Statistical Properties of a Sine Wave Plus Random Noise 



By S. O. RICE 

 Introduction 



TN some technical problems we are concerned with a current which 

 -*- consists of a sinusoidal component plus a random noise component. A 

 number of statistical properties of such a current are given here. The present 

 paper may be regarded as an extension of Section 3.10 of an earlier paper/ 

 "Mathematical Analysis of Random Noise", where some of the simpler 

 properties of a sine wave plus random noise are discussed. 

 The current in which we are interested may be written as 



/ = Qcos at + Is 



(3.4) 

 = i?cos {qt + B) 



where Q and q are constants, / is time, and In is a random (in the sense of 

 Section 2.8 of Reference A) noise current. When the second expression in- 

 volving the envelope R and the phase angle 6 is used, the power spectrum of 

 In is assumed to be confined to a relatively narrow band in the neighborhood 

 of the sine wave frequency /^ = q/i2x). This makes R and 6 relatively 

 slowly (usually) varying functions of time. 



In Section 1, the probabihty density and cumulative distribution of / are 

 discussed. In Section 2, the upward "crossings" of / (i.e., the expected 

 number of times, per second, / increases through a given value /i), are 

 examined. 



The probability density and the cumulative distribution of R are given in 

 Section 3.10 of Reference A. The crossings of R are examined in Section 4 

 of the present paper. 



The statistical properties of d', the time derivative of the phase angle 0j 

 are of interest because the instantaneous frequency of / may be defined to 

 be/g + 6' /(It). The probabihty density of d' is investigated in Section 5 

 and the crossings of 6' in Section 6. 6' is a function of time which behaves 

 somewhat like a noise current and may accordingly be considered to consist 

 of an infinite number of sinusoidal components. The problem of determin- 

 ing the "power spectrum" W(J) of 6', i.e., the distribution of the mean 

 square value of the components as a function of frequency, is attacked in 



^B.S.T.J. 23 (1944), 282-332 and 24 (1945), 46-156. This paper will be called 

 "Reference A". 



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