Fourier Analysis and Wavelet Analysis

658 NOTICES OF THE AMS VOLUME 44, NUMBER 6 I n this article we will compare the classical methods of Fourier analysis with the newer methods of wavelet analysis. Given a signal, say a sound or an image, Fourier analysis easily calculates the frequencies and the amplitudes of those frequencies which make up the signal. This provides a broad overview of the characteristics of the signal, which is important for theoretical considerations. However, although Fourier inversion is possible under certain circumstances, Fourier methods are not always a good tool to recapture the signal, particularly if it is highly nonsmooth: too much Fourier information is needed to reconstruct the signal locally. In these cases, wavelet analysis is often very effective because it provides a simple approach for dealing with local aspects of a signal. Wavelet analysis also provides us with new methods for removing noise from signals that complement the classical methods of Fourier analysis. These two methodologies are major elements in a powerful set of tools for theoretical and applied analysis. This article contains many graphs of discrete signals. These graphs were created by the computer program FAWAV, A Fourier–Wavelet Analyzer, being developed by the author. Frequency Information, Denoising As an example of the importance of frequency information, we will examine how Fourier analysis can be used for removing noise from signals. Consider a signal f (x) defined over the unit interval (where here x stands for time). The period 1 Fourier series expansion of f is defined by ∑ n∈Z cnei2πnx, with cn = ∫ 1 0 f (x)e−i2πnx dx. Each Fourier coefficient, cn , is an amplitude associated with the frequency n of the exponential ei2πnx. Although each of these exponentials has a precise frequency, they all suffer from a complete absence of time localization in that their magnitudes, |ei2πnx| , equal 1 for all time x. To see the importance of frequency information, let us examine a problem in noise removal. In Figure 1(a)[top] we show the graph of the signal