Fourier's theorem assumes we add sine waves of infinite duration.
As a consequence, the Fourier Transform is good at representing signals which are long and periodic.
But the Fourier Transform has problems when used with signals which are short, and not periodic.
Other transforms are possible - fitting the data with different sets of functions than sine waves.
The trick is, to find a transform whose base set of functions look like the signal with which we are dealing.
The diagram shows a signal that is not a long, periodic signal but rather a periodic signal with a decay over a short time. This is not very well matched by the Fourier Transform's infinite sine waves. But it might be better matched by a different set of functions - say, decaying sine waves. Such functions are called 'wavelets' and can be used in the 'wavelet transform'.
Note that the wavelet transform cannot really be used to measure frequency, because frequency only has meaning when applied to infinite sine waves. But, as with the Short Time Fourier Transform, we are always willing to stretch a point in order to gain a useful tool.
The Fourier Transform's real popularity derives not from any particular mathematical merit, but from the simple fact that some one (Cooley and Tukey) managed to write an efficient program to implement it - called the Fast Fourier Transform (FFT). And now there are lots of FFT programs around for all sorts of processors, so it is likely the FFT will remain the most popular method for many years because of its excellent support.
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