I have a 2 part question, one may be related to why I'm not understanding the other.
A while back, I remember some professor saying that for the $N$ point DFT frequency domain, the values for $k$ around $N/2$ (halfway point for the spectral domain - those around $\pi$) are considered high frequencies, and the end points $(0, N-1)$ (those around 0, $2\pi$) are considered low frequencies. I am having trouble seeing this mathematically given the definition for computing the coefficients of the DFT.
To me, it seems that the frequencies grow as $k$ increases right up until $k=N-1$, where the coefficients repeat due to periodicity. Perhaps I am mis-remembering and this may be an incorrect assumption?
This leads on to my second question. I have seen a few examples floating around online, where to interpolate in the time domain, one may pad in the frequency domain (running the IFFT after of course). The examples I have seen seem to decide to pad the coefficients around the middle, i.e. for $N=4$ and a desired sampling of $2N$, [1 2 3 4] become [1 2 0 0 0 0 3 4]. Is there a reason why the middle is targeted and not the end like [1 2 3 4 0 0 0 0]?
If my first question is correct, then this simply pads the higher frequencies as 0. I am aware that these zeroes add nothing to the frequency information as $X[k]=0$ cancels any effect during the inverse DFT. But wouldn't moving any of the coefficients change some frequency information, i.e. for [1 2 3 4] becoming [1 2 0 0 0 0 3 4]
$N=4, k=2, X[k]=3$, becomes $2N=8, k=6, X[k]=3$