The downsampling operation selects every sample of a signal:
In the DFT case, maps to , while for the DTFT, maps to .
The Aliasing Theorem states that downsampling in time corresponds to aliasing in the frequency domain:
where the operator is defined for
For (DTFT case), the operator is
where is a common notation for the primitive th root of unity, and as usual. This normalization corresponds to after downsampling. Thus, prior to downsampling.
The summation terms above for are called aliasing components.
The aliasing theorem points out that in order to downsample by factor without aliasing, we must first lowpass-filter the spectrum to . This filtering essentially zeroes out the spectral regions which alias upon sampling.