4 The unit step function

The unit step function is defined as follows:

Study this definition carefully. You will see that it is defined in two parts, with one expression to be used when $t$ is greater than or equal to 0, and another expression to be used when $t$ is less than 0. The graph of this function is shown in Figure 33. Note that the part of $u\left(t\right)$ for which $t<0$ lies on the $t$ -axis but, for clarity, is shown as a distinct dashed line.

Figure 33 :

There is a jump, or discontinuity in the graph when $t=0$ . That is why we need to define the function in two parts; one part for when $t$ is negative, and one part for when $t$ is non-negative. The point with coordinates (0,1) is part of the function defined on $t\ge 0$ .

The position of the discontinuity may be shifted to the left or right. The graph of $u\left(t-d\right)$ is shown in Figure 34.

Figure 34 :

In the previous two figures the function takes the value 0 or 1. We can adjust the value 1 by multiplying the function by any other number we choose. The graph of $2u\left(t-3\right)$ is shown in Figure 35.

Figure 35 :

Exercises

Sketch graphs of the following functions:

1. $u\left(t\right)$ ,
2. $-u\left(t\right)$ ,
3. $u\left(t-1\right)$ ,
4. $u\left(t+1\right)$ ,
5. $u\left(t-3\right)-u\left(t-2\right)$ ,
6. $3u\left(t\right)$ ,
7. $-2u\left(t-3\right)$ .

Answer