636 BELL SYSTEM TECHNICAL JOURNAL 



multi-channel peak factor, the fraction of such intervals during which 

 the amplifier will be unsatisfactory from the standpoint of overloading 

 may be determined. For a particular amplifier, the operating trans- 

 mission level must be so chosen that this fraction will be small enough 

 to make any adverse effects on transmission unimportant. For sys- 

 tems of very many channels the proper value of this fraction is probably 

 about 1 per cent. During the busiest hour, this corresponds to 36 

 seconds during which audible interference may occur and as this will 

 be broken up into many very short intervals, the total efifect should be 

 slight. For systems of very few channels, the equivalent volume may 

 reach objectionably high values during these intervals and it might be 

 necessary to make this fraction smaller than 1 per cent to secure good 

 performance. For illustrative purposes, the 1 per cent figure will be 

 used in what follows without implying that it may not need alteration 

 in some cases. The methods used are applicable no matter what value 

 is chosen for the fraction of time overloading is permitted. 



Controlled Volumes 



As the simplest case to which the above procedure may be applied, 

 and one that may occasionally be of practical interest, consider a 

 commercial system with all the channels controlled to the same volume. 

 If there are N channels in the system, the probability that exactly n 

 channels will be active at any given time is given by equation (1), 

 with r = 0.25. By computing the value of p{n) for all values of n, 

 and taking the cumulative sum, the value of n which makes the sum 

 0.99 (or the next greater n) is readily determined. This determines 

 the number n of active channels that is exceeded 1 per cent of the 

 time. A plot of these values of n is given by the curve of Fig. 5, as a 

 function of N, the number of channels in the system. For small values 

 of N this curve has been drawn in a manner to smooth out the steps 

 introduced because n must of necessity be an integer and when the 

 value of n read from the curve is not an integer, the next higher in- 

 tegral value is to be used. It is of interest to compare this curve with 

 the two straight lines of the figure. The lower straight line represents 

 the asymptote for sufiiciently large N and the upper straight line is for 

 the condition where all channels are active simultaneously {n = N). 



The average power for n channels is n times that of one channel, and 

 the equivalent volume expressed in db is 10 logio n above that in one 

 channel. The equivalent volume may thus be computed as a function 

 of n, and by means of Fig. 5, as a function of N. Curve A of Fig. 6 

 shows the values of equivalent volume so determined as a function of 

 N, the number of channels in the system; it applies specifically to the 



