132 BELL SYSTEM TECHNICAL JOURNAL 



When p is zero equation (5.16) agrees with a result given in Section 3.4 of 

 reference A, namely, for an ideal band pass filter 



ave I r — Ti I _ fb — fa 



Vl VS{fb+fa) 



where r is the interval between two successive zeros and n is its average 

 value, r is equal to /i — to of our equation (3.10) from which it follows that 



(r - ri)/ri - - d'/q (5.17) 



do 

 6. Expected Number of Crossings of 6 and — per Second 



at 



After a brief study of the expected number of times per second the phase 

 angle 6 increases through and through t (where it is assumed that — tt < 

 d < it) expressions are obtained for the expected number Ne' of times per 

 second the time derivative of 6 increases through the value d\ 



The point T shown in Fig. 4 of Section 3 wanders around, as time goes by, 

 in the plane of the figure. How many times may we expect it to cross some 

 preassigned section of the Hne OQ in one second? To answer this problem 

 we note that, from expression (2. IX the probability that 6 increases through 

 zero during the interval t, t -\- dt with the envelope lying between R and 

 R-\- dR is 



dtdRf e'p{R,0,d')de' (6.1) 



where the probability density in the integrand is obtained by setting 6 equal 

 to zero in equation (5.1). The expected number of such crossings per second 

 is • 



(liry'" {boBy"'R'dRe-'^''-''''''''' 



.« (6.2) 



/ dd'e'exp [-boR^e^'/ilB) + hR{R - Q)e'/B\ 



which may be evaluated in terms of error functions or the function (p-i{x) 

 defined by equation (1.8). For the special case in which the power spec- 

 trum of the noise current In is symmetrical about the sine wave frequency, 

 bi is zero and (6.2) yields 



{lirr'^-^'bTe-'''-''''"'''' dR (6.3) 



From equation (6.1) onwards we have tacitly assumed that the range of 6 

 is given by — tt < ^ < tt because setting B equal to any multiple of 27r in our 

 equations leads to the same result as setting B equal to zero. This is due to 

 B occurring only in cos B and sin B. When B increases through the value tt. 



