MATHEMATICAL AX A LYSIS OF RANDOM NOISE 61 



band pass filter whose range extends from/a toft . The correlation function 

 is given by (3.2-5). 



lAr = — sin TT(fb — fa) COS TTrifb + fa) 



'^^ (3.2-5) 



xl^Q = Woifb - fa) 



From physical considerations we know that in a narrow filter most of the 

 distances between zeros will be nearly equal to 



1 



Ti = 



fb+fa 



i.e., nearly equal to the distance between the zeros of a sine wave having 

 the mid-band frequency. We therefore expect (3.4-1) to have a peak very 

 close to n . We also expect peaks at 3ti , 5ti etc. but we shall not consider 

 these. We wish to examine the behavior of (3.4-1) near ri . 



It turns out that M23 is nearly equal to M22 so that H is large and (3.4-1) 

 becomes approximately 



dr r i/'o 1"' M23 



[^'] 



2i3/2 



2 L~'/'0 J hpO — 'At] 



where r is near n . 



In order to see that M23 is nearly equal to M22 we use the expressions 



M22 = — lAo ('Ao — "At) — Mr 



MiZ = ^'!{^l — lAr) + h^? 



M22 + M23 = (h - ^r)[{h + 4^t){4^'t - 'Ao') - 'Ar'] 



= (^0 - ^Pr)[B + C] 



M22 - M23 = (i/'o + 4^r)[{4^0 - ^t){.- lAr' - ^0') - ^7] 



= {h + rPr)[- B + C] 



B = Mr — 'At'/'O 



C = — l/'O'Ao + 'At'/'t — ^T 



From (3.2-5) it is seen that 4/r may be written as 



lA, = ^ cos /3t, /3 = x(/6+/a) 



where /3ri = tt and ,4 is a function of r which varies slowly in comparison 

 with cos ^r. We see that near n , \pr is nearly equal to — i/'o . Likewise 



