A sequence of zero mean unit variance independent random variables, Xn, n = 0, 1, 2, ,

Question:

A sequence of zero mean unit variance independent random variables, Xn, n = 0, 1, 2, …, N – 1 are input to a filter that produces an output sequence according to Yn = (Xn + Xn – 1)/ 2, for . For n = 0, 1, 2 … N – 1.For initialization purposes, X –1 is taken to be zero.
(a) Find the covariance (correlation) matrix of the Yn.
(b) Now let the variance of the Xn be σ2X. Find the covariance (correlation) matrix of the Yn.