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

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 .

• | Author: Volker Pohl
• | Publisher: Springer
• | Publication Date: Oct 16, 2009
• | Number of Pages: 241 pages
• | Binding: Hardback or Cased Book
• | ISBN-10: 3642036384
• | ISBN-13: 9783642036385