# Modified Tukey window applied in frequency domain

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.