Frequency Domain Equalization of Modulation Formats with Low Peak to Average Power Ratio

Dissertation

Frequency Domain Equalization of Modulation Formats with Low Peak to Average Power Ratio Frequenzbereichsentzerrung von Modulationsverfahren mit niedrigem Spitzen- zu Mittelwert Verh¨altnis

Der Technischen Fakultaet Fakultät der Universität Erlangen-Nürnberg zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von

Tuﬁk Buzid, MSc.

Erlangen, 2010 2

Als Dissertation genehmigt von der Technischen Fakultät der Universität Erlangen-Nürnberg

Tag der Einreichung: 20.07.2009 Tag der Promotion: 16.12.2009 Dekan: Prof.Dr.-Ing.habil.ReinhardGerman Berichterstatter: Prof. Dr. MarioHuemer Prof. Dr. Leonard Reindl Prof. Dr.-Ing. Dr.-Ing. habil. Robert Weigel

SC/FDE - Buzid ———————————————————- Zusammenfassung

Der Austausch von Informationen mit hohen Datenraten zwischen verschiedenen mo- bilen oder stationären Endgeräten benötigt Techniken, welche die Beschränkungen durch den Funkkanal überwinden. Ein Funkkanal, ein Kanal mit Mehrwegeausbreitung, wird durch die verschiedenen Signalwege bis hin zum Ziel beschrieben. Zusätzlich zum di- rekten Weg, falls dieser überhaupt existiert, können verzerrte Kopien des Signals auch durch Reflexion, Beugung und Streuung zum Empfänger gelangen. Eine der größten Herausforderungen der Datenübertragung mit hoher Datenrate ist die Uberwindung¨ der durch die Mehrwegeausbreitung verursachten Zeitdispersion. Eine Herangehensweise an das Problem der Zeitdispersion ist die Anwendung des Orthogonal Frequency Division Multiplexing (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-Leistungsverstärker durch Nichtlin- earität beeinträchtigt wird, was bis zu einem gewissen Maß praktisch immer der Fall ist. Zudem sind Verstärker für Signale mit hohem Spitzen-zu Mittelvert Verhältnis meist sehr ineffizient. Eine weitere Breitbandtechnik ist die Einträgerübertragung mit Frequenzbereichsentzerrung (engl. single carrier transmission with frequency domain equalisation, SC/FDE). SC/FDE bringt nachweislich die gleiche Leistungsfähigkeit wie OFDM und weist weniger PAPR auf als dieses. Besondere Beachtung fand SC/FDE in Kombination mit Quadratur-Amplitudenmodulation (QAM), die noch immer ein sub- optimales PAPR aufweist. Allerdings erlaubt SC/FDE im Gegensatz zu OFDM den Einsatz von Modulationsarten mit konstanter Einhüllender. Durch Verwendung dieser Modulationsart, welche zur Klasse der nichtlinearen Modulationsverfahren gehört und sich durch kontinuierliche Phaseneigenschaften auszeichnet, kann das PAPR des betra- chteten Systems auf den Wert Eins reduziert werden. Durch den Einsatz der Laurent Darstellung wird eine lineare Form des Signals mit kontinuierlicher Phasenmodulation (CPM) erreicht, die es ermöglicht das CPM-Signal linear zu demodulieren. Folglich wird die ursprünglich wegen seiner Trellis-Struktur große Komplexität des Empfängers deut- lich reduziert. Darüber hinaus können Techniken wie z.B. Matched Filtering und lineare Entzerrung, die bereits für lineare Modulation entwickelt wurden, ohne Schwierigkeiten für die nichtlineare Modulation verwendet werden. In dieser Arbeit wird die linearisierte Form der CPM auf SC/FDE (CPM-SC/FDE) angewendet. Eine andere Form der Modu- lation mit konstanter Amplitude, die als Tamed Frequenzmodulation (TFM) bekannt ist und vor einem Jahrzehnt eingeführt wurde, wird ebenfalls auf SC/FDE angewendet. Das resultierende System wird mit OFDM unter dem Aspekt der Nichtlinearität der HF-Stufe iii verglichen. Der Vergleich zeigt, dass die Leistungsfähigkeit der CPM-SC/FDE nicht so stark beeinträchtigt wird wie die der OFDM, deren Leistungsfähigkeit dramatisch nachlässt. Obwohl das erhaltene System im nichtlinearen Bereich des Verstärkers ar- beitet, bleibt seine Fehler Performanz aufgrund der inhärenten differentiellen Codierung unbefriedigend. 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 eingeführt. Dabei wird vor allem räumliches Multiplexing verwendet, da es die Transmissionseffizienz stark erhöht und somit die Systembandbreite linear mit der Anzahl der verwendeten Antennen ansteigt. Darüber hinaus wird SC/FDE mit linearen und nichtlinearen Mod- ulationsarten erweitert für ”Point to Multipoint”-Anwendungen und mit Spreizcodes (Code Division Multiple Access, CDMA) kombiniert. Schließlich wird SC/FDE-CDMA mit einer Kombination von OFDM und CDMA (MC-CDMA) verglichen und skizziert, dass eine Anwendung der Fourier Spreizcodes auf ein voll ausgelastetes SC/FDE-CDMA zu einem OFDM-System führt, während die Kombination von OFDM und CSMA ein SC/FDE-Verfahren erzeugt.

SC/FDE - Buzid Abstract

High data rates and the exchange of large amounts of information among various mobile or roaming and stationary terminals require techniques that conquer the restrictions imposed by the wireless channels. A wireless channel, known as multipath channel, is modeled by a number of paths that a signal travels to reach its destination. In addition to a direct path, if it exists, distorted copies of the signal may arrive at the receiver through different paths that are formed by reflections, diffractions and scattering. One of the most challenging problems in high data rate wireless transmission is to overcome the time dispersion caused by multipath propagation. An approach to overcome the problems of time dispersion is the use of orthogonal frequency division multiplexing (OFDM). A primary drawback is the large dynamic range of the OFDM signal. 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 is in practice always the case to some extent. Another broadband technique is the single carrier transmission with frequency domain equalization (SC/FDE). SC/FDE has been shown to exhibit similar performance as OFDM and shows less PAPR than OFDM. Mainly SC/FDE has been investigated in combination with quadrature amplitude modulation (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 constant amplitude type of modulation, which is nonlinear and marked by the continuous phase property, the PAPR of the concerned system can be reduced to one. By the deployment of the Laurent representation a linear form of the CPM signals is established, which enables the CPM signals to be linearly demodulated. Consequently, the receiver complexity which is originally high due to the trellis nature of the structure of the CPM receivers, is significantly reduced. Further, the techniques already developed for linear modulation, e.g. matched filtering and linear equalization can be straightforwardly adapted to non-linear modulation. In this work, the linearized form of the CPM is adapted to SC/FDE (CPM-SC/FDE). Another form of modulation with a constant amplitude, that is known as tamed frequency modulation (TFM), introduced a decade ago, is also adapted to SC/FDE. The resulting (emerged) system is compared to OFDM under the constraints of the non-linearity of the RF stage. The comparison shows that the performance of CPM-SC/FDE is not affected as OFDM whose performance deteriorates dramatically. Although the emerged system is effective in the non-linear region of the amplifier, its error performance remains unsatisfactory because of the inherent differential 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 multiplexing is particularly applied as it improves the transmission efficiency tremendously. There- fore, the system bandwidth efficiency increases linearly with the number of the deployed antennas. Further, SC/FDE with linear and non-linear modulation formats is extended to ”point to multipoint” applications. It is combined with code division multiple access (CDMA). Finally, SC/FDE-CDMA is compared to the combination of OFDM and CDMA (MC-CDMA), and it is outlined, that applying Fourier spreading codes to full load SC/FDE-CDMA yields to an OFDM system, whereas, in contrary, the combination of OFDM and CDMA produces an SC/FDE scheme.

SC/FDE - Buzid List of Abbreviations

ACI Adjacentchannelinterference ACTS Advanced communications technologies and services ADSL Asymmetricdigitalsubscriberline AGC Automaticgaincontrol AMPS Advancedmobilephoneservice ASK Amplitudeshiftkeying AWGN AdditivewhiteGaussiannoise B3G Beyond3G BER Biterrorrate BPSK Binaryphaseshiftkeying BT Bandwidthtimeproduct CDMA Codedivisionmultipleaccess CDPD Cellulardigitalpacketdata COHASK Coherent amplitude shift keying COHFSK Coherent frequency shift keying COHPSK Coherent phase shift keying CPM Continuousphasemodulation DCS1800 Digital cellular system 1800 DECT Digital European cordless telephone DEPSK DiﬀerentialencodedPSK DFT DiscreteFouriertransform DMSK DuobinaryMSK DPSK Diﬀerentialphaseshiftkeying DVB-C Digital video broadcasting-Cable DVB-S Digital video broadcasting-Satellite vii

DVB-T Digital video broadcasting-Terrestrial EDGE EnhanceddataratesforGSMevolution ETSI European telecommunications standard institute FDM Frequencydivisionmultiplexing FDMA Frequency division multiple access FDOSS Frequency domain orthogonal signature sequences FFT FastFouriertransform FPLMTS Future public land mobile telephone system FSK Frequencyshiftkeying GMSK Gaussianminimumshiftkeying GPRS Generalpacketradioservice GPS Globalpositioningsystem GSM Groupspecialmobile GSMSK GeneralizedSMSK GTFM Generalized tamed frequency modulation HPA High power amplifier IDFT InversediscreteFouriertransform IFFT InversefastFouriertransform IJF-QPSK Intersymbol-interference and jitter-free QPSK ISI Intersymbolinterference ITU International telecommunication union JTACS Japanese total access communication system LMDS Local multipoint distribution service MMDS Multichannel multipoint distribution service MMSE Minimummeansquareerror MSK Minimumshiftkeying 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-offkeying OQASK Offset quadrature amplitude shift keying

SC/FDE - Buzid viii

OQPSK Offset quadrature phase shift keying PA Poweramplifier PAM Pulseamplitudemodulation PAPR Peaktoaveragepowerratio PDC Personaldigitalcellular PDM Pulsedurationmodulation PHS Personalhandyphonesystem PPM Pulsepositionmodulation PSK Phaseshiftkeying PSTN Public switching telephone networks QAM Quadratureamplitudemodulation QPSK Quadraturephaseshiftkeying SAW Surfaceacousticwavedevice SC/FDE Single carrier with frequency domain equalization SFSK Sinusoidalfrequencyshiftkeying SMSK SimplifiedMSK SOQPSK Shaped offset quadrature phase shift keying SOQPSK Staggered offset quadrature phase shift keying TACS Total access communication systems TDMA Timedivisionmultipleaccess TETRA Terrestrialradioaccess TFM Tamedfrequencymodulation UMTS Universal mobile telecommunication system UW Unique word V-BLAST Vertical-Bell laboratories layered space- time VCO Voltagecontrolledoscillator WLAN Wirelesslocalareanetwork XPSK Cross-CorrelatedPSK ZF Zeroforcing

SC/FDE - Buzid Contents

1. Introduction 1 1.1. Evolution of Mobile Radio Communications ...... 1 1.1.1. FirstGenerationCellularSystems ...... 2 1.1.2. SecondGenerationCellularSystems ...... 2 1.1.3. ThirdGenerationCellularSystems ...... 2 1.1.4. ForthGenerationCellularSystems ...... 3 1.2. SC/FDETechnique...... 5 1.3. GoalsofthisWork ...... 5

2. Basics on Modulation Techniques 7 2.1. Modulation ...... 7 2.2. AnalogModulation ...... 9 2.3. HybridModulation ...... 10 2.3.1. PulseAmplitudeModulation(PAM) ...... 10 2.3.2. PulsePositionModulation(PPM)...... 11 2.3.3. PulseDurationModulation(PDM) ...... 11 2.4. DigitalModulation ...... 12 2.4.1. AmplitudeShiftKeying(ASK) ...... 13 2.4.2. FrequencyShiftKeying(FSK)...... 13 2.4.3. PhaseShiftKeying(PSK) ...... 18 2.4.4. ContinuousPhaseModulation(CPM) ...... 23 2.5. Modulation Schemes Selection Criteria ...... 24 Contents x

3. Signal Envelopes Choices 28

3.1. Non-ConstantEnvelopeSignals ...... 28

3.2. NearConstantEnvelopeSignals ...... 28

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

3.2.2. Cross-CorrelatedPSK(XPSK) ...... 30

3.3. ConstantEnvelopeSignals ...... 30

3.4. ConstantEnvelopeSignalGeneration ...... 30

3.4.1. ShapedOﬀsetQPSK(SOQPSK) ...... 31

3.4.2. GaussianMSK ...... 31

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

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

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

3.4.6. ContinuousPhaseModulation ...... 35

3.5. Modulation Indexand Signal Phase States ...... 43

4. Demodulation of CPM 46

4.1. CPMReceiverComplexity ...... 46

4.2. DecompositionofCPM...... 51

4.3. LinearRepresentationofCPM...... 53

4.3.1. Multi-levelCPM ...... 56

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

4.3.3. GeneralAspects...... 57

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

4.3.5. Precoding ...... 60

4.4. SimulationandResults ...... 63

4.5. Summary ...... 64

SC/FDE - Buzid Contents xi

5. The Concept of Single Carrier Transmission with Frequency Domain Equal- ization (SC/FDE) 66 5.1. Introduction...... 66 5.2. SystemDescription ...... 67 5.2.1. BasebandEquivalentModel ...... 68 5.2.2. Scrambling ...... 69 5.2.3. ChannelCoding...... 69 5.2.4. Interleaving ...... 70 5.2.5. Modulation ...... 70 5.3. The Mobile Radio Channel and Additive Noise...... 72 5.3.1. AdditiveWhiteNoise...... 72 5.3.2. TheMobileRadioChannel...... 73 5.3.3. TheIEEE802.11aChannelModel...... 75 5.4. Transmission Model and Optimum Receiver Structure ...... 76 5.4.1. Burst- and Block Structure and Guard Interval ...... 77 5.4.2. TheSystemBandwidthEﬃciency ...... 79 5.4.3. OptimumLinearReceiver ...... 80 5.4.4. Frequency Domain Equalization Concept ...... 82 5.5. ThePeakPowerProblem ...... 86 5.6. Complementary Cumulative Distribution Function Curves(CCDF) . . . 87

6. SC/FDE for Constant and Near Constant Envelope Modulation 88 6.1. SC/FDEforOQPSKModulation ...... 88 6.2. SC/FDE for Constant Envelope Modulation Schemes ...... 89 6.2.1. TheTransmissionModel ...... 89 6.2.2. TheReceiverModel ...... 90 6.3. Impact of Front-End Nonlinearity on CPM-SC/FDE System Performance 92 6.4. SpatialMultiplexingforCPM-SC/FDE ...... 97

SC/FDE - Buzid Contents xii

7. PointtoMultipointSystems,SC/FDE-CDMA 101 7.1. CDMACellularSystems ...... 101 7.2. SC/FDE-CDMA ...... 102 7.3. Multicarrier-CDMA versus SC/FDE-CDMA ...... 104

A. Appendix 108 A.1.FourierMatrix ...... 108 A.2.ConvolutionMatrix...... 109 A.3. Circulant Convolution and Toeplitz Matrix ...... 109 A.4.SamplingMatrix ...... 110 A.4.1. UpSamplingMatrix ...... 110 A.4.2. DownSamplingMatrix...... 110

SC/FDE - Buzid 1. Introduction

Today, if someone goes back in literature twenty years or more, he will surprisingly find that the fundamental motivation behind the introduction of the wireless communications is the idea of replacing the costly cabling systems. The drawback of cabling system comprises 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 important element in the society, if not its core. They take a more central role in our daily lives 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. The data at the end user is converted into a voice or picture or command or movies or a combination of all of that. The wireless technology profile is like a snowball; the more you roll it over and over, the larger it becomes. Today’s emerged systems could have been a science fiction twenty years ago. In the early eighties the cordless telephones were invented and provided the end user a mobility range of tens of meters what represented large scale mobility at this time. From cordless telephones that allowed a mobility range of few tens of meters and that were owned by wealthy individuals it took only about 20 years to cellular systems that support global mobility and multimedia services to millions 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 the ships sailing in the English channel in 1897. Further, other point to point applications like e.g., satellite communications, the half duplex communication, or the atmospheric communication (Ground-sky wave) have been introduced. But the wireless revolution has really started with the introduction of the digital cellular communication systems, which are divided in the ﬁrst generation (1G), the second generation (2G), the third generation (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 only the analog modulation techniques. Across Europe, many incompatible systems operated and a pan-European roaming was not possible. For example, the Nordic countries and the Netherlands deployed Nordic mobile telephones (NMT-450). The UK and Ireland introduced total access communication systems (TACS). Germany and Portugal had the C-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 were founded in Europe (the European telecommunication standards institute (ETSI) and its working groups) aiming to create a communication environment that provides Eu- ropeans a roaming across the European countries. The ETSI inherited all European telecommunication standardization activities from the European conference of postal and telecommunications administrations (CEPT) in 1989. A group called ”group special mobile GSM” (a French name) within the CEPT was charged with developing a pan-European standard in 1982. Later, the GSM has been accepted in many parts of the 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 a combination of time division multiplexing (TDMA) and frequency division multiplexing (FDMA) for multiple access. The US IS-95 system uses a CDMA technique. In the ﬁeld of cordless telephones, two systems belong to the second generation and make use of microcells which cover small distances. One is the cordless telephone (CT2), which is dedicated to a voice transmission and does not support handovers between base stations. The other one is the digital European cordless telephony (DECT) which accommodates data 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 digitized technology. Further, besides voice service the 2G systems enabled the user to roam over a 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 rates of 2 Mbit/s, which is at least required for todays applications such as the multimedia and others. An enhancement of the data rate in the GSM system by increasing the number of 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 scheme and 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 wireless multimedia. They are sometimes known as 2.5G or 2G+. The unique and worldwide 3G standard which was aimed at is the future public land mobile telephone system (FPLMTS) renamed as international mobile telecommunication system (IMT-2000). However, two standards were realized, the universal mobile telecommunication system (UMTS) in Europe and the cdma2000 in the US. Both standards use CDMA for multiplexing of multiple users. The high data rates offered by the two standards (voice, data and video) make the systems vulnerable to the wireless channel impairments and the 3G systems are classified as broadband systems, in contrast to 2G systems. Also, the UMTS system has been enhanced by the so called high speed downlink 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 local area networks (WLAN). WLANs offer high data rate services to both mobile and stationary users. Many developed standards targeted the speed rather than the roaming property for broadband services. These broadband techniques are the Bluetooth, IEEE 802.11a,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 a project 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 integration with other open standards.

Mobility

B Vehicular e y o n d 3 G

( Nomadic 4 DAB 3G/3G G

DVB-T )

(UMTS/IMT2000) 2G(e.g.,GSM)

Stationary HIPERLAN/IEEE802.11a

HIPERMAN/IEEE802.16a 0.1 1.0 10 100 Datarate

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 technology rather than the enhancement of 3G technology. The improvements are partially towards higher data rates to meet needs of future high-performance applications, e.g. 100

SC/FDE - Buzid 1. Introduction 4

Mbit/s for outdoors and a peak of 1 Gbit/s for indoor. Further, B3G will witness the integration of existing technologies in a common platform, implying a global mobility and service portability. Further, to ensure the success of 4G systems, the cost of a broadband transmission per bit must be reduced dramatically compared to the cost of the 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 the future development of IMT-2000 and other systems by setting up a study group and working 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 European vendors, contribute to the ITU standardization work. Further, new research projects are 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 an intranetworking procedure to manage and administrate the traffic among and within coexisting, diverse, heterogeneous and overlapping networks through commands and software rather than added hardware or physical tools. In this context, networks are on the move instead of a single mobile terminal is on the move. The vision of ambient

VAN

Ambient

Solution’s

Cellular OfficeLAN PAN service

AmbientCity’s Hotspotservice

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 communications and to enable the integration of coexisting future multi-technologies, self configured networks, with the constraints that people can move from one network to another without any effort or interruption. The ambient networks operate in a dynamic nature of environment and can be extended with new capabilities as well as operate over existing connectivity infrastructure. Moreover, ambient networks enable the concept of instant composition of networks belonging to different business entities. The viable and challenging element in ambient networks is that a diverse bulk of users (multi-technology

SC/FDE - Buzid 1. Introduction 5 mobile networks) accesses the diverse bulk of stationary and/or mobile networks which must 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) technique is 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 multiple carrier system, and both are possible methods of transmitting data from one point to another over a frequency-selective fading channel. The SC/FDE performance has shown to be similar to that of orthogonal frequency division multiplexing (OFDM) as a multicarrier technique [7, 8]. Broadband transmission means the bandwidth of data to be transmitted exceeds the coherence bandwidth of the multipath channel. The superiority of the SC/FDE concept to classical single carrier systems lies in the equalization that mitigates the wireless distortions and is carried out in frequency domain. The consequence of the use of frequency domain equalization is the significant complexity reduction of 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 discrete Fourier transformation (DFT) and the guard interval. Only a limited number of researchers world wide considered the SC/FDE topic at the beginning. Among them a group supervised by D. Falconer [8]. In late nineties, Czylwik [11] investigated channel estimation and synchronization for SC/FDE. Further investigations of the SC/FDE took place at Linz University, where the first dissertation on this topic was published by M. Huemer [12]. More publications followed and covered the system synchronization, channel 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-Dahir in [18] applied the Alamouti-principle to SC/FDE, and further enhancement by applying time space diversity and multiple input multiple output (MIMO) techniques as is achieved [19, 20, 21, 22]. In recent years, different proposals have been made for SC/FDE, especially in IEEE 802.16 Wireless MAN (Metropolitan Area Network) [8]. Further, SC/FDE combined with frequency devision multiple access (SCFDMA), is a promising 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, that single carrier transmission schemes show signal envelope characteristics with lower peak to average power ratios (PAPR). This relaxes the linearity requirements of the transmit power ampliﬁer, and thus improves the transceiver’s power eﬃciency. Mostly, SC/FDE

SC/FDE - Buzid 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 show rather large amplitude variations. The fundamental aim of this work is the adaption of SC/FDE for modulation formats with effectively low PAPR. One candidate is the linear OQPSK (offset QPSK) scheme, but the focus of the work is on CPM (continuous phase modulation) formats. Further the spatial multiplexing (SM) extension of CPM- SC/FDE as a method to improve the bandwidth efficiency shall be investigated. Finally the combination of SC/FDE schemes with CDMA shall be addressed.

SC/FDE - Buzid 2. Basics on Modulation Techniques

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

2.1. Modulation

What does modulation mean and why is it needed? To answer this and other related questions, popular modulation techniques are reviewed in the next sections. The term modulation comes from the process of modulating a carrier. The reverse of the actions and measures including modulation that are performed at the transmitter are carried out at the receiver. However, the detection may be distinguished from demodulation by deﬁning detection as a process of extracting the information from the baseband representation of the demodulated signal. In non-coherent systems, however, the demodulation and detection may not be distinguished in a straightforward manner. Modulating the carrier is implemented by the manipulation of one or more carrier parameters. The carrier parameters are the amplitude, the phase and the frequency. Therefore, the type of the 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 • 2. Basics on Modulation Techniques 8 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 carrier phase, whose relation to the instant frequency fi(t) is given as

1 dφ(t) f (t)= . (2.2) i 2π dt The frequency and phase modulation are closely related to each other and are also known as angle modulation. Another important classiﬁcation which is in accordance with the type of the information is:

Analog modulation • Digital modulation •

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

Modulation Digital Hybird Analog Hybird Ampl Angle Ampl Angle Phase Freq ASK PM FM BPSK PPM BFSK AM NB-FM ASK-PSK M-QAM QPSK M-FSK PWM DSB-SC WB-FM CFSK DPSK PCM SSB-SC M-PSK MSK PAM VSB JIFQPSK OQPSK GMSK PDM FeherQPSK XPSK SMSK CPM GSMSK Non-linearmodulation ConstantEnvelope

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

SC/FDE - Buzid 2. Basics on Modulation Techniques 9

2.2. Analog Modulation

Due to its historical signiﬁcance, analog modulation is brieﬂy reviewed. Although the use of the analog modulation is shifted towards the digital, it is still important and being in use in practice, since many signals generated in real life are analog in nature and transmitted in the analog form. In analog modulation category, frequency modulation is the most important. The carrier’s frequency varies with the amplitude of the message signal and hence the carrier’s amplitude remains constant. The FM scheme is important because 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 the instantaneous phase of the carrier changes with the transmitted information. However, both phase and frequency modulation are known as angle modulation and hence their performances are expected to be similar. Employing an integrator in front of a phase modulator yields a frequency modulator. In contrast to the angle modulation the amplitude modulation is less complex and requires less bandwidth, but comparably inefficient power amplifiers have to be used.

Coherent Demodulation

In most high performance applications, receivers are based on coherent detection. To perform coherent detection, the receiver must know the frequency and the phase of the received signal. In a receiver, a phase-locked loop (PLL) tracks the frequency and phase of 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 frequency and phase of the received signal

z(t) = 2cos(2πfct + θ0),

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

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Non-Coherent Demodulation

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

2.3. Hybrid Modulation

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

2.3.1. Pulse Amplitude Modulation (PAM)

PAM represents a form of modulation in which the amplitude of individual and regularly spaced pulses in a pulse train is varied in accordance with some characteristics of the modulating signal. The amplitude of the pulses conveys the information in this case. Pulse amplitude modulation is the simplest form of pulse modulation. It is generated in the same manner as analog amplitude modulation. The timing pulses are applied to a pulse amplifier in which the gain is controlled by the modulating waveform. Since these variations in amplitude actually represent the signal, this type of modulation is basically a form of AM. The distinctive difference is that the signal is now in the form of pulses. This signifies that PAM has the same built in weaknesses as any other AM signal, which is essentially the high susceptibility to noise and interference. The reason for susceptibility to noise is that any interference in the transmission path either adds to or subtracts directly from the wanted signal. That affects the signal amplitude which carries 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 not very popular. Techniques of pulse modulation other than PAM have been developed to overcome problems of noise interference. PAM is now rarely used and has been largely superseded by pulse position modulation. The following sections will discuss other types of pulse modulation that are shown in Figure 2.2.

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Clock

PAM t

PPM t

PDM

Data 4 5 3 2 6 1 7 t

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

2.3.2. Pulse Position Modulation (PPM)

Pulse position modulation is a form of signal modulation in which m message bits are encoded by transmitting a single pulse in one of 2m possible time shifts. This is repeated 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 position of each pulse, in relation to the position of a recurrent reference pulse is varied by each instantaneous sampled value of the modulating wave. PPM has the advantage of requiring constant transmitter power since the pulses are of constant amplitude and duration. It is primarily useful for optical communication systems where there is only little or no multipath interference. As this technique relies mainly on the time delays of the pulses within a period, it is a difficult task to maintain the synchronization and the alignments of the received pulses. Therefore, it is often implemented differentially and also known as differential pulse position modulation, whereby each pulse position is relatively encoded to the previous pulse in such a way the receiver must only measure the difference in the arrival time of successive pulses. One of the important advantages of the PPM is that it can be implemented non-coherently and the receiver does not need to track the phase of the carrier. This makes it a suitable candidate for optical communication systems.

2.3.3. Pulse Duration Modulation (PDM)

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

SC/FDE - Buzid 2. Basics on Modulation Techniques 12 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 in ship-to-shore communications. The synonyms are pulse length modulation (PLM) and pulse width modulation (PWM). It requires very little bandwidth and was employed in shortwave 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 exchange of information at very high data rates. Digital modulation beneﬁts from the huge data processing power provided by modern processors. A conceptual block diagram, shown in Figure 2.4, illustrates the modulator and demodulator that are covered in this chapter. 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 ﬁnite number of angles and amplitudes are used. Digital modulation is fundamentally categorized into amplitude shift keying (ASK), frequency shift keying (FSK) and phase shift keying (PSK). Each one or a combination of them are employed in diﬀerent applications according 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 units of 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 shift keying (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 or changes the amplitude of the carrier. The simplest and most common form of ASK operates as a switch. The presence of a carrier indicates a binary one and its absence indicates a binary zero. Generally, amplitude modulation has the property of translating the spectrum of the modulating signal to the carrier frequency. It is also known as On-Off keying (OOK) because of the twin states of the ASK signal. The dominant application of ASK is in control applications. This is partly due to its simplicity and low implementation costs. In addition, the detection process can be a simple envelope detector which is much simpler than a coherent approach. One of the disadvantages of ASK, when compared with its counterpart modulation formats, is that it does not have a constant envelope which makes the amplifications of the signal more difficult because of 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 information signal yields a frequency modulated signal. The instantaneous frequency is shifted between discrete values. A binary FSK (BFSK) modulation scheme consists of two sinusoidal pulses at two frequencies, f1 and f2 representing 1s ”mark” and 0s ”space”, respectively. 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 is called modulation index. The bipolar input to the modulator increases or decreases the signal 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. The BFSK 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 signals of diﬀerent carrier frequencies f1 and f2 [28]. The normalized BFSK signal may also be expressed 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 alternate instantaneously between 0s and 1s in a predetermined manner as the following:

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w 2 f

0.8

0.6

0.4 Halo

0.2

Normalized amplitude 0 Normalized amplitude Normalized -0.2 f1 f2 f3

-0.4 0.998 0.998 1 1.001 1.002 1.003 1.004 1.005 1.006 f kHz

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 compared to OOK.

2.4.2.1. M-Ary FSK

An M-ary FSK or Multi-FSK is an extension of a BFSK modulation where M carriers are utilized consecutively. The signal spectrum is indicated in Figure 2.5 for a rectangular pulse shape. The required bandwidth for M carriers is then

B − = 2[(M 1)∆f + w] (2.7) M F SK − = 2w [(M 1)~ + 1] . − In a non-coherent receiver, the received signal is presented across M parallel bandpass ﬁlters which are centered at frequencies f1,f2,...,fM , as shown in Figure 2.6, provided that ~ >> 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 the largest 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−F SK =(M + 1)w, which gives the total bandwidth for M orthogonal signals. As an example, the bandwidth 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 be determined as

actual 2w B − = (2.8) M orthogonal 2w[(M 1)~ + 1] − 1 = . (M 1)~ +1 − Assuming a 16-FSK orthogonal system with ~ =1/2,

actual 2 B − = , (2.9) M orthogonal 17 which represents 12% of the transmission bandwidth and shows how ineﬀecient 16-FSK is in terms of the spectrum. For this reason, orthogonal M-FSK is not popular.

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

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cos(wt)

I(t) I-generator

Bipolar data CPM signal +

Q(t) Q-generator

- sin(wt)

Figure 2.7.: Quadrature modulator (parallel approach). used during one time interval. Another counterpart of an orthogonal M-FSK is an orthogonal frequency division multiplexing (OFDM) system, where the carriers overlap orthogonally 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 the BFSK system? What is the criterion to establish such a requirement? The answer to these questions yields the MSK modulation. The minimum frequency separation is obtained when the spectrum of the two signals (carriers) are overlapped orthogonally 1 to each other. Therefore, the minimum separation frequency equals 2∆f = 2T and the modulation index is ~ = 0.5. The minimum frequency separation 2∆f is equivalent to a phase contribution of π/2 for each symbol. Note that the phase transitions between one phase state and the next state is forced to follow a circular path. Thus, the result is a constant envelope signal. MSK was ﬁrst reported by Doelz and Heald [31]. As is evident in the expression of the MSK signal

s(t)= I(t)cos(w t) Q(t)sin(w t), (2.10) c − c which is the sum of two pulse streams modulating the in-phase and quadrature channels of a single carrier, where I(t) and Q(t) are inphase and quadrature phase baseband shaped 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 this type of signals and they are divided into two categories. In addition to the parallel MSK modulators, the serial type of modulators (SMSK) is the other. SMSK was ﬁrst introduced by Amoros [32]. However, it was also reported under the name of simpliﬁed MSK (SMSK). One possible serial modulator is illustrated in Figure 2.8. The data in serial modulators, as the name suggests, are modulated serially. In another simple serial 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 the polar coordinates, instead of the Cartesian coordinates (parallel form). It has been practically used in GSM. A generalized form of SMSK has been proposed by Allan [33]. The concept of the GSMSK scheme is based on replacing the straight trellis arms of the phase trellis by generalized curved trellis arms, depicted in Figure 2.9. The impulse response of the conversion ﬁlter is generalized and deﬁned by [33]

d(φ (t)) h(t)= 1 sin(wt + φ (t)), (2.11) dt 1 where φ1(t) denotes the excess-phase of the GSMSK signal. For a particular data symbol φ1(t) is allowed to be any slow changing function in the range 0

0 t 0 t 2 t 3 ≤ φ1(t)= 3π( ) 2π( ) 0

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2. Basics on Modulation Techniques 18

0.0 -20.0

MSK -40.0

GMSK

-60.0

Normalized outofband powerdB

-80.0

-100.0 00.0 2.0 4.0 6.0 8.0 10.0 NormalizedBandwidthf T

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 bandwidth greater 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 accordance with 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 two opposite signal phases. As there are two possible wave phases, BPSK is sometimes called biphase modulation. The two phase states, i.e. 0 and π are projected on the carrier amplitude as 1 and -1, which polarizes the carrier. In addition, more sophisticated forms of the PSK are in use in todays communication systems. In an M-ary or multiple phase shift keying (M-PSK), there are M phase states. The term M-PSK is often replaced by quadrature phase shift keying or quaternary phase shift keying (QPSK) when M =4. In M-PSK where M =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 speciﬁcally for the symbol-set used by the modulator, determines the phase of the received signal and maps it back to the symbol it represents, thus recovering the original data.

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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 quadrature phase modulation as the name suggests [34]. OQPSK eliminates 180 degree phase changes. This is achieved by introducing a delay with the duration of half a symbol between the quadrature components of a QPSK signal. This implies that the quadrature 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. In DPSK, 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 of that 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 between successive 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 bits per QPSK symbol are transmitted. The bandwidth efficiency can further be improved by sending more bits per modulated symbol. M-PSK can improve the efficiency by a factor of ln (M). The symbol phase is then given by π φ =2m , m =0, 1, 2, ..., M 1, m M − where M N. Any increase in M raises the number of modulation states. However, as the number∈ of modulation states increases, the distance between phase states reduces and the likelihood of a demodulator error increases. At higher values of M, the scheme is not preferred any more. In practice, M = 8 (3 bits per modulated symbol) is the maximum 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 the expense of power efficiency. The M signal waveforms in digital phase modulation can be represented by [26]

j(φm+wct) sm(t) = Re g(t)e (2.13) = g(t) cos(wct + φm) = g(t) cos(φ ) cos(w t) g(t) sin(φ ) sin(w t) m c − m c

SC/FDE - Buzid 2. Basics on Modulation Techniques 20 for m = 0, 1, ..., M 1, where g(t) is the signal pulse shape whose energy ξ is given by T 2 − ξ = 0 g(t) 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 error R probability 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 and PQPSK are respectively given by

2ξ P = Q (2.14) BPSK N r 0 ! and ξ P = Q 2 , (2.15) QPSK N r 0 ! where Q(x) is the Q function deﬁned as

∞ 1 2 Q(x)= e−t /2dt x 0 (2.16) √2π ≥ Zx and N0 is the noise power spectral density. Coming back to large values of M, the exact analysis of the system bit error rate (BER) is often complicated and usually results in non-closed form solutions [36]. A closed form expression for the exact solution of up to 8-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 mapping for higher order than 8-ary PSK modulation [37, 38]. Irshid, in [35] advised a simple technique 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 into half and quadrant decision subspaces. Then it finds expressions for the probabilities that the constellation point lies in a given decision subspace. Jianhua in [38] reached the same results differently, the bit error probability of coherent M-PSK based on the signal space geometry was reached for Gray code bit mapping. The principle of the approach is to divide the signal space into two main axes and if necessary they can be subdivided to subspaces by rotating the main axes by an amount R (a rotated coordinate system). Bit error 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 on the bit-mapping used. Consequently, the determination of the BER of coherent M-ary PSK schemes has been performed by either calculating the symbol error probability or using lower/upper bounds particularly for higher values of M [37, 38, 39]. The application of the bounds estimations does not always ensure sufficient accuracy, while the transformation from SER to BER is, in general, not straightforward and accurate. The relation between binary natural code and Gray code bit mapping is demonstrated next.

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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, has the property that two m-bit symbols corresponding to adjacent symbols differ only in a single bit. As a result, an error in an adjacent symbol is accompanied by one and only one bit error. The name, Gray reflected binary code, is derived from the fact that it may be constructed from a binary code by a reflection process. Today, Gray codes are widely used in digital communications and many algorithms for conversion between binary code and binary reflected Gray code were developed. Agrell in [40] revised the original work of Gray and reported that Gray bit mapping for M-PSK is optimum in comparison with binary natural mapping. A comparison between M-PSK with Gray code and binary natural 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]. Let th the m-tuple b = bn,0...bn,i...bn,m denote the n symbol (codeword), where m = ln(M) and n =0, 1, ..., M 1. The following equation, −

Binarybits Graybits Binarybits Graybits

1 00 00 1 0000 0000 2 01 01 2 0001 0001 3 10 11 3 0010 0011 4 11 10 4 0011 0010

Q=6b Q=4g 5 0100 0110 Graybits Binarybits 6 0101 0111 1 000 000 7 0110 0101 2 001 001 8 0111 0100 3 010 011 9 1000 1100 4 011 010 10 1001 1101 5 100 110 11 1010 1111 6 101 111 12 1011 1110 7 110 101 13 1100 1010 8 111 100 14 1101 1011 15 1110 1001 16 1111 1000

Q=14b Q=8g Q=30b Q=16g

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

m M Q = b b (2.17) ni ⊕ (n+1) i i=1 n=1 X X gives a decimal number which represents the total number of possible transitions (change from one to zero or vice versa) for any M-PSK constellation, as illustrated in table 2.1, 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

SC/FDE - Buzid 2. Basics on Modulation Techniques 22 are tabulated in table 2.2. However, it is worth mentioning that, for m > 3 there exist several Gray code constellations (labeling) and as m increases the number of such constellation becomes rapidly very large [40]. Nevertheless, Qg is always optimum for any constellation (labeling). From the table, one can see Qb > Qg. The interpretation

m M Qb Qg 2 4 6 4 3 8 14 8 4 16 30 16 6 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 error is higher than in case of a Gray code. Letting F = Qb , a closer look at table 2.2 indicates Qg 2m 1 F = − , (2.18) 2m−1 which relates the error probability of both binary natural code mapping and Gray code mapping, as Pb = F Pg (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 the analytical result.

2.4.3.5. Quadrature AM (QAM)

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

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0 10

Binary mapping Gray mapping Obtained Binary mapping from Gray −1 10

−2 10 b P

−3 10

−4 10

0 1 2 3 4 5 6 7 8 9 10 E /N [dB] b 0

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 constellation such 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 CPM concept is its constant envelope which is essential in many wireless systems. The properties and performance (bandwidth/power) characteristics of this class of modulations are sufficiently voluminous to fill a textbook of their own [26, 30, 44]. For the sake of brevity, certain special cases of the CPM, that have gained popularity in the literature and have been used in practice, shall be covered. Following the usual custom of modulation classifications, the CPM concept does not belong entirely to the phase modulation category as the name may suggest and promotes some ambiguity to the perception of the 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 phase in successive symbol intervals [44]. The instantaneous frequency is constant over each symbol interval, therefore, CPM is a frequency modulation with superiority. However, the changing of the frequency and subsequently the phase or vice versa can be compared to a master and slave principle [45]. Furthermore, the definition of the phase and frequency 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 modems QPSK (M=4) (4-QAM) Satellite, CDMA, NADC, TETRA, PHS, PDC, LMDS and DVB-S 8PSK (M=8) EDGE,Telemetry pilots for monitoring broadband video systems 16 QAM (M=16) Microwave digital radio, DVB-C, DVB-T 32 QAM (M=32) Terrestrial microwave, DVB-T 64 QAM (M=64) DVB-C, MMDS 256 QAM (M=256) DVB-C (Europe), Digital Video (US)

Table 2.3.: The application of different modulation schemes in practical systems. For the abbreviations refer to the list of abbreviations. and hence, resulting in constant envelope signals. For this reason, phase and frequency modulation are also known as constant envelope type-modulation in some literature. M-FSK is achieved by shifting carrier frequency, and switching from one frequency to another is accomplished by having M = 2m separate oscillators, tuned to the desired frequencies. The transferring from one frequency to another causes the sidelobes in the spectrum to be large, consequently this method requires large bandwidth. To avoid this, a carrier whose frequency changes continuously with time, is modulated and the resulting modulated signal is called continuous phase FSK. This type of signal has a memory and therefore, the phase of the carrier has to be continuous. Continuous phase modulation implies that the carrier phase copes and acts in response to the changes of the transmitted information continuously and smoothly. In other words, the phase of the carrier ascending without abrupt changes that result in spectrum sidelobes. The data symbols modulate the instantaneous phase of the transmitted signal, where the phase is a continuous function in time [44]. As an example, the phases of a full response CPM signal for an antipodal input sequence and a modulation index of 1/2 with a linear as well 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 symbol affects 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 interval depends on the previous data symbols. However, the phase of a PSK signal is independent of the previous data symbols and this explains further the memory associated with a CPM scheme.

2.5. Modulation Schemes Selection Criteria

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

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0.2 2/64 1

0.8 + + + + + + + + 0.6 + + + + + + + + 0.4 0.2 + + + + + + + + 0.2 + + + + + + + + 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

Figure 2.12.: 64-QAM and 64-PSK constellation. scheme is the result of compromises on the bandwidth efficiency, performances, complexity and immunity to noise and other interferences. However, power efficiency, bandwidth efficiency and system complexity are major criteria for choosing an appropriate modulation scheme.

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

Power Efficiency The definition of the power efficiency is not only related to the signal energy, but also to the dc efficiency of the power supply which depends on the chosen modulation scheme.

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

Transmission rate η = [bits/s/Hz] . Bandwidth

SC/FDE - Buzid 2. Basics on Modulation Techniques 26

360 360

270 270

180 180

90 90

0 0

(t) in degrees −90 φ (t) in degrees −90 φ −180 −180 −270 −270 −360 −360 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 t / T t / T

(a) (b)

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

However, the spectrum compactness, where sidelobs are at a minimum, is another aspect of 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 the spectrum efficiency. Practically, there are several definitions or methods of determining the 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 bandwidth is the width between null to null of the power spectral density that represents the main lobe. Based on this deﬁnition, various modulation formats can be compared as follows: The minimum bandwidth needed to transmit a BPSK symbol which carries one bit 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 1 η − = log (M)w/w = log (M) bits/s/Hz. M QPSK 2 1 2 Further, in an M-FSK system, the needed bandwidth is

B − =2w [(M 1)~ + 1] , M F SK −

SC/FDE - Buzid 2. Basics on Modulation Techniques 27

therefore the eﬃciency is 1 η − = bits/s/Hz, M F SK 2[(M 1)~ + 1] − which is very poor.

2. Percentage bandwidth eﬃciency:

In other deﬁnitions instead of null to null bandwidth the band containing more than 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. For instance, coherent demodulation is far more complex compared to non-coherent demodulation and hence, non-coherent based systems are more cost eﬀective.

SC/FDE - Buzid 3. Signal Envelopes Choices

In the following sections, another classification of modulation techniques that differs from the usual custom of the modulation categories, is considered. This classification is based on the signal envelope nature. The power spectra of most modulated signals exhibit sidelobes that may interfere with adjacent 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 nonlinear effects of the transmitter power amplifier. These nonlinearities tend to restore the spectral sidelobes that have been previously removed by filtering. As a precaution against spectral spreading, the high power amplifier (HPA) in the transmitter in practice is forced to operate below saturation, approximately in a linear zone. This comes with the obvious penalty, that an HPA designed for a larger power has to be used with consequent high inefficiency. These problems can significantly be reduced when using constant envelope signals.

3.1. Non-Constant Envelope Signals

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

3.2. Near Constant Envelope Signals

Signals in this category reveal reduced envelope fluctuations. In fact, such a taxonomy is seldom used in literature. In practice, near constant envelope signals have received considerable attention and deployment. They show moderate peak to average power ratio (PAPR) compared to non-constant envelope signals. Such signals relax the linearity requirements of e.g. the handset power amplifier (PA). This is why they are preferred in 3. Signal Envelopes Choices 29 the uplink in cellular communication systems. The widely known near constant envelope signals are OQPSK and π/4-DQPSK. Figure 3.1 shows the signal constellation of the rotated 8-PSK modulation, where data symbols are shaped with the linearized Laurent pulse shape (presented in the next chapter). 8-PSK technique is used in the EDGE system [47, 48]. Remarkably, the 8-PSK signal constellation is improved by both the rotation of the symbols and by the shaping pulse. Other systems using near constant envelope signals are Satellite, TETRA, DVB-S and many others. The common signature of these modulation formats is that the carrier phase variation is always below 180 (π rad). In general, data shaping in this type of modulation does not play a fundamental role in determining the signal envelope. However, when half a sinusoidal shaping filter is applied to OQPSK, MSK signals can be obtained, which show constant envelope.

1.5

0.5

0 Imag part

Quadrature-Phase -0.5

-1

-1.5 -1.5 -1 -0.5 0 0.5 1 1.5 RealIn-Phase part

Figure 3.1.: Rotated 8-PSK signal in the I/Q plane shaped by Laurent linearized pulse shape 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-linearly ampliﬁed satellite channels in a densely packed adjacent channel interference (ACI) environment [51]. It is a quadrature type of modulation with overlapping between the inphase and quadrature arms, similar to OQPSK. The diﬀerence is that transition between any adjacent dissimilar antipodal data symbols 1, 1 takes half sinusoidal shape instead of an abrupt change. That reduces the∈ power { − spe} ctrum sidelobes and relieves the non-linearity constraints. However, IJF-QPSK signals can be alternatively generated by directly shaping the binary data sequence using a pulse shape, given as π(t + T ) T 3T g(t) = sin2 2 , t (3.1) 2T ! − 2 ≤ ≤ 2

SC/FDE - Buzid 3. Signal Envelopes Choices 30 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 modulation technique by introducing a cross-correlation between the in-phase and quadrature data streams and specific waveforms [52, 53]. The aim of cross-correlation is to reduce the IJF-QPSK signal fluctuation by 3 dB to approximately 0 dB, so that the envelope is nearly constant.

3.3. Constant Envelope Signals

Constant envelope modulation shows better compromises than other approaches and that 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 envelope modulation formats, particularly MSK and GMSK [54, 55]. The major features and the contributions of the constant envelope modulation to wireless transmission are summarized as follows:

The prospect to fully exploit the high eﬃciency of power ampliﬁers. • The optimum power consumption of the power supply, and hence, the maximum • battery lifetime. Therefore it is appropriate to mobile terminals.

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

The PAPR of this signal is unity. That explains the excessive study dedicated to this kind of signals over the last few decades and to the present day. Unity of PAPR totally eliminates 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 BER deteriorates compared to e.g. QAM schemes, and since typically no higher order constant envelope schemes are used, the bandwidth efficiency is limited. This means that low PAPR is not the only determinant criterion in system design. Ultimately, constant envelope 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. These are data mapping (data encoding) and data shaping. In practice there is a wide range

SC/FDE - Buzid 3. Signal Envelopes Choices 31 of methods of generation and reception of this type of signals. Generally, there are two categories, serial and quadrature modulators. Serial modulators require a control circuit to monitor the voltage controlled oscillator (VCO) sensitivity, which deteriorates due to large phase error accumulation. The quadrature (generalized) modulator is shown in Figure 2.7. In the following sections, methods of generation of constant envelope signals based 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 of the instantaneous phase of 180 degrees as a result to overlapping between the quadrature components, but it fails to avoid the abrupt change of 90 degrees. Passing the data through a half sine shape ﬁlter of a width twice the symbol duration enforces a circle path 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 pulse weighting [56].

3.4.2. Gaussian MSK

GMSK is a minimum shift keying modulation with data filtered by a Gaussian-shaped frequency response filter, aiming primarily to make the output power spectrum more compact. The width of the Gaussian filter is determined by the bandwidth-time product BT . With a BT value less than 1, a controlled intersymbol interference is then introduced and a partial response CPM signal is obtained. A relaxed or a wider impulse response Gaussian filter (BT is small) suppresses the higher frequency components. Furthermore, BT is a trade-off factor between bandwidth efficiency, power efficiency and detector complexity [57]. Practical values of BT are e.g. 0.3 for GSM and 0.5 for cellular digital packet data (CDPD). Further, GMSK can be generated by alteration of a VCO frequency directly via the Gaussian data stream. With this simple method, however, it is difficult to maintain the center frequency within the acceptable value under the restriction of preserving 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 the constant envelope property of the signal and maintains less out of band power.

SC/FDE - Buzid 3. Signal Envelopes Choices 32

3.4.4. Correlative Coded Minimum Shift Keying (Correlative MSK)

Introducing memory in the modulation process produces signals with better bandwidth efficiency at the expense of system performance [58]. The controlled intersymbol interference is a method of introducing the memory. Correlative coding cascaded with a pulse shaping with ideally zero ISI at the sampling moments is an approach to generate pulses that give rise to controlled ISI. In correlative MSK, the data symbols are correlated prior to the modulation process [59]. Such coding is specified by a coding polynomial given as N n F (D)= qnD n=0 X n where D corresponds to a delay of nT and qn are the taps coefficients as shown in Figure 3.2. Two particularly useful schemes, using first- and second order encoding, are duobinary MSK (DMSK) and tamed frequency modulation (TFM).

T T T

q0 q1 qN + 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 to data prior to modulation in Duobinary MSK. DMSK has less phase variation than MSK and should consequently have better bandwidth eﬃciency [60]. DMSK is known also as 2REC, which is a rectangular pulse that scans two consecutive data symbols required to shape data and given by 1 0 t 2T q(t)= 4T ≤ ≤ (3.2) (0 otherwise.

Tamed Frequency Modulation TFM is a minimum shift keying scheme that was ﬁrst introduced by Jager and Dekker [61]. It is a constant envelope modulation scheme that exhibits a narrow power spectrum with minimum sidelobes and hence, a low adjacent 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

SC/FDE - Buzid 3. Signal Envelopes Choices 33

π(1−r) 1 0 f 2T (fT −0.5)π π(1≤|−r) |≤ π(1+r) G(f)= 0.5 1 sin r 2T f 2T (3.7)  − ≤| |≤ 0 otherwise   802.16.1, as a possible co-existence with 4-QAM dual mode and it is also an option for residential terminals [62]. In general, TFM is ideal for low cost terminal solutions. TFM is 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 keeps maximum phase changes of the TFM signal during one bit period restricted to π/2 rad. The allowable phase shifts of the modulating carrier during the nth bit period± at t = nT are given by π γ − γ γ ∆φ(nT )= n 1 + n + n+1 , (3.3) 2 4 2 4 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. Various classes of signals can then be obtained by properly adjusting the coefficients a and b. This technique of signal generation is called generalized TFM (GTFM). Setting a=1/4 and b=1/2, GTFM reduces to the TFM scheme. The TFM modulator is basically a cascade of 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 response coding) and thus introduces the correlative encoding property into the premodulation filter 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]

2a W (f) = b 1+ cos(2πfT ) . (3.6) | | b It is, however, worth mentioning that increasing b raises the noise margin for detection at the expense of lowering the spectral efficiency [63]. On the other hand, a Nyquist filter G(f) satisfying the Nyquist criterion, e.g., a raised-cosine filter which is extensively used and expressed in equation (3.7), adds the rolloff factor (0 r 1) as another design parameter in GTFM when compared to TFM [63]. The premo≤ dulation≤ filter in TFM, where the rolloff factor is set to zero, aims to produce a sequence of quinary symbols as shown in Figure 3.3. This type is a serial implementation. Further, the parallel implementation, which was also investigated in [61], is always favored. The block 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

SC/FDE - Buzid 3. Signal Envelopes Choices 34 from the TFM modulator depicted in Figure 3.4 is basically data shaped by the overall induced pulse (premodulation ﬁlter), which is called TFM pulse and given as 1 1 1 h (t)= g(t T )+ g(t)+ g(t + T ), (3.8) p 8 − 4 8 where 2πt πt π2t2 πt 1 2 T cot( T ) T 2 g(t) sin − 3 − . (3.9) ≈ T πt − 24πt " T 2 #! The TFM pulse partially prescribes the phase path from the phase at nT to the phase

T T 1/4 1/2 1/4 +

Nyquist Filter VCO

Figure 3.3.: Serial TFM transmitter.

sin(wt) sin(wt)

F modulator TFM I(t) I(t) Delay LPF + 2T Input TFM Input data + Gate data signal Signal out Q(t) Q(t) Delay LPF + 2T cos(wt) cos(wt)

a) b)

Figure 3.4.: a) Parallel TFM transmitter b) TFM coherent receiver. For details of the receiver see reference [61]. at (n +1)T and partially determines the total amount of the phase increase or decrease during a sampling interval. The amount of increase or decrease in the phase in TFM is either of 0, π/4, π/2 . The signal phases belonging to MSK and TFM schemes are plotted{ in± Figure± 3.5,} which show the smoothness of phase transition of TFM signals in comparison with MSK. Obviously, TFM shows less phase transitions and more smoothness than MSK. Consequently, the TFM spectrum characteristic is better than MSK.

3.4.5. Sinusoidal Frequency Shift Keying (SFSK)

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

SC/FDE - Buzid

3. Signal Envelopes Choices 35 (rad)

(t)

(a) (n+2) /2

(n+1) /2 (b) n /2

(n-1) /2 /4 (c) (n-2) /2

timet

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: πt t g(t) = cos Usin(2π ) , (3.10) 2T − T 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 shaping pulse is illustrated by the variation of the parameter U. The pulse g(t) for diﬀerent values of U is plotted in Figure 3.6. Further, the signal power spectrum for both MSK and SFSK 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 modulation schemes 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 generally used for this class of modulation scheme is shown in Figure 3.8. The input data symbols denotedγ ˜ excite a premodulation filter whose impulse response is denoted by g(t). Thus the input to the FM modulator is defined as ∞ γ¯(t)= γ g(t iT ). (3.11) i − i=−∞ X The output of the FM modulator is a passband CPM signal denoted by a notation similar to [44, 45] and is given by

sˆ(t)=Re 2E/T exp(j[2πf t + φ( t, γ˜)+ φ ]) . (3.12) { c 0 } p

SC/FDE - Buzid 3. Signal Envelopes Choices 36

1.2

0.8

0.6

0.4

Amplitude U=0

0.2 U=0.4

U=0.25 0 U=.5 U=0.1

−0.2 0 0.2 0.4 0.6 0.8 1 t / T

Figure 3.6.: Pulses suitable for the MSK.

0 10 MSK SFSK

−1 10 Power in dB

−2 10

0.4 0.45 0.5 0.55 0.6 f T

Figure 3.7.: SFSK and MSK power spectral density.

SC/FDE - Buzid 3. Signal Envelopes Choices 37

s(t,) Txfilter FM-modulator g(t) X

2πh fc

Figure 3.8.: Schematic CPM modulator.

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

∞ φ ( t, γ˜ ) = 2π ~ γ q(t iT ). (3.13) i − i=0 X

The elements of the vectorγ ˜ = ...,γ− ,γ− ,γ ,γ ,... are independent and identically { 2 1 0 1 } distributed symbols from an M-ary information set, γi 1, 3,..., (M 1) with 1 ~ q` ∈ {∓ ∓ ∓ − } probability M . = p`, represents the modulation index. The phase pulse or slope function q(t) with the restriction:

0 t< 0 q(t)= (3.14) M−1 t LT ( 2 ≥ is related to the normalized frequency pulse g(t) as

t q(t)= g(τ) dτ (3.15) −∞ Z

(M−1)π and the maximum phase changes over any symbol interval is 2 . g(t) determines the smoothness of the transmitted information carrying phase. As discussed earlier, the phase continuity of a CPM signal imposes a memory on the system. The memory denoted by L is the number of symbols scanned by the frequency pulse and 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 complexity. But on the other hand the benefits are crucial in wireless communications. As a low-cost and non-linear power amplifier can be employed, CPM has been an attractive modulation scheme [44, 65, 66]. Let’s recall the baseband of the CPM signal given in equation (3.12),

2E s(t, γ˜)= ejφ(t,γ˜). (3.16) r T

SC/FDE - Buzid 3. Signal Envelopes Choices 38

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

φ(t, γ˜) = 2π~ γkq(t kT ) nT t (n + 1)T (3.17) −∞ − ≤ ≤ kX= n−L n = 2π~ γ +2π~ γ q(t kT ) k k − k=−∞ k=n−L+1 X n X

= θn +2π~ γkq(t kT ), − − k=Xn L+1 where − 2π n L θn = q` γk mod 2`p (3.18) p` ! ! Xk=0 can take onlyp ` possible distinct values. θn represent the accumulated phase of a CPM signal at the time interval [nT, (n + 1)T ], due to the data up to t = (n L)T and demonstrate the signal memory introduced by the modulation. Thep ` diﬀerent− values are 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 the≤ signal given by equation≥ (3.17) as

1 ∂φ(t) n ∂q(t nT ) = ~ γk − (3.20) 2π ∂t − ∂t k=Xn L+1 and the instantaneous frequency of a full response CPM signal is given as dq(t nT ) f(t)= f + ~γ − nT t< (n + 1)T. (3.21) c k dt ≤ Assuming the phase is a linear function, thus M 1 f(t)= f + ~γ − nT t< (n + 1)T. (3.22) c k 2T ≤ ~ + 1 Now if =1/2 and γn 1 , the two frequency components are f (t) = fc + 4T for − ∈ {±1} γn = +1 and f (t) = fc 4T for γn = 1. The maximum frequency deviation then is 1 − − 4T . Further, the instantaneous phase and the instantaneous frequency of a full response CPM signal for M=2, M=8 and ~ =1/2 are plotted in Figures 3.9, 3.10, 3.11 and 3.12 for both REC and RC frequency pulses. One can note from the ﬁgures the behavior of the CPM signals phase and frequency. Moreover, the instantaneous frequency when M=8

SC/FDE - Buzid 3. Signal Envelopes Choices 39

1/4T Instantaneous phase Instantaneous frequency

dφ(t)/dt (t)/dt φ 0 (t), d φ

φ(t)

−1/4T 0 5 10 15 20 25 30 t / T

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

Instantaneous phase Instantaneous frequency 7/4T

5/4T φ(t) 3/4T dφ(t)/dt 1/4T (t)/dt φ 0 (t), d φ −1/4T

−3/4T

−5/4T

−7/4T

0 5 10 15 20 25 30 t / T

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

SC/FDE - Buzid 3. Signal Envelopes Choices 40

5/4T Instantaneous phase Instantaneous frequency

3/4T

dφ(t)/dt

1/4T (t)/dt φ 0 (t), d φ

−1/4T

φ(t)

−3/4T

−5/4T 0 5 10 15 20 25 30 t / T

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

100 Instantaneuos phase 7/4T Instantaneous frequency

5/4T

3/4T

φ(t) 1/4T (t)/dt φ 0 (t), d φ −1/4T

−3/4T

−5/4T dφ(t)/dt −7/4T

0 5 10 15 20 25 30 t / T

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

SC/FDE - Buzid 3. Signal Envelopes Choices 41

−7 −5 −3 −1 1 3 5 7 oscillates between the values 4T , 4T , 4T , 4T , 4T , 4T , 4T and 4T , as shown in Figures 3.10 and 3.12. Furthermore, in order to determine the instantaneous phase of the received signals during any symbol interval, it is necessary to keep a record of the previous phase status. However, the instantaneous frequency of the received signal can be determined without the phase tracking. Therefore, the data sequence can be recovered from the instantaneous frequency of the received signal instead of the instantaneous phase. This property 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 and bandwidth efficiency of the CPM signal. Since the phase contribution of each individual symbol is constant and finite, the following holds ∞ q(t)= g(t)dt = constant. (3.23) −∞ Z In contrast to linear modulation schemes (non-constant envelope), where the pulse spans one symbol, q(t) spans one or more data symbols in a CPM scheme. The most common pulses are presented in the following sections.

Rectangular Pulse: • A rectangular frequency pulse (REC) with an L symbols duration is given as 1 0 t LT g(t)= 2LT ≤ ≤ (3.24) (0 otherwise. Letting L=1, the obtained modulation scheme is MSK. However, the rectangular pulse is employed only with L = 1 in practice. Further, q(t) is the integral of the frequency pulse g(t) and given by t 0 t LT q(t)= 2LT ≤ ≤ (3.25) (0 otherwise. 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 a wide range of applications in practice. The RC pulse is deﬁned by 1 1 cos( 2πt ) 0 t LT g(t)= 2LT − LT ≤ ≤ (3.26) (0 otherwise. The integral of g(t) is given as t + 1 sin( 2πt ) 0 t 2LT q(t)= 2LT 4π LT ≤ ≤ (3.27) (0 otherwise.

SC/FDE - Buzid 3. Signal Envelopes Choices 42

g(t) q(t) 0.5 0.5

t t 0 1 2 3 0 1 2 3 3RC

g(t) q(t) 0.5 0.5

t t -3 -1 1 3 -3 -1 1 3 3SRC

g(t) q(t) 0.5 0.5

t t -3 -1 1 3 -3 -1 1 3 TFM

g(t) q(t) 0.5 0.5

t t -3 -1 1 3 -3 -1 1 3 GMSK(BT=0.25)

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

SC/FDE - Buzid 3. Signal Envelopes Choices 43

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

1 sin( 2πt ) cos( 2πtβ ) g(t) = LT LT , 0 β 1. (3.28) 2LT 2πt 1 ( 4βt )2 ≤ ≤ LT − LT Tamed Frequency Modulation Pulse: • The TFM pulse is deduced from the nature of the TFM as discussed previously and is given by 1 g(t) = [ag (t T )+ bg (t)+ ag (t + T )], a =1,b =2 (3.29) 8 0 − 0 0 πt πt 2 πt 1 2 cot( T ) ( T ) g (t) = sin( ) − 3 − . o T πt − 24πt " T 2 #

Gaussian Pulse: • The Gaussian pulse can be expressed as

t T t + T g(t) = Q 2πB − 2 Q 2πB 2 0 BT 1, (3.30) " √ln2! − √ln2!# ≤ ≤ where ∞ τ 2 Q(t) = exp( )dτ. (3.31) − 2 Zt Contrary to the other type of pulses, the duration of the Gaussian pulse is determined by BT instead of the parameter L, and the pulse is truncated since it has an inﬁnite duration [30], where B represents the system bandwidth. Practically, the Gaussian pulse has been employed by a number of systems. For instance, GMSK scheme which applies Gaussian pulse, has been employed in GSM, GPRS, DECT and DSC1800. In GSM, BT = 0.3 implies the pulse duration is approximated to 4 symbols, which is equivalent to L=4. Both phase and frequency pulses are also depicted 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 variation of the carrier frequency that results from the changing modulating signal amplitude. Furthermore, ~ plays a signiﬁcant role in characterization and determination of a CPM system performance as well as its complexity. It is worth mentioning that for practical ~ q` purposes, is chosen to be a rational of the form p`, whereq ` andp ` are relatively prime

SC/FDE - Buzid 3. Signal Envelopes Choices 44 positive integers. Otherwise, the number of phase states of the CPM system rises and consequently the receiver becomes complex [67]. Nevertheless, the pulse shape and its duration L, the modulation index ~ and the size of the information alphabet M together determine the complexity and types of the CPM scheme [67]. However, due to the constraints 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 determined by both L and ~. The phase states or phase trajectories (phase≤ ≤ paths) which represent the signal elements [45] address the signal in any time interval uniquely. The possible phase 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) isp ` and the terminal phase states for a full response CPM signal at time instant t = nT are denoted by Θ and given whenq ` is even as πq` 2πq` (`p 1)πq` Θ= 0, , ,..., − (3.32) p` p` p` and whenq ` is odd as

πq` 2πq` (2`p 1)πq` Θ= 0, , ,..., − . (3.33) p` p` p` 2. Partial response CPM (L> 1); In this case the phase depends on both the current and the L 1 previous symbols. Hence, the number of states for a partial response CPM signal− at time instant t = nT is given as

pM` L−1 evenq ` κ = L−1 (3.34) (2`pM oddq. `

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

cˆn =[θn,γn,...,γn−L+1] , (3.35) which is a combination of the accumulated phase and the correlative state. The correlative state is inﬂuenced by the previous L-1 symbols (phase transition factors γ).c ˆn describe and address the signal elements in the interval nT t (n +1)T . The collection of all possible phase trajectories forms the phase tree,≤ and≤ correspondingly, there is a collection of possible frequency deviations. A collection of both, the instantaneous phase and frequency of the CPM signal for an embedded arbitrarily 8-ary sequence of symbols of length 4 and RC frequency pulse with a duration L=1 are plotted in Figures 3.14 and 3.15. Figure 3.14 represents the phase tree and shows the time variant phase at each interval. However, Figure 3.15 shows that in full response CPM, the signal in any time interval can also be addressed by only M instantaneous frequency deviations, and the ﬁgure hints also to the structure of the receiver.

SC/FDE - Buzid 3. Signal Envelopes Choices 45

0 (t) in degrees

φ −20

−40

−60 0 1 2 3 4 5 6 7 t / T

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

2/T

7/4T

5/4T

3/4T

1/4T

−1/4T

Instantaneous frequency −3/4T

−5/4T

−7/4T

−2/T 0 1 2 3 4 5 6 7 t / T

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

SC/FDE - Buzid 4. Demodulation of CPM

This chapter deals with CPM scheme’s error performance. The signal spectrum analysis is extensively covered in literature [26, 57] but disregarded in the thesis. The optimum demodulation of the linear type modulation was reviewed in the previous chapter. In this 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 the optimum detection of a simpliﬁed and linearized CPM is detailed.

4.1. CPM Receiver Complexity

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

δ = s (t) s (t) (4.1) CPM k i − j k NT 1/2 = [s (t) s (t)]2 dt . i − j Z0

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

2ξ NT δ2 = s 1 cos[φ(t, γ˜ ) φ(t, γ˜ )] dt, (4.2) CPM T { − i − j } Z0 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 the diﬀerence ∆ between the M-ary data and− hence

2ξ NT δ2 = s 1 cos[ψ(t, ∆)] dt (4.3) CPM T { − } Z0 2ln(M)ξ NT = 1 cos[ψ(t, ∆)] dt, T { − } Z0 4. Demodulation of CPM 47 where ∆ 2, 4, 6,..., 2(M 1) . The error probability for CPM in terms of Euclidean∈ distance {± ± is [57]± − } 2 ξsδCPM PCPM = Q . (4.4) s N0 

The possible number of CPM signal states κ given in equation (3.34) in the interval iT t (i+1)T dictates the complexity of the demodulator, which compares κ possible ≤ ≤ 2 Euclidean distances to select a minimum δCPM [68]. For this purpose a trellis structure (Viterbi trellis) is employed and realized by the Viterbi algorithm [69]. Though the 2 Viterbi algorithm searches for the minimum δCPM in the optimal sense, it is unpractical for a higher order trellis structure, which increases exponentially with the system memory L. 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 channel is 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)

N0 where n(t) is a Gaussian noise with double side power spectrum density 2 and s(t, γ˜) is the transmitted signal given in equation (3.16). In the absence of the noise, the received signal is given as 2E r(t, γ˜)= ejφ(t,γ˜). (4.6) r T The received signal is ﬁrst passed through a limiter that ﬁxes the signal amplitude

n(t)

CPM s(t , ) r(t , ) CPM transmitter + receiver

Figure 4.1.: CPM transmission model. without affecting the signal phase, then applied to a differentiator whose output is given as ∂r(t, γ˜) ∂φ(t, γ˜) = j ejφ(t,γ˜). (4.7) ∂t ∂t The above equation reveals that the data symbols are embedded in the envelope of the differentiator output. An envelope detector typically used for detecting a PAM signal is used in the case of a full response CPM. A modulus circuit can be utilized instead, when the M-ary information set γi 1, 3,..., (M 1) is transformed via the relation 1 ∈ {∓ ∓ ∓ − } 2 (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 in Figure 4.3. The figure compares 4-ary and 8-ary full response, with an RC pulse shape

SC/FDE - Buzid 4. Demodulation of CPM 48

g% g r(t,r(t,) ) Differentiator Matched filter | . | ˆ

Figure 4.2.: M-level CPM simple receiver.

+ + + + + + + -1 + +

+ -2 + M=8 + + -3 + + + Log(BER) M=4

-4

-5

0 5 10 15 20 25 30

E/NindBb 0

Figure 4.3.: Uncoded BER for M-level CPM demodulator, full response and an RC pulse shape, 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 of the instantaneous frequencies of the same signal is plotted in Figure 4.5. From the figures one can predict the receiver complexity and conclude that the receiver shown in Figure 4.2 can not be used for the partial response case. Further, the optimum detection of 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 over the last two decades. A complexity reduction was first proposed in [57, 70] by the use of a slightly mismatched receiver. By using shorter frequency pulses at the receiver than the 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 the system memory. Additionally, the number of the linear filters at the receiver is reduced as a result of underlying system memory reduction, as will become clear in the following sections. Rimoldi [71] showed that any CPM system can be decomposed into a continuous-phase encoder (CPE) and a memoryless modulator (MM). Huber in [72] exploited the dimen-

SC/FDE - Buzid 4. Demodulation of CPM 49

0 (t) in degrees φ −20

−40

−60 0 1 2 3 4 5 6 7 8 t / T

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

SC/FDE - Buzid 4. Demodulation of CPM 50

2/T

7/4T

5/4T

3/4T

1/4T

−1/4T

−3/4T Instantantenous frequency

−5/4T

−7/4T

−2/T 0 1 2 3 4 5 6 7 8 t / T

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

SC/FDE - Buzid 4. Demodulation of CPM 51 sion of the CPM signal space aiming to reduce the complexity of CPM receivers, and showed that four to six linear filters can be sufficient to implement almost all CPM receivers of interest in practice. However, Laurent’s representation (LR), introduced more than two decades ago, is a way to circumvent this problem. Laurent [73] demonstrated that any binary CPM signal can be represented linearly by a finite number of time limited amplitude modulated pulses. He showed the exact number of the pulses of which each represents a linear receiver filter.

4.2. Decomposition of CPM

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

Txfilter PM-modulator s(t,) q(t) X 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 exhibits an inherent encoder that resembles in many ways a convolutional encoder. In order to reduce the number of states which have to be traced in the receiver, the virtual encoder can be cascaded with an external convolution encoder [45]. Let’s recall equation (3.13) ∞ φ ( t, γ˜ ) = 2π ~ γ q(t iT ). (4.8) i − i=0 X The phase trellis of the physical phase φ(t, γ˜) for MSK, plotted again for comparison in Figure 4.8(a), appears to be time variant in the sense that the physical phase trajectories

s(t,) CPE Memoryless Modulator

Figure 4.7.: Decomposite CPM modulator.

SC/FDE - Buzid 4. Demodulation of CPM 52 in the even-numbered symbol intervals are not time shifted to those in the odd-numbered symbol intervals. However, adding the lowest possible phase trajectory to the physical phase, 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 Figure 4.8(b).

16 20

14 15

12 10

8 0 (t) in degrees φ (t) in degrees

φ 6

−10 4

−15 2

−20 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 t / T t / T

(a) (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.

i−L i

ψ ( t, γ˜ )=2π~ γk + 2π~ γkq(t kT ) + π~ (M 1))t/T, (4.10) −∞ − − − kX= k=Xi L+1 which written in terms of χ as

i−L i

ψ ( t, χ˜ ) = 2π~ χk + 2π~ χkq(t kT )+ (4.11) −∞ − − kX= k=Xi L+1 i π~ (M 1)t/T + π~(M 1)(L 1) 2π~(M 1) q(t kT ), − − − − − − − k=Xi L+1 where χ 0, 1, ..., M 1 . The CPM tilted-phase decomposition in equation (4.11) is illustrated∈ in{ Figure 4.9.− However,} it is found in this dissertation that the time invariant

SC/FDE - Buzid 4. Demodulation of CPM 53 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 and the rest of the equation is data independent.

4q(t-nT)πh X 4q(t-(n-1)T)πh X

4q(t-(n-L+1)T)πh X 2πh s(t) c T c T T c + T X + PM i i-1 i-l+1

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

4.3. Linear Representation of CPM

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

L−1 i−L 2E (jπ~ P γ ) (j2π~γ − q(t−(i−k)T )) s(t, γ˜) = e k=−∞ k e i k . (4.12) T r k=0 Y

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

~ − − sin(π~ 2π~q(t (i k)T )) ej2π γi−kq(t (i k)T ) = − − − + (4.13) sin(π~)

~ sin(2π~q(t (i k)T )) ejπ γi−k − − . sin(π~)

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

SC/FDE - Buzid 4. Demodulation of CPM 54

Let ψ(t) denote the generalized phase-pulse function that has nonzero values only for 0 t 2LT , given as ≤ ≤ 2π~q(t) 0 t

0.9

0.8

0.7

0.6 q (t) 0.5

Amplitude 0.4

0.3

0.2

0.1

0 0 0.5 1 1.5 2 t / T

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

It is therefore convenient to deﬁne the signal pulse as sin(ψ(t + iT ) s (t) = (4.15) i sin(π~)

= s0(t + iT ). Substituting equations (4.13) and (4.15) into equation (4.12), the baseband CPM signal is expressed as L−1 − 2E (jπ~ Pi L γ s(t, γ˜) = e k=−∞ k s (t +(k + L i)T )+ (4.16) T 0 − r k=0 ~ Y ejπ γk−i s (t (k i)T ). 0 − − The outcome of the product in the above equation are 2L terms, where a detailed analysis of 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 be expressed as 2L−1−1 2E ~ s(t, γ˜)= ejπ ak,i c (t iT ), (4.17) T k − r i k=0 X X

SC/FDE - Buzid 4. Demodulation of CPM 55 where the pulses c (t), for 0 k 2L−1 1, are deﬁned as k ≤ ≤ − L−1 s0(t) i=0 s0(t +(k + Lβk,i)T ) 0 t T min L(2 βk,i) i ck(t)= ≤ ≤ × { − − } (4.18) 0 otherwise. ( Q These functions accordingly have the following durations

c0(t) (L + 1)T (4.19) c (t) (L 1)T 1 − c (t),c (t) (L 2)T 2 3 − c4(t),c5(t),c6(t),c7(t) (L 3)T . −. . . c (t) ,c − T. M/2 ··· M 1 The coeﬃcients in the exponent in equation 4.17 are determined by

i L−1 a = γ γ − β (4.20) k,i m − i m k,m m=−∞ m=1 X X L−1 = a γ − β 0,i − i m k,m m=1 XL−1 = a − + γ − (1 β )+ γ , 0,i L i m − k,m n m=1 X where the βk,n are either 0 or 1 and are the coefficients in the binary representation of the index k, i.e., L−1 k = 2m−1β , k =0, 1,..., 2L−1 1. (4.21) k,m − m=1 X A detailed look at equation 4.17 reveals that the exponential term represents a weighting factor and varies with the transmitted data. The input bits are coded via the ak,i then th linearly modulated onto fixed and data independent pulses ck(t) for the k component. Therefore, the CPM signal is alternatively composed of 2L−1 superimposed, amplitude modulated 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< 5T k =1 ⇒ β =1, β =0, β =0 ⇒ c (t)= s (t)s (t)s (t)s (t) 0 ≤ t< 3T ⇒ 1,1 1,2 1,3 ⇒ 1 0 2 3 5 ≤ k =2 β2,1 =0, β2,2 =1, β2,3 =0 c2(t)= s0(t)s1(t)s3(t)s6(t) 0 t< 2T k =3 ⇒ β =1, β =1, β =0 ⇒ c (t)= s (t)s (t)s (t)s (t) 0 ≤ t< 2T ⇒ 3,1 3,2 3,3 ⇒ 3 0 3 5 6 ≤ k =4 β4,1 =0, β4,2 =0, β4,3 =1 c4(t)= s0(t)s1(t)s2(t)s7(t) 0 t

SC/FDE - Buzid 4. Demodulation of CPM 56

1 c (t), L=4 0 0.9 c (t), L=4 1 0.8

0.7

0.6

0.5

Amplitude 0.4

0.3

0.2

0.1

0 0 1 2 3 4 5 t/T

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

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 Laurent principles. According to Mengali, the M-ary symbols are mapped into the binary bits and each bit of the n-tuple is linearized by LR principles. Furthermore, an M-level CPM signal is the product of the n signals. However, in practice, multi-level CPM signals have not been attractive and rarely been deployed.

SC/FDE - Buzid 4. Demodulation of CPM 57

4.3.2. The Importance of Linear Representation

The beneﬁt of the linear form of the CPM is the considerable simplicity of handling this form 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 of the eﬀect of the CPM memory at the same time.

It converts the CPM memory from the phase function to the pulses lengths and • filter structure of ak,i. As it is noticeable from equation (4.20), the ak,i have both a feed-forward FIR filter and a feedback IIR filter structure capturing the memory effect 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, the • CPM 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 signal • components 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, the resultant pulse c0(t) is the well known half sine wave which evolved originally from a rectangular frequency pulse [73] πt c (t) = sin( ) 0 t 2T. (4.22) 0 MSK T ≤ ≤ Similarly, GMSK is obtained by choosing a Gaussian frequency pulse. MSK and GMSK are special cases of CPM. MSK is a special case of the sub-family of CPM known as CPFSK which is deﬁned by a rectangular frequency pulse (i.e. its phase pulse is linear) of one symbol-time duration (full response signaling). On the other hand, if the tilted angle is set to π/4 for 0 t T , the only pulse for a duration of 2T is given by [73] ≤ ≤ 1 c0(t)OQPSK = , 0 t 2T (4.23) √2 ≤ ≤ and the CPM signal becomes an OQPSK signal.

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

SC/FDE - Buzid 4. Demodulation of CPM 58

4.3.4. Linear Representation Approximation

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

a0,n PSK C(t) 0 s(t,) n(t) r(t)

a1,n PSK C(t) 1 + +

aK,n PSK C(t) K

Figure 4.12.: CPM is composed of superimposed, amplitude modulated and time limited pulses. 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 from a bank of matched filters, as shown in Figure 4.13, presents sufficient statistics for making a decision on which of thep ` signals was transmitted [26]. It is important to note that a decision at each time interval nT requires the knowledge of all possible a ; k =0, 1,..., 2L−1 1 . These in turn depend on the current symbol γ and a state k,n − n defined by the vector a − ,γ − ,...,γ − ,γ − . As a consequence, a trellis decoder 0,n L n L+1 n 2 n 1 is imposed for the detection{ process at the receiver} and a decision can be made using the Viterbi algorithm (VA). The complexity of the VA is proportional to the number of states κ given in equation (3.34). However, Kaleh in [75] explicitly showed that the complexity of this receiver can be reduced by considering a smaller number of PAM components (Laurent pulses). He further showed that for a GMSK with a BT = 0.25 and a 4T wide approximation of the Gaussian pulse (L= 4) two pulses approximation is 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) to the power of the pulse c1(t), called conventionally signal to noise ratio (S/N), is plotted versus ~ in Figure 4.14 for various values of BT. As the signal is approximated by ignoring c1(t), its contribution to the signal power appears as a noise, that causes an error. The figure shows this error increases with the modulation index and decreases as BT increases. Further, at a practical value of ~ = 0.5, the signal to noise ratio equals 25 and 35 dB at practical values of BT = 0.3 and 0.5, respectively. The conclusion

SC/FDE - Buzid 4. Demodulation of CPM 59

C(-t) 0

r(t) C(-t) Outputdata

1 Viterbi Algorithm C(-t) K

Figure 4.13.: Optimum linear CPM receiver. from the ﬁgure is that at the practical values of both BT and ~, considering only the primary Laurent pulse and ignoring the rest ensures a good signal approximation without system performance degradation. Another interesting frequency pulse, given in equation (3.8) and discussed in Chapter 4, is TFM. Based on a TFM frequency pulse, the approximation 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 the TFM 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 a suboptimal reduced-complexity receiver, whose complexity is reduced dramatically, can be achieved. Accordingly, equation 4.17 is approximated and the linear CPM signal is then expressed as ∞ s (t, γ˜) = 2E/T a c (t nT ), (4.25) 0,n 0 − n=0 p X which in terms of the input data is given as ∞ ~ n s (t, γ˜) = 2E/T ejπ Pm=−∞ γm c (t nT ), (4.26) 0 − n=0 p X 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 response CPM. 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 as follows:

The pre-modulation filter (shaping filter), whose impulse response is denoted by • c0(t), scans (L+1) symbols. The filter produces ISI in order to obtain continuous phase modulation signals. As stated earlier, c0(t) is derived from various frequency pulses 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 emphasized as follows:

SC/FDE - Buzid 4. Demodulation of CPM 60

50 BT=0.7 BT=0.5 40

X 35dB 30

X 25dB 20 BT=0.1

BT=0.3

S/NdB 10

-10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Modulationindex

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

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

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 approximated and linearized CPM is the significant reduction of receiver complexity in addition to the bandwidth efficiency of this system. On the other hand, the error performance suffers a degradation as a result to the differential encoder [26].

4.3.5. Precoding

The term ”precoding” refers to the pre-equalization which requires the knowledge of the channel state information (CSI) [78, 79, 80]. However, in this chapter precoding is restricted to data encoding at the transmitter and is isolated from channel inﬂuences, as demonstrated previously [41, 81]. To further illustrate the diﬀerential encoding associated with the CPM signal, the linearized form of the signal equation (4.26) is recalled: ∞ n jπ/2 P γi s(t, γ˜)= 2E/T e i=0 c (t nT ). (4.27) 0 − i=0 p X

SC/FDE - Buzid 4. Demodulation of CPM 61

40 TFM

GaussianpulseBT=0.5

S/NdB 10

-10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Modulationindex

Figure 4.15.: A comparison between TFM and Gaussian pulse (BT = 0.5). r(t) Outputdata C(-t)0 VA

Figure 4.16.: Simple linear optimum CPM receiver.

n jπ/2 P γi It is apparent that the exponential term e i=0 in equation (4.27) represents an inherent encoding. As the phase state is a function of all past symbols, the encoder has a diﬀerential structure and the CPM depends on the phase diﬀerences rather than the absolute values. In the following, the exponential term is more detailed and written also in the form [41] n n n jπ/2 P γi P γi =0 =0 e i =(j)i = j γi (4.28) i=0 Y which is equivalent to

n−1

an = jγn j γi (4.29) i=0 Y = jγnan−1. The above equation represents a 2-bit diﬀerential encoder. The decoding equation follows to an γn = j . (4.30) − an−1

SC/FDE - Buzid 4. Demodulation of CPM 62

Since α 1, +1, j, +j it follows that n ∈ {− − } ∗ γ = ja a − . (4.31) n − n n 1 Figure 4.17 illustrates both the differential modulator and demodulator which is an implementation of equations (4.29) and (4.31). The figure demonstrates the inherent differential encoding and the requirement for differential decoding at the receiver. Be- cause of this the system suffers a small performance degradation, which is a direct implication 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 in return avoids error probability degradation resulting from application of the differential encoding [26]. Nevertheless, from a spectral standpoint, the precoding operation has no effect on power spectrum density of the transmitted signal [82]. In what follows, the demonstration of how the differential decoding at the receiver is replaced by a differential encoding at the transmitter is presented [81]. Suppose the transmitted symbols z 1, 1 are constructed from the source data i ∈ {− }

j α α -j γ i i i γi

T T

a) b)

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

γ 1, 1 as follows: i ∈ {− } zi = γiγi−1 (4.32) and substituting in equation (4.28), given γ−1 =1

n n an = j zi (4.33) i=0 Yn n = j γi.γi−1 i=0 n Y = j γ0γ−1.γ1γ0....γnγn−1 n−1 n = j γnγ−1 γi.γi i=0 n Y = j γn.

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

−n γn = j an, (4.34)

SC/FDE - Buzid 4. Demodulation of CPM 63

−1

−2 Log(BER) −3

−4 non−differential GMSK, BT=0.2 differential GMSK, BT=0.2 non−differential GMSK, BT=0.5 differential GMSK, BT=0.5

−15 −10 −5 0 5 10 15 Eb/N in dB 0

Figure 4.18.: Uncoded BER versus S/N for differential and non-differential CPM signals with a Gaussian frequency pulse and various BT values in an AWGN channel. which, in fact, represents only a de-rotation of the received symbols. The meaning of the above equation is that the transmitted data, which is differentially encoded prior to the transmission, is recovered simply by a de-rotation process, and the degradation of system 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 precoded CPM signals, BER Monte Carlo simulations were conducted in an AWGN channel. In the simulation, the synchronization was assumed to be ideal. The comparison covers, in particular, the differential linearized CPM and the non-differential linearized coherent CPM with Gaussian pulse for various values of BT . Figure 4.18 compares error performance of both the differential and non-differential cases, where a Gaussian frequency pulse with BT values of 0.2 and 0.5 is used. From the figure it can be noted that non-differential CPM shows some improvement in lower signal to noise ratios. However, the two approaches tend to show similar performance at higher signal to noise ratios. Furthermore, the linearization principle is also extended to the TFM which has been discussed in Chapter 3. Error performance of the obtained system, a non-differential linearized CPM based on a TFM, which is conventionally called 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 4. Demodulation of CPM 64

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

−1

−2

−3

Log(BER) −4

−5

non−differential GMSK −6 differential GMSK non−differential TFM differential TFM

−7 −15 −10 −5 0 5 10 15 Eb/N in dB 0

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

It can be noted that a linearized TFM performs better than the others when the BT of the Gaussian pulse is equal to 0.3, which is equivalent to L=4. The comparison is repeated and a similar performance is obtained when BT is equal 0.5, as shown in Figure 4.20.

4.5. Summary

In the last two chapters, various signal envelopes were reviewed and a basis for sorting systems with respect to the transmitted signals envelope is set. The generation and reception of constant envelope signals were presented. In this chapter, the CPM demodulation was treated in detail, since it is the central theme of the coming chapters, in which the broadband techniques for multipath channels are the objective. Additionally, in this chapter non-differential CPM signals are developed from the differential CPM signal which suffer from error propagation and some performance penalty due to the phase continuity of the CPM signals. The non-differential signals show an improved performance as a direct result to the elimination of the differential decoding

SC/FDE - Buzid 4. Demodulation of CPM 65

−1

−2

−3

−4

−5 Log(BER)

−6

0 non−differential GMSK differential GMSK non−differential TFM 0 differential TFM

−15 −10 −5 0 5 10 15 Eb/N in dB 0

Figure 4.20.: Uncoded BER versus S/N for diﬀerential and non-diﬀerential CPM with a Gaussian pulse (BT = 0.5) and a linearized TFM signal, in an AWGN channel. and the avoidance of error propagation.

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

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

5.1. Introduction

The single carrier transmission with frequency domain equalization concept is basically a 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. the transmitted signal reaches its destination over many paths. The received signal is then a summation of all incoming signals which are attenuated and time delayed in accordance with the individual paths. The superposition can be destructive or constructive and thus the received signal is a distorted copy of the original. The extraction of the data from the distorted signal is, therefore, an impossible task without the use of some powerful measures or techniques. The SC/FDE technique is an elegant and effective solution for mitigating impairments of the dispersive channel [5, 6]. The antonym of a single carrier system as the name may suggest is the multiple carrier system, and both are possible methods of transmitting data from one point to another over such an environment. The single carrier transmission system has been used to transmit data since man started transmission. However, the superiority of the SC/FDE concept lies in the equalization that mitigates the multipath distortions which is carried out in frequency domain. The consequence of the use of frequency domain equalization is the significant complexity reduction of the receiver. Equalization using the discrete frequency domain was first reported 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 wireless channels and has a lower complexity than time domain equalization due to its use of the computationally-efficient fast Fourier transform (FFT). One implication of the frequency domain equalization is the imposed block structure transmission. 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 67 (SC/FDE) 5.2. System Description

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

Scrambling Input Guard Up- Transmit Coding Mapping data interval sampling filter Interleaving

Pilots Channel estimation Synchronisation aallt serial to Parallel payload output Descrambling Down Matched Demapping IFFT Equalizer FFT Remove GI data Decoding N sampling filter 2N Deinterleaving

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 0.5 generation 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-schemes is investigated in chapter 6. The principle diﬀerences between SC/FDE and OFDM concepts 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, where they are made in frequency domain, as depicted in Figure 5.2. A comparison and the resulting 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/FDE system are reviewed and associations to the appropriate blocks in an IEEE 802.11a OFDM based system are given in each case. Both SC/FDE and OFDM systems are block based transmission schemes and the reason is twofold: ﬁrst to ease the process of

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 68 (SC/FDE)

FFT equalizer IFFT SC/FDE transmitter SC/FDE receiver

IFFT FFT equalizer

OFDM transmitter OFDM 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 guard interval is necessary to prevent the intersymbol interference (ISI) by freshing or clearing the channel memory. The ISI is a result of the multipath components of the wireless channel.

5.2.1. Baseband Equivalent Model

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

sˆ(t) = Re a(t)ej(2πfct+θ(t)) (5.1) = s (t)cos(w t) s (t)sin(w t), I c − Q c

where a(t) and θ(t) are bearing data information and fc is the carrier frequency. sI (t) and sQ(t) describe the quadrature components. Because of time consumption and simulation complexity, 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 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 69 (SC/FDE) given as

jθ(t) s(t) = a(t)e = sI (t)+ jsQ(t). (5.2)

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

5.2.2. Scrambling

Scrambling limits the ﬂuctuations of the OFDM signal envelope by avoiding long sequences of bits with the same value. In OFDM, long sequences of bits with the same value would implicate constructively superposition of subcarriers and hence the resulting envelope 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 most of todays communication systems. Although, FEC means adding redundancy, today, it is seldom to implement a communication system without the application of FEC techniques. The redundant transmitted data enable error correction at the receivers. Two most known encoders among others are the linear block encoder and the convolution encoder. Block convolution encoders show powerful error correcting capabilities. They have 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. In our investigations the convolution encoder, shown in Figure 5.3, as deﬁned in the IEEE 802.11a standard is used. At the receiver, the maximum likelihood principle is used to decode the convolutional codes. In order to simplify the decoding process, the Viterbi algorithm is widely used.

K=7 2 3 5 6 7-Stage Shift Register g0=1338=1+x +x +x +x

x0 x1 x2 x3 x4 x5 x6

1 2 3 6 Two Symbols are output for g1=1718=1+x +x +x +x 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 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 70 (SC/FDE) 5.2.4. Interleaving

Interleaving is designed to arrange the transmitted symbols in a manner that improves the system performance. In OFDM the technique interleaves the data over frequency such that, if any frequency fading null damages neighboring subcarriers, the damaged data can be recovered with the help of FEC. For the SC/FDE, a block by block based interleaver 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 modulated. 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 rate pairs 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/FDE system. the modulation step is carried out in two stages. In the ﬁrst stage, the data is mapped onto 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 into continuous time signal, for which a pulse shaping ﬁlter is used in the SC/FDE case.

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

Modulation type γ¯ Bits/symbol BPSK +1,-1 1 QPSK +1,-1,+j,-j 2 16-QAM 3, 1+ j 3, 1 4 ± ± {± ± } 64-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 the mean transmit power is the same for each modulation. From the above table, one

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 71 (SC/FDE) can conclude that BPSK, QPSK and QAM modulation schemes in the baseband are simply a data transformation or mapping process. That explains the interchangeability of mapping and modulation in most literature. The baseband equivalent transmit signal can be written as N−1 s(t) = γ¯ g(t iT ), (5.3) i − i=0 X where g(t) is the impulse response of the pulse shaping ﬁlter.γ ¯ are complex multilevel input data and can be obtained for each speciﬁc modulation uniquely as summarized in the following table.

Modulation type Remarks BPSK 2γ 1 γ 0, 1 − ∈ { } QPSK 2γ 1+ j(2γ 1) γ ,γ 0, 1 ,i,k =1, 2 i − k − i k ∈ { } 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 be reviewed 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 a signal spectrum with acceptable sidelobs level that fulfills the international bodies constraints and regulations, e.g. the Federal Communications Commission’s (FCCs) spectrum mask. A typical shaping pulse g(t), known as a Nyquist pulse, is given in time domain as sin(πt/T ) cos(πrt/T ) g (t)= , (5.4) rc πt/T 1 4r2t2/T 2 − where r is a roll-off factor with 0 r 1, that influences the pulse bandwidth which is limited to f < 1+r . The RC frequency≤ ≤ response is given by | | 2T 1−r T 0 f < 2T T πT 1−r 1−≤|r | 1+r Grc(f)= 1+ cos f f < (5.5)  2 r | |− 2T 2T ≤| | 2T 0 f 1+r . | |≥ 2T Typically in both, the transmitter and the receiver root raised cosine filters (RRC) are used: √T 0 f < 1−r ≤| | 2T T πT 1−r 1−r 1+r Grrc(f)=  1+ cos f f < (5.6)  2 r | |− 2T 2T ≤| | 2T 0q f 1+r . | |≥ 2T The cascade of the transimeter RRC pulse shaping filter and the receiver RRC matched  filter is again an RC filter.

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 72 (SC/FDE) 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, a good knowledge of the communication channel should be available [86]. Furthermore, each channel dictates its own communication system. In wireless systems, a channel affects and degrades the system performance via a multipath phenomena, besides the Gaussian noise [87]. A multipath channel is basically a bond of a number of traveling rays that reach the destination through a multiplicity of different paths, with each path having 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 the wireless channel impulse response.

n(t) Linear channel h (t) s(t) b 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 noise produced by the receiver front end as a result of imperfection of the electronics components of the receiver. It is white noise because of its uniform frequency content. While it is independent of the paths over which the signal is being received, its spectral density N0 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

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 73 (SC/FDE)

2 with zero mean and variance σn [89, 90]. The channel capacity C, which is deﬁned as the maximum data rate at which a reliable communication is possible, is given with 2 bandwidth B, signal power P , and noise variance σn by the Shannon relation

2 C = B log2(1 + P/σn).

5.3.2. The Mobile Radio Channel

The radio (wireless) channel is characterized by a superposition of a number of signals propagating over various paths, among which may be a direct path from the transmitter to the receiver (the line of site, LOS). The loss of a direct path is determined by the free space inverse square law, where the received signal power is given by [91] λ P = P ( c )2G G , (5.7) r t 4πd t r where Pt and Pr are the transmitted and received power, respectively. λc is the wavelength and d is the range separation. Gt, Gr are the power gains of the transmit and receive antenna, respectively [92]. The signal also arrives at the receiver over other paths resulting from reflection, refraction and scattering. The contribution of the individual paths to the overall channel can be constructive or destructive, which results in amplitude and phase fluctuations as well as a time delay in the received signals. The multipath channel is space and time dependent, and modelling the channel unique is an impossible task. This makes the statistical models of typical scenarios or different areas of applications of relevance. Channel models for the area of indoor communications have been developed by ETSI for the standard HiperLan/2 and by IEEE for the standard IEEE 802.11a.

Channel Impulse Response

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

ν−1 h (t)= c ejΩi δ(t τ ). (5.8) c i − i i=0 X 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 of these 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 to the channel is a reduction of the channel capacity and in severe cases the creation of signal outage and loss of connection [87]. The multipath channel can be characterized by the following parameters:

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 74 (SC/FDE)

Multipath delay spread • Because of the late arrival of signals over different paths, the total signal will be dispersed and broadened over the time. The dispersion time of the channel is called multipath delay spread τd. It is also defined as the difference in the propagation time between the longest and shortest paths. Considering only the paths with a considerable energy, τd may be given as

τd = max τi(t) τj(t) . i,j | − |

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

ν−1 PDF = c2δ(t τ ). (5.9) i − i i=0 X The PDF can be determined by sounding a wireless channel by an impulse function δ(t) [86]. A useful measure of the channel delay spread is the RMS delay spread τrms. It can be determined via the PDF and is given by

− ν 1 2 2 i=0 (τi τ¯) ci τrms = −− , (5.10) ν 1 c2 sP i=0 i where P ν−1 2 i=0 τici τ¯ = − . (5.11) ν 1 c2 P i=0 i Coherence bandwidth P • Analogous to the delay spread parameters in time domain, the coherence bandwidth (correlation bandwidth) Bc is another parameter to characterize the channel in the frequency domain [91]. Coherence bandwidth is defined as the channel bandwidth that passes all spectral components with roughly equal gain and linear phase. Further, the coherence bandwidth is inversely proportional to the RMS delay spread, i.e., 1 Bc α , (5.12) τrms where the proportional constant depends on the particular definition. Typical values 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 if Bc is small compared with the bandwidth of the transmitted signal. Otherwise the channel is frequency non-selective or flat and a single channel filter tap represents the channel.

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 75 (SC/FDE)

Doppler spread and coherence time •

Doppler spread Bd and coherence time tc are parameters needed to describe the channel that suffers from a Doppler effect. The Doppler effect is due to the motion of one or both ends of the wireless communication system. The maximum Doppler frequency is given by the ratio of the mobile speed υ to the carrier wavelength λc, i.e. [88] υ Bd = . (5.13) λc Coherence time is the time domain dual of Doppler spread. The Doppler spread and coherence time are inversely proportional to one another, i.e. 1 tc α , (5.14) Bd

A time selectivity or fast fading is experienced when tc is much less than the symbol time T. On the other hand, if the symbol duration of the transmitted signal is much less 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 the bands 2.45 GHz ISM and 5 GHz, the IEEE working groups 802.11a and 802.11b have established what is called the IEEE indoor channel model. The reason for this is the need to have a fair comparison between different wireless system concepts. The model is time invariant for the duration of a burst and meant to simulate the small scale fading effects as a tapped delay line defined by

ν−1 h = c δ(m n), (5.15) m i − n=0 X where the −jΨn cn = bne (5.16) are based on the following statistical model:

σ2 σ2 c = N(0, n ) + jN(0, n ) (5.17) n 2 2 − ∆τ 2 τrms σ0 = 1 e (5.18) − ∆ −n τ 2 2 τrms σn = σ0e . (5.19) Here ∆τ represents the multipath resolution time and its value can be given as ∆τ < 1 . 2 B σn τrms indicates the channel delay spread and N(0, ) is a zero mean Gaussian random 2 2 σn variable with a variance 2 . Based on the above equations, it can be concluded that the

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 76 (SC/FDE) 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 for independent− actual impulse responses is

2 σn =1. (5.20) n X 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 well as OFDM. However, the linear modulation formats (BPSK, QPSK and M-QAM) are straightforwardly applied to a block of data. Moreover, the received signal has to be processed 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 written as N−1 s(t)= γ¯ g(t (n + lN)T ) , (5.21) n,l − n=0 ! Xl X

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 77 (SC/FDE)

th where the data stream is grouped in blocks andγ ¯n,l is the n modulated symbol in the lth block of size N. T is the symbol duration and g(t) is a pulse form which may take many possible shapes. The symbol spaced discrete time model of the signal at the nth interval is given as

s (n)=¯γ for n [0,N 1] . (5.22) l n,l ∈ − The complete signal is constructed by the concatenation of all SC/FDE symbols and expressed as

s(n)= s (n lN). (5.23) l − Xl As shown in Figure 5.1, following the mapping process is the insertion of a guard interval between the successive blocks. The purpose of the guard interval is to preclude the interblock interference. In SC/FDE, likewise in OFDM, the duration of the guard interval, in order to be effective, has to be longer than the duration Th of the channel impulse response hb(t) [94]. In [21], a closed form expression for the optimum guard interval duration is derived, subject to a total average energy constraint on the information and the guard symbols. Further, a zero-padding (zero stuffing) and a cyclic prefix are the most common guard sequence types. In a cyclic prefix, a part of the information symbols is copied from the block tail to the guard interval. At the receiver, the guard sequences are discarded since they do not carry information. Recently, a known sequence called unique word (UW) has come into practice instead of the cyclic prefix or the zero padding [14, 16]. The UW pilots can be used for channel estimation and synchronization purposes, in addition to the clearing of the channel memory after each block 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 for synchronization and channel estimation which are postponed in this work. The preamble is composed of two sections. The contents of the first section is a copy of 10 repeated short training symbols each denoted by A (constant time domain envelope signals are most suitable). The section is dedicated to frame-, coarse timing-, coarse frequency synchronization and automatic gain control (AGC). The second section of the preamble consists of two identical long training symbols (pilot one and pilot two) and a large guard interval as shown in Figure 5.6. The second section is used mainly for fine frequency offset estimation/synchronization and for a channel estimation. More details regarding the preamble can be found in references [95, 17]. Furthermore, the payload which is composed of a number of blocks, represents mostly the user data.

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 78 (SC/FDE)

Figure 5.6.: Transmission burst with preamble for synchronization and channel estimation 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 symbols with duration TFFT = NT and the cyclic extension with duration TG, which is occupied by the last Ng data symbols copied in front of the block as depicted in Figure 5.7.

Block l-1 Block l

copy

CP data CP data CP

NgT NT

TFFT

UW data UW data UW

NgT (N-Ng)T

TFFT

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

If sl(t) with t [0,TFFT ] denotes the continuous time representation of the original ∈ th symbol sequence of the l block, then the extended block denoted bys ˆl(t) is given by s (t) t [0,T ] l ∈ FFT sˆl(t)= s (t + T ) t [ T , 0] (5.24)  l FFT ∈ − G 0 otherwise.

The over all transmit signal may then be written as s(t)= sˆ (t l(T + T )). (5.25) l − FFT G Xl

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 79 (SC/FDE)

It is essential to mention that the other benefit of the cyclic prefix is that the linear convolution of the lth transmit block with the channel impulse response becomes a circular convolution. This, in turn, reduces the receiver complexity [12, 16] and the received blockr ˆl(t) fulfills the condition rˆ (t)=ˆs (t) h (t)= s (t) h (t) (5.26) l l ∗ b l ⊗ b within the interval t [ T + T ,T ], where and indicate linear convolution and ∈ − G h FFT ∗ ⊗ circular convolution process, respectively. The restriction ofr ˆl(t) to the interval [0,TFFT ] contains exactly one period of the cyclically extended signalr ˆl(t) and is denoted byr ˜l(t) in the following. Applying the circular convolution theorem to the above equation results in R˜l(nf0)= Sl(nf0)Hb(nf0) (5.27) for f =1/T and n Z, where R˜ (f), S (f) and H (f) are related to the time domain 0 FFT ∈ l l b signalsr ˜l(t), sl(t) and hb(t) by the continuous Fourier transform. It is worth mentioning that Sl(f) appearing in the above equation is the spectrum of the non-cyclic extended th l transmitted block sl(t). This means, R˜l(nf0) appearing in the above equation is equivalent to Rl(nf0), which denotes the Fourier transform of the received block that would result from transmission of the original non-cyclically extended block sl(t) over the 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 to cyclically 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 zero stuffing or a cyclic prefix which are thrown away at the receiver, a unique word or a predefined and deterministic sequence fills the guard interval as illustrated in Figure 5.7. It fulfills the theorem of cyclic convolution in exactly the same way as the CP concept [14]. At the receiver, in addition to the protection of the user data from the IBI, the sequence is used advantageously in equalization, synchronization and channel estimation, 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 bandwidth 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 assigned bandwidth. Since part of the transmitted data is consumed by the system to establish a communication link (overhead sequences) and/or a guard interval to clear channel memory, the efficiency may be defined as Bit rate η = . (5.28) Bandwidth

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 80 (SC/FDE)

In an AWGN channel, where a guard interval is not compulsory, the eﬃciency for an M-ary system is given as ld(M) η = , (5.29) 1+ r where r is the roll-oﬀ factor. For a dispersive channel with a memory Th, over which a guard interval TG is imposed by the block transmission where TG Th, the achievable channel block throughput is ≥

ld(M) T η = FFT . (5.30) CP 1+ r T + T FFT G For the UW concept, the guard interval or guard sequence is processed at the receiver as part of the useful signal and hence, the discrete Fourier transform interval overlaps the guard interval TG. Therefore, the achievable channel block throughput is

ld(M) T T η = FFT − G . (5.31) UW 1+ r T FFT A simple comparison between the system eﬃciency of a CP and a UW concepts reveals

T 2 η /η =1 G , (5.32) UW CP − T FFT which shows that the UW approach is less eﬃcient than CP. However, the UW is found to 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 system bandwidth eﬃciency.

5.4.3. Optimum Linear Receiver

For the sake of simplicity the blockwise transmission is disregarded for the upcoming considerations. Moving forward to the receiver side in the frame of the system description, the received signal corrupted by an AWGN as shown in the simpliﬁed transmission model shown in Figure 5.8 can be written as

r(t) = γ¯ g (t nT )+ n(t), (5.33) n T − n X where gT is the transmitted ﬁlter impulse response,γ ¯n are the transmitted complex symbols and n(t) is the Gaussian noise. The impulse response of the matched ﬁlter gR(t) that maximizes the signal to noise ratio (S/N) is given as [26]

g (t)= Kg∗ ( t), (5.34) R T −

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 81 (SC/FDE)

n(t)

s(t) r(t) x(t) gT(t) + gR(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 complex conjugate operation. In frequency domain, the transfer function of the matched filter GMF (f) is also given as ∗ GMF (f)= GR(f)= KGT (f), (5.35) where GR(f) denotes the receiver filter transfer function. Furthermore, the Nyquist criterion is another important criterion that has to be fulfilled to prevent ISI. The criterion is that at the sampling instants or multiple of sampling instants, the interference has to be zero (ISI free). The filter with a raised cosine impulse response satisfies the Nyquist criterion and its frequency response Xrc(f) is given by Equation (5.5). The overall frequency response from the transmit filter input to the receiver filter output is then

GR(f)GT (f)= Xrc(f), (5.36) which implies G (f) = G (f) = X (f). | R | | T | rc p Therefore, a root raised cosine (RRC) filter with a transfer function Xrc(f) can be utilized for both the transmit and receive filters. p In a multipath channel, the received signal suffers from the multipath effects, in addition to an AWGN. A simple model describing this phenomena is shown in Figure 5.9 and the baseband received signal may be written as

r(t) = γ¯ g (t nT ) h (t)+ n(t) (5.37) n T − ∗ b n X = γ¯ h(t nT )+ n(t), n − n X where the operator represents the convolution operation. hb(t) is the complex valued baseband channel impulse∗ response and h(t) = g (t) h (t), whose frequency response T ∗ b H(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 on hb(t). In the receiver, as shown in Figure 5.9, the output of the matched ﬁlter in the

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 82 (SC/FDE)

n(t)

x(t) y(t) gT(t) s(t) r(t) gR(t) Equalizer hb(t) G (f) + GR(f) E(f) T Hb(f) X(f) Y(f)

Figure 5.9.: Simpliﬁed communication system model including a linear channel model and a linear equalizer. frequency domain can be written as

X(f) =γ ¯nGT (f)Hb(f)GMF (f)+ GMF (f)N(f) (5.38) ∗ =γ ¯nGT (f)Hb(f)GT (f)+ NG(f) =γ ¯ H (f) G (f) 2 + N (f), n b | T | G

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

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

Y (f) =γ ¯nXrc(f)+ Ne(f). (5.40)

This type of linear equalization is called Zero-Forcing equalizer, since the equalizer output is distortionless (ISI free). Ne(f)= GMF (f)N(f)/Hb(f) is the equalizer output noise spectrum, which may lead to a noise enhancement problem. At this point, it is important to emphasis that the frequency response of the equalizer given in equation (5.39) is continuous. A dramatical complexity reduction is obtained once 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 equalization. Traditionally, time domain equalization is commonly utilized in wired communication systems as well as digital radio microwave. Nowadays applications, known for the high data rates and the demand of a bit rate of tens or hundreds of megabits per second or more, enforce the substitution for such equalization, since the complexity of the 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 than linearly. In Figure 5.10 a system based on the frequency domain equalization principle

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 83 (SC/FDE)

s(t) n(t) r(t) Txfilter Rxfilter IFFT g(t)/G(f) h(t)/H(f) + Equalizer T T b b g(t)/G(f)R R Decision Receiver data EqualizerCofficients

Sampler FFT Matchedfilter Sampling T/ G(f) ratereduction Multiplier

Figure 5.10.: Communication system with optimum linear reception and frequency domain equalization.

on block basis is shown. A discrete form r(µTµ) of r(t) is obtained at a rate µ/T , where µ = T/Tµ > 1 is the oversampling factor and Tµ is the sampling period. µ takes integer values, preferably a power of two. An µN -point FFT transforms the µN samples of the received data block into the frequency domain [13, 12, 98], since only data symbols are considered and the samples corresponding to cyclic prefix are discarded. Let R[κ] denote the discrete frequency domain of the discrete sequence r[µTµ], therefore, µN samples are obtained and given in frequency domain as µN−1 R[κ]= r[υ]e−j2πυκ/µN . (5.41) υ=0 X The receiver filter is matched to the channel distorted transmitted pulse h(t) and its ∗ impulse response is given by gMF (t) = Kh ( t). Because the output of the matched filter 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 ISI corrupted transmitted symbols. To improve the system performance, a compensator or an equalizer that eliminates or minimizes the ISI is needed. For the purpose of deriving the equalizer coeﬃcients e(n), the digital receiver illustrated in Figure 5.10 can be equivalently represented by a symbol rate model as depicted in Figure 5.11, where the input to the equalizer is the T -spaced sampled signal given as

x[n]= γ¯ q[n m]+ z[n], (5.43) m − m X where q[n]=[h(t) g (t)] is the over all impulse response seen at the input of ∗ MF t=nT the equalizer, its frequency response is Q(f), and z(n)=[n(t) gMF (t)]t=nT denotes T -spaced sampled noise sequence at the output of the matched ﬁ∗lter. Minimizing the

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 84 (SC/FDE) error power between the transmitted and the received symbols is the principle on which the operation of the equalizer is based. The equalizer error denoted by ǫn is deﬁned as the diﬀerence between the transmitted symbolsγ ¯n and the equalizer output y(n) as depicted in Figure 5.11.

z[n]

x[n] q[n] + e[n] y[n]

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

Therefore, the error may be given as

ǫ =γ ¯ y(n). (5.44) n n − The mean power in the equalizer error denoted σ2 and expressed as

σ2 = E ǫ 2 (5.45) | n| may be computed in the frequency domain as

1/T 2 σ = ϕǫǫ(0) = T Φǫǫ(f)df, (5.46) Z0 where ϕǫǫ(0) represents the autocorrelation function and Φǫǫ(f) is the power spectral density of the equalizer error. From Figure 5.11, Φǫǫ(f) is derived as [12, 13]

Φ (f)= σ2 Q(f)E(f) 1 2 + Φ (f) E(f) 2, (5.47) ǫǫ d| − | zz | | 2 where σd 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

1/T σ2 = T σ2 Q(f)E(f) 1 2 + Φ (f) E(f) 2 df. (5.48) d| − | zz | | Z0 The ﬁrst term in the above equation describes the residual ISI power and the second is the output noise power [12, 13]. Two criteria have found widespread use in optimizing the equalizer coeﬃcients. One already mentioned is the ZF and the other is minimum mean square error (MMSE) [99]. Accordingly, two types of linear equalizers are commonly used: The ZF equalizer and the minimum mean square error (MMSE) equalizer.

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 85 (SC/FDE)

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 equalizer given by 1 E (f)= . (5.49) ZF Q(f)

It is essential to mention that the transfer function of the equalizer EZF (f) is simply the inverse filter to the linear filter model Q(f). Consequently, the use of a ZF equalizer may 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

1/T Φ (f) σ2 = T zz df. (5.50) ZF Q(f) 2 Z0 | | The implication of the noise enhancement is the system performance degradation in terms of the BER. At last, the output of the equalizer as depicted in Figure 5.11, given by X(f) Y (f)= X(f)E (f)= , (5.51) ZF Q(f) 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 the noise ampliﬁcation and the ISI reduction. An MMSE equalizer can be optimized via the

2 EMMSE(f) = argmin E ǫi . (5.52) E | | Since the integrand in (5.48) is non negative, the criterion reduces to the frequency wise criterion 2 2 2 σd Q(f)E(f) 1 + Φzz(f) E(f) min (5.53) | − | | | → E(f) at every frequency f. Applying the gradient method with respect to E(f) yields the MMSE equalizer frequency response [12]

∗ [Q(f)] EMMSE(f)= . (5.54) 2 Φzz(f) Q(f) + σ2 | | d Note, that the equalizer transfer function is real valued, since all terms appearing at the right side in the above equation are real valued, too. Hence, the complex conjugation in the numerator can be omitted. The additive term in the denominator of the expression appearing in the above equation protects against an inﬁnite noise enhancement, and

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 86 (SC/FDE) 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

1/T Φ (f) σ2 = T zz df. (5.55) MMSE 2 Φzz(f) 0 Q(f) + σ2 Z | | d

2 Comparing equations (5.55) and (5.50), it is obvious that σZF is always equal or greater 2 than σMMSE. 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, which processes 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 is defined as the ratio between the peak to the average power of the signal and for a signal defined on the interval [0,T ] given by

max s(t) 2 P AP R = | | . (5.56) 1 T s(t) 2 dt T 0 | | R An unmodulated carrier has a PAPR of unity or 0 dB. A large PAPR is a serious drawback to many systems and increases the complexity of the analog to digital and digital to analog converters. On the other hand, large PAPR makes them susceptible to system nonlinearities [17]. Indeed, in spite of its attractiveness and popularity for future wireless communications systems, OFDM may be outweighed by its handicap of having a large PAPR [100]. In this regard, various costly techniques which aim to reduce rather than to eliminate PAPR have been investigated and developed. Basically, these are signal distortion techniques, including signal clipping, peak windowing and peak cancellation. The other techniques are scrambling and coding [101, 102]. Though PAPR reduction techniques can be quite eﬀective, they are rather expensive and add to system complexity, 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 and may be tolerable to some extent. Furthermore, the nature of SC/FDE systems allows the use of modulation schemes that are near constant or constant envelope modulation techniques, and hence, a much lower if not unit PAPR is obtained.

SC/FDE - Buzid 5. The Concept of Single Carrier Transmission with Frequency Domain Equalization 87 (SC/FDE) 5.6. Complementary Cumulative Distribution Function Curves (CCDF)

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

CCDF (P AP R(s)) = Pr(P AP R(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 Near Constant Envelope Modulation

It has been repeatedly mentioned, that the SC/FDE advantage over OFDM lies in the reduced PAPR. OFDM systems inherit the high PAPR from the fact, that the signal is composed of a number of superimposed and independently modulated coherent subcarriers, which individually contribute to the total signal ﬂuctuation. The PAPR in SC/FDE stems from the applied modulation scheme, which needs to be carefully selected in order to reduce the high PAPR. This chapter details the adaption of the constant envelope type modulation to SC/FDE systems. The objective is to eliminate the problems with the PAPR of the SC/FDE system. Ultimately, a comprehensive treatment of the problems of the PAPR is obtained. Contrary to SC/FDE, the CPM scheme can not be adapted to OFDM, because of phase continuity interruptions between the 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 of the baseband signal by half the symbol duration. This prevents the zero crossings in the IQ-plane. OQPSK ﬁts perfectly into the block-wise transmission, and also the receiver signal processing for QPSK can be easily applied with only little adaptations. Referring to SC/FDE layout and the model, shown in Figures 5.1 and 5.11, respectively, the straight forward approach to demodulate a received block is to substitute the N-point IFFT by a 2N-point IFFT and perform sampling rate reduction in time domain by

γˆ = Re y(2n) + j Im y(2n + 1) (6.1) n { } · { } for n = 0,...,N 1. An alternative solution is to reverse the time delay of the Q-path − samples Xu already in frequency domain. For that the following are deﬁned [106]

Xr = Re X , Xi = Im X (6.2) u { u} u { u} 6. SC/FDE for Constant and Near Constant Envelope Modulation 89 eilt parallel to Serial

Scrambling Input CPM Guard Up- Transmit Coding data Mapping interval sampling filter Interleaving

Pilots Channel estimation Synchronisation aallt serial to Parallel payload Descrambling output CPM Down Matched Decoding IFFTN Equalizer FFT2N Remove GI data Demapping sampling filter Deinterleaving

Figure 6.1.: SC/FDE system adapted to CPM technique for u =0,..., 2N 1, and the mirrored versions − r i Xˆ = Re X − ; Xˆ = Im X − (6.3) u { ( u) mod (2N)} u { ( u) mod (2N)} for u =0,..., 2N 1. The spectra of the I- and Q-path can then be separated by − 1 I = (Xr + Xˆ r)+ j(Xi Xˆ i ) (6.4) u 2 u u u − u h i 1 Q = (Xi + Xˆ i )+ j(Xˆ r Xr) . (6.5) u 2 u u u − u Reversing the time delay of the Q-pathh may then be performedi by

corr jπu/N Xu = Iu + jQue . (6.6)

Sampling rate reduction and application of an N-point IFFT complete the equalization/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 in Figure 6.1. The received signal CPM-SC/FDE in a multipath environment is expressed as r(t)= h (t) s(t)+ n(t), (6.7) b ∗ where is the convolution operator, hb(t) is the baseband channel impulse response given in∗ equation (5.8) and n(t) is an additive white Gaussian noise with two-sided

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 90

power spectral density N0/2. The transmit signal is organized in blocks and a block is given by N−1 s(t)= a c (t iT ), (6.8) i 0 − i=0 X where c0(t) is the prime Laurent pulse and ai are the coeﬃcients gained from the mapping process described by (see also equation (4.33))

i ai = j .γi, (6.9) with γi describing the coded data bit values γi 1, 1 . So the diﬀerences to BPSK are 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 a • block have to be dropped since c0(t) spans an interval of length (L +1)T . Due to that the matched filter will not perfectly match the first and the last symbols of a block.

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

6.2.2. The Receiver Model

The receiver block diagram is given in Figure 6.2. It looks similar to the one explained in Chapter 5. One difference is, that in the following simulation the receiver filter has T/µ been chosen to C (f) = TDFT c0( t) , where TDFT is the time discrete Fourier transform. That means the receive{ filter− is} matched to the transmit pulse as it is done in many of today’s systems. In most of the simulations µ=2 is chosen. The equalizer is implemented using Equation (5.49) in the ZF case and using Equation (5.54) in the MMSE 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 symbolsγ ˆ with the formula given by equation

−i γˆi = j .aˆi. (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 standard as in Table 5.1. The system error performance was simulated in a multipath channel and a linear MMSE equalizer is used. Cyclic preﬁx is used as a guard interval and for coding a convolutional encoder with a rate 1/2 and a generator polynomial (133,171) is

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 91

Matched Sampling r(t) FFTµN Filter rate Equalizer IFFTN T/µ a r(T/µ) C MF reduction i

Figure 6.2.: Linear receiver in the frequency domain adapted to the CPM-SC/FDE concept. applied. A non-diﬀerential CPM-SC/FDE with the Laurent shaping pulse derived from the Gaussian pulse of BT =0.3 as well as from the TFM pulse, as both pulses imply constant envelope signals, is simulated. For the purpose of comparison, SC/FDE with QPSK (RRC, α=.25) is included in the simulation and the results are shown in Figure 6.3. The results show that CPM-SC/FDE for the particular parameters of the applied modulation techniques have comparable performance. Note that due to the blockwise structure the CPM-SC/FDE transmit signal exhibits interrupted phase continuity at the block edges.

Log(BER)

E/NindBb 0

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

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 92

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 classifying the power amplifiers. It reflects the introduced distortion and is closely related to power consumption by the transmitter [86, 55]. A frequently used HPA model well suited for Matlab simulations is described in [109]. The impairments of the power amplifier nonlinearity 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-band component causes the system performance degradation in terms of BER, while the out- band component, also called spectrum regrowth, affects adjacent frequency bands and causes adjacent channel interference (ACI) [111, 112]. Traditionally, the nonlinearity of an RF amplifier is described by the HPA input-referred 3rd order intercept point IIP3 or by the 1 dB compression point P1dB [86, 113]. The compression point, shown in Figure 6.4, is the point at which the output power is 1dB less than that for the linear case, as a result to non-linearity. Remarkably, the distortion has joint effects, on one hand the BER, on the other hand the spectral spreading. The HPA is typically modeled as a memoryless nonlinear device which is described by two transfer functions representing the amplitude modulation/amplitude modulation (AM/AM) and amplitude modulation/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 amplifier as 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

j(FP [ρ(t)]+φ(t)) so(t)= FA[ρ(t)] e , (6.13) where FA[.] and FP [.] are envelope transfer functions that represent the AM/AM and AM/PM conversion of the nonlinear PA. The model described in [109] which has also been used in this thesis neglects the AM/PM eﬀect and is given by ρ FA[ρ]= FP [ρ]=0. (6.14) 2p 1/2p 1+[(ρ/A0) ]

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 93

30 +

Compression point P 26 1dB

Output Amplitude dB 24

20 10 15 20 25 30 35 40 Input Amplitude dB

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

Inputdata Outputdata Modulator HPA Rx s(t)i s(t)o

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

During the IEEE 802.11a standardization process the parameter which describes the order of the non-linearity has been chosen to p = 2. Figure 6.6 shows the characteristic of this model, whose non-linearity affects only the amplitude of the signal. Obviously, a linear 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 predistortion before the modulator [114]. The other solution is the use of a back-off power. The operating point of the amplifier is usually identified by the input back-off (IBO) as well as output back-off (OBO), which respectively can be given as

P IBO = 10log o,in indB (6.15) Pin P OBO = 10log o,out in dB Pout

SC/FDE - Buzid

6. SC/FDE for Constant and Near Constant Envelope Modulation 94 Output AmplitudedBm

Input AmplitudedBm

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 at high values of p.

where Pin is the mean power of the signal at the input of the HPA, Pout the mean power of the transmitted signal, Po,out the maximum output power (saturation power) of the HPA and Po,in the input power corresponding to the maximum output power. The use of the back-off power is to keep the amplifier operating in the linear region and hence to avoid signal distortion. Remarkably, the larger back-off implies the larger linear distance between the operating point and the saturation point which is usually determined by a compression point shown in Figure 6.4. Further, the larger the back-off, the less signal distortion and therefore, the back-off determines the PA linear dynamic range and also its power consumption [115]. Shown in Figure 6.7 is the OFDM spectrum according to OFDM based IEEE 802.11a standard 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 the compression point [116]. However, the spectrum of CPM-SC/FDE shown in Figure 6.8 is by far less affected by the amplifier [117]. The reason why the spectrum is affected at all is given by the phase discontinuity at the block edges in the blockwise structured transmit signal. On the other hand, the maximum dc efficiency of the PA is at the non-linear region and is related to the source power consumption. Operating the PA away from the non-linear region means the PA works with less efficiency. Consequently, the power consumption is high and not preferred for mobile terminals where only a limited power supply is available. Furthermore, a survey of the market reveals that the power supply life-time is crucial in marketing of such mobile equipments. Moreover, the back-off is also a measure of the system error performance, since the larger the back-off, the less added distortion and less system degradation is observed.

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 95

-5

-10

-15

-20

-25

-30

NormalizedpowerspectaldensityindB -35

-40 8 8 [MHz]

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

-5

-10

-15

-20

-25 AmplificationafterCP,L =1 -30 AmplificationbeforeCP,L =1 AmplificationafterCP,L =2

Normalized powerspectaldensity in dB -35 AmplificationbeforeCP,L =2

-40

[MHz]

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

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 96

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

Log (BER)

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 comes SC/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 the nonlinearity of the HPA.

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 97

6.4. Spatial Multiplexing for CPM-SC/FDE

In the previous chapters it has been shown that the adaptation of the nonlinear modulation to SC/FDE represents a comprehensive and an elegant solution to the PAPR problems and the system becomes quite effective against the nonlinearity effects. But it is also known that the system capacity in terms of the number of bits per transmitted symbol, which is one, is insufficient when compared to e.g, the OFDM system with the 64-QAM where 6-bit modulate a carrier. Therefore, CPM-SC/FDE in that sense remains dissatisfactory. One way of improving the bandwidth efficiency is to apply spatial multiplexing (SM), a MIMO scheme [118]. It has extensively been investigated for linear modulated SC/FDE signals in [119]. The concept of SM is that an arbitrary number of transmit antennas are transmitting different data streams called layers without space time coding. Further, all layers are detected and equalized simultaneously by 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

CPM GI

Mapping GT S/P Input Coding interleaving data CPM GI

Mapping GT MIMO-syncronization CPM G Rem

Demapping IFFT MF FFT Equalizer

/S P GI Output decoding data deinterleaving CPM Rem IFFT GMF FFT Demapping GI

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

ﬁgure, each layer is mapped individually to obtain the CPM symbols, via applying the non-diﬀerential mapping [120]. However, other possibility for obtaining the layers of the CPM symbols of input data is by obtaining the CPM symbols before structuring the input data into layers. The guard interval is added to each layer and a pulse shaping is next. At the receiver, each layer from each antenna is detected and processed separately. The SM-SC/FDE system can be described conveniently using a matrix notation [119]. The MIMO channel, which describes the transmission behavior from each

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 98 transmit antenna l to receive antenna k can be described with the channel matrix H

H11 H12 ... H1j e H H ... H  21 12 2j H = e. e . e......  e e e  H H ... H  e  i1 i2 ij    (6.16) e e e

Each individual Hij has the form

h (0) h (N 1) ... h (1) e ij ij − ij hij(1) hij(0) ... hij(2)  . .  Hij = . . ... hij(3) (6.17)  . . .   ......    e h (N 1) h (N 2) ... h (0)  ij ij ij   − −  where h (0),h (1),...,h (N 1) are the impulse response coeﬃcients from transmit { ij ij ij − } antenna i to receive antenna j. The frequency domain version of H is obtained from [119, 121] ˇ ˇ−1 H = F2N HF2N e (6.18) where e F2N 0 ... 0 0 F ... 0 ˇ  2N  F2N = . . . . (6.19) . . .. .    0 0 ... F2N    and   −1 F2N 0 ... 0 0 F−1 ... 0 ˇ−1  2N  F2N = . . . . , (6.20) . . .. .  −1   0 0 ... F   2N    where F2N is a DFT matrix, given in the Appendix. In [121], it is shown that the ZF and MMSE equations can be derived by

−1 EZF = HD , (6.21) and − σ2 1 E = HH H HH + n I , (6.22) MMSE D D D σ2 nN d H where (.) denotes the transpose conjugate of a matrix, and HD is given by

HD = Λˇ dGˇ MF HGˇ Λˇ u. (6.23)

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 99

The superscript ˇ, for example in Λˇ u, indicates a block-diagonal matrix with Λu as diagonal elements. Both, Λu and Λd are upsampling and downsampling matrices and given in the Appendix. Further, Gˇ and Gˇ MF are block-diagonal matrices having the frequency domain versions of the transmit filter and the matched filters, respectively, in their diagonals. The system shown in Figure 6.10 is simulated in a multipath MIMO channel. 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 2 multipath channels. SC/FDE with a QPSK scheme is also included in the× figure× for comparison. An MMSE equalizer is used. Further, the system with a Laurent pulse that is derived from both Gaussian and TFM pulses, is simulated and compared in the same multipath channel.

Log(BER)

E/NindBb 0

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

SC/FDE - Buzid 6. SC/FDE for Constant and Near Constant Envelope Modulation 100

-1

-2

-3

-4

Log(BER)

-5 1X1CPM-SC/FDE(TFM) 2X2CPM-SC/FDE(TFM) -6 1X1CPM-SC/FDE(Gaussian) 2X2CPM-SC/FDE(Gaussian) -7

-10 -5 0 5 10 15 S/NindB E/NindBb 0

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

It can be concluded that spatial multiplexing can be applied to CPM-SC/FDE similar as for QPSK-SC/FDE. The simulation results furthermore show similar performance results 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 systems are inspired by or are a direct or indirect application of the spread spectrum technique. Applications for commercial spread spectrum range from wireless cellular systems and phones, WLAN, to wide area network WiMAX. Another technology has been in the focus in the recent years. This technology is OFDM. It has attracted great attention and has been very popular in the last decade. It can be combined with TDMA, FDMA or CDMA for use in multiple access environments. OFDM is considered an effective modulation technique for high-speed digital transmission. It is used in a wide range of applications, including IEEE 802.11a, g, wireless LAN standard, digital audio broadcasting DAB and digital video broadcasting DVB [122]. In OFDM systems, the spectrum is divided into smaller bands or subcarriers. On the other hand, SC/FDE has been an emerging technique in the last years. Both SC/FDE and OFDM concepts can be combined with multiple access techniques and offer promising multiple access schemes for broadband radio applications. A potential multiple access technique for wireless networks is code division multiple access (CDMA). CDMA avoids the burst transmission or pulsed transmission which is strongly associated with time division multiple access schemes (TDMA). Pulsed transmission is clearly noticeable in narrowband TDMA based systems such as GSM, DECT and TETRA. In this chapter, the combination of SC/FDE and CDMA concepts is considered and the frequency 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 always related to the universal mobile telecommunication system (UMTS) which is also known as UMTS Terrestrial Radio Access (UTRA), in which two types of CDMA approaches were proposed. One type is the UTRA frequency division duplex (UTRA-FDD) which 7. Point to Multipoint Systems, SC/FDE-CDMA 102 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 standard provides a seamless upgrade for the users of 2G and 2.5G CDMA technology. It is expected that these two major 3G technologies cdma2000 and W-CDMA, will remain popular throughout the early part of the 21st century [91]. In Japan, the W-CDMA system 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 network to evolve through the addition of high data rate equipments. All these 3G standards are known in the International Telecommunications Union (ITU) as the members of IMT-2000. It is well known that the CDMA scheme is robust to frequency selective fading and has been successfully introduced in commercial cellular mobile communication systems [123]. Instead of dividing the spectrum between several independent and non-overlapping users in frequency division multiple access (FDMA) or assigning the time slots to diﬀerent, independent and non-overlapping users in time division multiple access (TDMA), the users 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 are diﬀerentiated through the distinct signatures or spreading codes.

7.2. SC/FDE-CDMA

Although the frequency domain equalization concept has been proposed for more than 3 decades, the application of FDE for CDMA detection has just begun to be considered recently [124]. The low complexity, eﬃcient and powerful receive structures of the SC/FDE can be combined advantageously with the concept of CDMA (SC/FDE-CDMA). The equalizer 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 frequency domain.

7.2.0.1. System Description

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

SC/FDE - Buzid 7. Point to Multipoint Systems, SC/FDE-CDMA 103

Binary Serial to Guard Mod CDMA intput spreading parallel interval

Binary Parallel CDMA Demod IFFT Equalizer FFT Remove GI output to serial despreading

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 be used. Generally, the SC/FDE-CDMA baseband signal for the ith user may be expressed as [126] N−1 Q−1 si(t)= γ¯i ci(q)g(t nT qT ), (7.1) n − − c n=0 q=0 X X The chip rate given by 1/Tc equals Q/T, where Q is the code length and T is the symbol i th duration. i = 0, 1,...,K 1 , where K represents the number of users.γ ¯n is the n symbol in a block{ of size N−, of} the ith user, g(t) represents the pulse shape and ci is the spreading code of the ith user. For brevity, the continuous time element is eliminated here and the Fourier codes are applied for illustrative purpose. Fourier codes are the columns or rows of the inverse i 1 j2πqi/Q discrete Fourier matrix. The matrix entities are given by c (q) = Q e . A spread symbol of the ith user at the qth chip can be expressed as 1 si (n)= γ¯i ej2πqi/Q. (7.2) q Q n The sum over K users is then given as

− 1 K 1 s (n)= γ¯i ej2πqi/Q, (7.3) q Q n i=0 X where K Q. The CDMA symbol is then given as ≤ s´(n)= s (n),s (n),...,s − (n) . (7.4) { 0 1 Q 1 } Letting K = Q, full load, the above equation represents an inverse discrete Fourier transform of the nth transmitted symbol of the K users. It represents an OFDM symbol. The complete 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 the SC/FDE-CDMA system.

SC/FDE - Buzid 7. Point to Multipoint Systems, SC/FDE-CDMA 104

1 BPSK Non−differential CPM

−1

−2

−3 Log(BER)

−4

−5

−6 0 2 4 6 8 10 12 14 16 18 Eb/No in dB

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

7.2.0.2. Simulation Results

The system SC/FDE-CDMA shown in the figure is implemented in Matlab and simulated for different modulation formats, including the non-differential CPM technique which is discussed 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) as spreading 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 a non-differential CPM, the Gaussian pulse with BT = 0.8 for the linearized Laurent pulse shape is used. The full load case (16 users) for BPSK and non-differential CPM is plotted in Figure 7.3. Further, the system with a non-differential CPM is also simulated and compared for different values of BT of the Gaussian frequency pulse and the result is shown 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 possibility is multicarrier direct sequence CDMA (MC-DS-CDMA), where each data symbol is spread over Q chips and then modulates the N subcarriers (N=Q) [127, 128]. Another possibility is multicarrier CDMA (MC-CDMA), in which the spread data of the active users are added before modulating the subcarriers [129, 130, 131]. In the MC-CDMA system, the CDMA symbols are transported on the orthogonal subcarriers within the usable frequency band of the channel and implemented through IDFT [126]. In this

SC/FDE - Buzid 7. Point to Multipoint Systems, SC/FDE-CDMA 105

−1

−2

Log(BER) −3

−4

BPSK Non−recursive CPM

0 3 6 9 12 15 18 21 24 27 30 33 Eb/No in dB

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

BT=.2 −1 BT=.3 BT=.5 BT=.8

−2

−3 Log(BER) −4

−5

0 2 4 6 8 10 12 Eb/No in dB

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 7. Point to Multipoint Systems, SC/FDE-CDMA 106

−1 10

−2 10

−3 10 Log(BER)

−4 10

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

0 3 6 9 12 15 18 21 24 27 30 33 Eb/No in dB

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

nubnr data Inputbinary C0 OFDM

Spreader S/P C1 + MC-CDMA

Spreader CK-1

Figure 7.6.: MC-CDMA down link transmitter. study, only MC-CDMA is considered and compared to an SC/FDE-CDMA system. For Fourier 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 of the DFT matrix whose entities are given as ci(q) = e−j2πqi/Q. The qth chip of the nth CDMA symbol is given as

K−1 i −j2πqi/Q xq(n) = γ¯ne . (7.5) i=0 X Again making K=Q, full load, the above equation represents a discrete Fourier transform. In order to obtain the MC-CDMA symbol, the CDMA symbol is next modulated on the subcarriers through an IDFT process. This results in two transformation processes; 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 7. Point to Multipoint Systems, SC/FDE-CDMA 107 ulating the resulting signals on the subcarriers through IDFT operations results in an SC/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 which are taken from the inverse discrete Fourier matrix turns into OFDM [133]. In contrast to that, the combination of OFDM and CDMA with the Fourier codes, taken this time from a discrete Fourier matrix, turns into an SC/FDE. This is summarized in Figure 7.7.

Inverse CDMA Fouriercodes

SC/FDE OFDM

CDMA Fouriercodes

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 expressed as − 1 N 1 X(n)= x(k)e−j2πnk/N . (A.1) √N Xk=0 Further, a DFT process can also be represented in a matrix form, denoted by FN CN×N , whose entities are determined via ∈

1 −j2π(k−1)(l−1)/N FN (k,l)= e . (A.2) √N The input discrete signal x(n) can also be written in a vector notation ~x and the DFT matrix can be applied to obtain X~ = FN ~x. (A.3) Further, the inverse discrete Fourier transform of X~ is given by −1 ~ ~x = FN X. (A.4)

−1 H −1 H It is necessary to note that FN = FN , and thus, (FN ) = FN . DFT can also be applied to matrices. Consider the linear equation ~y = A˘ ~x; ~x, ~y CN×1; A˘ CN×N . (A.5) ∈ ∈ Multiplying the linear equation with FN at the left side yields

FN ~y = FN A˘ ~x. (A.6)

−1 Since, FN FN = IN , the above equation can be written as ˘ −1 FN ~y = FN A FN FN ~x (A.7) ~ ˘ −1 ~ Y = FN A FN X, thus, the Fourier transformation A of the matrix A˘ is given as ˘ −1 A = FN A FN . (A.8) A. Appendix 109

A.2. Convolution Matrix

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

ν−1 y(n)= h(l)x(n l), (A.9) − Xl=0 where h(l) is a discrete time invariant channel impulse response, l = 0, 1,...,ν 1 and x(k) is the input where k =0, 1,...,N 1. The above equation can also be represented− in a compact form by using matrix notations− as

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

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(0) 0 0 0 ... 0 h(1) h(0) 0 0 ... 0  h(2) h(1) h(0) 0 ... 0   . . .. .   ...... 0  C(N+ν−1)×N H =   (A.11) h(ν 1) h(ν 2) ... h(0) ... h(0)  ∈  − −   0 h(ν 1) ... h(1) ... h(1)   . .− . . .   ......     0 0 00 ... h(ν 1)    −  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 as ∈ y = 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 ∈

h(0) h(N 1) h(N 2) ... h(1) h(1) h(0)− h(N − 1) ... h(2)  −  T = h(2) h(1) h(0) ... h(3) CN×N (A.14)  . . . .  ∈  ......    h(N 1) h(N 2) ...... h(0)  − −    A.4. Sampling Matrix

In realizing the digital part of communication systems, it is generally necessary to adjust the rate of digital signals by up and down sampling processes. Up- and downsampling by 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: 1 0 ... 0 0 0 ... 0   0 1 ... 0 0 0 ... 0 N2N×N Λu =   (A.15) . . . ∈ . .. .   f 0 0 ... 1   0 0 ... 0     So the upsampled version of a vector ~x is given by ~y = Λu~x. This matrix can be transformed to frequency domain by f −1 √2 IN Λu = F2N ΛuFN = (A.16) 2 IN e A.4.2. Down Sampling Matrix

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

1 0 0 ... 0 0 0 0 1 ... 0 0   NN×2N Λd = . . . . . (A.17) ...... ∈   0 0 0 ... 1 0 f    

SC/FDE - Buzid A. Appendix 111

√2 Λ = F Λ F−1 = I I (A.18) d N d 2N 2 N N e

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