If we already had Short-time Fourier transform for better analysis of a signal than Discrete Fourier Transform, then what was the need that leads to development of Wavelet Transform ?


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The short-time Fourier transform doesn’t offer better analysis of data than the discrete Fourier transform, it offers a different kind of analysis. The DFT offers an exact decomposition of data to a frequency representation. The STFT offers an approximate decomposition to a time/frequency representation. Which is better depends on what you are after. The same holds true of the Wavelet transform. Wavelet transforms can be thought of as decomposition to a time/frequency representation, but wavelet transforms generalize the concept of decomposition. Different wavelet functions have been created so you can choose a decomposition that suits your needs.

  • $\begingroup$ i know that different wavelets were developed by different people. they could have developed different STFT too then why they developed different windows for wavelets not for STFT ? $\endgroup$ Commented Mar 4, 2013 at 8:42
  • $\begingroup$ There are differen't windows available for the STFT (rectangle, triangle Welch, Hamming, Hanning, Blackman), but in all cases the STFT is still a sinusoidal decomposition. Step away from the sinusoids and you no longer are doing Fourier analysis. Wavelets fill this new space where other kinds of dunctions can be used for decomposition. Why? Because it exposes different features in the data. $\endgroup$
    – user2718
    Commented Mar 4, 2013 at 13:19
  • $\begingroup$ @BZ: There is overlap, though, with sinusoidal decompositions that vary in length: dsp.stackexchange.com/q/6266/29 $\endgroup$
    – endolith
    Commented Mar 4, 2013 at 19:02
  • $\begingroup$ @endolith Yes. I remember that discussion. It was a good one indeed. I didn't realize just how similar a wavelet transform and a windowed Fourier transform can be until someone on that thread made me look closely. It is great when someone on the site makes me see something I was missing. Hey, I think that was your doing :-) $\endgroup$
    – user2718
    Commented Mar 4, 2013 at 19:17

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