112 BELL SYSTEM TECHNICAL JOURNAL 



Second, consider the case in wliich 



V = Fjv + P cos pt (4.1-13) 



where P/2t hes near the noise band of IV • The probabiUty density of the 

 envelope R is 



-[-'-^1^3 



R 



From this and equations (3.10-12), (3.10-13), we find 



y aR- aP /. 1 1 A\ 



^dc = ^ = «^o -1- -^ (4.1-14) 



I'U = ^ i?^ = a' 



o P'' 



2^0 + 2PVo + J 



I}f = ru - Idc = (x-[h + P-]h (4.1-15) 



In (4.1-14) xpo is the mean square value of V^ and P"/2 is the mean 

 square value of the signal. These two equations show that Idc and the 

 rms value of 7^/ are independent of the distribution of the noise power 

 spectrum in IV as long as the input V is confined to a relatively narrow band. 

 In other words, although this distribution does affect the power spectrum 

 of the output, it does not affect the d.c. and rms 7^/ when xf/o and P are given. 

 That the same is also true for a large class of non-Unear devices was first 

 pointed out by Middleton (see end of Section 4.9). 



When the voltage is 



V = TV + P cos pt -\- Q cos qt, (4.1-4) 



p 9^ q, we obtain from equation (3.10-25) 



2 



lU=-fR' (4.1-16) 



l}^ =a'Ul + P'h + Q'h + 



p^ 



2 



^ These results are special cases, obtained by assuming no audio frequency filter, of 

 formulas given by F. C. Williams, Jour. Inst, of E. E., 80 (1937), 218-226. Williams also 

 discusses the response of a linear rectifier to (4.1-4) when P ^ Q -\- F,v • An account 

 of WilUaras' work is given by E. B. MouUin, "Spontaneous Fluctuations of Voltage," 

 Oxford (1938), Chap. 7. 



