FROM FOURIER TRANSFORMS TO WAVELET ANALYSIS: MATHEMATICAL CONCEPTS AND EXAMPLES

Abstract. This paper studies two data analytic methods: Fourier transforms and wavelets. Fourier transforms approximate a function by decomposing it into sums of sinusoidal functions, while wavelet analysis makes use of mother wavelets. Both methods are capable of detecting dominant frequencies in the signals; however, wavelets are more ecient in dealing with time-frequency analysis. Due to the limited scope of this paper, only Fast Fourier Transform (FFT) and three families of wavelets are examined: Haar wavelet, DaubJ, and CoifI wavelets. Examples for both methods work on one dimensional data sets such as sound signals. Some simple wavelet applications include compression of audio signals, denoising problems, and multiresolution analysis. These applications are given for comparison purposes between Fourier and wavelet analysis, as well as among wavelet families. Although wavelets are a recent discovery, their ecacy has been acknowl-edged in a host of elds, both theoretical and practical. Their applications can be expanded to two or higher dimensional data sets. Although they are omitted in this paper, more information is available in [7] or many other books on wavelet applications. 1