Which are the wavenumbers for the fast Fast Fourier Transform?

Question

I am having a similar problem to the one discussed here: Computing wavenumbers for discrete Fourier transform

Say I performed a FFT on $N = 256$ points, are my wavenumbers indexed as [-128,127] or [-127,128]?

Answer 1

$e^{2i \pi n (k+N)/N} = e^{2i \pi n k/N}$ for $n \in \mathbb{Z}$.

Thus it makes a difference only when you use the DFT to interpolate the signal at non-integer values.

In that case it seems legitimate to ask : the interpolation is the smoothest possible, and it is real for real signals.

Those conditions give for $N$ even :

$$X(k) = \sum_{n=0}^{N-1} x(n) e^{-2i \pi nk/N}, \qquad x(t) = \frac{1}{N}\sum_{k=-N/2+1}^{N/2-1} X(k) e^{2i \pi tk/N} \ \ + \ \ \frac{X(N/2)}{N} \cos(\pi t)$$

Note how $\int_0^N |x(t)|^2dt= \sum_{n=0}^{N-1} |x(n)|^2 - \frac{1}{2 N} |X(N/2)|^2$, so if you prefer to preserve instead the energy and linearity, the interpolation won't be real for real signals anymore.