590 BELL SYSTEM TECHNICAL JOURNAL 



we find that in addition to interference in channels having twice and 

 three times the fundamental carrier frequencies, there are modulation 

 terms appearing in channels having carriers which are various com- 

 binations of sums and differences of the original carrier frequencies. 

 The second order term gives rise to crosstalk products with carriers 

 {m + n)p and (w — n)p as well as Imp. The third order term 

 causes products with carriers (2m + n)p, {2m — n)p, {I -\- m -\- n)p, 

 (J -{- m — n)p, and {I — m — n)p as well as 2>mp. In the above /, m 

 and n are integers. For convenience we represent the tones asso- 

 ciated with the carriers Ip, mp, np by A, B and C and designate the 

 various types of products as A -{- B, A — B, 2A -\- B, 2A — B, 

 A -\- B + C, etc. It will be noted that the resultant modulation 

 falling in a particular channel at any instant depends on the particular 

 loading conditions prevailing on other channels at the same instant, 

 and that a wide variety of amplitudes, numbers, and types of products 

 are possible. Detailed study of these possibilities is necessary for the 

 solution of our problem, 



3. Nature of Multichannel Speech Load and 

 Resulting Modulation Products 



Considering any individual channel of the system, we note that (1) 

 it may be active or inactive and (2) if active, the signal power being 

 transmitted may vary throughout a considerable range of values. 

 With regard to (1), we may estimate from traffic data a probability r 

 that a channel is active.^ With regard to (2), data are available on 

 the distribution of volumes corresponding to different calls at the toll 

 switchboard. By "volume" is meant the reading of a volume indi- 

 cator of a standard type. The distribution is approximately normal 

 and hence may be expressed in terms of the average value Vo and 

 standard deviation a. In mathematical language, the probability that 

 the volume from any subscriber is in the interval dV at F is given by 



p(V)dV = -^ e-(^-i^o)^/2.^J7. (3.1) 



The value of Vo is about 16 db below reference volume, or about — 8 vu 

 when measured on the new volume indicator recently standardized in 

 the Bell System. The value of a is about 6 db. It is to be noted that 

 volume is proportional to the logarithm of average speech power and 

 hence Vo is not the volume corresponding to the mean of the average 

 speech powers of different talkers. The latter quantity, which will 



* A channel is said to be active whenever continuous speech is being introduced 

 into it. See Reference (1). 



