Linear Operators and Linear Systems

An Operator Perspective on Signals and Systems

Authors: Frazho, Arthur, Bhosri, Wisuwat

  • 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
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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)


Table of contents (15 chapters)

  • The Wold Decomposition

    Frazho, Arthur E. (et al.)

    Pages 3-22

  • Toeplitz and Laurent Operators

    Frazho, Arthur E. (et al.)

    Pages 23-40

  • Inner and Outer Functions

    Frazho, Arthur E. (et al.)

    Pages 41-53

  • Rational Inner and Outer Functions

    Frazho, Arthur E. (et al.)

    Pages 55-90

  • The Naimark Representation

    Frazho, Arthur E. (et al.)

    Pages 91-116

Buy this book

eBook $119.00
price for USA (gross)
  • ISBN 978-3-0346-0292-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $159.00
price for USA
  • ISBN 978-3-0346-0291-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
An Operator Perspective on Signals and Systems
Authors
Series Title
Linear Operators and Linear Systems
Series Volume
204
Copyright
2010
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-0346-0292-1
DOI
10.1007/978-3-0346-0292-1
Hardcover ISBN
978-3-0346-0291-4
Edition Number
1
Number of Pages
XI, 431
Topics