408 BELL SYSTEM TECHNICAL JOURNAL 



mined by the receiver characteristics. It will be shown that feedback 

 also reduces the noise level at the output of the receiver, provided that 

 the disturbance is not too great. Thus when the modulation level is 

 raised to offset the effect of feedback an improvement in signal-to-noise 

 ratio is realized. 



The mechanism by which noise is reduced can be described qualita- 

 tively as follows : Noise at the output terminals of the receiver is caused 

 to frequency-modulate the intermediate wave in such fashion as to 

 produce, upon detection, a component which tends to cancel that 

 which would exist in the absence of feedback. An analysis of this 

 process for the case where the carrier is large compared with the dis- 

 turbance responsible for the noise is developed ^ in Appendix B, It is 

 assumed that the disturbance can be represented by a continuous 

 spectrum of sinusoidal voltages of equal amplitude but phased at 

 random. Impressed along with the disturbance is the signaling wave 

 (2). Then if N"^ is the mean disturbing power per unit of band width 

 in the vicinity of the carrier frequency, and ri is the resistance of the 

 input circuit, it is shown that the output noise power is "* 



27VVi 



F2 



2 , ai^Acj^ ^ ai^qa- 



^0 + 27?? + 



qa (10) 



where ao and ai are, respectively, the gain and slope factor of the inter- 

 mediate amplifier and conversion system as defined by (47), and qa 

 represents the upper limit of frequency response of the output circuit, 

 or the upper limit of audibility as the case may be. F is the feedback 

 factor (1 — fJL0). The corresponding signal power is 



A^afAc^ (11) 



The reduction in signal level occasioned by feedback can be offset by 

 increasing the frequency shift of the transmitted wave. If it is in- 

 creased so as to have the value Ai2 = FAco then the shift of the inter- 



3 An analysis of the effect of feedback upon noise in this system was first developed 

 by J. R. Carson by methods similar to those used in "Variable Frequency Electric 

 Circuit Theory with Application to the Theory of Frequency Modulation," Carson 

 and Fry, Bell Sys. Tech. Jour., vol. 16, pp. 513-540, October 1937. This has been 

 embodied in a paper by Mr. Carson entitled, "Frequency Modulation: Theory of 

 the Feedback Receiving Circuit," published in this issue of the Bell Sys. Tech. Jour. 

 Carson's treatment is more general in that an arbitrary signal wave is postulated 

 whereas the analysis given in Appendix B is restricted to a sinusoidal signal wave. 

 The methods used here are more elementary and may therefore appeal to a somewhat 

 wider audience. 



^ The expressions for signal and noise power used in this section are relative. 

 Factors determining their absolute magnitude are given in the Appendix. In all 

 cases the symbol Aco^ is to be taken as signifying (Aco)^. 



