536 BELL SYSTEM TECHNICAL JOURNAL 



Appendix 2 

 We take the frequency modulated wave as 



cos ( ojc/ + X I sdt), (lb) 



f 



Jo 



where Wc is the carrier frequency and s = s(t) is the low frequency 

 signal. X is a real parameter, which fixes the amplitude of the frequency 

 spread. 



Correspondingly, we take the typical noise element as 



^„ cos ((co, + w„)/ + On). i2b) 



For reasons stated in the text, we take the more general formula for 

 the low frequency current as proportional to 



X5 + (wo + co„ + fj.s)An cos ( wj + ^n — X | sdt ) , (Sb) 



where wo, X, fx are real parameters. The term X5 is the recovered signal 

 and the second term is the low frequency noise corresponding to the 

 high frequency noise element (2b). 



We suppose that the noise is uniformly distributed over the frequency 

 spectrum, at least in the neighborhood of co = ojc, so that, corresponding 

 to the noise element 



^„cos (coj + dn), (46) 



the noise is representable as the Fourier integral 



N 



?/ 



cos (coj + dn)do}n (5b) 



and the corresponding wowe/>ower for the frequency interval coi < co„ < C02 

 is, by the Fourier integral energy theorem, 



' dcCn=- (C02 - C0i)iV2. (6b) 





The Fourier integral energy theorem states that, if in the epoch 

 < ^ ^ r, the function /(/) is representable as the Fourier integral 



/(/) =- r F(o:)-cos (oit + d(c^))do}, 



Ob) 



