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.
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!).