A Generalized Technique for Spectral Analysis

A technique is presented to implement a class of orthogonal transformations on the order of pN logp N operations. The technique is due to Good [1] and implements a fast Fourier transform, fast Hadamard transform, and a variety of other orthogonal decompositions. It is shown how the Kronecker product can be mathematically defined and efficiently implemented using a matrix factorization method. A generalized spectral analysis is suggested, and a variety of examples are presented displaying various properties of the decompositions possible. Finally, an eigenvalue presentation is provided as a possible means of characterizing some of the transforms with similar parameters.