FREQ UENC Y-MOD ULA TION 403 



Stated in words, 1 + w is the reciprocal of the ratio of the values of 

 0- without and with the low-frequency feedback into the oscillator. It 

 should be noted that this requires that the band-pass filter transmit the 

 frequency band 2X centered on coc. 



Appendix 



In formula (22), the expression {\s — /jLay has been replaced by 

 W/(l + m)^ its value when the noise is absent. When noise is 

 present, but small compared with the signal, this should still give a 

 good approximation for o--. We now propose to derive an exact solution 

 of (22) ; to this end we write 



X5 



\s — ixa ^ - — ^ nit) (la) 



1 + /« 



which is always possible. 



Now inspection of (la) shows that n{t) is the value of /jlct when 5 = 0; 

 consequently w^ = /jL^ao^ where ao^ is given by (34). Furthermore, 

 since 5 and n are entirely independent, sn = 0, and 



X252" 



(\s - ix(xY = + mVo^ . (2a) 



(1 -|- m)" 



Substitution of (2a) in (22) gives for Pn, instead of (23), 



3 \ \a;2/ (1 -f m)- 



+ o>aP.WN^^Ic{WM\ (3a) 



The second term is a second order quantity in the noise power and 

 may therefore be neglected when the noise is small, as is assumed 

 throughout this paper. 



