For a given narrowband Gaussian filter with a specific central frequency and filter width, I need corners of a bandpass Butterworth filter whose amplitude response is close enough to the Gaussian filter. How should I choose the order and corner frequencies of the Butterworth filter ? The reason is I need to calculate statistics of the initial part of a time-series filtered with a Gaussian filter. However, the Gaussian filter is zero-phase and always shifts energy back and contaminates the initial part of the time-series. With the Butterworth filter, I can have a one-pass filter so that the filtered time-series is causal and there is no contamination of the initial part of the time-series.

  • $\begingroup$ Define "close enough". Also, are you doing this in digital or analog? What was the reason (design objective) to pick a Gaussian Filter? $\endgroup$ – Marcus Müller Apr 20 '17 at 9:51
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    $\begingroup$ Why don't you design a causal filter with gaussian frequency response instead? $\endgroup$ – Jazzmaniac Apr 20 '17 at 9:58
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    $\begingroup$ You will never get a Butterworth filter to be a Gaussian filter. You need a transitional filter. A transitional filter allows you to adjust the filter's parameters to be as close to a Butterworth, or a Gaussian, as you please, but you cannot get a filter to give both responses at the same time. The "Adjustable Gauss Filter" is an example of such a filter. $\endgroup$ – user5108_Dan Apr 20 '17 at 23:23
  • $\begingroup$ @Jazzmaniac so do you think I should have a filter with amplitude response of a Gaussian filter but phase response of a simple causal Butterworth filter ? $\endgroup$ – Guddu Apr 22 '17 at 1:10
  • $\begingroup$ @MarcusMüller the task is narrowband filtering to measure frequency-dependent arrival time of a specific signal using the peak of the envelope of the filtered signal. i am not sure whether it is analog or digitial. But I take the signal to frequency domain using FFT, then apply the frequency-domain response of the Gaussian filter and then do IFFT. For Butterworth filter I calculate the poles of the filter and calculate the response $\endgroup$ – Guddu Apr 22 '17 at 1:11

The solution is to implement a FIR filter with the coefficients using the Gaussian response desired. The center dominant tap will set the delay of the filter, which will of course not be zero phase, and the result will be the causal one-pass filter desired.

Also related, see my post at the link below on how you can implement this filter as a cascade of 2 tap unity gain filters (it doesn't get much simpler than that!).

Gaussian FIR filter with no multipliers?


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