To find [C(X,Y,f) - iQ(X,Y,f)] we need only know R(X,Y,T) for continuous 

 T for the given value of X,Y) . Further, we have just stated that 



This is in the form of a Fourier transform; thus, we can write from 

 transform pairs 



(3.9) 



As has been stated, we cannot hope to have a continuous set of values 

 of (X,Y) . The solution is to find R(X,Y,T) for continuous T and 

 selected values of (X,Y) , and then employ the above to estimate S(J!,,m,f). 

 This is described in the next section. 



CROSS SPECTRAL MATRIX OF AN ARRAY 



Let us look at ri(x,y,t) at two fixed points in space, say 

 (xo> Yo) ^^"^ (^1» Yl) • This would give two stationary Gaussian pro- 

 cesses indexed on time alone because (xq, y^) and (x^, y^) are fixed. 

 Thus, we may write 



If X = (xi - Xq) and Y = (y^ - y^) Then we can say, since n(x,y,t) is 

 £issumed weakly stationary (see Equation (3.3)), that 



R(XXT) = eL7.(t) ^(*^T)] 



where the expected value is over the ensemble for some specific value 

 of t, where - » < t < «. Let us extend this idea by a change of notation 

 and let N(x,Y,t) be a two-dimensional (vector) process, double-indexed 

 on time; i.e., let N be a vector function 



N(XXt) = (7./*),7'W)='^W 



12 



