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Multiuser pre-filtered ultra-wideband systems Ahmadian, Zahra 2013

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Multiuser Pre-Filtered Ultra-Wideband SystemsbyZahra AhmadianB.Sc., Ajman University of Science and Technology, United Arab Emirates, 2005M.A.Sc., University of British Columbia, Canada, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2013c? Zahra Ahmadian, 2013AbstractUltra-wideband (UWB) communication enables license-exempt transmission withvery low power over large bandwidths. The technology can provide very high datarates over short transmission ranges to support applications such as real-time datastreaming, interchip communication and wireless memory.The work in this thesis considers a particular type of high data-rate multiuserdirect-sequence UWB (DS-UWB) with popular and commonly used binary UWBsignalling. The system consists of multiple low-complexity DS-UWB transceivers(nodes) and a central unit that is more powerful in terms of signal processing capa-bilities. We mostly focus on the transmission from the central unit to the nodes.We address the following main questions: (1) What signal processing should beapplied at the central unit to enable simple yet reliable detection at low-complexitynodes? (2) How can the system performance be optimized in the presence of im-perfect channel estimation? (3) Is it possible to improve the system performanceby incorporating the binary detector structure in the transmitter design? (4) Howcan the performance of a network of multiple UWB nodes communicating through acentral relay be optimized?For question (1), we propose to shift the signal processing load from the nodes tothe central unit via pre-filtering (the combination of pre-rake and pre-equalization)of the transmit signal at the central node, and we provide filter design strategies forthe downlink communication.iiAbstractQuestions (2) is addressed by studying the impact of errors in estimation of thechannel impulse response at the central unit. Two mathematical models are proposedto represent the channel estimation error and robust strategies are formulated for thedesign of downlink pre-equalization filters (PEFs). For the popular binary UWBsignalling, the real-part of the received signal contains sufficient statistics for signaldetection. Hence the widely linear design of PEFs is proposed to answer question(3).As for question (4), we extend our design methods to multi-way internode com-munication via a central relay. Two relaying strategies namely, detect-and-forwardrelaying and filter-and-forward relaying with partial and full self-interference cancel-lation are devised.iiiPrefaceHereby, I declare that I am the first author of this thesis. The following publicationshave resulted from the thesis research.Journal Papers1. Zahra Ahmadian, Lutz Lampe, and Jan Mietzner, ?Multiuser Two-Way Relay-ing Schemes for UWB Communication,? Submitted (Included in Chapter 4).2. Zahra Ahmadian and Lutz Lampe, ?Robust Design of Widely Linear Pre-Equalization Filters for Pre-Rake UWB Systems,? IEEE Transactions on Com-munications, vol. pp, no. 99, pp. 1-12, 2013 (Included in Chapter 3).3. Zahra Ahmadian, Michael Botros Shenouda, and Lutz Lampe, ?Design of Pre-Rake DS-UWB Downlink with Pre-Equalization,? IEEE Transactions on Com-munications, vol. 60, no. 2, pp. 400-410, Feb. 2012 (Included in Chapter 2).Conference Papers1. Zahra Ahmadian, and Lutz Lampe, ?Robust Pre-Equalization for Pre-RakeUWB Systems with Spectral Mask Constraints,? in IEEE Global Communi-cations Conference (GLOBECOM), Anaheim, USA, Dec. 2012 (Included inChapter 3).ivPreface2. Zahra Ahmadian, and Lutz Lampe, ?Widely Linear Design of Pre-EqualizationFilters for Multiuser Pre-Rake UWB Systems,? in IEEE International Con-ference on Ultra-Wideband (ICUWB), Bologna, Italy, Sept. 2011 (Included inChapter 3).3. Zahra Ahmadian, Michael Botros Shenouda, and Lutz Lampe, ?TransceiverDesign for Multiuser Broadcast in Pre-Rake UWB Communication,? in IEEEInternational Symposium on Antenna Technology and Applied Electromagnetics(ANTEM), Ottawa, Canada, Invited Paper , Jul. 2010 (Included in Chapter 2).4. Zahra Ahmadian, Michael Botros Shenouda, and Lutz Lampe, ?Design of Mul-tiuser Pre-Rake Systems for Reliable Ultra-Wideband Communications,? inIEEE International Conference on Communications (ICC), Cape Town, SouthAfrica, May 2010 (Included in Chapter 2).Unless stated differently, for all publications, I conducted the survey on relatedtopics, identified the challenges, formalized the suggested solution, performed theanalysis, and carried out all of the simulations. I also wrote all paper drafts. Mysupervisor, Prof. Lutz Lampe, guided my research, validated analysis and method-ology, and edited the manuscripts for papers co-authored by him. Parts of the thesisare a result of research collaboration with additional contributors. The collaborators?contribution is listed below.1. The co-author in journal paper 3 and conference paper 3 and 4, Dr. MichaelBotros Shenouda, helped with the analysis and editing the manuscripts.2. Part of the work in journal paper 1 (included in Chapter 4), in particularthe methodologies related to the detect-and-forward relaying scheme, was com-pleted in collaboration with Dr. Jan Mietzner.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 UWB Communication . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Pre-filtered UWB Systems . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Robust Pre-Equalization . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Multi-way Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 132 Pre-Equalization for DS-UWB Communication . . . . . . . . . . . 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18viTable of Contents2.2.1 Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Power Minimization Design . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.2 Problem Formulation and Convex Design . . . . . . . . . . . 252.3.3 Properties of the Optimal Design . . . . . . . . . . . . . . . . 272.4 Mean Squared Error Minimization Design . . . . . . . . . . . . . . . 282.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Downlink Transmission with Power Minimization . . . . . . . 332.5.2 Transceiver Design for Maximum Quality of Service . . . . . 352.5.3 Alternative Design Approaches . . . . . . . . . . . . . . . . . 392.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Robust Pre-Equalization for DS-UWB Communication . . . . . . 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2.1 Description in Matrix Form . . . . . . . . . . . . . . . . . . . 473.2.2 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . 503.3 Robust PEF Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.1 Uncertainty Models . . . . . . . . . . . . . . . . . . . . . . . 513.3.2 Design Objective and Optimization . . . . . . . . . . . . . . . 523.4 Widely Linear PEF Design . . . . . . . . . . . . . . . . . . . . . . . 603.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 WL Robust Design with Average MSE Constraints . . . . . . 623.4.3 WL Robust Design with Instantaneous MSE Constraints . . . 64viiTable of Contents3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.1 Advantages of WL Design . . . . . . . . . . . . . . . . . . . . 663.5.2 Advantages of Robust Design . . . . . . . . . . . . . . . . . . 683.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764 Multi-way Relaying for DS-UWB Communication . . . . . . . . . 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Multi-way Detect-and-Forward Relaying . . . . . . . . . . . . . . . . 814.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.2 Filter Design for DTF Relaying . . . . . . . . . . . . . . . . . 864.2.3 Filtering with User Specific Scaling Factor . . . . . . . . . . . 894.2.4 Widely Linear Filtering for DTF Relaying . . . . . . . . . . . 924.2.5 Downlink BER Analysis . . . . . . . . . . . . . . . . . . . . . 934.3 Multi-way Filter-and-Forward Relaying . . . . . . . . . . . . . . . . 934.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 944.3.2 Filter Design for FF Relaying . . . . . . . . . . . . . . . . . . 974.3.3 Iterative Design based on User Specific Scaling Factor . . . . 994.3.4 Widely Linear Filtering for FF Relaying . . . . . . . . . . . . 1004.3.5 BER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4.1 DTF Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4.2 FF Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Summary of the Thesis and Topics for Future Research . . . . . . 1145.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . 1145.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . 116viiiTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117AppendicesA Proof of Lemma 2.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 128B Proof of Lemma 2.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 130C Proof of Lemma 2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 132D Proof of Theorem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 133ixList of Figures1.1 FCC limits on power spectral density of transmitted UWB signal forindoor UWB communication. . . . . . . . . . . . . . . . . . . . . . . 21.2 European ETSI EN-302-065 limits on power spectral density of trans-mitted UWB signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The network setup for multi-user DS-UWB system. . . . . . . . . . . 72.1 Block diagram of a downlink multiuser pre-rake UWB broadcast system. 202.2 Block diagram of the equivalent uplink multiuser pre-rake UWB system. 292.3 Required transmission power versus PEF length (Lq) with M = 2transmit antennas for U = 2 users and given MSE (?1 = ?2 = ?).Averaged results for 400 channel realizations of CM2. Inner figure:Number of infeasible channel realizations versus PEF length. . . . . . 342.4 SINR for user u = 1 versus the spreading code length (N). Resultsaveraged over 500 channel realizations of CM2. M = 2 transmittingantennas, U = 2 users at Pmax = 14 dB. . . . . . . . . . . . . . . . . . 352.5 SINR for user u = 1 versus PEF length (Lq). Results averaged for 500channel realizations of CM2. M = 2 transmitting antennas, U = 2users, spreading code length of N = 8 and Pmax = 14 dB. . . . . . . . 37xList of Figures2.6 MSE for user u = 1 versus PEF length (Lq). Results averaged over500 channel realizations of CM2 and CM6, respectively. M = 2 trans-mitting antennas, U = 2 users, spreading code length of N = 8 andPmax = 14 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 BER versus the maximum transmission power Pmax. Results averagedfor 500 channel realizations of CM2. M = 2, 3 and 4 transmittingantennas, spreading code length N = 8 and U = 2 users. Solid lines:pre-equalization filter length of Lq = 10. Dashed lines: pre-rake trans-mission without PEF. Markers: simulation results, lines: analyticalresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 BER versus the maximum transmission power Pmax. Results averagedfor 500 channel realizations of CM2. M = 4 transmitting antennas,spreading code length N = 8 for U = 2, 3 and 4 users. Solid lines:pre-equalization filter length of Lq = 10. Dashed lines: pre-rake trans-mission without PEF. Dashed dot line: Single-user pre-equalizationfilter. Markers: simulation results, lines: analytical results. . . . . . . 402.9 Comparison of filter designs. U = 2 users, filter length of Lq = 10,M = 2 transmitting antennas and unequal receiver thermal noise of?2z,1 = 0.1 and ?2z,2 = 1. a) The first user?s equalizing receiver gain?1 versus second user?s receiver gain ?2 for 500 channel realizations ofCM2. b) Average MSE results for our proposed design and the designwith common receiver gains (?common-?? design). . . . . . . . . . . 41xiList of Figures3.1 Block diagram of a downlink multiuser pre-rake UWB broadcast sys-tem. Multiuser pre-equalization and pre-rake combining are applied atthe base station with M antennas. Single-antenna users with simpleslicing detector are considered. . . . . . . . . . . . . . . . . . . . . . . 463.2 Empirical CDF of the MSE of the first user for different training se-quence lengths. PEF length of Lf = 10 with M = 4 transmittingantennas and U = 2 users are considered. . . . . . . . . . . . . . . . . 533.3 Required transmit SNR = Pmin/?2z versus the BER, for widely linearand linear design schemes and (i) ideal CSI (ii) imperfect CSI (train-ing sequence length K = 2Lp) using the robust average MSE design.Results generated for U = 2 receiving users, M = 4 transmitting an-tennas and PEF of length Lf = 10. . . . . . . . . . . . . . . . . . . . 683.4 MSE1 and Pmin versus ratio of training sequence and pre-rake filterlengths for robust WL PEF design with average MSE constraints andnon-robust design. MSE threshold of ?1 = ?2 = 0.08 for U = 2 users,M = 4 transmitting antennas and PEF length of Lf = 10. . . . . . . 703.5 Empirical CDF of the instantaneous MSE of the first user for a pre-rake system with U = 1, 2 and 4 users. Results for robust WL PEFdesign with average MSE constraints and non-robust WL design. MSEthreshold set as ?u = 0.08 for u = 1, 2, 4,M = 4 transmitting antennas,PEF length of Lf = 10 and training sequence length of K = 2Lp areconsidered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71xiiList of Figures3.6 WL robust PEF design with instantaneous MSE constraints and boundon uncertainty region set as  = 0.1, 0.15, . . . , 0.3, for a pre-rake UWBsystem with U = 2 users. MSE threshold of ?1 = ?2 = 0.08, M = 4transmitting antennas, PEF length of Lf = 10, and training sequencelength of K = 2Lp are considered. Results for WL non-robust designare included for comparison. . . . . . . . . . . . . . . . . . . . . . . . 723.7 WL robust PEF design with instantaneous MSE constraints and chan-nel uncertainty model applied on a selected set of Ls maximum-energychannel taps Ls = 10, 20, Lh, for a pre-rake UWB system with U = 2users. MSE threshold of ?1 = ?2 = 0.08, M = 4 transmitting antennas,PEF length of Lf = 10, and training sequence length of K = 2Lp areconsidered. Results for WL non-robust design is included for comparison. 743.8 WL robust PEF design with instantaneous MSE constraints and boundon uncertainty region for the overall channel error vector ?u. Con-sidered are single UWB channel estimation vectors h?1 and h?2 and1000 different instances of channel error vectors ?1 and ?2 and ? =0.08, 0.12, 0.16, 0.2. Pre-rake UWB system with U = 2 users, MSEthreshold of ?1 = ?2 = 0.08, M = 4 transmitting antennas, PEFlength of Lf = 10, and training sequence length of K = 2Lp. Resultsfor WL non-robust design are included for comparison. . . . . . . . . 754.1 Block diagram of the nodes in transmit and receive modes. . . . . . . 824.2 Block diagram of the central unit for the two phase DTF relaying. . . 834.3 Block diagram of the central relay with pre/post-rake filtering andmulti-way FF relaying. . . . . . . . . . . . . . . . . . . . . . . . . . . 95xiiiList of Figures4.4 The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme withU = 4 users, M = 2 and M = 4 antennas at the relay and Lf = 10.Comparison between linear and widely linear designs. . . . . . . . . . 1034.5 The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme withU = 4 users, M = 4 antennas at the relay and Lf = 10. Comparisonof the BER for interference cancellation lengths of Lc = [1, 3, 5, 30]. . 1054.6 The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme withU = 4 users, M = 4 antennas at the relay. Comparison of the BER forpost/pre-rake scheme without post/pre-equalization, post/pre-equalizedDTF relaying with Lf = 1 and Lf = 10. . . . . . . . . . . . . . . . . 1064.7 The sum MSE versus relay transmit SNR, SNRTx,R = Pmax/?2d(u), forDTF relaying scheme with U = 4 users, M = 4 antennas at the relayand Lf = 10. Comparison between designs with same receiver scalingfactor for all users and the design with user specific scaling factor. . . 1074.8 The sum MSE versus user?s noise level difference ?22/?21 in dB, for DTFrelaying scheme with U = 2 users, M = 4 antennas at the relay andLf = 10. Comparison between designs with and without interferencecancellation and same receiver scaling factor for all users and the designwith user specific scaling factor with self-interference cancellation. . . 108xivList of Figures4.9 The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for FF relaying scheme withU = 4 users, M = 4 antennas at the relay and Lq = 20. Com-parison between linear and widely linear designs, with and withoutself-interference cancellation. . . . . . . . . . . . . . . . . . . . . . . . 1094.10 The Analytical BER between a pair of source and destination nodesversus relay transmit SNR, SNRTx,R = Pmax/?2d(u), for FF relayingscheme with U = 4 users, M = 4 antennas at the relay and Lq = 20.Comparison of the BER for self-interference cancellation lengths ofLc = [1, 3, 5, 7, 11, 42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.11 The sum MSE versus user?s noise level difference ?22/?21 in dB, forFF relaying scheme with U = 2 users, M = 4 antennas at the re-lay and Lq = 20. Comparison between designs with and withoutself-interference cancellation and same receiver scaling factor for allusers and the design with user specific scaling factor, and with self-interference cancellation. . . . . . . . . . . . . . . . . . . . . . . . . . 111xvList of Abbreviations and SymbolsAcronymsAF Amplify-and-ForwardAO Alternating OptimizationAWGN Additive White Gaussian NoiseBER Bit Error RateBPSK Binary Phase Shift KeyingCDF Cumulative Density FunctionCDMA Code Division Multiple AccessCM Channel ModelCSI Channel State InformationDAA Detect-and-AvoidDF Decode-and-ForwardDS-UWB Direct Sequence Ultra-WidebandDTF Detect-and-ForwardFCC Federal Communications CommissionFF Filter-and-ForwardFIR Finite Impulse ResponseIR-UWB Impulse Radio Ultra-widebandISI Inter-symbol InterferencexviList of Abbreviations and SymbolsKKT Karush Kuhn TuckerLDC Low Delay CycleLMI Linear Matrix InequalityLMMSE Linear Minimum Mean-Square ErrorMB-OFDM Multiband Orthogonal Frequency Division MultiplexingMIMO Multiple-Input Multiple-OutputMISO Multiple-Input Single-OutputMMSE Minimum Mean-Square ErrorMSE Mean-Square ErrorMUI Multiuser InterferencePEF Pre-Equalization FilterPSD Power Spectral DensityQCQP Quadratically Constrained Quadratic ProgrammingQoS Quality of ServiceRFID Radio Frequency IdentificationSC-UWB Single Carrier Ultra-WidebandSINR Signal-to-Interference-and-Noise RatioSISO Single-Input Single-OutputSNR Signal-to-Noise RatioTDMA Time Division Multiple AccessTH-UWB Time-Hopped Ultra-WidebandUWB Ultra-WidebandWL Widely LinearWPAN Wireless Personal Area NetworkWSN Wireless Sensor NetworkxviiList of Abbreviations and SymbolsNotations and OperatorsBold upper case and lower case letters denote matrices and vectors, respectively. Theremaining notation and operators used in this thesis are listed below:(?)T Transpose(?)H Hermitian transpose(?)? Complex conjugateIN N ?N identity matrixdiag{?} (Block) Diagonal matrixE{?} Statistical expectationtrace{X} Trace of square matrix XN (?, ?2) Gaussian distribution with mean ? and variance ?2Q(?) Gaussian Q-functionsgn(?) Sign function<{?} Real part of a complex number={?} Imaginary part of a complex number? ? ? Euclidean norm[.] ? [.] Linear convolution0n?m All-zero n?m matrixIn Identity matrix of dimension n? n0n All-zero column vector of length nxviiiAcknowledgmentsI would like to express my sincere gratitude to my supervisor Professor Lutz Lampe,for his continuous support during the course of my Ph.D. and master?s studies. Hehas been a patient and motivational mentor and an immensely knowledgeable advisorat all times.I am grateful to my colleagues at the data communications lab for creating aninspirational environment for learning and exchanging ideas. In particular, I wouldlike to thank Chris Snow, Jeebak Mitra, Anand Oka, Roee Diamant, Nasim Arian-poo, Pedram Samadi, Farzad Moghimi, Dana Hoffmann and Julien Renard for theirfriendship and support.Special thanks to my parents who have unconditionally loved and supported meat all times in achieving this and every other life milestone.Last but not least, I would like to thank my husband, Hani, for his unwaveringlove, patience and encouragement at every step towards completion of this work, andfor the many night and day shifts of caring for our lovely daughter, Kimia, duringthe write up of this thesis.xixDedicationTo my mother Nasrin for her love, support and countless days of daycare to mydaughter, Kimia, during the final months of writing this thesis.xxChapter 1IntroductionScarce access to the wireless spectrum and the cost associated with exclusive usage ofthe spectrum have been pushing system designers to consider wireless solutions with-out license requirements. The release of the Federal Communications Commission(FCC) ruling in United States [1], which allowed license-free ultra-wideband (UWB)transmission for data communication, radar and safety applications over the 3.1 -10.6 GHz band, opened the door for considering this technology for a wide range ofapplications. In this chapter, first the unique characteristics of UWB communicationare discussed. Then, as pertinent for the contributions of this thesis, the concept ofpre-filtering in UWB systems is motivated and the literature on robust filtering andmulti-way relaying is reviewed. Finally, an overview of the thesis is presented.1.1 UWB CommunicationAs the name implies, UWB communication is based on the transmission of signalsoccupying very large (ultra-wide) bandwidths. According to the FCC ruling [1], anUWB signal is specified as having a minimum bandwidth of 500 MHz or a minimumfractional bandwidth of 0.2. The fractional bandwidth is defined as 2(fH?fL)/(fH +fL), where fL and fH are the lower and higher ?3 dB points in the signal spectrum,respectively. According to [1], unlicensed use of UWB devices is generally permittedin the 3.1 GHz to 10.6 GHz range. However depending on the application and1Chapter 1. Introduction1 2 3 4 5 6 7 8 9 10 11?80?75?70?65?60?55?50?45?40?35GHzdBm/MHzFigure 1.1: FCC limits on power spectral density of transmitted UWB signal forindoor UWB communication.the communication environment, the transmissions are subject to stringent emissionlimits on the signal power spectral density (PSD), as low as -41.3 dBm/MHz asper the FCC ruling depicted in Figure 1.1 for indoor communication. In Europe,the UWB emission is regulated under the ETSI EN 302 065 standard [2] accordingto which, the minimum bandwidth for UWB signal is 50 MHz. Figure 1.2 showsthe limits on the UWB signal PSD set by ETSI EN 302 065 standard. The limitsapply to indoor and outdoor communication. UWB transmission is allowed in the 6- 8.5 GHz band with signal PSD limited to -41.3 dBm/MHz. Additional frequencybands in the ranges of 2.7 - 4.8 GHz and 8.5 - 10.6 GHz are available for UWBtransmission with the following restrictions. The UWB transmission in 2.7 - 4.8GHz is allowed with detect-and-avoid (DAA) or low duty cycle (LDC) interferencemitigation techniques. In the 8.5 - 10.6 GHz band, UWB transmission is allowed2Chapter 1. Introduction1 2 3 4 5 6 7 8 9 10 11?90?80?70?60?50?40GHzdBm/MHz With DAA or LDC With DAA Figure 1.2: European ETSI EN-302-065 limits on power spectral density of transmit-ted UWB signal.with DAA. In the DAA scheme the UWB device is required to sense the channeland if another system is detected within the operational bandwidth then the UWBsystem should avoid transmission until the detected system disappears. In the LDCmode, the standard limits the cumulative duration for which the UWB transmitteris turned on over a defined observation period. Such restrictions on the PSD of thetransmitted signal and the DAA and LDC requirements are due to the fact that UWBsystems operate as a secondary underlay system. The UWB radios are required tocoexist with primary technologies with licensed operation without causing harmfulinterference on the primary system.According to the Shannon theorem, the capacity, C, of the additive white Gaus-sian noise (AWGN) channel is related to the signal bandwidth, B, and the received3Chapter 1. Introductionsignal power, PRx, asC = B log2(1 + PRxB N0), (1.1)where N0 represents the PSD of the AWGN. From (1.1), considering the restrictionson the PSD of the transmitted UWB signal, the large bandwidth of UWB signalsgives UWB systems the potential for a) achieving high data rates (of the order ofhundreds to thousands of Mbps) over short distances, and b) requiring lower transmitpower compared to narrowband systems for a given data rate. The first propertymakes UWB systems an attractive candidate for high data rate wireless personalarea networks (WPANs) as per applications defined in [3, 4, 5, 6]. Lower transmitpower translates to longer battery life for UWB transceivers, a factor that is crucialfor wireless sensor networks (WSNs). Additionally, the narrow pulse width results invery fine timing resolution and resolvable multipath components which is desirablein localization, radar and imaging applications. Multiple reflections of the transmitsignal can be coherently combined at the receiver using rake combining schemes toimprove the signal-to-noise ratio (SNR) at the receiver.The UWB communication channel is characterized by a large number of multipathcomponents and a relatively long (in terms of transmission symbol intervals) delayspread. Some of the early works that proposed statistical models of UWB channelsinclude [7, 8, 9]. In reference [7] a stochastic tapped delay line model was developedbased on channel measurements from [9]. Also using these measurements, a differentmodel considering the clustering of the multipath components of the channel wasderived in [8]. These works were followed by efforts supported by the IEEE 802.15.3aand IEEE 802.15.4a task groups, which led to the development of a modified Saleh-Valenzuela model that was used during the standardization of UWB systems. We4Chapter 1. Introductionrefer to [10, 11, 12] for details of this model.There are two main categories of UWB signalling, a) carrier-free impulse radioUWB (IR-UWB), and b) carrier-modulated UWB signalling. The IR-UWB, whichis the original flavour of UWB, is based on the transmission of carrier-free narrowpulses [13], and is associated with low data rate applications such as WSNs, imaging,radar and localization. Carrier-modulated UWB signalling is tailored toward highdata rate UWB transmission and can realize the promise of very high data rates overshort distances (< 10 meters) [14]. The category includes two main variants, namely,multi-band orthogonal frequency division multiplexing (MB-OFDM) UWB [15], andsingle-carrier UWB (SC-UWB) [16, 17, 18].MB-OFDM UWB transmission is standardized via the ECMA-368 standard [19].According to the ECMA-368, the UWB spectrum of 3.1 - 10.6 GHz is divided into14 bands of width of about 500 MHz each. OFDM modulation using 122 sub-carriers(combined data, pilot and guard subcarriers) within each band is used for transmis-sion. However, MB-OFDM UWB does not benefit from resolvable multipath pro-cessing characteristics specific to UWB signalling, since the subcarriers are processedindividually. Results from [20] show that for a typical indoor environment, resolvablemultipath components at the receiver are observed at bandwidth of order of about200 MHz. Deep fading on any of the subcarriers can have a significant effect on theoverall BER. Therefore, techniques such as interleaving and forward error correctionare implemented for performance improvement [21, 22].SC-UWB transmission is based on transmission of carrier modulated pulses overfrequency bands of width greater than 500 MHz. Hence, the transmission retainsunique characteristics of UWB signalling outlined earlier and also includes the bene-fits associated with base-band signal processing such as reduced sampling rate. Using5Chapter 1. Introductionfrequency mixers for frequency up and down conversion reduces the sampling require-ment on the digital-to-analog converter at the transmitter and the analog-to-digitalconverter at the receiver. Direct-sequence UWB (DS-UWB) is one of the most pop-ular SC-UWB transmission schemes. In DS-UWB, the transmit data is spread overa pre-defined spreading sequence prior to transmission and de-spreading is appliedat the receiver [23]. Furthermore, rake combining is performed to improve the re-ceived SNR for symbol-by-symbol detection. We note that direct-sequence has alsobeen considered as an option for carrier-free IR-UWB communication [24, 25, 26].Another approach for SC-UWB signalling is cyclic prefix UWB [16], where the datais transmitted block-wisely in the time domain while equalization is carried out inthe frequency domain. Cyclic prefix blocks with a length similar to that of theUWB channel impulse length are inserted between the data blocks to mitigate theinter-block-interference. In terms of complexity, cyclic prefix UWB is comparable toMB-OFDM UWB but has the advantages of lower peak-to-average power ratio andlower power consumption [16]. In general, DS-UWB has the potential of achievingvery high data rates but requires implementation of rake receivers with large numberof taps and potential equalization to suppress inter-symbol interference (ISI). Rakecombining and equalization can be fairly costly for UWB systems. Hence the key torealizing the promise of low power and high data rate UWB links is in reducing thereceiver complexity.In this thesis a single-carrier multi-user DS-UWB system consisting of multi-ple low-complexity transceivers (nodes) equipped with a single antenna and a morecomplex central unit with multiple antennas is considered. For the downlink com-munication from the central unit to the nodes, according to the network setup shownin Figure 1.3, we propose to reduce the signal processing at the nodes by shifting6Chapter 1. IntroductionFigure 1.3: The network setup for multi-user DS-UWB system.the computations from the nodes to the central unit. This is accomplished via pre-filtering of the transmitted signals at the central unit. Furthermore, as relevantextensions of the work, robust pre-filtering in case of imperfect channel informationat the central unit, and relay-enabled communication between the nodes are studied.The proposed system structure is a viable solution for applications such as real-timestreaming in WPANs [3], wireless memory [27], intra-vehicle communication [28] andwireless inter-chip bus [29].1.2 Pre-filtered UWB SystemsPre-filtering refers to the filtering of a signal prior to its transmission. Through pre-filtering the phase and the amplitude of the transmit signal are adjusted such thatthe effects of the propagation channel, interference and distortion on the receivedsignal, after sampling, is minimized. Pre-filtering at the transmitter alleviates theequalization/filtering task at the receiver and hence reduces the complexity of the7Chapter 1. Introductionreceiver-side signal processing. Consider, for example, a wireless sensor networkwhich consists of a large number of relatively simple devices (sensor motes, radio-frequency identification (RFID) tags, etc.) with limited processing capabilities. Forsuch scenarios implementation of the rake, auto-correlation, or even energy detectorsat the nodes is challenging. Thus, it is desirable to move the computationally-heavytasks to a few dedicated devices, such as a fusion center or a base station, which havelarger processing capability compared to the nodes.One of the pre-filtering techniques that is considered in UWB systems is pre-rakecombining, also referred to as time-reversal in the IR-UWB literature. In pre-rakecombining the modulated signal is processed with the available channel informationprior to transmission. Two of the pioneering works that considered the use of pre-rake filters in UWB systems are [30] and [31]. However, the major drawback ofpre-rake combining is the residual ISI experienced at the receiver, which can resultin high error floors. For the single user case, reference [32] proposed channel phasepre-coding for ISI mitigation. Application of post-equalization (at the receiver) wasconsidered at [33]. Reference [34] proposed substituting the pre-rake filter with aminimum mean-square-error (MMSE) pre-filter of length greater than or equal tothe channel impulse response. In reference [35], combining the pre-rake filtering withpre-equalization at the transmitter was considered to reduce the residual ISI. Sincepre-equalization is performed at the transmitter (fusion center, base station, etc.),the advantage of employing simple receiver terminals (e.g. sensor nodes, tags, etc.) isretained. Combination of pre-rake combining at the transmitter with rake combiningat the receiver was considered in [36] for communication over multiple-input multiple-output (MIMO) channels. Multi-user operation of a pre-rake combined UWB systemvia time-division multiplexing was proposed in [37], where the ISI is avoided by8Chapter 1. Introductionimplementing large guard intervals, which of course affects the data rate.Outside the UWB context, the application of pre-filtering has been widely studiedfor code-division multiple access (CDMA) systems. For example, [38, 39, 40, 41, 42]consider finite impulse response (FIR) transmit filter design for multiple-input single-output (MISO) and single-input single-output (SISO) CDMA communication. Trans-mit filter optimization has also been extensively discussed in the MIMO transmissionsystem literature. Some examples of transmit filter design for frequency-flat channelscan be found in [43, 44, 45, 46]. The case of MIMO ISI channels, which is moreclosely related to UWB scenarios, is considered in, e.g., [47, 48, 49], albeit with morecomplex receiver structures than those desired for pre-rake UWB systems.In the work presented in this thesis, a serial concatenation of pre-filtering is usedfor suppressing the multiuser interference (MUI) and ISI in the downlink of themultiuser DS-UWB system described in Section 1.1. The pre-filter consists of a pre-equalization filter (PEF) and a pre-rake filter. The use of the pre-rake filter is apragmatic choice to reduce the PEF length required to effectively mitigate ISI andMUI. In UWB systems, different from most of the mentioned CDMA and MIMOworks, the long delay spread of UWB channels requires a multipath problem formu-lation for the pre-filter design. This and the presence of a pre-rake filter render thesystem description different from representations considered in the MIMO literatureon transmit filter design and necessitate new PEF design procedures for the proposedmultiuser DS-UWB system.1.3 Robust Pre-EqualizationPre-rake combining and PEF design for the downlink communication relies on theavailability of channel information at the central unit. Mismatch between the es-9Chapter 1. Introductiontimated and the actual channel impulse response between the central unit and thenodes, results in considerable performance degradation. Therefore, a practical im-plementation of pre-filter UWB systems should account for this mismatch due toimperfect channel estimation, which leads to the problem of robust PEF design.The robust design is based on a mathematical model of the error in the channelestimation which is incorporated into the PEF design procedure. One of the commonapproaches to robust design is a stochastic modelling of the error, which often lead todesign formulations that satisfy quality-of-service (QoS) constraints on average [50].An alternative robust design approach that provides a deterministic level of controlover satisfying the QoS constraints is based on bounding the norm of the error vector.The design based on bounded error guarantees that the QoS constraints are met forall instances of the error vector which have a vector norm lower than the specifiedthreshold [51].In the UWB literature, the case of imperfect channel state information (CSI)has been mostly studied for its effect on error rate performance at the receiver, e.g.[52, 53, 54]. In reference [55], multi-user precoding for a network of UWB sensorswith non-coherent receivers is considered, where ideal and statistical knowledge ofthe channel is assumed. However, the design of robust transmit filters and the effectof channel uncertainties in UWB systems with pre-filtering has not been addressedbefore.Outside the UWB literature, the design of robust transceivers has been studiedextensively in MIMO and cognitive radio literature [51, 56, 57, 58]. The robust designof linear and non-linear pre-coders for a multi-user system with QoS constraints aresuggested in [51] and [56], respectively. Reference [57] proposes multiuser precodingfor spectral sharing in cognitive radio applications. The design of robust precoders10Chapter 1. Introductionfor communication over frequency selective MIMO channels was considered in [58].However, the robust filter design procedures therein are not directly applicable tosystems with pre-rake filtering. Hence, new design formulations are required forrobust PEF design in pre-rake DS-UWB systems.1.4 Multi-way RelayingCooperative communication via a relay is one of the popular schemes for improvingthe throughput and extending the communication range in wireless networks. In arelaying scenario the message from a source is forwarded to the destination througha relay. When a direct link between the source and the destination exists, the relaychannel acts as an auxiliary channel for improving the throughput. In the absence ofa direct link between the source and the destination nodes, the relay channel becomesthe primary link between the nodes.Relaying protocols are defined according to the network setup and the type ofsignal processing performed at the relay. For example, based on the number ofsource and destination nodes transmitting simultaneously, one-way, two-way [59] ormulti-way relaying [60] schemes are defined. Examples of relay signal processinginclude amplify-and-forward (AF), filter-and-forward (FF), and detect-and-forward(DTF). In the AF scheme, the relay simply retransmits the scaled version of thereceived signal. The FF relaying is suited for frequency-selective channels. Thefiltering is optimized to partially resolve the ISI in the source to relay channel. Inthe DTF relaying, the signal received at the relay is processed to detect the messagetransmitted by the source. The message is then modulated for forwarding to thedestination. If channel or network encoding or pre-coding is applied at the source,the DTF relaying procedure is also referred to as decode-and-forward (DF) [61].11Chapter 1. IntroductionOther factors that define the specifics of the relaying protocol are the number ofrelay nodes and number of hops in each source to destination relay channel.Relaying can play a significant role in extending the range and throughput of UWBsystems, where the communication range is typically limited to less than 10 metersdue to restrictions on average PSD [62]. In particular, multi-way relaying which hashigher spectral efficiency compared to one-way and two-way relaying schemes, hasthe potential for enabling high data rate UWB links [60].Multi-way relaying involves multiuser interference cancellation. For frequency-flat fading channels with multiple antennas at the relay, techniques such as digital oranalog network coding [63, 64, 65], the combination of channel and network coding[66, 67], or MIMO precoding [68, 69, 70, 71, 72, 73, 74] have been considered tomitigate the MUI associated with multi-way relaying.Relaying over frequency-selective channels, a concept that is close to UWB relay-ing, is more challenging in general due to the presence of ISI. Some of the works thatinvestigate relaying over multipath fading channels with single carrier modulationinclude [75, 76, 77, 78, 79, 80]. One-way relaying with multiple DF relays was con-sidered in [75] for CDMA communication over multipath channels. In [76], one-wayFF relaying was proposed. The case for two-way FF relaying with multiple antennaswas considered in [78, 79].In the UWB literature, one of the earliest studies on relaying in IR-UWB wasperformed in [81]. It was shown that the full-duplex amplify-and-forward relayingscheme considered in narrowband communication is not suitable for UWB systemsdue to the backward coupling of the relay signal from transmit antenna to the receiveantenna. Therefore, in [82], a half-duplex DTF relaying was suggested to avoid thecoupling problem for UWB relaying. For the TH-UWB signalling scheme, [83, 84]12Chapter 1. Introductionproposed one-way IR-UWB relaying via space-time coding, and time-reversal, respec-tively. References [85, 86] consider relaying for transmit-reference UWB signallingwith analog network coding. AF relaying for differential transmit reference UWBsignalling is considered in [87, 88] for single and multiple-hop AF relaying schemes,respectively. One-way relaying with pre/post-rake combining at the relay was studiedin [89] for IR-UWB signalling with guard intervals of the order of channel spread.In [90] one-way decouple and forward relaying with rake receivers at the destinationnode was considered. To the best of our knowledge, there is no work in the literaturethat addresses multi-way relaying for high data rate DS-UWB communication.1.5 Overview of the ThesisConsidering the network setup described in Section 1.1, in this thesis we study thedesign of pre-filters and equalization at the central unit with the purpose of reducingthe complexity associated with signal detection at the nodes. The pre-filtering taskat the central unit is performed by the combination of pre-equalization and pre-rakefiltering.In Chapter 2, the PEF design problem is formulated for downlink communicationover multipath UWB channels. Two PEF design strategies, relevant to the consideredUWB system, are proposed: a) transmit power minimization subject to satisfying amean-square error (MSE) constraint, and b) sum MSE minimization while meetingan average transmit power constraint for the system. For both design schemes, arelationship between the MSE and the signal-to-interference-and-noise ratio (SINR)at the receiving node is established. The relationship is then used for analyticalperformance evaluation.As mentioned in Section 1.3, availability of accurate CSI is crucial to the PEF13Chapter 1. Introductiondesign. In Chapter 3 the effect of CSI uncertainty on the performance of the pre-filtered UWB system is investigated. Two uncertainty models, namely, a stochasticand a bounded uncertainty model are considered and robust PEF design strategies areproposed for the power minimization design problem from Chapter 2. Furthermore,in Chapter 3, widely linear (WL) design strategies matched to UWB transmissionwith real-valued (binary) modulation are devised and the benefits arising from sucha design are discussed.In Chapter 4 cooperative communication for pre-rake DS-UWB is considered. Asa relevant extension of the work in Chapter 2, we consider filter design for relaying thenode-to-node messages through the central unit. Multi-way relaying, which obtainsa higher spectral efficiency compared to one-way and two-way relaying schemes, isselected as the relaying protocol. Two multi-way relaying schemes, based on DTFand FF relaying that were introduced in Section 1.4, are considered for inter-nodecommunication via the central unit. In the DTF relaying scheme the ISI and MUIare mitigated using post/pre-rake and post/pre-equalization filters at the relay. Thefilter designs are based on the sum-MSE minimization and the constraints on therelay?s average transmit power. In the FF relaying, an equalization filter is combinedwith the post and pre-rake filtering. A sum-MSE criterion is used for design ofthe optimized filter at the relay. For both relaying schemes, partial and full self-interference cancellation is considered at the destination nodes. Moreover, the widelylinear counter-parts of the designs are devised for performance comparison.Chapter 5 summarizes the major results of the thesis and includes suggestions forfuture research directions.14Chapter 2Design of Pre-Equalization Filtersfor DS-UWB CommunicationIn this chapter the design of PEFs for the downlink of the multiuser DS-UWB sys-tem is addressed. We start by reviewing the relevant literature. Then we describethe details of the proposed multiuser DS-UWB system model and proceed with thederivation of the PEF design strategies, followed by numerical results and concludingremarks.2.1 IntroductionThe essential feature of UWB systems is the large signal bandwidth, which makesit possible to meet stringent constraints on transmission power mandated by regu-lators worldwide and to deploy battery-powered UWB transceivers with high life-expectancy. The unlicensed use and low-power consumption render UWB systemsparticularly useful for replacing the many wires in in-home applications and for pro-viding a communication infrastructure in wireless sensor networks [4, 5, 3].Notwithstanding the appeal of UWB technology, its widespread adoption is ham-pered by the relatively high receiver complexity. Hence, as discussed in Section 1.2,it is desirable to move computational complexity to a few dedicated devices, such asfusion center, base station, etc., which have essentially unlimited signal processing15Chapter 2. Pre-Equalization for DS-UWB Communicationpower. This can be accomplished by pre-rake UWB systems, in which rake combiningis performed at the transmitter rather than at the receiver.In the UWB literature, reference [30] considered the use of time division duplexingfor downlink and uplink transmission, which allows channel estimation via the reverselink, and compared different pre-rake configurations at the transmitter. In [31] it wasshown that partial-pre-rake combining at the transmitter can achieve a performanceclose to that of a partial-rake receiver. Following [30, 31], there have been manyworks on pre-rake UWB system design, e.g. [32, 91, 92, 93, 37, 33, 34, 35, 36]. Theseinclude pre-rake UWB with channel phase precoding [32], where simple receivers per-form phase estimation and send phase information to the central transmitter via afeedback channel, and the design and analysis of high data rate pre-rake DS-UWBmultiple access systems [91, 92]. Furthermore, the benefits of using pre-rake com-bining together with antenna-arrays at the central transmitter were experimentallyexplored in [93]. Channel reciprocity was verified by experimental measurements andit was shown that using channel estimation algorithms the downlink channel can ef-fectively be estimated from the uplink received signal. The case of pre-rake filteringwith multiple transmitting antennas for multiuser UWB has been considered in [37],in which time-offsetting of signals for different users has been proposed to suppressMUI.Pre-rake filtering over UWB channels, with large delay spread, suffers from ISI atthe receiver. To overcome the remaining ISI, the application of post-equalization (atthe receiver) was proposed in [33] and it was shown to significantly improve the systemperformance. Another approach to reduce the ISI was proposed in [34] by substitutingthe pre-rake filter with a MMSE pre-filter of length greater than or equal to thechannel impulse response. In [35], the combination of pre-rake combining and pre-16Chapter 2. Pre-Equalization for DS-UWB Communicationequalization was extended to DS-UWB transmitters equipped with multiple antennasfor single user communication. In particular, [35] provided closed-form solutions forthe PEFs used in the single-user MISO DS-UWB system setting. The application ofFIR transmit filters to DS modulation has widely been studied for CDMA systems.For example, [38], [39] and [40] consider FIR transmit filter design for MISO CDMAand transmission with ISI resulting from short multipath channel lengths. Transmitfiltering for SISO CDMA is considered in [41, 42] where again the ISI is negligible.Transmit filter optimization for frequency-flat MIMO channels were studied in [43,44, 45, 46]. Pre-filtering for frequency-selective MIMO channels is considered in,e.g., [47, 48]. However, the proposed schemes require complex receiver structures.Furthermore, non-linear precoding has been applied to MIMO ISI channels in, e.g.,[49].In this chapter, we consider a multiuser pre-rake UWB broadcast communicationsystem with multiple antennas at the base station and single antenna users. Dif-ferent from the above-mentioned UWB literature, we propose the use and design ofmultiuser PEFs to simultaneously suppress ISI and MUI. Different from most of thementioned CDMA and MIMO works, our proposed design scheme follows a multipathproblem formulation. This renders the system description different from representa-tions considered in MIMO literature on transmit filter design for transmission overfrequency flat channels (e.g. [46, 44, 45]), and therefore the procedures presentedthere cannot directly be applied to the design of pre-rake UWB systems. While werestrict the receiver structure to be a simple decision device, we allow user-specificscaling factors in the filter optimization (see e.g. [45] for frequency-flat MIMO trans-mission). This provides additional flexibility compared to the modified Wiener filterdesigns previously proposed in the CDMA literature [39, 40] which consider a com-17Chapter 2. Pre-Equalization for DS-UWB Communicationmon scaling factor for all users.For the described UWB-transmission setup, we apply two complementary filteroptimization paradigms. First, considering the transmit power restrictions for UWBsystems, we aim at minimizing the transmit power required to attain a desired level ofQoS, measured in terms of MSE between the transmitted and estimated message, ateach user. We formulate this design problem as a convex optimization problem, whichmakes it amenable for solution by computationally efficient algorithms. Second, weassume a preset level of available transmit power and maximize the QoS, i.e., minimizethe weighted sum of each user?s MSE. We propose an efficient method for solving thisproblem by exploiting the duality (cf. e.g. [94, 95, 96, 97, 50]) between the DS-UWBdownlink and the dual uplink that employs rake combining and post-equalizationfilters at a central receiver. Our results show that the obtained multiuser PEF designsare highly effective in mitigating both ISI and MUI and thus (i) in reducing totaltransmission power required to meet QoS targets and (ii) in improving QoS given afixed power budget. Using a Gaussian assumption for ISI and MUI signal componentsthe benefit of the proposed PEF design strategies is also verified in terms of bit-errorrate (BER) performance.2.2 System ModelFor the multiuser pre-rake DS-UWB communication system considered in this the-sis, the base station is equipped with M antennas to enable high data rates, whileeach of the U users employs a single-antenna receiver. Figure 2.1 shows the blockdiagram of the discrete-time system model. As in e.g. [33, 34, 35] we consider thecomplex baseband discrete-time model, and thus baseband and carrier-modulatedtransmission are included.18Chapter 2. Pre-Equalization for DS-UWB Communication2.2.1 TransmitterWe consider a DS-UWB transmission system similar to DS-UWB considered in the802.15.3a task group, which also includes UWB transmission without spreading asa special case, e.g. [35]. For this system, we propose to incorporate multiuser pre-equalization filters qu,m, u = 1, . . . , U , m = 1, . . . ,M , (see Figure 2.1) to mitigatethe effects of ISI and MUI at the receivers. We use au[n] to denote the data symbolintended for the uth user at symbol time n. We assume that au[n] is chosen from aconstellation with unit average power, i.e., E{|au[n]|2} = 1 and that data symbolsare independent of each other (with respect to u and n). The duration of eachsymbol will be denoted T . The uth user?s data symbols are processed by M PEFsqu,m = [qu,m[0], . . . , qu,m[Lq ? 1]], m = 1, . . . ,M , of length Lq each. The output ofeach PEF, vu,m[n], is given byvu,m[n] = au[n] ? qu,m[n] =Lq?1?`=0qu,m[`]au[n? `] . (2.1)This output is then upsampled by a factor of N and processed with the user?s uniquespreading sequence cu[k] of length N and chip duration Tc = T/N . The spreadingsequences are normalized such that?Nk=1 |cu[k]|2 = 1. Following the spreading se-quence, the signal undergoes pre-rake combining using filters pu,m[k], m = 1, . . . ,M ,of length Lp. Each pre-rake combining filter is assumed to be the time reversed con-jugate version of the corresponding discrete channel impulse response, hu,m[k], (notethat [.]H applied to a scalar variable indicates complex conjugate)pu,m[k] = hHu,m[Lp ? k ? 1] , 0 ? k < Lp .19Chapter 2. Pre-Equalization for DS-UWB CommunicationFigure 2.1: Block diagram of a downlink multiuser pre-rake UWB broadcast system.The partial-pre-rake combining with Lp ? Lh includes all-pre-rake combining as aspecial case when Lp = Lh, where Lh is the length of hu,m[k]. The transmitted signalof user u from the mth antenna can be written assu,m[k] =??`=??vu,m[`]p?u,m[k ? `N ] , (2.2)where p?u,m[k] = cu[k] ? pu,m[k] combines the effects of spreading and pre-rake com-bining. Finally, the signal transmitted from the mth antenna, sm[k], is the sum of allsu,m[k] (see Figure 2.1),sm[k] =U?u=1su,m[k].2.2.2 ChannelWe adopt the channel model developed during the standardization efforts by the IEEE802.15.3a/4a task groups [12], according to which the channel impulse response canbe modelled ashu,m(t) =Lc?`=0Lr?k=0?k,`ej?k,`?(t? T` ? ?k,`) ,20Chapter 2. Pre-Equalization for DS-UWB Communicationwhere Lc is the number of clusters, Lr is the number of rays within a cluster, ?k,` is thetap weight of the kth component in the `th cluster, T` is the delay of the `th cluster,?k,` is the delay of the kth multipath component relative to the `th cluster arrivaltime T`, and ?k,` is uniformly distributed in [0, 2pi). The statistics of the parameters(?k,`, T`, ?k,`) are collectively specified in different channel models (CMs) accordingto different propagation environments [12]. The equivalent baseband discrete timechannel impulse response can be written ashu,m[k] = gTx(t) ? hu,m(t) ? gRx(t)???kTc, (2.3)where gTx(t) and gRx(t) are the transmitter pulse-shaping filter and the receiver noise-rejection filter that satisfy the first Nyquist criterion, respectively.2.2.3 ReceiverAs shown in Figure 2.1, the discrete-time received signal at each user is filtered usinga time reversed version of its spreading sequence. The filtered received signal is thensampled at time indices k = Nn+k0 where k0 = Lp +N ? 2?Nb(Lp +N ? 2)/Nc isthe sampling phase that is chosen to be the time at which the sampled desired signalreaches its maximum, cf. [35]. Hence, the sampled output at the uth receiver can bewritten asru[n] =M?m=1U?i=1??`=??wi,m,u[N`+ k0]vi,m[n? `] + zu[n] , (2.4)where zu[n] is the sampled noise at the uth receiver that is assumed to be additivewhite Gaussian with variance ?2z , and wi,m,u[k] = p?i,m[k] ? hu,m[k] ? cu[N ? 1 ? k]is the combined impulse response of the ith user spreading sequence and pre-rakecombining, channel impulse response from the mth transmitter antenna to the uth21Chapter 2. Pre-Equalization for DS-UWB Communicationuser, and the time reversed version of the spreading sequence of the uth user. Weassume that each user employs a simple symbol-by-symbol detection according toa?[n? n0] = D{?uru[n]} , (2.5)where D denotes the slicer operation (i.e., decision towards the nearest signal point),n0 is the decision delay common to (pre-)equalization systems, and ?u is the receivergain of the uth user (see Figure 2.1).Note that having user specific gains allows more degrees of freedom in the designof the transmit filters compared to the case with common receiver gain (as for ex-ample considered in [40, 39]). While receivers need to be notified about the gains, ifmultilevel transmission is applied, for the case of binary phase-shift keying (BPSK)signalling, scaling at the receiver need not actually be implemented as is obvious from(2.5). We adjust the decision delay as n0 = dNLq+2N+2Lp?k0?4N e/2, which attemptsto maximize the desired signal component at the decision time. Numerical evidencesuggests that little can be gained by further optimizing n0 for each individual chan-nel realization. Finally, it is assumed that the actual information transmission startsafter the first n0 bits.In the following two sections, we present two designs for the joint optimizationof the multiuser pre-equalization filters, qu,m, u = 1, . . . , U , m = 1, . . . ,M , andeach user?s scalar equalization gains, ?u, u = 1, . . . , U , subject to different pertinentobjectives and constraints.22Chapter 2. Pre-Equalization for DS-UWB Communication2.3 Power Minimization DesignIn this section, we will present the first design, whose objective is the minimizationof the total amount of transmitted power from the UWB central unit subject toachieving a physical layer QoS requirement for each user. Since UWB systems operateas spectrum underlay systems and need to avoid harmful interference at receiversof incumbent (licensed) wireless systems, minimization of transmission power is ofimmediate relevance for UWB devices.2.3.1 PreliminariesIn order to facilitate the exposition of our design, we first re-write (2.4) in the compactmatrix formru[n] =U?i=1qiW i,uaTi + zu[n] , (2.6)where au =[au[n], . . . , au[n ? Lt + 1]]? R1?Lt is the vector of data symbols of theuth user to be transmitted at time instances n, . . . , n?Lt +1, qu = [qu,1, . . . , qu,M ] ?C1?MLq is the concatenated multiuser PEF of user u, andW i,u =[W Ti,1,uW Ti,2,u . . . W Ti,M,u]Tis an MLq ? Lt block matrix. Each block W i,m,u is an Lq ? Lt Toeplitz matrix withvector[wi,m,u[k0], . . . , wi,m,u[N(Lw ? 1) + k0], 0Lq?1]as first row, and vector [wi,m,u[k0], 0Lt?1]T as its first column, where Lw = d(Lp +Lh +2N ? 3? k0)/Ne is the length of the overall channel impulse response measuredin data-symbol intervals. Due to the effect of the PEFs qi,u, the dimensions of au23Chapter 2. Pre-Equalization for DS-UWB Communicationand W i,u in (2.6) depend on Lt = Lq + Lw ? 1.Using the matrix form in (2.6), we proceed to obtain an expression for the sumof transmitted signal power over one symbol interval. This can be expressed asPDL = E???M?m=1N(n+1)?1?k=Nn???sm[k]???2???(2.7)=U?u=1M?m=1N(n+1)?1?k=NnE{??su,m[k]??2}=U?u=1PDLu , (2.8)where the last step follows from the independence of users? messages and their zero-mean constellations, and PDLu is the power of the uth user?s transmitted signal. Usingderivations analogous to that of the special case of the single-user system in [35], itcan be shown that PDLu is given byPDLu = qu?uqHu , (2.9)where ?u = diag{?u,1, ?u,2, . . . , ?u,M} is a block diagonal matrix whose blocks?u,m are Hermitian Toeplitz matrices with the first row defined as[?u,m[0], ?u,m[?N ], . . . , ?u,m[?N(Lq ? 1)]],where ?u,m[k] = p?u,m[k] ? p?Hu,m[?k].Next, we proceed to obtain an expression of the QoS measure of each user. Tothis end, we will consider the MSE and we will show that the MSE can be related toother QoS measures such as the SINR and BER. The MSE of the uth user is definedas?2u = E{|au[n? n0]? ?uru[n]|2}.24Chapter 2. Pre-Equalization for DS-UWB CommunicationUsing the expression in (2.6), this can be evaluated as?2u =1 + |?u|2U?i=1qiW i,uWHi,uqHi ? ?Hu en0WHu,uqHu ? ?uquW u,ueTn0 + |?u|2?2z ,(2.10)where en0 is a unit row vector whose elements are all zero except the nth0 element whichis equal to 1. Setting 1 + |?u|2quW u,uWHu,uqHu ? ?Hu en0WHu,uqHu ? ?uquW u,ueTn0 =??uquW u,u ? en0?2, the MSE in (2.10), can further be written as?2u =???[?uq1W 1,u, . . . , ?uquW u,u ? en0 , . . . , ?uqUW U,u, ?u?z]???2. (2.11)2.3.2 Problem Formulation and Convex DesignUsing the expressions of the total transmitted power in (2.9), we can now formulatethe power minimization problem. Given MSE constraints ?2u ? ?u, 0 ? ?u < 1,u = 1, . . . , U , we would like to design the multiuser PEFs and scalar equalizationgains such that the total transmission power is minimized; that isminq1 ... qU?1 ... ?UU?u=1qu?uqHu (2.12a)s.t. ?2u ? ?u, 1 ? u ? U . (2.12b)In order to show the equivalence of the design problem in (2.12) to a convex problem,we first observe that there is no loss of generality in considering real values for thereceiver gains ?u, u = 1, . . . , U . In fact, it can be verified that if the optimal solutionof (2.12) is the set {(qu, ?u)|u = 1, . . . , U}, then {(quej??u , |?u|)|u = 1, . . . , U} willalso yield the same value of the objective function in (2.12a) and will still satisfy25Chapter 2. Pre-Equalization for DS-UWB Communicationeach of the constraints in (2.12b). Using the above observation, we now proceed byshowing that the set of non-convex constraints in (2.12b) can be formulated as a setwhich is convex in qu and ?u = 1/?u. Indeed, using (2.11) each constraint in (2.12b)can be written as???[q1W 1,u, . . . , quW u,u ? en0?u, . . . , qUW U,u, ?z]???2? ?u?2u . (2.13)Now that we have shown that the MSE constraints can be represented as convexquadratic constraints in {(qu, ?u)}, we conclude the equivalence of the problem for-mulation in (2.12) to convex optimization by observing that the objective functionin (2.12a) is a sum of quadratic terms of the form qu?uqHu , where each ?u is a pos-itive semi-definite matrix. Using this observation, and the representation of MSEconstraints in (2.13), we can formulate the design problem in (2.12) as the followingquadratically constrained quadratic programming (QCQP) problem:minq1 ... qU?1 ... ?UU?u=1qu?uqHu (2.14a)s.t.???[q1W 1,u, . . . , quW u,u ? en0?u, . . . , qUW U,u, ?z]???2? ?u?2u , 1 ? u ? U.(2.14b)The class of QCQP problems is a subclass of convex optimization problems forwhich the global optimal solution can be efficiently obtained using interior-pointmethod [98].26Chapter 2. Pre-Equalization for DS-UWB Communication2.3.3 Properties of the Optimal DesignIn this subsection two properties of the optimal solution of the design problem in(2.14) are presented.Lemma 2.3.1 If the optimization problem in (2.14) is feasible, then the optimalsolution satisfies the inequalities in (2.14b) with equality. That is, ?2u = ?u for u =1, . . . , U .Proof The proof is included in Appendix A.Lemma 2.3.2 Let {q~u , ?~u |u = 1, . . . , U} be the optimal solution of the power mini-mization problem design in (2.14). For each user, the achieved SINR is related to itsMSE requirement by ?u = 1SINRu+1 .Proof The proof is included in Appendix B.Lemma 2.3.2 shows that the selection of the MSE as a measure of the QoS of eachuser is not restrictive, since the SINR that is achieved by each user using the optimalsolution of the power minimization problem in (2.14) is a function of each MSE con-straint ?u. While the given relation between MSE and SINR is well-known for MMSEreceiver optimization problems, e.g. [99, 95], here we have shown that the relationholds for the designs from the power minimization in (2.14). The above result can beused to evaluate the BER of the system in terms of the readily available MSE. Assum-ing the received MUI and ISI are approximated as zero-mean Gaussian distributedrandom variables and for example, BPSK modulation, then the corresponding BERat receiver terminal u is given byBERu = Q(?2 SINRu)= Q(?2?u? 2), (2.15)27Chapter 2. Pre-Equalization for DS-UWB Communicationwhere Q(.) is Gaussian Q-function [100].Considering the relation between the MSE and SINR and the Gaussian approx-imation, we expect that gains achieved by the proposed pre-equalization scheme interms of uncoded BER according to (2.15) translate into BER improvements also for(interleaved) coded systems through some function BER = f(SINR) which is specificto the applied coding scheme.2.4 Mean Squared Error Minimization DesignIn this section we propose a design approach that minimizes a weighted sum ofthe MSE of each user subject to a total power constraint denoted as Pmax. Usingthe expressions for MSEs and total power in (2.10) and (2.9) respectively, we canformulate the problem asminq1 ... qU?1 ... ?UU?u=1?u[1 + |?u|2(U?i=1qiW i,uWHi,uqHi + ?2z)? 2<{?Hu en0WHu,uqHu}](2.16a)s.t.U?u=1qu?uqHu ? Pmax , (2.16b)where ?u is the QoS weighting coefficient for user u. Unlike the power minimizationdesign in (2.14), the objective in (2.16a) is not jointly convex in all the design variables{(qu, ?u)}. Hence, the optimization problem in (2.16) is non-convex. To arrive at aneffective solver, we will make use of duality by considering a DS-UWB uplink withpower loading at the transmitters, and combined rake front-end and equalizationfiltering at the common receiver. Duality between downlink and uplink has beena useful tool for the design of multiuser systems, e.g. [94, 97, 50], we generalize28Chapter 2. Pre-Equalization for DS-UWB CommunicationFigure 2.2: Block diagram of the equivalent uplink multiuser pre-rake UWB system.this approach to UWB systems with (pre-) rake combining and (pre-) equalizationfiltering.The dual DS-UWB uplink consists of U single antenna transmitters and a receiverwith M antennas as shown in Figure 2.2. At each transmitter the data symbol au[n]is upsampled by a factor of N and spread with sequence cu[N?1?k], and the outputis multiplied by the transmission gain ?u. Hence the per user transmit power is |?u|2and the total transmit power for all users of the uplink is PUL =?Uu=1 |?u|2. Weassume that the channel impulse response between the uth transmitter and the mthreceiving antenna is the conjugate of the corresponding downlink channel hu,m[k]. Atthe receiver, the signal from each antenna is processed by a rake combining filter,pu,m[k], and the spreading sequence, cu[k]. The output is then downsampled andprocessed by an equalization filter, fu,m = [fu,m[0] , . . . , fu,m[Lq ? 1]]T . The outputsof these equalization filters are combined to obtain the input to the detector for theuth user. Using derivations similar to those of Section 2.3, one can verify that thedetector input signal rULu [n] can be written in the following matrix formrULu [n] =U?i=1?iaiWHu,ifu + zULu [n] , (2.17)where fu = [fTu,1 , . . . , fTu,M ]Tand zULu [n] is the noise at the input of the detector.29Chapter 2. Pre-Equalization for DS-UWB CommunicationAssuming that ?2z is the noise variance at the input of each antenna, we find thevariance of zULu [n] as fHu ?ufu?2z . From the expression of rULu [n] the uplink MSE ofthe uth user can be given as?2u =1 +U?i=1|?i|2fHuW u,iWHu,ifu ? ?uen0WHu,ufu ? ?Hu fHuW u,ueTn0 + fHu ?ufu?2z .(2.18)Defining the relation between uplink and downlink parameters as ?u = ?Hu /gu andqu = gufHu , where gu is a positive scalar, and setting up the U equations ?2u = ?2u, weobtain (cf. [97])[g21, g22, . . . , g2U]A =[|?1|2, |?2|2, . . . , |?U |2], (2.19)where A is a square matrix of size U with off-diagonal entriesAi,j = ?|?i|2?2zfHj W j,iWHj,if jand diagonal entriesAi,i =U?k=1,k 6=i|?k|2?2zfHi W i,kWHi,kf i + fHi ?if i .It can be verified that A is a diagonally dominant matrix and, hence, non-singular.Hence, given a design of the transmitter and the receiver of the dual DS-UWB uplink,one can compute the transformation parameters gu using (2.19) and use them toobtain the corresponding design of the DS-UWB downlink that results in the same setof users? MSEs as the dual uplink. Furthermore, using the transformation parametersgu given in (2.19) the total transmitted power in both the uplink and downlink are30Chapter 2. Pre-Equalization for DS-UWB Communicationthe same, i.e., PDL = PUL.The DS-UWB uplink design problem that minimizes a weighted sum of users?MSEs subject to a constraint on the total power transmitted by all users is given asminf1...fU?1...?UU?u=1?u[1+U?i=1|?i|2fHuW u,iWHu,ifu?2<{?uen0WHu,ufu}+fHu ?ufu?2z](2.20a)s.t.U?u=1|?u|2 ? Pmax (2.20b)Unlike the downlink, the expression for uplink MSE of the uth user, ?2u, is afunction only of equalization filter coefficients fu of the uth user and is independentof the filter coefficients of other users, f i, i 6= u. This observation allows each term inthe summation in (2.20a), which is a convex quadratic function of fu, to be minimizedindependently. Hence, by setting the derivative of each ?2u with respect to fu to zerowe obtain an expression of the optimal receiver filter coefficientsfu = ?Hu T?1u W u,u eTn0 , (2.21)where T u =U?i=1|?i|2W u,iWHu,i + ?2z?u. Using the expression for optimal fu, theexpression for each MSE in (2.18) reduces to?2u = 1? |?u|2en0WHu,uT?1u W u,ueTn0 , (2.22)31Chapter 2. Pre-Equalization for DS-UWB Communicationand the design problem reduces tomin|?1|2,...,|?U |2U?u=1?u(1? |?u|2en0WHu,uT?1u W u,ueTn0)(2.23a)s.t.U?u=1|?u|2 ? Pmax . (2.23b)Hence, we have reduced the original design problem (2.16) with U(MLq+1) variablesto the design problem in (2.23) with only U variables. In particular note that thenumber of variables is independent of the PEF length Lq, which leads to significantsavings in computational complexity. The optimization problem in (2.23) can besolved using a gradient descent algorithm [98]. At the optimal solution of the prob-lems in (2.16) and (2.23), the downlink receiver gain and the uplink equalizing filterssatisfy the MMSE criterion. We thus have the following relation between minimumMSE and maximum achievable SINR (cf. e.g. [99]).Lemma 2.4.1 The optimal solution of the minimum weighted sum of MSE problemin (2.16) for the DS-UWB downlink achieves SINRu = 1/?2u ? 1. Similarly, for thedual DS-UWB uplink SINRULu = 1/?2u ? 1.Proof The proof is included in Appendix C.2.5 Results and DiscussionIn this section, we present and discuss numerical results to demonstrate the effec-tiveness of the two proposed multiuser PEF design strategies. The transmission andchannel model parameters are selected such that representative and insightful con-clusions can be drawn. More specifically, we consider BPSK transmission and twodifferent transmission environments, namely CM2 for the residential non-line-of-sight32Chapter 2. Pre-Equalization for DS-UWB Communicationenvironment and CM6 for the outdoor non-line-of-sight environment, cf. [12]. Weassume a center frequency of 6 GHz and a pulse bandwidth of 500 MHz using root-raised cosine pulses with roll-off of 0.6. The overall impulse response including UWBchannel impulse response normalized to unit energy, transmitter pulse shaping andreceiver noise rejection filtering, and sampling, are truncated such that about 99%of the energy is captured. This leads to Lh = 104 for CM2 and Lh = 430 for CM6.The normalized spreading codes are selected as Walsh-Hadamard sequences for ex-periments with N = 8 and as binary mutually orthogonal sequences for experimentswith varying N .2.5.1 Downlink Transmission with Power MinimizationWe start with the first design approach aiming at minimization of transmit powerwhile meeting user QoS constraints. As an equivalent measure of the transmissionpower, we show results in terms of the normalized power ? = PDLU?2z .Figure 2.3 shows ? averaged over 400 CM2 realizations versus the PEF lengthLq for a pre-rake DS-UWB broadcast system with M = 2 transmitting antennasand U = 2 users with equal MSE requirements ?1 = ?2 = ? and spreading codelength of N = 8. The results are presented for the two MSE requirements ? = 0.08and ? = 0.1. According to (2.15), these MSE constraints correspond to BERs ofapproximately 10?5 and 10?6, respectively. We note that the optimization problemin (2.14) can become infeasible, i.e., it is not possible to meet the QoS constraintsfor all users. In these cases, the channel is discarded and a new channel realization isdrawn. (In practice, when the optimization problem is infeasible, the transmitter cana) increase filter length, b) redesign for a lower QoS constraint, or c) temporarily turnoff the users with lower priority to maintain QoS constraints for the remaining users to33Chapter 2. Pre-Equalization for DS-UWB Communicationmake the problem feasible.) Hence, the results in Figure 2.3 apply to cases (channelsand values of Lq) for which the target MSEs could be achieved for all shown Lqlengths. The presented results clearly demonstrate the effectiveness of the proposedPEF design in decreasing the required transmission power. For example, consideringthe channel length of 104 taps, power savings of 11 dB for ? = 0.08 and 3 dB for? = 0.1 are achievable by increasing the filter length from 5 to about 10 taps. Since5 10 15 2081012141618202224Lq? [dB]  ? = 0.08? = 0.15 10 15 2002004006008001000Lq# of infeasible channels  ? = 0.08Figure 2.3: Required transmission power versus PEF length (Lq) with M = 2 trans-mit antennas for U = 2 users and given MSE (?1 = ?2 = ?). Averaged results for 400channel realizations of CM2. Inner figure: Number of infeasible channel realizationsversus PEF length.filter optimization and filtering are performed only at the broadcast transmitter,these gains come without increasing the complexity at the receivers. At the sametime, power minimization is highly desirable to avoid harmful interference of licensedcommunication systems by UWB. Furthermore, the inset in Figure 2.3 shows thenumber of infeasible channels from the 1000 randomly selected CM2 channel impulse34Chapter 2. Pre-Equalization for DS-UWB Communication2 4 6 8 10 12 14 16 18 20 2202468101214NSINR [dB]  All?pre?rake, PEF Lq = 10Partial?pre?rake Lp = Lh/5 = 20, PEF Lq = 10All?pre?rake onlyPartial?pre?rake only Lp = Lh/5 = 20Figure 2.4: SINR for user u = 1 versus the spreading code length (N). Resultsaveraged over 500 channel realizations of CM2. M = 2 transmitting antennas, U = 2users at Pmax = 14 dB.responses where QoS constraint of ? = 0.08 is imposed on both users. We observethat increasing the filter length from Lq = 5 to 10 taps, the number of infeasiblechannels drops from 600 to only 40 channels.2.5.2 Transceiver Design for Maximum Quality of ServiceWe now turn to our second design approach, which maximizes the throughput undera given power budget. For simplicity we assume equal weights ?u = 1, u = 1. . . . , U ,in (2.23). We start with the U = 2 user MISO pre-rake UWB system with M = 2transmitting antennas at the central unit, then extend the results to larger numbersof transmitting antennas and more users. Unless otherwise specified, the results areaveraged over 500 CM2 channel impulse responses.35Chapter 2. Pre-Equalization for DS-UWB CommunicationFigure 2.4 shows the effect of the spreading sequence length N on the SINR atPmax = 14 dB, assuming unit noise variance ?2z = 1. (Recall that channel impulseresponses are normalized to have unit energy). The results show the received SINRfor user index u = 1 with all-pre-rake (Lp = Lh) and partial-pre-rake (Lp = 0.2Lh)combining with and without pre-equalization. The receiver for the pre-rake UWBwithout pre-equalization is the same as before and the decision delay n0 is optimizedaccordingly. We observe that using PEFs results in an average gain of 2 to 3 dB inSINR at all considered spreading code lengths N = 2, 4, 6, . . . , 22. The upper boundwhich the achievable SINR approaches is due to the presence of residual interferencein the decision variable, which can further be reduced by increasing the PEF length.Interestingly, partial-pre-rake transmission with PEFs not only outperforms partial-pre-rake transmission without pre-equalization but it also achieves an about 1 dBhigher SINR compared to all-pre-rake transmission without pre-equalization.The effect of PEF length on the achievable SINR for all- and partial-pre-rakeis shown in Figure 2.5 for a spreading length of N = 8. It can be seen that pre-equalization with moderate filter lengths already achieves significant SINR improve-ments. Moreover, PEF together with partial-pre-rake UWB can achieve a more than5 dB gain compared to the partial-pre-rake transmission without PEF. These find-ings are reinforced by the MSE results shown in Figure 2.6, for the same scenario asconsidered in Figure 2.5, but also including the case of CM6 channels. Longer PEFlengths are required for partial-pre-rake UWB over CM6 channels, because of thesignificant spread of the channel impulse response. Note that the partial-pre-rake forCM6 applies longer pre-rake filters than the partial-pre-rake transmission for CM2,and therefore the corresponding MSE curves cross at about Lq = 30.In the following results we highlight the effects of number of transmitting antennas36Chapter 2. Pre-Equalization for DS-UWB Communication2 4 6 8 10 12 14 16 18 20 2201234567891011LqSINR [dB]  All?pre?rake with pre?equalizationPartial?pre?rake Lp = Lh/5 = 20 with pre?equalizationAll?pre?rake onlyPartial?pre?rake only Lp = Lh/5 = 20Figure 2.5: SINR for user u = 1 versus PEF length (Lq). Results averaged for 500channel realizations of CM2. M = 2 transmitting antennas, U = 2 users, spreadingcode length of N = 8 and Pmax = 14 dB.and number of users on system performance. In particular, the average bit-error rateBER = 1UU?u=1BERu, where BERu is the BER for user u, for all-pre-rake UWB isconsidered. Figure 2.7 shows BER versus transmit power for U = 2, N = 8, andpre-rake transmission without and with multiuser PEFs of length Lq = 10, anddifferent numbers of transmit antennas. CM2 is applied and the markers are thesimulated results and lines correspond to analytical results obtained using (2.15). Weobserve a good match between the simulated and analytical results, which confirmsthe Gaussian approximation for MUI and ISI used in (2.15). The error floor thatis present in the case of M = 2 antennas is due to the residual interference at PEFlength of Lq = 10. The addition of transmitter antennas is seen to be highly effectivein mitigating the effect of interference. We emphasize, however, that the devised37Chapter 2. Pre-Equalization for DS-UWB Communication5 10 15 20 25 30 35 40 45 5000.  CM2 All?pre?rakeCM2 Partial?pre?rake Lp = Lh/5 = 20CM6 All?pre?rakeCM6 Partial?pre?rake Lp = Lh/5 = 85Figure 2.6: MSE for user u = 1 versus PEF length (Lq). Results averaged over 500channel realizations of CM2 and CM6, respectively. M = 2 transmitting antennas,U = 2 users, spreading code length of N = 8 and Pmax = 14 dB.PEF design is crucial to benefit from the increased number of antennas as otherwiseespecially MUI remains unprocessed. This can be seen from the high error floor forthe pre-rake transmission without pre-equalization. Of course, larger M increaseshardware complexity.Finally, Figure 2.8 shows the BER versus Pmax for an all-pre-rake UWB systemwith M = 4 antennas. U = 1, 2, 3, 4 users and pre-rake transmission without andwith multiuser PEFs of length Lq = 10 is considered. The case with U = 1 usercorresponds to the single-user MISO transmission in [35]. We again observe that theproposed PEF design leads to significant performance improvements. In particular,the high error floor experienced by pre-rake UWB without pre-filtering, even forM = 4 > U = 2, is overcome using the optimized multiuser PEFs.38Chapter 2. Pre-Equalization for DS-UWB Communication2 4 6 8 10 12 14 16 18 20 2210?810?710?610?510?410?310?210?1100Pmax [dB]BER  M = 2, Lq = 10M = 3, Lq = 10M = 4, Lq = 10M = 2, Pre?rake onlyM = 3, Pre?rake onlyM = 4, pre?rake onlyFigure 2.7: BER versus the maximum transmission power Pmax. Results averaged for500 channel realizations of CM2. M = 2, 3 and 4 transmitting antennas, spreadingcode length N = 8 and U = 2 users. Solid lines: pre-equalization filter length ofLq = 10. Dashed lines: pre-rake transmission without PEF. Markers: simulationresults, lines: analytical results.2.5.3 Alternative Design ApproachesIn the following, we discuss alternative design approaches from the CDMA literature,namely [38, 39, 40], which consider the design of transmit filters for transmissionover multipath fading channels. The authors of [38] describe a zero-forcing designapproach that nulls a) MUI or b) MUI and ISI, and maximizes the useful receivedsignal power. While this approach works well for the case of mild ISI, it is inefficientfor severe ISI experienced in UWB transmission. In particular, a high error flooroccurs if ISI is neglected and only MUI is suppressed, and large PEF lengths andhigh SNRs are required for low BERs when joint MUI-and-ISI suppression is applied.References [39, 40] consider an MSE-based design approach in which all users39Chapter 2. Pre-Equalization for DS-UWB Communication2 4 6 8 10 12 14 16 18 20 2210?710?610?510?410?310?210?1100Pmax [dB]BER  U = 1, Lq = 10U = 2, Lq = 10U = 3, Lq = 10U = 4, Lq = 10U = 2, Pre?rake onlyU = 3, Pre?rake onlyU = 4, Pre?rake onlyFigure 2.8: BER versus the maximum transmission power Pmax. Results averagedfor 500 channel realizations of CM2. M = 4 transmitting antennas, spreading codelength N = 8 for U = 2, 3 and 4 users. Solid lines: pre-equalization filter length ofLq = 10. Dashed lines: pre-rake transmission without PEF. Dashed dot line: Single-user pre-equalization filter. Markers: simulation results, lines: analytical results.have a common receiver gain, i.e., ?1 = ?2 = . . . = ?U , referred to as transmitWiener filter [40]. This is a constrained version of our MSE-based PEF design andthus the objective function achieved by our proposed design is always at least asgood as that for the common-? design. We have found that allowing for user-specificreceiver scaling is especially beneficial in cases where the two users are operating atdifferent SNRs. Figure 9(a) presents a scatter plot of the two receiver gains obtainedwith the proposed and the common-? filter design for the case of U = 2 users,M = 2 transmitting antennas, PEF length of Lq = 10, Pmax = 14 dB, ?2z,1 = 0.1and ?2z,2 = 1, i.e., the two users have a 10 dB difference in their received SNR, and40Chapter 2. Pre-Equalization for DS-UWB Communication0.16 0.18 0.2 0.22 0.24 0.26 0.28 2  Proposed DesignCommon ? Design(a) Receiver Gain Comparison2 4 6 8 10 12 14 16 18 20 2200. [dB]MSE \ SUM MSE5 10 15 2010?610?510?410?310?210?1Pmax [dB]BER  Proposed DesignCommon ? Design User 1 SUM MSE User 2(b) MSE ComparisonFigure 2.9: Comparison of filter designs. U = 2 users, filter length of Lq = 10, M = 2transmitting antennas and unequal receiver thermal noise of ?2z,1 = 0.1 and ?2z,2 = 1.a) The first user?s equalizing receiver gain ?1 versus second user?s receiver gain ?2 for500 channel realizations of CM2. b) Average MSE results for our proposed designand the design with common receiver gains (?common-?? design).41Chapter 2. Pre-Equalization for DS-UWB Communication500 realizations of CM2. We observe that with our method ?1 and ?2 are notablydifferent, demonstrating the difference between our method and the optimizationproblem and solution from [39, 40]. The corresponding results for the sum MSEand the MSE for the individual users (averaged over the 500 channel realizations)versus the transmit power threshold Pmax are shown in Figure 2.9(b). It can be seenthat our design achieves a better sum MSE performance compared to the common-?design method according to [39, 40]. This MSE gain, as shown in the subfigure inFigure 2.9(b), results in a 4 dB SNR-gain at a BER of 10?4. From this discussionand the results presented in Sections 2.5.1 and 2.5.2, we conclude that the developedoptimization framework is highly effective in enhancing the performance of multiuserpre-rake UWB broadcast systems.2.6 ConclusionIn this chapter, we investigated the design of multiuser PEFs for pre-rake UWBbroadcast transmission with multiple antennas at the transmitter. Firstly, motivatedby reducing potential interference caused by the UWB system, we have consideredthe transceiver design to minimize the total transmission power for given QoS require-ments. We have shown that the corresponding optimization problem can be writtenin convex form and solved efficiently. Secondly, we have considered QoS maximiza-tion for a given maximal transmit power. In this case, since the direct optimizationof multiuser PEFs is complicated, we have made use of a downlink-uplink duality toformulate a numerically less demanding optimization problem. Hence, we have es-tablished a versatile framework for the design of pre-rake UWB broadcast multiusersystems. The presented numerical results confirm that the effects of ISI and MUIcan be successfully mitigated using our proposed design schemes.42Chapter 3Robust Pre-Equalization Filtersfor DS-UWB CommunicationIn this chapter, we extend the PEF design for multiuser pre-rake DS-UWB systemsfrom Chapter 2 to include channel uncertainties. To this end, we study the effectof channel uncertainty for the design of transmit filters and propose robust solutionsthat effectively mitigate ISI and MUI. After introducing the relevant literature, weproceed to describe the system model and mathematical representations of receivedsignal that is necessary for the design. Then the mathematical models selected torepresent the error in channel estimation are presented and details of the proposedrobust PEF designs are described. Next widely linear design strategies are proposedfor the binary DS-UWB signalling and WL counter-parts of the robust designs arederived. The advantages of WL design and the proposed robust design schemes arepresented in the results section, followed by concluding remarks.3.1 IntroductionThe design of PEFs has commonly been based on the assumption of accurate channelknowledge. In practice, errors in the channel estimation at the central unit, candeteriorate the system performance considerably.Robust transceiver design has been commonly studied in MIMO and cognitive43Chapter 3. Robust Pre-Equalization for DS-UWB Communicationradio literature. For example, the robust design of pre-coders for a multiuser systemwith QoS constraints specified in terms of MSE and SINR for communication overfrequency flat channels was studied in [51]. Similarly, the authors of [56] presentedlinear and non-linear pre-coders for multiuser MISO communication over frequencyflat fading channels with MSE constraints and channel uncertainties. In the cognitiveradio literature, robust beam-forming algorithms for multiuser spectral sharing wasconsidered in [57]. However, the multipath nature of the UWB channel and thepresence of pre-rake filters require a system description and a problem formulationthat are different from the procedures considered in MIMO literature. Perhaps thescenario that is closest to the UWB transmission scenario considered here is the onein [58], where robust precoders for communication over frequency selective MIMOchannels were designed. But due to the long delay spread of the UWB channel,the application of pre-rake combining is deemed necessary for pre-equalized UWBsystems to avoid the need for very long PEFs and the associated high optimizationand implementation complexities. Since the pre-rake filters become part of the overallchannel between PEF outputs and receiver inputs, the received signals of all userscannot be grouped in the way that is typically done for the design of precoders inthe MIMO literature, cf. e.g. [43]. In particular, the robust filter design proceduresfrom [58] are not directly applicable to systems with pre-rake filtering.Most literature on imperfect CSI in UWB systems study the effect of uncertaintyin estimation of CSI on the error rate performance. For example [52] presents anasymptotic error-rate analysis for a single user UWB communication system withrake reception. In [53] the effect of channel estimation error on performance of amaximum-likelihood detector is studied. A remedy for the performance loss of aUWB rake receiver due to the channel uncertainty was proposed in [54], where it was44Chapter 3. Robust Pre-Equalization for DS-UWB Communicationshown that implementation of a weighted maximal ratio combining rake receiver cansignificantly improve the system performance. Multiuser pre-coding with ideal andstatistical knowledge of the channel was considered for a network of UWB sensorswith non-coherent receivers in [55].Different from the mentioned UWB literature we study the effect of channel un-certainty for the design of transmit filters and propose robust designs to mitigateISI and MUI. Different from most of the literature on robust transceiver design, weconsider transmission over frequency selective channels. Furthermore, the presenceof a pre-rake filter, as already mentioned above, as well as the use of a modified MSEformulation from [101] change the design problem compared to that considered in [58]for MIMO transmission. Finally, we propose widely linear design strategies matchedto UWB transmission with real-valued (binary) modulation.3.2 System ModelSimilar to the system model introduced in Section 2.2, here we consider a pre-rakeUWB system, applying PEFs at the transmitter that is broadcasting data to multipleusers [102]. The block diagram of this transmission system is shown in Figure 3.1,and M and U are the number of transmit antennas and users, respectively.As typical for UWB and in particular pre-rake UWB systems using simple receiverstructures, the user specific data symbols au[n] are taken from a BPSK constellation.The uth user?s data symbols are processed by M PEFs fu,m = [fu,m[0], . . . , fu,m[Lf ?1]] of length Lf each. The output of each PEF is given by vu,m[n] =Lf?1?`=0fu,m[`]au[n?`]. This output is then upsampled by a factor of N and processed with the user?sunique spreading sequence cu[k] of length N . This spreading is often applied inUWB transmission to alleviate the equalization and interference suppression task,45Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationFigure 3.1: Block diagram of a downlink multiuser pre-rake UWB broadcast system.Multiuser pre-equalization and pre-rake combining are applied at the base stationwith M antennas. Single-antenna users with simple slicing detector are considered.cf. e.g. [35, 102]. The signal undergoes pre-rake combining using filters pu,m[k],m = 1, . . . ,M , of length Lp prior to transmission. Each pre-rake combining filteris assumed to be the time reversed conjugate version of the corresponding channelimpulse response, in this case hu,m[k], the equivalent baseband discrete time channelbetween the mth transmit antenna and user u. Denoting the available estimate ofhu,m[k] as h?u,m[k] (further details are given below), we have pu,m[k] = h?Hu,m[Lp ?k ? 1], 0 ? k < Lp, where Lp ? Lh and Lh is the length of h?u,m[k]. The casesLp < Lh and Lp = Lh correspond to partial and all-pre-rake combining, respectively.The extension to other combining techniques such as selective pre-rake combining isstraightforward.Denoting the equivalent baseband continuous-time UWB channel impulse re-sponse for transmission from antenna m to user u as hu,m(t), the equivalent basebanddiscrete time channel for this link can be written ashu,m[k] = gTx(t) ? hu,m(t) ? gRx(t)???kT/N, (3.1)where gTx(t) and gRx(t) are the transmitter pulse-shaping filter and the receiver noise-rejection filter, and T is the symbol duration.46Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationAt the receiver, the discrete-time received signal at each user is despread anddown sampled at time indices k = Nn + k0 where the sampling phase k0 = Lp +N ? 2?Nb(Lp +N ? 2)/Nc maximizes the magnitude of the desired signal, cf. [35].Hence, the sampled output at the uth user can be written asru[n] =M?m=1U?i=1??`=??wi,m,u[N`+ k0]vi,m[n? `] + zu[n] , (3.2)where zu[n] is the sampled noise at the uth user that is assumed to be AWGN withvariance ?2z , and wi,m,u[k] = p?i,m[k]?hu,m[k]?cu[N?1?k] with p?i,m[k] = ci[k]?pi,m[k].We assume that each user employs a simple symbol-by-symbol detection accordingtoa?u[n? n0] = sign[<{ru[n]}]. (3.3)The decision delay n0 is the same as the one considered in Section Description in Matrix FormFor the filter designs developed in the upcoming Sections 3.3 and 3.4, it is convenientto provide compact expressions for the system model as well as the relevant systemparameters. It can easily be verified that the received signal in (3.2) can be writtenasru[n] =U?i=1wHi,uFHi ai[n] + zu[n] (3.4)=U?i=1fHi WHi,uai[n] + zu[n] , (3.5)where in Eq. (3.4)47Chapter 3. Robust Pre-Equalization for DS-UWB Communication- wi,u = [wTi,1,u,wTi,2,u, . . . ,wTi,M,u]T ? CMLw?1 is the vector that contains the Mconcatenated vectors of overall channel responses wi,m,u,- wi,m,u = [wi,m,u[k0], wi,m,u[N + k0], . . . , wi,m,u[(Lw ? 1)N + k0]]H ,- F u = [F u,1, . . . ,F u,M ] ? CLt?MLw is formed by concatenation of M Toeplitzmatrices, F u,m, each of which contains the multiuser PEF coefficients for useru applied at transmit antenna m,- F u,m ? CLt?Lw is a Toeplitz matrix with first row defined as [fu,m[1]H ,0TLw?1]and first column [fu,m[0], . . . , fu,m[Lf ? 1],0TLw?1]H ,and in Eq. (3.5)- fu = [fTu,1,fTu,2, . . . ,fTu,M ]T ? CMLf?1 is the vector that contains the M con-catenated PEF coefficients,- fu,m = [fu,m[0], . . . , fu,m[Lf ? 1]]H ,- W i,u = [W i,1,u,W i,2,u, . . . ,W i,M,u] ? CLt?MLf is a block matrix with Toeplitzblock components, W i,m,u,- W i,m,u is an Lt ? Lf Toeplitz matrix with first row [wHi,u,m[k0],0TLf?1] and firstcolumn vector [wi,u,m[k0], . . . , wi,u,m[N(Lw ? 1) + k0],0TLf?1]H ,and- au[n] = [au[n] . . . au[n? Lt + 1]]T is the Lt ? 1 vector of data symbols of theuth user,- Lw = d(Lp +Lh +2N ? 3? k0)/Ne is the length of the overall channel impulseresponse.48Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationNote that the terms wHi,m,uFHi,m and fHi,mWHi,m,u represent the convolution of theoverall channel impulse response with its corresponding PEF and thus Lt = Lf +Lw ? 1. As it will be shown in Section 3.3, the two equivalent representations inEqs. (3.4) and (3.5) are required for the convex formulation of robust PEF designproblems.Furthermore, the average transmitted power can be written as [102]P =U?u=1fHu ?ufu , (3.6)where ?u = diag{?u,1,?u,2, . . . ,?u,M} is a block diagonal matrix whose blocks ?u,mare Hermitian Toeplitz matrices with the first row[?u,m[0], ?u,m[?N ], . . . , ?u,m[?N(Lf ? 1)]],where ?u,m[k] = p?u,m[k] ? p?Hu,m[?k].Finally, we can express the modified MSE [101, 45]?2u = E{|au[n? n0]? ?uru[n]|2} , (3.7)using Eqs. (3.4) and (3.5) as?2u =???[?uwH1,uFH1 , . . . , ?uwHu,uFHu ? eTn0 , . . . , ?uwHU,uFHU , ?u?z]???2, (3.8)and?2u =???[?ufH1 WH1,u, . . . , ?ufHuWHu,u ? eTn0 , . . . , ?ufHUWHU,u, ?u?z]???2, (3.9)49Chapter 3. Robust Pre-Equalization for DS-UWB Communicationwhere en0 is a unitary vector with zeros in all positions and one at position n0 and?u is a real-valued parameter. Optimization based on the modified MSE allows thetransmit filter to generate a received signal whose amplitude is scaled to minimizethe difference between the received and the original desired signal au[n? n0]. It wasshown in [101, 40] that the design based on the modified MSE, using the same ?for all users, outperforms that based on the conventional MMSE criterion. Resultsin [45, 102] demonstrate that considering independent values of ? for each user, canfurther improve the overall system performance. Hence we allow for user specificvalues ?u in our design. We note that in the considered case of BPSK transmission,the scaling factor ?u is only considered as part of the PEF design but it does notneed to be implemented at the receiver (see Eq. (3.3)).3.2.2 Channel EstimationSince our PEF design takes channel uncertainties into account, we need to brieflydiscuss channel estimation for pre-rake UWB systems. Channel estimation is per-formed at the broadcasting node when it receives known training signals from theindividual users by exploiting the reciprocity of the UWB channel [93]. While therobust PEF design procedures introduced in this chapter are applicable independentof the specific channel estimation algorithm used (cf. e.g. [103, 104, 105, 106]), forconcreteness we consider linear minimum mean-square error (LMMSE) estimation ofthe channel. Denoting Y u = [Y u,1, . . . ,Y u,M ] as the K ?M matrix containing thesamples received during the transmission of the training sequence of length K fromuser u at the M antennas, we haveY u = SuHu +Nu , (3.10)50Chapter 3. Robust Pre-Equalization for DS-UWB Communicationwhere Hu = [hu,1, . . . ,hu,M ] is the Lh ?M channel matrix with hu,m = [hu,m[0], . . . ,hu,m[Lh ? 1]]T , Su = [su,0, . . . , su,Lh?1] is the K ?Lh matrix whose `th column su,` =[su[n? `], su[n? `? 1], . . . , su[n? `?K + 1]]T is the shifted version of the trainingsequence and Nu ? CK?M contains the AWGN samples with variance ?2z at theM antennas. The LMMSE estimate of the channel and the corresponding errorcovariance matrix areh?u,m = (SHu Su + ?2zC?1hu,m)?1SHu Y u,m , (3.11)?u,m = (??2z SHu Su +C?1hu,m)?1 , (3.12)where Chu,m is the channel covariance matrix. We assume knowledge of the channelcovariance matrix at the transmitter side. This information can be in the form of thecovariance matrix averaged over sufficient number of channel realizations or can beobtained using a correlation-based estimate of the channel realization, as suggestedin [107].3.3 Robust PEF DesignIn this section, we first discuss the two approaches to robust design, which startingfrom Eq. (3.11) lead to two uncertainty models. Then, we derive the design objective,and formulate design problem formulations that incorporate the channel uncertaintymodels in the robust design of PEFs.3.3.1 Uncertainty ModelsThe first type of robust designs considers optimizing using averages with respect tochannel uncertainties in the objective function and constraints, e.g. [50]. Then, the51Chapter 3. Robust Pre-Equalization for DS-UWB Communicationso-called stochastic uncertainty modelhu,m = h?u,m + ?u,m , ?u,m ? N (0,?u,m) , (3.13)which directly follows from channel estimation (3.11) and (3.12), is applied in the op-timization. If, however, the optimization objective and/or constraints are formulatedfor a given channel realization, then the so-called bounded uncertainty model is moresuitable. The bounded uncertainty model is incorporated into our design by applyingthe bound on the cascaded channel impulse response vector hu = [hTu,1, . . . ,hTu,M ]Tas follows:hu = h?u + ??u , ??u? ? u , (3.14)where ?u = [?Tu,1, . . . , ?Tu,M ]H is an MLh ? 1 vector. That is, a PEF design is validfor a channel estimate h?u and all uncertainty vectors ?u satisfying (3.14).3.3.2 Design Objective and OptimizationOur design objective is to minimize the average transmit power subject to constraintson the MSE experienced at the U receiving users, e.g. [44, 51, 102]. Power mini-mization is desired for UWB systems, which operate as unlicensed spectrum underlaysystems, to reduce the level of interference caused to other (licensed) communicationsystems. Using the approach from [102] and the MSE formulation in Eq. (3.8), thepower minimization problem can be written in terms of the design parameters as52Chapter 3. Robust Pre-Equalization for DS-UWB Communication0.07 0.075 0.08 0.085 0.09 0.095 1 ? X) K = 3 Lh K = 5 Lh K = 2 Lh K = 4 Lh Target MSE Figure 3.2: Empirical CDF of the MSE of the first user for different training sequencelengths. PEF length of Lf = 10 with M = 4 transmitting antennas and U = 2 usersare considered.minf1 ... fU?1 ... ?UU?u=1fHu ?ufu (3.15a)s.t.???[wH1,uFH1 , . . . ,wHu,uFHu ? eHn0?u, . . . , wHU,uFHU , ?z]???2? ?u?2u, 1 ? u ? U ,(3.15b)where ?u is the bound on the MSE and ?u = 1/?u. Assuming perfect CSI and thusaccurate knowledge of the overall channel impulse responses wi,u, i = 1, . . . , U andu = 1, . . . , U , problem (3.15) is a convex QCQP, solvable using interior point methods[98]. However, applying (3.15) in the case of imperfect CSI can cause significant per-formance degradation. To illustrate this, Figure 3.2 shows the empirical cumulativedensity function (CDF) of the achieved MSE at the first user for a two-user systemand different training sequence lengths for channel estimation (we will detail the sim-ulation setup in Section 3.5). The PEFs are optimized using (3.15) and treating thechannel estimations as perfect CSI. In the figure, the dashed line represents the tar-get MSE, which indeed would have been achieved if channel estimation was perfect.Increasing the training sequence length shifts the CDF closer to the target MSE, but53Chapter 3. Robust Pre-Equalization for DS-UWB Communicationeven with a training sequence length that is five times the length of channel impulseresponses, the design achieves the required level of QoS in less than 30% of channelrealizations. Therefore, an effective PEF design requires robustness to uncertaintiesdue to imperfect channel estimation.To modify the optimization problem such that channel uncertainty is accountedfor, we write the overall channel aswi,u,m[n] = ci[k] ? pi,m[k] ?(h?u,m[k] + ?u,m[k])? cu[N ? 1? k]??k=Nn+k0= w?i,u,m[n] + ?i,u,m[n] , (3.16)where w?i,u,m[n] = h?u,m[k] ? Ri,u,m[k]|k=Nn+k0 , ?i,u,m[n] = ?u,m[k] ? Ri,u,m[k]|k=Nn+k0captures the effect of the estimation error, and Ri,u,m[k] = ci[k]?cu[N?1?k]?pi,m[k].Then, the received signal in Eqs. (3.4) and (3.5) can be expressed asru[n] =U?i=1(w?i,u +?i,u)HFHi ai[n] + zu[n] (3.17)=U?i=1fHi (W? i,u +?i,u)Hai[n] + zu[n] . (3.18)In Eqs. (3.17) and (3.18), w?i,u and W? i,u have the same structure as wi,u andW i,u inEqs. (3.4) and (3.5), with wi,m,u[n] being replaced with w?i,m,u[n], ?i,u = ?i,u?u where?i,u = diag{Ri,u,1, . . . ,Ri,u,M} is an MLw ?MLh rectangular block diagonal matrixwith block components Ri,u,m ? CLw?Lh whose kth column is given as[Ri,u,m[(k ?1)N+k0], Ri,u,m[(k?1)N+k0?1], . . . , Ri,u,m[(k?1)N+k0?(Lw?1)]]H. The matrix?i,u ? CLt?MLf in Eq. (3.18) is a cascaded block matrix. Its block components?i,u,m ? CLt?Lf are Toeplitz matrices with first row vector [?Ti,u,m[k0],0TLf?1], andfirst column vector [?i,u,m[k0], . . . , ?i,u,m[N(Lw ? 1)+ k0],0TLf?1]T . Using the received54Chapter 3. Robust Pre-Equalization for DS-UWB Communicationsignal representation from Eqs. (3.17) and (3.18), the MSE is re-written as follows?2u = ?2u???[(w?1,u +?1,u)HFH1 , . . . , (w?u,u +?u,u)HFHu ?eHn0?u,. . . , (w?U,u +?U,u)HFHU , ?z]???2(3.19)= ?2u???[fH1 (W? 1,u +?1,u)H , . . . ,fHu (W? u,u +?u,u)H ?eHn0?u,. . . ,fHU (W? U,u +?U,u)H , ?z]???2. (3.20)Using these expressions, we now present two robust PEF design formulations.Robust Design with Average MSE ConstraintsWe first consider power minimization under average MSE constraints (see e.g. [50]),where the average is taken with respect to the channel uncertainty and for whichwe can use the stochastic uncertainty model (3.13). While the objective functionis clearly not affected by channel uncertainty, using Eq. (3.20), the average MSEconstraint for the robust design is written as????z,fH1 W?H1,u, . . . ,fHu W?Hu,u ? en0?u, . . . ,fHU W?HU,u???2+ E{U?i=1fHi ?Hi,u?i,uf i}? ?2u?u . (3.21)Since the averaging is over the overall uncertainty vector ?i,u, the second termon the left-hand side of (3.21) can be re-written asU?i=1fHi E{?Hi,u?i,u}f i. Next,defining ?i,u = E{?Hi,u?i,u}and noting that the element of matrix ?Hi,u?i,u in row` and column k is[?Hi,u?i,u]`,k= ?Hu,m`VHi,u,` V i,u,k ?u,mk , where mk = dk/Lfe and55Chapter 3. Robust Pre-Equalization for DS-UWB Communicationthe matrix V i,u,k =[0Lh?k?1?(mk?1)Lf ,RHi,u,mk ,0Lh?mkLf?k]H , using (3.13) we have[?i,u]`,k = E{[?Hi,u?i,u]`,k}=?????trace{V Hi,u,`V i,u,k?u,m`} m` = mk0 m` 6= mk. (3.22)Setting Gi,u = [W? i,u,??i,u], the robust design problem with average MSE con-straints can be written asminf1 ... fU?1 ... ?UU?u=1fHu ?ufu (3.23a)s.t.???fH1 GH1,u, . . . ,fHu W?Hu,u ? ?uen0 ,fHu??Hu,u, . . . ,fHUGHU,u, ?z???2? ?2u?u,1 ? u ? U . (3.23b)Robust Design with Instantaneous MSE ConstraintsAs a second relevant robust design approach, we consider power minimization underMSE constraints for each channel realization (see e.g. [51, 56, 58]). In this case,the bounded uncertainty model (3.14) is applicable. The MSE in Eq. (3.19) can bewritten as ?2u = (?Hu + ?Hu ?Hu )(?u + ?u?u)/?2u, where ?u =[wH1,uFH1 , . . . ,wHu,uFHu ?eTn0?u, . . . ,wHU,uFHU , ?z]H and ?Hu =[?H1,uFH1 , . . . ,?HU,uFHU ,0MLh?1]. Thus the MSEconstraint can be expressed as ?2u?u ? (?Hu + ?Hu ?Hu )(?u + ?u?u) ? 0. Furthermore,using the Schur complement property [108], the uth user?s constraint is written as the56Chapter 3. Robust Pre-Equalization for DS-UWB Communicationlinear matrix inequality (LMI)????u??u ?Hu + ?Hu ?Hu?u + ?u?u ?u??uIULt+1??? 0 , ??u? ? u , (3.24)where IULt+1 is the identity matrix of dimension ULt+1. The LMI (3.24) includes aninfinite number of constraints for all possible values of ?u. We will use the followinglemma from [109] to rewrite the constraint in a computationally efficient form.Lemma 3.3.1 Given matrices A, B and D, with A = AH , thenA ? BHZD +DHZHB, ?Z : ?Z? ? ?if and only if there exists a ? such that???A? ?DHD ??BH??B ?I??? 0 . (3.25)Using Lemma 3.3.1 we propose the following theorem.Theorem 3.3.2 The LMI (3.24) is equivalent to???????u??u ? ?u ?Hu 01?MLh?u ?u??uIULt+1 ?u?Hu0MLh?1 ?u?u ?uIMLh?????? 0 . (3.26)Proof The proof is included in Appendix D.57Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationUsing Theorem 3.3.2, we can formulate the robust PEF design with instantaneousMSE constraints asminf1,...,fU?1,...?U?1,...,?UU?u=1fHu ?ufu (3.27a)s.t.???????u??u ? ?u ?Hu 01?MLh?u ?u??uIULt+1 ?u?Hu0MLh?1 ?u?u ?uIMLh?????? 0, ? u ? {1, . . . , U}. (3.27b)The above problem is a convex problem with quadratic objective function andLMI constraints and is solvable using optimization algorithms such as SeduMi [110].In our implementation to obtain the numerical results presented in Section 3.5, weused the YALMIP linear matrix inequality parser with SeduMi solver [111].Reducing the Problem DimensionThe left-side of the LMI in (3.27b) is a square matrix of order ULt + 2 + MLh.The large delay spread of the UWB channel and thus length Lh of UWB impulseresponses makes the problem computationally challenging. However, considering theclustered structure of these impulse responses [12], most of the channel energy isconcentrated in relatively few channel taps. For example, considering a set of 1000channel impulse responses generated according to CM2 from [12], which representsresidential non-line of sight environment, more than 80% of the channel energy isconcentrated in less than 10 taps. Therefore, the complexity of solving (3.27) canbe notably reduced by applying the uncertainty model only to a certain number ofmaximum-energy taps of h?u,m. To see this, we note that applying the bounded errormodel to the Ls selected channel taps is equivalent to having a vector ??u,m[k] = ?u,m[k]58Chapter 3. Robust Pre-Equalization for DS-UWB Communicationfor k ? S and ??u,m[k] = 0 otherwise, where the set S contains the index of the Lsmaximum-amplitude channel taps. Therefore, the problem dimension is reduced bynoting that ?i,u??u = ??i,u??u, where ??u ? CMLs?1 and contains the error vector forthe Ls maximum-amplitude channel taps across the M antennas and similarly ??i,uconsists of the Ls columns of ?i,u for different transmitting antennas. Therefore thedimensions of the LMI is reduced to ULt + 2 +MLs.An alternative design scheme that reduces the problem dimension is obtained byaccounting for the estimation error in the overall channel vector wu = w?u + ?uinstead of hu = h?u + ??u in the design, and hence applying bounded uncertainty onthe ??u?, where ?u = [?T1,u, . . . ,?TU,u]T . In this case, the MSE from (3.8) is writtenas?2u = (?Hu +[?Hu FH , 0])(?u +???F?u0???)/?2u ,where F = diag{F 1, . . . ,F U}. Similar to the robust design problem in terms of ?u,the MSE constraints can be written as the following LMI???????u??u ?Hu +[?Hu FH , 0]?u +???F?u0????u??uIULt+1?????? 0 , ??u? ? ?u, ?u ? {1, . . . , U} , (3.28)where ?u is the bound applied to ?u. Using the results of Lemma 3.3.1 it followsthat the instantaneous MSE robust design problem with uncertainty applied on theoverall channel impulse response can be written as59Chapter 3. Robust Pre-Equalization for DS-UWB Communicationminf1,...,fU?1,...?U?1,...,?UU?u=1fHu ?ufu (3.29a)s.t.??????????u??u ? ?u ?Hu 01?UMLw?u ?u??uIULt+1?????u F01?UMLw???0UMLw?1[??u FH ,0UMLw?1]?uIUMLw????????? 0 , ? u ? {1, . . . , U} .(3.29b)The order of the square matrix in (3.29b) is ULt +2+UMLw. Hence, as long asULw  Lh, which is the case for small number of users, the design problem in (3.29)has reduced dimension compared to the design problem in (3.27).3.4 Widely Linear PEF DesignThe new robust PEF designs introduced in the previous section have consideredconstraints on the MSE defined in (3.8) and (3.9). This has commonly been donein the literature on pre-rake UWB systems, cf. e.g., [35], [102], [112]. However, thedesign can be further improved by exploiting the fact that the real part of the receivedsignal is a sufficient statistic for the transmitted BPSK signal (see (3.3)), which isalso referred to as widely linear optimization [113]. WL processing has been used forreceiver-side equalization as in e.g. [113, 114, 115, 116, 17, 18]. In particular [17]and [18] propose WL equalizer design for BPSK UWB systems and for widely linearestimation. The performance enhancement from WL design has been demonstratedin our work in reference [117] for PEF design according to a sum MSE criterion,60Chapter 3. Robust Pre-Equalization for DS-UWB Communicationconsidering perfect CSI at the transmitter. In this section, we make use of the WLoptimization concept to enhance the robust PEF designs developed in Section PreliminariesStarting from Eqs. (3.4) and (3.5), we can write the real part of the received signalat user u asyu[n] = <{ru[n]}=U?i=1( ??wi,u + ??i,u)T F?Ti ai[n] + z?u[n] (3.30)=U?i=1f?Ti (??W i,u + ??i,u)Tai[n] + z?u[n] , (3.31)where we used the real-valued parameter variables F? i =[<{F i}, ={F i}], ??wi,u =[<{w?i,u}, ? ={w?i,u}], ??i,u =[<{?i,u}, ? ={?i,u}], and z?u[n] ? N (0, ?2r) isthe AWGN after the real operator with variance of ?2r =?2z2 . Similarly, f? i =[<{f i}, ={f i}], ??W i,u =[<{W? i,u}, ? ={W? i,u}], and ??i,u =[<{?i,u}, ?={?i,u}]. The average transmitted signal power in terms of real-valued parame-ters isP =U?u=1f?Tu ??uf?u , (3.32)where??u =???<{?u} ?={?u}={?u} <{?u}???.61Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationThe key feature of the WL design is to consider the MSE after the real-part operatorhas been applied to the received signal, i.e.,??2u = E{???au[n? n0]? ?uyu[n]???2}. (3.33)3.4.2 WL Robust Design with Average MSE ConstraintsTo arrive at the WL formulation of the robust design with average MSE constraints,we apply the signal representation from Eq. (3.31) to (3.9) to write??2u = ?2u(????r, f?T1??W T1,u, . . . , f?Tu??W Tu,u ?en0?u, . . . , f?TU??W TU,u???2+U?i=1f?Ti E{??Ti,u??i,u}f? i).(3.34)Next, we attempt to find the elements of the matrix??i,u = E{??Ti,u??i,u} = E????????<{?Ti,u}<{?i,u} ?<{?Ti,u}={?i,u}?={?Ti,u}<{?i,u} ={?Ti,u}={?i,u}????????. (3.35)The elements in row ` ?MLf and column k ?MLf of matrix ??Ti,u??i,u are given as[??Ti,u??i,u]`,k =[<{?i,u}T<{?i,u}]`,k =???????Tu,m`V?Ti,u,`V? i,u,k??u,mk m` = mk0 m` 6= mk,(3.36)where mk = dk/Lfe ??u,mk = [<{?Tu,mk},={?Tu,mk}]T and V? i,u,k = [02Lh?k?1?(mk?1)Lf ,<{Ri,u,mk}, ?={Ri,u,mk},02Lh?mkLf?k]T . The matrix ={?i,u}T={?i,u} has slightlydifferent elements. Hence the elements in row ` > MLf and column k > MLf of62Chapter 3. Robust Pre-Equalization for DS-UWB Communicationmatrix ??Ti,u??i,u are given as[??Ti,u??i,u]`,k =[={?i,u}T={?i,u}]`,k =???????Tu,m`??Ti,u,`??i,u,k??u,mk m` = mk0 m` 6= mk,(3.37)where ??i,u,k = [02Lh?k?1?(mk?1)Lf ,={Ri,u,mk},<{Ri,u,mk},02Lh?mkLf?k]T . Therefore,defining ??u,m` = diag{<{?u,m`}/2,<{?u,m`}/2}, the elements of the matrix ??i,u aregiven by[??i,u]`,k=???????????trace{V? Ti,u,`V? i,u,k??u,m`} m` = mk, ` ?MLf , k ?MLftrace{??Ti,u,`??i,u,k??u,m`} m` = mk, ` > MLf , k > MLf0 otherwise. (3.38)Finally, using the matrix ??i,u from (3.35) with elements given in (3.38) and defin-ing G?Ti,u =[ ??W Ti,u,???Ti,u], allows us to formulate the robust WL optimization prob-lem byminf?1 ... f?U?1 ... ?UU?u=1f?Tu ??uf?u (3.39a)s.t.?????r, f?T1 G?T1,u, . . . , f?Tu??W Tu,u ? ?uen0 , f?Tu???Tu,u, . . . , f?TUG?TU,u????2? ?2u?u,1 ? u ? U . (3.39b)63Chapter 3. Robust Pre-Equalization for DS-UWB Communication3.4.3 WL Robust Design with Instantaneous MSEConstraintsFor the WL formulation of the robust design with instantaneous constraints in (3.27),we need to determine the WL counterpart of the LMI (3.27b). Following the stepsfrom Section 3.3.2 for the MSE in (3.33), we note that the MSE constraint for useru, can be written as????u??u ??Tu + ??Tu ??Tu??u + ??u??u ?u??uIULt+1??? 0 , ???u? ? u , (3.40)where ??u =[<{?Tu},={?Tu}]T , ??u =[w?T1,uF?T1 , . . . , w?Tu,uF?Tu ?eTn0?u, . . . , w?TU,uF?TU , ?r],and ??u =[??T1,uF?T1 , . . . , ??TU,uF?TU ,01?Lh]with??i,u =???<{?i,u} ?={?i,u}?={?i,u} ?<{?i,u}???.Using Eq. (3.40) and the procedures in Section 3.3.2, the WL formulation of therobust MSE constraint for user u is???????u??u ? ?u ??Tu 01?2MLh??u ?u??uIULt+1 ?u??Tu02MLh?1 ?u??u ?uI2MLh?????? 0 . (3.41)From Eq. (3.41), formulation of the power minimization problem for WL design ofrobust PEFs is straightforward. Similar to the linear design case, the dimension ofthe matrix on the left hand-side of (3.41) can be reduced by applying the uncertaintymodel to a selected number of maximum-energy channel taps of h?u,m in which case64Chapter 3. Robust Pre-Equalization for DS-UWB Communicationthe dimension of the matrix ??u is reduced to (ULt + 2) ? 2MLs and Lh is replacedwith Ls in (3.41).Similarly, it can be shown that the WL formulation of the instantaneous MSErobust design in terms of the overall channel vector ?u from (3.29) is given byminf?1,...,f?U?1,...?U?1,...,?UU?u=1f?Tu ??uf?u (3.42a)s.t.???????????u??u ? ?u ??Tu 01?UMLw??u ?u??uIULt+1?????u F?01?UMLw???0UMLw?1[??u F?T ,0UMLw?1]?uIUMLw?????????? 0 ,? u ? {1, . . . , U},(3.42b)where F? = diag{F? 1, . . . , F? U}.3.5 Numerical ResultsIn this section, we present and discuss numerical results to demonstrate the effective-ness of the proposed robust multiuser PEF design strategies in presence of channeluncertainties. For concreteness, we consider the CM2 channel model for the residen-tial non-line-of-sight environment, cf. [12]. But we have confirmed that the proposeddesign schemes are effective for other channel models as well. For the following re-sults, we generate 500 channel realizations according to the procedure described in[10]. Note that we do not include path loss or shadowing in the generated channelimpulse responses, which however is not important for the performance evaluation ofour PEF design as it mostly scales the transmit power and/or the power expended for65Chapter 3. Robust Pre-Equalization for DS-UWB Communicationtransmitting to different users but does not affect robustness. For the transmission,we assume a center frequency of 6 GHz and a pulse bandwidth of 0.5 GHz using root-raised cosine pulses gTX(t) and gRX(t) with roll-off 0.7. Without loss of generality,we apply spreading with N = 8 and spreading codes are set as shifted versions ofcu = [1, 0, . . . , 0]. Such spreading codes are typical for DS-UWB systems to mitigateISI and MUI. The pre-rake filter length is set as Lp = Lh. Note that dependingon the channel condition and defined level of QoS, the power minimization problemin (3.15) and its robust counterparts introduced in Section 3.3 and Section 3.4 canbecome infeasible. In such scenarios the transmitter can redesign for a lower QoS,modify system parameters, e.g., increase filter length, or temporarily stop transmis-sion for lower priority users in order to maintain the desired level of QoS for highpriority users. In the following test results, for fair and valid comparison, we considera set of channels for which all considered design problems are feasible.3.5.1 Advantages of WL DesignWe first demonstrate the advantages of the proposed WL optimization for the PEFdesign. To this end, we establish a fair measure for comparison, namely the BERassociated with the MSEs considered for linear and WL optimizations, i.e., Eq. (3.7)and Eq. (3.33). From Lemma 2.3.1 and Lemma 2.3.2, it is known that when idealCSI is available, the MSE constraints (3.15b) are met with equality and the achievedSINR is related to the MSE threshold via SINRu = 1/?u ? 1. Using this result andassuming that residual ISI and MUI can be approximated as Gaussian distributeddisturbance, the BER for linear and WL designs can be expressed as (the superscripts66Chapter 3. Robust Pre-Equalization for DS-UWB Communication?WL? and ?L? indicate WL and linear design, respectively)BERu = Q(?SINRWLu)= Q(?2 SINRLu). (3.43)Hence, for equal BER, we set?Lu = 2?WLu /(1 + ?WLu ) . (3.44)While a similar relation between SINR and ?u cannot be established for the robustdesign, the approximationE{SINRu} ? 1/?u ? 1 , (3.45)where the average is taken with respect to channel uncertainty, is found to be fairlyaccurate for the robust optimization (3.23). Since channel uncertainty causes an extraadditive Gaussian distortion at the receiver, we can use (3.43) with the average SINRfrom (3.45) to approximate the BER averaged over channel uncertainty. Hence, weapply (3.44) for setting the average MSE thresholds also for the robust case (3.23).In generating the following results, we consider a scenario with U = 2 single-antenna receiving users, the transmitter is equipped with M = 4 antennas and usingPEFs with length Lf = 10. All users have the same MSE threshold that is setaccording to (3.44). Figure 3.3 shows the required transmit SNR defined as Pmin/?2z(recall that channel realizations do not include path loss and shadowing) versus theBER for two scenarios where (i) ideal CSI is available at the transmitter and PEFoptimization (3.15) is performed and (ii) channel estimation with K = 2Lp is doneand the average MSE robust design is applied. The BER is calculated using (3.43)and for the robust design we apply the approximation from (3.45). As can be seen67Chapter 3. Robust Pre-Equalization for DS-UWB Communication10?7 10?6 10?5 10?4 10?3 10?22468101214BER(?1)Required Transmit SNR [dB]  Ideal CSI, Linear DesignIdeal CSI, WL DesignImperfect CSI, Robust Linear DesignImperfect CSI, Robust WL DesignFigure 3.3: Required transmit SNR = Pmin/?2z versus the BER, for widely linearand linear design schemes and (i) ideal CSI (ii) imperfect CSI (training sequencelength K = 2Lp) using the robust average MSE design. Results generated for U = 2receiving users, M = 4 transmitting antennas and PEF of length Lf = 10.from the figure, the WL design approach is superior to its linear counterpart inthat it requires lower transmit SNR and hence lower average transmit power forachieving the desired BER for both the perfect and imperfect CSI case. For example,considering a BER of 10?4, the WL design results in a gain of approximately 0.7 dBand 1 dB in transmit SNR compared to its linear counterpart design for the case ofideal and imperfect CSI, respectively. Similar gains in transmit SNR were obtainedwhen considering WL design for other UWB channel models (not shown here forconciseness).3.5.2 Advantages of Robust DesignHaving demonstrated the superiority of the WL over the linear design scheme, inthe following we focus on the WL design strategy and present results that show the68Chapter 3. Robust Pre-Equalization for DS-UWB Communicationeffectiveness of our PEF optimization approach.Average MSE ConstraintsWe first consider the robust design with average MSE constraints. Again, we selecta scenario with multiple single-antenna users, and we set the number of antennas atthe broadcast transmitter to M = 4 and the PEF length to Lf = 10. The MSEthresholds are adjusted to ?u = 0.08, u = 1, . . . , U .Figure 3.4 shows the average MSE of the first user (averaged over simulatedchannel realizations) and the minimum power required versus the ratio of trainingsequence length to the pre-rake filter length (as a normalized measure of the trainingsequence length) for a UWB system with U = 2 users. From top part of the figure,it can be seen that the average MSE achieved with the non-robust design decreaseswith increasing training sequence length but consistently fails to achieve the targetMSE. Hence, it does not provide the expected QoS. Applying the WL robust design,however, the target MSE is met. Of course, the robustness with regards to channeluncertainty requires a somewhat increased transmit power, as seen in the bottompart of Figure 3.4. The extra power diminishes with increasing training-sequencelength, as channel uncertainty decreases.To gain some further insight into the performance of the robust design approachwith average MSE constraints, Figure 3.5 shows the empirical CDF of the instanta-neous MSE (i.e., without averaging over channel uncertainty) achieved by the firstuser. The results are shown for both robust and non-robust designs with U = 1, 2, 4users. We observe that in the single user scenario, the non-robust design is able toachieve the MSE in 30% of the channels and applying robust design the target MSE ismet in 70% of the channels. However, increasing the number of users, the non-robustdesign can meet the target MSE in less than 10% of the channels. In contrast to69Chapter 3. Robust Pre-Equalization for DS-UWB Communication2 2.5 3 3.5 4 4.5 50.0750.080.0850.09K/LpMSE 1  Target MSEWL Robust Design with Average MSE ConstraintsWL Non?Robust Design2 2.5 3 3.5 4 4.5 555.566.57K/LpP min [dB]  WL Robust Design with Average MSE ConstraintsWL Non?Robust DesignFigure 3.4: MSE1 and Pmin versus ratio of training sequence and pre-rake filter lengthsfor robust WL PEF design with average MSE constraints and non-robust design.MSE threshold of ?1 = ?2 = 0.08 for U = 2 users, M = 4 transmitting antennas andPEF length of Lf = 10.this, the robust design operates close to the target MSE regardless of the number ofusers. Of course, since the constraints are in terms of the average MSE constraints,the target MSE is met on average, but not for every instance of channel uncertaintyvector.Instantaneous MSE ConstraintsThe robust design with instantaneous MSE constraints is an attractive choice whenguarantees on MSE for every instance of channel estimation are required. For thefollowing results we consider the same system parameters as in the previous section.70Chapter 3. Robust Pre-Equalization for DS-UWB Communication0.06 0.07 0.08 0.09 0.1 0.11 0.1200. 1 ? X)  Target MSEU = 1, WL Robust DesignU = 1, WL Non?robust DesignU = 2, WL Robust DesignU = 2, WL Non?robust DesignU = 4, WL Robust DesignU = 4, WL Non?robust DesignFigure 3.5: Empirical CDF of the instantaneous MSE of the first user for a pre-rakesystem with U = 1, 2 and 4 users. Results for robust WL PEF design with averageMSE constraints and non-robust WL design. MSE threshold set as ?u = 0.08 foru = 1, 2, 4, M = 4 transmitting antennas, PEF length of Lf = 10 and trainingsequence length of K = 2Lp are considered.Figure 3.6(a) shows the first user?s MSE versus  (2 = 1 =  for all results),the bound on channel uncertainty region, for a single realization of the CM2 channelmodel. Note that increasing  results in a more conservative design and thus smallerMSE values. In particular, the robust design with instantaneous MSE constraintsguarantees that the MSE constraint for user 1 is met if ??1? ? . However, nothingcan be said for the cases in which ??1? > . In the particular case considered inFigure 3.6, ??1? = 0.96. However, the MSE constraints are already met for a bound = 0.14, which indicates that absolute channel estimation error ??u? is not a strictcriterion for whether the QoS in terms of MSE can be met. In order to gain moreinsight into the selection of the threshold , in Figure 3.6(b) we present the empiricalCDF of the first user?s MSE across 500 channel realizations for different  values.As expected, since increasing the bound on the uncertainty region includes moreinstances of estimation error vectors, a larger threshold  results in higher probabilityof meeting the target MSE, or in other words, the outage probability is reduced.71Chapter 3. Robust Pre-Equalization for DS-UWB Communication0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 1?  Target MSEWL Robust DesignWL Non?Robust Design(a) MSE1 versus  for a single UWB channel realization.0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.1200. 1<=X)  Target MSEWL Robust DesignWL Non?Robust Design ? = 0.3  ? = 0.1  Increasing ?(b) Empirical CDF of the first user?s MSE for  = 0.1, 0.15, . . . , 0.3 andPr(MSE1 ? X) evaluated across 500 UWB channel realizations.Figure 3.6: WL robust PEF design with instantaneous MSE constraints and boundon uncertainty region set as  = 0.1, 0.15, . . . , 0.3, for a pre-rake UWB system withU = 2 users. MSE threshold of ?1 = ?2 = 0.08, M = 4 transmitting antennas, PEFlength of Lf = 10, and training sequence length of K = 2Lp are considered. Resultsfor WL non-robust design are included for comparison.72Chapter 3. Robust Pre-Equalization for DS-UWB CommunicationIn the next set of results we consider applying the channel uncertainty model toa selected number (Ls) of maximum-energy taps of the estimated channel impulseresponse to reduce the dimension of the LMI in (3.41) and thus the computation time.In Figures 3.7(a) and 3.7(b), the first user?s MSE and the minimum required averagetransmit power are plotted for a single UWB channel realization and Ls = 10, 20, Lh,where Lh = 80. From figures, adjusting the value of , the robust design withuncertainty being applied to smaller set of channel taps can achieve the requiredQoS for the considered channel realization. We observe that considering a subset ofchannel taps, which relaxes the design constraints, leads to (i) the (undesired) effectof a larger actually achieved MSE1 (Figure 3.7(a)) and (ii) a (desirable) reductionin transmit power. It still provides notable robustness to channel uncertainty. This,together with its computational complexity advantage makes it an attractive designchoice.As proposed in Section 3.3.2, an alternative approach to reducing problem di-mension and hence the design computation time is applying the uncertainty modelto the overall channel error vector ?u. In the next set of results we study the ro-bustness of the proposed design from a different perspective by looking at a singlechannel estimation vector and 1000 different instances of the channel error vector.The performance is compared to that of WL non-robust design. Figure 3.8(a) showsthe empirical CDF of the first user?s MSE for different values of ?1 = ?2 = ?. Therequired transmit power and the average MSE for the first user are presented in Fig-ures 3.8(b) and 3.8(c). We observe a trade-off between outage probability and powerconsumption similar to that for the robust design based on h?u. But the advantageof the robust design based on the overall channel estimation error vector is its lowercomputational complexity. The CDF results in Figure 3.8(a) for fixed estimates h?u73Chapter 3. Robust Pre-Equalization for DS-UWB Communication0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 1  Target MSEWL Robust DesignWL Non?Robust DesignLs = 10Ls = 20Ls = Lh(a) MSE1 versus  for single UWB channel realization.0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.35678910?P min [dB]  WL Robust DesignWL Non?Robust Design Ls = Lh Ls = 20  Ls = 10(b) The required transmit power Pmin versus  for a single UWB channelrealization.Figure 3.7: WL robust PEF design with instantaneous MSE constraints and channeluncertainty model applied on a selected set of Ls maximum-energy channel tapsLs = 10, 20, Lh, for a pre-rake UWB system with U = 2 users. MSE threshold of?1 = ?2 = 0.08, M = 4 transmitting antennas, PEF length of Lf = 10, and trainingsequence length of K = 2Lp are considered. Results for WL non-robust design isincluded for comparison.74Chapter 3. Robust Pre-Equalization for DS-UWB Communication0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.1200. 1<=X)  Target MSEWL Robust DesignWL Non?Robust Design ?? = 0.2 ?? = 0.16  ?? = 0.12  ?? = 0.08(a) Empirical CDF of MSE1.0.08 0.1 0.12 0.14 0.16 0.18 0.245678??P min [dB]  WL Robust DesignWL Non?Robust Design(b) The required transmit power Pmin versus ?.0.08 0.1 0.12 0.14 0.16 0.18 1  Target MSEWL Robust DesignWL Non?Robust Design(c) Average MSE1 versus ?.Figure 3.8: WL robust PEF design with instantaneous MSE constraints and boundon uncertainty region for the overall channel error vector ?u. Considered are singleUWB channel estimation vectors h?1 and h?2 and 1000 different instances of channelerror vectors ?1 and ?2 and ? = 0.08, 0.12, 0.16, 0.2. Pre-rake UWB system withU = 2 users, MSE threshold of ?1 = ?2 = 0.08, M = 4 transmitting antennas,PEF length of Lf = 10, and training sequence length of K = 2Lp. Results for WLnon-robust design are included for comparison.75Chapter 3. Robust Pre-Equalization for DS-UWB Communicationprovides a possibility to select the threshold such that a specific outage rate is met.3.6 ConclusionIn this chapter, we proposed two procedures for design of robust multiuser PEFs forpre-rake UWB systems. The robust design strategies are based on applying aver-age and instantaneous MSE constraints for the system performance. Our presentedsimulation results show the effectiveness of the proposed methods in maintaining therequired level of QoS in presence of uncertainties in estimation of the channel impulseresponse. We have shown that while the design with average MSE constraints en-sures that QoS constraints are met on average, using instantaneous MSE constraintsand setting the design parameters e.g. , the uncertainty threshold limiting normof channel error vector, can adjust the percentage of the time for which the QoSare met. For the two robust design strategies we introduce WL robust design coun-terparts and show that for the multiuser pre-rake UWB system, using WL designstrategy can result in significant power saving which is of particular importance inUWB system with limited spectral power allowance.76Chapter 4Multiway Relaying for DS-UWBCommunicationIn this chapter the PEF design schemes from Chapter 2 are extended to cooperativecommunication between multiple low-complexity DS-UWB nodes through a centralunit with multi-way DTF relay processing. As an alternative lower-complexity ap-proach, multi-way FF relaying is also considered. After the introduction of the relatedliterature, for each of the two proposed multi-way relaying schemes, the system modelis described and corresponding filter design schemes are developed. Then numericalresults and concluding remarks are presented.4.1 IntroductionCooperative communication can play an important role in extending the communica-tion range in UWB systems, where the restrictions on the PSD of the transmit signallimit the available transmit power and the communication range of UWB devices [62].In particular, for the multiuser DS-UWB system considered in this thesis, the designsdeveloped in Chapter 2 can be extended to multi-way relaying. In particular, we focuson the multiuser two-way relaying which is a special case of multi-way relaying in thischapter. As introduced in Section 1.4, in a multi-way relaying scenario, the messagesfrom multiple pairs of source and destination nodes are forwarded simultaneously77Chapter 4. Multi-way Relaying for DS-UWB Communicationthrough a relay node. Multi-way relaying has the potential of achieving higher datarates compared to the alternate one-way and two-way relaying approaches [60].There exists a large body of literature on relaying and the many variants ofrelaying protocols. In the following we start by describing some of the relevant workfor relaying over frequency-flat channels and then narrow it down to methods tailoredfor frequency-selective channels and UWB relaying.Relaying over frequency-flat channels with MIMO precoding at the relay, has beenstudied in [68, 69, 71, 73, 74]. In reference [68], two-way AF relaying with transmitand receiver beam-forming at the source and destination and precoding at the centralrelay for a single pair of nodes is considered. Two design strategies based on zero-forcing and MMSE are developed for the relay pre-coding matrix. The multi-wayrelaying with MIMO precoding at the relay is considered in [69], where an iterativeapproach for the joint design of the relay pre-coder matrix and the receive beam-forming at the destination for multiple pairs of nodes is derived. The joint design ofsource and relay precoders for multi-way relaying has been considered in [73, 74]. In[73], the multi-way relaying network operates as a secondary system in the presenceof primary user transmission, and in [74], the effect of channel estimation error ina multi-way relaying system with pre-coding at the source nodes and the relay, andMSE filtering at the destination nodes is studied.From the above literature review we note that the multi-way relaying is performedby implementing a) precoders at the source nodes and the relay, b) precoding at therelay and filtering at the destination nodes, or c) pre-coding at the source nodesand the relay plus destination node filtering/beam-forming. Problem formulationsinvolving the joint design of precoders at the relay and pre-coding or receiver-sideprocessing at the source and destination are non-convex, and hence iterative ap-78Chapter 4. Multi-way Relaying for DS-UWB Communicationproaches that decouple the overall problems into a number of convex sub-problemshave been suggested. These iterative methods apply alternating optimization [118],which is a method based on optimizing a function jointly over a number of variablesby alternating restricted optimizations over non-overlapping subsets of variables.The presence of ISI differentiates relaying over frequency-selective channels fromthat over frequency-flat fading channels. We briefly revisit the literature as relevantto our scheme. An alternative to AF, for relaying over frequency-selective channels isthe FF method, which was first introduced in [76]. Reference [77] offers an extensionof this work for one-way relaying when a direct link between source and destinationexists and equalization is performed at the destination node. Two-way FF relayingwith multiple antennas was considered in [78, 79]. In reference [78], two-way FFrelaying was optimized according to a worst-case SINR criterion and an algorithmbased on bisection search was proposed to solve the relaxed problem. The design ofequalization filters at the destination nodes was also addressed as part of the design.In [79], two worst-case SINR maximization as well as transmit power minimizationdesign formulations were proposed. Similar to [78], for the worst-case SINR max-imization problem, a one-dimensional search approach was used to solve a relaxedversion of the problem.In the UWB literature, one of the early works that considered half-duplex AFrelaying for IR-UWB with pulse-position modulation is [82]. References [83] and[84] propose one-way UWB relaying with TH-UWB. Furthermore, [85, 86] considerrelaying for transmit-reference UWB communication. Differential transmit referenceschemes with non-coherent amplify-and-forward for single and multiple-hop relayingare developed in [87] and [88], respectively. One-way relaying with pre/post-rakecombining at the relay was considered in [89] for UWB signalling with guard in-79Chapter 4. Multi-way Relaying for DS-UWB Communicationtervals. In [90] one-way decouple and forward relaying with rake receivers at thedestination node was considered. All of the mentioned methods cannot handle multi-way relaying links. In fact, to the best of our knowledge, multi-way relaying for UWBcommunication has not been considered in the literature yet.In this chapter multi-way relaying schemes for UWB communication are pre-sented. Unlike the mentioned literature on UWB relaying, we consider relaying forhigh data rate DS-UWB. Different from the mentioned literature on relaying overfrequency-selective channels, we propose the combination of post/pre-rake filteringand optimized post/pre-equalization filters at the relay. Furthermore, we considerfilter optimization based on sum-MSE minimization as the design criterion whichallows us to develop convex problem formulations for maximizing the throughput.The first design scheme is based on DTF relaying, in which case we propose theuse of post/pre-equalizing filters in addition to the post/pre-rake combining to enablemulti-way relaying. The second design scheme is based on FF relaying where differentfrom the literature, a sum-MSE design optimization technique that enables writingthe multi-way FF relaying problem in convex form is proposed. Due to the large delayspread of the UWB channels, the FF strategy requires filter lengths of the order of theoverall impulse response between source node and destination node. The optimizationof such long filters can be fairly demanding in terms of computations involved. Hencewe propose reducing the filter length at the relay by the use of post-and-pre-rake filtersat the relay. Furthermore, for both relaying schemes, an alternative design based ona modified MSE formulation from [40] that allows for user-specific scaling, similarto the PEF design schemes from Section 2.4, is proposed. The user-specific scalingrenders the optimization problem non-convex. Hence, an iterative solution accordingto AO principle is developed. Finally, following the developments in Section 3.4, since80Chapter 4. Multi-way Relaying for DS-UWB Communicationthe real-part of the received signal contains sufficient statistics for signal detection,the performance can be improved by considering WL filter design procedures. Hence,we extend our multi-way design strategies to include the WL designs.4.2 Multi-way Detect-and-Forward RelayingIn this section the system model and the filter design for multi-way DTF relayingbetween U nodes, that form U source and destination pairs, via the central relay isdescribed. We use notations s(u) and d(u) to refer to the source and destination nodeof the uth pair, respectively.4.2.1 System ModelThe block diagram of the nodes for the considered multi-way relay network is shownin Figure 4.1. The nodes are single antenna units with limited signal processingcapability while the central relay node, depicted in Figure 4.2, is equipped withmultiple antennas. The network uses a two-phase multiple access and broadcastschedule as follows. During the first phase (uplink), all U nodes send their message tothe central relay simultaneously. In the second phase (downlink), the relay processesand broadcasts the sum of all node messages through its M antennas.Source/Destination Transceiver Node: The block diagram representing the nodeoperation in transmit and receive mode is shown in Figure 4.1. During the uplinkphase, the data symbols at the source node of the uth pair, as(u)[n], are upsampled andpassed through the pulse shaping filter gTx(t) before transmission. The upsamplingprocess is equivalent to having a spreading sequence consisting of a one followedby N ? 1 zeros and it reduces the average transmit power and the ISI. Similar toSection 3.2, the data symbols are from a BPSK constellation with unit average power.81Chapter 4. Multi-way Relaying for DS-UWB CommunicationFigure 4.1: Block diagram of the nodes in transmit and receive modes.The signal after upsampler is denoted as a?s(u)[k]. In the receive mode, the signal atthe node is processed with the noise rejection filter gRx(t), downsampled and passed tothe detector. Prior to detection it is possible to perform self-interference cancellationby subtracting the destination node?s own message from the received signal which willbe discussed in the upcoming sections. Similar to the nodes described in Section 3.2,for the BPSK modulated signal, the detection is performed according toa?s(u)[n? n0] = sign[<{rd(u)[n]}], (4.1)where the delay n0 is set depending on the processing at the relay and is describedin the upcoming sections.Channel: The assumptions on communication channel are same as the ones con-sidered in previous chapters e.g. in Section 2.2. For the sake of readability we describethe details as follows. Using the modified Saleh-Valenzuela multipath fading modelfrom [12] for communication over UWB channels, the equivalent baseband discretetime channel between the source node s(u) and themth antenna at the relay is written82Chapter 4. Multi-way Relaying for DS-UWB CommunicationFigure 4.2: Block diagram of the central unit for the two phase DTF relaying.ashs(u),m[k] = gTx(t) ? hs(u),R,m(t) ? gRx(t)|kT/N , (4.2)where T is the symbol duration, N is the chip upsampling factor and gTx(t) and gRx(t)are the transmitter pulse-shaping and the receiver noise rejection filters, respectively.The discrete-time channel between the mth antenna at the relay and the destinationnode d(u) is denoted as hd(u),m[k] and has a description similar to (4.2)Central Relay: The central relay structure in the two phase operation is shown inFigure 4.2. The received signal at the relay antennas contains the message transmit-ted from all source nodes. Considering the signal route for detection of the messagefrom the source user s(u), intended for destination user d(u), the received signal ateach of the relay antennas are passed through rake combining filter ps(u),m, down-sampler and an equalizing filter fULs(u),m, where m = 1, . . . ,M is the relay antennaindex. The detector then makes a decision on a?s(u)[n ? n0] based on the sum of theprocessed signals from all the antennas. n0 is the delay associated with equalizing.The detected bit is then processed as per block diagram in Figure 4.2, for transmis-83Chapter 4. Multi-way Relaying for DS-UWB Communicationsion to user d(u). It is passed through pre-equalizing filter f d(u),m, upsampled and ispre-rake combined with pre-rake filter pd(u),m. The sum of all outgoing messages isthen transmitted via the M relay antennas.Similar to the system model described in Section 2.2, post/pre-rake combiningfilters ps(u),m[k] and pd(u),m[k] are set as the time reversed conjugate of the estimatedchannel impulse response of length Lh, i.e., ps(u),m[k] = (hs(u),m[Lp ? k ? 1])? andpd(u),m[k] = (hd(u),m[Lp?k?1])?. Lp = Lh corresponds to all-post/pre-rake combining.The relaying method can be extended to other rake combining techniques as well.Let us consider the signal route for data symbol as(i)[n] received via relay?s mthantenna and processed by rake filter ps(u),m, the overall impulse response after down-sampling can be defined as wULs(i),m,s(u)[k] = hs(i),m[k] ? ps(u),m[k]. Using this definition,the received signal at the input of the detector for s(u) can be described asrULs(u)[n] =?PULU?`=1fHd(`)WHd(`),d(u)a?s(`)[n? n0] + zd(u)[n] , (4.3)where? f d(u) = [fTd(u),1, . . . ,fTd(u),M ]T is the vector containing the M concatenated pre-equalization filter coefficient vectors f d(u),m, for destination node d(u),? f d(u),m = [fd(u),m[0], . . . , fd(u),m[Lf ? 1]]H ,? W d(`),d(u) = [W d(`),1,d(u), . . . ,W d(`),M,d(u)] ? CLt?MLf is a block matrix withToeplitz block components,? W d(`),m,d(u) ? CLt?Lf is a Toeplitz matrix with first row [wHd(`),m,d(u)[k0],0TLf?1]and first column vector [wd(`),m,d(u)[k0], . . . , wd(`),m,d(u)[N(Lw?1)+k0],0TLf?1]H ,where Lw = d(Lp + Lh + 2N ? 3? k0)/Ne is the length of the overall channelimpulse response and Lt = Lw + Lf ? 1,84Chapter 4. Multi-way Relaying for DS-UWB Communication? a?s(`)[n? n0] = [a?s(`)[n? n0], . . . , a?s(`)[n? n0 ? Lt + 1]]T is the Lt ? 1 vector ofestimated data symbols of the source node s(`),? zd(u)[n] is the white Gaussian noise at the destination node with variance ?2d(u).The variance of zR[n] can be written in terms of the equalization filter coeffi-cients, as ?2zR =U?i=1(fULs(i))HPHs(i)P s(i)fULs(i), where P s(i) = diag{P s(i),1, . . . ,P s(i),M}and P s(i),m are Toeplitz matrices with first row [ps(i),m[0],0Lf?1] and first column[ps(i),m[0], . . . , ps(i),m[Lp ? 1],0Lf?1]H .Next we proceed to find the matrix form signal representation for the downlinkphase, which is similar to the downlink received signal representation from Section 2.2.Defining the overall impulse response including pre-rake combining for destinationnode d(i) and the channel impulse response between relay?s mth antenna and desti-nation node d(u) as wd(i),m,d(u)[k] = pd(i),m[k] ? hd(u),m[k], the received signal at thedestination node d(u) at second transmission phase is described asrd(u)[n] =U?`=1fHd(`)WHd(`),d(u)a?s(`)[n? n0] + zd(u)[n] , (4.4)where? f d(u) = [fTd(u),1, . . . ,fTd(u),M ]T is the vector containing the M concatenated pre-equalization filter coefficient vectors f d(u),m, for destination node d(u),? f d(u),m = [fd(u),m[0], . . . , fd(u),m[Lf ? 1]]H ,? W d(i),d(u) = [W d(i),1,d(u), . . . ,W d(i),M,d(u)] ? CLt?MLf is a block matrix withToeplitz block components,? W d(i),m,d(u) ? CLt?Lf is a Toeplitz matrix with first row [wHd(i),m,d(u)[k0],0TLf?1]and first column vector [wd(i),m,d(u)[k0], . . . , wd(i),m,d(u)[N(Lw ? 1)+k0],0TLf?1]H ,85Chapter 4. Multi-way Relaying for DS-UWB Communicationwhere Lw = d(Lp + Lh + 2N ? 3? k0)/Ne is the length of the overall channelimpulse response and Lt = Lw + Lf ? 1,? a?s(`)[n? n0] = [as(`)[n? n0], . . . , as(`)[n? n0 ? Lt + 1]]T is the Lt ? 1 vector ofdata symbols at the source node s(`).? zd(u)[n] is the white Gaussian noise at the destination node with variance ?2d(u).The average transmit power for the second transmit phase at the relay has asimilar formulation to that of downlink transmit power at central unit described inSection 2.2 and is written asP =U?u=1fHd(u)?d(u)f d(u) , (4.5)where ?d(u) = diag{?d(u),1,?d(u),2, . . . ,?d(u),M} is a block diagonal matrix whoseblocks ?d(u),m are Hermitian Toeplitz matrices with the first row[?d(u),m[0], ?d(u),m[?N ], . . . , ?d(u),m[?N(Lq ? 1)]],and ?d(u),m[k] = pd(u),m[k] ? pHd(u),m[?k].4.2.2 Filter Design for DTF RelayingIn this section we describe the design strategies for optimizing the uplink and down-link filter coefficients for the multi-way DTF relaying with self-interference cancella-tion at the nodes. For the uplink phase the equalizing filters are designed accordingto the MMSE criterion. In the downlink, similar to the design in Section 2.4, thesum-MSE criterion is applied for throughput maximization.During the uplink phase, the individual users transmit their message using average86Chapter 4. Multi-way Relaying for DS-UWB Communicationtransmit power PUL. Using the description of the received signal in (4.3), and definingthe uplink MSE as MSEULs(u) = E{|rULs(u) ? as(u)[n ? n0]|2}. The MSE in terms of theuplink filter coefficients is written asMSEUL =???[?PUL(fULs(u))H(W ULs(1),s(u))H , . . . ,?PUL(fULs(u))H(WULs(u),s(u))H ? en0, . . . ,?PUL(fULs(u))H(W ULs(U),s(u))H , ?2R(fULs(u))HPHs(1),s(u)]???2. (4.6)Applying the KKT conditions, the solution for the uplink MMSE filter design can beobtained asfULs(u) =?PUL(PULU?`=1(W ULs(i),s(u))HW ULs(i),s(u) + ?2zPHs(u)P s(u))?1WHs(u),s(u)en0 .(4.7)In phase II, if PEF design without interference cancellation is considered thenthe design procedures from Chapter 2 can be applied to obtain the PEF coefficients.However, since the signal received at the destination contains the node?s self message,it is possible to cancel the self-interference at the node and reduce the PEF task. Dueto the multipath nature of the UWB channel, the received signal at the destinationcontains self-interference from more than one bit of the destination?s transmitted sig-nal. Depending on the number of stored bits, it is possible to cancel the effect ofmultiple transmitted bits in the received signal. Note that self-interference cancella-tion requires feedback of the interference cancellation coefficients from the relay tothe nodes. The coefficients have to be updated each time the PEFs change.In the following we describe the PEF design for downlink phase of DTF relaying,where different from the design developed in Section 2.4 self-interference cancellationis considered. As mentioned earlier the PEF coefficients are optimized based on sum-87Chapter 4. Multi-way Relaying for DS-UWB CommunicationMSE minimization design criterion that aims at maximizing the network throughputwith constraints on relay?s average transmit power which is relevant to UWB systems.The MSE of the downlink transmission, at the destination node prior to detection isdefined asMSEICd(u) = E{|?rd(u)[n]? as(u)[n? 2n0]? ?Hd(u)a?d(u)[n]|2} , (4.8)where a?d(u)[n] is an Lt ? 1 vector containing the stored transmitted symbols at theLc indices selected for interference cancellation and zeros everywhere else, and thevector ?d(u) ? CLt?1 contains the interference cancellation coefficients at the Lcinterference cancellation indices and zeros elsewhere. The scaling factor ? is relatedto the modified MSE definition from [101]. Unlike the PEF design from Section 2.4,here we set the same scaling factor for all users. Therefore, the MSE from Eq. (4.8)isMSEICd(u) = ?[ ??d(u), ?fHd(1)WHd(1),d(u), . . . , ?fHd(u)WHd(u),d(u) ? e2n0 ,?fHs(u)WHs(u),d(u) ? ?Hd(u), . . . , ?fHd(U)WHd(U),d(u) ]?2 . (4.9)Using the MSE definition from Eq. (4.9), the sum-MSE minimization problem canbe formulated asminf1,...,fU ,?,?1,...,?UU?u=1MSEICd(u) , (4.10a)s.t.U?u=1fHd(u)?d(u)f d(u) ? Pmax . (4.10b)Setting ?d(u) = ?HALcW s(u),d(u)f s(u), where ALc = E{ad(u)[n]a?d(u)[n]T}, and apply-88Chapter 4. Multi-way Relaying for DS-UWB Communicationing the KKT conditions on the Lagrangian function for the above convex problemresults in the following closed-form solution for the design parameters? =????1PmaxU?u=1e2n0W d(u),d(u)T?Hd(u)?d(u)T?1d(u)WHd(u),d(u)e2n0 , (4.11)andf d(u) = T?1d(u)WHd(u),d(u)e2n0/? , (4.12)whereT d(u) =????U?i=1WHd(u),d(i)W d(u),d(i) +U?u=1?2d(u)Pmax?d(u) ?WHd(u),s(u)ALcW d(u),s(u)????? Filtering with User Specific Scaling FactorThe proposed design strategy in Section 4.2.2 is based on applying a single scalingfactor to all users. It was shown in Section 2.5.3 that assigning user specific scalingfactor can improve the system performance and the interference cancellation char-acteristics of the system when users operate at different SNR levels. In this sectionthe sum MSE filter design problem with self-interference cancellation from (4.10) isextended to allow for user specific scaling.Including a user specific scaling factor, the sum MSE optimization problem in Eq.(4.10) becomes non-convex. Therefore, we use the AO method from [118] to arriveat a solution. Note that one could utilize the relation between downlink transmissionand its equivalent uplink to find a downlink solution as suggested in Section 2.4. Herewe present filter design using the AO technique as an alternative approach.89Chapter 4. Multi-way Relaying for DS-UWB CommunicationSetting ?d(u) = ?Hd(u)ALcW s(u),d(u)f s(u), the Lagrangian function correspondingto the sum MSE minimization problem subject to a constraint on the total transmitpower at the central relay isL = U +U?u=1|?d(u)|2U?i=1fHd(i)WHd(i),d(u)W d(i),d(u)f d(i) ? ?d(u)fHd(u)WHd(u),d(u)e2n0? ?Hd(u)eT2n0W d(u),d(u)f d(u) + |?d(u)|2(?2d(u) ? fHs(u)WHs(u),d(u)ALcW s(u),d(u)f s(u))+ ?(U?i=1fHd(i)?d(i)f d(i) ? Pmax), (4.13)where ? is the Lagrange coefficient.The alternating optimization method is an iterative procedure for optimizing afunction jointly over a number of variables. The method is applicable to the prob-lems which are convex with respect to individual subset of the variables and is basedon dividing the parameter space into a number of subsets and alternating betweenrestricted minimizations over each subset of variables [118]. For the sum MSE min-imization problem, since the objective function is convex with respect to the indi-vidual subsets and is lower bounded by zero then the AO method always converges[69, 73, 74]. However, convergence to a globally optimum solution is not guaranteed.We derive closed-form solutions for updating the optimization parameters at eachstep until convergence to a local optima.In applying the alternating optimization method, the parameters from (4.13) aredivided into three partitions, ?1 = [?1, . . . , ?U ], ?2 = [f 1, . . . ,fU ] and ?3 = [?].Then in an iterative approach, for each of the partitions the restricted minimizeris computed while fixing all other parameters. The iterations stop if a maximumnumber of iterations is reached or if the change in the value of the parameters overall subsets and per iteration is smaller than a threshold . At each iteration t, the90Chapter 4. Multi-way Relaying for DS-UWB Communicationsolutions are computed by applying the KKT conditions on the Lagrangian functionin Eq. (4.13) as follows. The user specific scaling factors for each user are calculatedand updated as?(t+1)d(u) =<{f (t)Hd(u)WHd(u),d(u)e2n0}U?i=1f (t)Hd(i) WHd(i),d(u)W d(i),d(u)f(t)d(i) ? f(t)Hs(u)WHs(u),d(u)ALcW s(u),d(u)f(t)s(u) + ?2d(u).(4.14)The downlink PEF coefficients are updated by settingf (t+1)d(u) = T d(u)(WHd(u),d(u)e2n0)?(t+1)d(u) , (4.15)whereT d(u) =????(t+1)d(u)???2(U?i=1WHd(u),d(i)W d(u),d(i) ?WHd(u),s(u)ALcW d(u),s(u) + ?(t)?d(u))?1.The Lagrangian coefficient ? is updated as?(t+1) =U?u=1|?(t+1)d(u) |2?d(u)Pmax. (4.16)After each iteration the difference in the vector containing all of variables is calculatedas ? = [?(t+1)1 ,?(t+1)2 , ?(t+1)3 ] ? [?t1,?t2, ?t3]. The iterations stop if the maximumnumber of iterations is reached i.e., t > Niter or if ??? ? .91Chapter 4. Multi-way Relaying for DS-UWB Communication4.2.4 Widely Linear Filtering for DTF RelayingThe WL design relies on optimizing the filter coefficients based on the real part ofthe received signal. In the uplink phase, the WL uplink filter obtained according tothe MMSE design isf?ULs(u) =?1PUL(U?`=1(W? ULs(i),s(u))TW?ULs(i),s(u) + ?2z/2PULP?Ts(u)P? s(u))?1W? Ts(u),s(u)en0 ,(4.17)whereW? ULs(i),s(u) = [<{W ULs(i),s(u)},?={W ULs(i),s(u)}] ,andP? s(u) = [<{P s(u)},?={P s(u)}] .In the downlink phase, the MSE definition is modified asMSEICd(u) = E{|?yd(u)[n]? as(u)[n? 2n0]? ?Td(u)a?d(u)[n]|2} , (4.18)where yd(u)[n] is the real part of the signal rd(u)[n].The widely linear version of the downlink filters are obtained by replacing f d(u)with f? d(u) =[<{f d(u)},={f d(u)}], replacingW d(i),d(u) with W? d(i),d(u) =[<{W d(i),d(u)},?={W d(i),d(u)}], and ?d(u) with??d(u) =???<{?d(u)} ?={?d(u)}={?d(u)} <{?d(u)}???,92Chapter 4. Multi-way Relaying for DS-UWB Communicationin equations (4.12) through (4.15).4.2.5 Downlink BER AnalysisFor the BPSK DS-UWB signalling, the analytical BER can be written in terms ofthe SINR at the detector prior to the sign operator asBERd(u) = Q(?SINRd(u)). (4.19)The SINR between relay and the uth pair?s destination node d(u) is derived asSINRd(u) =|f?Td(u)W?Td(u),d(u)en0 |2U?i=1f?Td(i)W?Td(i),d(u)W? d(i),d(u)f? d(i) ? ?s(u),d(u) + ?2d(u)/2, (4.20)where ?s(u),d(u) = ?[f?Td(u)W?Td(u),d(u)en0 , f?Ts(u)W?Ts(u),d(u)ALc ]?2.Note that end-to-end analysis of the BER for a pair of source and destinationnodes requires analytical representation of the error propagation in uplink and down-link phase. Since stream packetization and coding are not considered here, we do notperform a error propagation analysis.4.3 Multi-way Filter-and-Forward RelayingIn this section filter design for multi-way FF relaying is developed. Similar to thesystem model described in Section 4.2.1, the nodes processing is relatively simple andinter-node communication is achieved through a central relay that is equipped withmultiple antennas. The relay estimates the CSI between itself and the nodes, andhandles the complexity associated with filtering. As it was mentioned in Section 4.1,93Chapter 4. Multi-way Relaying for DS-UWB Communicationthe FF relaying can be considered as an extension of the AF relaying over frequencyselective channels [76]. The received signal at the relay is passed through a filter thatis optimized to reduce the distortion caused by ISI and MUI in the multiple-accesscase [79]. We start by describing the relay structure and the signal representationsand then proceed to developing the filter design procedures.4.3.1 System ModelThe general system model is fairly similar to the one introduced in Section 4.2.1with the main difference being in the relay structure. In particular, the source anddestination node structure is the same as the one considered in Section 4.2.1 andshown in Figure 4.1.The block diagram of the central relay for FF processing is shown in Figure4.3. The received signal at the relay?s mth antenna is rake combined with availablechannel coefficients of the link between source and relay through ps(u),m, and is thenpassed through an optimized filter qm. Prior to re-transmission, the signal is pre-rake combined with the estimated channel impulse response of the link between therelay and the destination node, pd(u),m[k]. At each of the relay?s antennas, the receivedsignals from all transmitting users pass through the U pairs of source and destinationpre/post rake filtering. Hence, we define Rsd,m[k] =?U`=1(ps(`),m[k] ? pd(`),m[k]).Then, the overall channel consisting of the transmit channel between transceivernode s(i) and the mth antenna at the relay, post/pre-rake combining filters andtransmit channel between the relay?s mth antenna and the destination transceiverd(u) is defined as gs(i),m,d(u)[k] = hs(i),m[k] ?Rsd,m[k] ? hd(u),m[k].For the FF relaying scheme, the received signal at the destination node can be94Chapter 4. Multi-way Relaying for DS-UWB CommunicationFigure 4.3: Block diagram of the central relay with pre/post-rake filtering and multi-way FF relaying.written asrd(u)[n] =U?`=1qHGHs(`),d(u)as(`)[n] + zd(u)[n] + vd(u)[n] , (4.21)where? q = [qT1 , . . . , qTM ]T is the concatenated vector of the filter coefficients across allantennas,? qm = [qm[0], . . . , qm[Lq ? 1]]H is the vector of filter coefficients at the mth an-tenna,? Gs(`),d(u) = [Gs(`),1,d(u), . . . ,Gs(`),M,d(u)] is a block matrix with block componentsGs(`),m,d(u),? Gs(`),m,d(u) is formed by downsampling the rows of Toeplitz matrix G?s(`),m,d(u)by factor N ,? G?s(`),m,d(u) is defined by first row [(gs(`),m,d(u)[kf ])?,0Lq?1], where the sampling95Chapter 4. Multi-way Relaying for DS-UWB Communicationphase kf is set as kf = Lp + Lh ? 2?NbLp+Lh?2N c, and first column[gs(`),m,d(u)[kf ], gs(`),m,d(u)[kf + 1], . . . , gs(`),m,d(u)[kf + (2Lh + 2Lp ? 4)],0Lq?1]H ,? as(`)[n] =[as(`)[n], . . . , as(`)[n ? Lg + 1]]T is the Lg ? 1 vector of transmittedsymbols affecting the received signal, where Lg = d2Lp+2Lh+Lq?4?kfN e,? zd(u)[n] ? N (0, ?2d(u)) is the additive white Gaussian noise (AWGN) at thedestination node,? vd(u)[n] =M?m=1vd(u),m[n] is the colored noise that is added at the relay andis being processed and forwarded to the destination node. It is defined asvd(u)[n] = qH?Hd(u)zR, where ?d(u) = diag{?d(u),1, . . . ,?d(u),M} is MLv ?MLq and its block components ?d(u),m are Toeplitz matrices with first row[?d(u),m[0],0Lq?1] and first column [?d(u),m[kf ],?d(u),m[kf + 1], . . . ,?d(u),m[kf +(Lh + 2Lp ? 3)],0Lq?1]. ?d(u),m = Rsd,m[k] ? hd(u),m[k] and zR,m[n] = [zR,m[n],zR,m[n ? 1], . . . , zR,m[n ? Lv + 1]]. Lv is the length of the relay noise vectoraffecting the received colored noise at the destination node and is defined asLv = kf + Lh + 2Lp ? 3 + Lq.The average transmit power at the relay is the sum of the average power trans-mitted from individual relay antennas. The transmit signal at the mth antenna issR,m[k] = (zR,m[k] +U?i=1a?i[k] ? hi,m[k]) ?Rsd,m[k] ? qm[k], where zR,m[k] is the AWGNadded at the mth antenna of the relay with variance ?2zR . The average transmit powerat the relay can be written asPR =M?m=1E{sR,m[k]s?R,m[k]} = qH(?2zR?R +U?i=1??s(i))q , (4.22)96Chapter 4. Multi-way Relaying for DS-UWB Communicationwhere the matrices ?R, and ??s(i) are block diagonal matrices with HermitianToeplitz block component matrices ?R,m and ??s(i),m , respectively. The first row of?R,m is defined as [?R,m[0], ?R,m[?1], . . . , ?R,m[?Lq + 1]], where ?R,m[k] = Rsd,m[k] ?(Rsd,m[?k])?. The matrix ??s(i) is structured similarly to ?R by replacing ?R,m[k]with ??s(i) [k] = ?s(i),m[k] ??s(i),m[?k]?.4.3.2 Filter Design for FF RelayingThe FF strategy offers simpler relay design and reduced delay compared to the DTFprocessing, since the message from all nodes are transmitted simultaneously to therelay, filtered and retransmitted (there is no detection and remodulation prior totransmission from the relay to the destination node). In addition in the FF relayingscheme, all user messages are filtered by q, hence the design has a smaller number ofdegrees of freedom compared to the DTF relaying scheme. Therefore, self-interferencecancellation plays a critical role in improving the overall performance. Similar to thePEF design problems considered in Section 2.4 and the DTF relaying filter design inSection 4.2.2, the sum-MSE is the criterion of choice for the filter design.The MSE with self-interference cancellation at the destination node prior to de-tection is defined asMSEIC,FFd(u) = E{???rd(u)[n]? as(u)[n? nf ]? ?Hd(u)a?d(u)[n]??2}, (4.23)where a?d(u)[n] has the same definition as per (4.8), the vector ?d(u) ? CLg?1 containsthe interference cancellation coefficients at the Lc interference cancellation indices andzeros everywhere else, and delay, nf , is set as nf = dd2Lh+2Lp+Lf?4?k0N e/2e. Using the97Chapter 4. Multi-way Relaying for DS-UWB Communicationreceived signal representation from Eq. (4.21), the MSE is represented asMSEIC,FFd(u) =???[?qHGHs(1),d(u), . . . , ?qHGHs(u),d(u) ? enf , . . . , ?qHGd(u),d(u) ? ?Hd(u),. . . , ?qHGHs(U),d(u), ??d(u), ??zRqH?Hd(u)]???2, (4.24)where enf is a vector with the nf element set as 1 and zeros elsewhere. Assumingthat factor ? is the same for all users, and setting ?d(u) = ?HELcGd(u),d(u)q withELc = E{ad(u)[n]a?d(u)[n]}, then the sum-MSE design problem is written asminq,?U?u=1MSEIC,FFd(u) , (4.25a)s.t. qH(?2zR?R +U?u=1??s(u))q ? Pmax . (4.25b)The above problem is in convex form and applying KKT conditions, the closed-form solutions are found as? =?????eTnf(U?u=1Gs(u),d(u))(T FF)HD(T FF)(U?u=1GHs(u),d(u))enfPmax, (4.26)where D = ?2zR?R +U?u=1??s(u) ,T FF =(U?i=1U?j=1GHs(i),d(j)Gs(i),d(j) ?U?u=1GHd(u),d(u)ELcGd(u),d(u)+ ?2zRU?u=1?Hd(u)?d(u) +DU?u=1?2d(u)Pmax)?1,98Chapter 4. Multi-way Relaying for DS-UWB Communicationand the downlink PEF is given asq = (T FF)(U?u=1GHs(u),d(u))enf/? . (4.27)4.3.3 Iterative Design based on User Specific Scaling FactorIn this section the design from Section 4.3.2 is extended to a more general case withuser specific scaling for the nodes. Similar to Section 4.2.3, an iterative design schemeis proposed based on minimizing the sum-MSE.The MSE with self-interference cancellation and user specific scaling is defined asMSEICd(u) = E{|?d(u)rd(u)[n]? as(u)[n? nf ]? ?Hd(u)a?d(u)|2} . (4.28)Using the received signal representation from (4.21), the MSE is derived asMSEICd(u) =???[?d(u)qHGHs(1),d(u), . . . , ?d(u)qHGHs(u),d(u) ? enf , . . . , ?d(u)qHGHd(u),d(u)? ?Hd(u), . . . , ?d(u)qHGHs(U),d(u), ?d(u)?d(u), ?d(u)?zRqH?Hd(u)]???2(4.29)Next we use the alternating optimization approach from Section 4.2.3 to arrive at asolution for the sum MSE minimization problem subject to maximum relay transmitpower constraint. As mentioned in Section 4.2.3, for a non-convex optimizationproblem, the AO algorithm does not guarantee convergence to a global optimumpoint. The parameter space is divided as ?1 = [?1, . . . , ?U ], ?2 = [q] and ?3 = ?.Setting ?d(u) = ?d(u)qHGHd(u),d(u)ELc , using KKT conditions, the strict minimizers at99Chapter 4. Multi-way Relaying for DS-UWB Communicationeach iteration index, t, are found as?(t+1)d(u) =<{q(t)HGHs(u),d(u)enf}q(t)H?d(u)q(t) + ?d(u), (4.30)where?d(u) =U?i=1GHs(i),d(u)Gs(i),d(u) ?GHd(u),d(u)ELcGd(u),d(u) + ?2zR?Hd(u)?d(u) ,q(t+1) = T ?(t+1)(U?u=1?(t+1)d(u) GHs(u),d(u)en0), (4.31)whereT (t+1)? =(U?u=1|?(t+1)d(u) |2?d(u) + ?(t)D)?1,and?(t+1) =U?u=1|?(t+1)d(u) |2?2d(u)Pmax.For the initial solution, the parameters from optimal solution of the design withsame ? for all users from Eqs. (4.26) and (4.27) are used to solve Eq. (4.30),and (4.31) in the first iteration. After each iteration two conditions are checked asstopping criteria, (i) ??(t+1)??t? ? , where ?t = [?t1,?t2, ?t3], (ii) t < Niter, checkingif the number of iteration has reached the maximum allowable number of iterations.4.3.4 Widely Linear Filtering for FF RelayingThe widely linear counterpart of the filter design schemes introduced in Section 4.3.2and Section 4.3.3 are obtained by incorporating the real-part of the received signal100Chapter 4. Multi-way Relaying for DS-UWB Communicationin the MSE definitions of (4.23) and (4.28). The real-part of the received signal iswritten asyd(u)[n] =U?i=1q?T G?Ts(i),d(u)as(i)[n] + z?d(u)[n] + v?d(u)[n] , (4.32)where q? = [<{q},={q}], G?s(i),d(u) = [<{Gs(i),d(u)},?={Gs(i),d(u)}], the AWGN noiseterm z?d(u)[n] = N (0, ?2d(u)/2). The real part of the colored noise forwarded throughthe relay is written as v?d(u)[n] = q?T ??Td(u)z?R, where z?R = [<{zR},={zR}] and??d(u) =???<{?d(u)} ?={?d(u)}?={?d(u)} ?<{?d(u)}???.The matrices ?R and ??s(u) are replaced with??R =???<{?R} ?={?R}={?R} <{?R}???,and???s(u) =???<{??s(u)} ?={??s(u)}={??s(u)} <{??s(u)}???.4.3.5 BER AnalysisThe BER for each of the source and destination pairs can be obtained using the SINRfor the corresponding link. The SINR for the design with interference cancellation at101Chapter 4. Multi-way Relaying for DS-UWB Communicationthe destination node d(u), after the real operator is derived to beSINRd(u) =|q?T G?Ts(u),d(u)en0 |2U?i=1q?T G?Ts(i),d(u)G?s(i),d(u)q? ? ?[q?T G?Ts(u),d(u)en0 , q?T G?Td(u),d(u)ELc ]?2 + ?d(u),(4.33)where ?d(u) =?2zR2 q???Td(u)??d(u)q? +?2d(u)2 .Using the above definition for SINR, the BER for the BPSK DS-UWB signallingcan be evaluated asBERd(u) = Q(?SINRd(u)). (4.34)4.4 Numerical ResultsIn the following we describe and discuss a set of numerical results that demonstratethe performance of the two proposed relaying algorithms. Similar to the simulationsetup for Chapters 2 and 3, we consider the CM2 channel model for the residentialnon-line-of-sight environment, cf. [12], and the channel realizations are generated ac-cording to the procedure described in [10]. Note that the designs proposed in thischapter are also applicable to other UWB channel models. The signalling speci-fications include a center frequency of 6 GHz and a pulse bandwidth of 0.5 GHzusing root-raised cosine pulses gTx(t) and gRx(t) with roll-off 0.7. Unless otherwisespecified, results are averaged over 500 channel realizations, and it is assumed that?2R = ?2d(1) = . . . , ?2d(U).102Chapter 4. Multi-way Relaying for DS-UWB Communication0 5 10 15 2010?410?310?210?1100SNRTx,R [dB]BER S,D  M = 2 Linear DesignM = 4 Linear DesignM = 2 WL DesignM = 4 WL DesignFigure 4.4: The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme with U = 4 users,M = 2 and M = 4 antennas at the relay and Lf = 10. Comparison between linearand widely linear designs.4.4.1 DTF RelayingIn this section the simulated results for multi-way DTF relaying are presented. In allresults it is assumed that the filter lengths for uplink equalization and downlink pre-equalization are equal, i.e., LULf = Lf . We start by comparing the linear and widelylinear design schemes. It was shown in Chapter 3 that the WL (robust) design schemeis preferred to its linear counter-part in terms of requiring lower minimum averagetransmit power for meeting a pre-defined level of QoS. Here we compare the simulatedBER between a pair of source and destination nodes, obtained using the design from(4.10) with the same scaling factor for all users, for the linear and WL uplink anddownlink filter designs.In Figure 4.4, the simulated BER between a pair of source and destination nodes103Chapter 4. Multi-way Relaying for DS-UWB Communicationversus the relay transmit SNR, defined as SNRTx,R = Pmax/?2d(u), is shown for anetwork of U = 4 nodes communicating via the central relay. In the figure, theresults for linear and WL designs are plotted for scenarios in which the central relayis equipped withM = 2 andM = 4 antennas, respectively. Considering a BER of 1?10?3 as a point of reference for the comparison, when M = 2, the WL design achievesthe reference BER at a 5 dB lower transmit SNR than the linear design. Increasingthe number of antennas at the relay to M = 4, the difference in transmit SNR isapproximately 1.5 dB. The transmit SNR at the relay is a measure of the transmitpower required for achieving the reference BER. Considering the superiority of theWL design approach, we select WL as the preferred design scheme for generating thenext set of results in this section, that evaluates the performance of the DTF relayingscheme.Next, we evaluate the effect of self-interference cancellation on the BER. Figure4.5 shows BER between a pair of source and destination nodes versus the relay trans-mit SNR for U = 4 nodes and M = 2 and M = 4 antennas at the relay, respectively.Similar to the previous figure, the results are generated for the design problem from(4.10). The BER is simulated for the WL design without self-interference cancella-tion at the node (Lc = 0) and the WL design with self-interference cancellation forLc = 1, 3, 5, 30, where Lc = 30 corresponds to full self-interference cancellation, i.e.,cancelling the effect of the entire sequence of symbols which affect the received signal.The effect of self-interference cancellation is more pronounced in the scenario withM = 2 antennas at the relay compared to the case with M = 4. For example atreference BER of 1? 10?4, and for M = 2, the design with Lc = 30 requires 2.5 dBless transmit SNR compared to the design without self-interference cancellation. Inthe case of M = 4, the difference in the transmit SNR is on the order of 1 dB.104Chapter 4. Multi-way Relaying for DS-UWB Communication0 5 10 15 2010?610?510?410?310?210?1100SNRTx,R  [dB]BER  Lc = 0Lc = 1Lc = 3Lc = 5Lc = Lt = 30 M = 4  M = 2Figure 4.5: The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme with U = 4 users,M = 4 antennas at the relay and Lf = 10. Comparison of the BER for interferencecancellation lengths of Lc = [1, 3, 5, 30].Furthermore, it can be seen from the figure that the benefits of full self-interferencecancellation can be achieved by cancelling the self-interference from a small num-ber of bits. For the design scenarios in Figure 4.5, Lc = 5 achieves a performancethat is close to that for full self-interference cancellation. Note that self-interferencecancellation at the nodes requires i) feedback of interference cancellation coefficientsand ii) storing the transmitted bits at the node. Hence reducing the number ofself-interference cancellation bits affects the feedback channel and the operations (interms of storage and recovery of bits) at nodes.Figure 4.6 presents a comparison of our proposed DTF relaying scheme fromSection 4.2.2 with two other DTF schemes namely, post/pre-rake only and the de-sign with Lf = 1 which corresponds to single-tap beam-forming. We observe that105Chapter 4. Multi-way Relaying for DS-UWB Communication0 2 4 6 8 10 12 14 1610?610?510?410?310?210?1100SNRTx,R  [dB]BER S,D  Post/Pre?Rake OnlyLf = 1, w/o ICLf = 10, w/o ICFigure 4.6: The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for DTF relaying scheme with U = 4 users,M = 4 antennas at the relay. Comparison of the BER for post/pre-rake schemewithout post/pre-equalization, post/pre-equalized DTF relaying with Lf = 1 andLf = 10.rake combining in the uplink and pre-rake combining in the downlink are not suffi-cient to overcome the ISI and MUI. Furthermore, applying single tap beam-forming(Lf = 1), which is typically considered for relaying over flat fading MIMO channels,suffers from considerable residual interference, while our proposed design based onpost-equalization in uplink and pre-equalization in downlink can fully eliminate thedetrimental effects of ISI and MUI in UWB channels.Next we proceed to discuss and present results related to the DTF relaying withuser specific scaling factor from Section 4.2.3. In Figure 4.7, the downlink sum-MSEis plotted versus relay transmit SNR for U = 4 nodes and M = 4 antennas at therelay and Lf = 10. The noise variance at the relay and the nodes are set as follows?2R = ?d(1)2, and ?2d(u)/?2d(1) = 1, 2, 5, 10 for user indices u = 1, 2, 3, 4 respectively.106Chapter 4. Multi-way Relaying for DS-UWB Communication0 5 10 15 2000.511.522.533.5SNRTx,R [dB]? MSE  Lc = 5, ?1 = ? = ?ULc = 5, User Specific ScalingFigure 4.7: The sum MSE versus relay transmit SNR, SNRTx,R = Pmax/?2d(u), forDTF relaying scheme with U = 4 users, M = 4 antennas at the relay and Lf = 10.Comparison between designs with same receiver scaling factor for all users and thedesign with user specific scaling factor.We observe from the figure that applying the iterative design for user specific scalingcan improve the sum-MSE.For an in-depth study of the effect of user specific scaling, we consider a two userscenario. Figure 4.8 presents the sum-MSE as a function of the difference in node?snoise levels for several transmit SNR values set as SNRTx,R = 8, 12, 16 dB. As it canbe seen from the figure, the efficacy of the iterative design for user specific scalingvaries with the transmit SNR and also the difference in the receiving node noise levels.If users operate at similar noise levels then applying the convex design from (4.10)would be more efficient. The relay can switch to the iterative approach to improvethe throughput if the channel and system conditions change.107Chapter 4. Multi-way Relaying for DS-UWB Communication0 2 4 6 8 / ?12 [dB]? MSE  w/o ICLc = 5, ?1 = ?2Lc = 5, User Specific ? SNRTx,R = 8 dB SNRTx,R = 16 dB SNRTx,R = 12 dBFigure 4.8: The sum MSE versus user?s noise level difference ?22/?21 in dB, for DTFrelaying scheme with U = 2 users, M = 4 antennas at the relay and Lf = 10.Comparison between designs with and without interference cancellation and samereceiver scaling factor for all users and the design with user specific scaling factorwith self-interference cancellation.4.4.2 FF RelayingIn this section we present and discuss the results related to the multi-way FF relaying.Similar to Section 4.4.1, we start by comparing the linear and widely linear designapproaches and then proceed to a more detailed system performance evaluation.In Figure 4.9, the pairwise BER for linear and WL designs is shown for two designscenarios, with and without self-interference cancellation. For the simulations U = 4nodes, M = 4 antennas at the relay and an equalization filter length of Lq = 20 isconsidered. As per the relay block diagram in Figure 4.3, the received signal at therelay is processed with a combination of post-and-pre-rake filters in addition to theequalizing filter prior to re-transmission. In case of self-interference cancellation, Lc108Chapter 4. Multi-way Relaying for DS-UWB Communication0 5 10 15 2010?410?310?210?1100SNRTx,R [dB]BER S,D  WL Design, AnalyticalWL Design, SimulationLinear Design, AnalyticalLinear Design, Simulationw/o IC  Lc = Lg = 42  Figure 4.9: The BER between a pair of source and destination nodes versus relaytransmit SNR, SNRTx,R = Pmax/?2d(u), for FF relaying scheme with U = 4 users,M = 4 antennas at the relay and Lq = 20. Comparison between linear and widelylinear designs, with and without self-interference cancellation.is set to Lc = 42 which corresponds to full self-interference cancellation. The closematch of the simulated results with that of the analytical evaluation from (4.34),confirms the validity of the derivations in Section 4.3.5. It is observed that applyingWL design without self-interference cancellation achieves a BER that is comparableto the BER for the linear design with full self-interference cancellation. Consideringthat the gains achieved by WL design come without any transmission overhead,unlike the self-interference cancellation scheme that requires feedback of informationand storing the transmitted bits, applying the WL design is clearly advantageous.The effect of interference cancellation on WL design of filters for multi-way FFrelaying with U = 4 users and relaying viaM = 4 antennas at the relay, for the designwith same receiver scaling is depicted in Figure 4.10. Self-interference cancellation109Chapter 4. Multi-way Relaying for DS-UWB Communication0 5 10 15 2010?410?310?210?1100SNRTx,R [dB]BER S,D  WL Design w/o ICWL Design with IC Increasing LcLc = 1Lc = Lg = 42 Lc = 11Figure 4.10: The Analytical BER between a pair of source and destination nodesversus relay transmit SNR, SNRTx,R = Pmax/?2d(u), for FF relaying scheme with U = 4users, M = 4 antennas at the relay and Lq = 20. Comparison of the BER for self-interference cancellation lengths of Lc = [1, 3, 5, 7, 11, 42]..lengths of Lc = 1, 3, 5, 7, 11, 42 are considered for comparison. It can be seen fromthe figure that increasing the number of self-interference cancellation bits, improvesthe BER significantly. As an example, applying Lc = 42, the BER of 1 ? 10?3 isachieved at 5 dB lower relay transmit SNR, which is quite significant compared tothe marginal gains achieved from self-interference cancellation in the DTF scheme(referring to Figure 4.5). It is also observed that Lc = 11 results in a performancecomparable to Lc = Lg = 42.Next, we proceed to evaluate the effect of iterative design with user specific scalingon the throughput. In Figure 4.11, the sum-MSE is plotted versus the noise leveldifference for a FF relaying network consisting of U = 2 nodes, M = 4 antennas110Chapter 4. Multi-way Relaying for DS-UWB Communication0 2 4 6 8 / ?12 [dB]? MSE  w/o IC, ?1 = ? = ?ULc = 5, ?1 = ? = ?ULc = 5, User Specific Scaling SNRTx,R = 12 dB SNRTx,R = 16 dB SNRTx,R = 8 dBFigure 4.11: The sum MSE versus user?s noise level difference ?22/?21 in dB, forFF relaying scheme with U = 2 users, M = 4 antennas at the relay and Lq = 20.Comparison between designs with and without self-interference cancellation and samereceiver scaling factor for all users and the design with user specific scaling factor,and with self-interference cancellation.at the relay, and equalizing filter length set as Lq = 20. The sum-MSE is plottedfor three transmit SNR levels of SNRTx,R = 8, 12, 16 dB. It is observed that athigher transmit SNR levels, the iterative design with user specific scaling achievesa sum-MSE that is comparable to that achieved by the non-iterative convex designwith same receiver scaling for all users. The effect of user specific scaling factor ismore pronounced at lower transmit SNR levels and when the two users operate atdifferent receive SNR levels. Note that the difference in noise levels translates todifference in received SNR. Therefore, the convex design scheme with same scalingfactor for all users can be used as the default FF relaying design procedure. Once theoptimal solution of the design problem in (4.25) is obtained, the received SINR can111Chapter 4. Multi-way Relaying for DS-UWB Communicationbe evaluated analytically using (4.33). Combining the information about the relaytransmit SNR and the destination node SINR levels, the relay can make a decisionabout whether or not switching to the iterative design is feasible.4.5 Concluding RemarksIn this chapter we have extended the work from the previous chapters and developedtwo novel multi-way relaying techniques, for a network of low-complexity DS-UWBnodes communicating via a more powerful central relay. The designs are based onDTF and FF processing at the relay. Our design schemes are novel in that weconsider relaying over frequency-selective fading channels and the use of post/pre-rake combining. Considering the BPSK DS-UWB signalling, WL counter-parts ofthe proposed filter design schemes were devised and the superiority of WL design forboth relay schemes was demonstrated via numerical performance evaluation.For both relaying schemes we formulated convex optimization problems withclosed-form solution and also a more general formulation with an iterative designapproach using alternating optimization. Based on numerical evaluations, for bothschemes, the benefits of the iterative design are notable when users operate at dif-ferent SNR levels. Using our BER analysis, we suggest adopting the convex designswith identical receiver scaling for all users as the default design procedure for bothschemes and switching to the iterative design when signalling conditions change.The example results on interference cancellation, for channels randomly drawnfrom CM2 model, reveal that cancelling the self interference resulting from a smallnumber of transmitted bits, e.g. setting Lc = 5 for DTF relaying and Lc = 11 for FFrelaying, is sufficient to reduce the complexity at the receiving nodes and achieve aperformance close to the full self-interference cancellation. We note that the reported112Chapter 4. Multi-way Relaying for DS-UWB Communicationnumbers vary for different channel models. Reducing the number of self-interferencecancellation bits is desired since it affects the traffic on the feedback channel and theprocessing at the destination nodes.113Chapter 5Summary of the Thesis and Topicsfor Future ResearchIn this chapter, we first summarize the contributions of the thesis. Then, we proposepossible future research directions.5.1 Summary of ContributionsThis thesis has focused on the design of a multiuser DS-UWB system and reducingthe computational load of data detection at processing-constrained nodes by usingpre-filtering at a more powerful central unit. Through this work, in particular the de-veloped procedures for the design of multiuser PEFs in the context of communicationover pre-rake combined multipath channels, the state-of-the-art of methodologies forpre-filtering has been advanced.? In Chapter 2, the design of pre-equalization filters for the downlink of a mul-tiuser DS-UWB system is addressed. The proposed pre-filtering which is thecombination of pre-equalization and pre-rake combining alleviates the signalprocessing task at the lower-complexity receiver nodes. Two design strategies,based on average transmit power and sum-MSE minimization, were consideredto establish a framework for the design of pre-rake UWB multiuser systems.Numerical results demonstrated the efficacy of the proposed filter designs to114Chapter 5. Summary of the Thesis and Topics for Future Researchmitigate the effects of MUI and ISI experienced in the considered UWB sys-tems, which otherwise often fail to achieve satisfactory, e.g., BER performance.? In Chapter 3, the effect of non-ideal CSI at the central unit on the designof PEFs for the downlink of the multiuser DS-UWB system was studied androbust PEF design strategies were proposed. Two CSI uncertainty models wereconsidered for incorporating robustness into the design of PEFs. It was shownthat the robust design strategies, based on applying average and instantaneousMSE constraints, can maintain the required level of QoS in the presence of CSIuncertainties. Furthermore, widely linear robust design procedures based onreal-valued binary signalling were proposed. It was shown that the WL designcan result in significant power savings, which is of particular importance forUWB systems which need to coexist with incumbent narrowband systems.? In Chapter 4, two novel multi-way relaying schemes, namely DTF and FF wereproposed for cooperative communication through the central unit in a networkof low-complexity DS-UWB nodes. The novelty in the design is associatedwith the combination of post-and-pre-rake filtering with equalization filters inthe central relay to mitigate ISI and MUI. Partial and full self-interferencecancellation at the destination node was considered for both relaying strategies,and it was shown that cancelling the self-interference from a small number ofinterference bits is sufficient for achieving a performance close to that with full-interference cancellation. Moreover, the designs were extended to their WLcounter-parts, and it was shown that WL filter design is superior to the lineardesign in terms of lowering the transmit power at the central relay.115Chapter 5. Summary of the Thesis and Topics for Future Research5.2 Suggestions for Future WorkIn the following, we propose several interesting future research directions that arebased on the work in this thesis.Building a testbed for experimental validation of the design procedures describedin Chapters 2, 3 and 4 would be the immediate extension to the current work. Atestbed for UWB MIMO with pre-rake combining (time-reversal) is described in [119],where digital arbitrary waveform generator is implemented in the FPGA, coupledwith digital-to-analog converter and RF front end for single user MIMO and MISOconfigurations. From the experimental results obtained using the testbed in [119],increasing the data rate, the signal-to-interference ratio is saturated by the ISI. Ourfilter design strategies from Chapters 2 and 3 can be incorporated to the base bandprocessing block in the FPGA implementation in [119] to overcome the ISI and enablemultiuser operation (transmission of multiple streams).Inclusion of spectral mask constraints in the design procedures presented in Chap-ters 2, 3 and 4 is another extension that is useful for practical implementation. To-wards this end, in [112] a convex representation of the spectral mask constraints ispresented for the single user pre-rake DS-UWB. 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IEEE Global CommunicationsConference (GLOBECOM), 2012, pp. 4060?4064.127Appendix AProof of Lemma 2.3.1The proof uses a contradiction argument. Let {(q~u , ?~u ) | u = 1, . . . , U} be theoptimal solution of (2.14) such that one of the constraints in (2.14b) (say for the ithuser) is satisfied with strict inequality, i.e.,???[q~1W 1,i, . . . , q~i W i,i ? en0?~i , . . . , q~UW U,i, ?z]???2< ?i?~i2 . (A.1)We will show that one can construct a new solution {(q?u, ??u) | u = 1, . . . , U} that isfeasible and also results in a lower total power than the total power achieved by theacclaimed optimal solution {(q~u , ?~u )}. Let us construct {(q?u, ??u)} from {(q~u , ?~u )}by changing (q~i , ?~i ) only. In particular, let q?i = q~i /? and ??i = ?~i /?, for some? > 1, while q?u = q~u and ??u = ?~u for all u 6= i. By construction, the new solutionsatisfies the MSE constraint for each u 6= i in (2.14b). To show that the new solutionalso satisfies the MSE constraint for u = i, we first observe that (A.1) can be writtenas????[q?1W 1,i?, . . . , q?iW i,i ? en0 ??i, . . . ,q?UW U,i?, ?z?]????2< ?i ??2i , (A.2)128Appendix A. Proof of Lemma 2.3.1or equivalently,1?2??[q?1W 1,i, . . . , q?i?1W i?1,i, q?i+1W i+1,i, . . . , q?UW U,i, ?z]??2+ ?[q?iW i,i ? en0 ??i]?2 < ?i??2i , (A.3)which implies that there exists a value of ? > 1 such that (A.3) implies??[q?1W 1,i, . . . , q?i?1W i?1,i, q?i+1W i+1,i, . . . , q?UW U,i, ?z]??2+ ?[q?iW i,i ? en0 ??i]?2 = ?i??2i . (A.4)Hence the new solution satisfies the MSE constraint for the case of u = i.To show that the new solution {(q?u, ??u)} yields lower transmitted power than{(q~u , ?~u )}, we observe that q?i?iq?Hi = 1?2q~i ?iq~Hi < q~i ?iq~Hi , while q?u?uq?Hu =q~u?uq~Hu for u 6= i. Therefore, the total transmission power achieved by solution{(q?u, ??u)} is less than the one achieved by {(q~u , ?~u )}. This contradicts the optimalityof {(q~u , ?~u )} and completes the proof.129Appendix BProof of Lemma 2.3.2Using the Lagrangian and KKT conditions [98] associated with the optimizationproblem in (2.14), we obtain the relationsq~u = ?~u?~u en0WHu,u[?u +U?i=1?~i W i,uWHi,u]?1, (B.1)and?~u[?~u (1? ?u)?<{q~uW u,ueTn0}]= 0 . (B.2)where ?~u in (B.1) is the optimal dual variable associated with QoS constraint of useru. We observe that ?~u 6= 0 for all 1 ? u ? U , since from (B.1), if ?~u is zero, thenthe optimal filter design q~u is zero, which would violate the MSE constraints. Fromthe complementary slackness condition Lemma 2.3.1 follows.For the proof of Lemma 2.3.2 we need to obtain an expression for the receivedSINR. We first write the received signal ru[n] in (2.6) asru[n] = quW u,ueTn0au[n0] +U?i=1, i 6=uqiW i,uaTi + quW u,uaTu,n0 + zu[n] , (B.3)where aTu,n0 is the vector of the uth user?s message with its nth0 element replaced by azero. Using (B.3) the effective SINR at the receiver for the downlink transmission is130Appendix B. Proof of Lemma 2.3.2given as a function of the pre-equalization filter used (in our case q~u ) bySINRu =??q~uW u,ueTn0??2U?i=1??q~i W i,u??2 ???q~uW u,ueTn0??2 + ?2z. (B.4)Equivalently, (B.4) can be written as1SINRu + 1= 1 ?q~uW u,ueTn0en0WHu,uq~HuU?i=1q~i W i,uWHi,uq~Hi + ?2z. (B.5)Furthermore, from (B.2) we have?~u =<{q~uW u,ueTn0}1? ?u. (B.6)Substituting this result in (2.14b) and by using the result from Lemma 2.3.1, we have?u = 1?<{en0WHu,uq~Hu }2U?i=1q~i W i,uWHu,iq~Hi + ?2z. (B.7)Finally, to show that right hand side of (B.7) is equivalent to that of (B.5), we showthat q~uW u,ueTn0 is a real quantity. Indeed, using (B.1) we haveq~uW u,ueTn0 = ?~u?~u en0WHu,uG?1W u,ueTn0 , (B.8)where G = ?u +U?i=1?iW u,iWHu,i  0, and therefore, q~uW u,ueTn0 is a scalar realquantity. This completes the proof of Lemma CProof of Lemma 2.4.1For the DS-UWB downlink problem in (2.16), by constructing the Lagrangian asso-ciated with (2.16) and setting its derivative w.r.t. ?Hu to zero, we obtain the followingrelation?~u =en0WHu,uq~uHU?i=1q~i W i,uWHi,uq~iH + ?2z. (C.1)Substituting (C.1) in the expression of ?2u in (2.10) and comparing it to the expressionof 1/(SINRu + 1) in (B.5), we can obtain the required relation. For the DS-UWBuplink, we first write an expression for the SINR of each user in the uplink, similarto that of the downlink in (B.4), asSINRULu =|?u|2fHuW u,ueTn0en0WHu,ufuU?i=1|?i|2fHuW u,iWHu,ifu ? |?u|2fHuW u,ueTn0en0WHu,ufu + fHu ?ufu?2c.(C.2)By substituting the expression in (2.21) for optimal f~u in above equation and com-paring it to the expression of ?2u in (2.18), we obtain the desired relation in theDS-UWB uplink.132Appendix DProof of Theorem 3.3.2To prove the above theorem we first note that (3.24) can be written as????u??u ?Hu?u ?u??uIULt+1???????0 ??Hu ?Hu??u?u 0ULt+1???, ??u? ? u . (D.1)Then, setting Z = ?u, B = [0MLh?1 ?Hu ], D = [?1 01?ULt+1], ? = u, andA =????u??u ?Hu?u ?u??uIULt+1???and applying Lemma 3.3.1 to (D.1) directly leads to (3.26).133


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