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 =

    4.0000   -2.0000