# 7.1: Dirac delta (impulse) function

## 1.1 Introduction

The Dirac delta function δ(tt0) is a mathematical idealization of an impulse or a very fast burst of substance at t = t0. (Here we are considering time but the delta function can involve any variable.) The delta function is properly defined through a limiting process. One such definition is as a thin, tall rectangle, of width ε:

$\delta\left(t-t_{0}\right)=\frac{1}{\epsilon}$

for

$t_{0}-\frac{\epsilon}{2}<t<t_{0}+\frac{\epsilon}{2}$

and zero otherwise, in the limit that $$\epsilon \rightarrow 0$$.

Then, we have

$\int_{a}^{b} \delta\left(t-t_{0}\right) d t=\lim _{\epsilon \rightarrow 0}\left[\epsilon \cdot\left(\frac{1}{\epsilon}\right)\right]=1$

as long as a < t0 < b. When t0 is outside the range of (a,b), then the integral is zero.

Likewise, for any function $$f(t)$$ that is continuous and differentiable (analytic) at t0,

where the quantity in the square brackets above is just the average value of f(t) in the interval . Thus, when , it becomes just the value at t0.

## 1.3 Delta function at the initial time

Note: if one of the limits of the integral coincides exactly with $$t_0$$, then the result is usually cut in half:

$\int_{t_{0}}^{b} \delta\left(t-t_{0}\right) f(t) d t=f\left(t_{0}\right) / 2$

for b > t0. For example:

$\int_{2}^{\infty} \delta(t-2) t^{2} d t=(1 / 2) 2^{2}=2$

However, when we think of an impulse to a system at the initial time t0, then we really consider that the entire delta function is added to the system - that is, the actual time is an infinitesimal amount beyond $$t_0$$; that is, $$t=t_{0}^{+}$$. In that case

$\int_{t_{0}}^{b} \delta\left(t-t_{0}^{+}\right) f(t) d t=f\left(t_{0}\right)$

## 1.4 Definition in terms of a Gaussian

Another equivalent definition of the delta functions are as a Gaussian function:

$\delta\left(t-t_{0}\right)=\lim _{\sigma \rightarrow 0} \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{\left(t-t_{0}\right)^{2}}{2 \sigma^{2}}}$

All the properties work out to be the same where $$a=\sqrt{2} \sigma$$ is used.

## 1.5 Physical applications

In Control, the delta function is used an idealization of a very fast perturbation to the system. For example, if you dump a bucket of water into a tank, then the "flow rate" is essentially a delta function - a very highly peaked function, but with a net integral (the total amount of water in the bucket).

In mechanics, and example of the delta function is the force when hitting an object by a hammer. Say you hit a steel ball with a hammer. It moves with a certain velocity representing the total momentum transferred by the hammer. Rather than talk about the force x time (the net momentum transfer), one talks about an "impulse" which is the net momentum transferred in an infinitesimally short amount of time.

## 1.6 Relation to the step function

The step function, $$Θ(t − t_0)$$, is the integral of the delta function or alternatively, the delta function is the derivative of the theta function, where $$Θ(t − t_0)$$ is defined at 1 for $$t > t_0$$ and 0 for $$t < t_0$$:

.

Here, the smooth or gaussian definition of the delta function corresponds to a smooth representation of the Θ function as the integral of a gaussian or equivalently, the error function.