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

This paper is about the eigenvalues and eigenvectors of (↓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 (↓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 filter is governed by the eigenvalues. If the transfer function H(z)=Σh(k)z^-k has a zero of order p at z=-1, corresponding to ω=π, then (↓2)H has p special eigenvalues ½,¼...,(½)^p. We show how each additional “zero at π” divides all eigenvalues by 2 and creates a new eigenvector for λ=½. This eigenvector solves the dilation equation φ(t)=2Σh(k)φ(2t-k) at the integers t=n. The left eigenvectors show how 1,t,...,t^p-1 can be produced as combinations of φ(t-k). The dilation equation is solved by the cascade algorithm, which is an infinite iteration of M=(↓2)2H. Convergence in L² is governed by the eigenvalues of T=(↓2)2HH^T corresponding to the response 2H(z)H(z^-1 ). We find a simple proof of the necessary and sufficient condition for convergence