Model-matching design of sample-rate changers: asymptotic analysis

Multirate systems are characterized by a matrix-valued frequency response, the alias-component matrix rather than a scalar-valued frequency response, since they are not time-invariant. Recent work on multirate filter design emphasises a model-matching approach, in which a desired alias-component matrix must be approximated. Adopting this approach may require evaluating the maximum singular value of the alias-component matrix on a dense grid of frequencies, which is computationally expensive, particularly in cases where the dimensions of the matrix are large. An asymptotic analysis of the sample-rate changer, described here, shows that the alias-component matrix becomes Toeplitz as one of the dimensions becomes large. Closed form solutions for the asymptotic eigenvalue distribution of Toeplitz matrices may then be used to eliminate the computational bottleneck.