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
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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