432 BELL SYSTEM TECHNICAL JOURNAL 



lems has been of great value; and to Messrs. E. A. Krauth and O. E. 

 DeLange for assistance in the experimental work. Reference has al- 

 ready been made to the theoretical work of Dr. J. R. Carson. 



APPENDIX A 



Analysis of Distortion Reduction 



Assume that the transmitter is frequency-modulated with a signal 

 wave which we shall represent by the symbol 5 = S{t). Then the 

 instantaneous frequency of the transmitter will be 



coi + p,S. (30) 



The instantaneous phase of the transmitted wave is the integral of this 

 expression and the voltage delivered to the input of receiver can be 

 written • 



A cos ( coi/ + pi r Sdt + <^i ) . (31) 



Designating the low-frequency voltage delivered at the output of the 

 receiver as a = (j{t) the result of feeding back a portion of ka of the 

 output so as to frequency-modulate the local oscillator is the wave 



B cos I Wit -j- p2 I k<^dt + </>2 I . (32) 





Application of these two waves to the modulator produces the inter- 

 mediate-frequency product '" 



aAB cos coo/ + p\ I Sdt — Pi I kadt + <^o i^^) 



where 



Wo = coi — a>2 



00 = 01 — 02. 



Terms in the above which involve the integral sign represent phase 

 angles which vary with time. Hence we shall rewrite {2)2>) more 

 compactly 



aAB cos [coo/ + ^(0 + 0o]. (34) 



It has been shown by Carson and Fry ^ that the process of detecting 

 a frequency-modulated wave is, in effect, its differentiation. Since the 

 high-frequency wave itself exhibits the integral of the signal wave, see 



"* This expression constitutes a more general form of (4). 



