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
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
241
Advanced Topics in System and Signal Theory: A Mathematical Approach
241Hardcover(2010)
$109.99
109.99
In Stock
Product Details
ISBN-13: | 9783642036385 |
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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) |
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