The two fundamental operations in multirate signal processing are:

• upsampling by $$N$$, in which we increase the number of samples by inserting $$N-1$$ zeros after each original sample; the insertion of zeros causes the spectrum of the original sequence to contract by $$N$$, which "brings in" $$N$$ copies of the original spectrum in the $$[-\pi, \pi]$$ interval. To remove the extra copies we need to use a lowpass with cutoff $$\pi/N$$ after upsampling.

• downsampling by $$N$$, in which we decrease the number of samples by keeping only 1 sample out of $$N$$ in the original sequence; the operation causes the spectrum of the original sequence to expand by $$N$$ and produces an overlap of spectral copies that may result in aliasing. To prevent aliasing we need to use a lowpass with cutoff $$\pi/N$$ before downsampling.

Upsampling and downsampling change the underlying sampling rate by an integer factor. For a rational rate change, we can cascade an upsampler by $$N$$, a lowpass filter and a downsampler by $$M$$; the cutoff of the filter should be $$\pi/K$$ where $$K$$ is the max between $$M$$ and $$N$$. However, when $$M$$ and $$N$$ are large coprime numbers, the theoretical approach may be too onerous; in this case we can use local interpolation techniques to perform these sampling rate changes on the fly at a very low computational cost.