Filter Banks on Discrete Abelian Groups

TL;DR

This paper develops a unified theoretical framework for filter banks on discrete abelian groups, covering multidimensional, cyclic, and various structured signal spaces, with conditions for perfect reconstruction.

Contribution

It provides a comprehensive analysis of filter banks on discrete abelian groups, including polyphase, modulation, and frame theory, unifying various multidimensional and cyclic cases.

Findings

Derived perfect reconstruction conditions

Established frame bounds for filter banks

Unified analysis for multidimensional and cyclic filter banks

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.