“Rewiring” Filterbanks for Local Fourier Analysis: Theory and Practice

This paper 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 - as well as signal recovery from sampled sequences based on their filterbank characterizations. 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, the likes of which are often encountered in practical signal processing applications.