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BELL SYSTEM TECHNICAL JOURNAL 



where Xi = Awi -^ p = pEi -^ p, and Acoi is Itt times the maximum 

 frequency shift. The local oscillation impresses the wave 



where 



B cos {wit + X2 sin pt + ^2) 



X2 = Aw2 -^ p- 



(3) 



Application of these waves to a square-law modulator will yield a 

 difference frequency wave proportional to 



AB cos [(coi — C02)/ + (xi — X2) sin pt + 4)1 — 02] (4) 



for the case where coi > 0)2, or when the reverse is true 



AB'cosJ_{w2 — ooi)i — (xi — Xi) sln'pt + 02 — <^i]. (5) 



In either case the resultant modulation index of the intermediate wave 

 is the numerical difference of the original indexes, the difference in sign 



MODULATOR 



INTERMEDIATE FREQUENCY 

 AMPLIFIER 



FREQUENCY 

 DETECTOR 



FREQUENCY 

 MODULATED 

 OSCILLATOR 



VvWVW" 

 -www- 



Fig. 1 — Basic feedback circuit. 



in the two cases signifying that the detected outputs will be of opposite 

 phase. If Xi = X2 the modulation is reduced to zero, and if X2 > xi 

 modulation reappears with a phase reversal. It might be noted that 

 if X2 were originally made negative, thus causing the two oscillators to 

 be frequency-modulated in opposite phase, the apparent modulation 

 of the incoming wave could be increased indefinitely. 



Suppose, now, that instead of frequency-modulating the local oscil- 

 lator from an independent source, the equivalent is accomplished in a 

 practical way. For this purpose a voltage from the output of the 

 receiver is impressed upon the local oscillator as shown in Fig. 1. The 

 transmitted wave will then have a modulation index Xi = piEi -^ p, 

 while the local oscillator, being acted upon by a portion of the output 

 voltage £0, will have an index ^2 = kpiEo -^ p. If the frequency de- 

 tector 2 is assumed to be linear the amplitude of the detected output 



^ The term frequency detector is used in this paper to designate the combination of 

 conversion circuit and amplitude detector. A more extended discussion of modula- 

 tion and detection is given in Appendix A. 



