# up sample and down sample

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?

The downsampling by a factor of 4 can introduce aliasing (because you upsampled only by a factor 2 in the first stage). If this is the case, it cannot be undone by the following upsampling operation. So in general this system will not produce at its output the original input signal $x[n]$.