Wvelets and Sianal U Processin 14 OLlVlER RlOUL and MARTIN VETTERLI avelet theory provides a unified framework for a number of techniques which had been developed inde- pendently for various signal processing applications. For ex- ample, multiresolution signal processing, used in computer vision; subband coding, developed for speech and image compression; and wavelet series expansions, developed in applied mathematics, have been recently recognized as different views of a single theory. In fact, wavelet theory covers quite a large area. It treats both the continuous and the discrete-time cases. It provides very general techniques that can be applied to many tasks in signal processing, and therefore has numerous potential applications. In particular, the Wavelet Transform (WT) is of inter- est for the analysis of non-stationary signals, because it provides an alternative to the classical Short-Time Fourier Transform (STFT) or Gabor transform [GAE346, ALL77, POR801. The basic difference is as follows. In contrast to the STET, which uses a single analysis window, the WT uses short windows at high frequencies and long windows at low frequencies. This is in the spirit of so-called “constant-Q” or constant relative bandwidth frequency analysis. The WT is also related to time-frequency analysis based on the Wigner-Ville distribution [FLA89, FLASO, RIOSOal. For some applications it is desirable to see the WT as a signal decomposition onto a set of basis functions. In fact, basis functions called wauelets always underlie the wavelet analysis. They are obtained from a single prototype wavelet by dilations and contractions (scal- IEEE SP MAGAZINE OClOBER 1991 Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 29,2010 at 16:23:44 EDT from IEEE Xplore. Restrictions apply.ings) as well as shifts. The prototype wavelet can be thought of as a bandpass filter, and the constant-Q property of the other bandpass filters (wavelets) follows because they are scaled versions of the prototype. Therefore, in a WT, the notion of scale is introduced as an alternative to frequency, leading to a so-called time-scale representation. This means that a signal is mapped into a time-scale plane [the equivalent of the time-frequency plane used in the STFT). There are several types of wavelet transforms, and, depending on the application, one may be preferred to the others. For a continuous input signal, the time and scale parameters can be continuous [GR089], leading to the Continuous Wavelet Transform (CWT). They may as well be discrete (DAU88, MAL89b, MEY89, DAUSOa], leading to a Wavelet Series expansion. Finally, the wavelet transform can be defined for discrete-time sig- nals [DAU88, RI090b. VETSOb], leading to a Discrete Wavelet Transform (DWT). In the latter case it uses multirate signal processing techniques [CRO83] and is related to subband coding schemes used in speech and image compression. Notice the analogy with the (Con- tinuous) Fourier Transform, Fourier Series, and the Discrete Fourier Transform. Wavelet theory has been developed as a unifying framework only recently, although similar ideas and constructions took place as early as the beginning of the century [HAAlO, FRA28, LIT37, CAL641. The idea of looking at a signal at various scales and analyzing it with various resolutions has in fact emerged inde- pendently in many different fields of mathematics, physics and engineering. In the mid-eighties, re- searchers of the “French school,” lead by a geophysicist, a theoretical physicist and a mathematician (namely, Morlet, Grossmann, and Meyer), built strong mathe- matical foundations around the subject and named their work “Ondelettes” (Wavelets). They also interacted considerably with other fields. The attention of the signal processing community was soon caught when Daubechies and Mallat, in ad- dition to their contribution to the theory of wavelets, established connections to discrete signal processing results [DAU88], [MAL89a]. Since then, a number of theoretical, as well as practical contributions have been made on various aspects of WT’s, and the subject is growing rapidly [wAV89], [IT92]. The present paper is meant both as a review and as a tutorial. It covers the main definitions and properties of wavelet transforms, shows connections among the various fields where results have been developed, and focuses on signal processing applications. Its purpose is to present a simple, synthetic view of wavelet theory, with an easy-to-read, non-rigorous flavor. An extensive bibliography is provided for the reader who wants to go into more detail on a particular subject. NON-STATIONARY SIGNAL ANALYSIS The aim of signal analysis is to extract relevant information from a signal by transforming it. Some methods make a priori assumptions on the signal to be analyzed: this may yield sharp results if these assump- tions are valid, but is obviously not of general ap- plicability. In this paper we focus on methods that are applicable to any general signal. In addition, we con- sider invertible transformations. The analysis thus un- ambiguously represents the signal, and more involved operations such as parameter estimation, coding and pattern recognition can be performed on the “transform side,” where relevant properties may be more evident. Such transforms have been applied to stationay signals, that is, signals whose properties do not evolve in time (the notion of stationarity is formalized precisely in the statistical signal processing literature). For such signals dt), the natural “stationary transform” is the well-known Fourier transform (FOU88l: The analysis coefficients Xu define the notion of global frequencyfin a signal. As shown in (l), they are computed as inner products of the signal with sinewave basis functions of infinite duration. As a result, Fourier analysis works well if x( t) is composed of a few stationary components (e.g., sinewaves). However, any abrupt change in time in a non-stationary signal x(t) is spread out over the whole frequency axis in xyl. Therefore, an analysis adapted to nonstationay signals requires more than the Fourier Transform. The usual approach is to introduce time dependency in the Fourier analysis while preserving linearity. The idea is to introduce a “local