Stability characteristics of a new inverse filter which is optimum in an error-distribution sense

Abstract

An inverse filter which is optimum in an error-distribution sense is a filter which will transform a given wavelet into a sequence of widely spaced spikes. The errors between the actual output and desired output, being the off-centered spikes, may be decreased and moved away from the region of interest by increasing the number of subfilters used in the design of the main filter. In this paper it is shown that in general the filter is stable. In the special cases which arise, when the z-transform of the given wavelet has roots which lie on the unit circle, the off-central subfilter weights acquire cyclical values and hence remain finite as the number of subfilters used is increased. The resulting inverse filter is stable in the sense that its weights remain finite as time is increased, however, it is infinitely long with significant weights all the way to infinity. The conventional method of obtaining the weights of an inverse filter by first obtaining 2 M spectral points of the input with the Fast Fourier Transform method and then performing 2 M spectral divisions may also be looked upon as a special case for the filter described in this paper. It is shown that this is the case in which M subfilters are used. The time-domain filter derived from the inverse Fourier transform is in reality a time-domain aliased version of the filter derived entirely in the time-domain by convolution operations. The two filters are thus not identical and their output spike sequences differ in their magnitudes. If at least one of the spectral values of the input spectrum is zero, the spectrum of the inverse as obtained by spectral-divisions is completely indeterminate. If, however, we depart from the special case and instead choose N subfilters where N is less than M, division by zero is avoided and meaningful spiking filters may still be designed in the frequency domain by multiplying the spectra of the subfilters.