# Creating a 4th order Butterworth filter R

I have sampled my data at 2000 Hz and want to generate a fourth order low-pass zero-lag filter with a cut-off frequency of 150 Hz.

I have put the code below that I have created but I have the following questions. Given that I have used the filtfilt function do I need to change the number to 2 so that it is a second-order and should I also double the cut-off frequency i.e change W to 0.3 from 0.15.

rawdata = data$$Moment unfiltered bf = butter(4,0.15,type = c("low"), plane = c("z")) filtereddata = filtfilt(bf, data$$Moment unfiltered)


do I need to change the number to “2” so that it is a second-order

Yes

and should I also double the cut-off frequency i.e change “W” to 0.3 from 0.15.

No

Strictly speaking, doubling up the filter will make the gain at the cutoff frequency -6dB instead of -3dB. So you have created a Linkwitz-Riley and not a Butterworth filter.

If you want a gain of exactly -3dB at 150 Hz you would need to increase the frequency slightly. You can work out the math by calculating the -1.5dB point of a Butterworth filter, but trial and error is probably quicker. 185Hz should get you fairly close.

In any case, the shape will still NOT be that of a Butterworth. Cascading two Butterworth filters does not result in another Butterworth filter. The zeros are in the right location but the poles are simply duplicated and are not Butterworth-distributed any more.

• Ahhh I did not know that the frequency had to be slightly different, brilliant, thank you very much. I will change the order to 2 and keep as 0.15 as I am not too confident with the maths side of things. Commented Mar 18, 2021 at 17:18