LOAD RATING THEORY 639 



and examining how the distribution curve of equivalent volume may 

 be obtained for this fixed number of channels. The relation between 

 volume and average speech power given in equation (2) may be re- 

 written for this case in the form 



Wi 

 Ft = 10 logio^T^db, 

 vvo 



where Wo = 1.66 milliwatts, Wi is the average speech power in milli- 

 watts, and Ft is the volume in db for any one of the active channels, 

 all at a point of zero transmission level. 



Likewise, the relation between equivalent volume and average 

 speech power for n active channels is given by the expression 



HWi 



V = w-channel equivalent volume = 10 logio -^ttt — db. 



Wq 



Since the distribution of the channel volumes Ft is known and the 

 volumes of the various channels are independent, the straightforward 

 procedure to obtain the distribution of the w-channel equivalent 

 volume F would involve the following steps: (1) the obtaining of the 

 distribution function of Wi by a transformation of that of Ft; (2) the 



n 



calculation of the distribution function for the quantity Yin) = II PFj; 



1 



(3) the transformation of the Y{n) distribution to that of F by invert- 

 ing the process used in step (1). 



The difficulties in this process are all in the second step, where, 

 having given p\{W), the distribution of average powers for a single 

 channel, it is required to obtain pniY), the distribution for n active 

 channels, with Y defined in terms of W by the relation given im- 

 mediately above. The formal solution requires the evaluation of 

 integrals of the following type: 



Pn{Y) = f pn-k{W)Pk(Y- W)dW. 



Jo 



By successive calculation of such integrals for w = 2, 4, 8 . . . .taking 

 k each time equal to w/2, the required distributions may be obtained 

 for the necessary range of values of n. 



As in the case of the instantaneous voltage distributions, it has not 

 proved feasible to perform the integrations analytically. It was 

 necessary to resort to numerical evaluation of these integrals; by 

 combining the transformations in steps (1) and (3) with the process 



