Problem 1 Suppose Yt = St + Nt; t E R where (St; t e R} and (It/t; l E R} are zero mean WSS and orthogonal. Suppose that we wish to mintage the proccaa X, - f 1.(t - r).9,4r. t E It with an tatimate of the ham oo = 7 h(t - r)lcdr. f 12. where k and h are impulse responses of linear-time invariant systems. -oo Show that:
12{(X, - it)] = EIK(w) - Hhell2Ss(w)+ ill(w)17S,v(w)f
where K and H are the transfer function of k and h, nespetilvely, and Sc and SN are the power spectral densities of (St; I E R) and (Ns; t E R), respectively.
Problem 2 Let (X,; I E 12) be a continuous time, zero mean Gaussian random proem with spectral density Sx (co) = Vw E R. Lot HP) and CP) be the transfer functions of the two linear systems (time-invariant) with impulse responses h(t) and g(t), respectively. The proems (X1) is passed through the filter h(t) to obtain (Vi; t e RI and it is also passed through the filter g(t) to obtain (V,; t c a) Find &w(t. s) b) Under what assumptions Vs and Vt are independent ?
