Applications of Multiplicative Series and Transforms to Digital Information Processing

Abstract

In the last decade interest has increased significantly in applications of the Walsh system and its generalizations, especially applications to digital information processing. This interest stems from a peculiarity of the Walsh functions, namely, that each one of them takes on only two values +1 and -1. A consequence of this peculiarity is that multiplication can be avoided when utilizing high speed computers for certain problems. In such cases a discrete Hadamard transform can be computed almost 10 times faster than the corresponding discrete Fourier transform. Moreover, with the discrete Hadamard transform (DHT) and the discrete transform with respect to multiplicative systems (DMT), one can perform parallel calculations obtaining output of simultaneous calculations on single instruction processors better than can be done using of the discrete Fourier transform (DFT). This allows one to carry out basic calculations using only addition thereby avoiding the more costly operation of multiplication.