(Show all your work for full credit) Z(1) is a binary random process with T-3, received in a communication system. a) Find the best LMS estimate of the signal at t=0 when it is a given that Z(1)=-1 and Z(2) = +1. *KT b) Find the mean square error. Note: R(T)= 0 JDT

(b) Let X₁, X₂, X3 be uncorrelated random variables, having the same variance ². Consider the linear transformations Y₁ = X₁ + X₂, Y₂ = X₁ + X3 and Y3 = X₂ + X3 . Find the correlations of Yi, Y; for i #j. (5 marks)

Let {e} be a zero-mean, unit-variance white noise process. Consider a process {Y} which starts at time t = 0, and is defined recursively as follows: Let Yo = cieo, Y₁ = c₂Yo + e₁, and Y₁ = ¢₁Yt-1 + 2Yt-2 + et for t≥ 2 (as in an AR(2) process). a) Show that the process has zero mean. 1 b) For values of 01, 02 in the stationarity region for an AR (2) process, show how to choose C₁, C₂ such that both Var(Y) = Var(Y₁) and the lag 1 autocorrelation cov(Yo, Y₁) equals that of a stationary AR(2) process with parameters 01, 02. c) Once the process {Y} has been generated, show how one can transform in into a new process which has any desired mean and variance. (Remark. This gives a method for simu- lating stationary AR(2) processes.)

Show that the maximum likelihood estimation for the error variance oin linear regression is given by: (see attached)