Consider two random processes and have zero mean, and they are individually stationary. The random process is = + . Now when stationary processes are uncorrelated then power spectral density of is given by

Consider two random processes and have zero mean, and they are indiv

1.	Consider two random processes and have zero mean, and they are individually stationary. The random process is = + . Now when stationary processes are uncorrelated then power spectral density of is given by
Answer» The autocorrelation function of z(t) is given by Rz(t, u) = E[Z(t)Z(u)] = E[(x(t) + y(t)) (x(u) + y(u))] = E[x(t) x (u)] + E[x(t) y(u)] + E[y(t) x(u)] + E[y(t) y(u)] = Rx(E, u) + Rxy(t, u) + Ryx(t, u) + Ry(t, u) Defining t = t - u, we may therefore write Rz(t) = Rx(t) + Rxy(t) + Ryx(t) + Ry(t). When the random process x(t) and y(t) are also jointly stationary. Accordingly, taking the fourier transform of both sides of equation we get Sz(f = Sx(f) + Sxy(f) + Syx(f) + Sy(f) We thus see that the cross spectral densities Sxy(f) and Syx(f) represent the spectral components that must be added to the individual power spectral densities of a pair of correlated random processes in order to obtain the power spectral density of their sum. When the stationary process x(t) and y(t) are uncorrelated the cross-sectional densities Sxy(f) and Syz(f) are zero ∴ Sz(f) = Sx(f) + Sy(f).