Walsh Summing and Differencing Transforms

Analogous to Fourier frequency transforms of the integration and differentiation of a continuous-time function, Walsh sequency transforms of the summing and differencing of an arbitrary discrete-time function have been derived. These transforms can be represented numerically in the form of matrices of simple recursive structure. The matrices are not orthogonal, but they are the inverse of each other, and the value of their determinants is one. (To be published in IEEE Trans. on Electromagnetic Compatibility) *Work supported by the U. S. Atomic Energy Commission. INTRODUCTION The existence of Fourier frequency transforms of the integration and differentiation of continuou&ime functions suggests seeking Walsh sequency transforms of the summing and differencing of discrete-time (or other discretedata) functions. In brief, if s and d represent the summing Z and differencing A of an arbitrary discrete-time function f, and if S, D, and F represent their respective Walsh transforms, we seek transforms E and H that carry F into S and D respectively, as depicted in Fig. 1. A Fig. 2: Relations between existing and desired sequency transforms. The desired transforms are easily derived, and can be represented numerically in matrix form. If the Walsh transform is defined as a matrix W of sequency-ordered Walsh functions, then the desired matrices 2 and & are of simple recursive structure. Although not orthogonal, they are the inverse of each other, and the value of their determinants is 1. DEFINITIONS Let fi denote the value of an arbitrary discrete-time function f in the ith subinterval (i = 0, 1, , . . 2n 1) of the finite discrete-time interval (0, T). -2Let j sj= Cfi (j=O, 1, . . . Zn-1) i=O (1) denote the forward sum of f, so that sj is the value of the sum at the end of the j th subinterval. Let di = fi fi 1 (f -1 = 0) (2) denote the backward difference of f, so that di is the value of the difference at the beginning of the ith subinterval. In matrix form (1) and (2) are simply s=z. f (3) d= A. f (4) where s, d, and f are all column matrices of 2n elements, and 2 and A are square matrices of order 2n. The summing matrix 2 is a lower triangular matrix: