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The Gaussian curve (sometimes called the normal distribution) is the familiar bell shaped curve that arises all over mathematics, statistics, probability,
engineering, physics, etc. We will look at a simple version of the Gaussian, given by equation [1]:
 [1]
The Gaussian is plotted in Figure 1:
Figure 1. The Gaussian Bell-Curve.
We will now evaluate the Fourier Transform of the Gaussian function in Figure 1. Let G(f) be the Fourier Transform of g(t), so that:
 [2]
To resolve the integral, we'll have to get clever and use some differentiation and then differential equations. Take the derivative of both sides of
equation [2] with respect to f, so that:
 [3]
Now we have this set up so that we can use integration by parts. That is, let
 [4]
Recall the formula for integration by parts:
 [5]
The uv term in equation [5] becomes zero, because the limits are evaluated from -infinity to +infinity, where
the product is zero. Using this fact and equations [3-5], we get:
 [6]
This is a first order simple differential equation for G(f). The solution for this differential equation is given by:
 [7]
All we need to do now to find G(f) is figure out what G(0) is. G(0) is simply the average value of g(t), because if you substitute f=0
into the equation for G(f), equation [2], the complex exponential term goes away (equation [8]). The integral in equation 8 has actually
an elegant solution, developed by Euler (the proof of which won't be given here), but the result is:
 [8]
Hence, we have found the Fourier Transform of the gaussian g(t) given in equation [1]:
 [9]
Equation [9] states that the Fourier Transform of the Gaussian is the Gaussian! The Fourier Transform operation returns
exactly what it started with. This is a very special result in Fourier Transform theory.
The Fourier Transform of a scaled and shifted Gaussian can be found
here.
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