54 



BELL SYSTEM TECHNICAL JOURNAL 



The first step is to find the probabihty density function of the two random 

 variables 



k = Lj Cn cos {oOnk — <Pn) 



71=1 



N 

 7] = I'ih) = — X/ ^«<^n sin (cOnh — (fn) 



(3.3-7) 



where the prime denotes differentiation with respect /. From section 2.10 

 Mil = ^^ = "Ao 



M22 = 1?^ = Z-i Cni^n Sln (cO„ /l — <Pn) 

 n=l 



= E (2TfnYw{fMf 

 ri=l 



-> 4/ [ fwif) df = -^'o' 

 Jo 



Ml2 



= ^V ~ ~Z^(^~n W„ cos (oJn h — <pn) sin (cOn ^1 — ^n) 



= 



The expression for /X22 arises from (2.1-6) by differentiation. In this expres- 

 sion i/'o denotes the second derivative of i/'(r) with respect to t at r = 0: 



(3.3-8) 



,p"{r) = -4/ [ fw(f) cos iTrfrdf 

 Jo 



' is 



TT L 2l/'o 2l/'oJ 



Hence the probabihty density is 



/-(^^ r,t) = 



where i/'o is negative. It will be observed that the expression on the right 

 is independent of t. Hence the probability of having a zero in ^i , /i + dt, 



27r 



which follows from {3.3-3), is independent of/. 



The expected number of zeros per second, which may be obtained from 

 (3.3-3) by integrating (3.3-10) over an interval of one second, is 



- -co -jl/2 



/ /«;(/) ^/ 



1 \_rm" = 2 



TT L ^(0) J 



[ iv{f)df 

 L Jo -J 



(3.3-11) 



