Mathematics of the Discrete Fourier Transform (DFT)

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Decimation Theorem (Aliasing Theorem)

Theorem: For all $$x\$ ,

Proof: Let $$k^\$ denote the frequency index in the aliased spectrum, and let $$Y(k^\$ . 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 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.
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
An illustration of aliasing in the frequency domain is shown in Fig. 8.10.
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