FREQ UENC Y-MOD ULA TION 399 



We write for convenience in the subsequent analysis 



a„ = AnE. (15) 



^„ is then the relative amplitude of the noise element, referred to the 

 amplitude E of the high-frequency signal. We shall suppose through- 

 out that ^n is small compared with unity; that is, the noise is small 

 compared with the signal. 



We further suppose that at terminals 2, 2 between the band-pass filter 

 and the frequency detector there is introduced a second typical noise 

 element 



bn exp (i(aj, + co„)/ -f i/3„), (16) 



which is entirely independent of the noise element (14). This may be 

 regarded as caused by tube-noise in amplifiers (not shown in sketch). 

 We write 



hn = B,,E (17) 



so that Bn is the relative amplitude of the noise element, referred to 

 the amplitude E of the incoming signal wave. It also is assumed small 

 compared with unity. 



The total input to the frequency detector, neglecting the random 

 phase angles, is then 



CiCiEM 



whf 



exp ( / I J/ j +^„exp ( i I (S2 -f i]„«)^/ j 



expT/ r (0 + ^n'')dt\\ , (18) 



_>_ Bn 



ciCiM 



^ = (j^c -\- \s — yL(X 



^n" = COn - \S (19) 



0„* — cOn — \s -{- yLcr. 



The output of the frequency detector is then (see Ref.' 

 CidCzEMexp lit U dt 



X\ 1 -^~{\s - txa) 



L wi 



+ ^ „ M + — (co„ - M0-) exp ( * I Un^i 



