88 SAMPLED-DATA CONTROL SYSTEMS 
simply the transfer function of the two continuous elements in cascade. 
The relation which is being sought is the one between the final sampled 
output and the sampled input. The final synchronous switch establishes 
this pulse sequence, and its Laplace transform C}(s) is obtained simply by 
replacing all s in C2(s) by s + njwo and summing over alln. For simplic- 
ity, Gi(s)G2(s) will be replaced by G(s). Using the result of (5.7) and 
making this substitution, 
+o +0 
Hie = > HOE njvso) 7 » Ris -+ (m = n)\jad 
A typical term C>2,,(s) of this summation is 
aes 
er 
Cras) = G0 + giv) 7) Rist (m+ ahind — 6) 
m=—0 
It is noted that the summation in (5.9) is over all integral values of m 
ad infinitum, and since q is also an integer, the summation is unaffected in 
the limit by the choice of g. Hence, the summation is not a function of 
q, or m, for that matter. The summation can be written as 
+. 
TY, Ris+ (m+ aie = RG (5.10) 
m=—2 
regardless of the value of g. Thus, in (5.8) R*(s) can be taken out as a 
common factor and the expression for C}(s) can be rewritten as 
+2 
CAG) = l7 y G(s + nie) | R*(s) (5.11) 
n=— 
The transfer function G(s + njwo) will now be examined. By defini- 
tion, it is simply 
G(s + njwo) = Gils + njwo)Geo(s + njwo) (5.12) 
The Laplace transform of the output C¥(s) contains the summation of 
these terms over all integral values of ; hence, 
+o +0 
» Gnas > Gils + njwo)Ge(s + njwo) (5-18) 
It is noted that the summation of the product of terms is not equal to 
the product of the summations, except under very special conditions. 
