MATHEMATICAL ANALYSIS OF RANDOM NOISE 293 



for the case in which V is Viv + P ccs pt, we find 



h 



(4.3-17) 



. B ,P , B' + h 



2 TT ZTrr" 



72- P- - B, 

 Iff ~ ■ — :r^^ ¥'0 



when P » I J5 I , P" » i/'o where i/'o is the mean square value of Fat . 



4. When V is confined to a relatively narrow band and there are no 

 audio-frequency filters, the probability density and all the associated sta- 

 tistical properties of Id may be obtained by expressing lu as a function 

 of the envelope R of V and then using the probabiUty density of R. When 

 V is Vf] + P ccs pi -{- Q cos qt this probability density is given by the in- 

 tegral, (3.10-21) (which is the integral containing three Bessel functions 

 stated in the above summary of Part III). When V consists of three sine 

 waves plus noise there are four /o's in the integrand, and so on. Expres- 

 sions for i?" when R has the above distribution are given by equations 

 (3.10-25) and (3.10-27). 



\\T:en audio-frequency filters remove part of the low-frequency band the 

 statistical properties, except the mean square value, of the resulting cur- 

 rent are hard to compute. In section 4.3 it is shown that as the output band 

 is chosen narrower and narrower, the statistical properties of the output 

 current approach those of a random noise current. 



5. The sections in Part IV from 4.4 onward are concerned with the 

 problem: Given a non-Hnear device and an input voltage consisting of noise 

 alone or of a signal plus noise. What is the power spectrum of the output? 

 A survey of the methods available for the solution of this problem is given 

 in section 4.4. 



6. "Wlien a noise voltage TV with the power spectrum w(/) is applied to 

 the square-law device 



/ = aV^ (4.1-1) 



the power spectrum of the output current / is, when/ ?^ 0, 



Z+00 

 w{x)w{f - x) dx (4.5-5) 



where w{—x) is defined to equal wix). The power spectrum of / when V 

 is either P cos pt + TV or 



Q(l + k cos pt) cos qt ■{■ Vff 



is considered in the portion of section 4.5 containing equations (4.5-10) to 

 (4.5-17). 



