The Fast Wavelet Transform

The Fourier transform has been used as a reliable tool in signal analysis for many years. Invented in the early 1800s by its namesake, Jean Baptiste Joseph Fourier, the Fourier transform has become a cornerstone of modern signal analysis. The Fourier transform has proven incredibly versatile in applications ranging from patten recognition to image processing. Nevertheless, it suuers from certain limitations. Recently, a new kind of transform, the wavelet transform, has been shown to be as powerful and versatile as the Fourier transform, yet without some of its limitations. The wavelet transform is the result of work by a number of researchers. Initially, a French geophysicist, Jean Morlet, came up with an ad hoc method to model the process of sound wavelet traveling through the earth's crust. Unlike Fourier analysis, he did not use sine and cosine curves, but simpler ones which he called "wavelets." Yves Meyer, a French mathematician, recognized this work to be part of the eld of harmonic analysis, and came up with a family of wavelets that he proved were most eecient for modeling complex phenomena. This work was improved upon by two researchers in America, Stephane Mallat of New York University and Ingrid Daubechies of Bell Labs. Since 1988, there has been a small explosion of activity in this area, as engineers and researchers apply the wavelet transform to applications ranging from image compression to ngerprint analysis. The wavelet transform has even been implemented in silicon, in the form of a chip from Aware Inc. But before discussing how the wavelet transform works, let's rst examine its predecessor. A function in the time domain is translated by the Fourier transform into a function in the frequency domain, where it can be analyzed for its frequency content. This translation occurs because the Fourier transform expands the original function in terms of orthonormal basis functions of sine and cosine waves of innnite duration. The Fourier coeecients of the transformed function represent the contribution of each sine and cosine wave at each frequency. The Fourier transform works under the assumption that the original time-domain function is periodic in nature. As a result, the Fourier transform has diiculty with functions that have transient components, that is, components which are localized in time. This is especially apparent when a signal has sharp transitions. Another problem is that the Fourier transform of a signal does not convey any information pertaining to …