frequency domain equalization of modulation … · dissertation frequency domain equalization of...

Dissertation

Frequency Domain Equalizationof Modulation Formats

with Low Peak to Average PowerRatio

Frequenzbereichsentzerrung

von Modulationsverfahren

mit niedrigem Spitzen- zu Mittelwert Verhaltnis

Der Technischen Fakultaet Fakultat derUniversitat Erlangen-Nurnberg

zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Tufik Buzid, MSc.

Erlangen, 2010

2

Als Dissertation genehmigt vonder Technischen Fakultat derUniversitat Erlangen-Nurnberg

Tag der Einreichung: 20.07.2009Tag der Promotion: 16.12.2009Dekan: Prof. Dr.-Ing. habil. Reinhard GermanBerichterstatter: Prof. Dr. Mario Huemer

Prof. Dr. Leonard ReindlProf. Dr.-Ing. Dr.-Ing. habil. Robert Weigel

SC/FDE - Buzid

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Zusammenfassung

Der Austausch von Informationen mit hohen Datenraten zwischen verschiedenen mo-bilen oder stationaren Endgeraten benotigt Techniken, welche die Beschrankungen durchden Funkkanal uberwinden. Ein Funkkanal, ein Kanal mit Mehrwegeausbreitung, wirddurch die verschiedenen Signalwege bis hin zum Ziel beschrieben. Zusatzlich zum di-rekten Weg, falls dieser uberhaupt existiert, konnen verzerrte Kopien des Signals auchdurch Reflexion, Beugung und Streuung zum Empfanger gelangen. Eine der großtenHerausforderungen der Datenubertragung mit hoher Datenrate ist die Uberwindung derdurch die Mehrwegeausbreitung verursachten Zeitdispersion. Eine Herangehensweise andas Problem der Zeitdispersion ist die Anwendung des Orthogonal Frequency DivisionMultiplexing (OFDM). Der Hauptnachteil hierbei ist allerdings der große Dynamikbere-ich des OFDM-Signals. Dieser ist definiert durch das Peak to Average Power Ratio(PAPR). Dies ist ein wichtiges Thema falls der HF-Leistungsverstarker durch Nichtlin-earitat beeintrachtigt wird, was bis zu einem gewissen Maß praktisch immer der Fallist. Zudem sind Verstarker fur Signale mit hohem Spitzen-zu Mittelvert Verhaltnismeist sehr ineffizient. Eine weitere Breitbandtechnik ist die Eintragerubertragung mitFrequenzbereichsentzerrung (engl. single carrier transmission with frequency domainequalisation, SC/FDE). SC/FDE bringt nachweislich die gleiche Leistungsfahigkeit wieOFDM und weist weniger PAPR auf als dieses. Besondere Beachtung fand SC/FDE inKombination mit Quadratur-Amplitudenmodulation (QAM), die noch immer ein sub-optimales PAPR aufweist. Allerdings erlaubt SC/FDE im Gegensatz zu OFDM denEinsatz von Modulationsarten mit konstanter Einhullender. Durch Verwendung dieserModulationsart, welche zur Klasse der nichtlinearen Modulationsverfahren gehort undsich durch kontinuierliche Phaseneigenschaften auszeichnet, kann das PAPR des betra-chteten Systems auf den Wert Eins reduziert werden. Durch den Einsatz der LaurentDarstellung wird eine lineare Form des Signals mit kontinuierlicher Phasenmodulation(CPM) erreicht, die es ermoglicht das CPM-Signal linear zu demodulieren. Folglich wirddie ursprunglich wegen seiner Trellis-Struktur große Komplexitat des Empfangers deut-lich reduziert. Daruber hinaus konnen Techniken wie z.B. Matched Filtering und lineareEntzerrung, die bereits fur lineare Modulation entwickelt wurden, ohne Schwierigkeitenfur die nichtlineare Modulation verwendet werden. In dieser Arbeit wird die linearisierteForm der CPM auf SC/FDE (CPM-SC/FDE) angewendet. Eine andere Form der Modu-lation mit konstanter Amplitude, die als Tamed Frequenzmodulation (TFM) bekannt istund vor einem Jahrzehnt eingefuhrt wurde, wird ebenfalls auf SC/FDE angewendet. Dasresultierende System wird mit OFDM unter dem Aspekt der Nichtlinearitat der HF-Stufe

iii

verglichen. Der Vergleich zeigt, dass die Leistungsfahigkeit der CPM-SC/FDE nichtso stark beeintrachtigt wird wie die der OFDM, deren Leistungsfahigkeit dramatischnachlasst. Obwohl das erhaltene System im nichtlinearen Bereich des Verstarkers ar-beitet, bleibt seine Fehler Performanz aufgrund der inharenten differentiellen Codierungunbefriedigend. Um diese Fehler-Performanz zu verbessern, wird eine vereinfachte Vari-ante des Laurent Mapping angewendet. Zur Verbesserung der CPM-SC/FDE Band-breiteneffizienz wird die Multiple Input Multiple Output (MIMO)-Technik eingefuhrt.Dabei wird vor allem raumliches Multiplexing verwendet, da es die Transmissionseffizienzstark erhoht und somit die Systembandbreite linear mit der Anzahl der verwendetenAntennen ansteigt. Daruber hinaus wird SC/FDE mit linearen und nichtlinearen Mod-ulationsarten erweitert fur ”Point to Multipoint”-Anwendungen und mit Spreizcodes(Code Division Multiple Access, CDMA) kombiniert. Schließlich wird SC/FDE-CDMAmit einer Kombination von OFDM und CDMA (MC-CDMA) verglichen und skizziert,dass eine Anwendung der Fourier Spreizcodes auf ein voll ausgelastetes SC/FDE-CDMAzu einem OFDM-System fuhrt, wahrend die Kombination von OFDM und CSMA einSC/FDE-Verfahren erzeugt.

SC/FDE - Buzid

Abstract

High data rates and the exchange of large amounts of information among various mobileor roaming and stationary terminals require techniques that conquer the restrictionsimposed by the wireless channels. A wireless channel, known as multipath channel, ismodeled by a number of paths that a signal travels to reach its destination. In ad-dition to a direct path, if it exists, distorted copies of the signal may arrive at thereceiver through different paths that are formed by reflections, diffractions and scat-tering. One of the most challenging problems in high data rate wireless transmissionis to overcome the time dispersion caused by multipath propagation. An approach toovercome the problems of time dispersion is the use of orthogonal frequency divisionmultiplexing (OFDM). A primary drawback is the large dynamic range of the OFDMsignal. The signal dynamic range is defined by the peak to average power ratio (PAPR).This is an important topic when RF power amplifiers suffer from nonlinearity, which isin practice always the case to some extent. Another broadband technique is the singlecarrier transmission with frequency domain equalization (SC/FDE). SC/FDE has beenshown to exhibit similar performance as OFDM and shows less PAPR than OFDM.Mainly SC/FDE has been investigated in combination with quadrature amplitude mod-ulation (QAM) formats which still show non-optimum PAPR. But, in contrast to OFDM,SC/FDE also allows the use of constant amplitude type of modulations. By using a con-stant amplitude type of modulation, which is nonlinear and marked by the continuousphase property, the PAPR of the concerned system can be reduced to one. By thedeployment of the Laurent representation a linear form of the CPM signals is estab-lished, which enables the CPM signals to be linearly demodulated. Consequently, thereceiver complexity which is originally high due to the trellis nature of the structure ofthe CPM receivers, is significantly reduced. Further, the techniques already developedfor linear modulation, e.g. matched filtering and linear equalization can be straightfor-wardly adapted to non-linear modulation. In this work, the linearized form of the CPMis adapted to SC/FDE (CPM-SC/FDE). Another form of modulation with a constantamplitude, that is known as tamed frequency modulation (TFM), introduced a decadeago, is also adapted to SC/FDE. The resulting (emerged) system is compared to OFDMunder the constraints of the non-linearity of the RF stage. The comparison shows thatthe performance of CPM-SC/FDE is not affected as OFDM whose performance deteri-orates dramatically. Although the emerged system is effective in the non-linear regionof the amplifier, its error performance remains unsatisfactory because of the inherentdifferential encoding. To improve the error performance, a simplified approach of Lau-

v

rent mapping is applied. In order to improve the CPM-SC/FDE bandwidth efficiency,a multiple input multiple output (MIMO) technique is introduced. Spatial multiplexingis particularly applied as it improves the transmission efficiency tremendously. There-fore, the system bandwidth efficiency increases linearly with the number of the deployedantennas. Further, SC/FDE with linear and non-linear modulation formats is extendedto ”point to multipoint” applications. It is combined with code division multiple ac-cess (CDMA). Finally, SC/FDE-CDMA is compared to the combination of OFDM andCDMA (MC-CDMA), and it is outlined, that applying Fourier spreading codes to fullload SC/FDE-CDMA yields to an OFDM system, whereas, in contrary, the combinationof OFDM and CDMA produces an SC/FDE scheme.

SC/FDE - Buzid

List of Abbreviations

ACI Adjacent channel interference

ACTS Advanced communications technologies and services

ADSL Asymmetric digital subscriber line

AGC Automatic gain control

AMPS Advanced mobile phone service

ASK Amplitude shift keying

AWGN Additive white Gaussian noise

B3G Beyond 3G

BER Bit error rate

BPSK Binary phase shift keying

BT Bandwidth time product

CDMA Code division multiple access

CDPD Cellular digital packet data

COHASK Coherent amplitude shift keying

COHFSK Coherent frequency shift keying

COHPSK Coherent phase shift keying

CPM Continuous phase modulation

DCS1800 Digital cellular system 1800

DECT Digital European cordless telephone

DEPSK Differential encoded PSK

DFT Discrete Fourier transform

DMSK Duobinary MSK

DPSK Differential phase shift keying

DVB-C Digital video broadcasting-Cable

DVB-S Digital video broadcasting-Satellite

vii

DVB-T Digital video broadcasting-Terrestrial

EDGE Enhanced data rates for GSM evolution

ETSI European telecommunications standard institute

FDM Frequency division multiplexing

FDMA Frequency division multiple access

FDOSS Frequency domain orthogonal signature sequences

FFT Fast Fourier transform

FPLMTS Future public land mobile telephone system

FSK Frequency shift keying

GMSK Gaussian minimum shift keying

GPRS General packet radio service

GPS Global positioning system

GSM Group special mobile

GSMSK Generalized SMSK

GTFM Generalized tamed frequency modulation

HPA High power amplifier

IDFT Inverse discrete Fourier transform

IFFT Inverse fast Fourier transform

IJF-QPSK Intersymbol-interference and jitter-free QPSK

ISI Intersymbol interference

ITU International telecommunication union

JTACS Japanese total access communication system

LMDS Local multipoint distribution service

MMDS Multichannel multipoint distribution service

MMSE Minimum mean square error

MSK Minimum shift keying

NMT-450 Nordic mobile telephones

NONCASK Non-coherent amplitude shift keying

NONCFSK Non-coherent frequency shift keying

OFDM Orthogonal frequency division multiplexing

OFDMA Orthogonal frequency division multiple access

OOK On-off keying

OQASK Offset quadrature amplitude shift keying

SC/FDE - Buzid

viii

OQPSK Offset quadrature phase shift keying

PA Power amplifier

PAM Pulse amplitude modulation

PAPR Peak to average power ratio

PDC Personal digital cellular

PDM Pulse duration modulation

PHS Personal handyphone system

PPM Pulse position modulation

PSK Phase shift keying

PSTN Public switching telephone networks

QAM Quadrature amplitude modulation

QPSK Quadrature phase shift keying

SAW Surface acoustic wave device

SC/FDE Single carrier with frequency domain equalization

SFSK Sinusoidal frequency shift keying

SMSK Simplified MSK

SOQPSK Shaped offset quadrature phase shift keying

SOQPSK Staggered offset quadrature phase shift keying

TACS Total access communication systems

TDMA Time division multiple access

TETRA Terrestrial radio access

TFM Tamed frequency modulation

UMTS Universal mobile telecommunication system

UW Unique word

V-BLAST Vertical-Bell laboratories layered space- time

VCO Voltage controlled oscillator

WLAN Wireless local area network

XPSK Cross-Correlated PSK

ZF Zero forcing

SC/FDE - Buzid

Contents

1. Introduction 1

1.1. Evolution of Mobile Radio Communications . . . . . . . . . . . . . . . . 1

1.1.1. First Generation Cellular Systems . . . . . . . . . . . . . . . . . . 2

1.1.2. Second Generation Cellular Systems . . . . . . . . . . . . . . . . 2

1.1.3. Third Generation Cellular Systems . . . . . . . . . . . . . . . . . 2

1.1.4. Forth Generation Cellular Systems . . . . . . . . . . . . . . . . . 3

1.2. SC/FDE Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. Goals of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Basics on Modulation Techniques 7

2.1. Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2. Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3. Hybrid Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1. Pulse Amplitude Modulation (PAM) . . . . . . . . . . . . . . . . 10

2.3.2. Pulse Position Modulation (PPM) . . . . . . . . . . . . . . . . . . 11

2.3.3. Pulse Duration Modulation (PDM) . . . . . . . . . . . . . . . . . 11

2.4. Digital Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1. Amplitude Shift Keying (ASK) . . . . . . . . . . . . . . . . . . . 13

2.4.2. Frequency Shift Keying (FSK) . . . . . . . . . . . . . . . . . . . . 13

2.4.3. Phase Shift Keying (PSK) . . . . . . . . . . . . . . . . . . . . . . 18

2.4.4. Continuous Phase Modulation (CPM) . . . . . . . . . . . . . . . 23

2.5. Modulation Schemes Selection Criteria . . . . . . . . . . . . . . . . . . . 24

Contents x

3. Signal Envelopes Choices 28

3.1. Non-Constant Envelope Signals . . . . . . . . . . . . . . . . . . . . . . . 28

3.2. Near Constant Envelope Signals . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1. Intersymbol-Interference and Jitter-free QPSK (IJF-QPSK) . . . 29

3.2.2. Cross-Correlated PSK (XPSK) . . . . . . . . . . . . . . . . . . . 30

3.3. Constant Envelope Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4. Constant Envelope Signal Generation . . . . . . . . . . . . . . . . . . . . 30

3.4.1. Shaped Offset QPSK (SOQPSK) . . . . . . . . . . . . . . . . . . 31

3.4.2. Gaussian MSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.3. Generalized Serial Minimum Shift Keying (GSMSK) . . . . . . . 31

3.4.4. Correlative Coded Minimum Shift Keying (Correlative MSK) . . . 32

3.4.5. Sinusoidal Frequency Shift Keying (SFSK) . . . . . . . . . . . . . 34

3.4.6. Continuous Phase Modulation . . . . . . . . . . . . . . . . . . . . 35

3.5. Modulation Index and Signal Phase States . . . . . . . . . . . . . . . . . 43

4. Demodulation of CPM 46

4.1. CPM Receiver Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2. Decomposition of CPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3. Linear Representation of CPM . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1. Multi-level CPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.2. The Importance of Linear Representation . . . . . . . . . . . . . . 57

4.3.3. General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.4. Linear Representation Approximation . . . . . . . . . . . . . . . . 58

4.3.5. Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4. Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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Contents xi

5. The Concept of Single Carrier Transmission with Frequency Domain Equal-ization (SC/FDE) 66

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2. System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.1. Baseband Equivalent Model . . . . . . . . . . . . . . . . . . . . . 68

5.2.2. Scrambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.3. Channel Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.4. Interleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.5. Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3. The Mobile Radio Channel and Additive Noise . . . . . . . . . . . . . . . 72

5.3.1. Additive White Noise . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2. The Mobile Radio Channel . . . . . . . . . . . . . . . . . . . . . . 73

5.3.3. The IEEE 802.11a Channel Model . . . . . . . . . . . . . . . . . . 75

5.4. Transmission Model and Optimum Receiver Structure . . . . . . . . . . . 76

5.4.1. Burst- and Block Structure and Guard Interval . . . . . . . . . . 77

5.4.2. The System Bandwidth Efficiency . . . . . . . . . . . . . . . . . . 79

5.4.3. Optimum Linear Receiver . . . . . . . . . . . . . . . . . . . . . . 80

5.4.4. Frequency Domain Equalization Concept . . . . . . . . . . . . . . 82

5.5. The Peak Power Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6. Complementary Cumulative Distribution Function Curves (CCDF) . . . 87

6. SC/FDE for Constant and Near Constant Envelope Modulation 88

6.1. SC/FDE for OQPSK Modulation . . . . . . . . . . . . . . . . . . . . . . 88

6.2. SC/FDE for Constant Envelope Modulation Schemes . . . . . . . . . . . 89

6.2.1. The Transmission Model . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.2. The Receiver Model . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.3. Impact of Front-End Nonlinearity on CPM-SC/FDE System Performance 92

6.4. Spatial Multiplexing for CPM-SC/FDE . . . . . . . . . . . . . . . . . . . 97

SC/FDE - Buzid

Contents xii

7. Point to Multipoint Systems, SC/FDE-CDMA 101

7.1. CDMA Cellular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2. SC/FDE-CDMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.3. Multicarrier-CDMA versus SC/FDE-CDMA . . . . . . . . . . . . . . . . 104

A. Appendix 108

A.1. Fourier Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.2. Convolution Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.3. Circulant Convolution and Toeplitz Matrix . . . . . . . . . . . . . . . . . 109

A.4. Sampling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.4.1. Up Sampling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 110

A.4.2. Down Sampling Matrix . . . . . . . . . . . . . . . . . . . . . . . . 110

SC/FDE - Buzid

1. Introduction

Today, if someone goes back in literature twenty years or more, he will surprisingly findthat the fundamental motivation behind the introduction of the wireless communicationsis the idea of replacing the costly cabling systems. The drawback of cabling system com-prises both the cost of the wire (copper) and the consumed time for cabling. Nowadays,the motivations have gone further than that. The wireless services have become an im-portant element in the society, if not its core. They take a more central role in our dailylives and can be seen as a stepping-stone to improve quality of life in the coming years.The applications range from the voice services to numerous forms of data services. Thedata at the end user is converted into a voice or picture or command or movies or acombination of all of that. The wireless technology profile is like a snowball; the moreyou roll it over and over, the larger it becomes. Today’s emerged systems could havebeen a science fiction twenty years ago. In the early eighties the cordless telephones wereinvented and provided the end user a mobility range of tens of meters what representedlarge scale mobility at this time. From cordless telephones that allowed a mobility rangeof few tens of meters and that were owned by wealthy individuals it took only about20 years to cellular systems that support global mobility and multimedia services tomillions if not billions of people.

1.1. Evolution of Mobile Radio Communications

As mentioned wireless communication has been dedicated to point to point transmission,since Guglielmo Marconi first demonstrated radio’s ability to provide contact with theships sailing in the English channel in 1897. Further, other point to point applicationslike e.g., satellite communications, the half duplex communication, or the atmosphericcommunication (Ground-sky wave) have been introduced. But the wireless revolutionhas really started with the introduction of the digital cellular communication systems,which are divided in the first generation (1G), the second generation (2G), the thirdgeneration (3G) and beyond 3G (B3G) or forth generation (4G) wireless systems.

1. Introduction 2

1.1.1. First Generation Cellular Systems

Pagers and cordless radio phones are the main products of 1G. These products used onlythe analog modulation techniques. Across Europe, many incompatible systems operatedand a pan-European roaming was not possible. For example, the Nordic countries andthe Netherlands deployed Nordic mobile telephones (NMT-450). The UK and Irelandintroduced total access communication systems (TACS). Germany and Portugal had theC-net. Further, other international standards are the advanced mobile phone system(AMPS) in the US and Japanese total access communication system (JTACS) in Japan.

1.1.2. Second Generation Cellular Systems

In the late eighties new communication organization bodies for standardization werefounded in Europe (the European telecommunication standards institute (ETSI) andits working groups) aiming to create a communication environment that provides Eu-ropeans a roaming across the European countries. The ETSI inherited all Europeantelecommunication standardization activities from the European conference of postaland telecommunications administrations (CEPT) in 1989. A group called ”group spe-cial mobile GSM” (a French name) within the CEPT was charged with developing apan-European standard in 1982. Later, the GSM has been accepted in many parts ofthe world and enjoys world wide recognition. Besides, there are the standards IS-95(later known as cdmaOne) and IS-136 in the US and the personal digital cellular (PDC)in Japan. All these systems apply digital communication techniques. The GSM uses acombination of time division multiplexing (TDMA) and frequency division multiplexing(FDMA) for multiple access. The US IS-95 system uses a CDMA technique. In thefield of cordless telephones, two systems belong to the second generation and make useof microcells which cover small distances. One is the cordless telephone (CT2), which isdedicated to a voice transmission and does not support handovers between base stations.The other one is the digital European cordless telephony (DECT) which accommodatesdata as well as voice transmissions.

1.1.3. Third Generation Cellular Systems

2G systems are mainly characterized by the transition of analog towards a fully digitizedtechnology. Further, besides voice service the 2G systems enabled the user to roam overa few kilometers, but only with a voice service and maximum data rates of some kbits/s.In contrast to that, the 3G systems enabled the users to transmit data at maximum ratesof 2 Mbit/s, which is at least required for todays applications such as the multimedia andothers. An enhancement of the data rate in the GSM system by increasing the numberof the used time slots turns the GSM to a system known general packet radio service(GPRS). A further enhancement of the data rate is by using a new modulation schemeand the system is then known as enhanced data rate for global evolution (EDGE). Those

SC/FDE - Buzid

1. Introduction 3

two systems in terms of the offered data rates were only a first step towards wirelessmultimedia. They are sometimes known as 2.5G or 2G+.The unique and worldwide 3G standard which was aimed at is the future public land mo-bile telephone system (FPLMTS) renamed as international mobile telecommunicationsystem (IMT-2000). However, two standards were realized, the universal mobile telecom-munication system (UMTS) in Europe and the cdma2000 in the US. Both standardsuse CDMA for multiplexing of multiple users. The high data rates offered by the twostandards (voice, data and video) make the systems vulnerable to the wireless channelimpairments and the 3G systems are classified as broadband systems, in contrast to 2Gsystems. Also, the UMTS system has been enhanced by the so called high speed down-link packet access (HSDPA) and high speed uplink packet access (HSUPA) techniques,which are called 3.5th generation systems. HSDPA supports data rates up to 14.4 Mbit/s.Additionally, wireless services are offered to the users in selected spots via wireless localarea networks (WLAN). WLANs offer high data rate services to both mobile and sta-tionary users. Many developed standards targeted the speed rather than the roamingproperty for broadband services. These broadband techniques are the Bluetooth, IEEE802.11a,b and 802.16 a,b (known also WiMAX), HiperLan/2 and HiperMan/2. As Fig-ure 1.1 shows, the mobility and the data rates in bits/s are two compromising objectives.Further improvements to the UMTS to cope with future requirements are intended in aproject named third generation partnership project long term evolution (3GPP LTE).These improvements include but are not limited to improving efficiency, lowering costs,improving services, making use of new spectrum opportunities and better integrationwith other open standards.

HIPERMAN/IEEE802.16a

Data rate0.1 1.0 10 100

HIPERLAN/IEEE802.11a

DVB-T

B e yo

nd

3G

(

4G

)

3G

/3G

(UM

TS

/IM

T2

00

0)

2G

(e

.g.,

GS

M)

DAB

Vehicular

Nomadic

Stationary

Mobility

Figure 1.1.: Data rate versus mobility in wireless standards [1].

1.1.4. Forth Generation Cellular Systems

The 4G or B3G fundamental objective is not only the creation of a new technologyrather than the enhancement of 3G technology. The improvements are partially to-wards higher data rates to meet needs of future high-performance applications, e.g. 100

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1. Introduction 4

Mbit/s for outdoors and a peak of 1 Gbit/s for indoor. Further, B3G will witness theintegration of existing technologies in a common platform, implying a global mobilityand service portability. Further, to ensure the success of 4G systems, the cost of abroadband transmission per bit must be reduced dramatically compared to the cost ofthe existing services. Examples of broadband transmissions are office and home LANs,asymmetric digital subscriber line (ADSL) and optical fiber access systems.Moreover, the anticipated services will be based on internet protocol (IP) networks,which manifest the efficient transmission of IP packets over wireless networks.Around 2000, the international telecommunication union (ITU) began research on thefuture development of IMT-2000 and other systems by setting up a study group andworking parties leading the research sharing a global vision for the wireless future [2].However, in Europe, national research programmes, which are led by the main Europeanvendors, contribute to the ITU standardization work. Further, new research projectsare promoted and established, e.g. the ambient network (AM) [3] and the WINNER[4] projects. As an example, the ambient networks as demonstrated in Figure 1.2 is anintranetworking procedure to manage and administrate the traffic among and withincoexisting, diverse, heterogeneous and overlapping networks through commands andsoftware rather than added hardware or physical tools. In this context, networks areon the move instead of a single mobile terminal is on the move. The vision of ambient

Ambient City’sHotspot service

Ambient

Solution’s

Cellular

servicePAN

VAN

Office LAN

Figure 1.2.: Ambient networks, on the fly, on the run, on the move.

networks is to facilitate and to cope with the growing and expanding wireless com-munications and to enable the integration of coexisting future multi-technologies, selfconfigured networks, with the constraints that people can move from one network toanother without any effort or interruption. The ambient networks operate in a dynamicnature of environment and can be extended with new capabilities as well as operate overexisting connectivity infrastructure. Moreover, ambient networks enable the concept ofinstant composition of networks belonging to different business entities. The viable andchallenging element in ambient networks is that a diverse bulk of users (multi-technology

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1. Introduction 5

mobile networks) accesses the diverse bulk of stationary and/or mobile networks whichmust be geographically reliable. The bulk of users are e.g. personal area networks(PAN), vehicle area networks (VAN), home area networks and future networks.

1.2. SC/FDE Technique

The single carrier transmission with frequency domain equalization (SC/FDE) techniqueis an elegant and effective solution for mitigating impairments of a dispersive channel[5, 6]. The counterpart of a single carrier system as the name may suggest is a multiplecarrier system, and both are possible methods of transmitting data from one point toanother over a frequency-selective fading channel. The SC/FDE performance has shownto be similar to that of orthogonal frequency division multiplexing (OFDM) as a mul-ticarrier technique [7, 8]. Broadband transmission means the bandwidth of data to betransmitted exceeds the coherence bandwidth of the multipath channel. The superiorityof the SC/FDE concept to classical single carrier systems lies in the equalization thatmitigates the wireless distortions and is carried out in frequency domain. The conse-quence of the use of frequency domain equalization is the significant complexity reductionof the receiver. Equalization using the discrete frequency domain was first reported in[9]. However, Sari et al. [7, 10] combined the frequency domain equalization via dis-crete Fourier transformation (DFT) and the guard interval. Only a limited number ofresearchers world wide considered the SC/FDE topic at the beginning. Among them agroup supervised by D. Falconer [8]. In late nineties, Czylwik [11] investigated channelestimation and synchronization for SC/FDE. Further investigations of the SC/FDE tookplace at Linz University, where the first dissertation on this topic was published by M.Huemer [12]. More publications followed and covered the system synchronization, chan-nel estimation and the use of cyclic prefix and later the unique word [13, 14, 15, 16, 17].Further work aimed in performance enhancement via diversity techniques. Al-Dahirin [18] applied the Alamouti-principle to SC/FDE, and further enhancement by ap-plying time space diversity and multiple input multiple output (MIMO) techniques asis achieved [19, 20, 21, 22]. In recent years, different proposals have been made forSC/FDE, especially in IEEE 802.16 Wireless MAN (Metropolitan Area Network) [8].Further, SC/FDE combined with frequency devision multiple access (SCFDMA), is apromising technique for high data rate uplink communications in future cellular systems.SCFDMA has been chosen for the uplink multiple access scheme in LTE [3, 4, 23, 24, 25].

1.3. Goals of this Work

One of the often mentioned advantages of SC/FDE compared to OFDM is the fact, thatsingle carrier transmission schemes show signal envelope characteristics with lower peakto average power ratios (PAPR). This relaxes the linearity requirements of the transmitpower amplifier, and thus improves the transceiver’s power efficiency. Mostly, SC/FDE

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1. Introduction 6

has been investigated in combination with quadrature amplitude modulation (QAM).But single carrier QAM modulation schemes (e.g. QPSK, 16-QAM, 64-QAM) also showrather large amplitude variations. The fundamental aim of this work is the adaptionof SC/FDE for modulation formats with effectively low PAPR. One candidate is thelinear OQPSK (offset QPSK) scheme, but the focus of the work is on CPM (continuousphase modulation) formats. Further the spatial multiplexing (SM) extension of CPM-SC/FDE as a method to improve the bandwidth efficiency shall be investigated. Finallythe combination of SC/FDE schemes with CDMA shall be addressed.

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2. Basics on Modulation Techniques

In order to transfer information or messages from one point to another, most probablya carrier is required. This work focuses on wireless communications, and the differentcommon modulation techniques are reviewed in this chapter.

2.1. Modulation

What does modulation mean and why is it needed? To answer this and other relatedquestions, popular modulation techniques are reviewed in the next sections. The termmodulation comes from the process of modulating a carrier. The reverse of the actionsand measures including modulation that are performed at the transmitter are carriedout at the receiver. However, the detection may be distinguished from demodulation bydefining detection as a process of extracting the information from the baseband represen-tation of the demodulated signal. In non-coherent systems, however, the demodulationand detection may not be distinguished in a straightforward manner. Modulating thecarrier is implemented by the manipulation of one or more carrier parameters. The car-rier parameters are the amplitude, the phase and the frequency. Therefore, the type ofthe modulation is determined according to these parameters and categorized as follows:

• Amplitude modulation (AM)

• Frequency modulation (FM)

• Phase modulation (PM)

• Hybrid modulation

In general, modulation schemes differ in

• spectral efficiency

• resistance to noise and other disturbances

• the peak to average power ratio


and more. The transmitted signal can generally be expressed as [26]

s(t) = Re[a(t)ej(2πfct+θ(t))

](2.1)

= a(t)cos(2πfct+ θ(t))

= a(t)cos(φ(t)),

where fc is the carrier frequency, a(t) is the carrier amplitude and θ(t) is the carrierphase, whose relation to the instant frequency fi(t) is given as

fi(t) =1

2π

dφ(t)

dt. (2.2)

The frequency and phase modulation are closely related to each other and are also knownas angle modulation. Another important classification which is in accordance with thetype of the information is:

• Analog modulation

• Digital modulation

Demodulation can also be divided according to the type of the detection into coherentand non-coherent demodulation. Another classification is linear and nonlinear modula-tion. Figure 2.1 shows a number of different types of modulation techniques and relatesthem one to the other. It can be noted from the figure that the schemes overlap andconsequently any particular classification can not be unique.

Modulation

Digital

Ampl Angle

PhaseASK

Analog

Ampl Angle

WB-FM

Hybird

Freq

BFSK

M-FSK

CFSK

MSK

GMSK

BPSK

QPSK

M-PSK

OQPSK

DPSK

PWM

PPM

PCM

PAM

PDM

AM

DSB-SC

SSB-SC

VSB

FM

NB-FM

PM

Non-linear modulation

Hybird

ASK-PSK

XPSK

GSMSK

SMSKFeher QPSK

M-QAM

JIFQPSK

CPM

Constant Envelope

Figure 2.1.: Categorization of types of modulation (for further information about abbre-viations and details see following sections or the list of abbreviations).

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2.2. Analog Modulation

Due to its historical significance, analog modulation is briefly reviewed. Although the useof the analog modulation is shifted towards the digital, it is still important and beingin use in practice, since many signals generated in real life are analog in nature andtransmitted in the analog form. In analog modulation category, frequency modulationis the most important. The carrier’s frequency varies with the amplitude of the messagesignal and hence the carrier’s amplitude remains constant. The FM scheme is importantbecause of:

• Its noise immunity (compared to amplitude modulation),

• The possibility to use efficient power amplifiers,

• Its possibility of having a trade-of between bandwidth and performance.

In conjunction with a frequency modulation there is a phase modulation in which theinstantaneous phase of the carrier changes with the transmitted information. However,both phase and frequency modulation are known as angle modulation and hence theirperformances are expected to be similar. Employing an integrator in front of a phasemodulator yields a frequency modulator. In contrast to the angle modulation the ampli-tude modulation is less complex and requires less bandwidth, but comparably inefficientpower amplifiers have to be used.

Coherent Demodulation

In most high performance applications, receivers are based on coherent detection. Toperform coherent detection, the receiver must know the frequency and the phase of thereceived signal. In a receiver, a phase-locked loop (PLL) tracks the frequency and phaseof the received signal. The local oscillator signal is then mixed with the received signal.If for example, the received signal is amplitude modulated

r(t) = a(t)cos(2πfct+ θ0), (2.3)

and the local oscillator signal carries frequency and phase that correspond to frequencyand phase of the received signal

z(t) = 2cos(2πfct+ θ0),

then the mixed signal follows to 2a(t)cos2(2πfct+θ0), which equals a(t)+a(t)cos[2(2πfct+θ0)]. A lowpass filter eliminates the signal components at the frequency 2fc yielding thedesired source signal a(t). An attractive alternative to coherent detection for some digitalschemes is differential detection which avoids the need for carrier phase synchronization.In differential detection, simplicity and robustness of implementation take precedenceover achieving the best possible performance.

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Non-Coherent Demodulation

As the name suggests, the non-coherent demodulation contradicts with the coherentdemodulation. In non-coherent demodulation, the received signal is not mixed with alocal oscillator’s signal as typically done in coherent demodulation, instead it is processedand the data is extracted from the carrier by using an envelope detector. Non-coherentdemodulation is mostly associated with amplitude modulation.

2.3. Hybrid Modulation

Hybrid modulation is also known as pulse modulation, which involves modulating a car-rier comprising regularly recurrent pulses. The pulse is modulated or altered accordingto the information signal. The pulse amplitude, width or position is adjusted accordingto the information and the resultant schemes are pulse amplitude modulation PAM,pulse width modulation PWM (pulse duration modulation PDM) and pulse positionmodulation PPM, respectively. Additionally, pulse code modulation (PCM) is anotherform of pulse modulation.

2.3.1. Pulse Amplitude Modulation (PAM)

PAM represents a form of modulation in which the amplitude of individual and regularlyspaced pulses in a pulse train is varied in accordance with some characteristics of themodulating signal. The amplitude of the pulses conveys the information in this case.Pulse amplitude modulation is the simplest form of pulse modulation. It is generatedin the same manner as analog amplitude modulation. The timing pulses are appliedto a pulse amplifier in which the gain is controlled by the modulating waveform. Sincethese variations in amplitude actually represent the signal, this type of modulation isbasically a form of AM. The distinctive difference is that the signal is now in the formof pulses. This signifies that PAM has the same built in weaknesses as any other AMsignal, which is essentially the high susceptibility to noise and interference. The reasonfor susceptibility to noise is that any interference in the transmission path either addsto or subtracts directly from the wanted signal. That affects the signal amplitude whichcarries the data, and in return, causes a distortion to transmitted data. The distortion,hence, deteriorates the transmission system performance and for this reason, PAM is notvery popular. Techniques of pulse modulation other than PAM have been developed toovercome problems of noise interference. PAM is now rarely used and has been largelysuperseded by pulse position modulation. The following sections will discuss other typesof pulse modulation that are shown in Figure 2.2.

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PAM

PDM

PPM

t

4 5 3 2 6 1 7

Clock

Data

t

t

t

Figure 2.2.: Pulse modulation schemes: pulse amplitude modulation, pulse positionmodulation and pulse duration modulation.

2.3.2. Pulse Position Modulation (PPM)

Pulse position modulation is a form of signal modulation in which m message bitsare encoded by transmitting a single pulse in one of 2m possible time shifts. This isrepeated every T seconds which yields the transmitted bit rate m/T bits per second.The amplitude and width of the pulse is kept constant in the system. The positionof each pulse, in relation to the position of a recurrent reference pulse is varied byeach instantaneous sampled value of the modulating wave. PPM has the advantageof requiring constant transmitter power since the pulses are of constant amplitude andduration. It is primarily useful for optical communication systems where there is onlylittle or no multipath interference. As this technique relies mainly on the time delaysof the pulses within a period, it is a difficult task to maintain the synchronization andthe alignments of the received pulses. Therefore, it is often implemented differentiallyand also known as differential pulse position modulation, whereby each pulse position isrelatively encoded to the previous pulse in such a way the receiver must only measurethe difference in the arrival time of successive pulses. One of the important advantagesof the PPM is that it can be implemented non-coherently and the receiver does notneed to track the phase of the carrier. This makes it a suitable candidate for opticalcommunication systems.

2.3.3. Pulse Duration Modulation (PDM)

Another kind of pulse modulation is pulse duration modulation (PDM), in which theduration of the pulses is varied in accordance with some characteristics of the modulating

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signal. Because of its constant amplitude, it is desirable in control and telemetry systems.A familiar example of PDM is the international Morse code, shown in Figure 2.3, used inship-to-shore communications. The synonyms are pulse length modulation (PLM) andpulse width modulation (PWM). It requires very little bandwidth and was employed inshortwave transmitters.

Figure 2.3.: PDM telemetry frame.

2.4. Digital Modulation

Digital modulation is the trend of the era as it enables the transmission and exchangeof information at very high data rates. Digital modulation benefits from the huge dataprocessing power provided by modern processors. A conceptual block diagram, shownin Figure 2.4, illustrates the modulator and demodulator that are covered in this chap-ter. Any form of digital modulation necessarily uses a finite number of distinct signals

Figure 2.4.: Digital communication system model for modulation and demodulation,s(t), r(t) and n(t) are transmit, receive and noise signals, respectively.

to represent digital data. The digital data (message) manipulates amplitude, phase,frequency or a combination of a reference signal appropriately. Consequently, a finitenumber of angles and amplitudes are used. Digital modulation is fundamentally catego-rized into amplitude shift keying (ASK), frequency shift keying (FSK) and phase shiftkeying (PSK). Each one or a combination of them are employed in different applicationsaccording to their performances. The number of times of signal parameter (amplitude,frequency, phase) changes per second is called the signaling rate, which is given in unitsof baud; where 1 baud = 1 change per second. With binary modulations such as ASK,FSK and PSK, e.g. the signaling rate equals the bit-rate and in quadrature phase shiftkeying (QPSK), the bit-rate exceeds the baud rate.

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2.4.1. Amplitude Shift Keying (ASK)

Amplitude shift keying is a very popular modulation. The input binary data shifts orchanges the amplitude of the carrier. The simplest and most common form of ASKoperates as a switch. The presence of a carrier indicates a binary one and its absenceindicates a binary zero. Generally, amplitude modulation has the property of translatingthe spectrum of the modulating signal to the carrier frequency. It is also known asOn-Off keying (OOK) because of the twin states of the ASK signal. The dominantapplication of ASK is in control applications. This is partly due to its simplicity andlow implementation costs. In addition, the detection process can be a simple envelopedetector which is much simpler than a coherent approach. One of the disadvantages ofASK, when compared with its counterpart modulation formats, is that it does not havea constant envelope which makes the amplifications of the signal more difficult becauseof the nonlinearity of the amplifiers. Also, it requires high signal to noise ratio.

2.4.2. Frequency Shift Keying (FSK)

The variation of the reference signal (carrier) frequency in accordance with the informa-tion signal yields a frequency modulated signal. The instantaneous frequency is shiftedbetween discrete values. A binary FSK (BFSK) modulation scheme consists of two si-nusoidal pulses at two frequencies, f1 and f2 representing 1s ”mark” and 0s ”space”, re-spectively. Accordingly a normalized BFSK signal may be described by its pre-envelope[27]

s(t) = cos(2πfc t+ 2πγ∆f t), (2.4)

where γ ∈ ±1 is non-return to zero (NRZ) or bipolar binary input data. ∆f =|f1 − f2| /2 is the frequency offset from the carrier frequency fc and the frequency sep-aration is 2∆f . Thus the bandwidth B of a BFSK signal is given as

B = 2(∆f + w) (2.5)

= 2w(~+ 1).

The dimensionless parameter ~ given by ~ = 2∆f/w with the symbol rate w = 1/T iscalled modulation index. The bipolar input to the modulator increases or decreases thesignal phase by π~. ~ determines the class of the BFSK [28]. A wideband FSK (WBFSK)is obtained when ~ >> 1, otherwise the narrowband FSK (NBFSK) is attained. TheBFSK requires less bandwidth than any other FSK and is hence known as ′′fast FSK′′,too.The modulated signal may be regarded also as a sum of two amplitude modulated signalsof different carrier frequencies f1 and f2 [28]. The normalized BFSK signal may also beexpressed as

s(t) = m1(t)sin(2πf1t+ φ0) +m2(t)sin(2πf2t+ φ0), (2.6)

where φ0 is the initial phase. m1(t) and m2(t) are the baseband signals which alternateinstantaneously between 0s and 1s in a predetermined manner as the following:

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Halo

0.998 0.998 1 1.001 1.002 1.003 1.004

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.005 1.006

2 fwN

orm

ali

zed a

mpli

tude

f kHz

f1f3f

2

No

rma

lize

da

mp

litu

de

Figure 2.5.: M -FSK signal spectrum, the frequency separation is 2∆f .

γ m1(t) m2(t)+1 +1 0-1 0 +1

Further, FSK transmitter are more complex and less bandwidth efficient when comparedto OOK.

2.4.2.1. M-Ary FSK

AnM -ary FSK or Multi-FSK is an extension of a BFSK modulation whereM carriers areutilized consecutively. The signal spectrum is indicated in Figure 2.5 for a rectangularpulse shape. The required bandwidth for M carriers is then

BM−FSK = 2 [(M − 1)∆f + w] (2.7)

= 2w [(M − 1)~+ 1] .

In a non-coherent receiver, the received signal is presented across M parallel bandpassfilters which are centered at frequencies f1, f2, ..., fM , as shown in Figure 2.6, providedthat ~ >> 1. Each of the bandpass filters is followed by an envelope detector. The

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Figure 2.6.: M-FSK non-coherent receiver [30].

envelope detectors apply their outputs to a logic circuit that selects the detector with thelargest output [29]. Non-coherent detection has the advantages of a relative simplicity.Referring to Figure 2.5 and letting ~ = 1/2, it follows that

BM−FSK = (M + 1)w,

which gives the total bandwidth for M orthogonal signals. As an example, the band-width of an orthogonal BFSK signal is BBFSK = 3w. Further, it is worth mentioning,that when M carriers are used, the occupied bandwidth fraction of the carrier can bedetermined as

BactualM−orthogonal =

2w

2w[(M − 1)~+ 1](2.8)

=1

(M − 1)~+ 1.

Assuming a 16-FSK orthogonal system with ~ = 1/2,

BactualM−orthogonal =

2

17, (2.9)

which represents 12% of the transmission bandwidth and shows how ineffecient 16-FSKis in terms of the spectrum. For this reason, orthogonal M-FSK is not popular.

For further comparison, frequency division multiplexing (FDM) which is a techniqueused in multiplexing is recalled. An FDM in principle is similar to an M -ary FSK. Theviable difference between them is that the bandwidth of an FDM can be completelyoccupied by the users, whereas in an M -ary FSK only a fraction of its bandwidth is

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I(t)

+

Q(t)

CPM signal

- sin(wt)

cos(wt)

I-generator

Q-generator

Bipolar data

Figure 2.7.: Quadrature modulator (parallel approach).

used during one time interval. Another counterpart of an orthogonal M -FSK is an or-thogonal frequency division multiplexing (OFDM) system, where the carriers overlaporthogonally and the assigned bandwidth for OFDM can also be completely occupied.That among other reasons explains again its popularity.

2.4.2.2. Minimum Shift Keying (MSK)

Is there a minimum or optimum separation frequency between the two carriers in theBFSK system? What is the criterion to establish such a requirement? The answerto these questions yields the MSK modulation. The minimum frequency separation isobtained when the spectrum of the two signals (carriers) are overlapped orthogonallyto each other. Therefore, the minimum separation frequency equals 2∆f = 1

2Tand the

modulation index is ~ = 0.5. The minimum frequency separation 2∆f is equivalent toa phase contribution of π/2 for each symbol. Note that the phase transitions betweenone phase state and the next state is forced to follow a circular path. Thus, the resultis a constant envelope signal. MSK was first reported by Doelz and Heald [31]. As isevident in the expression of the MSK signal

s(t) = I(t) cos(wct)−Q(t) sin(wct), (2.10)

which is the sum of two pulse streams modulating the in-phase and quadrature channelsof a single carrier, where I(t) and Q(t) are inphase and quadrature phase basebandshaped data. This approach, shown in Figure 2.7, is known as parallel MSK system.In practice there is a wide range of methods for the generation and reception of thistype of signals and they are divided into two categories. In addition to the parallelMSK modulators, the serial type of modulators (SMSK) is the other. SMSK was firstintroduced by Amoros [32]. However, it was also reported under the name of simplifiedMSK (SMSK). One possible serial modulator is illustrated in Figure 2.8. The datain serial modulators, as the name suggests, are modulated serially. In another simpleserial modulator, the serial bipolar data stream inputs a voltage controlled oscillators

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Figure 2.8.: Conceptual serial CPM modulator

Figure 2.9.: Phase trellis.

(VCO). A further form of a modulator is the polar modulator that makes use of thepolar coordinates, instead of the Cartesian coordinates (parallel form).It has been practically used in GSM. A generalized form of SMSK has been proposed byAllan [33]. The concept of the GSMSK scheme is based on replacing the straight trellisarms of the phase trellis by generalized curved trellis arms, depicted in Figure 2.9. Theimpulse response of the conversion filter is generalized and defined by [33]

h(t) =d(φ1(t))

dtsin(wt+ φ1(t)), (2.11)

where φ1(t) denotes the excess-phase of the GSMSK signal. For a particular data symbolφ1(t) is allowed to be any slow changing function in the range 0 < t < T . A functionwhich retains this property is a cubic function and given by

φ1(t) =

0 t ≤ 0

3π( tT)2 − 2π( t

T)3 0 < t < T

π ≥ T.

(2.12)

A comparison with the standard MSK out-of-band power versus bandwidth is illustratedin Figure 2.10. As can be noted, standard MSK has a more compact spectrum for band-

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0.0

-20

.0-4

0.0

-60.0

-80.0

-10

0.0

00.0 2.0 4.0 6.0 8.0 10.0

Normalized Bandwidth f T

No

rma

lize

do

ut of ba

nd

po

we

r dB

MSK

GMSK

Figure 2.10.: Out of band power for MSK and GSMSK schemes [33].

width less than about 2/T , while GSMSK has less out-of-band power for a bandwidthgreater than 2/T [33].

2.4.3. Phase Shift Keying (PSK)

In PSK, the phase of the carrier wave is varied between discrete values in accordancewith the digital data. There are several methods that can be used to accomplish a PSK.The simplest PSK technique is called binary phase shift keying (BPSK), which uses twoopposite signal phases. As there are two possible wave phases, BPSK is sometimes calledbiphase modulation. The two phase states, i.e. 0 and π are projected on the carrieramplitude as 1 and -1, which polarizes the carrier. In addition, more sophisticated formsof the PSK are in use in todays communication systems. In an M-ary or multiple phaseshift keying (M-PSK), there are M phase states. The term M-PSK is often replaced byquadrature phase shift keying or quaternary phase shift keying (QPSK) whenM = 4. InM-PSK whereM = 2m and m is the number of bits per symbol, and compared to BPSK,data can be transmitted m times faster. The demodulator, which is designed specificallyfor the symbol-set used by the modulator, determines the phase of the received signaland maps it back to the symbol it represents, thus recovering the original data.

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2.4.3.1. Offset Quadrature Phase Shift Keying (OQPSK)

The Offset QPSK, sometimes also called staggered QPSK is in principle a quadraturephase modulation as the name suggests [34]. OQPSK eliminates 180 degree phasechanges. This is achieved by introducing a delay with the duration of half a symbolbetween the quadrature components of a QPSK signal. This implies that the quadra-ture phase components of the a OQPSK can not be zero simultaneously. As a result,the range of the fluctuations in the signal is smaller than that in QPSK.

2.4.3.2. Differential PSK (DPSK)

Differential phase shift keying is basically a common form of the phase modulation. InDPSK, the information is packed in the phase difference between the successive symbols’intervals. Then, in the receiver, it is not required to estimate the carrier phase. Instead ofthat only the phase changes between the consecutive symbols’ intervals are determined.Thus, the receiver complexity is significantly reduced as compared to ordinary PSK.The most prominent DPSK scheme is π/4-DQPSK, where the phase difference betweensuccessive symbols is ±45o or ±135o.

2.4.3.3. M-PSK

In the BPSK scheme only one bit per symbol is transmitted and in the QPSK two bitsper QPSK symbol are transmitted. The bandwidth efficiency can further be improvedby sending more bits per modulated symbol. M-PSK can improve the efficiency by afactor of ln (M). The symbol phase is then given by

φm = 2mπ

M, m = 0, 1, 2, ...,M − 1,

where M ∈ N. Any increase in M raises the number of modulation states. However, asthe number of modulation states increases, the distance between phase states reducesand the likelihood of a demodulator error increases. At higher values of M , the schemeis not preferred any more. In practice, M = 8 (3 bits per modulated symbol) is themaximum used e.g. in the enhanced data rates for GSM evolution (EDGE) system.Higher bandwidth efficiencies are achieved by a combination of both, PSK and ASK,which is then called the quadrature amplitude modulation (QAM) scheme. Nevertheless,the highly efficient modulation techniques achieve better bandwidth efficiency at theexpense of power efficiency. The M signal waveforms in digital phase modulation canbe represented by [26]

sm(t) = Re[g(t)ej(φm+wct)

](2.13)

= g(t) cos (wct+ φm)

= g(t) cos(φm) cos(wct)− g(t) sin(φm) sin(wct)

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for m = 0, 1, ...,M − 1, where g(t) is the signal pulse shape whose energy ξ is given by

ξ =∫ T

0|g(t)|2 dt. The φm given above are the M possible phases of the carrier. For

large M , the evaluation of the error probability is not simple [35]. However, the errorprobability P of the optimum receiver in the additive white Gaussian noise (AWGN)channel has been evaluated in the case of BPSK (M=2) and QPSK (M=4). PBPSK andPQPSK are respectively given by

PBPSK = Q

(√2ξ

N0

)(2.14)

and

PQPSK = Q

(2

√ξ

N0

), (2.15)

where Q(x) is the Q function defined as

Q(x) =1√2π

∫∞

x

e−t2/2dt x ≥ 0 (2.16)

and N0 is the noise power spectral density. Coming back to large values of M , the exactanalysis of the system bit error rate (BER) is often complicated and usually results innon-closed form solutions [36]. A closed form expression for the exact solution of up to8-ary PSK modulation order with Gray code mapping was presented in [37]. However,tight upper and lower bounds on BER are also reached for the Gray code mappingfor higher order than 8-ary PSK modulation [37, 38]. Irshid, in [35] advised a simpletechnique for evaluating the bit error probability of M-PSK with any bit-mapping (Gray,Folded binary and Natural binary). Basically, the technique divides the signal space intohalf and quadrant decision subspaces. Then it finds expressions for the probabilities thatthe constellation point lies in a given decision subspace. Jianhua in [38] reached the sameresults differently, the bit error probability of coherent M-PSK based on the signal spacegeometry was reached for Gray code bit mapping. The principle of the approach is todivide the signal space into two main axes and if necessary they can be subdivided tosubspaces by rotating the main axes by an amount R (a rotated coordinate system). Biterror rate is more efficient as a performance measure than symbol error rate (SER) [36].However, the computation of exact BER is rather tedious because of its dependence onthe bit-mapping used. Consequently, the determination of the BER of coherent M-aryPSK schemes has been performed by either calculating the symbol error probabilityor using lower/upper bounds particularly for higher values of M [37, 38, 39]. Theapplication of the bounds estimations does not always ensure sufficient accuracy, whilethe transformation from SER to BER is, in general, not straightforward and accurate.The relation between binary natural code and Gray code bit mapping is demonstratednext.

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2.4.3.4. Gray and Binary Natural Code Mapping

Gray reflected binary code, as Frank Gray first named it when he introduced it, hasthe property that two m-bit symbols corresponding to adjacent symbols differ only in asingle bit. As a result, an error in an adjacent symbol is accompanied by one and onlyone bit error. The name, Gray reflected binary code, is derived from the fact that it maybe constructed from a binary code by a reflection process. Today, Gray codes are widelyused in digital communications and many algorithms for conversion between binary codeand binary reflected Gray code were developed. Agrell in [40] revised the original workof Gray and reported that Gray bit mapping for M-PSK is optimum in comparison withbinary natural mapping. A comparison between M-PSK with Gray code and binarynatural code bit mapping shown in table 2.1 reveals an interesting and useful relation,which surprisingly, has never been reported and further, finalizes the work in [40]. Letthe m-tuple b = bn,0...bn,i...bn,m denote the nth symbol (codeword), where m = ln(M)and n = 0, 1, ...,M − 1. The following equation,

0 0 0 0

0 0 0 1

0 0 1 0

0 0 1 1

0 1 0 0

0 1 0 1

0 1 1 0

0 1 1 1

1 0 0 0

1 0 0 1

1 0 1 0

1 0 1 1

1 1 0 0

1 1 0 1

1 1 1 0

1 1 1 1

0 0 0 0

0 0 0 1

0 0 1 1

0 0 1 0

0 1 1 0

0 1 1 1

0 1 0 1

0 1 0 0

1 1 0 0

1 1 0 1

1 1 1 1

1 1 1 0

1 0 1 0

1 0 1 1

1 0 0 1

1 0 0 0

Binary bits Gray bits

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Q = 16

0 0

0 1

1 0

1 1

0 0

0 1

1 1

1 0


1

2

3

4

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

0 0 0

0 0 1

0 1 1

0 1 0

1 1 0

1 1 1

1 0 1

1 0 0

1

2

3

4

5

6

7

8


gQ = 8g Q = 30bQ = 14b

Q = 4gQ = 6b

Table 2.1.: Binary code bit mapping versus Gray code bit mapping.

Q =m∑

i=1

M∑

n=1

bn i ⊕ b (n+1) i (2.17)

gives a decimal number which represents the total number of possible transitions (changefrom one to zero or vice versa) for any M-PSK constellation, as illustrated in table2.1, where ⊕ is an xor operation. Here, b (M+1) i is set to b 0 i. Based on equation(2.17), an exhaustive search for the optimum values of Q was launched and the results

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are tabulated in table 2.2. However, it is worth mentioning that, for m > 3 thereexist several Gray code constellations (labeling) and as m increases the number of suchconstellation becomes rapidly very large [40]. Nevertheless, Qg is always optimum forany constellation (labeling). From the table, one can see Qb > Qg. The interpretation

m M Qb Qg

2 4 6 43 8 14 84 16 30 166 64 126 64

Table 2.2.: Optimum number of state transitions for different constellations.

of such a difference is, that the probability of a binary code mapping falling in an erroris higher than in case of a Gray code.Letting F = Qb

Qg, a closer look at table 2.2 indicates

F =2m − 1

2m−1, (2.18)

which relates the error probability of both binary natural code mapping and Gray codemapping, as

Pb = FPg (2.19)

where Pb and Pg are the error probabilities of binary natural and Gray codes, respectively.Figure 2.11 shows the probability of error of both bit mapping techniques [41]. Further,the probability of error of Gray code, which is obtained by the application of equation(2.19), is also included. The figure shows an agreement between the simulated and theanalytical result.

2.4.3.5. Quadrature AM (QAM)

M-PSK is an important type of modulation and has been very attractive practicallyas summarized in table 2.3. Quadrature amplitude modulation is a composition ofamplitude and phase modulation methods. It is a hybrid type of modulation that variesboth phase and amplitude of the carrier. QAM is a way to achieve better bandwidthefficiency. Bandwidth efficiency (also known as spectral efficiency) is measured as thenumber of bits per second per Hertz. Because of this reason, QAM, i.e. 64-QAM is anefficient modulation scheme and widespread in practice. However, M-ary PSK is anotherway of improving the efficiency. Now the question is, which of M-QAM or M-PSK hasa better performance than the other? To answer the question, the signal constellationfor both schemes is plotted in Figure 2.12. In M-PSK, symbols are mapped on a unitcircle and as M increases, the premise of each symbol shrinks. At higher values of M , ittherefore becomes impossible to detect the signal in the receiver. For this reason, 8-PSK

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0 1 2 3 4 5 6 7 8 9 10

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

Pb

Binary mappingGray mappingObtained Binary mapping from Gray

Figure 2.11.: BER performance of both Gray and Binary Natural Code mapping for M=4.

is the largest constellation adapted in practice. Alternatively, multilevel constellationsuch as M-QAM and different labeling strategies are introduced for higher M [42, 43].

2.4.4. Continuous Phase Modulation (CPM)

This class of schemes is referred to as continuous phase frequency modulation (CPFM)or more simply continuous phase modulation (CPM). The significant attribute of a CPMconcept is its constant envelope which is essential in many wireless systems. The prop-erties and performance (bandwidth/power) characteristics of this class of modulationsare sufficiently voluminous to fill a textbook of their own [26, 30, 44]. For the sake ofbrevity, certain special cases of the CPM, that have gained popularity in the literatureand have been used in practice, shall be covered. Following the usual custom of modula-tion classifications, the CPM concept does not belong entirely to the phase modulationcategory as the name may suggest and promotes some ambiguity to the perception ofthe scheme. The signal phase is a continuous time function within any symbol interval,but it is steady at the interval edges that is to allow the coupling between the phasein successive symbol intervals [44]. The instantaneous frequency is constant over eachsymbol interval, therefore, CPM is a frequency modulation with superiority. However,the changing of the frequency and subsequently the phase or vice versa can be com-pared to a master and slave principle [45]. Furthermore, the definition of the phase andfrequency modulation intimates that the amplitude of the carrier remains unchanged,

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Modulation Applications

BPSK (M=2) GPS, deep space telemetry and cable modemsQPSK (M=4) (4-QAM) Satellite, CDMA, NADC, TETRA, PHS, PDC,

LMDS and DVB-S8PSK (M=8) EDGE,Telemetry pilots for monitoring

broadband video systems16 QAM (M=16) Microwave digital radio, DVB-C, DVB-T32 QAM (M=32) Terrestrial microwave, DVB-T64 QAM (M=64) DVB-C, MMDS256 QAM (M=256) DVB-C (Europe), Digital Video (US)

Table 2.3.: The application of different modulation schemes in practical systems. Forthe abbreviations refer to the list of abbreviations.

and hence, resulting in constant envelope signals. For this reason, phase and frequencymodulation are also known as constant envelope type-modulation in some literature.M-FSK is achieved by shifting carrier frequency, and switching from one frequency toanother is accomplished by having M = 2m separate oscillators, tuned to the desiredfrequencies. The transferring from one frequency to another causes the sidelobes in thespectrum to be large, consequently this method requires large bandwidth. To avoidthis, a carrier whose frequency changes continuously with time, is modulated and theresulting modulated signal is called continuous phase FSK. This type of signal has amemory and therefore, the phase of the carrier has to be continuous. Continuous phasemodulation implies that the carrier phase copes and acts in response to the changes ofthe transmitted information continuously and smoothly. In other words, the phase of thecarrier ascending without abrupt changes that result in spectrum sidelobes. The datasymbols modulate the instantaneous phase of the transmitted signal, where the phase isa continuous function in time [44]. As an example, the phases of a full response CPMsignal for an antipodal input sequence and a modulation index of 1/2 with a linear aswell as a raised cosine (RC) slope function (phase pulses) are plotted in Figure 2.13(a)and 2.13(b), respectively. From the figure, one can notice that a single data symbolaffects the phase of the transmitted CPM signal over more than one symbol interval.Although the scheme is a full response, the actual phase in any specific symbol intervaldepends on the previous data symbols. However, the phase of a PSK signal is indepen-dent of the previous data symbols and this explains further the memory associated witha CPM scheme.

2.5. Modulation Schemes Selection Criteria

Naturally, there is no unique and ultimate modulation format suitable for a variety ofdata transmission systems, because different applications requirements impose differentcriteria and demand different modulation techniques [46]. The selection of a modulation

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+ + + + + + ++

+ + + + + + + +

+

+

+

+

+

+

+ + + + + + +

+ + + + + + +

+ + + + + + +

+ + + + + + +

+ + + + + + +

+ + + + + +

-1 -0.8 -0.6 -0.2-0.4 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

+

0.2

0.2 2 /64

Figure 2.12.: 64-QAM and 64-PSK constellation.

scheme is the result of compromises on the bandwidth efficiency, performances, complex-ity and immunity to noise and other interferences. However, power efficiency, bandwidthefficiency and system complexity are major criteria for choosing an appropriate modu-lation scheme.

Bit Error Performance The quality of service provided by the communication system isassessed by, among others, the amount of errors in the received signal which is measuredby the bit error rate (BER). The relation between the transmitted signal power and thereceived errors is determined by the probability Pe which is related to the signal averageenergy per bit Eb and to the noise power spectral density N0. Optimum receivers tendto minimize the noise associated with the received signal.

Power Efficiency The definition of the power efficiency is not only related to the signalenergy, but also to the dc efficiency of the power supply which depends on the chosenmodulation scheme.

Bandwidth Efficiency The bandwidth efficiency or sometimes called spectrum effi-ciency is defined by the amount of information being transmitted per Hertz. To be moreprecise, it is determined by the number of transmitted data bits per second per Hertzbandwidth and is given by [47]

η =Transmission rate

Bandwidth[bits/s/Hz] .

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0 0.5 1 1.5 2 2.5 3 3.5 4−360

−270

−180

−90

0

90

180

270

360

t / T

φ(t)

in

degr

ees

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4

−360

−270

−180

−90

0

90

180

270

360

t / T

φ(t)

in d

egre

es

(b)

Figure 2.13.: A collective of CPM signal phases with (a) Rectangular filter (b) Raisedcosine filter.

However, the spectrum compactness, where sidelobs are at a minimum, is another aspectof the spectrum efficiency. This is applicable particularly in CPM signals. Further,nowadays, in cellular systems, the number of users per cell is another measure of thespectrum efficiency. Practically, there are several definitions or methods of determiningthe bandwidth efficiency of the various modulation techniques.

1. Null to null bandwidth efficiency:

Most of the transmitted signal spectra contain null points. The null to null band-width is the width between null to null of the power spectral density that representsthe main lobe. Based on this definition, various modulation formats can be com-pared as follows:The minimum bandwidth needed to transmit a BPSK symbol which carries onebit at rate w = 1/T is w and hence, BPSK efficiency can be expressed as

ηBPSK = w/w = 1 bits/s/Hz.

In QPSK, there are two bits per symbol, and hence, the resultant efficiency is

ηQPSK = 2w/w = 2 bits/s/Hz.

In M-PSK, there are log2(M) bits per symbol, therefore the related efficiency is

ηM−QPSK = log2(M)w/w =1

1log2(M) bits/s/Hz.

Further, in an M-FSK system, the needed bandwidth is

BM−FSK = 2w [(M − 1)~+ 1] ,

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therefore the efficiency is

ηM−FSK =1

2 [(M − 1)~+ 1]bits/s/Hz,

which is very poor.

2. Percentage bandwidth efficiency:

In other definitions instead of null to null bandwidth the band containing morethan 95% or 99% of the total power us used.

System Complexity The system complexity is in fact related to the system produc-tion cost, which is partially based on the chosen modulation/demodulation scheme. Forinstance, coherent demodulation is far more complex compared to non-coherent demod-ulation and hence, non-coherent based systems are more cost effective.

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3. Signal Envelopes Choices

In the following sections, another classification of modulation techniques that differsfrom the usual custom of the modulation categories, is considered. This classification isbased on the signal envelope nature.The power spectra of most modulated signals exhibit sidelobes that may interfere withadjacent channels, so that a certain amount of filtering is necessary at the transmitter.However, the filtering results in a great amount of envelope fluctuation in the signal,which leads to a considerable spectrum spreading. The spreading is due to the non-linear effects of the transmitter power amplifier. These nonlinearities tend to restorethe spectral sidelobes that have been previously removed by filtering. As a precautionagainst spectral spreading, the high power amplifier (HPA) in the transmitter in prac-tice is forced to operate below saturation, approximately in a linear zone. This comeswith the obvious penalty, that an HPA designed for a larger power has to be used withconsequent high inefficiency. These problems can significantly be reduced when usingconstant envelope signals.

3.1. Non-Constant Envelope Signals

All the fundamental modulation schemes, apart from the AM scheme, yield constantamplitude signals, since the carrier amplitude conveys no information. Unfortunately,this is only true when using rectangular pulse shapes, which show unfeasible spectralproperties. Ultimately, phase and amplitude modulated signals are non-constant enve-lope signals in nature, therefore, without additional measures, these schemes fall underthis category.

3.2. Near Constant Envelope Signals

Signals in this category reveal reduced envelope fluctuations. In fact, such a taxonomyis seldom used in literature. In practice, near constant envelope signals have receivedconsiderable attention and deployment. They show moderate peak to average powerratio (PAPR) compared to non-constant envelope signals. Such signals relax the linearityrequirements of e.g. the handset power amplifier (PA). This is why they are preferred in


the uplink in cellular communication systems. The widely known near constant envelopesignals are OQPSK and π/4-DQPSK. Figure 3.1 shows the signal constellation of therotated 8-PSK modulation, where data symbols are shaped with the linearized Laurentpulse shape (presented in the next chapter). 8-PSK technique is used in the EDGEsystem [47, 48]. Remarkably, the 8-PSK signal constellation is improved by both therotation of the symbols and by the shaping pulse. Other systems using near constantenvelope signals are Satellite, TETRA, DVB-S and many others. The common signatureof these modulation formats is that the carrier phase variation is always below 180 (πrad). In general, data shaping in this type of modulation does not play a fundamentalrole in determining the signal envelope. However, when half a sinusoidal shaping filteris applied to OQPSK, MSK signals can be obtained, which show constant envelope.

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Real part

Imag

pa

rt

In-Phase

Qu

ad

ratu

re-P

ha

se

Figure 3.1.: Rotated 8-PSK signal in the I/Q plane shaped by Laurent linearized pulseshape which will be discussed later.

3.2.1. Intersymbol-Interference and Jitter-free QPSK (IJF-QPSK)

IJF-QPSK is an improved OQPSK scheme developed by Feher [49, 50] for non-linearlyamplified satellite channels in a densely packed adjacent channel interference (ACI) en-vironment [51]. It is a quadrature type of modulation with overlapping between theinphase and quadrature arms, similar to OQPSK. The difference is that transition be-tween any adjacent dissimilar antipodal data symbols ∈ 1,−1 takes half sinusoidalshape instead of an abrupt change. That reduces the power spectrum sidelobes andrelieves the non-linearity constraints. However, IJF-QPSK signals can be alternativelygenerated by directly shaping the binary data sequence using a pulse shape, given as

g(t) = sin2

(π(t+ T

2)

2T

), −T

2≤ t ≤ 3T

2(3.1)

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where T is the symbol duration.

3.2.2. Cross-Correlated PSK (XPSK)

XPSK, also known as Feher QPSK, is a modified version of the IJF-QPSK modulationtechnique by introducing a cross-correlation between the in-phase and quadrature datastreams and specific waveforms [52, 53]. The aim of cross-correlation is to reduce theIJF-QPSK signal fluctuation by 3 dB to approximately 0 dB, so that the envelope isnearly constant.

3.3. Constant Envelope Signals

Constant envelope modulation shows better compromises than other approaches andthat is why it is used in some of the most prominent standards around the world. Ex-amples are GSM, DECT, CDPD and DCS1800. All of them use constant envelopemodulation formats, particularly MSK and GMSK [54, 55]. The major features and thecontributions of the constant envelope modulation to wireless transmission are summa-rized as follows:

• The prospect to fully exploit the high efficiency of power amplifiers.

• The optimum power consumption of the power supply, and hence, the maximumbattery lifetime. Therefore it is appropriate to mobile terminals.

• The least possible out of band power; that implies dense channel spacing andincreased multi-users capability.

The PAPR of this signal is unity. That explains the excessive study dedicated to thiskind of signals over the last few decades and to the present day. Unity of PAPR totallyeliminates the signal degradation and distortion induced by the power amplifier. Unfor-tunately, there is a price to be paid for such performance. Performance in terms of BERdeteriorates compared to e.g. QAM schemes, and since typically no higher order con-stant envelope schemes are used, the bandwidth efficiency is limited. This means thatlow PAPR is not the only determinant criterion in system design. Ultimately, constantenvelope signals do not always prevail in a system design trade-off.

3.4. Constant Envelope Signal Generation

The generation of constant envelope signals comprises mostly two cascaded stages. Theseare data mapping (data encoding) and data shaping. In practice there is a wide range

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of methods of generation and reception of this type of signals. Generally, there are twocategories, serial and quadrature modulators. Serial modulators require a control circuitto monitor the voltage controlled oscillator (VCO) sensitivity, which deteriorates dueto large phase error accumulation. The quadrature (generalized) modulator is shown inFigure 2.7. In the following sections, methods of generation of constant envelope signalsbased on these two basic structures are illustrated as well as the combination of both,data mapping and data shaping.

3.4.1. Shaped Offset QPSK (SOQPSK)

OQPSK is categorized near constant envelope signals and avoids the abrupt changes ofthe instantaneous phase of 180 degrees as a result to overlapping between the quadraturecomponents, but it fails to avoid the abrupt change of 90 degrees. Passing the datathrough a half sine shape filter of a width twice the symbol duration enforces a circlepath of the instantaneous phase alternation, and hence, SOQPSK collapses to MSK.Therefore, MSK can be thought of as a special case of OQPSK with sinusoidal pulseweighting [56].

3.4.2. Gaussian MSK

GMSK is a minimum shift keying modulation with data filtered by a Gaussian-shapedfrequency response filter, aiming primarily to make the output power spectrum morecompact. The width of the Gaussian filter is determined by the bandwidth-time productBT . With a BT value less than 1, a controlled intersymbol interference is then introducedand a partial response CPM signal is obtained. A relaxed or a wider impulse responseGaussian filter (BT is small) suppresses the higher frequency components. Furthermore,BT is a trade-off factor between bandwidth efficiency, power efficiency and detectorcomplexity [57]. Practical values of BT are e.g. 0.3 for GSM and 0.5 for cellular digitalpacket data (CDPD). Further, GMSK can be generated by alteration of a VCO frequencydirectly via the Gaussian data stream. With this simple method, however, it is difficultto maintain the center frequency within the acceptable value under the restriction ofpreserving the linearity and the sensitivity for the required FM modulation.

3.4.3. Generalized Serial Minimum Shift Keying (GSMSK)

As discussed earlier, GSMSK is a generalized version of serial MSK, which retains theconstant envelope property of the signal and maintains less out of band power.

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3.4.4. Correlative Coded Minimum Shift Keying (Correlative

MSK)

Introducing memory in the modulation process produces signals with better bandwidthefficiency at the expense of system performance [58]. The controlled intersymbol inter-ference is a method of introducing the memory. Correlative coding cascaded with a pulseshaping with ideally zero ISI at the sampling moments is an approach to generate pulsesthat give rise to controlled ISI. In correlative MSK, the data symbols are correlated priorto the modulation process [59]. Such coding is specified by a coding polynomial givenas

F (D) =N∑

n=0

qnDn

where Dn corresponds to a delay of nT and qn are the taps coefficients as shown inFigure 3.2. Two particularly useful schemes, using first- and second order encoding, areduobinary MSK (DMSK) and tamed frequency modulation (TFM).

q0

T T T

+

q1qN

Nyquist Filter

Figure 3.2.: A filter implementing a shaping pulse with a defined intersymbol interference(ISI).

Duobinary MSK First order encoding given by the polynomial (1+D) is applied todata prior to modulation in Duobinary MSK. DMSK has less phase variation than MSKand should consequently have better bandwidth efficiency [60]. DMSK is known also as2REC, which is a rectangular pulse that scans two consecutive data symbols required toshape data and given by

q(t) =

14T

0 ≤ t ≤ 2T

0 otherwise.(3.2)

Tamed Frequency Modulation TFM is a minimum shift keying scheme that was firstintroduced by Jager and Dekker [61]. It is a constant envelope modulation schemethat exhibits a narrow power spectrum with minimum sidelobes and hence, a low adja-cent channel power. It is therefore suitable for use in wireless communication systems.Moreover, TFM was proposed as an optional modulation scheme for the uplink IEEE

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G(f) =

1 0 ≤ |f | ≤ π(1−r)2T

0.5(1− sin

((fT−0.5)π

r

))π(1−r)

2T≤ |f | ≤ π(1+r)

2T

0 otherwise

(3.7)

802.16.1, as a possible co-existence with 4-QAM dual mode and it is also an option forresidential terminals [62]. In general, TFM is ideal for low cost terminal solutions. TFMis a correlative coded MSK signal using an arbitrary coding polynomial of second order,whose coefficients sum to unity. Unity of the coefficients sum is mandatory and keepsmaximum phase changes of the TFM signal during one bit period restricted to ±π/2rad. The allowable phase shifts of the modulating carrier during the nth bit period att = nT are given by

∆φ(nT ) =π

2

(γn−1

4+γn2

+γn+1

4

), (3.3)

where γn are the independent data symbols which take their values from the set 1,−1.Chung in [63] investigated the generalization of these coefficients sustaining the unity.The polynomial is given by

F (D) = a+ bD + aD2, (3.4)

where, a and b are the tap coefficients that satisfy the condition (2a+b)=1. Variousclasses of signals can then be obtained by properly adjusting the coefficients a and b. Thistechnique of signal generation is called generalized TFM (GTFM). Setting a=1/4 andb=1/2, GTFM reduces to the TFM scheme. The TFM modulator is basically a cascadeof a 3-tap transversal filter W (f) and a Nyquist filter G(f) [61]. On the one hand,W (f) extends the influence of an input data bit over three bit periods (partial responsecoding) and thus introduces the correlative encoding property into the premodulationfilter Hp(f) which can then be given by

Hp(f) = W (f)G(f), (3.5)

where W (f) derived from equation (3.4) can be expressed by [63]

|W (f)| = b

1 +

2a

bcos(2π f T )

. (3.6)

It is, however, worth mentioning that increasing b raises the noise margin for detection atthe expense of lowering the spectral efficiency [63]. On the other hand, a Nyquist filterG(f) satisfying the Nyquist criterion, e.g., a raised-cosine filter which is extensivelyused and expressed in equation (3.7), adds the rolloff factor (0 ≤ r ≤ 1) as anotherdesign parameter in GTFM when compared to TFM [63]. The premodulation filter inTFM, where the rolloff factor is set to zero, aims to produce a sequence of quinarysymbols as shown in Figure 3.3. This type is a serial implementation. Further, theparallel implementation, which was also investigated in [61], is always favored. Theblock diagrams of both the transmitter and the receiver are plotted in Figure 3.4.However, it is apparent from Figure 3.3 that the input to the VCO or the output

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from the TFM modulator depicted in Figure 3.4 is basically data shaped by the overallinduced pulse (premodulation filter), which is called TFM pulse and given as

hp(t) =1

8g(t− T ) +

1

4g(t) +

1

8g(t+ T ), (3.8)

where

g(t) ≈ sin

(πt

T

[1

πt− 2− 2πt

Tcot(πt

T)− π2t2

T 2

24πt3

T 2

]). (3.9)

The TFM pulse partially prescribes the phase path from the phase at nT to the phase

1/2

T T

+

1/4

Nyquist Filter VCO

1/4

Figure 3.3.: Serial TFM transmitter.

TF

Mm

od

ula

tor

I(t)

+

Q(t)

Input TFM

signaldata

sin(wt)

+

Q(t)

Input

Signal

cos(wt)

LPFDelay

2T

Gate

+LPFDelay

2T

I(t)

sin(wt)

cos(wt)

data

out

a) b)

Figure 3.4.: a) Parallel TFM transmitter b) TFM coherent receiver. For details of thereceiver see reference [61].

at (n+ 1)T and partially determines the total amount of the phase increase or decreaseduring a sampling interval. The amount of increase or decrease in the phase in TFMis either of 0,±π/4,±π/2. The signal phases belonging to MSK and TFM schemesare plotted in Figure 3.5, which show the smoothness of phase transition of TFMsignals in comparison with MSK. Obviously, TFM shows less phase transitions andmore smoothness than MSK. Consequently, the TFM spectrum characteristic is betterthan MSK.

3.4.5. Sinusoidal Frequency Shift Keying (SFSK)

SFSK is an alternative to MSK that enables further improvements in the spectrum andfurther spectral sidelobes reduction while preserving the constant envelope property and

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(ra

d)

time t

/4

(a)

(b)

(c)

(t)

(n+2)

n /2

(n-1) /2

(n-2) /2

(n+1) /2

/2

Figure 3.5.: Signal phase transitions a) MSK b) Half sine MSK c) TFM [61].

the efficiency of the MSK [64]. Instead of half a sine wave pulse shape, used in MSK,the following pulse is applied for SFSK:

g(t) = cos

[πt

2T− Usin(2π

t

T)

], (3.10)

where SFSK scheme is obtained by setting the parameter U to 0.25. Nevertheless,setting U to 0 yields the MSK. Generally, the sensitivity of the spectrum to a shapingpulse is illustrated by the variation of the parameter U. The pulse g(t) for different valuesof U is plotted in Figure 3.6. Further, the signal power spectrum for both MSK andSFSK schemes is plotted in Figure 3.7.

3.4.6. Continuous Phase Modulation

It has long been known, that the bandwidth of constant envelope digital modulationschemes could be reduced by smoothing the variations of the information carrying phase.This can be done by shaping the phase using an analog filter [44, 65]. The model generallyused for this class of modulation scheme is shown in Figure 3.8. The input data symbolsdenoted γ excite a premodulation filter whose impulse response is denoted by g(t). Thusthe input to the FM modulator is defined as

γ(t) =∞∑

i=−∞

γi g(t− iT ). (3.11)

The output of the FM modulator is a passband CPM signal denoted by a notationsimilar to [44, 45] and is given by

s(t) = Re√2E/T exp(j[2π fct+ φ( t, γ) + φ0]). (3.12)

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0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t / T

Am

plitu

de

U=.5 U=0.1

U=0.25

U=0.4

U=0

Figure 3.6.: Pulses suitable for the MSK.

0.4 0.45 0.5 0.55 0.6

10−2

10−1

100

f T

Pow

er in

dB

MSKSFSK

Figure 3.7.: SFSK and MSK power spectral density.

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Tx filterXg(t)

FM-modulator

2πh fc

s (t , )

Figure 3.8.: Schematic CPM modulator.

Here fc is the carrier frequency, E is the energy per bit and φ0 is an arbitrary constantphase shift which can be set to zero without loss of generality. The information bearingphase function φ(t, γ) is defined by

φ ( t, γ ) = 2π ~

∞∑

i=0

γi q(t− iT ). (3.13)

The elements of the vector γ = ..., γ−2, γ−1, γ0, γ1, ... are independent and identicallydistributed symbols from an M-ary information set, γi ∈ ∓1,∓3, ...,∓(M − 1) withprobability 1

M. ~ = q

p, represents the modulation index. The phase pulse or slope

function q(t) with the restriction:

q(t) =

0 t < 0M−12

t ≥ LT(3.14)

is related to the normalized frequency pulse g(t) as

q(t) =

∫ t

−∞

g(τ) dτ (3.15)

and the maximum phase changes over any symbol interval is (M−1)π2

. g(t) determinesthe smoothness of the transmitted information carrying phase.As discussed earlier, the phase continuity of a CPM signal imposes a memory on the sys-tem. The memory denoted by L is the number of symbols scanned by the frequency pulseand dictates the extent to which a single input symbol can effect surrounding symbols.Nevertheless, the introduction of the memory, in contrast to other types of modulation,manifests itself as intersymbol interference (ISI) and increases the demodulator com-plexity. But on the other hand the benefits are crucial in wireless communications. As alow-cost and non-linear power amplifier can be employed, CPM has been an attractivemodulation scheme [44, 65, 66].Let’s recall the baseband of the CPM signal given in equation (3.12),

s(t, γ) =

√2E

Tejφ(t,γ). (3.16)

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To stress the phase states, their transition and the time explicitly, the phase functionφ(t, γ) within the time interval nT ≤ t < (n+ 1)T, n ≥ L can be given as

φ(t, γ) = 2π~n∑

k=−∞

γkq(t− kT ) nT ≤ t ≤ (n+ 1)T (3.17)

= 2π~n−L∑

k=−∞

γk + 2π~n∑

k=n−L+1

γkq(t− kT )

= θn + 2π~n∑

k=n−L+1

γkq(t− kT ),

where

θn =2π

p

((q

n−L∑

k=0

γk

)mod 2p

)(3.18)

can take only p possible distinct values. θn represent the accumulated phase of a CPMsignal at the time interval [nT, (n+ 1)T ], due to the data up to t = (n − L)T anddemonstrate the signal memory introduced by the modulation. The p different valuesare necessary to describe partially the signal in each time interval.It is important to note that θn can be evaluated recursively since

θn = θn−1 + π~γn−L. (3.19)

The instantaneous frequency of the transmitted signal denoted f(t) in the interval nT ≤t < (n + 1)T, n ≥ L, can be determined by obtaining the gradiant of the phase of thesignal given by equation (3.17) as

1

2π

∂φ(t)

∂t= ~

n∑

k=n−L+1

γk∂q(t− nT )

∂t(3.20)

and the instantaneous frequency of a full response CPM signal is given as

f(t) = fc + ~γkdq(t− nT )

dtnT ≤ t < (n+ 1)T. (3.21)

Assuming the phase is a linear function, thus

f(t) = fc + ~γkM − 1

2TnT ≤ t < (n+ 1)T. (3.22)

Now if ~=1/2 and γn ∈ ±1, the two frequency components are f+(t) = fc +14T

forγn = +1 and f−(t) = fc − 1

4Tfor γn = −1. The maximum frequency deviation then is

14T. Further, the instantaneous phase and the instantaneous frequency of a full response

CPM signal forM=2,M=8 and ~ =1/2 are plotted in Figures 3.9, 3.10, 3.11 and 3.12for both REC and RC frequency pulses. One can note from the figures the behavior of theCPM signals phase and frequency. Moreover, the instantaneous frequency when M=8

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0 5 10 15 20 25 30−1/4T

0

1/4T

t / T

φ(t)

, dφ(

t)/d

t

Instantaneous phaseInstantaneous frequency

φ(t)

dφ(t)/dt

Figure 3.9.: Instantaneous frequency and phase for full response CPM signal with a RECpulse shape M=2.

0 5 10 15 20 25 30

−7/4T

−5/4T

−3/4T

−1/4T

0

1/4T

3/4T

5/4T

7/4T

t / T

φ(t)

, dφ(

t)/d

t


φ(t)

dφ(t)/dt

Figure 3.10.: Instantaneous frequency and phase for full response CPM signal with aREC pulse shape M=8.

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0 5 10 15 20 25 30−5/4T

−3/4T

−1/4T

0

1/4T

3/4T

5/4T

t / T

φ(t)

, d

φ(t)

/dt


dφ(t)/dt

φ(t)

Figure 3.11.: Instantaneous frequency and phase for full response CPM signal with a RCpulse shape M=2.

0 5 10 15 20 25 30

−7/4T

−5/4T

−3/4T

−1/4T

0

1/4T

3/4T

5/4T

7/4T

100

t / T

φ(t)

, dφ

(t)/

dt

Instantaneuos phaseInstantaneous frequency

dφ(t)/dt

φ(t)

Figure 3.12.: Instantaneous frequency and phase for full response CPM signal with a RCpulse shape M=8.

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oscillates between the values −74T, −54T, −34T, −14T, 14T, 34T, 54T

and 74T, as shown in Figures 3.10

and 3.12. Furthermore, in order to determine the instantaneous phase of the receivedsignals during any symbol interval, it is necessary to keep a record of the previous phasestatus. However, the instantaneous frequency of the received signal can be determinedwithout the phase tracking. Therefore, the data sequence can be recovered from theinstantaneous frequency of the received signal instead of the instantaneous phase. Thisproperty may be useful in the receiver.

3.4.6.1. Various Pulse Shapes

Pulse shaping plays a crucial and significant role in resolving the jointly power andbandwidth efficiency of the CPM signal. Since the phase contribution of each individualsymbol is constant and finite, the following holds

q(t) =

∫∞

−∞

g(t)dt = constant. (3.23)

In contrast to linear modulation schemes (non-constant envelope), where the pulse spansone symbol, q(t) spans one or more data symbols in a CPM scheme. The most commonpulses are presented in the following sections.

• Rectangular Pulse:A rectangular frequency pulse (REC) with an L symbols duration is given as

g(t) =

1

2LT0 ≤ t ≤ LT

0 otherwise.(3.24)

Letting L=1, the obtained modulation scheme is MSK. However, the rectangularpulse is employed only with L = 1 in practice. Further, q(t) is the integral of thefrequency pulse g(t) and given by

q(t) =

t

2LT0 ≤ t ≤ LT

0 otherwise.(3.25)

Both phase and frequency pulses are shown in Figure 3.13

• Raised Cosine Pulse:The raised cosine (RC) pulse shown in Figure 3.13 is very popular and gains awide range of applications in practice. The RC pulse is defined by

g(t) =

1

2LT

[1− cos(2πt

LT)]

0 ≤ t ≤ LT

0 otherwise.(3.26)

The integral of g(t) is given as

q(t) =

t

2LT+ 1

4πsin(2πt

LT) 0 ≤ t ≤ 2LT

0 otherwise.(3.27)

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g(t)

g(t)

g(t)

g(t) q(t)

q(t)

q(t)

q(t)

-3 -1 1 3

-3 -1 1 3

-3 -1 1 3 -3 -1 1 3

-3 -1 1 3

-3 -1 1 3

0 1 2 30 1 2 3

0.50.5

tt

0.50.5

tt

0.50.5

tt

0.50.5

tt

GMSK (BT=0.25)

TFM

3SRC

3RC

Figure 3.13.: Different frequency g(t) and phase q(t) pulses.

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• Spectrally Raised Cosine pulse:The pulse is illustrated in Figure 3.13 and expressed as

g(t) =1

2LT

sin(2πtLT

)2πtLT

cos(2πtβLT

)

1− (4βtLT

)2, 0 ≤ β ≤ 1. (3.28)

• Tamed Frequency Modulation Pulse:The TFM pulse is deduced from the nature of the TFM as discussed previouslyand is given by

g(t) =1

8[ag0(t− T ) + bg0(t) + ag0(t+ T )], a = 1, b = 2 (3.29)

go(t) = sin(πt

T)

[1

πt− 2− cot(πt

T)− (πt

T)2

24πt3

T 2

].

• Gaussian Pulse:The Gaussian pulse can be expressed as

g(t) =

[Q

(2πB

t− T2√

ln 2

)−Q

(2πB

t+ T2√

ln 2

)]0 ≤ BT ≤ 1, (3.30)

where

Q(t) =

∫∞

t

exp(−τ2

2)dτ. (3.31)

Contrary to the other type of pulses, the duration of the Gaussian pulse is deter-mined by BT instead of the parameter L, and the pulse is truncated since it has aninfinite duration [30], where B represents the system bandwidth. Practically, theGaussian pulse has been employed by a number of systems. For instance, GMSKscheme which applies Gaussian pulse, has been employed in GSM, GPRS, DECTand DSC1800. In GSM, BT = 0.3 implies the pulse duration is approximated to4 symbols, which is equivalent to L=4. Both phase and frequency pulses are alsodepicted in Figure 3.13.

3.5. Modulation Index and Signal Phase States

As already seen, the modulation index ~ is a design parameter in communication systems.For example, in a frequency modulation, ~ determines the amount of the variationof the carrier frequency that results from the changing modulating signal amplitude.Furthermore, ~ plays a significant role in characterization and determination of a CPMsystem performance as well as its complexity. It is worth mentioning that for practicalpurposes, ~ is chosen to be a rational of the form q

p, where q and p are relatively prime

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positive integers. Otherwise, the number of phase states of the CPM system rises andconsequently the receiver becomes complex [67]. Nevertheless, the pulse shape and itsduration L, the modulation index ~ and the size of the information alphabet M togetherdetermine the complexity and types of the CPM scheme [67]. However, due to theconstraints imposed by the continuous phase property and thus the induced memory L,the overall phase states of such a signal in the interval nT ≤ t ≤ (n+1)T are determinedby both L and ~. The phase states or phase trajectories (phase paths) which representthe signal elements [45] address the signal in any time interval uniquely. The possiblephase states are subject to:

1. Full response CPM (L=1), i.e. the phase depends only on the current symbol.The number of phase states (phase trajectories) is p and the terminal phase statesfor a full response CPM signal at time instant t = nT are denoted by Θ and givenwhen q is even as

Θ =

0,πq

p,2πq

p, ...,

(p− 1)πq

p

(3.32)

and when q is odd as

Θ =

0,πq

p,2πq

p, ...,

(2p− 1)πq

p

. (3.33)

2. Partial response CPM (L > 1); In this case the phase depends on both the currentand the L−1 previous symbols. Hence, the number of states for a partial responseCPM signal at time instant t = nT is given as

κ =

pML−1 even q

2pML−1 odd q.(3.34)

The state of the CPM signal at time nT is given as

cn = [θn, γn, ..., γn−L+1] , (3.35)

which is a combination of the accumulated phase and the correlative state. The cor-relative state is influenced by the previous L-1 symbols (phase transition factors γ). cndescribe and address the signal elements in the interval nT ≤ t ≤ (n+ 1)T . The collec-tion of all possible phase trajectories forms the phase tree, and correspondingly, thereis a collection of possible frequency deviations. A collection of both, the instantaneousphase and frequency of the CPM signal for an embedded arbitrarily 8-ary sequence ofsymbols of length 4 and RC frequency pulse with a duration L=1 are plotted in Figures3.14 and 3.15. Figure 3.14 represents the phase tree and shows the time variant phaseat each interval. However, Figure 3.15 shows that in full response CPM, the signal inany time interval can also be addressed by only M instantaneous frequency deviations,and the figure hints also to the structure of the receiver.

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0 1 2 3 4 5 6 7−60

−40

−20

0

20

40

60

t / T

φ(t)

in d

egre

es

Figure 3.14.: A collection of instantaneous signal phases (phase tree) for the RC filterwith L = 1, ~ = 0.5 and M = 8.

0 1 2 3 4 5 6 7−2/T

−7/4T

−5/4T

−3/4T

−1/4T

0

1/4T

3/4T

5/4T

7/4T

2/T

t / T

Inst

anta

neou

s fr

eque

ncy

Figure 3.15.: Collection of instantaneous signal frequencies for the RC filter with L =1,~ = 0.5 and M = 8.

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4. Demodulation of CPM

This chapter deals with CPM scheme’s error performance. The signal spectrum analysisis extensively covered in literature [26, 57] but disregarded in the thesis. The optimumdemodulation of the linear type modulation was reviewed in the previous chapter. Inthis chapter, the principle of the optimum demodulation is extended to the CPM scheme.At first, the optimum detection of the conventional CPM scheme is illustrated, then theoptimum detection of a simplified and linearized CPM is detailed.

4.1. CPM Receiver Complexity

Likewise, the maximum likelihood receiver requires the determination of the Euclideandistance between a pair of signals si(t) and sj(t) over an interval of length iT [57]. TheEuclidean distance denoted by δCPM is given by

δCPM = ‖si(t)− sj(t)‖ (4.1)

=

[∫ NT

0

[si(t)− sj(t)]2 dt

]1/2.

Substituting for si(t) and sj(t) from equation (3.16) in equation (4.1) and squaring theEuclidean distance to allow for power comparisons,

δ2CPM =2ξsT

∫ NT

0

1− cos [φ(t, γi)− φ(t, γj)] dt, (4.2)

where ξs represents the energy of the transmitted symbol. Given φi(iT, γ) in equation(3.13), the phase difference φi(iT, γ) − φj(iT, γ) in equation (4.2) is reduced to thedifference ∆ between the M-ary data and hence

δ2CPM =2ξsT

∫ NT

0

1− cos [ψ(t,∆)] dt (4.3)

=2ln(M)ξ

T

∫ NT

0

1− cos [ψ(t,∆)] dt,


where ∆ ∈ ±2,±4,±6, ..., 2(M − 1). The error probability for CPM in terms ofEuclidean distance is [57]

PCPM = Q

√ξsδ2CPM

N0

. (4.4)

The possible number of CPM signal states κ given in equation (3.34) in the intervaliT ≤ t ≤ (i+1)T dictates the complexity of the demodulator, which compares κ possibleEuclidean distances to select a minimum δ2CPM [68]. For this purpose a trellis structure(Viterbi trellis) is employed and realized by the Viterbi algorithm [69]. Though theViterbi algorithm searches for the minimum δ2CPM in the optimal sense, it is unpracticalfor a higher order trellis structure, which increases exponentially with the system memoryL. In general, optimum receivers for CPM signals have a high degree of complexity.In the following the receiving mechanism for full response CPM in an AWGN channelis illustrated. The received signal r(t, γ), as shown in Figure 4.1, is given as

r(t, γ) = s(t, γ) + n(t), (4.5)

where n(t) is a Gaussian noise with double side power spectrum density N0

2and s(t, γ) is

the transmitted signal given in equation (3.16). In the absence of the noise, the receivedsignal is given as

r(t, γ) =

√2E

Tejφ(t,γ). (4.6)

The received signal is first passed through a limiter that fixes the signal amplitude

CPM

transmitter

CPM

receiver

n(t)

+s(t , ) r(t , )

Figure 4.1.: CPM transmission model.

without affecting the signal phase, then applied to a differentiator whose output is givenas

∂r(t, γ)

∂t= j

∂φ(t, γ)

∂tejφ(t,γ). (4.7)

The above equation reveals that the data symbols are embedded in the envelope of thedifferentiator output. An envelope detector typically used for detecting a PAM signal isused in the case of a full response CPM. A modulus circuit can be utilized instead, whenthe M-ary information set γi ∈ ∓1,∓3, ...,∓(M − 1) is transformed via the relation12(γi +M − 1) to 0, 1, ..., (M − 1). Further, for an optimum detection a matched filter

can be used. The receiver is given in Figure 4.2 and its error performance is depicted inFigure 4.3. The figure compares 4-ary and 8-ary full response, with an RC pulse shape

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r(t, )g% | . | ˆgMatched filterDifferentiatorr(t , )

Figure 4.2.: M-level CPM simple receiver.

+

+

++++++++

+

++

++

+

-1

-2

-3

0

-4

-5

M = 4

M = 8

5 10 15 20 25 30

0bE / N in dB

Log(B

ER

)

Figure 4.3.: Uncoded BER for M-level CPM demodulator, full response and an RC pulseshape, in a Gaussian channel.

CPM receiver.The partial response CPM scheme is next. A phase tree of the partial response (L=2)signal with a raised cosine frequency pulse is plotted in Figure 4.4 and a collection ofthe instantaneous frequencies of the same signal is plotted in Figure 4.5. From thefigures one can predict the receiver complexity and conclude that the receiver shown inFigure 4.2 can not be used for the partial response case. Further, the optimum detectionof one symbol requires that the received signal is observed over a number of symbols,as opposed to symbol by symbol detection. This is known as correlation detection. Al-ternatives to circumvent the high complexity of CPM receivers have been proposed overthe last two decades. A complexity reduction was first proposed in [57, 70] by the use ofa slightly mismatched receiver. By using shorter frequency pulses at the receiver thanthe frequency pulses at the transmitter, the number of phase states can be reduced.Basically, the reduction of the number of phase states is directly proportional to thesystem memory. Additionally, the number of the linear filters at the receiver is reducedas a result of underlying system memory reduction, as will become clear in the followingsections.Rimoldi [71] showed that any CPM system can be decomposed into a continuous-phaseencoder (CPE) and a memoryless modulator (MM). Huber in [72] exploited the dimen-

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0 1 2 3 4 5 6 7 8−60

−40

−20

0

20

40

60

t / T

φ(t)

in d

egre

es

Figure 4.4.: Instantaneous signal phase for the RC filter with L=2, ~=0.5 and M=8.

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0 1 2 3 4 5 6 7 8−2/T

−7/4T

−5/4T

−3/4T

−1/4T

0

1/4T

3/4T

5/4T

7/4T

2/T

t / T

Inst

anta

nten

ous

freq

uenc

y

Figure 4.5.: Instantaneous signal frequency for the RC filter with L=2, ~=0.5 andM=8.

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sion of the CPM signal space aiming to reduce the complexity of CPM receivers, andshowed that four to six linear filters can be sufficient to implement almost all CPM re-ceivers of interest in practice. However, Laurent’s representation (LR), introduced morethan two decades ago, is a way to circumvent this problem. Laurent [73] demonstratedthat any binary CPM signal can be represented linearly by a finite number of time lim-ited amplitude modulated pulses. He showed the exact number of the pulses of whicheach represents a linear receiver filter.

4.2. Decomposition of CPM

As already mentioned, the CPM system can be decomposed into a continuous phaseencoder and a memoryless modulator. The motivations are to obtain alternative realiza-tions of CPM modulators and to reduce the complexity of optimum detection associatedwith CPM demodulators. Figure 4.6 demonstrates a simple CPM modulator, which isbasically a shaping filter for the purpose of smoothing the information-carrying phasefollowed by a phase modulator [71]. However, this representation does not demonstrate

Tx filterX Q(t)

PM-modulator s (t , )q(t)

2πh

Figure 4.6.: M -level CPM simple modulator.

the memory associated with the CPM. The alternative is shown in Figure 4.7. It ex-hibits an inherent encoder that resembles in many ways a convolutional encoder. Inorder to reduce the number of states which have to be traced in the receiver, the virtualencoder can be cascaded with an external convolution encoder [45].Let’s recall equation (3.13)

φ ( t, γ ) = 2π ~

∞∑

i=0

γi q(t− iT ). (4.8)

The phase trellis of the physical phase φ(t, γ) for MSK, plotted again for comparison inFigure 4.8(a), appears to be time variant in the sense that the physical phase trajectories

CPEMemoryless

Modulator

s (t , )

Figure 4.7.: Decomposite CPM modulator.

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in the even-numbered symbol intervals are not time shifted to those in the odd-numberedsymbol intervals. However, adding the lowest possible phase trajectory to the physicalphase, the result is the tilted phase, given as [71]

ψ ( t, γ ) = φ ( t, γ ) + π~ (M − 1))t/T. (4.9)

The phase trellis of the the tilted-phase given in Equation (4.9) is plotted in Figure4.8(b).

0 2 4 6 8 10 12 14−20

−15

−10

0

10

15

20

t / T

φ(t)

in d

egre

es

(a)

0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

16

t / T

φ(t)

in d

egre

es

(b)

Figure 4.8.: A collection of instantaneous signal phases for REC pulse with L = 1, ~= 0.5 and M = 2 (phase tree) (a) Input data are antipodal (bipolar) (b)Input data are unipolar.

ψ ( t, γ ) = 2π~i−L∑

k=−∞

γk + 2π~i∑

k=i−L+1

γkq(t− kT ) + π~ (M − 1))t/T, (4.10)

which written in terms of χ as

ψ ( t, χ ) = 2π~i−L∑

k=−∞

χk + 2π~i∑

k=i−L+1

χkq(t− kT ) + (4.11)

π~ (M − 1)t/T + π~(M − 1)(L− 1)− 2π~(M − 1)i∑

k=i−L+1

q(t− kT ),

where χ ∈ 0, 1, ...,M − 1. The CPM tilted-phase decomposition in equation (4.11) isillustrated in Figure 4.9. However, it is found in this dissertation that the time invariant

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property as shown in Figure 4.8(b) can be simply obtained by applying the relation(γ − (M − 1))/2 to get 0, 1, ..., (M − 1). The first term of the right side of equation(4.11) represents the cumulative, the second term represents the partial response andthe rest of the equation is data independent.

T T T + Ti-1i i-l+1

c

X

X

X

X

4 q(t-(n-L+1)T)

+ PMs(t)

4 q(t-(n-1)T)

4 q(t-nT)

2

cc

πh

πh

πh

πh

Figure 4.9.: CPM decomposition into a continuous phase encoder CPE and a memorylessmodulator [71].

4.3. Linear Representation of CPM

CPM is a non-linear scheme in which the relation between the modulating (transmitteddata) and modulated signal is not linear. The prime implication is the complexityof the receiver. Further, the nonlinear relation complicates the analysis of the CPMsignals. Laurent in [73] described an exact representation for CPM in the form of asuperposition of a number of time-/phase-shifted amplitude modulation pulse (AMP)streams. It follows an illustration of the LR principles: The baseband CPM signalassociated with the ith symbol in equation (3.16) and φ(t, γ) given in equation (3.17) issimplified and stressed in the following as a product of complex exponentials [26, 73]

s(t, γ) =

√2E

Te(jπ~

∑i−Lk=−∞

γk)

L−1∏

k=0

e(j2π~γi−kq(t−(i−k)T )). (4.12)

Taking into account that |γi| = 1, the exponential term is replaced by an equivalent sumof two terms as

ej2π~γi−kq(t−(i−k)T ) =sin(π~− 2π~q(t− (i− k)T ))

sin(π~)+ (4.13)

ejπ~γi−ksin(2π~q(t− (i− k)T ))

sin(π~).

The above representation is not valid for an integer modulation index. The right side ofthe equation, excluding the exponential term, is independent of the transmitted data.

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Let ψ(t) denote the generalized phase-pulse function that has nonzero values only for0 ≤ t ≤ 2LT , given as

ψ(t) =

2π~q(t) 0 ≤ t < LT

π~− 2π~q(t− LT ) LT ≤ t < 2LT

0 otherwise,

(4.14)

which is obtained by the concatenation of q(t) in the interval 0 < t < LT and itsreflection in the interval LT < t < 2LT about the t =LT axis, as depicted in Figure4.10.

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t / T

Am

plitu

de

q (t)

Figure 4.10.: q(t) and its reflection about t = T axis.

It is therefore convenient to define the signal pulse as

si(t) =sin(ψ(t+ iT )

sin(π~)(4.15)

= s0(t+ iT ).

Substituting equations (4.13) and (4.15) into equation (4.12), the baseband CPM signalis expressed as

s(t, γ) =

√2E

Te(jπ~

∑i−Lk=−∞

γk

L−1∏

k=0

s0(t+ (k + L− i)T ) + (4.16)

ejπ~γk−is0(t− (k − i)T ).

The outcome of the product in the above equation are 2L terms, where a detailed analysisof the equation reveals that 2L−1 of these are distinct and the other 2L−1 terms are time-shifted versions of the distinct terms. Finally, the binary baseband CPM signal can beexpressed as

s(t, γ) =

√2E

T

∑

i

2L−1−1∑

k=0

ejπ~ak,ick(t− iT ), (4.17)

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where the pulses ck(t), for 0 ≤ k ≤ 2L−1 − 1, are defined as

ck(t) =

s0(t)

∏L−1i=0 s0(t+ (k + Lβk,i)T ) 0 ≤ t ≤ T ×min L(2− βk,i)− i

0 otherwise.(4.18)

These functions accordingly have the following durations

c0(t) (L+ 1)T (4.19)

c1(t) (L− 1)T

c2(t), c3(t) (L− 2)T

c4(t), c5(t), c6(t), c7(t) (L− 3)T...

...

cM/2(t) · · · , cM−1 T.

The coefficients in the exponent in equation 4.17 are determined by

ak,i =i∑

m=−∞

γm −L−1∑

m=1

γi−mβk,m (4.20)

= a0,i −L−1∑

m=1

γi−mβk,m

= a0,i−L +L−1∑

m=1

γi−m(1− βk,m) + γn,

where the βk,n are either 0 or 1 and are the coefficients in the binary representation ofthe index k, i.e.,

k =L−1∑

m=1

2m−1βk,m, k = 0, 1, ..., 2L−1 − 1. (4.21)

A detailed look at equation 4.17 reveals that the exponential term represents a weightingfactor and varies with the transmitted data. The input bits are coded via the ak,i thenlinearly modulated onto fixed and data independent pulses ck(t) for the k

th component.Therefore, the CPM signal is alternatively composed of 2L−1 superimposed, amplitudemodulated and time limited pulses.To clarify the expansion, let L=4, which results in 8 different component functions ck(t)where k = 0, 1, ..., 7 and are obtained in Laurent representation and given as

k = 0 ⇒ β0,1 = 0, β0,2 = 0, β0,3 = 0 ⇒ c0(t) = s0(t)s1(t)s2(t)s3(t) 0 ≤ t < 5Tk = 1 ⇒ β1,1 = 1, β1,2 = 0, β1,3 = 0 ⇒ c1(t) = s0(t)s2(t)s3(t)s5(t) 0 ≤ t < 3Tk = 2 ⇒ β2,1 = 0, β2,2 = 1, β2,3 = 0 ⇒ c2(t) = s0(t)s1(t)s3(t)s6(t) 0 ≤ t < 2Tk = 3 ⇒ β3,1 = 1, β3,2 = 1, β3,3 = 0 ⇒ c3(t) = s0(t)s3(t)s5(t)s6(t) 0 ≤ t < 2Tk = 4 ⇒ β4,1 = 0, β4,2 = 0, β4,3 = 1 ⇒ c4(t) = s0(t)s1(t)s2(t)s7(t) 0 ≤ t < Tk = 5 ⇒ β5,1 = 1, β5,2 = 0, β5,3 = 1 ⇒ c5(t) = s0(t)s2(t)s5(t)s7(t) 0 ≤ t < T

SC/FDE - Buzid


0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/T

Am

plitu

de

c0(t), L=4

c1(t), L=4

Figure 4.11.: Laurent pulses based on rectangular frequency pulse for L=4.

k = 6 ⇒ β6,1 = 0, β6,2 = 1, β6,3 = 1 ⇒ c6(t) = s0(t)s1(t)s6(t)s7(t) 0 ≤ t < Tk = 7 ⇒ β7,1 = 1, β7,2 = 1, β7,3 = 1 ⇒ c7(t) = s0(t)s5(t)s6(t)s7(t) 0 ≤ t < T

each of which is a product of the basic generalized pulse given in equation (4.18) andthree other time shifts of this pulse.A rectangular frequency pulse, as an example, is considered next to underline the dif-ferences among Laurent pulses. The system memory L is a design parameter, however,BT is the tool in case of a Gaussian pulse. The functions (Laurent pulses) for L = 4are plotted in Figure 4.11. Remarkably, the primary pulse c0(t) and the first pulsec1(t) are visible while the rest are too small to plot and notice. The linearization of theCPM signal appears reasonably in literature [74, 75, 76] and yet, the implementation ofsuch a technique in modern systems is rarely done. But the prime pulse of the linearrepresentation (Laurent pulse) is part of many practical systems today.

4.3.1. Multi-level CPM

So far, only the binary alphabet is considered, with M = 2 (two levels). Multi-level(M > 2) signals (M -ary) retaining the continuous phase property are also possible.Mengali in [77] has linearized multiple level CPM, and his work is based on the Laurentprinciples. According to Mengali, the M -ary symbols are mapped into the binary bitsand each bit of the n-tuple is linearized by LR principles. Furthermore, anM -level CPMsignal is the product of the n signals. However, in practice, multi-level CPM signals havenot been attractive and rarely been deployed.

SC/FDE - Buzid


4.3.2. The Importance of Linear Representation

The benefit of the linear form of the CPM is the considerable simplicity of handling thisform in the following aspects:

• The linear relation between the modulator input and its output enables the ap-plication of the known linear techniques, such as the forward error correction.Additionally, it separates the data shaping and data encoding with the capture ofthe effect of the CPM memory at the same time.

• It converts the CPM memory from the phase function to the pulses lengths andfilter structure of ak,i. As it is noticeable from equation (4.20), the ak,i have botha feed-forward FIR filter and a feedback IIR filter structure capturing the memoryeffect of the CPM signal [74].

• The correlation function and the signal spectrum are easily obtained.

• Design parameters, such as L, M and ~ emerge. Setting these parameters, theCPM signal emerges or evolves to different systems.

4.3.3. General Aspects

• Full response CPM: The simplest case is when L=1, for which the number of signalcomponents is one. Therefore, only the prime pulse c0(t), whose duration is 2T ,is considered. Choosing the modulation index to be the half of an integer, theresultant pulse c0(t) is the well known half sine wave which evolved originally froma rectangular frequency pulse [73]

c0(t)MSK = sin(πt

T) 0 ≤ t ≤ 2T. (4.22)

Similarly, GMSK is obtained by choosing a Gaussian frequency pulse. MSK andGMSK are special cases of CPM. MSK is a special case of the sub-family of CPMknown as CPFSK which is defined by a rectangular frequency pulse (i.e. its phasepulse is linear) of one symbol-time duration (full response signaling). On the otherhand, if the tilted angle is set to π/4 for 0 ≤ t ≤ T , the only pulse for a durationof 2T is given by [73]

c0(t)OQPSK =1√2, 0 ≤ t ≤ 2T (4.23)

and the CPM signal becomes an OQPSK signal.

• Partial response CPM: In this case L > 1. As mentioned earlier, the CPM signalis a composite of the sum of 2L−1 amplitude modulated pulses.

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4.3.4. Linear Representation Approximation

Figure 4.12 illustrates the structure of the linearized CPM. The structure is a linearcombination of 2L−1 components. In each branch, a selected point of PSK constellationof size p points is modulated on ck(t), where k is the branch index. The received signal

a

+

PSK

+

n(t) r(t)

0,n C (t)0

a1,n C (t)

1

aK,n C (t)

K

PSK

PSK

s (t , )

Figure 4.12.: CPM is composed of superimposed, amplitude modulated and time limitedpulses.

at the input of a coherent receiver is given as

r(t, γ) = s(t, γ) + n(t), (4.24)

where s(t, γ) is a transmitted signal and n(t) is a zero mean Gaussian noise with double-sided power spectral density N0/2. The optimum receiver, which is constructed froma bank of matched filters, as shown in Figure 4.13, presents sufficient statistics formaking a decision on which of the p signals was transmitted [26]. It is important tonote that a decision at each time interval nT requires the knowledge of all possibleak,n; k = 0, 1, ..., 2L−1 − 1

. These in turn depend on the current symbol γn and a state

defined by the vector a0,n−L, γn−L+1, ..., γn−2, γn−1. As a consequence, a trellis decoderis imposed for the detection process at the receiver and a decision can be made usingthe Viterbi algorithm (VA). The complexity of the VA is proportional to the numberof states κ given in equation (3.34). However, Kaleh in [75] explicitly showed that thecomplexity of this receiver can be reduced by considering a smaller number of PAMcomponents (Laurent pulses). He further showed that for a GMSK with a BT = 0.25and a 4T wide approximation of the Gaussian pulse (L= 4) two pulses approximationis exact for all practical purposes [75]. Further, the prime pulse carries more than 99%of the total signal power [73, 76]. The ratio between the power of the pulse c0(t) tothe power of the pulse c1(t), called conventionally signal to noise ratio (S/N), is plottedversus ~ in Figure 4.14 for various values of BT. As the signal is approximated byignoring c1(t), its contribution to the signal power appears as a noise, that causes anerror. The figure shows this error increases with the modulation index and decreases asBT increases. Further, at a practical value of ~ = 0.5, the signal to noise ratio equals25 and 35 dB at practical values of BT = 0.3 and 0.5, respectively. The conclusion

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C (-t)0

C (-t)1

C (-t)K

Vit

erbiA

lgori

thm

r(t) Output data

Figure 4.13.: Optimum linear CPM receiver.

from the figure is that at the practical values of both BT and ~, considering onlythe primary Laurent pulse and ignoring the rest ensures a good signal approximationwithout system performance degradation. Another interesting frequency pulse, given inequation (3.8) and discussed in Chapter 4, is TFM. Based on a TFM frequency pulse, theapproximation error is compared to a CPM case with a Gaussian pulse whose BT = 0.5,as shown in Figure 4.15. The comparison reveals that the signal approximation of theTFM is similar to the Gaussian frequency pulse with BT = 0.5. As a conclusion,the approximation implied a reduction in the number of states required in VA and asuboptimal reduced-complexity receiver, whose complexity is reduced dramatically, canbe achieved. Accordingly, equation 4.17 is approximated and the linear CPM signal isthen expressed as

s (t, γ) =√

2E/T∞∑

n=0

a0,n c0(t− nT ), (4.25)

which in terms of the input data is given as

s (t, γ) =√2E/T

∞∑

n=0

ejπ~∑n

m=−∞γm c0(t− nT ), (4.26)

which is identical to a PSK signal except that the input data is passed through a feed-forward filter and the duration of the shaping pulse is two symbols even for a full responseCPM. The linear CPM optimum receiver is depicted in Figure 4.16. Furthermore,equation (4.25) reveals, besides the structure of the modulator, two cascade stages asfollows:

• The pre-modulation filter (shaping filter), whose impulse response is denoted byc0(t), scans (L+1) symbols. The filter produces ISI in order to obtain continuousphase modulation signals. As stated earlier, c0(t) is derived from various frequencypulses shown in Figure 3.13.

• Data encoder, known in literature as differential encoder or multiple symbol ob-servation whose output denoted by a0,n is basically a differential process.

In comparison with the PSK modulation, the similarity and dissimilarity may be em-phasized as follows:

SC/FDE - Buzid


BT = 0.3

BT = 0.1

BT = 0.5

BT = 0.7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-10

0

10

20

30

40

50

Modulation index

S/N

d

B

X

X

35 dB

25 dB

Figure 4.14.: S/N ratio after Laurent approximation for a Gaussian pulse with variousvalues of BT .

• The pulse shape has to scan a minimum of two symbols and a controlled ISI isintroduced.

• The encoding of transmitted data, which is desired in PSK is necessary.

Finally, it is important to emphasize at this point that the benefit of the approximatedand linearized CPM is the significant reduction of receiver complexity in addition to thebandwidth efficiency of this system. On the other hand, the error performance suffers adegradation as a result to the differential encoder [26].

4.3.5. Precoding

The term ”precoding” refers to the pre-equalization which requires the knowledge ofthe channel state information (CSI) [78, 79, 80]. However, in this chapter precoding isrestricted to data encoding at the transmitter and is isolated from channel influences,as demonstrated previously [41, 81]. To further illustrate the differential encodingassociated with the CPM signal, the linearized form of the signal equation (4.26) isrecalled:

s(t, γ) =√

2E/T∞∑

i=0

ejπ/2

n∑

i=0

γic0(t− nT ). (4.27)

SC/FDE - Buzid


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-10

0

10

20

30

40

50

Modulation index

S/N

d

BTFM

Gaussian pulse BT= 0.5

Figure 4.15.: A comparison between TFM and Gaussian pulse (BT = 0.5).

C (-t)0

r(t)VA

Output data

Figure 4.16.: Simple linear optimum CPM receiver.

It is apparent that the exponential term ejπ/2

n∑

i=0

γiin equation (4.27) represents an in-

herent encoding. As the phase state is a function of all past symbols, the encoder hasa differential structure and the CPM depends on the phase differences rather than theabsolute values. In the following, the exponential term is more detailed and written alsoin the form [41]

ejπ/2

n∑

i=0

γi= (j)

n∑

i=0

γi=

n∏

i=0

j γi (4.28)

which is equivalent to

an = jγn

n−1∏

i=0

j γi (4.29)

= jγnan−1.

The above equation represents a 2-bit differential encoder. The decoding equation followsto

γn = −j anan−1

. (4.30)

SC/FDE - Buzid


Since αn ∈ −1,+1,−j,+j it follows that

γn = −jana∗n−1. (4.31)

Figure 4.17 illustrates both the differential modulator and demodulator which is animplementation of equations (4.29) and (4.31). The figure demonstrates the inherentdifferential encoding and the requirement for differential decoding at the receiver. Be-cause of this the system suffers a small performance degradation, which is a directimplication of the continuous phase.A simple fix to this implication is to precode the input data with a differential encoder,which in effect eliminates the need for a differential decoding at the receiver [82] and inreturn avoids error probability degradation resulting from application of the differentialencoding [26]. Nevertheless, from a spectral standpoint, the precoding operation has noeffect on power spectrum density of the transmitted signal [82].In what follows, the demonstration of how the differential decoding at the receiver isreplaced by a differential encoding at the transmitter is presented [81].Suppose the transmitted symbols zi ∈ −1, 1 are constructed from the source data

γ γ

a) b)

i ii i

j

T

-j

T

α α

Figure 4.17.: a) Differential modulator b) Differential demodulator.

γi ∈ −1, 1 as follows:zi = γiγi−1 (4.32)

and substituting in equation (4.28), given γ−1 = 1

an = jnn∏

i=0

zi (4.33)

= jnn∏

i=0

γi.γi−1

= jnγ0γ−1.γ1γ0....γnγn−1

= jnγnγ−1

n−1∏

i=0

γi.γi

= jnγn.

The recovering of the data at the receiver is carried out simply by

γn = j−nan, (4.34)

SC/FDE - Buzid


−15 −10 −5 0 5 10 15

−4

−3

−2

−1

1

Eb/N0 in dB

Log(

BE

R)

non−differential GMSK, BT=0.2differential GMSK, BT=0.2non−differential GMSK, BT=0.5differential GMSK, BT=0.5

Figure 4.18.: Uncoded BER versus S/N for differential and non-differential CPM sig-nals with a Gaussian frequency pulse and various BT values in an AWGNchannel.

which, in fact, represents only a de-rotation of the received symbols. The meaning ofthe above equation is that the transmitted data, which is differentially encoded prior tothe transmission, is recovered simply by a de-rotation process, and the degradation ofsystem error performance that results from the differential decoding is avoided.

4.4. Simulation and Results

In order to compare systems based on the Laurent representation and the precodedCPM signals, BER Monte Carlo simulations were conducted in an AWGN channel. Inthe simulation, the synchronization was assumed to be ideal. The comparison covers, inparticular, the differential linearized CPM and the non-differential linearized coherentCPM with Gaussian pulse for various values of BT .Figure 4.18 compares error performance of both the differential and non-differentialcases, where a Gaussian frequency pulse with BT values of 0.2 and 0.5 is used. Fromthe figure it can be noted that non-differential CPM shows some improvement in lowersignal to noise ratios. However, the two approaches tend to show similar performanceat higher signal to noise ratios. Furthermore, the linearization principle is also extendedto the TFM which has been discussed in Chapter 3. Error performance of the obtainedsystem, a non-differential linearized CPM based on a TFM, which is conventionallycalled hereafter linearized TFM, was also included in the simulation.Shown in Figure 4.19 is the BER for a differential and non-differential linearized TFM.

SC/FDE - Buzid


Again, at low signal to noise ratios, a non-differential linearized TFM performs betterthan a differential TFM. At higher signal to noise ratios both approaches tend to performsimilarly. In the same figure, error performance of a differential and non-differential CPM(Gaussian frequency pulse) are plotted for comparison reasons.

−15 −10 −5 0 5 10 15−7

−6

−5

−4

−3

−2

−1

1

Eb/N0 in dB

Log(

BE

R)

non−differential GMSKdifferential GMSKnon−differential TFMdifferential TFM

Figure 4.19.: Uncoded BER versus S/N for differential and non-differential CPM witha Gaussian pulse (BT = 0.3) and a linearized TFM signal, in an AWGNchannel.

It can be noted that a linearized TFM performs better than the others when the BTof the Gaussian pulse is equal to 0.3, which is equivalent to L=4. The comparison isrepeated and a similar performance is obtained when BT is equal 0.5, as shown in Figure4.20.

4.5. Summary

In the last two chapters, various signal envelopes were reviewed and a basis for sortingsystems with respect to the transmitted signals envelope is set. The generation andreception of constant envelope signals were presented. In this chapter, the CPM demod-ulation was treated in detail, since it is the central theme of the coming chapters, inwhich the broadband techniques for multipath channels are the objective.Additionally, in this chapter non-differential CPM signals are developed from the differ-ential CPM signal which suffer from error propagation and some performance penaltydue to the phase continuity of the CPM signals. The non-differential signals show animproved performance as a direct result to the elimination of the differential decoding

SC/FDE - Buzid


−15 −10 −5 0 5 10 15

0

0

−6

−5

−4

−3

−2

−1

Eb/N0 in dB

Log(

BE

R)

non−differential GMSKdifferential GMSKnon−differential TFMdifferential TFM

Figure 4.20.: Uncoded BER versus S/N for differential and non-differential CPM witha Gaussian pulse (BT = 0.5) and a linearized TFM signal, in an AWGNchannel.

and the avoidance of error propagation.

The overall advantage of the simplification with respect to the system performanceis the transformation of the nonlinear modulation into a linear modulation without theinherent differential encoding. This is beneficial for broadband systems as will be seenin the coming chapters.

SC/FDE - Buzid

5. The Concept of Single CarrierTransmission with FrequencyDomain Equalization (SC/FDE)

5.1. Introduction

The single carrier transmission with frequency domain equalization concept is basicallya broadband technique that is developed for high data rates over a wireless channel.The distinctive feature of the wireless channel is the multipath propagation, i.e. thetransmitted signal reaches its destination over many paths. The received signal is then asummation of all incoming signals which are attenuated and time delayed in accordancewith the individual paths. The superposition can be destructive or constructive and thusthe received signal is a distorted copy of the original. The extraction of the data fromthe distorted signal is, therefore, an impossible task without the use of some powerfulmeasures or techniques. The SC/FDE technique is an elegant and effective solution formitigating impairments of the dispersive channel [5, 6]. The antonym of a single carriersystem as the name may suggest is the multiple carrier system, and both are possiblemethods of transmitting data from one point to another over such an environment. Thesingle carrier transmission system has been used to transmit data since man startedtransmission. However, the superiority of the SC/FDE concept lies in the equalizationthat mitigates the multipath distortions which is carried out in frequency domain. Theconsequence of the use of frequency domain equalization is the significant complexityreduction of the receiver. Equalization using the discrete frequency domain was firstreported in [9]. Indeed, it marks the begin of the SC/FDE technique. However, Sari et al.[7, 10] combined the frequency domain equalization via discrete Fourier transformation(DFT) and the guard interval. SC/FDE is an attractive technique for broadband wirelesschannels and has a lower complexity than time domain equalization due to its use of thecomputationally-efficient fast Fourier transform (FFT). One implication of the frequencydomain equalization is the imposed block structure transmission.

5. The Concept of Single Carrier Transmission with Frequency Domain Equalization(SC/FDE)

67

5.2. System Description

For performance comparison purposes the SC/FDE parameters in this thesis are adaptedto the OFDM based IEEE 802.11a standard. The conceptual physical layer layout of anSC/FDE system is as shown in Figure 5.1 and the main physical layer parameters areas given in Table 5.1. However, for more details it can be referred to [83, 84]. Note that

Scrambling

Coding

Interleaving

MappingGuard

interval

Up-

sampling

Transmit

filter

Down

sampling

Matched

filterFFT2N Remove GI

Synchronisation

IFFTNDemappingDescrambling

Decoding

Deinterleaving

Equalizer

Channel estimation

Input

data

output

data

Pilots

payload

Se

rialto

pa

ralle

lP

ara

llelto

se

rial

Figure 5.1.: Conceptual physical layer layout of an SC/FDE system.

Net symbol rate 12 Msps

N (symbols per block) 64

NG (cyclic extension block) 12

Block duration 5.33µs

Guard time TG 842 ns

Duration of a processed block (FFT integration time) 4.49µs

Duration T of one individual symbol 70 ns

Modulation BPSK, QPSK, 16-QAM and 64-QAM

Pulse shaping Root raised cosine roll off = 0.25

Convolutional encoder with code rate 0.5generation polynomial (133,171)

Table 5.1.: Main PHY parameters of the SC/FDE system.

in this chapter linear modulation schemes are assumed, the application of CPM-schemesis investigated in chapter 6. The principle differences between SC/FDE and OFDMconcepts are the formation of the transmitted signal and the decision making process.In SC/FDE, decisions are made in the time domain as opposed to the OFDM, wherethey are made in frequency domain, as depicted in Figure 5.2. A comparison and theresulting discrepancies between SC/FDE and OFDM systems are shown in Table 5.2.In the following subsections, the baseband processing steps of the investigated SC/FDEsystem are reviewed and associations to the appropriate blocks in an IEEE 802.11aOFDM based system are given in each case. Both SC/FDE and OFDM systems areblock based transmission schemes and the reason is twofold: first to ease the process of

SC/FDE - Buzid


68

FFT IFFTequalizer

FFTIFFT equalizer

OFDM transmitter

SC/FDE transmitter

OFDM receiver

SC/FDE receiver

Figure 5.2.: SC/FDE and OFDM transceiver comparison.

SC/FDE OFDM

1 Less sensitive to RF impairments

2 Adaptive subcarrier allocation is possible

3 Has reduced PAPR Adaptive modulation is possible

5 Less costly power amplifier

6 Appropriate to constant envelope modulation Only non constant envelope modulation is possible

7 Coding is desirable Coding is necessary

Table 5.2.: A comparison between SC/FDE and OFDM concepts.

the frequency domain transformation and second, to insert a guard interval. The guardinterval is necessary to prevent the intersymbol interference (ISI) by freshing or clearingthe channel memory. The ISI is a result of the multipath components of the wirelesschannel.

5.2.1. Baseband Equivalent Model

The system layout, shown in Figure 5.1, is a baseband data transmission system model.However, the RF signal s(t) is expressed as

s(t) = Rea(t)ej(2πfct+θ(t))

(5.1)

= sI(t)cos(wct)− sQ(t)sin(wct),

where a(t) and θ(t) are bearing data information and fc is the carrier frequency. sI(t) andsQ(t) describe the quadrature components. Because of time consumption and simulationcomplexity, the baseband model is more attractive and useful in simulations. Generally,the baseband model is always adapted in the investigations by excluding the RF part.By dropping the RF term in equation (5.1), the baseband signal can be obtained and

SC/FDE - Buzid


69

given as

s(t) = a(t)ejθ(t) = sI(t) + jsQ(t). (5.2)

The above equation will appear frequently and more will be said about it in the comingsections.

5.2.2. Scrambling

Scrambling limits the fluctuations of the OFDM signal envelope by avoiding long se-quences of bits with the same value. In OFDM, long sequences of bits with the samevalue would implicate constructively superposition of subcarriers and hence the resultingenvelope may have quite high peaks [13]. For SC/FDE scrambling is not a must.

5.2.3. Channel Coding

Forward error correction (FEC) is one of the powerful concepts that are utilized in mostof todays communication systems. Although, FEC means adding redundancy, today,it is seldom to implement a communication system without the application of FECtechniques. The redundant transmitted data enable error correction at the receivers.Two most known encoders among others are the linear block encoder and the convolutionencoder. Block convolution encoders show powerful error correcting capabilities. Theyhave been adapted by todays dominant standards as well as by many IEEE standards.The convolution encoder encodes the entire data stream into one long code word. Inour investigations the convolution encoder, shown in Figure 5.3, as defined in the IEEE802.11a standard is used. At the receiver, the maximum likelihood principle is used todecode the convolutional codes. In order to simplify the decoding process, the Viterbialgorithm is widely used.

x0

x1

x2

x3 x

4x

5x

6

g1=1718=1+x1+x

2+x

3+x

6

g0=1338=1+x2+x

3+x

5+x

6

K=7

7-Stage Shift Register

Two Symbols are output for

each Bit input to the

Encoder. The output Rate is

Twice the Input Rate

;

Figure 5.3.: Convolutional coder as defined by the IEEE 802.11a standard.

SC/FDE - Buzid


70

5.2.4. Interleaving

Interleaving is designed to arrange the transmitted symbols in a manner that improvesthe system performance. In OFDM the technique interleaves the data over frequencysuch that, if any frequency fading null damages neighboring subcarriers, the damageddata can be recovered with the help of FEC. For the SC/FDE, a block by block basedinterleaver is less significant, while for OFDM it forms an essential element [13].

5.2.5. Modulation

After the encoding and interleaving as shown in Figure 5.1, the data is ready to be mod-ulated. The linear modulation techniques used in SC/FDE concept are BPSK, QPSK,16 QAM and 64 QAM as shown in table 5.3 [83, 85]. Again the modulation/coding ratepairs are adapted to the IEEE 802.11a standard. As detailed in the previous chapters,

Net symbol rate Modulation Code rate bits/processed block

6 Mb/s BPSK 1/2 32

9 Mb/s BPSK 3/4 48

12 Mb/s QPSK 1/2 64

18 Mb/s QPSK 3/4 96

24 Mb/s 16-QAM 1/2 128

36 Mb/s 16-QAM 3/4 192

48 Mb/s 64-QAM 2/3 256

54 Mb/s 64-QAM 3/4 288

Table 5.3.: PHY linear modulation schemes/coding rates for the investigated SC/FDEsystem.

the modulation step is carried out in two stages. In the first stage, the data is mappedonto a constellation or labeling, where each element in the labeling is known as a symbol.The second stage is the shaping of the symbols and the conversion of the discrete intocontinuous time signal, for which a pulse shaping filter is used in the SC/FDE case.

Data MappingThe mapped symbols are tabulated below according to the used modulation scheme.

Modulation type γ Bits/symbolBPSK +1,-1 1QPSK +1,-1,+j,-j 216-QAM ±3,±1 + j ±3,±1 464-QAM ±7,±5,±3,±1 + j ±7,±5,±3,±1 6

Note that in the simulation set up a normalization has been conducted, such that themean transmit power is the same for each modulation. From the above table, one

SC/FDE - Buzid


71

can conclude that BPSK, QPSK and QAM modulation schemes in the baseband aresimply a data transformation or mapping process. That explains the interchangeabilityof mapping and modulation in most literature. The baseband equivalent transmit signalcan be written as

s(t) =N−1∑

i=0

γig(t− iT ), (5.3)

where g(t) is the impulse response of the pulse shaping filter. γ are complex multilevelinput data and can be obtained for each specific modulation uniquely as summarized inthe following table.

Modulation type RemarksBPSK 2γ − 1 γ ∈ 0, 1QPSK 2γi − 1 + j(2γk − 1) γi, γk ∈ 0, 1 , i, k = 1, 2

M-QAM 2u− 1−√M + j(2v − 1−

√M) v, u = 1, 2, ...,

√M

Moreover, in most digital transmission systems, a Gray code bit mapping, that will bereviewed later, is applied instead of a binary natural bit mapping.

Pulse Shaping

After the mapping, the modulated symbols are typically shaped in order to obtain asignal spectrum with acceptable sidelobs level that fulfills the international bodies con-straints and regulations, e.g. the Federal Communications Commission’s (FCCs) spec-trum mask. A typical shaping pulse g(t), known as a Nyquist pulse, is given in timedomain as

grc(t) =sin(πt/T )

πt/T

cos(πrt/T )

1− 4r2t2/T 2, (5.4)

where r is a roll-off factor with 0 ≤ r ≤ 1, that influences the pulse bandwidth which islimited to |f | < 1+r

2T. The RC frequency response is given by

Grc(f) =

T 0 ≤ |f | < 1−r2T

T2

1 + cos

[πTr

(|f | − 1−r

2T

)]1−r2T

≤ |f | < 1+r2T

0 |f | ≥ 1+r2T.

(5.5)

Typically in both, the transmitter and the receiver root raised cosine filters (RRC) areused:

Grrc(f) =

√T 0 ≤ |f | < 1−r

2T√T2

1 + cos

[πTr

(|f | − 1−r

2T

)]1−r2T

≤ |f | < 1+r2T

0 |f | ≥ 1+r2T.

(5.6)

The cascade of the transimeter RRC pulse shaping filter and the receiver RRC matchedfilter is again an RC filter.

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5.3. The Mobile Radio Channel and Additive Noise

”He or she who does not know the channel, can never be a good radio

engineer”said a wise man.

It is a fact that in order to design and establish a reliable communication system, agood knowledge of the communication channel should be available [86]. Furthermore,each channel dictates its own communication system. In wireless systems, a channelaffects and degrades the system performance via a multipath phenomena, besides theGaussian noise [87]. A multipath channel is basically a bond of a number of travelingrays that reach the destination through a multiplicity of different paths, with each pathhaving a different length and characteristics in terms of fading, phase and time of arrival[88]. Figure 5.4 shows a model of the used channel model where s(t), n(t) and r(t)represent the transmit, noise and receive signals, respectively, and hb(t) describes thewireless channel impulse response.

Linear

channel hb(t)s(t)

n(t)

r(t)

Figure 5.4.: Channel model.

5.3.1. Additive White Noise

The additive noise at the receiver is a collective of man made noise and thermal noiseproduced by the receiver front end as a result of imperfection of the electronics compo-nents of the receiver. It is white noise because of its uniform frequency content. Whileit is independent of the paths over which the signal is being received, its spectral densityN0 is temperature dependent and given as

N0 = kT [W/Hz] ,

in which k is the Boltzmann constant and T is the receiver system noise temperature.Without going into detail, a standard assumption is that the noise is Gaussian distributed

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with zero mean and variance σ2n [89, 90]. The channel capacity C, which is defined as

the maximum data rate at which a reliable communication is possible, is given withbandwidth B, signal power P , and noise variance σ2

n by the Shannon relation

C = B log2(1 + P/σ2n).

5.3.2. The Mobile Radio Channel

The radio (wireless) channel is characterized by a superposition of a number of signalspropagating over various paths, among which may be a direct path from the transmitterto the receiver (the line of site, LOS). The loss of a direct path is determined by the freespace inverse square law, where the received signal power is given by [91]

Pr = Pt(λc4πd

)2GtGr, (5.7)

where Pt and Pr are the transmitted and received power, respectively. λc is the wave-length and d is the range separation. Gt, Gr are the power gains of the transmit andreceive antenna, respectively [92]. The signal also arrives at the receiver over otherpaths resulting from reflection, refraction and scattering. The contribution of the indi-vidual paths to the overall channel can be constructive or destructive, which results inamplitude and phase fluctuations as well as a time delay in the received signals. Themultipath channel is space and time dependent, and modelling the channel unique is animpossible task. This makes the statistical models of typical scenarios or different areasof applications of relevance. Channel models for the area of indoor communications havebeen developed by ETSI for the standard HiperLan/2 and by IEEE for the standardIEEE 802.11a.

Channel Impulse Response

In a complex representation, the time invariant channel impulse response, which re-lates the gain or attenuation ci, the phase shift Ωi and the time delay τi per path overν paths, can be given as [93]

hc(t) =ν−1∑

i=0

ciejΩiδ(t− τi). (5.8)

To be more precise, these parameters as well as the number of paths are time variant.However, for most of the applications, including indoor channels, the rate of change ofthese parameters is reasonably slow in comparison to the system data rate, and hence,equation (5.8) represents a static or quasi-static channel [86]. The impairment due tothe channel is a reduction of the channel capacity and in severe cases the creation ofsignal outage and loss of connection [87]. The multipath channel can be characterizedby the following parameters:

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• Multipath delay spread

Because of the late arrival of signals over different paths, the total signal willbe dispersed and broadened over the time. The dispersion time of the channel iscalled multipath delay spread τd. It is also defined as the difference in the prop-agation time between the longest and shortest paths. Considering only the pathswith a considerable energy, τd may be given as

τd = maxi,j

|τi(t)− τj(t)| .

The time distribution of the received signal can be given by the power delay profile(PDF) which is expressed as

PDF =ν−1∑

i=0

c2i δ(t− τi). (5.9)

The PDF can be determined by sounding a wireless channel by an impulse functionδ(t) [86]. A useful measure of the channel delay spread is the RMS delay spreadτrms. It can be determined via the PDF and is given by

τrms =

√∑ν−1i=0 (τi − τ)2c2i∑ν−1

i=0 c2i

, (5.10)

where

τ =

∑ν−1i=0 τic

2i∑ν−1

i=0 c2i

. (5.11)

• Coherence bandwidth

Analogous to the delay spread parameters in time domain, the coherence band-width (correlation bandwidth) Bc is another parameter to characterize the chan-nel in the frequency domain [91]. Coherence bandwidth is defined as the channelbandwidth that passes all spectral components with roughly equal gain and linearphase. Further, the coherence bandwidth is inversely proportional to the RMSdelay spread, i.e.,

Bc α1

τrms

, (5.12)

where the proportional constant depends on the particular definition. Typicalvalues of the constant are in the range of 1/5 to 1/7 [91][93]. Furthermore, Bc

defines the type of the wireless channel, e.g., the channel is frequency selective ifBc is small compared with the bandwidth of the transmitted signal. Otherwise thechannel is frequency non-selective or flat and a single channel filter tap representsthe channel.

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• Doppler spread and coherence time

Doppler spread Bd and coherence time tc are parameters needed to describe thechannel that suffers from a Doppler effect. The Doppler effect is due to the motionof one or both ends of the wireless communication system. The maximum Dopplerfrequency is given by the ratio of the mobile speed υ to the carrier wavelength λc,i.e. [88]

Bd =υ

λc. (5.13)

Coherence time is the time domain dual of Doppler spread. The Doppler spreadand coherence time are inversely proportional to one another, i.e.

tc α1

Bd

, (5.14)

A time selectivity or fast fading is experienced when tc is much less than the symboltime T. On the other hand, if the symbol duration of the transmitted signal is muchless than tc, the channel is time non-selective or slow.

5.3.3. The IEEE 802.11a Channel Model

In the frame of the standardization process of the high data rate WLAN system in thebands 2.45 GHz ISM and 5 GHz, the IEEE working groups 802.11a and 802.11b haveestablished what is called the IEEE indoor channel model. The reason for this is theneed to have a fair comparison between different wireless system concepts. The model istime invariant for the duration of a burst and meant to simulate the small scale fadingeffects as a tapped delay line defined by

hm =ν−1∑

n=0

ciδ(m− n), (5.15)

where thecn = bne

−jΨn (5.16)

are based on the following statistical model:

cn = N(0,σ2n

2) + jN(0,

σ2n

2) (5.17)

σ20 = 1− e−

∆ττrms (5.18)

σ2n = σ2

0e−n ∆τ

τrms . (5.19)

Here ∆τ represents the multipath resolution time and its value can be given as ∆τ < 1B.

τrms indicates the channel delay spread and N(0, σ2n

2) is a zero mean Gaussian random

variable with a variance σ2n

2. Based on the above equations, it can be concluded that the

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bn follow a Rayleigh distribution and Ψn follow the uniform distribution in the interval[−π, π]. Further, the mean power decays exponentially and the mean receive power forindependent actual impulse responses is

∑

n

σ2n = 1. (5.20)

Figure 5.5 shows an exemplary channel snapshot in the time and frequency domain.

Figure 5.5.: Exemplary channel snapshot in the time and frequency domain.

5.4. Transmission Model and Optimum Receiver

Structure

As mentioned earlier, the block transmission is indeed the core of SC/FDE as wellas OFDM. However, the linear modulation formats (BPSK, QPSK and M-QAM) arestraightforwardly applied to a block of data. Moreover, the received signal has to beprocessed block by block because of the use of the discrete Fourier transformation.For any linear modulation, the baseband signal in block-based transmission can be writ-ten as

s(t) =∑

l

(N−1∑

n=0

γn,lg(t− (n+ lN)T )

), (5.21)

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where the data stream is grouped in blocks and γn,l is the nth modulated symbol in the

lth block of size N . T is the symbol duration and g(t) is a pulse form which may takemany possible shapes. The symbol spaced discrete time model of the signal at the nth

interval is given as

sl(n) = γn,l for n ∈ [0, N − 1] . (5.22)

The complete signal is constructed by the concatenation of all SC/FDE symbols andexpressed as

s(n) =∑

l

sl(n− lN). (5.23)

As shown in Figure 5.1, following the mapping process is the insertion of a guard in-terval between the successive blocks. The purpose of the guard interval is to precludethe interblock interference. In SC/FDE, likewise in OFDM, the duration of the guardinterval, in order to be effective, has to be longer than the duration Th of the channelimpulse response hb(t) [94]. In [21], a closed form expression for the optimum guardinterval duration is derived, subject to a total average energy constraint on the informa-tion and the guard symbols. Further, a zero-padding (zero stuffing) and a cyclic prefixare the most common guard sequence types. In a cyclic prefix, a part of the informa-tion symbols is copied from the block tail to the guard interval. At the receiver, theguard sequences are discarded since they do not carry information. Recently, a knownsequence called unique word (UW) has come into practice instead of the cyclic prefixor the zero padding [14, 16]. The UW pilots can be used for channel estimation andsynchronization purposes, in addition to the clearing of the channel memory after eachblock transmission.

5.4.1. Burst- and Block Structure and Guard Interval

The burst structure shown in Figure 5.6 is in accordance with the IEEE 802.11a standard[95]. As can be seen, the data blocks are preceded by a preamble that is designated forsynchronization and channel estimation which are postponed in this work. The preambleis composed of two sections. The contents of the first section is a copy of 10 repeatedshort training symbols each denoted by A (constant time domain envelope signals aremost suitable). The section is dedicated to frame-, coarse timing-, coarse frequencysynchronization and automatic gain control (AGC). The second section of the preambleconsists of two identical long training symbols (pilot one and pilot two) and a large guardinterval as shown in Figure 5.6. The second section is used mainly for fine frequencyoffset estimation/synchronization and for a channel estimation. More details regardingthe preamble can be found in references [95, 17]. Furthermore, the payload which iscomposed of a number of blocks, represents mostly the user data.

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Figure 5.6.: Transmission burst with preamble for synchronization and channel estima-tion tasks followed by the payload (user data).

5.4.1.1. The Concept of Cyclic Prefix

The structure of one transmitted block consists of the original sequence of N symbolswith duration TFFT = NT and the cyclic extension with duration TG, which is occupiedby the last Ng data symbols copied in front of the block as depicted in Figure 5.7.

Block l-1

CP CP CP

UW UW UW

data

data data

data

copy

NgT NT

(N-Ng)TNgT

Block l

TFFT

TFFT

Figure 5.7.: Transmission block and guard interval occupied by either cyclic prefix CPor unique word UW with duration Tg = NgT .

If sl(t) with t ∈ [0, TFFT ] denotes the continuous time representation of the originalsymbol sequence of the lth block, then the extended block denoted by sl(t) is given by

sl(t) =

sl(t) t ∈ [0, TFFT ]

sl(t+ TFFT ) t ∈ [−TG, 0]0 otherwise.

(5.24)

The over all transmit signal may then be written as

s(t) =∑

l

sl(t− l(TFFT + TG)). (5.25)

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79

It is essential to mention that the other benefit of the cyclic prefix is that the linear con-volution of the lth transmit block with the channel impulse response becomes a circularconvolution. This, in turn, reduces the receiver complexity [12, 16] and the receivedblock rl(t) fulfills the condition

rl(t) = sl(t) ∗ hb(t) = sl(t)⊗ hb(t) (5.26)

within the interval t ∈ [−TG +Th, TFFT ], where ∗ and ⊗ indicate linear convolution andcircular convolution process, respectively. The restriction of rl(t) to the interval [0, TFFT ]contains exactly one period of the cyclically extended signal rl(t) and is denoted by rl(t)in the following. Applying the circular convolution theorem to the above equation resultsin

Rl(nf0) = Sl(nf0)Hb(nf0) (5.27)

for f0 =1/TFFT and n ∈ Z, where Rl(f), Sl(f) and Hb(f) are related to the time domainsignals rl(t), sl(t) and hb(t) by the continuous Fourier transform. It is worth mentioningthat Sl(f) appearing in the above equation is the spectrum of the non-cyclic extendedlth transmitted block sl(t). This means, Rl(nf0) appearing in the above equation isequivalent to Rl(nf0), which denotes the Fourier transform of the received block thatwould result from transmission of the original non-cyclically extended block sl(t) overthe channel hb(t) [95]. This result implies that the frequency transformation for non-cyclically extended block transmission required by a linear receiver can be applied tocyclically extended block transmission, too.

5.4.1.2. The Concept of Unique Word

Another important concept emerged recently is the unique word [14]. Instead of a zerostuffing or a cyclic prefix which are thrown away at the receiver, a unique word or apredefined and deterministic sequence fills the guard interval as illustrated in Figure5.7. It fulfills the theorem of cyclic convolution in exactly the same way as the CPconcept [14]. At the receiver, in addition to the protection of the user data from theIBI, the sequence is used advantageously in equalization, synchronization and channelestimation, aiming to improve the system performance [16, 20, 96, 97].

5.4.2. The System Bandwidth Efficiency

The limited spectrum resource and the demand for high transmission rates impose band-width constraints. The system bandwidth efficiency η, discussed in Chapter 2, is rede-fined here again by the ratio between the amount of transmitted data to the assignedbandwidth. Since part of the transmitted data is consumed by the system to establisha communication link (overhead sequences) and/or a guard interval to clear channelmemory, the efficiency may be defined as

η =Bit rate

Bandwidth. (5.28)

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80

In an AWGN channel, where a guard interval is not compulsory, the efficiency for anM-ary system is given as

η =ld(M)

1 + r, (5.29)

where r is the roll-off factor. For a dispersive channel with a memory Th, over which aguard interval TG is imposed by the block transmission where TG ≥ Th, the achievablechannel block throughput is

ηCP =ld(M)

1 + r

(TFFT

TFFT + TG

). (5.30)

For the UW concept, the guard interval or guard sequence is processed at the receiveras part of the useful signal and hence, the discrete Fourier transform interval overlapsthe guard interval TG. Therefore, the achievable channel block throughput is

ηUW =ld(M)

1 + r

(TFFT − TGTFFT

). (5.31)

A simple comparison between the system efficiency of a CP and a UW concepts reveals

ηUW/ηCP = 1−[TGTFFT

]2, (5.32)

which shows that the UW approach is less efficient than CP. However, the UW is foundto be useful in equalization as well as in synchronization [20, 14, 96]. Nevertheless,for short guard intervals, both CP and UW approaches tend to show similar systembandwidth efficiency.

5.4.3. Optimum Linear Receiver

For the sake of simplicity the blockwise transmission is disregarded for the upcomingconsiderations. Moving forward to the receiver side in the frame of the system descrip-tion, the received signal corrupted by an AWGN as shown in the simplified transmissionmodel shown in Figure 5.8 can be written as

r(t) =∑

n

γngT (t− nT ) + n(t), (5.33)

where gT is the transmitted filter impulse response, γn are the transmitted complexsymbols and n(t) is the Gaussian noise. The impulse response of the matched filtergR(t) that maximizes the signal to noise ratio (S/N) is given as [26]

gR(t) = Kg∗T (−t), (5.34)

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81

gT(t) gR(t)

n(t)

+s(t) x(t)r(t)

Figure 5.8.: A simplified transmission model in an additive white Gaussian noise channel.

where K stands for a real constant. The superscript asterisk ()∗ indicates a complexconjugate operation. In frequency domain, the transfer function of the matched filterGMF (f) is also given as

GMF (f) = GR(f) = KG∗

T (f), (5.35)

where GR(f) denotes the receiver filter transfer function. Furthermore, the Nyquistcriterion is another important criterion that has to be fulfilled to prevent ISI. The cri-terion is that at the sampling instants or multiple of sampling instants, the interferencehas to be zero (ISI free). The filter with a raised cosine impulse response satisfies theNyquist criterion and its frequency response Xrc(f) is given by Equation (5.5). Theoverall frequency response from the transmit filter input to the receiver filter output isthen

GR(f)GT (f) = Xrc(f), (5.36)

which implies

|GR(f)| = |GT (f)| =√Xrc(f).

Therefore, a root raised cosine (RRC) filter with a transfer function√Xrc(f) can be

utilized for both the transmit and receive filters.

In a multipath channel, the received signal suffers from the multipath effects, in ad-dition to an AWGN. A simple model describing this phenomena is shown in Figure 5.9and the baseband received signal may be written as

r(t) =∑

n

γngT (t− nT ) ∗ hb(t) + n(t) (5.37)

=∑

n

γnh(t− nT ) + n(t),

where the operator ∗ represents the convolution operation. hb(t) is the complex valuedbaseband channel impulse response and h(t) = gT (t) ∗ hb(t), whose frequency responseH(f) = GT (f)Hb(f) is denoted as the channel distorted transmitted pulse. Since gT (t)is a deterministic function, the statistical analysis of the h(t) is directly dependent onhb(t). In the receiver, as shown in Figure 5.9, the output of the matched filter in the

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82

gT(t) Equalizer

n(t)

+s(t) hb(t)

gR(t) x(t)r(t)

GR(f)Hb(f)GT(f) X(f) E(f)Y(f)

y(t)

Figure 5.9.: Simplified communication system model including a linear channel modeland a linear equalizer.

frequency domain can be written as

X(f) = γnGT (f)Hb(f)GMF (f) +GMF (f)N(f) (5.38)

= γnGT (f)Hb(f)G∗

T (f) +NG(f)

= γnHb(f) |GT (f)|2 +NG(f),

NG denotes the matched filter output colored noise. Further, the output of the matchedfilter is also corrupted by an ISI introduced by the channel. An equalizer E(f) thatremoves the distortion resulted from the channel may be given as

E(f) = 1/Hb(f), (5.39)

thus,

Y (f) = γnXrc(f) +Ne(f). (5.40)

This type of linear equalization is called Zero-Forcing equalizer, since the equalizeroutput is distortionless (ISI free). Ne(f) = GMF (f)N(f)/Hb(f) is the equalizer outputnoise spectrum, which may lead to a noise enhancement problem.At this point, it is important to emphasis that the frequency response of the equalizergiven in equation (5.39) is continuous. A dramatical complexity reduction is obtainedonce a discrete frequency domain signal representation is available [13].

5.4.4. Frequency Domain Equalization Concept

As illustrated in the previous section, the need to combat ISI enforces the use of equal-ization. Traditionally, time domain equalization is commonly utilized in wired commu-nication systems as well as digital radio microwave. Nowadays applications, known forthe high data rates and the demand of a bit rate of tens or hundreds of megabits persecond or more, enforce the substitution for such equalization, since the complexity ofthe time domain equalization grows quadratically with the bit rate [13]. Alternatively,frequency domain equalization, proposed over two decades ago [9, 7], forms a substitute,since the complexity of frequency domain equalization grows only slightly more thanlinearly. In Figure 5.10 a system based on the frequency domain equalization principle

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83

Tx filter+g (t)/G (f) h (t)/H (f)

Rx filterg (t)/G (f)

data

IFFTDecision

Equalizer Cofficients

Receiver

n(t)

T T b R R

r(t)s(t)

Matched filterG ( f)

Samplingrate reduction

SamplerFFT

T/ Multiplier

Equalizerb

Figure 5.10.: Communication system with optimum linear reception and frequency do-main equalization.

on block basis is shown. A discrete form r(µTµ) of r(t) is obtained at a rate µ/T , whereµ = T/Tµ > 1 is the oversampling factor and Tµ is the sampling period. µ takes integervalues, preferably a power of two.An µN -point FFT transforms the µN samples of the received data block into the fre-quency domain [13, 12, 98], since only data symbols are considered and the samplescorresponding to cyclic prefix are discarded. Let R[κ] denote the discrete frequencydomain of the discrete sequence r[µTµ], therefore, µN samples are obtained and givenin frequency domain as

R[κ] =

µN−1∑

υ=0

r[υ]e−j2πυκ/µN . (5.41)

The receiver filter is matched to the channel distorted transmitted pulse h(t) and itsimpulse response is given by gMF (t) = Kh∗(−t). Because the output of the matchedfilter is at a rate of µ/T , the equalizer has be to preceded by a sampling rate reduction,which is described in the frequency domain as

x[κ] = 1/µ [x(κ) + x(κ−N) + ...+ x(κ− (µ− 1)N)] . (5.42)

As mentioned previously, the signal samples at the matched filter output represent ISIcorrupted transmitted symbols. To improve the system performance, a compensator oran equalizer that eliminates or minimizes the ISI is needed. For the purpose of derivingthe equalizer coefficients e(n), the digital receiver illustrated in Figure 5.10 can beequivalently represented by a symbol rate model as depicted in Figure 5.11, where theinput to the equalizer is the T -spaced sampled signal given as

x[n] =∑

m

γmq[n−m] + z[n], (5.43)

where q[n] = [h(t) ∗ gMF (t)]t=nT is the over all impulse response seen at the input ofthe equalizer, its frequency response is Q(f), and z(n) = [n(t) ∗ gMF (t)]t=nT denotesT -spaced sampled noise sequence at the output of the matched filter. Minimizing the

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84

error power between the transmitted and the received symbols is the principle on whichthe operation of the equalizer is based. The equalizer error denoted by ǫn is definedas the difference between the transmitted symbols γn and the equalizer output y(n) asdepicted in Figure 5.11.

q[n] e[n]

z[n]

+ y[n]x[n]

Figure 5.11.: A symbol rate model of the overall system

Therefore, the error may be given as

ǫn = γn − y(n). (5.44)

The mean power in the equalizer error denoted σ2 and expressed as

σ2 = E[|ǫn|2

](5.45)

may be computed in the frequency domain as

σ2 = ϕǫǫ(0) = T

∫ 1/T

0

Φǫǫ(f)df, (5.46)

where ϕǫǫ(0) represents the autocorrelation function and Φǫǫ(f) is the power spectraldensity of the equalizer error. From Figure 5.11, Φǫǫ(f) is derived as [12, 13]

Φǫǫ(f) = σ2d|Q(f)E(f)− 1|2 + Φzz(f)|E(f)|2, (5.47)

where σ2d is the variance of the transmitted symbols and Φzz(f) denotes the power

spectral density of the matched filter output noise. Substituting for Φǫǫ(f) in equation(5.46) yields

σ2 = T

∫ 1/T

0

σ2d|Q(f)E(f)− 1|2 + Φzz(f)|E(f)|2

df. (5.48)

The first term in the above equation describes the residual ISI power and the second is theoutput noise power [12, 13]. Two criteria have found widespread use in optimizing theequalizer coefficients. One already mentioned is the ZF and the other is minimum meansquare error (MMSE) [99]. Accordingly, two types of linear equalizers are commonlyused: The ZF equalizer and the minimum mean square error (MMSE) equalizer.

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85

5.4.4.1. Zero-Forcing Equalizer

A ZF equalizer, also known as orthogonal restoring combining, aims to eliminate the ISI.Setting the ISI term in the integrand of equation (5.48) to zero leads to a ZF equalizergiven by

EZF (f) =1

Q(f). (5.49)

It is essential to mention that the transfer function of the equalizer EZF (f) is simplythe inverse filter to the linear filter model Q(f). Consequently, the use of a ZF equalizermay result in a significant noise enhancement, when Q(f) has very deep spectral fades.Substituting equation (5.49) into (5.48) yields the output noise power of the ZF equalizer

σ2ZF = T

∫ 1/T

0

Φzz(f)

|Q(f)|2df. (5.50)

The implication of the noise enhancement is the system performance degradation interms of the BER. At last, the output of the equalizer as depicted in Figure 5.11, givenby

Y (f) = X(f)EZF (f) =X(f)

Q(f), (5.51)

is transfered into the time domain for further detection.

5.4.4.2. Minimum Mean Square Error Equalizer

The MMSE equalizer avoids the problem of the noise enhancement by compromising thenoise amplification and the ISI reduction. An MMSE equalizer can be optimized via the

EMMSE(f) = arg minE

E[|ǫi|2

]. (5.52)

Since the integrand in (5.48) is non negative, the criterion reduces to the frequency wisecriterion

σ2d|Q(f)E(f)− 1|2 + Φzz(f)|E(f)|2 → min

E(f)(5.53)

at every frequency f. Applying the gradient method with respect to E(f) yields theMMSE equalizer frequency response [12]

EMMSE(f) =[Q(f)]∗

|Q(f)|2 + Φzz(f)

σ2

d

. (5.54)

Note, that the equalizer transfer function is real valued, since all terms appearing at theright side in the above equation are real valued, too. Hence, the complex conjugation inthe numerator can be omitted. The additive term in the denominator of the expressionappearing in the above equation protects against an infinite noise enhancement, and

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86

hence, the MMSE is more reliable than the ZF equalizer. Substituting equation (5.54)into (5.48) yields the output noise power of the MMSE equalizer as

σ2MMSE = T

∫ 1/T

0

Φzz(f)

|Q(f)|2 + Φzz(f)

σ2

d

df. (5.55)

Comparing equations (5.55) and (5.50), it is obvious that σ2ZF is always equal or greater

than σ2MMSE. Furthermore, at high signal to noise ratios, the above mentioned additive

term vanishes and the two equalizers become equivalent.Ultimately, in the same manner as in a ZF equalizer, a memoryless detector, whichprocesses the corresponding time domain sequence, concludes the procedures.

5.5. The Peak Power Problem

A measure for the signal fluctuation is the peak to average power ratio (PAPR). It isdefined as the ratio between the peak to the average power of the signal and for a signaldefined on the interval [0, T ] given by

PAPR =max |s(t)|2

1T

∫ T

0|s(t)|2 dt

. (5.56)

An unmodulated carrier has a PAPR of unity or 0 dB. A large PAPR is a seriousdrawback to many systems and increases the complexity of the analog to digital anddigital to analog converters. On the other hand, large PAPR makes them susceptibleto system nonlinearities [17]. Indeed, in spite of its attractiveness and popularity forfuture wireless communications systems, OFDM may be outweighed by its handicap ofhaving a large PAPR [100]. In this regard, various costly techniques which aim to reducerather than to eliminate PAPR have been investigated and developed. Basically, theseare signal distortion techniques, including signal clipping, peak windowing and peakcancellation. The other techniques are scrambling and coding [101, 102]. Though PAPRreduction techniques can be quite effective, they are rather expensive and add to systemcomplexity, i.e. coding achieves satisfactory results, but it reduces the useful data rate[100]. However, in SC/FDE systems, the PAPR problems are not that extreme andmay be tolerable to some extent. Furthermore, the nature of SC/FDE systems allowsthe use of modulation schemes that are near constant or constant envelope modulationtechniques, and hence, a much lower if not unit PAPR is obtained.

SC/FDE - Buzid


87

5.6. Complementary Cumulative Distribution Function

Curves (CCDF)

The CCDF curves specify completely and without ambiguity the power characteristicsof signals [103]. It is defined as [100]

CCDF (PAPR(s)) = Pr(PAPR(s) > threshold value), (5.57)

which indicates the probability that the PAPR of the signal s exceeds a threshold value.

SC/FDE - Buzid

6. SC/FDE for Constant and NearConstant Envelope Modulation

It has been repeatedly mentioned, that the SC/FDE advantage over OFDM lies inthe reduced PAPR. OFDM systems inherit the high PAPR from the fact, that thesignal is composed of a number of superimposed and independently modulated coherentsubcarriers, which individually contribute to the total signal fluctuation. The PAPRin SC/FDE stems from the applied modulation scheme, which needs to be carefullyselected in order to reduce the high PAPR. This chapter details the adaption of theconstant envelope type modulation to SC/FDE systems. The objective is to eliminatethe problems with the PAPR of the SC/FDE system. Ultimately, a comprehensivetreatment of the problems of the PAPR is obtained. Contrary to SC/FDE, the CPMscheme can not be adapted to OFDM, because of phase continuity interruptions betweenthe subcarriers.

6.1. SC/FDE for OQPSK Modulation

As discussed in the previous chapters, OQPSK, a near constant envelope modulation,is a simple alternative to QPSK or QAM modulation with comparably lower PAPR[105, 106]. It is transformed from QPSK simply by delaying the Q-phase components ofthe baseband signal by half the symbol duration. This prevents the zero crossings in theIQ-plane. OQPSK fits perfectly into the block-wise transmission, and also the receiversignal processing for QPSK can be easily applied with only little adaptations. Referringto SC/FDE layout and the model, shown in Figures 5.1 and 5.11, respectively, thestraight forward approach to demodulate a received block is to substitute the N -pointIFFT by a 2N -point IFFT and perform sampling rate reduction in time domain by

γn = Rey(2n)+ j · Imy(2n+ 1) (6.1)

for n = 0, ..., N − 1. An alternative solution is to reverse the time delay of the Q-pathsamples Xu already in frequency domain. For that the following are defined [106]

Xru = ReXu, X i

u = ImXu (6.2)


Scrambling

Coding

Interleaving

CPM

MappingGuard

interval

Up-

sampling

Transmit

filter

Down

sampling

Matched

filterFFT2N Remove GI

Synchronisation

IFFTNCPM

Demapping

Descrambling

Decoding

Deinterleaving

Equalizer

Channel estimation

Input

data

output

data

Pilots

payload

Se

rialto

para

llel

Pa

ralle

lto

seria

l

Figure 6.1.: SC/FDE system adapted to CPM technique

for u = 0, ..., 2N − 1, and the mirrored versions

Xru = ReX(−u) mod (2N); X i

u = ImX(−u) mod (2N) (6.3)

for u = 0, ..., 2N − 1. The spectra of the I- and Q-path can then be separated by

Iu =1

2

[(Xr

u + Xru) + j(X i

u − X iu)]

(6.4)

Qu =1

2

[(X i

u + X iu) + j(Xr

u −Xru)]. (6.5)

Reversing the time delay of the Q-path may then be performed by

Xcorru = Iu + jQue

jπu/N . (6.6)

Sampling rate reduction and application of an N -point IFFT complete the equaliza-tion/demodulation process in the same manner as for QPSK schemes.

6.2. SC/FDE for Constant Envelope Modulation

Schemes

6.2.1. The Transmission Model

The block diagram of the SC/FDE system adapted to CPM technique is as illustrated inFigure 6.1. The received signal CPM-SC/FDE in a multipath environment is expressedas

r(t) = hb(t) ∗ s(t) + n(t), (6.7)

where ∗ is the convolution operator, hb(t) is the baseband channel impulse responsegiven in equation (5.8) and n(t) is an additive white Gaussian noise with two-sided

SC/FDE - Buzid


power spectral density N0/2. The transmit signal is organized in blocks and a block isgiven by

s(t) =N−1∑

i=0

aic0(t− iT ), (6.8)

where c0(t) is the prime Laurent pulse and ai are the coefficients gained from the mappingprocess described by (see also equation (4.33))

ai = ji.γi, (6.9)

with γi describing the coded data bit values γi ∈ −1, 1. So the differences to BPSKare the following:

• Instead of RRC pulse shaping appropriate prime Laurent pulses are used.

• Laurent pulses already introduce ISI.

• The first samples of the first symbols and the samples of the last symbols of ablock have to be dropped since c0(t) spans an interval of length (L+ 1)T . Due tothat the matched filter will not perfectly match the first and the last symbols of ablock.

• The coefficients ai are not equal to the original data, but the result of a mappingprocess as described.

6.2.2. The Receiver Model

The receiver block diagram is given in Figure 6.2. It looks similar to the one explainedin Chapter 5. One difference is, that in the following simulation the receiver filter hasbeen chosen to CT/µ(f) = TDFT c0(−t), where TDFT is the time discrete Fouriertransform. That means the receive filter is matched to the transmit pulse as it is donein many of today’s systems. In most of the simulations µ=2 is chosen. The equalizeris implemented using Equation (5.49) in the ZF case and using Equation (5.54) in theMMSE case. Note that Q(f) is now given as the time discrete Fourier transform of[c0(t) ∗ hb(t) ∗ c0(−t)]t=nT/µ. The equalizer output is transfered back to time domain,and mapped to the estimated data symbols γ with the formula given by equation

γi = j−i.ai. (6.10)

Finally decoding using the Viterbi algorithm is used.The baseband model of a CPM-SC/FDE system shown in Figure 6.1 was assembled

in Matlab. Most of the system parameters are adapted to the IEEE 802.11a standardas in Table 5.1. The system error performance was simulated in a multipath channeland a linear MMSE equalizer is used. Cyclic prefix is used as a guard interval and forcoding a convolutional encoder with a rate 1/2 and a generator polynomial (133,171) is

SC/FDE - Buzid


Matched

Filter

CT/µMF

FFTµN

Sampling

rate

reduction

Equalizer IFFTN

r(T/µ) ai

r(t)

Figure 6.2.: Linear receiver in the frequency domain adapted to the CPM-SC/FDE con-cept.

applied. A non-differential CPM-SC/FDE with the Laurent shaping pulse derived fromthe Gaussian pulse of BT=0.3 as well as from the TFM pulse, as both pulses implyconstant envelope signals, is simulated. For the purpose of comparison, SC/FDE withQPSK (RRC, α=.25) is included in the simulation and the results are shown in Figure6.3. The results show that CPM-SC/FDE for the particular parameters of the appliedmodulation techniques have comparable performance.Note that due to the blockwise structure the CPM-SC/FDE transmit signal exhibitsinterrupted phase continuity at the block edges.

Lo

g(B

ER

)

E /N in dBb 0

Figure 6.3.: BER for non-differential CPM-SC/FDE system with both a Gaussian fre-quency pulse (BT = 0.3) and TFM and SC/FDE with QPSK, with MMSE-Equalizer in a Multipath SISO channel.

SC/FDE - Buzid


6.3. Impact of Front-End Nonlinearity on

CPM-SC/FDE System Performance

Generally, nonlinear distortions are primarily due to the transmitter high-power amplifier(HPA) [109, 110]. The HPA linearity is an essential aspect in designing and classifyingthe power amplifiers. It reflects the introduced distortion and is closely related to powerconsumption by the transmitter [86, 55]. A frequently used HPA model well suitedfor Matlab simulations is described in [109]. The impairments of the power amplifiernonlinearity at the transmitter are

• Spectral-spreading of the transmitted signal

• Intermodulation effects and

• Warping of the signal constellation.

Hence, the implications are both inside and outside the signal bandwidth. The in-bandcomponent causes the system performance degradation in terms of BER, while the out-band component, also called spectrum regrowth, affects adjacent frequency bands andcauses adjacent channel interference (ACI) [111, 112]. Traditionally, the nonlinearityof an RF amplifier is described by the HPA input-referred 3rd order intercept pointIIP3 or by the 1 dB compression point P1dB [86, 113]. The compression point, shownin Figure 6.4, is the point at which the output power is 1dB less than that for thelinear case, as a result to non-linearity. Remarkably, the distortion has joint effects, onone hand the BER, on the other hand the spectral spreading. The HPA is typicallymodeled as a memoryless nonlinear device which is described by two transfer functionsrepresenting the amplitude modulation/amplitude modulation (AM/AM) and amplitudemodulation/phase modulation (AM/PH) characteristics, respectively [109, 111, 114].Let si(t) designate the complex envelope of the signal at the input of the power amplifieras shown in Figure 6.5. It can be expressed as

si(t) = x(t) + jy(t) (6.11)

= ρ(t) ejφ(t). (6.12)

The output of the nonlinear PA is given as

so(t) = FA[ρ(t)] ej(FP [ρ(t)]+φ(t)), (6.13)

where FA[.] and FP [.] are envelope transfer functions that represent the AM/AM andAM/PM conversion of the nonlinear PA. The model described in [109] which has alsobeen used in this thesis neglects the AM/PM effect and is given by

FA[ρ] =ρ

1 + [(ρ/A0)2p]1/2p

FP [ρ] = 0. (6.14)

SC/FDE - Buzid


10 15 20 25 30 35 4020

22

24

26

28

30

32

Input Amplitude dB

Out

put A

mpl

itude

dB

+

Compression point P1dB

Figure 6.4.: The compression point of an amplifier model.

HPAModulator Rxs (t)i s (t)o

Input data Output data

Figure 6.5.: System impaired by the high power amplifier (HPA).

During the IEEE 802.11a standardization process the parameter which describes theorder of the non-linearity has been chosen to p = 2. Figure 6.6 shows the characteristicof this model, whose non-linearity affects only the amplitude of the signal. Obviously, alinear power amplifier should be used for non-constant envelope signals. Alternatively,compensation techniques are applied, one of which is the predistortion in the transmitter,that is either continuous-waveform predistortion before the PA or data predistortionbefore the modulator [114]. The other solution is the use of a back-off power. Theoperating point of the amplifier is usually identified by the input back-off (IBO) as wellas output back-off (OBO), which respectively can be given as

IBO = 10 logPo,in

Pin

in dB (6.15)

OBO = 10 logPo,out

Pout

in dB

SC/FDE - Buzid


Input Amplitude dBm

Outp

utA

mplit

ude d

Bm

Figure 6.6.: Normalized characteristics(AM/AM) of an IEEE power amplifier model(Rapp model) for various values of p; the amplifier becomes a clipper athigh values of p.

where Pin is the mean power of the signal at the input of the HPA, Pout the mean powerof the transmitted signal, Po,out the maximum output power (saturation power) of theHPA and Po,in the input power corresponding to the maximum output power. The useof the back-off power is to keep the amplifier operating in the linear region and hence toavoid signal distortion. Remarkably, the larger back-off implies the larger linear distancebetween the operating point and the saturation point which is usually determined by acompression point shown in Figure 6.4. Further, the larger the back-off, the less signaldistortion and therefore, the back-off determines the PA linear dynamic range and alsoits power consumption [115].Shown in Figure 6.7 is the OFDM spectrum according to OFDM based IEEE 802.11astandard at the output of the PA. The spectrum is compact at large values of the back-off and becomes distorted as the operating point of the amplifier moves closer to thecompression point [116]. However, the spectrum of CPM-SC/FDE shown in Figure 6.8is by far less affected by the amplifier [117]. The reason why the spectrum is affectedat all is given by the phase discontinuity at the block edges in the blockwise structuredtransmit signal.On the other hand, the maximum dc efficiency of the PA is at the non-linear region andis related to the source power consumption. Operating the PA away from the non-linearregion means the PA works with less efficiency. Consequently, the power consumptionis high and not preferred for mobile terminals where only a limited power supply isavailable. Furthermore, a survey of the market reveals that the power supply life-time iscrucial in marketing of such mobile equipments. Moreover, the back-off is also a measureof the system error performance, since the larger the back-off, the less added distortionand less system degradation is observed.

SC/FDE - Buzid


88

[ MHz ]

0

-5

-10

-15

-20

-25

-30

-35

-40

Norm

aliz

ed p

ow

er

specta

l density in d

B

Figure 6.7.: Transmitted spectrum of OFDM signal for amplification around the com-pression point of the power amplifier.

[ MHz ]

Amplification after CP, L = 1

Amplification before CP, L = 1

Amplification after CP, L = 2

Amplification before CP, L = 2

-20

-25

-30

-35

-40

0

-5

-10

-15

No

rma

lize

dp

ow

er

sp

ecta

l d

en

sity

ind

B

Figure 6.8.: Transmitted spectrum of CPM-SC/FDE signal for amplification around thecompression point of the power amplifier.

SC/FDE - Buzid


To further examine the effectiveness of the CPM-SC/FDE system to the non-linearity ofthe HPA, the system error performance was simulated in the presence of the HPA andcompared to linear modulation based systems. For this purpose, the IEEE model withp chosen to be 2 is used to model the HPA in a baseband system simulation. Shown inFigure 6.9 is the simulated BER (for a constant SNR value) versus the input backoffpower for the CPM-SC/FDE with a Gaussian frequency pulse of L=1,2 and 3, where Lis the number of input symbols scanned by the pulse. The OFDM system with parame-ters of IEEE 802.11a and QPSK modulation scheme and the SC/FDE with both QPSKand OQPSK are included in the simulations. The x-axis represents the input backoffpower and the zero point corresponds to the compression point. The minus sign meansthe operating point of the amplifier is beyond the compression point. From the figure

Lo

g(B

ER

)

Figure 6.9.: Uncoded BER rate vs input backoff power for both SC/FDE and OFDM(IEEE 802.11a standard).

one can conclude that OFDM with QPSK shows the worst performance. Then comesSC/FDE with QPSK as expected. SC/FDE with OQPSK performers better than both,nevertheless, the performance of non-differential CPM-SC/FDE system with a Gaus-sian frequency pulse shape remains unchanged and demonstrates its effectiveness to thenonlinearity of the HPA.

SC/FDE - Buzid


6.4. Spatial Multiplexing for CPM-SC/FDE

In the previous chapters it has been shown that the adaptation of the nonlinear mod-ulation to SC/FDE represents a comprehensive and an elegant solution to the PAPRproblems and the system becomes quite effective against the nonlinearity effects. Butit is also known that the system capacity in terms of the number of bits per transmit-ted symbol, which is one, is insufficient when compared to e.g, the OFDM system withthe 64-QAM where 6-bit modulate a carrier. Therefore, CPM-SC/FDE in that senseremains dissatisfactory. One way of improving the bandwidth efficiency is to apply spa-tial multiplexing (SM), a MIMO scheme [118]. It has extensively been investigated forlinear modulated SC/FDE signals in [119]. The concept of SM is that an arbitrarynumber of transmit antennas are transmitting different data streams called layers with-out space time coding. Further, all layers are detected and equalized simultaneouslyby the frequency domain equalizer. Figure 6.10 illustrates a block diagram of a CPM-SC/FDE architecture with SM. The input blocks constitute layers. As shown in the

S / P

MIM

O-sy

ncro

nizatio

n

GI

GI

FFT

G

RemGI

G

T

T

Input

data

CPMMapping

Codinginterleaving

GMFEq

ualizer

RemGI

GMF

P/ S

Output

data

decodingdeinterleaving

FFT

CPMMapping

CPMDemapping

CPMDemapping IFFT

IFFT

Figure 6.10.: Block diagram of a CPM-SC/FDE transceiver using a spatial multiplexingscheme.

figure, each layer is mapped individually to obtain the CPM symbols, via applying thenon-differential mapping [120]. However, other possibility for obtaining the layers ofthe CPM symbols of input data is by obtaining the CPM symbols before structuringthe input data into layers. The guard interval is added to each layer and a pulse shap-ing is next. At the receiver, each layer from each antenna is detected and processedseparately. The SM-SC/FDE system can be described conveniently using a matrix no-tation [119]. The MIMO channel, which describes the transmission behavior from each

SC/FDE - Buzid


transmit antenna l to receive antenna k can be described with the channel matrix H

H =

H11 H12 . . . H1j

H21 H12 . . . H2j...

. . ....

Hi1 Hi2 . . . Hij

.

(6.16)

Each individual Hij has the form

Hij =

hij(0) hij(N − 1) . . . hij(1)hij(1) hij(0) . . . hij(2)

...... . . . hij(3)

...... . . .

...hij(N − 1) hij(N − 2) . . . hij(0)

(6.17)

where hij(0), hij(1), ..., hij(N − 1) are the impulse response coefficients from transmit

antenna i to receive antenna j. The frequency domain version of H is obtained from[119, 121]

H = F2NHF−12N (6.18)

where

F2N =

F2N 0 . . . 00 F2N . . . 0...

.... . .

...0 0 . . . F2N

(6.19)

and

F−12N =

F−12N 0 . . . 00 F−1

2N . . . 0...

.... . .

...0 0 . . . F−1

2N

, (6.20)

where F2N is a DFT matrix, given in the Appendix. In [121], it is shown that the ZFand MMSE equations can be derived by

EZF = H−1D , (6.21)

and

EMMSE = HHD

[HDH

HD +

σ2n

σ2d

InN

]−1

, (6.22)

where (.)H denotes the transpose conjugate of a matrix, and HD is given by

HD = ΛdGMFHGΛu. (6.23)

SC/FDE - Buzid


The superscript , for example in Λu, indicates a block-diagonal matrix with Λu asdiagonal elements. Both, Λu and Λd are upsampling and downsampling matrices andgiven in the Appendix. Further, G and GMF are block-diagonal matrices having thefrequency domain versions of the transmit filter and the matched filters, respectively, intheir diagonals. The system shown in Figure 6.10 is simulated in a multipath MIMOchannel. Shown in Figure 6.11 are the simulation results of a non-differential CPM-SC/FDE with a Laurent pulse that is derived from a Gaussian pulse in 1×1 and 2×2multipath channels. SC/FDE with a QPSK scheme is also included in the figure forcomparison. An MMSE equalizer is used. Further, the system with a Laurent pulsethat is derived from both Gaussian and TFM pulses, is simulated and compared in thesame multipath channel.

Lo

g(B

ER

)

E /N in dBb 0

Figure 6.11.: BER for non-differential CPM-SC/FDE system with a Gaussian frequencypulse (BT = 0.3) and SC/FDE with QPSK in multipath channel.

SC/FDE - Buzid


-10 -5 0 5 10 15

-7

-6

-5

-4

-3

-2

-1

0

S/N in dB

1X1 CPM-SC/FDE (TFM)

2X2 CPM-SC/FDE (TFM)

1X1 CPM-SC/FDE (Gaussian)

2X2 CPM-SC/FDE (Gaussian)

Lo

g (

BE

R)

E /N in dBb 0

Figure 6.12.: BER for non-differential CPM-SC/FDE system with both a Gaussian fre-quency pulse (BT = 0.3) and TFM, an MMSE-Equalizer in a multipathMIMO channel.

It can be concluded that spatial multiplexing can be applied to CPM-SC/FDE similaras for QPSK-SC/FDE. The simulation results furthermore show similar performanceresults for both schemes.

SC/FDE - Buzid

7. Point to Multipoint Systems,SC/FDE-CDMA

The spread spectrum technology (SS) has been widely used since its release and avail-ability for civil use and a commercial market has emerged. Many of today’s systemsare inspired by or are a direct or indirect application of the spread spectrum technique.Applications for commercial spread spectrum range from wireless cellular systems andphones, WLAN, to wide area network WiMAX. Another technology has been in thefocus in the recent years. This technology is OFDM. It has attracted great attentionand has been very popular in the last decade. It can be combined with TDMA, FDMAor CDMA for use in multiple access environments. OFDM is considered an effectivemodulation technique for high-speed digital transmission. It is used in a wide range ofapplications, including IEEE 802.11a, g, wireless LAN standard, digital audio broadcast-ing DAB and digital video broadcasting DVB [122]. In OFDM systems, the spectrumis divided into smaller bands or subcarriers. On the other hand, SC/FDE has beenan emerging technique in the last years. Both SC/FDE and OFDM concepts can becombined with multiple access techniques and offer promising multiple access schemesfor broadband radio applications. A potential multiple access technique for wireless net-works is code division multiple access (CDMA). CDMA avoids the burst transmissionor pulsed transmission which is strongly associated with time division multiple accessschemes (TDMA). Pulsed transmission is clearly noticeable in narrowband TDMA basedsystems such as GSM, DECT and TETRA.In this chapter, the combination of SC/FDE and CDMA concepts is considered and thefrequency domain equalization is particularly emphasized. Further, the system SC/FDE-CDMA is simulated and compared for different modulation formats, including the non-differential CPM.

7.1. CDMA Cellular Systems

All kinds of 3G systems use one or another variant of CDMA. In Europe, the 3G is alwaysrelated to the universal mobile telecommunication system (UMTS) which is also knownas UMTS Terrestrial Radio Access (UTRA), in which two types of CDMA approacheswere proposed. One type is the UTRA frequency division duplex (UTRA-FDD) which


is a wideband CDMA (W-CDMA) system and the other is UTRA time division duplex(UTRA-TDD) which is a CDMA/TDMA system. In USA, the cdma2000 standardprovides a seamless upgrade for the users of 2G and 2.5G CDMA technology. It isexpected that these two major 3G technologies cdma2000 and W-CDMA, will remainpopular throughout the early part of the 21st century [91]. In Japan, the W-CDMAsystem is adopted. China pushes forward its own time division synchronous CDMA (TD-SCDMA), which relies on the existing core GSM infrastructure and permits a 3G networkto evolve through the addition of high data rate equipments. All these 3G standardsare known in the International Telecommunications Union (ITU) as the members ofIMT-2000.It is well known that the CDMA scheme is robust to frequency selective fading and hasbeen successfully introduced in commercial cellular mobile communication systems [123].Instead of dividing the spectrum between several independent and non-overlapping usersin frequency division multiple access (FDMA) or assigning the time slots to different,independent and non-overlapping users in time division multiple access (TDMA), theusers in CDMA access the channel in the same frequency channel and in the same time.Further, CDMA supports low-to-high data rate of multimedia services and the users aredifferentiated through the distinct signatures or spreading codes.

7.2. SC/FDE-CDMA

Although the frequency domain equalization concept has been proposed for more than 3decades, the application of FDE for CDMA detection has just begun to be considered re-cently [124]. The low complexity, efficient and powerful receive structures of the SC/FDEcan be combined advantageously with the concept of CDMA (SC/FDE-CDMA). Theequalizer at the receiver has two assignments in this case:

• It removes the ISI that is produced by the multipath channel and

• It restores the properties of the spreading codes.

Both advantage from the low complexity and powerful equalizer that functions in fre-quency domain.

7.2.0.1. System Description

An SC/FDE-CDMA system is shown in Figure 7.1. As shown in the figure, the inputdata is mapped according to a specific modulation. Following is the CDMA spreadingprocess. For a block-wise transmission, the CDMA symbols are structured into blocksand guard intervals e.g., CP symbols are added. However, the pulse shaping is not ofa concern and dropped here. In the receiver, the CDMA symbols are equalized and

SC/FDE - Buzid


Guard

interval

CDMA

despreadingEqualizer FFT Remove GI

Binary

outputDemod IFFT

Binary

intputMod

Serial to

parallelCDMA

spreading

Parallel

to serial

Figure 7.1.: An FDE detector for an SC/FDE-CDMA system.

the properties of the spreading codes are restored. A ZF or an MMSE equalizer can beused. Generally, the SC/FDE-CDMA baseband signal for the ith user may be expressedas [126]

si(t) =N−1∑

n=0

Q−1∑

q=0

γinci(q)g(t− nT − qTc), (7.1)

The chip rate given by 1/Tc equals Q/T, where Q is the code length and T is the symbolduration. i = 0, 1, ..., K − 1, where K represents the number of users. γin is the nth

symbol in a block of size N , of the ith user, g(t) represents the pulse shape and ci is thespreading code of the ith user.For brevity, the continuous time element is eliminated here and the Fourier codes areapplied for illustrative purpose. Fourier codes are the columns or rows of the inversediscrete Fourier matrix. The matrix entities are given by ci(q) = 1

Qej2πqi/Q. A spread

symbol of the ith user at the qth chip can be expressed as

siq(n) =1

Qγine

j2πqi/Q. (7.2)

The sum over K users is then given as

sq(n) =1

Q

K−1∑

i=0

γinej2πqi/Q, (7.3)

where K ≤ Q. The CDMA symbol is then given as

s(n) = s0(n), s1(n), ..., sQ−1(n) . (7.4)

LettingK = Q, full load, the above equation represents an inverse discrete Fourier trans-form of the nth transmitted symbol of the K users. It represents an OFDM symbol. Thecomplete signal is constructed by the concatenation of all OFDM symbols. Therefore,combining the SC/FDE concept and CDMA with Fourier codes and making Q = K,the system collapses to an OFDM system. It shows that OFDM is a special form of theSC/FDE-CDMA system.

SC/FDE - Buzid


0 2 4 6 8 10 12 14 16 18−6

−5

−4

−3

−2

−1

1

Eb/No in dB

Log(

BE

R)

BPSKNon−differential CPM

Figure 7.2.: SC/FDE-CDMA single user with OVSF spreading codes and BPSK andnon-differential modulation formats.

7.2.0.2. Simulation Results

The system SC/FDE-CDMA shown in the figure is implemented in Matlab and simulatedfor different modulation formats, including the non-differential CPM technique which isdiscussed in Chapter 4. Figure 7.2 depicts the performance of a single user, SC/FDE-CDMA system. The system applies orthogonal variable spreading factors (OVSF) asspreading codes and a ZF equalizer. The modulation schemes are BPSK and non-differential CPM. An RRC pulse shape with a roll-off = 0.5 is used for BPSK. For anon-differential CPM, the Gaussian pulse with BT = 0.8 for the linearized Laurent pulseshape is used. The full load case (16 users) for BPSK and non-differential CPM is plottedin Figure 7.3. Further, the system with a non-differential CPM is also simulated andcompared for different values of BT of the Gaussian frequency pulse and the result isshown in Figure 7.4 for a single user, and in Figure 7.5 for a full load (16 users) system.

7.3. Multicarrier-CDMA versus SC/FDE-CDMA

Spread spectrum techniques and OFDM can be combined advantageously. One possibil-ity is multicarrier direct sequence CDMA (MC-DS-CDMA), where each data symbol isspread over Q chips and then modulates the N subcarriers (N=Q) [127, 128]. Anotherpossibility is multicarrier CDMA (MC-CDMA), in which the spread data of the activeusers are added before modulating the subcarriers [129, 130, 131]. In the MC-CDMAsystem, the CDMA symbols are transported on the orthogonal subcarriers within theusable frequency band of the channel and implemented through IDFT [126]. In this

SC/FDE - Buzid


0 3 6 9 12 15 18 21 24 27 30 33

−4

−3

−2

−1

Eb/No in dB

Log(

BE

R)

BPSKNon−recursive CPM

Figure 7.3.: SC/FDE-CDMA full load (16 uers) with OVSF spreading codes and BPSKand non-differential modulation formats.

0 2 4 6 8 10 12

−5

−4

−3

−2

−1

Eb/No in dB

Log(

BE

R)

BT=.2BT=.3BT=.5BT=.8

Figure 7.4.: Uncoded BER of a single user SC/FDE-CDMA with OVSF spreading codes,Gaussian frequency pulse for different BT values.

SC/FDE - Buzid


0 3 6 9 12 15 18 21 24 27 30 33

10−4

10−3

10−2

10−1

Eb/No in dB

Log(

BE

R)

BT = 0.2BT = 0.3BT = 0.5BT = 0.8

Figure 7.5.: Uncoded BER of a full load SC/FDE-CDMA with OVSF spreading codes,a Gaussian frequency pulse for different BT values.

Spreader

Spreader

Spreader

S / P

O F

D M

Input b

inary

data

C

C

C

0

1

K-1

+MC-CDMA

Figure 7.6.: MC-CDMA down link transmitter.

study, only MC-CDMA is considered and compared to an SC/FDE-CDMA system. ForFourier spreading codes the spreading matrix is the inverse fast Fourier matrix. How-ever, for a reason that will become clear later, the spreading codes are now the rows ofthe DFT matrix whose entities are given as ci(q) = e−j2πqi/Q. The qth chip of the nth

CDMA symbol is given as

xq(n) =K−1∑

i=0

γine−j2πqi/Q. (7.5)

Again making K=Q, full load, the above equation represents a discrete Fourier trans-form. In order to obtain the MC-CDMA symbol, the CDMA symbol is next modulatedon the subcarriers through an IDFT process. This results in two transformation pro-cesses; each cancels the other. The resultant signal corresponds to an SC/FDE symbol.It is therefore highlighted that the spreading of the data with Fourier codes and mod-

SC/FDE - Buzid


ulating the resulting signals on the subcarriers through IDFT operations results in anSC/FDE structure. It is then concluded that SC/FDE is a special form of the MC-CDMA [132]. Thus, the combination of SC/FDE and CDMA with Fourier codes whichare taken from the inverse discrete Fourier matrix turns into OFDM [133]. In contrastto that, the combination of OFDM and CDMA with the Fourier codes, taken this timefrom a discrete Fourier matrix, turns into an SC/FDE. This is summarized in Figure7.7.

SC/FDE

InverseFourier codes

OFDM

Fourier codes

CDMA

CDMA

Figure 7.7.: SC/FDE and OFDM irreversible cycle.

SC/FDE - Buzid

A. Appendix

A.1. Fourier Matrix

The discrete Fourier transform of a discrete signal x(k) of a length N may be expressedas

X(n) =1√N

N−1∑

k=0

x(k)e−j2πnk/N . (A.1)

Further, a DFT process can also be represented in a matrix form, denoted by FN ∈C

N×N , whose entities are determined via

FN(k, l) =1√Ne−j2π(k−1)(l−1)/N . (A.2)

The input discrete signal x(n) can also be written in a vector notation ~x and the DFTmatrix can be applied to obtain

~X = FN ~x. (A.3)

Further, the inverse discrete Fourier transform of ~X is given by

~x = F−1N

~X. (A.4)

It is necessary to note that F−1N = FH

N , and thus, (F−1N )H = FN .

DFT can also be applied to matrices. Consider the linear equation

~y = A ~x; ~x, ~y ∈ CN×1; A ∈ C

N×N . (A.5)

Multiplying the linear equation with FN at the left side yields

FN ~y = FN A ~x. (A.6)

Since, F−1N FN = IN , the above equation can be written as

FN ~y = FN A F−1N FN ~x (A.7)

~Y = FN A F−1N

~X,

thus, the Fourier transformation A of the matrix A is given as

A = FN A F−1N . (A.8)

A. Appendix 109

A.2. Convolution Matrix

The discrete output of a linear time invariant system (LTI), which models a channel asshown in Figure 1.1, may be expressed as

y(n) =ν−1∑

l=0

h(l)x(n− l), (A.9)

where h(l) is a discrete time invariant channel impulse response, l = 0, 1, ..., ν − 1 andx(k) is the input where k = 0, 1, ..., N − 1. The above equation can also be representedin a compact form by using matrix notations as

h(l) y(n)x(k)

Figure 1.1.: LTI Model.

~y = H ~x, (A.10)

where H is the convolution matrix, also known by Toeplitz matrix, and given as

H =

h(0) 0 0 0 . . . 0h(1) h(0) 0 0 . . . 0h(2) h(1) h(0) 0 . . . 0...

.... . .

... . . . 0h(ν − 1) h(ν − 2) . . . h(0) . . . h(0)

0 h(ν − 1) . . . h(1) . . . h(1)...

.... . .

... . . ....

0 0 0 0 . . . h(ν − 1)

∈ C(N+ν−1)×N (A.11)

A.3. Circulant Convolution and Toeplitz Matrix

Denoting x and h ∈ CN as two vectors of a size N . The circular convolution of x with

h is given asy = x⊗ h, (A.12)

which can be expressed in a compact matrix form as

~y = T ~x (A.13)

SC/FDE - Buzid

A. Appendix 110

where T ∈ CN×N is given by

T =

h(0) h(N − 1) h(N − 2) . . . h(1)h(1) h(0) h(N − 1) . . . h(2)h(2) h(1) h(0) . . . h(3)...

.... . . . . .

...h(N − 1) h(N − 2) . . . . . . h(0)

∈ CN×N (A.14)

A.4. Sampling Matrix

In realizing the digital part of communication systems, it is generally necessary to adjustthe rate of digital signals by up and down sampling processes. Up- and downsamplingby integer factors can also be expressed using a matrix notation:

A.4.1. Up Sampling Matrix

For an upsampling factor of two one can define:

Λu =

1 0 . . . 00 0 . . . 00 1 . . . 00 0 . . . 0...

. . ....

0 0 . . . 10 0 . . . 0

∈ N2N×N (A.15)

So the upsampled version of a vector ~x is given by ~y = Λu~x. This matrix can betransformed to frequency domain by

Λu = F2NΛuF−1N =

√2

2

(ININ

)(A.16)

A.4.2. Down Sampling Matrix

The downsampling matrix (for a downsampling factor of two) for time and frequencydomain can be derived equivalently to

Λd =

1 0 0 . . . 0 00 0 1 . . . 0 0...

.... . .

......

0 0 0 . . . 1 0

∈ N

N×2N (A.17)

SC/FDE - Buzid

A. Appendix 111

Λd = FNΛdF−12N =

√2

2

(IN IN

)(A.18)

SC/FDE - Buzid

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