524 BELL SYSTEM TECHNICAL JOURNAL 



which agrees with the result arrived at from the idea of instantaneous 

 frequency. On the other hand, suppose we make X so small that 

 X/co <C 1. Then (40) becomes to a first order 



. , , 1 /X\ ., , ,, 1 /X\ ., 



so that the frequencies coc, coc + co, coc — co are present in the pure 

 frequency modulated wave. 



It is possible to generalize the foregoing and build up a formal 

 steady-state theory by supposing that 



M 

 S{t) = H Am COS {Oimt + dm). (41) 



On this assumption, it can be shown that the frequency modulated 

 wave (38) is expressible as 



00 



W = exp {iwct) n Y. Jn(vm) cxp [in{(x,mt + Qm)'], (42) 



m re= — 00 

 Vm = \Amlo:m. 



The corresponding current is then 



00 



exp {iwct) n Y. Jn{vr^Y{iwc + wcom) exp [in{wrnt + 6m)']- (43) 



Formulas (42) and (43) are purely formal and far too complicated 

 for profitable interpretation. Consequently this line of analysis will 

 not be carried farther.* 



If we compare the pure frequency modulated wave, as given by (38), 

 with the pure amplitude modulated wave, as given by (39), it will be 

 observed that, in the latter, the low frequency signal s{t)y which is 

 ultimately wanted in the receiver, is explicit and methods for its 

 detection and recovery are direct and simple. In the pure frequency 

 modulated wave, on the other hand, the low frequency signal is 

 implicit; indeed it may be thought of as concealed in minute phase or 

 frequency variations in the high frequency carrier wave. 



If we differentiate (38) with respect to time /, we get 



dWldt = [coc + X5(/)] exp ( ioict + *X j sdt\ - (44) 



* See Appendix 1. 



