A class of generalized continuous orthogonal transforms

Abstract

This paper presents a class of generalized continuous transforms for the orthogonal decomposition of signals. Base functions for the continuous transform range from Walsh functions of order two to stair-like functions which resemble approximations to sinusoids and which are distinct from the generalized Walsh functions. Standard desirable properties which are shown to hold for the generalized continuous transform operator include orthogonality of the base functions, linearity of the transform operator, inverse transformability, and admissibility to fast transform representation. The transform class is governed by a definition of time translation in terms of signed-bit dyadic time shift. Mathematical properties leading to this definition are discussed and the impact of the definition is assessed. Properties of the continuous class of generalized transforms make feasible analysis which could be extremely tedious using matrix representations of the operations actually mechanized in a sampled-data system. Analysis techniques are illustrated with a target detection system which is conceptually designed using the generalized continuous transform and implemented using fast transform algorithms to perform correlation operations. Since the correlation operations are valid for inputs which include signals represented in terms of Walsh functions, the example illustrates one instance in which the binary Fourier representation (BIFORE) transform can be used for practical pattern recognition.