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I have some complex data in the frequency domain and want to multiply by an appropriate function to get complex data in the frequency domain that is appopriately filtered.

I think such a function should be, in terms of frequency:

$$ 1 - \frac{1}{\sqrt{1 + (\frac{fW}{f^2 - f_0 ^2})^{2n}}} $$

where W is the width of the frequency band to pass, and $f_0$ is the center frequency.

Is the above correct, or what would be the correct function to accomplish this?

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  • $\begingroup$ that looks to me more like the magnitude of a band-reject filter, not a bandpass filter. do you want the actual transfer function (with powers pf $s$)? there's a bit more to do, to get this. $\endgroup$ – robert bristow-johnson Jun 12 '20 at 16:12
  • $\begingroup$ I am admittedly not too familiar with signal processing, although I don't believe I am looking for one with powers of s. I am basically looking for some function that when multiplied by complex data in the frequency domain, yields new complex data in the frequency domain that has been filtered. $\endgroup$ – theta Jun 12 '20 at 16:14
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okay, the magnitude frequency response for a general $N$th-order Butterworth lowpass filter is:

$$ \Big|H_\mathrm{LP}(f)\Big|^2 = \frac{1}{1 + \left(\frac{f^2}{W^2} \right)^N} $$

for bandpass filtering, centered at frequency $f_0$, this substitution is made to the LPF:

$$ f \leftarrow \frac{f}{f_0} - \frac{f_0}{f} $$

so then it comes out as:

$$ \Big|H_\mathrm{BP}(f)\Big|^2 = \frac{1}{1 + \left(\frac{\left(\frac{f}{f_0} - \frac{f_0}{f}\right)^2}{W^2} \right)^N} $$

blast that out and you'll have the frequency response for an $2N$th-order Butterworth bandpass filter.

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