Exact wave-shaping with a time-domain digital filter of finite length

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 (F0, F1, F2 ⋯ FN) 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 o...