University of Illinois
Spring 2015
ECE 562: Problem Set 3: Problems and Solutions
Probability of Error for Linear Modulation, Orthogonal Modulation
Due:
Reading:
Tuesday September 22 in class
Lecture Notes 8-10; Sections 4.1 and 5.4 of Proakis & Salehi; C
University of Illinois
Spring 2013
ECE 361: Problem Set 4: Solutions
Orthogonal modulation, Noncoherent communication
1. [Semi-Orthogonal Signal Set]
(a) The functions cfw_gk (t) form a basis for our complex signal space. By the principle of irrelevance,
ECE 562
Fall 2011
HOMEWORK ASSIGNMENT 2 SOLUTIONS
1. Consider the following three waveforms:
f2(t)
f1(t)
1
2
1
2
0
4
2
0
t
4t
1
2
f3(t)
1
2
0
1
3
4
t
1
2
(a) Show that these waveforms are orthonormal.
(b) Express the following waveform s(t) as a linear co
ECE 562
Fall 2011
September 8, 2011
HOMEWORK ASSIGNMENT 2
Reading: Madhow, Sections 3.3, 2.5, 2.6
Due Date: September 20, 2010 (in class)
1. Consider the following three waveforms:
f2(t)
f1(t)
1
2
1
2
0
4
0
t
2
4t
1
2
f3(t)
1
2
0
1
3
4
t
1
2
(a) Show that
ECE 562
Fall 2011
HOMEWORK ASSIGNMENT 2 SOLUTIONS
1. Consider the following three waveforms:
f2(t)
f1(t)
1
2
1
2
0
4
2
0
t
4t
1
2
f3(t)
1
2
0
1
3
4
t
1
2
(a) Show that these waveforms are orthonormal.
(b) Express the following waveform s(t) as a linear co
ECE 562
Fall 2011
September 22, 2011
HOMEWORK ASSIGNMENT 3
Due Date: October 4, 2011 (in class)
1. For a complex random vector Y , show the following:
= (I + Q ) + j (QI IQ )
and
= (I Q ) + j (QI + IQ )
2. Consider the signal s(t) = [sin(t) + j cos(t)]1
University of Illinois
Spring 2013
ECE 361: Problem Set 5
Dierential Modulation, Channel Capacity, Error Control Coding
Due:
Reading:
Tuesday November 5 in class
Lecture Notes 11-14; Sections 4.5-4.6 and 7.1-7.6 of Proakis & Salehi; Section 2.7 of Madhow.
University of Illinois
Spring 2013
ECE 361: Problem Set 6
Equalization, Wireless Communication, Fading Channels, Diversity
Due:
Reading:
Thursday, November 21 in class
Lecture Notes 15-18; Chapter 5 of Madhow.
Reminder: Exam 2 will be held on Monday, Dece
PASSBAND DIGITAL MODULATION TECHNIQUES
Consider the following passband digital communication system model.
cos(ct )
message
source
mi
signal
encoder
si
modulator
si (t )
communication
channel
x(t )
transmitter
x(t )
detector
X
signal
decoder
mi
receiver
P
University of Illinois
Spring 2013
ECE 361: Problem Set 6: Solutions
Equalization, Wireless Communication, Fading Channels, Diversity
Reminder: Exam 2 will be held on Monday, December 2 from 7-8:20 PM in 245 Everitt Lab. You will be
allowed two sheets of
ECE 562
PRACTICE EXAM 2
Fall 2013
November 22, 2013
1. True or False. Determine if the following statements are True or False. You need to
provide a brief justication for your answer to get credit.
(a) A binary DPSK system in AWGN has Pb = 104 . It is pos
ECE 562
PRACTICE EXAM 2
Fall 2013
November 22, 2013
1. True or False. Determine if the following statements are True or False. You need to
provide a brief justication for your answer to get credit.
(a) A binary DPSK system in AWGN has Pb = 104 . It is pos
ECE 562
Fall 2013
October 8, 2013
Coherent Detection of Orthogonally Modulated Signals
The signal set is described as:
sm (t) =
Egm (t),
m = 0, 1, . . . , M 1, 0 t Ts
Here we consider the case where cfw_gm are orthonormal signals, i.e., k = 0, for k = (
ECE 562
Fall 2013
October 10, 2013
Noncoherent Communication
Rayleigh Random Variable. If Y = YI + jYQ is distributed as CN (0, 2 2 ), then we showed in class
that Z = |Y | has a Rayleigh pdf,
pZ (z) =
z
z2
1
exp 2 1 cfw_z0
2
2
Furthermore = Y = tan1 (YQ
University of Illinois
Spring 2013
ECE 361: Problem Set 4
Orthogonal modulation, Noncoherent communication
Due:
Reading:
Tuesday October 22 in class
Lecture Notes 9,10; Section 5.4 of Proakis & Salehi; Chapter 4 of Madhow.
1. [Semi-Orthogonal Signal Set]
Multipath Channel between pair of Tx & Rx Antennas
v
Y
Mobile
d
BS Antenna
X
Multipath channel seen at location (d, )
2
Channel Gain (dB)
Fading on link between pair of Tx & Rx Antennas
Distance between Tx & Rx Antennas
3
Channel Model for Point-to-Point
ECE 562
Fall 2013
September 10, 2013
Digital Modulation
After possible source and error control encoding, we have a sequence cfw_m of message symbols
to be transmitted on the channel. The message symbols are assumed to come from a nite
alphabet, say cfw
ECE 562
Fall 2013
September 24, 2013
Optimum Reception in AWGN
Restricting to the case of memoryless modulation with no ISI (ideal AWGN channel), we can
focus on one symbol interval [0, Ts ] without loss of optimality. We will also assume perfect synchro
University of Illinois
Spring 2013
ECE 361: Problem Set 3: Solutions
Complex WGN, Optimum Receiver in WGN, Probability of Error for
Linear Modulation
Reminder: Exam 1 will be held on Thursday, October 10 from 7-8:20 PM in 245 Everitt Lab. You will be
allo
ECE 562
Fall 2013
August 27, 2013
Gaussian Random Variables and Vectors
The Gaussian Probability Density Function
This is the most important pdf for this course. It also called a normal pdf.
(x m)2
1
exp
fX (x) =
.
2 2
2
It can be shown this fX integrat
ECE 562
Fall 2013
August 29, 2013
Random Processes and White Gaussian Noise (WGN)
A random process is cfw_X(t), t T simply a random signal or a random function of t (which will
usually denote time). Just as in the denition of a random variable, there is
ECE 562
Fall 2013
August 29, 2013
Complex Baseband Representation
Channel Model for Point-to-Point Communications
Point-to-point communications systems are well modeled using a bandpass additive noise channel
model of the form shown in Figure 1.
s(t)
h(t)
University of Illinois
Spring 2013
ECE 361: Problem Set 1: Solutions
Review of Probability, Random Processes, White Gaussian Noise, Complex
Baseband
1. [Gaussian pdf ].
(a) The PDF can be written as:
fX (x) =
1
2(2)
e
(x3)2
2(2)
Therefore 2 = 2.
(b) By in
University of Illinois
Spring 2013
ECE 361: Problem Set 1
Review of Probability, Random Processes, White Gaussian Noise, Complex
Baseband
Due:
Reading:
Tuesday September 10 in class
Lecture Notes 1-3, Chapters 1-3 of Wozencraft & Jacobs, Sections 3.1 and
University of Illinois
Spring 2013
ECE 361: Problem Set 1
Signal Space, Digital Modulation
Due:
Reading:
Tuesday September 24 in class
Lecture Notes 4,5; Sections 3.1 and 3.2 of Proakis & Salehi; Sections 3.3, 2.5, 2.6. of Madhow.
1. [Signal Space]
Consid
University of Illinois
Spring 2013
ECE 361: Problem Set 3
Complex WGN, Optimum Receiver in WGN, Probability of Error for
Linear Modulation
Due:
Reading:
Tuesday October 8 in class
Lecture Notes 6,7,8; Sections 4.14.3 of Proakis & Salehi; Chapter 4 of Woze
ECE 562
Fall 2013
October 15, 2013
Dierential Phase Modulation and Detection
Consider MPSK signaling on an ideal AWGN channel with phase oset (t) that may change
with time. Over N symbol intervals
N 1
r(t) =
Eejn g(t nTs ) ej(t) + w(t) .
n=0
Suppose (t)
ECE 562
Fall 2013
October 29, 2013
Signaling on Bandlimited Channels
We restrict our attention to linear modulation schemes. The information bearing signal s(t) is
given by
smn g(t nT )
s(t) =
n=0
with T = Ts , the symbol period, and g being a unit energ
University of Illinois
Spring 2015
ECE 562: Problem Set 1
Random Processes, White Gaussian Noise, Complex Baseband, Signal
Space Concepts
Due:
Reading:
Tuesday September 8 in class
Lecture Notes 1-4, Chapters 1-3 of Wozencraft & Jacobs, Sections 2.2, 3.1,