318 BELL SYSTEM TECHNICAL JOURNAL 



order of integration and using the second of (2.4-2) to replace F2 by G2 gives 

 the right hand side. 



We now set Gi(t) and G^iO equal to zero except for intervals of length T. 

 These intervals and the corresponding values of Gi and G2 are 



Gi(/) = /(/), < / < r (2.4-3) 



G2(t) = /(-/ + r), T - T <t < T 



From (2.4 3) it follows that Fi(f) is the spectrum S(f) of I{t) given by equa- 

 tion (2.1-2). Since /(/) is real it follows from the first of equations (2.4-2) 

 that 



S{-f) = S*if), (2.4-4) 



where the star denotes conjugate complex, and hence that | S(f) | ^ is an 

 even function of/. 



The first of equations (2.4-2) also gives 



F2{f) = r I{-t + T)e- '''''' 



J T—T 



= f Hi)e 

 Jq 



dt 



i2rf(t-T) 7, (2.4-5) 



JO 



= S*{f)e- 

 When these G's and F's are placed in (2.4-1) we obtain 



f " I Sif) I V'' dj = f ^ 1(1)1(1 + r) dt (2.4-6) 



where we have made use of the fact that G2(—t) is zero except in the inter- 

 val — T<t<T—T and have assumed r > 0. If r < the limits of 

 integration on the right would be — t and T. 



Since | S(f) \ is an even function of/ we may write (2.4-6) as 



1 jf ' IU)I(t -^r)dt + (^-pj = jf 2M^' COS 27r/r dJ (2.4-7) 



If we now define the correlation function \I/(t) as the hmit, as T — > oc , of the 

 left hand side and define w(f) as the function 



w(f) = Limit ^^ , / > (2.1-3) 



we obtain the second, (2.1-6), of the fundamental inversion formulas. As 

 before, the first may be obtained from Fourier's integral theorem. 



