The relative motion between the transmitter and the receiver modifies the nonstationarity properties of the transmitted signal. In particular, the almost-cyclostationarity property exhibited by almost all modulated signals adopted in communications, radar, sonar, and telemetry can be transformed into more general kinds of nonstationarity. A proper statistical characterization of the received signal allows for the design of signal processing algorithms for detection, estimation, and classification that significantly outperform algorithms based on classical descriptions of signals.Generalizations of Cyclostationary Signal Processing addresses these issues and includes the following key features: 

• Presents the underlying theoretical framework, accompanied by details of their practical application, for the mathematical models of generalized almost-cyclostationary processes and spectrally correlated processes; two classes of signals finding growing importance in areas such as mobile communications, radar and sonar.
• Explains second- and higher-order characterization of nonstationary stochastic processes in time and frequency domains.
• Discusses continuous- and discrete-time estimators of statistical functions of generalized almost-cyclostationary processes and spectrally correlated processes.
• Provides analysis of mean-square consistency and asymptotic Normality of statistical function estimators.
• Offers extensive analysis of Doppler channels owing to the relative motion between transmitter and receiver and/or surrounding scatterers.
• Performs signal analysis using both the classical stochastic-process approach and the functional approach, where statistical functions are built starting from a single function of time.

Dedication iii

Acknowledgements xiii

Introduction xv

1 Background 1

1.1 Second-Order Characterization of Stochastic Processes 1

1.1.1 Time-Domain Characterization 1

1.1.2 Spectral-Domain Characterization 2

1.1.3 Time-Frequency Characterization 4

1.1.4 Wide-Sense Stationary Processes 5

1.1.5 Evolutionary Spectral Analysis 5

1.1.6 Discrete-Time Processes 7

1.1.7 Linear Time-Variant Transformations 8

1.2 Almost-Periodic Functions 10

1.2.1 Uniformly Almost-Periodic Functions 11

1.2.2 AP Functions in the Sense of Stepanov,Weyl, and Besicovitch 12

1.2.3 Weakly AP Functions in the Sense of Eberlein 13

1.2.4 Pseudo AP Functions 14

1.2.5 AP Functions in the Sense of Hartman and Ryll-Nardzewski 15

1.2.6 AP Functions Defined on Groups and with Values in Banach and Hilbert Spaces 16

1.2.7 AP Functions in Probability 16

1.2.8 AP Sequences 17

1.2.9 AP Sequences in Probability 18

1.3 Almost-Cyclostationary Processes 18

1.3.1 Second-OrderWide-Sense Statistical Characterization 18

1.3.2 Jointly ACS Signals 20

1.3.3 LAPTV Systems 24

1.3.4 Products of ACS Signals 27

1.3.5 Cyclic Statistics of Communications Signals 29

1.3.6 Higher-Order Statistics 30

1.3.7 Cyclic Statistic Estimators 32

1.3.8 Discrete-Time ACS Signals 32

1.3.9 Sampling of ACS Signals 33

1.3.10 Multirate Processing of Discrete-Time ACS Signals 37

1.3.11 Applications 37

1.4 Some Properties of Cumulants 38

1.4.1 Cumulants and Statistical Independence 38

1.4.2 Cumulants of Complex Random Variables and Joint Complex Normality 392 Generalized Almost-Cyclostationary Processes 43

2.1 Introduction 43

2.2 Characterization of GACS Stochastic Processes 47

2.2.1 Strict-Sense Statistical Characterization 48

2.2.2 Second-OrderWide-Sense Statistical Characterization 49

2.2.3 Second-Order Spectral Characterization 59

2.2.4 Higher-Order Statistics 61

2.2.5 Processes with Almost-Periodic Covariance 65

2.2.6 Motivations and Examples 66

2.3 Linear Time-Variant Filtering of GACS Processes 70

2.4 Estimation of the Cyclic Cross-Correlation Function 72

2.4.1 The Cyclic Cross-Correlogram 72

2.4.2 Mean-Square Consistency of the Cyclic Cross-Correlogram 76

2.4.3 Asymptotic Normality of the Cyclic Cross-Correlogram 80

2.5 Sampling of GACS Processes 84

2.6 Discrete-Time Estimator of the Cyclic Cross-Correlation Function 87

2.6.1 Discrete-Time Cyclic Cross-Correlogram 87

2.6.2 Asymptotic Results  91

2.6.3 Asymptotic Results  95

2.6.4 Concluding Remarks 102

2.7 Numerical Results 104

2.7.1 Aliasing in Cycle-Frequency Domain 105

2.7.2 Simulation Setup 105

2.7.3 Cyclic Correlogram Analysis with Varying N 105

2.7.4 Cyclic Correlogram Analysis with Varying N and T 106

2.7.5 Discussion 111

2.7.6 Conjecturing the Nonstationarity Type of the Continuous-Time Signal 114

2.7.7 LTI Filtering of GACS Signals 116

2.8 Summary 116

3 Complements and Proofs on Generalized Almost-Cyclostationary Processes 123

3.1 Proofs for Section 2.2.2 “Second-OrderWide-Sense Statistical Characterization” 123

3.2 Proofs for Section 2.2.3 “Second-Order Spectral Characterization” 125

3.3 Proofs for Section 2.3 “Linear Time-Variant Filtering of GACS Processes” 129

3.4 Proofs for Section 2.4.1 “The Cyclic Cross-Correlogram” 131

3.5 Proofs for Section 2.4.2 “Mean-Square Consistency of the Cyclic Cross-Correlogram” 136

3.6 Proofs for Section 2.4.3 “Asymptotic Normality of the Cyclic Cross-Correlogram” 147

3.7 Conjugate Covariance 150

3.8 Proofs for Section 2.5 “Sampling of GACS Processes” 151

3.9 Proofs for Section 2.6.1 “Discrete-Time Cyclic Cross-Correlogram” 152

3.10 Proofs for Section 2.6.2 “Asymptotic Results as 158

3.11 Proofs for Section 2.6.3 “Asymptotic Results as 168

3.12 Proofs for Section 2.6.4 “Concluding Remarks” 176

3.13 Discrete-Time and Hybrid Conjugate Covariance 177

4 Spectrally Correlated Processes 181

4.1 Introduction 182

4.2 Characterization of SC Stochastic Processes 186

4.2.1 Second-Order Characterization 186

4.2.2 Relationship among ACS, GACS, and SC Processes 194

4.2.3 Higher-Order Statistics 195

4.2.4 Motivating Examples 200

4.3 Linear Time-Variant Filtering of SC Processes 205

4.3.1 FOT-Deterministic Linear Systems 205

4.3.2 SC Signals and FOT-Deterministic Systems 207

4.4 The Bifrequency Cross-Periodogram 208

4.5 Measurement of Spectral Correlation – Unknown Support Curves 215

4.6 The Frequency-Smoothed Cross-Periodogram 222

4.7 Measurement of Spectral Correlation – Known Support Curves 225

4.7.1 Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram 225

4.7.2 Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram 229

4.7.3 Final Remarks 231

4.8 Discrete-Time SC Processes 233

4.9 Sampling of SC Processes 236

4.9.1 Band-Limitedness Property 237

4.9.2 Sampling Theorems 239

4.9.3 Illustrative Examples 243

4.10 Multirate Processing of Discrete-Time Jointly SC Processes 256

4.10.1 Expansion 257

4.10.2 Sampling 260

4.10.3 Decimation 262

4.10.4 Expansion and Decimation 265

4.10.5 Strictly Band-Limited SC Processes 267

4.10.6 Interpolation Filters 268

4.10.7 Decimation Filters 270

4.10.8 Fractional Sampling Rate Converters 271

4.11 Discrete-Time Estimators of the Spectral Cross-Correlation Density 272

4.12 Numerical Results 273

4.12.1 Simulation Setup 273

4.12.2 Unknown Support Curves 273

4.12.3 Known Support Curves 274

4.13 Spectral Analysis with Nonuniform Frequency Resolution 281

4.14 Summary 2865 Complements and Proofs on Spectrally Correlated Processes 291

5.1 Proofs for Section 4.2 “Spectrally Correlated Stochastic Processes” 291

5.2 Proofs for Section 4.4 “The Bifrequency Cross-Periodogram” 292

5.3 Proofs for Section 4.5 “Measurement of Spectral Correlation – Unknown Support Curves” 298

5.4 Proofs for Section 4.6 “The Frequency-Smoothed Cross-Periodogram” 306

5.5 Proofs for Section 4.7.1 “Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram” 309

5.6 Proofs for Section 4.7.2 “Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram” 325

5.7 Alternative Bounds 333

5.8 Conjugate Covariance 334

5.9 Proofs for Section 4.8 “Discrete-Time SC Processes” 337

5.10 Proofs for Section 4.9 “Sampling of SC Processes” 339

5.11 Proofs for Section 4.10 “Multirate Processing of Discrete-Time Jointly SC Processes” 3426 Functional Approach for Signal Analysis 355

6.1 Introduction 355

6.2 Relative Measurability 356

6.2.1 Relative Measure of Sets 356

6.2.2 Relatively Measurable Functions 357

6.2.3 Jointly Relatively Measurable Functions 358

6.2.4 Conditional Relative Measurability and Independence 360

6.2.5 Examples 361

6.3 Almost-Periodically Time-Variant Model 361

6.3.1 Almost-Periodic Component Extraction Operator 361

6.3.2 Second-Order Statistical Characterization 363

6.3.3 Spectral Line Regeneration 365

6.3.4 Spectral Correlation 366

6.3.5 Statistical Function Estimators 367

6.3.6 Sampling, Aliasing, and Cyclic Leakage 369

6.3.7 FOT-Deterministic Systems 371

6.3.8 FOT-Deterministic Linear Systems 372

6.4 Nonstationarity Classification in the Functional Approach 374

6.5 Proofs of FOT Counterparts of Some Results on ACS and GACS Signals 3757 Applications to Mobile Communications and Radar/Sonar 381

7.1 Physical Model for the Wireless Channel 381

7.1.1 Assumptions on the Propagation Channel 381

7.1.2 Stationary TX, Stationary RX 382

7.1.3 Moving TX, Moving RX 383

7.1.4 Stationary TX, Moving RX 387

7.1.5 Moving TX, Stationary RX 388

7.1.6 Reflection on Point Scatterer 388

7.1.7 Stationary TX, Reflection on Point Moving Scatterer, Stationary RX (Stationary Bistatic Radar) 390

7.1.8 (Stationary)Monostatic Radar 391

7.1.9 Moving TX, Reflection on a Stationary Scatterer, Moving RX 392

7.2 Constant Velocity Vector 393

7.2.1 Stationary TX, Moving RX 393

7.2.2 Moving TX, Stationary RX 394

7.3 Constant Relative Radial Speed 395

7.3.1 Moving TX, Moving RX 395

7.3.2 Stationary TX, Moving RX 398

7.3.3 Moving TX, Stationary RX 401

7.3.4 Stationary TX, Reflection on a Moving Scatterer, Stationary RX (Stationary Bistatic Radar) 404

7.3.5 (Stationary)Monostatic Radar 406

7.3.6 Moving TX, Reflection on a Stationary Scatterer, Moving RX 406

7.3.7 Non synchronized TX and RX oscillators 407

7.4 Constant Relative Radial Acceleration 407

7.4.1 Stationary TX, Moving RX 408

7.4.2 Moving TX, Stationary RX 408

7.5 Transmitted Signal: Narrow-Band Condition 409

7.5.1 Constant Relative Radial Speed 411

7.5.2 Constant Relative Radial Acceleration 414

7.6 Multipath Doppler Channel 416

7.6.1 Constant Relative Radial Speeds – Discrete Scatterers 416

7.6.2 Continuous Scatterer 416

7.7 Spectral Analysis of Doppler-Stretched Signals – Constant Radial Speed 417

7.7.1 Second-Order Statistics (Continuous-Time) 417

7.7.2 Multipath Doppler Channel 422

7.7.3 Doppler-Stretched Signal (Discrete-Time) 427

7.7.4 Simulation of Discrete-Time Doppler-Stretched Signals 430

7.7.5 Second-Order Statistics (Discrete-Time) 432

7.7.6 Illustrative Examples 437

7.7.7 Concluding Remarks 443

7.8 Spectral Analysis of Doppler-Stretched Signals – Constant Relative Radial Acceleration 448

7.8.1 Second-Order Statistics (Continuous-Time) 449

7.9 Other Models of Time-Varying Delays 452

7.9.1 Taylor Series Expansion of Range and Delay 452

7.9.2 Periodically Time-Variant Delay 454

7.9.3 Periodically Time-Variant Carrier Frequency 454

7.10 Proofs 4558 Bibliographic Notes 463

8.1 Almost-Periodic Functions 463

8.2 Cyclostationary Signals 463

8.3 Generalizations of Cyclostationarity 464

8.4 Other Nonstationary Signals 464

8.5 Functional Approach and Generalized Harmonic Analysis 464

8.6 Linear Time-Variant Processing 465

8.7 Sampling 465

8.8 Complex Random Variables, Signals, and Systems 465

8.9 Stochastic Processes 465

8.10 Mathematics 466

8.11 Signal Processing and Communications 466

References 467

List of Abbreviations 475