Digital Processing of Random Signals : Theory and Methods

Preface viii Introduction 1(10) References 9(2) The Structure of Stationary Processes 11(45) Discrete-time Random Processes 11(2) Gaussian Random Processes 13(1) Stationarity and Ergodicity 13(1) The Covariances of Stationary Processes 14(1) Some Examples 15(2) Hilbert Spaces of Stationary Processes 17(4) The Wold Decomposition 21(6) Spectral Distributions and Spectral Densities 27(6) Spectral Factorization 33(1) Properties of the Best Linear Predictor 34(4) Wide Sense Ergodicity of Gaussian Processes 38(3) Parametric Models for Stationary Processes 41(7) Mathematica Packages 48(8) References 49(1) Problems 50(6) Parameter Estimation Theory 56(42) Principles of Parameter Estimation 56(5) Properties of Estimates 61(4) Covariance Inequalities and Efficiency 65(5) Sequences of Estimates 70(3) Maximum Likelihood Estimation 73(5) The Method of Moments 78(10) Least-squares Estimation 88(2) Maximum Entropy Estimation 90(1) Model Order Selection 91(7) References 93(1) Problems 94(4) Nonparametric Spectrum Estimation 98(35) Introduction 98(2) Estimation of the Mean and the Covariances 100(6) The Periodogram 106(6) Periodogram Averaging 112(2) Smoothed Periodograms 114(5) Windowed Periodograms 119(8) Mathematica Packages 127(6) References 128(1) Problems 129(4) Parameter Estimation Theory for Gaussian Processes 133(19) Introduction 133(1) The Fisher Information of Stationary Gaussian Processes 134(5) Maximum Likelihood Parameter Estimation 139(3) The Relative Efficiency of the Sample Covariances 142(2) Parameter Estimation from the Sample Covariances 144(3) Mathematica Packages 147(5) References 148(1) Problems 148(4) Autoregressive Parameter Estimation 152(35) The Yule-Walker Estimate 152(4) The Levinson-Durbin Algorithm 156(7) Algorithms Related to Levinson-Durbin 163(4) Lattice Filters 167(3) Maximum Entropy Estimation 170(1) Least-squares Estimation 171(2) Maximum Likelihood Estimation 173(2) Estimation of the Partial Correlation Coefficients 175(3) Model Order Selection 178(1) Estimation of the Spectral Density 179(2) Mathematica Packages 181(6) References 182(1) Problems 183(4) Moving Average and ARMA Parameter Estimation 187(31) Introduction 187(1) Moving Average Parameter Estimation: Elementary Methods 188(3) Moving Average Parameter Estimation: Advanced Methods 191(5) ARMA Estimation: The Modified Yule-Walker Method 196(3) ARMA Estimation from the Sample Covariances 199(3) Approximate Maximum Likelihood ARMA Estimation 202(3) Exact Maximum Likelihood ARMA Estimation 205(4) An Example: EMG Signal Analysis 209(2) Mathematica Packages 211(7) References 213(2) Problems 215(3) Adaptive AR and ARMA Estimation 218(42) Introduction 218(1) The Recursive Least-squares Algorithm 219(4) The Extended Least-squares Algorithm 223(2) The Recursive Maximum Likelihood Algorithm 225(4) Stochastic Gradient Algorithms 229(2) Projection Operators in Euclidean Spaces 231(4) Lattice Algorithms for Autoregressive Estimation 235(7) Lattice Algorithms for ARMA Estimation 242(4) Extensions of Lattice Algorithms 246(6) Mathematica Packages 252(8) References 254(1) Problems 255(5) Estimation of Deterministic Processes 260(34) Introduction 260(2) The Cramer-Rao Bound for Sinusoids in White Noise 262(4) Maximum Likelihood Estimation 266(4) The Prony Method 270(2) The Truncated Singular Value Decomposition Method 272(5) Estimation from the Sample Covariances 277(4) Estimation from the Sample Covariance Matrix 281(4) Concluding Remarks 285(1) Mathematica Packages 286(8) References 288(2) Problems 290(4) High-order Statistical Analysis 294(38) Introduction 294(1) Definition and Properties of Cumulants 295(5) Cumulants and Polyspectra of Stationary Linear Processes 300(5) The Cumulants of ARMA Processes 305(3) Estimation of the Cumulants 308(5) Moving Average Parameter Estimation: Linear Methods 313(4) ARMA Parameter Estimation: Linear Methods 317(1) MA and ARMA Parameter Estimation: Nonlinear Methods 318(3) Deconvolution 321(3) Estimation of Sinusoids in Gaussian Noise 324(1) Mathematica Packages 325(7) References 327(2) Problems 329(3) Time-frequency Signal Analysis: Linear Transforms 332(37) Introduction 332(3) The Short-time Fourier Transform 335(2) The Gabor Representation: Elementary Discussion 337(5) The Gabor Representation: Advanced Discussion 342(7) The Wavelet Transform 349(4) Orthonormal Wavelet Bases 353(8) Implementation of Linear Transforms for Discrete-time Signals 361(3) Mathematica Packages 364(5) References 365(1) Problems 366(3) Time-frequency Signal Analysis: Nonlinear Transforms 369(41) Introduction 369(1) The Wigner-Ville Distribution: Part I 370(7) The Wigner-Ville Distribution: Part II 377(6) The Ambiguity Function 383(4) The Cohen Class of Distributions 387(5) High-order Ambiguity Function 392(5) Estimation Using the High-order Ambiguity Function 397(7) Mathematica Packages 404(6) References 405(2) Problems 407(3) Appendix A. Notations and Facts 410(4) References 413(1) Appendix B. Hilbert Spaces 414(7) References 420(1) Appendix C. Asymptotic Theory 421(8) References 428(1) Appendix D. Kronecker Products and Liapunov Equations 429(10) References 437(2) Author Index 439(4) Subject Index 443