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Exact Discrete Gaussian Window

It can be shown [44] that

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

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