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University of Illinois at Urbana-Champaign
ECE 434: Random Processes
Spring 2003 Final Exam
Monday, May 12, 2003
Name:
- You have three hours for this exam. The exam is closed book and closed note, except that you may consult both sides of three sheets of notes, typed in font size 10 or equivalent handwriting size.
- Calculators, laptop computers, Palm Pilots, two-way e-mail pagers, etc. may not be used.
- Write your answers in the spaces provided.
- Please show all of your work. Answers without appropriate justification will receive very little credit. If you need extra space, use the back of the previous page.
Score:
- (21 pts.)
- (12 pts.)
- (12 pts.)
- (7 pts.)
- (6 pts.)
- (10 pts.)
- (12 pts.)
Total: (80 pts.)
Problem 1 (21 points) Indicate true or false for each statement below and justify your answers. (One third credit is assigned for correct true/false answer without correct justification.)
(a) If H(z) is a positive type z-transform, then so is cosh(H(z)). (Recall that cosh(x) = e x+e−x 2 .)
(b) If X is a m.s. differentiable stationary Gaussian random process, then for t fixed, Xt is independent of the derivative at time t: X t′.
(c) If X = (Xt : t ∈ IR) is a WSS, m.s. differentiable, mean zero random process, then X is mean ergodic in the mean square sense.
Problem 2 (12 points) Let N be a Poisson random process with rate λ > 0 and let Yt =
∫ (^) t 0 Nsds. (a) Sketch a typical sample path of Y and find E[Yt].
(b) Is Y m.s. differentiable? Justify your answer.
(c) Is Y Markov? Justify your answer.
(d) Is Y a martingale? Justify your answer.
Problem 3 (12 points) Let Xt = U
2 cos(2πt) + V
2 sin(2πt) for 0 ≤ t ≤ 1, where U and V are independent, N (0, 1) random variables, and let N = (Nτ : 0 ≤ τ ≤ 1) denote a real-valued Gaussian white noise process with RN (τ ) = σ^2 δ(τ ) for some σ^2 ≥ 0. Suppose X and N are independent. Let Y = (Yt = Xt + Nt : 0 ≤ t ≤ 1). Think of X as a signal, N as noise, and Y as an observation. (a) Describe the Karhunen-Lo`eve expansion of X. In particular, identify the nonzero eigenvalue(s) and the corresponding eigenfunctions.
There is a complete orthonormal basis of functions (φn : n ≥ 1) which includes the eigenfunctions found in part (a) (the particular choice is not important here), and the Karhunen-Lo`eve expan- sions of N and Y can be given using such basis. Let N˜i = (N, φi) =
0 Ntφi(t)dt^ denote the^ i
th
coordinate of N. The coordinates ( N˜ 1 , N˜ 2 ,.. .) are N (0, σ^2 ) random variables and U, V, N˜ 1 , N˜ 2 ,... are independent. Consider the Karhunen-Lo`eve expansion of Y , using the same orthonormal basis. (b) Express the coordinates of Y in terms of U, V, N˜ 1 , N˜ 2 ,... and identify the corresponding eigen- values (i.e. the eigenvalues of (RY (s, t) : 0 ≤ s, t ≤ 1)).
(c) Describe the minimum mean square error estimator Û of U given Y = (Yt : 0 ≤ t ≤ 1), and find the minimum mean square error. Use the fact that observing Y is equivalent to observing the random coordinates appearing in the KL expansion of Y.
Problem 4 (7 points) Let (Xk : k ∈ ZZ) be a stationary discrete-time Markov process with state
space { 0 , 1 } and one-step transition probability matrix P =
. Let Y = (Yt : t ∈ IR)
be defined by Yt = X 0 + (t × X 1 ). (a) Find the mean and covariance functions of Y.
(b) Find P [Y 5 ≥ 3].
Problem 6 (10 points) Let X be a real-valued, mean zero stationary Gaussian process with RX (τ ) = e−|τ^ |. Let a > 0. Suppose X 0 is estimated by X̂ 0 = c 1 X−a + c 2 Xa where the constants c 1 and c 2 are chosen to minimize the mean square error (MSE). (a) Use the orthogonality principle to find c 1 , c 2 , and the resulting minimum MSE, E[(X 0 − X̂ 0 )^2 ]. (Your answers should depend only on a.)
(b) Use the orthogonality principle again to show that X̂ 0 as defined above is the minimum MSE estimator of X 0 given (Xs : |s| ≥ a). (This implies that X has a two-sided Markov property.)
Problem 7 (12 points) Suppose X and N are jointly WSS, mean zero, continuous time random processes with RXN ≡ 0. The processes are the inputs to a system with the block diagram shown, for some transfer functions K 1 (ω) and K 2 (ω):
K 1 + K 2 X Y=X^ out+Nout
N Suppose that for every value of ω, Ki(ω) 6 = 0 for i = 1 and i = 2. Because the two subsystems are linear, we can view the output process Y as the sum of two processes, Xout, due to the input X, plus Nout, due to the input N. Your answers to the first four parts should be expressed in terms of K 1 , K 2 , and the power spectral densities SX and SN. (a) What is the power spectral density SY?
(b) What is the signal-to-noise ratio at the output (equal to the power of Xout divided by the power of Nout)?
(c) Suppose Y is passed into a linear system with transfer function H, designed so that the output at time t is X̂ t, the best (not necessarily causal) linear estimator of Xt given (Ys : s ∈ IR). Find H.
(d) Find the resulting minimum mean square error.
(e) The correct answer to part (d) (the minimum MSE) does not depend on the filter K 2. Why?
