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 .
| Autorius: | Holger Boche, Volker Pohl, |
| Serija: | Foundations in Signal Processing, Communications and Networking |
| Leidėjas: | Springer Berlin Heidelberg |
| Išleidimo metai: | 2012 |
| Knygos puslapių skaičius: | 252 |
| ISBN-10: | 3642260969 |
| ISBN-13: | 9783642260964 |
| Formatas: | Knyga minkštu viršeliu |
| Kalba: | Anglų |
| Žanras: | Maths for engineers |
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