Fixed Polarity Quaternary Transforms Derived from Linearly Independent Transform over GF(2) Structure

Abstract

A fixed polarity quaternary linearly independent (FPQLI) transform is introduced in this paper where the basic transforms are derived from the recursive structure of some linearly independent transforms over Galois Field (2) (GF(2)). For some polarities the FPQLI transform for n-variable quaternary functions directly corresponds to the binary fixed polarity Reed-Muller (FPRM) transforms for 2n-variable binary functions. In this paper, the fast flow graph and recursive equations for the FPQLI transform are given together with the underlying basis functions. Formulae for converting the FPQLI spectral coefficient vector from one polarity to another are also given and used to generate a recursive algorithm to obtain the optimal FPQLI expansion with reduced computational cost. Experimental results of the FPQLI transform have been obtained for a set of quaternary test files. Comparison of the obtained results with FPRM over GF(4) as well as the existing recursive quaternary linearly independent transforms show the advantage of applying the concept of fixed polarity to the basic transforms for function minimization in terms of smaller number of nonzero spectral coefficients.