2 PERFORMANCE BOUNDS FOR BLIND CHANNEL ESTIMATION 41 2.1 Introduction 42 2.2 Problem Statement and Preliminaries 42 2.2.1 The Blind Channel Identification Problem 42 2.2.2 Ambiguity Elimination 44 2.2.3 The Unconstrained FIM 46 2.2.4 Achievability of the CRB 47 2.3 CRB for Constrained Estimates 48 2.4 CRB for Estimates of Invariants 49 2.5 CRB for Projection Errors 52 2.6 Numerical Examples 53 2.7 Concluding Remarks 58 Appendix 2.A Proof of Proposition 2 59 Bibliography 61
3 SUBSPACE METHOD FOR BLIND IDENTIFICATION AND DECONVOLUTION 63
3.1 Introduction 63 3.2 Subspace Identification of SIMO Channels 65 3.2.1 Practical Considerations 69 3.2.2 Simplifications in the Two-Channel Case 70 3.3 Subspace Identification of MIMO Channels 71 3.3.1 Rational Spaces and Polynomial Bases 72 3.3.2 The Structure of the Left Nullspace of a Sylvester Matrix 76 3.3.3 The Subspace Method 78 3.3.4 Advanced Results 82 3.4 Applications to the Blind Channel Estimation of CDMA Systems 84 3.4.1 Model Structure 84 3.4.2 The Structured Subspace Method: The Uplink Case 88 3.4.3 The Structured Subspace Method: The Downlink Case 89 3.5 Undermodeled Channel Identification 92 3.5.1 Example: Identifying a Significant Part of a Channel 99 3.5.2 Determining the Effective Impulse Response Length 100 Appendix 3.A 102 3.A.1 Proof of Theorem 1 103 3.A.2 Proof of Proposition 3 104 3.A.3 Proof of Theorem 4 105 3.A.4 Proof of Proposition 5 106 Bibliography 108
4 BLIND IDENTIFICATION AND EQUALIZATION OF CHANNELS DRIVEN BY COLORED SIGNALS 113 4.1 Introduction 114 4.2 FIR MIMO Channel 115 4.2.1 Original Model 115 4.2.2 Slide-Window Formulation 115 4.2.3 Noise Variance and Number of Input Signals 116 4.3 Identifiability Using SOS 117 4.3.1 Identifiability Conditions 117 4.3.2 Some Facts of Polynomial Matrices 118 4.3.3 Proof of the Conditions 120 4.3.4 When the Input is White 121 4.4 Blind Identification via Decorrelation 121 4.4.1 The Principle of the BID 121 4.4.2 Constructing the Decorrelators 126 4.4.3 Removing the GCD of Polynomials 128 4.4.4 Identification of the SIMO Channels 130 4.5 Final Remarks 135 Bibliography 135
5 OPTIMUM SUBSPACE METHODS 139 5.1 Introduction 139 5.2 Data Model and Notations 140 5.2.1 Scalar Valued Communication Systems 140 5.2.2 Multi Channel Communication Systems 141 5.2.3 A Stacked System Model 143 5.2.4 Correlation Matrices 145 5.2.5 Statistical Assumptions 147 5.3 Subspace Ideas and Notations 148 5.3.1 Basic Notations 149 5.4 Parameterizations 151 5.4.1 A Noise Subspace Parameterization 151 5.4.2 Selection Matrices 153 5.5 Estimation Procedure 154 5.5.1 The Signal Subspace Parameterization 155 5.5.2 The Noise Subspace Parameterization 156 5.6 Statistical Analysis 156 5.6.1 The Residual Covariance Matrices 157 5.6.2 The Parameter Covariance Matrices 159 5.7 Relation to Direction Estimation 161 5.8 Further Results for the Noise Subspace Parameterization 162 5.8.1 The Results 163 5.8.2 The Approach 163 5.9 Simulation Examples 164 5.10 Conclusions 171 Appendix 5.A 173 Bibliography 174
6 LINEAR PREDICTIVE ALGORITHMS FOR BLIND MULTICHANNEL IDENTIFICATION 179 6.1 Introduction 179 6.2 Channel Identification Based on Second Order Statistics: Problem Formulation 181 6.3 Linear Prediction Algorithm for Channel Identification 183 6.4 Outer-Product Decomposition Algorithm 185 6.5 Multi-step Linear Prediction 188 6.6 Channel Estimation by Linear Smoothing (Not Predicting) 189 6.7 Channel Estimation by Constrained Output Energy Minimization 192 6.8 Discussion 195 6.8.1 Channel Conditions 195 6.8.2 Data Conditions 196 6.8.3 Noise Effect 196 6.9 Simulation Results 197 6.10 Summary 198 Bibliography 207
7 SEMI-BLIND METHODS FOR FIR MULTICHANNEL ESTIMATION 211 7.1 Introduction 212 7.1.1 Training Sequence Based Methods and Blind Methods 212 7.1.2 Semi-Blind Principle 213 7.2 Problem Formulation 214 7.3 Classification lf Semi-Blind Methods 217 7.4 Identifiability Conditions for Semi-Blind Channel Estimation 218 7.4.1 Identifiability Definition 218 7.4.2 TS Based Channel Identifiability 219 7.4.3 Identifiability in the Deterministic Model 219 7.4.4 Identifiability in the Gaussian Model 222 7.5 Performance Measure: Cramer-Rao Bounds 224 7.6 Performance Optimization Issues 226 7.7 Optimal Semi-Blind Methods 227 7.8 Blind DML 229 7.8.1 Denoised IQML (DIQML) 230 7.8.2 Pseudo Quadratic ML (PQML) 231 7.9 Three Suboptimal DML Based Semi-Blind Criteria 232 7.9.1 Split of the Data 232 7.9.2 Least Squares-DML 232 7.9.3 Alternating Quadratic DML (AQ-DML) 233 7.9.4 Weighted-Least-Squares-PQML (WLS-PQML) 235 7.9.5 Wimulations 236 7.10 Semi-Blind Criteria as a Combination of a Blind and a TS Based Criteria 236 7.10.1 Semi-Blind SRM Example 237 7.10.2 Subspace Fitting Example 239 7.11 Performance of Semi-Blind Quadratic Criteria 242 7.11.1 MU and MK infinite 243 7.11.2 MU infinite, MK finite 243 7.11.3 Optimally Weighted Quadratic Criteria 247 7.12 Gaussian Methods 247 7.13 Conclusion 249 Bibliography 250
8 A GEOMETRICAL APPROACH TO BLIND SIGNAL ESTIMATION 255 8.1 Introduction 256 8.2 Design Criteria for Blind Estimators 258 8.2.1 The Constant Modulus Receiver 260 8.2.2 The Shalvi-Weinstein Receiver 261 8.3 The Signal Space Property and Equivalent Cost Functions 263 8.3.1 The Signal Space Property of CM Receivers 263 8.3.2 The Signal Space Property of SW Receivers 264 8.3.3 Equivalent Cost Functions 265 8.4 Geometrical Analysis of SW Receivers: Global Characterization 266 8.4.1 The Noiseless Case 268 8.4.2 The Noisy Case 270 8.4.3 Domains of Attraction of SW Receivers 275 8.5 Geometrical Analysis of SW Receivers: Local Characterizations 277 8.5.1 Local Characterization 277 8.5.2 MSE of CM Receivers 281 8.6 Conclusion and Bibliography Notes 282 8.6.1 Bibliography Notes 283 Appendix 8.A Proof of Theorem 5 285 Bibliography 288
9 LINEAR PRECODING FOR ESTIMATION AND EQUALIZATION OF FREQUENCY-SELECTIVE CHANNELS 291 9.1 System Model 293 9.2 Unifying Filterbank Precoders 296 9.3 FIR-ZF Equalizers 301 9.4 Jointly Optimal Precoder and Decoder Design 306 9.4.1 Zero-order Model 306 9.4.2 MMSE/ZF Coding 308 9.4.3 MMSE Solution wit Constrained Average Power 309 9.4.4 Constrained Power Maximum Information Rate Design 311 9.4.5 Comparison Between Optimal Designs 313 9.4.6 Asymptotic Performance 317 9.4.7 Numerical Examples 318 9.5 Blind Symbol Recovery 320 9.5.1 Blind Channel Estimation 322 9.5.2 Comparison with Other Blind Techniques 324 9.5.3 Statistical Efficiency 330 9.6 Conclusion 332 Bibliography 332
10. BLIND CHANNEL IDENTIFIABILITY WITH AN ARBITRARY LINEAR PRECODR 339 10.1 Introduction 339 10.2 Basic Theory of Polynomial Equations 344 10.2.1 Definition of Generic 344 10.2.2 General Properties of Polynomial Maps 344 10.2.3 Generic and Non-Generic Points 346 10.2.4 Invertibility Criteria 347 10.3 Inherent Scale Ambiguity 348 10.4 Weak Identifiability and the CRB 348 10.5 Arbitrary Linear Precoders 349 10.6 Zero Prefix Precoders 351 10.7 Geometric Interpretation of Precoding 354 10.7.1 Linear Precoders 354 10.7.2 Zero Prefix Precoders 355 10.8 Filter Banks 355 10.8.1 Algebraic Analysis of Filter Banks 357 10.8.2 Spectral Analysis of Filter Banks 358 10.9 Ambiguity Resistant Precoders 360 10.10 Symbolic Methods 361 10.11 Conclusion 362 Bibliography 363
11 CURRENT APPROACHES TO BLIND DECISION FEEDBACK EQUALIZATION 367 11.1 Introduction 367 11.2 Notation 370 11.3 Data Model 373 11.4 Wiener Filtering 374 11.4.1 Unconstrained Length MMSE Receivers 375 11.4.2 Constrained Length MMSE Receivers 377 11.4.3 Example: Constrained Versus Unconstrained Length Wiener Receivers 379 11.5 Blind Tracking Algorithms 380 11.5.1 DD-DFE 381 11.5.2 CMA-DFE 388 11.5.3 Algorithmic and Structural Modifications 389 11.5.4 Summary of Blind Tracking Algorithms 391 11.6 DFE Initialization Strategies 391 11.6.1 Generic Strategy 391 11.6.2 Multistage Equalization 395 11.6.3 CMA-IIR Initialization 397 11.6.4 Local Stability of Adaptive IIR Equalizers 398 11.6.5 Summary of Blind Initialization Strategies 399 11.7 Conclusion 400 Appendix 11.A Spectral Factorization 402 Appendix 11.B CL-MMSE-DFE 403 Appendix 11.C DD-DFE Local Convergence 405 Appendix 11.D Adaptive IIR Algorithm Updates 406 Appendix 11.E CMA-AR Local Stability 409 Bibliography 411
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