and 



/ 



Q(X,Y,f) = Qoi(f) = / Roi(T) Sin(2^fT)dT. (4.9) 



We also have (see Equations (4.2) and (4.7)) CoQ(f) = Cuif) = ^qqCO = 

 Pll(f). 



Qoo(f) = Qii(f) = 0; 



P8i(f) = coi(f) - 1 Qoi(f)- (^-10) 



The phase of Pq, (f ) is given by 



Qoi(f) = Arctan -^^^J^ • (^-H) 



This is the expected phase lead of the signal at (x , y ) over the 

 signal at (xj^, y^) for f where - » < f < «> . 



For an array of N detectors located at (x]^, y^) , (x2, y2) » (xo, yo) 

 ...(xj^, y^j) we can find N spectra [Pj^-s] i = l,N,j=l,N. This gives 

 a unique spectrum P^^ (Pn = ... = P^jvj; and since P^^ = 'P^, (see 

 Equation (4.6) and 4.7)) J Ji 



we have 



NOhl) (4,12) 



2 



unique cross spectra or a total of [N(N-l) + 2]/2 unique spectra. Thus, 

 for real n(x,y,t) we have Pi^(f) = P=^j,(f) and P^.(f) = P* (-f) allowing 

 us to define the information about PXA,Y,f) obtainable from an array by 

 a cross spectral matrix 





• 



(4.13) 



16 



