Abstract

We review the general properties of the wavelet transform, both in its continuous and its discrete versions, in one or more dimensions. We also indicate some generalizations and applications in physics.

What is wavelet analysis?

Wavelet analysis is a particular time- or space-scale representation of signals which has found a wide range of applications in physics, signal processing and applied mathematics in the last few years. In order to get a feeling for it and to understand its success, let us consider first the case of one-dimensional signals.

It is a fact that most real life signals are nonstationary and usually cover a wide range of frequencies. They often contain transient components, whose apparition and disparition are physically very significant. In addition, there is frequently a direct correlation between the characteristic frequency of a given segment of the signal and the time duration of that segment. Low frequency pieces tend to last a long interval, whereas high frequencies occur in general for a short moment only. Human speech signals are typical in this respect: vowels have a relatively low mean frequency and last quite long, whereas consonants contain a wide spectrum, up to very high frequencies, especially in the attack, but they are very short.

Clearly standard Fourier analysis is inadequate for treating such signals, since it loses all information about the time localization of a given frequency component.