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

using the closed form expression for a geometric series derived in §6.1. We see that the sum over effectively

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University