
Example 4.3
Find the Laplace transform of the signal f(t) shown in Figure 4.4.
The piecewise continuous function f(t)isdefined in different intervals by
f(t) ¼
0, t < 1
2(t 1), 1 t < 2
3, 2 t < 4
t þ 7, 4 t < 7
0, 7 t
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
(4:26)
Next, we write the function f( t) in a single expression using unit step functions. The procedure is
straightforward. The first nonzero term in Equation 4.26 is multiplied by the appropriate step
function, so that it ‘‘turns on’’ at the correct time. This gives
f(t) ¼ 2(t 1)
^
u(t 1) (4:27)
that is valid for the first two intervals t < 1 and 1 t <
2. The description of f(t)
changes for t 2
necessitating ...