Download ECE 459: Communications I Problem Set 6 at University of Illinois at Urbana-Champaign and more Assignments Digital Communication Systems in PDF only on Docsity!
University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering
ECE 459: Communications I
Fall 2005
Problem Set 6 MMSE Estimation, Linear MMSE Estimation, Jointly Gaussian R.V.’s
Issued: Thursday, Oct. 20st. Due: Thursday, Oct. 27th (beginning of lecture).
Reading from Lathi: Chapter 10.
Announcement: The second Mid-Semester Exam will be held on Thursday, November 10th, from 7:00pm to 9:00pm in 165 Everitt. The exam will cover all material from the beginning of the term up to and including the lecture on Thursday, November 3rd. The corresponding material includes Problem Sets 1 through 7 and Chapters 1, 2, 3, 4, 5, 10 and 11 (excluding Section 11.6) from Lathi. Emphasis will be placed on the material not covered in the first Mid-Semester Exam. For the exam, you can bring two 8. 5 × 11-inch double-sided sheets of handwritten notes. Cal- culators are allowed but will not be necessary.
Problem 6.
Random variables X and Y have joint pdf fX,Y (x, y) that is constant in the shaded region (and zero elsewhere).
y
1
1
x
(a) Make fully labeled sketches of the densities fX (x) and fY (y).
(b) Are X and Y statistically independent? Explain.
(c) Determine X̂ M M SE (y), the minimum mean square error estimator for X, given the ob- servation Y = y.
Problem 6.
Random variables X and Y have joint pdf fX,Y (x, y) that is constant in the shaded region (and zero elsewhere).
y
x
1
1/
1/2 1
(a) Make a fully labeled sketch of the density fX (x). What is the mean and variance of X?
(b) Are X and Y uncorrelated? Are X and Y statistically independent?
(c) Determine X̂ M M SE (y), the minimum mean square error estimator for X, given the ob- servation Y = y.
(d) Determine X̂ LM M SE (y), the linear minimum mean square error estimator for X, given the observation Y = y.
Problem 6.
In a certain wireless communication system, the transmitted value X is attenuated by a random attenuation and is corrupted by channel noise so that the available measurement Y at the receiving end is related to X as Y = W X + N.
The transmitted value X is a uniform random variable in the interval [− 1 , 1], the attenuation W is a uniform random variable in the interval [ 12 , 1], and the additive noise N is a Gaussian random variable with zero mean and unit variance. Furthermore, X, W and N are mutually independent.
Given that you observe the value Y = y at the receiving end, find the linear minimum mean square error (LMMSE) estimate for the transmitted value, i.e., find α and β so that
XˆLM M SE (y) = αy + β
and E[( XˆLM M SE (y) − X)^2 ] is minimized.
Hint: The following may make your calculation easier: E[X] = 0, E[X^2 ] = 1/3, E[W ] = 3/4, E[W 2 ] = 7/12.
