NLOS QPSK–OFDM with Concatenated Reed-Solomon /Convolution Coding for Data

Transmission over Fading Channels

Dushantha Nalin Kumara Jayakody Arachchilage

Submitted to the Institute of Graduate Studies and Research

in partial fulfillment of the requirements for the Degree of

Master of Science in

Electrical and Electronic Engineering

Eastern Mediterranean University August 2010

Gazimağusa, North Cyprus

Approval of the Institute of Graduate Studies and Research

Prof. Dr. Elvan Yılmaz Director (a)

I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Electrical and Electronic Engineering.

Assoc. Prof. Dr. Aykut Hocanın

Chair, Department of Electrical Electronic and Engineering

We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Electrical and Electronic Engineering.

Assoc. Prof. Dr. Erhan A. İnce Supervisor

Examining Committee

1. Assoc. Prof. Dr.Aykut Hocanın

2. Assoc. Prof. Dr.Erhan A. İnce

3. Assoc. Prof. Dr. Hasan Demirel

iii

ABSTRACT

The work presented herein, provides a link level (physical level) performance

analysis for non line of sight (NLOS) QPSK-OFDM for data transmission over

Rayleigh fading channels. During simulations three different scenarios has been

considered. First, the performance of un-coded and coded 1024 point OFDM over

the AWGN channel was obtained. Second, using Jake’s sum of sinusoids fading

channel model together with delay and power parameters for the COST 207 Typical

Urban (TU) channel and Winner vehicular NLOS (RS-MS-NLOS) channel link level

performance over the two fading channels was obtained and compared with results in

the literature. In the second scenario, a rate ½ convolutional encoder and its

corresponding Viterbi decoder on the receiver side was used to improve the decibel

gain for a target BER. Finally, in the third scenario an outside Reed Solomon (RS)

encoder was serially concatenated with the convolutional encoder and BER

performance was obtained over the COST207 TU channel.

The basic blocks of the physical (PHY) layer included the following; an outside Reed

Solomon encoder [RS(11,15,4)], a rate ½ convolutional inner encoder, a

constellation mapper, an Orthogonal Frequency Division Multiplexing (OFDM)

transmitter, Jake’s sum of sinusoids fading channel model, OFDM receiver, Viterbi

decoder and Reed Solomon decoder.

iv

Since in mobile wireless access the location of the end user is changing and different

users may be moving at different speeds, in this work performance was obtained for

Doppler shifts of 100, 400 and 833 Hz ( 30Km/hr, 120 Km/hr, 250 Km/hr). It was

observed that the Doppler shifts due to relative motion will degrade the performance

and also cause an error floor in multipath fading channels. Improvement of the

performance was observed after the inclusion of a rate ½ convolutional coder of

constraint length K = 7 and generator polynomials of 171 and 133 .

Also inclusion of the outside RS encoder will further improve the system

performance.

The thesis also points out that in practice due to frequency mismatch between the

frequency of the Local Osciallator (LO) at the transmitter and the receiver there will

be degradation in BER/SER performance of the system. It also provides some

example curves to show the amount of degradation for different SNR values.

Finally, the work points out how the spectral efficiency of the system can be obtained

using bit error rate, number of bits in each block, number of bits per symbol, and the

overall code rate of the system.

Keywords: OFDM transceiver, Convolutional Coding, Jake’s sum of sinusoids

channel model, Viterbi decoder, Reed Solomon encoder/decoder, frequency offset,

spectral efficiency.

v

ÖZ

Bu tez, dördün faz kaydırmalı kiplenim kullanan ve sönümlemeli Rayleigh kanalları

üzerinde veri iletişimi yapan dikgen frekans bölüşümlü çoğullama sisteminin fiziki

katmandaki başarım çözümlemelerini sunmaktadır. Benzetimler esnasında üç farklı

senaryo denenmiştir. İlk olarak kodlanmamış ve kodlanmış 1024 taşıyıcılı dikgen

frekans bölüşümlü çoğullama sisteminin toplanır beyaz Gauss gürültülü kanal

üzerindki başarım analizi MATLAB benzetimleri ile elde edilmiştir. İkinci iş olarak

Jake’in sinüslerin toplamı tabanlı sönümlemeli kanal modeli ve COST 207 TU ve

Wiener kanallarındaki gecikme ve güç parametreleri de kullanılarak fiziki

katmandaki başarım analizleri elde edilmiş ve literatürdeki benzer çalışmalarla

kıyaslanmıştır. Bu ikinci senaryoda hedef bir bit hata oranında (BHO) daha iyi

desibel kazanç sağlayabilmek amacı ile verici tarafında ½ hızlı evrişimsel bir

kodlayıcı ve alıcı tarafında da bir Viterbi kodçözücüden yararlanılmıştır. Son olarak,

üçüncü senaryoda evrişimsel kodlayıcı bir dış Reed-Solomon kodlayıcısı ile art arda

bağlanmış ve COST 207 TU sönümlemeli kanalı üzerindeki BHO başarım analizleri

yapılmıştır.

Benzetimlerde kullanılan fiziki katmanı oluşturan bloklar sırasıyla: bir dış Reed-

Solomon kodlayıcı [RS(11,15,4)], bir ½ hızlı evrişimsel iç kodlayıcı, bir işaret

kümesi eşleştiricisi, bir dikgen frekans bölüşümlü çoğullama vericisi, Jake’in

sinüslerin toplamı tabanlı sönümlemeli kanal modeli, bir dikgen frekans bölüşümlü

çoğullama alıcısı, bir Viterbi kodçözücüsü ve bir Reed-Solomon kod çözücüsünden

oluşmaktadır.

vi

Mobil iletişim esnasında kullanıcıların yeri ve hızları değiştiğinden bu çalışmada

100, 400, ve 833 Hz Doppler kaymasına neden olan üç farklı hızda (30Km/hr, 120

Km/hr, 250 Km/hr) başarım analizleri yapılmıştır. Benzetim sonuçları incelendiğinde

bağıl devinimden kaynaklanan Doppler kaymasının başarımı kötüleştirdiği ve

sönümlemeli kanallarda bir hata eşiğine sebep olduğu görülmüştür. Bu durumlarda,

½ hız ve K=7 kısıt uzunluklu bir evrişimsel kodlayıcı kullanıldığında (üreteç

polinomları G1 = 171oct ve G2 = 133oct) iyileşme elde edilebilmektedir. İç kodlayıcı

bir Reed-Solomon dış kodlayıcısı ile art arda bağlandığında ise başarım daha da

artmaktadır.

Bu çalışma ayrıca verici ve alıcılarda kullanılan yerel salınıcıların frekanslarının

pratikte hiçbir zaman ayni olmadığını ve genelde salınıcılar arasında bir frekans

kayması bulunduğunu ve bu frekans kaymasının sistemlerin BHO ve/veya SHO

başarımlarını kötüleştirdiğini işaret etmekte ve farklı sinyal gürültü oranlarında

frekans kayma oranına göre meydana gelebilecek kötüleşmeyi gösteren bazı örnek

eğriler vermektedir.

Çalışmamız son olarak sistemin spektrumsal verimliliğinin bit hata oranı, her blok

içindeki bit sayısı, sembol başına düşen bit sayısı ve sistemin toplam kod hızı

kullanılarak nasıl hesaplanabileceğini belirtmektedir.

Anahtar Kelimeler: DFBÇ vericisi, evrişimsel kodlayıcı, Jake’s sönümlemeli kanal

modeli, Viterbi kodçözücüsü, Reed Solomon kodlayıcı ve kod çözücüsü, frekans

kayması, spectrum verimliliği.

vii

To:

Mum, Dad & my angels Sana, Sadew and Kusal

viii

ACKNOWLEDGEMENTS

First and foremost I would like to offer my sincerest gratitude to my supervisor

Assoc Prof. Dr. Erhan A. İnce, who has supported me throughout my research work

& write-up of this thesis with patience all along. This thesis would not have been

possible without his support and guidance. I couldn’t have been imagining having a

better supervisor and a mentor for my postgraduate studies.

My sincere thanks definitely go to Assoc. Prof. Dr. Aykut Hocanın, the Chair of our

department, for his great assistance with my early enrolment process to the

university. He facilitated me in a number of ways to have a smooth running of

academic life in North Cyprus.

I convey my special thanks to my friends and colleagues specially Obina Iheme,

Nadeeshani, Niluka, Harsha, Gayan, Usman, Nazzal, Pouya and Nasser.

Word would fail to express my greatest appreciation towards my parents & my

siblings Gayan, Anu, and Sana for their inseparable support, gentle love and prayers.

ix

TABLE OF CONTENTS

ABSTRACT ................................................................................................................ iii

ACKNOWLEDGEMENTS ...................................................................................... viii

LIST of FIGURES .................................................................................................... xiv

LIST of SYMBOLS .................................................................................................. xvi

LIST of ABBREVIATIONS ................................................................................... xviii

1 INTRODUCTION ................................................................................................... 1

1.1 Generations of Wireless Cellular Networks ..................................................... 1

1.2 IEEE 802.11 Standards .................................................................................... 4

1.3 Introduction to OFDM ..................................................................................... 6

1.4 Thesis Outline .................................................................................................. 7

2 MULTIPATH FADING CHANNELS ................................................................... 9

2.1 Multipath Fading Basics ................................................................................... 9

2.2 Multipath .......................................................................................................... 9

2.3 Time Dispersion ............................................................................................. 10

2.4 Doppler Spread ............................................................................................... 12

2.5 Rayleigh Fading ............................................................................................. 15

2.6 Frequency Selective and Non Frequency Selective Channels ....................... 17

2.7 Simulations of Fading .................................................................................... 18

3 ORTHOGONAL FREQUENCY DIVISION MULTIPLEXING ......................... 21

x

3.1 Introduction .................................................................................................... 21

3.2 Multicarrier Communication System ............................................................. 22

3.3 Orthogonal Signals and OFDM ...................................................................... 24

3.4 Bandwidth Efficiency ..................................................................................... 26

3.5 Modulation ..................................................................................................... 27

3.6 OFDM Transceiver ........................................................................................ 28

3.6.1 The IDFT and the DFT ........................................................................... 29

3.6.2 Guard Interval ......................................................................................... 31

3.7 Coded OFDM (CODFM) ............................................................................... 34

4 CHANNEL CODING FOR OFDM ...................................................................... 35

4.1 Introduction .................................................................................................... 35

4.2 Convolutional Codes ...................................................................................... 35

4.2.1 Convolutional Encoder ............................................................................ 38

4.2.2 State Diagram of a Convolutional Code ................................................. 38

4.2.3 Trellis Diagram ....................................................................................... 39

4.2.4 Decoding ................................................................................................. 40

4.2.5 Viterbi Algorithm .................................................................................... 41

4.3 Reed Solomon Codes ..................................................................................... 41

4.3.1 Reed Solomon Encoder ........................................................................... 43

4.3.2 Reed Solomon Decoder ........................................................................... 45

4.3.3 Syndrome Calculation ............................................................................. 48

4.3.4 Error Locator Polynomial Calculation .................................................... 49

xi

5 LINK LEVEL BER PERFORMANCE OF QPSK-OFDM OVER AWGN AND

FADING CHANNELS .......................................................................................... 51

5.1 Introduction .................................................................................................... 51

5.2 Simulation of OFDM ..................................................................................... 52

5.2.1 Decoding Un-Coded OFDM over AWGN Channel ............................... 53

5.2.2 CC Coded QPSK-OFDM Over the AWGN Channel ............................. 55

5.2.3 RS-CC Coded QPSK-OFDM Over AWGN Channel ............................. 56

5.3 Performance of QPSK-OFDM Over Multipath Rayleigh Fading Channels .. 58

5.3.1 BER Performance of Un-Coded QPSK-OFDM Over a Non Frequency

Selective Fading Channels ...................................................................... 60

5.3.1.1 BER Performance of Un-Coded QPSK-OFDM Over COST 207 Channel ............................................................................................. 61

5.3.1.2 Performance of un-Coded OFDM Performance Over Winner Scenario 2.8 ...................................................................................... 65

5.3.2 Link Level Performance of Convolutional Coded (CC) QPSK-OFDM

Over Rayleigh Fading Channels ............................................................. 66

5.3.3 Performance of Reed Solomon (RS) Coded QPSK-OFDM Over Rayleigh

Fading Channels ..................................................................................... 67

5.3.4 Performance of QPSK-OFDM using RS/CC Concatenated Coding Over

Rayleigh Fading Channels ...................................................................... 69

5.4 Degradation In System Performance Due to Frequency Offset ..................... 70

5.5 Computation of Spectral Efficiency of the System ........................................ 73

5.6 Analysis of Bit Error Correction Capacity for Individual and Combined

Coding ............................................................................................................ 74

xii

6 CONCLUSIONS AND FUTURE WORK ............................................................ 77

6.1 Conclusions .................................................................................................... 77

6.2 Future Work ................................................................................................... 78

REFERENCES ........................................................................................................... 79

xiii

LIST OF TABLES

Table 1.1 : 3G Characteristics by cell size and mobile speed [1] ............................... 4

Table 1.2: IEEE 802.11a-1999 technical standard ...................................................... 6

Table 2.1: Measured delay spread in frequency range of 800 MHz to 1.5 GHz ...... 11

Table 5.1: OFDM simulation parameters ................................................................. 52

Table 5.2: OFDM timing related parameters ............................................................ 53

Table 5.3: COST 207 reference models .................................................................... 61

Table 5.4: Coherence bandwidth and r.m.s.delay spreads of used channels ............ 61

Table 5.5: COST 207 TU channel parameters .......................................................... 62

Table 5.6: COST 207 BU channel parameters .......................................................... 63

Table 5.7: Winner scenario 2.8 channel .................................................................... 65

Table 5.8: System correction capacity for CC only coded sequence ........................ 74

Table 5.9: System Correction Capacity for RS only coded inputs .......................... 75

Table 5.10: System Correction Capacity for RS/ CC coding scheme ...................... 76

xiv

LIST of FIGURES

Figure 2.1: Multipath encountered while transmitting a signal through an outdoor

wireless environment [2] ....................................................................... 10

Figure 2.2: Sample Power delay Profile [2] .............................................................. 12

Figure 2.3: A typical Rayleigh fading envelope [2] ................................................... 16

Figure 2.4: Jakes’ Fading Simulator [34] ................................................................... 20

Figure 3.1: The frequency-selective channel response and the relatively flat response

on each sub-channel [10] ........................................................................ 22

Figure 3.2: Block diagram of a multi-carriers modulation system with L subcarriers

[3] ............................................................................................................ 23

Figure 3.3: Spectrum for an OFDM system with 16 subcarriers [3]. ....................... 27

Figure 3.4: Simplex point-to-point transmission using OFDM [12]. ....................... 29

Figure 3.5: Multi Carrier Modulation using IDFT .................................................... 31

Figure 3.6: Cyclic extension and windowing of the OFDM symbol [12] ................. 32

Figure 3.7: Guard intervals by cyclic extension [11]. ................................................ 33

Figure 4.1: Block diagram of a constraint length 7 convolutional encoder [16]. ...... 37

Figure 4.2: Generation of a convolutional code [16]. ............................................. 38

Figure 4.3:State diagram of a rate ½ convolutional encoder [16] .............................. 39

Figure 4.4: A trellis diagram corresponding to the encoder on the Figure [17] ......... 40

Figure 4.5: Typical block diagram of Reed Solomon encoder/decoder..................... 42

Figure 4.6: A typical Reed Solomon codeword ......................................................... 43

Figure 4.7: Reed Solomon encoding circuit [2] ......................................................... 45

Figure 5.1: OFDM Performance over AWGN Channel ............................................ 54

xv

Figure 5.2: Performance of rate ½ Convolutional Coded OFDM-QPSK transmission

over AWGN Channel ............................................................................. 56

Figure 5.3: RS-CC Coded QPSK-OFDM BER Performance over AWGN channel. 57

Figure 5.4: RS-CC Coded QPSK-OFDM BER Performance over AWGN channel. 58

Figure 5.5: Theoretical Un-coded OFDM Performance over .................................... 60

Figure 5.6: BER vs. Eb/N0 over COST 207 TU channel (12 taps) ............................. 62

Figure 5.7: BER vs. Eb/N0 over COST 207 BU channel (12 taps)............................. 63

Figure 5.8: Bit Error Rate Comparison for COST 207 TU and BU channels ........... 64

Figure 5.9: BER Performance of Un-Coded QPSK-OFDM over Winner Scenario. 2.8

............................................................................................................... 66

Figure 5.10: BER for CC Coded QPSK-OFDM Over COST207 TU ....................... 67

Figure 5.11: BER for Reed Solomon Coded QPSK-OFDM Over COST207 TU ..... 68

Figure 5.12: BER for QPSK-OFDM With RS and CC Coding Over COST207 TU 69

Figure 5.13: BER for QPSK-OFDM With RS-CC Coding Over COST207 TU ....... 70

Figure 5.14: Illustration of ICI ................................................................................... 71

Figure 5.15: Performance degradation due to frequency offset and variations in Eb/No

............................................................................................................... 72

xvi

LIST of SYMBOLS

Bandwidth

Coherence bandwidth

c(x) Codeword polynomial

d(x) Raw information polynomial

Frequency degradation

Doppler spread

Carrier frequency

Doppler shift

Maximum Doppler frequency

Rayleigh probability density function

Ratio of cyclic prefix time to useful symbol time

g(x) Generator polynomial

k Length of data symbols

Total number of subcarriers

m Number of bits per symbol

n Symbol codeword

Number of paths

AWGN term

P(e) bit error rate

p(x) Parity polynomial

q(x) quotient polynomial

R code rate

xvii

Mean value of the Rayleigh distribution

Rs Symbol rate of main data stream

Received signal

r(x) Remainder polynomial

rem(x) Remainder

Si Syndrome

Transmitted signal

t Symbol error correction capacity

Delay spread

Symbol duration

Velocity

Multiplicative gain of the kth path ∆ Subcarrier frequency spacing ∆Ω Frequency spaced

r.m.s vale of the received voltage

, Variance of the Rayleigh distribution

Phase shiftof the kth path

Wavelength of carrier frequency

Phase of carrier

Random phase

Channel delay spread

Delay of kth path

Efficiency

r.m.s. delay spread

xviii

LIST of ABBREVIATIONS

1G 1st Generation

2G 2nd Generation

3G 3rd Generation

2.5G 2.5 Generation

4G 4th Generation

AMPS Advanced Mobile Phone System

AWGN Additive White Gaussian Noise

BER Bit Error Rate

BPSK Binary Phase Shift Keying

BU Bad Urban

CC Convolution Code

CDMA Code Division Multiple Access

CP Cyclic Prefix

CDPD Cellular Digital Packet Data

DAB Digital Audio Broadcasting

DFT Discrete Fourier Transform

DVB-H Digital Video Broadcasting-Handheld

DVB-H Digital Video Broadcasting-Terrestrial

EU European Union

FDD Frequency Division Duplexing

FDM Frequency Division Multiplexing

FEC Forward Error Correction

FFT Fast Fourier Transform

xix

GPRS General Packet Radio Service

GSM Global System for Mobile

HD High Definition

HSCSD High Speed Circuit Switched Data

ICI Inter Carrier Interference

IDFT Inverse Discrete Fourier Transform

IEEE Institute of Electrical and Electronics Engineers

IFFT Inverse Fast Fourier Transform

IQ In-phase and Quadrature-phase

ISI Inter Symbol Interference

ITU International Telecommunications Union

JTACS Japanese Total Access Communication System

LAN Local Area Network

LO Local Oscillator

LOS Line Of Sight

MS Mobile Station

NLOS Non Line Of Sight

NMT Nordic Mobile Telephone

NTACS Narrowband Total Access Communications Systems

NTT Nippon Telegraph and Telephone

OFDM Orthogonal Frequency Division Multiplexing

PHY Physical

PSK Phase Shift Keying

QAM Quadrature Amplitude Modulation

QPSK Quadrature Phase Shift Keying

xx

RS Reed Solomon

SFBC Space Frequency Block Coding

SNR Signal to Noise Ratio

STBC Space Time Block Coding

TACS Total Access Communications Systems

TDMA Time Division Multiple Access

TIA Telecom Industry Association

TU Typical Urban

UMTS Universal Mobile Telecommunications System

WiFi Wireless Fidelity

1

Chapter 1

1 INTRODUCTION

1.1 Generations of Wireless Cellular Networks

Since the deployment of the wireless communication, which dates almost as far back

as two decades, the cellular communication has grown at a phenomenal rate.

Inevitably, cellular mobile phone companies become more famous & rapidly

developed. In the beginning of the cellular era, communication companies used

analog technologies that were referred to as 1st generation (1G) [1].

All 1G cellular systems exploited analog frequency modulation schemes together

with separate bands for downlink (base station to mobile) and uplink (mobile to base

station). Such systems would be known as frequency division duplex (FDD) systems.

Also to have higher system capacity in each band frequency division multiplexing

(FDM) was used. One of the very earliest 1G systems developed was the Advanced

Mobile Phone System (AMPS) which began to operate in the 800 MHz band.

Numerous other analog 1G technologies were also developed and some include:

Total Access Communications Systems (TACS), NMT cellular, NTT Cellular,

JTACS and NTACS [1].

All 2G cellular networks are known to convert a signal from analog to digital form

and apply some form of digital modulation before transmission. This conversion to

digital form aids the accommodation of more than one user at a time (multiplexing)

2

per radio channel. The two most popular multiplexing schemes used in 2G systems

are time division multiple access (TDMA) and code division multiple access

(CDMA). TDMA systems which include GSM, North American TDMA and PDC,

all use timeslots to allocate a fixed periodic time for a subscriber. The CDMA

cellular systems on the other hand use a digital modulation scheme which is referred

to as the spread spectrum. In CDMA systems each user’s digitally encoded signal is

further encoded by a special code that converts each bit of the original message into

many bits. At the receiver the same special code would be used to recover the

original bit stream.

Soon after the deployment of 2G cellular systems there came an increasing desire for

increased data rate and mobile data delivery. Modifications to the existing 2G

installed systems enabled the so called 2.5G data rates that included bit rates up to

115.2 kbps (IS-95B revision). Cellular digital packet data (CDPD), high speed circuit

switched data (HSCSD) and general packet radio services (GPRS) are a few of the

proprietary systems developed for 2.5G [1].

3G cellular systems have the ability to support high data rate services, advanced

multimedia services and global roaming. Furthermore they need to support advanced

digital services in various different operating environments as shown by Fig 1.1.

Corresponding size, mobility rate and minimum supported data rates for the various

cell types developed for 3G are also given in Table 1.1. A popular 3G scheme known

as Universal Mobile Telecommunications Systems (UMTS) Terrestrial Radio Access

Networks have evolved from GSM. CDMA 2000 on the other hand is an enhanced

wideband version of CDMA. It has been supported by USA’s Telecom Industry

3

Association (TIA) and has the advantage of being backward compatible with CDMA

IS-95B, yet another 2.5G technology.

The goal of 4G is to combine wireless mobile with wireless access communication

technologies. 4G mobile networking will be based on an all-IP architecture and

connectivity for anyone, anywhere. 4G systems are mostly based on a multi user

version of Orthogonal Frequency Division Multiplexing (OFDM).

Figure 1.1: 3G Opereating Environments [1]

4

Table 1.1 : 3G Characteristics by cell size and mobile speed [1]

1.2 IEEE 802.11 Standards

The 802.11a standard is designed to operate in the 5 GHz band (typically 5.15 to

5.35 and 5.725 to 5.825 GHz). The band is not unlicensed, although the devices

operating in a majority of countries may not require a license. The devices operating

with 802.11a are sometimes called WiFi5 to denote the frequency band of the

operation.

The band of 5 GHz is characterized by high loss with distance from the transmitter

and the inability of signals to pass through obstructions such as walls. In order to

overcome these, the 802.11a standard made a major improvement in the physical

layer by using the OFDM modulation. In OFDM, the data to be transmitted is

divided into a large number of orthogonal subcarriers. Each subcarrier now carries

only a few kbps of data as against tens of megabits if a single carrier was to be used.

Because of this, the symbol time of each subcarrier is very large and the subcarriers

are not individually affected by frequency selective fading common in NLOS

environments. The OFDM systems are very resistant to frequency selective fading

5

and propagation path delay variations when the number of subcarriers is large. In the

case of 802.11a, the number of subcarriers used is 52, of which 48 are for data and 4

are for pilots.

IEEE 802.11a with 52 subcarriers (48 for data and 4 as pilot subcarriers) has a carrier

spacing of 0.3125 MHz. Each of these subcarriers can be a BPSK, QPSK, 16-QAM,

or 64-QAM. The total bandwidth is 20 MHz with an occupied bandwidth of

16.6 MHz Symbol duration is 4 microseconds, which includes a guard interval of 0.8

microseconds. The actual generation and decoding of orthogonal components is done

in baseband using DSP which is then unconverted to 5 GHz at the transmitter. Each

of the subcarriers could be represented as a complex number. The time domain signal

is generated by taking an IFFT. Correspondingly the receiver down converts samples

at 20 MHz and does an FFT to retrieve the original coefficients. The advantages of

using OFDM include reduced multipath effects in reception and increased spectral

efficiency. Table 1.2 below shows the spectral efficiency for various

modulation/coding schemes.

The 802.11a devices can operate from 100 to 250 feet (35 to 80 meters) in indoor and

outdoor environments, respectively. The 802.11a standard was also approved for use

in the European Union (EU) in late 2002.

6

Table 1.2: IEEE 802.11a-1999 technical standard Mod. Gross Net FEC Efficiency T1472 B

(Mbit/s) (Mbit/s) rate (Bit/sym.) (µs)

BPSK 06 12 1/2 024 2012

BPSK 09 12 3/4 036 1344

QPSK 12 24 1/2 048 1008

QPSK 18 24 3/4 072 0672

16-QAM 24 48 1/2 096 0504

16-QAM 36 48 3/4 144 0336

64-QAM 48 72 2/3 192 0252

64-QAM 54 72 3/4 216 0224

1.3 Introduction to OFDM

Orthogonal frequency-division multiplexing, or OFDM, is a process of digital

modulation that is used in computer technology today. Essentially, OFDM is

configured to split a communication signal in several different channels. Each of

these channels is formatted into a narrow bandwidth modulation, with each channel

operating at a different frequency. The process of OFDM makes it possible for

multiple channels to operate within close frequency levels without impacting the

integrity of any of the data transmitted in any one channel [2].

Conceptually, it has been known since at least the 1970s. Originally known as

multicarrier modulation, as opposed to the traditional single-carrier modulation,

OFDM was extremely difficult to implement with the electronic hardware of the

time. So, it remained a research curiosity until semiconductor and computer

technology made it a practical method.

7

One major advantage of OFDM is that it is bandwidth efficient. What that really

means is that you can transmit more data faster in a given bandwidth in the presence

of noise. The measure of spectral efficiency is bits per second per Hertz, or bps/Hz.

For a given chunk of spectrum space, different modulation methods will give widely

varying maximum data rates for a given bit error rate (BER) and noise level [2].

One disadvantage of OFDM is its sensitivity to carrier frequency variations. To

overcome this problem, OFDM systems transmit pilot carriers along with the

subcarriers for synchronization at the receiver. Another disadvantage is that an

OFDM signal has a high peak to average power ratio. As a result, the complex

OFDM signal requires linear amplification. That means greater inefficiency in the

RF power amplifiers and more power consumption.

OFDM has been used for digital radio broadcasting—specifically Europe’s DAB and

Digital Radio Mondial. It is used in the U.S.’s HD Radio. It is used in TV

broadcasting like Europe’s DVB-T and DVB-H. You will also find it in wireless

local-area networks (LANs) like Wi-Fi. The IEEE 802.11a/g/n standards are based

on OFDM.

1.4 Thesis Outline

The contents of this thesis are organized as follows. Following the general

introduction to wireless cellular systems, WiFi and OFDM in chapter 1, chapter 2

will focus on the multipath channel and show how Rayleigh fading can be generated

using Jake’s sum of sinusoids model together with measured power delay profile

parameters. A brief description of the channel models used for simulation in this

thesis will conclude chapter 2. OFDM, orthogonality principle, spectral efficiency,

8

the use of cyclic prefix and coding for OFDM will be introduced in chapter 3. The

chapter will end with a mathematical description of OFDM with supporting

equations. Chapter 4 will in general introduce the various channel coding techniques

that can be used together with OFDM. First, the use of a convolutional encoder

together with its corresponding Viterbi decoder at the receiver will be discussed. The

concept of state diagram, trellis diagram, metric computation and locating the most

likely path will be discussed. Secondly concatenated codes where the outside code is

a Reed Solomon and the inner one is a convolutional code will be discussed and

details of Reed Solomon encoding, decoding, syndrome computation and error

locator polynomial calculation will be explained. Chapter 5 will investigate the

performance of QPSK OFDM over AWGN and fading channels (Rayleigh channel)

using convolutional and/or concatenated coding. Simulation results will be provided

particularly for the COST 207 Typical Urban (TU), Bad Urban (BU) channels and

the Winner vehicular NLOS (RS-MS-NLOS) channel. The chapter will also outline

the frequency mismatch between the frequency of the Local Osciallator (LO) at the

transmitter and the receiver and provides some example curves to show the amount

of degradation for different SNR values. Finally Chapter 6 will present the

conclusions and future research directions for the thesis.

9

Chapter 2

2 MULTIPATH FADING CHANNELS

2.1 Multipath Fading Basics

Multipath fading is a feature that needs to be taken into account when designing or

developing a radio communications system. In any terrestrial radio communications

system, the signal will reach the receiver not only via the direct path, but also as a

result of reflections from objects such as buildings, hills, ground, water, etc that are

adjacent to the main path. The overall signal at the radio receiver is a summation of

the variety of signals being received. As they all have different path lengths, the

signals will add and subtract from the total dependent upon their relative phases.

At times, there will be changes in the relative path lengths. This generally results

from motion of the radio transmitter, the receiver, or any of the objects that provides

a reflective surface. Changes in path lengths will result in changes in the phases of

the signals arriving at the receiver, and in turn in the signal strength as a result of the

different ways in which the signals will sum together. This process is what causes the

fading that is present on many signals.

2.2 Multipath

The mobile radio channel places fundamental limitations on the performance of

wireless communication systems. The path between the transmitter and the receiver

can vary from simple line-of-sight (LOS) to one that is severely obstructed by

10

buildings, mountains, and foliage. Unlike wired channels, which are stationary and

predictable, radio channels are extremely random and do not offer easy analysis.

Even, the speed of motion impacts how rapidly the signal level fades as mobile

terminal moves in space. Modeling the radio channel has historically been one of the

most difficult parts of mobile radio system design, and is typically done in a

statistical fashion, based on measurements made specifically for intended

communication system or spectrum allocation [3].

Figure 2.1 depicts an example for the multipath encountered while transmitting a

signal through an outdoor environment.

Figure 2.1: Multipath encountered while transmitting a signal through an outdoor wireless environment [2]

2.3 Time Dispersion

Time dispersion is a manifestation of multipath propagation phenomena. The time

period of the received signal is higher than that of transmitted signal as a result of

distended signal in time due to multipath propagation. Because of multipath

11

propagation the impulse response of wireless systems seems to be series of pulses.

Generated pulses depend on time resolution of the communication system.

One of the key parameter in the design of a transmission system is the maximum

delay spread value that it has to tolerate [4].

Table 2.1: Measured delay spread in frequency range of 800 MHz to 1.5 GHz Median delay spread [ns]

Maximum delay spread [ns]

Reference Remakes

25 50 [13] Office Building 30 56 [14] Office Building 27 43 [15] Office Building 11 58 [16] Office Building 35 80 [17] Office Building 40 90 [17] Shopping centre 80 120 [17] Airport 120 180 [17] Factory 50 129 [18] Warehouse 120 300 [18] Factory

In most of WLAN applications, delay spread is always promotional to the size of

particular indoor area. The median delay spread is 50% value, meaning that 50% of

all channels have a delay spread that is lower than the median value. Clearly, the

median value is not interesting for designing a wireless link, because we have to

guarantee that the link works for at least 90% to 99% [5].

The second column of Table 2.1 gives the measured maximum delay spread values.

The reason to use maximum delay spread instead of a 90% or 99% value is that

many papers only mention the maximum value. From papers listed above that do

present cumulative distribution functions of their measured delay spreads. We can

deduce that the 99% value is only a few percent smaller than the maximum measured

delay spread [5].

12

The maximum delay time spread can me defined as total time interval during which

reflections with significant energy arrive. Mean excess delay and r.m.s-delay

spread ( ) are the time dispersive properties of wide band multipath channels that

most commonly quantify these channels [3].

Figure 2.2: Sample Power delay Profile [2]

2.4 Doppler Spread

The Doppler shift can be defined as a shift in frequency of the signal when an

observer is moving at a particular speed. Signals transmitted in different path may

have different amount of Doppler shift as each signal path has different signal phase.

The Doppler spread is the difference of foresaid different Doppler shifts for different

signals. As fading depend on the constructive or destructive nature of the composite

signals, these channels may have small coherence time.

13

Further, coherence time is inversely related to Doppler spread

(2.1)

Where is the coherence time, is the Doppler spread, and is a constant.

Delay spread and coherence-bandwidth are parameters which describe the time

dispersive nature of the channel in a local area. However it don’t provide the

information about the time varying nature of the channel caused by either relative

motion between the mobile and base station, or by movement of objects [3].

Doppler spread Ds , is a measure of the spectral broadening caused by the time rate of

change of the mobile radio and is defined as the range of frequencies over which the

received Doppler spectrum is essentially non zero. When a pure sinusoidal tone of

frequency is transmitted the received signal spectrum, called Doppler spectrum

will have components in the range where, is the

Doppler shift.

The amount of the spectral broadening depends on which is a function of the

relative velocity of the mobile, and angle between the direction of motion of the

mobile and direction of arrival of the scattered waves. If the baseband signal is much

greater than Ds, the effects of Doppler spread are negligible at the receiver [3].

If the reciprocal bandwidth of the baseband signal is greater than the coherence time

of the channel, then the channel will change during the transmission of the base band

message, thus it cause distortion at the receiver, if the coherence time is defined as

14

the time over which the time correlation function is above 0.5, then the coherence

time can be written as 916 (2.2)

Where, is the maximum Doppler shift and can be computed based on the speed of

the mobile subscriber and the wavelength of the signal ( / ).

A popular rule of thumb for modern digital communication is to define the coherence

time as the geometric mean of eq. (3.2) as shown below:

916 (2.3)

Definition of coherence time implies that two signals arriving with a time separation

greater than are affected differently by the channel [3].

The COST 207 channel models developed for the GSM system, are based on four

different Doppler spectra, . Defining , , as in eq. (2.4) below the four

different spectra are:

, , 12 (2.4)

a) CLASS is used for path delays less than 500 ns ( 500 ns)

CLASS = / | | ≤

(2.5)

15

b) GAUS1 is used for path delays from 500 ns to 2 µs; (500 2 )

(GAUS1) , 0.8 ,0.05 , 0.4 ,0.1 (2.6)

where, is 10 dB below A.

c) GAUS2 is used for path delays exceeding 2 µs ( >2 )

(GAUS2) = , 0.7 ,0.1 + , 0.4 ,0.15 (2.7)

where, is 15 dB below B.

d) RICE is a sometimes used for the direct ray

(RICE) = . / 0.91 0.7 | | ≤ (2.8)

2.5 Rayleigh Fading

Rayleigh fading is a statistical model for the effect of propagation environment in

wireless communication. It commonly used to describe the statistical time varying

nature of the received envelop of a flat fading signal. It’s a known fact that envelop

of the sum of the two independent Gaussian noise signals obey a Rayleigh

distribution as shown in eq. (2.9).

, 0 ∞0, 0

(2.9)

Here, denoted the r.m.s vale of the received voltage signal before envelop

detection, and is the time-average power of the signal before envelops detection

16

[3]. Figure 2.3 provides a sample for a Rayleigh distributed signal envelope over

time.

Figure 2.3: A typical Rayleigh fading envelope [2]

The probability that the envelop of the received signal does not exceed the specified

value R is given by the cumulative distribution function

1 (2.10)

The mean value of the Rayleigh distribution can be written as

∞ 2 1.2533

(2.11)

All the variance of the Rayleigh distribution , is given as

(2.12)

17

∞ 2

(2.13)

2 2 0.4292

(2.14)

The rms value of the envelope is the square root of the mean square, or √2 , where is the standard deviation of the original complex Gaussian signal prior to envelope

detection.

2.6 Frequency Selective and Non Frequency Selective Channels

Based on the parameters of the channel and the characteristics of the signal to be

transmitted, time-varying fading channels can be classified as frequency non-

selective versus frequency selective and slow fading versus fast fading.

In this thesis we mainly focus of frequency non-selective and frequency selective. If

the bandwidth of the transmitted signal is small compared with ∆ then all

frequency components of the signal would roughly undergo the same degree of

fading. The channel is then classified as frequency non-selective (also called flat

fading). We notice that because of the reciprocal relationship between ∆ and ∆ and the one between bandwidth and symbol duration, in a frequency non-

selective channel, the symbol duration is large compared to ∆ . In this case,

delays between different paths are relatively small with respect to the symbol

duration. We can assume that we would receive only one copy of the signal, whose

gain and phase are actually determined by the superposition of all those copies that

come within ∆ .

18

On the other hand, if the bandwidth of the transmitted signal is large compared with ∆ , then different frequency components of the signal (that differ by more

than ∆ ) would undergo different degrees of fading. The channel is then classified

as frequency selective. Due to the reciprocal relationships, the symbol duration is

small compared with ∆ . Delays between different paths can be relatively large

with respect to the symbol duration. We then assume that we would receive multiple

copies of the signal [6].

If the symbol duration is small compared with ∆ , then the channel is classified as

slow fading. Slow fading channels are very often modeled as time-invariant channels

over a number of symbol intervals. Moreover, the channel parameters, which are

slow varying, may be estimated with different estimation techniques.

On the other hand, if ∆ is close to or smaller than the symbol duration, the

channel is considered to be fast fading (also known as time selective fading). In

general, it is difficult to estimate the channel parameters in a fast fading channels.

2.7 Simulations of Fading

From the definition of Rayleigh fading as given above, it is possible for one to

generate this model by generating two independent Gaussian random variables

of . However, it is sometimes the case that it is simply the amplitude

fluctuations that are of interest There are two main approaches to this. In both cases,

the aim is to produce a signal which has the Doppler power spectrum given above

and the equivalent autocorrelation properties. There are two main approaches to this.

In both cases, the aim is to produce a signal which has the Doppler power spectrum

given above and the equivalent autocorrelation properties [7]. In his book [8], Jakes

19

popularized a model for Rayleigh fading based on summing sinusoids. Let the

scattered be uniformly distributed around a circle at angles with k rays emerging

from each scatter. The Doppler shift on ray n is

(2.15)

Jakes’ model is based on summing sinusoids as defined by the following equations:

√2 2 cos cos 22 cos cos 2 √2 sin cos 2

(2.16)

(2.17)

Where

(2.18)

is the random phase given by:

2 (2.19)

Where:

20

is the maximum Doppler frequency, and is the carrier frequency. From

the above development, the fading simulator shown in Fig 2.4 can be developed.

There are M low frequency oscillators with frequency as follows

cos 2 , 1,2,3 … … . , (2.20)

Where M = 1 where N is the number of sinusoids. The amplitudes of

the oscillators are all unity except for the oscillator at frequency which has

amplitude1/√2. Note that Fig 2.4 implements eq. (2.6) except for the scaling factor

of √2. It is desirable that the phase of be uniformly distributed.

This can be accomplished using time averaging described in [8].

Figure 2.4: Jakes’ Fading Simulator [34]

21

Chapter 3

3 ORTHOGONAL FREQUENCY DIVISION

MULTIPLEXING

3.1 Introduction

The principles of orthogonal frequency division multiplexing (OFDM) modulation

have been in existence for several decades. However, in recent years these

techniques have quickly moved out of textbooks and research laboratories and into

practice in modern communications systems. The techniques are employed in data

delivery systems over the phone line, digital radio and television, and wireless

networking systems.

One of the biggest advantages that an OFDM modem offers is that it converts a

dispersive broadband channel into parallel narrowband sub-channels, thus

significantly simplifying equalization at the receiver side. Another intrinsic feature of

OFDM is its flexibility in allocating power and rate, optimally among narrowband

sub-carriers. This ability is particularly important for broadband wireless where

multipath are “frequency selective“(due to cancellation of primary and etched

signals) [9]. In a conventional serial data system, the symbols are transmitted

sequentially, one by one, with the frequency spectrum of each data symbol allowed

to occupy the entire available bandwidth [10].

22

A high rate data transmission supposes very short symbol duration, conducing at a

large spectrum of the modulation symbol. There are good chances that the frequency

selective channel response affects in a very distinctive manner the different spectral

components of the data symbol, hence introducing undesired inter symbol

interference (ISI) [10].

One can assume that the frequency selectivity of the channel can be mitigated if,

instead of transmitting a single high rate data stream, we transmit the data

simultaneously, on several narrow-band sub-channels (with a different carrier

corresponding to each sub-channel), on which the frequency response of the channel

looks “flat” (see Fig. 3.1). Hence, for a given overall data rate, increasing the number

of carriers reduces the data rate that each individual carrier must convey, therefore

lengthening the symbol duration on each subcarrier. Slow data rate on each sub-

channel merely means that the effects of ISI are severely reduced.

Figure 3.1: The frequency-selective channel response and the relatively flat response on each sub-channel [10]

3.2 Multicarrier Communication System

Figure 3.2 shows a general representation of a multicarrier system. The transmitter

can be viewed as consisting of several single carrier transmitters and the receiver as

23

one with the same amount of single receivers. All the transmitters and receivers have

different carrier frequencies. The data stream is multiplexed into N parallel data

stream and modulates each carrier simultaneously.

Let Rs denote the symbol rate of main data stream. Then the data rate on each

subcarrier will be reduced by a factor of L i.e.

And the symbol period, Tn, on each data bank will be,

(3.1)

1

(3.2)

Figure 3.2: Block diagram of a multi-carriers modulation system with L subcarriers [3]

24

Now, let us consider the above example with terrestrial digital TV parameters. DT

uses 6817 active carriers to convey data simultaneously. If the maximum number of

symbols likely to interfere is calculated once again,

(3.3)

And recalling that 224 for such a channel, then, . . . 0.16

symbol will be found which means only a portion of each successive symbol will

interfere. Compared to the number calculated in the case of a single carrier system,

this amount of ISI can be taken care of by the receiver with relative ease. However,

as far as the circuit complexity is concerned, the use of such very large number of

carriers with the same number of modulator/demodulators and filters on both sides is

not practical as all. Also to mention is the increase in the bandwidth required to

accommodate all the carriers. The guard band inserted between each adjacent carrier

to prevent RF interference among carriers (termed Inter carrier Interference –ICI)

causes a significant increase in the amount of the bandwidth required and lowers the

overall spectral efficiency.

3.3 Orthogonal Signals and OFDM

Two signals are said to be orthogonal, if,

0 (3.3)

Frequency division multiplexed signals are inherently orthogonal, since they are

spaced well apart such that they do not overlap. In other words, they have no

25

common components in the frequency spectrum. (A complementary example is the

Time Division Multiplexing signals, which are separated in the time domain.). There

is yet another method of obtaining orthogonal signals without having to separate

them in frequency or time domain. Let and represent two sinusoidal

signals: cos (3.5)

Where k = 1, 2. If the product of these two signals is integrated over some period, Tu,

12 cos 12 cos

(3.6)

The result is heterodyning (mixing) of the two signals. Now, to establish

orthogonality between these two signals, the conditions under which the r.h.s. of the

above equation is zero must be set. For this integral to equal zero, the integration

period must contain an integral number of cosine periods. Defining,

∆ (3.7)

2 , 1,2 (3.8)

1∆ , 1, 2, .. (3.9)

satisfies this condition. In other words, the separation ∆ between two subcarriers

must be exactly equal to the integral multiples of the reciprocal of . Here, is

defined as the integral period (period over which the receiver integrates the received

26

signal) and hence is actually the symbol period. As for the first term on the r.h.s of

Eq. (3.6), simply setting ∆ , it is easily seen that the this term will integrate to

zero as well. We therefore conclude that, simply by defining the separation between

each successive subcarrier equal to the reciprocal of all the subcarriers are made

orthogonal to each other. Noting also that, 1 (3.10)

Actually gives the symbol rate, , or the data rate on each parallel bank (symbol

rate on each subcarrier ), we further notice from Eq. (3.7) and (3.8) that the

bandwidth of each subcarrier is equal to the separation frequency and hence they

overlap. It is obvious that the total data rate of the system. The spectrum is used

efficiency, as opposed to conventional FDM system with guard bands.

3.4 Bandwidth Efficiency

One important consideration about OFDM is related to bandwidth efficiency. Using

a serial system with one carrier, the minimum bandwidth required is 1/ and

the bandwidth efficiency is 1 / because the spectrum of this pulse is

represented by the sinc function whose first zero is at 1/ . The approximate

bandwidth of a (2 1 carriers system using an impulse as well as M sine and

cosine carriers of length (2 1 become as follows [11]. 12 1

(3.11)

27

Yielding a bandwidth efficiency of

2 1 /1 (3.12)

For large M, the efficiency tends to 2Bd/Hz. For a value of M = 512, the bandwidth

efficiency = 1.998Bd/Hz.

Figure 3.3: Spectrum for an OFDM system with 16 subcarriers [3].

3.5 Modulation

In an OFDM link, the data bits are modulated on the subcarriers by some form of

phase shift keying (PSK) or quadrature amplitude modulation (QAM). To estimate

the bits at the receiver, knowledge is required about the reference phase and

amplitude of the constellation on each subcarrier. In general, the constellation of

28

each subcarrier shows a random phase shift and amplitude change, caused by carrier

frequency offset, timing ffset and frequency-selective fading. To cope with these

unknown phase and amplitude variations, two di_erent approaches exist. The first

one is coherent detection, which requires estimates of the reference amplitudes and

phases to determine the best possible decision boundaries for the constellation of

each subcarrier.

The second approach is differential detection, which does not use absolute reference

values, but only looks at the phase and/or amplitude differences between two QAM

values. Differential detection can be done both in the time domain and in the

frequency domain. Iin the first case, each subcarrier is compared with the subcarrier

of the previous OFDM symbol. In the case of differential detection in the frequency

domain, each subcarrier is compared with the adjacent subcarrier within the same

OFDM symbol.

3.6 OFDM Transceiver

Fig 3.2 shows the block diagram of a simple point-to-point transmission system

using OFDM and FEC coding.

29

Figure 3.4: Simplex point-to-point transmission using OFDM [12].

3.6.1 The IDFT and the DFT

The IDFT and the DFT are used for, respectively, modulating and demodulating the

data constellations on the orthogonal sub carriers. These signal processing algorithms

replace the banks of I/Q modulators and demodulators that would otherwise be

required. Note that at in the fig 3.2 the input of the IDFT, N data constellation points

xi, k are present, where N is the number of DFT points. (i is an index on the

subcarrier, k is an index on the OFDM symbol). These constellations can be taken

according to any phase shift keying (PSK) or QAM signaling set (symbol mapping).

The N output samples of the IDFT, being in time domain, form the baseband signal

carrying the data symbols on a set of N orthogonal SCs. In a real system, however,

not all of these N possible subcarriers can be used for data [12].

FFT merely represents a rapid mathematical method for computer applications of

Discrete Fourier Transform (DFT). The ability to generate and to demodulate the

signal using a software implementation of FFT algorithm is the key of OFDM

current popularity. In fact, the signal is generated using the Inverse Fast Fourier

Transform (IFFT), the fast implementation of Inverse Discrete Fourier Transform

30

(IDFT). There is a mysterious connection between this transform and the concept of

multicarrier modulation. According to its mathematical distribution, IDFT

summarizes all sine and cosine waves of amplitudes stored in X[k] array, forming a

time domain signal [13].

. (3.13)

cos , 0,1, . 1 (3.14)

It’s simply observe that IDFT takes a series of complex exponential carriers,

modulate each of them a different symbol from the information array X[k], and

multiplexes all this to generate N samples of a time domain signal (Figure 3.3). And

what is really important, the complex exponential carriers are orthogonal to each

other, as we know from the Fourier decomposition. These carriers are frequency

spaced ∆Ω . If we consider that the N data symbols X[k] come from sampling

analog information with a frequency of fs, an easy to make discrete to analog

frequency conversion indicates a ∆ 1/ spacing between the subcarriers of the

transmitted signal. The N samples of the time domain signals are synthesized from

sinusoids and cosinusoids of frequencies . The weight with which each complex

exponential contributes to the time domain signal. Waveform is given by the

modulation symbol X[k]. Therefore, the information X[k] to be transmitted could be

regarded as being defined in the frequency domain.

31

Figure 3.5: Multi Carrier Modulation using IDFT

At the receiver, the inverse process is realized, the time domain signal constitutes the

input to a DFT “signal analyzer ”, implemented of course using the FFT algorithm.

The FFT demodulator takes the N time domain transmitted samples and determines

the amplitudes and phases of sine and cosine waves forming the received signal,

according to the equation below [13].

(3.15)

cos , 0,1, . 1 (3.16)

3.6.2 Guard Interval

The second key principle is the introduction of a cyclic prefix as a guard interval,

whose length should exceed the maximum excess delay of the multipath propagation

channel. Due to the cyclic prefix, the transmitted signal becomes periodic, and the

effect of the time-dispersive multipath channel becomes equivalent to a cyclic

convolution, discarding the guard interval at the receiver [12]. Due to the properties

of the cyclic convolution, the effect of the multipath channel is limited to a point

32

wise multiplication of the transmitted data constellations by the channel time frame,

or the Fourier transform of the channel IR (radio impulse) that is, the sub carriers

remain orthogonal [12].

Its duration Tguard is simply selected to be larger than the maximum excess delay of

the (worst case) radio channel. Therefore, the effective part of the received signal can

be seen as the cyclic convolution of the transmitted OFDM symbol by the channel IR

[12].

Figure 3.6: Cyclic extension and windowing of the OFDM symbol [12]

By dividing the input data stream in N subcarriers, the symbol duration is made N

time longer, which reduces the relative multipath delay spread, relative to the symbol

time, by the same factor. To eliminate Inter-Symbol Interference (ISI) almost

completely, a guard time is introduced for each OFDM symbol. The guard interval is

chosen larger than the expected delay spread, such that multipath components from

33

one symbol cannot interfere with the next symbol. The method is best explained with

reference to figure 3.4 Every block of N samples as obtained by IFFT is quasi-

periodically extended by a length Ng simply repeating Ng samples of the useful

information block [11].

Figure 3.7: Guard intervals by cyclic extension [11].

The total sequence length becomes N + Ng samples, corresponding to duration of Ts

+ Tg. Trailing and leading samples of this extended block are corrupted by the

channel transient response, hence the receiver should demodulate only the central N

number of samples, essentially unaffected by the channel's transient response. It must

be appreciated that cyclic extension actually wastes channel capacity as well as

transmitted power; however, if the useful information blocks are long, the extension

length can be kept low relative to the useful information block length. Thus the

efficiency in terms of bit rate capacity can be expressed as shown in equation 3.5

[11].

(3.17)

34

3.7 Coded OFDM (CODFM)

OFDM avoids the problem of inter-symbol interference by transmitting a number of

narrow-band subcarriers together with using a guard time. This gives rise to another

problem, however, which is the fact that in a multipath fading channel, all subcarriers

will arrive at the receiver with different amplitudes. In fact, some subcarriers may be

completely lost because of deep fades. Hence, even though most subcarriers may be

detected without errors, the overall bit-error ratio (BER) will be largely dominated by

a few subcarriers with the smallest amplitude, for which the bit-error probability is

close to 0.5. To avoid this domination by the weakest subcarriers, forward-error

correction coding is essential. By using coding across the subcarriers, error of weak

ones can be corrected up to a certain limit that depends on the code and the channel.

35

Chapter 4

4 CHANNEL CODING FOR OFDM

4.1 Introduction

In coding theory, concatenated codes form a class of error-correcting codes that are

derived by combining an inner code and an outer code. They were conceived in 1966

by Dave Forney [31] as a solution for the problem of finding a code that has both

exponentially decreasing error probability with increasing block length and

polynomial-time decoding complexity.

Concatenated codes were first implemented for deep space communication in the

Voyager program, which launched their first probe in 1977. Typically, the inner code

is not a block code but a soft-decision convolutional Viterbi-decoded with a short

constraint length. For the outer code, a longer hard-decision block code, frequently

Reed Solomon with 8-bit symbols is selected. The larger symbol size makes the

outer code more robust to burst errors that may occur due to channel impairments.

The combination of an inner Viterbi/convolutional code with an outer Reed-Solomon

code (known as an RSV code) was first used on Voyager 2, and became a popular

construction both within and outside of the space sector. It is still notably used today

for satellite communication, such as the DVB-S digital television broadcast standard.

4.2 Convolutional Codes

Convolutional codes are intensively using to ensure reliable data transfer. A

convolutional code maps each k-bits of a continuous input stream on n output bits,

where the mapping is performed by convolving the input bits with a binary impulse

36

response. The convolutional encoding can be implemented by a simple shift register

and modulo-2 adders. As an example, figure 4.1 shows the encoder for a rate 1/2

code which is actually one of the most frequently applied convolutional codes. This

encoder has a single data input and two outputs G1 and G2, which are interleaved to

form the coded output sequence A B A B … [14].

Each pair of output bits G1 and G2, which depends on seven input bits, being the

current input bit plus six previous input bits that are stored in the length-6 shift

register. The value of 7- or in general the shift register length plus 1- is called the

constraint length. The shift register taps are often specified by the corresponding

polynomials or generator vectors. For the example of figure 4.1, the generator

vectors are 1111001, 1011011 or 171, 133 octal. The ones in the generator

vectors correspond with taps on the shift register [14].

Decoding of convolutional codes is most often performed by soft-decision Viterbi

decoding, which is an efficient way to obtain the optimal maximum likelihood

estimate of the encoded sequence. The complexity of Viterbi decoding grows

exponentially with the constraint length. Hence, practical implementations do not go

further than a constraint length of about 10. Decoding of convolutional codes with

larger constraint length is possible by using suboptimal decoding techniques like

sequential decoding [15]. Since convolutional codes do not have a fixed length, it is

more difficult to specify their performance in terms of Hamming distance and a

number of correctable errors [14].

37

One measure that is used is the free distance, which is the minimum Hamming

distance between arbitrarily long different code sequences that begin and end with

the same state of the encoder, where the state is defined by the contents of the shift

register of the encoder. For example, the previously shown code has a free distance

of 10. When hard decision decoding is used, this code can correct up to floor ((10-1)

/2) = 4 bit errors within each group of encoded bits with a length of about 3 to 5

times the constraint length. When soft decision decoding is used, however, the

number of correctable errors does not really give a useful measure anymore. A better

performance measure is the coding gain, which is defined as the gain in the bit

energy-to-noise density ratio Eb/N0 relative to an un-coded system to achieve a

certain bit error ratio. The Eb/N0 gain is equivalent to the gain in input signal-to-noise

ratio (SNR) minus the rate loss in dB because of the redundant bits [14].

Figure 4.1: Block diagram of a constraint length 7 convolutional encoder [16].

38

4.2.1 Convolutional Encoder

At the encoder, the input is fed continually through a shift register of length m. The memory

of the code is equal to the “constraint length” plus one (m+1). Each time a new bit enters into

the register, several modulo-2 sums of the present and past bits are formed. The choice of

which bits are operated on is designated as a polynomial P(z) with binary coefficient. n such

modulo-2 sums are formed and multiplexed to form the output of the “mother code”. Since

n-bits are generated for each bit, the rate of the code is 1/n.

The rate of the code can be increased by the process of puncturing. This involves deleting

some of the bits generated by the mother code. The process of generating a convolutional

code is shown in Fig. 4.2.

Figure 4.2: Generation of a convolutional code [16].

4.2.2 State Diagram of a Convolutional Code

A convolution code can also be described by a state diagram. Since the output of the

39

encoder is determined by the input and the current state of the encoder, state diagram can

be used to represent the encoding process. The state diagram depicts the possible states of

the encoder and the possible transitions from one state to another [03]. For a binary

convolutional code, the number of state is s = 2m, where m is the number of shift register

stages in the transmitter. In the state diagram shown in figure 4.3, the transition from a

current state to a next state are determined by the current input bit, and each transition

produces an output of n bits [17].

Figure 4.3:State diagram of a rate ½ convolutional encoder [16]

4.2.3 Trellis Diagram

A tree diagram generally shows the structure of the encoder in the form of a tree where the

branches of the tree represent the various states and the output of the encoder. Close

observation of a tree reveals that the structure repeats itself once the number of stages is

greater than the constraint length. It is observed that all branches emanating from two

nodes having the same state are identical in the sense that they generate identical output

40

sequences. This means that two nodes having the same label can be merged. By doing this

throughout the tree diagram, we can obtain what is called the trellis diagram which is a

more compact representation [03].

As shown in Fig 4.4 a solid line corresponds to an input of 0 and a dotted line to an input

of 1.Each path in the trellis diagram corresponds to a valid sequence from the encoder’s

output.

Figure 4.4: A trellis diagram corresponding to the encoder on the Figure [17]

4.2.4 Decoding

The function of the decoder is to estimate the encoded input information using a rule or

method that results in the minimum possible number of errors. There is a one-to-one

correspondence between the information sequence and the code sequence. Further, any

information and code sequence pair is uniquely associated with a path through the trellis.

Thus, the job of the convolutional decoder is to estimate the path through the trellis that

was followed by the encoder [03].

41

There are a number of techniques for decoding convolutional codes. The most

important of these methods is the Viterbi algorithm which performs maximum

likelihood decoding of convolutional codes. The algorithm was first described by

A. J. Viterbi [30]. Even though both hard and soft decision decoding can be

implemented, it has been shown that soft decision decoding is would provide

approximately 2-3 dB better gain [03].

4.2.5 Viterbi Algorithm

A Viterbi decoder uses the Viterbi algorithm for decoding a bit stream that has

been encoded using a convolutional code. It was developed by Andrew J. Viterbi

and was published in an IEEE transaction in 1967 [18]. According to Viterbi’s

descriprion the algorithm consists of three major parts:

I. Branch Matric Calculation

Calculation of a distance between the input pair of bits and the four possible

“ideal” pairs (“00”, “01”, “10”, “11”)

II. Path Metric Calculation

For every encoder state, calculate a metric for the survivor path ending in this state

(a survivor path is a path with the minimum metric).

III. Back Tracing

This step is necessary for hardware implementations that don't store full

information about the survivor paths, but store only one bit decision every time

when one survivor path is selected from the two.

4.3 Reed Solomon Codes

Reed-Solomon codes are block-based error correcting codes with a wide range of

applications in digital communications and storage. The characteristics of an RS

42

code will be decided based on the number and type of the errors that it is aiming to

correct. Reed Solomon codes are non-binary cyclic codes which are used to correct

errors that appear in burst. A typical reed Solomon system is as shown below:

Figure 4.5: Typical block diagram of Reed Solomon encoder/decoder

A Reed-Solomon code is specified as RS (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. The block length for a

(n,k) RS codes is: 2 1 (4.1)

A Reed-Solomon decoder can correct up to t symbol errors where t is defined as

below: /2 (4.2)

Data Source Reed Solomon Encoder

Communication actions channel

or storage device

Reed Solomon decoder Data Sink

43

Figure 4.6 provides a representation for a typical RS codeword where 2t parity bits

are appended at the end of each data block.

Figure 4.6: A typical Reed Solomon codeword

Large values of t means large number of errors can be corrected.

4.3.1 Reed Solomon Encoder

While describing a Reed Solomon encoder, the following polynomials are frequently

used:

d(x): raw information polynomial,

p(x) parity polynomial,

c(x) codeword polynomial,

g(x) generator polynomial,

q(x) quotient polynomial,

r(x) remainder polynomial.

The information polynomial d(x) can be written as, … …

And the parity polynomial p(x) as,

(4.3)

44

… … … … … (4.4)

The encoded RS polynomial can thus be expressed as

(4.5)

vector of n field elements , , , is taken as a codeword if and only if it is

a multiple of the generator polynomial g(x). The generator polynomial for a Reed

Solomon code correcting up to t-errors has the form:

(4.6)

A common way for encoding a cyclic code is to derive p(x) by dividing d(x) by g(x).

This yields an irrelevant quotient polynomial q(x) and an important polynomial r(x)

as follows

(4.7)

Thus, the codeword polynomial can be expressed as

(4.8)

If the parity polynomial is defined as being equal to the negatives of the coefficient

of r(x), then it follows that

(4.9)

Thus, by ensuring that the code word polynomial is a multiple of the generator

polynomial, a Reed Solomon encoder can be constructed to obtain p(x).

45

The key to encoding and decoding is to find r(x), the remainder polynomial, which

maps to the transmitted data [03].

Initially, all registers are set to 0, and the switch is set to the data position. Code

symbols cn-1 through cn-k are sequentially shifted into the circuit and simultaneously

transmitted to the output line. As soon as code symbol cn-k enters the circuit, the

switch is flipped to the parity position, and the gate to the feedback network is

opened so that no further feedback is provided [03].

Figure 4.7: Reed Solomon encoding circuit [2]

4.3.2 Reed Solomon Decoder

The decoder for a RS (n,k) code would look at all possible subsets of k symbols from

the set of n symbols that were received. For the code to be correctable in general, at

least k symbols had to be received correctly, and k symbols are needed to interpolate

46

the message polynomial. An ideal decoder would interpolate a message polynomial

for each subset, and it would keep track of the resulting polynomial candidates.

Unfortunately, due to the fact that there exist a lot of subsets, the algorithm is

impractical.

A practical solution to this problem was proposed by Peterson as the syndrome

decoding and was later developed by Berlekamp-Massey.

Assuming that a codeword c(x) is represented as, … (4.10)

and the channel errors results in the received codeword … (4.11)

Then, the error pattern e(x) is the difference between c(x) and r(x). Using Equation

4.10 and 4.11

e(x) = r(x) – c(x) = + + (4.12)

The 2t partial syndromes Si, 1 2 , can be defined as . Since , , … , are roots of each transmitted codeword c(x), it follows that c ( ) = 0

and Si ) + e ) = e ). Thus, it is clear that the 2t partial syndrome

depend only on the error pattern e(x) and not on the specific codeword r(x) [03].

47

Supposing that the error pattern contains k errors (k ) at locations, , , … , ,where 0 … … 1. If the error magnitude at

each location is denoted as e . Then e (x) has the form

+ + + (4.13)

Defining the set of error locators as , 1,2, … … , the set of 2t partial

syndromes yields the following system of equations:

… (4.14)

… (4.15)

… (4.16)

Any algorithm which solves this system of equations can be considered as a Reed

Solomon decoding algorithm. The error magnitudes are found directly, and the

error locations can be determined from .

A typical Reed-Solomon decoder uses five distinct algorithms. The first algorithm

calculates the 2t partial syndrome Si. The second step in RS decoding process is the

Berlekamp-Massey algorithm. Which calculate the error locator polynomial . This polynomial is a function of the error location in the received codeword r(x), but

it does not directly indicate which symbols of the received codeword are in error at

48

each location. Finally, knowing both the error locations in the received codeword and

the magnitude of the error at each location, a correction algorithm may be

implemented to correct up to t errors successfully [03].

4.3.3 Syndrome Calculation

The syndrome of a cyclic code is typically defined as the remainder obtained by

dividing the received codeword r(x) by the generator polynomial g(x). However, for

Reed-Solomon codes, 2t partial syndromes are computed. Each partial syndrome Si

is defined as the remainder when dividing the received codeword r(x) by .

, 1,2, … ,2 (4.17)

The division of two polynomial results in a quotient polynomial q(x) and a remainder

polynomial rem(x). The degree of the remainder rem(x) must be less than the degree

of the dividing polynomial p(x). Thus, if p(x) has degree 1 (i.e., p(x) = ), rem(x)

must have degree 0. In other words, rem(x) is simply a field element and can be

denoted as rem. Thus, the determination of the 2t partial syndromes begins with the

calculation

(4.18)

Letting x = and rearranging the above equation yields . (4.19)

Thus, the calculation of the 2t partial syndromes can be simplified from a full-

blown polynomial division (which is computationally intense) to merely evaluating

the received polynomial r(x) at :

49

, 1,2, … .2 (4.20)

Since

(4.21)

then has the following form: r r x … … … . . r x (4.22)

The evaluation of can be implemented very efficiently in software by

arranging the function so that it has the following form r r r … … … . . r α (4.23)

The evaluation of can be implemented very efficiently in software by

arranging the function so that it has the following form … . r r r r (4.24)

4.3.4 Error Locator Polynomial Calculation

The Reed Solomon decoding process is any implementation which solves equations

(4.14) through (4.16). These 2t equations are symmetric function in , , … , , known as power-sum symmetric function. We now define the polynomial 1 1 … 1 … (4.25)

The roots of are , , … , , which are the inverse of the error locations

numbers . Thus, is called the error locator polynomial because it indirectly

contains the exact locations of each of the error in r(x). Note that is an unknown

50

polynomial whose coefficient must also determined during the Reed Solomon

decoding process.

The coefficient and error location numbers are related by the following

equations 1 (4.26)

(4.27)

(4.28)

… … … … (4.29)

The unknown quantities and can be related to the known partial syndromes

by the following set of equations known as Newton’s Identities.

0 (4.30)

2 0 (4.31)

3 0 (4.32)

… . 0 (4.33)

The most common method for determining is the Berlekamp-Massey algorithm.

51

Chapter 5

5 LINK LEVEL BER PERFORMANCE OF QPSK-OFDM

OVER AWGN AND FADING CHANNELS

5.1 Introduction

The aim of this chapter is to show the link-level BER performance of un-coded and

coded QPSK-OFDM over AWGN and fading channels. For channel coding a rate ½

convolutional code and a concatenated code of Reed Solomon/Convolutional Code

will be used. Simulation results will be provided particularly for the COST 207

Typical Urban (TU), COST 207 Bad Urban (BU) and the Winner Vehicular NLOS

(RS-MS-NLOS) channels.

Standardized models are important tools for the development of new radio systems.

They allow assessing the benefits of different techniques (signal processing, multiple

access, etc.) for enhancing capacity and improving performance, in a manner that is

unified and agreed on by many parties. To achieve this a common reference for

evaluating different MIMO concepts in outdoor environments was developed by

3GPP/3GPP2 groups and this is referred to as the spatial channel model (SCM) [19].

SCM and extended SCM (SCME) [19] are both geometry based stochastic channel

models. The Winner channel model adopted in this thesis is of the SCM type. The

COST 207 model is used for SISO and SIMO channel modeling. It has a wideband

power delay profile and has been used in the development of GSM, and used as a

basis for the decision on modulation and multiple-access methods. Even though in

52

COST 207 four different scenarios can be considered in this thesis only the Typical

Urban (TU) and the Bad Urban (BU) will be considered.

Together with the performance analysis results for coded QPSK OFDM the chapter

will also highlight the performance degradation that can be incurred due to frequency

offset at the local oscillators of the transmitter and the receiver. Chapter will

conclude by a discussion on how to compute spectral efficiency for concatenated

QPSK-OFDM.

5.2 Simulation of OFDM

Parameters used in the simulation of OFDM in this thesis are summarized in Table

5.1. The performance of OFDM with concatenated Reed Solomon/Convolutional

Codes was simulated over different channel conditions. The effect of the speed of the

receiver was also taken into consideration while simulating the multipath

performance of the QPSK-OFDM.

Theoretical and simulated results would be compared in order to show conformance

of the simulation to already developed theory. The BER is the number of received

bits that have been altered due to noise, interference and distortion, divided by the

total number of transferred bits during a studied time interval.

Table 5.1: OFDM simulation parameters Parameter Value FFT Size 1024

Constellation Mapping QPSK Symbol duration 102.4µs Length of Cyclic Prefix 1/8 of Symbol duration (12.8µs)

Channel Coding R = ½ CC and Viterbi Decoding

53

Multipath Channels • Winner Channel • COST 207

Reed Solomon code RS(11,15,4)

Table 5.2: OFDM timing related parameters Number of data subcarriers 48

Number of pilot subcarriers 4

Total number of subcarriers 52

Subcarrier frequency spacing 0.3125MHz

IFFT/FFT period 3.2 µs

Preamble duration 16 µs

Signal duration BPSK_OFDM

symbol 4

Guard interval (GI) duration 0.8

Training symbol GI duration 0.4

Symbol interval 4

Short training duration 8

Long training sequence duration 8 2

Power delay profile

∑ P τ τ∑ P τ

r.m.s. delay spread

Coherence Bandwidth 15

5.2.1 Decoding Un-Coded OFDM over AWGN Channel

It has been stated in [55] that the additive white Gaussian noise in the time domain

channel corresponds to AWGN of the same average power in the frequency domain.

54

Hence the performance of OFDM in AWGN channel should be identical to that of a

serial modem’s. For a serial modem the bit-error-rate (BER) versus signal-to-noise

ratio (SNR) characteristics are determined by the modulation scheme used. Fig 5.1

shows the link level BER performance for the un-coded QPSK-OFDM system over

the AWGN channel. For QPSK the theoretical BER is given by 0.5 / (5.1)

Figure 5.1: OFDM Performance over AWGN Channel

The observed SNR loss of approximately 0.5 dB in the simulated BER curve is as a

result of the cyclic prefix introduced by OFDM [20]. The SNR loss can be calculated

as:

55

10 1 (5.2)

Where, denotes the length of cyclic prefix and T = is the length of

the transmitted symbol. Assuming 102.4 and 12.8 and using them

in Eq. (5.2) the SNR loss will be around 0.5 . As can be seen this theoretical value

and the SNR loss that was incurred closely agrees.

5.2.2 CC Coded QPSK-OFDM Over the AWGN Channel

One way to improve system performance is the make use of error coding schemes

over the carriers [21]. In this thesis we applied two forms of coding: 1) rate ½

convolutional coding (CC), 2) concatenated Reed Solomon/Convolutional coding.

Firstly, the data is coded using a rate ½ convolutional encoder with a constraint

length of 7 and decoded using a corresponding Viterbi decoder with a trace back

length of 32 (approximately 5 times the constraint length). In the concatenated

coding a Reed Solomon code of RS (11, 15, 4) was used and again the rate for the

CC was taken as ½.

The simulation was repeated 400 times for 15 OFDM symbols and the results were

averaged. It can be seen from Fig 5.2 that the system BER performance for CC coded

QPSK-OFDM would reach a much lower bit error rate at an earlier signal to noise

ratio. As an example the un-coded QPSK-OFDM will achieve a target BER of 10

at an SNR of 8.5 dB whereas the coded system will achieve the same BER at 6.2 dB.

56

Figure 5.2: Performance of rate ½ Convolutional Coded OFDM-QPSK transmission

over AWGN Channel

Increasing the number of bits or symbols in a frame to be transmitted, we can have

lower BER. The significant reduction in the SNR in order to achieve a required BER

is known as coding gain [22]. Fig 5.2 shows that with convolutional coding of rate R

= ½ and constraint length of K = 7, at a BER of 10-5 there is a coding gain of 2.4 dB

over the un-coded performance.

5.2.3 RS-CC Coded QPSK-OFDM Over AWGN Channel

The concatenated coding scheme originally proposed by Forney [23], is an error

correcting coding scheme in which two different error correcting codes called the

57

inner code and outer code are concatenated, achieving comparatively high coding

gain by using codes that are relatively easy to decode. The concatenated coding

scheme using the convolutional coding with Viterbi decoding for the inner code and

the Reed-Solomon (RS) code for the outer code is a special case of the general

concatenated coding scheme proposed by Forney. In this thesis we also chose to use

this special coding scheme as proposed by Forney.

Using an outer Reed Solomon code of RS (11,15,4) and an inner rate ½

convolutional code of constraint length 7 we have obtained the physical layer BER

performance depicted in Fig. 5.3.

Figure 5.3: RS-CC Coded QPSK-OFDM BER Performance over AWGN channel.

0 0.5 1 1.5 2 2.5 3 3.5 410

-5

10-4

10-3

10-2

10-1

100

Eb/No (dB)

BE

R

Sim RS-CC-QPSK-OFDM-AWGN

58

To show the advantage of concatenated codes theoretical QPSK, simulated CC coded

and RS-CC coded QPSK-OFDM performances have all been plotted on the same set

of axis in Fig. 5.4 ,[31].

Figure 5.4: RS-CC Coded QPSK-OFDM BER Performance over AWGN channel.

Clearly for a BER target rate of 10-4 using a RS code of (11,15,4) brings in a 2.2 dB

extra gain over the CC coded system.

5.3 Performance of QPSK-OFDM Over Multipath Rayleigh Fading

Channels

Performance of the multipath channel can be much worse than that in an AWGN

channel. Hereafter in this simulation mobility of the receiver will be taken into

consideration. In this section of the thesis we will compare the system performances

0 1 2 3 4 5 6 710

-4

10-3

10-2

10-1

100

Eb/No (dB)

BE

R

Sim RS-CC-QPSK-OFDM-AWGN

Sim CC-QPSK-OFDM-AWGNTheor BER

59

over the COST 207 Typical Urban (TU), COST 207 Bad Urban (BU) and the Wiener

channel model at different Doppler frequencies.

As to justify the correctness of the simulated results some theoretical BER curves

were also provided using eq. (5.3) obtained from [33].

12 34 1 1 1 tan 1 (5.2)

Where,

∆ (5.3)

From the theoretical curves depicted in Fig 5.5, it can be observed that the BER is

only slowly decaying with increasing Eb/No . Intuitively, this can be understood by

considering the significant differences in the signal to noise ratio between different

carriers. The dips in the frequency responses of typical indoor multipath channels

lead to low SNR values causing high bit error probabilities on the bad carriers which

dominate the average bit error rate.

60

Figure 5.5: Theoretical Un-coded OFDM Performance over

Rayleigh Multipath Fading Channel

5.3.1 BER Performance of Un-Coded QPSK-OFDM Over a Non Frequency

Selective Fading Channels

In this section we simulate the BER performance of un-coded QPSK-OFDM over the

COST 207 and Wiener channels. With the help of the relative power delay profile

values provided in Tables 5.2, 5.3 and 5.4 and using Jake’s sum of sinusoids model,

three different performance curves were simulated for each channel using Doppler

shifts of 100Hz, 400Hz, and 833 Hz (corresponding to speeds of 30 Km/hr, 125

Km/hr, and 250 Km/hr respectively).

61

5.3.1.1 BER Performance of Un-Coded QPSK-OFDM Over COST 207

Channel

COST 207 has developed wideband propagation models that have gained widespread

usage not only through their adaptation in the GSM standard but also because of their

simplicity [24]. The COST 207 project had proposed reference models using an

exponential profile with one or two decaying peaks. Table 5.3 below shows the

different power delay profiles for the various environment types considered under

COST project[45].

Table 5.3: COST 207 reference models Environment Power Delay Profiles Typical Urban (TU) (non hilly) exp /1Rural Area (RA) (non hilly) exp 9.2 / 1Bad Urban (BU) (hilly) exp /1 for 0 50.5exp 5 /1 for 0 10Hilly Terrain (HT) 0.1exp 15 /1 for 15 20

Coherence bandwidth is a statistical measure of the range of frequencies over which

the channel can be considered flat (i.e., a channel which passes all spectral

components with approximately equal gain and linear phase). In order to be

frequency non selective fading channel coherence bandwidth should be less than the

bandwidth of the channel which is 5 MHz. Computed values of coherence bandwidth

and r.m.s. delay profile of fading channels are listed in Table 5.4.

Table 5.4: Coherence bandwidth and r.m.s.delay spreads of used channels

Fading Channel Coherence bandwidth (MHz)

r.m.s delay spread

COST 207 Typical Urban 0.1216 1.65 µs

COST 207 Bad Urban 0.1044 1.915µs

Winner scenario 2.8 2.10 95.13 ns

62

The power and delay parameters for the COST 207 TU channel has been listed for

the 12 tap scenario in Table 5.5. These values has have been adopted from [25] and

the BER vs the Eb/No curve obtained for the TU channel is provided in Fig 5.6.

Table 5.5: COST 207 TU channel parameters Tap Number

Relative Delay ( ) Fading (dB)

0 0 -4 1 0.2 -3 2 0.4 0 3 0.6 -2 4 0.8 -3 5 1.2 -5 6 1.4 -7 7 1.8 -5 8 2.4 -6 9 3.0 -9 10 3.2 -11 11 5.0 -10

Figure 5.6: BER vs. Eb/N0 over COST 207 TU channel (12 taps)

63

Similarly Table 5.5 shows the power and delay parameters for the COST 207 Bad

Urban (BU) channel.

Table 5.6: COST 207 BU channel parameters

Tap Number Relative Delay ( ) Fading (dB)

0 0 -7 1 0.2 -3 2 0.4 -1 3 0.8 0 4 1.6 -2 5 2.2 -6 6 3.2 -7 7 5.0 -1 8 6.0 -2 9 7.2 -7 10 5.0 -10 11 10.0 -15

Figure 5.7: BER vs. Eb/N0 over COST 207 BU channel (12 taps)

64

As shown in the Figures 5.6 and 5.7 the COST 207 TU and BU channels has been

simulated over Doppler frequencies of 100 Hz, 400Hz and 833Hz respectively. Note

that in each case as the Doppler frequency increases the error floor in the BER will

move higher.

The BER performances for the COST 207 TU and BU channels plotted against each

other in Fig 5.8 indicate that as expected the QPSK-OFDM will perform better in the

Typical Urban case. This is expected because the maximum relative delay for the

Bad Urban channel is twice that of the typical urban channel.

Figure 5.8: Bit Error Rate Comparison for COST 207 TU and BU channels

65

5.3.1.2 Performance of un-Coded OFDM Performance Over Winner Scenario 2.8

With the help of the relative power delay profile parameters provided in Table 5.7

and Jake’s sum of sinusoids model, the following performance curves were obtained

over the Winner scenario 2.8 for Doppler shifts of 100Hz, 400Hz and 833 Hz.

Table 5.7: Winner scenario 2.8 channel Tap index

Relative Delay(ns)

Average Power(dB)

1 0 -1.25 2 10 0 3 40 -0.38 4 60 -0.1 5 85 -0.73 6 110 -0.63 7 135 -1.78 8 165 -4.07 9 190 -5.12 10 220 -6.34 11 245 -7.35 12 270 -8.86 13 300 -10.1 14 325 -10.5 15 350 -11.3 16 375 -12.6 17 405 -13.9 18 430 -14.1 19 460 -15.3 20 485 -16.3

From Fig. 5.9 we can observe that the QPSK-OFDM performance goes below a BER

value of 10-4 for 100Hz Doppler shift. This in comparison to the COST 207 is a

much better performance and is also expected because even though the Winner

scenario 2.8 has 20 taps the max delay it inures is 485 ns. This is only 10 times

smaller in comparison to the 5µs for the COST207 TU channel.

66

Figure 5.9: BER Performance of Un-Coded QPSK-OFDM over Winner Scenario. 2.8

5.3.2 Link Level Performance of Convolutional Coded (CC) QPSK-OFDM

Over Rayleigh Fading Channels As indicated in [23], coding is essential in order to achieve good BER and/or SER

performance over fading channels. Since the wireless channel has the multipath

effects some form of coding is inevitable. In this thesis, we first study the QPSK-

OFDM performance making use of convolutional coding and Viterbi decoding. This

section demonstrates the advantage of using a rate ½ convolutional code with a

constraint length of 7 as opposed to not using coding at all.

The BER performance for the CC coded QPSK-OFDM system is shown in Fig. 5.10.

The channel assumed is the COST 207 TU.

67

Figure 5.10: BER for CC Coded QPSK-OFDM Over COST207 TU

5.3.3 Performance of Reed Solomon (RS) Coded QPSK-OFDM Over Rayleigh

Fading Channels

In this section the BER performance of QPSK-OFDM only with RS coding is

demonstrated. In order to be consistent the COST207 TU channel has been assumed.

Figure 5.11 depicts the BER performance achievable over this channel when the RS

code is RS (11, 15,4).

68

Figure 5.11: BER for Reed Solomon Coded QPSK-OFDM Over COST207 TU

Comparison curve of rate ½ with constraint length 7 Convolutional code and Reed

Solomon code RS (11, 15, 4) over COST 207 Typical Urban is shown in the Fig

5.12. As shown in the Fig below convolutional code system having better

performances for higher SNR values, typically above 17 dB convutional code system

is better than that of Rees Solomon code system. We have also put the uncoded, CC

coded and RS coded system performances in Fig 5.12 for comparative purposes. As

can be seen from this figure for lower SNR values RS coded QPSK-OFDM has

better performance than the CC-coded version and for higher Eb/No values

(exceeding 18dB) the CC coded system performance supersedes that of RS coded

QPSK-OFDM system.

69

Figure 5.12: BER for QPSK-OFDM With RS and CC Coding Over COST207 TU

5.3.4 Performance of QPSK-OFDM using RS/CC Concatenated Coding Over

Rayleigh Fading Channels

As explained in the section 4.3 Reed Solomon codes can correct up to floor((n-k)/2)

erroneous symbols. Since each symbol contains m-bits, a maximum of m×floor ((n-

k)/2) erroneous bits may be corrected for each codeword. Reed Solomon codes are

particularly useful for correcting error that may result over bursty channels.

The main advantage of concatenated coding is that it can provide large coding gain

with less implementation complexity, instead of using a single block code or a

convolutional code. Fig 5.13 shows the performance of the Reed Solomon/CC coded

QPSK-OFDM system. Clearly the using of an outside RS encoder brings a big

70

improvement to the system performance. For example at an Eb/N0 of 20dB the BER

gain is approximately two and a half orders of magnitude better then the CC coded

version and RS coded version.

Figure 5.13: BER for QPSK-OFDM With RS-CC Coding Over COST207 TU

5.4 Degradation In System Performance Due to Frequency Offset

Frequency offset takes place mainly due to Doppler shift and frequency drift in the

modulator and demodulator oscillators. Due to the relative motion between

transmitter and receiver (mobile scenario) an error in system performance will result

due to non perfect synchronization. The other main source of frequency offset is due

to the frequency errors in the oscillators. According to the IEEE 802.11a standards

the oscillators may have frequency errors within 20 ppm 20 10 . For 6

GHz carriers maximum frequency error allowed is computed as below: |∆ | 2 20 10 5 10 200 (5.4)

71

Frequency offset destroys the Orthogonality between carriers and introduces inter

carrier interference between them. Fig 5.14 shows the loss of orthogonality causes by

the ICI (non zero values at the sampling points).

Figure 5.14: Illustration of ICI

All OFDM subcarriers are orthogonal as long as they have a different integer number

of cycles within the FFT interval. If there is a frequency offset, then the number of

cycles in the FFT interval is not an integer anymore which causes ICI to occurs after

the FFT. The FFT output for each subcarrier will contain interfering terms from all

other subcarriers, with an interference power that is inversely proportional to the

frequency spacing. The amount of ICI for subcarriers in the middle of the OFDM

spectrum is approximately twice as large as that for subcarriers at the band edges,

because the subcarriers in the middle have interfering subcarriers on both sides, so

there are more interferers within a certain frequency distance [26]. In [27], it was

shown that the degradation in system performance can be approximated using:

72

103 ln 10 ∆ (5.5)

Where ∆ is the frequency offset, T is the symbol duration in seconds and Es/No is

the Signal-to-noise ratio (SNR) under which the performance is tested. Fig 5.15

shows the calculated degradation in dB for various frequency offset and Es/No values

respectively. As can be seen from the figure for smaller variations in SNR, the

degradation is much less than for larger variations. Authors of [26] also point this

out.

Figure 5.15: Performance degradation due to frequency offset and variations in Eb/No

73

5.5 Computation of Spectral Efficiency of the System

If we denote the inner and outer code rates of the serial concatenated block coding

scheme as / and / , respectively, then the overall system code

rate, R, can be calculated using eq. (5.6).

. (5.6)

Hence, with respect to parameters used in this thesis, the overall code rate of

concatenated Reed Solomon/convolutional code system would be 11/15 1/2 0.36 . The overall code rate that results is lower than both the Reed

Solomon rate and the rate of the convolutional code individually.

Using this new concatenated rate value (R) one can compute the spectral efficiency

for the concatenated system using Eq. (5.7) of [32],[33].

1 × × (5.7)

Here is spectral efficiency, is the bit error rate of the tested system, n is the

number of bits in the block, m is the number of bits per symbol and r is the overall

code rate of the system.

74

5.6 Analysis of Bit Error Correction Capacity for Individual and

Combined Coding

In order to get an understanding on how the concatenated RS/CC channel coding

improves the system performance in comparison to CC and RS only coded versions

the following analysis was carried out. Firstly, a 64 bit input sequence was coded by

a rate ½ non systematic convolutional code (128 bits per block after coding) and the

experiment was repeated 5000 times (identical to having 5000 blocks of coded data).

In the simulation both scattered and burst type bit errors were introduced into each

block. From the literature it is known that for a rate ½ non-systematic CC with

constraint length of 7 the minimum free distance, df, would be 10. Accordingly the

minimum number of bit error the CC generally would guarantee to correct would be

≈ 4. The number of bit errors introduced per block for scattered and burst

scenarios and the percentage of corrections in all 5000 iterations (blocks) was as

follows:

Table 5.8: System correction capacity for CC only coded sequence Scattered Bit Errors

# of bit errors introduced per block Correction capacity (%)

4 bits 100 %

5 bits 100 %

6 bits 100 %

7 bits 99.62 %

10 bits 99.14 %

Burst Bit Errors

# of bit errors introduced per block Correction capacity (%)

2 bits 100 %

3 bits 50.76 %

4 bits 50.40 %

75

It is worth mentioning that these results were obtained for a CC with G1 = 171oct and

G2 = 133oct.

Secondly, the RS (11, 15, 4) coded input sequences were individually tested. Each

block was designed to have 44 information bits and 5000 iterations were carried out.

RS(11,15,4) is expected to correct up to 8-bit errors (2 symbols) per block. Since

RS is known to be good for burst errors (consecutive bits in errors) we have tested

the RS only scenario with 2 and 3 consecutive symbol errors. The results obtained

were as follows:

Table 5.9: System Correction Capacity for RS only coded inputs Burst Bit Errors

# of bit errors introduced per block Correction capacity (%)

4 bits 100 %

8 bits 100 %

12 bits 7.7 %

It can be seen from Table 5.9 that as expected the RS can correct up to 8 consecutive

bit errors per block and can do this consistently for 5000 blocks. However when the

number of bit errors per block is made more than 8 (3 symbols) the correction

capacity drastically reduces.

Finally, the concatenated RS/CC coding scheme was tested. The trace back length

used by the Viterbi decoder was taken as 35 (5 × K). For scattered and for mixed

type bit errors the correction capacity of the system with RS/CC scheme is as shown

in Table 5.10.

76

Table 5.10: System Correction Capacity for RS/ CC coding scheme

Scattered Bit Errors

# of bit errors introduced per block Correction capacity (%)

4 bits 100 %

5 bits 100 %

6 bits 100 %

7 bits 100 %

8 bits 100 %

9 bits 100 %

10 bits 100 %

11 bits 99.32 %

12 bits 94.30 %

14 bits 93.28 %

Half Scattered and half Burst Errors (total of 14 bits)

Scattered Burst Correction capacity (%)

10 bits 4 bits 83.2400

9 bits 5 bits 86.9000

7 bits 7 bits 90.04

The table indicates that for scattered type bit errors the system would be able to

perfectly correct each block for as long as the number of bit errors were less than or

equal to 10. For more than 10 bit errors the correction capacity will start to drop. As

for mixed type of errors the best performance was obtained when bit errors were half

by half.

By comparing the correction capacities in all three cases we can say that for scattered

bit errors the combined RS/CC correction capacity is 40 % more than that of CC only

(10 bits as opposed to 6 bits correction). Also, for the RS/CC when mixed type of bit

errors were assumed, the best performance was obtained when 7 bit errors of each

type were introduces. This is because the RS is good for burst but not for scattered

errors and CC vice versa.

77

Chapter 6

6 CONCLUSIONS AND FUTURE WORK

6.1 Conclusions

In this thesis we studied the performance of un-coded, convolutional coded and

concatenated Reed Solomon/Convolution coded QPSK-OFDM over AWGN and non

frequency selective multipath fading channels. The channels assumed were the

Winner and the COST 207. For the COST 207 the two environments simulated were

the Typical Urban (TU) and the bad urban (BU).

Simulations carried out over the AWGN channel indicated that both the CC coding

and RS/CC coding will bring extra gains to the system. When a rate ½ Convolutional

code of constraint length 7 was used together with a Viterbi decoder of traceback

length of 32 the QPSK-OFDM system performance at a BER of 10-4 would give a

2dB gain. It was also observed that when the RS (11,15,4) was used as an outer code

together with the rate ½ CC another 2dB gain was attained.

Second set of simulations carried out over the fading channels were done for

uncoded and coded scenarios also. The best BER performance was obtained over the

Winner channel (smallest delay spread) and this was followed by COST 207 TU and

COST 207 BU. As expected the BU showed a bit more inferior performance in

comparison to the COST 207 TU due to its larger delay spread (10 micro seconds as

opposed to 5).

78

Further it was observed that when CC and or RS/CC coding was applied to the

QPSK-OFDM over the fading channels then the system performance would improve

drastically. Since in our study no equalization is assumed the effect of coding is

observed much more easily. For example, when CC is used for QPSK-OFDM

transmission over the COST207 TU fading channel the error floor occurs close to a

BER of 10-4 as opposed to 10-2 for an un-coded system.

6.2 Future Work

In the future this work can be expanded by assuming multiple antenna scenarios and

using space time and space frequency block codes. It is also possible to expand the

work so that instead of OFDM we have its multiple user versions, the OFDMA.

79

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