What is the right way to downsample using Fourier method?
Note that the result of a DFT is inherently periodic, so the second half of the resulting output vector does not correspond to the highest frequencies, but to the negative frequencies. The highest frequency (Nyquist) is in the middle of the vector, after that you get the most negative frequency and the last element is the smallest negative frequency just below DC. That's why when taking out the middle part of the vector you actually remove the highest frequencies.