An Operator Perspective on Signals and Systems

✍ Scribed by Arthur E. Frazho, Wisuwat Bhosri (auth.)

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 430
• Series: Operator Theory: Advances and Applications 204
• Edition: 1
• Category: Library

✦ Synopsis

In this monograph, we combine operator techniques with state space methods to solve factorization, spectral estimation, and interpolation problems arising in control and signal processing. We present both the theory and algorithms with some Matlab code to solve these problems. A classical approach to spectral factorization problems in control theory is based on Riccati equations arising in linear quadratic control theory and Kalman ?ltering. One advantage of this approach is that it readily leads to algorithms in the non-degenerate case. On the other hand, this approach does not easily generalize to the nonrational case, and it is not always transparent where the Riccati equations are coming from. Operator theory has developed some elegant methods to prove the existence of a solution to some of these factorization and spectral estimation problems in a very general setting. However, these techniques are in general not used to develop computational algorithms. In this monograph, we will use operator theory with state space methods to derive computational methods to solve factorization, sp- tral estimation, and interpolation problems. It is emphasized that our approach is geometric and the algorithms are obtained as a special application of the theory. We will present two methods for spectral factorization. One method derives al- rithms based on ?nite sections of a certain Toeplitz matrix. The other approach uses operator theory to develop the Riccati factorization method. Finally, we use isometric extension techniques to solve some interpolation problems.

✦ Table of Contents

Front Matter....Pages i-xi
Front Matter....Pages 1-1
The Wold Decomposition....Pages 3-22
Toeplitz and Laurent Operators....Pages 23-40
Inner and Outer Functions....Pages 41-53
Rational Inner and Outer Functions....Pages 55-90
The Naimark Representation....Pages 91-116
The Rational Case....Pages 117-141
Front Matter....Pages 143-143
The Levinson Algorithm and Factorization....Pages 145-178
Isometric Representations and Factorization....Pages 179-208
Signal Processing....Pages 209-244
Front Matter....Pages 245-245
Riccati Equations and Factorization....Pages 247-282
Kalman and Wiener Filtering....Pages 283-314
Front Matter....Pages 315-315
Tangential Nevanlinna-Pick Interpolation....Pages 317-339
Contractive Nevanlinna-Pick Interpolation....Pages 341-371
Front Matter....Pages 373-373
A Review of State Space....Pages 375-397
The Levinson Algorithm....Pages 399-412
Back Matter....Pages 413-429

✦ Subjects

Operator Theory; Systems Theory, Control; Computational Science and Engineering