## 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 theorem^{D.2}provides 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
near its tip. Iterating
convolutions then corresponds to
, which becomes
[2]

(D.27) |

for large , by the definition of [264]. This proves that the th power of approaches the Gaussian function defined in §D.1 for large .

Since the inverse Fourier transform of a Gaussian is another Gaussian (§D.8), we can define a time-domain function as being ``sufficiently regular'' when its Fourier transform approaches in a sufficiently small neighborhood of . That is, the Fourier transform simply needs a ``sufficiently smooth peak'' at that can be expanded into a convergent Taylor series. This obviously holds for the DTFT of any discrete-time window function (the subject of Chapter 3), because the window transform is a finite sum of continuous cosines of the form in the zero-phase case, and complex exponentials in the causal case, each of which is differentiable any number of times in .

### 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