Introduction to redundancy rules: the continuous wavelet transform comes of age

Abstract

Redundancy: it is a word heavy with connotations of lacking usefulness. I often hear that the rationale for not using the continuous wavelet transform (CWT)-even when it appears most appropriate for the problem at hand-is that it is 'redundant'. Sometimes the conversation ends there, as if self-explanatory. However, in the context of the CWT, 'redundant' is not a pejorative term, it simply refers to a less compact form used to represent the information within the signal. The benefit of this new form-the CWT-is that it allows for intricate structural characteristics of the signal information to be made manifest within the transform space, where it can be more amenable to study: resolution over redundancy. Once the signal information is in CWT form, a range of powerful analysis methods can then be employed for its extraction, interpretation and/or manipulation. This theme issue is intended to provide the reader with an overview of the current state of the art of CWT analysis methods from across a wide range of numerate disciplines, including fluid dynamics, structural mechanics, geophysics, medicine, astronomy and finance.This article is part of the theme issue 'Redundancy rules: the continuous wavelet transform comes of age'.

Keywords: continuous wavelet transform; signal processing; time-frequency representation.

Figure 1.

Wavelet transform of a frequency drifting and modulated test signal. (a) The complex Morlet wavelet (real part: continuous line. Imaginary part: dashed). (b) Test signal modulated and drifting in frequency. (c) Fourier transform power spectrum of the test signal and (d) wavelet transform modulus of the test signal. (Greyscale: black to white = lowest to highest amplitude.) Band ridge shown as a superimposed line. (e) Instantaneous ridge signal (frequency against time) extracted from the band maxima locus. (f) Three-dimensional view of the CWT modulus surface for panel (d). (The cover illustration to this themed issue has a three-dimensional plot the CWT modulus for the same signal but with added Gaussian noise.) (g) A secondary wavelet transform of the band ridge signal in (e).

Figure 2.

Manipulating the transform. (a) Test signal containing two sinusoidal components. (b) Wavelet transform modulus. (Greyscale: black to white = lowest to highest amplitude.) (c) Transform phase. (Greyscale: black to white = −π to π.) (d) Synchrosqueezed transform. (e) Test signal containing underlying component with modulations and a break in the signal, high-frequency noise and large amplitude artefact. (f) CWT modulus of (e). (g) Running wavelet archetype of (f).