Ultra-wideband based Communications and Localization in Wireless Sensor Networks Vom Fachbereich Elektrotechnik und Informatik der Universität Siegen zur Erlangung des akademischen Grades Doktor der Ingenieurwissenschaften (Dr.-Ing.) genehmigte Dissertation von M.Sc. Tech. Gustave Franck Tchere 1. Gutachter: Prof. Dr.-Ing. habil. O. Loffeld 2. Gutachter: Prof. Dr.-Ing. H. Roth Vorsitzender: Univ.-Prof. Dr. Christoph Ruland Tag der mündlichen Prüfung: 28.10.2010 Acknowledgments This work was conducted under the framework of the International Postgraduate Programme (IPP) Multi Sensorics of the Center for Sensorsystems (ZESS) and the Research Center for Multidisciplinary Analysis and Applied System Optimization (FOMAAS) at the University of Siegen. I would like to take this opportunity to express my greatest thanks to all of you who have supported me in various ways. It is a great pleasure for me to express my sincere gratitude to Prof. Dr.-Ing. habil. Otmar Loﬀeld, who supervised this work. During the last three years, I have been inspired by his broad knowledge of applied estimation theory, his brilliant insights and his scientiﬁc rigorousness. I am also grateful to Prof. Dr.-Ing. Hubert Roth, who is my second supervisor. I have beneﬁtted from his useful comments and advice throughout my thesis work. I would like to extend my gratefulness to Dr.Ing. Stefan Knedlik for his continuous encouragement and support during the wireless sensor network related activities. It was a pleasure for me working with him. I am grateful to the thesis committee for their careful reading of this manuscript. I would like to express my gratitude to all colleagues in the Center for Sensor Systems (ZESS) and members of International Postgraduate Programme (IPP), especially to M.Sc. Pakorn Ubolkosold and M.Sc. Miao Zhang for numerous discussions on both scientiﬁc and non-scientiﬁc matters and their friendly helps with various problems. I am grateful to Silvia Niet-Wunram, Renate Szabó and Ira Dexling, and for their excellent administrative services. Finally, I dedicate this thesis to my parents and friends, whose unconditional love and support has been a great source of inspiration to me. i Kurzfassung Die Hauptbeweggründe für das Verwenden von Ultrabreitband (UWB) Kommunikationssysteme ist ihre Fähigkeit für das Bereitstellen drahtloser Kommunikationen mit hoher Kapazität, sowie die Verwendbarkeit der Technologie, um UWB Signale mit verhältnismäßig niedriger Komplexität einzuführen und zu erzeugen. Nicht kohärente Empfänger ohne die Schätzung des Kanals sind vorgeschlagen worden, um die UWB Technologie attraktiv für die Anwendungen zu machen, in denen niedrige Kosten und niedrige Stromaufnahme eine wichtige Rolle spielen, wie es bei drahtlosen Sensor-Netzen (WSNs) der Fall ist. Die Signalübermittlung der Sendereferenz (TR), in Verbindung mit einem Autokorrelations Empfänger (AcR) sind in diesem Kontext besonders verwendbar. Die große Bandbreite der UWB Technologie liefert nicht nur die Möglichkeit, um mit einer sehr hohen Datenrate zu übertragen, sondern auch sehr genaue zeitliche und räumliche Informationen, die für exakte Schätzung der Zeitverschiebung verwendet werden können. Das UWB Impuls-Radio (IR) verwendet Sub-Nanosekunden - Impulse, die eine hoch auﬂösende Fähigkeit in der Zeitdomäne liefern und ist dadurch attraktiv für die genaue drahtlose Lokalisation. In der Tat macht seine Fähigkeit, Mehrwegbestandteile zu beheben möglich, genaue Positionsbestimmung ohne die Notwendigkeit komplizierter Schätzungsalgorithmen zu erhalten. Dieses erleichtert viele Anwendungen wie das Position-bewußte Sensor-Netzwerk. Diese Doktorarbeit trägt zur Entwicklung der UWB Technologie für Kurzstrecken niedriger bis zu durchschnittlicher Datenraten der WSN Anwendungen bei. TR-UWB Empfänger niedriger Komplexität werden zunächst unter Annahme Gauß’scher Rauschmodelle analysiert, danach werden äquivalente nicht lineare System- und Rauschmodelle betrachtet. Beide Ansätze führen zu der Leistungsanalyse der Bit Fehler, die auch in dieser Dissertation dargestellt werden. Die TR-UWB Systeme wurden als Polynome nichtlinearer Systeme modelliert. Da die Statistiken des Funkkanals in der Systemmodellierung sowie in der Leistungsanalyse verwendet werden, werden passende Kanalmodelle für UWB Impuls-Radiosysteme in dieser Doktorarbeit besprochen. Weiter wird die Synchronisierung angesprochen. Zu diesem Zweck wird eine neue datengestützte Zeiterfassungstechnik für die Rahmen-Niveau Synchronisierung der DTRUWB Systeme vorgeschlagen. Sie basiert auf dem Einfügen der parallelen Integration-anddump Schaltkreise innerhalb der Impuls-Paare der Korrelatorzweige, um die Energieerfassung bei der Präsenz der Zeitverschiebung beträchtlich zu verbessern. Außerdem wird ein einfacher Algorithmus für die Feinsynchronisierung der TR-UWB Systeme, keine Intersymbolstörung annehmend, vorgeschlagen. Er verwendet die Energie, die mit der Symbolrate iii gesammelt wird und verringert so beträchtlich die Implementierungskomplexität. Wenn der Taktzeitfehler bekannt ist, sind Schätzungsprobleme der Zeitverschiebung analog zu Entfernungsmessungsproblemen, und der vorgeschlagene Algorithmus kann leicht als neue Abstandmaßtechnik verwendet werden. Es wird in dieser Dissertation gezeigt, dass der Entfernungsansatz eine Lokalisierungsauﬂösung im Zentimeterbereich mit TR-UWB Systemen bei Datenraten bis zu 5 Mb/s erlaubt. Executive Summary The key motivations for using ultra-wideband (UWB) communication systems are their capability for providing high capacity wireless communications, as well as the availability of technology to implement and generate UWB signals with relatively low complexity. Non-coherent receivers with no channel estimation have been proposed to make the UWB technology attractive for applications where low cost and low power consumption are playing an important role, as in the case of wireless sensor networks (WSNs). Transmitted Reference (TR) signaling, in combination with an autocorrelation receiver (AcR) is especially suitable in this context. The large bandwidth of UWB technology does not only provide the possibility to transmit at a very high data rate, but also provides very accurate temporal and spatial information that can be used for precise timing oﬀset estimation. UWB Impulse Radio (IR) uses sub-nanosecond pulses which provide a high resolution capability in the time domain, making it attractive for accurate wireless localization. Indeed, its ability to resolve multipath components makes it possible to obtain accurate location estimates without the need for complex estimation algorithms. This facilitates many applications such as location-aware sensor networking. This thesis presents several contributions towards developing UWB technology for short-range low to medium data rate WSN applications. First, diﬀerent low complexity TR-UWB receivers are thoroughly analyzed, using the Gaussian approximation on the noise terms in the receiver statistics, and then using the equivalent system and noise models of these receivers. Both approaches lead to the bit error performance analysis, which is also presented in this thesis. The TR-UWB systems were modelled as polynomial nonlinear systems. Since the statistics of the radio channel are used in the system modelling as well as in the performance analysis, appropriate channel models for UWB impulse radio systems are discussed in this thesis. Further, synchronization issues are addressed. To this end, a novel data-aided timing acquisition technique for frame-level synchronization of DTR-UWB systems is suggested. It is based on incorporating parallel integration-and-dump circuits within pulse-pair correlator branches to improve considerably the energy capture in the presence of timing oﬀset. Moreover, a simple algorithm for ﬁne synchronization of low complexity TR-UWB systems, assuming no inter-symbol interference, is proposed. It uses energy collected at the symbol rate, thus reducing considerably the implementation complexity. If the hardware clock timing error is known, timing oﬀset estimation problems are analogical to ranging problems, and the proposed algorithm can readily be used as a new distance measurement technique. It is shown, in this thesis, that the proposed ranging approach allows localization accuracy in the centimeter range using TR-UWB systems with data rates up to 5 Mb/s. Contents Acknowledgements i Kurzfassung ii Executive Summary iv Contents v List of Figures viii List of Tables x Abbreviations xi Mathematical Symbols xiv 1 Introduction 1.1 Ultra-Wideband Technology . . . . . . . . . . . . . . . . . . . 1.1.1 UWB History and Regulatory Issues . . . . . . . . . . 1.1.2 Basics of UWB Techniques . . . . . . . . . . . . . . . . 1.1.3 Open Research Problems . . . . . . . . . . . . . . . . . 1.2 Ultra-Wideband Wireless Sensor Networks . . . . . . . . . . . 1.2.1 Wireless Sensor Network Requirements . . . . . . . . . 1.2.2 Ultra-Wideband Approach to Wireless Sensor Networks 1.2.3 Conceptual Design of the Envisaged Application . . . . 1.3 Goal of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . 1.5 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . 1.5.1 Systems Modelling . . . . . . . . . . . . . . . . . . . . 1.5.2 Performance Analysis . . . . . . . . . . . . . . . . . . . 1.5.3 BER derivation . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Coarse Synchronization . . . . . . . . . . . . . . . . . . 1.5.5 Fine Synchronization . . . . . . . . . . . . . . . . . . . 1.5.6 Distance and Location Estimation . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 4 9 10 10 11 12 15 15 16 16 16 16 16 16 17 vi 2 Polynomial Nonlinear Systems: Overview 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Homomorphic Systems . . . . . . . . . . . . . 2.1.2 Order Statistic Filters . . . . . . . . . . . . . 2.1.3 Morphological Filters . . . . . . . . . . . . . . 2.1.4 Neural Networks . . . . . . . . . . . . . . . . 2.1.5 Polynomial Filters . . . . . . . . . . . . . . . 2.2 Volterra Series Expansion . . . . . . . . . . . . . . . 2.2.1 Continuous-time Systems . . . . . . . . . . . . 2.2.2 Discrete-time Systems . . . . . . . . . . . . . 2.2.3 Algebraic Representation of Quadratic Filters 2.2.4 Realization of Quadratic Filters . . . . . . . . 2.3 Wiener & Hammerstein Models . . . . . . . . . . . . 2.3.1 Hammerstein-Wiener Model . . . . . . . . . . 2.3.2 Hammerstein Model . . . . . . . . . . . . . . 2.3.3 Wiener Model . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 18 19 20 20 21 22 22 25 26 28 28 30 30 32 32 3 Channel Characterization 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characterization of the Mobile Radio Channel . . . . . . 3.2.1 Components of a Multipath Channel Model . . . 3.2.2 Channel Impulse Response . . . . . . . . . . . . . 3.2.3 Channel Parameters . . . . . . . . . . . . . . . . 3.2.4 Channel Description . . . . . . . . . . . . . . . . 3.3 Standardized Channel Models for UWB Communications 3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The IEEE 802.15.4a Channel Model . . . . . . . . 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 33 33 33 34 35 35 39 39 40 46 4 Transmitted Reference Systems 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 TR-UWB Signal and Systems Models . . . . . . . . 4.2.1 TR-UWB Transmission Schemes . . . . . . 4.2.2 TR-UWB Receiver Front-End . . . . . . . . 4.3 Equivalent TR-UWB Systems Models . . . . . . . . 4.3.1 System with ISI: Volterra Equivalent Model 4.3.2 Systems without ISI . . . . . . . . . . . . . 4.4 Bit Error Rate Performance Analysis . . . . . . . . 4.4.1 Gaussian Approximations approach . . . . . 4.4.2 System Modelling Approach . . . . . . . . . 4.4.3 Derivation of Average BER . . . . . . . . . 4.4.4 Simulation Results . . . . . . . . . . . . . . 4.5 Optimization of the Integration Time Interval TI . 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 48 49 49 51 53 53 60 61 62 67 67 71 73 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 5 Synchronization of Transmitted Reference Systems 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Data-Aided Timing Acquisition . . . . . . . . . . . . . 5.2.1 Timing Acquisition Technique . . . . . . . . . . 5.2.2 Simulation Results . . . . . . . . . . . . . . . . 5.3 Timing Oﬀset Estimation . . . . . . . . . . . . . . . . 5.3.1 Normalization of the decision variable . . . . . . 5.3.2 Classical Approach to Timing Oﬀset Estimation 5.3.3 Bayesian Approach to Timing Oﬀset Estimation 5.3.4 Determination of Es , A and B . . . . . . . . . . 5.3.5 Simulation Results . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 6 Localization with Transmitted Reference 6.1 Introduction . . . . . . . . . . . . . . . . 6.2 Localization Systems . . . . . . . . . . . 6.2.1 Ranging-based Systems . . . . . . 6.2.2 Directionality-based Systems . . . 6.2.3 System Architecture . . . . . . . 6.3 Location Estimation . . . . . . . . . . . 6.3.1 The LS algorithm . . . . . . . . . 6.3.2 The SI Method . . . . . . . . . . 6.3.3 Simulation results . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 76 77 77 79 82 82 83 84 85 85 91 . . . . . . . . . . 92 92 93 93 96 97 98 98 98 100 105 7 Conclusions and Outlook 106 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.3 Discussion and Recommendations . . . . . . . . . . . . . . . . . . . . . . . 107 Appendices 109 Publications 123 Bibliography 125 List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 FCC spectral mask for indoor commercial applications. Proposed spectral mask for Europe. . . . . . . . . . . . Gaussian pulses and spectra . . . . . . . . . . . . . . . UWB modulation schemes. . . . . . . . . . . . . . . . . Typical TH-PPM waveform. . . . . . . . . . . . . . . . Transmit PSD corresponding to PPM and THPPM . . UWB based wireless sensor network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 7 7 8 13 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 A homomorphic ﬁlter employing a logarithmic nonlinearity. An artiﬁcial neuron. . . . . . . . . . . . . . . . . . . . . . Nonlinearities commonly employed in neural networks. . . A quadratic ﬁlter . . . . . . . . . . . . . . . . . . . . . . . Realization of a quadratic operator using Simulink. . . . . The Hammerstein-Wiener model. . . . . . . . . . . . . . . The Hammerstein model. . . . . . . . . . . . . . . . . . . . The Wiener model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 20 21 27 29 30 31 32 3.1 3.2 3.3 3.4 3.5 3.6 Model of the average power delay proﬁle . . . . . . . Typical channel realization . . . . . . . . . . . . . . . Channel realizations from the IEEE 802.15.4a model Average power delay proﬁle. . . . . . . . . . . . . . . RMS delay spreads for diﬀerent CIR realizations. . . Number of signiﬁcant paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 38 44 45 45 46 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 Diagram of the transmitted and received signals in DTR-UWB. . . . Diagram of transmitted signals for LDC DTR-UWB systems . . . . . Received signal for a DP TR-UWB system. . . . . . . . . . . . . . . . Receiver front-end for a DTR-UWB system. . . . . . . . . . . . . . . 2nd order Volterra equivalent system model of the DTR-UWB system. Simulated receiver output 1 . . . . . . . . . . . . . . . . . . . . . . . Simulated receiver output 2 . . . . . . . . . . . . . . . . . . . . . . . Normalized average delay-dependent noise-free decision variable . . . Average and variance of the output noise versus delay . . . . . . . . . Diagram of the received signals in DTR-UWB (without IFI). . . . . . Simulated noise autocorrelation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 51 51 52 57 58 59 59 60 62 63 viii . . . . . . . . . . . . . . . . . . . . . . . . . ix 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 Autocorrelation function of the received prototype Noise autocorrelation function Rν (κ). . . . . . . . Probability density function of zτ [i]. . . . . . . . . BER performance of the DTR-UWB system 1 . . BER performance of the DTR-UWB system 2 . . BER performance of the LDC DTR-UWB system BER performance of the DP TR-UWB system . . Optimal choice for TI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 65 68 72 73 73 74 74 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 Illustration of the timing acquisition technique . . . . . Illustration of the outputs Yk for diﬀerent timing oﬀsets Normalized timing acquisition MSE vs. SNR 1 . . . . . Normalized timing acquisition MSE vs. SNR 2 . . . . . Probability of detection vs. SNR . . . . . . . . . . . . Average BER vs. SNR . . . . . . . . . . . . . . . . . . Normalized timing oﬀset estimation MSE vs. SNR 1 . Normalized timing oﬀset estimation MSE vs. SNR 2 . Normalized timing oﬀset estimation MSE vs. SNR 3 . Timing oﬀset estimation rmse vs. SNR . . . . . . . . . Timing oﬀset estimation rmse vs. delay . . . . . . . . . Biasedness of timing oﬀset estimator 1 . . . . . . . . . Biasedness of timing oﬀset estimator 2 . . . . . . . . . Biasedness of timing oﬀset estimator 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 79 80 81 81 82 87 87 88 88 89 89 90 90 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 TOA localization principle. . . . . . . . . . . . . . . . . . . . . . . TDOA localization principle. . . . . . . . . . . . . . . . . . . . . . Normalized average distance-dependent noise-free decision variable AOA localization principle. . . . . . . . . . . . . . . . . . . . . . . Illustration of the spherical interpolation approach. . . . . . . . . Distance estimation rmse vs. distance TX-RX . . . . . . . . . . . Distance estimation variance vs. distance TX-RX . . . . . . . . . 2D localization scenario. . . . . . . . . . . . . . . . . . . . . . . . Location estimation with DTR-UWB systems. . . . . . . . . . . . Location estimation with LDC DTR-UWB systems. . . . . . . . . Location estimation with DP TR-UWB systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 94 95 96 99 101 101 102 103 103 104 1 2 3 4 Experimental design of wireless sensor nodes Algorithm ﬂowchart . . . . . . . . . . . . . . Interaction application-Home RF functions . Relation DLL functions-HomeRF functions . . . . . . . . . . . . . . . . . . . . . 118 120 121 122 . . . . . . . . . . . . pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Tables 1.1 FCC spectral mask for UWB systems (EIRP in dBm). . . . . . . . . . . . 3 3.1 3.2 Simulation values of the channel parameters. . . . . . . . . . . . . . . . . . Channel parameters for indoor oﬃce environment. . . . . . . . . . . . . . . 38 44 4.1 4.2 DTR-UWB system and simulation parameters . . . . . . . . . . . . . . . . TR-UWB systems and simulation parameters. . . . . . . . . . . . . . . . . 57 72 5.1 TR-UWB systems and simulation parameters . . . . . . . . . . . . . . . . 86 6.1 Localization accuracy of diﬀerent TR-UWB systems. . . . . . . . . . . . . 104 7.1 Performance of diﬀerent TR-UWB systems. . . . . . . . . . . . . . . . . . 108 x Abbreviations AcR ADC A-GPS APDP AOA ASIC BER BPF BPPM BPSK CEPT CIR CMOS CRC CRLB DAA dB DDP DLL DP DTI DTR DOA e.g. EIRP ETSI FBO FCC FIFO FIR FT GHz GNSS GPS Autocorrelation Receiver Analog-to-Digital Converter Aided Global Positioning System Average Power Delay Proﬁle Angle of Arrival Application Speciﬁc Integrated Circuit Bit Error Rate Band Pass Filter Biorthogonal Pulse Position Modulation Binary Phase Shift Keying European Conference of Postal and Telecommunications Administrations Channel Impulse Response Complementary MetalOxideSemiconductor Cyclic redundancy check Cramér-Rao Lower Bound Detection-and-Avoidance Decibel Dominant direct path Dynamic-link Library Dual-Pulse Discrete Time-invariant Diﬀerential Transmitted Reference Direction of Arrival Abbreviation for exempli gratia (for example) Equivalent Isotropically Radiated Power European Telecommunications Standards Institute Feedback Block-oriented Federal Communications Commission First In, First Out Finite Impulse Response Fourier Transform Gigahertz Global Navigation Satellite System Global Positioning System xi xii Hz I&D i.e. IF IFI IIR IMU IPP IR ISI kb/s kHz LDC LOS LS LSE LTI MA MAC MAI Mb/s MHz ML MSE MU NBI NDDP NLOS NMEA OOK PAM PC pdf PDP PPM PSD PVT RF RMS RSS SI SNR TAIP Hertz Integrate-and-dump Abbreviation for id est (that is) Intermediate Frequency Inﬁnite Impulse Response Inter-frame Interference Inertial Measurement Unit International Postgraduate Programme Impulse Radio Inter-symbol Interference Kilobit per second Kilohertz Low-Duty-Cycle Line-of-sight Least Square Least Square Error Linear Time Invariant Multi-user Access Medium Access Control Multiple Access Interference Megabit per second Megahertz Maximum Likelihood Mean Square Error Multiuser Narrowband Interference Non-dominant Direct Path Non-line-of-sight National Marine Electronics Association On-oﬀ Keying Pulse Amplitude Modulation Personal Computer Probability Density Function Power Delay Proﬁle Pulse Position Modulation Power Spectral Density Position, Velocity and Time Radio Frequency Root Mean Square Received Signal Strength Spherical-interpolation Signal-to-Noise Ratio Trimble ASCII Interface Protocol xiii TDOA TH TOA TR TSIP UART UDP UWB WPAN w.r.t. WSN ZESS Time-diﬀerence of Arrival Time Hopping Time of Arrival Transmitted Reference Trimble Standard Interface Protocol Universal Asynchronous Receiver/Transmitter Undetected Direct Path Ultra-Wideband Wireless Personal Area Network With respect to Wireless Sensor Network Center for Sensor Systems Mathematical Symbols |·| ⌊·⌋ ⌈·⌉ (·)T ∀ ∝ ⊗ ! !! 7→ cjk ∂ δ(·) diag(· · · ) E{·} e(·) , exp(·) Eb Es f fc H h(t) iN IN L{(·)} log(·) ln(·) Ncr Nf n(t), ν(t) Q(·) Rν (t) r(t) Absolute value Floor function Ceiling function Transpose For all Proportional to Kronecker product factorial double factorial maps to TH code sequence of the k th user Partial diﬀerentiation Dirac delta function Diagonal matrix Expectation Exponential Bit energy Symbol energy Frequency Center frequency Homomorphism Channel impulse response All-ones vector of length N Identity matrix of size N × N Linear operator Logarithm to base 10 Natural logarithm Number of pulse-pair correlators Number of frames in a symbol White Gaussian noise Complementary Gaussian cumulative distribution function Noise autocorrelation function Received signal xiv xv σ2 si (t) t τ τm tm Tf Tm Ts Tω Ts var{·} ω(t) x X z[i] Z Noise variance Transmitted signal containing the ith symbol Time Delay Optimal modulation index in M -ary PPM Pulse parameter Frame duration Channel maximum excess delay Symbol duration Pulse duration Symbol duration Variance Gaussian pulse Vector Matrix Decision variable Field of integers Chapter 1 Introduction 1.1 1.1.1 Ultra-Wideband Technology UWB History and Regulatory Issues The term Ultra-WideBand or UWB signal has come to signify a number of synonymous terms such as impulse, carrier-free, baseband, time domain, nonsinusoidal, orthogonal function and large-relative-bandwidth radio/radar signals. According to modern deﬁnition [1], UWB characterizes transmission systems with instantaneous spectral occupancy in excess from 500 MHz, or fractional bandwidth of more than 20 %. The fractional bandwidth is deﬁned as B/fc , where B = fH − fL denotes the -10 dB bandwidth and center frequency fc = (fH + fL )/2 with fH being the upper frequency of the -10 dB emission point and fL the lower frequency of the -10 dB emission point. UWB systems with fc > 2.5 GHz need to have a -10 dB bandwidth of at least 500 MHz, while UWB systems with fc < 2.5 GHz need to have a fractional bandwidth of at least 0.20. Contributions to the development of a ﬁeld addressing UWB radio frequency (RF) signals commenced in the late 1960’s with the pioneering contributions of Harmuth at the Catholic University of America, Ross and Robbins at Sperry Rand Corporation, Paul van Etten at the USAF’s Rome Air Development Center and in Russia [2]. The Ross and Robbins (R&R) patents (1972-1987) pioneered the use of UWB signals in a number of application areas, including communications and radar and also using coding schemes. After the 1970s, the only innovations in the UWB ﬁeld came from improvements in particular instantiations of subsystems, but not in the overall system concept itself, nor even in the overall subsystems’ concepts. The basic components were known, e.g., pulse train generators, pulse train modulators, switching pulse train generators, detection receivers and wideband antennas. In 1994, T.E. Mc Ewan invented the Microwave Impulse Radar (MIR) which provided, for the ﬁrst time a UWB operating at ultra low power, besides being extremely compact and inexpensive [3]. This was the ﬁrst UWB radar to operate on only microwatts of battery drain. The UWB radar and communications approach is a shift in emphasis with respect to the use of the available time-bandwidth-power product. As the signal duration decreases, the 1 Chapter 1. Introduction 2 bandwidth increases and the signal-to-noise ratio (SNR) per frequency decreases. Moreover, the SNR per frequency decreases below the threshold of frequency selective receivers, which is a major argument made by UWB proponents that UWB systems are able to operate in the presence of frequency selective receivers without interference. The methods used to reliably receive a UWB signal with such low SNR per frequency are: • a high sampling rate receiver to capture in a non-synchronous way all the signal energy in a minimum number of sampling bins, summing across all the signal bandwidth which implies a receiver front-end open to that instantaneous ultra-wide bandwidth and thus also open to noise; or • signal averaging or matched ﬁltering which lowers the data rate; or • counteracting the low power per frequency by increasing to high signal transmit power, which implies interference to other receivers. As in the case of more conventional communications systems, the UWB wireless system designer must balance trade-oﬀs among high bandwidth eﬃciency, low transmission peak power, low complexity, ﬂexibility in supporting multiple rates and reliable performance as expressed in bit error rates. A UWB radar system is more unconventional than a UWB communications system, not only in system components, but also in the physics involved in the signal-target interactions. Indeed, conventional radars use signals longer in distance length than the distance length of the target, thus the target is a point scatterer. In the UWB radar case, the transmitted signal is shorter in distance length than the target, so the target is not a point scatterer. Even though the concept on which UWB transmission is based is the overlay of lowpower signals over existing spectrum and the peaceful coexistence with narrow-band systems already in operation, UWB devices will emit radiation as any other radio system. In order to minimize the level of interference and to coordinate its ordered deployment, a set of regulations is necessary to indicate limits and restrictions in the implementation of this technology and introduction into the real market. The Federal Communications Commission (FCC) in the United States released a huge frequency band 3.1 - 10.6 GHz with equivalent isotropically radiated power (EIRP) below -43 dBm, where UWB radios overlaying coexistent RF systems can operate using low-power ultra-short information bearing pulses. The applications of UWB technology approved by the FCC today include communications (short-range very high speed broadband communications, covert communication links), vehicular radar systems, imaging systems and distance measurement systems (localization at centimeter-level accuracy, precision navigation and asset tracking). The FCC’s spectral masks assigned to these applications are listed in Tab. 1.1. In particular, the FCC’s assigned bandwidth and spectral mask for indoor communications is illustrated in Fig. 1.1. Far from being universally accepted and welcomed by everyone, the FCC’s initial work raised the discussion levels also in Europe, where very important coexistence studies started in early 2000, ﬁrst in the CEPT (European Conference of Postal and Telecommunications Administrations) working group SE24 and ETSI (European Telecommunications Standards Institute) working group TG31a, and later also in the newly formed working group TG3. The recommended spectral mask for Europe Chapter 1. Introduction Frequency GHz 0.96 - 1.61 1.61 - 1.9 1.9 - 1.99 1.99 - 3.1 3.1 - 10.6 10.6 - 22 22 - 29 29 - 31 > 31 Indoor Comm. -75.3 -53.3 3 Hand held Comm. -75.3 -63.3 -61.3 -51.3 -41.3 Low Frq. Imaging -65.3 -53.3 -41.3 High Frq. Imaging -65.3 -53.3 Med. Frq. Imaging -53.3 -51.3 Vehic. Radar -75.3 -51.3 -41.3 -41.3 -61.3 -51.3 -51.3 -41.3 -51.3 -61.3 -51.3 -51.3 -61.3 Table 1.1: FCC spectral mask for UWB systems (EIRP in dBm). −40 UWB EIRP Emission Level in dBm −45 −50 3.1 1.99 −55 −60 10.6 Indoor Limit −65 GPS Band −70 −75 1.61 0.96 0 10 1 Frequency in GHz 10 Figure 1.1: FCC spectral mask for indoor commercial applications. is shown in Fig. 1.2, where power levels are equivalent to those of the FCC regulations in the frequency band 3.1 - 10.6 GHz. However, European regulators have introduced the concept of “Detection-and-Avoidance” (DAA) technology to ensure coexistence with existing wireless communications technologies [4]. It means that UWB devices operating in the 3.1-4.2 GHz band should be ﬁtted with DAA systems, which automatically search for nearby broadband wireless signals and then switch frequency to prevent interference. DAA is not required in the 4.2-4.8 GHz band until 2010, and no DAA is required in the 6-10.6 GHz band. In parallel to the regulatory eﬀorts, another fundamental part of the introduction of this new technology into the market was the deﬁnition of an industry inter-operability standard Chapter 1. Introduction 4 −40 3.1 UWB EIRP Emmission Level in dBm −50 −60 −70 2.8 −80 2.1 −90 10.6 1.5 0.5 −100 0 10 1 Frequency in GHz 10 Figure 1.2: Proposed spectral mask for Europe. based on the best properties of UWB technology. Given the nature of UWB systems, which can be suitable for high-speed links, low to medium data rate and positioning applications, standardization eﬀorts were carried out in parallel in two international standardization groups: the IEEE 802.15.3a (late 2001) and in the IEEE 802.15.4a (late 2003). 1.1.2 Basics of UWB Techniques Shape and Spectrum of UWB Signals Impulse Radio (IR) based UWB systems rely on ultra-short sub-nanosecond waveforms that can be free of sine-waves carriers and do not require Intermediate Frequency (IF) processing because they can operate at baseband. Generally adopted spectrum shapers ω(t) for UWB communications include the Gaussian pulse, the Gaussian monocycle (ﬁrst derivative of Gaussian pulse), and the Gaussian doublet (second derivative of the Gaussian pulse), as depicted in Fig. 1.3, along with their Fourier Transforms (FT). The reason behind the popularity of these pulses is twofold: • Gaussian pulses come with the smallest possible time-bandwidth product of 0.5, which maximizes range-rate resolution and • the Gaussian pulses are readily available from the antenna pattern [5]. Let Tω be the duration of the transmitted pulse ω(t). With Tω at the sub-nanosecond scale, ω(t) occupies an UWB with bandwidth B ≈ 1/Tω . The mathematical expression of Chapter 1. Introduction 5 10 Gaussian doublet 1 Gaussian monocycle 0 Gaussian pulse 0.8 −10 0.6 −20 0.2 Power [dB] Amplitude [dB] 0.4 0 −0.2 −30 −40 −0.4 Gaussian doublet −50 −0.6 Gaussian monocycle −0.8 Gaussian pulse −60 −1 0 0.1 0.2 0.3 0.4 Time [ns] 0.5 0.6 0.7 0.8 −70 1 Frequency (GHz) 10 (b) (a) Figure 1.3: (a) Adopted pulse shapes in UWB communications; (b) Fourier transform of the pulse shapes. Pulse duration: Tω = 0.7 ns. the Gaussian pulse, Gaussian monocycle and the Gaussian doublet are given, respectively, as ωg (t) = K1 e−2π(t/tm ) , 2 ωg1 (t) = dωg (t)/dt = K1 and −4πK1 ωg2 (t) = d ωg (t)/dt = t2m 2 2 (1.1) −4πt −2π(t/tm )2 e , t2m ( 4πt2 1− 2 tm (1.2) ) e−2π(t/tm ) , 2 (1.3) where tm is the pulse parameter. It can be seen that by utilizing one of the properties of the pulses (i.e., by diﬀerentiating or integrating them) another pulse can be created, with the order of the pulse being one more or less than the original pulse, respectively. In this thesis, the energies associated with the Gaussian pulse and its derivatives are expressed analytically as K12 tm , 2 (1.4) K12 π = , tm (1.5) Eg = Eg1 and Eg2 = 6π 2 K12 . t5m (1.6) Chapter 1. Introduction 6 The derivations can be found in Appendix II. UWB Modulation Techniques Being real, baseband UWB transmissions neither have to entail frequency modulation nor phase modulation. Consequently, symbol values can be transmitted by modulating the position and/or the amplitude of ω(t). { In M -ary Pulse } Position Modulation (PPM), M distinctly delayed pulses are employed, M −1 ω(t − τm )m=0 , each representing one symbol value. Generally, the modulation indices τm are chosen such that τm = mτ with τ ≥ Tω , which corresponds to orthogonal PPM. In ∫ binary PPM, the delay τ can also be chosen to minimize the correlation ω(t)ω(t − τ )dt [6]. As bandwidth eﬃciency drops with increasing modulation size M , PPM is suitable for power-limited applications. In fact, PPM was almost exclusively adopted in the early development of UWB radios because negating ultrashort pulses were diﬃcult to implement. Biorthogonal pulse position modulation (BPPM) was introduced in [7]. It is shown that N-ary BPPM has better performance than N-ary PPM with the same throughput and half the computational complexity. Another modulation scheme that does not require pulse negation is the so termed OnOﬀ Keying (OOK), where symbol “1” is represented by transmitting a pulse, and “0” by transmitting nothing. The main advantage of the OOK scheme is its ease of physical implementation. As pulse negation became easier to implement, Pulse Amplitude Modulation (PAM) attracted more attention. In particular, when M = 2, antipodal pulses are used to represent binary symbols, as in Binary Phase Shift Keying (BPSK) or bipolar signaling. The bit error rate (BER) performance of the BPSK scheme is superior to that of the OOK since OOK has a smaller symbol separation (transmitted symbols 0 and 1) compared to BPSK (transmitted symbols 1 and -1). The diﬀerent modulations schemes are illustrated in Fig. 1.4 for a bit sequence b = {1, 0, 1}. Multiple Access Techniques To allow for multi-user access (MA) to the UWB channel, Time Hopping (TH) was introduced [6]. With TH, each pulse is positioned with each frame duration Tf according to a user-speciﬁc TH sequence ckj . Speciﬁcally, dividing each frame into Nc chips each of duration Tc , the kth user’s TH code ckj ∈ [0, Nc − 1] corresponds to a time shift of ckj Tc during the kth frame [8]. A typical TH-PPM transmitted waveform from the kth user is given by sk (t) = +∞ ∑ j=0 ω(t − jTf − ckj Tc − τ bk⌊j/N ⌋ ), f (1.7) where Nf is the number of pulses transmitted per symbol and ⌊x⌋ is the ﬂoor function of a real number x. It is deﬁned as ⌊x⌋ = max{n ∈ Z|n ≤ x}. With TH codes, MA is achieved by altering the pulse position from frame to frame, according to the sequence ckj . Chapter 1. Introduction 7 1 1 BPPM PPM 0.5 0.5 0 0 −0.5 0 2 4 Time [ns] 6 1 −0.5 0 2 1 OOK 4 6 4 6 BPSK 0.5 0.5 Time [ns] 0 0 −0.5 −0.5 0 2 4 Time [ns] 6 −1 0 2 Time [ns] Figure 1.4: UWB modulation schemes. Fig. 1.5 depicts a TH-PPM transmitted waveform, where Nf = 10, Tf = 10 ns, Tc = 0.1 ns and the code sequence is given by cj = {0, 5.3, 0.4, 3, 7.3, 0.8, 5, 1.6, 1.8, 0}, j = {0,1,· · · ,9}. 1.5 15.3 20.4 Amplitude [dB] 1 33 47.3 50.8 65 71.6 81.8 90 0.5 0 −0.5 −1 0 20 40 time [ns] 60 80 Figure 1.5: Typical TH-PPM waveform. 100 Chapter 1. Introduction 8 In addition to facilitating multiple access, spreading codes also shape the transmit spectrum, as shown in Fig. 1.6. TH−PPM 40 40 20 20 0 0 Power [dB] Power [dB] PPM −20 −20 −40 −40 −60 −60 1 Frequency [GHz] (a) 10 1 Frequency [GHz] 10 (b) Figure 1.6: Transmit power spectral density (PSD) corresponding to (a) no spreading code, (b) random TH codes. Applications • Wireless Personal Area Networks (WPANs): Also known as in-home networks, WPANs address short-range (within 10 - 20 m) ad hoc connectivity among portable consumer electronics and communication devices. UWB technology emerges as a promising physical layer candidate for WPANs, because it oﬀers high-rates over short-range, with low cost, high power eﬃciency and low duty cycle. • Wireless Sensor Networks (WSNs): WSNs consist of a large number of nodes spread across a geographical area. Key requirements for WSNs operating in challenging environments include low cost, low power, and multi-functionality. Emerging applications of UWB are foreseen for sensor networks as well. Such networks combine low to medium rate communications with positioning capabilities. UWB signaling is especially suitable in this context because it allows centimeter accuracy in ranging, as well as low-power and low-cost implementation of communication systems. These features allow a new range of applications, including logistics (package tracking), security applications (localizing authorized persons in high-security areas), medical applications (monitoring of patients), family communications/supervision of children, search and rescue missions (communications with ﬁre ﬁghters, or avalanche/earthquake victims), control of home appliances, and military applications [9]. High data-rate UWB communication systems are well motivated for gathering and exchanging a vast quantity of sensory data in a timely manner. Chapter 1. Introduction 9 • Radar Systems: UWB-based sensing has the potential to improve the resolution of conventional proximity and motion sensors. Relying on the high ranging accuracy and target diﬀerentiation capability enabled by UWB, intelligent collision-avoidance and cruise-control systems may be envisioned. These systems can also improve airbag deployment and adapt suspension/braking systems depending on road conditions. UWB technology can also be integrated into vehicular entertainment and navigation systems by downloading high-rate data from airport oﬀ ramp, road-side gas station UWB transmitters [10]. • Imaging Systems: Diﬀerent from conventional radar systems where targets are typically considered as point scatterers, UWB radar pulses are shorter than the target dimensions. UWB reﬂections oﬀ the target exhibit not only changes in amplitude and time shift but also changes in the pulse shape. As a result, UWB waveforms exhibit pronounced sensitivity to scattering relative to conventional radar signals. This property has been readily adopted by radar systems [2] and can be extended to additional applications, such as underground, through-wall and ocean imaging, as well medical diagnostics and border surveillance devices [11]. 1.1.3 Open Research Problems UWB is an emerging new technology that is expected to enable low-cost and low-power devices. As already mentioned in 1.1.2, streams of ultra-short pulses (< 1 ns) are used for wireless data transmission instead of a modulated sinusoidal carrier, yielding signals of huge bandwidths (> 1 GHz) but at very low power densities. In principle, the baseband nature of the signal transmission makes the UWB technology suitable for low-cost implementations in standard CMOS (complementary metal-oxide-semiconductor) technology. Indeed, unlike conventional radio systems, the transmitted pulse is able to propagate without the need for an additional RF mixing stage. Then the UWB receiver can bypass the need for complex phase tracking loops. However, before UWB systems can be produced at a large scale and low cost, there are numerous open research issues to be solved. UWB systems have opened up new dimensions of antenna design. Antennas have become an organic part of RF systems, providing ﬁltering and other custom-designed frequency-dependent properties. The wide bandwidths of UWB antennas present new challenges for design, simulation, and modelling. Optimizing UWB antennas to meet the demands of UWB propagation channels is similarly challenging. Designers are meeting these challenges with novel antenna designs, novel materials and using concepts like polarization diversity, directivity arrays, and electric-magnetic element combinations. Still more research at fundamental and applied levels is needed to make cheap and power-eﬃcient UWB chips available, going beyond the existing state-of-the-art solutions in several areas of transceiver architecture design and signal processing. Due to the extremely large bandwidth of UWB systems, which prevents direct sampling of the received signal at suﬃcient accuracy, it is expected that a straight-forward downscaling of signal processing algorithms for conventional receivers will not lead to practical solutions for UWB devices. This implies new algorithms for channel estimation, synchronization and other typical receiver tasks that have to be developed for UWB devices. The derivation of appropriate Chapter 1. Introduction 10 system models including the accurate modelling of the multi-path radio channel is as well necessary. Additionally, the shared nature of the medium means that UWB receivers must contend with a variety of interference sources. Traditional interference mitigation techniques are not amenable to UWB due to the complexity of straight-forward translations to UWB bandwidths. Thus, multi-user signal detection, multiple access interference (MAI) cancellation, narrowband interference (NBI) detection and cancellation are open research issues that must be met in order to exploit the potential beneﬁts of UWB systems. UWB radio technology is a well-suited physical layer candidate for the robust, low power and ubiquitous data communication requirements of WSNs. However, the fundamental role of UWB technology in WSNs is still open and a wide range of research questions continue to present challenges. The characterization of the UWB radio channel in industrial areas and large open spaces has to be addressed. The design of signaling schemes that make eﬃcient use of the channel capacity at low complexity and low power consumption is necessary, as well as eﬃcient medium access control (MAC) protocols for a large number of distributed sensor nodes. The derivation of algorithms to provide location information of wireless sensor nodes and geographic information about the environment has equal interest. 1.2 Ultra-Wideband Wireless Sensor Networks In this section, the basic challenges in designing WSNs are revisited, followed by the introduction of the UWB approach for radio communications among the sensor nodes in a WSN, that addresses each one of the design challenges. Finally, the overview of our envisaged application is presented. 1.2.1 Wireless Sensor Network Requirements WSNs have emerged as a new wireless networking paradigm allowing distributed sensing and cooperative communication strategies. They also represent a new generation of realtime embedded systems with diﬀerent signiﬁcant communication constraints when compared with traditional networked systems. The main design challenges in WSNs can be categorized into the following areas: • Scalability: As the number of sensor nodes in a WSN increases, scalability imposes diﬃculties in transferring data. In order to send information to far away nodes, signals with higher transmission power should be employed which can cause internode interference or a multi-hop approach need to be considered. • Power conservation: The nodes in WSNs have limited energy resources, so to extend the lifetime of the entire network power conservation in individual nodes is of signiﬁcant importance. In WSNs, radio communications is the major consumer of energy. Hence, minimizing the radio transmission power or avoiding unnecessary communications can considerably save power in sensor nodes. • Synchronization: During radio communications between diﬀerent wireless sensor nodes of a WSN, nodes continuously listen to transmissions and consume power Chapter 1. Introduction 11 if they are not time synchronized with each other. While global synchronization is unrealistic, especially when there is large sensor population, node-by-node synchronization becomes a necessity in WSN design. • Channel estimation: Channel estimation plays a critical role in WSNs, since sensor nodes communicate over wireless channels and have to overcome the eﬀects of a wireless link, such as noise, multipath eﬀect, intentional jamming and inter-node interference. Estimating the wireless link between a speciﬁc transmitter and receiver pair provides directionality and reliable data transfer between the nodes. • Self-organization: When large number of sensor nodes are deployed in remote environments, the ability of individual sensor nodes to self-organize is vital. Selforganization should be done in a way to improve the performance while reducing the power consumption of the entire sensor network. 1.2.2 Ultra-Wideband Approach to Wireless Sensor Networks WSNs are characterized by devices with low complexity that have limitations on processing power and memory, and severe restrictions on power consumption. By the very nature of the application, traﬃc in WSNs is often bursty with long periods of no activity (low duty cycle). For event detection operations, a device may remain idle for long periods, then suddenly be required to send signiﬁcant amounts of information when an event occurs. For devices deployed in the ﬁeld, this has signiﬁcant implications for the design of eﬃcient MAC protocols, radio communications technology, and the reliability of information transfer. For devices involved in continuous monitoring, the ﬂow of traﬃc will be more stable. However, eﬃcient multiple access techniques, reliability, and battery life are still major considerations. Since 2002, interest in UWB-based applications has increased greatly, especially concerning the use of UWB for WSNs. IR-based UWB technology has a number of inherent properties that are well suited for WSN applications. In particular, UWB-IR systems have potentially low complexity and low cost. This arises from the essentially baseband nature of the signal transmission. Unlike conventional radio systems, the UWB transmitter produces a very short time domain pulse that is able to propagate without the need for an additional RF mixing stage. The RF mixing stage translates the signal to a frequency that has desirable propagation characteristics. The signal will propagate well without need for additional upconversion and ampliﬁcation. The UWB receiver also does not require the reverse process of downconversion. Again, this means a local oscillator in the receiver can be omitted, which means the removal of associated complex delay and phase tracking loops. Using UWB technology for inter-node communication in WSNs will oﬀer not only low complexity and cost factors, but also provide high performance for communication over the wireless channels in spite of multipath distortions. Furthermore, transmission of short duration UWB pulses requires much lower power compared to strong narrowband signal transmission. In UWB-based WSNs, nodes can only communicate with their close-by Chapter 1. Introduction 12 neighbors due to low transmission power and avoid the inter-node interference issue that exists in narrowband techniques. Despite all the beneﬁts that UWB technology oﬀers to the design of WSNs, it can create a unique set of challenges too. Employing the low powered UWB pulses for inter-node communications, introduces the scalability problem in WSNs. As the distance between nodes or the number of nodes increases, weak UWB pulses cannot transfer information between nodes in a reliable manner. Also, the short duration of UWB pulses introduces a major challenge in time synchronization for sensor nodes in a wireless network. In order to synchronize sub-nanosecond pulses, very high-speed analog-to-digital converter (ADC) components are needed. Another problem with using UWB technology for WSNs is the performance degradation due to interference from strong narrowband signals that share the spectrum with low powered UWB pulses. Moreover, detection of UWB pulses is commonly performed using classical matched ﬁltering technique, where the received signal is correlated with a UWB pulse template. Thus, wireless channel multipath eﬀects on the received signal can signiﬁcantly degrade the detection process due to low correlation between the predeﬁned template and the distorted received signal. The challenge posed by synchronization and channel degradation to narrow and low powered UWB pulses has been addressed by the transmitted-reference (TR) signaling which is extensively presented in this thesis. TR receivers correlate the received signal with a delayed replica of itself. Using TR-UWB transceivers for radio communication of WSNs oﬀers simplicity, low transmission power, and capability to reduce the stringent UWB synchronization requirements, as well as channel estimation. On the other hand the performance of TR receivers is considerably limited by the severity of noise-noise products introduced by the correlation of noise in the received signal. Therefore, scalability becomes a major issue in designing WSNs with TR-UWB transceivers. On the other hand, channel estimation is not required with TR-UWB transceivers. Finally, UWB-IR systems have a very good time domain resolution, allowing for localization and tracking applications [12]. 1.2.3 Conceptual Design of the Envisaged Application The ranging network presented in this subsection can be viewed as a WSN where the physical parameter to be sensed is location. The radio module of each node is based on low-power low-complexity TR-UWB transceivers since sensor nodes are inherently resource constrained: limited processing capabilities, storage capacity, and power consumption. Moreover, they need to be designed with low complexity to meet the low cost requirements. Within the overall objective of investigating the communication requirements in WSNs our primary focus was the design of some experimental sensor nodes, which are capable of a tremendous diversity of functionality such as sensing capabilities, signal processing, network protocol functions, short-range radio communication at low to medium bit rates (up to 1 Mbps) and localization capabilities. The experimental design of the ZESS sensor nodes, as well as the asynchronous transfer of GPS data to those sensor nodes are described in Appendix VI. An overview of the diﬀerent activities involved in our envisaged WSN, with communi- Chapter 1. Introduction 13 Beacon Nodes Central station GPS sensor Serial interface GPS - MCU MCU UWB Radio Extraction of position information Sensor Node GPS sensor sensor MCU DSP MCU UWB Radio Dissection of GPS message string Radio Channel UWB Radio GPS sensor Timing offset estimation Decision Decoder MCU UWB Radio Text Image Video Coarse Acquisition LOS/NLOS detection Central station Location estimation Distance estimation Estimated Distance Estimated Location Figure 1.7: UWB based wireless sensor network. cation and localization capabilities using IR TR-UWB systems, is depicted in Fig. 1.7. In this application, the sensor node communicates with diﬀerent beacon nodes. The beacon nodes are assumed to be aware of their position by interfacing them with GPS sensors and allowing them to be mobile, or by keeping them at ﬁxed known positions. The signals received at the beacon nodes are processed in order to localize the sensor node. Using a centralized tightly-coupled system architecture, the beacon nodes are connected to a central station, through a wire or wirelessly, where the location of the sensor node is estimated. Tracking of a mobile sensor node is also possible using appropriate tracking algorithms. The UWB radio module in Fig. 1.7 contains two diﬀerent subsystems: the data subsystem and the distance measurement subsystem. All the diﬀerent blocks in both subsystems, that is, coarse acquisition, timing oﬀset estimation, symbol decision, decoding, and distance estimation are thoroughly addressed in this thesis. The line-of-sight (LOS)/non line-of-sight (NLOS) detection, which is beyond the scope of this thesis, is brieﬂy addressed hereafter. The radio channel presented in Fig. 1.7 may generally have three diﬀerent proﬁles. Chapter 1. Introduction 14 The ﬁrst one is the dominant direct path (DDP) case, in which the strongest path of the channel estimate corresponds to the LOS. Those channel proﬁles, where the ﬁrst path is not the strongest, however still detectable by an appropriate receiver architecture, are called non-dominant direct path (NDDP). Finally and only regarded as NLOS, in this thesis, are the situations in which the receiver architecture is not able to detect the direct path anymore. This case has been deﬁned as undetected direct path (UDP) and leads to ranging errors. Consequently, NLOS channels are one of the major drawbacks for accurate ranging and localization with UWB technology. To mitigate this drawback, many LOS/NLOS detection algorithms are proposed in the literature. In [13] a simple estimator has been proposed for NLOS detection using the running variance on N subsequent range estimates dˆn . Another LOS/NLOS detection solely uses the estimate of one channel impulse response and therefore introduces no additional delay. The so called conﬁdence metric, as deﬁned in [12] and used for LOS/NLOS detection, is based on the idea that later multipath components should have less power than the direct component in the LOS case and vice versa in the NLOS case. This reasoning is valid for distinguishing between DDP and UDP channel proﬁles. Some authors have stated that the root mean square (RMS) delay spread could be used as an LOS/NLOS detection method [14], however, they have not investigated the proposal further since no threshold was proposed. Finally, LOS/NLOS detection could be based on the assumption that a sudden decrease of signal-to-noise ratio (SNR) could indicate the movement from a LOS into an NLOS condition, and vice versa. In this case, changes in the power of subsequent maximum paths are detected and normalized. Many of the described algorithms are diﬃcult to apply with real UWB localization systems. For a suitable hypothesis testing of ranging error distributions, many range estimations must be evaluated at the same position to create an error distribution estimate, and valid error distributions for the LOS and NLOS case must be available. Both demands are rather seldom in reality. In addition, the necessity of a statistical signiﬁcance contrasts the need for a small latency of detection for a robust localization system. Similar obstacles occur when channel estimates are to be tested against predeﬁned channel models. Even if these methods work in real environments, the existence of enough LOS channels in realistic indoor applications is probably quite seldom. Consequently, distance measuring techniques which are less prone to LOS/NLOS conditions are useful. In this thesis, a novel distance measurement approach based on the modelling of the energy collected at the output of the TR-UWB receivers as a function of delay is proposed. Since the total energy is the sum of signal strengths of each individual path in a multipath environment, this measurement technique takes advantage of the multipath diversity in the channel. Furthermore, the timing requirement in our method is less rigorous than when using time-based distance measuring techniques, and it is more tolerant in UDP cases. This implies that the NLOS/NLOS detection can be omitted while using our distance measurement technique, at the expense of loosing some accuracy in distance measurement. This is the tradeoﬀ complexity/accuracy. Chapter 1. Introduction 1.3 15 Goal of the Thesis The main objective of this doctoral thesis is to present the conceptual design of UWB-IR transceivers that have dual functionality in providing robust, low power data communication and accurate location information in WSNs. The intended application would be the monitoring and localization of wireless sensor nodes within a closed area (indoor and outdoor environments). The emphasis of this study lays on the physical layer: modelling and analysis of the radio transceivers, simulation of receiver algorithms and characterization of the physical radio link. Wireless transmission using diﬀerent TR-UWB IR systems allows peak data rates up to 5 Mbit/s and location estimation with around 30 cm accuracy for a transmitterreceiver separation within 10 m. The TR signalling was used due to its simple receiver structure. 1.4 Organization of the Thesis This doctoral thesis is structured as follows: • In Chapter 1 the UWB technology was presented highlighting the history behind it and the regulatory issues. The basic techniques in UWB systems were introduced and an insight was provided on the open research issues in this new emerging technology. The envisaged application related to this work was presented. This chapter ends with emphasis on the goal and contributions of this thesis. • Chapter 2 deals with polynomial nonlinear systems. The Volterra model is also presented in this chapter. Further, the Hammerstein and Wiener models, which are the simpliﬁed forms of the Volterra model, are addressed as well. • Chapter 3 presents an important tool for the design and analysis of communication systems: the radio channel model. Diﬀerent channel models for UWB applications are revisited. • Diﬀerent TR systems are introduced in chapter 4. System models are derived for evaluating more eﬀectively the BER performance of the low complexity TR-UWB systems considered through simulation. • Chapter 5 addresses the synchronization issues. Coarse and ﬁne synchronization algorithms are developed and analyzed. • Based on the results from chapter 5, the performance of diﬀerent location estimation algorithms is evaluated in chapter 6. • Finally, general conclusions, recommendations and possible directions for future works are summarized in chapter 7. Chapter 1. Introduction 1.5 16 Contributions of the Thesis This thesis presents several contributions towards developing UWB-IR technology for short-range low to medium data rate WSN applications. The contributions can be classiﬁed into two classes. The ﬁrst class of contribution concerns the design through simulation of low complexity TR-UWB systems capable of providing data rates up to 5 Mbit/s. Based on previous work done on modelling TR-UWB systems, delay-dependent systems models for diﬀerent low complexity TR-UWB systems are proposed. These models are aimed in facilitating the performance analysis of the various systems considered, designing eﬃcient detectors and developing new algorithms. The second class of contribution consists of the development of simple algorithms for synchronization and distance estimation using the proposed delay-dependent systems models. 1.5.1 Systems Modelling In [15] a discrete-time Volterra equivalent system model was derived for DTR-UWB systems in a multipath channel, where both inter-frame interference (IFI) and inter-symbol interference (ISI) are present. Using this model, delay-dependent discrete time equivalent system and noise models for diﬀerent low complexity TR-UWB systems are proposed in this thesis. 1.5.2 Performance Analysis The diﬀerent TR-UWB receiver statistics are analyzed, ﬁrst, using the Gaussian approximation on the noise terms in the receiver statistics, then, using the equivalent system and noise models of the diﬀerent TR-UWB receivers. 1.5.3 BER derivation Both perspectives for receiver statistics analysis allow for bit error rate (BER) performance analysis to be performed. The derivation of an analytic expression for the average BER is provided in this thesis using the concept of minimum distance receiver. 1.5.4 Coarse Synchronization In this thesis, a novel data-aided timing acquisition technique for diﬀerential TR-UWB systems is presented. It achieves eﬃcient multipath energy collection even in presence of timing oﬀset and noise. 1.5.5 Fine Synchronization Another contribution of this thesis in the ﬁeld of synchronization concerns a simple timing oﬀset estimator for low complexity TR-UWB systems. The proposed algorithm uses energy collected at the symbol rate, thus reducing considerably the implementation complexity. Chapter 1. Introduction 1.5.6 17 Distance and Location Estimation Assuming coarse synchronization achieved and hardware clock timing error known, the timing oﬀset estimation algorithm can readily be used for distance and location estimation. To my knowledge, the distance measuring approach proposed in this thesis has never been dealt with before. It is similar to the signal strength measurement technique, but instead of using a path loss model, the modelling of the energy collected at the output of the UWB receivers as a function of delay is used. Chapter 2 Polynomial Nonlinear Systems: Overview 2.1 Introduction All linear systems obey the superposition principle, which implies that the output of a linear combination of input signals to a linear system is the same linear combination of the outputs of the system corresponding to the individual components. That is, L{αx1 (n) + βx2 (n)} = αL{x1 (n)} + βL{x2 (n)}, (2.1) where L{(·)} denotes the output of the linear system, x1 (n) and x2 (n) are two diﬀerent input signals, and α and β are two arbitrary constants. The superposition principle is a powerful mechanism that allows the study of all linear systems in a uniﬁed manner. Nonlinear systems do not satisfy the superposition principle. Furthermore, since every system that does not satisfy the superposition principle is nonlinear, it is impossible to develop a framework that is applicable to all nonlinear systems. Consequently, the traditional approach for modelling and studying nonlinear systems is to consider one or more classes of such systems and to develop a theory for analysis, design and realization as well as applications of such classes individually. In the next subsections several classes of nonlinear systems are brieﬂy described [16]. 2.1.1 Homomorphic Systems A homomorphic system satisﬁes the generalized superposition principle stated as H{x1 (n) ∗ x2 (n)} = H{x1 (n)} + H{x2 (n)}, (2.2) where “∗” and “+” are two operations deﬁned on the input signal and the output signal, respectively. As an example, the intensity of light reﬂected from an object can be modelled as the product of the intensity of the light that shines on the object at each location and the reﬂection coeﬃcient of the object [17], i.e., an image x(n1 , n2 ) may be expressed as 18 Chapter 2. Polynomial Nonlinear Systems: Overview 19 x(n1 , n2 ) = I(n1 , n2 )r(n1 , n2 ), (2.3) where I(n1 , n2 ) is the light intensity that falls on the object at location (n1 , n2 ) and r(n1 , n2 ) denotes the reﬂection coeﬃcient of the object at the same location. Homomorphic ﬁlters are particularly useful in situations, where the information-bearing function r(n1 , n2 ) needs to be separated from the image, or when it is necessary to process the two components in diﬀerent ways using an appropriate nonlinear transformation and a linear ﬁlter following the nonlinearity. Fig. 2.1 shows the block diagram of a homomorphic ﬁlter, where the ﬁrst block transforms the input signal using a logarithmic nonlinearity. The transformed signal is then processed by a linear ﬁlter. The output of the logarithmic nonlinearity is given by y(n1 , n2 ) = ln (x(n1 , n2 )) = ln (I(n1 , n2 )) + ln (r(n1 , n2 )) . (2.4) Thus, the linear ﬁlter will see the multiplicative components as additive components at its input. Consequently, linear ﬁltering techniques can be used to discriminate the two components. x(n) Log Nonlinearity Linear Filter y(n) Figure 2.1: A homomorphic ﬁlter employing a logarithmic nonlinearity. 2.1.2 Order Statistic Filters Order statistic ﬁlters are employed in applications in which the input signal is corrupted by impulsive noise. The class of order statistic ﬁlters includes the median ﬁlter and its generalizations. A one-dimensional (2K+1)-point median ﬁlter is deﬁned by the inputoutput relationship y(n) = median{x(n + K), x(n + K − 1), . . . , x(n − K)}, (2.5) where the median of the 2K+1 samples within the curly brackets is the (K + 1)th value among the samples rearranged in the ascending or descending order of magnitude. A generalization of the median ﬁlter computes the output signal as a linear combination of samples arranged in ascending or descending order of magnitude. The output of such a ﬁlter may be expressed as y(n) = N −1 ∑ hi xi (n), (2.6) i=0 where xi (n) is the ith sample in a rearrangement of the set x(n), x(n − 1), . . . , x(n − N + 1) in the ascending order of magnitude and hi denotes the coeﬃcients of xi (n). This type Chapter 2. Polynomial Nonlinear Systems: Overview 20 of generalization provides the ﬁlter with a combination of properties associated with the linear ﬁlters and the median ﬁlters. Other types of order statistic ﬁlters include weighted median ﬁlters, ﬁnite impulse response (FIR)-median hybrid ﬁlters, weighted order statistic ﬁlters, rank selection ﬁlters and stack ﬁlters [18] [19]. 2.1.3 Morphological Filters In applications such as pattern recognition and robotics, it is necessary to partition an image into segments on the basis of the geometric properties of the objects depicted in the image. This can be achieved through the use of mathematical transformations of the input images using operators known as morphological transforms. Signal processing systems that employ morphological transforms are known as morphological ﬁlters [20]. In addition to pattern and shape recognition and image decomposition using skeletal representation, morphological ﬁlters have also been used in image compression systems [21]. 2.1.4 Neural Networks Neural networks model nonlinear systems using interconnections of simple nonlinear devices known as artiﬁcial neurons. Artiﬁcial neurons are typically multiple-input, single-output systems with input-output relationship in the form { N } ∑ y(n) = f wi xi (n) − θ , (2.7) i=1 where wi is the weight associated with the ith input xi (n) to the device and θ is a constant term that controls the operating point of the nonlinearity f . An artiﬁcial neuron with this characteristic is depicted in Fig. 2.2. w1 x1(n) w2 x2(n) ... f( .) y(n) wN xN(n) - Figure 2.2: An artiﬁcial neuron. Many diﬀerent types of nonlinear functions have been employed, and some of the common ones are shown in Fig. 2.3. Chapter 2. Polynomial Nonlinear Systems: Overview (a) Hard limiter (b) Threshold logic 21 (c) Sigmoid Figure 2.3: Nonlinearities commonly employed in neural networks. In typical applications, the weights wi used in the network are selected by training the neural network on data that are representative of what the network will encounter in normal applications. The training is accomplished using an adaptation algorithm such as the back propagation algorithm. Algorithms for neural network design and their applications are thoroughly described in [22]. The advantage of artiﬁcial neural networks is their ability to model most nonlinear systems. However, in order to perform the modelling accurately, the network might require a very large number of artiﬁcial neurons. Another disadvantage of neural networks is that global convergence of training algorithms such as back propagation is not guaranteed. 2.1.5 Polynomial Filters Causal, discrete-time polynomial ﬁlters satisfy the input-output relationship of the form y(n) = P ∑ fi [x(n), x(n − 1), . . . , x(n − N ), y(n − 1), . . . , y(n − M )] , (2.8) i=0 where the function fi [. . . ] is an ith-order polynomial in the variables within the bracket. This deﬁnition contains the linear ﬁlters as special case since the system in (2.8) is linear if fi [. . . ] = 0 for all i ̸= 1. If the system in (2.8) is stable in the bounded-input boundedoutput sense, it admits a convergent Volterra series expansion of the form y(n) = h0 + + N∑ 1 −1 h1 (m1 )x(n − m1 )+ m1 =0 N∑ 1 −1 N∑ 2 −1 m1 =0 m2 =0 ... N∑ r −1 N∑ 1 −1 N∑ 2 −1 h2 (m1 , m2 )x(n − m1 )x(n − m2 ) + . . . m1 =0 m2 =0 hr (m1 , m2 , . . . , mr )x(n − m1 )x(n − m2 ) . . . x(n − mr ) mr =0 + . . ., (2.9) where hr (m1 , m2 , . . . , mr ) denotes the r -order Volterra kernel of the nonlinear system. The term involving the rth -order kernel looks like an r-dimensional convolution. Consequently, polynomial systems are considered as generalizations of linear systems. th Chapter 2. Polynomial Nonlinear Systems: Overview 2.2 22 Volterra Series Expansion Volterra series expansions form the basis of the theory of polynomial nonlinear systems. In this section, the Volterra series expansions are introduced and their properties are discussed. 2.2.1 Continuous-time Systems A system is deﬁned mathematically as a rule for transformation of an input signal x into another signal y by means of an operator S so that y = S{x}. (2.10) The input and output signals are usually functions of one or more independent variables such as position or time. If they are functions only of time t, and the variable t is deﬁned over a continuous range of values, (2.10) deﬁnes a continuous one-dimensional system whose input-output relationship can be represented as y(t) = S{x(t)}. (2.11) Linear Shift-invariant Systems A shift-invariant system is characterized by the invariance of its output with respect to a shift in the independent variable. As an example, a time-invariant system satisﬁes the relationship y(t + τ ) = S{x(t + τ )} (2.12) for all values of τ . A linear system is one that satisﬁes the superposition principle. This implies that S{αx1 (t) + βx2 (t)} = αS{x1 (t)} + βS{x2 (t)} (2.13) for all arbitrary constants α and β and arbitrary x1 (t) and x2 (t). The output of a continuous time linear and time-invariant (LTI) system is related to the input signal through the convolution integral ∫∞ h(τ )x(t − τ )dτ , y(t) = (2.14) −∞ where h(t) is the impulse response of the system. Equation (2.14) implies that the impulse response completely characterizes an LTI system. A system is said to be causal if its output at any given time does not depend on the future values of its input. An LTI system is causal if and only if h(t) = 0 for t < 0. (2.15) Chapter 2. Polynomial Nonlinear Systems: Overview 23 The unit impulse response signal represents the memory of the LTI system since the contribution to the current value of the output signal from the value of the input signal T seconds prior to the present time is determined by h(T ). The output of a memoryless LTI system is given by y(t) = c(x(t)), (2.16) which is obtained by convolving the input x(t) with cδ(t), where δ(t) represents the Dirac function and c is a constant that determines the extent of ampliﬁcation or attenuation of the input signal at the output of the system. Volterra Series Expansion for Nonlinear Systems A nonlinear system without memory can be often described by means of an appropriate series expansion. A power series expansion such as the Taylor series expansion may be used to describe the output of such systems as y(t) = ∞ ∑ cp xp (t). (2.17) p=0 A nonlinear system with memory can be represented by means of an extension of this expression. Such an extension, known as the Volterra series expansion [23], relates the input and output signals of the system as y(t) = h0 + + ∫∞ ∫∞ −∞ −∞ ... h1 (τ1 )x(t − τ1 )dτ1 + ∫∞ −∞ ∫∞ ∫∞ −∞ −∞ h2 (τ1 , τ2 )x(t − τ1 )x(t − τ2 )dτ1 dτ2 + . . . hp (τ1 , τ2 , . . . , τp )x(t − τ1 )x(t − τ2 ) . . . x(t − τp )dτ1 . . . dτp . (2.18) The nonlinear system represented by a Volterra series expansion is completely characterized by the multidimensional functions hp (t1 , t2 , . . . , tp ), called the Volterra kernels. The zeroth order kernel h0 is a constant. The higher-order kernels can be assumed, without loss of generality, as symmetric functions of their arguments so that any of the p! possible permutations of t1 , t2 , . . . , tp leaves hp (t1 , t2 , . . . , tp ) unchanged. This symmetry is the direct result of the invariance of the products of the delayed input functions {x(t−τi ), i = 1, . . . , p} with respect to their permutations. The system is causal if and only if hp (t1 , . . . , tp ) = 0 for any ti < 0 and i = 1, . . . , p. (2.19) The lower limits in the integrals in (2.18) are therefore set to zero for causal nonlinear systems. The upper limits of the integrals in (2.18) given as ∞ indicate that the system may have inﬁnite memory. If the upper limits are all ﬁnite, the system possesses ﬁnite memory. Each integral in (2.18) has the form of a multidimensional convolution. By deﬁning the pth -order Volterra operator h̄p [x(t)] as Chapter 2. Polynomial Nonlinear Systems: Overview ∫∞ h̄p [x(t)] = ∫∞ hp (τ1 , τ2 , . . . , τp )x(t − τ1 )x(t − τ2 ) . . . x(t − τp )dτ1 . . . dτp . ... −∞ 24 (2.20) −∞ Equation (2.18) may be written more compactly as y(t) = h0 + ∞ ∑ h̄p [x(t)]. (2.21) p=1 A truncated Volterra series expansion is obtained by setting the upper limit of the summation in (2.21) to a ﬁnite integer value P . The parameter P is called the order, or the degree, of the Volterra series expansion. (2.21) reveals the similarity of Volterra series expansions with the Taylor series expansions. When the input signal is multiplied by a constant factor c, the output y(t) becomes y(t) = h0 + ∞ ∑ h̄p [cx(t)] = h0 + p=1 ∞ ∑ cp h̄p [x(t)]. (2.22) p=1 which is a power series expansion in c. Limitations of Volterra Series Expansions As a consequence of its power series characteristic, there are some limitations associated with the application of the Volterra series expansion to nonlinear system modeling. These are: • Convergence issues: Because of its close relationship with Taylor series expansions, Volterra series expansions exhibit convergence problems when the nonlinear systems to be modeled include strong nonlinearities such as saturating elements. Therefore, the Volterra series approach can be applied with good results only to systems with mild nonlinearities. • Nonlinear systems not completely charaterized by the impulse response function: In contrast to the LTI systems, the impulse response function does not fully characterize a nonlinear system. Indeed, the response of the nonlinear system to the impulse cδ(t) is given by h(t) = h0 + ch1 (t) + c2 h2 (t, t) + . . . + cp hp (t, . . . , t) + . . . . (2.23) This indicates that the impulse response is determined only by diagonal values of he kernel hp , that is for t = t1 = t2 = . . . = tp . Thus, it does not completely specify the pth -order Volterra kernels of higher order than 1. • Multiple-valued nonlinearities: Since the Volterra series expansion is a single-valued representation, it cannot be used to represent multiple-valued nonlinearities. Chapter 2. Polynomial Nonlinear Systems: Overview 2.2.2 25 Discrete-time Systems Consider a system for which the input and output values are known only at given time instants t = nT , where n is an integer variable. By assuming a normalized time interval T = 1, (2.11) becomes y(n) = S{x(n)}. (2.24) This equation describes a general one-dimensional discrete-time system. Many classiﬁcations of discrete-time systems can be made in a manner similar to those of continuous-time systems. A discrete time shift-invariant system is characterized by the property that it is invariant to shifts in the independent variable: y(n + m) = S{x(n + m)}, (2.25) where m is an integer shift. A discrete time shift-invariant system is said to be discrete time-invariant (DTI) when the independent variable is time. A discrete-time is said to be linear if it obeys the superposition principle: S[αx1 (n) + βx2 (n)] = αS[x1 (n)] + βS[x2 (n)], (2.26) for all arbitrary constants α and β and arbitrary input signals x1 (n) and x2 (n). If a discretetime system is both linear and time-invariant its output can be evaluated by convolving its input signal with its unit impulse response function hn as y(n) = ∞ ∑ h(m)x(n − n). (2.27) m=−∞ A discrete-time LTI system is causal if and only if h(n) = 0 for n < 0. (2.28) DTI Nonlinear Systems In a manner similar to continuous case, it is possible to describe DTI nonlinear systems with memory by means of the discrete-time Volterra series expansion y(n) = h0 + ∞ ∑ h̄p [x(n)], (2.29) p=1 where y(n) and x(n) are the output and input signals, respectively, and h̄p [x(n)] = ∞ ∑ m1 =−∞ ... ∞ ∑ mp =−∞ hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ), (2.30) Chapter 2. Polynomial Nonlinear Systems: Overview 26 where hp (m1 , . . . , mp ) is the pth -order Volterra kernel of the system. If hp (m1 , . . . , mp ) = 0 for all mi < 0, and i = 1, . . . , p, the DTI nonlinear system is causal, and (2.30) becomes h̄p [x(n)] = ∞ ∑ ... m1 =0 ∞ ∑ hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ). (2.31) mp =0 The discrete-time Volterra kernels can be interpreted in a manner similar than the continuous-time systems. The constant h0 is an oﬀset term, h1 (m1 ) is the impulse response of a discrete-time LTI ﬁlter, and the pth -order kernel hp (m1 , . . . , mp ) can be considered as a generalized pth -order impulse response characterizing the nonlinear behavior of the system. The upper limit in the summations in (2.31) given as inﬁnity indicates that the discrete system may have inﬁnite memory. The diﬃculties that arise because of the inﬁnite summations in (2.31) may be avoided by using recursive polynomial system models. This approach is similar to the use of inﬁnite impulse response (IIR) ﬁlters in applications involving linear system models. Usually, the memory required to adequately approximate a nonlinear system is ﬁnite. In such situations, a Volterra series expansion involving only the input signal is suﬃcient to model the system. This simpler class of nonlinear system models is derived by limiting all the summations in (2.31) to ﬁnite values. In this case, h1 (m1 ) represents the impulse response of a ﬁnite impulse response (FIR) ﬁlter, and the eﬀect of the nonlinearity on the output depends only on the present and past values of the input signals deﬁned on the extent of the ﬁlter support. If the discrete Volterra series expansion is truncated by limiting the summation in (2.29) to a ﬁnite value P , ﬁnite-order expansion is obtained as y(n) = h0 + P ∑ h̄p [x(n)], (2.32) p=1 where h̄p [x(n)] = N −1 ∑ m1 =0 ... N −1 ∑ hp (m1 , . . . , mp )x(n − m1 )x(n − m2 ) . . . x(n − mp ). (2.33) mp =0 Equation (2.32) represents a general discrete-time Volterra model of order P and memory length N . The upper limits in all the summations of (2.33) are made identical only for convenience. They may be set to arbitrary values to obtain a more general expression. The simplest polynomial ﬁlter of this class is the quadratic ﬁlter obtained by choosing P = 2 in (2.32). This type of ﬁlter is discussed in the next two subsections. 2.2.3 Algebraic Representation of Quadratic Filters A causal, discrete and time-invariant quadratic ﬁlter with ﬁnite support for its kernels is described by means of the ﬁrst three terms of the Volterra series expansion as Chapter 2. Polynomial Nonlinear Systems: Overview y(n) = h0 + N −1 ∑ h1 (m1 )x(n − m1 ) + m1 =0 N −1 N −1 ∑ ∑ 27 h2 (m1 , m2 )x(n − m1 )x(n − m2 ), (2.34) m1 =0 m2 =0 where h0 represents a constant oﬀset term, {h1 (m1 ); 0 ≤ m1 ≤ N − 1} denotes the coeﬃcients of a linear time-invariant FIR ﬁlter and {h2 (m1 , m2 ); 0 ≤ m1 , m2 ≤ N −1} represents the coeﬃcient of a homogenous quadratic ﬁlter. The quadratic ﬁlter can be realized as a parallel combination of these three components as shown in Fig. 2.4. h0 h1 (m) x (n) y (n) h2 ( m 1 ,m 2 ) Figure 2.4: A quadratic ﬁlter realized as a parallel combination of three components. Matrix-Vector Representation It is usual to arrange the input values and the ﬁlter coeﬃcients of one-dimensional linear FIR ﬁlters into two vectors x(n) and h1 deﬁned as x(n) = [ and h1 = x(n) x(n − 1) · · · x(n − N + 1) [ h1 (0) h1 (1) · · · h1 (N − 1) ]T ]T . (2.35) (2.36) h̄1 [x(n)] is expressed as h̄1 [x(n)] = xT (n)h1 = hT1 x(n). (2.37) In similar way, the input-output relation of a quadratic ﬁlter can be written as h̄2 [x(n)] = xT (n)H2 x(n), (2.38) where H2 is an N × N matrix in which the coeﬃcients of the quadratic kernel h2 (m1 , m2 ) are arranged as Chapter 2. Polynomial Nonlinear Systems: Overview h2 (0, 0) h2 (0, 1) h2 (1, 0) h2 (1, 1) H2 = .. .. . . h2 (N − 1, 0) h2 (N − 1, 1) 2.2.4 ··· h2 (0, N − 1) ··· h2 (1, N − 1) . .. ... . · · · h2 (N − 1, N − 1) 28 (2.39) Realization of Quadratic Filters The homogeneous quadratic term in (2.34) may be equivalently represented using a triangular kernel as h̄2 [x(n)] = N −1 ∑ N −1 ∑ h2t (m1 , m2 )x(n − m1 )x(n − m2 ), (2.40) m1 =0 m2 =m1 where m1 = m2 h2 (m1 , m2 ); h2 (m1 , m2 ) + h2 (m2 , m1 ); m1 < m2 h2t (m1 , m2 ) = 0; otherwise. (2.41) h̄2 [x(n)] can be implemented by means of a nonlinear combiner, a number of multipliers, and a summing bus as shown in Fig. 2.5. The nonlinear combiner computes all the necessary products of the input samples. 2.3 Wiener & Hammerstein Models One of the most frequently studied classes of polynomial nonlinear models are the so called block-oriented models, which consist of the interconnection of LTI systems and static (memoryless) nonlinearities. Within this class, four of the more common model structures are the: • Hammerstein model, which consists of the cascade connection of a static nonlinearity followed by a LTI system (see for instance [24] for a review on identiﬁcation of Hammerstein models) • Wiener model, in which the order of the linear and the nonlinear blocks in the cascade connection is reversed (see for instance [25] and [26] for diﬀerent methods for the identiﬁcation of Wiener models) • LNL cascade model, which is represented by a combination of Hammerstein and Wiener systems. Indeed, it is a cascade of a LTI ﬁlter followed by a static nonlinearity, followed by another LTI ﬁlter. In [27] a three steps adaptive algorithm for identiﬁcation of the LNL model is outlined. Chapter 2. Polynomial Nonlinear Systems: Overview 1 [x(n)] 1 z input signal z Unit Delay1 x(n)*x(n) 29 Unit Delay2 x(n)*x(n-1) x(n)*x(n-2) K h_2(0,2) K h_2(0,0) Unit Delay3 y_out Add1 Add3 1 Add4 Add2 z K K h_2(1,1) Unit Delay4 1 Add5 Unit Delay5 1 h_2(0,1) z K z K h_2(1,2) h_2(2,2) Figure 2.5: Realization of a quadratic operator using Simulink. y(n) Chapter 2. Polynomial Nonlinear Systems: Overview 30 • Feedback block-oriented (FBO) model, which consists of a static nonlinearity in the feedback path around a LTI system (see for instance [28] for an identiﬁcation algorithm for this type of model). These models have been successfully used to represent nonlinear systems in a number of practical applications in the areas of signal processing [29], communications and control theory [30]. In the next subsections, the Hammerstein and Wiener models will be discussed. 2.3.1 Hammerstein-Wiener Model The Hammerstein-Wiener structure models dynamic systems using up to two static nonlinear blocks in series with a linear block. The input signal passes through the ﬁrst nonlinear block, a linear block, and a second nonlinear block to produce the output signal, as shown in Fig. 2.6. x(n) Input Nonlinearity u(n) Linear Block w(n) Output Nonlinearity y(n) Figure 2.6: The Hammerstein-Wiener model. This model structure represents a nonlinear system as a linear system that is modiﬁed by static input and output nonlinearities. Thus, the linear model provides a reference point for estimating the nonlinear contributions in a system. The following general equation describes the Hammerstein-Wiener structure: u(n) = f (x(n)) −1 ) w(n) = B(q u(n) −1 A(q ) y(n) = g (w(n)) (2.42) where x(n) and y(n) are the input and output for the system, respectively. f and g are nonlinear functions corresponding to the input and output nonlinearities, respectively. For multiple inputs and multiple outputs, f and g are deﬁned independently for each input and output channel. u(n) and w(n) are internal variables. u(n) has the same dimension as x(n), whereas w(n) has the same dimension as y(n). B(q −1 ) and A(q −1 ) in the linear dynamic block are polynomials in the unit delay operator q −1 . For multiple inputs and outputs, the linear block is a transfer function matrix. If only the input nonlinearity is present, the model is called the Hammerstein model. If only the output nonlinearity is present, the model is called the Wiener model. 2.3.2 Hammerstein Model The Hammerstein model of a nonlinear system is shown in Fig. 2.7. Chapter 2. Polynomial Nonlinear Systems: Overview 31 f(u) Taylor series expansion x(n) u(n) H(q -1) y(n) LTI filter Static nonlinearity Figure 2.7: The Hammerstein model. It consists of a static nonlinear mapping or gain followed by a linear dynamic block. The input signal x(n) goes through the nonlinear static mapping block and gives an unobservable intermediate signal u(n). Then u(n) goes through the linear dynamic block to give the output signal y(n), with f [x(n)] denoting the nonlinear static gain function and the dynamics being modelled by a linear discrete transfer function H(z). The Hammerstein model models the nonlinear eﬀects as an input-dependent gain nonlinearity. If the static nonlinear function is assumed to be approximated by a ﬁnite polynomial expansion, the Hammerstein model can be described by the following equations y(n) + a1 y(n − 1) + . . . + ak y(n − k) = b1 u(n − 1) + . . . + bk u(n − k) (2.43) and u(n) = γ1 x(n) + γ2 x2 (n) + . . . + γm xm (n). (2.44) The intermediate variable u(n) cannot be measured, but it can be eliminated from the equations. Substituting (2.44) into (2.43) we get y(n) + a1 y(n − 1) + . . . + ak y(n − k) = b1 γ1 x(n − 1) + . . . + b1 γm xm (n − 1) +b2 γ1 x(n − 2) + . . . + b2 γm xm (n − 2) . (2.45) .. . +bk γ1 x(n − k) + . . . + bk γm xm (n − k) Equation (2.45) can be written in operator form as ( m ) m ∑ ∑ B(q ) γi xi (n) = H(q −1 ) γi xi (n) y(n) = −1 A(q ) i=1 i=1 −1 (2.46) where the polynomials A(q −1 ) and B(q −1 ) are expressed as A(q −1 ) = 1 + a1 q −1 + . . . + ak q −k . B(q −1 ) = b1 q −1 + . . . + bk q −k (2.47) The orders of the polynomials are assumed to be the same, but this needs not be the case Chapter 2. Polynomial Nonlinear Systems: Overview 32 in general. 2.3.3 Wiener Model The Wiener system has the same two type of blocks as the Hammerstein system, but in the reverse order, as shown in Fig. 2.8. g(x) Taylor series expansion x(n) w(n) H(q -1) LTI filter y(n) Static nonlinearity Figure 2.8: The Wiener model. In the Wiener structure, the input signal x(n) is transformed by the linear dynamic block to get w(n), and then w(n) goes through the static nonlinear block to produce y(n). Mathematically, x(n) and y(n) can be expressed as w(n) = H(q −1 )x(n) (2.48) y(n) = g (w(n)) . (2.49) and If the static nonlinear function g is assumed to be approximated by a ﬁnite polynomial expansion, the Wiener model can be described by the following equation y(n) = m ∑ [ ]i γi H(q −1 )x(n) . (2.50) i=1 2.4 Summary Polynomial nonlinear systems were described in this chapter. They are used, in this thesis, to model diﬀerent TR-UWB systems. After a brief introduction on the diﬀerent classes of nonlinear systems, Volterra series expansions were presented. They constitute the basis of the theory of polynomial nonlinear systems. Further, the Wiener and Hammerstein models were revisited. They are part of the most frequently studied classes of polynomial nonlinear models. Chapter 3 Channel Characterization 3.1 Introduction The goal of this chapter is the description and discussion of an appropriate channel model for mobile radio systems, in general, and UWB radio systems, in particular. Radio propagation in a mobile radio channel is determined mainly by its multipath nature. Multiple reﬂections, and sometimes a LOS component of the transmitted signal arrive at the receiver via diﬀerent propagation paths and therefore with diﬀerent amplitudes and delay times. As an eﬀect of this, the narrowband received power ﬂuctuates dramatically when observed as a function of location (or time) and frequency. In the early days of mobile systems, the communications engineer was mainly interested in the time-variability of narrowband channels, which were thus studied extensively [31]. By that time, transmission bandwidths were small, hence ﬂat-fading was a reasonable assumption. As the systems evolved, demand for higher transmission rates has been increasing, making the channels time dispersion (which is equivalent to its frequency-selectivity) a major issue. UWB systems transmit signals with very low power levels. Unavoidably, this constraint limits the range of UWB wireless links to values typically in few meters, making such systems suitable to short range indoor applications. This is the reason why most of the research works on the characterization and modelling of UWB channels have focused on indoor environments with both LOS and NLOS settings. The following sections review the propagation mechanisms that have to be characterized by the channel model. Diﬀerent UWB channels are introduced with emphasis on the standardized IEEE 802.15.4a realistic channel model. 3.2 3.2.1 Characterization of the Mobile Radio Channel Components of a Multipath Channel Model Three mechanisms should be distinguished in order to describe mathematically a multipath radio channel, namely: • path loss 33 Chapter 3. Channel Characterization 34 • shadowing • multipath interference. The ﬁrst two are described by large-scale channel models, which essentially provide information about the average received power at a certain location. Path loss strictly describes the dependency of this average power on the distance between transmitter and receiver, while shadowing accounts for the ﬂuctuations observed at a ﬁxed distance, due to geometric features of the propagation environment. These ﬂuctuations occur for instance because of the blocking of relevant propagation paths, e.g., the LOS component, as the mobile moves around. For mobile radio applications, the description of the eﬀects of multipath interference is required, since the air-interface has to cope with them. These eﬀects are often referred to as small-scale fading. Small-scale models are valid within (small) local areas, where the signal ﬂuctuations due to shadowing and path loss can be neglected. The dimension of such a local area is therefore limited to approximately 5 · · · 40λ, where λ is the wavelength of the RF carrier [32]. The channel model considered in this section is limited to the description of smallscale eﬀects. A set of average parameters speciﬁes the channels behavior within a local area. These parameters are the normalized received power, P0 , the Ricean K-factor, K, and the RMS delay spread, τrms . P0 is the ratio of the received power Prx and the transmitted power Ptx . It is important noting that each realization obtained from the model has varying instantaneous parameters denoted {P̂0 , K̂, τ̂rms }, since the model is a stochastic one. The amount of variation of these parameters from the local-area parameters depends in particular on the observed bandwidth. When the bandwidth is much greater than the coherence bandwidth, then the multipath is completely resolved and the channel parameters vary little, since the individual multipath amplitudes do not change rapidly within a local area. However, if the system is narrowband, then the multipath is not resolved, and the path amplitudes at each resolvable delay time-bin (being spaced by the reciprocal of the bandwidth) vary due to multipath interference. This leads to the ﬂuctuation of the instantaneous channel parameters within the local area [33]. 3.2.2 Channel Impulse Response In complex low-pass equivalent notation, the channel impulse response (CIR) is given as h(t) = ∞ ∑ αn e−jθn δ(t − τn ), (3.1) n=0 where {αn }, {θn } and {τn } are the the propagation paths’ amplitudes, phases, and delays, respectively. Normally, the delay of the ﬁrst (shortest) ray is deﬁned as τ0 = 0. Since τn > 0 for i > 0, the channel impulse response is causal. Note that in a real environment, the parameters {αn }, {θn } and {τn } are time-variant. For the sake of simplicity, this time dependency was omitted in (3.1). Chapter 3. Channel Characterization 3.2.3 35 Channel Parameters The channel parameters P0 ,K and τrms are deﬁned from the static power delay proﬁle (PDP), which is a function derived from the CIR. The PDP speciﬁes the ray-power versus time delay, and is given as ph (t) = ∞ ∑ αn2 δ(t − τn ). (3.2) n=0 As the ray phases are dropped in this equation, the channel parameters must be constant within the local area, provided that the propagation paths are fully resolvable. The normalized received power is deﬁned as the sum of the ray powers P0 = ∞ ∑ αn2 . (3.3) n=0 The Ricean K-factor is the ratio of the dominant path’s power to the power in the scattered paths, and it is deﬁned as K= 2 αn,max , 2 P0 − αn,max (3.4) 2 where αn,max = max{αn2 }. The K-factor speciﬁes the depth of the fades within a local area. n Larger K-factors relate to shallower fades, and smaller K-factors relate to deeper fades. In the presence of a LOS, the ﬁrst ray is the dominant one, implying that αn,max = α0 at τ0 = 0. The RMS delay spread, deﬁned as the second central moment of the PDP, is written as √ (3.5) τrms = τ¯2 − τ̄ 2 , ∞ ∑ where τ¯m = τnm αi2 /P0 , for m = {1, 2}. τrms is an important parameter for specifying n=0 the time extent of the dispersive channel. It also characterizes the frequency-selectivity, since τrms is related to the average number of fades per bandwidth, and to the average bandwidth of the fades [32]. 3.2.4 Channel Description The channel is characterized by its delay power spectrum (or average power delay Proﬁle ∞ ∑ - APDP), Ph (t) = E{h2 (t)} = E{ αn2 δ(t − τn )}. The expected power of ray at delay τn n=0 given by E { αn2 |τn } { = ρ2 Ph (τn ) λ(τn ) n=0 , n>0 (3.6) Chapter 3. Channel Characterization 36 where λ(τn ) [rays/s] is the density of arriving rays, being a function of the excess delay time t. ρ2 is the power of the dominant path, given by K . (3.7) K +1 In agreement with measurements reported in [34], the shape of the APDP is deﬁned as shown in Fig. 3.1. 2 ρ2 = αn,max = P0 Ph(t) r 2 P e-g (t-tc) 0 tc Excess delay [s] Figure 3.1: Model of the average power delay proﬁle It is speciﬁed by four parameters: • ρ2 - the normalized power of the direct ray • Π [1/s] - the normalized power density of the constant-level part • tc [s] - the duration of the constant level part • γ [1/s] - the decay exponent of the exponentially decaying part. Mathematically, the APDP [W/s] can be written as 0 t<0 ρ2 δ(t) t=0 Ph (t) = . Π 0 < t ≤ tc Πe−γ(t−tc ) t>t (3.8) c The exponentially decaying APDP is a good approximation for most practical channels, which is implemented by letting tc = 0. The existence of a LOS ray at t = 0 implies that the fading envelope distribution is Ricean. Rayleigh fading channels have ρ = 0. The relationship between channel parameters deﬁned in subsection 3.2.3 and the parameters {ρ, Π, γ, tc } is presented below. P0 relates to the APDP as Chapter 3. Channel Characterization ∫∞ 37 [ ] 1 Ph (t)dt = ρ + Π tc + . γ 2 P0 = (3.9) 0 For the special case of the exponentially decaying APDP (tc = 0), we have ∫∞ Ph (t)dt = ρ2 + P0 = Π . γ (3.10) 0 The K-factor characterizes the amplitude distribution of Ricean channels, relating the power of the direct path to the power of the scattered paths. K= ρ2 ρ2 [ ]. = P 0 − ρ2 Π tc + γ1 (3.11) The RMS delay spread τrms was deﬁned in (3.5) as a function of τ̄ and τ¯2 . τ̄ is expressed as ∫∞ τ̄ = [ 2 ] Ph (t) tc tc 1 τ dt = Π + + 2 , P0 2 γ γ (3.12) 0 and τ¯2 can be written as ∫∞ τ¯2 = τ 2 Ph (t) P0 [ ] t3c t2c 2tc 2 dt = Π + + 2 + 3 . 3 γ γ γ (3.13) 0 From (3.12), (3.13) and (3.11), τrms is expressed as √ 1 1 u3 1 u22 τrms = − , γ K + 1 u1 (K + 1)2 u21 where u1 = tc γ + 1, u2 = t2c γ 2 /2 + tc γ + 1 and u3 = t3c γ 3 /3 + t2c γ 2 + 2tc γ + 2. If tc = 0, we have √ 1 2K + 1 . τrms = γ K +1 (3.14) (3.15) Finally, from (3.7) and (3.11), we have Π= If tc = 0, we have P0 γ . K + 1 u1 (3.16) Chapter 3. Channel Characterization 38 P0 γ. (3.17) K +1 Let us deﬁne the maximum excess delay as the delay time where the exponentially decaying part has decreased by about 43 dB. Such attenuation is reached if the duration of the exponentially decaying part is exactly texp = 10/γ, leading to the maximum delay spread tmax = tc + texp = tc + 10/γ. Expressed in terms of channel parameters, we have Π= u1 (K + 1) , tmax = trms (u + 10) √ u1 u3 (K + 1) − u22 (3.18) which simpliﬁes for tc = 0 to K +1 tmax = 10trms √ . (3.19) 2K + 1 It can be seen that tmax and trms are related by a factor which is function of K, tc and γ. A typical realization of the CIR (ℜe{h(t)}) and its associated PDP are illustrated in Fig. 3.2 (a) and (b) respectively, with channel parameters shown in Tab. 3.1. 0.25 0.025 0.2 0.02 0.1 Normalized ray powers Multipath ray amplitudes 0.15 0.05 0 −0.05 −0.1 −0.15 0.015 0.01 0.005 −0.2 −0.25 0 20 40 60 80 Multipath channel delays [ns] 100 120 0 0 20 40 60 80 Multipath channel delays [ns] (a) 100 120 (b) Figure 3.2: A typical channel realization (a) Channel impulse response; (b) Associated power delay proﬁle. tc [ns] 0 P0 1 K NLOS 0 λ [rays/ns] 5 τrms [ns] 10 ρ2 0 γ [ns−1 ] 0.1 Π 0.1 Table 3.1: Simulation values of the channel parameters. Chapter 3. Channel Characterization 3.3 3.3.1 39 Standardized Channel Models for UWB Communications Overview In UWB systems, the intended radiation can cover a bandwidth of almost 10 GHz. This large bandwidth can give rise to new eﬀects. For example, only few multipath components overlap within each resolvable delay bin, so the central limit theorem is no more applicable, and the amplitude fading statistics are no longer Rayleigh. Also, there can be delay bins into which no multipath components fall, and thus are empty. It is then necessary to characterize the likelihood that this happens, and that an empty bin is followed by a full one. A channel model taking into account those characteristics is the well-known SalehValenzuela (S-V) indoor channel model [35]. It is based on measurements utilizing low power ultra-short pulses (of width 10 ns and center frequency 1.5 GHz) in a medium-size, two-storey oﬃce building. In the S-V model, multipath components arrive at the receiver in groups (clusters). Cluster arrivals are Poisson distributed with rate Λ. Within each cluster, subsequent arrivals are also Poisson distributed with rate λ > Λ. With αl,k denoting the gain of the kth multipath component of the lth cluster, having phase θl,k , the CIR can be expressed as h(t) = L−1 K−1 ∑ ∑ αl,k ejθl,k δ(t − Tl − τl,k ). (3.20) l=0 k=0 The gains αl,k of the kth component in the lth cluster are independent Rayleigh random 2 2 variables with power E{αl,k }=E{α0,0 }e−Tl /Γ e−τl,k /γ , where Γ is the cluster decay factor, γ is the ray decay factor. Tl is the delay of the lth cluster, τl,k is the delay of the kth multipath component relative to the lth cluster arrival time Tl . The phases θl,k are uniformly distributed, i.e., for a bandpass system, the phase is taken as a uniformly distributed random variable from the range [0,2π). K is the number of multipath components within a cluster, while L is the number of clusters. To come up with a statistical model, channel realizations are identiﬁed either in the frequency domain by frequency sweeping or in the time domain using impulsive signals. In November 2002, the channel modelling subcommittee of the IEEE 802.15.3a Task Group recommended a channel which captures the aforementioned works, as well as recent reﬁnements [36]. Because the clustering phenomenon has been experimentally conﬁrmed, the standardized channel model is basically a modiﬁed version of the S-V model. To reach an analytically manageable channel model, the total number of paths is deﬁned as the number of multipath arrivals with expected power within 10 dB from that of the strongest arrival path. The Rayleigh distribution in the S-V channel model is replaced by the lognormal distribution. The phases θl,k are also constrained to take values 0 or π with equal probability to account for signal inversion due to reﬂection, yielding a real-valued channel model. With path gains normalized to have unit energy, a log-normal random variable is introduced to account for shadowing (large-scale fading). The Task Group 802.15.4a has the mandate to develop an alternative physical layer Chapter 3. Channel Characterization 40 for sensor networks and similar devices, working with the IEEE 802.15.4 MAC layer. The main goals for this new standard are energy-eﬃcient data communications with data rates between 1kbit/s and several Mbit/s; additionally, the capability for geolocation plays an important role. The channel modelling subgroup started its activities at the meeting in September 2003 (Singapore), and submitted its ﬁnal report in September 2004 (Berlin). The proposed IEEE 802.15.4a channel model was aimed at modelling attenuation and delay dispersion [37]. The former includes both shadowing and average path loss, while the latter describes the PDP and the small-scale fading statistics, with parameters such as RMS delay spread, number of multipath components carrying x% of the energy, etc... This channel model can be used for UWB systems spanning frequency range 100-1000 MHz and 2-10 GHz. 3.3.2 The IEEE 802.15.4a Channel Model The key features of the 802.15.4a Channel Model are • d−n law for the path loss • frequency dependence of the path loss • modiﬁed S-V model: – arrival of paths in clusters – Poisson distribution for ray arrival times – possible delay dependence of cluster decay times – some NLOS environments have ﬁrst increase, then decrease of power delay proﬁle • Nakagami-distribution of small-scale fading, with diﬀerent m-factors for diﬀerent components • block fading: channel stays constant over data burst duration. Path Loss Model By deﬁnition, the attenuation undergone by an electromagnetic wave in transit between a transmitter and receiver in a communication system is called path loss or path attenuation. Path loss may be due to many eﬀects, such as: free space loss, refraction, reﬂection, diﬀraction, clutter, aperture-medium coupling loss and absorption. The path loss in a narrowband system is conventionally deﬁned as E{Prx (d, fc )} , (3.21) Ptx where d is the distance between transmitter and receiver, fc is the center frequency, and the expectation E{·} is taken over an area that is large enough to allow averaging out of P L(d) = Chapter 3. Channel Characterization 41 the shadowing as well as the small-scale fading. A frequency-dependent path loss related to wideband path loss, as suggested in [38] is deﬁned as f +∆f ∫ /2 P L(f, d) = E{ ( )2 ˜ H f , d df˜}, (3.22) f −∆f /2 where H(f, d) is the transfer function from transmitter antenna connector to receiver antenna connector, and ∆f is chosen small enough so that diﬀraction coeﬃcients, dielectric constants, etc., can be considered constant within that bandwidth. The total path loss is obtained by integrating over the whole bandwidth of interest. To simplify computations, it is assumed that the path loss as a function of the distance and frequency can be written as a product of the terms P L(f, d) = P L(f )P L(d). The frequency dependence of the path loss is given as [39] √ P L(f ) ∝ f −κ . The distance dependence of the path loss in dB is described by ( ) d P L(d) = P L0 + 10n log10 , d0 (3.23) (3.24) (3.25) where the reference distance d0 is set to 1 m, and P L0 is the path loss at the reference distance. n is the path loss exponent. The path loss exponent also depends on the environment, and on whether a LOS connection exists between the transmitter and receiver or not. LOS path loss exponents in indoor environments range from 1.0 in a corridor to about 2 in an oﬃce environment. NLOS exponents typically range from 3 to 7. The above model includes the eﬀects of the transmit and the receive antenna, as it deﬁnes the path loss as the ratio of the received power at the receiver antenna connector, divided by the transmit power (as seen at the transmitter antenna connector). Fading Statistics 1. Shadowing Shadowing, or large-scale fading, is deﬁned as the variation of the local mean around the path loss. Also this process is fairly similar to the narrowband fading. The path loss in dB can be written as ( ) d + S, (3.26) P L(d) = P L0 + 10n log10 d0 where S is a Gaussian-distributed random variable with zero mean and standard deviation σS . Chapter 3. Channel Characterization 42 2. Power Delay Profile The discrete-time impulse response, in complex baseband, of the S-V model was expressed in (3.20). The number of clusters L is an important parameter of the model, and it is assumed to be Poisson-distributed (L̄)L e−L̄ pL (L) = . L! The distributions of the cluster arrival times are given by a Poisson process p(Tl /Tl−1 ) = Λe[−Λ(Tl −Tl−1 )] , l > 0 (3.27) (3.28) where Λ is the cluster arrival rate, assumed to be independent of l. The classical S-V model also uses a Poisson process for the ray arrival times. Due to the divergence in the ﬁtting for the indoor residential, and indoor and outdoor oﬃce environments, ray arrival times are modelled with mixtures of two Poisson processes as follows p (τl,k |τl,k−1 ) = βλ1 e[−λ1 (τl,k −τl,k−1 )] + (β − 1)λ2 e[−λ2 (τl,k −τl,k−1 )] , k > 0 (3.29) where p (τl,k |τl,k−1 ) is the conditional probability density function conditioning consecutive ray arrival times within a cluster. By deﬁnition τl,0 = 0. β is the mixture probability, while λ1 and λ2 are the ray arrival rates. For some environments, like industrial areas, a “dense” arrival of multipath components is observed, i.e., each resolvable delay bin contains signiﬁcant energy. In that case, the concept of ray arrival rates loses its meaning, and a realization of the CIR based on a tapped delay line model with regular tap spacings is to be used. Next, the cluster powers and cluster shapes are determined. The average power delay proﬁle (APDP), which is the mean power of the diﬀerent paths, is exponential within each cluster 2 E{|αl,k |} = El 1 e−τl,k /γl , γl [(1 − β)λ1 + βλ2 + 1] (3.30) where El is the integrated energy of the lth cluster, and γl is the intra-cluster decay time constant. The cluster decay rates are found to depend linearly on the arrival time of the cluster γl ∝ kγ Tl + γ0 , (3.31) where kγ describes the increase of the decay constant with delay and γ0 is a constant. The mean energy (normalized to γl ), of the lth cluster follows in general an exponential decay ) ( 10 log(El ) = 10 log e−Tl /Γ + Xcl , (3.32) where Xcl is a normally distributed variable with standard deviation σcl . Γ is the cluster decay factor, and is typically around 10-30 ns, while widely diﬀering values (between 1 and 60 ns) have been reported for the intra-cluster constant γl . Chapter 3. Channel Characterization 43 For the NLOS case of some environments (oﬃce and industrial), the shape of the PDP can be diﬀerent, namely 2 E{|α1,k |} = (1 − χe−τl,k /γr )e(−τl,k /γ1 ) E1 γ1 + γr . γ1 γ1 + γr (1 − χ) (3.33) Here, the parameter χ describes the attenuation of the ﬁrst component, the parameter γr determines how fast the APDP increases to its local maximum, and γ1 determines the decay at late times. 3. Small-scale fading The distribution density of the small-scale amplitudes is Nakagami 2 ( m )m 2m−1 − mx2 x e Ω , px (x) = Γ(m) Ω (3.34) where m ≥ 1/2 is the is the Nakagami m-factor, Γ(m) is the gamma function, and Ω is the mean-square value of the amplitude. A conversion to a Rice distribution is approximately possible with the conversion equations [40] (K + 1)2 m= 2K + 1 (3.35) and √ K= (m2 − m) √ , m − (m2 − m) (3.36) where K and m are the Ricean factor and Nakagami-m factor, respectively. The parameter Ω corresponds to the mean power, and its delay dependence is thus given by the APDP above. The m-parameter is modelled as a lognormally distributed random variable, whose logarithm has a mean µm (τ ) and standard deviation σm . Both of these can have a delay dependence µm (τ ) = m0 − km τ σm (τ ) = m̂0 − k̂m τ. (3.37) For the ﬁrst component of each cluster, the Nakagami factor is modelled diﬀerently. It is assumed to be deterministic and independent of delay m = m̃0 . (3.38) A realization of the CIR of the IEEE 802.15.4a Channel Model for indoor oﬃce environment with NLOS scenario is illustrated in Fig. 3.3 (a). It can be observed a certain delay before receiving the ﬁrst path with signiﬁcant energy. Fig. 3.3 (b) depicts 100 CIR realizations with random ﬁrst path arrival delays. The channel parameters are shown in Tab. 3.2. The APDP, averaged over 100 channel realizations, is shown in Fig. 3.4. Chapter 3. Channel Characterization 44 1.5 0.6 0.5 1 Multipath ray amplitudes Multipath ray amplitudes 0.4 0.3 0.2 0.1 0 0.5 0 −0.1 −0.5 −0.2 −0.3 −0.4 0 20 40 60 80 Multipath channel delays [ns] 100 120 140 −1 0 20 40 60 80 100 Multipath channel delays [ns] 120 140 160 (b) (a) Figure 3.3: Channel impulse response from the IEEE 802.15.4a Channel Model (a) a single realization; (b) multiple realizations. Path loss n σS P L0 Power delay proﬁle L̄ Λ [ns−1 ] λ1 , λ2 [rays/ns], β Γ[ns] kγ , γ0 [ns] σcl [dB] Small-scale fading m0 [dB], km m̂0 , k̂m χ γr , γ1 Oﬃce LOS NLOS 1.63 1.9 36.6 3.07 3.9 51.4 5.4 0.016 0.19, 2.97, 0.0184 14.6 0, 6.4 3 3.1 0.19 0.11, 2.09, 0.0096 19.8 0, 11.2 3 0.42, 0 0.31, 0 0.5, 0 0.25, 0 0.78 15.21, 11.84 Table 3.2: Channel parameters for indoor oﬃce environment. The diﬀerent channel realizations are studied independently in terms of their RMS delay spreads and the number of signiﬁcant paths they contain with more than 85% of total energy, as shown in Fig. 3.5 and Fig. 3.6, respectively. The average RMS delay Chapter 3. Channel Characterization 45 0 −10 Average power (dB) −20 −30 −40 −50 −60 0 20 40 60 80 100 Multipath channel delay [ns] 120 140 160 Figure 3.4: Average power delay proﬁle. spread is τ̄rms = 12 ns. The average number of signiﬁcant paths capturing more than 85% of total energy is 40.5. 25 Average RMS delay RMS delays RMS delay [ns] 20 15 10 5 0 10 20 30 40 50 60 Channel realizations 70 80 90 100 Figure 3.5: RMS delay spreads for diﬀerent CIR realizations. Chapter 3. Channel Characterization 46 80 Average number of significant paths Significant paths per realization Number of significant paths capturing > 85% energy 70 60 50 40 30 20 10 0 0 10 20 30 40 50 60 Channel realizations 70 80 90 100 Figure 3.6: Number of signiﬁcant paths. 3.4 Summary This chapter presented ﬁrst a generic mobile radio channel, identifying the components inﬂuencing its description. The CIR and the APDP were expressed analytically. The channel parameters characterizing this channel model are: • P0 - the normalized received power • K - the Ricean K-factor • τrms - the RMS delay spread • ρ2 - the normalized power of the direct ray • Π - the normalized power density of the constant-level part • tc - the duration of the constant level part • γ - the decay exponent of the exponentially decaying part. Further, standardized channels models are introduced, especially the IEEE 802.15.4a channel model. The statistical modelling of path loss, path inter-arrival times, large-scale and small-scale fadings was presented, as well as the determination of the cluster powers and cluster shapes. The considered parameters characterizing the IEEE 802.15.4a channel model are: • P L0 - the path loss at 1m distance Chapter 3. Channel Characterization 47 • n - the path loss exponent • σS - the shadowing standard deviation • Aant - the normalized power of the direct ray • L̄ - the mean number of clusters • Λ - the inter-cluster arrival rate • λ1 , λ2 , β - the ray arrival rates (mixed Poisson model parameters) • Γ - the inter-cluster decay constant • kγ , γ0 - the intra-cluster decay time constant parameters • σcl - the cluster shadowing variance • m0 , km - the Nakagami-m factor mean • m̂0 , k̂m - the Nakagami-m factor variance • m̃0 - the Nakagami-m factor for strong components • γ - the decay exponent of the exponentially • γr , γ1 , χ - parameters for alternative PDP shape The standardized IEEE 802.15.4a UWB channel model, as it is claimed, match better the measurements, making it a more realistic channel model. However, the log-normal distribution and the shadowing factor render this model less tractable for theoretical performance analysis and quantiﬁcation of the channel-induced diversity. For this reason the channel model presented in section 3.2 will be used whenever performance analysis is concerned. For most other simulations in this thesis the standardized model will be used. Chapter 4 Transmitted Reference Systems 4.1 Introduction The key motivations for using UWB communications systems are the ability of UWB signals to ﬁnely resolve multipath components, as well as the availability of technology to implement and generate UWB signals with relatively low complexity. Non-coherent receivers with no channel estimation have been proposed in [41, 42] to make the UWB technology attractive for applications where low cost and low power consumption are playing an important role, as in the case of WSNs. Compared with coherent receivers that are facing great design challenges, non-coherent autocorrelation receivers (AcR) represent an interesting solution for realizing low complexity UWB systems. They can also detect data with synchronization inaccuracy as large as one frame duration [43]. In AcR front-ends for UWB communications systems, the received signal consisting of a train of pulses is delayed and correlated with itself. Hence, each information-conveying pulse is coupled with an unmodulated pulse. Generally, the ﬁrst pulse is known as the “reference pulse” and the second pulse is the “data pulse”. The reference pulse acts as a template for the AcR. Data can be applied, for instance, by diﬀerentially modulating the polarity of the data pulse with respect to the reference pulse. The delay line in an AcR front-end, also known as “pulse-pair ” correlator, is matched to the lag between the reference and the data pulses. The overall system is called a Transmitted-Reference (TR) AcR [41], [43] - [49]. Generally, in AcRs the reference pulse is corrupted by noise and interference, which is an inherent disadvantage. If the data rate of such systems is increased, interference among multiple pulses becomes one of the most fundamental deteriorating eﬀects, due to multipath propagation. This is known as Inter-Frame Interference (IFI) if interference between multiple pulses of one data symbol is referred to. It is termed Inter-Symbol Interference (ISI) if interference among consecutive data symbols is considered. The TR-UWB systems considered, in this thesis, are the Diﬀerential TransmittedReference(DTR) UWB system [50], the Low-Duty-Cycle (LDC) DTR-UWB system [51] and the Dual-Pulse (DP) TR-UWB system [52]. Discrete-time equivalent system models for these TR-UWB systems are derived in [15], [51] and [52]. In this chapter, their delay-dependence is studied. The delay-dependent system models accurately relate the 48 Chapter 4. Transmitted Reference Systems 49 transmitted data bits di and the timing oﬀset τ to the test statistics at the decision device ẑ[i]. The data model can be written as ẑ[i] = zτ [i] + zτ ν [i]. (4.1) The decision variable ẑ[i] comprises a data and delay dependent term zτ [i] and a noise term zτ ν [i]. Using the data model and the concept of minimum distance receiver, the BER of lowcomplexity TR-UWB systems was derived, where ISI is minimal or absent. 4.2 4.2.1 TR-UWB Signal and Systems Models TR-UWB Transmission Schemes Narrowband signals (i.e. sinusoidal and quasi-sinusoidal signals) have the unique property of keeping their sinusoidal shape during forms of signal conversions such as addition, subtraction, diﬀerentiation and integration. Shape is deﬁned here as the law of change of a signal in time. The waveforms of sinusoidal and quasi-sinusoidal signals keep a shape identical to that of the original function and may diﬀer only in their amplitude and time shift, or phase. On the contrary, the UWB signal has a nonsinusoidal waveform that can change shape while processing the above speciﬁed transformations. When a UWB signal is generated and transmitted to the antenna in a form of a current pulse, the ﬁrst change of the UWB signal shape occurs during pulse radiation, since the intensity of the radiated electromagnetic ﬁeld varies proportionally with the derivative of the antenna current. The second change occurs during signal propagation through the multipath channel. The third change of the shape occurs during signal reception, for the same reason as for signal radiation. Thus, when a Gaussian pulse is transmitted, the second derivative of the Gaussian pulse is expected at the UWB receiver front-end. Hence, the standard pulse-shape assumed in theoretical work is the second derivative of a Gaussian pulse (Gaussian doublet), given by [ ] 2 ω(t) = 1 − 4π(t/tm )2 e−2π(t/tm ) . (4.2) This pulse is conﬁned in time to an interval [0,Tω ). A DTR-UWB system sends data as a stream of very narrow pulses. Let us assume each data symbol is transmitted via Nf consecutive pulses/frames, where i is the symbol index. A known random sequence bj ∈ {−1, +1} is diﬀerentially modulated on the time-hopped pulses, where j ∈ {0, 1, ..., Nf − 1} is the pulse index within a symbol. The diﬀerentially modulated pulse-polarities are obtained as ai,j+1 = ai,j bj di and ai+1,0 = ai,Nf −1 bNf −1 di , where di is the ith data symbol, di ∈ {−1, +1}. The transmitted signal is written as s(t) = f −1 ∞ N ∑ ∑ i=0 j=0 ai,j ω̃(t − ti,j ), (4.3) Chapter 4. Transmitted Reference Systems 50 where ω̃(t) is the transmitted pulse shape and ti,j = (j + iNf )Tf + cj = Cj + iTs . Tf is the average spacing between two pulses, i.e., the average frame duration, cj is the known TH sequence and Cj = cj + jTf . {Cj } are the relative pulse timings within a symbol. Ts is the symbol duration. The time shifts between consecutive pulses are given as Dj = ti,j+1 − ti,j . Since d2i = 1, every second pulse of a symbol is not modulated by data. Each pulse is reused as a reference and data pulse to increase the power eﬃciency. Indeed, the diﬀerential TR scheme transmits half the number of pulses that would transmit a conventional TR scheme. Let si (t) be the ith transmitted symbol waveform containing the Nf pulses, i.e., Nf −1 si (t) = ∑ ai,j ω̃(t − iTs − jTf − cj ). (4.4) j=0 The transmitted signal is visualized in Fig. 4.1(a). symbol i+1 symbol i s(t): timing: D0 D1 j=0 ... D2 2 1 3 DNf-1 4 N f-1 D0 0 D1 1 ... D2 2 3 4 t modulation dib1 dib0 dib2 dibNf-1 ... di+1b0 di+1b1 di+1b2 ... (a) r(t): l=0 1 2 3 ... 4 t r(t+D2 ): l'=0' 1' 2' 3' ... 4' t TI Integration interval of correlator 3 (b) Figure 4.1: Diagram of the transmitted and received signals in DTR-UWB. The burst-oriented LDC DTR-UWB system is fundamentally similar to a conventional DTR-UWB system. The diﬀerence is that the random sequence bj is of length Nf − 1. This is because the polarity of the ﬁrst pulse within a symbol is no more dictated by the polarity of the last pulse of the previous symbol. Hence, the diﬀerentially modulated pulse-polarities are obtained as ai,j+1 = ai,j bj di , j ∈ {0, 1, ..., Nf −2} and ai+1,0 ∈ {−1, +1}. The LDC DTR-UWB system, as conceived in [51], was designed for eﬃcient narrowband Chapter 4. Transmitted Reference Systems 51 interference (NBI) mitigation and robustness in multiuser scenarios. Pulses are transmitted in short bursts, separated by comparably longer idle periods, making up a “low-duty-cycle” as shown in Fig. 4.2. symbol i, duty cycle timing: D0 D1 j=0 modulation dib0 ... D2 2 1 dib1 symbol i+1 symbol i, idle D0 Nf 3 di+1 b0 ... D2 1 0 ... dib2 D1 2 di+1 b1 3 di+1 b2 t ... Figure 4.2: Diagram of transmitted signals for the burst oriented LDC DTR-UWB systems. For a DP TR-UWB system, the unmodulated data pulse is followed at a delay D by the data-modulated pulse, as illustrated in Fig. 4.3. The timing of pulse j = 0 is c0 = 0 and for j = 1 it is c1 = D. 1 2 0 D r(t) 1' Tmax t 2' r(t-D) t TI Figure 4.3: Received signal for a DP TR-UWB system. 4.2.2 TR-UWB Receiver Front-End ∑ Let g̃(t) = h(t) ∗ ω̃(t) = ∞ n=0 αn ω̃(t − τn ) be the overall channel capturing both pulse shaping by linear system components and multipath eﬀects. h(t) is a realization of a simulated CIR, generated according to the channel models presented in sections 3.2 and 3.3. It can also be a measured channel impulse response. The received noisy signal is expressed as Chapter 4. Transmitted Reference Systems 52 r(t) Filter frx(t) y0[i] xb0 yj[i] xbj yNf-1[i] xbNf-1 I&D TI Delay D0 Delay c0+D0 I&D z[i] TI Delay Dj Delay jTf+cj+Dj I&D TI Delay DSP: MUD, sync., etc. DNf-1 Delay (Nf-1)Tf+cNf-1+DNf-1 Symb. clk. 1/Ts Figure 4.4: Receiver front-end for a DTR-UWB system. r(t) = f −1 ∞ N ∑ ∑ ai,j g̃(t − iTs − jTf − cj − τ ) + n(t), (4.5) i=0 j=0 where Ts is the symbol period, τ ∈ [0, Ts ) is the timing delay and n(t) represents noise. A conventional DTR-UWB receiver front-end is shown in Fig. 4.4. It has at its input a bank of pulse-pair correlators, whose lags are matched to the time shifts Dj . While a conventional DTR-UWB receiver has Ncr = Nf pulse-pair correlators, a LDC DTR-UWB receiver has Ncr = Nf − 1 pulse-pair correlators. In a DP TR-UWB receiver, a single correlator (Ncr = 1) is used for the detection of the pulse-pairs. Integration is performed over a time interval TI , which can be selected to optimize the signal-to-noise ratio (SNR) of the AcR outputs [49, 53, 54]. The outputs of the Integrate-and-Dump (I&D) blocks, after being sampled at the symbol rate, are coherently combined by removing the chip-level modulation bj . We obtain Chapter 4. Transmitted Reference Systems 53 Nf −1 z[i] = ∑ yj [i] bj , (4.6) j=0 where ∫ ti,j +TI yj [i] = r̂(t)r̂(t + Dj )dt, (4.7) ti,j and r̂(t) = r(t) ∗ frx (t) = rτ (t) + ν(t), (4.8) where rτ (t) is the delay-dependent noise-free received signal. frx (t) denotes a front-end analog ﬁlter in the receiver. The ﬁltered noise process ν(t) = n(t) ∗ frx (t) is characterized by its autocorrelation function N0 (4.9) frx (κ) ∗ frx (−κ), 2 where N0 /2 is the double-sided power spectral density of the receiver front-end noise. Rν (κ) = E{ν(t)ν(t + κ)} = Using (4.7) and (4.8), we obtain ∫ t +T ∫ t +T yj [i] = ti,ji,j I rτ (t)rτ (t + Dj )dt+ ti,ji,j I ν(t)rτ (t + Dj )dt ∫ t +T ∫ t +T + ti,ji,j I rτ (t)ν(t + Dj )dt + ti,ji,j I ν(t)ν(t + Dj )dt = yτ j [i] + ντ 1j [i] + ντ 2j [i] + ν3j [i]. 4.3 (4.10) Equivalent TR-UWB Systems Models Among all the TR-UWB systems presented in section 4.2, only the DTR-UWB system is prone to ISI in the case the average frame duration Tf is less than the maximum delay spread of the multipath channel Tm . In this thesis, the LDC DTR-UWB and DP TR-UWB are chosen in such a way that they mitigate ISI eﬀects. 4.3.1 System with ISI: Volterra Equivalent Model In [15] a discrete-time Volterra equivalent system model is derived for DTR-UWB systems in a multipath channel, where both IFI and ISI are present. In this subsection, a review of the system modelling is presented, introducing the delay parameter. Equivalent System Model without Noise The noise-free delay-dependent received signal can be written as Chapter 4. Transmitted Reference Systems p −1 ∞ N ∑ ∑ rτ (t) = ai,l g(t − ti,l − τ ), 54 (4.11) i=−∞ l=0 where g(t) = g̃(t) ∗ frx (t) is the ﬁltered overall channel. yτ j [i] is expressed as N∑ f −1 ∞ ∑ yτ j [i] = ai+n,l g(t − ti+n,l − τ ) n=−∞ l=0 N∑ f −1 ∞ ∑ · = ai+n′ ,l′ g(t n′ =−∞ l′ =0 N∑ f −1 ∑ f −1 1 N∑ 1 ∑ n=−η l=0 n′ =−η l′ =0 − ti+n′ ,l′ − τ )dt (4.12) ai+n,l ai+n′ ,l′ ·Ig (ti,j − ti+n,l − τ, ti,j − ti+n,l − τ + TI ; Dj + ti+n,l + cl − cl′ ), where Ig (t1 , t2 ; ζ) = ∫t2 g(t)g(t + ζ)dt. Equation (4.12) accounts for the interference from t1 all pulses of η previous and one consecutive symbol. η = ⌈Tm /Ts ⌉ is the memory length, where ⌈·⌉ represents the ceiling function deﬁned by ⌈x⌉ = min {n ∈ Z|x ≤ n}. This IFI was depicted in Fig. 4.1(b). Equation (4.12) can be written as yτ j [i] = aT [i]Yτ j a[i], (4.13) where a[i] = [ai,0 , ai,1 , . . . , ai,Nf −1 , ai,Nf ]T represents the polarities of all pulses. The elements of the Np (η + 2) × Np (η + 2) matrix Yτ j are given by [Yτ j ](n+η)Np +l+1,(n′ +η)Np +l′ +1 = Ig (cj − nTs − cl − τ, cj − nTs − cl − τ + TI ; Dj + (n − n′ )Ts + cl − cl′ ) (4.14) ∀ n,n′ ∈ {−η, −η + 1, . . . , 1} and l,l′ ∈ {0, 1, . . . , Np − 1}. N∑ f −1 The noise-free signal term zτ [i] = yτ j [i]bj can be modelled as j=0 Nf −1 zτ [i] = ∑ bj aT [i]Yτ j a[i] =aT [i]Zτ a[i], (4.15) j=0 where Zτ = N∑ f −1 bj Yτ j . It is demonstrated in [15] that (4.15) constitutes a discrete-time j=0 Volterra model of order two, having the transmitted symbols {di } at its input. The data and code dependent vector a[i] is expressed as a[i] = ai−η,0 B(p + Pd[i]), (4.16) Chapter 4. Transmitted Reference Systems 55 where d[i] = [di−η , di−η+1 , . . . , di , di+1 ]T , B = diag[1, b0 , b0 b1 , . . .], [B]k,k = k−2 ∏ bµ mod Nf , µ=0 k = 1, 2, . . . , Nf . P = Iη+2 ⊗ s is a block-diagonal matrix of dimension Nf (η + 2) × (η + 2), with the length Nf selection vectors s = [0, 1, 0, 1, . . . , 1]T at its main diagonal. ⊗ denotes the Kronecker product. The length Nf (η + 2) × (η + 2) vector p = iη+2 ⊗ (iNf − s) contain ones at the elements of a[i] that are not data dependent. IN and iN denote an identity matrix of size N × N and an all-ones vector of length N , respectively. Using (4.15) and (4.16), a second-order Volterra model is obtained zτ [i] = (p + Pd[i])T BZτ B(p + Pd[i]) (4.17) = hτ 0 + hTτ1 d[i] T + d [i]Hτ 2 d[i], where hτ 0 = pT BZτ Bp, hτ 1 = PT B(Zτ + ZTτ )Bp and Hτ 2 = PT BZτ BP. The bias term hτ 0 is due to the interference among the (channel’s response to) ﬁxed reference pulses plus interference between equally modulated data pulses. The linear model coeﬃcient hτ 1 relates to the interference between unmodulated reference and modulated data pulses. The nonlinear product terms Hτ 2 are attributed to interference between diﬀerently modulated pulses of any two diﬀerent data symbols. Thus, nonlinear product terms appear only if signiﬁcant ISI is present. Multiuser Equivalent System Model In a multiuser (MU) equivalent system model, nonlinear terms also exist if data symbols of various users overlap [15]. An equivalent system model can be derived for the MU case by including the interfering users’ pulse polarities in the a[i]i -vectors and extending Yj matrices with all channels’ auto- and cross-correlation values for any pulse-pairs. The MU Volterra model comprises d[i]-vectors extended by the other users’ data and correspondingly extended coeﬃcient vectors and matrices. Equivalent Noise Model The three noise terms in (4.10) are considered for the noise analysis. We encounter two “linear ” terms ντ 1j [i] and ντ 2j [i] which depend on the received signal, and a “product” noise term ν3j [i]. Following [15] and [55], all three noise components at the output of the correlators are assumed to be zero-mean Gaussian noise variables. This requires that the noise process ν(t) is zero mean, that Rν (Dj ) ≈ 0, ∀j, and that the integration-time by noise-bandwidth product TI W ≫ 1. The noise component of the decision variable z[i] is deﬁned by zτ ν [i] = ẑ[i]−zτ [i]. Using (4.6) and (4.10), we obtain Chapter 4. Transmitted Reference Systems E{zτ ν [i]zτ ν [i + ι]} = N∑ f −1 N∑ f −1 j=0 56 bj bj ′ [E{ντ 1j [i]ντ 1j ′ [i + ι]} j=0 +E{ντ 1j [i]ντ 2j ′ [i + ι]} + E{ντ 2j [i]ντ 1j ′ [i + ι]} (4.18) +E{ντ 2j [i]ντ 2j ′ [i + ι]} + E{ν3j [i]ν3j ′ [i + ι]}], since ν3j [i] is uncorrelated from ντ 1j ′ [i] and ντ 2j ′ [i], ∀ j, j ′ . The delay and data-dependent covariance of the linear noise terms is expressed by quadratic forms N0 T (νm νm′ ) a[i], a [i]Vτ ι,j,j (4.19) E{ντ mj [i]ντ m′ j ′ [i + ι]} = ′ 2 for m, m′ ∈ {1, 2}. It is zero for |ι| > 1 [15]. We derived the expression of the matrix (νm νm′ ) Vτ ι,j,j as ′ (ν ν ) m m′ [Vτ ι,j,j ](n+η)Nf +l+1,(n′ +η)Nf +l′ +1 ′ = · 2 N0 ti,j∫+TI g(t + (−1)m+1 Dj − ti+n,l − τ ) (4.20) ti,j ti+ι,j ′ −t ∫ ti+ι,j ′ −t ′ g(t + κ + (−1)m +1 Dj ′ − ti+n′ ,l′ − τ )Rn (κ)dκdt. These expressions can be evaluated numerically if the channel response g(t) is known. For the noise product we obtain E{ν3j [i]ν3j ′ [i + ι]} ≈ δ[ι]δ[j − j ′ ]TI ≈ δ[ι]δ[j − ∫∞ −∞ N2 j ′ ]TI 40 Rn2 (κ)dκ ∫∞ −∞ (4.21) |Frx (f )|4 df, where Frx (f ) is the Fourier transform of the receiver front-end. Finally, the noise model for the decision variable can be expressed by E{zτ ν [i]zτ ν [i + ι]} = where Wτ ι = N2 = Nf TI ∫∞ −∞ ∑ ∑ m,m′ j,j ′ (z z ) N0 T N2 a [i]Wτ ι a[i] + δ[ι] 0 N2 , 2 4 (4.22) m m′ bj bj ′ Vτ ι,j,j , ι = 0 (noise variance) or ι = 1 (noise covariance), and ′ |Frx (f )|4 df . Chapter 4. Transmitted Reference Systems 57 Model Illustration The equivalent system model is depicted in Fig. 4.5, where the upper branch with the “Volterra System I” represents the data model, while the lower branch introduces the data dependency of the noise variance, described by the “Volterra System II”. delay data d[i] Volterra System I ẑ [i] z [i] z [i] Channel h(t) System parameters Volterra System II sqrt N0 / 2 (N0 )2N2 / 4 Unit variance Gaussian noise Noise density N0 Figure 4.5: 2nd order Volterra equivalent system model of the DTR-UWB system. Model Validation We select the pulse ω(t) as the second derivative of the Gaussian pulse with unit energy and duration Tω = 0.7 ns. The pulse parameter τm = 0.2877. The pulse is sampled using the sampling period Tsamp = 0.05 ns. The precise position of the pulses in each channel impulse response has been oversampled in order to represent better the unquantized ray arrival times. The pulse position resolution is Tres = 0.005 ns. The DTR-UWB system’s key parameters are Nf = 10, Tf = 10 ns, Ts = 100 ns, η = 1 and TI = 20 ns. Random time hopping (TH) codes were used. All the system parameters are summarized in Tab. 4.1 DTR-UWB system parameters Tω [ns] 0.7 τm [ns] 0.2877 Tsamp [ns] 0.05 Tf [ns] 10 TI [ns] 20 Ts [ns] 100 Nf Ncr 10 10 Table 4.1: DTR-UWB system and simulation parameters A NLOS channel has been simulated with an exponentially decaying delay proﬁle at RMS delay spread of τrms = 10 ns using the channel model presented in section 3.2. The channel simulator generates an unquantized vector of ray arrival times according to a Poisson process with a mean ray arrival rate of λ = 5 ns−1 and a vector of Rayleigh distributed ray magnitudes with random polarity. The SNR is deﬁned as the ratio between the symbol energy and the noise variance, i.e., SNR= Es /σ 2 = Nf Eb /σ 2 , where Eb = +∞ ∫ 2 g (t)dt and σ 2 is the noise variance. Es is the symbol energy. −∞ Chapter 4. Transmitted Reference Systems 58 The validation of the delay-dependent equivalent system and noise models is shown in Fig. 4.6 and Fig. 4.7. The data dependence of the models is ﬁrst studied, followed by the delay dependence. (a) 15 10 zτ[i] 5 0 −5 −10 −15 10 Volterra model conv. sim. w/o noise data sequence 15 20 25 30 35 40 (b) 1.2 E{zτ ν[i]zτ ν[i+ι]} 1 0.8 noise var.; Volterra m. 0.6 noise var.; conv. sim. 0.4 noise cov.; Volterra m. 0.2 0 −0.2 10 15 20 25 Symbol index i 30 35 40 Figure 4.6: Simulated receiver output compared with the output of the second-order Volterra equivalent system. (a) Noise-free case (τ = 0). (b) Data dependence of the noise (co)-variance at SNR= 30 dB (τ = 0). We compare the noise-free receiver output, simulated by processing the received signal according to the elements of the receiver front-end shown in Fig. 4.4 (conventional simulations) with the output of the second-order Volterra equivalent system. In Fig. 4.6(a) the sampled output of the conventional simulation is marked by “◦”. Plus characters “+” show the output of the equivalent system model at these sampling instants. Perfect agreement between these two demonstrates the suitability of the proposed equivalent system model. Fig. 4.6(b) compares the variance of the additive noise var{zτ ν [i]} at the same set of observed decision variables, for a conventional simulation “◦” and for the noise model in (4.22) “+”. In the conventional simulation runs, one of 1000 independent noise processes has been added to the receiver front-end at a time, while using a constant data sequence and channel realization. The simulation results show a signiﬁcant data-dependence of the noise variance, whereas the noise covariance, illustrated by asteriks “∗” is insigniﬁcant. After the data dependence, the delay-dependence is illustrated. Fig. 4.7 shows the simulated receiver output compared with the output of the second-order Volterra equivalent system model zτ [i], with and without delay. It can be seen that zτ [i] decreases with increase in time delay, as expected. In fact, zτ [i] decays exponentially with respect to τ as is shown in Fig. 4.8, where E{zτ [i]/Es } is obtained by averaging zτ [i]/Es over 1000 channel realizations, for di = 1 and a constant data sequence. On the other hand, under the same simulation Chapter 4. Transmitted Reference Systems 59 (a) τ = 0 ns 10 zτ[i] 5 0 −5 −10 10 15 20 25 τ = 15 ns 30 35 (b) 10 Conv. sim. w/o noise Full Volterra model 5 zτ[i] 40 0 −5 −10 10 15 20 25 Symbol index [i] 30 35 40 Figure 4.7: Simulated receiver output compared with the output of the second-order Volterra equivalent system. (a) Noise-free case without delay. (b) Noise-free case with delay τ = 15 ns. conditions, it can be observed that E{zτ ν [i]} and E{zτ2ν [i]} do not exhibit such signiﬁcant delay-dependence, as shown in Fig. 4.9. 0.9 0.8 0.7 E{zτ[i]/Es} 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 delay [ns] 30 35 40 45 50 Figure 4.8: Normalized average delay-dependent noise-free decision variable Chapter 4. Transmitted Reference Systems 60 (a) 1 E{zτ ν[i]} 0.5 0 −0.5 −1 0 2 4 6 8 12 14 16 18 20 12 14 16 18 20 (b) 1.5 E{zτ2 ν[i] } 10 delay [ns] 1 0.5 0 0 2 4 6 8 10 delay [ns] Figure 4.9: (a) Average output noise versus delay (SNR= 30 dB). (b) Variance of output noise versus delay (SNR= 30 dB). 4.3.2 Systems without ISI In this subsection, low complexity TR-UWB systems without ISI are addressed. DTR-UWB Systems For DTR-UWB systems with no ISI the memory length η = 0. The noise-free delay dependent equivalent system model zτ [i] and the variance of the noise variable zτ ν [i] for ISI-free DTR-UWB systems are expressed as and zτ [i] = hτ 0 + hτ 1 di , (4.23) N0 N02 var{zτ ν [i]} = (hτ ν,0 + hτ ν,1 di ) + Nf TI W, 2 4 (4.24) where hτ 0 = pT BZτ Bp + trace(qT BZτ Bq) (4.25) T hτ 1 = p B(Zτ + ZT τ )Bq. Let b be a vector containing the code sequence {bj }. In the diagonal matrix B, the spreading code b is represented in the form of an accumulated product, B=diag[1 b0 b0 b1 b0 b1 b2 . . .]. q is a vector of length Nf , which in this case is identical to the selection vector s = ∑ cr −1 bj Yτ j , [0, 1, 0, 1, . . . , 1]T . The length Nf vector p is deﬁned as p = iNf − s. Zτ = N j=0 Chapter 4. Transmitted Reference Systems 61 where Yτ j is a Ncr × Ncr matrix containing the autocorrelation integrals Ig (t1 , t2 ; ζ). Yτ j is expressed as [Yτ j ]l+1,l′ +1 = Ig (ti,j − ti,l − τ, ti,j − ti,l + TI − τ ; Dj + ti,l − ti,l′ ), (4.26) ∫ ∞ ∀ l, l′ ∈ {0, 1, . . . , Ncr − 1}. W = −∞ |Frx (f )|4 df is the equivalent input bandwidth of the receiver, where Frx (f ) is the Fourier transform of the front-end ﬁlter frx (t). The coeﬃcients of the noise model, hτ ν,0 and hτ ν,1 (4.24), are obtained in a similar way like hτ 0 and hτ 1 in (4.25), replacing the matrix Zτ by a matrix Wτ = ∑∑ (m,m′ ) bj bj ′ Vτ,j,j ′ , (4.27) m,m′ j,j ′ where (m,m) [Vτ,j,j ′ ]l+1,l′ +1 = 2 N0 · ti,j∫+TI g(t + (−1)(m+1) Dj − ti,l − τ ) ti,j ti+,j ′ −t ∫ (4.28) (m+1) g(t + κ + (−1) ti,j ′ −t Dj ′ − ti,l′ − τ )Rν (κ)dκdt, for m, m′ ∈ {1, 2}. DP TR-UWB Sytems The equivalent system and noise models for DP TR-UWB systems have been derived in [52]. The expressions of the noise-free delay dependent equivalent system model zτ [i] and variance of the noise variable zτ ν [i] for DP TR-UWB systems are the same as in (4.23) and (4.24), respectively. For these systems the coeﬃcients hτ 0 , hτ 1 , are given as hτ 0 = Ig (−τ, TI − τ ; D) + Ig (−D − τ, TI − D − τ ; D) hτ 1 = Ig (−τ, TI − τ ; 0) + Ig (−D − τ, TI − D − τ ; 2D). (4.29) For D ≥ TI , hτ ν,0 and hτ ν,1 are expressed as hτ ν,0 ≈ 2Ig (−τ, TI − τ ; 0) = 2hτ 1 hτ ν,1 ≈ 2Ig (−τ, TI − τ ; D) = 2hτ 0 . 4.4 (4.30) Bit Error Rate Performance Analysis In this section, the receiver statistics are analyzed from two perspectives; both perspectives allow for BER performance analysis to be performed. The ﬁrst (widely accepted) perspective is to use a Gaussian approximation on the noise terms in the receiver statistics and determine their respective variances. The second approach is to use the equivalent system models of the receiver. The average BER is veriﬁed through simulation for diﬀerent SNR Chapter 4. Transmitted Reference Systems r(t) aij aij+1 aij+2 62 aij+3 Dj ... ... aij r(t-D j ) aij+1 aij+2 aij+3 ... ... aij+1 a ij r(t) r(t-D j ) ... TI Figure 4.10: Diagram of the received signals in DTR-UWB (without IFI). values. It is also compared to the average BER derived using the equivalent system model. Since the derivation of an analytic expression for the average BER would need to solve a complicated integral, we adopt another approach for computing the average BER, namely using the concept of minimum distance receiver [56]. 4.4.1 Gaussian Approximations approach As we presented in [57], the goal is to derive the receiver decision statistic in the absence of any kind of interference. Fig. 4.1(b) shows the correlator output for a duration of TI in the presence of IFI and Fig. 4.10 shows the correlator output in the absence of noise and IFI. Equation (4.7) can be rewritten as ∫ ti,j +TI yj [i] = [ai,j g(t − ti,j ) + ν(t)][ai,j+1 g(t − ti,j+1 ) + ν(t + Dj )]dt. (4.31) ti,j Since ti,j+1 = Dj + ti,j , (4.31) can be expressed as ∫ ti,j +TI yj [i] = [ai,j g(t − ti,j ) + ν(t)][ai,j+1 g(t − ti,j − Dj ) + ν(t + Dj )]dt. (4.32) ti,j Making the appropriate change of variable in (4.32) we ﬁnally obtain yj [i] = = ∫ TI 0 ∫ TI 0 ∫ TI 0 [ai,j g(t) + ν(t + ti,j )][ai,j+1 g(t − Dj ) + ν(t + ti,j + Dj )]dt ai,j+1 ai,j g(t)g(t − Dj )dt+ ∫ TI 0 ai,j+1 g(t − Dj )ν(t + ti,j )dt+ = y1j [i] + y2j [i] + y3j [i] + y4j [i]. ai,j g(t)ν(t + ti,j + Dj )dt+ ∫ TI 0 (4.33) ν(t + ti,j )ν(t + ti,j + Dj )dt. Chapter 4. Transmitted Reference Systems 63 Equation (4.6) can be rewritten as z[i] = N∑ f −1 bj y1j + j=0 N∑ f −1 + N∑ f −1 bj y2j + N∑ f −1 j=0 bj y3j j=0 (4.34) bj y4j j=0 = z1 [i] + z2 [i] + z3 [i] + z4 [i]. In (4.34) z1 [i] is the signal term, while z2 [i], z3 [i] and z4 [i] are all noise terms. Let Tm be the channel maximum excess delay. Since there is no IFI, Dj > Tm ≫ Tω , then {ν(t + ti,j } and {ν(t + ti,j + Dj } can be regarded as uncorrelated and thus independent zero mean Gaussian random variables. The validity of this assumption is illustrated by Fig. 4.11, where the noise autocorrelation function is simulated for time delays in the interval [−Tm , Tm ]. Hence z4 [i] can be approximated as a zero mean random variable. g(t) is zero mean for t > Tω [50], thus z2 [i] and z3 [i] are independent zero mean Gaussian random variables. Finally, all the three noise terms in (4.34) can be approximated as independent zero-mean random variables whose variances, given by V2 , V3 and V4 , are equal to the second moments of z2 [i], z3 [i] and z4 [i], respectively. Normalized noise autocorrelation function 1 0.8 0.6 0.4 0.2 0 −0.2 −100 −50 0 Time delay [ns] 50 100 Figure 4.11: Simulated noise autocorrelation function. Analysis of z1 [i] In subsection 4.2.1 the diﬀerentially modulated pulse-polarities were expressed as ai,j+1 = ai,j bj di , which implies that ai,j+1 ai,j = bj di and Chapter 4. Transmitted Reference Systems z1 [i] = di N∑ f −1 ∫ j=0 TI 0 g(t − Dj )g(t)dt = di z1 . 64 (4.35) z1 is actually a random variable, considering several realizations of g(t). Since the channel is quasi-static, g(t−Dj ) is the exact replica of g(t) delayed by Dj . Thus, z1 can be expressed as ∫ TI z1 = Nf g 2 (t)dt. (4.36) 0 Analysis of z2 [i] z2 [i] = N∑ f −1 ∫ j=0 Let z2 [i] = N∑ f −1 I2j , then V2 = j=0 ∫ 2 E(I2j ) N∑ f −1 ai,j bj g(t)ν(t + ti,j + Dj )dt. (4.37) 2 E(I2j ), where j=0 ∫ TI TI 0 TI E{ai,j bj g(t)ν(t + ti,j + Dj )ai,j bj g(ϱ)ν(ϱ + ti,j + Dj )}dtdϱ. = (4.38) 0 0 Considering that g(t) and ν(t + ti,j + Dj ) are uncorrelated, we can split up the expectation term in two separate terms. ∫ 2 E(I2j ) TI ∫ TI = E{g(t)g(ϱ)}E{ν(t + ti,j + Dj )ν(ϱ + ti,j + Dj )}dtdϱ. 0 (4.39) 0 Equation (4.39) can be rewritten as ∫ 2 E(I2j ) TI ∫ TI −t = 0 −t E{g(t)g(t + κ)}E{ν(t + ti,j + Dj )ν(t + κ + ti,j + Dj )}dtdκ, (4.40) where ϱ = t + κ. E{g(t)g(t + κ)} ≈ ρ2 ω(t)ω(t + κ) + P̃h (t)ϕω (κ) [50], with P̃h (t) = Πe−γt ∫ Tω and ϕω (κ) = 0 ω(µ)ω(µ + κ)dµ is the autocorrelation of the received prototype pulse ω(t). We deﬁne ϕω (0) = 1, i.e., the prototype pulse is assumed to have unit energy. E{ν(t + ti,j + Dj )ν(t + κ + ti,j + Dj )} = Rν (κ) is the noise autocorrelation function. If we assume that the received signal is passed through an ideal bandpass ﬁlter (BPF) with one-side bandwidth W and center frequency f0 , where W equals the bandwidth of the received signal, then Rν (κ) = W N0 sin(πW κ) cos(2πf0 κ), πW κ (4.41) Chapter 4. Transmitted Reference Systems 65 where N0 is the noise power. The normalized prototype pulse and noise autocorrelation functions are shown in Fig. 4.12 and Fig. 4.13, respectively. Normalized autocorrelation function φω 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 Time delay [ns] 0.4 0.6 0.8 Figure 4.12: Autocorrelation function of the received prototype pulse. Normalized noise autocorrelation function Rν 1 0.8 0.6 0.4 0.2 0 −0.2 −30 −20 −10 0 Time delay [ns] 10 20 30 Figure 4.13: Noise autocorrelation function Rν (κ). Since ρ2 = 0 (NLOS channel), ∫ 2 E(I2j ) = 0 TI −γt Πe ∫ Tω dt −Tω ϕω (κ)Rν (κ)dκ. (4.42) Chapter 4. Transmitted Reference Systems 66 Finally, V2 = = N∑ f −1 Π γ ( 1 − e−γTI j=0 ΠNf (1−e−γTI ) γ )∫ Tω −Tω ∫ Tω −Tω ϕω (κ)Rν (κ)dκ (4.43) ϕω (κ)Rν (κ)dκ. Analysis of z3 [i] Since bj ai,j+1 = ai,j di , z3 [i] = di N∑ f −1 ∫ TI 0 j=0 N∑ f −1 Let z3 [i] = di I3j then V3 = j=0 N∑ f −1 ai,j g(t − Dj )ν(t + ti,j )dt. (4.44) 2 2 E(I3j ), where E(I3j ) is determined the same way j=0 2 as E(I2j ). E ( 2 I3j ) ∫ TI ∫ TI = E{g(t − Dj )g(ν − Dj )}E{n(t + ti,j )n(ν + ti,j )}dtdν. (4.45) 0 0 2 Using the same arguments as the ones used to derive E(I2j ), we have E{g(t−Dj )g(ϱ−Dj )} = E{g(t)g(ϱ)}, and ﬁnally V3 = = Π γ N∑ f −1 ( 1 − e−γTI j=0 ΠNf (1−e−γTI ) γ ∫ Tω −Tω ) ∫ Tω −Tω ϕω (κ)Rν (κ)dκ (4.46) ϕω (κ)Rν (κ)dκ. Analysis of z4 [i] V4 is derived as follows: V4 = N∑ f −1 ∫ TI 0 ∫ TI 0 E{ν(t + ti,j )n(ϱ + ti,j )}E{ν(t + ti,j + Dj )ν(ϱ + ti,j + Dj )}dtdϱ ∫ T ∫ T −t = Nf 0 I −tI Rν2 (κ)dtdκ ∫T = Nf TI −TI I Rν2 (κ)dκ. j=0 (4.47) Bit Error Probability The conditional bit error probability is expressed as Chapter 4. Transmitted Reference Systems 67 √ z12 P {e|z1 }DT R = Q V2 + V3 + V4 (4.48) √ ∫∞ 2 where Q(x) = 1/ 2π x e−t /2 dt . P {e|z1 }DT R is conditioned by the channel realization. 4.4.2 System Modelling Approach Using the equivalent system and noise models derived in subsections 4.3.2, the conditional BER can also be determined as ) ( (zτ [i]−hτ 0 )di Pz {e|zτ [i]} = Q √ 2 E{zn [i]} ( = Q (4.49) ) √ hτ 12 E{zn [i]} , where hτ 0 is the bias term and hτ 1 is the linear model coeﬃcient. 4.4.3 Derivation of Average BER The BER of our TR-UWB systems can be obtained by averaging over several conditional BER values. This average BER can be expressed as ∫∞ P {e|x}fz1 (x)dx, P̄e,T R = (4.50) 0 where fz1 (·) is the probability density function (pdf) of z1 . The derivation of an analytic expression for P̄e,T R by solving the integral in (4.50) is complex. Instead, we propose to determine the average BER P̄e for low complexity TR-UWB systems, with minimal ISI (η = 1) or no ISI (η = 0), using the concept of minimum distance receiver. Probability Density Function of the Equivalent System Model zτ [i] The equivalent system model zτ [i] is a random process as it depends on the channel realization and data. It is assumed, in this thesis, that zτ [i] is a Gaussian random process. This assumption is justiﬁed through simulation as shown in Fig. 4.14, which is obtained after 1000 channel simulation runs, for positive data. The histogram depicts the probability of occurrence of the diﬀerent values of zτ [i], whereas the solid line is obtained by plotting a Gaussian probability density function with mean E{zτ [i]} and variance E{zτ2 [i]} − (E{zτ [i]})2 . Chapter 4. Transmitted Reference Systems Probability of occurence Probability of occurence 0.25 68 0.5 p(zτ[i]) p(zτ[i]) 0.45 τ =τ12 = 0nsns τ = 0 ns 0.2 0.4 0.35 0.15 P(zτ[i]=a) P(zτ[i]=a) 0.3 0.1 0.25 0.2 0.15 0.05 0.1 0.05 0 −10 0 10 a 20 30 0 −10 0 10 a 20 30 Figure 4.14: Probability density function of zτ [i]. Formulation as a Hypothesis Testing Problem Let us assume a binary detector with the following hypothesis testing problem for detecting zτ [i] H−1 : ẑ[i] = s−1τ [i] + zτ ν [i] (4.51) H1 : ẑ[i] = s1τ [i] + zτ ν [i], where s−1τ [i] and s1τ are respectively the equivalent system model for negative and positive data transmitted at time instant i. s−1τ [i] and s1τ are expressed as s1τ [i] = hτ 0 + hTτ1 d1 [i] + d1 [i]Hτ 2 d1 [i] s−1τ [i] = hτ 0 + hTτ1 d−1 [i] + d−1 [i]Hτ 2 d−1 [i] d1 [i] = [di−η , di−η+1 , . . . , di−1 , 1, di+1 ]T d−1 [i] = [di−η , di−η+1 , . . . , di−1 , −1, di+1 ]T . (4.52) Low complexity TR-UWB systems with minimal ISI (η = 1) are ﬁrst considered. Since ∀i, di ∈ {−1, 1}, s1τ and s−1τ can each have 2η+1 possible values skjτ [i], where k ∈ {−1, 1} and j = 1, 2, . . . , 2η+1 . Thus, at the symbol rate, one of the 2η+2 variables skjτ [i] is transmitted. This is a classical signal classiﬁcation problem, which motivates the use of a minimum distance receiver. Since zτ [i] is a Gaussian random process, the skjτ [i] are also Gaussian. Moreover they 2 are independent of each other. At time i let we decide Hkj for which Dkj = (ẑ[i] − skjτ [i])2 is minimum. Because ẑ 2 [i] is the same ∀(i, j), we decide Hjk for which Tkj (ẑ[i]) = −2ẑ[i]skjτ [i]+s2kj [i] is minimum, i.e., Tkj (ẑ[i]) = ẑ[i]skjτ [i] − 21 s2kjτ [i] is maximum. Since Hk is decided when Chapter 4. Transmitted Reference Systems 69 one of the Hkj is decided, and Hkj are independent elements of Hk , p(ẑ[i]|Hk ) = η+1 2∑ p(ẑ[i]|Hkj ). (4.53) j=1 This motivates us to decide Hk for which Tk (ẑ[i]) = η+1 2∑ 1 (ẑ[i]skjτ [i] − s2kjτ [i]) 2 j=1 (4.54) is maximum. Thus, H1 is decided if T1 (ẑ[i]) > T−1 (ẑ[i]). Let T (ẑ[i]) = T1 (ẑ[i]) − T−1 (ẑ[i]), T (ẑ[i]) = η+1 2∑ ( j=1 ) 1 ẑ[i](s1jτ [i] − s−1jτ [i]) − (s21jτ [i] − s2−1jτ [i]) . 2 (4.55) Following the condition that T (ẑ[i]) > 0, we have T ′ (ẑ[i]) = η+1 2∑ ẑ[i](s1jτ [i] − s−1jτ [i]) j=1 > − 21 η+1 2∑ (4.56) (s21jτ [i] − j=1 s2−1jτ [i]) = γτ . Equation (4.56) shows that the assumed detector consists of a test statistics T ′ (ẑ[i]) and a threshold γτ . For the case of ISI-free TR-UWB systems, we have s1τ [i] = hτ 0 + hTτ1 s−1τ [i] = hτ 0 − hTτ1 . (4.57) Th (ẑ[i]) = ẑ[i](s1τ [i] − s−1τ [i]) > − 21 (s21τ [i] − s2−1τ [i]) = ζτ . (4.58) H1 is decided if After simpliﬁcation we obtain Th (ẑ[i]) = ẑ[i] ζτ = hτ 0 . (4.59) Analysis of the Test Statistics T ′ (ẑ[i]) and Th (ẑ[i]) are Gaussian random variables conditioned on either hypothesis. We already considered zτ ν in subsection 4.3.1 to be zero-mean Gaussian. Using this approxi- Chapter 4. Transmitted Reference Systems 70 mation we express analytically the moments of T ′ (ẑ[i]) and Th (ẑ[i]). E {T ′ |H−1 } = E{ = E{ η+1 2∑ (s−1jτ [i] + zτ ν [i])(s1jτ [i] − s−1jτ [i])} j=1 η+1 2∑ s−1jτ [i](s1jτ [i] j=1 η+1 2∑ − s−1jτ [i])} (4.60) zτ ν [i](s1jτ [i] − s−1jτ [i])} +E{ j=1 = E{s−1τ (s1τ − s−1τ )T } = ε−1τ , where skτ = [sk1τ [i], sk2τ [i], . . . , skN τ [i]] and N = 2η+1 . E {Th |H−1 } = E{s−1τ }. var(T ′ |H−1 ) = var(s−1τ (s1τ − s−1τ )T ) + var(zτ ν [i] (4.61) η+1 2∑ (s1jτ [i] − s−1jτ [i])) j=1 = 2 σ−1τ + (4.62) στ2sν , where η+1 2∑ (s1jτ [i] − s−1jτ [i]))2 } × E{zτ2ν }. (4.63) var(Th |H−1 ) = var(zτ ) + E{zτ2ν } = στ2 + στ2ν . (4.64) στ2sν = E{( j=1 Following the same steps as above, we derive { } E {T ′ |H1 } = E s1τ (s1τ − s−1τ )T = ε1τ . (4.65) E {Th |H1 } = E{s1τ }. (4.66) 2 var(T ′ |H1 ) = σ1τ + στ2sν , (4.67) 2 = var(s1τ (s1τ − s−1τ )T ). where σ1τ var(Th |H1 ) = στ2 + στ2ν . (4.68) Bit Error Probability The probability of error P̄e is deﬁned as P̄e = P (H−1 |H1 )P (H1 ) + P (H1 |H−1 )P (H−1 ), (4.69) Chapter 4. Transmitted Reference Systems 71 where P (H1 |H−1 ) is the conditional probability of deciding H1 given that H−1 is true. Assuming that the prior probabilities are equal, i.e., P (H1 ) = P (H−1 ) = 21 we have P̄e = = 1 [P (H−1 |H1 ) 2 1 [Pr (T ′ (ẑ[i]) 2 P̄e = = 1 [1 2 + P (H1 |H−1 )] < γτ |H1 ) + Pr (T ′ (ẑ[i]) > γτ |H−1 )] (4.70) − Q( √γτ2−ε1τ2 ) + Q( √γτ2−ε−1τ2 )] σ−1τ +στ sν σ1τ +στ sν 1 [Q( √ε1τ2 −γτ2 2 σ +σ τ sν 1τ ) + Q( √γτ2−ε−1τ2 )], (4.71) σ−1τ +στ sν √ ∫∞ 2 2 2 where Q(x) = 1/ 2π x e−t /2 dt and Q(−x) = 1 − Q(x). Since σ1τ ≈ σ−1τ , Pe can be expressed as [ ( )] ε1τ − ε−1τ 1 P̄e = Q √ 2 . (4.72) 2 σkτ + στ2sν For the non-ISI case, we obtain 1 P̄e = [Pr (Th (ẑ[i]) < ζτ |H1 ) + Pr (Th (ẑ[i]) > ζτ |H−1 )], 2 (4.73) and ﬁnally P̄e = = 4.4.4 1 2 [ ( )] E{s1τ −s−1τ } √ 2 2 Q στ +στ ν [ ( )] 2E{hτ 1 } 1 Q √ 2 2 . 2 (4.74) στ +στ ν Simulation Results The channel model is the same as in subsection 4.3.1. The diﬀerent TR-UWB systems considered are shown in Table 4.2 with their simulation parameters. The bit error performance of the diﬀerent considered TR-UWB systems, in the presence and absence of timing delay, is demonstrated in Fig. 4.15-Fig. 4.18. The BER curves are presented as functions of SNR and delay (τ = {0 ns,20 ns}). The solid lines denote the simulated average BER obtained using Monte Carlo simulation, the dash-dot lines denote the average BER obtained from (4.72) or (4.74), and the dotted line in Fig. 4.15 denotes the average BER computed numerically by averaging P {e|z1 } in (4.48). There is a perfect matching between both BER curves at τ = 0 ns, and good agreement at higher delay values for the ISI-free DTR-UWB system (η = 0)). Next, the bit error performance of DTR-UWB system with minimal ISI is illustrated in Fig. 4.16, in the presence and absence of timing delay (τ = 20 ns and τ = 0 ns, respectively). It can be observed, for both values of τ , that the solid lines (denoting the simulated average Chapter 4. Transmitted Reference Systems TR-UWB systems ⇒ Parameters ⇓ Tω [ns] τm [ns] Tsamp [ns] Tf [ns] TI [ns] Ts [ns] Nf Ncr Data rates [Mb/s] DTR UWB η = 1/η = 0 0.7 0.2877 0.05 10/100 20 100/1000 10 10 10/1 72 LDC DTR-UWB DP TR-UWB 0.7 0.2877 0.05 2 20 100 9 8 10 0.7 0.2877 0.05 20 20 200 2 1 5 Table 4.2: TR-UWB systems and simulation parameters. DTR−UWB no IFI (η = 0), τ = 0 ns 0 10 Avg P{e|z1} Avg Pe sim. Avg. BER −1 Average BER 10 −2 10 −3 10 −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 4.15: BER performance of the DTR-UWB system (η =0) versus SNR. BER obtained using Monte Carlo simulation) and the dash-dot lines (denoting the average BER obtained from (4.72)) are in good agreement. Finally, the ISI-free systems LDC DTR-UWB and DP TR-UWB are considered. Fig. 4.17 and Fig. 4.18 show the bit error performance of both systems, in the presence and absence of timing delay. It can again be observed, for both values of τ , that the solid lines (denoting the simulated average BER obtained using Monte Carlo simulation) and the dash-dot lines (denoting the average BER obtained from (4.74)) are in good agreement. Chapter 4. Transmitted Reference Systems 73 DTR−UWB (η = 1) 0 10 −1 Average BER 10 −2 10 sim. Avg. BER (τ = 0 ns) −3 10 sim. Avg. Pe (τ = 0 ns) sim. Avg. BER (τ = 20 ns) sim. Avg. Pe (τ = 20 ns) −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 4.16: BER performance of the DTR-UWB system (η =1) versus SNR. LDC DTR−UWB 0 10 −1 Average BER 10 −2 10 sim. Avg. BER ( τ = 0 ns) sim. Avg. Pe (τ = 0 ns) −3 10 sim. Avg. Pe (τ = 20 ns) sim. Avg. BER ( τ = 20 ns) 0 5 10 15 SNR [dB] 20 25 30 Figure 4.17: BER performance of the LDC DTR-UWB system versus SNR. 4.5 Optimization of the Integration Time Interval TI From (4.36) it is clear that by increasing TI more energy is collected. Simultaneously, more noise is also collected as the sum of the noise variances V2 + V3 + V4 is increased. Chapter 4. Transmitted Reference Systems 74 DP TR−UWB 0 10 −1 Average BER 10 −2 10 sim. Avg. Pe (τ = 0 ns) sim. Avg. BER ( τ = 0 ns) −3 10 sim. Avg. Pe (τ = 20 ns) sim. Avg. BER ( τ = 20 ns) 0 5 10 15 SNR [dB] 20 25 30 Figure 4.18: BER performance of the DP TR-UWB system versus SNR. 90 80 70 Average f(T I) 60 50 40 30 20 10 0 10 20 30 40 50 TI [ns] 60 70 80 Figure 4.19: Optimal choice for TI . 90 100 Chapter 4. Transmitted Reference Systems 75 Equation (4.48) can be rewritten as P {e|z1 }DT R v u u = Q t 2ΠN = Q )2 ( ∫T Nf 0 I g 2 (t)dt −γTI f 1−e γ ( ) ∫ Tω −Tω ϕω (κ)Rn (κ)dκ+Nf TI ∫ TI −TI 2 (κ)dκ Rn (4.75) (√ ) f (TI ) . The optimal choice of TI for optimal BER can be found by optimizing the function TI 7→ f (TI ), which is out of scope of this thesis. Instead, the optimal choice for TI is made graphically by plotting f (TI ) versus TI , as shown in Fig. 4.19, where the function TI 7→ f (TI ) is averaged over 1000 channel realizations. This motivates the choice for TI = 20 ns in all the simulations. 4.6 Summary Signal and systems models for diﬀerent TR-UWB systems were presented in this chapter. The diﬀerent TR-UWB transmission schemes were introduced, followed by the presentation of the receiver front-end. Delay-dependent discrete time equivalent system and noise models for diﬀerent TR-UWB systems were also reviewed in this chapter for TR-UWB systems with and without ISI. Further, the diﬀerent receiver statistics are analyzed from two perspectives: • Use of Gaussian approximation on the noise terms in the receiver statistics. • Use of the equivalent system models of the receiver. Both perspectives allow for BER performance analysis to be performed. The average BER is veriﬁed through simulation for diﬀerent SNR values. It is also compared to the average BER derived using the equivalent system models. The derivation of an analytic expression for the average BER was provided using the concept of minimum distance receiver. Finally, the optimization of the integration time interval TI was dealt with. Chapter 5 Synchronization of Transmitted Reference Systems 5.1 Introduction The large bandwidth of UWB technology besides providing the possibility to transmit at a very high data rate also provides very accurate temporal and spatial information that can be used for precise timing oﬀset estimation. This high accuracy also allows new perspectives in terms of synchronization, positioning and tracking [9, 58]. Synchronization is a particulary acute problem faced by UWB systems due to the fact that they employ low-power ultra-short (sub-nanoseconds) pulses. Thus, timing requirements are stringent because minor misalignments may result in a lack of energy capture which renders symbol detection impossible [59]. Synchronization is typically performed in two stages: the ﬁrst stage achieves coarse synchronization to within a reasonable amount of accuracy, and the second stage is responsible for achieving ﬁne synchronization (timing oﬀset estimation). Synchronization schemes for UWB systems in the literature can be roughly classiﬁed into detection-based schemes [60] - [65] and estimation-based schemes [66] - [70]. The synchronization methods which employ a detection-based approach evaluate a candidate delay by ﬁrst obtaining a measure of correlation between the received signal and a locally generated template signal oﬀset by the candidate delay. This measure of correlation is then compared to a threshold in order to make a decision. The candidates could be evaluated in a serial, parallel or hybrid manner. For threshold based synchronization schemes in multipath fading channels, it was shown in [64] that, no matter how large the SNR is or how the threshold is chosen, it may not be possible to make the probabilities of detection and false alarm arbitrarily large and small. Moreover, the large search space in UWB signal acquisition poses signiﬁcant challenges in the design and implementation of practical systems for high data rate applications. In the estimation-based methods, an estimate of the delay is obtained through maximizing a statistic over a set of candidate delays. This statistic is usually obtained from the correlation of the received signal with a template. These schemes do not involve a threshold comparison. They rather exploit the cyclostationarity inherent in UWB signaling due to pulse repetition. Most of these schemes are based on the Nyquist or higher sampling 76 Chapter 5. Synchronization of Transmitted Reference Systems 77 rate of the signal and may not be feasible for low cost UWB devices. To avoid the high sampling rate and computational complexity associated with channel estimation and time synchronization, there has been a renewed interest in the TR systems for impulse radio due to their simple receiver structure. In this thesis, a novel data-aided timing acquisition technique for frame-level synchronization of DTR-UWB systems is suggested [71]. It is based on incorporating parallel I&D circuits within pulse-pair correlator branches to improve considerably the energy capture in the presence of timing oﬀset. Since this technique performs perfectly well when the noise variance is much less than the signal energy, symbol-long segments of the received signal are averaged before timing acquisition. A training sequence is chosen to ensure that pulse polarities within consecutive symbol-long segments of the received signal do not get cancelled through the averaging operation. The good performance of the proposed timing acquisition technique is conﬁrmed through simulation results. Furthermore, a simple timing oﬀset estimator for low complexity TR-UWB systems, assuming no ISI, is considered. Both the classical and bayesian approaches are used for estimating the timing oﬀset. It is shown, in this chapter, that both approaches yield the same timing oﬀset estimate. The proposed algorithm uses energy collected at the symbol rate, thus reducing considerably the implementation complexity. If coarse synchronization is achieved (e.g. [71]) and hardware clock timing error is known, timing oﬀset estimation problems are analogical to ranging problems, and the proposed algorithm can readily be used for distance measurement. The theoretical performance limit of such a timing oﬀset estimation is illustrated through the derived Cramér-Rao Lower Bound (CRLB) for the variance of the timing oﬀset estimates. 5.2 Data-Aided Timing Acquisition The DTR-UWB receiver illustrated in Fig. 4.4 has at its input a bank of pulse-pair correlators, whose lags are matched to the time shifts Dj . The I&D blocks are triggered at the arrival-times of the respective pulses, which requires time-synchronization and knowledge of the TH-sequence. 5.2.1 Timing Acquisition Technique In this section, we perform a parallel search of the timing oﬀset by incorporating parallel I&D circuits within pulse-pair correlator branches and using the outputs of these I&D blocks to determine the frame index that matches the timing oﬀset. It is worth noting that, in practice, the data obtained at the output of the I&D circuits are actually acquired serially since the parallel I&D circuits within each pulse-pair correlator branch are actually fractional parts of a single I&D circuit (fractional sampling). The parallel structure shown in Fig. 5.1 is used to help understand our timing acquisition technique. This novel timing acquisition technique relies on judiciously choosing the diﬀerent triggering times of the I&D circuits and the time interval TI over which the integration operations are performed, as shown in Fig. 5.1. Each pulse-pair correlator branch with correlator lag Dj is associated with Nf parallel I&D blocks which are triggered at diﬀerent times than the respective Chapter 5. Synchronization of Transmitted Reference Systems Integrate-and-Dump blocks ( triggering times) I&D ( t 0,0 +n 0 ) r ' (t ) I&D ( t 0,0 +n 1 ) X ... D0 I&D ( t 0,0 +n Nf-1 ) I&D ( t 0,1 +n 0 ) X I&D ( t 0,Nf-1 +n 1 ) I&D (t 0,Nf-1 +n Nf-1 ) ( )² y0,Nf-1 ( )² y1,0 ( )² Y0 + y1,1 Y1 + ( )² Timing offset estimate y1,Nf-1 + ( )² YN f-1 yNf-1,0 ( )² yNf-1,1 ( )² . . . DNf-1 ... X y0,1 . . . . . . I&D ( t 0,Nf-1 +n 0 ) ( )² . . . I&D ( t 0,1 +n Nf-1 ) y0,0 ... ... D1 I&D ( t 0,1 +n 1 ) 78 yNf-1,Nf-1 ( )² Figure 5.1: Illustration of the timing acquisition technique pulses arrival times. Without loss of generality, the synchronization to symbol i = 0 is considered. Thus, the diﬀerent pulse arrival times are given by t′j,k = t0,j + nk , (5.1) where nk = kTf for k ∈ {0, 1, ..., Nf −1 }. Our timing acquisition technique, through the parallel I&D operations, aims to achieve eﬃcient multipath energy collection even in presence of timing oﬀset and noise. The accuracy of timing acquisition is enhanced by reducing the noise level, i.e., by averaging over diﬀerent symbol-long segments of the received signal r(t). The shortcomings of such averaging stem from the fact that the diﬀerentially modulated pulse polarities (±1) are diﬀerent from one symbol to another. Thus, averaging could result on the cancellation of some pulses within a symbol waveform and therefore altering the energy collection within some frames. This is the reason why we use a training sequence, given by d′p = (−1)p , where p ∈ {0, 1, ..., P − 1} and P is the length of the training sequence. This sequence ensures that pulse polarities within the P/2 ﬁrst odd numbered symbol-long segments of r(t) are identical, as well as pulse polarities within the P/2 ﬁrst even numbered symbol- Chapter 5. Synchronization of Transmitted Reference Systems 79 long segments of r(t). Thus, consecutive segments of r(t) of duration 2Ts are summed together before we apply our timing acquisition technique. Indeed,( let us assume )the mth segment of the received signal r(t) of duration 2Ts is rm (t) = r t + 2(m − 1)Ts for t ∈ [0, 2Ts ), and rm (t) = 0 elsewhere. Then the waveform used for timing acquisition is r′ (t) = 2 P P/2 ∑ (5.2) rm (t). m=1 The outputs of the Nf × Nf I&D blocks are given by ∫ t0,j +nk +TI ′ yj,k = t0,j r (t + Dj )r′ (t)dt. +nk Let Yk = N∑ f −1 j=0 2 yj,k . Fig. 5.2 shows the magnitude of the Yk for diﬀerent timing oﬀsets. 5 τ = 10 ns 0 Y0 5 0 Y0 5 0 Y0 Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 5 Y9 τ = 20 ns Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 τ = 30 ns Y1 Y2 Y3 Y4 Y5 Y6 Y7 5 Y8 Y9 τ = 40 ns 0 Y0 5 Y1 0 Y0 Y1 (5.3) Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y3 Y4 Y5 Y6 Y7 Y8 0 Y0 5 Y9 Y9 τ = 60 ns Y1 0 Y0 5 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 τ = 70 ns Y1 τ = 80 ns 0 Y0 5 τ = 50 ns Y2 0 Y0 5 Y1 τ = 90 ns Y1 τ = 0 ns 0 Y0 Y1 Figure 5.2: Illustration of the outputs Yk for diﬀerent timing oﬀsets The ﬁrst step towards estimating the timing oﬀset is achieved by ﬁnding the index n given by: n = arg max {Yk } . k (5.4) Finally, the coarse estimate of τ is given as τ̃ = nTf , (5.5) where n ∈ {0, 1, . . . , Nf −1 }. 5.2.2 Simulation Results In this subsection, the simulation results are presented. The timing acquisition performance is discussed by using the following performance metrics: the normalized acquisition esti- Chapter 5. Synchronization of Transmitted Reference Systems 80 { } mation mean square error (MSE) given by E (τ̃ − τ )2 /Ts2 , the probability of detecting correctly the timing oﬀset PCD and the average bit-error-rate (BER). Fig. 5.3 depicts the MSE of our timing acquisition technique plotted versus SNR for values of M = P/2 = 16, 32, 64, 128. As M increases, the normalized MSE decreases monotonically. The same trend can be observed for increasing SNR. For all the M values, the MSE curves ﬂatten at high SNR towards a certain error ﬂoor. This is due to the frame resolution limit imposed by the coarse estimation. This frame resolution limit can be removed through tracking. 0 10 M=128 M=64 M=32 M=16 −1 Normalized MSE 10 −2 10 −3 10 −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 5.3: Normalized timing acquisition MSE vs. SNR for M = {16, 32, 64, 128}. This error ﬂoor is illustrated in Fig. 5.4 for diﬀerent timing oﬀsets τ = {41 ns, 42 ns, 45 ns, 48 ns, 49 ns} where the expected estimated timing oﬀsets are given respectively as τ̃ = {40 ns, 40 ns, 40 ns, 50 ns, 50 ns}. From the frame resolution limit it is clear that the best performance of our timing acquisition technique is achieved when the timing oﬀset to be estimated is an integer multiple of the frame duration. Thus, we have tested the performance of our timing acquisition technique in terms of its probability of detecting correctly the timing oﬀset PCD , when the timing oﬀset is an integer multiple of the frame duration. This is shown in Fig. 5.5 for values of M = 16, 32, 64, 128. It can be observed that the PCD increases monotonically as M increases. The same trend is observed for increasing SNR. With M = 128, we achieve PCD = 1 from SNR = 10 dB. Finally, the BER performance of our timing acquisition technique is tested for values of M = 16, 128. This BER performance is compared to the cases without timing acquisition, and with perfect timing. It can be observed from Fig. 5.6 that, as M increases, the BER performance improves monotonically. It is also observed that there is no signiﬁcant performance loss when we compare the BER performance of our timing acquisition technique to Chapter 5. Synchronization of Transmitted Reference Systems M = 64 0 10 τ=41 ns τ=42 ns τ=45 ns τ=48 ns τ=49 ns −1 10 Normalized MSE 81 −2 10 −3 10 −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 5.4: Normalized timing acquisition MSE vs. SNR for τ = {41 ns, 42 ns, 45 ns, 48 ns, 49 ns}. τ = 40 ns Probability of detection (PCD) 1 0.8 M=128 M=64 M=32 0.6 M=16 0.4 0.2 0 −5 0 5 10 SNR [dB] 15 20 25 30 Figure 5.5: Probability of detection vs. SNR for M = {16, 32, 64, 128}. the case where there is perfect timing. Indeed, the performance loss is less than 1 dB for M = 128 and around 1.2 dB when M = 16. This is due to a combination of the robustness Chapter 5. Synchronization of Transmitted Reference Systems 82 of the DTR-UWB systems against timing oﬀset and the good performance of our timing acquisition technique. 0 10 −1 Average BER 10 −2 10 −3 10 Perfect timing With timing acquisition (M=128) With timing acquisition (M=16) Without timing acquisition −4 10 0 5 10 15 SNR[dB] 20 25 30 Figure 5.6: Average BER vs. SNR for M = {16, 168 }. 5.3 Timing Oﬀset Estimation The received signal with time delay τ , expressed in (4.5), will start within a window of duration Td that represent the uncertainty on the arrival time of the received signal. This time window is allocated in the time line by a previous coarse acquisition process, presented in the previous section. In this section, two approaches to timing oﬀset estimation are presented, using the statistical analysis of the delay-dependent equivalent system and noise models presented in section 4.3 for low complexity TR-UWB systems, and the analysis in subsection 4.4.3. 5.3.1 Normalization of the decision variable Let zs [i] = ẑ[i]/Es be the normalized decision variable. This normalization is necessary to minimize the channel dependence of the decision variable. We obtain zs [i] = zeτ [i] + zeτ ν [i]. (5.6) where zeτ [i] = zτ [i]/Es and zeτ ν [i] = zτ ν [i]/Es . At a time instant i, zeτ [i] is a Gaussian ran2 . The data-dependence of zeτ [i] impose dom variable with mean E{zeτ [i]} and variance σze a prior knowledge of the data sequence for eﬃcient characterization of zeτ . This implies the Chapter 5. Synchronization of Transmitted Reference Systems 83 need of a training sequence in order to obtain an eﬃcient timing oﬀset estimation. When ﬁtted with an exponential function, E{zeτ } can be expressed as E{zeτ } = A1 exp(Bτ ), di = 1 E{zeτ } = A2 exp(Bτ ), di = −1. (5.7) Generally, A2 is slightly diﬀerent from −A1 , due to the bias term hτ 0 . Hence a time instant i the realizations of zeτ [i]|di are asymmetric. Nevertheless, for most of the systems considered in this work, hτ 0 ≈ 0, since the interference among ﬁxed reference pulses plus interference between equally modulated data pulses is reduced by allowing larger average pulse separations. Hence, we can write E{zeτ [i]} = Adi exp(Bτ ). (5.8) B takes approximately the same value for all TR-UWB systems considered in this work. The value of A is proportional to the ratio Ncr /Nf . 5.3.2 Classical Approach to Timing Oﬀset Estimation Let us consider zeτ to be a deterministic but unknown function of τ . We assume that the normalized output energy of the TR-UWB systems, collected at the symbol rate, is composed of the equivalent system model zeτ with zero mean gaussian modelling uncer2 tainty plus system noise zeτ ν . The variance of the modelling error is given by σze and 2 var(zeτ ν ) = σνe . In [15] it is indicated that the noise zτ ν can be considered as a zero-mean Gaussian process. Thus, we have ] [ P −1 ∑ 1 1 p(zs ; τ ) = exp − (zs [i] − Adi exp(Bτ ))2 , 2 + σ 2 )]P/2 2 + σ2 ) [2π(σze 2(σ ze νe νe i=0 (5.9) where P indicate the number of data symbols in the training sequence with the same polarity. Taking the logarithm of (5.9) and ﬁnding its derivative w.r.t τ , we obtain ) (P −1 ∑ ABdi exp(Bτ ) zs [i]−P Adi exp(Bτ ) ∂ ln p(z; τ ) i=0 = . (5.10) 2 + σ2 ) ∂τ (σze νe p(z;τ ) Solving ∂ ln∂τ = 0 for τ , we obtain the maximum likelihood (ML) estimate of the timing oﬀset as (P −1 / ) ∑ 1 zs [i] (P Adi ) τ̂ = B ln (5.11) ( i=0) = B1 ln |z̄As | , Chapter 5. Synchronization of Transmitted Reference Systems where z̄s = (1/P ) P∑ −1 84 zs [i], and |z̄s | = z̄s /di = zs di . i=0 The Cramér-Rao Lower Bound (CRLB) for the variance of the timing oﬀset estimates is derived by ﬁnding the derivative of (5.10) w.r.t τ [72]. P AB 2 exp(Bτ ) [|z̄s | − 2A exp(Bτ )] ∂ 2 ln p(z; τ ) . = 2 + σ2 ) ∂τ 2 (σze νe { 2 } ) Finally, CRLB(τ̂ ) = −E ∂ ln∂τp(z;τ is expressed as 2 2 2 (σze + σνe ) . 2 P AB exp(Bτ ) [2A exp(Bτ ) − |z̄s |] CRLB(τ̂ ) = 5.3.3 (5.12) (5.13) Bayesian Approach to Timing Oﬀset Estimation zeτ [i] is now considered to be a stochastic process with pdf ] [ 1 1 2 p(zeτ [i]) = √ exp − 2 (zeτ − Adi exp(Bτ )) . 2 2σze 2πσze (5.14) The pdf of zs conditioned on zeτ is given by ] [ P −1 1 1 ∑ p(zs |zeτ ) = (zs [i] − zeτ )2 . exp − 2 2 )P/2 (2πσνe 2σνe i=0 (5.15) In the Appendix III, we have proven that zeτ |zs is a Gaussian random variable with variance σz2eτ |zs = and mean ( mzeτ |zs = 1 P/ 2 + 1/ 2 σνe σze P z̄s Adi exp(Bτ ) + 2 2 σνe σze )( , (5.16) P 1 + 2 2 σνe σze )−1 . (5.17) Equation (5.17) can be written as mzeτ |zs = α + β exp(Bτ ), (5.18) where ( α= β= ( P z̄s 2 σνe Adi 2 σze )( )( P 2 σνe + 1 2 σze P 2 σνe + 1 2 σze )−1 )−1 (5.19) . The timing oﬀset estimate τ̂ is derived from (5.18) through functional mapping, by miniP∑ −1 mizing the least square (LS) error criterion J(τ ) = [zs [i] − (α + β exp(Bτ ))]2 . Diﬀereni=0 Chapter 5. Synchronization of Transmitted Reference Systems 85 tiating J(τ ) w.r.t. τ , we obtain ∂J(τ ) = 2P Bβ exp(Bτ )[α + β exp(Bτ ) − z̄s ]. ∂τ Setting ∂J(τ ) ∂τ = 0 we obtain 1 τ̂ = ln B Since z̄s − α = (5.20) 2 z̄s σνe 2 +σ 2 ) (P σze νe and β = ( ) z̄s − α . β (5.21) 2 Adi σνe 2 +σ 2 ) , (P σze νe (5.21) can be expressed as (P −1 / ) ∑ 1 τ̂ = B ln zs [i] (P Adi ) ( i=0) = B1 ln |z̄As | , (5.22) which is equal to the ML estimate of τ in (5.11). 5.3.4 Determination of Es , A and B Estimating the symbol energy An estimation of the symbol energy at the receiver front-end can be obtained by integrating the received signal over a reasonably long time interval as follows KN s −1 ∑ Ês = rk2 − σ 2 KNs k=0 , (5.23) K where K is the number of symbols within the time interval used for symbol energy estimation and Ns = Ts /Tsamp is the number of samples per symbol. rk = r(kTsamp ) is the k th sample of r(t). Determining A and B The mathematical procedure for ﬁnding the best-ﬁtting curve to a given set of points by minimizing the sum of the squares of the oﬀsets (“the residuals”) of the points from the curve is known as least squares ﬁtting [73] [74]. A and B can be determined oﬄine through least squares exponential ﬁtting of the average of the normalized equivalent system model zeτ over several channel realizations, as is shown in Appendix IV. For distance measurement applications, A and B are tuning parameters that can be determined during the calibration phase for making accurate measurements. 5.3.5 Simulation Results In this subsection, the simulation results are presented. The performance of our dataaided timing oﬀset estimator is demonstrated for diﬀerent TR-UWB systems by using the Chapter 5. Synchronization of Transmitted Reference Systems 86 following performance metrics: the normalized estimation mean √square error (MSE) given by mse= E{(τ̂ − τ )2 }/Ts2 , the root mean square error rmse= E{(τ̂ − τ )2 } and the bias (estimation error). We deﬁne the bias of our estimator to be Bias(τ̂ ) = E{τ̂ −τ } = E{τ̂ }−τ . The channel model used, is the IEEE 802.15.4a CM3 model for indoor oﬃce environment [37], with RMS delay spread τrms = 12 ns. The diﬀerent systems used and their simulation parameters are shown in Table 5.1. TR-UWB systems ⇒ Parameters ⇓ Tω [ns] τm [ns] Tsamp [ns] Tf [ns] TI [ns] Ts [ns] Nf Ncr DTR UWB 0.7 0.2877 0.05 100 20 1000 10 10 LDC DTR-UWB 0.7 0.2877 0.05 16 20 200 5 4 BLDC DTR-UWB 0.7 0.2877 0.05 2 20 100 9 8 DP TR-UWB 0.7 0.2877 0.05 20 20 200 2 1 Table 5.1: TR-UWB systems and simulation parameters Fig. 5.7 - Fig. 5.9 depict the MSE of our timing oﬀset estimator plotted versus SNR for values of P = 64, 256 and for diﬀerent TR-UWB systems. Solid lines show the performance with known Es and dash-dot lines show the performance with estimated Es . For all the TR-UWB systems considered, it can be observed that, as P increases, the normalized MSE decreases monotonically. The same trend can be observed for increasing SNR. At SNRs above 20 dB the MSE of our timing oﬀset estimator with estimated Es get closer to the MSE of our timing oﬀset estimator with known Es . This conﬁrms the good accuracy of our energy estimator at SNR>20 dB. For all the TR-UWB systems, the MSE curves ﬂatten at high SNR towards a certain error ﬂoor. This error ﬂoor is related to P (amount of averaging) and Tf (average pulse separation). IFI plays an important role in the observed error ﬂoor as shown in Fig. 5.10, where the rmse of our timing oﬀset estimation technique is plotted versus SNR for diﬀerent TRUWB systems. BLDC DTR-UWB system is the burst-oriented type of the low duty cycle (LDC) DTR-UWB, with pulse-burst of 16 ns compared to the pulse presence of 64 ns for LDC DTR-UWB system. It can be observed that the BLDC DTR-UWB system, which has severe IFI, has the highest level of error ﬂoor. For other TR-UWB systems, as Tf is increased, the error ﬂoor decreases monotonically. Fig. 5.11 conﬁrms (5.13) since the performance of our timing oﬀset estimator is related to the delay. This is due to the exponential ﬁtting, where zeτ values get closer to each other as delay increases, making it diﬃcult to resolve them. Chapter 5. Synchronization of Transmitted Reference Systems 87 τ = 5 ns −3 10 P = 256, Es known P = 256, Es estimated P = 64, Es known P = 64, Es estimated −4 Normalized MSE 10 −5 10 −6 10 0 5 10 15 SNR [dB] 20 25 30 Figure 5.7: Normalized timing oﬀset estimation MSE vs. SNR for DTR-UWB systems, P = {64, 256}. τ = 5 ns P = 256, Es known P = 256, Es estimated P = 64, Es known P = 64, Es estimated −2 Normalized MSE 10 −3 10 −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 5.8: Normalized timing oﬀset estimation MSE vs. SNR for LDC DTR-UWB systems, P = {64, 256}. Chapter 5. Synchronization of Transmitted Reference Systems 88 τ = 5 ns P = 256, Es known P = 256, Es estimated P = 64, Es known −2 P = 64, Es estimated Normalized MSE 10 −3 10 −4 10 0 5 10 15 SNR [dB] 20 25 30 Figure 5.9: Normalized timing oﬀset estimation MSE vs. SNR for DP TR-UWB systems, P = {64, 256}. τ = 5 ns P = 256 DTR−UWB LDC DTR−UWB BLDC DTR−UWB DP TR−UWB 1 rmse [ns] 10 0 10 10 12 14 16 18 20 SNR [dB] 22 24 26 28 30 Figure 5.10: Timing oﬀset estimation rmse vs. SNR for diﬀerent TR-UWB systems. Chapter 5. Synchronization of Transmitted Reference Systems 89 P = 128 SNR = 30 dB 2 10 DTR−UWB LDC DTR−UWB DP DTR−UWB 1 rmse [ns] 10 0 10 −1 10 0 5 10 15 20 25 Delay [ns] 30 35 40 45 50 Figure 5.11: Timing oﬀset estimation rmse vs. delay for diﬀerent TR-UWB systems. (a) 10 (b) 55 Mean est. delays 50 Unbiased line 45 P=128 SNR = 30 dB 5 Mean estimated delays Bias of delay estimation 40 0 35 30 25 20 15 10 P=256, Es estimated 5 P=64, Es estimated −5 0 10 SNR [dB] 20 30 0 0 10 20 30 Delay [ns] 40 50 Figure 5.12: Biasedness of timing oﬀset estimator for DTR-UWB systems, (a) vs. SNR (b) vs. delay. Chapter 5. Synchronization of Transmitted Reference Systems (a) 10 (b) 55 P=256, Es estimated P=64, Es unknown Mean est. delays 50 Unbiased line 45 P=128 SNR = 30 dB Mean estimated delays Bias of delay estimation 40 5 0 90 35 30 25 20 15 10 5 −5 0 10 SNR [dB] 20 0 30 0 10 20 30 Delay [ns] 40 50 Figure 5.13: Biasedness of timing oﬀset estimator for LDC DTR-UWB systems, (a) vs. SNR (b) vs. delay. (a) 10 (b) 55 P=256, Es estimated P=64, Es estimated Mean est. delays 50 Unbiased line 45 P=128 SNR=30 dB 5 Mean estimated delays [ns] Bias of delay estimation 40 0 35 30 25 20 15 10 5 −5 0 10 SNR [dB] 20 30 0 0 10 20 30 Delay [ns] 40 50 Figure 5.14: Biasedness of timing oﬀset estimator for DP TR-UWB systems, (a) vs. SNR (b) vs. delay. Chapter 5. Synchronization of Transmitted Reference Systems 91 Fig. 5.12 - Fig. 5.14 depict the biasedness of our timing oﬀset estimator plotted versus SNR and delay for values of P = 64, 256 and for diﬀerent TR-UWB systems. This knowledge contributes in deciding the operating SNR and range (looking also at Fig. 5.11) of our timing oﬀset estimator. For conventional DTR-UWB and dual pulse (DP) TR-UWB systems the biasedness of our estimator is negligible from 20 dB onwards, which is not the case for LDC DTR-UWB system which exhibits some small biasedness. 5.4 Summary In this chapter, a novel data-aided timing acquisition technique for DTR-UWB systems was presented. The main signal processing techniques used were: • Averaging of the received signal • Fractional sampling • Maximum selection criterion The technique proposed is limited by the frame resolution which can be overcome through timing oﬀset estimation, which was also presented, in this chapter, for low complexity TRUWB systems. The proposed data-aided timing oﬀset estimator uses energy collected at the symbol rate, thus reducing considerably the implementation complexity. Simulation results provided a thorough performance analysis of the timing acquisition technique in terms of timing acquisition MSE, probability of correct detection and average BER. For the timing oﬀset estimator, the performance metrics used were the normalized estimation MSE and estimation error (biasedness). The CRLB for such timing oﬀset estimation approach was also derived. Chapter 6 Localization with Transmitted Reference Systems 6.1 Introduction Emergence of location based services requires a technical solution for accurate indoor localization of persons and objects. As satellite based positioning systems are not readily applicable to indoor localization, alternative solutions are needed. Indoor wireless systems have to cope with severe multipath scenarios. Due to their extremely large bandwidth, UWB signals oﬀer a good multipath resolution and enable accurate localization. UWB localization approaches can be divided into Time of Arrival (TOA), Time-diﬀerence of Arrival (TDOA), Angle of Arrival (AOA) and Received Signal Strength (RSS) based systems. There are various existing and potential applications of high precision localization using UWB technology. An UWB precision location system [75] can be used to identify and locate valuable assets, for example, in hospitals, industrial ﬁelds and government oﬃces. In [12] a low cost and small size UWB ASIC (application speciﬁc integrated circuit) is proposed for outdoor sport and recreational activities. The architecture and implementation status of a low-cost experimental UWB localization system is reported in [76]. It is designed to enable practical evaluations of UWB based positioning concepts. Another potential application is in WSN. Awareness of sensor positions may eﬀectively improve network performance. For instance, location-aware routing protocols can reduce routing overhead and save energy by avoiding route search [77]. In [78] a novel packet routing scheme for UWB ad-hoc network is proposed and evaluated. This routing scheme employs UWB’s features and controls spreading factor according to the distance between the nodes. When controlling the maximum transmission distance and adopting multi-hop transmission, the throughput performance can be improved as compared to single-hop transmission. Another position based routing strategy which exploits the high precision ranging capabilities oﬀered by UWB is presented in [79]. It is shown that if a position-aware routing protocol and a power-aware routing metric based on ranging measurements are adopted, emitted power levels as well as multi-user interference are signiﬁcantly reduced. The system proposed in this chapter deals with UWB localization based on the timing 92 Chapter 6. Localization with Transmitted Reference Systems 93 oﬀset estimation algorithm presented in section 5.3. This algorithm can readily be used for distance measurement since c = d/τ , where c is the speed of light and d is the distance travelled by light or other electromagnetic waves during a time span τ . Our algorithm provides a new aspect in distance measurement and is similar to the RSS technique. The diﬀerence is that the RSS technique relies on the knowledge of the path loss model for accurate and reliable distance estimation, whereas our algorithm uses the statistical analysis of the energy collected at the output of the receiver. Generally, the RSS metric is relatively simple to detect and no high synchronization eﬀort and protocols are needed like in the TOA based solutions. Moreover, TOA based systems are sensitive to the available bandwidth, and also to the occurrence of UDP channel conditions, whereas RSS based systems are less sensitive to the available bandwidth and more resilient to UDP conditions. Nevertheless, it is not expected that RSS methods can match the precision of time-based methods. The achievable accuracy in RSS localization decreases with the distance, which has to been taken into consideration in the system design. 6.2 Localization Systems Locating a node in a wireless system involves the collection of location information from radio signals traveling between the target node and a number of beacon nodes. Diﬀerent measurement techniques for localization are discussed in this section from the viewpoint of UWB systems. The general system architecture of localization systems will be brieﬂy discussed. 6.2.1 Ranging-based Systems The most popular measurement technique for localization is ranging. There are two methods to obtain range measurements: timing and signal strength. Timing Time-based techniques rely on measurements of travel times of signals between nodes. If two nodes have a common clock, the node receiving the signal can determine the TOA of the incoming signal that is time-stamped by the reference node. The measured time can provide a circle of radius representing the distance between the node to be localized and the receiving node, centered at the latter. In two-dimensional localization, at least three circles are required as shown in Fig. 6.1. This technique, also known as trilateration, is easy to implement; however, it requires knowledge of the transmission time of the emitted signal as well as synchronization of the transmitter and receivers clocks. Otherwise, considerable localization errors can occur. For instance, a clock inaccuracy of just 1 µs will lead to a position error of 300 m. Furthermore, this technique can suﬀer from multipath propagation eﬀects. In the TDOA technique the diﬀerence in time at which the signal from the node to be localized arrives at two diﬀerent receivers is measured. Each time diﬀerence is then converted into a hyperbola with a constant distance between the two receivers. In two-dimensional Chapter 6. Localization with Transmitted Reference Systems 94 Location Receiver 1 d1 = ct1 d2 = ct2 d3 = ct3 Receiver 2 Receiver 3 Figure 6.1: TOA localization principle. localization, at least two pairs of receivers are required; the location is the intersection of the hyperbolas. This technique, known as multilateration or hyperbolic positioning, requires synchronization of the receivers’ clocks; however, knowledge of the absolute transmission time of the emitted signal is no longer required. Here again, multipath propagation eﬀects can inﬂuence the accuracy of the location of the receivers. Fig. 6.2 shows an example of TDOA localization. For both TOA and TDOA techniques, the wider the bandwidth of the signal the lower the measurement error becomes. This property allows extremely accurate location estimates using time-based techniques via UWB radios. Location Receiver 1 Hyperbola d1,2 Receiver 2 Receiver 3 Hyperbola d1,3 Hyperbola d2,3 Figure 6.2: TDOA localization principle. Signal Strength With this technique the signal strength or energy of the node to be localized is measured at several stationary receivers. Ideally, each measurement will provide a circle of radius representing the distance between the node to be localized and the receiver that made this Chapter 6. Localization with Transmitted Reference Systems 95 measurement, centered at the corresponding receiver. The location of the node is then given by the intersection of these circles. In two-dimensional localization, and assuming no measurement error occurs, at least three circles are required in order to resolve the ambiguities arising from multiple crossings of the circles. The accuracy of the estimated location can be improved by increasing the number of measurements and then averaging the results. However, this approach requires an exact knowledge of the path loss in order to get an accurate estimation of the signal strength at the results. Our Approach We proposed a RSS-like technique that uses the statistical analysis of the distance-dependent discrete-time TR-UWB system models, instead of the path loss model, to estimate the distance d between the transmitter and the receiver. Indeed, considering c = d/τ , where c is the speed of light in vacuum, (5.6) can be written as zs [i] = czeτ [i] + czeτ ν [i] = zed [i] + zedν [i]. (6.1) At a time instant i, zed [i] is a Gaussian random variable with mean E{zed [i]} = cE{zeτ [i]} 2 and variance c2 σze . Hence, (5.8) becomes E{zed [i]} = Adi exp(Bd/c). (6.2) 0.9 0.8 0.7 E{zed} 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 distance [m] 10 15 Figure 6.3: Normalized average distance-dependent noise-free decision variable The distance dependence of E{zed [i]} is shown in Fig. 6.3, and is obtained by averaging zed over 1000 channel realizations. The maximum likelihood (ML) estimate of the distance between the transmitter and the receiver can be expressed as Chapter 6. Localization with Transmitted Reference Systems c dˆ = ln B ( |z̄s | A 96 ) . (6.3) The CRLB for the variance of the distance estimates is derived from (5.13) and is given by ˆ = CRLB(d) 6.2.2 2 2 c2 (σze + σνe ) . 2 P AB exp(Bd/c) [2A exp(Bd/c) − |z̄s |] (6.4) Directionality-based Systems Another way of estimating location is to measure the signal direction or angle of arrival (DOA/AOA) of the target node seen by reference nodes by steering the main lobe of a directional antenna or an adaptive antenna array. To determine the location of a node in a two-dimensional plane, it is suﬃcient to measure the angles of the straight lines that connect the node and two reference nodes, as shown in Fig. 6.4. Location y Receiver 2 Receiver 1 x Figure 6.4: AOA localization principle. This method is also called triangulation and it has the advantage of not requiring synchronization of the receivers nor an accurate timing reference. On the other hand, receivers require regular calibration in order to compensate for antenna mismatches. The AOA approach is not suited to UWB localization for the following reasons: • The use of antenna arrays increases the system cost, annulling the main advantage of a UWB radio equipped with low-cost transceivers. • Due to the large bandwidth of a UWB signal, the number of paths may be very large, especially in indoor environments. Therefore, accurate angle estimation becomes very challenging due to scattering from objects in the environment. • Time-based approaches can provide very precise location estimates, and therefore they are better motivated for UWB over the most costly AOA-based techniques. Chapter 6. Localization with Transmitted Reference Systems 97 Ranging and directionality-based systems can be termed as Network-based systems. 6.2.3 System Architecture Localization systems using similar measurement techniques can diﬀer considerably in their system architecture. We basically have centralized, hierarchical or decentralized architectures. The design of a localization system can also be largely inﬂuenced by application requirements. Certain applications require tightly coupled systems using beacon nodes that are wired to a centralized controller and placed at ﬁxed positions, whereas for other applications loosely coupled systems are more appropriate. They use beacon nodes that are wireless and coordinate in a completely decentralized manner without central control. Tightly coupled Systems Several of the traditional localization technologies have a tightly coupled system architecture, motivated by application requirements. These applications generally require high accuracy and real-time tracking. Problems of time synchronization and coordination among beacon nodes are easily resolved because these systems are wired and have a centralized controller. These systems achieve high accuracy, but their drawback is that the centralized location estimation limits the number of devices they can simultaneously track. Secondly, wiring signiﬁcantly impedes deployment. A key research challenge in these systems is achieving similar performance with outdoor applications, where deployment cannot be controlled and wiring may be infeasible. Loosely coupled Systems Motivated by deployment concerns, recently proposed localization systems are decentralized and completely wireless [80] - [84]. From all these systems the three approaches that meet the basic requirements for self-organization, robustness, and energy-eﬃciency are: • Ad-hoc positioning [82] • N-hop multilateration [84] • Robust positioning [83]. These three algorithms are fully distributed and use local broadcast for communication with immediate neighbors. This last feature allows them to be executed before any multihop routing is in place, hence, they can support eﬃcient location-based routing schemes. They use a three-phase approach for determining the individual node positions: 1. Determination of the distances between nodes and anchor nodes. 2. Derivation for each sensor node of its location from its beacon distances. 3. Reﬁnement of the node locations using information about the range to, and locations of neighboring nodes. The reﬁnement phase is optional and may be included to obtain more accurate locations. Chapter 6. Localization with Transmitted Reference Systems 6.3 98 Location Estimation In this section, the standard Least Squares (LS) [81] [82] and spherical-interpolation (SI) [85] methods for location estimation are described. For convenience, we assume that the nodes with known positions, (xk , yk , zk ), k = 1, 2, . . . , N , are beacons while the nodes with unknown positions, (x, y, z) are sensors. 6.3.1 The LS algorithm Let ti be the TOA of the signal at anchor k, t0 be the transmit time at the sensor and dk = c(tk − t0 ) be the range between the sensor and the k th beacon node. Then, we obtain d2k = (xk − x)2 + (yk − y)2 + (zk − z)2 . (6.5) Subtracting (6.5) for k = 2, 3, . . . , N by (6.5) for k = 1 produces xk,1 x + yk,1 y + zk,1 z = bk,1 , k = 2, . . . , N , (6.6) where xk,1 = xk − x1 , yk,1 = yk − y1 , zk,1 = zk − z1 and bk,1 = 0.5[(x2k + yk2 + zk2 ) − (x21 + y12 + z12 ) + d21 − d2k ]. Equation (6.6) can be written in a compact form as Ap=b, (6.7) where x2,1 y2,1 z2,1 b2,1 x x3,1 y3,1 z3,1 b3,1 A= , p = y , b = .. .. ... ... . . z xN,1 yN,1 zN,1 bN,1 . (6.8) The standard LS solution to (6.7) is derived, as shown in Appendix V, as ( )−1 T p̂LS = AT A A b, (6.9) where the matrix inverse is supposed to exist and dk are replaced by their corresponding estimates. 6.3.2 The SI Method When the transmit time t0 is unknown, TDOA measurements can be employed as for centralized networks. In this case, the SI method would be a suitable candidate. First, the spatial origin is mapped receiver√1, as shown in Fig. 6.5. √to one of the beacon nodes, say T 2 2 2 By deﬁning Rk = xk + yk + zk , pk = [xk , yk , zk ] , R = x2 + y 2 + z 2 and dk,l = dk − dl , dk can be expressed using the Pythagorean theorem as d2k = R2 + Rk2 − 2pTk p. (6.10) Chapter 6. Localization with Transmitted Reference Systems 99 Location z d1 =R di Receiver 1 ( x1 , y1 , z1 ) =(0,0,0) x R i y Receiver i ( x i , yi , z i ) Figure 6.5: Illustration of the spherical interpolation approach. where p is the location of the sensor node to be estimated. Since d1 = R, R − d1 + dk = dk ⇒ dk = R + dk1 . (6.10) can be rewritten as (R + dk1 )2 = R2 + Rk2 − 2pTk p, k = 2, 3, . . . , N. (6.11) In the presence of measurement errors, (6.11) becomes (R + dk1 )2 + ek = R2 + Rk2 − 2pTk p ⇒ ek = (Rk2 − dk1 )2 − 2Rdk1 − 2pTk p, (6.12) where ek is the equation error. (6.12) can be written in compact form as e = rd − 2Rd − 2Ap. (6.13) 2 − d2N 1 ]T A is given by (6.8), e is the equation error vector, rd = [R22 − d221 , R32 − d231 , . . . , RN T and d = [d21 , d31 , . . . , dN 1 ] . The standard LS solution for p, given R, is 1 ( T )−1 T A A A (rd − 2Rd). (6.14) 2 A closed-form solution can be derived by substituting (6.14) in (6.13) and minimizing the equation again with respect to R. Rewriting the equation error (6.13) to eliminate p by substituting the value from (6.14), we get p̂ = Chapter 6. Localization with Transmitted Reference Systems ( )−1 T ẽ = rd − 2Rd − A AT A A (rd − 2Rd) = (IN −1 − As )(rd − 2Rd), 100 (6.15) ( )−1 T where As = A AT A A . The error criterion can be expressed as J(R) = ẽT ẽ = (rd − 2Rd)T B(rd − 2Rd), (6.16) where B = (IN−1 −As )T (IN−1 −As ) and IN−1 denotes an identity matrix of size N −1×N −1 . Minimizing J with respect to R is a form of weighted LS in which the weighting matrix B is idempotent of rank N − 4 [85]. For this reason more than four beacon nodes are needed for the SI method. ∂J Diﬀerentiating (6.16) with respect to R and setting ∂R = 0 we obtain 1 dT Brd . (6.17) 2 dT Bd Substituting this solution back into (6.14) yields the sensor node location estimate R̃ = p̂SI 6.3.3 1 ( T )−1 T dT Brd d = A A A (rd − T ). 2 d Bd (6.18) Simulation results The simulation results are presented in this subsection. The performance of the proposed distance estimator is demonstrated √ for diﬀerent TR-UWB systems by using the following ˆ = E{(dˆ − d)2 } and the distance estimation variance which performance metrics: rmse(d) ˆ E{(dˆ − d)2 } − [Bias(d)] ˆ 2. can be deﬁned from the MSE and the bias of d as Var(d)= The IEEE 802.15.4a CM3 channel model for indoor oﬃce environment, with RMS delay spread τrms = 12 ns, is used once again. The diﬀerent systems used and their simulation parameters are shown in Table 5.1. Using a training sequence of length P = 128 and operating the TR-UWB receivers at a SNR of 30 dB, the rmse of the distance estimator in (6.3) is shown in Fig. 6.6. It can be observed that, for all TR-UWB systems considered, the rmse increase monotonically with distance. At distance within 15 m the DTR-UWB system has a superior distance measuring performance. The TR-UWB system with relatively worst distance measuring performance is the low duty cycle (LDC) DTR-UWB system. This is because it has the lowest average frame duration, which means it is more aﬀected by IFI. It should also be noted that the DTR-UWB system is able to provide centimeter accuracy for a transmitter-receiver separation up to 10 m. The variance of the distance estimator for the three TR-UWB systems considered is shown in Fig. 6.7. Next, the localization capability of our three TR-UWB systems is analyzed. For this purpose the proposed simulation setup is shown in Fig. 6.8. A typical laboratory room or ﬁeld size of 10 m × 6 m is illustrated. A local 2D reference system is used. Chapter 6. Localization with Transmitted Reference Systems P = 128 5 101 SNR = 30 dB DTR−UWB LDC DTR−UWB DP DTR−UWB 4.5 4 3.5 rmse [m] 3 2.5 2 1.5 1 0.5 2 4 6 8 distance [m] 10 12 14 Figure 6.6: Distance estimation rmse vs. distance transmitter-receiver for diﬀerent TR-UWB systems. P = 128 11 SNR = 30 dB DTR−UWB 10 LDC DTR−UWB DP DTR−UWB 9 8 Variance [m2] 7 6 5 4 3 2 1 2 4 6 8 distance [m] 10 12 14 Figure 6.7: Distance estimation variance vs. distance transmitter-receiver for diﬀerent TRUWB systems. Chapter 6. Localization with Transmitted Reference Systems 102 (10,6) (3,5) 3.6 6.4 0m 6m 1m 5m (5,2) (0,2) 3m 4.12 m m (0,0) 2.2 5.38 (9,1) (4,0) 10 m (xk,yk) (x,y) kth beacon node sensor node Figure 6.8: 2D localization scenario. Both location estimation methods described in section 6.3 are used to localize the sensor node. The diﬀerent beacon nodes are positioned in such a way that the matrix A is nonsingular, hence invertible. This is ensured by not allowing more than two beacon nodes to have the same X or Y coordinates. For simulation purpose, the sensor node is localized at the location (x = 5, y = 2). Using both the LS and SI localization methods, diﬀerent location estimates for diﬀerent TR-UWB systems are shown in Fig. 6.9 - Fig. 6.11. The dots “·” represent the location estimates of the sensor node using 100 Monte Carlo runs, whereas the star “⋆” indicates the true sensor node location. It can be observed, for all TR-UWB systems considered, that the SI method exhibits less dispersion of the location estimates than the LS method. Indeed, with the LS method, more outliers are observed. Thus, the SI method is more precise than the LS method. Moreover, the SI method is more accurate than the LS method as can be seen in Tab. 6.1 where the localization accuracy is illustrated. This is because the SI method is analogous to the weighted LS method. It can also be observed that the DTR-UWB and dual pulse (DP) TR-UWB systems exhibit less outliers than the LDC DTR-UWB system, and they √ are more suitable for localization. In Tab. 6.1 the localization error is deﬁned by ϵ = (x̂ − x)2 + (ŷ − y)2 , where x̂ and ŷ are the estimates of the X and Y coordinates of the sensor node, respectively. Chapter 6. Localization with Transmitted Reference Systems DTR−UWB P = 128 SNR = 30 dB 6 Y coordinate [m] Sensor true location Location estimates LS Method 5 4 3 2 1 0 0 1 2 3 4 5 6 X coordinate [m] 7 6 Y coordinate [m] 9 10 Sensor true location Location estimates SI Method 5 8 4 3 2 1 0 0 1 2 3 4 5 6 X coordinate [m] 7 8 9 10 Figure 6.9: Location estimation with DTR-UWB systems. LDC DTR−UWB 6 SNR = 30 dB Sensor true location Location estimates LS Method 5 Y coordinate [m] P = 128 4 3 2 1 0 0 1 2 3 4 5 6 X coordinate [m] 7 8 9 10 6 Y coordinate [m] 5 Sensor true location SI Method Location estimates 4 3 2 1 0 0 1 2 3 4 5 6 X coordinate [m] 7 8 9 10 Figure 6.10: Location estimation with LDC DTR-UWB systems. 103 Chapter 6. Localization with Transmitted Reference Systems DP TR−UWB 6 Sensor true location Location estimates LS Method 5 Y coordinate [m] P = 128 SNR = 30 dB 4 3 2 1 0 0 1 2 3 4 5 6 X coordinate [m] 7 6 Y coordinates [m] 9 10 Sensor true location Location estimates SI Method 5 8 4 3 2 1 0 0 1 2 3 4 5 X coordinates 6 7 8 9 10 Figure 6.11: Location estimation with DP TR-UWB systems. TR-UWB systems ⇒ Performance ⇓ Average p̂LS Average p̂SI Average ϵLS [cm] Average ϵSI [cm] DTR UWB η=0 (5.041,1.944) (4.982,1.999) 48.43 28.57 LDC DTR-UWB DP TR-UWB (4.886,1.341) (4.822,1.802) 127.6 58.702 (5.035,1.948) (4.976,2.015) 59.95 33.60 Table 6.1: Localization accuracy of diﬀerent TR-UWB systems. 104 Chapter 6. Localization with Transmitted Reference Systems 6.4 105 Summary Localization with diﬀerent TR-UWB systems was dealt with in this chapter. Diﬀerent distance measurement techniques were presented, namely the TOA, TDOA, AOA and RSS methods. A new approach to distance measurement, similar to the RSS technique was introduced. It uses the statistical analysis of the energy collected at the output of the receiver rather than relying on the knowledge of the path loss model in RSS technique. The performance of the proposed distance estimator was demonstrated for diﬀerent TR-UWB systems. Further, the localization capability of the considered TR-UWB systems were analyzed using the LS and SI localization methods. Simulation results have shown the superiority of the SI method over the LS method. It was also shown that the DTR-UWB and DP TR-UWB are more suitable for localization than LDC DTR-UWB system. Chapter 7 Conclusions and Outlook 7.1 Conclusions The main objective of this doctoral thesis is to present the conceptual design and analysis of a transmitted reference (TR)-UWB impulse response (IR) receiver that has dual functionality in providing robust, low power data communication and accurate location information in wireless sensor networks (WSNs). TR signaling in combination with an autocorrelation receiver (AcR) is especially suitable in this context because it allows a lower design complexity compared to coherent receivers. It allows higher accuracy in distance measurement as well. In this thesis, the theory of polynomial nonlinear systems was revisited. Volterra series expansions were also presented. Further, the Wiener and Hammerstein models, which are the simpliﬁed forms of the Volterra model, were discussed. The eﬃcient design of UWB communications systems requires an appropriate channel model that accurately describes the UWB propagation. The description and discussion of appropriate channel models for mobile radio systems and UWB radio systems, in particular, were provided in this thesis. Emphasis was made on the IEEE 802.15.4a standardized channel model for UWB Communications. The polynomial nonlinear models and the channel models are important in the simulation and analysis of low complexity TR-UWB systems. The TR-UWB systems considered, in this thesis, are the diﬀerential transmitted reference (DTR)-UWB system, the low duty cycle (LDC) DTR-UWB system and the dual pulse (DP) TR-UWB system. Discrete-time equivalent system models for these TR-UWB systems were derived and their delay-dependence studied. TR-UWB receiver statistics were analyzed from two perspectives: Gaussian approximation on the noise terms in the receiver and the analysis using the equivalent system and noise models of the receiver. Both perspectives led to the BER performance analysis. Synchronization is a particularly acute problem faced by UWB systems due to the fact that they employ low-power ultra-short pulses. Coarse synchronization and ﬁne synchronization algorithms were developed and evaluated in this thesis, utilizing a known training sequence. A novel data-aided timing acquisition technique for frame-level synchronization of DTR-UWB systems was suggested. Furthermore, a simple timing oﬀset estimator for 106 Chapter 7. Conclusions and Outlook 107 low complexity TR-UWB systems, assuming no ISI, was considered. Both the classical and bayesian approaches were used for estimating the timing oﬀset. It was shown that both approaches yield the same timing oﬀset estimate. The theoretical performance limit of such timing oﬀset estimation was illustrated through the derived Cramer-Rao Lower Bound (CRLB) for the variance of the timing oﬀset estimates. Assuming that coarse synchronization has been achieved and hardware clock timing error is known, the proposed timing oﬀset algorithm was used as a new approach for distance measurement. It is a RSS-like technique that uses the statistical analysis of the distance-dependent discrete-time TR-UWB system models, instead of the path loss model, to estimate the distance between the transmitter and the receiver. Diﬀerent measurement techniques for localization were discussed in this thesis from the viewpoint of UWB systems. The general system architectures of localization systems were presented. Further, the performance of the proposed distance estimator was demonstrated for diﬀerent TR-UWB systems. Finally, the localization capability of the considered TR-UWB systems were analyzed using the LS and SI localization methods. 7.2 Summary of Contributions The main contributions of this thesis are summarized below Systems modelling { Gaussian approximation Low complexity TR systems Performance analysis using derived models Average BER derivation Coarse synchronization Fine synchronization Synchronization and Localization Distance and location estimation 7.3 Discussion and Recommendations The equivalent system models for TR-UWB systems presented in this thesis constitute an important step towards developing an emulator for TR-UWB systems. Such emulator would be very useful for developing optimal and suboptimal detection algorithms for diﬀerential UWB communications systems under ISI or without ISI. It would also be a powerful tool for developing signal processing algorithms for equalization and synchronization. The performance of the diﬀerent TR-UWB systems studied is summarized in Tab. 7.1. DTR-UWB systems are recommended for short-range radio location-aware WSN applications with lower data rate requirement but higher localization accuracy requirement. DP TR-UWB systems have lower complexity, but provide higher data rate, when compared to DTR-UWB systems, at the expense of loosing some localization accuracy. Thus, they are recommended for short-range radio location-aware WSN applications with stringent low cost requirements. TR-UWB systems ⇒ Performance ⇓ Symbol rate [Mb/s] Timing oﬀset rmse [ns] (τ = 5 ns, P = 256, SNR = 30 dB) Mean localization error [cm] (LS method) Mean localization error [cm] (SI method) DTR UWB η=0 1 1.65 LDC DTR-UWB DP TR-UWB 5 1.9 5 1.7 48.43 28.57 127.6 58.702 59.95 33.60 Table 7.1: Performance of diﬀerent TR-UWB systems. Sensor network localization has been an active area of research for the last few years. For WSNs, the ability for sensor nodes to determine their position through automatic means is recognized as an essential capability. The research community has made great strides in ranging technologies, systems for infrastructure-based localization, and algorithms and techniques for ad-hoc localization. UWB systems are being investigated for ad-hoc and wireless sensor networks, where it is not necessary to have ﬁxed base stations like with conventional radio systems. Hence, each node need to have a routing function. A multihop transmission will also be necessary if the area cover is larger than the permitted distance of communication between nodes. As already discussed in this thesis, the very large bandwidth of UWB technology, besides providing the possibility to transmit at very high data rate, also provides very accurate temporal and spatial information. This high accuracy allows for new prospectives in terms of synchronization, positioning and tracking. In the case of tracking a mobile sensor node in a wireless system, two recursive tracking algorithms, namely the extended Kalman ﬁlter (EKF) and the unscented Kalman ﬁlter (UKF), can be employed as smoothing algorithms on the location estimates. The Kalman ﬁlter is one of the best optimal recursive Bayesian estimator for linear tracking problems. The design of an UWB indoor network as a local augmentation to the global navigation satellite system (GNSS) positioning, can be envisaged. It would enable a seamless absolute positioning and ensure a smooth outdoor-indoor transition. It would also help overcoming signal attenuation and extending coverage, by pseudo-ranging from low-level ground based UWB beacon nodes. Further in this context is the development of a joint UWB-IR/inertial measurement unit (IMU) system for joint communication and positioning/location in industry and logistics scenarios. Such system be based on peer-to-peer mechanisms with network routing functionality. The system elements might be either mobile system elements or ﬁxed system elements. The major diﬀerence is that ﬁxed nodes are furnished with their absolute or relative position, whereas mobile nodes use ﬁxed nodes to obtain their absolute or relative. Whether mobile or ﬁxed, systems elements would comprise two loosely coupled parts: the UWB subsystem and the IMU subsystem. Both would provide data into a data fusion module computing enhanced position co-ordinates. 108 Appendix I Gaussian Integral The Gaussian integral, or probability integral, is an integral of the Gaussian function e−x . It is expressed analytically as 2 ∫∞ e−x dx = 2 √ π. (1) −∞ 2 bx+c The integral of any Gaussian function ae−( f ) is reducible in terms of the Gaussian integral. Let make the change of variable y = bx + c ⇒ dx = dy/b, ∫∞ ae−( ) dx = a b bx+c 2 f −∞ ∫+∞ y 2 e−( f ) dy. (2) −∞ Substituting y with f z, and using (1) gives ∫∞ ae−( bx+c 2 f ) dx = −∞ = af b ∫∞ e−z dz 2 (3) −∞ af √ π. b The integrals of the exponential functions x2 e−bx and xn e−bx are deﬁnite and are expressed analytically as 2 ∫∞ x2 e−bx dx = −∞ ∫∞ xn e −∞ −bx2 dx = 2 1 2 (2k−1)!! √ π b 2k+1 bk k! 2bk+1 √π b3 2 b > 0, (4) b > 0, n = 2k, k ∈ Z , (5) n = (2k + 1) where Z is the ﬁeld of integers, n!! denotes the double factorial of n and is deﬁned recursively by 109 { n!! = 1 ifn ∈ {−1, 0, 1} . n(n − 2)!! ifn ≥ 2 (6) For example, 8!!= 2 · 4 · 6 · 8 = 384 and 9!!= 1 · 3 · 5 · 7 · 9 = 945. Some identities involving double factorials are (2n)!! = 2n n! . (2n + 1)!! = (2n+1)! 2n n! (7) Following the same change of variables as in (2) and (3), in this thesis, more general analytical expressions to (2) and (3) are given as ∫∞ ) ( bx+c2 2 − f ax e −∞ ∫∞ −∞ axn e−( bx+c 2 f ) dx = dx = a 2 ( f )3 √ b π b > 0, (8) ( )2k+1 √ (2k−1)!! π n = 2k a 2k+1 bk fb a k!2 110 ( f )2k+2 b . n = 2k + 1 (9) Appendix II Energy of Gaussian pulses ∫∞ ∫∞ −∞ K12 e−4π(t/tm ) dt. 2 ωg2 (t)dt = Eg = (10) −∞ √ Using (3) with a = K12 , b = 2 π and f = tm , we have Eg = ∫∞ ∫∞ ωg21 (t)dt Eg1 = Using (8) with a = K12 = −∞ −∞ 16π 2 K12 t4m K12 tm . 2 (11) 16π 2 −4π(t/tm )2 e dt. t4m (12) √ , b = 2 π and f = tm , we have K12 π . tm ∫∞ 2 ∫∞ 16π2 K12 ( 1− ωg2 (t)dt = t4 (13) Eg1 = Eg2 = = −∞ ∫∞ −∞ + = m −∞ 16π 2 K12 t4m ∫∞ 2 e−4π(t/tm ) dt − −∞ 16π 2 Eg t4m − 8π E t2m g1 − 8π E t2m g1 2 e−4π(t/tm ) dt 128π 3 K12 2 −4π(t/tm )2 te dt t6m (14) + Eg21 . 162 π 4 K12 , t8m Eg21 = 16π 2 Eg t4m −∞ )2 162 π 4 K12 4 −4π(t/tm )2 te dt t8m Eg21 is computed using (9) with a = Since ∫∞ 4πt2 t2m √ b = 2 π, f = tm , n = 4 and k = 2. 6π 2 K12 . t5m = 0, we ﬁnally have 111 (15) Eg2 = 6π 2 K12 . t5m 112 (16) Appendix III Posterior Probability Density Function Using the Bayes’ rule, we can express p(zeτ |zs ) as p(zs |zeτ )p(zeτ ) p(zs ) p(zs |zeτ )p(zeτ ) . ∞ ∫ p(zs |zeτ )p(zeτ )dzeτ p(zeτ |zs ) = = (17) −∞ Using (5.14) and (5.15), (17) can be expressed as [ ] ] [ P∑ −1 1 2 (zs [i] − zeτ ) exp − 2σ12 (zeτ − θ(τ ))2 exp − 2σ2 νe ze i=0 [ ] p(zeτ |zs ) = ∞ , ] [ P −1 ∫ ∑ 1 1 2 2 exp − 2σ2 (zs [i] − zeτ ) exp − 2σ2 (zeτ − θ(τ )) dzeτ νe ∞ (18) ze i=0 where θ(τ ) = Adi exp(Bτ ). After simpliﬁcation, we obtain p(zeτ |zs ) = [ ( )] 1 2 −2P z z̄ )+ 1 (z −θ(τ ))2 exp −1 (P z eτ s eτ eτ 2 2 σ2 σze )] [ ( νe ∞ ∫ −1 1 1 2 −2P z z̄ )+ 2 exp 2 (P z (z −θ(τ )) dzeτ eτ s eτ eτ 2 2 σνe ∞ = ( P 2 σνe 1 2 σze νe ze f (zeτ ) = + ( Let v = σP2 + σ12 ) 2 zeτ )−1 −2 ( ∞ ∫ ∞ P z̄s 2 σνe exp[ −1 f (zeτ )] 2 exp[ −1 f (zeτ )]dzeτ 2 + ( and m = f (zeτ ) = = = θ(τ ) 2 σze P z̄s 2 σνe ) σze (19) . 2 ) zeτ + θ(τ 2 . This function is quadratic in zeτ . σze ) ) + θ(τ v, then we have 2 σ ze 2 ) 1 2 z − 2 mv zeτ + θσ(τ 2 v eτ ze 2 1 2 2 (z − 2mzeτ + m2 ) − mv v eτ 2 ) 2 1 (z − m)2 − mv + θσ(τ 2 . v eτ ze Putting the expression of (20) back into (17), we have 113 + θ2 (τ ) 2 σze (20) p(zeτ |zs ) = [ ( )] 2 θ 2 (τ ) 1 exp[− 2v (zeτ −m)2 ] exp − 21 − mv 2 σze [ ( )] . ∞ ∫ 1 θ 2 (τ ) m2 1 2 − dzeτ exp[− 2v (zeτ −m) ] exp − 2 2 v (21) σze ∞ Following the requirement that p(zeτ |z) integrate to 1 [72], we ﬁnally have [ ] 1 1 2 p(zeτ |zs ) = √ exp − (zeτ − m) . (22) 2v 2πv This shows that )z(eτ conditioned on zs is a Gaussian( random )variable with mean ( ) mzeτ |zs = P z̄s 2 σνe + θ(τ ) 2 σze P 2 σνe + 1 2 σze −1 and variance σz2eτ |zs = 114 P 2 σνe + 1 2 σze −1 . Appendix IV Least Squares Exponential Fitting Let E{zeτ } = z¯e (n) = A exp(Bn), where n ∈ {0, 1, . . . , N − 1} are the diﬀerent delay values in ns, as shown in Fig. 4.8 for N = 51. Since ln(z¯e (n)) = ln(A) + Bn = ε + γn, to determine A and B we start by minimizing the function χ2 (ε, γ) = N −1 ∑ z̄e (n)[ln(z̄e (n)) − ε − γn]2 . (23) n=0 Applying least squares ﬁtting gives N −1 N N∑ −1 ∑ ∑ z̄e (n) ln(z̄e (n)) nz̄e (n) = z̄e (n) + γ ε n=1 n=0 n=0 (24) N N∑ −1 N∑ −1 ∑ ε nz̄e (n) ln(z̄e (n)) n2 z̄e (n) = nz̄e (n) + γ n=1 n=0 n=0 The system of equation in (24) can be rewritten in matrix format N −1 N −1 N∑ −1 ∑ ∑ [ ] z̄e (n) ln (z̄e (n)) z̄ (n) nz̄ (n) e e ε n=0 n=0 n=0 = −1 −1 N∑ −1 γ N∑ N∑ nz̄e (n) n2 z̄e (n) nz̄e (n) ln (z̄e (n)) n=0 (25) n=0 n=0 or [ M ε and γ are obtained by solving (26). [ ε γ ε γ ] = m. (26) ] = M−1 m. (27) Finally, A and B are expressed as A = exp(ε) B = γ. 115 (28) Appendix V Linear Least Squares Let us consider a data model expressed as x[n] = s[n] + w[n], n = 0, 1, · · · , N − 1. (29) w[n] is a zero mean noise process and s[n] is the signal model. (29) can be expressed in compact form as x = s + w. (30) Let us consider a unknown vector parameter θ= [θ1 , θ2 , . . . , θp ] of dimension p × 1 to be estimated. For the signal s=[s[0], s[1], . . . , s[N − 1]]T to be linear in the unknown parameters, it is assumed, using matrix notation, that s = Hθ, (31) where H is a known N × p matrix (N > p) of rank p. The matrix H is known as the observation matrix. The least square error (LSE) is found by minimizing J(θ) = N∑ −1 (x[n] − s[n])2 n=0 = (x − Hθ)T (x − Hθ) = xT x − xT Hθ − θ T HT x + θ T HT Hθ = xT x − xT − 2xT Hθ + θ T HT Hθ. (32) It is important noting that xT Hθ is a scalar and J is a quadratic function of θ. The gradient ∂J(θ)/∂θ is expressed as ∂J(θ) = −2HT x + 2HT Hθ. ∂θ Setting the gradient equal to zero yields the LSE θ̂ = (HT H)−1 HT x. 116 (33) (34) Appendix VI Design of ZESS wireless sensor nodes Design of sensor nodes 1. Hardware Selection In our completed experimental implementation of a sensor node we feed the digital input data from a digital source such as a personal computer (PC) via a USB interface to an 8051-compatible microcontroller. The sensor module is represented by a GPS module. The signal processing and network protocol functions are handled by the Cypress Semiconductor’s EZ-USB which is an enhanced 8051 core with fast execution time and added features. It uses internal RAM for program and data storage, making it a soft solution. The EZ-USB chip operates at 3.3V. The radio module used is a single chip 2.4 GHz Gaussian Frequency Shift Keying (GFSK) transceiver based on the HomeRF technology and capable of delivering data rate up to 1 Mbps. An illustration of the experimental design of a sensor node with wireless communication capabilities is shown in Fig. 1. 2. Firmware Development Generally, a ﬁrmware is software (programs or data) that is embedded in a hardware device, allowing reading and execution of the software, but does not allow modiﬁcation, as for example, writing or deleting data by an end user. A great feature of the ﬁrmware we developed is that it can be modiﬁed. This is because of the soft (RAMbased) feature of the EZ-USB microcontroller that allows unlimited conﬁguration and upgrades. Integration of GPS data The main objective of this work is to provide our sensor nodes with positioning information for localization purposes in WSN. This work consists of three tasks: 1. Serial Interface of a Handheld GPS Receiver Based on the strictly low energy consumption requirements of sensor nodes, we have R decided to select the Lassen⃝ IQ GPS module whose main features are: ultra low power consumption - economical price - Aided GPS (A-GPS) capability through TSIP 117 Clock signal 8051 simulation tools C51 compiler EZ-USB control panel Matlab DLL Data Sensor Node GPS Sensor USB 2.0 dio Ra ne an Ch Programming Configuration Cypress EZ-USB FX2 µC Control Packetization l Home RF UWB Zigbee Verifying the clock and data timing diagrams with an oscilloscope USB 2.0 Figure 1: Experimental design of wireless sensor nodes (Trimble Standard Interface Protocol) for faster acquisition - possibility to switch the module automatically to higher sensitivity when satellite signals are week - 12 channel simultaneous operation for complete position, velocity and time (PVT) solutions - supports standards NMEA (National Marine Electronics Association ) 0183, TSIP, TAIP (Trimble ASCII Interface Protocol) and Diﬀerential GPS (DGPS). R As an embedded design, the Lassen⃝ IQ GPS module provides direct CMOS compatible TTL level serial I/O. The RX and TX signals are driven directly by the dual R universal asynchronous receiver/transmitter (UART) on the Lassen⃝ IQ GPS receiver. Interfacing these signals directly to the UART of our 8051-Microcontroller provides direct serial communication without the complication of RS-232 or RS-422 line drivers. 2. The NMEA Protocol The hardware interface for GPS units is designed to meet the NMEA requirements. R The Lassen⃝ IQ GPS receiver normally operates using one of the three protocols - TSIP, TAIP or NMEA 0183. Our program to dissect the desired GPS message string and extract the required information is based on the NMEA 0183 which is an industry standard protocol common to marine applications. The signal characteristics of NMEA 0183 serial data transmission are: Baud rate=4800, Data bits=8, Stop bits=1, Parity bits=none. The entire protocol encompasses over 50 messages, but only a subset of these messages apply to a particular GPS receiver. Indeed, at 4800 baud we can only send 480 characteres per seconds. Since NMEA 0183 messages can have up to 79 characters, 118 we are limited to 6 diﬀerent messages. 3. Dissection of the desired GPS message The NMEA protocol is designed to run as a process in the background spitting out messages which are then captured as needed by a program. Thus our program, written in C51, will sample the data stream, use that data for screen display, and then sample the data again, based on our information requirements. Depending on the time needed to use the data there can be easily a lag of 4 seconds in the responsiveness to changed data. As an example, a dissected NMEA message structure, based on some GPS measurements (June 11th 2005) at ZESS is shown and explained below: $ GPGGA,215354.463,5054.2113,N,00801.7561, E,1,07,1.1,370.4,M,,,1.1, 0000*2A The “$” sign indicates the start of a message. The GP talker identiﬁcation signiﬁes a GPS source (source of navigation information. The GGA message identiﬁcation describe the message content and the number and order of the data ﬁelds. GGA describes the Global Positioning System ﬁxed data The “,” serve as delimiters for the data ﬁelds. 215354.463 is a speciﬁc data ﬁeld in the GPGGA message. In fact it describes the UTC Time given as 21h53mn54.463s. The other following ﬁelds describe, respectively, the latitude, the North/South indicator, the longitude, the East/West indicator, the position ﬁx indicator, the satellites used (Range 0 to 12), the Horizontal Dilution of Precision (HDOP), the Mean Sea Level (MSL) altitude, the unit used (M=meters). 0000 is the diﬀerential referential station ID. The “*” sign serves as a checksum delimiter and 2A are the two ASCII characters which indicates the hexadecimal value of the checksum. Not seen in the message above are the carriage return [CR] and line feed [LF] whose combination terminate the message. 4. Extraction of the required information Upon dissection of the required GPS message, our C51 program extract the desired ﬁelds of interest (UTC Time, Position etc...) and store the data in dedicated buﬀers for further use including data display, localization solutions, Aided GPS experiments, emulation of position-based routing algorithms, etc... The algorithm ﬂowchart for extracting the required position information is shown in Fig. 2. Communication Radio Module - Personal Computer A dynamic-link library (DLL) named USB HRF DLL and its corresponding source-code for the microcontroller were developed with the objective of enabling a relatively simple communication capability between the Home RF hardware and a Personal Computer (PC) via the USB interface. Since the USB Framework in the microcontroller exhibits a relatively complex interrupt processing system, it was advantageous to make a change on the hardware. This concerns the Interrupt signal INT0 for signaling that Home RF has received data. This was set to INT5 (Trigger in). This change is absolutely necessary for the signaling of a receive packet to occur. 119 Start No Check for rx GPS msg string Yes Read GPS msg string and store in buffer Is the stored string a <$GPGGA>? Refresh buffer No Yes Extract latitude longitude and UTC msgs Display Figure 2: Algorithm ﬂowchart A special driver for the USB system must be installed. It can be found in the folder Treiber USB CAM of the provided CD. Loading the DLL and corresponding microcontroller programs, will allow the so-called renumeration to take place. This causes the PC to detect the microcontroller as a new sensor interface. It is important noting that the hardware and the DLL can be utilized fully only after the driver is correctly installed. Two types of DLL can be exported, one for use under Matlab (Macro “Matlab” in USB HRF DLL.h) and the other one under C++. Using the Matlab environment, the program HomeRF Matlab.m was written as a test-program which depicts the DLL principal functionality. Home RF Functionalities The DLL and the programs which run on the microcontroller are closely interrelated to each other. Each exported DLL function calls certain functions in the microcontroller via the USB Protocol. Thus it is possible for the source code of the microcontroller to adapt itself to new tasks without knowledge of the USB protocols or the entire USB Frameworks. The basic idea in implementing the functions in the microcontroller is to try to separate the USB speciﬁc functions from the Home RF functions. In this way, for any user of the DLL and the frameworks in the microcontroller, only two ﬁles are important: HRF.c and HRF.h. Both can be found in the directory . . . /Config USB HRF/Control8051 (see provided CD), including also all relevant functions addressed by the HomeRF radio module. The block diagram in Fig. 3 shows the communication between the Home RF device, the microcontroller, the DLL and the software application. For the communication between the DLL and the Home RF hardware via the microcontroller four DLL functions with diﬀerent tasks are provided: 120 I2C EEPROM Hardware PC with application software Application USB_HRF_DLL USB framework Home RF functionality Home RF Microcontroller USB connection ICs Figure 3: Interaction between the application and the microcontroller functions used by the radio module. • USB Dll HRF Set Config: this function allows for the conﬁguration settings to be sent to the Home RF device (length of data payload, address width, destination address for received data, enable/disable cyclic redundancy check (CRC), communication mode (Direct or ShockBurstTM ), RF data rate, crystal frequency, RF output power, frequency channel and receive/transmit operation ). • USB Dll HRF Transfer Data: this function allows the Home RF radio module to operate in Active Receive/Transmit modes (Direct or ShockBurstTM ). The ShockBurst technology uses on-chip FIFO (First In, First Out) to clock in data at a low data rate and transmit at a high rate (1 Mbps) thus enabling extremely power reduction. In direct mode the radio module functions like a traditional RF device. Data must be sent at 1Mbps or 250kbps at low data rate setting, for the receiver to detect the signals. • USB Dll HRF Receive Init: this function prepares the microcontroller programs to accept data from the Home RF chip, at the rising edge of the DR signal, and to store it internally. No data are passed on to the DLL and thus to the PC. • USB Dll HRF Receive Data: this function allows to read in the received data via the DLL. These four DLL functions are related to the following three functions: HomeRF Set Config, HomeRF Transmit Data and HomeRF Receive Data in the microcontroller, as shown in Fig. 4. When the exported functions of the DLL are called (through the framework of the microcontroller), the necessary Home RF functions belonging to them are activated. 121 HomeRF_Set_Config USB_Dll_HRF_Set_Config HomeRF_Transmit_Data USB_Dll_HRF_Transfer_Data USB_Dll_Receive_Data HomeRF_Receive_Data USB_Dll_Receive_Init DLL Microcontroller Figure 4: Relation between the DLL functions and the HomeRF functions in the microcontroller 122 List of Publications Journal Papers 1. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loﬀeld., “Timing Oﬀset Estimation for Transmitted-Reference Impulse Radio UWB Systems”, IEEE Journal of Communications and Networks, submitted October 2008. Conference Papers 1. P. Ubolkosold, G. F. Tchere, S. Knedlik, O. Loﬀeld, “A New Closed-Form Frequency Estimator in the Presence of Fading-Induced Multiplicative Noise”, 67th IEEE Vehicular Technology Conference, Singapore, May 2008. 2. P. Ubolkosold, G. F. Tchere, S. Knedlik, O. Loﬀeld, “An Improved Frequency Oﬀset Estimator for SIMO-OFDM with Maximum Estimation Range”, 67th IEEE Vehicular Technology Conference, Singapore, May 2008. 3. P. Ubolkosold, G.F. Tchere, S. Knedlik, O. Loﬀeld, “Simple Carrier Frequency Oﬀset Estimators in Frequency Flat-fading Channels”, IEEE International Conference on Communications (ICC 2007), Glasgow, Scotland, June 2007. 4. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loﬀeld, “Bit Error Performance of Diﬀerential Impulse Radio UWB systems”, 65th IEEE Vehicular Technology Conference, VTC Spring 2007, Dublin, Ireland, April 2007. 5. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loﬀeld, “Communication and Localization in Wireless Sensor Networks using Diﬀerential Impulse Radio UWB systems”, 4th Workshop on Positioning, Navigation and Communication (WPNC 2007), Hannover, Germany, March 2007. 6. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loﬀeld, “Bit Error Performance of UWB Diﬀerential Transmitted Reference systems”, IEEE International Symposium on Communications an Information Technologies (ISCIT), Bangkok, Thailand, October 2006. 7. G.F. Tchere, P. Ubolkosold, S. Knedlik, O. Loﬀeld, K. Witrisal, “Data-Aided Timing Acquisition in UWB Diﬀerential Transmitted Reference systems”, IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Helsinki, Finland, September 2006. 123 8. P. Ubolkosold, G.F. Tchere, S. Knedlik, O. Loﬀeld, “Nonlinear Least-Squares Frequency Oﬀset Estimation and its Simpliﬁed Versions for Flat fading Channels ”, IEEE International Symposium on Communications an Information Technologies (ISCIT), Bangkok, Thailand, October 2006. 9. P. Ubolkosold, G.F. Tchere S. Knedlik , O. 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