Generalizations of Cyclostationary Signal Processing
Spectral Analysis and Applications
Wiley - IEEE
1. Edition October 2012
492 Pages, Hardcover
Wiley & Sons Ltd
Short Description
Addressing the problem of consistent statistical-function estimation for two classes (GACS and SC) of nonstationary processes, this book is organized for readers with different prerequisites. The first three chapters are intended for readers to grasp the main ideas, and contain results in the form of theorems with sketches of proofs and illustrative examples. The last chapters present two-part mathematical proofs: the first part consists of formal manipulations, aimed at advanced readers such as engineering graduate students; the second consists of justification of the formal manipulations, for specialists such as mathematicians.
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.
Acknowledgements xiii
Introduction xv
1 Background 1
1.1 Second-Order Characterization of Stochastic Processes 1
1.2 Almost-Periodic Functions 10
1.3 Almost-Cyclostationary Processes 18
1.4 Some Properties of Cumulants 38
2.1 Introduction 43
2.2 Characterization of GACS Stochastic Processes 47
2.3 Linear Time-Variant Filtering of GACS Processes 70
2.4 Estimation of the Cyclic Cross-Correlation Function 72
2.5 Sampling of GACS Processes 84
2.6 Discrete-Time Estimator of the Cyclic Cross-Correlation Function 87
2.7 Numerical Results 104
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
4.3 Linear Time-Variant Filtering of SC Processes 205
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
4.8 Discrete-Time SC Processes 233
4.9 Sampling of SC Processes 236
4.10 Multirate Processing of Discrete-Time Jointly SC Processes 256
4.11 Discrete-Time Estimators of the Spectral Cross-Correlation Density 272
4.12 Numerical Results 273
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
6.3 Almost-Periodically Time-Variant Model 361
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
7.2 Constant Velocity Vector 393
7.3 Constant Relative Radial Speed 395
7.4 Constant Relative Radial Acceleration 407
7.5 Transmitted Signal: Narrow-Band Condition 409
7.6 Multipath Doppler Channel 416
7.7 Spectral Analysis of Doppler-Stretched Signals - Constant Radial Speed 417
7.8 Spectral Analysis of Doppler-Stretched Signals - Constant Relative Radial Acceleration 448
7.9 Other Models of Time-Varying Delays 452
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
