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I'm still in the first stages of learning about these different transformations. But one thing that came to my mind while learning about them is the question above. I learned that FT can only detect the frequencies in a signal without giving us much knowledge about when these frequencies happen. Therefore, WT came here to be a great solution. However, does WT do any better job than FT in detecting the frequencies, or is the time problem the only improvement?

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  • $\begingroup$ FT is a frequency transformation, CWT and STFT are time-frequency transformations. $\endgroup$ – endolith Jun 28 '17 at 16:50
  • $\begingroup$ Yes, you are right! But If I read the frequency from a CWT plot, would it be more accurate from reading it from an FT plot? $\endgroup$ – Robot0110 Jun 28 '17 at 17:48
  • $\begingroup$ @ I'm not sure they can be compared, since CWT gives you many measurements of the frequency, and FT only gives you one. If the frequency is constant for the entire signal, FT would be more accurate because it covers more of the signal. $\endgroup$ – endolith Jun 28 '17 at 18:01
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Since they seem more fashionable than Fourier transforms, some persons tend to overuse wavelets in contexts where they do not belong, or at least don't do a much better job.

For "knowing the frequencies", Fourier transforms are the correct tools to use first: it is so easy to badly select a wavelet transformation (including its shape and its sampling) that will give you outcomes difficult to interpret, and even erroneous information. To be more specific, among fast orthogonal transforms (DWT and FFT), due to the restrictions on dyadic wavelets, it is unlikely to have better results with wavelets than Fourier (because of shift-variance, aliasing and badly-shaped FIR wavelets).

does WT do any better job than FT in detecting the frequencies

It really depends on what you expect in detection, and the nature of your signals, and it often takes more than a transform. However, I would recommend to use wavelets or time-frequency tools if Fourier reveals not sufficient, or in parallel to address statonnarity, etc., but not instead.

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