Downsampling Theorem (Aliasing Theorem)

**Theorem: **For all
,

*Proof: *Let
denote the frequency index in the
aliased spectrum, and
let
. Then is length ,
where is the downsampling factor. We have

Since , the sum over becomes

Since the above derivation also works in reverse, the theorem is proved.

An illustration of aliasing in the frequency domain is shown in Fig.7.12.

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