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An Operator Perspective on Signals and Systems

  • Book
  • © 2010

Overview

Authors:
  1. Arthur E. Frazho

    1. School of Aeronautics and Astronautics, Purdue University, West Lafayette, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Wisuwat Bhosri

    1. Chiang Mai, Thailand

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    You can also search for this author in PubMed   Google Scholar

  • This book uses geometric techniques to develop both the theory and the algorithms behind many factorization, interpolation, and signal processing problems arising in engineering and operator theory
  • Some "new" algorithms based on finite section methods are presented
  • Numerical examples using MATLAB are included
  • Includes supplementary material: sn.pub/extras

Part of the book series: Operator Theory: Advances and Applications (OT, volume 204)

Part of the book sub series: Linear Operators and Linear Systems (LOLS)

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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xi
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  2. Basic Operator Theory

    1. Front Matter

      Pages 1-1
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    2. The Wold Decomposition

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 3-22

    3. Toeplitz and Laurent Operators

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 23-40

    4. Inner and Outer Functions

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 41-53

    5. Rational Inner and Outer Functions

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 55-90

    6. The Naimark Representation

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 91-116

    7. The Rational Case

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 117-141

  3. Finite Section Techniques

    1. Front Matter

      Pages 143-143
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    2. The Levinson Algorithm and Factorization

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 145-178

    3. Isometric Representations and Factorization

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 179-208

    4. Signal Processing

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 209-244

  4. Riccati Methods

    1. Front Matter

      Pages 245-245
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    2. Riccati Equations and Factorization

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 247-282

    3. Kalman and Wiener Filtering

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 283-314

  5. Interpolation Theory

    1. Front Matter

      Pages 315-315
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    2. Tangential Nevanlinna-Pick Interpolation

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 317-339

    3. Contractive Nevanlinna-Pick Interpolation

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 341-371

  6. Appendices

    1. Front Matter

      Pages 373-373
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    2. A Review of State Space

      • Arthur E. Frazho, Wisuwat Bhosri
      Pages 375-397
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Keywords

About this book

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.

Reviews

From the reviews:

“This monograph combines operator techniques with state space methods to solve factorization, spectral estimation and interpolation problems arising in control and signal processing. … In general, the book lays out intricate ideas in a relaxed one-step at time style and hence should be accessible for beginners in the subject. In addition, the novel way in which known material is interrelated should make the book informative for experts as well.” (Joseph A. Ball, Mathematical Reviews, Issue 2011 a)

“This book covers a number of fundamentals of signal and system theory mostly concentrating on the state space methods combined with application of operator techniques. … It is of interest for studying and teaching mathematical fundamentals of engineering solutions developed for signals and systems analysis, and can be recommended for both engineers interested in deeper understanding of modern and classical mathematical results … and pure mathematicians for taking a wider view on solutions recently deployed for signals and systems.” (Denis Sidorov, Zentralblatt MATH, Vol. 1218, 2011)

Authors and Affiliations

  • School of Aeronautics and Astronautics, Purdue University, West Lafayette, USA

    Arthur E. Frazho

  • Chiang Mai, Thailand

    Wisuwat Bhosri

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