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

(DFT case) as

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.

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