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1 CHANNEL ESTIMATION AND EQUALIZATION USING HIGHER-ORDER STATISTICS 1

1.1 Introduction 1

1.2 Single-User Systems:Baud Rate Sampling 4

1.2.1 Cumulant Matching 4

1.2.2 Inverse Filter Criteria 8

1.2.3 Equation Error Formulations 8

1.2.4 Simulation Examples 8

1.3 Single-User Systems:Fractional Sampling 12

1.3.1 Cumulant Matching 13

1.3.2 Simulation Example 20

1.4 Multi-user Systems 24

1.4.1 Inverse Filter Criteria 26

1.4.2 Cumulant Matching 28

1.4.3 Simulation Examples 31

1.5 Concluding Remarks 35

Bibliography 37

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

3.4.1 Model Structure 84

3.4 Applications to the Blind Channel Estimation of CDMA Systems 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

4.2.1 Original Model 115

4.2.2 Slide-Window Formulation 115

4.2 FIR MIMO Channel 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

Bibliography 135

4.5 Final Remarks 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 Minirmization 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

7.1.1 Training Sequence Based Methods and Blind Methods 212

7.1 Introduction 212

7.1.2 Semi-Blind Principle 213

7.2 Problem Formulation 214

7.3 Classification of 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:Cramér-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 Simulations 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

7.12 Gaussian Methods 247

7.11.3 Optimally Weighted Quadratic Criteria 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 with 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 PRECODER 339

10.1 Introduction 339

10.2.2 General Properties of Polynomial Maps 344

10.2 Basic Theory of Polynomial Equations 344

10.2.1 Definition of Generic 344

10.2.3 Generic and Non-Generic Points 346

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 EQU ALIZATION 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