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Embedded SystemsFPGAElectronics

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Gaussian Window and Transform
          Exact Discrete Gaussian Window

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Exact Discrete Gaussian Window

It can be shown [43] 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} $

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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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