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Imagine a filterbank with N channels. A possible implementation is a sliding short-time FFT. If you want to capture frequencies as low as 100 Hz, the length of your sliding window should be at least 10 ms. However, because of this long window length (or in general filter length), you cannot track fast temporal fluctuations in the energy at high frequencies.

Is this ever considered as a problem? How would it be adressed? A possible solution would be to different filter lengths for different channels (i.e. shorter filters for longer channels). However,

  • I guess that in this case you lose the efficiency of the FFT algorithm?
  • Does it even make sense to consider fast energy fluctuations at higher frequencies, while assuming that the energy in lower frequencies keeps constant?

Thanks in advance!

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  • $\begingroup$ Take a quick google of 'wavelet transform'. This transform attempts to address the problem you state. $\endgroup$ – Izzo Apr 20 '17 at 14:23
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Using different filter lengths for different channels is basically what the Wavelet Transform does. I think it basically addresses this issue. Wavelet transforms divide the spectrum up into spectral bins of unequal width. There are efficient options for computing the Wavelet Transform (a tree structured filter bank approach usually).

In the linked Wikipedia Article, you can see a picture comparing the time-frequency tiling of the STFT to that of the Wavelet Transform. It is directly applicable to your question.


I realize that I addressed the "How can I" part of your question, but not the "Should I" part... I'm not sure I can answer the "Should I" part unless I am given more information regarding the target application. So, I will briefly say that Wavelets have found widespread use in compression and image recognition. They are often helpful for multiresolution analysis which (in rough terms) is when one wishes to find something of a certain shape but has no idea how large it will be (for example a pulse of a certain shape that may have been compressed or expanded significantly). If you don't require these types of properties, then a uniform filter bank is often the way to go.

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  • $\begingroup$ Is the basis of wavelets also complete (in the discrete time domain), in the sense that summing the wavelet components also results in the exact signal you started from? $\endgroup$ – BNJMNDDNN Apr 20 '17 at 17:53
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    $\begingroup$ Wavelets are filter banks with a certain prototype filter. As with filter banks, the general answer is no. Lest this discourage you though, wavelets that satisfy certain conditions (the perfect reconstruction conditions from filter bank theory) are complete. A quick google search of wavelets perfect reconstruction resulted in a lot of useful material on the subject. $\endgroup$ – hops Apr 20 '17 at 18:28
  • $\begingroup$ In filter bank theory, the perfect reconstruction conditions apply to the prototype filter in case that was unclear. $\endgroup$ – hops Apr 20 '17 at 18:29

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