Beyond Fourier: The Wavelet Transform

Abstract

Wavelet analysis is a relatively recent signal processing tool that has been successfully used in a number of fields. This chapter presents a general overview of wavelet analysis by emphasizing its relationship to the Fourier transform. Although formulas are used to support important concepts, mathematical rigor is left aside in the interest of simplicity and clarity. The reader may refer to the bibliography for a more complete and rigorous description of the subject. The continuous wavelet transform can be introduced through the concept of time-frequency analysis. In chapter 2 we discussed the concept of windows to isolate short records of a long sequence. A generalization of this technique is the windowed Fourier transform. Because the window may be placed anywhere in the signal, the windowed Fourier transform is a time-frequency extension of the usual Fourier transform. Is is used here to link the concepts of the Fourier transform and wavelet transform. Multiresolution analysis is then introduced and to leads to an efficient algorithm for computing the wavelet transform of a discrete signal. Finally, some applications of wavelet analysis in the biomedical domain are discussed.