Laplace transform - Continuous and Discrete Time Signals and Systems - page 295

Fig. 6.8. Locations of zeros and
poles of LTIC systems specified
in Example 6.18. The ROCs for
causal LTIC systems are
highlighted by the shaded
regions. Parts (a)-(c) correspond
to parts (i)-(iii) of Example 6.18.

Im{ s } Im{ s }

double poles
at s = 0

j3

x

Re{ s }

x x x x

Re{ s }

−8 −6 −4 −2 0 2 468 0

−8 −6 −4 −2 2 6

−j3

x

(a) (b)

Im{ s }

j3 p x

j p

x

Re{ s }

0

−8 −6 −4 −2 2 6

x

−j p

x

−j3 p

(c)

Solution
(i) The zeros are the roots of the quadratic equation ( s + 4)( s + 5) = 0, which
are given by s =− 4 , − 5. The poles are the roots of the fourth-order equation
s 2 ( s + 2)( s − 2) = 0, and are given by s = 0 , 0 , − 2 , 2. Figure 6.8(a) plots the
location of poles and zeros in the complex s-plane. The poles are denoted by
the “ ” symbols, while the zeros are denoted by the “ ◦ ” symbols.
(ii) The zeros are the roots of the equation s + 4 = 0, which are given by
s =− 4. The poles are the roots of the third-order equation s 3 + 5 s 2 + 17 s +
13 = 0, and are given by s = − 1, − 2 j3. Figure 6.8(b) plots the location of
poles and zeros in the complex s-plane.
(iii) Since the numerator is a constant, there is no zero for the LTIC system.
The poles are the roots of the characteristic equation e s + 0 . 1 = 0. Following
the procedure shown in Appendix B, it can be shown that there are an infinite
number of roots for the equation e s + 0 . 1 = 0. The locations of the poles are
given by

s = ln 0 . 1 + j(2 m + 1) π ≈− 2 . 3 + j(2 m + 1) π.

The poles are plotted in Fig. 6.8(c).

6.7 Properties of the ROC

In Section 3.7.2, we showed that the impulse response h ( t ) of a causal LTIC
system satisfies the following condition:

h ( t ) = 0 for t < 0 .