SELF-TIMING REGENERATIVE REPEATERS 923 



purposes be disregarded, with root-sum-square combination as assumed 

 above. The value of C obtained from (6.11) will differ from that obtained 

 from (6.9) when 7,, is given by (6.5), by a small fraction of one per cent. 

 Although root-sum-square combination appears justified for reasons 

 given before, it is of interest to determine an upper limit to the cumula- 

 ! tion based on direct addition of random timing deviations. The maxi- 

 mum cumulation factor thus obtained is 



I If 



1 



n = l 



Cxnax = Z 7. . (6.12) 



Employing (6.8) for 7„ and neglecting the terms in ai', the upper limit 

 to the cumulation factor for A^ = x becomes 



Cmax = :; ■ (6.13) 



With Pr < 0.6 for ^ < ±60°, C„,ax < 2.5. 



If the above maximum cumulation factor is applied to random timing 

 deviations resulting from amplitude variations in the timing wave, as 

 given in Table I^' of Section 5.4, the resultant rms phase deviation at 

 the end of a long repeater chain could be as great as 25°, rather than 10° 

 for a single repeater, when i^ = 60° and Q = 100. To attain satisfactory 

 performance it would in this case be necessary to limit the maximum 

 fixed phase shift to substantially less than ±60°, which would entail 

 greater freciuency precision than indicated in Sections 5.1 and 5.2. 



li \p < ±15°, Pr < 0.40 and Cmax < 1-7- Ii^ this case the rms phase 

 deviation as given in Table I for a single repeater is ^r = 4°, and the 

 rms phase deviation in a long repeater chain would be less than 7°. In 

 a long repeater chain the rms phase deviation resulting from pulse dis- 

 tortion would be greater than given in Table II by an rms cumulation 

 factor C = 1.08 for pr = 0.4, and would thus be about 8° when \{/ < 

 ±15°. The total rms phase deviation would thus be about (7" + 8')^ " = 

 11°. Random phase deviations exceeding 4 times the latter value, or 

 about 45°, would be rather unlikely. The sum of the fixed and random 

 phase deviations would thus be limited to about 60°, so that satisfactory 

 performance would be expected when the fixed phase deviation is 

 limited to about ±15°. 



With the approximations for 7,, employed above, the rms cumulation 

 factor for a chain of A" repeaters as obtained from (6.9) is less than for 

 N = oc by the factor (1 - p.'-'Y'" ^ 0.99 for p. = 0.5 and A' = 3. The 

 maximum cumulation factor obtained from (6.12) is less than for A' = y- 

 by the factor 1 — p/^ = 0.99 for A' = (i. Thus, cumulation of random 



