598 BELL SYSTEM TECHNICAL JOURNAL 



and shown in the table. The number of these is to be subtracted from 

 the total of the {A + B — C)- or {A — B — C)-types to obtain the 

 number of products not involving the listening channel. It should be 

 pointed out also that kp is the derived carrier frequency throughout 

 and that the products may extend over into adjacent channels. The 

 principal component of the product usually falls in the channel with 

 carrier frequency kp, but in some cases the amount of energy falling in 

 adjacent channels may be quite considerable, as may be seen from 

 Figs. 1 and 2. 



The average number of products falling simultaneously in one 

 channel is found by multiplying the total possible number by t^ for 

 two-frequency products and t^ for three-frequency products, these 

 factors being the probability that any particular product is present. 

 The average number present is not affected by the dependence of the 

 products arising from the fact that one talker may participate in the 

 formation of more than one of the products falling in a channel. For 

 convenience in making use of the results of Table II in evaluating the 

 amplifier requirements, we shall represent the number of x-type prod- 

 ucts falling in channel number k when it is idle and all other channels 

 are active by the symbol Vxk- We shall also let y.{x) represent the num- 

 ber of distinct fundamental components required to produce an x-type 

 product, e.g., m(^ + B) = 2, fx{2A + B) = 2, fx{A + B - C) = 3, 

 etc. It follows that the probability that any particular product is 

 present is t''(*\ since r is the probability that any one required com- 

 ponent is present. The average number of x-type products present 

 in the i^-channel is therefore VxkT''^'\ and may be considered as de- 

 termined since Vxk is the quantity tabulated in Table II. 



7. Modulation Requirement in Terms of Average 

 Total Noise Permissible in a Channel 



From Section 3 we have a result for the volume of one product of 

 arbitrary type, averaged on a power basis for a distribution of funda- 

 mental talker volumes, referred to the product of the same type pro- 

 duced by zero volume talkers. From Section 6 we have the average 

 number of products of each type appearing in a channel. Combining 

 these two results should give the average total modulation of each type 

 present in a channel. A difficulty occurs however inasmuch as it is 

 not certain how the interfering effect of superimposed modulation adds. 

 The noise caused by one modulation product is of an irregular nature 

 and it is probable that its most disturbing effect is associated with 

 infrequent peak values. When two products are superimposed their 

 individual peaks are not apt to coincide and hence the resultant dis- 



