Eigenvalues of (↓2)H and convergence of the cascade algorithm

Abstract

This paper is about the eigenvalues and eigenvectors of (/spl darr/2)H. The ordinary FIR filter H is a convolution with a vector h=(h(O),...,h(N)), which is the impulse response. The operator (/spl darr/2) downsamples the output y=h*x, keeping the even-numbered components y(2n). Where H is represented by a constant-diagonal matrix, this is a Toeplitz matrix with h(k) on its kth diagonal, the odd-numbered rows are removed in (/spl darr/2)H. The result is a double shift between rows, yielding a block Toeplitz matrix with 1/spl times/2 blocks. Iteration of the filter is governed by the eigenvalues. If the transfer function H(z)=/spl Sigma/h(k)z/sup -k/ has a zero of order p at z=-1, corresponding to /spl omega/=/spl pi/, then (/spl darr/2)H has p special eigenvalues 1/2 , 1/4 ...,( 1/2 )/sup p/. We show how each additional "zero at /spl pi/" divides all eigenvalues by 2 and creates a new eigenvector for /spl lambda/= 1/2 . This eigenvector solves the dilation equation /spl phi/(t)=2/spl Sigma/h(k)/spl phi/(2t-k) at the integers t=n. The left eigenvectors show how 1,t,...,t/sup p-1/ can be produced as combinations of /spl phi/(t-k). The dilation equation is solved by the cascade algorithm, which is an infinite iteration of M=(/spl darr/2)2H. Convergence in L/sup 2/ is governed by the eigenvalues of T=(/spl darr/2)2HH/sup T/ corresponding to the response 2H(z)H(z/sup -1/). We find a simple proof of the necessary and sufficient condition for convergence.