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Doctoral Thesis

Multicarrier modulation techniques for bandwidth efficient fixedwireless access systems

Author(s): Hunziker, Thomas

Publication Date: 2002

Permanent Link: https://doi.org/10.3929/ethz-a-004469984

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Diss. ETH No 14818

Multicarrier ModulationTechniques for Bandwidth Efficient

Fixed Wireless Access Systems

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGYZURICH

for the degree ofDoctor of Technical Sciences

presented by

Thomas HunzikerDipl. El.-Ing. ETH

born September 11th, 1966citizen of Staffelbach, AG

accepted on the recommendation of

Prof. Dr. Dirk Dahlhaus, examinerProf. Dr. Andreas Polydoros, co-examiner

2002

Acknowledgments

In the first place I would like to thank Prof. Dirk Dahlhaus for supporting mywork in all the years I spent in his group at the Communication Technology Lab-oratory. I thank him particularly for guiding my research activities, for writingmany papers with me, and for being the referee of this thesis. Furthermore, Iwould like to thank Prof. Andreas Polydoros for the time he sacrificed to reviewmy thesis as a co-examiner, and for his valuable comments.

The Communication Technology Laboratory has been an ideal environmentfor me to work, and I would like to thank all the people who supported me inthe past years. In particular, I thank Prof. P. E. Leuthold for inviting me to jointhe laboratory as a research assistant and my colleague Mauro Pesce for shar-ing an office with me. Additionally, I would like to thank the current, former,or external laboratory members Urs Bernhard, Zhenlan Cheng, Simeon Furrer,Bruno Haller, Jan Hansen, Dirk Hugen, Andreas Jarosch, Jürgen Kemper, Chris-tian Mauz, Jürg Meierhofer, Carlo Mutti, Rico Schwendener, Pascal Truffer, andMartin Tschudin for their support and the many fruitful discussions.

Abstract

Fixed wireless access (FWA) technologies are of great interest as a means toprovide broadband data services to households and businesses without requir-ing any existing cable infrastructure. Multicarrier modulation techniques likeorthogonal frequency-division multiplexing (OFDM) facilitate the high-speeddata transmission over time-dispersive channels. What makes the physical layerdesign in FWA systems demanding is the limited frequency spectrum and thefact that, unlike in some cable based systems, the transmission medium needs tobe shared among multiple users. Bandwidth efficient transmission schemes arethus a central issue.

In this thesis, two potential techniques for achieving higher spectral efficien-cies than with a conventional OFDM modulation are investigated. The first tech-nique exploits available channel state information (CSI) at the transmitter sidefor accomplishing an adaptive modulation. In particular, the spectral power al-location of a bit-interleaved coded broadband OFDM transmission over slowlytime-variant channels is optimized. The presented adaptation policies aim tominimize the bit error rates at the decoder output of hard-decision and soft-decision based receivers, assuming either perfect or outdated CSI. The achiev-able increase in the spectral efficiency with respect to nonadaptive transmissionschemes is investigated by means of computer simulations.

The second part of the thesis focuses on alternative multicarrier transmissiontechniques without bandwidth wasting guard periods. Multicarrier modulationschemes in which the elementary signal pulses relate to the elements of a Weyl-Heisenberg system, i.e. resulting from a prototype function shifted in time andfrequency, are studied. The overlapping of the information-bearing signal partsafter dispersive channels and the resulting interference are bounded by utilizinga prototype pulse whose energy is concentrated in both time and frequency. Re-

i

ceiver schemes are presented in which first a sufficient statistic for the unknowndata symbols is calculated from the output signals of a filter bank, and secondan iterative maximization of the likelihood function or a decorrelation is per-formed. The bounded pulse overlapping allows for limiting the computationaleffort in the receivers. The computational complexity, error rate performanceand information capacity are analyzed and compared to the characteristics of aconventional coded OFDM transmission.

ii

Kurzfassung

Drahtlose Festanschlüsse sind eine Alternative zu drahtgebundenen Lösun-gen zur Versorgung von Haushalten und Geschäften mit breitbandigen Daten-diensten. Der Vorteil der drahtlosen Systeme liegt darin, dass keine Kabel-Infrastruktur vorausgesetzt werden muss und die Endgeräte einfach installiertwerden konnen. Mit Mehrtrager-Modulationsverfahren wie OFDM (OrthogonalFrequency-Division Multiplexing) lassen sich Daten mit hoher Geschwindigkeituber Funkkanäle mit Mehrwegausbreitungserscheinungen übertragen. Anderer-seits sind wegen der Knappheit an Frequenzen, und weil sich die Benutzer ineiner Zelle das Spektrum teilen müssen,Ubertragungsverfahren mit hoher spek-traler Effizienz gefordert.

Gegenstand dieser Arbeit ist die Untersuchung zweier Ansätze zur Verbes-serung der Bandbreite-Effizienz gegenüber einer gewöhnlichen OFDMUbert-ragung. Die erste Technik beruht auf der Ausnutzung von Kanalinformation imSender. Es werden insbesondere Verfahren zur Adaptierung der Leistung derUntertragersignale an den gegenwärtigen Kanalzustand betrachtet. Die herge-leiteten Algorithmen minimieren die Fehlerraten am Decoder-Ausgang bei Sy-stemen mit Codierung und einer Verschachtelung (Interleaving) auf Bit-Ebene,wobei auch der Fall einer wegen eines zeitvarianten Kanals veralteten Kanalin-formation untersucht wird. Die mittels adaptiver Leistungsanpassung erzielba-ren Gewinne werden durch Computersimulationen ermittelt.

Im zweiten Teil der Arbeit wird die Eignung von alternativen Mehrträger-Modulationstechniken ohne die für OFDM notwendigen Schutzintervalle unter-sucht, welche die Bandbreite-Effizienz vermindern. Es wird eine Klasse von Mo-dulationsverfahren betrachtet, bei welchen die elementaren Pulsformen durchVerschiebung einer Prototypfunktion in der Zeit und der Frequenz resultierenund somit den Elementen eines Weyl-Heisenberg-Systems entsprechen. Durch

iii

Verwendung von Prototypen mit hoher Signalenergiekonzentration im Zeit-und Frequenzbereich lässt sich dieUberlappung der informationstragenden Si-gnalanteile am Kanalausgang - und damit die Interferenz am Empfänger - be-schranken. Empfängerkonzepte werden vorgestellt, welche mittels einer Filter-bank zuerst eine suffiziente Statistik für die unbekannten Datensymbole erzeu-gen und auf dieser Basis eine iterative Maximierung der Likelihood-Funktionoder eine Dekorrelation durchführen. Die erforderliche Rechenkomplexität lasstsich durch die Beschränkung der Pulsüberlappungen begrenzen. Neben der Re-chenkomplexität werden die Fehlerraten und die Kapazitäten untersucht, und eswird ein Vergleich zu konventionellen OFDM-Systemen ohne und mit Codie-rung angestellt.

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Fixed Wireless Access Systems. . . . . . . . . . . . . . . . . . 31.3 Multicarrier Modulation . . .. . . . . . . . . . . . . . . . . . 41.4 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . 61.5 Outline of this Thesis .. . . . . . . . . . . . . . . . . . . . . . 7

2 OFDM for Fixed Wireless Access Systems 92.1 The Outdoor Radio Channel .. . . . . . . . . . . . . . . . . . 102.2 Stochastic Modeling of the Channel . . . . . . . . . . . . . . . 112.3 The OFDM Concept . . . . . . . . . . . . . . . . . . . . . . . 132.4 DFT-Based Transceiver Structure . . . . . . . . . . . . . . . . . 162.5 Coding and Interleaving Techniques . . . . . . . . . . . . . . . 182.6 Synchronization and Channel Estimation . . . . . . . . . . . . . 212.7 Error Performance and Spectral Efficiency . . . . . . . . . . . . 222.8 Information Theoretic Considerations . . . . . . . . . . . . . . 26

3 Adaptive OFDM Transmission 313.1 Maximizing Capacity: The Water-Filling Principle. . . . . . . 323.2 Adaptive Modulation Techniques . . .. . . . . . . . . . . . . . 343.3 Adaptive Power Allocation for BICM-OFDM Systems . . . . . 37

3.3.1 Optimum Power Allocation for Hard-Decision Decoding 383.3.2 Outdated Channel State Information . . . . . . . . . . . 443.3.3 Power Adaptation for Soft-Decision Decoding . . . . . 47

3.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . 51

v

3.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 55

4 Multicarrier Transmission with Nonorthogonal Pulses 574.1 Alternative Multicarrier Transmission Techniques . . . . . . . . 584.2 Weyl-Heisenberg System Related Multicarrier Modulation . . . 594.3 Receiver Structures . .. . . . . . . . . . . . . . . . . . . . . . 63

4.3.1 Iterative Detection Methods .. . . . . . . . . . . . . . 664.3.2 Computational Complexity and Residual Interference

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.3 Matched Filtering . . .. . . . . . . . . . . . . . . . . . 73

4.4 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . 764.4.1 Uncoded Transmission . . . . . . . . . . . . . . . . . . 774.4.2 Coded Transmission . . . . . . . . . . . . . . . . . . . 81

4.5 Channel Capacity with a Zero-Forcing Receiver . .. . . . . . . 834.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . 85

5 Outlook 87

A The Gaussian WSSUS Channel Model 89

B The Matched Filter Bound 93

C List of Abbreviations 97

D List of Symbols 101

Bibliography 109

Chapter 1

Introduction

There is a rapidly increasing demand for high-speed Internet access. A numberof digital subscriber line (DSL) standards, as a whole referred to as xDSL, haveemerged for connecting households at data rates of up to several Mbit/s overunshielded twisted-pair wires. Besides, Internet access is offered over coaxialcables and also over power lines. Unlike a fiber-to-the-home solution all thesetechniques rely on existing cable infrastructures and do therefore not cause muchextra installation costs.

Fixed wireless access (FWA) systems are another cost-efficient possibilityfor providing multimedia services to households and businesses. While thewireless solutions are clearly advantageous in rural areas with underdevelopedcable infrastructures, the simple installation could make them also an interest-ing alternative in suburban and urban areas. Organized in cells of up to a fewkilometers in diameter, FWA systems could in the future coexist with the thirdgeneration (3G) of personal communication systems, which are built up for mo-bile users in particular, and wireless local area networks (WLANs), which areprimarily intended for in-house use.

Apart from their distinct positions within the world of wireless communica-tions, these three systems also differ from an engineering viewpoint. Comparedto 3G mobile radio systems, channel estimation and tracking is less demandingand there are no handovers in FWA scenarios. On the other hand, outdoor sys-tems have to deal with much larger delay spreads due to multipath propagationthan indoor WLANs. A state-of-the-art physical layer is a prerequisite for FWA

2 Chapter 1: Introduction

systems to achieve the envisaged data rates far beyond the maximum2 Mbit/s inthe UMTS (universal mobile telecommunications system) standard.

1.1 Motivation

To be successful in the market with a FWA technology, the terminal stationsmust be cheap and easy to install at the subscribers’ premises. This rules outusing directive antennas high above the roofs for establishing exclusive line-of-sight (LOS) links between the subscribers and a base station. Rather, it shouldbe possible to mount the antennas of the terminals in-house or to adjust themjust outside of a window. This implies that the system must also function undernon-line-of-sight (NLOS) conditions. Moreover, a cellular structure becomesnecessary in which the radio channel is a shared medium. In order to deliverbroadband services to a multitude of users an efficient usage of the scarce band-width resources is a central issue.

Bandwidth efficient techniques are demanded on the physical layer as well ason the higher layers. In this work, we shall primarily focus on appropriate mod-ulation techniques for FWA systems. Orthogonal frequency-division multiplex-ing (OFDM) is employed as the underlying modulation method in many high-speed communication systems encountering multipath channels as this evadesthe need for complex equalizers in the receivers. We shall investigate two con-cepts which may achieve a more favorable performance-complexity ratio thana conventional OFDM transmission. The first technique aims to exploit knowl-edge about the channel state in the transmitter for increasing the spectral effi-ciency. The considered adaptation of the spectral power allocation to the instan-taneous channel state is particularly useful in the downlink, i.e. the link from thebase station to the subscriber, since it involves an additional complexity only inthe transmitter while a simple demodulation and decoding scheme can be ap-plied in the receiver.

To enable an interference-free demodulation the OFDM signal contains so-called guard periods. The signal energy within these guard periods is not usedby the conventional OFDM receiver, and the favorable complexity is in fact onlyenabled by a certain sacrifice of signal power. This motivates the search foralternative multicarrier (MC) modulation techniques which do not require band-width consuming guard periods while still allowing for receiver structures withacceptable complexities. In the second part of this work, a class of MC modu-

1.2 Fixed Wireless Access Systems 3

lation schemes based on nonorthogonal pulse shapes is discussed. The receivercomplexity can be limited by choosing pulses with superior time-frequency (TF)localization properties. Additionally, there is a greater freedom in the choice ofthe signaling intervals and the number of subchannels, which can be used to re-lax the requirements of the hardware components in the transceiver front ends interms of linearity and frequency stability.

1.2 Fixed Wireless Access Systems

Bridging the last mile by a wireless technology is not a new concept. Wirelesslocal loop systems are in operation since quite a while, connecting subscribers tothe PSTN (public switched telephone network). These systems are particularlyinteresting for underdeveloped countries as they evade a deployment of hugeamounts of copper wires. While voice services require data rates of at most64 kbit/s, much larger bandwidths are necessary for offering Internet access ata satisfactory speed or even video-on-demand. The local multipoint distributionservice (LMDS) and the multichannel multipoint distribution service (MMDS)can deliver broadband services. They have originally been intended to providesubscribers with video data and assume an asymmetric bandwidth allocation forthe uplink and downlink. The today available LMDS and MMDS products areproprietary, rather than based on a common standard. However, the major reasonwhy these systems have not been successful so far is that they require LOS linksfrom the subscribers to the base station. This necessitates the installation of highrooftop antennas before the links can go into operation.

Broadband FWA systems operating under NLOS conditions have a higherpotential to attract a broad interest since e.g. they allow the subscribers to placethe terminal antenna just next to a desktop computer. The most suitable fre-quency bands for a NLOS operation are between2 and about6 GHz. Higherradio frequencies would only allow for short range services due to the largepath loss. Furthermore, both licensed and unlicensed frequency bands like e.g.the ISM (instrumentation, scientific, and medical) band in the2.4 and5.8 GHzranges are conceivable. In unlicensed bands the transmitted signal must complywith certain regulations and the receivers must be prepared to deal with interfer-ence from other systems using the same band.

Antennas with small beam-widths are usually not practical since in NLOSscenarios the various reflected and diffracted signal parts may impinge from


different angles. Adaptive antennas which accomplish a beam-forming to max-imize the ratio of the gathered signal power over the power of the interferenceand the noise are of great value. They can substantially enhance the system ca-pacity but also increase the complexity in both the analog front ends and thedigital baseband processing units. In the scope of this work we assume the useof conventional antennas.

Efforts towards creating standards for broadband FWA systems are currentlyin progress, in Europe in the HIPERACCESS project within the ETSI (Euro-pean telecommunications standards institute) BRAN (broadband radio accessnetwork) group, and in the United States within the 802.16 working group atthe IEEE (institute of electrical and electronics engineers). Overviews of FWAtechnologies can be found in e.g. [1, 2].

1.3 Multicarrier Modulation

In traditional single carrier systems, time domain equalizers are employed tocope with intersymbol interference (ISI) caused by multipath propagation in theradio channel. The complexity of these equalizers grows with the relative extentof the echo delays with regard to the signaling intervals. With a MC modulationthe equalization complexity can be kept at an acceptable level even for highdata rates. The idea behind MC transmission is to allocate multiple subchannelswith uniformly spaced center frequencies for conveying a large number of bitsin parallel. For a given bandwidth, the signaling rate on each subchannel ismuch lower and the signaling intervals are by the same factor longer than for acorresponding single carrier transmission. A common form of MC modulationwith overlapping subchannel spectra is OFDM. Here, cyclic signal extensionsin guard periods preserve the orthogonality of the information-bearing signalpulses, making the data detection in the receivers a particularly straightforwardand simple task.

The principle of employing multiple frequency-spaced subchannels has itsorigin at least forty years ago, when it was applied in some military systems. In1971, Weinstein and Ebert showed that a digital implementation of an OFDMsystem can be based on computationally efficient fast Fourier transform (FFT)methods [3]. It nevertheless took more than a decade from that time until OFDMstarted to attract a broad interest, boosted by the demand for broadband com-munication systems and the vast development in the digital signal processing

1.3 Multicarrier Modulation 5

technology.

Today, OFDM is utilized both in wireless communications and for cablebased data transmission. In the latter field the technique is normally referredto as discrete multitone (DMT) and applied in the new xDSL standards likee.g. ADSL (asymmetric DSL). A comprehensive introduction to the characteris-tics of the DSL transmission medium and the application of DMT is containedin [4]. In wireless data transmission, OFDM was adopted by the IEEE 802.11aand the European HIPERLAN/2 standards1 for high-speed WLANs. Moreover,OFDM was chosen for starting up terrestrial digital broadcasting services likethe DAB (digital audio broadcasting) system and the DVB (digital video broad-casting) project. Here, the challenge for the receivers is to deal with the long de-lays of the signal replicas from synchronously transmitting distant base stations.For 3G mobile radio, which will soon be available under the acronym UMTSin Europe, W-CDMA (wideband code-division multiple-access) was chosen asthe underlying modulation technique. However, OFDM is today often namedthe prospective candidate for the fourth generation of mobile communicationsystems.

In view of the high data rates and the significant delay spread in outdoorenvironments, OFDM is also an auspicious modulation method for future FWAsystems. A combination with a time-division duplex (TDD) scheme is favor-able since this provides the flexibility to choose the uplink and downlink burstsizes according to the traffic requirements. The reciprocity of the uplink and thedownlink channel characteristics in a TDD transmission and their only marginalvariations from burst to burst facilitate an adaptation of the OFDM modulationto the actual channel state. Specifically, the transmitter may use the channelparameters that have been assessed for the demodulation of a previous burst tooptimize the modulation methods and the power allotment to the subchannels.An overview of adaptive modulation methods in general with many referencesis part of [5]. A survey of adaptation techniques for OFDM systems is alsoprovided in Sect. 3.2.

While OFDM is by far the most considered MC modulation scheme therehave also been a number of proposals for alternative MC transmission tech-niques. Receivers which are capable of dealing with the ISI resulting fromsimply omitting the guard periods were studied in [6, 7]. In addition to sav-

1HIPERLAN stands for HIgh PERformance LAN. Only the HIPERLAN type 2 is OFDM based,offering higher data rates than the HIPERLAN type 1.


ing bandwidth an elimination of the guard periods also increases the freedomin the design of the elementary pulse shapes. An overview of MC modulationand receiver schemes which exploit this freedom, e.g. to limit ISI, is given inSect. 4.1.

1.4 Contributions of this Thesis

The main subjects of the thesis areadaptive power control for OFDM systems,andnonorthogonal MC transmission schemes. Both techniques can be viewedas efforts to achieve a better spectral efficiency than with a conventional OFDMmodulation. The first part of the thesis provides a review of the concepts behinda typical OFDM system according to e.g. today’s IEEE 802.11a and HIPER-LAN/2 standards. The following analysis of the information theoretic capacityof an OFDM transmission without and with knowledge of the channel state inthe transmitter can also be found in other literature in a similar form. Its purposein this work is to contrast the information theoretical gains by a power adap-tation with the gains of the later on presented adaptation strategies for systemswith practical encoding and decoding schemes.

The main unique contributions of the thesis are the following:

• Adaptive power loading strategies are derived in Sect. 3.3 which aimto optimize the error rate performance of an OFDM system with bit-interleaved coded modulation (BICM). The optimization criterion is di-rectly the bit error rate (BER) at the decoder output, rather than the BERof an uncoded system or the capacity.

• It is shown in Sect. 3.4 that adaptive power control is particularly benefi-cial in combination with a hard-decision decoding. As the power adapta-tion involves only the transmitter side, the presented adaptation strategiescan optimize the downlink performance in systems where the complexityof the subscriber stations is limited, i.e. where a soft-decision decoding isnot feasible.

• In Sect. 4.3, iterative detection and decorrelation schemes are investigatedfor the reception of MC signals based on nonorthogonal pulse shapes.Receiver structures are presented in which the use of appropriate TF con-centrated pulses allows for a significant complexity reduction.

1.5 Outline of this Thesis 7

• A comparison of a MC transmission in which ISI is eliminated by a decor-relation in the receiver, and an ISI-free OFDM transmission is drawn inSect. 4.4 in terms of the bit error performance, and in Sect. 4.5 in terms ofthe capacity.

1.5 Outline of this Thesis

Chapter 2 contains a discussion about the characteristics and the modeling ofthe radio channel encountered in FWA systems. This is followed by an intro-duction to the basics of OFDM and an overview of the main elements in OFDMtransceivers. Particular emphasis will be on systems with forward error cor-rection (FEC) coding, specifically on the use of a BICM scheme. Finally, theachievable error rate performance is discussed, and the transmission techniqueis reviewed from an information theoretic view point.

Chapter 3 presents adaptive OFDM modulation techniques. The water-filling principle is first addressed, defining the optimum spectral power allo-cation with respect to the information theoretic capacity. Then, adaptive mod-ulation schemes which aim to optimize the spectral efficiency or the error rateperformance are discussed. Techniques for only adapting the power allotmentto the subchannels are derived in particular, since they may be applied withoutadditional complexity in the receiver. The performance gain from the poweradaptation is investigated for BICM-OFDM transmission schemes at the end ofthe chapter.

Chapter 4 addresses alternative MC modulation techniques. In particular,bandwidth efficient MC modulation schemes in which the overlapping of theinformation-bearing signal parts is limited by employing pulses with superiorTF localization properties are investigated. Appropriate receiver structures arepresented and their performance compared against a conventional OFDM re-ceiver.

Chapter 5 gives an outlook to some further research activities, which may beof interest in the context of next-generation ubiquitous wireless communicationsystems.

Chapter 2

OFDM for Fixed WirelessAccess Systems

Severe signal distortion in the transmission medium can make reliable data re-covery in wireless communication receivers a challenging task. In outdoor wire-less data transmission using fixed antennas, a major cause for the signal distor-tion is multipath propagation, i.e., the original signal gets superimposed by anumber of echoes. Complex equalizers or other sophisticated signal processingtechniques are necessary to deal with the resulting ISI in single carrier broad-band systems.

With an OFDM signaling, on the contrary, the channel is subdivided into alarge number of narrowband channels and interference thereby completely elim-inated. As a drawback, the narrowband subchannels are exposed to large-scalegain variations. Applying an appropriate FEC coding technique is therefore cru-cial for achieving acceptable performance figures in environments with multi-path propagation.

This chapter first addresses the characteristics of the outdoor radio channel aswell as suitable models, and then provides an introduction to OFDM. Besides themodulation and demodulation principles, the channel estimation, synchroniza-tion and FEC coding aspects are discussed. Finally, the error rate performancewith different decoding schemes and the information capacity are studied.

10 Chapter 2: OFDM for Fixed Wireless Access Systems

2.1 The Outdoor Radio Channel

The signal attenuation in a LOS link including the loss in the transmitter andreceiver antenna subsystems can be computed by the Friis free space equation,which can be found in [8]. The so-called path loss increases with the radiofrequency (RF) since the antenna aperture is linked to the wavelength. An addi-tional attenuation due to rain and clouds must be taken into account in outdoorsystems operating in frequency bands higher than a few GHz.

Apart from the direct LOS path the information-bearing signal may alsoreach the receiver antenna via reflections at buildings and other objects. Manywireless communication systems are intended to function even if the direct sig-nal path from the transmitter to the receiver is totally obstructed, i.e. under NLOScondition. In this case, the signal at the receiver antenna may be composed ofa number of overlapping transmit signal replicas with quite similar power butdifferent delays. As a result of the scattering, diffraction and reflections, thepath loss over a certain distance is normally higher in NLOS scenarios than ina LOS transmission. The actual signal strength at the receiver depends on therelative differences in the delays of the overlapping essential signal parts, as thisdetermines whether the combining of the waves takes place in a constructive ordestructive fashion.

If the transceivers, scatterers or reflectors are in motion, the signal attenua-tion changes with time. Power variations due to appearing or disappearing signalpaths are termedlarge-scale fading, whereas the faster fluctuations due to smallchanges in the relative delays associated with the paths are termedsmall-scalefading. There may be rapid variations in the instantaneous received signal powerin systems where the transceiver stations are mobile or if e.g. speedy vehicles actas reflectors. Although the latter cannot be ruled out, FWA systems are in gen-eral subject to much slower channel state variations. In [2] the fade rates in fixedwireless environments are claimed to typically range between0.1 and2 Hz.

Outdoor high-speed transmission employing non-directive antennas often in-volves afrequency-selectivechannel. That is, the bandwidthW over which thechannel is used clearly exceeds its coherence bandwidthWcoh. The coherencebandwidth defines the maximum frequency spacing for which two narrowbandsignals are exposed to correlated fading, and depends on the signal dispersion asa consequence of the multipath propagation. IfWcoh is defined as the bandwidthover which the normalized correlation coefficient exceeds the value0.5, and the

2.2 Stochastic Modeling of the Channel 11

so-called root mean-square (RMS) delay spreadτd represents the standard de-viation associated with the distribution of the spread signal power in the delaydimension, the approximation

Wcoh ≈1

5τd

can be used to estimateWcoh from τd or vice versa [8].Results from delay spread measurements in outdoor environments were pub-

lished in [9, 10, 11, 12, 13, 14], for frequencies between1.8 and29 GHz andtransmitter-receiver separations between tens of meters and a few kilometers.The reported values forτd range from16 to 330 nanoseconds. This gives riseto the assumption that the coherence bandwidth in typical FWA systems will bein the order of0.6 to 13 MHz. The ratio ofW andWcoh is significant for awireless communication system since it characterizes the frequency-selectivityof the channel, which allows for achieving diversity and thereby reducing thenegative effects of the fading.

2.2 Stochastic Modeling of the Channel

Channel models are important to predict the achievable performance with aprospective transmission technique by analytical or numerical means. To cre-ate an appropriate model, a statistical description of the diverse effects in theradio channel is needed. For the large-scale fading it has turned out that thedistribution of the signal power level after a certain distance closely matchesa lognormal distribution, provided that the observation interval is long enough.The appropriate distribution for the description of the small-scale fading dependson whether the LOS signal path is present or not. In NLOS scenarios, the as-sumption of a large number of impinging signals having mutually independentrandom complex amplitudes leads to a Rayleigh distribution for the magnitudeof the combined narrowband signal.

To regard the wideband channel characteristics, the impulse response or,equivalently, the transfer function needs to be modeled. We will always consideraslowly time-variantchannel, by which we mean that the impulse response maybe assumed as invariant for the duration of a burst transmission. The equivalentbaseband version of the time-invariant impulse response is represented by thecomplex-valued functionc(τ). The corresponding transfer function is defined


as the Fourier transform ofc(τ), i.e.C(f) =∫∞−∞ c(τ)e−j2πfτdτ , and deter-

mines the attenuation and phase rotation of a narrowband signal aroundf .

Some wideband channel models simply include one or a set of representativeimpulse responses. To be independent of specific realizations ofc(τ), the lattercan also be viewed as a random process. The notion that there is no positionalinterdependence between the reflectors or scatterers motivates the assumption ofa nonstationary zero-mean white complex random process characterized by

E [c(τ)c∗(τ ′)] = φc(τ)δ(τ − τ ′), (2.1)

whereE[·] andδ(·) represent the expectation and the Dirac delta function, re-spectively, and the asterisk in the superscript denotes complex conjugation. Inthis commonuncorrelated scattering(US) model the power delay profile (PDP),represented byφc(τ) in (2.1), provides second-order a priori information aboutthe impulse response.

To reflect the situation in a NLOS environment with small-scale fading, therandom processc(τ) may be modeled as white complex Gaussian. In this case,any random variable|C(f)| with a fixedf ∈ R exhibits a Rayleigh distribu-tion. Furthermore, theoretical considerations and results from measurementcampaigns suggest an exponentially decaying PDP [10], given as

φc(τ) =τ−1d exp(−τ/τd), τ ≥ 0

0, τ < 0.(2.2)

As mentioned above, larger RMS delay spreadsτd imply faster fluctuations inthe realizations|C(f)| per bandwidth, i.e. a smallerWcoh.

In this work we will characterize the error rate performance of different MCtransmission schemes in environments with a fading channel by the mean BERversus the mean signal-to-noise ratio (SNR). The mean is always computed oversome distributions which reflect the random channel variations due to the small-scale fading. That is, the performance figures include the impact of the small-scale fading, whereas the frequency-nonselective large-scale fading must be con-sidered separately together with the other pass loss terms.

2.3 The OFDM Concept 13

2.3 The OFDM Concept

The complex envelope of an OFDM transmit signal may be expressed as

s(t) =√S0

NB−1∑k=0

NC−1∑n=0

xk,n gk,n(t), (2.3)

whereNC andNB determine the numbers of subchannels and time slots, respec-tively, xk,n is a complex-valued data symbol,gk,n(t) represents the elementarysignal pulse being modulated by the information inxk,n, andS0 is a real positiveconstant. The pulse associated with thekth time slot and thenth subchannel hasthe waveform

gk,n(t) =

T− 1

2F ej2π(n−NC/2)F∆(t−kT∆), t ∈ [kT∆ − TG, kT∆ + TF)

0, t /∈ [kT∆ − TG, kT∆ + TF)(2.4)

with TF =1/F∆ andT∆ =TF+TG, where the role ofTF andTG will become clearbelow. For the moments(t) may be regarded as a superposition ofNC linearlymodulated single carrier signals with rate1/T∆ and carrier frequency separationF∆ and overlapping(sinx/x)2-power spectra. The data symbols, which are typ-ically from anM -ary phase-shift keying (PSK) or quadrature amplitude modu-lation (QAM) signal set, are viewed in the following as independent randomvariables subject toE

[|xk,n|2

]=1 ∀k∈0, . . . , NB−1∀n∈0, . . . , NC−1.

If NC is large, the composite signals(t) exhibits a rather flat power spectrumwithin the occupied frequency band with a power density ofS0.

Fig. 2.1 illustrates the synthesis of the OFDM signal within thekth time slotfor an exemplary case withNC = 8 and quadrature PSK (QPSK) modulation.The upper section shows the real parts of the subchannel signals, and the lowersection contains the real part and the magnitude of the resultings(t).

Within the interval[kT∆, kT∆ +TF) the harmonic waveforms differ by oneperiod and the subchannel signal parts are hence mutually orthogonal, i.e.∫ kT∆+TF

kT∆

xk,ngk,n(t)x∗k,mg∗k,m(t)dt = δn,m|xk,n|2,

whereδn,m represents the Kronecker delta. The main point in OFDM is that thisorthogonality remains valid at the output of a time-invariant multipath channel


Re[xk,0gk,0(t)]

Re[xk,1gk,1(t)]

Re[xk,2gk,2(t)]

Re[xk,3gk,3(t)]

Re[xk,4gk,4(t)]

Re[xk,5gk,5(t)]

Re[xk,6gk,6(t)]

Re[xk,7gk,7(t)]

Re[s(t)]

|s(t)|t0

0

kT∆−TG kT∆ kT∆+TF

Figure 2.1: Composition of an OFDM transmit signal.

as a result of the cyclic signal extensions in the preceding guard periods, pro-vided that their lengthTG exceeds the excess delay spreadτexcess defined as themaximum delay difference between the overlapping multipath signal parts.

To reveal this property and proceed with the description of the conventionalOFDM demodulation, we assume that the received signal is given as

y(t) = (c ∗ s)(t) + v(t), (2.5)

where the asterisk denotes convolution andc(τ) = 0 for τ /∈ [0, TG). The addi-tive white Gaussian noise (AWGN) processv(t) with the power spectral densityN0 models the thermal noise in the receiver front end. Now, an orthonormalprojection of the received signal parts within the intervals of lengthTF onto thetime-limited exponentials

φk,n(t) =

T− 1

2F ej2π(n−NC/2)F∆(t−kT∆), t ∈ [kT∆, kT∆ + TF)

0, t /∈ [kT∆, kT∆ + TF)

2.3 The OFDM Concept 15

yields the coefficients

uk,n =∫ ∞−∞

y(t)φ∗k,n(t)dt,k=0, . . . , NB−1n=0, . . . , NC−1,

which serve as decision variables for the detection. With (2.5),uk,n can bewritten as

uk,n =∫ ∞−∞

∫ TG

0

c(τ)s(t − τ)dτ φ∗k,n(t)dt+∫ ∞−∞

v(t)φ∗k,n(t)dt,

and using (2.3) this expression can be transformed into

uk,n=√S0

NB−1∑`=0

NC−1∑m=0

x`,m

TG∫0

c(τ)

∞∫−∞

g`,m(t−τ)φ∗k,n(t)dtdτ+

∞∫−∞

v(t)φ∗k,n(t)dt.

As a consequence of∫ ∞−∞

g`,m(t− τ)φ∗k,n(t)dt = δk,`δn,me−j2π(m−NC/2)F∆τ

for τ ∈ [0, TG), uk,n can be expressed as

uk,n =√S0αnxk,n + vk,n

with the complex gain factor

αn =∫ ∞−∞

c(τ)e−j2π(n−NC/2)F∆τdτ (2.6)

and the random noise term

vk,n =∫ ∞−∞

v(t)φ∗k,n(t)dt.

The integral in (2.6) suggests that the gain factor can also be expressed as

αn = C

((n− NC

2

)F∆

).

Furthermore, the random variables resulting from the orthonormal projectionof the AWGN processv(t) are independent and zero-mean complex Gaus-sian distributed with varianceN0. Speaking of acomplexGaussian dis-tributed random variableX with meanm and varianceσ2 it is meant that


Re[X ] and Im[X ], i.e. the real and imaginary parts ofX , are indepen-dent with Re[X ] ∼ N (Re[m], σ2/2) and Im[X ] ∼ N (Im[m], σ2/2), whereN (m0, σ

20) represents a (real) Gaussian or normal distribution with meanm0

and varianceσ20 . From the independence ofRe[X ] andIm[X ] it follows that

E[(X−m)(X−m)∗

]=σ2.

We may finally express the decision variables in vector notation. The columnvectoruk = (uk,0, . . . , uk,NC−1)T is actually given as

uk =√S0Cxk + vk, (2.7)

where xk = (xk,0, . . . , xk,NC−1)T and (·)T stands for transposition. Ad-ditionally, C is the diagonal matrix with the elementsα0, . . . , αNC−1, andvk = (vk,0, . . . , vk,NC−1)T represents a zero-mean jointly complex Gaus-sian random vector with the covariance matrixN0INC , with IN denoting theN×N -identity matrix.

The diagonal form ofC and the covariance matrix results from the preservedsignal orthogonality. Interference between symbols transmitted on different sub-channels or within consecutive time slots, in the sequel termed interchannel andinterblock interference (ICI and IBI), respectively, is avoided and a straightfor-ward data detection enabled.

The mapping of a vector√S0xk ontouk according to (2.7) is also described

by the equivalent OFDM channel model in Fig. 2.2. The model includesNC

independent (narrowband) AWGN channels, each preceded by a multiplication.For a more detailed introduction to OFDM modulation with a number of

applications the reader is referred to [15]. Further introductory articles are [16,17, 18, 19].

2.4 DFT-Based Transceiver Structure

The modulation and demodulation concepts described in the previous section arewell suited for digital signal processors. The computation of the sample valuesof the transmit signal for a certain time slot from the associated data symbolsactually corresponds to performing an inverse discrete Fourier transform (IDFT),for which efficient FFT algorithms may be employed. To see this, let the vectorsk=(sk,0, . . . , sk,NFFT−1) contain theNFFT sample values ofs(t) taken at thetimeskT∆ + iTF/NFFT, i=0, . . . , NFFT−1. With (2.3) and (2.4) and assuming

2.4 DFT-Based Transceiver Structure 17

+...

+

+ +

...... α0

αNC−1

vk,0

vk,NC−1

√S0xk,0

√S0xk,NC−1

uk,0

uk,NC−1

Figure 2.2: Equivalent OFDM channel model.

an evenNC we obtain

sk,i =√S0

TF

NC/2−1∑n=−NC/2

xk,n+NC/2 ej2πni/NFFT , i = 0, . . . , NFFT−1,

which coincides with the definition of the IDFT of the vector(xk,NC/2, . . . , xk,NC−1, xk,0, . . . , xk,NC/2−1) for NFFT = NC, except ofa scaling factor. The discrete time signal for thekth time slot is finallyconstructed by adding the cyclic prefix, i.e. preceding the vectorsk with thevector(sk,NFFT−NG , . . . , sk,NFFT−1) with NG appropriately chosen. Similarly,a discrete Fourier transform (DFT) of the sampledy(t) within [kT∆, kT∆ + TF)yields the vectoruk.

Normally,NFFT is chosen larger thanNC and zeros are inserted in the mid-dle of the actual IDFT input vector. This oversampling prevents aliasing in thesignal produced by the digital-to-analog conversion (DAC). The recurring com-ponents in the spectrum around multiples of the sampling frequency are removedby a subsequent lowpass filter with the frequency responseGTX(f) before thetransmit signal is transfered to the RF. In the receiver an antialiasing filter withthe frequency responseGRX(f) is employed after the down-conversion, fol-lowed by the analog-to-digital conversion (ADC) and the cyclic prefix removal.Oversampling may also be applied in the receiver by using more thanNC points


signalencoding

up-conversionto RF

cylic signalextension

IDFTprocessing

DAC

signaldecoding

cyclic prefixremoval

DFTprocessing

ADC

datain

dataout

transmitter path

receiver path

down-conv.from RF

GTX(f)

GRX(f)

Figure 2.3: OFDM transceiver elements.

in the DFT computation. Fig. 2.3 summarizes the order of the transmitter and re-ceiver path elements. The signal encoding and decoding is discussed in Sect. 2.5.

The impact of the filtering by the lowpass and the antialiasing filters maybe taken into account by usingC(f) = GRX(f)C(f)GTX(f) as the overalltransfer function. If its inverse Fourier transformc(τ) is zero forτ /∈ [0, TG)the matrixC in (2.7) keeps its diagonal from. We shall further assume that thefrequency responsesGTX(f) andGRX(f) have the value1 within the essentialfrequency band and that therefore the filters have no impact on the diagonalelements ofC and the covariance matrix of the noise vector.

2.5 Coding and Interleaving Techniques

In the conventional OFDM receiver the subchannels appear totally frequency-nonselective, and hence the situations where the gain of a subchannel is in adeep fade have a dominant effect on the mean BER of an uncoded data transmis-sion over a fading channel. The application of error control coding is thereforemandatory for achieving acceptable error rates. Unlike in a single carrier trans-mission where the frequency diversity is exploited by the equalizer, the FECcoding introduces the necessary redundancy for achieving frequency diversityin OFDM systems.

Trellis-coded modulation (TCM) is an advantageous technique for band-width efficient data transmission. By accomplishing the encoding and the map-ping of the encoded bit sequence onto signals jointly, the trellis codes can bedesigned to maximize the Euclidean distance in the signal space rather than theHamming distance of different code words. Traditional TCM schemes, how-ever, like the original ones proposed in [20] were conceived for AWGN chan-

2.5 Coding and Interleaving Techniques 19

nels. Their performance is unsatisfactory in the presence of fading since they donot aim to achieve diversity. Improved trellis codes were proposed in the follow-ing for a serial data transmission over time-variant channels. They require theemployment of a time domain interleaver after the TCM encoder and a deinter-leaver before the decoder to avoid bursts of unreliable decoder input symbols.An overview of the design methods for such improved trellis codes can be foundin [21], Chap. 5. TCM can also be used with an OFDM transmission. In broad-band systems, the interleaver normally operates in the frequency domain in orderto avoid burst errors due to fades in the channel transfer function.

It was pointed out in [22] that the diversity order can be maximized for agiven decoder complexity by an interleaving at bit-level rather than at symbol-level. The proposed technique in which the encoding and the modulation (andthe demodulation and decoding in the receiver) are decoupled and a bit inter-leaver interposed is today called BICM (bit-interleaved coded modulation). In-deed, a simple BICM outperforms a TCM with comparable complexity in manytypical transmission scenarios involving fading channels.

Fig. 2.4 illustrates a BICM-OFDM transmission in a block diagram. Afteran initial convolutional encoder, the bit stream passes through an interleaver.Its output is transformed into a sequence ofL-tuples, and each of them is en-coded into an element from a given signal set of sizeM = 2L by the mappingµ :0, 1L→C . The obtained signals constitute the vectorsx0, . . . ,xNB−1, fromwhich the transmit signal is generated by means of IDFTs as discussed in theprevious section.

convolutionalencoder

IDFT,front end

serial/parallel-conversion

bit-wiseinterleaver

bits-to-signalmapper

Viterbidecoder

parallel/serial-conversion

deinterleaver signalde-mapper

front end,DFT

frequency-selectivechannel

datain

dataout

......

transmitter

receiverCSI

Figure 2.4: BICM-OFDM transmission.

In the receiver, the de-mapper transforms every complex value from the DFTback into bit information. This can be in the form of hard-decisions, obtained by


a so-called slicer which simply chooses the nearest signal point in the complexplane. Specifically, the slicer yields

bk,n = arg minb∈0,1L

∣∣∣uk,n −√S0αnµ(b)∣∣∣

as the bit tuple for the DFT outputuk,n, which is actually the maximum-likelihood (ML) estimate. For this coherent detection the de-mapper requireschannel state information (CSI), i.e. knowledge of the subchannel gain factors.The binary values are then deinterleaved, and a Viterbi decoder delivers the finalbit estimates.

Alternatively, the de-mapper may provide soft bit information to a subse-quent soft value deinterleaver and a soft input Viterbi decoder. The bit log-likelihood ratios (LLRs) serve as the branch metrics for the ML sequence de-coder (MLSD). The LLR of the th bit encoded in a signal associated with thenth subchannel is defined as

Λn,`(u) = lnp(u|b` = 0)p(u|b` = 1)

, (2.8)

whereb` denotes theth bit in b, and the conditional probability density func-tions (PDFs) for the complex-valued DFT outputu are given as

p(u|b` = i) =∑

b∈0,1Lb`=i

p(b|b` = i)p(u|b)

= 2−L+1∑

b∈0,1Lb`=i

1πN0

exp(−|u−

√S0αnµ(b)|2N0

)(2.9)

for i = 0, 1. The expression in the second line is obtained by assuming that allbit combinations are equiprobable. This implies independence of the encodedbits in a signal, resulting from an ideal interleaver. We note that here the CSI inthe receiver must include the AWGN power spectral densityN0.

There is a significant performance gain from using a soft-decision based re-ceiver architecture instead of a slicer, at the cost of higher complexities in thechannel estimator, de-mapper, deinterleaver and decoder. The LLR computa-tion complexity can be reduced with a minor impact on the performance by theapproximation

p(u|b` = i) ≈ 2−L+1 maxb∈0,1Lb`=i

p(u|b)

2.6 Synchronization and Channel Estimation 21

for the conditional PDFs in (2.8), which is equivalent to using thesimplifiedbranch metricsdescribed in [23].

As opposed to a ML trellis decoder the separate calculation of the LLR forevery bit under the assumption of independent co-bits with equiprobable bit val-ues in a signal, and the ML decoding based on these metrics does not necessarilyyield the code word for which the likelihood function of the corresponding sig-nal sequence is maximized. A joint ML detection of the bits encoded in everysignal is impractical for a MLSD since the interposed interleaver would let thenumber of states explode. Iterative decoding techniques are a suboptimum ap-proach for taking a priori information about the co-bits into account. With amaximum a posteriori decoding employing e.g. the well-known Bahl-Cocke-Jelinek-Raviv algorithm [24], a posteriori probabilities can be computed for theencoded bits, and these can be used as the a priori probabilities for the co-bitsin the following iteration. An appropriate engine for updating the a priori prob-abilities was described in [25], and its employment for the decoding of BICMsignals was proposed in [26]. It turns out that by applying thisturbo principleinthe decoding a data transmission close to the capacity limit is possible. However,convergence of the probabilities in the low SNR region is only attainable withmapping schemes that differ from the usual Gray encoding [27]. Additionally,complex interleavers are required, causing significant latency of the data.

2.6 Synchronization and Channel Estimation

Before starting the detection process, the receiver must synchronize its clocksand the received signal with respect to the time slots and the carrier frequency,where the latter is more demanding. Special preamble signals, which e.g. alwaysprecede the information-bearing signal parts in a bursty transmission, facilitatethe initial synchronization. The carrier frequency and phase can in the followingbe tracked by means of pilot signals on dedicated subchannels as available e.g.in HIPERLAN/2 systems, or by a more sophisticated signal processing on thebasis of the DFT output.

In OFDM receivers, particular attention needs to be paid to phase noise. Itwas shown in [28] that phase noise leads to both a common phase error in thecoefficients from the DFT and to ICI. The latter is much more difficult to dealwith and increases with the number of subchannels in the OFDM scheme. Amethod for the compensation of both frequency offsets and phase noise was


proposed in [29].

The estimation of the parametersα0, . . . , αNC−1 is another central task in anOFDM receiver. A simple method involves the transmission of pilot symbols,i.e. symbols which are known to the receiver, in dedicated time slots. Estimatesfor the channel parameters are then obtained by multiplying the observed DFToutput in these time slots with the inverse of the pilot symbols. More powerfultechniques take into account that the fading processes on neighboring subchan-nels are correlated. In the presence of AWGN and if the cross-correlations ofthe channel parameters are available, which is e.g. the case for US channels withknown PDP, Wiener filters are an obvious choice as they minimize the mean-square error. To save bandwidth, pilot symbols may be transmitted in only asubset of the subchannels, and a Wiener filter used to obtain estimates for allα0, . . . , αNC−1. Of course, such an interpolation must be extended to the timedimension if the channel is time-variant. Channel estimation for OFDM systemsemploying Wiener filters is discussed in e.g. [30] for the time-invariant case, andin e.g. [31] for time-varying channels. Robust estimators are derived in [32, 33]for receivers without exact knowledge of the PDP.

2.7 Error Performance and Spectral Efficiency

We now address the error rate performance of an OFDM receiver applying thecoherent demodulation techniques described in the preceding sections. Idealtransmitter and receiver front ends are assumed as well as perfect CSI in thereceiver. The analysis is restricted to QPSK, and the 16-QAM and 64-QAM sig-nal sets with square lattice signal constellations, using a Gray encoding schemefor the mapping of theL/2 bits onto the real and the same number of bits ontothe imaginary signal parts. For convenience, the one-dimensional labeling ofGray encoded signals is illustrated in Fig. 2.5. The information bits are assumedmutually independent with equiprobable values.

Due to the separability of the bits mapped onto the quadrature signal partsin coherent detection, we can assess the error probabilities by focusing on eitherof the one-dimensional real signal spaces containing

√M signal points. For a

given channel realization and a QPSK signal set, the BER on thenth subchannelafter a slicer is given as

P(n)b = Q(

√γn) ,

2.7 Error Performance and Spectral Efficiency 23

L/2=1:

L/2=2:

L/2=3:

0 1

00 01 11 10

000 001 011 010 110 111 101 100

Figure 2.5: One-dimensional signal spaces for Gray encoded QPSK (L=2),16-QAM (L=4), and64-QAM (L=6).

whereγn = S0|αn|2/N0 represents the actual SNR in the decision variablesu0,n, . . . , uNB−1,n. The corresponding formulas for 16-QAM and 64-QAMare [34]

P(n)b =

34

Q

(√15γn

)+

12

Q

(√95γn

)− 1

4Q(√

5γn)

and

P(n)b =

712

Q

(√121γn

)+

12

Q

(√37γn

)− 1

12Q

(√2521γn

)

+112

Q

(√277γn

)− 1

12Q

(√16921

γn

),

respectively. These two expressions actually represent the average error proba-bilities of the bits in anL-tuple. The averaged overall BER is given as

Pb =1NC

NC−1∑n=0

P(n)b .

We now turn to the case of a random, slowly time-variant channel. If therandom factorsα0, . . . , αNC−1 are identically distributed, which is usually thecase in wireless OFDM transmission, the mean bit error probabilityPb =E[Pb]is obtained by simply picking out one of the subchannels for the analysis. ForRayleigh distributed|αn| with E

[|αn|2

]= 1 ∀n∈ 0, . . . , NC−1, the actual

SNR values are chi-square distributed and

Pb =∫ ∞

0

Q (√γ)

1γe−γ/γdγ =

12

(1−

√γ

2 + γ

)(2.10)


results for the QPSK case, whereγ=S0/N0 denotes the mean SNR. Note thathereγ corresponds to the averagetransmit signal power spectral densityoverN0, rather than thebit energyoverN0 ratio. The respective expressions for theQAM schemes are

Pb =38

(1−

√γ

10 + γ

)+

14

(1−

√γ

10/9 + γ

)− 1

8

(1−

√γ

2/5 + γ

)(2.11)

for 16-QAM, and

Pb =724

(1−√

γ

42+γ

)+

14

(1−√

γ

14/3+γ

)− 1

24

(1−√

γ

42/25+γ

)+

124

(1−√

γ

14/27+γ

)− 1

24

(1−√

γ

42/169+γ

)(2.12)

for 64-QAM.

The above expressions also represent the limits for the average BERPb inbroadband systems asNC → ∞ and in the same timeW/Wcoh→∞. Moreprecisely,limNC→∞N−1

C

∑NC−1n=0 P

(n)b = Pb holds if the process|α0|, |α1|, . . .

is stationary and mean-ergodic. We shall call an OFDM channel with a verylarge number of subchannels and with the associated gain factors revealing theergodic characteristics of the fading anideal broadband(IB) OFDM channel1.

An analytical calculation of the exact error rate performance of a transmis-sion with FEC is very difficult in general. For convolutional and trellis codes,performance upper bounds can be derived by identifying the diverse error eventsand using the union bound. We do not bother here about such analytical methodsbut rather rely on numerical results. A BICM-OFDM scheme withNC = 256is assumed, employing the rateRC = 1/2 convolutional code with the genera-tors133oct and171oct. This widely used code has 64 states and Hamming freedistanceDfree = 10. The subchannel gain factors are modeled as independentwith Rayleigh fading characteristics in the simulations. Using a random numbergenerator, a new channel and bit interleaver permutation are generated for eachburst comprising20 time slots. The averaged BER results are shown in Fig. 2.6for providing the Viterbi decoder with either hard-decisions or the LLRs defined

1The slowly time-variant OFDM channel can be viewed as ablock-fading channel, which wasintroduced in [35] and thoroughly studied in [36]. The particular case of an infiniteNC correspondsto the case withno delay constraintsin [36].

2.7 Error Performance and Spectral Efficiency 25

0 5 10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

me

an

BE

R

uncoded, QPSKuncoded, 16-QAMuncoded, 64-QAMBICM QPSK, hard-decisionBICM 16-QAM, hard-decisionBICM 64-QAM, hard-decisionBICM QPSK, soft-decisionBICM 16-QAM, soft-decisionBICM 64-QAM, soft-decision

Figure 2.6: Mean BERs versus the mean SNR of an uncoded OFDM transmis-sion and a BICM-OFDM transmission over a slowly time-variant Rayleigh fad-ing channel, employing QPSK, 16-QAM and 64-QAM, and either hard-decisionor soft-decision decoding in the BICM case.

in (2.8) and (2.9). Additionally, the figure contains the mean bit error probabili-ties (2.10), (2.11) and (2.12) of a corresponding uncoded transmission.

The utilized FEC, bits-to-signal mapping and modulation schemes complywith the definitions in the IEEE 802.11a and HIPERLAN/2 standards. The largeNC and the assumed independence among the subchannel fading processes,however, let the results be representative mainly for systems where the inter-leaver depth by far exceedsWcoh. Otherwise, the performance gain from theFEC coding is smaller as a consequence of the reduced diversity.

The spectral efficiency of a certain transmission scheme is defined as theinformation bit rateR perW . The bit rate equalsR = RCNCL/T∆, and abandwidth usage ofW =NCF∆ can be assumed neglecting surplus bandwidth


usually needed at the band edges. We hence obtain

R

W=

RCL

T∆F∆.

The factor(T∆F∆)−1, which corresponds to the ratio ofTF andT∆, reflects theloss due to the guard periods. ChoosingTG equal toτexcess and a very largeT∆

keeps the loss of bandwidth negligible. However, this may necessitate a hugenumber of subchannels.

2.8 Information Theoretic Considerations

The information capacity of a multipath channel with AWGN and a certain inputsignal power spectrum is normally computed by subdividing the channel into alarge number of parallel narrowband channels, similar to applying an OFDMscheme, and summing up the respective capacities [37]. This shows that a re-liable data transmission arbitrarily close to the multipath channel capacity cantheoretically be attained with an OFDM modulation with a sufficiently largeNC

and adequate signal set and coding complexities.The capacity of the OFDM channel shown in Fig. 2.2 is given as

COFDM =1NC

NC−1∑n=0

CAWGN(γn),

whereCAWGN(γ) = ld(1 + γ) represents the capacity of an AWGN channel2

with SNRγ. The capacityCOFDM is normalized byNC as there areNC symbolstransmitted in a time slot.

For a channel with time-variant SNR, being used over a very long timeexceeding the coherence time and revealing the ergodic properties of the fad-ing process, the capacity3 is obtained by viewingγ as a random variable withE[γ]= γ and computing the expectation ofCAWGN(γ) [39]. We can express thecapacity of an IB OFDM channel accordingly, i.e.

COFDM,IB(γ) =∫ ∞

0

CAWGN(γ)1γe−γ/γdγ =

∫ ∞0

ld(1 + γ)1γe−γ/γdγ

2Due to the usage of complex-valued information symbols, such an AWGN channel is equivalentto two Gaussian channels [38], each having capacity1

2ld(1 + γ).

3This capacity is also sometimes calledergodic capacity.

2.8 Information Theoretic Considerations 27

for Rayleigh fading. By Jensen’s inequality, the capacity of any fading channelis smaller thanCAWGN(γ).

To achieveCAWGN(γ) the source must exhibit a Gaussian distribution. Thisis not the case for the assumed QPSK or QAM constellations with equiprobablesignals. The separate de-mapping of every encoded bit without using informa-tion about the co-bits in a BICM scheme implies a further degradation of themutual information. The actual BICM channel capacity can be expressed as

CBICM(γ) =L∑`=1

(h(u)− h(u|b`)) ,

whereh(·) andh(·|·) denote the differential entropy and the conditional dif-ferential entropy, respectively, and the observationu is given asu = µ(b)+v

with v being a complex Gaussian noise term with varianceσ2n =E

[|µ(b)|2

]/γ.

Using the formulas for the entropy and the normal distribution, e.g.h(Z) =−∫∞−∞

∫∞−∞ pX(x)pY (y)ld(pX(x)pY (y)) dxdy for the differential entropy of a

complex random variableZ=X+jY with the PDFspX(·) andpY (·), we have

CBICM(γ) =L∑`=1

(−∞∫−∞

∞∫−∞

∑b∈0,1L

1M

1πσ2

n

exp(−|(x+jy)− µ(b)|2

σ2n

)×

ld

( ∑b∈0,1L

1M

1πσ2

n


σ2n

))dxdy

+12

∞∫−∞

∞∫−∞

∑b∈0,1Lb`=0

2M

1πσ2

n


σ2n

)×

ld

( ∑b∈0,1Lb`=0

2M

1πσ2

n

exp(−|(x+jy)−µ(b)|2

σ2n

))dxdy

+12

∞∫−∞

∞∫−∞

∑b∈0,1Lb`=1

2M

1πσ2

n


σ2n

)×

ld

( ∑b∈0,1Lb`=1

2M

1πσ2

n

exp(−|(x+jy)−µ(b)|2

σ2n

))dxdy

).


−5 0 5 10 15 20 25 300

1

2

3

4

5

6

γ [dB]

cap

acity

[bit/

sym

bo

l]

CAWGN(γ)CBICM(γ) for 64-QAMCBICM(γ) for 16-QAMCBICM(γ) for QPSK

Figure 2.7: AWGN channel capacity and BICM channel capacities for QPSK,16-QAM and 64-QAM.

The capacity of the IB BICM-OFDM channel with Rayleigh fading character-istics, denoted byCBICM,IB(γ), is again found by numerically computing theexpectation ofCBICM(γ) with γ being a chi-square distributed random variablewith mean valueγ.

In Fig. 2.7, the resultingCBICM(γ) for employing QPSK, 16-QAM and64-QAM with Gray encoding are contrasted with the AWGN channel capacityCAWGN(γ). Fig. 2.8 shows the corresponding curves for an IB OFDM channelwith Rayleigh fading. Obviously, the BICM channel capacitiesCBICM(γ) andCBICM,IB(γ) are quite close toCAWGN(γ) andCOFDM,IB(γ), respectively, upto certain (mean) SNR values beyond which they flatten out approachingL.

Other bits-to-signal mapping schemes achieve smaller capacities, as shownin [23]. This paper also considers the capacity for accomplishing the de-mappingjointly for all the bits encoded in a signal (referred to ascoded modulation), and

2.8 Information Theoretic Considerations 29

−5 0 5 10 15 20 25 300

1

2

3

4

5

6

γ [dB]

cap

acity

[bit/

sym

bo

l]

COFDM,IB(γ)CBICM,IB(γ) for 64-QAMCBICM,IB(γ) for 16-QAMCBICM,IB(γ) for QPSK

Figure 2.8: IB OFDM and IB BICM-OFDM channel capacities for Rayleighfading.

the respective cutoff rates.

Chapter 3

Adaptive OFDMTransmission

Adaptability is of great importance to let FWA systems cope with the varying en-vironmental conditions and the specific quality-of-service (QoS) requirements ofa subscriber. On the physical layer there are actually various forms of adaptationconceivable. A basic version involves the definition of a set of coding and mod-ulation schemes offering different data rates and FEC capabilities, and the selec-tion of the most expedient among them for every link. If the transmitter has CSIavailable, the adaptation can be accomplished on a subchannel-by-subchannelbasis. That is, every subchannel is assigned a distinct modulation scheme andtransmit power level depending on its actual gain factor.

In most systems there is a limit to the overall transmitted power. It is well-known that the optimization of the information theoretic capacity leads to apower allotment which relates to a ”water-filling”. However, for practical sys-tems operating far from the theoretical capacity other optimization criteria likee.g. the BER or the data rate are more adequate. A variety of techniques foradapting the signaling to the channel state under different constraints have beenproposed in recent years, and some are already in use in xDSL equipment.

If the channel characteristics frequently change, which is normally the casein wireless systems, assigning every subchannel its own modulation scheme re-sults in a considerable additional complexity as the receiver needs to be informedall the time about the applied modulation pattern. Varying only the power while

32 Chapter 3: Adaptive OFDM Transmision

utilizing identical signal sets on all subchannels is a simpler form of adaptationby which the aforementioned complexity increase is avoided.

This chapter focuses on appropriate power control techniques for BICM-OFDM systems. In particular, adaptation policies which directly aim to optimizethe BER at the decoder output are derived. To start, the water-filling principle isaddressed.

3.1 Maximizing Capacity: The Water-Filling Prin-ciple

For the considerations in this chapter, the OFDM transmit signal is assumed tohave the form

s(t) =NB−1∑k=0

NC−1∑n=0

√ωnxk,ngk,n(t)

with the real positive parametersω0, . . . , ωNC−1 defining the power al-lotment. Specifically, ωn determines the mean energy in the signals√ωnx0,n, . . . ,

√ωnxNB−1,n and at the same time the transmit signal power

spectral density at the subchannel frequency(n−NC/2)F∆. The parametersare subject to

1NC

NC−1∑n=0

ωn = S0, (3.1)

whereS0 represents the average power spectral density ofs(t) andWS0 theoverall transmit power.

Now, the SNR for the decision variablesu0,n, . . . , uNB−1,n associated withthenth subchannel equalsγn=ωn|αn|2/N0. The OFDM channel capacity withtransmitter side CSI is given as

COFDM,CSIT = maxω0+...+ωNC−1≤NCS0

1NC

NC−1∑n=0

ld(

1 +ωn|αn|2N0

).

Finding the optimized power allocation achievingCOFDM,CSIT is a typical prob-lem that can be solved by Lagrange’s method. This translates the problem to the

3.1 Maximizing Capacity: The Water-Filling Principle 33

maximization of the cost function

J(ω0, . . . , ωNC−1, λ) =NC−1∑n=0

ld(

1 +ωn|αn|2N0

)− λ

(NC−1∑n=0

ωn −NCS0

)with the Lagrange multiplierλ∈ R. DifferentiatingJ(ω0, . . . , ωNC−1, λ) withrespect to every argument yields the conditions

(ln 2)−1 1N0/|αn|2 + ωn

− λ = 0, n = 0, . . . , NC−1

andNC−1∑n=0

ωn −NCS0 = 0

for the unconstrained case. Using the Kuhn-Tucker conditions to handle thecases where some energies would become negative we finally obtain

ωn = max

(λ ln 2)−1−N0/|αn|2, 0, n = 0, . . . , NC−1, (3.2)

where the term(λ ln 2)−1 has to be chosen such that the power constraint (3.1)is fulfilled.

Optimizing the power according to (3.2) and (3.1) can be depicted as fillingup a basin with a shaped bottom with water until the content reaches a certainvalue, as illustrated in Fig. 3.1. This is called thewater-filling(or water-pouring)principle. In the present case, the bottom shape corresponds toN0/|αn|2,n = 0, . . . , NC− 1, i.e. the AWGN power spectral density over the sampledsquared channel transfer function magnitude, and the depth of the water definesthe subchannel signal power.

Fig. 3.2 shows the capacity of an IB OFDM channel with transmitter sideCSI, denoted byCIB,CSIT(γ), for the Rayleigh fading case. The figure addition-ally presents the capacitiesCBICM,IB,WF(γ) of the considered IB BICM-OFDMchannels with Rayleigh fading, with the power allotment defined by (3.2) and(3.1). All capacity curves are found by numerical methods since analytical ex-pression are not available.

Comparison of the curves in Fig. 3.2 and Fig. 2.8 leads to the conclusionthat ”water-filling” yields only a marginal capacity increase. The larger the SNRthe closer the coincidence of the curves in the two figures. It is noteworthy thatcapacities beyond those in Fig. 2.7 of a non-fading channel are achieved by thepower optimization in the low SNR region in Fig. 3.2.


ωn

f−8F∆ +7F∆

N0

/∣∣C2(f)∣∣

WS0

n= . . . 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Figure 3.1: Power optimization by ”water-filling” for an OFDM transmissionwith NC =16 over a frequency-selective channel.

Similar results as for the IB OFDM channel are presented in [39] for theequivalent scenario of a serial transmission over a frequency-nonselective fad-ing channel with transmitter side CSI and a constraint on the long-term averagetransmit power. Further power adaptation strategies motivated by informationtheoretic considerations are presented in [40]. Specifically, power control poli-cies which minimize the information outage probability, i.e. the probability thatthe instantaneous capacity falls below a certain level, are derived in [40] for boththe case of constraining the average power within finite time periods, which cor-responds to having a limited number of subchannels, and the case of constrain-ing the average power over an infinite time, which corresponds to having an IBOFDM channel.

3.2 Adaptive Modulation Techniques

Practical systems usually employ transmission schemes with moderate complex-ities and operate far below the channel capacity. Adaptation techniques shouldtherefore be elaborated on the basis of the predetermined coding and modulation

3.2 Adaptive Modulation Techniques 35

−5 0 5 10 15 20 25 300

1

2

3

4

5

6

γ [dB]

cap

acity

[bit/

sym

bo

l]

CIB,CSIT(γ)CBICM,IB,WF(γ) for 64-QAMCBICM,IB,WF(γ) for 16-QAMCBICM,IB,WF(γ) for QPSK

Figure 3.2: IB OFDM and IB BICM-OFDM channel capacities for Rayleighfading and applying ”water-filling”.

methods rather than accomplishing a ”water-filling” for marginally increasingthe theoretical capacity. The optimization of the error rate performance throughcontrolling the power and the modulation level of amplitude modulated signalstransmitted over frequency-nonselective fading channels was already studied in1974 by Hentinen [41]. Much more recently, the development of broadband MCcommunication systems has motivated proposals for adapting the signaling on asubchannel-by-subchannel basis. Algorithms for optimizing the signal set sizesin DMT applications, like for instance the rather heuristic procedures suggestedin [42] and [43], are often calledbit-loading algorithms, though they can ofcourse also be employed in wireless transmission. Techniques for OFDM sys-tems aiming to minimize the transmit power by adapting jointly theM -QAMschemes and the power while providing a fixed data rate and BER level are pre-sented in [44, 45, 46]. The algorithm proposed in [47] minimizes the BER under


a constant data rate and transmit power constraint. Maximizing the spectral effi-ciency subject to a mean power and BER constraint for a variable-rate adaptiveserialM -QAM transmission over fading channels is discussed in [48, 49, 50],and also in [51] for imperfect outdated CSI.

Substantial gains are achieved by all these techniques due to avoiding ahighly unreliable data transmission in frequency or time slots which are affectedby a deep channel fade. The same effect, however, also results from a FECcoding via achieving time/frequency diversity. Hence, the performance gainsclaimed in the above cited publications for uncoded systems do not translate to atransmission with FEC. Moreover, the potential of adaptive modulation dependson the nature of the encoding and decoding methods, and these should there-fore be taken into account in the algorithm design. This was done in [52] andalso in [51] for certain TCM systems. Furthermore, a suboptimum strategy for aBICM transmission, according to which the actual channel fading level directlydetermines the signal set size, is presented in [53].

Adaptive modulation is certainly useful for cable based communication sys-tems, but the much swifter state variations of radio channels make the necessaryexchange of control information in wireless transmission systems a quite com-plex and bandwidth consuming task. This overhead can be avoided by onlyvarying the power while utilizing the same modulation on all subchannels andthereby eliminating the need to inform the receiver about the instantaneous com-bination of utilized signal sets. Since the receiver may interpret the variationsin the subchannel power levels as being caused entirely by the channel, there isno necessity for modifications in the channel estimation and equalization proce-dures, and the power adaptation technique can be readily employed within theexisting standards. In a TDD transmission, an extra control channel is not evennecessary in the reverse link. Instead of being informed about the channel stateby the opposite station the transmitter may assume channel reciprocity and relyon the parameters that have been estimated for the coherent demodulation of aforegoing reverse link burst. Depending on the coherence time of the channel,estimation errors have to be taken into account in the subsequent power alloca-tion.

In the next section we will derive power optimization schemes for BICM-OFDM systems on the basis of either perfect, or outdated CSI due to channelvariations between the uplink and the downlink periods.

3.3 Adaptive Power Allocation for BICM-OFDM Systems 37

3.3 Adaptive Power Allocation for BICM-OFDMSystems

The adaptation policies proposed in [44, 45, 46, 47, 48, 52] control the powersuch that a certain target BER is attained on every subchannel. This leads to achannel inversion strategy, turning the fading channel into an ideal channel withidentical SNR values on all subchannels at the receiver side. If the objectiveis to let the receivers manage without real channel estimation and equalizationparts, such an approach is also calledpre-equalization. An unlimited channelinversion is, however, normally not feasible as e.g. the inverse of a Rayleighdistributed random variable has infinite variance. Pre-equalization techniquespaying respect to this fact are discussed in [54, 55].

Choosing the parametersω0, . . . , ωNC−1 inversely proportional to|α0|2, . . . , |αNC−1|2 is not optimal with respect to the average BER. Thesame attributes to the power optimization by Lagrange’s method proposedin [56], which relies on a rough approximation for the error probabilities.Our objective in the following is to minimize the BER at the decoder outputof the BICM-OFDM transmission depicted in Fig. 3.3. Note that the blockdiagram is equivalent to the one in Fig. 2.4 except of the additional powerallocation block with CSI in the transmitter part. The number of bits conveyedper symbol may vary from subchannel to subchannel. We letMn= 2Ln denotethe predetermined size of the signal set associated with thenth subchannel,and restrict the possible signal sets to QPSK andM -QAM schemes withM ∈ 16, 64, 256, . . . and square lattice signal constellations. In any case,the firstLn/2 bits are mapped onto the real signal part, and the secondLn/2onto the imaginary signal part, both by a Gray encoding scheme. To limit thecomplexity or be compatible with some existing standards, a unique modulationscheme may be used on all subchannels, i.e.Ln = L. Alternatively, the belowderived power adaptation techniques may be combined with an appropriatealgorithm for adaptingL0, . . . , LNC−1. The definition of such an algorithm isaddressed in [57], but not considered in this work.

In [23] an equivalent channel model was introduced for the analysis of aBICM system with ideal interleaving, consisting of a set of parallel independentbinary input channels and a switch which randomly selects one of the channelsfor every bit at the encoder output. We employ a similar model for the consideredadaptive BICM-OFDM transmission. As sketched in Fig. 3.4, every subchannel


convolutionalencoder

IDFT,front end

serial/parallel-conversion

bit-wiseinterleaver

adaptive power allocation

bits-to-signalmapper

Viterbidecoder

parallel/serial-conversion

deinterleaver signalde-mapper

front end,DFT


datain

dataout

... ... .........

transmitter

receiverCSI

CSI

Figure 3.3: BICM-OFDM transmission with adaptive power control.

is represented by a number of parallel binary input channels, each modeling themapping, transmission and de-mapping of the bits from a specific position inthe respectiveLn-tuples. The totalNBC =

∑NC−1n=0 Ln parallel channels are

binary inputbinary output for the case of performing hard-decision decoding,and binary inputcontinuousoutput for the case of soft-decision decoding. Thepreceding switch, which selects a channel for every encoded bit, models the in-terleaver. We let(φi, ψi) index the specific binary channel chosen for theith bit,whereφi∈0, . . . , NC−1 andψi∈1, . . . , Lφi. The interleaver depth corre-sponds to the number of encoded bits in a burst, and the permutation is randomlygenerated and unknown to the power allocation unit. The burst size shall be largeenough to justify the assumptions of an ideal interleaver, and a sequence of inde-pendent random variable pairs(φ1, ψ1), (φ2, ψ2), . . . with uniform distribution,i.e.

Pr [(φi, ψi) = (n, `)] = N−1BC ∀n∈0, . . . , NC−1 ∀`∈1, . . . , Ln.

Based on this model, the optimum power allocation is found in the followingfirst for a receiver employing a slicer and perfect transmitter side CSI, then forthe case of outdated CSI, and finally for using a soft de-mapping and a soft inputMLSD, assuming again perfect CSI.

3.3.1 Optimum Power Allocation for Hard-Decision Decod-ing

If the receiver performs hard-decision decoding, a slicer accomplishes the de-mapping and every binary input binary output channel in Fig. 3.4 can be as-sociated with a certain BER. Let us analyze the error probabilities for theLnbinary channels forming thenth subchannel. The ideal interleaving perfectly


......

...

(φi, ψi)(φi, ψi)

binary channel(0, 1)

binary channel(0, L0)

binary channel(NC−1, 1)

binary channel(NC−1, LNC−1)

Figure 3.4: Equivalent BICM-OFDM channel model.

cancels the correlations between successive bits in the stream and makes all bitcombinations in theLn-tuples transmitted on this subchannel equiprobable. Weagain focus on either of the one-dimensional real signal spaces with

√Mn Gray

encoded signals. Their average energy before the slicer isωn|αn|2/2 and thedistance between adjacent signal points equals

dn=√

6ωn|αn|2/(Mn − 1). (3.3)

The exact error probability for a bit transmission over a specific binary channelcan be computed by first expressing the probabilities of a corresponding bit er-ror event conditioned on each of the

√Mn signals, and second averaging over

these conditional probabilities. ForLn ∈ 4, 6, . . . this yields a weighted sumof Q-functions similar to the expressions (2.11) and (2.12). We shall rely on ap-proximate error expressions for these QAM signal sets for the sake of simplicity,obtained by considering only the dominant error events where the slicer choosesa neighboring signal. The resultingLn/2 approximate error probabilities for thebinary channels associated with the real (resp. the imaginary) signal part bits ofthenth subchannel are

P(n,`)b = 2−(`−1)Q

(dn/2√N0/2

)

= 2−(`−1)Q

(√3 ωn|αn|2

(Mn−1)N0

), ` = 1, . . . , Ln/2. (3.4)


The factor before the Q-function reflects the a priori probability for the`th bit tohave a neighboring signal with opposite bit value in Gray encoding.

As a consequence of the ideal interleaving and the random permutation, thesuperchannel constituted by all the entities from the interleaver in the transmitterup to the deinterleaver in the receiver may be viewed as a memoryless binarysymmetric channel (BSC) with the transition probability

PSC =2

NBC

NC−1∑n=0

Ln/2∑`=1

P(n,`)b

=4

NBC

NC−1∑n=0

√Mn − 1√Mn

Q

(√3 ωn|αn|2

(Mn−1)N0

), (3.5)

corresponding to the averaged error probabilities of the binary channels. Theasserted symmetry actually does not hold for the exact bit error probabilitiesfor Gray encoded QAM. However, the superchannel can be made symmetricby accomplishing complementary bits-to-signal mappings for the real and theimaginary signal part bits on each subchannel.

Since the BER of a coded transmission over a memoryless BSC de-creases with the transition probability of the channel, we find the optimumω0, . . . , ωNC−1 by an optimization with respect toPSC. Provided that the fac-tors |α0|, . . . , |αNC−1| are perfectly known, the expression in the second lineof (3.5) can directly be minimized subject to the power constraint in (3.1).We again use Lagrange’s method for this optimization problem and obtainJ(ω0, . . . , ωNC−1, λ) =

NC−1∑n=0

√Mn − 1√Mn

Q

(√3 ωn|αn|2

(Mn−1)N0

)− λ

(NC−1∑n=0

ωn −NCS0

)

as the cost function and∂J(ω0, . . . , ωNC−1, λ)/∂ωn = 0, n = 0, . . . , NC−1as necessary conditions for the optimum mean signal energies. Computing thepartial derivatives yields

√Mn − 1√2π√Mn

exp(−1

2ωnBn

) √Bn

2√ωn

+ λ = 0, n=0, . . . , NC−1,

whereBn=3|αn|2/((Mn−1)N0). After some simple algebra and the cancella-tion of constant terms the aboveNC conditions may finally be expressed in the


0 1 2 3 4 5 6 7 8 9 100

1

2

100

101

102

103

0123456

x

x

W(x)

W(x)

Figure 3.5: Plots of the W-function.

form

An exp(ωnBn)ωn = λ0, n=0, . . . , NC−1,

whereAn = Mn(Mn−1)/((√Mn−1)2|αn|2) andλ0 is a positive constant.

Solving these equations forω0, . . . , ωNC−1 yields

ωn =1Bn

W(BnAn

λ0

), n = 0, . . . , NC−1. (3.6)

Here,W(·) denotes the real-valued Lambert’s W-function defined as the inverseof the functionf(w)=wew for w≥0.

As can be seen from Fig. 3.5,W : [0,∞)→ [0,∞) is a concave and un-bounded function withW(0) = 0 andW(x)≤x. As a consequence, the uniquesolution forω0, . . . , ωNC−1 can be found by the following simple iterative pro-cedure:


(i) Choose a small positiveλ0 which fulfills

1NC

NC−1∑n=0

λ0

An≤ S0.

(ii) Calculate

S0 =1NC

NC−1∑n=0

1Bn

W(BnAn

λ0

).

(iii) If S0 is not yet sufficiently close toS0, multiply λ0 by S0/S0 and go backto step (ii).

(iv) Computeω0, . . . , ωNC−1 according to (3.6).

If the channel possesses the IB property andM0 = . . .=MNC−1 =M , theoptimum power assignment to the subchannels can be expressed as

ωn = ω(|αn|2

), n=0, . . . , NC−1,

where the functionω(·) is independent of the channel realization. To findω(·)we note that the transition probabilityPSC can now be formulated as the ex-pectation of the subchannel bit error probability over the random fading level,i.e.

PSC =4

ldM

√M−1√M

∫ ∞0

Q

(√3 ω(y)y

(M − 1)N0

)p|α|2(y)dy, (3.7)

with p|α|2(y) representing the PDF of the squared gain factor magnitude. Fur-thermore, the power constraint (3.1) can be replaced by∫ ∞

0

ω(y)p|α|2(y)dy = S0. (3.8)

The minimization of (3.7) subject to (3.8) can be carried out analytically usingcalculus of variations. For Rayleigh fading withp|α|2(y) = e−y this leads to theEuler-Lagrange differential equation

(1 + βω(y)y) (ω′(y)y + ω(y))− 2ω(y) = 0

with β=3/((M−1)N0). Its solution is given as

ω(y) =1βy

W(λ0y

2),


whereλ0 has to chosen such that (3.8) is fulfilled. As expected, the above for-mula corresponds to (3.6).

Fig. 3.6 displays the normalized mappings|αn|2 7−→ω(|αn|2)/S0 forM=4and variousγ. Apparently, the required transmit power decreases as|αn|2→∞.In fact, W(·) can be approximated by a constant for large arguments, causingω(|αn|2) to decay with about1/|αn|2. On the other hand, it is not worth ex-pending much power on subchannels with a very poor gain. A high BER has tobe accepted on these subchannel. In the neighborhood of0 the W-function maybe approximated asW(x)≈x, so that hereω(|αn|2)≈(λ0/β)|αn|2.

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

|αn|2

ω(|α

n|2

)/S

0

γ=3 dB

γ=4.5 dB

γ=6 dB

γ=7.5 dB

γ=9 dB

Figure 3.6: Optimized mappings|αn|2 7−→ ω(|αn|2)/S0 for the IB Rayleighfading channel with different mean SNRs, employing QPSK and hard-decisiondecoding.


3.3.2 Outdated Channel State Information

We now consider a channel which changes its state from burst to burst.As proposed in [51], the time-variant channel parameters are modeled as astationary complex Gaussian random process, which isNC-dimensional inour case, denoted byc(t) = (c0(t), . . . , cNC−1(t))T. The random vectorc=(α0, . . . , αNC−1)T =c(T1) represents the channel state at the timeT1 of theactual burst transmission, and the CSI in the transmitter is now reduced to the ob-servationc = (α0, . . . , αNC−1)T = c(T0) of the random process at the timeT0,i.e. during a foregoing reverse link burst reception.

Let Rc =E[ccH

]=E

[ccH

]represent the covariance matrix for bothc and

c, andRcc =E[ccH

]the cross-covariance matrix of the two random vectors,

where(·)H denotes Hermitian transposition. As shown in Appendix A,Rc andRcc can be derived from the scattering function if the channel complies with thewide-sense stationary US (WSSUS) model. Furthermore, the composite randomvector

(cT, cT

)Tis jointly complex Gaussian. As a consequence, the random

vectorc exhibits a noncentral jointly complex Gaussian distribution when con-ditioned onc, with the mean

(µ0, . . . , µNC−1)T = E[c∣∣c] = RccR−1

c c

and the conditional covariance matrix

Rc|c = Rc −RccR−1c RH

cc

with the diagonal elementsσ20 , . . . , σ

2NC−1 [58].

The squared gain factor magnitudes are thus noncentral chi-square dis-tributed with two degrees of freedom, with the conditional PDFs [37]

p|αn|2(y|c) =1σ2n

e−(|µn|2+y)/σ2n

∞∑k=0

1(k!)2

(|µn|σ2n

)2k

yk, y ≥ 0,

n = 0, . . . , NC−1. The bit error probabilities defined in (3.4) as functions of|α0|2, . . . , |αNC−1|2, andPSC are now also random variables. We choose tominimize the conditional meanPSC = E[PSC|c], although the resulting powerallocation scheme is not strictly optimal with respect to the mean error proba-bility after the decoder due to the nonlinear relationship between the BER ofthe coded and the uncoded systems. However, if the conditional variance of


PSC =(2/NBC)∑NC−1

n=0

∑Ln/2`=1 P

(n,`)b is sufficiently small as a consequence of

a largeNC the excursion from the optimum solution may actually be assumedas negligible.

With (3.5),PSC can be written as

PSC =4

NBC

NC−1∑n=0

√Mn−1√Mn

E

[Q

(√3 ωn |αn|2

(Mn−1)N0

)∣∣∣∣∣ c]

=4

NBC

NC−1∑n=0

√Mn−1√Mn

∫ ∞0

Q

(√3 ωn y

(Mn−1)N0

)p|αn|2(y|c)dy. (3.9)

Taking this expression as the starting point, the optimization can be accom-plished in a similar fashion as in Sect. 3.3.1. The partial derivatives of the termsin (3.9) with respect toω0, . . . , ωNC−1 can be computed before the integrationis carried out. After some calculus we obtain the conditions

σ2n

Mn+√Mn

L

(3σ2

n ωn2(Mn−1)N0

;|µn|2σ2n

)= λ0, n = 0, . . . , NC − 1,

whereλ0 is again a positive constant,

L(x; a) =

√1

x(x+ 1)3e−a

∞∑k=0

Γ(k + 32 )

(k!)2

(a

x+ 1

)k,

x ∈ (0,∞)a ∈ [0,∞)

andΓ(·) denotes the gamma function. LetL−1(·; a) be the inverse of the func-tion L(·; a) with respect to the first argument and defined on(0,∞). The opti-mizednth subchannel mean signal energy is now given as

ωn =2(Mn−1)N0

3σ2n

L−1

(Mn+

√Mn

σ2n

λ0;|µn|2σ2n

), n = 0, . . . , NC − 1.

(3.10)The inverse functionsL−1(·; a) may be computed off-line for variousa andstored in a lookup table. Fig. 3.7 depicts the mappingsx 7−→ L−1(x; a) fora number of differenta. The functions are unbounded and strictly decreasingfor positivex. Hence, there again exists a unique solution for the parametersω0, . . . , ωNC−1, which can be found by evaluatingλ0 in an appropriate iterativeprocedure as in Sect. 3.3.1.

Fig. 3.8 shows the resulting mappings|αn|2 7−→ω(ρ)(|αn|2)/S0 for a QPSKtransmission over an IB Rayleigh fading channel withγ equal to6 dB. As the


10−10

10−8

10−6

10−4

10−2

100

102

10−3

10−1

101

103

x

L−1(x; a)

a = 0a = 2a = 4a = 6a = 8a = 10

Figure 3.7: Plots of the functionL−1(·; a) for variousa.

covariance and cross-covariance matricesRc = INC andRcc = ρINC , respec-tively, are used withρ = 1, 0.95, 0.9, 0.8, whereρ = 1 implies perfect CSI.That is, uncorrelated subchannel fading is assumed and the coefficientρ de-fines the autocorrelation between a gain factor and its outdated observation. Asshown in Appendix A, the interrelationRcc = ρRc and thusµn = ραn andσ2n=(1−ρ2)E

[|αn|2

], n=0, . . . , NC−1, result in the general case of a WSSUS

channel with a separable scattering function.

The figure is generated by a channel realization with a huge number of sub-channels with gains distributed according to a Rayleigh PDF, and computing thepower allocation by (3.10) forρ < 1 and (3.6) forρ= 1. The power allotmentturns out to be more balanced for smallerρ. Approaching a uniform power al-location as the gain predictability decreases seems plausible. We also note that

lim|αn|2→0

ω(ρ)(|αn|2)>0 if ρ<1.


0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

|αn|2

ω(ρ

)(|α

n|2

)/S

0

ρ=1ρ=0.95

ρ=0.9ρ=0.8

Figure 3.8: Optimized mappings|αn|2 7−→ω(ρ)(|αn|2)/S0 for the IB Rayleighfading channel withγ equal to6 dB,M = 4 and hard-decision decoding, basedon perfect CSI (ρ=1) and outdated CSI (ρ<1).

3.3.3 Power Adaptation for Soft-Decision Decoding

We now return to the case of having perfect CSI in the transmitter, and aim tofind the optimum power allotment if the receiver performs soft-decision decod-ing. The idea of directly linking the system BER to the BER of the uncodedtransmission subsystem via the BSC does not work out if the de-mapper pro-vides soft bit information, and we have to find another way here for obtaining anappropriate cost function.

In fact, the dominant error events in the low BER region are those wherethe decoder chooses an erroneous code word which is a Hamming distance ofDfree off the correct encoded bit sequence. Minimizing the probability of theseerror events is thus a reasonable though not the optimum approach. Rememberthat the soft input MLSD simply adds the branch metrics, i.e. the LLRs, of the


bits and finally chooses the code word for which the sum is maximum. As aconsequence of the random bit permutation, the probability of any specific errorevent with a corresponding Hamming distance ofDfree does not depend on theactual positions of the erroneous bits and can be expressed as

Pe = Pr

[Dfree∑i=1

Λφi,ψi(ui) < 0

∣∣∣∣∣ ψith bit encoded inui is 0for i=1, . . . , Dfree

], (3.11)

whereu1, . . . , uDfree is a sequence of signals from the DFT associated with thesubchannelsφ1, . . . , φDfree . Here, the probability space is constituted by thepermutation in the interleaver, represented by(φ1, ψ1), . . . , (φDfree , ψDfree), andthe co-bits and noise inu1, . . . , uDfree . Taking the all-zero sequence as a refer-ence is possible because of the complementary bits-to-signal mapping for thequadrature signal parts1.

A DFT outputu associated with thenth subchannel represents a noisy ver-sion of a signal in

αn√ωnµ(b) : b∈0, 1Ln

. We first consider a trans-

mission with QPSK on all subchannels. In this case the set has the elements±αn

√ωn/2± jαn

√ωn/2

. The LLR from the de-mapper for e.g. the real

part bit is a linear function ofRe[(α∗n/|αn|)u], where the factor(α∗n/|αn|) com-pensates for the phase rotation in the subchannel. Specifically,

Λn,1(u)=lnexp

(− (Re[(α∗n/|αn|)u]−dn/2)2

N0

)exp

(− (Re[(α∗n/|αn|)u]+dn/2)2

N0

) =2dnN0

Re[(α∗n/|αn|)u] (3.12)

with dn = |αn|√

2ωn. As the noise term inRe[(α∗n/|αn|)u] is Gaussian dis-tributed with varianceN0/2 we haveΛn,1(u) ∼ N (d2

n/N0, 2d2n/N0) provided

that the first bit encoded inu is zero. If conditioned on the second bit the sameapplies toΛn,2(u). The probabilityPe expressed in (3.11) can now be writtenas

Pe = E

Q

∑Dfreei=1 d2

φi

/N0√∑Dfree

i=1 2d2φi

/N0

= E

Q

√√√√ 1

2N0

Dfree∑i=1

d2φi

(3.13)

1A mapping for which the all-zero sequence may be used as a reference for the derivation of theBER via the Euclidean distance is called regular [59]. In our case, however, we only have regularityin the mean sense due to the random permutation.


and

Pe = E

Q

√√√√ 1N0

Dfree∑i=1

|αφi |2ωφi

.With higher-level QAM schemes, the LLR defined in (2.8) and (2.9) is a

non-linear function ofu, i.e.

Λn,`(u) = ln

∑b∈0,1Ln

b`=0

exp(− |u−αn

√ωnµ(b)|2N0

)∑

b∈0,1Lnb`=1

exp(− |u−αn

√ωnµ(b)|2N0

) , ` = 1, . . . , Ln.

Fig. 3.9 depicts the three LLR functionalsΛn,1(u), Λn,2(u) andΛn,3(u) for thereal part bits of a 64-QAM constellation with Gray encoding and a certain signalpoint separation and noise variance. For the samedn andN0 the additionaldashed curve represents the LLR of a QPSK bit computed in (3.12).

Apparently, the LLR curves of the 64-QAM bits take a rather linear runwithin the intervals between two neighboring signal points with opposite valuesin the respective bit position. Furthermore, within such an interval the curvesclosely match with the LLR of a QPSK bit with coinciding two signal points.For the 16-QAM and 64-QAM cases and a sufficiently high SNR we may assertthat the dominant error events result in erroneous bit sequences which corre-spond to signals located at mostdn off the correct signals in the complex planes.Consequently, power adaptation in the general case of having mixed modulationschemes may also rely on a minimization of the cost function (3.13). With (3.3),the general cost function now reads

P ′e = E

Q

√√√√ 3N0

Dfree∑i=1

|αφi |2ωφiMφi − 1

. (3.14)

Unlike in the hard-decision case, the real time calculation ofω0, . . . , ωNC−1

by minimizing (3.14) subject to (3.1) is a difficult task. The computation ofP ′efor a certain subchannel power allocation actually involves an averaging over alltheNDfree

C possible combinations ofφ1, . . . φDfree .

For an IB channel withM0 = . . .=MNC−1 =M the optimization problem


64-QAM:

QPSK:

Re[µ

(000···)]

Re[µ

(001···)]

Re[µ

(011···)]

Re[µ

(010···)]

Re[µ

(110···)]

Re[µ

(111···)]

Re[µ

(101···)]

Re[µ

(100···)]

Re[µ

(0·)]

Re[µ

(1·)]

0

Re[

uαn√ωn

]

Λn,`(u)

`=1 (64-QAM)`=2 (64-QAM)

`=3 (64-QAM)

`=1 (QPSK)

Figure 3.9: LLR functionals for64-QAM and QPSK signal sets with Gray en-coding, and some exemplaryαn, ωn andN0.

can be stated as follows: Find the mappingω : [0,∞)→ [0,∞) which minimizes

P ′e =

∞∫0

· · ·∞∫

0

Q

√√√√ 3N0

Dfree∑i=1

yiω(yi)M − 1

p|α|2(y1)· · · p|α|2(yDfree)dy1· · ·dyDfree

subject to (3.8). This problem can be tackled analytically using e.g. calculus ofvariations, or by extensive numerical methods. The latter approach leads to theresults shown in Fig. 3.10 for the Rayleigh fading case, i.e. forp|α|2(y)=e−y.The figure displays the signal amplification versus the gain in a subchannel forvariousγ.

As could be expected, the required transmit power decreases also here as|αn|2 →∞. On the other hand, no power at all is expended if|αn|2 falls be-low a certain threshold, and a BER of0.5 is deliberately accepted on such asubchannel.

3.4 Performance Analysis 51

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

|αn|2

ω(|α

n|2

)/S

0

γ=3 dB

γ=4.5 dB

γ=6 dB

γ=7.5 dB

γ=9 dB

Figure 3.10: Optimized mappings|αn|2 7−→ ω(|αn|2)/S0 for the IB Rayleighfading channel with different mean SNRs,M=4 and soft-decision decoding.

3.4 Performance Analysis

We now assess the achievable performance gains by utilizing the power adapta-tion techniques derived in the previous section. Computer simulations are car-ried out for this purpose, utilizing the same FEC, interleaving, bits-to-signalmapping, burst dimensions, and Rayleigh fading characteristics as in the nu-merical performance analysis in Sect. 2.7. First, we let the receiver accomplishhard-decision decoding and provide the transmitter with perfect CSI. The signalset sizes are uniform, i.e.M0 = . . . = MNC−1 = M , with M = 4, 16, or 64.Fig. 3.11 shows the resulting error rates for either applying the power controlpolicy defined in (3.6), or allocating the power in a uniform fashion, where inthe latter cases the curves coincide with the respective error rates in Fig. 2.6.

Obviously, the performance gain by the power adaptation grows with the


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

me

an

BE

R

QPSK, uniform power allocationQPSK, power adaptation16-QAM, uniform power alloc.16-QAM, power adaptation64-QAM, uniform power alloc.64-QAM, power adaptation

Figure 3.11: Error rate performance of a BICM-OFDM transmission with hard-decision decoding, with either an adaptive power allocation based on perfect CSIor a uniform power allocation.

SNR and attains approximately4 dB at a BER level of10−6 for all three mod-ulation schemes. As the gain factors for the256 subchannels are generated in-dependently, these simulation results actually reflect the situation in a nearly IBtransmission. In systems where the assumptionW Wcoh does not hold, theachievable performance gain depends on the actual channel state, specifically onthe range over which the magnitudes|α0|, . . . , |αNC−1| vary.

Fig. 3.12 compares the achieved BER with QPSK and perfect CSI againstthe corresponding error rates with outdated CSI, assuming againRc = INC andRcc =ρINC . Both the power control schemes according to (3.6) and (3.10) areapplied in the scenarios with outdated CSI.

The results indicate that the achievable performance gain is significantlysmaller for outdated CSI, even with an autocorrelation coefficient ofρ= 0.95.


8 9 10 11 12 13 14 15 16 17 1810

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

me

an

BE

R

no power adaptationρ=0.80, adaptation policy for perfect CSIρ=0.80, adaptation policy for outdated CSIρ=0.95, adaptation policy for perfect CSIρ=0.95, adaptation policy for outdated CSIρ=1.00, adaptation policy for perfect CSI

Figure 3.12: Error rates with a slowly time-variant channel, applying both thepower adaptation policies designed for perfect and outdated CSI.

Additionally, it turns out that taking the channel variations into account, i.e.employing the adaptation scheme (3.10), is clearly beneficial. Forρ= 0.8, anadaptive subchannel power control based on the assumption of having perfectCSI results in even higher error rates than a nonadaptive transmission.

Finally, in Fig. 3.13, we investigate the potential of an adaptive power load-ing if the receivers are equipped with a soft input MLSD. Every power coef-ficient is determined from the respective subchannel gain by the numericallyoptimized mappings depicted in Fig. 3.10. A large number of subchannels withindependent Rayleigh fading characteristics are employed in the simulations toclosely match the IB assumption.

Obviously, the error rate performance can only be marginally improved inthis case. There is a gain of approximately1 dB for a QPSK transmission at aBER of10−6. Surprisingly, a uniform power allocation yields even better results


0 2 4 6 8 10 12 14 16 18 20 2210

−8

10−7

10−6

10−5

10−4

10−3

10−2

γ [dB]

me

an

BE

R

QPSK, uniform power allocationQPSK, power adaptation16-QAM, uniform power alloc.16-QAM, power adaptation64-QAM, uniform power alloc.64-QAM, power adaptation

Figure 3.13: Error rate performance of a BICM-OFDM transmission over anIB channel with soft-decision decoding, with either power adaptation based onperfect CSI or a uniform power allocation.

with 64-QAM above a BER level of10−7. This has to be attributed to the onlyloose approximation of the non-linear LLR functionals.

The benefits of the derived power adaptation strategies were also investi-gated in [60] in the context of the ADAMAS2 project. The performance analysisin [60] was based on a reference channel realization found as representative foran urban environment and the ADAMAS RF and bandwidth of5.8 GHz and7 MHz, respectively, rather than on a stochastic channel model.

2ADAMAS stands for ADAptive Multicarrier Access System, being a research project in the IST(Information Society Technologies) framework funded by the European Union.

3.5 Summary and Discussion 55

3.5 Summary and Discussion

Strategies for adapting the power spectrum of the transmitted OFDM signal tothe channel state have been derived in this chapter in different contexts. If theinformation capacity is to be maximized, a ”water-filling” procedure yields theoptimum distribution of the power among the subchannels. Other adaptationschemes result if the objective is to optimize the error rate performance. Thebit error probability at the decoder output in a BICM-OFDM system has beenchosen as the optimization criterion in the main part of the chapter, assuming anideal interleaving with a random bit permutation.

If the de-mapping is accomplished by a simple slicer, the subsystem com-posed of all transmission chain elements from the interleaver in the transmitterup to the deinterleaver in the receiver can be viewed as a BSC. As a consequence,the optimum power allocation scheme can be found by minimizing the transitionprobability of this BSC, which can be accomplished by Lagrange’s method. Anequivalent procedure yields the optimum power allotment if only outdated CSIis available, and the channel state from burst to burst is described by a stationarycomplex Gaussian random process.

It has turned out that a considerable transmit power saving in the order ofseveral decibels can be achieved by the power loading if the receiver performshard-decision decoding, whereas only a minor gain is attainable in the case ofa soft-decision decoding. This result is in line with the findings that the infor-mation capacity of a fading channel can only be increased marginally by poweradaptation. The elaborated technique is therefore particularly attractive for thedownlink in systems where the subscriber units need to be cheap, favoring asimple hard-decision decoding technique. Like the recently proposed transmitdiversity schemes, power adaptation allows for a performance improvement byadding complexity at the transmitter side.

Chapter 4

Multicarrier Transmissionwith Nonorthogonal Pulses

Extending the signal cyclically in inserted guard periods in the transmitter, andobserving only the transition-free signal intervals in the receiver is a simple yeteffective means to evade both ICI and IBI. However, while being favorable interms of receiver complexity the OFDM concept implies a degradation of thebandwidth efficiency by the factor(T∆−TG)/T∆. A value close to1 can beattained for this factor by dimensioning the time slots long compared to thechannel delay spread, i.e.T∆ τexcess, but with a largeT∆ and a smallF∆

the system becomes vulnerable to slow channel variations and to phase noiseand frequency offsets caused by hardware imperfections1. Furthermore, a hugenumber of subchannels are necessary for achieving high data rates in e.g. outdoorenvironments with severe multipath propagation. As a consequence, the poweramplifiers must be operated with large back-offs to cope with the high powerpeaks in the transmit signals.

Alternative MC modulation schemes without guard periods, along with ap-propriate receivers which deal with resulting ICI/IBI, have the potential to ex-ploit the available bandwidth resources even with only a moderate number ofsubchannels. A further benefit of abandoning the OFDM concept is the in-

1A unified analysis of the impact of these three phenomena on the effective SNR can be foundin [61].

58 Chapter 4: Multicarrier Transmission with Nonorthogonal Pulses

creased flexibility in the pulse design. In this chapter we review proposals forsuch an alternative MC transmission, and study a class of modulation schemesin which the elementary signal pulses relate to a so-called prototype functionshifted in time and frequency. It is shown that the transceivers can still rely onefficient FFT methods for the signal synthesis and analysis. In Sect. 4.3, we de-vise receiver structures incorporating an iterative detection or decorrelation, andassess their computational complexities. Finally, the nonorthogonal MC trans-mission is compared against a conventional OFDM system in terms of the errorrate performance and the capacity.

4.1 Alternative Multicarrier Transmission Tech-niques

With the guard periods being omitted, the receiver may compensate for theinterference caused by dispersive channels by means of appropriate equalizerstructures. Multiple input multiple output (MIMO) equalizers can be employedto perform the equalization in the frequency domain after the DFT. Minimummean-square error (MMSE) criterion based MIMO equalizers are derived in ageneral context in [62] and investigated for the symbol detection in MC systemswithout guard periods in [7]. For DMT systems with short guard periods, timedomain equalizers were also considered which aim to shorten an over-lengthchannel impulse response such that it fits into the guard period [63, 64, 65].

With a MC signaling using the elementary pulses defined in (2.4), multi-path propagation leads to significant ICI even among data symbols conveyed onsubchannels at distant locations in the frequency ifτexcess clearly exceedsTG.Both the extent of ICI and the vulnerability to narrowband noise can be limitedby the use of alternative pulse shapes exhibiting steeper slopes in the frequencydomain. This is proposed in [66] for a bandwidth efficient DMT modulationsystem. In [67], sets of orthonormal pulse shapes with superior TF localizationcharacteristics, i.e. concentration of the energy in both time and frequency, arepresented for limiting both IBI and ICI. With this approach, the pulse overlap-ping can be bounded and the amount of hampering interference reduced com-pared to a single carrier transmission with an equivalent data rate without wast-ing bandwidth by inserting guard periods.

The MC signal synthesis using waveforms that differ from those in (2.4) is

4.2 Weyl-Heisenberg System Related Multicarrier Modulation 59

sometimes represented by a bank ofNC pulse shaping filters. Another filterbank (FB) may also be employed in the receiver for the signal analysis. In [68],the filters in such a configuration are designed to minimize ICI/IBI at the outputof the receiver FB for WSSUS channels. Other FB based transmission systemswere investigated using the framework of discrete time FB theory. In the pa-pers [69, 70], optimized finite impulse response (FIR) filters are derived on thebasis of a perfectly known channel impulse response. In [71], the receiver FBis extended into a MIMO decision feedback equalizer structure, and the filtercoefficients are again found by applying the MMSE criterion.

In this work, the FB based MC transmission and receiver techniques pro-posed in [72, 73] are reviewed and further elaborated. The filters and the channelare modeled in continuous time. The receiver consists of an initial FB yielding avector representation of the distorted signal, and a subsequent signal processingstructure incorporating the channel estimation and information recovery.

4.2 Weyl-Heisenberg System Related MulticarrierModulation

It is well-known that from an information theoretic point of view orthogonal ba-sis functions are the optimum choice for the pulse shapes for data transmissionover AWGN channels. With a multipath channel, however, the situation is differ-ent since anyway the orthogonality of the transmitted waveforms normally getsdestroyed. It was pointed out in [68] that for minimizing ICI/IBI after dispersivechannelsnonorthogonalpulse shapes are the best choice. In the same paper, thepulse shaping and mismatched filters for the transmitter FB and the receiver FB,respectively, are derived from optimized prototype functions by uniform shiftsin the frequency domain. Such a configuration is of particular interest since itallows for an efficient digital realization using FFT methods.

The discrete time synthesis of the transmit signals(t) defined in (2.3) canactually be accomplished by IDFT computations if the not necessarily orthog-onal waveforms of the elementary pulses can be expressed as the TF-translatedversions

gk,n(t) = g(t− kT∆)ej2π(n−NC/2)F∆(t−kT∆),k=0, . . . , NB−1n=0, . . . , NC−1

(4.1)


of a prototype functiong(t) with energy∫∞−∞ |g(t)|2dt= 1. Note that the con-

stantsT∆ andF∆ again define the spacings of the possibly overlapping pulsesin time and frequency, respectively. The samples of the signal part composed ofthe modulated pulses with time indexk now take the values

sk,i =√S0 g(TFi/NFFT)

NC/2−1∑n=−NC/2

xk,n+NC/2 ej2πni/NFFT ,

where i is taken from the seti∈Z : TFi/NFFT∈Ig and the time intervalIg=[T0, T1] chosen such that it contains the temporal support ofg(t). As shownin [74], the composite discrete time transmit signalsi, i∈Z can be generated bytheuniform DFT FBin Fig. 4.1, comprising an IDFT computation and a subse-quent unit withNC FIR filters with the transfer functionsG0(z), . . . , GNC−1(z).The FIR filters represent the so-called polyphase components and they are de-termined by the shape ofg(t).

...

xk,0

xk,1

xk,NC−1

IDFT

G0(z)

G1(z)

GNC−1(z) ↑NC

↑NC

↑NC

z−1

z−1

z−1

si

IDFT processor polyphase filtering

Figure 4.1: Discrete time signal synthesis using a uniform DFT FB.

The systemg(t−kT∆)ej2πnF∆t : k, n∈Z

is termed a Weyl-

Heisenberg (WH) system in the mathematical literature. A necessary conditionfor the system to form a Riesz basis [75] (for the spanned space), i.e. beingcomposed of linearly independent elements, is that the density of the functionsin the TF plane fulfillsT∆F∆ ≥ 1 [76]. If the WH system forms a Riesz basisthere exists a biorthogonal WH system

ξ(t−kT∆)ej2πnF∆t : k, n∈Z

derived

4.2 Weyl-Heisenberg System Related Multicarrier Modulation 61

from a prototype functionξ(t) such that⟨g(t−kT∆)ej2πnF∆t, ξ(t−`T∆)ej2πmF∆t

⟩=δk,`δn,m,

where〈f, g〉 =∫∞−∞ f(t)g∗(t)dt represents the scalar product. If the channel

is ideal, an interference-free data transmission with a subchannel symbol rateof 1/T∆ can be realized by employing a bank of pulse shaping filters with theresponsesg(t)ej2πnF∆t, n = 0, . . . , NC− 1 in the transmitter, and a bank offilters matched to the functionsξ(t)ej2πnF∆t, n= 0, . . . , NC−1 in the receiver.In discrete time FB theory a corresponding set-up is called perfect reconstructionsystem, composed of a synthesis FB in the transmitter and an analysis FB in thereceiver.

Furthermore, forT∆F∆ > 1 any such WH system is incomplete in theHilbert space of the finite-energy signals. Consequently, a corresponding MCtransmission does not fully exploit the available time-bandwidth resourcesfor the information transfer. An OFDM signaling with guard periods be-longs to this category. If on the contraryT∆F∆ < 1, the functions ing(t−kT∆)ej2πnF∆t : k, n∈Z

get linearly dependent. A reliable detection

may therefore be difficult in such a MC system, especially when employing ahigh-order modulation scheme.

We restrict our attention to the case where the WH system relating to thetransmit pulse shapes represents a Riesz basis, and thusT∆F∆≥1. Additionally,we consider the transmission over multipath channels in which the pulses getdistorted, with the channel state being unknown to the transmitter. It is shownin [73] that the pulse overlapping at the receiver side can be confined by using aTF concentrated prototype functiong(t), unless the channel is extremely disper-sive. This simplifies the work of the iterative symbol detectors discussed below.The Gaussian functions

fG(t) = (2β)14 exp

(−πβt2

)with β > 0 are an obvious choice forg(t), since they provide optimum TFconcentration of the signal energy in the sense that the uncertainty product [77]is minimized. As a property of the Gaussian functions, their shape does notchange by the Fourier transform, given as

FG(f) = (2/β)14 exp

(−πf2/β

),


except for a scaling. We note that by adjustingβ the slope offG(t) can beincreased at cost of the slope ofFG(f) or vice versa, where choosingβ=F∆/T∆

leads to a symmetrical pulse overlapping in time and frequency. In practice, thefunctionfG(t) must of course be truncated such that its support fits some finitetime intervalIg.

The extent of the pulse overlapping in the transmitted signal can be studiedby the ambiguity function ofg(t) defined as [78]

Ag(τ, ν) =∫ ∞−∞

g(t)g∗(t− τ)e−j2πν(t−τ/2)dt.

It reflects the decay of the autocorrelation ofg(t) at increasing lagsτ andνin time and frequency, respectively. Fig. 4.2 depicts the ambiguity functionsof fG(t) with β = 1 and of the pulse with a rectangular shape used in OFDMsystems, i.e.

fR(t) =

T− 1

2∆ , t∈ [−T∆/2, T∆/2)

0, t /∈ [−T∆/2, T∆/2)

with T∆ =1. ForfG(t), the ambiguity function is given as

AfG(τ, ν) = exp(−π

2(βτ2 + β−1ν2

))and exhibits an exponential decay inτ2 andν2, whereas forfR(t) the ambiguityfunction turns out to decay with only1/ν in the frequency dimension.

−3−2

−1 0

1 2

3

−3−2

−1 0

1 2

310

−2

10−1

100

τν−2

−1 0

1 2

−16−12

−8 −4

0 4

8 12

1610

−2

10−1

100

τν

Figure 4.2: Logarithmic plots of the ambiguity functions offG(t) (left) andfR(t) (right).

4.3 Receiver Structures 63

4.3 Receiver Structures

Let us again consider a burst transmission in whichNB information symbolsare conveyed on each of theNC subchannels. The baseband equivalent transmitsignal is given as

s(t) =√S0

NB−1∑k=0

NC−1∑n=0

xk,n gk,n(t),

with xk,n denoting a complex-valued data symbol from a certain signal setΩwith the squared magnitudes of the elements equal to1 on the average. Notethat in this chapterS0 defines the average symbol energy rather than the powerspectral density ofs(t). The signal at the receiver reads

y(t) =√S0

NB−1∑k=0

NC−1∑n=0

xk,nhk,n(t) + v(t),

wherehk,n(t)=(c∗gk,n)(t), i.e., the channel is again modeled as time-invariantfor the duration of the burst.

We aim to estimate the information symbols from the signaly(t) under theassumption thatc(t) is perfectly known. In fact, the functions from the set

hk,n(t) : k=0, . . . , NB−1; n=0, . . . , NC−1 (4.2)

still form a Riesz basis if the signal mapping by the channel can be representedby a bounded invertible operator [75], which is e.g. the case for the assumedtime-invariant channel ifinff∈R |C(f)| > 0. As a consequence of the propertiesof the AWGN processv(t), the coefficients

uk,n = 〈y, hk,n〉,k=0, . . . , NB−1n=0, . . . , NC−1

(4.3)

represent a sufficient statistic2 for the estimation problem. Using vector notationwe can expressu = (u0,0, . . . , u0,NC−1, u1,0, . . . , uNB−1,NC−1)T as

u =√S0Ax + v

2The coefficients defined in (4.3) constitute asufficient statisticbecause they contain all theinformation about the data symbols iny(t) [79].


with x = (x0,0, . . . , x0,NC−1, x1,0, . . . , xNB−1,NC−1)T. The matrixA containsthe cross-correlations between any two distorted pulse shapes, i.e.

ai(k,n),i(`,m) = 〈h`,m, hk,n〉, (4.4)

wherea`,m is themth element in the th row in A and i(k, n)=kNC+n+1translates the TF index(k, n) into the index of the corresponding vector element.Furthermore, thei(k, n)th element in the noise vectorv equalsvk,n= 〈v, hk,n〉.Consequently,v is a zero-mean complex Gaussian random vector with the co-variance matrix

E[vvH

]= N0A. (4.5)

The Hermitian matrixA is also known as the Gram matrix of the functionsin (4.2), and nonsingular positive definite if they form a Riesz basis.

The ML estimate forx based on the sufficient statisticu may be formulatedas

xML = arg maxx∈ΩNBNC

Λ(x; u),

whereΛ(x; u) represents the log-likelihood function (LLF). As a consequenceof (4.5), the conditional joint PDF ofu can be expressed in the form [80]

pu(y|x) =1

πNBNC |N0A|exp(−(y−√S0Ax

)H(N0A)−1

(y−√S0Ax

))if A is nonsingular. For the LLF we hence find

Λ(x; u) = 2Re[xHu

]−√S0xHAx (4.6)

after discarding the constant term and factor. This expression actually turns outto hold even with a singularA.

The calculation ofxML has a huge complexity in the general case withnonzero cross-correlation terms between the pulse shapes in both the time andthe frequency dimension, since for accomplishing a ML sequence estimation inthe time dimension the number of states increases exponentially withNC. Asuboptimum two-step approach in which the interference in one dimension isignored in the first step is investigated in [81]. Performing the LLF maximiza-tion iteratively, as will be discussed in Sect. 4.3.1, is another way for achievinga substantial complexity reduction.


Alternatively, the receiver may eliminate the interference terms inu and esti-matex by simply multiplying with the inverse cross-correlation matrix, providedthat the latter exists, i.e.

d = S− 1

20 A−1u = x + S

− 12

0 A−1v.

We will refer to this approach as performing a zero-forcing (ZF) or decorrela-tion3. Like ZF equalizers this method suffers from noise amplification, and theerror rate performance of a receiver in which the signals ind are passed to adecision device is inferior to the one of a ML detector.

In a system with FEC, a direct decoding on the basis of the sufficient statisticu is generally impractical. The number of states would explode in a MLSD as aconsequence of the interference terms contained inu. To avoid this, the receivermay first accomplish a ML detection assuming an uncoded bit stream, and sec-ond apply a decoder to the obtained hard-decision bit estimates. However, it hasturned out in the preceding chapters that hard-decisions prior to the decoder leadto a significant performance degradation. Performing a ZF and passing the ob-tained soft bit estimates (from a subsequent de-mapper) to a soft input decoderas in the MC system sketched in Fig. 4.3 could be the better solution. Still, thisreceiver is suboptimal since the decoder regards the additive noise terms indas being white. The error rate performance of these two receiver schemes forsystems with FEC will be discussed in Sect. 4.4.


datain

dataout

transmitter receiver

MC signalsynthesis

signalencoding

decorrelationmatchedfiltering+

soft-decisionsignal

decoding

x s(t)c(τ)

v(t)

y(t) u d

Figure 4.3: MC transmission with FEC, performing a ZF prior to a soft inputdecoding at the receiver.

3A MC system in which the receiver accomplishes a ZF by directly employing an appropriateFB is investigated in [69]. Conditions for the existence and rules for the design of pairs of optimizedtransmit and ZF equalizing FBs are given in the context of discrete FB theory, assuming perfectknowledge of the channel impulse response in both the transmitter and the receiver.


4.3.1 Iterative Detection Methods

In a straightforward approach, the maximization of (4.6) can be accomplishedwith respect to one data symbol per iteration, passing through all elements in thevectorx within an iteration cycle and employingNcyc of such cycles. Startingfrom an initial estimatex(0) we thus obtain the sequencex(1), . . . , x(NcycNBNC)

of estimates, with each of them differing from the predecessor by at most onevector component. The updating rule for the`th element in the vectorx(µ),denoted byx(µ)

` , reads

x(µ)` = arg max

x`∈ΩΛ((x

(µ−1)1 , · · · , x(µ−1)

`−1 , x`, x(µ−1)`+1 , · · · , x(µ−1)

NBNC

)T

; u).

Using (4.6) and ignoring all terms which are independent ofx` we obtain

x(µ)` = arg max

x`∈Ω

2Re

x∗ù` −√S0

NBNC∑m=1m 6=`

a`,mx(µ−1)m

−√S0a`,`|x`|2

,whereu` is the`th element inu. Introducing a vectorr(µ) = u −

√S0Ax(µ)

and lettingr(µ)` denote its th element, the above expression can be rewritten as

x(µ)` = arg max

x`∈Ω

2Re

[x∗`

(r

(µ−1)` +

√S0a`,`x

(µ−1)`

)]−√S0a`,`|x`|2

.

(4.7)

If a`,`=0 thenx(µ)` =arg max

x`∈ΩRe[x∗` r

(µ−1)`

], whereas otherwise (4.7) is equiv-

alent to

x(µ)` =arg max

x`∈Ω

−a`,`∣∣∣∣∣x`−

(r

(µ−1)`√S0a`,`

+x(µ−1)`

)∣∣∣∣∣2

−∣∣∣∣∣ r

(µ−1)`√S0a`,`

+x(µ−1)`

∣∣∣∣∣2.

Discarding the constant term and factor the updating rule can finally be writtenas

x(µ)` = arg min

x`∈Ω

∣∣∣∣x` − ((√S0a`,`

)−1

r(µ−1)` + x

(µ−1)`

)∣∣∣∣2 . (4.8)

Hence, we choose the element inΩ with minimum distance to(√S0a`,`

)−1r

(µ−1)` + x

(µ−1)` in the complex plane. Fig. 4.4 provides a

flow diagram of the obtained detection procedure, neglecting the case wherea`,` = 0. Step (a) corresponds to (4.8), followed by the updating ofr(µ) in


x(0) = initial estimater(0) = u−

√S0Ax(0)

µ := 0

q := 1, . . . , Ncyc

k := 1, . . . , NB

n := 1, . . . , NC

µ := µ+ 1` := i(k, n)

x(µ) =(x

(µ−1)1 , . . . , x

(µ−1)−1 ,

arg minx`∈Ω

∣∣∣x` −((√S0a`,`)−1

r(µ−1)` + x

(µ−1)`

)∣∣∣2, (a)

x(µ−1)+1 , . . . , x

(µ−1)NBNC

)T

r(µ) = r(µ−1) −√S0A

(x(µ) − x(µ−1)

)(b)

Figure 4.4: Iterative LLF maximization procedure.

step (b). Apparently, this detection scheme can be regarded as accomplishingsuccessive symbol estimation in step (a) and interference cancellation (IC) instep (b).

Alternatively, the updating of the symbol estimates according to (4.8) couldbe carried out for all elements inx before recalculatingr(µ) only at the end ofan entire iteration cycle. However, it turns out that thisparallel symbol estima-tion and IC procedure provides less accurate results compared to the consideredserial scheme in Fig. 4.4 after an equal number of iteration cycles [72].

The sequenceΛ(x(0); u), . . . ,Λ(x(NcycNBNC); u) is non-decreasing as a re-


sult of the iterative maximization of the LLF4. Nevertheless, reaching the globalmaximum of the LLF and simultaneously a convergence ofx(µ) towardsxML

is not guaranteed even with an infinite number of iterations. The chance thatiteration ends up in a local maximum of the LLF can actually be reduced byproviding a useful initial estimate.

To this end, let us return to the above discussed decorrelation, and assumethatA is nonsingular. For the solution of the linear equation systemu=

√S0Ad

a large number of standard iterative methods are available [83]. It turns out thatapplication of the basic Gauss-Seidel iteration leads to exactly the proceduredefined in Fig. 4.4 when removing the hard-decision in the step (a) and using

x(µ)` =

(√S0a`,`

)−1

r(µ−1)` + x

(µ−1)` (4.9)

as the updating rule forx(µ)` . As opposed to the hard-decision iterations, this

method provides soft symbol estimates converging towardsd. The convergenceproperties of the Gauss-Seidel iteration are discussed in [83]. Instead of a di-rect element-wise hard-decision based ond, the estimates after a certain numberNdec of Gauss-Seidel iteration cycles can be taken as initial values for a subse-quent iterative LLF maximization with the procedure in Fig. 4.4.

The magnitudes of the data symbols are upper bounded by the valuemaxx∈Ω |x|. It turns out that the convergence of the iterations can be accel-erated by adding a corresponding constraint to the updating rule, replacing (4.9)by e.g.

x(µ)` =

((√S0a`,`

)−1

r(µ−1)` +x(µ−1)

`

)min

1,maxx∈Ω |x|∣∣∣(√S0a`,`)−1

r(µ−1)` +x(µ−1)

`

∣∣∣.

(4.10)Faster convergence may also be attained by utilizing more sophisticated itera-tions like for instance the conjugate gradients method [83, 84].

4The described algorithm is in fact a variant of the space-alternating generalized expectation-maximization algorithm (SAGE), presented in [82]. The monotonicity of the LLF sequence is oneproperty of the SAGE algorithm.


4.3.2 Computational Complexity and Residual InterferenceAnalysis

A TF concentrated prototype functiong(t) bounds the pulse overlapping, whichresults in an essentially sparse structure of the matrixA. The detection pro-cedure in Fig. 4.4 can exploit this property ofA by ignoring its essentiallyzero-valued entries for the IC in step (b), achieving a significant reduction ofthe computational effort per iteration. In the following, we study the effects ofconsidering only the interference between two symbols placed at mostDIC,T

time slots andDIC,F subchannels apart. That is, we replaceA by A, with theelementsan,m defined as

ai(k,n),i(`,m) =ai(k,n),i(`,m), |`−k|≤DIC,T and|m−n|≤DIC,F

0, otherwise.

Keeping in mind that(x(µ)−x(µ−1)

)contains at most one nonzero element, i.e.(

x(µ)` − x

(µ−1)`

), we find that step (b) in Fig. 4.4 requires the computation of

(2DIC,T +1)(2DIC,F +1) multiplications and an equal number of subtractions,all operations involving complex numbers. Additionally, the updating of the datasymbol estimate in step (a) can be carried out with one division and one addition,not taking the hard-decision into account. Consequently, the total numbers ofcomplex multiplications/divisions (nM) and additions/subtractions (nA) for theiterative detection including the initial decorrelation are

nM = nA = (Ndec +Ncyc) (1 + (2DIC,T + 1)(2DIC,F + 1)) (4.11)

per transmitted data symbol.ReplacingA by A leads to perturbations inr(µ), affecting the symbol es-

timation in (4.8), (4.9) and (4.10). Supposing that all data symbols in theburst have been correctly estimated afterµ0 iterations, i.e.x(µ0) = x, we haver(µ0) =u−

√S0Ax=v+

√S0(A−A)x. Hence, the remaining parts inr(µ0) are

due to the AWGN and the residual interference. The remainder in thei(k, n)th

element inr(µ0) equals

r(µ0)i(k,n) = vk,n +

√S0

∑(`,m)∈0,...,NB−1×0,...,NC−1

(| −k|,|m−n|)/∈0,...,DIC,T ×0,...,DIC,F

ai(k,n),i(`,m)x`,m.

We now study the mean-square values of the terms in this expression and com-pare the residual interference part against the noise part. The analysis is based


on an US channel. Specifically, the impulse responsec(τ) is modeled as a zero-mean complex white Gaussian random process being independent of the noiseand subject to a given PDP. The data symbols are also viewed as random, mu-tually independent withE[x`,m] = 0 andE[|x`,m|2] = 1 ∀` ∈ 0, . . . , NB−1∀m∈0, . . . , NC−1, and independent of the channel and noise. For the PDPwe finally assume that ∫ ∞

−∞φc(τ)dτ = 1,

such that the mean received signal energy per data symbol equalsS0.First, we focus on the mean energy in the noise term, which can be expressed

as

E[|vk,n|2

]= E

[|〈v, hk,n〉|2

]=

E

∞∫−∞

v(t)

∞∫−∞

c∗(τ)g∗k,n(t−τ)dτ dt

∞∫−∞

v∗(t′)

∞∫−∞

c(τ ′)gk,n(t′−τ ′)dτ ′ dt′ .

Exchanging the order of the expectation and integration,E[|vk,n|2

]=

∞∫−∞

∞∫−∞

∞∫−∞

∞∫−∞

E [c∗(τ)c(τ ′)v(t)v∗(t′)] g∗k,n(t−τ)gk,n(t′−τ ′)dτdτ ′dtdt′.

As a consequence of (2.1) and sinceE[v(t)v∗(t′)]=N0δ(t−t′) we obtain

E[|vk,n|2

]= N0

∫ ∞−∞

φc(τ)dτ = N0.

Hence, the ratio of the mean received signal energy per symbol and the meannoise energy inr(µ0)

i(k,n) equalsS0/N0.

Next, we analyze the mean residual interference energy inr(µ0)i(k,n), given as

ε(k,n)IC =

∑(`,m)∈0,...,NB−1×0,...,NC−1

(| −k|,|m−n|)/∈0,...,DIC,T ×0,...,DIC,F

S0E[∣∣ai(k,n),i(`,m)

∣∣2] .The above expectation can be expressed as

E[∣∣ai(k,n),i(`,m)

∣∣2] = E[|〈h`,m, hk,n〉|2

]=


E

[∣∣∣∣ ∫ ∞−∞

∫ ∞−∞

c(τ)g`,m(t− τ)dτ∫ ∞−∞

c∗(τ ′)g∗k,n(t− τ ′)dτ ′ dt∣∣∣∣2]

=∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

E [c(τ)c∗(τ ′)c∗(τ ′′)c(τ ′′′)]×∫ ∞−∞

g`,m(t−τ)g∗k,n(t−τ ′)dt∫ ∞−∞

g∗`,m(t−τ ′′)gk,n(t−τ ′′′)dtdτdτ ′dτ ′′dτ ′′′.

With the fourth moment formula for Gaussian processes [85],E [c(τ)c∗(τ ′)c∗(τ ′′)c(τ ′′′)] can be written as

E [c(τ)c∗(τ ′)]E [c∗(τ ′′)c(τ ′′′)] +

E [c(τ)c∗(τ ′′)]E [c∗(τ ′)c(τ ′′′)] +

E [c(τ)c(τ ′′′)]E [c∗(τ ′)c∗(τ ′′)] .

It follows from (2.1) that the first two of these three terms equalφc(τ)φc(τ ′′)δ(τ−τ ′)δ(τ ′′−τ ′′′) andφc(τ)φc(τ ′)δ(τ−τ ′′)δ(τ ′−τ ′′′), respec-tively, while the third term is zero. This leads to the expression

E[∣∣ai(k,n),i(`,m)

∣∣2] = |Ag((k−`)T∆, (n−m)F∆)|2 +∫ ∞−∞

∫ ∞−∞

φc(τ)φc(τ ′) |Ag((k−`)T∆+τ ′−τ, (n−m)F∆)|2 dτdτ ′, (4.12)

where the inner products are replaced by the ambiguity function ofg(t).Note that (4.12) depends on the relative offset(k−`, n−m) rather than on

the absolute TF slot positions(k, n) and(`,m) within the burst. Clearly,ε(k,n)IC

is largest when interference emanates from adjacent symbols on both sides inthe time and the frequency dimensions. We therefore consider the limit

εIC = limNB,NC→∞

ε(NB/2,NC/2)IC ,

which represents the supremum of the mean residual interference energyε(k,n)IC

for given g(t), T∆, F∆, φc(τ), DIC,T andDIC,F . The valueεIC grows forincreasingly dispersive channels when keepingDIC,T andDIC,F constant. Thisis reflected in Fig. 4.5 for the case of the exponentially decaying PDP definedin (2.2). The figure shows the resultingS0/εIC as a function ofτd for T∆F∆ =1and six possible choices ofDIC,T andDIC,F , using as prototype function either


0

20

40

60

80

100

120

τd

0 T∆/8 T∆/4 3T∆/8 T∆/2

S0/ε I

C[d

B]

g(t)=fG(t)

g(t)=fR(t)

DIC,T=2,DIC,F=2; nA = nM = 26(Ndec+Ncyc)DIC,T=3,DIC,F=1; nA = nM = 22(Ndec+Ncyc)DIC,T=1,DIC,F=3; nA = nM = 22(Ndec+Ncyc)DIC,T=2,DIC,F=1; nA = nM = 16(Ndec+Ncyc)DIC,T=1,DIC,F=2; nA = nM = 16(Ndec+Ncyc)DIC,T=1,DIC,F=1; nA = nM = 10(Ndec+Ncyc)

Figure 4.5: Mean signal to mean residual interference energy ratio versus theRMS delay spread, in slowly time-variant channels with an exponentially de-caying PDP.

fG(t) or fR(t). For everyτd the scalableβ in the definition offG(t) is chosensuch that the residual interference is minimized.

As expected, much smaller amounts of residual interference result for TFconcentrated pulses derived fromfG(t) than for a prototype function with arectangular shape, unlessτd is close to zero. In this case, the negligible channeldispersion does not destroy the orthogonality of the signal pulses, enabling anearly interference-free transmission.

Like in an OFDM system, the data rate can be increased by just addingsubchannels in the frequency domain while leaving the detection complexityper data symbol as well asεIC unchanged. This is different in a single carriersystem, where the equalizer complexity per transmitted data symbol generallygrows when the symbol rate is increased.


4.3.3 Matched Filtering

A direct calculation of the sufficient statisticu would require an initial estima-tion of everyhk,n(t), and a correlation ofy(t) with every estimated waveform.A more suitable method for assessingu, based on a distinct set of channel pa-rameters that need to be estimated, results from an appropriate series expansionof the waveforms of the distorted signal pulses using fixed basis functions.

Let us first focus on a specific pulsehk,n(t) and adopt an US channelmodel. We assume that the temporal support of the pulse, which depends onIg and the time dispersion induced by the channel, is contained within the

finite time intervalI(k)h of lengthTh. Now, hk,n(t) represents a zero-mean

nonstationary random process withint ∈ I(k)h , with the covariance function

Rk,n(t, t′) = E[hk,n(t)h∗k,n(t′)]. With hk,n(t) = (c∗gk,n)(t), exchanging theorder of the expectation and integration, and using (2.1) the covariance functioncan be expressed as

Rk,n(t, t′) =∫ ∞−∞

φc(τ)gk,n(t− τ)g∗k,n(t′ − τ)dτ, t, t′ ∈ I(k)h . (4.13)

Furthermore, the Karhunen-Loève expansion of the random processhk,n(t)reads [86]

hk,n(t) =∞∑p=1

ω(p)k,nψ

(p)k,n(t), (4.14)

where ψ(1)k,n(t), ψ(2)

k,n(t), . . . are the orthonormal eigenfunctions of the pos-itive semidefinite, conjugate complex symmetric kernelRk,n(t, t′), and

ω(p)k,n=

⟨hk,n, ψ

(p)k,n

⟩. The fixed eigenfunctions depend on the transmit pulse

shape and the PDP, whereas the channel parametersω(1)k,n, ω

(2)k,n, . . . are random

variables, being uncorrelated as a property of the Karhunen-Loève expansion.

The eigenfunctions shall be ordered such that their corresponding realnon-negative eigenvaluesλ(1)

k,n, λ(2)k,n, . . . form a non-increasing sequence. An

approximate reconstruction ofhk,n(t) is obtained by an expansion accord-

ing to (4.14) using only the eigenfunctionsψ(1)k,n(t), . . . , ψ(K)

k,n (t) with theKlargest eigenvalues. These functions actually represent the optimum projec-tion filters for hk,n(t) in the sense that the mean energy in the error signal

ek,n(t)=∑∞

p=K+1 ω(p)k,nψ

(p)k,n(t) is minimum for usingK filters. Finally, we in-


sert the truncated series into (4.3) and find

uk,n =K∑p=1

(ω

(p)k,n

)∗ ⟨y, ψ

(p)k,n

⟩(4.15)

as an approximation foruk,n.As c(τ) is time-invariant, the same applies to the channel parameters, i.e.

ω(p)k,n = ω

(p)n ∀k ∈ 0, . . . , NB−1, p = 1, 2, . . .. There are hence a total of

KNC parameters to be estimated in the receiver, and the sufficient statistic canbe approximated as sketched in Fig. 4.6.

......

+

Σ... .........

++

+

Σy(t)

〈·, ψ(K)k,1 〉

〈·, ψ(K)k,NC〉

〈·, ψ(1)k,1〉

〈·, ψ(1)k,NC〉

(ω

(K)1

)∗

(ω

(K)NC

)∗(ω

(1)1

)∗

(ω

(1)NC

)∗

uk,1

uk,NC

Figure 4.6: FB structure for calculating the sufficient statistic.

Furthermore, it can be seen by expanding the expression (4.13) using (4.1)that the eigenvalue sequence is identical for all TF slots, i.e.λ

(p)k,n = λ(p)

∀k∈0, . . . , NB−1∀n∈0, . . . , NC−1, p=1, 2, . . .. Additionally, the setψ

(p)k,n(t) : k = 0, . . . , NB− 1; n = 0, . . . , NC− 1

with the pth eigenfunc-

tions represents a subset of a WH system. As a consequence, the discrete timeprojection ofy(t) onto the basis functions can be accomplished byK paralleluniform DFT FB blocks, replacing theK correlator blocks in Fig. 4.6.

As a further property of the Karhunen-Loève expansion,E[∣∣ω(p)

k,n

∣∣2]=λ(p).


Consequently, the ratio of the mean energies of a signal pulsehk,n(t) and theerror signalek,n(t) from the reconstruction ofhk,n(t) equals

γKL =E[〈hk,n, hk,n〉]E[〈ek,n, ek,n〉]

=

∑∞p=1 λ

(p)∑∞p=K+1 λ

(p).

The valueγKL quantifies the impact of the series truncation asek,n(t) may beviewed as an additional noise signal. Tab. 4.1 summarizes the resultingγKL forthe PDP defined in (2.2) withτd = T∆/5 and pulses derived from eitherfR(t)or fG(t). In the latter caseβ = 0.68F∆/T∆ is chosen as this has turned out toresult in minimum residual interferenceεIC for the assumedτd among the ICschemes investigated in Fig. 4.5. The figures in the table suggest that utilizing afew basis functions may be sufficient for a transmission based onfG(t), whereasfor fR(t) a significant number of basis functions are necessary to keep the errorenergy reasonably low.

K γKL for g(t)=fG(t) γKL for g(t)=fR(t)2 22.1 dB 10.9 dB3 31.0 dB 12.6 dB4 39.2 dB 13.9 dB5 46.9 dB 14.9 dB6 54.2 dB 15.7 dB10 80.6 dB 18.1 dB

Table 4.1: Signal to truncation error energy ratio for variousK.

TheK uniform DFT FBs replacing the correlator blocks in Fig. 4.6 each in-corporate an FFT computation and polyphase components in reverse order thandepicted in Fig. 4.1, i.e. here the polyphase components precede the FFT com-putation. Performing anNFFT-point FFT, withNFFT ≥NC, requires no morethan(NFFT/2)ldNFFT multiplications andNFFTldNFFT additions ifNFFT isa power of two [87]. An extraNh multiplications andNh−NFFT additions arenecessary per FFT for the polyphase components, whereNh is the length of aninterval I(k)

h in samples, i.e.,Nh is aboutTh/T∆ times larger thanNFFT forT∆F∆≈1. Together with the subsequent combining in (4.15) using the channel


parameters, the overall computational complexity per element inu amounts to

n′M = K(NFFT/2)ldNFFT +Nh

NC+K (4.16)

complex multiplications and

n′A = KNFFTldNFFT +Nh −NFFT

NC+K − 1 (4.17)

complex additions.The estimation of the time-invariant channel parameters may be accom-

plished based on a received preamble signal, which is correlated with theK

basis functions on every subchannel. Then, unlike with an OFDM signaling,the observed coefficients contain interference terms from overlapping pulses.The interference could, however, be averaged out by employing sequences ofpilot symbols with appropriate cross-correlation characteristics. Furthermore, itseems likely that small-scale frequency offsets due to synchronization errorshave a less dramatic impact on the performance than in an OFDM receiver,where they cause ICI. Within the scope of this work, however, we do not botherabout channel estimation and restrict our attention to the case of perfect syn-chronization and channel parameters.

The actual elements of the matrixA can be estimated from the channel pa-rameters using (4.4) and (4.14), i.e.

ai(k,n),i(`,m) =K∑p=1

K∑q=1

ω(q)`,m

(ω

(p)k,n

)∗〈ψ(q)`,m, ψ

(p)k,n〉,

where the scalar products having the predetermined eigenfunctions as argumentsare constants.

4.4 Performance Analysis

In this section, we investigate the achievable error rate performance with thediscussed iterative LLF maximization and decorrelation procedures. Computersimulations are carried out for this purpose since analytical derivations of the ex-act error probabilities are not available. Burst transmissions with the dimensions


NC = 24 andNB = 50, over a severely time-dispersive US channel with an ex-ponentially decaying PDP and a RMS delay spread ofτd =T∆/5 and Rayleighfading characteristics are simulated. To obtain the mean BER, a random im-pulse response satisfying (2.2) is generated for each burst, and the error rates areaveraged over a large number of burst transmissions.

4.4.1 Uncoded Transmission

At first, we assume a QPSK modulation and use the prototype functionfG(t)with β = 0.68F∆/T∆ and a support limited to the intervalIg = [−2T∆, 2T∆].Starting fromx(0) = 0, the detector accomplishes a partial decorrelation byNdec iteration cycles using the updating rule (4.10), followed byNcyc iterationcycles with hard-decisions according to the procedure in Fig. 4.4. We chooseT∆F∆ =33/32, i.e. a value slightly above 1, in order to attain linear indepen-dence in the underlying WH system. Also, a perfect calculation of the sufficientstatisticu is assumed, i.e.K1.

Fig. 4.7 shows the mean BER for using different numbers of iteration cycleswith DIC,T =3 andDIC,F =1, as a function of the mean SNR

γ = (T∆F∆)−1 S0

N0.

For every given number of total iteration cycles the most favorable combinationof Ndec andNcyc is chosen. The figure is complemented by the matched filterbound (MFB), which represents a lower bound for the attainable detector perfor-mance with uncoded signals. The MFB is derived in Appendix B and based onthe hypothetical assumption that every data symbol can be detected by an idealmatched filtering without having to deal with contingent ICI/IBI. Additionally,the analytical bit error probabilityPb computed in (2.10) is shown, reflecting theachievable mean BER by a conventional OFDM transmission over a Rayleighfading channel, using guard periods which perfectly absorb the channel disper-sion.

Obviously, all the BER curves decay with an offset of about 1-2 dB fromthe MFB and saturate whenγ increases beyond a certain value. The error floorcan be reduced by successively increasing the number of iteration cycles. Theimprovement with respect toPb results from exploiting the inherent frequencydiversity of the proposed transmission scheme.


5 10 15 20 25 30 3510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

me

an

BE

R

Ndec =0,Ncyc =1Ndec =1,Ncyc =1Ndec =2,Ncyc =1Ndec =4,Ncyc =2Ndec =7,Ncyc =2Ndec =11,Ncyc =2Ndec =15,Ncyc =3Ndec =20,Ncyc =3

Pb for QPSK-OFDMMFB

Figure 4.7: Mean BER for a QPSK transmission employing pulses derived froma Gaussian prototype function, carrying out different numbers of iteration cycleswith DIC,T =3 andDIC,F =1, based on a perfectly calculatedu.

For comparison, we also investigate the error rate performance when em-ploying pulses derived fromfR(t) with T∆F∆ = 1. Fig. 4.8 depicts the meanerror rates forDIC,T = 1 and different choices ofDIC,F , carrying out 27 initialdecorrelating iteration cycles followed by 3 cycles with hard-decisions. Here,the pulse energies are much less concentrated in the frequency domain. Thecorresponding MFB exhibits a steeper decay forγ→∞ than for the Gaussiancase, as an increased amount of frequency diversity is achieved. However, theperformance of the iterative detections remains poor even for e.g.DIC,F =7 dueto the considerably higher residual interference from other subchannel signals.

Next, the error rate performance of a receiver which uses the FB structuredescribed in Sect. 4.3.3 for obtaining an approximate version of the sufficientstatistic is investigated. Fig. 4.9 shows the error rates for a similar set-up as


5 10 15 20 25 30 3510

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100

γ [dB]

me

an

BE

R

DIC,T=1,DIC,F=1DIC,T=1,DIC,F=2DIC,T=1,DIC,F=3DIC,T=1,DIC,F=5DIC,T=1,DIC,F=7

Pb for QPSK-OFDMMFB

Figure 4.8: Mean BER for a QPSK transmission employing pulses derived froma prototype function with a rectangular shape, choosingNdec = 27, Ncyc = 3,DIC,T =1, and differentDIC,F , based on a perfectly calculatedu.

used for finding the results in Fig. 4.7, except of employing only two, three orfour parallel uniform DFT FB blocks for estimatingu, where the subsequentcombiner again has perfect channel parameters available. The legend of the fig-ure additionally provides the total numbers of complex multiplications/divisionsnM = nM +n′M and additions/subtractionsnA = nA +n′A per data symbol asassessed in (4.11), (4.16) and (4.17), forNFFT =32 andNh=184.

ForK=2, a high error floor persists even with many iteration cycles. Choos-ing K = 3 seems appropriate for e.g. achieving a mean BER of10−3 with 6iteration cycles in total, requiring aγ of about21 dB. Here, the offset from theMFB is about2 dB, whereas the gain with respect to a corresponding uncodedOFDM transmission is approximately5 1

2 dB.

Finally, the behavior of the nonorthogonal MC transmission and receiver


5 10 15 20 25 30 3510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

γ [dB]

me

an

BE

R

(Ndec, Ncyc,K, nM, nA)=(2, 1, 2, 90, 93)(Ndec, Ncyc,K, nM, nA)=(4, 2, 2, 156, 159)(Ndec, Ncyc,K, nM, nA)=(15, 3, 2, 420, 423)(Ndec, Ncyc,K, nM, nA)=(2, 1, 3, 102, 107)(Ndec, Ncyc,K, nM, nA)=(2, 1, 4, 114, 121)(Ndec, Ncyc,K, nM, nA)=(4, 2, 3, 168, 173)(Ndec, Ncyc,K, nM, nA)=(4, 2, 4, 180, 187)(Ndec, Ncyc,K, nM, nA)=(15, 3, 3, 432, 437)(Ndec, Ncyc,K, nM, nA)=(15, 3, 4, 444, 451)

Pb for QPSK-OFDMMFB

Figure 4.9: Mean BER for a QPSK transmission employing pulses derived froma Gaussian prototype function, carrying out different numbers of iteration cycleswith DIC,T =3 andDIC,F =1, estimatingu usingK parallel uniform DFT FBblocks.

technique is studied for the case of a 64-QAM with square lattice constellation ofthe elements inΩ and the usual Gray encoding. With this high-order modulationscheme the detectors actually require a very accurate version of the sufficientstatisticu. Moreover,Ndec needs to be chosen much larger. This is apparentin Fig. 4.10, showing the error rates for employing five parallel uniform DFTFB blocks and different numbers of iteration cycles. The curves again illustratethe mean BER as a function ofγ. Here,Pb from (2.12) is shown in addition,and a modified MFB computed by considering only those error events wherethe detector decides in favor of a neighboring signal point in the constellation,resulting in one erroneous bit within a Gray encoded bit sequence.

Clearly, a larger gap appears between the MFB and the error rates from the


20 25 30 35 40 45 5010

−7

10−6

10−5

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10−3

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10−1

100

γ [dB]

me

an

BE

R

(Ndec, Ncyc, nM, nA)=(4, 1, 170, 179)(Ndec, Ncyc, nM, nA)=(13, 2, 390, 399)(Ndec, Ncyc, nM, nA)=(22, 3, 610, 619)(Ndec, Ncyc, nM, nA)=(47, 3, 1160, 1169)(Ndec, Ncyc, nM, nA)=(97, 3, 2260, 2269)(Ndec, Ncyc, nM, nA)=(197, 3, 4460, 4469)

Pb for OFDM with 64-QAMMFB

Figure 4.10: Mean BER for a 64-QAM transmission employing pulses derivedfrom a Gaussian prototype function, carrying out different numbers of iterationcycles withDIC,T = 3 andDIC,F = 1, estimatingu using 5 parallel uniformDFT FB blocks.

iterative detectors. For achieving a mean BER of10−3 a γ of 38 dB is requiredfor 50, and36 dB for 100 iteration cycles in total, being5 dB and3 dB, respec-tively, away from the theoretical MFB.

4.4.2 Coded Transmission

It has been seen in Chapter 2 that a FEC coding is essential for a data transmis-sion over fading channels. Moreover, it is in general very difficult to estimate theachievable error rate performance of a system with FEC from the BER resultsof a corresponding uncoded transmission. In the following we extend the abovestudy to systems employing a BICM for effectively exploiting the frequency di-


versity. In all the simulated scenarios, the same64-state convolutional encoderas chosen for the studies in Sect. 2.7 and Sect. 3.4, and a 16-QAM with Grayencoding are utilized. There are hence two information bits encoded in each datasymbol, resulting in a spectral efficiency of2/(T∆F∆) bit/s/Hz.

At first, the elementary transmit pulses are derived fromfG(t) as inSect. 4.4.1. The receiver uses the FB structure in Fig. 4.6 withK = 3, andperforms a LLF maximization withDIC,T = 3, DIC,F = 1, Ndec = 30 andNcyc = 3. The obtained bit estimates are deinterleaved and passed to a Viterbidecoder.

Additionally, a transmission with a ZF receiver as illustrated in Fig. 4.3 isinvestigated, where only the30 iteration cycles for the decorrelation are carriedout, followed by a soft-decision signal decoding. The latter consists of a de-mapper which computes LLR values according to (2.8), and a soft input MLSD.

The BER results for these two receivers are shown in Fig. 4.11 versusγ, to-gether with the error rates from a conventional OFDM transmission using eitherhard-decision or soft-decision decoding. To cope with the large delay spread,T∆ andNC are chosen by a factor32 larger for the OFDM system than for thenonorthogonal signaling, and the subchannel spacing is reduced by the samefactor. This keeps the bandwidth usage and the spectral efficiency equal to theother scenarios while allowing for guard periods which fulfill a common rule ofthumb by whichTG should at least be in the order of5τd.

Obviously, the ZF with a subsequent soft-decision decoding outperforms thecombination of a LLF maximization and hard-decision decoding for the signal-ing scheme based onfG(t) by about4 dB at a BER level of10−6. This again em-phasizes the benefits of a soft-decision approach. The error rates are, however,by about3 dB inferior to those with a conventional BICM-OFDM transmissionusing soft-decision decoding. Unlike with an uncoded transmission, there is aclear performance degradation here as a consequence of the suboptimum ZF.Nevertheless, the MC system in Fig. 4.3 is an advantageous alternative in envi-ronments where channel variations or hardware components impose limitationson the choice ofT∆ andNC. For using a slicer, there is obviously no significantdiscrepancy between the performance with the nonorthogonal signaling and LLFmaximization and with an OFDM transmission.

4.5 Channel Capacity with a Zero-Forcing Receiver 83

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γ [dB]

me

an

BE

R

g(t)=fG(t); LLF maxim.+hard-decision dec.OFDM transmission with hard-decision dec.g(t)=fG(t); ZF+soft-decision decodingOFDM transmission with soft-decision dec.

Figure 4.11: Mean BER for a BICM MC transmission using a 16-QAM signalset and different signaling and receiver schemes.

4.5 Channel Capacity with a Zero-Forcing Re-ceiver

It was claimed in Sect. 2.8 that any data rate below the multipath channel ca-pacity can theoretically be achieved by an OFDM scheme withT∆F∆ chosensufficiently close to1. In this section, we consider the capacity with a perfectZF receiver in which the decoder regards the additive noise terms in the decor-relator outputd as being uncorrelated. To this end, we model the channel asNBNC parallel AWGN channels, each of them utilized for transmitting one datasymbol, and sum up the respective capacities.

As v is a zero-mean jointly complex Gaussian random vector with thecovariance matrix given in (4.5),S−1

0 N0A−1 results as the covariance ma-trix for the zero-mean jointly complex Gaussian transformed random vector


v′=S− 1

20 A−1v, which represents the additive noise contained ind. The noise

amplification from the ZF is hence determined by the diagonal elements inA−1,and the SNR in the equivalent AWGN channel associated with the TF slot(k, n)equals

γk,n = ζi(k,n)S0

N0,

whereζ` denotes the inverse of the`th diagonal element inA−1. The normalizedcapacity with the ZF receiver is now given as

CZF =1

NBNC

NB−1∑k=0

NC−1∑n=0

ld(1 + γk,n).

The factorsζ1, . . . , ζNBNC depend on the prototype function, the pulse spac-ings, the burst dimensions, and the channel. With an ideal channel the factorsconverge towards a valueζ0 = lim

NB,NC→∞ζi(NB/2,NC/2) for growing burst di-

mensions, provided thatA is always nonsingular. For orthonormal pulse shapesζ0 = 1, whereas otherwiseζ0 turns out to be smaller than1. This reflects thefact that sets of orthonormal pulse shapes are optimal for signaling over an idealchannel. With nonorthogonal waveforms and a ZF, the limit for the attainable bitrate is below the actual channel capacity. For the Gaussian prototype function,ζ0 goes towards zero as the pulse densityT∆F∆ approaches the critical value1from above. In fact, a WH system derived fromfG(t) forms a Riesz basis onlyfor T∆F∆ strictly greater than1.

A dispersive channel changes the cross-correlation properties and destroysthe merits of orthonormal pulse shapes. For the pulse design, it would be ofinterest to determine the average capacity over the fading statistic of a slowlytime-variant channel with the ZF receiver and a certain signaling scheme. How-ever, an analytical study seems impossible due to the non-trivial distribution ofthe diagonal elements inA−1.

What can be shown is that if applied to an OFDM signaling, i.e. employingthe elementary pulses defined in (2.4) withTG≥ τexcess, the capacity with a ZFreceiver exceeds (or at least achieves) the capacity with a conventional OFDMreceiver. To see this, letA andA denote the cross-correlation matrices of theorthogonal waveforms

hk,n(t) =hk,n(t), t ∈ [kT∆, kT∆ + TF)0, otherwise

4.6 Summary and Discussion 85

and the waveforms

hk,n(t) =

0, t ∈ [kT∆, kT∆ + TF)hk,n(t), otherwise,

respectively, with(k, n) ∈ 0, . . . , NB−1×0, . . . , NC−1. For the OFDMreceiver, the SNR associated with the TF slot(k, n) equals

γk,n = ζi(k,n)S0

N0,

whereζ` represents theth diagonal element in the diagonal matrixA. The ma-trix A is Hermitian positive semidefinite, and may be factorized intoA=UUH.We again rule out zeros in the channel transfer function such thatA is nonsin-gular. AsA=A+A and using the matrix inversion lemma [79],

A−1 =(A + UUH

)−1

= A−1−A−1U(INBNC + UHA−1U

)−1

UHA−1.

Clearly,(INBNC +UHA−1U

)is positive definite, Hermitian, and nonsingular.

The same holds for its inverse. The entire expression on the right side of theminus sign is therefore Hermitian positive semidefinite, having non-negative di-agonal elements. This proofs thatζ−1

` ≤ ζ−1` , and hence

γk,n≥ γk,n ∀(k, n)∈0, . . . , NB−1×0, . . . , NC−1.

4.6 Summary and Discussion

Alternative MC transmission schemes without bandwidth consuming guard pe-riods have been investigated, which offer more freedom in the choice of thenumber of subchannels and the elementary pulse shapes. In particular, the useof waveforms with superior TF localization properties has been suggested inorder to confine the essential signal overlapping to information-bearing pulsesnearby in time and frequency, and thereby limiting the extent of ICI/IBI.

Receiver structures have been devised which can exploit the bounded pulseoverlapping for limiting the computational effort. A sufficient statistic is first cal-culated for the unknown data by weighting the output of an appropriate FB usinga distinct number of channel parameters. The FB allows for an efficient discrete


time implementation by FFT techniques. Iterative methods are employed in thefollowing for the IC, accomplishing either a likelihood maximization or a ZF.

The performance analysis has revealed that a mean BER close to the MFBcan be attained with a sufficient number of iteration cycles. More iterationsare actually necessary for the higher-order modulation schemes. There is alsoa trade-off between the computational complexity and the residual interferencedegrading the error rate performance. As a consequence of the higher diversityorder achieved by the iterative detection, a conventional OFDM transmissionwithout FEC can be clearly outperformed.

This is different for systems with a FEC coding. To keep the demodulationand decoding complexity within limits, the receiver should resort to suboptimumtechniques like e.g. performing a ZF prior to the decoder. A certain performancedegradation with respect to a coded OFDM transmission with negligible guardperiods is inevitable. We may thus conclude that the propounded alternativeMC transmission techniques are primarily favorable for systems where the en-vironmental conditions impose restrictions on the length of the time slots or thenumber of subchannels.

Chapter 5

Outlook

In the longer term there will be efforts towards a unification of the different wire-less communication standards and building up one system for providing a widerange of services to both mobile and fixed users. Employing a MC techniqueon the physical layer will be beneficial, not least due to the powerful spectrummanagement capabilities. A resource allocation on a TF slot basis would infact provide the utmost flexibility for accommodating links with highly vari-able data rates and QoS requirements. An OFDM signaling combined with sucha two-dimensional multiple access technique is sometimes termed orthogonalfrequency-divisionmultiple access (OFDMA), and may serve as a universal plat-form for both the uplink and the downlink and for services ranging from voicetelephony to video-on-demand.

Novel adaptation mechanisms will have to be elaborated to optimize the us-age of the TF slots, involving the physical as well as the higher layers. TheOFDMA scheme allows for assigning only a small subset of the subchannelsto those users who request moderate data rates. A smart subchannel allocationpolicy can prevent utilizing unfavorable frequency bands, being subject to highsignal attenuation or hampered by narrowband noise. Additionally, latency canbe kept small by an individual allocation of the TF slots and, if combined with anadequate duplexing scheme, fast feedback of CSI can be provided to the trans-mitter without wasting much bandwidth.

High-speed transmissions involving fixed terminal stations have a greateroptimization potential through an adaptation of the power or the modulation

88 Chapter 5: Outlook

schemes than links to mobile terminals, as the latter exhibit much more rapidchanges in the channel state. To avoid sudden violations of agreed QoS criteria,anticipating channel prediction techniques will have to be devised as part of thetransmitter functionality. As a consequence of the sometimes unreliable CSI,however, achieving diversity is still crucial, and BICM thus an expedient FECscheme.

In a first effort, adaptation techniques could be elaborated which aim to op-timize the power allocation in BICM-OFDMA systems while taking the varyingCSI accuracy from user to user into account. At this point, the additional bene-fit from adapting the modulation schemes on a subchannel-by-subchannel basisversus an adaptation on a user-by-user basis should be investigated. Again, theremight be a trade-off between performance and complexity.

Equipping the base stations with two or more antennas is another methodto increase the spectral efficiency of future cellular wireless communication sys-tems. Techniques for achieving transmit diversity have recently attracted a broadinterest as they can improve the performance by adding complexity mainly inthe transmitting station, without necessitating CSI. If, on the other hand, CSI isavailable in either perfect or outdated form, the transmitter has control over thecombining of the signals it sends out from the multiple antennas, and no particu-lar space-time coding design is required. Instead, a diversity gain can be realizedby synchronously sending out individually phase and power adapted replicas ofe.g. a BICM-OFDM signal. With this sort of a beam-forming approach, the re-ceiver needs to estimate just the parameters defining the state of one equivalentchannel, i.e. the receiver side is not involved by the extension at all. Employinga space-time code may, however, still be beneficial in situations where only out-dated CSI is available. Optimizing the transmission scheme and the adaptationpolicy for such scenarios will be a challenging task.

The dimensioning of the TF slots in a MC based air interface poses anotherchallenge in the development of a universal wireless communication systemsince it must satisfy the requirements of both remote and nearby fast movingusers. An OFDM scheme designed for the worst case delay spread and thushaving a huge number of subchannels may not be favorable. The use of ade-quate pulse shapes could let the receivers more easily cope with a wide range ofdifferent channel types, while achieving a high spectral efficiency.

Appendix A

The Gaussian WSSUSChannel Model

The input-output behavior of a linear, time-variant channel can be describedby the time-variant impulse responsec(τ, t), or also by means of its Fouriertransform with respect tot, the delay-Doppler-spread functionU(τ, ν). Withthe former the noise-free channel output signalr(t) is given as

r(t) =∫ ∞−∞

c(τ, t)s(t− τ)dτ,

while using the latterr(t) can be written as

r(t) =∫ ∞−∞

∫ ∞−∞

U(τ, ν)s(t− τ)ej2πνtdνdτ.

Wide-sense stationarity is a frequently adopted assumption for randomlytime-variant channels. That is, the cross-correlation of the two impulse responsesamplesc(τ, T0) andc(τ ′, T1) is assumed to depend onτ , τ ′ and(T1−T0) only,rather than on the absolute timesT0 andT1. As shown in [88], this wide-sensestationarity translates to the property thatU(τ, ν) andU(τ ′, ν′) are uncorrelatedfor ν 6=ν′. In combination with the US assumption with respect toτ this definesthe WSSUS channel, for which

E[U(τ, ν)U∗(τ ′, ν′)] = S(τ, ν)δ(τ − τ ′)δ(ν − ν′). (A.1)

90 Appendix A: The Gaussian WSSUS Channel

The scattering functionS(τ, ν) represents a second-order moment, providingstatistical information about the spreading in the delay and the Doppler dimen-sions. A scattering function with an exponential shape in the delay dimensionand a two-sided exponential shape in Doppler dimension is sometimes assumed,given as

S(τ, ν) =(√

2τdνD

)−1

exp(−τ/τd −

√2∣∣ν∣∣/νD

), τ ≥ 0 (A.2)

with νD denoting the Doppler spread.The cross-correlation between the gain on themth subchannel at the time

T0 and the gain on thenth subchannel at the timeT1, denoted byαm andαn,respectively, can be expressed as

E [αnα∗m] = E [C((n−NC/2)F∆, T1)C∗((m−NC/2)F∆, T0)] , (A.3)

whereC(f, t) represents the time-variant transfer function. At this point we ofcourse again assume that the channel is only slowly time-variant, as this was anecessary assumption for the definition of the gain factor in Sect. 2.3. The time-variant transfer function is obtained fromU(τ, ν) via the time-variant impulseresponse by a two-dimensional Fourier transform according to

C(f, t) =∫ ∞−∞

∫ ∞−∞

U(τ, ν)e−j2π(fτ−tν)dνdτ.

From (A.1) it follows that

E[C(F1, T1)C∗(F0, T0)] =∫ ∞−∞

∫ ∞−∞

S(τ, ν)e−j2π((F1−F0)τ−(T1−T0)ν)dνdτ,

and inserting into (A.3) yields

E[αnα∗m] =∫ ∞−∞

∫ ∞−∞

S(τ, ν)e−j2π((n−m)F∆τ−(T1−T0)ν)dνdτ (A.4)

for the gain factor cross-correlation. Applying the model (A.2) results in

E[αnα∗m] =1

1 + j2πτd(n−m)F∆· 1

1 + 2π2νD2(T1 − T0)2

.

If in addition to (A.1) the delay-Doppler-spread functionU(τ, ν) is modeledas a two-dimensional zero-mean complex Gaussian random process, a time-variant transfer function valueC(F0, T0) represents a complex Gaussian ran-dom variable. Furthermore,(α0, . . . , αNC−1)T and(α0, . . . , αNC−1)T are jointly

91

complex Gaussian random vectors, and their covariance matrixRc and cross-covariance matrixRcc can be derived fromS(τ, ν) by applying (A.4). If thescattering function is separable, i.e. it can be expressed as the product of a func-tion depending only onτ and a function depending only onν as is the casein (A.2), the cross-covariance matrix can be written asRcc = ρRc, whereρdepends on(T1−T0). In the case of the scattering function (A.2) we have

ρ =1

1 + 2π2νD2(T1 − T0)2

.

Appendix B

The Matched Filter Bound

The derivation of the exact performance of sophisticated ML or equalizer basedreceiver algorithms in fading channel environments is most often a tedious task.For uncoded signals the MFB represents a theoretical lower bound for the at-tainable receiver performance. The MFB is obtained by assuming an isolated,interference-free transmission of each modulated signal pulse, and averagingthe error probability from a perfect matched filter detector over the fading statis-tic1. The slope of the MFB curve reflects the intrinsic diversity resulting fromthe channel frequency and time selectivity. For time-discrete, time-invariantRayleigh fading channels the MFB was derived in [90] and [91]. For time-continuous WSSUS channels where the scattering function can be factorizedinto two components depending on only the delay and the Doppler shift, respec-tively, the MFB was calculated in [92, 93]. In this appendix we derive the MFBusing the method presented in [94], which is practicable for general doubly dis-persive Gaussian WSSUS channels characterized by the scattering function.

First, let s1(t) ands2(t) denote two deterministic signals transmitted withprobability0.5 ands(t) = s1(t)−s2(t). At the output of a time-variant, linearchannel described by the delay-Doppler-spread functionU(τ, ν) the signals read

ri(t) =∫ ∞−∞

∫ ∞−∞

U(τ, ν)si(t− τ)ej2πνtdνdτ, i = 1, 2.

1By focusing on the pairwise error probabilities, the same technique can also be employed forthe performance analysis of a coded transmission over fading channels [89].

94 Appendix B: The Matched Filter Bound

We employ the Gaussian WSSUS model, according to whichU(τ, ν) representsa two-dimensional, zero-mean white complex Gaussian random process. Themean probability of an error in the detection ofs1(t) ands2(t) using an optimumdecision device equals

PMF = E

[Q(√

εr2N0

)], (B.1)

where the random variableεr holds the energy inr(t) = r1(t)−r2(t) andN0

again determines the power spectral density of the AWGN corrupting the signalin the receiver. The covariance function of the zero-mean nonstationary randomprocessr(t) is given as

Rr(t, t′) =E[r(t)r∗(t′)] = E

[ ∫ ∞−∞

∫ ∞−∞

U(τ, ν)s(t − τ)ej2πνtdνdτ ×

(∫ ∞−∞

∫ ∞−∞

U(τ, ν)s(t′ − τ)ej2πνt′dνdτ

)∗ ].

To simplify the following derivations, it is assumed thatr(t) = 0 for t /∈ Irwith Ir ⊂R being a finite time interval, and thatRr(t, t′) is continuous withinIr×Ir. By exchanging the order of expectation and integration in the aboveequation and using the WSSUS assumption (A.1), the covariance function canbe expressed as

Rr(t, t′) =∫ ∞−∞

∫ ∞−∞

S(τ, ν)s(t− τ)s∗(t′ − τ)ej2πν(t−t′)dνdτ, t, t′ ∈ Ir.

The Karhunen-Loève expansion of the processr(t) reads

r(t) =∞∑p=1

zpψp(t), t ∈ Ir,

wherezp=∫Ir r(t)ψ

∗p(t)dt. The orthonormal functionsψ1(t), ψ2(t), . . . are the

eigenfunctions of the positive semidefinite conjugate complex symmetric kernelRr(t, t′), and ordered such that the corresponding real non-negative eigenval-uesλ1, λ2, . . . form a non-increasing sequence. The energy inr(t) can now beexpressed asεr=

∑∞p=1 |zp|2. Further properties of the Karhunen-Loève expan-

sion areE[zpz∗q

]= 0 for p 6= q andE

[|zp|2

]= λp. Since the processU(τ, ν)

95

is complex Gaussian, the random variablesz1, z2, . . . are independent complexGaussian, while|z1|2, |z2|2, . . . are independent and chi-square distributed withtwo degrees of freedom.

The PDFpyK (y) of the truncated sumyK =∑Kp=1 |zp|2 can easily be found,

e.g. via the characteristic function. In the case of distinct eigenvalues,

pyK (y) =K∑p=1

βpλp

exp(− y

λp

), y ≥ 0,

where

βp =K∏q=1q 6=p

λpλp − λq

.

For the expectation in (B.1) withεr replaced byyK we obtain

P(K)MF =

∫ ∞0

Q(√

y

2N0

)pyK (y)dy =

K∑p=1

βp2

(1−

√λp

λp + 4N0

).

Expressions for the case that not all eigenvalues are distinct can be found in [89].Since0≤y1≤y2≤ . . . and as the mappingQ(

√·) is monotonically decreasing,

it follows that 0.5 ≥ P(1)MF ≥ P

(2)MF ≥ . . .. Finally, PMF = lim

K→∞P

(K)MF can be

deduced from the convergence ofyK towardsεr in mean by convergence theo-rems in probability theory (see e.g. [95]). The approximate error rates approachthe MFBPMF from above. An alternative sequence of true performancelowerbounding probabilities converging towardsPMF from below is derived in [94].For both approaches it usually turns out to be sufficient to consider only a fewdominant eigenvalues ofRr(t, t′) for receiving stable values ofP (K)

MF .

Appendix C

List of Abbreviations

ADAMAS adaptive multicarrier access systemADC analog-to-digital conversionADSL asymmetric digital subscriber lineAWGN additive white Gaussian noiseBER bit error rateBICM bit-interleaved coded modulationBRAN broadband radio access networkBSC binary symmetric channelCSI channel state informationDAB digital audio broadcastingDAC digital-to-analog conversionDFT discrete Fourier transformDMT discrete multitoneDSL digital subscriber lineDVB digital video broadcastingETSI European telecommunications standards instituteFB filter bankFEC forward error correctionFFT fast Fourier transformFIR finite impulse responseFWA fixed wireless accessHIPERLAN/2 high performance LAN type 2

98 Appendix C: List of Abbreviations

IB ideal broadbandIBI interblock interferenceIC interference cancellationICI interchannel interferenceIDFT inverse discrete Fourier transformIEEE institute of electrical and electronics engineersISI intersymbol interferenceISM instrumentation, scientific, and medicalIST information society technologiesLLF log-likelihood functionLLR log-likelihood ratioLMDS local multipoint distribution serviceLOS line-of-sightMC multicarrierMFB matched filter boundMIMO multiple input multiple outputML maximum-likelihoodMLSD maximum-likelihood sequence decoderMMDS multichannel multipoint distribution serviceMMSE minimum mean-square errorNLOS non-line-of-sightOFDM orthogonal frequency-division multiplexingOFDMA orthogonal frequency-division multiple accessPDF probability density functionPDP power delay profilePSK phase-shift keyingPSTN public switched telephone networkQAM quadrature amplitude modulationQoS quality-of-serviceQPSK quadrature phase-shift keyingRF radio frequencyRMS root mean-squareSAGE space-alternating generalized expectation-maximization

algorithmSNR signal-to-noise ratioTCM trellis-coded modulation

99

TDD time-division duplexTF time-frequencyUMTS universal mobile telecommunications systemUS uncorrelated scatteringW-CDMA wideband code-division multiple-accessWH Weyl-HeisenbergWLANs wireless local area networksWSSUS wide-sense stationary uncorrelated scatteringZF zero-forcing3G third generation

Appendix D

List of Symbols

Latin symbols

A pulse cross-correlation matrixA sparse pulse cross-correlation matrixA matrixA−AAf (τ, ν) ambiguity function off(t)An factor equal toMn(Mn−1)/((

√Mn−1)2|αn|2)

a`,m themth element in the th row in Aa`,m approximate reconstruction ofa`,ma`,m themth element in the th row in ABn factor equal to3|αn|2/((Mn−1)N0)b bit tuplebk,n bit tuple estimate from slicerb` `th bit value

C set of the complex numbersC diagonal matrix with the subchannel gain factorsC(f) time-invariant transfer functionC(f, t) time-variant transfer functionCAWGN(γ) capacity of an AWGN channel with SNRγCBICM(γ) BICM channel capacity as a function ofγCBICM,IB(γ) IB BICM-OFDM channel capacityCBICM,IB,WF(γ) IB BICM-OFDM channel capacity with water-fillingCIB,CSIT(γ) IB OFDM channel capacity applying water-filling

102 Appendix D: List of Symbols

COFDM OFDM channel capacityCOFDM,CSIT OFDM channel capacity with transmitter side CSICOFDM,IB(γ) IB OFDM channel capacityCZF channel capacity with the ZF receiverC(f) overall transfer functionc channel parameter vector for burst transmissionc(t) channel parameter vector random processc observed outdated channel parameter vectorc(τ) time-invariant impulse responsec(τ, t) time-variant impulse responsecn(t) nth subchannel gain factor random processc(τ) overall impulse responseDfree Hamming free distanceDIC,F number of adjacent subchannels included in the ICDIC,T number of adjacent time slots included in the ICd decorrelated symbol vectordn signal point distanceE[·] expectation operatorek,n(t) reconstruction error signalFG(f) Fourier transform offG(t)F∆ subchannel frequency spacing

fG(t) Gaussian function defined as(2β)14 exp

(−πβt2

)fR(t) pulse with a rectangular shapeGn(z) nth FIR filter transfer functionGRX(f) antialiasing filter frequency responseGTX(f) lowpass filter frequency responseg(t) prototype functiongk,n(t) waveform of an elementary signal pulse(·)H Hermitian transpositionh(·) differential entropyh(·|·) conditional differential entropyhk,n(t) elementary signal pulse at the channel outputhk,n(t) partial signal pulse at the channel outputhk,n(t) signal pulsehk,n(t)−hk,n(t)Ig time interval containing the support ofg(t)I(k)h time interval containing the support ofhk,n(t)

103

Ir time interval containing the support ofr(t)IN N×N -identity matrixIm[·] imaginary part operatori indexing a bit or sample in a sequencei(k, n) index mapping(k, n) 7−→kNC+n+1J(·) cost functionK number of dominant eigenfunctions consideredk indexing a time slotL number of coded bits mapped onto a signalLn number of coded bits per signal on thenth subchannel

L(x; a) function defined as√

1x(x+1)3 e

−a∞∑k=0

Γ(k+ 32 )

(k!)2

(ax+1

)kL−1(·; a) the inverse ofL(·; a) with respect to the first argument` indexing a bit in a tupleld(·) logarithm to the base2ln(·) logarithm to the baseeM signal set sizeMn signal set size on thenth subchannelmax· · · maximum real-valued element in a setmin· · · minimum real-valued element in a setN (m0, σ

20) normal distribution with meanm0 and varianceσ2

0

NB number of time slots in a burstNBC number of encoded bits per time slotNC number of subchannelsNcyc number of LLF maximization iteration cyclesNdec number of Gauss-Seidel iteration cyclesNFFT FFT sizeNG number of guard period sample values

Nh length of the intervalI(k)h in samples

N0 AWGN power spectral densityn indexing a subchannelnA number of additions/subtractions for the detectionnM number of multiplications/divisions for the detectionn′A number of additions for the matched filteringn′M number of multiplications for the matched filteringnA total number of additions/subtractions


nM total number of multiplications/divisionsPb averaged burst BER

P(n)b BER on thenth subchannel

P(n,`)b BER on the binary equivalent channel(n, `)Pe free distance error event probabilityPMF error probability with an optimum decision device

P(K)MF approximation ofPMF usingK eigenvaluesPSC BSC transition probabilityPr [· · ·] probability of an eventP ′e approximate free distance error event probabilityPb mean BERPSC conditional expectation ofPSC

p(·), px(·) PDF (of a random variablex)p(·|·), px(·|·) conditional PDF (of a random variablex)Q(·) Q-function defined asQ(x)=

∫∞x (2π)−

12 e−t

2/2dtq indexing the iteration cycle

R set of the real numbersRc covariance matrix of channel parameter vectorcRcc cross-covariance matrix ofc andcRc|c conditional covariance matrix ofcR information bit rateRC code rateRk,n(t, t′) covariance function ofhk,n(t)Rr(t, t′) covariance function ofr(t)Re[·] real part operatorr(µ) residual vectoru−

√S0Ax(µ)

r(t) noise-free signal at the channel output

r(µ)` `th element inr(µ)

r1(t), r2(t) noise-free channel output signals fors1(t), s2(t)r(t) reconstructed signalr(t)S(τ, ν) scattering functionS0 transmit power spectral densityS0 variable converging towardsS0 in iterative proceduresk vector withkth time slot signal sampless(t) transmitted signalsi discrete time transmit signal

105

sk,i kth time slot signal sample values1(t), s2(t) deterministic transmit signals(·)T transpositionTF FFT window lengthTG guard period length

Th length of the intervalI(k)h

T0, T1 time constantsT∆ spacing of the time slotsU factorization ofAU(τ, ν) delay-Doppler-spread functionu vector containing the sufficient statisticuk kth time slot DFT output vectoru DFT output signaluk,n matched filtering coefficientui DFT output signal associated with theith bitu` `th element inuuk,n approximate reconstruction ofuk,nv noise vectorvk kth time slot noise vectorv′ noise vector after the decorrelationv complex Gaussian noise termv(t) AWGN processvk,n noise term corruptinguk,nW channel bandwidthW(·) real-valued Lambert’s W-functionWcoh coherence bandwidthx vector with the data symbolsxk vector with the data symbols conveyed inkth time slotx(µ) estimate ofx afterµ iterationsxML ML estimate ofxxk,n complex-valued data symbol

x(µ)` `th element inx(µ)

y(t) received signalyK sum

∑Kp=1 |zp|2

Z set of the integerszp projection coefficient, Gaussian random variable


Greek symbols

α random subchannel gain factorαn nth subchannel gain factorαn observed outdatednth subchannel gain factorβ positive factor determining the slope offG(t)

βp constant equal toK∏q=1q 6=p

λpλp−λq

Γ(·) the gamma functionγ instantaneous SNRγKL signal to truncation error energy ratioγk,n (k, n)th TF slot equivalent AWGN channel SNRγn SNR on thenth subchannelγk,n respective SNR value forγk,n with an OFDM receiverγ mean SNRδ(·) Dirac delta functionδn,m Kronecker delta

εIC supremum ofε(k,n)IC for NB, NC→∞

ε(k,n)IC mean residual interference energy inr(µ0)

i(k,n)

εr received signal energyζ` the inverse th diagonal element inA−1

ζ0 limes forζ` with an ideal channel asNB, NC→∞ζ` the`th diagonal element inAΛ(·; ·) LLFΛn,`(·) function yielding the LLRλ Lagrange multiplierλp pth eigenvalue ofRr(t, t′)λ(p) pth TF slot independent eigenvalue

λ(p)k,n pth eigenvalue ofRk,n(t, t′)λ0 positive constantµ indexing the iterationµ(·) bits-to-signal mappingµn conditional mean ofαnµ0 number of iterations after whichx(µ0) =xν frequency lagνD Doppler spread

107

ξ(t) prototype function of the biorthogonal WH systemρ cross-correlation betweenαn andαnσ2

n noise term varianceσ2n conditional variance ofαnτ time lagτd RMS delay spreadτexcess excess delay spreadφc(τ) PDPφi subchannel chosen for theith bitφk,n(t) basis functionψi bit tuple position chosen for theith bitψp(t) pth eigenfunction ofRr(t, t′)ψ

(p)k,n(t) pth eigenfunction ofRk,n(t, t′)

Ω signal setω(·) power assignment for an IB channel

ω(p)k,n channel parameter defined as

⟨hk,n, ψ

(p)k,n

⟩ωn mean signal energy on thenth subchannel

ω(p)n time-invariant channel parameterω(ρ)(·) IB channel power assignment with outdated CSI

Other symbols

(·)∗ complex conjugation∗ convolution〈·, ·〉 scalar product

Bibliography

[1] W. Webb, “Broadband fixed wireless access as a key component of thefuture integrated communications environment,”IEEE Commun. Mag.,vol. 39, pp. 115–121, Sept. 2001.

[2] H. Bolcskei, A. J. Paulraj, K. V. S. Hari, R. U. Nabar, and W. W. Lu,“Fixed broadband wireless access: State of the art, challenges, and futuredirections,”IEEE Commun. Mag., vol. 39, pp. 100–108, Jan. 2001.

[3] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-divisionmultiplexing using the discrete fourier transform,”IEEE Trans. Commun.Technology, vol. 19, pp. 628–634, Oct. 1971.

[4] J. A. C. Bingham,ADSL, VDSL, and Multicarrier Modulation. New York,NY: John Wiley and Sons, 2000.

[5] A. Duel-Hallen, S. Hu, and H. Hallen, “Long-range prediction of fadingsignals,”IEEE Sig. Processing Mag., vol. 17, pp. 62–75, May 2000.

[6] V. Demoulin and M. Pecot, “Vector equalization: An alternative approachfor OFDM systems,”Annales des Telecommunications, vol. 52, no. 1–2,pp. 4–11, 1997.

[7] L. Vandendorpe, “Fractionally spaced linear and DF mimo equalizers formultitone systems without guard time,”Annales des Telecommunications,vol. 52, no. 1–2, pp. 21–30, 1997.

[8] T. S. Rappaport,Wireless Communications. Upper Saddle River, NJ:Prentice-Hall, 1996.

109

110 BIBLIOGRAPHY

[9] N. Patwari, G. D. Durgin, T. S. Rappaport, and R. J. Boyle, “Peer-to-peerlow antenna outdoor radio wave propagation at 1.8 GHz,” inIEEE Proc.Veh. Technol. Conf. (VTC) ’99-Spring, (Houston, TX), pp. 371–375, May1999.

[10] V. Erceg, D. G. Michelson, S. S. Ghassemzadeh, L. J. Greenstein, A. J.Rustako, Jr., P. B. Guerlain, M. K. Dennison, R. S. Roman, D. J. Barnickel,S. C. Wang, and R. R. Miller, “A model for the multipath delay profile offixed wireless channels,”IEEE J. Select. Areas Commun., vol. 17, pp. 399–410, Mar. 1999.

[11] M. Pettersen, P. H. Lehne, J. Noll, O. Rostbakken, E. Antonsen, andR. Eckhoff, “Characterisation of the directional wideband radio channelin urban and suburban areas,” inIEEE Proc. Veh. Technol. Conf. (VTC)’99-Fall, (Amsterdam, The Netherlands), pp. 1454–1459, Sept. 1999.

[12] M. Mizuno, S. Sekizawa, and K. Taira, “Measurement of spatiotemporalpropagation characteristics in urban microcellular environment,” inIEEEProc. Veh. Technol. Conf. (VTC) ’99-Fall, (Amsterdam, The Netherlands),pp. 2263–2267, Sept. 1999.

[13] A. Bohdanowicz, G. J. M. Janssen, and S. Pietrzyk, “Wideband indoor andoutdoor multipath channel measurements at 17 GHz,” inIEEE Proc. Veh.Technol. Conf. (VTC) ’99-Fall, (Amsterdam, The Netherlands), pp. 1998–2003, Sept. 1999.

[14] P. Karlsson, N. Löwendahl, and J. Jordana, “Narrowband and widebandpropagation measurements and models in the 27-29 GHz band,” inCOST259 TD(98)17, COST 259 Workshop, (Berne, Switzerland), Feb. 1998.

[15] R. van Nee and R. Prasad,OFDM for Wireless Multimedia Communica-tions. Boston, MA: Artech House, 2000.

[16] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digitalterrestrial TV broadcasting,”IEEE Commun. Mag., vol. 33, pp. 100–109,Feb. 1995.

[17] J. A. C. Bingham, “Multicarrier modulation for data transmission: An ideawhose time has come,”IEEE Commun. Mag., vol. 28, pp. 5–14, May 1990.

BIBLIOGRAPHY 111

[18] Z. Wang and G. B. Giannakis, “Wireless multicarrier communications,”IEEE Sig. Processing Mag., vol. 17, pp. 29–48, May 2000.

[19] U. P. Bernhard and J. Meierhofer, “Coded OFDM transmission for high-speed wireless access to ATM networks,” inProc. Wireless ’96, (Calgary,Canada), pp. 102–115, July 1996.

[20] G. Ungerboeck, “Channel coding with multilevel/phase signals,”IEEETrans. Inform. Theory, vol. 28, pp. 55–67, Jan. 1982.

[21] S. H. Jamali and T. Le-Ngoc,Coded-Modulation Techniques for FadingChannels. Norwell, MA: Kluver Academic Publishers, 1994.

[22] E. Zehavi, “8-PSK trellis codes for a rayleigh channel,”IEEE Trans. Com-mun., vol. 40, pp. 873–884, May 1992.

[23] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,”IEEE Trans. Inform. Theory, vol. 44, pp. 927–946, May 1998.

[24] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of lin-ear codes for minimizing symbol error rate,”IEEE Trans. Inform. Theory,vol. 20, pp. 284–287, Mar. 1974.

[25] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input soft-output APP module for iterative decoding of concatenated codes,”IEEECommun. Let., vol. 1, pp. 22–24, Jan. 1997.

[26] X. Li and J. A. Ritcey, “Trellis-coded modulation with bit interleaving anditerative decoding,”IEEE J. Select. Areas Commun., vol. 17, pp. 715–724,Apr. 1999.

[27] S. ten Brink, “Designing iterative decoding schemes with the extrinsic in-formation transfer chart,”Int. J. Electron. Commun. (AEU), vol. 54, no. 6,pp. 389–398, 2000.

[28] P. Robertson and S. Kaiser, “Analysis of the effects of phase-noise in or-thogonal frequency division multiplex (OFDM) systems,” inIEEE Proc.Int. Conf. on Commun. (ICC) ’95, (Seattle, WA), pp. 1652–1657, June1995.

112 BIBLIOGRAPHY

[29] K. Nikitopoulos and A. Polydoros, “Compensation schemes for phasenoise and residual frequency offsets in OFDM systems,” inIEEE Proc.Globecom 2001, (San Antonio, TX), pp. 330–333, Nov. 2001.

[30] O. Edfors, M. Sandell, J.-J. van de Beek, S. K. Wilson, and P. O. Börjesson,“OFDM channel estimation by singular value decomposition,”IEEE Trans.Commun., vol. 46, pp. 931–939, July 1998.

[31] Y. Li, L. J. Cimini, Jr., and N. R. Sollenberger, “Robust channel estimationfor OFDM systems with rapid dispersive fading channels,” inIEEE Proc.Int. Conf. on Commun. (ICC) ’98, (Atlanta, GA), pp. 1320–1324, June1998.

[32] Y. Li, “Pilot-symbol-aided channel estimation for OFDM in wireless sys-tems,”IEEE Trans. Veh. Technol., vol. 49, pp. 1207–1215, July 2000.

[33] S. Furrer and D. Dahlhaus, “Mean bit-error rates for OFDM transmissionwith robust channel estimation and space diversity reception,” inIEEEProc. Int. Zurich Seminar on Broadband Commun. 2002, (Zurich, Switzer-land), pp. 41–1 – 41–6, Feb. 2002.

[34] M. P. Fitz and J. P. Seymour, “On the bit error probability of QAM mod-ulation,” Int. J. Wireless Information Networks, vol. 1, pp. 131–139, Apr.1994.

[35] G. Kaplan and S. Shamai, “Error probabilities for the block-fading gaus-sian channel,”Int. J. Electron. Commun. (AEU), vol. 49, no. 4, pp. 192–205, 1995.

[36] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of block-fading channels with multiple antennas,”IEEE Trans. Inform. Theory,vol. 47, pp. 1273–1289, May 2001.

[37] J. G. Proakis,Digital Communications. New York, NY: McGraw-Hill,3rd ed., 1995.

[38] T. M. Cover and J. A. Thomas,Elements of Information Theory. New York,NY: John Wiley and Sons, 1991.

BIBLIOGRAPHY 113

[39] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with chan-nel side information,”IEEE Trans. Inform. Theory, vol. 43, pp. 1986–1992,Nov. 1997.

[40] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fadingchannels,”IEEE Trans. Inform. Theory, vol. 45, pp. 1468–1489, July 1999.

[41] V. O. Hentinen, “Error performance for adaptive transmission on fadingchannels,”IEEE Trans. Commun., vol. 22, pp. 1331–1337, Sept. 1974.

[42] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discretemultitone transceiver loading algorithm for data transmission over spec-trally shaped channels,”IEEE Trans. Commun., vol. 43, pp. 773–775,Feb./Mar./Apr. 1995.

[43] R. V. Sonalkar and R. R. Shively, “An efficient bit-loading algorithm forDMT applications,”IEEE Commun. Let., vol. 4, pp. 80–82, Mar. 2000.

[44] A. Czylwik, “Adaptive OFDM for wideband radio channels,” inIEEEProc. Globecom ’96, (London, UK), pp. 713–718, Nov. 1996.

[45] S. K. Lai, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Adaptive trelliscoded MQAM and power optimization for OFDM transmission,” inIEEEProc. Veh. Technol. Conf. (VTC) ’99-Spring, (Houston, TX), pp. 290–294,May 1999.

[46] S. K. Lai, R. S. Cheng, K. B. Letaief, and C. Y. Tsui, “Adaptive tracking ofoptimal bit and power allocation for OFDM systems in time-varying chan-nels,” in IEEE Proc. Wireless Commun. and Networking Conf. (WCNC)’99, (New Orleans, LA), pp. 776–780, Sept. 1999.

[47] R. F. H. Fischer and J. B. Huber, “A new loading algorithm for discretemultitone transmission,” inIEEE Proc. Globecom ’96, (London, UK),pp. 724–728, Nov. 1996.

[48] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAMfor fading channels,”IEEE Trans. Commun., vol. 45, pp. 1218–1230, Oct.1997.

[49] C. Kose and D. L. Goeckel, “On power adaptation in adaptive signalingsystems,”IEEE Trans. Commun., vol. 48, pp. 1769–1773, Nov. 2000.

114 BIBLIOGRAPHY

[50] B. S. Krongold, K. Ramchandran, and D. L. Jones, “Computationally ef-ficient optimal power allocation algorithm for multicarrier communicationsystems,” inIEEE Proc. Int. Conf. on Commun. (ICC) ’98, (Atlanta, Ga),pp. 1018–1022, June 1998.

[51] D. L. Goeckel, “Adaptive coding for time-varying channels using outdatedfading estimates,”IEEE Trans. Commun., vol. 47, pp. 844–855, June 1999.

[52] A. J. Goldsmith and S.-G. Chua, “Adaptive coded modulation for fadingchannels,”IEEE Trans. Commun., vol. 46, pp. 595–602, May 1998.

[53] P. Ormeci, X. Liu, D. L. Goeckel, and R. D. Wesel, “Adaptive bit-interleaved coded modulation,”IEEE Trans. Commun., vol. 49, pp. 1572–1581, Sept. 2001.

[54] T. Keller and L. Hanzo, “Sub-band adaptive pre-equalised OFDM trans-mission,” inIEEE Proc. Veh. Technol. Conf. (VTC) ’99-Fall, (Amsterdam,The Netherlands), pp. 334–338, Sept. 1999.

[55] E. A. Al-Susa and R. F. Ormondroyd, “An improved channel inversionbased adaptive OFDM system in the presence of channel errors and rapidtime variations,” in IEEE Proc. Veh. Technol. Conf. (VTC) 2000-Fall,(Boston, MA), pp. 1114–1119, Sept. 2000.

[56] A. Scaglione and S. Barbarossa, “Optimal power loading for OFDM trans-missions over underspread rayleigh time-varying channels,” inIEEE Proc.Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP) 2000,(Istanbul, Turkey), pp. 2969–2972, June 2000.

[57] C. Mutti, D. Dahlhaus, T. Hunziker, and M. Foresti, “Bit and power loadingprocedures for OFDM systems with bit-interleaved coded modulation,” inIEEE Int. Conf. on Telecommunications (ICT) 2003, (submitted).

[58] H. V. Poor,An Introduction to Signal Detection and Estimation. New York,NY: Springer, 2nd ed., 1994.

[59] C. Schlegel,Trellis Coding. New York, NY: IEEE Press, 1997.

[60] T. Hunziker and D. Dahlhaus, “Adaptive OFDM transmission for broad-band fixed wireless access systems,” inProc. 6th International OFDM-Workshop, (Hamburg, Germany), pp. 5–1 – 5–4, Sept. 2001.

BIBLIOGRAPHY 115

[61] B. Stantchev and G. Fettweis, “Time-variant distortions in OFDM,”IEEECommun. Let., vol. 4, pp. 312–314, Sept. 2000.

[62] A. Duel-Hallen, “Equalizers for multiple input/multiple output channelsand PAM systems with cyclostationary input sequences,”IEEE J. Select.Areas Commun., vol. 10, pp. 630–639, Apr. 1992.

[63] N. Al-Dhahir and J. M. Cioffi, “Optimum finite-length equalization formulticarrier transceivers,”IEEE Trans. Commun., vol. 44, pp. 56–64, Jan.1996.

[64] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response short-ening for discrete multitone transceivers,”IEEE Trans. Commun., vol. 44,pp. 1662–1672, Dec. 1996.

[65] G. Arslan, B. L. Evans, and S. Kiaei, “Optimum channel shorteningfor discrete multitone transceivers,” inIEEE Proc. Int. Conf. on Acous-tics, Speech, and Signal Processing (ICASSP) 2000, (Istanbul, Turkey),pp. 2965–2968, June 2000.

[66] S. D. Sandberg and M. A. Tzannes, “Overlapped discrete multitone modu-lation for high speed copper wire communications,”IEEE J. Select. AreasCommun., vol. 13, pp. 1571–1585, Dec. 1995.

[67] R. Haas and J.-C. Belfiore, “A time-frequency well-localized pulse formultiple carrier transmision,”Wireless Personal Communications, vol. 5,pp. 1–18, July 1997.

[68] W. Kozek and A. F. Molisch, “Nonorthogonal pulseshapes for multicar-rier communications in doubly dispersive channels,”IEEE J. Select. AreasCommun., vol. 16, pp. 1579–1589, Oct. 1998.

[69] A. Scaglione, G. B. Giannakis, and S. Barbarossa, “Redundant filterbankprecoders and equalizers part i: Unification and optimal designs,”IEEETrans. Sig. Processing, vol. 47, pp. 1988–2006, July 1999.

[70] Y.-P. Lin and S.-M. Phoong, “Perfect discrete multitone modulation withoptimal transceivers,”IEEE Trans. Sig. Processing, vol. 48, pp. 1702–1711, June 2000.

116 BIBLIOGRAPHY

[71] L. Vandendorpe, J. Louveaux, B. Maison, and A. Chevreuil, “About theasymptotic performance of MMSE MIMO DFE for filter-bank based mul-ticarrier transmission,”IEEE Trans. Commun., vol. 47, pp. 1472–1475,Oct. 1999.

[72] T. Hunziker and D. Dahlhaus, “Iterative symbol detection for bandwidth ef-ficient nonorthogonal multicarrier transmission,” inIEEE Proc. Veh. Tech-nol. Conf. (VTC) 2000-Spring, (Tokyo, Japan), pp. 61–65, May 2000.

[73] T. Hunziker and D. Dahlhaus, “Iterative detection for multicarrier transmis-sion employing time-frequency concentrated pulses,”IEEE Trans. Com-mun., to appear.

[74] P. P. Vaidyanathan,Multirate Systems and Filter Banks. Englewood Cliffs,NJ: Prentice-Hall, 1993.

[75] R. M. Young,An Introduction to Nonharmonic Fourier Series. New York,NY: Academic Press, 1980.

[76] H. G. Feichtinger and T. Strohmer,Gabor Analysis and Algorithms: The-ory and Applications. Boston, MA: Birkhauser, 1998.

[77] T. W. Parks and R. G. Shenoy, “Time-frequency concentrated basis func-tions,” in IEEE Proc. Int. Conf. on Acoustics, Speech and Signal Processing(ICASSP) ’90, (Albuquerque, NM), pp. 2459–2462, Apr. 1990.

[78] F. Hlawatsch,Time-Frequency Analysis and Synthesis of Linear SignalSpaces. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1998.

[79] J. M. Mendel,Lessons in Estimation Theory for Signal Processing, Com-munications, and Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.

[80] A. van den Bos, “The multivariate complex normal distribution – a gener-alization,” IEEE Trans. Inform. Theory, vol. 41, pp. 537–539, Mar. 1995.

[81] K. Matheus, K. Knoche, M. Feuersänger, and K.-D. Kammeyer, “Two-dimensional (recursive) channel equalization for multicarrier systems withsoft impulse shaping (MCSIS),” inIEEE Proc. Globecom ’98, (Sydney,Australia), pp. 956–960, Nov. 1998.

BIBLIOGRAPHY 117

[82] J. A. Fessler and A. O. Hero, “Space-alternating generalized expectation-maximization algorithm,”IEEE Trans. Sig. Processing, vol. 42, pp. 2664–2677, Oct. 1994.

[83] O. Axelsson,Iterative Solution Methods. Cambridge, UK: Cambridge Uni-versity Press, 1994.

[84] G. H. Golub and C. F. Van Loan,Matrix Computations. Baltimore, MD:John Hopkins University Press, 3rd ed., 1996.

[85] A. D. Whalen,Detection of Signals in Noise. New York, NY: AcademicPress, 1971.

[86] A. Papoulis,Probability, Random Variables, and Stochastic Processes.Singapore: McGraw-Hill, 3rd ed., 1991.

[87] A. V. Oppenheim and R. W. Schafer,Digital Signal Processing. Engle-wood Cliffs, NJ: Prentice-Hall, 1975.

[88] P. A. Bello, “Characterization of randomly time-variant linear channels,”IEEE Trans. Commun. Syst., vol. 11, pp. 360–393, Dec. 1963.

[89] C. Schlegel, “Error probability calculation for multibeam rayleigh chan-nels,” IEEE Trans. Commun., vol. 44, pp. 290–293, Mar. 1996.

[90] V.-P. Kaasila and A. Mämmela, “Bit error probability of a matched filterin a rayleigh fading multipath channel,”IEEE Trans. Commun., vol. 42,pp. 826–828, Feb./Mar./Apr. 1994.

[91] F. Ling, “Matched filter-bound for time-discrete multipath rayleigh fadingchannels,”IEEE Trans. Commun., vol. 43, pp. 710–713, Feb./Mar./Apr.1995.

[92] K.-W. Yip and T.-S. Ng, “Karhunen-loève expansion of the WSSUS chan-nel output and its application to efficient simulation,”IEEE J. Select. AreasCommun., vol. 15, pp. 640–646, May 1997.

[93] K.-W. Yip and T.-S. Ng, “Matched filter bound for multipath rician-fadingchannels,”IEEE Trans. Commun., vol. 46, pp. 441–445, Apr. 1998.

[94] T. Hunziker and D. Dahlhaus, “Bounds on matched filter performance indoubly dispersive gaussian WSSUS channels,”IEE Elec. Lett., vol. 37,pp. 383–384, Mar. 2001.

[95] Y. S. Chow and H. Teicher,Probability Theory: Independence, Inter-changeability, Martingales. New York: Springer-Verlag, 2nd ed., 1988.

Curriculum Vitae

Thomas Philipp Hunziker, born 11. September 1966, in Zurich

Education

1981 - 1985 College in Bülach, ZH

1985 - 1992 Studies in Electrical Engineering at theSwiss Federal Institute of Technology (ETH) Zurich,granted ”Dipl. Ing. ETH” Degree

1996 - 2001 Postgraduate studies in Information Technology at theSwiss Federal Institute of Technology (ETH) Zurich,granted Post Graduate Diploma

Work

1992 - 1996 Software Engineer withNCR/AT&T Global Information Solutions Switzerland

1997 - 2002 Research Assistant at theCommunication Technology Laboratory,Swiss Federal Institute of Technology (ETH) Zurich

from Sept. 2002 Visiting Researcher at the Advanced TelecommunicationsResearch Institute International (ATR), Kyoto, Japan

Documents

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