= Discrete Fourier conjugate symmetry
{title2=$X_{(-k)\bmod N}=\overline{X_k}$}
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>.
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