530 BELL SYSTEM TECHNICAL JOURNAL 



Corresponding to the noise element (60), the output of the ideal 

 frequency detector is 



- i\ r sdt\\. (61) 

 Since the expression 



exp ( iwnt -\- idn — i\ i sdt ] 



occurs so frequently in the analysis which is to follow, it is convenient 

 to adopt the notation 



fin = W„ — \s(t), 



JQ,ndt = Wnt — X I 

 Jq 



sdt. (61a) 



With this notation and on the assumption that ^„ <C 1 and coi real, 

 the amplitude of the wave (61) is 



l+^+(l +'^\AnCoJ C\jt\. (62) 



In this formula the argument of the cosine function should be 

 strictly 



r Q^ndt + On. 

 Jo 



The phase angle dn is random however and does not affect the final 

 formulas; it may therefore be omitted at the outset. Consequently, 

 if the wave (61) is passed through a straight line rectifier, the rectified 

 or low frequency current is proportional to 



\s{t) + (coi + co„)^„ cos ( I ' ^ndt ] . (63) 



Jo 



The first term is the recovered signal and the second term the low 

 frequency noise or interference corresponding to the high frequency 

 element (60). 



Now the wave (63) , before reaching the receiver proper, is transmitted 

 through a low-pass filter, which cuts off all frequencies above coai coa is 

 the highest essential frequency in the signal s(t). Consequently, in 

 order to find the noise actually reaching the receiver proper, it is 



