Next: Inverse Theory for signal Up: Background and definitions Previous: An example of calculating

Prediction-error filtering in the frequency domain

If the convolution is expressed in matrix form as , where is the convolution matrix of ,the filter can be solved for to get the least-squares minimum of .The normal equations expression for the least-squares inverse is ,or .This expression for may be decomposed into simpler expressions in the frequency domain since may be expressed as .The expression may be transformed into the frequency domain as or .(Here indicates a component of the Fourier transform of the data, and indicates the complex conjugate of ). Canceling out gives .Thus in the frequency domain, where filtering is described as a multiplication such as ,inversion is simply division, or .The values of , , and are scalars (although they are complex numbers).

The term in the denominator is the Fourier transform of the autocorrelation of .If is the identity matrix , will be constant. This corresponds to an input with a white spectrum. If all the terms of are constant, will be non-zero only at ,and the inversion will be unstable. This corresponds to a data series containing a constant. It can be seen that is a measure of the information available at , and is a function of the uncertainty, or variance, at .The original autocorrelation matrix is the information matrix, and its inverse is the covariance matrixStrang (1986).

The expression will generally have a stabilizer in the denominator to avoid having approach infinity when gets small. Adding this stabilizer in the frequency domain corresponds to adding a small value to the diagonal of the autocorrelation matrix. In the cases discussed here, the stabilizer will seldom be needed since random noise in the data generally keeps from going to zero.

Next: Inverse Theory for signal Up: Background and definitions Previous: An example of calculating
Stanford Exploration Project
2/9/2001