E-raamat: Advanced Topics in System and Signal Theory: A Mathematical Approach

In an in-depth analysis of selected methods in signal and system theory that apply to problems in communications, stochastic processes and optimal filter theory, the authors take a functional analysis and operator theoretic approach to linear system theory

The requirement of causality in system theory is inevitably accompanied by the appearance of certain mathematical operations, namely the Riesz proj- tion,theHilberttransform,andthespectralfactorizationmapping.Aclassical exampleillustratingthisisthedeterminationoftheso-calledWiener 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 .