Recursive relationships, fast transforms, generalisations and VLSI iterative architecture for Gray code ordered Walsh functions

Two Walsh transforms in Gray code ordering are introduced. The generation of two Walsh transforms in Gray code ordering from the binary code is shown. Recursive relationship between higher and lower matrix orders for Gray code ordering of Walsh functions, using the concepts of operator matrices with symmetric and shift copy, are developed. The generalisation of the introduced Gray code ordered Walsh functions for arbitrary polarity is shown. Another recursive algorithm for a fast Gray code ordered Walsh transform, which is based on the new operators on matrices, joint transformations and a bisymmetrical pseudo-Kronecker product, is introduced. The latter recursive algorithm is the basis for the implementation of a constant-geometry iterative architecture for the Gray code ordered Walsh transform. This architecture can be looped n times or cascaded n times to produce a useful VLSI integrated circuit