Generic Feasibility of Perfect Reconstruction with Short FIR Filters in Multi-channel Systems

TL;DR

This paper investigates the minimal length of FIR synthesis filters needed for perfect reconstruction in multi-channel filter banks, providing generic necessary and sufficient conditions with practical verification.

Contribution

It offers the first generic conditions for the shortest FIR synthesis filters guaranteeing perfect reconstruction, with explicit prescriptions related to oversampling factors.

Findings

Prescribed filter length is as short or shorter than analysis filters.

Conditions hold for almost all filter banks, excluding special nongeneric cases.

Numerical results show small gap between prescribed and true minimum filter length.

Abstract

We study the feasibility of short finite impulse response (FIR) synthesis for perfect reconstruction (PR) in generic FIR filter banks. Among all PR synthesis banks, we focus on the one with the minimum filter length. For filter banks with oversampling factors of at least two, we provide prescriptions for the shortest filter length of the synthesis bank that would guarantee PR almost surely. The prescribed length is as short or shorter than the analysis filters and has an approximate inverse relationship with the oversampling factor. Our results are in form of necessary and sufficient statements that hold generically, hence only fail for elaborately-designed nongeneric examples. We provide extensive numerical verification of the theoretical results and demonstrate that the gap between the derived filter length prescriptions and the true minimum is small. The results have potential applications in synthesis FB design problems, where the analysis bank is given, and for analysis of fundamental limitations in blind signals reconstruction from data collected by unknown subsampled multi-channel systems.