### 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

*samples*every samples. This can be expressed in the previous formula by defining which ranges only over the nonzero samples:

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.

#### Illustration of the Downsampling/Aliasing Theorem in Matlab

>> N=4; >> x = 1:N; >> X = fft(x); >> x2 = x(1:2:N); >> fft(x2) % FFT(Downsample(x,2)) ans = 4 -2 >> (X(1:N/2) + X(N/2 + 1:N))/2 % (1/2) Alias(X,2) ans = 4 -2

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