MATHEMATICAL AXALYSIS OF RANDOM NOISE 65 



The .vi and .V2 i^lay the role of parameters in (3.4-12). This result may be 

 established in much the same way as (3.3-5). 



When we identify F with one of our representations, (2.8-1) or (2.8-6), 

 of the noise current /(/) it is seen that p is normal in four dimensions. We 

 may obtain the second moments directly from this representation, as has 

 been done in the equations just below (3.3-7). The same results may be 

 obtained from the definition of \I/(t), and for the sake of variety we choose 

 this second method. We set 0:1 = /i , Xo = /i + r. Then 



i = ^2 = fU) =h 



lia = mnt + r) =xPr (3.4-13) 



--=^30L=^e^r^'(^+^)^'(^)^^ 





where primes denote differentiation with respect to the arguments. Inte- 

 grating by parts: 



r nt + r) dl{t) = [fit + r)I{t)]^ - [ r(t + t)/(/) 

 Jo Jo 



dt 



We assume that / and its derivative remains finite so that the integrated 

 portion vanishes, when divided by T, in the limit. Since 



^2 



we have 



Setting T = gives 



I"{t + T) = ^J{t + T) 



Vim = ~Q-o^{'^) = —^r 



2 2 ," 



r?! = TJ2 = — lAo 



in agreement with the value of H22 obtained from (3.3-7). In the same 

 way 



i^, = Limit I [ I\i + t)/(0 dt = ^ iir) 



^,r,, = Limit ^ [ I'{t)I{t + T) 

 r— w i Jo 



dt 

 dt 



= -^r 



