Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 102

j ω

ROC

σ

- a

FIGURE 3.1 ROC of X ( s ) .

The convergence of X(s) requires that lim t!1 e (sþa)t ! 0. Thus, the ROC is the set

of points in the complex s-plane for which Re(s) > a, as shown in Figure 3.1.

Notice that the function X(s) has a single pole located at s ¼a, which is outside
ROC of X(s).

Example 3.4

Find the Laplace transform and ROC of the signal x(t) ¼ u(t), where u(t) is the unit

step function.

S OLUTION
This is a special case of Example 3.3 where a ¼ 0. Therefore,

1
s

X(s) ¼ ROC : Re(s) > 0

Example 3.5

Find the Laplace transform and ROC of the unit impulse signal x(t) ¼ d (t).

S OLUTION

1 1

0 d

x(t)e st dt ¼ (t)e st dt ¼ d (t)e st

X(s) ¼ j t¼0 ¼ 1

0

Since the above integral converges for all values of s, the ROC is the entire

complex s-plane.

The Laplace transform of some elementary signals and their ROCs are listed in
Table 3.1. Properties of Laplace transform are given in Table 3.2. Interested readers
are referred to Ref. [1] for proofs of these properties.