Kronecker products of matrices and their implementation on shuffle/exchange-type processor networks

Abstract

In generalized spectral analysis an orthogonal transformation provides a set of spectral coefficients by a vector-matrix-multiplication. If the pⁿxpⁿ-transformation matrix H_n is a Kronecker product of n pxp-matrices M_r, r=0,1,...,n−1, then — as has been shown by Good — there exists a factorization $\mathop \prod \limits_{r = 0}^{n - 1}$G_r of the transformation matrix H_n in such a way that the vector-matrix-multiplication can be reduced from 0(N²) to 0(NlogN) operations, where N=pⁿ. The specific structure of the matrix factors G_r, r=0,1...,n−1, permits an implementation of the multiplication on processor networks, in case of p=2 on permutation networks of the Shuffle/Exchange-type.