Walsh-like functions and their relations

Abstract

A new discrete transform, the ‘Haar–Walsh transform’, has been introduced. Similar to well known Walsh and non-normalised Haar transforms, the new transform assumes only +1 and –1 values, hence it is a Walsh-like function and can be used in different applications of digital signal and image processing. In particular, it is extremely well suited to the processing of two-valued binary logic signals. Besides being a discrete transform on its own, the proposed transform can also convert Haar and Walsh spectra uniquely between themselves. Besides the fast algorithm that can be implemented in the form of in-place flexible architecture, the new transform may be conveniently calculated using recursive definitions of a new type of matrix, a ‘generator matrix’. The latter matrix can also be used to calculate some chosen Haar–Walsh spectral coefficients which is a useful feature in applications of the new transform in logic synthesis.