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 .
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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