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University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 434: Random Processes Spring 2004 Final Exam
Friday, May 7, 1:30–4:30pm, 106B1 Engineering Hall
READ THESE COMMENTS BEFORE STARTING THE EXAM!
- This is a closed-book exam! You are allowed three sheets of handwritten notes (both sides). Calculators should not be necessary, but feel free to use one.
- Write your name on the answer booklet.
- There are eight unequally weighted problems for a total of 60 points. A bonus prob- lem worth 5 points is also included. Problems are not necessarily in order of difficulty.
- A correct answer does not guarantee credit; an incorrect answer does not guarantee loss of credit. Provide clear explanations, show all relevant work and justify your answers! If we cannot make sense of your writing or reasoning, you may loose points.
- Read each problem carefully and think before performing detailed calculations.
- Only the supplied answer booklet is to be handed in. No additional pages will be considered in the grading. You may want to work things through in the blank areas of the exam and then neatly transfer to the answer sheet the work you would like us to look at.
Formulas:
- cos(a ± b) = cos a cos b ∓ sin a sin b, sin(a ± b) = sin a cos b ± sin b cos a
- sin a sin b = 12 [cos(a − b) − cos(a + b)], cos a cos b = 12 [cos(a − b) + cos(a + b)]
- sin a cos b = 12 [sin(a − b) + sin(a + b)], cos(2θ) = cos^2 θ − sin^2 θ
- FT {e−αtu(t)} = (^) α+j^12 πf , α > 0
- FT {e−αt 2 } =
√ π α e
− ω 4 α^2
Problem 1 (12/60, equally weighted parts)
This problem has twelve independent true/false questions.
(a) If the matrix
[ 2 r r 3
] is the covariance matrix of a zero-mean random vector X = [ X 1 X 2
] , then r necessarily satisfies |r| ≤ 2.
(b) If Xt and Yt are independent wide-sense stationary (WSS) random processes, then Zt = Xt · Yt is a WSS random process.
(c) The function SX (ω) = 1 − ω^2 1 + ω^2
is a valid power spectral density (PSD) for a wide-sense stationary random process Xt.
(d) If A ⊂ B and P [B] 6 = 0, then P [A|B] ≥ P [A].
(e) If X and Y are i.i.d. Gaussian random variables with mean 0 and variance 1, then Z 1 = X − Y and Z 2 = X + Y are independent, identically distributed (i.i.d.) Gaussian random variables.
(f) Let the random process Yt be defined as Yt = X 1 sin(2πt) + X 2 cos(2πt), where X 1 and X 2 are zero-mean random variables that satisfy E[X 1 X 2 ] = 0 and E[X^21 ] = E[X 22 ] = σ^2. Then, Yt is a wide-sense stationary random process.
Problem 2 (6/60, equally weighted parts)
Suppose that the random variable Y is Gaussian with mean 0 and variance 1. We know that random variable X satisfies X = |Y | and we are interested in estimating X based on the observation that Y = y. (Hint: To answer the following questions, you may want to use the following numerical values: E[|Y |] = 0.8, E[|Y |^3 ] = 1.6, E[|Y |^4 ] = 3.)
(a) Find the minimum mean square error (MMSE) estimator X̂ M M SE (y)? What is the asso- ciated mean square error MSE 1?
(b) Find the linear minimum mean square error (LMMSE) estimator X̂ LM M SE (y) = α + βy? What is the associated mean square error MSE 2?
(c) Find the minimum mean square error estimator of the form
X̂ (y) = a + by + cy^2.
Problem 3 (8/60, equally weighted parts)
This problem has two independent parts.
Part A: Let Xt, 0 ≤ t < +∞, be a Poisson random process with X 0 = 0 and rate λ = 9. Let
Yn =
n
Xn − 9
n
for n = 1, 2 , 3 , .... To what random variable, if any, does the sequence Yn converge in distribution (i.d.)?
Part B: Let Xn for n = 1, 2 , 3 , ... be a sequence of independent, identically distributed (i.i.d.) Poisson random variables with parameter λ = 4. Let
Wn = αnX 1 X 2 X 3 ...Xn
for n = 1, 2 , 3 , .... For what values of α, if any, is Wn a Martingale?
Problem 5 (8/60, equally weighted parts)
Consider the following estimation problem: a Gaussian random variable X with mean 0 and variance 1 is used to modulate the amplitude of a deterministic signal s(t), 0 ≤ t ≤ T , that satisfies (^) ∫ T 0
|s(t)|^2 dt = 1.
The modulated waveform is corrupted by an additive white Gaussian noise process Nt with mean 0 and autocorrelation function RN (τ ) = σ^2 δ(τ ). Assume that Nt and X are independent. The resulting random process Yt satisfies
Yt = Xs(t) + Nt
and the receiver needs to find the minimum mean square (MMSE) estimate for X based on Yt, 0 ≤ t ≤ T.
(a) By considering an appropriate KL expansion of Yt with basis functions φ 1 (t) = s(t) and φi(t) orthogonal to φ 1 (t) for i > 1, show that all but one coefficients are orthogonal to X. Express this coefficient in terms of X, s(t) and Nt.
(b) Using part (a) or otherwise, derive X̂ , the MMSE estimate of X given Yt for 0 ≤ t ≤ T.
Problem 6 (4/60, equally weighted parts)
Let Xt be a zero-mean wide-sense stationary (WSS) random process with autocorrelation func- tion RX (τ ) = e−|τ^ |. Suppose that Xt is processed via a linear time-invariant (LTI) system as shown below.
- h(t) -
Xt Yt
(a) For this part, assume that h(t) = e−btu(t) for some positive constant b. What is SY (ω), the power spectral density (PSD) of the output random process Yt?
(b) For this part, assume that you have no direct information about the impulse response of the LTI system, but that you know the cross-correlation between the output and input
RY X (τ ) = e−τ^ u(τ ) − 2 e−^2 τ^ u(τ ) + e−^3 τ^ u(τ ).
Find a possible impulse response h(t). Is your answer unique? If so, explain why. If not, specify another possible h(t).
Problem 8 (8/60, equally weighted parts)
Let Xt and Nt be independent zero-mean wide-sense stationary (WSS) random processes with autocorrelation functions RX (τ ) = e−^3 |τ^ |^ and RN (τ ) = δ(τ ). Suppose that Xt is processed in the fashion shown below where both systems are linear time-invariant (LTI) with h 1 (t) = 2δ(t) and h 2 (t) = 2e−^5 tu(t).
6
h 1 (t)
Nt
h 2 (t)
Xt Yt
?
(a) Find SY (ω), the power spectral density (PSD) at the output Yt.
(b) Find the frequency response of the noncausal Wiener filter H(ω) that takes Yt, −∞ < t < +∞, as input and produces the minimum mean square error (MMSE) estimate for Xt given Yt.
Bonus Problem (5/60)
Suppose that you randomly place two marks on a stick of length one. More specifically, one mark is placed at distance X from the left end of the stick and the other mark is placed at distance Y from the left end of the stick, with X and Y being i.i.d. random variables uniformly distributed in [0, 1]. What is the probability that if you cut the stick at both marks, you can form a triangle? [Hint: For a triangle with sides of lengths a, b and c we need a + b > c and a + c > b and b + c > a.]
