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Lattices in MIMO Spatial Multiplexing Lattices in MIMO Spatial Multiplexing: Detection and Geometry Francisco A. T. B. N. Monteiro (Fitzwilliam College) Department of Engineering University of Cambridge May 2012 To Fotini and To my mother iii iv Abstract Multiple-input multiple-output (MIMO) spatial multiplexing (SM) allows unprecedented spectral efficiencies at the cost of high detection complexity due to the fact that the underlying detection problem is equivalent to the closest vector problem (CVP) in a lattice. Finding better algorithms to deal with the problem has been a central topic in the last decade of research in MIMO SM. This work starts by introducing the most prominent detection techniques for MIMO, namely linear filtering, ordered successive interference cancellation (OSIC), lattice- reduction-aided, and the sphere decoding concept, along with their geometrical interpretation. The geometric relation between the primal and the dual-lattice is clarified, leading to the proposal of a pre-processing technique that allows a number of candidate solutions to be efficiently selected. A sub-optimal quantisation-based technique that reduces the complexity associated with exhaustive search detection is presented. Many of the detection algorithms for MIMO have roots in the fields of algorithmic number theory, theoretical computer science and applied mathematics. This work takes some of those tools originally defined for integer lattices and investigates their suitability for application to the rational lattices encountered in MIMO channels. Looking at lattices from a group theory perspective, it is shown that it is possible to approximate the typical lattices encountered in MIMO by a lattice having a trellis representation. Finally, this dissertation presents an alternative technique to feedback channel state information to the transmitter that shifts some of the processing complexity from the receiver to the transmitter while also reducing the amount of data to be sent in the feedback link. v vi Declaration This dissertation is submitted for the degree of Doctor of Philosophy. I hereby declare that this dissertation is not substantially the same as any that I have submitted for a degree or diploma or other qualification at any other university. I further state that no part of my dissertation has already been or is being concurrently submitted for any such degree, diploma or other qualification. I also declare that this dissertation is the result of my own work carried out at the University of Cambridge and includes nothing which is the of work done in collaboration, except where specified explicitly in the text and Acknowledgments. The length of this dissertation does not exceed 65,000 words and contains fewer than 150 figures, according to the limits stipulated by the Department of Engineering. Francisco A. T. B. N. Monteiro May 2012 vii viii Acknowledgements I thank first of all Dr. Ian Wassell for his patience in supervising me and allowing me to attend the large number of lectures, talks, and courses in fields so much beyond the scope of my research topic, in the unique intellectual environment that Cambridge provides. Dr. Wassell’s fast and reliable feedback to anything asked is unique. This thesis could not exist without his endless support and the confidence he showed in me. I am thankful to both Prof. Alister Burr and Dr. Jossy Sayir for their insightful questions and comments during an intellectually stimulating and pleasant viva exam. Not only was this dissertation improved by their comments, but I was also happy in getting to know further links between current MIMO topics and earlier related problems. My departmental affiliation in Cambridge was always a complicated affair; with an application to a group in the process of moving between departments, I ended up affiliated with both: the Department of Engineering and the Computer Laboratory. Given the scope of my research, this could not have proved more appropriate. Moreover, I was lucky in having full access to both departments, taking the best from the opportunities provided by both. I thank Professor Frank Kschischang for the privilege of learning from his immense knowledge. I shall never forget the invitation to have Christmas Dinner at his house with his family when I was far from all. Also at Toronto, I thank Dr. Danilo Silva for his kindness and his example as a researcher. In Cambridge I was lucky to have met a group of friends. I will never forget the intellectual brilliance of Dr. Karen Su, Dr. Ioannis Chatzigeorgiou for his example as ix ACKNOWLEDGEMENTS well-rounded academic and person and Dr. Bogdan Roman, whose theoretical and practical skills never cease to surprise me. I have no words to express how deeply thankful I am for the unshakeable confidence that Karen, Yannis and Bogdan showed in me, and how much I learnt from them. I have also been privileged to learn from the academic example of Dr. Miguel Rodrigues and Dr. Ford Wong in the group. I thank Dr. William Carson, Dr. Weisi Guo, Dr. Jaime Adeane, Ruoshui Liu and to Dr. Vaughan Wittorff for all the pleasant conversations we had over the years. The friendship and human example of Dr. Yan Wu will be always remembered. I thank Dr. Oded Regev for the long discussions on the orthogonal sublattice problem from the algorithmic perspective, and to Dr. Frédérique Oggier for discussing the same problem from the algebraic point of view. I also thank to Dr. Steven Galbraith for interesting conversations on the Babai algorithm and to Dr. Keith Matthews for discussions on the Hermite Normal Form. I am thankful to all the libraries I have used in Cambridge: Engineering Department Library for supporting the borrowing from the British Library and for still allowing the pleasure of exploring the latest IEEE issues on paper; the Computer Laboratory Library, Fitzwilliam College Library, University Library, Betty and Gordon Moore Library, and Library of the Cambridge Philosophical Society, for the endless hours of discoveries and for having papers that “no other” has. In Toronto I am thankful to the libraries of the Electrical and Computer Engineering Department and the Mathematics Department, which also have “everything”. The Portuguese B-on service was also important for accessing some of the papers. My time in Cambridge was possible thanks to the scholarship from the Foundation for Science and Technology. The several months spent in Toronto were possible thanks to grants from the Gulbenkian Foundation, the Royal Academy of Engineering, the Cambridge Philosophical Society, the Digital Technology Group of The Computer Laboratory and Fitzwilliam College. Some other partial grants were obtained from the Instituto de Telecomunicações and the University Institute of Lisbon. x ACKNOWLEDGEMENTS I am thankful to Professor Nick Kinsgbury and Dr. Albert Guillén i Fàbregas for their wise and timely advice on research paths and working methodologies. I thank the students at Fitzwilliam College for their example, their talent, and for creating the most admirable of all environments to live in. I am thankful for the teaching opportunity given to me by the University Institute of Lisbon. No words are enough to thank Dr. Fotini Hadjittofi, now my wife, for all her support in the most difficult times. And I thank my mother Lourdes for all the support throughout all my life. xi ACKNOWLEDGEMENTS xii Contents Abstract v Declaration vii Acknowledgements ix Contents xiii List of Figures xvii List of Tables xxiii Acronyms xxv Symbols and Notation xxix Chapter 1 – Introduction 1 1.1 – MIMO in Context 1 1.1.1 – From Shannon to Codes on Graphs 1 1.1.2 – The First Appearance of Lattices: Coding for The Band-limited Channel 4 1.1.3 – The Advent of MIMO 5 1.1.4 – MIMO in Wireless Standards 7 1.2 – The Different Faces of MIMO 8 1.2.1 – Diversity and Multiplexing 9 1.2.2 – Space-time Codes (STC) 12 1.2.3 – Spatial Multiplexing (SM) 13 1.2.4 – Spatial Diversity Versus Spatial Multiplexing 17 1.2.5 – Multi-user MIMO 17 1.2.6 – Single-user Closed loop (Water-filling) 18 1.2.7 – Beamforming 19 xiii CONTENTS 1.2.8 – Channel Feedback 19 1.3 – Motivation and Scope 21 1.3.1 – Limitations of Scope 23 1.4 – Publications 23 1.5 – Dissertation Outline 23 Chapter 2 – Fundamentals on Lattices and Spatial Multiplexing 25 2.1 – Lattices 25 2.1.1 – Context 26 2.1.2 – Basic Definitions 27 2.1.3 – The Dual Lattice 34 2.2 – MIMO Spatial Multiplexing 38 2.2.1 – System Model 38 2.2.2 – The Real Equivalent Model 43 2.2.3 – Capacity with CSIR 44 2.3 – Detection in MIMO SM 47 2.3.1 – The Complexity of Optimal Detection 48 2.4 – Summary 51 Chapter 3 – Geometry and Detection in Spatial Multiplexing 53 3.1 – Linear Receivers 54 3.1.1 – Zero-forcing Detection 55 3.1.2 – The Geometry of ZF Detection 58 3.1.3 – Algebraic Analysis of ZF 60 3.1.4 – Minimum Mean Squared Error Detection 61 3.1.5 – Projection Matrices 66 3.2 – Ordered Successive Interference Cancellation 68 3.2.1 – The Geometry of Optimal Ordering 68 3.3 – Gram-Schmidt Orthogonalisation and QR Decomposition 75 3.4 – Lattice-Reduction-Aided Detection 77 3.5 – Sphere Decoding 82 xiv CONTENTS 3.6 – Dual-Lattice-Aided Detection 86 3.6.1 – Successive Minima in the Dual Lattice 86 3.6.2 – Projections Onto Hyperplanes 87 3.6.3 – List of Candidate Solutions 88 3.7 – Performance Comparison 90 3.8 – Summary 95 Chapter 4 − Exhaustive Search in Quantised Spaces 97 4.1 – Quantised Spaces 98 4.2 – Quantisation Error 99 4.2.1 – Uncorrelated Noise and Uncorrelated Data 100 4.2.2 – Saturation Does not Impair Detection 100 4.2.3 – Uniform Error Per Component
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