Problem 2. PSD and Poisson. a. Let X(n) be an independent and identically distributed random se- quence with each X(n) having mean 0 and variance 2. Suppose we form where Y(n) = h(k)X(nh) k=0 h(n) = a" u(n) where 0 < a < 1. Find Sy (f), the power spectral density of Y. b. Recall that at time t the Poisson process N(t) is a Poisson random variable with parameter At. We can think of N(t) as counting the number of points in an interval (0, t] if we start at 0 as the origin (the meaning of these points depends on the application). Now define a new random process as X(t) = 1 if N(t) is even and X(t) = -1 if N(t) is odd. Note that N(0) = 0 so X(0) = 1. == i. Compute E[X(t)]. Simplify your result as much as possible. ii. As noted above X(0) = 1 (since N(0) = 0) and is not random. To remove this certainty we form a new random process Y(t) A X(t) where A is a random variable taking values +1 and -1 with equal probability. One can show that Y(t) is WSS with covariance function Ky (7) = e-2. Determine (mathematically) whether or not Y(t) is ergodic in mean.