SELF-TIMING REGEXERATIVE REPEATERS 



933 



ture u 

 write 



rith the steady state response as given by (21), it is convenient to 



uo = ui — Aw, 

 [(^{t - nT) -yp +yp - Ao:(t - nT)] 



- Aco{t - nT)] (23) 



cos uo(t — n T) = cos 



= cos [o}(t — nT) — lA] cos [\l/ 



- nT) - xP\ sin [xP - Acc{t - nT)] 



^\ 



— sin [co(/ ... , ^ J . .- ^-r -- s 

 k'ith / - NT + to , and w7' = tt, (22) can be written: 



V 



.4.(/) = cos (c/o - lA) E ± ^os [,Ai - Ac.7^(-V - ,,)]e-"of^)(^-")/2e 



n = 



-a)o(r)(A--nW2Q 



(24) 



If 

 - sin {cctu - lA) H ± sin [^i - Au}T{N - n)]e 



where \pi = \p — Aw/,, = i/'(l— ,^) = i/', since a)/n/2Q < 7r/2Q « 1. 



With eciual probabilities of a plus or a minus sign in the summations, 

 the rms ^•alue of the in-phase component becomes 



.1/ = 



— u,-or( .V— /(I :q 



1/2 



(25) 



Ar" = 



- \ 



X cos'- [.A - Aco7X.V - n)]e~ 



i: hi + cos 2[,A - Acor(.V - /Ojc-"'"^^-"^''') 

 _ >,=o 2 



The rms value of the (juadrature component becomes 



Z shr [4. - Ac.T{N - n)]e-"°^*-^'-"^'*^]''" 



i; ^ (1 - cos 2[,A - Acor(.V - ;OF""'*''~"^'')T "• 



_ n=0 2 



These expressions can be transformed into sums of geometric series 

 by writing 



cos.r = i(e'' + e-'O, x = 2[rP - AvoT{X - n)]. 



Evaluation of (25) and (26) by this method gives 



1 



(26) 



, . P(0) 1 



2 21/2 |_1 - e 



g-wor/Q ' D 



1/2 



, // _ P(0) 1 

 (2) 2''2 



1 



1 - e-"or/Q D 



12 



(27) 

 (28) 



