please submit a Matlab scripts that print the solutions to assignments and exercise

= In the simple random walk process, the process w[n] is recursively generated as x[n] = x[n 1] + w[n], where w[n] is N(0,1) AWGN. The filtering introduces a correlation between successive samples of x[n]. The first task is to theoretically characterize this correlation. Assignment. Assume that the process is started with x[0] = 0, so that x[1] = w[1] and E[x?[1]] = 0% = 1. Verify that Mx[n] = 0 for all n, and derive a formula for recursively computing Px[n] = E[22[n]] in terms of Pz[n 1] (Exploit the fact that x[n 1] is uncorrelated to w[n] for any 1 > 1). What happens as n + oo? Assignment. Compute the auto-correlation rx(n,n 1), continue with re(n,n - 2) and generalize to rx(n, n l) for any l > 0. Is the process wide-sense stationary? Finally, give an explicit expression for the normalized correlation coefficient px(n, n 1). What happens as n + (for fixed 1)? Assignment. Generate a 512 x 256 matrix X, where each column is generated as a random walk process. Plot all realizations in the same figure. Explain what you see! Is it consistent with your theoretical calculations? Next, generate scatter-plots for pairs (x[nl], x[na]) with (ni, n2) = {(10,9), (50,49), (100,99), (200, 199)} and (n1, n2) {(50, 40), (100,90), (200, 190)}. Comment on the plots in light of the theoretical computations! Assignment. Compute the sample ensemble auto-correlation fx(n,n 1) as a function of n (n = 2 : 512). Plot f(n, n 1) versus n, together with the theoretical values r(n, n 1) in the same plot. Note the agreement with the theoretical values. In this experiment you used K 256 realizations of the same process to compute the ensemble auto-correlation. Would it be possible to estimate the auto-correlation rx(n,n 1) from one realization only, for this process? = 2, n