Orthogonal designs and generalized Fourier-Walsh transforms

A Walsh transform matrix is an Hadamard matrix in which the rows are permuted to be in a specific order. In this note it is shown that certain generalized Walsh transforms are orthogonal designs which, in turn, are generalizations of Hadamard matrices. We also show that a Fast Fourier Transform algorithm for an orthogonal design transform is structurally equivalent to that of the corresponding Walsh transform permitting more efficient computational algorithms for certain generalized Fourier-Walsh transforms.