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Advanced Topics in System and Signal Theory: A Mathematical Approach by Volker Pohl, Holger Boche

Advanced Topics in System and Signal Theory: A Mathematical Approach

by Volker Pohl , Holger Boche
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Book cover type: Hardcover
  • ISBN13: 9783642036385
  • Binding: Hardcover
  • Subject: N/A
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Publication Date:
  • Pages: 241
  • Original Price: USD 109.99
  • Language: English
  • Edition: 2010
  • Item Weight: 531 grams
  • BISAC Subject(s): Applied

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 .

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