Advanced Topics in System and Signal Theory: A Mathematical Approach
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 .
1101509904
Advanced Topics in System and Signal Theory: A Mathematical Approach
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 .
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Advanced Topics in System and Signal Theory: A Mathematical Approach

Advanced Topics in System and Signal Theory: A Mathematical Approach

Advanced Topics in System and Signal Theory: A Mathematical Approach

Advanced Topics in System and Signal Theory: A Mathematical Approach

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.
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