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.