Question

Ⅳ、X()=X(t)+X(,) is a complex random process, X(t) is the Hilbert transform of X(1), find that ETX(1)X(1-r)] and ElX(1)X.(t-t)]. (15 points)

Answer 1

"Given X(t) = x(t) + x(t) is a complex random process x(t) is the Hilbert transform of x(t) E[x(t) x(t - tau)] = sigma_j = 1^alpha sigma_k = 1^alpha R_jk(t- (t - tau))exp(ir_jt - ir_k(t - tau)) x_~(t) = sigma_j = 1^k(n) eta_j, k(N)(t) exp(ir_j t) converges uniformly in t to x(t) is norm bounded, then for every (t-(t-tau)), rho_n(t(t-tau)) = sigma_j = 1 ^K(N) sigma_k = 1 ^K(N) R_jk,k(N)(t- (t - tau))exp(ir_jt - ir_k(t - tau)) converges uniformly to R(t,(t - tau)) as N rightarrow alpha where R_jk,k(N)(t- (t - tau)) = E{eta_j,k(N) (t) cdot eta_k,k(N)(t-tau)} E[x(t) x(t-tau)] = R(t(t-tau)) = sigma_j sigma_k f_j(t)f*_k(t times tau) rho_jk(t-(t-tau)) where {F_j(t)} are UAP functions with sigma_j