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Exact Discrete Gaussian Window

It can be shown [44] that

$\displaystyle e^{j\frac{2\pi}{N}\frac{1}{2}n^2} \;\longleftrightarrow\; e^{-j\frac{2\pi}{N}\frac{1}{2}k^2}$ (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$ DFT. In other words, the DFT of this particular sampled Gaussian pulse is exactly a sampled Gaussian pulse. (The proof is nontrivial.)


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