A Kotel'nikov Representation for Wavelets

TL;DR

This paper introduces a wavelet representation based on Kotel'nikov's theory, linking wavelet analysis to band-limited signals and filter banks, providing conditions for orthogonality and methods to construct non-overlapping filter banks.

Contribution

It offers a novel wavelet representation using baseband signals and Kotel'nikov's results, and establishes simple conditions for orthogonality and filter bank construction.

Findings

Orthogonal analysis guaranteed if $f_M \,\leq 3f_m$

Method to construct non-overlapping filter banks for certain wavelets

Revisits wavelet analysis as a filter bank with constant quality factor

Abstract

This paper presents a wavelet representation using baseband signals, by exploiting Kotel'nikov results. Details of how to obtain the processes of envelope and phase at low frequency are shown. The archetypal interpretation of wavelets as an analysis with a filter bank of constant quality factor is revisited on these bases. It is shown that if the wavelet spectral support is limited into the band $[f_m,f_M]$, then an orthogonal analysis is guaranteed provided that $f_M \leq 3f_m$, a quite simple result, but that invokes some parallel with the Nyquist rate. Nevertheless, in cases of orthogonal wavelets whose spectrum does not verify this condition, it is shown how to construct an "equivalent" filter bank with no spectral overlapping.