54 SAMPLED-DATA CONTROL SYSTEMS 
in Fig. 4.1. Because points similarly located in each of the strips plot 
as the same point in the z plane, each pole of F(s) produces one pole in 
F(z), and the singularities of F(s) and F*(s) are represented as a finite 
number of poles (and zeros) in the z plane. This makes the use of the 
z plane very desirable and is only one of many reasons why the use of the 
auxiliary variable z is desirable. 

Strip 1 

s-plane z-plane 
Fia. 4.1. Relation between s plane and z plane. 
It is noted that all points in the right half of the s plane map in the 
region of the z plane outside the unit circle, while those points in the left 
half of the s plane map into the region inside the unit circle. This simple 
observation indicates that, if a system is being examined for stability 
as evidenced by the presence of poles in the right half of the s plane, this 
condition could be studied by testing for the presence of poles of F(z) 
outside the unit circle of the z plane. 
4.2 Mathematical Derivation of the Z Transform 
The characteristics of the z transformation can be obtained on a more 
rigorous basis by the use of contour integration. Using the impulse 
approximation, the sampled function f*(¢) has been expressed as 
HO = fOorO (4.6) 
where it is recalled that 67(¢) is an infinite impulse sequence with a 
sampling period of 7. As expressed in (4.6), the sampled time function 
is represented as the product of two time functions whose Laplace trans- 
form is obtained by a process of complex convolution. The complex- 
convolution integral is given by 
c+jo 
eth] = gh [ F@MGEs - P) ap (4.7) 
where h(t) is a time function which is the product of f(‘) and g(t), F(p) 
is the Laplace transform of f(t), and G(p) is the Laplace transform of 
