Cut-off frequencies for fractional sample rate adjustment

Question

We have a signal sampled att 22 kHz that we want to interpolate to 40 kHz.

So we can do this by upsampling by a factor 20, then downsampling by a factor 11. My question regards the choice of cut-off frequencies after the upsampling and the downsampling.

The old Nyquist frequency was at $11$ kHz so it would make sense to me to low-pass frequencies above $11$ kHz after upsampling - we're basically replicating our signal 20 times across the new spectra, right? The correct cut-off is apparently 20 kHz, but I can't quite wrap my head around that fact.

We then downsample by a factor 11 to 40 kHz. Now the correct LP cut-off is 11 kHz. I am seriously confused. Can anyone shed some wisdom here?

Answer 1

Don't be confused; you're doing everything correctly here:

$$\begin{align*} f_{sample,in} &&\overset{\text{interpolate}}{\rightarrow}&& f_{sample,intermediate} &&\overset{\text{decimate}}{\rightarrow}&& f_{sample,out}\\ 22 \mathrm{kHz}&&\overset{\uparrow 20}{\rightarrow}&& 440 \mathrm{kHz} &&\overset{\downarrow 11}{\rightarrow}&& f_{sample,out} \end{align*}$$

1. To avoid images after interpolating, you will have to suppress all signal above 11kHz, which demands low pass with a cut off of $\frac1{2\cdot20}f_{sample,intermediate}$.
2. To avoid aliases during decimation, you will have to suppress all signal above 20kHz, which demands a low pass filter with a cut off frequency of $\frac1{2\cdot11}f_{sample,intermediate}$.

Since the filter from 1. implies 2., using one filter is sufficient.

This is a very commonly applied optimization.