VARIABLE FREQUENCY ELECTRIC CIRCUIT THEORY 523 

 the instantaneous frequency varies between the limits 



OJc ± X. 



In all cases it will be postulated that X <<C coc- 



With the method of producing the frequency modulated wave (38) 

 we are not here concerned beyond stating that it may be gotten by 

 varying the capacity or inductance of a high frequency oscillating 

 circuit by and in accordance with the signal s(t). 



Corresponding to (38), the pure amplitude modulated wave (carrier 



suppressed) is of the form 



5(/)-e'"c'. (39) 



If the maximuni essential frequency in the signal s{t) is Wa, the wave 

 (39) occupies the frequency band lying between Wc — "a and Wc + (^a, 

 so that the band width is 2coa. In the pure frequency modulated wave 

 the "instantaneous" frequency band width is 2X. In practical 

 applications X ;;:>> coa. We shall now examine in more detail the concept 

 of "instantaneous" frequency and the conditions under which it has 

 physical significance. 



The instantaneous frequency is, as stated, coc + \s(t) ; a steady-state 

 analysis is of interest and importance. To this end we suppose 

 s(t) = cos o)t so that oj is the frequency of the signal. Then the wave 

 (38) may be written 



^iwct ) ^QQ I - sin wt ) -\- i sin I - sin cot y > , 



and, from known expansions, 



00 



W ^ Y. /„(X/co)g'("'+"")', (40) 



n=— 00 



where /„ is the Bessel function of the first kind. Thus the frequency 

 modulated wave is made up of sinusoidal components of frequencies 



Wc zt WW, W = 0, 1, 2, • • •, CO. 



If X/oj >>> 1 (the case in which we shall be interested in practice) the 

 terms in the series (40) beyond n = X/co are negligible; this follows 

 from known properties of the Bessel functions. In this case the 

 frequencies lie in the range 



COc ± nco = OJc ± X, 



