Downsampling Theorem (Aliasing Theorem)
Theorem: For all ,
Since , the sum over becomes
Since the above derivation also works in reverse, the theorem is proved.
>> 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
Zero Padding Theorem (Spectral Interpolation)
Stretch Theorem (Repeat Theorem)