Let's say that I have a sampled signal x[n], it is being, in this exact order, up sampled by 2, down sampled by4, up sampled by 4 and down sampled by 2 to produce y[n].
It seems to me that it should be pretty self evident that since we up sampled the signal by 2 and down sampled it by 2, then up sampled it by 4 and down sampled by 4, I should just get the original x[n] back.
Am I right?
So the real question is, can the various up/down sampling pieces be readily swapped?
Answer
To make Matt L.'s answer more precise: whenever a downsampling operation results in a sampling frequency that is less than twice the maximum frequency in your signal, you'll end up losing some of the signal's energy.
In the downsampling implementations I've seen, the signal is low-pass filtered prior to downsampling, so no actual aliasing occurs.
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