120 BELL SYSTEM TECHNICAL JOURNAL 



same properties as the random noise current studied there. For example, 

 the distribution of J^ tends towards a normal law. In our discussion we 

 had to assume that /3r <C 1. If the voltage V applied to the non-linear 

 dfevice is confined to a relatively narrow frequency band, say /& — /a , it 

 appears that the interval r (chosen above so that /(/) and /(/ + r) are sub- 

 stantially independent) may be taken to be of the order of \/{fb — fa)- 

 In this case Jn tends to behave like a random noise current if /3/(/6 — fa) is 

 much smaller than unity. 



We now turn our attention to the second statement made at the begin- 

 ning of this section. Let the applied voltage be confined to a relatively 

 narrow band so that it mav be represented by equation (4.1-8) of Section 

 4.1, 



V = R cos {ccj -\- d), R > 0, (4.1-8) 



where fm = oom/i^ir) is some representative frequency within the band 

 and R and 6 are functions of time which vary slowly in comparison with 

 cos comt. We call R the envelope of V. 

 From equation (4A-1) 



I = -^ f F{iu)e''"' ""' ^"'"'+'' du (4.3-7) 



27r J c 



We expand the integrand by means of 



^ix cos V _ g ^jn ^^g n^j^{^x) (4.3-8) 



71 = 



where eo is 1 and €„ is 2 when w > and Jr,{x) is a Bessel function. 

 Thus 



eo 



I = Yj An{R) COS («w^/ + ne) (4.3-9) 



n=0 



where 



Ar^iR) = €n^ [ F{iu)Jn{uR) du (4.3-10) 



27r Jc 



Since i? is a relatively slowly varying function of time we expect the 

 same to be true of An{R), at least for moderately small values of n. Thus 

 from (4.3-9) we see that the power spectrum of / will consist of a suc- 

 cession of bands, the n^^ band being clustered around the frequency «/,„ . 

 If we eliminate all of the bands except the n^ by means of a filter we 

 see that the output will have the envelope An{R) when n ^ 1. Taking 

 n to be zero, shows that the low frequency output is simply 



A^{R) =^ [ F(iu)MuR)du (4.3-11) 



27r J c 



