Home / ECE / Exam Questions Paper :: Discussion

Discussion :: Exam Questions Paper

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

  2. A.
    Sx(f) + Sy(f) + 2Sxy(f)
    B.
    Sx(f) + Sy(f) + 2Sxy(f) + 2Syx(f)
    C.
    Sx(f) + Sy(f)
    D.
    Sx(f) + Sy(f) - 2Sxy(f) - 2Syx(f)

    View Answer

    Workspace

    Answer : Option C

    Explanation :

    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).


Be The First To Comment