Signals and Systems - Electrical Engineering

3.4 Inverse Laplace Transform 205

− 2 0 2

−2.5

− 2

−1.5

− 1

−0.5

0

0.5

1

1.5

2

2.5

σ

jΩ

−0.5 0 5 10

0

0.5

1

1.5

2

2.5

t

x(t)

(a) (b)

FIGURE 3.14
Inverse Laplace transform ofX(s)=( 2 s+ 3 )/(s^2 + 2 s+ 4 ): (a) poles and zeros and (b) inversex(t).

ezplot(x,[0,12]); title(’x(t)’)

axis([0 12 -0.5 2.5]); grid

The results are shown in Figure 3.14. n

Double Real Poles

If a proper rational function has double real poles

X(s)=

N(s)

(s+α)^2

=

a+b(s+α)

(s+α)^2

=

a

(s+α)^2

+

b

s+α

(3.26)

then its inverse is

x(t)=[ate−αt+be−αt]u(t) (3.27)

whereacan be computed as

a=X(s)(s+α)^2 |s=−α

After replacing it,bis found by computingX(s 0 )for a values 0 6=α.