514 BELL SYSTEM TECHNICAL JOURNAL 



(1) To secure any advantage by frequency modulation as distin- 

 guished from amplitude modulation, the frequency band width must 

 be much greater in the former than in the latter system. 



(2) Frequency modulation in combination with severe amplitude 

 limitation for the received wave results in substantial reduction of the 

 noise-to-signal power ratio. Formulas are developed which make 

 possible a quantitative estimate of the noise-to-signal power ratio in 

 frequency modulation, with and without amplitude limitation, as 

 compared with amplitude modulation. 



It is a pleasure to express our thanks to several colleagues who have 

 been helpful in various ways: to Dr. Ralph Bown who in a brief but 

 very incisive memorandum, which was not intended to be a mathe- 

 matical study, disclosed all the essential ideas of the quasi-stationary 

 method of attack; to Mr. J. G. Chaffee,* who has been conducting 

 experimental work on frequency modulation in these Laboratories for 

 some years past, by means of which quantitative checks on the 

 accuracy of some of the principal results have been possible; and to 

 various associates, especially Mr. W. R. Bennett and Mrs. S. P. 

 Mead, for detailed criticism of certain portions of the work. 



I 



In the well-known steady-state theory of alternating currents, the 

 e.m.f. and the currents in all the branches of a network in which 

 the e.m.f. is impressed involve the time / only through the common 

 factor e'"' where i = V— 1 and co is the constant frequency. To this 

 fact is attributable the remarkable simplicity of alternating current 

 theory and calculation, and also the fact that the network is completely 

 specified by its complex admittance Yiioi). Thus, if the e.m.f. is 

 £g'"', the steady-state current is 



7,3 = EY{ioi)e''^K (1) 



In the present paper we shall deal with the case where the frequency 

 is variable, and write the impressed e.m.f. as 



E^xpli f'n{t)dt). (2) 



: r n{t)dt j . 



^(t) will be termed the instantaneous frequency. This agrees with 

 the usual definition of frequency when il is a constant; it is the rate of 

 change of the phase angle at time /; and in addition the interval T 

 between adjacent zeros of sin Xil{t)dt or cos J'^{t)dt is approximately 

 7r/i2(/) in cases of practical importance. 



* .See Bibliography, No. 11. 



