RECURSIVE RELATIONSHIPS, FAST TRANSFORMS, GENERALIZATIONS AND VLSI ITERATIVE ARCHITECTURE FOR GRAY CODE ORDERED WALSH-FUNCTIONS

Two Walsh transforms in Gray code ordering are introduced. The generat ion of two Walsh transforms in Gray code ordering from the binary code is shown. Recursive relationship between higher and lower matrix orde rs for Gray code ordering of Walsh functions, using the concepts of op erator matrices with symmetric and shift copy, are developed. The gene ralisation of the introduced Gray code ordered Walsh functions for arb itrary polarity is shown. Another recursive algorithm for a fast Gray code ordered Walsh transform, which is based on the new operators on m atrices, joint transformations and a bisymmetrical pseudo-Kronecker pr oduct, is introduced. The latter recursive algorithm is the basis for the implementation of a constant-geometry iterative architecture for t he Gray code ordered Walsh transform. This architecture can be looped n times or cascaded n times to produce a useful VLSI integrated circui t.