# 7.1: Dirac delta (impulse) function

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- 22774

## 1.1 Introduction

The Dirac delta function δ(*t* − *t*_{0}) is a mathematical idealization of an impulse or a very fast burst of substance at *t* = *t*_{0}. (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* < *t*_{0} < *b*. When *t*_{0} 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 *t*_{0},

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 *t*_{0}.

## 1.2 Examples

## 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* > *t*_{0}. 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.