Properties and ways of calculation of multi-polarity generalized Walsh transforms

A new formulation of the multi-polarity generalized Walsh transform has been introduced. The formulation allows a uniform representation of Boolean functions by a set of orthogonal and generalized Walsh spectral coefficients. Forward and inverse transformation kernels and the method of recursive generation of transform matrices by using Kronecker products of elementary matrices have been given. Mutual relations among transform matrices and spectra for arbitrary polarities have been investigated. Fast forward and inverse generalized Walsh transforms have been developed. An algorithm to calculate the generalized Walsh transform from a cube array specification of incompletely specified Boolean functions has been shown. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. The computer implementation drastically reduces the required time of computation by allowing calculation of only chosen coefficients in specified polarity. The entire spectrum-if required-can be computed incrementally for groups of coefficients.<
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