EXACT WAVE‐SHAPING WITH A TIME‐DOMAIN DIGITAL FILTER OF FINITE LENGTH

Abstract

When the signs of alternate terms of a symmetric discrete time series are reversed and the newly created series is then convolved with the original series, the resultant time‐series will have alternate values equal to zero. This property of symmetric functions may be exploited to design a new deconvolution and wave‐shaping time‐domain filter which is capable of transforming a given wavelet into an output made up of a sequence of spikes separated by zeros, or a sequence of wavelets, whose shapes are identical to that of any desired wavelet. In its design, no Z-transform polynomials are factored or divided and no equations are solved. The weights are derived entirely in the time domain from a series of successively derived subfilters ([Formula: see text], [Formula: see text], [Formula: see text] ⋯ [Formula: see text]) which, when convolved with the original wavelet, creates the spike sequence output. These subfilters may be conveniently grouped into a symmetric component which is derived from the autocorrelation function, a component which depends upon the characteristics of the original wavelet and a component which depends upon the desired wavelet. The number of zeros separating the spike outputs may be controlled by increasing the number of sub‐filters N according to the formula [Formula: see text]. The Wiener filter is an optimum filter in the least‐squares sense but its errors occur across the output. The new filter is an optimum filter in an “error‐distribution” sense. Its errors are in reality the noncentral spikes of the spike sequence. By choosing the length properly, the errors may be moved away from the region of interest leaving that region effectively “error‐free”. A limitation to this procedure is the computational round‐off error which increases as the filter length is increased. In a series of experiments with various types of wavelets it was found that the spike position always occurs at the center of the filter, with the anticipation and memory components automatically falling into place. A very important property of the filter is the fact that the input parameters required for its design are identical to those needed for the normal equations of the Wiener filter. Initial tests with a noisy time‐series showed that the new filter could be effectively employed using the statistical properties of the noise in the same manner that the Wiener filter is applied.