The Fourier Transform of the Gaussian

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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]:

gaussian function       [1]

The Gaussian is plotted in Figure 1:

plot of gaussian function

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:

Fourier Transform of Gaussian       [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:

derivative       [3]

Now we have this set up so that we can use integration by parts. That is, let

integration by parts       [4]

Recall the formula for integration by parts:

using 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:

we end up with a differential equation       [6]

This is a first order simple differential equation for G(f). The solution for this differential equation is given by:

solution of differential equation       [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:

dc value is the fourier transform at f=0       [8]

Hence, we have found the Fourier Transform of the gaussian g(t) given in equation [1]:

fourier transform of the gaussian is the gaussian       [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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