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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