The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz proj- tion, the Hilberttransform, andthespectralfactorizationmapping.Aclassical exampleillustratingthisisthedeterminationoftheso-called Wiener?lter(the linear, minimum means square error estimation ?lter for stationary stochastic sequences [88]). If the ?lter is not required to be causal, the transfer function of the Wiener ?lter is simply given by H(?)=? (?)/? (?), where ? (?) xy xx xx and ? (?) are certain given functions. However, if one requires that the – xy timation ?lter is causal, the transfer function of the optimal ?lter is given by 1 ? (?) xy H(?)= P , ?? (??, ?] . + [? ] (?) [? ] (?) xx + xx? Here [? ] and [? ] represent the so called spectral factors of ? , and xx + xx? xx P is the so called Riesz projection. Thus, compared to the non-causal ?lter, + two additional operations are necessary for the determination of the causal ?lter, namely the spectral factorization mapping ? ? ([? ] , [? ] ), and xx xx + xx? the Riesz projection P .
Mục lục
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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