Electronics and Communication Engineering - Exam Questions Papers - Discussion

The autocorrelation function of z(t) is given by

R_z(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)]

= R_x(E, u) + R_xy(t, u) + R_yx(t, u) + R_y(t, u)

Defining t = t - u, we may therefore write R_z(t) = R_x(t) + R_xy(t) + R_yx(t) + R_y(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

S_z(f = S_x(f) + S_xy(f) + S_yx(f) + S_y(f)

We thus see that the cross spectral densities S_xy(f) and S_yx(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 S_xy(f) and S_yz(f) are zero

∴ S_z(f) = S_x(f) + S_y(f).