The Fourier Transform of the Box Function

Fourier Transform Pairs
Next: The Triangle Function

On this page, the Fourier Transform for the box function, or square pulse, is given. This page is pretty much a rehash of the page in the Fourier Transform theory section.

The box function is a square pulse, as shown in Figure 1:

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. We will write the square pulse or box function as rect_T(t), indicating that the rectangle function is equal to 1 for a period of T (from -T/2 to +T/2) and 0 elsewhere:

definition of rectangle function for Fourier Transform

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

fourier transform calculation         [1]

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

sinc function
[Equation 2]

[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).

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

Next: The Triangle Function

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