EECS 455: Solution to Problem Set 10
1. Consider the finite field of size 32 generated with polynomial x5 + x2 + 1 = 0. Let be a
solution to this equation
(a) Build a table of powers of represented as bits and as polynomials in of degree 4 or
less. The ta

EECS 455: Solutions to Problem Set 2
1. (a) Consider two signals of duration T seconds.
r
2
cos(2 f0t)pT (t)
0 (t) =
T
r
2
1 (t) =
cos(2 f1t)pT (t).
T
Determine the minimum separation between f0 and f1 so that 0 (t) and 1 (t) are orthogonal.
Solution:
(0

EECS 455: Solutions to Problem Set 3
1. (a) Consider a complex sequence of length N=10;
x = (1 j, 1 + j, 1 + j, 1 + j, 1 + j, 1 + j, 1 + j, 1 j, 1 j, 1 + j)
Now consider a matched filter h(n) = x (N n) Determine the output of the matched filter
when the i

EECS 455: Solutions to Problem Set 1
Due: Wednesday, September 13, 2015.
1. A communication system transmits one of 8 equally likely signals. The signal (waveforms)
are represented by the vectors shown below by some. suitable set of orthonormal signals.
(

(c) Now consider the effect with noise. Determine the probaiblity of error with and without
multipath. Your answers should be in terms of the Q function and Eb /N0 .
Solution: Without multipath
Pe = Pcfw_y(T ) < 0|b0 = +1
= Pcfw_ E + < 0|b0 = +1
= Pcfw_ <

EECS 455: Solutions to Exam I, Fall 2012
1. (a) A communication system has (null-to-null)bandwidth of 20MHz available. The requirement is for a system with error probability Q( 16). The received power is P = 1016 Watts.
The noise power spectral density is

EECS 455: Solution to Problem Set 5
1. (a) A communication system with data rate of 30Mbps is desired using a bandwidth of
10MHz. What is the minimum received signal-to-noise ratio (Eb /N0 ) required to achieve
reliable (arbitrarily small error probabilit

Lecture Notes 16
Lecture Notes 16
Lecture 16
Block Diagram
Data
! Encoder
Goals
Coded
Data
! Channel
Rcvd
Data
! Decoder
Decoded
Data
!
Be able to encode using a linear block code
Be able to decode a linear block code received over a binary
symmetric chan

Lecture Notes 14
Lecture Notes 14
Lecture 14
Frequency Shift Keying (FSK)
Frequency shift keying communicates information by transmitting
different frequencies.
Goals
Be able to demodulate frequency modulated signals without
phase reference.
It can be gen

Lecture Notes 15
Lecture Notes 15
Lecture 15
In this lecture we examine a number of different simple modulation
schemes.
Goals:
We examine the implementation of the optimum receiver, the error
probability and the bandwidth occupancy.
Be able to modulate a

Introduction to Digital Communications
Wayne E. Stark
c
Copyright Wayne
E. Stark, 2015
September 11, 2015
2
Contents
1 Introduction
1.1 Communication System Coat of Arms . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Signals as Vectors . . . . .

Lecture Notes 17
Lecture Notes 17
Lecture Notes 17
Convolutional Codes
Convolutional codes are an alternative way of introducing
redundancy into the data stream.
They are preferred to block codes in many situations because of
the ease with which soft deci

Lecture Notes 13
Lecture Notes 13
Lecture Notes 13
Orthogonal Frequency Division Multiplexing (OFDM)
OFDM is a method of multiplexing several different modulated
carriers into one signal
Goals
Understand Orthogonal Frequency Division Multiplexing (OFDM)
U

EECS 455: Solutions to Problem Set 4
1. A data signal consists of an infinite sequence of rectangular pulses of duration T . That is
s(t) =
l=
bl pT (t lT )
where pT (t) is 1 for 0 t T and zero elsewhere. The data is represented by bl and is either
+1 or

Appendix A
Maximal length sequences: m-sequences
Maximal length shift register sequences are used in many applications including spread-spectrum systems. They
are sometimes called pseudo-noise (PN) sequences because they seem to have noise like properties

EECS 455 Exam II: Fall 2016
Instructions: Print your name and sign the honor code.
Print Name:
Honor Code: I have neither given nor received unauthorized aid on this examination, nor have
I concealed any violations of the Honor Code
Signature
Problem 1 is

EECS 455: Solutions to Problem Set 6
1. A binary communications system operates over an AWGN channel with spectral density
N0 /2. The transmitted signals are given by
s0 (t) = ApT /2 (t) ApT /2 (t T /2)
s1 (t) = 0
(a) Give an expression (in terms of A, T

EECS 455:Solutions to Problem Set 11
1. (a) Consider the convolutional code with encoder shown below. Assume that the code is
truncated at depth 3 so that three information bits enter the encoder (which is initially in the
all zero state) followed by two

Chapter 5
Bandwidth of Digital Signals
The goals of this chapter are to first to determine the bandwidth of digital signals. Because the signals
are random (due to the random nature of the data) we need to determine the power spectral density
of the rando

Chapter 1
Introduction
The goal of communication systems is to transmit information from one location and reliably receive that
information at a second location. There are two important resources that are needed for reliable communications. These resource

Chapter 2
Background on Signals and Systems
In this chapter we review the basic concepts of signals and systems that are needed in understanding
digital communication systems. First we show how points in space as described in Chapter 1 can be
mapped into

Chapter 6
Binary Communication: Optimum
Receiver, Optimum Signals
The goal of this chapter is to derive the optimal threshold, filter, signals for a binary communications
system that consists of transmitter that communicates one bit of information using o

Chapter 3
Simulation of Signals and Noise
The goal of this chapter is to be able to generate signals and noise in time and frequency domain in
MATLAB
3.1 Review of Signals
Consider a signal x(t) with Fourier Transform X ( f ). Suppose X ( f ) is zero for

Chapter 4
Error Probability for Binary Signals
The goal of this chapter is to be able to determine the error probability in a system with two signals and a
receiver filter in the presence of additive white Gaussian noise. The error probability is a functi

Lecture Notes 12
Lecture Notes 12
Lecture 12
Lecture 11: Global Positioning System
(Location, Location, Location)
Uses 24 satellites
20,200km (12,000 miles) with half day orbit
Goals
Uses one-way ranging from GPS
Understand the GPS System
Uses two frequen

Lecture Notes 11
Lecture Notes 11
Lecture 11
Propagation Equation: Free Space
Suppose a transmitter radiates with an antenna a power of P watts.
The receiver is located a distance r meters from the transmitter.
Then the received power is
Goals
Pr =
Be abl

Lecture Notes 9
Lecture Notes 9
Lecture 9
Orthogonal Signals
A set of signals cfw_i (t) : 0 t T , 0 i M 1 are said to be
orthogonal (over the interval [0, T ]) if
Be able to determine bandwidth efficiency and energy efficiency of
orthogonal signals.
Be ab