THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 53 
it is in closed or open form, the Laplace transform contains the complex 
frequency s as an exponent in e7*. This suggests the use of an auxiliary 
variable to replace e7*, the one chosen being z.!7_ Thus, if the change of 
variable is made such that 
B= GE (4.2) 
the transform of a sampled sequence can be expressed in terms of the 
auxiliary variable z. 
The transform given in (4.1) is expressed as a power series in 2 as 
follows: 
F*(s) = F(z) = > f(nT)e- (4.3) 
n=0 
where F(z) is known as the ‘‘z transform” of f*({). The variable z may 
be thought of either as e7*, as defined in (4.2), or as an ordering variable 
whose exponent represents the position of the particular pulse in the 
sequence f*(t). When viewed in the latter light, F(z) is a ‘‘ generating 
function”” and may be treated without identification with a Laplace 
transform. 
In studying the z transform, use is made of the complex z plane, just 
as in the case of Laplace transforms the complex s plane is used. In 
view of the relationship between the z transform and the Laplace trans- 
form, a brief review of common characteristics of the two planes will be 
given. Referring to Fig. 4.1, the standard s plane and the z plane are 
illustrated side by side. All the points comprising the imaginary axis 
of the s plane lie on the unit circle of the z plane because, for s = ju, 
B= Go (4.4) 
which is a complex number whose magnitude is unity and whose phase 
angle iswT’. Thus, the z plane reflects the periodicity of e7* by repeating 
the same values every time the angle w7’ advances 27 radians. The 
z plane may be thought of as an infinite series of planes overlaid on each 
other, or, more formally, as a Riemann surface. 
When applied to the sampled-data systems, this apparently ambiguous 
medium for representation of the properties of F(z) presents no difficulties. 
It is recalled that 
+2 
7 > F(s + njwo) = > f@M)E 24 (4.5) 
n=—2 n=0 
and that F*(s) is periodic in jwo. The poles and zeros of F*(s) are those 
of F(s), with infinite repetitions displaced by jw». The poles and zeros 
of F*(s) repeat themselves in each of the strips of the s plane, as sketched 
