where g(X,Y) is a set of Dirac delta functions standing on the separa- 

 tion points of the probe array, and zero elsewhere. 



Thus, we have the estimate 



Let 



G(Jl.m')=:j|§(x,Y)Exp(-ietT(!lX + v,Y))Axc)iY (,3^ 



Using this and Equation (3.8) we have for a given (ilQ,niQ,fQ) that 



"O » O » •^O'' 



A 



S 



-co -aa*>a,>«0 ^ 



eKp(-Xt1T(Jl.X-»->noY))Jix<lY 



S(]i>,£)^(x,Y)Exp(-iaiT(x(i--l)+Y(rno-v.)))<lxAJ^^^ 



= r(^(i>n.gRj^(K,Y)EXP(-iatT(x(i-Jl)+Y(in,->n)^^ 



=[rs(iM.-f.)6(«.-s,m.-»«)Jiu 



(6.4) 



As expected, S(£,m,f) is a two-dimensional convolution of the true 

 directional spectrum S()!-,m,f) with G(£,m) the Fourier transform of 

 g(X,Y) . We see then that S(£,m,f) is a weighted average of S(£,m,f) 



24 



