23 - Short-Time Fourier Transform and Introduction to Wavelet Analysis

Summary

This chapter discusses the drawback of Fourier analysis and the methods that can overcome its limitations. In general, Fourier analysis does not include information about time, particularly events. A slight modification of Fourier analysis can allow the addition of a dimension in time: by dividing the time series into smaller segments and doing the Fourier Transform for each segment, a method called short-time Fourier Transform (STFT) is introduced. Wavelet analysis is then discussed as a much better alternative to or replacement for STFT. It involves scaled and translated convolution with a short base function (short in the sense that it is essentially non-zero only in a finite interval). Wavelet analysis uses different base functions than the Fourier Transform. They are limited in time (unlike the infinitely long sinusoidal functions) and can be stretched or compressed to represent different scales (equivalent to frequencies). This method will allow the resolution of events at different times and different scales.