Advanced Topics in System and Signal Theory A Mathematical Approach

Synopsis

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 .

Book details

Edition:

2010

Series:

Foundations in Signal Processing, Communications and Networking (Book 4)

Author:

Volker Pohl, Holger Boche

ISBN:

9783642036392

Related ISBNs:

9783642036385

Publisher:

Springer Berlin Heidelberg

Pages:

N/A

Reading age:

Not specified

Includes images:

No

Date of addition:

2022-08-20

Usage restrictions:

Copyright

Copyright date:

2010

Copyright by:

N/A 

Adult content:

No

Language:

English

Categories:

Communication, Computers and Internet, Nonfiction, Technology