(nd) Q   √√√√2 d∑ l=1 µ̃2(nl)   · pµ(µ(n1), µ(n2), · · · , µ(nd))dµ(n1)dµ(n2) · · · dµ(nd), (5.48) where µ̃(nl) = µ(nl)/(µ(nl)− µ2(nl)) and pµ(µ(n1), µ(n2), · · · , µ(nd)) is the joint prob- ability density function (PDF) of the equivalent amplitudes in the given pattern, [µ(n1), µ(n2), · · · , µ(nd)]. Q(x) , 1√2pi ∫∞ x e −t2/2dt, x ≥ 0 is the Gaussian Q-function. Now following a similar approach as for analyzing the performance turbo codes over Rayleigh fading channels in e.g., [25], it can be shown that the unconditional PEP, PPEP (d) can 185 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . be given as PPEP (d) = 1 pi ∫ pi/2 0 [∫ µ e− µ̃2 sin2 θ p(µ)dµ ]d dθ. (5.49) Note that since {µ(n)}Ncn=1 are assumed to be i.i.d., the code bit index n is of no impor- tance and hence, omitted in (5.49). In order to perform the integration in (5.49), we need the probability distribution function (pdf) of µ, p(µ). However, in the proposed scheme, it is very difficult (or even impossible) to obtain p(µ) analytically. Hence, we numerically evaluate the expectation E { e− µ̃2 2 sin2 θ } = ∫ µ e − µ̃2 2 sin2 θ p(µ)dµ in (5.49) for a given value of θ by taking the average over many realizations of amplitude µ. Then the average PEP PPEP (d) can be obtained by evaluating (5.49) through numerical integra- tion over variable θ. We can avoid the numerical integration over θ in (5.49) by using the Chernoff upper bound [25] for Q-function Q(x) ≤ 12e−x 2/2. Using the Chernoff upper bound, we can easily show that PPEP (d) can be expressed as follows: PPEP (d) ≤ E { 1 2 e Pd l=1 µ̃ 2(l) } = 1 2 [∫ µ e−µ 2 p(µ)dµ ]d . (5.50) Similarly in (5.49), since the PDF of µ, p(µ) is unknown, the expectation E { e−µ̃2 } =∫ µ e −µ̃2p(µ)dµ, is numerically approximated by taking the average over a large number of realizations of µ via simulations. Substituting the unconditional PEP with distance d to all-zero code sequence, PPEP (d) given in (5.49) in the upper bound for the BER performance in (5.47) can be expressed as Pb < 1 κpi Nc∑ d=dfree βd ∫ pi/2 0 E { e− µ̃2 2 sin2 θ } dθ. (5.51) 5.5 Simulation Results In this section, we investigate the performance of the proposed turbo equalization scheme for ST block coded downlink transmissions via (Monte-Carlo) simulations. We use a 186 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . frame length of Nu = 1024 information bits. Unless otherwise specified, a rate-1/2 constraint length ν = 5 recursive systematic convolutional channel code with generator (17, 35) in octal notation is employed. A block size of M = 8 symbols is chosen. Each mobile user is assigned a distinct Walsh-Hadamard spreading code of length G = 8. A complex scrambling code constructed using two binarym-sequences as described in [6] is used. For simplicity in simulations, we assume that both subchannels (i.e., subchannels between two transmit antennas and the receiver antenna) of the user of interest exhibit the same power-delay profile. We use square-root raised cosine (SRC) filters with a roll-off factor of 0.22 as the transmit pulse-shaping filters [29]. We also use a SRC filter with roll-off factor of 0.22 as the receiver input filter. The delay spreads of the combined channel impulse responses (i.e., convolution of the transmit filter, multipath channel and receive filter) are Lk0 = 3, Lk0 = 10 and Lk0 = 9 in chip intervals for OI&PTE-A, OI&PTE-B and VTE-A channel models, respectively. The power-delay profiles of the equivalent channel models are shown in Figure 5.3. After sampling at the chip rate, multipath components of the discrete-time combined channel impulse response, which have an average power less than ≈ 1% of the (normalized) total average power, are neglected. Hence, in order to satisfy the postfix condition, the maximum channel order is set to L = 3, L = 10 and L = 9 (in chip intervals) for OI&PTE- A, OI&PTE-B, and VTE-A channel models, respectively.8 Note that unless otherwise mentioned, we use the OI&PTE-B channel model. According to the specifications of UMTS-WCDMA systems, the duration of two symbol blocks is much less than the coherence time for moderate mobile speeds, which justifies our assumption that channels are invariant during two block times. Unless otherwise mentioned, we also assume that perfect channel information is available at the mobile receiver of interest. For channel decoding, we use the BCJR-MAP algorithm with the modifications found in [13]. 8Note that in practice the maximum channel order should be chosen considering the delay spread of all mobile user channels. 187 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Relative Delay (in chip intervals) Av er ag e Po we r ( No rm ali ze d)   IO&PTE−A Channel Model IO&PTE−B Channel Model VTE−A Channel Model Figure 5.3: Power-delay profiles of the equivalent channel models 5.5.1 Performance obtained using the (symbol-by-symbol) ExactImpl with SBIS In this subsection, we investigate the performance obtained using the (symbol-by-symbol) ExactImpl with SBIS (i.e., with knowledge of the SSs of interfering users). It should be noted that the chip-by-chip equalizer is derived only for the case of BIS. In Figure 5.4, we compare the BER performance obtained for BS and CS with K = 8 users. As a lower bound to the actual performance, the performance obtained with the perfect a priori information at the SoftISoftO equalizer for a single user (i.e., without any interfering user) is shown. Note that this performance corresponds to the matched filter bound (except that the effect of the noise correlation introduced by the MMSE-based filtering), as ISI is completely removed from the soft interference cancelation with the perfect a 188 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . priori knowledge. It is observed that with BS at BER = 10−4 we can obtain ≈ 0.6 dB improvements through turbo equalization for K = 8 users. We observe that the per- formance of CS obtained after single-pass equalization (i.e., after the first iteration) is almost identical to that of BS. Nevertheless, as expected, we can clearly see that it is not possible to achieve considerable performance improvements for CS through turbo equalization. Therefore, in this chapter, we mainly focus on the performance obtained for BS. Note that we will further investigate the performance of CS and BS as a function of the number of users for the case of BIS. For comparison purposes, in Figure 5.4 also included the performance of the MMSE-based iterative equalization is. It is observed that turbo equalization clearly outperforms iterative equalization. In Figure 5.5 and Figure 5.6, we compare the performance of BW-STBC and STTD with K = 8 and K = 4 users, respectively. As the reference performance, the per- formance of the scheme proposed by Petre et al. [12] with channel coding is shown. For comparison purposes, we also show the performance obtained for SISO transmission (i.e., without STBC) with BS. We can see that the BW-STBC scheme clearly outper- forms the Alamouti STBC-based STTD scheme proposed for WCDMA systems. The performance gains obtained using BW-STBC compared with STTD is more prominent with K = 8 users. Note that we compare the performances of the STTD and BW-STBC schemes obtained using hard-decision based (i.e., non iterative) equalization in CDMA systems without channel coding in Appendix A. It is also apparent from Figure 5.5 and Figure 5.6 that substantial performance improvements are obtained using BW-STBC, compared with the SISO transmission. For example, after four iterations at BER = 10−4, it is possible to obtain roughly 1 dB gain using STBC compared with SISO, for both cases of K = 4 and K = 8 users. It should be noted that for simulation, we assume that coefficients of different sub-channels are statistically independent. However, in practice, channel coefficients of different sub-channels may be correlated. In such a situation, the performance improvements obtained using BW-STBC over SISO will be less. 189 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−5 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   ExactImpl, BS, It1 ExactImpl, BS, It2 ExactImpl, BS, It3 ExactImpl, PA, K=1 ExactImpl, CS, It1 ExactImpl, CS, It3 Iterative Equalization, It1 Iterative Equalization, It2 Figure 5.4: Comparison of the BER performance of CS and BS obtained using the symbol-by-symbol ExactImpl for K = 8 users These results clearly justify the use of BW-STBC for downlink transmission in CDMA systems. Furthermore, we can see that the performance obtained after achiev- ing convergence is within less than one-tenth and 0.2 dB away from the lower-bound performance, with K = 4 and K = 8 users, respectively. We also observe that better performance can be obtained using the proposed receiver scheme compared with the Petre et al. scheme, even though MAI is removed deterministically (i.e., user signals are separated deterministically) in [12]. This result is mainly due the fact that in [12], maximum likelihood transmit stream separation is not used; instead, transmit stream separation and equalization are combined into a single liner equalization step. Obvi- 190 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   BW−STBC, It1 BW−STBC, It2 BW−STBC, It3 BW−STBC, PA, K=1 Petre et al. Scheme [10] STTD, It1 STTD, It2 STTD, It3 SISO, It1 SISO, It2 SISO, It3 Figure 5.5: BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users ously, we can obtain even higher improvements using our proposed scheme for a lower number of users compared with the Petre et al. scheme. In Figure 5.7, we show the performance obtained without scrambling, for K = 8 users. It is clear from Figure 5.7 that similar observations can be made regarding the performance obtained using the proposed symbol-by-symbol ExactImpl in CDMA sys- tems without scrambling as in systems with complex scrambling. Moreover, it has been found that similar performance improvements can be obtained using the proposed symbol-by-symbol turbo equalization scheme for the VTE-A channel model as for the OI&PTE-B channel model. However, we have observed that for the OI&PTE-A channel 191 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10−3 10−2 10−1 Eb/N0 dB BE R   BW−STBC, It1 BW−STBC, It2 BW−STBC, It3 BW−STBC, PA, K=1 Petre et al. Scheme [10] STTD, It1 STTD, It2 STTD, It3 SISO, It1 SISO, It2 SISO, It3 Figure 5.6: BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 4 users model it is not possible to obtain considerable performance improvements using turbo equalization. This is due to the fact that the impact of soft ISI cancelation on perfor- mance is low for channels with short delay spreads. The performances obtained for the VTE-A and OI&PTE-A channel models are not shown here due to space limitations. The frame error rate (FER) performance obtained using the proposed turbo equal- ization scheme is shown in Figure 5.8. For comparison purposes, we also show the FER performance obtained for the STTD, SISO and Petre et al. schemes. We can clearly see that BW-STBC outperforms the STTD scheme. At FER = 10−2 in particular, we can obtain a performance gain of about 0.4 dB using BW-STBC, compared with STTD. 192 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−5 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   BW−STBC, It1 BW−STBC, It2 BW−STBC, It3 Petre et al. Scheme [10] STTD, It1 STTD, It2 STTD, It3 SISO, It1 SISO, It2 SISO, It3 Figure 5.7: BER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users without complex scrambling Note also that the performance of the coded Petre et al. scheme [10] is worse that that obtained using our proposed receiver scheme after the first iteration. Similarly to the BER performance, we can see from Figure 5.8 that it is possible to obtain about 1 dB gain at FER = 10−2 using BW-STBC, compared with SISO. Moreover, we can achieve a performance improvement of around 0.7 dB in the performance at moderate Eb/N0 val- ues using the proposed turbo equalization scheme compared with conventional one-pass equalization and decoding. Usually in practical packet-switched-based wireless systems, an automatic repeat request (ARQ) scheme is employed to handle retransmission of data frames (or packets) when the FER is above the target FER. It is apparent using the 193 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . proposed turbo equalization scheme with ST block coded transmission that the average number of retranslations needed can be considerably reduced. If we assume that the target FER is 10−2, we can reach this FER threshold using the proposed turbo equalizer at around 5.8 dB, while for one-pass equalization without STBC we reach this threshold only around 8 dB. Obviously, a reduction in the number of retransmissions will increase the system throughput and reduce the delay incurred in retransmissions. 2 2.5 3 3.5 4 4.5 5 5.5 6 10−2 10−1 100 Eb/N0 dB FE R   BW−STBC, It1 BW−STBC, It2 BW−STBC, It3 Petre et al. Scheme [10] STTD, It1 STTD, It2 STTD, It3 SISO, It1 SISO, It2 SISO, It3 Figure 5.8: FER performance obtained using the ExactImpl of the symbol-by-symbol SoftISoftO equalizer for K = 8 users 194 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 5.5.2 Performance obtained using the ExactImpls for BIS Let us now investigate the performance obtained using the ExactImpls of symbol-by- symbol and chip-by-chip equalization schemes for the case of BIS. In Figure 5.9 we show the BER of BS for K = 8 users. For comparison purposes, we also show the performance of SISO for the case of BIS and the performance obtained using the symbol- by-symbol ExactImpl with SBIS. Since in this case of BIS the impact of interfering user signals is higher, we can see a reduction in the improvements obtained using turbo equalization compared with the case of SBIS. It can be seen from Figure 5.9 that the performance obtained using the proposed symbol-by-symbol scheme is almost identical to that obtained using the chip-by-chip scheme. We see that an improvement of about 0.4 dB can be obtained using turbo equalization with BIS for K = 8 users. We also observe that the performance obtained using either the symbol-by-symbol scheme with BIS or the chip-by-chip scheme after reaching convergence is 0.5 dB away that obtained using the symbol-by-symbol scheme with SBIS at BER = 10−4. Furthermore, we see that the performance of the Petre et al. scheme is almost identical to that of the symbol-by-symbol scheme with BIS and the chip-by-chip scheme obtained after the first iteration. Figure 5.10 shows the FER performance obtained using the symbol-by-symbol and chip-by-chip ExcatImpls for the case of BIS and K = 8 users. It can be seen that the performance of the chip-by-chip turbo equalizer is almost identical to that of the symbol-by-symbol turbo equalizer. Similarly in the case of SBIS, we see that substan- tial improvements can be obtained in the FER performance using the proposed turbo equalization schemes in this case of BIS as well. Particularly at FER = 0.1, we can get an improvement of about 0.5 dB through turbo equalization. Moreover, it can be seen that considerable performance improvements can be obtained using BW-STBC com- pared with SISO. Hence, it is clear from these results that even for the case of BIS, we can obtain substantial improvements in the FER performance using the proposed turbo 195 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   symbol−by−symbol, BIS, It1 symbol−by−symbol, BIS, It3 chip−by−chip, It1 chip−by−chip, It3 Petre et al. Scheme [10] symbol−by−symbol, SBIS, It1 symbol−by−symbol, SBIS, It3 symbol−by−symbol, PA, K=1 SISO, symbol−by−symbol, BIS, It1 SISO, symbol−by−symbol, BIS, it3 Figure 5.9: BER Performance obtained using the exact implementations of the symbol- by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users equalization schemes in combination with BW-STBC. In Figure 5.11, we compare the BER of BS and CS obtained using the proposed chip- by-chip turbo equalization scheme for K = 8 users. We can see that for BS it is possible to get substantial improvements using turbo equalization in systems with or without scrambling. We also note that slightly higher performance improvements are obtained for systems without scrambling compared with systems that use scrambling. On the other hand, for CS we see that when scrambling is used, it is not possible to achieve sizeable improvements even using the chip-by-chip equalization scheme. Nevertheless, when scrambling is not used, we see that at BER = 10−4 we can achieve an improvement 196 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 2 2.5 3 3.5 4 4.5 5 5.5 6 10−2 10−1 100 Eb/N0 dB FE R   symbol−by−symbol, BIS, It1 symbol−by−symbol, BIS, It3 chip−by−chip, It1 chip−by−chip, It3 Petre et al. Scheme [10] symbol−by−symbol, SBIS, It1 symbol−by−symbol, SBIS, It3 SISO, symbol−by−symbol, BIS, It1 SISO, symbol−by−symbol, BIS, It3 Figure 5.10: FER Performance obtained using the ExactImpls of the symbol-by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users of about 0.25 dB using the chip-by-chip turbo equalization scheme for CS. Note that the improvement obtained for BS is considerably larger than that obtained for CS. It should also be noted that for the case of SBIS as well, it is not possible to obtain substantial improvements using the symbol-by-symbol scheme for CS in systems with or without scrambling. In Figure 5.12, we investigate the BER variation of BS and CS obtained using the symbol-by-symbol and chip-by-chip schemes with the number of users at Eb/N0 = 4 dB for CDMA systems with scrambling. Note that here a processing gain of G = 16 is used. After the first iteration, it can be observed that the performance of CS obtained 197 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   BS, It1 BS, It3 CS, It1 CS, It3 BS, Without Scram., It1 BS, Without Scram., It3 CS, Without Scram., It1 CS, Without Scram., It3 Figure 5.11: Comparison of the BER performance of BS and CS obtained using the chip-by-chip SoftISoftO equalizer using either the symbol-by-symbol or chip-by-chip scheme is better than that of BS for low-to-moderate system loads. The difference between the performance of CS and BS is more pronounced for fewer interfering users. Hence, these results indicate that the performance of CS obtained using one-pass (non-iterative) equalization outperforms that of BS for low-to-moderate system loads. Nevertheless, due to the high impact of soft ISI cancelation, in the subsequent iterations, BS clearly outperforms CS. In particular, for CS we can not see considerable improvements obtained using turbo equalization even without MAI (i.e., for K = 1 users). Moreover, it can be seen that the difference between the performance of BS and CS is more significant for a larger number of users. 198 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . For example, in order to achieve approximately the same performance with CS as for BS for a fully-loaded system (i.e., K = 16 users), we can have only about K = 7 users. Therefore, these results clearly demonstrate the advantage of using BS rather than CS when turbo equalization is used at the receiver. 2 4 6 8 10 12 14 16 10−3 No of Users (K) BE R   symbol−by−symbol, BS, It1 symbol−by−symbol, BS, It3 chip−by−chip, BS, It1 chip−by−chip, BS, It3 symbol−by−symbol, CS, It1 symbol−by−symbol, CS, It3 chip−by−chip, CS, It1 chip−by−chip, CS, It3 Figure 5.12: Comparison of the BER performance of BS and CS for BIS with different number of users at Eb/N0 = 4 dB 5.5.3 Performance obtained for time-variant channels It is note worthy that quasi-static fading assumption may not hold for propagation channels encountered in practical systems. As we mentioned before, the duration of two blocks considered in this chapter (withM = 8 andG = 8) is much less than the coherence 199 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . time for low-to-moderate mobile speeds according to the radio-link specifications of WCDMA systems. Hence, time variation of the channel is negligible and there is no considerable impact on the performance obtained using the proposed turbo equalization schemes for low-to-moderate mobile speeds. However, for very high speeds (over 250 km/h), we have observed slight error floor behavior at high SNR. By considering the length of the paper, the performance obtained with time-varying channels is not shown. 5.5.4 Performance obtained using approximate implementations In Figure 5.13 and Figure 5.14, we show the performance obtained using the (symbol- by-symbol) Hybrid ExactImpl & AprxImpl for the case of SBIS with K = 4 and K = 8 users, respectively. In the Hybrid scheme, we use the ExactImpl in the first iteration. Note that we show the performance of the AprxImpl with and without use of the ap- proximation in computing the variance of the equivalent AWGN system model. It is clear that for K = 4 users the performance obtained using the Hybrid ExactImpl & AprxImpl scheme with and without the approximation used in computing the variance of the equivalent AWGN system model is nearly the same. Particularly at BER = 10−3, we achieve about 0.4 dB improvement using the Hybrid ExactImpl & AprxImpl scheme. We observe that for K = 4 users there is no significant degradation in the performance using the Hybrid ExactImpl & AprxImpl scheme compared with the ExactImpl. For K = 8 users it can be seen that we are not able to achieve considerable improvements using the Hybrid ExactImpl & AprxImpl scheme with the approximation in computing the variance of the equivalent AWGN model. Nevertheless, we can still get substantial improvements using the Hybrid ExactImpl & AprxImpl scheme for K = 8 users without this approximation. Hence, these results indicate that the Hybrid ExactImpl & Aprx- Impl scheme (without the approximation in computing the variance) provides a better trade-off between performance and complexity in the case of SBIS. In Figure 5.15, we show the BER performance for the case of BIS obtained using 200 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   Hybrid ExactImpl & AprxImpl, It1 Hybrid ExactImpl & AprxImpl, It2 Hybrid ExactImpl & AprxImpl, It2 (with aprx. in computing var) ExactImpl, It2 ExactImpl, It3 ExactImpl PA, K=1 Figure 5.13: BER performance obtained using the AprxImpl of the symbol-by-symbol SoftISoftO equalizer with SBIS for K = 4 users the (symbol-by-symbol) Hybrid ExactImpl & AprxImpl and the chip-by-chip AprxImpl schemes for K = 8 users. For comparison purposes, the performance of the symbol- by-symbol ExactImpl with SBIS is also shown. For the (symbol-by-symbol) Hybrid ExactImpl & AprxImpl scheme, the ExactImpl was used in the first iteration. We found that when knowledge of the interfering users’ SSs is not used, there is no performance degradation due to use of the approximation in computing the variance of the equiva- lent AWGN model in the symbol-by-symbol AprxImpl. Hence, in this case with BIS, we used this approximation for computing the variance in the AprxImpl scheme. We observe that after two iterations (i.e., after achieving convergence) the performances 201 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−5 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   Hybrid ExactImpl & AprxImpl, It1 Hybrid ExactImpl & AprxImpl, It2 Hybrid ExactImpl & AprxImpl, It2 (with aprx. in computing var) ExactImpl, It2 ExactImpl, It3 ExactImpl, PA, K=1 Figure 5.14: BER performance obtained using the AprxImpl of the symbol-by-symbol SoftISoftO equalizer with SBIS for K = 8 users of both the (symbol-by-symbol) Hybrid ExactImpl & AprxImpl and the chip-by-chip AprxImpl schemes are almost identical. We also observe that at BER = 10−4 the per- formance obtained for these schemes using the approximate implementations are about 0.4 dB worse than that of the (symbol-by-symbol) ExactImpl with SIBS. Furthermore, interestingly, we see that there is almost no difference in the performances of the chip-by- chip AprxImpl and the chip-by-chip ExactImpl or the symbol-by-symbol ExactImpl with BIS (shown in Figure 5.9). Hence, in this case of BIS, the proposed chip-by-chip Aprx- Impl is mostly attractive due to its low-complexity. It should be noted that in the case of BIS similar observations can be made regarding the performance of the approximate 202 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . implementations for different numbers of K users. 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 10−5 10−4 10−3 10−2 10−1 Eb/N0 dB BE R   Hybrid ExactImpl & AprxImpl, It1 Hybrid ExactImpl & AprxImpl, It2 chip−by−chip, AprxImpl, It1 chip−by−chip, AprxImpl, It2 symbol−by−symbol, ExactImpl, SBIS, It1 symbol−by−symbol, ExactImpl, SBIS, It2 symbol−by−symbol, ExactImpl, SBIS, It3 symbol−by−symbol, ExactImpl, PA, K=1 Figure 5.15: BER performance obtained using the approximate implementations of the symbol-by-symbol and chip-by-chip SoftISoftO equalizers with BIS for K = 8 users 5.5.5 Comparison of the semi-analytical upper bounds with the simulation results In this subsection, we investigate the accuracy of the semi-analytical upper bounds derived for the BER obtained using the proposed symbol-by-symbol turbo equalization scheme (ExactImpl). A rate-1/2 linear convolutional code with generators (7, 5) in octal notation provided in [30] is employed.9 The semi-analytical upper bounds obtained 9This convolutional code was chosen for the simplicity of obtaining the code enumeration function. 203 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . for the cases with both no a priori information and perfect a priori information are compared with the BER obtained from the simulations after the first iteration (i.e., with no a priori information delivered to the SoftISoftO equalizer) and after 4 iterations (i.e., after achieving convergence), respectively. Note that the upper bounds are averaged over 100 frames.10 In Figure 5.16 and Figure 5.17, we compare the semi-analytical upper bounds with the BER obtained directly from simulations for K = 8 and K = 4 users, respectively. It is known that upper bounds on the BER performance of turbo codes derived based on the union bound technique are usually very loose at the low BER region (BER < 10−3). A similar behavior is observed in this case for turbo equalization in the ST block coded CDMA downlink. Nevertheless, for high Eb/N0 values, we can clearly see that the analytical bounds are tight enough with the BER obtained through simulations to be used for practical purposes. Therefore, these semi-analytical bounds can be used to obtain a good estimation of the actual BER in the high SNR region without using lengthy simulations. Moreover, these results show that at BER ≈ 10−6 we get an improvement of ≈ 1 dB using the proposed MMSE-based (symbol-by-symbol) turbo equalization scheme. 10We learned from simulation that taking the average over 100 frames is sufficient. It should be noted that the sufficient number of frames required for taking the average, however, depends on the number of independent fading blocks per frame. Hence, when there is a large number of blocks per frame, taking the average over even only 10 frames should be accurate enough. 204 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 2 3 4 5 6 7 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  dB BE R   Simulation, It1 Simulation, It4 SemiAna Upper Bound, NA SemiAna Upper Bound, PA Figure 5.16: Comparison of the semi-analytical upper bounds with the simulated BER performance for K = 8 users 205 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 1 2 3 4 5 6 7 8 10−6 10−5 10−4 10−3 10−2 10−1 Eb/N0  dB BE R   Simulation, It1 Simulation, It4 SemiAna Upper Bound, NA SemiAna Upper Bound, PA Figure 5.17: Comparison of the semi-analytical upper bounds with the simulated BER performance for K = 4 users 206 Chapter 5. MMSE-based Turbo Equalization for Space-Time Block-Coded. . . 5.6 Conclusions In this chapter, we have considered frequency-domain MMSE-based turbo equalization for downlink transmission in ST block coded CDMA systems. A novel unified CDMA system model, which considers both conventional spreading and block spreading, has been proposed. Simulation results have corroborated the fact that high performance im- provements can be obtained using BW-STBC compared with SISO transmission. More- over, it has been shown that the ZP-only BW-STBC scheme remarkably outperforms the STTD scheme proposed for WCDMA systems. Frequency-domain symbol-by-symbol and chip-by-chip MMSE-based equalization schemes have been proposed for ST block coded downlink transmission. We have considered two scenarios: semi-blind interfer- ence suppression (SBIS) and blind interference suppression (BIS). We have observed that substantial performance improvements can be obtained using MMSE-based turbo equalization compared with conventional one-pass equalization. It has been shown that BS is more suitable than CS for achieving substantial performance improvements using the proposed turbo equalization schemes. We have also shown that that superior per- formance can be obtained using our turbo equalization schemes with less computational complexity than with iterative MMSE-equalization. Two approximate implementations have been proposed for symbol-by-symbol and chip-by-chip schemes in order to achieve better trade-offs between performance and complexity. In particular, simulation results have established that the AprxImpl of the symbol-by-symbol equalization scheme and the AprxImpl of the chip-by-chip equalization scheme are more suitable for the cases of SBIS and BIS, respectively. We have derived semi-analytical upper bounds on the BER performance of the proposed turbo receiver. 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Technol., vol. 19, pp. 751–772, Oct. 1971. 211 Chapter 6 Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink1 6.1 Introduction High-speed downlink packet access (HSDPA) has been introduced in wideband code division mutiple access (WCDMA) systems to increase user data rates and spectral effi- ciency, and to improve the quality-of-service (QoS) of broadband packet-based services. In WCDMA systems with HSDPA, multicode transmission, which is one of the key features of the HSDPA technology, is used to increase the data rate. In multicode trans- mission, multiple channelization codes are allocated to a user, and data is transmitted simultaneously using the channelization codes assigned to the user. The HSDPA technol- ogy uses 16-quadrature amplitude modulation (QAM) as well as quadrature phase-shift keying (QPSK) modulation. Both phase and amplitude information are required to de- tect 16-QAM symbols. In order to achieve high performance with 16-QAM modulation, we need more precise estimation of phase information than with QPSK modulation. Also, space-time block coding (STBC) based transmit diversity (STTD) has been con- sidered in WCDMA systems to reduce the detrimental effects of channel fading. Hence, it 1A version of this chapter has been submitted for publication. Wavegedara, K. C. B. and Bhargava, V. K. Turbo Multicode Detection Schemes for Space-Time Block Coded CDMA Downlink. 212 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink is essential to develop advanced and efficient receiver schemes for space-time (ST) block coded CDMA systems with multicode transmission to improve link-level performance. Typically, wideband transmission over wireless channels is subject to frequency- selective fading due to multipath propagation, destroying the orthogonality of channel- ization codes. Due to use of a high chip rate, the number of distinguishable multipaths is especially high in WCDMA systems, resulting in severe multipath interference (MPI). Under such a condition, multicode interference (MCI) has a significant impact on perfor- mance. Therefore, in order to obtain high performance, suppression of the impact of MCI is imperative. Initially, multistage MCI cancelation schemes with a RAKE receiver were considered for multicode transmissions (see [1] and references therein). Chip equalization can partially restore the orthogonality among the channelization codes and hence, later minimum mean square error (MMSE) chip equalizers were used. Recently, frequency- domain MMSE-based equalization schemes were developed for multicode CDMA sys- tems, in e.g.,[2]. In [3], an MMSE-based turbo chip equalization scheme was proposed for multicode downlink transmission in DS-CDMA systems. More recently, in [4] a sim- ilar turbo receiver scheme was proposed for multicode transmission in multiple-input multiple-output (MIMO) DS-CDMA systems. STBC was, however, not considered in any of these schemes. Recently, STBC has received tremendous attention as an effective technique to com- bat fading. ST block codes were originally designed for frequency-flat fading channels [5, 6]. Later, burst-wise (BW)-STBC schemes were proposed for frequency-selective channels, e.g., the time-reversal BW-STBC scheme [7] and the zero padding (ZP)-only BW-STBC scheme [8]. In [9], it was shown that BW-STBC schemes can yield the maxi- mum diversity order, defined as the multiple of the number of transmit antennas, receive antennas, and multipath components. Previously, receiver structures (and/or equaliza- tion schemes) were developed for STTD-WCDMA systems with [10, 11] and without cell-specific scrambling [12]. On the other hand, the ZP-only BW-STBC scheme was 213 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink adapted for downlink transmission in CDMA systems in [13]. In [14], the ZP-only BW- STBC scheme was also used for the downlink transmission of block-spread (BS)-CDMA systems with frequency-domain receiver precessing. These schemes are, however, limited to one-pass (hard-decision based) equalization. In channel-coded broadband wireless systems, additional performance improvements can be obtained by using turbo equalization, rather than traditional one-pass (hard- decision based) equalization. In [15], we proposed an MMSE-based turbo equalization scheme for the downlink of ST block coded CDMA systems. However, in [15] neither multicode transmission nor high-level modulation was considered. To this end, in this chapter, we develop an MMSE-based turbo multicode detection scheme for the ST block coded CDMA downlink. In [15], the ZP-only BW-STBC scheme [8] is shown to outperform the STTD scheme for frequency-selective fading channels in the downlink of CDMA systems. Therefore, in this chapter, we consider the ZP-only BW-STBC scheme [8] instead of the Alamouti SW- STBC scheme considered for STTD in WCDMA systems. Our proposed turbo multicode detection scheme is significantly different from the existing schemes, such as [3, 4, 16], since in our scheme BW-STBC is considered, and receiver processing is performed in the frequency domain. In the proposed turbo multicode detection scheme, ST block decoding is performed in the frequency domain prior to soft-in soft-out (SoftISoftO) multicode detection. We propose employing either a symbol-estimation-wise or chip-estimation-wise MMSE-based SoftISoftO multicode detector. In each iteration, code-bit soft information is exchanged between the SoftISoftO multicode detector and SoftISoftO channel decoder. The Soft- ISoftO symbol mapper computes the soft estimates of symbols in the physical channels assigned to the desired user using the a priori code-bit information delivered by the SoftISoftO channel decoder in the previous iteration. The frequency-domain multipath interference due to other symbols (or other chips) of the same physical channel and sym- 214 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink bols in other physical channels of the desired user is regenerated, and then subtracted from the ST block decoded signal in the symbol-estimation-wise (or chip-estimation- wise) multicode detector. We consider general linear modulation, with special attention paid to the QPSK and 16-QAM modulation schemes, as these schemes are used in WCDMA systems with HSDPA. Approximate implementations are developed for both symbol-estimation-wise and chip-estimation-wise SoftISoftO multicode detectors in or- der to reduce their computational complexities. The computational complexities of the different proposed algorithms are evaluated and compared. We also propose a least square (LS)-based iterative channel estimation scheme. The performance of the pro- posed turbo multicode detection scheme is investigated for a WCDMA system with HSDPA through simulation. Notation: Bold lower-case letters represent vectors, while bold upper-case letters de- note matrices; E{.}, Var{.}, Cov{.} stand for the expected value, variance and covari- ance operators, respectively; (.)T , (.)∗, and (.)H denote the transpose, complex con- jugate (component-wise), and Hermitian transpose operations, respectively; ∗ denotes the convolution operation and ⊗ represents the Kronecker product; IM and 0M×N de- note the M ×M identity matrix and an all-zero matrix of size M × N , respectively; FM stands for an FFT matrix of size M ×M , where its (i, j)th element is given by (1/ √ M)e−j2pi(i−1)(j−1)/M ,∀i, j ∈ [1,M ]; PnJ is the J × J permutation matrix carrying the reverse operation followed by a right cyclic shift of over n positions of a given vector of length J ; diag{s} represents a diagonal matrix with vector s on its diagonal. 6.2 System Model We consider ST block coded multicode transmission in the downlink of a DS-CDMA system. The transmitter and receiver sections of the equivalent discrete-time system model are shown in Fig.6.1 and Fig.6.2, respectively. In this system configuration, the 215 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink base-station transmitter is equipped with NT = 2 transmit antennas, while the mobile receiver of the desired user have NR ≥ 1 receive antennas. We consider that J physical channels (i.e., channelization codes) are assigned to the desired user, while a total of K physical channels are assigned to the other users (i.e., interfering users). In WCDMA systems, a common pilot channel (CPICH), which carries the pre-defined bit sequence, is transmitted in the downlink to facilitate channel estimation and synchronization. Hence, a CPICH is included in the system considered. We use superscripts (.)D, (.)I and (.)P to represent the desired user, the interfering users, and the CPICH, respectively. In this chapter, for simplicity, we assume that all physical channels including those assigned to the other users and the common-pilot-input channel (CPICH) are ST block encoded prior to transmission. No channel information is assumed at the transmitter and perfect channel information is assumed at the receiver of the desired user unless otherwise mentioned. Intracell interference is not considered. 6.2.1 Transmitter First, the cyclic redundancy check (CRC) header is added to each block of binary infor- mation bits receiving from the medium access control layer (MAC). If the packet size exceeds the maximum code block size Nmax (i.e., (Ns > Nmax), the CRC-coded data sequence of Ns length is segmented into code blocks of size Nu. The number of code blocks is given by NCB = dNs/Nue. Each code block is fed into a turbo encoder. Note that we use the same turbo coding scheme with the code rate Rc to encode every code block of the desired user. The resulting encoded blocks of length Lc = Lu/Rc+Nt, where Nt denotes the length of the trellis terminating sequence, are serially concatenated into a block of Ncb = NcbLc. Rate matching is then performed so that the number of output bits N rm = JNd, where Nd denotes the number of bits per physical channel per radio frame. In rate matching, either punturing or bit repeating is performed according to the given pattern depending on whether Ncb > N rm or Ncb < N rm, respectively. After 216 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink Figure 6.1: Transmitter section of the equivalent discrete-time system model rate matching, the resulting data sequence is segmented into J subsequences. Each sub- sequence {bdj (a)}Nd−1a=0 , j = 1, 2 · · · , J , is interleaved using a block interleaver and let us express the interleaved subsequence as {bdj (a′)}Nd−1a′=0 , where a′ = Π(a), a = 0, 1, · · · ,Nd and Π denotes the interleaver function. Let Q denote the modulation order (e.g., Q = 2 for QPSK and Q = 4 for 16-QAM). The (interleaved) code bits are grouped into sets of Q bits, bDj (n) = {bDj ((n − 1)Q + 1), bj((n−1)Q+2), · · · , bDj (Qn)}, n = 0, 1, · · · ,Ns−1, whereNs = Nc/Q is the number of symbols per physical channel per radio channel. Each set of Q modulating bits bDj (n) is mapped intoM-ary symbols sDj (n) ∈ {α0, α1, · · · , αM−1}, where αν = ej2piν/M, ∀ν, ν = 217 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink Figure 6.2: Receiver section of the equivalent discrete-time system model 0, 1, · · · ,M− 1, and M = 2Q is the size of the signal constellation (e.g., M = 4 for QPSK andM = 16 for 16-QAM). Note that the signal constellation has zero mean Es , 1/M∑M−1ν=0 αν = 0 and unit variance Vs , E{sDj (n)(sDj (n))∗} = 1/M∑M−1ν=0 |αν |2 = 1. We assume that data symbols are uncorrelated such that E{sDj (n)(sDj (n′))∗} = 0, n 6= n′). The symbol sequence in each physical channel is serial-to-parallel (S/P) converted into blocks of length MD, sDj (q) = [s D j (iM D), sDj (iM D + 1), · · · , sDj (iMD +MD − 1)]T , for i = 0, · · · , B−1, B = dNc/MDe denotes the number of blocks. Each physical channel 218 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink is assigned a distinct channelization code of length GD, cDj = [c D j (0), c D j (1), · · · , cpc(GD− 1)]T . Note that we assume the same processing gain GD for all physical channels (i.e., for all multicodes). Let {cs(ι)} be the base-station specific long scrambling code sequence. We define a diagonal matrix φ(i) = diag(cs(q)), where ss(i) = [s s(i, 0), ss(i, 1), · · · , ss(i,Q− 1)]T is the scrambling code vector in the ith block interval and Q =MDGD. The spread- ing matrix of the jth physical channel of the desired user in the ith subblock interval, CDj (i) is constructed as C D j (i) = φ(i) ( IMD ⊗ cDj ) for i = 1, 2, · · · , B − 1. Note that although in this chapter we consider only conventional spreading, block spreading can easily be adapted. The chip blocks of the physical channels assigned to the desired user can be expressed as uDj (i) = C D j (i)s D j (i), j = 1, 2, · · · , J , and i = 0, 1, · · · , B − 1. Let us now consider the physical channels assigned to the interfering users. We denote the symbol sequences of the interfering user channels as sIk(n), k = 0, 1, · · · ,K−1. In the system model considered, it is not necessary to use the same modulation scheme for the other (interfering) users as used in the the physical channels of the desired user. Hence, in the physical channels assigned to the other users different modulation schemes can be used. The data sequences of interfering users are serial-to-parallel converted into BIk blocks of size MIk . Note that the block length M I k is chosen so that Q = M I kG I k, where GIk is the processing gain of the kth physical channel assigned to interfering users. The spreading matrix for the kth physical channel in the ith chip block can be constructed as CIp(i) = φ(i) ( IMI ⊗ cIp ) , where the channelization code of the kth interfering physical channel cIk = [c I k (0), c I k (1), · · · , cIk (G − 1)]T . We can express the chip blocks of the interfering users as uIk(i) = α I kC I k(i)b I k (i), k = 1, 2, · · · ,K, and i = 0, 1, · · · , B − 1. The spreading matrix of the CPICH can be given as CP(i) = φ(i) ( IMP ⊗ cPp ) , where cP = [cP (0), cP (1), · · · , cP(G − 1)]T is the channelization code assigned to the CPICH. The chip block of the CPICH in the ith block interval can be expressed as uP(i) = αPCP(i)sP (i), i = 0, 1, · · · , B − 1. Following spreading, the chip blocks of all physical channels are added together and the resulting chip block in the ith chip-block interval 219 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink can be expressed as u(i) = αD P∑ p=1 uIp (i) + K∑ k=1 αIku I k(i) + α PuP(i), (6.1) where αD, αIk and α P denotes the weighting factors of the desired user channels, the kth interfering user channel and the CPICH, respectively. Two consecutive chip blocks, u(2i),u(2i+1), are fed into the ST block encoder, which performs ST encoding at the chip-block level. The output of the ST block encoder can be given as [8],   ū1(2i) ū1(2i+ 1) ū2(2i) ū2(2i+ 1)   =   u(2i) −P0Bu∗(2i+ 1) u(2i + 1) P0Bu ∗(2i)   . (6.2) Let us define the maximum channel order L as L , maxκ L̃κ,nt,nr , ∀κ, nt, nr, κ = 1, 2, · · · , K̄, nt = 1, 2, and nr = 1, 2, · · · ,NR, where L̃κ,nt,nr is the order of the equivalent discrete-time finite impulse response (FIR) subchannel between transmit antenna nt and receive antenna nr of the κth user, and K̄ is the number of active mobile users. The chip blocks obtained at the output of the ST block encoder are zero padded as ũnt(i) = TZP ūnt(i), nt = 1, 2, where TZP = [IQ,0Q×L] T is the zero-padding matrix of size P×M and P = Q + L. After parallel to serial (P/S) conversion of ũnt(i) = [ũnt,i(0), ũnt,i(1), · · · , ũnt,i(P − 1)], multiuser chip sequences {ũnt,i(n)}P−1n=0 , nt = 1, 2, are filtered using the transmit chip-pulse shaping filter ϕT (t). The resulting continuous- time signal unt(t) transmitted through antenna nt can be expressed as unt(t) = B∑ i=0 P∑ n=0 unt,i(n)ϕT (t− (iP + n)Tc), (6.3) where Tc is the chip period. 220 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink 6.2.2 Channel Model In this chapter, we consider a wideband frequency-selective Rayleigh fading channel model with Lnt,nr resolvable multipath components for each subchannel between the ntth transmit antenna and the nrth the receive antenna of the desired user. The equivalent impulse response of the multipath subchannel between the ntth transmit antenna and the nrth the receive antenna is given as gnt,nr(t) = Lnt,nr∑ l=0 αnt,nr(l)δ(t − τnt,nr(l)), (6.4) where αnt,nr(l) and τnt,nr(l) denote the complex-valued fading coefficient and the delay of the lth resolvable path, respectively and δ(t) denotes the Dirac-delta function. αnt,nr(l), ∀nt, nr, l, are assumed statistically independent. Each Rayleigh distributed channel coefficient may have different average gains. We also assume that multipath subchannels are invariant over two consecutive symbol blocks of lengthM and channel fading between units of two consecutive blocks is independent (i.e., quasi-static fading). We assume that the receive filter ϕR(t) is matched to the chip pulse shaping transmit filter (i.e., ϕR(t) = ϕ ∗ T (−t)). Let hnt,nr,i(l) = [hnt,nr,i(0), hnt,nr ,i(1), · · · , hnt,nr,i(Lnt,nr)], where hnt,nr,i(l) , ϕT (t) ∗ gnt,nr(t) ∗ ϕR(t)|t=lTc , l = 0, 1, · · · , Lnt,nr , be the chip-rate sampled discrete-time equivalent FIR channel between the ntth transmit antenna and the nrth receive antenna in the ith chip block interval. Without loss of generality, we assume that the equivalent subchannels between both transmit antennas and receive antenna nr of the desired user have the same channel order Lnr . It should be noted that hnt,nr,2i(l) = hnt,nr,2i+1(l), ∀l, l = 0, 1, · · · , Lnr , as we assume that subchannels are time invariant over two consecutive chip blocks. For notational simplicity, from this point onwards, we drop the block index from channel coefficients. 221 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink 6.2.3 Receiver The continuous-time received signal at the nrth receive antenna of the desired user rnr(t) = ∑2 nt=1 gnt,nr(t) ∗ unt(t) + ηnr(t), nr = 1, 2, · · · ,NR, is first, receive filtered and then, sampled at the chip rate, where ηnr(t) denotes the complex additive white Gaussian noise (AWGN) process. The resulting discrete-time received signal at receive antenna nr can be expressed as xnr,i(n) , ∫ ∞ −∞ r(τ)ϕR(τ − (iP + n)Tc)dτ = Lnr∑ l=0 [h1,nr(l)ũ1(n− l) + h2,nr(l)ũ2(n− l)] + ηnr(n), (6.5) for n = 0, 1, · · · , P − 1 and i = 0, 1, · · · , B − 1, where ηnr(n) , ϕR(t) ∗ ηnr(t)|t=nTc is the discrete-time zero-mean AWGN with variance σ2η/2 per dimension. Note that we assume the same noise variance σ2η at all receive antennas. Received chip sequences {xnr ,i(n)}P−1n=0 , nr = 1, 2, · · · ,NR, are serial-to-parallel (S/P) converted into blocks of length P . The received chip-block of the nrth receive antenna in the ith block interval, xnr(i) , [xnr ,i(0), xnr,i(1) · · · , xnr ,i(P − 1)]T can be expressed as xnr(i) = Hnr ,1(i)ũ1(i) +Hnr ,2(i)ũ2(i) + ηnr(i), (6.6) i = 0, 1, · · · , B − 1, where Hnt,nr(i), nt = 1, 2, are the P × P lower triangular Toeplitz matrices with [hnt,nr(0), hnt,nr(1), · · · , hnt,nr(Lnr), 01×(P−Lnr−1)]T as the first column and ηnr(i) , [ηnr(iP ), ηnr (iP + 1), · · · , ηnr(iP + P − 1)]T denotes the AWGN vector with the covariance matrix E{ηnr(i)ηHnr(i)} = σ2ηIP . We perform ST block decoding on the received chip-block vector at each receive antenna separately. In this chapter, we adopt a similar frequency-domain ST block de- coding approach as proposed in [8]. We can easily see that Hnt,nr(i)Tzp = H̃nt,nr(i)Tzp, nt = 1, 2, where H̃nt,nr(i) is the P × P circulant matrix having [hnt,nr(0),01×(Q−1), 222 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink hnt,nr(L), · · · , hnt,nr(1)]T as the first row. We can replace Topelitz matrices in (7.2) with the corresponding circulant matrices and hence, xk0(i) can be reexpressed as xk0(i) = H̃1,nr(i)ũ1(i) + H̃2,nr(i)ũ2(i) + ηnr(i), (6.7) We take two consecutive chip blocks xnr(2i) and xnr(2i + 1) and transform into the frequency domain using the DFT transform matrix FP as ynr(2i) = FPxnr(2i) and ynr(2i+1) = FPP Q Px ∗ nr(2i+1). Note that before transforming into the frequency domain, the complex conjugate of the second chip block is reverse-cyclic shifted by Q positions us- ing the permutation matrix PQP . Similarly, we define the frequency-domain noise vectors as η̃nr(2i) , FPηnr(2i) and η̃nr(2i+ 1) , FPP Q Pη ∗ nr(2i+ 1). The circulant subchannel matrices H̃nt,nr(i), nt = 1, 2, are diagonalized using the DFT operation as H̃nt,nr(i) = FHP Dnt,nr(i)FP , where Dnt,nr(i) = diag{h̃nt ,nr}. Here, h̃nt,nr denotes the P -point DFT of the channel impulse response hnt,nr = [hnt,nr(0), hnt ,nr(1), · · · , hnt,nr(L)]. After simple manipulations, we can show that [8],   ynr(2i) ynr(2i + 1)   ︸ ︷︷ ︸ ,y̌nr (i) =   D1,nr(i) D2,nr(i) D ∗ 2,nr(i) −D∗1,nr(i)   ︸ ︷︷ ︸ ,Ďnr (i)   FP ũ1(2i) FP ũ1(2i + 1)  +   η̃nr(2i) η̃∗nr(2i + 1)   ︸ ︷︷ ︸ ,η̌nr (i) . (6.8) We define a real-valued P × P diagonal matrix D̄nr(i) , [|D1,nr(i)|2 + |D2,nr(i)|2]1/2. Let us also define a unitary matrix Unr(i) , Ďnr(i)(I2 ⊗ (D̄nr(i))−1). After some manipulations, the output of the ST decoder [zTnr(2i), z T nr (2i+ 1)] T = UHnr(i)y̌nr (i) can be shown as [8],   znr(2i) znr(2i + 1)   =   D̄nrFPTzpu(2i) D̄nrFPTzpu(2i+ 1)  +   η̄nr(2i) η̄nr(2i+ 1)   , (6.9) where the resulting noise vectors are defined as [η̄Tnr(2i), η̄ T nr(2i + 1)] T , UHnr(i)η̌nr(i). 223 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink It is clear from (6.9) that the frequency-domain ST block decoded signal vector (corre- sponding to the chip-block u(i) transmitted in the ith block interval) can be separated and expressed as znr(i) = D̄nr(i)uf (i) + η̄nr(i), (6.10) where the frequency-domain multiuser (zero-padded) chip block uf (i) , FPTzpu(i). Following ST block decoding, receive antenna combining is performed. We follow a similar approach used in [9]. First, by stacking the ST block decoded vectors znr(i) given in (6.11), we obtain the combined vector z(i) = [zT1 (i), z T 2 (i), · · · , zTNR(i)]T = B(i)uf (i) + η̄(i), where B(i) , [D̄1(i), D̄2(i), · · · , D̄NR(i)]T and η̄(i) , [η̄Tnr(i), η̄Tnr(i), · · · , η̄TNR(i)]T . Let us now define a diagonal matrix B̄(i) , [∑NR nr=1 D̄ 2 nr(i) ]1/2 . We also construct a matrix Ū(i) , B(i)B̄(i)−1, which has orthonormal columns. We then multiply z(i) by Ū H (i) to obtain the receive-combined vector as r(i) = Ū H (i)z(i) = B̄(i)uf (i) + η̄(i), (6.11) where η̄(i) = Ū H (i)η̄(i). Note that since the columns of Ū(i) are orthonormal, η̄(i) is still white [9]. 6.3 MMSE-based Turbo Multicode Detection In this section, the proposed MMSE-based turbo multicode detection scheme for ST block coded multicode transmission in the downlink of CDMA systems is described. The structure of the proposed turbo receiver is depicted in Fig.6.2. At the beginning of each iteration (except in the first iteration), we provide the SoftISoftO-symbol mapper with the a priori log likelihood ratio (LLR) values of code bits of the physical channels assigned the desired user, La1(b D j (a ′)) , ln P [bDj (a ′)=1] P [bDj (a′)=0] , ∀a′, j, a′ = 0, 1, · · · ,Nd − 1 and j = 0, 1, · · · , J−1, which were computed by the SoftISoftO turbo decoder in the previous iteration. The symbol a priori probabilities P ( sDj (n) = αν ) , for ν = 0, 1, · · · ,M− 1, 224 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink can be obtained using the a priori LLR values of code bits as P ( sDj (n) = αν ) = Q−1∏ ι=0 P ( bDj ((n− 1)Q + ι)=bν(ι) ) , = Q−1∏ ι=0 1 2 ( 1 + (2bν(ι)− 1)tanh ( La1(b D j ((n − 1)Q+ ι))/2 )) . (6.12) The symbol mapper computes the mean s̄Dj (n) and the variance υ 2 j (n) of every symbol sDj (n), n = 0, 1, · · · , Ns−1, j = 0, 1, · · · , J−1, of the desired user and input them to the SoftISoftO MMSE-based multicode detector. s̄Dj (n) and b D j (n) can be computed using the symbol a priori probabilities P ( sDj (n) = αν ) as follows: s̄Dj (n) , E{sDj (n)} = M−1∑ ν=0 ανP ( sDj (n) = αν ) , (υDj (n)) 2 , Cov{sDj (n), sDj (n)} = M−1∑ ν=0 |αν |2 − |s̄Dj (n)|2. In particular, we can show that s̄Dj (n) = − 1√ 2 tanh ( La1(b D j (2n − 1)) + jLa1(bDj (2n)) ) (υDj (n)) 2 = 1− |s̄Dj (n)|2 for QPSK and s̄Dj (n) = − 1√ 2   ( 1 + 3eL a 1(b D j (4n−1)) )( 1− eLa1(bDj (4n−3)) ) ( 1 + eL a 1(b D j (4n−3)) )( 1 + eL a 1(b D j (4n−1)) ) + √−1 ( 1 + 3eL a 1(b D j (4n)) )( 1− eLa1(bDj (4n−2)) ) ( 1 + eL a 1(b D j (4n−2)) )( 1 + eL a 1(b D j (4n)) )   (υDj (n)) 2 = 1 2 ( 2 + 10 ( eL a 1(b D j (4n−1)) + eL a 1(b D j (4n)) ) + 18e(L a 1(b D j (4n−1))+La1 (bDj (4n))) ) ( 1 + eL a 1(b D j (4n−1)) )( 1 + eL a 1(b D j (4n)) ) − |s̄Dj (n)|2 225 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink for 16-QAM. Note that since in the first iteration there is no a priori information de- livered to the SoftISoftO multicode estimator by the turbo code decoder, we assume that coded bit sequence {bDj (m)}Nd−1n=0 , and j = 0, 1, · · · , J − 1, are equiprobable and i.i.d., and hence, we have Lp1(b D j (m)) = 0, ∀m, j. Therefore, in the first iteration we can easily show that s̄Dj (n) = 0 and (υ D j (n)) 2 = 1, ∀n, j. It is also noteworthy that since in the proposed turbo multicode detection scheme soft values of code bits of the physical channels assigned to the interfering users are not computed, we consider that s̄Ik(n) = 0 and (υIk (n)) 2 = 1, ∀n, k, n = 0, 1, · · · ,Nd − 1, and k = 0, 1, · · · ,K − 1. 6.3.1 Symbol-by-Symbol Estimation wise MMSE-based SoftISoftO Multicode Detector First, we describe the symbol-by-symbol estimation wise (from here onwards we referred to as symbol-by-symbol) MMSE-based multicode detector. Let us consider soft estima- tion of the mth symbol of the j′th physical channel (or multicode), sDi,j(m). Prior to MMSE filtering, frequency-domain soft inter-symbol interference (ISI) and MCI cance- lations are performed using the soft estimates of the other symbols in the same phys- ical channel (i.e., s̄Di,j(m ′), ∀m′, m′ = 0, 1, · · · ,MD − 1, m′ 6= m) and the soft esti- mates of the symbols in the other physical channels of the desired user (i.e., s̄Di,j′(m), ∀m, j′, m = 0, 1, · · · ,M − 1, and j′ = 0, 1, · · · , J − 1, j′ 6= j), respectively. Let us define vectors s̄Dj (i) = [ s̄Di,j(0), s̄ D i,j(1), · · · , s̄Di,j(M − 1) ]T , for j = 0, 1, · · · , J − 1. We also define vector s̄Dm′,j′(i) , s̄Dj′(i) − s̄Di,j′(m′)em′ , m′ = 0, 1, · · · ,MD − 1, where em′ is an all zero column vector of MD except for a one at the m′th position. After soft ISI and MCI cancelations (with respect to symbol si,j(m′)), 226 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink we have r̃m′,j′(i) = r(i)−Hf (i)   J∑ j=1, 6=j′ CDj (i)s̄ D j (i) +C D j′(i)s̄ D m′,j′(i)   , = Hf (i)  αD   J∑ j=1, 6=j′ CDj (i) ( sDj (i)− s̄j(i) ) +CDj′(i) ( sDj′(i)− s̄m′,j′(i) ) + K∑ k=1 αIkC I k(i)s I k(i) ] + η̃(i), (6.13) where for notational brevity, we define a matrix Hf (i) , D̄(i)FPTzp. Note that since in the first iteration no a priori information is delivered by the turbo decoder, we have s̄Dj (i) = 0MD) and s̄ D m′,j′(i) = 0MD), and hence, we can skip the step of soft ISI and MCI cancelations. Soft ISI and MCI cancelations are followed by MMSE filtering. The linear MMSE filter is designed to minimize the mean square error between the filter output and the m′th code symbol in the ith block sDi,j′(m ′) as wi,j′(m ′) = arg min wi,j′ (m′) E {∣∣wHi,j′(m′)r̃m′,j′(i)− sDi,j′(m′)∣∣2} , ∀m′, i, j′, m′ = 0, · · · ,MD − 1, j′ = 0, · · · ,M − 1, and i = 0, · · · , B− 1. Using the well- known Wiener solution based on the orthogonal principle, the optimum MMSE filter can be given by wi,j′(m ′) = [ E { r̃m′,j′(i)r̃ H m′,j′(i) }]−1 E { r̃m′,j′(i)(s D i,j′(m ′))∗ } . (6.14) For notation simplicity, let us define a vector h̃i,j′(m ′) , Hf (i)CDj′(i)em′ . It can be 227 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink shown that E { r̃m′,j′(i)r̃ H m′,j′(i) } = Hf (i)  (αD)2 J∑ j=1 CDj (i)V D j (i)(C D j (i)) H +RMAI  HHe (i) + (αD)2(1− ῡDi,j′(m′))h̃i,j′(m′)h̃Hi,j′(m′) + σ2ηIP , (6.15) where the MD ×MD covariance matrix of sDj (i), VDj (i) , Cov{sDj (i)},∀j is given by VDj (i) = diag[υ D i,j(0), υ D i,j(0), · · · , υDi,j(MD − 1)]. Note that in (7.15) term (αD)2(1− ῡDi,j′(m′))h̃i,j′(m′)h̃Hi,j′(m′) is due the fact that we set the variance of the m′th code symbol υDi,j′(m ′) = 1 in Vj(i) based on the turbo principle. Let RMUI denote the covariance matrix of the chip blocks of K − 1 interfering users, which is defined as RMUI , E   [ K∑ k=1 αIku I k(i) ] [ K∑ k=1 αIku I k(i) ]H  . (6.16) Note that in the first iteration since υDi,j(m) = 1 ,∀m, i, j, we have E { r̃m′,j′(i)r̃ H m′,j′(i) } = Hf (i)  (αD)2 J∑ j=1 CDj (i)(C D j (i)) H +RMAI  HHf (i) + σ2ηIP , (6.17) We can also show that E { r̃m′,j′(i)(s D i,j′(m ′))∗ } = (αD)2He(i)CDj′(i)em′ = α Dh̃i,j′(m′). (6.18) 228 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink Using (7.15) and (6.18), we can express the MMSE filter as wi,j′(m ′) = αD  Hf (i)  (αD)2 J∑ j=1 CDj (i)V D j (i)(C D j (i)) H +RMAI  HHf (i) + (αD)2(1− ῡDi,j′(m′))h̃i,j′(m)h̃Hi,j′(m′) + σ2ηIP ]−1 h̃i,j′(m ′). (6.19) It is clear from (6.19) that in order to obtain the MMSE-based estimator, ex- cept in the first iteration, it is required to compute the inverse of the P × P co- variance matrix E { r̃m′,j′(i)r̃ H m′,j′(i) } separately for each symbol. This can be bur- den for a mobile receiver especially with a large P . Hence, in the following, we de- rive a computationally-efficient implementation. Applying the matrix inversion lemma,[ E { r̃m′,j′(i)r̃ H m′,j′(i) }]−1 can be reexpressed as [ E { r̃m′,j′(i)r̃ H m′,j′(i) }]−1 = Ψ(i)−1 − (α D)2(1− ῡDi,j′(m′)) γi,j′(m′) Ψ(i)−1h̃i,j′(m′)h̃Hi,j′(m ′)Ψ(i)−1, (6.20) where we define a P × P matrix Ψ(i) = Hf (i)  (αD)2 J∑ j=1 CDj (i)V D j (i)(C D j (i)) H +RMAI  HHf (i) + σ2ηIP , and a scalar γi,j′(m ′) = 1 + (αD)2(1 − ῡDi,j′(m′))h̃Hi,j′(m′)Ψ−1(i)h̃i,j′(m′). After simple manipulations, we can show that the MMSE filter can be given by wi,j′(m ′) = αD γi,j′(m′) Ψ−1(i)h̃i,j′(m′). (6.21) Note that the matrix Ψ(i) is the same for all symbols of the physical channels in a given block. Hence, in this implementation it is necessary to compute Ψ−1(i) only once per block. Hence, it is clear that the computational complexity can be reduced significantly 229 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink using this implementation given in (6.21). Note that since in the first iteration no a priori code-bit information is delivered to the SoftISoftO multicode detector, we have υi,j′(m ′) = 1, ∀m′, j′, and hence, γi,j′(m′) = 1, ∀m′, i, j′. Semi-Blind Multiuser Interference Suppression In this case we consider that the SSs of the physical channels assigned to the interfering users are known to the receiver of interest and hence, knowledge of the SSs of the interfering users is used in MMSE filtering. However, soft multiuser interference (MUI) cancelation is not performed using the soft estimates of symbols in the physical channels assigned to the interfering users.2 Therefore, we refer to this case as semi-blind multiuser interference suppression (SBMUIS). We can show that RMUI = K∑ k=1 (αIk ) 2CIk(i)V I k (i)(C I k (i)) H , (6.22) where VIk (i) , Cov{sIk (i)} is the covariance matrix of sIk (i). Since soft information of interfering users’ symbols are not available, we have Vk(i) = IMIk , ∀k, k = 1, 2, · · · ,K. Blind Multiuser Interference Suppression In this case, we assume that even knowledge of SSs of the physical channels assigned to the interfering users is not available at the mobile receiver of interest. Since neither soft estimates of interfering users’ symbols nor knowledge of interfering users’ SSs are used in MMSE-based SoftISoftO multicode detection, this case is referred to as blind multiuser interference suppression (BMUIS). We assume that the chip blocks of the physical chan- nels assigned to the interfering users, uIk(i) ∀k, k = 1, · · · ,K, are mutually uncorrelated (i.e., uIk(i)(u I k(i)) H = 0, k 6= k′). We also assume that uIk(i), k = 1, 2, · · · ,K, are white with variance 1/GIk such that E { uk(i)u H k′ (i) } = 1/GIkIMIkG I k . This is a reasonable 2It should be noted that in the proposed receiver scheme symbols of the physical channels assigned to the interfering users are not estimated at the mobile receiver of interest. 230 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink assumption in practice due to use of a long pseudo-random scrambling code. Hence, we can easily see that RMAI = βIQ, where β , ∑K k=1(α I k) 2/GIk. An unbiased estimate for symbol si,j′(m ′) can be obtained using the proposed symbol- by-symbol MMSE-based multicode detector as ŝi,j′(m ′) = 1 µi,j′(m′) wHi,j′(m ′)r̃i,j′(m′), (6.23) where µi,j′(m ′) , E{wHi,j′(m′)r̃i,j′(m′)(sDi,j′(m′))∗} is the bias value. It can be shown that µi,j′(m ′) = αDwHi,j′(m ′)h̃i,j′(m′). (6.24) As will be seen, the computation of the LLR vales of code bits is uncomplicated if unbiased symbol estimates {ŝi,j′(m′)}M−1m′=0 obtained using the MMSE filter is modeled using a Gaussian distribution. Hence, we assume that the unbiased estimates ŝDi,j′(m ′) obtained using the MMSE-based detector can be modeled as the output of an additive white Gaussian noise (AWGN) model with unit mean and variance σ2i,j′(m) as ŝDi,j′(m ′) = sDi,j′(m ′) + ηi,j′(m′), (6.25) for m′ = 0, 1, · · · ,M − 1 and j′ = 0, 1, · · · , J − 1, where ηi,j′(m′) denotes the zero-mean complex-valued white Gaussian noise with variance σ2i,j′(m ′). Note that previously, a Gaussian distribution was used to model the output of the MMSE filter in, e.g., turbo interference cancelation in the uplink of CDMA systems [16] and turbo equalization [17]. We can show that σ2i,j′(m ′) = 1− µi,j′(m′) µi,j′(m′) . (6.26) 231 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink Approximate Implementations The computational complexity of MMSE filtering is dominated by the computation of[ E { r̃m,j′(i)r̃ H m,j′(i) }]−1 . As was seen previously, in the simplified implementation, in order to obtain [ E { r̃m,j′(i)r̃ H m,j′(i) }]−1 we need to compute the inverse of P ×P matrix Ψ(i) once per a block. Computation of Ψ−1(i) even once per a block is a burden for a practical mobile receiver for a large value of P . Note that a large P is preferable to achieve high-spectral efficiency. Hence, in the following we propose two approximate implementations. Approximate Implementation-1 (AprxImpl-1) In AprxImpl-1 we consider BMUIS and hence, we have RMAI = βIQ. Let us reexpress the covariance matrix as follows: E { r̃m′,j′(i)r̃ H m′,j′(i) } = Hf (i) [ (αD)2CDj′(i)V D j′ (i)(C D j′(i)) H + E   J∑ j=1, j 6=j′ ũDj (i)(ũ D j ) H(i)   + βIQ  HHf (i) + (αD)2(1− ῡDi,j′(m′))h̃i,j′(m′)h̃Hi,j′(m′) + σ2ηIP , where ũDj = u D j (i) − CDj′(i)s̄Dj′ (i). We assume that the chips of the physical channels assigned to the desired user are statistically independent, i.e., E{uDj,i(q)(uDj,i(q′))∗} = 0, q = 0, 1, · · · , Q−1, q 6= q′ and j = 1, 2 · · · , J . We can justify this assumption as a long- scrambling sequence is used. Hence, we have E{ũDj (i)(ũDj (i))H} = diag{υcj,i(0), υcj,i(1), · · · , υcj,i(Q − 1)}, ∀j, j 6= j′, where υcj,i(q) , E{ũDj,i(q)(ũDj,i(q))∗} = E{uDj,i(q)(uDj,i(q))∗} − |ūDj,i(q)|2. For simplicity, we also assume that E{ũDj,i(q)(ũDj,i(q))∗} = 1/G and υcj,i(q) = ῡcj,i = 1/G− 1/Q ∑Q−1 q=0 |ūDj,i(q)|2. Hence, we have E{ũDj (i)(ũDj (i))∗} ≈ ῡcj,iIQ. The covariance matrix can be approximately expressed as E { r̃m′,j′(i)r̃ H m′,j′(i) } ≈ (αD)2He(i)CDj′(i)VDj′ (i)CDj′(i))HHHf (i) + (αD)2 · ·(1− ῡDi,j′(m′))h̃i,j′(m′)h̃Hi,j′(m′) + ζHf(i)HHf (i) + σ2ηIP , 232 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink where ζ = (αD)2 ∑J j=1, j 6=j′ ῡ c j,i+β. Let us now consider termHf (i)H H f (i) = D̄(i)FPTzp TTzpF H P D̄(i). Using the approximation TzpT T zp ≈ (Q/P )IP and the fact that FPF H P = IP , we can easily see that Hf (i)H H f (i) ≈ (Q/P )D̄2(i). Hence, the covariance matrix can be reexpressed as E { r̃m′,j′(i)r̃ H m′,j′(i) } ≈ (αD)2Hf (i)CDj′(i)VDj′ (i)CDj′(i))HHHf (i) + (αD)2(1− ῡDi,j′(m′))h̃i,j′(m′)h̃Hi,j′(m′) + ζ̃D̄2(i) + σ2ηIP , where ζ̃ = (Q/P )ζ. Let us define a P × P diagonal matrix Φ(i) , ζ̃D̄2(i) + σ2ηIP . Applying the matrix inversion lemma, we can rewrite [ E{r̃m′,j′(i)r̃Hm′,j′(i)} ]−1 as [ E { r̃m′,j′(i)r̃ H m′,j′(i) }]−1 ≈ (ΨAI1(i))−1−(ΨAI1(i))−1h̃i,j′(m′) [ 1 (αD)2(1− ῡDi,j′(m′)) − h̃Hi,j′(m′)(ΨAI1(i))−1h̃i,j′(m′) ] , (6.27) where ΨAI1(i) = (αD)2Hf (i)CDj (i)V D j (i)(C D j (i)) HHHf (i) +Φ. Substituting (6.27) and (6.18) in (6.14), after some manipulations the MMSE filter can be given as wAI1i,j′ (m) = αD γAI1i,j′ (m) (ΨAI1)−1(i)h̃i,j′(m), (6.28) where γAI1i,j′ (m ′) , 1+(αD)2(1− ῡDi,j′(m′))h̃Hi,j′(m′)(ΨAI1)−1(i)h̃i,j′(m′) is scalar. Apply- ing the matrix inversion lemma (ΨAI1)−1(i) can be computed as follows ΨAI1(i) = Φ−1 −Φ−1HHf (i)CDj′ (i) [ (αD)−2VDj (i) + (CDj′(i)) HHHf (i)Φ −1Hf (i)CDj′(i) ]−1 (CDj′(i)) HHHf (i)Φ −1. (6.29) Note that Φ−1 is a real-valued diagonal matrix. Hence, in AprxImpl-1 it is necessary to compute the inverse of a MD ×MD matrix per each physical channel assigned to the desired user. Therefore, it is clear that AprxImpl-1 is computationally efficient 233 Chapter 6. Turbo Multicode Detection for Space-Time Block-Coded CDMA Downlink for a small number of physical channels J assigned to the desired user and for a large processing gain G, compared with the ExactImpl. Similarly in the ExactImpl, we assume that the unbiased estimates ŝDi,j′(m ′) = 1 µAI1 i,j′ (m ′) (w AI1 i,j′ (m ′))H r̃i,j′(m′) obtained using AprxImpl-1 can be modeled using a Gaus- sian distribution with the mean sDi,j′(m ′) and the variance (σAI1i,j′ (m ′))2, where µAI1i,j′ (m ′) = αD(wAI1i,j′ (m ′))H h̃i,j′(m′) is the bias value. It can be shown that the variance (σAI1i,j′ (m ′))2 , Cov { ŝDi,j′(m ′) } of the equivalent Gaussian model can be given as (σAI1i,j′ (m ′))2 = (wAI1i,j′ (m ′))H(E { r̃m′,j′(i)r̃ H m,j′(i) } )wAI1i,j′ (m ′)− (µAI1i,j′ (m′))2. (6.30) It