MULTICHANNEL AMPLIFIERS BELOW OVERLOAD 591 



be designated by Vop, may be calculated by averaging the distribution 

 according to power, thus 



Vqp = 10 logio antilogio F/10 



= Fo + 1^ loge 10 (3.2) 



= Fo + .115 a\ 



when Fo and <t are expressed in db. The method of obtaining this 

 result is indicated in Appendix A. 



It will be convenient to extend the term "volume" to apply to 

 modulation products by designating the modulation product produced 

 by 0-vu talkers as a "zero volume modulation product" of its par- 

 ticular type. This is not its absolute volume as read by a volume 

 indicator, but a reference value to which modulation products of the 

 same type produced by talkers of other volumes may be referred. We 

 assume on the basis of a power law of modulation that the volume of a 

 product will increase one db for each one db increase in volume of a 

 fundamental appearing once in the product, two db for each db increase 

 in volume of a fundamental appearing twice, etc. Thus a {2A — B)- 

 product should increase two db for one db increase in the volume of 

 the ^-component, and one db for one db increase in the volume of the 

 5-component. If the fundamental talker volumes producing a par- 

 ticular product are normally distributed on a db scale, it follows from 

 established relations concerning the distributions of sums ^ of normally 

 distributed quantities that the volume of the product is also normally 

 distributed. The relations between average and standard deviation 

 for the modulation product and the corresponding quantities Fo and 

 0- for the fundamental are shown in Table I. 



TABLE I 



Modulation Average in db Referred to Standard Deviation 



Product Product from 0-vu Talkers in db 



2A 2 Fo 2a- 



A ±B 2 Fo V2(r 



iA 3 Fo 2,a 



2A ± B, B - 2A 3Fo VSa 



A + B±C,A-B-C 3Fo -^Ja 



That is, if the fundamental talker volumes are normally distributed 

 with average value — 8 vu, and standard deviation 6 db, the 



2 Multiplying amplitudes of fundamental components is equivalent to adding 

 logarithms of amplitudes; hence the volumes of the fundamental components add 

 to determine product volumes. For a derivation of the distribution function of the 

 sum of two independent normally distributed quantities, see Cramer, Random 

 Variables and Probability Distribution, Cambridge Tract No. 36, 1937, p. 50. 



