EE226: Random Processes in Systems Fall’06Problem Set 5 — Due Oct, 19Lecturer: Jean C. Walrand GSI: Assane GueyeThis problem set essentially reviews estimation theory and the Kalman filter. Not allexercises are to be turned in. Only those with the sign F are due on Thursday, October19that the beginning of the class. Although the remaining exercises are not graded, you areencouraged to go through them.We will discuss some of the exercises during discussion sections.Please feel free to point out errors and notions that need to be clarified.Exercise 5.1. Let Y = H × X + Z where X is some Gaussian random vector in Rnwithzero mean and covariance matrix KX, H is a non-singular n × n known matrix, and Z is agaussian random noise with zero mean and non-singular covariance matrix KZuncorrelatedto X.(a) Find the MMSE estimator of X given Y .(b) Explain what happens in the case where |KZ| = 0.(c) Repeat part (b) when both H and KZare singular.Exercise 5.2. FThe values of a random sample, 2.9, 0.5, -0.1, 1.2, 3.5, and 0, are obtained from a ran-dom variable X uniformly distributed over the interval [a, b]. Find the maximum-likelihoodestimates of a and b (assume that the samples are independent).Exercise 5.3. FFor the estimation problem modeled by the equations:xk= xk−1+ wk−1, wk∼ N (0, 30), white noisezk= xk+ vk, vk∼ N (0, 20), white noiseσ20= 150find σ2k, sk, and rkfor k = 1, 2, 3, 4 and σ2∞(the steady value).(σ2k, skare the estimation and prediction square error updates, and rkis the Kalman gain.)Exercise 5.4. F Parameter Estimation (recursive)Let x be a zero-mean Gaussian random variable with covariance matrix P0, and let zk= x+vkbe an observation of x with white noise vk∼ N (0, R).(a) Find a recursive (MMSE) estimator of x given the observations zkand compute theestimation error.Hint: Example 4.3 Gallager’s notes.(b) What is the value of x1if R = 0?(c) What is the value of x1if R = ∞?(d) Explain the results of (b) and (c) in terms of measurement uncertainty.5-1EE226 Problem Set 5 — Due Oct, 19 Fall’06Exercise 5.5. F Parameter Estimation (using KF)Let us consider the estimation of the value of an (unknown) constant x given measurementsyn= x + vnthat are corrupted (but uncorrelated) with a zero mean white noise vnthat hasvariance σ2v.(a) Write the estimation problem as a Kalman filter problem and compute the Kalman gain(rn) and the variance of the estimation error (e2n).(You are asked to find close forms of enand rnas functions of n, e20, σ2v.(b) What is the Kalman filter as n → ∞?(c) What is the Kalman filter as σ2v→ ∞?(d) Now suppose that we do not have no a priori information about x (i.e ˆx0= 0 ande20→ ∞).Show that the Kalman filter simply becomes the sample meanˆx =1nnXi=1ynExercise 5.6. Consider a system consisting of two sensors, each making a simple measure-ment of an unknown constant x. Each measurement is noisy and may be modeled as followsy(1) = x + v(1)y(2) = x + v(2)where vi, i = 1, 2’s are zero mean, uncorrelated random variables with variances σ2i, i = 1, 2.(a) We want to compute the best linear estimate of x of the formˆx = k1y(1) + k2y(2).Find the values of k1and k2that produce an unbiased estimate of x that minimizes themean-square error E[(x − ˆx)2].(b) Repeat part (a) for the case where the measurement are correlated,E[v(1)v(2)] = ρσ1σ2where ρ is the correlation coefficient.(c) Repeat part (a) in the framework of a Kalman filtering, treating the measurements y(1)and y(2) sequentially.Exercise 5.7. A vector discrete-time random sequence xkis given byxk=·1 10 1¸xk−1+ wk−1wk∼ N (0, 1), white noiseThe observation equation is given byzk= [1|0]xk+ vkvk∼ N (0, 2 + (−1)k), white noise5-2EE226 Problem Set 5 — Due Oct, 19 Fall’06Compute the Kalman filter updates for the recursive estimation of the process xkgiven theobservation zk.Assume thatΣ0=·10 00 10¸Exercise 5.8. FEstimation of an autoregressive processAn autoregressive process of order 1 is described by the difference equationxn= 0.5xn−1+ wnwhere wnis zero-mean white noise with a variance σ2w= 0.64. The observed process ynisgiven byyn= xn+ vnwhere vnis zero-mean white noise with a variance σ2v= 1.(a) Write the Kalman filter equations to find the LLSE estimate ˆxn(yn1) of xngiven theobservations yi, i = 1, . . . , n.The initial conditions are ˆx0(·) = 0, σ20= 1, (σ2i= E[(xi− ˆxi)2])(b) Assuming that the filter reaches a steady state solution, find the steady Kalman gainand the limiting form of the estimation equation for ˆxn(yn1).Exercise 5.9. FIn class we have derived the Kalman filter equations of the estimation problem,Xn+1= AXn+ Vnand Yn= CXn+ Wn, n ≥ 1where {X1, Vn, Wn, n ≥ 1} are all orthogonal and are zero-mean with cov(Vn) = KVandcov(Wn) = KW. (ref. Theorem 10.2 of the course notes for Rn, Sn, Σn)In this exercise, we will derive the following expression for the Kalman gain,Rn= ΣnCTK−1W(5.1)(a) By substituting the expression given for the Kalman gain Rninto that of the estimationerror covariance matirx Σn, show thatΣn= Sn− SnCT×£CSnCT+ KW¤−1CSTn(b) Using the matrix inversion Lemma, show that that inverse covariance matrix can bewritten asΣ−1n= S−1n+ CTK−1WCTheorem: Matrix Inversion LemmaSuppose A is n × n, B is n × m, and D is m × n with A and C nonsingular matrices. Then(A + BCD)−1= A−1− A−1B(C−1+ DA−1B)−1DA−1(c) By using the results in part (b) show that the expression of Rngiven in the note can bewritten as eq. 5.1.5-3EE226 Problem Set 5 — Due Oct, 19 Fall’06Exercise 5.10. F F BonusDerive the Kalman filter equations for the general caseXn+1= AnXn+ Vnand Yn= CnXn+ Wn, n ≥ 1where E[VnVTk] = KV(n)δ(n − k) and E[WnWTk] = KW(n)δ(n − k), with the usual orthogo-nality assumptions.Exercise 5.11. F In the Kalman filter setting, we are always interested in the limitingbehavior of the updates.Let’s consider the KF problemXn+1= AXn+ Vnand Yn= CXn+ Wnwith the usual orthogonality assumptions.Assume that KV= QQTand that (A, Q) is reachable and (A, C) is observable.(a) For A = Inand cov(X1− L(X1|Y0)) = 0, show that there exists a positive semi-definitematrix M such thatMCT(CM CT+ KW)CM − QQT= 0(b) Now suppose that A is general.Show that if the prediction error covariance matrix Snconverges to some limit S, then Smust satisfyS = (A − ARC)S(A − ARC)T+ ARKWRTA+KVwhere R is some matrix that you should specify.Exercise 5.12. FGive an example of ( A, B) not reachable with state
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