MATHEMATICAL AX A TVS IS OF RA.XDOM NOISE 153 



Then 



f ^i{r)<p.{r)e-'''"^Ur = f Fi(.v)Fo(/ - x) dx (4C-3) 



•A— 00 V— 00 



i.e., the spectrum of the product <(Ji(t)<^2(7-) is the integral on the right. 

 If <^i(t) and ifiir) are real even functions of r, (4C-3) may be written as 



\ sri(r).^2(r) cos ItJt dr = \\ F,{x)F,{f - x) dx (4C-4) 



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In order to obtain Gi{j^ we set ip\{j) and ipiij) equal to ■^{j). We may 

 then use (4C-4) since i/'(t) is an even real function of r. When ^prij) is an 

 even real function of r we see, from the Fourier integral for i^r(/), that Frij) 

 must be an even real function of/. We therefore set 



IFrij) = w{f), r= 1,2 



and define w(f) for negative/ by 



w{-f) = w{f) (4C-5) 



Equation (4C-4) then gives 



1 /•+" 

 <^2(/) = e / w{x)w(f - x) dx 



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= - / w{x)w{f — x) dx (4C-6) 



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1 r 



+ - / w(x)w{f + .-v) <f:i; 

 4 Jo 



where in the second equation only positive values of the argument of w(/) 

 appear. 



In order to get Gz(f) we set ^i(r) equal to \P(t), 2Fi{f) equal to w(f), and 

 <Pi{t) equal to xj/ (r). Then 



■^2(/) = 2 / <P2(t) cos 27r/r </r 

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= 2G2(/) 



and from (4C-4) we obtain 



Gzif) =1 ( ^ w{x)G,{J - x) dx 



L J— 00 



(4C-7) 



. -+» -+00 "■ ^ 



= — / w(x) dx I w{y)w{f — y) dy 



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