Let's say I have a multi-stage interpolator at the transmitter, which interpolates by 2, 4 and 8 (that is, 64). When decimating (at the receiver) why should I start decimating by 8, then 4, then 2, instead of 2, 4 and 8? I mean what's the reason behind? What makes it better than doing it the other way round?
The idea is that you'll be saving computational steps in the filtering part of the decimation. Think about it, if you have 800 samples then downsample by a factor of 8, now you only have to low pass filter 100 samples for the first stage. If you instead tried to downsample by a factor of 2 first, then you'd have 400 samples to filter after the first stage. The order matters because of the number of computations taken in the filtering step.