920 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



resulting from amplitude variations at subsequent repeaters will be modi- 

 fied by N-1, N-2 etc. intermediate resonant circuits. The situation is 

 similar to that of applying identical noise waves at the input of each of 

 A'' resonant circuits in tandem. At the output the A'' noise waves will have 

 different shapes owing to restriction of the l)and and increasing phase 

 distortion as the number of resonant circuits in tandem increases. For 

 this reason combination on a root-sum-square basis appears justified also 

 in this case, particularly with various degrees of mistuning of the res- 

 onant circuits, so that the amplitude \'ariations in the timing waves 

 will differ in phase among repeaters. 



6.2 Propagation of Timing Deviations 



To determine the cumulation of timing deviations along a repeater 

 chain, it is convenient to first consider a single repeater as a source of 

 timing deviations, and to determine the propagation of these timing 

 deviations along a repeater chain. In the following, 7,, will designate the 

 rms propagation factor for /? repeaters in tandem; i.e., the factor by 

 which the rms timing deviations at the end of a chain of n repeaters is 

 smaller than at the first repeater, with timing deviations originating at 

 the first repeater only. 



Let the rms timing deviation at the output of the first repeater as 

 given by (5.14) for convenience be taken as unity. At the output of the 

 second repeater the squared rms timing deviation is then reduced by the 

 factor 



2 2 , 22 



72 = Pr + Qi Vr , 



ai 



a. 



(6.1) 



As indicated symbolically in Fig. 8, the first term represents the reduc- 

 tion owing to partial retiming. The second term is the additional devia- 



-RESONANT CIRCUITS- 



Fig. 8 — Propagation of random timing deviations along repeater chain. 



