### 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 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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