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Channel state information feedback, prediction and scheduling for the downlink of MIMO-OFDM wireless systems

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Channel state information feedback, prediction and scheduling for the downlink of MIMO-OFDM wireless systems

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Content CHANNEL STATE INFORMATION FEEDBACK, PREDICTION AND SCHEDULING FOR THE DOWNLINK OF MIMO-OFDM WIRELESS SYSTEMS by Hooman Shirani-Mehr A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2010 Copyright 2010 Hooman Shirani-Mehr Dedication To my parents, grandparents and to my wife for all their support and love ii Acknowledgements I would like to express my most profound gratitude to my advisor, Professor Giuseppe Caire, for his patience and guidance during my Ph.D. program at the University of Southern California (USC). Working with Professor Caire for the last ¯ve years has been one of the most valuable experiences. I am particularly grateful for the generous amount of time he dedicated towards my research work, and for the enthusiasm and interest he showed in my work. His comments and ideas have shaped this thesis in many ways. It has been an honor and a privilege for me to be Professor Caire's Ph.D. student. I need to thank Professors Peter Baxendale, C.-C. (Jay) Kuo, Andreas Molisch, and Michael Neely for having been on my dissertation committee, and Professors Todd Brun, KeithM.Chugg, RobertM.Gagliardi, SolomonW.Golomb, WilliamC.Lindsey, Urbashi Mitra, Shrikanth S. (Shri) Narayanan, Antonio Ortega, Robert A. Scholtz and Zhen Zhang for their support and guidance. My deepest appreciation and thanks are dedicated to my colleagues at the Com- munication Sciences Institute (CSI), Ozgun Bursalioglu, Hoon Huh, Dr. Raj Kumar Krishna Kumar, Terry Lewis, Dr. Daniel Liu, Dr. Majid Nemati, Dr. Reza Omrani, iii Rahul Urgaonkar and Dr. Satish Vedantam for their discussions and insightful talks and for all the good times both on and o®-campus. My deepest gratitude also goes to the CSI sta®, especially Anita Fung, Milly Mon- tenegro, Mayumi Thrasher and Gerrielyn Ramos, for their exceptional administrative help along the way, and more importantly, their great and unique friendship. The pro- fessional, yet friendly environment they create in CSI is outstanding. My appreciation to them can not be expressed in words. I also would like to thank Diane Demetras and Tim Boston for their great help. They are always ready to help. I'm honored to have the opportunity of studying at USC. Last but not least, I would like to thank my family for their continuous support, encouragement and love. My deepest thanks to my parents Dr. Homayoun Shirani- Mehr and Dr. Zahra Emami-Naeini, for all the sacri¯ces they've made that enabled me toearnmyPh.D.degree. MySpecialthankstomylovelywife,FarnooshMoshir-Fatemi for always being there and supporting me. She walked beside me in this long journey. I also would like to thank my brothers Mr. Houtan Shirani-Mehr and Mr. Houshamnd Shirani-Mehr for their love and support. None of my achievements would be possible without them. Hooman Shirani-Mehr Los Angeles, California May 2010 iv Table of Contents Dedication ii Acknowledgements iii List of Tables vii List of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2: OFDM Reference Channel Model 13 Chapter 3: Channel State Feedback Schemes for Multiuser MIMO- OFDM Downlink 21 3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Analog Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Directional Vector Quantization . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Time-Domain Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.1 Rate-Distortion Limit . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4.2 Scalar Uniform Quantization . . . . . . . . . . . . . . . . . . . . 38 3.5 Exploiting the Physical Channel Structure . . . . . . . . . . . . . . . . . 41 3.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Chapter 4: Parametric Channel Estimation and Prediction 52 4.1 SCM channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 1-D Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Maximum Likelihood. . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.2 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v 4.2.3 Sparse Sampling of Signal Innovation . . . . . . . . . . . . . . . 68 4.3 2-D Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.1 ESPRIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.2 Wiener Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 CRLB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Chapter 5: Scheduling Under Non-perfect CSI 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 System set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Optimal downlink scheduling . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.1 System stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3.2 System optimization . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3.3 Proportional fairness and hard fairness scheduling . . . . . . . . 100 5.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.5.1 Rayleigh fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5.2 3GPP channel model and actual channel prediction schemes . . . 111 Chapter 6: Joint Scheduling and Hybrid-ARQ for MU-MIMO Down- link in the Presence of Inter-Cell Interference 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.2 System setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.3 Scheduling with adaptive variable-rate coding and ARQ-LLC . . . . . . 125 6.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.4 Scheduling with incremental redundancy HARQ . . . . . . . . . . . . . 130 6.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4.2 Extremal ICI distributions . . . . . . . . . . . . . . . . . . . . . 139 6.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 7: Conclusions 152 Bibliography 156 Appendices 165 A Analog feedback proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 B RVQ proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 C KL proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 D KL high SNR proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 E Overestimating and underestimating model order . . . . . . . . . . . . . 171 F Nonperfect CSI proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 vi List of Tables 4.1 System Speci¯cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Maximum Doppler frequency and duration of validity of model for user with di®erent speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Optimal prediction method . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1 System parameters for simulation . . . . . . . . . . . . . . . . . . . . . . 112 5.2 Angles of arrival for well-separated case (in radians), µ v =4:4780 radians 113 5.3 Angles of arrival for packed case (in radians), µ v =0:6939 radians . . . . 113 vii List of Figures 3.1 Comparison of lowerbounds and upperbounds on the sum rate for di®er- ent feedback schemes with the discrete-time, uncorrelated path channel model when SNR is 10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Comparison of lowerbounds and upperbounds on the sum rate for di®er- ent feedback schemes with the discrete-time, uncorrelated path channel model when SNR is 10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Comparison of lowerbounds and upperbounds on the sum rate for di®er- ent feedback schemes with the continuous-time, uncorrelated path chan- nel model when SNR is 10dB. . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Comparison of lowerbounds and upperbounds on the sum rate for di®er- ent feedback schemes with the continuous-time, uncorrelated path chan- nel model when SNR is 10dB. . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Comparisonofupperboundsonthesumratefordi®erentfeedbackschemes with the continuous-time, uncorrelated path channel model for known masking matrix vs. unknown matrix when SNR is 10dB. . . . . . . . . . 50 4.1 Pilot pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Packed Doppler frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3 Well-separated Doppler frequencies . . . . . . . . . . . . . . . . . . . . . 77 4.4 Snapshot, Well-separated Doppler frequencies . . . . . . . . . . . . . . . 78 4.5 Predicting channel 1T sym ahead . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Snapshot, Packed Doppler frequencies . . . . . . . . . . . . . . . . . . . 79 4.7 Predicting channel 1T sym ahead . . . . . . . . . . . . . . . . . . . . . . . 79 4.8 Snapshot, High speed (75 km/h), well-separated, D t = 20, N t = 100, SNR=20dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 viii 4.9 Predicting channel D t T sym ahead . . . . . . . . . . . . . . . . . . . . . . 80 4.10 Snapshot, High speed (75 km/h), packed, D t =20, N t =100, SNR=20dB 81 4.11 Predicting channel D t T sym ahead . . . . . . . . . . . . . . . . . . . . . . 81 4.12 Snapshotofpredictedchannelforwell-separatedDopplerfrequenciesand low speed user with v =5km/h . . . . . . . . . . . . . . . . . . . . . . . 82 4.13 Meansquarederrorforwell-separatedDopplerfrequenciesandlowspeed user with v =5km/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.14 Snapshot of predicted channel for packed Doppler frequencies and low speed user with v =5km/h . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.15 Mean squared error for for packed Doppler frequencies and low speed user with v =5km/h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.1 New HFS, A max =100, V =100 vs. V =1000. . . . . . . . . . . . . . . 108 5.2 Ergodic sum rate, Rayleigh fading. . . . . . . . . . . . . . . . . . . . . . 109 5.3 Sum log ergodic rate, Rayleigh fading. . . . . . . . . . . . . . . . . . . . 109 5.4 Activity fractions at SNR = 20 dB, outage rate assumption, Rayleigh fading (Black: Mismatched PFS, Grey: New PFS; White: New HFS). . 110 5.5 Average sum rate, SCM channel model, ESPRIT prediction, optimistic rates.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.1 Qualitative plot of the mutual information level-crossing process that determines the decoding events of the HARQ protocol. The jumps of the accumulated mutual information process correspond to slot times at which user (k;c) is active. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Average-throughput, proportional fairness. . . . . . . . . . . . . . . . . . 145 6.3 Average-throughput, max-min fairness. . . . . . . . . . . . . . . . . . . . 146 6.4 Averageratevs. decodingdelaywithproportionalfairnessfortwosample users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.5 Average rate vs. decoding delay with max-min fairness for two sample users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.6 Throughput-delaytradeo®sversusUTlocationfor theHARQsystemfor di®erent scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 ix E1 Conditional number of I(µ) as a function of ! 1 ¡ ! 2 when N t = 10, D t =10 and N 0 =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 E2 Comparison between CRLB for 1 and 2 sinusoids and the limit as a functionof! 1 ¡! 2 whenN t =10, D t =10, N 0 =1andk =(N t ¡1)D t + 1=91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 E3 Comparison between CRLB for 1 sinusoid and 2 sinusoids in limit ! 1 ! ! 2 as a function of k when N t =10, D t =10, N 0 =1 . . . . . . . . . . . 176 x Abstract Multiple input multiple output (MIMO) communication systems and multiuser diver- sity play important rolls in future wireless systems and enable them to achieve high data rates. Earlier studies on multi-user MIMO (MU-MIMO) systems were based on some ideal assumptions on system such as perfect channel state knowledge, perfect feedback channel and known inter-cell interference. This thesis investigates design and performance of MU-MIMO systems under more practical assumptions. We ¯rst consider channel state information (CSI) feedback schemes for MIMO- OFDM broadcast channel. By considering simple zero-forcing beamforming (ZFBF) precoder and ignoring user selection problem, we compare achievable ergodic rates un- derdi®erentchannelstatefeedbackschemes. Weproposeanovel"time-domain"channel quantized feedback which is inspired by rate-distortion theory of Gaussian correlated sources and takes advantage of the channel frequency correlation structure. Next, we consider spacial channel model (SCM) which is used as a benchmark in standardization and examine some alternatives for channel estimation and prediction scheme. We show that a parametric method based on ESPRIT is able to accurately xi predict the channel even for relatively high user mobility as long as the angular spread is large. Subsequently, we study the problem of user scheduling in MIMO broadcast systems under non-perfect CSI. We propose a dynamic novel opportunistic scheme that, de- pending on the channel state quality of user terminals, categorizes user terminals into "predictable" and "non-predictable" users and at each scheduling slot, serves one non- predictable user (transmit diversity) or several predictable users (multi-user diversity). Inour¯nalanalysis,weinvestigateuserschedulinginthepresenceofunknowninter- cell interference. We present a systematic method for joint scheduler/ARQ design. We de¯ne"optimistic"throughputasthethroughputthatcanbeachievedifthetransmitter knewtheinstantaneoususerICIandtransmittedtoeachscheduleduseratthemaximum possible instantaneous rate and show that properly designed incremental-redundancy hybrid ARQ schemes can achieve any desired fraction of the "optimistic" throughputs with ¯nite average decoding delay where optimistic. Results in this work have been published in the following papers: ² H.Shirani-Mehr,H.Papadopoulos,S.A.RamprashadandG.Caire,"JointSchedul- ing and Hybrid-ARQ for MU-MIMO Downlink in the Presence of Inter-Cell In- terference", submitted to IEEE Transactions on Communications. ² H. Shirani-Mehr, G. Caire and M. J. Neely, MIMO Downlink Scheduling with Non-PerfectChannelStateKnowledge",submittedtoIEEETransactionsonCom- munications. xii ² H.Shirani-Mehr,H.Papadopoulos,S.A.RamprashadandG.Caire,"JointSchedul- ing and Hybrid-ARQ for MU-MIMO Downlink in the Presence of Inter-Cell In- terference", Submitted to ICC 2010. ² H. Shirani-Mehr and G. Caire, "Channel State Feedback Schemes for Multiuser MIMO OFDM Downlink", IEEE Transactions on Communications, September 2009. ² H. Shirani-Mehr, and G. Caire, "MIMO Downlink Scheduling with Non-perfect ChannelStateKnowledge",InformationTheoryWorkshop,2009,Taormina,Italy. ² H. Shirani-Mehr, D. N. Liu and G. Caire, "Parametric Channel Estimation and Prediction with Applications to Channel State Feedback for MIMO Downlink Schemes", 42nd Asilomar Conference on Signals, Systems and Computers, 2008. ² G. Caire and H. Shirani-Mehr, "Feedback Schemes for Multiuser MIMO-OFDM Downlink", Information Theory and Applications Workshop, 2008. xiii Chapter 1 Introduction Since the ¯rst demonstration of wireless telegraphy by Marconi in about a century ago, wireless communications has steadily advanced. Cell phones and laptop computers are now fundamental to wireless communication systems and provide a wide range of functions from voice calls and short messaging to internet browsing and multimedia streaming. Typically, in order to increase system throughput and allow multiplexing of users in multi-user wireless systems, the wireless medium is divided into multiple bands (chan- nels) where each one of these channels, consists of speci¯ed regions in frequency, time, and/or code dimensions. Consequently, communicating between each user and the base station can only be performed within that user's channel. On the other hand, band- width and power are limited resources in wireless systems. Therefore, the challenge is to achieve the desired performance goals despite the fact that wireless system is a resources limited medium shared among all users. 1 Recently, two new concepts have been introduced to improve throughput of wireless systems: multiple-input multiple-output (MIMO) and multiuser diversity. The idea of MIMO system is to deploy multiple antennas at the transmitter and receiver to exploit the new dimension known as "space". This new dimension can be used both in single userscenariowhereallspatialdimensions(antennas)areassignedtooneusertoachieve higher rates, or in multi-user setting where antennas are shared among several users. ThelatterisreferredtoasmultiuserMIMO(MU-MIMO)system. Theideaofmultiuser diversityistoonlyservetheuserswithgoodinstantaneouschannelconditions. Inother words, as channel gains °uctuate over time due to fading and mobility, users with good instantaneous channel conditions are selected for signal transmission. These two techniques, when combined with advanced signal processing and coding techniques, are capable of delivering higher data rates. In the next generation of cellular systems (e.g., the so-called LTE-Advanced [1]), high-rate data-oriented downlink schemes will be combined with MU-MIMO transmission techniques, supporting spectral e±ciencies in the 10's of bits/sec/Hz [72, 34]. Performance limits of MU-MIMO systems have been successfully characterized by the means of information theoretic approaches. In this thesis, we investigate MU- MIMOcommunicationsystemsbyapplyingsimilartools. However,weinvestigatesmore practical systems by incorporating implementation issues such as non-perfect channel state feedback and unknown inter-cell interference. 2 The remainder of this chapter is organized as follows. Some background on MIMO and multiuser diversity is given in Section 1.1. Section 1.2 explains motivations that led to our thesis. Finally, Section 1.3 provides an outline of the thesis. 1.1 Background The concept of channel capacity was ¯rst introduced by Shannon in 1948 as the maxi- mumdataratethatcanbetransmittedoverthechannelwithasymptoticallysmallerror probability [78]. In the simple case of additive white Gaussian noise (AWGN) channel the Shannon capacity is in the form of C =W log µ 1+ P N 0 W ¶ (1.1) whereP is the transmit power, N 0 is the one-sided noise power spectral density, and W is the available bandwidth. Signal-to-noise power ratio (SNR) is de¯ned as the ratio of received signal to noise power, i.e., P N 0 W . The use of multiple transmit and receive antennas introduced a new dimension (an- tenna or spatial domain) that was unnoticed previously. It is shown that using multiple antennas increases capacity linearly with the number of antennas [33, 88]. Speci¯cally, with M transmit antennas and N receive antennas, multiplexing gain of minfM;Ng can be exploited i.e., the capacity of MIMO system is roughly minfM;Ng times that of a single-input single-output (SISO) system given by (1.1). This multiplexing gain is achieved by transmitting minfM;Ng independent data streams in parallel. 3 However, the multiplexing gain can only be achieved in a single user MIMO system if both the transmitter and receiver are equipped with multiple antennas. In typical wireless systems, user terminals are very small and cannot be equipped with multiple antennas. On the other hand, multiple antennas can easily be deployed in the base station. ThismotivatedtheresearchonMU-MIMObroadcastchannels(pointtomulti- point communications). MIMO broadcast (MIMO-BC) schemes are primarily motivated by the very signif- icant capacity increase associated with multi-user MIMO techniques. It is well-known thatifatraditionalorthogonalizationtechniquesuchasTDMAisused,thebasestation (BS) transmits to a single receiver on each time-frequency resource and thus is limited to point-to-point MIMO techniques [33, 88]. Alternatively, the BS can use multi-user MIMO (also commonly referred to as SDMA, or space-division multiple access) to si- multaneously transmit to multiple receivers on the same time-frequency resource by appropriate utilization of spatial dimensions. Multi-user MIMO can exploit multi-user diversityi.e., itisabletousethetransmitantennaarrayandknowledgeoftheinstanta- neous channel to e±ciently direct signals/energy towards di®erent receivers. However, multiuser diversity can only be exploited if channel state information (CSI) is perfect at the BS. To emphasize the importance of CSI, note that in the extreme case of no CSI at the BS and identical fading statistics at all receivers, the multiuser multiplexing gain will be lost [19]. 4 The capacity region of MIMO-BC under perfect CSI at transmitter (CSIT) is fully characterized in [95]. It is shown that, under perfect CSI at the transmitter and re- ceivers, sum-rate capacity of MIMO-BC can be achieved by dirty paper coding (DPC) [19, 92, 94, 105, 95]. However, implementation of DPC that approaches capacity while maintaining reasonable system complexity still remains a formidable challenge [28, 11, 86]. Simpler schemes such as linear beamforming, Tomlinson-Harashima pre- coding [15, 30] or vector precoding [45, 97], combined with single user coding and decoding, have emerged as a low-complexity, near-capacity transceiver design options. It has been shown that linear beamforming schemes asymptotically approach DPC per- formance i.e., in a system with M transmit antennas at the BS and K single antennas users, the sum rate scales as Mlogsnr when snr goes to in¯nity for ¯xed K À M and scales as MloglogK when K goes to in¯nity for ¯xed snr [102, 42]. The resource allocation problem in a MU-MIMO downlink, under perfect knowledge of the down- link channels, has been widely investigated under various precoding and beamfomring schemes [106, 96, 15, 24, 93, 59, 65]. 1.2 Motivations Given the widespread applicability of the MIMO downlink channel model to wireless systems, therehasbeenaremarkableamountofresearchactivity,bothinacademiaand industry,withthegoalofdesigningpracticalsystemsthatcanoperatenearthecapacity limits of the MIMO downlink channel. In this thesis we focus on two main issues in MIMO-BC systems: CSI and scheduling. 5 As mentioned earlier, when perfect CSI is available at transmitter and receiver, the capacity region of MIMO-BC is achieved by DPC [19, 95]. For a systems with M transmit antennas at BS and K user terminals (UTs), at high SNR, the sum capacity can be approximated as Mlogsnr i.e., the multiplexing gain is M [69]. The situation is di®erent if instantaneous CSI is not known at the BS. In this case, if all UTs su®er from fading with same statistics, any codeword that can be decoded by one user can be decoded by any other user. Therefore, the optimal transmission scheme is TDMA [23]. Hence, the sum capacity is equal to the capacity of a point-to-point channel from BS to any individual UT i.e., the multiplexing gain is 1. MIMO-BC systems with perfect CSIT has been investigated thoroughly [19, 95, 19, 92, 94, 105, 95]. However, in practice, CSIT is obtained through some form of training and feedback. In Time-Division Duplexing (TDD) systems, the BS can learn the downlink channel coe±cients in \open-loop" mode, by exploiting the uplink pilot symbolsandchannelreciprocity(e.g.,[53,51]). InFrequency-DivisionDuplexing(FDD) systems, since uplink and downlink take place in widely separated frequency bands, the downlink channel coe±cients must be learned in \closed loop" mode, via some explicit CSI feedback scheme (e.g., [16, 81] and references therein). Mostifnotallpreviousworkshavedealtwiththecaseofafrequency-°atchanneland proposeddi®erentCSIfeedbackmethodssuchasanalogfeedback,randombeamforming, random vector quantization [16, 65, 69]. The frequency-selective (OFDM) case is more directly relevant to 4-th Generation wireless systems. A trivial solution consists of operatingoneindependentCSITfeedbackpercarrier. Thissolutionissuboptimalsince 6 it does not take advantage of the fact that the channel vectors at di®erent carriers are correlated. Therefore, designing an e±cient scheme to feedback CSI coe±cients from UTs to the BS is very important. Delay in CSI feedback causes performance degradation in practical MIMO-BC sys- tems. Infact,althoughmostpreviousworksassumethatCSIfeedbackisinstantaneous, inpracticalsystemssomenonzerodelayisalwaysassociatedwithfeedbacktransmission from UTs to the BS. Hence, the CSI when received at the BS is out-dated. Therefore, in order to compensate for the delay, each UT needs to either somehow predict its own future channel and feed it back to the BS or, by sending some information to the BS, enable it to extrapolate the channel model in future. Another problem of interest in MIMO-BC systems is scheduling and user selection. In general, downlink scheduling aims at making the system operate at a desired point on the ergodic (or long-term average) achievable rate region of the system, for a given physical layer signaling scheme. The operating point re°ects some form of \fairness," corresponding the maximization of a concave non-decreasing utility function of the ergodic rates. Downlink scheduling under perfect CSIT has been well-studied [24, 101, 93]. Asmentionedearlier,inpractice,theCSIisobtainedthroughsomeformoftraining and feedback (TDD or FDD). In both cases, the CSI available to the BS can be seen as some sort of \noisy" version of the true channel coe±cients. Downlink scheduling for MIMO-BC in the presence of non-perfect CSI is only scantly treated in the literature and has been treated mainly in the case where all users have the same CSI quality. Static mode-switching criteria have been studied for example in [107, 56] where the 7 number of users to be simultaneously served is optimized depending on the CSI quality and channel SNR. Finally,thelevelofinter-cellinterference(ICI)isveryimportantinmulti-cellMIMO- BC.Forasingle-celloperatinginisolation,CSITisadequateandprovidesaccuraterate information for the scheduler. However, in the practical and most relevant case of a multi-cellsystem, theBSineachcellrunsitsownMU-MIMOandschedulingalgorithm independently of the other BSs. Consequently, the ICI level of each UT changes at each time-frequency slot in an unpredictable manner. Therefore, the achievable rates in each cell are random variables even under the assumption of perfect CSIT. In clas- sical theoretical \downlink scheduling" papers, it is assumed that the BS knows the rate that each UT can support. With MU-MIMO, the rate supported by each UT is generally a complicated function of all the user channel vectors, and of the set of the transmitterbeamformingsteeringvectors, powerallocation, andprecodingschemeused [19, 95, 15, 16, 25]. Motivated by the aforementioned limitations, in this thesis we address and ¯nd an- swers to the following questions: ² Can we design an e±cient feedback scheme for MIMO-OFDM systems? (Chapter 3) ² IsitpossibletopredictthechannelstateinfuturetocompensatetheCSIfeedback delay? (Chapter 4) 8 ² If the CSIT in not perfect, what is the e±cient scheduling scheme? (Chapter 5) ² In multi-cell setting with unknown inter-cell interference, what signaling scheme is optimal? (Chapter 6) 1.3 Outline of the Thesis This report is organized as follows: In Chapter 2, we derive an accurate discrete time-frequency model for the OFDM signalthatallowsustoformulatetheprobleminthefollowingchaptersinanexactway. In Chapter 3, we compare the achievable ergodic rates of MIMO-OFDM systems under di®erent feedback schemes. For the sake of analytical tractability, we restrict ourselves to the case of zero-forcing beamforming (ZFBF). Moreover, we neglect user selection and scheduling by assuming same number of UTs and BS antennas. The key di®erence with respect to the widely treated frequency-°at case is that in MIMO- OFDM the frequency-domain channel transfer function is a Gaussian correlated source. Weconsiderthreechannelstatefeedbackschemes: analogfeedback, directionquantized feedback and \time-domain" channel quantized feedback. In analog feedback, unquan- tized channel coe±cients are transmitted from UTs to the BS as real and imaginary parts of a complex modulation symbol. In direction quantized feedback, we consider random vector quantization (RVQ) where each UT quantizes its channel at a speci¯c subcarrier to B bits and feeds back the bits to the BS. In this scheme, each UT has a randomly generated quantization codebook consisting of 2 B codewords independently 9 and isotropically distributed on a unit complex sphere and quantizes its channel to the codeword that forms the smallest angle with the channel vector. In \time-domain" channel quantized feedback, instead of feeding back the frequency-domain channel co- e±cients, users feed back the time-domain channel coe±cients. The ¯rst two schemes are direct extensions of previously proposed schemes. The third scheme is novel, and it is directly inspired by rate-distortion theory of Gaussian correlated sources. For each schemewederivetheconditionsunderwhichthesystemachievesfullmultiplexinggain. The new time-domain quantization scheme takes advantage of the channel frequency correlation structure and outperforms the other schemes. Moreover, it is much sim- pler than complicated spherical vector quantization to implement since no structured codebook design and vector quantization is required for e±cient CSI feedback. In Chapter 4, we focus on the channel prediction problem. We consider the para- metric model obtained by superposition of rays coming from random angles as speci¯ed in the 3GPP SCM channel model. Considering this model, each UT can estimate its channelparametersusingthestreamofpilotssentfromtheBS.Theseestimatedparam- eters are then sent to the BS and BS uses this information to extrapolate the channel model to computes future channel coe±cients required for beamforming. In this chap- ter, ¯rst it is shown that the 2-D sum-of-sinusoids (SOS) parameter estimation problem can be written as two 1-D problem parameter estimation problems. In the ¯st part, we investigate several techniques available in the literature for 1-D parameter estima- tion such as maximum likelihood (ML), subspace methods (ESPRIT and MUSIC) and 10 sparse sampling of signal innovation. It will be shown that for channels with large an- gular spread (well-separated Doppler frequencies), ESPRIT and ML outperform other methods and work very well. On the contrary, in the case that angular spread is small (packed Doppler frequencies) , none of these methods work well. These results are then applied to solve the original 2-D problem and ESPRIT is used to solve each one of the two 1-D parameter estimation problems. The other possible solution is to model time- domain channel coe±cients as independent Gaussian random variables and use Wiener ¯lter to predict channel coe±cients in future. In this method the autocorrelation and cross-correlation matrices are computed by estimating second order statistics from ob- servations. Finally, we categorize channels based on the speed of the user (High speed or low speed) and the distribution of scatterers in the environment (well-separated or packed Doppler frequencies) and ¯nd the optimal prediction method for each category. It clearly appears that with the SCM model, Doppler is not so important and even fast varying channels can be handled if the angular spread is large. Chapter 5 treats the combination of multiuser MIMO downlink scheme with user selectionandscheduling. InthischapterweconsiderZFBFandassumethatthenumber of users is more than the number of transmit antennas at the BS and scheduling is applied to serve the users. For the sake of simplicity, we assume that scheduling is performed on each subband independently. We solve this problem for the case of MU- MIMOwithnon-perfectCSI¯rst. Then,motivatedbyresultsinChapter4thattheusers can be partitioned into two classes with either small or large channel prediction MSE and based on the general solution, we ¯nd a practical simpli¯ed scheduling policy. The 11 resultingschedulingalgorithmcanberegardedasanopportunisticMIMO\multi-mode" schemethat, ateachschedulingslot, eitherselectsaMU-MIMOdownlinkbeamforming mode that performs spatial multiplexing to a subset of predictable users, or a single- user space-time coding mode that serves a single selected non-predictable user. Results based on a realistic channel model and actual channel state prediction algorithms show that the proposed approach achieves very signi¯cant improvement with respect to a conventional mismatched scheme that treats the available CSI as if it was perfect. Finally, in Chapter 6, we consider joint operation of MU-MIMO with opportunis- tic user scheduling and ARQ in the presence of a random ICI power. We present a systematic method for joint scheduler/ARQ design and show that properly designed incremental-redundancy hybrid ARQ (HARQ) schemes, with ¯nite average decoding delay, can achieve any desired fraction of the throughputs that would be achieved if the transmitter knew the instantaneous user ICI and transmitted to each scheduled user at the maximum possible instantaneous rate. 12 Chapter 2 OFDM Reference Channel Model We consider an OFDM system with N subcarriers. Let x[k] = [x[k;0];:::;x[k;N ¡1]] be the kth OFDM symbol. Then the corresponding symbol in time domain, e x[k], is in the form of e x[k] = F H x[k] where F is a unitary N £ N discrete Fourier trans- form (DFT) matrix, with (l;k) elements [F] l;k = 1 p N e ¡j2¼kl=N . Letting e x cp [k] = [e x cp [k;0];:::;e x cp [k;L+N¡1]]bethetime-domainsymbolafterinsertingcyclicpre¯c(CP) andLbethelengthofCP,wehavethate x cp [k]=[e x[k;N¡L];e x[k;N¡L+1];:::;e x[k;N¡ 1];e x[k;0];e x[k;1];:::;e x[k;N¡1]]. The continuous-time transmitted signal can be written as e x(t)= 1 X k=¡1 e x k (t¡kT sym ) (2.1) whereT sym =(N+L)T isthelengthofanOFDMsymbolandT isthesamplingperiod, and e x k (t)= N+L¡1 X i=0 e x cp [k;i]'(t¡iT) (2.2) 13 where '(t) is the transmit ¯lter impulse response and e x cp [k;i] is the ith element of e x cp [k]. Therefore, e x(t)= 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i]'(t¡iT ¡kT sym ) (2.3) If the channel has P discrete multipath components, the continuous-time baseband channel can be written as c(t;¿)= P X p=1 c p (t)±(¿¡¿ p (t)) (2.4) where ¿ p is the delay in the pth path and c p (t) is the complex amplitude of the pth path. We can make the assumption that the time delay ¿ p (t) varies slowly in time in comparison to the OFDM symbol duration and therefore it can be considered as locally time-invariant. Also, the complex amplitude of the pth path c p (t) can be considered as constant over an OFDM symbol, otherwise we are in the presence of a fast-fading channel and OFDM would be a®ected by inter-carrier interference, that is not taken into account in the standard model. Therefore, in a neighborhood of the k-th OFDM symbol we can write the locally time-invariant channel impulse response as c(t;¿)¼ P X p=1 c p (kT sym )±(¿¡¿ p ) (2.5) 14 The received signal at the receiver front-end is in the form of r(t) = e x(t)¤c(t;¿)+e z(t) = 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i] P X p=1 c p (kT sym )'(t¡¿ p ¡iT ¡kT sym )+e z(t) = 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i]' c (kT sym ;t¡iT ¡kT sym )+e z(t) (2.6) where ¤ represents convolution, e z(t) is complex circularly symmetric Gaussian noise with mean zero and variance N 0 and ' c (kT sym ;t¡iT ¡kT sym )= P X p=1 c p (kT sym )'(t¡¿ p ¡iT ¡kT sym ) (2.7) We consider a standard receiver front-end formed by a receiving ¯lter matched to the transmit pulse only (notice: this is *not* the correct matched ¯lter that should be matched to ' c , but in standard OFDM receiver design this option is not generally used since channel estimation is performed after matched ¯lter, and in general we do not 15 know in advance the pulse ' c ). Denoting the timing o®set by t 0 , then the output of the transmit-pulse matched ¯lter can be written as e y(t) = r(t)¤' ¤ (¡t+t 0 ) = 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i] Z 1 ¡1 ' c (kT sym ;¿¡iT ¡kT sym )Ã ¤ (¿¡t+t 0 )d¿ +noise term = 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i] Z 1 ¡1 ' c (kT sym ;u+t¡t 0 ¡iT ¡kT sym )' ¤ (u)du +noise term = 1 X k=¡1 N+L¡1 X i=0 e x cp [k;i]h(kT sym ;t¡t 0 ¡iT ¡kT sym )+noise term (2.8) where noise is Gaussian and h(d;t)= Z 1 ¡1 ' c (d;u+t)' ¤ (u)du (2.9) After sampling the signale y(t) at rate 1=T, e y cp (kT sym +iT) = 1 X k 0 =¡1 N+L¡1 X i 0 =0 e x cp [k 0 ;i 0 ]h(k 0 T sym ;(i¡i 0 )T +(k¡k 0 )T sym ¡t 0 ) +e z(kT sym +iT) = N+L¡1 X i 0 =0 e x cp [k;i 0 ]h(kT sym ;(i¡i 0 )T ¡t 0 )+e z(kT sym +iT) (2.10) 16 where we used the fact that h(k 0 T sym ;(i¡i 0 )T +(k¡k 0 )T sym ¡t 0 )=0 for any k 0 6=k. Furthermore, we assume h(kT sym ;(i¡i 0 )T ¡t 0 ) = 0 for (i¡i 0 ) ¸ L. If we denote h(kT sym ;lT ¡t 0 ) by h[k;l], (2.10) can be written in the equivalent matrix form as, e y[k]=h[k]e x[k]+e z[k] (2.11) wheree y[k]=[e y cp [k;L];:::;e y cp [k;L+N¡1]] withe y cp [k;l]=e y cp (kT sym +lT) and h[k]= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 h[k;0] 0 ¢¢¢ h[L¡1;k] ¢¢¢ h[1;k] h[k;1] h[0;k] 0 . . . . . . . . . h[1;k] h[L¡1;k] h[k;L¡1] . . . 0 0 h[L¡1;k] h[0;k] . . . . . . . . . 0 ¢¢¢ 0 h[L¡1;k] ¢¢¢ h[0;k] 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (2.12) Then by taking DFT of (2.11) y[k] = Fh[k]F H x[k]+Fe z[k] = FF H H[k]FF H x[k]+z[k] = H[k]x[k]+z[k] (2.13) 17 where H[k] is a diagonal matrix with elements that are DFT coe±cient of h[k;l] for l =0;:::;L i.e., H[k;n]= L¡1 X l=0 h[k;l]e ¡j2¼ln=N (2.14) The discrete-time Fourier transform (DTFT) of h(kT sym ;lT ¡t 0 ) can now be written as H[k;¸)= L¡1 X l=0 h(kT sym ;lT ¡t 0 )e ¡j2¼¸l ¸2[¡0:5;0:5] (2.15) where ¸ is the digital frequency. On the other hand, h(kT sym ;lT ¡t 0 )= Z 1 ¡1 H(kT sym ;f)e j2¼f(lT¡t 0 ) df (2.16) where H(kT sym ;f) is the Fourier transform (FT) of h(kT sym ;t) From (2.7) and (2.9) H(kT sym ;f)= P X p=1 c p (kT sym )e ¡j2¼f¿ p j©(f)j 2 (2.17) where ©(f) = F['(t)]. To simplify these equations, we ¯rst consider an arbitrary continuous time signal g(t) and with FT G(f). Let g[l] be the corresponding discrete- time signal when it is sampled with sampling rate 1=T and G(¸) be its DTFT. Then we have the followings: G(¸)= 1 X l=¡1 g[l]e ¡j2¼¸l (2.18) and g[l]=g(lT)= Z 1 ¡1 G(f)e j2¼flT df (2.19) 18 Now if we combine these two equations, G(¸) = 1 X l=¡1 µZ 1 ¡1 G(f)e j2¼flT df ¶ e ¡j2¼¸l = Z 1 ¡1 G(f) 1 X l=¡1 e j2¼lT(f¡ ¸ T ) df = Z 1 ¡1 G(f) 1 T 1 X l=¡1 ± µ f¡ ¸ T + l T ¶ = 1 T 1 X l=¡1 G µ ¸¡l T ¶ (2.20) From (2.20), and by using (2.15), (2.16) and (2.17) we arrive at H[k;¸) = 1 T 1 X l=¡1 ¯ ¯ ¯ ¯ © µ ¸¡l T ¶¯ ¯ ¯ ¯ 2 e ¡j2¼ ¸¡l T t 0 2 4 P X p=1 c p (kT sym )e ¡j2¼ ¸¡l T ¿p 3 5 (2.21) By letting ¸= n N mod[¡0:5;0:5] in (2.21) for n=0;:::;N¡1 H[k;n] = 1 T 1 X l=¡1 ¯ ¯ ¯ ¯ © µ n¡lN NT ¶¯ ¯ ¯ ¯ 2 e ¡j 2¼ NT t 0 (n¡lN) P X p=1 c p (kT sym )e ¡j 2¼ NT ¿ p (n¡lN) = P X p=1 " 1 T 1 X l=¡1 ¯ ¯ ¯ ¯ © µ n¡lN NT ¶¯ ¯ ¯ ¯ 2 e ¡j 2¼ N t 0 +¿p T (n¡lN) # c p (kT sym ) If the ¯lter '(t) is considered to be root-raised-cosine (RRC) ¯lter and the symbols are allocated in the °at portion of the frequency band, the folding e®ect can be neglected 19 and instead of in¯nite summation, we can only consider the term in summation which corresponds to l =0. Therefore, y[k;n]=H[k;n]x[k;n]+z[k;n] (2.22) where H[k;n] = ®[n] P P p=1 e ¡j 2¼ N ¿p T n c p (kT sym ) with ®[n] = 1 T ¯ ¯ © ¡ n NT ¢¯ ¯ 2 e ¡j 2¼ N t 0 T n . We can consider perfect timing (t 0 = 0) and since the masking ¯lter is known to both transmitter and receiver and moreover it is constant over the used frequency band, we can neglect ®[n] and H[k;n] = P X p=1 e ¡j 2¼ N ¿ p T n c p (kT sym ) (2.23) 20 Chapter 3 Channel State Feedback Schemes for Multiuser MIMO-OFDM Downlink Channel state feedback schemes for the MIMO broadcast downlink have been widely studied in the frequency-°at case. In this chapter, we focuses on the more relevant frequency selective case, where some important new aspects emerge. We consider a MIMO-OFDM broadcast channel and compare achievable ergodic rates under three channel state feedback schemes: analog feedback, direction quantized feedback and \time-domain" channel quantized feedback. The ¯rst two schemes are direct extensions of previously proposed schemes. The third scheme is novel, and it is directly inspired by rate-distortion theory of Gaussian correlated sources. For each scheme we derive the conditions under which the system achieves full multiplexing gain. The key di®er- ence with respect to the widely treated frequency-°at case is that in MIMO-OFDM the frequency-domain channel transfer function is a Gaussian correlated source. The new 21 time-domain quantization scheme takes advantage of the channel frequency correlation structure and outperforms the other schemes. Furthermore, it is by far simpler to im- plement than complicated spherical vector quantization. In particular, we observe that no structured codebook design and vector quantization is actually needed for e±cient channel state information feedback. WeconsideraMIMO-OFDMbroadcastchannelwithonebasestation(BS),equipped with M antennas, and K ¸ M single-antenna user terminals (UT). MIMO broadcast channels have been widely studied in the recent past (see for example [19, 92, 94, 105, 95]). Underperfecttransmitterchannelstateinformation(CSIT)attheBSandreceiver channelstateinformation(CSIR)attheUTs,itscapacitywasfullycharacterizedin[95] and e±cient resource allocation algorithms have been proposed in order to operate at desired points in the capacity region (e.g., [104, 54, 55]). In the current standardization of the 4-th Generation of wireless communication systems (e.g., IEEE802.16m), MIMO broadcast schemes are going to play a fundamental role in order to achieve high data rates in the downlink. In practice, CSIT must be provided to the BS by some form of feedback. CSIT feedback schemes are a very active area of research (see for example [16] and the special issue [2] for a fairly complete list of references). In brief, we may iden- tify three broad families: 1) open-loop schemes based on channel reciprocity and uplink trainingsymbols,applicabletoTime-DivisionDuplexing(TDD);2)closed-loopschemes basedonfeedingbacktheunquantizedchannelcoe±cients(analogfeedback); 3)closed- loop schemes based on explicit quantization of the channel vectors and on feeding back 22 quantization bits, suitably channel-encoded (digital feedback). Closed-loop schemes are suitable for Frequency-Division Duplexing (FDD), where channel reciprocity cannot be exploited. Most if not all present works deal with the case of a frequency-°at channel. In particular, it was recognized that the most important information about the channel vectors consists of their directions. Directional quantization is obtained by using vector quantization codebooks formed by unit vectors distributed on the M dimensional com- plex sphere. In [69], ergodic achievable rates are analyzed assuming linear zero-forcing beamforming (ZFBF) and random ensembles of spherical quantization codebooks, uni- formly distributed on the unit sphere. These results have been extended in [16] to a variety of cases including realistic feedback channels with noise, fading and delay, and to non-perfect CSIR at the UTs obtained by explicit downlink training. In particular, these works show that the sum-rate scales optimally, as Mlogsnr+O(1), provided that the number of quantization bits per UT increases with SNR as B = ®(M¡1)log 2 snr for some ®¸ 1. For example, at SNR of 10 dB a codebook of size 1024 is needed for M =4 antennas, and a codebook of size 2 24 =16777216 is needed for M =8 antennas. Clearly, such channel vector quantizers involve an enormous computational complexity unless some special structure is exploited. Structured spherical vector quantizers for direction quantization have been studied, for example, in [7]. The frequency-selective (OFDM) case is more directly relevant to 4-th Generation wireless systems. A trivial solution consists of operating one independent CSIT feed- back per carrier. This solution is suboptimal since it does not take advantage of the fact that the channel vectors at di®erent carriers are correlated. In this chapter we 23 compare three channel state feedback schemes for the MIMO-OFDM downlink: ana- log feedback, digital direction quantized feedback and a new \time-domain" channel quantized feedback inspired by rate-distortion theory. For each scheme we derive the conditions under which the system achieves full multiplexing gain (i.e., the pre-log fac- tor of the sum-rate is equal to M). The new rate-distortion inspired scheme takes full advantage of the channel frequency correlation structure and it is shown to outperform the ¯rst two. Furthermore, time-domain quantization is by far simpler to implement than complicated spherical vector quantization. In particular, it is seen that no struc- tured codebook design for vector quantization is actually needed for e±cient channel state information feedback. 3.1 System Model Typically, in a cellular downlink the number of UTs K is larger than the number of BS antennas M. Since with linear beamforming at most M users can be served at the same time, some user selection and scheduling algorithm is advocated (see for example [24, 101, 102]). Scheduling and user selection based on CSIT make the channel vectors oftheusersthatareservedateachtimestatisticallydependent. Forexample,algorithms such as those proposed in [24, 101, 102] try to ¯nd subsets of mutually quasi-orthogonal users. This makes an analytical characterization of the rate gap with respect to the ideal CSIT case very di±cult if not impossible. For the sake of analytical simplicity, we assume here that scheduling is performed independently of the CSIT. Therefore, without loss of generality, we may assume that a set of M out of K users is randomly 24 selected at each time slot or, equivalently, that the system contains K =M users with statistically independent channels. Also, in order to focus solely on the comparison of thedi®erentchannelstatefeedbackschemes, werestricttothecasewheretheUTshave perfect CSIR. In particular, this means also that each UT has perfect knowledge of the coupling coe±cient between its channel and its own beamforming steering vector. Our results can be generalized following in the footsteps of [16] by considering more realistic training-based downlink channel estimation schemes. Channels are identically distributed for all users, and spatially independent (no an- tennacorrelation). Therefore,atthebeginningofthissectionwefocusonthedescription ofthescalarchannelbetweenanyBSantennaandagenericuser, droppingantennaand user index for the sake of notation simplicity. A standard assumption in OFDM is that channels behave locally as linear time-invariant ¯nite impulse response ¯lters of length L. We assume block-fading channels, constant on blocks of duration T À L symbols, and changing according to some ergodic statistics from block to block. In this work we consider zero-delay CSIT feedback and block-by-block estimation. Therefore, we are not concerned with the time-correlation from block to block of the channel (the case of delayed feedback and explicit channel prediction is considered in [9]). Using the standard cyclic-pre¯x method, blocks of N = T ¡L+1 information symbols can be transmitted without inter-block interference at the cost of a small dimensionality loss factor of ¡ 1¡ L¡1 T ¢ ¼ 1, that shall be neglected in the achievable rate expressions of this chapter since it applies to all such OFDM schemes in the same way. 25 After cyclic pre¯x insertion and removal the resulting channel model is de¯ned by a block transmission of N symbols per transmit antenna, over the N OFDM subcarriers. Let h = [h[0];h[1];:::;h[L¡ 1]] T denote the discrete-time channel impulse response, and H = [H[0];:::;H[N¡1]] T be the channel in the DFT frequency domain given by (2.14). Acommonassumptionconsistsofmodelingthetime-domainchannelcoe±cients h[l]'s as independent Gaussian random variables»CN(0;¾ 2 l ), where the path variances f¾ 2 0 ;:::;¾ 2 L¡1 g forms the Delay Intensity Pro¯le (DIP) of the channel. We follow this model here, and re-discuss it in Section 3.5 where we show how to take advantage of a more physically motivated channel model. The frequency-domain channel covariance matrix is given by § H = E[HH H ]=F 2 6 6 4 N§ h 0 0 0 3 7 7 5 F H (3.1) where § h =diag(¾ 2 0 ;:::;¾ 2 L¡1 ). Furthermore, the diagonal elements of § H are equal to ¾ 2 H =E £ jH[n]j 2 ¤ = P L¡1 l=0 ¾ 2 l . IntheMIMOcase,thechannelfromtheBStoUTkisde¯nedbythevectordiscrete- time impulse response [h k [0];h k [1];:::;h k [L¡1]] where h k;i [l] is the channel coe±cient from the BS antenna i to the UT k at discrete-time delay l. By applying OFDM modulation and demodulation, the received signal at UT k on the n-th subcarrier can be written as y k [n]=H H k [n]x[n]+z k [n] (3.2) 26 where k = 1;:::;K; n = 0;:::;N¡1, x[n]2C M is the transmitted vector of frequency- domainsymbolsontheM BSantennas,atsubcarriern,andH k [n]=[H k;1 [n];:::;H k;M [n]] T is the channel vector of UT k at subcarrier n. The average transmit power constraint is given by 1 N P N¡1 n=0 E[jx[n]j 2 ]·P. Forsimplicityofanalysis,wetreatonlythecaseoflinearZero-ForcingBeamforming (ZFBF). It is well-known that ZFBF performs at a ¯xed gap from the optimal capacity achievingstrategyunderperfectCSIT.Hence,ourgoalisto¯ndconditionsunderwhich ZFBF performs at a ¯xed rate gap from the perfect CSIT case, which implies ¯xed rate gap from optimal. For perfect CSIT, the ZFBF transmitted signal at subcarrier n is given by x[n] = V[n]u[n] where V[n] 2 C M£K is a zero-forcing beamforming matrix with unit norm columns such that each k-th column v k [n] is orthogonal to the subspace spanned by fH j [n] : j 6= kg, and u[n] 2 C K denotes the vector of coded symbols, independently generated for the di®erent UTs. In high SNR the uniform powerallocationyieldsa¯xedrategapfromtheoptimal(water¯lling)powerallocation. Therefore, following [69] and [16], we restrict to this case and let E[u[n]u[n] H ] = P M I. Under these assumptions, the achievable rate at each UT k under ZFBF with perfect CSIT is given by R k;CSIT = 1 N N¡1 X n=0 E " log Ã 1+ ¯ ¯ H H k [n]v k [n] ¯ ¯ 2 P N 0 M !# =exp µ N 0 M P¾ 2 H ¶ E i µ 1; N 0 M P¾ 2 H ¶ (3.3) where E i (n;x)= R 1 1 e ¡xt t n dt; x>0, is the exponential-integral function. 27 In the case of non-ideal CSIT, the BS uses the available channel information b H k [n], k =1;:::;K; n=0;:::;N¡1, and computes the ZFBF matrix b V[n] by treating b H k [n] as if it was the true channel. Hence, b V[n] 2 C M£K is a ZF matrix with unit norm columnsb v k [n] orthogonal to the subspace spanned by b H j [n], for j6=k. The resulting received signal at the k-th UT is y k [n] = H H k [n]b v k [n]u k [n]+ X j6=k H H k [n]b v j [n]u j [n]+z k [n] = a k;k [n]u k [n]+ X j6=k a k;j u j [n]+z k [n] (3.4) where a k;j [n] denotes the coupling coe±cient between the user channel h k [n] and the beamforming vectorb v j [n]. By following in the footsteps of the achievable rate bound in [16, Theorem 2] we obtain that the achievable ergodic rate for user k is lowerbounded by R k ¸R k;CSIT ¡¢R k , where the rate-gap is upperbounded by ¢R k · 1 N N¡1 X n=0 log Ã 1+ E[jI k [n]j 2 ] N 0 ! (3.5) withI k [n]= P j6=k a k;j u j [n]indicatingthemultiuserinterferenceterm. Anupperbound ontherateR k achievablewithGaussianrandomcodingisalsoobtainedin[16,Theorem 3] by assuming that a genie provides each UT k with exact knowledge of the signal and 28 interference coe±cients a k;j for j = 1;:::;M. This upperbound is referred to as the "genie-aided upperbound" and takes on the form R k · 1 N N¡1 X n=0 E " log Ã 1+ ja k;k j 2 P=M N 0 + P j6=k ja k;j j 2 P=M !# (3.6) Bydividingbothlowerandupperboundtotheachievableratebylog(P=N 0 )andletting P=N 0 !1, it is clear that a su±cient condition for achieving full multiplexing gain is that ¢R k is a bounded function of the SNR P=N 0 . 1 We shall examine this condition under di®erent CSIT feedback schemes in the following sections. 3.2 Analog Feedback Analog feedback consists of sending back the unquantized channel coe±cients, trans- mitted as real and imaginary parts of a complex modulation symbol [61]. We model the feedback channel as AWGN, with the same SNR of the downlink, equal to P=N 0 . The more involved case of a fading MIMO multiple-access (uplink) feedback channel is treated, for the frequency-°at case, in [16, 61]. In order to take advantage of the channel frequency correlation, we partition the N subcarriers into J clusters such that N 0 = N=J is an integer, and feed back only the channel measured at frequencies n 0 = iN 0 for i = 0;1;:::;J¡1. Each UT transmits its channelcoe±cientsatfrequencyn 0 byusingM 0 ¸M feedbackchannelusesperchannel 1 This condition is actually stronger, since it requires constant rate gap from optimal. Strictly speak- ing, full multiplexing gain is achieved if ¢R k is o(log(P=N 0 )). However, in the cases analyzed in this work either ¢R k is bounded, or it is O(log(P=N0)), therefore this option is irrelevant in this context. 29 coe±cient, for a total of M 0 J channel uses. This is achieved by modulating the channel vector H[n 0 ] by a M 0 £M unitary spreading matrix [16, 61]. After despreading, the noisy analog feedback observation for UT k at frequency n 0 =iN 0 is given by g k [i]= p ¯PH k [iN 0 ]+w k [iN 0 ] (3.7) where ¯ =M 0 =M ¸1 and wherew k [n 0 ]2C M£1 is the AWGN in the feedback channel, with i.i.d. components »CN(0;N 0 )). The BS performs linear MMSE \interpolation" based on the observation (3.7) for i=0;:::;J¡1 and compute the beamforming b V[n] for each subcarrier based on the estimated channel. Since channels are spatially i.i.d., theBScanestimateindependentlyeachantennaforeachUT.Therefore,withoutlossof generality, we focus on the side information and estimation of antenna m of UT k. By stackingthefeedbackobservations, weformthevectorg k;m =[g k;m [0];:::;g k;m [J¡1]] T given by g k;m = p ¯PSH k;m +w k;m (3.8) where H k;m = [H k;m [0];H k;m [1];:::;H k;m [N ¡1]] T , w k;m contains the AWGN samples and S is a J£N sampling matrix de¯ned by [S] i;n = ± n=iN 0, for i = 0;:::;J¡1 and n = 0;:::;N ¡1. By letting ½ = ¯P=N 0 , the MMSE estimator of H k;m from g k;m is given by b H k;m = r ½ N 0 § H S H ³ ½S§ H S H +I ´ ¡1 g k;m (3.9) 30 where§ H is de¯ned by (3.1). The corresponding MMSE covariance matrix is given by § e = § H ¡½§ H S H ³ I+½S§ H S H ´ ¡1 S§ H (3.10) Our main result on analog feedback is summarized by the following: Theorem 3.2.1. The achievable rate gap of MIMO-OFDM ZFBF with analog CSIT feedback as described above is upperbounded by ¢R AF k = log Ã 1+ M¡1 M P N 0 " L¡z¡1 X i=0 ¾ 2 [l] + L¡1 X l=L¡z ¾ 2 [l] 1+ N¯P N 0 ¸ (l¡L+z) #! (3.11) where f¾ 2 [l] : l = 0;:::;L¡1g are the DIP components arranged in decreasing order, z = minfJ;Lg and f¸ (i) : i = 0;:::;zg are the non-zero eigenvalues of the matrix ®§ h ® H arranged in increasing order, where ® is the leftmost J£L block of the matrix SF. Proof. See Appendix A. In particular, if z = minfJ;Lg = L, as P=N 0 !1 the rate gap is upper bounded by the constant ¢R AF k =log Ã 1+ M¡1 MN L¡1 X l=0 ¾ 2 [l] ¯¸ (l) ! (3.12) A fair comparison of digital and analog CSIT feedback schemes is provided by the achievablerategapversusthenumberofCSITfeedbackchanneluses. Forexample, the above analog feedback scheme makes use of M 0 J feedback channel uses. We generally express our results in terms of the normalized number of feedback channel uses per 31 antenna. In particular, we introduce the coe±cient ® fb =M 0 J=M ¸ 1 such that ® fb M is the total number of feedback channel uses per user per frame. 3.3 Directional Vector Quantization Weconsiderdirectionalquantizationbasedonrandomvectorquantization(RVQ)code- book ensembles, as in [69]. Each UT has a randomly generated quantization codebook C =fc 1 ;:::;c 2 Bgconsistingof2 B codewordsindependentlyandisotropicallydistributed on the M-dimensional unit complex sphere. In order to reduce the number of feedback bits, several current system proposals consider to cluster the subcarriers and feedback the quantized channel only for one representative frequency for each cluster, as done for the analog feedback scheme considered in Section 3.2 (see for example [22] in the single-user MIMO-OFDM case). Since it is not clear how to interpolate the direction information over the subcarriers, a common approach consists of assuming that the channel is constant over clusters spanning less that the channel coherence bandwidth, and use a piece-wise constant beamforming matrix, computed from the CSIT at the center subcarrier in each cluster. We analyze this \piecewise constant" approach in terms of achievable rate gap. We consider again a grid of J equally spaced frequencies as before. On each such frequency n 0 , the quantization of the channel vector H k [n 0 ] obeys the rule b H k [n 0 ]=argmax c2C ¯ ¯ H H k [n 0 ]c[n 0 ] ¯ ¯ 2 jH k [n 0 ]j 2 (3.13) 32 The binary indices corresponding the selected quantization codewords f b H k [n 0 ] : n 0 = iN 0 ;i=0;:::;J¡1garefedbacktotheBSoveraperfect(error-free, delayfree)digital feedback link, for a total of B tot feedback bits per UT. The total number of feedback bits per UT per frame is given by B tot =BJ. Using the MMSE decomposition, the channel vector at a subcarrier n 6= n 0 in the same cluster of n 0 can be written as H k [n]= · H k [n]+· e k [n;n 0 ] (3.14) where · H k [n] = c[n;n 0 ]H k [n 0 ] and where we de¯ne the channel correlation coe±cient between subcarriers n and n 0 as c[n;n 0 ]= E[H k;m [n]H k;m [n 0 ] ¤ ] E[jH k;m [n 0 ]j 2 ] = P L¡1 l=0 ¾ 2 l e ¡j2¼l(n¡n 0 )=N ¾ 2 H The corresponding MMSE is given by ¾ 2 · e [n;n 0 ] = ¾ 2 H (1¡jc[n;n 0 ]j 2 ). The ZFBF ma- trix b V[n 0 ] computed from the quantized channels b H 1 [n 0 ];:::; b H K [n 0 ] is used for all subcarriers n in the cluster of adjacent frequency indices fn 0 ¡ a;:::;n 0 + bg, taken modulo N because of the circulant statistics of the frequency-domain channels, where a=N 0 =2¡1;b=N 0 =2 if N 0 is even and a=b=bN 0 =2c if N 0 is odd. Our main result with this form of quantized feedback is given by the following: 33 Theorem 3.3.1. The achievable rate gap of MIMO-OFDM ZFBF with digital channel state feedback based on RVQ as described above is upperbounded by ¢R RVQ k = J N b X ±=¡a log µ 1+¾ 2 H P N 0 · jc(±)j 2 2 ¡ B M¡1 + M¡1 M (1¡jc(±)j 2 ) ¸¶ (3.15) where a;b have been de¯ned above and where we de¯ne c(±)= P L¡1 l=0 ¾ 2 l e ¡j2¼l±=N ¾ 2 H Proof. See Appendix B. In order to express the total number of feedback bits B tot in terms of feedback channel uses, we make the optimistic assumption that the feedback link can operate error-free at capacity within the strict one-frame delay constraint. This assumption is justi¯ed in light of the achievability results of [16], where it is shown that a rate gap very close to this case can be achieved by using very simple practical codes and taking into account the feedback error probability. It follows that B tot bits can be transmitted in ® fb (M¡1) channel uses, 2 where ® fb = B tot (M¡1)log 2 (1+P=N 0 ) . Expressing the rate gap in terms of ® fb , we obtain ¢R RVQ k = J N b X ±=¡a log µ 1+¾ 2 H P N 0 · jc(±)j 2 (1+P=N 0 ) ® fb =J + M¡1 M (1¡jc(±)j 2 ) ¸¶ (3.16) 2 Forsimplicity,wenormalizeherebyM¡1insteadofM. Thisisjusti¯edbythefactthatdirectional quantization does not include any information on the channel magnitude. Furthermore, our numerical resultsshowthatthisslightbiasagainstdirectionalquantizationdoesnotyieldanysigni¯cantdi®erence in the performance comparisons. 34 We observe that the rate gap grows linearly with log(P=N 0 ) unless we let J = N. Hence, providing only one direction quantized channel per subcarrier cluster does not takeadvantageofthechannelfrequencycorrelationinane±cientway,sincethechannel is not exactly piecewise constant in frequency. Eventually, for su±ciently large SNR, the channel frequency variations are such that the residual interference will dominate on all frequencies n6=n 0 . Letting J =N and using B =B tot =N bits per carrier yields ¢R RVQ k · log Ã 1+¾ 2 H µ P N 0 ¶ 1¡® fb =N ! (3.17) which is bounded (or even vanishing with increasing SNR) as long as ® fb =N ¸ 1. However, this choice may not be optimal for a given SNR. In practice, for given ® fb and SNR, the system performance can be optimized by choosing the number of clusters J. TheoptimizationofJ iscarriedoutnumericallyandgenerallydependsontheoperating SNR and on the channel DIP, that determines the correlation coe±cient c(±). 3.4 Time-Domain Quantization The frequency-domain channel vectorH k;m for a given BS antenna m and UT k can be regarded as a correlated Gaussian source with covariance matrix § H . Letting H k;m = p NFh k;m , where h k;m is the time-domain channel impulse response for UT k and BS antenna m, and noticing that F is an isometry, it follows that E · ¯ ¯ ¯H k;m ¡ b H k;m ¯ ¯ ¯ 2 ¸ =NE · ¯ ¯ ¯h k;m ¡ b h k;m ¯ ¯ ¯ 2 ¸ (3.18) 35 where we let b H k;m = p NF b h k;m . It follows that the mean-square distortion for H k;m is minimized by minimizing the mean-square distortion for h k;m . In the next section we assume optimistically that the rate-distortion limit for the channel vector can be achieved, and carry out the analysis of the corresponding channel statefeedbackscheme. Achievingtherate-distortionlimitrequires, ingeneral,grouping many source symbols into large blocks and performing optimal vector quantization. On the other hand, the CSIT feedback must have very low delay, and the L channel path coe±cients must be quantized and sent back on each slot of T channel uses in order to enable the BS to compute the downlink beamforming matrix. Hence, optimal source coding and low feedback delay are two contrasting issues. Then, in Section 3.4.2 we consider scalar quantization and show that a carefully optimized scalar quantizer operating on the time-domain channel coe±cients achieves essentially the same rate- gap performance of the rate-distortion limit. 3.4.1 Rate-Distortion Limit Since the components of h k;m are independent, we are in the presence of a set of L \parallel" Gaussian sources. The rate-distortion function for parallel Gaussian sources and mean-square distortion is given by [23] R(D)= L¡1 X l=0 · log 2 ¾ 2 l ° ¸ + (3.19) 36 where ° is the solution of P L¡1 l=0 minf°;¾ 2 l g = D. The number of bits per symbol allocated to the quantization of the l-th path is given by B l = h log 2 ¾ 2 l ° i + . Notice that if ° ¸ ¾ 2 l , then B l = 0. This corresponds to the appealing and intuitive fact that more quantization bits should be allocated to the dominant paths. The bit allocation is usually referred to as reverse water¯lling (RWF). Under the (optimistic) assumption that the CSIT feedback can operate at the rate- distortion limit, our main result with this form of quantized feedback is given by the following: Theorem 3.4.1. The achievable rate gap of MIMO-OFDM ZFBF with digital channel state feedback based on time-domain quantization described above is given by ¢R KL,RWF,Limit k = log µ 1+ M¡1 M P N 0 D ¶ (3.20) where D = E h jh k;1 ¡ b h k;1 j 2 i and the number of quantization bits per UT given by B tot =MR(D) and R(D) is given in (3.19). Proof. See Appendix C. The superscript \KL" indicate the fact that the above approach corresponds to quantizing the Karhunen-Loeve transformed channel which corresponds to quantizing thetime-domainchannelvectorh k;m , undertheassumptionofindependentcoe±cients. We wish to study the high-SNR behavior of the rate gap upperbound in Theorem 3.4.1, inordertodetermineconditionsunderwhichthefullmultiplexinggaincanbeattained. We have the following result: 37 Corollary 3.4.2. In high SNR regime, the rate gap (3.20) can be relaxed to: ¢R KL,RWF,Limit k = log µ 1+¾ 2 H P N 0 M¡1 M 2 ¡R(D)=L ¶ (3.21) Proof. See Appendix D. In order to relate the number of feedback bits to the number of feedback channel uses, we make again the assumption that the feedback link can operate error-free at capacity. With a total of ® fb M = B tot log 2 (1+P=N 0 ) feedback channel uses per UT per frame, we let R(D)=B tot =M in (3.21) and obtain ¢R KL,RWF,Limit k · log Ã 1+¾ 2 H M¡1 M µ P N 0 ¶ 1¡® fb =L ! (3.22) It follows that the rate gap is bounded if ® fb =L¸1 and it vanishes when the inequality is strict. 3.4.2 Scalar Uniform Quantization It is well-known that simple scalar quantization with suitable bit-allocation achieves the rate-distortion limit of parallel Gaussian sources under quadratic distortion within a constant gap factor. Here we exploit this fact and consider a simple practical im- plementation of the above time-domain quantization scheme, where each UT performs uniform scalar quantization on real and imaginary part of its channel coe±cients. Real and imaginary parts of each channel coe±cient h k;m [l] are quantized independently withbB l =2c bits, where B l is obtained, for example, by RWF or by some bit-allocation 38 scheme aimed at minimizing the total distortion. The uniform scalar quantizer Q l has Q l = 2 bB l =2c quantization intervals of size ¢ l > 0 where Q l is an even integer, with thresholds 0;§¢ l ;§2¢ l ;:::;§(Q l ¡ 2)¢=2 and midpoint reconstruction levels §¢ l =2;§3¢ l =2;:::;§(Q l ¡1)¢ l =2. Thel-thpathquantizerisobtainedbychoosing¢ l in order to minimize the quadratic distortion D(Q l ;¢ l ) = 2 Q l =2¡2 X i=0 Z (i+1)¢ l i¢ l µ ´¡i¢ l ¡ ¢ l 2 ¶ 2 f(´)d´ +2 Z 1 (Q l ¡2) ¢ l 2 µ ´¡(Q l ¡1) ¢ l 2 ¶ 2 f(´)d´ where f(´)= 1 p ¼¾ 2 l e ¡ ´ 2 ¾ 2 l . The corresponding rate gap is upperbounded by ¢R KL,RWF,SUQ k = Ã 1+ M¡1 M P N 0 L¡1 X l=0 2D SUQ l ! (3.23) where D SUQ l = min ¢ l >0 D(Q l ;¢ l ). While for any ¯nite B l the optimization of ¢ l must be carried out numerically and amounts to a simple line search, we can follow the analysis in [17] in order to capture the high-SNR behavior in closed form. If the total bit budget for quantization is large, we can assume that B l À1 for all l =0;:::;L¡1. Then, our goal is to set ¢ l such that D(Q l ;¢ l ) : = 2 ¡B l , in order to have the same asymptotic behavior of the rate-distortion limit analyzed before. For a real Gaussian source with variance ¾ 2 l =2 the following asymptotic upperbound holds [17] D(Q l ;¢ l ) · ¢ 2 l 12 +(Q l ¢ l ) 2 P over +o(¢ 2 l ) (3.24) 39 where the ¯rst term accounts for the so-called \granular distortion" and the second term accounts for the overload distortion, where the overload probability is given by P over = Z 1 (Q l ¡2) ¢ l 2 f(´)d´·exp µ ¡((Q l ¡2)¢ l ) 2 4¾ 2 l ¶ By choosing ¢ l = q 4B l ¾ 2 l log 2 e 2 ¡B l =2 we obtain the desired mean-square distortion behavior thatdecreasesasD SUQ l = ¾ 2 l 2 ·B l 2 ¡B l +o(B l 2 ¡B l )where·¼6isaconstantindependent of B l . In particular, for uniform bit allocation B l = B tot =(LM) and letting B tot = ® fb Mlog 2 (1+P=N 0 ) we obtain the upperbound ¢R KL,RWF,SUQ k = log µ 1+·¾ 2 H M¡1 M P N 0 2 ¡B tot =(LM) B tot LM ¶ · log Ã 1+· ® fb ¾ 2 H L M¡1 M µ P N 0 ¶ 1¡® fb =L log 2 µ 1+ P N 0 ¶ ! (3.25) Hence,simplescalaruniformquantizationyieldsavanishingrategapaslongas® fb =L> 1, whichcoincides withthecondition fortherate-distortion limit of Corollary 3.4.2. On the other hand, this bound is not tight enough to capture the behavior for ® fb = 1 (indeed, for ® fb =1 the bound yields a loglog(P=N 0 ) increase of the rate gap). In our numerical results we considered the optimization of the bit-allocation B l subject to the constraint P L¡1 l=0 B l = B tot . This is a classical integer programming problem, for which greedy solutions have been considered (e.g., see [39]). We omit the details of the allocation algorithm for the sake of space limitation here. However, it is apparent from the results of Section 3.6 that RWF allocation comes very close to the 40 more computational intensive greedy bit-allocation, and therefore it can be safely used in practice. 3.5 Exploiting the Physical Channel Structure While most analysis of OFDM systems assumes that the discrete-time channel impulse response h is formed by L independent Gaussian coe±cients, this does not generally hold exactly. The commonly accepted Wide-Sense Stationary Uncorrelated Scattering (WSSUS) fading channel model [91] postulates that multipath components at di®er- ent delays are uncorrelated. However, the delays of the physical channel are not, in general, integer multiples of the OFDM sampling frequency. In other words, while the continuous-time physical channel may obey the WSSUS model, the corresponding discrete-time channel has correlated coe±cients. In this section we remove this unrealistic assumption and take advantage of the physical channel model. As in (2.4), the continuous-time baseband channel impulse response can be written as c(t;¿)= P¡1 X p=0 c p (t)±(¿¡¿ p (t)) (3.26) wherec p (t)isastationaryGaussianproperprocesswith¯rst-orderdistributionCN(0;¹ 2 p ) and ¿ p (t) is the p-th path delay [91]. Under the slowly time-varying assumption, ¿ p (t) is assumed to be independent of t for time intervals several order of magnitudes larger 41 than the OFDM symbol duration, while c p (t) is assumed to be locally time-invariant over the channel coherence time, larger than the OFDM symbol duration. Let Ã(t) denote the convolution of the transmit and receiving front-end ¯lters (included in the D/A and A/D conversion). Then, the concatenation of ¯lters and physical propagation channel around a reference time t is given by the convolution h(t;¿)=Ã(¿)c(t;¿). From (2.7) and (2.9), by uniform sampling at rate 1=W, focus- ing on an arbitrary reference time t=0 and neglecting the time-dependence because of the locally time-invariance assumption, we arrive at the discrete-time channel impulse response h[l]= P¡1 X p=0 c p Ã([l¡¿ p W]=W) (3.27) In matrix form, this can be written as h = ªc where ª 2C L£P , c, (c 0 ;:::;c P¡1 ) T and h , (h[0];:::;h[L¡ 1]) T as de¯ned before. It is clear that in this case the co- variance of h is not diagonal any longer, and it is given by § h = ª§ c ª H where § c =diag(¹ 2 0 ;:::;¹ 2 P¡1 ). We would like to notice here that the channel physical delays f¿ p g and tap coef- ¯cients fc p g can be estimated from a grid of time-frequency downlink pilot symbols using parametric channel model ¯tting such as in [98]. Although more complex than the conventional estimation of h, this shows that the assumption of knowing the phys- ical channel parameters at the receiver is not unrealistic. Furthermore, feeding back the channel delays costs only a very small feedback overhead since, as said before, they varies very slowly in time. 42 Next, we state our main results on the achievable rate gap for analog feedback and \time-domain" quantized feedback by considering this more realistic channel statistics. We omit the proofs since they follow almost trivially into the footsteps of the previous results. It is however interesting to notice that the main e®ect of this more re¯ned channelmodelistoreplaceL(thelengthofthediscrete-timechannelimpulseresponse) by P (the number of physical multipath components). In practice, depending on the shapeofÃ(t), wemayhaveP signi¯cantlylessthanL. Hence, exploitingtheknowledge of the physical channel (in terms of coe±cients fc p g and delays f¿ p g) yields a clear advantage. In general, we assume that the multipath delays f¿ p g are known to the BS since they vary at a much slower rate and can be reliably tracked by the UTs and fed backatmuchlowerdutycycle. Furthermore,thedelayssatisfyreciprocityeveninFDD, and can be estimated by the BS using the uplink pilot symbols. For the case of analog feedback, we have: Theorem 3.5.1. The achievable rate gap of MIMO-OFDM ZFBF with analog channel state feedback as described in Section 3.2 is upperbounded by ¢R AF k = log 0 @ 1+ M¡1 M P N 0 2 4 P¡z¡1 X p=0 ± 2 [p] + P¡1 X p=P¡z ± 2 [p] 1+ N¯P N 0 ¸ (p¡P+z) 3 5 1 A (3.28) where z =minfJ;Pg, f¸ (i) :i=0;:::;zg are the non-zero eigenvalues of ®ª§ c ª H ® H arranged in increasing order, and where f± 2 [p] : p = 0;:::;P ¡1g are the eigenvalues of ª§ c ª H arranged in decreasing order. 43 In particular, the rate gap is bounded as P=N 0 ! 1 if J ¸ P. In this case, it is upper bounded by the constant ¢R AF k =log 0 @ 1+ M¡1 MN P¡1 X p=0 ± 2 [p] ¯¸ (p) 1 A (3.29) At this point it is worthwhile to notice that directional RVQ quantization scheme, that operates in the frequency domain under the assumption of piecewise constant channel, cannot take advantage from the physical channel knowledge. The application of the time-domain quantization approach presented before to the physical model h = ªc requires the projection of h onto the appropriate Karhunen- Loeve basis, since under this physical model the vector h is correlated. We decompose § H as § H = U©U H where U is a unitary matrix and © is the diagonal matrix of eigenvalues. It follows that § H has rank P < N, and we let Á 2 0 ;:::;Á 2 P¡1 denote its non-zero eigenvalues. Without loss of generality we can take U to be the tall N £P matrixoftheeigenvectorsof§ H correspondingtothenon-zeroeigenvalues. First,H k;m is K-L transformed resulting in ~ c k;m =U H H k;m . Then, RWF bit allocation is applied to the quantization of ~ c k;m . From the application of rate-distortion theory we have: Theorem 3.5.2. The achievable rate gap of MIMO-OFDM ZFBF with K-L quantiza- tion described above, operating at the rate-distortion limit, is given by ¢R KL,RWF,Limit k = log µ 1+ M¡1 NM P N 0 D ¶ (3.30) 44 where D =E h j~ c k;1 ¡ b ~ c k;1 j 2 i , the number of quantization bits per UT given by B tot = MR(D), andR(D)= P P¡1 p=0 h log Á 2 p ° i + suchthat° isthesolutionof P P¡1 p=0 minf°;Á 2 p g= D. Inhigh-SNR,usinganapproachsimilartowhatwasdoneinSection3.4.1andletting B tot =® fb Mlog 2 (1+P=N 0 ), we ¯nd the rate gap upper bound in the following simple and appealing form: ¢R KL,RWF,Limit k ·log Ã 1+¾ 2 H M¡1 M µ P N 0 ¶ 1¡® fb =P ! (3.31) As already noticed, this shows that further performance improvement can be obtained by exploiting the structure of the physical channel. In particular, this is the case where L is considerably larger than P. SincetheK-Ltransformrequiresansingularvaluedecomposition(SVD)ofanN£N matrix, which may be computationally demanding for practical values of the OFDM symbol length N, we also consider quantizing directly the time domain coe±cients, c k;m = [c k;m [0];:::;c k;m [P ¡1]] T . This approach disregards the fact that ª has not mutually orthogonal columns. Nevertheless, for commonly used Nyquist pulses Ã(t) it follows that Ã(t) and Ã(t¡ ¿) are quasi-orthogonal if the delay ¿ is of the same order of one symbol interval or larger. Hence, we expect that the proposed suboptimal quantization approach yields almost optimal results. As we shall see, this is indeed con¯rmed by simulation results. 45 Lettingb c k;m denote the quantized version of c k;m , we have ¢R TQ,RWF,Limit k · log µ 1+ M¡1 M P N 0 E h jªc k;1 ¡ªb c k;1 j 2 i ¶ = log µ 1+ M¡1 M P N 0 D ¶ (3.32) where D = P P¡1 p=0 Ã p D p with D p = E £ jc k;1 [p]¡b c k;1 [p]j 2 ¤ and Ã p is the p-th diagonal element of ª H ª. Theoptimaltime-domainquantizationshouldconsideramodi¯edRWFbit-allocation that minimizes the weighted sum of distortions D = P P¡1 p=0 Ã p D p . This can be straight- forwardly done, and also a greedy bit-allocation can be applied to the case of scalar quantization. We omit the details for the sake of space limitation. It is interesting to notice that by applying the geometric-arithmetic mean inequality as in the proof of Corollary 3.4.2 and noticing that ¾ 2 H = P P¡1 p=0 Ã p ¹ 2 p , the rate gap achieved by time- domain quantization is upperbounded by the same expression (3.31) that holds for the K-L approach. This shows that the use of a K-L transform can only yield marginal improvements to the rate gap for high SNR. Therefore, we conclude that the time- domain quantization of the physical path coe±cients provides a very attractive and low complexity solution for the CSIT feedback implementation. 3.6 Numerical results We considered a MIMO-OFDM system with M = 4 transmit antennas at the BS, K = 4 single antenna UTs and N = 64 carriers. We assumed a discrete-time channel 46 model with 5 independent taps and DIP of f0:5;0:24;0:17;0:06;0:03g. Figs. 3.1 and 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 α fb Sum Rate(bps/Hz) SNR = 10 dB ZF Perfect CSIT Analog,UB Analog,LB RVQ,UB RVQ,LB Figure 3.1: Comparison of lowerbounds and upperbounds on the sum rate for di®erent feedback schemes with the discrete-time, uncorrelated path channel model when SNR is 10dB. 3.2 compare the lowerbounds and upperbounds on the sum rates for di®erent feedback schemes as a function of ® fb , that quanti¯es the amount of total feedback channel uses per frame when SNR= 10dB. The lowerbound on the sum rate is calculated by R ¸ K(R k;CSIT ¡¢R k ) where upperbound on the rate gap is computed from (3.11) for analog feedback, (3.16) for RVQ, (3.20) for time-domain quantization and (3.23) for scalar uniform quantization with both RWF and greedy bit-allocation (GBA). The upperboundsarecomputedbyMonteCarlosimulation. ThecurveforRVQcorresponds to the optimal value of J obtained numerically for a given ® fb . We notice that RVQ achieves the worst performance. We interpret this fact qualita- tivelybyobservingthatwithRVQitisnotclearhowtoexploitfrequencycorrelationin 47 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 α fb Sum Rate(bps/Hz) SNR = 10 dB ZF Perfect CSIT KL,RWF,SUQ,UB KL,RWF,SUQ,LB KL,RWF,Limit,LB KL,GBA,SUQ,UB KL,GBA,SUQ,LB Figure 3.2: Comparison of lowerbounds and upperbounds on the sum rate for di®erent feedback schemes with the discrete-time, uncorrelated path channel model when SNR is 10dB. an e±cient way since the \interpolation" of the direction information over the subcarri- ers is not easily accomplished. On the other hand, if we augment direction information with (quantized) channel magnitude, we cannot outperform the rate-distortion inspired time-domain quantization, which treats directly the corresponding parallel Gaussian source in terms of mean-square distortion. In terms of order of decay for high SNR, scalar quantization of the time domain channel coe±cients yields a very simple scheme that performs very close to perfect CSIT. Furthermore, time-domain scalar quantiza- tion is very simple to implement, and requires no complicated construction of spheri- cal codebooks and vector quantization algorithms. Overall, also analog feedback with frequency-domainMMSEinterpolationyieldsverycompetitiveperformanceatlowcom- plexity, although its rate gap remains bounded and does not vanish as SNR increases. 48 Next we considered the same system with SUI-4 channel model given in [27] and omnidirectional antennas where the continuous-time channel model has 3 taps with path delaysf0;1:5;4g ¹s and path variancesf1;0:3162;0:1585g. Ã(t) is assumed to be atriangularpulseresultingfromconvolutionofrectangularpulsescorrespondingtoD/A and A/D (sample-hold) with width 1=W = 1¹s. The lowerbounds and upperbounds on the sum rate can be computed similar to above. Figs. 3.3 and 3.4 compare the 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 α fb Sum Rate(bps/Hz) SNR = 10 dB ZF Perfect CSIT Analog,UB Analog,LB RVQ,UB RVQ,LB Figure 3.3: Comparison of lowerbounds and upperbounds on the sum rate for di®erent feedbackschemeswiththecontinuous-time,uncorrelatedpathchannelmodelwhenSNR is 10dB. lowerbounds and upperbounds on the sum rates for di®erent CSIT feedback schemes as a function of ® fb when SNR=10dB. We observe that time-domain quantization and K-L domain quantization perform very similar, in accordance with the rate-gap bound analysis done before. This shows that for any practical purpose there is no need of K-L transform. 49 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 α fb Sum Rate(bps/Hz) SNR = 10 dB ZF Perfect CSIT KL,RWF,SUQ,UB KL,RWF,SUQ,LB KL,RWF,Limit,LB TQ,RWF,SUQ,UB TQ,RWF,SUQ,LB TQ,RWF,Limit,LB Figure 3.4: Comparison of lowerbounds and upperbounds on the sum rate for di®erent feedbackschemeswiththecontinuous-time,uncorrelatedpathchannelmodelwhenSNR is 10dB. 0 10 20 30 40 50 60 0 1 2 3 4 5 6 7 8 α fb Sum Rate(bps/Hz) SNR = 10 dB ZF Perfect CSIT Analog,Known masking matrix RVQ KL,RWF,SUQ,Known masking matrix Analog,Unknown masking matrix TQ,RWF,SUQ,Unknown masking matrix Figure 3.5: Comparison of upperbounds on the sum rate for di®erent feedback schemes with the continuous-time, uncorrelated path channel model for known masking matrix vs. unknown matrix when SNR is 10dB. 50 Finally, weconsidered thesameSUI-4channelmodel andcompare twocases: 1)the transmit/receive pulse-shaping ¯lter matrix ª is known and 2) the matrix is unknown andthediscrete-timechannelcoe±cientsare assumedtobeindependentwhiletheyare, indeed, correlated. Fig. 3.5 compares the upperbounds on the sum rates corresponding todi®erentfeedbackschemesforthesetwocases asafunctionof ® fb whenSNR=10dB. As it can be observed, knowledge of masking matrix indeed improves the performance, even for such simple channel model and pulse-shaping ¯lter. 51 Chapter 4 Parametric Channel Estimation and Prediction In the previous chapter we discussed the importance of CSI at the BS and studied dif- ferent feedback schemes. In all those feedback schemes, we assumed that the feedback link is delay-free. This is not usually a valid assumption in practical systems. In re- ality, due to the feedback delay, the CSI which is sent from UT to the BS is usually out-dated when received at the BS. To overcome this issue, we need an algorithm to predict the future channel at the UT and feed the predicted channel back to the BS. Several channel prediction methods have been investigated in recent years. While there exists an exceedingly large number of algorithms, these algorithms can be classi¯ed as two main categories: Bayesian estimation approach where the channel is modeled as an autoregressive wide-sense stationary stochastic process and prediction is performed by using linear mean square error to extrapolate the fading channel values in future 52 (e.g., [26]) and parametric estimation approach where the channel is modeled as a sum of complex sinusoids (ray-based model) and prediction is performed by ¯rst estimat- ing the parameters of the model and then using that model to extrapolate the future fading channel values (e.g., [48] and [21]). Here we consider the 3GPP spatial channel model(SCM) which is a ray-based model and investigate di®erent methods to predict the channel. This Chapter is organized as follows: In Section 4.1 3GPP SCM is introduced and the 2-D estimation problem is described. In Section 4.2 we focus on 1-D parameter estimationproblemandinvestigateseveraldi®erentestimationmethodsavailableinthe literature. In Section 4.3 we again consider the original parameter estimation problem and generalize the results from the previous section to 2-D problem. In Section 4.4 Cramer-Rao lowerbound (CRLB) for channel prediction is derived and ¯nally, Section 4.5 provides some simulation results and compares the performance of the di®erent algorithms. 4.1 SCM channel model The SCM is a system level model for simulating urban micro-cell, urban macro-cell and suburban macro-cell fading environments [3]. It considers P cluster of scatterers. Each cluster corresponds to a path and each path consists of R p subpaths. The channel coe±cients are generated as follows: First, we choose the environment which can be eithersuburbanmacro-cell,urbanmacro-cell,orurbanmicro-cell. Thentheparameters associated with that environment is obtained as explained in detail and tabulated in [3] 53 and ¯nally the channel coe±cients are generated based on the parameters and by using (4.1). c p (t) = s P p ¾ SF R p Rp X r=1 q G BS (µ p;r;AoD )G UT (µ p;r;AoA )e j©p;r e j 2¼F c v C cos(µ p;r;AoA ¡µv)t (4.1) where P p is the power of the pth path, ¾ SF is the lognormal shadow fading, R p is the number of subpaths in the pth path, µ p;r;AoD is the the angle of departure for the rth subpath of the pth path, µ p;r;AoA is the the angle of arrival for the rth subpath of the pth path, G BS (µ p;r;AoD ) is the BS antenna gain of each array element, G UT (µ p,r,AoA ) is the UT antenna gain of each array element, F c is the carrier frequency, C denotes the light speed, v is the velocity absolute value of the relative movement between the UT and the BS and µ v is the angle of the UT velocity vector. Generally speaking, the quantity F c v=C is known as the maximum Doppler frequency shift, measured in Hz. This model can be written in the simpli¯ed form of c p (kT sym )= R p X r=1 a r;p e jÁ r;p e j2¼³ r;p kT sym (4.2) where a r;p and Á r;p are amplitude and phase of the coe±cient and ³ r;p is the Doppler frequency corresponding to rth subpath in pth path. From (2.23) and by letting ¢f = 1 NT denote the subcarrier spacing H[k;n]= P X p=1 R p X r=1 a r;p e jÁr;p e ¡j2¼¿p¢fn e j2¼³r;pkTsym (4.3) 54 We consider the pilot allocation similar to which illustrated in Fig. 4.1. Every D t Figure 4.1: Pilot pattern OFDM symbols form an OFDM frame with the total number of N t available frames and the pilots are located at the beginning of each frame. The frame preamble consists ofN f pilot subcarriers inserted every D f subcarriers in the OFDM symbol. In Fig. 4.1, the pilots locations are represented by blue and we want to predict the future channels which is illustrated by red. Note that to avoid aliasing we need that D f · 1 ¢f¿max and D t · 1 2T sym ³ max where ¿ max is the maximum delay and ³ max is the maximum Doppler frequency. 55 Let k = qD t ; q = 0;:::;N t ¡1 and n = mD f ; m = 0;:::;N f ¡1. At the N t £N f pilot locations, the pilot symbols x[qD t ;mD f ] are known and hence, from (2.22) and (5.23), the estimate of the channel can be obtained from observations y[qD t ;mD f ] as b H[qD t ;mD f ] = y[qD t ;mD f ] x[qD t ;mD f ] = P X p=1 R p X r=1 a r;p e jÁ r;p e ¡j2¼¿ p ¢fmD f e j2¼³ r;p qD t T sym +z 0 [q;m] (4.4) where z 0 [q;m] = z[qD t ;mD f ] x[qD t ;mD f ] is AWGN with variance N 0 mathcalP . By de¯ning ! r;p = 2¼³ r;p D t T sym , À p =¡2¼¿ p D f ¢f and ¯ r;p = a r;p e jÁ r;p , (4.4) can be written in the form of 2-D sum of sinusoids in additive white Gaussian noise as b H[qD t ;mD f ]= P X p=1 R p X r=1 ¯ r;p e j(! r;p q+À p m) +z 0 [q;m] (4.5) This is a 2-D parameter estimation problem but can be reduced to 1-D parameter estimation problem by decomposing (4.5) as b H[qD t ;mD f ]= P X p=1 ° p [qD t ]e jÀ p m +z 0 q;m (4.6) where the complex gain for the pth path is in the form of ° p [qD t ]= R p X r=1 ¯ r;p e j!r;pq (4.7) 56 The goal is now to ¯nd the estimates of P, À p , ° p [qD t ], R p , ! r;p and ¯ r;p represented by b P,b À p ,b ° p [qD t ], b R p ,b ! r;p and b ¯ r;p . Inthefollowingsectionwefocuson1-Dparameterestimationproblemgivenas(4.7) by considering single tap channel model (P =1) which can be rewritten as ° =E¯ (4.8) where ¯ = [¯ 1 ;:::;¯ R ] T , ° = [°[0];:::;°[(N t ¡ 1)D t ]] T and [E] m;r = e j!rm for m = 0;:::;N t ¡ 1, r = 1;:::;R. The goal is now to ¯nd the estimates of R, ! r and ¯ r represented by b R, b ! r and b ¯ r . We assume that by solving (4.6), estimate of °[qD t ] is available as b °[qD t ]= R X r=1 ¯ r e j!rq +w q (4.9) and equivalently b ° =E¯+w (4.10) whereb ° =[b °[0];:::;b °[(N t ¡1)D t ]] T andw =[w 0 ;:::;w Nt¡1 ] T . Notice that the statistics of the estimation noise w q is not necessarily Gaussian but here we assume that it is Gaussian with mean zero and variance ¾ 2 w . 57 4.2 1-D Problem 4.2.1 Maximum Likelihood Suppose that a set of observations are available as (4.9) and we know that the order is not larger than R 0 . Then we have R 0 hypothesis as H Q : °[qD t ;Q;¯ Q ;E Q ]= Q X r=1 ¯ r e j! r q (4.11) where Q=1;2;:::;R 0 and q =0;:::;N t ¡1. This can be equivalently written as H Q : °[Q;¯ Q ;E Q ]=E Q ¯ Q (4.12) where ¯ Q = [¯ 1 ;¯ 2 ;:::;¯ Q ] T , [E Q ] m;r = e j! r m for m = 0;:::;N t ¡1, r = 1;:::;Q and °[Q;¯ Q ;E Q ]=[° Q [0;Q;¯ Q ;E Q ];° Q [D t ;Q;¯ Q ;E Q ];:::;° Q [(N t ¡1)D t ;Q;¯ Q ;E Q ]] T . Then we have the following problem max Q max ¯ Q ;E Q p(b °jQ;¯ Q ;E Q ) (4.13) where, due to the Gaussian nature of noise, the conditional pdf above is given as p(b °jQ;¯ Q ;E Q )= 1 (¼¾ 2 w ) N t exp µ ¡ 1 ¾ 2 w ¯ ¯ b °¡°[Q;¯ Q ;E Q ] ¯ ¯ 2 ¶ (4.14) 58 By plugging (4.14) in (4.13) and after simpli¯cation, we have the following LS problem min Q min ¯ Q ;E Q ¯ ¯ b °¡°[Q;¯ Q ;E Q ] ¯ ¯ 2 (4.15) From (4.12), for a given model order Q, the inner minimization can be written as f(Q)= min ¯ Q ;E Q ¡ b °¡E Q ¯ Q ¢ H ¡ b °¡E Q ¯ Q ¢ (4.16) By de¯ning! Q =[! 1 ;! 2 ;:::;! Q ] T , the solution to (4.16) is given by b ! Q = argmax ! Q b ° H E Q ³ E H Q E Q ´ ¡1 E H Q b ° b ¯ Q = ³ E H Q E Q ´ ¡1 E H Q b °j ! Q = b ! Q (4.17) Notice that the optimal model order Q can not be computed by minimizing f(Q) since theobservationsarenoisyandincreasingthemodelorderalwaysdecreasesf(Q). There- forethemodelorderisestimatedbyusingminimumdescriptionlength(MDL)asfollows: 59 Model Order Estimation From (4.9), for i=I¡1;:::;N t ¡1 2 6 6 6 6 6 6 6 6 6 4 b °[iD t ] b °[(i¡1)D t ] . . . b °[(i¡I +1)D t ] 3 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 4 1 1 ::: 1 e ¡j! 1 e ¡j! 2 ::: e ¡j! R . . . . . . . . . . . . e ¡j(I¡1)! 1 e ¡j(I¡1)! 2 ::: e ¡j(I¡1)! R 3 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 4 ¯ 1 e j! 1 i ¯ 2 e j! 2 i . . . ¯ R e j! R i 3 7 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 6 4 w 1 w 2 . . . w I 3 7 7 7 7 7 7 7 7 7 5 (4.18) where I must be chosen to be larger than the maximum possible number of rays and less than N t . By de¯ning e ¯[i]= £ ¯ 1 e j! 1 i ;¯ 2 e j! 2 i ;:::;¯ R e j! R i ¤ T and e °[i]=[b °[iD t ];b °[(i¡1)D t ];:::;b °[(i¡I +1)D t ]] T I£RVandermondematrix e E= · e e(! 1 ) ::: e e(! R ) ¸ withe e(!)= · 1 ::: e ¡j(I¡1)! ¸ T , and e w =[w 1 ;w 2 ;:::;w I ] T , (4.18) can be written as e °[i]= e E e ¯[i]+e w (4.19) The sample covariance matrix is now given by [84] b R= 1 2 ³ e R+J e R T J ´ (4.20) 60 whereJ is the exchange matrix with 1 antidiagonal elements and 0 everywhere else and e R= 1 N t N t ¡1 X i=I¡1 e °[i]e ° H [i] (4.21) R can be estimated by applying MDL as [100] b R=arg min 1·½·I¡1 ¡N t (I¡½)log 0 B @ ³ Q I i=½+1 b ¸ i ´ 1 I¡½ 1 I¡½ P I i=½+1 b ¸ i 1 C A+ 1 4 ½(2I¡½+1)logN t (4.22) where b ¸ i s are eigenvalues of b R respectively in non-increasing order, b ¸ 1 ¸:::¸ b ¸ I . The e®ectofmodelorderover-estimationandunder-estimationareinvestigatedinAppendix E. Levenberg-Marquardt method In[83]Gauss-Newtonmethodisusedtoestimateparametersofsum-of-sinusoidssignal. Here we use Levenberg-Marquardt method which is known to be more robust. First we compute the initial condition with an approach similar to [73]. (4.16) can be written as f( b R)= min ¯ b R ;E b R ¡ b °¡E b R ¯ b R ¢ H ¡ b °¡E b R ¯ b R ¢ (4.23) 61 Using the fact thatjb °j 2 is independent of the parameters and from (4.12), we have the following problem max ¯ b R ;! b R 2 b R X r=1 Re 8 < : ¯ ¤ r N t ¡1 X q=0 b °[qD t ]e ¡j!rq 9 = ; ¡N t b R X r=1 j¯ r j 2 ¡ b R X r=1 b R X r 0 =1;r 0 6=r ¯ r ¯ ¤ r 0 N t ¡1 X q=0 e j(!r¡! r 0)q (4.24) By ignoring the cross-term, and de¯ning b ¡ as DTFT ofb °, b ¡= N t ¡1 X q=0 b °[qD t ]e ¡j!q (4.25) (4.24) can be written as max ¯ b R ;! b R 2 b R X r=1 j¯ r jRe n e ¡\¯ rb ¡(! r ) o ¡N t b R X r=1 j¯ r j 2 (4.26) Each frequency estimate, b ! r , maximizes j b ¡(!)j and j¯ r j =j b ¡(b ! r )j and\¯ r =\ b ¡(b ! r ). Therefore, the initial values of a r , Á r and ³ r for r =1;:::; b R can be computed as a 0 r = j b ¡(b ! r )j Á 0 r = \ b ¡(b ! r ) ³ 0 r = ! r 2¼D t T sym (4.27) Aftercomputingtheinitialvaluefortheparameters,wecanapplyLevenberg-Marquardt algorithm to solve our problem. We ¯rst de¯ne the 2N t £ 1 real observation vector 62 b ° a =[b ° T Re b ° T Im ] T whereb ° Re andb ° Im are real and imaginary parts ofb ° respectively. In a similar way, ° a (µ) = [° T Re (µ) ° T Im (µ)] T where µ = [a 1 ;:::;a b R ;Á 1 ;:::;Á b R ;³ 1 ;:::;³ b R ] T is the set of parameters and °(µ)=[°[0;µ];°[D t ;µ];:::;°[(N t ¡1)D t ;µ]] T with °[qD t ;µ]= b R X r=1 a r e jÁ r e j2¼³ r qD t T sym (4.28) Here we are trying to minimize the cost function J(µ)= 2N t ¡1 X i=0 jb ° a [i]¡° a [q(i)D t ;µ]j 2 (4.29) whereq(i) is the function that returns the valueof q corresponding toi. More precisely, q(i)=imodN t . Let²[i;µ]=b ° a [i]¡° a [q(i)D t ;µ]and b µ k denotetheparameterestimate at iteration k. The updated estimate b µ k+1 is computed by b µ k+1 = b µ k ¡ Ã @² k (µ) @µ T @² k (µ) @µ +¸I ! ¡1 @² k (µ) @µ ²(µ k ) (4.30) where @² k (µ) @µ = @²(µ) @µ ¯ ¯ ¯ µ=µ k . Theelementsofthegradientvectorcanbecomputedeasily. The coe±cient ¸ in (4.30) is the non-negative damping factor and is adjusted at each iteration. The algorithm works as follows: [71] 1. Compute J(µ 0 ) 2. Start with a modest initial value of ¸ such as ¸=0:001 3. At each iteration k: 63 (a) Solve the equation (4.30) and obtainµ k (b) If J(µ k )¸J(µ k¡1 ), increase ¸ by a factor of 10 and go back to (a) (c) If J(µ k ) < J(µ k¡1 ), decrease ¸ by a factor of 10, update the solution with µ k and go back to (a) 4.2.2 Subspace Methods MUSIC In this section we consider multiple signal classi¯cation (MUSIC) method proposed in [77]. If we assumef\¯ r g are independent random variables uniformly distributed over [¡¼;¼], from (4.19), the true covariance matrix R can be written as R = E h e °[i]e ° H [i] i = e E¨ e E H +¾ 2 w I (4.31) where ¨ = diag(j¯ 1 j 2 ;:::;j¯ R j 2 ). It is clear that rank( e E¨ e E H ) = R and by letting ¸ i s be eigenvalues of R in nonincreasing order, ¸ 1 ¸:::¸¸ I , we have 8 > > < > > : ¸ i >¾ 2 w i=1;2;:::;R ¸ i =¾ 2 w i=R+1;:::;I (4.32) 64 Let u 1 ;:::;u I¡R be orthonormal eigenvectors corresponding to ¸ R+1 ¸ :::¸ ¸ I . Then, clearly, RU = N 0 U. Moreover, from (4.31), RU = ³ e E¨ e E H +¾ 2 w I ´ U. By combining these two equalities, e E¨ e E H U=0 (4.33) Nowbyusingthefactthat e E¨isfullcolumnrank, e E H U=0. Therefore,foranyI >R, the true frequencies ! 1 ;! 2 ;:::;! R are the solutions of equation e H (!)UU H e(!)=0 (4.34) Furthermore, if(4.34)hasanyothersolution! 0 6=! r , thensinceUU H istheorthogonal projection onto column space of U, e H (! 0 ) has to be in the R-dimensional null space of U H . However, e H (! 0 ) is linearly independent of fe(! r )g R r=1 and this is not possible becausen+1linearlyindependentvectorscannotbelongtoan n-dimensionalsubspace. Hence, ! 1 ;! 2 ;:::;! R are the only solutions of (4.34). After estimating frequencies ! 1 ,...,! R , we can estimate¯ by using LS approach to solve (4.10) as : b ¯ = ³ b E H b E ´ ¡1 b E H b ° (4.35) where [ b E] m;r = e jb !rm for m = 0;:::;N t ¡ 1, r = 1;:::;R. Therefore, MUSIC can be summarized as follows: 1. Compute b R from (4.20) 2. Compute b R from (4.22) 65 3. Thefrequencyestimatescorrespondtothelocationof b Rhighestpeaksof 1 e H (!) b U b U H e(!) , !2[¡¼;¼] where b U=[b u 1 ;:::;b u I¡ b R ] is a matrix of orthonormal eigenvectors cor- responding to b ¸ b R+1 ¸:::¸ b ¸ I 4. Use (4.35) to compute b ¯ ESPRIT First note that if e E 1 =[I I¡1 0] e E and e E 2 =[0 I I¡1 ] e E with e E de¯ned in the previous section, it can be easily shown that e E 2 = e E 1 (4.36) where = diag(e ¡j! 1 ;:::;e ¡j! R ). Now let V = [v 1 :::v R ] be a matrix of orthonormal eigenvectors v 1 ;:::;v R corresponding to R largest eigenvectors of R, ¸ 1 ¸ :::¸ ¸ R and V 1 = [I I¡1 0]V and V 2 = [0 I I¡1 ]V. It can be clearly seen that RV =V¤ where ¤=diag(¸ 1 ;:::;¸ R ). Moreover, from (4.31), RV = ³ e E¨ e E H +¾ 2 w I ´ V. Therefore, V = e E¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 (4.37) 66 Then V 2 = e E 2 ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 = e E 1 ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 = V 1 ³ ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 ´ ¡1 ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 = V 1 © (4.38) where the second line follows from (4.36) and the third line follows from (4.37). Notice that since e E and V are full column rank, ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 is nonsingular and its inverse exists. Consequently, ©= ³ V H 1 V 1 ´ ¡1 V H 1 V 2 (4.39) Now since © = ³ ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 ´ ¡1 ³ ¨ e E H V ¡ ¤¡¾ 2 w I ¢ ¡1 ´ , © and are re- lated with a similarity transformation and have the same eigenvalues. Hence, we have the following algorithm 1. Compute b R from (4.20) 2. Compute b R from (4.22) 3. b ! r is obtained fromb ! r =¡arg(²) where ² is eigenvalue of b ©= ³ b V H 1 b V 1 ´ ¡1 b V H 1 b V 2 4. Use (4.35) to compute b ¯ 67 4.2.3 Sparse Sampling of Signal Innovation In this section we use an approach similar to [14]. By de¯ning u r =e j! r (4.40) as in [14], it can be shown that any ¯lterfg q g q=0;1;:::;Q with Q¸R that has u 1 ,...,u R as zeros, has the property that it annihilates °. This annihilating ¯lter can be written as G(z)= Q X q=0 g q z ¡q = Q Y q=1 (1¡u q z ¡1 ) (4.41) From (4.7) and (4.40), °[qD t ]= P R r=1 ¯ r u q , g q ¤°[qD t ] = Q X i=0 g i Ã R X r=1 ¯ r u q¡i r ! = R X r=1 ¯ r u q r 0 @ Q X q=0 g i u ¡i r 1 A = R X r=1 ¯ r u q r G(u r ) = 0 (4.42) 68 where¤ represents convolution. Let ¡ be (N t ¡Q)£(Q+1) Toeplitz matrix ¡= 2 6 6 6 6 6 6 6 6 6 4 °[QD t ] °[(Q¡1)D t ] ¢¢¢ °[0] °[(Q+1)D t ] °[QD t ] ¢¢¢ °[D t ] . . . . . . . . . . . . °[(N t ¡1)D t ] °[(N t ¡2)D t ] ¢¢¢ °[(N t ¡1¡Q)D t ] 3 7 7 7 7 7 7 7 7 7 5 (4.43) and g =[g 0 ;g 1 ;:::;g Q ] T . Then, from (4.42), ¡g =0 (4.44) Hence, to solve the general noisy problem we have the following approach: 1. Compute b R from (4.20) 2. Compute b R from (4.22) 3. Use Cadzow's iterative denoising as follows: (a) Build the Toeplitz matrix ¡ with Q= N t ¡1 2 and observations b °[0];b °[D t ];:::;b °[(N t ¡1)D t ] as in (4.43). (b) PerformSVDof¡=USV H whereUandVare N t +1 2 £ N t +1 2 unitarymatrices and S is Nt+1 2 £ Nt+1 2 diagonal matrix. 69 (c) Compute the low rank approximation of ¡ as ¡ 0 = US 0 V H where S 0 is the diagonal matrix derived from S by keeping only its b R most signi¯cant diagonal elements. (d) Compute a "denoised" approximation of b °[l] by averaging the subdiagonal elements of ¡ 0 . (e) Repeatsteps(a)-(e)untilthe b R+1thlargestdiagonalelementofSissmaller than its b Rth largest element by some factor. 4. Build the Toeplitz matrix ¡ with Q= b R as (4.43). 5. Solve (4.44) by performing SVD of ¡ and picking the eigenvector [b g 0 ;b g 1 ;:::;b g b R ] T corresponding to the smallest eigenvalue of ¡. 6. Obtainb u r by computing the roots of G(z)= P b R r=0 b g r z ¡r , . 7. From (4.40),b ! r =\b u r . 8. Use (4.35) to compute b ¯ 4.3 2-D Problem We de¯ne the channel parameter vector µ 2 R M£1 with M = P +3 P P p=1 R p as µ = [µ T 1 ;µ T 2 ;:::;µ T P ] T where µ p = [a p;1 ;:::;a p;R p ;Á p;1 ;:::;Á p;R p ;³ p;1 ;:::;³ p;R p ;¿ p ] T . If we form 70 theobservationvectorb ¹=[ b H[0;0];::::; b H[0;(N f ¡1)D f ];:::; b H[(N t ¡1)D t ;0];:::; b H[(N t ¡ 1)D t ;(N f ¡1)D f ]] T , the cost function to be minimized, J(µ), is given by J(µ)= N t N f ¡1 X i=0 ¯ ¯ ¯ ¯ ¯ ¯ b ¹[i]¡ P X p=1 R p X r=1 ¯ r;p e j! r;p q(i)+À p m(i) ¯ ¯ ¯ ¯ ¯ ¯ 2 (4.45) where q(i) and m(i) are functions that return the values of q and m corresponding to i respectively. More precisely, q(i)=bi=N f c and m(i)=i mod N f . (4.45) can be written as J(µ)= Nt¡1 X q=0 (q+1)N f ¡1 X i=qN f ¯ ¯ ¯ ¯ ¯ ¯ b ¹[i]¡ P X p=1 ° p [qD t ]e jÀ p m(i) ¯ ¯ ¯ ¯ ¯ ¯ 2 (4.46) Hence, the optimal set of parameters can be obtained by minimizing N t 1-D cost func- tions given by J q (µ)= (q+1)N f ¡1 X i=qN f ¯ ¯ ¯ ¯ ¯ ¯ b ¹[i]¡ P X p=1 ° p [qD t ]e jÀpm(i) ¯ ¯ ¯ ¯ ¯ ¯ 2 q =0;1;:::;N t ¡1 (4.47) Inthisway, oneneedstominimizeeachcostfunctionindependentlytoobtainestimates of° p [qD t ],P andÀ p andthensomehowcombinetheseinformationtogettheestimations of all the parameters. It is not clear how this can be done. Therefore, we use a suboptimal solution for this problem given in the following section [98]. 71 4.3.1 ESPRIT By de¯ning e H q [i] = h b H[qD t ;iD f ]; b H[qD t ;(i¡1)D f ];:::; b H[qD t ;(i¡I +1)D t ] i T with I greater than maximum possible value of P and less than N f , the sample covariance matrix is given by [84] b R= 1 N t N t ¡1 X q=0 1 2 ³ e R q +J e R T q J ´ (4.48) where e R q = 1 N f N f ¡1 X i=I¡1 e H q [i] e H H q [i] (4.49) P can be estimated by applying MDL as [100] b P =arg min 1·½·I¡1 ¡N f (I¡½)log 0 B @ ³ Q I i=½+1 b ¸ i ´ 1 I¡½ 1 I¡½ P I i=½+1 b ¸ i 1 C A+ 1 4 ½(2I¡½+1)logN f (4.50) where b ¸ i s are eigenvalues of b R respectively in non-increasing order, b ¸ 1 ¸ ::: ¸ b ¸ I . Hence, we have the following algorithm 1. Compute b R from (4.48) 2. Compute b P from (4.50) 3. b À p is obtained fromb À p =¡arg(²) where ² is eigenvalue of b ©= ³ b V H 1 b V 1 ´ ¡1 b V H 1 b V 2 where b V 1 = [I I¡1 0] b V and b V 2 = [0 I I¡1 ] b V and b V = [b v 1 :::b v b P ] is matrix of orthonormal eigenvectorsb v 1 ;:::;b v b P corresponding to b P largest eigenvectors of R, ¸ 1 ¸:::¸¸ b P . 72 4. Use LS approach to solve (4.6) and obtainb ° p [qD t ] for q = 0;:::;N t ¡1 and p = 0;:::;P ¡1 5. Use ESPRIT to solve P 1-D parameter estimation problems as discussed in the previous section 4.3.2 Wiener Prediction In this section we model time-domain channel coe±cients as independent Gaussian random variables and use Wiener ¯lter to predict channel coe±cients in future i.e., we consider a scheme which performs prediction on each tap h[k;0],h[k;1],...,h[k;L¡ 1] where L is the ¯nite duration of channel impulse response in time domain and h[k;l]= h(kT sym ;lT) is related to the frequency domain channel H[k;n] by H[k;n]= L¡1 X l=0 h[k;l]e ¡j2¼ln=N (4.51) Tomakeafaircomparisonwiththepreviouspilotallocationscheme,wescaletheampli- tude of transmitted pilot in time domain by q N f L to maintain same energy. Therefore, we have observations in the form b h[qD t ;l]=h[qD t ;l]+z 00 l =0;:::;L¡1; q =0;:::;N t ¡1 (4.52) 73 wherez 00 »CN(0; N 0 L N f P ). Nowbystackingtheobservationsas b h l =[ b h[0;l]; b h[D t ;l];:::; b h[(N t ¡ 1)D t ;l]] T , h[q 0 D t ;l] can be estimated by using LMMSE prediction as b h[q 0 D t ;l]=E h h[q 0 D t ;l] b h H l i E h b h l b h H l i ¡1 b h l (4.53) where the cross covariance matrix has elements in the form ofE h h[q 00 D t ;l] b h ¤ [qD t ;l] i = R h ((q 00 ¡q)D t ) with R h l ((q 00 ¡q)D t )=E[h[q 00 D t ;l]h ¤ [qD t ;l]] and the auto covariance matrixhaselementsintheformofE h b h[qD t ;l] b h ¤ [q 0 D t ;l] i =R h l ((q¡q 0 )D t )+ N 0 L N f P ± q¡q 0. Notice that the correlations are usually unknown and one needs to estimate them from data as R h l (mD t )¼ 1 N t N t ¡1¡m X q=0 b h[(q+m)D t ;l] b h ¤ [qD t ;l] (4.54) 4.4 CRLB Thissectionprovidestheso-calledCramer-Raolowerbound(CRLB)whichisauniversal lowerboundonthevarianceoftheMLestimation. Asbefore, channelparametervector is represented by µ whereµ =[µ T 1 ;:::;µ T P ] T with µ p =[a p;1 ;:::;a p;R p ;Á p;1 ;:::;Á p;R p ;³ p;1 ;:::;³ p;R p ;¿ p ] T From [76, p. 230] we know that for a given set of parameters µ, CRLB can be obtained from e[k;n;µ]= @H H [k;n;µ] @µ I ¡1 (µ) @H[k;n;µ] @µ (4.55) 74 where @H[k;n;µ] @µ = · @H[k;n;µ] @µ 1 ;:::; @H[k;n;µ] @µ M ¸ T (4.56) where M = P +3 P P p=1 R p and I(µ) is the M£M Fisher information matrix for 2-D sinusoid parameter estimation. The Fisher information matrix for a set of observations givenby¹(µ)=[H[0;0;µ];::::;H[(N t ¡1)D t ;(N f ¡1)D f ;µ]] T inadditivewhiteGaussian noise is given as [63, p.144] [I(µ)] m;n = 2 N 0 =P Re · @¹ H (µ) @µ m @¹(µ) @µ n ¸ = 2 N 0 =P Re 2 4 NtN f ¡1 X i=0 @¹ ¤ [i;µ] @µ m @¹[i;µ] @µ n 3 5 (4.57) where ¹[i;µ] = H[q(i)D t ;m(i)D f ;µ] and, as before, q(i) and m(i) are functions that return the values of q and m corresponding to i respectively. Notice that, depending on µ, FIM can become singular. For those cases, instead of CRLB, asymptotic CRLB (ACRLB) can be computed as · e[k;n;µ] = lim N t ;N f !1 e[k;n;µ] = @H H [k;n;µ] @µ · I ¡1 (µ) @H[k;n;µ] @µ (4.58) where · I ¡1 (µ)=lim Nt;N f !1 I ¡1 (µ) [98]. 75 4.5 Simulation Results The system speci¯cations are given in Table 5.1. We ¯rst consider a single tap channel Table 4.1: System Speci¯cations Description Value 1=T, Sampling frequency 3:84MHz F c , Carrier frequency 2:6GHz N, Number of subcarriers 256 N a , Number of active subcarriers 200 ¢f, Subcarrier frequency spacing 15KHz Length of CP 64 T sym , OFDM symbol with CP 83:33¹s N f , Number of Pilot subcarriers 50 N t , Number of Pilot OFDM symbols 100 D f , Pilot subcarrier spacing 4 D t , Pilot OFDM symbol spacing 20 ¿ max , Maximum delay spread 16:67¹s with 20 subpaths generated according to the SCM model. The user is assumed to be high speed user with speed v =75km/h. Maximum Doppler frequency for this user can be computed from ³ max = F c v C . In [98], a rule-of-thumb for the duration of validity of this model is given by T valid = r Cr min 3F c v 2 (4.59) where r min is the distance from user to nearest scatterer and we assume r min = 600m. Maximum Doppler frequency and duration of validity of model for this user is given in Table4.2. AsillustratedinFigs4.2and4.3,dependingonthedistributionofscatterers in the enlivenment, the Doppler frequencies may be well-separated or packed. Figs. 4.4 and 4.5 show snapshots of the predicted channel and MSE in predicting channel 76 Table 4.2: Maximum Doppler frequency and duration of validity of model for user with di®erent speeds v (Km/h) ³ max (Hz) T valid (s) T valid (OFDM symbols) High speed 75 180.56 0.231 2700 Low speed 5 12.04 3.458 41500 Building 2 BS Building 2 Tree Tree Building 1 -0.5 0 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Normalized Frequency Amplitude Figure 4.2: Packed Doppler frequencies Building 2 BS Building 2 Tree Tree Building 1 -0.5 0 0.5 0 0.05 0.1 0.15 0.2 0.25 Normalized Frequency Amplitude Figure 4.3: Well-separated Doppler frequencies 1T sym ahead resulting from applying di®erent prediction methods on a channel with well-separated Doppler frequencies. Figs. 4.6 and 4.7 show similar simulation results for a channel with packed Doppler frequencies. We observe that for well-separated Doppler frequencies ESPRIT and ML outperform other methods while in the case of 77 0 10 20 30 40 50 60 70 80 90 100 −30 −25 −20 −15 −10 −5 0 5 10 Symbol ahead ESPRIT MUSIC Vetterli ML Actual channel Figure 4.4: Snapshot, Well-separated Doppler frequencies 10 15 20 25 30 −40 −35 −30 −25 −20 −15 −10 −5 0 SNR(dB) MSE(dB) 1 symbol ahead ESPRIT MUSIC Vetterli ML CRLB Figure 4.5: Predicting channel 1T sym ahead packedDopplerfrequencies,noneofthesemethodsworkverywell. Noticethatalthough MLperformsslightlybetterthanESPRIT,itscomplexityismuchhigherthanESPRIT. Next we consider the exact SCM model with 6 taps. First we consider high speed user with speed v =75km/h. Here we compare using ESPRIT and Wiener for channel 78 0 10 20 30 40 50 60 70 80 90 100 −35 −30 −25 −20 −15 −10 −5 0 5 Symbol ahead ESPRIT MUSIC Vetterli ML Actual channel Figure 4.6: Snapshot, Packed Doppler frequencies 10 15 20 25 30 −40 −35 −30 −25 −20 −15 −10 −5 0 SNR(dB) MSE(dB) 1 symbol ahead ESPRIT MUSIC Vetterli ML CRLB Figure 4.7: Predicting channel 1T sym ahead prediction. Notice that, as mentioned earlier, Wiener prediction can only be used to predictchannelcoe±cientsattimeinstancesequaltointegermultiplesofD t T sym . Figs. 4.8 and 4.9 illustrate snapshots of the predicted channel and MSE in predicting channel D t T sym ahead resulting from applying di®erent prediction methods on a channel with 79 0 10 20 30 40 50 60 70 80 90 100 −35 −30 −25 −20 −15 −10 −5 0 5 10 OFDM symbol ahead ESPRIT Wiener Actual channel Figure 4.8: Snapshot, High speed (75 km/h), well-separated, D t = 20, N t = 100, SNR=20dB 10 15 20 25 30 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 SNR(dB) MSE ESPRIT Wiener CRLB ACRLB Figure 4.9: Predicting channel D t T sym ahead well-separatedDopplerfrequencies. Figs. 4.10and4.11demonstratesimilarsimulation results for a channel with packed Doppler frequencies. These results show that for well- separated Doppler frequencies the channel can be predicted very well by ESPRIT while channels with packed Doppler frequencies are not predictable. 80 0 10 20 30 40 50 60 70 80 90 100 −12 −10 −8 −6 −4 −2 0 2 4 OFDM symbol ahead ESPRIT Wiener Actual channel Figure 4.10: Snapshot, High speed (75 km/h), packed, D t =20, N t =100, SNR=20dB 10 15 20 25 30 −45 −40 −35 −30 −25 −20 −15 −10 −5 SNR(dB) MSE ESPRIT Wiener ACRLB Figure 4.11: Predicting channel D t T sym ahead Next we consider a low speed user with speed v = 5km/h. Maximum Doppler frequency and duration of validity of model for this user is given in Table 4.2. Notice that in this case, T valid is much larger than the previous case due to low mobility of user and for ESPRIT, everything can be scaled by down-sampling the observations 81 and prediction can be done very well if Doppler frequencies are well separated. As 0 10 20 30 40 50 60 70 80 90 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 OFDM symbol ahead ESPRIT Wiener Actual channel Figure 4.12: Snapshot of predicted channel for well-separated Doppler frequencies and low speed user with v =5km/h 10 12 14 16 18 20 22 24 26 28 30 −40 −35 −30 −25 −20 −15 −10 SNR(dB) ESPRIT Wiener ACRLB Figure 4.13: Mean squared error for well-separated Doppler frequencies and low speed user with v =5km/h before, Figs. 4.12 and 4.13 illustrate snapshots of the predicted channel and MSE in predicting channel D t T sym ahead resulting from applying di®erent prediction methods onachannelwithwell-separatedDopplerfrequencies. Figs. 4.14and4.15demonstrate similar simulation results for a channel with packed Doppler frequencies. These results 82 0 10 20 30 40 50 60 70 80 90 100 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 OFDM symbol ahead ESPRIT Wiener Actual channel Figure 4.14: Snapshot of predicted channel for packed Doppler frequencies and low speed user with v =5km/h 10 15 20 25 30 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 SNR(dB) MSE ESPRIT Wiener ACRLB Figure 4.15: Mean squared error for for packed Doppler frequencies and low speed user with v =5km/h show that for well-separated Doppler frequencies the channel can be predicted very well by ESPRIT while for channels with packed Doppler frequencies, Wiener is optimal. The results of this chapter are summarized in Table. 4.3. 83 Table 4.3: Optimal prediction method Well-separated Dopplers Packed Dopplers Low speed ESPRIT Wiener High speed ESPRIT No prediction 84 Chapter 5 Scheduling Under Non-perfect CSI 5.1 Introduction Under perfect knowledge of the downlink channels, the resource allocation problem in a Multiuser MIMO (MU-MIMO) downlink has been widely investigated under vari- ous precoding and beamfomring schemes (see for example [106], [96], [15], [24], [93], [59], [65], and references therein). In practice, the Channel State Information (CSI) is obtained through some form of training and feedback. In Time-Division Duplexing (TDD) systems, the base station (BS) can learn the downlink channel coe±cients in \open-loop" mode, by exploiting the uplink pilot symbols and channel reciprocity (e.g., [53], [51]). In Frequency-Division Duplexing (FDD) systems, since uplink and downlink take place in widely separated frequency bands, the downlink channel coe±cients must be learned in \closed loop" mode, via some explicit CSI feedback scheme (e.g., [16], [81] and references therein). In both cases, the CSI available to the BS can be seen as some sort of \noisy" version of the true channel coe±cients. 85 The key impact of CSI quality on the performance of MU-MIMO is evidenced in the relevant regime of medium-to-high SNR. In [16] it is shown that the gap between the sum capacity under perfect CSI and the sum-rate achievable by a simple linear Zero-Forcing Beamforming (ZFBF) scheme with non-perfect CSI takes on the form ¢R=minfM;Kglog(1+· 1 ¾ 2 e snr)+· 2 , where · 1 ;· 2 are constants that depend on the particular CSI training and feedback scheme used, and where ¾ 2 e denotes the Mean- Square Error (MSE) between the true channel coe±cients and the CSI available to the BS. Since the sum-rate of a MU-MIMO downlink channel in high-SNR behaves like minfM;Kglog(1+·snr), for some constant ·, it follows that ¾ 2 e must decrease at least as fast as snr ¡1 in order to preserve the optimal O(minfM;Kglogsnr) increase of the sum-rate with SNR. Under both TDD and FDD, the main source of CSI estimation error consists of the delay introduced by the estimation/feedback scheme in the pres- ence of time-varying wireless channels [53], [16], [81], [9], [56]: even after neglecting all other sources of suboptimality, such as channel state quantization, feedback errors, and so on, the MSE ¾ 2 e cannot be less than the channel prediction error from noisy pilot symbols. This estimation-theoretic quantity represents a fundamental lower bound to the accuracy of CSI. We performed an extensive study of MIMO channel prediction based on the 3GPP Spatial Channel Model given in [3]. Our results, summarized in Section 5.5.2, show that channel prediction is generally quite accurate with the exception of a speci¯c class of channels characterized by a large Doppler spread (high user mobility) and clustered angles of arrival. For such channels, the channel prediction MSE is very large, no 86 matter which prediction method is used, as re°ected by a Cramer-Rao bound analysis. These results suggest that users can be classi¯ed according to their channel prediction MSE,andthatthisclassi¯catione®ectivelyreducestoonlytwoextremeclassesof\non- predictable" (high-mobility and clustered angle of arrivals) and \predictable" (all other cases). This simpli¯cation is instrumental to the main contribution of this chapter: a simple and e±cient MU-MIMO downlink scheduling scheme that takes explicitly into account the CSI quality. Downlink scheduling aims at making the system operate at a desired point on the ergodic (or long-term average) achievable rate region of the system, for a given physical layer signaling scheme. The operating point re°ects some form of \fairness," corre- sponding the maximization of a concave non-decreasing utility function of the ergodic rates. Although a direct maximization is typically hopelessly complicated, the opti- mal point is implicitly achieved using a stochastic optimization approach [38, 66, 68]. We solve this problem for the case of MU-MIMO with non-perfect CSI in Section 5.3. Then, based on the general solution, we ¯nd a practical simpli¯ed scheduling policy un- der the assumption, motivated before, that the users can be partitioned into two classes with either small or large channel prediction MSE. The resulting scheduling algorithm can be regarded as an opportunistic MIMO \multi-mode" scheme that selects at each scheduling slot either a MU-MIMO downlink beamforming mode that performs spatial multiplexing to a subset of predictable users, or a single-user space-time coding mode that serves a single selected non-predictable user. 87 With respect to existing literature, we notice that downlink scheduling with non- perfect CSI has been treated mainly in the case where all users have the same CSI quality. Static mode-switching criteria have been studied for example in [107, 56] where the number of users to be simultaneously served is optimized depending on the CSI quality and channel SNR. In contrast, the present work presents a dynamic schedul- ing policy that can handle users with very di®erent CSI qualities at the same time, and allocatesopportunisticallythesignalingmodes(namely: spatialmultiplexingandspace- timecoding)overtimeandacrosstheusers. Thefundamentalroleofchannelprediction in downlink scheduling schemes was noticed before, e.g., in [56], [9]. In particular, [9] proposesachannel-predictiveproportionalfairschedulingrule,withoutanalyticalproof, forthescalar(notMIMO)case. Incomparisonwiththeseworks,hereweprovideagen- eralframeworkfordownlinkschedulingwithnon-perfectCSIthatappliestoMU-MIMO and to a wide class of fairness utility functions. Also, we present novel rigorous results on system stability and performance bounds of the proposed scheduling algorithms. Numerical results are provided for two relevant fairness utility functions re°ecting proportional fairness and max-min fairness (referred to as \hard-fairness"). It should be noticed, though, that our framework can be applied to any concave non-decreasing utility function. Results based on a realistic channel model [3] and actual channel statepredictionalgorithms(seedetailsinSection 6.5)showthattheproposedapproach achievesverysigni¯cantimprovementwithrespecttoaconventionalmismatchedscheme that treats the available CSI as if it was perfect. 88 5.2 System set-up We consider a MIMO downlink channel with a BS equipped with M antennas and K single-antennaUTs. Thechannelisassumedfrequency°at 1 andconstantover\slots"of length T À 1 symbols (block-fading model). The received complex baseband discrete- time signal at the k-th UT during block t is described by y k;i [t]=h H k [t]x i [t]+z k;i [t]; i=1;:::;T (5.1) where H denotes Hermitian transpose, t tics at the slot rate, i tics at the symbol rate, k denotes the user index, h k [t]2C M is the channel vector from the BS antenna array to the k-th receiver antenna, x i [t] 2 C M is the transmit signal vector transmitted at symbol interval i of slot t, and z k;i [t]»CN(0;N 0 ) is the corresponding additive white Gaussian noise (AWGN). We collect all channel vectors into a channel state matrix H[t]=[h 1 [t];:::;h K [t]]2C M£K . At the beginning of each slot t, the BS has knowledge of the CSI b H[t]=[ b h 1 [t];:::; b h K [t]]2C M£K , obtained by some form of training, channel prediction and feedback, as discussed in Section 6.1. We assume thatH[t] and b H[t] are jointlystationaryandergodicmatrix-valuedprocesses. Forconvenience,wealsoassume that b H[t] is a su±cient statistic for the causal estimation of H[t] from the CSI process f b H[t]g. While the capacity region of the MIMO-BC in the perfect CSI case (i.e., for b H[t]= H[t]) is well-known [95], the case of imperfect CSI is still open although outer bounds 1 The generalization to MIMO-OFDM and frequency selective fading is immediate. 89 and achievability lower bounds exist. In this work we focus on a simple physical layer signaling scheme based on linear precoding and independently generated Gaussian user codes. Nevertheless, the general scheduling framework developed in this work can be easily extended to other MU-MIMO downlink schemes, such as Tomlinson-Harashima precoding [15], Vector Precoding [44] and Dirty-Paper Coding [95], [19]. Withlinearprecoding,eachk-thusercodewordisaM£T space-timearraydenoted by U k [t] =fu k;i [t] : i = 1;:::;Tg. The signal vector transmitted at symbol interval i of slot t is given by x i [t] = P K k=1 u k;i [t]. In the following, we let (H; b H) denote a pair of jointly distributed random matrices with the same joint distribution of (H[t]; b H[t]) (independent of t by stationarity). A linear precoding signaling scheme is de¯ned as a possibly randomized function ° such that °( b H) ¢ =(§ 1 ( b H);:::;§ K ( b H);r( b H)) where§ k ( b H) is the spatial-domain transmit covariance matrix of user k, andr( b H) is a transmit rate allocation vector. Then, upon observation of the CSI b H[t], the signaling scheme ° chooses for each user k a Gaussian generated codebook of rate r k ( b H[t]), where the codewords U k [t] have i.i.d. columns generated according to the Gaussian distribution CN(0;§ k ( b H[t])). We say that a scheme ° is feasible with respect to the power constraintP if P K k=1 tr ³ § k ( b H) ´ ·P with probability 1. The set of all feasible schemesisdenotedby¡(P). Foragiven° andCSIvalue b H, thesetofusersk suchthat 90 tr(§ k ( b H))>0 is called the active set, and will be denoted byU ° ( b H). The complement setU c ° ( b H) of all users k with tr(§ k ( b H))=0 is referenced to as the idle set. The above de¯nition of ° encompasses in full generality all linear precoding strate- giesbasedonGaussianrandomcoding,rangingfrombeamformingtospace-timecoding. Forlateruse,werecallheretwowell-knownchoicesforthetransmitcovariancematrices that will be essential in the practical scheduling policy of Section 5.4: 1)ApopularchoiceforMU-MIMOlinearprecodingconsistsofcomputingZFBF\steer- ing vectors" by treating the CSI b H as if it was the true channel matrix (see for example [53], [16], [25] and references therein). In our notation, this corresponds to choosing an active set U ° ( b H) of size not larger than rank( b H) and, rank-1 transmit covariance matrices§ k ( b H)=p k v k v H k where p k >0 is the transmit power allocated to user k, and v k is a unit-length vector obtained by calculating the Moore-Penrose pseudo-inverse b H y (U ° )= b H(U ° ) ³ b H H (U ° ) b H(U ° ) ´ ¡1 (5.2) of the matrix b H(U ° ) with columns f b h j : j 2 U ° g, and taking the normalized column of b H y (U ° ) corresponding to user k. In particular, v k is orthogonal to all f b h j : j 2 U ° ( b H);j6=kg. 2) At the other extreme of the range of possible linear precoding signaling schemes we ¯nd the classical space-time coding to a single user [56], [87], [6]. In our notation, this correspondstochoosinganactivesetU ° ( b H)ofsize1andthetransmitcovariancematrix § k ( b H) = (P=M)I for the only k 2U ° ( b H). Interestingly, ZFBF serves simultaneously 91 up to M active users, each with a rank-1 transmit covariances matrix, while space-time coding serves just one active user, with a rank-M transmit covariance matrix. For a ¯xed set of transmit covariance matrices (§ 1 ;:::;§ K ), a linear precoding scheme yields a Signal-to-Interference plus Noise Ratio (SINR) at receiver k given by sinr k (H;§ 1 ;:::;§ K ) ¢ = h H k § k h k N 0 + P j6=k h H k § j h j (5.3) We let R k [t] denote the e®ective rate of user k on slot t. In general R k [t] is di®erent from the allocated rate r k ( b H[t]) since CSI is not perfect. As far as rate allocation is concerned, we consider the following two cases: 1) Outage rates: following standard information theoretic arguments (see [13] and references therein), under slot-by-slot coding and decoding, receiver k can reliably de- code a rate r k provided that no information-outage occurs, i.e., provided that r k is smallerthanthemutualinformationI k (H;§ 1 ;:::;§ K ) ¢ =log(1+sinr k (H;§ 1 ;:::;§ K )). Asaconsequence,foragivensignalingscheme° wede¯netheoutagerateastherandom variable: R k (H;°( b H))=r k ( b H)£1 n r k ( b H)<I k (H;§ 1 ( b H);:::;§ K ( b H)) o (5.4) where 1fAg is the indicator function of an eventA. 92 2) Optimistic rates: in this case, we assume that some genie-aided rate adaptation schemeisabletoachieveane®ectiveinstantaneousrateequaltothemutualinformation: R k (H;°( b H))=I k (H;§ 1 ( b H);:::;§ K ( b H)) (5.5) The system model underlying the outage rate assumption consists of standard ARQ protocol that removes R k [t] = r k ( b H[t]) bits/channel use from the transmission bu®er of active user k if no outage occurs. The system model underlying the optimistic rate assumption corresponds to an idealized fast rate adaptation scheme (see for example [10], [46]). Any practical rate adaptation scheme yields performance in between the outage and the optimistic rates de¯ned above. Therefore, these two extreme cases are relevant in the sense that they provide upper and lower bounds to practical adaptive rate schemes. Under either one of the above assumptions, we let the e®ective rate be R k [t]=R k (H[t];°( b H[t])), given by (5.4) or by (5.5). 5.3 Optimal downlink scheduling The achievable ergodic rate regionR, for a given set of feasible physical layer signaling schemes, isde¯nedastheclosureoftheconvexhullofallachievableergodicratepoints. Under a ¯xed signaling scheme ° 2 ¡(P), user k is served with a long-term average rate ¹ R k = lim t!1 1 t P t¡1 ¿=0 R k (H[¿];°( b H[¿])) = E[R k (H;°( b H))], where convergence is 93 with probability 1 because of ergodicity. Since time-sharing between any set of feasible signaling schemes is also a feasible scheme, we have: R=coh [ °2¡(P) n ¹ R2R K + : ¹ R k ·E h R k (H;°( b H)) i ; 8k o (5.6) where \coh" denotes \closure of the convex hull" and where the expectation is with respect to the joint probability distribution of (H; b H) and ° (for randomized signaling schemes). We consider an \in¯nite backlog" situation where all the data to be transmitted are available at the BS. The goal of the downlink scheduler is to maximize some concave entrywise non-decreasing utility function g(¢) of the user individual ergodic rates ¹ R = ( ¹ R 1 ;:::; ¹ R K ). 2 The problem that we wish to solve is: maximize g( ¹ R); subject to ¹ R2R (5.7) Suppose that the solution ¹ R ? of (5.7) is found. Then, by de¯nition, there exists a (possiblyrandomized)signalingstrategythatachieves ¹ R ? . Afeasibleschedulingpolicyis analgorithmthatchoosesateachtimetsomephysicallayersignalingscheme°2¡(P), based on the history of all past transmissions and arrivals, on the observation of the CSI and on the knowledge of the joint statistics of all variables in the system. We are interested in ¯nding an explicit scheduling policy (denoted for brevity by ° ? ) that 2 By entrywise non-decreasing we mean that for all r 2R K + and ± 2R K + , g(r) ¸ g(r+±). Also, recall that a concave function is continuous in the interior of its domain. Without loss of generality, we consider g(¢) with domainR K + . 94 achieves ¹ R ? . Despite the fact that (5.7) is a convex optimization problem, a direct solution is generally overly complicated since R does not admit in general a simple characterization. For example,R is generally not a polytope, and may be described by an uncountable number of linear constraints (supporting hyperplanes). Fortunately, we can use the framework of [38] and obtain a dynamic scheduling policy that operates arbitrarily closely to the optimal point ¹ R ? . This is obtained in two steps: ¯rst, a dynamic scheduling policy that achieves the stability of transmission queues whenever the arrival rates are inside R is obtained. Then, we build \virtual queues" driven by appropriate \virtual arrival processes," such that their arrival rates are as close as desired to the desired rate point ¹ R ? . Our analysis extends the results in [38] to this new context and also provides a new and tighter bounding analysis for the queues, particularly for general (possibly negative) concave utilities that include the proportional fairness utility. We note that it may be possible to pursue utility optimization using the alternative stochastic approximation and °uid transformation approaches in [58, 5, 85, 29], although these approaches may not yield explicit queue bounds. Further, the stochastic approximation techniques in [58, 5] use an in¯nite running time average of transmission rates, whereas our approach does not require an in¯nite running time average and can thus adapt to system changes. 5.3.1 System stability Suppose that the data to be transmitted to user 1;:::;K arrive to the BS according to a stationary and ergodic vector-valued process A[t] = (A 1 [t];:::;A K [t]), with rate 95 vector ¸ = E[A[t]] (expressed in bit/channel use) and such that 0 · A k [t] · A max , 8 t, for some constant A max < 1. The BS maintains a transmission queue for each user, and we let Q k [t] denote the size of the k-th queue bu®er at the beginning of slot t. As described in Section 5.2, R k [t] bit/channel use are removed from queue k during slot t, i.e., R k [t] represents the instantaneous \service rate" of the k-th queue. De¯ning Q[t] = (Q 1 [t];:::;Q K [t]) and R[t] = (R 1 [t];:::;R K [t]), the queues evolution is described by the stochastic di®erence equation 3 Q[t+1]=maxf0;Q[t]¡R[t]g+A[t] (5.8) We have the following de¯nition [38]: De¯nition1. Adiscrete-timequeueQ k [t]is stronglystableiflimsup t!1 1 t P t¡1 ¿=0 E[Q k [¿]]< 1. The system is strongly stable if all queues k =1;:::;K are strongly stable. § For convenience, throughout this work we use the term \stability" to refer to strong stability. It can be shown [38] that if Q k [t] is strongly stable and A k [t] is uniformly bounded by a ¯nite constant A max , as in our case, then lim t!1 Q k [t]=t = 0 with prob- ability 1 and lim t!1 E[Q k [t]]=t = 0. These properties are referred to as rate stability and mean-rate stability, respectively. In particular, rate stability implies that the time average rate of bits going into the queue is equal to the time average rate of bits going out of the queue. 3 The function maxf¢;¢g is applied componentwise to vectors. 96 The system stability region is the the closure of the convex hull of all arrival rate points¸forwhichthereexistsafeasibleschedulingpolicythatachievessystemstability [38]. The following result yields both the system stability region and the dynamic scheduling policy that stabilizes the system for any arrival rate point inside the region: Theorem 5.3.1. Suppose the arrival vector A[t] is i.i.d. over slots with each entry uniformly bounded by some ¯nite constant A max , and that the joint channel state and CSI pair fH[t]; b H[t]g is i.i.d. over slots. 4 For the system de¯ned in Section 5.2, the system stability region coincides with the ergodic rate region R given in (6.6). Further- more, for any arrival rate point ¸ in the interior of R, the system is stabilized by the dynamic scheduling policy ° ¤ de¯ned as follows. ForQ2R K + and b H2C M£K , consider the signaling scheme with covariance matrices (§ ¤ 1 ( b H);:::;§ ¤ K ( b H)) and rate allocation vector r ¤ ( b H) given by the solution of the weighted sum-rate maximization: maximize P K k=1 Q k E h R k (H;§ 1 ;:::;§ K ;r)j b H i subject to P K k=1 tr(§ k )·P; § k ¸08k; r¸0 (5.9) Then, the dynamic scheduling policy ° ¤ chooses at each time t the signaling scheme de¯ned by (6.11) for the current queue states (i.e., for Q =Q[t]) and the current CSI (i.e., for b H= b H[t]). Proof. See Appendix F. 4 This result and the result of Theorem 5.3.2 are stated for the i.i.d. case, but they can be extended to jointly ergodic processes subject to some mild technical conditions by following the technique of [68]. We omit this extension for brevity. 97 Interestingly, the weighted sum-rate maximization in (6.11) de¯ning ° ¤ involves the conditional expected service rates for given CSI: in the absence of perfect CSI the BS schedules the users on the basis of the MMSE estimation (conditional mean) of their instantaneous service rates. We conclude this section with a note on the optimal rate allocationinthestabilitypolicy° ¤ . Undertheoptimisticrateassumption,risirrelevant since (5.5) does not depend on r. Under the outage rate assumption, using (5.4) we obtain the optimal rate allocation r ¤ for a given set of covariance matrices and CSI as the solution of (see also [56]): r ¤ k ( b H)=argmax r¸0 r h 1¡P ³ log(1+sinr k (H;§ 1 ;:::;§ K ))·rj b H ´i (5.10) 5.3.2 System optimization Going back to the original problem (5.7) and following [38], we build \virtual queues" with arrival rate K-tuple ¸ arbitrarily close to the desired optimal point ¹ R ? , although thelatterisnotknownapriori. Then,usingthestabilitypolicy° ¤ appliedtothevirtual queues, the system necessarily operates at a throughput point ¹ R¸¸ (componentwise domination). From the monotonicity of g(¢) we are guaranteed that the system will operate arbitrarily close to the optimal point. Speci¯cally, we de¯ne ° ? as follows: let V;A max > 0 be suitable constants. At each time t, let A k [t]=a k , where a=(a 1 ;:::;a K ) is the solution of max a:0·a k ·A max ;8k Vg(a)¡ K X k=1 a k Q k [t] (5.11) 98 Then, for given Q[t] and CSI b H[t] the signaling scheme ° ¤ given in (6.11) is applied, resulting in the service rates R[t]. Finally, the virtual queues are updated according to (6.10), with arrivalsA[t] given by (6.12) and service ratesR[t]. The performance of the scheduling policy ° ? is given by the following: Theorem 5.3.2. Suppose the joint channel state and CSI pairfH[t]; b H[t]g is i.i.d over slots. Consider the scheduling policy ° ? de¯ned above, for given constants V > 0 and A max > 0. Assume that g(¢) is concave and entry-wise non-decreasing and that there exists at least one point r 2 R with strictly positive entries such that g(r=2) > ¡1. Then: (a) The utility associated with the time average transmission rates achieved by ° ? satis¯es: liminf t!1 g Ã 1 t t¡1 X ¿=0 E[R[¿]] ! ¸g( ¹ R ? (A max ))¡C=V (5.12) where C ¢ = 1 2 Ã KA 2 max + K X k=1 E[log 2 (1+jh k [t]j 2 P=N 0 )] ! (5.13) and where ¹ R ? (A max ) denotes the solution of the problem (5.7) with the additional con- straint 0· ¹ R k ·A max for all k =1;:::;K. (b) For any point ¹ R 2 R such that 0 · ¹ R k · A max for all k, and for any value ¯2[0;1] we have: limsup t!1 1 t t¡1 X ¿=0 K X k=1 ¹ R k E[Q k [¿]]· C +V[g( ¹ R ? (A max ))¡g(¯ ¹ R)] 1¡¯ (5.14) 99 Thus, all queues Q k [t] are strongly stable. (c) Suppose that g(r)= P K k=1 g k (r k ), where for each k the function g k (r) is concave, non-decreasing, and satis¯es g k (0) > ¡1 and g k (r) · g k (0) + rº k , for some ¯nite constant º k that represents the maximum right derivative of g k (r). If Q k [0] = 0 for all k, then we have the deterministic backlog bound Q k [t]·Vº k +A max for all k and all t. Proof. See Appendix F. Theorem 5.3.2 implies that if A max is su±ciently large, such that A max ¸ ¹ R ? k for all k, then: liminf t!1 g Ã 1 t t¡1 X ¿=0 E[R[¿]] ! ¸g( ¹ R ? )¡C=V: Hence, the control parameter V can be chosen as large as desired to make the achieved utility arbitrarily close to the optimal utility g( ¹ R ? ) for the problem (5.7). This comes withatradeo®inthevirtualqueueaveragesizesthat, asseenfrom(6.15), growlinearly with V. The virtual queue sizes represent the di®erence between the virtual bits ad- mitted into the queues and the actual bits transmitted, and thus a®ect the time-scales required for the time averages to become close to their limiting values. 5.3.3 Proportional fairness and hard fairness scheduling We shall focus on two particularly relevant special cases for the system utility function g(¢) that re°ect useful forms of fairness. The proportional fairness schedulling (PFS) is 100 de¯ned by the utility function g( ¹ R) = P k log ¹ R k [93, 59]. In this case, the solution of (6.12) is given by: A k [t]=min ½ V Q k [t] ;A max ¾ (5.15) The hard fairness scheduling (HFS), uses the utility function g( ¹ R)=minf ¹ R 1 ;:::; ¹ R K g. Making use of an auxiliary variable ½, (6.12) can be re-stated as max 0·½·a k ·A max V½¡ K X k=1 a k Q k [t] (5.16) Solving ¯rst with respect to a for ¯xed ½ and then solving with respect to ½ we obtain A k [t]= 8 > > < > > : A max if V > P K k=1 Q k [t] 0 else (5.17) An interesting and new aspect of Theorem 5.3.2 is that part (b) allows us to ¯nd explicitly the tighter upper bound on the virtual queue sizes. For example, consider the proportionalfairnessutility,chooseasu±cientlylargeA max suchthatA max ¸ ¹ R ? k forall k,andconsiderthevector ¹ R= ¹ R ? in(6.15). Then,wehaveg(¯ ¹ R ? )=Klog(¯)+g( ¹ R ? ) and the bound in Theorem 5.3.2 part (b) becomes C¡VKlog(¯ ¤ ) 1¡¯ ¤ , where ¯ ¤ is the unique solution in [0;1] of the equation log¯+ 1 ¯ =1+ C KV . For the hard fairness utility, choosing again A max ¸ ¹ R ? k for all k and letting ¹ R= ¹ R ? in the bound, we have g(¯ ¹ R ? ) = ¯g ? , where g ? is the max-min ergodic per-user rate. Hence, the bound in Theorem 5.3.2 part (b) becomes simply C +Vg ? . 101 5.4 Algorithms Inordertoimplement° ? , asimpleandeasilycomputableexpressionfortheconditional average service ratesE[R k (H;§ 1 ;:::;§ K ;r)j b H] is needed. Unfortunately, this is gen- erally not available even in the case where the joint statistics of H and b H is known and easy to characterize (e.g., when H and b H are jointly Gaussian). Another di±culty in the implementation of ° ? is that the weighted sum rate maximization in (6.11) is generally a non-convex problem for the linear precoding signaling schemes at hand. To overcome these di±culties, we introduce some approximations. We start by considering the ZFBF scheme reviewed in Section 5.2. In this case, the maximization in (6.11) reduces to the maximization over all active sets U µ f1;:::;Kg such that jUj·M of the weighted sum-rate S(U;p;Q; b H)= X k2U Q k E " log Ã 1+ jh H k v k j 2 p k N 0 + P j2U:j6=k jh H k v j j 2 p j !¯ ¯ ¯ ¯ ¯ b H # (5.18) where p = fp k g and v k are the ZFBF vectors de¯ned in Section 5.2. Notice that fv k : k 2 Ug and p are functions of b H, even though we omit the explicit dependence for the sake of notation simplicity. Also, it is understood that p k =0 for all k = 2U. Motivatedbythe¯ndingsonchannelstatepredictionerrormentionedinSection 6.1 and illustrated in Section 6.5, we make the working assumption that the users can be partitioned into two classes: the subset K pr of users with very small channel state pre- diction error (\predictable users") and the subsetK npr of users with very large channel state prediction error (\non-predictable users"). In order to develop some intuition, we 102 assume the CSI model h k = b h k +e k where the CSI estimation error e k is statistically independent of b h k , with mean zero and variance ¾ 2 e per component. Furthermore, we restrict to the on-o® power allocation p k = P=jUj£1fk 2 Ug, that is known to yield near-optimal sum-rate for the optimal choice of the active user set U and su±ciently large P=N 0 . We wish to understand whether a given user should be included in the active set in the maximization of (5.18). For this purpose, we evaluate the gap between the actual service rate of user k and the service rate user k would achieve if the BS has perfect knowledge of h k . We will evaluate this gap under both assumptions k 2 K pr (corresponding to ¾ 2 e ¼ 0) and k 2 K npr (corresponding to ¾ 2 e ¼ 1), and eventually conclude that if a user k2K npr is to be served, then no other user should be served in the same slot. Suppose that a genie provides the true channel vector h k to the BS. Then perfect zero-forcingtouserk ispossible. Wewilldenoteby b H genie k thegenie-aidedCSIobtained by replacing b h k with h k in the CSI matrix b H. The beamforming vectors computed using the genie-aided CSI are denoted by fv genie j : j 2Ug, and have the property that h H k v genie j = 0 for j 6= k, since h k is known perfectly. In general, v genie k 6= v k unless jUj=M. 103 The conditional expected service rate under the augmented CSI for user k is given by R k ( b H genie k ) = E " log Ã 1+ jh H k v genie k j 2 P N 0 jUj+ P j2U:j6=k jh H k v genie j j 2 P !¯ ¯ ¯ ¯ ¯ b H;h k # = log Ã 1+ jh H k v genie k j 2 P N 0 jUj ! ; (5.19) while in the case of the actual CSI we have R k ( b H)=E " log Ã 1+ jh H k v k j 2 P N 0 jUj+ P j2U:j6=k je H k v j j 2 P !¯ ¯ ¯ ¯ ¯ b H # (5.20) De¯ning the conditional rate-gap as ¢R k ( b H;h k ) = R k ( b H genie k )¡R k ( b H), using (5.19) and (5.20), the monotonicity of log(¢) and Jensen's inequality, after simple algebra we obtain the rate-gap upper bound: ¢R k ( b H;h k ) · log Ã 1+ jh H k v genie k j 2 P N 0 jUj ! ¡E · log µ 1+ jh H k v k j 2 P N 0 jUj ¶¯ ¯ ¯ ¯ b H ¸ | {z } £ k +log µ 1+ ¾ 2 e (jUj¡1)P N 0 jUj ¶ By the properties of the Moore-Penrose pseudo-inverse (5.2), we have thatE[£ k ] ¸ 0 where equality holds exactly whenjUj=M and approximately when ¾ 2 e ¼0. It follows that if k 2K pr , then ¢R k ( b H;h k )¼ 0 with high probability, i.e., for predictable users the gap between perfect and non-perfect CSI is very small, as we may expect. In contrast, if k 2 K npr , then h k ¼ e k , independent of b h k . In this case, ¢R k ( b H;h k ) ¼ 104 £ k +log ³ 1+ ¾ 2 e (jUj¡1)P N 0 jUj ´ grows on average like log(P=N 0 ), unless we letU =fkg, i.e., we schedule only user k. The above argument leads to the following conclusions: a near-optimal scheduling policy should select at each slot t either a group of predictable users and serve them using ZFBF spatial multiplexing mode, or a single non-predictable user and serve it using space-time coding, that does not require CSI at the transmitter apart from the rate allocation. Operating along these guidelines, in all cases the rate-gap with respect to perfect CSI is a constant that does not grow withP=N 0 . ForU µK pr , the objective function in (5.18) becomes S pr (U;p;Q; b H)¼ X k2U Q k log µ 1+ jh H k v k j 2 p k N 0 ¶ (5.21) where, for each such subset, the ZFBF vectors fv k g are obtained as the normalized columnsoftheMoore-Penrosepseudo-inverse(5.2). Then,thepowerallocationvectorp is obtained from the standard water¯lling formula [96, 24]. If the number of predictable users is large, the near-optimal user selection algorithm of [24] can be used to avoid the combinatorialsearchoverallU µK pr . Forthenon-predictableusers, thecorresponding objective function is given by S npr (fkg;Q)= 8 > > < > > : Q k E h log ³ 1+ jh k j 2 P MN 0 ´i for optimistic rates Q k max r¸0 n r h 1¡P ³ jh k j 2 · 2 r ¡1 P=(MN 0 ) ´io for outage rates (5.22) 105 Notice that the rate allocation in the outage rate case is actually very simple: it is su±cient to know the CDF ofjh k j 2 , which is either known a priori or it can be learned \on-line" from the channel measurements at each UT. The the proposed simpli¯ed scheduling policy can be summarized as follows: for a desired concave non-decreasing utility function g(¢) of the ergodic rates, the virtual arrival processes and the corresponding virtual queues are de¯ned in Section 5.3.2, yielding queue bu®ersQ[t] at each scheduling slot t. The scheduler computes S pr max [t]= max UµK pr S pr (U;p;Q[t]; b H[t]) and S npr max [t] = max k2K npr S npr (fkg;Q[t]) and chooses to servethebestsubsetofpredictableusersifS npr max [t]·S pr max [t],orthebestnon-predictable user if S npr max [t]>S pr max [t]. 5.5 Results and discussion In this section we illustrate the performance advantages of the proposed MU-MIMO scheduling policies over a conventional \mismatched" PFS scheme that treats the exist- ing CSI as if it was perfect. The mismatched scheme computes the ZFBF vectors fv k g from the CSI matrix b H[t] as described in Section 5.2, and selects the active user subset by maximizing the mismatched weighted sum rate P K k=1 r k [t]=T k [t], where r k [t] = log ³ 1+ j b h k [t] H v k j 2 p k N 0 ´ and where the powers fp k g are computed by water¯lling [24]. The coe±cientsT k [t] represent time-averaged rates, that are updated according to the rule [93, 59] T k [t+1]=(1¡1=t c )T k [t]+(1=t c )R k [t] 106 where R k [t] denotes the actual service rate of user k, under the outage or optimistic rate assumption as de¯ned in (5.4) and in (5.5). It is well-known that this algorithm approximately maximizes P k log ¹ R k in the case of perfect CSI (i.e., when b H[t] = H[t] for all t), when t c is very large. InSection5.5.1,weconsideranidealizedsettingwhereallchannelsarei.i.d. Rayleigh fading, and where a subsetK pr of users have perfect CSI while the complement setK npr has channels completely unknown to the BS. Then, in Section 5.5.2 we considered the SCM channel model used in 3GPP standardization [3], and actual channel estimation and prediction schemes. Interestingly, even in this very realistic setting, similar perfor- mance trends are observed. 5.5.1 Rayleigh fading We consider a BS with M = 4 antennas and K = 8 users, with K npr =f1;2g. In this section we consider the extreme case where the channel of users 1 and 2 is completely unknown to the BS, while the channels of the other users are perfectly known. All channel vectors are i.i.d. across scheduling slots and in the antenna domain, with elements » CN(0;1) (independent block-fading with spatially white Rayleigh fading). Inthiscase, themismatchedschemecomputestheZFBFbeamformingvectorsforusers 3;:::;8 without any orthogonality constraint with respect to the channels of users 1 and 2, since the latter are unknown and isotropically distributed. We start by illustrating the e®ect of the constants V and A max on the scheduling performance. Fig. 5.1 shows the time evolution of the long-term time-average rates 107 achieved by the proposed approximation of the policy ° ? in the HFS case (max-min throughput), forA max =100 and V =100, V =1000 when SNR is 20dB. In agreement 0 2000 4000 6000 8000 10000 12000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t (Time) Rk(t) HFS V=1000 V=100 Figure 5.1: New HFS, A max =100, V =100 vs. V =1000. with Theorem 5.3.2, by increasing V the time response of the algorithm becomes slower whiletheachievedutilityfunctionvalueimproves. Ingeneral,thetwoparametersV and A max should be tuned using the bounds of Theorem 5.3.2 and are functions of channel statistics, of K and M and of the SNRP=N 0 . Next, we examine the ergodic sum rate and sum log-rate achieved by the new algo- rithmsunderPFSandHFS,andcomparetheirperformancewiththatofthemismatched PFSscheme. Figs. 5.2and5.3showtheschedulingalgorithmsperformanceversusSNR in dB, for both the optimistic and the outage rate assumption. The gain of the novel algorithms over mismatched PFS is very large, especially under the outage rate as- sumption. This fact is understood by considering Fig. 5.4, showing the users \activity 108 10 15 20 25 30 0 5 10 15 20 25 30 SNR (dB) ΣkRk New PFS−Optimistic New PFS−Outage New HFS−Optimistic New HFS−Outage Mismatched PFS−Optimistic Mismatched PFS−Outage Figure 5.2: Ergodic sum rate, Rayleigh fading. 10 15 20 25 30 −35 −30 −25 −20 −15 −10 −5 0 5 10 SNR (dB) Σ k log(R k ) New PFS−Optimistic New PFS−Outage New HFS−Optimistic New HFS−Outage Mismatched PFS−Optimistic Mismatched PFS−Outage Figure 5.3: Sum log ergodic rate, Rayleigh fading. fractions",i.e.,thefractionoftimeslotsinwhichagivenuserisactive. Themismatched PFS allocates a very large fraction of slots to the non-predictable users. This is because 109 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Users index Activity fractions (SNR = 20 dB), Outage Rates Assumption Figure5.4: ActivityfractionsatSNR=20dB,outagerateassumption,Rayleighfading (Black: Mismatched PFS, Grey: New PFS; White: New HFS). if some users have poor quality CSI and the scheduler does not take this explicitly into account, then the fairness induced by the PFS utility function forces the system to serve these users very often. Hence, the unpredictable users \drain" a large fraction of the system capacity despite the fact that there might be a large number of users with very good quality CSI. In contrast, the novel schemes treat the non-predictable users separately, and this has a very signi¯cant impact not only on the ergodic rates of these users, but also on the whole system sum rate. It is also interesting to notice that, under the proposed scheduling policies, the gap between optimistic rates and outage rates is very small. This indicates that any suitable fast rate adaptation (e.g., based on rateless 110 coding and/or incremental redundancy ARQ) has only a minor impact on the system performance with respect to a much simpler conventional ARQ scheme. 5.5.2 3GPP channel model and actual channel prediction schemes We run extensive experiments based on the so-called \Spatial Channel Model" (SCM) [3]. Thischannelmodelisnotblock-fadingandthechannelcoe±cientsvarycontinuously over time. Although this model is frequency selective, we considered a frequency-°at version of the channel corresponding to a single subcarrier of an OFDM system, for consistency with the rest of this chapter. For a generic user and antenna (indices are omitted), this channel model yields the time-varying channel coe±cients in the form h[i]= ´ X r=1 A r e j2¼³ r i (5.23) where´ isthenumberofimpingingscatteredwavefrontsarrivingatthereceiver(´ =20 is speci¯ed in [3]), A r are random complex amplitude coe±cients, ³ r is the Doppler frequency shift corresponding to the r-th wavefront, normalized by the signal band- width and i ticks at the symbol rate. In turns, the Doppler shifts are given by ³ r = f c v c T s cos(µ r ¡µ v ),wheref c isthecarrierfrequency,v isthemobilespeed,cdenoteslight speed, T s is the symbol interval, µ r is the angle of arrival (AoA) of the r-th wavefront, and µ v is the mobile azimuth direction. We assume that a set of M orthogonal downlink pilot symbols are sent by the BS every slot of T symbols. Each user estimates and predicts the channel on the next slot using the pilot symbols. After thorough comparisons of various channel estimation and 111 prediction schemes, not reported here for the sake of space limitation, we report here theresultsforthetwomostpromisingschemesintermsofperformanceversuscomplex- ity. The ¯rst scheme consists of a block-by-block prediction based on the parametric estimation of the parameters f´;A r ;³ r g in (5.23) using ESPRIT applied to blocks of N À 1 pilot symbols, as described in [98]. The second scheme is a classical Recursive Least-Squares (RLS), approximating a Wiener MMSE predictor for the channel vector sampled at the pilot-insertion rate 1=T [99], [75], [43]. In our simulations we considered a system with parameters given in Table 5.1, that corresponds to a single subcarrier of an OFDM system with 256 subcarriers and bandwidth 256£15KHz =3:84MHz. Table 5.1: System parameters for simulation Description Value 1=T s , symbol rate 15KHz f c , Carrier frequency 2:6GHz N, Number of pilot symbols 200 T, Pilot symbol spacing 20 d min , scattering distance 600m We compared the two prediction methods by considering the four possible di®erent scenarios of: 1) High speed (v = 75km/h) vs. low speed (v = 5km/h) mobiles, and 2) well-separatedandpackedAoAsoftheimpingingwavefronts. Aknownlimitationofany estimator of a linear combination of sinusoids in noise (see [73]) is that the estimation error increases sharply when the separation between some of the frequency components falls below some minimum resolution that depends on the number of pilots N. On the otherhand,theRLSpredictionerrordegradesasmax r j³ r jisnon-negligiblewithrespect to the pilot insertion rate 1=T. It follows that there exists a class of channels with both 112 high mobility and clustered AoAs for which all prediction methods essentially fail. This corresponds to the \non-predictable" users said before. We considered a BS with M = 4 antennas and K = 8 UTs. We report only the results for one scenario because of space limitation, but the same trend is observed in a variety of cases. We consider high-mobility users with ESPRIT parameter estima- tion/prediction, where users 1 and 2 have clustered AoAs (given in Table 5.3) and users 3;:::;8 have well-separated AoAs (given in Table 5.2). The simulation results are Table 5.2: Angles of arrival for well-separated case (in radians), µ v =4:4780 radians µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 µ 10 4.8328 5.2210 5.4479 5.6090 5.7340 5.8360 5.9223 5.9970 6.0629 6.1219 µ 11 µ 12 µ 13 µ 14 µ 15 µ 16 µ 17 µ 18 µ 19 µ 20 6.1765 6.2356 6.3015 6.3762 6.4625 6.5644 6.6895 6.8505 7.0774 7.4657 Table 5.3: Angles of arrival for packed case (in radians), µ v =0:6939 radians µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 µ 9 µ 10 3.7263 3.6717 3.7854 3.6127 3.8513 3.5468 3.9260 3.4721 4.0123 3.3858 µ 11 µ 12 µ 13 µ 14 µ 15 µ 16 µ 17 µ 18 µ 19 µ 20 4.1142 3.2838 4.2393 3.1588 4.4003 2.9977 4.6272 2.7708 5.0155 2.3826 obtained by keeping the AoAs ¯xed, and by averaging with respect to the amplitudes of the SCM model. Fig. 5.5 shows the average sum-rate for the various scheduling algorithms in this case. We notice that the results for these realistic channel models and actual channel estimation and prediction schemes are in agreement with those for the i.i.d. Rayleigh fading case. 113 10 15 20 25 30 2 3 4 5 6 7 8 9 10 11 SNR (dB) ΣkRk New PFS−Optimistic New HFS−Optimistic Mismatched PFS−Optimistic Figure 5.5: Average sum rate, SCM channel model, ESPRIT prediction, optimistic rates. 114 Chapter 6 Joint Scheduling and Hybrid-ARQ for MU-MIMO Downlink in the Presence of Inter-Cell Interference 6.1 Introduction High-rate data-oriented downlink schemes [10, 50] have been successfully deployed as an extension of 3G cellular standards (WCDMA and CDMA2000). These schemes are based on the results of [89, 90, 93], showing that the throughput (or \ergodic") sum-capacity of single-antenna multi-access (uplink) and broadcast (downlink) fading Gaussian channels is achieved by allocating opportunistically each time-frequency slot to the user with the best instantaneous channel conditions. In a multiuser setting, the sum-capacity is usually not the most meaningful measure of the system performance. Instead, maximizing the sum-throughput subject to some fairness constraint is more 115 desirable [93]. To this purpose, a downlink scheduling policy can be designed in order to maximize a suitable concave and component-wise monotonically increasing network utilityfunctionoverthesystem'sachievablethroughputregion(i.e.,theregionofachiev- able long-term average user rates). The network utility function is designed in order to capture the desired notion of \fairness" (e.g., proportional fairness, max-min fairness and, more in general, ®-fairness [64]). In the next generation of cellular systems (e.g., the so-called LTE-Advanced [1]), high-ratedata-orienteddownlinkschemeswillbecombinedwithmultiusermulti-antenna (MU-MIMO)transmissiontechniques[19,95],supportingspectrale±cienciesinthe10's of bits/sec/Hz [72, 34]. With MU-MIMO, the rate supported by each user is generally a function of all the user channel vectors, and depends on the type of MU-MIMO pre- coding [19, 95, 15]. In order to compute the transmitter precoder parameters (e.g., the beamforming steering vectors and the transmitted rates and powers), channel state in- formation at the transmitter (CSIT) is required. This can be accurately obtained using open and closed loop channel estimation and feedback schemes (the literature on this subject is overwhelming, for example, see [62, 16, 25, 57, 81] and references therein). In particular, scheduling with MU-MIMO and non-perfect CSIT was considered in [80],particularizingthegeneralstochasticoptimizationframeworkof[38]tothecaseofa single-cellsystemwithlinearZero-ForcingBeamforming(ZFBF)MU-MIMOprecoding, where CSIT is obtained via noisy channel estimation and prediction. 116 In this work we focus on a multi-cell environment with no inter-cell cooperation. For su±ciently slowly-moving user terminals it is possible to design training and feed- back schemes that achieve almost perfect CSIT [57, 81, 80]. Therefore, for simplicity we shall assume that each BS has perfect CSIT for its own users. In contrast, in a multi-cell system, inter-cell interference (ICI) emerges as another source of unavoidable uncertainty. (see[31,49]andreferencestherein). WhentheschedulersateachBSmake their own decisions independently, based only on the locally available CSIT relative to their own users, the ICI power seen at each user receiver changes on a slot-by slot basis in a random and unpredictable manner, depending on the scheduling decision made at all the interfering BSs. As a consequence, the instantaneous Signal to Interference plus Noise Ratio (SINR) \seen" at any given user receiver is a random variable. Thedecentralizedschedulingprobleminamulti-cellenvironmentcanbeformulated as a non-cooperative game: each BS (player) wishes to maximize its own utility func- tion over its own feasible throughput region. The players' strategies are all feasible scheduling policies. In addition, the throughput region of any given cell depends on the ICI power statistics seen at the users' receivers, which in turn depend on the schedul- ing policies applied at the interfering BSs. We show that when the individual network utility functions are concave the multi-cell decentralized scheduling game is a concave game and therefore Nash equilibria exist. InordertosolvethenetworkutilitymaximizationateachBS,forgivenICIstatistics, we apply the stochastic optimization framework of [38, 66, 68, 80]. A straightforward application of this approach yields a scheme based on variable-rate adaptive coding 117 at the physical layer, and conventional ARQ at the Logical Link Control (LLC) layer. We notice that similar approaches are included in several wireless standards such as EV-DO and HSDPA [12, 35, 60], and therefore this can be regarded as the base-line \conventional" approach. In order to improve upon the conventional approach, we propose a new method based on combining incremental redundancy Hybrid Automatic Retransmission reQuest (HARQ) [20] and MU-MIMO opportunistic scheduling. In the proposedscheme,eachuserfeedsbackthevalueoftheinstantaneousmutualinformation observed in the previous slot, that is used by the scheduler to update recursively the scheduler weights. We show that the throughput achieved by the proposed HARQ scheme approaches the throughput of a \virtual system", as if a genie provided non- causally the ICI values at each scheduling slot. However, we stress that the proposed scheme makes use of strictly causal information, and therefore requires no genie. 6.2 System setup We consider the downlink of a system with C > 1 cells. In each cell, a BS equipped with with M antennas transmits to K single-antenna users. The channel is assumed frequency °at 1 and constant over \slots" of length T À1 symbols (block-fading model 1 The generalization to MIMO-OFDM and frequency selective fading is immediate. 118 [13]). Any given channel use of the complex baseband discrete-time signal at the k-th user in cell c during slot t is described by y k;c [t]= p g k;c;c h H k;c;c [t]x c [t] | {z } desired BS + X c 0 6=c p g k;c;c 0h H k;c;c 0[t]x c 0[t] | {z } inter-cell interference +z k;c [t]; (6.1) where t ticks at the slot rate, (k;c) denotes user k in cell c, h k;c;c 0[t] 2 C M is the channelvectorfromthec 0 -thBSantennaarraytothe(k;c)-threceiverantenna,x c 0[t]2 C M is the signal transmitted by c 0 -th BS and z k;c [t] » CN(0;1) is the additive white Gaussian noise (AWGN) sample. The coe±cients g k;c;c 0 are distance-dependent path gains [40] that are assumed to be time-invariant over many slots. The BSs are sum- power constrained such that tr(§ c [t])· 1 for all t, where § c [t] =E[x c [t]x H c [t]] denotes the transmit covariance matrix. The actual channel SNR is included as a common scaling factor in the coe±cients g k;c;c 0. The channel vectors of users in cell c form the columns of the channel matrix H c [t] = [h 1;c;c [t];:::;h K;c;c [t]]2C M£K . We assume that all vectorsh k;c;c 0[t] are mutually independent with i.i.d. components»CN(0;1), for all distinct 4-tuples (t;k;c;c 0 ). Each BS c knows all time-invariant quantities relative to its own users and has perfect knowledge of H c [t] immediately before the beginning of slot t (perfect CSIT for the own users). Afeasibleschedulingpolicy° c forBScisapossiblyrandomizedstationaryfunction 2 that mapsH c [t] into the pair ° c (H c [t])=(§ c [t];r c [t]), wherer c [t]=(r 1;c [t];:::;r K;c [t]) is a rate allocation vector. We assume that the MU-MIMO precoder is based on linear 2 Using the theory developed in [67] we can show that restricting to stationary policies does involve any suboptimality in terms of the achievable throughput region. 119 ZFBF.Thisyieldsthetransmittedsignalvectorintheformx c [t]= P k2Sc[t] v k;c [t]u k;c [t], where S c [t] denotes the set of active users, i.e., users that are selected to be served on slot t and where u k;c [t] 2 C denotes the coded symbol for user (k;c), with power E[ju k;c [t]j 2 ] = P k;c [t]. The ZFBF steering vectors fv k;c [t] : k 2 S c [t]g are given by the unit-norm (normalized) k-th column of the Moore-Penrose pseudoinverse (e.g., see [16, 52, 25, 57, 80] and references therein) of the channel matrix restricted to the active users, i.e., to the columns fh k;c;c [t] : k2S c [t]g. It follows that the transmit covariance matrix takes on the form § c [t]= X k2S c [t] v k;c [t]v H k;c [t]P k;c [t]: (6.2) where non-negative coe±cientsfP k;c [t]:k2S c [t]g de¯ne the power allocation over the active users in cell c, and satisfy the power constraint P k2S c [t] P k;c [t]· 1. A necessary and su±cient condition for perfect zero-forcing of the intra-cell multiuser interference is that jS c [t]j · minfM;Kg. Without loss of generality, in the following we identify the set of active users S c [t] with those users with positive powers, i.e., P k;c [t] > 0 for k2S c [t] and P k;c [t]=0 for k = 2S c [t]. The ICI power at user (k;c) receiver in slot t is given by Â k;c [t]= X c 0 6=c g k;c;c 0h H k;c;c 0[t]§ c 0[t]h k;c;c 0[t] (6.3) 120 with mean given by Â k;c =E[Â k;c [t]]= P c 0 6=c g k;c;c 0tr(§ c 0[t]). The SINR at user (k;c) is given by sinr k;c [t] = g k;c;c ¯ ¯ ¯h H k;c;c [t]v k;c [t] ¯ ¯ ¯ 2 P k;c [t] 1+Â k;c [t] (6.4) We let R k;c [t] denote the instantaneous service rate of user (k;c) on slot t, measured in bits/channeluse. Thisisingeneralafunctionofsinr k;c [t],andthereforeofH c [t];§ c [t];Â k;c [t], andoftheallocatedrater k;c [t]. Wede¯nethek-thuserserviceratefunctionR k (g;H;Â;§;r), suchthatR k;c [t]=R k (g k;c;c ;H c [t];Â k;c [t];§ c [t];r c [t]). Let¡denotethesetofallfeasible scheduling policies and let ° ¡c =f° c 0 : c 0 6= cg denote the set of scheduling policies at all cells c 0 6=c. For ¯xed ° ¡c 2¡ C¡1 , the throughput of user (k;c) under the scheduling policy ° c is given by R k;c (° c ;° ¡c ) = liminf t!1 1 t t X ¿=1 R k (g k;c;c ;H c [¿];Â k;c [¿];° c (H c [¿])) = E[R k (g k;c;c ;H c ;Â k;c ;° c (H c ))] (6.5) where the e®ect of the policies at the interfering BSs is captured by the statistics of the ICI power process Â k;c [t], the limit holds almost surely because of stationarity 121 and ergodicity, and expectation is with respect to the joint distribution of the triple (H c [t];Â k;c [t];° c ). 3 The region of achievable throughputs for cell c is given by R c (° ¡c )=coh [ ° c 2¡ © R2R K + :R k ·E[R k (g k;c;c ;H c ;Â k;c ;° c (H c ))]; 8k ª (6.6) \coh" denotes \closure of the convex hull". Notice that R c (° ¡c ) depends on the other cells'schedulingpolicies° ¡c throughthejointprobabilitydistributionoftheICIpowers fÂ k;c :k =1;:::;Kg). Under our assumptions, the BSs operate in a decentralized way and in°uence each other only in terms of the generated ICI statistics (i.e., the joint cdfs fÂ k;c : k = 1;:::;Kg). Each BS wishes to maximize its own network utility function. This multi- objective optimization problem is formulated as a non-cooperative game [37, 36] that we nickname the multi-cell decentralized scheduling game, where each player (i.e., BS) c seeks to achieve maximize U c (R) subject to R2R c (° ¡c ) (6.7) where we assume that U c (¢) is a continuous, strictly concave and component-wise in- creasing utility function, re°ecting some suitable fairness criterion [64]. 3 With some abuse of notation, we denote by H c andfÂ k;c : k = 1;:::;Kg random variables whose joint distribution coincides with the ¯rst-order joint distribution of the processes Hc[t] and fÂ k;c [t] : k =1;:::;Kg, which is time-invariant by stationarity. 122 By de¯nition, for any given joint statistics of H c and of fÂ k;c : k = 1;:::;Kg, the maximum in (6.7) is achieved by some scheduling policy ° ? c . A Nash equilibrium of the decentralized scheduling game is a set of scheduling policies (also denoted, with some abuse of notation, by f° ? c : c = 1;:::;Cg) such that ° ? c is the solution to (6.7) when ° ¡c =° ? ¡c , for all c=1;:::;C. We have: Theorem 6.2.1. The decentralized scheduling game de¯ned above is a concave game and therefore has a Nash equilibrium. Proof. All players have the same strategy set ¡. This is a compact convex set due to the covariance trace constraint and to the fact that we can assume that the rate allocation vector is bounded in r c 2 [0;r max ] K for some constant r max . 4 Also, each c-th utility is a concave function of ° c for ¯xed ° ¡c . In order to see this, let R(° c ;° ¡c ) denote the throughput point of R c (° ¡c ) achieved by policy ° c for ¯xed ° ¡c , consider any two policies ° 0 c ;° 00 c 2¡ and de¯ne ° (¸) c as the policy that applies ° 0 c with probability ¸2 [0;1] and ° 00 c with probability ¹ ¸ = 1¡¸. Then, from the convexity ofR c (° ¡c ) and the concavity of U c (¢) we have that ¸U c (R(° 0 c ;° ¡c ))+ ¹ ¸U c (R(° 00 c ;° ¡c ))·U c (¸R(° 0 c ;° ¡c )+ ¹ ¸R(° 00 c ;° ¡c ))=U c (R(° (¸) c ;° ¡c )) Now, let ° = f° c : c = 1;:::;Cg and ° 0 = f° 0 c : c = 1;:::;Cg denote two vectors of scheduling policies and de¯ne the sum-utility function ½(°;° 0 ) = P C c=1 U c (R(° c ;° 0 ¡c )). SincethefunctionsU c (¢)arecontinuous(byassumption)andthethroughputvectorsare 4 This limitation does not involve any signi¯cant loss of generality if r max is su±ciently large, and always holds in practice since practical variable-rate coding has a ¯nite maximum rate. 123 continuous functions of the scheduling policies, it follows that ½(°;° 0 ) is a continuous function of (°;° 0 )2¡ C £¡ C and, for what said before, it is concave in° for any ¯xed ° 0 . These properties match exactly the assumption of Rosen Theorem [74]. Therefore, as a direct consequence of [74], the existence of a Nash equilibrium is proved. Since U c (¢) is component-wise increasing, it follows that the maximum of (6.7) is obtained for some ° ? c such that R(° ? c ;° ¡c ) is on the Pareto boundary of R c (° ¡c ). If the service rate function R k (g;H;Â;§;r) is strictly increasing in the power allocated to user k, then the Pareto boundary of R c (° ¡c ) is achieved by policies that satisfy tr(§ c [t]) = P k2S c [t] P k;c [t] = 1 with probability 1. In this case, any Nash equilibrium f° ? c : c = 1;:::;Cg must correspond to scheduling policies that achieve the power constraint with equality for all BSs. In Sections 6.3 and 6.4 we will focus on reference cell c, assuming that all other in- terferingcellsapplya¯xedarbitrarypolicy° ¡c (i.e., for¯xedandknownjointstatistics of the ICI powers at all users of cell c). We shall apply the theory developed in [38, 80] and provide a stochastic optimization algorithm that solves (6.7) to any desired level of approximation, for any given ICI powers statistics. 124 6.3 Schedulingwithadaptivevariable-ratecodingandARQ- LLC From now on we shall assume Gaussian random coding and consider speci¯c cases of service rate functions. In this case, we de¯ne the k-th user mutual information function as I k (g;H;Â;§)=log Ã 1+ g ¯ ¯ h H k v k ¯ ¯ 2 P k 1+Â ! (6.8) The mutual information at user (k;c) receiver on slot t is given by I k;c [t] ¢ =I k (g k;c;c ;H c [t];Â k;c [t];§ c [t]) We approximate the decoding error probability by the corresponding information outage probability (see [70, 13] for the information-theoretic motivations underlying this very common and very useful approximation). Namely, if the mutual information I k;c [t] is less than the scheduled coding rate r k;c [t], the decoder makes a decoding error with probabilitycloseto1,whileifI k;c [t]>r k;c [t]therandomcodingaverageerrorprobability is very close to 0. Therefore, for slot length T large enough, there exist \good" codes drawn from a Gaussian ensemble such that their block error probability is close to the information outage probabilityP(r k;c [t] > I k;c [t]). In this case, the user k service rate function is given by \outage rate" function [80] R k (g;H;Â;§;r)=r k £1fr·I k (g;H;Â;§)g (6.9) 125 In order to obtain the desired near-optimal scheduling policy, we apply the framework of [80]. We de¯ne the virtual queues 5 with bu®er state Q c [t] = (Q 1;c [t];:::;Q K;c [t]) and virtual arrival processes A c [t] = (A 1;c [t];:::;A K;c [t]). The virtual queues evolve according to the stochastic di®erence equations Q k;c [t+1]=maxf0;Q k;c [t]¡R k;c [t]g+A k;c [t]; k =1;:::;K (6.10) Then, we consider the adaptive policy de¯ned by: 1. For any given t, let the transmit covariance matrix § c [t] and the rate allocation vector r c [t] be the solution of maximize K X k=1 Q k;c [t]E[r k;c [t]£1fr k;c [t]·I k (g k;c;c ;H c [t];Â k;c [t];§ k;c [t])gjH c [t]] subject to tr(§ c [t])·1; r k;c [t]¸0 8k (6.11) 2. For suitable constants V;A max > 0, let the virtual arrival processes at time t be given by the solution of max 0·A k;c [t]·A max ;8k VU c (A c [t])¡ K X k=1 A k;c [t]Q k;c [t] (6.12) 5 ItisimportanttokeepinmindthatthevirtualqueueshavenothingtodowiththeARQtransmission bu®ers: they are used here as a tool to recursively update the weights of the the scheduling policy. 126 3. Update the virtual queues according to (6.10), with arrivals A c [t] given by (6.12) and service rates R k;c [t] given by (6.9) calculated for § c [t] and r c [t] solutions of (6.11). As stated in Theorem 6.3.1 below, the policy de¯ned above achieves the optimal point R ? c solution of (6.7) within any desired accuracy, depending on the constants V and A max . Neglecting the (small) degradation due to stochastic adaptation and quanti¯ed by Theorem 6.3.1, we shall refer to this policy as ° ? . Theorem 6.3.1. Assume i.i.d. channels and ¯xed joint statistics of the ICI powers fÂ k;c :k =1;:::;Kg. Assume that U c (¢) is concave and entry-wise non-decreasing, and that there exists at least one point r 2 R c (° ¡c ) with strictly positive entries such that U c (r=2)>¡1. Then, the scheduling policy ° ? c de¯ned above, for given constants V >0 and A max >0, has the following properties: (a) The utility achieved by ° ? satis¯es: liminf t!1 U c Ã 1 t t X ¿=1 R c [¿] ! ¸U c (R ? (A max ))¡·=V (6.13) where · ¢ = 1 2 Ã KA 2 max + K X k=1 E · log 2 µ 1+ g k;c;c jh k;c;c j 2 1+Â k;c ¶¸ ! (6.14) and where R ? c (A max ) denotes the solution of the problem (6.7) with the additional con- straint 0·R k;c ·A max for all k =1;:::;K. 127 (b) For any point R c 2R c (° c 0 :c 0 6=c) such that 0·R k;c ·A max for all k, and for any value ¯2[0;1] we have: limsup t!1 1 t t X ¿=1 K X k=1 R k;c E[Q k;c [¿]]· ·+V[U c (R ? c (A max ))¡U c (¯R c )] 1¡¯ (6.15) Thus, all virtual queues Q k;c [t] are strongly stable. 6 Proof. The proof follows verbatim from the results in [80] and it is not repeated here for brevity. As a corollary of Theorem 6.3.1, if A max is su±ciently large such that A max ¸ R ? k;c for all k, then ° ? c satis¯es liminf t!1 U c Ã 1 t t X ¿=1 R c [¿] ! ¸U c (R ? c )¡·=V: (6.16) Hence, the control parameter V can be chosen su±ciently large in order to make the achieved utility as close as desired to the optimal value U c (R ? ) of problem (6.7). This comes with a tradeo® in the virtual queue average sizes that, as seen from (6.15), grow linearlywithV. Thevirtualqueuesizesrepresentthedi®erencebetweenthevirtualbits admittedintothequeuesandtheactualbitstransmitted,andthusa®ectthetime-scales required for the time averages to become close to their limiting values. 6 A discrete-time queue Q k [t] is strongly stable if limsup t!1 1 t P t ¿=1 E[Q k [¿]] <1. The system is strongly stable if all queues k =1;:::;K are strongly stable. 128 6.3.1 Implementation The policy ° ? found before computes recursively the \weights" Q c [t] via (6.12) and (6.10) and, for each t, solves the weighted conditional average rate sum maximization (6.11). Problem (6.12) is a standard convex optimization problem the solution of which does not present any major conceptual di±culty and is found in closed form for the important cases of proportional fairness and max-min fairness (see [80]), corresponding to the choices U c (R) = P K k=1 logR k and U c (R) = min k R k , respectively. In contrast, solving(6.11)presentssomedi±culties. LettingF k;c (¢)denotethemarginalcdfofÂ k;c [t] and using (6.8), the objective function in (6.11) can be rewritten as P k2S c [t] Q k;c [t]r k;c [t]F k;c 0 B @ g k;c;c ¯ ¯ ¯h H k;c;c [t]v k;c [t] ¯ ¯ ¯ 2 P k;c [t] 2 r k;c [t] ¡1 ¡1 1 C A (6.17) The optimization in (6.11) is generally a non-convex problem that involves a combina- torial search over all subsets S c [t] µ f1;:::;Kg of cardinality · minfK;Mg and, for each candidate subset, the maximization of (6.17) with respect to r c [t] and the power allocation fP k;c [t] : k2S c [t]g. Since this optimization may be di±cult to compute, we propose the following suboptimal low-complexity two-step approach: Step 1) the active user subset and the corresponding power allocation are selected by assuming deterministic ICI powers, equal to their mean value Â k;c . Under this assumption, the problem is reduced to the well-known user selection with ZFBF, that can be solved using standard techniques based on quasi-orthogonal user selection and water¯lling (e.g., [24, 103, 47]). 129 Step2)forthetransmitcovariance§ c [t]obtainedinstep1, (6.17)isoptimizedwith respect to the rate allocation. This reduces to optimizing the outage rate separately for each k2S k;c [t] by letting r k;c [t]=arg max r¸0 ( r F k;c Ã g k;c;c ¯ ¯ h H k [t]v k;c;c [t] ¯ ¯ 2 P k;c [t] 2 r ¡1 ¡1 !) (6.18) where g k;c;c ¯ ¯ ¯h H k;c;c [t]v k;c [t] ¯ ¯ ¯ 2 P k;c [t] is ¯xed by Step 1. Notice that, both in the original problem and in the proposed low-complexity two- step approximated solution, only the marginal statistics of the ICI powersfÂ k;c [t]:k = 1;:::;Kgarerelevant. Thesemarginalstatisticscanbemeasuredbyeachuserterminal individually and fed back to the BS scheduler by some very low-rate feedback scheme. 6.4 Scheduling with incremental redundancy HARQ If a genie provides the BS scheduler with the values of the the mutual information fI k;c [t]:k =1;:::;Kg in a non-causal fashion, just before the beginning of slot t, then the optimal rate allocation would be, trivially, r k;c [t] = I k;c [t] for all k2S c [t], yielding zero outage probability. This \genie-aided" case was considered in [80] and referred to as\optimisticrate"allocation, althoughnoactualalgorithmtoapproachtheoptimistic throughput was given. Since for any non-negative random variable I and r >0 we have E[r1fr > Ig]·E[I], then the optimistic service rates provide an upper bound to the throughput of any system with the same signaling scheme (ZFBF and Gaussian codes) and given rate allocation. 130 In this section we show how to achieve the \optimistic" throughput without the aid of any genie. As a preliminary step, let's consider the following incremental redundancy HARQ scheme. The BS scheduler maintains a bu®er of information packets for each user in the cell. The size of user (k;c) packets is equal to b k;c bits per packet. Each packet is encoded into an in¯nite-length sequence of complex symbols. 7 The encoded sequence is partitioned into blocks of length T symbols. At each slot t, the scheduling policy computes § c [t] according to some rule to be found later. For all active users k2S c [t], if the most recent HARQ feedback message from user k is \NACK" (negative acknowledgement),thenthe¯rstnot-yettransmittedcodedblockofthecurrentpacketis transmitted. Otherwise, if the most recent received HARQ feedback message is \ACK" (positive acknowledgement), then the current packet is removed from the transmission bu®er of user k and the ¯rst coded block of next packet in the bu®er is transmitted. The (k;c)-th receiver stores in memory all the received slots for times ft : k 2 S c [t]g and attempts to decode the current packet at every newly received slot, by using all the available received slots. If decoding fails, NACK is sent back, otherwise ACK is sent back and the decoder memory is reset. Notice that the scheme does not require any genie-aided \look-ahead" of the instantaneous ICI power Â k;c [t], and makes use of time- invariant packet sizes b k;c . These may di®er from user to user but are independent of t. Forlateruse, wede¯nethe\¯rst-blockcodingrate"astheratio r k;c = b k;c T bits/channel use. 7 In practice, this rateless coding can be implemented by using Raptor codes [82]. 131 Next, we describe a scheduling rule, denoted again by ° ? c , that operates arbitrar- ily closely to the genie-aided throughput when combined with the HARQ scheme de- scribed above. At the end of each slot t, the active users k2S c [t] feed back both their ACK/NACKmessageandthemutualinformationI k;c [t]\seen"attheirreceiver. Then, ° ? c coincides with what given in Section 6.3, after the following two changes. 1) The virtual queues evolution equation (6.10) is replaced by Q k;c [t+1]=maxf0;Q k;c [t]¡I k;c [t]g+A k;c [t]; 8k (6.19) 2) The transmitter optimization (6.11) is replaced by maximize K X k=1 Q k;c [t]E[I k (g k;c;c ;H c [t];Â k;c [t];§ c [t])jH c [t]] subject to tr(§ c [t])·1 (6.20) In brief, the scheduler updates recursively its weights Q c [t] and computes the trans- mitted signal covariance § c [t] according to (6.20), as if it was operating on a virtual \genie-aided" system with instantaneous service rates I k;c [t]. The throughput region of the virtual genie-aided system, denoted by R genie c (° ¡c ), is given by (6.6), after re- placing the general rate function R k (¢¢¢) with the mutual information function I k (¢¢¢) de¯ned in (6.8). The performance of ° ? c for the genie-aided system is again given by Theorem 6.3.1, where R c [¿] in (6.13) is replaced by the vector of mutual informations I c [¿] = (I 1;c [¿;:::;I K;c [¿]) and where R ? (A max ) denotes the solution of (6.7) when 132 R c (° ¡c ) is replaced byR genie c (° ¡c ), with the additional constraint 0·R k;c ·A max for all k =1;:::;K. For su±ciently large A max , ° ? c yields: liminf t!1 U c Ã 1 t t X ¿=1 I c [¿] ! ¸U c (R genie;? c )¡·=V; (6.21) where R genie;? c is the utility-maximizing throughput point in the region R genie c (° ¡c ). At this point, it remains to be shown that the combination of the policy ° ? c with the incremental redundancy HARQ scheme yields a network utility as close as desired to the limit in (6.21). This is shown by the following: Theorem 6.4.1. Let R harq;? c = (R harq;? 1;c ;:::;R harq;? K;c ) denote the throughput achievable by the incremental redundancy HARQ protocol under scheduling policy ° ? c de¯ned above. For each user (k;c) and ² k;c >0 there exists a su±ciently large ¯rst-block rate r k;c such that R harq;? k;c ¸(1¡² k;c )R genie;? k;c . Proof. Consider user (k;c). Following the argument in [20], we can model the event of successful decoding as a \mutual information level-crossing event". Suppose that the transmissionofthecurrentpacketforuser(k;c)startsatslott start (i.e.,anACKwasfed backatslottimet start ¡1). Then,thecurrentpacketcanbesuccessfullydecodedatslot t ¸ t start if P t ¿=t start I k;c [¿] ¸ r k;c . Otherwise, a decoding error occurs with very high probability. As shown in [20, 32], the probability of undetected decoding error vanishes exponentially with T. Therefore, in the regime of large T, if P t ¿=t start I k;c [¿] < r k;c the decoding error is detected with arbitrarily high probability and a NACK is sent 133 back. Fig. 6.1 shows, qualitatively, the mutual information level-crossing and the corresponding successful decoding events of the (k;c) decoder. Notice that the mutual informationincrementisnon-negative,anditisexactlyzeroforall tsuchthatk = 2S c [t], i.e., when user (k;c) is not scheduled. k,c r k,c Accumulated mutual information ACK ACK ACK W (1) W (2) W (3) k,c k,c Figure 6.1: Qualitative plot of the mutual information level-crossing process that de- termines the decoding events of the HARQ protocol. The jumps of the accumulated mutual information process correspond to slot times at which user (k;c) is active. Consider the transmission of a long sequence of packets. Without loss of generality, assumethatthesystemstartsattimet start =1,denotebyN k;c [t]thenumberofsuccess- fuldecodingeventsofdecoder(k;c)uptotimetandletW k;c (1);W k;c (2);:::;W k;c (N k;c [t]) denote the corresponding \inter-ACK" times (see Fig. 6.1). Since at each successful decoding a \reward" of r k;c bit per channel use is delivered to the destination, the throughput of the HARQ protocols is given by R harq;? k;c = lim t!1 r k;c N k;c [t] P N k;c [t] n=1 W k;c (n)+¢ k;c [t] (6.22) 134 where¢ k;c [t]=t¡ P N k;c [t] n=1 W k;c (n)denotesthedi®erencebetweenthecurrenttimetand thetimeatwhichtheN k;c [t]-thsuccessfuldecodingoccurred. Undertheassumptionsof this paper, the system with HARQ protocol and scheduling policy ° ? c evolves according to a discrete-time, continuous-valued vector Markov process with state given by Q c [t] andbythevectorofaccumulatedmutualinformationsateachreceiver. Sincethevirtual queuesarestronglystable(seeTheorem6.3.1)andtheaccumulatedmutualinformations are bounded in [0;r k;c ], the process is stationary and ergodic. Therefore, the limit in (6.22) holds almost surely, and can be explicitly computed as follows: R harq;? k;c = lim t!1 r k;c 1 N k;c [t] P N k;c [t] n=1 W k;c (n)+ ¢ k;c [t] N k;c [t] = r k;c lim t!1 1 N k;c [t] P N k;c [t] n=1 W k;c (n)+lim t!1 ¢ k;c [t] N k;c [t] = r k;c E[W k;c ] (6.23) where W k;c is an integer-valued random variable with the same marginal distribution of the inter-ACK times. In order to determineE[W k;c ], consider the case t start =1 and de¯ne the event A k;c [t]= ( t X ¿=1 I k [¿] · r k;c ) (6.24) 135 Since the accumulated mutual information between two ACKs is non-decreasing, the following nesting condition holds: A k;c [t]µA k;c [t¡1]; 8t whereA k;c [0]=f0·r r;c g has probability 1. It follows that P(W k;c =t)=P(A k;c [t¡1];A k;c [t])=P(A k;c [t¡1])¡P(A k;c [t]); yielding the average inter-ACK time in the form E[W k;c ] = 1 X t=1 tP(W k;c =t) = 1+ 1 X t=1 P(A k;c [t]) (6.25) Owing to the complete formal analogy of results (6.23) and (6.25) with the throughput of HARQ considered in [20]), we can directly apply the limit proved in [20]: 8 lim r k;c !1 r k;c E[W k;c ] =E[I k (g k;c;c ;H c ;Â k;c ;§ c )] (6.26) 8 This result is indeed quite intuitive: when r k;c becomes large, then E[W k;c ] in- creases. Therefore, the accumulated mutual information divided by the number of slots 1 W k;c P W k;c ¿=1 I k (g k;c;c ;H c [¿];Â k;c [¿];§ c [¿]) converges to an ensemble average. It follows that in this limit the level crossing condition tends to become deterministic, and satis¯es (approximately) W k;c X ¿=1 I k (g k;c;c ;Hc[¿];Â k;c [¿];§c[¿])=W k;c r k;c Of course, this argument can be made rigorous by following in the footsteps of [20]. 136 In particular, as r k;c ! 1 the average inter-ACK time E[W k;c ] diverges to in¯nity linearly with r k;c . The analysis in [20] shows that, for any ´ k;c >0, R harq;? k;c ¸(1¡´ k;c )E[I k (g k;c;c ;H c ;Â k;c ;§ c )] (6.27) for all su±ciently large r k;c . The proof of Lemma 6.4.1 is ¯nally concluded by combining the result (6.26) with (6.21). By stationarity and ergodicity, under ° ? c we have that lim t!1 1 t t X ¿=1 I k;c [¿]=E[I k (g k;c;c ;H c ;Â k;c ;§ c )] holds almost surely. Since U c (¢) is component-wise increasing, (6.21) implies that for any ± k;c >0 there exist su±ciently large A max and V for which E[I k (g k;c;c ;H c ;Â k;c ;§ c )]¸(1¡± k;c )R genie;? k;c (6.28) By letting (1¡² k;c )=(1¡´ k;c )(1¡± k;c ) and using (6.27) and (6.28) Theorem 6.4.1 is proved. From the above proof it follows that the delay-throughput operating point of the incremental redundancy HARQ protocol can be chosen individually for each user by settingthethresholdvaluer k;c (or,equivalently,thesizeb k;c oftheinformationpackets). By making r k;c large, the average decoding delay D k;c =E[W k;c ] becomes large and the throughput approaches R genie;? k;c . 137 Also, we wish to stress the di®erence between the ARQ-LLC scheme described in Section 6.2 and the incremental-redundancy HARQ protocol illustrated in this section. TheARQ-LLCprotocolmakesuseofadaptivevariable-ratecodingatthephysicallayer, and removes or keeps in the transmission bu®er packets of information bits of variable size b k;c [t] = Tr k;c [t]. In contrast, the HARQ protocol make use of a ¯xed packet size b k;c (equivalent to ¯xed ¯rst-block rate r k;c ), but the e®ective service rate is adaptive by varying the decoding delay through the ACK/NACK mechanism. 6.4.1 Implementation The scheme previously proposed requires that each active user, at the end of each slot t, feedsbackamessageformedbyonebitforACK/NACKandbythevalueof I k;c [t]or, equivalently, the value of sinr k;c [t]. We notice that feeding back the instantaneous SINR is widely proposed in the literature on opportunistic downlink scheduling [79, 102] and it is referred to as Channel Quality Indicator (CQI). However, in the current literature the CQI is relative to the current slot, and it is used to select users and allocate the rate of a variable-rate coding scheme. In contrast, here the CQI refers to the past slot, and it is used to update the scheduler weights according to (6.19). Denoting again by F k;c (¢) the marginal cdf of Â k;c [t], the objective function in (6.20) can be rewritten as P k2S c [t] Q k;c [t] Z 1 0 log 0 B @1+ g k;c;c ¯ ¯ ¯h H k;c;c [t]v k;c [t] ¯ ¯ ¯ 2 P k;c [t] 1+z 1 C A dF k;c (z) (6.29) 138 While for any ¯xed user subset S c [t] the maximization of (6.29) with respect to the powers fP k;c [t] : k 2S c [t]g is a convex problem, the solution is not generally given by the simple water¯lling formula and it may be di±cult to compute since the cdfs F k;c (¢) are typically not known in closed form. A near-optimum low-complexity approximation consists of choosing § c [t] that maximizes the objective function lower bound P k2S c [t] Q k;c [t]log 0 B @1+ g k;c;c ¯ ¯ ¯h H k;c;c [t]v k;c [t] ¯ ¯ ¯ 2 P k;c [t] 1+Â k;c 1 C A (6.30) obtained by applying Jensen's inequality to (6.29). Notice that the maximization of (6.30) with respect to the transmit covariance matrix coincides with step 1 in the low- complexity approximation of the variable-rate coding/ARQ-LLC case of Section 6.3.1, and can be solved e±ciently using the methods in [24, 103, 47]. 6.4.2 Extremal ICI distributions The throughput performance of the HARQ scheme depends on the statistics of the ICI powers, which in turns depend on the scheduling policies ° ¡c at the interfering BSs. In this section we ¯nd extremal marginal statistics for the ICI powers that provide non- trivialinnerandouterboundstoR genie c (° ¡c )thatareindependentof° ¡c . Herewedrop the slot index t since all processes are stationary. We start with the following: 139 Lemma6.4.2. Forallfeasiblepolicies° c 0 :c 0 6=cthatsatisfytheinputpowerconstraint with equality and for all users k =1;:::;K, we have E[I k (g k;c;c ;H c ;Â k;c ;§ c )]·E[I k (g k;c;c ;H c ;Â k;c ;§ c )]·E[I k (g k;c;c ;H c ;e Â k;c ;§ c )] (6.31) where Â k;c = E[Â k;c ] = P c 0 6=c g k;c;c 0 and where e Â k;c = P c 0 6=c g k;c;c 0 ¯ ¯ ¯h H k;c;c 0 v 1;c 0 ¯ ¯ ¯ 2 is the ICI power at the (k;c) receiver when all interfering BSs c 0 6= c 0 schedule a single user in their own cell. Proof. The ¯rst inequality (lower bound) follows immediately from Jensen's inequal- ity applied to the convex function f(x) = log(1 + a b+x ) with a;b > 0, and by the fact that, by assumption, the interfering BSs use all their available power. In or- der to show the second inequality (upper bound), we use (6.2) in (6.3) and write Â k;c = P c 0 6=c g k;c;c 0 P j2S c 0 ® k;c;c 0 ;j P j;c 0, where ® k;c;c 0 ;j ¢ = jh H k;c;c 0 v j;c 0j 2 are random vari- ables independent of the SINR numerator jh H k;c;c v k;c j 2 P k;c . Since the ZFBF steering vectors v j;c 0 have unit norm and are independent of h k;c;c 0, the variables ® k;c;c 0 ;j are marginally identically distributed as central chi-squared with 2 degrees of freedom [41]. Also, notice that the ® k;c;c 0 ;j 's are statistically dependent for the same index c 0 , while f® k;c;c 0 ;j : j 2 S c 0g and f® k;c;c 00 ;j : j 2 S c 00g are group-wise mutually independent for c 0 6= c 00 . By assumption, P j2S c 0 P j;c 0 = 1 for all c 0 . Therefore, P j2S c 0 ® k;c;c 0 ;j P j;c 0 is a convex combination of identically distributed, possibly dependent, random variables. 140 The second inequality in (6.31) follows by repeated application of Jensen's inequality. Choose c 00 6=c. Then, using (6.8), we have E 2 6 4log 0 B @1+ g k;c;c ¯ ¯ ¯h H k;c;c v k;c ¯ ¯ ¯ 2 P k;c 1+Â k;c 1 C A 3 7 5· X j2S c 00 P j;c 00E 2 6 4log 0 B @1+ g k;c;c ¯ ¯ ¯h H k;c;c v k;c ¯ ¯ ¯ 2 P k;c 1+g k;c;c 00® k;c;c 00 ;j + P c 0 6=c;c 00g k;c;c 0 P j2S c 0 ® k;c;c 0 ;j P j;c 0 1 C A 3 7 5 = E 2 6 4log 0 B @1+ g k;c;c ¯ ¯ ¯h H k;c;c v k;c ¯ ¯ ¯ 2 P k;c 1+g k;c;c 00® k;c;c 00 ;1 + P c 0 6=c;c 00g k;c;c 0 P j2S c 0 ® k;c;c 0 ;j P j;c 0 1 C A 3 7 5 (6.32) where the equality in (6.32) follows from the fact that the ® k;c;c 00 ;j 's are identically distributed with respect to the index j. Next, pick c 000 6=c;c 00 , and apply the same steps to the last line of (6.32). After eliminating all convex combinations, the ¯nal upper bound coincides with the right most term in (6.31). As a corollary, we have the following interesting \robustness" result: Theorem 6.4.3. For any choice of the scheduling policies ° ¡c that satisfy the input power constraint with equality, we have R c µR genie c (° ¡c )µ e R genie c (6.33) 141 where R c is the region with deterministic ICI powers fÂ k;c g, 9 and where e R genie c is the region corresponding to random ICI powers fe Â k;c g. Furthermore, the gap between the inner and outer bounds in (6.33) is bounded by a constant that does not depend on the channel path coe±cients. Proof. The proof (6.33) follows directly as a consequence of Lemma 6.4.2. In order to show the bounded gap, we have to ¯nd some constant ¢, independent of fg k;c;c 0g, such that maxfr¡¢1;0g2R c for all points r2 e R genie c . To this purpose, pick a point r2 e R genie c correspondingtosomefeasibleschedulingpolicy° c forthegenie-aidedsystem. Applying the same sequence of input covariance matrices as determined by ° c , to the system with deterministic ICI powers, we certainly ¯nd a point R c (° c )2R c . Consider thethroughputofthek-thuserandletforconvenienceA ¢ =g k;c;c ¯ ¯ ¯h H k;c;c v k;c ¯ ¯ ¯ 2 P k;c . Then, by applying Jensen's inequality we have E h log ³ 1+ A 1+ P c 0 6=c g k;c;c 0® k;c;c 0 ;1 ´¯ ¯ ¯A i ¡log ³ 1+ A 1+ P c 0 6=c g k;c;c 0 ´ · log ³ 1+ P c 0 6=c g k;c;c 0 ´ ¡E h log ³ 1+ P c 0 6=c g k;c;c 0® k;c;c 0 ;1 ´i (6.34) The RHS in the above inequality is easily seen to be non-negative and component-wise increasing with respect to any coe±cient g k;c;c 0. Therefore, its maximum is obtained in the limit for all g k;c;c 0 !1 (in passing, we notice that this corresponds to considering 9 Notice that if the ICI powers were deterministic, then no genie or HARQ is needed and the system reduces to a collection of isolated cells, where each cell c has modi¯ed channel path gain coe±cients g k;c;c = g k;c;c 1+Â k;c . In this case, the throughput regionR c is achieved by the standard scheduling/resource allocation schemes with perfect state information and zero outage probability. 142 the interference-limited regime where SNR ! 1). In order to see that this limit is ¯nite, let g max =maxg k;c;c 0, then we have RHS of (6.34) · log(1+(C¡1)g max )¡E 2 4 log 0 @ 1+g max X c 0 6=c ® k;c;c 0 ;1 1 A 3 5 · ¡E 2 4 log 0 @ 1 C¡1 X c 0 6=c ® k;c;c 0 ;1 1 A 3 5 (6.35) · ¡E £ log ¡ ® k;c;c 0 ;1 ¢¤ (6.36) · °=ln(2) (6.37) where(6.35)followsbylettingg max !1,(6.36)followsbyapplyingJensen'sinequality to the convex function ¡logx and (6.37) follows by using the fact that ® k;c;c 0 ;1 is chi- squaredwith2degreesoffreedom,andusingthelimitlim ²#0 R 1 ² lnxe ¡x dx=¡°,where ° denotes the Euler-Mascheroni constant [4]. Theorem 6.4.3 has the following interesting consequence: consider the multi-cell de- centralizedschedulinggameundertheproposedincrementalredundancyHARQscheme, achieving the genie-aided throughput region in each cell. The performance of any given cell c (in terms of its network utility value) at any Nash equilibrium (° ? 1 ;:::;° ? C ) is bounded below and above by the solutions of (6.7) whenR genie c (° ? ¡c ) is replaced byR c and e R genie c , respectively. Thisfollowsfromthefactthat, asarguedattheendofSection 6.2, all Nash equilibria must achieve the power constraints with equality at each BS. 10 10 Notice that the mutual information function is strictly increasing with the SINR. 143 6.5 Numerical results We considered a simple one-dimensional cellular layout with unit width cells arranged on a line. BSs are located at integer positions c 2Z. In each cell c, users are placed on a uniform grid in positions u(k;c) = (2k¡ K ¡ 1)=(2K) + c, for k = 1;¢¢¢ ;K. The channel path gains are given by g k;c;c 0 = G 0 1+(ju(k;c)¡c 0 j C =±) º , where the modulo-C distanceju¡cj C = minfju¡c+zCj :z2Zg induces a torus topology that eliminates border e®ects and where º and ± are the propagation exponent and the 3dB breakpoint distance, respectively, and G 0 determines the received SNR at the cell edge [40]. We present results for a system with C = 18 cells, M = 2 antennas per BS, K = 36 users per cell and parameters G 0 = 60dB, ® = 3:0 and ± = 0:05. For the implementation of the policy ° ? c we chose parameters A max = 50, V = 50 and suboptimal low-complexity approximationsasexplainedinSections6.3.1and6.4.1,respectively. Asforthenetwork utility functions, we considered both proportional fairness and max-min fairness (see Section 6.3.1 and [64, 80] and references therein). In order to gather the ICI statistics, we run the same scheduling algorithm in all BSs and measure the empirical cdf of the ICIpowerateachuserlocationinthereferencecellc=0(sincethesystemiscompletely symmetric, all cells see the same ICI statistics). Figs. 6.2 and 6.3 compare user throughputs in cell c=0 under proportional fairness and max-min fairness, respectively. Thick dashed lines illustrate the throughput upper bounds of Theorem 6.4.3. Thin dashed lines correspond to the actual \genie-aided" rates achievable by the proposed HARQ scheme in the limit of in¯nite decoding delay. Solid lines show the throughput achieved by the HARQ scheme operating at ¯nite 144 −0.5 −0.25 0 0.25 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 User Location ¯ R k,0 (bits/channel use) Proportional Fairness Genie−aided Upper Bound (Th. 4) Genie−aided, Simulated ICI HARQ (97% of Genie−aided), Simulated ICI Mean ICI Lower Bound (Th. 4) ARQ−LLC, Simulated ICI Figure 6.2: Average-throughput, proportional fairness. average decoding delay for all users, by setting the parameters fr k;0 g such that each userachieves97%ofthegenie-aidedrates(in¯nitedelay). The\triangle"marksindicate the throughput lower bounds of Theorem 6.4.3. Finally, the \square" marks indicate the throughputs achieved by the conventional adaptive variable-rate coding with ARQ- LLC.Weobservethatunderbothfairnessobjectivefunctions,thethroughputsachieved by HARQ achieve a gain of more than 100% for the users at the edge of the cell in the proportional fairness case, and a throughput gain of more than 40% for all users in the max-min fairness case, with respect to the ARQ-LLC scheme. Figs. 6.4 and 6.5 illustrate the average throughput as a function of the average decoding delay for the HARQ scheme in the case of two speci¯c users: user (1;0) at 145 −0.5 −0.25 0 0.25 0.5 0 0.1 0.2 0.3 0.4 0.5 User Location ¯ R k,0 (bits/channel use) Max−min Fairness Genie−aided Upper Bound (Th. 4) Genie−aided, Simulated ICI HARQ (97% of Genie−aided), Simulated ICI Mean ICI Lower Bound (Th. 4) ARQ−LLC, Simulated ICI Figure 6.3: Average-throughput, max-min fairness. the left cell edge and (18;0) at the cell center, under proportional fairness and max- min fairness, respectively. The thick dashed lines show genie-aided rates. The solid lines are obtained by increasing ¯rst-block coding rate parameter r k;0 and computing averagedecodingdelayfrom(6.25)withP(A k;0 [t])obtainedbyMonteCarlosimulation. Notethatasr k;0 increases, also the delayE[W k;0 ] increasesand theHARQ throughputs approach the genie-aided throughputs, in agreement with Theorem 6.4.1. The \o" marks indicate the throughput-delay points at which the HARQ protocol achieves 70%, 80% and 90% of the genie-aided throughput based on simulations. For example, under proportional fairness, 90% of the genie-aided throughput can be achieved at users (1;0) and (18;0) with average decoding delays of about 57 and 126 slots, respectively. These 146 20 40 60 80 100 120 140 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E[W k,0 ] (slots) ¯ R k,0 (bits/channel use) Proportional Fairness 70% 70% 80% 80% 90% 90% User close to BS User at left cell edge Genie−aided Semi−analytic, HARQ Simulation, HARQ Figure 6.4: Average rate vs. decoding delay with proportional fairness for two sample users. points (obtained by full system simulation) are accurately predicted by the analytical formulas of Section 6.4 ¯tted with the Monte Carlo estimation of the probabilities P(A k;0 [t]). For K = 36 users per cell and M = 2 BS antennas, assuming that exactly M = 2 users are served in each slot, a round-robin scheduling with no outage (genie-aided rate allocation) would take an average delay of 18 slots. Remarkably, under proportional fairness, 90% of the genie-aided throughput can be achieved with about 57 slots of averagedelayforcenteruser. Thisisonly¼3timesthatofthegenie-aidedround-robin scheduling. For edge users, this is achieved with ¼ 126 slots of average delay for the 147 0 50 100 150 200 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 E[W k,0 ] (slots) ¯ R k,0 (bits/channel use) Max−min Fairness 70% 70% 80% 80% 90% 90% User close to BS User at left cell edge Genie−aided Semi−analytic, HARQ Simulation, HARQ Figure6.5: Averageratevs. decodingdelaywithmax-minfairnessfortwosampleusers. edge users, which is only 7 times that of round-robin. Under max-min fairness, both users (1;0) and (18;0) achieve genie-aided throughputs close to 0:25 bits/channel use. The decoding delay for the center user is larger than for the edge user due to the fact that center users are scheduled very rarely. For the 70% point, edge users achieve 0:16 bits/channel use with average delay of 18 slots while center users achieves a similar throughput of 0:18 bits/channel use with delay of 44 slots. Fig.6.6showstheexpecteddelay(top)andpercentageoftheoptimisticthroughput (bottom) versusthe UT location inthe reference cell for the proposedHARQ scheme in two cases: 1) a system where the HARQ rate parameters fr k;0 g are chosen in order to 148 −0.5 −0.25 0 0.25 0.5 0 20 40 60 80 100 120 140 User Location D k,0 (slots) 90% Delay Constrained Semi−analytic, HARQ Simulation, HARQ −0.5 −0.25 0 0.25 0.5 0.6 0.7 0.8 0.9 1 90% Delay Constrained User Location Rate Percentage Figure 6.6: Throughput-delay tradeo®s versus UT location for the HARQ system for di®erent scenarios. achieve 90% of the optimistic PFS UT throughputs; 2) a delay-constrained system, in whichtheHARQrateparametersfr k;0 garechosensoastomaximizeT arq k;0 subjecttoan average decoding delay not exceeding ¹ D k;0 = 40 slots. Again, simulation and analysis are in perfect agreement. Furthermore in the delay-constrained system, for some UTs, the choice of r k;0 that maximizes the UT throughput yields an average decoding delay thatisstrictlylessthan ¹ D k;0 . Thisisindirectagreementwiththenonmonotonicnature of the function T arq k;0 (D k;0 ). 6.6 Concluding remarks Inthisworkweconsidereddecentralizeddownlinkschedulinginamulti-cellenvironment withmulti-antennaBSs,wheretheschedulerateachBShasperfectCSITaboutitsown 149 users and statistical information about the ICI caused by the other cells. Since each BS modi¯es its transmit covariance matrix at every slot, the ICI powers experienced at the users' receivers are random variable. We addressed the scheduling problem in the presence of uncertain ICI powers in the framework of stochastic network optimization. A straightforward application of this framework yields a conventional scheme based on adaptive variable-rate coding at the physical layer, and ARQ at the Logical Link Control layer. Then, a new combination of the same stochastic network optimization framework with incremental redundancy Hybrid ARQ at the physical layer was shown to improve over the conventional scheme, and achieve a network utility arbitrarily close totheperformanceofagenie-aidedsystemthatcanscheduletheuserratesequaltothe (non-causally known) instantaneous mutual information on each slot. For this scheme, we also showed that all Nash equilibria of the multi-cell decentralized scheduling game yield network utility values that can be uniformly upper and lower bounded by virtual systems corresponding to \extremal" ICI statistics, where the lower bound corresponds tothecaseofdeterministicICIpowersequaltotheirmeanvalues, andtheupperbound corresponds to the case where all interfering BSs transmit to a single user at full power (rank 1 interfering covariance matrices). These bounds stay at a ¯xed gap that is inde- pendent of the cellular system con¯guration, i.e., of the channel path gain coe±cients and operating SNR. The proposed incremental redundancy HARQ can be implemented in practice by using Raptor codes [82] at the physical layer, and need no protocol over- head to communicate slot-by-slot rate allocation as in adaptive variable-rate coding. 150 Hence, the proposed HARQ scheme is both easier to implement and performs signif- icantly better than the conventional variable-rate coding scheme. Also, we hasten to say that our approach applies directly to a variety of possible con¯gurations, including di®erent MU-MIMO precoding schemes and network MIMO schemes with clusters of coordinated cells [18]. In this paper we considered the case of linear ZFBF and no cell clustering for the sake of clarity of exposition. The approach can also be extended to the case of non-perfect CSIT, following [80]. Here we focused on perfect CSIT for its simplicity and in order to focus on the random nature of ICI as the fundamental source of uncertainty in a multi-cell environment. 151 Chapter 7 Conclusions At the beginning of the thesis we have raised three questions, for which the subsequent chapters have provided answers. These questions were: ² Can we design an e±cient feedback scheme in MIMO-OFDM systems? ² IsitpossibletopredictthechannelstateinfuturetocompensatetheCSIfeedback delay? ² If the CSIT in not perfect, what is the e±cient scheduling scheme? ² In multi-cell setting with unknown inter-cell interference, what signaling scheme is optimal? InChapter3,wecomparedtheachievableergodicratesofthesystemunderdi®erent feedback schemes by restricting ourselves to ZFBF without user scheduling and assum- ing perfect feedback link. We proposed the \time-domain" channel quantized feedback, 152 in which users feed back the time-domain channel coe±cients. This scheme takes ad- vantage of the channel frequency correlation structure and outperforms other schemes. IntermsoforderofdecayforhighSNR,scalarquantizationofthetimedomainchannel coe±cients yields a very simple scheme that performs very close to perfect CSIT. Fur- thermore, time-domain scalar quantization is very simple to implement, and requires nocomplicatedconstructionofsphericalcodebooksandvectorquantizationalgorithms. Analog feedback with frequency-domain MMSE interpolation also yields very compet- itive performance at low complexity, although its rate gap remains bounded and does not vanish as SNR increases. In this chapter, a few important issues are not considered. One assumption was that the channel feedback is instantaneous which is not true in practice due to the delay in transmitting CSI from UTs to the BS. Another assumption wasthattheCSIisperfectatUTswhiletherewillbealwaysanonzeroestimationerror associated with measuring CSI at UTs. These issues have been considered in [16] for frequency°atchannelsandcanbetopicsoffutureworkonMIMO-OFDMCSIfeedback. Subsequently, in Chapter 4, we focused on channel prediction problem. We con- sidered 3GPP SCM channel model with di®erent estimation/prediction methods and showed that, as long as the angular spread is large, Doppler is not very important and even fast varying channels can be handled easily. For these channels, the parameters of the model can be estimated by using ESPRIT and the channel can be predicted very reliably. On the contrary, for small angular spread, if the user is low speed then the 153 channel varies slowly and Wiener prediction or even very simple piecewise constant ap- proximationworkverywellwhileforhighspeedusers,apparentlynoneofthesemethods work well. Next, in Chapter 5, we considered user selection and scheduling in MIMO-BC un- der non-perfect (noisy) CSIT. For the sake of simplicity, we assumed that scheduling is performed on each subband independently. We used the results from Chapter 4 and assumed that UTs can by classi¯ed into only two extreme classes of \non-predictable" (high-mobility and clustered angle of arrivals) and \predictable" (all other cases) ac- cording to their channel prediction MSE. We considered two di®erent fairness criteria, PFS and HFS, and solved the associated maximization problem by using a stochastic optimization approach. The resulting scheduling algorithm selects at each scheduling slot either a MU-MIMO downlink beamforming mode that performs spatial multiplex- ing to a subset of predictable users, or a single-user space-time coding mode that serves a single selected non-predictable user. The work in this chapter can be extended by considering multiple classes of UTs with di®erent CSIT qualities. Another possible ex- tension to this chapter is to consider other precoding strategies in contrastto the ZFBF assumption. Last, in Chapter 6, we considered joint operation of MU-MIMO with opportunistic user scheduling and ARQ in the presence of a random ICI power. We provided a new method to optimize the HARQ parameters jointly with a downlink scheduler and a MU-MIMO scheme. Our approach consists of running an ideal system model in parallel with the actual system and letting the ideal system compute the scheduler 154 parameters according to \optimistic" instantaneous rates. We showed that the long- term average rates achieved by the actual system, using HARQ, can indeed approach those of the ideal system at the cost of computable decoding delay. Since the scheduler parametersarecomputedonthebasisoftheideal\free-running"system,schedulingand ARQ operations are decoupled, making analysis possible. 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Submitted to IEEE Transactions on Wireless Communications, December 2008. 164 Appendices A Analog feedback proof For each UT k the channel estimation error on subcarrier n is given bye k [n]=H k [n]¡ b H k [n]. Since the noise and the fading process are spatially uncorrelated, we have that E £ e k [n]e H k [n] ¤ =¾ 2 e [n]I,where¾ 2 e [n]isthen-thdiagonalelementof§ e de¯nedin(3.10). In particular, 1 N P N¡1 n=0 ¾ 2 e [n]= 1 N tr(§ e )=¾ 2 e . From (3.1) we have that S§ H S H = N®§ h ® H where ® is the leftmost J£L block of the J£N matrix SF. Using this in (3.10) we can write 1 N tr(§ e ) = tr µ § h ¡½N§ h ® H ³ I+½N®§ h ® H ´ ¡1 ®§ h ¶ = tr µ § h h I+½N§ 1=2 h ® H ®§ 1=2 h i ¡1 ¶ (A1) wherethelastlinefollowsfromthematrixinversionlemma. Noticethat§ h isdiagonal. We let f¾ 2 [l] : l = 0;:::;L¡ 1g denote the sorted diagonal elements in decreasing order. Then, we let f¸ (i) : i = 0;:::;z¡ 1g, with z = minfJ;Lg, denote the non- zero eigenvalues of ®§ h ® H sorted in increasing order. The eigenvalues of the L£L matrix h I+½N§ 1=2 h ® H ®§ 1=2 h i ¡1 , sorted in decreasing order, are given by 1;:::;1 | {z } L¡z ; 1 1+N½¸ (0) ;:::; 1 1+N½¸ (z¡1) 165 Now, we use result H.1.g in [8, Ch. 9], stating that for anytwo n£n Hermitian positive semide¯nite matrices A and B, we have tr(AB)· P n i=1 ¸ i (A)¸ i (B) where ¸ i (A) and ¸ i (B) are eigenvalues of A and B sorted in the same order. It follows that 1 N tr(§ e )· L¡z¡1 X l=0 ¾ 2 [l] + L¡1 X l=L¡z ¾ 2 [l] 1+N½¸ (l¡L+z) (A2) For each UT k the channel estimation error on subcarrier n is given bye k [n]=H k [n]¡ b H k [n]. Since the noise and the fading process are spatially uncorrelated, we have that E £ e k [n]e H k [n] ¤ =¾ 2 e [n]I,where¾ 2 e [n]isthen-thdiagonalelementof§ e de¯nedin(3.10). In particular, 1 N P N¡1 n=0 ¾ 2 e [n]= 1 N tr(§ e )=¾ 2 e . We use the rate-gap expression (3.5), and ¯nd E h jI k [n]j 2 i = X j6=k E · ¯ ¯ ¯H H k [n]b v j [n] ¯ ¯ ¯ 2 ¸ P M = X j6=k E · ¯ ¯ ¯ b H H k [n]b v j [n]+e H k [n]b v j [n] ¯ ¯ ¯ 2 ¸ P M = M¡1 M P¾ 2 e [n] (A3) where the last line follows from the fact that b H H k [n]b v j [n]=0 for any j6=k from ZFBF, and thatb v j [n] and e k [n] are independent, due to the fact that b v j [n] is a deterministic 166 function of b H i [n] for i 6= j, and jb v j [n]j 2 = 1. Using this in (3.5) and using Jensen's inequality we obtain ¢R AF k · 1 N N¡1 X n=0 log µ 1+ M¡1 M P N 0 ¾ 2 e [n] ¶ · log µ 1+ M¡1 M P N 0 ¾ 2 e ¶ (A4) The desired expression (3.11) follows from (A2). 167 B RVQ proof Wecomputethevarianceoftheinterferencetermatfrequency n, whereweassumethat n;n 0 areinthesamecluster. UsingknownresultsontheaveragedistortionofRVQ[69], we can write E h jI k [n]j 2 i = X j6=k E · ¯ ¯ ¯H H k [n]b v j [n 0 ] ¯ ¯ ¯ 2 ¸ P M (a) = X j6=k E · ¯ ¯ ¯ ¡ c[n;n 0 ]H k [n 0 ]+· e k [n;n 0 ] ¢ H b v j [n 0 ] ¯ ¯ ¯ 2 ¸ P M (b) = X j6=k Ã ¯ ¯ c[n;n 0 ] ¯ ¯ 2 E h ¯ ¯ H k [n 0 ] ¯ ¯ 2 i E " ¯ ¯ H H k [n 0 ]b v j [n 0 ] ¯ ¯ 2 jH k [n 0 ]j 2 # +¾ 2 · e [n;n 0 ] ! P M (c) · X j6=k Ã ¯ ¯ c[n;n 0 ] ¯ ¯ 2 M¾ 2 H 2 ¡B=(M¡1) M¡1 +¾ 2 H (1¡jc[n;n 0 ]j 2 ) ! P M = ¾ 2 H P µ ¯ ¯ c[n;n 0 ] ¯ ¯ 2 2 ¡ B M¡1 +(1¡jc[n;n 0 ]j 2 ) M¡1 M ¶ (B1) where (a) follows from (3.14), (b) follows from the fact· e k [n;n 0 ] is zero mean Gaussian independent of H H k [n 0 ] andb v j [n 0 ] and that norm and direction of H k [n 0 ] and iondepen- dent, and (c) from ( Lemma 2 in [69]), the expression of the MMSE in terms of the correlation coe±cient c[n;n 0 ] and the fact thatE h jH k [n 0 ]j 2 i =M¾ 2 H since channels are spatially i.i.d.. The ¯nal result follows from (3.5) and from the fact thatjc(n;n 0 )j 2 depends only on the di®erence ± =n¡n 0 and it is periodic of period N 0 . 168 C KL proof LetH k [n]denotethevectorchannelofUTk atfrequencyn,and b H k [n]denoteitsrecon- structed version obtained from the quantization of h k;1 ;h k;2 ;:::;h k;M . By replicating what was done for the analog feedback case, we have that E h jI k [n]j 2 i = (M¡1)P M ¾ 2 e [n] (C1) where ¾ 2 e [n] denotes the quantization error per antenna at frequency n. The rate gap for this case can be upperbounded by ¢R KL,RWF,Limit k · 1 N N¡1 X n=0 log µ 1+ M¡1 M P N 0 ¾ 2 e [n] ¶ (a) · log Ã 1+ M¡1 M P N 0 1 N N¡1 X n=0 ¾ 2 e [n] ! = log µ 1+ M¡1 M P N 0 1 N E h jH k;1 ¡ b H k;1 j 2 i ¶ (b) = log µ 1+ M¡1 M P N 0 E h jh k;1 ¡ b h k;1 j 2 i ¶ = log µ 1+ M¡1 M P N 0 D ¶ (C2) where (a) follows from Jensen's inequality and (b) from (3.18). 169 D KL high SNR proof InhighSNRregimewehavethatalargenumberofquantizationbitspersymbolcanbe used, therefore ° becomes small so that, eventually, ° <min l ¾ 2 l for all l =0;:::;L¡1. In this limit, all path coe±cients are quantized with equal distortion °. Therefore, D =L° and from (3.20) we get ¢R KL,RWF,Limit k · log µ 1+ P N 0 M¡1 M L° ¶ (D1) where ° can be obtained from (3.19) as ° = 2 ¡R(D)=L Ã L¡1 Y l=0 ¾ 2 l ! 1=L (D2) Next, we use the geometric-arithmetic mean inequality and write the loser, but more appealing, upper bound Ã L¡1 Y l=0 ¾ 2 l ! 1=L · 1 L L¡1 X l=0 ¾ 2 l = 1 L ¾ 2 H Using this into (D1), we arrive at (3.21). 170 E Overestimating and underestimating model order Inthissectionweinvestigatethee®ectofoverestimatingthemodelorder. Firstconsider a single sinusoid as: H[k;µ]=A 1 e j! 1 k where A 1 is the real amplitude which is assumed to be known and the parameter vector to be estimated is µ =[! 0 ]. The CRLB can be obtained from e[k;µ]=varf b H[k;µ]g= @H H [k;µ] @µ I ¡1 (µ) @H[k;µ] @µ (E1) where @H[k;µ] @µ = @H[k;! 1 ] @! 1 =jkA 1 e j! 1 k (E2) The FIM in this case is 1£1 as I(µ) = 2 N 0 Re 2 4 N t ¡1 X q=0 @H ¤ [qD t ;µ] @! 1 @H[qD t ;µ] @! 1 3 5 = 2 N 0 Re 2 4 N t ¡1 X q=0 (¡jqD t A 1 )e ¡j! 1 qDt (jqD t A 1 )e j! 1 qDt 3 5 = 2 N 0 (D t A 1 ) 2 S 1 (E3) where S 1 = N t ¡1 X q=0 q 2 (E4) 171 Then, form (E1), (E2) and (E3), e[k;µ] = ³ ¡jkA 1 e ¡j! 1 k ´ µ 2 N 0 (D t A 1 ) 2 S 1 ¶ ¡1³ jkA 1 e j! 1 k ´ = N 0 2 k 2 D 2 t 1 P N t ¡1 q=0 q 2 (E5) Now consider two sinusoids as: H[k;µ]=A 1 e j! 1 k +A 2 e j! 2 k where again the amplitudes A 1 and A 2 are assumed to be known and the parameter vector isµ =[! 1 ;! 2 ]. The CRLB can be obtained from (E1) and now @H[k;µ] @µ = · @H[k;µ] @! 1 ; @H[k;µ] @! 2 ¸ T = jk h A 1 e j! 1 k ;A 2 e j! 2 k i T (E6) The FIM in this case is 2£2 with the following elements: [I(µ)] 1;1 = 2 N 0 Re 2 4 N t ¡1 X q=0 @H ¤ [qD t ;µ] @! 1 @H[qD t ;µ] @! 1 3 5 = 2 N 0 Re 2 4 N t ¡1 X q=0 (¡jqD t A 1 )e ¡j! 1 qDt (jqD t A 1 )e j! 1 qDt 3 5 = 2 N 0 D 2 t A 2 1 S 1 (E7) 172 and [I(µ)] 1;2 = 2 N 0 Re 2 4 N t ¡1 X q=0 @H ¤ [qD t ;µ] @! 1 @H[q;µ] @! 2 3 5 = 2 N 0 Re 2 4 N t ¡1 X q=0 (¡jqD t A 1 )e ¡j! 1 qDt (jqD t A 2 )e j! 2 qDt 3 5 = 2 N 0 D 2 t A 1 A 2 S 2 (E8) where S 2 = N t ¡1 X q=0 q 2 cos((! 1 ¡! 2 )qD t ) (E9) In a similar way, [I(µ)] 2;1 = 2 N 0 D 2 t A 1 A 2 S 2 (E10) and [I(µ)] 2;2 = 2 N 0 D 2 t A 2 2 S 1 (E11) Therefore, I(µ)= 2 N 0 D 2 t 2 6 6 4 A 2 1 S 1 A 1 A 2 S 2 A 1 A 2 S 2 A 2 2 S 1 3 7 7 5 (E12) Note that the Fisher information for the estimation of the frequencies tends to be singular as ! 1 ! ! 2 since from (E4) and (E9), S 2 ! S 1 and therefore CRLB for for the parameter estimation does not exist. This fact is shown in Fig. E1. As it can 173 be seen from the this ¯gure when the frequency separation is less than 1=(N t D t ) the FIM becomes singular. Note that although the FIM becomes singular and there order −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0 100 200 300 400 500 600 700 800 ω 1 − ω 2 Cond(I) Figure E1: Conditional number ofI(µ) as a function of ! 1 ¡! 2 when N t =10, D t =10 and N 0 =1 reductionintheprobleminthiscase,thechannelitselfcanstillbeestimated/predicted. In the following we try to ¯nd the limit of the CRLB as ! 1 ! ! 2 . The inverse of FIM can be computed as I ¡1 (µ)= N 0 2 1 D 2 t 1 A 2 1 A 2 2 (S 2 1 ¡S 2 2 ) 2 6 6 4 A 2 2 S 1 ¡A 1 A 2 S 2 ¡A 1 A 2 S 2 A 2 1 S 1 3 7 7 5 (E13) From (E1), (E6) and (E13), e[k;µ] = N 0 k 2 D 2 t (S 1 ¡S 2 cos((! 1 ¡! 2 )k)) S 2 1 ¡S 2 2 (E14) 174 Now lim ! 1 !! 2 e[k;µ] = lim ! 1 !! 2 N 0 k 2 D 2 t S 1 ¡S 2 cos((! 1 ¡! 2 )k) S 2 1 ¡S 2 2 = N 0 k 2 D 2 t lim ! 1 !! 2 ³ P N t ¡1 q=0 q 2 ´ ¡ ³ P N t ¡1 q=0 q 2 cos((! 1 ¡! 2 )qD t ) ´ cos((! 1 ¡! 2 )k) ³ P N t ¡1 q=0 q 2 ´ 2 ¡ ³ P N t ¡1 q=0 q 2 cos((! 1 ¡! 2 )qD t ) ´ 2 = N 0 k 2 D 2 t lim h!0 ³ P N t ¡1 q=0 q 2 ´ ¡ ³ P N t ¡1 q=0 q 2 cos(hqD t ) ´ cos(hk) ³ P Nt¡1 q=0 q 2 ´ 2 ¡ ³ P Nt¡1 q=0 q 2 cos(hqD t ) ´ 2 = N 0 k 2 D 2 t lim h!0 ¡D t ³ P N t ¡1 q=0 q 3 sin(hqD t ) ´ cos(hk) ¡2D t ³ P Nt¡1 q=0 q 3 sin(hqD t ) ´³ P Nt¡1 q=0 q 2 cos(hqD t ) ´ +N 0 k 2 D 2 t lim h!0 ¡k ³ P N t ¡1 q=0 q 2 cos(hqD t ) ´ sin(2¼hkT sym ) ¡2D t ³ P Nt¡1 q=0 q 3 sin(hqD t ) ´³ P Nt¡1 q=0 q 2 cos(hqD t ) ´ = N 0 2 k 2 D 2 t 1 P N t ¡1 q=0 q 2 + N 0 2 k 3 D 3 t lim h!0 sin(hk) P N t ¡1 q=0 q 3 sin(hqD t ) = N 0 2 k 2 D 2 t 1 P Nt¡1 q=0 q 2 + N 0 2 k 3 D 3 t lim h!0 kcos(hk) D t ³ P N t ¡1 q=0 q 4 cos(hqD t ) ´ = N 0 2 k 2 D 2 t 1 P N t ¡1 q=0 q 2 + N 0 2 k 4 D 4 t 1 P N t ¡1 q=0 q 4 where the forth and seventh line follow from l'Hpital's rule. Note that the last line makes sense since it increases by increasing N 0 and k and decreases by increasing N t and D t . This shows that even though the Fisher information for the estimation of the frequencies tends to be singular as ! 1 ! ! 2 , the estimation of the channel coe±cient itself should not diverge. If (E15) is compared to (E5), one could see that there is a penaltyterm N 0 2 k 4 D 4 t 1 P N t ¡1 q=0 q 4 iftheorderofthesystemisoverestimatedanditisassumed that thesignal consistsof2 sinusoidswhileindeed it is onlyone. Figs. E2 comparesthe 175 CRLB for 1 and 2 sinusoids and the computed limit for di®erent frequency separations and Figs. E3 represent the CRLB for 1 sinusoid and 2 sinusoids when ! 1 ! ! 2 as a function of k. −0.05 −0.04 −0.03 −0.02 −0.01 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ω 1 − ω 2 MSE in prediction MSE, one sinusoid MSE, two sinusoid Limit, ω 1 goes to ω 2 FigureE2: ComparisonbetweenCRLBfor1and2sinusoidsandthelimitasafunction of ! 1 ¡! 2 when N t =10, D t =10, N 0 =1 and k =(N t ¡1)D t +1=91 90 100 110 120 130 140 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 k MSE in prediction MSE, one sinusoid Limit, ω 1 goes to ω 2 Figure E3: Comparison between CRLB for 1 sinusoid and 2 sinusoids in limit ! 1 !! 2 as a function of k when N t =10, D t =10, N 0 =1 176 Next we investigate the e®ect of underestimating the model order. We assume that the signal is sum of two sinusoids but it is estimated with a single sinusoid. In this case, the goal is to ¯t s[k]=Ae jÁ e j!k to H[k]=A 1 e j! 1 k +A 2 e j! 2 k based on the observations H[qD t ] for q =0;1;:::;N t ¡1. min A;! N t ¡1 X q=0 jH[qD t ]¡s[qD t ]j 2 This minimization can be written as min A;Á;! Nt¡1 X q=0 jH[qD t ]¡s[qD t ]j 2 = min A;Á;! Nt¡1 X q=0 jH[qD t ]j 2 ¡2Re 8 < : Nt¡1 X q=0 s[qD t ] ¤ H[qD t ] 9 = ; + N t ¡1 X q=0 js[qD t ]j 2 (E15) Therefore, the problem is to maximize 2Re n P N t ¡1 q=0 s[qD t ] ¤ H[qD t ] o ¡ P N t ¡1 q=0 js[qD t ]j 2 . This can be written as 2Re 8 < : N t ¡1 X q=0 s[qD t ] ¤ H[qD t ] 9 = ; ¡ N t ¡1 X q=0 js[qD t ]j 2 = 2ARe 8 < : e ¡jÁ N t ¡1 X q=0 H[qD t ]e ¡j!qD t 9 = ; ¡A 2 N t (E16) 177 To maximize this, Á should match the angle of P Nt¡1 q=0 H[qD t ]e ¡j!qD t . Therefore, the optimal value of Á, Á 0 , can be obtained from Á 0 = \ N t ¡1 X q=0 H[qD t ]e ¡j!qDt = \ N t ¡1 X q=0 ¡ A 1 e j! 1 qDt +A 2 e j! 2 qDt ¢ e ¡j!qDt = \ 8 < : A 1 N t ¡1 X q=0 e j(! 1 ¡!)qDt +A 2 N t ¡1 X q=0 e j(! 2 ¡!)qDt 9 = ; = \ ( A 1 1¡e j(! 1 ¡!)N t D t 1¡e j(! 1 ¡!)D t +A 2 1¡e j(! 2 ¡!)N t D t 1¡e j(! 2 ¡!)D t ) (E17) Consequently, the the optimal value of !, ! 0 , is obtained from ! 0 = argmax ! ¯ ¯ ¯ ¯ ¯ ¯ N t ¡1 X q=0 H[qD t ]e ¡j!qDt ¯ ¯ ¯ ¯ ¯ ¯ = argmax ! ¯ ¯ ¯ ¯ ¯ A 1 1¡e j(! 1 ¡!)N t D t 1¡e j(! 1 ¡!)D t +A 2 1¡e j(! 2 ¡!)N t D t 1¡e j(! 2 ¡!)D t ¯ ¯ ¯ ¯ ¯ (E18) Finally the optimal value of A, A 0 is obtained by setting the derivative of (E16) with respect to A to zero and A 0 = 1 N t ¯ ¯ ¯ ¯ ¯ A 1 1¡e j(! 1 ¡! 0 )N t D t 1¡e j(! 1 ¡! 0 )D t +A 2 1¡e j(! 2 ¡! 0 )N t D t 1¡e j(! 2 ¡! 0 )D t ¯ ¯ ¯ ¯ ¯ (E19) 178 F Nonperfect CSI proofs Proof of Theorem 5.3.1. First, we show that any arrival rate for which the system is strongly stable must be in R. Suppose that for some uniformly bounded i.i.d. process A(t)withrate¸,thereexistsapolicythatstabilizesthesystem. Usingthequeuebu®er evolution equation (6.10), assumingQ(0)=0 for simplicity, and summing with respect to ¿ =0;:::;t¡1 we obtain: Q(t)¸ t¡1 X ¿=0 A(¿)¡ t¡1 X ¿=0 R(¿) (F1) where R(t) denotes the service rate achieved by the policy. Dividing by t, taking expectations and rearranging terms we arrive at: 1 t t¡1 X ¿=0 E[A(¿)]· 1 t t¡1 X ¿=0 E[R(¿)]+ E[Q(t)] t (F2) Using the fact thatE[A(¿)] = ¸ for all ¿, we see that the left hand side of the above boundisequalto¸. Sincestrongstabilitywitha¯niteA max impliesmean-ratestability [38], it follows thatE[Q(t)=t]! 0, and so the ¯nal term in (F2) converges to the zero vector. Finally, we haveE[R(¿)]2R for all ¿, and hence 1 t P t¡1 ¿=0 E[R(¿)]2R for all t (as this is a convex combination of the vectors E[R(¿)], and R is a convex set). It followsthat¸isarbitrarilyclosetoapointinR. BecauseRisclosed, weconclude that ¸2R. 179 Then, inordertoshowthat° stab stabilizesthesystemforany¸intheinteriorofR, we will use the Lyapunov drift approach. Let L(Q) = 1 2 P K k=1 Q 2 k denote a Lyapunov function de¯ned onR K + . The corresponding one-step Lyapunov drift is given by ¢(Q(t))=E[L(Q(t+1))¡L(Q(t))jQ(t)] (F3) The following result is standard (see [38] and references therein): Fact F.1. If there exist constants C >0 and ²>0 such that ¢(Q(t))·C¡² K X k=1 Q k (t) (F4) then limsup t!1 1 t t¡1 X ¿=0 K X k=1 E[Q k (¿)]· C ² and hence each queue Q k (t) is strongly stable. § In order to show that (F4) holds in our case, we use (6.10) and write Q k (t+1) 2 · [Q k (t)¡R k (t)] 2 +A 2 k (t)+2A k (t)maxf0;Q k (t)¡R k (t)g · Q k (t) 2 +R k (t) 2 +A k (t) 2 ¡2Q k (t)[R k (t)¡A k (t)] (F5) Summing with respect to k and applying conditional expectationE[¢jQ(t)] we arrive at ¢(Q(t))· 1 2 K X k=1 E[R k (t) 2 +A k (t) 2 jQ(t)]¡ K X k=1 Q k (t)E[R k (t)¡A k (t)jQ(t)] (F6) 180 Observing that R k (t)·log(1+jh k (t)j 2 P=N 0 ), where the latter is the maximum achiev- able instantaneous rate for user k under perfect CSI as if it was alone in the system, it follows that 1 2 K X k=1 E[R k (t) 2 +A k (t) 2 jQ(t)]· 1 2 Ã KA 2 max + K X k=1 E £ log 2 (1+jh k (t)j 2 P=N 0 ) ¤ ! ¢ =C <1 (F7) Next, we shall use the following: LemmaF.2. Let the service ratesfR k (t)g be obtained by the application of the schedul- ing policy ° stab . Then, for any ¹ R2R, we have that K X k=1 Q k (t)E[R k (t)jQ(t)]¸ K X k=1 Q k (t) ¹ R k : (F8) Proof. Notice that R is a convex compact region in R K + . For any ¯xed non-negative weight vector Q, the maximum of the linear function P K k=1 Q k r k of r2R is achieved bysome°2¡(P). Hence, forany ¹ R2RandweightvectorQ(t), thereexists°2¡(P) such that K X k=1 Q k (t) ¹ R k · K X k=1 Q k (t)E[R k (H(t);°( b H(t)))] = K X k=1 Q k (t)E h E[R k (H(t);°( b H(t)))j b H(t);°] i · E " max § 1 ;:::;§ K ;r K X k=1 Q k (t)E h R k (H(t);§ 1 ;:::;§ K ;r)j b H(t) i ¯ ¯ ¯ ¯ ¯ Q(t) # = K X k=1 Q k (t)E h E h R k (H(t);° stab ( b H(t))) ¯ ¯ ¯ b H(t) i¯ ¯ ¯Q(t) i (F9) 181 where the second equality follows from the de¯nition of ° stab in (6.11). Since we assumed that the service ratesfR k (t)g are obtained by applying the policy ° stab , then, by de¯nition, E h E h R k (H(t);° stab ( b H(t))) ¯ ¯ ¯ b H(t) i¯ ¯ ¯Q(t) i = E[R k (t)jQ(t)], and the Lemma is proved. Now, let ¸ be in the interior of R and let the service rates fR k (t)g be obtained by ° stab . Then, there exists a ²>0 such that¸+²12R. Letting ¹ R=¸+²1 in (F8) and using Lemma F.2 we have K X k=1 Q k (t)E[R k (t)¡A k (t)jQ(t)] = K X k=1 Q k (t)(E[R k (t)jQ(t)]¡¸ k )¸² K X k=1 Q k (t) (F10) Using (F7) and (F10) in (F6) we ¯nd that the condition (F4) is satis¯ed under ° stab . ProofofTheorem5.3.2. Itisconvenienttode¯nethequantitiesA(t)= 1 t P t¡1 ¿=0 E[A(¿)] and R(t) = 1 t P t¡1 ¿=0 E[R(¿)], where A(t) and R(t) are the virtual arrival process and theserviceratevectorinducedbypolicy ° ? . Westartwithapreliminaryfact, theproof of which uses the general bound (F2) and the fact that strong stability and uniformly bounded arrival processes imply mean-rate stability (i.e.,E[Q(t)]=t!0) [38]. FactF.3. SupposequeuesQ(t)arestronglystableandthereisa¯niteupperboundA max on arrivals every slot. If g(¢) is a continuous and entry-wise non-decreasing function, then: liminf t!1 g(A(t)) · liminf t!1 g(R(t)) (F11) limsup t!1 g(A(t)) · g( ¹ R ? (A max )) (F12) 182 § From (F6), (F7) and Lemma F.2, we can write ¢(Q(t))·C¡ K X k=1 Q k (t) ¹ R k + K X k=1 Q k (t)E[A k (t)jQ(t)] (F13) where¢(Q(t))istheLyapunovdriftde¯nedin(F3),C istheconstantgivenin(F7)and ¹ R = ( ¹ R 1 ;:::; ¹ R K ) is any vector in R. Following the technique of [38, 66], we subtract a term related to the utility function from both sides of (F13) to yield: ¢(Q(t))¡VE[g(A(t))jQ(t)] · C¡ K X k=1 Q k (t) ¹ R k +E " K X k=1 Q k (t)A k (t)¡Vg(A(t)) ¯ ¯ ¯ ¯ ¯ Q(t) # Note from (6.12) that A(t) is chosen for every t to minimize the right hand side of the above inequality over all vectors a that satisfy 0· a k · A max for all k. Let z be any vector inR that satis¯es 0·z k ·A max for all k. Thus: ¢(Q(t))¡VE[g(A(t))jQ(t)] · C¡ K X k=1 Q k (t) ¹ R k + K X k=1 Q k (t)z k ¡Vg(z) Taking expectations of both sides of the above inequality and using the law of iterated expectations yields: E[L(Q(t+1))]¡E[L(Q(t))]¡VE[g(A(t))] · C¡ K X k=1 E[Q k (t)]( ¹ R k ¡z k )¡Vg(z) 183 Forsimplicity,assumethatQ(0)=0. Theaboveinequalityholdsforallt. Summingthe aboveover¿ 2f0;:::;t¡1g, dividingbyt, rearrangingterms, andusingnon-negativity ofL(¢) gives: 1 t t¡1 X ¿=0 K X k=1 E[Q k (¿)]( ¹ R k ¡z k )·C +Vg(A(t))¡Vg(z) (F14) where we have used Jensen's inequality in the concave function g(¢). The above holds for all t, all ¹ R 2 R, and all z 2 R such that 0 · z k · A max for all k. Parts (a) and (b) of Theorem 5.3.2 are proven by plugging di®erent values into (F14). We ¯rst prove part (b). Proof of part (b). Take any point x2R such that 0·x k ·A max for all k. Choose ¹ R=x and z=¯x, for any ¯2[0;1]. Then from (F14) we have: 1 t t¡1 X ¿=0 K X k=1 x k E[Q k (¿)]· C +Vg(A(t))¡Vg(¯x) 1¡¯ (F15) Atthispoint, we¯rstprovethatthequeuesarestronglystableandthen, usingFact F.3, we obtain part (b). Notice that g(A(t))· g(A max ), where A max is a vector with all entries equal to A max . Using this bound in (F15) and taking a limsup yields: limsup t!1 1 t t¡1 X ¿=0 K X k=1 x k E[Q k (¿)]· C +Vg(A max )¡Vg(¯x) 1¡¯ (F16) 184 By assumption, there exists at least one point r2R that has all positive entries and such that g(r=2)>¡1. Choosing ¯ =1=2 andx=r, it follows that the right-end side of (F16) is ¯nite and hence all queues are strongly stable. Because of strong stability and since the arrival processes are uniformly bounded by A max <1 by construction, we can apply inequality (F12) of Fact F.3 to the right-end side of (F15) and obtain the result of part (b). Proof of part (a). We plug ¹ R=z= ¹ R ? (A max ) into (F14) and obtain: g(A(t))¸g( ¹ R ? (A max ))¡C=V Taking liminf and using (F11) in Fact F.3 yields the result of part (a). Proof of part (c). In the special case when g(r) = P k g k (r k ), we have from (6.12) that for a given queue k and slot t, the arrivals A k (t) are chosen to maximize Vg k (A k (t))¡A k (t)Q k (t) subject to 0 · A k (t) · A max . Suppose that Q k (t) > Vº k . Then, we have Vg k (A k (t))¡A k (t)Q k (t)·Vg k (0)+(Vº k ¡Q k (t))A k (t)·Vg k (0) where equality is achieved if and only if A k (t)=0. Thus, queue backlog starts at 0 and cannotincreaseonanyslotwhenitcurrentlyexceedslevelVº k . Sincethequeuebacklog at any time cannot increase by more than A max , it follows that Q k (t)·Vº k +A max for all t. 185

Abstract (if available)

Abstract Multiple input multiple output (MIMO) communication systems and multiuser diversity play important rolls in future wireless systems and enable them to achieve high data rates. Earlier studies on multi-user MIMO (MU-MIMO) systems were based on some ideal assumptions on system such as perfect channel state knowledge, perfect feedback channel and known inter-cell interference. This thesis investigates design and performance of MU-MIMO systems under more practical assumptions.

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Asset Metadata

Creator Shirani-Mehr, Hooman (author)

Core Title Channel state information feedback, prediction and scheduling for the downlink of MIMO-OFDM wireless systems

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 04/30/2010

Defense Date 11/25/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag channel state information,feedback,hybrid ARQ,MIMO,OAI-PMH Harvest,scheduling,wireless

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Caire, Giuseppe (committee chair), Baxendale, Peter H. (committee member), Kuo, C.-C. Jay (committee member), Molisch, Andreas F. (committee member), Neely, Michael J. (committee member)

Creator Email hooman.shiranimehr@gmail.com,shiranim@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2976

Unique identifier UC1321209

Identifier etd-ShiraniMehr-3520 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-314564 (legacy record id),usctheses-m2976 (legacy record id)

Legacy Identifier etd-ShiraniMehr-3520.pdf

Dmrecord 314564

Document Type Dissertation

Rights Shirani-Mehr, Hooman

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu