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Iterated Convolutions

Any ``reasonable'' probability density function (PDF) (§C.1.3) has a Fourier transform that looks like $ S(\omega) = 1 - \alpha \omega^2$ near its tip. Iterating $ N$ convolutions then corresponds to $ S^N(\omega)$ , which becomes [#!Abel06!#]

$\displaystyle S^N(\omega) = (1-\alpha \omega^2)^N = \left(1-\frac{N\alpha \omega^2}{N}\right)^N \approx e^{-N\alpha\omega^2}$ (D.27)

for large $ N$ , by the definition of $ e$ [#!MDFT!#]. This proves that the $ N$ th power of $ 1-\alpha\omega^2$ approaches the Gaussian function defined in §D.1 for large $ N$ .

Since the inverse Fourier transform of a Gaussian is another Gaussian (§D.8), we can define a time-domain function $ s(t)$ as being ``sufficiently regular'' when its Fourier transform approaches $ S(\omega)\approx 1-\alpha\omega^2$ in a sufficiently small neighborhood of $ \omega
= 0$ . That is, the Fourier transform simply needs a ``sufficiently smooth peak'' at $ \omega
= 0$ that can be expanded into a convergent Taylor series. This obviously holds for the DTFT of any discrete-time window function $ w(n)$ (the subject of Chapter 3), because the window transform $ W(\omega)$ is a finite sum of continuous cosines of the form $ w(n)\cos(n\omega T)$ in the zero-phase case, and complex exponentials in the causal case, each of which is differentiable any number of times in $ \omega$ .

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2019-10-29 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University