Math

# Intuition behind the Delta Dirac function.

## Particular function that is zero everywhere except at zero where it approaches infinity.

Today we are going to look closely at a very interesting function, called the delta Dirac function. This function has a particularity that is zero everywhere except at zero where it approaches infinity. This function was introduced by the theoretical physicist Paul Dirac. In the context of signal processing it is often referred to as the unit impulse symbol. Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

and which is also constrained to satisfy the identity

In this post we will try to think about the intuition behind this definition. Because in mathematics, we can’t define a function with theses properties on real numbers.

First, why to have such a function? Let’s think about a system where nothing happens during a long period of time, and then there is an impulse that hits hard but vanish quickly, and again nothing happens during a long period of time. So we need a function that can model this kind of behavior, a function that can mimic this behavior where nothing happens except for t = 0.

So how can a function defined this way have an integral of 1 over the entire real numbers?

For this purpose, we are going to define another function that makes a better intuition of how this delta Dirac function can be constructed.

Let’s consider $\delta_{\tau}$ define as the following:

This function has the particularity to look more like a real function since we can calculate the integral for this function. In the same time, this can turn into the delta Dirac function when when we narrow the interval $[-\tau, \tau]$, i.e. when $\tau \to 0$

Let’s calculate the integral of this new function:

And since, we said that :

We can do something like :

And intuitively, we can say that :