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