3.3 The unit impulse function

The unit impulse function

The unit impulse is very useful in the analysis of signals, linear systems , and sampling . Consider the plot of a rectangular pulse in [link] . Note the height of the pulse is $1/\tau$ and the width of the pulse is $\tau$ . So we can write

As we let $\tau$ get small, then the width of the pulse gets successively narrower and its height gets progressively higher. In the limit as $\tau$ approaches zero, we have a pulse which has infinite height, and zero width, yet its area is still one. We define the unit impulse function as

The area under $\delta \left(t\right)$ is one, and so we can write

If we multiply the unit impulse by a constant, $K$ , its area is now equal to that constant, i.e.

The area of the unit impulse is usually indicated by the number shown next to the arrow as seen in [link] .

Suppose we multiply the signal $x\left(t\right)$ with a time-shifted unit impulse, $\delta (t-\tau )$ . The product is a unit impulse, having an area of $x\left(\tau \right)$ . This is illustrated in [link] .

In other words,

Equation [link] is called the sifting property of the unit impulse. As we will see, the sifting property of the unit impulse will be very useful.