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

This paper is about the eigenvalues and eigenvectors of (# 2)H. The ordinary FIR lter H is convolution with a vector h = (h(0); : : : ; h(N)), the impulse response. The operator (# 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 (# 2)H. The result is a double shift between rows, yielding a block Toeplitz matrix with 1 2 blocks. Iteration of the lter is governed by the eigenvalues. If the transfer function H(z) = P h(k)z ?k has a zero of order p at z = ?1, corresponding to ! = , then (# 2)H has p special eigenvalues 1 2 ; 1 4 : : : ; ? 1 2 p. We show how each additional \zero at " divides all eigenval-ues by 2 and creates a new eigenvector for = 1 2. This eigenvector solves the dilation equation (t) = 2 P h(k)(2t ?k) at the integers t = n. The left eigenvectors show how 1; t; : : :; t p?1 can be produced as combinations of (t ? k). The dilation equation is solved by the cascade algorithm, an innnite iteration of M = (#2)2H. Convergence in L 2 is governed by the eigenvalues of T = (# 2)2HH T , corresponding to the response 2H(z)H(z ?1). We nd a simple proof of the necessary and suucient condition for convergence.