EIGENVALUES OF (DOWN-ARROW-2)H AND CONVERGENCE OF THE CASCADE ALGORITHM

This paper is about the eigenvalues and eigenvectors of (down arrow 2) H. The ordinary FIR filter H is convolution with a vector h = (h(0),.. ., h(N)), which is the impulse response. The operator (down arrow 2) d ownsamples 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 ar e removed in (down arrow 2)H. The result is a double shift between row s, yielding a block Toeplitz matrix with 1 x 2 blocks. Iteration of th e filter is governed by the eigenvalues. If the transfer function H(z) = Sigma h(k)z(-k) has a zero of order p at z = -1, corresponding to o mega = pi, then (down arrow 2)H has p special eigenvalues 1/2, 1/4..., (1/2)(p). We show how each additional ''zero at pi'' divides all eigen values by 2 and creates a new eigenvector for lambda = 1/2. This eigen vector solves the dilation equation phi(t) = 2 Sigma h(k)phi(2t-k) at the integers t = n. The left eigenvectors show how 1, t,..., t(p-1) fa n be produced as combinations of phi(t - k). The dilation equation is solved by the cascade algorithm, which is an infinite iteration of M = (down arrow 2)2H. Convergence in L(2) is governed by the eigenvalues of T = (down arrow 2)2H H-T corresponding to the response 2H(z)H(z(-1) ). We find a simple proof of the necessary and sufficient condition fo r convergence.