SAMPLED-DATA SYSTEMS 99 
unit circle is transformed into the right half of the \ plane. Similarly, the 
entire region which lies inside the unit circle of the z plane is transformed 
into the left half \ plane. This relationship is shown graphically in Fig. 
5.8. The regions labeled A and B are corresponding regions in their 
respective planes. 
The procedure which can be used to ascertain whether or not a system 
is unstable is to express the over-all pulse transfer function G(z) as the 




Unit circle d-plane 
# 
yyy 
y YY 
Li 
LLL. 
i 



Le 
iY R 
Yj e 
Yi 
7 
yyy 
f 4 
Fia. 5.8. Corresponding regions in z and X planes. 
ratio of two polynomials. The denominator polynomial, when set equal 
to zero, is the characteristic equation of the system. Each z in this 
polynomial is then treated by applying the relation for z given by (5.46), and 
a resultant characteristic equation in \ obtained. The Routh-Hurwitz 
criterion is then applied directly to this equation in X. 
EXAMPLE 
The over-all pulse transfer function G(z) of a system relates the input 
and output pulse sequence and is given by 
Zon la Onaem) 
— 1.627! + 0.4822 

G(z) = i 
Multiplying both numerator and denominator by 2?, 
z2— 0.5 
NO) = eres ac ais 

The denominator of G(z) contains the polynomial leading to the charac- 
teristic equation 
27 — 1.62 + 0.48 = 0 
While this simple equation can be solved directly for its roots, the 
bilinear transformation will be applied to illustrate the method. Using 
(5.46), 
(NEELI2 eal 
