Laplace transform - Continuous and Discrete Time Signals and Systems - page 263

To derive the synthesis equation for the bilateral Laplace transform, consider
the inverse transform of the CTFT pair, x ( t )e

CTFT
←−−→ X ( σ

−σ t

+ j ω ) = X ( s ).

Based on Eq. (6.1), we obtain

∞

1
2 π

−σ t =

X ( s )e j ω t d ω.

x ( t )e (6.6)

−∞

Multiplying both sides of Eq. (6.6) by e σ t

and changing the integral variable ω

to s using the relationship s = σ + j ω yields

σ − j ∞

1
2 π j

X ( s )e st d s . x ( t ) = Laplace synthesis equation (6.7)

σ − j ∞

Solving Eq. (6.7) involves the use of contour integration and is seldom used
in the computation of the inverse Laplace transform. In Section 6.3, we will
consider an alternative approach based on the partial fraction expansion to
evaluate the inverse Laplace transform. Collectively, Eqs. (6.5) and (6.7) form
the bilateral Laplace transform pair, which is denoted by

←→

x ( t ) X ( s ) . (6.8)

To illustrate the steps involved in computing the Laplace transform, we consider
the following examples.

Example 6.1
Calculate the bilateral Laplace transform of the decaying exponential function:
x ( t ) = e

− at u ( t ).

Solution
Substituting x ( t ) = e

− at u ( t ) in Eq. (6.5), we obtain

∞
0

∞ ∞

1
( s + a ) e

− at u ( t )e − st d t − ( s + a ) t d t − ( s + a ) t

X ( s ) = e = e =− .

−∞ 0

− ( s + a ) t − ( s + a ) t

At the lower limit, t → 0, e = 1. At the upper limit, t →∞ ,e = 0

− ( s + a ) t

if Re { s + a > 0orRe { s > − a .IfRe { s }≤− a , then the value of e is

infinite at the upper limit, t →∞ . Therefore,

1
( s + a )

for Re s > − a

X ( s ) =

for Re s ≤− a . undefined

The set of values of s over which the bilateral Laplace transform is defined
is referred to as the region of convergence (ROC). Assuming a to be a real
number, the ROC is given by Re { s > − a for the Laplace transform of the
decaying exponential function, x ( t ) = e

− at u ( t ). Figure 6.1 highlights the ROC
by shading the appropriate area in the complex s-plane.