EN 257 : Applied Stochastic Processes Problem Set 6

Let W [n] be an independent random sequence with constant mean µ W = 0 and variance σ 2 W. Define a new random sequence X[n] as follows: X[0] = 0 X[n] = ρX[n − 1] + W [n] for n ≥ 1. (a) Find the mean value of X[n] for n ≥ 0. (b) Find the autocovariance of X[n], denoted as K XX [m, n]. (c) For what values of ρ does K XX [m, n] tend to G[m − n], for some finite-valued function G, as m and n become large? (This situation is known as asymptotic stationarity.) Let's begin by determining the general form for X[n]. Following the derivation presented in class, we can evaluate the first few terms in the sequence directly. X[1] = ρX[0] + W [1] X[2] = ρ(ρX[0] + W [1]) + W [2] = ρ 2 X[0] + ρW [1] + W [2] X[3] = ρ(ρ 2 X[0] + ρW [1] + W [2]) + W [3] = ρ 3 X[0] + ρ 2 W [1] + ρW [2] + W [3] By inspection, we conclude that the general form for X[n] is given by X[n] = ρ n X[0] + n m=1 ρ n−m W [m], where ρ n X[0] is the homogeneous solution to X[n] = ρX[n − 1]. Substituting the initial condition X[0] = 0 yields the specific solution for X[n]. X[n] = n m=1 ρ n−m W [m] (1) At this point we recall, from page 319 in [5], that the mean function of a random sequence is given by the following expression. µ X [n] E{X[n]} Substituting Equation 1 and exploiting the linearity of the expectation operator, we find µ X [n] = E n m=1 ρ n−m W [m] = n m=1 ρ n−m E{W [m]} = n m=1 ρ n−m µ W = 0.