110 BELL SYSTEM TECHNICAL JOURNAL 



Sections 7 and 8. The correlation function of B' is expressed in terms of 

 exponential integrals in Section 7, the power spectrum of In being assumed 

 symmetrical and centered onfq. In Section 8, values of W{f) are obtained 

 for the special case in which the power spectrum of In is centered on/^ and is 

 of the normal-law type. 



It is beheved that some of the material presented here may find a use in 

 the study of the effect of noise in frequency modulation systems. For 

 example, the curves in Section 8 yield information regarding the noise power 

 spectrum in the output of a primitive type of system. Also, the procedure 

 employed to obtain the expression (5.7) for 6' may be used to show that if 



Qcos[(^/coo) cos oiQt -\- qt] + /at = i?cos {qt + d) 



the sinusoidal component of dO/dt is^ 



— ^ (1 — e~^) sinojo/ 



where p is the ratio QV(2 1%)- This illustrates the ''crowding effect" of the 

 noise. The statistical analysis associated with R and 6 of equations (3.4) 

 (when the sine wave is absent) is similar to that used in the examination of 

 the current returned to the sending end of a transmission line by reflections 

 from many small irregularities distributed along the line. This suggests 

 another application of the results. 



Acknowledgment 



I am indebted to a number of my associates for helpful discussions on the 

 questions studied here. In particular, I wish to thank Mr. H. E. Curtis for 

 his suggestions regarding this subject. As in Reference A, all of the compu- 

 tations for the curves and tables have been performed by Miss M. Darville. 

 This work has been quite heavy and I gratefully acknowledge her contribu- 

 tion to this paper. 



1. Probability Distribution of a Sine Wave Plus Random Noise 



A current consisting of a sine wave plus random noise may be represented 

 as 



/ = (2cos qt + In (1.1) 



where Q and q are constants, t is the time, and In is a random noise current. 

 The frequency, in cycles per second, of the sine wave is/g = q/{2'n). In all 



*The first person to obtain this result was, I believe, W. R. Young who gave it in an 

 unpublished memorandum written early in 1945. He took the output of a frequency 

 modulation limiter and discriminator to be proportional to either the signal frequency or 

 to the insUntaneous frequency (assumed to be distributed uniformly over the input band) 

 of Is according to whether Q is greater or less than the envelope of In • His memorandum 

 also contains results which agree well with several obtained in this paper. 



