Filter Banks on Discrete Abelian Groups

In this work we provide polyphase, modulation, and frame theoretical analyses of a filter bank on a discrete abelian group. Thus, multidimensional or cyclic filter banks as well as filter banks for signals in $\ell^2(\mathbb{Z}^d\times \mathbb{Z}_s)$ or $\ell^2(\mathbb{Z}_r \times \mathbb{Z}_s)$ spaces are studied in a unified way. We obtain perfect reconstruction conditions and the corresponding frame bounds.Comment: This version is the same as the previous one except some comments on the polyphase transform and some references have been adde

Semi-direct product of groups, filter banks and sampling

An abstract sampling theory associated to a unitary representation of a countable discrete non abelian group $G$, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples and the use of a filter bank formalism allows to fix the mathematical problem to be solved: the search of appropriate dual frames for $\ell^2(G)$. An example involving crystallographic groups illustrates the obtained results by using average or pointwise samples.Comment: 16 pages, 1 figur