
Theorem: For all

,
Proof: Let
![$ k^\prime \in[0,M-1]$](http://www.dsprelated.com/josimages_new/mdft/img1440.png)
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
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.
>> 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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