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University of Illinois at Urbana-Champaign
ECE 534: Random Processes
Fall 2004 Final Exam
Monday, December 13, 2004
Name:
- You have three hours for this exam. The exam is closed book and closed note, except you may consult both sides of three 8. 5 ′′^ × 11 ′′^ sheets of notes in ten point font size or larger, 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:
- (9 pts.)
- (12 pts.)
- (12 pts.)
- (9 pts.)
- (8 pts.)
- (6 pts.)
- (6 pts.)
Total: (62 pts.)
Problem 1 (9 points) Let X, Y be jointly Gaussian random variables with mean zero and covariance matrix
Cov
X
Y
You may express your answers in terms of the Φ function defined by Φ(u) =
∫ (^) u −∞ √^1 2 π e
−s^2 / (^2) ds.
(a) Find P [|X − 1 | ≥ 2].
(b) What is the conditional density of X given that Y = 3? You can either write out the density in full, or describe it as a well known density with specified parameter values.
(c) Find P [|X − E[X|Y ]| ≥ 1].
Problem 3 (12 points) Let U 1 , U 2 ,... be a sequence of independent random variables, each uniformly distributed on the interval [0, 1]. Let Y 0 = 1, and Yn = U 1 U 2 · · · Un for n ≥ 1. (a) Find the variance of Yn for each n ≥ 1.
(b) Find E[Yn|Y 0 ,... , Yn− 1 ] for n ≥ 1.
(c) Find Ê [Yn|Y 0 ,... , Yn− 1 ] for n ≥ 1.
(d) Find the linear innovations sequence Y˜ = ( Y˜ 0 , Y˜ 1 ,.. .).
(e)(4 points) Fix a positive integer M and let XM = U 1 +... + UM. Using the answer to part (d), find Ê [XM |Y 0 ,... , YM ], the best linear estimator of XM given (Y 0 ,... , YM ).
Problem 4 (9 points) Let X be a mean zero, WSS random process with power spectral density SX (ω) = (^) ω (^4) +13^1 ω (^2) +. (a) Find the positive type, minimum phase rational function S X+ such that SX (ω) = |S X+ (ω)|^2.
(b) Let T be a fixed known constant with T ≥ 0. Find X̂ t+T |t, the MMSE linear estimator of Xt+T given (Xs : s ≤ t). Be as explicit as possible. (Hint: Check that your answer is correct in case T = 0 and in case T → ∞).
(c) Find the MSE for the optimal estimator of part (b).
Problem 6 (6 points) Let a > 0 and let g be a nonnegative function on IR which is zero outside of the interval [a, 2 a]. Suppose X is a narrowband WSS random process with power spectral density function SX (ω) = g(|ω|), or equivalently, SX (ω) = g(ω) + g(−ω). The process X can thus be viewed as a nar- rowband signal for carrier frequency ωc, for any choice of ωc in the interval [a, 2 a]. Let U and V be the baseband random processes in the usual complex envelope representation: Xt = Re((Ut + jVt)ejωct). (a) Express SU and SU V in terms of g and ωc.
(b) Describe which choice of ωc minimizes
−∞ |SU V^ (ω)|
2 dω 2 π. (Note: If^ g^ is symmetric arround some frequency ν, then ωc = ν. But what is the answer otherwise?)
Problem 7 (6 points) Suppose X is a continuous-time Markov process with the transition rate diagram shown, for a positive integer B and positive constant λ.
1 1 1 1 1
λ λ λ λ λ 0 1 2 B− 1 B
(a) Give the Q matrix for X.
(b) Find the equilibrium probability distribution.
