The Fourier Transform

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We will begin the analysis of the Fourier Transform on this page. This section gives an introduction to the Fourier Transform and then we take a look at the fundamental properties of the Fourier Transform.

Introduction

Fourier Transform of Square Pulse (Box Function)

Fourier Transform Properties

Parseval's Theorem

The Integration Property


Introduction to the Fourier Transform

The Fourier Transform is a magical mathematical tool. The Fourier Transform decomposes any function into a sum of sinusoidal basis functions. Each of these basis functions is a complex exponential of a different frequency. The Fourier Transform therefore gives us a unique way of viewing any function - as the sum of simple sinusoids.

The Fourier Series showed us how to rewrite any periodic function into a sum of sinusoids. The Fourier Transform is the extension of this idea to non-periodic functions.

While the the Fourier Transform is a beautiful mathematical tool, its widespread popularity is due to its practical application in virtually every field of science and engineering. It's hard to understand why the Fourier Transform is so important. But I can assure you it enables the solution to difficult problems be made simpler (and also makes previously unsolved problems solvable). In addition, the Fourier Transform gives us a new method of viewing the world, which is fantastic for giving a more intuitive feel for our universe.

To begin the study, it's best to jump right in to the definition and study of the Fourier Transform. Then a bunch of applications can be presented, which will justify the hooplah surrounding the Transform.

Without further adieu, the Fourier Transform of a function g(t) is defined by:

definition of Fourier Transform
[Equation 1]

The result is a function of f, or frequency. As a result, G(f) gives how much power g(t) contains at the frequency f. G(f) is often called the spectrum of g. In addition, g can be obtained from G via the inverse Fourier Transform:

definition of inverse Fourier Transform
[Equation 2]

Equation [2] states that we can obtain the original function g(t) from the function G(f) via the inverse Fourier transform. As a result, g(t) and G(f) form a Fourier Pair: they are distinct representations of the same underlying identity. We can write this equivalence via the following symbol:

Fourier pair
[Equation 3]

In the next section, we'll look at the Fourier Transform of the box function, and then discuss properties of the transform.


The Fourier Transform of the Box Function

Continuing the study of the Fourier Transform, we'll look at the box function (also called a square pulse or square wave):

box function or square pulse wave

Figure 1. The box function.

In Figure 1, the function g(t) has amplitude of A, and extends from t=-T/2 to t=T/2. For |t|>T/2, g(t)=0.

Using the definition of the Fourier Transform (Equation [1] above), the integral is evaluated:

fourier transform calculation
[Equation 4]

The solution, G(f), is often written as the sinc function, which is defined as:

sinc function
[Equation 5]

[While sinc(0) isn't immediately apparent, using L'Hopitals rule or whatever special powers you have, you can show that sinc(0) = 1]

The Fourier Transform of g(t) is G(f),and is plotted in Figure 2 using the result of equation [2].

plot or graph of sinc function

Figure 2. The sinc function is the Fourier Transform of the box function.

To learn some things about the Fourier Transform that will hold in general, consider the square pulses defined for T=10, and T=1. These functions along with their Fourier Transforms are shown in Figures 3 and 4, for the amplitude A=1.

Fourier pair of box function when T=10

Figure 3. The Box Function with T=10, and its Fourier Transform.

Fourier pair of box function when T=1

Figure 4. The Box Function with T=1, and its Fourier Transform.

A fundamental lesson can be learned from Figures 3 and 4. From Figure 3, note that the wider square pulse produces a narrower, more constrained spectrum (the Fourier Transform). From Figure 4, observe that the thinner square pulse produces a wider spectrum than in Figure 3. This fact will hold in general: rapidly changing functions require more high frequency content (as in Figure 4). Functions that are moving more slowly in time will have less high frequency energy (as in Figure 3).

Further, notice that when the box function is shorter in time (Figure 4), so that it has less energy, there appears to be less energy in it's Fourier Transform. We'll explore this equivalence later.

In the next section, we'll look at properties of the Fourier Transform.


Next: Properties of the Fourier Transform

Up: Fourier Transform Analysis

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