Discrete Fourier transform with negative frequencies

Question

In the standard definition of Discrete Fourier transform, as in Wikipedia, only positive frequencies exist. I want zero frequency to be at the center of the spectrum. In such a case, how should I change formula for Discrete Fourier transform?

Answer 1

Please don't change the formula, people have given it some thought. If you look at it

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$

you can see that

$$X[k]=X[k+N]\tag{2}$$

holds (because $e^{-j2\pi nk/N}$ is $N$-periodic in $k$ and $n$). Consequently, values corresponding to negative indices $k$ can simply be derived from positive values by increasing the negative index by the DFT length $N$. E.g.,

$$X[-1]=X[N-1]\tag{3}$$

et cetera.

Answer 2

Negatives exist also; $k=0$ to $N - 1$ spans positive frequencies up to $N/2$, then negatives.

To center the zero bin, shift indexing into the complex sinusoid:

$$ X[k] = \sum_{n=0}^{N-1} x[n] e^{-j2\pi n(k + N/2)/N} $$

equivalent to fftshift. Python code below. (For "positives only", there's real DFT.)

import numpy as np
from numpy.fft import fft, fftshift

def dft(x, center_zero=False):
    N = len(x)
    offset = np.ceil(N / 2) if center_zero else 0
    out = np.zeros(N, dtype='complex128')
    
    for k in range(N):
        for n in range(N):
            out[k] += x[n] * np.exp(-2j*np.pi * n * (k + offset) / N)
    return out

for N in (128, 129):  # check even & odd case
    x = np.random.randn(N)
    xf0 = fftshift(fft(x))
    xf1 = dft(x, center_zero=True)
    assert np.allclose(xf0, xf1)