SELF-TIMING REGENERATIVE REPEATERS 899 



i By adding and subtracting P(/(/) + R{tu') and rearranging terms, (2.4) 

 can also be written 



[P(/o') - P{h)\ + [R{un - Hit,)] 



I = [7^(/n') - P{h' - T,)\ + [R{h') - R{h' - Tr)]. 



; For small values of Tp and r^ , such that br = to' — ^o is sufficiently small, 

 I both sides of (2.8) can be represented in differential form as 



' 8r[P'ito) + R'ito)] = T,P'{h) + TrR'ih) (2.6) 



where P'{h) = dPo{t)/dt at t = to , and R' is correspondingly defined. 

 Equation (2.9) can be written in the form 



5t = PrTp + l-rTr (2.7) 



where 



P'(to) R'ito) .28) 



^' P'ito) ^ R'itoV '' P'{to) + R'itoY ^'^ 



and 



Vr + /v = 1. (2.9) 



With random uncorrelated displacements of rms values fp and f, , 

 the rms value of 5t is 



8r = iprri + rrrry" (2.10) 



Eciuation (2.9) and (2.10) give the timing deviations in regenerated 

 pulses in terms of the deviations tp and r^ in the received pulses and 

 in the timing wave. To limit timing deviations in the regenerated pulses, 

 it is necessary to make pr and the product VrTr small. This will entail 

 the use of a timing wave comparable in amplitude to that of the pulses, 

 or greater, in conjunction with a small timing deviation r^ in the timing 

 wave. 



2.3 Conversion of Amplitude Variations Into Timing Deviations 



With partial retiming there is a conversion of amplitude variations 

 I in the received pulses and in the timing wave into timing deviations in 

 the regenerated pulses. 



Let the pulses have an amplitude variation Op and the timing wave Or 

 expressed as fractions of the normal values. Pulses will then be regen- 

 erated at a time /(/ given by 



(1 + ap)P{t,') + (1 + ar)R{t^') = L. (2.11) 



