Advanced Topics in System and Signal Theory: A Mathematical Approach / Edition 1

by Volker Pohl, Holger Boche

Overview

The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz projection,the Hilberttrans form, and the spectral factorization mapping. A classical example illustrating this is the determination of the so-calledWiener filter (the linear, minimum means square error estimation filter for stationary shastic sequences [88]). If the filter is not required to be causal, the transfer function of the Wiener filter is simply given by H(ω)=Φ xy(ω)/Φxx (ω),whereΦ xx(ω) and Φxy(ω) are certain given functions. However, if one requires that the - xy timation filter is causal, the transfer function of the optimal filter is given by Φxy (ω) 1 P+ H(ω) = , ω ∈ (−π, π] . [Φxx ]+ (ω) [Φxx ]− (ω) Here [Φxx ]+ and [Φxx ]− represent the so called spectral factors of Φxx, and P is the so called Riesz projection. Thus, compared to the non-causal filter, + two additional operations are necessary for the determination of the causal filter, namely the spectral factorization mapping Φxx → ([Φxx]+ ,[Φxx]− ),and the Riesz projection P .

Product Details

ISBN-13:	9783642036385
Publisher:	Springer Berlin Heidelberg
Publication date:	10/14/2009
Series:	Foundations in Signal Processing, Communications and Networking , #4
Edition description:	2010
Pages:	241
Product dimensions:	6.20(w) x 9.30(h) x 0.80(d)

Table of Contents

I Mathematical Preliminaries.- Function Spaces and Operators.- Fourier Analysis and Analytic Functions.- Banach Algebras.- Signal Models and Linear Systems.- II Fundamental Operators.- Poisson Integral and Hilbert Transformation.- Causal Projections.- III Causality Aspects in Signal and System Theory.- Disk Algebra Bases.- Causal Approximations.- On Algorithms for Calculating the Hilbert Transform.- Spectral Factorization.

Editorial Reviews

From the reviews:

“It is intended primarily for engineers working in the area of robustness problems under a casualty constraint. At the beginning it presents the mathematical methods, necessary to approaching theses problems. Then, some related classical results concerning the boundedness and continuity of the Hilbert transform and Riesz projection are presented. Finally, these methods and results are applied to selected topics from signal processing.” (Tzvetan Semerdjiev, Zentralblatt MATH, Vol. 1179, 2010)

“The monograph gives an elaborate treatment of some tools used in signal and system theory, which have applications in communications, shastic processes and optimal filter theory. … Appropriate references are provided for the reader interested in details. … The book is a valuable addition to the mathematical literature related to signal processing, and should be useful to students in engineering as well as to mathematicians interested in applications.”­­­ (Amol J. Sasane, Mathematical Reviews, Issue 2010 j)

From the Publisher