922 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



2 2(/i-l) I {n — 1) 2(n-2) 2 2 

 Pn = Pr + Z-, Pr Vr «■> 



1! (().()) 



, , „ 2(,i-l) 2 2 2 



The rms deviation at the output of repeater n thus becomes 



A,,' = (Ap' + A,.')7„" + a"/v'fp'p-,'"- (6.7) 



In the case of repeaters with partial retiming the last term in (6.7) can 

 be neglected, in which case the cumulation of timing deviation will be 

 virtually the same when the timing wave is derived from the regenerated 

 as when it is derived from the received pulse train. 



The above expressions apply for resonant circuits consisting of a coil 

 and capacitor which have a gradual cut-off. If resonant circuits with a 

 flat pass-band and sharp cut-offs were used, a2 = as = «„ and (6.5) can 

 be simplified to 



7;' = (i - a{)vi""'' + ahp; -f uT'"'- (6.8) 



6.3 Cumulation of Timing Deviations 



The cumulation of random timing deviations from various repeaters 

 in a chain can be determined from the propagation constant given above 

 for any prescribed law of combination of timing deviations from various 

 repeaters. When equal rms deviations are contributed by each of A'^ 

 repeaters, and they are combined on a root-sum-square basis, the rms :! 

 deviation at the end of a repeater chain is greater than for a single 

 repeater by the cumulation factor ! 



/ N \ 1/2 



c = ( E yn . (6.9) i 



H=l 



An upper limit to (' is obtained by taking a-2 = a-i = a„ = 1 in (6.5) 

 in which case 7,," is given by (6.8) ; (6.9) then becomes for A^ = oc 



C 



(1 — ai") :; -, + af 



1/2 



(6.10) I 



- Pr^ 1 - Pr- - >-_ 



1/2 



(6.11) 



where the terms in gf have been neglected in (6.11), since gi" = a « 1, 

 about 0.03 for Q = 100. 



From Fig. 5 it will be seen that when \p < ±60°, pr < 0.6. Hence | 

 (' < 1 .25. Cumulation of random timing deviations can thus for practical 



