TRANSIENT RESPONSE OF AN FM RECEIVER 



729 



the output voltage of the frequency detector, when a phase variation, 

 B'if) = xq cos g/, is impressed on the carrier at the input, then is 



[00 -|l/2 r « -|l/2 



^0 + 2 22 a„2 cos nqt\ — flo + 2 2Z ^m cos nqt (49) 



where a„i is given by (47) and On2 is given by the same formula with the 

 sign of A reversed. 



An approximation that is permissible when the frequency swing xq does 

 not approach the available frequency range A is 



^o(0 = 2 ^n cos nqt 



where 



An = 



an2 ~~ O'nl 



V 



(h 



n odd; = 0, w even. 



(SO) 



(51) 



The table following gives the results of a computation of the first four 

 coefl&cients from formulas (45) and (51) for the case of a frequency deviation 

 ratio, a; = 5, for q/lir = 3000 cycles per second, for A/27r = 30,000 cycles 

 per second and for a/A = J, 1 and 2. 



Note: To obtain volts, multiply all values by 



Rha 



(a2 + A2)i/2 



The coefficients for even values of n vanish, which confirms what can be 

 inferred from physical considerations, namely, that the balanced construc- 

 tion of the frequency detector eliminates the d.c. component and all even 

 harmonics. From the ratio of ^3 to ^i we obtain the following ratios, ex- 

 pressed in db's, of the third harmonic distortion to the fundamental signal 

 for the three circuit designs: 



a/A 201ogl0U3/^i| 



The results for the sinusoidal signal, when considered in conjunction with 

 those for the impulse modulation, also permit certain conclusions regarding 

 signal-to-noise ratios for impulsive interference in FM reception. 



