Rowan University Department of Electrical and Computer Engineering

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1 Rowan University Department of Electrical and Computer Engineering Estimation and Detection Theory Fall 2013 to Practice Exam II This is a closed book exam. There are 8 problems in the exam. The problems are not in order of difficulty. We recommend that you read through all the problems, then do the problems in whatever order suits you best. A correct answer does not guarantee full credit, and a wrong answer does not guarantee loss of credit. You should clearly but concisely indicate your reasoning and show all relevant work. Your grade on each problem will be based on our assessment of your level of understanding as reflected by what you have written in the space provided. Please be neat and box your final answer, we cannot grade what we cannot decipher. Name 1

2 Problem 1 The heart rate h of a patient is automatically recorded by a computer every 100 ms. In 1 s the measurements ĥ 1, ĥ2,, ĥ10 are averaged to obtain ĥ. If E(ĥi) = h, and var(ĥi) = 1 for each i, determine the mean and variance of the average, i.e., determine E(h) and var(h). Does averaging improve the estimation? Assume each measurement is uncorrelated. See solution to Hw 2, Problem

3 Problem 2: DC level in WGN Consider a DC level in white Gaussian noise (WGN): x[n] = A + w[n], n = 0, 1,, N, where var(w[n]) = σ 2. Determine the CRLB for A. See to Problem 2 in Practice Exam I with r = 1. We then obtain V arâ σ2 N. (1) 3

4 Problem 3: Fourier Analysis Consider a data model consisting of sinusoids in white Gaussian noise: x[n] = M k=1 a k cos( 2πkn N )+ M k=1 b k sin( 2πkn )+w[n], n = 0, 1,, N, N (2) where w[n] is WGN. We want to estimate the amplitudes a k, b k of the cosines and sines. Therefore, the unknown vector parameter, θ, is given by θ = [a 1, a 2,, a M, b 1, b 2,, b M ] T. 1. Write Equation (2) in vector form, in terms of the linear model: x = Hθ + w. 2. Find the MVU estimator of θ, ˆθ MV U. 3. The columns of the matrix H are orthogonal, so that H T H = N 2 I. Simplify the expression of ˆθ MV U, and find the estimators â k and ˆb k. 4. Find the covariance matrix of ˆθ MV U, Cˆθ. See Class notes! The problem has been solved in class! 4

5 Problem 4 Consider the following model: x[n] = Ar n + w[n], n = 0, 1,, N, (3) where A is an unknown parameter, r is a known constant, and w[n] is zero mean white (not necessarily Gaussian) with variance σ Write equation (3) in the linear model form. 2. Find the BLUE of A. 3. Find the minimum variance. Does it approach zero as N? Hint: Recall that ˆθ BLUE = (H t C 1 H) 1 H T C 1 x, where C is the noise covariance matrix. See solution to Problem 2 in Practice Exam I. 5

6 Problem 5 We observe N IID samples from an exponential distribution { λe p(x; λ) = λx, x > 0; 0, x < 0. Find the MLE of the unknown parameter λ. estimator make sense? Does this The joint distribution of the data (given independence) is p(x, λ) = λ N exp λ N 1 n=0 x[n] (4) Taking the derivative of p(x, λ) with respect to the parameter λ and setting it to zero, we obtain dp dλ = NλN 1 exp λ N 1 N 1 n=0 x[n] +λ N ( x[n]) exp λ N 1 n=0 x[n] = 0 n=0 (5) = ˆλ = 1 x = N N 1 (6) n=0 x[n]. This result is reasonable since the mean of x is 1 λ. 6

7 Problem 6 If the data set x[n] = As[n] + w[n], n = 0, 1,, N is observed, where s[n] is known and w[n] is WGN with known variance σ 2, find the MLE of A. The likelihood of the data is given by p(x, A) = 1 (2πσ 2 ) N/2 exp 1 2σ 2 N 1 n=0 (x[n] As[n])2 (7) Maximizing the likelihood with respect to A is thus equivalent to minimizing N 1 n=0 (x[n] As[n])2. (x[n] As[n])s[n] = 0 = ÂMLE = N 1 2 n=0 N 1 n=0 x[n]s[n] N 1. (8) n=0 s[n]2 7

8 Problem 7 In the below table, fill in the criterion of optimality and the statistical assumptions about the noise, for each estimator. Model Criterion of optimality Assumptions about the noise x = Hθ + w MVU White and Gaussian ˆθM x = Hθ + w Linear unbiased minimum variance estimator White noise (unknown pdf) ˆθBL x = Hθ + w maximum likelihood White and Gaussin ˆθM x = Hθ + e Least squares None ˆθL Table 1: Comparison of the MVU, BLUE, MLE and LS estimators 8

9 Problem 8 For the signal model s[n] = p A i cos(2πf i n) (9) i=1 where the frequencies f i are known and the amplitudes A i are to be estimated. Therefore, we have A 1 θ = A 2... A p 1. Write model (9) in the linear form 2. Find the LS estimate of θ. The linear model is s = Hθ + w, (10) where H = cos(2πf 1 ) cos(2πf 2 ) cos(2πf p ).... cos(2πf 1 (N 1)) cos(2πf 2 (N 1) cos(2πf p (N 1) The LS estimator is given by (since the noise is white): ˆθ = (H t H) 1 H t s (11) 9

10 If f i = i/n, the columns of H are orthogonal, and we have Hence, ˆθ = 2 N Ht x = Âi = 2 N H t H = N 2 I (12) N 1 n=0 x[n] cos(2πf i n). (13) 10