← Lecture 17 | Lecture 19 →
The Impulse Function
In each field of engineering, there is a question as to what will occur to the system
if a sudden but brief force is applied to a system. For example, when a charged
body is brought close to ground, at some point the charge will overcome the natural
resistance of the air and produce a spark. This produces a sudden and brief electric
force. Lightning is nothing more than a similar spark caused by the friction between
the air and the ground.
In general, such forces are referred to as impulse forces and we must
represent such forces mathematically.
Consider the function
where is a very small number. We can shift this function to start at
an arbitrary point by
. Some examples are shown in Figure 1.
Figure 1. , , and .
The integral of this function for any values of and may be calculated as follows:
By our definition, the area under the impulse is independent of
our choice of and therefore we can make
it arbitrarily small. In the limit, as ,
we are left with a function which is defined as
We will represent this infinite spike at
as an arrow, as shown in Figure 2.
Figure 2. The representation of an infinite spike at time .
The limit of this sequence of functions does not exist in the strict
definition of a function; however, it is still an object where the
integral is defined and that integral is 1.
This bizarre function was first conceived by the British
mathematical physicist Paul A.M. Dirac in 1925 in his investigations of quantum mechanics.
He used the symbol which is
today called the Dirac delta by physicists and called
the impulse function by engineers.
The Integral of the Impusle Function
Next, ask yourself what is the integral of the impulse function
Because the area under the square wave is always one, the integral
must grow linearly from 0 to 1 along the non-zero region of the
This is shown in Figure 3 and it becomes clear that the integral
approaches the unit step function.
Figure 3. The integral of the peicewise constant approximations of
the impulse function.
While, in the limit, the value of the integral is undefined
, it is conventional to use the
unit impulse function and define
The Sifting Property
One of the most significant uses of the impulse function is the ability
to select the value of a function at a point. Assuming that a
function is continuous at a point , it follows that
When Dirac introduced his impulse function, he did not focus on the rigorous
aspects: it gave the correct results. Twenty years later, mathematicians
introduced the theory of distributions of which the impulse function is
In a similar manner, Oliver Heaviside
pioneered the use of the Laplace
transform methods for solving linear differential equations which were
later made rigorous by Thomas Bromwich.
It is not necessary to start with the step function used in the
above example. For example, one could use
which is symmetric around the origin, or a tent function as demonstrated
in Figure 4.
Figure 4. Three tent functions corresponding to the three functions
shown in Figure 1.