PROPERTIES OF SINE WAVE PLUS NOISE 117 



Using 



jf" z-'" dz [ <p{x) dz = j[" ^(x) dx f 2-'" dz = 2"\-"' r(|) 



we obtain 



r Pi(y) dy ~ 2-"V-'"r(f)a-"- = 0.185- • -a-'" (1.19) 



For use in computations we list the following values 



r(|) = 3.62561, r(f) = 1.22542, r(f) = 0.90640 

 2. Expected Number of Crossings of I per Second 



In this section, we shall study two questions. First, what is the proba- 

 bility P(/i, t^dt of / increasing through the value 7i (i.e. of / passing 

 through the value /i with positive slope) during the infinitesimal interval 

 /i, ^1 + dO Second, what is the expected number N{Ii) of times per second 

 I increases through the value /i. When /i is zero, 2N(0) is the expected 

 number of zeros per second, and when /i is large /V(/i) is approximately 

 equal to the expected number of maxima lying above the value /i in an 

 interval one second long. 



We start on the first question by considering the random function 



2 = F(fli, ^2, • • • cln; i) 



where the a's are random variables. The probabiUty that the random curve 

 obtained by plotting 2 as a function of / increases through the value z = zi 

 in the interval /i, /i + dt is 



■ dt f 7jp(zuv;h)dr, (2.1) 



Jo 



where p(^, t/; ti) denotes the probability density of the random variables 

 ^ = F{ai, 02, • • • , djf-, h) 



[-1 



In our case z becomes the current I defined by equation (1.1). The 

 method used to obtain equation (3.3-9) of Reference A may also be used to 

 show that the quantity p{Ii, rj, ti) (which now appears in (2.1)) is given by 



/)(/i, 77, h) = — 4f (fiy - a cos qti)(p{x + b sin qh) (2.2) 



1° This result is a straightforward generalization of expression (3.3-5) in Section 3.3 of 

 Reference A where references to related results by M. Kac are given. A formula equiva- 

 lent to (2.1) has also been given by Mr. H. Bondi in an unpublished paper written in 1944. 

 He applies his formula to the problem studied in Section 4. 



