4 BELL SYSTEM TECHNICAL JOURNAL 



wave in which tlie aniphLude of the radio frequency is an audio 

 frequency function. In this case the radio frequency phase angle is 

 constant. In the case of a single frequency modulation with one side- 

 band eliminated there are impressed on the detector input only two 

 frequencies. These may be combined in a well known manner. ■• 

 Thus, if the impressed voltages are of the form acosx and bcosy, then 

 the amplitude is given by 



^la- + b'^ + lab cos (.r - y). (3) 



The expression for the phase angle will not be given here as it can be 

 shown that if a and b are unequal and the difference between the 

 frequencies xllir and yjlir is small compared with either frequency, 

 then the variation of the phase angle with time may be neglected in 

 computing the audio frequency components. In the present case we 

 have two radio frequency waves the amplitudes of which are not 

 constants but are slow functions of time and these may be substituted 

 for a and b in (3). Thus the effective amplitude of the total input 

 signal may be taken to be 



5 - V^- + B' -\- 2AB cos Id, (4) 



in which 



A = E{\ + AI cos pt), 

 B = e{l + m cos qt), 

 and 



U — 0)1 — C02. 



The problem then resolves itself into an analysis of the detection, by a 

 straight line detector, of a single radio frequency component. The 

 results of such an analysis are well known and it can be readily shown 

 that the audio frequency output may be obtained, except for a factor 

 of proportionality, by resolving the amplitude into its audio frequency 

 components. In the present case the amplitude to be resolved is 

 given by (4) which may be written 



S = V(^ + B)^ - 2AB{\ - cos ut). 



The interfering signal B will be taken to be always less than the de- 

 sired signal A, and hence A- -\- B- > 2AB, from which it follows that 

 {A + By > 2AB{1 — cos «/.) Hence the radical may be expanded 

 by the binominal theorem, giving 



.45(1 - cos ut) 



S = A -\- B - 



A + B 



A'^B~[\ - cos Kty- A'B\\ - cos ut) 



2 (.4 + By 2iA + By 



^ \'ide: Lord Rayleigh, "Theory of Sound," i)age 23, sec. ed. 



C^) 



