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I sort of understand that when removing values from a signals DFT is not the same as a brick wall filter since it will only zero the frequencies that are perfectly periodic. Other spectrum frequencies who are not exactly periodic will have their energies spread over the whole DFT result. Is this right?

I'd like to know how you can mathematically prove that zeroing frequencies is not a brick wall filter using the DFT and inverse DFT.

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  • $\begingroup$ when you are "using the DFT and inverse DFT", are you filtering a signal much longer than the DFT length using something like Overlap-Add or Overlap-Save? $\endgroup$ – robert bristow-johnson May 6 '17 at 4:48
  • $\begingroup$ yeah the signal would be much larger than the length of the DFT. $\endgroup$ – user3485148 May 6 '17 at 5:00
  • $\begingroup$ so you're using Overlap-Add or Overlap-Save? $\endgroup$ – robert bristow-johnson May 6 '17 at 6:47
  • $\begingroup$ neither I think. I'm not looking to make a filter really I am more curious as to why zeroing certain frequencies of a frequency spectrum does not work as a brick wall filter. If I had a signal and took the DFT of a section of the signal, zeroed the result at certain frequencies and then took the IDFT why is the final result not a brick wall filter and how does the maths explain it. $\endgroup$ – user3485148 May 6 '17 at 7:43
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    $\begingroup$ i disagree with your premise. remember the DFT (and iDFT) inherently periodically extend the data passed to it. what comes out of the DFT are literally the Fourier coefficients of the periodically extended function. when you zero those frequency components, those frequencies have zero amplitude. if you do that to all frequency components above some index, it's a brick wall filter applied to your periodic function. $\endgroup$ – robert bristow-johnson May 6 '17 at 7:48
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A brick-wall filter (any filter that includes a piecewise continuous step function fragment in any portion of its frequency domain response) would be able to remove a signal of some frequency that is not integer periodic in the DFT width (somewhere is the flat portion of the step function).

By (waving hands, left as an exercise for the student, etc.) inspection, the only way to completely zero a non-zero sinusoid that has, say, an irrational frequency, is to zero the entire DFT of that signal. Therefore zeroing any subset of DFT bins (unless you zero all of them) can never perform a brick wall filtering.

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A true brickwall filter would require an infinite impulse response in the time domain, since the inverse Fourier Transform of a rectangular window (brickwall frequency response) is a Sinc function. Since for the DFT the number of samples in the frequency domain is equal to the number of samples in the time domain, this would require you to have an infinite number of samples in the frequency domain and then require you to null an infinite number of samples.

For anything less, the impulse response in time becomes an aliased Sinc function which no longer corresponds to a true brickwall frequency response.

If you restrict your frequency space to just the DFT samples however, you will of course achieve a brickwall response following the rules that all samples outside of your rectangular window (in frequency) are zero. However in the implementation of a filter we are often concerned with the frequency response of signals that exist on a continuous frequency domain, in which case as described above you will not be able to achieve a perfect brickwall frequency response.

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