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The Short Time Fourier Transform (STFT) is used to identify time localized frequency content of a signal. The STFT operates by chunking an input signal into blocks and performing FFT on the block, and then recording the frequency coefficients for that point in time. However, the STFT enforces fixed time and frequency resolution which is not always desired.

A wavelet transform is an alternative method for identifying the frequency content of a signal where the window size typically varies based on the frequency, thus creating an inverse relation ship between time localization and frequency localization. That is: high-frequencies have good time localization but poor frequency localization, and low-frequencies have good frequency localization but bad time localization.

Let's say I want to use a wavelet transform as an alternative to the STFT. What specific wavelet should I use? Do some wavelets work better for encoding frequency content?

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Just because the math is easier, I might recommend the Morlet or possibly the Chirplet as the mother wavelet.

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As already answered by @robert bristow-johnson, the Morlet one is quite efficient for many purposes, and the Gabor wavelets are close cousins.

The question of the best continuous wavelet transform is a long standing debate. What is almost settled is that it is preferable to use a complex-valued wavelet, or at least append its dual-Hilbert pair to get complex scalograms.

I remember people promoting the use of wavelets with Poisson kernels, especially for signals modelsof a superposition of delayed dampled sinusoids (eg authors from Identification of sources of potential fields with the continuous wavelet transform: Basic theory).

Recently, J.-M. Lilly and Sofia Olhede described the two-parameter wavelet family of Generalized Morse wavelets. With the two parameters, you can emulate a quantity of known wavelets

Generalized Morse wavelets

"subsuming eight apparently distinct types of analysis filters into a single common form" (L” for the lognormal wavelets, “C” for the Cauchy wavelets, “G” for the Derivative of Gaussian wavelets, “A” for the Airy wavelets, “e” for complex exponentials, “S” for the Shannon wavelet, “a” for the analytic filter, and “B” for the Bessel wavelet). Generalized Morse wavelets are implemented in Matlab, and I would probably go for them.

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  • $\begingroup$ This answer is better than mine. And Izzo, if you choose to, you can change your check mark. $\endgroup$ – robert bristow-johnson Nov 9 '20 at 3:30
  • $\begingroup$ well, i don't understand how this answer isn't more complete and simply better than mine. $\endgroup$ – robert bristow-johnson Nov 9 '20 at 4:20

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