(8.18) Q(x'y') = lim \ \ '^^ (x, y),?2.(x + x', y + y')dxdy 



Xhoo 



2 2 

 and 



OO CO 



(8.19) [A*(c*,p*)]^ + [A*(-c*,-p*)]2=-^ \ \ Q(x',y')cos(cx' + (3y')dx'dy' 



OO -OO 



The analogous summation formula for the covariance surface over the 

 set of leveled readings N-^ is given by 



m-l-lql n-l-p]vrr tst* , , 

 (8.20) Q(p,q)= S S J^ J+P' ^+q 



k=0 j = (n-p)(m-|q|) 



p = 0. 1. . . . ., 20 



q = -20, -19, -18, .„., -1, 0, 1, „.., 20 

 This determines the estimates of the covariance surface for the first and 

 fourth quadrants of the q, p plane (really x', y'). Since Q(p, q) = Q(-p3 -q)s the 

 results can be extended into all four quadrants of the q, p plane. 



The function must no-w be extended into the entire q, p plane so that its 

 Fourier coefficients can be determined, and the property that Q(p,q) = Q(-p,-q) 

 must be preserved. This is accomplished by simply translating the covari- 

 ance surface parallel to itself to fill the whole plane. 



As a consequence, the Q's have to be redefined slightly in order to 

 weight them properly. The definitions are that 



67 



