1 Introduction.- 1.1 Introduction.- 1.2 Organization of the Book.- 1.3 Notations and Preliminaries.- 2 Power Spectra and Positive Functions.- 2.1 Stationary Processes and Power Spectra.- 2.1.1 Linear Prediction.- 2.2 Positive Functions.- 2.2.1 Bounded Functions.- 2.2.2 Schur Algorithm and Line Extraction.- Problems.- 3 Admissible Spectral Extensions.- 3.1 Introduction.- 3.2 Geometrical Solution.- 3.3 Parametrization of Admissible Extensions.- 3.3.1 Levinson Polynomials and Reflection Coefficients.- 3.3.2 Reflection Coefficients and Characteristic Impedances.- 3.4 Two-Step Predictor.- Appendix 3.A Maximization of the k-Step Minimum Mean-Square Prediction Error.- Appendix 3.B Uniqueness of ?(z).- Appendix 3.C Negative Discriminant.- Appendix 3.D Existence of the Second Bounded-Real Solution.- Problems.- 4 ARMA-System Identification and Rational Approximation.- 4.1 Introduction.- 4.1.1 Exact Autocorrelations Case.- 4.1.2 Known Data Case.- 4.2 ARMA-System Identification - A New Approach.- 4.2.1 ARMA-System Characterization.- 4.2.2 ARMA-Parameter Estimation.- 4.2.3 Model Order Selection.- 4.2.4 Simulation Results.- 4.3 Rational Approximation of Nonrational Systems.- 4.3.1 Padé-like Approximation.- 4.3.2 Numerical Results.- Appendix 4.A A Necessary Condition for Padé-like Approximation.- Appendix 4.B Diagonal Padé Approximations of e-p.- Problems.- 5 Multichannel System Identification.- 5.1 Introduction.- 5.2 Multichannel Admissible Spectral Extensions.- 5.2.1 Multichannel Schur Algorithm and a Weak Form of Richards' Theorem.- 5.2.2 Left-Inverse Form.- 5.2.3 Right-Inverse Form.- 5.2.4 Multichannel Spectral Extension - Youla's Representation.- 5.3 Multichannel Rational System Identification.- 5.3.1 MARMA System Characterization.- 5.3.2 MARMA Parameter Estimation.- 5.3.3 Model Order Selection.- 5.3.4 Simulation Results.- 5.4 Multichannel Rational Approximation of Nonrational Systems.- 5.4.1 Multichannel Padé-like Approximation.- Appendix 5.A Reflection Coefficient Matrices and Their Left and Right Factors.- Appendix 5.B Renormalization of dn
(z).- Problems.- References.
Spectrum estimation refers to analyzing the distribution of power or en ergy with frequency of the given signal, and system identification refers to ways of characterizing the mechanism or system behind the observed sig nal/data. Such an identification allows one to predict the system outputs, and as a result this has considerable impact in several areas such as speech processing, pattern recognition, target identification, seismology, and signal processing. A new outlook to spectrum estimation and system identification is pre sented here by making use of the powerful concepts of positive functions and bounded functions. An indispensable tool in classical network analysis and synthesis problems, positive functions and bounded functions are well and their intimate one-to-one connection with power spectra understood, makes it possible to study many of the signal processing problems from a new viewpoint. Positive functions have been used to study interpolation problems in the past, and although the spectrum extension problem falls within this scope, surprisingly the system identification problem can also be analyzed in this context in an interesting manner. One useful result in this connection is regarding rational and stable approximation of nonrational transfer functions both in the single-channel case and the multichannel case. Such an approximation has important applications in distributed system theory, simulation of systems governed by partial differential equations, and analysis of differential equations with delays. This book is intended as an introductory graduate level textbook and as a reference book for engineers and researchers.
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