204 SAMPLED-DATA CONTROL SYSTEMS 
The input is now expressed in terms of the double-rate auxiliary variable 
z2 by the simple process of squaring all z’s and then replacing them with 
zo’s. Physically, this means that every other double-rate sample is zero, 
as expected. 
The double-rate output pulse transform C(z2) is related to the input 
double-rate pulse transform R(z:”) by the double-rate pulse transfer 
function G(z2), 
C(é2) a G(z2) R(22?) (8.11) 
The double-rate output pulse transform is inverted in the usual manner 
to obtain the double-rate pulse sequence. This inversion includes an 
extra sample midway between two sampling instants of the basic rate. 
This procedure is easy to apply but, at the same time, it produces informa- 
tion on the output ripple only midway between the basic sampling 
instants. If this is considered sufficient, the slightly increased complexity 
involved in determining the double-rate output pulse transform is well 
worthwhile. 
EXAMPLE 
To illustrate the technique, the same example given in the previous 
section will be used. In this system, 
G(s) = 

s+ 4a 
and R(s) = 
The z transform of the input at the basic rate is given by 
1 
he) ane 
The double-rate z transform is obtained by replacing z by 2z.’. 
it 
1-— Zou 
R(Z2”) = 
The double-rate pulse transfer function G(z2) is given by 
Ge) = 
ft [ete 
The output double-rate z transform is given by (8.11) and becomes 
plished is. Lelie), eo iain 
(1 aers Zan wen Ca ae) 
which, upon inversion by the normal procedure, gives 
cx(t) = 1 + e976 — T/2) + (1 — e-*7)6¢ — 27/2) 
+ e-9T/2(] — e-*T)6(¢ — 87/2) + (1 — e+ e777) 6 — 47/2) + °-- 
C(z2) = 
