Exact Discrete Gaussian Window

It can be shown [44] that

$$e^{-j\frac{\pi}{8}} \, e^{j\frac{2\pi}{N}\frac{1}{2}n^2} \;\longleftrightarrow\; e^{j\frac{\pi}{8}} \, e^{-j\frac{2\pi}{N}\frac{1}{2}k^2}$$ (4.58)

where $n\in[0,N-1]$ is the time index, and $k\in[0,N-1]$ is the frequency index for a length $N$ (even) normalized DFT (DFT divided by $\sqrt{N}$ ). In other words, the Normalized DFT (NDFT) of this particular sampled Gaussian pulse is exactly the complex-conjugate of the same Gaussian pulse. (The proof is nontrivial.)