# 16x interpolation using polyphase FIR - sum of the coefficients

I have an issue with polyphase implementation during interpolation. Lets assume I have 256 taps long FIR lowpass sinc filter. In order to interpolate by a factor of 2 I do use two FIR filters (polyphase filter) out of the main one since I don't need to calculate zero stuffed samples. The same goes with interpolation by a factor of 4 in which case four sub-filters (polyphase filters) are used.

However, there is something which bothers me a lot and I don't know what to think about it - the sum of the coefficients within sub-filters should always equal to 1 (if the gain is 1). That is the case with polyphase filters for interpolation factors of 2 and 4. However, if I set interpolation to a factor of 8 or greater sum of coefficients within each polyphase does no longer create a correct sum (which should be close to 1).

## 1 Answer

Basically, you are up-sampling to a higher sample rate. The purpose of the interpolation filter is to eliminate the mirror-spectra, that you get from up-sampling and inserting zeros.

You need to chose the cutoff frequency for the interpolation filter accordingly. If you upsample by two you want a cutoff of fs/4 (or thereabouts). If you want to upsample by 8, your cutoff frequency needs to be lower, i.e. fs/16.

• Thank you! That is actually the case - by lowering the cutoff frequency accordingly all polyphase sub-filters and their coefficients do sum up to 1. Commented Jul 23, 2018 at 13:42
• @texan: I don't understand why you would do 16 up and 8 down instead of just doing 2 up. Commented Jul 24, 2018 at 15:12
• @texan Depends on why you have these different input rates. Your approach will work, if all input rates have the same bandwith and are low pass filtered below 24 kHz. If your 384 kHz signal has content above 24 kHz (and it should be able to have content up to 192 kHz), that content will be removed by your proposal. You have interpolated up to 768 kHz but the highest frequency in any of the upsampled signals would only be 24 kHz regardless of the original rate. Commented Jul 29, 2018 at 11:43