TY - GEN

T1 - There exists no always convergent algorithm for the calculation of spectral factorization, Wiener filter, and Hilbert transform

AU - Boche, Holger

AU - Pohl, Volker

PY - 2006

Y1 - 2006

N2 - Spectral factorization, Wiener filtering, and many other important operations in information theory and signal processing can be lead back to a Hilbert transform and a Poisson integral. Whereas the Poisson integral causes generally no problems, the Hilbert transform has a much more complicated behavior. This paper investigates the possibility to calculate the Hilbert transform f̃ of a given continuous function f based on a finite set of sampling points of f. It shows that even if f is continuous, no linear approximation operator exists which approximates f arbitrary well from a finite number of sampling points of f, in general. Moreover, the paper characterizes the set of all functions for which such linear approximation operators exist and discusses some consequences for practical applications.

AB - Spectral factorization, Wiener filtering, and many other important operations in information theory and signal processing can be lead back to a Hilbert transform and a Poisson integral. Whereas the Poisson integral causes generally no problems, the Hilbert transform has a much more complicated behavior. This paper investigates the possibility to calculate the Hilbert transform f̃ of a given continuous function f based on a finite set of sampling points of f. It shows that even if f is continuous, no linear approximation operator exists which approximates f arbitrary well from a finite number of sampling points of f, in general. Moreover, the paper characterizes the set of all functions for which such linear approximation operators exist and discusses some consequences for practical applications.

UR - http://www.scopus.com/inward/record.url?scp=34547885230&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2006.261686

DO - 10.1109/ISIT.2006.261686

M3 - Conference contribution

AN - SCOPUS:34547885230

SN - 1424405041

SN - 9781424405046

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 118

EP - 122

BT - Proceedings - 2006 IEEE International Symposium on Information Theory, ISIT 2006

T2 - 2006 IEEE International Symposium on Information Theory, ISIT 2006

Y2 - 9 July 2006 through 14 July 2006

ER -