SELF-TIMING REGENERATIVE REPEATERS 



935 



With the following further approximations 



cos uoit — nT) — cos wo[f — nT + 5(n)] 



= sin wo[/ + 5(w)/2] 2 sin M(n)/2], 

 ^ coo5(w) sin cooi, 

 expression (36) becomes: 



A,{t) = P(0)coo sin coi E 5(n)e-"°^'-"^^^''^ 



(37) 



71 = 



N 



(38) 



= P(0)a;o sin coo^ Z 5(n)e-"''^^^-"^''^ 



n=0 



where the substitution t = NT -\- to has been made as in previous ex- 

 pressions. 



The above expression shows that the resonant circuit response will be 

 at quadrature vnih. the steady state timing wave cos w^o • 



In the above expressions, the dipulses are assumed to be present at 

 intervals T, whereas in a random pulse train they will be present at 

 average intervals 2T. The rms value of the quadrature component ^^'ith 

 randomly positive and negative dipulses at intervals 2T, with an rms 

 displacement 5, is 



11/2 



Al = P(0)a;o5 

 P(0) 



JV 



y^ ^-2uQT{N-n)IQ 



H=0 



] 



- C005 (^ 



1/2 



(39) 



In (38) the function e ""''"^ will be recognized as the impulse response 

 function of a circuit with impedance 



/3 + iw 



Z(2co) = 



^ 



.1 + CO-//32J 



1/2 



tan i/' = co//3. 



(40) 

 (41) 

 (42) 

 (43) 



It will also be recognized that (39) corresponds to the rms response of 

 such a circuit, when impulses 5(n) of random amplitude with an rms value 

 h are applied to average intervals 2T. Thus (39) can alternately be ob- 

 tained from 



