Question: Consider a zero-mean random vector X = (X X) corrupted by a zero- mean noise vector y = (y y2) that is statistically independent

Consider a zero-mean random vector X = (X X) corrupted by a

Consider a zero-mean random vector X = (X X) corrupted by a zero- mean noise vector y = (y y2) that is statistically independent of x. The random vector z = x + y is observed. Let Rx - 3 a B3 Ry 21 (a) Specify 3 in terms of a and find the set of admissible values for a and 3. (b) For a 1, find the inhomogeneous MMSE estimator of x. (c) Find the corresponding MMSE in (d) For arbitrary a, find the eigenvalues A and orthonormal eigenvectors u of R.. Show that for U= (uus) the random vector w = UT x consists of orthogonal elements. (e) Calculate the correlation matrix of the random vector v = UTz. Are the elements of v orthogonal? Justify your answer.

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