EE 230A
Fall 2013
K. Yao
Final I (Closed Book; 90 min.) - 25 pts
1. Quickies. Give a brief explanation or denition of each of the following terms (2 pts each)
a.
b.
c.
d.
Neyman-Pearson Criterion for binary hypothesis testing.
Karhunen-Lo`ve Expansion.
e
EE 230A
Homework #1 Solutions
Fall 2014
K. Yao
1. a. For R = 1 G bits/sec., the new power requirement becomes PT = 5.68 105 watts.
b. For R = 100 bits/sec., the new power requirement becomes PT = 5.54105 watts. If this power is too
demanding, we can get c
EE230A
Fall 2014
K. Yao
Class Project
Due Tuesday Dec. 19th noon time.
(4.5 Points)
In our class, we have studied the performances (e.g., Pes) of various digital communication
systems in the presence of AGN. In wireless communications (e.g., in a cellular
EE 230A Homework #1 Fall 2014
Due October 15th, 2014 K. Yao
Read Chapter 1 and review probability and random processes in Chapter 2 of the textbook.
Please note, all odd number homework problem solutions can be found in the Cambridge Univer
sity Book webs
EE 230A
Midterm Exam
(90 minutes/25 pts - An 8.5x11 sheet(two-sided) is allowed)
Please write your Last Name:
UID:
Fall 2013
K. Yao
; First Name:
;
and hand-in this sheet as the rst page of your exam.
1. Answer True or False and give a brief explanation (
EE 230A
Fall 2013
K. Yao
Final II (Open Book Solutions)
1. Let X = [X1 , . . . , XM ]T . Under the three hypothesis, the LH functions are given by
M
p1 (x) =
(xi )2 /(2 2 )
1
(2 2 )M/2
e
, H1 ,
i=1
M
p2 (x) =
(xi 1)2 /(2 2 )
1
(2 2 )M/2
e
, H2 ,
i=1
M
p3
EE 230A
Fall 2014
K. Yao
Homework #8
Due Dec. 3rd
1. We know how to generate independent uniformly distributed r.v.s cfw_Ui between 0
and 1. Now, we want to extend that technique to the generation of Gaussian random
variables. Consider the following bina
EE 230A
Fall 2014
K. Yao
Homework #7
Due Nov. 26th
Read Chapter 7 (pp. 238 - 257)
1. Problem 7.1 (p. 264).
2. Problem 7.2 (p. 264).
3. Problem 7.4 (p. 265).
4. Problem 7.8 (p. 267).
1
EE 230A
Fall 2014
K. Yao
Homework #8 Solution
1. For the simple binary detection problem of
x=
1 + n , H0 ,
+1 + n , H1 ,
where n is the realization of a zero-mean unit variance Gaussian r.v., if each hypothesis occurs
with equal probability and the thres
EE 230A
Homework #4 Solutions
Fall 2014
K. Yao
1. a. Let
(t) =
s1 s1 (t)
s0 s0 (t),
(1)
then
T
N0
s1 s0
ln 0 +
=
2
2
H1
(t)y(t) dt >
<
H0
z=
0
(2)
and z is the sucient statistic.
T
PD = Prcfw_z | H1 = Pr
(t)( s1 s1 (t) + n(t) dt
.
0
But
T
(t) s1 s1 (t)
EE 230A
Fall 2008
K. Yao
Final II (Open Book)- 25 pts
1. Consider a binary non-coherent detection system
n(t), 0 t T, H0 ,
A sin(0 t + 1 ) + n(t), 0 t T/2, H1 ,
x(t) =
A sin(0 t + 1 ) + n(t), T/2 t T, H1 ,
where n(t) is the realization of a a WGN process
EE 230A
Fall 2008
K. Yao
Final I (Closed Book) - 25 pts
1. For each part of Question 1, rst answer True (T) or False (F) (1 pt) and then give a brief
explanation (the minimum amount needed to provide the explanation) for your choice of
T or F (1 pt).
a. U
EE 230A
Fall 2013
K. Yao
Final II (Open Book; 90 minutes) - 25 pts
1. Let the observed data be modeled by
N,
H1 ,
1 + N , H2 ,
X =
1 + N , H3 ,
where N is a zero-mean Gaussian r.v. with 2 = 0.25 , and all three hypothesis are equally
likely.
a. From M ind
EE 230A
Homework #3
Fall 2014
Due October 29th, 2014 (Midnight PST)
K. Yao
Read Chapter 3 (pp. 65 - 79; 90 - 96), Chapter 4 (pp. 97 - 129; 135 - 148) of the textbook.
1. a. Design a LR test to choose between the hypotheses:
H1 : The sample X is chi-square
EE 230A
Homework #3 Solutions
Fall 2014
K. Yao
2
2
1. Let X = X1 + .Xn with Xi N (0, 1). Then X is a chi-square r.v. of degree n with a pdf given by
n/2
pX (x) = 2
(1/(n/2)x1+n/2 ex/2 , x 0. Furthermore, 0 = 1.
x/2
a. Under H0 , p0 (x) = (1/2)ex/2 , x 0;
UCLA
Dept. of Electrical Engineering
EE 114, Winter 2013
Problem Set 5
Due: February 20, 2013
EE114, Winter 2011
1. Find the 2D Fourier Transform of the two-dimensional function
Problem Set 4
_
f (x, y ) = rect(ax + b) sinc(cy ) .
Problem Set #4
2. Let f
UCLA
Dept. of Electrical Engineering
EE 114, Winter 2013
Problem Set 6
Due: February 27, 2013
1. Consider a continuous 2D signal of theform
f (x, y ) = cos (2 (4x + 3y ) .
Suppose you wish to design a sampling/reconstruction system that fails to satisfy t
UCLA
Dept. of Electrical Engineering
EE 114, Winter 2013
Problem Set 7
Due: March 6, 2013
1. The four basis vectors of the N = 4 1D DFT form an orthonormal set, i.e.,
1, k = ,
0, otherwise,
aH a =
k
where
H
denotes conjugate transpose.
a) Conrm that the a
Interference Rejection based on Statistical Decorrelation
1 Introduction
There are various practical problems in which a main sensor designed for the reception of a desired
weak signal 30(t) also receives a strong interfering signal s(t) plus a random the
EE 230A
Fall 2014
K. Yao
Homework #9
You do not need to turn in HW9. However, you should try to solve these problems by
yourself before looking at the solutions
1. 1. Consider the problem of detecting a noise-like signal in noise.
x(t) = s(t) + n(t), H1
x
EE 230A
Homework #4
Fall 2014
Due Nov. 5th, 2014 (Midnight PST)
K. Yao
Read Chapter 4 (pp. 97 - 129; 135 - 148); Chapter 5 (pp.149 - 168) of the textbook.
1. Problem 4.2, p. 142.
2. Problem 4.3, p. 142.
3. Problem 4.4, p. 142 -143.
4. Problem 4.8, p. 144.
EE 230A
Homework #2
Due October 22nd, 2014
Read Chapter 2 - Chapter 3 of the textbook.
Fall 2014
K. Yao
1. Problem 2.5, p. 62.
2. A random process is dened by X(t) = K cos(t), > 0, < t < , where K is a
uniform r.v. between 0 and 2.
a. Find Ecfw_X(t).
b. F
EE 230A
Fall 2014
K. Yao
Homework #9 Solution
1. For the bandlimited white signal and noise Gaussian processes of radian bandwidth
= 2B, sampling at the Nyquist interval spacing of / = 1/(2B), yields un2
correlated and thus independent Gaussian random va
EE 230A
Fall 2014
K. Yao
Homework #5
Due November 12th
Read Chapter 6 (pp. 190 - 211)
1. Problem 4.10 (Pp. 144 - 145).
2. Problem 5.2 (P. 187).
3. Problem 5.7 (P. 189)
1