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THE DETERMINATION OF DIGITAL WIENER FILTERS FROM AN ILL‐CONDITIONED SYSTEM OF NORMAL EQUATIONS *
Author(s) -
TREITEL S.,
WANG R. J.
Publication year - 1976
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1111/j.1365-2478.1976.tb00928.x
Subject(s) - wiener filter , mathematics , system of linear equations , digital filter , conjugate gradient method , filter (signal processing) , digital computer , matrix (chemical analysis) , computer science , mathematical analysis , algorithm , chemistry , chromatography , computer engineering , computer vision
A bstract The system of normal equations associated with the discrete Wiener filter is sometimes ill‐conditioned. The purpose of this paper is to show that in such cases the solutions obtained vary drastically with the particular choice of an algorithm and of the computer used for its implementation. A review of the basic mathematical theory behind an ill‐conditioned matrix is first presented. Numerical examples are then given to illustrate that the solutions of the normal equations are sensitive to the word length of a given computer. Finally, two possible remedies are described: (1) The well‐known method of prewhitening and (2) the use of the conjugate‐gradient algorithm for solving the normal equations.