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I have a transfer function in Fourier space with $N=2028$ frequencies $(\frac {0, 1}{(N\cdot dx)} \dots ) $

Where $dx = 0.1m$.

I need to apply this transfer function to a signal with 20000 samples (also $dx=0.1m$). When I transform this signal to Fourier space I get 20000 frequencies. So, the sizes don't match. What do I need to do to be able to apply the transfer function (i.e. to multiply the Fourier transforms)? I guess, I could just remove the first $20000-2028=17952$ frequencies from the Fourier transform of the input signal, since they are not present in the transfer function.

But is that correct?

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That is not correct. The frequency range is still the same with 20000 bins, the bins are just closer. What you need to do is:

  • Apply a Window Function with lenght 2028 (such as Hamming Window) with an appropriate overlap between windows.
  • Fourier-transform each window.
  • Multiply each window by the transfer function.
  • Inverse-F-transform each window.
  • Overlap-Add the windows again to gain the filtered signal.
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For the signal that is shorter than the other, before you FFT it, zero-pad it so that the length is the same as the other.

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