MATHEMATICAL ANALYSIS OF RANDOM NOISE 63 



Our expression for B and C become approximately 



B = F cos I3t - lAoAo^T sin j3t 



C = F cos' /3r - ^o.lo' sin' (3t - AoAo^V 



When T is near n , /3r is approximately x. Hence both C -{- B and C — B 

 are approximately — .4o^'lo tt" and are of the same order of magnitude. Con- 

 sequently M-2'2 and Mn are both nearly equal and 



M23 = UC + B] 



= —AoAoTT 



When this expression for M23 is used our approximation to (3.4-1) gives 

 us the result: If the correlation function is of the form 



\I/t = A cos /?r 



where A is a slowly varying function of t, the probability that the distance 

 between two successive zeros lies between t and t -\- dr is approximately 



dr a 



2" [1 + c2(t - Ti)2J3/2 



where a is positive and 



a = ," 2 , Ti — - 



For our ideal band pass filter with the pass band/^ — fa , 



fb —fa fb+ fa 



and the average value of | t — n | is a . Thus 



ave. I T — n I _ 1 _ fb — fa _ 1 band width 

 Ti UTi V3 (/6 + fa) 2-V/3 mid-frequency 



Wlien the correlation function cannot be put in the form assumed above 

 but still behaves like a sinusoidal wave with slowly varying amplitude we 

 may use our first approximation to (3.4-1). Thus, the probability that the 

 distance between two successive zeros lies between t and t -{- dr is approxi- 

 mately 



bdr 



when r lies near ti where n is the smallest value of r which makes \pT 

 approximately equal to —\po. This probability is supposed to approach 



