Overview

Back in the early 1800s, the French mathematician Joseph Fourier discovered that any periodic function can be expressed as a (possibly infinite) sum of sines and cosines. This surprising fact is now known as Fourier expansion and it has many applications in engineering, mainly in the analysis of signals. It can isolate the various frequencies that underlie a signal and thereby enable the user to study the signal and also edit it by deleting or adding certain frequencies. The downside of Fourier expansion is that it does not tell us when (at which point or points in time) each frequency is active in a given signal. We therefore say that Fourier expansion offers frequency resolution but no time resolution.

Wavelet analysis (or the wavelet transform) is a successful approach to the problem of analyzing a signal both in time and in frequency. Given a signal that varies with time, we select a time interval, and use the wavelet approach to identify and isolate the frequencies that constitute the signal in that interval. The interval can be wide, in which case we say that the signal is analysed on a large scale. As the time interval gets narrower, the scale of analysis is said to become smaller and smaller. A large scale analysis illustrates the global behavior of the signal, while each small scale analysis illuminates the way the signal behaves at a short interval of time; it is like zooming in the signal in time, instead of in space.