1.1 Introduction
1.2 Stochastic Processes
1.2.1 Continuous-Time Processes
1.2.2 Discrete-Time Processes
1.3 Complex Signals
1.4 Nonzero-Mean Signals
1.5 Jointly ACS Signals
1.5.1 Symmetry Relationships
1.6 Representations by Stationary Components
1.6.1 Continuous-Time
1.6.2 Discrete-Time
1.7 Special Topics
1.8 Summary
1.9 Proofs
2.1 Introduction
2.2 Fraction-of-Time Probability
2.2.1 Continuous Time
2.2.2 Discrete Time
2.3 Almost-Cyclostationary Time Series
2.3.1 Continuous-Time
2.3.2 Discrete-Time
2.3.3 Nonstationarity Classification in the FOT Approach
2.4 Stochastic Versus Fraction-of-Time Approach
2.4.1 AP Component of PAM Time Series
2.5 Summary
2.6 Proofs
3.1 Introduction
3.2 Linear Filtering
3.2.1 Linear Almost Periodically Time Variant Systems
3.2.2 Input/Output Relations in Terms of Cyclic Statistics
3.3 Products of ACS Signals
3.4 Supports of Cyclic Spectra of Band Limited Signals
3.5 Rice’s Representation
3.6 Sampling and Aliasing
3.6.1 Cyclic Statistics of the Sampled Signal
3.6.2 Sampled Cyclostationary Signal
3.6.3 Bandpass Sampling
3.7 Multirate Processing of Discrete-Time ACS Signals
3.7.1 Expansion (Upsampling)
3.7.2 Sampling
3.7.3 Decimation (Downsampling)
3.8 Special Topics
3.9 Summary
3.10 Proofs
4.1 Introduction
4.2 Continuous-Time Signals
4.2.1 Moments
4.2.2 Cumulants
4.2.3 Moments versus Cumulants
4.3 Discrete-Time Signals
4.4 Input/Output Relations for MIMO LAPTV Systems
4.4.1 Continuous-Time
4.4.2 Discrete-Time
4.5 Sampling
4.6 Rice’s Representation
4.7 Higher-Order Hybrid Temporal-Spectral Cyclic Statistics
4.8 Stochastic Processes
4.9 Developments and Applications
4.10 Summary
4.11 Proofs
5.1 Introduction
5.2 Second-Order Cyclic Statistic Estimators
5.2.1 Cyclic Cross-Correlogram
5.2.2 Cyclic Cross-Periodogram
5.2.3 Frequency-Smoothed Cyclic Cross-Periodogram
5.2.4 Time-Smoothed Cyclic Cross-Periodogram
5.2.5 Median-Filtering Based Smoothing
5.2.6 Bifrequency Cross-Spectrum Density Estimation
5.3 Supplementary Analysis
5.3.1 Time Versus Frequency Smoothing
5.3.2 Alternative Assumptions
5.3.3 Cyclic Cross-Spectral Analysis
5.3.4 Cycle Leakage
5.3.5 Combined Effects of Aliasing and Cycle Leakage
5.4 Implementation of Cyclic Statistic Estimators
5.4.1 Cyclic Cross-Correlogram
5.4.2 Frequency-Smoothed Cyclic Cross-Periodogram
5.4.3 Time-Smoothed Cyclic Cross-Periodogram
5.4.4 Computationally Efficient Estimators
5.4.5 Matlab/Octave Code for Cyclic Spectral Analysis
5.4.6 Numerical Results
5.5 Estimators with Estimated Cycle Frequencies
5.5.1 Cyclic Correlogram
5.5.2 Frequency-Smoothed Cyclic Periodogram
5.6 Statistical Function Estimators in the Functional Approach
5.7 Higher-Order Cyclic Statistic Estimators
5.7.1 Estimators of Time-Domain Statistics
5.7.2 Estimators of Frequency-Domain Statistics
5.7.3 Estimators Based on Median Filtering
5.7.4 Estimators of Hybrid Temporal-Spectral Statistics
5.7.5 Numerical Results
5.8 Summary
5.9 Proofs
6.1 Introduction
6.2 Finite-Energy Signals: Correlations and Spectra
6.3 Spectrogram
6.3.1 Expected Value of Spectrogram of ACS Signals
6.4 Quadratic TFDs
6.4.1 Expected Values of Quadratic TFDs of ACS Signals
6.5 Filtered Quadratic TFDs
6.5.1 Kernels
6.6 Filtered Quadratic TFDs of ACS Signals
6.6.1 Expected Values of Filtered Quadratic TFDs
6.7 Cohen’s Class
6.8 Proofs
7.1 Introduction
7.2 Double Side-Band Amplitude-Modulated Signal
7.3 Pulse-Amplitude-Modulated Signal
7.4 Direct-Sequence Spread-Spectrum Signal
7.5 Higher-Order Cyclic Spectra of Modulated Signals
7.5.1 PAM Signals
7.5.2 QAM Signals
7.5.3 DSB-AM Signals
7.5.4 SSB Signals
7.5.5 ASK Signals
7.6 Cyclic Spectral Analysis of Man-Made Signals
7.7 Proofs
8.1 Introduction
8.2 Spectral Line Regeneration
8.3 Maximum Likelihood Detection and Source Location
8.4 Detection of Signals Exhibiting Cyclostationarity
8.4.1 ACS Signal Detection
8.4.2 Statistical Test for Presence of Cyclostationarity
8.4.3 Performance Analysis
8.5 Detection of Signals Exhibiting Spectral Correlation
8.6 Statistical Test for Presence of Spectral Coherence
8.7 Subsampling-Based Significance Test
8.8 Robust Detectors
8.9 Higher-Order Statistic Based Detectors
8.10 Cycle Frequency Estimation
8.11 Detection of a Moving Source
8.12 Spectrum Sensing and Signal Classification
8.12.1 Spectrum Sensing
8.12.2 Cyclic Spectral Analysis of Man-Made Signals
8.12.3 Hiding the Modulation Format
8.13 Summary
9.1 Introduction
9.2 Signal Selectivity Property
9.3 Cyclic Wiener Filtering
9.4 Synchronization
9.5 System Identification
9.5.1 General Aspects
9.5.2 LTI-System Identification by Noisy-Measurements
9.5.3 Blind LTI-System Identification and Equalization
9.5.4 Nonlinear-System Identification
9.6 Applications
9.6.1 Signal Parameter Estimation
9.6.2 Source Location
9.6.3 Beamforming
9.6.4 Source Separation
9.6.5 Miscellaneous
9.7 Performance of Cyclostationarity-Based Algorithms
9.8 Summary
9.9 Proofs
10.1 PARMA Systems
10.2 Compressive Sensing
10.3 Random Fields
10.4 Level Crossing
10.5 Applications to Systems, Circuits, and Control
10.6 Applications to Acoustics and Mechanics
10.7 Applications to Econometrics
10.8 Applications to Biology
10.9 Other Applications
11.1 Introduction
11.2 Doppler Effect on ACS Signals
11.3 Mismatch to the ACS Model
11.4 Irregular Statistical Cyclicity
12.1 Introduction
12.2 Strict-Sense Characterization
12.3 Second-Order Characterization
12.3.1 Time Domain
12.3.2 Frequency Domain
12.4 Discrete-Time Processes
12.5 Jointly GACS Processes
12.6 Estimation of the Cyclic Cross-Correlation Function
12.6.1 Continuous Time
12.6.2 Discrete Time
12.7 Examples and Applications
12.7.1 Constant Relative Radial Acceleration
12.7.2 ACS Model Mismatch
12.7.3 Signal Detection
12.8 Summary
13.1 Introduction
13.2 Second-Order Characterization
13.3 Discrete-Time Processes
13.4 Jointly SC Processes .
13.5 Estimation of the Spectral Cross-Correlation Density
13.5.1 Unknown Support Curves
13.5.2 Known Support Curves
13.6 Examples and Applications
13.6.1 Multipath Doppler Channel
13.6.2 Moving Source Location
13.6.3 Signal Detection
13.6.4 Fractional Brownian Motion
13.6.5 Multirate Processing
13.6.6 Nonuniform Frequency Spacing
13.7 Summary
14.1 Introduction
14.2 Second-Order Characterization
14.2.1 LTV Filtering of ACS Processes
14.2.2 Modulated Cyclical Processes
14.3 Amplitude-Modulated Time-Warped ACS Processes
14.3.1 Probabilistic Characterization
14.3.2 Time Warping
14.3.3 Estimation
14.3.4 De-Warping
14.4 Cyclostationarity Restoral
14.5 Monolateral ACS Signals
14.6 Electrocardiogram
14.7 Summary
14.8 Proofs
15.1 Introduction
15.2 Oscillatory Spectrally Correlated Processes
15.3 Relationships Among Classes of Nonstationary Processes
15.3.1 ACS, CS, and WSS Processes
15.3.2 GACS, SC, and ACS Processes
15.3.3 OSC, OACS, SC, and ACS Processes
15.3.4 Oscillatory Processes
A.1 Introduction
A.2 Second-Order Processes
A.3 Harmonizable Processes
A.4 Time-Frequency Representations
A.5 Wide-Sense Stationary Processes
A.5.1 Time-Averaged Autocorrelation
A.6 Discrete-Time Nonstationary Stochastic Processes
A.7 Proofs
B.1 Almost-Periodic Functions
B.2 Uniformly Almost-Periodic Functions
B.3 Almost-Periodic Sequences
B.4 Generalizations of AP Functions
B.4.1 AP in the Sense of Stepanov, Weyl, Besicovitch
B.4.2 Other Generalizations of AP Functions
B.5 Synchronized Averaging for Periodic Functions
B.6 Proofs
C.1 Sampling in Time
C.2 Sampling in Frequency
C.3 Poisson’s Summation Formulas
C.4 LTI Filtering of Continuous-Time Periodic Signals
C.5 Sampling of Discrete-Time Signals
C.6 LTI Filtering of Discrete-Time Periodic Signals
D.1 Rice’s Representation
D.2 Polar Representation
D.3 Non-Uniqueness of QAM Representation
D.4 Linear Time-Invariant Systems
D.4.1 Incoherent Channel with Amplitude Fading
D.5 Inner Product
D.6 Miscellaneous Results
E.1 Complex (Normal) Random Variables and Vectors
E.1.1 Complex Random Variables
E.1.2 Complex Random Vectors
E.1.3 Multivariate Complex Normal Distribution
E.1.4 Cumulants of Complex Normal Vectors
E.2 Chi Squared Distribution and Complex Normality
E.2.1 Chi Squared Distribution
E.2.2 Real Normal Vector
E.2.3 Complex Normal Vector
E.2.4 Asymptotic Results
E.3 Nonlinear Transformation of Two Complex Random Variables
F.1 Almost-Periodic Functions
F.2 Cyclostationary Signals
F.3 Generalizations of Cyclostationarity
F.4 Other Nonstationary Signals
F.5 Functional Approach and Generalized Harmonic Analysis
F.6 Linear Time-Variant Processing
F.7 Sampling
F.8 Complex Random Variables, Signals, and Systems
F.9 Stochastic Processes
F.10 Mathematics