SWITCHED NETWORK FOR TIME MULTIPLEX SYSTEMS 1427 



Thus, it always happens that for t = r/2, i.e., at the middle of switch 

 closure time, 62(0 — es(t) = 0. 



Therefore if we consider the time function 62(0 ~ ^3(0 we have for all 



A:'s (-a>, ... 0, ••• + ^), 



62 (kT + l)-es (jcT + ^) = 0. 



If, for simplicity of analysis, we assume that the switch is closed during 

 the intervals -(r/2) + kT ^ t ^ +r/2 + kl\ then for all A-'s, 



62(^7^) - e.ikT) = 0. 



Using (17), this condition implies that [V{p)]* - 2[Iro{p)Z{p)]* = 0. 



Now, remembering that irn{t) couvsists of a sequence of half sine waves 

 whose shape is defined by So{t) (which is by definition identical to Si{t) 

 except for an advance in time of t/2) it follows that 7ro(p) = B{p)So{p), 

 where B{p) is the .^^-transform of the seciuence of impulses whose measure 

 is eciual to the charge interchanged between A''i and A^2 at each switch 

 closure. Since [B(p)Soip)Z(p)]* = Bip)[Soip)Z{p)]*, then 



j.(. ^ [Zn{p)h{p)\* 

 ^^^ 2[So{p)Z{p)\*- 



From which it immediately follows that 



[Zr,ip)Up)]*So{p) 



Iroip) = 



2[So{p)Z{p)y 



and 



,,./!_ [Za(p)Up)]*Up)Zn(p) 

 ^'°<P' 2[S„(p)Z(p)l* ^ 



where 



+» 



[So(p) Zip)]* = ^ Z Soip + jnco.) Zip + jno^s) 

 1 »» = —00 



REFERENCES 



1. L. A. MacColl, Fundamental Theory of Servomechanisms, D. Van Nostrand Co. 

 Inc., New York, 1945. 



