PROPERTIES OF SINE WAVE PLUS NOISE 133 



as it does when the point T crosses, in the downward direction, the extension 

 of the Hne OQ lying to the left of the point in Fig. 4, we imagine the value 

 of 6 to change discontinuously to the value — w. 



The expected number of times per second 6 increases through r may be 

 obtained from (6.2) and, in the symmetrical case, from (6.3) by changing the 

 sign of Q since this produces the same effect as changing 6 from to tt in 



p{R, e, e'). 



The expected number of crossings per second when R lies between two 

 assigned values may be obtained by integrating the above equations. For 

 example, the number of times per second B increases through zero with R 

 between Q and Ri is, from (6.3) for the symmetrical case, 



{^^)-Kh/h,yi^ erf [{2h,)-^i^ \Ri-Q\] (6.4) 



where we have used the absolute value sign to indicate that Ri may be either 

 less than or greater than Q and 



erf X 



-1/2 



f e-'" dt (6.5) 



Expressions for ^o and 62 are given by equations (A2-1) of Appendix II. 

 The mean square value of In is ^0, and when the power spectrum of I^ is 

 constant over a band of width j(3, Z>2 = T'^^^bo/3. 



In much the same way it may be shown that the expected number of times 

 per second increases through tt with R between and i?i is 



{^T)-'{b2/boy" {erf l{2bo)-'iKRi + Q)] - erf l{2bo)-'^V]} (6.6) 



A check on these equations may be obtained by noting that the expected 

 number of zeros per second of 7^, given by equation (4.11), is equal to twice 

 the number of times 6 increases through zero plus twice the number of times 

 increases through tt. Setting Ri equal to zero in (6.4), to infinity in both 

 (6.4) and (6.6), and adding the three quantities obtained gives half of (4.11), 

 as it should. 



Now we shall consider the crossings of 0'. The equations in the first part 

 of the analysis are quite similar to those encountered in Section 3.8 of 

 Reference A where the maxima of R, for noise alone, are discussed. We 

 start by introducing the variables 0:1, 3:2, • • • x^ where 



x^ = I, = Rcos - Q, xa = Is = Rsin (6.7) 



and the remaining x^s are defined in terms of the derivatives of Ic and Is and 

 are given by the equations just below (3.8-4) of Reference A. 



Here we shall consider the noise band to be symmetrical about the sine 



