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## Decimation Theorem (Aliasing Theorem)

Theorem:For all ,

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

Since , the sum over becomes

using the closed form expression for a geometric series derived earlier. We see that the sum over effectivelysamplesevery 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.Here is an illustration of the Decimation Theorem in Matlab:

>> N=4; >> x = 1:N; >> X = fft(x); >> x2 = x(1:2:N); >> fft(x2) % FFT(Decimate(x,2))ans =

`4 -2`

>> (X(1:N/2) + X(N/2 + 1:N))/2 % (

^{1}⁄_{2}) Alias(X,2)ans =

`4.0000 -2.0000</pre></p><p>An illustration of aliasing in the frequency domain is shown in`

Fig. 8.10.