Phase Noise Effect in OFDM Based Communication Systems

To Study The Phase Noise Effect In
OFDM Based Communication System
A THESIS
submitted by
Ashutosh Maithani
for the award of the degree
of
Master of Technology
Department of Electronics & Communication Engineering
Graphic Era University, Dehradun, India.
August , 2012

ACKNOWLEDGEMENTS
I would like to acknowledge the contribution of all those people who have been blessed
to be associated with me. I would like to thank my guide and mentor Er. Navita Sajwan ,
Assistant Professor, Department of Electronics and Communication Engineering, GEU
Dehradun Uttrakhand, for her supervision, knowledge, support and persistent
encouragement during my research work. She steered me through this journey with her
invaluable advice, positive criticism, stimulating discussions and consistent
encouragement.
With a grateful heart, I acknowledge the noble and gentle hand of support lent to me by
Dr. Anamika Bhatia, HOD, Department of Electronics and Communication Engineering,
, GEU Dehradun Uttrakhand, , for her valuable guidance at every step and cooperation to
carry out simulations. Her enthusiasm and engagement in giving guidance and sharing
knowledge cannot be valued.
I also express my deep sense of gratitude to Dr. Rajarshi Mahapatra , Project Coordinator
Department of Electronics and Communication Engineering, GEU Dehradun
Uttrakhand. He provided me continuous help and guidance to complete my dissertation.
I also express my deep sense of gratitude to other staff members of the department have
given me help and valuable advice during this period. My studies would not have been
complete without the help and friendship of colleagues. They will always have a place in
my fond memories.
Date : Ashutosh Maithani
ii i

iv
DECLARATION
I certify that,
a) the work contained in this thesis is original and has been done by me under
the guidance of my supervisor.
b) the work has not been submitted to any other institute for any degree or diploma.
c) I have followed the guidelines provided by the institute in preparing the thesis.
d) I have conformed to the norms and guidelines given in the ethical code of conduct
of the institute.
e) whenever I have used materials (data, theoretical analysis, figures, and text) from
other sources, I have given due credit to them by citing them in the text of the
thesis and giving their details in the references. Further, I have taken permission
from the copyright owners of the sources, whenever necessary.
Name of the student
Ashutosh Maithani

THESIS CERTIFICATE
This is to certify that the thesis titled TITLE submitted to the Graphic Era University,
Dehradun, by Author, for the award of the degree of Master of Technology (Full
time/Part time), is a bona fide record of the research work done by him under my
supervision. The contents of this thesis, in full or in parts, have not been submitted to any
other Institute or University for the award of any degree or diploma.
v
Name of the Prof. Dr. Rajarshi Mahapatra
Research Guide- Navita Sajwan
Designation-Asistant Professor
Department- ECE
GEU-Dehradun, 248 002
Place: Dehradun
Date:

CERTIFICATE OF APPROVAL
v i
16th Aug. 2012
Certified that the thesis entitled Title submitted by name to Graphic Era University,
Dehradun for the award of the degree of Master of Technology has been accepted by the
external examiners and that the student has successfully defended the work carried out, in
the final examination.
Signature:
Name: Er. Navita Sajwan
(Supervisor)
Signature:
Name: Dr. Rajarshi Mahapatra.
(Internal examiner)
Signature:
Name:
(External Examiner)
Signature:
Name: Dr. Anamika Bhatia
(Head of the department)

ABSTRACT
Orthogonal frequency division multiplexing (OFDM) is being successfully used in many
applications. It was chosen for IEEE 802.11a wireless local area network (WLAN)
standard, and it is being considered for the fourth-generation mobile communication
systems. Along with its many attractive features, OFDM has some principal drawbacks.
Sensitivity to frequency errors and phase noise between the transmitted and received
signals is the most dominant of these drawbacks. In this thesis, phase noise effects on
OFDM based communication systems are investigated under Rayleigh fading
environment. Phase noise has two main effects. First, it causes a random phase variation
common to all sub-carriers. The effects of this common phase error(CPE) are minimized
by employing phase tracking techniques or differential decoding. Second, it introduces
Inter carrier interference (ICI).In OFDM system, when subjected to fading extremely
high signal to noise ratio(SNR) are required to achieve resonable error probability.Coding
becomes obvious choice to achieve higher possible rate in presence of crosstalk,
impulsive and other interferences. This form of OFDM is called coded OFDM
(COFDM). Reed-Solomon codes can compensate these two dimensional errors.
Channel estimation in OFDM based communication system is a technique use to
minimize common phase error(CPE) occurred due to phase noise. Least square with
averaging (LSA) is block-type pilot symbol aided channel estimation technique used to
multiplex reference symbols, so-called pilot symbols, into the data stream. The receiver
estimates the channel state information based on the received, known pilot symbols. The
pilot symbols can be scattered in time and/or frequency direction in OFDM frames.
This thesis analyzed Uncoded, Reed-Solomon coded and Reed-Solomon coded with LSA
channel estimated OFDM based communication system in presence of phase noise by
using MATLAB୘୑ Simulink. Various Simulink modal of OFDM based communication
system is developed in this thesis.The LSA channel estimation scheme is use to remove
common phase error (CPE) occured due to phase noise and then Reed-Solomon coding is
use to improve BER performance of OFDM system with phase noise.The simulation
performance results of the OFDM system for Rayleigh fading with QPSK modulation is
discuss in this thesis.
vi i

TABLE OF CONTENTS
DEDICATION ii
ACKNOWLEDGEMENTS iii
DECLARATION BY THE CANDIDATE iv
CERTIFICATE BY THE SUPERVISOR v
CERTIFICATE OF APPROVAL vi
ABSTRACT vii
LIST OF TABLES ix
LIST OF FIGURES x
ABBREVIATIONS xi
NOTATIONS xii
1. INTRODUCTION 1
1.1. MS Word features………………………………….……………….......... 2
1.2. MS Word figures………………………………………………………… 2
1.3. MS Word options………………………………………………………… 2
BRIEF BIO DATA OF THE CANDIDATE
PUBLICATIONS OUT OF THIS WORK
REFERENCES
ix
A. A SAMPLE APPENDIX

LIST OF TABLES
x
TABLE
NO.
TITLE PAGE
NO.
5.1 Simulation Parameters 52
5.2 Uncoded OFDM with Rayleigh fading in absence of PHN 54
5.3 Uncoded OFDM system with Rayleigh fading at different values
of PHN
56
5.4 Comparison table between R-S coded and uncoded OFDM system
at different values of phase noise
59
5.5 Comparison table between R-S coded OFDM and R-S coded with
LSA channel Estimated OFDM system
62

LIST OF FIGURES
x i
FIGURE
NO.
TITLE PAGE
NO.
2.1 Delayed Signals 11
2.2 Representation of a Symbol in a Frequency Selective Channel 11
2.3 Illustration of ISI 12
2.4 Representation of a Symbol in Flat Fading Channel 12
2.5 OFDM Splits a Data Stream into N Parallel Data Streams 13
2.6 Frequency spectrum of OFDM transmission 14
2.7 Carrier signals in an OFDM transmission 15
2.8 OFDM Transmitter 17
2.9 Serial to Parallel conversion 18
2.10 Parallel to Serial conversion 19
2.11 Guard period insertion in OFDM 20
2.12 OFDM Receiver 21
2.13 Constellation Diagram 25
2.14 Constellation Diagram for QPSK 28
2.15 Timing diagram for QPSK 30
3.1 Oscillator Phase Noise 35
3.2 Phase Noise 36
4.1 Channel Estimation 39
4.2 R-S System 43
4.3 R-S codeword 44
4.4 Architecture of a R-S (n – k) Encoder 47
4.5 Architecture of a R-S(n-k) Decoder 48
5.1 Uncoded OFDM System 53
5.2 BER vs. Eb/No plot of uncoded OFDM 54

5.3 Uncoded OFDM with PHN 55
5.4 BER performance curve of uncoded OFDM system at different
xi i
PHN
57
5.5 R-S coded OFDM system with PHN 58
5.6 Comparision curve between R-S coded and uncoded at PHN=
-70 dBc/Hz
60
5.7 R-S coded with LSA channel estimated OFDM system
61
5.8 Comparison curve between uncoded, R-S coded and , R-S
coded with LSA channel Estimated OFDM system at PHN=
-70 dBc/Hz
63

ABBREVIATIONS
ADSL Asymmetric Digital Subscriber Line
ADC Analog to Digital Converter
BER Bit Error Rate
BPSK Binary Phase Shift Keying
CP Cyclic Prefix
CIR Carrier to Interference Power Ratio
CPE Common Phase Error
CDMA Code Division Multiple Access
DAB Digital Audio Broadcast
DVB-T Digital Video Broadcasting-Terrestrial
DAC Digital To Analog Converter
DSP Digital Signal Processing
DFT Discrete Fourier Transform
DUT Device Under Test
EDGE Enhanced Data Rates for Global Evolution
FFT Fast Fourier Transform
FDM Frequency Division Multiplexing
GMSK Gaussian Minimum Shift Keying
GSM Global System for Mobile Communication
GPRS General packet Radio Service
HDSL High speed Digital Subscriber Line
HDTV High Definition Television
xi ii

ICI Inter Carrier Interference
ISI Inter Symbol Interference
IFFT Inverse Fast Fourier Transform
IEEE Institute for Electrical and Electronic Engineers.
IDFT Inverse Discrete Fourier Transform
LSA Least Square With Averaging
MCM Multi Carrier Modulation
MC Multicarrier Communication
NTT Nippon Telephone and Telegraph
OFDM Orthogonal Frequency Division Multiplexing
PSK Phase Shift Keying
PHN Phase Noise
PSD Power Spectral Density
QPSK Quadrature Phase Shift Keying
QAM Quadrature Amplitude Modulation
R&D Research and Development
R-S OR RS Reed Solomon
SNR Signal to Noise Ratio
SIR Signal to Interference Ratio
TACS Total Access Communications system
TDMA Time Division Multiple Access
UMTS Universal Mobile Telecommunication System
VLSI Very Large Scale Integration
WLAN Wireless Local Area Network
xi v

SYMBOLS & NOTATIONS
xv i
Ts- Symbol Period
Td- Delay Spread
Bc- Coherence Bandwidth
Bs- Symbol Bandwidth
M- number of points in the constellation

CHAPTER 1
INTRODUCTION
1.1 INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is successfully used in various
applications, such as European digital audio broadcasting and digital video broadcasting
systems [1,2]. In 1999, the IEEE 802.11a working group chose OFDM for their 5-GHz
band wireless local area network (WLAN) standard, which supports a variable bit rate
from 6 to 54 Mbps. OFDM was also one of the promising candidates for the European
third-generation personal communications system (universal mobile telecommunication
system). However, it was not approved since the code division multiple access(CDMA)
based proposals received more support. OFDM is now being considered for the fourth-generation
mobile communication systems [3]. Therefore, OFDM’s performance in
mobile and fading environments is the topic of many current studies.
Orthogonal Frequency Division Multiplexing (OFDM) is a special form of multi carrier
modulation technique which is used to generate waveforms that are mutually orthogonal.
In an OFDM scheme, a large number of orthogonal, overlapping, narrow band sub-carriers
are transmitted in parallel. These carriers divide the available transmission
bandwidth. The separation of the sub-carriers is such that there is a very compact spectral
utilization. With OFDM, it is possible to have overlapping sub channels in the frequency
domain, thus increasing the transmission rate. In order to avoid a large number of
modulators and filters at the transmitter and complementary filters and demodulators at
the receiver, it is desirable to be able to use modern digital signal processing techniques,
such as fast Fourier transform (FFT). After more than forty years of research and
development carried out in different places, OFDM is now being widely implemented in
high-speed digital communications. OFDM has been accepted as standard in several wire
line and wireless applications. Due to the recent advancements in digital signal
processing (DSP) and very large-scale integrated circuits (VLSI) technologies, the initial
obstacles of OFDM implementations do not exist anymore. In a basic communication

system, the data are modulated onto a single carrier frequency. The available bandwidth
is then totally occupied by each symbol. This kind of system can lead to inter-symbol-interference
(ISI) in case of frequency selective channel. The basic idea of OFDM is to
divide the available spectrum into several orthogonal sub channels so that each
narrowband sub channels experiences almost flat fading. The attraction of OFDM is
mainly because of its way of handling the multipath interference at the receiver.
Multipath phenomenon generates two effects
(a) Frequency selective fading and
(b) Intersymbol interference (ISI).
The "flatness" perceived by a narrowband channel overcomes the frequency selective
fading. On the other hand, modulating symbols at a very low rate makes the symbols
much longer than channel impulse response and hence reduces the ISI. Use of suitable
error correcting codes provides more robustness against frequency selective fading. The
insertion of an extra guard interval between consecutive OFDM symbols can reduce the
effects of ISI even more. The use of FFT technique to implement modulation and
demodulation functions makes it computationally more efficient. OFDM systems have
gained an increased interest during the last years. It is used in the European digital
broadcast radio system, as well as in wired environment such as asymmetric digital
subscriber lines (ADSL). This technique is used in digital subscriber lines (DSL) to
provides high bit rate over a twisted-pair of wires.
1.2 HISTORY OF MOBILE WIRELESS COMMUNICATIONS
The history of mobile communication [4,5] can be categorized into 3 periods:
(1) The pioneer era
(2) The pre-cellular era
(3) The cellular era In the pioneer era,
A great deal of the fundamental research and development in the field of wireless
communications took place. The postulates of electromagnetic (EM) waves by James
Clark Maxwell during the 1860s in England, the demonstration of the existence of these
2

waves by Heinrich Rudolf Hertz in 1880s in Germany and the invention and first
demonstration of wireless telegraphy by Guglielmo Marconi during the 1890s in Italy
were representative examples from Europe. Moreover, in Japan, the Radio Telegraph
Research Division was established as a part of the Electro technical Laboratory at the
Ministry of Communications and started to research wireless telegraph in 1896. From the
fundamental research and the resultant developments in wireless telegraphy, the
application of wireless telegraphy to mobile communication systems started from the
1920s. This period, which is called the pre-cellular era, began with the first land-based
mobile wireless telephone system installed in 1921 by the Detroit Police Department to
dispatch patrol cars, followed in 1932 by the New York City Police Department. These
systems were operated in the 2MHz frequency band. In 1946, the first commercial mobile
telephone system, operated in the 150MHz frequency band, was set up by Bell Telephone
Laboratories in St. Louis. The demonstration system was a simple analog communication
system with a manually operated telephone exchange. Subsequently, in 1969, a mobile
duplex communication system was realized in the 450MHz frequency band. The
telephone exchange of this modified system was operated automatically. The new
system, called the Improved Mobile Telephone System (IMTS), was widely installed in
the United States. However, because of its large coverage area, the system could not
manage a large number of users or allocate the available frequency bands efficiently.
The cellular zone concept was developed to overcome this problem by using the
propagation characteristics of radio waves. The cellular zone concept divided a large
coverage area into many smaller zones. A frequency channel in one cellular zone is used
in another cellular zone. However, the distance between the cellular zones that use the
same frequency channels is sufficiently long to ensure that the probability of interference
is quite low. The use of the new cellular zone concept launched the third era, known as
the cellular era. So far, the evolution of the analog cellular mobile communication system
is described. There were many problems and issues, for example, the incompatibility of
the various systems in each country or region, which precluded roaming. In addition,
analog mobile communication systems were unable to ensure sufficient capacity for the
increasing number of users, and the speech quality was not good. To solve these
problems, the R&D of cellular mobile communication systems based on digital radio
3

transmission schemes was initiated. These new mobile communication systems became
known as the second generation (2G) of mobile communication systems, and the analog
cellular era is regarded as the first generation (1G) of mobile communication systems
[6,7].
1G analog cellular systems were actually a hybrid of analog voice channels and digital
control channels. The analog voice channels typically used Frequency Modulation (FM)
and the digital control channels used simple Frequency Shift keying (FSK) modulation.
The first commercial analog cellular systems include Nippon Telephone and Telegraph
(NTT) Cellular – Japan, Advanced Mobile Phone Service (AMPS) – US, Australia,
China, Southeast Asia, Total Access Communications system (TACS) - UK, and Nordic
Mobile Telephone (NMT) – Norway, Europe.
2G digital systems use digital radio channels for both voice (digital voice) and digital
control channels. 2G digital systems typically use more efficient modulation
technologies, including Global System for Mobile communications (GSM), which uses a
standard 2-level Gaussian Minimum Shift Keying (GMSK). Digital radio channels offer a
universal data transmission system, which can be divided into many logical channels that
can perform different services. 2G also uses multiple access (or multiplexing)
technologies to allow more customers to share individual radio channels or use narrow
channels to allow more radio channels into a limited amount of radio spectrum band.
The 3 basic types of access technologies used in 2G are:
(1) Frequency division multiple access (FDMA)
(2) Time division multiple access (TDMA)
(3) Code division multiple access (CDMA)
The technologies either reduce the RF channel bandwidth (FDMA), share a radio channel
by assigning users to brief time slot (TDMA), or divide a wide RF channel into many
different coded channels (CDMA). Improvements in modulation techniques and multiple
access technologies amongst other technologies inadvertently led to 2.5G and 3G. For
example, EDGE can achieve max 474 kbps by using 8-PSK with the existing GMSK.
This is 3x more data transfer than GPRS.
4

1.3 GENERATIONS OF TELECOMMUNICATION
First Generation (1G) is described as the early analogue cellular phone technologies. 1G
analog cellular systems were actually a hybrid of analog voice channels and digital
control channels. The analog voice channels typically used Frequency Modulation (FM)
and the digital control channels used simple Frequency Shift keying (FSK) modulation.
The first commercial analog cellular systems include Nippon Telephone and Telegraph
(NTT) Cellular – Japan, Advanced Mobile Phone Service (AMPS) – US, Australia,
China, Southeast Asia, Total Access Communications system (TACS) - UK, and Nordic
Mobile Telephone (NMT) – Norway, Europe. NMT and AMPS cellular technologies fall
under this categories.
Second Generation (2G) described as the generation first digital fidely used cellular
phones systems. 2G digital systems use digital radio channels for both voice (digital
voice) and digital control channels. GSM technology is the most widely used 2G
technologies. 2G digital systems typically use more efficient modulation technologies,
including Global System for Mobile communications (GSM), which uses a standard 2-
level Gaussian Minimum Shift Keying (GMSK). This gives digital speech and some
limited data capabilities (circuit switched 9.6kbits/s). Other 2G technologies are IS-95
CDMA, IS-136 TDMA and PDC. 2G also uses multiple access (or multiplexing)
technologies to allow more customers to share individual radio channels or use narrow
channels to allow more radio channels into a limited amount of radio spectrum band. The
3 basic types of access technologies used in 2G are: frequency division multiple access
(FDMA), time division multiple access (TDMA), and code division multiple access
(CDMA). The technologies either reduce the RF channel bandwidth (FDMA), share a
radio channel by assigning users to brief timeslot (TDMA), or divide a wide RF channel
into many different coded channels (CDMA).
Two and Half Generation (2.5G) is an enhanced version of 2G technology. 2.5G gives
higher data rate and packet data services. GSM systems enhancements like GPRS and
EDGE are considered to be in 2.5G technology. The so-called 2.5G technology represent
an intermediate upgrade in data rates available to mobile users.
5

Third Generation (3G) mobile communication systems often called with names 3G,
UMTS and WCDMA promise to boost the mobile communications to the new speed
limits. The promises of third generation mobile phones are fast Internet surfing advanced
value-added services and video telephony. Third-generation wireless systems will handle
services up to 384 kbps in wide area applications and up to 2 Mbps for indoor
applications.
Fourth Generation (4G) is intended to provide high speed, high capacity, low cost per bit,
IP based services. The goal is to have data rates up to 20 Mbps. Most probable the 4G
network would be a network which is a combination of different technologies, for
example, current cellular networks, 3G cellular network and wireless LAN, working
together using suitable interoperability protocols.
1.4 MOTIVATION
OFDM is robust in adverse channel conditions and allows a high level of spectral
efficiency. Multiple access techniques which are quite developed for the single carrier
modulations (e.g. TDMA, FDMA) had made possible of sharing one communication
medium by multiple number of users simultaneously. The sharing is required to achieve
high capacity by simultaneously allocating the available bandwidth to multiple users
without severe degradation in the performance of the system. FDMA and TDMA are the
well known multiplexing techniques used in wireless communication systems.
While working with the wireless systems using these techniques, various problems
encountered are
(1) Multi-path fading
(2) Time dispersion which lead ISI
(3) Lower bit rate capacity
(4) Requirement of larger transmit power for high bit rate and
(5) Less spectral efficiency
Disadvantage of FDMA technique is its Bad Spectrum Usage. Disadvantages of TDMA
technique is Multipath Delay spread problem. In a typical terrestrial broadcasting, the
6

transmitted signal arrives at the receiver using various paths of different lengths. Since
multiple versions of the signal interfere with each other, it becomes difficult to extract the
original information.
Orthogonal Frequency Division Multiplexing (OFDM) has recently gained fair degree of
prominence among modulation schemes due to its intrinsic robustness to frequency
selective Multipath fading channels. OFDM system also provides higher spectrum
efficiency and supports high data rate transmission. This is one of the main reasons to
select OFDM a candidate for systems such as Digital Audio Broadcasting (DAB), Digital
Video Broadcasting (DVB), Digital Subscriber Lines (DSL), and Wireless local area
networks (HiperLAN/2), and in IEEE 802.11a, IEEE 802.11g. The focus of future fourth-generation
(4G) mobile systems is on supporting high data rate services such as
deployment of multi-media applications which involve voice, data, pictures, and video
over the wireless networks. At this moment, the data rate envisioned for 4G networks is 1
GB/s for indoor and 100Mb/s for outdoor environments.Orthogonal frequency division
multiplexing (OFDM) is a promising candidate for 4G systems because of its robustness
to the multipath environment.
1.5 RELATED RESEARCH
Due to its many attractive features, OFDM has received much attention in the wireless
communications research communities. Numerous studies have been performed to
investigate its performance and applicability to many different environments. Below are
some of the many studies conducted concerning the effect of frequency errors and Phase
Noise on OFDM systems.
Weinstein and Ebert proposed a modified OFDM system [8] in which the discrete Fourier
Transform (DFT) was applied to generate the orthogonal subcarriers waveforms instead
of the banks of sinusoidal generators. Their scheme reduced the implementation
complexity significantly, by making use of the inverse DFT (IDFT) modules and the
digital-to-analog converters. In their proposed model, baseband signals were modulated
by the IDFT in the transmitter and then demodulated by DFT in the receiver. Therefore,
7

all the subcarriers were overlapped with others in the frequency domain, while the DFT
modulation still assures their orthogonality.
Cyclic prefix (CP) or cyclic extension was first introduced by Peled and Ruiz in 1980 [9]
for OFDM systems. In their scheme, conventional null guard interval is substituted by
cyclic extension for fully-loaded OFDM modulation. As a result, the orthogonality
among the subcarriers was guaranteed. With the trade-off of the transmitting energy
efficiency, this new scheme can result in a phenomenal ISI (Inter Symbol Interference)
reduction. Hence it has been adopted by the current IEEE standards. In 1980, Hirosaki
introduced an equalization algorithm to suppress both inter symbol interference (ISI) and
ICI [10], which may have resulted from a channel distortion, synchronization error, or
phase error. In the meantime, Hirosaki also applied QAM modulation, pilot tone, and
trellis coding techniques in his high-speed OFDM system, which operated in voice-band
spectrum.
Many of the published studies about the frequency errors use two main references.The
first is the study of Pollet on sensitivity of OFDM systems to frequency offset and
Wiener phase noise [11], and the second is the study of Moose on a technique for OFDM
frequency offset correction [12].
Other related studies include the study of Armada on the phase noise and subcarrier
spacing effects on OFDM system’s performance [13], the study of Xiong about the effect
of Doppler frequency shift, frequency offset, and phase noise on OFDM receiver’s
performance [14] and the study of Zhao on the sensitivity of OFDM systems to Doppler
shift and carrier frequency errors [15].
Other related studies include the study of Mohammad Reza Gholami on the phase noise.
In his paper [16] he discussed about the LS Filter approach to suppress phase noise in
OFDM system.
Other related studies include the study of Ana Garcia Armada on the Phase Noise. In the
paper [17] Author Analyzes the performance of OFDM system under phase noise and
its dependence on the no of sub-carriers both in the presence and absence of a phase
correction mechanism.
8

1.6 OBJECTIVE AND OUTLINE OF THESIS
The main objective of this thesis is to compensate the effects of phase noise in OFDM
based communication system and enhanced the performance of the system in terms of
bit error rate (BER) by using R-S coding with LSA channel estimation technique. Some
other objectives are
(1) To analysis the BER Performance of Uncoded OFDM System without considering
phase noise.
(2) To analysis the BER Performance of Uncoded OFDM System at different values of
phase noise.
(3) To analysis the Comparison between Uncoded OFDM and R-S Coded OFDM
System at different values of phase noise .
(4) To analysis the Comparison between R-S Coded OFDM and R-S coded with LSA
Channel Estimated OFDM System at different values of phase noise.
This report is organized as follows:
In Chapter 2, the basics of OFDM, its transmitter and receiver,its advantages and
application are discussed. Digital modulation, quadrature phase-shift keying ,radio
propagation,rayleigh fading and doppler shift are also present in this chapter.
In Chapter 3, phase noise problem in OFDM based communication system is discussed.
Its theortical analysis is also present in this chapter.
In Chapter 4, Reed-Solomon coding and decoding process, least square with averaging
channel estimation technique is discussed.
In Chapter 5, simulation parameters and steps, Simulation results is discussed. Differents
simulink models of OFDM based communication system and results in tabular as well as
graphical form is also present in this chapter .
In Chapter 6, conclude the report and future works are also outline.
9

CHAPTER 2
ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING
2.1 INTRODUCTION
The rapid growth of the applications utilizing digital communication systems increased
the need for high-speed data transmission. New multi-carrier modulation techniques are
being proposed and implemented to keep up with the demand of higher data rates. Of
these multi-carrier techniques, OFDM is the method of choice for high-speed
communication due to its many attractive features. This chapter attempts to justify the
choice of OFDM among other communication techniques.
Orthogonal Frequency Division Multiplexing (OFDM) is a multicarrier transmission
technique, which divides the bandwidth into many carriers, each one is modulated by a
low rate data stream [18, 19]. In term of multiple access technique, OFDM is similar to
FDMA in that the multiple user access is achieved by subdividing the available
bandwidth into multiple channels that are then allocated to users. However, OFDM uses
the spectrum much more efficiently by spacing the channels much closer together. This is
achieved by making all the carriers orthogonal to one another, preventing interference
between the closely spaced carriers.
2.2 FUNDAMENTALS OF OFDM
2.2.1 Multi-path ( Delay-spread or time dispersion )
In general, high data rate means short symbol time compared to the delay spread
(TSYMBOL<TDELAY) . Delay-spread greatly affects the communication system and the
signal might not be recovered at the receiver.
This section addresses the effects of delay spread which occurs as the surfaces between a
transmitter and a receiver reflect a transmitted signal. The receiver obtains the transmitted
signals with random phase offsets and this causes random signal fades as reflected signals
destructively or constructively affect each other [20], as seen in Figure (2.1).
1 0

Figure (2.1)-Delayed Signals [21]
When TSYMBOL< TDELAY (BC<BS) as in Figure (2.2), the signal faces frequency selective
fading and this causes time dispersion. The effect of this is intersymbol interference (ISI),
where the energy of one symbol leaks into another symbol, as can be viewed from Figure
(2.3). As a result, the bit error rate (BER) increases, this in turn degrades the
performance. ISI is one of the biggest problems of digital communication and OFDM
deals with this problem very effectively.
(a) (b)
Figure (2.2)-Representation of a Symbol in a Frequency Selective Channel
1 1

(a) Time domain (b) Frequency domain
Figure (2.3)-Illustration of ISI [22]
A way to deal with frequency selective fading is to decrease the data rate and thus change
the frequency selective fading to flat fading. The desired scheme is illustrated in Figure
(2.4). OFDM systems mitigate the ISI by changing the frequency selective fading channel
to flat fading channel as discussed below
(a) (b)
1 2

Figure (2.4)-(a) Time Domain Representation, (b) Frequency Domain
Representation of a Symbol in Flat Fading Channel.
OFDM modulates user data onto tones by using either phase shift keying (PSK) or
quadrature amplitude modulation (QAM). An OFDM system takes a high data rate
stream, splits it into N parallel data streams and transmits them simultaneously. As can be
observed from Figure (2.5), each of these parallel data streams has a rate of R N, where R
is the original data rate. The data streams are modulated by different carriers and
combined together by inverse fast Fourier transform (IFFT) to generate the time-domain
signal to be transmitted [20]
Figure(2.5)-OFDM Splits Data Stream into N Parallel Data Streams[23]
By creating a slower data stream, the symbol duration becomes larger than the channel’s
impulse response. In this way, each carrier is subject to flat fading
2.2.2 Orthogonality
OFDM is simply defined as a form of multi-carrier modulation where the carrier spacing
is carefully selected so that each sub carrier is orthogonal to the other sub carriers. Two
signals are orthogonal if their dot product is zero. That is, if you take two signals multiply
them together and if their integral over an interval is zero, then two signals are orthogonal
1 3

in that interval. Orthogonality can be achieved by carefully selecting carrier spacing, such
as letting the carrier spacing be equal to the reciprocal of the useful symbol period. As the
sub carriers are orthogonal, the spectrum of each carrier has a null at the centre frequency
of each of the other carriers in the system. This results in no interference between the
carriers, allowing them to be spaced as close as theoretically possible. Mathematically,
suppose we have a set of signals ψ then
1 4
(2.1)
The signals are orthogonal if the integral value is zero over the interval [a a+T], where T
is the symbol period. Since the carriers are orthogonal to each other the nulls of one
carrier coincides with the peak of another sub carrier. As a result it is possible to extract
the sub carrier of interest.
Figure (2.6)-Frequency spectrum of OFDM transmission
OFDM transmits a large number of narrowband sub channels. The frequency range
between carriers is carefully chosen in order to make them orthogonal each another. In

fact, the carriers are separated by an interval of 1/T, where T represents the duration of an
OFDM symbol. The frequency spectrum of an OFDM transmission is illustrated in
Figure (2.6). This Figure indicates the spectrum of carriers significantly over laps over
the other carrier. This is contrary to the traditional FDM technique in which a guard band
is provided between each carrier. Each sinc of the frequency spectrum in the Figure (2.6)
corresponds to a sinusoidal carrier modulated by a rectangular waveform representing the
information symbol.
Figure (2.7)-Carrier signals in an OFDM transmission
It is easily notice that the frequency spectrum of one carrier exhibits zero-crossing at
central frequencies corresponding to all other carriers. At these frequencies, the
intercarrier interference is eliminated, although the individual spectra of subcarriers
overlap. It is well known that orthogonal signals can be separated at the receiver by
correlation techniques. The receiver acts as a bank of demodulators, translating each
carrier down to baseband, the resulting signal then being integrated over a symbol period
to recover the data. If the other carriers beat down to frequencies which, in the time
domain means an integer number of cycles per symbol period (T), then the integration
1 5

process results in a zero contribution from all these carriers. The waveforms of some of
the carriers in an OFDM transmission are illustrated in Figure (2.7).
2.3 INTERSYMBOL AND INTERCARRIER INTERFERENCE
In a multipath environment, a transmitted symbol takes different times to reach the
receiver through different propagation paths. From the receiver‘s point of view, the
channel introduces time dispersion in which the duration of the received symbol is
stretched. Extending the symbol duration causes the current received symbol to overlap
previous received symbols and results in intersymbol interference (ISI).
In OFDM, ISI usually refers to interference of an OFDM symbol by previous OFDM
symbols. For a given system bandwidth the symbol rate for an OFDM signal is much
lower than a single carrier transmission scheme. For example for a single carrier BPSK
modulation, the symbol rate corresponds to the bit rate of the transmission. However for
OFDM the system bandwidth is broken up into N subcarriers, resulting in a symbol rate
that is N times lower than the single carrier transmission. This low symbol rate makes
OFDM naturally resistant to effects of Inter-Symbol Interference (ISI) caused by
multipath propagation. Multipath propagation is caused by the radio transmission signal
reflecting off objects in the propagation environment, such as walls, buildings,
mountains, etc. These multiple signals arrive at the receiver at different times due to the
transmission distances being different. This spreads the symbol boundaries causing
energy leakage between them.
In OFDM, the spectra of subcarriers overlap but remain orthogonal to each other. This
means that at the maximum of each sub-carrier spectrum, all the spectra of other
subcarriers are zero. The receiver samples data symbols on individual sub-carriers at the
maximum points and demodulates them free from any interference from the other
subcarriers. Interference caused by data symbols on adjacent sub-carriers is referred to
intercarrier interference (ICI).
The orthogonality of subcarriers can be viewed in either the time domain or in frequency
domain. From the time domain perspective, each subcarrier is a sinusoid with an integer
number of cycles within one FFT interval. From the frequency domain perspective, this
1 6

corresponds to each subcarrier having the maximum value at its own center frequency
and zero at the center frequency of each of the other subcarriers. The orthogonality of a
subcarrier with respect to other subcarriers is lost if the subcarrier has nonzero spectral
value at other subcarrier frequencies. From the time domain perspective, the
corresponding sinusoid no longer has an integer number of cycles within the FFT
interval. ICI occurs when the multipath channel varies over one OFDM symbol time.
When this happens, the Doppler shift on each multipath component causes a frequency
offset on the subcarriers, resulting in the loss of orthogonality among them.This situation
can be viewed from the time domain perspective, in which the integer number of cycles
for each subcarrier within the FFT interval of the current symbol is no longer maintained
due to the phase transition introduced by the previous symbol. Finally, any offset
between the subcarrier frequencies of the transmitter and receiver also introduces ICI to
an OFDM symbol.
2.4 OFDM TRANSMITTER
A block diagram of the OFDM transmitter module is presented in Figure (2.8). Each of
the blocks is explained in detail in the following subsections.
Figure (2.8)-OFDM Transmitter
2.4.1 Channel Coding
A sequential binary input data stream is first encoded by the channel coder. Error
correction coding is important for OFDM systems used for mobile communications.
1 7

When channel coding is used to improve its performance, OFDM is referred to as coded
OFDM (COFDM).
2.4.2 Signal Mapping
A large number of modulation schemes are available allowing the number of bits
transmitted per carrier per symbol to be varied. Digital data is transferred in an OFDM
link by using a modulation scheme on each subcarrier. A modulation scheme is a
mapping of data words to a real (In phase) and imaginary (Quadrature) constellation, also
known as an IQ constellation. For example 256-QAM (Quadrature Amplitude
Modulation) has 256 IQ points in the constellation constructed in a square with 16 evenly
spaced columns in the real axis and 16 rows in the imaginary axis.
The number of bits that can be transferred using a single symbol corresponds to
where M is the number of points in the constellation, thus 256-QAM transfers
8 bits per symbol. Increasing the number of points in the constellation does not change
the bandwidth of the transmission, thus using a modulation scheme with a large number
of constellation points, allows for improved spectral efficiency. For example 256-QAM
has a spectral efficiency of 8 b/s/Hz, compared with only 1 b/s/Hz for BPSK. However,
the greater the number of points in the modulation constellation, the harder they are to
resolve at the receiver.
2.4.3 Serial to Parallel and Prallel to Serial conversion
1 8

Figure (2.9)-Serial to Parallel conversion
Data to be transmitted is typically in the form of a serial data stream. In OFDM, each
symbol transmits a number of bits and so a serial to parallel conversion stage is needed to
convert the input serial bit stream to the data to be transmitted in each OFDM symbol.
The data allocated to each symbol depends on the modulation scheme used and the
number of subcarriers. At the receiver the reverse process takes place, with the data from
the subcarriers being converted back to the original serial data stream.
1 9

Figure (2.10)-Parallel to Serial conversion
2.4.4 Inverse Fast Fourier Transform
The OFDM message is generated in the complex baseband. Each symbol is modulated
onto the corresponding subcarrier using variants of phase shift keying (PSK) or different
forms of quadrature amplitude modulation (QAM).The data symbols are converted from
serial to parallel before data transmission. The frequency spacing between adjacent
subcarriers is Nπ/2, where N is the number of subcarriers. This can be achieved by using
the inverse discrete Fourier transform (IDFT), easily implemented as the inverse fast
Fourier transform (IFFT) operation [26].
The OFDM baseband sub-carrier is
2 0
(2.3)
Where ݂௞ is the ݇௧௛ sub-carrier frequency An OFDM symbol consists of N modulated
sub-carriers. The OFDM signal not including a cyclic prefix is given by [24]
(2.4)
Where is the complex data symbol and NT is the OFDM symbol duration. The
sub-carriers in Eq. (2.3) and (2.4) have frequencies
(2.5)
In the sense that ensures orthogonality
(2.6)

If the signal s (t) is sampled with a sampling period of T, the following is obtained:
2 1
(2.7)
This Eq. (2.7) is IDFT { } and was proposed by [25]. As can be seen from Eq. (2.7), a
baseband OFDM transmission symbol is an N-point complex modulation sequence. It is
composed of N complex sinusoids, which are modulated with z (k)
2.4.5 Guard Period
The effect of ISI on an OFDM signal can be reduced by the addition of a guard period to
the start of each symbol. This guard period is a cyclic copy that extends the length of the
symbol waveform. Each subcarrier, in the data section of the symbol, (i.e. the OFDM
symbol with no guard period added, which is equal to the length of the IFFT size used to
generate the signal) has an integer number of cycles.
Figure (2.11)-Guard period insertion in OFDM
Figure (2.11) shows the insertion of a guard period. The total length of the symbol is TS=
TG+TFFT, where TS is the total length of the symbol in samples, TG is the length of the
guard period in samples, and TFFT is the size of the IFFT used to generate the OFDM
signal. In addition to protecting the OFDM from ISI, the guard period also provides
protection against time-offset errors in the receiver.
A Guard time is introduced at the end of each OFDM symbol in form of cyclic prefix to
prevent Inter Symbol Interference (ISI).

The Guard time is cyclically extended to avoid Inter-Carrier Interference (ICI) - integer
number of cycles in the symbol interval. Guard Time > Multipath Delay Spread, to
guarantee zero ISI & ICI.
2.5 OFDM RECEIVER
A block diagram of the OFDM RECEIVER module is presented in Figure (2.12).
Figure (2.12)-OFDM Receiver
2.5.1 Removing Guard Interval and FFT Processing
At the OFDM receiver end, the first step is to remove the guard interval to obtain the
information portion of the symbol for further processing. Next, the time domain samples
are transformed into the frequency domain by the FFT process. This also makes it
possible to recover the OFDM frequency tones.
2.5.2 Decoding
The next step in the receiver is the time or frequency differential decoding. Following the
differential decoding, the inverse mapping of each received complex modulation value
into a corresponding N-ary symbol is accomplished.
2.6 ADVANTAGES OF OFDM
2 2

(1) OFDM Is less sensitive to sample timing offsets than single carrier systems.
(2) It Provides good protection against co channel interference and impulsive
2 3
parasitic noise.
(3) Eliminates ISI through use of a cyclic prefix.
(4) By dividing the channel into narrowband flat fading sub channels, OFDM is more
resistant to frequency selective fading than single carrier systems are. i.e.
robustness to frequency selective fading channels.
(5) Channel equalization becomes simpler than by using adaptive equalization
techniques with single carrier systems.
(6) Using adequate channel coding and interleaving one can recover symbols lost due
to the frequency selectivity of the channel.
(7) It is possible to use maximum likelihood decoding with reasonable complexity.
(8) OFDM is computationally efficient by using FFT techniques to implement the
modulation and demodulation functions.
2.7 APPLICATIONS OF OFDM
(1) OFDM is used in European Wireless LAN Standard – HiperLAN/2.
(2) OFDM is used in IEEE 802.11a and 802.11g Wireless LANs.
(3) OFDM is used in IEEE 802.16 or WiMax Wireless MAN standard.
(4) OFDM is used in IEEE 802.20 or Mobile Broadband Wireless Access (MBWA)
standard.
(5) OFDM is used in Digital Audio Broadcasting (DAB).
(6) OFDM is used in Digital Video Broadcasting (DVB) & HDTV.
(7) OFDM is used in Used for wideband data communications over mobile radio
channels such as
(7.1) High-bit-rate Digital Subscriber Lines (HDSL at 1.6Mbps).
(7.2) Asymmetric Digital Subscriber Lines (ADSL up to 6Mbps).
(7.3) Very-high-speed Digital Subscriber Lines (VDSL at 100 Mbps).
(7.4) ADSL and broadband access via telephone network copper wires.
(8) OFDM is used in Point-to-point and point-to-multipoint wireless applications .

(9) OFDM is under consideration for use in 4G Wireless systems.
2.7 MODULATION
In communication, modulation is the process of varying a periodic waveform, in order to
use that signal to convey a message over a medium. Normally a high frequency
waveform is used as a carrier signal. The three key parameters of a sine wave are
frequency, amplitude, and phase, all of which can be modified in accordance with a low
frequency information signal to obtain a modulated signal. There are 2 types of
modulations
2 4
(1) Analog modulation.
(2) Digital modulation.
In analog modulation, an information-bearing analog waveform is impressed on the
carrier signal for transmission whereas in digital modulation, an information-bearing
discrete-time symbol sequence (digital signal) is converted or impressed onto a
continuous-time carrier waveform for transmission.
2.8.1 Digital Modulation
Nowadays, digital modulation is much popular compared to analog modulation. The
move to digital modulation provides more information capacity, compatibility with
digital data services, higher data security, better quality communications, and quicker
system availability. The aim of digital modulation is to transfer a digital bit stream over
an analog band pass channel or a radio frequency band. The changes in the carrier signal
are chosen from a finite number of alternative symbols. Digital modulation schemes have
greater capacity to convey large amounts of information than analog modulation
schemes. There are three major classes of digital modulation techniques used for
transmission of digitally represented data
(1) Amplitude-shift Keying (ASK).
(2) Frequency-shift keying (FSK).
(3) Phase-shift keying (PSK).

All convey data by changing some aspect of a base-band signal, the carrier wave, (usually
a sinusoid) in response to a data signal. In the case of PSK, the phase is changed to
represent the data signal. There are two fundamental ways of utilizing the phase of a
signal in this way
(1) By viewing the phase itself as conveying the information, in which case the
demodulator must have a reference signal to compare the received signal's phase
against or
(2) By viewing the change in the phase as conveying information — differential
schemes, some of which do not need a reference carrier (to a certain extent)
A convenient way to represent PSK schemes is on a constellation diagram. This shows
the points in the Argand plane where, in this context, the real and imaginary axes are
termed the in-phase and quadrature axes respectively due to their 90° separation. Such a
representation on perpendicular axes lends itself to straightforward implementation. The
amplitude of each point along the in-phase axis is used to modulate a cosine (or sine)
wave and the amplitude along the quadrature axis to modulate a sine (or cosine) wave.
2.9 PHASE SHIFT KEYING (PSK)
PSK is a modulation scheme that conveys data by changing, or modulating, the phase of
a reference signal (i.e. the phase of the carrier wave is changed to represent the data
signal) [27]. A finite number of phases are used to represent digital data. Each of these
phases is assigned a unique pattern of binary bits; usually each phase encodes an equal
number of bits. Each pattern of bits forms the symbol that is represented by the particular
phase.
A convenient way to represent PSK schemes is on a constellation diagram (as shown in
figure (2.13) below). This shows the points in the Argand plane where, in this context,
the real and imaginary axes are termed the in-phase and quadrature axes respectively due
to their 90° separation. Such a representation on perpendicular axes lends itself to
straightforward implementation. The amplitude of each point along the in-phase axis is
used to modulate a cosine (or sine) wave and the amplitude along the quadrature axis to
modulate a sine (or cosine) wave.
2 5

Figure (2.13)-Constellation Diagram
In PSK, the constellation points chosen are usually positioned with uniform angular
spacing around a circle. This gives maximum phase-separation between adjacent points
and thus the best immunity to corruption. They are positioned on a circle so that they can
all be transmitted with the same energy. In this way, the moduli of the complex numbers
they represent will be the same and thus so will the amplitudes needed for the cosine and
sine waves. Two common examples are binary phase-shift keying (BPSK) which uses
two phases, and quadrature phase-shift keying (QPSK) which uses four phases, although
any number of phases may be used. Since the data to be conveyed are usually binary, the
PSK scheme is usually designed with the number of constellation points being a power of
2. Notably absent from these various schemes is 8-PSK. This is because its error-rate
performance is close to that of 16-QAM it is only about 0.5 dB better but its data rate is
only three-quarters that of 16-QAM. Thus 8-PSK is often omitted from standards and, as
seen above, schemes tend to 'jump' from QPSK to 16-QAM (8-QAM is possible but
difficult to implement).
Any digital modulation scheme uses a finite number of distinct signals to represent digital
data. PSK uses a finite number of phases,each assigned a unique pattern of binary bits.
2 6

Usually, each phase encodes an equal number of bits. Each pattern of bits forms the
symbol that is represented by the particular phase. The demodulator, which is designed
specifically for the symbol set used by the modulator, determines the phase of the
received signal and maps it back to the symbol it represents, thus recovering the original
data. This requires the receiver to be able to compare the phase of the received signal to a
reference signal such a system is termed coherent (and referred to as CPSK).
Alternatively, instead of using the bit patterns to set the phase of the wave, it can instead
be used to change it by a specified amount. The demodulator then determines the
changes in the phase of the received signal rather than the phase itself. Since this scheme
depends on the difference between successive phases, it is termed differential phase-shift
keying (DPSK). DPSK can be significantly simpler to implement than ordinary PSK
since there is no need for the demodulator to have a copy of the reference signal to
determine the exact phase of the received signal (it is a non-coherent scheme). In
exchange, it produces more erroneous demodulations. The exact requirements of the
particular scenario under consideration determine which scheme is used.
 Applications of PSK
Owing to PSK's simplicity, particularly when compared with its competitor quadrature
amplitude modulation, it is widely used in existing technologies.
The wireless LAN standard, IEEE 802.11b-1999, uses a variety of different PSKs
depending on the data-rate required. At the basic-rate of 1 Mbit/s, it uses DBPSK
(differential BPSK). To provide the extended-rate of 2 Mbit/s, DQPSK is used. In
reaching 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK is employed, but has to be
coupled with complementary code keying. The higher-speed wireless LAN standard,
IEEE 802.11g-2003 has eight data rates: 6, 9, 12, 18, 24, 36, 48 and 54 Mbit/s. The 6 and
9 Mbit/s modes use OFDM modulation where each sub-carrier is BPSK modulated. The
12 and 18 Mbit/s modes use OFDM with QPSK. The fastest four modes use OFDM with
forms of quadrature amplitude modulation.
Because of its simplicity BPSK is appropriate for low-cost passive transmitters, and is
used in RFID standards such as ISO/IEC 14443 which has been adopted for biometric
2 7

passports, credit cards such as American Express's ExpressPay, and many other
applications. IEEE 802.15.4 (the wireless standard used by ZigBee) also relies on PSK.
IEEE 802.15.4 allows the use of two frequency bands: 868–915 MHz using BPSK and at
2.4 GHz using OQPSK.
For determining error-rates mathematically, some definitions will be needed
ܧ௕ = Energy-per-bit
ܧ௦ = Energy-per-symbol = kܧ௕ with k bits per symbol
ܶ௕ = Bit duration
ܶ௦ = Symbol duration
N0 / 2 = Noise power spectral density (W/Hz)
ܲ௕ = Probability of bit-error
ܲ௦ = Probability of symbol-error
Q(x) will give the probability that a single sample taken from a random process with
zero-mean and unit-variance Gaussian probability density function will be greater or
equal to x. It is a scaled form of the complementary Gaussian error function
2 8
√૛࣊ ∫ ࢋି࢚૛/૛ ࢊ࢚ ஶ
Q(x) = ૚
࢞ = ૚
૛ ࢋ࢘ࢌࢉ ቀ ࢞
√૛ቁ , x≥0 (2.8)
The error-rates quoted here are those in additive white Gaussian noise (AWGN).
QPSK digital modulation schemes for OFDM system is use in this thesis . Hence a study
on QPSK has been carried out in next section.
2.9.1 Quadrature Phase Shift Keying (QPSK)
QPSK is a multilevel modulation techniques, it uses 2 bits per symbol to represent each
phase. Compared to BPSK, it is more spectrally efficient but requires more complex
receiver.

Fig (2.14)-Constellation Diagram for QPSK
Figure (2.14) shows the constellation diagram for QPSK with Gray coding. Each adjacent
symbol only differs by one bit. Sometimes known as quaternary or quadric phase PSK or
4-PSK, QPSK uses four points on the constellation diagram, equispaced around a circle.
With four phases, QPSK can encode two bits per symbol, shown in the diagram with
Gray coding to minimize the BER- twice the rate of BPSK. Analysis shows that QPSK
may be used either to double the data rate compared to a BPSK system while maintaining
the bandwidth of the signal or to maintain the data-rate of BPSK but halve the bandwidth
needed. Although QPSK can be viewed as a quaternary modulation, it is easier to see it as
two independently modulated quadrature carriers. With this interpretation, the even (or
odd) bits are used to modulate the in-phase component of the carrier, while the odd (or
even) bits are used to modulate the quadrature-phase component of the carrier. BPSK is
used on both carriers and they can be independently demodulated.
The implementation of QPSK is more general than that of BPSK and also indicates the
implementation of higher-order PSK. Writing the symbols in the constellation diagram in
terms of the sine and cosine waves used to transmit them:
2 9

3 0
(2.9)
This yields the four phase‘s π/4, 3π/4, 5π/4 and 7π/4 as needed. This results in a two-dimensional
signal space with unit basis functions.
∅૚(࢚) = √૛/√ࢀ࢙ ܋ܗܛ (2࣊ࢌࢉ ࢚)
∅૛(࢚) = √૛/√ࢀ࢙ ܛܑܖ (2࣊ࢌࢉ ࢚) (2.10)
The first basis function is used as the in-phase component of the signal and the second as
the quadrature component of the signal. Hence, the signal constellation consists of the
signal-space 4 points ±ඥ۳ܛ
√૛ ,±ඥ۳ܛ
√૛
The factors of 1/2 indicate that the total power is split equally between the two carriers.
QPSK can be viewed as two independent BPSK signals.
 Bit error rate
Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two
independently modulated quadrature carriers. With this interpretation, the even (or odd)
bits are used to modulate the in-phase component of the carrier, while the odd (or even)
bits are used to modulate the quadrature-phase component of the carrier. BPSK is used on
both carriers and they can be independently demodulated. As a result, the probability of
bit-error for QPSK is the same as for BPSK:
۾܊ = ۿ( √ ૛۳܊
ඥۼ۽
) (2.11)
However, in order to achieve the same bit-error probability as BPSK, QPSK uses
twice the power (since two bits are transmitted simultaneously). The symbol error rate is
given by:
ࡼ࢙ = ૚ − (૚ − ࡼ࢈)૛ = 2ࡽ ൬ √ࡱ࢈
ඥࡺࡻ
൰ − ࡽ૛ ൬ √ࡱ࢈
ඥࡺࡻ
൰ (2.12)

If the signal-to-noise ratio is high (as is necessary for practical QPSK systems) the
probability of symbol error may be approximated.
3 1
ࡼࡿ ≈ 2ࡽ ൬ √ࡱ࢈
ඥࡺࡻ
൰ (2.13)
The modulated signal is shown below for a short segment of a random binary data-stream.
The two carrier waves are a cosine wave and a sine wave, as indicated by the
signal-space analysis above. Here, the odd-numbered bits have been assigned to the in-phase
component and the even-numbered bits to the quadrature component (taking the
first bit as number 1)
The total signal ,the sum of the two components is shown at the bottom. Jumps in phase
can be seen as the PSK changes the phase on each component at the start of each bit-period.
Figure (2.15)-Timing diagram for QPSK
In figure (2.15) binary data stream is shown on the time axis. The two signal components
with their bit assignments are shown the top and the total, combined signal at the bottom.
Note the abrupt changes in phase at some of the bit-period boundaries.
The binary data that is conveyed by this waveform is: 1 1 0 0 0 1 1 0.
The odd bits, highlighted here, contribute to the in-phase component: 1 1 0 0 0 1 1 0
The even bits, highlighted here, contribute to the quadrature-phase component:

1 1 0 0 0 1 1 0
2.10 RADIO PROPAGATION
In an ideal radio channel, the received signal would consist of only a single direct path
signal, which would be a perfect reconstruction of the transmitted signal. However in a
real channel, the signal is modified during transmission in the channel. The received
signal consists of a combination of attenuated, reflected, refracted, and diffracted replicas
of the transmitted signal [28]. On top of all this, the channel adds noise to the signal and
can cause a shift in the carrier frequency if the transmitter or receiver is moving (Doppler
Effect). Understanding of these effects on the signal is important because the
performance of a radio system is dependent on the radio channel characteristics
2.10.1 ATTENUATION
Attenuation is the drop in the signal power when transmitting from one point to another.
It can be caused by the transmission path length, obstructions in the signal path, and
multipath effects. Any objects that obstruct the line of sight signal from the transmitter to
the receiver can cause attenuation. Shadowing of the signal can occur whenever there is
an obstruction between the transmitter and receiver. It is generally caused by buildings
and hills, and is the most important environmental attenuation factor. Shadowing is most
severe in heavily built up areas, due to the shadowing from buildings. However, hills can
cause a large problem due to the large shadow they produce. Radio signals diffract off the
boundaries of obstructions, thus preventing total shadowing of the signals behind hills
and buildings. However, the amount of diffraction is dependent on the radio frequency
used, with low frequencies diffracting more than high frequency signals. Thus high
frequency signals, especially, Ultra High Frequencies (UHF), and microwave signals
require line of sight for adequate signal strength. To overcome the problem of shadowing,
transmitters are usually elevated as high as possible to minimize the number of
obstructions
2.11 FADING EFFECTS
3 2

Fading is about the phenomenon of loss of signal in telecommunications. Fading
channels refers to mathematical models for the distortion that a carrier modulated
telecommunication signal experiences over certain propagation media. Small scale fading
also known as multipath induced fading is due to multipath propagation. Fading results
from the superposition of transmitted signals that have experienced differences in
attenuation, delay and phase shift while travelling from the source to the receiver.
2.11.1 Rayleigh Fading
Rayleigh fading with AWGN is use in this thesis , so in this section we will discuss
about the Rayleigh fading
Rayleigh fading channel are useful models of real-world phenomena in wireless
communication. These phenomena include multipath scattering effects, time dispersion,
and Doppler shifts that arise from relative motion between the transmitter and receiver. It
is a statistical model for the effect of a propagation environment on a radio signal, such as
that used by wireless devices
Rayleigh fading models assume that the magnitude of a signal that has passed through
such a transmission medium (also called a communications channel) will vary randomly,
or fade, according to a Rayleigh distribution.
Rayleigh fading is viewed as a reasonable model for troposphere and ionospheric signal
propagation as well as the effect of heavily built-up urban environments on radio signals.
Rayleigh fading is most applicable when there is no dominant propagation along a line of
sight between the transmitter and receiver.
2.12 DOPPLER SHIFTS
When a wave source and a receiver are moving relative to one another the frequency of
the received signal will not be the same as the source. When they are moving toward each
other the frequency of the received signal is higher than the source, and when they are
moving away each other the frequency decreases. This is called the Doppler Effect. An
3 3

example of this is the change of pitch in a car‘s horn as it approaches then passes by. This
effect becomes important when developing mobile radio systems. The amount the
frequency changes due to the Doppler Effect depends on the relative motion between the
source and receiver and on the speed of propagation of the wave. The Doppler shift in
frequency can be written
3 4
Δࢌ = ±ࢌ࢜
ࢉ ܋ܗܛ ࣂ (2.14)
Where f is the change in frequency of the source seen at the receiver, f is the frequency of
the source, v is the speed difference between the source and receiver, c is the speed of
light and is the angle between the source and receiver. For example: Let
f = 1 GHz, and v = 60km/hr (16.67m/s) and = 0 degree, then the Doppler shift will be
ࢌ = ૚૙ૢ . ૚૟.૟ૠ
૜×૚૙ૡ = ૞૞. ૞ ࡴࢠ (2.15)
This shift of 55Hz in the carrier will generally not affect the transmission. However,
Doppler shift can cause significant problems if the transmission technique is sensitive to
carrier frequency offsets (for example OFDM) or the relative speed is very high as is the
case for low earth orbiting satellites.

CHAPTER 3
PHASE NOISE PROBLEM IN OFDM SYSTEM
3 5
3.1 PHASE NOISE
Phase noise is the frequency domain representation of rapid, short-term, random
fluctuations in the phase of a waveform, caused by time domain instabilities ("jitter").
Generally speaking radio frequency engineers speak of the phase noise of an oscillator,
whereas digital system engineers work with the jitter of a clock.
Historically there have been two conflicting yet widely used definitions for phase noise.
The definition used by some authors defines phase noise to be the Power Spectral Density
(PSD) of a signal's phase the other one is based on the PSD of the signal itself. Both
definitions yield the same result at offset frequencies well removed from the carrier. At
close-in offsets however, characterization results strongly depends on the chosen
definition. Recently, the IEEE changed its official definition to ∅(݊) = ݏ∅/2 where ݏ∅ is
the (one-sided) spectral density of a signal's phase fluctuations.
An ideal oscillator would generate a pure sine wave. In the frequency domain, this would
be represented as a single pair of delta functions (positive and negative conjugates) at the
oscillator's frequency, i.e., all the signal's power is at a single frequency. All real
oscillators have phase modulated noise components. The phase noise components spread
the power of a signal to adjacent frequencies, resulting in noise sidebands. Oscillator
phase noise often includes low frequency flicker noise and may include white noise.
Consider the following noise free signal v (t) = Acos(2πf0t).
Phase noise is added to this signal by adding a stochastic process represented by φ to the
signal as v(t) = Acos(2πf0t + φ(t)).
Phase noise is a type of cyclostationary noise and is closely related to jitter. A particularly
important type of phase noise is that produced by oscillators.

Phase noise (∅(݊)) is typically expressed in units of dBc/Hz, representing the noise
power relative to the carrier contained in a 1 Hz bandwidth centered at a certain offsets
from the carrier. For example, a certain signal may have a phase noise of -80 dBc/Hz at
an offset of 10 kHz and -95 dBc/Hz at an offset of 100 kHz. Phase noise can be measured
and expressed as single sideband or double sideband values, but as noted earlier, the
IEEE has adapted as its official definition, one-half the double sideband PSD.
Phase noise cannot be removed by filtering without also removing the oscillation signal.
And since it is predominantly in the phase, it cannot be removed with a limiter. so phase
noise removing is a major problem in OFDM.
3 6

Figure (3.1)-Oscillator phase noise
Figure (3.1) shows that how the oscillator phase noise is introduced in the OFDM system.
A local oscillator produces common phase error (CPE). The signal transmit at transmitter
side have phase rotation at receiver side.
Phase noise can be measured using a spectrum analyzer if the phase noise of the device
under test (DUT) is large with respect to the spectrum analyzer's local oscillator.
Spectrum analyzer based measurement can show the phase-noise power over many
decades of frequency from 1 Hz to 10 MHz. The slope with offset frequency in various
offset frequency regions can provide clues as to the source of the noise.
3 7

Figure (3.2)-Phase Noise
Figure (3.2) shows the OFDM carriers in frequency domain and the effect of phase noise
on these carriers.
The phase noise in the local oscillator of transmitter and receiver affects on the
orthogonality between the adjacent subcarriers. This introduce two main effects First, it
causes a random phase variation common to all sub-carriers. Second, it introduces ICI.
This ICI degrades the bit error rate (BER) performance of the system.
Based on the model defined in [11], the degradation D in SNR, i.e., the required increase
in SNR to compensate for the phase noise is
3 8
۲܌۰ ≅ ૚૚
૟ ܔܖ ૚૙ (૝ૈۼ ઺
܀) ۳܁
ۼ۽
(3.1)
Since R= N/T = NRୗ , where N is the total number of sub-carriers and ܴௌ is the subcarrier
symbol rate, Equation (3.1) can be rewritten as
ࡰࢊ࡮ ≅ ૚૚
૟ ࢒࢔ ૚૙ (૝࣊ ࢼ
ࡾ࢙
) ࡱࡿ
ࡺࡻ
(3.2)
3.2 THEORTICAL ANALYSIS OF PHASE NOISE
A theoretical analysis of phase noise effects in OFDM signals can be found in [29]. The
complex envelope of the transmitted OFDM signal for a given OFDM symbol sampled
with sampling frequency ݂௦ = B
S(n)=Σ ࢆ࢑
ࡺି૚
࢑ୀ૙ ࢋ࢐(૛࣊/ࡺ)࢑࢔ (3.3)
with This symbol is actually extended with a Time Guard in order to cope with multipath
delay spread, For the sake of simplicity, we will not consider this prefix since it is
eliminated in the receiver. Assuming that the channel is flat, the signal is only affected by
phase noise ∅(݊)
r(n)= S(n) .ࢋ࢐ ∅(࢔) (3.4)

The received signal is Orthogonal Frequency Division Demultiplexed (OFDD) by means
of a Discrete Fourier Transform. In order to separate the signal and noise terms, let us
suppose that ∅(݊) is smaller so that
܍ܒ∅ܖ ≈ ૚ + ܒ∅(ܖ) (3.5)
In this case, the demultiplexed signal is
ۼି૚
ܚୀ૙ (3.7)
3 9
܇۹ ≈ ࢆ࢑ + ࢐
ࡺି૚
࢘ୀ૙ ࢋ࢐ቀ૛࣊
Σ ࢆ࢘ Σࡺି૚ ∅(݊)
ࡺ ࢔ୀ૙
ࡺ ቁ(࢘ି࢑)࢔
܇۹ = ࢆ࢑ + ࢋ࢑ (3.6)
Thus we have an error term ݁௞ for each sub-carrier which results from some
combination of all of them and is added to the use signal.
If r=k: Common Phase Error
ܒ
Σ ܈ܚ Σۼି૚ ۼ ܖୀ૙
∅(ܖ) = ܒ. ܈ܓ. ∅
If r≠k : Inter-Carrier Interference
ܒ
ۼ Σ ܈ܚ Σ ∅(ܖ) ܍ܒቀ૛ૈ
ۼ ۼି૚ ቁ(ܚିܓ)ܖ
ܖୀ૙
ۼି૚
ܚୀ૙ (3.8)

CHAPTER 4
METHODOLOGY USED TO COMPENSATE PHASE NOISE
4.1 LEAST SQUARE WITH AVERAGING CHANNEL ESTIMATION
TECHNIQUE
A wideband radio channel is normally frequency selective and time variant. For an
OFDM mobile communication system, the channel transfer function at different
subcarriers appears unequal in both frequency and time domains. Therefore, a dynamic
estimation of the channel is necessary. Pilot-based approaches are widely used to
estimate the channel properties and correct the received signal.
There are two types of pilot-based channel estimation
(1) Block-type pilot channel estimation
(2) Comb-type pilot channel estimation
Figure (4.1)-Channel Estimation ([30])
In Figure (4.1) the first kind of pilot arrangement is block-type pilot arrangement. The
pilot signal assigned to a particular OFDM block, which is sent periodically in time-domain.
This type of pilot arrangement is especially suitable for slow-fading radio
channels. Because the training block contains all pilots, channel interpolation in
4 0

frequency domain is not required. Therefore, this type of pilot arrangement is relatively
insensitive to frequency selectivity.
The second kind of pilot arrangement is comb-type pilot arrangement. The pilot
arrangements are uniformly distributed within each OFDM block. Assuming that the
payloads of pilot arrangements are the same, the comb-type pilot arrangement has a
higher re-transmission rate. Thus the comb-type pilot arrangement system is provides
better resistance to fast-fading channels. Since only some sub-carriers contain the pilot
signal, the channel response of non-pilot sub-carriers will be estimated by interpolating
neighboring pilot sub-channels. Thus the comb-type pilot arrangement
is sensitive to frequency selectivity when comparing to the block-type pilot arrangement
system.
LS with averaging channel estimation technique is use in this thesis to remove common
phase error. It is a block-type channel estimation technique. In this channel estimation
technique we consider the data carried by the k୲୦ subcarrier of an OFDM symbol is
X୩ = c୩ + p୩ where c୩ is the information symbol with varience σଶ and p୩ is the
superimposed pilot symbol with varience σ୮ଶ
defined
ۺି૚
ܔୀ૙ (4.2)
4 1
૛ / ો܋
િ = ો܋
૛ + ોܘ૛
(4.1)
is the ratio of information symbol power to total transmitted symbol power. In the
superimposed pilot scheme, the power ratio η can take values 0<η < 1whereas in a
conventional scheme η = 1 when information symbols are transmitted ( X୩ = c୩) and
η = 0 for pilot transmission ( X୩ = p୩) Consider a frequency-selective channel with
memory L, and channel tap value vector h=[ h଴ ……. h୐ିଵ]. The received OFDM sample
y୬ is given by
ܡܖ = Σ ܐܔ ܠܖିܔ ܍ܒ∅(࢔) + ܟܖ
where ∅(݊) is the time domain phase error due to phase noise introduced at the receiver
and w୬ is the channel noise which is gaussian distributed N(0,σ୵ଶ
) in Eq.(4.2)

x=[x଴ , xଵ, xଶ …. . x୒ିଵ] is the IFFT of the data symbol X=[X଴ , Xଵ , Xଶ ……X୒ିଵ]. The
post FFT signal at the receiver (FFT of y୬ , 0 ≤ n ≤ N − 1) is
ۼି૚
ܔୀ૙ (4.3)
4 2
܇۹ = ۶۹ ܆۹ ܁૙ + Σ ۶ܔ ܆ܔ ܁ܔିܓ + ܅ܓ
Where H୏ and S୪ are the channel frequency response and intercarrier interference (ICI),
respectively. The ICI term ܵ௟ is a function of the phase noise ∅(݊) given by
ࡿ࢒ = ૚
ࡺ Σࡺି૚ ࢋ࢐૛࣊࢔࢒/ࡺ ࢋ∅(࢔)
࢔ୀ૙ , ࢒=0……N-1 (4.4)
From Eq. (4.3) it can be seen that the phase noise cause common phase error as well as
ICI. The received post-FFT signal given in (4.3) can be written as
܇۹ = ۶۹ ۱۹ ܁૙ + ۶۹ ۾۹ ܁૙ + ܅ܓ + ۷ܓ (4.5)
Where I୩ is the ICI term . the effect of S୭ on the post-FFT data symbol C୩′
s is a common
phase rotation. The least squares estimation with averaging scheme treats the contribution
of the unknown information symbol C୏ in the received signal (post-FFT) Y୩ as noise.
This means that the term H୏ C୏ S଴ is the noise term in Eq. (4.5) thus Y୩ can be
expressed as
܇۹ = ۶۹ ۾۹ ܁૙ + ܈ܓ (4.6)
Where Z୩= H୏ C୏ S଴ +W୩ + I୩ is the total noise The least squares (LS) estimate of the
phase rotation term S଴ based on k୲୦ subcarrier signal is
⋀(k) = ࢅ࢑
ࡿ࢕
ࡴࡷ
ࡼࡷ (4.7)
Substitute Eq. (4.5) in Eq. (4.7)
⋀(k) = ࡿ૙ + ࡯࢑ ࡿ૙
ࡿ࢕
ࡼ࢑
+ ࢂ࢑
ࡴ࢑ ࡼ࢑
(4.8)

⋀(k) is the initial estimate obtained only using
⋀(ܓ) ࢑∈ࡵ (4.9)
࢑∈ࡵ (4.10)
4 3
Where V୩ = I୏ + W୩ In Eq. (4.8), S୭
k୲୦ post-FFT signal. In a frequency selective channel, different subcarriers experience
different fading according to the channel conditions. In the conventional techniques of
phase estimation, if a dedicated pilot subcarrier falls in deep fade, the phase estimation
accuracy would be adversely affected. However, in superimposed pilot scheme since
pilots are present in all the subcarriers, it is advantageous to use subcarriers that have
better channel response for phase estimation instead of using all the subcarriers. This can
be effectively implemented as the channel state information is present at the receiver
(Since the preamble can be used to estimate the channel). Thus we can use subcarrier
selection for phase estimation as follows:
Compute Ω = {|ܪ௜|ଶ | 0 ≤ ݅ ≤ ܰ − 1} and select set of Indices I={ܭ଴, ܭଵ,…ܭேబିଵ }
corresponding to the ܰ଴ highest elements of Ω. Some assumptions about the noise terms
in Eq. (4.8) can be made in the presence of above mentioned subcarrier selection. The
second and the third terms in Eq. (4.8) are noise terms and it is valid to assume that the
variance of third term in Eq. (4.8), ୚ౡ
ୌౡ ୔ౡ
is negligible compared to the variance of the
second term େౡ ୗబ
୔ౡ
due to following reasons.
(i) With the subcarrier selection the lower values of |H୩|ଶ are eliminated and
(ii) The variance of the transmitted symbols C୩, which is contributing towards the
noise term, is higher than the sum of variances of the ICI term and channel
noise,V୩. With this assumption, it can be noted that the variance of the noise
term in Eq. (4.8) is approximately constant irrespective of channel and the
subcarrier.
Since variance of the noise terms is constant over the subcarriers, an equal weight
averaging scheme is proposed to improve the estimate of S଴ as
ࡿ⋀ = ૚
ࡺ૙
Σ ࡿ࢕
⋀(k) in Eq. (4.9) gives
Substituting for ܵ௢
⋀ = ࡿ૙ + ૚
ࡿ࢕
ࡺ૙
Σ ࡯࢑ ࡿ૙
࢑∈ࡵ + ૚
ࡼ࢑
ࡺ૙
Σ ࢂ࢑
ࡴ࢑ ࡼ࢑
= ࡿ૙ + ࡿ૙ࢻ + ࢼ (4.11)

4 4
Here α = ଵ
୒బ
Σ େౡ
୩∈୍ β = ଵ
୔ౡ
୒బ
Σ ୚ౡ
ୌౡ ୔ౡ
୩∈୍
And ܵ௢ߙ + ߚ denotes the total estimation error.
4.2 REED-SOLOMON CODING
Reed-Solomon codes are block-based error correcting codes with a wide range of
applications in digital communications and storage. Reed-Solomon codes are used to
correct errors in many systems including:
(1) Storage devices (including tape, Compact Disk, DVD, barcodes, etc).
(2) Wireless or mobile communications (including cellular telephones, microwave
links, etc).
(3) Satellite communications.
(4) Digital television / DVB.
(5) High-speed modems such as ADSL, xDSL, etc.
An R-S code was invented by Irving S. Reed and Gustave Solomon. They described a
systematic way of building codes that could detect and correct multiple random symbol
errors. By adding t check symbols to the data, an R-S code can detect any combination of
up to t erroneous symbols, and correct up to ⌊t/2⌋ symbols. In Reed-Solomon coding,
source symbols are viewed as coefficients of a polynomial
over a finite field. The original idea was to create n code symbols from k source symbols
by oversampling at n > k distinct points, transmit the sampled points, and use
interpolation techniques at the receiver to recover the original message.
A typical system is shown here:

Figure (4.2)-R-S System
4 5
4.2.1 Properties of Reed-Solomon Codes
Reed Solomon codes are a subset of BCH codes and are linear block codes. A Reed-
Solomon code is specified as R-S (n,k) with s-bit symbols. This means that the encoder
takes k data symbols of s bits each and adds parity symbols to make an n symbol
codeword. There are n-k parity symbols of s bits each. A Reed-Solomon decoder can
correct up to t symbols that contain errors in a codeword, where 2t = n-k.
Figure (4.3) shows a typical Reed-Solomon codeword (this is known as a Systematic
code because the data is left unchanged and the parity symbols are appended):
Figure (4.3)-R-S codeword
For example a popular Reed-Solomon code is R-S (15, 11) with 4-bit symbols. Each
codeword contains 15 code word bytes, of which 11 bytes are data and 4 bytes are parity.
For this code:
n = 15, k = 11, s = 4 , 2t = 4, t = 2
The decoder can correct any 2 symbol errors in the code word: i.e. errors in up to 2 bytes
anywhere in the codeword can be automatically corrected.
Given a symbol size s, the maximum codeword length (n) for a Reed-Solomon code is n
= 2s – 1
For example, the maximum length of a code with 4-bit symbols (s=4) is 15 bytes.
 Symbol error

One symbol error occurs when 1 bit in a symbol is wrong or when all the bits in a symbol
are wrong. for example R-S (15,11) can correct 2 symbol errors. In the worst case, 2 bit
errors may occur, each in a separate symbol (byte) so that the decoder corrects 2 bit
errors. In the best case, 2 complete byte errors occur so that the decoder corrects 2 x 4 bit
errors.
4 6
 Decoding
Reed-Solomon algebraic decoding procedures can correct errors and erasures. An erasure
occurs when the position of an erred symbol is known. A decoder can correct up to t
errors or up to 2t erasures. Erasure information can often be supplied by the demodulator
in a digital communication system, i.e. the demodulator "flags" received symbols that are
likely to contain errors.
When a codeword is decoded, there are three possible outcomes:
(1) If 2s + r < 2t (s errors, r erasures) then the original transmitted code word will always
be recovered,
(2) Otherwise the decoder will detect that it cannot recover the original code word and
indicate this fact.
(3) OR the decoder will mis-decode and recover an incorrect code word without any
indication
 Coding Gain
The advantage of using Reed-Solomon codes is that the probability of an error remaining
in the decoded data is (usually) much lower than the probability of an error if Reed-
Solomon is not used. This is often described as coding gain
4.2.2 Reed-Solomon Encoding and Decoding Process
(1) Encoding Process
The amount of processing "power" required to encode and decode Reed-Solomon codes
is related to the number of parity symbols per codeword. A large value of t means that a

large number of errors can be corrected but requires more computational power than a
small value of t. In digital communication systems that are both bandwidth-limited and
power-limited, error-correction coding (often called channel coding) can be used to save
power or to improve error performance at the expense of bandwidth [31]. The R-S
encoding and decoding require a considerable amount of computation and arithmetical
operations over a finite number system with certain properties, i.e. algebraic systems,
which in this case is called fields. R-S’s initial definition focuses on the evaluation of
polynomials over the elements in a finite field (Galois field GF) [32]. The k information
symbols that form the message to be encoded as one block can be represented by a
polynomial M(x) of order k – 1, so that:
ࡹ(࢞) = ࡹ࢑ି૚ ࢞࢑ି૚ + ………ࡹ૚࢞ + ࡹ૙ (4.12)
where each of the coefficients M୩ିଵ,…….. Mଵ, M଴ is an m-bit message symbol, that is an
element of GF(2୑). M୩ିଵ is the first symbol of the message. To encode the message, the
message polynomial is first multiplied by X୬ି୩ and the result is divided by the generator
polynomial, g(x). Division by g(x) produces a quotient q(x) and a remainder r(x), where
r(x) is of degree up to n – k– 1.Thus
4 7
ۻ(ܠ) × ܠܖିܓ
܏(ܠ)
ܚ(ܠ)
܏(ܠ) +
ܚ(ܠ)
܏(ܠ) (4.13)
Having produced r(x) by division, the transmitted code word T(x) can then be formed by
combining M(x) and r(x) as follows
܂(ܠ) = ۻ(ܠ) × ܠܖିܓ + ܚ(ܠ)
= ۻܓି૚ ܠܖି૚ + ⋯ +ۻ૙ ܠܖିܓ + ܚܖିܓି૚ + ⋯ +ܚ૙ (4.14)
Which shows that the code word is produced in the required systematic form. Adding the
remainder, r(x), ensures that the encoded message polynomial will always be divisible by
the generator polynomial without remainder. This can be seen by multiplying Eq. (4.13)
by g(x)
M(x)× ܠܖିܓ = ܏(ܠ) × ܙ(ܠ) + ܚ(ܠ) (4.15)

and rearranging
M(x)× ܠܖିܓ + ܚ(ܠ) = ܏(ܠ) × ܙ(ܠ) (4.16)
Here we, note that the left-hand side is the transmitted code word, T(x), and that the
right-hand side has g(x) as a factor. Also, because the generator polynomial. The code
generator polynomial takes the form
g(x)= (x+ࢻ࢈) (x+ ࢻ࢈ା૚)………..(x+ࢻ࢈ା૛࢚ି૚) (4.17)
Eq. (4.17), has been closer to consist of a number of factors, each of these is also a factor
of the encoded message polynomial and will divide it without remainder. Thus, if this is
not true for the received message, it is clear that one or more errors have occurred [33].
To visualize hardware that implements Eq. (4.13), one must understand the operations
M(x)× x୬ି୩ and r(x). As known, for systematic encoding, the information symbols must
be placed as the higher power coefficients.
So means that information symbols toward the higher powers of x, from n
– 1 down n – k. The remaining positions from power n – k – 1 to 0 fill with zeros.
Consider, for example, the same polynomial as above:
4 8
(4.18)
Multiplying the above equation by yields
(4.19)
The second term of Eq. (4.13), r(x) is the remainder when it divides polynomial
by the polynomial g(x). Therefore, it needs designing a circuit that
performs two operations: a division and a shift to a higher power of x. Linear-feedback
shift registers enable one to easily implement both operations. Figure (4.4) shows a
general diagram of the encoder for Reed-Solomon (n,k) code. The main design task is to

implement the GF( ) multiplication and addition circuits, apart from some control
circuitry or logic. It can add any two elements from the GF( ) field by modulo 2
adding their binary notations, which resembles the XOR hardware operation [34].
Figure (4.4)-Architecture of a R-S (n – k) Encoder
(2) Decoding Process
A general architecture for decoding Reed-Solomon codes is shown in the Figure (4.5)
Figure (4.5)-Architecture of a R-S(n-k) Decoder
4 9

ܓି૚
ܑୀ૙ ܠܑ (4.20)
ܑୀ૚ (4.21)
5 0
Here
C(x) Received codeword
Syndromes
Λ(x) Error locator polynomial
Error locations
Error magnitudes
Recovered code word
The received codeword C(x) is the original (transmitted) codeword plus errors:
C(x) = + e(x) A Reed-Solomon decoder attempts to identify the position and
magnitude of up to t errors (or 2t erasures) and to correct the errors or erasures.
 Peterson decoder
Peterson developed a practical decoder based on syndrome decoding. Peterson decoder
contains the following processes,
 Syndrome decoding
The transmitted message is viewed the coefficients of a polynomial M(x) that is divisible
by a generator polynomial g(x)
ۻ(ܠ) = Σ ۻܑ
܏(ܠ) = Π܊ା૛ܜି૚ (ܠ + હܑ)
where ߙ is a primitive root. Since M(x) is divisible by generator g(x), it follows that
M(α୧)=0, i=1, 2,….n-k
The transmitted polynomial is corrupted in transit by an error polynomial e(x) to produce
the received polynomial C(x).

C(x) = M(x) + e(x) (4.22)
e(x)=Σܖ−૚ ܍ܑܠܑ
ܑ=૙ (4.23)
where ei is the coefficient for the i୲୦ power of x. Coefficient ei will be zero if there is no
error at that power of x and nonzero if there is an error. If there are ν errors at distinct
powers ik of x, then
ܑୀ૙ (4.24)
5 1
e(x)= Σ ܍ܑܓ ܖି૚ ܠܑܓ
The goal of the decoder is to find ν, the positions i୩ and the error values at those
positions. The syndromes s୨ are defined as
ܛܒ = ۱(હܒ) + ܍(હܒ) = ૙ + ܍(હܒ) = ܍(હܒ) , ܒ = ૚, ૛,…. . ܖ − ܓ
= Σ ܍ܑܓܞ હܒ ܑܓ
ܓୀ૚ (4.25)
The advantage of looking at the syndromes is that the message polynomial drops outs
 Error locators and error values
For convenience, define the error locators X୩and error values Y୩ as
X୩ = α୧ౡ , Y୩ = e୧ౡ
Then the syndromes can be written in terms of the error locators and error values as
ܞ
ܓୀ૚ ܆ܓܒ
ܛܒ = Σ ܇ܓ
(4.26)
The syndromes give a system of n-k ≥ 2ν equations in 2ν unknowns, but that system of
equations is nonlinear in the X୩ and does not have an obvious solution. However, if the
X୩ were known (see below), then the syndrome equations provide a linear system of

equations that can easily be solved for the Yk error values
࢑ୀ૚ ܆ܓ) = ૚ + ઩૚࢞૚ + ઩૛ ࢞૛ + …………઩࢜࢞࢜ (4.28)
ା୴ and it will still be zero
ା࢜ࢄ࢑
5 2
⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡
૚ ܆૛૚
………. . ܆ܞ
܆૚
૚
⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ૛
ܖିܓ⎤
܆૛૚
܆૛૛
…………. ܆ܞ
...
ܖିܓ ܆૛
܆૚
ܖିܓ ……܆ܞ
⎣ ⎢ ⎢ ⎢ ⎢ ⎡
܇૚
܇૛
ܞ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
...܇
=
⎣ ⎢ ⎢ ⎢ ⎢ ⎡
܁૚
܁૛
܁ܖିܓ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
...
(4.27)
 Error locator polynomial
Peterson found a linear recurrence relation that gave rise to a system of linear equations.
Solving those equations identifies the error locations. Define the Error locator polynomial
Λ(x) as
Λ(x) = Π࢜ (૚ − ࢞
ିଵ
The zeros of Λ(x) are the reciprocals X୩
ି૚) = 0
Λ(ࢄ࢑
ି૚) =૚ + ઩૚ࢄ࢑
Λ(ࢄ࢑
ି૚ + ઩૛ ࢄ࢑
ି૛ + …………઩࢜ࢄ࢑
ି࢜ =0 (4.29)
Multiply both sides by Yk X୩୨
ା࢜Λ(ࢄ࢑
ࢅ࢑ ࢄ࢑࢐
ା࢜ + ઩૚ࢅ࢑ ࢄ࢑࢐
ି૚)= ࢅ࢑ ࢄ࢑࢐
ା࢜ࢄ࢑
ି૚ + ઩૛ ࢅ࢑ ࢄ࢑࢐ା࢜ ࢄ࢑
ି૛
….+઩࢜ࢅ࢑ ࢄ࢑࢐
ି࢜ =0 (4.30)
ା࢜ +઩૚ࢅ࢑ ࢄ࢑࢐
= ࢅ࢑ ࢄ࢑࢐
ା࢜ି૚ +઩૛ ࢅ࢑ ࢄ࢑࢐
ା࢜ି૛ +………+ ઩࢜ࢅ࢑ ࢄ࢑࢐
=0 (4.31)
Σ ࢅ࢑ ࢄ࢑࢐࢜ ା࢜
࢑ୀ૚ + ઩૚ Σ ࢅ࢑ ࢄ࢑࢐
࢜ ା࢜ି૛
ࡷୀ૚ + …………+ ઩࢜ Σ ࢅ࢑ ࢄ࢑࢐
࢜
࢑ୀ૚
= ૙ (4.32)
Which reduces to

ܛܒାܞ + ઩૚ ܛܒାܞି૚ + ………….+઩࢜ି૚ ܛܒା૚ + ઩࢜ ܛܒ = 0 (4.33)
ܛܒ ઩࢜+ ܛܒା૚઩࢜ି૚ + ……. . ܛܒାܞି૚ ઩૚ + = − ܛܒାܞ (4.44)
Now have system of linear equations that can be solved for the coefficients Λi of the error
location polynomial
5 3
⎣ ⎢ ⎢ ⎢ ⎢ ⎡
࢙૚ ࢙૛ …. . ࢙࢜
࢙૛ ࢙૜ …࢙࢜ା૚ ...
࢙࢜ ࢙࢜ା૚ …࢙૛࢜ା૚⎦ ⎥ ⎥ ⎥ ⎥ ⎤
⎣ ⎢ ⎢ ⎢ ⎢ ⎡
઩࢜
઩࢜ି૚
઩૚ ⎦ ⎥ ⎥ ⎥ ⎥ ⎤
...
=
⎣ ⎢ ⎢ ⎢ ⎢ ⎡
−࢙࢜ା૚
− ࢙࢜ା૛
− ࢙࢜ା࢜⎦ ⎥ ⎥ ⎥ ⎥ ⎤
...
(4.45)
 Obtain the error locations from the error locator polynomial
Use the coefficients Λi found in the last step to build the error location polynomial. The
roots of the error location polynomial can be found by exhaustive search. The error
locators (and hence the error locations) can be found from those roots. Once the error
locations are known, the error values can be determined and corrected.

CHAPTER 5
SIMULATION RESULTS AND DISCUSSION
5.1 SIMULATION PARAMETERS AND STEPS
This chapter presents simulation of an OFDM communication system with phase noise ,
operating under Rayleigh channel conditions. The Simulation parameters of an OFDM
system are shown in Table (5.1)
Table (5.1)-Simulation Parameters
PARAMETERS VALUE
Modulation type QPSK
FFT length nFFT 128
Number of data subcarriers 102
Number of guard and pilot carriers 22
Doppler Shift 200 Hz
Frequency offset 100 Hz
Samples per frame 44
R-S code rate 0.73
5 4
 SIMULATION’S STEPS
(1) Generate the information bits randomly.
(2) Encode the information bits using a R-S encoder.
(3) Use QPSK to convert the binary bits 0 and 1, into complex signals.
(4) Insert pilot training bits for channel estimation.
(5) Perform serial to parallel conversion.
(6) Use IFFT to Generate OFDM signals, zero padding has been done before IFFT.
(7) Use parallel to serial convertor to transmit signal serially.

(8) Introduce phase noise.
(9) Introduce noise to simulate channel errors.
(10)At the receiver side, perform reverse operation to decode the received sequence.
(11)Estimate the channel by using LSA technique.
(12)Calculate BER and plot it.
5.2 BER PERFORMANCE OF UNCODED OFDM SYSTEM WITHOUT
CONSIDERING PHASE NOISE
BER Multipath Channel
Figure (5.1)-Uncoded OFDM System
Figure(5.1) shows the MATLAB୘୑ Simulink model of uncoded OFDM System. Bernoulli
Binary has been used as a signal generator and samples per frame=44. Rayleigh fading
has been used as a channel fading and AWGN used as a channel Noise. Maximum
5 5
OFDM Transmitter
OFDM Receiver
and AWGN
.
BER
To Workspace
QPSK Mapping
QPSK Demapping
guianrsde irntitoenrv al
. S/P
P/S
OFDM Baseband
Demodulator
Remove Zero & CP
OFDM Baseband
Modulator
Add Zero & CP
BER
Calculation
.
Remove
Zero
Selector
Multipath
Rayleigh Fading
0.03363
Display2
Bernoulli
Binary
AWGN

dopper shift=200 Hz and sample time = ( 8e-5)/180. On simulating this model the
following Results has been obtained.
Table (5.2)-Uncoded OFDM with Rayleigh fading in absence of PHN
SNR 0 2 4 6 8 10 12 14 16 18 20
BER of
uncoded
OFDM
without
PHN
.2818 .2111 .1488 .0995 .0643 .0423 .0261 .0158 .0104 .0070 .0042
Table (5.2) shows the BER performance of uncoded OFDM system at different values of
SNR (Eb/No).
5 6

0 2 4 6 8 10 12 14 16 18 20
Figure (5.2)-BER vs. Eb/No plot of uncoded OFDM
Figure (5.2) shows the graphical representation of BER performance of uncoded OFDM.
This is the BER plot of OFDM system when effect of phase noise and frequency offset is
not considered.
5.3 BER PERFORMANCE OF UNCODED OFDM SYSTEM AT DIFFERENT
VALUES OF PHASE NOISE
Figure (5.3) shows the MATLAB୘୑ Simulink model of uncoded OFDM system with
phase noise. Bernoulli Binary is use as a signal generator.
5 7
10
-3
10
-2
10
-1
10
0
Eb/No
B E R
BER vs Eb/No plot for rayleigh fading in OFDM system
Uncoded OFDM without PHN

Figure (5.3)-Uncoded OFDM with PHN
Here also Rayleigh fading used as a channel fading and AWGN used as a channel Noise.
Frequency offset is fixed to 100Hz .On simulation of this model at different values of
phase noise following results has been obtained.
Table (5.3)-Uncoded OFDM system with Rayleigh fading at different values of PHN
5 8
OFDM Transmitter
OFDM Receiver
and AWGN
BER
To Workspace
QPSK Mapping
QPSK Demapping
.
. S/P
P/S
OFDM Baseband
Demodulator
Remove Zero & CP
OFDM Baseband
Modulator
Add Zero & CP
SER
Calculation
.
Remove
Zero
Selector
Phase
Noise
Phase
Noise
Multipath
Rayleigh Fading
0.3492
Display2
Bernoulli
Binary
AWGN

SNR 0 2 4 6 8 10 12 14 16 18 20
BER
AT
PHN=
-90
dBc/Hz
.3502 .2729 .2030 .1431 .0951 .0630 .0389 .0240 .0143 .0078 .0048
5 9
BER
AT
PHN=
-80
dBc/Hz
.3509 .2731 .2035 .1435 .0957 .0636 .0395 .0244 .0150 .0084 .0054
BER
AT
PHN=
-70
dBc/Hz
.3510 .2735 .2040 .1440 .0961 .0642 .0400 .0251 .0153 .0091 .0064
BER
AT
PHN=
-60
dBc/Hz
.3577 .2822 .2111 .1518 .1029 .0690 .0430 .0274 .0176 .0103 .0070
BER
AT
PHN=
-55
dBc/Hz
.3692 .2968 .2284 .1670 .1237 .0847 .0548 .0369 .0246 .0170 .0119
BER
AT
PHN=
-50
dBc/Hz
.4070 .3438 .2825 .2326 .1914 .1567 .1286 .1083 .0970 .0861 .0777
BER
AT
PHN=
-45
.4984 .4547 .4193 .3839 .3622 .3399 .3277 .3173 .3074 .3025 .2982

6 0
dBc/Hz
Table(5.3) shows that, uncoded OFDM system without phase noise have better BER
performance in comparatively with uncoded OFDM system with phase noise.On
increasing the value of phase noise in OFDM system, its BER performance degrade
respectively.The reason behind it is that due to phase noise, common phase error(CPE)
occurred in the OFDM system and this breaks the orthogonallity of the OFDM symbols
and produce inter carrier interference (ICI).
Table (5.4) also shows that, BER performance of uncoded OFDM system at PHN =
-70 dBc/Hz, -80 dBc/Hz, -90 dBc/Hz have approximately same. So ,at the simulation
parameters shown in Table (5.1), PHN= -70 dBc/HZ is considered as the optimum value
of phase noise. It means, effect of phase noise on OFDM system is consider negligible at
the PHN< -70 dBc/Hz .This limit may varied on varying the simulation parameters
especially the guard interval, number of OFDM sub-carriers and frequency offset.

0 2 4 6 8 10 12 14 16 18 20
Figure (5.4)-BER performance curve of uncoded OFDM system at different PHN
Figure (5.4) shows the graphical representation of BER performance of uncoded OFDM
system at different values of phase noise.The effect of phase noise may de reduced by
using some methods ,who have already disccus in previous chapters.
6 1
10
-3
10
-2
10
-1
10
0
Eb/No
B E R
BER vs Eb/No plot for OFDM system with different phase noise at frequency offset=100Hz
Uncoded OFDM without PHN
curve at PHN= -70 dBc/Hz

5.4 COMPARISON ANALYSIS BETWEEN UNCODED OFDM SYSTEM AND
REED-SOLOMON CODED OFDM SYSTEM AT DIFFERENT VALUES OF
PHASE NOISE
Figure (5.5) shows the MATLAB୘୑ Simulink model of coded OFDM System. In this
model Reed-Solomon coding having code rate of 0.73 is use as a channel coding. QPSK
mapping is use as a symbol mapping. In Reed-Solomon coding, code rate is the ratio of
message length (K) and codeword length (N). Here K=11 and N=15 is use to achieve
code rate of 0.73, Rayleigh fading is use as a channel fading and AWGN used as a
channel noise.
Figure (5.5)-R-S coded OFDM system with PHN
6 2
BER
OFDM Transmitter
OFDM Receiver
and AWGN
BER1
To Workspace1 BER
To Workspace
QPSK Mapping
QPSK Demapping
.
.
BER
Calculation
S/P
P/S
OFDM Baseband
Demodulator
Remove Zero & CP
OFDM Baseband
Modulator
Add Zero & CP
BER
Calculation
RS(15,11) Decoder
RS(15,11) Encoder
Remove
Zero
Selector1
Selector
Phase
Noise
Phase
Noise Multipath
Rayleigh Fading
0.06007
Display2
0.0569
Display1
Bernoulli
Binary
AWGN

On simulation of this model at different values of phase noise following Results has been
obtained.
Table (5.4)-Comparison table between R-S coded and uncoded OFDM system at
different values of phase noise
6 3
SNR
(dB)
0 2 4 6 8 10 12 14 16 18 20
BIT ERROR RATE
OFDM
without
PHN
.2818 .2111 .1488 .0995 .0643 .0423 .0261 .0158 .0104 .0070 .0042
OFDM with
PHN= -70
dBc/Hz
.3510 .2735 .2040 .1440 .0961 .0642 .0400 .0251 .0153 .0091 .0064
OFDM with
PHN= -60
dBc/Hz
.3577 .2822 .2111 .1518 .1029 .0690 .0430 .0274 .0176 .0103 .0070
OFDM with
PHN= -55
dBc/Hz
.3692 .2968 .2284 .1700 .1237 .0847 .0548 .0369 .0246 .0170 .0119
OFDM with
PHN= -50
dBc/Hz
.4070 .3438 .2825 .2326 .1914 .1567 .1286 .1083 .0970 .0861 .0777
OFDM with
PHN= -45
dBc/Hz
.4984 .4547 .4193 .3839 .3622 .3399 .3277 .3173 .3074 .3025 .2982
OFDM with
RS coding
at PHN=
-70 dBc/Hz
.3656 .2811 .2115 .1516 .0983 .0603 .0328 .0201 .0094 .0037 .0025
OFDM with
RS coding
at PHN=
-60 dBc/Hz
.3745 .2962 .2192 .1587 .1057 .0646 .0371 .0217 .0122 .0041 .0025

Phase Noise Effect in OFDM Based Communication Systems

Phase Noise Effect in OFDM Based Communication Systems

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