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Why Gaussian?

This section lists some of the points of origin for the Gaussian function in mathematics and physics.

Central Limit Theorem

The central limit theoremD.2provides that many iterated convolutions of any ``sufficiently regular'' shape will approach a Gaussian function.

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 [2]

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

for large $ N$ , by the definition of $ e$ [264]. 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$ .

Binomial Distribution

The last row of Pascal's triangle (the binomial distribution) approaches a sampled Gaussian function as the number of rows increases.D.3 Since Lagrange interpolation (elementary polynomial interpolation) is equal to binomially windowed sinc interpolation [301,134], it follows that Lagrange interpolation approaches Gaussian-windowed sinc interpolation at high orders.

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Gaussian Probability Density Function
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Fourier Transform of Complex Gaussian