Wavelets and Time‐dependent Spectral Analysis

Abstract

One of the key features of wavelet analysis is its potential use for effecting time-frequency decompositions of non-stationary signals. The relationship between wavelet analysis and time-dependent spectral analysis has so far rested mainly on heuristic reasoning: in this paper we examine the relationship in a more precise mathematical form. A crucial feature of this analysis is the need to define carefully the notion of "frequency" when applied to non-stationary signals.The recent explosion of interest in wavelet analysis has stimulated renewed studies of time-dependent spectral analysis. The basic aim of wavelet analysis is to represent a function as a linear superposition of 'wavelets' centred on a sequence of time points, and in this sense it forms a natural tool for the investigation of 'local' properties of time varying functions. The literature on wavelets is replete with terminology such as 'discrete wavelet transforms', 'time-frequency localisation', 'windowed Fourier transforms', all of which indicate that, at some level, there should be a close connection between wavelet analysis and time-dependent spectra. Indeed, phrases such as 'local power spectra' are widely used in the wavelet literature, but so far the motivation for the use of this type of terminology has been highly intuitive and there has been little attempt to investigate the relationship between the two concepts in a formal and mathematically sound fashion. In pursuing this latter approach we encounter two basic difficulties: (a) almost all the literature on wavelet analysis deals with the representation of deterministic square-integrable functions, and (b) any form of time-dependent spectral analysis requires a very carefully constructed background theoretical model if the results of numerically calculated spectra are to have any meaningful physical interpretation. To put it quite simply, what meaning do sample spectra calculated from different time intervals possess? What functions are they estimating? In this paper we try to address these questions and establish a more analytical link between wavelet analysis and time-dependent spectral analysis. We first review briefly the basic ideas underlying wavelet analysis.

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