Filter Banks on Discrete Abelian Groups

Abstract: 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.