Distortion Produced in a Noise Modulated 



FM Signal by Nonlinear Attenuation 



and Phase Shift 



By S. O. Rice 



(Manuscript received December 6, 1956) 



An expression is given for the FM distortion introduced by a transducer 

 whose attenuation and phase shift depend upon the frequency in an arbitrary 

 way. This expi'ession appears to be difficult to evaluate, but it yields useful 

 approximations for the second and third order modulation terms. In all of 

 the work, it is assumed that the distortion is S7nall compared to the signal, 

 and that the signal can be represented by a random noise having the same 

 power spectrum. 



INTRODUCTION 



A number of workers have been concerned with the problem of com- 

 puting the distortion introduced by a transducer when an FM wave 

 passes through it. Some of the earUest results were published by Carson 

 and Fry and by van der Pol. Several contributions to the subject have 

 been made recently in connection with studies of microwave radio 

 systems. 



An excellent paper on this subject has been published recently by 

 R. G. Medhurst and G. F. Small. Although their results differ consider- 

 ably in form from those given here, they are nevertheless closely related 

 to ours — their "sinusoidal variations of transmission characteristics" 

 being special cases of our "nonlinear attenuation and phase shift." 



Here we treat the problem by applying a method used in a recent 

 paper to study the distortion produced by an echo. Two assumptions 

 are made, (1) that the distortion is small compared to the signal, and (2) 

 that the signal can be represented by a random noise which has the 

 same power spectrum as the signal. In Section I, we review some known 

 results and put them in a form suited to our needs. Sections II and III 

 are devoted to the derivation of our main formulas. The principal result 

 is given by the triple integral (3.2) for the power spectrum of the dis- 



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