128 BELL SYSTEM TECHXICAL JOURNAL 



Now consider the modulation products of the sum t^'pe. The terms of 

 this type in (4.5-8) which give rise to the frequency cojt are those for which 

 m + n is equal to k. Let « be 1 then m = k — 1. The phase of this term 

 is random with respect to all the other terms except the one given by n = 

 k — \, m = 1 which has the same phase. The average power contributed 

 by these two terms in (-i.5-8) is, as in (4.5-9), 



This disposes of two terms for which m + n is equal to k. Taking n to be 2 

 and going through the same process gives two more. Thus, assuming for 

 the moment that k is an odd number, the power contributed to the interval 

 fk , fk + A/ by the sum modulation products is 



- S (aCr^Ck-nf = 7 S (aCnChS' "> Ol'A/ / w(f)w(Jh - f) df 

 I 71=1 4 n=l Jo 



and this leads to the second term in (4.5-7). 



\\lien the voltage T" applied to the square law device is the sum of a noise 

 voltage T> and a sine wave : 



F = P cos ^^ + TV, (4.1-13) 



we have 



Y- = P^ cos^ pi + IPVxcos pt + Vl (4.5-10) 



From the two equations 



2 1 1 



cos pt = -z -{• - cos 2pt 



ave. Vl = ^cl,--^ / w{f) df 



1 2 JQ 



we see that 7, or aV , has a dc component of 



+ a f wif) df (4.5-11) 



Jo 



i2 



2 JO 



which agrees with (4.1-14), and a sinusoidal component 



^ cos 2pt (4.5-12) 



The continuous power spectrum Wc(f) of the remaining portion of I may 

 be computed from 



2PVn cos pi + T'a- . 



