The z-transform - Continuous and Discrete Time Signals and Systems - page 601

(i) Determine the z-transfer function of the system.
(ii) Determine the equivalent Laplace transfer function of the system.
(iii) Using the Laplace domain approach, determine if the system is stable.

Im s

j40 p

j20 p

Re
48

s s

8 4

Solution
(i) H ( z ) =

j20 p
j40 p

1 − 0 . 5 z − 1 , or z

1

0 . 5 k u [ k ]

z − 0 . 5 , ROC: z > 0 . 5 .
(ii) Using Eq. (13.41b), the Laplace transfer function is given by

Z =

Fig. 13.8. Location of poles in
the s-plane for the system in
Example 13.15 with T

e sT

H ( s ) = H ( z ) =

− 0 . 5 ,

(13.43)

z = e sT

e sT

= 0.1.

where T is the sampling interval.
(iii) A causal LTIC system is stable if all the poles corresponding to the
Laplace transfer function lie in the left-hand half of the s-plane. Therefore, we
will first calculate the pole locations in the s-plane, and then determine if the
system is stable. The poles of the transfer function, Eq (13.43), are calculated
from the characteristic equation as follows:

e sT − 0 . 5 = 0 ⇒ e sT = 0 . 5 ⇒ e ( sT j2 π m ) = 0 . 5 ,

= 0, 1, 2, . . . Solving for the roots of this equation yields

where m

= 1

T [ln 0 . 5 j2 π m ] ≈ 1 T [ − 0 . 693 j2 π m ] .

It is observed that an LTID system has an infinite number of poles in the
s-domain. The locations of these poles for T = 0 . 1 are shown in Fig. 13.8. It
is clear that these poles would lie in the left-half of the s-plane, irrespective of
the value of the sampling interval T . The LTID system is, therefore, causal and
stable.
Alternatively, the stability of the LTID system can be determined from its
impulse response by noting that
∞

s

k =−∞ 0 . 5 k

∞

h [ k ] = = 2 < ∞,

k =−∞

which satisfies the BIBO stability requirement derived in Chapter 10.

13.8 Stabilty analysis in the z-domain

In Example 13.15, the stability of an LTID system was determined by trans-
forming its z-transfer function H ( z ) to the Laplace transfer function H ( s )ofan
equivalent LTIC system and observing if the poles of H ( s ) lie in the left-half
s-plane. In this section, we derive a z-domain condition to check the stability
of a system directly from its z-transfer function.