Signals and Systems - Electrical Engineering

3.3 The One-Sided Laplace Transform 183

FIGURE 3.8
Poles (top right) of the Laplace
transform of the autocorrelation
c(t)=e−^2 |t|(top left), which is
noncausal. The ROC ofC(s)is the
region in between the poles, which
includes thejaxis. The spectrum
|C()|^2 corresponding toc(t)is
shown in the bottom plot—this is the
magnitude square of the Fourier
transform ofc(t).

− 20 − 10 0 10 20

0

0.2

0.4

0.6

0.8

1

Ω(rad/sec)

|C(

Ω

(^2) )|
− 5 0 5
0
0.2
0.4
0.6
0.8
1
t
c(
t)
− 2 0 2
− 1
−0.5
0
0.5
1
σ
jΩ
Thus, the system function is
H(s)=

1

s+ 1

+

1

−s+ 2

=

− 3

(s+ 1 )(s− 2 )

with a region of convergence the intersection of the regions of convergence of its components, or

the intersection of{(σ,):σ >−1,−∞<  <∞}and{(σ,):σ <2,−∞<  <∞}, or

{(σ,):− 1 < σ <2,−∞<  <∞}

which is a sector of thes-plane that includes thejaxis. Thus,H(j)can be computed from its

Laplace transform. n