Discrete Fourier conjugate symmetry ($X_{(-k)\mathrm{mod}N}=\overline{X_k}$)

For real samples, taking the complex conjugate of the forward discrete Fourier transform changes the exponent's sign and proves the displayed symmetry. Hence $X_0$ is real; for even $N$, $X_{N/2}$ is also real. In the eight-sample example, $X_7=\overline{X_1}=4$ and $X_6=\overline{X_2}=2i$. This is the finite counterpart of conjugate symmetry of trigonometric polynomial coefficients. It reduces storage for real signals and provides a quick consistency check on a computed discrete Fourier transform.