How to design a resampling filter?

Question

All the references I can find only describe how to design an idealized lowpass filter for doing resampling. For example, MATLAB's documentation describes how to choose a "cutoff frequency" (for rational-ratio resampling):

Something like this:

We can't implement an ideal filter, so a slightly better approximation would be to introduce a transition band:

So, how should I set the passband edge $f_{pass}$ and stopband edge $f_{stop}$?

The only idea I have is to consider halfband filters. In halfband filters, there is symmetry about $f_c$. So, for example, in a halfband decimator, we could set $f_{pass} = 0.9f_c$ and we would get $f_{stop} = 1.1f_c$. This works rather nicely because anything input between $f_c$ and $f_{stop}$ will only alias back into the transition band (so the passband $0$ - $f_{pass}$ is clean).

Is that a reasonable approach in general? We just place $f_{pass}$ and $f_{stop}$ symmetrically about the theoretical cutoff frequency $f_c$?

Answer 1

Like most filter designs, choosing the "best" resampling filter involves a fairly complicated tradeoff that depends on the requirements of your specific application and the properties of the signals you want to resample.

These tradeoffs include

1. Usable bandwidth
2. Passband artifacts in magnitude, phase and group delay
3. Amount and frequency range of residual aliasing
4. Causality and pre-ringing in the time domain.
5. Transient preservation in the time domain.
6. Edge effects
7. Latency
8. Implementation properties (CPU, memory, real-time requirements, etc.)

You typically need to figure out what's the best combination of all of these depending on what you need and, even more so, which type of artifacts or residual error you are most sensitive to.