898 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



The conversion of these variations into timing deviations in the re- 

 generated pulses depends on certain relationships between the pulse , 

 train and the timing wave, discussed in the following sections. I 



2.1 Tolerance to Noise 



From Fig. 2 it can be seen that if the timing wave is displaced by to , " 

 the value of P{t) -{- R{t — to) in the presence of a pulse exceeds the 

 triggering level by a maximum amount 



[P(0 + Rit - to) - L]„,ax ^ [P(ro) - L]. (2.1) j 



It will be recognized that the right-hand side of this equation represents i 

 the tolerance to noise of negative amplitudes with instantaneous sam- ' 

 pling at t = To , as in an ideal repeater with complete retiming. 



With partial retiming, the tolerance to noise will be less than the above 

 maximum value. However, it will be greater than the average of P{t) + 

 R{t — To) — L in the range where the latter difference is positive. Let it 

 be assumed that it is smaller than the maximum by a factor k somewhat 

 smaller than unity. The tolerance to noise with a displacement to in the 

 timing wave is then smaller than without a displacement (i.e., to = 0) 

 by the factor 



^ k[P(ro) - L] ^ P(to) - L ..^ .^ 



^ k[P{0) - L] P(0) - L ■ 



The tolerance to noise will thus be reduced in a way similar to that 

 for an ideal repeater with complete retiming. The absolute tolerance to 

 noise will be less than for a repeater with complete retiming by a factor 

 A; somewhat smaller than unity, say in the order 0.8, corresponding to 

 about 2 db. 



2.2 Conversion of Timijig Deviations 



With partial retiming, timing deviations in received pulses and in the 

 timing wave are converted into smaller deviations in regenerated pulses. 



Let Tp be a time displacement in a received pulse and Tr in the timing 

 wave, both in the positive direction. Pulses will then be regenerated . 

 at a time to' given by I 



P(to' - T;0 + R{t,' - Tr) = L (2.3) 



where the minus signs are used since this corresponds to a displacement 

 of P and R in the positive direction. Subtracting (1.8) from (2.8), 



P(/o' - Tp) - P{to) + R{to' - Tr) - R{to) = 0. (2.4) 



