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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?

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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.

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