opus4.kobv.de€¦ · Acknowledgements1 First of all, I would like to thank my supervisor Prof. Robert Fischer, for his patient guidance in my time as his student, his advice and

Coding, Modulation, and Detection for

Impulse-Radio Ultra-Wideband

Communications

Der Technischen Fakultät der

Friedrich-Alexander-Universität Erlangen-Nürnberg

zur Erlangung des Grades

DOKTOR–INGENIEUR

vorgelegt von

ANDREAS SCHENK

Erlangen — 2013

Codierung, Modulation und Detektion für

die Impulsradio-Ultrabreitband-

Kommunikation

Der Technischen Fakultät der

Friedrich-Alexander-Universität Erlangen-Nürnberg

zur Erlangung des Grades

DOKTOR–INGENIEUR

vorgelegt von

ANDREAS SCHENK

Erlangen — 2013

Als Dissertation genehmigt von der Technischen Fakultät

der Friedrich-Alexander-Universität Erlangen-Nürnberg

Tag der Einreichung: 30.10.2012

Tag der Promotion: 20.12.2012

Dekanin: Prof. Dr.-Ing. Marion Merklein

Berichterstatter: Prof. Dr.-Ing. Robert Fischer

Prof. Dr.-Ing. Lutz Lampe

Prof. Dr.-Ing. Johannes Huber

Acknowledgements1

First of all, I would like to thank my supervisor Prof. RobertFischer, for his patient guidance inmy time as his student, his advice and suggestions in uncountable precious discussions, and hisprompt responds to all my queries and questions. Furthermore, I am deeply indebted to Prof.Johannes Huber for arousing my interest in communication theory, for his constant advice inmany aspects, and for providing me with the possibilty to work at his lab. I also would like tothank Prof. Lutz Lampe for his interest in my work and his support during my diploma thesis.

Special thanks goes to all my colleagues for the joyful time Ihad at the lab, and in particularto Clemens Stierstorfer, Christian Siegl, Martin Hoch, Ümit Abay, Andreas Lehmann, MathisSeidl, Fabian Schuh, Michael Cyran, Alexander Onic, and Benjamin Lämmle for numerousdiscussions. Parts of this thesis are based on results emerged in collaboration with studentspreparing their theses at the LIT under my supervision, namely Susanne Sparrer, ChristophRachinger, Bilal Amin, and Matthias Hafner.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the frame-work UKoLoS (Ultra-Wideband Radio Technologies for Communications, Localization andSensor Applications) under grant FI 982/3.

I dedicate this booklet to my love, Veronika, and to my familyfor their constant support.

1In view of §4(1) of the doctoral examination regulations, parts of these acknowledgements may be seen as alisting of all aids obtained from persons and other sources not listed in the bibliography.

i

Abstract

Impulse-radio ultra-wideband (IR-UWB) communication systems are especially well-suited forshort-range low-data-rate communications, such as, e.g.,in wireless-sensor networks. The mainadvantages of IR-UWB include its robustness to severe multi-path fading even in indoor envi-ronments, the potential to provide accurate localization,and its low cost and complexity.

Whereas the design of low-complexity low-power-consumingIR-UWB transmitters using po-wer-efficient modulation schemes such as pulse-position modulation (PPM) or variants of dif-ferential binary phase-shift keying (so-called differential transmitted reference (DTR) signaling)is well known, in the first part of this thesis detection schemes for low-complexity and power-efficient IR-UWB receivers are derived. In the second part, the impact of this low-complexitydemand on the design of coded IR-UWB communication systems is addressed.

Noncoherent detection schemes, which directly use the receive signal as reference for demodu-lation, are widely regarded as the key for low-complexity IR-UWB receiver design, since costlychannel estimation required for coherent detection is avoided. To this end, based on a genericmodel of IR-UWB communication, which describes a number of popular variants of (coded)IR-UWB modulation including PPM and DTR IR-UWB, a generic noncoherent maximum-likelihood receiver is presented. This approach avoids explicit channel estimation. Based onthis unified treatment, energy detection and autocorrelation are identified as two equivalent de-tection principles underlying such receivers. The autocorrelation principle is preferable froman implementation point of view, as it allows to separate an (analog) autocorrelation receiverfront-end from (digital) signal processing implementing the decision process.

Consistently incorporating the application of observation window lengths larger than one sym-bol duration in the derivation of the receiver, a comprehensive comparison of very low-com-plexity symbol-wise detection schemes and their extensions to advanced noncoherent detectionschemes, which process multiple symbols jointly, is provided. It is shown, that the inherent lossin performance of traditional noncoherent symbol-wise detection, as compared to coherent de-tection based on ideal channel estimation, can be alleviated by such multiple-symbol-detection-based schemes. These operate on the output of an extended autocorrelation receiver, whichdelivers correlation coefficients of symbols separated by several symbol durations.

Focusing on DTR IR-UWB as the most attractive variant of IR-UWB modulation, block-baseddetection schemes, which partition the receive symbol stream into (possibly overlapping) blocksand thus process multiple symbols jointly, are proven to outperform detection schemes im-plementing sequence estimation in both uncoded and coded IR-UWB transmission systems.Especially improved variants of block-wise decision-feedback differential detection (DFDD)

ii

employing an optimum decision order in combination with techniques termed “multiple-obser-vations combining” and “virtually increased block size” enable an excellent trade-off betweencomputational complexity and error-rate performance. Moreover, it is shown how a simple formof reliability information can be obtained at the output of such detectors. This enables to im-prove the decoding performance of a subsequent (soft-input) channel decoder at only marginalincrease in computational complexity.

In the second part of this thesis, an information-theoreticanalysis of coded IR-UWB transmis-sion in combination with the above noncoherent detection schemes is presented. The investiga-tions are restricted to the common approach of bit-interleaved coded modulation (BICM), i.e.,the serial concatenation of coding, interleaving, and modulation. Based on the capacity of thetransmission chain including modulation, transmission over multipath propagation channels,and autocorrelation-based detection, as presented in the first part of this thesis, design rules forrealistic IR-UWB systems are deduced, and optimum rates forthe employed channel code areidentified.

Despite being in the order of2/3 to 4/5, and thus leading to increased data rate compared tothe more common setting of a rate-1/2 code, these optimum code rates improve the power effi-ciency of coded IR-UWB transmission. The design rules are validated by means of numericalresults in a coded IR-UWB system employing convolutional codes.

The insights gained demonstrate, that, especially if noncoherent receivers are employed, coding,modulation, and detection should be considered jointly in order to successfully design low-complexity power-efficient IR-UWB communication systems.

iii

Kurzfassung

Impulsradio-Ultrabreitband-Systeme (engl., impulse-radio ultra-wideband, IR-UWB) eignensich besonders für die niederratige Kommunikation über kurze Distanzen, wie beispielweise indrahtlosen Sensornetzwerken. Die wesentlichen Vorteile von IR-UWB liegen in seiner Robust-heit gegenüber ausgeprägter Mehrwegeausbreitung auch in Innenraumszenarien, der Möglich-keit zur akkuraten Lokalisierung sowie den geringen Kostenund der geringen Komplexität.

Während die Entwicklung von aufwandsgünstigen energiesparenden IR-UWB Sendern mittelsleistungseffizienten Modulationsformaten wie Pulspositionsmodulation (PPM) oder Variantenvon differentiellem binärem phase-shift keying (sog. differential transmitted reference (DTR))wohl bekannt ist, werden im ersten Teil der Arbeit Detektionsverfahren für aufwandsgünstigeund leistungseffiziente IR-UWB Empfänger hergeleitet. Im zweiten Teil der Arbeit wird dieAuswirkung der Forderung nach geringer Komplexität auf denEntwurf von codierten IR-UWBSystemen behandelt.

Inkohärente Detektionsverfahren, welche direkt Teile desEmpfangssignals als Referenz zur De-modulation heranziehen, werden weithin als Schlüssel zum Erfolg für den Entwurf aufwands-günstiger IR-UWB Empfänger gesehen, da hierdurch aufwändige Kanalschätzung, wie siefür eine kohärente Detektion benötigt wird, vermieden wird. Aufbauend auf einem allge-meinen Modell der IR-UWB Kommunikation, welches eine Vielzahl populärer Varianten von(codierter) IR-UWB Modulation wie u.a. PPM und DTR beschreibt, wird ein allgemeinerinkohärenter maximum-likelihood Empfänger vorgestellt.Dieser Ansatz vermeidet expliziteKanalschätzung. Mit Hilfe dieser allgemeinen Beschreibung werden Energiedetektion undAutokorrelation als zwei zugrundeliegende äquivalente Detektionsprinzipien herausgestellt. DasAutokorrelationsprinzip ist hierbei aus Implementierungssicht zu bevorzugen, da es ermöglichtein (analoges) Autokorrelationsempfänger-Front-end vonder (digitalen) Signalverarbeitung,welche die Symbolentscheidung implementiert, zu trennen.

Ein konsequentes Mitführen der Möglichkeit in der Herleitung des Empfängers Beobachtungs-zeiträume zu verwenden, die die Dauer eines Symbols überschreiten, ermöglicht einen umfassen-den Vergleich von besonders aufwandsgünstigen symbolweise arbeitenden Detektionsverfahrenmit Erweiterungen in Richtung verbesserter inkohärenter Detektionsverfahren, welche mehrereSymbole gemeinsam verarbeiten. Es kann gezeigt werden, dass der inhärente Verlust in Leis-tungsfähigkeit von traditioneller inkohärenter symbolweise arbeitender Detektion im Vergleichzu kohärenter Detektion basierend auf idealer Kanalschätzung durch solche Mehrsymbolde-tektionsverfahren verringert wird. Letztere arbeiten aufdem Ausgangssignal eines erweitertenAutokorrelationsempfängers, der Korrelationskoeffizienten von Symbolen liefert, die mehrereSymbolintervalle auseinander liegen.

iv

Für die Verwendung von DTR IR-UWB, das die attraktivste Variante von IR-UWB Modu-lation darstellt, wird gezeigt, dass blockbasierte Detektionsverfahren, welche den Strom anEmpfangssymbolen in (möglicherweise überlappende) Blöcke unterteilen und damit mehrereSymbole gemeinsam verarbeitenim Vergleich zu Ansätzen, die eine Sequenzschätzung durch-führen, sowohl in uncodierten als auch in codierten IR-UWB Übertragungssystemen überlegensind. Vor allem verbesserte Varianten von blockweisen entscheidungsrückgekoppelten De-tektionsverfahren (sog. decision-feedback differentialdetection (DFDD)), die eine optimalenEntscheidungsreihenfolge verwenden, in Verbindung mit Verfahren wie sog. „multiple-observa-tions combining” und „virtually increased block size”, erzielen einen hervorragenden Austauschzwischen benötigtem Berechnungsaufwand und Leistungsfähigkeit bzgl. Fehlerrate. Weiter-hin wird eine einfache Möglichkeit dargestellt, wie Zuverlässigkeitsinformation am Ausgangsolcher Detekoren bereitgestellt werden kann. Dies erlaubt die Decodierfähigkeit des anschlies-senden (soft-input) Kanaldecoders bei nur unwesentlich erhöhtem Berechnungsaufwand zuverbessern.

Im zweiten Teil der Arbeit wird eine informationstheoretische Analyse von codierter IR-UWBÜbertragung in Verbindung mit den obigen inkohärenten Detektionsverfahren präsentiert. DieseUntersuchungen beschränken sich auf den üblichen Ansatz der „bit-interleaved coded modula-tion”, d.h. der seriellen Verknüpfung von Kanalcodierung,interleaving und Modulation. Aus-gehend von der Kapazität der Übertragungskette von Modulation und Übertragung über einenMehrwegeausbreitungskanal bis hin zur autokorrelationsbasierten Detektion, wie sie im erstenTeil der Arbeit vorgestellt wurde, werden Entwurfsregeln für realistische IR-UWB Systemeabgeleitet und optimale Raten für den verwendeten Kanalcode aufgezeigt.

Obwohl sich diese optimalen Raten im Bereich von2/3 bis4/5 bewegen, und damit gegenüberder üblicheren Verwendung von Codes mit Rate1/2 deutlich höhere Datenraten erlauben, kannhiermit die Leistungseffizient von codierter IR-UWB Übertragung gesteigert werden. DieseEntwurfsregeln werden mit Hilfe von numerischen Simulationen von codierter IR-UWB Über-tragung unter Verwendung von Faltungscodes verifiziert.

Die erzielten Ergebnisse untermauern, dass, gerade wenn inkohärente Empfängerstrukturen ver-wendet werden, Codierung, Modulation und Detektion gemeinsam betrachtet werden sollten,um aufwandsgünstige und leistungseffiziente IR-UWB Kommunikationssysteme zu entwerfen.

v

Contents

Abstract i

Kurzfassung (in German) iii

1 Introduction and Outline 1

2 Impulse-Radio UWB Communication 7

2.1 Generic Model of IR-UWB Modulation . . . . . . . . . . . . . . . . . . . 7

2.1.1 Channel Coding and Interleaving . . . . . . . . . . . . . . . . . . 8

2.1.2 Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Time-Hopping and Code-Division Multiple Access . . . . . . . . . 9

2.2 Variants of IR-UWB Modulation . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Pulse-Position Modulation . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Transmitted-Reference IR-UWB . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Differential Transmitted-Reference IR-UWB . . . . . . . . . . . . . 13

2.2.4 Further Variants of IR-UWB Modulation . . . . . . . . . . . . . . . 14

2.3 IR-UWB Receive Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.2 Sampled Representation . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Receiver Input Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.4 Signal-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Channel Models for UWB Transmission . . . . . . . . . . . . . . . . . . . 17

2.5 Key Figures for IR-UWB Communication Systems . . . . . . . . . . . . . . 19

3 Receiver Design for Generic IR-UWB Modulation 21

vi Contents

3.1 Detection Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.2 Approaches to IR-UWB Receiver Design . . . . . . . . . . . . . . 23

3.2 Noncoherent Maximum-Likelihood Detection . . . . . . . . . . . . . . . . 25

3.2.1 Derivation of Maximum-Likelihood Decision Metric . . . . . . . . 25

3.2.2 Discussion of Maximum-Likelihood Noncoherent Detection . . . . 27

3.2.3 Discussion of Weighting Coefficients . . . . . . . . . . . . . . . . 29

3.3 Computation of Reliability Information . . . . . . . . . . . . . . . . . . . 30

3.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.2 Soft-Output Receiver for IR-UWB Transmission . . . . . . . . . . . 31

3.4 Autocorrelation-Based Detection . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Autocorrelation Device . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Implementation Aspects . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Discrete-Time Model for Autocorrelation-Based Detection . . . . . . . . . 35

3.5.1 Analysis of Captured Pulse Energy . . . . . . . . . . . . . . . . . . 36

3.5.2 Analysis of Equivalent Noise . . . . . . . . . . . . . . . . . . . . . 37

3.5.3 Approximate Discrete-Time Model . . . . . . . . . . . . . . . . . . 39

3.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Comparison to GLRT-Based Detection . . . . . . . . . . . . . . . . . . . . 43

3.6.1 GLRT-Based Receiver Design . . . . . . . . . . . . . . . . . . . . . 43

3.6.2 GLRT-Based Soft-Output Detection . . . . . . . . . . . . . . . . . 44

3.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Detection Schemes for PPM IR-UWB 47

4.1 Energy Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


4.1.2 Error-Rate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.3 Influence of Weighting Coefficients . . . . . . . . . . . . . . . . . 50

4.2 Multiple-Symbol Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 52



4.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

vii

5 Detection Schemes for DTR IR-UWB 57

5.1 Motivation and Classification . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Differential Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62


5.2.2 Influence of Weighting Coefficients . . . . . . . . . . . . . . . . . 63

5.3 Block-Wise Detection Schemes . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3.1 Block-Wise Processing . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3.2 Multiple-Symbol Differential Detection . . . . . . . . . . . . . . . 67

5.3.3 Decision-Feedback Differential Detection . . . . . . . . . . . . . . 78

5.3.4 Optimum Decision Order . . . . . . . . . . . . . . . . . . . . . . 78

5.3.5 Numerical Results I . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.6 Combining Multiple Observations . . . . . . . . . . . . . . . . . . 88

5.3.7 Numerical Results II . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3.8 Virtually Increased Block Size . . . . . . . . . . . . . . . . . . . . 93

5.4 Sequence Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4.1 DFDD with Virtually Increased Block Size . . . . . . . . . . . . . 95

5.4.2 Reduced-State Sequence Estimation . . . . . . . . . . . . . . . . . 96

5.4.3 Sliding-Window Decision-Feedback Differential Detection . . . . . 97

5.4.4 Comparison to Block-Wise Detection Schemes . . . . . . . . . . . 99

5.5 Related Receiver Concepts for DTR IR-UWB . . . . . . . . . . . . . . . . . 104

5.5.1 Decision-Directed Autocorrelation-Based Differential Detection . . 104

5.5.2 Compressed-Sensing-Based Detection . . . . . . . . . . . . . . . . 109

5.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Design Rules for Coded IR-UWB Communication 117

6.1 The Minimum Required Energy per Bit . . . . . . . . . . . . . . . . . . . 117

6.1.1 Power-Efficient Noncoherent Detection . . . . . . . . . . . . . . . 117

6.1.2 Excursus: Different Looks at the Capacity . . . . . . . . . . . . . . 118

6.2 Design Rules for Coded PPM IR-UWB . . . . . . . . . . . . . . . . . . . . 122

6.2.1 Soft-Output Energy Detection . . . . . . . . . . . . . . . . . . . . 122

6.2.2 Hard-Output Energy Detection . . . . . . . . . . . . . . . . . . . . 125

6.3 Design Rules for Coded DTR IR-UWB . . . . . . . . . . . . . . . . . . . . 128

viii Contents

6.3.1 Coded DTR IR-UWB . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.3.2 Information-Theoretic Analysis . . . . . . . . . . . . . . . . . . . . 129

6.4 Validation for Convolutional Coded Transmission . . . . . . . . . . . . . . 148

6.4.1 Energy Detection of PPM IR-UWB . . . . . . . . . . . . . . . . . . 149

6.4.2 Symbol-Wise Differential Detection of DTR IR-UWB . . . . . . . . 152

6.4.3 Block-Wise Detection of DTR IR-UWB . . . . . . . . . . . . . . . . 154

6.4.4 Power-Efficient Coded DTR IR-UWB Transmission . . . . . . . . . 157

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7 Conclusions 163

A Supporting Material and Derivations 167

A.1 Derivation of (3.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

A.2 The Chi-Squared Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 170

A.3 Convolutional Codes for the Binary-Input AWGN Channel . . . . . . . . . 172

B Notation 177

B.1 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

B.2 Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

B.3 Detection Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Bibliography 183

1

1. Introduction and Outline

Ultra-wideband (UWB) technology offers attractive solutions to a broad class of problems inimaging, localization, identification, and communication[YG04, AR03, TWZ+12]. Such solu-tions have gained increasing attention, especially since regulatory authorities approved license-free operation in a wide frequency range subject to moderateoperational conditions (first, in theUnited States [FCC02] from roughly3 to 10GHz, followed by similar regulations worldwide,cf., e.g., [Bun08] for Germany).

In this thesis, we focus on the application of UWB technologyfor communication over shortto moderate distance. To this end, essentially two conceptsare discussed in literature. Forhigh-data-rate communication, e.g., between consumer electronic devices, so-called multibandorthogonal frequency-division multiplexing is often investigated, cf., e.g., [SLS07]. If low com-plexity and low power consumption are the major design objectives—as we aim for in thisthesis—impulse-radio UWB (IR-UWB) in combination with power-efficient coded modulationand low-complexity detection schemes is a promising solution [WS98].

Very low complexity and low power consumption are demanded especially in the design ofwireless sensor networks, which are employed, e.g., for monitoring of operating conditions inindustrial facilities or in cars, tracking of vital signs ofinpatients in health care, or surveillanceof environmental conditions [Fon04, ZOS+09]. Here, low-cost low-complexity transceiver de-vices with battery-powered lifetimes of up to several yearsare required. Clearly, low complexityis desired especially at the sensors (usually the transmitter side), but also the complexity of theso-called data fusion center (the receiver side) should be kept as low as possible in order to en-able processing, e.g., on a standard laptop computer. Moreover, if bidirectional communicationshall be implemented, e.g., to transmit control signals to the sensors or to enable sensor-to-sensor communication, also the receivers built into the sensors have to fulfill strict complexityrestrictions.

IR-UWB potentially meets exactly these demands. Basically, IR-UWB modulation is a variantof conventional digital (usually baseband) pulse-amplitude modulation [PS08]. The major dif-ference to the latter is represented in the very short duration of the employed transmit pulses

2 1. Introduction and Outline

(hence, the “ultra-wide” signal bandwidth), when comparedto the symbol duration.

The relatively large signaling bandwidth of IR-UWB modulation enables a reduced transmitpower spectral density and, in turn, coexistence to established narrow-band systems [CG09] aswell as support of a large number of simultaneous users [WS00]. Moreover, IR-UWB com-munication ensures robustness to severe multipath fading even in indoor environments andprovides the potential to incorporate accurate localization within the communication system[CNBL06, WS98].

Naturally, the low-complexity and low-power-consumptiondemands impose restrictions on theapplied modulation and detection schemes. For IR-UWB communications simple, but power-efficient modulation schemes, such as pulse-position modulation or binary phase-shift keying,are employed. Thus, the IR-UWB transmitter architecture can be implemented with low costcomposed of simple analog circuits. In combination with thelow-duty-cycle operation, thisleads to extremely low power consumption.

The ramifications at the receiver side due to the low-complexity demand are manifold. Mostimportantly, due to the large signal bandwidth and the rich multipath propagation usually ob-served in IR-UWB application scenarios, a large number of (possibly distorted) echos of eachtransmitted pulse is received. Thus, performing explicit channel estimation in order to en-able matched-filtering (e.g., implemented using a so-called rake receiver) and/or equalizationof the resulting inter-symbol interference is regarded overly complex [LDM02]. Instead, sim-pler methods avoiding any kind of explicit channel estimation—so-called truly noncoherentdetection schemes—are required [WLJ+09, DB04, YG04].

Such noncoherent detection schemes usually build upon the principle of energy and/or auto-correlation-based detection. E.g., in case of pulse-position-modulated (PPM) IR-UWB, theleast-complex variant of noncoherent detection is implemented by symbol-wise energy detec-tion (ED), which compares the received energy of the PPM intervals and decides for the onewith the largest energy [DMR07, CM06, FM06]. Symbol-wise autocorrelation-based differen-tial detection (DD) represents the analogon to the latter for (differential) transmitted reference(DTR) IR-UWB, a variant of IR-UWB employing differentiallyencoded binary phase-shiftkeying [CS03, CS02, QWD07, PJ07]. Here, the sign of the correlation of adjacent pulses ismeasured and serves as a decision variable for the transmitted bit.

The application of these variants of noncoherent detectionschemes, however, leads to signifi-cant performance degradation compared to the idealistic benchmark case of coherent detectionwith perfect channel state information. Of course, increasing the transmit power in order tostill operate at a desired error rate conflicts with the desired power efficiency and low powerconsumption of the IR-UWB transmitter. Instead advanced noncoherent detection schemes arediscussed, which decide multiple symbols jointly by processing the receive symbol stream ina block-wise fashion [GQ06, LT08, TY08, ZM12a, ZML10, ZMR10a, SF10a]. Many con-cepts applied for these methods originate from well-known concepts developed for detection ofdifferential phase-shift keying, such as multiple-symboldifferential detection (MSDD) [Mac92,DS90, HF92, LSPW05] and variants of decision-feedback differential detection (DFDD) [LP88,Edb92, SGH98]. These blockwise detection schemes have to bedistinguished from methods

3

implementing sequence estimation, such as the benchmark case of ideal noncoherent sequenceestimation (INSE) or reduced-state sequence estimation (RSSE) [SF11c, LT06b].

Figure 1.1 depicts an overview of the detection schemes discussed in this thesis and may thusserve as an outline especially for Chapter 4 and 5 (for brevity, we refer to the given sectionsfor a definition of all acronyms). The detection schemes are classifed according to their wayof processing the burst of receive symbols (symbol-wise, block-wise, sequence estimation);relations between the detection schemes are indicated. Theproposed variant of detection witha virtually increased block size represents a hybrid schemein-between both classes, block-wisedetection and sequence estimation.

We conduct a comprehensive study of the above variants of noncoherent detection. Receiverconcepts for symbol-wise as well as advanced multiple-symbol detection schemes are derivedin a unified framework. On the one hand, this generic approachenables to study the trade-off between receiver complexity and error-rate performance resulting from the application ofvarious receiver types consistently. On the other hand, thederived principles are applicableto several prominent variants of IR-UWB modulation. These insights are then utilized for thedesign of optimized variants of multiple-symbol/decision-feedback detection schemes for thecase of PPM IR-UWB and DTR IR-UWB, leading to detection schemes with improved error-rate performance and/or smaller computational complexity.

As a further means to guarantee the desired power efficiency,the application of channel codingschemes in IR-UWB communications is studied. To this end, two aspects are investigated.First, in order to fully exploit the benefits of channel coding, so-called soft-output detectionschemes are called for, which deliver reliability information on the estimated code symbols[WJ65] (applicable schemes are highlighted in Figure 1.1).In order to enable energy-efficientlow-cost receiver structures, the receive signal processing required for and the computationalcomplexity of such advanced (soft-output) detection schemes have to be kept as low as possible[SF10b].

Secondly, the application of low-complexity noncoherent detection schemes has to be takeninto account for the design of the coded IR-UWB communication system [SF11d]. Therebywe restrict ourselves to the common approach of bit-interleaved coded modulation, i.e., theserial concatenation of coding, interleaving, and modulation. We derive the capacity of thetransmission chain including modulation, IR-UWB transmission over multipath propagationchannels, and noncoherent detection. Based on these capacity considerations, optimum coderates are identified, which increase the power efficiency of the entire IR-UWB communicationsystem.

Due to its relatively low complexity and low structural delay, in this thesis, we focus on convo-lutional-coded transmission [Wic95]. Moreover, convolutional codes are also applied in stan-dardized IR-UWB systems [IEE07]. However, it has to be notedthat most of the results trans-late to other coding schemes, cf., e.g., [SF11d] for the caseof low-density-parity-check-codedtransmission.

Despite being motivated by the application in wireless sensor networks, in this thesis our inves-


tigations are restricted to a single-user point-to-point transmission. The impact of interferencecaused by the simultaneous operation of multiple users and/or the presence of other commu-nication systems (usually with significantly smaller bandwidth compared to the UWB system,thus so-called narrowband interference) on the quality of transmission and the performanceof the entire sensor network is not studied. Multi-user and narrowband interference is simplyincorporated into the conventional additive zero-mean white Gaussian noise process. The de-sign and investigation of a joint multi-user detector [Ver98] and/or interference cancellation arebeyond the scope of this thesis.

This thesis is organized as follows:

� In Chapter 2, a generic system model of IR-UWB communicationis introduced. Wedescribe how prominent variants of IR-UWB modulation are included in this genericmodel, review channel models for UWB propagation, and give an overview of realisticsystem parameters for such systems.

� In Chapter 3, a noncoherent receiver is derived for this generic IR-UWB system. Wediscuss different approaches to IR-UWB receiver design as well as the problem of incor-porating the computation of reliability information on theestimates in a general view. Weshow that noncoherent detection of IR-UWB signals can be implemented in two equiv-alent variants: based on the principle of energy detection or (preferably) based on auto-correlation of the receive signal. For the latter receiver type we review implementationaspects and present an equivalent and an approximate discrete-time model of the IR-UWBtransmission.

� In Chapter 4, this generic noncoherent IR-UWB receiver is narrowed down to the spe-cial case of pulse-position-modulated IR-UWB. For this modulation scheme, we reviewsymbol-wise energy detection and multiple-symbol detection (MSD), cf. right hand sideof Figure 1.1. For both cases, accurate analytical error-rate expressions are given andcompared to numerical results for realistic UWB channel models.

Previously published papers related to this chapter are [SF10a, SF11a].

� In Chapter 5, again based on the generic noncoherent receiver derived before, we presentand discuss different approaches to noncoherent detectionof differential transmitted-reference IR-UWB. More precisely, conventional symbol-wise differential detection iscompared to advanced detection schemes, which operate on the output of an extendedautocorrelation device and decide multiple symbols jointly. For the latter we distin-guish block-wise approaches (such as multiple-symbol differential detection (MSDD)and block-wise decision-feedback differential detection(DFDD)) and detection schemesimplementing sequence estimation (such as reduced-state sequence estimation (RSSE)and sliding-window decision-feedback differential detection), cf. Figure 1.1. We showthat the block-wise detection schemes, especially in combination with several suggestedimprovements, enable power-efficient, yet low-complexitydetection for both uncodedand coded IR-UWB transmission systems.

5

virt. block size= burst length

feedbackdecision opt. decision

order

truncatedmemory

blocksoverlapping

sorting crit.

multiple-observations comb.vir

t.in

cre

ase

db

lock

size

blo

ck-w

ise

nr. of states = 1

sym

bo

l-w

ise

of entire burstjoint decision

sequ

en

ce

est

imatio

n

PPMDTR

of

mu

ltip

lesy

mb

ols

virt. increased block size

join

td

ecis

ion

symbol-wisedecisions

Sec. 5.3.2MSDD

Sec. 5.3.3 Sec. 5.3.4

Sec. 5.4.3swDFDD

Sec. 5.3.6mocMSDD

Sec. 3.2

Sec. 5.2DD

Sec. 4.1ED

Sec. 5.4.2RSSE

INSESec. 5.1

Sec. 4.2MSD

Sec. 5.3.6

Sec. 5.4.1vbMSDDSec. 5.3.8

bDFDD sbDFDD

generic noncoherent receiver

vbDFDD

INSE

mocDFDD

Figure 1.1: Overview and classification of detection schemes for PPM IR-UWB and DTR IR-UWB derivedfrom the generic noncoherent maximum-likelihood receiver (cf. referenced section for details). Schemes that

do not deliver soft output are indicated with a dashed box.


These approaches are compared to two related receiver concepts for differential trans-mitted-reference IR-UWB, namely decision-directed autocorrelation-based detection andcompressed-sensing-based autocorrelation-based detection.

The results presented in this chapter have in parts been published in the following papers:[SFL09a, SFL09b, SF10b, SF11b, SF11c, SF12b, SF12a, RSH13].

� In Chapter 6, based on an information-theoretic analysis, design rules for coded IR-UWB transmission are derived, which explicitly take into account the presented non-coherent detection schemes. Again we focus on the special cases of PPM in combinationwith symbol-wise energy detection and DTR IR-UWB in combination with MSDD andDFDD. For both schemes, these design rules are validated by means of numerical simu-lations of convolutional coded IR-UWB transmission.

Parts of this chapter have previously been published in [SF11a, SF11d, SF12a].

� Chapter 7 provides a summary and concludes this thesis with an outlook to open problemsfor future research.

The appendices collect supporting material, mathematicalbackground and derivations, and alist of mathematical symbols and abbreviations.

7

2. Impulse-Radio UWB Communication

In this chapter, the system model employed in the investigations conducted in the subsequentchapters is introduced. To this end, we introduce a generic model of impulse-radio UWB (IR-UWB) modulation in Section 2.1, which is related to popular variants of IR-UWB modulationin Section 2.2. The receive signal of IR-UWB communication is defined in Section 2.3. Inorder to make quantitative statements for realistic IR-UWBsystems, channel models for UWBtransmission and key figures for IR-UWB systems are summarized in Sections 2.4 and 2.5,respectively.

2.1 Generic Model of IR-UWB Modulation

Motivated by the application in wireless sensor networks, where data is typically transmitted inrelatively short bursts, a finite-length sequence ofKsource binary information symbolsqu ∈ F2,u = 0, . . . , Ksource − 1, (F2 denotes the Galois field of size two) originating from a discretememoryless i.i.d. uniform source, shall be transmitted to the receiver (as shown in Figure 2.1).The sequence of information symbols is encoded using an encoder of a rate-Rc channel code andpassed through a bit-interleaver (not shown in Figure 2.1),yielding a sequence of binary codesymbolscv ∈ F2, v = 0, . . . , Ncode − 1, with Ncode = Ksource/Rc. The interleaved sequence ofcoded bits is mapped to a sequence of transmit symbolsbk ∈ B, k = 0, . . . , Nburst − 1, takenfrom anM-ary signal constellationB. The length of the transmit symbol sequence is given asNburst = Ncode/Rm, whereRm is the rate of the modulation scheme. The second moment of thetransmit symbols is denoted asσ2

b . The mapping and the signal constellation are specific to theconsidered variant of IR-UWB modulation.

The transmit symbols are serially transmitted using conventional digital pulse-amplitude mod-ulation [Hub05, PS08]. The transmit signal of IR-UWB reads

s(t) =

Nburst−1∑

k=0

bk pTX(t− kT ) (2.1)

8 2. Impulse-Radio UWB Communication

pTX(t) RX DEChRX(t)ENC Mc b s(t) qRXq

hCH(t)r(t)

n0(t)

Figure 2.1: Block diagram of generic IR-UWB transmission.

wherepTX(t) is the transmit pulse, to be specified in Section 2.1.2, andT is the transmit symbolduration.

In order to enable low power consumption and implementationcomplexity in particular at thetransmitter side, but also at the receiver side, two major restrictions have to be regarded forIR-UWB communication. Most importantly, IR-UWB systems avoid up-/down-conversion toa carrier frequency, i.e., they operate in the baseband. Hence, only real-valued amplitude co-efficientsbk and pulse shapes are allowed. Furthermore, in order to enable the application oflow-complexity receiver structures, higher-order amplitude modulation is infeasible. Only signinversions of the transmit pulse can be realized. Consequently, the set of transmit symbols isrestricted toB ⊆ {−1, 0, +1}.

For UWB transmission systems regulatory issues allow only arelatively low duty-cycle for theoperation of the IR-UWB transmitter [FCC02, Bun08]. Hence,different from conventional dig-ital pulse-amplitude modulation, the symbol duration is usually significantly larger comparedto the duration of the transmit pulse. This is also necessarydue to the low-complexity demandat the receiver side. The subsequently discussed low-complexity detection schemes require aninter-symbol-interference-free transmission even in case of severe multipath propagation be-tween transmitter and receiver.

2.1.1 Channel Coding and Interleaving

In this thesis, we focus on the application of convolutionalcodes for channel coding, as theyoffer relatively low-complexity encoding and decoding schemes [Wic95], as well as good per-formance, especially if low latency is desired [HH09]. Moreover, convolutional codes are alsoapplied in standardized IR-UWB systems [IEE07] and represent a well-studied field of research.For this reason, a comprehensive review of convolutional codes is omitted in this thesis. Apartfrom a brief discussion of necessary basics, we refer to the standard text books on channelcoding, cf., e.g., [Wic95, PS08].

For numerical simulations, maximum-free-distance convolutional codes with a constraint lengthof ν are employed [PS08]. For code rates aboveRc = 0.5 puncturing is applied to the rate-1/2

maximum-free-distance parent code [PS08]. The code parameters and puncturing patterns arelisted in Appendix A.3. The encoding is performed in a non-systematic non-recursive form.Fixed information symbols are appended in order to terminate the convolutional code in a well-defined state (for the moment, the rate loss due to termination is neglected).

For decoding the soft- and hard-input Viterbi algorithm is employed [Wic95]. The decoding ofpunctured convolutional codes is performed in the same manner as the decoding of the rate-1/2

parent code. When one or more bits are punctured, the corresponding metrics passed to the

2.1. Generic Model of IR-UWB Modulation 9

decoder are set such that the punctured bits do not contribute to the decoding process [PS08].

The performance of convolutional codes increases with larger constraint lengthν (or equiva-lently, its memory lengthν − 1). However, since the decoder operates in a trellis with2ν−1

states, also the decoder complexity strongly depends on theconstraint length.

For the considered slowly time-variant propagation scenarios, the application of an interleaveris not required at first glance. However, the later-on investigated detection schemes potentiallylead to error bursts spanning over several adjacent code symbols. Since convolutional codesare susceptible to such clustered error events, an interleaver can be applied in order to removethe memory within the error process visible at the decoder input. It is thus included in thedescription of the system model. This aspect is briefly discussed in Section 6.3.2.4.

2.1.2 Pulse Shaping

The IR-UWB transmit pulse has to fulfill regulatory issues, i.e., the spectrum of the transmitsignal must fit into a specified frequency mask (in Germany, e.g., specified by the “Bundes-netzagentur” [Bun08]). There are several approaches to optimize the pulse shape under suchconstraints, cf., e.g., [CK02, HB04].

Neglecting such regulations and noting that the particularpulse shape has minor influence onthe system performance, in this thesis the transmit pulse ischosen as the second derivative ofthe Gaussian function, as depicted in Figure 2.2(a) and given by

pTX(t) =

√

4√2

3τ·(

1− 2π

(

t

τ

)2)

e−π( tτ )

2

(2.2)

where the scaling in front of the brackets is a normalizing constant to ensure unit energy andτis the pulse-width parameter. This represents a common choice in literature (cf., e.g., [YG04]).The corresponding spectrum

PTX(f) =

∫ +∞

−∞

pTX(t)e−j2πftdt (2.3)

is depicted in Figure 2.2(b).

2.1.3 Time-Hopping and Code-Division Multiple Access

In realistic IR-UWB systems, usually a so-called frame structure including a time-hopping(TH) and/or a code-division multiple access component, implemented via direct-sequence (DS)spreading, is employed [AR03, WS98, WS00, RMMW03]. This is necessary for a number ofreasons. First, given the typically desired operating points of IR-UWB communication systems,the required transmit power to ensure sufficient quality andreliability of the overall transmissionusually would be to high to meet regulatory requirements, inparticular for uncoded transmis-sion. Employing TH/DS, the transmit power can be reduced at the cost of lower data rate.

The second reason is related to the simultaneous operation of a large number of users in the IR-UWB system. Applying TH/DS ensures that the multiple-access interference from other userscan be seen as statistically independent of the signal of thedesired user.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.5

0

0.5

1

1.5

t/τ →

p TX(t)/√τ→

(a) Pulse shape

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2−50

−40

−30

−20

−10

0

fτ →

20log(

|PTX(f)|)

[dB]→

(b) Normalized spectrum (maximum at0 dB)

Figure 2.2: Transmit pulse parameterized by pulse width τ .

Moreover, for IR-UWB modulation schemes which generate a transmit signal with non-zeromean (e.g., PPM and TR), these techniques avoid harmonics atmultiples of the symbol rate[Hub05].

Incorporating this frame structure1 is achieved by transmitting multiple (Nframe) transmit pulsesfor each amplitude coefficientbk at the positions given by the time-hopping sequenceTTH

n andamplitudes given by the direct-sequence spreading codecDS

n . This can be taken into account bya redefinition of the transmit signal as

sTH/DS(t) =

Nburst−1∑

k=0

bk

Nframe−1∑

n=0

cDSn pTX(t− TTH

n − kT ) . (2.4)

Since both sequences are known at the receiver, there are essentially three strategies to cope withthis frame structure. The first option is to average out the frame structure as a first processingstep prior to further receive signal processing at the receiver [TL08, LT08]. The second optionis given by considering the frame structure after detectionas an additional repetition code. Inprinciple, as a last option, the TH/DS can be used to restrictthe search space of the detectionproblems described in Section 3.

Since only point-to-point transmission in a single-user system is considered in this thesis, forclarity TH/DS is not explicitly taken into account.

2.2 Variants of IR-UWB Modulation

The generic model defined in Section 2.1 can be used to describe a number of modulationschemes typically employed for IR-UWB communication. These variants can be defined byspecifying the applied signal constellation and the way thecode symbols are mapped to thetransmit symbols.

1Here, the term “frame” is used as it is common in UWB literature, although it is counter-intuitive from theperspective of conventional digital communication systems.

2.2. Variants of IR-UWB Modulation 11

To this end, the sequence of coded symbols is partitioned into non-overlapping blocksc of mcode symbols, i.e.,

c = [c0, c1, . . . cm−1] ∈ Fm2 (2.5)

which are mapped to aD-dimensional hyper-symbol

b = [b0, b1, . . . bD−1] ∈ BD . (2.6)

These hyper-symbols are then concatenated and serially transmitted using conventional pulse-amplitude modulation (i.e., a conventional multidimensiomal mapping following [Wei87] isapplied). This mapping is specified by a labeling rule

M : c ∈ Fm2 7→ b ∈ BD . (2.7)

The rate of the modulation calculates toRm = m/D. For later use we also defineTD = DT asthe duration of theD-dimensional hyper-symbols.

In this thesis, we focus on IR-UWB communication employing orthogonal pulse-position mod-ulation (PPM) and so-called differential transmitted-reference (DTR) IR-UWB, a variant basedon differential binary phase-shift-keying. The popular schemes transmitted-reference (TR) IR-UWB and IR-UWB using biorthogonal PPM are discussed briefly.Variants of IR-UWB mod-ulation, such as TR in frequency domain, where the referencepulse and the data-modulatedpulse are separated in frequency domain rather than in time,or on/off-keying are not considered[AR03, YG04].

2.2.1 Pulse-Position Modulation

In pulse-position modulation (PPM), the code symbols are encoded in the position of a singletransmit pulse per hyper-symbol duration (cf. Example 2.1)[PS08].D-dimensional (orD-ary)PPM is modeled by restricting the set of transmit symbols to

BPPM = {ed|d = 0, . . . , D − 1} ⊂ BD (2.8)

whereed denotes thedth unit vector inD-dimensional space. EachD-dimensional transmitsymbol representsm = log2(D) code symbols, thusRm = log2(D)

D. The second moment of the

transmit symbols calculates toσ2b = 1/D. Due to the orthogonality of the modulation, the ap-

plied labeling rule does not influence the overall performance, such as bit error rate or capacity;hence, we apply a so-called natural labeling rule where eachblock c of binary code symbolsis interpreted as its corresponding integer numberd, which then selects the pulse positiondTwithin the hyper-symbol of durationTD = DT .

Example 2.1: Transmit Signal of Binary PPM IR-UWB

Consider a sequence of three code symbols [0, 1, 0], yielding b0 = [1, 0], b1 = [0, 1], and

b2 = [1, 0]. Using a transmit pulse as depicted in Figure 2.2 (for clarity the pulse has been

shifted by 1.5τ ), the transmit signal of PPM IR-UWB is given as:


t

00 1

s(t)

0 T TD 2TD 3TD

Clearly, at the transmitter PPM requires to only trigger thetime instant when a transmit pulseis sent; sign inversions of the transmit pulse are not required. At the receiver side it enablesto employ very low-complexity energy detection (cf. Section 4.1). Furthermore, the powerefficiency of PPM increases with increasing orderD of the modulation [Gur08, SF10a]. Theseadvantages come at the drawback of an even lower duty cycle compared to TR and DTR IR-UWB and a reduced data rate (or equivalently, spectral efficiency) compared to DTR.

2.2.2 Transmitted-Reference IR-UWB

In TR IR-UWB, each code symbol is encoded in the phase difference relative to a referencepulse, i.e.,m = 1 (cf. Example 2.2) [CS03, WS98, FM06]. For clarity, we assumethat eachdata-modulated pulse is preceded by a reference pulse2 and an equal spacing between refer-ence and data-modulated pulse. The code symbols are first mapped to information symbolsaaccording to

a =

{

+1 for c = 0

−1 for c = 1. (2.9)

The hyper-symbols are then taken from the two-dimensional constellation

BTR = {[1, a]|a ∈ {−1, +1}} ⊂ B2 . (2.10)

The rate and second moment of transmit symbols of TR IR-UWB are given asRm = 1/2 andσ2b = 1, respectively.

Example 2.2: Transmit Signal of TR IR-UWB

Given a sequence of three code symbols [0, 1, 0], the transmit symbols of TR IR-UWB are

b0 = [1, 1], b1 = [1, −1], and b2 = [1, 1]. Similar to Example 2.1, we have:

c0 = 0 c1 = 1 c2 = 0

t

s(t)

0 T TD 2TD 3TD

2It can easily be verified, that the extension of using a singlereference pulse for multiple data-modulatedsymbols is equivalent to DTR IR-UWB.

2.2. Variants of IR-UWB Modulation 13

TR IR-UWB enables to use the reference pulse for one-shot channel sounding and, subse-quently, as a noisy matched filter for demodulation of the succeeding data-modulated pulse.Thus, simple receiver structures based on correlation of reference and data-modulated pulsecan be applied, which avoid explicit channel estimation.

The major drawback of TR IR-UWB is its waste of energy required for transmitting the ref-erence pulses (similar to double-sideband amplitude modulation with carrier [Hub05]) and itsreduced data rate compared to DTR (asymptotically a factor of two smaller). These drawbacksare avoided by DTR IR-UWB, as described next.

2.2.3 Differential Transmitted-Reference IR-UWB

In DTR IR-UWB the code symbols are encoded in the phase difference of adjacent pulses (cf.Example 2.3). This avoids the waste of energy in TR IR-UWB required for transmitting areference pulse for each data-modulated symbol and increases the data rate (asymptotically) bya factor of two [CS03]. Only a single reference is transmitted at the beginning of each burst.

Effectively, DTR is equivalent to conventional differential binary phase-shift-keying (cf., e.g.,[FH06a]). A different name is adopted to emphasize the low duty cycle and the (here notconsidered) combination with the TH/DS frame structure. Asin differential binary phase-shift-keying, the code symbols are first mapped to binary information symbolsa, cf. (2.9). Theresulting sequence is then differentially encoded via

bk = ak bk−1 (2.11)

where the first reference symbol is set tob0 = 1. SinceNburst = Ncode + 1, the rate of DTR IR-UWB calculates toNburst/Ncode. Neglecting the reference symbol, which is valid for sufficientlylargeNburst, we setRm = 1. The second moment of the transmit symbols calculates toσ2

b = 1.

The mapping from code symbols to transmit symbols can be described in essentially twoways. On the one hand, it can be viewed as a symbol-wise process with memory, i.e., theone-dimensional set of transmit symbols is given as

BDTR ∈ {±1} (2.12)

and the memory is introduced through the differential encoding (cf. (2.11)).

On the other hand, the mapping can be described for the entireburst at once. In this case, theset of transmit symbols is given as

BDTR,Nburst= {b = [1, b1, . . . , bNburst−1]|bk ∈ {±1}} . (2.13)

Equivalently, in-between both variants, a block-wise mapping with memory can be introduced,e.g., matched to the block-wise processing at the receiver side [FLMH00], discussed in Sec-tion 5.3. Due to the memory introduced by the differential encoding, the last symbol of thepreceding block then serves as a reference for the current block.

The latter two options enable to apply different labeling rules for mapping blocks of code sym-bols to blocks of transmit symbols. In this thesis, we mainlyconsider the conventional case of


symbol-wise mapping followed by differential encoding anddiscuss the application of block-wise mappings only briefly.

In comparison to PPM and TR, cf. Example 2.1 and Example 2.2, respectively, it can be ob-served, that for the same symbol intervalT the data rate is increased by (asymptotically forlargeNburst) a factor of two, yet, similar to TR IR-UWB, a correlation-based receiver avoidingchannel estimation can be employed (cf. Chapter 5). Hence, we will mainly focus on DTRIR-UWB in this thesis.

Example 2.3: Transmit Signal of DTR IR-UWB

Consider a sequence of five code symbols [0, 1, 0, 0, 1], yielding b0 = [1, 1, −1, −1, −1, 1].Similar to the above examples, we obtain the following transmit signal:

c0 = 0 c1 = 1 c2 = 0 c3 = 0 c4 = 1

t

s(t)

0 T 2T 3T 4T 5T 6T

2.2.4 Further Variants of IR-UWB Modulation

Finally, we describe two further variants of IR-UWB modulation. These are not considered inthis thesis.

A possible combination of DTR and PPM IR-UWB is obtained by biorthogonal PPM. Here,both the polarity and the position of the pulses represent the modulated data symbols. In orderto enable noncoherent detection at the receiver, a reference pulse precedes the symbol streamsimilar to DTR (for simplicity, the position of the reference pulse may be set tot = 0, i.e., with-out PPM component), and data represented by the polarity of the transmit signal is differentiallyencoded with respect to this reference.

A variant of biorthogonal pulse-position modulation is employed in the IEEE standard for IR-UWB [IEE07]. Receiver design for such systems has been studied in [SF10a, AL09].

The second often considered variant is well-known on-/off-keying. Here, the information isrepresented in the presence or absence of a transmit pulse ineach interval. A simple thresholddetector can be applied at the receiver, that decides for thepresence of a receive pulse, if thereceived energy of the symbol interval exceeds a predetermined threshold. The problem withon-/off-keying is the accurate setting of this threshold. This setting depends on the expectedchannel characteristics and especially the signal-to-noise ratio, since, if no pulse is present,only noise energy is accumulated.

2.3. IR-UWB Receive Signal 15

2.3 IR-UWB Receive Signal

2.3.1 Definition

The IR-UWB signal is transmitted over a linear dispersive channel with impulse responsehCH(t). The channel is assumed time-invariant for the entire burstof Nburst symbols of du-rationT . In Section 2.4 the channel is discussed in detail.

At the receiver input, a noise signaln0(t) is superimposed to the IR-UWB signal. This additivenoise consists of the self-noise of the receiver signal processing (in particular, the noise of theamplifier), as well as the interference from different usersand/or systems. For simplicity, thisnoise process is modeled as a zero-mean additive white Gaussian noise (AWGN) process withtwo-sided power spectral density ofN0/2.

A receiver input filter with impulse responsehRX(t) is applied to select the relevant signal bandand to limit the noise power. Defining the overall receive pulse shape asp(t) = pTX(t)∗hCH(t)∗hRX(t), the receive signal at the output of this filter is given as

r(t) =

Nburst−1∑

k=0

bk p(t− kT ) + n(t) (2.14)

wheren(t) denotes the filtered noise process (i.e., white Gaussian noise having passed thereceiver input filter,n(t) = n0(t) ∗ hRX(t)).

It is assumed that the overall pulse has decayed before the next pulse is received, i.e., inter-symbol interference is not present (cf. Example 2.4). This is achieved by choosing a sufficientlylow duty cycle, or equivalently, the symbol intervalT larger than the (relevant) duration of theoverall pulse shape.

Example 2.4: Receive Signal of DTR IR-UWB

For the same setup as in Example 2.3, an exemplary inter-symbol-interference-free receive

signal of DTR IR-UWB corrupted by AWGN with high SNR (10 log10(Ec/N0) = 30 dB)

and a realization of the channel impulse response according to the indoor non-line-of-sight

channel model of [MFP03] is shown below:

c0 = 0 c1 = 1 c2 = 0 c3 = 0 c4 = 1

t

r(t)

0 T 2T 3T 4T 5T 6T


2.3.2 Sampled Representation

For convenient representation, we introduce a sampled representation of the receive signal. Forthekth symbol interval we define the row-vector

rk = [r(kT ), r(kT + Ts), . . . , r(kT + (Ns − 1)Ts)] (2.15)

whereTs andfs = 1/Ts denote the sampling period and sampling rate, respectively, andNs =

T/Ts is the number of samples per symbol. For sufficiently large sampling frequency, this yieldsan equivalent description of the receive signal.

As we assume inter-symbol-interference-free transmission and the channel impulse response toremain constant within the transmission of one burst, we have

rk = bk p+ nk (2.16)

where

p = [p(0), p(Ts), . . . , p((Ns − 1)Ts)] (2.17)

nk = [n(kT ), n(kT + Ts), . . . , n(kT + (Ns − 1)Ts)] (2.18)

are the sampled receive pulse and the samples of the noise of thekth symbol interval, respec-tively.

2.3.3 Receiver Input Filter

In this thesis, we assume that the receiver input filter fulfills the square-root Nyquist propertywith respect to the above sampling frequency (i.e., in its most simple form, represents an ideallow-pass filter of bandwidthfs/2). Hence, the noise samples ofnk are independent zero-meanGaussian distributed with identical varianceσ2

n. Anticipating the next section, the noise varianceis given as

σ2n =

σ2b‖p‖2Rm

1

Ns

· N0/2 · fsEc/T

(2.19)

whereEc is the received energy per code symbol.3

2.3.4 Signal-to-Noise Ratio

The key parameter for performance assessment of IR-UWB communication is the receiver-sidesignal-to-noise ratio given by the ratio of received energyper information bitEb and noise-power spectral densityN0. This ratio calculates to [Hub05]

Eb/N0 =‖p‖2RmRc

σ2b

2σ2n

. (2.20)

3 For sake of completeness note thatα = Ns − 1 is the bandwidth-excess or roll-off factor, i.e., the relativeoverhead of the used bandwidth over the minimum required bandwidth (clearly, for UWB communicationα≫ 1).

2.4. Channel Models for UWB Transmission 17

For uncoded transmission,Rc = 1. For this case we define the signal-to-noise ratio as the ratioof energy per code symbolEc and noise-power spectral densityN0, i.e.,

Ec/N0 =‖p‖2Rm

σ2b

2σ2n

(2.21)

in order to clearly distinguish uncoded and coded transmission.

2.4 Channel Models for UWB Transmission

Due to the high multipath resolution of UWB signals and short-range indoor communicationsas the major field of application of UWB communication systems, new techniques to modelthe physics of UWB signal propagation—in a mathematical as well as numerically treatableway—have been studied in the literature, cf., e.g., [SV87, CSW02, CWM02, SB07, Mol05] toname only a few. Attempts for standardized UWB channel models are described in [MFP03]and [MCC+06]. Here, we briefly summarize channel modeling for UWB signal propagationand its main parameters.

Already early measurements of UWB propagation channels in [SV87] report strong clusteringof the multipath components with respect to time of arrival.This is taken into account in theso-called Saleh-Valenzuela model, where the channel impulse response is described as a tappeddelay line

hCH(t) =

+∞∑

l=0

+∞∑

k=0

αk,lδ(t− τ clusterl − τk,l) (2.22)

with the path gainsαk,l and the starting positionτ clusterl of thelth cluster and intra-cluster arrivalsτk,l. These inter-arrival distances follow exponential distributions with arrival rateΛCM andλCM,respectively, i.e.,

Pr{

τ clusterl |τ clusterl−1

}

= ΛCMe−ΛCM(τ cluster

l−τ cluster

l−1 ) (2.23)

Pr {τk,l|τk−1,l} = λCMe−λCM(τk,l−τk−1,l) (2.24)

whereτ cluster0 = 0ns andτ0,l = 0ns. Note that clusters may overlap in general.

The power of each multipath component decays exponentially, determined by cluster and intra-cluster power-decay time constantsΓCM andγCM, respectively, yielding

E{

α2k,l

}

= e−

τclusterlΓCM e

−τk,lγCM . (2.25)

The decay of the power-delay profile is dominated by the cluster power-decay time constantΓCM.

The number of clusters is either set according to a suitable power threshold [MFP03] or explic-itly given, e.g., as a realization of a random variable [MCC+06].

Since IR-UWB transmission operates in the baseband, the path gainsαk,l may be restricted toreal values. In contrast to narrowband channels, due to the high multipath resolution it can be


Table 2.1: Channel model parameters from [MCC+06].

Model identifier CM 1 CM 2 CM 3Characteristics indoor indoor office

line-of-sight non-line-of-sight line-of-sightAverage number of clusters 3 3.5 5.4Cluster decay time ΓCM 22.61 ns 26.27 ns 14.6 nsCluster arrival rate ΛCM 0.047 ns−1 0.12 ns−1 0.016 ns−1

Intra-cluster decay time γCM 12.53 ns 17.50 ns 6.4 nsIntra-cluster arrival rate λCM 1.54 ns−1 1.77 ns−1 0.19 ns−1

Average delay spread 44.91 ns 51.26 ns 25.42 ns

argued that the central limit theorem can not be invoked and aGaussian distribution does notmatch the probability density function of the path gains. Instead Nagakami [MCC+06], log-normal [MFP03], or Rayleigh [LT08] distributions are considered for the amplitude, while thesign takes values±1 with equal probability.

Although no explicit measurements have been devoted to the time-variance of UWB channels,reports in the literature given above agree on the fact that UWB channel scenarios can be as-sumed to remain constant over several milliseconds. This isalso supported by the fact thatIR-UWB applications focus on short-range indoor communications and hence, mobility of thecomponents is limited. Thus, coherence time is not an issue for IR-UWB channels.

Due to shadowing and fading, the overall received energy is arandom variable. Usually, thisfading process is assumed to be slowly time-varying and the attenuation is modeled as a log-normal distributed random variable [MCC+06]. Since such fading effects can easily be incorpo-rated into the system performance evaluation by averaging the results according to the specificfading distribution, in contrast to [MCC+06], here each channel realization is normalized tounit energy and no explicit shadowing term is taken into account, i.e.,‖p‖2 = 1.

Exemplarily, the major parameters (with slight simplifications4) and properties of three channelmodels of [MCC+06], representing short-range transmission (up to4m) in line-of-sight (CM 1of [MCC+06]) and non-line-of-sight (CM 2) indoor scenarios, and line-of-sight office (CM 3)scenarios, respectively, are summarized in Table 2.1. For these models, the amplitude of thepath gains is Nakagami-distributed (the Nakagami-m parameter is a random variable itself) andthe sign takes values±1 with equal probability.

The conducted numerical simulations in this thesis are mainly based on this channel model.Different from [MCC+06], a frequency-dependent path loss is not included in the model con-sidered in this thesis.

4The channel model of [MCC+06] involves more parameters with some influence on the process of constructinga realization of the power-delay profile and the channel impulse response; for brevity the description in this sectionis limited to the major effects and parameters.

2.5. Key Figures for IR-UWB Communication Systems 19

2.5 Key Figures for IR-UWB Communication Systems

Finally, in order to obtain key figures for a realistic IR-UWBcommunication system,5 an exem-plary parameter setting for the considered IR-UWB communication system is specified. Pos-sible application examples include monitoring of operating conditions in industrial facilities,surveillance of vital signs of inpatients in health care, orrecording of environmental conditions[Fon04, AR03, PH03].

We choose the pulse-width parameter asτ = 0.25 ns, yielding a pulse duration of approximately1 ns, a10 dB-bandwidth of3.1GHz and a center frequency of2.3GHz (cf. Figure 2.2(b)). Forthis setting, the sampling rate may be chosen tofs = 10GHz.

Taking into account the considered propagation scenarios described above (i.e., indoor, short-range), the symbol duration should be chosen in the order ofT = 50 ns to 150 ns to ensurean inter-symbol-interference-free transmission. This results in a symbol rate from1/T =

6Msymbols/s to 20Msymbols/s. This net-symbol rate is reduced by the application of TH/DSby a factor of10 to 100 for realistic IR-UWB systems in order to enable simultaneous operationof multiple users and to reduce the transmit power, yieldingeffective symbol rates in the orderof 60 ksymbols/s up to2Msymbols/s. The net-data rate is further reduced by a factor ofRc

andRm, depending on the applied modulation and channel coding scheme.

We assume that transmission takes places in relatively short bursts of approximately one mi-crosecond duration, i.e., a few hundreds of symbols, which represent data packets of only50 bitup to2 kbit. This is motivated by the application in wireless sensor networks, for which usuallyrelatively small data packets are sufficient. Since, the coherence time of typical UWB propaga-tion scenarios is expected to be in the order of several hundreds of microseconds, time varianceof the channel hardly impacts the system performance. Its influence may thus be neglected.

Again motivated by the application in wireless sensor networks, acceptable error rates for sat-isfactory performance of the communication system are in the order of10−3 to 10−4. Sig-nificantly lower error rates are not necessary, due to the presence of observations of multiplesensors and/or the update frequency of the measurements, such that erroneous messages can becorrected at the data-fusion center.

5In this thesis, all physical signals are normalized to represent dimensionless signals. Parameters related totime and frequency domain values are usually specified in nanoseconds and Hertz, respectively.

20

21

3. Receiver Design for Generic IR-UWB

Modulation

The key problem in receiver design is to develop efficient detection schemes, which deliver anestimate for the most-likely sequence of transmitted symbols. Moreover, especially for codedtransmission, reliability information on the estimates, so-called soft output, is desired as well.This soft output can be used to improve the performance of theoverall system.

For IR-UWB communication systems, these soft-output receivers have to fulfill strict complex-ity constraints, which renders coherent detection infeasible due to the required accurate chan-nel estimation. Instead, noncoherent detection schemes avoiding any kind of explicit channelestimation—so-called truly noncoherent receivers—appear to be a promising solution.

In this chapter, we derive an optimum (truly) noncoherent receiver for generic IR-UWB trans-mission, as introduced in the previous chapter. To this end,in Section 3.1, we first motivate thedetection problem and point out the specific requirements and problems as well as differences toconventional detection in digital transmission systems, induced by the low-complexity demandfor receiver design in IR-UWB communications. Based on these requirements, we derive thenoncoherent maximum-likelihood IR-UWB receiver in Section 3.2 and extend it to incorporatethe computation of reliability information on the estimates in Section 3.3, as required for soft-decision channel decoding. The receive signal processing required for these detection schemesis efficiently implemented using a so-called autocorrelation device, which is introduced in Sec-tion 3.4 and discussed further in Section 3.5. Finally, our approach is compared to receiverdesign based on the principle of generalized likelihood-ratio testing in Section 3.6. The chapteris summarized in Section 3.7.

22 3. Receiver Design for Generic IR-UWB Modulation

3.1 Detection Problem

3.1.1 Problem Formulation

Without loss of generality, we consider the detection of thecode/transmit symbols correspond-ing to an arbitrary block ofN receive symbols,1 i.e., the decision is based on the receive signalin an interval spanning overN symbol durationsT . The length of the observation window isused to model different detection schemes, with symbol-wise detection schemes forN = D asone extreme, and ideal sequence estimation for the entire burst with N = Nburst as the otherextreme. Detection schemes based on observation windows in-between these two extremesare of particular interest, as they enable a flexible trade-off between receiver complexity andperformance.

To this end, we assume perfect timing synchronization between transmitter and receiver, i.e., thesymbol rate1/T and the timing offset are known exactly. Robustness of the derived detectionschemes with respect to mismatched timing offset is discussed later-on.

For compact notation, we consider an arbitrary observationwindow of durationNT , i.e., an ar-bitrary block ofN receive symbols. Thus, absolute symbol/time indices are omitted, and indiceshave to be understood as intra-block indices. For convenience we introduce a vector/matrix no-tation by grouping the sampled receive signal of theN symbol intervals (cf. (2.15)) into theNs ×N matrix

R =[

rT0, rT1, . . . , r

TN−1

]

. (3.1)

Assuming inter-symbol-interference-free transmission,i.e., the receive pulses of adjacent sym-bols do not overlap, using (2.16), we have

R = pT b+N (3.2)

whereb = [b0, b1, . . . , bN−1] groups the transmit symbols of the observation window into avector,p is the sampled receive pulse (cf. (2.17)), andN =

[

nT0, n

T1, . . . , n

TN−1

]

collects thenoise samples (cf. (2.18)).

As the receive signal is corrupted by additive white Gaussian noise (cf. Section 2.3), the prob-ability density function of the receive signal conditionedon a hypothesis for the sequence oftransmitted symbolsb and a hypothesis for the receive pulse shapep is given as

fR(R|b, p) = const. · e−1

2σ2n‖R−pTb‖2F (3.3)

where‖ · ‖F denotes the Frobenius norm and all factors irrelevant for the subsequent derivationare summarized in a constant.

The first point to tackle for receiver design is to derive the marginal distributionfR(R|b) from(3.3) taking into account the characteristics of the receive pulse shape. This point is addressed in

1The ratio of window length and dimensionality of the modulation,N/D, is restricted to natural numbers.

3.1. Detection Problem 23

Section 3.2 and delivers the maximum-likelihood (ML) marginal distributionfMLR (R|b). Based

on this marginal distribution, the remaining maximum-likelihood detection problem is given as

bML = argmaxb∈B

fMLR (R|b) (3.4)

whereB ⊂ BN is the set of possible transmit symbols, which is specific to the consideredvariant of IR-UWB modulation.

Subsequently, we mainly consider an identical detection problem, obtained by maximizing theso-called log-likelihood functionln

(

fR(R|b))

instead of the above problem, i.e.,

bML = argmaxb∈B

ln(

fMLR (R|b)

)

. (3.5)

Efficient schemes solving (3.5) are derived and discussed inChapter 4 and 5 for PPM IR-UWBand DTR IR-UWB, respectively.

In order to fully exploit the benefits of channel coding in coded IR-UWB communications, reli-ability information on the estimates is desired; this so-called soft-output detection is addressedin Section 3.3.

3.1.2 Approaches to IR-UWB Receiver Design

At this point it is insightful to briefly review detection in conventional digital transmissionschemes (cf., e.g., [PS08]) and contrast this approach to the approach pursued for IR-UWBreceiver design.

In conventional digital transmission, usually sufficient knowledge of the receive pulse shapeis assumed, i.e., the pulse hypothesis can be replaced by a relatively accurate estimate (forsimplicity we assume perfect channel state information andset p = p). For the consideredinter-symbol-interference-free transmission, this leads to the well-known matched filter receiver[Hub05], e.g., implemented using a so-called rake receiver[CWM02].

Neglecting addends, which do not depend onb, and defining the sampled matched-filter outputd = pR

‖p‖2(sampling rateT ), the detection problem (3.5) reduces to

bCSI = argmaxb∈B

2bdT − ‖b‖2 . (3.6)

For the modulation schemes mainly employed for IR-UWB, namely DTR, or equivalently dif-ferential binary phase-shift keying, and PPM, we have

‖b‖2 = const. , ∀b ∈ B (3.7)

i.e., all (hypotheses for the) transmit symbols have equal norm. ForD-ary PPM symbol-wisedecisions are optimal, i.e.,N = D. In case of DTR IR-UWB, forN ≥ 2 this is equivalentto well-known MSDD of differential phase-shift keying (DPSK) transmitted over the AWGNchannel [DS90, Mac94], which reduces to symbol-wise (conventional) differential detection forN = 2.


All these detection schemes use parts of the receive signal as a reference for demodulation (e.g.,the matched-filter output of the previous symbol for symbol-wise differential detection or theenergy of the matched-filter output of the current symbol forenergy detection) and allow aresidual ambiguity of the polarization of the receive pulseshape, i.e., the results are equivalentfor p = −p instead ofp = p. For clarity, we use the term semi-noncoherent to describe suchdetection schemes, since some form of explicit channel estimation is required (the pulse shapehas to be known only up top = ±p).

In IR-UWB communications, however, an explicit channel estimation device is consideredoverly complex due to the large (and random) number of multipath components caused by thetypical channel conditions with severe multipath propagation and the high multipath resolutiondue to the large signal bandwidth [WLJ+09, YG04, LDM02, CWM02]. Incorporating a suffi-ciently accurate statistical model into the derivation, which describes the behavior of the receivepulse shape, e.g., taking into account typically employed channel models (cf. Section 2.4), hastwo major drawbacks. First, the resulting detection schemes are only applicable in the scenariofor which the particular channel model is valid. Second, theamount of randomness typicallyinvolved in UWB channel modeling results in detection schemes with prohibitively high com-plexity [TY08].

For these reasons, in this thesis, we pursue an approach which does not require the receiver tohave access to any kinds of channel state information—the resulting detection schemes maythus be referred to astruly noncoherent schemes. Thus, we aim for detection of the informa-tion represented in signals with unknown shape in the presence of white noise, in the spirit of[Urk67].

Following this approach, subsequently, low complexity noncoherent detection schemes for theconsidered variants of IR-UWB modulation are derived and proven to be optimal subject tothese requirements. This low-complexity demand however requires to avoid inter-symbol inter-ference. To ensure this property, the symbol duration has tobe chosen sufficiently large suchthat the receive pulse has decayed prior to reception of the next pulse. The only information onthe channel characteristics required to ensure this property is the worst-case (relevant) length ofthe receive pulse shape.

Two approaches are reviewed and contrasted: the first approach (the well-known Bayesian ap-proach [Kay98]) performs maximum-likelihood detection based on best-guess assumptions onthe statistical properties of the receive pulse shape. The second approach is based on generalizedlikelihood-ratio testing, a variant of intuitive statistical test procedures [Kay98]. For the consid-ered modulation schemes, both approaches lead to the same energy-detection-/autocorrelation-based receiver. Thus, as opposed to the rather heuristic motivation of many low-complexityIR-UWB receiver designs (cf. [FS10] for a discussion), optimality of the energy-detection-/autocorrelation principle is shown for truly noncoherent receivers.

The approaches, however, differ, e.g., in the way soft output is calculated and in the way differ-ent channel characteristics—if known beforehand to the receiver—can be incorporated. More-over, as an intuitive statistical test, the generalized likelihood-ratio testing approach does notensure optimality [Kay98, FS10], but is included due to its wide application in literature, cf.,

3.2. Noncoherent Maximum-Likelihood Detection 25

e.g., [ZM12a, LT08, DM05, SF10a].

A different approach, which is based on the principle of statistical invariance, has been sug-gested in [FS10]. For the considered IR-UWB variants, the resulting detection schemes areagain equivalent to those of the maximum-likelihood and thegeneralized likelihood-ratio test-ing approach, which supports the approach considered in this thesis.

3.2 Noncoherent Maximum-Likelihood Detection

Maximum-likelihood detection is based on the marginal distribution

fMLR (R|b) = Ep{fR(R|b, p)} (3.8)

=

∫ +∞

−∞

fR(R|b, p) · fp(p)dp (3.9)

where the statistical properties of the receive pulse shapeare described via its probability den-sity functionfp(p). In order to derive a (truly) noncoherent maximum-likelihood receiver, weconduct the following simplifications of the statistical properties of the receive pulse shape. Weassume uncorrelated zero-mean Gaussian distributed elements and a given power-delay profileof the underlying channel impulse response. These assumptions can be seen as a best-guessmodeling of the statistical properties of the receive pulseshape given almost no knowledge onits actual characteristics.2 In particular, the Gaussian assumption of the samples of thereceivepulse shape should be considered as a mathematical tool which enables to derive a truly non-coherent receiver, instead of being motivated by the effective channel modeling (compare toSection 2.4).

A more general derivation based on Nakagami-distributed uncorrelated elements ofp (as un-derlying the channel models of [MCC+06]) has been presented in [TY08]. This approach,however, results in relatively complex receiver processing steps, and is thus narrowed down tothe special case of the Gaussian distribution in [TY08], as well. Here we directly assume aGaussian distribution.

Different assumptions on the underlying power-delay profile are used to derive weighting of thereceive signal prior to further signal processing, cf. Section 3.2.3. Motivated by the modelingof the channel impulse response, we consider two particularpower-delay profiles, the specialcase of an exponential decay and a uniform power-delay profile.

3.2.1 Derivation of Maximum-Likelihood Decision Metric

More precisely, we require the elements ofp being uncorrelated zero-mean Gaussian distributedaccording to the probability density function

fpm(p) =1

√

2πσ2pm

· e− p2

2σ2pm , for m = 0, . . . , Ns − 1 (3.10)

2This approach can be compared, e.g., to the assumption of a uniform phase distribution over±π in the deriva-tion of ML-MSDD of DPSK [HF92].


with varianceσ2pm given by the considered/assumed power-delay profile. The probability den-

sity function of the receive pulse hypothesis is then given as

fp(p) =Ns−1∏

m=0

fpm(pm) =Ns−1∏

m=0

1√

2πσ2pm

e−

p2m2σ2

pm . (3.11)

Using this probability density function in (3.9), the marginal distribution calculates to (for adetailed derivation, cf. Appendix A.1)

fMLR (R|b) = const. ·

(

Ns−1∏

m=0

σ2pm

σ2n

‖b‖2 + 1

)−Ns/2

e− 1

2σ2n

(

‖R‖2F−‖bRT

Σ‖2)

(3.12)

where all terms irrelevant for the detection are grouped into a constant. In (3.12) we haveintroduced the weighting matrix

Σ =

w0 0. . .

0 wNs−1

(3.13)

with the weighting coefficients

wm =2

√

‖b‖2 + σ2n

σ2pm

(3.14)

on its main diagonal. The weighting coefficients depend on the noise variance and the assumedpower-delay profile. Optimum setting of the weighting coefficients for given channel charac-teristics is discussed below.

Finally, considering the log-likelihood function and neglecting terms which do not depend onthe hypothesisb, the detection problem reads

bML = argmaxb∈B

1

2σ2n

∥

∥bRTΣ∥

∥

2 − Ns

2

Ns−1∑

m=0

ln

(

σ2pm

σ2n

‖b‖2 + 1

)

. (3.15)

For the considered modulation schemes the energy of the transmit sequences is constant, i.e.,‖b‖2 = const. holds,∀b ∈ B. In this case, the scaling of the first summand in (3.15) and theentire second summand can be neglected; the detection problem thus reduces to

bML = argmaxb∈B

∥

∥bRTΣ∥

∥

2. (3.16)

This truly noncoherent maximum-likelihood detection scheme can be used to derive noncoher-ent receivers for a large class of IR-UWB variants, such as PPM IR-UWB (cf. Chapter 4 andthe following examples) and DTR IR-UWB (cf. Chapter 5). Of course, it can be used for TRIR-UWB and IR-UWB and biorthogonal PPM, as well, and is also applicable, e.g., for IR-UWBmodulation using on/off keying or any form of unipolar amplitude shift-keying.


3.2.2 Discussion of Maximum-Likelihood Noncoherent Detection

From (3.16) it can be seen, that the maximum-likelihood estimate for the transmit symbols isgiven as the sequence of symbols leading to the (weighted) signal with the largest energy amongall signals resulting from coherently combining the receive symbol intervals according to thedifferent hypotheses.

For symbol-wise detection of binary PPM IR-UWB transmission, the receive signal processingis illustrated in Example 3.1.

Example 3.1: Symbol-Wise Energy Detection of Binary PPM IR-UWB

Considering binary PPM IR-UWB (D = 2), for an observation window of one PPM symbol,

i.e., N = 1 · D = 2, the optimum receiver performs symbol-wise energy detection. For

simplicity, we set Σ = I , i.e., apply uniform weighting within T . The energy of the two

symbol intervals is computed for the two possible transmit symbols candidates according to

∥

∥

∥

∥

[1 0] ·[

r0r1

]∥

∥

∥

∥

2

= ‖r0‖2 and

∥

∥

∥

∥

[0 1] ·[

r0r1

]∥

∥

∥

∥

2

= ‖r1‖2 .

For each PPM symbol, the noncoherent maximum-likelihood estimate is given as the symbol

corresponding to the interval with largest captured energy.

This receiver processing is visualized below for two PPM symbols (assuming b = [1 0]followed by b = [0 1] and 10 log10(Ec/N0) = 18dB):

First PPM symbol:

t

r(t)

0 T TD

=>t

energ

y0 T

t

r(t)

0 T TD

=>t

energ

y

0 T

Second PPM symbol:

t

r(t)

0 T TD

=>t

energ

y

0 T

t

r(t)

0 T TD

=>t

energ

y

0 T

The plots on the left hand side depict the receive signal with the parts relevant for the

decision highlighted. On the right hand side, the squared signal is shown. Its integral gives

the desired decision variable, i.e., the energy of the PPM interval, as indicated by a filled bar

(the filling level represents the captured energy).


In order to interpret the receive signal processing and thusexplain the potential performanceimprovement of multiple-symbol detection schemes (N > D) over symbol-wise detectionschemes, note that the receive pulse shape remains constantwithin the observation interval.For observation windows greater than one symbol duration this receiver aims to exploit thisfact by inherently implementing an improved noise averaging and implicitly estimating the re-ceive pulse shape. However, this noise averaging is only possible for a given hypothesis of thetransmit signals. The maximum-likelihood receiver thus estimates the transmit symbols and thereceive pulse jointly. This is illustrated in Example 3.2.

Example 3.2: Multiple-Symbol Energy Detection of Binary PPM IR-UWB

Again considering binary PPM IR-UWB as in Example 3.1, the maximum-likelihood decision

based on an observation interval of two PPM symbols is obtained from so-called multiple-

symbol detection (cf. Section 4.2), i.e., the two PPM symbols are decided jointly. This

receiver processing is visualized below. Due to the window length of N = 2 ·D = 4, there

are 22 = 4 hypotheses. For each hypothesis, two symbol intervals of the receive signal are

added and the energy of the resulting signal is computed, i.e.,

∥

∥

∥

∥

∥

∥

∥

∥

[1 0 1 0] ·

r0r1r2r3

∥

∥

∥

∥

∥

∥

∥

∥

2

= ‖r0 + r2‖2 ,

∥

∥

∥

∥

∥

∥

∥

∥

[1 0 0 1] ·

r0r1r2r3

∥

∥

∥

∥

∥

∥

∥

∥

2

= ‖r0 + r3‖2 ,

∥

∥

∥

∥

∥

∥

∥

∥

[0 1 1 0] ·

r0r1r2r3

∥

∥

∥

∥

∥

∥

∥

∥

2

= ‖r1 + r2‖2 , and

∥

∥

∥

∥

∥

∥

∥

∥

[0 1 0 1] ·

r0r1r2r3

∥

∥

∥

∥

∥

∥

∥

∥

2

= ‖r1 + r3‖2 .

Again, the noncoherent maximum-likelihood detector decides for the hypothesis with largest

energy, in this case, however, representing two PPM symbols jointly.

For the example above, we now have the following situation:

t

r(t)

0 T TD

=>t

0

energ

y

T

t

r(t)

0 T TD

=>t

0

energ

y

T

t

r(t)

0 T TD

=>t

0

energ

y

T

t

r(t)

0 T TD

=>t

0

energ

y

T


Apparently, since the energy of the correct hypotheses (i.e., the one corresponding to the

actual transmitted symbols) can clearly be distinguished from the other hypotheses, the de-

cision is more reliable compared to symbol-wise energy detection above. This is achieved

by averaging the receive signal prior to computation of the energy, which reduces the con-

tribution of the noise to the sum energy.

3.2.3 Discussion of Weighting Coefficients

There are different options to set the weighting coefficients of the receive signal in (3.16) priorto computing the energy of the coherently combined signal. Related approaches have beenconducted in [LV04, CS04, RW06].

We consider three variants, which differ mainly in the amount of required knowledge on thechannel characteristics. The first two options assume an exponentially decaying and a uniformpower-delay profile, whereas the last option represents themost general scenario with no ad-ditional assumptions on the channel characteristics. The different options for the weightingcoefficients are depicted in Figure 3.1.

Clearly, a timing offset can be compensated by estimating the relevant part of the support ofthe receive pulse shape and correspondingly shifting the correlation window, e.g., in uniformweighting withinTi. This synchronization task could be implemented in an adaptive fashion,e.g., using the captured receive energy as target criterion.

wei

ghts

tTTi

Figure 3.1: Illustration of different weighting coefficients. Solid line: exponential weighting (3.17), dashed:

simplified exponential weighting (3.18), dash-dotted: uniform weighting within Ti, dotted: uniform weight-ing within T .

3.2.3.1 Exponential Weighting

Motivated by the channel modeling (cf. Section 2.4), the first setting is obtained by assuming anexponentially decaying power-delay profile. In this case, we haveσ2

pm = 1ΓCM

e−mTs/ΓCM , whereΓCM represents the power-decay time constant. Using (3.14), the squared weighting coefficientsare given as

w2m =

4

‖b‖2 + ΓCMσ2n · emTs/ΓCM

. (3.17)

Since 11+αex

≈ 1αe−x for x≫ 0 (such thatαex ≫ 1), this expression is well approximated by

w2m =

4

ΓCMσ2n

e−mTs/ΓCM (3.18)


as shown in Figure 3.1. The weighting coefficients for computation of the captured energywithin a symbol interval hence resemble the exponential decay of the power-delay profile. Sincethe factor 4

ΓCMσ2n

does not depend onm, it can be factored out and neglected in (3.16), makingthe weighting coefficients independent of the noise variance.

3.2.3.2 Uniform Weighting within Correlation Interval

In case of a non-exponentially-decaying power-delay profile and/or only coarse knowledge ofthe channel characteristics (i.e.,ΓCM unknown), exponential weighting is not applicable. In-stead, as a second option, a more general setting is considered. To this end, a uniform power-delay profile of some durationTi ≤ T is assumed. Defining the time-bandwidth productNi = fsTi, we haveσ2

pm = σ2p = 1

Nifor m = 0, . . . , Ni − 1 and zero elsewhere. The squared

weighting coefficients are thus given as

w2m =

{

4‖b‖2+Niσ2

n= const. for 0 ≤ m ≤ Ni − 1

0 else. (3.19)

Effectively the elements of the receive signal are uniformly weighted within the support set ofthe assumed power-delay profile, i.e., only the relevant elements are considered. Clearly, for‖b‖2 = const., the constant scaling factor can be neglected in (3.16).

3.2.3.3 Uniform Weighting within Entire Symbol Interval

The third option considered is the most general scenario, where no additional assumptions onthe channel characteristics are drawn. It is thus widely applicable and becomes necessary, e.g.,if only coarse synchronization is achieved (i.e., the exactsupport of the receive pulse shapewithin the symbol interval is not known). In this case all samples have to be weighted equally.This is equivalent to a uniform power-delay profile and setting Ti = T andNi = Ns, i.e., theentire symbol interval is used to compute the receive signalenergy.

3.3 Computation of Reliability Information

In case of coded IR-UWB transmission, it is well known that estimates on the reliability ofthe decisions, so-called soft output, can be used to improvethe performance of the subsequentsoft-input channel decoder, i.e., soft-output detection schemes in combination with soft-inputdecoding should be employed [WJ65]. Moreover, detection schemes based on the principleof multiple-observations combining (cf. Section 5.3.6 and[SF10a]) benefit from soft-outputdetection schemes, as well.

3.3.1 Problem Formulation

This soft output is commonly obtained from calculating the probability that the correspondingtransmitted code symbol has taken the value0 or 1 given the observation of the receive signal.For thekth code symbolck (prior to deinterleaving) of the sequenceb taking the value0, this is

3.3. Computation of Reliability Information 31

given as

Pr{ck =M−1k (b) = 0|R} = 1

∑

b∈B fMLR (R|b) ·

∑

b∈B

M−1k

(b)=0

fMLR (R|b) (3.20)

whereM−1k (b) is the inverse mapping function which delivers thekth code symbol associated

to the sequenceb.

In many cases, computing (3.20) exactly is not required and only minor performance degra-dation results by applying a so-called nearest-neighbor approximation. In this case, (3.20) isapproximated by

Pr{ck = 0|R} ≥≈ 1∑

b∈B fMLR (R|b) · max

b∈B

M−1k

(b)=0

fMLR (R|b) . (3.21)

Many channel decoders can be implemented more efficiently, when using equivalent metrics onthe reliability of the code symbols. In particular, log-likelihood ratios (LLR) are often consid-ered [LH06]. The LLR of thekth code symbol is defined as

LLRk = ln

(

Pr{ck = 0|R}Pr{ck = 1|R}

)

(3.22)

= ln

∑

b∈B

M−1k

(b)=0

fMLR (R|b)

− ln

∑

b∈B

M−1k

(b)=1

fMLR (R|b)

. (3.23)

Similarly, the nearest-neighbor approximation of (3.21) results in3

LLRk≈= max

b∈B

M−1k

(b)=0

ln(

fMLR (R|b)

)

− maxb∈B

M−1k

(b)=1

ln(

fMLR (R|b)

)

. (3.24)

Subsequently, we will mainly focus on this approximate way of soft-output computation (andomit the indicator for the approximation).

3.3.2 Soft-Output Receiver for IR-UWB Transmission

Using the marginal distribution of (3.12) and assuming‖b‖2 = const., ∀b ∈ B, we obtain

LLRk =1

2σ2n

·

maxb∈B

M−1k

(b)=0

∥

∥bRTΣ∥

∥

2 − maxb∈B

M−1k

(b)=1

∥

∥bRTΣ∥

∥

2

. (3.25)

3In case of LLRs, this approximation is called max-log approximation and motivated by the fact that the loga-rithm of the sum of exponentials is dominated by the term withthe largest argument in the exponential function;the summation is simply replaced by this argument, i.e.,ln

∑

x ex ≈ max x.


Clearly, one of the two maximization problems in (3.25) (or equivalently (3.24)) is equivalentto the hard-output maximum-likelihood detection problem (3.16) [SB08]. Soft-output com-putation employing the nearest-neighbor/max-log approximation thus boils down to findingthe maximum-likelihood sequencebML (and its decision metric‖bMLRT

Σ1/2‖2) and the best

sequence for each code symbolcMLk associated to the maximum-likelihood sequence in a re-

stricted search space, obtained from fixing thekth code symbol to the complementcMLk of the

corresponding code symbol estimate. This metric is specificfor each code symbol and given by

maxb∈B

M−1k

(b)=cMLk

∥

∥bRTΣ∥

∥

2. (3.26)

The LLRs are thus equivalently computed via

LLRk = ± 1

2σ2n

·

∥

∥bMLRTΣ∥

∥

2 − maxb∈B

M−1k

(b)=cMLk

∥

∥bRTΣ∥

∥

2

(3.27)

where the sign depends on the particular value ofcMLk (“+” for cML

k = 0). Equation (3.27)represents the intuitive result that reliability information of the maximum-likelihood estimateis given as the (scaled) distance to the next-best counter-hypothesis (distance measured withrespect to decision metric).

Assuming a uniform weighting over the entire symbol interval Ts, a constant scaling can befactored out, which yields

LLRk =4

2σ2n‖b‖2 + 2Ns(σ2

n)2·

maxb∈B

M−1k

(b)=0

∥

∥bRT∥

∥

2 − maxb∈B

M−1k

(b)=1

∥

∥bRT∥

∥

2

. (3.28)

Basically, the scaling factor of the LLRs thus resembles thevariance of the equivalent noise onthe autocorrelation coefficients (derived later-on, cf. Section 3.5.2). In particular, the product oftime-bandwidth productNs and the squared original noise variance, which originates from the“noise×noise” multiplication, is represented.

In order to compute this scaling coefficient, an estimate of the signal-to-noise ratio has to beavailable at the receiver. Usually, this is not the case in low-complexity receivers, as investi-gated here. Depending on the employed channel decoder, the scaling factor of the LLR valuesmay however be neglected, if the decoding process is scale-invariant [WHW00]. This is thecase, e.g., in a convolutional coded system employing the Viterbi algorithm for soft-input/hard-output decoding. In this case, for DTR IR-UWB the resulting LLR computation is equivalentto the approach presented in [SF10b] (cf. Section 3.6) and the approach of [ZM12b] based on aGaussian approximation of the correlation coefficients (cf. Section 3.5).

3.4 Autocorrelation-Based Detection

The preprocessing of the receive signal required for noncoherent maximum-likelihood detec-tion and computation of soft output is efficiently implemented using a so-called autocorrelation

3.4. Autocorrelation-Based Detection 33

device following the receiver input filter [GQ06, LT08]. This autocorrelation device calculatesthe correlation coefficients of different symbol intervalsof the receive signal. In this section, wemotivate the use of this autocorrelation device and commenton implementation aspects. Opti-mization of key parameters of this autocorrelation device are discussed along with an equivalentdiscrete-time model for autocorrelation-based detectionin Section 3.5.

3.4.1 Autocorrelation Device

To this end, we note that the decision metric in (3.16) is equivalently computed as∥

∥bRTΣ∥

∥

2= bRT

Σ2Rb

T= bZb

T(3.29)

where we have introduced the matrix

Z = RTΣ

2R =

z0,0 z0,1 . . . z0,N−1

z1,0 z1,1 z1,N−1...

. . ....

zN−1,0 zN−1,1 . . . zN−1,N−1

(3.30)

composed of the correlation coefficients of the weighted receive signal in symbol intervali andsymbol intervalk, given by

zk,i = rkΣ2rTi . (3.31)

Of course, sincezk,i = zi,k holds for alli, k, the matrixZ is symmetric.

The correlation coefficients form a set of sufficient statistics for noncoherent maximumlike-lihood detection. Thus, as opposed to energy detection of (3.16), this autocorrelation-baseddetection enables to separate the receiver into two major building blocks: the autocorrelationdevice, which marks the transition from analog to sampled representation of the relevant partsof the receive signal, and a block evaluating the decision metric, operating on the output of theformer.

Figure 3.2 depicts a block diagram of this generic IR-UWB autocorrelation-based receiver as-suming an all-analog implementation of the autocorrelation device which computes the corre-lation coefficients. In each branch of the autocorrelation device, the receive symbol intervalsare correlated by multiplying the current receive signal with the signal delayed by multiples ofthe symbol duration. The resulting signalr(t) · r(t − iT ) is then weighted (w2

Σ(t) denotes theperiodically repeated analog equivalent of the squared signal weights, periodT ) and integrated;for each symbol interval the integrator puts out one coefficient and is reset afterwards, yielding

zk,i =

∫ T

0

w2Σ(t)r(t− kT )r(t− iT )dt (3.32)

which is the non-sampled equivalent of (3.31).

These correlation coefficients are then passed to a block, which evaluates the decision metricand decides for the most-likely sequence of transmitted symbols (or computes reliability infor-mation).


hRX(t)∫

(T )·dt

∫

(T )·dt

∫

(T )·dt

T

(N−1)T

r(t) zk,k

zk−1,k

zk−(N−1),k

kT

kT

kT

bML

argm

axbbZbT

n0(t)

autocorrelation device

w2Σ(t)

w2Σ(t)

w2Σ(t)

Figure 3.2: Block diagram of generic IR-UWB receiver with autocorrelation device (highlighted).

Assuming uniform weighting and neglecting a constant scaling, the calculation simplifies to

zk,i =

∫ Ti

0

r(t− kT )r(t− iT )dt . (3.33)

Clearly, the assumed maximum duration of the power-delay profile directly translates into theduration of the time interval relevant for correlation. Subsequently, we thus refer toTi as thecorrelation interval, which represents one of the major receiver design parameters with signifi-cant influence on the overall receiver performance. Its influence and optimal setting is addressedalong with the discussion of the specific detection schemes in Chapter 4 and 5.

Autocorrelation-based detection is illustrated for TR IR-UWB in Example 3.3.

Example 3.3: Autocorrelation-Based Detection of TR IR-UWB

Exemplarily, the receiver processing is illustrated for TR IR-UWB. Clearly, a window length

of at least N = 2 is required. For N = 2, we have the two transmit signal hypotheses

b = [+1 + 1] and b = [+1 − 1].

Again assuming Σ = I, energy-based detection computes

∥

∥

∥

∥

[+1 + 1] ·[

r0r1

]∥

∥

∥

∥

2

= ‖r0 + r1‖2 and

∥

∥

∥

∥

[+1 − 1] ·[

r0r1

]∥

∥

∥

∥

2

= ‖r0 − r1‖2 .

If b = [+1 − 1] has been transmitted, we expect that ‖r0 − r1‖2 > ‖r0 + r1‖2.

Expanding the energy computations, we have

∥

∥

∥

∥

[+1 ± 1] ·[

r0r1

]∥

∥

∥

∥

2

= ‖r0 ± r1‖2 = ‖r0‖2 ± 2r0rT1 + ‖r1‖2

Apparently, ‖r0‖2 and ‖r1‖2 do not influence the final decision. Thus, a decision is equiv-

alently obtained from evaluating the sign of the correlation coefficient z0,1 = r0rT1. If

b = [+1 − 1] has been transmitted, we expect that z0,1 < 0.

3.5. Discrete-Time Model for Autocorrelation-Based Detection 35

3.4.2 Implementation Aspects

The all-analog implementation depicted in Figure 3.2, has some drawbacks from implementa-tion point of view [PH03, CD05]. The major drawback of an all-analog implementation arethe required accurate analog delay lines realizing delays in multiples of the symbol duration.Additionally, analog components often require tuning, show a time-varying behavior, and aresusceptible to variations in the operating conditions, e.g., the temperature.

In order to avoid these drawbacks, a second implementation variant is obtained by samplingthe receive signal similar to (2.15). Depending on the system parameters, this all-digital im-plementation might lead to relatively high sampling rates in the order of several hundreds ofMegasamples per second. Noteworthy, the detection schemesconsidered in this thesis are ableto operate with relatively low resolution of the quantizer in the analog-to-digital conversion atonly marginal loss in performance [LT08, Sch09], which to some extend reduces the implemen-tation complexity of high-rate analog-to-digital converters.

A further implementation variant in-between these two extremes is based on the principleof compressed sensing [EK12]; this approach is considered,e.g., in [SF11b, GLL10, OL09,LZLZ10, WAPS07] and is discussed in detail in Section 5.5.2.It enables to significantly reducethe number of samples per second by acquiring measurements not in time domain, but in someother domain. The inherent loss in quality of the detection is manageable.

Although, an all-digital implementation is clearly favorable and the above problems with re-spect to its implementation may be regarded as a question of technology solved within less thana decade according to the recent rate of progress in microelectronics [Wal99], nevertheless weaim to design power-efficient detection schemes based on correlation coefficients computed forsymbols separated as few as possible, i.e., we aim to employ an autocorrelation device withminimum maximum delay(N − 1)T to still achieve the desired detection quality. This restric-tion accounts for the possibility of an all-analog implementation and alleviates the demand thatthe channel may not vary within the interval correlations are recorded. Moreover, a reduceddelay in the autocorrelation device also reduces the structural delay of the receiver processingand memory demands in an all-digital implementation.

3.5 Discrete-Time Model for Autocorrelation-Based Detection

In this section, an equivalent discrete-time model for IR-UWB transmission with an autocorre-lation-based receiver is introduced and simplified to an approximate discrete-time model. Thekey parameters of the model are discussed and related to the channel and system parameters.Under certain conditions, the approximate discrete-time model enables to analyze IR-UWBtransmission in a generic setup avoiding the need for an explicit definition of system parameters,the transmit pulse, and channel characteristics.

For clarity, we assume correlation over an interval of duration Ti with uniform weighting (cf.Section 3.2.3). In order to reduce notation overhead, we omit the weighting matrixΣ; in thissection, all vectors are restricted to the corresponding support set ofNi elements. Differentweighting coefficients render the analysis cumbersome and yield results, which strongly depend


z−1

zN−1

‖p‖2

‖p‖2

‖p‖2

ηk,k

ηk−1,k

ηk−(N−1),k

zk,k

zk−1,k

zk−(N−1),k

bML

argm

axbbZbT

bk

Figure 3.3: Equivalent system model of IR-UWB transmission (including modulation, transmission over prop-

agation channel, and autocorrelation-based detection) from transmit symbols b to corresponding estimates

.

on the assumed channel model (which disagrees with the main motivation for developing thediscrete-time model).

From (2.16) it can be seen, that the correlation coefficientsare composed of the scaled phasetransition frombk to bi superimposed by equivalent noise, i.e.,

zk,i = rkrTi = bkbi‖p‖2 + ηi,k (3.34)

where the equivalent noise

ηk,i = bkpnTi + bipn

Tk + nkn

Ti (3.35)

collects the “information×noise” and “noise×noise” terms. This enables to introduce an equiv-alent discrete-time model of IR-UWB transmission in combination with autocorrelation-baseddetection, as depicted in Figure 3.3. This equivalent discrete-time model replaces a large partof the transmission chain of Figure 3.2, from pulse shaping at transmitter side, to propagationover the multipath channel and the receiver input filter, as well as the autocorrelation device.

Subsequently, we investigate the respective terms which form the correlation coefficients. Webegin with the captured receive pulse energy‖p‖2 and proceed to the equivalent noise term inSection 3.5.2.

3.5.1 Analysis of Captured Pulse Energy

Due to correlation within a finite correlation interval of durationTi, p is composed of the firstNi = Tifs samples of the receive pulse shape. The useful signal energy‖p‖2 captured in theautocorrelation device thus depends on the correlation intervalTi and the channel characteris-tics. The randomness involved in the construction of a particular realization of the power-delayprofile of the employed channel models (cf. Section 2.4) renders an analytical analysis almost


impossible. However, in most UWB channel models, the power decay as a function of thedelay is dominated by an exponential terme−τ/ΓCM , parameterized by the cluster-power decayparameterΓCM (specific for the particular channel model, cf. Section 2.4 and Table 2.1).

We resort to a numerical evaluation of the captured pulse energy as a function of the integrationintervalTi, exemplary considering two indoor channel models (cf. Section 2.4 and [MCC+06])and a transmit pulse as depicted in Figure 2.2 withτ = 0.25 ns. The results are depicted inFigure 3.4 and have been averaged over a large number of channel realizations. The underlyingshaded areas represent histograms indicating the statistical fluctuations. For comparison thecaptured energy

Ep =

∫ Ti

0

1

ΓCMe−τ/ΓCMdτ = 1− e−Ti/ΓCM (3.36)

is depicted assuming a perfectly exponential decay. Additional lines indicate the average root-mean-square delay spread and the regimes, where on average at least99% and99.9% of thepulse energy is captured.

Despite the random and clustered structure of the considered model of the power-delay profile,the average captured energy well matches the assumption of aperfectly exponential decay.Hence, neglecting the statistical fluctuations, we approximate the captured receive pulse energyby

‖p‖2 ≈ Ep = 1− e−Ti/ΓCM . (3.37)

Of course, forTi ≫ ΓCM, the effect of imperfect capturing of the receive pulse energy can beneglected. It is then reasonable to setEp = 1.

3.5.2 Analysis of Equivalent Noise

The analysis of the equivalent noise has to distinguish two cases: on the one hand, equivalentnoise originating from different symbol intervals, i.e.,ηk,i, wherek 6= i, and, on the other hand,equivalent noise samplesηi,i corresponding to only one symbol interval.

In the latter case, from (3.35) we have

ηi,i = 2 · bipnTi + ‖ni‖2 . (3.38)

Since the components ofni are independent zero-mean Gaussian random variables with vari-anceσ2

n, depending on the value ofbi, the first summand either is zero forbi = 0, or may bemodeled as a zero-mean Gaussian random variable with variance 4Epσ

2n for bi = ±1. The

second summand, as the sum of the squares ofNi independent zero-mean Gaussian randomvariables, is given as aχ2-distributed random variable withNi degrees of freedom; its meanand variance are given byNiσ

2n and2Ni(σ

2n)

2, respectively (a short summary of the propertiesand parameters of theχ2-distribution can be found in Appendix A.2). Since, the different termscontributing to the equivalent noise are uncorrelated (thefirst term is randomized by the trans-mit symbol), the mean and variance of the random variableηi,i are given asE{ηi,i} = Niσ

2n and

E{

(ηi,i − E{ηi,i})2}

= b2i · 4Epσ2n + 2Ni(σ

2n)

2, respectively.


0 25 50 75 100 125 1500

0.25

0.5

0.75

1

indoor non-line-of-sight (ΓCM = 26.27 ns)

Ti [ns]

‖p‖2

avg.

dela

ysp

read

‖p‖2≥

0.99

‖p‖2≥

0.999

0 25 50 75 100 125 1500

0.25

0.5

0.75

1

office line-of-sight (ΓCM = 14.6 ns)

Ti [ns]

‖p‖2

avg.

dela

ysp

read

‖p‖2≥

0.99

‖p‖2≥

0.999

Figure 3.4: Average captured pulse energy E{‖p‖2} (solid) vs. correlation interval Ti for two channel models

taken from [MCC+06] compared to perfectly exponential decay Ep (dashed). The underlying shaded areas

depict the histogram of ‖p‖2 indicating its statistical fluctuations.

Turning to the noise originating from the correlation of different symbol intervals, again, thetwo “information×noise” terms in (3.35) are each zero-mean Gaussian random variables withvarianceEpσ

2n (assuming|bi| = |bk| = 1, different cases forbi andbk are not relevant for the

considered modulation and detection schemes).

The “noise×noise” term, as the sum ofNi products of independent Gaussian random variables,4

is zero-mean with varianceNi(σ2n)

2. Consequently,ηk,i may be modeled as a zero-mean randomvariable with variance2Epσ

2n +Ni(σ

2n)

2.

In summary, we have

σ2η = E

{

(ηi,i − E{ηi,i})2}

=

2Epσ2n +Ni(σ

2n)

2 for i 6= k and|bi| = |bk| = 1

4Epσ2n + 2Ni(σ

2n)

2 for i = k and|bi| = 1

2Ni(σ2n)

2 for i = k andbi = 0

. (3.39)

We observe, that the contribution of the terms in case of correlation of noise originating from thesame symbol interval is twice as large as the contribution ofthe noise originating from different

4To the knowledge of the author, an established phrase for thedistribution and a simple analytical expressionfor its probability density function does not exist for suchrandom variables.


symbol intervals.

Since eachηk,i results from the multiplication of different parts of noiseand symbols, the equiv-alent noise at different time instances and autocorrelation-device branches is uncorrelated.

3.5.3 Approximate Discrete-Time Model

Combining the results for the captured pulse energy and the equivalent noise, the mean valuesof the correlation coefficients, i.e., the elements ofZ, are given as

E{zi,k} =

bibkEp for i 6= k

Ep +Niσ2n for i = k and|bi| = 1

Niσ2n for i = k andbi = 0

. (3.40)

The variances are given by (3.39). More precisely, the diagonal elements follow aχ2 distribu-tion withNi degrees of freedom, i.e.,

zi,i ∼{

χ2(Ni, Ep/σ2n) for |bi| = 1

χ2(Ni, 0) for bi = 0. (3.41)

The off-diagonal elements can not be characterized by a distribution with an established name.

Invoking the law of large numbers, already for moderate time-bandwidth productsNi it is rea-sonable to approximate the correlation coefficients as Gaussian random variables with corre-sponding mean and variance [Urk67, PJ07, QWD07, SF11b]. In summary, we approximate

zi,k≈∼

N (bibkEp, 2Epσ2n +Ni(σ

2n)

2) for i 6= k and|bi| = |bk| = 1

N (Ep +Niσ2n, 4Epσ

2n + 2Ni(σ

2n)

2) for i = k and|bi| = 1

N (Niσ2n, 2Ni(σ

2n)

2) for i = k andbi = 0

. (3.42)

The resulting approximate discrete-time model of IR-UWB transmission with autocorrelation-based detection is shown in Figure 3.5. Compared to the equivalent discrete-time model ofFigure 3.3 only the scaling of the correlations of the transmit symbols and the additive noisehas changed.

These approximations are validated by means of numerical simulations. Figures 3.6 and 3.7depict recorded (and normalized) histograms of the correlation coefficients compared to theGaussian approximation and, in case of the diagonal elements, to the exact model of theχ2

distribution for different time-bandwidth productsNi adjusted via the correlation intervalTi

and a fixed sampling frequency offs = 10GHz. The results have been averaged over a largenumber of realizations of the indoor non-line-of-sight channel model of [MCC+06].

We conclude that the Gaussian approximation is very reasonable for this setting; a difference totheχ2 distribution is hardly visible. Notable deviations from the recorded histograms occur onlyfor low time-bandwidth products; this is due to the simplified assumption of an exponentiallydecaying power-delay profile, instead of the effective modeling via clusters.


z−1

zN−1

zk,k

zk−1,k

zk−(N−1),k

bML

argm

axbbZbT

bk

Ep

Ep

Ep

N (Niσ2n, σ

2η)

N (0, σ2η)

N (0, σ2η)

Figure 3.5: Approximate system model of IR-UWB transmission in combination with autocorrelation-based

detection.

3.5.4 Discussion

The approximate discrete-time model for IR-UWB transmission with autocorrelation-based de-tection is mainly characterized by the time-bandwidth productNi. For a given signal bandwidth,the key parameter for the performance of an autocorrelationdevice is the correlation intervalTi

reflected in the captured pulse energy in (3.37) and in the time-bandwidth productNi = Tifscontributing to the variance of the equivalent noise in (3.39). Clearly, a shorter correlation in-terval leads to less accumulated noise. However, as explained above, the integration intervalshould be chosen sufficiently large to capture a sufficient amount of the receive pulse energy.The resulting trade-off has been studied extensively in theliterature (cf. in particular [WLPK05]for a comprehensive overview) and is discussed in Section 4.1.3 and 5.2.2 for energy detectionof PPM and differential detection of DTR IR-UWB, respectively.

Subject to reasonable assumptions, the approximate discrete-time model offers to analyze theIR-UWB receivers under very generic conditions; the results are independent of the particularpulse shape employed at the transmitter and the channel model for multipath propagation. Therequirements are a uniform weighting prior to correlation,a sufficiently large time-bandwidthproduct, uncorrelated noise (in the Nyquist-rate sampled noise process) at the output of thereceiver input filter,5 and, in case the effect of imperfect energy capturing shall be covered, an(essentially) exponential decaying power-delay profile. These requirements are usually met intypical IR-UWB systems.

The approximate discrete-time model enables an analyticalevaluation of the error-rate per-formance of IR-UWB receivers as well as an information-theoretic analysis. Based on theseresults, design rules for realistic IR-UWB systems can be derived. Additionally, it may serve asa tool for efficient numerical simulations of the IR-UWB transmission chain.

5This assumption can be alleviated to incorporate correlations by replacing the noise variance with an equivalentnoise variance obtained from the equivalent noise bandwidth of the receiver input filter.


0 5 10 15 20 25 30 35 40 4510

−3

10−2

10−1

100

101

zi,i for bi = 0

z

hist(z)

0 5 10 15 20 25 30 35 40 4510

−3

10−2

10−1

100

101

zi,i for |bi| = 1

z

hist(z)

(a) −10 log10(σ2n) = 13 dB

0 5 10 1510

−3

10−2

10−1

100

101

zi,i for bi = 0

z

hist(z)

0 5 10 1510

−3

10−2

10−1

100

101

zi,i for |bi| = 1

z

hist(z)

(b) −10 log10(σ2n) = 17 dB

Figure 3.6: Normalized histograms of zi,i (markers) compared to χ2 distribution (dashed, cf. (3.41)) and

Gaussian approximation (gray, cf. (3.42)) for different time-bandwidth products Ni = 100, 500, 1000, and1500 (left-to-right) for the indoor non-line-of-sight channel model of [MCC+06], fs = 10GHz. Note the

different scaling of the axes.


0 0.5 1 1.5 2 2.5 3 3.510

−3

10−2

10−1

100

101

zi,k for bibk = 1

z

hist(z)

(a) −10 log10(σ2n) = 13 dB

0 0.5 1 1.5 2 2.5 3 3.510

−3

10−2

10−1

100

101

zi,k for bibk = 1

z

hist(z)

(b) −10 log10(σ2n) = 17 dB

Figure 3.7: Normalized histograms of zi,k (markers) compared to Gaussian approximation (gray, cf. (3.42))for different time-bandwidth products Ni = 100, 500, 1000, and 1500 (left-to-right) for the indoor non-line-

of-sight channel model of [MCC+06], fs = 10GHz.

3.6. Comparison to GLRT-Based Detection 43

3.6 Comparison to GLRT-Based Detection

Finally, the above maximum-likelihood approach is compared to the generalized likelihood-ratio testing (GLRT) approach, which is widely used for the motivation of various IR-UWBreceivers, cf., e.g., [FM06, CS03, DM05, CS02] and [SF10a, LT08, ZM12a] for approachesdirectly related to the later-on considered IR-UWB systems. Different to maximum-likelihooddetection, in the GLRT approach the derivation is not based on a marginal distribution. Instead,a maximization over all unknown parameters is performed [Kay98]. The GLRT approach rep-resents an intuitive statistical test procedure; in general its optimality can not be guaranteed[FS10, Kay98].

3.6.1 GLRT-Based Receiver Design

In our case, assuming a finite-length and finite-energy receive pulse shape within the symbolinterval, we thus need to calculate

fGLRTR (R|b) = max

pfR(R|b, p) . (3.43)

With slight abuse of notation, we adopt the same notation forthe resulting functionfGLRTR ,although it does not necessarily represent a probability density function. In particular, normal-ization to unity cannot be ensured in general.

The maximization in (3.43) implicitly assumes uniform weighting over the entire symbol inter-val, i.e., a uniform power-delay profile; uniform weightingwithin a shorter correlation intervalcan easily be incorporated. However, taking into account different profiles is cumbersome inthe GLRT approach.

Clearly, maximization offR(R|b, p) (cf. (3.3)) overp is equivalent to minimization of

∥

∥R− pT b∥

∥

2

F= tr

(

(

R− pT b)T (

R− pT b)

)

(3.44)

= tr(

RTR)

− 2tr(

RTpTb)

+∥

∥b∥

∥

2 ∥∥p∥

∥

2(3.45)

=∥

∥R∥

∥

2

F− 2bRTpT +

∥

∥b∥

∥

2 ∥∥p∥

∥

2(3.46)

where the last step follows from the cyclic shift property ofthe trace operator [HJ90]. It isstraight-forward to show that the optimum pulse shape is given aspGLRT = bRT

‖b‖2, i.e., the co-

herently combined receive signal, which however depends onthe hypothesis for the transmitsymbols. Hence, we have

minp

∥

∥R− pT b∥

∥

2

F=∥

∥R∥

∥

2

F−∥

∥bRT∥

∥

2

∥

∥b∥

∥

2 (3.47)

which yields the (pseudo-)probability density function

fGLRTR (R|b) = const. · e

− 1

2σ2n

‖R‖2F−‖bRT‖2‖b‖2

. (3.48)


Consequently, again neglecting terms irrelevant for the detection, the GLRT detection rule reads

bGLRT = argmaxb∈B

∥

∥bRT∥

∥

2

∥

∥b∥

∥

2 (3.49)

which, in case‖b‖2 = const. holds, is equivalent to the final detection rule of a truly noncoher-ent maximum-likelihood IR-UWB receiver in (3.16) assuminguniform weighting.

3.6.2 GLRT-Based Soft-Output Detection

Simply using the (pseudo-)probability density function (3.48) in the definition of the LLRs (cf.(3.24)), the reliability information for GLRT-based detection [SF10b] is given as

LLRGLRTk =

1

2σ2n

∥

∥b∥

∥

2 ·

maxb∈B

M−1k

(b)=0

∥

∥bRT∥

∥

2 − maxb∈B

M−1k

(b)=1

∥

∥bRT∥

∥

2

. (3.50)

This soft output is equivalent to (3.28) up to the scaling factor in front of the square brackets.In particular, compared to (3.28) the termNs(σ

2n)

2, reflecting the noise amplification due to the“noise×noise” products, is not represented. As already mentioned,depending on the employedchannel decoder, scaling of the soft output may however be neglected, if the decoding processis scale-invariant, i.e., especially in convolutional coded transmission.

3.7 Summary and Discussion

In this chapter, we have derived the optimum receiver for generic IR-UWB signals with un-known receive pulse shape in inter-symbol-interference-free transmission. We have shown, thatdifferent approaches lead to the same detection principle,which can be implemented eitherbased on energy detection or (and preferably) based on autocorrelation of the receive signal.Both approaches are equivalent. The latter is favorable from an implementation point of view,as it enables to separate processing of the analog receive signal from digital signal processingperforming the decision of the information symbols.

Moreover, weighting of the receive signal matched to the channel characteristics prior to detec-tion and computation of reliability information on the decided code symbols has been addressed.

We have discussed the implementation of autocorrelation-based detection and have developedan equivalent discrete-time model for IR-UWB transmissionfor this receiver type. Based onGaussian approximation of the correlation coefficients, anapproximate discrete-time modelfor the transmission has been derived, which enables analytical performance evaluation. Thismodel can further be utilized to efficiently perform numerical simulations of the IR-UWB trans-mission.

Our approach consistently includes the joint decision of multiple symbols based on observationwindows of several symbol durations. This generic framework may serve as a basis for the

3.7. Summary and Discussion 45

design of a large class of low-complexity energy-efficient detection schemes for various IR-UWB modulation schemes. It enables a flexible trade-off between receiver complexity anddetection performance, i.e., the simplest receiver types are based on an observation windowof a single symbol, whereas the most powerful ones operate onthe entire burst interval. Inthe subsequent chapters, this detection principle is applied in the design and optimization ofreceivers for PPM and DTR IR-UWB communications.

46

47

4. Detection Schemes for PPM IR-UWB

In this chapter, the generic concept for noncoherent detection of IR-UWB presented in theprevious chapter is utilized to derive power-efficient noncoherent detection schemes for PPMIR-UWB, as introduced in Section 2.2.1. In particular, we review symbol-wise energy detec-tion in Section 4.1 [SK03, DMR07, CM06] and advanced noncoherent detection based on theprinciple of multiple-symbol detection in Section 4.2 [TY05, SF10a] (for a general overview ofthe discussed detection schemes, we refer to Figure 1.1 in the introduction). For both schemes,accurate analytical error-rate expressions are derived and verified by means of numerical sim-ulations. Optimum setting of the correlation interval is addressed and compared to a weightedenergy detector, that weights the receive signal prior to computation of the received energy. Abrief summary and a discussion of the presented approach concludes this chapter in Section 4.3.

4.1 Energy Detection

Symbol-wise energy detection (ED) of PPM IR-UWB has alreadybeen discussed as an illus-trative application example of the optimum generic IR-UWB receiver in Chapter 3 (cf. Ex-ample 3.1). Recall that energy detection of PPM is based on anobservation window of onehyper-symbol, i.e.,N = D. For each of the symbol intervals within this hyper-symbol theenergy is computed according to

zd,d = ‖rdΣ‖2 for d = 0, . . . , D − 1 . (4.1)

The maximum-likelihood estimate for the transmit symbol isgiven as the one with largest cap-tured energy, i.e.,

bED = edED with dED = argmaxd∈{0,..., D−1}

zd,d . (4.2)

The corresponding code symbols are given as the binary labelof this transmit symbol, i.e., incase of natural labeling the position indexdED in binary notation.

48 4. Detection Schemes for PPM IR-UWB

Following (3.25), soft output in terms of LLRs for thelog2(D) code symbols per PPM symbolis given as

LLREDm = ln

∑

d

M−1m (ed)=0

e− 1

2σ2n(∑D−1

i=0 zi,i−zd,d) − ln∑

d

M−1m (ed)=1

e− 1

2σ2n(∑D−1

i=0 zi,i−zd,d)

= ln∑

d

M−1m (ed)=0

e− 1

2σ2n

∑D−1i=0, i6=d

zi,i − ln∑

d

M−1m (ed)=1

e− 1

2σ2n

∑D−1i=0, i6=d

zi,i (4.3)

for m = 0, . . . , log2(D) − 1. For binary PPM, there is only a single code symbol per PPMsymbol. The above equation simplifies to

LLRED =1

2σ2n

· [z0,0 − z1,1] . (4.4)

Note that in particular for uniform weighting over the correlation intervalTi, a constant scalingfactor is hidden in the computation of the energy of the symbol intervals (cf. Section 3.3).


The autocorrelation device required for energy detection of PPM reduces to a simple concate-nation of a square-law device (e.g., implemented as a Gilbert cell [LTS11]) and a (weighted)accumulate-and-dump device, as shown in Figure 4.1. Computation of the cross correlations ofthe PPM symbol intervals are not required for detection. Thedecision device is easily imple-mented using a comparator.

z−1

hRX(t)∫

(T )·dtr(t)

kT

n0(t)

cED

z 0,0≷

z 1,1

zk,k

w2Σ(t)

Figure 4.1: Block diagram of energy detection of PPM IR-UWB.

Noting that the transmitter of PPM IR-UWB mainly consists ofa triggered pulse generator,PPM IR-UWB in combination with energy detection thus represents an IR-UWB communi-cation system with very low hardware implementation complexity. This is also supportedby several available hardware demonstrators of energy detection for PPM IR-UWB, cf., e.g.,[KO11, LTS11] to pick only two examples.

4.1.2 Error-Rate Analysis

We review the performance of energy detection of PPM IR-UWB by deriving an analyticalerror-rate expression (following [DMR07, DB04, CM06]), which is then verified for realisticUWB channel models by means of numerical simulations.

Based on the results of Section 3.5, depending on the presence or absence of the information-bearing pulse in the interval, the captured energy in the symbol intervals can be modeled as a

4.1. Energy Detection 49

non-central and centralχ2 distribution, respectively, withNi degrees of freedom (cf. (3.41)).For brevity and clarity, the exposition is restricted to binary PPM, i.e.,D = 2; the extensionto higher-order PPM is straight-forward. Moreover, we employ uniform weighting within thecorrelation intervalTi (cf. (3.19)). The key parameter characterizing the performance is thus thetime-bandwidth productNi = fsTi.

In this case, due to symmetry of the orthogonal modulation, the error rate is given as

SERED = Pr{z0,0 < z1,1|c = 0} (4.5)

wherec is the transmitted code symbol corresponding to the observation window. Since, forc = 0, b = [1, 0], the receive pulse is present in the first symbol interval; inthe second intervalthere is noise only. Hence, the symbol error rate corresponds to the probability, that a realizationof a centralχ2-distributed random variable (energy of noise only) exceeds a realization of a non-centralχ2-distributed random variable (energy of pulse and noise), which is easily computedvia numerical integration.

A simplified error-rate expression is obtained by employinga Gaussian approximation of thetwo involvedχ2-distributed random variables, i.e., the equivalent noiseon the correlation coef-ficients is assumed to be Gaussian (cf. (3.42)). Since the decision variables for the two symbolintervals are independent, the average error rate can be approximated as

SERED ≈ Pr{N (Ep +Niσ2n, 4Epσ

2n + 2Ni(σ

2n)

2) < N (Niσ2n, 2Ni(σ

2n)

2)} (4.6)

= Pr{N (Ep, 4Epσ2n + 4Ni(σ

2n)

2) < 0} (4.7)

= Q

(√

E2p

4Epσ2n + 4Ni(σ2

n)2

)

. (4.8)

For comparison recall that in the presence of perfect channel state information, the error rate ofcoherent detection of PPM is given as [PS08]

SERPPM,CSI = Q

(√

1

2σ2n

)

. (4.9)

Even for high signal-to-noise ratio (σ2n → 0) and capturing of the entire receive pulse energy

(Ep = 1), truly noncoherent energy detection of PPM thus induces a loss of at least3 dBcompared to detection with channel state information.

Figure 4.2 depicts the error rate of energy detection of binary PPM IR-UWB averaged over alarge number of channel realizations for two channel modelsof [MCC+06] compared to the ex-act and Gaussian-approximated analytical error-rate expressions. The symbol duration has beenset toT = 150 ns to preclude inter-symbol interference. We assume uniform weighting withinthe correlation intervalTi = T = 150 ns (right) and for optimizedTi (left, setting according toFigure 4.3, i.e.,Ti = 30 ns for line-of-sight and50 ns for non-line-of-sight conditions), corre-sponding to a time-bandwidth product ofNi = 1500 in the first case. For reference, detectionwith perfect channel state information is included.


10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1

indoor non-line-of-sight

10 log10(Ec/N0) [dB]

SER

10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1

office line-of-sight


SER

Figure 4.2: SER of ED of binary PPM IR-UWB vs. Ec/N0 in dB for CM 2 (left) and 3 (right) for uniformweighting within the correlation intervals Ti. Solid-black: simulation results, solid-gray: exact analytical

results, dashed-gray: approximate analytical results, left-most solid-black: detection with CSI. Left group of

plots: optimized setting of Ti according to Figure 4.3, right: Ti = T = 150 ns.

The analytical results match the numerical simulations; the Gaussian approximation slightlyoverestimates the performance for the longer correlation interval, and underestimates it for theshorter correlation interval. The predicted significant loss compared to detection with perfectchannel state information is verified. Since the same symboldurationT is applied for bothchannel models, forTi = T there is almost no difference between the two channel models.

4.1.3 Influence of Weighting Coefficients

We now address different settings for the weighting coefficientswm prior to correlation (cf.Section 3.2.3). This optimization step is only possible, ifaccurate synchronization has beenachieved, i.e., the timing offset between transmitter and receiver is perfectly known, and someknowledge on the channel characteristics is available at the receiver.

First, we consider uniform weighting over a correlation interval of durationTi. Figure 4.3depicts theSER of energy detection of PPM as a function of the correlation interval Ti (orequivalently as a function of the time-bandwidth productNi = fsTi, where, in our setup,fs =10GHz) for different signal-to-noise ratiosEc/N0. The numerical results are compared to theanalyticalSER expressions.

The results emphasize that the correlation interval shouldbe carefully selected for energy detec-tion of PPM; clearly, the minimum is more pronounced in the line-of-sight channel model. Theoptimum correlation interval roughly equals the average delay spread of the underlying channelmodel (cf. Table 2.1); with increasingEc/N0 it shifts to larger values. The analytical resultsare accurate for largeTi, but tend to underestimate the optimum correlation interval in case of

4.1. Energy Detection 51

0 25 50 75 100 125 15010

−6

10−5

10−4

10−3

10−2

10−1


Ti [ns]

SER

0 25 50 75 100 125 15010

−6

10−5

10−4

10−3

10−2

10−1


Ti [ns]

SER

Figure 4.3: SER of ED of binary PPM IR-UWB vs. correlation interval Ti of uniform weighting for CM 2 (left)and 3 (right) for different 10 log10(Ec/N0) from 16 dB to 21 dB (top-to-bottom, steps of 1 dB). Solid-black:

simulation results, solid-gray: exact analytical results, dashed-gray: approximate analytical results, markers

indicate optimum setting for correlation interval.

the non-line-of-sight channel. This is due to the fact, thatthe analytical results do not take intoaccount the random and clustered structure of the power-delay profile and the variations in thecaptured receive pulse energy (cf. Figure 3.4).

The problem of suitably adopting the correlation interval to the channel characteristics at handcan be avoided if the power-decay time constantΓCM of the power-delay profile is known at thereceiver. Since the power decay with respect to arrival timeof the considered channel modelsis dominated by this exponential term, weighting of the signals prior to integration according to(3.17) is applicable. With slight simplifications, the weights can also be computed according to(3.18). For the same scenario as above, Figure 4.4 depicts the resulting error-rate performancecompared to uniform weighting with optimum correlation interval (cf. Figure 4.3) andTi = T .Both exponential weightings outperform the best setting ofuniform weighting (at least in awide regime of signal-to-noise ratio). The simplified computation of the weighting coefficientsachieves even better performance compared to the exact computation.

The results support that, despite the random and clustered structure, essentially, the power-delayprofile of the UWB channel models of [MCC+06] decays exponentially with respect to arrivaltime, especially in case of a dominant line-of-sight component. For a pragmatic optimizationof the receiver parameters it is thus sufficient to neglect the clustered structure with specificintra-cluster decay factors.


10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1



SER

10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1



SER

Figure 4.4: SER of weighted ED of binary PPM IR-UWB vs. Ec/N0 in dB for CM 2 (left) and 3 (right). Solid-black: exponential weighting according to (3.17), dashed-black: simplified exponential weighting (3.18),

solid-gray: uniform weighting with optimized setting of Ti and Ti = T , left-most solid-black: detection with

CSI.

4.2 Multiple-Symbol Detection

The detection performance can be improved by considering larger observation windowsN =

L·D, whereL ≥ 1 denotes the number of PPM-symbols within the observation window [SF10a,TY05], and jointly deciding the transmit symbols within this window. The maximum-likelihooddecision rule is given by (3.16). Equivalently, based on theoutput of an autocorrelation receiverfront-end, we have

bMSD = argmaxb∈BL

PPM

bZbT. (4.10)

The receiver then proceeds to the next block of symbols by shifting the observation window byN symbol intervals. For brevity, at this point, a detailed description of the block-wise processingis omitted; it is described more precisely in Section 5.3.1 for the related concept of block-wisedetection of DTR IR-UWB (for multiple-symbol detection of PPM a block shift ofS = N issufficient, i.e., no overlap is required; larger overlaps are not considered).

The performance improvement of multiple-symbol detectionof PPM over symbol-wise en-ergy detection can be explained by looking at the utilized correlation coefficients of the matrixZ, which affect the decision of each PPM symbol. This is illustrated in Figure 4.5. Clearly,multiple-symbol detection utilizes significantly more correlation coefficients, which, however,leads to a more reliable decision.

An illustrative example for multiple-symbol detection of binary PPM has already been given inExample 3.1 based on the viewpoint of extended energy detection. Here, Example 4.1 shows

4.2. Multiple-Symbol Detection 53

ZMSD =

ZED =

Figure 4.5: Illustration how block-wise MSD (with L = 3) and symbol-wise ED operate on the correlation

matrix Z. The used correlation coefficients are highlighted, the ones which influence the decision of thesecond PPM symbol are marked with a dashed line. Left: MSD, right: ED. Nburst = 6, D = 2.

the equivalent processing steps of autocorrelation-basedmultiple-symbol detection of binaryPPM.

Example 4.1: Autocorrelation-Based Multiple-Symbol Detection of Binary PPM

Consider multiple-symbol detection of binary PPM (D = 2) with a window length of L = 2(N = 2 ·D = 4) and the correlation matrix

Z =

0.6 x 0.3 1.1x 0.7 0.2 0.50.3 0.2 0.1 x

1.1 0.5 x 0.9

.

The correlation coefficients marked with “x” are not required for the detection and do not

have to be computed in the autocorrelation device.

The decision metric for the transmit signal candidate b = [1, 0, 1, 0] computes to

[1, 0, 1, 0]Z[1, 0, 1, 0]T = [0.6 + 0.3, x, 0.3 + 0.1, x][1, 0, 1, 0]T

= 0.6 + 0.3 + 0.3 + 0.1 = 1.3 .

Similarly, for the other candidates one obtains

[1, 0, 0, 1]Z[1, 0, 0, 1]T = 0.6 + 1.1 + 1.1 + 0.9 = 3.7

[0, 1, 1, 0]Z[0, 1, 1, 0]T = 0.7 + 0.2 + 0.2 + 0.1 = 1.2

[0, 1, 0, 1]Z[0, 1, 0, 1]T = 0.7 + 0.5 + 0.5 + 0.9 = 2.6 .

The maximum-likelihood estimate is given by the sequence with largest decision metric

(here 3.7) which is given as bMSD = [1, 0, 0, 1].

For this example, symbol-wise energy detection (i.e., maximum-likelihood based on L = 1)

delivers a different estimate given by bED = [0, 1, 0, 1] (with corresponding MSD decision

metric 2.6 < 3.7), since for the decision only the diagonal of Z is considered.



The correlation coefficients required for detection can be computed using an autocorrelationdevice, as depicted in Figure 3.2. However, already the joint decision of two binary PPMsymbols, i.e., an observation window ofL = 2, requires a maximum delay in the autocorrelationdevice spanning over four PPM intervals (4T ), which is twice as large as the required delay forjoint decision of two DTR IR-UWB symbols (cf. Chapter 5).

An efficient method to solve the high-dimensional multiple-symbol detection problem (4.10) isnot available in literature; methods reducing the computational complexity, similar to MSDD ofDTR IR-UWB, e.g., based on the sphere decoder algorithm, seem not applicable. Hence, onehas to resort to a brute-force computation of the decision metrics of all DL possible transmitsequence hypotheses. Clearly, for largeL (and/orD) the computational complexity increasesdramatically, and prohibits implementation in low-power-consuming receivers.


Based on nearest-neighbor arguments and the Gaussian approximation discussed in Section 3.5,the error rate of multiple-symbol detection of binary PPM IR-UWB can easily be estimated.Again we assume uniform weighting within the correlation intervalTi.

To this end, note that the decision metric (energy of the combined receive signal) of the actualtransmitted symbols calculates to

∥

∥bRT∥

∥

2=

∥

∥

∥

∥

∥

Lp+L−1∑

l=0

nl

∥

∥

∥

∥

∥

2

(4.11)

= L2 ‖p‖2 + 2LpL−1∑

l=0

nTl +

∥

∥

∥

∥

∥

L−1∑

l=0

nl

∥

∥

∥

∥

∥

2

(4.12)

For simplicity, in the above equation, the respective noisecontributions are simply enumeratedby l = 0, . . . , L − 1. The most-likely error event corresponds to the case that one of L PPMsymbol is decided wrongly, i.e., if the energy∥

∥

∥

∥

∥

(L− 1)p+

L−2∑

l=0

nl + n′

∥

∥

∥

∥

∥

2

= (L− 1)2 ‖p‖2 +∥

∥

∥

∥

∥

L−2∑

l=0

nl

∥

∥

∥

∥

∥

2

+ ‖n′‖2

+ 2(L− 1)pL−2∑

l=0

nTl + 2(L− 1)n′pT + 2n′

L−2∑

l=0

nTl (4.13)

exceeds the above decision metric∥

∥bRT∥

∥

2of the transmitted symbols (n′ denotes noise from an

interval different to those ofnl, l = 0, . . . , L− 1). In the above equation, instead of the energyof L “pulse+noise” intervals as in (4.11), only the energy ofL − 1 “pulse+noise” intervalsand one “noise-only” interval is captured. Clearly, the noise in the “noise-only” interval (n′) isdifferent from that of the “pulse+noise” intervals.

Using a Gaussian approximation and simple algebraic reformulations (similar to the case of en-ergy detection of binary PPM), it can be shown that the error rate of multiple-symbol detection

4.3. Summary and Discussion 55

is well-approximated by the probability that a Gaussian random variable takes a value less thanzero. The mean of this random variable is given as(L2−(L−1)2)Ep = (2L−1)Ep. Its varianceis given as the sum of the variance of various constellationsof “pulse×noise”, “noise×noise”of different symbol intervals, and “noise×noise” of the same symbol interval. The number ofcontributions of each case is given in Table 4.1. Since each contribution originates from differ-ent symbol intervals, the sum of the individual variances yields the desired variance. Of course,for L = 1, i.e., symbol-wise energy detection, the result is equivalent to (4.8).

Table 4.1: Number of contributions of the individual variances to the overall variance of the random variable

used for calculating the error rate of multiple-symbol detection of binary PPM IR-UWB.

variance number of contributions

“pulse×noise” Epσ2n 4L(2L− 1)

“noise×noise” of different intervals Ni(σ2n)

2 8(L− 1)of the same interval 2Ni(σ

2n)

2 2

Figure 4.6 depicts the error rate of multiple-symbol detection for various block sizesL = 2,3, 4, 5, 7, and10 compared to symbol-wise energy detection (L = 1), and detection withperfect channel estimation. The analytical results based on the Gaussian approximation wellmatch the simulation results. Due to the restriction to nearest-neighbor error events, we observeminor deviations at low SNR. Already performing a joint decision ofL = 2 adjacent PPMsymbols leads to a gain of approximately2 dB. With increasing block size, the performanceapproaches detection with perfect channel state information; for comparison, analytical resultsfor L = 50 and500 are also shown (of course, realization and even numerical simulations ofmultiple-symbol detection using such a large block size is impossible and thus not included inFigure 4.6).

Optimization of the performance of multiple-symbol detection by introducing suitable weight-ing of the receive signal prior to correlation is possible, as well, and leads to additional gains asin the case of symbol-wise energy detection. For brevity, this is not discussed here.

4.3 Summary and Discussion

In this chapter, we have reviewed and analyzed the performance of noncoherent detectionschemes for PPM IR-UWB, in particular, symbol-wise energy detection and multiple-symboldetection. The optimum setting of the correlation intervalas well as the performance of aweighted computation of the correlations in energy detection have been addressed. The expo-sition has focused on binary PPM; the results can be extendedto higher-order PPM, cf., e.g.,[SF10a]. Higher-order PPM, however, has the drawback of further reduced spectral efficiency(for D > 4) and increased peak-to-average power ratio.

In combination with symbol-wise energy detection, binary PPM IR-UWB is an attractive trans-mission scheme, enabling low-complexity low-power-consuming transmitter as well as receiverdesign. This low complexity demand, however, leads to significant loss in performance com-pared to detection with perfect channel state information,which can be alleviated, on the one


8 10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1



SER

Figure 4.6: SER of MSD of binary PPM IR-UWB vs. Ec/N0 in dB for CM 2 of [MFP03] for different windowlength L = 1 (ED) 2, 3, 4, 5, 7, 10 (right to left), and 50, 100 (only analytical results). Solid-black: simulation

results, solid-gray: approximate analytical results, dashed-black: detection with CSI. fsTi = 800.

hand by an optimized weighting (if channel characteristicsare known at the receiver) prior toenergy detection, and on the other hand by employing multiple-symbol detection. Whereasin multiple-symbol detection the required coefficients canbe obtained relatively efficientlyby an autocorrelation device, a computational-efficient algorithm solving the resulting high-dimensional search problem is not available.

Compared to noncoherent detection schemes for DTR IR-UWB, as considered in the next chap-ter, for the same channel characteristics, i.e., equal worst-case delay of the channel impulseresponse, binary PPM IR-UWB suffers a loss in data rate of a factor of two. Moreover, forDTR IR-UWB, more efficient advanced noncoherent detection schemes with lower hardwarecomplexity, and significantly reduced computational complexity compared to multiple-symboldetection of PPM can be derived. Taking this into consideration, multiple-symbol detectionof PPM IR-UWB is not included in subsequent investigations of this thesis. Due to its verylow complexity, symbol-wise energy detection, however, isconsidered later-on; in particular,design rules for coded IR-UWB transmission using PPM in combination with energy detectionare derived in Chapter 6.

57

5. Detection Schemes for DTR IR-UWB

Based on the generic concept of optimum truly-noncoherent detection of IR-UWB, a variety ofdetection schemes for DTR IR-UWB is derived in this chapter.For the same autocorrelationfront-end, the proposed detection schemes offer a flexible trade-off between desired perfor-mance (required signal-to-noise ratio for a target error rate) and computational complexity.

In this chapter, we restrict this evaluation to uncoded IR-UWB transmission; the performanceof these detection schemes in coded IR-UWB transmission (inparticular, of their respectivesoft-output variants) is discussed in Chapter 6 along with design rules for coded IR-UWB trans-mission.

After a brief motivation and classification of the detectionschemes in Section 5.1, we discusssymbol-wise differential detection in Section 5.2. In Sections 5.3 and 5.4, advanced detectionschemes for DTR IR-UWB are presented. These IR-UWB receivers are compared to relatedconcepts in Section 5.5. Final conclusions are drawn in Section 5.6.

5.1 Motivation and Classification

Recall that so-called ideal noncoherent sequence estimation (INSE) would jointly decide for themost-likely sequence ofNburst symbols, taking into account the receive signal in the entire burstinterval0 ≤ t ≤ NburstT [GQ06, SF11c]. The decision rule of INSE is obtained from (3.16) bysettingN = Nburst and noting that the first symbol is the fixed reference, i.e.,

bINSE = argmaxb∈B

NburstDTR

b0=1

bZbT. (5.1)

The optimum sequence ofNburst − 1 information and code symbols is obtained via differentialdecoding (aINSE

k = bINSEk bINSE

k−1 ) and demapping.

The diagonal elementszk,k of Z represent the energy of thekth receive symbol. Sinceb2k = 1

holds for DTR IR-UWB, these correlation coefficients do not influence the decision. Thus,

58 5. Detection Schemes for DTR IR-UWB

the first branch of the autocorrelation device depicted in Figure 3.2 can be omitted. Still, anautocorrelation device with a total ofNburst − 1 branches is required.

Further exploiting the symmetryzk,i = zi,k, we have [SF11c]

bINSE = argmaxb∈B

NburstDTR

b0=1

Nburst−1∑

k=1

(

bk

k−1∑

l=0

blzl,k

)

. (5.2)

It is evident that already for moderate burst lengthsNburst INSE becomes infeasible for threemajor reasons:

� in order to compute the required correlation coefficients, correlations of the receive signalover time delays of(Nburst − 1)T have to be performed, which, e.g., in an all-analogimplementation of the autocorrelation device, requires accurate delay lines over possiblyhundreds of symbols, making hardware implementation impossible (cf. Section 3.4.2),

� given these correlation coefficients, the computational complexity of finding the best se-quence, i.e., solving (5.2), is exponential inNburst, and thus intractable, since—at least atworst case—any algorithm must perform an exhaustive searchover all2Nburst−1 possiblesequences (note that in DTR, the first symbol is the fixed referenceb0 = 1), and

� a final decision even for the first symbols of the burst is available only after the last symbolhas been received, i.e., the structural delay of INSE equalsNburstT .

Hence, methods which are based on an autocorrelation devicewith onlyL≪ Nburst−1 branchesand reduced computational complexity—at best linear inL—are called for. Figure 5.1 depictsa DTR IR-UWB receiver employing thisL-branch autocorrelation device (for simplicity, anall-analog implementation is considered and uniform weighting within Ti prior to correlation isassumed).

The least complex DTR receiver is based onL = 1. In this case, the differential encoding isreverted directly based on the analog (or sufficiently sampled) receive signal, i.e., symbol-wiseautocorrelation-based differential detection is performed. Thus, a single information symbol isdecided in each step (L = 1) based on an observation window length ofN = L+ 1 = 2.

Based on the output of an(L > 1)-branch autocorrelation device, larger observation win-dows (N > 2) are possible, which enable the application of more advanced detection schemes.Naturally, with increasingL performance improvement is expected at the cost of increasedcomplexity. Essentially, two different concepts to approximate INSE can be distinguished[SF11c, LT06b, LT08] (for a general overview of the discussed detection schemes, we againrefer to Figure 1.1 in the introduction):

On the one hand, there are block-based detection schemes, which partition the stream of re-ceive symbols into (possibly overlapping) blocks ofN receive symbols, and process each blockindividually. Usually, the block size is matched to the number of branches of the autocorrela-tion device, but larger block sizes are also possible (so-called virtually increased block size).

5.1. Motivation and Classification 59

zk−(N−1),k

kT

kT

kT

T

r(t)hRX(t)

n0(t)

LT

2T

b·∫

(Ti)·dt

∫

(Ti)·dt

∫

(Ti)·dt

zk−1,k

zk−2,k

autocorrelation device

Dete

ctio

n

Figure 5.1: Block diagram of DTR IR-UWB receiver with L-branch autocorrelation device.

The most prominent example of this kind is multiple-symbol differential detection (MSDD)[GQ06, LT08], but block-wise decision-feedback differential detection (DFDD) [SF11c] andvariants of MSDD and DFDD, as well as extensions, e.g., basedon the concept of multiple-observations-combining [SF12b], belong to this class, as well. In general these detectionschemes share a varying (usually linearly increasing) memory length, i.e., a non-constant num-ber of adjacent symbols affect the decision of each symbol.

On the other hand, there are detection schemes, which perform sequence estimation on the entireburst of receive symbols. Examples for such schemes are reduced-state sequence estimation(RSSE) based on the Viterbi algorithm [LT06b] and sliding-window DFDD [SF11c]. Here,each symbol is decided based on its relation to a constant number of adjacent symbols, i.e.,the memory length is constant. A third example of this kind isobtained, if the block size ofblock-wise detection schemes is no longer matched to the autocorrelation receiver, but set tothe length of the burst (virtual block size ofNburst − 1).

Of course, the maximum memory length of both concepts is limited by the maximum span overwhich correlations are recorded in the autocorrelation device, i.e., byL symbols, since infor-mation on the relation to symbols out of this interval is not available. For sake of completeness,note that symbol-wise differential detection (with a blocksize and constant memory length ofL = 1) and INSE (withL = Nburst − 1) are the most simple and most complex representativesof both classes, respectively.

The different concepts are contrasted in Figure 5.2 for the case ofL = 2 (burst lengthNburst =

7). The correlation coefficients used for the particular scheme of the entireNburst × Nburst

correlation matrixZ, as it would be required by INSE, are highlighted. In each case, thecorrelation coefficients, which influence the decision of the fourth transmit symbol are marked.Clearly, in INSE all correlation coefficients are utilized for the joint decision of all symbolswithin the burst. However, note that the diagonal elements do not influence the decision (asopposed to detection schemes for PPM, cf. Figure 4.5).


Z INSE =

ZDD =

ZMSDD =

ZRSSE =

Figure 5.2: Illustration how different DTR detection schemes operate on the correlation matrix Z. The

utilized correlation coefficients are highlighted, the ones which influence the decision of the fourth transmitsymbol are marked with a dashed line. Top-left: INSE, top-right: DD, bottom-left: block-wise MSDD,

bottom-right: RSSE. Nburst = 7, L = 2 for MSDD and RSSE.

Symbol-wise differential detection decides each symbol based on the relation to its predecessoronly, thus exploits only the correlation coefficientszk−1,k (= zk,k−1, due to symmetry).

Next to INSE and DD, MSDD and RSSE represent block-wise detection and sequence estima-tion, respectively. The block-wise processing of MSDD is clearly visible. Note that the fifthsymbol would be decided based on the same correlation coefficients as of the fourth one. Thisis different in RSSE, where for the decision of the fifth symbol the block of utilized correlationcoefficients shifts one symbol down to the right in a sliding-window fashion.

Clearly, for short burst length,Nburst = L+ 1, INSE, MSDD, and RSSE coincide. In this case,the implementation should be based on MSDD, as various complexity reduction techniques leadto superior performance over its competitors.

Based on this illustration, a first statement on the relativeperformance of the different schemescan be deduced already at this point. The best scheme approximating INSE is RSSE, followedby MSDD and DD. For low-complexity energy-efficient IR-UWB receivers, however, MSDDand RSSE have prohibitively large computational complexity. Nevertheless, they may serve asa reference and—more importantly—as an excellent startingpoint for the derivation of low-complexity detection schemes, in particular based the principle of DFDD. In this chapter, suchlow-complexity schemes are presented in more detail and a comprehensive comparison with

5.1. Motivation and Classification 61

respect to error-rate performance and computational complexity is conducted.

It will be shown that block-wise detection schemes offer distinct advantages compared to theconstant-memory schemes. For a given autocorrelation device, they enable an excellent trade-off between performance and computational complexity. Following this reasoning, in this the-sis, we focus on block-wise detection schemes, in particular on DFDD and MSDD as well astheir soft-output variants.


5.2 Differential Detection

In symbol-wise differential detection a single information/code symbol is decided in each timestep. Due to the memory in the mapping from code to transmit symbols introduced through thedifferential encoding, the minimum observation window length applicable for DTR IR-UWB isN = 2. Hence, an autocorrelation device withL = 1 is sufficient.

From (3.16), the optimum symbol-wise detection rule is given as

[bDDk−1, b

DDk ] = argmax

b∈B2DTR

b0=1

(

2b0b1zk−1,k + zk−1,k−1 + zk,k)

. (5.3)

Sincebkbk−1 = ak holds due to the binary alphabet and the differential encoding of (2.11), andthe last two summands are not influenced by the hypothesis, this is equivalent to

aDDk = argmax

a∈{−1,+1}

azk−1,k = sign(zk−1,k) . (5.4)

Using (3.25), reliability information for this estimate isgiven as

LLRDDk =

1

2σ2n

· [zk−1,k−1 + 2zk−1,k + zk,k − (zk−1,k−1 − 2zk−1,k + zk,k)] (5.5)

=2

σ2n

· zk−1,k (5.6)

which supports the intuitive result that the reliability ofthe estimated information symbol, ob-tained from evaluating the sign of the correlation coefficient, is directly proportional to themagnitude of the correlation coefficient.

In case of uniform weighting over the symbol interval of length T , a constant scaling can befactored out (cf. (3.28)), yielding

LLRDDk =

8

2σ2n +Ns(σ2

n)2· rk−1r

Tk (5.7)

since‖b‖2 = 2, and similar for uniform weighting within the correlation intervalTi. The de-nominator thus exactly resembles the noise amplification due to the “pulse×noise” and “noise-×noise” products caused by the correlation process (compareto (3.39)).

The receive signal processing for autocorrelation-based symbol-wise differential detection issummarized in Figure 5.3. The block diagram is based on an all-analog implementation of theautocorrelation device.


Similar to energy detection of PPM IR-UWB, the error rate of differential detection can beestimated by employing a Gaussian approximation (similar to [PJ07, CS02, QWD07] and basedon the results of Section 3.5). Recall, that for uniform weighting within the correlation intervalTi, the correlation coefficientszk−1,k can be modeled as Gaussian random variables according

5.2. Differential Detection 63

scalingkT

T

r(t)hRX(t)

n0(t)

LLRDDk

aDDksign(·)

∫

(Ti)·dt zk−1,k

Figure 5.3: Block diagram of DTR IR-UWB receiver for autocorrelation-based differential detection (uniform

weighting within Ti).

to (3.42), i.e., the mean is given by the captured receive pulse energyEp (scaled byak) and thevariance as the noise varianceσ2

η of the equivalent noise on the correlation coefficients.

Since an estimate for the information symbol is obtained from the sign of the correlation coef-ficient, the symbol error rate is directly given as

SERDD = Pr{zk−1,k < 0|ak = 1} = Q

(√

E2p

σ2η

)

= Q

(√

E2p

2Epσ2n +Ni(σ2

n)2

)

. (5.8)

In comparison to (semi-noncoherent or so-called differentially coherent) detection of differen-tial binary phase-shift keying with perfect channel knowledge, where the symbol error rate iswell approximated by [FH06a]

SERDTR,CSI = 2 ·Q(√

1

σ2n

)

(5.9)

autocorrelation-based differential detection suffers a loss of at least3 dB even for perfect cap-turing of the receive pulse energy (Ep = 1) and large signal-to-noise ratio (σ2

n → 0). Comparedto energy detection of PPM IR-UWB (cf. (4.8)), we observe a gain of exactly3 dB.

The error rate of differential detection of DTR IR-UWB is depicted in Figure 5.4 for channelmodels 2 and 3 of [MCC+06] and two settings of the correlation interval, i.e.,Ti = T and theoptimizedTi according to Figure 5.5 (Ti = 30 ns for line-of-sight and50 ns for non-line-of-sight conditions). The symbol duration has been set toT = 150 ns to preclude inter-symbolinterference. The analytical error rates agree with the numerical results. The relations to co-herent detection and to energy detection of binary PPM IR-UWB are also verified. We observea tremendous loss of truly-noncoherent autocorrelation-based differential detection to coherentdetection, which amounts to almost8 dB.

5.2.2 Influence of Weighting Coefficients

Similar to energy detection of PPM IR-UWB, different weightings can be employed in thecorrelation process in order to reduce the amount of noise and increase the amount of usefulsignal in the correlation coefficients. Again, we emphasizethat this step is only applicable, ifsufficient synchronization between transmitter and receiver has been achieved, i.e., the timingoffset is perfectly known, and the receiver has some knowledge on the channel characteristics.

The most simple way for optimized weighting is to compute thecorrelation coefficients notover the entire symbol duration, but over a shorter intervalof durationTi < T , i.e., uniform


8 10 12 14 16 18 20 2210

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1



SER

8 10 12 14 16 18 20 2210

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1



SER

Figure 5.4: SER of DD of DTR IR-UWB vs. Ec/N0 in dB for CM 2 (left) and 3 (right) of [MCC+06] foruniform weighting within the correlation interval Ti. Solid-black: simulation results, solid-gray: analytical

results, dashed-gray: energy detection of binary PPM IR-UWB, left-most solid-black: detection with CSI. Left

(within each subplot): optimized setting of Ti according to Figure 5.5, right: Ti = T = 150 ns. Note that therange of the y-axis is increased compared to the other figures.

weighting withinTi. Optimum setting of the correlation interval is an important optimizationtask for power-efficient differential detection of DTR IR-UWB.

The influence of the correlation interval is studied in Figure 5.5. The conclusions are the sameas for energy detection of PPM (cf. Figure 4.3), i.e., a distinct optimum is observed forTi inthe order of the average channel delay spread. The simplifications conducted for the analyticalerror rate (the random clustered structure is neglected), however, lead to some mismatch com-pared to the numerical results. Especially for the non-line-of-sight model the analytical resultssuggest significantly lower correlation intervals. For both models, the behavior at very low cor-relation intervals significantly differs from the numerical results, which is due to the fact thatthe variations in captured signal energy are not covered.

In order to avoid the problem of setting the correlation interval, non-uniform weighting of thesignals prior to correlation can be employed (cf. Section 3.2.3). Approximating the power-delayprofile of the underlying multipath channel as an exponential decay parameterized by the powerdecay time constantΓCM (cf. Table 2.1), the weighting coefficients are given by (3.17), and, withslight simplifications, by (3.18). From Figure 5.6, we conclude that both methods show similarperformance improvement compared to their application in energy detection of PPM IR-UWBand, especially for the line-of-sight channel model, outperform the best setting of Figure 5.5. Ofcourse, this optimization step requires rather detailed knowledge of the channel characteristics.

5.2. Differential Detection 65

0 25 50 75 100 125 15010

−6

10−5

10−4

10−3

10−2

10−1


Ti [ns]

SER

0 25 50 75 100 125 15010

−6

10−5

10−4

10−3

10−2

10−1


Ti [ns]

SER

Figure 5.5: SER of DD of DTR IR-UWB vs. correlation interval Ti for CM 2 (left) and 3 (right) of [MCC+06] for

different 10 log10(Ec/N0) from 13 dB to 18 dB (top-to-bottom, steps of 1 dB). Solid-black: simulation results,solid-gray: analytical results, markers indicate optimum setting for correlation interval.

8 10 12 14 16 18 20 2210

−6

10−5

10−4

10−3

10−2

10−1



SER

8 10 12 14 16 18 20 2210

−6

10−5

10−4

10−3

10−2

10−1



SER

Figure 5.6: SER of weighted DD of DTR IR-UWB vs. Ec/N0 in dB for CM 2 (left) and 3 (right) of [MCC+06].

Solid-black: exponential weighting according to (3.17), dashed-black: simplified exponential weighting

(3.18), solid-gray: no weighting with optimized setting of Ti and Ti = T , left-most solid-black: detectionwith CSI.


5.3 Block-Wise Detection Schemes

The performance of symbol-wise autocorrelation-based differential detection shows significantloss compared to detection with perfect channel estimation. Detection schemes which decidemultiple symbols jointly, i.e., based on the principle of multiple-symbol differential detection(MSDD) [DS90, HF92, Mac92], have been recognized as an efficient tool to bridge this gapwithout performing explicit costly channel estimation [GQ06, LT08].

These detection schemes partition the stream of receive symbols into (possibly overlapping)blocks ofN receive symbols and decide theN − 1 information symbols within each blockjointly. Usually, the block size is matched to the number of branches of the applied autocorrela-tion device. Thus, in case anL-branch autocorrelation device is employed (cf. Figure 5.1), theblock size is set toN = L + 1 receive symbols. At first, we follow this common (but to someextend artificial) restriction and comment on the application of larger block sizes at the end ofthis section. This so-called virtually increased block size1 is closely related to algorithms forsequence estimation as dicussed in Section 5.4.

We review variants of block-wise detection schemes. In particular (block-wise-) optimumMSDD and the related concept of block-wise decision-feedback differential detection (bDFDD)are considered. These schemes are extended according to theprinciple of multiple-observationcombining [SF12b], where approaches presented for MSDD of DPSK [PL07, PHL09] are com-bined with concepts of information combining [LH06]. The resulting detection schemes enablelow-complexity power-efficient detection and are able to very efficiently compute reliabilityinformation of the estimates. To this end, we first illustrate the block-wise processing of thereceive symbols.

5.3.1 Block-Wise Processing

In block-wise detection schemes, the stream ofNburst receive symbols is partitioned into blocksof N symbols. In order to cope with the differential encoding of DTR IR-UWB and to employmultiple-observations combining, these blocks overlap byat least one symbol. This overlap isimplemented by shifting the block edges byS symbols, withS ≤ N − 1. The symbol indicesof thenth block are thus given by

[nS, nS + 1, . . . , nS +N − 1] . (5.10)

Due to the differential encoding, each block ofN receive symbols representsL = N − 1

information/code symbols, the indices of which are given as

[nS + 1, nS + 2, . . . , nS + L] . (5.11)

With respect to blocks of information symbols, the block overlap is given byL−S informationsymbols.

1Of course, the opposite direction of choosing the block sizes smaller than the number of branches of theautocorrelation device hardly is trivial.

5.3. Block-Wise Detection Schemes 67

S = L

info symbols:

receive symbols:

N = L+ 1

L

receive symbols:

N = L+ 1

S = 1

L

info symbols:

Figure 5.7: Illustration of block-wise processing of the receive symbol stream for a block size of N = 3,L = N − 1 = 2, and the two block shift S = L = 2 (top) and S = 1 (bottom).

This block-wise processing is illustrated forN = 3, L = 2, and the two different block shiftsS = L andS = 1 in Figure 5.7. For a block shift ofS = L, processing of each blockleads to estimates for a new set of information symbols. Thisis different forS = 1. In this casemultiple,L−S (here two), estimates are available for each information symbol. In Section 5.3.6we propose techniques to suitably utilize these multiple observations with the aim of improvingdetection performance and/or computing reliability information.

Based on these definitions, subsequently, we consider an arbitrary block ofN receive, respec-tively L = N −1 information symbols, similar to Section 3.2; all indices have to be understoodas intra-block indices. For the description of the employedblock-wise detection schemes, it isthus sufficient to specify the block shiftS and block sizeL.

5.3.2 Multiple-Symbol Differential Detection

The conventional application of MSDD with a block size ofL information symbols employsa non-overlapping block structure with respect to information symbols (equivalently, a blockoverlap of one receive symbol, cf. top part of Figure 5.7), i.e., a block shift ofS = L, cf.,e.g., [GQ06, LT08] in the field of MSDD of DTR IR-UWB, but also,e.g., [DS90, HF92] forMSDD of DPSK. For each block, MSDD estimates the optimum sequence of symbols withinthis block. Based on the generic maximum-likelihood noncoherent detector (cf. Section 3.2),the MSDD estimate is obtained from solving

bMSDD = argmaxb∈BN

DTR

b0=1

bZbT. (5.12)

Usually, one is interested in theL = N − 1 information symbolsaMSDD only, which are givenby differentiating the symbolsbMSDD, i.e., after differential decoding. It is thus sufficient tosetone component ofb to a fixed value; for simplicity, we setb0 = 1.


MSDD is applicable for the special case of transmission of very short burst, i.e., ifNburst =

L+1. Here only a single block ofL receive symbols is available at the receiver. This emphasizesthe need for efficient block-wise detection schemes, as all concepts discussed later-on, such ascombining of multiple-observations, RSSE, and block-wisedetection with virtually increasedblock size are no longer applicable.

Clearly, a full search among all2L = 2N−1 candidate sequencesb ∈ BNDTR, with b0 = 1,

is infeasible already for moderate values ofL, as the computational complexity required forcomputing all decision metrics quickly becomes prohibitively large. There are several optionsto (approximately) solve the MSDD problem with reduced computational complexity.

In [ZMR10a, ZMR10b, ZM12a] different relaxations of the discrete (thus, non-convex) searchspace underlying (5.12) have been suggested in order to avoid a full search among all candidates.A relaxation to the reals, i.e.,b ∈ R

N , b0 = 1, enables to approximately solve (5.12) bycalculating the dominant eigenvector of the (slightly modified) correlation matrixZ. A secondrelaxation leads to a reformulated search problem, which can be solved using a semi-definiteprogram [ZMR10b]. Both approaches yield close-to-optimumperformance at a complexitywhich is polynomial (more precisely, at least cubic) in the block sizeL.

In this thesis, we favor the approach of [LT06a, LT08], whichis based on the application of thewell-known sphere decoder algorithm [AEVZ02] and inspiredby the application of the spheredecoder for MSDD of DPSK transmitted over time-varying flat-fading channels [LSPW05].This approach has two distinct advantages compared to concurrent approaches, in particularcompared to [ZM12a].

First, the arithmetic operations required for sphere-decoder-based detection are of very lowcomplexity, since, essentially, only additions and sign inversions are performed. Operations ofsignificantly higher complexity, such as calculating dominant eigenvectors or solving a semi-definite program as in [ZM12a] are not required.2

Moreover, exploiting relations and principles well-knownfrom lattice theory and tree-searchdecoding [AEVZ02, Bab86, MGDC06, Fis11], the study of sphere-decoder-based detection di-rectly leads to powerful low-complexity detection schemes, based on the principle of decision-feedback detection [LP88, Fos96, HH89], which achieve close-to-optimum performance atcomputational complexity being only linear in the block size [SF11c] (cf. Section 5.3.3 and5.4.1).

5.3.2.1 Tree-Search Algorithms and the Sphere Decoder

Since MSDD can be transformed into an equivalent tree-search problem which is efficientlysolved using the sphere decoder algorithm [LT06a], we first briefly review the general concept oftree search decoding and in particular the sphere decoder algorithm3 (cf. also [Sch09, Pau07]).

2The implementation of such operations in digital signal processors optimized for low cost and low powerconsumption is highly questionable, whereas it is well-known to efficiently implement simple arithmetic operationsin combination with simple logical routines.

3Here, we limit ourselves to the description of the sphere decoder as a tree-search algorithm, as opposed to themore geometric lattice-based point of view usually adoptedin sphere-decoder-based detection in MIMO systems,


These algorithms belong to the general framework of sequential decoding, cf. [AM84] for anoverview.

Closely following the terminology in [Pau07], a tree consists of a single root at depth zero,branches, and nodes at different depths of the tree. From every node at depthi− 1 a number ofbranches emanate, each ending at a node in depthi, i = 1, 2, .... For the scope of this work itis sufficient—but by no means comprehensive—to only consider binary trees with leafs (nodeswithout emanating branches) at depthi = L and two branches at all intermediate nodes (atdepthi = 0, 1, ..., L− 1), as shown in Figure 5.8.

root→← path

node→← branch

Figure 5.8: Binary tree of depth L = 3.

A path is a sequence of branches connecting a node with the root of the tree; the number ofbranches constituting a path is referred to as the length of the path. Branches emanating from anode are labeled with two different symbolsb, as suggested by the application, and consequentlyeach path of lengthi / node in depthi is uniquely defined by the sequence of correspondingbranch labelsbi = [b1, · · · , bi]. Each branch shall be associated with a real-valued branchmetric/cost functionδi(bi). Likewise each path has an associated additive path metric given bythe sum of the corresponding branch metrics∆i(bi) =

∑in=1 δn(bi).

Tree search algorithms for minimization problems have in common to search the path from rootto leaf with the smallest path metric.4 To guarantee that a tree search algorithm finds the optimalsolution of a minimization problem, it has to fulfill two requirements: it must only compare pathmetrics to a threshold and further all branch metrics must benon-negative [Pau07].

As a measure for the computational complexity of a tree search algorithm, the number of visitednodes/branches during the tree search process can be used. This measure directly translates intomore practical complexity measures, such as energy consumption and chip area of the signalprocessing device, and thus represents a descriptive and commonly adopted complexity measure[SB08, HV05, Pau07, JO05a]. Consequently, an algorithm checking all2L leafs to find the onewith minimum path metric has a maximum possible complexity of 2 · (2L − 1).

The Schnorr-Euchner sphere decoder applied for MSDD of IR-UWB is a variant of so-calleddepth-first tree-search algorithms [Pau07, AM84].

Starting at the root of the tree, the sphere decoder extends the current path into the most

cf., e.g., [CSB87, AEVZ02, DGC03, Fis02, Win04, Fis11].4Maximization problems are not considered in this work, but can be treated equivalently.


promising direction given by the branch with the smallest metric, hence reaching a leaf ofthe tree “as quickly as possible”. The resulting estimate isequal to decision-feedback detection[Bab86, Fis11]. It is stored as a preliminary result.

The Schnorr-Euchner sphere decoder continues its search process after having reached the firstleaf and now back-traces the path into the direction of the next-best branch in lower depth,5

which is only extended, if its path metric is smaller than thepath metric of the current prelimi-nary result. If during this process a new leaf is reached, it has to have a smaller path metric andis stored as the new preliminary result. This process is repeated until no paths can be extendedanymore without exceeding the path metric of the latest preliminary result; the back-tracingprocess is back at the root of the tree.

In contrast, e.g., to the Fincke-Pohst sphere decoder [FP85], a variant of breadth-first tree-search algorithms, the Schnorr-Euchner sphere decoder does not need an initial sphere radius.6

In general it is less complex in terms of the average number ofconsidered nodes during thesearch process, as the search radius is adaptively tightened. This comes at the cost of increasedlogical routines compared to [FP85] (it needs to decide for the (next-)best branch, i.e., thedirection in which to extend a path). Another advantage of the Schnorr-Euchner sphere decoderis that it finds a preliminary result (the decision-feedbackdetection sequence) as fast as possible(after considering onlyL nodes) and then continues to improve it. This allows to terminatethe search process at any time, still preserving a—possiblynon optimal, but often sufficientlygood—solution. Consequently, early termination criteria[SFL09b, SFL09a] can directly beincorporated into the Schnorr-Euchner sphere decoder in order to reduce the average and/orrestrict the worst-case run-time.

5.3.2.2 Application of the Sphere Decoder for MSDD

The detection problem (5.12) is not directly suited for the application of the sphere decoder, asit does not fulfill the necessary conditions for tree-searchalgorithms to be applicable. In orderto translate the MSDD problem into an equivalent tree-search problem suitable for the spheredecoder algorithm, the decision metric in (5.12) has to be reformulated [LT06a].

The first step results from extending the decision metric as

bZbT=

N−1∑

i=0

N−1∑

k=0

bk bizk,i (5.13)

and noting that, sinceb2k = 1, the diagonal elements ofZ do not influence the decision andmay thus be set to zero. Exploiting the symmetryzk,i = zi,k, we obtain the equivalent detectionproblem

bMSDD = argmaxb∈BN

DTR

b0=1

N−1∑

i=1

i−1∑

k=0

bk bizk,i . (5.14)

5For non-binary trees, the order the paths are extended is crucial to guarantee optimality and has to followthe so-called Schnorr-Euchner (or zig-zagging) strategy,which can be summarized to extend paths only along the(next-)best branch in every step. For binary trees, as considered here, there is only one other branch at each depth.

6However, it is sometimes helpful to “guide” the sphere decoder via limiting the initial search radius [Pau07].


The second step is obtained by transforming the maximization problem into an equivalent min-imization problem with non-negative branch metrics. This is achieved by subtracting an upperbound on

∑N−1i=1

∑i−1k=0 bk bizk,i, given by

∑N−1i=1

∑i−1k=0 |zk,i|. Finally, we obtain

bMSDD = argminb∈BN

DTR

b0=1

Λ(b, Z) (5.15)

where we have defined the decision metric

Λ(b, Z) =

N−1∑

i=1

i−1∑

k=0

(

|zk,i| − bk bizk,i)

. (5.16)

In the view of tree-search decoding, the decision metric nowrepresents a valid path metric in abinary tree. To see this, note that theith increment of the decision metric,

δi =

i−1∑

k=0

(

|zk,i| − bi bk zk,i)

(5.17)

(i.e., in the view of tree-search decoding, the so-called branch metric) is always non-negativeand solely depends on thei preceding symbolsbk, k = 0, . . . , i − 1. This allows to check thedecision metric component-wise, and thus fits into the framework of tree-search decoding. Inturn, the application of the sphere decoder algorithm is possible.

We briefly describe the sphere decoder search process:7 employing the Schnorr-Euchner searchstrategy, at some node at depthi − 1 the sphere decoder chooses the branch labeled bybi withminimum branch metric. As the sphere decoder operates on thetransmit symbols, using (5.17),this is directly given as

bi = argminbi∈±1

δi = argmaxbi∈±1

bi

i−1∑

k=0

bk zk,i = signi−1∑

k=0

zk,ibk . (5.18)

The path is only extended along this branch, if its path metric of the current partial sequence,∑i

ι=1 δι, is less than the current sphere decoder search radiusRsd. At the beginning this searchradius can be chosen arbitrarily large, but is updated whenever a new (preliminary) best se-quence is found.

The sphere decoder algorithm for MSDD, abbreviated as multiple-symbol differential spheredecoder (MSDSD), is summarized in pseudo-code representation in Figure 5.9 (including tech-niques for complexity reduction as described below). For brevity we definedδi = qi − bipi(qi =

∑i−1k=0 |zk,i|, pi =

∑i−1k=0 bk zk,i) and∆i =

∑iι=1 δι, and omitted the indicator for a hy-

pothesis. Additionally, the counterni has been introduced, which is used to check if the twobranches emanating from each node have been checked. Note that this pseudo-code representa-tion is based on the pseudo-code representation of [SF11c] and the sphere decoder for MSDDof DPSK given in [LSPW05].

7Note that in contrast to [LT08] the presented sphere decoderoperates on the transmit symbolsbi rather thanon the data symbolsai, yielding certain benefits as described later-on.


bMSDD = MSDSD(Z, Rstop)

1: Rsd := +∞; ∆0 := 0 // initialization2: b0 := 1; i := 1 // fixed reference3: p1 := z0,1; q1 := |z0,1|4: b1 := sign(p1); n1 := 15: while i > 0 {6: ∆i := ∆i−1 + qi − bi pi // compute current path metric7: if ∆i < Rsd {8: if i < L {9: i := i+ 1 // move down

10: pi :=∑i−1

k=0 zk,i bk; qi :=∑i−1

k=0 |zk,i|11: bi := sign(pi); ni := 112: } else {13: bMSDD := b; Rsd := ∆i // update14: if Rsd < Rstop { break and return bMSDD } // stopping radius15: i := i− 1 // move up again16: while ni > 1 { i := i− 1 }17: bi := −bi; ni := ni + 118: }19: } else {20: i := i− 1 // move up21: while ni > 1 { i := i− 1 }22: bi := −bi; ni := ni + 123: }24: }

Figure 5.9: Pseudo-code representation of the sphere decoder algorithm for MSDD of DTR IR-UWB [SF11c].

5.3.2.3 Techniques for Complexity Reduction of Sphere-Decoder-Based MSDD

The computational complexity of the sphere decoder dependson the particular channel realiza-tion, i.e., the realization ofZ, and thus on the signal-to-noise ratio. Especially the worst-casecomputational complexity might prohibit the implementation in low-cost energy-efficient re-ceivers. To alleviate this drawback several techniques have been proposed to reduce the averagerun-time of the sphere decoder search process, and/or to upper-bound its worst-case complexityat guaranteed optimality of the output (or at least with the goal of close-to-optimum perfor-mance).

In literature mainly the problem of sphere-decoder-based (close-to-) maximum-likelihood de-tection in MIMO systems is considered, cf., e.g., [GT08, GH07, GN04, YK08, SVH08, TTP04,BT08] to name only a few; unfortunately, most of the techniques derived for this scenario relyon the specific geometric structure of the underlying MIMO detection problem and are thus notdirectly applicable in MSDD of IR-UWB.

However, it is well recognized that in general an optimized decision order within the searchprocess helps to significantly reduce the average run-time of the sphere decoder [MGDC06,AEVZ02, ZG06]. This is motivated by the fact that an optimized decision order leads to abetter first preliminary sequence in the search process, which in turn enables to significantly


tighten the search radius for the subsequent search process. Thus, a large number of candidatescan be discarded already at early steps in the search processand fewer preliminary sequenceshave to be investigated, until the best sequence is found.

An optimized decision order is directly applicable in MSDD,since the block-wise processingenables to break causality in processing of the stream of receive symbols, as, anyway, a jointdecision of all symbols within the block is only possible, ifthe entire block has been receivedand the required correlation coefficients to build upZ have been computed.

Since the MSDSD operates on the transmit symbolsb rather than directly on the informationsymbolsa, a different decision order can easily be incorporated by interchanging the rows andcolumns ofZ prior to the call of the sphere decoder according to the new decision order andcorrespondingly resorting the output sequence of the sphere decoder [SF11c]. More precisely,for a given decision order represented in theN × N permutation matrixP (i.e., a matrix withonly a single one per column and row), the sphere decoder algorithm is called via

bSD = MSDSD

(

PZPT, Rstop

)

(5.19)

and the final MSDD estimate is obtained from the sphere decoder output viabMSDD = bSDP .The remaining problem is to find a good (or even optimal) decision order. This task is addressedon its own in Section 5.3.4.

Of course, early termination of the sphere decoder search process decreases the average run-time [SFL09a], and may also be used to upper-bound the worst-case run-time [SB08]. Wediscuss two strategies for early termination.

First, the search process may be terminated early if the metric of any preliminary-best sequenceduring the sphere decoder search process is less than a precomputed stopping radiusRstop, cf.Line 14. Choosing this stopping radius according to

Rstop = L · mini,k, i 6=k

|zk,i| (5.20)

preserves the optimality of the sphere decoder output [SFL09a]. Although, the effectivenessof this setting is limited to the regime of moderate-to-highsignal-to-noise ratio [SFL09a] thecomputational overhead of employing this technique is marginal and thus well-worth spending.

A second option for early termination can be implemented by terminating the sphere decoder,if the number of investigated nodes in the tree-search process (i.e., the iterations of the outerwhile-loop in the MSDSD algorithm) exceeds a given threshold. This stopping criterion caneffectively be used to limit the worst-case complexity of the sphere decoder. In [SB08], acumulative worst-case complexity limit has been suggested. This is motivated by the fact that,in order to ensure a constant throughput of the receiver device especially in the considered burst-wise communication, run-time demands have to be fulfilled for the entire burst rather than foreach processed block. Consequently, the maximum number of visited nodes is defined per burst;the sphere decoder runs processing the first blocks of the burst are then allowed to potentiallyconsume a larger part of this amount compared to processing of the last blocks of the burst. Ofcourse, it has to be ensured that at least one hypothesis can be investigated for processing of


each block (i.e., at least DFDD can be performed). This is achieved by ensuring that even forprocessing of the final block at leastL nodes are available. Here, we do not study the effect of aworst-case complexity limit, since power-efficient detection is achieved with DFDD at low andin particular constant complexity.

A further often-considered technique is to start the spheredecoder with a finite initial radius—although the Schnorr-Euchner search strategy allows to choose the initial search radius arbitrarylarge. The initial radius may be set, e.g., to a precomputed value depending on the signal-to-noise ratio. In this case, the sphere decoder algorithm mustbe modified to catch the case that nosequence falls within the initial search radius; the spheredecoder then has to be restarted with anappropriately increased initial search radius. This can beavoided by choosing the initial radius,e.g., as the decision metric of the sequence of symbols obtained by differential detection. In thiscase, if the sphere decoder does not find a better sequence,bMSDD = bDD. If already the metricof the differential-detection sequence meets the above stopping criterion a sphere decoder callis not necessary at all.

In [SF11c] it has been shown, that the computational overhead of finding a good initial searchradius is not compensated by a sufficiently large search complexity reduction. Especially incombination with an optimized decision order the sphere decoder anyway finds a very goodinitial radius itself. Setting the initial radius according to the signal-to-noise ratio requires anestimate of the noise power, that is usually not available inour receiver. Consequently, anoptimized initial radius is not employed subsequently.

5.3.2.4 Soft-Output MSDD

Reliability information on the MSDD estimates, i.e., so-called soft-output MSDD, is easilyderived based on the results of Section 3.3. For the estimateof theith information symbol, softoutput calculates to

LLRMSDDi =

1

2σ2n

·{

ΛMSDD − ΛMSDDi for aMSDD

i = +1

ΛMSDDi − ΛMSDD for aMSDD

i = −1 (5.21)

=1

2σ2n

· aMSDDi

(

ΛMSDD − ΛMSDDi

)

(5.22)

whereaMSDDi is the estimate of theith information symbol, obtained via differential decoding

of bMSDD, ΛMSDD is the associated decision metric, i.e.,ΛMSDD = Λ(bMSDD, Z), andΛMSDDi is

the decision metric of the best sequence out of the restricted set.

This restricted set is obtained by fixing theith information symbol to the opposite value of theMSDD estimate. This metric is thus specific to the consideredsymbol and is defined as

ΛMSDDi = min

b∈BNDTR

ai=−aMSDDi

Λ(b, Z) . (5.23)

Scaling of the LLRs in (5.22) depends on the noise variance and the applied weighting prior tocorrelation (hidden in the scaling of the correlation coefficients, which appear in the decision


metric, cf. Section 3.3). For the later employed convolutional codes, the performance of thedecoder, however, is independent of a constant scaling at the input. Thus, estimation of thenoise variance is not necessary.

We note that due to the one-to-one mapping from information symbols to code symbols, softoutput on the information symbols equals reliability information on the code symbols. More-over, the soft-output computation is preferably based on the nearest neighbor approximation (asemployed above, cf. (3.24)), since already for moderateL the exact computation according to(3.23) results in prohibitively large computational complexity.

Finally, soft-output MSDD is easily extended to incorporate a-priori information on the infor-mation symbols (so-called soft-input/soft-output MSDD),which enables iterative detection anddecoding for coded IR-UWB transmission [ZM12b], in the spirit of “Turbo-DPSK” [HL99,PLS06]. Since every iteration of the detector and the channel decoder adds to the overallreceiver complexity and its energy consumption, and here weaim to design low-complexityenergy-efficient IR-UWB receivers, we do not consider iterative detection and decoding.

5.3.2.5 Application of the Sphere Decoder for Soft-Output MSDD

Calculating the LLRs according to (5.22) resorts to finding the minimum of an unrestrictedtree search (cf. (5.15)), the corresponding sequence, andL “next-best” minima (5.23). Onecould solve these minimization problems subsequently by rerunning the sphere decoder foreach counterhypothesis with correspondingly restricted search space. This requires to run thesphere decoderL + 1 times per block ofL information symbols, and hence, imposes a highcomplexity burden.

This can be alleviated by a modified, so-called single-tree-search soft-output sphere decoderalgorithm, as introduced for soft-output detection in MIMOsystems in [JO05b], and furtherrefined in [SBB08], which ensures that every node in the search tree is visited at most once.8 In[SF10b], it has been shown how this single-tree-search principle has to be incorporated into thesphere decoder for hard-output MSDD in order to generate thedesired soft output. Thus, therequiredΛMSDD, ΛMSDD

i , andaMSDD result from a single tree-search process.

This soft-output multiple-symbol differential sphere decoder (SO-MSDSD) algorithm is givenin pseudo-code representation in Figure 5.10. Compared to the hard-output MSDSD algorithm(cf. Figure 5.9), the basic structure remains unchanged. The required modifications for softoutput generation (including LLR clipping as described below) are highlighted. For compactnotation the counterhypothesis metricsΛMSDD

i are grouped into the vectorΛ, a Matlab-styleindexing is adopted, and the indicator for a hypothesis is omitted.

Similar to the sphere decoder for hard-output MSDD, and different from [SF10b], the presentedsoft-output sphere decoder mainly operates on the transmitsymbolsb. However, in order tofind the corresponding counter-hypotheses, differential decoding has to be performed withinthe search process, i.e., the information symbolsa have to be computed fromb. This is done

8Note that this single-tree-search soft-output sphere decoder represents a very general concept, which hasproven to be very effective for soft-output MSDD of DPSK transmitted over time-varying flat-fading channels[RSH13] and for soft-output detection of unique-word OFDM [OSHH12], too.


by the functiondiffdec.

The major difference compared to hard-output MSDD is represented by the update of the searchradius. The search radiusRsd is not updated, whenever a new (preliminary) best sequence hasbeen found, but the search radius update is based on the current valuesΛMSDD andΛMSDD

l ,l = 1, ..., L. It is set such, that only branches in the search tree are considered, which can leadto an update of eitherΛMSDD orΛMSDD

l , l = 1, ..., L. This is achieved by setting

Rsd = max

{

ΛMSDD, maxl=i,...,L

ΛMSDDl , max

l=1,...,i−1

with al 6=aMSDDl

ΛMSDDl

}

. (5.24)

SinceΛMSDD ≤ ΛMSDDl , ∀l, the current best metric can be removed from this evaluation

[SBB08].

Further, in the case a sequenceb with path metricΛ(b) is investigated, i.e., a leaf in the searchtree has been reached, two cases are distinguished: IfΛ(b) = ∆L < ΛMSDD, a new (preliminary)best sequence has been found. Then allΛMSDD

l , l = 1, ..., L, whereal 6= aMSDDl , are set to

ΛMSDD, followed by the usual sphere decoder update of the current best sequencebMSDD = b

and metricΛMSDD = Λ(b). As opposed to the hard-output sphere decoder, the sphere decoderremains at the lowest level in the tree (it does not move up), since counterhypotheses at thelowest level have to be considered as well.

In the opposite case, ifΛ(b) = ∆L ≥ ΛMSDD, only the counterhypotheses have to be checked,i.e., allΛMSDD

l , l = 1, ..., L, whereal 6= aMSDDl andΛMSDD

l > Λ(b), are set toΛ(b).

Similar to the MSDSD algorithm (cf. Section 5.3.2.3), various techniques can be employedin order to speed up the search process of the soft-output sphere decoder and thus reduce itscomputational complexity.

As shown in [SBB08, SF10b], a crucial part for complexity reduction of the soft-output spheredecoder is to limit the maximum LLR values during the sphere decoder search process. ThisLLR clipping enables a trade-off between the power efficiency of optimal soft-output detectionand the complexity of hard-output detection. LLR clipping with maximum LLR valueLLRmax

is implemented by limiting the counterhypotheses metricsΛMSDDl via

ΛMSDDl = min

{

ΛMSDDl ,ΛMSDD + Λmax

}

, ∀l . (5.25)

after each update, whereΛmax = LLRmax · 2σ2n. From (5.24) it can be seen that this LLR

clipping limits the search radius toRsd ≤ ΛMSDD +Λmax and, together with (5.22), ensures that|LLRl| ≤ LLRmax, ∀l, after the sphere decoder search. Consequently, only counter-hypotheseswithin this reduced search radius are considered. Basically, if the nearest neighbor is too farapart, the exact distance does not matter anymore; the estimate is already known to be veryreliable (clearly, withLLRmax = 0, no reliability information is computed).

Similar to the MSDSD algorithm, a significant complexity reduction can be achieved for sphere-decoder-based soft-output MSDD by employing an optimized decision order. Again we utilizethe optimum decision order obtained from the DFDD process (cf. Section 5.3.4). However,


[aMSDD, ΛMSDD, Λ] = SO-MSDSD(Z, Λmax)

1: Rsd := +∞; ∆0 := 0; Λ :=∞; ΛMSDD :=∞ // initialization2: b0 := 1; i := 13: p1 := z0,1; q1 := |z0,1|4: b1 := sign(p1); n1 := 15: while i > 0 {6: ∆i := ∆i−1 + qi − bi pi7: if ∆i < Rsd {8: if i < L {9: i := i+ 1

10: pi :=∑i−1

k=0 zk,i bk; qi :=∑i−1

k=0 |zk,i|11: bi := sign(pi); ni := 112: } else {13: a := diffdec(b) // diff. dec. fromb to a

14: if ∆i < ΛMSDD {15: Λa 6=aMSDD := ΛMSDD // update counterhypotheses

16: bMSDD := b; aMSDD := a; ΛMSDD := ∆i

17: } else {18: Λ

a 6=aMSDD and Λ>∆i:= ∆i // update counterhypotheses

19: }20: Λ

Λ>Rsd+Λmax:= Rsd + Λmax // LLR Clipping

21: }22: } else {23: while ni > 1 { i := i− 1 }24: bi := −bi; ni := ni + 125: }26: a := diffdec(b)27: Rsd := max{Λi,...,L, Λ1,...,i−1 and a 6=aMSDD} // update search radius28: }

a = diffdec(b)

1: for i = 1, 2, . . . , L {2: ai := bi−1bi // differential decoding3: }

Figure 5.10: Pseudo-code representation of the soft-output sphere decoder algorithm for soft-output MSDD

of IR-UWB. Only modifications required for soft output generation compared to the MSDSD algorithm arehighlighted.

incorporating a different decision order into the SO-MSDSDalgorithm requires slight modifi-cations to the SO-MSDSD algorithm. Since the entire sequence of transmit symbolsb is onlyavailable ifi = L, in case of an arbitrary decision order the corresponding information symbolsa can only be obtained at this point. Thus, a search radius update is no longer possible at theend of each while-loop iteration, but only ifi = L, i.e., after the LLR-clipping step (line 20). Ofcourse, this potentially leads to slower tightening of the search radius, which, however, is shownto be compensated by the amount of reduction achieved through applying a good decision order(cf. Section 5.3.5).


Of course, the worst-case run-time can be limited effectively by imposing a limit on the max-imum number of visited nodes (again equal to the iterations of the outer while-loop in theSO-MSDSD algorithm) in the same way as for the hard-output MSDSD (at the cost of soft-output quality). Employing a stopping-radius-based earlytermination criterion enables a fur-ther performance-complexity trade-off [SF10b]; for sake of brevity, this is not considered in thisthesis.

The resulting algorithm enables efficient computation of the reliability of the MSDD estimates,which can be used to improve the performance of a subsequent channel decoder and/or ofmultiple-observations combining (cf. Section 5.3.6).

5.3.3 Decision-Feedback Differential Detection

Despite its efficient implementation using the sphere decoder and its techniques for complexityreduction, block-wise MSDD has two major drawbacks from implementation point of view.First, the computational complexity of the MSDSD algorithmis not constant, but depends onthe channel realization and the signal-to-noise ratio. Second, the worst-case complexity is inthe order of2L, i.e., exponential in the block size. It is desirable to havedetection schemes withlow and in particular constant computational complexity, at best linear in the block sizeL.

This can be achieved by the closely-related detection scheme block-wise decision-feedbackdifferential detection (bDFDD), cf. [LP88, Edb92, SGH98] and its modifications for IR-UWBdetection [SF11c]. Block-wise DFDD decides the symbols within each block in a successivemanner taking into account the feedback from already decided symbols within the block.

Exploiting the relation of the Schnorr-Euchner sphere decoder to the Babai algorithm [Bab86](both are depth-first tree-search algorithms), the decision rule for block-wise DFDD is directlyobtained from sphere-decoder-based block-wise MSDD [SFL09a]. The depth-first search strat-egy in the sphere decoder for MSDD ensures that the first estimate in the sphere decoder searchprocess equals DFDD. Thus, terminating the sphere decoder after the first point found, resultsin DFDD with a linearly increasing feedback window length (from 1 to L). This is achieved,e.g., by calling the sphere decoder withRstop =∞, or, equivalently, choosingbbDFDD

0 = 1, and,similar to (5.18),

bbDFDDi = sign

(

i−1∑

k=0

zk,ibbDFDDk

)

. (5.26)

For sake of completeness, the algorithm for block-wise DFDDis given in pseudo-code in Fig-ure 5.11. The processing of the matrixZ of block-wise DFDD is compared to block-wiseMSDD in Figure 5.12. Clearly, the decision of the first symbolwithin each block is equal todifferential detection (here the fourth transmit symbol).The decisions of the other symbols arethen based on a linearly increasing number of symbols fed back.

5.3.4 Optimum Decision Order

Since the first estimate of DFDD equals the one of symbol-wisedifferential detection, whichthen strongly affects the upcoming decisions, it is expected that the performance of the straight-


bbDFDD = bDFDD(Z)

1: bbDFDD0

:= 12: for i = 1, . . . , L {

3: bbDFDDi

:= sign(

∑i−1k=0 zk,i b

bDFDDk

)

4: }

Figure 5.11: Pseudo-code representation of the bDFDD algorithm for block-wise DFDD of DTR IR-UWB.

ZMSDD =

ZbDFDD =

Figure 5.12: Illustration how block-wise MSDD and DFDD operate on the correlation matrix Z. The used

correlation coefficients are highlighted, the ones which influence the decision of the fourth transmit symbolare marked with a dashed line. Left: MSDD, right: DFDD. Nburst = 7, L = 2.

forward application of block-wise DFDD does not lead to satisfactory results. It is well known—especially from decision-feedback equalization in multi-antenna systems, also known as BLAST[MGDC06, Fos96]—that taking the decisions in an optimized order, i.e., employing some sort-ing, improves the performance. Similarly, in the context ofIR-UWB, interchanging the decisionorder within a block is enabled through the block-wise processing of DFDD. Again, an arbitrarydecision order can easily be implemented by reordering the columns and rows ofZ accordingto the given decision order and correspondingly resorting the estimated sequence.

Moreover, an optimized decision order within the processing of each block enables to signifi-cantly reduce the computational complexity of sphere-decoder-based MSDD.

The problem of finding a good (or even optimum) decision orderis non-trivial.

5.3.4.1 Optimum Decision Order for DFDD

A reasonable sorting criterion can be derived form the DFDD process itself (then denoted assorted block-wise DFDD, sbDFDD) [SF11c]. This procedure isillustrated in the following.After the derivation, we explain why this decision order maybe termed optimum for DFDD.

The reasoning behind this derivation is, that for reliable decisions in each step, the magnitudeof the argument of thesign-function in (5.26) is desired to be as large as possible.

Different from [SF11c],9 we start with selecting the pair of transmit symbols with index k0 and

9In [SF11c], the reference symbol is the one with indexk0 = 1. This restriction, however, leads to some


k1, which can be decided most reliable. Since these two symbolsare connected via the corre-lation coefficientzk0,k1, we are looking for the correlation coefficient with largestmagnitude,i.e.,

[

k0, k1

]

= argmax[k0, k1]∈{0,...,L}2

k0<k1

|zk0,k1| . (5.27)

As we later are interested in the information symbols only, we may fix one of these arbitrarilyto bsbDFDD

k0= 1. An estimate for the other symbol is obtained from

bsbDFDD

k1= sign

(

zk0,k1

bsbDFDD

k0

)

. (5.28)

Taking these decisions into account, the remaining symbolsbk, k ∈ {0, . . . , L}/{k0, k1}, maybe estimated from

sign(

zk0,k

bsbDFDD

k0+ z

k1,kbsbDFDD

k1

)

. (5.29)

Thus, the symbol which can be decided most reliable next is given by

k2 = argmaxk∈{0,...,L}/{k0, k1}

∣

∣

∣zk0,k

bsbDFDD

k0+ z

k1,kbsbDFDD

k1

∣

∣

∣. (5.30)

In summary, givenk0, the optimum decision order can be derived successively from

ki = argmaxk∈{0,...,L}/{k0,...,ki−1}

∣

∣

∣

∣

∣

i−1∑

l=0

zkl,k

bsbDFDD

kl

∣

∣

∣

∣

∣

(5.31)

wherei = 1, ..., L. The symbol estimatesbsbDFDD

kiare required for finding this order. With

bsbDFDD

k0= 1, they are given as

bsbDFDD

ki= sign

i−1∑

l=0

zkl,ki

bsbDFDD

kl. (5.32)

The decision process of sorted DFDD is illustrated in Figure5.13 for a block size ofL = 4.The decision order of this example is given by[1, 3, 4, 2, 0].

The algorithm of sorted block-wise DFDD is given in the pseudo-code of Figure 5.14. HereIdenotes the set of symbols indices which have already been decided, andI the set of symbolsindices which have not yet been decided (the decision order is given by the sequence in whichthe indices have been added toI). Since in each iteration of the while-loop one element isremoved fromI, the algorithm terminates afterL− 1 iterations.

Basically, this sorting criterion forces reliable decisions for the first decided symbols, whichthen strongly influence the upcoming decisions. Thus, for the decision of each symbol, allcorrelation coefficients of the block are utilized.

performance degradation (cf. Figure 5.16).


Z(1) =

0 1 2 3 4

0

1

2

3

4

Z(2) =

0 1 2 3 4

0

1

2

3

4

Z(3) =

0 1 2 3 4

0

1

2

3

4

Z(4) =

0 1 2 3 4

0

1

2

3

4

Figure 5.13: Illustration of decision process of sorted block-wise DFDD with a block size for L = 4 (the

superscript of the correlation matrix indicates the iteration). Level of gray indicates the magnitude of zk,l,processed correlation coefficients are shaded. A square indicates already decided symbols which are usedas feedback (dotted lines) for the decision of the current symbol (diamond).

bsbDFDD = sbDFDD(Z)

1: I := {0, . . . , L}; I := {}2: [m, n] := argmax[k, l]∈I |zk,l|3: bsbDFDD

m := 14: bsbDFDD

n := sign(

zm,n

)

5: I := I/{m, n}; I := I ∪ {m, n}6: while I 6= ∅ {

7: n := argmaxn∈I

∣

∣

∣

∑

k∈I zk,n bsbDFDDk

∣

∣

∣

8: bsbDFDDn := sign

(

∑

k∈I zk,n bsbDFDDk

)

9: I := I/{n}; I := I ∪ {n}10: }

Figure 5.14: Pseudo-code representation of the sbDFDD algorithm for block-wise DFDD of DTR IR-UWB

with optimum decision order.

We emphasize that, in contrast to similar concepts in MIMO detection, cf., especially the


BLAST algorithm [Fos96, Fis11] or lattice-reduction-aided detection [Fis02, Win04, GLM09,Fis11], in this approach sorting is done per block based on the actual receive symbols andtaking the previous decisions into account, rather than on the channel realization. Since allavailable information is used and in each step the most-reliable symbol is decided next, forblock-wise DFDD this decision order is optimal.10 Moreover, as shown in [SF11c], for the spe-cial case ofL = 2, sorted block-wise DFDD and block-wise-optimum MSDD are equivalent,i.e.,bsbDFDD = bMSDD in all cases.

5.3.4.2 Application to MSDD

It is straight-forward to incorporate the computation of this decision order into the sphere de-coder algorithms for MSDD and soft-output MSDD, such that the optimum DFDD decisionorder can be found within the sphere decoder search process.Note that for the latter, reorderinghas to take place prior to inversion of the differential encoding. However, the optimum DFDDdecision order can not be guaranteed to be optimal with respect to smallest run-time of thesphere decoder search process; however, as shown in [SF11c]and below, it leads to significantreduction of computational complexity of the sphere decoder search process.

5.3.5 Numerical Results I

At this point, we compare the performance of the detection schemes presented up to now forthe case of uncoded transmission by means of numerical simulations.

5.3.5.1 Simulation Setup

We consider transmission over the non-line-of-sight channel model 2 of [MCC+06] and applyuniform weighting within the correlation interval of duration Ti = 50 ns, which represents agood compromise between noise suppression and capturing ofreceive signal energy for thischannel model (cf. Figure 5.5). All results have been averaged over a large number of channelrealizations (at least 1000). The symbol duration has been set toT = 100 ns, which enables toneglect the effect of inter-symbol interference.11 The simulations have been performed using asampling rate offs = 10GHz. These parameters yield a time-bandwidth product ofNi = 500.

The burst length is set toNburst = 101 transmit symbols, for which the channel impulse responseis assumed to be invariant. However, the results in this section are also valid for the case of veryshort bursts ofNburst = L + 1 symbols. In this case MSDD equals INSE. All other detectionschemes discussed subsequently rely on the fact that the burst length significantly exceeds theblock size (or equivalently, the autocorrelation-device parameter)L.

As we mainly focus on relative statements regarding the performance of the detection schemesfor systems operating under the same channel conditions, i.e., on the same output of anL-branch autocorrelation device, it is sufficient to consideronly a single channel model. Very

10This is also supported by the fact, that different sorting criteria, e.g., based on thel1- andl∞-norm (column/rownorm are equivalent) of the matrixZ, or according to the first row ofZ, show some loss with respect to error rateof DFDD compared to successive sorting during the DFDD process (up to1 dB for thel∞-norm and the first-rowcriterion, and only marginal loss for thel1-norm).

11In order to entirely remove the influence of this effect, the receive pulse shapes have been shortened accord-ingly.


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Figure 5.15: SER of block-wise detection of DTR IR-UWB vs. Ec/N0 in dB for different block sizes L = 2(squares), 5 (diamonds), and 10 (circles), right-to-left. Solid-gray: MSDD, dashed-black: sorted DFDD,

dotted-black: DFDD, left-most solid-gray: detection with CSI, right-most dashed-gray: DD. Non-line-of-sight

channel model 2 of [MCC+06] with Ti = 50 ns.

similar results and the same conclusions are obtained for other channel models, cf. also [SF11c,SFL09a].

5.3.5.2 Error-Rate Performance

Figure 5.15 depicts the error rate of the above block-wise detection schemes for different blocksizesL = 2, 3, 5, and10 (right-to-left) compared to differential detection (L = 1) and detectionwith perfect channel knowledge. Employing MSDD (implemented using the MSDSD algo-rithm), significant gains are achieved over symbol-wise differential detection with increasingblock size. Already forL = 2 the required signal-to-noise ratio can be significantly reducedcompared to differential detection in order to guarantee the same error rate.

Apparently, the straight-forward application of block-wise DFDD does not lead to satisfactoryresults. Only minor improvement over DD is achieved, since the first decision in block-wiseDFDD equals that of DD. However, employing an optimum decision order within the DFDDprocess (sorted DFDD), yields an error rate very close to that of MSDD. Especially for low tomoderate block sizes, there is hardly a difference to be observed. Recall that forL = 2, sortedDFDD and MSDD are equivalent [SF11c].

Due to the binary signaling, error propagation is not an issue in the detection process of DFDD.This is also verified by the excellent performance of sorted DFDD compared to (block-wise-optimal) MSDD.

For sake of completeness, in Figure 5.16 the presented optimum decision order of DFDD, asproposed in Section 5.3.4, is compared to the decision orderearlier proposed in [SF11c]. The


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Figure 5.16: SER of block-wise detection of DTR IR-UWB vs. Ec/N0 in dB for different block sizes L = 2(squares), 3 (triangles), 5 (diamonds), and 10 (circles), right-to-left. Solid-black: sorted DFDD, solid-gray:

sorted DFDD with decision order as in [SF11c], left-most solid-gray: detection with CSI, right-most dashed-

gray: DD. Non-line-of-sight channel model 2 of [MCC+06] with Ti = 50 ns.

only difference between the two variants is the starting point of the decision-feedback process.Whereas the variant of [SF11c] always begins with deciding the first two symbols of each block,the variant proposed here optimally selects its starting point by selecting the pair of symbolswhich can be decided most reliable in the first step. This improvement leads to additional gainsfor L > 2 as clearly visible from Figure 5.16.

5.3.5.3 Computational Complexity of Sphere-Decoder-Based MSDD

The optimum decision order of DFDD can additionally be utilized to reduce the computationalcomplexity of the sphere decoder for MSDD. Different to [SF11c], but similar to [SFL09a,SB08, SBB08, JO05a], we adopt the number of visited nodes persymbol in the sphere-decodertree-search process as a measure for complexity (cf. Section 5.3.2.1).

Figure 5.17 depicts the average computational complexity of MSDD12 using the MSDSD algo-rithm forL = 5 and10 vs.Ec/N0. Two variants are compared: the conventional sphere decoder(cf. [LT08, SFL09a]) and the sphere decoder with optimized decision order (obtained from theDFDD process). Both variants employ the stopping-radius-based early termination criterion(5.20).

As expected, the computational complexity of the sphere decoder is higher for lowEc/N0. Forhard-output MSDD, the average complexity quickly decreases to its lowest value of one nodeper symbol, i.e., the search is terminated after the first sequence has been found and the search

12For simplicity, the results on computational complexity have been obtained by considering the approximatediscrete-time model withNi = 500 andEp = 1.


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Figure 5.17: Average computational complexity (number of visited nodes per symbol) of sphere-decoder-based MSDD vs. Ec/N0 in dB for block sizes L = 5 (left) and 10 (right). Solid-black: MSDSD with optimized

decision order, solid-gray: conventional MSDSD, dashed-gray: bDFDD. Approximate discrete-time model

with Ni = 500. Note the different scaling of the axes.

is terminated due to the early termination criterion [Sch09, SFL09a]. Employing an optimizeddecision order significantly reduces the average computational complexity.

Often not only the average complexity, but also its variation has to be considered for receiverdesign. To this end, Figure 5.18 depicts recorded (and normalized) histograms of the com-putational complexity of the MSDSD algorithm for MSDD forL = 5 and 10. Additionallines indicate the average complexity and the99-percentile (i.e.,99% of the sphere decoderruns require a complexity less than this value), which represents an indicator for the worst-casecomplexity.

The optimized decision order significantly reduces the worst-case complexity as well as theamount of variation. Especially forL = 5, the complexity concentrates at a value of(2L −1)/L, which corresponds to the case, that the first sequence foundby the sphere decoder is theoptimum sequence, and all alternatives are pruned as soon aspossible [Sch09, SFL09a]. Thepeak visible forL = 5 at a computational complexity of one node per symbol is caused by theearly termination criterion.

Figures 5.19 and 5.20 depicts corresponding results for soft-output MSDD employing the SO-MSDSD algorithm.

Although an optimized decision order requires some modifications of the SO-MSDSD algo-rithm, which potentially lead to a slower tightening of the search radius, this technique effec-tively reduces the computational complexity of soft-output MSDD, too (cf. Figure 5.19, con-sidering the LLR-clipping level ofLLRmax = 10). Clearly, the average complexity is increased


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Figure 5.18: Recorded histograms of the computational complexity (in nodes per symbol) for sphere-

decoder-based MSDD with block sizes L = 5 (left) and 10 (right). Solid-black: MSDSD with optimized de-cision order, solid-gray: conventional MSDSD, dashed line: average complexity, dotted line: 99-percentile.

Approximate discrete-time model with Ni = 500 at 10 log10(Ec/N0) = 8 dB (top) and 11 dB (bottom). Note

the different scaling of the axes.

compared to hard-output sorted MSDD. Moreover, for soft-output MSDD (cf. Figure 5.20) theworst-case complexity, as well as the amount of variation, is increased compared to hard-outputMSDD.

As shown in [SFH12, SF12b], LLR clipping is a crucial part forlow-complexity soft-outputMSDD. Compared to no LLR clipping, choosingLLRmax = 10 leads to significantly reducedcomputational complexity (cf. Figure 5.19). This reduction comes at almost no loss in perfor-mance, as shown in [SF10b, SF12b] and Section 6.4.

A comprehensive analysis of the worst-case complexity of sphere-decoder-based MSDD is notrequired, since with DFDD powerful detection schemes with low and in particular constantcomplexity are available.


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Figure 5.19: Average computational complexity (number of visited nodes per symbol) of sphere-decoder-

based soft-output MSDD vs. Ec/N0 in dB for block sizes L = 5 (left) and 10 (right). Solid-black: SO-MSDSD

with optimized decision order and LLRmax = 10, solid-gray: conventional SO-MSDSD and LLRmax = 10,dash-dotted: no LLR clipping, dashed-gray: hard-output sorted MSDD. Approximate discrete-time model

with Ni = 500. Note the different scaling of the axes.

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Figure 5.20: Recorded histograms of the computational complexity (in nodes per symbol) for sphere-

decoder-based soft-output MSDD with block sizes L = 5 (left) and 10 (right). Solid-black: SO-MSDSD

with optimized decision order, solid-gray: SO-MSDSD without optimized decision order, dashed line: av-erage complexity, dotted line: 99-percentile. LLR-clipping level LLRmax = 10. Approximate discrete-time

model with Ni = 500 at 10 log10(Ec/N0) = 11 dB. Note the different scaling of the axes.


5.3.6 Combining Multiple Observations

In conventional block-wise MSDD a relatively large part of correlation coefficients computedby the autocorrelation device is not used for the detection,13 as can be seen in particular inFigures 5.2 and 5.22. Combining multiple observations alleviates this drawback and thus offersa further possibility to improve the performance without increase of the block size and, in turn,the maximum delay of the autocorrelation device [SF12b].

Moreover, based on the output of a hard-output detection scheme, it enables to compute a simpleform of reliability information. Thus, the additional gains of soft- vs. hard decision decodingcan be exploited at only slightly increased computational complexity.

This method utilizes an overlapping block-structure, i.e., a block shift ofS < L (cf. Figure 5.7).Since multiple blocks thus contain the same information symbol, processing of each blockdelivers (possibly different) beliefs on the same information symbol, i.e., multiple observationsare available. Suitably combining the observations obtained from processing of each blockresults in a (possibly more reliable) final decision. This process is illustrated in Figure 5.21.

info symbols:

final estimate:

L

S = 3

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info symbols:

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info symbols:S = 1

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L

Figure 5.21: Illustration of multiple-observations combining for block-wise processing of the information

symbols (block size L = 3) for block shifts S = L = 3 (top), S = L− 1 = 2 (center), and S = 1 (bottom).

Depending on the applied block-wise detection scheme (heresoft-output MSDD and DFDDare considered), there are different options how to combinemultiple soft/hard observations

13Clearly, these do not have to be computed by the autocorrelation device, which potentially leads to lowerpower consumption. However, note that the entire front-endhas to be implemented anyway.


ZMSDD =

ZmocMSDD =

Figure 5.22: Illustration how block-wise MSDD and block-wise mocMSDD operate on the correlation matrix

Z. The used correlation coefficients are highlighted, the ones which influence the decision of the fourthtransmit symbol are marked with a dashed line. Left: MSDD, right: mocMSDD. Nburst = 7, L = 2.

to deliver a final hard and/or soft decision for the respective information symbol [SF12b], asdescribed below.

Of course, the generation of multiple observations, i.e., processing of overlapping blocks, leadsto an increased computational complexity and structural delay; however, the autocorrelationfront-end remains unchanged. Moreover, whereas increasing the block size leads to an in-crease of computational complexity, which is exponential in L, processing overlapping blocksincreases the complexity only linearly.

As we aim for a flexible trade-off between performance and computational complexity, we varythe mutual shiftS = 1, . . . , L of adjacent blocks (i.e., overlap ofL − S symbols) to adjustthe number of multiple observations for each information symbol (cf. Figure 5.21). Thus, weobtain on averageL/S observations per information symbol (for finite symbol streams thisvalue can be ensured by reducing the block size at the beginning and at the end), with maximumoverlap forS = 1, and the traditional single observation per information symbol forS = L (cf.Figure 5.21).

Figure 5.22 depicts the utilized correlation coefficients of multiple-observations combining withMSDD (mocMSDD) compared to conventional MSDD. It can clearly be seen that more cor-relation coefficients affect the decision of each symbol. Compared to RSSE (cf. Figure 5.2),multiple-observations combining uses the same correlation coefficients, however, these are notprocessed jointly, but in separate processing steps, whoseindividual results are then combined.

5.3.6.1 Combining Multiple Soft Decisions

If soft decisions are available for each individual observation, as obtained from soft-outputMSDD, we may consider these observations as independent (though independence cannot beassured), and obtain the resulting final estimate accordingto the principle of parallel informationcombining [LH06]. In terms of LLRs, the resulting LLR for each symbol is then directly givenas the sum of the corresponding individual LLRs. This process is illustrated in the left part ofFigure 5.23.


combining selection

info symbols:

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L

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Figure 5.23: Illustration of combining multiple soft decisions (left) and subset MSDD (right) for block-wise

processing of the information symbols (block size L = 3, block shift S = 1).

In the case of coded transmission, the resulting LLR can directly be passed to the subsequent(soft-input) channel decoder. Since LLR clipping affects each SO-MSDSD algorithm run, themaximum absolute input value isL·LLRmax. Final hard-decisions can be obtained by evaluatingthe sign of the sum of the LLRs, corresponding to combining the individual hard decisionsaccording to their reliability.

5.3.6.2 Combining Multiple Hard Decisions

Based on multiple hard decisions, here obtained from DFDD (so-called mocDFDD—of course,hard-output MSDD is also possible, but not considered for brevity), a final hard decision isdirectly given following the majority of the individual decisions (i.e., evaluating the sign ofthe sum of the individual estimates), which effectively treats each observation with equal reli-ability (in case of a draw, we choose±1 at random). Clearly, this strategy leads to improvedperformance only if more than two observations are available per symbol, i.e., only ifS < L/2.

The most interesting option is to combine multiple hard decisions of the same symbol to forma single soft decision. Exemplarily, consider the case thatthree hard decisions for the sameinformation symbol are available. If these coincide, the reliability of this symbol is three timesas large as the reliability of only a single observation, whereas if only two coincide and onediffers, the reliability is reduced to the case of a single observation.

This method can be implemented by using the sum of the individual hard-decisions as (quan-tized and scaled) “soft” output. Depending on the employed channel decoder, scaling of thesesoft-output values may be neglected if the decoding metric is scale-invariant (e.g., in a convo-lutional coded system employing the Viterbi algorithm for soft-decision decoding). However,e.g., in an LDPC-coded system, the soft output should be scaled to match the decoder inputrange.

5.3.6.3 Comparison to Subset MSDD

Multiple-observations combining can be seen as a development of subset MSDD introduced fornoncoherent detection of DPSK (or more general differential space-time modulation) [PLS06,PL07, PHL09, RSH13], where similarly multiple observations are generated with an overlap-ping block-structure. There, motivated by the observationthat the reliability of the estimatesdegrades at the block edges, only the symbols of the middle ofeach block are processed fur-


ther.14 The estimates for symbols from the block edges, which are assumed to be unreliable, aresimply discarded. This process is illustrated in the right part of Figure 5.23.

The drawback of subset MSDD is that not all available information is utilized. Moreover, thistechnique is not applicable in MSDD when an optimized decision order is employed, since inthis case, the most reliable decisions are in general not limited to the block center.

5.3.7 Numerical Results II

The performance evaluation of the system setup as describedin Section 5.3.5.1 is extended toanalyze the effectiveness of multiple-observations combining.


Figure 5.24 depicts the error rate of multiple observationscombining for maximum block over-lap (S = 1) for the cases of soft-output MSDD and sorted DFDD employed as block-wisedetection scheme. For soft-output MSDD, the LLR clipping level is set toLLRmax = 10, whichinduces only negligible loss compared toLLRmax → ∞ [SF12b]. For reference, block-wiseMSDD without multiple-observations combining is also shown (no overlap,S = L).

Multiple-observations combing leads to notable gains of a few tenths ofdB. Interestingly, sim-ply combining multiple hard decisions (majority decision over the estimates, i.e., sign of sumof estimates, hard-output mocDFDD), as obtained from sorted DFDD as block-wise detectionscheme, leads to almost the same performance as combining multiple soft decisions (i.e., signof sum of LLRs, mocMSDD), as obtained from soft-output MSDD,where the different obser-vations are weighted according to their individual reliability. Note that forL = 2, combining ofmultiple hard decisions cannot lead to additional gains.

However, the reasoning for multiple-observations combining is the possibility to compute re-liability information on the estimates with very low-complexity, especially when combiningmultiple hard decisions to a single soft decision (especially in the case of sorted DFDD asblock-wise detection scheme, so-called soft-output mocDFDD). This benefit is not reflectedin Figure 5.24, and will be shown later in the design of coded IR-UWB communications (cf.Section 6.4.3.1).

In Figure 5.25 we evaluate the performance of sorted block-wise DFDD with multiple-observa-tions combining when simplified exponential weighting prior to autocorrelation is applied (cf.Section 3.2.3). As already stated, this technique requiresadditional detailed knowledge of thechannel characteristics. Similar to symbol-wise differential detection, also in case of advancedautocorrelation-based detection schemes this additionalknowledge slightly improves the error-rate performance over uniform weighting within an optimized correlation interval (cf. also Fig-ure 5.6) for low-to-moderate signal-to-noise ratio. For high signal-to-noise ratio, however, thissimplified exponential weighting performs even worse compared to uniform weighting. This isdue to the fact that neither of the variants exactly resembles the underlying power-delay profile.

14Of course, in case of soft-output MSDD as block-wise detection scheme, this approach is preferably combinedwith the repeated-tree-search concept as described in Section 5.3.2.5, since reliability information is only requiredfor the centered symbols within each block.


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Figure 5.24: SER of block-wise detection of DTR IR-UWB vs. Ec/N0 in dB for different block sizes L = 2(squares), 5 (diamonds), and 10 (circles), right-to-left. Solid-black: mocMSDD (maximum overlap, S = 1),

dashed-black: hard-output mocDFDD (S = 1), solid-gray: conventional MSDD (S = L), left-most solid-

gray: detection with CSI, right-most dashed-gray: DD. Non-line-of-sight channel model 2 of [MCC+06] withTi = 50 ns.

5.3.7.2 Trade-Off Computational Complexity vs. Error-Rate Performance

Of course, multiple-observations combining leads to increased computational complexity, sincefor each individual observation block-wise detection has to be performed. The number of in-dividual observations can be adjusted by the block shiftS, thus, also the increase is roughlyproportional to the number of observations per symbolL/S. Recall that computational com-plexity is measured as the number of visited nodes per symbolin the sphere-decoder searchprocess.

The resulting trade-off is investigated in Figure 5.26 for different target error rates and blocksizes.15 Again, we consider the two detection schemes combining multiple soft decisions ob-tained from soft-output MSDD (sign of sum of LLRs, mocMSDD),and combining multiplehard decisions obtained from DFDD (sign of sum of estimates,mocDFDD).

For both cases the upper-left point corresponds to the maximum block-overlap (maximum num-ber ofL observations,S = 1), and the lower-right point to traditional detection with only asingle observation per symbol (no overlap,S = L). The performance of soft-output MSDDimproves with increasing number of combined individual soft decisions, with gains of up to0.8 dB compared to traditional MSDD; the increase in complexity isproportional to the aver-age number of observations per symbol,L/S.

15In order to handle the increased amount of required simulations, the simulations have been performed on theapproximate discrete-time model of Section 3.5 withNi = 500 andEp set according to (3.37) for the line-of-sight channel model 2 of [MCC+06] with Ti = 50 ns. Whereas the relative statements remain the same, absoluteperformance estimates are quite optimistic for largeL, sayL = 10.


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Figure 5.25: SER of sorted block-wise DFDD with MOC (S = 1) of DTR IR-UWB vs. Ec/N0 in dB fordifferent block sizes L = 2 (squares) and 5 (diamonds). Solid-black: simplified exponential weighting,

solid-gray: uniform weighting within Ti = 50 ns, dashed-black: DD with simplified exponential weighting,

dashed-gray: DD with uniform weighting within Ti = 50 ns, left-most solid-gray: detection with CSI. Non-line-of-sight channel model 2 of [MCC+06].

Combining multiple hard decisions (mocDFDD), improvements are only achieved forS < L/2,as otherwise only up to two observations are available. ForS ≥ L/2, multiple-observationscombining with DFDD achieves even larger gains at lower and,in particular, fixed complexity,as opposed to the gains and the varying complexity of sphere-decoder-based soft-output MSDD.The increase in complexity is exactly proportional to the average number of observations persymbol, hence the number of nodes visited per information symbol equalsL− S + 1.

5.3.8 Virtually Increased Block Size

Finally, we note that the restriction to a block size matchedto the number of branches of theautocorrelation device is somewhat artificial and can be abandoned. The block size can be“virtually” increased toLvb (> L) by neglecting the contribution of the correlation coefficientsnot computed by the autocorrelation device in the decision process (the opposite direction ofLvb < L is trivial).

A virtually increased block size can be used with all previously described block-wise detectionschemes, including multiple-observations combining. Especially for block-wise DFDD thisenables a larger degree of freedom in finding the optimum decision order and thus potentiallyleads to improved performance.

This technique is easily implemented using the same algorithms as given above, i.e., in particu-lar, the algorithm of Figure 5.14 for DFDD with optimized decision order. However, now thesealgorithms operate on an extended matrixZvb where the correlation coefficients not computed


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Figure 5.26: Trade-off required (average) computational complexity vs. required Ec/N0 for a target SER =10−3 of block-wise detection with multiple-observations combining obtained by adjusting the block shift Sfrom 1 (left-most upper end of each curve) to L (right-most lower end) for different block sizes L. Dashed-black: mocDFDD, solid-gray: mocMSDD. Approximate discrete-time model with Ni = 500 and Ep set

according to (3.37).

by the autocorrelation device are simply set to zero.

The drawback of this method is the increased structural delay of the detection device and thehigher memory requirements. In case of DFDD the computational complexity remains almostunchanged. Clearly, this technique is applicable only if the burst length exceedsL.

For sake of clarity, we do not evaluate the effectiveness of the above approach, but note thatthe extreme case of a virtual block size equal to the burst length resembles sequence estimationof the entire burst of receive symbols, as it similarly searches for the best sequence of symbolswithin the entire burst. This is discussed in detail in the next section.

5.4. Sequence Estimation 95

5.4 Sequence Estimation

As already motivated in Section 5.1, a different approach toapproximate ideal noncoherentsequence estimation (INSE) is obtained by restricting the detection process to a fixed mem-ory length ofL symbols and processing the receive symbol stream, e.g., in asliding-windowfashion. Again, these detection schemes operate on the output of anL-branch autocorrelationdevice (cf. Figure 5.1).

This approach is closely connected to block-wise detectionwith a virtually increased block size,as discussed above. Hence, one could simply employ the very same algorithms as presentedabove, with the difference that they operate on a virtually increased block size of

Lvb = Nburst − 1 . (5.33)

In this view, based on the output of anL-branch autocorrelation device one could directly em-ploy sphere-decoder-based MSDD to estimate the optimum sequence of symbols within theburst. Compared to INSE, this corresponds to truncating thememory length toL, and isthus equivalently implemented by so-called reduced-statesequence estimation (RSSE) [LT06b,LT08], which, in turn, can be solved using the well-known Viterbi algorithm.

Similarily, in order to approximate optimum sequence estimation at reduced computationalcomplexity, block-wise DFDD can be employed with a virtual block size ofLvb = Nburst − 1.

Based on the relation of the Viterbi algorithm and decision-feedback sequence estimation,which has been established in [HH89, EQ88], a further variant of DFDD which operates ina sliding-window fashion is discussed [SF11c].

Subsequently, we briefly review these concepts and compare them to the previously describedblock-wise detection schemes. We begin with DFDD with a virtually increased block size.

5.4.1 DFDD with Virtually Increased Block Size

DFDD with a virtually increased increased block size is implemented using the same algo-rithms as given above, i.e., in particular, the algorithm ofFigure 5.14 for DFDD with optimizeddecision order. However, now these algorithms operate on anextended matrixZvb where thecorrelation coefficients not computed by the autocorrelation device are simply set to zero. Es-pecially for block-wise DFDD this enables a larger degree offreedom in finding the optimumdecision order.

The processing of sorted DFDD with virtually increased block size is illustrated in Figure 5.27.For this example, we setLvb = 10 (corresponding toNburst = 11) and consider autocorrelationdevices with differentL (the illustrations have been computed for exactly the same receivesignal). For illustration purposes, we restrict the starting point of sorted DFDD tok0 = 0, asopposed to the optimized starting point in Section 5.3.4 andFigure 5.13. When implementingthis detector, the starting pointk0 is, however, optimized according to (5.27).

ForL = 1, this method falls back to symbol-wise differential detection where each symbol isdecided based on its predecessor.


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1

2

3

4

5

6

7

8

9

10

Z(L=2)vb =

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

Z(L=3)vb =

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

Z(L=10)vb =

0 1 2 3 4 5 6 7 8 9 10

0

1

2

3

4

5

6

7

8

9

10

Figure 5.27: Illustration of decision order of sorted DFDD with virtually increased block size for L = 1(DD), 2, 3, and 10. Virtually increased block size Lvb = 10, k0 = 0.

Starting withL = 2, different (potentially better) decision orders are possible. Whereas allschemes start in the top-left corner, at some point a decision is made not for the next symbol,but for a symbol up toL time steps apart. The decision of the symbol that has been left outis catched up in later processing steps, then based on the feedback of more previously decidedsymbols. With increasingL more look-ahead is possible, which allows to postpone decisionsof currently unreliable symbols.

Although the structural delay of the detection device is increased and the memory requirementsare higher, the computational complexity remains almost unchanged compared to conventionalblock-wise DFDD. In both cases, only up toL correlation coefficients are summed up for thedecision of each symbol.

5.4.2 Reduced-State Sequence Estimation

Based on Figure 5.2, it can be seen that RSSE aims to approximate ideal noncoherent sequenceestimation by similarly searching for the optimum sequenceof symbols within the entire burst.Different to ideal sequence estimation, RSSE is based on theoutput of anL-branch autocorre-lation device only.


The decision metric for RSSE is thus obtained from (5.1) by truncating the memory length toL, yielding the detection problem

bRSSE = argmaxb∈B

NburstDTR

b0=1

Nburst−1∑

i=1

bi

i−1∑

l=max(0,i−L)

blzl,i

. (5.34)

Depending onL, RSSE will trade-off between symbol-wise differential detection (L = 1) andideal noncoherent sequence estimation (L = Nburst − 1).

This detection problem can efficiently be solved using the well-known Viterbi algorithm op-erating on a trellis with2L states. As usual for the Viterbi algorithm, the final estimate is thesequence with maximum path metric.16 The Viterbi algorithm ensures a fixed computationalcomplexity (exponential inL, but linear inNburst).

Similarly, taking the point of view that ideal noncoherent sequence estimation is a search in abinary tree of depthNburst, restricting the memory length toL, which is a direct consequence ofthe restriction to anL-branch autocorrelation device, results in nodes, which become equivalentstarting from a depth greater thanL. Merging these nodes, the trellis structure with a total of2L states is obtained. This procedure is in the spirit of delayed decision-feedback sequenceestimation [EQ88, HH89].

In this framework, a soft-output sequence estimation couldbe realized using the well-knownBCJR algorithm [BCJR74]. For sake of brevity, and due to the effectiveness and the flexibleperformance-complexity trade-off of block-wise detection schemes (i.e., soft-output MSDD andDFDD with multiple-observations combining), this is not considered in this thesis.

5.4.3 Sliding-Window Decision-Feedback Differential Detection

Another variant of DFDD with a symbol-wise processing (in a sliding-window fashion) canbe derived from RSSE making use of its relation to DFDD, whichhas been established in[HH89, EQ88]. In this view, DFDD corresponds to a reduced-state sequence estimation withonly a single state. In contrast to block-wise DFDD, cf. (5.26), now for each symbolL previousdecisions are fed back to improve the decision of the currentsymbol, thus, fixing the memorylength toL as in RSSE. In detail, so-called sliding-window DFDD (swDFDD) chooses

bswDFDDi = sign

i−1∑

l=max(0, i−L)

zl,ibswDFDDl . (5.35)

Due to the symbol-wise processing, sorting is not applicable for sliding-window DFDD. Thetransient behavior at the beginning of the stream leads to sliding-window DFDD and block-wiseDFDD being equivalent whenL = Nburst − 1.

16Note that, in principal, the structural delay of RSSE is again NburstT , since a final decision can only bedelivered after the entire burst has been received. However, it is well-known that delivering the preliminary bestsequence for the symbols roughly5L steps prior to the current symbol leads to negligible loss inperformance. Thestructural delay is thus approximately5LT .


ZRSSE =

ZswDFDD =

Figure 5.28: Illustration how RSSE and swDFDD operate on the correlation matrix Z. The used correlation

coefficients are highlighted, the ones which influence the decision of the fourth transmit symbol are markedwith a dashed line. Left: RSSE, right: swDFDD. Nburst = 7, L = 2.

The operation of sliding-window DFDD is compared to RSSE in Figure 5.28. Whereas inDFDD the current symbol is decided based on the previous decisions, i.e., the window extendsonly into the past, in RSSE the window also has a look-ahead component. This is due to thefact that the current symbol is still present in future states, such that its decision is affected alsoby future symbols.

Finally, similar to block-wise sorted DFDD, due to the binary signaling, we expect that errorpropagation hardly impacts the performance of sliding-window DFDD.


5.4.4 Comparison to Block-Wise Detection Schemes

The performance of the detection schemes described in this section is compared to their block-wise counterparts MSDD and sorted DFDD with multiple-observations combining, the systemparameters are the same as described in Section 5.3.5.1. Recall that the burst length is set toNburst = 101.

For the same block size/memory lengthL these schemes are directly comparable, as they op-erate on the output of the same autocorrelation device (cf. Figure 5.1). The schemes differ inperformance and computational complexity.


Again we start with an evaluation of the error rates in Figure5.29 and 5.30 for RSSE vs.MSDD and the three variants of DFDD. These are sliding-window DFDD (swDFDD), sortedblock-wise DFDD with multiple-observations combining andmaximum overlap (mocDFDD)and sorted block-wise DFDD with virtually increased block size of Lvb = Nburst − 1 = 100

(vbDFDD).

Whereas RSSE outperforms MSDD with multiple-observationscombining for all parametersL,the variants of block-wise DFDD significantly outperform sliding-window DFDD. Especiallyfor larger block sizes, the slopes of the error rate curves diverge. Interestingly, for a blocksize ofL = 2, sorted block-wise DFDD (which in this case is equivalent tohard-output MSDD)shows very similar performance compared to sliding-windowDFDD. Hence, deciding a symbolbased on the feedback ofL = 2 previous decisions is almost equivalent to finding the optimumsequence of three symbols, where each symbol decision is affected by two others, as well.

In case of DFDD with virtually increased block size the increased degrees of freedom for findingthe optimum decision order significantly help to improve theerror-rate performance comparedto DFDD with multiple-observations combining. The performance is very close to optimalRSSE, as seen in Figure 5.31. ForL = 2 both, RSSE and vbDFDD, perform almost equal andyield a gain over symbol-wise differential detection of2 dB.

For sake of completeness, we consider the application of exponential weighting instead of uni-form weighting within an optimized correlation interval for DFDD with virtually increasedblock size in Figure 5.32. Similar to its application in sorted block-wise DFDD with multiple-observations combining (cf. Figure 5.25), this technique however does not yield significantperformance improvement compared to uniform weighting andeven leads to some loss at highsignal-to-noise ratio.


6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.29: SER of autocorrelation-based detection of DTR IR-UWB vs. Ec/N0 in dB for different ACRs

with L = 2 (squares), 5 (diamonds), and 10 (circles). Black: RSSE, gray: block-wise mocMSDD (maximumoverlap, S = 1), left-most solid-gray: detection with CSI, right-most dashed-gray: DD. Non-line-of-sight

channel model 2 of [MCC+06] with Ti = 50 ns.

6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.30: SER of autocorrelation-based detection of DTR IR-UWB vs. Ec/N0 in dB for different ACRs

with L = 2 (squares), 5 (diamonds), and 10 (circles). Solid-black: swDFDD, dashed-black: vbDFDD, gray:

block-wise mocDFDD (maximum overlap, S = 1), left-most solid-gray: detection with CSI, right-most solidgray: DD. Non-line-of-sight channel model 2 of [MCC+06] with Ti = 50 ns.


6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.31: SER of autocorrelation-based detection of DTR IR-UWB vs. Ec/N0 in dB for different ACRs with

L = 2 (squares), 5 (diamonds), and 10 (circles). Solid-gray: RSSE, dashed-black: block-wise DFDD withvirtually increased block size Lvb = 100 = Nburst − 1, left-most solid-gray: detection with CSI, right-most

dashed-gray: DD. Non-line-of-sight channel model 2 of [MCC+06] with Ti = 50 ns.

6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.32: SER of sorted DFDD with virtually increased block size of DTR IR-UWB vs. Ec/N0 in dBfor different block sizes L = 2 (squares) and 5 (diamonds). Solid-black: simplified exponential weighting,

solid-gray: uniform weighting within Ti = 50 ns, dashed-black: DD with simplified exponential weighting,

dashed-gray: DD with uniform weighting within Ti = 50 ns, left-most solid-gray: detection with CSI. Non-line-of-sight channel model 2 of [MCC+06].


5.4.4.2 Trade-Off Computational Complexity vs. Error-Rate Performance

As above, for the block-wise detection schemes computational complexity can be measured asthe number of nodes visited per symbol with block-wise DFDD visitingL nodes perL symbols,due to its relation to sphere-decoder-based MSDD, cf. Section 5.3.2.1 (there is no need to con-sider MSDD for uncoded transmission, due to the excellent performance of sorted DFDD). Thismeasure may directly be compared to the number of states per symbol,2L, investigated in theViterbi algorithm employed for RSSE [SF10b]. In this view, sliding-window DFDD examinesa single state per symbol. Due to its poor performance, it is not included in this discussion, too.

When compared to conventional block-wise DFDD, DFDD with virtually increased block sizevisits Lvb nodes perLvb symbols. The virtually increased block size does not increase thecomputational complexity, since similarly only up toL correlation coefficients are summed upfor the decision of each symbol.

This complexity measure is different to the one adopted in [SF11c], where the number of re-quired real-valued additions is taken as a complexity measure (real-valued multiplications arenot required by any of the discussed algorithms, apart from simple scaling by±1 or by powersof 2). Although being more general, the nodes/states measure ismore informative and descrip-tive, as the operations per node and state performed by the different algorithms are directlycomparable. Furthermore, the conclusions remain unchanged compared to [SF11c].

Figure 5.33 depicts the trade-off of computational complexity vs. error-rate performance forvarious target error rates and block sizes, respectively memory lengthsL. We conclude thatsorted block-wise DFDD with a virtually increased block size outperforms all other investigatedschemes either in terms of performance or in terms of computational complexity.

Due to its low complexity (similar to that of conventional block-wise or sliding-window DFDD),thus sorted DFDD operating on a virtually increased block size should be preferred over itscompetitors in case of uncoded transmission. For a given autocorrelation device it delivers thebest performance at very low computational complexity.

These benefits come at the cost of increased structural delayand larger memory requirements,since the entire burst has to be available for the decision process to start.


9 10 11 12 13 14 15 16 17 180

5

10

15

20

25

30

200

400

600

800

1000

required 10 log10(Ec/N0) [dB]

requ

ired

co

mp

uta

tio

nal

co

mp

lexity

DD

mocDFDD

vbDFDD

RSSE

Figure 5.33: Trade-off required computational complexity vs. required Ec/N0 for a target SER = 10−2 (gray)

and 10−4 (black) of autocorrelation-based detection of DTR IR-UWB for different ACRs with L = 2–10, and

15 (right-to-left). Squares: RSSE (only L = 2 to 5 and 10), empty circles: sorted DFDD with MOC, filledcircles: sorted DFDD with virtually increased block size, diamond: DD. Non-line-of-sight channel model 2

of [MCC+06] with Ti = 50 ns.


5.5 Related Receiver Concepts for DTR IR-UWB

In this section, two related receiver concepts for DTR IR-UWB are discussed and compared tothe methods presented above. The first approach aims to improve the performance of symbol-wise differential detection by employing a so-called decision-directed autocorrelation device.The second approach can be seen as a means to mitigate the drawback of the high implemen-tation complexity of the autocorrelation device, by utilizing the novel concept of compressedsensing.

5.5.1 Decision-Directed Autocorrelation-Based Differential Detection

Similar to DFDD, in decision-directed autocorrelation-based differential detection, previousdecisions are fed back in order to improve the error-rate performance [DB04, ZLT06]. Tothis end, this approach employes a so-called decision-directed [ZLT06] (or adaptive [DB04])autocorrelation device, where previous decisions are fed back into the autocorrelation devicein order to improve the quality of the correlation template.Similar to the methods developedin this thesis, this method relies on the assumption that thechannel impulse response remainsunchanged over several symbol durations.

5.5.1.1 Decision-Directed Autocorrelation Device

In autocorrelation-based detection, the decision variable zk−1,k is obtained by correlating thecurrent receive signal with a correlation template (i.e., the delayed signal or state of the auto-correlation device). In conventional autocorrelation-based detection, as described above, thistemplate signal is given as the receive signal of the previous symbol interval (cf. (3.31)).

The quality of the template can be improved by feeding back previous decisions of the transmitsymbols [ZLT06, DB04]. This process is illustrated in Figure 5.34.

kThRX(t)

n0(t)

aDDk

∫

(Ti)·dtr(t)

sign(·)zddk (t)

T

1−βddβdd

zk−1,k

Figure 5.34: Block diagram of DTR IR-UWB receiver for decision-directed autocorrelation-based differential

detection (uniform weighting within Ti).

The conventional computation of the correlation coefficients given by (3.31), is replaced by

zddk−1,k =

∫ Ti

0

r(t− kT )zddk (t)dt (5.36)

wherezddk (t) is the correlation template of the autocorrelation device in thekth time step. As inconventional symbol-wise differential detection, in decision-directed differential detection the

5.5. Related Receiver Concepts for DTR IR-UWB 105

ZDD =

ZddDD =

Figure 5.35: Illustration how DD and ddDD operate on the correlation matrix Z. The used correlation

coefficients are highlighted, the ones which influence the decision of the fourth transmit symbol are markedwith a dashed line. Left: DD, right: ddDD. Nburst = 7, L = 1.

estimate for the information symbols is given by

addDDk = sign

(

zddk−1,k

)

. (5.37)

Different from conventional differential detection, the correlation template is recursively up-dated by feeding back the decisions via

zddk (t) = βdd · r(t− (k − 1)T ) + (1− βdd) · addDDk−1 zddk−1(t) (5.38)

whereβdd is an optimization parameter. Forβdd = 1, the feedback of previous decisions iseffectively switched off, i.e., the performance is equal toconventional autocorrelation-baseddifferential detection (this setting is required for initialization). Forβdd = 0, after suitableinitialization, all symbols are decided based on their correlation to the first symbol interval(neglecting error propagation, the performance of this scheme is again equal to differentialdetection). This processing basically implements a first-order IIR filter and thus leads to alow-pass filtering characteristic.

Assuming correct decisions, the amount of noise in the recursively updated correlation templateis reduced step-by-step, which yields improved decisions on the information symbols. Eventu-ally, the correlation template represents an almost noise-free estimate of the receive pulse shape,i.e., essentially blind channel estimation is performed.

5.5.1.2 Related Approaches

In [ZLT06] two further implementations of a decision-directed autocorrelation device have beenproposed, one operating in a sliding-window fashion, and the second one based on a least-mean-squares (LMS) approach. Since for optimized setting of the parameterβdd the recursivedecision-directed autocorrelation device outperforms both approaches, and the implementationcomplexity is higher (more delay elements required for the sliding-window decision-directedautocorrelation device and higher computational complexity for the LMS approach) [ZLT06],for sake of brevity, we only consider the recursive decision-directed approach. Additional per-formance improvement can be achieved by weighting the feedback symbols according to theirreliability [ZLT06].


A different decision-directed approach has been presentedin [ZML10]. Here, the correlationtemplate is iteratively updated according to preliminary decisions on multiple symbols. Thismethod however requires to record the receive signal over multiple symbol intervals and re-peatedly correlate it with the updated template. Clearly, this represents a challenging task forhardware implementation and is thus not considered, too.


The error-rate performance of decision-directed autocorrelation-based differential detection iscompared to conventional autocorrelation-based differential detection in Figure 5.36 for theindoor non-line-of-sight channel model 2 of [MCC+06]. For both cases uniform weightingover Ti = 50 ns is assumed (corresponding toNi = 500). For comparison, a genie-aideddecision-directed autocorrelation-based differential detection is included, i.e., the correct trans-mitted symbols are fed back in (5.38), which allows to study the effect of error propagation. Aburst length ofNburst = 101 symbols is assumed, for which the channel remains constant;thetransient behavior at the beginning of each burst is reflected in the results.

For each signal-to-noise ratio, the optimum parameterβdd (found out of a finite set by numericalsimulations) is selected. The influence ofβdd can be deduced from Figure 5.37. The optimumβdd lies in the order of0.1 to 0.4. We observe significant gains for moderate error rates (3 dB at10−2), which however are reduced when considering low error rates (2 dB for 10−4). The lossdue to error propagation is marginal.

For reference, the performance of sorted DFDD with an extended block size ofLvb = 100 is alsoshown, which represents the best detection scheme operating on the output of an autocorrelationdevice withL = 2. This autocorrelation device thus requires one additionaldelay elementcompared to the decision-directed autocorrelation device.

Of course, decision-directed differential detection can be combined with different weightingsprior to autocorrelation, too. In Figure 5.38 (simplified) exponential weighting is applied (cf.Section 3.2.3). However, only marginal additional gains are obtained over decision-directeddifferential detection with uniform weighting in an optimized correlation interval.

5.5.1.4 Discussion

In summary, we conclude that decision-directed autocorrelation-based detection is an effectivetechnique for power-efficient receivers for DTR IR-UWB. However, it does not offer the flexi-bility with respect to the performance-complexity trade-off of the block-wise detection schemesemploying an extended autocorrelation device, and similarily is not able to mitigate the draw-back of accurate delaying of the receive signal.

Finally, the combination of a decision-directed autocorrelation device and block-wise detectionschemes, as well as the extension of this concept to different variants of IR-UWB (in particulardecision-directed energy detection of PPM), appears to be an interesting task for future work.


6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

DD

ddDD

vbDFDD (L = 2)

Figure 5.36: SER of decision-directed autocorrelation-based DD of DTR IR-UWB vs. Ec/N0 in dB compared

to conventional autocorrelation-based DD. Solid-black: ddDD, dashed-dotted-black: ddDD with genie-aided feedback, dashed-gray: DD, solid-gray: sorted DFDD with virtually increased block size Lvb = 100,

left-most solid-gray: detection with CSI. For all cases: uniform weighting over Ti = 50 ns. For each Ec/N0,

the optimum parameter βdd (found out of a finite set by numerical simulations, cf. Figure 5.37) is selected.Non-line-of-sight channel model 2 of [MCC+06].

0 0.2 0.4 0.6 0.8 110

−4

10−3

10−2

10−1

βdd

SER

Figure 5.37: SER of decision-directed autocorrelation-based DD of DTR IR-UWB vs. βdd for

10 log10(Ec/N0) = 10 dB, 13 dB, and 15 dB (top-to-bottom). Solid-black: ddDD, dashed-dotted-black:

ddDD with genie-aided feedback. For all cases: uniform weighting over Ti = 50 ns. Non-line-of-sightchannel model 2 of [MCC+06].


6 8 10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.38: SER of decision-directed autocorrelation-based DD (solid lines, βdd = 0.2) of DTR IR-UWB vs.

Ec/N0 in dB compared to DD (dashed). Black: simplified exponential weighting, gray: uniform weightingwithin Ti = 50 ns, left-most solid-gray: detection with CSI. Non-line-of-sight channel model 2 of [MCC+06].


5.5.2 Compressed-Sensing-Based Detection

The major—and most likely crucial—drawback of autocorrelation-based detection is the auto-correlation device required for computation of the correlation coefficients on which subsequentdetection schemes operate on (cf. Section 3.4.2). Recall, that this autocorrelation device couldbe implemented either in the analog or in the digital domain based on a sufficiently high sampledreceive signal. However, both options incur demanding challenges for hardware implementa-tion (cf. Section 3.4.2). While realization of the latter requires high sampling rates of up to10GHz, an analog implementation is hard to realize due to the required accurate delaying ofthe receive signal over multiples of the symbol duration, i.e., delays in the order of severalhundered nanoseconds.

Recent approaches for IR-UWB detection aim to avoid this drawback by exploiting the inher-ent sparsity of the IR-UWB receive signal via techniques known as compressed sensing (CS)[CW08, Don06, EK12]. The idea of compressed sensing is to represent a high-dimensionalvector by a vector of reduced dimension. If the original vector is sufficiently sparse (in somedomain) and the reduced vector consists of linear combinations of its elements, it is possible toreconstruct the original vector [SSF13].

For IR-UWB systems, the receive signal can be seen as the high-dimensional, but, due to theimpulsive nature of IR-UWB signals, sparse signal (in time domain). The most-common ap-proach in compressed-sensing-based detection is then to first obtain a relatively small numberof measurements of the receive signal—which can be done withrelatively low-complexity, inparticular compared to an autocorrelation front-end—followed by (joint or separate) recon-struction and detection, which then involves solving relatively complex optimization problems[GLL10, OL09, LZLZ10, WAPS07].

In [SF11b], we have combined both approaches, autocorrelation-based and compressed-sensing-based detection, in such a way, that we pick only the respective processing steps of low com-plexity, i.e., the measurement step of compressed sensing and the DFDD step of autocorrelation-based detection (cf. Section 5.3.3), and avoid the respective drawbacks, i.e., the autocorrela-tion front-end and algorithms for reconstruction. Thereby, the low hardware complexity ofcompressed-sensing-based detection and the low computational complexity of DFDD is re-tained, yet their excellent performance retrieved. This isachieved by avoiding the reconstruc-tion step and performing correlation directly based on the measurements [GLL10]—motivatedby the fact that, at the receiver, one is typically only interested in the detection of the informationsymbols;17 the reconstruction step is only a means to obtain this decision.

5.5.2.1 Compressed-Sensing-Based Receiver

To this end, similar to [GLL10], a compressed-sensing front-end is introduced, as shown inFigure 5.39, which performsNcs (≪ Ns) measurements of the receive signal, i.e., computes thecorrelation coefficient of the receive signal andNcs functionsdl(t), l = 0, . . . , Ncs − 1, whichare periodical in the symbol durationT . Again we represent these functions as their sampled

17Of course, the reconstructed signal delivers important information, e.g., for localization purposes, which arenot considered in this thesis.


∫

(T )·dt

∫

(T )·dt

∫

(T )·dt

kT

kT

kT

r(t)hRX(t)

n0(t)

d1(t)

d0(t)

dNcs−1(t)

yk,0

yk,1

yk,Ncs−1

CS front-end

Figure 5.39: Block diagram of compressed sensing receiver front-end for DTR IR-UWB.

equivalent (for one symbol duration)

dl = [dl(0), dl(Ts), . . . , dl((Ns − 1)Ts)] (5.39)

and thus compactly write

yk = rkD = bkpD + nkD (5.40)

whereD is theNs×Ncs measurement matrix, withD =[

dT0, . . . , dTNcs−1

]

. The actual (analogor sampled) receive signal is not directly observable, but veiled in these measurementsyk.

From the perspective of compressed sensing, the measurements areNcs randomly selected co-efficients of the receive signal represented in a different basis [CW08]. We assume thatD hasfull rank Ncs, and callD an orthogonal measurement matrix, ifDTD = const. · I.

Clearly, forNcs < Ns, the particular choice of the basis will influence the performance of thereceiver. In [SF11b], it has been shown, that binary random measurement functions (so-calledBernoulli matrices with elements from±1) are preferred for IR-UWB receivers as they offerexcellent performance and can be generated reproducibly and with low complexity.

5.5.2.2 Reconstruction and Detection

The most-common approach for compressed-sensing-based detection is to try to reconstruct thereceive signal based on the measurementsyk, which is then used to detect the information sym-bols usually via differential detection [GLL10], as shown in Figure 5.40. Taking into accountthe sparsity ofrk, the reconstruction involves solving the so-calledl0-reconstruction problem[CW08]

rcs−l0k = argmin ‖x‖0 , subject toyk = xD . (5.41)

There are various options to relax thel0-reconstruction problem, resulting in computationallymore tractable optimization problems. E.g., the reconstruction problem can be relaxed to a


r(t)hRX(t)

n0(t)

Dig

ital

AC

R

CS

fro

nt-en

d

zcsk−1,k

zcsk−2,k

zcsk−L,k argm

axbZ

csbT

Reco

nst

ructio

n

yk rcsk bcs

Figure 5.40: Block diagram of compressed-sensing-based DTR IR-UWB with separate reconstruction and

detection.

so-calledl1-reconstruction [CW08, OL09], which under certain conditions on the sparsity andthe number of measurements still ensures to find the optimal solution. A different approachis to employ the greedy algorithm orthogonal matching pursuit (OMP) [TG07], which deliv-ers a possibly non-optimal solution to (5.41). Additionally, it can be taken into account that,different from the classical compressed-sensing framework, reconstruction is based on noisymeasurements of the receive signal. Here, we do not considerjoint reconstruction and detec-tion [GLL10], as it involves solving relatively complex optimization problems and thus leads toprohibitively high computational complexity.

The approach considered here is different to the above process in two key aspects. First, moti-vated by the fact, that one is typically interested only in detection of the information symbolsand the reconstruction step often is only a means to this end,reconstruction should be kept ascoarse as possible, or even be dropped entirely, as long as detection is possible with sufficientperformance. In addition, in realistic IR-UWB scenarios the receive signal typically is “not sosparse at all”, due to the rich multipath propagation, such that the well-known conditions forreconstruction [Don06] do not hold and reconstruction fails in many cases. Second, the inherentloss due to the skipped reconstruction is then compensated by advanced autocorrelation-baseddetection schemes, similar to those described in Section 5.3.

The correlation coefficients thus are obtained from direct correlation of the measurements, cf.[GLL10],

zcsk−l,k = yk−lyTk . (5.42)

Noteworthy, CS-based detection with direct correlation ofthe measurements can be seen as aspecial case of weighted autocorrelation-based detectionsimilar to Section 3.2.3. Since

zcsk−l,k = rk−lDDTrTk (5.43)

holds, the weighting matrixΣ fulfills Σ2 = DDT. As opposed to conventional weighted

autocorrelation-based detection, the off-diagonal elements of this matrix are non-zero.

5.5.2.3 Autocorrelation-Based CS-Detection

Based on these correlations, i.e., the output of an all-digital compressed-sensing autocorrelationdevice following the (analog) compressed-sensing front-end, the very same detection schemes


can be used as in the case of detection based on the output of a conventional autocorrelation de-vice, i.e., symbol-wise compressed-sensing-based DD (csDD) [GLL10], compressed-sensing-based MSDD, and, in particular, compressed-sensing-based(sorted) DFDD (csDFDD).

There is a major advantage compared to a conventional autocorrelation device: Whereas, e.g.,the analog autocorrelation device requires accurate delaying of the analog receive signal, in theall-digital compressed-sensing autocorrelation device,delays of theNcs measurement samplesper symbol can easily be realized over several symbols. Limiting factors are only given if thechannel impulse response changes within this interval. Of course, as opposed to the approachin [GLL10], also the measurement matrix must not change for csDFDD to work. If the assump-tions are fulfilled, relatively large block sizes can be usedfor csDFDD, resulting in significantperformance gains over symbol-wise csDD. In addition, since direct correlation of the mea-surements is applied, the particular choice of the measurement matrix, i.e., knowledge of theunderlying basis, is not required for subsequent detectionschemes.

More precisely, instead of operating on the correlation coefficients z·,·, the detection schemessimply operate on the compressed-sensing counterpartszcs·,·, which are obtained from (5.42).


A detailed evaluation of CS-based IR-UWB receivers is beyond the scope of this thesis. Here,we only show selected numerical results and refer the interested reader to [SF11b] for exhaustivenumerical results and an analytical error-rate analysis ofCS-based differential detection.

Figures 5.41, 5.42, and 5.43 depict the error rate of the CS-based detection schemes differ-ential detection, sorted DFDD withL = 10, and approximate INSE (achieved through theapplication of sorted DFDD withL = Nburst − 1 = 100) operating onNcs = 100, 250, and500 non-orthogonal binary random measurements, respectively. For reference, conventionalautocorrelation-based differential detection is shown. The correlation interval is set to the en-tire symbol duration, in order to enable a fair comparison with CS-based detection (thus, neitherof the schemes requires knowledge of the channel characteristics). The remaining parametersare not changed compared to the previous setup (described inSection 5.3.5.1).

For this setup, an all-digital implementation of the conventional autocorrelation device requiresfs = 10GHz. Hence, withT = 100 ns, Ns = 1000 samples per symbol have to be recorded.The depicted CS-based receiver is based on only one half (Ncs = 500), a fourth (Ncs = 250),down to a tenth (Ncs = 100) of this number. Nevertheless, satisfactory performance is achieved.Interestingly, the loss of CS-based differential detection (cf. Figure 5.41) is not directly pro-portional to the compression ratioNcs/Ns. In [SF11b] we have shown, that (for orthogonalmeasurements) this loss is approximately proportional to

√

Ncs/Ns, only.

Already a CS-based receiver withNcs = 250 in combination with sorted DFDD withL = 10

(cf. Figure 5.42) performs almost equal to conventional differential detection. Additional gainsare achieved when further increasing the block size toL = 100, i.e., when approximately CS-based INSE is performed, as, here,L = Nburst − 1 (cf. Figure 5.43). Recall that for CS-baseddetection large block sizes are feasible due to direct correlation of the measurements only andthe low computational complexity of DFDD.


8 10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.41: SER of CS-based DD of DTR IR-UWB vs. Ec/N0 in dB for Ncs = 100, 250, and 500 (right-to-left)binary random measurements. Dashed-gray: conventional DD, solid-gray: detection with CSI. Non-line-of-

sight channel model 2 of [MCC+06], Ns = 1000.

5.5.2.5 Discussion

Compressed sensing represents a promising approach for low-complexity receiver design forIR-UWB systems. It alleviates the drawbacks associated to autocorrelation-based detection.In particular, it avoids the high sampling rates of an all-digital implementation and the accu-rate analog delay lines of an all-analog implementation. Wehave shown that autocorrelation-based detection schemes may simply operate on direct correlations of measurements. Thuscomputational-complex reconstruction algorithms, usually employed in CS-based detection, arenot required.

Beyond the scenario considered in this thesis, under certain channel conditions, CS additionallyenables efficient channel estimation, which allows to dispense with the requirement of inter-symbol-interference-free transmission and in turn leads to increased data rates. The interestedreader is referred to [OL09, LDL09, WAS+07].


8 10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.42: SER of CS-based sorted block-wise DFDD with MOC (block size L = 10) of DTR IR-UWB

vs. Ec/N0 in dB for Ncs = 100, 250, and 500 (right-to-left) binary random measurements. Dashed-gray:conventional DD, solid-gray: detection with CSI. Non-line-of-sight channel model 2 of [MCC+06], Ns =1000.

8 10 12 14 16 18 20 22 2410

−6

10−5

10−4

10−3

10−2

10−1


SER

Figure 5.43: SER of CS-based approximate INSE (via sorted block-wise DFDD operating with L = Nburst− 1)

of DTR IR-UWB vs. Ec/N0 in dB for Ncs = 100, 250, and 500 (right-to-left) binary random measure-

ments. Dashed-gray: conventional DD, solid-gray: detection with CSI. Non-line-of-sight channel model2 of [MCC+06], Ns = 1000.

5.6. Summary and Conclusions 115

5.6 Summary and Conclusions

In this chapter, various concepts for power-efficient low-complexity detection of DTR IR-UWBcommunications have been discussed and contrasted. All presented schemes operate on theoutput of an autocorrelation receiver and require the channel conditions to remain constant overseveral symbol durations.

For a givenL-branch autocorrelation device, we have seen that decision-feedback-based detec-tion schemes represent the best trade-off between computational complexity and performance.

In case of uncoded transmission of sufficiently long bursts of transmit symbols, and if memoryrequirements and structural delay play a minor role, sortedDFDD operating on the entire burstof receive symbols (so-called sorted DFDD with virtually increased block size) outperforms allother investigated detection schemes, as it achieves close-to-optimal performance (the optimumbeing achieved with RSSE) at very low computational complexity. However, for this scheme adecision is only available after the entire burst has been received.

If this structural delay has to be kept small and/or only veryshort bursts are transmitted, sortedblock-wise DFDD, where the block size is matched to the autocorrelation device, offers a verygood alternative. In case of coded transmission (and again for sufficiently large burst length),by processing overlapping blocks and combining the individual observations, DFDD can beextended to efficiently deliver reliability information for its estimates, which can be utilized inthe subsequent channel decoder in order to increase the power-efficiency of coded IR-UWBtransmission.

The remaining question to answer is, whichL should be implemented in theL-branch autocor-relation device. This number should be carefully selected,as it dictates the maximum requireddelayLT in the autocorrelation device, cf. Figure 5.1, and consequently strongly affects therequired hardware implementation complexity as well as theoverall receiver complexity.

This point is addressed in Figure 5.44. Clearly, if one is notwilling to or able to implement anydelay elements (L = 0), only energy detection, preferably in combination with binary PPM,is possible. If a single delay element with a delay of one symbol duration can be spent, bestperformance is achieved employing a so-called decision directed autocorrelation device.

For anL-branch autocorrelation device, i.e., if more delay elements are available, the twoDFDD schemes described above represent the most efficient detection schemes of its class.However, based on Figure 5.44 we conclude that the gain achieved with each additional auto-correlation branch saturates quickly. Hence, a good compromise between power efficiency andoverall receiver complexity is achieved at moderate delaysin the autocorrelation device, sayL = 5.

In order to avoid the implementation of an autocorrelation device operating directly on thereceive signal, we have proposed an approach, which is basedon the principle of compressedsensing. Here sampling of the receive signal at a (possibly prohibitively) high sampling rateis replaced by correlation of the receive signal with a moderate number of fixed measurementfunctions. Based on this receiver preprocessing, all presented detection schemes are applicable


0 1 2 3 4 5 6 7 8 9 10 157

9

11

13

15

17

19

21

DD

ddDD

perfect CSI

ED of PPM

sorted block-wise DFDD

DFDD with virt. inc. block size

requ

ired10

log 1

0(E

c/N

0)

[dB

]

maximum delay of autocorrelation device L in symbol durations T

Figure 5.44: Power efficiency (required Ec/N0 for target SER = 10−4) vs. maximum delay of autocorrelationdevice in symbol durations. Squares: block-wise DFDD, circles: DFDD with virtually increased block size

(Lvb = Nburst − 1), diamonds: decision-directed autocorrelation device (only L = 1). The points at L = 0represent energy detection of binary PPM, those at L = 1 differential detection of DTR. Dashed lines indicatethe performance of detection with CSI. Non-line-of-sight channel model 2 of [MCC+06] with Ti = 50 ns.

and yield satisfactory results already at small fractions of the Nyquist sampling rate (in the orderof 10 to 25%).

Finally, regarding the impact of multiuser interference, we refer to [ZM12a]. There it hasbeen shown that in particular sorted DFDD is robust to interference caused by the simultaneousoperation of up to hundreds of users, when simply treating the interference as additional additivenoise.

117

6. Design Rules for Coded IR-UWB

Communication

In this chapter, coded IR-UWB communication systems, whichemploy channel coding forincreased noise-robustness, are studied. We derive designrules which take into account the pre-viously discussed (truly) noncoherent detection schemes.To this end, we consider information-theoretic limits, in particular the channel capacity, as a well-suited measure for performanceassessment. Based on these evaluations, optimum rates for the applied channel code are identi-fied, that improve the overall power efficiency of the system.These design rules are verified bymeans of numerical simulations using convolutional codes in combination with the previouslypresented soft-output detection schemes.

After a brief motivation of our approach in Section 6.1, design rules for PPM IR-UWB with en-ergy detection and DTR IR-UWB in combination with autocorrelation-based detection schemesare derived in Section 6.2 and 6.3, respectively. In Section6.4 these design rules are validatedby means of numerical simulations of convolutional coded IR-UWB transmission.

6.1 The Minimum Required Energy per Bit

6.1.1 Power-Efficient Noncoherent Detection

Recall, that the minimum energy per information bitEb required for reliable communicationover an AWGN channel with (one-sided) noise-power spectraldensityN0 is given as the ratioEb/N0 = ln(2) (or, equivalently,−1.59 dB) [Hub05, Sha48]. This operating point correspondsto highest power-efficiency, but infinitely small spectral efficiency, i.e., either infinitely largebandwidth or close-to-zero rate of the transmission scheme. Thus, in principle, this point canbe approached using, e.g., BPSK with an ideal code of code rate approaching zero.

However, following results for noncoherent detection in various digital communication schemes,such as noncoherent detection of PPM with finiteD [BDL+76, Sta85, SF11a] and differentialdetection of DPSK [PS98, CR01], the restriction to noncoherent detection schemes leads to

118 6. Design Rules for Coded IR-UWB Communication

a very different behavior: highest power-efficiency is achieved at non-zero rate. Hence, alsoin coded IR-UWB systems employing noncoherent detection weexpect that highest power-efficiency is achieved at non-zero code rate.

These optimum rates can be found by computing the average mutual information, i.e., the chan-nel capacity1 for the modulation and detection scheme, between the outputof the channel en-coder and the input of the channel decoder (e.g., log-likelihood ratios or hard decisions) asshown in Figure 6.1 [SF11d].

c

DetectionModulation

I(c; LLR)

Channel

IR-UWB communication channel

ENC DEC

LLR / cRX

Figure 6.1: Block diagram of coded IR-UWB transmission and illustration of conducted approach.

Building upon the method of performance estimation using information processing characteris-tics [SFH12], these capacity results for the inner communication channel (from encoder outputto decoder input) can directly be used to compute performance estimates (bit error rate andcapacity) for the overall coded channel given the information processing characteristic of thechannel coding scheme for a binary-input AWGN channel.

6.1.2 Excursus: Different Looks at the Capacity

In order to explain the reasoning behind optimum operating points, we first illustrate the dif-ference of looking at the capacity plotted vs. the ratio of energy per symbol2 over noise-powerspectral density,Es/N0, and the capacity plotted vs. the ratio of energy per information bit overnoise-power spectral density,Eb/N0. We consider the well-defined example of coded modu-lation of (D = 16)-ary PPM transmitted over the AWGN channel with coherent detection, cf.Figure 6.2 [SF11a].

Clearly, the capacity is a monotonically increasing function of the signal-to-noise ratioEs/N0,i.e., with increasingEs/N0 more information bits can be reliably transmitted over the channel,

1We note that the capacity ensures error-free transmission,whereas usually finite target error rates are relevantfor the design of coded IR-UWB systems. Following rate-distortion theory, our approach can be extended toincorporate design for finite target error rates. This extension, however, hardly changes the conclusions and is thusnot pursued subsequently.

2The notation adopted in this excursus is more generic and to some extend isolated from the rest of this thesis;it closely follows the common notation for digital communications. In particular, as common for digital commu-nications we defineEs as the energy per modulation symbol, which has to be distinguished from the energy percode symbolEc, as defined in (2.21).

6.1. The Minimum Required Energy per Bit 119

cf. the top part of Figure 6.2. Here the multi-level coding3 (MLC) or constellation-constraintcapacity and the bit-interleaved coded modulation4 (BICM) capacity [SF11a] are shown as afunction ofEs/N0 in semi-logarithmic and linear scale.

From an information-theoretic point of view, it often shedsmore insight to consider plots of thecapacity vs. the ratio of energy perinformation bitand noise-power spectral density,Eb/N0, asdepicted in the bottom part of Figure 6.2 (again in semi-logarithmic and linear scale). In thiscontext, for a given capacityC, the energy per information bit is given as

Eb = Es/C . (6.1)

−15 −13 −11 −9 −7 −5 −30

1

2

3

4

cap

acity

[bit/

sym

bo

l]

10 log10(Es/N0) [dB]0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

4

cap

acity

[bit/

sym

bo

l]

Es/N0

−1.59 0 1 2 3 4 50

1

2

3

4

cap

acity

[bit/

sym

bo

l]

10 log10(Eb/N0) [dB]1 1.5 2 2.5 3

0

1

2

3

4

cap

acity

[bit/

sym

bo

l]

Eb/N0

Figure 6.2: Capacity of 16-PPM with coherent detection vs. 10 log10(Es/N0) (top left), Es/N0 (top right),

10 log10(Eb/N0) (bottom left), and Eb/N0 (bottom right). Solid-black: BICM capacity, solid-gray: MLCcapacity, dashed-black: operating point of Eb/N0 = 1.5, square marker: optimum operating point, shaded

area: gap to capacity.

Whereas the MLC capacity shows the usual behavior of highestpower efficiency achieved atzero rate—here obtained at [SF11a]

Eb/N0|C=0 = ln(2)D

D − 1(6.2)

3A detailed classification of coded modulation schemes can befound in [WFH99, GMC08, Sti10] and inSection 6.3.1.

4The BICM capacity of orthogonal PPM constellations is independent of the choice of the binary labeling rule.


which is slightly larger compared to the absolute minimum ofln(2) due to the non-zero meanof the constellation—the BICM capacity shows a very different shape, similar to that we laterexpect for noncoherent detection schemes (although here coherent detection is performed).

The optimum operating point is achieved at non-zero rate as indicated (more precisely, atroughly1.9 bit per symbol andEb/N0 ≈ 1.35). At this operating point, both options, decreasingand increasing the code rate, lead to operating points whichdo not allow reliable transmission.

This behavior can be explained as follows (motivated by [AAS09]): Consider a fixed operatingpoint of Eb/N0 and an arbitrary rateR, as indicated in Figure 6.2 forEb/N0 = 1.5. SinceR = Es/Eb, this operating point translates into a linear function ofR in Es, i.e., for differentratesR we move along a line with slope1/Eb as shown in the top right part of Figure 6.2.Clearly, operating points which, in principle, allow reliable transmission must lie below therespective capacity curve. For MLC using16-PPM this is the case up to a signal-to-noise ratioof Es/N0 = 0.33 and a rate of roughly3.5 bit per symbol. According to the channel codingtheorem, selecting code rates below this threshold allows error-free transmission.

This is different for BICM using16-PPM. In this case, since the BICM capacity is a non-concave function inEs/N0, only values ofEs/N0 between0.1 and0.3 enable reliable transmis-sion, i.e., the rate must be in the interval from1 to 3 bit per symbol. This region is indicatedin Figure 6.2. For code rates out of this interval, reliable transmission is not possible at anoperating point ofEb/N0 = 1.5.

For this setting, we observe an extreme point, i.e., the minimum possibleEb/N0 for error-freetransmission of BICM using16-PPM, at roughly1.9 bit per symbol andEb/N0 ≈ 1.35. Here,the corresponding lineR = Es/Eb just touches the capacity plotted vs.Es/N0 in a single point.

For comparison, based on so-called wideband analysis [Ver02, MGCW08], note that the ratiocorresponding to zero rate computes to [SF11a]

Eb/N0|C=0 = 4 · ln(2) ≈ 2.77 (6.3)

due to the non-zero mean of the constellation. This point is directly related to the slope of thecapacity curve vs.Es/N0 at the pointEs/N0 = 0, which calculates to

dC

dEs/N0

∣

∣

∣

∣

Es/N0=0

=1

4 · ln(2) . (6.4)

Unfortunately, as opposed to the point of zero rate, as givenabove, the optimum operating pointcan not be computed analytically. One has to resort to a numerical evaluation of the capacityand read out the optimum operating point from the numerical results.

To see this, note that this point fulfills

dEb/N0

dC= 0 (6.5)

and thus is the solution to the equation

dEs/N0

dC=

Es/N0

C(6.6)

6.1. The Minimum Required Energy per Bit 121

where we have usedEb/N0 = Es/N0

Cand the quotient rule of differentiation. This equation is

illustrated in Figure 6.3. However, for channels of interest, the first derivative of the capacityand the capacity itself evaluated at an arbitrary point, which both appear in (6.6), can not begiven in closed-form.

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

Es/N0

Eb/N

0

Figure 6.3: Numerical evaluation of (6.6) for BICM 16-PPM. Solid-black:Es/N0

C = Eb

N0

(right hand side of

(6.6)), dashed-black:dEs/N0

dC (left hand side).

Finally, we note that different shapes of the capacity-plotvs. Eb/N0, compared to the con-ventional shape of the MLC capacity or theC-like shape of the BICM capacity, as depictedin Figure 6.2, are also possible. E.g., for BICM using8-ary biorthogonal PPM (or higher) incombination with near-Gray labeling we observe an almostS-like shape [SF11a].


6.2 Design Rules for Coded PPM IR-UWB

In this section, we investigate the application of channel coding for IR-UWB communicationemploying binary PPM. We show that the restriction to low-complexity noncoherent detectionhas to be taken into account in the design of coded PPM IR-UWB in order to increase power-efficiency of the overall system.

Again, we restrict ourselves to binary PPM IR-UWB in combination with symbol-wise energydetection as it represents the IR-UWB system with lowest transmitter and receiver complexity.In order to facilitate our analysis, we consider uniform weighting prior to energy detectionwithin the correlation interval of durationTi, only. Thus, the time-bandwidth productNi = fsTi

represents the major parameter describing the channel conditions.

6.2.1 Soft-Output Energy Detection

First, we consider soft-output energy detection of PPM IR-UWB, as shown in Figure 6.4. Re-call that according to (4.4) soft output is proportional to the difference of the energies of thetwo PPM intervals. Since the bijective mapping and the calculation of the LLRs thus do notcause information loss, it is sufficient to consider the mutual information from the output of themapper to the output of the energy detector, i.e.,

IEDsoft = I(c; LLRED) = I(c; z0,0 z1,1) = I(b0 b1; z0,0 z1,1) (6.7)

wherec is the code symbol,bk are the associated PPM symbols, andzk,k are the correlationcoefficients used for energy detection of this symbol (cf. Figure 4.5). Collecting the correlationcoefficients in a vectorz = [z0,0 z1,1], we have

I(c; z) = Ez,c

{

log2

(

fz(z|c)fz(z)

)}

(6.8)

wherefz(z) represents the probability density function of the correlation coefficientsz andfz(z|c) the one of the correlation coefficientsz given the codesymbolc.

scaling

ENC

M

DEC

z−1

kTpTX(t)

∫

(Ti)·dt

IEDsoft LLREDc

br(t)

zk,khCH(t)

n0(t)

hRX(t)

Figure 6.4: System model of coded PPM IR-UWB with energy detection.

Using the equivalent discrete-time model of the IR-UWB system for this receiver type, i.e., as-suming the energies of the two PPM intervals being central and non-centralχ2-distributed with

6.2. Design Rules for Coded PPM IR-UWB 123

Ni degrees of freedom, depending on the presence or absence of the receive pulse, respectively(cf. Section 3.5), the capacity is easily computed via numerical integration. The computationcan be simplified further when applying the Gaussian approximation (cf. Section 3.5). In thiscase, calculation of the capacity boils down to the well-known case of two-dimensional signal-ing over an additive Gaussian noise channel with equivalentnoise variance.

The probability density functions underlying the capacitycalculations are shown in Figure 6.5for the exemplary case ofNi = 32 and variousEc/N0. With increasingEc/N0 two peaks takeshape, which correspond to the conditional casesfz(z|c = 0) andfz(z|c = 1), respectively. Forcomparison, the Gaussian approximation is shown, as well.

0 1 2 3 40

1

2

3

4

z0,0

z 1,1

Ec/N0: 5 dB

0 1 2 3 40

1

2

3

4

z0,0

z 1,1

Ec/N0: 7 dB

0 1 2 3 40

1

2

3

4

z0,0

z 1,1

Ec/N0: 9 dB

0 1 2 3 40

1

2

3

4

z0,0

z 1,1

Ec/N0: 11 dB

Figure 6.5: Illustration of probability density function fz(z) of energy detection in binary PPM for various

10 log10(Ec/N0). Gray: Gaussian approximation. Ep = 1, Ni = 32.


−20 −10 0 10 200

0.1

0.2

0.3

0.4

0.5

IED

soft

[bit/

dim

.]


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

IED

soft

[bit/

dim

.]

Ec/N0

Figure 6.6: Capacity of PPM IR-UWB with energy detection vs. 10 log10(Ec/N0) (left) and Ec/N0 (right) fordifferent time-bandwidth products Ni = 2x, x = 0 to 15 (steps of 1, left to right). Solid-gray: Exact capacity,

dashed-black: Gaussian approximation, solid-black: coherent detection.

The resulting capacity5 (in bit per dimension, or equivalently per PPM intervalT ) is plotted vs.10 log10(Ec/N0) andEc/N0 in Figure 6.6 for various time-bandwidth productsNi. Althoughnot directly applicable for IR-UWB, very low (down toNi = 1, i.e., the case of matched-filter-based energy detection) and very large (up toNi > 10000) time-bandwidth products are shownfor comparison. Already for moderate (and more realistic) time-bandwidth productsNi ≥ 25

there is hardly a difference between the exact results and those obtained from the Gaussianapproximation.

We observe the expected different shape of the curve of noncoherent detection compared tothat of coherent detection. In particular, the slope at which the pointEc/N0 = 0 is approacheddiffers significantly from that of coherent detection. Thiseffect can be explained by lookingat the Gaussian approximation. Here, the original noise varianceσ2

n occurs squared in theequivalent noise varianceσ2

η. Hence, forEc/N0 → 0, or equivalentlyσ2n → ∞, the effective

signal-to-noise ratio of energy detection,E2p/σ

2η, decreases faster compared to the signal-to-

noise ratio of coherent detection,1/(2σ2n).

This translates into theC-like shape of the capacity curves plotted vs.Eb/N0, as shown inFigure 6.7. The close-to-zero slope of the capacity curve vs. Ec/N0 leads to a ratioEb/N0 atzero capacity which tends to infinity (compared toEb/N0|I=0 = 2ln(2) for coherent detection).

Consequently, the minimum ratio ofEb/N0 is achieved at non-zero rate, as indicated in Fig-ure 6.7. We note that for low time-bandwidth products, the Gaussian approximation signifi-

5For simplicity and in order to obtain results independent from a specific channel model, the captured receivepulse energy has been set toEp = 1.


0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5IED

soft

[bit/

dim

.]

10 log10(Eb/N0) [dB]

Figure 6.7: Capacity of PPM IR-UWB with energy detection vs. Eb/N0 for different time-bandwidth prod-ucts Ni = 2x, x = 0 to 15 (steps of 1, left to right). Solid-gray: Exact capacity, dashed-black: Gaussian

approximation, solid-black: coherent detection, markers indicate optimum operating point.

cantly underestimates this ratio and the respective capacity, but yields valid estimates for time-bandwidth products of interest for IR-UWB communications,sayNi ≥ 25.

An intuitive explanation of this effect goes as follows: Fixa constant ratio ofEb/N0. SinceEc = Eb · Rc, with decreasing code rate the underlying signal-to-noiseratioEc/N0 decreases,too. However, in noncoherent detection parts of the receivesignal are directly used as referencefor demodulation. Thus, the effective signal-to-noise ratio decreases even faster (see above).For too small code rates, the effective signal-to-noise ratio is too small to guarantee reliablenoncoherent detection—the receive signal itself may no longer serve as a valid reference for de-modulation. Thus, an operating point is desired, which achieves the optimum trade-off betweenthe gain resulting from the application of low-rate channelcodes, yet still ensures a sufficientlyhigh signal-to-noise ratio to successfully perform noncoherent detection.

The corresponding code rates are summarized in Table 6.1 (second row). Interestingly, withincreasing time-bandwidth product higher code rates should be chosen, i.e., less redundancyshould be spent for channel coding. The results suggest thatfor Ni → ∞ the optimum coderates tends toln(2). For reasonable time-bandwidth products a code rate of roughly Rc = 2/3

should be preferred over the perhaps more common choice ofRc = 1/2, as adopted, e.g., in[IEE07].

6.2.2 Hard-Output Energy Detection

We now extend the information-theoretic analysis to the case of hard-output energy detection,i.e., for the capacity calculation the hard decision device, as shown in Figure 6.8, is taken into


account.

The overall channel from encoder output to decoder input is thus a binary symmetric channel,whose capacity is given as

IEDhard = I(c; cED) = 1− e2(

SERED)

(6.9)

whereSERED is the code symbol error rate (as derived in Section 4.1.2) and e2(·) denotes thebinary entropy function, defined ase2(x) = −x log2(x)− (1− x) log2(x).

ENC

M

DEC

hRX(t)

n0(t)

kTpTX(t)

∫

(Ti)·dt

c

b

cEDIEDhard

r(t)

z0,0 ≷ z1,1

hCH(t)zk,k

z−1

Figure 6.8: System model of coded PPM IR-UWB with hard-output energy detection.

Figure 6.9 depicts the capacity of hard-output energy detection compared to soft-output energydetection vs.Eb/N0; the calculation is based on the exact capacity and error rate expressions.The capacity curves show a similarC-like shape.

Clearly, the decision device leads to reduced mutual information. E.g., for the minimumEb/N0

a loss of up to1 dB is observed which slowly decreases with increasingNi. The differencedecreases for lower rates. The optimum code rates, summarized in Table 6.1 (last row), areslightly increased compared to soft-output energy detection.

This loss can be visualized in the information processing characteristic of the decision device[HH03], as shown in Figure 6.10. For comparison the information loss for hard-output coherentdetection is also shown. For noncoherent energy detection the information loss due to the hard-decision device is lower compared to the case of coherent detection.

Table 6.1: Optimum code rates for binary PPM IR-UWB with soft- and hard-output energy detection and

different time-bandwidth products Ni.

Ni 21 22 23 24 25 26 27 28 29 210 211

Ropt.−softc 0.47 0.51 0.54 0.57 0.59 0.62 0.64 0.64 0.65 0.66 0.67

Ropt.−hardc 0.48 0.53 0.56 0.59 0.61 0.64 0.65 0.65 0.66 0.69 0.69


0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

IED

hard

[bit/

dim

.]


Figure 6.9: Capacity of PPM IR-UWB with hard-output energy detection (exact results) vs. Eb/N0 for different

time-bandwidth products Ni = 2x, x = 1 to 10 (steps of 1, left to right). Solid-gray: soft-output energydetection, dashed-black: hard-output energy detection, solid-black: coherent detection, markers indicate

optimum operating point.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

IED

hard

[bit/

dim

.]

IEDsoft [bit/dim.]

Figure 6.10: Information loss due to hard-output detection vs. soft-output detection for energy detection of

PPM IR-UWB for different time-bandwidth products Ni = 2x, x = 1 to 15 (steps of 1, unsorted ordering).Dashed-gray: Information loss of hard-decision coherent detection vs. soft-decision coherent detection.


6.3 Design Rules for Coded DTR IR-UWB

Similar to the approach in PPM IR-UWB, in this section, we derive design rules for coded DTRIR-UWB systems which explicitly take into account the noncoherent autocorrelation-based de-tection. Different to PPM IR-UWB, here we also consider advanced noncoherent detectionbased on the output of anL-branch autocorrelation receiver. Clearly, the applied detectionscheme plays an important role for this analysis. Here, we consider the block-wise detectionschemes MSDD and (sorted) DFDD and their extensions based onmultiple-observations com-bining. Based on an information-theoretic analysis, optimum rates for the applied channel codeare derived. Moreover, the influence of the interleaver and the loss compared to the optimumapproach of multi-level coding is discussed.

These insights are validated for DTR IR-UWB employing convolutional codes in combinationwith the Viterbi algorithm as channel coding scheme (cf. Section 2.1.1) by means of numeri-cal simulations in Section 6.4. However, it has to be noted, that the subsequent information-theoretic analysis is independent of the applied channel coding scheme.

Before investigating design rules for coded DTR IR-UWB, we introduce and discuss the appli-cation of channel coding in DTR IR-UWB in more detail.

6.3.1 Coded DTR IR-UWB

Despite being a binary modulation scheme, for DTR IR-UWB in combination with block-wiseautocorrelation-based detection schemes as presented in the previous chapter, various codedmodulation schemes are applicable due to the memory introduced by the differential encoding.This approach follows the reasoning of MSDD of DPSK in [FCLM99, LFCM99]. To see this,note that a block of symbols can be mapped jointly (cf. Section 2.2.3), and thus can be seen as a(hyper-)symbol taken from a higher-order multi-dimensional modulation alphabet. Clearly, theblock size of the mapping should be matched to the block-wiseprocessing in autocorrelation-based block-wise detection. For blocks ofL information symbols mapped and decided jointly,the cardinality of the symbol alphabet is2L.

Hence, the optimum combination of channel coding with DTR IR-UWB is achieved followingthe multi-level coding (MLC) principle [WFH99]. In this thesis, however, we refrain fromthe application of multi-level coded modulation and instead consider the somewhat simplerapproach of bit-interleaved coded modulation (BICM) [CTB98, GMC08], as already describedin Section 2.1.

6.3.1.1 Bit-Interleaved Coded DTR IR-UWB Modulation

In general, in BICM coding and modulation are viewed as separate (almost independent) build-ing blocks: a binary channel code (usually optimized for thebinary input AWGN channel) isconnected via (possibly) an interleaver and a mapper with a higher-order modulation scheme.For reasonable system design, this suboptimum setup induces only a small loss in capacity[CTB98, WFH99].

6.3. Design Rules for Coded DTR IR-UWB 129

The advantages of BICM (especially compared to MLC) are its robustness to varying channelconditions (fading), its low complexity especially at the transmitter side, and its flexibility insystem design. Apart from this, BICM has become a de-facto standard for most digital commu-nications systems, including IR-UWB systems (cf. [IEE07]), and is thus of major interest.

The system model of bit-interleaved coded DTR IR-UWB modulation with autocorrelation-based detection is shown in Figure 6.11.

6.3.1.2 Comments on the Binary Labeling Rule

It is well understood, that the applied binary labeling rulestrongly affects the performance ofBICM (cf. [Sti10] for a comprehensive overview). Since in BICM effectively each coded bitis treated equally, all coded bits should see approximatelythe same channel. This is usuallyachieved by using a binary labeling rule following the Gray principle.

As shown in [LCF+99], however, for block-wise mapping withL > 1 of differentially encodedbinary modulation, Gray labeling of the differentially encoded symbols is not possible. Hence,in this analysis we mainly resort to the conventional approach of symbol-wise labeling. Thisapproach is contrasted to block-wise labeling rules. Compared to the latter, the symbol-wiseapproach has the advantage that the transmitter does not need to be aware of the block size usedat the receiver for receive signal processing.

6.3.2 Information-Theoretic Analysis

For the derivation of design rules for coded DTR IR-UWB, we compute the average mutualinformation between encoder output and decoder input, as indicated in Figure 6.11. Clearly,this analysis is specific to the applied autocorrelation-based detection scheme. In order to quan-tify the loss induced by low-complexity suboptimum detection schemes, we first consider themutual information from the encoder output to the output of the autocorrelation device (sim-ilar to [SF11d]). This measure serves as an upper bound for the case of an ideal detectionscheme, which causes no further loss in information processing, apart from that introduced bythe noncoherent autocorrelation-based detection.

Advanced low-complexity autocorrelation-based detection schemes, which operate on the out-put of thisL-branch autocorrelation device (cf. Chapter 5), with symbol-wise differential detec-tion as the scheme with lowest complexity forL = 1, are compared to this benchmark. Here,we only take into account the application of block-wise detection schemes, such as MSDD andDFDD, as they offer an excellent trade-off between performance and complexity, in particu-lar, if soft-output detection is desired. Sliding-window DFDD, DFDD with virtually increasedblock size, and RMSE are not considered.

6.3.2.1 Capacity of Autocorrelation-Based Detection

Similar to the case of PPM IR-UWB, we collect all correlationcoefficients relevant for thedetection in a vectorz. Considering a block ofL code symbolsc, the MLC capacity pertransmitted bit of DTR IR-UWB with autocorrelation-based detection is given as the mutual


ENC

M

DEC

hRX(t)kT

n0(t)

pTX(t)∫

(Ti)·dt

c

br(t)

∫

(Ti)·dt

kT

LT

T

zk−1,k

zk−L,k

Detection

IACR(L)

hCH(t)

Figure 6.11: System model of BICM DTR IR-UWB with autocorrelation-based detection.

information fromc to z, i.e.,

IACRMLC(L) =1

L· I(c; z) (6.10)

whereI(c; z) = Ez,c

{

log2

(

fz(z|c)fz(z)

)}

andfz(z) is the probability density function of the cor-

relation coefficients andfz(z|c) is the one of the correlation coefficients given the code symbolsc.

The capacity of BICM DTR IR-UWB with autocorrelation-baseddetection is given as the sumof the individual capacities from each code symbol to the autocorrelation device output (so-called bit-level capacities), i.e.,

IACRBICM(L) =1

L

L−1∑

l=0

I(cl; z) . (6.11)

whereI(cl; z) = Ez,cl

{

log2

(

fz(z|cl)fz(z)

)}

. Clearly, the BICM capacity usually is smaller than theMLC capacity [WFH99]. Moreover, forL = 1 we obtain the capacity of DTR IR-UWB withsymbol-wise differential detection (in this case MLC and BICM are equivalent).

An open point to be specified is which correlation coefficients influence the decision of theblock of code symbolsc, i.e., in particular the number of elements ofz. For a block ofL codesymbols (or equivalently,N = L+ 1 transmit symbols), the autocorrelation device delivers

DACR = L · (L+ 1)/2 + 2 · L2 = L · 5L+ 1

2(6.12)

correlation coefficients, which can be exploited for the decision. This is illustrated in Fig-ure 6.12 (recall, the correlation coefficients are symmetric). The first summand in the aboveequation originates from the correlation coefficients connecting theL + 1 symbols within theblock. The second summand represents the remaining correlation coefficients which relate the


ZACR =

ZMSDD =

Figure 6.12: Illustration of correlation coefficients computed by an L-branch autocorrelation device (left)

and those utilized for MSDD (right). Block size L = 2.

symbols of the block to those of future and past symbols (L symbols in the future and the pastconnected to the symbols of the block viaL coefficients each).

Since an exact evaluation of the MLC/BICM capacity of the IR-UWB system at hand, andin particular the respective conditional probability density functions, is overly complex, weutilize the discrete-time system model. Moreover, the correlation coefficients are Gaussianapproximated (cf. Section 3.5), uniform weighting within the correlation intervalTi is assumedand, in order to obtain results independent of the applied channel model, the captured pulseenergy is set toEp = 1. Thus, the time-bandwidth productNi = fsTi represents the majorparameter characterizing transmission, propagation, andautocorrelation-based detection. Theresulting system model is shown in Figure 6.13.

Thus, calculating the capacity boils down to the well-knowncase ofDACR-dimensional sig-naling over an AWGN channel with noise varianceσ2

η [SF11d]. More precisely, we employ aGaussian approximation of the probability density functions, yielding, e.g.,

fz(z|c) = fz(z|z) =1

(2πσ2η)

DACR/2· e−

(z−z)T(z−z)

2σ2η (6.13)

andfz(z) = Ec {fz(z|c)}, wherez represents the noise-free correlation coefficients, i.e.,thecorrelated transmit symbolsb, as shown in Figure 6.13.

Figure 6.14 depicts the BICM and MLC capacity6 of a DTR IR-UWB system with autocor-relation-based detection vs.Ec/N0 for different block sizes and a time-bandwidth product ofNi = 500, as applied in the numerical simulations of Chapter 5 and later-on. For this time-bandwidth product we expect only small deviation from an exact calculation of the capacity ofthe IR-UWB system induced by using the Gaussian approximation. This is motivated by theresults of energy detection of PPM IR-UWB (cf. Figure 6.6).

Clearly, already symbol-wise differential detection (L = 1, in this case MLC and BICM areidentical) outperforms energy detection of PPM IR-UWB, since each symbol carries twice thenumber of information bit. Note that the rate loss due to the required first reference symbol is

6ForL > 1, the capacities have been evaluated via Monte-Carlo integration.


DECENC

Mc

b

Detection

zk−1,k

zk−L,k

z−1

z−L

zk−1,k

zk−L,k

ηk−L,k

ηk−1,k

IACR(L)

Figure 6.13: Discrete-time system model of BICM DTR IR-UWB with autocorrelation-based detection (as-suming Ep = 1).

neglected, which is reasonable for large burst lengths as considered in this thesis. The capacityincreases with increasing block sizeL, resulting in gains of4–5 dB for L = 5 compared tosymbol-wise differential detection. ForL > 1 we also observe that MLC outperforms BICM.The loss due to the restriction to BICM increases for increasing L, but vanishes for sufficientlylargeEc/N0. Thus, especially forIACR ≥ 2/3 using BICM instead of MLC induces only smallloss.

Especially for lowEc/N0, the autocorrelation-operation causes a significant gap compared tocoherent detection, which is mainly caused by the squared original noise varianceσ2

n in theequivalent noise varianceσ2

η. Thus, reliable transmission at very low signal-to-noise ratio isalmost impossible, even for very close-to-zero rates.

Transferring the capacity curves into a plot vs.Eb/N0, usingEb = Ec/IACR, this behavior is

even better visible in Figure 6.15. The capacity curves of IR-UWB with autocorrelation-baseddetection and MLC or BICM plotted vs.Eb/N0 have aC-like shape, similar to those of PPMIR-UWB with energy detection (cf. Figure 6.7).

Thus, as opposed to coherent detection, the minimum ratioEb/N0, which still guarantees re-liable transmission, is obtained at non-zero rates (indicated with markers). At this operatingpoint, both options, decreasing and increasing the code rate, lead to operating points which donot allow reliable transmission.

Consequently, in BICM/MLC IR-UWB systems employing noncoherent autocorrelation-baseddetection the code rate should be carefully selected. Especially for increasingL this minimumgets more and more pronounced, and higher code rates should be favored compared to theprobably more common choice ofRc = 0.5. For allL, the optimum rate in the BICM setup isslightly higher than in the MLC setup.

These capacities may serve as a reference for the performance of coded DTR IR-UWB trans-


−2 0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1IACR

[bit/

sym

bo

l]


Figure 6.14: Capacity of DTR IR-UWB with autocorrelation-based detection vs. 10 log10(Ec/N0) for differentL = 1 (DD, left-triangle), 2 (square marker), 3 (triangle), and 5 (diamond) (right-to-left). Solid-gray: MLC,

solid-black: BICM, left-most solid-gray: coherent detection, right-most solid-gray: energy detection of PPM

IR-UWB. Gaussian approximation with time-bandwidth product Ni = 500.

mission employing reduced-state sequence estimation (RSSE) or soft-output detection schemeswith multiple-observations combining (e.g., mocMSDD), which operate on exactly these cor-relation coefficients.


−2 0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

IACR

[bit/

sym

bo

l]


Figure 6.15: Capacity of DTR IR-UWB with autocorrelation-based detection vs. 10 log10(Eb/N0) for differentL = 1 (DD, left-triangle), 2 (square marker), 3 (triangle), and 5 (diamond) (right-to-left). Solid-gray: MLC,

solid-black: BICM, left-most solid-gray: coherent detection, right-most solid-gray: energy detection of PPM

IR-UWB. Gaussian approximation with time-bandwidth product Ni = 500.


6.3.2.2 Capacity of Soft-Output MSDD

Block-wise soft-output MSDD, as shown in Figure 6.16, operates on a smaller part of the avail-able correlation coefficients in its decision process only [SF11d]. The number of utilized corre-lation coefficients calculates to

DMSDD = L · L+ 1

2(6.14)

as shown in Figure 6.12. For largeL only a fifth of the available correlation coefficients is used.

More precisely, MSDD operates on the autocorrelation-device output

zMSDD = [z0,1 z1,2 . . . zL,L+1 z0,2 . . . zL−1,L+1 . . . z0,L+1] (6.15)

representing the correlation coefficients ordered according to time index and delay (i.e., alongthe off-diagonals from inner-most to outer-most in Figure 6.12). Correspondingly,zMSDD isdefined as the noise-free correlation coefficients.

The mutual information from a block ofL code symbols to theDMSDD correlation coefficientsused for block-wise detection for the cases of MLC and BICM isthus given as

IMSDDMLC (L) =

1

L· I(c; zMSDD) (6.16)

IMSDDBICM (L) =

1

L

L−1∑

l=0

I(cl; zMSDD) (6.17)

respectively. For this case, the BICM capacity represents the mutual information from en-coder output to the output of the soft-output MSDD device, asshown in Figure 6.16. We notethat—strictly speaking—soft-output MSDD as implemented in this thesis (cf. Section 5.3.2.4)introduces an additional loss due to the application of the max-log (or nearest-neighbor approx-imation) and LLR clipping. This loss is not taken into account.

Employing the Gaussian approximation of the correlation coefficients, based on very similardefinitions of the underlying probability density functions as in (6.13), these expressions areeasily evaluated using Monte-Carlo integration. The resulting capacities are compared to thoseof autocorrelation-based detection in Figures 6.17 and 6.18.

We observe a significant loss due to the restriction to block-wise detection schemes, whichincreases for increasing block sizeL. This loss is in accordance to the error-rate results of un-coded transmission, where similarly block-wise MSDD is significantly outperformed by RMSE,which exploits all recorded correlation coefficients. However, the general shape of the curvesis preserved. Thus, also the conclusions on highest power efficiency through the application ofan optimally selected code rate remain unchanged (the optimum rates for block-wise detectiondiffer only slightly from that of autocorrelation-based detection). Again, the optimum rates forBICM are slightly larger compared to those for MLC.

6.3.2.3 Influence of the Binary Labeling Rule

A point to be thoroughly investigated in every communication system employing BICM is theapplied binary labeling rule [GMC08, Sti10]. As pointed outin Sections 6.3.1.2 and 2.2.3, in


DECENC

Mc

bzk−1,k

zk−L,k

z−1

z−L

zk−1,k

zk−L,k

ηk−L,k

ηk−1,k

SO-MSDD

IMSDD(L)LLRMSDD

Figure 6.16: Discrete-time system model of BICM DTR IR-UWB with autocorrelation-based soft-outputMSDD (assuming Ep = 1).

DTR IR-UWB systems this degree of freedom is available due tothe memory introduced by thedifferential encoding. Hence mapping multiple code symbols c to multiple transmit symbolsb jointly, i.e., a block-wise mapping, can be applied. This approach is inspired by the block-wise processing of MSDD at the receiver side, and hence suggests to set the block size of themapping to the block size of MSDDL.

Usually, in a BICM setup, Gray labeling is favorable [Sti10,CTB98]. However, for our setupandL > 1, Gray-labeling is not possible due to the differential encoding [LCF+99]. Hence,other binary labelings (e.g., with the aim of approximatingGray labeling) can be adopted andmight lead to performance improvement. However, since there exist various possibilities forbinary labeling rules, here we restrict ourselves to two promising strategies.

The first strategy is the (up-to-now solely) considered conventional symbol-wise labeling rule.Here, the code symbols are first mapped to information symbols a (cf. (2.9)), which are thendifferentially encoded according to (2.11), yielding the transmit symbolsb (cf. Example 6.1 andExample 6.2).

A second strategy is inspired by the signal space spanned by the symbols at the output of theautocorrelation device. The received signal points after autocorrelation-based detection,zMSDD,representing the block of code symbols are signal points inDMSDD-dimensional Euclidean space(cf. (6.15)).

Soft-output MSDD directly operates on this set of symbols (cf. [GQ06, ZM12b] for derivationsof the MSDD decision metric based on this point of view). For this strategy we consider threevariants: labeling according to the (strictly regular) set-partitioning (SP) rule [Sti10], and twovariants which aim to approximate Gray labeling. The latterdiffer in the way an iterative processis implemented which aims to approximate Gray labeling as close as possible.


−2 0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1I M

LC

[bit/

sym

bo

l]


Figure 6.17: MLC capacity of DTR IR-UWB with block-wise autocorrelation-based detection compared toautocorrelation-based detection vs. 10 log10(Eb/N0) for different L = 1 (DD, left-triangle), 2 (square marker),

3 (triangle), and 5 (diamond) (right-to-left). L = 10 (circle) included only for block-wise detection. Solid gray:

IACRMLC, solid black: IMSDDMLC , left-most solid-gray: coherent detection. Gaussian approximation with time-

bandwidth product Ni = 500.

Both variants7 start at an arbitrary signal point and assign labels cyclingthrough the remainingpoints in a greedy fashion. The first variant (so-called near-Gray labeling (next)) in each stepassigns the least different label to the closest neighbor and thus aims for minimum number ofcode symbol errors in case of the most-likely symbol error events. Note that this method yieldsa valid Gray labeling in case of conventional QAM/PSK constellations.

The second variant (so-called near-Gray labeling (far)) ineach step assigns the most differ-ent label (i.e., the complementary) to the signal point at maximum distance and thus aims toconcentrate a large number of code symbol errors on unlikelysymbol error events.

It can be verified that for the case ofL = 2, all considered strategies lead to the same binarylabels and hence the same BICM capacity (cf. Example 6.1 and Figure 6.20). To see this, notethat allzMSDD differ by only two symbols, hence each symbol is a nearest neighbor to the other.

Starting withL = 3, significant differences can be observed, both, when looking at the binarylabels (cf. Example 6.2) as well as the BICM capacities (cf. Figure 6.20).

7The algorithms of near-Gray labeling are not presented in detail due to the excellent performance of symbol-wise labeling. Examples of the constructed near-Gray labelings can be found in Example 6.1 and Example 6.2 forL = 2 and3, respectively.


−2 0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1I B

ICM

[bit/

dim

.]


Figure 6.18: BICM capacity of DTR IR-UWB with block-wise autocorrelation-based detection compared

to autocorrelation-based detection vs. 10 log10(Eb/N0) for different L = 1 (DD, left-triangle), 2 (squaremarker), 3 (triangle), and 5 (diamond) (right-to-left). L = 10 (circle) included only for block-wise detection.

Solid gray: IACRBICM, solid black: IMSDDBICM , left-most solid-gray: coherent detection. Gaussian approximation with

time-bandwidth product Ni = 500.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

IMSDD

BICM

[bit/

dim

.]

IACRBICM [bit/dim.]

Figure 6.19: Information loss due to block-wise autocorrelation-based detection vs. autocorrelation-based

detection for different L = 2 (square marker), 3 (triangle), and 5 (diamond) (left-to-right). Gaussian approxi-mation with time-bandwidth product Ni = 500.


10 11 12 13 14 150

0.2

0.4

0.6

0.8

1IMSDD

BICM

[bit/

sym

bo

l]


Figure 6.20: Capacity of DTR IR-UWB with block-wise autocorrelation-based detection for L = 2 (square

marker), 3 (triangle), and 5 (diamond) with different binary labeling rules. Solid: symbol-wise labeling,dotted: SP labeling, dashed: near-Gray labeling (next), dash-dotted: near-Gray labeling (far), right-most

solid-gray: DD. Gaussian approximation with time-bandwidth product Ni = 500.

Example 6.1: Symbol-wise binary labeling for L = 2

For a block size of L = 2, this example lists the mapping from a block of codesymbols

c via symbol-wise mapping to information symbols a, followed by differential encoding

yielding the transmit symbols b. Finally, autocorrelation-based detection yields the (noise-

free) correlation coefficients zMSDD = [z0,1 z1,2, z0,2] (±1 abbreviated as ±):

c a b zMSDD

0 0 ++ +++ +++01 +− ++− +−−1 0 −+ +−− −+−1 1 −− +−+ −−+

z0,2

z1,2

z0,1

[0 1]

[1 0]

[1 1][0 0]

It can be observed, that the correlation coefficients zMSDD differ at only two positions, each.

Hence, in the signal space depicted on the right hand side, all points have equal mutual

distance.


Example 6.2: Binary labeling rules for L = 3

For L = 3 different labeling rules are investigated: The symbol-wise labeling as in Exam-

ple 6.1, a labeling rule constructed from the set-partitioning principle operating on the auto-

correlation device output zMSDD, and two variants approximating Gray labeling, both again

operating on zMSDD.

Symbol-wise labeling:

c a b zMSDD

0 0 0 + + + ++++ ++++++001 + +− +++− ++−+−−0 1 0 +−+ ++−− +−+−−−0 1 1 +−− ++−+ +−−−++100 −++ +−−− −++−+−1 0 1 −+− +−−+ −+−−−+110 −−+ +−++ −−++−+111 −−− +−+− −−−++−

SP labeling:

c a b zMSDD

0 0 0 + + + ++++ ++++++001 + +− +++− ++−+−−0 1 0 +−+ ++−− +−+−−−0 1 1 +−− ++−+ +−−−++100 −+− +−−+ −+−−−+101 −++ +−−− −++−+−1 1 0 −−− +−+− −−−++−1 1 1 −−+ +−++ −−++−+

near-Gray labeling (next):

c a b zMSDD

0 0 0 + + + ++++ ++++++001 + +− +++− ++−+−−0 1 0 +−− ++−+ +−−−++011 +−+ ++−− +−+−−−1 0 0 −++ +−−− −++−+−1 0 1 −+− +−−+ −+−−−+110 −−+ +−++ −−++−+111 −−− +−+− −−−++−

near-Gray labeling (far):

c a b zMSDD

0 0 0 + + + ++++ ++++++001 −++ +−−− −++−+−0 1 0 +−− ++−+ +−−−++011 −+− +−−+ −+−−−+100 −−+ +−++ −−++−+101 + +− +++− ++−+−−1 1 0 −−− +−+− −−−++−1 1 1 +−+ ++−− +−+−−−

Considering, e.g., near-Gray labeling (next), it can be seen, that the correlation coefficients

corresponding to binary labels that differ at one position from c = [0 0 0] (Hamming distance

1) differ at only three positions from the corresponding correlation coefficients. Thus, the

nearest neighbors w.r.t. correlation coefficients are labeled with least different labels. How-

ever, there is another nearest neighbor (namely “−−++−+”, with label [1 1 0]), for which

no label with Hamming distance 1 is available anymore. Thus, Gray labeling is not possible.

However, from Figure 6.20 we conclude that the conventionalsymbol-wise labeling rule signif-icantly outperforms all other investigated labeling rulesin terms of BICM capacity. Moreoverthis labeling offers increased flexibility, since the receiver can easily switch between differentblock sizes independent of the transmitter. Hence, this labeling rule should thus be implementedin BICM DTR IR-UWB systems.

6.3.2.4 Influence of an Interleaver

As long as a non-fading scenario is investigated, as considered in this thesis, the applicationof an interleaver in a BICM setup is only motivated by so-called constellation-inherent fading[Sti10, SFH10, WFH99]. This fading process is expressed by differing bit-level capacities,yielding a well-structured memory in the quality of symbolsat the output of the demapper (here,


10 11 12 13 14 150

0.2

0.4

0.6

0.8

1I(c

l;z)

[bit/

sym

bo

l]


Figure 6.21: Bit-level capacities of BICM DTR IR-UWB with block-wise autocorrelation-based detection for

L = 2 (square marker), 3 (triangle), and 5 (diamond). Solid: Symbol-wise labeling, dotted: SP labeling,right-most solid-gray: DD. Gaussian approximation with time-bandwidth product Ni = 500.

the SO-MSDD device). In BICM with a pseudo-random interleaver, this inherent memoryis neglected. Especially in combination with convolutional coding, this potentially leads toinferior performance compared to an optimized interleaveror even no interleaver at all (cf.[SFH10]).

In order to investigate the impact of an interleaver in our system setup, Figure 6.21 depicts thebit-level capacities of symbol-wise-labeled BICM DTR IR-UWB with block-wise autocorrela-tion-based detection. We observe, that all bit-level capacities are equal. Hence, an interleaver isnot necessary in BICM DTR IR-UWB using conventional symbol-wise labeling.

This would be different, if SP labeling was applied in BICM DTR IR-UWB with block-wisedetection. Here, the bit-level capacities differ significantly (cf. Figure 6.21, note that still someof the bit-level capacities are equal, e.g., forL = 3 only one differs from the other two),suggesting potential gains when using an optimized insteadof a pseudo-random interleaver.This point is not further investigated due to the focus on symbol-wise labeling.


6.3.2.5 Capacity of DFDD

For uncoded transmission, sorted block-wise DFDD has proven to be very competitive to hard-output MSDD. Hence, especially if one aims for coding schemes employing hard-decision de-coding (i.e., a hard-input decoder and a hard-output detection device), this DFDD variant is anattractive candidate for coded transmission, too.

In this case, the output of the detection device is restricted to±1. Hence, the channel from en-coder output to decoder input can be modeled as a memoryless binary symmetric channel. Theproperties memoryless8 and symmetric result from the optimized decision order within DFDD,which effectively acts as a random block interleaver. This simplifies capacity calculations to

IDFDD(L) =1

L·L−1∑

l=0

I(cl; cDFDDl ) = 1− e2(SER

DFDD) (6.18)

wherecDFDD are the estimates andSERDFDD is the code symbol error rate of DFDD, respec-tively. At this point, there is no need to distinguish between MLC and BICM, since DFDD,as presented here, is applicable in BICM scenarios only. Clearly, for L = 1 the capacity ofhard-output symbol-wise differential detection is obtained.

The resulting capacities are compared to that of soft-output MSDD in Figure 6.22; the lossinduced by the application of DFDD instead of soft-output MSDD is shown in Figure 6.23.

We begin the discussion with symbol-wise differential detection (L = 1). Similar to hard-outputenergy detection in PPM IR-UWB (cf. Figure 6.8), for symbol-wise differential detection theloss amounts to less than1 dB for this setup. Again, the optimum rates of hard-output detectionare higher than that of soft-output detection.

This is slightly different for hard-output DFDD compared tosoft-output MSDD. Here, this trendcan only be observed for largeL. Interestingly, the difference between hard- and soft-outputdetection almost diminishes forL = 2 and then starts to increase again with increasing blocksize (cf. also Figure 6.23).

In general, we observe that code rates in the order of2/3 and4/5 should be chosen for BICMDTR IR-UWB in combination with hard-output DFDD and hard-decision decoding.

Up to now, for our capacity calculations we have relied on theGaussian approximation of thecorrelation coefficients and have neglected imperfect capturing of the receive pulse energy, i.e.,Ep < 1, as well as variations of the latter (cf. Section 3.5.1). However, for the case of DFDD itis straight-forward to estimate the capacity from numerical results obtained from simulation ofIR-UWB transmission over a more realistic channel model andplug in the resulting error rateinto (6.18). For the system setup as considered in Chapter 5 (non-line-of-sight channel modelof [MCC+06] with uniform weighting withinTi = 50 ns, cf. Section 5.3.5.1), the resultingcapacity estimates are shown in Figure 6.24.

8This property can be verified by computing the bit-level capacities, which turn out to be all equal.


8 10 12 14 16 180

0.2

0.4

0.6

0.8

1I

[bit/

sym

bo

l]


Figure 6.22: BICM capacity of DTR IR-UWB with autocorrelation-based detection using block-wise DFDD

vs. 10 log10(Eb/N0) for different L = 1 (gray), 2 (square marker), 3 (triangle), and 5 (diamond), and 10 (circle)(right-to-left). Solid-black: IMSDD, dashed-black: IDFDD. Gaussian approximation with time-bandwidth

product Ni = 500.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

IDFDD

[bit/

dim

.]

IMSDDBICM [bit/dim.]

Figure 6.23: Information loss due to block-wise hard-output sorted DFDD cs. soft-output MSDD for different

L = 1 (left-triangle, gray), 2 (square), 3 (triangle), 5 (diamond), and 10 (circle). Gaussian approximation withtime-bandwidth product Ni = 500.


8 10 12 14 16 180

0.2

0.4

0.6

0.8

1I

[bit/

sym

bo

l]


Figure 6.24: BICM capacity of DTR IR-UWB using block-wise DFDD based on Gaussian approximation(solid gray) compared to simulation results for channel model 2 of [MCC+06] (markers) vs. 10 log10(Eb/N0)for different L = 1 (triangle), 2 (square marker), and 5 (diamond), and 10 (circle) (right-to-left). Time-

bandwidth product Ni = 500, uniform weighting within Ti = 50 ns.

The results are compared to the Gaussian-approximated capacity. In order to account for imper-fect capturing of the receive pulse energy, we approximate the loss induced by this ramificationvia

−10 log10(

1− e−Ti/ΓCM)

(6.19)

as motivated in Section 3.5.1, and simply shift the capacitycurves by this amount to the right.Note that this neglects the variations in the captured receive pulse energy and also the occurrenceof the captured receive pulse energy in the “pulse×noise” cross-terms in the equivalent noisevariance.

These simplifications and the Gaussian approximation lead to some deviance compared to thesimulated capacity results. However, the general statements and conclusions—in particular,regarding optimum rates—remain unchanged.

6.3.2.6 Capacity of DFDD with Multiple-Observations Combining

Based on the principle of multiple-observations combining, the hard-output scheme block-wiseDFDD can be extended to additionally provide reliabilitiesfor its estimates (cf. Section 5.3.6).This enables the application of soft-decision decoding schemes, although a hard-output detec-tion scheme is applied. Thus, low complexity is preserved, whereas potential gains in bit errorrate are expected.

To this end, overlapping blocks (in case of a block shift ofS symbols we have an overlap ofL−S code symbols, cf. Figure 5.21) are processed independentlyand the individual results are


combined. This combination is simply implemented as the sumof the individual hard decisions.Hence, forS = 1 (and thusL observations for each symbol are available), the output of DFDDwith multiple-observations combining (mocDFDD),LLRmocDFDD, takes the values from the set

{−L, −L+ 2, . . . , L− 2, L} (6.20)

with cardinalityL + 1. For a single observation per code symbol (i.e.,S = L), conventionalDFDD with

{−1, +1} (6.21)

is obtained. For other block shiftsS ∈ {2, . . . , L − 1}, all integer values between−L andLare possible, since the number of observations isL/S on average only, i.e., some symbols areobservedL times, others only once (cf. Figure 5.21).

Clearly, this output does not represent a true log-likelihood ratio, but can be seen as a quantizedversion of the latter. Thus, we use the same notation for this“soft output” as for soft-outputMSDD, since both share an important property: the sign of theestimate can be used as a finalhard decision for the information symbols. The corresponding estimate for the code symbols isdenoted ascmocDFDD.

Thus, the capacity of DFDD with multiple-observations combining is given as

ImocDFDDsoft (L) =

1

L·L−1∑

l=0

I(cl; LLRmocDFDDl ) . (6.22)

The corresponding hard-output capacity calculates to

ImocDFDDhard (L) =

1

L·L−1∑

l=0

I(cl; cmocDFDDl ) = 1− e2(SER

mocDFDD) (6.23)

wherecmocDFDDl = sign

(

LLRmocDFDDl

)

.

Both capacity expressions can easily be computed numerically by simulation of DFDD withmultiple-observations combining.

The results are shown in Figure 6.25 for the case of maximum block overlap, i.e.,S = 1, andthusL observations per code symbol. It can be seen that “soft-output” DFDD with multiple-observations combining leads to significant gains in capacity already forL = 2. This is dueto the fact that forL = 2, different from hard-output DFDD with multiple-observations com-bining, ternary estimates are passed to the channel decoderinstead of binary ones. Thus, thedetection scheme effectively marks unreliable decisions as an erasure. As known from otherapplications, cf. [FH06b], this—although very limited—reliability information already offerssignificant gains in capacity.

Again, forL = 5, the optimum rate of the hard-output scheme is slightly larger compared to therespective soft-output variant.


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1I

[bit/

sym

bo

l]


Figure 6.25: BICM capacity of DTR IR-UWB with autocorrelation-based detection using block-wise DFDDwith multiple-observations combining vs. 10 log10(Eb/N0) for L = 2 (square marker) and 5 (diamond).

Solid-black: ImocDFDDsoft , dashed-black: ImocDFDD

hard , solid-gray: IDFDD. Right-most gray (triangle): DD. Gaussian

approximation with time-bandwidth product Ni = 500.

6.3.2.7 Summary and Discussion

Based on the numerical evaluations of the capacity presented above, we conclude that in orderto increase the power efficiency of BICM DTR IR-UWB with autocorrelation-based detection(operating at a time-bandwidth product ofNi = 500), the code rate of the applied channelcoding scheme should be carefully selected. The results aresummarized in Table 6.2.

For soft-output symbol-wise differential detection, a rate slightly below2/3 is optimum. In caseof hard-output differential detection a rate slightly above2/3 should be chosen.

For block-wise detection schemes, the optimum rate dependson the block sizeL. Apart fromthe general tendency that for hard-output schemes the rate should be chosen a little larger com-pared to the respective soft-output variants, the optimum rate is almost independent from theapplied detection scheme. For low block sizes, sayL = 2 and3, a rate of roughly0.7–0.8should be chosen. With increasing block size, the optimum shifts to rates between0.8 and0.9.

These results emphasize the need for a joint design of coding, modulation, and detection, al-though the BICM setup suggests to design these building blocks almost independently. For oursetup, rates in the order of2/3 to 4/5 should be preferred over the straight-forward and perhapsmore common choice of rate-1/2 codes, as, e.g., found in [IEE07].

Hence, the amount of redundancy required for channel codingto operate is rather small. For afixed amount of information bits to be communicated, the length of the transmit symbol burstcan be kept relatively small. In turn, the energy consumption of the transmitter is reduced,


which potentially leads to increased battery-powered lifetimes.

The given optimum code rates assume an ideal channel coding scheme and target error-freetransmission. Non-ideal, i.e., finite-length codes (possibly in combination with non-optimalchannel decoders) could be taken into account when considering error exponents instead ofcapacity considerations [FH06b]. Moreover, based on rate-distortion theory a target bit errorrate (greater than zero) could be incorporated, as well. In order to obtain results almost inde-pendent9 of the applied coding scheme and due to the effectiveness of our approach (cf. thefollowing Section and [SF11d]), we refrain from such extensions.

In [SF11d], we have verified the derived optimum code rates for the case of coded DTR IR-UWB employing low-density parity-check codes. In this thesis, we focus on convolutionalcoded BICM IR-UWB—mainly due to the lower encoder and decoder complexity of convolu-tional codes compared to low-density parity-check codes.

In any case, the optimum code rates have to be quantized to reasonable rates supported by thecoding scheme. For convolutional codes, in particular, thecomplexity of encoder and decoderhave to be taken into account for this selection. In this case, reasonable code rates are fractionsof low-to-moderate natural numbers, i.e., rates of2/3, 3/4, and4/5 are of interest. Clearly, forconvolutional codes, rates above1/2 should preferably be constructed by puncturing a rate-1/2

parent code [PS08].

In this context, the application of polar codes [Arı09] appears to be an interesting point forfuture research, as they offer to flexibly select code rates with fine granularity without the needto change the code construction or increasing encoder/decoder complexity.

Table 6.2: Optimum code rates for BICM DTR IR-UWB with autocorrelation-based detection and different

block sizes L.

L 1 2 3 5 10soft-output DD 0.65 — — — —hard-output DD 0.68 — — — —soft-output MSDD 0.65 0.74 0.77 0.81 0.82hard-output DFDD 0.68 0.70 0.77 0.81 0.83hard-output mocDFDD 0.68 0.70 0.74 0.81 —soft-output mocDFDD 0.68 0.72 0.73 0.79 —

9Obviously, efficient channel codes for rates above0.5 should be chosen.


6.4 Validation for Convolutional Coded Transmission

The design rules obtained from the above capacity considerations are validated by means ofnumerical simulations for convolutional coded IR-UWB transmission. Maximum-free-distanceconvolutional codes with nonrecursive nonsystematic encoding in combination with soft- andhard-input Viterbi decoding are considered (cf. Section 2.1.1). We restrict ourselves to codesof low constraint lengthν in order to keep the additional complexity due to the application ofchannel coding, especially decoding at the receiver side, as low as possible. For code ratesRc > 0.5, puncturing of the rate-1/2 parent code is applied, cf. [PS08, Table 8.3-1, -2, -3, and8.4-1] and Appendix A.3.

Exemplarily, we consider the non-line-of-sight channel model 2 of [MCC+06] and uniformweighting prior to energy and/or autocorrelation-based detection withinTi = 50 ns, which rep-resents a good compromise between energy capturing and noise suppression, cf. Figures 4.3 and5.5. For our simulations, this yields a time-bandwidth product ofNi = 500. The duration of thesymbol/PPM interval is set toT = 100 ns, which allows to neglect inter-symbol interference.

Again, in order to enable a fair comparison, the capacity results are shifted by (6.19) to accountfor imperfect capturing of the receive pulse energy.

In case of DTR IR-UWB, forKsource information bits to be transmitted, a transmitted burst iscomprised of

Nburst = 1 + ⌈(Ksource + ν − 1)/Rc⌉ (6.24)

transmit symbols (one reference symbol andν − 1 additional bits for termination). In case ofbinary PPM IR-UWB this number calculates to

Nburst = 2 · ⌈(Ksource + ν − 1)/Rc⌉ (6.25)

as a reference at the beginning of each burst is not required,but two transmit symbol intervalsare used for each code symbol. The overall rate of coding and modulation in bit/dimension,including rate loss due to termination, is thus given as

R = Ksource/Nburst

≈< Rc . (6.26)

We setKsource = 100, since for significantly shorter burst length the rate loss due to the termi-nation of convolutional codes would be too high.

For even shorter burst length, short block codes spanning over the entire burst in combinationwith maximum-likelihood decoding appear to be the more reasonable choice (e.g., binary codeswith length equal to the burst length, but also non-binary codes with very short length seempromising).

We begin with energy detection of PPM IR-UWB and proceed withtransmission of DTR IR-UWB. For the latter, symbol-wise differential detection and variants of block-wise detection areconsidered.

6.4. Validation for Convolutional Coded Transmission 149

6.4.1 Energy Detection of PPM IR-UWB

Figures 6.26 and 6.27 depict the overall rate of transmission vs. the power efficiency (measuredas the requiredEb/N0 for a target error rate ofBER = 10−3). The considered systems of eachfigure have approximately equal complexity and are thus directly comparable.

Maximum-free-distance codes with constraint lengthν = 3 (Figure 6.26) andν = 5 (Fig-ure 6.27) with nonrecursive nonsystematic encoding and code ratesRc = 1/4, 1/3, 1/2, 2/3,3/4, and4/5 are considered. We distinguish between soft- and hard-output energy detection (ofcourse, in combination with soft- and hard-input Viterbi decoding, respectively) and comparethe results with the capacity, as derived in Section 6.2.

For both settings, the measured power vs. spectral efficiency well agrees with the capacityresults. The plots of the capacity and those of the convolutional coded system have a verysimilar shape. Thus, even for the relatively simple codes with constraint lengthν = 3, thecapacity is shown to be a very good means for performance assessment. The gap to capacityamounts to only1.6 dB, yet, it has to be noted that a target error rate ofBER = 10−3 isconsidered, whereas the capacity ensures error-free transmission.

As expected from the capacity considerations, for soft-output energy detection the rate-2/3 code(i.e., overall rate of modulation and channel coding of roughly R = 1/3 bit/dim.) outperformsall other investigated rate settings (cf. also Figure 6.28). In particular, as opposed to the con-ventional behavior in coherent detection (cf. Appendix A.3), decreasing the code rate yieldsreduced overall power efficiency of the coded PPM IR-UWB system. The additional rate lossis not compensated for by the additional coding gain.

For low code rates this effect is to some extend augmented by the behavior of the consideredconvolutional codes itself (cf. Appendix A.3). However, taking the results of Figures A.2 andA.4 into consideration allows us to conclude that in the region of interest (rates of and above1/2), the major source for the increased power efficiency of codes of rate2/3 compared to1/2is caused by the capacity of the underlying noncoherent communication channel.

For hard-decision decoding the gains over uncoded transmission almost vanish. This can alsobe seen from Figure 6.28, which depicts the bit error rate of coded PPM IR-UWB for rate-1/2and rate-2/3 convolutional codes in combination with soft- and hard-output energy detection.In case of hard-output energy detection, but also for soft-output energy detection with a rate-1/2

code, we observe almost no gains in bit error rate over uncoded transmission in the signal-to-noise region of interest. This emphasizes the need for the application of soft-output detectionschemes as well as for carefully selecting the rate of the applied channel code. For low signal-to-noise ratio, this drawback could be alleviated by employing systematic recursive encodingof the convolutional codes.

Clearly, more powerful (i.e., almost capacity-achieving)coding schemes lead to larger gains,as indicated by the rate-distortion bound in Figure 6.28. This, however, conflicts with the mainobjective in this thesis, namely the low complexity at transmitter and receiver.


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0.5

IED

soft/hard

/R[b

it/d

im.]

required10 log10(Eb/N0) [dB]

Figure 6.26: Rate vs. power efficiency (required Eb/N0 for target BER = 10−3) of convolutional coded PPM

IR-UWB (constraint length ν = 3) with soft- and hard-output energy detection compared to capacity of PPMIR-UWB with ED. Solid-gray: IEDsoft, dashed-gray: IEDhard, solid-black: soft-output ED, dashed-black: hard-output

ED, circle: uncoded transmission. Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).

16 17 18 19 20 21 22 23 240

0.1

0.2

0.3

0.4

0.5

IED

soft/hard

/R[b

it/d

im.]


Figure 6.27: Rate vs. power efficiency (required Eb/N0 for target BER = 10−3) of convolutional coded PPM

IR-UWB (constraint length ν = 5) with soft- and hard-output energy detection compared to capacity of PPM

IR-UWB with ED. Solid-gray: IEDsoft, dashed-gray: IEDhard, solid-black: soft-output ED, dashed-black: hard-outputED, circle: uncoded transmission. Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).


15 17 19 2110

−5

10−4

10−3

10−2

10−1

BER


15 17 19 2110

−5

10−4

10−3

10−2

10−1

BER


Figure 6.28: BER vs. Eb/N0 of convolutional coded PPM IR-UWB (constraint length ν = 3 (left) and 5 (right))for rate 1/2 (square markers) and rate 2/3 (circles). Solid-black: soft-output ED, dashed-black: hard-output

ED, solid-gray with marker: rate-distortion bound for Rc = 2/3, solid-gray: uncoded transmission. Channel

model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).


6.4.2 Symbol-Wise Differential Detection of DTR IR-UWB

Similar to the case of energy detection of PPM, Figure 6.29 depicts the trade-off between powerefficiency and code rate for soft- and hard-output symbol-wise differential detection of DTRIR-UWB. Again, the results of numerical simulations of convolutional coded transmission arecompared to the capacity of the underlying noncoherent communication channel.

As expected, also for this modulation and detection scheme the numerical results validate thedesign rules obtained from the information-theoretic analysis. The conclusions hardly changecompared to the case of energy detection of PPM IR-UWB, all results are basically shifted by3 dB (note the different range of the x-axis).

Figure 6.30 shows the bit error rate of a rate-1/2 reference system and a rate-2/3 system. Thereference system is clearly outperformed by the system withan optimally selected code rate interms of power efficiency.

Again, the results clearly show the additional gains obtained from the application of soft-outputdetection compared to hard-output detection and the need for optimally selecting the appliedcode rate, in order to achieve notable gains over uncoded transmission.

13 14 15 16 17 18 19 20 210

0.2

0.4

0.6

0.8

1

IDD

soft/hard

/R[b

it/sy

mb

ol]


Figure 6.29: Rate vs. power-efficiency (required Eb/N0 for target BER = 10−3) of convolutional coded DTR

IR-UWB (constraint length ν = 5) with soft- and hard-output differential detection compared to capacity of

DTR IR-UWB with DD. Solid-gray: IDDsoft, dashed-gray: IDD

hard, solid-black, square markers: convolutional codewith soft-output DD, dashed-black, diamonds: convolutional code with hard-output DD, circle: uncoded

transmission. Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).


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−5

10−4

10−3

10−2

10−1

BER


12 14 16 1810

−5

10−4

10−3

10−2

10−1

BER


Figure 6.30: BER vs. Eb/N0 of convolutional coded DTR IR-UWB (constraint length ν = 3 (left) and 5 (right))for rate 1/2 (square markers) and rate 2/3 (circles). Solid-black: soft-output DD, dashed-black: hard-output

DD, solid-gray with marker: rate-distortion bound for Rc = 2/3, solid-gray: uncoded transmission. Channel

model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).


6.4.3 Block-Wise Detection of DTR IR-UWB

Before validating design rules for autocorrelation-basedblock-wise detection schemes, we in-vestigate the performance of the latter when applied for detection in convolutional coded trans-mission. To this end, we restrict ourselves to a rate-2/3 convolutional code with constraintlengthν = 5.

We distinguish soft-output MSDD and hard-output DFDD. Bothschemes can be extended ac-cording to the concept of multiple-observations combiningby processing overlapping blocksand combining the individual observations (mocMSDD and mocDFDD). Here we considermultiple-observations combining with maximum overlap only, i.e., a block shift ofS = 1 andL observations per code symbol. In all cases, the decisions are taken in the optimum DFDDorder.


Figure 6.31 depicts the bit error rate of the above schemes for different block sizes. The LLR-clipping level for soft-output MSDD has been set toLLRmax = 5 (setting of this parameter isdiscussed in Section 6.4.3.2). Clearly, with increasing block size performance improves oversymbol-wise differential detection.

As expected, multiple-observations combining is very effective in combination with DFDD. Ofcourse, by application of multiple-observations combining the estimate itself is improved (cf.the results for uncoded transmission in Section 5.3.7). However, different from conventionalDFDD, now reliability information—although in a very coarse form—is available, as well,which can be delivered to a soft-decision channel decoder. Thus, additionally, the gains of soft-vs. hard-decision decoding can be exploited. Consequently, the error rate of mocDFDD is closeto that of soft-output mocMSDD and even exceeds that of conventional soft-output MSDD.

However, especially forL = 5 and10, the restriction to the relatively low-complexity convo-lutional code leads to almost vanishing gains over uncoded transmission (visible only at targeterror rates belowBER = 10−4, block-wise DFDD with multiple-observations combining isshown as a reference). Hence, the application of channel coding and the channel code param-eters have to be thoroughly investigated in order to justifythe complexity increase due to theapplication of channel coding. In particular, for a given constraint length, the parameter coderate has to be carefully selected as motivated in the capacity considerations above. This task isaddressed in the following section.

For reference, the rate-distortion bound for a rate-2/3 code and autocorrelation-based block-wise detection with the respective block sizeL is included, representing the minimum achiev-able error rate of an ideal coding scheme. AtBER = 10−4 the gap to this benchmark amountsto approximately3 dB independent of the block size.


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−6

10−5

10−4

10−3

10−2

10−1

BER


L = 2

10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1

BER


L = 3

10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1

BER


L = 5

10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1

BER


L = 10

Figure 6.31: BER vs. Eb/N0 of convolutional coded DTR IR-UWB (constraint length ν = 5, rate Rc = 2/3) for

block-wise detection with various L. Solid-black: soft-output MSDD, dashed-black: DFDD, filled markers:with MOC, empty markers: without MOC, left-most solid-gray: rate-distortion bound for Rc = 2/3, solid-

gray: uncoded mocDFDD, right-most dashed-gray: coded DD. Channel model 2 of [MCC+06] with Ti =50 ns (Ni = 500).


6.4.3.2 LLR Clipping

LLR clipping is an important means to reduce the computational complexity of soft-outputMSDD. In order to avoid the need for estimation of the noise variance, required in the scalingfactor of the LLRs, different from [SF10b], we neglect this scaling in (5.22) and simply definethe LLR-clipping level asLLRmax = Λmax (cf. (5.25)). This simplification is directly appli-cable in case of convolutional coded transmission and leadsto no loss in performance, sincethe Viterbi algorithm employed for decoding the convolutional code is invariant to a constantscaling factor of the input LLRs.

Figure 6.32 depicts the trade-off between required computational complexity and power effi-ciency of block-wise detection with different block sizes and LLR-clipping levels. The targetbit error rate isBER = 10−3 and the code rate isRc = 2/3. Similar to Chapter 5, we adopt thenumber of visited nodes in the sphere-decoder search process as a measure for computationalcomplexity.

Again, we consider the two variants of soft-output MSDD, theconventional application op-erating on non-overlapping blocks of information symbols (gray) and soft-output mocMSDD(black). The latter operates with a maximum block-overlap (block shiftS = 1), thusL obser-vations are available for each code symbol. These results are compared to the respective DFDDvariants with and without multiple-observations combining.

From Figure 6.32 we conclude, that LLR clipping enables a flexible trade-off between the lowcomputational complexity of hard-output detection (withLLRmax = 0)10 and the high powerefficiency of optimum (up to the applied max-log approximation) soft-output detection (withLLRmax = ∞). For the considered block sizes, a reasonable setting for the LLR-clipping levelis obtained atLLRmax = 5.

Clearly, the impact of LLR-clipping is reduced for soft-output MSDD with multiple-observa-tions combining. To see this, note that in this case the effective LLR-clipping level at the inputof the decoder is given byL · LLRmax.

Without LLR clipping (LLRmax = ∞), the computational complexity of soft-output MSDDincreases dramatically already for moderate block size. This can also be seen in Figure 6.33,which depicts the average computational complexity of the single-tree-search sphere decodervs.Eb/N0 for different LLR-clipping levels andL = 5 (here, without multiple-observationscombining; the results for multiple-observations combining are simply scaled by the numberof observations,L). Different from the expected behavior of the sphere-decoder complexity,without LLR clipping the computational complexity is almost constant (but still significantlysmaller compared to an exhaustive search, which amounts to2 · (25 − 1)/5 = 12.4 for L = 5

and204.6 for L = 10).

These results emphasize that DFDD with multiple-observations combining represents a verycompetitive soft-output detection scheme. It offers significant gains over conventional soft-output MSDD and almost the same power efficiency as soft-output MSDD with multiple-

10The soft-output sphere decoder of Figure 5.10 delivers the information symbols and the decision metrics,which are then used to compute the LLRs via (5.22); thus hard-output detection can be modeled withLLRmax = 0.


12 12.5 13 13.5 14 14.5 15 15.50

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700

required 10 log10(Eb/N0) [dB]

avera

ge

co

mp

uta

tio

nal

co

mp

lexity

L = 3L = 5

L = 10

Figure 6.32: Trade-off required average computational complexity (number of visited nodes per symbol) vs.

required Eb/N0 in dB for a target BER = 10−3 for coded DTR IR-UWB transmission (Rc = 2/3) with block-

wise detection. Block size L = 3 (triangles), 5 (squares), and 10 (diamonds). Black: with MOC, gray: withoutMOC, solid lines: soft-output MSDD with different LLR-clipping levels LLRmax = 0, 1, 2.5, 5, 10, and ∞(right to left), single markers: DFDD. Line-of-sight channel model 2 of [MCC+06] with uniform weighting

within Ti = 50 ns.

observations combining, at comparable but constant computational complexity. Thus, withDFDD and multiple observations combining, the low computational complexity of hard-outputdetection is retained, yet, increased power efficiency is achieved. Note that this gain mainlyoriginates from the fact, that soft-decision decoding can be applied, even though a hard-outputdetection scheme is applied.

6.4.4 Power-Efficient Coded DTR IR-UWB Transmission

Finally, we investigate the impact of the code rate on the power efficiency of convolutionalcoded DTR IR-UWB transmission. Based on the above capacity considerations, even highercode rates, as compared to the optimum of symbol-wise differential detection attained atRc =

2/3, are expected to outperform the default setting ofRc = 1/2.

However, we note that the employed convolutional codes (andthe puncturing patterns forRc >

0.5) are optimized for the memoryless binary-input AWGN channel. ForL > 1, an optimizationof the employed codes for the considered setup might lead to further gains. However, theresults of Section 6.3.2.4 suggest, that—in combination with symbol-wise labeling—the inner


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tio

nal

co

mp

lexity

Figure 6.33: Average computational complexity (number of visited nodes per symbol) vs. Eb/N0 in dB forcoded DTR IR-UWB transmission (Rc = 2/3) with block-wise soft-output MSDD (block size L = 5 (left) and

10 (right)) and different LLR-clipping levels LLRmax = 0 (dashed), 1, 2.5, 5, 10, and∞ (bottom to top). Dash-

dotted: exhaustive search (only L = 5). Line-of-sight channel model 2 of [MCC+06], uniform weightingwithin Ti = 50 ns. Note the different scaling of the axes.

communication channel behaves like a memoryless binary-input AWGN channel. Thus, theclass of codes considered seems reasonable.

6.4.4.1 Block-Wise Detection

First, receivers for DTR IR-UWB that employ block-wise DFDDand soft-output MSDD withand without multiple-observations combining are considered. Exemplarily, the trade-off be-tween power efficiency and code rate is shown in Figures 6.34 and 6.35 forL = 2, and5,respectively. Note that, different to the symbol-wise detection schemes (energy detection forPPM and differential detection for DTR), here, we consider atarget bit error rate of10−4. ForL = 2, the constraint length is set toν = 5, as above. Since a system with larger block sizeanyway requires more complexity, we chooseν = 7 for L = 5. Corresponding bit-error-ratecurves are shown in Figure 6.36.

The results are compared to the respective capacity results, obtained from the Gaussian ap-proximation of the correlation coefficients, i.e., soft-output MSDD is compared toIACR(L) andIMSDD(L), and DFDD toImocDFDD(L) andIDFDD(L), for multiple-observations combining andconventional detection, respectively. For reference, thecapacity of soft-output differential de-tection is also shown (L = 1).

As expected, optimum power efficiency is achieved with code rates above1/2 and even largerthan2/3 representing the optimum rate for symbol-wise differential detection. However, otherthan expected (cf. Table 6.2), forL = 2 the code rate3/4 does not represent the optimum oper-ating point. This is due to the bad performance of the appliedrate-3/4 code (cf. Appendix A.3).


12 14 16 180

0.2

0.4

0.6

0.8

1

IMSDD

/IDFDD

/R[b

it/sy

mb

ol]


12 14 16 180

0.2

0.4

0.6

0.8

1


IACR

/ImocDFDD

/R[b

it/sy

mb

ol]


(constraint length ν = 5) with block-wise detection (L = 2) compared to capacity. Solid lines: soft-outputMSDD, dashed: DFDD, left: without MOC, right: with MOC, R = 1: uncoded soft-output MSDD without

MOC. Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).

10 12 14 160

0.2

0.4

0.6

0.8

1

IMSDD

/IDFDD

/R[b

it/sy

mb

ol]


10 12 14 160

0.2

0.4

0.6

0.8

1


IACR

/ImocDFDD

/R[b

it/sy

mb

ol]


(constraint length ν = 7) with block-wise detection (L = 5) compared to capacity. Solid lines: soft-output

MSDD, dashed: DFDD, left: without MOC, right: with MOC, R = 1: uncoded soft-output MSDD withoutMOC. Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).


10 12 14 16 1810

−5

10−4

10−3

10−2

10−1

BER


L = 2

10 12 14 16 1810

−5

10−4

10−3

10−2

10−1

BER


L = 5

Figure 6.36: BER vs. Eb/N0 of convolutional coded DTR IR-UWB for block-wise soft-output MSDD withMOC. Left: L = 2, ν = 5, right: L = 5, ν = 7. Code rates: Rc = 1/3, 1/2, 2/3, 3/4, 4/5, 5/6, and 7/8(light to dark gray). Dashed-black: uncoded mocMSDD, dashed-gray: coded DD (Rc = 2/3), left-most solid

gray: uncoded transmission with perfect channel state information. Channel model 2 of [MCC+06] withTi = 50 ns (Ni = 500).

Instead, forL = 2, out of the investigated settings the best performance is achieved withRc =

4/5, which results in an overall rate (including reference symbol and termination) ofR = 0.76

bit/dim. For a block size ofL = 5, too, the optimum operating point is achieved atRc = 4/5

(R = 0.75). Yet, slightly smaller and larger rates lead to almost the same performance.

6.4.4.2 Detection with Virtually Increased Block Size

In case of uncoded transmission, virtually increasing the block size has proven to be the mostattractive detection scheme operating on the output of a given autocorrelation device (cf. Sec-tion 5.4.1). This technique aims to approximate ideal noncoherent sequence estimation byDFDD operating on the entire burst of symbols (so-called vbDFDD, whereLvb = Nburst − 1).However, as opposed to the proposed block-wise schemes, this scheme does not deliver softoutput as desired for coded transmission. Here, we investigate if the lower error rate of harddecided code symbols at the input of the decoder is sufficientto compensate the loss due to therestriction to hard-decision decoding.

To this end, we consider an autocorrelation device withL = 5 and a code withν = 5.Figure 6.37 compares the bit error rate of soft-output mocMSDD, mocDFDD, and vbDFDD.For reference, symbol-wise soft-output differential detection is also shown. For the advancedschemes a rate of4/5 has been selected, whereas DD operates withRc = 2/3. These choicescorrespond to the optimum rates for the respective detection scheme, as derived above.

This comparison leads to the conclusion, that, although vbDFDD achieves better performancewith respect to code symbol error rate, it is not able to compensate for the lack of soft output.

6.5. Discussion 161

10 12 14 16 1810

−6

10−5

10−4

10−3

10−2

10−1

BER


Figure 6.37: BER vs. Eb/N0 of convolutional coded DTR IR-UWB (constraint length ν = 5, block sizeL = 5). Diamonds: vbDFDD, circles: soft-output mocMSDD, squares: mocDFDD, solid-gray: DD. Dashed-

gray: uncoded transmission of detection scheme corresp. to marker. Code rate of Rc = 4/5 (DD: Rc = 2/3).

Channel model 2 of [MCC+06] with Ti = 50 ns (Ni = 500).

Soft-output MSDD with MOC in combination with soft-decision decoding is still superior interms of bit error rate. This gain comes at the cost of significantly increased computationalcomplexity.

The complexity of mocDFDD and vbDFDD is comparable. Here, the simplified computation ofsoft output as in DFDD with MOC is not capable to achieve better performance than hard-outputvbDFDD.

Again, for a wide regime of signal-to-noise ratio, uncoded transmission outperforms codedtransmission (especially when considering vbDFDD). This is due to the restriction to relativelylow-complexity convolutional codes (here,ν = 5).

6.5 Discussion

In this chapter, we have studied the application of channel coding for PPM and DTR IR-UWBcommunications. We have derived and verified design rules for such systems, in particularoptimum code rates were identified, which take into account the noncoherent detection at thereceiver side.

For symbol-wise detection schemes, such as energy detection of PPM and differential detectionof DTR, it is shown that the use of codes with rates in the orderof 2/3 leads to highest powerefficiency. For advanced multiple-symbol detectors even higher rates are optimal.

For the considered convolutional codes, the gains over uncoded transmission are relatively small(in the order of1 dB). Hence, spending the additional complexity for the application of such


channel coding schemes depends on the application. Clearly, this would be different, whenmore powerful (i.e., almost capacity-achieving) codes were applied (which, in turn, leads toeven higher complexity), e.g., convolutional codes with larger constraint length or low-densityparity check codes [SF11d].

In conclusion, spending additional complexity in order to enable the application of advanced de-tection schemes, i.e., in particular providing an additional branch in the autocorrelation receiver,always pays off. Optimally selecting the code rate and employing a powerful (soft-output) de-tection scheme operating on the output of this advanced autocorrelation device in any case leadsto significant gains over coded transmission using conventional symbol-wise differential detec-tion. Hence, the achievable performance is only limited by the amount of complexity one isable to invest for the implementation of the receiver.

In case the receiver is located in the data-fusion center of the wireless sensor network, the im-plementation of an advanced autocorrelation device seems to be worth spending the additionalcomplexity. In case each sensor has to be equipped with a transmitter and a receiver, a fairevaluation of the power efficiency has to take into account the resources required for the entiresensor device. In order to increase battery-powered life times of the sensors, apart from thetransmit power required to guarantee sufficient transmission quality, also the energy consump-tion of the receiver chain has to be regarded. A comprehensive study of this trade-off is beyondthe scope of this thesis.

163

7. Conclusions

In this thesis, we have investigated coding, modulation, and detection for IR-UWB communi-cation systems. The design goals, e.g., demanded by wireless sensor networks, of these studieshave been low cost and low complexity of the transceivers as well as power- and energy-efficientoperation.

Based on a generic model of IR-UWB communication, a number ofpopular variants of (coded)IR-UWB modulation, such as pulse-position modulation (PPM) and differential transmitted-reference (DTR) IR-UWB, to name only the most important examples, have been treated in auniform manner. We have derived a generic truly noncoherentmaximum-likelihood receiver,which avoids any kind of explicit channel estimation. The conducted maximum-likelihood de-tection approach has been contrasted to generalized-likelihood ratio testing, an approach oftenpursued in literature.

We have shown, that for both cases the receiver basically implements the principle of energydetection or, equivalently, performs an autocorrelation of the receive signal prior to detection.The latter is to be preferred from an implementation point ofview, as it allows to separate the(analog) receiver front-end from the (digital) signal processing without loss in performance.First, the receiver front-end computes correlation coefficients, which are then used by variousdetection schemes for estimating the transmitted information symbols.

An approximate discrete-time model of IR-UWB transmissionin combination with autocorre-lation-based detection has been introduced, that serves asa basis for the analytical performanceanalysis conducted in this thesis. In particular, approximate error-rate expressions and capacityresults have been calculated from this model.

The presented derivation of the noncoherent receiver for generic IR-UWB modulation con-sistently incorporates the application of observation window lengths larger than one symbolduration. This enables to study very low-complexity symbol-wise detection schemes and theirextensions to advanced noncoherent detection schemes, which process multiple symbols jointly,simultaneously in a general view. The latter schemes offer improved performance due to im-

164 7. Conclusions

plicit noise averaging, however, rely on the fact, that the channel impulse response remainsconstant within the observation window—a condition usually satisfied in typical UWB scenar-ios.

This concept has been applied for the design of efficient detection schemes for pulse-position-modulated (PPM) IR-UWB and differential-transmitted reference (DTR) IR-UWB modulation,representing the most prominent and most attractive modulation schemes for IR-UWB com-munications (please refer to Figure 1.1 in the introductionfor an overview of the discusseddetection schemes).

For both variants, next to well-known symbol-wise detection schemes, such as energy detec-tion and autocorrelation-based differential detection, advanced detection schemes processingmultiple symbols jointly have been presented. These operate on the output of an extended auto-correlation receiver and aim to approximate ideal noncoherent sequence estimation of the entireburst of transmit symbols.

For PPM IR-UWB, the only attractive detection scheme is given by symbol-wise energy detec-tion. Here, both, modulation and detection have extremely low implementation complexity andare thus the candidate of choice, if cost and complexity are the major design goals. This comesat the cost of reduced data rate and inferior error-rate performance. In principle, increased per-formance could be achieved with multiple-symbol detectionof PPM IR-UWB. Here, however,significantly higher implementation complexity, both, forthe (analog) autocorrelation receiverfront-end and the (digital) signal processing of the decision unit has to be invested. Similarperformance gains are achieved for DTR IR-UWB at even lower complexity of the receiverfront-end and especially of the digital signal processing.Hence, multiple-symbol-detection-based schemes are promising candidates especially for DTR IR-UWB.

Here, we distinguish block-wise detection schemes from those performing sequence estimation.Starting from block-wise optimal detection according to the well-known concept of multiple-symbol differential detection (MSDD) (preferably implemented using the sphere decoder algo-rithm), block-wise decision-feedback differential detection (DFDD) schemes are derived. It isshown, that the performance of block-wise detection schemes can be improved by combiningmultiple observations resulting from processing of slightly shifted blocks.

Similarly, for the sequence-estimation schemes, a sliding-window variant of DFDD is derivedbuilding upon the signal processing of optimum sequence estimation. However, whereas theimplementation of the latter in IR-UWB receivers is infeasible due to its very high computa-tional complexity, sliding-window DFDD achieves only marginal performance improvementover conventional symbol-wise differential detection.

A comprehensive comparison of these detection schemes for DTR IR-UWB with respect tocomputational complexity and error-rate performance leads to the conclusion that block-wiseDFDD represents the most attractive candidate for low-complexity power-efficient DTR re-ceivers. It offers excellent error-rate performance at lowand, more importantly, constant com-putational complexity. This excellent performance mainlyresults from the application of anoptimum decision order within the detection process.

165

If sufficiently large bursts are transmitted, a proposed hybrid version of the above approaches,so-called DFDD with virtually increased block size, enables further performance gains at mod-erately increased structural delay and memory requirements. These gains are achieved by anincreased number of degrees of freedom for finding the optimum decision order, and thus thisvariant of DFDD closely and efficiently approximates ideal noncoherent sequence estimationof the entire transmitted burst.

Moreover, we have presented extensions of the block-wise detection schemes in order to incor-porate the computation of reliability information on the estimates. These so-called soft-outputdetectors deliver log-likelihood ratios for the estimatedcode symbols, which can be utilized inthe decoding process of a subsequent channel decoder in coded IR-UWB transmission.

In this context, again especially DFDD in combination with multiple-observations combiningoffers very good performance at only marginally increased computational complexity com-pared to conventional hard-output DFDD. By simply adding the individual hard decisions, thisso-called “soft-output DFDD” forms a simple form of—although very coarse but sufficientlymeaningful—reliability information. Thus, the additional gains of soft- vs. hard-decision de-coding in power efficiency can be exploited without the need to significantly increase the com-putational complexity of the receive signal processing.

All presented detection schemes operate on the output of an autocorrelation receiver. Thisautocorrelation device could be implemented either in the analog or in the digital domain basedon a sufficiently high sampled receive signal. In case of advanced multiple-symbol-detection-based schemes, both options incur demanding challenges forhardware implementation. Whilerealization of the latter requires (possibly prohibitively high) sampling rates of up to10GHz,an analog implementation appears to be hard to realize due tothe required accurate delayingof the receive signal over multiples of the symbol duration,i.e., delays in the order of severalhundred nanoseconds.

In order to avoid the implementation of an autocorrelation device operating directly on thereceive signal, we have proposed an approach, which is basedon the principle of compressedsensing. Here, sampling of the receive signal at a high sampling rate is replaced by correlationof the receive signal with a moderate number of fixed measurement functions. Based on thisreceiver preprocessing, all presented detection schemes are applicable and yield satisfactoryresults already at small fractions of the Nyquist sampling rate (in the order of10 to 25%).

Finally, we have addressed the impact of noncoherent autocorrelation-based detection schemeson the overall performance of coded IR-UWB transmission. Tothis end, we have conductedan information-theoretic analysis of PPM and DTR IR-UWB communications, which explicitlytakes into account the noncoherent detection schemes. Thereby we have restricted ourselves tothe common approach of bit-interleaved coded modulation, i.e., the serial concatenation of cod-ing, interleaving, and modulation. As expected from long-known principles for other commu-nication systems with noncoherent detection, optimum coderates for power-efficient IR-UWBtransmission could be identified.

For the considered system parameters, these rates are in theorder of2/3 to 4/5, and thus, for

166 7. Conclusions

the same channel conditions, enable higher data rates at lower required signal-to-noise ratioto guarantee a desired error rate compared to the probably straight-forward choice of a rate-1/2 code, as, e.g., employed in [IEE07]. Moreover, the influenceof the binary label and theinterleaver in bit-interleaved coded DTR modulation has been investigated. These observationshave been verified by means of numerical simulations of convolutional coded transmission.

For these and all other conducted numerical simulations in this thesis, up-to-date channel mod-els of UWB signal propagation (especially those of [MCC+06]) have been employed, whichvalidates the results and conclusions for realistic operating conditions. However, some impor-tant aspects have been blanked out in this analysis and/or simply taken to be ideal. These aspectsinclude the synchronization of transmitter and receiver (i.e., clock rate and timing offset) andthe effects caused by non-ideal implementation of the analog receiver front-end. Moreover, thesystem parameters considered in this thesis do not fulfill the regulatory requirements for UWBcommunications of [FCC02, Bun08].

In order to ensure low implementation and computational complexity, the detection schemespresented in this thesis avoid any kind of explicit channel estimation. A comparison of thisapproach to detection based on coarse channel estimation, such as rake receivers with smallnumber of rake fingers [CS02] or detection based on compressed-sensing-based channel esti-mation [OL09, LDL09], has not been conducted.

Moreover, the impact of narrow-band interference and of thesimultaneous (and asynchronous)operation of multiple users in the UWB system has not been investigated. The extension ofthe noncoherent detection schemes presented in this thesiswith the aim of incorporating jointmulti-user detection and/or interference cancellation appears to be an interesting task for futureresearch.

In summary, this thesis has layed an information-theoreticfoundation for the implementationof power-efficient low-complexity IR-UWB communication systems.

167

A. Supporting Material and Derivations

A.1 Derivation of (3.12)

In this appendix, a detailled derivation of the marginal distribution required for noncoherentmaximum-likelihood detection is given. Parameters and variables of this derivation refer to thedefinitions as in Section 3.2.

The derivation is based on the definition of the marginal distribution in (3.9). In order to obtaina receiver structure valid for a large class of typical UWB channels, the following assumptionsare drawn:

� the elements of the receive pulse hypothesisp are uncorrelated zero-mean Gaussian dis-tributed, i.e.,

fpm(p) =1

√

2πσ2pm

e− p2

2σ2pm (A.1)

� the variance of the elements ofp are given byσ2pm , representing an arbitrary power-delay

profile underlying the modeling of the channel impulse response.

The reasoning behind these assumptions is explained in Section 3.2.

In this case, the probability density function of the receive pulse can be written as

fp(p) =Ns−1∏

m=0

fpm(pm) =Ns−1∏

m=0

1√

2πσ2pm

e−

p2m2σ2

pm . (A.2)

Substituting these assumptions into the definition of the marginal distribution, one obtains

fMLR (R|b) =

∫ +∞

−∞

fR(R|b, p) · fp(p)dp (A.3)

=

Ns−1∏

m=0

∫ +∞

−∞

ft(tm|b, pm) · fpm(pm)dpm (A.4)

168 A. Supporting Material and Derivations

wheretm = emR denotes themth row of R, i.e., the collection of theN samples at thetime instancemTs within each symbol duration. Again, this follows a multivariate Gaussiandistribution, i.e.,

ft(tm|b, p) = ct · e−1

2σ2n‖tm−pmb‖2

(A.5)

wherect denotes a constant, irrelevant for the detection.

Substituting (A.2) and (A.5) into (A.4), we have

fMLR (R|b) =

Ns−1∏

m=0

∫ +∞

−∞

ct√

2πσ2pm

· e−1

2σ2n‖tm−pmb‖2 · e

−p2m

2σ2pm dpm . (A.6)

This expression can be simplified using the relation∫ +∞

−∞e−αp2+2βpdp =

√

παe

β2

α [GR00, Equa-tion 3.462–2.], yielding

fMLR (R|b) =

Ns−1∏

m=0

ct√

σ2pm

σ2n‖b‖2 + 1

e

− 1

2σ2n

‖tm‖2−(2btTm)2

‖b‖2+σ2n

σ2pm

(A.7)

=

c2t∏Ns−1

m=0

(

σ2pm

σ2n‖b‖2 + 1

)

Ns/2

e

− 1

2σ2n

∑Ns−1m=0

‖tm‖2−(2btTm)2

‖b‖2+σ2n

σ2pm

(A.8)

=

c2t∏Ns−1

m=0

(

σ2pm

σ2n‖b‖2 + 1

)

Ns/2

e

− 1

2σ2n

‖R‖2F−∑Ns−1

m=04

‖b‖2+σ2n

σ2pm

(btTm)2

. (A.9)

In order to simplify notation, we introduce the squared weighting factors

w2m =

4

‖b‖2 + σ2n

σ2pm

(A.10)

and the diagonal matrix

Σ =

w0 0. . .

0 wNs−1

. (A.11)

Thus, the second summand of the exponent can be simplified to

Ns−1∑

m=0

4

‖b‖2 + σ2n

σ2pm

(

btTm)2

=

Ns−1∑

m=0

(

wmbtTm

)2=∥

∥bRTΣ∥

∥

2. (A.12)

Finally, we obtain

fMLR (R|b) =

c2t∏Ns−1

m=0

(

σ2pm

σ2n‖b‖2 + 1

)

Ns/2

e− 1

2σ2n

(

‖R‖2F−‖bRTΣ‖2

)

(A.13)

A.1. Derivation of (3.12) 169

which is the desired marginal distribution of (3.12).

Based on this result, a truly noncoherent maximum-likelihood receiver for IR-UWB communi-cations is derived in Section 3.2.


A.2 The Chi-Squared Distribution

In this appendix, the properties of a chi-squared (in short,χ2) distribution are summarizedbriefly. GivenL independent real-valued Gaussian distributed random variablesxl, l = 0, . . . , L−1, with (possibly different) meanml and equal varianceσ2

x, the random variabley given as thesum of the squared values, i.e.,

y =L−1∑

l=0

|xl|2 (A.14)

is non-centralχ2-distributed withL degrees of freedom. Defining the non-centrality parameterλ =

∑L−1l=0 m2

l , its probability density function, its mean, and its variance are given, respectively,as

fy(y) =1

2σ2x

(

y

λσ2x

)L4− 1

2

e−

y+λσ2x

2σ2x · IL

2−1(√

λy/σ2x) (A.15)

my = σ2x(L+ λ) (A.16)

σ2y = (σ2

x)2(2L+ 4λ) . (A.17)

Here,Iα(·) denotes the modified Bessel function of first kind and orderα. Noteworthy, theseparameters depend only on the sum of the squared mean values of xl, i.e., the non-centralityparameterλ, and not on the particular distribution ofml overl.

For compact notation, we introduce the notationy ∼ χ2(L, λ), for a χ2-distributed randomvariabley with L degrees of freedom and non-centrality parameterλ.

Only if ml = 0, for all l, we haveλ = 0; in this case the distribution reduces to the so-calledcentralχ2 distribution with

fy(y) =1

(2σ2x)

L/2(

L2− 1)

!y

L2−1e

− y

2σ2x (A.18)

my = σ2xL (A.19)

σ2y = 2L(σ2

x)2 . (A.20)

For sufficiently large degrees of freedom, following the lawof large numbers,y is well ap-proximated as a Gaussian random variable, with mean and variance according to the respectiveparameters.

Exemplarily, Figure A.1 depicts the centralχ2 distribution for differentL andσ2x = 1. For

comparison, the Gaussian approximation is included starting fromL ≥ 10.

A.2. The Chi-Squared Distribution 171

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 7510

−4

10−3

10−2

10−1

100

L = 13 5 10 20 30 40 50

y

f y(y)

Figure A.1: Central χ2 distribution for different degrees of freedom L with σ2x = 1 compared to Gaussian

approximation (gray, only for L ≥ 10).


A.3 Convolutional Codes for the Binary-Input AWGN

Channel

For reference and sake of completeness, in this appendix theparameters of the applied convolu-tional codes are listed and the trade-off between power efficiency and code rate is reviewed forthe case of convolutional coded transmission over the symmetric binary-input AWGN channelwith coherent detection.

Figures A.2, A.3, A.4, and A.5 depict this trade-off for the convolutional codes considered inChapter 6 with constraint lengthν = 3, 4, 5, and7, respectively. Code ratesRc = 1/4, 1/3,1/2, 2/3, 3/4, 4/5, 5/6, and7/8 are shown (forRc > 0.5, puncturing of the rate-1/2 parentcode has been applied). The code parameters are taken from [PS08, Table 8.3-1, -2, -3, and8.4-1] and summarized in Tables A.1 and A.2.

In both cases, soft- and hard-decision decoding are considered and the target bit error rate hasbeen set to10−4. The results are compared to the capacity of the binary-input AWGN channelwith coherent detection. Based on capacity considerations, for decreasing code rate the requiredEb/N0 to guarantee the target bit error rate is expected to shift tosmaller values. This is thecase for the code with constraint lengthν = 5 and7 and for a wide regime also for the case ofν = 4.

For the convolutional code withν = 3 andν = 7, however, we observe aC-like shape similarto that obtained for energy- and autocorrelation-based noncoherent detection of IR-UWB. Thisis caused by the difficulty of designing “good” low-rate convolutional codes with low constraintlength [PS08]. Yet, also for this case the optimum operatingpoint with respect to highest powerefficiency is clearly shifted to lower code rates compared tothe case of IR-UWB. For the latter,code rates above2/3 are preferred, whereas here the optimum code rate is1/2.

For soft-decision decoding we observe a relatively strict line-up, that closely follows the ca-pacity curve of binary phase-shift keying. The only exeption is the code withν = 5 and therate3/4, is can be explained by jumps in the maximum free distance at this point (from4 atRc = 2/3 to 3 atRc = 3/4, cf. Table A.2).

We also observe that for high rates, the shape slightly differs from that of the capacity. Espe-cially for ν < 7, the loss caused by selecting higher rates is less than suggested by the capacity.

For hard-decision decoding significant deviation from thisstrict line-up can be observed. Areason for this behavior might be that the codes are optimized for soft-decision decoding.

A.3. Convolutional Codes for the Binary-Input AWGN Channel 173

Table A.1: Convolutional codes applied in this thesis. Generator polynomials in octal representation.Adapted from [PS08, Table 8.3-1, -2, -3].

Rc ν generator polynomials free distance1/4 3 5, 7, 7, 7 10

4 13, 15, 15, 17 135 25, 27, 33, 37 167 135, 135, 147, 163 20

1/3 3 5, 7, 7 84 13, 15, 17 105 25, 33, 37 127 133, 145, 175 15

1/2 3 5, 7 54 15, 17 65 23, 35 87 133, 171 10

Table A.2: Punctured convolutional codes applied in this thesis. Rate-1/2 parent code as in Table A.1.

Adapted from [PS08, Table 8.4-1].

Rc ν puncturing pattern free distance2/3 3 1 1 0 1 3

4 1 1 1 0 45 1 1 1 0 47 1 1 1 0 6

3/4 3 1 1 0 1 1 0 34 1 1 1 0 0 1 45 1 1 0 1 1 0 37 1 1 1 0 0 1 5

4/5 3 1 1 0 1 1 0 1 0 24 1 1 0 1 1 0 1 0 35 1 1 0 1 1 0 0 1 37 1 1 1 0 1 0 1 0 4

5/6 3 1 1 0 1 1 0 1 0 1 0 24 1 1 0 1 1 0 0 1 0 1 35 1 1 0 1 1 0 1 0 1 0 37 1 1 1 0 0 1 1 0 1 1 4

7/8 3 1 1 0 1 1 0 1 0 1 0 1 0 1 0 24 1 1 0 1 0 1 0 1 0 1 1 0 0 1 25 1 1 0 1 1 0 0 1 0 1 1 0 1 0 37 1 1 1 0 1 0 1 0 0 1 1 0 0 1 3


−2 0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Rc

[bit/

sym

bo

l]


Figure A.2: Rate vs. power-efficiency (required Eb/N0 for target BER = 10−4) of convolutional coded

transmission over the binary-input AWGN channel with coherent detection. Constraint length ν = 3. Solidlines: soft-decision decoding, dashed: hard-decision decoding, gray: capacity.

−2 0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Rc

[bit/

sym

bo

l]




A.3. Convolutional Codes for the Binary-Input AWGN Channel 175

−2 0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Rc

[bit/

sym

bo

l]




−2 0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Rc

[bit/

sym

bo

l]




176

177

B. Notation

B.1 Abbreviations

Acronym Meaning

ACR autocorrelation receiver

AWGN additive white Gaussian noise

BER bit error rate

BICM bit-interleaved coded modulation

BPSK binary phase-shift keying

CH channel

CM channel model

CS compressed sensing

CSI channel state information

DD differential detection

dd decision-directed

DEC decoder

DFDD decision-feedback differential detection

DPSK differential phase-shift keying

DS direct-sequence spreading

DTR differential transmitted-reference

ED energy detection

ENC encoder

GLRT generalized likelihood-ratio testing

hist histogram

178 B. Notation

Acronym Meaning

ISI inter-symbol-interference

INSE ideal noncoherent sequence estimation

IR-UWB impulse-radio ultra-wide band

LDPC low-density parity check

LLR log-likelihood ratio

MIMO multiple-input/multiple-output

ML maximum-likelihood

MLC multi-level coding

MOC multiple-observations combining

MSD multiple-symbol detection

MSDD multiple-symbol differential detection

MSDSD multiple-symbol differential sphere decoder

opt. optimum

PAM pulse-ampitude modulation

PDF probablity density function

PPM pulse-position modulation

req. required

RX receiver

RSSE reduced-state sequence estimation

SER symbol error rate

SO-MSDSD soft-output multiple-symbol differential sphere decoder

TH time-hopping

TR transmitted-reference

TX transmitter

UWB ultra-wide band

vb virtually-increased block size

B.2. Mathematical Symbols 179

B.2 Mathematical Symbols

Frequently used variables, vectors, matrices and sets.

Variable Meaning

a information symbols

b transmit symbols

b transmit symbols of observation window grouped into vector

BER bit error rate

B set of transmit symbols

βdd weighting factor in decision-directed autocorrelation device

c binary code symbols

C capacity

const. constant factor

D dimensionality of modulation

D sampled measurment functions collected in matrix

dl(t) measurment functions of CS-based detection

Eb energy per information bit

Ec energy per code symbol

Es energy per transmit symbol

Ep captured receive pulse energy

fs sampling rate

fx probability density function of random variablex

F2 Galois field of size two

γCM intra-cluster power-decay factor

ΓCM cluster power-decay factor

hCH(t) channel impulse response

hRX(t) receiver input filter

I mutual information

I identity matrix

k symbol index

Ksource number of binary source symbols per burst

L number of information symbols in observation window

λCM intra-cluster arrival rate

ΛCM cluster arrival rate

Λ MSDD decision metric

180 B. Notation

Variable Meaning

Λmax MSDD decision metric corresp. toLLRmax

LLR log-likelihood ratio

LLRmax LLR clipping level

m number of code symbols per hypersymbol

M cardinality of signal constellationBn(t) filtered AWGN

n0(t) AWGN

nk sampled filtered noise of symbol intervalk

ηk,l equivalent noise on correlation coefficientzk,l

ν constraint length of convolutional code

N number of transmit symbol in observation window

N0 one-sided noise-power spectral density

Nburst number of transmit symbols per burst

Ncs number of measurments of CS-based detection

Ncode number of code symbols per burst

Ni time-bandwidth product

Ns number of samples per symbol

p(t) receive pulse

pTX(t) transmit pulse

p sampled receive pulse

q binary source symbols

r(t) receive signal

rk sampled receive signal of symbol intervalk

R overall rate

R grouped sampled receive signal of observation window

Rc code rate

Rm rate of modulation

Rsd sphere-decoder search radius

Rstop stopping radius

s(t) transmit signal

σ2b variance of transmit symbols

σ2n noise variance

σ2η variance of equivalent noise

B.2. Mathematical Symbols 181

Variable Meaning

S block shift

SER symbol error rate

Σ matrix of weighting factors

t continuous time

τ pulse-width parameter

T symbol duration

TD hypersymbol duration

Ti duration of correlation interval

Ts sampling period

wm weighting factors

wΣ(t) weighting functions

y measurments of CS-based detection

zk,l correlation coefficients of receive signal intervalsk andl

Z matrix of correlation coefficients

Operators, mappings, etc.

Meaning

·T transpose of vector/matrix

‖ · ‖F Frobenius norm of matrix

‖ · ‖ Euclidean norm of vector

‖ · ‖0 l0-“norm” (number of non-zero elements)

argmax returns value maximizing argument

argmin returns value minimizing argument

e2(·) binary entropy functione2(x) = −x log2(x)− (1−x) log2(1−x)

Ex{·} expectation operator w.r.t. random variablex

I(·; ·) mutual information

logx(·) logarithm to basex

ln(·) natural logarithm to basee, ln(·) = loge(·)M mapping from code to transmit symbols

max returns maximum value of argument

min returns minimum value of argument

N (m, σ2) normal distribution with meanm and varianceσ2

Pr{·} probability

182 B. Notation

Meaning

Q(·) Q-functionQ(z) = 12π

∫∞

ze−

τ2

2 dτ

x hypothesis for variablex

xscheme estimate/result forx obtained fromscheme

χ2(L, λ) χ2 distribution withL degrees of freedom and non-centrality pa-rameterλ

z variablez of Z-transform

sign(x) sign operator (+1 for x ≥ 0,−1 for x < 0)

B.3 Detection Schemes

Overview of analyzed detection schemes.

Acronym Detection scheme

CSI detection with perfect channel-state information

cs· compressed-sensing-based detection scheme “·”DD differential detection

ddDD decision-directed differential detection

DFDD decision-feedback differential differential detection

bDFDD block-wise decision-feedback differential differential detection

sbDFDD sorted block-wise decision-feedback differential detection

swDFDD sliding-window decision-feedback differential detection

ED energy detection

moc· block-wise detection scheme “·” with multiple-observationscombining

MSD multiple-symbol detection

MSDD multiple-symbol differential detection

RSSE reduced-state sequence estimation

vb· block-wise detection scheme “·” with virtually increased blocksize

183

Bibliography

[AAS09] A. Alvarado, E. Agrell, and A. Svensson. On the Capacity of BICM withQAM Constellations. InProceedings of the International Conference on Wire-less Communications and Mobile Computing, pages 573–579, New York, USA,June 2009.

[AEVZ02] E. Agrell, T. Eriksson, E. Vardy, and K. Zeger. Closest Point Search in Lattices.IEEE Transactions on Information Theory, 48(8):2201–2214, August 2002.

[AL09] Z. Ahmadian and L. Lampe. Performance Analysis of theIEEE 802.15.4a UWBSystem.IEEE Transactions on Communications, 57(5):1474–1485, May 2009.

[AM84] J. Anderson and S. Mohan. Sequential Coding Algorithms: A Survey and CostAnalysis. IEEE Transactions on Communications, 32(2):169–176, February1984.

[AR03] G. Aiello and G. Rogerson. Ultra-Wideband Wireless Systems. IEEE Mi-crowave Magazine, 4(2):36–47, June 2003.

[Arı09] E. Arıkan. Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels. IEEETransactions on Information Theory, 55(7):3051–3073, July 2009.

[Bab86] L. Babai. On Lovász’ Lattice Reduction and the Nearest Lattice Point Problem.Combinatorica, 6(1):1–13, March 1986.

[BDL+76] S. Butman, I. Bar David, B. Levitt, R. Lyon, and M. Klass. Design Criteriafor Noncoherent Gaussian Channels with MFSK Signaling and Coding. IEEETransactions on Communications, 24(10):1078–1088, October 1976.

[BCJR74] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv. OptimalDecoding of Linear Codesfor Minimizing Symbol Error Rate (Corresp.).IEEE Transactions on Informa-tion Theory, 20(2):284–287, March 1974.

[BT08] L. Barbero and J. Thompson. Fixing the Complexity of the Sphere Decoder forMIMO Detection.IEEE Transactions on Wireless Communications, 7(6):2131–2142, June 2008.

184 Bibliography

[Bun08] Bundesnetzagentur. Allgemeinzuteilung von Frequenzen für die Nutzung durchAnwendungen geringer Leistung der Ultra-Wideband (UWB) Technologie.Vfg1/2008 geändert mit Vfg. 17/2008, 2008.

[CD05] M. Casu and G. Durisi. Implementation Aspects of a Transmitted-ReferenceUWB Receiver. Wireless Communications and Mobile Computing, 5(5):537–549, August 2005.

[CNBL06] R. Cardinali, L. De Nardis, M. Di Benedetto, and P. Lombardo. UWB RangingAccuracy in High- and Low-Data-Rate Applications.IEEE Transactions onMicrowave Theory and Techniques, 54(4):1865–1875, June 2006.

[CG09] M. Chiani and A. Giorgetti. Coexistence Between UWB and Narrow-BandWireless Communication Systems.Proceedings of the IEEE, 97(2):231–254,February 2009.

[CK02] X. Chen and S. Kiaei. Monocycle Shapes for Ultra Wideband System.In Pro-ceedings of the IEEE International Symposium on Circuits and Systems, vol. 1,pages 597–600, May 2002.

[CM06] C. Carbonelli and U. Mengali. M-PPM Noncoherent Receivers for UWB Ap-plications. IEEE Transactions on Wireless Communications, 5(8):2285–2294,August 2006.

[CR01] G. Colavolpe and R. Raheli. The Capacity of the Noncoherent Channel.Euro-pean Transactions on Telecommunications, 12(4):289–296, April 2001.

[CS02] J. Choi and W. Stark. Performance of Ultra-Wideband Communications withSuboptimal Receivers in Multipath Channels.IEEE Journal on Selected Areasin Communications, 20(9):1754–1766, September 2002.

[CS03] Y. Chao and R. Scholtz. Optimal and Suboptimal Receivers for Ultra-WidebandTransmitted Reference Systems.In Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM), vol. 2, pages 759–763, December2003.

[CS04] Y. Chao and R. Scholtz. Weighted Correlation Receivers for Ultra-WidebandTransmitted Reference Systems.In Proceedings of the IEEE Global Telecom-munications Conference (GLOBECOM), vol. 1, pages 66–70, Nov./Dec. 2004.

[CSB87] J. Conway, N. Sloane, and E. Bannai.Sphere Packings, Lattices, and Groups.3rd edition, Springer-Verlag, Inc., New York, NY, USA, December 1998.

[CSW02] R. Cramer, R. Scholtz, and M. Win. Evaluation of an Ultra-Wide-Band Propa-gation Channel.IEEE Transactions on Antennas and Propagation, 50(5):561–570, May 2002.

185

[CTB98] G. Caire, G. Taricco, and E. Biglieri. Bit-Interleaved Coded Modulation.IEEETransactions on Information Theory, 44(3):927–946, March 1998.

[CW08] E. Candes and M. Wakin. An Introduction To Compressive Sampling. IEEESignal Processing Magazine, 25(2):21–30, February 2008.

[CWM02] D. Cassioli, M. Win, and A. Molisch. The Ultra-Wide Bandwidth Indoor Chan-nel: From Statistical Model to Simulations.IEEE Journal on Selected Areas inCommunications, 20(6):1247–1257, August 2002.

[DB04] G. Durisi and S. Benedetto. Performance of Coherent and Noncoherent Re-ceivers for UWB Communications. InProceedings of the IEEE InternationalConference on Communications, vol. 6, pages 3429–3433, June 2004.

[DGC03] M. Damen, H. El Gamal, and G. Caire. On Maximum-Likelihood Detectionand the Search for the Closest Lattice Point.IEEE Transactions on InformationTheory, 49(10):2389–2402, October 2003.

[HH89] A. Duel Hallen and C. Heegard. Delayed Decision-Feedback Sequence Estima-tion. IEEE Transactions on Communications, 37(5):428–436, May 1989.

[DM05] A. D’Amico and U. Mengali. GLRT Receivers for UWB systems. IEEE Com-munications Letters, 9(6):487–489, June 2005.

[DMR07] A. D’Amico, U. Mengali, and E. Arias-de Reyna. Energy-Detection UWB Re-ceivers with Multiple Energy Measurements.IEEE Transactions on WirelessCommunications, 6(7):2652–2659, July 2007.

[Don06] D. Donoho. Compressed Sensing.IEEE Transactions on Information Theory,52(4):1289–1306, April 2006.

[DS90] D. Divsalar and M. Simon. Multiple-Symbol Differential Detection of MPSK.IEEE Transactions on Communications, 38(3):300–308, March 1990.

[Edb92] F. Edbauer. Bit Error Rate of Binary and Quaternary DPSK Signals with Mul-tiple Differential Feedback Detection.IEEE Transactions on Communications,40(3):457–460, March 1992.

[EK12] Y. Eldar and G. Kutyniok, editors.Compressed Sensing: Theory and Applica-tions. Cambridge University Press, New York, 1st edition, May 2012.

[EQ88] M. Eyuboglu and S. Qureshi. Reduced-State Sequence Estimation with SetPartitioning and Decision Feedback.IEEE Transactions on Communications,36(1):13–20, January 1988.

[FCC02] Federal Communications Commission (FCC). First Report and Order: In thematter of Revision of Part 15 of the Commission’s Rules Regarding Ultra-Wideband Transmission Systems. 02–48, April 2002.

186 Bibliography

[FCLM99] R.F.H. Fischer, S. Calabro, L. Lampe, and S. Müller-Weinfurtner. Performanceof Coded Modulation Employing Differential Encoding over Rayleigh FadingChannels.Electronics Letters, 35(2):122–123, January 1999.

[FH06a] R.F.H. Fischer and J.B. Huber. Digitale Übertragung. Lecturenotes, Lehrstuhl für Informationsübertragung, Friedrich-Alexander-UniversitätErlangen-Nürnberg, April 2006.

[FH06b] R.F.H. Fischer and J.B. Huber. Informationstheorie und deren Anwendungen inder Nachrichtenübertragung. Lecture notes, Lehrstuhl fürInformationsübertra-gung, Friedrich-Alexander-Universität Erlangen-Nürnberg, April 2006.

[Fis02] R.F.H. Fischer.Precoding and Signal Shaping for Digital Transmission. JohnWiley & Sons, Inc., New York, NY, USA, August 2002.

[Fis11] R.F.H. Fischer. Multiuser Communications and MIMOSystems. Lecturenotes, Lehrstuhl für Informationsübertragung, Friedrich-Alexander-UniversitätErlangen-Nürnberg, April 2011.

[FLMH00] R.F.H. Fischer, L. Lampe, S. Müller-Weinfurtner,and J.B. Huber. Coded Mod-ulation for Differential Encoding and Non-Coherent Reception on Fading Chan-nels. InProceedings of the International ITG Conference on Source and Chan-nel Coding (SCC), pages 139–146, München, Germany, January 2000.

[FM06] S. Franz and U. Mitra. Generalized UWB Transmitted Reference Systems.IEEEJournal on Selected Areas in Communications, 24(4):780–786, April 2006.

[Fon04] R. Fontana. Recent System Applications of Short-Pulse Ultra-Wideband(UWB) Technology.IEEE Transactions on Microwave Theory and Techniques,52(9):2087–2104, September 2004.

[Fos96] G. Foschini. Layered Space-Time Architecture for Wireless Communicationin a Fading Environment When Using Multiple Antennas.Bell LaboratoriesTechnical Journal, 1(2):41–59, 1996.

[FP85] U. Fincke and M. Pohst. Improved Methods for Calculating Vectors of ShortLength in a Lattice, Including a Complexity Analysis.Mathematics of Compu-tation, 44(170):463–471, April 1985.

[FS10] M. Farhang and J. Salehi. Optimum Receiver Design forTransmitted-ReferenceSignaling. IEEE Transactions on Communications, 58(5):1589–1598, May2010.

[GH07] R. Gowaikar and B. Hassibi. Statistical Pruning for Near-Maximum LikelihoodDecoding. IEEE Transactions on Signal Processing, 55(6):2661–2675, June2007.

187

[GMC08] A. Guillén i Fábregas, A. Martinez, and G. Caire. Bit-Interleaved Coded Mod-ulation. Foundations and Trends in Communications and Information Theory,5(1-2):1–153, 2008.

[GLL10] S. Gishkori, G. Leus, and V. Lottici. Compressive Sampling Based Differen-tial Detection of Ultra Wideband Signals. InProceedings of the IEEE Interna-tional Personal Indoor and Mobile Radio Communications (PIMRC) Commu-nications, pages 194–199, September 2010.

[GLM09] Y. Gan, C. Ling, and W. Mow. Complex Lattice Reduction Algorithm for Low-Complexity Full-Diversity MIMO Detection.IEEE Transactions on Signal Pro-cessing, 57(7):2701–2710, July 2009.

[GN04] Z. Guo and P. Nilsson. Reduced Complexity Schnorr-Euchner Decoding Algo-rithms for MIMO Systems.IEEE Communications Letters, 8(5):286–288, May2004.

[GQ06] N. Guo and R. Qiu. Improved Autocorrelation Demodulation Receivers Basedon Multiple-Symbol Detection for UWB Communications.IEEE Transactionson Wireless Communications, 5(8):2026–2031, August 2006.

[GR00] I. Gradshtein and I. Ryzhik.Table of Integrals, Series, and Products. AcademicPress, San Diego, 6th edition, August 2000.

[GT08] A. Ghaderipoor and C. Tellambura. A Statistical Pruning Strategy for Schnorr-Euchner Sphere Decoding.IEEE Communications Letters, 12(2):121–123,February 2008.

[Gur08] M. Gursoy. On the Energy Efficiency of Orthogonal Signaling. InProceedingsof the IEEE International Symposium on Information Theory (ISIT), pages 599–603, July 2008.

[HB04] B. Hu and N. Beaulieu. Pulse Shaping in UWB Communication Systems. InProceedings of the IEEE Vehicular Technology Conference, vol. 7, pages 5175–5179, September 2004.

[HF92] P. Ho and D. Fung. Error Performance of Multiple-Symbol Differential De-tection of PSK Signals Transmitted over Correlated Rayleigh Fading Channels.IEEE Transactions on Communications, 40(10):1566–1569, October 1992.

[HH03] J.B. Huber and S. Huettinger. Information Processing and Combining in Chan-nel Coding. InProceedings of the International Symposium on Turbo Codes &Related Topics, pages 95–102, Brest, France, September 2003.

[HH09] T. Hehn and J.B. Huber. LDPC Codes and Convolutional Codes with EqualStructural Delay: A Comparison.IEEE Transactions on Communications,57(6):1683–1692, June 2009.

188 Bibliography

[HJ90] R. Horn and C. Johnson.Matrix Analysis. Cambridge University Press, NewYork, NY, USA, February 1990.

[HL99] P. Hoeher and J. Lodge. “Turbo DPSK”: Iterative Differential PSK De-modulation and Channel Decoding.IEEE Transactions on Communications,47(6):837–843, June 1999.

[Hub05] J.B. Huber. Nachrichtenübertragung. Lecture notes, Lehrstuhl für Informa-tionsübertragung, Friedrich-Alexander-Universität Erlangen-Nürnberg, October2005.

[HV05] B. Hassibi and H. Vikalo. On the Sphere-Decoding Algorithm I. Expected Com-plexity. IEEE Transactions on Signal Processing, 53(8):2806–2818, August2005.

[IEE07] IEEE Std 802.15.4a-2007, IEEE Standard for PART 15.4: Wireless MAC andPHY Specifications for Low-Rate Wireless Personal Area Networks, 2007.

[JO05a] J. Jalden and B. Ottersten. On the Complexity of Sphere Decoding in DigitalCommunications.IEEE Transactions on Signal Processing, 53(4):1474–1484,April 2005.

[JO05b] J. Jalden and B. Ottersten. Parallel Implementation of a Soft Output SphereDecoder. InConference Record of the Asilomar Conference on Signals, Systemsand Computers, pages 581–585, Oct./Nov. 2005.

[Kay98] S. Kay. Fundamentals of Statistical Signal Processing: Volume II -DetectionTheory. Prentice-Hall, New Jersey, USA, February 1998.

[KO11] D. Kreiser and S. Olonbayar. Design and ASIC Implementation of an IR-UWB-Baseband Transceiver for IEEE 802.15.4a. InProceedings of the InternationalConference on Wireless Communications and Signal Processing, Nov. 2011.

[LCF+99] L. Lampe, S. Calabro, R.F.H. Fischer, S. Müller-Weinfurtner, and J.B. Huber.On the Difficulty of Bit-Interleaved Coded Modulation for Differentially En-coded Transmission over Fading Channels. InProceedings of the IEEE Infor-mation Theory and Communications Workshop, June 1999.

[LDL09] T. Liu, X. Dong, and W. Lu. Compressed Sensing Maximum Likelihood Chan-nel Estimation for Ultra-Wideband Impulse Radio. InProceedings of the IEEEInternational Conference on Communications (ICC), June 2009.

[LDM02] V. Lottici, A. D’Andrea, and U. Mengali. Channel Estimation for Ultra-Wideband Communications.IEEE Journal on Selected Areas in Communica-tions, 20(9):1638–1645, December 2002.

[LFCM99] L. Lampe, R.F.H. Fischer, S. Calabro, and S Müller-Weinfurtner. Coded Mod-ulation DPSK on Fading Channels. InProceedings of the Global Telecommuni-cations Conference (GLOBECOM), vol. 5, pages 2540–2544, December 1999.

189

[LH06] I. Land and J.B. Huber. Information Combining.Foundation and Trends inCommunication and Information Theory, 3:227–330, November 2006.

[LP88] H. Leib and S. Pasupathy. The Phase of a Vector Perturbed by Gaussian Noiseand Differentially Coherent Receivers.IEEE Transactions on Information The-ory, 34(6):1491–1501, June 1988.

[LSPW05] L. Lampe, R. Schober, V. Pauli, and C. Windpassinger. Multiple-Symbol Differ-ential Sphere Decoding.IEEE Transactions on Communications, 53(12):1981–1985, December 2005.

[LT06a] V. Lottici and Z. Tian. A Sphere Decoding Approach toMultiple SymbolsDifferential Detection for UWB Systems.Proceedings of the IEEE GlobalTelecommunications Conference (GLOBECOM), November 2006.

[LT06b] V. Lottici and Z. Tian. Reduced-Complexity Multiple Symbol Differential De-tection for UWB Communications. InProceedings of the European Signal Pro-cessing Conference (EUSIPCO), September 2006.

[LT08] V. Lottici and Z. Tian. Multiple Symbol DifferentialDetection for UWB Com-munications.IEEE Transactions on Wireless Communications, 7(5):1656–1666,May 2008.

[LTS11] D. Lin, A. Trasser, and H. Schumacher. A Fully Differential IR-UWB Front-End for Noncoherent Communication and Localization. InProceedings of theIEEE International Conference on Ultra-Wideband, pages 116 –120, Bologna,Italy, September 2011.

[LV04] G. Leus and A. van der Veen. Noise Suppression in UWB Transmitted Refer-ence Systems. InIEEE 5th Workshop on Signal Processing Advances in WirelessCommunications, pages 155–159, July 2004.

[LZLZ10] D. Li, Z. Zhou, B. Li, and W. Zou. Noncoherent Detection Based on Com-pressed Sensing for Ultra-Wideband Impulse Radio. InProceedings of the In-ternational Communications and Information Technologies(ISCIT) Symposium,pages 783–787, October 2010.

[Mac92] K. Mackenthun. A Fast Algorithm for Maximum Likelihood Detection of QPSKor π/4-QPSK Sequences with Unknown Phase. InProceedings of the IEEEInternational Symposium on Personal, Indoor and Mobile Radio Communica-tions, pages 240–244, October 1992.

[Mac94] K. Mackenthun. A Fast Algorithm for Multiple-Symbol Differential Detectionof MPSK. IEEE Transactions on Communications, 42(234):1471–1474, March1994.

190 Bibliography

[MCC+06] A. Molisch, D. Cassioli, C. Chong, S. Emami, A. Fort, B. Kannan, J. Karedal,J. Kunisch, H. Schantz, K. Siwiak, and M. Win. A Comprehensive Standard-ized Model for Ultrawideband Propagation Channels.IEEE Transactions onAntennas and Propagation, 54(11):3151–3166, November 2006.

[MFP03] A. Molisch, J. Foerster, and M. Pendergrass. Channel Models for UltrawidebandPersonal Area Networks.IEEE Wireless Communications Magazine, 10(6):14–21, December 2003.

[MGDC06] A. Murugan, H. Gamal, M. Damen, and G. Caire. A Unified Framework for TreeSearch Decoding: Rediscovering the Sequential Decoder.IEEE Transactions onInformation Theory, 52(3):933–953, March 2006.

[MGCW08] A. Martinez, A. Guillén i Fábregas, G. Caire, and F.Willems. Bit-InterleavedCoded Modulation in the Wideband Regime.IEEE Transactions on InformationTheory, 54(12):5447–5455, December 2008.

[Mol05] A. Molisch. Ultrawideband Propagation Channels — Theory, Measurement,and Modeling.IEEE Transactions on Vehicular Technology, 54(5):1528–1545,September 2005.

[OL09] A. Oka and L. Lampe. A Compressed Sensing Receiver forUWB Impulse Ra-dio in Bursty Applications like Wireless Sensor Networks.Elsevier PhysicalCommunication, Special Issue on Advances in Ultra-Wideband Wireless Com-munications, 2:248–264, December 2009.

[OSHH12] A. Onic, A. Schenk, M. Huemer, and J.B. Huber. Soft-Output Sphere Detectionfor Coded Unique Word OFDM. Accepted for presentation at theAsilomarConference on Signals, Systems and Computers, Pacific Grove, CA, November2012.

[Pau07] V. Pauli. Design and Analysis of Low–Complexity Noncoherent DetectionSchemes, vol. 16 ofErlanger Berichte aus Informations– und Kommunikation-stechnik. Shaker Verlag, Aachen, Germany, 2007. PhD Thesis.

[PH03] D. Porcino and W. Hirt. Ultra-Wideband Radio Technology: Potential andChallenges Ahead.IEEE Communications Magazine, 41(7):66–74, July 2003.

[PHL09] V. Pauli, J.B.Huber, and L. Lampe. Decision-Feedback Subset Multiple-Symbol Differential Detection for Unitary Space-Time Modulation. IEEETransactions on Vehicular Technology, 58(2):1022–1026, February 2009.

[PJ07] M. Pausini and G. Janssen. Performance Analysis of UWB AutocorrelationReceivers Over Nakagami-Fading Channels.IEEE Journal of Selected Topicsin Signal Processing, 1(3):443–455, October 2007.

191

[PL07] V. Pauli, L. Lampe, and R. Schober. Tree-Search Multiple-Symbol DifferentialDecoding for Unitary Space-Time Modulation.IEEE Transactions on Commu-nications, 55(8):1567–1576, August 2007.

[PLS06] V. Pauli, L. Lampe, and R. Schober. “Turbo DPSK” Using Soft Multiple-Symbol Differential Sphere Decoding.IEEE Transactions on Information The-ory, 52(4):1385–1398, April 2006.

[PS98] M. Peleg and S. Shamai. On the Capacity of the Blockwise Incoherent MPSKChannel.IEEE Transactions on Communications, 46(5):603–609, May 1998.

[PS08] J. Proakis and M. Salehi.Digital Communications. McGraw-Hill, New York,NY, USA, 5. edition, 2008.

[QWD07] T. Quek, M. Win, and D. Dardari. Unified Analysis of UWB Transmitted-Reference Schemes in the Presence of Narrowband Interference. IEEE Trans-actions on Wireless Communications, 6(6):2126–2139, June 2007.

[RMMW03] P. Runkle, J. McCorkle, T. Miller, and M. Welborn. DS-CDMA: The Modula-tion Technology of Choice for UWB Communications. InProceedings of theIEEE Conference on Ultra Wideband Systems and Technologies, pages 364–368, 2003.

[RSH13] C. Rachinger, A. Schenk, and J.B. Huber. Efficient Sphere-Decoding for Soft-Output Multiple-Symbol Differential Detection of DPSK. Accepted for presen-tation at theInternational ITG Conference on Systems, Communications andCoding (SCC), January 2013.

[RW06] J. Romme and K. Witrisal. Transmitted-Reference UWBSystems UsingWeighted Autocorrelation Receivers.IEEE Transactions on Microwave The-ory and Techniques, 54(4):1754–1761, June 2006.

[SB07] U. Schuster and H. Bölcskei. Ultrawideband Channel Modeling on the Basisof Information-Theoretic Criteria.IEEE Transactions on Wireless Communica-tions, 6(7):2464–2475, July 2007.

[SB08] C. Studer and H. Bölcskei. Soft-Input Soft-Output Sphere Decoding. InPro-ceedings of the IEEE International Symposium on Information Theory (ISIT)2008, pages 2007–2011, July 2008.

[SBB08] C. Studer, A. Burg, and H. Bölcskei. Soft-Output Sphere Decoding: Algorithmsand VLSI Implementation.IEEE Journal on Selected Areas in Communications,26(2):290–300, February 2008.

[Sch09] A. Schenk. Application of the Sphere Decoder for Low-Complexity Detec-tion in Ultra-Wideband Systems. Diploma thesis at the Lehrstuhl für Informa-tionsübertragung, Universität Erlangen–Nürnberg, Erlangen, Germany, March2009.

192 Bibliography

[SF10a] A. Schenk and R.F.H. Fischer. Multiple-Symbol-Detection-Based Noncoher-ent Receivers for Impulse-Radio Ultra-Wideband. InProceedings of the 2010International Zürich Seminar on Communications (IZS), pages 70–73, Zürich,Switzerland, March 2010.

[SF10b] A. Schenk and R.F.H. Fischer. Soft-Output Sphere Decoder for Multiple-Symbol Differential Detection of Impulse-Radio Ultra-Wideband. InProceed-ings of the IEEE International Symposium on Information Theory (ISIT), pages2258–2262, Austin (TX), U.S.A., June 2010.

[SF11a] A. Schenk and R.F.H. Fischer. Capacity of BICM Using(Bi-)Orthogonal SignalConstellations in the Wideband Regime. InProceedings of the IEEE Interna-tional Conference on Ultra-Wideband, Bologna, Italy, September 2011.

[SF11b] A. Schenk and R.F.H. Fischer. Compressed-Sensing (Decision-Feedback) Dif-ferential Detection in Impulse-Radio Ultra-Wideband Systems. InProceed-ings of the IEEE International Conference on Ultra-Wideband, Bologna, Italy,September 2011.

[SF11c] A. Schenk and R.F.H. Fischer. Decision-Feedback Differential Detection inImpulse-Radio Ultra-Wideband Systems.IEEE Transactions on Communica-tions, 59(6):1604–1611, June 2011.

[SF11d] A. Schenk and R.F.H. Fischer. Design Rules for Bit-Interleaved Coded Impulse-Radio Ultra-Wideband Modulation with Autocorrelation-Based Detection. InProceedings of the International Symposium on Wireless Communication Sys-tems, pages 146–150, Aachen, Germany, November 2011.

[SF12a] A. Schenk and R.F.H. Fischer.UKoLoS—Ultra-Wideband Radio Technologiesfor Communications, Localization and Sensor Applications, chapter Coding,Modulation, and Detection for Power-Efficient Low-Complexity Receivers inImpulse-Radio Ultra-Wideband Transmission Systems. Intech, 2012, to be pub-lished.

[SF12b] A. Schenk and R.F.H. Fischer. Low-Complexity Soft-Output Detection forImpulse-Radio Ultra-Wideband Systems Via Combining Multiple Observa-tions. InProceedings of the International Zürich Seminar on Communications(IZS), pages 119–122, Zurich, Switzerland, Feb./March 2012.

[SFH10] C. Stierstorfer, R.F.H. Fischer, and J.B. Huber. Optimizing BICM with Convo-lutional Codes for Transmission over the AWGN Channel. InProceedings ofInternational Zurich Seminar (IZS), Zurich, Switzerland, March 2010.

[SFH12] A. Schenk, R.F.H. Fischer, and J.B. Huber. Performance Estimation of Bit-Interleaved Coded Modulation Using Information Processing Characteristics.In Proceedings of the IEEE International Symposium on Information Theory(ISIT), June 2012.

193

[SFL09a] A. Schenk, R.F.H. Fischer, and L. Lampe. A New Stopping Criterion for theSphere Decoder in UWB Impulse-Radio Multiple-Symbol Differential Detec-tion. In Proceedings of the IEEE International Conference on Ultra-Wideband,pages 606–611, Vancouver, Canada, September 2009.

[SFL09b] A. Schenk, R.F.H. Fischer, and L. Lampe. A StoppingRadius for the SphereDecoder and its Application to MSDD of DPSK.IEEE Communications Letters,13(7):465–467, July 2009.

[SGH98] R. Schober, W. Gerstacker, and J.B. Huber. Decision–Feedback DifferentialDetection of MDPSK for Flat Rayleigh Fading.IEEE Transactions on Commu-nications, 47(7):1025–1035, July 1998.

[Sha48] C.E. Shannon. A Mathematical Theory of Communication. Bell System Tech-nical Journal, 27:379–423, July 1948.

[SK03] Y. Souilmi and R. Knopp. On the Achievable Rates of Ultra-Wideband PPMwith Non-Coherent Detection in Multipath Environments. InProceedings of theIEEE International Conference on Communications (ICC), vol. 5, pages 3530–3534, May 2003.

[SLS07] C. Snow, L. Lampe, and R. Schober. Performance Analysis and Enhancement ofMultiband OFDM for UWB Communications.IEEE Transactions on WirelessCommunications, 6(6):2182 –2192, June 2007.

[SSF13] S. Sparrer, A. Schenk, and R.F.H. Fischer. Communication over ImpulsiveNoise Channels: Channel Coding vs. Compressed Sensing. Accepted for pre-sentation at theInternational ITG Conference on Systems, Communications andCoding (SCC), January 2013.

[Sta85] W. Stark. Capacity and Cutoff Rate of Noncoherent FSK with NonselectiveRician Fading. IEEE Transactions on Communications, 33(11):1153–1159,November 1985.

[Sti10] C. Stierstorfer.A Bit-Level-Based Approach to Coded Multicarrier Transmis-sion, vol. 28 ofErlanger Berichte aus Informations– und Kommunikationstech-nik. Shaker Verlag, Aachen, Germany, 2010. PhD Thesis.

[SV87] A. Saleh and R. Valenzuela. A Statistical Model for Indoor Multipath Prop-agation. IEEE Journal on Selected Areas in Communications, 5(2):128–137,February 1987.

[SVH08] M. Stojnic, H. Vikalo, and B. Hassibi. Speeding up the Sphere Decoder WithH∞ and SDP Inspired Lower Bounds.IEEE Transactions on Signal Processing,56(2):712–726, February 2008.

194 Bibliography

[TG07] J. Tropp and A. Gilbert. Signal Recovery From Random Measurements ViaOrthogonal Matching Pursuit. IEEE Transactions on Information Theory,53(12):4655–4666, 2007.

[TL08] T. Thiasiriphet and J. Lindner. A Novel Comb Filter Based Receiver with En-ergy Detection for UWB Wireless Body Area Networks. InProceedings ofthe IEEE International Symposium on Wireless Communication Systems, pages498–502, October 2008.

[TTP04] J. Tang, A. Tewfik, and K. Parhi. Reduced Complexity Sphere Decoding andApplication to Interfering IEEE 802.15.3a Piconets. InProceedings of the IEEEInternational Conference on Communications, vol. 5, pages 2864–2868, June2004.

[TWZ+12] R. Thomä, H. Willms, T. Zwick, R. Knöchel, and J. Sachs, editors. UKoLoS—Ultra-Wideband Radio Technologies for Communications, Localization andSensor Applications. Intech, 2012, to be published.

[TY05] Y. Tian and C. Yang. Multi-Symbol Detection for Ultra-Wideband PPM Com-munications in Multipath Environments. InProceedings of the IEEE Interna-tional Conference on Communications (ICC), vol. 2, pages 939–943, May 2005.

[TY08] Y. Tian and C. Yang. Noncoherent Multiple-Symbol Detection in Coded Ultra-Wideband Communications.IEEE Transactions on Wireless Communications,7(6):2202–2211, June 2008.

[Urk67] H. Urkowitz. Energy Detection of Unknown Deterministic Signals.Proceedingsof the IEEE, 55(4):523–531, April 1967.

[Ver98] S. Verdú.Multiuser Detection. Cambridge University Press, August 1998.

[Ver02] S. Verdú. Spectral Efficiency in the Wideband Regime. IEEE Transactions onInformation Theory, 48(6):1319–1343, June 2002.

[Wal99] R. Walden. Analog-to-Digital Converter Survey andAnalysis. IEEE Journalon Selected Areas in Communications, 17(4):539–550, April 1999.

[WAPS07] Z. Wang, G. Arce, J. Paredes, and B. Sadler. Compressed Detection for Ultra-Wideband Impulse Radio. InProceedings of the IEEE Workshop on SignalProcessing Advances in Wireless Communications (SPAWC), June 2007.

[WAS+07] Z. Wang, G. Arce, B. Sadler, J. Paredes, and X. Ma. Compressed Detectionfor Pilot Assisted Ultra-Wideband Impulse Radio. InProceedings of the IEEEInternational Conference on Ultra-Wideband, pages 393–398, September 2007.

[Wei87] L. Wei. Trellis-Coded Modulation with Multidimensional Constellations.IEEETransactions on Information Theory, 33(4):483–501, July 1987.

195

[WFH99] U. Wachsmann, R.F.H. Fischer, and J.B. Huber. Multilevel Codes: Theoreti-cal Concepts and Practical Design Rules.IEEE Transactions on InformationTheory, 45(5):1361–1391, May 1999.

[WHW00] A. Worm, P. Hoeher, and N. Wehn. Turbo-Decoding Without SNR Estimation.Communications Letters, IEEE, 4(6):193–195, June 2000.

[Wic95] S. Wicker. Error Control Systems for Digital Communications and Storage.Prentice-Hall, NJ, USA, 1st edition, July 1994.

[Win04] C. Windpassinger.Detection and Precoding for Multiple Input Multiple OutputChannels, vol. 9 of Erlanger Berichte aus Informations– und Kommunikation-stechnik. Shaker Verlag, Aachen, Germany, 2004. PhD Thesis.

[WJ65] J. Wozencraft and I. Jacobs.Principles of Communication Engineering. Wiley,New York, USA, June 1965.

[WLJ+09] K. Witrisal, G. Leus, G. Janssen, M. Pausini, F. Troesch,T. Zasowski, andJ. Romme. Noncoherent Ultra-Wideband Systems.IEEE Signal ProcessingMagazine, 26(4):48–66, July 2009.

[WLPK05] K. Witrisal, G. Leus, M. Pausini, and C. Krall. Equivalent System Model andEqualization of Differential Impulse Radio UWB Systems.IEEE Journal onSelected Areas in Communications, 23(9):1851–1862, September 2005.

[WS98] M. Win and R. Scholtz. Impulse Radio: How It Works.IEEE CommunicationsLetters, 2(2):36–38, February 1998.

[WS00] M. Win and R. Scholtz. Ultra-Wide Bandwidth Time-Hopping Spread-Spectrum Impulse Radio for Wireless Multiple-Access Communications.IEEETransactions on Communications, 48(4):679–689, April 2000.

[YG04] L. Yang and G. Giannakis. Ultra-Wideband Communications: An Idea WhoseTime Has Come.IEEE Signal Processing Magazine, 21(6):26–54, November2004.

[YK08] R. Yazdi and T. Kwasniewski. Low Complexity Sphere Decoding Algorithms.In Proceedings of the IEEE International Symposium on Wireless Communica-tion Systems 2008 (ISWCS), pages 438–442, October 2008.

[ZG06] W. Zhao and G. Giannakis. Reduced Complexity ClosestPoint Decoding Algo-rithms for Random Lattices.IEEE Transactions on Wireless Communications,5(1):101–111, January 2006.

[ZLT06] S. Zhao, H. Liu, and Z. Tian. Decision Directed Autocorrelation Receivers forPulsed Ultra-Wideband Systems.IEEE Transactions on Wireless Communica-tions, 5(8):2175–2184, August 2006.

196 Bibliography

[ZM12a] Q. Zhou and X. Ma. Designing Low-Complexity Near-Optimal Multiple-Symbol Detectors for Impulse Radio UWB Systems.IEEE Transactions onSignal Processing, 60(5):2460 –2469, May 2012.

[ZM12b] Q. Zhou and X. Ma. Soft-Input Soft-Output Multiple Symbol Differential De-tection for UWB Communications.IEEE Communications Letters, 16(8):1296–1299, August 2012.

[ZML10] Q. Zhou, X. Ma, and V. Lottici. Fast Multi-Symbol Based Iterative Detectors forUWB Communications.EURASIP Journal on Advances in Signal Processing,article ID 903161, June 2010.

[ZMR10a] Q. Zhou, X. Ma, and R. Rice. A Near-Optimal Multi-Symbol Based Detector forUWB Communications. InProceedings of the IEEE International Conferenceon Ultra-Wideband (ICUWB), September 2010.

[ZMR10b] Q. Zhou, X. Ma, and R. Rice. Near-ML Detection Basedon Semi-definiteProgramming for UWB Communications. InProceedings of the IEEE Inter-national Symposium on Information Theory (ISIT), pages 2253–2257, Austin(TX), U.S.A., June 2010.

[ZOS+09] J. Zhang, P. Orlik, Z. Sahinoglu, A. Molisch, and P. Kinney. UWB Systems forWireless Sensor Networks.Proceedings of the IEEE, 97(2):313–331, February2009.

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opus4.kobv.de€¦ · Acknowledgements1 First of all, I would like to thank my supervisor Prof. Robert Fischer, for his patient guidance in my time as his student, his advice and