PROPERTIES OF SINE WA VE PLUS NOISE 123 



quency is approximately equal to twice the distance between two successive 

 zeros which is 2(/i — to). 



The probability density of R is^^ 



where /o (RQ/\po) denotes the Bessel function of order zero with imaginary 

 argument. In Section 3.10 of Reference A, it is shown that the average 

 value of 7?" is* 



(3.13) 



^ = (2^orr(|+l),/r,(-|; 1; -p), 



where p = (3V(2^o), of which special cases are 



R = e-'\^,llf' [(1 + p)/o(p/2) + p/i(p/2)] 

 — (3.14) 



Curves showing the distribution of R are also given there. 



4. Expected Number of Crossings of R per Second 



Here we shall obtain expressions for the expected number iV/j of times, 

 per second, the envelope passes through the value R with positive slope. 

 When R is large, N r is approximately equal to the expected number of 

 maxima of the envelope per second exceeding R and when R is small iV« is 

 approximately equal to the expected number of minima less than R. For 

 the special case in which the noise band is symmetrical and is centered on 

 the sine wave frequency /g N e is given by the relatively simple expression 

 (4.8). 



The probability that the envelope passes through the value R during the 

 interval t,t-\- dt with positive slope is, from (2.1), 



f R'p{R,R',t)dR' (4.1) 



where p{R, R\ t) denotes the probability density of R and its time derivative 

 R\ t being regarded as a parameter. 



An expression for p{R, R' , t) may be obtained from the probability density 

 of /c, /s, I'c, I's. From our representation of a noise current and the central 

 Hmit theorem it may be shown (as is done for similar cases in Part III of 

 Reference A) that the probabihty distribution of these four variables is 



1^ In equation (60- A) of an unpublished appendix to his paper appearing in the B.S.TJ. 

 Vol. 12 (1933), 35-75, Ray S. Hoyt gives an integral, obtained by integrating (3.12) with 

 respect to R, for the cumulative distribution of R. 



*The correlation function for the envelope of a signal plus noise, together with associated 

 probability densities of the envelope and phase, is given by D. Middleton in a paper 

 appearing soon in the Quart. Jl. of Appl. Math. 



dt 



