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?
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.
