  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.
