## Abstract

In the previous chapter, we showed that a key issue with the short-time Fourier transform (STFT) is the choice of the window length, given the basic limitation imposed by the uncertainty principle of signal analysis. Short windows give good time (but bad frequency) resolution, and conversely, long windows give good frequency (but bad time) resolution (see Sect. 3.3). In the late 1970s, Jean Morlet, a geophysicist working for a French oil company, realized that the STFT was not suitable for the study of his seismic data. He observed that a good compromise between time and frequency resolution was not possible because high-frequency patterns had a shorter duration compared to the low-frequency ones. So, a single window for all frequencies would not do. His solution was quite straightforward: he just took different window sizes for different frequencies, or more accurately, he took a cosine function tapered with a Gaussian (a Gabor function, see Sect. 3.2) and compressed it or stretched it in time to get the different frequencies (see Fig. 4.1). Then, instead of always having the same window size, he had the same wave shape at different scales, that is, with a variable size. With this simple trick, he created the basis of wavelets!

## Keywords

Wavelet Coefficient Digital Filter Wavelet Function Mother Wavelet Haar Wavelet## Supplementary material

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