ECE 531: Detection and Estimation Theory, Spring 2011
Homework 11 Solutions
Problem1. (5.14 Shu Wang)
From Eq 5.5 and 5.6, we have:
T (x) = xT Cs (Cs + 2 I )1 x
s = Ah
2
Cs = E [ssT ] = E [AhAhT ] = E
ECE 531 Detection and Estimation - Final Exam
May 8, 2009. 8 am - 10 am in ERF 1003.
This exam has 6 questions. You will be given 2 hours. You may use the 2 course textbooks but no other aides/notes.
ECE 531 - Detection and Estimation Theory
Homework 2
Solutions
3.3 (Luke Vercimak) The data x[n] = Arn + w[n] for n = 0, 1, . . . , N 1 are observed, where
w[n] is WGN with variance 2 and r > 0 is kno
Lecture 6
o Sufcient Statistics
0 Factorization
o Likelihood Function
o Receiver Operating Curve (ROC)
o Randomization o Sufcient Statistics
. A statistic is any random variable that can be computed f
Lecture 1
Introduction to Detection Theory
Elements of Decision Theory
Examples
Types of Detection Problems
Introduction to Detection Theory
Statistical Decision Theory
- Also known as Decision Theory
EEL 6537 Detection Theory
Homework #1
Fall 2017
SOLUTION
1. Problem 1-1
Let cfw_1 , 2 , , be a sequence of independent and identically distributed
random
variable with cdf () and pdf ().
= cfw_1 , 2
Lecture 5
o Bayes Criterion
o Neyman-Pearson Criterion
o Minimax Criterion 5-1
0 Bayes Criterion
. Let C ,1 = costs assigned to deciding in favor of hypothesis H .- When H ,- is the true hypothesis.
.
EEL 6537 - Detection Theory
Homework #2
Fall 2017
1. Problem 4.3
2. Problem 4.9
3. Obtain a minimum probability of error decision rule to discriminate between two
hypotheses 0 and 1 based on the obser
EEL 6537 - Detection Theory
1. Problem 4.3
Homework #2
Solution
Problem 4.4
Problem 4.6
Problem 4.8
2. Problem 4.9
3. Obtain a minimum probability of error decision rule to discriminate between two
hy
EEL 6537 Detection Theory
Homework #1
Fall 2017
1. Problem 1-1
2. Problem 1-5
Hint:
The following integration result may be useful:
1
=
,
/ ()
0 [1 + ]
where
() = 0 1 is the gamma function
( + 1
Lecture 4
o Hypothesis Testing
o Classifying Statistical Tests
0 Optimality Criteria
- MAP ctriterion 4.2.
o Hypothesis Testing
Hypothesis Testing is a statistical tool for making decisions.
In all as
Lecture 3
o Expectations
o Limit Theorems
o Characteristic Function
o Random Processes
o Examples 32
o Expectation (Mean Value)
E(X) = 7 = f x fx(x)dx for continuous random variables
= 2m P(X = x,-) f
Lecture 7
o Multiple Measurements
o Multiple Hypotheses
o Composite Hypotheses
o GLRT 7.2
0 Multiple Measurements
In practice, decisions are made based on multiple measurements.
X = cfw_X.,X2,.,X,
Ho
Lecture 2
o Review of Probability
o Random Variables
0 Functions of Random Variables - Review of Probability
There are two Approaches to the denition of probability:
Relative Frequency Approach
- Axi
University of Illinois at Chicago
Department of Electrical and Computer Engineering
EC 531: Detection and Estimation Theory
Spring 2009
Final exam: PRACTICE
NAME:
This exam has 6 questions.
You will
ECE 531: Detection and Estimation Theory, Spring 2011
Homework 1
ALL SOLUTIONS BY Shu Wang thanks for submitting latex le in 1st homework!
Problem1 (2.1)
Solution:
1
E [ 2 ] = E [
N
=
1
N
N 1
x2 [n]
n
ECE 531 - Detection and Estimation Theory
Homework 4
February 5, 2011
6.7 (Shu Wang) Assume that x[n] = As[n] + w[n] for n = 0, 1, . . . , N 1 are observed, where
w[n] is zero mean noise with covarian
ECE 531: Detection and Estimation Theory, Spring 2011
Homework 6
Problem 1. (8.20) (Shu Wang)
Solution:
From the problem, we know that:
H =
1
r
.
.
.
r N 1
So we have h[n] = r n . According to 8.46, w
ECE 531: Detection and Estimation Theory, Spring 2011
Homework 8
Problem1. (13.4) (Shu Wang)
Solution:
From eq 13.5, we have:
n
2
2
Cs [m, n] = am+n+2 s + u amn
a2k
k=0
n
2
2
= amn (a2n+2 s + u
a2k )
ECE 531: Detection and Estimation Theory, Spring 2011
Homework 10
Problem 1 (4.6 Luke Vercimak)
The this is a known signal in WGN. Per eq 4.3, the test statistic will be:
T (x) =
N 1
x[n]s[n] >
n=0
I
University of Illinois at Chicago
Department of Electrical and Computer Engineering
EC 531: Detection and Estimation Theory
Spring 2009
Midterm 1
NAME:
This exam has 4 questions.
You will be given t
University of Illinois at Chicago
Department of Electrical and Computer Engineering
EC 531: Detection and Estimation Theory
Spring 2009
Midterm 2
NAME:
This exam has 4 questions.
You will be given t
University of Illinois at Chicago
Department of Electrical and Computer Engineering
EC 531: Detection and Estimation Theory
Spring 2009
Midterm 2
NAME:
This exam has 4 questions.
You will be given t
University of Illinois at Chicago
Department of Electrical and Computer Engineering
EC 531: Detection and Estimation Theory
Spring 2009
Practice for Midterm 1
1. (30 points) You are a lawyer. In a com
WW; ,3 Programs
32 Chapter 1 Review of Probability
1.13 BIBLIOGRAPHICAL NOTES
An excellent reference for material in this chapter is Papoulis [111]Wha1en [164] also provides
important results for Ga