



SELF-TIMING REGENERATIVE REPEATERS 937 



1 



^.^3 ...^„- = (l-,){l-,)(l-^.J...(^l-2-^— -^j (51) 



1-3-5-7 ••• [2{n - 1) - 1] 

 2-4-G-8 2(w - 1) 



(2«) ! 



22"(n!)2 

 When n » 1, (51) approaches the value 



(52) 

 (53) 



o •> 



1 



1/2 



a2'ai ■ ■ ■ a,- ^ — . (54) 



Vxn/ 



The latter approximation is based on the following expression, for 

 X = —^, giveninWhittaker and Watson's: "Modern Analysis" page 259: 



lim (1 + .r)(l + .r/2)(l + .r/3) •••(!+ x/n) = — ^V^' (55) 



n~^x r(l + X) 



where T is the gamma fmiction, r(~j + 1) = tt''". 



The above analysis assumes that the timing wave at each resonant 

 circuit is applied directly to the next resonant circuit, except for the 

 amphfication between resonant circuits. This would be the case if the 

 timing wave were transmitted on a separate pair, in which case A'l^n 

 would be the rms cjuadrature component owing to noise in the timing 

 circuit. 



In regenerative repeaters, deviations in the timing wave resulting 

 from the cjuadrature component are imparted at intervals T into the 

 next repeater section as deviations in the spacing of pulses. These 

 timing deviations occurring at intervals T will have a certain random 

 amplitude distribution, which can he regarded as having a certain 

 frequency spectrum. When the deviations are discrete and occur at inter- 

 vals T, the spectrum will extend to a maximum frequency /max = l/2r, 

 or co„,ax = T^/T = wo/2. In this case the upper and lower limits of the 

 integrals above would be replaced by ±coo/2, except for the first repeater 

 section. The recurrence relation (48) is then no longer exact, but the 

 resultant modification is insignificant and can be disregarded. This will 

 be seen when the value wn/2 is inserted for co in the integrand of (47), 

 which then becomes 1/(1 + Q ), as compared with 1 for w = 0. Thus 

 the contribution to the integrals for w > a)n/2 can for practical purposes 

 be disregarded. 



