Gaussian Window and Transform

The Gaussian window for FFT analysis was introduced in §3.11, and complex Gaussians (``chirplets'') were utilized in §10.6. For reference in support of these topics, this appendix derives some additional properties of the Gaussian, defined by

$$\zbox { \frac{1}{\sigma\sqrt{2\pi}}e^{-t^2 \left / 2\sigma^2\right.} \;\longleftrightarrow\; e^{-\omega^2 \left/ 2\left(1/\sigma\right)^2\right.} } % zbox$$ (D.1)

and discusses some interesting applications in spectral modeling (the subject of §10.4). The basic mathematics rederived here are well known (see, e.g., [202,5]), while the application to spectral modeling of sound remains a topic under development.