MATHEMATICAL ANALYSIS OF RAX DOM NOISE 89 



Thus 



III 2 = ^o(l + 2 COS'ip) 



(3.9-7) 



Incidentally, this gives an expression for the correlation function of I^(t). 

 Replacing t by its value of /2 — /i and returning to (3,9-4), 



/r/2 -7'/2 



dh / dtif'ik - h) (3.9-8) 



r/2 J—TI2 



When we introduce ar , the standard deviation of E, and use 



o — 2 



a'r = Er — ntr 



we obtain 



.r/2 -r/2 



^/l / dt2XP\t2 - h) 



Til J— Til 



= 4 f (r - :v)iA'(a;) Jx 



^0 



where the second line may be obtained from the first either by changing the 

 variables of integration, as in (3.9-27), or by the method used below in 

 dealing with E^. I am indebted to Prof. Kac for pointing out the advantage 

 obtained by reducing the double integral to a single integral. It should be 

 noted that the limits of integration — Tjl, T/2 in the double integral may 

 be replaced by 0, T by making the change of variable t = t' — T/2 for both 

 h and h. 

 When we use 



Kt) = f w{f) cos 27r/T df (2.1-6) 



Jo 



we obtain the result stated in the paper, namely, 



4 = r »(/0 df, f » W df, [" ^'"'"■^■+{^>^ (3.9-9) 



Jo Jo L 7^2 (/i 4-/2)2 



sin' Tjfi - f2)T l 



^Kh-hY J 



If this formula is applied to a relatively narrow band-pass filter and if 

 T{fb —fa) > > 1 the contribution of the/i +/2 term may be neglected and 

 we have the approximation 



(3.9-10) 



