Laplace transform - Continuous and Discrete Time Signals and Systems - page 276

Similarly, comparing the coefficients of s yields

k 2 + k 3 + 1 . 6 = 1 ⇒ k 3 =− 0 . 2 .

The partial fraction expansion of X 2 ( s ) reduces to

2
5( s + 1)

2 s + 1
( s + 2) 2 + 9 ,

X 2 ( s ) = − 0 . 2

which is expressed as

2
5( s + 1)

2( s + 2)
( s + 2) 2 + 9

3
( s + 2) 2 + 9 .

X 2 ( s ) = − 0 . 2 + 0 . 2

Based on entries (4) and (13) in Table 6.1, the inverse Laplace transform is
given by

− t − 2 t cos(3 t ) + 0 . 2e − 2 t sin(3 t )) u ( t ) .

x 1 ( t ) = (0 . 4e − 0 . 4e

6.4 Properties of t he Laplace transform

The unilateral and bilateral Laplace transforms have several interesting prop-
erties, which are used in the analysis of signals and systems. These properties
are similar to the properties of the CTFT covered in Section 5.4. In this section,
we discuss several of these properties, including their proofs and applications,
through a series of examples. A complete listing of the properties is provided
in Table 6.2. In most cases, we prove the properties for the unilateral Laplace
transform. The proof for the bilateral Laplace transform follows along similar
lines and is not included to avoid repetition.

6.4.1 Linearity

If x 1 ( t ) and x 2 ( t ) are two arbitrary functions with the following Laplace trans-
form pairs:

←→

x 1 ( t ) X 1 ( s ) with ROC: R 1

and

←→

with ROC: R 2 , x 2 ( t ) X 2 ( s )

then

←→

a 1 x 1 ( t ) + a 2 x 2 ( t ) a 1 X 1 ( s ) + a 2 X 2 ( s )

∩ R 2
(6.17)

with ROC: at least R 1

for both unilateral and bilateral Laplace transforms.