Page 122 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS 111

where 0-= lim,,,(O - E). Clearly the bilateral and unilateral transforms are equivalent
only if x(t) = 0 for t < 0. The unilateral Laplace transform is discussed in Sec. 3.8. We will
omit the word "bilateral" except where it is needed to avoid ambiguity.
Equation (3.3) is sometimes considered an operator that transforms a signal x(t) into a
function X(s) symbolically represented by

and the signal x(t) and its Laplace transform X(s) are said to form a Laplace transform
pair denoted as

B. The Region of Convergence:
The range of values of the complex variables s for which the Laplace transform
converges is called the region of convergence (ROC). To illustrate the Laplace transform
and the associated ROC let us consider some examples.
EXAMPLE 3.1. Consider the signal

x(t) =e-O1u(t) a real
Then by Eq. (3.3) the Laplace transform of x(t) is

because lim, ,, e-("'")' = 0 only if Re(s +a) > 0 or Reb) > -a.

Thus, the ROC for this example is specified in Eq. (3.9) as Re(s) > -a and is displayed
in the complex plane as shown in Fig. 3-1 by the shaded area to the right of the line
Re(s) = -a. In Laplace transform applications, the complex plane is commonly referred to
as the s-plane. The horizontal and vertical axes are sometimes referred to as the a-axis and
the jw-axis, respectively.

EXAMPLE 3.2. Consider the signal
~(t) a real
= -e-"u( -t)
Its Laplace transform X(s) is given by (Prob. 3.1)

Thus, the ROC for this example is specified in Eq. (3.11) as Re(s) < -a and is displayed
in the complex plane as shown in Fig. 3-2 by the shaded area to the left of the line
Re(s) = -a. Comparing Eqs. (3.9) and (3.11), we see that the algebraic expressions for X(s)
for these two different signals are identical except for the ROCs. Therefore, in order for the
Laplace transform to be unique for each signal x(t), the ROC must be specified as par1 of the
transform.