Discrete Convolution by Means of Forward and Backward Modeling

Abstract

The standard methods of performing discrete convolution, that is, directly in the time domain or by means of the fast Fourier transform in the frequency domain, implicitly assume that the signals to be convolved are zero outside the observation intervals. Often this assumption produces undesirable end effects which are particularly severe when the signals are short in duration. This paper presents an approach to discrete convolution which obviates the zero assumption. The method is structurally similar to the Burg method [l], which estimates the autocorrelation coefficients of a series in a manner which does not require a predefinition of the behavior of the signal outside of the known interval. The basic principle of the present approach is that each term of the convolution is recursively determined from previous terms in a manner consistent with the optimal modeling of one signal into the other. The recursion uses forward and backward modeling together with the Morf et al. [2] algorithm for computation of the prediction error filter. The method is illustrated by application to the computation of the analytic signal and the derivative.