Question

4. (a) Show the following result when Ui, U2, Us and U^ are jointly Gaussian random variables (zero mean) [15 points] where U,U E[U,U (b) Using the above result, show that if U1, U2 are complex circular Gaussian random variables [15 points]: 1201U2 (c) Using the above result, show the following result for Intensity Correlation Interfer- ometery [15 points] where J12 is the mutual intensity between the radiation at points 1 and 2, say U1 and Δ1,-11-11, and Δ12-12-12

solve part b and c

Answer 1

"Given that U_1, U_2, U-3, U-4 are Jointly Gaussian random variables. Let ""U"" be a random vector i.e, U = [U_1 U_2 U_3 U_4] Now, denote phi_U (x) and the characteristic function of ""U"" where x = [x_1, x_2, x_3, x_4]^T Therefore, phi_U (x) = E [exp degrees 0 ((j_y T_U)] = E[Exp degrees 0(j (x_1 U_1 + x_2 U_2 + x_3 U_3 + x_4U_4))] sigma^infinity_k = 0 j^k/rightangle k E [(x_1 U_1 + x_2 U_2 + x_3 U_3 + x_4 U_4)] Now it can be written as partial differential U/partial differential x_1 partial differential x_2 partial differential x_3 partial differential x_4 phi_U (x) Vertical-bar _x_1 x_2 x_3 x_4 = 0 = E(U_1 U_2 U_3 U_4) equation (1) \r\n since ""U"" has jointly Gassion distribution with zero mean, phi_U (x) = exp^0 (- 1/2 x T Kx) equation (2) where K is the Cavarlance Matrix Such that K_ij (ijth element of k) satisfies. K_ij = C_0v (U_i, U_j) equation "