# Interpolation FIR filter Remez design criteria

I want to upsample an audio signal by a factor of 8.

The technique I was going to use was to upsample by zero stuffing and then interopolate using a low pass anti-aliasing FIR filter designed using the Remez exchange algorithm.

My question is: what are the design constraints for the Remez algorithm?

My initial thoughts are:

If I have an initial sampling rate of 44100Hz then my new sampling rate will be 352800‬Hz. When I zero stuff, my spectral content in the range 0-22050Hz will be folder around the new upsampled Nyquist of 176400Hz in the range ‭154350‬-176400Hz.‬ So I have the following constraints:

• Sample rate: 352800‬Hz
• Pass band: 0-22050Hz (0dB)
• Stop band: 154350-176400Hz (-120dB)

This does not produce an very good filter, so my assumption that I only have one spectral image folded around 176400Hz must be incorrect.

• you usually wouldn't use a remez design for a simple low-pass filter like this. Also, -120 dB is probably way more attenuation than you need. Then, your first spectral image is the original signal spectrum (-22.05 kHz to +22.05 kHz) repeated around 44.1 kHz, so your transition width is far, far too wide. I think the best advice I can give here is: Make a drawing of your spectrum after inserting the zeros, and carefully label the axes. Make sure you realize that your spectrum is symmetric to 0 Hz, because your signal is real-valued. – Marcus Müller Aug 28 '19 at 15:17
• (so, yes, your assumption is incorrect, but I'm not sure where that assumption comes from – you don't explain that. The easiest way to discuss this, as said, is probably by looking at the spectrum you'd draw.) – Marcus Müller Aug 28 '19 at 15:19
• @MarcusMüller, I took the information from this paper: lavryengineering.com/pdfs/… where it shows the spectrum fold around the new Nyquist (page 3). – keith Aug 28 '19 at 15:33
• Really not quite sure how you apply this to your interpolation problem - can you really draw the spectrum of the zero-stuffed signal for us? – Marcus Müller Aug 28 '19 at 15:58
• Any qualified lowpass filter will work for your interpolation problem. This of course includes the Remez (Parks-McClellan Equiripple) FIR filter too. What's your allowed / required filter order ? – Fat32 Aug 28 '19 at 20:22