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Lecture 18

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.

Derivation

Consider the function tex:$$f_\delta(t) = \frac{1}{\delta}(u(t) - u(t - \delta))$$ where tex:$$\delta$$ is a very small number. We can shift this function to start at an arbitrary point tex:$$a$$ by tex:$$f_\delta(t - a)$$. Some examples are shown in Figure 1.


Figure 1. tex:$$f_{0.5}(t - 2)$$, tex:$$f_{0.2}(t - 2)$$, and tex:$$f_{0.1}(t - 2)$$.

The integral of this function for any values of tex:$$delta$$ and tex:$$a$$ may be calculated as follows:

tex:$$\int_{-\infty}^\infty f_\delta(t - a) dt = \int_a^{a + \delta} \frac{1}{\delta} dt = \left . \frac{t}{\delta} \right|_{t = a}^{a + \delta} = \frac{a + \delta - a}{\delta} = \frac{\delta}{\delta} = 1$$.

By our definition, the area under the impulse is independent of our choice of tex:$$\delta$$ and therefore we can make it arbitrarily small. In the limit, as tex:$$\delta \rightarrow 0$$, we are left with a function which is defined as

tex:$$\lim_{\delta \rightarrow \infty} f_\delta(t - a) = \cases{ $\infty$ & $t = a$ \cr $0$ & otherwise }$$.

We will represent this infinite spike at tex:$$t = a$$ as an arrow, as shown in Figure 2.


Figure 2. The representation of an infinite spike at time tex:$$t = 2$$.

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 tex:$$\delta(t - a)$$ 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 tex:$$\delta(t)$$. 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 square pulse:

tex:$$\int_{-\infty}^t f_\delta(\tau) d\tau = \left \{ \matrix{ 0 & t \le 0 \cr \frac{1}{\delta}t & 0 \le t \le \delta \cr 1 & t \ge \delta } \right .$$.

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 at tex:$$t = 0$$, it is conventional to use the unit impulse function tex:$$u(t) = \left \{ \matrix{ 0 & t < 0 \cr 1 & t \ge 0 } \right .$$ and define

tex:$$\int_{-\infty}^t \int \delta(\tau) d\tau = u(t)$$

and

tex:$$\frac{d}{dt} u(t) = \delta(t)$$.

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 tex:$$a$$, it follows that

tex:$$f(a) = \int_{-\infty}^\infty f(t)\delta(t - a) dt = f(a)$$.

Remark

When Dirac introduced his impulse function, he did not focus on the rigorous aspects: it gave the correct results. Twenty years later, mathematicians (e.g., Laurent Schwartz) introduced the theory of distributions of which the impulse function is an example.

Remark

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.

Remark

It is not necessary to start with the step function used in the above example. For example, one could use

tex:$$f_\delta(t) = \frac{1}{\delta}(u(t + \delta/2) - u(t - \delta/2))$$

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.