A Kotel'nikov Representation for Wavelets
For researchers in wavelet theory and signal processing, this work offers a new perspective on orthogonality conditions for wavelets, though it is incremental as it builds on existing Kotel'nikov sampling theory.
This paper derives a condition for orthogonal wavelet analysis based on spectral support limits, showing that if the wavelet spectral support is within [f_m, f_M] and f_M ≤ 3f_m, orthogonal analysis is guaranteed. It also provides a method to construct an equivalent filter bank with no spectral overlap for wavelets not meeting this condition.
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.