Modified Tukey window applied in frequency domain

Question

In analysis of transient signals (ground motion time-series) in seismology, many operations are performed in frequency domain and it is common practice to apply a filter which is like a modified Tukey window in frequency domain to keep data band-limited before applying an inverse IFFT to go back to time-domain. Essentially, 4 frequencies f1, f2, f3 and f4 are specified and the filter is a combination of high pass and low pass taper. Here quoting from a web manual- "The taper is unity between f2 and f3 and zero below f1 and above f4. Frequencies f1 and f2 specify the high-pass filter at low frequencies, while frequencies f3 and f4 specify the low-pass filter at high frequencies. Both f3 and f4 should be less than the Nyquist frequency: 0.5/DELTA. The filters applied between f1 and f2 and between f3 and f4 are quarter cycles of a cosine wave. To avoid ringing in the output time series, a suggested rule-of-thumb is f1 = f2/2 and f4 >= 2*f3".

Can someone please explain what is the reason behind the rule of thumb mentioned at the end ? Is there any specific advantage of using such a filter compared to other filters like an acausal Butterworth filter ?

I have added links to 2 figures (for some reason, upload image is not working for me) which show time and frequency domain response of 4 filters (cosine tapers with f2=2f1 and f4=2f3 [black], and f2=1.25f1 and f4=1.25f3 [red] with f2=0.2 Hz and f3=0.9 Hz) and 2-pass bandpass Butterworth filters with 4 and 8 poles applied at frequencies f2 and f3.

Answer 1

Regarding the 2:1 frequency ratio recommendation for the transition band (f2:f1 or f4:f3):

The steeper the filter response (i.e. the smaller the frequency ratio between pass band and stop band limits), will necessarily produce a much longer impulse response and produce ringing artifacts in the time domain whenever there is a transient.

These filters, which are applied in the frequency domain, also seem to have linear phase response, which is very desirable for analysing this type of signal.

Butterworth filters have a compact and nice representation as difference equations (for filtering in the time domain), but do not have linear phase response, and as you see from your plots, have not very good transition between pass band and stop band.

(I don't understand how you plotted a symmetric impulse response (time domain) for Butterworth filters; their impulse response is causal and asymmetric in general.)