912 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



The resultant rms time deviation is (T'/2x)(^5 or 



h = 52, (4.9) 



and 



« = {T^/Qf- (4.10) 



The above factor a appHes to a single resonant circuit. When the rms 

 timing deviations represented by (4.9) are present in the regenerated 

 pulse train, the rms deviation at the output of the second resonant 

 circuit is 



^5,2 = 5aia2 , 

 where 



Si = ^• 



With n resonant circuits in tandem, 



Ss.n = daiaoQCs ■ ■ ■ a,, . (4.11) 



The factors q:„ are given by 



si - « = (tt/QY", (4.12) 



^j 



[ai-ao-as • • • a„J = a 



= a 



«2 = (1 - hr\ 

 «3 = (1 - ir, 



Qi = (I - iY'^, etc. 



2 1-3-5 ••• [2(n - 1) - 1] 



2.4.6 ••• 2(w - 1) 



(2n)! 



(4.14) 

 (4.15) 



22"(n!)- 



= a' ( — ) when n » 1. (4.16) 



The factors ay for j ^ 2 represent the reduction in timing deviations 

 resulting from the reduction in bandwidth as resonant circuits are 

 added in tandem. If resonant circuits with a narrow flat pass-band were 

 used, the bandwidth of any number of resonant circuits in tandem 

 would be the same as for a single resonant circuit. In this case 22 = 

 23 = a„ = 1. 



