436 BELL SYSTEM TECHNICAL JOURNAL 



^ =^ ^ [ 1 - '''^'^'^''''^^ j lA'a^p.S + Q'{a, + aico„) cos ^] (53) 



where F = \ -{- aiA'kpz = 1 — /x/3 as before. Finally, neglecting 

 terms in Q'"^, we have 



a = i [^'aipi5 + (2' (ao + aico„ - ^^^ll^P^^ cos ^ ] . (54) 



The first term is the recovered signal while the remaining terms repre- 

 sent noise. Both signal and noise are modified by feedback. If we let 



PiS = Aco cos pt (55) 



then the noise becomes 



— (ao + aiojn) - -p {A'ai^kp2^oi cos pt) 



X cos {iOnt — X sin pt + 4>n). (56) 

 By means of the Jacobi expansions this can be put in the form 

 Q' ^ \ r , N A'ai^kpifnpl J. . , 



% L («0 + fllWn) -p Jm{x) 



X cos [(aj„ - mp)t + </)„] (57) 



where Jm{x) is the Bessel coefficient of the first kind. 



Now let it be assumed that the disturbance consists of a very large 

 number of sinusoidal components of like amplitude Q, random phase, 

 and uniformly distributed along the frequency scale. The summation 

 of this series of voltages can be represented by the very general 

 expression 



/(/) cos [a,/ + </>(/)]. (58) 



So long asf(t), the equivalent amplitude of the high-frequency disturb- 

 ance, is small compared with the carrier amplitude A, the approxima- 

 tion (49) will be valid and the total output noise can be obtained by 

 summing up the effects of the individual elements which constitute the 

 disturbance. 



The effect of a single disturbing element is given by (57). Any term 

 of this expression can be made to have the frequency qii m and aj„ are 

 so chosen that 



co„ = mp ± q. (59) 



Then for each value of m in (57) there will be available values of co„ 



