606 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX A 



Evaluation of Volume Corresponding to Mean Power When 

 Volume Distribution is Normal 



The successive steps in the evaluation of (3.2) are as follows: 



/Flog. 10 \ 



antilogioF/10 = lO^/^" = exp ( ^~ ) 



(x^-Fo-^log. 10 ) 



g 10 V 20 ' \ e 2a2 dV 



U — 00 



= g 10 \ • 20 ° / = IQl 



Hence 



V 0-2 



Fop = 10 logio antilogiOjQ = Fo + 20^°^' •^^• 



APPENDIX B 



Counting of Modulation Products 



Consider an iV-channel system with carrier frequencies »i/>, («i + \)P, 

 • ' ', flip, where n2 = fii -\- N — 1. Let g„ represent the frequency of 

 the signal impressed on the voice frequency channel associated with 

 the carrier frequency np. The wave to be amplified is then 



ea = ZQn COS l{np + qn)t + ^n]. (1) 



n =ni 



The phase angles are of no consequence if the signal frequencies are 

 incommensurable; hence we shall simplify the notation by setting 

 0„ = 0. The square of the single series in n may be written as a double 

 series in m and n, thus : 



