292 BELL SYSTEM TECHNICAL JOURNAL 



WTien F is a sine wave plus noise, Fa- + P cos pt, 



Tac = a^l^J ' iF,(-i; 1; -x) (4.2-3) 



2 



lit =-AP' + 2^o) (4.2-6) 



where i^i is a hypergeometric function and 



P Ave. sine wave power ,^„ ,, 



a: = — = — , ^ (4.2-4) 



2^0 Ave. noise power 



When X is large 



1-1...] (4.2-7) 



2 aVo 



1(! ~ TT 



If F consists of two sine waves plus noise, Idc consists of a h}'pergecmetric 

 function of two variables. The equations running from (4.2-9) to (4.2-15) 

 are concerned with this case. About the only simple equation is 



2 



fit = -. [21^0 + P' + Q'\ (4.2-14) 



3. The expressions (4.1-6) and (4.2-2) for I ,( in terms of the envelope 

 R of F, namely 



al^ , aR 



— - and — , 



2 T 



are special cases of a more general result 



lit = Aq{R) = ~ f F{iu)Jo{uR) du. (4.3-11) 



Ztt J c 



In this expression Jq{uR) is a Eessel function. The path of integration C 

 and the function F{iu) are chosen so that the relation between I and F may 

 be expressed as 



I = ^ [ F{iu)e''"' du. (4A-1) 



27r Jc 



A table giving F(iu) and C for a number of common non-linear devices is 

 shown in Appendix 4A. 



If this relation is used to study the biased linear rectifier. 



/ 0, V <B 



\V - B, V > B 



