SELF-TIMING REGENERATIVE REPEATERS 911 



and 



/ \ 1/2 I I I 



(4.4) 



4Q/ cos xp 



These relations apply for small values of x// and for t/Q « 1. 



The resultant rms amplitude variation in the timing wave is Qr = g/ 

 as given by (4.3). 



The rms phase error ipr resulting from the quadrature component a," 

 is given by 



tan ^r = ^r = a/'. (4.5) 



The corresponding rms time deviation is {T/2Tr)ipr or 



rr / \l/2 



2ir \4Q/ cos \l/' 



(4.6) 



With regard to the probability of exceeding the above rms values by 

 various factors the normal law can probably be invoked with reasonable 

 accuracy. As mentioned before, the maximum possible amplitudes are 

 Ar{t) = ±As which would correspond to a peak factor {'IQ/tY '. With 

 Q = 100, the factor is about 8, while with Q = 1000 it is about 25. 

 Based on the normal law the probability of exceeding the rms value 

 by a factor of 4 is about 5 X 10~^ and by a factor of 5, about 10~'. The 

 normal law would be expected to apply, since the limiting peak values 

 are substantially greater than the peak values expected with significant 

 probabilities. 



4-3 Resonant Circuit Response to Pulse Displaceme7its 



Because of the random components given by (4.3) and (4.4), the 

 timing wave will contain small random amplitude and phase deviations 

 from a sinusoidal wave represented by (4.1). This will result in small 

 random deviations in the positions of regenerated pulses triggered from 

 the timing wave, which is represented by the third component shown in 

 Fig. 7. When the rms deviation in the pulse positions is 5, there will be 

 an additional random cjuadrature component in the timing wave which, 

 when taken in relation to the steady state component, is given by 



a/' = As" /A. = co8 (|^Y . (4.7) 



The corresponding rms phase de^'iation is given by 



ip, ^ a,". (4.8) 



