Generalizations of Cyclostationary Signal Processing

Antonio Napolitano, University of Napoli "Parthenope", ItalyDr. Napolitano is currently Professor of Telecommunications at the University of Napoli "Parthenope". He is a Senior Member of IEEE. He has been a member of various journal editorial boards such as Signal Processing (Elsevier) and Journal of Electrical and Computer Engineering (Hindawi). He is the principal investigator of a NATO Grant on Communications Security. Recent awards include 2008 Most Cited Paper Award from Elsevier for a paper co-authored with W. A. Gardner and L. Paura entitled "Cyclostationarity: Half a century of research" (Signal Processing, vol. 86, April 2006), the most cited article in the last 5 years on Signal Processing (Elsevier); and in the 10 top most downloaded articles in the journal. Dr. Napolitano has held visiting appointments including: University Jeann Monnet, France; University of South Australia and Centro de Investigacion en Matematicas (CIMAT), Mexico. His research interests include statistical signal processing, system identification, the theory of higher order statistics of nonstationary signals and wireless systems.

• Dedication iiiAcknowledgements xiiiIntroduction xv1 Background 11.1 Second-Order Characterization of Stochastic Processes 11.1.1 Time-Domain Characterization 11.1.2 Spectral-Domain Characterization 21.1.3 Time-Frequency Characterization 41.1.4 Wide-Sense Stationary Processes 51.1.5 Evolutionary Spectral Analysis 51.1.6 Discrete-Time Processes 71.1.7 Linear Time-Variant Transformations 81.2 Almost-Periodic Functions 101.2.1 Uniformly Almost-Periodic Functions 111.2.2 AP Functions in the Sense of Stepanov,Weyl, and Besicovitch 121.2.3 Weakly AP Functions in the Sense of Eberlein 131.2.4 Pseudo AP Functions 141.2.5 AP Functions in the Sense of Hartman and Ryll-Nardzewski 151.2.6 AP Functions Defined on Groups and with Values in Banach and Hilbert Spaces 161.2.7 AP Functions in Probability 161.2.8 AP Sequences 171.2.9 AP Sequences in Probability 181.3 Almost-Cyclostationary Processes 181.3.1 Second-OrderWide-Sense Statistical Characterization 181.3.2 Jointly ACS Signals 201.3.3 LAPTV Systems 241.3.4 Products of ACS Signals 271.3.5 Cyclic Statistics of Communications Signals 291.3.6 Higher-Order Statistics 301.3.7 Cyclic Statistic Estimators 321.3.8 Discrete-Time ACS Signals 321.3.9 Sampling of ACS Signals 331.3.10 Multirate Processing of Discrete-Time ACS Signals 371.3.11 Applications 371.4 Some Properties of Cumulants 381.4.1 Cumulants and Statistical Independence 381.4.2 Cumulants of Complex Random Variables and Joint Complex Normality 392 Generalized Almost-Cyclostationary Processes 432.1 Introduction 432.2 Characterization of GACS Stochastic Processes 472.2.1 Strict-Sense Statistical Characterization 482.2.2 Second-OrderWide-Sense Statistical Characterization 492.2.3 Second-Order Spectral Characterization 592.2.4 Higher-Order Statistics 612.2.5 Processes with Almost-Periodic Covariance 652.2.6 Motivations and Examples 662.3 Linear Time-Variant Filtering of GACS Processes 702.4 Estimation of the Cyclic Cross-Correlation Function 722.4.1 The Cyclic Cross-Correlogram 722.4.2 Mean-Square Consistency of the Cyclic Cross-Correlogram 762.4.3 Asymptotic Normality of the Cyclic Cross-Correlogram 802.5 Sampling of GACS Processes 842.6 Discrete-Time Estimator of the Cyclic Cross-Correlation Function 872.6.1 Discrete-Time Cyclic Cross-Correlogram 872.6.2 Asymptotic Results  912.6.3 Asymptotic Results  952.6.4 Concluding Remarks 1022.7 Numerical Results 1042.7.1 Aliasing in Cycle-Frequency Domain 1052.7.2 Simulation Setup 1052.7.3 Cyclic Correlogram Analysis with Varying N 1052.7.4 Cyclic Correlogram Analysis with Varying N and T 1062.7.5 Discussion 1112.7.6 Conjecturing the Nonstationarity Type of the Continuous-Time Signal 1142.7.7 LTI Filtering of GACS Signals 1162.8 Summary 1163 Complements and Proofs on Generalized Almost-Cyclostationary Processes 1233.1 Proofs for Section 2.2.2 “Second-OrderWide-Sense Statistical Characterization” 1233.2 Proofs for Section 2.2.3 “Second-Order Spectral Characterization” 1253.3 Proofs for Section 2.3 “Linear Time-Variant Filtering of GACS Processes” 1293.4 Proofs for Section 2.4.1 “The Cyclic Cross-Correlogram” 1313.5 Proofs for Section 2.4.2 “Mean-Square Consistency of the Cyclic Cross-Correlogram” 1363.6 Proofs for Section 2.4.3 “Asymptotic Normality of the Cyclic Cross-Correlogram” 1473.7 Conjugate Covariance 1503.8 Proofs for Section 2.5 “Sampling of GACS Processes” 1513.9 Proofs for Section 2.6.1 “Discrete-Time Cyclic Cross-Correlogram” 1523.10 Proofs for Section 2.6.2 “Asymptotic Results as 1583.11 Proofs for Section 2.6.3 “Asymptotic Results as 1683.12 Proofs for Section 2.6.4 “Concluding Remarks” 1763.13 Discrete-Time and Hybrid Conjugate Covariance 1774 Spectrally Correlated Processes 1814.1 Introduction 1824.2 Characterization of SC Stochastic Processes 1864.2.1 Second-Order Characterization 1864.2.2 Relationship among ACS, GACS, and SC Processes 1944.2.3 Higher-Order Statistics 1954.2.4 Motivating Examples 2004.3 Linear Time-Variant Filtering of SC Processes 2054.3.1 FOT-Deterministic Linear Systems 2054.3.2 SC Signals and FOT-Deterministic Systems 2074.4 The Bifrequency Cross-Periodogram 2084.5 Measurement of Spectral Correlation – Unknown Support Curves 2154.6 The Frequency-Smoothed Cross-Periodogram 2224.7 Measurement of Spectral Correlation – Known Support Curves 2254.7.1 Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram 2254.7.2 Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram 2294.7.3 Final Remarks 2314.8 Discrete-Time SC Processes 2334.9 Sampling of SC Processes 2364.9.1 Band-Limitedness Property 2374.9.2 Sampling Theorems 2394.9.3 Illustrative Examples 2434.10 Multirate Processing of Discrete-Time Jointly SC Processes 2564.10.1 Expansion 2574.10.2 Sampling 2604.10.3 Decimation 2624.10.4 Expansion and Decimation 2654.10.5 Strictly Band-Limited SC Processes 2674.10.6 Interpolation Filters 2684.10.7 Decimation Filters 2704.10.8 Fractional Sampling Rate Converters 2714.11 Discrete-Time Estimators of the Spectral Cross-Correlation Density 2724.12 Numerical Results 2734.12.1 Simulation Setup 2734.12.2 Unknown Support Curves 2734.12.3 Known Support Curves 2744.13 Spectral Analysis with Nonuniform Frequency Resolution 2814.14 Summary 2865 Complements and Proofs on Spectrally Correlated Processes 2915.1 Proofs for Section 4.2 “Spectrally Correlated Stochastic Processes” 2915.2 Proofs for Section 4.4 “The Bifrequency Cross-Periodogram” 2925.3 Proofs for Section 4.5 “Measurement of Spectral Correlation – Unknown Support Curves” 2985.4 Proofs for Section 4.6 “The Frequency-Smoothed Cross-Periodogram” 3065.5 Proofs for Section 4.7.1 “Mean-Square Consistency of the Frequency-Smoothed Cross-Periodogram” 3095.6 Proofs for Section 4.7.2 “Asymptotic Normality of the Frequency-Smoothed Cross-Periodogram” 3255.7 Alternative Bounds 3335.8 Conjugate Covariance 3345.9 Proofs for Section 4.8 “Discrete-Time SC Processes” 3375.10 Proofs for Section 4.9 “Sampling of SC Processes” 3395.11 Proofs for Section 4.10 “Multirate Processing of Discrete-Time Jointly SC Processes” 3426 Functional Approach for Signal Analysis 3556.1 Introduction 3556.2 Relative Measurability 3566.2.1 Relative Measure of Sets 3566.2.2 Relatively Measurable Functions 3576.2.3 Jointly Relatively Measurable Functions 3586.2.4 Conditional Relative Measurability and Independence 3606.2.5 Examples 3616.3 Almost-Periodically Time-Variant Model 3616.3.1 Almost-Periodic Component Extraction Operator 3616.3.2 Second-Order Statistical Characterization 3636.3.3 Spectral Line Regeneration 3656.3.4 Spectral Correlation 3666.3.5 Statistical Function Estimators 3676.3.6 Sampling, Aliasing, and Cyclic Leakage 3696.3.7 FOT-Deterministic Systems 3716.3.8 FOT-Deterministic Linear Systems 3726.4 Nonstationarity Classification in the Functional Approach 3746.5 Proofs of FOT Counterparts of Some Results on ACS and GACS Signals 3757 Applications to Mobile Communications and Radar/Sonar 3817.1 Physical Model for the Wireless Channel 3817.1.1 Assumptions on the Propagation Channel 3817.1.2 Stationary TX, Stationary RX 3827.1.3 Moving TX, Moving RX 3837.1.4 Stationary TX, Moving RX 3877.1.5 Moving TX, Stationary RX 3887.1.6 Reflection on Point Scatterer 3887.1.7 Stationary TX, Reflection on Point Moving Scatterer, Stationary RX (Stationary Bistatic Radar) 3907.1.8 (Stationary)Monostatic Radar 3917.1.9 Moving TX, Reflection on a Stationary Scatterer, Moving RX 3927.2 Constant Velocity Vector 3937.2.1 Stationary TX, Moving RX 3937.2.2 Moving TX, Stationary RX 3947.3 Constant Relative Radial Speed 3957.3.1 Moving TX, Moving RX 3957.3.2 Stationary TX, Moving RX 3987.3.3 Moving TX, Stationary RX 4017.3.4 Stationary TX, Reflection on a Moving Scatterer, Stationary RX (Stationary Bistatic Radar) 4047.3.5 (Stationary)Monostatic Radar 4067.3.6 Moving TX, Reflection on a Stationary Scatterer, Moving RX 4067.3.7 Non synchronized TX and RX oscillators 4077.4 Constant Relative Radial Acceleration 4077.4.1 Stationary TX, Moving RX 4087.4.2 Moving TX, Stationary RX 4087.5 Transmitted Signal: Narrow-Band Condition 4097.5.1 Constant Relative Radial Speed 4117.5.2 Constant Relative Radial Acceleration 4147.6 Multipath Doppler Channel 4167.6.1 Constant Relative Radial Speeds – Discrete Scatterers 4167.6.2 Continuous Scatterer 4167.7 Spectral Analysis of Doppler-Stretched Signals – Constant Radial Speed 4177.7.1 Second-Order Statistics (Continuous-Time) 4177.7.2 Multipath Doppler Channel 4227.7.3 Doppler-Stretched Signal (Discrete-Time) 4277.7.4 Simulation of Discrete-Time Doppler-Stretched Signals 4307.7.5 Second-Order Statistics (Discrete-Time) 4327.7.6 Illustrative Examples 4377.7.7 Concluding Remarks 4437.8 Spectral Analysis of Doppler-Stretched Signals – Constant Relative Radial Acceleration 4487.8.1 Second-Order Statistics (Continuous-Time) 4497.9 Other Models of Time-Varying Delays 4527.9.1 Taylor Series Expansion of Range and Delay 4527.9.2 Periodically Time-Variant Delay 4547.9.3 Periodically Time-Variant Carrier Frequency 4547.10 Proofs 4558 Bibliographic Notes 4638.1 Almost-Periodic Functions 4638.2 Cyclostationary Signals 4638.3 Generalizations of Cyclostationarity 4648.4 Other Nonstationary Signals 4648.5 Functional Approach and Generalized Harmonic Analysis 4648.6 Linear Time-Variant Processing 4658.7 Sampling 4658.8 Complex Random Variables, Signals, and Systems 4658.9 Stochastic Processes 4658.10 Mathematics 4668.11 Signal Processing and Communications 466References 467List of Abbreviations 475

“This book is written both for advanced readers with the background of graduate students in engineering and for specialists (e.g., mathematicians).”  (Zentralblatt MATH, 1 May 2013)