MATHEMATICAL ANALYSIS OF RANDOM NOISE 141 



where eo = 1 and e,,, = 2 for m > 1. When the product of the cosines is 

 expressed as a sum of cosines of the angles ;wo xo ± ffii Xi • • • zLhinXn , it is 

 seen that the coefficient of the typical term is /Imo-mjv > except when all 

 the w's are zero in which case it is ^.lo-o • Thus 



^yioo-.-o = dc component of / 

 I Amo---mx I = amplitude of component of frequency (4.9-16) 



;r- I niopo =b niipi ± • • • ± niffpff \ 

 Ztt 



For all values of the m's, 



IT J c r=0 



(4.9-17) 



M = mo -}- nil + " • + Mn 



Following Bennett's procedure, we identify V as given by (4.9-14), with 



V = Pcospt -\- V^ (4.1-13) 



by setting Po = P, po ^ p, and representing the noise voltage I'at by the sum 

 of the remaining terms. Since this makes Pi , P^ all very small, Laplace's 

 process indicates that in (4.9-17) we may put 



n MPrti) = exp - ^ (PI + • • . + Pi) 



r=i 4 (4.9-18) 



__ ^-l^ou2/2 

 — ^ o 



We have used the fact that \{/o is the mean square value of V^ . It follows 

 from these equations that 



dc component oi I = ~ [ F{iu)MPu)e^~'^°'^^"^ du 

 Component of frequency-^ = - / F{iu)Jn{Pu)e''^°" '' du 



ZTT TT Jc 



These results are identical with those of (4.9-9). 



The equations just derived show that h„Q is to be associated with the n 

 harmonic of p. In much the same way it may be shown that hnk is to be 

 associated with the modulation products arising from the n harmonic of 

 p and k of the elementary sinusoidal components representing IV . We 

 consider only combinations of the form pi ± p2 ± ps , taking ^ = 3 for ex- 

 ample, and neglect terms of the form 3pi and 2pi ± p2 . The former t>'pe 

 is much more numerous, there being about N of them while there are only 

 about N and N^ , respectively, of the latter type. 



