"Rewiring" Filterbanks for Local Fourier Analysis: Theory and Practice

TL;DR

This paper introduces a theoretical framework for understanding filterbank rewirings and their effects on local Fourier analysis, enabling better analysis of signals with subsampling, multiplexing, or noise.

Contribution

It develops a general theory linking filterbank rewirings to convolution, modulation, and downsampling, enhancing analysis of complex, real-world signals.

Findings

Provides a framework for analyzing aliasing and modulation in subband signals

Enables closed-form analysis of signals with missing data or multiplexed acquisition

Improves understanding of local signal properties beyond traditional Fourier analysis

Abstract

This article describes a series of new results outlining equivalences between certain "rewirings" of filterbank system block diagrams, and the corresponding actions of convolution, modulation, and downsampling operators. This gives rise to a general framework of reverse-order and convolution subband structures in filterbank transforms, which we show to be well suited to the analysis of filterbank coefficients arising from subsampled or multiplexed signals. These results thus provide a means to understand time-localized aliasing and modulation properties of such signals and their subband representations--notions that are notably absent from the global viewpoint afforded by Fourier analysis. The utility of filterbank rewirings is demonstrated by the closed-form analysis of signals subject to degradations such as missing data, spatially or temporally multiplexed data acquisition, or signal-dependent noise, such as are often encountered in practical signal processing applications.