Maestretlo and Linden 



Consider a homogeneous , stationary, random process X-=».^ 

 (Rosenblatt writes this as X^^(co) to explicitly indicate that it is a 

 function defined on a sample space) with a cross -correlation defined 



R 



(T,t;T',t') = (X-^^Xtt^^. ) (24) 



where ( ) is the expectation operator, i.e. , R is defined through 

 the ensemble average. Because the process is homogeneous and 

 stationary, 



R(T,t;T',t') = R(T-T', t-t') 



R(T - T' ,t - t') dT dt = ( dM (x-*,) dM (X-«r ., )> 



where dM (x'^i ) is the Stieljes measure of the process. The pro- 

 cedure may be' simply stated as the problem of finding a Fredholm 

 expansion of R and subsequently representing X^^ by such as 

 expansion. Such an expansion is provided by the eigenfunctions and 

 eigenvalues of the integral equation 



4j(r,t) = \\ R{r- r',t-t')L|;(r',t') dr' dt (25) 



The spectrum, is of course, continuous. The eigenfunctions are 

 plane waves and the eigenvalues the inverse of the power spectral 

 density as can be seen by applying the Fourier transform. Thus the 

 desired expansion for R is 



Now let 



2^ = ' ^ r e'^'"^-'-*^dM (X-^, ) (26) 



^" (2Tr)2Vfe(K,co) ^-^ ''' 



It follows from (24) and (26) that 



(Z-^^^Z-^,^^) = 6(K -K')6(co-a)') 



496 



