Convolution

Think That Was Convoluted? You Ain't Seen Nothing Yet

You were warned...

In we go with another mathematical operation! This one is called convolution (the * operator), and it is defined as:

What is it? I don't know, because it doesn't really represent anything geometrical, so I can't draw you a picture. However, convolution has some useful properties:

• The Fourier transform of a convolution x(t) * y(t) is X(ω) Y(ω).
• The Fourier transform of a product x(t) y(t) is (1 / 2π) X(ω) * Y(ω). This is known as modulation.

These really help when trying to transform weird functions that can be expressed as products. If you look at the Laplace transform properties, you'll find that these hold for Laplace transforms as well.

A Convolved Recipe

(or, I'm going to get as much mileage out of 'convolution' as I can)

So, let's see what happens when we convolve a function with itself: let's use the pie function we used in the previous section, π(x).

Obviously, π(x) is only non-zero for |x| < 1/2, so this is equivalent to:

This is a strange thing to integrate analytically, so let's think about it: when x = 0, this integral will be 1, since both π(x) and π(x - ξ) occupy the same space. When x > 1 or x < -1, π(ξ) and π(x - ξ) do not intersect, so the integral will be zero. At other times, the integral will be equal to the area of intersection of these two functions:

which is obviously equal to 1/2 + (1/2 - x) = 1 - x for x > 0, and 1 + x for x < 0. Therefore, the convoluted function is 1 - |x|, known as the triangle function λ(x).

So, how do we go about finding the Fourier transform of such a function? It's easy - we already have the transform of π(t), so the transform of a convolution of itself with itself is merely the square of that transform. Since τ = 1:

Easy!