The Fourier Transform of the Triangle Function

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The Fourier Transform of the triangle function is derived on this page. The unit triangle function is given in Figure 1:

triangle function

Figure 1. The triangle function.

Mathematically, the triangle function can be written as:

mathematical representation of triangle function
[Equation 1]

We'll give two methods of determining the Fourier Transform of the triangle function.

Method 1. Integration by Parts

We can simply substitute equation [1] into the formula for the definition of the Fourier Transform, then crank through all the math, and then get the result. This is pretty tedious and not very fun, but here we go:

integration by parts to get the Fourier Transform of the triangle function

The Fourier Transform of the triangle function is the sinc function squared.

Now, you can go through and do that math yourself if you want. It's a complicated set of integration by parts, and then factoring the complex exponential such that it can be rewritten as the sine function, and so on. It's an ugly solution, and not fun to do. Method 2, using the convolution property, is much more elegant.

Method 2. Using the Convolution Property

The convolution property was given on the Fourier Transform properties page, and can be used to find Fourier Tranforms of functions. In some cases, as in this one, the property simplifies things. If you've studied convolution, or you've sat down and thought about it, or you are very clever, you may know that the triangle function is actually the convolution of the box function with itself. That is, let's simplify box function on the previous page, g(t) such that the amplitude A=1 and T=1, then the box function is simply:

unit box function
[Equation 2]

The triangle function can be mathematically expressed as the unit box function convolved with itself:

the triangle function is the box function convolved with itself
[Equation 3]

If you recall the convolution property of Fourier Transforms, we know that the Fourier Transform of the convolution of functions g1 and g2 is just the product of Fourier Transform of the individual functions: (G1 times G2). Since we know the Fourier Transform of the box function is the sinc function, and the triangle function is the convolution of the box function with the box function, then the Fourier Transform of the triangle function must be the sinc function multiplied by the sinc function. That is:

using convolution to get the fourier transform
[Equation 4]

So we arrive at the same solution as the brute-force calculus method, but we get there using a much simpler and more intelligent method. Either way, at the end of the day the Fourier Transform of the triangle function is the sinc function squared. This pair is shown in Figure 2.

fourier transform triangle is sinc squared

Figure 2. The triangle function and its Fourier Transform the sinc squared function.


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