218 BELL SYSTEM TECHNICAL JOURNAL 



Since it seems certain that g and 6 must remain small to make the system 

 operative, we investigate the nature of (50) when expanded in powers of 

 g and 6. The leading terms are : 



i,Mr = ^^^|^^v) + ^^(co) + ... (51) 



Hence for g and 9 small, we have independent of any correlation which may 

 exist between g and 6, 



hMl^ = {^^^)^) + ^) (52) 



Let 



0-1 = g^(o}), 0-2 = ^(w) (53) 



Then from (46), (49), (52), 



_ y ioge 10 V 



'" ~ L\ 20 / 



U,-k = 



2 I 2 

 0-1 + (72 



N (54) 



In (54) (Ti is the r.m.s. departure of the gain in db from a constant and 0-2 

 is the r.m.s. departure of the phase shift in radians from a straight line. 

 If 0-2 is expressed in degress instead of radians, (54) becomes 



Ujk = 10~' (13.25 0-1 + .3046 al)/N (55) 



The total interference received in any one channel is the sum of the 

 individual contributions from the other A^ — 1 channels. The addition 

 factor required to express the total in terms of the interference from one 

 channel depends on the nature of the individual loads. Thus if the proba- 

 bility that any one channel is transmitting a signal wave is r, the average 

 total interference power received in one channel is 



X = t{N - l)X^k = USi f ' WiMdoii, (56) 



where 



U = {N - l)TU,h = ^^ ~ ^^'" 10~\l3.25<xl + .3046(72) (57) 



For large values of N, the ratio (N — l)/N approaches unity, and the 

 average interference becomes independent of the number of channels. 

 The average interference may not be the most significant quantity, however. 

 For example, if there is a considerable probability that all channels are 



* This method of avoiding any assumption concerning correlation of attenuation and 

 phase was suggested by Dr. T. C. Fry. 



