EE 562a
Homework Set 2
Due Monday 5 Feb 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. Let y(u) be a random vector with second-moment description 3 -1 0 3

Homework 2 Due Monday June 12, 2006 Work 3 problems. Problem 1. Let X= (X1 , X2 , X3 , X4 ) be a Gaussian random vector where E[Xi ] = 0 for i = 1, 2, 3, 4. Show that E[X1 X2 X3 X4 ] = K12 K34 + K13 K24 + K14 K23 where Kij is the i, j element of the

EE 562a
Homework Set 3
Due Wednesday 14 Feb 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. Let y(u) be a real random vector with mean vector my and covarian

EE 562a
Homework Set 1
Due Monday, 22 Jan 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. A function is called concave (or equivalently convex (cap) or conv

EE 562a 1. Solution:
Homework Solutions 2
September 22, 2009
1
This problem is a straightforward problem involving moment calculations of the quantity y (u) = At x(u) + B . my = E cfw_y (u) = At E cfw_x(u) + B = At + B
2 y
E cfw_(y (u) my )(y (u) my ) =

Problem RAS0103
1
1. Compute the mean value, correlation, and covariance functions of the following random processes x(u, t), u U , t T : (a) x(u, t) = ai (u) t (i, i + 1], i cfw_0, 1, 2, . . . Here T is the real line with x(u, t) taking on a dierent valu

EE 562a 1. Solution:
Homework Solutions 3
February 14, 2007
1
(a) In this problem we will use the shorthand notation of mi = the ith component of my First consider event A:
y(u,2) Event A
kij = the ij th element of Ky
y(u,1)
In order to bound

EE 562a
Homework Set 7
Due Monday 16 April 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. Hypothesis testing in discrete time. (Modified Final Exam Problem

Homework 3 Due Monday June 19, 2006 Work all 12 problems. Problem 1. Suppose X is a mean-zero matrix 1 4 KX = 4 20 3 22 random vector with covariance 3 22 . 36
EE 562a
Find a transformation A such that Y = AX has covariance matrix KY = I. Pro

EE 562a 1. Solution:
Homework Solutions 3
October 1, 2009
1
(a) In this problem we will use the shorthand notation of mi = the ith component of my First consider event A:
y(u,2) Event A
kij = the ij th element of Ky
y(u,1)
In order to bound the probabilit

EE 562a Self-Test Solution
1
Topic 1: Linear Algebra
1. To nd M, you must solve two linear equations, namely [1 1]M = [2 0] and [5 4]M = [3 7].
Equivalently in partitioned form you must solve the matrix equation
1 1
5 4
2 0
3 7
M=
,
yielding the solution

EE 562a
Homework Set 8
Due Wednesday 25 April 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. Consider an LTI system, M, characterized by the following diffe

EE 562a Midterm Exam
Name:
1
EE562a MIDTERM EXAM Wednesday, October 14, 2009, 11 AM (1 hour, 20 minutes) Mudd Hall of Philosophy (MHP), Room 101 Open Book and Notes Special Instructions. Check to make sure that this exam contains 5 pages, including the co

EE 562a
Homework Set 4
Due Monday 26 February 2007
1
(The following well-defined problems come from different sources, and the notation used may vary. Don't let that bother you!)
1. Let x(u) and y(u) be random vectors that are related to each ot

Supplemental Notes on Random Processes (A Work in Progress) ( marks incomplete sections)
Robert Scholtz and Keith Chugg January 3, 2007
ii
Contents
1 A Roadmap 1.1 The Big Questions . . . 1.2 The Starting Point . . . 1.3 The Trip . . . . . . . . .

Chapter 10
The Fourier Realm
The value of describing a linear transformation H from an inner-product space L into itself in terms of its eigenvalues and eigenvectors has been illustrated in the prior chapters when the space L is the finite-dimensional inn

Supplemental Notes on Random Processes (A Work in Progress) ( marks incomplete sections)
Robert Scholtz (with contributions and corrections by faculty and students in the Ming Hsieh Department of Electrical Engineering at the University of Southern Califo

EE 562a
Homework Set 3
1. Chapter 8, Problem 5
2. Chapter 8, Problem 6
3. Chapter 8, Problem 9
4. Chapter 9, Problem 1
5. Chapter 9, Problem 2
6. Chapter 9, Problem 5
Due Tuesday, February 15, 2011
1

EE 562a
Homework Solutions 8
April 20, 2011
1
1. (a) If the input to the system is D (t), then the input to the integrator is D (t) D (t ).
Since the integral of a Dirac delta is a step function,
h(t) = U(t) U(t ) = rect
t /2
.
Since we only need the magn

EE 562a
Midterm Solution
1. (a) Kx = Rx mx mx t =
March 4, 2011 1
2 1
1
2
(b) The eigen-solutions of Kx e = e can be found by standard means, or can be determined
using the specialized results for circulant matrices. The solutions are
1 = 3 ,
1
e1 =
2
1

EE 562a
Midterm Solution
October 13, 2010 1
1. (a) Tabulating sample values gives
1
1/ 2
0
Ry = 1/ 2
1
1/ 2
0
1/ 2
1
(b) There are dierent solutions to this problem. The request for a lower triangular solution
for G implies that
1
0
g11 0
0
g11 g21 g31

EE 562a
Final Solution
May 10, 2011
1
1. (a)
1/10
1/2
ei2f df =
Sx (f )ei2f df =
Rx ( ) =
1/10
1/2
ei2f
i2
1/10
=
f =1/10
sin( /5)
(b) Uncorrelated Gaussian random variables are independent, and hence
sin(t/5)
=0
t
t/5 = n where n Z cfw_0
x(u, 0) and x(u

EE 562a Final Exam
1
EE562a FINAL EXAM
Tuesday, May 10, 2011; 11 am (2 hours)
Place: MHP 106 (Mudd Hall of Philosophy)
Open Notes
Special Instructions.
INFORMATION THAT YOU CAN BRING TO THE EXAM: Any material that has been posted
on the EE562a website, i

Problem RAS0350
1
1. Let z(u) and x(u) be vectors with non-singular covariance matrices. the MMSEs for ane
and linear estimate of z(u) from an observation of x(u) are given by
Ecfw_|e(u)|2 AMMSE = Trcfw_Kzz Kzx K1 Kxz ,
xx
Ecfw_|e(u)|2 LMMSE = Trcfw_Rzz

EE 562a Midterm Exam
Name & Location:
1
EE562a MIDTERM EXAM
Thursday, March 3, 2011, 11 AM (1 hour, 20 minutes)
Location: MHP 106 (Mudd Hall of Philosophy)
Closed Book, one crib sheet allowed.
Special Instructions.
Check to make sure that this exam conta

EE 562a Midterm Exam
Name & Location:
1
EE562a MIDTERM EXAM
Wednesday October 13, 2010, 11 AM (1 hour, 20 minutes)
Location for students with family names beginning A-F: EEB 248
Location for students with family names beginning G-Z: OHE 100C
Closed Book,

Homework Policy: You can work alone or in groups on the homework. As an
important part of the learning process, it is imperative that you attempt all
homework problems and turn the homework in, even if late.
More on the Grading Policy: This policy has evo