This page intentionally left blank
Cooperative Cellular Wireless Networks A selfcontained guide to the stateoftheart in cooperative communications and networking techniques for nextgeneration cellular wireless systems, this comprehensive book provides a succinct understanding of the theory, fundamentals, and techniques involved in achieving efficient cooperative wireless communications in cellular wireless networks. It consolidates the essential information, addressing both theoretical and practical aspects of cooperative communications and networking in the context of cellular design. This onestop resource covers the basics of cooperative communications techniques for cellular systems, advanced transceiver design, relaybased cellular networks, and gametheoretic and microeconomic models for protocol design in cooperative cellular wireless networks. Details of ongoing standardization activities are also included. With contributions from experts in the field divided into five distinct sections, this easytofollow book delivers the background needed to develop and implement cooperative mechanisms for cellular wireless networks. Ekram Hossain is a Professor in the Department of Electrical and Computer Engineering at the University of Manitoba, Canada, where his current research interests lie in the design, analysis, and optimization of wireless/mobile communications networks. He serves as an Editor for IEEE Transactions on Mobile Computing, IEEE Communications Surveys & Tutorials, and IEEE Wireless Communications, and is an Area Editor for IEEE Transactions on Wireless Communications. Dong In Kim is a Professor and SKKU Fellow in the School of Information and Communication Engineering at Sungkyunkwan University (SKKU), Korea, and Director of the Cooperative Wireless Communications Research Center. He is currently an Editor for IEEE Transactions on Communications, an Area Editor for IEEE Transactions on Wireless Communications and coEditorinChief for Journal of Communications and Networks. Vijay K. Bhargava is a Professor in the Department of Electrical and Computer Engineering at the University of British Columbia, Canada. He has served on the Board of Governors of the IEEE Information Theory Society and the IEEE Communications Society and was President of the IEEE Information Theory Society. He is now the PresidentElect of the IEEE Communications Society and will serve as its President during 2012 and 2013.
“Edited by three of the most prominent experts in the field of cooperative communications, this is the defining book on this topic. It is a must have for any practicing researcher/engineer in this field.” Vahid Tarokh, Harvard University “Cooperative communications has been one of the most active areas of research in the communications field over the past decade. This research effort has now produced a significant body of work in the area, and this book is a valuable resource for students or practitioners wanting to enter the field, or simply to understand the scope and implications of the research.” H. Vincent Poor, Princeton University
Cooperative Cellular Wireless Networks Edited by EKRAM HOSSAIN University of Manitoba, Canada
DONG IN KIM Sungkyunkwan University, Korea
VIJAY K. BHARGAVA University of British Columbia, Canada
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521767125 C
Cambridge University Press 2011
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Cooperative cellular wireless networks / edited by Ekram Hossain, Dong In Kim, Vijay K. Bhargava. p. cm. Includes index. ISBN 9780521767125 (hardback) 1. Wireless communication systems. 2. Cell phone systems. I. Hossain, Ekram, 1971– II. Kim, Tongin, 1958– III. Bhargava, Vijay K., 1948– IV. Title. TK5103.2.C6625 2011 621.384 – dc22 2010048066 ISBN 9780521767125 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
For our families
Contents
List of contributors Preface
Part I Introduction 1
Network architectures and research issues in cooperative cellular wireless networks
page xvi xx 1
3
Aria Nosratinia and Ahmadreza Hedayat
1.1 1.2
1.3
1.4 1.5 2
Introduction Base station cooperation 1.2.1 Downlink cooperation 1.2.2 Uplink cooperation Dedicated wireless relays 1.3.1 IEEE 802.16j 1.3.2 Highspectraleﬃciency relay channels Mobile relays Conclusion
Cooperative communications in OFDM and MIMO cellular relay networks: issues and approaches
3 4 4 6 7 7 8 10 11
13
Mohammad Moghaddari and Ekram Hossain
2.1 2.2
Introduction Cooperative relay networks 2.2.1 Cooperative communication 2.2.2 Relay channel 2.2.3 Overview of relay protocols 2.2.4 Strategies of relayassisted transmission 2.3 General system model of cellular relay networks 2.4 General system model for virtual antenna arrays (VAAs) 2.5 RRA in OFDMAbased relay systems: general form 2.6 Dynamic RA RRA in OFDMA relay networks 2.6.1 Centralized RA RRA schemes in singlecell OFDMA relay networks
13 15 15 16 17 18 19 20 21 23 24
viii
Contents
2.6.2
Centralized RA RRA schemes in multicell OFDMA relay networks 2.6.3 Distributed RA RRA schemes in OFDMA relay networks 2.6.4 RA RRA schemes with fairness in OFDMA relay networks 2.7 Dynamic centralized margin adaptive RRA schemes in OFDMA relay networks 2.8 MIMO communications systems 2.8.1 RRA in MIMO relay networks 2.8.2 Optimal design and power allocation in singleuser singlerelay systems 2.8.3 Optimal design and power allocation in singlerelay multiuser systems 2.9 RRA in MIMO multihop networks 2.10 Conclusion
26 27 28 30 31 31 32 35 36 38
Part II Cooperative base station techniques
45
3
47
Cooperative base station techniques for cellular wireless networks Wibowo Hardjawana, Branka Vucetic, and Yonghui Li
3.1
3.2
3.3
3.4 3.5 3.6 3.7 3.8
3.9
4
Introduction 3.1.1 Related work 3.1.2 Description of the proposed scheme 3.1.3 Notation System model 3.2.1 Transmitter structure 3.2.2 THP precoding structure Cooperative BS transmission optimization 3.3.1 Iterative weight optimization (ﬁrst step) 3.3.2 Power allocation (second step) Modiﬁcation of the design of R Geometric mean decomposition Adaptive precoding order (APO) The complexity comparison of the proposed and other known schemes Numerical results and discussions 3.8.1 Convergence study 3.8.2 Performance of the individual links 3.8.3 Overall system performance Conclusion Appendix
Turbo base stations
47 47 48 49 49 49 51 53 55 57 58 59 60 61 63 64 65 66 70 72 77
Emre Aktas, Defne Aktas, Stephen Hanly, and Jamie Evans
4.1
Introduction
77
Contents
4.2
4.3
4.4
4.5
4.6 5
Review of message passing and belief propagation 4.2.1 Factor graph review 4.2.2 Factor graph examples Distributed decoding in the uplink: onedimensional cellular model 4.3.1 Hidden Markov model and the factor graph 4.3.2 Gaussian symbols Distributed decoding in the uplink: twodimensional cellular array model 4.4.1 The rectangular model 4.4.2 Earlier methods not based on graphs 4.4.3 Statebased graph approach 4.4.4 Decomposed graph approach 4.4.5 Convergence issues: a Gaussian modeling approach 4.4.6 Numerical results 4.4.7 Adhoc methods utilizing turbo principle 4.4.8 Hexagonal model Distributed transmission in the downlink 4.5.1 Main results for the downlink of a singlecell network 4.5.2 Main results for downlink of a multicellular network 4.5.3 BS cooperation schemes with message passing Current trends and practical considerations
Antenna architectures for network MIMO
ix
81 82 85 88 89 91 96 96 98 98 102 102 107 109 110 110 110 114 115 122 128
LiChun Wang and ChuJung Yeh
5.1 5.2 5.3
5.4
5.5
5.6 5.7
Introduction System model Network MIMO 5.3.1 ZF network MIMO transmission 5.3.2 ZFDPC network MIMO transmission Eﬀects of intergroup interference 5.4.1 SINR analysis 5.4.2 Example of IGI: network MIMO with omnidirectional cell planning 5.4.3 Unbalanced signal quality caused by IGI Frequencypartitionbased threecell network MIMO 5.5.1 Fractional frequency reuse (FFR) 5.5.2 FFRbased network MIMO with regular frequency partition 5.5.3 FFRbased network MIMO with rearranged frequency partition 5.5.4 Eﬀect of frequency planning among coordinated cells 5.5.5 Eﬀect of cell planning with diﬀerent sectorization Simulation setup numerical results Conclusion
128 130 132 133 133 134 134 134 135 136 136 138 140 142 143 145 147
x
Contents
Part III Relaybased cooperative cellular wireless networks
151
6
153
Distributed spacetime block codes Matthew C. Valenti and Daryl Reynolds
6.1 6.2 6.3 6.4
6.5
6.6
6.7 7
Introduction System model Spacetime block codes (STBCs) DF distributed STBC 6.4.1 Performance analysis 6.4.2 Numerical results AF distributed STBC 6.5.1 Performance analysis 6.5.2 Practical distributed STBC for AF systems The synchronization problem 6.6.1 Delay diversity 6.6.2 Delaytolerant spacetime codes 6.6.3 Spacetime spreading (STS) Conclusion
Collaborative relaying in downlink cellular systems
153 155 157 162 162 165 168 169 170 170 171 171 172 173 176
Chandrasekharan Raman, Gerard J. Foschini, Reinaldo A. Valenzuela, Roy D. Yates, and Narayan B. Mandayam
7.1
7.2 7.3 7.4
7.5
7.6
7.7
Introduction 7.1.1 Research challenges 7.1.2 Related work 7.1.3 Overview of contribution System model Collaborative relaying in cellular networks CPA with peak power transmissions (PCPA) 7.4.1 Principle of operation 7.4.2 User discarding methodology 7.4.3 Network operation and simulation aspects 7.4.4 Simulation results Powercontrolbased collaborative relaying (PCCPA) 7.5.1 Principle of operation 7.5.2 Optimization framework 7.5.3 User discarding methodology 7.5.4 Network operation and simulation aspects 7.5.5 Simulation results Orthogonal relaying 7.6.1 Network operation and simulation aspects 7.6.2 User discarding method 7.6.3 Simulation results Conclusion
176 176 178 179 180 184 186 186 188 190 190 192 192 194 197 198 199 199 200 201 202 202
Contents
8
Radio resource optimization in cooperative cellular wireless networks
xi
205
Shankhanaad Mallick, Praveen Kaligineedi, Mohammad M. Rashid, and Vijay K. Bhargava
8.1 8.2
8.3
8.4 8.5 9
Introduction Networks with single source–destination pair 8.2.1 Threenode relay network 8.2.2 Dualhop relay networks Multiuser cooperation 8.3.1 System model 8.3.2 Centralized power allocation 8.3.3 Distributed power allocation Relay selection Conclusion
Adaptive resource allocation in cooperative cellular networks
205 206 207 213 220 221 222 223 228 230 233
Wei Yu, Taesoo Kwon, and Changyong Shin
9.1 9.2
Introduction System model 9.2.1 Orthogonal frequencydivision multiplexing (OFDM) 9.2.2 Adaptive power, spectrum, and rate allocation 9.2.3 Cooperative networks Network optimization 9.3.1 Singleuser waterﬁlling 9.3.2 Network utility maximization 9.3.3 Proportional fairness 9.3.4 Rate region maximization Network with base station cooperation 9.4.1 Problem formulation 9.4.2 Joint scheduling and power allocation 9.4.3 Performance evaluation Cooperative relay network 9.5.1 Problem formulation 9.5.2 Joint routing and power allocation 9.5.3 Performance evaluation Conclusion
233 235 235 237 237 238 238 240 241 242 244 244 245 248 250 251 254 254 256
Crosslayer scheduling design for cooperative wireless twoway relay networks
259
9.3
9.4
9.5
9.6 10
Derrick Wing Kwan Ng and Robert Schober
10.1 Introduction 10.2 Crosslayer scheduling design – some basic concepts 10.2.1 Utility functionbased crosslayer optimization 10.2.2 Qualityofservice (QoS) measure 10.2.3 Multiuser diversity gain
259 263 264 266 266
xii
11
Contents
10.3 Network model for relayassisted OFDMA system 10.3.1 System model 10.3.2 Channel model 10.3.3 Channel state information (CSI) 10.4 Crosslayer design for twoway relayassisted OFDMA systems 10.4.1 Instantaneous channel capacity and system goodput 10.4.2 Crosslayer design problem 10.5 Crosslayer optimization solution 10.5.1 Transformation of the optimization problem 10.5.2 Dual problem formulation 10.5.3 Distributed solution – subproblem for each relay station 10.5.4 Solution of the master problem at the BS 10.6 Asymptotic performance analysis and computational complexity reduction scheme 10.6.1 Asymptotic analysis of system goodput 10.6.2 Scheme for reducing computational burden at each relay 10.7 Results and discussions 10.7.1 Convergence of the distributed resource allocation algorithm 10.7.2 Average system goodput vs. transmit power and user mobility 10.7.3 Asymptotic system goodput performance of PF scheduling 10.8 Conclusion Appendix
270 270 271 273 274 274 275 276 276 279 279 280
Green communications in cellular networks with ﬁxed relay nodes
300
281 281 283 283 284 284 289 290 292
Peter Rost and Gerhard Fettweis
11.1 Introduction 11.1.1 Two motivating examples 11.1.2 Scope and key problems 11.1.3 Outline and contributions 11.2 System model 11.2.1 Propagation scenarios 11.2.2 Air interface and scheduling 11.3 System and protocol design 11.3.1 Nonrelaying protocols 11.3.2 Relayonly protocol 11.3.3 An integrated approach 11.3.4 Simpliﬁcations 11.4 Numerical analysis 11.4.1 Simulation methodology 11.4.2 Throughput performance in the widearea scenario 11.4.3 Throughput performance in the Manhattanarea scenario 11.4.4 Femtocells vs. relaying 11.4.5 Computationtransmissionpower tradeoﬀ
300 300 302 303 303 303 305 306 307 307 308 309 309 309 310 312 313 315
Contents
12
xiii
11.4.6 Reduced backhaul requirements 11.4.7 Cost–beneﬁt tradeoﬀ 11.5 Conclusion
316 317 320
Network coding in relaybased networks
324
Hong Xu and Baochun Li
12.1 Introduction 12.2 Network coded cooperation 12.2.1 Simple network coded cooperation 12.2.2 Joint network and channel coding/decoding 12.3 Physicallayer network coding 12.4 Scheduling and resource allocation: crosslayer issues 12.5 Conclusion
324 326 327 331 334 337 341
Part IV Game theoretic models for cooperative cellular wireless networks
345
13
347
Coalitional games for cooperative cellular wireless networks Walid Saad, Zhu Han, and Are Hjørungnes
14
13.1 Introduction 13.2 A brief introduction to coalitional game theory 13.3 A coalition formation game model for distributed cooperation 13.3.1 Motivation and basic problem 13.3.2 Distributed virtual MIMO coalition formation game 13.4 Coalitional graph game among relay stations 13.4.1 Motivation and basic problem 13.4.2 A network formation game among relay stations 13.5 Conclusion
347 348 350 351 355 368 368 369 378
Modeling malicious behavior in cooperative cellular wireless networks
382
Ninoslav Marina, Walid Saad, Zhu Han, and Are Hjørungnes
14.1 Introduction 14.2 Cooperating jammers 14.2.1 System model 14.2.2 The game 14.2.3 Simulation results 14.3 Cooperating relays 14.3.1 System model 14.3.2 Secrecy capacity 14.3.3 Simulation results 14.4 Eavesdroppers cooperative model
382 384 385 386 393 398 399 400 404 407
xiv
Contents
14.4.1 Coalition formation games for distributed eavesdroppers cooperation 14.4.2 Simulation results 14.4.3 Conclusion
411 416 418
Part V Standardization activities
423
15
425
Cooperative communications in 3GPP LTEAdvanced standard Hichan Moon, Bruno Clerckx, and Farooq Khan
15.1 Introduction 425 15.2 LTE and LTEAdvanced 426 15.2.1 Carrier aggregation 428 15.2.2 Latency improvements 429 15.2.3 DL multiantenna transmission 430 15.2.4 UL multiantenna transmission 431 15.3 Cooperative multipoint transmission 431 15.3.1 Interference mitigation techniques in previous releases of LTE 432 15.3.2 Overview of CoMP techniques 433 15.3.3 Release 10 of LTEAdvanced 450 15.4 Wireless relay 451 15.4.1 Key technologies 452 15.4.2 Standard trends on Release 10 and future works 454 15.5 Heterogeneous network 454 15.5.1 Key technologies 454 15.5.2 Standard trends on Release 10 and future work 457 15.6 Conclusion 457 16
Partial information relaying and relaying in 3GPP LTE
462
Dong In Kim, Wan Choi, Hanbyul Seo, and ByoungHoon Kim
16.1 Introduction 16.2 Partial information relaying with multiple antennas 16.2.1 Perantenna superposition coding (PASC) 16.2.2 Multilayer superposition coding (MLSC) 16.2.3 Rate matching for superposition coding 16.2.4 Overall rate capacity 16.2.5 Features of partial information relaying 16.3 Analysis of PASC with zeroforcing decorrelation 16.4 Multinode partial information relaying 16.4.1 Twostage superposition coding 16.4.2 Successive decoding in cooperating phase 16.4.3 Relay selection for maximum capacity 16.5 Concluding remarks on partial information relaying 16.6 Relaying in 3GPP LTEAdvanced
462 463 465 466 468 469 470 470 474 475 477 477 479 480
Contents
16.6.1 Functionality of RNs 16.6.2 Separation of the backhaul and access links 17
Coordinated multipoint transmission in LTEAdvanced
xv
481 488 495
SungRae Cho, Wan Choi, YoungJo Ko, and JaeYoung Ahn
17.1 Introduction 17.2 CoMP architecture 17.2.1 Joint processing and transmission (JPT) 17.2.2 Coordinated scheduling and beamforming (CS/CB) 17.2.3 Cell clustering 17.2.4 IntereNodeB and intraeNodeB coordination 17.3 CoMP design parameters 17.3.1 Reference signal (RS) 17.3.2 Precoding 17.3.3 Feedback 17.4 CoMP performance evaluation methodologies 17.4.1 Link level simulation 17.4.2 System level simulation 17.5 Conclusion
495 496 497 497 497 499 499 499 501 502 504 504 506 509
Index
514
Contributors
JaeYoung Ahn Electronic Telecommunication and Research Institute (ETRI), Korea Defne Aktas Bilkent University, Turkey Emre Aktas Hacettepe University, Turkey Vijay K. Bhargava The University of British Columbia, Canada SungRae Cho Korea Advanced Institute of Science and Technology (KAIST), Korea Wan Choi Korea Advanced Institute of Science and Technology (KAIST), Korea Bruno Clerckx Samsung Electronics, Korea Jamie Evans University of Melbourne, Australia Gerhard Fettweis Technische Universit¨at Dresden, Germany Gerard J. Foschini Bell Laboratories, USA Zhu Han University of Houston, USA
List of contributors
Stephen Hanly National University of Singapore, Singapore Wibowo Hardjawana University of Sydney, Australia Ahmadreza Hedayat CISCO Systems, USA Are Hjørungnes UNIK – University of Oslo, Norway Ekram Hossain University of Manitoba, Canada Praveen Kaligineedi The University of British Columbia, Canada Farooq Khan Samsung Electronics, Korea ByoungHoon Kim LG Electronics, Inc., Korea Dong In Kim Sungkyunkwan University (SKKU), Korea YoungJo Ko Electronic Telecommunication and Research Institute (ETRI), Korea Taesoo Kwon Samsung Electronics, Korea Baochun Li University of Toronto, Canada Yonghui Li University of Sydney, Australia Shankhanaad Mallick The University of British Columbia, Canada Narayan B. Mandayam Rutgers University, USA
xvii
xviii
List of contributors
Ninoslav Marina Princeton University, USA Mohammad Moghaddari University of Manitoba, Canada Hichan Moon Samsung Electronics, Korea Derrick Wing Kwan Ng The University of British Columbia, Canada Aria Nosratinia University of Texas at Dallas, USA Chandrasekharan Raman Rutgers University, USA Mohammad M. Rashid The University of British Columbia, Canada Daryl Reynolds West Virginia University, USA Walid Saad UNIK – University of Oslo, Norway Robert Schober The University of British Columbia, Canada Hanbyul Seo LG Electronics, Inc., Korea Changyong Shin Samsung Electronics, Korea Matthew C. Valenti West Virginia University, USA Reinaldo A. Valenzuela Bell Laboratories, USA Peter Rost NEC Euro Labs, Germany
List of contributors
Branka Vucetic University of Sydney, Australia LiChun Wang National Chiao Tung University, Taiwan Hong Xu University of Toronto, Canada Roy D. Yates Rutgers University, USA ChuJung Yeh National Chiao Tung University, Taiwan Wei Yu University of Toronto, Canada
xix
Preface
Cooperative communications and networking represent a new paradigm which uses both transmission and distributed processing to signiﬁcantly increase the capacity in wireless communication networks. Current wireless networks face challenges in fulﬁlling users’ everincreasing expectations and needs. This is mainly due to the following reasons: lack of available radio spectrum, the unreliable wireless radio link, and the limited battery capacity of wireless devices. The evolving cooperative wireless networking paradigm can tackle these challenges. The basic idea of cooperative wireless networking is that wireless devices work together to achieve their individual goals or one common goal following a common strategy. Wireless devices share their resources (i.e., radio link, antenna, etc.) during cooperation using shortrange communications. The advantages of cooperation are as follows: ﬁrst, the communications capability, reliability, coverage, and qualityofservice (QoS) of wireless devices can be enhanced by cooperation; second, the cost of information exchange (i.e., transmission power, transmission time, spectrum, etc.) can be reduced. Cooperative communication and networking will be a key component in next generation wireless networks. In this book we particularly focus on cooperative transmission techniques in cellular wireless networks. Although cellular wireless systems are regarded as a highly successful technology, their potential in throughput and network coverage has not been fully realized. Cooperative communication is a key technique to harness the potential throughput and coverage gains in these networks. Cooperation is possible among mobile stations (MSs) inside a cell as well as among base stations (BSs). In addition, specialized relay stations (RSs) can be installed in the network to facilitate cooperative communications. In addition to improving throughput and coverage, cooperative communication can improve the energy saving performance at the mobile devices, increase reliability in transmission, and decrease the overall interference in the network. However, successful deployment and operation of a cooperative cellular wireless network hinges on the development of advanced radio transmission and resource management techniques and optimization of these techniques considering the diﬀerent network parameters. This has spurred a vibrant ﬂurry of research on diﬀerent aspects of cooperative communication during the last few years.
Preface
xxi
With BS cooperation, neighboring cells can exchange information aiming at mitigating intercell interference by coordinating the multicell transmission to a mobile or reception from a mobile. In a relaybased cooperative cellular wireless network, an MS may communicate with a BS via potential relays, and similarly, a BS can send data to distant nodes through relays. The potential relays could be either preinstalled ﬁxed RSs or relaycapable MSs. The RSs are much cheaper than conventional BSs because they have far fewer functionalities compared to BSs. If the relay is positioned suitably, it is possible to increase the data rate (especially at the cell boundaries) and the reliability of the system. Similar to multiantenna transceivers, relays provide diversity by creating multiple replicas of the signal of interest. By properly coordinating diﬀerent spatially distributed nodes in the system, a virtual antenna array can be synthesized that emulates the operation of a multiantenna transceiver. With cooperation at all layers of the protocol stack, the network can achieve higher throughput, higher system reliability, higher energy eﬃciency, a lower biterror rate, and a smaller packet loss rate. For cooperation at physical, medium access control (MAC), network, and application layers, various cooperative signaling methods are being widely explored and many new mechanisms are under development with respect to medium access, routing, location management, scheduling, energy management, etc. This book provides a comprehensive treatment of the stateoftheart of cooperative communications and networking techniques for cellular systems (e.g., Beyond 3G, LongTerm evolution systems). It consists of chapters covering different aspects of cooperative cellular wireless network design which include the following: architectures and protocols for cooperative cellular wireless networks; cooperative BS techniques (e.g., cooperative beamforming technique); radio resource management protocol design and network planning for relaybased cooperative cellular wireless systems (e.g., relaying strategies and protocols, resource allocation, energy management, network coding, and crosslayer issues); and the latest IEEE standardization activities pertaining to cooperative cellular wireless systems. The chapters are written by the distinguished researchers in these areas. This book is targeted at graduate students, or researchers working in the area of cellular wireless networks. It can also be used for selfstudy to become familiar with the stateoftheart in cooperative communications for cellular wireless systems. This book contains 17 chapters which have been organized into ﬁve parts. A brief account of each chapter in each of these parts is given next. Part I: Introduction In Chapter 1, Nosratinia and Hedayat outline the trends in research into cooperative cellular wireless networks, as well as some of the outstanding problems in this area. In particular, the issues related to BS cooperation (for both
xxii
Preface
downlink and uplink transmission) and cooperation through dedicated wireless relays (ﬁxed and mobile) are discussed from the physical layer perspective. In Chapter 2, Moghaddari and Hossain focus on the resource allocation problem for cooperative communications in relaybased orthogonal frequencydivision multiple access (OFDMA) and multipleinput multipleoutput (MIMO) wireless networks. Starting with the basics of cooperative relay networks and strategies for relayassisted transmission, a survey on the diﬀerent approaches for radio resource allocation in OFDMA relay networks is provided. To this end, research issues on resource allocation in multihop MIMO relay networks and some related work in the literature are discussed. Part II: Cooperative base station techniques In Chapter 3, Hardjawana, Vucetic, and Li focus on BS cooperation for interference cancelation where each BS transmitter uses the transmitted signal information from other BS and channel state information to precode its own signal. A spectrally eﬃcient cooperative downlink transmission scheme is designed by employing precoding and beamforming. The proposed scheme achieves fairness among diﬀerent users in terms of symbol error rate. In Chapter 4, Aktas et al. focus on an approach for implementing BS cooperation in a distributed manner via message passing in network MIMO systems. This approach is based on a graphical model (in particular, a factor graph) of the network MIMO communication processes. Both uplink and downlink transmissions are considered. As an example, a graphbased approach for distributed beamforming and power allocation is discussed. In Chapter 5, Wang and Yeh discuss the antenna architectures for the network MIMO schemes based on BS cooperation in a multicellular system. One fundamental question when applying the network MIMO technique in such a high interference environment is: how many BSs should cooperate together to provide satisfactory signaltointerferenceplusnoise ratio (SINR) performance? Considering the interferences from the other cooperating groups, it is found that on top of the trisector directional antenna and fractional frequency reuse (FFR), the network MIMO based on the threecell coordination strategy can outperform sevencellbased network MIMO with omnidirectional antenna. The authors also consider the eﬀect of diﬀerent cell sectorizations by using 120◦ and 60◦ beamwidths directional antennas. Part III: Relaybased cooperative cellular wireless networks In Chapter 6, Valenti and Reynolds focus on spacetime block coding (STBC) strategies in a cooperative system to forward signals eﬃciently from multiple relays to the destination by exploiting the spatial diversity present in a multirelay network. Both decodeandforward distributed STBC and
Preface
xxiii
amplifyandforward distributed STBC are considered for a twophase transmission protocol in a network with one source node, one destination node, and a set of relay nodes. The endtoend outage probability, coding gain, and achievable diversity as well as the optimal ratio of the power used in the two phases of transmission are analyzed for diﬀerent spacetime codes. Unlike conventional spacetime codes, the distributed spacetime codes have to deal with the synchronization problem at the destination receiver. Delay diversity, delaytolerant distributed spacetime codes, and spacetime spreading are some eﬀective methods of dealing with this problem. In Chapter 7, Raman et al. present a simulation study of the downlink of a cellular system with relays in order to evaluate peak and average power savings for a given target common rate requirement for users. In particular, three schemes, namely, the collaborative power addition (CPA) scheme, the power controlbased collaborative power addition (PCCPA) scheme, and an orthogonal relaying scheme are simulated. In the CPA scheme, when a relay receives the complete message, it collaborates with the BS to transmit the complete message to the user using its peak power. In the PCCPA scheme, power control is performed jointly at the BS and RS. In the orthogonal relaying scheme, the BSs and the RSs transmit in orthogonal time slots. The peak power savings (at the BSs) are rate gains (for the users) and are observed to be better with the PCCPA scheme. In Chapter 8, Mallick et al. study the radio resource (i.e., bandwidth, transmit power) allocation problem in relaybased cooperative cellular wireless networks. For the diﬀerent relaying schemes (i.e., amplifyandforward, decodeandforward) in single and multiuser network scenarios, diﬀerent optimization models for resource allocation and their solution approaches are described. Also, the problem of relay selection for individual communication between source and destination nodes is discussed. The problem of joint optimization of resource allocation and relay selection is an open research issue. In Chapter 9, Yu, Kwon, and Shin study the resource (i.e., power, spectrum, and rate) allocation problem for OFDMAbased cooperative cellular wireless networks. Two types of cooperative networks are considered: the multicell network with BS cooperation where multiple BS cooperatively allocate power to the diﬀerent frequency subchannels, and networks with RSs. With a view to maximizing the sum of utilities of multiple uses in a multicell network, a network utility maximization (NUM) framework is used to solve the scheduling, and the power, frequency, and rate allocation problem. A key observation here is that, with OFDMA, the network utility maximization problem often decomposes into a tonebytone optimization problem, which is easier to solve. In Chapter 10, Ng and Schober focus on the problem of crosslayer scheduling design for twoway halfduplex amplifyandforward relayassisted OFDMA cellular networks. Such a scheduling scheme has to satisfy the diﬀerent data rate and outage probability requirements of diﬀerent users. Starting with the basics of crosslayer scheduling design and the related implementation challenges, the
xxiv
Preface
problem considered is formulated as a mixed combinatorial and nonconvex optimization problem. In the problem formulation, the objective is to obtain the optimal power, rate, and subcarrier allocation policies while taking the imperfect channel state information (CSI) as well as heterogeneous QoS requirements of the users into account. The problem is then solved by dual decomposition. Also, a distributed iterative algorithm is designed to reduce the computational load at the relays. In Chapter 11, Rost and Fettweis focus on the systemwide energy consumption in cooperative cellular networks. Two deployment scenarios are considered, namely, a macrocellular deployment and a microcellular deployment, both of which use OFDMA air interface for uplink/dowlink transmission in timedivision duplex (TDD) mode. The system performance is simulated considering multicell MIMO transmission only (nonrelaying protocol), a relay only protocol, and an integrated approach (which supports both multicell MIMO and relaying). The achievable throughput performance, the energy saving potentials, and the deployment costs are compared. In addition, performance comparison is carried out between femtocells and relaying. In Chapter 12, Xu and Li study the potential application of network coding and the related issues in relaybased networks. Also, the idea of physicallayer network coding is discussed; this has the potential to improve the throughput performance of relaybased networks signiﬁcantly. One key observation is that, since network coding is mostly applied at the lower layers of the protocol stack, the scheduling and resource allocation at the upper layers have to be codingaware. Such a crosslayer approach for network coded cooperation may reap the beneﬁts of network coding, however, at the expense of increased complexity. Part IV: Game theoretic models for cooperative cellular wireless networks In Chapter 13, Saad, Han, and Hjørungnes explore the application of coalitional game theory to model the various aspects of cooperative behavior in cellular wireless networks. For example, cooperation among the BSs can be modeled by a class of coalitional games, known as coalition formation games, and thereby, algorithms can be derived which help in analyzing the groups of cooperating BSs that will emerge in a given network scenario. As another example, network formation games, a class of coalitional graph games, can be used to model the interactions among RSs. The key message is that coalitional game theory provides a rich framework to design eﬃcient, fair, and robust models for resource allocation and sharing in cooperative cellular wireless networks. In Chapter 14, Marina et al. use game theory to analyze the secrecy capacity in cooperative networks in the presence of malicious users (e.g., eavesdroppers). The secrecy capacity refers to the maximum reliable data rate at which a perfectly secret communication is possible between a sender and a receiver. Three
Preface
xxv
diﬀerent communications scenarios are considered. In the ﬁrst scenario, several friendly jammers help the source in transmitting data to a destination by jamming the eavesdropper. The interaction between the source and the friendly jammers is analyzed using a Stockholder type of game. In the second scenario, several relay nodes help the source by relaying the transmitted data in the presence of a malicious node, and this cooperation improves the secrecy capacity. In this cooperative system, the number and locations of the relay nodes determine the secrecy region, i.e., the geometric area in which the secrecy capacity is positive. In the last scenario, the eavesdroppers cooperate to improve their reception performance. A coalitional gamebased model is proposed for forming cooperative groups among the eavesdroppers. This modeling will be useful to develop defense mechanisms against the eavesdroppers’ cooperation. Part V: Standardization activities In Chapter 15, Moon, Clerckx, and Khan discuss the standard trends on cooperative communications in the Third Generation Partnership Project (3GPP) LongTerm Evolution Advanced (LTEAdvanced) system. In particular, an overview of the key technical features of LTEAdvanced Release 10 including carrier aggregation, cooperative multipoint transmission/reception (CoMP), extended multiantenna systems, and wireless relays is provided. Carrier aggregation provides wider transmission bandwidth and makes full use of the existing fractional spectrum bands. The other techniques provide higher cell spectrum eﬃciency, better coverage, and lower handover interruption time. CoMP transmission refers to a new class of intercell interference mitigation technique, which is also called multicell MIMO, collaborative MIMO (CoMIMO), or network MIMO. The basic idea is to extend the conventional singlecelltomultipleuser transmission to a multiplecelltomultipleuser transmission through BS cooperation. In Chapter 16, Kim et al. develop methods for partial information relaying in multiantenna decodeandforward relay networks. These methods use a twophase transmission strategy and exploit the asymmetric link conditions in cellular networks, where the source–relay link and the relay–destination link are relatively better than the source–destination link. With multiple antennas available at source, relay, and destination, multiple parallel data streams are transmitted which consist of basic data streams and superposition coded (SC) data streams. The relay forwards only the SC streams (i.e., partial information in the second hop). Two methods, namely, perantenna superposition coding (PASC) and multilayer superposition coding (MLSC), are proposed for power allocation among basic and superposed layers, and across the spatial layers. It is observed that partial information relaying results in signiﬁcant capacity gain over full information relaying. To this end, the authors summarize the issues, discussions, and current conclusions on relaying in the LTEAdvanced standard.
xxvi
Preface
In Chapter 17, Cho et al. discuss the proposals and current conclusions on the CoMP technique in the 3GPP LTEAdvanced standard. Many companies and research groups are conﬁdent that CoMP systems are feasible in real systems and have put forth eﬀort to ﬁnd and evaluate what type of cooperation scheme should be standardized. The authors outline the issues discussed in the LTEAdvanced study group, for downlink CoMP, and present related simulation methodologies.
Part I Introduction
1
Network architectures and research issues in cooperative cellular wireless networks Aria Nosratinia and Ahmadreza Hedayat
1.1
Introduction The systematic study of relaying and cooperation in the context of digital communication goes back to the work of Van der Meulen [1] and Cover and El Gamal [2]. The basic relay channel of [1, 2] consists of a source, a destination, and a relay node. The system models in [1, 2] are either discrete memoryless channels (DMC), or continuousvalued channels which are characterized by constant (nonrandom) links and additive white Gaussian noise. The study of cooperative wireless communication is a more recent activity that started in the late 1990s, and since then has seen explosive growth in many directions. Our focus is speciﬁcally on aspects of cooperative communication related to cellular radio. Aside from the fading model, the deﬁning aspects of a cellular system are base stations that are connected to an infrastructure known as the backhaul, which has a much higher capacity and better reliability than the wireless links. The endpoints of the system are mobiles that operate subject to energy constraints (battery) as well as constraints driven by the physical size of the device that lead to bounds on computational complexity and the number of antennas, among other considerations. There are multiple mobiles in each cell as well as frequency reuse, leading to intracell interference and intercell interference, respectively. The exponential pathloss laws lead to signiﬁcant variations in signal power at various points in the cell. In this chapter we are concerned with cooperative radio communication that speciﬁcally engages one or more of these deﬁning aspects. Within the context of cellular radio, cooperative communication may be used to enhance capacity, improve reliability, or increase coverage. It may be used in the uplink or the downlink. In the communication between a base station and a mobile, the cooperating entity may be another base station, another mobile, or a dedicated (often stationary) wireless relay node. The cooperating entity may have various amounts of information about the source data and channel state information. Cooperation may happen in the physical layer, data link layer, network layer, transport layer, or even higher layers. The large number of diﬀerent Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
4
Network architectures and research issues
ways that cooperation may be exploited to improve the quality of service in cellular radio has given rise to much research and a rich and expanding literature. There are many questions that remain unanswered, among them the relation between the various forms of cooperation and their relative merits are in general not fully known. In this chapter, we aim to catalog some of the directions of research in this area, and outline some of the open questions.
1.2
Base station cooperation Base station cooperation can take multiple forms. The simplest form of base station cooperation, especially with multiantenna base stations, involves the exchange of information among neighboring cells regarding their celledge nodes and remotecell aware processing at each of the base stations. Then, each of the base stations can put a null on the channel gain vector of the nodes that generate and/or are harmed by the most cochannel interference. This and other simple scenarios like it are in the realm of interference management, and are possible without fully coordinated action from base stations. Speciﬁcally, this form of action does not require the base stations to know the traﬃc for other base stations (therefore the issue of a wideband backbone and its delay does not come into play), nor is it required to know the codebooks used by the other base station, and nor does it require the base stations to be synchronized.
1.2.1
Downlink cooperation For downlink base station cooperation, base stations can generate a virtual multiantenna array with zeroforcing beamforming. There are a variety of ways to exploit this general idea. Somekh et al. [3] used the circular Wyner cellular model [4] to ﬁnd expressions for downlink (and uplink) capacities with base station cooperation, which is also sometimes called multicell processing. In [5] the same model and a zeroforcing beamforming approach were used for data transmission in multiple cells. In particular, the approach is to transmit to the best user in each cell, and the highload asymptotics are derived in an information theoretic approach. Mundarath et al. [6] considered the scheduling aspects of distributed downlink zeroforcing beamformers under ﬁnite loads. Such downlink strategies require certain assumptions about sharing of information among cells. To begin with, the data must be shared among the base stations. Secondly, the base station transmitters must be synchronized. Finally, the channel state information of the users must be shared among the base stations, and must be kept uptodate, so that beamforming vectors can be reliably determined. The nascent area of distributed beamforming has seen much activity; we brieﬂy mention a representative sample of the results.
1.2 Base station cooperation
5
The eﬀect of channel state statistics on distributed beamforming was investigated in [7]. Ng et al. [8] developed a distributed method of beamforming by message passing between adjacent base stations instead of sharing the data between many base stations. Mudumbai et al. [9] investigated the feasibility of distributed beamforming from the viewpoint of local oscillator phase errors, showing encouraging results. However, the results were tempered by the fact that in this study it was assumed that the frequency of the oscillators is stable. Brown and Poor [10] proposed a method of carrier synchronization for distributed beamforming. Other works in this area include [11–18]. A tutorial on the challenges and progress in distributed beamforming is given in [19]. There is a key diﬃculty that limits the usefulness of conventional beamforming techniques in the distributed scenario: because the antennas are not colocated, the diﬀerence of propagation delay from diﬀerent antennas to a node will become a relevant parameter. For beamforming to a single node (singlebeam solution) any diﬀerence in delays can be absorbed into the beamforming coeﬃcients. Multiple beams may also be generated in the same manner, but only as long as the target nodes are close to each other, i.e., the delay vectors for various nodes must be approximately similar. If the target nodes are far from each other and the transmit antennas are also far from each other, then the conventional methods of beamforming may not work. The degradation of distributed beamforming systems due to the distribution of the antennas and the target nodes is a subject that, to our knowledge, has not been systematically investigated. Also, the design of new optimization techniques to develop beamforming vectors in the presence of diﬀerent delays remains an open problem. To study this issue further, it is useful to separate the two diﬃculties raised by this delay variance. The ﬁrst one is the induced change in phase, which can be addressed via, e.g., systems of equations that impose the constraints at various nodes. The second, more challenging, problem is due to the variance in the time of arrival of the leading edge of each symbol at various nodes. Assuming that the arrival of symbols from the distributed antennas is calibrated to be cotimed at node A, the leading edge of the symbols may not arrive cotimed at node B. To our knowledge, no comprehensive method is known to address this problem. Intuitively, if the symbol duration is much larger than the diﬀerences in time of arrival, then there is a chance that if the phases can be appropriately adjusted, the overall eﬀect can be brought under control. This also suggests the use of systems with longer symbol duration, e.g., OFDM. It would be interesting to investigate whether variations of techniques used in OFDM, such as cyclic preﬁx, can be used to “collect” the various components of the distributed beamforming that may be out of phase and out of time. Aside from the abovementioned fundamental issue, there are also practical issues that require careful consideration and design. In particular, distributed beamforming requires delicate accounting for various types of delay, not just
6
Network architectures and research issues
in the channel but also within the base station and signal delivery path. Any unaccounted for delay, or delay variation anywhere within the transmit or receive path, would cause transmitted signals to arrive at the mobile station unaligned, causing loss in beamforming gain. These are requirements that often do not appear in theoretical studies, but in fact play an important role in any practical implementation. To summarize, base station cooperation oﬀers the opportunity for improvent in two opposing directions: incorporating more theoretical results and addressing practical issues. Among various aspects that call for further investigation, one may name:
r possibilities with multiantenna mobiles, among them the extension of results r r r r 1.2.2
from multipleinput multiple output (MIMO) broadcast channels to the distributed MIMO case; the eﬀect of partial channel state information or partially outdated channel state information; quantized channelstate feedback for distributed beamforming has been addressed in [16], but there is room for much more work in this area; the eﬀect of uneven channel state information across a set of heterogeneous mobiles; investigation of limits on the backbone, e.g., limits on sharing of traﬃc data.
Uplink cooperation The problem of uplink cooperation is rather diﬀerent from its downlink counterpart. In the uplink cooperation scenario, a mobile might be in a situation where no single base station can decode its data alone. However, the signals received at two or more base stations may be suﬃcient to decode the mobile data. The collection of information at various base stations and their combination present new issues. In particular, since each of the base stations cannot decode the received signal alone, these signals must be sampled and exchanged among base stations, which requires signiﬁcantly larger bandwidth than the data do. Thus, considering the eﬀect of base station cooperation on the backhaul capacity becomes an important issue. The capacity of the uplink linear cellular networks with base station cooperation via ﬁnite capacity links was broached in [20], and bounds on the rate of the system under the Wyner model were obtained from an information theoretic viewpoint. This work generated broad insights into the general capabilities of uplink base station cooperation, but the speciﬁcs of coding and signal design for such systems remain open problems. Thankfully, the uplink does not suﬀer in quite the same way from the timing problem that plagues the downlink distributed beamforming (see Section 1.2.1). The varying propagation times from the mobiles to base stations can be compensated in the algorithm that combines the data from multiple base stations, since the signal of each of the mobiles can be extracted separately. However, the
1.3 Dedicated wireless relays
7
problem will remain if we wish to listen to mobile A while simultaneously nulling the interference from mobile B, and these two mobiles have signiﬁcantly diﬀerent delay vectors to the set of base stations participating in multicell processing. The scope for future work in the area of uplink multicell processing is in two directions: incorporation of communication theoretic results into uplink cooperation and addressing issues related to practical limitations. The potential areas of work include:
r practical implications of the restrictions on the backhaul capacity and delay; r iterative methods based on belief propagation; r computeandforward (hashing) methods to reduce the cooperation bandwidth requirement between the base stations;
r analysis of nonideal conditions, including uncertainties in channel state information;
r the eﬀect of nonideal synchronization and sampling; r investigating possibilities presented by multiantenna mobiles. 1.3
Dedicated wireless relays Traditional cellular networks provide ﬁxed throughput for all subscribers where a basic voice service can be supported. Unlike such networks, broadband wireless cellular networks promise a high data rate throughout the coverage area. While such promise is feasible for the inner coverage area, at the celledge data rates are limited for various reasons. Decreasing the cellsize is one way to satisfy the required data rate; however, it is a costly solution because it requires the installation of additional base stations. In contrast, deploying relay stations provides a costeﬀective solution. Compared with a fullscale base station, a dedicated relay can save on equipment costs, backhaul link, and deployment. A relay station assists the main base station to improve its coverage or throughput. A relay station can be used to extend the coverage area of a base station, or to provide coverage in socalled holes.1 In addition, thanks to the advances in antenna array techniques, relays can also be used to improve throughput and capacity. Due to above facts, the IEEE 802.16 Working Group has developed the IEEE 802.16j standard with techniques that are compatible with the WiMAX standard.
1.3.1
IEEE 802.16j The IEEE 802.16j standard was created to be backward compatible with the 802.16e standard known as WiMAX. Various modes of operation of 802.16j ﬁt within the WiMAX OFDMA frame. In all the various modes of 802.16j, the base 1
For example, in tunnels or in certain areas inside a building.
8
Network architectures and research issues
station allocates part of its uplink and downlinks subframes for communicating with the relay station(s). It is envisioned that legacy mobile nodes are able to integrate seamlessly into 802.16j, therefore the mobiles are oblivious to the relays, i.e., they cannot distinguish between the relay and base station. Two of the modes of operation considered in the IEEE 802.16j standard are transparent and nontransparent modes. The usage of these modes depends on whether it is intended to increase throughput for celledge mobile stations, or to extend coverage to the mobile stations unreachable by base station. The transparent mode allows for onetime relaying while the nontransparent mode allows for multipletime relaying. The transparent mode can be used, e.g., for inbuilding coverage, while the nontransparent mode can be used to access remote areas which might possibly require multiple relaying. There are also diﬀerences within the downlink and uplink frames of these two modes. In transparent mode only the traﬃc portion is exchanged between relay and mobile stations, while in nontransparent mode in addition to the traﬃc portion other signals such as synchronization, downlink and uplink maps, and ranging need to be exchanged as well. Both the transparent and nontransparent modes are multihop modes, i.e., the relay intervenes between the source and destination, and the data portions of source–relay and relay–destination communications are performed separately in time. Another mode of operation in the IEEE 802.16j standard is cooperative relaying, where the base station and relay transmit the same signal, or two copies of the same signal, to the mobile simultaneously. As long as the timing diﬀerence between the signal of the base and relay stations as seen by mobile stations is within the acceptable range, the mobile station sees the combined signals as a signal with high diversity. Operational cooperative relaying has multiple requirements. First, traﬃc data need to be exchanged between the relay and base stations. Second, the deployment of the relays needs to be such that the time diﬀerence seen by the mobile station does not cause OFDMA intersymbol interference. With these practical requirements fulﬁlled, research eﬀorts have been focused on the various MIMO techniques that can be used to achieve cooperation between base and relay stations. The IEEE 802.16j standard says very little about the techniques for implementing cooperative relaying, therefore there is signiﬁcant room for innovation by the wireless industry within the context of the standard. It must be noted that other broadband wireless standards, such as LongTerm Evolution (LTE) and LTEAdvanced, consider the use of relay stations. For details, the readers are referred to Chapters 15 and 16 of this book.
1.3.2
Highspectraleﬃciency relay channels One of the main drawbacks of relaying is that in eﬀect it requires duplicate transmissions (e.g., base station to relay and relay to mobile), unlike standard singlehop transmission. For instance, in both transparent and nontransparent
1.3 Dedicated wireless relays
9
modes of the IEEE 802.16j there are multiple zones of uplink and downlink subframes that are assigned for the exchange of traﬃc between base station and relay stations. The loss of time and/or bandwidth is more pronounced for multihop relaying. However, the advantages of relaying often more than oﬀset the loss of bandwidth. The obvious advantages are the reduction of path loss and shadowing, as well as providing path diversity, but the advantages of relaying go beyond the obvious. For example, both the base station and the relay are stationary (nonmoving) and the base station is often elevated, so it is possible to estimate the channel gains between the base station and the relay with a very high degree of accuracy and stability, far beyond what is possible with mobiles. In addition, it is possible to use antenna arrays in both the base station and the relay, which combined with the accurate channel state information leads to a higher spectral eﬃciency as well as more eﬃcient frequency reuse. More speciﬁcally, the base station can employ spacedivision multiple access (SDMA) techniques to transmit traﬃc to several relay stations using the same bandwidth. For eﬃcient implementation of SDMA, the channel state information of each relay must be known by the base station with high precision. In fact, the more precise the channel state information, the larger the number of relay stations that can be served within the same bandwidth. Thanks to the ﬁxed location of relay and base stations, it is possible to obtain a highquality channel state information of each relay station. Likewise, it is possible to use a similar technique in the uplink to increase uplink spectral eﬃciency. For instance, collaborative spatial multiplexing, which is also introduced in the IEEE 802.16e and other wireless broadband standards, allows multiple relay stations to transmit uplink traﬃc utilizing the same burst location, i.e., the same bandwidth. If relay stations are equipped with antenna arrays, other MIMO techniques such as spatial multiplexing can also be used to further enhance spectral eﬃciency of downlink and uplink. Improving spectral eﬃciency of relay channels can further increase the utilization of relays in broadband cellular networks, and therefore is one of the important research topics in this area. For instance, the design of adaptive MIMO precoding techniques that can provide robustness as well as higher throughput can facilitate cellular relays. This could lead to precoding weights that adapt to spatial multiplexing, beamforming, or nulling. The various costs related to the backhaul channel constitute a big portion of the cost of cellular network deployment, operation, and maintenance. Increased utilization of wireless resources will result in shrinking cell sizes and an increase in the number of base stations, therefore it is expected that backhaul link expenses will be in future broadband networks, generating interest in backhaul links that are cheaper than laying cable or dedicated microwave links. It has been noted that one can use the same family of broadband wireless technologies that are servicing endusers to connect to some base stations. The similarity of operation and use of wireless backhaul links and relaying technologies suggests a potential
10
Network architectures and research issues
uniﬁcation of these two applications. Thus, spectrally eﬃcient relay channels could eventually also pave the way to lessexpensive and easily deployable backhaul links, and might even lead to uniﬁed technologies or standards for both.
1.4
Mobile relays In the previous two sections, we considered the outlook for base station cooperation and dedicated relays. It has been noted that there is nothing in principle that precludes the possibility of mobile relays, although technologically their implementation is much more diﬃcult. In this section, we give a brief overview of mobile relays. We begin by considering the distinguishing characteristics of mobile relaying. One of the fundamental diﬀerences between mobile and ﬁxed relays is the limitations on power/energy in the mobile nodes. Fixed relays can be connected to the power grid, while the mobiles are dependent on resident energy storage, which with the presentday technologies is a severe limiting factor. This situation seems a fairly stable one, since neither energy storage devices with ordersofmagnitude higher energy densities, nor an eﬃcient means of tetherless delivery of energy to mobiles, is visible on the technological horizon. Another distinguishing factor of most mobile nodes, compared to ﬁxed relays, is one of size. Mobile nodes are no more than several centimeters in each dimension, and this puts fundamental restrictions on the number of antennas that can be deployed on a mobile. Polarization diversity antennas oﬀer some improvement, but a ﬁxed relay has much more ﬂexibility in the number of antennas. An aspect of mobile relaying, which is not unrelated to size and energy, is computational complexity. However, unlike size, this is less of a fundamental limitation. In the past, advances in computational power, measured in MIPS/mm3 , have been much more rapid than, e.g., advances in battery technology measured in terms of energy density (joules/mm3 ). Thus it is not unreasonable to assume that nearfuture technologies in mobiles will become ever more computationally complex, while the power available to them will grow at a much slower pace. Due to limited resources, any mobile relay must balance the needs of the node itself with relaying for other nodes. This includes not only power and computation, but also the total spectral eﬃciency available to a node. The fundamental tradeoﬀ between a node’s own communications and relayed bandwidth was addressed in [21], which showed that the competition between its own bandwidth and the relayed bandwidth does not constitute a zerosum game. However, there will still be questions of the motivation of a relay node to use local resources for other nodes. The speciﬁc aspects of networkwide management of resources, and the development of network control algorithms that guide the action of mobile relay nodes that also have individual incoming/outgoing traﬃc, have attracted some attention but a complete understanding of these algorithms has still not been achieved. Some interesting advances have been made using game theory and pricing analysis. However, a good part of the work in this area is not relevant
1.5 Conclusion
11
to the design of cooperative cellular networks, because the nodes in a cellular network are not autonomous agents and most decisions are, in fact, made in a centralized manner. Resource allocation and network control for mobile cooperative nodes (including aspects such as local rate splitting) are interesting problems that, under the practical conditions that are of interest to the designers of cellular systems, remain for the most part unsolved.
1.5
Conclusion Cooperative communication is one of the promising wireless technologies that has been reintroduced in the last decade, and it has promising applications in the context of cellular networks. This chapter has presented a brief overview of some of the aspects of research in cooperative cellular networks. To summarize, several important directions of future work seem to beckon wireless researchers. One direction points to the more sophisticated algorithms and harnessing the full power of the coding and signaling methods that are emerging from information theory. Another direction is to reﬁne the models in order to get a more realistic grip on the practical aspects of operation in a cellular network. Finally, most systematic study and analysis of cooperation has concentrated on the physical layer. However, cooperative action at other layers of the communication hierarchy is also possible, but has not been studied quite as much, and this may present a host of opportunities to future wireless engineers.
References [1] E. C. Van der Meulen, “Three terminal communication channels,” Adv. Appl. Probab., 3, 1971, 120–154. [2] T. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inform. Theory, 25, 1979, 572–584. [3] O. Somekh, B. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cellsite processing,” IEEE Trans. Inform. Theory, 53, 2007, 4473– 4497. [4] A. Wyner, “Shannontheoretic approach to a Gaussian cellular multipleaccess channel,” IEEE Trans. Inform. Theory, 40, 1994, 1713–1727. [5] O. Somekh, O. Simeone, Y. BarNess, A. M. Haimovich, and S. Shamai, “Cooperative multicell zeroforcing beamforming in cellular downlink channels,” IEEE Trans. Inform. Theory, 55, 2009, 3206–3219. [6] J. Mundarath, P. Ramanathan, and B. Van Veen, “A distributed downlink scheduling method for multiuser communication with zeroforcing beamforming,” IEEE Trans. Wireless Commun., 7, 2008, 4508–4521. [7] V. H. Nassab, S. Shahbazpanahi, A. Grami, and Z.Q. Luo, “Distributed beamforming for relay networks based on secondorder statistics of the channel state information,” IEEE Trans. Signal Processing, 56, 2008, 4306–4316.
12
Network architectures and research issues
[8] B. L. Ng, J. Evans, S. Hanly, and D. Aktas, “Distributed downlink beamforming with cooperative base stations,” IEEE Trans. Inform. Theory, 54, 2008, 5491–5499. [9] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility of distributed beamforming in wireless networks,” IEEE Trans. Wireless Commun., 6, 2007, 1754–1763. [10] D. Brown and H. Poor, “Timeslotted roundtrip carrier synchronization for distributed beamforming,” IEEE Trans. Signal Processing, 56, 2008, 5630– 5643. [11] H. Ochiai, P. Mitran, H. Poor, and V. Tarokh, “Collaborative beamforming for distributed wireless ad hoc sensor networks,” IEEE Trans. Signal Processing, 53, 2005, 4110–4124. [12] V. H. Nassab, S. Shahbazpanahi, and A. Grami, “Optimal distributed beamforming for twoway relay networks,” IEEE Trans. Signal Processing, 58, 2010, 1238–1250. [13] H. Chen, A. B. Gershman, and S. Shahbazpanahi, “Filterandforward distributed beamforming in relay networks with frequency selective fading,” IEEE Trans. Signal Processing, 58, 2010, 1251–1262. [14] K. Zariﬁ, A. Ghrayeb, and S. Aﬀes, “Distributed beamforming for wireless sensor networks with improved graph connectivity and energy eﬃciency,” IEEE Trans. Signal Processing, 58, 2010, 1904–1921. [15] Z. Ding, W. H. Chin, and K. Leung, “Distributed beamforming and power allocation for cooperative networks,” IEEE Trans. Wireless Commun., 7, 2008, 1817–1822. [16] E. Koyuncu, Y. Jing, and H. Jafarkhani, “Distributed beamforming in wireless relay networks with quantized feedback,” IEEE J. Select. Areas Commun., 26, 2008, 1429–1439. [17] M. Ahmed and S. Vorobyov, “Collaborative beamforming for wireless sensor networks with Gaussian distributed sensor nodes,” IEEE Trans. Wireless Commun., 8, 2009, 638–643. [18] E. Zacarias, S. Werner, and R. Wichman, “Distributed Jacobi eigenbeamforming for closedloop MIMO systems,” IEEE Commun. Lett., 10, 2006, 825–827. [19] R. Mudumbai, D. Brown, U. Madhow, and H. Poor, “Distributed transmit beamforming: challenges and recent progress,” IEEE Commun. Mag., 47, 2009, 102–110. [20] O. Simeone, O. Somekh, H. V. Poor, and S. Shamai, “Local base station cooperation via ﬁnitecapacity links for the uplink of linear cellular networks,” IEEE Trans. Inform. Theory, 55, 2009, 190–204. [21] R. Tannious and A. Nosratinia, “Relay channel with private messages,” IEEE Trans. Inform. Theory, 53, 2007, 3777–3785.
2
Cooperative communications in OFDM and MIMO cellular relay networks: issues and approaches Mohammad Moghaddari and Ekram Hossain
2.1
Introduction The continually increasing number of users and the rise of resourcedemanding services require a higher link data rate than the one that can be achieved in current wireless networks [1]. Wireless cellular networks, in particular, have to be designed and deployed with unavoidable constraints on the limited radio resources such as bandwidth and transmit power [2]. As the number of new users increases, ﬁnding a solution to meet the rising demand for high data rate services with the available resources has became a challenging research problem. The primary objective of such research is to ﬁnd solutions that can improve the capacity and utilization of the radio resources available to the service providers [3]. While in traditional infrastructure networks the upper limit of the source– destination (S–D) link’s data capacity is determined by the Shannon capacity [4], advances in radio transceiver techniques such as multipleinput multipleoutput (MIMO) architectures and cooperative or relayassisted communications have led an enhancement in the capacity of contemporary systems. In the MIMO technique the diversity relies on uncorrelated channels, and is achieved by employing multiple antennas at the receiver side, the transmitter side, or both, and by suﬃciently separating the multiple antennas (of same polarization) [5]. The MIMO technique can be used to increase the robustness of a link as well as the link’s throughput. Unfortunately, the implementation of multiple antennas in most modern mobile devices may be challenging due to their small sizes [3]. Cooperative diversity or relayassisted communication has been proposed as an alternative solution where several distributed terminals cooperate to transmit/receive their intended signals. In this scheme, the source wishes to transmit a message to the destination, but obstacles degrade the S–D link quality. The message is also received by the relay terminals, which can retransmit it to a desired destination, if needed. The destination may combine the transmissions received by the source and relays in order to decode the message.
Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
14
Cooperative communications in OFDM and MIMO cellular relay networks
The limited power and bandwidth resources of the cellular networks and the multipath fading nature of the wireless channels have also made the idea of cooperation particularly attractive for wireless cellular networks [6]. Moreover, the desired ubiquitous coverage demands that the service reaches the users in the most unfavorable channel conditions (e.g., celledge users) by eﬃcient distribution of the high data rate (capacity) across the network [7, 8]. In conventional cellular architectures (without relay assistance) increasing capacity along with coverage extension dictates dense deployment of base stations (BSs) which turns out to be a costwise ineﬃcient solution for service providers [9]. A relay station (RS), which has less cost and functionality than the BS, is able to extend the high data rate coverage to remote areas in the cell under power and spectral constraints [10–13]. By allowing diﬀerent nodes to cooperate and relay each other’s messages to the destination, cooperative communication also improves the transmission quality [14]. This architecture exhibits some properties of MIMO systems; in fact a virtual antenna array is formed by distributed wireless nodes each with one antenna. Since channel impairments are assumed to be statistically independent, in contrast to conventional MIMO systems, the relayassisted transmission is able to combat these impairments caused by shadowing and path loss in S–D and relay–destination (R–D) links. To this end, an innovative system has been proposed in which the communication between transmitter and receiver is done in multiple hops through a group of relay stations. This cooperative MIMO relaying scheme creates a virtual antenna array (VAA) [15] by using the antennas of a group of RSs. These RSs transmit the signal received from the BS (or previous hops) cooperatively on diﬀerent channels to the receiving terminal (downlink case) or the signal that the transmitting terminal wants to send to the BS (uplink case). This system can be modeled as a MIMO system although the real receiver (downlink) or transmitter (uplink) only has one antenna. Since the relaying mobile stations (MSs) introduce additional noise and there is a double Rayleigh channel eﬀect, the scheme is expected to perform below the corresponding MIMO diversity gain when used for spatial multiplexing. The combination of relaying and orthogonal frequencydivision multiple access (OFDMA) techniques also has the potential to provide high data rate to user terminals everywhere, anytime [7, 8, 16]. Interest in orthogonal frequencydivision multiplexing (OFDM) is therefore growing steadily, as it appears to be a promising airinterface for the next generation of wireless systems due, primarily, to its inherent resistance to frequencyselective multipath fading and the ﬂexibility it oﬀers in radio resource allocations. Likewise, the use of multiple antennas at both ends of a wireless link has been shown to oﬀer signiﬁcant improvements in the quality of communication in terms of both higher data rates and better reliability at no additional cost of spectrum or power [17]. These essential properties of OFDMA and MIMO, along with the eﬀectiveness of cooperative relaying in combating largescale fading and enhancing system capacity immediately motivate the integration of these technologies into one network architecture.
2.2 Cooperative relay networks
15
Figure 2.1. Example of a traditional infrastructure network.
However, adequate and practical radio resource allocation (RRA) strategies have to be developed to exploit the potential gain in capacity and coverage improvement in the integration of relaying, OFDMA, and MIMO techniques [18]. This chapter surveys algorithms proposed in the literature to adaptively allocate the available resources in a relayenhanced OFDMAbased and MIMO wireless networks. We also brieﬂy review resource management approaches in distributedMIMO multihop systems.
2.2
Cooperative relay networks Wireless networks can be classiﬁed into two major categories: traditional infrastructure networks and multihop networks. In traditional networks the communication is performed directly between the BS and the MS and vice versa, so, there is only one hop. Though obstacles may degrade the lineofsight (LoS) S–D link quality in this scheme, the source makes no use of the cooperation potential of other terminals in the network to compensate for the impairments (Figure 2.1).
2.2.1
Cooperative communication The method of relaying, which was introduced by Van der Meulen in 1971 [19], was studied from an information theoretical point of view by Cover and El
16
Cooperative communications in OFDM and MIMO cellular relay networks
R S
D
Figure 2.2. The relay channel, source (S), relay (R), and destination (D).
Gamal in [20]. In these contributions, a source MS communicates with a target MS directly and via a relaying MS. In [20], the capacity of such a relaying conﬁguration was shown to exceed the capacity of a simple direct link. In later studies, a very simple but eﬀective user cooperation protocol was suggested to boost the uplink capacity and lower the uplink outage probability for a given rate [21]. The designed protocol stipulates an MS to broadcast its data frame to the BS and to a spatially adjacent MS, which then retransmits the frame to the BS. Such a protocol certainly yields a higher degree of diversity because the channels from both MSs to the BS can be considered uncorrelated. Cooperative communication was introduced in [10] using the scenario depicted in Figure 2.2 but with the relay terminal being another source. Both sources (associated partners) are also responsible for transmitting the information of their partners. It was assumed that the sources are working in fullduplex mode, so that both sources are transmitting to the destination and receiving a noisy version of the partner’s transmission. Results in terms of ergodic achievable rate regions and outage probability of the cooperative and noncooperative transmission show the beneﬁts of this scheme.
2.2.2
Relay channel The relay channel was introduced in [19]. It assumes that there is a source that wants to transmit information to a single destination. However, there is a relay terminal that is able to help the destination (relayassisted transmission). Based on the previously received symbols, it can transmit an additional message to the destination if needed. Figure 2.2 illustrates the channel model. When the channel to relay terminal is in a better condition than the LoS channel, this scheme is able to improve the S–D transmission. In general, it is assumed that the relay works in fullduplex mode, i.e., receiving and transmitting simultaneously. The duplex communication problem can be solved by assuming that the frequency bands for the main link (S–D) and the relaying links (R–D) diﬀer. Since full duplex terminals are currently unrealistic in practical systems, relays are forced to work in halfduplex mode. However, the halfduplex constraint impacts negatively on the theoretical spectral eﬃciency provided by an ideal fullduplex assisting relay. Multiplexing gains are not possible in halfduplex relaying, although signiﬁcant additive capacity gains are still possible [15, 22].
17
2.2 Cooperative relay networks
S
Protocol I
Protocol II
Protocol III
R
R
R
D
S
D
S
Figure 2.3. Halfduplex relay protocols. Solid lines correspond to the transmission during the relayreceive phase and dashed lines to the transmission during the relaytransmit phase.
2.2.3
Overview of relay protocols In the halfduplex mode there is an orthogonal duplexing (in time or frequency) between the phase that the relay is receiving (relayreceive phase) and the one it is transmitting in (relaytransmit phase). This phase separation allows the deﬁnition of several halfduplex relay protocols with various degrees of broadcasting and receiving collision in each relayreceive and relaytransmit phase among the three terminals (source, destination, and relay). The number of options leads to the four protocol deﬁnitions presented in Figure 2.3, (called protocols I, II, and III) and in Figure 2.4 (called forwarding) [23, 24]. In protocol I, the source communicates with the relay and destination during the relayreceive phase (solid lines in Figure 2.3). Then, in the relaytransmit phase, the relay terminal communicates with the destination (dashed line in Figure 2.3). On the other hand, in protocol II, during the relayreceive phase the source only transmits to the relay (solid line in Figure 2.3). It is assumed that the destination is not able to receive the message from the source in that phase. In the relaytransmit phase, the source and relay transmit simultaneously to the destination (dashed lines in Figure 2.3). Hence in the relaytransmit phase the channel becomes a multiple access channel. Protocol III can be seen as a combination of protocols I and II. The source transmits to the relay and the destination (solid lines in Figure 2.3) in the relayreceive phase. Then, in the relaytransmit phase, the source and the relay transmit to the destination (dashed lines in Figure 2.3). Notice that the relay is transmitting during the second phase, so that it cannot be aware of the signal transmitted by the source in the second phase. This protocol can achieve a better spectral eﬃciency than the previous ones. For example, this protocol was considered for obtaining the achievable rates of the relay channel in [22]. The traditional forwarding protocol consists of a transmission from the source to the relay during the relayreceive phase and a transmission from the relay to the destination in the relaytransmit phase, as in Figure 2.4. It should be
D
18
Cooperative communications in OFDM and MIMO cellular relay networks
R S
D
Figure 2.4. Halfduplex forwarding protocol. emphasized that the halfduplex relay protocols deﬁned in Figure 2.3 make good use of the S–D link in contrast to the forwarding protocol. Likewise, if that link presents very bad quality compared with the S–R and R–D links, the performance obtained by protocols I, II, and III converges to the forwarding one.
2.2.4
Strategies of relayassisted transmission The paradigm of the conventional S–D communication is now changed to S–R–D, and the role played by the relay can be selected from diﬀerent modes of operation inﬂuencing the total achievable rate of the system. Additionally, when there is a halfduplex relay, the resources allocated for each phase of the relayassisted transmission also have an important impact on the achievable rate. Therefore, the strategies of relayassisted transmission have to consider the decoding mode at the relay and the resource allocation. Basically, the three decoding modes analyzed in the literature are: amplifyandforward (AF), decodeandforward (DF), and compressandforward (CF).
Amplifyandforward (AF) This is the simplest strategy that can be used at the relay. The relay ampliﬁes the received signal from the source and transmits it to the destination without doing any decoding. For this reason, it is also called nonregenerative relaying. The main drawback of this strategy is that the relay terminal ampliﬁes the received noise at the same time. Applying this strategy to cooperative communication leads to a better bit error rate (BER) than direct transmission [1]. The outage probability of the cooperative communication was derived in [1], demonstrating that a diversity order of 2 is obtained for two cooperative users. When the relay is equipped with multiple antennas and there is channel state information (CSI) available for the S–R/R–D links, the AF strategy can attain signiﬁcant gains over the direct transmission by means of optimum linear ﬁltering of the data to be forwarded [25].
Decodeandforward (DF) In this strategy the complexity at the receiver increases in comparison to that in the AF strategy. Now the relay terminal has to estimate the message received
2.3 General system model of cellular relay networks
19
from the source, therefore the total performance depends on the success of this message decoding. Depending on the type of symbols retransmitted, the strategy at the relay is repetition coding (RC) or unconstrained coding (UC). In RC, the relay retransmits the same symbols previously estimated, while in UC the symbols transmitted are not the same as the received ones, but are related to the same information sent by the source (source and relay are using diﬀerent codebooks). Hence, this protocol is also called regenerative relaying: the terms are used interchangeably in this chapter.
Compressandforward (CF) In this strategy, the relay does not decode the data but uses Wyner–Ziv lossy source coding [26] on the estimated symbols of the received signal. Then, the compressed signal is transmitted to the destination by the relay. This strategy was suggested in [20]. Depending on the channel gains of the diﬀerent links, the CF strategy can be superior to the DF strategy. However, it adds more complexity to the system.
2.3
General system model of cellular relay networks A partial network in a multicellular relay network is shown in Figure 2.5, which is the general topology for reviewed algorithms and strategies in this chapter. It is assumed that the BS could continuously measure the quality of link, e.g., signaltointerferenceplusnoise ratio (SINR) per subcarrier when OFDM is applied. Furthermore, for slowly varying channels, BSs and RSs can be assumed to have accurate estimates of the channel states. Despite being a common assumption in literature, this may not always be the case, especially for relay station to subscriber station (RS–SS) links. RRA based on partial CSI has been addressed in a number of references such as [27]. The BSs can optimize the resource allocation or they can involve a central network controller as shown in Figure 2.5. Consider a twohop multiuser OFDM system with one BS at the center of each cell, K users, and N subcarriers, where each transceiver has only a single antenna. The RSs are assumed to be ﬁxed and evenly distributed on a circle with a radius equal to one half of the cell radius in order to eliminate the eﬀect of RS placements. The RS forwards the received signal to the BS (uplink) or SS (downlink) by employing either the amplifyandforward (AF) or the decodeandforward (DF) strategy on the same subcarrier. We consider a twotimeslot transmission pattern which is halfduplex in the sense that transmission and reception at any station do not occur simultaneously in the same frequency band. In the ﬁrst time slot, the BS (SS) transmits while the RS and the SS (BS) receive. In the second time slot, only the RS transmits to the SS (BS). When the regenerated message is encoded to provide additional error protection for the original message, this is referred to as coded cooperation [2].
20
Cooperative communications in OFDM and MIMO cellular relay networks
RS SS
SS
BS
RS RS
SS S SS SS RS
BS
Network controller
RS
SS
SS
BS RS
Subscriber station SS
RS
Relay station SS Backbone link
Figure 2.5. A partial network in a multicellular relay network.
2.4
General system model for virtual antenna arrays (VAAs) The underlying principle for cellular deployment of VAAs is depicted in Figure 2.6. A BS array, consisting of several antenna elements, transmits a spacetime encoded data stream to the associated mobile terminals which can form several independent VAA groups. Each mobile station within a group receives the entire data stream, extracts its own information, and concurrently relays further information to the other mobile terminals. It then receives more of its own information from the surrounding mobile terminals and, ﬁnally, processes the entire data stream. The wired links within a traditional receiving antenna array are thus replaced by wireless links. The same principle is applicable to the uplink [15]. In this situation, the VAA accomplishes a special type of network which bridges cellular and adhoc concepts to establish a heterogeneous network with increased capacity. It calls for intelligent synchronization, relaying, and data scheduling algorithms, the exact realization of which depends on the access scheme, the choice of main link technology, the choice of relaying technology, the technological limits, the number of antennas within a given geographical area, and other factors, e.g., the ability of the cellular system to synchronize users, etc. [15]. The deployment of VAAs creates various challenges which need to be addressed, for instance, how to enable the terminals to transmit and receive simultaneously and thus to operate in fullduplex mode. Of greater importance is the actual relaying process. Like in satellite transponders, the signal can be retransmitted using a transparent or regenerative relay. A transparent relay is generally
2.5 RRA in OFDMAbased relay systems
21
VAA MS
MS MS
MS
MS MS MS
MS
MS
MS BS
VAA
VAA cell
VAA
Figure 2.6. VAAs in cellular deployment.
easier to deploy since only a frequency translation is required. However, additions to the current standards are required. For a simple adaptation of VAA to current standards, regenerative relays should be deployed. This generally requires more computational power, but has been shown to increase the capacity of the network [28]. A generic realization of VAAs, which is henceforth referred to as distributedMIMO multihop relaying network was introduced in [29]. An example realization is depicted in Figure 2.7. Here, a source MS communicates with a target MS via a number of relaying MSs. Spatially adjacent relaying MSs form a VAA; each MS receives data from the previous VAA and relays data to the next VAA until the target MS is reached. Note that each of the terminals involved may have more than one antenna element. Furthermore, an arbitrary number of MSs of the same VAA may cooperate with each other. The suggested topology, depicted in Figure 2.7, encompasses a variety of communications scenarios. For instance, a cellular system operating on the downlink is obtained by replacing the source MS by the BS antenna array which communicates directly with the VAA containing the target MS. It may also represent a system where a BS array communicates with a VAA formed somewhere in the cell, which in turn relays the data to another VAA containing the target MS. This allows the coverage area of the BS to be extended. But as mentioned before, RRA algorithms should be designed to exploit the potential gains in all aforementioned wireless relay networks, by adaptively distributing scarce communication resources to either maximize or minimize some network performance metrics. We review the RRA algorithms proposed in the literature for relayenhanced OFDMAbased and MIMO wireless networks as well as distributedMIMO multihop systems.
2.5
RRA in OFDMAbased relay systems: general form In relayenhanced OFDMAbased wireless networks, a typical resource allocation problem statement might be as follows [30]:
22
Cooperative communications in OFDM and MIMO cellular relay networks
1st hop
2nd hop
Lth hop
D
S
Figure 2.7. DistributedMIMO multistage communications system. How many information bits, how much transmit power (for SSs, RSs, or BSs), which and how many subcarriers (subcarrier assignment and allocation, respectively) should be assigned to SSs to either maximize or minimize a desired performance metric, e.g., system throughput (capacity) or total transmit power in the network, respectively?
Assume Ku = {1, 2, ..., K} and Nu = {1, 2, ..., N } are the sets of users and subcarriers respectively. The data rate of the kth user Rk is given by [30] Rk =
N B ck ,n log2 (1 + γk ,n ), N n =1
(2.1)
where B is the total bandwidth of the system and ck ,n is the subcarrier assignment index indicating whether the kth user occupies the nth subcarrier. ck ,n = 1 only if subcarrier n is allocated to user k; otherwise it is zero. The bandwidth of each subchannel is B/N = 1/T where T is the OFDM symbol duration. γk ,n is the signaltonoise ratio (SNR) of the nth subcarrier for the kth user and is given by [30] γk ,n = pk ,n Hk ,n =
pk ,n h2k ,n , N0 B/N
(2.2)
where pk ,n is the power allocated for user k in subchannel n, and hk ,n and Hk ,n denote the channel gain and channeltonoise ratio for user k in subchannel n, respectively. From (2.1), the total data rate RT of a zero margin system is given by N K B RT = ck ,n log2 (1 + γk ,n ). N n =1
(2.3)
k =1
Knowing the modulation scheme, γk ,n , the eﬀective SNR, is adjusted accordingly to meet the BER requirements. The general form of the subcarrier and power allocation problem is [30]: max
c k , n ,p k , n
RT =
N K B ck ,n log2 (1 + γk ,n ) N n =1 k =1
(2.4)
2.6 Dynamic RA RRA in OFDMA relay networks
23
or min
c k , n ,p k , n
PT =
N K B ck ,n pk ,n , N n =1
(2.5)
k =1
subject to: C1: C2:
ck ,n ∈ {0, 1}, ∀k, n, K ck ,n = 1, ∀n, k =1
C4:
pk ,n ≥ 0, ∀k, n, N K ck ,n pk ,n ≤ Ptotal ,
C5:
user rate requirement.
C3:
(2.6)
k =1 n =1
The problem can be formulated with two possible objectives followed by various constraints (C1–C5). The ﬁrst two constraints are on subcarrier allocation to ensure that each subchannel is assigned to only one user. C4 is only eﬀective in problems where there is a power constraint Ptotal on the total transmit power of the system PT (e.g., rate adaptive algorithms). C5 determines the ﬁxed or variable rate requirements of the users. In each class, the problem is formulated accordingly and the optimal solution is derived using diﬀerent optimization techniques. Due to the high computational complexity of the optimal solutions, they may not be practical in realtime applications. As a result, suboptimal algorithms have been developed which diﬀer mostly in the approach they choose to split the procedure into several (preferably independent) steps to make the problem tractable and, in their simplifying assumptions to reduce the complexity of the allocation process. The performance of each algorithm greatly depends on the formulation of the problem and the validity of these simplifying assumptions. Two major classes of dynamic resource allocation schemes have been reported in literature [30]: margin adaptive (MA) schemes [31–33], and rate adaptive (RA) schemes [34–36]. The optimization problem in MA allocation schemes is formulated with the objective of minimizing the total transmit power while providing each user with its required QoS in terms of data rate and BER. The objective of the RA scheme is to maximize the total data rate of the system with the constraint on the total transmit power. Both classes are overviewed and discussed in the following sections.
2.6
Dynamic RA RRA in OFDMA relay networks Diﬀerent scenarios of centralized or distributed algorithms, along with singlecell and multicell network topologies have been considered in the literature for the RA RRA problem.
24
Cooperative communications in OFDM and MIMO cellular relay networks
2.6.1
Centralized RA RRA schemes in singlecell OFDMA relay networks A downlink singlecell network with a single ﬁxed RS was considered in [37], while such a network with multiple ﬁxed RSs was studied in [38]. In both [37] and [38], timedivisionbased halfduplex transmission and adaptive modulation and coding (AMC) were assumed. In [37] two algorithms (ﬁxed timedivision and adaptive timedivision) were proposed to improve the cell throughput and coverage while minimizing complexity and overhead requirements. However, the objective function in [38] was the total average throughput of both the direct and relayed links presented as a function of SNRs and some indicator (optimization) variables, with constraints as in [37]. In both the proposed algorithms, the BS transmission frame is followed by the transmission of an RS frame. The ﬁrst algorithm performs subcarrier allocation with a predetermined equal power allocation (the same level for both BS and RS). The second algorithm achieves an optimal joint power and subcarrier allocation. Simulation results showed that as the number of RSs increases, the sum rate increases, while the joint allocation algorithm continues to outperform the ﬁxed power allocation algorithm. In [39], the authors used the concept of utility functions to formulate the problem of RA in multiuser OFDM systems. Utility maps the network resources a user utilizes into a real number and is a function of the user’s data rate. The utilitybased dynamic resource allocation problem is formulated as: max
c k , n ,p k , n
K
Uk (Rk ),
(2.7)
k =1
subject to: C1:
Si ∩ Sj = ∅, ∀i, j ∈ Ku , i = j,
C2:
∪k Sk ⊆ {1, 2, ..., N },
C3:
pk ,n ≥ 0, ∀k, n,
C4:
N K
ck ,n pk ,n ≤ Ptotal ,
(2.8)
k =1 n =1
where Uk (Rk ) is the utility function for the kth user. Rk is deﬁned as in (2.1), Sk is the set of subcarriers assigned to user k for which ck ,n = 1. In [39] the extreme case of an inﬁnite number of orthogonal subcarriers each with an inﬁnitesimal bandwidth was investigated by introducing two theorems: theorem I gave the optimal subcarrier allocation assuming a ﬁxed power allocation on all the subcarriers and theorem II gave the optimal power allocation given a ﬁxed subcarrier allocation. Combining the results of the two theorems, the optimal frequency set and the power allocation for the extreme case were obtained. It is obvious from (2.7) and (2.8) that fairness among users was not considered. For RA schemes for OFDMA without relaying, it has been shown in literature, e.g., [40], that optimization can be achieved when a subcarrier is assigned to only one user who has the best channel gain for that subcarrier, and also that
2.6 Dynamic RA RRA in OFDMA relay networks
25
equal power allocation among subcarriers has almost the same performance as waterﬁlling transmit power adaptation but with less complexity. In [34], the authors imposed proportional rate constraints in the OFDMA relaying network to ensure that each user could achieve the required data rate. However, all the transmitters, including the BS and RSs, are limited to one ﬁxed transmitting power, which is not ﬂexible and this is not practical in a relaying network. To tackle the mixed integer and continuous variable optimization problem in (2.7)–(2.8) with Uk (Rk ) = Rk , the authors of [41] proposed a greedy subcarrier and power coallocation algorithm based on a new theorem. The theorem states that in general situations, i.e., when all link gains between RSs and SSs are diﬀerent from each other, for each SS and subcarrier allocated, there is only one RS among all RSs that has pk ,n = 0. In other words, for each symbol transmitted from the BS to the SS, only one RS needs to relay this symbol, given the power constraint on each RS. This also suggests that using subcarrier allocation without considering power allocation may not work well because many subcarriers at each RS may not have been allocated power even though they have been chosen [41]. The centralized subcarrier and power allocation algorithm which maximizes system capacity with a constraint on the overall transmission power when there is an LoS between the BS and the SS was studied in [42]. Hence, the aggregate system throughput was deﬁned as RT =
i∈D
Ri +
K
Rk ,
(2.9)
k =1
where D represents the set of direct links. Similar to the approach taken in [34], the problem was broken down into two steps. In the ﬁrst step, a heuristic algorithm was proposed to assign the subcarriers to each link based on the channel condition. In the second step, an iterative power allocation algorithm was proposed to balance the twohop links for maximizing the endtoend capacity of twohop SSs. In the proposed algorithm, ﬁrstly, the subcarriers are allocated to the links with the best channel gain as an initialization. The overall capacity of each transmission slot is maximized in this step. In the next step, the subcarriers are reallocated to improve the system capacity by iteration. In other words, the basic idea is to take out the worst subcarrier allocated to the ﬁrsthop (secondhop) link of the richest (poorest) RS and to reassign it to the ﬁrsthop (secondhop) link of the poorest (richest) RS. Two cases are compared and the one that can achieve more gain is exploited. The subcarrier reassigning procedure is repeated until no improvement can be achieved [42]. Another interesting technique to deal with the mixed integer optimization problem in (2.7)–(2.8) was proposed in [18]: an OFDMA relay network with multiple sources, multiple relays, and a single destination was investigated and resource allocation was considered with fairness (load balancing) constraints on relay nodes. The authors’ approach was to transform the integer optimization problem into a linear distribution problem in a directed graph to allow the use of
26
Cooperative communications in OFDM and MIMO cellular relay networks
Relay (multihop) region
Inner (singlehop) region
S1
S1
S2
S2
BS
Figure 2.8. Network layout and spatial reuse pattern.
the linear optimal distribution algorithms available in the literature. The required information is collected at the central unit for allocation decisions (Figure 2.5). It was assumed that the subcarriers allocated in the ﬁrst hop (S–R) are the same as those in the second hop (R–D). This reduction in available frequency diversity gain results in a potential system performance loss. To address this issue, the concept of subcarrier pairing was employed in [43] to attain extra gain in the average system rate. It was shown that a higher performance in terms of mutual information can be achieved if the subchannels of both links, S–R and R–D, are paired according to the actual magnitude of the link gains, i.e., a strong ﬁrst hop subchannel is coupled with a strong second hop subchannel and not with a weak one.
2.6.2
Centralized RA RRA schemes in multicell OFDMA relay networks A centralized downlink OFDMA scenario in a multicellular network enhanced with six ﬁxed relays per cell was considered in [44] and [45]. The proposed scheme allows eﬃcient use of subcarriers via opportunistic spatial reuse within the same cell, as shown in Figure 2.8 (i.e., a set of subcarriers used in a BS–RS link (S1) can be reused after 180degree angular spacing in a RS–SS link), even when no directional antennas are employed. These are probably among the ﬁrst papers to consider such spatial reuse in a multicell environment, although, with a greatly simpliﬁed model. Fading was not considered except for independent lognormal shadowing on links, which means that subcarriers are similar on any particular link. Consequently, the problem was formulated as the minimum number of subcarriers required to satisfy a SS’s QoS. This was used in the transmit scheme selection algorithm (TSSA) that switches among singlehop, multihop, and multihop with spatial reuse [7].
2.6 Dynamic RA RRA in OFDMA relay networks
27
In the TSSA, the only scheme allowed in the interior hexagon is the single hop directly from the BS, whereas outside this region (relay region), the TSSA can choose the multihop with spatial reuse scheme or the multihop scheme, which requires fewer subcarriers to satisfy the QoS requirements. With this strategy, an integer programming optimization problem is formulated to maximize the number of SSs with satisﬁed QoS requirements. It was observed that a significant increase in the number of supported SSs is achieved when applying the TSSA compared with the case where the transmission scheme is restricted in the cell region to a particular scheme, regardless of the channel conditions. This suboptimal scheme potentially reduces the performance gains. By considering latency, overhead and system, and computational complexity, it may be seen that centralized RRA schemes are not the best option for future wireless networks. This has led to the importance of distributed schemes being recognized. However, there has not been much progress on this front.
2.6.3
Distributed RA RRA schemes in OFDMA relay networks A semidistributed downlink OFDMA scheme in the form of two algorithms of separate and sequential allocation (SSA) and separate and reuse allocation (SRA), in a single cell enhanced by M halfduplex ﬁxed relays was considered in [46]. The scheme divides the SSs into disjoint sets located in the neighborhoods of the BS and RSs, an approach that is common in the literature. The SSs attached to the BS and relays are referred to as the BS–SS and RS–SS clusters, respectively. The BS allocates some resources to the BS–SS cluster directly and to the RS–SS clusters through the RS. Implicitly, it is assumed that all routes have been established prior to resource allocation, regardless of the channel conditions, and that the same subcarrier is used on the two hops, BS–RS and RS–SS. The starting point of both the twostep SSA and the SRA algorithm allocation schemes is basically the same. In the ﬁrst step, each RS, along with its SS cluster, is treated as a large SS with a required minimum rate equal to the sum of all the minimum required rates of the SSs in its cluster. The BS allocates the resources among its own SSs and these virtual large SSs. In the second step, the RS allocates resources to the SSs in its cluster based on one of two allocation schemes:
r Resources assigned to that BSRS link in the ﬁrst step are allocated among the connected SSs (SSA).
r The RS reallocates all the N subcarriers to its connected SSs regardless of the BS assignments (SRA). Simulation results for a single cell with one relay showed that the semidistributed scheme, SSA in particular, has a comparable capacity and outage probability performance to the centralized scheme [7]. The SSA algorithm showed signiﬁcant performance stability over the SRA. Since both RS and BS may
28
Cooperative communications in OFDM and MIMO cellular relay networks
assign the same subcarriers to their respective SSs in the SRA algorithm, intracell interference may occur, which results in a considerable increase in outage probability. In general, the proposed semidistributed schemes reduce the amount of overhead required to feedback the CSI and minimum rates to the BS. However, in the case of SRA, there is no need to communicate such information to the BS. These schemes fail to exploit the interference avoidance and traﬃc diversity gains. In addition, there is an inherent loss in performance due to the decoupling of routing and scheduling processes.
2.6.4
RA RRA schemes with fairness in OFDMA relay networks Looking back at utilityfunctionbased optimizations, such as in [39], one way to accomplish both eﬃciency and fairness is to use utility functions that are both increasing and marginally decreasing. As a result, the slope of the utility curve decreases with an increase in the data rate. Choosing a marginally decreasing utility function also guarantees its strict concavity which ensures global optimality as well as the uniqueness of the optimal solution. A logarithmic utility function U (R) = ln(R) is both increasing and marginally decreasing. Therefore, a resource allocation policy using a logarithmic utility function is said to be proportionally fair. Diﬀerent types of utility functions were proposed in [36], [47], and [48], depending on the type of application. A utility function that ensures both eﬃciency and fairness is better obtained through subjective survey than by applying theory. For the general resource allocation problem formulated in (2.4)–(2.6), the optimization with variable rate constraints is given in (2.10)–(2.11), in which the objective is to maximize the total rate within the total power constraint of the system while maintaining rate proportionality among the users indicated in C5: K N pk ,n h2k ,n B , (2.10) RT = ck ,n log2 1 + max B c k , n ,p k , n N N0 N n =1 k =1
subject to: C1: C2:
ck ,n ∈ {0, 1}, ∀k, n, K
ck ,n = 1, ∀n,
k =1
C3: C4:
pk ,n ≥ 0, ∀k, n, N K
ck ,n pk ,n ≤ Ptotal ,
k =1 n =1
C5:
R1 : R2 : . . . : RK = α1 : α2 : . . . : αK .
(2.11)
2.6 Dynamic RA RRA in OFDMA relay networks
29
Here {α1 , α2 , . . . , αK } is a set of predetermined proportional constraints in which αk is a positive real number with αm in = 1 for the user with the lowest required proportional rate. When all αk terms are equal, the objective function in (2.10) is similar to the objective function of the max–min problem introduced in [49]. The authors of [49] studied the max–min problem, where by maximizing the worst user’s capacity, it was ensured that all users achieve the same data rate. It was reported that in a singleuser waterﬁlling solution the total data rate of a zero margin system is close to capacity, even with ﬂat transmit power spectral density (PSD), as long as the energy is poured only into subchannels with good channel gains. Hence, in a system with N subchannels, a ﬂat transmit power over all the subcarriers would always give close to optimum performance. Though acceptable fairness amongst users is achieved in [49] by allocating power uniformly across all subcarriers, the frequency selective nature of a user’s channel is not fully exploited. To improve its performance, Shen et al. [34] added a second adaptive power allocation step to further enforce the rate proportionality among the users. The twostep approach adopted in [34] is as follows: in the ﬁrst step, the modiﬁed version of the algorithm outlined in [49] is employed for subcarrier allocation to achieve coarse proportional fairness. Hence, instead of giving priority to the user with the lowest achieved data rate Rk , priority is given to the user with the lowest achieved proportional data rate, i.e., Rk /αk . In this step, the achieved rate is calculated assuming equal power on all the subcarriers. After subcarrier allocation is carried out, the problem is simpliﬁed into maximization over continuous variables of power. In the second step, the power is reallocated among the users and then among the subcarriers through the use of waterﬁlling to enforce the rate proportionality among the users. Another suboptimal algorithm to solve RA RRA was formulated in [50] as a binary integer programming problem to maximize the Nash product of all users r eq in the form K k =1 (Rk − Rk ) and, at the same time, satisfy users’ minimum rate requirements. It was demonstrated in [51] that the solution achieves a Nash bargaining solution (NBS) fairness, which is a generalization of the wellknown proportional fairness. The algorithm can be divided into two steps. In the ﬁrst step, a subcarrier is randomly selected and then allocated to the user whose required minimum rate is not satisﬁed and whose achievable rate on this subcarrier is the largest, either by direct or by relayed transmission. The process loops until all users achieve their required rates. In the second step, the NBS algorithm assigns the remaining subcarriers to users in such a way that the Nash product for each assignment is maximized. While this scheme is fair in the sense that the user’s rate is determined only by its own channel condition, and not by other competing users’ conditions, it was also shown that the algorithm provides a good tradeoﬀ between the overall system performance and the fairness among users.
30
Cooperative communications in OFDM and MIMO cellular relay networks
2.7
Dynamic centralized margin adaptive RRA schemes in OFDMA relay networks In deriving this group of algorithms, a given set of user data rates is assumed with a ﬁxed QoS requirement. The optimization problem can then be formulated as [30]: min
c k , n ,p k , n
PT =
N K
ck ,n pk ,n ,
(2.12)
k =1 n =1
subject to: C1: C2:
ck ,n ∈ {0, 1}, ∀k, n, K
ck ,n = 1, ∀n,
k =1
C3: C4:
pk ,n ≥ 0, ∀k, n, K N
ck ,n pk ,n ≤ Ptotal ,
k =1 n =1
C5:
Rk ≥ Rk ,m in , k = 1, 2, . . . , K
(2.13)
with the rate requirements indicated in C5. This problem was ﬁrst addressed in [52], where the focus was only on subcarrier allocation, and further in [32], where adaptive power allocation was also considered. With the help of constraint relaxation and to make the problem tractable, the authors of [31] introduced a new parameter to the cost function, taking values within the interval [0,1], which can be interpreted as the sharing factor for each subcarrier; the same technique was also used in [3] and [53] to simplify the optimization problem. With the help of the new parameter, it was shown that the optimization problem can be reformulated as a convex minimization problem over a convex set [30]. Using the standard optimization techniques, the Lagrangian of the new problem is obtained along with the necessary conditions under which not only the minimum total transmit power occurs but also the data rate constraint of each user is satisﬁed. Lagrange multipliers which satisfy the individual data rate constraints can be found using an iterative search algorithm. Each subcarrier is then assigned to only the user that has the largest sharing factor on that subcarrier, using the set obtained in the previous step. However, the iterative computation and search for this algorithm make it prohibitively expensive and highly complex. One solution to simplify the algorithm is to assume that the channel is ﬂat for a certain number of subcarriers, as in [54]. Adaptive resource allocation methods have been shown to oﬀer higher user data rates due to the additional degree of freedom provided by multichannel systems [30]. One way to create multiple channels in a frequency domain is to
2.8 MIMO communications systems
31
use multiple carrier frequencies with the methods and algorithms discussed in this chapter. The other way is with multiple transmit and receive antennas in the spatial domain. The latter is also referred to as MIMO.
2.8
MIMO communications systems MIMO systems have been shown to hold the promise of providing capacity and data rates far exceeding those oﬀered by conventional singleinput singleoutput (SISO) communications systems, and hence are being widely studied for use in wireless systems. MIMO channels have a number of advantages over traditional SISO channels such as beamforming (or array) gain, diversity gain, and multiplexing gain. The beamforming and diversity gains are not exclusive for MIMO channels and also exist in singleinput multipleoutput (SIMO) and multipleinput singleoutput (MISO) channels. The multiplexing gain, however, is a unique characteristic of MIMO channels. Some gains can be simultaneously achieved, while others compete and establish a tradeoﬀ. An excellent overview of the gains of MIMO channels is avaiable in [55]. In a nutshell, the use of multiple dimensions at both ends of a communication link oﬀers signiﬁcant improvements in terms of spectral eﬃciency and link reliability.
2.8.1
RRA in MIMO relay networks The main challenge in pointtomultipoint ﬁxed relaying is to provide a highcapacity link between the BS and the RS, while at the same time providing multiple data links to multiple users [56]. A natural solution to this problem is to exploit the advantages of MIMO systems. A wireless MIMO relay can be regenerative or nonregenerative. It was suggested in [57] that a nonregenerative MIMO relay has the following potential advantages over a regenerative MIMO relay. First, a nonregenerative relay can relay signals faster than a regenerative relay if separate frequency channels are used for the relay’s input and output. Second, deployment of a nonregenerative relay can have little eﬀect on the operations at the source and the destination as there is no handshaking requirement for each packet going through the nonregenerative relay using two frequency channels. Third, a nonregenerative relay contains virtually no information for decoding the source and hence exposes no security information even if it is stolen by an enemy. There has been much research into nonregenerative MIMO relay systems [56– 61]. The focus in the literature has been on the optimal design and power allocation of these systems. In the context of MIMO relays, the power allocation problem is to determine the source covariance matrix and the relay matrix that maximize the system performance. Here we present a review of work on two main categories of MIMOenhanced relaying systems, singleuser singlerelay MIMO
32
Cooperative communications in OFDM and MIMO cellular relay networks
1
Source
Relay
HSR
HRd
2
2 Encode
1
(Decode /encode
Q
Destination
F
M
N
Figure 2.9. DF MIMO relaying system. and multiuser singlerelay MIMO systems as well as distributedMIMO multihop systems.
2.8.2
Optimal design and power allocation in singleuser singlerelay systems For a singleuser twohop MIMO relay system, an optimal structure of the relay matrix that maximizes the S–D mutual information was presented in [57] and [58], and an optimal structure for both the source precoder matrix and the relay matrix was established in [59]. Optimal designs of beamforming and forwarding matrices for both the DF and the AF relaying protocols are presented in the following subsections.
DF relay system We ﬁrst consider a singleuser DF MIMO downlink relay system as illustrated in Figure 2.9, where s ∈ CM ×1 denotes the signal transmitted from the source equipped with M antennas, Q ∈ CM ×M the source precoder (beamforming) matrix, and HS R ∈ CM ×M the S–R channel matrix. Then the signal received at the relay, yR , is given by yR = HS R Qs + nR , where nR is the zeromean Gaussian noise at the relay. The achievable rate at the ﬁrst hop (S–R) can be represented as [62] (2.14) R1 (Q) = 12 log det σR2 IM + HS R QE[ssH ]QH HH SR 2 1 H H (2.15) = 2 log det σR IM + HS R QQ HS R , 1 2 where E[ssH ] = IM , and E[nR nH R ] = σR IM . The 2 represents the rate loss due to transmission in two hops. In the second phase, the relay decodes data (with the assumption of no outage) and forwards them to the destination using the forwarding matrix F. The received signal at the destination is given by yD = HR D Fs + nD . The achievable rate at the second hop (R–D) can be represented as 2 (2.16) R2 (F) = 12 log det σD IN + HR D FFH HH RD .
Finally, the achievable rate of the relay system is bounded by the minimum of R1 and R2 , i.e., R(Q, F) =min(R1 (Q), R2 (F)). Similarly to Section 2.6, the RA optimization problem can be formulated as max Q ,F
R(Q, F),
(2.17)
2.8 MIMO communications systems
33
subject to: C1:
tr(QQH ) ≤ PS ,
C2:
tr(FFH ) ≤ PR ,
(2.18)
where PS and PR are the total power constraints at the source and relay, respectively and tr(·) represents the trace of the matrix. From (2.15) and (2.17), we note that the optimization problem can be divided into two MIMO channel rate maximization problems as: max Q
R1 (Q)
max F
R2 (F)
subject to: tr(QQH ) ≤ PS
tr(FFH ) ≤ PR .
(2.19)
It is well known that the optimal solution of (2.19) is singular value decomposition (SVD) with waterﬁlling power allocation.
AF relay system Consider the singleuser AF MIMO downlink relay system as illustrated in Figure 2.10. The achievable rate at the ﬁrst hop (S–R) can be obtained as [62] R1 (Q) =
1 2
log det σR2 IM + HS R QQH HH SR .
(2.20)
In the second phase, the relay simply ampliﬁes its received data and forwards the data to the destination using the forwarding matrix F. The received signal at the destination is given by yD = HR D FyR + nD
(2.21)
= HR D F(HS R Qs + nR ) + nD
(2.22)
= HR D FHS R Qs + HR D FnR + nD .
(2.23)
From (2.23), it is obvious that relay ampliﬁes not only data vector s, but the noise of the ﬁrst hop nR . Though amplifying the noise causes rate loss in the AF relay protocol, the implementation is easy and delay is much smaller than in DF. The achievable rate of the second phase can be derived as H H H H −1 . R2 (Q, F)= 12 log det I+HR D FHS R QQH HH F H (I+H FF H ) RD SR RD RD (2.24) As with DF, the achievable rate of the relay system is bounded by the minimum of R1 and R2 , i.e., R(Q, F) =min(R1 (Q), R2 (Q, F)). Hence, the RA optimization
34
Cooperative communications in OFDM and MIMO cellular relay networks
1
Source
Relay
HSR
HSR
2
2 Encode
1
Destination
F
Q M
N
Channel state information
Figure 2.10. AF MIMO relaying system.
problem can be formulated as max Q ,F
R(Q, F),
subject to: C1: C2:
tr(QQH ) ≤ PS , 2 H tr F(HS R QQH HH ≤ PR . S R + σR I)F
(2.25)
This is more complex than the problem for the DF case and cannot be divided into individual optimization problems. To tackle (2.25), the authors in [57–59] proposed diﬀerent approaches though with the same basic idea. In [57] it was proved that for a relay system without a direct S–D link the optimal relay matrix is given by F = V2 ΛF UH 1 ,
(2.26)
H H H where ΛF is a diagonal matrix, and H1 HH 1 = U1 Σ1 U1 and H2 H2 = V2 Σ2 V2 are the eigenvalue decomposition (EVD) of channel matrices H1 = HS R and H2 = HR D . Hence, F can be considered as a matched ﬁlter along the singular vectors of the channel matrices. When the weighting matrix F obeys a set of canonical coordinates given by (2.26), the MIMO relay channel is decomposed into several parallel SISO channels. If we substitute (2.26) into (2.23) with Q = IM , the signal received at the destination is
˜D , ˜R + n ˜ D = Λ2 ΛF Λ1 ˜s + Λ2 ΛF n y
(2.27)
˜D = where we used the SVD of H1 = U1 Λ1 V1H and H2 = U2 Λ2 V2H . Also, y H H H ˜ ˜ ˜ y , s = U s, n = U n , n = U n . UH D R 2 D 1 2 D 1 R Since the MIMO channel is decomposed into (orthogonal) parallel SISO subchannels, the problem now is how to allocate the total power to those subchannels. The optimization objective is now concave and since the constraint is convex, the problem is easily transformed into a standard convex optimization problem.
2.8 MIMO communications systems
2.8.3
35
Optimal design and power allocation in singlerelay multiuser systems A generalized convex optimization problem was introduced in [63] as J = − log  I + HQHH ,
min Q ≥0
tr(Bi QBH i ) ≤ Pi ,
subject to:
∀i ∈ {1, . . . , m},
(2.28)
(2.29)
where H and Bi are complex matrices, Q is a complex positive semideﬁnite matrix, and Pi are positive numbers. If m = 1, the solution to the above problem can be found by a wellknown waterﬁlling algorithm. It can be proved that the solution of (2.28)–(2.29) is given by Q = W−H V(I − Σ−2 )+ VH W−1 , where W =
m i=1
µi BH i Bi
12
(2.30)
(assumed to be nonsingular), V and Σ are deter
−H
mined by the SVD HW = UΣVH , (.)+ replaces all negative diagonal elements by zeros and leaves all nonnegative diagonal elements unchanged, and µ = (µ1 , . . . , µm ) is the solution to the following dual problem: min µ≥0
− log  I + HQHH  +
subject to:
m
µi (tr(Bi QBH i ) − Pi ),
(2.31)
i=1
Q = W−H V(I − Σ−2 )+ VH W−1 .
(2.32)
A multiuser MIMO relay downlink/uplink system was treated in [63], where rate adaptive and margin adaptive algorithms were presented to maximize the system throughput (i.e., sum rate) under a power constraint, and to minimize the system power consumption under individual user rate constraints, respectively. Similarly, the problem of maximizing the sum rate for all users under a power constraint for the downlink case was considered in [56] where each user has a single antenna. In [56] and [63], the authors showed that with the use of zeroforcing dirty paper coding (ZFDPC) [64] the multiuser interference could be eliminated in a consecutive way, so the problem could eventually be reformulated as (2.28)– (2.29) and solved using (2.30). In other words, with dirty paper coding and QR decomposition, the interference from the ﬁrst stream to the second stream can be virtually eliminated, and so forth. In this way, the source covariance matrix is matched to the right singular vectors of the S–R channel matrix [63], the optimality of which for a singleuser relay system was shown in [59]. Also, the relay matrix is matched to the left singular vectors of the S–R channel matrix.
36
Cooperative communications in OFDM and MIMO cellular relay networks
2.9
RRA in MIMO multihop networks In this section we review the resource allocation strategies in a wireless multihop network, where a source communicates with the destination via a number of relays. In order to avoid interference between the relaying hops, orthogonal access schemes such as frequencydivision multiple access (FDMA) or timedivision multiple access (TDMA) are usually used. However, it can be shown that both access schemes achieve the same capacities [15]. At each relaying node the DF relaying protocol is applied, whereby the data are ﬁrst detected and decoded completely, then reencoded and transmitted to the next relaying nodes [65]. The endtoend connection is therefore accomplished through a number of topologically imposed VAAs. In [66], a suboptimal bandwidth and power allocation algorithm was proposed to maximize the ergodic endtoend link capacity for FDMAbased multihop networks. However, its allocated bandwidth was assumed to be consecutive, which makes it impractical in an OFDMAbased system. In each hop, the capacity can be considered as parallel narrowband subchannels. So the capacity of the ith hop is
βi P γi , i = 1, . . . , L, (2.33) Ci = αi BEγ i log2 1 + αi σR2 B where L is the number of hops (Figure 2.7), αi and βi are the ith hop bandwidth and power fractions, respectively, and P is the source transmit power. Also since the statistics of channel gain do not depend on the frequency, we use γi to denote the channel gain of the ith hop. The capacity of the endtoend link is dictated by the smallest Ci and the goal is to ﬁnd the fractional bandwidth and power allocations for given γi for all i, such as to maximize the minimum capacity. In other words, an optimal resource allocation strategy should assign fractional power and bandwidth to each hop in such a way that the endtoend capacity is maximized: C = max min{C1 , . . . , CL }, α i ,β i
subject to:
L i=1
αi = 1,
L
βi = 1.
(2.34)
i=1
The strategy was shown to be of low complexity and to achieve nearmaximum endtoend ergodic capacity [66]. Similar resource allocation strategies were introduced in [28] to maximize the endtoend data throughput over ergodic fading channels under the assumption of ﬁxed total transmit power. For the same power constraint, an allocation strategy for minimizing the error rate was given in [67], where a power allocation solution to reduce pairwise error probability (PEP) for a twohop wireless network with an AF relaying protocol was introduced. It was shown that the optimal power allocation assigns half the total power to the source and the other half is shared
2.9 RRA in MIMO multihop networks
37
between the relays. However, as the majority of today’s wireless communications happen over slowfading channels, i.e., nonergodic in the capacity sense, the consideration of the endtoend outage probability is of greater practical relevance. Outage probability can be expressed as the probability that the channel cannot support an errorfree transmission at a speciﬁc data rate. In addition, approaches for minimizing the transmit power are desired. The task of minimizing the total power while meeting a given endtoend outage probability constraint was analyzed in [68] and [69]. In [68], the authors introduced an eﬃcient nearoptimal power allocation strategy for symmetric distributed MIMO networks by solving a highorder equation in one variable. For the case of a large number of relaying nodes per VAA, a closedform solution was proposed by approximating the highorder equation to a quadratic equation. Based on these results an eﬃcient closedform solution for an arbitrary number of nodes per VAA was presented in [69]. According to the capacity of a MIMO channel determined in [4], the capacity of a MISO system is given by Ck ,j = αk B log2
tk 1 Pk ,i 1+  hk ,i,j 2 αi σR2 B i=1 dk ,i,j
(2.35)
where Pk ,i is the transmission power of the ith node at the kth hop. The channel from the tk transmit nodes to the jth receive node at the kth hop is expressed as hk ,j and its elements hk ,i,j obey the same uncorrelated Rayleigh fading statistics. It is assumed that the relaying nodes belonging to the same VAA are spatially suﬃciently close to justify a common pathloss between the two VAAs; this pathloss is given by dk ,i,j , where dk is the distance between two nodes and is the pathloss exponent, which is within the range of 2–5 for most wireless channels. The outage probability Pou t,k ,j can be expressed as Pou t,k ,j = Pr{R > Ck ,j }. If all of the receiving nodes of a hop cannot decode the message, the corresponding hop is in outage. Consequently, the endtoend connection is in outage if any hop is broken, and the endtoend outage probability corresponds to [70] Pe2e = 1 −
rk L
(1 − Pou t,k ,j ),
(2.36)
k =1 j =1
where rk is the number of successful receivers at the kth hop. Therefore, the optimization task could be deﬁned as min
Ptotal =
L
Pk (1 − Pou t,k ,j ),
k =1
subject to: Pe2e ≤ e,
L k =1
αk = 1.
(2.37)
38
Cooperative communications in OFDM and MIMO cellular relay networks
The assumption that should be noted and modiﬁed here is that the number of nodes at each VAA is considered constant and equal to tk . Optimally assigning resources in terms of fractional time and transmission power to the hops on the basis of network geometry, i.e., the number of nodes per VAA and the distances of the hops was considered in [70] and [71]. A power allocation problem and a joint power and time allocation (JPTA) problem were formulated. As shown in [70], the optimization problems are convex and can therefore be solved by common optimization tools. All of the resource allocation schemes require the solution of the optimization task at a central position and the distribution of the time fractions and power allocations to the diﬀerent nodes. However, if the geometry changes, i.e., the number of nodes per VAA or the hop distance changes, the resource allocation solution has to be adapted.
2.10
Conclusion In this chapter, we have ﬁrst presented a brief overview of relaying and its basic protocols, and introduced the concept of VAA as the main idea behind the MIMO multihop systems. This has been followed by a survey of algorithms in the literature which adaptively allocate the available resources in a singleuser/multiuser relayenhanced OFDMAbased, MIMO, and distributed MIMO wireless networks. Diﬀerent classes of algorithms consider diﬀerent objectives and attempt to obtain a solution that is close to optimum but at the same time simple enough to be implemented. Numerous publications have highlighted the need for eﬃcient resource management in such networks. Although RRA in OFDMAbased relay networks has begun to attract attention, only a few papers have investigated OFDMAbased relay networks in multicellular environments. Some of the proposed algorithms are designed to reduce the required signaling overhead compared with the optimal solution, when the optimal solution involves prohibitive complexity. There are ongoing research eﬀorts towards ﬁnding more eﬃcient centralized RRA algorithms, since most of those proposed in the literature are overly complex. Furthermore, signiﬁcant savings in overhead and system complexity can be obtained through distributed resource allocation schemes. Therefore, research into distributed RRA algorithms in OFDMAbased relay networks has begun to attract attention.
References [1] J. N. Laneman, Cooperative diversity in wireless networks: algorithms and architectures, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 2002.
References
39
[2] S. Mallick, P. Kaligineedi, M. M. Rashid, and V. K. Bhargava, “Radio resource optimization in cooperative wireless communication networks,” Chapter 8, in Cooperative Cellular Wireless Networks. Cambridge University Press, 2011. [3] L. B. Le, S. A. Vorobyov, K. Phan, and T. L. Ngoc, “Resource allocation and QoS provisioning for wireless relay networks,” in QualityofService Architectures for Wireless Networks: Performance Metrics and Management. IGI Global, 2009. [4] I. E. Telatar, “Capacity of multiantenna Gaussian channels,” Eur. Trans. Telecommun. (ETT), vol. 10, pp. 585–595, Nov.–Dec. 1999. [5] A. Goldsmith, “Capacity limits of MIMO Channels,” IEEE J. Selected Areas in Commun., vol. 21, no. 5, June 2003. [6] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior,” IEEE Trans. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [7] M. Salem et al., “An overview of radio resource management in relayenhanced OFDMAbased networks,” IEEE Commun. Surveys and Tutorials, to appear. [8] M. Salem, A. Adinoyi, H. Yanikomeroglu, and D. Falconer, “Opportunities and challenges in OFDMAbased cellular relay networks: A radio resource management perspective,” IEEE Trans. Vehicular Technology, to appear. [9] R. Pabst et al., “Relaybased deployment concepts for wireless and mobile broadband radio,” IEEE Wireless Communication Magazine, vol. 42, no. 9, pp. 80–89, Sept. 2004. [10] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversitypart I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927– 1938, Nov. 2003. [11] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversitypart II: Implementation aspects and performance analysis system,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1939–1948, Nov. 2003. [12] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Magazine, vol. 42, no. 10, pp. 74–80, Oct. 2004. [13] Y. W. Hong, W. J. Huang, F. H. Chiu, and C. C. J. Kuo, “Cooperative communications in resourceconstrained wireless networks,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 47–57, May 2007. [14] Y. Liang and V. V. Veeravalli, “Gaussian orthogonal relay channel: Optimal resource allocation and capacity,” IEEE Trans. Inform. Theory, vol. 51, no. 9, pp. 3284–3289, Sept. 2005. [15] M. Dohler, Virtual antenna arrays, Ph.D. dissertation, King’s College London, London, UK, 2003. [16] C. Bae and D.H. Cho, “Fairnessaware adaptive resource allocation scheme in multihop OFDMA systems,” IEEE Commun. Letters, vol. 11(2), pp. 134– 136, Feb. 2007.
40
Cooperative communications in OFDM and MIMO cellular relay networks
[17] O. Oyman and A. J. Paularj, “Design and analysis of linear distributed MIMO relaying algorithms,” IEE Proc. on Commun., vol. 153, no. 4, pp. 565–572, Aug. 2006. [18] G. Li and H. Liu, “Resource allocation for OFDMA relay networks with fairness constraints,” IEEE J. on Selected Areas in Commun., vol. 24(11), pp. 2061–2069, Nov. 2006. [19] E. C. Van der Meulen, “Threeterminal communication channels,” Adv. Appl. Prob., vol. 3, pp. 120–154, 1971. [20] T. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inform. Theory, vol. IT25, no. 5, pp. 572–584, Sep. 1979. [21] T. Cover and J. A. Thomas, Elements of Information Theory. John Wiley & Sons, Inc., 1991. [22] A. H. Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Trans. Inform. Theory, vol. 51, no. 6, pp. 2020–2040, June 2005. [23] R. Nabar, H. Bolcskei, and F. Kneubuhler, “Fading relay channels: Performance limits and spacetime signal design,” IEEE J. Selected Areas in Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004. [24] H. Ochiani, P. Mitran, and V. Tarokh, “Variable rate two phase collaborative communications protocols for wireless networks,” IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 4299–4313, Sep. 2006. [25] O. Munoz, J. Vidal, and A. Agustin, “Nonregenerative MIMO relaying with channel state information,” in Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP’05), Philadelphia, USA, Mar. 2005 IEEE, 2005. [26] A. Wyner and J. Ziv, “The ratedistortion function for source coding with side information at the decoder,” IEEE Trans. Inform. Theory, vol. 22, no. 6, pp. 1986–1992, Nov. 1976. [27] H. W. Je, B. Lee, S. Kim, and K. B. Lee, “Design of nonregenerative MIMOrelay system with partial channel state information,” in Proc. of IEEE International Conference on Communications, pp. 4441–4445, May 2008. IEEE, 2008. [28] M. Dohler, A. Gkelias, and H. Aghvami, “A resource allocation strategy for distributed MIMO multihop communication systems,” IEEE Commun. Letters, vol. 8, no. 2, pp. 98–101, Feb. 2004. [29] M. Dohler, F. Said, A. Ghorashi, and H. Aghvami, “Improvements in or relating to electronic data communication systems,” Publication no. WO 03/003672, priority date 28 June 2001. [30] S. Sadr, A. Anpalagan, and K. Raahemifar, “Radio resource allocation algorithms for the downlink of multiuser OFDM communication systems,” IEEE Commun. Surveys and Tutorials, vol. 11(3), pp. 92–106, 3rd Quarter 2009. [31] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit and power allocation,” IEEE J. Select. Areas in Commun., vol. 17, pp. 1747–1758, Oct. 1999.
References
41
[32] G. Zhang, “Subcarrier and bit allocation for realtime services in multiuser OFDM systems,” in Proc. of IEEE International Conference on Communications (ICC’07), vol. 5, pp. 2985–2989, June 2004. IEEE, 2004. [33] L. Xiaowen and Z. Jinkang, “An adaptive subcarrier allocation algorithm for multiuser OFDM system,” in Proc. of IEEE Vehicular Technology Conference (VTC’03), vol. 3, pp. 1502–1506, Oct. 2003. IEEE, 2003. [34] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Trans. Wireless Commun., vol. 4, pp. 2726–2737, Nov. 2005. [35] Z. Shen, J. G. Andrews, and B. L. Evans, “Optimal power allocation in multiuser OFDM systems,” in Proc. of IEEE Globecom, vol. 1, pp. 337–341, Dec. 2003. IEEE, 2003. [36] G. Song and Y. G. Li, “Crosslayer optimization for OFDM wireless networkspart II: Algorithm development,” IEEE Trans. Wireless Commun., vol. 4, pp. 625–634, Mar. 2005. [37] M. Kaneko and P. Popovski, “Radio resource allocation algorithm for relayaided cellular OFDMA system,” in Proc. of IEEE International Conference on Communications (ICC’07), pp. 4831–4836, June 2007, IEEE, 2007. [38] W. Nam, W. Chang, S.Y. Chung, and Y. Lee, “Transmit optimization for relaybased cellular OFDMA systems,” in Proc. of IEEE International Conference on Communications (ICC’07), pp. 5714–5719, June 2007. IEEE, 2007. [39] G. Song and Y. G. Li, “Utilitybased joint physicalMAC layer optimization in OFDM,” in Proc. IEEE Global Communications Conference (Globecom), vol. 1, pp. 671–675, Nov. 2002. IEEE, 2002. [40] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE J. Selected Areas in Commun., vol. 21, pp. 171–178, 2003. [41] Y. Pan, A. Nix, and M. Beach, “Resource allocation techniques for OFDMAbased decodeandforward relaying networks,” in Proc. of IEEE Vehicular Technology Conference (VTC’08), pp. 1717–1721, May 2008, IEEE, 2008. [42] L. Huang, M. Rong, L. Wang, Y. Xue, and E. Schulz, “Resource allocation for OFDMAbased relay enhanced cellular networks,” in Proc. of IEEE Vehicular Technology Conference (VTC’07), pp. 3160–3164, April 2007, IEEE, 2007. [43] I. Hammerstrom and A. Wittneben, “Power allocation schemes for amplifyandforward MIMOOFDM relay links,” IEEE Trans. Wireless Communications, vol. 6, no. 8, pp. 2798–2802, Aug. 2007. [44] J. Lee, S. Park, H. Wang, and D. Hong, “QoSguaranteed transmission scheme selection for OFDMA multihop cellular networks,” in Proc. of IEEE International Conference on Communications (ICC’07), pp. 4587– 4591, June 2007. IEEE, 2007. [45] J. Lee, S. Park, H. Wang, and D. Hong, “QoSguaranteed transmission mode selection for eﬃcient resource utilization in multihop cellular networks,” IEEE Trans. Wireless Commun., vol. 7, no. 10, pp. 3697–3701, Oct. 2008.
42
Cooperative communications in OFDM and MIMO cellular relay networks
[46] M. Kim and H. Lee, “Radio resource management for a twohop OFDMA relay system in downlink,” in Proc. of IEEE Symposium on Computers and Communications, pp. 25–31, July 2007, IEEE, 2007. [47] G. Song and Y. G. Li, “Crosslayer optimization for OFDM wireless networkspart I: Theoretical framework,” IEEE Trans. Wireless Commun., vol. 4, pp. 614–624, March 2005. [48] Z. Jiang, Y. Ge, and Y. G. Li, “Maxutility wireless resource management for besteﬀort traﬃc,” IEEE Trans. Wireless Commun., vol. 4, pp. 100–111, Jan. 2005. [49] W. Rhee and J. M. Cioﬃ, “Increase in capacity of multiuser OFDM system using dynamic subchannel allocation,” in Proc. of IEEE Vehicular Technology Conference (VTC’00), vol. 2, pp. 1085–1089, May 2000. IEEE, 2009. [50] K. Chen, B. Zhang, D. Liu, J. Li, and G. Yue, “Fair resource allocation in OFDMA twohop cooperative relaying cellular networks,” in Proc. of IEEE Vehicular Technology Conference (VTC’09), pp. 1–5, Sept. 2009, IEEE, 2009. [51] Z. Han, Z. Ji, and K. J. R. Liu, “Fair multiuser channel allocation for OFDMA networks using Nash bargaining solutions and coalitions,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1366–1376, Aug. 2005. [52] C. Y. Wong, C. Y. Tsui, R. S. Cheng, and K. B. Letaief, “A realtime subcarrier allocation scheme for multiple access downlink OFDM transmission,” in Proc. of IEEE Vehicular Technology Conference (VTC’99), vol. 2, pp. 1124–1128, Sept. 1999. IEEE, 1999. [53] T. C.Y. Ng and W. Yu, “Joint optimization of relay strategies and resource allocations in cooperative cellular networks,” IEEE J. Selected Areas in Commun., vol. 25, no. 2, pp. 328–339, Feb. 2007. [54] H. Yin and H. Liu, “An eﬃcient multiuser loading algorithm for OFDMbased broadband wireless systems,” in Proc. of IEEE Global Communications Conference (GLOBECOM), vol. 1, pp. 103–107, Nov. 2000. IEEE, 2000. [55] S. Barbarossa, Multiantenna Wireless Communication Systems. Artech House, 2004. [56] C. Chae, T. Tang, R. Heath, and S. Cho, “MIMO relaying with linear processing for multiuser transmission in ﬁxed relay networks,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 727–738, Feb. 2008. [57] X. Tang and Y. Hua, “Optimal design of nonregenerative MIMO wireless relay,” IEEE Trans. Wireless Commun., vol. 6, pp. 1398–1407, Apr. 2007. [58] O. MunozMedina, J. Vidal, and A. Agustin, “Linear transceiver design in nonregenerative relays with channel state information,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2593–2604, June 2007. [59] Z. Fang, Y. Hua, and J. Koshy, “Joint source and relay optimization for a nonregenerative MIMO relay,” IEEE Workshop Sensor Array MultiChannel Processing, Waltham, MA, Jul. 2006, IEEE, 2006.
References
43
[60] Y. Fan and J. Thompson, “MIMO conﬁgurations for relay channels: Theory and practice,” IEEE Trans. Wireless Commun., vol. 6, pp. 1774–1786, May 2007. [61] Y. Rong, X. Tang, and Y. Hua, “A uniﬁed framework for optimizing linear nonregenerative multicarrier MIMO relay communication systems,” IEEE Trans. Signal Process., vol. 57, no. 12, pp. 4837–4852, Dec. 2009. [62] N. Varanese, O. Simeone, Y. BarNess, and U. Spagnolini, “Achievable rates of multihop and cooperative MIMO amplifyandforward relay systems with full CSI,” in Proc. of Signal Processing Advances in Wireless Communications (SPAWC’06), July 2006. IEEE, 2006. [63] Y. Yu and Y. Hua, “Power allocation for a MIMO relay system with multipleantenna users,” IEEE Trans. Signal Process., vol. 58, no. 5, May 2010. [64] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multipleinput multipleoutput broadcast channel,” IEEE Trans. Inform. Theory, vol. 52, pp. 3936–3964, Sept. 2006. [65] Y. Lang, D. Wubben, C. Bockelmann, and K. D. Kammeyer, “A closed power allocation solution for outage restricted distributed MIMO multihop networks,” in Proc. of Workshop on Resource Allocation in Wireless Networks (RAWNET), Berlin, Germany, Mar. 2008, pp. 65–70. IEEE, 2008. [66] M. Dohler, “Resource allocation for FDMAbased regenerative multihop links,” IEEE Trans. Wireless Commun., vol. 3, no. 6, pp. 1989–1993, Nov. 2004. [67] Y. Jing and B. Hassibi, “Distributed spacetime coding in wireless relay networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524–3536, Dec. 2006. [68] Y. Lang, D. Wubben, and K. D. Kammeyer, “Eﬃcient power allocation for outage restricted asymmetric distributed MIMO multihop networks,” in Proc. of IEEE International. Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), Cannes, France, Sept. 2008, IEEE, 2008. [69] D. Wubben and Y. Lang, “Nearoptimum power allocation solutions for outage restricted distributed MIMO multihop networks,” in Proc. of IEEE Global Communications Conf. (GLOBECOM), New Orleans, LA, Nov. 2008. IEEE, 2008. [70] Y. Lang, D. Wubben, and K. D. Kammeyer, “Joint power and time allocations for adaptive distributed MIMO multihop networks,” in Proc. of IEEE Vehicular Technology Conference (VTC), Spring, Barcelona, Spain, Apr. 2009. IEEE, 2009. [71] D. Wubben, “High quality endtoendlink performance,” IEEE Vehicular Technology Magazine, vol. 4, no. 3, pp. 26–32, Sept. 2009.
Part II Cooperative base station techniques
3
Cooperative base station techniques for cellular wireless networks Wibowo Hardjawana, Branka Vucetic, and Yonghui Li
3.1
Introduction The spectral eﬃciency of existing cellular networks [1] is limited by interference. In cellular mobile networks, the dominant interference comes from adjacent cells [2]. This is especially true when the users are located near the cell edges where the interference from the adjacent cells is very strong. By getting the adjacent base stations (BS) to cooperate, spatial antenna diversity in each BS can be utilized to cancel the interference. To obtain BS cooperation, multiple BSs share information about the transmitted messages to their respective users and wireless channels via a backbone network. Each BS can transmit either a single symbol stream or multiple symbol streams to its respective mobile station (MS). Individual BSs and MSs are equipped with multiple transmit and receive antennas, respectively. Each BS transmitter uses the transmitted signal information from other BSs and wireless channel conditions to precode its own signal. The precoded signal for each BS is broadcast through all BS transmit antennas in the same frequency band and time slots. The precoding operation and transmit– receive antenna coeﬃcients are chosen in such a way as to minimize the interference coming from other BS transmissions. The calculated receive antenna coeﬃcients are then sent from the transmitter to the receiver through a wireless channel prior to the data transmission. In this chapter, we consider the use of a cooperative BS system to eliminate the interference in cellular networks. We will ﬁrst start by reviewing related work in the literature, and follow this by presenting ﬁrst a system model, and then a cooperative BS system design and its advantages over other existing schemes. Lastly, numerical results and conclusions are presented.
3.1.1
Related work Most of the published papers in this area consider only a downlink multiuser multipleinput singleoutput (MISO) broadcast channel. A variety of methods to suppress interference from other users have been developed. In [3], a method to Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
48
Cooperative base station techniques for cellular wireless networks
maximize individual signaltointerferenceplusnoise ratios (SINRs) by jointly adjusting the transmit weights and transmission powers was developed. A different approach was proposed in [4] where a combination of a zeroforcing (ZF) method, which determines transmit weights by forcing part of the interference to zero, and dirty paper coding (DPC) [5] was used to suppress interference from other users. A more practical approach than the one in [4] was considered in [6] by replacing DPC with Tomlinson–Harashima precoding (THP) [7, 8]. These algorithms, however, only consider single receive antenna scenarios, which are not directly applicable to multipleinput multipleoutput (MIMO) systems. These works on downlink multiuser MISO systems have been extended to incorporate multiple receive antennas. In [9], the authors showed how a ZF method can be used to exploit the availability of multiple receive antennas. Here transmit– receive antenna weights are ﬁrst jointly optimized by a ZF diagonalization technique and then a waterﬁlling power allocation method is applied to allocate power to each user. The scheme in [9] was further improved by using an iterative method in [10]. Nonlinear methods, utilizing a combination of a ZF method with DPC and a combination of a ZF method with THP [7, 8] for a multiuser MIMO system were considered in [11, 12], respectively. The authors use the ZF method to eliminate part of the interlink interference. DPC or THP is then used to cancel the remaining interference. These schemes, however, are not practical for cooperative MIMO systems, since their symbol error rate (SER) performance varies from user to user. In particular, this SER variation is not desirable since the MIMO systems can be deployed by diﬀerent operators and they expect the systems to have a similar performance.
3.1.2
Description of the proposed scheme In this chapter, we propose a cooperative transmission scheme employing precoding and beamforming for the downlink of a MIMO system. In this algorithm, the THP cancels part of the interference while the transmit–receive antenna weights cancel the remaining interference. A novel iterative method based on [13] is used to generate the transmit–receive antenna weights. The receive and transmit weights are optimized iteratively until the SINR for each user converges to a ﬁxed value. In addition to the iterative joint transmit–receive antenna weights optimization and THP above, we also employ SINR equalization, and an adaptive precoding ordering (APO) in the algorithm. The SINR equalization process [3] is used to allocate power to users in such a way that all users have the same SINRs. This ensures SER fairness among all users. We consider two types of power constraints. The ﬁrst one is a total BS power constraint, where the total power for all BSs is constrained to a particular value. The second one is a per BS power constraint where the power for each BS is constrained to a particular value. We derive expressions for optimum power allocation under these two power constraints. The APO is then used to improve the performance of MIMO systems further by maximizing the minimum SINR for each user [14]. The proposed method oﬀers a signiﬁcant improvement over a nonlinear cooperative precoding
3.2 System model
49
algorithm presented in [9–12]. The improvements are a signiﬁcant enhancement of the SER performance and a signiﬁcant computational complexity reduction. These features allow the proposed algorithm to be applied to a wider range of scenarios than the schemes in [9–12], while providing a capacityapproaching performance in cellular mobile systems.
3.1.3
Notation The notation used in this chapter is as follows. We use boldface lower case letters to denote vectors and boldface upper case letters to denote matrices. The superscripts .H , .T , I and Diag() denote the conjugate transpose, transpose, an identity matrix, and a diagonal matrix, respectively. C a×b indicates a complex matrix with a rows and b columns. x , · and  ·  are the greatest integer smaller than x, the Euclidean distance, and the absolute value, respectively. LoT(A) is deﬁned as the operation to extract the lower triangular components of A and to set the other components to zero. UpT(A) is deﬁned as the operation to extract the upper triangular components of A and to set the other components to zero. DiT(A) is deﬁned as the operation to extract the diagonal components of A and to set the other components to zero. Lastly, we deﬁne 1 as a column vector with all entries equal to 1.
3.2
System model In this chapter, we consider a multiuser MIMO system, where K BSs transmit to K MSs. Each BS and MS is equipped with NB S k and NM S k antennas, for k = 1, ..., K, respectively. All BSs cooperate with each other to transmit S sym K bol streams to their respective MSs via N B S = kk = =1 NB S k antennas. Each of these transmissions is deﬁned as a link. In a practical cellular wireless network, if frequencydivision multiple access (FDMA) or orthogonal frequencydivision multiple access (OFDMA) is used, a frequency or time slot can only be allocated to one user at a given time instant. Therefore, only one user can be served by a BS at any one time or frequency slot. Thus, to simplify the analysis, we consider a scenario where each BS transmits at a given time slot to a single mobile station. In a practical system, the proposed scheme enables a BS to communicate with multiple MSs through FDMA and timedivision multiple access (TDMA). The proposed method aims to enable K such base stations deployed by different network operators in the same location to communicate simultaneously with K respective mobile stations, using the same frequency band at a given time slot.
3.2.1
Transmitter structure The proposed transmitter structure with joint precoding and beamforming is shown in Figure 3.1.
50
Cooperative base station techniques for cellular wireless networks
Cooperative transmitter for K BSs
BS 1
x1
y1
NR
1
u1
–
+
v1
n1
1 R1
x2
u2
+
–
modM' (•)
ˆ1 u
MS 1
n2
1
BS 2
y1
H1
NBS
d1
+
1
y2
v2 NBS
1 NR
H2
R2
+
y2
modM' (•)
ˆ2 u
MS 2
2
Mperm
d2
P
T K non cooperative MSs
BS K
dK
uK –
xK
+
vK
1 NBS
K
HK
nK
1 NR
RK
+
yK
modM' (•)
ˆK u
MS K
MTHP APO
THP precoding
(a)
Power allocation
Transmit–receive antenna weights optimization
THP decoding
(b)
Figure 3.1. (a) Nonlinear cooperative precoding transmitter structure; c 2009 IEEE). (b) receiver structure (adapted from [27] Let xk = [x1,k · · · xs,k · · · xS,k ]T represent the modulated signal vector, consisting of K M ary quadrature amplitude modulation (MQAM) modulated symbols, where xs,k is the sth modulated symbol stream from BS k intended for MS k. Thus, we have a multistream transmission where S symbol streams are transmitted from BS k to MS k simultaneously. The constellation √points for √ M QAM are drawn from the signal set A = {±1 ± j, ... ± M ± j M }. The modulated symbols for K MSs can then be written as x = [xT1 · · · xTk · · · xTK ]T . The transmitted symbols for each user are ﬁrst permuted by a block diagonal permutation matrix Mper m = [m1 ...mK ], where mi = [0...1S (i−1)+i ...1iS ...0]T , i = 1, ..., K, is a KS × 1 vector with its elements S(i − 1) + i to iS set to 1 and its other elements set to 0. The permutation operation is done by changing the location of mi in Mper m . By doing this, we have K! possible Mper m . The operation of selecting Mper m is referred to as the APO. The APO adaptively selects the precoding order of x that maximizes the minimum SINR of K users. It selects a suitable permutation matrix Mper m to permute x. Let u = Mper m x = [uT1 · · · uTj · · · uTK ]T be the permutated transmitted symbol vector, where uj = [u1,j · · · us,j · · · uS,K ]T . Thus, after the APO, xk for MS k is permuted into uj , which will be transmitted in link j. We will explain the APO in more detail later in Section 3.6. Let us ﬁrst assume that we do not use the THP scheme at the BS transmitter. Thus, we omit THP precoding and decoding in Figure 3.1. The SINR equalization module then allocates powers to each symbol in u in such a way that the received SINRs for all KS symbols are equal. This is done by multiplying u with √ √ the matrix P = Diag(P1 , ..., PK ), Pj = Diag( p1,j , ..., pS,j ) where ps,j is the downlink power allocated to the sth symbol in link j, denoted by us,j . The signal for K links is then given as Pu.
3.2 System model
51
The interference in each link needs to be suppressed by multiplying the signal from each link by the transmit antenna weights of all BSs, T ∈ C N B S ×K S , and by the receive antenna weights matrix at the receiver of link j, Rj ∈ C N M S j ×S . The transmitted signal is thus given as xT = TPu. The receiver for each link is shown in Figure 3.1(b). Note that there is no cooperation among the receivers. Let yj ∈ C N M S j represent the received signal matrix for link j. The received signal matrix for K links, denoted by y, y = T T ] , can be written as [y1T ...yK y = HTPu + N,
(3.1)
where H = [H1 ...Hj ...HK ]T in which Hj ∈ C N M S j ×N B S is the channel matrix for link j. N = [nT1 , ..., nTK ]T and T = [T1 , ..., TK ]. nj ∈ C N M S j ×1 is the noise vector for link j. The transmit weight for link j is deﬁned as Tj = [t1,j ...tS,j ] ∈ C N B S ×S , where ts,j is the transmit weight vector for the sth symbol transmitted in link j. After multiplying y by the receive weights matrix R, the received signal vector becomes y = RHTPu + RN,
(3.2)
H H H where R = Diag(RH 1 , ..., RK ) is a block diagonal matrix with R1 , ..., RK as its block diagonal components. Rj is the receive signal matrix for link j and is deﬁned as Rj = [r1,j ...rS,j ], where ri,j is the receive weight vector for stream i in link j. The signal received at MSs is deﬁned as y = [y1 ...yj ...yK ]T ,yj =
[y 1,j , ..., y S,j ]T , where y s,j is the received signal at the input of the THP decoder for sth symbol transmitted in link j. The received signal y, can be further written as y = RHTPu + RN = (D + F + B)Pu + RN,
(3.3)
where D = DiT(RHT), B = UpT(RHT), and F = LoT(RHT); DPu is a vector of scaled replicas of the transmitted symbols for K links. We deﬁne interlink interference as the interference between symbol streams in diﬀerent links and interstream interference as the interference between symbol streams in the same link. FP is deﬁned as the frontchannel interference matrix, since the rows j = 1, ..., KS of FP represent the interlink interference caused by links 1, ..., j − 1 and interstream interference caused by symbol streams 1, ..., s − 1 transmitted in link j to symbol stream s in link j. Similarly, BP is deﬁned as the rearchannel interference matrix, since the rows j = 1, ..., KS of BP represent the interlink interference caused by rear links j + 1, ..., K and the interstream interference caused by symbols s + 1, ..., S in link j to symbol stream s in link j.
3.2.2
THP precoding structure To further improve system performance, we presubtract some of the interference prior to transmission instead of using the transmit–receive weights for symbol
52
Cooperative base station techniques for cellular wireless networks
stream s in link j to cancel frontchannel and rearchannel interference. Because of this, the transmit–receive weight matrix needs to suppress less interference. We will use the THP scheme proposed in [7, 8] and choose to presubtract the frontchannel interference FP. Thus, prior to the power allocation module, THP precodes u into v = [v1 · · · vj · · · vK ]T ∈ C K S , where vj = [v1,j · · · vS,j ]T . The frontchannel interference is then subtracted from u as in [15], v = u + d − (DP)−1 FP v, v ˜
MT
(3.4)
HP
where (DP)−1 is used to normalize√the frontchannel interference with respect to u, d = [d1,1 , ..., dS,K ]T , ds,j = 2 M ∆, and ∆ is a complex number whose real and imaginary parts are suitable integers√selected √ to ensure the real and imaginary parts of vs,j are constrained into ( M , M ]. Here, the integers for ∆ can be found by an exhaustive search across √all integers [15]. d is an oﬀset √ to ensure the energy of v lies between (− M , M ], since the value of v after presubtraction of the frontchannel interference can be very large and exceed √ √ ( M , M ]. Note that if ds,j is selected as above, adding ds,j to us,j is equivalent to performing a modulo operation on ds,j +us,j [1, 12, 15]: us,j = mod√M (˜ vs,j ) = mod√M (ds,j + us,j ), s = 1, ..., S,j = 1, ..., K, where the modulo operation is deﬁned as [6] √ √ √ M √ M us,j + mod M (us,j ) = us,j − M 2
(3.5)
(3.6)
for s = 1, ..., S and j = 1, ..., K. In addition, if we apply √ to (3.4), the oper√ (3.6) ation in (3.6) actually maps v into the interval of (− M , M ] [15]. Thus, by using the modulo operations, we implicitly ﬁnd d that forces the THP precoded symbols v to lie within this interval. In addition as mentioned √ in [15], √ the variance 2 ] = 2M/3 and it is distributed uniformly in (− M , M ]. Note that of v is E[vs,j E[x2s,j ] is 2(M − 1)/3. Thus, there is a power enhancement of M/(M − 1) due to THP. This enhancement needs to be taken into consideration when designing the transmit–receive weights and power allocation. In general, the transmit 2 ] = 1. To ensure this, we scale down the symbol energy is normalized, i.e., E[vs,j ds,j and xs,j by 3/2M . This is discussed in detail in [15]. By taking this into consideration, the operation of the THP precoder in (3.4) can be rewritten as ⎧ 3 ⎪ ⎪ [u]j , j = 1, ⎨ (2M − 1) [v]j = (3.7) ⎪ 3 ⎪ ⎩ modM ( [u]j − aj ), j = 2, .., KS, 2M √ j −1 [MT H P ]j,l [v]j and M = 3/2M M . [MT H P ]j,l denotes the where aj = l=1 (j, l)t component of MT H P and [a]l is the lth component of vector a. Note that the ﬁrst line of (3.7) comes from the fact that, when precoding [v]1 , there is no
3.3 Cooperative BS transmission optimization
53
frontchannel interference to be canceled. Thus, we can simply set [v]1 as the normalized u1,1 . Since now we are using the THP scheme, we are transmitting THP precoded symbol streams v instead of u. After THP precoding, the received signal y in (3.3) can be rewritten by replacing u by v. By using (3.4) and (3.3), the received signal now becomes v + BPv + RN. y = (D + F)Pv + BPv + RN = DP˜
(3.8)
ˆj = The estimates of the transmitted symbols for link j, denoted by u uS,j ]T , can be recovered from yj , by applying an elementwise modulo [ˆ u1,j ...ˆ operator in (3.6) to each ys,j , as u ˆs,j =
y s,j , s = 1, j = 1, modM (y s,j ) , otherwise,
(3.9)
where u ˆs,j is the estimate of us,j . Here, the eﬀect of oﬀset vector d on the desired transmitted signal is removed at the MS receiver by applying the modulo operation in (3.6) to each y s,j in (3.8). This is shown in (3.5) and in Figure 3.1(b). In the proposed scheme, THP cancels the interference caused by the frontchannel, while the interference caused by the rearchannel is eliminated by the transmit–receive antenna weight optimization process. Note that no modulo operation is performed on the ﬁrst stream to be transmitted in link 1. Thus, MSs need to know which one of them will be scheduled in the ﬁrst link. This information can be appended in the signal preamble of wireless systems such as cellular networks, WiMax or WLANs. However, if high modulation rates are used, (e.g., M = 16, 64), the power enhancement of THP is negligible. The THP performance loss is given as M/(M − 1) [15] and decreases as the modulation rate increases. Thus the THP performance loss for M = 16 and M = 64 is 0.28 and 0.07 dB, respectively. Thus, if a higher modulation rate is used, we could apply the modulo operation to the ﬁrst stream of the ﬁrst link in (3.7) and (3.9).
3.3
Cooperative BS transmission optimization In this section, we propose a joint iterative transmit–receive antenna weight optimization and power allocation method based on ZF to cancel the rearchannel interference, while maximizing the SINR for each link and maintaining the same SER for all links. The transmitted signal estimate of stream s in link j at each
54
Cooperative base station techniques for cellular wireless networks
receiver, y s,j , can be obtained from (3.8) and expressed as y s,j =
K S √ √ ps,j rH H t v ˜ + pl,i rH s,j j s,j s,j s,j Hj tl,i vl,i l=1 i= j +1
+
S
√
H pl,j rH s,j Hj tl,j vl,j + rs,j nj .
(3.10)
l= s+1
By using the fact that the eﬀect of vector d on the received signal is completely removed by the THP decoder modulo operator, the SINR for the sth transmitted symbol in link j, can be written as SIN Rs,j =
H ps,j rH s,j Hj ts,j (Hj ts,j ) rs,j
rH s,j Rs,j rs,j
,
(3.11)
where Rs,j =
K S
pl,i Hj tl,i (Hj tl,i )H +
l=1 i= j +1
S
pl,j Hj tl,j (Hj tl,j )H + σ 2 I
(3.12)
l= s+1
is the interference in link j. Maximizing the minimum SINR for each link, while maintaining its equality for all links, can be formulated as follows: max
min
R,T,P 1≤i≤K ,1≤s≤S
subject to:
SIN Rs,i T (1) TH T = I, (2) rH s,j rs,j = 1, (3) 1 p = Pm ax H (4) rH s,j Hj ts,i = 0, (5) rs,j Hj ts ,j = 0
(3.13)
for j = 1, ..., K, i = j + 1, ..., K, s = 1, ..., S, s = s + 1, ..., S, where Pm ax and p = [p1,1 , ..., pS,K ]T = P2 1 are the power constraint at the cooperative transmitter and the set of powers assigned to each link, respectively. Here the objective of (3.13) is to maximize the minimum SINR for each link. The ﬁrst, second, and third constraints in (3.13) are to ensure that the transmit–receive weight vectors are unitary vectors and the sum of the power allocated to each link does not exceed the maximum power available at the transmitter (total BSs power constraint). These constraints bound the possible solution for R, T, and P and ensure the convergence of (3.13) to a solution. Finally, the fourth and ﬁfth constraints are the ZF constraints which ensure the interlink interference from links j + 1, ..., K to link j and the interstream interference from symbol stream s + 1, ..., S in link j are fully canceled. Here, to maximize the minimum SINR in (3.13), we reduce the SINR of the best symbol streams until the SINRs of all links are equal. Thus, the optimal solution is reached when all symbol streams attain the same SINR [3]. This optimization problem, however, is diﬃcult to solve as it is not jointly convex in variables R,T, and p. To solve (3.13), we propose a suboptimal solution that splits the problem into a twostep optimization. The ﬁrst step is to solve R and T iteratively, when p is ﬁxed. Hence in this step we simply ignore the equalization of SINRs among all links. The second step is to
3.3 Cooperative BS transmission optimization
55
R(i)
R(0)
f1(R(i–1))
T(i)
gj(T(i))
(i) (i) R ,T
MTHP = (DP)–1FP
SINR equalization
p,R(i),T(i)
Feedback matrix
Figure 3.2. Iterative weight optimization, SINR equalization, and feedback c 2009 IEEE). matrix process (adapted from [26] solve p in a way that equalizes SINRs for all links under ﬁxed R and T. Once R, T, and p are obtained, MT H P = DP−1 FP is computed. The process is described in Figure 3.2, where i, f1 (·) and gs,j (·) are the iteration number, a function to generate transmit antenna weights for K links, and a function to generate the receive antenna weights vector for symbol stream s in link j, respectively.
3.3.1
Iterative weight optimization (ﬁrst step) In the ﬁrst step, we assume an equal power allocation for each link by setting P = I. Then (3.13) can be simpliﬁed as max R,T
SIN Rs,i ,
subject to: (1) TH T = I, (2) rH s,j rs,j = 1, H (3) rH s,j Hj ts,i = 0, (4) rs,j Hj ts ,j = 0
(3.14)
for j = 1, ..., K, i = j + 1, ..., K, s = 1, ..., S, s = s + 1, ..., S. To solve (3.14), we propose to alternately optimize R and T until they converge, under the ZF constraints in (3.14). We ﬁrst assign the initial value of the receive antenna (0) weights for K links. The initial receive weights of K links are given as rs,j = vsv d (Hj ) , j = 1, ..., K, s = 1, ..., S, where vsv d (·) is the singular value decomposition operation (SVD) [16], to select the S left singular vectors of Hj , corresponding to the Sth largest singular value. We then transform the system into (0) H
a downlink multilink MISO system by ﬁxing R = Diag(R1 (3.3) can be written as y = RHTv + RN = He Tv + N.
(0) H
, ..., RK
). Then
(3.15)
Here, we know from (3.8) that the interference from the frontchannel does not exist at the receiver. The remaining interference at symbol stream s in link j that needs to be canceled is the rearchannel interference, coming from links j + 1, ..., K to links j = 1, ..., K and stream s + 1, ..., S in link j, respectively. At each iteration, we apply a QR decomposition [16] to HH e to ﬁnd T that forces
56
Cooperative base station techniques for cellular wireless networks
this interference to zero: T = f1 (R) , f1 (R) = [QQR(HH e )].
(3.16)
We choose the unitary matrix Q obtained from the QR decomposition of HH e in (3.16) as T. We need to compute R that gives a maximum SIN Rs,j for each link for the derived T. This can be calculated as rs,j = gs,j (T),
(3.17)
where j = 1, ..., K. gs,j (T) is a function that generates the receive weight vector rj for the derived T such that the gain SIN Rs,j for each link is maximized. We now describe how gs,j (T) operates. By using (3.11), the SINR maximization for the sth symbol in link j can be written as max rs , j
˜ ˜H ps,j rH s,j hs,j hs,j rs,j rH s,j Rs,j rs,j
,
(3.18)
˜ s,j = Hj ts,j and Rs,j is the interference in link j as given in (3.12). To where h obtain rs,j that maximizes SINR in (3.18) we use the spectral/eigenvalue decomposition [16]. Thus the functions to generate r1,1 , ..., rS,K , gs=1,...,S,j =1,...,K (T) can now be written as −1 ˜ s,j h ˜ H ) , s = 1, ..., S, j = 1, ..., K, gs,j (T) = vE V D (ps,j Rs,j h s,j
(3.19)
where vE V D (·) is the spectral/eigenvalue decomposition operation [16] to select −1 ˜ s,j h ˜ H , corresponding to the largest singular the left singular vector of ps,j Rs,j h s,j value. We can obtain a simpler expression for rs,j that gives the same maximum SINR, as in (3.18), by using the following fact: max rs , j
˜ ˜H rH s,j hs,j hs,j rs,j rH s,j Rs,j rs,j
= max rs , j
˜ rH s,j hs,j rH s,j Rs,j rs,j
.
(3.20)
Here we state that the optimum SIN Rs,j obtained by using the term on the left hand side of (3.20) is equal to the optimum SIN Rj obtained by using the term on the right hand side of (3.20). The proof of SINR equivalence is shown in the Appendix. By solving the term on the right hand side of (3.20), the normalized receive antenna weight vector for stream s in link j can be obtained as [17] −1
rs,j = gs,j (T) =
Rs,j Hj ts,j Rs,j Hj ts,j
.
(3.21)
It is straightforward to show that the SIN Rs,j generated by using the receive antenna weight vector from (3.21) yields the optimum SIN Rj given in (3.20). We can conclude from this fact and (3.20) that the normalization process of the receive weight vector in (3.21) will not aﬀect the SINR. Note that this receiver design is also known in the literature as the minimum variance distortionless response (MVDR) design [18]. The iterative calculations of R and T continue
3.3 Cooperative BS transmission optimization
57
by ﬁxing one and optimizing the other one, until they converge to a ﬁxed solution. The proof of convergence of the proposed iterative method is shown in the Appendix.
3.3.2
Power allocation (second step) In the second step, we use the R and T obtained in the ﬁrst step to ﬁnd p. By using the fact that at the optimal solution all links will attain equal SINR, (3.11) can be written as S K
rs ,j Hj ts ,i 2 ps ,i +
s =1 i= j +1
S
rs ,j Hj ts ,j 2 ps ,j + σ 2 =
s = s+1
ps,j rs,j Hj ts,j 2 SIN R
(3.22) for s = 1, ..., S, j = 1, ..., K. Equation (3.22) can be represented in a matrix format as p , (3.23) A−1 Bp + σA−1 1 = SIN R where A = DiT(M), B = UpT(M), and M is a KS by KS matrix with entries rs,j Hj ts ,j 2 , s = s = 1, ..., S, j = j = 1, ..., K in row S(j − 1) + s and column S(j − 1) + s . By multiplying both sides of (3.23) with 1T , we obtain [3] 1 Pm ax
(1T A−1 Bp + σ1T A−1 1) =
1 . SIN R
(3.24)
By deﬁning the extended power vector pe = [pT 1]T , we can then combine (3.23) and (3.24) to obtain [3]
pe σA−1 1 A−1 B . (3.25) pe = 1T A−1 B/Pm ax σ1T A−1 1/Pm ax SIN R Hence the optimum p can be obtained by selecting the pe that corresponds to the maximum eigenvalue of Ψ. This is the only possible solution of (3.25) satisfying ps,j ≥ 0 for j = 1, ..., K and SINR ≥ 0. The proof is described in detail in Theorems 1 and 2 of [19]. The above SINR equalization process can be further simpliﬁed. By using the proof of SINR convergence in the Appendix, we can assume that after i iterations, we are very close to the local optima (i.e., the rearchannel interference is zero, UpT(M) = 0). The solution for (3.25) can be given as ps,j =
Pm ax , s = 1, ..., S, j = 1, ..., K. K rs,j Hj ts,j 2 s =1 i=1 rs ,i Hi ts ,i 2
S
(3.26)
Even though the power constraint used above allocates the power to each link in an optimal manner, it is not very practical. This power constraint does not take into account the power constraint for each BS in a real system. A more practical power allocation that takes into account the power constraint for each BS can also be derived. We refer to this power constraint as the per BS power
58
Cooperative base station techniques for cellular wireless networks
constraint. We denote the maximum power available at each BS j as Pm ax,j and the modiﬁed transmit weights ti,s,j ∈ C N B S ×1 as the entries of ti,s,j used as antenna weights by BS i. A power allocation formulation that takes into account the power constraint for each BS can be derived as σ 2 SIN R,
max p
subject to: (1) σ 2 SIN R ≤ pj rs,j Hj ts,j 2 , K S tj,s ,i 2 ps ,i ≤ Pm ax,j (2)
(3.27)
s =1 i=1
for j = 1, ..., K, s = 1, ..., S. After some manipulations, the solution for (3.27) can be given as rs,j Hj ts,j 2 ps,j =
min
j =1,...,K ,s=1,...,S
Pm ax,j K tj,s ,i 2 s =1 i=1 rs ,i Hi ts ,i 2
S
(3.28)
for j = 1, ..., K, s = 1, ..., S.
3.4
Modiﬁcation of the design of R In this section, we modify the receive antenna weight calculations in (3.21) to speed up the convergence to the local maxima and improve the SINR during the (i) (i) iteration process. We deﬁne rs,j and ts,j as the receive and transmit antenna weights found in step 1 in Section 3.3.1 at the ith iteration, for link j. The entries of the rearchannel interference matrix BP at the ith step of the iterative (i) process, εl,k can be written as follows: (i)
εl,k =
√
(i)
(i)
H ps ,j (HH j rs,j ) ts ,j ,
(3.29)
where l = jS + s − S, k = j S + s − S. The interlink and interstream components of the rearchannel interference in stream s in link j correspond to row l and column k of BP when s = 1, ..., S, j = j + 1, ..., K and when s = s + 1, ..., S, respectively. Similarly, the diagonal entries of matrix D at the ith step of the iterative pro(i) cess, denoted by βl= j S + s−S , represent the signal gain for stream s in link j. This (i)
(i)
(i)
H can be written as βl= j S + s−S = (HH j rs,j ) ts,j , s = 1, ..., S, j = 1, ..., K at the ith iteration. In the Appendix, we show the proof of transmit–receive weight opti (i) (i) (i) H (i) mality. There we prove that l βl ≤ det(R∗ HT∗ ), where βl = (HH j rs,j ) ts,j and (R)∗ and (T)∗ are the optimal transmit–receive antenna weight vectors for K links satisfying the proof of convergence in the Appendix. From the proof of convergence and transmit–receive weight optimality in the (i) appendix, we know that at the local maximum: (1) l βl achieves the maximum value equal to det(R∗ HT∗ ); (2) the frontchannel interference BP converges to
3.5 Geometric mean decomposition
59
0 since (3.16) forces R∗ HT∗ to have a lower triangular structure. Hence, we (i) could simply maximize βl to achieve the local maxima. By using the Matrix (i) (i) Inversion Lemma [20], (3.12), and (3.21) in βl , we can rewrite βl= j S + s−S for stream s in link j as cβl= j S + s−S = (Hj ts,j )H (σ −1 I − (Z−1 + σI)−1 )Hj ts,j , (i)
(i)
(i)
(3.30)
Hj ts ,j (Hj ts ,j )H
(3.31)
where Z=
S K
(i)
(i)
Hj ts ,a (Hj ts ,a )H +
s =1 a= j +1
S
(i)
(i)
s = s+1
and c is a scaling/normalization factor given by c = Rs,j Hj ts,j . It is obvious (i) (i) (i) that (Hj ts,j )H (Z−1 + σI)−1 Hj ts,j in (3.30) reduces the value of βl . Therefore, if we omit this term in calculating the receive antenna weights, we can (i) reach the maximum βl faster. Thus, we can simply ignore this term to speed up the convergence of the iterative process. Therefore by omitting the term (i) (i) (Hj ts,j )H (Z−1 + σI)−1 Hj ts,j , we have βl= j S + s−S ∼ (Hj ts,j )H (σ −1 I)ts,j = σ −1 (rs,j )H Hj ts,j . (i)
(i)
(i)
The maximum βl
(i)
(i)
(i)
(i)
(3.32) (i)
can be obtained by aligning rs,j in the direction of Hj ts,j . (i)
The total power of the receive weight vector, rs,j is normalized to 1 to ensure it satisﬁes the second constraint in (3.13), (i)
(i) rs,j
=
Hj ts,j (i)
Hj ts,j
.
(3.33)
We refer to this receiver structure as a matched ﬁlter (MF) design.
3.5
Geometric mean decomposition In [22, 23], geometric mean decomposition (GMD) has been shown to give the best BER performance for multistream transmission while at the same time maintaining an equal SINR across S symbols within each link and the lower triangular structure required by THP [12]. We want to further maximize the channel gain for each stream in each link j by modifying the found iterative transmit and receive weights using GMD. We ﬁrst deﬁne a new eﬀective channel ¯ j as matrix for each link j H ¯ j = Hj Tj , H
(3.34)
where Tj is the transmit weight matrix for link j previously deﬁned in Section ¯ j into a lower triangular matrix with 3.2.1. The GMD is used to decompose H ¯j an equal diagonal component. The process is as follows. We ﬁrst decompose H
60
Cooperative base station techniques for cellular wireless networks
Algorithm 3.1. Cooperative transmission algorithm 1 2 3 4 5 6 7 8
Initialize receive weights and set Maxiteration For i=2 to Mit Find transmit weights using (3.16) Find receive weights using (3.21) or (3.33) end Use GMD to modify transmit–receive weights (3.37) and (3.38) Equalize SINR for all links using (3.28) or (3.26) or (3.25) THP precoding operation using (3.7)
by using the SVD [24] as follows: ¯ j Tj = [US U0 ] H
DS 0
0 D0
[VS V0 ]T ,
(3.35)
¯ j Tj where US and VS consist of the ﬁrst S left and right singular vectors of H and DS is a diagonal matrix with entries that are the ﬁrst S nonnegative square ¯ jH ¯ H . The GMD takes US , VS , and DS as inputs roots of the eigenvalues of H j j, V S , and D S . The GMD transforms DS into a lower triangular and produces U S , by rotating US and VS . This is given matrix with equal diagonal entries, D as ˜ jD ¯j = U ˜SV ˜H. ¯jT H S
(3.36)
The goal of our transmit–receive design is to create a lower triangular structure within each link. Thus, by using (3.36), the new transmit–receive weights denoted ¯ j and R ¯ j can be written as by T ˜ S , j = 1, ..., K ¯ j = Tj V T
(3.37)
j , j = 1, ..., K. ¯j = U R
(3.38)
and
The cooperative transmission algorithm is shown in Algorithm 3.1, where i represents the iteration number and Mit is the maximum number of iterations.
3.6
Adaptive precoding order (APO) In the algorithm in Table 3.1, we ﬁx the order of uj , resulting in a ﬁxed permutation matrix Mper m . The performance of the system, however, diﬀers when a diﬀerent Mper m is used. In addition, the performance of the system also depends on the weakest link. In this section we propose an APO scheme. APO arranges the order of links, x, by selecting on Mper m that maximizes the minimum SINR
3.7 Complexity comparison of the proposed and other known schemes
61
for each link. We formulate the optimization process to ﬁnd a permutation matrix ˘ per m ∈ Mper m that gives the maximum SINR as M ˘ per m = argM max min(SIN R1 (Mper m ), ..., SIN RK (Mper m )), M perm
(3.39)
where SIN Rj (Mper m ) is the SINR of link j, given that the permutation matrix Mper m is used. Note that here, due to the use of GMD, the SINRs for each ˘ per m without searching K! possible stream in link j are equal. To ﬁnd the M orderings, we use the idea of the myopic optimization method proposed in [14], which has been proven to be optimal. Using this idea we now only need to search K −1 i=0,i = 1 K − i possible orderings.
3.7
The complexity comparison of the proposed and other known schemes In this section, we discuss the advantages of the proposed scheme over other existing schemes. We ﬁrst compare the proposed method with the scheme in [9] and the iterative scheme in [10] that work by ﬁnding transmit–receive weights that diagonalize the receive signal matrix of K users without the receiver noise in (3.3). To have a fair comparison with the proposed method, we replace the waterﬁlling power allocation in [9] with (3.26) and (3.28), which equalize the SINR for all links under total BS and per BS power constraints, respectively. This is required as the waterﬁlling power allocation used in [9] tends to assign more power to stronger links and less power to weaker links. Hence, the performance of a weaker link will decrease the overall SINR. The main diﬀerences between the methods in [9, 10] and the proposed method are (1) [9, 10] suppress both the frontchannel and rearchannel interference using transmit–receive weights, while the proposed method suppresses the rearchannel and frontchannel interference using THP and iterative transmit–receive weights, respectively. (2) Unlike [9, 10], the proposed scheme does not calculate null spaces. To compute these null spaces, the iterative scheme in [10] and the noniterative scheme in [9] perform K SVD operations per iteration and K SVD operations, respectively. (3) Within a single iteration, a QR decomposition [16] and K matched ﬁlter (MF) receiver calculations are required to ﬁnd all transmit–receive antenna weights, while in [10] K SVD operations per iteration are required to ﬁnd the transmit–receive weights of all links. Note that [9] requires K SVD operations to ﬁnd the transmit–receive weights of all links. The complexity order requirements in terms of the number of ﬂoating point operations (ﬂops), for the proposed method and the methods in [9, 10] are listed in Table 3.1, where Mit and Mit,P an denote the total number of iterations for the proposed method and the scheme in [10], respectively. Hence for K = 3, NM S = 2, and NB S = 2, the number of ﬂops for the proposed method is approximately 93% less than the number of ﬂops for [9] per iteration.
Proposed algorithm Pan et al. [10] Spencer et al. [9] Liu and Witold [12] Foschini et al. [11]
Schemes
O(K 3 SMit N B S ) 2 2 3 O(Mit,P an K(4KSN B S + 8(KS)2 N B S + 9N B S )) 2 2 3 O(K(4KSN B S + 8(KS)2 N B S + 9N B S )) 2 2 3 2 2 2 O(0.5K 2 (4KNM S N B S + 8KNM S N B S + 9KN B S + KNM S N B S (K − 1) + KNM S + KNM S N B S )) 2 2 3 2 2 O(0.5K 2 (4KNM S N B S + 8KNM S N B S + 9KN B S + KNM S N B S (K − 1)))
2
Computational complexity order (in ﬂops)
Table 3.1. Computational complexity of linear and nonlinear precoding algorithms (in ﬂops)
3.8 Numerical results and discussions
63
The second comparison is done using the nonlinear precoding methods in [11, 12]. Again to have a fair comparison with the proposed method, after the algorithm in [11, 12], we apply (3.25) to equalize the SINR, instead of using the original power allocation. Unlike [11, 12], we do not require the constraint of (K − 1)NM S < KNB S because we do not create null spaces. Thus, there is no relationship between the required number of transmit antennas and receive antennas. This is a deﬁnite advantage, since to support say ﬁve users with NM S = 4 receiving one stream each, the proposed method only needs ﬁve transmit antennas while [9] needs 12 transmit antennas. Another important diﬀerence is in the ZF condition deﬁnition. In our scheme when S = 1 for example we have rH 1,j Hj t1,i = 0 , j = 1, ..., K, i = j + 1, ..., K, while [11, 12] have Hj t1,i = 0 , j = 1, ..., K, i = j + 1, ..., K for the ZF condition. By using the ZF condition in our scheme, the proposed algorithm allows some interlink interference to be transmitted (e.g., Hj t1,i = 0), and cancels the interference by steering Hj t1,i to be perpendicular with the receive antenna weights vector r1,j . Hence the receive and the transmit antenna weights jointly cancel the interference. The ZF constraint in [11, 12], on the other hand, does not allow any interlink interference to be transmitted. Here, the receive antenna weights are not used at all to cancel the interference. The computational complexity required for the methods in [11, 12] is shown in Table 3.1. The complexity of the methods in [11, 12] for a system with K = 3, NM S = 2, S = 1 and NB S = 2, is approximately 47 628 and 47 844 ﬂops, respectively.
3.8
Numerical results and discussions For convenience, in our simulations, we will use the notation (NB S , NM S , S, K) in all ﬁgures to denote a system with NB S transmit antennas per BS, NM S receive antennas per MS, S data streams transmitted in each link, and K BSs. Monte Carlo simulations have been carried out to assess the performance of the proposed method. We investigate its performance and compare it with [9, 11, 12] and with an interferencefree performance. An interferencefree performance is deﬁned as the performance of any random single link i assuming there is no interference from other links at all. In this case, the received signal of the cooperative transmission system is given as yi = rH i (Hi ti xi + ni ), where ri and ti are the left and right eigenvectors associated with the maximum eigenvalue of Hi HH i . To generate an interferencefree performance for multistream transmission with S symbols transmitted in each link, we use the left and right eigenvectors associated with the S largest eigenvalues of Hi HH i found by using the SVD. We then maximize the minimum SINR of S symbols by applying the power allocation method given in [25]. The comparison of the schemes is performed at SER = 10−4 . We use a ﬁxed permutation matrix that orders MSs 1, ..., K as links K, ..., 1, when we are not using APO, for all the simulation results except when stated otherwise. Perfect
64
Cooperative base station techniques for cellular wireless networks
channel state information (CSI) is assumed to be available at both ends. Rectangular 64QAM (M = 64) modulation is used for all transmissions. The wireless channel model we used is a Rayleigh fading channel. This channel model is commonly used for cellular networks or WLANs, since in most cases there is no lineofsight path between the transmitter and receiver in these networks. In order to simulate the wireless channel, we set each entry of the Hj channel matrix as an independent and identically distributed (i.i.d.) complex Gaussian variable with a zero mean and unit variance. In all simulations, we ﬁx the signaltonoise ratio H H vs,j ]/2σ 2 , where E[vs,j vs,j ] of each THP precoded symbol to be SN R = E[vs,j is normalized to 1, Pm ax = K and Pm ax,i=1,...,K = Pm ax /K . In all simulation ﬁgures, the proposed method refers to the algorithm with THP, joint iterative transmitreceive weight optimization, SINR equalization (SINRE) under a total BS power constraint unless stated otherwise and APO.
3.8.1
Convergence study Figure 3.3 shows the convergence characteristics of the proposed method with a total BS power constraint for (2, 2, 1, 3) and (1, 2, 1, 4) systems. Note that here it does not matter whether total power or per BS constraints are used. This is because, as shown in Figure 3.2, the SINR equalization is not an iterative process. We plot the number of iterations versus the average error and scaled output SINR (after SINR equalization), while ﬁxing the SNR at 21 dB. The average error is deﬁned as the average of the maximum entries of the front(i) channel interference BP, ε(i) = maxj,l εj,l  deﬁned previously in (3.29) over all channel realizations. The output SINRs for (1, 2, 1, 4) and (2, 2, 1, 3) systems are scaled up by 4 dB and 0 dB to ﬁt in one ﬁgure. The scaling does not matter here since we only want to observe the convergence rate. The ﬁgure also shows the convergence characteristics when the MF receiver design, represented by (3.33), and the MVDR receiver design, represented by (3.21) are used. An interesting observation is that during the ﬁrst few iterations the MVDR design outperforms the MF design. This improvement is due to the smaller errors (i) obtained using MVDR and is not due to a higher signal gain βj . During the ﬁrst few iterations, the second term of (3.30) for the MF receiver design is larger than for the MVDR receiver design, thus leading to a higher average error for the MF receiver. This happens because MF ignores the interference when calculating the receive antenna weights. However, its average error decreases rapidly. The average error for the (1,2,1,4) and (2,2,1,3) systems, denoted by (1,2,1,4)MF and (2,2,1,3)MF in Figure 3.3, approaches the average error of the MVDR method. The SINR using the MF method from that point onwards is always greater than the SINR using the MVDR method. This is shown in the analysis in Section 3.4. This analysis is consistent with the results shown in Figure 3.3. MF converges much faster to the optimal SINR solution than MVDR. This is shown in Figure 3.3 and agrees with the proof of convergence in the Appendix. MF’s SINR reaches a plateau after eight iterations, since it almost converges
3.8 Numerical results and discussions
65
Figure 3.3. SINR and average error convergence comparison for proposed scheme c 2009 IEEE). (adapted from [26]
to the optimal solution while MVDR’s SINR is still rising. A similar conclusion is found for all other conﬁgurations. Not much performance improvement can be obtained by increasing the number of iterations further. In all simulations for SER comparison, we set the maximum iteration number for the proposed scheme to ten.
3.8.2
Performance of the individual links Figure 3.4 shows the SER of the worst user and the best user versus the SNR in a (2, 2, 1, 3) system. Note that it does not matter whether total power or per BS constraints are used to compare the performance of the individual links. This is because, as shown in Section 3.2.2, the SINR for all links is the same under both constraints. As shown in Figure 3.4, when the proposed method does not perform SINRE and APO (denoted by w/o in the ﬁgures), MS 3 has the best performance while MS 1 has the worst performance. The SER performance diﬀerence between links 1 and 3 exceeds 3 dB. When SINRE is used, the SER diﬀerence between links disappears. This is shown in Figure 3.4. Here, the performance of link 1 is improved at the expense of links 2 and 3. This results in a similar SER across all links. An interesting point here is that APO tends to equalize the performance of K users even without the use of SINRE. This is shown in Figure 3.4. Hence, it seems suﬃcient to use APO without SINR equalization to maximize the
66
Cooperative base station techniques for cellular wireless networks
Figure 3.4. SER performance comparison of individual links for the (2, 2, 1, 3) system when SINRE is not used and when SINRE and APO are used (adapted c 2009 IEEE). from [26] minimum SINR in the system. The performance of APO without SINRE, however, is still worse than when the proposed method does not perform APO. This is denoted as “Proposed w/o APO” in Figure 3.4. This suggests that SINR equalization plays a more important role than APO in performance improvement. In other words, using a good power allocation scheme might be more beneﬁcial than searching for the best order of the users to achieve a higher diversity gain.
3.8.3
Overall system performance We ﬁrst show the overall SER of the proposed scheme and compare it with other available methods [9–12]. Overall SER is deﬁned as the average SER of K links. The overall SER performances for a (2, 2, 1, 3) system for both the proposed method with or without APO and for [9–12] are shown in Figure 3.5. The proposed method without APO outperforms the methods in [11, 12] and [9] by 5 dB and 3 dB, respectively, and is only 1 dB away from an interferencefree performance when SER = 10−4 . The large improvement in performance in the proposed scheme with respect to [11, 12] comes from an increase in the degree of freedom and the iteration process used in determining the transmit– receive antenna weights. In addition, the proposed method without APO is able to achieve much better performance with much less complexity (we only use
3.8 Numerical results and discussions
67
Figure 3.5. Average SER performance comparison for (2, 2, 1, 3) system using c 2009 IEEE). various nonlinear precoding algorithms (adapted from [26] ten iterations). The computational complexity of the proposed method for a (2, 2, 1, 3) system is on average about 75% less than the complexity of methods in [10–12]. In essence, the proposed method fully utilizes THP, transmit antennas, and receive antennas in a more optimal way with much less complexity to create noninterfering spatial channels. Figure 3.5 also shows the performance of the proposed method with APO. APO moves the SER performance of the proposed method to within 0.25 dB of an interferencefree transmission when SER = 10−4 . Thus, APO gives about 1 dB gain over the proposed method without APO. This gain, however, comes at the cost of complexity since now the proposed method −1 needs to do a search over K l=0,l = 1 K − l possible user orderings. As a result, −1 the complexity of the proposed method is K i=0,i = 1 K − i times more than the proposed method without APO. This is shown in the last column of Table 3.1. The performance of the iterative scheme in [10] depends on the number of iterations. To show that the proposed method performs better than the scheme in [10], we set the iteration number for the scheme in [10] to ﬁve, giving a computational complexity of around 74 520 ﬂops for a (2, 2, 1, 3) system. The computational complexity of the proposed method using ten iterations is 4860 ﬂops. The performance of [10] is shown in Figure 3.5. Here, we can see clearly that [10] is worse than the proposed method with or without APO. It is not possible to get much improvement in performance in [10] by raising the SNR above 21 dB. We refer to this SNR value, above which there is no further SER decrease, as the saturation point. Here, we must stress that the performance of [10] can be further improved by increasing the number of iterations. This is shown in Figure 3.5
68
Cooperative base station techniques for cellular wireless networks
Figure 3.6. Average SER performance comparison for a (2, 2, 2, 3) system using various nonlinear precoding algorithms.
when we increase the number of iterations to 11. However, the performance of the scheme in [10] is worse than the performance of the proposed method and the scheme in [10] is much more complex than the proposed method. Lastly, we also plot the performance of the proposed method under a per BS power constraint. There is only a 1 dB performance degradation by switching from total BS to per BS power constraints. The performance of the proposed method under a per BS power constraint is much better than the performance of other schemes under both total BS and per BS power constraints. In Figure 3.6, we illustrate how the proposed method performs under a diﬀerent conﬁguration. We show the performance when the number of users, K, the number of transmit antennas per BS, NB S , the number of streams per user, S, and the number of receive antennas per MS, NM S , are three, two, two, and two, respectively. Here, the total number of transmit antennas KNB S is equal to the number of MSs. Even when the proposed method does not perform APO, it still signiﬁcantly outperforms the one in [10, 11]. This improvement is even greater than that shown in Figure 3.6 (>4 dB). Interestingly, the performance in [12] is exactly the same as the performance of the proposed method. Note that here, the complexity of the proposed scheme is 80% less than that in [12], making the proposed method most practical for implementation. We also plot the performance of the proposed method under per BS power constraints for a (2, 2, 2, 3) system. Its performance is still better than the
69
3.8 Numerical results and discussions
40 35 30
bps/Hz
25 Proposed (total power constraint) Scheme [10] 11 iterations (total BS constraint) Proposed (per BS constraint) Scheme [10] 11 iterations (per BS constraint) Interference free
20 15 10 5 0 12
14
16
18
20
22
24
26
28
30
SNR
Figure 3.7. Capacity performance comparison for a (2, 2, 1, 3) system using various nonlinear precoding algorithms.
performance of the other schemes under both the total BS and per BS power constraints. As the system has an error performance close to an interferencefree system, its capacity should approach the capacity of individual interferencefree links. In typical cellular networks, only one user signal can be transmitted in a frequency band at a given time slot. The proposed method enables K base stations in the same location to simultaneously transmit to K users using the same frequency band and time slots. By using the proposed method, instead of transmitting to one user at a time with a power of 1, we can simultaneously transmit to K users with a power of K with the performance of each user approaching an interferencefree performance. For example, the capacity of cellular mobile networks for a (2, 2, 1, 3) system can be increased by up to K times, as shown in Figure 3.7, by using BS cooperation. Note that the capacity of the proposed method is higher than the capacity of its best competitor [10]. An interesting point here is that the capacity of a (2, 2, 2, 3) system using the method in [12] is the same as the capacity of the proposed method. Note that these two signiﬁcantly outperform any other methods. This is shown in Figure 3.8. In a (2, 2, 2, 3) system, we have six cooperative transmit antennas at BSs transmitting to users each with two receive antennas. The maximum possible number of independent transmissions for each user is two. Thus, if the number of streams is two, we fully use the transmission space and it is not possible to use this transmission space for other users without violating the ZF constraint. On
70
Cooperative base station techniques for cellular wireless networks
60 Proposed (total power constraint)
55
Scheme [10] (11 iterations) (per BS constraint) Proposed (per BS constraint) Scheme [10] (11 iterations) (per BS constraint)
50
Scheme [11] (total power constraint) Scheme [11] (per BS constraint)
45
Scheme [12] (total power constraint) Scheme [12] (per BS constraint) Interference free
bps/Hz
40 35 30 25 20 15 10 12
14
16
18
20
22
24
26
28
30
SNR
Figure 3.8. Capacity performance comparison for a (2, 2, 2, 3) system using various nonlinear precoding algorithms.
the other hand, for a (3, 3, 2, 3) system, we have three receive antennas for each link and only use two out of three possible transmission spaces for each user. It is possible to use the remaining available transmission space without violating the ZF constraint. In our proposed method, we allow the transmission space for each user to overlap. Our proposed scheme will outperform that in [12] in this type of conﬁguration. This is shown in Figure 3.9 where the capacity of the proposed scheme is 20% more than the capacity of its nearest competitor, i.e., that in [12].
3.9
Conclusion In this chapter, we have proposed a method to design a spectrally eﬃcient cooperative downlink transmission scheme by employing precoding and beamforming. THP and iterative transmit–receive weight optimization have been used to cancel multiuser interference. A new method to generate transmit–receive antenna weights has been proposed. SINRE and APO have been used to achieve SER fairness among diﬀerent users and further improve the system performance, respectively. The error performance for two sets of system parameters (NB S , NM S , S, K) has been shown. For a (2, 2, 1, 3) cooperative system, the proposed method outperforms existing schemes by at least 3 dB and is only 0.25 dB away from an interferencefree performance when SER = 10−4 . For a (3, 3, 2, 3) system, the proposed method has a 20% higher spectral eﬃciency than existing
71
3.9 Conclusion
80 Proposed (total power constraint) Proposed (per BS constraint) Scheme [12] (total power constraint) Scheme [12] (per BS constraint) Interference free
70
bps/Hz
60
50
40
30
20
10 12
14
16
18
20
22
24
26
28
30
SNR
Figure 3.9. Capacity performance comparison for a (3,3,2,3) system using various nonlinear precoding algorithms. schemes. In addition, the proposed method eliminates the dependency between the numbers of transmit and receive antennas. The complexities of the proposed method have been shown to be on average 75% less than the complexities of the schemes in [9–12] with the same conﬁgurations. Thus, we can conclude that our proposed scheme has the best capacity, the lowest SER, and the lowest computational complexity compared to those in [9–12] in most cases. These features make the proposed method suitable for practical implementation.
Appendix
Proof of SINR equivalence We ﬁrst calculate SIN Rs,j using the term on the left hand side of (3.20). We −1 ˜ s,j h ˜ H only has one eigenvalue. Let us assume ﬁrst need to prove that Rs,j h s,j −1 H ˜ H by b, where a = [a1 ...aN ]T ∈ ˜ = 0. We denote R h ˜ s,j by a and h that h s,j
s,j
s,j
−1
M S
˜ s,j h ˜ H as C N M S ×1 and b = [b1 ...bN M S ] ∈ C 1×N M S . We then express Rs,j h j −1
˜ s,j h ˜ H = ab = [b1 a...bN a]. Rs,j h s,j M S
(3.40)
−1 ˜ s,j h ˜ H is a matrix that has NM S columns and rows. We can see that Here, Rj h s,j −1 ˜ s,j h ˜ H can be rewritten the vectors represented by each column of matrix R h j
s,j
using vector a as a basis. This indicates that the rank of this matrix is 1 and as a consequence, there is only one eigenvalue. The receive weight vector computation in (3.19) can be written as −1 ˜ s,j h ˜ H rs,j = λs,j rs,j , where λs,j is the eigenvalue for stream s in link j. Rs,j h s,j ˜ H , we have By multiplying both sides of this equation by h s,j ˜H ˜ ˜ H R−1 h (h s,j s,j s,j − λs,j )hs,j rs,j = 0.
(3.41)
−1 ˜ H R−1 h ˜ ˜ ˜H The eigenvalue of h s,j s,j s,j is the same as the eigenvalue of Rs,j hs,j hs,j . −1 ˜ ˜H R h As a consequence, h s,j s,j s,j only has one eigenvalue. This eigenvalue is the solution for the term on the left hand side of (3.20). Thus, the SIN Rs,j for it ˜ H R−1 h ˜ is given as SIN Rs,j = λs,j = h s,j s,j s,j . We now ﬁnd the SIN Rs,j by solving the term on the right hand side of (3.20). The optimum receive weights vector −1 ˜ H R−1 h ˜ s,j /c where c = h ˜ is given by [17] as rs,j = Rs,j h s,j s,j s,j . By inserting this receive weights vector into (3.11) and replacing its numerator with rH s,j Rs,j rs,j , ˜ H R−1 h ˜ s,j . we obtain SIN Rs,j = h s,j
s,j
Proof of convergence In order to calculate SIN Rs,j in (3.11), we need to know the receive weight vector for stream s in link j, rs,j and all transmit weight vectors t1,1 ,...,tS,K , obtained
73
Appendix
by using (3.17) and (3.16), respectively. We write SIN Rs,j as SIN Rs,j (rs,j , T) since it is a function of rs,j and T. Since in the proposed scheme, we optimize one variable at a time, while ﬁxing the other one, we have SIN Rs,j (gs,j (T), T) = max SIN Rs,j (a, T), gs,j (T) ∈ A1 , a∈A 1
(3.42)
where T is ﬁxed while the best rs,j = gs,j (T) in the set A1 is searched and SIN Rs,j (rs,j , f1 (R)) = max SIN Rs,j (rs,j , a), f1 (R) ∈ A2 , a∈A 2
(3.43)
where rs,j is ﬁxed while the transmit weight vectors for K links, T = f1 (R), in the solution set A2 are searched. To describe the proposed alternating optimization process, we denote the number of iterations by i, the receive weight (i) (0) vector by rs,j and transmit weight vectors by T(i) . First, rs,j , j = 1, ..., K, are arbitrarily chosen as initial vectors. T(1) is then calculated by using the (i) function in (3.16), f1 (R0 ). For i ≥ 1, we then have, rs,j = gs,j (T(i) ) , s = (i)
(i)
1, ..., S, j = 1, ..., K, where T(i) = [t1,1 ...tS,K ], and T(i) = f1 (R(i−1) ), where R(i) = Diag(rH 1,1 (0)
(i)
, ..., rH S,K
(i)
(i)
). Here, rs,j and T(i) are generated in the order
(1)
rs=1,...,S,j =1,...,K , T(1) , rs=1,...,S,j =1,...,K , T(2) and so on. From (3.42) and (3.43), and by using the fact that SIN Rs,j (rs,j , T) is nondecreasing and bounded from above by constraints in (3.14), we can write (i)
SIN Rj (rj , T(i) )
(i)
≥ SIN Rj (rj , T(i−1) ) (i−1)
≥ SIN Rj (rj
, T(i−1) ).
(3.44)
The terms on the right hand side of (3.44) come from the fact that since we are performing an alternate optimization of the transmit–receive weights by using (3.42) and (3.43), the SINR obtained at iteration i − 1 could only be either equal or less than the SINR obtained at iteration i. This shows that as the (i) number of iterations increases, the SIN Rs,j (rs,j , T(i) ) will converge to a local maximum and simultaneously satisfy (3.17) and (3.16). The former will also cause the remaining interference to converge to 0 as the number of iterations goes to ∞.
Proof of transmit–receive weight optimality We know from the proof of convergence in this Appendix that we can write the optimal solution as det(R∗ HT∗ ) = det(Z) = l zl,l , where Z and zi,i with i = S(j − 1) + s, are a lower triangular matrix and the entry of the diagonal of Z, respectively. Thus the channel gain for stream s in link j corresponds to the ith diagonal entry of Z. R∗ and T∗ indicate the optimal transmit–receive weight vectors for K links. We also need (3.16) to be satisﬁed for the optimal solution ∗ H ∗ for each stream s in link j, where (HH j rs,j ) ts,l = 0, s = 1, ..., S, l = j + 1, ..., K,
74
Cooperative base station techniques for cellular wireless networks
∗ H ∗ and (HH j rs,j ) ts ,j = 0, s = s + 1, ..., S. The vector created by multiplying the channel matrix by the receive antenna weight vector is perpendicular to the transmit weight vector for links j + 1, ..., K and stream s + 1, ..., S in link j. As a result, there is no interference at all at link j. This is so since the transmission spaces of link j + 1, ..., K and stream s + 1, ..., S in link j do not overlap with the transmission space of stream s in link j. The interference from link 1, ..., j − 1 and stream 1, ..., s − 1 in link j to stream s in link j is canceled by THP. However, prior to ﬁnding the optimal solution, the receiver design from (3.17) destroys the orthogonality created by QR decomposition in (3.16). As a (i) H (i) result at the ith iteration, for link j, we have (HH j rj ) tl = 0, l = j + 1, ..., K H and (HH j rs,j ) ts ,j = 0, s = s + 1, ..., S. This means the transmission space for stream s in link j intersects with the transmission spaces of link j + 1, ..., K and stream s + 1, ..., S in link j prior to convergence. In other words, the vector (i) generated by HH j rs,j also has components in other directions. This reduces the optimal signal gain for stream s in link j, zi,i where i = S(j − 1) + s. Thus, we (i) can then conclude that j βs,j  ≤ j zj,j . (i)
(i)
References [1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2003. [2] R. Morrow, Wireless Network Coexistence. The McGrawHill Companies Inc., 2004. [3] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Transactions on Vehicular Technology, 53, 2004, 18–28. [4] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Transactions on Information Theory, 49, 2003, 1691–1706. [5] M. Costa, “Writing on dirty paper,” IEEE Transactions of Information Theory, 29, 1983, 439–441. [6] C. Windpassinger, R. F. H. Fischer, T. Vencel, and J. B. Huber, “Precoding in multiantenna and multiuser communications,” IEEE Transactions on Wireless Communications, 3, 2004, 1305–1316. [7] M. Tomlinson, “New automatic equalizer employing modulo arithmetic,” IEEE Electronics Letters, 7, 1971, 138–139. [8] M. Miyakawa and H. Harashima, “New automatic equalizer employing modulo arithmetic,” IECE Transactions, 52A, 1971, 272–273. [9] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zeroforcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Transactions on Signal Processing, 52, 2004, 461–471.
References
75
[10] Z. Pan, K. Wong, and T. Ng, “Generalized multiuser orthogonal spacedivision multiplexing,” IEEE Transactions on Wireless Communications, 3, 2004, 1969–1973. [11] G. J. Foschini, K. Karakayali, R. A. Valenzuela, “Coordinating multiple antenna cellular networks to achieve enormous spectral eﬃciency,” IEE Proceedings on Communications, 153, 2006, 548–555. [12] J. Liu and A. K. Witold, “A novel nonlinear joint transmitterreceiver processing algorithm for the downlink of multiuser MIMO systems,” IEEE Transactions on Vehicular Technology, 57, 2008, 2189–2204. [13] J. C. Bezdek and R. J. Hathaway, “Some notes on alternating optimization,” Lecture Notes on Computer Science, 2275, 4, 2002, 187–195. [14] G. J. Foschini, G. D. and Golden, R. A. Valenzuela, and P. W. Wolniansky, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Journal on Selected Areas in Communications, 17, 11, 1999, 1841–1852. [15] R. F. H. Fischer, Precoding and Signal Shaping for Digital Transmission. John Wiley and Sons Inc., 2002. [16] G. H. Golub and C. F. Van Loan, Matrix Computations. The John Hopkins University Press, 1996. [17] W. Hardjawana, B. Vucetic, and A. Jamalipour, “Adaptive beamforming and modulation for OFDM in coworking WLANs with ack eigensteering,” in Proc. of IEEE PIMRC 2006, 1–5. IEEE, 2006. [18] L. C. Godara, “Application of antenna arrays to mobile communications. II. Beamforming and directionofarrival considerations,” Proceedings of IEEE, 85, 8, 1997, 1195–1245. [19] W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in Proc. of IEEE ICASSP 1998, vol. 6. 3337–3340. IEEE, 1998. [20] S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall Press, 1993. [21] Y. Jiang, J. Li, and W. Hager, “Uniform channel decomposition for MIMO communications,” IEEE Transactions on Signal Processing, 53, 11, 4283– 4294, 2005. [22] Y. Jiang, J. Li, and W. Hager, “Joint transreceiver design for MIMO communications using geometric mean decomposition,” IEEE Transactions on Signal Processing, 53, 10, 2005, 3791–3803. [23] Y. Jiang, J. Li, and W. Hager, “The geometric mean decomposition,” Linear Algebra and Its Applications, 396, 2005, 373–384. [24] J. R. Schott, Matrix Analysis for Statistics. John Wiley and Sons, Inc., 1997. [25] D. P. Palomar, J. M. Cioﬃ, and M. A. Lagunas, “Joint TxRx beamforming design for multicarrier MIMO channels: A uniﬁed framework for convex optimization,” IEEE Transactions on Communications, 51, 9, 2003, 2381– 2401.
76
Cooperative base station techniques for cellular wireless networks
[26] W. Hardjawana, B. Vucetic, Y. Li, and Z. Zhou, “Spectrally eﬃcient wireless systems with cooperative precoding and beamforming,” IEEE Transactions on Wireless Communications, 8, 12, 2009, 5871–5882. [27] W. Hardjawana, B. Vucetic, Y. Li, and Z. Zhou, “Multiuser cooperative base station systems with joint precoding and beamforming,” IEEE Journal of Selected Topics in Signal Processing, 3, 6, 2009, 1079–1093.
4
Turbo base stations Emre Aktas, Defne Aktas, Stephen Hanly, and Jamie Evans
4.1
Introduction Cellular communication systems provide wireless coverage to mobile users across potentially large geographical areas, where base stations (BSs) provide service to users as interfaces to the public telephone network. Cellular communication is based on the principle of dividing a large geographical area into cells which are serviced by separate BSs. Rather than covering a large area by using a single, highpowered BS, cellular systems employ many lowerpowered BSs each of which covers a small area. This allows for the reuse of the frequency bands in cells which are not too close to each other, increasing mobile user capacity with a limited spectrum allocation. Traditional narrowband cellular systems require the cochannel interference level to be low. Careful design of frequency reuse among cells is then crucial to maintain cochannel interference at the required low level. The price of low interference, however, is a low frequency reuse factor: only a small portion of the system frequency band can be used in each cell. More recent wideband approaches allow full frequency reuse in each cell, but the cost of that is increased intercell interference. In both approaches, the capacity of a cell in a cellular network, with six surrounding cells, is much less than that of a single cell operating in an intercell interferencefree environment. In this chapter, we survey an approach that allows the cell with neighbors to achieve essentially the same capacity as the interferencefree cell. In a conventional cellular system, each mobile user is serviced by a single BS, except for the softhandoﬀ case – a temporary mode of operation where the mobile is moving between cells and is serviced by two base stations. A contrasting idea is to require each mobile station to be serviced by all BSs that are within its reception range. In this approach all the BSs in the cellular network are components of a single transceiver with distributed antennas, an approach known as “network multipleinput multipleoutput (MIMO).” Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
78
Turbo base stations
Network MIMO requires cooperation between BSs. On the uplink, the BSs must cooperate to jointly decode the users, whilst on the downlink, the BSs must cooperate to jointly broadcast signals to all the users in the network. This approach may appear unrealistically complex, but informationtheoretic studies have highlighted the potentially huge capacity gains from such an approach [23, 49, 59]. In a nutshell, these works (and others) have shown that such cooperation eﬀectively eliminates intercell interference. In other words, the percell capacity of a network of interfering cells is roughly the same as a noninterfering system where the cells are isolated and do not interfere at all (in fact, there is a diversity advantage for the interfering system, which means its capacity is higher than the capacity of the isolated cell model). In the network of interfering cells there is no wasted interference: all received signals contain useful information. Crucially, to obtain this advantage, it is necessary for there to be intercell interference: it was shown in [23, 59] that full frequency reuse in each cell is required in order to achieve the full informationtheoretic capacity. This is in contrast to the conventional cellular model with single cell processing which usually requires fractional frequency reuse. The question then arises: how can such cooperation be realized in practice? It is natural to conceive ﬁrst a centralized system in which a central processor is connected to all the base stations, so that the network is operated as a single cell MIMO system, but with distributed antennas. Such an architecture is, however, expensive to build, has a single point of failure, and does not satisfactorily address issues of complexity and delay. A more feasible and desirable solution is to distribute the processing among the base stations. In this chapter we present distributed BS cooperation methods for joint reception and transmission, which allow the desired network MIMO behavior to emerge in a distributed manner. For distributed processing, communication among the BSs is mandatory. The desired properties of a feasible distributed method are: (1) communication should only be required between neighboring BSs, as opposed to message passing among all BSs, and (2) the processing per BS and message passing delay should remain constant with increasing network size. In this chapter, we survey an approach to BS cooperation (and provide new results for this approach) based on a graphical model of the networkMIMO communication processes. In essence, we show that both uplink and downlink modes of communication reduce to belief propagation on graphs derived from the way BSs are interconnected in the backhaul, and from the signal propagation between BSs and mobiles, and vice versa, across the air interface. To give a simple picture of what we mean by message passing between BSs, consider a cellular network where the BSs and the cells are placed on a line. In this model, every cell has two neighboring cells. Although this simple model is far from being realistic, it provides a framework where the main concepts of distributed processing with message passing can be developed and explained, and it can then be generalized to less restrictive models. The onedimensional cellular array is illustrated in Figure 4.1.
79
4.1 Introduction
y1
y2
y3
yn
BS 1
BS 2
BS 3
BS n
x1
x2
x3
xn
MS 1
MS 2
MS 3
MS n
Figure 4.1. Linear cellular array. The cells are positioned on a line. Each cell has one active mobile station (MS). Dashed lines show boundaries between cells. At cell i, xi and yi represent the transmitted symbol and the received signal. Let xi denote the data symbol transmitted by mobile station (MS) i and yi denote the channel output observed at BS i. In the linear cellular array model, the relationship between the transmitted symbols and the received signals is yi = hi (−1)xi−1 + hi (0)xi + hi (1)xi+1 + zi ,
(4.1)
where hi (j) is the channel coeﬃcient from MS i + j to BS i, and zi is the additive Gaussian noise with variance σ 2 . We assume that the channel coeﬃcients hi (j) and the noise variance are known at BS i. For convenience, for the cells at the edges of the network, add dummy symbols x0 and xn +1 , and set the corresponding hi (j)s to zero. The signal model for the onedimensional cellular array model is depicted in Figure 4.2.
xi−1
)
h
−1 i(
hi
−1 (1
hi (0)
zi−1
)
xi
zi
)
hi
1 (−
+1
hi
(1
)
xi+1
yi+2
zi+1
)
1 (−
2
h i+
hi
+1
(1
)
zi+2
hi+2 (0)
yi+1
hi+1 (0)
yi
hi−1 (0)
yi−1
xi+2
Figure 4.2. Linear cellular array signal model. The symbol transmitted in one cell is received at that cell, and also in the two neighboring cells (one neighboring cell if it is one of the two edge cells). In the traditional singlecell processing (SCP) approach, BS i tries to detect symbol xi based on yi alone. Using a frequencyreuse factor of 1/2 avoids the intercell interference, but this halves the capacity of the system. With full frequency reuse, MS i receives interference from MSs i − 1 and i + 1, as is clear in (4.1). One could treat this interference as Gaussian noise, and use a mismatched decoder to decode the desired signal, but information theory tells us that intercell interference can be completely eliminated via multicell processing (MCP) [23, 49, 59].
80
Turbo base stations
MCP requires cooperation between the BSs, but how much cooperation is required in the simple model we are considering here? At ﬁrst sight, it might seem suﬃcient for BS i to use (yi−1 , yi , yi+1 ) in the detection of symbol xi , as these are the only outputs to which xi actively contributes. This is not the case, but it is certainly true that BS i can do a much better job of detecting xi in this scenario. The BS’s task is ﬁrst to compute the conditional distribution p(xi yi−1 , yi , yi+1 ) and then to pick the maximum a posteriori estimate for xi . One approach to realize this detection strategy would be for each BS to pass the observed channel output to its immediate neighbors: thus, BS i sends yi to BSs i − 1 and i + 1 respectively. This strategy involves one single message passing between adjacent BSs. Considering this further, however, we see that intercell interference has not been eliminated after a single message passing step. For example, yi−1 receives a contribution from data symbol xi−2 and the uncertainty in xi−2 must be accounted for in the above probabilistic model. Again, it could be treated as Gaussian noise, or it could be modeled more accurately than that, depending on what is measured or known by the BSs, and what information is passed from one to the other. For example, BS i may know the constellations from which the interfering symbols xi−2 and xi+2 have been chosen. The BS may also have phase information (the coherent case) or the phase may be unknown (incoherent). The exact model used by BS i depends on which particular assumptions best describe the realworld scenario, but in all these possible models, intercell interference remains after one message passing step in the eﬀect of the unknown symbols xi−2 and xi+2 , which cannot be reliably detected. The above interference model may remind the reader of standard intersymbol interference (ISI) channels that arise in frequencyselective digital communication scenarios. Such models are linear, and if we assume in addition that the a priori distributions on the input symbols are Gaussian, then the optimal equalizer is to apply the matched ﬁlter (in this case, the linear minimum mean squared error (LMMSE) ﬁlter) to the observed symbols y1 , y2 , ..., yn . This makes it clear that it is not optimal for BS i to have access only to (yi−1 , yi , yi+1 ): to be optimal, BS i requires all the channel outputs y1 , y2 , ..., yn , as well as all the channel gains, and information about the a priori distributions on the symbols. With that information, it can apply the optimal ﬁlter, and obtain an optimal estimate of xi . In other words, there is a systemwide coupling of the interference between cells. This approach might be called centralized MCP . The problem with centralized MCP is that it requires a huge amount of message passing. All BSs require global channel knowledge in order to each apply the globally optimal ﬁlter. Note, however, that distributed methods can be used in ISI equalization. In the Gaussian case, the LMMSE estimates can be obtained by the recursive Kalman smoother . In the case of discrete input constellations, the maximum a posteriori (MAP) detector can be obtained by the forward– backward or BCJR algorithm [8] . Such methods are special cases of Bayesian estimation for graphical models. This suggests the idea of representing the
4.2 Review of message passing and belief propagation
81
cellular network by a graphical model, and obtaining distributed versions of MCP that do not require each BS to obtain the complete global channel state information (CSI). Further, these methods will allow us to investigate how well performance improves with the number of message passing steps. For example, in some scenarios, we will see that a single message passing step is suﬃcient to get most of the gains of MCP, whereas in other scenarios, many more message passing steps are required. The challenge in the area of turbo BSs is to distribute the computations of the conditional distributions of the xi s, so that they can be obtained by message passing between neighboring BSs only. We do this for the uplink in Sections 4.3, and 4.4. In Section 4.5, we apply similar ideas to the downlink broadcast channel problem in which the BSs are sending data symbols to the MSs. To initiate this study, our ﬁrst step will be to review message passing and belief propagation methods in a more generic framework, and then to apply the results from this theory to the cellular models of interest in this chapter.
4.2
Review of message passing and belief propagation The distributed algorithms presented in this chapter are built on the key concepts of factor graphs and the sumproduct algorithm. We begin with a brief review of these concepts. The use of iterative, or turbo, receiver methods deﬁned on graphs has become an important focus of research in communications since the success of turbo codes and the rediscovery of lowdensity paritycheck codes. Both the turbo decoder [38] and the lowdensity paritycheck code decoder [20] are instances of belief propagation on associated graphs. A factor graph is a graphical representation on which message passing algorithms are deﬁned. There are at least two other popular graphical representations employed in the communications literature. Firstly, there are graphs on which codes are deﬁned. These graphs represent sets of constraints which describe a code and include Tanner graphs [51], Tanner–Wiberg–Loeliger (TWL) graphs [58], and Forney graphs [17]. These graphs also provide iterative decoding of the associated codes via message passing algorithms. Secondly, there are probabilistic structure graphs including Markov random ﬁelds [29] and Bayesian networks [44]. These graphs represent statistical dependencies among a set of random variables. Markov random ﬁelds are based on local Markov properties, whereas Bayesian networks are based on causal relationships among the variables and factoring the joint distribution into conditional and marginal probability distribution functions. The message passing algorithms deﬁned on these structures provide methods of probabilistic inference: compute, estimate, and make decisions based on conditional probabilities given an observed subset of random variables.
82
Turbo base stations
Factor graphs are not speciﬁcally based on describing code constraints or probabilistic structures. They indicate how a joint function of many variables factors into a product of functions of smaller sets of variables. They can be used, however, for describing codes and decoding codes, and in describing probabilistic models and statistical inference. In fact, factor graphs are more general than Tanner, TWL, and Forney graphs for describing codes [34], and they are more general than Markov random ﬁelds and Bayesian networks in terms of expressing factorization of a global distribution [19].
4.2.1
Factor graph review In this subsection, we provide just enough review for the uninitiated reader to be able to grasp the BS cooperation material presented in this chapter. For further information, the reader may refer to [30] and the excellent tutorials [32, 33]. The reader experienced in factor graphs may skip this section. Let g(x1 , x2 , . . . , xn ) be a function of variables x1 , . . . , xn , where for each i, xi takes on values in a set Ai .
Deﬁnition of marginal function and summary notation We are interested in a numerically eﬃcient computation of the marginal function gi (xi ) = g(x1 , x2 , . . . , xn ) (4.2) ∼{x i }
for some i. The right hand side of (4.2) denotes the summation for xi of function g as deﬁned in [30]: for each a ∈ Ai the value of gi (a) is obtained by summing the value of g(x1 , x2 , . . . , xn ) over all (x1 , . . . , xn ) ∈ A1 × · · · × An such that xi = a. For example, for n = 3, the summation for x2 of g is g2 (x2 ) = g(x1 , x2 , x3 ) = g(x1 , x2 , x3 ). ∼{x 2 }
x 1 ∈A 1 x 3 ∈A 3
Relationship to the APP For probabilistic models, the computation of the marginal in (4.2) is related to the computation of the a posteriori probability (APP), a quantity of particular interest to us in this chapter. Let (x1 , . . . , xn ) denote the realization of some random variables in a probabilistic model, let (y1 , . . . , ym ) denote some observed variables in the model, and let p(x1 , . . . , xn , y1 , . . . , ym ) denote the joint distribution. Taking a given (y1 , . . . , ym ) as ﬁxed, (i.e., observed) deﬁne the global function g: g(x1 , . . . , xn ) = p(x1 , . . . , xn , y1 , . . . , ym ). Typically, g is factorized into two as g(x1 , . . . , xn ) = p(y1 , . . . , ym x1 , . . . , xn )p(x1 , . . . , xn ),
(4.3)
83
4.2 Review of message passing and belief propagation
where the ﬁrst term is the likelihood function and the second term is the a priori distribution of (x1 , . . . , xn ). Depending on the probabilistic model, these two factors themselves are further factorized. The APP of xi for any desired i ∈ {1, . . . , n} is proportional to the marginal of g for xi : p(xi y1 , . . . , ym ) ∝ gi (xi ),
(4.4)
where gi (xi ) is the marginal of the joint distribution in (4.3), and the notation “∝” means “proportional to”, i.e., the right hand side of “∝” is scaled by a constant to obtain the left side. If the left hand side is a probability function, this scaling constant can be found using the fact that this function adds up to unity over all possible values of its argument.
Deﬁnition of factor graph Suppose that g(x1 , . . . , xn ) is in the form of a product of local functions fj : g(x1 , . . . , xn ) =
J
fj (Xj ),
(4.5)
j =1
where Xj is a subset of {x1 , . . . , xn }, and the function fj (Xj ) has the elements of Xj as arguments. A factor graph represents the factorization of g(x1 , . . . , xn ) as in (4.5). The corresponding factor graph has two types of nodes: variable nodes and factor nodes. For each variable xi there is a variable node shown by a circled xi , and for each local function fj there is a factor node shown by a solid square in the graph. Thus there are n variable nodes and J factor nodes in the graph. There is an undirected edge connecting variable node xi to factor node fj if and only if xi is an argument of fj . Thus connections are only between variable and factor nodes; two factor nodes are never connected, and two variable nodes are never connected. We deﬁne the neighbors of a variable node to be those factor nodes to which it is directly connected in the graph. We correspondingly deﬁne the neighbors of a factor node to be those variable nodes in the graph to which it is directly connected.
Deﬁnition of sum–product algorithm The goal of the sum–product algorithm is to obtain the marginal function in (4.2) for some i ∈ {1, . . . , n}. This is done in a numerically eﬃcient manner, based on the factorization in (4.5) using the distributive law to simplify the summation. The algorithm is deﬁned in terms of messages between connected factor and variable nodes. A message from node a to node b is computed based on previously received messages at node a from all its neighbors except for node b. A message from variable node xi to factor node fj is a function with argument xi that can take on values in Ai . A message from factor node fj to variable node xi is also a function of xi . After the messages from all nodes propagate through the factor graph in a sequential manner, at termination, the incoming messages
84
Turbo base stations
at desired variable nodes are combined in order to obtain the associated marginal function. The rules for message updates are given below. Message from variable node x to factor node f : µh−x (x). µx−f (x) =
(4.6)
h∈n (x)\{f }
Message from factor node f to variable node x: ⎛ ⎝f (n(f )) µf −x (x) = ∼{x}
⎞ µy −f (y)⎠ ,
(4.7)
y ∈n (f )\{x}
where n(x) : n(x)\{f } : n(f ) :
set of all factor node neighbors of variable node x in the factor graph, set of all neighbors of x except for f , set of all variable node neighbors of factor node f in the factor graph.
We make the following observations on the messages in the sum–product algorithm. The computations done by variable nodes in (4.6) are a simple multiplication of incoming messages, whereas the computations done by the factor nodes in (4.7) are more complex. A variable node of degree 2 (i.e., a node with two neighbors) simply replicates the message received on one edge onto the other edge. A factor node of degree 1 simply outputs the function of the variable that it is connected to as the message. The computation typically starts at the leaf nodes of the factor graph. Each leaf variable node sends a trivial identity function. If the leaf node is a factor node, it sends a description of f . If the computation is started from nonleaf nodes, it is assumed that it has received trivial identity messages during initiation. Each node remains idle until it receives all required messages based on which it can compute outgoing messages. To terminate the computations, the messages are combined at the desired variable nodes. The rule for combining messages at a variable node is to take the product of all incoming messages: µh−x (x). (4.8) µx (x) = h∈n (x)
Equivalently, µx (x) can be computed as the product of the two messages that were passed in opposite directions over any single edge incident on x: µx (x) = µf −x (x)µx−f (x)
for any f ∈ n(x).
(4.9)
If the factor graph is a tree, then µx (x) will be the marginal function g(x) deﬁned in (4.2). If the factor graph has loops, then the message passings can be repeated, and at termination µx (x) will be an approximation of the marginal
4.2 Review of message passing and belief propagation
85
function g(x). In many cases, scaled versions of the messages are computed, which results in a µx (x) scaled by a constant. Thus the ﬁnal µx (x) is obtained after a proper normalization.
Deﬁnition of [P ] notation If P is a Boolean proposition involving some set of variables, then [P ] is the {0, 1}valued truth function 1, if P is true, [P ] = (4.10) 0, if P is false.
4.2.2
Factor graph examples Example 1 Hidden Markov model Consider a probabilistic model where we have the states vector s = (s1 , s2 , . . . , sn ) and output variables vector u = (u1 , u2 , . . . , un ). The states s1 , . . . , sn form a Markov chain, and the transition from si−1 to si produces an output variable ui . The local function Ti computes the conditional probability of transitioning from si−1 to si , and the output ui : Ti (si−1 , ui , si ) = p(si si−1 )p(ui si , si−1 )
for i = 1, . . . , n.
(4.11)
In several examples, ui is a function of only si , so in those examples, Ti (si−1 , ui , si ) = p(si si−1 )[ui = d(si )], where d is the function that determines ui . Corresponding to each output variable ui is the “noisy” observation yi , where the relationship between the output variable and its observation is characterized by the conditional distribution p(yi ui ). The global function of (s, u) is g(s, u) = p(ys, u)p(s, u) n n = p(yi ui ) Ti (si−1 , ui , si ) . i=1
(4.12)
i=1
Note that y is ﬁxed for any realization of observation, so we consider g(s, u) to be a function of (s, u) only, and regard y as a vector of parameters. The factor graph corresponding to the factorization in (4.12) is given in Figure 4.3 for n = 3. The dummy nodes added in this graph do not alter the function g nor the resulting algorithm, but they allow a convenient description of the algorithm. For T1 , the state transition from s0 to s1 is independent of s0 . During initialization, each pendant factor node sends the messages, which are their function descriptions to their corresponding variable nodes. Then, since the corresponding variable nodes are all of order 2, they replicate the messages at the other edge. Afterwards, forward (si−1 to si for i = 1, . . . , n) and backward (si to
86
Turbo base stations
f (s0 ) = 1
s0
α(s0 ) T1 α(s1 ) β(s1 )
s1
α(s1 ) T2 α(s2 ) β(s1 )
β(s2 )
γ(u1 )
s2
α(s2 ) T3 α(s3 ) β(s2 )
β(s3 )
γ(u2 )
s3
f (s3 ) = 1
γ(u3 )
u1
u2
u3
f (y1 u1 )
f (y2 u2 )
f (y3 u3 )
Figure 4.3. Factor graph for hidden Markov model for n = 3. Dummy nodes f (s0 ), s0 , and f (s3 ) are added to handle the initialization of the algorithm at the edges of the Markov chain. Since the variable nodes in this graph have degree 2, they simply replicate the message received on one edge on the other edge. si−1 for i = n, . . . , i + 1) message passing occurs along the chain. The resulting algorithm is known as the forward–backward or BCJR algorithm [8]. In the literature, the message µu i −T i (ui ) is denoted by γ(ui ), the message µT i −s i (si ) is denoted by α(si ), and the message µT i −s i −1 (si−1 ) is denoted by β(si−1 ). Using that notation, at initialization, we have γ(ui ) = p(yi ui ) = f (yi ui )
for i = 1, . . . , n,
α(s0 ) = 1, β(sn ) = 1. Then the forward recursion is computed as the message from Ti to si , using (4.7): Ti (si−1 , ui , si )α(si−1 )γ(ui ) for i = 1, . . . , n (4.13) α(si ) = ∼{s i }
and the backward recursion is computed as the message from Ti to si−1 β(si−1 ) = Ti (si−1 , ui , si )β(si )γ(ui ) for i = n, . . . , 2. (4.14) ∼{s i −1 }
This is the general form of the forward–backward algorithm. For diﬀerent speciﬁc cases, the local functions are diﬀerent but the general structure of the algorithm is the same, outlined by the forward and backward recursions in (4.13) and (4.14). After the forward and backward recursions are complete, at termination, for each state variable node si , the incoming messages are combined as µs i (si ) = α(si )β(si )
for i = 1, . . . , n.
(4.15)
Since the factor graph is a tree, µs i (si ) is, in fact, the true marginal gi (si ) and a scaled version of the APP p(si y). This model is directly applicable to the uplink of the simple onedimensional cellular network that we examine in Section 4.3. It is the simplest model of turbo BS cooperation that we encounter in this chapter.
4.2 Review of message passing and belief propagation
p(x1 )
p(x2 )
p(x3 )
p(x4 )
p(x5 )
x1
x2
x3
x4
x5
87
µx1 −y1 (x1 ) µy1 −x1 (x1 ) p(y1 ·) p(y2 ·) p(y3 ·) p(y4 ·)
Figure 4.4. Factor graph for the interference channel model for n = 5, m = 4, ny 1 = {x1 , x2 }, ny 2 = {x1 , x3 }, ny 3 = {x2 , x3 , x4 , x5 }, and ny 4 = {x3 , x4 , x5 }. The notation p(yi ·) refers to the conditional distribution of yi given the neighbor variable nodes: p(yi ·) = p(yi ny i ). In the following sections, the prior distribution factor nodes, p(xi ), will not be shown in the graphs.
Example 2 Interference channel Consider a channel with n input variables x = {x1 , . . . , xn } and m output variables y = {y1 , . . . , ym }. Each output variable is a noisy observation of a linear combination of the elements in a subset of the inputs, indexed by ni ⊂ {1, . . . , n}, hi,j xj + zi , (4.16) yi = j ∈n i
where hi,j is the complex channel coeﬃcient of input xj at the channel output yi , and zi is the additive white circularly symmetric complex Gaussian noise. Suppose that the channel coeﬃcients and the variance of zi (σ 2 ) are known. Let ny i denote the set of the input variables indexed by ni : {xj : j ∈ ni }. Then the distribution of yi conditioned on ny i is ⎧ " "2 ⎫ " " ⎬ ⎨ " " 1 1 " " exp − − h x . (4.17) p(yi ny i ) = y i i,j j " " 2 2 ⎩ σ " πσ " ⎭ j ∈n i
Suppose that the inputs are independent, then the joint distribution of {x1 , . . . , xn } is g(x1 , . . . , xn ) = p(x1 , . . . , xn , y1 , . . . , ym ) m n p(yi ny i ) p(xj ). = i=1
(4.18)
j =1
We can use the (loopy) factor graph corresponding to the factorization in (4.18) and the sum–product algorithm on that graph to compute (an approximation of) the APP p(xi y1 , . . . , ym ) ∝ gi (xi ). The factor graph corresponding to (4.18) is given in Figure 4.4. There are two types of messages in Figure 4.4: xtoy messages, and ytox messages. Let nx j denote the set of yi nodes such that yi is a neighboring factor node of the variable node xj in the graph. If yi ∈ nx j , the message from variable
88
Turbo base stations
node xj to factor node yi is, from (4.6), µx j −y i (xj ) = p(xj )
µy k −x j (xj ).
(4.19)
y k ∈n x j \{y i }
If xj ∈ ny i , the message from factor node yi to variable node xj is, from (4.7), ⎛ ⎞ ⎝f (yi ny i ) µx l −y i (xl )⎠ . (4.20) µy i −x j (xj ) = ∼{x j }
x l ∈n y i \{x j }
During initialization, the pendant factor nodes p(xj ) send their description to variable nodes xj . In addition, the factor nodes p(yj ·) send trivial messages to their neighboring variable nodes: µy i −x j (xj ) = 1 for i = 1, . . . , m, and xj ∈ ny i . Afterwards, we have an iterative algorithm, where at each iteration we compute (1) xtoy messages for each j ∈ {1, . . . , n} and yi ∈ nx j in (4.19); (2) ytox messages for each i ∈ {1, . . . , m} and xj ∈ ny i in (4.20). Notice that the graph in Figure 4.4 is loopy, and this means that the algorithm will not terminate in a ﬁnite number of steps, nor will it be guaranteed to ﬁnd the correct marginalizations. If the algorithm does converge, however, then it can be terminated after a suﬃciently large number of steps, and then an approximation to the marginal distribution on the variable nodes can be obtained as follows. The messages at variable node xj for j ∈ {1, . . . , n} are combined as µy k −x j (xj ). (4.21) µx j (xj ) = p(xj ) y k ∈n x j
Models that lead to factor graphs with loops like this simple interference channel example will arise when we turn our attention to twodimensional cellular network models in Section 4.4. First, however, we will look at onedimensional cellular networks where the corresponding factor graphs are loopfree.
4.3
Distributed decoding in the uplink: onedimensional cellular model Consider again the cellular network where the BSs and the cells are placed on a line, as depicted in Figure 4.1. In this model, every cell has two neighboring cells. Although this simple model is far from being realistic, it provides a framework in which the main concepts of distributed processing with message passing can be developed and explained, and it can then be generalized to less restrictive models. Let xi denote the symbol transmitted by MS i and yi denote the channel output observed at BS i, as depicted in Figure 4.2. In the linear cellular array model, the relationship between the transmitted symbols and the received signals is described by (4.1). As discussed in Section 4.1, the goal is to obtain optimal detection of any particular xi given all observations y1 , . . . , yn , in a distributed
4.3 Distributed decoding in the uplink: onedimensional cellular model
89
manner with cooperating BSs, as an alternative to the traditional approach of SCP. In SCP, BS i has access to the channel output yi only. In contrast, we are interested here in distributed, messagepassing algorithms to accomplish MCP, based on probabilistic graphical models.
4.3.1
Hidden Markov model and the factor graph The linear cellular array model is highly reminiscent of a standard linear ISI model in digital communications, and hence we expect to be able to apply the BCJR algorithm [8]. In [8], a statebased hidden Markov model is used, as described in Example 1 in Section 4.2.2. In a statebased model, several input variables are combined to form a state such that each channel output is only a function of that state, and the state sequence forms a Markov chain. The key idea in [21] is to treat the onedimensional cellular model as an ISI channel. In fact, this idea goes back to [59]. The state for cell i is si = (xi−1 , xi , xi+1 ) and we assume the symbols from diﬀerent mobiles are independent, taking values in some ﬁnite alphabet (which can be diﬀerent for the diﬀerent users). Thus, there are several possible values for the state si , so we will write (xi−1 (si ), xi (si ), xi+1 (si )) for the values of the data symbols corresponding to a particular state value si . It is clear that the state sequence is a Markov chain, with the following transition probabilities: p(s1 ) = p(x0 (s1 ))p(x1 (s1 ))p(x2 (s1 )), p(si+1 si ) = [xi (si ) = xi (si+1 )][xi+1 (si ) = xi+1 (si+1 )]p(xi+2 (si+1 )), where the [P ] notation was deﬁned in (4.10). Note that [xi (si ) = xi (si+1 )][xi+1 (si ) = xi+1 (si+1 )] indicates whether state si+1 conforms with state si , i.e., whether a transition from si to si+1 is possible. Note that each cell has one channel output, yi , which is dependent only on the state si , as in the hidden Markov model of Section 4.2.2. To complete the match with the model in that section, we deﬁne the output variable corresponding to the transition from si−1 to si to be ui , where ui = d(si ) := hi (−1)xi−1 (si ) + hi (0)xi (si ) + hi (+1)xi+1 (si ), and we note that the conditional distribution of the observation, yi , given ui , is f (yi ui ) = N (yi ; σ 2 , ui ), where N (x; σ 2 , M ) denotes the Gaussian distribution with mean M and variance σ 2 . The corresponding factor graph is shown in Figure 4.5, where the function node Ti computes the function: Ti (si−1 , ui , si ) = p(si si−1 )[ui = d(si )].
(4.22)
It follows that the forward–backward algorithm can be applied to obtain the APP p(si y1 , . . . , yn ), which can be further marginalized to obtain p(xi y1 , . . . , yn ), the APP of the mobile data symbols. The implementation of the forward–backward recursions is distributed among the BSs, where BS i performs the computations done by node Ti in the algorithm.
90
Turbo base stations
s0
f (s0 ) T1
s1
f (sn ) T2
s2
T3
s3
Tn
u1
u2
u3
un
f (y1 u1 )
f (y2 u2 )
f (y3 u3 )
f (yn un )
sn
Figure 4.5. Factor graph for the hidden Markov model for the linear cellular array. Dashed lines show boundaries between cells. The computations of the nodes within a cell are done by the BS of that cell. Any message passing through a cell boundary corresponds to actual message passing between corresponding BSs.
For example, upon receiving the message α(si−1 ) from cell i − 1, BS i computes α(si ) and forwards it to cell i + 1. Thus, α messages ripple across the BSs from left to right, and β messages ripple in the reverse direction. After the forward and backward recursion is complete, the APP p(si y1 , . . . , yn ) is obtained as a scaled version of (4.15). In this formulation, the middle BS is the ﬁrst to be able to decode its mobile. This serial formulation of the forward–backward algorithm is the natural one to use in solving an ISI equalization problem. It is not natural, however, in cellular radio networks to designate a leftmost or rightmost BS. In fact, we cannot do that at all for an inﬁnite linear array model. Fortunately, the sum– product algorithm has ﬂexibility in terms of node activation schedules [30]. Initial conditions can be arbitrary, and each node can operate in parallel. This allows all BSs to immediately begin computing their messages starting with the a priori distributions on the input symbols. At each iteration, a BS passes an α message to the right, and a β message to the left. In a ﬁnite linear array, this parallel version of the forward–backward algorithm converges to the same solution as obtained from the serial implementation, but an important point is that it can be terminated early giving a suboptimal estimate of the mobile’s data symbol at an earlier time. In the inﬁnite linear array, the algorithm must be terminated at some point in time. This approach allows an investigation of estimation error versus delay, as can be found in [39]. The actual values that the variables can take have not been speciﬁed. In this section, we have in mind that each xi takes a value from a discrete constellation, and, as such, the BSs are engaged in the demodulation of the users’ data symbols. If the symbol xi is replaced by the transmitted codeword of mobile i and yi is replaced by the channel outputs corresponding to a codeword, i.e., if we include the time dimension, then we can use the described method for decoding as opposed to the detection of individual symbols, as considered in [21]. In the present section, the forward–backward algorithm is accomplishing joint
4.3 Distributed decoding in the uplink: onedimensional cellular model
91
multiuser detection (MUD) of the users’ data symbols, prior to singleuser decoding. After the detection of the symbols, each BS can decode its own user using a singleuser decoder. Note that the complexity of MUD is typically exponential in the number of users [54], but it is known that in some special cases the complexity can be much reduced [47, 52], for example when the signature sequences have constant crosscorrelation [48]. In the present section, we have a distributed MUD that is linear in the number of users, and this is due to the highly localized interference model: the crosscorrelations of most signature sequences are zero. Indeed, the BCJR algorithm implements the optimal MAP detection of the users’ symbols, and this is known to have a complexity that is linear in time [8], i.e., in the number of symbols. To approach Shannon capacity at high SNR, it is required to send many bits per symbol, which requires a large alphabet size (large signal constellations), and the BCJR algorithm is exponential in the alphabet size. So even if the complexity is linear in the number of users, the overall complexity can be very high. This observation also applies to the decoding of codewords in the model considered in [21]. A standard approach to limit the complexity of MUD is to restrict attention to suboptimal linear techniques, which we consider further in Section 4.3.2. Unfortunately, this does not avoid the complexity of the overall decoding problem, but at least one can then focus attention on wellestablished techniques for decoding singleuser codes.
4.3.2
Gaussian symbols A standard approach in MUD is ﬁrst to estimate the individual symbols from diﬀerent users using linear MUD techniques. Once the BS has estimated symbol xi from mobile i it then passes this soft estimate to a singleuser decoder for mobile i. The decoder waits until it receives the estimates of all symbols in the codeword, and then it attempts to decode the codeword. This approach limits the complexity of the MUD component of the receiver. It is well known that optimal MUD is in fact linear if the underlying symbols being estimated are jointly Gaussian. In this section, we assume that the input symbols are drawn from joint Gaussian distributions (independent across mobiles) and then we apply the corresponding optimal linear ﬁlters, and the task of the present section is to show how these ﬁlters can be implemented via message passing between the BSs in the cellular network. Another motivation for this section is that the developed methods will prove useful in designing iterative messagepassing algorithms to accomplish beamforming on the downlink of a cellular system, as we will see in Section 4.5. When the input symbols are modeled as Gaussian random variables, we can still employ factor graph methods. The global function is now a continuous function and the marginalization is done by integrating (as opposed to summing) with respect to unwanted variables. Since the messages are now continuous
92
Turbo base stations
functions, each message in general corresponds to a continuum of values. However, if the message functions can be parameterized, they can be represented by a ﬁnite number of parameter values. For example, if a message function is the probability density function of a Gaussian vector, then it is characterized by a mean vector and covariance matrix pair, which is the case for the Gaussian input model. We will now describe the Kalmansmoothingbased distributed algorithm in [39] for the linear cellular array. The model is the same as in (4.1) except that now the xi s are independent zeromean Gaussian distributed with variance p. We are going to use matrixvector notation, so deﬁne the state for cell i to be the column vector si = [xi−1 , xi , xi+1 ]T . The states again form a Markov chain, but we now express the transition from state si to si+1 as si+1 = Af si + bf xi+2 , where ⎡
0 Af = ⎣0 0
1 0 0
⎤ 0 1⎦ , 0
⎡ ⎤ 0 bf = ⎣0⎦ . 1
Then the state transition is characterized by the conditional distribution T
f (si+1 si ) = N (si+1 ; pbf bf , Af si ),
(4.23)
where we use the notation
* + 1 N (s; M, m) ∝ exp − (s − m)T M−1 (s − m) 2
to denote a Gaussian distribution, scaled by an arbitrary constant that is not a function of the argument of the function. Here, s is the argument of the function and M and m are parameters. Deﬁne the column vector T , hi = hi (−1) hi (0) hi (1) , then the observation in cell i can be expressed in vector form as yi = hTi si + zi . The factorization of the joint distribution again has the form in (4.12). The corresponding factor graph is shown in Figure 4.6. Since the si s and yi s are jointly Gaussian, all of the messages turn out to be Gaussian distributions. Thus the actual messages will be the mean vector and covariance matrix pairs. Before deriving the messages, let us present some useful results for the Gaussian distribution. Remember that in our notation the distribution is scaled by
93
4.3 Distributed decoding in the uplink: onedimensional cellular model
p10 (s1 ) f (s1 )
s1
p11 (s1 ) p21 (s2 ) f (s2 s1 )
f (y1 s1 )
s2
p22 (s2 ) p32 (s3 ) f (s3 s2 )
f (y2 s2 )
s3
p33 (s3 ) p43 (s4 ) f (s4 s3 )
f (y3 s3 )
s4
f (y4 s4 )
Figure 4.6. Factor graph for a hidden Markov model for n = 4 used for the linear cellular array with Gaussian inputs. Dashed lines show boundaries between cells. The computations of the nodes within a cell are done by the base station of that cell. Any message passing through a cell boundary corresponds to an actual message passing between corresponding base stations. an arbitrary constant. N (s; M, m) = N (m; M, s), −1
(4.24) −1 T
−1
, A (m − s)),
(4.25)
−1 −1 −1 −1 M3 = (M−1 1 + M2 ) , m3 = M3 (M1 m1 + M2 m2 );
(4.26)
N (As + b; M, m) = N (s; A MA
N (s; M1 , m1 )N (s; M2 , m2 ) = N (s; M3 , m3 ), where −1
N (s; M1 , m1 )N (s; M2 , m2 )
= N (s; M4 , m4 ),
where .
−1 −1 −1 −1 M4 = (M−1 1 − M2 ) , m4 = M4 (M1 m1 − M2 m2 );
N (s; M1 , m1 )N (As; M2 , t) ds = N (t; AM1 AT + M2 , Am1 ).
(4.27) (4.28)
We know that the messages are going to be Gaussian. Denote them by pii−1 (si ) = N (si ; Mii−1 , ˆsii−1 ), pii (si ) = N (si ; Mii , ˆsii ).
(4.29) (4.30)
From the observation node, we have the message f (yi si ) = N (yi ; σ 2 , hTi si ). From (4.6), the message from variable node si to factor node f (si+1 si ) is pii (si ) = pii−1 (si )f (yi si ) *
+ * 0+ 1/ 1 1 T 2 ˆ ∝ exp − (si − ˆsii−1 )T M−1 (s − s ) exp − (y − h s ) i i i ii−1 i ii−1 2 2 σ2 + * / 0 1 sii ) , ∝ exp − (si − ˆsii )T M−1 ii (si − ˆ 2
94
Turbo base stations
where
Mii =
M−1 ii−1
1 + 2 hi hTi σ
−1
ˆsii = Mii M−1 sii−1 + ii−1 ˆ
,
(4.31)
1 hi yi . σ2
Thus the pair (4.31)–(4.32) is the message from si to f (si+1 si ). This equations is another form of the more familiar Kalman ﬁlter correction [27]: Mii = I − Ki hTi Mii−1 , ˆsii = ˆsii−1 + Ki yi − hTi ˆsii−1 ,
(4.32) pair of update (4.33) (4.34)
where Ki =
Mii−1 hi . σ 2 + hTi Mii−1 hi
The equivalence of (4.31)–(4.32) and (4.33)–(4.34) can be shown using inversion of matrix sum identities. Next, let us obtain the message function pii−1 (si ) using (4.7): . pii−1 (si ) = f (si si−1 )pi−1i−1 (si−1 ) dsi−1 . (4.35) Note that the summation in (4.7) becomes integration in (4.35) since we are dealing with continuous variables. From (4.23) and (4.30): . T pii−1 (si ) = N (si ; pbf bf , Af si−1 )N (si−1 , Mi−1i−1 , ˆsi−1i−1 ) dsi−1 ∝ N (si ; pbf bf
T
T
+ Af Mi−1i−1 Af , Af ˆsi−1i−1 ),
(4.36)
where (4.36) is due to (4.24) and (4.28). As a result, the message function pii−1 (si ) is represented by the meancovariance pair: ˆsii−1 = Af ˆsi−1i−1 , fT
Mii−1 = pbf b
(4.37)
+ Af Mi−1i−1 A
fT
.
(4.38)
Equations (4.37)–(4.38) are Kalman ﬁlter prediction updates [27]. Note that the message pii (si ) is the posterior distribution of si given {y1 , . . . , yi }, and pii−1 (si ) = f (si y1 , . . . , yi−1 ). We desire the posterior distribution of si given all observations: f (si y1 , . . . , yn ). For that purpose, form a graph similar to Figure 4.6 but in the backward direction: states are ordered from sN to s1 and connected by the transition nodes f (si−1 si ), where [39] T
f (si−1 si ) = N (si−1 ; pbb bb , Ab si ), ⎡ ⎤ ⎡ ⎤ 0 0 0 1 Ab = ⎣1 0 0⎦ , bb = ⎣0⎦ . 0 1 0 0
4.3 Distributed decoding in the uplink: onedimensional cellular model
95
For the backward graph, denote the message from factor node f (si si+1 ) to variable node si by pii+1 (si ) = N (si ; Mii+1 , ˆsii+1 ), which will be the posterior distribution of si given {yi + 1, . . . , yn }. Combination of the backward message pii+1 (si ) with the forward message pii (si ) to obtain f (si y1 , . . . , yn ) can be done as follows: f (si y1 , . . . , yn ) ∝ f (si , y1 , . . . , yn ) = f (y1 , . . . , yi si , yi+1 , . . . , yn )f (si , yi+1 , . . . , yn ) ∝ f (y1 , . . . , yi si )f (si yi+1 , . . . , yn )
(4.39) −1
∝ f (si y1 , . . . , yi )f (si yi+1 , . . . , yn )f (si ) = pii (si )pii+1 (si )f (si )−1
= N (si ; Mii , ˆsii )N (si ; Mii+1 , ˆsii+1 )N (si ; pI, 0)−1 −1
(4.40)
= N (si ; M3 , m3 )N (si ; pI, 0)
(4.41)
= N (si ; Mi , ˆsi ).
(4.42)
Equation (4.39) is due to the fact that given si , {y1 , . . . , yi } and {yi+1 , . . . , yn } become independent. In (4.40) the fact that the prior distribution of si is zeromean Gaussian with covariance pI is used. Equation (4.41) is from (4.26), where −1 −1 M3 = (M−1 ii + Mii+1 ) ,
(4.43)
m3 = M3 (M−1 sii + M−1 sii+1 ). ii ˆ ii+1 ˆ
(4.44)
Equation (4.42) is from (4.27), where −1 −1 Mi = (M−1 3 − p I) ,
ˆsi =
(4.45)
Mi (M−1 3 m3 ).
(4.46)
Combining (4.43)–(4.46), we obtain the result Mi =
1 −1 M−1 ii + Mii+1 − I p
−1 ,
ˆsi = Mi (M−1 sii + M−1 sii+1 ). ii ˆ ii+1 ˆ For the onedimensional cellular network we have seen how message passing algorithms can be applied on the uplink to detect discrete data symbols and estimate Gaussian data symbols. The onedimensional nature of these models leads to underlying factor graphs without loops and thus to guaranteed convergence of the sumproduct algorithm on these factor graphs. In the sequel, we will see that the situation is quite diﬀerent when we move to twodimensional cellular networks.
96
Turbo base stations
BS 11
MS 11
BS 12
MS 12
BS 21
MS 21
BS 22
MS 22
BS 31
MS 31
BS 32
MS 32
BS 41
MS 41
BS 42
MS 42
BS 13
MS 13
BS 14
MS 14
BS 23
MS 23
BS 24
MS 24
BS 33
MS 33
BS 34
MS 34
BS 43
MS 43
BS 44
MS 44
Figure 4.7. Rectangular cellular array model. The cells are positioned on a rectangular grid. Each cell has one active MS. The signal transmitted in one cell is received at that cell, and also four neighboring cells (except for edge cells). Dashed lines show boundaries between cells.
4.4
Distributed decoding in the uplink: twodimensional cellular array model
4.4.1
The rectangular model A model that is more general than the linear array model is the model where BSs are positioned on a twodimensional grid. For example, consider the rectangular model where the BSs are on a rectangular grid. Again, assume ﬂat fading and orthogonal multiple access channels within a cell. The received signal at the BS of any cell, in any channel, is the superposition of the signal from its own MS, and the signals of the four adjacent cell cochannel users. The positioning of the BSs and MSs is shown in Figure 4.7. For cell (i, j) in the rectangular grid, let xi,j denote the symbol transmitted by the MS, yi,j the signal received at the BS, hi,j (xm ,n ) the channel from mobile (m, n) to BS (i, j), and zi,j additive Gaussian noise. The relationship between the observations yi,j and the transmitted symbols xi,j is expressed as yi,j =
x m , n ∈n y i , j
hi,j (xm ,n )xm ,n + zi,j ,
(4.47)
4.4 Distributed decoding in the uplink: twodimensional model
BCJR columns
y
APP
97
APP
BCJR rows
Figure 4.8. Iterative implementation of BCJR algorithm along the columns and c 2008 IEEE). rows of the rectangular array ( where ny i , j = {xi,j , xi−1,j , xi+1,j , xi,j −1 , xi,j +1 }
(4.48)
is the set of transmitted symbols that can be heard at BS (i, j). For the cells at the edges of the rectangular network, dummy symbols x0,j , xn +1,j , xi,0 , xi,n +1 are added and the corresponding hi,j (xm ,n ) are set to zero. The goal is again to obtain the global APP p(xi,j y), where y = {y1,1 , . . . , y1,n , . . . , yn ,1 , . . . , yn ,n } is the set of all observations. It is still possible to obtain exact inference by forming a Markov chain via clustering (e.g., states obtained by clustering along the rows of the twodimensional array) and then apply the BCJR algorithm, but the complexity grows exponentially with n (the number of columns or rows in the rectangular array) and is intractable as the network size grows. It is possible that the inherent complexity is only polynomial in the network size (we have not investigated this issue) but in any case, we are looking for distributed approaches using message passing between neighboring base stations. Encouraged by the elegance of the implementation of the BCJR algorithm for the onedimensional array, one can be tempted to use this approach along the columns and rows of a rectangular array in an iterative manner. The APP outputs of the BCJR along one direction will be used as a priori probabilities for the BCJR along the other direction. Thus the global decoder is built as an iterative decoder where the two modules of the iterative decoder are the BCJR in each direction (Figure 4.8 from [5]). The details of this approach, and a discussion of its implementation are in [4]. The idea of running BCJR along the rows and columns of a rectangular cellular array was also proposed for twodimensional ISI channels by Marrow and Wolf in [35]. Although applying the BCJR algorithm along the rows and columns of the rectangular array seems to work [5, 35], we see that it does not directly exploit the twodimensional structure of the problem but instead imposes a onedimensional structure on parts of it. As this is an adhoc iterative method, it will result in only an approximation to the desired APPs. However, if we are looking for an approximation of the APP, there is no need to impose the use of the BCJR algorithm which gives the optimum result only if the problem is onedimensional.
98
Turbo base stations
Thus we can accept an approximate APP and form loopy graphs that reﬂect the true twodimensional nature of the problem.
4.4.2
Earlier methods not based on graphs Research has considered distributed global demodulation in twodimensional cellular channels [53, 57]. In [53], the authors considered BSs computing soft estimates of the symbols, and then sharing and combining them to obtain a ﬁnal soft estimate. This strategy was compared with BSs sharing channel outputs and performing maximumratio combining of the channel outputs. In [57], a reduced complexity maximumlikelihood (ML) decoder was developed, which was motivated as an extension of the Viterbi algorithm which exploits the limited interference structure. Although the general large twodimensional cellular structure was not treated, it seems that the algorithm, if applied to that structure, would result in increasing complexity per symbol with growing network size. Alternatively, graphbased iterative message passing methods for distributed detection for twodimensional cellular networks were proposed in [3–5, 50].
4.4.3
Statebased graph approach One way to model the twodimensional case is to adapt the statebased graphical idea from the onedimensional case. Remember that in a statebased graph, each channel output depends only on one state variable, and everything else in the system is modeled by the transitions among the states. For the onedimensional case, the states form a Markov chain, but in the twodimensional case they do not. For cell (i, j), deﬁne the state to be si,j = ny i , j ,
(4.49)
where ny i , j is the set of symbols deﬁned in (4.48), upon which yi,j depends, as in yi,j = mi,j (si,j ) + zi,j , where the conditional mean mi,j (si,j ) is a deterministic function of si,j , hi,j (xm ,n )xm ,n . (4.50) mi,j (si,j ) = x m , n ∈n y i , j
It can be observed that the states form a Markov random ﬁeld, a fact that we will exploit in (4.51). As in the onedimensional model, the variables in the sum–product algorithm are not the channel input symbols xi,j , but the states si,j . The goal of BS (i, j) is to obtain the APP p(si,j y), from the marginalization of which it can obtain the APP p(xi,j y). Let s = {si,j } denote the set of all states. The global function
4.4 Distributed decoding in the uplink: twodimensional model
p(s22 s12 , s21 ) s22
p(s23 s13 , s22 )
p(s24 s14 , s23 )
s23
p(y22 ·)
s24
p(y23 ·)
p(s32 s22 , s31 ) s32
p(y24 ·)
p(s33 s23 , s32 )
p(s34 s24 , s33 )
s33
p(y32 ·)
s34
p(y33 ·)
p(s42 s32 , s41 ) s42
p(y34 ·)
p(s43 s33 , s42 )
p(s44 s34 , s43 )
s43
p(y42 ·)
p(y43 ·)
99
s44 p(y44 ·)
Figure 4.9. Factor graph for the statebased probabilistic model for the rectangular cellular array. Dashed lines show boundaries between cells. The computations of the nodes within a cell are done by the BS of that cell. Any message passing through a cell boundary corresponds to actual message passing between corresponding BSs.
to be marginalized to compute the p(si,j y) ∝ gi (si ) is g(s) = p(y, s) n n = p(yi,j si,j )p(si,j si−1,j , si,j −1 ),
(4.51)
i=1 j =1
where s0,j and si,0 are dummy states: p(s1,j s0,j , s1,j −1 ) = p(s1,j s1,j −1 ) and p(si,1 si−1,1 , si,0 ) = p(si,1 si−1,1 ). The factor graph for the factorization in (4.51) is depicted in Figure 4.9. Note that this is a loopy graph and hence the sum–product algorithm is not guaranteed to give the exact APP. Nevertheless, we now describe the algorithm which will be observed to provide good performance in practical settings. Uniform prior distributions p(xi,j ) are assumed on the input symbol constellations. Deﬁne the
100
Turbo base stations
set of states pi,j = {p1i,j , p2i,j }, where p1i,j = si−1,j , p2i,j = si,j −1 . The system is modeled by transitions from pi,j to si,j . Note that not every transition pi,j → si,j is possible. Similarly to the onedimensional case, we say that si,j is conformable with pi,j if there is transition from pi,j to si,j , i.e., p(si,j pi,j ) > 0 for some prior distribution on the xi,j s. The set of all conﬁgurations of pi,j that are conformable with si,j will be denoted by pi,j : si,j , and the set of all conﬁgurations of si,j that are conformable with pi,j will be denoted by si,j : pi,j . The message computations described next for cell (i, j) can be implemented at BS (i, j), simultaneously in parallel by all BSs. The message from p(yi,j ·) to si,j is µy i , j −s i , j (si,j ) = p(yi,j si,j ) = CN (yi,j ; m(si,j ), σ 2 ), where CN (y; m, σ 2 ) is the distribution function of the complex Gaussian with mean m and variance σ 2 . This message is computed only once given an observation yi,j at BS (i, j). Next, for convenience, deﬁne the following factor nodes corresponding to the conditional distribution functions in the factorization in (4.51): c1i,j : factor node p(si,j +1 si−1,j +1 , si,j ); c2i,j : factor node p(si+1,j si,j , si+1,j −1 ); di,j : factor node p(si,j si−1,j , si,j −1 ). The messages between nodes si,j and di,j are internal calculations in BS (i, j). The message from factor node di,j to variable node si,j , from (4.7), is
µd i , j −s i , j (si,j ) =
p(si,j si−1,j , si,j −1 )
∼s i , j
=
∝
2 p i , j :s i , j k =1
µp ki, j −d i , j (pki,j )
k =1
p(si,j pi,j )
pi , j
2
2
µp ki, j −d i , j (pki,j )
k =1
µp ki, j −d i , j (pki,j )
(4.52)
4.4 Distributed decoding in the uplink: twodimensional model
2 p 1i , j p 2i , j
=
µp ki, j −d i , j (pki,j )
101
(4.53)
:s i , j k =1 :s i , j
2
µ ¯p ki, j −d i , j (ski,j ),
(4.54)
k =1
where (4.52) is because p(si,j pi,j ) is a constant if pi,j is conformable with si,j , as prior distribution of xi,j s is uniform, and in (4.54) µp ki, j −d i , j (pki,j ) µ ¯p ki, j −d i , j (ski,j ) = p ki, j :s i , j
can be considered as a preprocessed message from pki,j to di,j . Note that (4.52) and (4.53) are not, in general, equal because in (4.52) the summation is over p1i,j and p2i,j , which conform with each other as well as with si,j , whereas in (4.53) we also include p1i,j and p2i,j , which do not conform with each other. Speciﬁcally, xi−1,j −1 s in p1i,j and p2i,j should be the same for the summation in (4.52), but they may be diﬀerent in (4.53). The additional terms in (4.53) lead to the simpliﬁcation in (4.54), which results in a considerable saving in complexity. The message from variable node si,j to factor node di,j , from (4.6), is 2
µs i , j −d i , j (si,j ) = µy i , j −s i , j
µc ki, j −s i , j (si,j ).
(4.55)
k =1
The message from variable node si,j to factor node cki,j for k = 1, 2 is µs i , j −c mi, j (si,j ) = µd i , j −s i , j µy i , j −s i , j µc li , j −s i , j (si,j ),
(4.56)
where l = {1, 2}\k. The message in (4.56) is an actual message from BS (i, j) to the corresponding BS. Finally, we need the message from di,j to pki,j for k = 1, 2, which also should be implemented as an actual message from BS (i, j) to adjacent cells. Using (4.7), p(si,j pki,j , sli,j )µs i , j −d i , j (si,j )µp li , j −d i , j (pli,j ) µd i , j −p ki, j (pki,j ) = ∼p ki, j
=
µs i , j −d i , j (si,j )
si , j
∝
p li , j
µs i , j −d i , j (si,j )
s i , j :p ki, j
α
p(si,j pki,j , pli,j )µp li , j −d i , j (pli,j )
p li , j p li , j
µp li , j −d i , j (pli,j )
(4.57) (4.58)
:s i , j :p ki, j
µs i , j −d i , j (si,j )¯ µp li , j −d i , j (si,j ),
(4.59)
s i , j :p ki, j
where l = {1, 2}\k, (4.58) is due to the fact that p(si,j pki,j , pli,j ) is a constant for all si,j : pki,j and pli,j : si,j . The approximation in (4.59) is similar to (4.53)
102
Turbo base stations
in that we have extra terms in the summation with nonconforming pki,j and pli,j . Combining everything, the approximation (due to loops) of the marginal function gi (si ) is, from (4.8), µs i (si ) = µy i , j −s i , j (si,j )µd i , j −s i , j (si,j )µc 1i , j −s i , j (si,j )µc 2i , j −s i , j (si,j ).
(4.60)
For some related work on statebased method for rectangular grids the reader is referred to [50].
4.4.4
Decomposed graph approach As an alternative to the statebased graph approach, let us remember the signal model in (4.47). It is seen that this model is the same as (4.16) in Example 2 of Section 4.2.2. Therefore, for each observation yi,j , the conditional distribution given the contributing symbols has the same form as (4.17). As the input symbols are independent, the joint distribution of all symbols and observations has the same form as (4.18). Thus, a factor graph, in the form of Figure 4.4 can be obtained for the purpose of obtaining APPs of the symbols. For the rectangular array model, corresponding to cell (i, j), there will be variable node xi,j and factor node yi,j in this graph, and each factor node yi,j will be connected to the variable nodes in the neighborhood: ny i , j deﬁned as in (4.48). Such a graph for the rectangular model in Figure 4.7 is depicted in Figure 4.10. Having the factor graph in Figure 4.10, we can perform message passing as described in Example 2 in Section 4.2.2 in order to obtain estimates of the transmitted symbols (the graph clearly has loops). Equations (4.19) and (4.20) are the xtoy and ytox messages, respectively, except that now we have two indices for each variable, denoting the twodimensional location of the cell. Any time a message is passed along an edge that crosses a dashed line in Figure 4.10, an actual message passing among corresponding BS is required. At termination, the posterior probability of the transmitted symbol xi,j at cell (i, j) is computed by combining all incoming messages, using (4.21). In addition to its conceptual simplicity, the decomposed graph approach has the advantage that it does not require the regular positioning of the cells. The method can be applied to any irregular network shape, where each cell has an arbitrary number of neighbors in arbitrary directions.
4.4.5
Convergence issues: a Gaussian modeling approach Unlike in the onedimensional cellular models, where the graphs are trees, we cannot provide deﬁnitive convergence results for our twodimensional cellular models, in general. It is well known that the sum–product algorithm is not guaranteed to converge when there are loops in the graph. This is the Achilles’ heel of our approach to BS cooperation, and it is an area that requires much further study. However, some insights into the convergence properties can be obtained from the Gaussian model, which we consider next.
4.4 Distributed decoding in the uplink: twodimensional model
x1,1
x1,2
p(y1,1 ·) x2,1
p(y1,2 ·)
x4,1
x2,3
p(y2,2 ·)
x3,3
p(y3,2 ·)
x2,4 p(y2,4 ·) x3,4
p(y3,3 ·)
x4,2
p(y4,1 ·)
p(y1,4 ·)
p(y2,3 ·)
x3,2
p(y3,1 ·)
x1,4
p(y1,3 ·)
x2,2
p(y2,1 ·) x3,1
x1,3
x4,3
p(y4,2 ·)
103
p(y3,4 ·) x4,4
p(y4,3 ·)
p(y4,4 ·)
Figure 4.10. Factor graph for the decomposed probabilistic model for the rectangular cellular array model. Dashed lines show boundaries between cells. The computations of the nodes within a cell are done by the BS of that cell. Any message passing through a cell boundary corresponds to actual message passing between corresponding base stations.
In Example 2 in Section 4.2.2, modeling the source symbols xj as circularly symmetric complex Gaussian in (4.16) also leads to a tractable solution. In that case, all yi s and xj s are jointly Gaussian, and also every local function in the factorization in (4.18) is a Gaussian distribution. As a result, the messages on the graph in Figure 4.10 will also be Gaussian. Let the pair (µx j −y i , σx j −y i ) denote the meanvariance pair of the message from xj to yi , and (µy i −x j , σy i −x j ) denote the meanvariance pair of the message from yi to xj . Note that they are both means and variances of the variable xj . Using the properties of complex Gaussian distributions, the mean and variance of the message from yi to xj can be shown to be yi − µy i −x j =
σ2 + σy i −x j =
hi (xl )µx l −y i
x l ∈n y i \{x j }
, hi (xj ) hi (xl )2 σx l −y i
x l ∈n y i \{x j }
hi (xj )2
(4.61)
.
(4.62)
104
Turbo base stations
The mean and variance of the message from xj to yi can be shown to be µπx j σxπj µx j −y i =
k ∈n x j \{i}
1 + σx j ⎛
σx j −y i = ⎝
+
µy k −x j σy k −x j
k ∈n x j \{i}
1 + σxπj
1
,
(4.63)
σy k −x j
k ∈n x j \{i}
⎞−1 1 ⎠ σy k −x j
,
(4.64)
where (µπx j , σxπj ) denotes the prior mean and variance of the symbol xj . At termination, estimates of the mean and variance of the posterior distribution are µy i −x j µπx j + σxπj σy i −x j i µx j = (4.65) 1 , 1 + σx j σy i −x j i −1 1 1 σx j = + . (4.66) σx j σy i −x j i The goal here is to analyze how the means and variances evolve, as the ytox message updates in (4.61)–(4.62) and xtoy message updates in (4.63)–(4.64) are computed iteratively. Next, we provide results from [40] which say that the convergence of the variances is always guaranteed, whereas the convergence of the means is closely related to the spectral radius of an iteration matrix Ω. First, consider the convergence of the variances. Equations (4.62) and (4.64) show that the evolution of the variances is independent of the means or the observations themselves. To simplify notation deﬁne: σi,j = σy i −x j . Let the graph be for a system with n input variables {x1 , . . . , xn } and also n output variables {y1 , . . . , yn }. For iteration t deﬁne the vector of variances of messages from y to x nodes T , v(t) = σ1,1 . . . σ1,n σ2,1 . . . σ2,n . . . σn ,1 . . . σn ,n . The following result is proven in [40], and is described in detail (but without proof) in [41]. This result, and the next one, make the simplifying assumption that the variances of all the observation variables (the yi s) are unity. Theorem 4.1 The sequence of vectors v(t) always converges to a unique ﬁxed point for any v(0) , i.e., lim v(t) = v∗ .
t→∞
4.4 Distributed decoding in the uplink: twodimensional model
105
Moreover, the sequence is monotonically decreasing under the initialization (0) σx j −y i = 1. Next, consider the convergence of the means in the updates in (4.61) and (4.63) assuming that the variance messages are ﬁxed to the converged values: v(t) = v∗ . Deﬁne µi,j = µy i −x j
(4.67)
and for iteration t , m(t) = µ1,1
...
µ1,n
. . . µn ,1
...
µn ,n
T
the channel matrix ˜ 1,n , . . . , h ˜ n ,1 . . . , h ˜ n ,n }, ˜ 1,1 , . . . , h ˜ = diag{h H ˜ i,j is the channel coeﬃcient from source xj to observation yi . Let Ω be where h ˜ −1 (Σx − I)H(I ˜ + D(Σf V∗ −1 Σf ) − V∗ −1 )−1 (Σf − I)V∗ −1 , Ω=H
(4.68)
where Σx = diag{1n ×n , . . . , 1n ×n }, V∗ = diag{v∗ }, ⎡ In I n . . . ⎢ .. .. Σf = ⎣ . . In
In
...
⎤ In .. ⎥ , . ⎦ In .
1n ×n is an n × n block of ones, In is the n × n identity matrix, and D(·) is the operator deﬁned as D(A) = diag{A11 , A22 , . . . , An n } for a n × n matrix A. The following theorem provides a necessary and suﬃcient condition for the convergence of the means. It was proven in [40] where it was shown to follow from Theorem 5.3 in [7]. Theorem 4.2 The sequence of vectors m(t) converges to the ﬁxed point ˜ −1 y m∗ = (I + Ω)−1 H for any m(0) if and only if the spectral radius ρ(Ω) < 1. This theorem conﬁrms that in some scenarios, where the spectral radius condition is not met, the sum–product algorithm will not converge. When the condition is violated it is necessary to switch to another form of preprocessing, perhaps at a lower data rate using SCP – or one can declare an outage. Note that the
106
Turbo base stations
same issue arises for power control algorithms, where instability can also occur if the system loading is too high. Preliminary investigation of practical methods to deal with this issue have been examined in [40], but much further work, both practical and theoretical, is required to characterize properly the conditions of convergence in more general settings, and to determine what to do about lack of convergence when it arises. We have undertaken extensive numerical experiments using diﬀerent channel parameter values. For a deterministic channel model, where the channel coeﬃcient for a mobile user is 1 to its own BS and α (crosscoupling factor) to an adjacent cell’s BS (symmetric deterministic channel model), we have observed the following for the case of Gaussian symbols: at realistic SNRs less than 30 dB, and crosscoupling factors less than 1/2 (the typical crosscoupling between two adjacent cells is quite low, usually less than 1/2), we have never observed lack of convergence of the sum–product algorithm. However, convergence is not guaranteed at higher levels of the crosscoupling factor, when the SNR is high. Similar convergence problems were also observed for the discrete symbol model, when the deterministic channel described above was utilized. On the other hand, the convergence problems are greatly mitigated when we replace the deterministic channel gains with independent, Rayleigh fading gains with the same means as above. Thus, even if the network is symmetric with respect to average gains, convergence problems almost completely disappear if the instantaneous gains are random (e.g., Rayleigh distributed). From our numerical experiments, we found that when the sum–product algorithm failed to converge, it was typically due to a symmetric network realization of channel coeﬃcients. However, symmetry can be perturbed by noise. If the SNR is suﬃciently low, the realizations of noise amplitudes are large enough to perturb the symmetry of the network. On the other hand, when the channel coefﬁcients are modeled as independent random variables (e.g., complex Gaussian), the system lacks symmetry with high probability. In other words, the probability of a realization of h that is symmetric enough to cause failure of convergence is extremely small. In a practical system, the channels of diﬀerent mobile users are indeed independent, due to the fact that they are located in diﬀerent cells. The simulations in Figures 4.11–4.13 are for such a scenario: in each simulation realization, independent samples of channel coeﬃcients hi,j (m, n) are generated. In those ﬁgures, no error ﬂoors are observed, even for very low uncoded bit error rates. There may still be an error ﬂoor, but it is too low to be detected. Note that if there were a set of channel realizations that always cause convergence failure, irrespective of SNR, then the probability of such a set must be very small, as it would provide an error ﬂoor at this probability. We emphasize that here convergence means the convergence to the true posterior probabilities of the symbols, not to the true values of the symbols. The true APP can still result in an incorrect estimate of the symbol, however if the true APP can be attained, the lowest possible error rate is obtained.
4.4 Distributed decoding in the uplink: twodimensional model
107
0
10
SP: state−based SP: decomposed Ind. optimum SU lower bound −1
10
−2
Pb
10
α=0.5, SNR = 7 dB −3
10
α=1, SNR = 5 dB −4
10
α=0.5, SNR = 11 dB
−5
10
0
4
8
12 16 20 Number of serial real number passings
24
28
Figure 4.11. Probability of error of the algorithms and the singleuser lower bound as a function of the number of serial real number transmissions between c 2008 IEEE). BSs for node (2, 2) of a 4 × 4 network (
In the next section, we present some numerical results that indicate the relative performance of the diﬀerent schemes we have proposed so far.
4.4.6
Numerical results We now present some numerical results [5] comparing the aforementioned approaches for rectangular cellular arrays given in Figures 4.11, 4.12, and 4.13. In these simulations, a fading channel model is considered where each hi,j (m, n) is complex Gaussian distributed with zero mean and variance 1 if (m, n) = (0, 0) and variance α2 otherwise. Thus α2 represents the average power of the intercell interference (ICI) from each neighbor. The additive noise zi,j is complex Gaussian with zero mean and variance σ 2 . The signal to noise ratio (SNR) is deﬁned as 1/σ 2 . The transmitted symbols are binary and from the set {−1, 1}. For comparison, the performance of individually optimum multiuser detection and the singleuser lower bound is also given. The singleuser lower bound is the performance of a system with a single user (with no interference) and multiple receiving base stations. Each message passing requires an amount of serial real number transmissions among base stations, and in Figure 4.11 the performance is shown as a function of those transmissions. Figure 4.11 illustrates that performance very close to the interferencefree case can be achieved after 3–4 message passing steps, with the decomposed model outperforming the clustered approach. In Figure 4.12, the performance of the optimum receiver for a base station that cannot communicate with other base stations is also shown. For the cooperative
Turbo base stations
0
10
α =1 α =0.5
−1
10
α =0.25 α=0 −2
10
Pb
α=0.25 −3
10
α=0.5 α=1
Single−cell processing BCJR BP−clustered BP−decomposed Ind. optimum SU lower bound
−4
10
−5
3
0
5
10
15
20
SNR (dB)
Figure 4.12. Probability of error of the algorithms and the singleuser lower c 2008 IEEE). bound as a function of the SNR for cell (2, 2) of a 4 × 4 network (
0
0
10
10 BCJR SP: state−based SP: decomposed SU lower bound
SP: state−based SP: decomposed SU lower bound
−1
−1
10
10
−2
Pb
Pb
108
10
−3
10
−2
10
−3
α = 0.5
10
α =0.5, SNR = 9 dB
−4
10
−5
−4
0
5 SNR (dB)
10
10
0
4 8 12 16 20 24 28 Number of serial real number passings
Figure 4.13. Probability of error of the algorithms and the singleuser lower bound as a function of the SNR and as a function of number of real number c 2008 IEEE). transmissions in series for cell (10, 10) of a 20 × 20 network ( cases, the values plotted are the error probabilities after enough iterations have occurred to satisfy the convergence criterion. As the ICI power increases, it is observed that the distributed decoding algorithms not only can handle the ICI, but they can also exploit the extra energy and diversity provided by it.
4.4 Distributed decoding in the uplink: twodimensional model
109
The error rate performance of the sum–product algorithms on the clustered and decomposed graphs is very close to the singleuser lower bound at low error probabilities. Thus, we observe a very large gain is possible from local message passing to reduce the ICI. The BCJR algorithm, implemented iteratively on the columns and rows of the rectangular array, does not perform as well as the sum–product algorithms; a 0.5 dB gap is observed. In Figure 4.13 performance is shown for a larger network. We observe that the performance on this 20 × 20 network is essentially the same as in the smaller network of size 4 × 4. The observation that the speed of convergence remains roughly the same is explained by the fact the ICI is a local eﬀect even though the overall network size is growing.
Simpliﬁcation of messages sent by factor nodes In the decomposed graph approach, there are two types of computations: computation of variabletofactor node messages in (4.19), and the computation of factortovariable node messages in (4.20). Upon examining these two types of messages, one sees that the main cause of computation complexity of this approach is the computation of the latter: factortovariable node messages. In [9], a simpliﬁcation of the messages sent by the factor nodes was proposed. The key idea here is to recognize that the message µy i −x j (xj ) is the individually optimal soft MUD of xj given the observation yi and the prior distributions of ny i \{xj }. If this is computationally unacceptable, suboptimal MUD methods can be substituted for this purpose. An arbitrary choice of MUD may, however, not be suitable. The MUD should be able to incorporate the prior distributions of ny i \{xj } to produce a posterior distribution of xj . If the MUD uses prior distributions of all ny i , then the prior information xj should be canceled in the message to xj . This is because the information received from a node is not fed back to that node in factor graph methods. In [9], an iterative groupwise MUD was considered.
4.4.7
Adhoc methods utilizing turbo principle One approach to BS cooperation is to have the BSs share their soft (or hard) bit estimates and reconstruct the interference components at the output. After subtracting the interference components from the observation, the BSs repeat the decoding. This approach was suggested in [37] for a twocell model. In this model, the received signal at a BS comprises a desired signal component and an interference component from othercell mobiles. For a convolutionally coded system, the BSs perform singleuser decoding, ignoring the interference component completely. Then the soft bit estimates are shared between the BSs, which use this information to reconstruct and subtract the interference components from their received signals. Repeating this procedure, an iterative “turbo” BS
110
Turbo base stations
cooperation method is obtained. The performance of this method was investigated for various quantization strategies and constellation sizes in [37]. Note, however, that this iterative approach is ad hoc, and is not based on graph algorithms. The turbo principle and sharing soft information with the adjacent BSs was also considered in [28]. Again, coding is included in the interference reconstruction and cancellation in [28]. The turbo principle is utilized for a general cellular network. The BS cooperation methods in [28, 37] are not explicitly based on algorithms on graphs. They are adhoc implementations of the turbo principle among BSs which exchange soft decisions in order to improve their decisions.
4.4.8
Hexagonal model All of the methods described for the rectangular cellular array model can be extended to the more realistic hexagonal array model. The positioning of the cells is shown in Figure 4.14: for example, the decomposed factor graph for this model is as in Figure 4.15.
4.5
Distributed transmission in the downlink So far our focus has been on BS cooperation for the uplink of a wireless communication network. In this section, we will shift our attention to the downlink of a wireless network. The scenario where a BS simultaneously transmits independent information to multiple uncoordinated users over the wireless channel can be classiﬁed as a broadcast channel (BC) . We will ﬁrst summarize the main informationtheoretic results for a MIMO (vector) BC and present some practical transmission techniques proposed in the literature. Since wireless communication networks for commercial applications are multicellular, we will then brieﬂy discuss the impact of ICI and some possible solutions. Finally, we will present turbo BS cooperation in the downlink as a practical approach for providing high spectral eﬃciency in the presence of ICI.
4.5.1
Main results for the downlink of a singlecell network Consider the downlink of a singlecell MIMO network where a BS with n antennas is transmitting information to m users each with a single antenna. Assuming the channel is ﬂat fading with no mobility, the vector of received signals at the users is modeled as y = Hx + z,
(4.69)
, T where y = y1 y2 · · · ym denotes the vector of received signals at the T , users, x = x1 x2 · · · xn denotes the transmitted signal vector, and H is the m × n channel matrix with the (i, j) entry representing the channel gain
111
4.5 Distributed transmission in the downlink
BS 11
BS 12
MS S 11 BS 21
MS 12 BS 22
MS 21 BS 31
MS 51
MS 52
MS 35 BS 44
MS 43 BS 52
BS 35
MS 34 BS 43
MS 42 BS 51
MS 24 BS 34
MS 33 BS 42
MS 41
BS 24
MS 23 BS 33
MS 32 BS 41
MS 13 BS 23
MS 22 BS 32
MS 31
BS 13
MS 44 BS 53
MS 53
Figure 4.14. Hexagonal cellular array model. The cells are positioned on a hexagonal grid. Each cell has one active MS. The signal transmitted in one cell is received at that cell, and also six neighboring cells (except for edge cells). Dashed lines show boundaries between cells. from the jth antenna of the BS to the ith user. Let hi denote the 1 × n channel vector for user i. Then the (i, j) entry of H can be written hi (j). The elements of the noise vector, z, are assumed to be independent zeromean Gaussian distributed random variables. Unless stated otherwise, we assume that both the base stations and the users have perfect CSI. This system model is classiﬁed as a BC in network information theory [14]. Even though the capacity region of a general BC is still an open problem, for the special case of a degraded BC superposition coding [10] is shown to achieve the capacity region. However, when the transmitter has more than one antenna, i.e., n > 1, the system can in general no longer be modeled as a degraded BC. Rather, the capacity region of this vector Gaussian BC is shown to be equal to the dirty paper coding (DPC) [12] rate region [56]. DPC is a nonlinear coding technique based on the observation that if the Gaussian interference is known noncausally at the transmitter but not the receiver, under a transmit power constraint, the eﬀect of the interference can be precanceled. In the case of a vector Gaussian BC
112
Turbo base stations
x11
x12
x13
p(y1,1 ·)
p(y1,2 ·)
p(y1,3 ·)
x21
x22
x23
x24
p(y2,1 ·)
p(y2,2 ·)
p(y2,3 ·)
p(y2,4 ·)
x31
x32
x33
x34
x35
p(y3,1 ·)
p(y3,2 ·)
p(y3,3 ·)
p(y3,4 ·)
p(y3,5 ·)
x41
x42
x43
x44
p(y4,1 ·)
p(y4,2 ·)
p(y4,3 ·)
p(y4,4 ·)
x51
x52
x53
p(y5,1 ·)
p(y5,2 ·)
p(y5,3 ·)
Figure 4.15. Factor graph for the decomposed probabilistic model for the hexagonal cellular array model. Dashed lines show boundaries between cells. The computations of the nodes within a cell are done by the BS of that cell. Any message passing through a cell boundary corresponds to actual message passing between corresponding BSs.
under perfect CSI the BS can precalculate noncausally the interference created by one user to the other. In this case for an arbitrary ordering of the users, using the DPC technique it is possible to encode the information of a user such that it is not aﬀected by the interference caused by previously encoded users. The capacity region of a vector Gaussian BC described in (4.69) under transmit covariance constraint Q = E[xxH ] S for positive semideﬁnite S (where A B implies that B − A is a positive semideﬁnite matrix) is given as [56] C = qhull
3
4 R(π, S, σ , H) , 2
(4.70)
π ∈Π
where qhull denotes the convex closure of the sets, π is an arbitrary permutation of user indices, Π corresponds to the set of all possible user permutations, σ 2
113
4.5 Distributed transmission in the downlink
denotes the variance of the elements of the noise vector z, and " i " 2 hi ( l=1 Bπ (l) )hH " i +σ 2 R(π, S, σ , H) = (R1 , . . . , Rm )"Ri = log H 2 " hi ( i−1 l=1 Bπ (l) )hi + σ for some {B1 , . . . , Bm } such that Bi 0 ∀i and S
m
4 Bi ,
i=1
(4.71) where hi is the 1 × n channel vector for user i and π(l) denotes the index of the user encoded in the lth position. Correspondingly, the sum capacity of the vector Gaussian BC under total transmit power constraint P is computed as Csum =
max
Q :Q 0, tr(Q )≤P
log(σ 2 Im + HQHH ),
(4.72)
where Im denotes an identity matrix of size m,  ·  denotes a matrix determinant and tr(·) denotes the trace operator. In [24] it was shown that CSI at the transmitter is essential in achieving the capacity gains of a vector Gaussian BC and at high SNR, with perfect CSI at the BS, the sum capacity scales linearly with n provided that m > n. The capacity achieving transmission strategy is a combination of nonlinear DPC and linear precoding (beamforming). The diﬃculties in implementing a practical DPC encoder have led to most attention being focused on suboptimal linear precoding schemes. In linear precoding the user data symbols are mapped to the transmitted signal vector via x = Td,
(4.73)
where d is the m × 1 vector of data symbols with dj , the data symbol intended for user j, taken from a ﬁnite constellation with E[dj 2 ] = 1 and where the ith column of the n × m beamforming matrix, ti , is the beamforming vector for user i. It should be noted that the power allocated to the ith user is given by P i = tH i ti . One approach for dealing with multiuser interference is to use zeroforcing (ZF) beamforming, where T satisﬁes HT = D with D being a diagonal matrix. One solution is T = HH (HHH )−1 D,
(4.74)
where D is determined according to the power constraint. The drawback of ZF beamforming is that transmit power might not be used as eﬃciently when attempting to cancel multiuser interference completely. An alternative linear precoding scheme is regularized channel inversion (RCI) where the beamforming matrix is of the form T = HH (HHH + βI)−1 D,
(4.75)
114
Turbo base stations
where D is determined according to the power constraint and β is a regularization parameter. For small β the RCI beamformer approaches the ZF beamformer while for large β the RCI beamformer tends towards the maximal ratio combining beamformer, the beamformer that maximizes the received signal strength while ignoring the resultant interference. The power allocation to users is done based on an optimization criterion under a power constraint. Optimization problems considered in the literature typically include maximization of the sum rate under a total power constraint and minimization of the total transmit power under individual minimum rate requirements for each user. However, a total transmit power constraint might result in an unbalanced power output over the transmit antennas, which is undesirable in practical systems since each antenna is limited by the linear region of the power ampliﬁers in its own RF chain. Power allocation based on more practical per antenna power constraints has been considered [60]. Optimization problems involving RCI and related beamformers are challenging since the beamformer vectors of the users are coupled. An alternative design criterion is to maximize the signaltoleakageandnoise ratio (SLNR), where leakage refers to the interference caused by the user considered to other users [46]. Leakagebased precoding is attractive since the optimization problem becomes decoupled. There are several works proposing practical nonlinear precoding techniques based on the DPC idea [16, 25, 31]. These works mainly use vector quantization or trellis/lattice precoding approaches to achieve a performance close to DPC.
4.5.2
Main results for downlink of a multicellular network In a multicellular network where several BSs simultaneously serve the users in their respective cells, cochannel interference is the major performance limiting factor. This is especially true for users near cell boundaries. This communication scenario is referred to as the interference channel [2]. The capacity region of a general interference channel is still an open problem. There are some results for the extreme cases of strong interference [13] and weak interference [6]. There are also several inner and outer bounds, the bestknown inner bound for discrete memoryless interference channels being the Han–Kobayashi bound [22]. As discussed in Section 4.1, the traditional method for mitigating cochannel interference in multicellular networks has been to use frequency planning such that neighboring cells use diﬀerent frequency bands for transmission. If a suﬃciently low frequencyreuse factor is used, one can ignore the eﬀect of cochannel interference and the communication scenario is reduced to a number of independent singlecell downlink problems. However, as the data rate requirements for next generation wireless networks increase, it has become evident that new paradigms to mitigate cochannel interference are required. In order to increase the spectral eﬃciency of the system a frequencyreuse factor of 1 can be used, i.e., the whole frequency band is utilized simultaneously
4.5 Distributed transmission in the downlink
115
in every cell. However, this results in a reduction in the achievable data rates due to interference, especially for users near cell edges. At this point BS cooperation in the downlink is an attractive solution which actually takes advantage of the interference to increase the spectral eﬃciency of the system. Assuming that the BSs are connected to each other with high capacity links over the backhaul, the cooperative system can be viewed as a virtual MIMO BC with macrodiversity, i.e., transmit antennas are distributed geographically. In that case the problem is reduced to downlink transmission in a singlecell network as discussed in Section 4.5.1, the only diﬀerence being that the transmission schemes have to be implemented in a distributed manner. Several works have considered simple and suboptimal linear precoding schemes for the cooperative downlink. In [18], performance gains of cooperative zeroforcing beamforming were analyzed. In [26] several linear precoding techniques for downlink BS cooperation were compared in terms of sum rate per cell in the asymptotic regime. Most of the literature on downlink BS cooperation assumes that the backhaul is very high capacity. In [36], however, a ﬁnite capacity backhaul was considered and the performances of several transmission schemes involving DPC compared. Another assumption made in most of the literature is that the BSs can perfectly synchronize their transmissions. In practice, networkwide synchronization is very diﬃcult to achieve. In [62] it was demonstrated that even though BSs can perform timing advancement perfectly such that the transmissions from diﬀerent BSs arrive at the user at the same time, the interference is inevitably asynchronous. This asynchronicity causes signiﬁcant performance degradation for linear precoding techniques. The techniques considered are then modiﬁed to better handle asynchronous interference.
4.5.3
BS cooperation schemes with message passing As we have highlighted previously, cooperative schemes that can be implemented in a distributed manner oﬀer several advantages over schemes requiring a central processing unit. Firstly, distributed schemes are more robust since they do not have a central processing unit as a single point of failure. Secondly, they are more scalable since schemes requiring a central processing unit require new BSs to be connected to the central processing unit as the network expands. Therefore it is of great interest to develop cooperative transmission and resource allocation schemes that only require local information exchange between BSs. In this section, we discuss several iterative BS cooperation algorithms that require information exchange between BSs. We ﬁrst focus on distributed beamforming and power allocation schemes in the literature. We then present a graphbased approach that requires only local information exchange between BSs.
Distributed beamforming and power allocation schemes In [15] an iterative algorithm for computing the optimal beamforming vectors and power allocation was presented for a cooperative multicellular system. The
116
Turbo base stations
algorithm iteratively ﬁnds the solution of the following optimization problem: minimize
b
αj tr(TH j Tj ),
(4.76)
j =1
subject to: Γi ≥ γj ,
i = 1, 2, . . . , m,
(4.77)
where Tj is the n × mj beamforming matrix used at the BS in the jth cell with mj users, b is the number of BSs (cells) in the network each with n transmit antennas, αj is the weighting factor of the transmit power of the jth BS, γj denotes the target SINR constraint for user i, and Γi is the SINR of user i expressed as Γi = m k =1 k = i
2 hi,j (i) tH j (i),i 
.
(4.78)
2 2 hi,j (k ) tH j (k ),k  + σ
In (4.78) hi,j denotes the 1 × n channel vector between user i and BS j, j(i) denotes the index of the cell at which user i is located, tH j (i),i is the beamforming 2 vector used for user i by the BS in its cell, and σ is the variance of the Gaussian noise at the user i. Note that in this scheme BSs do not jointly transmit data to all the users in the network, rather each BS transmits data to the users in its cell. However, the beamforming vectors used by BSs are jointly chosen to achieve the target SINRs of the users, taking the ICI into account while minimizing the weighted transmit power. The iterative algorithm utilizes uplink–downlink duality by solving the Lagrangian dual of the optimization problem. Alternatively, in [11] an iterative joint transmission and power allocation scheme was proposed. In the joint transmission scheme considered, a multicell precoding approach is employed where the data of each user are available at all BSs and all BSs simultaneously serve each user using a beamforming vector. In this case the transmitted signal vector at BS j is expressed as xj = Tj d, where d is the data vector of all the users in the network and the ith column of beamforming matrix Tj is the beamforming vector used by BS j for user i. The optimization problem of maximizing the sum rate of the users under individual transmit power constraints at each BS is diﬃcult to implement in a distributed manner and requires channel knowledge to be shared by all BSs. As a result, the authors proposed a heuristic beamforming and power allocation scheme that is fully distributed and requires only statistical channel knowledge. The scheme utilizes the leakagebased beamforming approach described in Section 4.5.1 to simplify the optimization of the beamforming vectors. Furthermore, the beamforming vectors were selected to maximize the average SLNR averaged over channel realizations. The power allocation proposed is also heuristic, allocating more of the transmit power to the user with the best channel gain.
4.5 Distributed transmission in the downlink
117
In [61], an iterative beamforming scheme for multicellular networks is proposed. The system model assumed is similar to the one in [15], where BSs do not employ multicell precoding, each BS serves a single user in its cell using a beamforming vector and the beamforming vectors are chosen such that observed ICI is taken into account. The proposed scheme can be implemented in a distributed manner as it only requires local channel knowledge. The idea is to select the beamforming vector for each BS as a linear combination of the zeroforcing beamformer and maximum ratio combining beamformer (which maximizes the received signal power ignoring the interference generated at other users). The combining weights are iteratively updated based on feedback from the users, until a Pareto optimum point is reached.
Cooperative beamforming based on factor graphs We now present a cooperative downlink beamforming approach that can be implemented in a truly distributed manner based on message passing on graphs. We will ﬁrst demonstrate how the downlink beamforming problem can be formulated as a virtual LMMSE estimation problem [42]. Once this is done we will be able to make use of the message passing algorithms developed for the uplink in the earlier parts of this chapter. We focus on a cooperative downlink beamforming scenario where a cellular network with n cells employs an orthogonal multiple access scheme within each cell so that users do not experience intracell interference. However, since a frequency reuse factor of 1 is used, users will experience interference from neighboring cells. For simplicity, it is assumed that each BS and user has only one antenna. The vector of received signals at the users is as given in (4.69) where the data vector of the users is mapped to the transmitted signal vector using joint linear precoding as in (4.73). We wish to implement regularized channel inversion beamforming as discussed in Section 4.5.1. Deﬁne ˆ = HH (HHH + βI)−1 , T
(4.79)
where β is a regularization parameter [45], and the beamforming matrix is written as ˆ T = TD.
(4.80)
The diagonal matrix D denotes the power allocation matrix whose diagonals correspond to power allocated to the corresponding user under a given power (total or per BS) constraint. ˆ = Dd, the transmitted signal x = T ˆ ˆd The key observation is that deﬁning d can be seen as the LMMSE estimate of some vector u under the signal model ˆ = Hu + w, d
(4.81)
where u and w are n × 1 vectors of i.i.d. random variables with zero mean and variance 1 and β, respectively. Note that u and w have no physical relationship
118
Turbo base stations
with the original beamforming problem and also that the problem in (4.81) is quite diﬀerent to the uplink–downlink duality concept used to simplify the downlink problem [55]. With the observation that the downlink beamforming problem can be seen as a virtual LMMSE estimation problem, one can employ distributed LMMSE estimation techniques such as the distributed Kalman smoothing algorithm described in Section 4.3.2 for onedimensional cellular networks and techniques based on the sum–product algorithms for twodimensional cellular networks presented in Section 4.4.5. We ﬁrst consider the onedimensional linear cellular array depicted in Figure 4.1 which models the communication scenario with BSs placed evenly on the side of a highway or subway tunnel or access points along a corridor. The channel matrix entries hi,j are of the form hi,j =
hi (k),
j = i + k,
0,
i − j > 1,
(4.82)
where i, j ∈ {1, 2, . . . , n} and k ∈ {−1, 0, +1}. One can take advantage of the local connectivity of the network to implement a distributed beamforming algorithm. Assuming that appropriate power allocation is already performed, the ˆ is treated as the observation vector from which the transdata symbol vector d mitted signal vector x is obtained as the LMMSE estimate of u. Due to the Markov structure of the problem, one can apply the message passing algorithm described in Section 4.3.2 for the uplink problem with Gaussian input T , symbols. Deﬁning the state vector for cell i as si = ui−1 ui ui+1 , the statespace model is si+1 = Af si + bf ui+2 , b
b
si−1 = A si + b ui−2 , ˆ i si + wi , ˆi = h d
(4.83) (4.84) (4.85)
f b f b ˆi = are deﬁned in Section 4.3.2, h ,where A , A , b  , and b with u0 = un +1 = h1 (−1) = hn (+1) = 0. Running hi (−1) hi (0) hi (1) a forward and backward Kalman estimator based on this statespace model and combining the two estimates, we obtain a forward–backward beamforming algorithm for the cooperative downlink. Assuming the virtual data vector u is Gaussian distributed (an assumption we are free to make since u does not have a physical meaning), the LMMSE ˆ i.e., the conditional mean vector of the jointly Gaussian ˆ = E[ud], estimate is u ˆ As a result, BS i can ﬁnd the signal that it should conditional density f (ud). ˆ using the transmit by computing the mean of the marginal distribution, f (ui d), sum–product algorithm described in Section 4.3.2 running on the factor graph depicted in Figure 4.6. The steps of the algorithm are summarized below [43].
4.5 Distributed transmission in the downlink
119
Forwardbackward cooperative downlink beamforming algorithm
r Initialization: The BSs at the two edges of the cellular array compute the mean vector and the covariance matrix of the distribution information, (ˆsii , Mii ) for i = 1 and n, as ˆ H (h ˆi h ˆ H + β)−1 dˆi ˆsii = h i i ˆ H (h ˆi h ˆ H + β)−1 h ˆi , Mii = ˆIi − h i
where
⎡
0 ˆI1 = ⎣0 0
0 1 0
(4.86) (4.87)
i
⎡
⎤ 0 0⎦ , 1
1 ˆIn = ⎣0 0
⎤ 0 0⎦ . 0
0 1 0
(4.88)
r Pass the estimates: BS i, i ∈ {1, . . . , n} computes the forward and the backward estimates as the required information becomes available as ˆsfi+1i = Af ˆsfii ,
(4.89)
Mfi+1i = Af Mfii Af
T
T
+ [i = n]bf bf ,
ˆsbi−1i = Ab ˆsbii ,
(4.90) (4.91)
Mbi−1i = Ab Mbii A
bT
bT
+ [i = 1]bb b
,
(4.92)
where ˆsf11 = ˆs11 and ˆsbn n = ˆsn n are described in (4.86). Then message (ˆsfi+1i , Mfi+1i ) is passed to the neighboring BS on the right and (ˆsbi−1i , Mbi−1i ) is passed to the neighboring BS on the left. r Correct the estimates: BS i, i ∈ {1, . . . , n} corrects the received forward and backward estimates using the observation dˆi as ˆ i ˆsf ), ˆsfii = ˆsfi+1i + kfi (dˆi − h i+1i Mfii
= (I3 −
f ˆ kfi h)M i+1i
(4.93) (4.94)
with kfi
ˆH Mfi+1i h = ˆ f h ˆH β + hM
(4.95)
i+1i
and ˆ i ˆsb ), ˆsbii = ˆsbi−1i + kbi (dˆi − h i−1i
(4.96)
b ˆ Mbii = (I3 − kbi h)M i−1i
(4.97)
ˆH Mbi−1i h . ˆ b h ˆH β + hM
(4.98)
with kbi =
i−1i
120
Turbo base stations
r Combine the estimates: Having computed both ˆsf and ˆsb , they are comi−1i ii bined using
ˆsi = Mi (Mfii )−1 ˆsfii + (Mbi−1i )−1 ˆsbi−1i , −1 Mi = (Mfii )−1 + (Mbi−1i )−1 − ˆIi ,
(4.99) (4.100)
where ˆIi = I3 for n = 2, . . . , n − 1 and ˆI1 and ˆIn are deﬁned in (4.88). The transmitted signal from the ith BS, xi , is set to be the middle element of the vector ˆsi . It should be pointed out that the algorithm described above has a fully distributed nature requiring only local information exchange between the BSs. Prior to running the algorithm, however, the power allocation to users is assumed to have been done (D is known), and this might well require networkwide knowledge of quantities such as the channel gains. Fortunately, since the channel gains typically change much more slowly than the data symbols, global sharing of the network knowledge might still be feasible at this slower time scale. Another point we would like to emphasize is that, as stated in Section 4.3.2, since the factor graph depicted in Figure 4.1 is free of loops, the message passing algorithm described above is guaranteed to converge to the optimal solution. It should further be noted that the delay experienced by the BSs at the edges of the cellular array grows linearly with the size of the array. In fact, the precoding delay experienced by BSs is locationdependent with the minimum delay experienced by the BS in the middle of the array. In [43] a suboptimal limited extent distributed beamforming algorithm was proposed based on the observation that due to the local connectivity structure of the channel, the information sent by BS i is expected to be less important for BS j if i − j is suﬃciently large. Therefore, one can achieve a ﬁxed delay, if the extent of the information exchange between BSs is limited. In the proposed limited extent beamforming algorithm, each BS starts by computing a ‘selfestimate’ based on dˆi using (4.86) and (4.87). Then this information is shared with both neighbors and received estimates are corrected using forward and backward correction equations (4.93)–(4.98). After τ phases of information exchange between the BSs, forward, backward and selfestimates are combined and the middle element of the state vector ˆsi is then an approximation to xi . With this algorithm, the precoding delay experienced by all BSs is τ . Numerical results in [43] demonstrate the clear tradeoﬀ between the performance and the precoding delay. Finally, we will discuss the extension of the proposed distributed beamforming algorithm to twodimensional networks. As an example, we will consider a hexagonal network with n = 7 cells, a special case of the hexagonal network depicted in Figure 4.14. As in the onedimensional linear array case, the downlink beamforming problem can be posed as the virtual LMMSE problem of estimating a virtual ˆ Assuming the virtual data vector data vector u from the observation vector d. is Gaussian distributed, the LMMSE estimate corresponds to the mean vector
4.5 Distributed transmission in the downlink
121
ˆ As of the MAP estimate, i.e., u maximizing the conditional distribution f (ud). a result, one can use the sum–product algorithm on the factor graph in Figure 4.15. For the downlink, the same factor graph, in which variable nodes now represent ui and the factor nodes represent the local function fdˆi = f (dˆi u). In addition, assume that for each variable node there is a pendant factor node that is connected only to that variable node, fu i = p(ui ) = N (ui ; σf u i −u i , µf u i −u i ), denoting the prior distribution of the virtual data symbols. Since all the distributions are Gaussian, the messages passed between variable and factor nodes are actually the mean vector and covariance matrix of the underlying distribution. Assuming a ﬂooding schedule [30] where at each iteration the messages on the graph are updated simultaneously, the sum–product update equations for kth iteration are summarized as follows [43]: dˆi − (k ) µf ˆ −u j d
u l ∈n f
=
i
u l ∈n f
(k )
σf ˆ
di
−u j
(k ) µu j −f ˆ d
=
dˆi
= i
=⎝ i
hi,l µu k −f ˆ
di
,
hi,j
\{u j }
i
(k )
⎝
(4.101)
(k −1) d −u j
(hi,l )2 σf ˆ
(hi,j )2
⎛ (k ) σu j −f ˆ d
µf −u j (k )
f ∈n u j \{f dˆ } σf −u j i ⎞−1 1 ⎠
⎛ (k ) σu j −f ˆ d
dˆi
β+
(k −1)
\{u j }
i
(k ) f ∈n u j \{f dˆ } σf −u j
,
(4.102)
⎠,
(4.103)
⎞
(4.104)
i
with initialization as (0)
(0)
µu j −f ˆ = 0,
σu j −f ˆ = 1
di
(4.105)
di
for all i, j for which uj is connected to fdˆi , nk denoting the set of nodes connected to node k on the factor graph and hi,j denoting the channel gain between the jth BS and the user in cell i. After the termination conditions are met at step m, the transmitted signal from BS j is obtained as ⎛ ⎞ (m ) µ f −u j (m ) (m ) ⎠ xj = σu j −f ˆ ⎝ (4.106) (m ) d f ∈n u σf −u j j
with ⎛
1
f ∈n u j
(m ) σf −u j
σu j −f ˆ = ⎝ (m )
d
⎞−1
⎠
.
(4.107)
122
Turbo base stations
Since the factor graph in Figure 4.15 contains loops, the convergence of the sum–product algorithm is not guaranteed. However, since the underlying distributions are all Gaussian, following the discussion in Section 4.4.5 it can be shown that the variance updates always converge [40, 41]. The necessary and suﬃcient condition for the convergence of the mean updates is that the spectral radius of the iteration matrix in (4.68) is strictly less than 1. As observed for the uplink problem, the convergence conditions are violated when the SNR is high and the interference created by the channel is strong. In [40], two approaches to improve the convergence of the algorithm for the downlink were proposed. In the ﬁrst approach, a tunable parameter is introduced that multiplies the regularization parameter β. The value of is chosen so that the sum–product algorithm converges or converges at the desired rate. The disadvantage of this approach is that the sum–product algorithm no longer computes the desired LMMSE estimates resulting in some loss in performance. In the second approach, a switched beam system is used where a cell is divided into angular sectors and each sector is covered by a narrow beam generated by a directional antenna. Comparing the signal strength from all sectors, the BS selects the best beam to transmit data to a user. In this way interference is reduced, and the loops in the factor graph can be reduced or eliminated.
4.6
Current trends and practical considerations Much work has demonstrated the huge performance gains that are possible when BSs cooperate to receive signals from mobile users on the uplink, and to send data to mobile users on the downlink. In this chapter we have examined methods for implementing BS cooperation in a distributed manner via message passing. These message passing techniques require communication between neighboring BSs only and have processing and communication requirements that remain constant with increasing network size. They are, in some sense, the natural algorithms to use in large cellular networks. However, in terms of both theoretical analysis and practical implementation of these turbostyle approaches, we have only scratched the surface. We list below some exciting areas where much work is still to be done. (1) Convergence issues. Many questions are still unanswered regarding the convergence of the sum–product algorithm on graphs with loops, as mentioned in Section 4.4.5. A more thorough treatment of convergence is required in order to understand the limitations of the distributed BS processing techniques presented in the chapter and to ﬁnd methods to improve the convergence properties. (2) Limited backhaul traﬃc. Backhaul traﬃc is the traﬃc between BSs required to implement cooperative schemes. For a practical system, the backhaul trafﬁc is not unlimited. Given a certain limit on the backhaul traﬃc, can we still
References
(3)
(4)
(5)
(6)
123
get the performance gains we see in this chapter? How should the message passing techniques be modiﬁed to reduce the backhaul traﬃc load? Synchronization. Perfect synchronization of BSs in the downlink has been assumed in this chapter. We have also assumed it for the uplink, although the assumption may be less critical there. What is the impact of synchronization errors on the performance? It may be possible that a synchronization imperfection that is tolerable in conventional singlecell processing may have more degrading eﬀects in the case of BS cooperation. Imperfect channel information. We have assumed that perfect CSI is available at the BS transceiver. What will be the impact of imperfections in channel estimates on the attainable gains due to base station cooperation? What will be the impact on the convergence of the proposed methods? Coded systems. We have not considered error control coding in this chapter. In practice, common decoders are often based on graphical methods and it would be possible to combine the factor graphs we use for joint detection of mobile users and the graphs which model the code constraints, to form one large graphical model on which message passing algorithms could be applied. What will happen to the convergence properties of the sum–product algorithm on this spacetime factor graph? What are the other issues to consider for distributed decoding in a cellular system? Distributed resource allocation. If we are allocating channels to mobile users in order to maximize the throughput, how can this be done in a BScooperating network in a distributed manner?
Some excellent work has already begun to answer some of these questions: channel estimation and coding [63], limited backhaul traﬃc and coding [28, 37], and resource allocation [1]. However, many questions remain unanswered and numerous challenges must be overcome if we are to realize the full potential of turbobased methods for BS cooperation.
References [1] A. Abrardo, P. Detti, and M. Moretti, “Message passing resource allocation for the uplink of multicarrier systems,” in Proc. of IEEE International Conference on Communications, 2009, pp. 1–6. IEEE, 2009. [2] R. Ahlswede, “The capacity region of a channel with two senders and two receivers,” Annals of Probability, 2, 1974, 805–814. [3] E. Aktas, J. Evans, and S. Hanly, “Distributed decoding in a cellular multipleaccess channel,” in Proc. of IEEE International Symposium on Information Theory, 2004, p. 484. IEEE, 2004. [4] E. Aktas, J. Evans, and S. Hanly, “Distributed base station processing in the uplink of cellular networks,” in Proc. of IEEE International Conference on Communications, 2006, pp. 1641–1646. IEEE, 2006.
124
Turbo base stations
[5] E. Aktas, J. Evans, and S. Hanly, “Distributed decoding in a cellular multipleaccess channel,” IEEE Transactions on Wireless Communications, 7, 2008, 241–250. [6] V. S. Annapureddy and V. V. Veeravalli, “Gaussian interference networks: Sum capacity in the low interference regime and new outer bounds for the capacity region,” IEEE Transactions on Information Theory, 55, 2009, 3032–3050. [7] O. Axelsson, Iterative Solution Methods. Cambridge University Press, 1994. [8] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Transactions on Information Theory, 20, 1974, 284–287. [9] S. Bavarian and J. K. Cavers, “Reducedcomplexity belief propagation for systemwide MUD in the uplink of cellular networks,” IEEE Journal on Selected Areas in Communications, 26, 2008, 541–549. [10] P. P. Bergmans, “Random coding theorem for broadcast channels with degraded components,” IEEE Transactions on Information Theory, 19, 1973, 197–207. [11] E. Bjornson and B. Ottersten, “On the principles of multicell precoding with centralized and distributed cooperation,” in Proc. of International Conference on Wireless Communication and Signal Processing, 2009, pp. 6–10. IEEE, 2009. [12] M. Costa, “Writing on dirty paper,” IEEE Transactions on Information Theory, 29, 1983, 439–441. [13] M. H. M. Costa and A. E. Gamal, “The capacity region of the discrete memoryless interference channel with strong interference,” IEEE Transactions on Information Theory, 33, 1987, 710–711. [14] T. Cover, “Broadcast channels,” IEEE Transactions on Information Theory, 18, 1972, 2–14. [15] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multiantenna wireless system,” in Proc. of Conference on Information Sciences and Systems, 2008, pp. 429–434. John Hopkins University Press, 2008. [16] U. Erez and S. ten Brink, “A close to capacity dirty paper coding scheme,” IEEE Transactions on Information Theory, 51, 2005, 3417–3432. [17] G. D. Forney Jr., “Codes on graphs: Normal realizations,” IEEE Transactions on Information Theory, 47, 2001, 520–548. [18] G. J. Foschini, M. K. Karakayali, and R. A. Valenzuela, “Coordinating multiple antenna cellular networks to achieve enormous spectral eﬃciency,” IEE Proceedings – Communications, 153, 2006, 548–555. [19] B. J. Frey, Graphical Models for Machine Learning and Digital Communication. MIT Press, 1998. [20] R. G. Gallager, LowDensity ParityCheck Codes. MIT Press, 1963. [21] A. Grant, S. Hanly, J. Evans, and R. M¨ uller, “Distributed decoding for Wyner cellular systems,” in Proc. of Australian Communications Theory
References
[22]
[23] [24]
[25]
[26]
[27] [28]
[29] [30]
[31]
[32] [33]
[34] [35]
[36]
125
Workshop, 2004, pp. 77–81. The University of Newcastle, NSW, Australia, 2004. T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Journal on Selected Areas in Communications, 27, 1981, 49–60. S. V. Hanly and P. Whiting, “Informationtheoretic capacity of multireceiver networks,” Telecommunication Systems, 1, 1993, 1–42. B. Hassibi and M. Sharif, “Fundamental limits in MIMO broadcast channels,” IEEE Journal on Selected Areas in Communications, 25, 2007, 1333– 1344. B. M. Hochwald, C. B. Peel, and A. L. Swindlehurst, “A vectorperturbation technique for nearcapacity multiantenna multiuser communication – Part II: Perturbation,” IEEE Transactions on Communications, 53, 2005, 537– 544. S. Jin, D. N. C. Tse, J. B. Soriaga, J. Hou, J. E. Smee, and R. Padovani, “Downlink macrodiversity in cellular networks,” in Proc. of IEEE International Symposium on Information Theory, pp. 2007, 1–5. IEEE, 2007. S. M. Kay, Statistical Signal Processing: Estimation Theory. Prentice Hall, 1993. S. Khattak, W. Rave, and G. Fettweis, “Distributed iterative multiuser detection through base station cooperation,” EURASIP Journal on Wireless Communications and Networking, 2008. R. Kinderman and J. Snell, Markov Random Fields and Their Applications. American Mathematical Society, 1980. F. R. Kschischang, B. J. Frey, and H. Loeliger, “Factor graphs and the sumproduct algorithm,” IEEE Transactions on Information Theory, 47, 2001, 498–519. S.C. Lin and H.J. Sun, “Practical vector dirty paper coding for MIMO Gaussian broadcast channels,” IEEE Journal on Selected Areas in Communications, 25, 2007, 1345–1357. H.A. Loeliger, “An introduction to factor graphs,” IEEE Signal Processing Magazine, 21, 2004, 28–41. H.A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F. R. Kschischang, “The factor graph approach to modelbased signal processing,” Proceedings of the IEEE, 95, 2007, 1295–1322. Y. Mao and F. R. Kschischang, “On factor graphs and the Fourier transform,” IEEE Transactions on Information Theory, 51, 2005, 1635–1649. M. Marrow and J. K. Wolf, “Iterative detection of 2dimensional ISI channels,” in Proc. of IEEE Information Theory Workshop, 2003, pp. 131–134. IEEE, 2003. P. Marsch and G. Fettweis, “On base station cooperation schemes for downlink network MIMO under a constrained backhaul,” in Proc. of IEEE Global Telecommunications Conference, 2008, pp. 1219–1224. IEEE, 2008.
126
Turbo base stations
[37] T. Mayer, H. Jenkac, and J. Hagenauer, “Turbo basestation cooperation for intercell interference cancellation,” in Proc. of IEEE International Conference on Communications, 2006, pp. 4977–4982. IEEE, 2006. [38] R. J. McEliece, D. J. C. MacKay, and J.F. Cheng, “Turbo decoding as an instance of Pearl’s belief propagation algorithm,” IEEE Journal on Selected Areas in Communications, 16, 1998, 140–152. [39] B. L. Ng, J. S. Evans, and S. V. Hanly, “Distributed linear multiuser detection in cellular networks based on Kalman smoothing,” in Proc. of IEEE Global Communications Conference, 2004, pp. 134–138. IEEE, 2004. [40] B. L. Ng, Cellular networks with Cooperating base stations: Performance analysis and distributed algorithms, Ph.D. Thesis, University of Melbourne, Melbourne, Australia, 2007. [41] B. L. Ng, J. S. Evans, and S. V. Hanly, “Distributed downlink beamforming in cellular networks,” in Proc. of IEEE International Symposium on Information Theory, 2007, pp. 6–10. IEEE, 2007. [42] B. L. Ng, J. S. Evans, S. V. Hanly, and D. Aktas, “Transmit beamforming with cooperative base stations,” in Proc. of IEEE International Symposium on Information Theory, 2005, pp. 1431–1435. IEEE, 2005. [43] B. L. Ng, J. S. Evans, S. V., Hanly, and D. Aktas, “Distributed downlink beamforming with cooperative base stations,” IEEE Transactions on Information Theory, 54, 2009, 5491–5499. [44] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 1988. [45] C. B. Peel, B. M. Hochwald, and A. L. Swindlehurst, “A vectorperturbation technique for nearcapacity multiantenna multiuser communication – Part I: Channel inversion and regularization,” IEEE Transactions on Communications, 53, 2005, 195–202. [46] M. Sadek, A. Tarighat, and A. H. Sayed, “A leakagebased precoding scheme for downlink multiuser MIMO channels,” IEEE Transactions on Wireless Communications, 6, 2007, 1711–1721. [47] C. Sankaran and A. Ephremides, “Solving a class of optimum multiuser detection problems with polynomial complexity,” IEEE Transactions on Information Theory, 44, 1998, 1958–1961. [48] C. Schlegel and A. Grant, “Polynomial complexity optimal detection of certain multipleaccess systems,” IEEE Transactions on Information Theory, 46, 2000, 2246–2248. [49] S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via coprocessing at the transmitting end,” in Proc. of IEEE Vehicular Technology Conference, 2001, pp. 1745–1749. IEEE, 2001. [50] O. Shental, A. J. Weiss, N. Shental, and Y. Weiss, “Generalized belief propagation receiver for nearoptimal detection of twodimensional channels with memory,” in Proc. of IEEE Information Theory Workshop, 2004, pp. 225–229. IEEE, 2004.
References
127
[51] R. Tanner, “A recursive approach to low complexity codes,” IEEE Transactions on Information Theory, 27, 1981, 533–547. [52] S. Ulukus and R. D. Yates, “Optimum multiuser detection is tractable for synchronous CDMA systems using msequences,” IEEE Communications Letters, 2, 1998, 89–91. [53] M. Valenti and B. Woerner, “Iterative multiuser detection, macrodiversity combining, and decoding for the TDMA cellular uplink,” IEEE Journal on Selected Areas in Communications, 19, 2001, 1570–1583. [54] S. Verd´ u, Multiuser Detection. Cambridge University Press, 1998. [55] P. Vishwanath and D. N. C. Tse “Sum capacity of vector Gaussian broadcast channel and uplinkdownlink duality,” IEEE Transactions on Information Theory, 49, 2003, 1912–1921. [56] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian multipleinput multipleoutput broadcast channel,” IEEE Transactions on Information Theory, 52, 2006, 3936–3964. [57] L. Welburn, J. K. Cavers, and K. W. Sowerby, “A computational paradigm for spacetime multiuser detection,” IEEE Transactions on Communications, 52, 2004, 1595–1604. [58] N. Wiberg, H. Loeliger, and R. Koetter, “Codes and iterative decoding on general graphs,” European Transactions on Telecommunications, 6, 1995, 513–525. [59] A. Wyner, “Shannontheoretic approach to a Gaussian cellular multipleaccess channel,” IEEE Transactions on Information Theory, 40, 1994, 1713– 1727. [60] W. Yu and T. Lan, “Transmitter optimization for the multiantenna downlink with perantenna power constraints,” IEEE Transactions on Signal Processing, 55, 2007, 2646–2660. [61] R. Zakhour, Z. K. M. Ho, and D. Gesbert, “Distributed beamforming coordination in multicell MIMO channels,” in Proc. of IEEE Vehicular Technology Conference, 2009, pp. 1–5. IEEE, 2009. [62] H. Zhang, N. B. Mehta, A. F. Molisch, J. Zhang, and H. Dai, “Asynchronous interference mitigation in cooperative base station systems,” IEEE Transactions on Wireless Communications, 7, 2008, 155–165. [63] Y. Zhu, D. Guo, and M. Honig, “Joint channel estimation and cochannel interference mitigation in wireless networks using belief propagation,” in Proc. of IEEE International Conference on Communications, 2008, pp. 2003–2007. IEEE, 2008.
5
Antenna architectures for network MIMO LiChun Wang and ChuJung Yeh
5.1
Introduction In conventional cellular systems, cochannel interference is a serious issue that degrades system performance. The spectral eﬃciency of cellular networks is fundamentally limited by the interference between cells and users sharing the same channel for both downlink and uplink. Generally, there are two kinds of cochannel interference: intracell interference and intercell interference. The intracell interference can be resolved by allocating orthogonal frequency resources. To mitigate intercell interference, there are several general approaches such as frequency reuse, cell sectoring, and spread spectrum transmission. The most commonly used technique is to avoid using the same set of frequencies in neighboring cells. This approach leads to the decrease of the number of available channels within each cell. Universal frequency reuse, i.e., reuse factor of 1, is preferred for future broadband wireless communications systems, such as the Third Generation Partnership Project (3GPP) longterm evolution (LTE) and worldwide interoperability for microwave access (WiMAX). In the orthogonal frequencydivision multiple access (OFDMA) systems, which do not have processing gain as the codedivision multiple access (CDMA) system, how to achieve the goal of both universal frequency reuse and reducing intercell interference is a key challenge. The concept of fractional frequency reuse (FFR) has been suggested to improve spectrum eﬃciency by applying reuse partition techniques, in which the inner region of the cell is assigned the whole frequency spectrum and the outer region is only assigned a small fraction of the frequency spectrum [1]. The capacity and outage rate of an FFRbased OFDMA cellular system with proportional fair scheduling was studied in [2]. To analyze the joint eﬀect of the FFR factor, the cell inner region, and the bandwidth assignment on cell throughput, an optimization problem was formulated in [3]. The resource allocation problem in an FFRbased multicell OFDMA system was translated to a graph coloring problem in [4]. In [5] FFR was applied to a trisector cellular OFDMA system. It was shown that the intercell interference is reduced and cell throughput was improved by the Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
5.1 Introduction
129
proposed subcarrier scheduling according to location or signaltointerferenceplusnoise ratio (SINR) of users. Under a similar FFRbased trisector cellular OFDMA system, an interference mitigation scheme using combining partial reuse and soft handover was proposed in [6]. The network multipleinput multipleoutput (MIMO) technique has become a hot topic which aims to mitigate the intercell interference by coordinating the multicell transmission for downlink or reception for uplink among a few geographically separated antennas (base stations, BSs). To eﬀectively reduce the intercell interference, the network MIMO requires a reliable and highspeed backhaul connection between BSs to pass the channel state information (CSI) and mobile messages between those cooperating cells. With full BS cooperation, the downlink transmission can be modelled as a multiuser MIMO broadcast system [7–9]. The interstream interferencefree transmission of a MIMO broadcast system is now translated to intercell interference cancelation in a multicell network MIMO system. For downlink transmission in a cellular network, the concept of coprocessing at the transmitting end was proposed in [10]. In [11] a distributed network beamforming in a cellular system was proposed and its performance was analyzed based on Wyner’s circular array model [12]. Coordinated strategies with grouped interior and edge users based on Wyner’s circular array model were studied in [13]. Downlink network MIMO transmission with multiple transmit and receive antennas under a general channel model was analyzed in [14–17]. In [18] downlink network MIMO coordination was compared with a denser BS deployment. For uplink network MIMO, coordinated BSs perform the reception of user signals within their coverage area and suppress interference between users by means of coherent linear (received) beamforming across BSs. The spectral eﬃciency gain with diﬀerent numbers of neighboring rings with which a BS is coordinated was investigated in [19]. In [20], the data rates of users assigned to each coordinated cluster were further chosen to be proportionally fair. A fundamental question for network MIMO is: how many cells should be coordinated to provide adequate SINR performance. Most of the studies on network MIMO assumed a global coordination which can eliminate the intercell interference completely. However, it is impractical to have cooperation (or coordination) among too many cells. The huge computational complexity and synchronization needed with a large number of cells are quite challenging. In practice, only a limited number of BSs can coordinate and jointly process the received or transmit signals. For uplink network MIMO, an isolationbased user grouping algorithm to optimize the capacity under a strongly constrained backhaul between seven coordinated sites was proposed in [21]. An uplink BS coordination with a dynamic BS clustering approach was investigated in [22]. This approach leads to signiﬁcant sum rate gain compared with the static BS clustering schemes proposed in [19, 20]. For the downlink network MIMO, the isolationbased user grouping algorithm used in [21] was modiﬁed to the downlink scenario for capacity improvement under a limited backhaul [23]. In [24], downlink coordination with limited distributed antenna arrays was compared with centralized antenna
130
Antenna architectures for network MIMO
arrays. Under a ﬁxed number of antennas per cell, the impact of diﬀerent numbers of sectoring per cell was evaluated in [25]. Also the impact of the number of coordinated cells was investigated. The objective of this chapter is to discuss a FFRbased network MIMO interference cancelation scheme for a downlink multicell system. Although both FFR and interBS coordination are being considered as possible intercell interference cancelation techniques in both WiMAX and LTE, combining network MIMO with FFR to mitigate the intercell interference is still an interesting problem. Due to the geographical distribution of cells and mobiles, the potential advantages of network MIMO can be further explored by utilizing diﬀerent frequency partition and cell sectorization instead of complex joint multicell transmission techniques. In addition, a group of coordinated cells still cause interference in the neighboring coordinated groups. This intergroup interference (IGI) should be taken into account for network MIMO under arbitrary coordinated sizes. In this chapter, we will also explore the potential gain of network MIMO by using a near minimum number of coordinated cells, i.e., only three cells. In a cellular system with an omnidirectional antenna, the received signal quality is severely aﬀected by IGI under three and sevencell coordinated network MIMO. However, the proposed FFRbased threecell network MIMO architecture can approach the traditional sevencell network MIMO with omnidirectional antenna. The rest of this chapter is organized as follows. In Section 5.2 we deﬁne the system model. In Section 5.3, we brieﬂy review the concept of network MIMO. In Section 5.4 we examine the IGI issue for network MIMO systems. In Section 5.5, we present the proposed threecell network MIMO architecture with FFR. In Section 5.6, we show numerical results and give concluding remarks in Section 5.7.
5.2
System model We consider a cellular system with Ncell = 19 cells, where the center cell has twotier neighboring cells, as shown in Figure 5.1. For simplicity, we assume that each BS of a cell has a single transmit antenna and each mobile user has a single receive antenna. For a cellular system with threesectored cells, the number of transmit antennas per sector is also 1. The sectors are created using directional antenna at the BS. The antenna gain pattern used for each BS sector is speciﬁed as 6 5 2 θ A(θ)dB = − min 12 (5.1) , Am , θ3dB where A(θ)dB is the antenna gain measured in decibels in the direction θ and θ ∈ [−180◦ , 180◦ ] is the direction of the mobile user with respect to the broadside direction of the considered BS. The 3 dB beamwidth θ3dB is the angle at which the antenna gain is 3 dB lower than the peak (the broadside direction). The parameter Am = 20 dB is the maximum attenuation for the sidelobe.
5.2 System model
131
B1 B1 8
B1 9
B2
2
10
B3
18
B3
B3
B3
B3
B3
B2 B1
B2
B2
B3
B3
B2
15
14
Regular frequency partition for cell B1
B1
B2 B1
B2 B1
B3
B3
B3
B2
16
5
13
B1
B B3 2 B1
B2 B1
B2
B3
17
6
4
12
B1
B2 B1
B2 B1
B2
1
B B3 2 B1
0
3
11
B3
B B3 2 B1
B2 B1
B3
B B3 2 B1
B B3 2 B1
B1
7
B3 Considered frequency band Coordinated cells Interferers by directional antenna
B2
Interferers by side lobe
Figure 5.1. Interference example for FFRbased threecell network MIMO systems with regular partition (consider cell 0). The general transmission model for mobile user k served by BS j is hk ,i xi + nk , yk = hk ,j xj +
(5.2)
i = j, i∈I
where hk ,j is the channel response between mobile k and BS j, xj is the transmitted signal from BS j, nk is the additive noise at the kth mobile, and I is the interference set for mobile k. The detailed channel response is 7 −µ dk ,j , (5.3) hk ,j = αk ,j βk ,j A(θk ,j ) dref where αk ,j is the fast (Rayleigh) fading between BS j and mobile k, βk ,j is the lognormal shadowing from BS j to mobile k, A(θk ,j ) is the antenna gain for mobile k with respect to BS j, dk ,j is the distance between BS j and mobile k, dref is a reference distance, and µ is the pathloss coeﬃcient. The value of A(θk ,j ) is a function of θk ,j based on (5.1) for the sectorized cell and A(θk ,j ) = 1 for an omnidirectional transmission cell. Note that the reference distance dref is the distance between the cell center and the cell vertex. The received SNR of mobile k from BS j is −µ dk ,j 2 SNR k ,j = αk ,j  βk ,j A(θk ,j ) Γ, (5.4) dref where the reference SNR Γ represents the interferencefree SNR deﬁned as the SNR measured at the cell boundary (i.e., at the reference distance dref ) for only considering pathloss and ignoring shadowing and fast fading. The parameter Γ captures the eﬀect of various channel and antenna parameters including transmit power, cable loss, transmit and receive antenna heights, thermal noise power, and other link budget parameters. Noise power is normalized to unity. The reference
132
Antenna architectures for network MIMO
SNR Γ = 18 dB for a BStoBS distance RB2B = 2 km macrocellular system with 30 watts transmit power in 1 MHz bandwidth [26]. In this case, the reference distance dref ∼ = 1.1547 km. For mobile k served by BS j, the corresponding SINR of mobile k can be expressed as γk ,j =
1+
SNR k ,j i = j, i∈I SNR k ,i
(5.5)
and the corresponding Shannon capacity is given by log2 (1 + γk ,j ) bps/Hz.
5.3
Network MIMO Network coordination is a means to eliminate the intercell interference and improve spectral eﬃciency in a downlink multicell system [14, 15]. With a highspeed backhaul, the BSs can be connected and synchronized. Assume that all BSs can cooperate and report CSI at each BS via the central coordinator. With full BS cooperation, the downlink transmission can be modeled as a multiuser MIMO broadcast system. Consider a network MIMO system with M coordinated BSs (each site with a single transmit antenna), each of which transmits a data stream to its own target mobile station. The received signal is given by Y = Hx + n,
(5.6)
where H = [hk ,j ]M ×M denotes the channel matrix with element hk ,j being the channel response between mobile k and BS j as deﬁned in (5.3), and n denotes the noise vector. The transmitted signals vector is denoted by x = Ws = [w1 . . . wM ][s1 . . . sM ]T , where sk is the kth mobile user’s data symbol, wk = [w1,k . . . wM ,k ] is the corresponding precoding weight column vector, and (·)T is the transpose operation. For a network MIMO system, the antenna output of the jth transmission site is a linear combination of M data symbols, i.e., xj = M k =1 wj,k sk . The received signal of mobile k can be written as yk = sk hk wk  +
M
si hk wi  + nk ,
(5.7)
i=1, i = k
where hk = [hk ,1 . . . hk ,M ] is the channel vector of the kth user. Let pk = E[sk 2 ] denote the average power of symbol sk . According to the network MIMO principle, the following constraint should be satisﬁed: E[xj 2 ] ≤ PBS for each BS j = 1, . . . , M . ⎡ ⎤⎡ ⎤ w1,1 2 · · · w1,M 2 p1 ⎢ ⎥ ⎢ .. ⎥ .. .. .. (5.8) ⎣ ⎦ ⎣ . ⎦ ≤ PBS 1M , . . . wM ,1 2
···
wM ,M 2
pM
where PBS is the maximum transmit power at each BS and IM is an M × 1 vector will all one elements.
5.3 Network MIMO
133
It is well known that dirty paper coding (DPC) can achieve the capacity of a multiuser MIMO broadcast system. Due to the high complexity of DPC, some suboptimal but practical schemes were proposed. In this chapter, we consider two suboptimal network MIMO schemes, including zeroforcing (ZF) and zeroforcing dirty paper coding (ZFDPC).
5.3.1
ZF network MIMO transmission The goal of ZF transmission is to invert the channel to obtain HW = I. This function can be achieved by using the pseudoinverse of the channel matrix as the weight matrix W. The received signal vector is hence given by Y = Hx + n = HWs + n = s + n.
(5.9)
For the network MIMO system with perbase power constraint (5.8), the objective is to maximize the minimum rate among the coordinated cells subject to the power constraint. The corresponding solution is the maximum equal rate assignment [15] pk =
PBS P BS = , for all k. maxj [WW∗ ](j,j ) maxj k wj,k 2
(5.10)
Therefore, the corresponding mobile’s rate is given by log2 (1 + pk /σ 2 ) bps/Hz among all coordinated cells, where σ 2 represents the noise power. Note that there is an excessive transmission power penalty for ZF transmission due to the required interference cancelation power for constructing weight matrix W.
5.3.2
ZFDPC network MIMO transmission ZFDPC transmission constructs the linear weight matrix W = Q∗ through the QR decomposition of the channel matrix H = LQ, where L is a lower triangular matrix, Q is a unitary matrix with QQ∗ = Q∗ Q = IM , and (·)∗ is the conjugate transpose operation. The received signal is written as Y = Hx + n = LQQ∗ s + n = Ls + n
(5.11)
and the corresponding kth mobile’s received signal is yk = Lk ,k sk + ∗ i< k Lk ,i si + nk . Note that the weight matrix W = Q ensures that no interference is from symbols with indices i > k. The remaining interference from symbols i < k is taken care of by DPC successive interference cancelation. Based on the result of ZF transmission, if we set pk = p for all k, the per site power constraint (5.8) becomes E[xj 2 ] = [WW∗ ](j,j ) p = k wj,k 2 p ≤ PBS for all transmitted signal xj . Recall that the weight matrix W = Q∗ . We have [WW∗ ](j,j ) = [Q∗ Q](j,j ) = [I](j,j ) = 1. Thus, we can set pk = PBS for each coordinated BS. Given the received signal in (5.11), mobile k can achieve the rate log2 (1 + Lk ,k 2 pk /σ 2 ) bps/Hz.
134
Antenna architectures for network MIMO
5.4
Eﬀects of intergroup interference
5.4.1
SINR analysis When applying network MIMO to group coordinated cells in a multicellular system, the eﬀects of IGI need to be taken into account. That is, a group of coordinated cells still cause interference in the neighboring coordinated groups of cells even if the intragroup interference has been canceled via network MIMO transmission. Consider a cell with a twotier surrounding cells layout under M cell network MIMO coordination. There are some interfering neighboring cells which belong to its corresponding M cell network MIMO group Gi for i ∈ IG , where IG is the index set of the interference group. For a M cell network MIMO group Gi , the transmit antenna output of BS a ∈ Gi is i xG a =
M
Gi Gi wa,b sb ,
(5.12)
b=1 Gi where [wa,b ]M ×M is the precoding weight designed for M cell network MIMO i group Gi and sG is the data symbol of mobile b ∈ Gi . The SINR of mobile k b considered in (5.7) becomes
γkIGI =
1+
i∈IG
pk hk wk 2 M G i G i G i 2 , a∈G i b=1 pb hk ,a wa,b 
(5.13)
i i where pG is the data symbol power for mobile b ∈ Gi and hG b k ,a is the channel response between the considered mobile k and the interfering BS a ∈ Gi . As in (5.5), the noise power has been normalized to unity. The kth mobile’s rate is now log2 (1 + γkIGI ) when considering the eﬀect of IGI in a multicellular system. By applying the ZFbased network MIMO coordination, the SINR of (5.13) becomes pk GI , (5.14) γkI ,Z F = Gi 2 Gi Gi 1 + i∈IG a∈G i M b=1 p hk ,a wa,b 
Gi i where pG for all b can be found via (5.10). For network MIMO with b =p ZFDPC transmission, the corresponding SINR is
γkIGI ,ZFDPC =
5.4.2
1+
i∈IG
PBS Lk ,k 2 . M Gi Gi 2 a∈G i b=1 PBS hk ,a wa,b 
(5.15)
Example of IGI: network MIMO with omnidirectional cell planning In this subsection, we show examples of IGI in the network MIMO system. We consider a conventional cellular network with omnidirectional cell planning. Figure 5.2 shows IGI for the threecell and sevencell network MIMO systems. Take cell 0 as an example, it forms a threecell network MIMO group G0 with cells 4 and 5. The intragroup interference coming from cells 4 and 5 is canceled via network MIMO transmission. However, the remaining twotier cells still
5.4 Eﬀects of intergroup interference
135
G7 G1
7 18
8 G8
G1
9
G6
1
9
16 G0
3
5 15
G3
13
G4
(a)
17
G0
3
16 G5
5 4
11 G3
14
12
0
10
G6
6
2
G5
4
11
18 1
G2
0
10 G2
17
6
2
7
8
15 14
12 13
G4
(b)
Figure 5.2. An example of IGI for: (a) the threecell and (b) sevencell network MIMO systems. cause interference to cell 0. In this case, IG = {G1 , G2 , . . . , G8 }, where the corresponding cells in each group are G1 = {1, 2, 8}, G2 = {3, 10, 11}, G3 = {12, 13}, G4 = {14}, G5 = {15, 16}, G6 = {6, 17, 18}, G7 = {7}, and G8 = {9}. There are still four ﬁrsttier interfering cells and all the secondtier interfering cells. Note that all interferers transmit coordinated signaling via M cell network MIMO. With sevencell network MIMO, the coordinated partners of cell 0 are cells {1, 2, 3, 4, 5, 6}. The IGI from the secondtier cells is still unavoidable even if the interferences from surrounding cells {1, 2, 3, 4, 5, 6} are canceled. Figure 5.3 shows the eﬀects of IGI for network MIMO with ZFDPC transmission. We ﬁrst consider the advantage of using network MIMO. The dotted line represents the SINR for cells with omnidirectional antenna. The SINR is evaluated according to (5.5) with I = 18 neighboring cells. We consider 19cell, 7cell, and 3cell network MIMO without IGI (denoted by solid lines). In this case, the received signal quality is averaged over M coordinated cells. Clearly, the received signal quality is largely improved via network MIMO. More importantly, we ﬁnd that the SINR performance of sevencell network MIMO is close to that of 19cell network MIMO. This result implies that the coordination size M = 7 is suﬃcient when designing network MIMO systems. However, this result is obtained by ignoring the eﬀects of IGI. The received signal quality will be greatly aﬀected when interference from the other groups is considered. The dashed lines denote the SINRs for 3cell and 7cell network MIMO with IGI as shown in Figure 5.2. The SINR degradation for 7cell network MIMO is about 14 dB and that for 3cell network MIMO is about 23 dB. The received signal quality for M = 3 is almost equivalent to the case in which there is no multicell coordination.
5.4.3
Unbalanced signal quality caused by IGI Based on the observations for the sevencell network MIMO system, the IGI results in an unbalanced signal quality when using larger coordination size. That
136
Antenna architectures for network MIMO
1 0.9 0.8
19cell with reuse one 3cell coord. (with IGI)
0.7 7cell coord. (with IGI) CDF
0.6 0.5 0.4 3cell coord. (no IGI) 0.3 0.2
7cell coord. (no IGI)
0.1 19cell coord. (no IGI) 0 −30
−20
−10
0
10 20 30 Received SINR (dB)
40
50
60
Figure 5.3. Eﬀect of IGI for the threecell and sevencell network MIMO systems. (CDF: cumulative density function.)
is, the central cells in a coordinated group have better signal quality than the cells at the edge. Take the sevencell network MIMO as the example: the IGI of cell 0 is from the secondtier interferers as mentioned before. However, for the edge cells {1, 2, 3, 4, 5, 6}, each of them has three ﬁrsttier IGIs and 15 secondtier IGIs. As a result, in a cooperating group of M cells, the edge cells suﬀer more serious IGI than the central cells. The signal quality among M coordinated cells is unbalanced in conventional network MIMO coordination. This unfair performance metric in a group becomes more obvious as coordination size M increases. To sum up, the IGI causes not only severe performance degradation but also results in unfairness in multicell network MIMO systems. Note that threecellbased network coordination does not suﬀer from this unbalanced service since each member of a coordinated group has the same geographic condition. In other words, each cell can be considered to be an “edge cell” within the threecell group with the same interference distribution. However, the performance improvement induced by a small group of coordinating cells is poor. In the following section, we propose novel network MIMO architectures to improve the signal quality and simultaneously achieve fairness among a group.
5.5
Frequencypartitionbased threecell network MIMO
5.5.1
Fractional frequency reuse (FFR) FFR, also called reuse partition, allows diﬀerent frequency reuse factors to be applied over diﬀerent frequency partitions during the period of transmission. Figure 5.4 shows the considered FFR partition for a trisector cellular system.
5.5 Frequencypartitionbased threecell network MIMO
Power level
fB1
fB2
137
fB3
N resource units (RU)
fA
Frequency
fB
B1 B1
CellGai B3
2
B3
1
6
Cell
B1
B1
B2
Regular
B3
B2
2
B1
B3
CellGci
B2
B1
B1
5
B2
B3
B2
4
3
B3
6 B3
B2
B1
Network MIMO group for fB1 Network MIMO group for f B2 Network MIMO group for fB
B2
B1
0 3
B2
B1
B3 B3
5 4
B3
B2
B2
B2 Gi b
B3
1
B2
B1
0 B3
B1
B2
B2 B1
3
B3
B1
B3
Rearranged
Figure 5.4. Frequency partition and cell planning for proposed threecell network MIMO coordination. The FFR partitions the frequency into inner frequency bands fA and outer frequency bands fB , where fB is further partitioned into three subbands fB 1 , fB 2 , and fB 3 (the gray, slash, and crosshatched areas, respectively). In general, the inner frequency band fA adopts a reuse factor of 1 and is used by interior cell users; the outer frequency band fB adopts a reuse factor of 1/3 for each sector for cell edge users. By means of the trisector FFR, the intracell interference can be avoided due to four orthogonal subbands, and the intercell interference can be signiﬁcantly reduced. For use in a wireless broadband system (like the OFDM system), the frequency partitions can be assumed to be such that fB 1 , fB 2 , and fB 3 have the same bandwidth and there are N frequency resource units (RU) in each outer subband, i.e., fB p = {fB p , 1 , . . . , fB p , N } for p = 1, 2, 3. Note that FFR is usually integrated with other functions such as power control or antenna technologies for adaptive control. In this chapter, we consider two kinds of frequency planning: regular and rearranged frequency partitions. In regular frequency partition, each cell has the same frequency planning in the outer region (e.g., the left hand cellular system in Figure 5.4). Similar designs can be found in [5, 6]. In rearranged frequency partition, each cell has diﬀerent frequency planning compared with its surrounding cells as shown in the right hand cellular system in Figure 5.4. Later we will design network MIMO systems based on the two frequency partitions. We know intuitively that the rearranged frequency partition will suﬀer from more severe cochannel intercell interference than the regular frequency partition. With the network MIMO technique, we ﬁnd that the threecell network MIMO under the rearranged frequency partition has the better performance.
138
Antenna architectures for network MIMO
5.5.2
FFRbased network MIMO with regular frequency partition In this section, we consider a novel multicell architecture combining FFR with network MIMO. As mentioned before, the entire frequency band is partitioned into diﬀerent zones by FFR frequency planning. Compared with the conventional 19cell layout, the trisector cellular system combined with FFR can signiﬁcantly reduce the interference sources, while fully utilizing the frequency band in each cell. As an example, in Figure 5.1 when a mobile user at cell 0 uses an RU fB 1 , n (n = 1, . . . , N ) in frequency band fB 1 , the interference comes from cells 4, 5, 12, 13, 14, 15, and 16 under the assumption of perfect 120◦ sectoring by directional antenna. In conventional systems, the mobile receives interference from all the other 18 cells. Here, the question is: how can we further improve the SINR on top of FFR? The solution proposed in this chapter is to integrate FFR with the network MIMO technique. From Figure 5.1, we ﬁnd that among the seven interfering sources, two critical interferers are ﬁrsttier neighbors (i.e., neighboring cells 4 and 5) and ﬁve weaker interferers (due to larger pathloss) are secondtier neighbors. Therefore, we use the network MIMO technique to cancel out the two most severe interferences. Instead of a huge number of coordinated cells, we propose a coordination scheme with only three coordinated cells. We deﬁne those coordinated cells as a group i shown in Figure 5.4. For arbitrary group Gi , we label the three cells as CellG a , Gi Gi Cellb , and Cellc . For the threecell coordination structure, we can apply network MIMO transmission to each subband fB p , n for p = 1, 2, 3 and n = 1, . . . , N . For the example on cell 0, under the assumption of perfect sectoring by directional antenna, the channel matrix of fB 1 , n in the group {0, 4, 5} is ⎡
h(0),0 H(fB 1 , n ) = ⎣ 0 0
h(0),4 h(4),4 0
⎤ h(0),5 0 ⎦, h(5),5
(5.16)
where (x) denotes the corresponding served user in cell x. We eliminate the interference caused by h(0),4 and h(0),5 (coming from cells 4 and 5) by network MIMO. As a result, the two most severely interfering sources are canceled through a small (3 × 3) matrix computation. Importantly, for system design, this cooperation can be applied to all cells. For example, at a certain time slot we have many cooperating groups among the 19 cell in the layout: cells {0, 4, 5}, {8, 2, 1}, {10, 11, 3}, and {18, 6, 17}. Not only is the middle cell (cell 0) coordinated with its neighboring cells, but the cells in the outer layer are also coordinated simultaneously.
Cell regrouping and partner selection We now discuss a cell regrouping and partner selection scheme that addresses the service fairness issue for the regular partitionbased network MIMO system. We
139
5.5 Frequencypartitionbased threecell network MIMO
B1
B1
7
B1
8
B1 B
B3
B2 B1
B 10 B3 B2 1
B2 B1
11
B3
B3
B2
B3 B2
B2 Slot one: group with cells 4 and 5 Primary band
B3
B1
B3 B2 B1
B3 B2 B1
B1
18 B3 B 1
B3 B2 B1
17 B3 6
B3 B2 B1
5
B3 B2 B1
1
10
B
B3 B2 B1
0 B3 B2 1 3
B3 B2 B1
B2 11 B3 B 1
4
B2
9
B3
10
13 B3 B2 B2
Slot two: group with cells 1 and 6 Primary band
B3 B2 B1
3
B3 B2 B1
B3
1
B3 B2 B1
B1
18 B3B
1
0
B2 11 B3 B 1
15 B3 14 B3 B2
2 B3 B2 B1
B2 B1
7 B3 B2 B1
B3 B2 B1
B2 B1
16
B3 B2 B1
B2 12 B3 B 1
B2
8
B1
1
2 2 B3 B B
B2 B1
15 B3 14 B3 B2
13
B3
B3 B2 B1
B2 B1
B2 12 B3 B 1
9 B2 B1
16 5
4
B3 B2 B1
B3 B2 B1
B2 B1
B3 B2 B1
B2
17 B3 6
0 3
B3 B2 B1
8
B1
1
B1
7
B1
18 B3B 1
2
B1
B3 B2 B1
9 B3 B2 1 B2 B1
B3
4
B2
B3 B2 B1
5
B3 B2 B1
B3 B2 B1
16 B3
B3 B2 B1
B2 12 B3 B 1
B2
17 B3 6
15 B3 14 B3 B2
13 B3 B2 B2
Slot three: group with cells 2 and 3 Primary band
Figure 5.5. Example of cell regrouping and partner selection for cell 0.
consider the service fairness issue in the following. Among an arbitrary group i Gi , the cell labeled “CellG a ” can be free from intragroup interference within the whole subband fB 1 = {fB 1 , 1 , . . . , fB 1 , N } under network MIMO. That is, the two most severely interfering sources are canceled via network MIMO. However, subbands fB 2 and fB 3 still suﬀer interference from seven interferers, i.e., two ﬁrsttier cells and ﬁve secondtier cells. We therefore deﬁne fB 1 as the primary bands i of CellG a for group Gi . Similarly, we deﬁne fB 2 and fB 3 as the primary bands Gi i of Cellb and CellG c , respectively. Therefore, the cell users served by diﬀerent subbands from a group will have diﬀerent signal quality. Note that this service fairness issue is diﬀerent from the unbalanced signal quality issue mentioned in Section 5.4.3. The service fairness issue can be resolved by the proposed regrouping and partner selection scheme. Assume that cell 0 is grouped with cells 4 and 5 with primary band fB 1 at time slot 1 as shown in Figure 5.5. Cell 0 regroups with cells 1 and 6 at the next time slot, slot 2 by rotating counterclockwise to reselect coordinating partner. After this regrouping, the primary band of cell 0 becomes fB 2 . Similarly, cell 0 regroups with cells 2 and 3 at time slot 3 by another counterclockwise selection of the coordinated partner and the corresponding primary band becomes fB 3 . In this way, all cells simultaneously “rotate” and regroup with two new neighboring cells at each new time slot, where “rotate” means selecting with the coordinated partner reselection procedure. Take the cells {0, 4, 5} as an example: those cells form a group at time slot 1. At time slot 2, cell 0’s regroup set is now {1, 0, 6}, cell 4’s regroup set is {3, 12, 4}, and cell 5’s regroup set is {5, 14, 15}. Similarly, at time slot 3 cell 0’s regroup set becomes {2, 3, 0}, cell 4’s regroup set is {4, 13, 14}, and cell 5’s regroup set is {6, 5, 16}. Each cell has the chance to cooperate with its neighboring six cells in order and each sector has the opportunity to become the primary band under this kind of timedivision multiple access (TDMA) based regrouping and partner selection scheme.
140
Antenna architectures for network MIMO
Three kinds of frequency partition for cell
B2 7
B1 8
B3 9
B B2 2 B2
B1 B1 10
11
B1
B1 B1
B3
12
B3
B1
B2
B3
B2
B2 15
14 13
B3
B3
B B2 2 B2
B1 B1
B3 B3
B2 B1
16 5
4
B2
B2
B B1 1 B1
B B3 3 B3
B B2 2 B2
B3
B3 17
6
0 3
B1
B3
B B2 2 B2
B B1 1 B1
B B3 3 B3
B2 B2
B1 18
1
2
B1
B B3 3 B3
B3
B3
B1
B1
Considered frequency band Coordinated cells
B2
Interferers by directional antenna
B1
Interferers by sidelobe
Figure 5.6. Interference example for the FFRbased threecell network MIMO systems with rearranged frequency partition (consider cell 0).
5.5.3
FFRbased network MIMO with rearranged frequency partition We now propose another multicell architecture with rearranged frequency partitions among cells. As shown in Figure 5.4, there are in total three kinds of frequency partitions for cell planning, for example, the diﬀerent frequency partitions for cell 0, cell 1, and cell 2. After this rearranged frequency partition among a multicell system, a cell coordinates with six neighboring cells to form three individual network MIMO groups for each subband. Take cell 0 as an example: it coordinates with cells 1 and 2 for subband fB 1 , with cells 3 and 4 for subband fB 2 , and with cells 5 and 6 for subband fB 3 . In other words, we perform threecell network MIMO transmission in each subband (or frequency partition). As the example in Figure 5.6 shows when a mobile user of cell 0 uses an RU fB 1 , n (n = 1, . . . , N ) in frequency band fB 1 , the interference sources are cells 1, 2, 11, 12, 14, 16, and 17 under the assumption of perfect 120◦ sectoring by a directional antenna. Similarly we use the network MIMO technique to cancel the two severest interferences, i.e., the only two ﬁrsttier interferences from cells 1 and 2 (the dotted area in Figure 5.6). The channel matrix for the group {0, 1, 2} in fB 1 , n is ⎡
h(0),0 H(fB 1 , n ) = ⎣ h(1),0 h(2),0
h(0),1 h(1),1 h(2),1
⎤ h(0),2 h(1),2 ⎦ . h(2),2
(5.17)
For the user in cell 0, we eliminate the interference caused by h(0),1 and h(0),2 (coming from cells 1 and 2) by network MIMO. As a result, the two severest sources of interference are canceled through a small (3 × 3) matrix computation. Ideally, there are only ﬁve secondtier interferers for each subband. As with
5.5 Frequencypartitionbased threecell network MIMO
141
1 0.9 Omnicell with reuse 1
0.8 0.7
Diamondshaped (120°) sector FFR
CDF
0.6
FFRbased 3cell network MIMO (ZFB)
0.5 0.4
FFRbased 3cell network MIMO (ZF–DPC)
0.3 0.2 0.1
Regular frequency partition 0 −30
−20
−10
0 10 20 Received SINR (dB)
30
40
50
Figure 5.7. The CDFs of received SINR for the omnidirectional cellular systems with universal frequency reuse 1, trisector FFR cellular systems and the proposed regular partitionbased threecell network MIMO systems (120◦ cell sectoring).
1 0.9 0.8 0.7
CDF
0.6
Omnicell with reuse 1
Diamondshaped (120o) sector FFR FFRbased 3cell networ k MIMO (ZFB)
0.5 0.4
FFRbased 3cell network MIMO (ZF−DPC)
0.3 0.2 0.1
Rearranged frequency partition 0 −30
−20
−10
0 10 20 Received SINR (dB)
30
40
50
Figure 5.8. The CDFs of received SINR for the omnidirectional cellular systems with universal frequency reuse 1, trisector FFR cellular systems and the proposed rearranged partitionbased threecell network MIMO systems (120◦ cell sectoring).
the regular partitionbased network MIMO systems, this cooperation can also be applied to all cells not only to a particular cooperating group. Compared with the regular partitionbased network MIMO, there is no service fairness issue for network MIMO systems with a rearranged partition. As a result, we avoid the necessity for cell regrouping which reduces the complexity of system design.
142
Antenna architectures for network MIMO
Table 5.1. Spectral eﬃciency for threecell network MIMO with regular frequency partition
(bps/Hz)/base Improvement
5.5.4
Reuse 1
Trisector FFR
ZFB
ZFDPC
2.5437
3.7844 48.78%
2.4977 1.81%
4.4275 74.06%
Eﬀect of frequency planning among coordinated cells We next present the received SINR performance for the proposed threecell coordination architectures. The regular and rearranged partition frequency partition scenarios are shown in Figures 5.7 and 5.8, respectively. The cell planning is 120◦ sectoring. The results for the conventional 19cell layout with universal frequency reuse 1 (denoted by the dotted line) and the trisector FFR layout with directional antenna (denoted by the bold black line) are also presented for comparison. From Figure 5.7, by comparing the dotted line with the bold line we see that the gain in interference sources reduction (from 18 reduce to 7) of the trisector FFR layout is quite signiﬁcant: about 10 dB improvement at 90th percentile (i.e., CDF = 0.1) of the received SINR. However, the gain is not so obvious (2 dB improvement) for the rearranged partitionbased cellular layout. This is because there are two neighboring interferers for each RU in the rearranged partitionbased cellular layout. In this case the advantage of using just frequency partition and directional antenna is limited. With the threecell network MIMO techniques, both ZF and ZFDPC transmissions, can indeed improve the received SINR, especially for ZFDPC scheme as shown in Figure 5.8. At the 90th percentile of the received SINR, the improvement (compared to the trisector FFR layout) is about 9 dB for ZF and 15 dB for ZFDPC. Although rearranged frequency partition causes interference from neighboring cells, network MIMO can enhance signal quality eﬀectively by interferences cancelation. As for cells with regular partition, the gain of executing network MIMO shown in Figure 5.7 is not as signiﬁcant as that in Figure 5.8. The gain of ZFbased network MIMO is actually lower than that of the trisector FFR layout using the regular partitionbased cellular system. Recall the TDMAbased regrouping and partner selection scheme for regular partitionbased network MIMO. Equivalently, the signal quality of only one third of the resources can be enhanced in each time slot. Additionally, because of the transmission power penalty of ZF transmission, the actual data symbol power (5.10) becomes weaker even if there is no ﬁrsttier interference (due to coordination). Note that there is no power penalty for ZFDPC transmission. From the perspective of capacity improvement, Tables 5.1 and 5.2 show the spectral eﬃciencies of regular and rearranged partitionbased cellular systems under 120◦ cell sectoring, respectively.
5.5 Frequencypartitionbased threecell network MIMO
143
Table 5.2. Spectrum eﬃciency for threecell network MIMO with rearranged frequency partition (120◦ sectoring)
(bps/Hz)/base Improvement
Reuse 1
Trisector FFR
ZFB
ZFDPC
2.5437 –
3.2890 29.30%
3.2553 27.97%
5.7534 126.18%
Figure 5.9. Interference example for FFRbased threecell network MIMO systems with 60◦ cell sectoring (consider cell 0).
5.5.5
Eﬀect of cell planning with diﬀerent sectorization Cell planning for 60◦ cell sectorization To address the eﬀect of cell planning with diﬀerent sectorization, we design threecell network MIMO systems with 60◦ cell sectoring. Here we consider rearranged partitionbased frequency planning to avoid the cell regrouping procedure. Figure 5.9 shows an example of the proposed rearranged partitionbased 60◦ sectoring network MIMO systems. In this design, each cell has three hexagonshaped sectors with diﬀerent frequency partitions. Cells with 60◦ and 120◦ sectoring are also called cloverleafshaped cells and diamondshaped cells, respectively [27, 28]. Due to the antenna pattern, the cloverleafshaped cells can match the sector contour better than cells with diamondshaped sectors [27]. Additionally,
Antenna architectures for network MIMO
1 0.9 0.8 0.7 0.6 CDF
144
Omni cell with reuse 1
Cloverleaf shaped (60°) sector FFR FFRbased 3cell network MIMO (ZFB)
0.5 0.4
FFRbased 3cell network MIMO (ZF–DPC)
0.3 0.2 0.1
Rearranged frequency partition 0 −30
−20
−10
0 10 20 Received SINR (dB)
30
40
50
Figure 5.10. The CDFs of received SINR for omnidirectional cellular systems with universal frequency reuse 1, trisector FFR cellular systems and the proposed rearranged partitionbased threecell network MIMO systems (60◦ cell sectoring). cells with 60◦ sectoring can use spectrum more eﬃciently [28]. Similarly to 120◦ sectorized cell planning, each cell forms three individually threecell network MIMO coordinations with its six neighboring cells for each frequency partition. The two ﬁrsttier interferers (the dotted areas in Figure 5.9) are canceled via network MIMO transmissions. The threecell coordination structure can be implemented for all cells, not only for a particular cell. Note that actual cell sectorization is not perfect when using a directional antenna pattern (5.1). The other cells also aﬀect the received signal quality of cell 0. However, the harm caused by sidelobe and backlobe transmissions (the crosshatched areas in Figures 5.1, 5.6, and 5.9) is not signiﬁcant compared to the cell planning with omnidirectional transmission. Finally, the features of the proposed FFRbased threecell network MIMO systems can be summarized as: (i) they use a small coordination size M = 3 to obtain low complexity network MIMO systems, (ii) they reduce the eﬀect of IGI via the combination of FFR and directional antennas, (iii) they avoid the unbalanced signal quality issue caused by a larger coordination size M .
Performance comparison under 60◦ cell sectorization Next, we consider the eﬀect of cell planning with diﬀerent directional antennas. Figure 5.10 shows the corresponding received SINR performance using 60◦
5.6 Simulation setup numerical results
145
Table 5.3. Spectrum eﬃciency for threecell network MIMO with a rearranged frequency partition (60◦ sectoring)
(bps/Hz)/base Improvement
Reuse 1
Trisector FFR
ZFB
ZFDPC
2.4933 –
4.0360 61..88%
2.9478 18.23%
5.6337 125.96%
sectorized cell planning. We ﬁnd that the advantage of using pure sectorization is obvious under 60◦ cell sectoring. Traditionally, a cell with 60◦ sectoring has better performance than one with 120◦ sectoring. The poor signal quality of 120◦ trisector FFR with rearranged partition in Figure 5.8 can be improved using 60◦ cell sectoring (see Figure 5.10). When applying threecell network MIMO, the SNR performance of 60◦ sectorized cells is similar to that of 120◦ sectorized cells by the ZFDPC transmission. However, the gain of the ZF scheme is not very signiﬁcant at the 90th percentile of the received SINR, and is actually lower than that obtained by using the trisector FFR scheme. In other words, under rearranged partitionbased cellular systems, the proposed threecell network MIMO enhances signal quality more eﬃciently on cell planning with 120◦ sectoring. Note that the performance of the 19cell layout with universal frequency reuse 1 will be diﬀerent to that in Figure 5.8 due to the diﬀerent cell planning. However, the diﬀerence is not very signiﬁcant. Table 5.3 shows the average spectrum eﬃciency improvements of threecell network MIMO with a rearranged frequency partition under 60◦ cell sectoring.
5.6
Simulation setup numerical results We have shown the SINR performance of the proposed FFRbased threecell network MIMO schemes. In our simulation environment, we consider a multicell system with BStoBS distance RB2B = 2 km. The interferencefree SNR at the cell edge is Γ = 18 dB for 120◦ sectorized cells. The same values of BStoBS distance and Γ (for same transmission power comparison) are used for cloverleafshaped cells. The standard deviation of shadowing is 8 dB, the pathloss exponent µ = 4. Mobile users are uniformly distributed within each sector/cell. The channel response between any userandcell pair is represented by (5.3), where the angledependent antenna pattern is taken into consideration. In this chapter, we do not consider the eﬀect of the inner cell. In fact, how to design the inner region is an important issue for a FFR multicell broadband system. We do not discuss this issue here and leave it as a future work.
Antenna architectures for network MIMO
Table 5.4. Spectrum eﬃciency improvement for network MIMO with FFR and directional antenna Proposed threecell network MIMO
Conventional setup
Threecell 7cell network 60◦ network MIMO MIMO sectorization (bps/Hz)/base Improvement
3.0946
5.1579 66.67%
5.6337 82.05%
120◦ sectorization 5.7534 85.92%
1 0.9 0.8 0.7 0.6 CDF
146
3cell omnicoordination (with IGI) 7cell omnicoordination (with IGI)
0.5 0.4 0.3 Diamond: FFRbased 3cell network MIMO under 120o−shaped cell
0.2
Square: FFRbased 3cell network o MIMO under 60 −shaped cell
0.1 0
−20
−10
0
10 20 Received SINR (dB)
30
40
50
Figure 5.11. Performance comparison of proposed threecell FFRbased network MIMO with general omnidirectional threecell and sevencell network MIMO systems. Finally, we show the potential gains of combining frequency partition and network MIMO in Figure 5.11. Take ZFDPC as an example. As studied in Section 5.4.2, the performance degradation caused by IGI for threecell and sevencell network MIMO systems is signiﬁcant. However, taking advantage of joint frequency partition and network MIMO, the proposed FFRbased threecell network MIMO architecture (for both cell sectorizations) can even outperform conventional sevencell network MIMO system using omnidirectional cell planning. The improvement is about 2 dB for 60◦ sectorized cell planning and 3 dB for 120◦ sectorized cell planning at the 90th percentile of the received SINR. This result indicates that network MIMO systems with a small number of coordinated cells are also comparable to network MIMO systems with a large number of coordinated cells as long as directional antennas and sector frequency arrangement techniques can be adopted appropriately. Table 5.4 shows the spectrum eﬃciency
5.7 Conclusion
147
improvement of the threecell network MIMO with the proposed frequency rearranged method for diﬀerent sector antennas.
5.7
Conclusion We have discussed joint FFR and network MIMO architecture for multicellular systems. Taking advantage of FFR to reduce cochannel interference by partitioning the frequency band into diﬀerent zones, we have exploited the performance gain of the coordinated network MIMO techniques. We have provided a threecell coordinated scheme for network MIMO based on a sector frequency rearrangement method, rather than a huge number of coordinated cells. We have shown that using a small number of coordinated cells can outperform the sevencellbased coordination with omnidirectional transmission. Future research topics for small network MIMO could include the design of the inner region of the FFR system and the lowcomplexity joint multicell transmission techniques. We hope that this study will focus the attention of future researchers on BS architecture and system deployment for network MIMO systems.
References [1] WiMAX Forum. (2006, Aug.) Mobile WiMAX  Part I: A technical overview and performance evaluation. Online: available at www.wimaxforum.org. [2] H. Fujii and H. Yoshino, “Theoretical capacity and outage rate of OFDMA cellular system with fractional frequency reuse,” in Proc. of IEEE Vehicular Technology Conference, pp. 1676–1680, May 2008. IEEE, 2008. [3] M. Assaad, “Optimal fractional frequency reuse (FFR) in multicellular OFDMA system,” in Proc. of IEEE Vehicular Technology Conference, Sep. 2008. IEEE, 2008. [4] R. Y. Chang, Z. Tao, J. Zhang, and C. C. J. Kuo, “A graph approach to dynamic fractional frequency reuse (FFR) in multicell OFDMA networks,” in Proc. of IEEE International Conference of Communications, June 2009. IEEE, 2009. [5] H. Lei, L. Zhang, X. Zhang, and D. Yang, “A novel multicell OFDMA system structure using fractional frequency reuse,” in Proc. of IEEE International Symposium Personal, Indoor and Mobile Radio Communications, pp. 1– 5, Sep. 2007. IEEE, 2007. [6] C.S. Chiu and C.C. Huang, “Combined partial reuse and soft handover in OFDMA downlink transmission,” in Proc. of IEEE Vehicular Technology Conference, pp. 1707–1711, May 2008. IEEE, 2008. [7] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Trans. on Information Theory, vol. 49, no. 7, pp. 1691–1706, July 2003.
148
Antenna architectures for network MIMO
[8] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sumrate capacity of gaussian MIMO broadcast channels,” IEEE Trans. on Information Theory, vol. 49, no. 10, pp. 2658–2668, Oct. 2003. [9] P. Viswanath and D. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplinkdownlink duality,” IEEE Trans. on Information Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003. [10] S. Shamai and B. M. Zaidel, “Enhancing the cellular downlink capacity via coprocessing at the transmitting end,” IEEE Vehicular Technology Conference, vol. 3, pp. 1745–1749, May 2001. IEEE, 2001. [11] O. Somekh, O. Simeone, Y. BarNess, and A. M. Haimovich, “Distributed multicell zeroforcing beamforming in cellular downlink channels,” in Proc. of IEEE Global Telecommunications Conference, pp. 1– 6, Nov. 2006. IEEE, 2006. [12] A. D. Wyner, “Shannontheoretic approach to a Gaussian cellular multipleaccess channel,” IEEE Trans. on Information Theory, vol. 40, pp. 1713– 1727, Nov. 1997. [13] S. Jing, D. N. C. Tse, J. Hou, J. B. Spriag, J. E. Smee, and R. Padovani, “Multicell downlink capacity with coordinated processing,” in Proc. of Inform. Theory and Application Workshop, Jan. 2007. University of California, San Diego, 2007. [14] M. K. Karakayali, G. J. Foschini, R. A. Valenzuela, and R. D. Yates, “On the maximum common rate achievable in a coordinated network,” in Proc. of IEEE International Conference of Communications, vol. 9, pp. 4333– 4338, June 2006. IEEE, 2006. [15] M. K. Karakayali, G. J. Foschini, and R. A. Valenzuela, “Network coordination for spectrally eﬃcient communications in cellular systems,” IEEE Trans. on Wireless Commun., vol. 13, no. 4, pp. 56–61, Aug. 2006. [16] G. J. Foschini, M. K. Karakayali, and R. A. Valenzuela, “Coordinating multiple antenna cellular networks to achieve enormous spectral eﬃciency,” IEE Proc. Commun., vol. 153, no. 4, pp. 548–555, Aug. 2006. [17] H. Zhang and H. Dai, “Cochannel interference mitigation and cooperative processing in downlink multicell multiuser MIMO networks,” EURASIP Journal on Wireless Commun. and Networking, pp. 222–235, Feb. 2004. [18] Y. Liang, A. Goldsmith, G. Foschini, R. Valenzuela, and D. Chizhik, “Evolution of base stations in cellular networks: Denser depolyment versus coordination,” in Proc. of IEEE International Conference on Communications, pp. 4128–4132, Jan. 2008. IEEE, 2008. [19] S. Venkatesan, “Coordinating base stations for greater uplink spectral eﬃciency in a cellular network,” in Proc. of IEEE International Symposium Personal, Indoor and Mobile Radio Communications, Sep. 2007. IEEE, 2007. [20] S. Venkatesan, “Coordinating base stations for greater uplink spectral eﬃciency: Proportionally fair user rates,” in Proc. of IEEE International
References
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
149
Symposium Personal, Indoor and Mobile Radio Communications, Sep. 2007. IEEE, 2007. P. Marsch and G. Fettweis, “A framework for optimizing the uplink performance of distributed antenna systems under a constrained backhaul,” in Proc. of IEEE International Conference on Communications, pp. 975–979, June 2007. IEEE, 2007. A. Papadogiannis, D. Gesbert, and E. Hardouin, “A dynamic clustering approach in wireless networks with multicell cooperative processing,” in Proc. of IEEE International Conference on Communications, pp. 4033– 4037, May 2008. IEEE, 2008. P. Marsch and G. Fettweis, “A framework for optimizing the downlink of distributed antenna systems under a constraint backhaul,” in Proc. of 13th European Wireless Conference, Apr. 2007. F. Boccardi and H. Huang, “Limited downlink network coordination in cellular networks,” in Proc. of IEEE International Symposium Personal, Indoor and Mobile Radio Communications, Sep. 2007. IEEE, 2007. H. Huang, M. Trivellato, A. Hottinen, M. Shaﬁ, P. J. Smith, and R. Valenzuela, “Increasing downlink cellular throughput with limited network MIMO coordination,” IEEE Trans. on Wireless Commun., vol. 8, no. 6, pp. 2983– 2989, June 2009. H. Huang and R. A. Valenzuela, “Fundamental simulated performance of downlink ﬁxed wireless cellular networks with multiple antennas,” in Proc. of IEEE International Symposium Personal, Indoor and Mobile Radio Communications, vol. 1, pp. 161–165, Sep. 2005. IEEE, 2005. L.C. Wang, K. C. Chawla, and L. J. Greenstein, “Performance studies of narrow beam trisector cellular systems,” International Journal of Wireless Information Networks, vol. 5, no. 2, pp. 89–102, July 1998. L.C. Wang, “A new cellular architecture based on an interleaved cluster concept,” IEEE Trans. on Vehicular Technology, vol. 48, no. 6, pp. 1809– 1818, Nov. 1999.
Part III Relaybased cooperative cellular wireless networks
6
Distributed spacetime block codes Matthew C. Valenti and Daryl Reynolds
6.1
Introduction In this chapter, we consider spacetime coding strategies for multiplerelay cooperative systems that eﬀectively harness available spatial diversity. More specifically, the goal is to examine ways to forward signals eﬃciently from multiple relays to the destination while addressing the important practical issue of synchronization among the relays. We assume a general twophase transmission protocol as illustrated in Figure 6.1. In the ﬁrst phase of the protocol, the source broadcasts a message which is received by the relays and (possibly) the destination. During the second transmission phase, a subset of the relays, possibly in conjunction with the source, transmits additional information to the destination. This protocol is useful in practical scenarios where signals received at the destination due to transmissions directly from the source (Phase 1) will not carry enough useful information because of noise, fading, and/or interference.
Phase 1
Phase 2
Figure 6.1. Illustration of the twophase transmission protocol using a distributed spacetime code. In the ﬁrst phase (left) the source transmits to several relays, while in the second phase (right), the relays simultaneously transmit to the destination. Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
154
Distributed spacetime block codes
It is expected that Phase 2 will dramatically increase the reliability of the system, but if the symbols cannot be decoded correctly after the second phase, the protocol can restart by returning to Phase 1 or Phase 2. The primary problem associated with forwarding information from multiple relays to the destination is determining how the information should be spread out among the relays over space and time. This is analogous to the classic spacetime coding problem in pointtopoint multipletransmitantenna systems, and so it is often called the distributed spacetime coding problem. Similar to the pointtopoint scenario, performance can vary dramatically based on the coding scheme, so care must be taken to achieve high performance in terms of diversity and coding gain while accounting for the practical limitations of cooperative systems. In fact, poorly designed distributed spacetime coding schemes may perform worse than pointtopoint systems, especially considering that the twostage protocol consumes additional degrees of freedom (time resources) relative to direct transmission. A conservative solution to the distributed spacetime coding problem is for each terminal that participates in the second phase to transmit using orthogonal subchannels, which could be implemented using disjoint time slots, diﬀerent frequency subbands, or orthogonal spreading codes (assuming that they can be synchronized). When disjoint time slots are used, this strategy is a distributed form of delay diversity [22]. However, there are two problems associated with this approach: (1) it is known that orthogonalizing system resources is suboptimal [18], and (2) enforcing orthogonalization in time, frequency, or code space would require signiﬁcant additional signaling overhead, e.g., feedback and/or synchronization, and, in many cases, may not even be possible. A more eﬃcient, albeit more aggressive, approach is for the terminals transmitting in the second phase to use a distributed spacetime block code1 [10, 15]. There are two main varieties of forwarding mechanisms that can be used with distributed spacetime coded systems: decodeandforward (DF) [10] and amplifyandforward (AF) [8]. In DF protocols, the source’s signal is encoded with a forward error correcting code that must be successfully decoded by a relay before that relay may participate in the transmission of the distributed spacetime code. After decoding, the information is reencoded and remodulated. A linear combination of the remodulated symbols and their conjugates is transmitted by each participating relay. While the DF protocol avoids error propagation or the ampliﬁcation of noise received over the source–relay channel, its implementation is complicated by fact that the set of relays that are able to decode and so participate in the spacetime code in the second phase is random and unknown to the system. Therefore, unlike with pointtopoint multipleantenna systems, extra
1
Distributed spacetime trellis codes can be used as well, but we focus here on block codes and refer the interested reader to, for example, [13].
6.2 System model
155
care must be taken to ensure that the relays coordinate in the transmission of the spacetime codeword. With AF protocols, a ﬁxed number of relays participate in the second phase by transmitting a linear combination of the time samples of the noisy received signal (not decoded symbols) they receive from the source. Because the number of relays is ﬁxed, AF does not have to manage which relays transmit which part of the spacetime codeword, unlike DF. However, with AF the noise received over the source–relay channels is ampliﬁed, which has the potential to degrade performance. There are a number of challenges associated with practical distributed spacetime code implementation, some of which are shared with conventional spacetime coding used in pointtopoint multipleantenna systems, but some are unique to the distributed scenario. As with conventional spacetime codes, distributed codes typically (but not always) require the destination receiver to have knowledge of the channels between the relays and the destination. Unlike conventional spacetime codes, however, distributed spacetime codes are subject to some challenging synchronization issues. Because the transmitters are widely separated and have diﬀerent time references, and due to diﬀerences in the propagation delay between the relays and the destination, the diﬀerent signals received by the destination will generally be oﬀset in time. Although this problem can perhaps be overcome with appropriate synchronization protocols, it can be handled more eﬀectively with delay diversity, delaytolerant distributed spacetime codes, or spacetime spreading. The remainder of this chapter is as follows. In Section 6.2, we give a system model that will be used throughout the chapter. In Section 6.3, we review concepts related to ﬁxed spacetime block codes (STBCs). In Section 6.4, we describe the DF protocol, and in Section 6.5 we describe the AF protocol. We discuss synchronization issues in Section 6.6, and conclude in Section 6.7.
6.2
System model Consider a wireless network consisting of a source node, a destination node, and a set of R relay nodes N = {Ni : 1 ≤ i ≤ R}. Each node has a halfduplex radio and a single antenna. Messages are transmitted according to a twophase protocol. During the ﬁrst phase, a signal of duration T1 symbol periods is broadcast by the source and received by the relays. During the second phase, a subset of the relays simultaneously, but perhaps not synchronously, transmits signals of duration T2 symbol periods, and the destination receives a noisy sum of the relay signals. After approximately T = T1 + T2 consecutive symbol periods (depending upon transmission and channel delays), the source moves on to the next message (or a retransmission of a failed message). For ease of exposition, we assume that there
156
Distributed spacetime block codes
is no direct link between the source and destination, although the protocols and performance analyses can easily be generalized to allow for such a link. We adopt a discretetime model, whereby the signal transmitted by the source during the ﬁrst phase is represented by the vector s = [s1 , ..., sT 1 ]t . The individual symbols s , 1 ≤ ≤ T1 are each drawn from a complex constellation X of M symbols. The signal constellation is normalized so that its average energy is unity, i.e., (1/M ) s∈X s2 = 1. The normalized signal is ampliﬁed and transmitted by the source with power P1 during the ﬁrst phase. Let fi represent the complex gain of the channel between the source and node Ni . Then the signal received by node Ni during the ﬁrst phase is ri = fi P1 s + vi , (6.1) where vi = [vi,1 , ..., vi,T 1 ]t is a noise vector containing independent circularly symmetric complex Gaussian random variables with zero mean and unit variance. In the second phase, a subset K ⊆ N of the relays simultaneously, but perhaps without symbollevel synchronization, transmits to the destination. During this phase, node Ni ∈ K transmits a signal represented by the discretetime vector ti = [ti,1 , ..., ti,T 2 ]t with power P2 . When the signals are perfectly synchronized, the signal received at the destination is x = gi P2 ti + w, (6.2) i:N i ∈K
where gi is the complex gain of the channel between node Ni and the destination, and the noise vector w = [w1 , ..., wT 2 ]t contains independent circularlysymmetric complex Gaussian random variables with zero mean and unit variance. When the signals are not synchronized, the model must be generalized to account for time oﬀsets. In general, the powers E[fi 2 ] and E[gi 2 ] of the channel gains fi and gi will depend on the topology of the network and the propagation characteristics of the wireless channel, and will usually be unequal. However, for ease of exposition, we make the simplifying assumption that the fi s and gi s are independent and identically distributed (i.i.d.). In particular, each fi and gi is assumed to be a circularly symmetric complex Gaussian with zero mean and unit variance, so that their envelopes fi  and gi  are Rayleigh distributed. The coeﬃcients fi are held ﬁxed for the transmission of the signal s, and the coeﬃcients gi are held constant for the transmission of the ti , i.e., we assume a Rayleigh block fading model. Just as the channel gains might have unequal powers, the powers P2 transmitted by the relays can, in general, be selected such that they are unequal, in which case our notation needs to be modiﬁed to indicate the diﬀerent powers. However, in the following discussion we impose the simplifying limitation that all relays transmit with the same power P2 , which is optimal when the average
6.3 Spacetime block codes (STBCs)
157
channel powers are all equal and the transmitters operate without channel state information. Note that this is a fairly general model which leaves unaddressed several critical design and implementation issues. For example, the composition of the set K depends on the protocol being used. In DF protocols, K contain only relays that successfully decoded the source’s transmission, while in AF protocols, K may contain any (or all) of the relays. Another key issue is the selection of the signals ti that are to be transmitted by the nodes in K during the second phase of the protocol. These signals can be jointly coded, but in a distributed way, using a spacetime code, or they can use simpler strategies, e.g., delay diversity or spacetime spreading. We will discuss all of these options in this chapter. Finally, the allocation of power between P1 and P2 for the two transmission phases and the allocation of time over the two time phases are protocoldependent optimization problems that must be solved to maximize performance.
6.3
Spacetime block codes (STBCs) As described in the introduction, one of the primary problems associated with forwarding information from relays to a destination in a cooperative wireless network is how information is transmitted from the relays over time, i.e., the spacetime transmission scheme. One natural strategy is to extend the concept of STBCs, typically used for pointtopoint multiple transmit antenna systems, to relay networks, where they are called distributed spacetime block codes. We begin with a description of conventional STBCs under the quite general linear dispersion paradigm [6]. Suppose for the moment that the ﬁrstphase transmission is perfectly received by all R relays. Under the linear dispersion paradigm, the ith relay transmits a linear combination of the T1 symbols in s and their complex conjugates, ti
= Ai s + Bi ¯s,
(6.3)
where ¯s is the column vector containing the complex conjugates of s and the complex T2 × T1 matrices Ai and Bi are called dispersion matrices. These matrices deﬁne the spacetime code. Each nonzero matrix Ai or Bi is constrained to be unitary, and since the signal set X is normalized to unit energy, the symbols transmitted by the relays will also have unit energy. The family of spacetime block codes that can be represented by (6.3) are called linear dispersion (LD) codes [6]. This family of codes includes many wellknown spacetime codes as special cases. For example, one linear dispersion code that has been proposed for cooperative communications with R = 2 relays is
158
Distributed spacetime block codes
described by the dispersion matrices [9]
A1 =
+1 0 0 +1
, A2 = 02×2 ,
B1 = 02×2 , B2 =
0 −1 +1 0
,
(6.4)
where 0m ×n is an m × n matrix of allzeros. This code is simply a transpose of the wellknown Alamouti spacetime block code [1]. For many codes of interest, including the one speciﬁed by (6.4), either Ai or Bi is a matrix of zeros for every i. This means that a particular relay will transmit a linear combination of the symbols in s or ¯s, but not both. If we deﬁne Ci
=
Ai ,
if Bi = 0,
(6.5)
Bi , if Ai = 0
and s
(i)
=
s, if Bi = 0,
(6.6)
¯s, if Ai = 0
then we can write (6.3) more compactly as = Ci s(i) .
ti
(6.7)
Assume that nodes N1 , ..., NR participate in the secondphase transmission with the same power P2 . The signal received at the destination is x
=
P2 Sh + w,
(6.8)
where S
=
,
C1 s(1)
...
CR s(R )
...
gR

(6.9)
is the T2 × R spacetime codeword, and h
=
,
g1
t
(6.10)
is the channel vector. The maximumlikelihood (ML) detector at the destination estimates the source signal as 8 8 8 8 ˆs = arg min 8x − P2 Sh8 , T s∈X
1
where · indicates the Frobenius norm.
(6.11)
159
6.3 Spacetime block codes (STBCs)
Unless the distributed spacetime code, described by the set of all possible codewords S, has some special structure, ML detection will have exponential complexity in the number of source symbols T1 . Fortunately, several classes of codes admit reduced complexity ML decoding, including the wellknown orthogonal design family [15], whose orthogonal structure allows decoupling of the symbols in the codeword, permitting symbolbysymbol ML detection with linear complexity in T1 . The (Alamouti) code given by (6.4) is one example of an orthogonal design. Another orthogonal design which has been applied to cooperative diversity with R = 4 relays is described by the dispersion matrices [9] ⎡
1 ⎢ 0 A1 = ⎢ ⎣ 0 0 ⎡ 0 ⎢ 0 A4 = ⎢ ⎣ 0 −1
0 0 0 0 0 0 1 0
⎤ 0 0 ⎥ ⎥, 0 ⎦ 0 ⎤ 0 1 ⎥ ⎥, 0 ⎦ 0
⎡
0 ⎢ 0 A2 = ⎢ ⎣ 0 0 ⎡ 0 ⎢ 0 B1 = ⎢ ⎣ 0 0 ⎡ 0 ⎢ 0 B3 = ⎢ ⎣ −1 0
⎤ 1 0 0 0 ⎥ ⎥, 0 0 ⎦ 0 0 ⎤ 0 0 −1 0 ⎥ ⎥, 0 1 ⎦ 0 0 ⎤ 0 0 0 0 ⎥ ⎥, 0 0 ⎦ −1 0
⎡ ⎢ A3 = ⎢ ⎣ ⎡ ⎢ B2 = ⎢ ⎣
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 1
⎤ ⎥ ⎥, ⎦ ⎤ ⎥ ⎥ , (6.12) ⎦
B4 = 04×3 .
Note that for this code, only B4 is allzero and thus the model given by (6.5)– (6.10) must be generalized slightly. See [8, 9] for details. The rate of a STBC is T1 /T2 , and a code is said to be full rate if it has a rate of unity. While the rate of the code speciﬁed by (6.4) is unity, the rate of the R = 4 code speciﬁed by (6.12) is only 3/4. No fullrate orthogonal STBC exists for R > 2 when complex symbols are used [11], although reducedrate orthogonal codes can be designed for any number of transmit antennas. Thus, for R > 2, the convenience (linear ML decoding complexity) of using an orthogonal design comes at the cost of reduced spectral eﬃciency. An alternative to using orthogonal designs is to use quasiorthogonal designs [7], which can achieve full rate with four antennas with higher complexity than orthogonal designs, but much lower than worstcase ML complexity. The additional complexity over orthogonal designs is because quasiorthogonal codes reduce the ML detection problem to the joint detection of pairs of complex symbols, whereas orthogonal designs reduce it to the detection of individual complex symbols. The ML detector for a quasiorthogonal code thus requires that each of the T1 /2 pairs of symbols be compared against M 2 hypothesis. An example quasiorthogonal STBC
160
Distributed spacetime block codes
considered for cooperativediversity with R = 4 relays is given by [9] ⎡
1 0 ⎢ 0 1 A1 = ⎢ ⎣ 0 0 0 0 ⎡ 0 0 ⎢ 0 0 A4 = ⎢ ⎣ 0 −1 1 0
⎤ 0 0 0 0 ⎥ ⎥, A2 = 04×4 , 1 0 ⎦ 0 1 ⎤ 0 1 −1 0 ⎥ ⎥ , B1 = 04×4 , 0 0 ⎦ 0 0 ⎡ 0 ⎢ 0 B3 = ⎢ ⎣ 1 0
A3 = 04×4 , ⎡
0 ⎢ 1 B2 = ⎢ ⎣ 0 0 0 −1 0 0 0 −1 0 0 0 1 0 0
−1 0 0 0 ⎤ ⎥ ⎥, ⎦
⎤ 0 0 0 0 ⎥ ⎥ , (6.13) 0 −1 ⎦ 1 0 B4 = 04×4 .
In Figure 6.2, four systems are compared by plotting the bit error rate (BER) of each system as a function of the signaltonoise ratio (SNR). Three values of R are considered, R = {1, 2, 4}, and all systems transmit with a spectral eﬃciency of 3 (bits per second) per Hertz (bps/Hz). The signals are transmitted over Rayleigh fading channels, and the power is split evenly across the R transmit antennas. Since the noise power is unity, the transmitted power is P2 = SNR/R. The R = 1 system represents conventional pointtopoint communications between a pair of terminals, each with a single antenna and no spacetime coding. The R = 2 system uses the transposed Alamouti code given by (6.4). Two systems are compared for use with R = 4 transmitting antennas, the orthogonal code of (6.12) and the quasiorthogonal code (6.13). The rate of the orthogonal spacetime code used with R = 4 antennas is 3/4, while the rates of the other STBC are all unity. In order for the spectral eﬃciency to be maintained at 3 bps/Hz, graylabeled 8PSK modulation is used for the fullrate systems (including the system with no spacetime coding), while graylabeled 16QAM modulation is used for the rate 3/4 system. The worstperforming system is the one that uses just one transmit antenna (R = 1), while the next worstperforming system is the one with two transmit antennas (R = 2). The two systems’ four transmit antennas (R = 4) exhibit the best performance. The most signiﬁcant feature to notice in Figure 6.2 is the steepness of the curves. At high SNR, these curves become straight lines, implying that asymptotically there is a linear relationship between the logarithm of the error probability and the SNR expressed in decibels. The negative slope is called the diversity of the system, also known as the diversity order or diversity gain. Inspection of the diagram reveals that the diversity of the R = 1 system is equal to 1 (i.e., the BER drops by an order of magnitude with every decade of SNR), while the diversity of the R = 2 system is equal to 2. Although the SNR is not suﬃciently high in the ﬁgure to show it, the diversity of both two R = 4 systems is equal to 4. A system with R antennas is said to have full diversity if its diversity order is
161
6.3 Spacetime block codes (STBCs)
R R
Figure 6.2. BER performance of four systems: An uncoded system with R = 1 transmit antenna, an Alamouticoded system with R = 2 transmit antennas, an orthogonal STBC with R = 4 transmit antennas, and a quasiorthogonal STBC with R = 4 transmit antennas. In each case, the spectral eﬃciency is 3 bps/Hz and the signals are transmitted over independent Rayleigh fading channels.
R. Thus, all four systems shown in Figure 6.2 exhibit full diversity, as seen by the fact that their error probabilities decay proportional to 1/SNRR . The performance of a spacetime coded system can be determined by analyzing the pairwise error probability (PEP) between all pairs of distinct spacetime codewords Sk and S . The PEP can be bounded by, for example, a Chernoﬀ bound. By taking the limit with respect to the SNR, the diversity order is then determined. Full diversity is achieved by ensuring that Sk − S is full rank for all k = [15]. Linear dispersion codes need not be orthogonal or quasiorthogonal. For instance, the linear dispersion codes presented in [6] were designed to maximize the mutual information between the transmitter and receiver under a power constraint. However, such codes do not lend themselves to the very simple decoder structures that are possible with orthogonal or quasiorthogonal codes. While a bruteforce ML decoder requires comparison against all M T 1 hypotheses, the complexity can be greatly reduced by using a sphere decoder [2]. Another option is to use the designs in [17], which permit decoupled symbolbysymbol decoding and fullrate at the cost of reduced diversity.
162
Distributed spacetime block codes
6.4
DF distributed STBC Returning to the twophase relay network conﬁguration, consider the case that the ﬁrstphase transmission must take place over a channel that is corrupted by noise and fading. Now, there is no guarantee that any particular node will receive the transmission correctly. With a DF protocol, the source encodes its transmissions with a channel code. Each relay attempts to decode using the signal it receives, and can only participate in the secondphase transmission if it successfully decodes the message sent by the source. The condition that a node can only transmit in the second phase if it successfully decodes the message requires that the code be used not only as an error correcting code, but also as an error detecting code (i.e., the relay needs to detect the existence of uncorrectable errors). The set of nodes K that successfully decode the source’s message and may transmit during the second phase is called the decoding set [10], and the number of nodes in the decoding set is K = K. During the second phase, nodes in K transmit using a distributed spacetime code. A major complicating factor is that the size of the decoding set is random, yet the spacetime code must be designed with a certain number of transmitting antennas in mind. Because of this, the number of relays that actually transmit should be limited to the maximum number of antennas supported by the spacetime code, which we denote Km ax . It is possible that K < Km ax , which means that there are not enough relays participating in the second phase to use the entire spacetime code. This implies that the spacetime code should be “scale free”, meaning that it still oﬀers the maximum possible diversity even if some of the transmitting antennas are not used. When K < Km ax , the maximum possible diversity order is reduced from Km ax to K. It is known that orthogonal spacetime codes have this scalefree property [9]. If relay Ni ∈ K transmits during the second phase, then it does so by transmitting a signal vector ti of the form given by (6.3), where s is the signal obtained by decoding, reencoding, and remodulating, and the Ai and Bi are the dispersion matrices currently assigned to that relay. Note that since the composition of the set of transmitting nodes changes after each source transmission, the set of dispersion matrices assigned to a particular relay may also change. The protocol must be careful to make sure that each transmitting relay is allocated a distinct set of dispersion matrices. When either Ai or Bi is all zeros for all i, the received signal at the destination is as given by (6.8), where the columns of the spacetime codeword S will be the signals transmitted by the relays. When K < Km ax , fewer relays transmit than there are columns in S and Km ax − K columns in S will be all zeros.
6.4.1
Performance analysis The performance of a DF system depends on the error control code or codes used. If the system uses a capacityapproaching code, such as a turbo code or a
6.4 DF distributed STBC
163
lowdensity paritycheck (LDPC) code, then the codeword error rate over a particular link may be approximated by the informationoutage probability of that link. The informationoutage probability is the probability that the conditional mutual information between the channel input and output is below some threshold. For the ﬁrst phase, the channel between the source and each relay is an additive white Gaussian noise (AWGN) channel when it is conditioned on the fading gain fi . The conditional mutual information between the signal transmitted by the source and the signal received by the ith relay is given by I(s, ri fi ) = log2 (1 + P1 fi 2 ),
(6.14)
where P1 fi 2 is the “instantaneous” SNR of the link between the source and the ith relay. Note that (6.14) represents the mutual information when the source– relay channel is used all the time. However, in the DF protocol, the relay source– relay link is only used for T1 out of every T symbol periods. Thus, the mutual information needs to be scaled by the ratio T1 /T when computing the probability that a relay is in an outage. The informationoutage probability of the link from the source to node Ni is
T1 I(s, ri fi ) < r , (6.15) pi = Pr T where r is the rate of the error control code. Substituting (6.14) into (6.15) gives
T1 log2 (1 + P1 fi 2 ) < r pi = Pr T , 2 (6.16) = Pr fi  < Γ1 , where Γ1
=
2r T /T 1 − 1 . P1
(6.17)
Equation (6.16) is the cumulative distribution function (CDF) of the random variable fi 2 evaluated at Γ1 . If we assume that fi is circularly symmetric complex Gaussian with zero mean and unit variance, then fi 2 will be exponential with unit mean. By recalling the CDF of an exponential random variable, the informationoutage probability is pi
=
1 − e−Γ 1 .
(6.18)
Because the fi s are i.i.d. and the threshold Γ1 is common to all relays, pi is the same at all R relays and may be denoted as p. Let Zi = {0, 1} be an indicator variable that equals unity when the ith relay is in an outage. Zi is a Bernoulli random variable with P [Zi = 1] = p. The number of relays K = K that successfully decode the ﬁrstphase transmission is K = R i=1 Zi . Because the channels are independent, so are the Zi s, and it follows that K is a binomial random variable. The probability that the random variable
164
Distributed spacetime block codes
K is equal to k is given by the probability mass function of K, pK [k]
= P r[K = k] R (1 − p)k pR −k . = k
(6.19)
The secondphase transmission may also be characterized in terms of an outage probability. However, the outage probability at the destination depends on the number of nodes K in the decoding set as well as the maximum number of nodes Km ax that may transmit during the second phase of the protocol. Deﬁne the conditional endtoend informationoutage probability for the second phase of the DF protocol as " , Pr[Outagek] = Pr I(S, xh) < r"K = k , (6.20) where S is the spacetime codeword and h is a length min(k, Km ax ) vector containing the coeﬃcients gi corresponding to those relays that transmit during the second phases. Equation (6.20) represents the probability that the destination is in an outage during the second phase given that the decoding set has k relays in it. When using a rate T1 /T2 orthogonal STBC over a pointtopoint link, the mutual information is [11] I(S, xh)
T1 log2 1 + P2 h2 . T2
=
(6.21)
This mutual information expression assumes full use of the channel. However, in the DF protocol, the relay–destination link is only active for T2 out of every T channel uses, and thus (6.21) must be scaled by T2 /T . Substituting the scaled version of (6.21) into (6.20) results in
" T1 2 " log2 1 + P2 h < r K = k Pr[Outagek] = Pr T " , = Pr h2 < Γ2 "K = k , (6.22) where Γ2
=
2r T /T 1 − 1 . P2
(6.23)
This is the CDF of h2 evaluated at Γ2 . When there are min(k, Km ax ) relays transmitting in the second phase, then h2 is Erlangm with min(k, Km ax ) degrees of freedom. Using the CDF of an Erlangm distribution, the outage probability becomes
m in(k ,K m a x )−1
Pr[Outagek]
=
1−
n =0
Γn2 −Γ 2 e . n!
(6.24)
6.4 DF distributed STBC
165
Of course, when k = 0, Pr[Outagek] = 1 since the system is always in an endtoend outage when no relays receive the source transmission (recall that we assume no direct link from source to destination). From the theorem on total probability, the overall endtoend outage probability may be found from the conditional outage probabilities as Pr[Outage]
=
R
pK [k]P r[Outagek].
(6.25)
k =0
By substituting (6.19) and (6.24) into (6.25), we get the following expression for endtoend outage probability ⎛ ⎞ m in{k ,K m a x }−1 n R R Γ2 −Γ 2 ⎠ R k R −k ⎝ (1 − p) p e 1− . pD = p + k n! n =0 k =1
(6.26)
6.4.2
Numerical results By using (6.26), we can determine the outage probability for a network comprising R relays that uses a particular spacetime code. Consider two examples, one that uses the transposed Alamouti code with dispersion matrices given by (6.4), and another that uses the orthogonal STBC with dispersion matrices given by (6.12). Both systems use a rate r = 1/2 error control code. With the Alamouticoded system, no more than Km ax = 2 relays may transmit during the second phase, while with the other orthogonal system, no more than Km ax = 4 relays may transmit. Let K = min(K, Km ax ) be the number of relays that actually transmit during the second phase, where K is the number of relays that successfully decode the source’s transmission. The total transmitted power of all relays is P = P1 + K P2 . As explained later in this section, the powers P1 and P2 are selected to minimize the outage probability subject to the total power constraint. Figure 6.3 shows the informationoutage probability as a function of SNR for both spacetime codes and a variable number of relays. Since the noise power is unity, the SNR is equal to the total power P . For the Alamouticoded system, the number of relays is between 2 and 8, while for the other orthogonal system, the number of relays is between 4 and 10. For both systems, performance improves with increasing R. Even though no more than Km ax relays can be used during the second transmission phase, it is still advantageous to have more than Km ax relays present in the network. This is due to the diversity present in the ﬁrst phase transmission. Having more than Km ax relays makes it more likely that at least Km ax relays will be able to decode the source’s transmission, and thus it is very likely that Km ax relays will transmit the entire spacetime codeword during the second phase. From the curves, it is seen that the Alamouti code provides a diversity order of Km ax = 2 while the other orthogonal system provides a
166
Distributed spacetime block codes
Kmax Kmax
Figure 6.3. Comparison of the informationoutage probability of several systems: direct, DF using the Alamouti code and up to Km ax = 2 transmitting relays, DF using orthogonal STBC and up to Km ax = 4 transmitting relays. For each value of Km ax , a set of seven curves is shown corresponding to a diﬀerent number of relays R. For the Km ax = 2 system, the total number of relays R is between 2 and 8, while for the Km ax = 4 system there were between 4 and 10 relays. For each value of Km ax performance improves with increasing R.
diversity order of Km ax = 4. Using additional relays does not improve the overall system’s diversity order, but it does provide an additional coding gain. Also shown in Figure 6.3 is the performance of a direct pointtopoint link transmitting with transmission power P = SNR and using a single antenna at each end of the communication link. Because this is just a singleinput singleoutput (SISO) system, the diversity order is only equal to 1, and thus asymptotically the performance of the direct transmission is worse than the considered DF protocols. However, at very low SNR, the performance of the direct link is actually better than the systems that use a distributed STBC. This is because the system using a direct link may concentrate all of its power into the single transmission rather than diluting the energy across the transmissions of the two hops. Thus, at very low SNR, a direct link might be more eﬀective than using the distributed STBC. However, keep in mind that these results assume that all channels have unity power gain. When relays are placed between the source and destination, then it is possible that channels used by the system with the
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.5
0.4
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
10 20 SNR (in dB)
0
30 (b)
167
0.6
0.3
0 (a)
Optimal power ratio
Optimal power ratio
6.4 DF distributed STBC
0
5
10 SNR (in dB)
15
20
Figure 6.4. The optimal power ratio for: (a) the Km ax = 2 system and between 2 and 8 relays and (b) the Km ax = 4 system with between 4 and 10 relays. The optimal power ratio decreases with increasing R.
distributed STBC (i.e., the gains from source to relays and the gains from relays to destination) will have a higher gain than the channel used by the system that uses direct transmission. Let P1 /(K P2 ) be the ratio of the power used during the ﬁrst phase to the power used during the second phase. The results shown in Figure 6.3 assume that the power is selected to minimize the informationoutage probability subject to the total power constraint P = P1 + K P2 . To ﬁnd the optimal power ratio for each SNR point, we compute the informationoutage probability for all ratios between 10−4 and 1 in increments of 10−4 , and pick the ratio that minimizes the informationoutage probability. The result of this optimization is shown in Figure 6.4. Figure 6.4(a) shows the optimal power ratios for the Km ax = 2 system (Alamouti coded) and Figure 6.4(b) shows the optimal power ratios for the Km ax = 4 system. For each system, a family of curves is shown corresponding to the diﬀerent values of R considered in Figure 6.3. The power ratio decreases with increasing R and with increasing SNR. This is because as R grows, the likelihood that Km ax relays can decode the source’s signal improves as a function of P1 , and thus the system can aﬀord to decrease P1 and devote more power to the secondphase transmission.
168
Distributed spacetime block codes
6.5
AF distributed STBC In contrast to DF protocols, AF protocols do not require that each relay fully demodulates and decodes the signal it receives from the source. Instead, relay Ni obtains the vector ri given in (6.1) by downconverting the received signal to baseband and passing it through a pair of ﬁlters matched to the inphase and quadrature basis functions. The matched ﬁlters are sampled at the symbol rate, resulting in a set of T1 complex samples that are placed into the vector ri . Rather than demodulating and decoding ri , the relay transmits a linear combination of the samples in ri and its conjugates at power P2 [8]. We can write the normalized signal transmitted by node Ni in vector form as 1 (Ai ri + Bi ¯ri ) , ti = (6.27) P1 + 1 where Ai and Bi are the dispersion matrices assigned to node Ni . The two major diﬀerences between (6.27) and the DF transmitted signal in (6.3) are: (1) in AF, the transmitted signal is a linear combination of the samples in the received vector ri (and its conjugates) rather than a linear combination of the remodulated symbols in the vector s (and its conjugates), and (2) because the noise power is unity, the average received signal power is P1 + 1, and the scaling 1/(P1 + 1) is required to normalize the signal. For the important special case where either Ai or Bi is zero, we can simplify the transmitted signal similar to (6.7) as 1 Ci r(i) , ti = (6.28) P1 + 1 where
* r(i) =
r, ¯r,
if Bi = 0, if Ai = 0
(6.29)
and Ci is as given by (6.5). Using (6.2) and assuming perfect symbollevel synchronization, the resulting received signal vector at the destination can be written as x =
R gi P2 ti + w.
(6.30)
i=1
Substituting (6.1) into (6.29) and (6.30) results in R R P1 P2 P2 (i) x = Ci s + gi fi gi Ci vi + w. P1 + 1 P1 + 1 i=1 i=1 n This may be represented compactly as P1 P2 x = Sh + n, P1 + 1
(6.31)
(6.32)
6.5 AF distributed STBC
where the T2 × R spacetime codeword S is as is ⎡ f1 g1 ⎢ f2 g2 ⎢ h = ⎢ . ⎣ ..
169
given in (6.9), the channel vector ⎤ ⎥ ⎥ ⎥, ⎦
(6.33)
fR gR and the complex noise vector n is Gaussian when conditioned on the {gi } and will generally be colored because the signal transmitted by the relay will contain a linear combination of the elements of the white noise vector vi . Compared with the DF case (6.8), the AF received signal in (6.32) diﬀers in three ways: (1) all R relays transmit a signal during the second phase, not just those that can decode the source’s transmission, (2) the channel vector consists of the product of the source–relay and relay–destination channel gains instead of just the relay–destination channel gains, and (3) the additive noise will have a higher power and will generally be colored. As was the case for DF, the T2 × R matrix S in (6.32) plays the same role as a spacetime code matrix in a conventional pointtopoint multipleinput multipleoutput (MIMO) system, except that, in the distributed spacetime code scenario, the matrix is generated without access to s. For this reason, we say that S deﬁnes a distributed spacetime code operating in AF mode. We can think of h as the equivalent channel matrix and n as additive noise, although n is clearly a function of the spacetime code. Because of the similarities to conventional STBCs, we can analyze the diversity gain and coding gain performance of this family of distributed AF spacetime codes using the same technique we use for conventional codes, i.e., bounding the pairwise error probability.
6.5.1
Performance analysis The achievable diversity of LD codes operating in an AF system can be determined using the same technique that is often used for pointtopoint spacetime coded systems, i.e., by bounding the pairwise error probability. The exact results are complex and we refer the reader to [8] for details. The main result, however, is that the achievable diversity is log log P , (6.34) d=R 1− P which is achieved whenever the Sk − Sl is full rank for all l = k. This result converges to R for very large power P , so LD coding operating in AF systems achieves approximately the same diversity as a pointtopoint system with R antennas.2 2
Note that here we assume that no information is passed directly between the source and destination. If such a link exists, the diversity result simply increases by 1.
170
Distributed spacetime block codes
Interestingly, when the number of relays is large and the power is also large, the coding gain for distributed LD codes is the same as for LD codes operating in pointtopoint systems. On the other hand, when the power P is moderate, the code matrices should be designed such that the code is “scalefree,” i.e., it should perform well when some relays are not working. Mathematically, this requires the codeword diﬀerence matrices to remain full rank when some columns are deleted.
6.5.2
Practical distributed STBC for AF systems Although arbitrary LD codes can achieve almost full diversity with mild conditions on the Ai matrices, they are generally diﬃcult to decode because ML decoding, i.e., 8 8 8 8 P1 P2 8 8 Sh8 , (6.35) arg min 8x − s 8 P1 + 1 8 has high computational complexity for the general case. This problem can be alleviated by using extending wellknown orthogonal [15] or quasiorthogonal [7] code designs for pointtopoint systems to the distributed scenario, as discussed in [9]. The results are distributed codes that are fully diverse, allow lowcomplexity decoding, and are scalefree, yielding good coding gain for moderate transmit powers. Because the noise vector n is not generally white, trueML detection cannot be achieved through linear processing methods such as the decoupleddecoding approach commonly used for orthogonal codes operating over pointtopoint links. However, as reported in [9], the performance when using decoupled decoding is only slightly inferior to that of using true ML detection (i.e., around 0.5 dB).
6.6
The synchronization problem One of the key challenges when designing highperformance distributed spacetime coded systems is symbollevel synchronization among the relay nodes. In conventional pointtopoint spacetime coded MIMO systems, colocated antennas obviate this issue. In cooperative systems, sometimes described as virtual MIMO, the antennas are separated by wireless links. One approach is simply to use appropriate hardware and higherlayer protocols to ensure that transmissions from every participating relay are synchronized. Unfortunately, this may not be possible in practice and, in any case, it would require signiﬁcant signaling overhead that may dramatically increase bandwidth requirements. Other approaches that eﬀectively circumvent the synchronization problem include delay diversity, delaytolerant distributed spacetime codes, and spacetime spreading (STS). We will next consider each of these approaches brieﬂy.
6.6 The synchronization problem
6.6.1
171
Delay diversity It is well known that pointtopoint communication over multipath fading channels provides diversity that can be exploited by appropriate receiver design [18]. In cases where intersymbol interference (ISI) is negligible, as is common in spreadspectrum systems, RAKE reception is suﬃcient. When ISI cannot be ignored, ML sequence detection can be performed using the Viterbi algorithm to extract full diversity in the number of resolvable paths. Mathematically, this involves transforming the frequency selective SISO system into an equivalent ﬂatfading multipleinput singleoutput (MISO) system that uses a particular spacetime code induced by the frequency selective channel. Interestingly, the reverse is also possible. That is, we can transform a ﬂatfading MISO system into a virtual frequency selective SISO system by using a spacetime code described by the following scheme: in the ﬁrst time slot, the symbol x[1] is transmitted on antenna 1 and all other antennas are silent. In the second time slot, x[1] is transmitted from antenna 2 and x[2] is transmitted by antenna 1 and all other antennas remain silent. At time slot m, x[m − l] is transmitted on antenna l + 1 for l = 0, 1, . . . , L − 1. This transmission scheme yields a received signal that is identical to that received in a SISO frequency selective channel with L paths. This special pointtopoint spacetime coding scheme is called delay diversity [22]. Delay diversity cannot be implemented in cooperative communication systems in exactly this way without requiring what we are trying to avoid, i.e., synchronization to determine which relay transmits which symbol in which order. Fortunately, it is straightforward to implement delay diversity in a distributed manner. The simplest way to do this is simply for the relays to wait a random amount of time before they retransmit the symbol or signal they have most recently received. The destination will receive a signal that is equivalent to that received in a SISO multipath channel, so full diversity will be achievable (with probability 1), assuming ML detection at the destination. Linear detectors/equalizers can also be used at the destination, e.g., minimum meansquared error (MMSE), or decorrelating equalization, with some diversity loss. Interestingly, MMSE detection in conjunction with serial interference cancellation (decision feedback implementation) achieves full diversity [21] with much lower complexity than ML detection when the number of relays is large.
6.6.2
Delaytolerant spacetime codes Another approach to distributed spacetime coding without synchronization involves the use of socalled delaytolerant distributed spacetime codes whose performance is insensitive to delays among the received signals from each relay. It is well known that the diversity order of a STBC is equal to the minimum rank of the diﬀerence matrix over all pairs of distinct code matrices [15]. A spacetime code is said to be τ delay tolerant if for all distinct code matrices Sk and
172
Distributed spacetime block codes
S , the diﬀerence matrix Sk − S retains full rank even though the columns of the code matrices are transmitted or received with arbitrary delays of duration at most τ symbols. Let S be a codeword matrix from a synchronized STBC, as in (6.9), and let ∆S be the code matrix received at the destination due to transmission or propagation delays. Then ∆S can be written as ⎤ ⎡ 0∆ 2 ··· 0∆ R 0∆ 1 (6.36) ∆S = ⎣ C1 s(1) C2 s(2) · · · CR s(R ) ⎦ . 0τ −∆ 1 0τ −∆ 2 · · · 0τ −∆ R The collection of all such codewords ∆S constitutes a τ delaytolerant spacetime code if for all delay proﬁles {∆k }R k =1 such that ∆k ≤ τ for all k, it achieves the same diversity as the synchronized code. Work on delaytolerant codes under this framework includes [3, 5, 13, 16]. Although delay diversity extracts full diversity in the number of relays, delaytolerant spacetime codes promise better coding gain and, in some circumstances, lower decoding complexity.
6.6.3
Spacetime spreading (STS) Delay diversity is successful in achieving full diversity in part because the distinct delays for the received signals from each relay provide a unique signature enabling the receiver to separate each resolvable path before cophasing and combining. A similar unique signature can be implemented with coding, i.e., STS. One of the simplest STS strategies is to assign the source and each relay a unique spreading code, as in codedivision multiple access (CDMA) communications. When the relays are not synchronized, the signal received at the destination is similar to that obtained in a conventional (noncooperative) asynchronous CDMA uplink, so that the transmissions from the source and each relay can be separated using wellknown multiuser detection (MUD) signal processing strategies, cophased, and recombined to extract full diversity without symbollevel synchronization. Note that, although CDMA is a spreadspectrum signaling format, it does not need to operate in a spectrally ineﬃcient mode. In fact, it was shown in [19] that the information outage probability of an asynchronous cooperative CDMA uplink under decorrelating MUD is minimized when the system is slightly overloaded, i.e., when the number of relays is slightly larger than the processing gain. This is not surprising because an overloaded CDMA system is operating at high spectral eﬃciency. A more sophisticated STS strategy was designed in [14]: it does not require synchronization among the relays, channel estimation, or complex multiuser signal processing at the destination or relays. The necessity for channel information is obviated by the use of diﬀerentially encoded symbols from each source, as in [4], during the ﬁrst transmission phase. During the second transmission phase, dedicated relays use an STS AF strategy described in [4] that allows low complexity decoding and large diversity gain without channel estimation. Because highcomplexity MUD strategies are not used here, the residual multiple access
References
173
interference (MAI) and ISI must be mitigated by the use of specially designed spreading codes that provide an “interferencefree window” (IFW), where the oﬀpeak aperiodic autocorrelation and crosscorrelation values become zero, resulting in zero MAI and ISI, provided the maximum asynchronous delay is within the IFW [20]. The resulting system extracts full diversity without channel knowledge or complex MUD at the destination or relays.
6.7
Conclusion Distributed STBCs are able to eﬀectively exploit the spatial diversity present in a multirelay network. With a distributed spacetime code, each relay transmits a particular column of a spacetime codeword. The DF strategy is appropriate when there are more relays than there are columns in the spacetime codeword, since only a subset of relays may participate in the transmission of the distributed spacetime codeword, namely those that receive the source’s transmission. However, DF protocols require coordination among the relays to ensure that each relay transmits a speciﬁc column of the spacetime codeword. AF protocols are well suited to the case that the number of relays is equal to the number of columns in the spacetime code, since with AF protocols every relay participates in the transmission of the spacetime codeword regardless of the quality of the source–relay transmission. In addition to the implementation challenges that are common to conventional MIMO systems, the lack of synchronization at the destination receiver imposes additional challenges to systems that use distributed spacetime codes. The synchronization problem can be alleviated by using delay diversity, STS, or delaytolerant spacetime codes.
References [1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal on Selected Areas in Communications, 16, 1998, 1451–1458. [2] M. O. Damen, A. Chkeif, and J. Belﬁore, “Lattice code decoder for spacetime codes,” IEEE Communications Letters, 4, 2000, 161–163. [3] M. O. Damen and A. R. Hammons, “Delaytolerant distributed TAST codes for cooperative diversity,” IEEE Transactions on Information Theory, 53, 2007, 3755–3773. [4] M. ElHajjar, O. Alamri, S. X. Ng, and L. Hanzo, “Turbo detection of precoded sphere packing modulation using four transmit antennas for differential spacetime spreading,” IEEE Transactions on Wireless Communications, 7, 2006, 943–952.
174
Distributed spacetime block codes
[5] A. R. Hammons and M. O. Damen, “On delaytolerant distributed spacetime codes,” in Proc. of IEEE Military Communications Conference (MILCOM), 2007. IEEE, 2007. [6] B. Hassibi and B. M. Hochwald, “Highrate codes that are linear in space and time,” IEEE Transactions on Information Theory, 48, 2002, 1804–1824. [7] H. Jafarkhani, “A quasiorthogonal spacetime block code,” IEEE Transactions on Communications, 49, 2001, 1–4. [8] Y. Jing and B. Hassibi, “Distributed spacetime coding in wireless relay networks,” IEEE Transactions on Wireless Communications, 5, 2006, 3524– 3536. [9] Y. Jing and H. Jafarkhani, “Orthogonal and quasiorthogonal designs in wireless relay networks,” IEEE Transactions on Information Theory, 53, 2007, 4106–4118. [10] J. N. Laneman and G. W. Woernell, “Distributed spacetimecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, 49, 2003, 2415–2425. [11] E. G. Larsson and P. Stoica, Spacetime Block Coding for Wireless Communications. Cambridge University Press, 2008. [12] R. Nabar, H. Bolcskei, and F. Kneubuhler, “Fading relay channels: Performance limits and spacetime signal design,” IEEE Journal on Selected Areas in Communications, 22, 2004, 1099–1109. [13] Y. Shang and X. G. Xia, “Shiftfullrank matrices and applications in spacetime trellis codes for relay networks with asynchronous cooperative diversity,” IEEE Transactions on Information Theory, 52, 2006, 3153–3167. [14] S. Sugiura, S. Chen, and L. Hanzo, “Cooperative diﬀerential spacetime spreading for the asynchronous relay aided CDMA uplink using interference rejection spreading code,” IEEE Signal Processing Letters, 17, 2010, 117– 120. [15] V. Tarokh, H. Jafarkhani, and A. Calderbank, “Spacetime block codes from orthogonal designs,” IEEE Transactions on Information Theory, 45, 1999, 1456–1467. [16] M. Torbatian and M. O. Damen, “On the design of delaytolerant distributed spacetime codes with minimum length,” IEEE Transactions on Wireless Communications, 8, 2009, 931–939. [17] D. Torrieri and M. C. Valenti, “Eﬃciently decoded fullrate spacetime block codes,” IEEE Transactions on Communications, 58, 2010, 480–488. [18] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. [19] K. Vardhe, D. Reynolds, and M. C. Valenti, “The performance of multiuser cooperative diversity in an asynchronous CDMA uplink,” IEEE Transactions on Wireless Communications, 7, 2008, 1930–1940. [20] H. Wei, L. Yang, and L. Hanzo, “Interferencefree broadband single and multicarrier DSCDMA,” IEEE Communications Magazine., 43, 2005, 68– 73.
References
175
[21] S. Wei, D. L. Goeckel, and M. C. Valenti, “Asynchronous cooperative diversity,” IEEE Transactions on Wireless Communications, 5, 2006, 1547– 1557. [22] A. Wittneben, “A new bandwidth eﬃcient transmit antenna modulation diversity scheme for linear digital modulation,” in Proc. of IEEE International Conference on Communications (ICC), 1993. IEEE, 1993.
7
Collaborative relaying in downlink cellular systems Chandrasekharan Raman, Gerard J. Foschini, Reinaldo A. Valenzuela, Roy D. Yates, and Narayan B. Mandayam
7.1
Introduction The deployment of relays in cellular system has been standardized in the WiMAX, IEEE 802.16j [1] standard and is a topic of discussion in the advanced speciﬁcations of Third Generation Partnership Project (3GPP) longterm evolution (LTE) [3]. Although commercial relay deployments in cellular systems are not prominent at present, future wireless cellular systems will involve operation with dedicated relays to improve coverage, increase celledge throughput, deliver high data rates, and assist group mobility. The proposed architecture is such that relays would be placed at certain locations (planned or unplanned) in the cell to help in forwarding the message from the base station to the user in the downlink, and from the user to the base station in the uplink. Relays will be more sophisticated than simple repeaters and could perform some digital base band processing to help the destination terminal get better reception. These relays will rely on air interfaces, and hence avoid the considerable backhaul costs involving data aggregation and infrastructure costs associated with backbone connectivity. However, there are a lot of open issues that require research to answer.
7.1.1
Research challenges Some of the major research issues in relaybased cellular systems are as follows: (1) Throughput gains due to relay deployments In cellular networks that are coverage limited, deploying relays can help in multihop transmission and provide power gains due to a reduction of distance attenuation [4]. These power gains, in turn, translate to throughput improvements for the edge users. However, in interference limited settings, as is common in cellular systems, uncoordinated transmission by relays leads to an increase in the overall interference levels in the cell and could be counterproductive by reducing the signaltointerferenceplusnoise (SINR) levels of users in the system. Coordination of transmissions in the system would require centralized control and incur high costs and overheads, especially in the uplink. Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
7.1 Introduction
177
Thus, there is a need for a thorough evaluation of throughput improvements in a cellular system. In the cellular systems literature, there have been simulation studies to evaluate throughput gains in cellular systems, e.g., [5, 6, 8]. Even though the studies were conducted under diﬀerent sets of (idealized) assumptions, throughput improvements in interferencelimited cellular systems were shown to be around 30–40% for the edge users. In this chapter, we evaluate the gains due to relay deployment by two diﬀerent relaying strategies and the results indicate that throughput gains are of the same order. However, there may exist better practical schemes – which remain open – or speciﬁc scenarios where relays provide larger throughput improvements. A simple case where relays provide throughput improvements is the downlink scenario where the edge user is in a deep canyon and the relay is placed in the line of sight of both the base station and the shadowed user. (2) Relay placement The beneﬁts from relay deployment depend on where the relays are placed in the cell. Throughput improvements depend on the transmit power, relay antenna pattern, and location of the relays in the system. Placing relays closer to an edge user helps the edge user. However, when relay transmissions are uncoordinated, the relays may cause near lineofsight interference to an edge user of the neighboring cell. The optimal relay placement depends on the transmission and scheduling strategies, transmit power of the relays, etc. An issue closely related to the relay placement problem is the choice of height of the deployed relays. In macrocellular environments, propagation characteristics of the base–relay link and the relay–user link could be completely diﬀerent, depending on whether the relays are mounted on tall poles or on roof tops. These factors may very well aﬀect the system performance due to relay deployments. There are not many measurementbased models to cover all the scenarios of relay placement; some empirical models were described in [2]. These issues apart, service providers often do not have much choice in placing the relays in a given geographical area. (3) Lack of good models for relaying in cellular systems Multihopping in wireless networks has been studied in the context of adhoc networks and peertopeer networks [7]. The main issue addressed in such networks is the routing problem. Interference constraints are abstracted as combinatorial constraints and many insightful results and good algorithms have been proposed to improve the throughput of such networks. Cellular networks, however, are unique in that the traﬃc is onetomany in the downlink and manytoone in the uplink. Direct application of the solutions obtained in the context of adhoc networks is not optimal for cellular systems. Hence, performance evaluation of relays in cellular system requires fresh thought to be given to the problem. On the other hand, the information theoretic relay channel [9] has been an active area of research since the 1980s. However, for some coding strategies proposed by Cover and El Gamal in [10] for special cases of the
178
Collaborative relaying in downlink cellular systems
singlerelay channel, the capacity of the general relay channel is still unknown [11]. Though most of the earlier work assumes that the relay can transmit and listen over the same band, the halfduplex constraint (the relay cannot simultaneously transmit and receive in the same band) is taken into account in later work, for example [12, 13]. Information theoretic studies reveal that when there are one or two relays, the best strategy is to make use of both the source and relay transmissions at the user location, rather than multihopping from the source to the destination through the relay(s). The user can make use of signals from both the source and the relay to get a better signal strength and hence a better rate. Multihopping, on the other hand, ignores the signal from the source, however strong it is. The information theoretic relaying protocols mentioned above often involve complicated multiuser coding and decoding techniques, which are far from practical implementation. There has been some work trying to bridge the gap between the information theoretic and practical multihopping schemes, e.g., [14, 15]. Most of the results in these works correspond to the case of a linear network of nodes, where there is a single commodity ﬂow of message from the source node to the sink node through a set of relay nodes. Any interference is due only to simultaneous transmissions from diﬀerent relay nodes. This can be completely eliminated by multiuser coding/decoding techniques. Such an analysis does not carry over directly to the cellular systems since there are multiple simultaneous ﬂows and multiuser techniques may incur signiﬁcant overheads. (4) Fairness Service level agreements between cellular service providers entail certain fairness requirements. For example, in the cellular system, the edge users and the users near the cell require the same level of service. Many other fairness schemes, including proportional fairness [16] and max–min fairness [17, Chapter 6] have been proposed for cellular systems serving voice and data. Presentday cellular systems implement schedulers in the MAC layer to provide various degrees of fairness to users. In this work, we assume that 90% of users are required to be served at a common rate. When relays are present in the system, designing distributed scheduling schemes to provide fairness is an active area of research. In this chapter, we evaluate the performance of lowcost halfduplex relays in the downlink of a cellular system. The deployment scenario we consider is to mount a lowcost (preferably lowpowered) device per sector over rooftops of buildings. Such devices can relay the information from the base station to users in the cell.
7.1.2
Related work Relay deployment in a cellular system has been proposed to solve the issue of a lack of coverage over a large area [1]. The use of relays in cellular systems has also
7.1 Introduction
179
been proposed to bring capacity improvements [18]. Viswanathan and Mukherjee [5] studied the performance of a centralized throughputoptimal scheduler on a cellular network with relays. They presented a centralized downlink scheduling scheme that guarantees the stability of user queues for the largest set of arrival rates into the system. Each user has a queue at the base station and a queue at its serving relay and the objective of the scheduler is to stabilize both queues while maximizing the throughput. The throughput results obtained by simulations in [5] suggest that simultaneous transmissions (due to multihopping) exploiting spatial reuse could lead to cellwide throughput gains in a cellular network.
7.1.3
Overview of contribution In addition to a multihopping model, wherein the message travels to the destination in two hops, in this chapter we evaluate the performance of a collaborative power addition (CPA) scheme with a single relay available per user. We bring an additional dimension to the beneﬁts of relays in a cellular system, by quantifying the power savings due to the deployment of relays. Peak power savings in cellular networks are very important elements of ampliﬁer costs in base stations. Significant peak power savings can reduce the cost of ampliﬁers and hence the capital expenses for deploying cellular networks. Also, average power savings while operating cellular networks can save operational expenses such as electricity bills for the cellular operators. In the peak CPA (PCPA) scheme, we ﬁrst consider a hypothetical model where 90% of the users are required to be served a ﬁle (henceforth, we use the terms message and ﬁle interchangeably) of a ﬁxed size before a certain deadline. Depending on the interference seen by each user, the mutual information (MI) or the instantaneous “rate” of the users varies over time. Users leave the system as they get the complete ﬁle before the deadline. We use an oﬄine computation to ﬁnd the 10% of users who are least able to get the whole ﬁle before the deadline and discard them at the beginning. We run the real system without the users in outage. When relays are present in the system, we evaluate the peak power savings at the base station to deliver a ﬁle of the same size to the same number of users in the system as when the base stations and relays are transmitting at their peak power limits. We also ﬁnd the improvements in common rate for the users in the presence of the relays when the peak power of the base stations is ﬁxed for both the baseline and the system with relays. We then consider the power control capability to the base stations and relays. We evaluate the power savings and throughput improvement in the power control CPA scheme (PCCPA). For a desired common rate requirement for 90% of users in the system, we ﬁnd the common peak power constraint in the baseline case and in the system with relays to guarantee the common rate. When the relays get the complete message, they collaborate with the base station to transmit the message to the users. Each time a relay becomes eligible to transmit, the optimal set of powers are found to satisfy the desired rate requirement. The
180
Collaborative relaying in downlink cellular systems
set of users that violate the peak power constraint are discarded at the beginning. The improvements in common rate are also evaluated through a similar procedure. The second relaying scheme we evaluate is simple multihopping, wherein the base stations and relays transmit in orthogonal time slots. The baseline system is similar to the baseline in the PCPA scheme described above. When relays are present in the system, we simulate a timeslotted system. For a common rate requirement for 90% of users in the system, the base stations transmit at peak power in odd time slots. The relays and users are in the receive mode. In the even time slots, the base stations are turned oﬀ and relays transmit to the respective users. The relays employ power control to target the remainder of the user population to provide the residual rate to the users. We describe the system in detail in Section 7.2. We do not consider multiuser scheduling gains, MIMO gains, or any other complex interference mitigation techniques. Thus the gains shown in the network arise purely from the power gains at the user location due to the relay transmissions.
7.2
System model Our work evaluates the power savings and improvement in common rate among users due to relay deployments in a cellular system. However, to model and simulate all dynamics of a cellular system may be too complicated. In order to overcome such diﬃculties, we make some reasonable simplifying assumptions and take an idealized look at the model and operation of a cellular system. In order to make a fair comparison, the assumptions are kept consistent across systems with and without relays. We consider a cellular system with idealized hexagonal cells with a base station at the center of each cell. The topology is shown in Figure 7.1. The ﬁrst two tiers of interferers are considered and the activities of the farther tiers of cells are mirrored by the central ring of 19 cells. The sitetosite distance (distance between any two base stations) is taken to be 1 mile. The cells are divided into 120 degree sectors, with each sector illuminated by a base station antenna pattern given by 2 θ (7.1) , Amax A(θ) = − min 12 θ3dB where A(θ) is the antenna gain in dBi in the direction θ, −180 ≤ θ ≤ 180, min(.) denotes the minimum function, θ3dB = 70 degrees is the 3 dB beamwidth and Amax = 20 dB is the maximum attenuation. The antenna gain pattern is shown in Figure 7.2. At the receiving terminal (relay or user), the transmitted power undergoes attenuation due to the distance traveled and shadowing eﬀects around the
7.2 System model
181
Figure 7.1. Wraparound simulation model. The central ring of 19 cells is used for the simulation. The surrounding cell activity is mirrored in the center ring. The directions of the arrows represent the directions of the main lobe of the sectorized antenna.
receiver. The propagation attenuation between a transmitting terminal (base station or relay) and a receiving terminal (relay or user) consists of the pathloss and the shadowing component. At any receiving terminal, the transmitted power is attenuated in dB as PL(d) = −31.5 − 38 log10 d, where d is in meters. The shadowing is modeled as lognormal with mean 0 dB and a standard deviation of 8 dB. The shadowing is assumed to be spatially uncorrelated and ﬁxed for a given set of user locations. The base station and the relay antenna gains are taken to be 15 dB (at zero degree horizontal angle) and user antenna gain as −1 dB. Other losses account for 10 dB. Together with the above losses, we include the antenna pattern loss to calculate the received power. The receiver noise ﬁgure is set at 5 dB, and the thermal noise power at each receiving terminal
182
Collaborative relaying in downlink cellular systems
Figure 7.2. Antenna gain pattern (from [2]) as a function of the horizontal angle in degrees. The mathematical expression for the gain is given in (7.1). (relay or user) is assumed to be −102 dBm. The eﬀect of multipath smallscale fading is ignored in our simulations. All users share the same band of frequencies and hence simultaneous transmissions can interfere with each other. The total interference at each receiving terminal from all transmitters in the system is modeled as Gaussian noise and assumes that other users use Gaussian codebooks. The achievable rate to a user i at time t is calculated as the Shannon rate Ri (t) = log2 (1 + ρi (t)),
(7.2)
where ρi (t) denotes the SINR for user i at time t The parameters used in the abovementioned simulation setup are summarized in Table 7.1. We use this simulation setup to evaluate all relaying methodologies proposed in this chapter. We simulate a downlink OFDMlike system wherein users in orthogonal time or frequency slots do not interfere with each other. However, users in the same resource unit interfere with the other transmissions in the band. We simulate the worstcase scenario where the system is fully loaded, i.e., users are present in all available resource units (or time–frequency slots) in all the sectors. The time– frequency slots are reused in each sector. We assume that the time–frequency slots are orthogonal, and focus only on a particular time–frequency slot within which we simulate the complete cellular system such that there is one active user per sector at a given time. Hence, in a 19cell network with three sectors
7.2 System model
183
Table 7.1. Simulation parameters Network topology Sitetosite distance Bandwidth Pathloss model Pathloss exponent Shadowing Multipath fading Antenna pattern Antenna gains Other losses Thermal noise power at the receiver Outage
19 cells, three sectors per cell with wraparound 1 mile 5 MHz COST231 Hata model α = 3.8 Lognormal, with zero mean, 8 dB standard deviation for access and backhaul None Sectorized for base stations; omnidirectional for relays 15 dB (for base station and relays); −1 dB for users 10 dB −102 dBm 10% for baseline and relays
per base antenna, at most 57 users are served in a given time–frequency slot. In our simulations, we use the following heuristic to create a random user population along with an association rule. Users are placed onebyone in a uniformly random fashion across the network until all 57 base station sectors are occupied. For each random realization of a user location, the base station sector with the highest received signal strength is chosen to associate with the user. If the base station sector is already occupied by another user, the user is not allowed into the system and a new user location is generated. Along with a random realization of a user location, independent lognormal random variables are also instantiated to account for the shadow fading gains between each base station and the user in the baseline system. If relays are present in the system, the fading gains are also generated for base station–relay links and relay–relay links. In this way, random placement is carried out until all 57 sectors are occupied by exactly one user per sector. Each user is equipped with an omnidirectional antenna. A relay with an omnidirectional antenna is placed in the direction of the main lobe of each base station sector antenna as shown in Figure 7.3. The relays always associate with the corresponding base station sector. The relay placement is an important parameter to be considered since the power gains and throughput improvements depend on the interference generated by the relays, which in turn depends on the transmit power, geographic location of the relays, and the propagation environment. In our simulations, we experiment with various relay placements and the simulation results are presented for the relay locations for which the gains are found to be maximum. The relay powers are also varied so that we get the maximum peak power savings.
184
Collaborative relaying in downlink cellular systems
Figure 7.3. Position of relay location in a cell. The relays (represented by small circles) are placed at half the cell radius in the direction (given by the arrows) of the main lobe of the sector antenna. The base station at the center of the cell is represented by a square.
7.3
Collaborative relaying in cellular networks In the CPA scheme devised in [19], the relay collaborates with the base station to help the message reach the destination. In our simulation model, each base station sector has a single user to be served and a relay that may help the source to deliver the message to the user associated with the source. In what follows, we focus our attention on an isolated triplet of base station (source), relay, and user in a single sector. Gaussian encoding is used across all other sectors, the interference from other sectors is considered as if it were additive Gaussian noise. Suppose the source wants to transmit one of the M messages to the destination, under a power constraint P . The source transmits a Gaussian codeword of length N = (log M )/R, where R is the rate of the code. By Shannon’s channel coding theorem [20, Chapter 9], if N is large enough, the message can be decoded reliably at the destination provided R < log(1 + ρ), where ρ is the received SINR. In our simulations, we are interested in achievable rates and assume that the instantaneous mutual information at the receiver is exactly R = log(1 + ρ). Assume that the source picks a rate R code C1 and sends one of M equally probable messages to the destination, using a codeword of length N . Let the received SINR ρS R between the source and relay be greater than the received SINR ρS D at the destination. Then, there exists some β > 1 such that log(1 + ρS R ) = β log(1 + ρS D ),
(7.3)
i.e., the capacity of the channel from source to relay is β times greater than that of the channel from source to destination. We can now construct codebook C2 derived from C1 by observing only the ﬁrst N/β symbols of every codeword. The relay can then reliably decode the received message since the rate of C2 is R =
log M < log(1 + ρS D ). N/β
(7.4)
7.3 Collaborative relaying in cellular networks
Source
Relay
185
Destination
Figure 7.4. Collaborative relaying: before the relay decodes the message.
Source
Relay
Destination
Figure 7.5. Collaborative relaying: after the relay decodes the message.
In [21, Appendix F], the authors discuss the coding interpretation of a similar collaborative strategy. They also discuss the connection of such a coding setting with coding for an arbitrary varying channel (AVC), which was ﬁrst dealt with in [22] and subsequently studied in [23]. We simulate a similar collaborative coding strategy wherein before the relay decodes the message as shown in Figure 7.4, the received power at the destination node is only due to the base station transmission. After it decodes the message, the relay joins the base station to help the base station in delivering the message to the destination as shown in Figure 7.5. At this point, if we assume that transmit symbol time slots at the relay and base station are synchronized and the code books are shared, the system can be viewed as a 2 × 1 multipleinput singleoutput (MISO) system without channel information at the transmitter. There is an eﬀective power addition of the base station and relay transmissions at the destination [24, Chapter 3]. A similar scheme has been proposed in the literature as the dynamic decodeandforward (DDF) scheme [25]. We simulate this collaborative relaying strategy in two ways:
r Base station and relay transmit at their respective peak powers. In this case, the transmit power is ﬁxed and the users get variable rates depending on SINR at the user locations. When a target rate is obtained by a user, the user leaves the system and the corresponding base station sector is turned oﬀ, thus reducing the amount of interference in the system. We term this
186
Collaborative relaying in downlink cellular systems
the peak collaborative power addition (PCPA) scheme. This is described in Section 7.4. r Base station and relay operate with power control so that the users obtain a target desired rate. In the baseline case, for a given desired rate requirement r0 bps/Hz, a feasible set of powers are found to better satisfy the rate requirement, allowing certain users to be in outage. When the relays decode the message in the collaborative scheme, the optimal powers are recalculated to ﬁnd another feasible set of powers to satisfy the rate requirement at the same outage level. This is the PCCPA scheme. It is described in Section 7.5.
7.4
CPA with peak power transmissions (PCPA)
7.4.1
Principle of operation PCPA baseline In the baseline of the PCPA scheme, each base station sector transmits at its peak power to its own intended user. Since all users share the same band of frequencies, they receive interference from all the base station sectors in the system. If at time t, pi (t) is the peak power of the transmitting base station sector corresponding to the ith user, hij is the channel gain, including pathloss and shadowing, from the jth base to the ith user, and σ 2 is the variance of the noise power at the receiver, the instantaneous received SINR for user i is given by hii pi (t) . 2 j = i hij pj (t) + σ
ρi (t) =
(7.5)
Since we assume Gaussian signaling, the mutual information (MI) or the instantaneous “rate” to each user is Ri (t) = log2 (1 + ρi (t)) bits/symbol.
(7.6)
At time t = 0, all base stations simultaneously transmit to their associated user. As time progresses, for any small time interval [t, t + ∆t], user i accumulates MI Ii (∆t) = Ri (t)∆t. The MI for user i at time t is given by . Ii (t) =
t
log2 (1 + ρi (ξ)) dξ.
(7.7)
0
If user i accumulates MI corresponding to the required amount L of data before the deadline T , i.e., τi = min {t : Ii (t) = L}, 0≤t≤T
(7.8)
7.4 CPA with peak power transmissions (PCPA)
187
then the user leaves the system and the associated base station sector is turned oﬀ at time τi , reducing the overall interference levels in the system. Hence, * pi (t) =
P, t < min(τi , T ), 0, t ≥ min(τi , T ),
(7.9)
where P is the peak power of the base station transmission. Note that the ρi (t) of user i and the rate Ri (t) are timevarying quantities. At time t = T , the users that remain in the system are those users that did not get the complete ﬁle. It is these remaining users that are in outage.
PCPA system with relays The operation of the PCPA system with relays is as follows. The requirement is the same as for the baseline case: to deliver a ﬁle of size L to as many users within the time T . At time t = 0, the base stations transmit at peak power to users associated with them. The relay node placed in the sector also receives the data sent to the user by the base station. If the relay receives the complete ﬁle before the user does, the relay can potentially be useful to the user by helping it receive the message faster. On the other hand, the relay transmission can create additional interference for the other users in the system. In our simulations, we follow a myopic1 policy on whether to turn on the relay or not: the relay transmits at peak power to help its user only if the instantaneous sumrate of the whole system increases by turning the relay on. The sumrate of the system is calculated as the sum total of the instantaneous rates of the existing users in the system and is a natural systemwide metric to use in order to decide whether the relays should transmit or not. At every epoch, one relay among the set of all relays that are eligible to be turned on, receives the message, the myopic sumrate metric is applied, and those relays that increase the sumrate are turned on to help the users in the system. If the relay increases the sumrate of the system, the relay is turned on and helps the user with a transmission reinforcing the same message as the base station using the code described in Section 7.3. If qi (t) is the power transmitted from relay i at time t and gij is the channel gain from user i to relay j, the eﬀective SINR at ith user location when the relay is active is given by
ρri elay (t) =
1
hii pi (t) + gii qi (t) . 2 j = i hij pj (t) + gij qj (t) + σ
(7.10)
The policy is myopic since, at the time when the relay receives the message, the global optimal decision whether the relay should transmit or not is unknown.
188
Collaborative relaying in downlink cellular systems
The instantaneous rate and the mutual information for user i at time t are given by Rir elay (t)
= log2 (1 + ρri elay (t)), . t = Rir elay (ξ)) dξ.
Iir elay (t)
(7.11) (7.12)
0
If Hij denotes the channel gain from the jth base station to the ith relay, . t Hii pi (ξ) dξ (7.13) log2 1 + Ji (t) = 2 0 j = i Hij pj (ξ) + σ represents the cumulative MI at the relay at time t. Suppose relay i becomes eligible to transmit at time t, i.e., Ji (t) > L, then denote the sumrate of the system at time t as a function of qi (t) as hii pi (t) + gii qi (t) . (7.14) log2 1 + SR(t, qi (t)) = 2 j = i hij pj (t) + gij qj (t) + σ i The relay power at time t is given by * Q, if Ji (t) > L, SR(t, Q) > SR(t, 0) and t < T, qi (t) = 0, otherwise,
(7.15)
where Q is the peak power constraint of the relays. Each user sees a timevarying SINR and the timevarying rate given by Ri (t) = log2 (1 + ρri elay (t)). As with the baseline case, for any interval of time [t, t + ∆t], user i accumulates MI amounting to Ii (∆t) = Ri (t)∆t and the MI for user i at time t is . t Ii (t) = log2 (1 + ρri elay (ξ)) dξ. (7.16) 0
Similarly to the baseline case, if the user accumulates MI amounting to the full ﬁle size L within the stipulated time T , the user leaves the system and the associated base station and relay are switched oﬀ. Thus the eﬀective interference in the system is reduced. At time t = T , the users that remain in the system are those users that did not receive the complete ﬁle.
7.4.2
User discarding methodology The user discarding procedure can be divided in two phases: (i) A learning phase where we learn the power threshold, which is used as a criterion to determine the users in outage. All the users that are not in outage require their corresponding base stations to have peak powers lower than the power threshold. The network will be operated with peak powers of all base stations capped at the power threshold. (ii) After the power threshold is found, we assume the availability of a very fast computing facility and perform an oﬀline computation to ﬁnd out the
7.4 CPA with peak power transmissions (PCPA)
189
set of users that are in outage. Such users could be discarded before the start of the simulations so that the other users beneﬁt from the absence of interference from these users. Hence, this takes care of the causality of the discarding phase from the operation of the real network. We conduct Monte Carlo simulation runs for the baseline case as well as the case with relays in this chapter. For each simulation run random instantiations of 57 user locations (as per Section 7.2) and associated statistically independent shadow fading values are generated. Once the random values are instantiated, they are stored in our simulation software program. The same set of user locations and shadow fading values serve as inputs to the baseline and the system with relays. We now explain the learning phase. Consider that a single instance of the simulation runs in the baseline case. For the given instantiation, there are 57 users, one in each sector. We ﬁx a peak power threshold P for the base stations and also ﬁx the desired common rate for users as r0 bps/Hz. When the baseline system operation is over, the users that are in outage remain in the system at time T . Let the number of users in outage for the kth instantiation when the power threshold is P and desired common rate r0 be Ok (P, r0 ). For the same power threshold P and common rate r0 , we run a large number K of instantiations. We then ﬁnd the total number of users in outage as O(P, r0 ) =
K
Ok (P, r0 ).
(7.17)
k =1
The percentage of users in outage for the threshold P and desired rate r0 is then O(P, r0 ) × 100%. 57K
(7.18)
If O(P, r0 ) > 10%, we increase the power threshold to P > P . On the other hand, if O(P, r0 ) < 10%, we decrease the power threshold to P < P . Proceeding in this fashion, the base station peak power thresholds are adjusted such that exactly 90% of the users are guaranteed the desired rate of r0 bps/Hz and the remaining 10% of the users are in outage. We could improve the performance of the system by discarding the users in outage upfront, since the interference due to the presence of these users will be eliminated at time t = 0. In our simulations, for a large user population over K instantiations, we identify 10% users in outage2 by ﬁrst running K instantiations of the system with all the users present in the system. We store the coordinates of all the users that were in outage at the end of each of the K instantiations. We then eliminate the outage users from the system (by preserving the coordinates of the user locations of only those users not in outage for all the K instantiations) 2
We remark that the eliminated set of 10% of the users in outage is not claimed to be the optimum set as would be obtained by evaluating all possible subsets amounting to 10% of the users. The latter is computationally prohibitive.
190
Collaborative relaying in downlink cellular systems
at time t = 0 in the real network simulations. Thus, the existing users in the system experience less interference due to the absence of those users in outage when the real network is simulated.
7.4.3
Network operation and simulation aspects Our objective is to obtain power savings and throughput improvement due to deployment of relays in a cellular system. To compare systems with and without relays in the CPAbased relaying scheme, we simplify the operation of a cellular downlink system such that 90% of the users in the system are guaranteed to be delivered a ﬁle of ﬁxed size L, within a ﬁxed period of time T . The ﬁle could be diﬀerent for all users but the ﬁle sizes are ﬁxed. Such an operation brings in the notion of a common rate for the users in the system. In order that the system beneﬁts from the users that receive the message within the ﬁxed time T , the satisﬁed users leave the system, and thus no longer cause interference to the remaining users. The remaining 10% of the users that are not guaranteed the ﬁle of size L are in outage. In our simulations, for the sake of simplicity, all base stations are assumed to have the same peak power threshold values. We run K = 200 (amounting to 11 400 user instantiations) diﬀerent user instantiations in the system. The common rate requirement is set as 1 bps/Hz. We divide the total time T into 1000 minislots and at the end of each minislot, we keep track of the cumulative MI Ii (t) of each user i. If at the end of a minislot, a particular user’s cumulative MI exceeds the ﬁle size L, the base station corresponding to that user is turned oﬀ. We run the baseline for diﬀerent peak power values of the base station (5 W to 30 W in increments of 5 W). For each peak power value, the relay powers are varied as a factor of the base station power. Figure 7.6 shows the variation of outage probability for various base station powers and various relay powers. For the case when there are no relays in the system (ratio of relay power to base station power is zero), increasing the peak powers of the base station decreases the outage. The percentage of outage saturates below a certain threshold as the interference limit sets in. As we increase the relay powers by increasing the ratio of relay power to base station power, the outage reduces but quickly saturates to a certain threshold outage value, because of the interference limit. From Figure 7.6 it is clear that the interference limit is quickly reached and this limits the performance of a system with relays. This is because there is no interference management and peak power transmissions from the base stations and relays lead to a highly interference limited scenario.
7.4.4
Simulation results Power savings The peak power required to guarantee the remaining 90% of the users (after the 10% of users in outage have been removed) a rate of 1 bps/Hz is 21 W in the
191
7.4 CPA with peak power transmissions (PCPA)
Table 7.2. PCPA relaying (base station and relays transmit at peak power) Peak power required to guarantee 1 bps/Hz at 10% outage; baseline (no relays)
Peak power required to guarantee 1 bps/Hz at 10% outage; with relays
Savings in dB
21 W
15 W
1.46
Common rate for 90% users; baseline (no relays)
Common rate for 90% users; with relays
Percentage rate increase
1 bps/Hz
1.21 bps/Hz
21 %
14 Base station power = 15 W Base station power = 20 W Base station power = 25 W
13
Outage percentage at 1 bps/Hz
12 11 10 9 8 7 6 5
0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Ratio of relay power to base station power
0.8
0.9
1
Figure 7.6. Variation of outage with relay powers and base station powers. As we increase the base station powers with no relays in the system (ratio = 0), the outage decreases and saturates at around 5%, due to the inteference limit. The interference limit sets in very quickly even for smaller values of relay powers. baseline case and it requires 15 W for a system with relays. The relays transmit 1 W of peak power. Hence the peak power saving at the base station locations in this case is 1.46 dB as shown in Table 7.2.
Rate gains In order to evaluate the throughput improvement, we ﬁnd how much the common rate of 90% of users can be improved with the peak power of the base stations
192
Collaborative relaying in downlink cellular systems
being ﬁxed. For the baseline, we ﬁx the power of the base stations to 21 W, so that 90% of the users are guaranteed to receive 1 bps/Hz (as obtained in the previous section); 10% of the users in outage are eliminated as explained in Section 7.4.2. For the PCPA system with relays, the peak power threshold of the base stations is ﬁxed to 21 W (the same value as in the baseline case). For the same peak power for the base stations and with relays present in the system, we expect the common rate to be better than 1 bps/Hz. To ﬁnd the improvement in common rate, we ﬁx a desired common rate r > 1 bps/Hz and run the sytem with relays. If this desired common rate is feasible,3 we double the desired common rate and run the simulations again. Or, if the desired common rate is not feasible, we ﬁx the new desired common rate at half the diﬀerence between the highest feasible common rate and the lowest not feasible common rate and rerun the simulations. In this manner, we converge to the achievable common rate in the presence of relays. In our simulations, we ﬁnd that the common rate can be improved to 1.21 bps/Hz in the CPAbased relaying scheme. Hence the common rate improvement is 21%.
7.5
Powercontrolbased collaborative relaying (PCCPA) In Section 7.4.3, we observed that for PCPA the interference from the other relays and base station sectors limited the peak power savings in the system with relays. The reason for that is that when the relays transmit to help the users, they transmit with peak powers and hence increase the interference levels in the system. If we could ﬁnd the optimal set of powers to transmit for the base station and the relays, we could reduce the overall interference levels in the system. This may improve the gains in the system. In the following, we describe a framework for power control in the downlink of a cellular system with relays. When the relays are not present, downlink power control in a cellular system has been well studied and understood [26]. When relays are present in the system, power control, if performed jointly at the base stations and relay locations, can provide power savings and throughput improvement. We describe the PCCPA relaying scheme in the following sections.
7.5.1
Principle of operation PCCPA baseline For the PCCPA baseline, the aim is to deliver a desired common rate for 90% of the user population by employing a simple power control scheme. Each base station sector powers down its transmitted power within the peak power limitations so that 90% of users are guaranteed a desired common rate of r0 bps/Hz. 3
The common rate is feasible if all the users present in the system are able to get the desired rate.
7.5 Powercontrolbased collaborative relaying (PCCPA)
193
Since all users share the same band of frequencies, they observe interference from all the base station sectors in the system. We use a common subscript for a base station or a user in a particular sector. If at time t, pi is the power of the transmitting base antenna corresponding to the ith user (we drop the argument t), hij is the channel gain, including pathloss and shadowing, from the jth base to the ith user, and σ 2 is the noise power at the receiver, the instantaneous SINR of the ith user in the system is given by ρi (t) =
hii pi . 2 j = i hij pj + σ
(7.19)
Since the transmission uses Gaussian codebooks, the corresponding instantaneous rate for the user i is given by Ri (t) = log2 (1 + ρi (t)) bps/Hz.
(7.20)
The set of feasible powers such that the users not in outage are guaranteed a rate r0 is obtained by solving for the feasibility of instantaneous rates subject to peak power constraints, speciﬁed by log2 (1 + ρi (t)) i.e., ρi (t) subject to: pi
≥ r0 , ≥ 2
r0
(7.21) − 1,
≤ pi,max
(7.22) (7.23)
for all users i not in outage. In practice, each base station increases its power autonomously in small increments, until it hits the peak power limit or until the user associated with it attains the desired rate r0 . Users that achieve the power constraint to go active before attaining the desired rate are discarded. We simulate the system without the users in outage such that all users achieve the desired rate. Since the transmit powers of the base stations are such that all users attain a common rate, none of the users leave the system.
PCCPA system with relays In the PCCPA system with relays, 10% of users are discarded in a manner similar to that in the baseline system. At time t = 0, the relays do not have the complete message required to relay to the user. Hence, the system starts out as it does for the baseline case. The base stations increase their powers autonomously in small increments targeting the user rates to increase. Users that do not meet the peak power constraints in (7.23) are eliminated one after the other. The remaining users attain the desired rate without violating the peak power constraint at the base stations. While the base station transmissions are targeted to the users, the relay in each sector also listens to the transmission by the base stations. Depending on the channel conditions and coupling of interference from the adjacent sectors, the relays receive their message at diﬀerent points in time. When the relay in the sector decodes the message from the base station, the relay collaboratively helps the base station such that the user gets a rate corresponding to the total SINR from the relay and the base station. As described
194
Collaborative relaying in downlink cellular systems
in Section 7.3, the code books at the base stations and relays are designed such that the mutual information at the receiver corresponds to the sum of received powers at the user location [19]. In order to maintain the common desired rate for all users, the relay and base station jointly adjust their powers so that the user receives the desired rate. This ensures that the base station and relay transmit just enough power to the user to obtain the desired rate. Let pi denote the power transmitted by the base station sector i at time t and hij be the channel gain, including pathloss and shadowing, from the jth base to the ith user. Let qi be the power transmitted from the relay i at time t and gij be the channel gain to the user i from the relay j. Then, when the relay and base station transmit simultaneously, the eﬀective SINR at the ith user location when the relay is active is given by ρri elay (t) =
hii pi + gii qi . 2 h j = i ij pj + gij qj + σ
(7.24)
As with the baseline case, the set of feasible powers (for both base station antennas and relays) such that the users not in outage are guaranteed a rate r0 is obtained by solving for the feasibility of instantaneous rates subject to peak power constraints log2 (1 + ρri elay (t)) i.e.,
ρri elay (t)
≥ r0 , ≥ 2
r0
(7.25) − 1,
(7.26)
subject to: pi
≤ pi,m ax
(7.27)
and qi
≤ qi,m ax
(7.28)
for all users i not in outage. If we consider the transmit powers of the base stations and relays to be variables of optimization, we have a total of 2N variables, for N base station sectors in the system. Thus, power control in cellular systems in the presence of relays gives us an additional N degrees of freedom over which to optimize. The transmit powers in the system can be optimized to reduce the maximum peak power transmission in the system, to reduce total energy in the system, etc. In what follows, we assume that a central controller has knowledge of all the channel gains between the base stations as well as relays and the users. We explain ways to achieve various aforementioned objectives using linear program (LP) formulations.
7.5.2
Optimization framework Minimizing the total instantaneous transmit power We are interested in evaluating the beneﬁts of relays in minimizing the total instantaneous sum power in the system while delivering the common rate r0 with 10% of the users being omitted from the system. The practical beneﬁt of minimizing the total sum of transmit powers in a cellular system is to save the
195
7.5 Powercontrolbased collaborative relaying (PCCPA)
energy costs in the network. Saving energy costs translates to saving electricity bills at the cell sites for the cellular service provider. The desired common rate for the users is ﬁxed at r0 bits/symbol. We deﬁne A(t) as the set of all active relays at time t, i.e., the set of relays that have obtained the message and are ready to help the base station. Ac (t) denotes the complementary set of all inactive relays. For simplicity, we drop the argument and write pi and qi for the base station powers pi (t) and qi (t) at time t, respectively. At a given time t, the central controller solves the following optimization problem: min pi + q i , (7.29a) p ,...,p 1
N
q 1 ,...,q N
subject to:
i
log2
hii pi + gii qi 1+ 2 h j = i ij pj + gij qj + σ
≥ r0 , i = 1, . . . , N,
(7.29b)
0 ≤ pi ≤ pi,m ax , i = 1, . . . , N,
(7.29c)
0 ≤ qi ≤ qi,m ax , i ∈ A(t),
(7.29d)
qi = 0, i ∈ A (t).
(7.29e)
c
The solution to the optimization problem (7.29), p∗i , qi∗ , i = 1, . . . , N deﬁnes the powers pi (t) = p∗i and qi (t) = qi∗ that are used at time t. The optimization problem (7.29) is an LP, since we can write the constraint (7.29b) as 1 (hii pi + gii qi ) − (hij pj + gij qj ) ≥ σ 2 (7.30) r 0 2 −1 j = i
for i = 1, . . . , N . Rewriting (7.29) in vector form, we have s∗ (t) = min p,q
subject to:
where
⎛
⎜ ⎜ A=⎜ ⎝ ⎛ ⎜ ⎜ B=⎜ ⎝
1T (p + q),
(7.31a)
Ap + Bq ≤ −σ 2 1,
(7.31b)
0 ≤ p ≤ pmax ,
(7.31c)
0 ≤ q ≤ qmax ,
(7.31d)
−h11 /(2r 0 − 1) h21 .. .
h12 −h22 /(2r 0 − 1) .. .
··· ··· .. .
h1N h2N .. .
hN 1
hN 2
···
−hN N /(2r 0 − 1)
−g11 /(2r 0 − 1) g21 .. .
g12 −g22 /(2r 0 − 1) .. .
··· ··· .. .
g1N g2N .. .
gN 1
gN 2
···
−gN N /(2r 0 − 1)
⎞ ⎟ ⎟ ⎟ , (7.32) ⎠ ⎞ ⎟ ⎟ ⎟, ⎠
(7.33)
196
Collaborative relaying in downlink cellular systems
and p(t)
=
[p1 (t) . . . pN (t)]T ,
q(t)
=
[q1 (t) . . . qN (t)]T ,
pmax
=
[p1,max . . . pN ,max ]T ,
qmax
(7.34)
T
=
[q1,max . . . qN ,max ] ,
with qi,max = 0, i ∈ Ac (t). The solution to the above LP provides the optimal power values that minimize the instantaneous total power in the system. We take a myopic approach of minimizing the total sum power of the system at time t in order to reduce the total average power transmission in the system. Each time a relay becomes eligible for transmission, the LP is solved to ﬁnd the best set of powers by minimizing the instantaneous powers in the system. Note that in some cases, when a relay is eligible to help the base station, turning oﬀ the base station may be the optimal thing to do. This choice comes out as a solution to the optimization program.
Minimizing the peak transmit power Minimizing the peak transmit power leads to peak power savings in the system. A practical beneﬁt of peak power savings is the signiﬁcant savings in the cost of power ampliﬁers for the cellular base stations. If by deploying lowpower relays in the system, we save on the cost of the power ampliﬁers of the base stations, cellular operators could save on capital expenses. To this end, we solve the following optimization problem of minimizing the maximum instantaneous transmit powers at the base stations: min
p 1 ,...,p N q 1 ,...,q N
subject to:
max pi , i
(7.35a)
log2
hii pi + gii qi 1+ 2 j = i hij pj + gij qj + σ
≥ r0 , i = 1, . . . , N,
(7.35b)
0 ≤ pi ≤ pi,max , i = 1, . . . , N,
(7.35c)
0 ≤ qi ≤ qi,max , i ∈ A(t),
(7.35d)
qi = 0, i ∈ A (t).
(7.35e)
c
Rewriting the above LP in vector form yields, p∗ (t) = min p,q
subject to:
α,
(7.36a)
Ap + Bq ≤ −σ 2 1,
(7.36b)
0 ≤ p ≤ pmax ,
(7.36c)
0 ≤ q ≤ qmax ,
(7.36d)
α1 ≥ pmax ,
(7.36e)
where A and B are given by (7.32) and (7.33), respectively.
7.5 Powercontrolbased collaborative relaying (PCCPA)
197
Improving the common rate As a corollary to the above approaches, if we keep the peak power constant across both the baseline and the system with relays, we can increase the common targeted rate in the system with relays. The problem of maximizing the common rate can be posed as an optimization program with the transmit powers of the base station and relays as the variables. A central controller then solves the optimization program: max
p 1 ,...,p N
subject to:
r0 ,
(7.37a)
log2
hii pi + gii qi 2 j = i hij pj + gij qj + σ
1+
≥ r0 , i = 1, . . . , N,
(7.37b)
0 ≤ pi ≤ pi,max , i = 1, . . . , N,
(7.37c)
0 ≤ qi ≤ qi,max , i ∈ A(t),
(7.37d)
qi = 0, i ∈ A (t).
(7.37e)
c
The optimization program can be viewed as a sequence of linear feasibility problems because the constraint set is nonconvex. We solve this program by an iterative approach. We start with a low easily achievable target rate r0 so that the constraint set (7.37b)–(7.37e) is feasible. We increase the target rate in small increments until the constraint set becomes infeasible. In each step, we get a set of feasible power assignments. The last set of feasible power assignments is the solution to the optimization program. The method converges, since the iterations generate a bounded sequence of increasing rates.
7.5.3
User discarding methodology In our simulations, we eliminate 10% of users (over a large number of user realizations) in the following way. The procedure is identical for both the baseline and the system with relays. We simply assume the same peak power constraints for all base stations across the network. We ﬁx the peak power threshold pmax for each base station. Consider a single instantiation, where there are 57 users in the system. We increase the transmit power in all base stations in small incremental steps to improve the rate of the users in the system. As we do so, we discard the user associated with the base station whose power constraint goes active ﬁrst. The base station is also turned oﬀ. This reduces the interference coupled with other users. Within the remaining set of users, we can increase the transmit powers further. We then discard the next user causing the power constraint to become active and continue in this fashion until all remaining users in the system are guaranteed the desired rate of r0 bps/Hz, without violating the peak power constraints. This procedure is repeated for a large number K of users instantiations. Let the number of users in outage for the kth instantiation when the power threshold is pmax and desired common rate r0 be Ok (pmax , r0 ).
198
Collaborative relaying in downlink cellular systems
We then ﬁnd the total number of outage users for K instantiations when the peak power threshold is pmax and the desired rate is r0 is calculated as O(pmax , r0 ) =
K
Ok (pmax , r0 ).
(7.38)
k =1
The percentage of users in outage for the threshold pmax and desired rate r0 is then O(pmax , r0 ) × 100 %. 57K
(7.39)
If O(pmax , r0 ) > 10%, we increase the power threshold to pmax > pmax . On the other hand, if O(pmax , r0 ) < 10%, we decrease the power threshold to pmax < pmax . Proceeding in this fashion, the base station peak power threshold pmax is adjusted such that exactly 90% of the users are guaranteed the desired rate of r0 bps/Hz and the remaining 10% of users are in outage. The coordinates of the discarded users are stored and the same set of users are discarded when relays are present in the system too. We remark here that the order in which the users are discarded results in diﬀerent power levels from the base stations, due to variations in the interference coupling among the users. Hence, the order in which the users are dismissed must be chosen carefully depending on the peak powers limitations at the base stations.
7.5.4
Network operation and simulation aspects Baseline operation We operate the baseline system as well as the system with relays such that, over a large number of user loading iterations, 90% of users obtain a common average rate of 1 bps/Hz. We follow the approach described in Section 7.5.3 to discard users in the system. For the PCCPA baseline, we solve a series of linear feasibility problems to obtain the base station powers that guarantee the desired common rate. One after another, we discard users that would cause the peak power constraint to became active. Hence we ﬁnd the feasible set of powers p1 , . . . , pN for the baseline such that 90% of the users receive exactly 1 bps/Hz.
PCCPA with relays: average power savings The peak power constraint of the base stations is ﬁxed at pmax such that the baseline can deliver 1 bps/Hz at 10% outage. Since the relays are assumed to be inexpensive, we assume small peak power constraints for the relays. In our work, the peak power of the relays is ﬁxed at 1 W. Let us consider a single instantiation of 57 users in the system. We start oﬀ similarly to the baseline system after discarding the same set of users. The base stations target the users to deliver the common rate of 1 bps/Hz. Only relays that have a better SINR to the base stations than the user are eligible to help the user. The other relays are always inactive. At time t = 0, all relays are inactive. The aim in this experiment is to
7.6 Orthogonal relaying
199
maintain a constant rate of 1 bps/Hz throughout the course of the simulation. When relay i is eligible to transmit at time t, we include relay i in the set of active relays A(t) and solve the LP (7.29). We stop when all the eligible relays are included in the set of active relays. The total power in the system when all eligible relays are active is noted for this instantiation. We repeat this experiment for all the K instantiations.
PCCPA with relays: peak power savings The peak power constraints of the base stations are ﬁxed at a value smaller than the baseline, say pmax . We assume that inexpensive relays are deployed in the system. Thus, the peak power constraints of the relays are ﬁxed at 1 W. Since the peak power value of the base stations is reduced from the baseline and the common rate is ﬁxed at 1 bps/Hz, the outage will be more than 10%. Let us consider a single instantiation of 57 users in the system. We start similarly to the baseline after discarding the same set of users. The base stations target the users to deliver the common rate of 1 bps/Hz. Only relays that have a better SINR to the base stations than the user are eligible to help the user. The other relays are always inactive. At time t = 0, all relays are inactive. Let relay i be eligible to transmit at time t. We include relay i in the set of active relays A(t) and solve the LP (7.35). We stop when all the eligible relays are included in the set of active relays. We repeat this experiment for all the K instantiations and the outage is calculated. If the outage is less than 10%, the peak power of the base stations reduces to pmax < pmax , otherwise the peak power value of the base stations increases to pmax > pmax and the above procedure is repeated until the outage is close to 10%.
7.5.5
Simulation results Power savings The average power saving over K = 200 instantiations is 3 dB and the peak power saving in the downlink when power control is employed is close to 2.6 dB.
Rate gains We have observed 34% improvement in the throughput for 90% of users in the system, with the baseline system being served at 1 bps/Hz. The results are summarized in Table 7.3.
7.6
Orthogonal relaying In Sections 7.4 and 7.5, we saw that in an interferencelimited setting, the improvement in throughput was limited by the interference due to multiple transmissions in the cell. In some cases the transmissions from the base stations are redundant. For instance, for a user located at the edge of the cell, the received
200
Collaborative relaying in downlink cellular systems
Table 7.3. PCCPAbased relaying (base station and relays employ power control) Peak power required to guarantee 1 bps/Hz at 10% outage; baseline (no relays)
Peak power required to guarantee 1 bps/Hz at 10% outage; with relays
Savings in dB
10 W
5.5 W
2.6
Common rate for 90% users; baseline (no relays)
Common rate for 90% users; with relays
Percentage rate increase
1 bps/Hz
1.34 bps/Hz
34%
power from the base station may be weak and the base station’s signal may be of little use. In that case, it might be better to turn the base station oﬀ since this may beneﬁt the system overall in terms of reducing the interference levels. Moreover, the practical implementation of such collaborative schemes can be complex with the state of the art technology. Hence, we investigate how much gain due to collaborative addition can be obtained if we just use simple multihopping, where the base station transmits to a relay in one slot and then that relay passes on the message to the destination in the next time slot. In this section, we exploit the halfduplex property of the relays in a downlink cellular system to stagger the transmissions of the base station and relays over two time slots. A natural way to operate these relays is for them to receive in one time slot and transmit in another time slot. This gives us a natural orthogonality in the transmission scheme. Henceforth, we term this scheme orthogonal relaying.
7.6.1
Network operation and simulation aspects The simulation setup is the same as that described in Section 7.2. Unlike CPA schemes, where the relay can start transmitting immediately after it decodes the message, relays can start transmitting only at speciﬁc times in the orthogonal relaying scheme. The system is assumed to be synchronous and time is divided into equal slots. The baseline and the system with relays are operated as follows.
Baseline The baseline system operates similarly to the PCPA baseline as described in Section 7.4.1. All base stations transmit with peak powers and the users are required to receive a ﬁxed sized ﬁle with a speciﬁed deadline. Satisﬁed users leave the system as soon as they receive the ﬁle. The associated base station sector is turned oﬀ. The users that do not receive the ﬁle are in outage. The peak transmit powers of the base stations are ﬁxed such that 10% of the users over a large population of users are in outage. However, in the orthogonal relaying case, we do not discard the users in outage and rerun the simulations.
7.6 Orthogonal relaying
201
System with relays When relays are present in the system, time is divided into slots of equal durations, and the duration of a time slot is half of that in the baseline system. The operation of the system is periodic with odd and even time slots recurring at regular intervals. The base stations transmit in the odd time slots and the relays transmit in the even time slots. The peak power of the base stations is ﬁxed as in the baseline and the peak power of the relays is ﬁxed as 1 W.
r In the odd time slots, the base stations transmit at peak power. Relays are in receive mode in this time slot. The users and relays in each sector accumulate mutual information, depending on their channel qualities. If some of the users receive the desired rate from the base station transmission itself, those users are satisﬁed users and leave the system as soon as they receive the desired rate. The corresponding base stations and relays are turned oﬀ. Let us denote the 57 × 1 vector of rates obtained by users in the odd time slots by ro . r In the even time slots, only those users that have yet to receive the desired rate of 1 bps/Hz remain in the system. The base stations are turned oﬀ in this time slot. The relays that are required to help the users start transmitting simultaneously at the beginning of the even time slots. Simple power control is employed at the relay locations to reduce the interference caused to the other sectors. The power control is performed to achieve the desired residual common rate re = 1 − ro , where 1 is a 57 × 1 vector all 1’s vector (representing the 1 bps/Hz desired common rate). The users that require the relays to transmit more than their peak power constraint are discarded at the beginning of the even time slots. There may be cases where the user has a better channel to the base station than to the relay. Such users are not given the beneﬁt of receiving the complete message from the base station. The base stations are switched oﬀ on the even time slots and are in outage if they do not receive the message at the end of the even time slots.
7.6.2
User discarding method Baseline The users that remain in the system are in outage. However, the users are not discarded in the baseline scheme.
System with relays The user elimination procedure is the same as that explained in Section 7.5.3. The users that violate the peak power constraint of the relays are discarded at the beginning of the even time slots. The discarded users do not receive the desired common rate at the end of odd and even time slots, and hence are said to be in outage. The system is operated such that there is 10% outage in the system over a large number of user instantiations. The peak power threshold of the relay nodes is adjusted such that the outage percentage is exactly 10%.
202
Collaborative relaying in downlink cellular systems
Table 7.4. Summary of gains due to relaying
7.6.3
Baseline
System with relays
57 sectors 1 user/sector
57 sectors 1 user/sector and 1 relay/sector
Peak power transmissions by base stations
Peak power transmissions by base stations and relays (CPA)
Base station power control
Base station and relays power control (PCCPA)
Peak power transmissions by base stations
Relays power control to users (orthogonal relaying)
Results Power saving (baseline at 1 bps/Hz)
Common rate gain (baseline at 1 bps/Hz)
1.46 dB (peak)
25%
2.6 dB (peak) 3 dB (average)
34%
3 dB (average)
35%
Simulation results The average power saving in the base station locations is 3 dB, since the base stations transmit only for half the time. There is no peak power saving since the base stations transmit at peak power in the odd time slots. We obtain 35% rate gain due to orthogonal relaying when there is 10% outage in the system. It is interesting to note that simpler relaying methods, such as orthogonal relaying do nearly as well as the more complex forms of relaying, such as CPA schemes, in obtaining throughput gains and power savings. This observation is in agreement with the studies in simple linear settings [19, 27].
7.7
Conclusion We have presented a simulation study of the downlink of a cellular system with relays. We evaluated the power saving and common rate increase for users when a common rate of 1 bps/Hz is required by 90% of users in the system. We ﬁrst described the CPA scheme of relay collaboration. In the CPA scheme, whenever the relay receives the complete message from the base station, it collaborates with the base station such that the mutual information at the user location corresponds to the sum of the received power at the user location, thus boosting the average rate. We observed that when the system is interferencelimited the peak power savings are hard to come by. Consequently, the PCCPA scheme along with a framework for power control was proposed. The power control framework can be posed as a formulation when the objective is to minimize peak power or to minimize average energy in the system. This formulation can be used to evaluate the average and peak power savings in the system. The peak power savings and the rate gains improve when power control is employed. We
References
203
then evaluated a simple multihopping scheme where the base stations and the relays transmit in orthogonal time slots. In the odd time slots, the base stations transmit at peak power and in the even time slots, the base stations are turned oﬀ and the relays employ simple power control to deliver the residual rate to the users. A summary of the results is given in Table 7.4.
References [1] Harmonized Contribution on 802.16j (Mobile Multihop Relay) Usage Models. http://ieee802.org/16. [2] Multihop relay system evaluation methodology. http://ieee802.org/16. [3] Y. Yang, H. Hu, J. Xu, and G. Mao, “Relay technologies for WiMAX and LTEadvanced mobile systems,” IEEE Commun. Magazine, 47(10): 100– 105, Oct. 2009. [4] L. Le and E. Hossain, “Multihop cellular networks: Potential gains, research challenges, and a resource allocation framework,” IEEE Commun. Magazine, 45(9): 66–73, Sep. 2007. [5] H. Viswanathan and S. Mukherjee, “Performance of cellular networks with relays and centralized scheduling,” IEEE Trans. Wireless Commun., 4(5): 2318–2328, Sep. 2005. [6] O. Oyman, J. N. Laneman, and S. Sandhu, “Multihop relaying for broadband wireless mesh networks: From theory to practice,” IEEE Commun. Magazine, 45(11): 116–122, Nov. 2007. [7] C. E. Perkins, Ad Hoc Networking. AddisonWesley, 2001. [8] O. Oyman, “Oppurtunistic scheduling and spectrum reuse in relaybased cellular OFDMA networks,” in Proc. of IEEE Globecom, Washington DC, USA, 2007. IEEE, 2007. [9] E. C. van der Meulen, Transmission of Information in a Tterminal discrete memoryless channel. PhD thesis, University of California, Berkeley, 1968. [10] T. M. Cover and A. A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, 25(5): 572–584, Sep. 1979. [11] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorems for relay networks,” IEEE Trans. Inf. Theory, 51: 3037–3063, Sep. 2005. [12] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “On the capacity of ‘cheap’ relay networks,” in Proc. of 37th CISS, Baltimore, MD, USA, 2003. The John Hopkins University, 2003. [13] G. Kramer, “Models and theory for relay channels with receive constraints,” in Proc. of Allerton Conf. on Commun., Control, and Comp., UIUC, IL, USA, 2004. University of Illinois at UrbanaChampaign, 2004. [14] D. Chen, M. Haenggi, and J. N. Laneman, “Distributed spectrumeﬃcient routing algorithms in wireless networks,” IEEE Trans. Wireless Commun., 7(12):5297–5305, Dec. 2008.
204
Collaborative relaying in downlink cellular systems
[15] M. Sikora, J. N. Laneman, M. Haenggi, Jr., D. J. Costello, and T. E. Fuja, “Bandwidth and powereﬃcient routing in linear wireless networks,” IEEE Trans. Inf. Theory, 52(6): 2624–2633, June 2006. [16] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, and A. Viterbi. “CDMA/HDR: A bandwidtheﬃcient high speed wireless data service for nomadic users,” IEEE Commun. Magazine, 38(7): 70–77, July 2000. [17] D. Bertsekas and R. Gallager, Data Networks. PrenticeHall, 1992. [18] R. Pabst, B. H. Walke, D. C. Schultz, et al., “Relaybased deployment concepts for wireless broadband radio,” IEEE Commun. Magazine, 42(9): 80–89, Sep. 2004. [19] G. Foschini, A. Tulino, and R. Valenzuela, “Performance comparison for basic relay systems,” Tech. Memo, Bell Labs NJ, 2008. [20] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley, 2006. [21] M. Katz and S. Shamai, “Transmitting to colocated users in wireless ad hoc and sensor networks,” IEEE Trans. Inf. Theory, 51: 3540–3563, Oct. 2005. [22] D. Blackwell, L. Breiman, and A. J. Thomasian, “Capacity of a class of channels,” Ann. Math. Stat., 30: 1229–1241, 1968. [23] I. Csiszar and P. Narayan, “Capacity of the gaussain arbitrarily varying channel,” IEEE Trans. Inf. Theory, 37: 18–26, Jan. 1991. [24] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. [25] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversityvsmultiplexing tradeoﬀ in halfduplex cooperative channels,” IEEE Trans. Inf. Theory, 51: 4152–4172, Dec. 2005. [26] M. Chiang, P. Hande, T. Lan, and C. W. Tan, “Power control in wireless cellular networks,” Foundations and Trends in Networking, 4(2): 381–533, July 2008. [27] N. Jacobsen, “Practical cooperative coding for halfduplex relay channels,” in Proc. of CISS 2009, March 2009. The John Hopkins University, 2009.
8
Radio resource optimization in cooperative cellular wireless networks Shankhanaad Mallick, Praveen Kaligineedi, Mohammad M. Rashid, and Vijay K. Bhargava
8.1
Introduction Wireless cellular networks have to be designed and deployed with unavoidable constraints on the limited radio resources such as bandwidth and transmit power. With the boom in the number of new users and the introduction of new wireless cellular services that require a large bandwidth or data rate, the demand for these resources, however, is rising exponentially. Finding a solution to meet this increasing demand with the available resources is a challenging research problem. The primary objective of such research is to ﬁnd solutions that can improve the capacity and utilization of the radio resources that are available to the service providers. Based on the concept of relay channels, cooperative communication1 has been found to greatly enhance the performance of a resourceconstrained wireless network [2–6]. It can achieve beneﬁts similar to those of the multipleinput multipleoutput (MIMO) system without the need for multiple antennas at each node. By allowing users to cooperate and relay each other’s messages to the destination, cooperative communication also improves the transmission quality [7]. Because of the limited power and bandwidth resources of the cellular networks and the multipath fading nature of the wireless channels, the idea of cooperation is particularly attractive for wireless cellular networks. Proposed cooperative schemes or strategies, such as decodeandforward (DF), amplifyandforward (AF), and coded cooperation [4, 8–10], usually involve two steps of operation. In the ﬁrst step, a user (called the source node), transmits its message to both the assigned partner (called the relay node) and to the destination. If the relay node employs a DF scheme, it will ﬁrst decode and regenerate the message and then transmit it to the destination in the second step. When the regenerated message is encoded to provide additional error protection to the original message, it is referred to as coded cooperation. If an AF scheme is
1
The term cooperation was introduced in [1] in which the capacity of a threenode relay network was analyzed.
Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
206
Radio resource optimization in cooperative cellular wireless networks
employed, the relay simply ampliﬁes the received message and forwards it to the destination in the next step. At the destination, the signals from both the source and the relay are then combined for detection, using methods such as maximal ratio combining (MRC) or by selection combining. However, the beneﬁts of a relaybased cooperation often depend on the channel quality between the source, relay, and destination involved in the transmission. This is because the transmission quality of a relayed message is limited by the channel signaltonoise ratio (SNR). For example, a relay can fail to forward a message reliably to the destination if the received message from the source is already corrupted due to poor channel quality. Therefore, cooperation may not be always useful. In general, when the complete channel state information (CSI) between the source, relay, and destination is known to the system, relays should be selected to forward the source message only if the quality of the channels between source and relay and relay and destination meet certain criteria. Whenever cooperation is beneﬁcial, the next goal is to allocate the limited resources (e.g., power, bandwidth) appropriately among the source and relay. Therefore, resource allocation among the cooperative nodes can be formulated as an optimization problem to exploit the maximum possible advantage of cooperation [11, 12]. The focus of this chapter is to formulate the resource allocation problems of diﬀerent cooperative networks and strategies as optimization problems and to discuss their solution approaches. The chapter is organized as follows: Section 8.2 discusses the resource allocation problem for networks with a single source–destination pair. First the optimal power allocation is studied for the most basic threenode topology using diﬀerent cooperation schemes, and the capacity or throughput gain is investigated. Then we consider the dual hop relay networks which contain multiple relays and study the relevant resource optimization problems in detail. Section 8.3 deals with the resource allocation problem of a cellular network with multiple source–destination pairs. Solution approaches using both centralized and distributed optimization algorithms are studied. The relay selection strategies for diﬀerent networks are also discussed brieﬂy in Section 8.4. Finally, a chapter summary is provided in Section 8.5.
8.2
Networks with single source–destination pair In this chapter, our strategy is to start with the problem of resource optimization in a simple cooperative architecture involving a single source–destination pair and then progress towards a more general system architecture. The simplest topology for a single source–destination is the threenode relay network shown in Figure 8.1. A more generic topology is the dual hop relay network which consists of multiple relays.
8.2 Networks with single source–destination pair
207
User 1 Feedback
(S)
(D)
hSD
Destination hSR hRD ack db e e F
(R)
User 2
Figure 8.1. A threenode relay network.
8.2.1
Threenode relay network Consider the threenode relay network shown in Figure 8.1, where we let user 1 be the source node (S) that intends to transmit a message to the destination (D) while user 2 serves as the relay node (R). Cooperation requires two time slots to send a message to destination, D. In the ﬁrst time slot, source S transmits its message XS to both R and D. The received signals at R and D can be expressed as XR = hS R XS + ZR , XD 1 = hS D XS + ZD 1 , respectively, where hS R and hS D are the complex channel coeﬃcients for the S–R and S–D links, and ZR and ZD 1 denote the independent and identically distributed (i.i.d.) circularly symmetric additive white Gaussian noise (AWGN) with zero mean and variance N0 . In the second time slot, relay R transmits the message (XR ) received in the ﬁrst slot after either decoding and reencoding it or simply amplifying it. Depending on the cooperative scheme (DF or AF), let Y = f (XR ) be the transmitted message2 in the second slot. The received signal in D can be written as XD 2 = hR D Y + ZD 2 = hR D f (XR ) + ZD 2 , where hR D is the channel coeﬃcient between the R–D pair and ZD 2 is the zero mean AWGN with variance N0 . In the following analysis, we assume that these coeﬃcients are known (i.e., full CSI is available) to S, R, and D and the AWGN noise has unit variance, i.e., N0 = 1. The transmit powers of S and R are given by PS and PR . 2
For the AF scheme Y is X R corrupted with AWGN and for the DF scheme Y is simply X R .
208
Radio resource optimization in cooperative cellular wireless networks
Quality of Service (QoS), a measure of performance, reﬂects the transmission quality and the service availability of a transmission system. For example, channel capacity, SNR, outage probability or bit error rate (BER) can be one of the QoS measures. Our objective is to formulate the optimization problem so that the QoS performance of the network is optimal subject to some resource constraint. In the following we use the total power budget as the constrained resource of the network and the channel capacity (or, maximum achievable data rate) as the QoS measure. Therefore, the problem is to maximize the channel capacity given some total power constraint and the solution will provide the optimal allocation of PS and PR among the cooperative nodes. The solution for optimal power allocation depends on whether the direct S–D link is taken into account. If the messages from both the source and the relay are combined at the destination for detection, this is referred to as the case with diversity [13]. For the case without diversity, only the message from the relay (i.e., XD 2 ) is considered. In the following we study optimal power allocation for both the cases.
Case 1 With direct link First, let us formulate the capacity maximization problem taking the direct link into account to examine the optimal power allocation for the threenode network with DF cooperative strategy. In a twotimeslot operation, the capacity of the S–R link is given as [14] Chop1 =
1 2
log2 (1 + hS R 2 PS ).
(8.1)
In the second time slot, the destination combines the messages from both the S–D and R–D links using MRC technique, and the capacity is given by Chop2 =
1 2
log2 (1 + hR D 2 PR + hS D 2 PS ).
(8.2)
The channel capacity in the DF cooperative scheme is limited by the minimum of the two hops: ; < CD F , div er sity = min 12 log2 (1 + hS R 2 PS ), 12 log2 (1 + hR D 2 PR + hS D 2 PS ) . (8.3) Therefore the optimization problem becomes a standard max–min problem: ; maximize {P S ,P R }
min
1 2
log2 (1 + hS R 2 PS ),
1 2
log2 (1 + hS D 2 PS + hR D 2 PR )
hS D 2 , the capacity is maximized with equal capacity for the two hops, i.e. 1 2
log2 (1 + hS R 2 PS ) =
1 2
log2 (1 + hS D 2 PS + hR D 2 PR ).
(8.5)
Since the objective is to maximize the capacity, the ﬁrst inequality constraint of (8.4) should be met at equality. Using PS + PR = P0 in (8.5), we get the optimal power allocation as PS = P0
hR D 2 hS R 2 + hR D 2 − hS D 2
(8.6)
P R = P0
hS R 2 − hS D 2 . hS R 2 + hR D 2 − hS D 2
(8.7)
and
From (8.6) and (8.7), we see that more power is allocated to the source than to the relay, which is justiﬁed since PS contributes both to the direct path and to the relay path. However, if hS D 2 ≥ hR D 2 we see that the capacity expressions (8.1) and (8.2) are both monotone increasing functions and the optimal power allocation that maximizes the objective function in (8.4) is PS = P0 and PR = 0. This result indicates that, if the direct channel has better quality than any of the S–R or R–D links, it is intuitive to allocate all available power to S alone. Now let us formulate the problem of optimal power allocation for the AF cooperative scheme. In this case, the message forwarded by the relay contains an ampliﬁed version of the noise along with the message transmitted originally by the source. As a result, the SNR at the destination node plays an important role in the optimal power allocation problem. The channel capacity of the twohop AF scheme can be expressed as CA F ,div er sity =
1 2
log2 (1 + SN RD ),
(8.8)
where SN RD is the SNR at destination node given as [15] SN RD =
hS R 2 PS hR D 2 PR + hS D 2 PS . 1 + hS R 2 PS + hR D 2 PR
(8.9)
For AF cooperation, optimizing the channel capacity is equivalent to SNR maximization. Therefore, the optimal power allocation problem for the AF scheme can be formulated as maximize {P S , P R }
hS R 2 PS hR D 2 PR + hS D 2 PS 1 + hS R 2 PS + hR D 2 PR
subject to : PS + PR ≤ P0 , PS ≥ 0, PR ≥ 0.
(8.10)
210
Radio resource optimization in cooperative cellular wireless networks
Applying the Lagrange multiplier, the optimal power allocation of (8.10) is obtained as
PS =
hS R 2 hR D 2 P0 + hS D 2 hR D 2 P0 + hS D 2 7 hS R 2 P0 + 1 A+ hS R 2 hR D 2 A 2 hR D  P0 + 1
(8.11)
and
PR =
hS R 2 hR D 2 P0 − hS D 2 hS R 2 P0 − hS D 2 7 , hR D 2 P0 + 1 2 2 A+ hS R  hR D  A hS R 2 P0 + 1
(8.12)
where A = hS R 2 hR D 2 + hS D 2 hR D 2 − hS R 2 hS D 2 . Optimal power allocation holds if and only if A > 0 and hS R 2 hR D 2 P0 > hS D 2 hS R 2 P0 + hS D 2 . Otherwise all available power should be allocated to the source alone, which indicates that the S–D link is better than the S–R or R–D link and that cooperation is not beneﬁcial in such channel conditions. When hS R 2 ≈ hR D 2 and both are suﬃciently larger than hS D 2 , the ratio of PS and PR can be approximated as hS R 2 hR D 2 P0 + hS D 2 hR D 2 P0 + hS D 2 PS ≈ . PR hS R 2 hR D 2 P0 − hS D 2 hS R 2 P0 − hS D 2
(8.13)
Example 1 In this example, we compare the outage probabilities of the AF and DF schemes with equal and optimal power allocation methods, as shown in Figure 8.2. We consider that an outage occurs when the achieved rate is less than 1 (C < 1) at the destination. For simplicity, we assume that the relay node is located in the middle of the source and the destination and the complex channel coeﬃcients hS R , hS D , and hR D are i.i.d. circularly symmetric Gaussian random variables with zero mean and variances, i.e., σS2 R = 1, σR2 D = 1, σS2 D = 1/(2α ), where α = 3 is the path loss coeﬃcient. From Figure 8.2, we can see that optimal power allocation has signiﬁcant SNR gain over the equal power allocation method. It is interesting to note that the outage probability of the AF scheme outperforms that of the DF scheme when the SNR value is suﬃciently high (over 15 dB). This is because the DF scheme does not provide additional diversity gain. From the results we can conclude that, whenever there is diversity, it is always better to use the AF scheme over the DF cooperation.
211
8.2 Networks with single source–destination pair
Figure 8.2. Comparison of outage probabilities of AF and DF schemes with diversity.
Case 2 Without direct link Without diversity, the destination node receives only the relayed message from R. For DF cooperation, the channel capacity can be written as ;
< log2 (1 + hS R 2 PS ), 12 log2 (1 + hR D 2 PR ) . (8.14) The capacity maximization problem is thus a max–min problem. The maximum capacity is reached when the capacities of both hops are equal, i.e., CD F , w ithou t div er sity = min
1 2
1 2
log2 (1 + hS R 2 PS ) =
1 2
log2 (1 + hR D 2 PR ).
(8.15)
The optimal power allocation is obtained using PS + PR = P0 in (8.15) as P S = P0
hR D 2 hS R 2 + hR D 2
(8.16)
P R = P0
hS R 2 . hS R 2 + hR D 2
(8.17)
and
This result is also veriﬁed directly, by putting hS D  = 0 in (8.6) and (8.7).
212
Radio resource optimization in cooperative cellular wireless networks
Figure 8.3. Comparison of outage probabilities of AF and DF schemes without diversity. Similarly for AF cooperation without diversity, the optimal power allocation is obtained by putting hS D  = 0 in (8.11) and (8.12) as 7
PS = 1+
P0
(8.18)
hS R 2 P0 + 1 hR D 2 P0 + 1
and 7
PR = 1+
P0 hR D 2 P0 + 1 hS R 2 P0 + 1
Thus the ratio between PS and PR is given by 7 hR D 2 P0 + 1 PS = . PR hS R 2 P0 + 1
.
(8.19)
(8.20)
Example 2 The outage probabilities for AF and DF cooperation without diversity are compared in Figure 8.3. For each scheme, we plot the results obtained from equal and optimal power allocation methods. It is clear that optimal power allocation has approximately 3 dB SNR gain over the equal power allocation method for both the AF and DF schemes. Since there is no diversity, the DF scheme performs better than the AF scheme at low
8.2 Networks with single source–destination pair
X1 hs1 X2 X0
Source (S)
213
U1=f1(X1)
Relay 1 (R1)
hs2
h1D
U2=f2(X2)
h2D Relay 2 (R2) XN
hND
hSN
Z
Destination (D)
Relay N (RN) UN =fN (XN )
Feedback Figure 8.4. A dualhop cooperative relay network.
to moderate SNR. However, at a very high SNR (over 30 dB), the performances of both the schemes are almost identical.
8.2.2
Dualhop relay networks Now let us consider a twostage cooperative network (also known as a parallel relay network) consisting of N relay nodes, denoted by Rk , k = 1, 2, . . . , N as shown in Figure 8.4. The complex channel coeﬃcients from the source S to relay Rk and from Rk to destination D are given by hS k and hk D respectively. In such networks, S broadcasts its message in the ﬁrst time slot and the set of relays {Rk , k = 1, 2, · · · , N } transmits simultaneously in the second slot. Here we assume all the channels are orthogonal and perfect CSI is available to all the nodes. The transmit powers of S and Rk are denoted by PS and Pk respectively. In this section, we focus on the power allocation among the relay nodes only. The power allocation among PS and PR can be determined using techniques similar to those derived in the previous section. The optimization problem is formulated as a capacity maximization problem but the total power constraint is imposed on the summation of relay powers, i.e., N k =1 Pk ≤ PR . In the following, we analyze the optimal power allocation for both AF and DF cooperation.
Case 1 AF scheme For the AF scheme, the capacity of the parallel relay channel is given as [16] CA F =
1 2
log(1 + SN RD ),
(8.21)
214
Radio resource optimization in cooperative cellular wireless networks
where
SN RD = PS
2
hS D  +
N k =1
hS k 2 hk D 2 Pk hS k 2 PS + hk D 2 Pk + 1
.
(8.22)
The goal here is to optimize the power allocation among the parallel relay nodes so that the capacity is maximized. Optimizing CA F is equivalent to maximizing the SN RD . For a constant source power PS , we deﬁne 7 hS k 2 hk D 2 γk = , (8.23) hS k 2 PS + 1 so that we can write N k =1
hS k 2 γ 2 Pk hS k 2 hk D 2 Pk k = hS k 2 PS + hk D 2 Pk + 1 k =1 hS k 2 + γk2 Pk N
(8.24)
and formulate the following optimization problem: maximize Pk
subject to:
N hS k 2 γk2 Pk hS k 2 + γk2 Pk k =1 N
Pk ≤ P R ,
k =1
Pk ≥ 0.
(8.25)
It can be easily shown that the optimization problem is convex since the objective function is now a concave increasing function of Pk and the constraints are linear. The solutions of any convex optimization problem can be easily obtained by available convex programming algorithms [17]. Among these algorithms, subgradient methods are the simplest and so are widely used. However, interiorpoint methods, bundle methods, and cuttingplane methods are also well known. The solution obtained using these algorithms does not provide any insight into the optimal power allocation among the relays though. Therefore, we solve the optimization problem of (8.25) using the Lagrange dual method. The Lagrangian of the problem is given by N N N hS k 2 γk2 Pk (8.26) + λ k Pk − η Pk − PR , L(Pk , λ, η) = hS k 2 + γk2 Pk k =1 k =1 k =1 where λ = λk ≥ 0, k = 1, 2, ..., N and η are the Lagrange multipliers corresponding to the inequality constraints on the relay power. From the Karush–Kuhn– Tucker (KKT) conditions, hS k 2 γk2 (hS k 2 + γk2 Pk ) − hS k 2 γk4 Pk ∂L = + λk − η = 0 ∂Pk (hS k 2 + γk2 Pk )2
(8.27)
8.2 Networks with single source–destination pair
215
and λk Pk = 0. By eliminating λk from (8.28), we get hS k 2 γk2 (hS k 2 + γk2 Pk ) − hS k 2 γk4 Pk Pk = 0. η− (hS k 2 + γk2 Pk )2
(8.28)
(8.29)
From (8.29) we see that if Pk = 0, then the expression hS k 2 γk2 (hS k 2 + γk2 Pk ) − hS k 2 γk4 Pk η− (hS k 2 + γk2 Pk )2 must be equal to zero. After simpliﬁcation, we get the optimal power allocation among the relay nodes as
+ 1 hS k 2 1 Pk = , (8.30) √ − γk η γk where [.]+ denotes the projection onto the feasible set of nonnegative orthants. Therefore, the optimal power allocation problem results in the following waterﬁlling solution [18]: ⎧ 1 1 1 hS k 2 1 ⎪ ⎪ ( √ − ), if √ > ⎨ γk η γk η γk (8.31) Pk = ⎪ ⎪ ⎩ 0, else. The Lagrange multiplier, η, is chosen to meet the total power constraint of the relay nodes. Also note that the relay node Rk is allowed to transmit if and only if λk > η. Example 3 The sum capacities (or maximum achievable data rate) of the AF scheme with diﬀerent numbers of relays are compared in Figures 8.5 and 8.6. For both the ﬁgures, we plot the results obtained from equal and optimal power allocation methods. In Figure 8.5, we assume that all the S–R and R–D channel coeﬃcients are i.i.d. circularly symmetric Gaussian random variables with zero mean and unit variances. As in the previous examples, the direct S–D channel coeﬃcient, hS D is assumed to have zero mean and variance σS2 D = 1/(2α ). In Figure 8.6, we assume that S–R and R–D channel coeﬃcients are non i.i.d. Gaussian random variables. From Figures 8.5 and 8.6, it is clear that the rates achieved are much higher for optimal power allocation than for equal power allocation. As we increase the number of relays, the sum capacity increases. However, for the cases where some relay channels are better than others (in Figure 8.6), it is interesting to note that two relays with optimal power allocation can achieve a better rate than four relays with equal power allocation.
Radio resource optimization in cooperative cellular wireless networks
3
Sum capacity (bps/Hz)
2.5
2
1.5
1
4 relay equal power 4 relay optimal power 2 relay optimal power 2 relay equal power
0.5
0
1
2
3
4
5
6
7
8
9
10
Total relay power (watt)
Figure 8.5. Comparison of sum capacities of the AF scheme with i.i.d. channels.
3
2.5
Sum capacity (bps/Hz)
216
2
1.5
1
4 relay equal power 4 relay optimal power 2 relay optimal power 2 relay equal power
0.5
0
1
2
3
4
5
6
7
8
9
10
Total relay power (watt)
Figure 8.6. Comparison of sum capacities of the AF scheme with non i.i.d channels.
217
8.2 Networks with single source–destination pair
Case 2 DF scheme A relay node can execute the DF scheme only if it is able reliably to decode the message received from the source node. Let us consider a set of relays denoted by ND F , among the N relay nodes, which are able to correctly decode the messages transmitted by the source. For given ND F , the dualhop relay network with orthogonal channels is analogous to a multiple antenna system. The capacity of such networks can be written as [19]: CD F =
1 2
log(1 + hS D 2 PS ) +
k ∈N D F
1 2
log(1 + hk D 2 Pk ).
(8.32)
For optimal power allocation the DF scheme, the ﬁrst task is to ﬁnd the reliable relay nodes k ∈ ND F . We assume that a relay will be able to execute the DF strategy if it is able to reliably communicate with the source at a desired rate, rS . Therefore, a relay node k is reliable if the following condition is met: 1 2
log(1 + hS k 2 PS ) ≥ rS .
(8.33)
Our goal is to ﬁnd the optimal power allocation that maximizes the capacity of the network. However, the problem of maximizing the capacity under the relay power constraint, N k =1 Pk ≤ PR has its dual problem with the objective to communicate with the destination at the desired rate, rS using minimum relay power, Pk∗ . The analogous solutions of the two problems become identical when Pk∗ = PR . We use binary variables xk to indicate the reliable relays and formulate the following dual problem:
minimize
N
Pk
k =1
subject to:
N
1 2
log(1 + hS D 2 PS ) +
1 2
log(1 + hS k 2 PS ) > xk rS ,
k =1
1 2
log(1 + hk D 2 Pk )xk ≥ rS ,
xk ∈ {0, 1}, Pk ≥ 0.
(8.34)
The formulated optimization problem is a mixed integer problem which is generally very diﬃcult to solve. However, for severely restricted transmitter power or low SNR cases, i.e. Pk hk D 2 1, the dual problem simpliﬁes to the wideband
218
Radio resource optimization in cooperative cellular wireless networks
DF relay problem [20]: minimize
N
Pk
k =1
subject to:
N
1 2
hS D 2 PS +
1 2
hS k 2 PS > xk rS ,
1 2
k =1
hk D 2 Pk xk ≥ rS ,
xk ∈ {0, 1}, Pk ≥ 0.
(8.35)
The solution of the above optimization problem is to choose a single relay which has the best channel gain towards the destination and allocate the total relay power to it. This is due to the fact that, at low SNR (Pk hk D 2 1), a waterﬁlling solution ﬁnds the relay which has the best channel gain to the destination node. This means that the selective relaying scheme is optimal for the DF strategy for dualhop relay networks with multiple relays. Therefore, the optimal power allocation among multiple relays becomes a single relay optimization problem: minimize
P S + Pk
subject to:
1 2
hS D 2 PS +
1 2
hS k 2 PS ≥ xk rS ,
1 2
hk D 2 Pk xk ≥ rS ,
xk ∈ {0, 1}, Pk ≥ 0, PS ≥ 0.
(8.36)
As in the threenode case, one can show that cooperation with relay k is beneﬁcial if and only if hS k 2 > hS D 2 and hk D 2 > hS D 2 . Otherwise it is optimal to allocate all the power to the source for direct transmission. The optimal power allocation is obtained using the ﬁrst and second constraints of (8.36) with equality: rS hS k 2
(8.37)
rS − PS∗ hS D 2 . hk D 2
(8.38)
PS∗ = and Pk∗ =
219
8.2 Networks with single source–destination pair
0.7
0.6
Sum capacity (bps/Hz)
0.5
0.4
0.3
0.2
DF optimal power (best relay) DF equal power (2 relay) DF equal power (4 relay)
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Total relay power (watt)
Figure 8.7. Comparison of sum capacities of DF scheme in the low SNR regime. The optimal total power is then 5 PS∗ + Pk∗ = rS
6 hS D 2 . + − hS k 2 hk D 2 hS k 2 hk D 2 1
1
(8.39)
The best relay will be the one among the set of useful relays which minimizes the total power PS∗ + Pk∗ . If the set of useful relays RU = {k ∈ ND F  hS k 2 > hS D 2 , hk D 2 > hS D 2 } is nonempty, then the optimal relay selection criterion is 5 6 1 1 hS D 2 ∗ k = argmin . + − hS k 2 hk D 2 hS k 2 hk D 2 k ∈R U
(8.40)
The maximum capacity of the primal problem for the dualhop DF scheme in the lowSNR regime can thus be approximated as CD F ≈
1 2
hS D 2 PS +
1 2
hk ∗ D 2 PR .
(8.41)
Example 4 Similarly to the previous example, we compare the sum capacities of DF scheme for the dualhop network with optimal and equal power allocation in Figure 8.7. The results are shown for the cases of severely restricted transmitter power or the lowSNR regime. We assume that the diﬀerent S–R and R–D channels have diﬀerent channel variances.
220
Radio resource optimization in cooperative cellular wireless networks
Relay Base station (BS)
Relay
User near BS
User near cell edge Figure 8.8. Typical cooperative cellular network with multiple users.
Unlike that for the AF scheme, the sum capacity of the DF scheme does not depend on the number of relays at low SNR. Since the DF scheme does not provide additional diversity gain, it is optimal to ﬁnd the best relay and allocate total relay power to it.
8.3
Multiuser cooperation In the previous section, we analyzed the problem of resource allocation for a single source–destination pair. Now we examine the resource allocation problem for multiuser (or multisource) cooperative networks. Consider the case of cellular networks shown in Figure 8.8, where some relays are deployed to assist the users located at cell edges for both uplink and downlink transmissions. Since the number of relays is smaller than the number of users in a practical scenario, each relay is assigned to assist more than one user. To formulate the power allocation problem for multiuser networks we choose one of two QoS measures, namely the minimum rate of the users (max–min fairness) or the weighted sum rates of the users (weightedsum fairness). Therefore, the problem is to maximize either the minimum rate of the users or the weightedsum rates given total relay power as the constraint. In the following, we will see that both the problems are convex and the problem can be solved using centralized or distributed algorithms.
8.3 Multiuser cooperation
8.3.1
221
System model Consider a multiuser network where M source nodes Si , i ∈ {1, 2, ..., M } transmit messages to their corresponding destination nodes Di , i ∈ {1, 2, ..., M }. There are N relay nodes Rk , k ∈ {1, 2, ..., N } in the network. The set of relays assisting the transmission of Si is denoted by R(Si ). The set of sources using the Rk relay is denoted by S(Rk ). Thus we can write S(Rk ) = {Si  Rk ∈ R(Si )},
(8.42)
which means one relay can forward the messages of several users. For this section we consider that no direct link exists between any source–destination (Si –Di ) pair. We also assume all the channels are orthogonal and perfect CSI is available to all the nodes. In a multiuser cooperative network with the AF scheme, each source Si transmits a message to its chosen relays in the set R(Si ) in the ﬁrst time slot, and each relay ampliﬁes and forwards its received message to Di in the second time slot. The transmit powers of the source Si and its assisting relays Rk ∈ R(Si ) are denoted by PS i and PRS ki respectively. Consider that in the ﬁrst time slot, source Si broadcasts the message Xi with unit energy to the relays Rk ∈ R(Si ). The received message at the relay Rk can be written as XRS ik = PS i hSRik Xi + ZR k , Rk ∈ R(Si ). (8.43) In the second time slot, the relay Rk ampliﬁes the received message and relays it to the destination node Di . The received message at Di is given by [21]
i XD Rk
= > > => ?
PRS ki PS i Si i ˆ hD " " R k hR k Xi + Z D i , " S i "2 PS i "hR k " + 1
Rk ∈ R(Si ),
(8.44)
i where hSRik and hD R k denote the complex channel coeﬃcients for the Si –Rk and Rk –Di links, and ZR k and ZD i are the zero mean AWGN with unit variance. The modiﬁed AWGN at Di is denoted by ZˆD i with equivalent variance " " " i "2 PRS ki "hD Rk " . 1+ " " " S i "2 PS i "hR k " + 1
Assuming that MRC is employed at the destination node Di , the combined SNR at Di can be written similarly, as given by [22]: " " " " " S i "2 " D i "2 S i "hR k " "hR k " PR k PS i SN RD i = . (8.45) " " " " " S i "2 " D i "2 S i R k ∈R(S i ) "hR " PS i + "hR " PR + 1 k k k
222
Radio resource optimization in cooperative cellular wireless networks
The above expression for SN RD i is similar to that of (8.22) used in the previous section for the dualhop relay network with the AF scheme. The rate ri of user i is given by ⎛ ⎞ " " " " " S i "2 " D i "2 S i "hR k " "hR k " PR k PS i 1 1 ⎜ ⎟ ri = log2 (1 + SN RD i ) = log2 ⎝1 + " " " " ⎠. " D i "2 S i " S i "2 2 2 R k ∈R(S i ) "hR " PS i + "hR " PR + 1 k k k (8.46) Similarly to the singleuser case, it can be shown that, for a constant source power PS i , the rate expression (8.46) is a concave increasing function with respect to PRS ki , Rk ∈ R(Si ). Using this fact, the optimal power allocations for both the max–min fairness and weightedsum fairness schemes can be calculated as shown in the following subsection.
8.3.2
Centralized power allocation Max–min fairness For the rate expression given in (8.46), the power allocation problem under the max–min rate can be formulated as min ri
maximize
Si
subject to:
PRS ki ≤ PRmax , k
k = 1, 2, ..., N,
S i ∈S(R k )
PRS ki ≥ 0,
(8.47)
where PRmkax is the maximum (or total) power of the relay Rk . The ﬁrst constraint can also be written equivalently as a constraint on the maximum sum of powers transmitted by the corresponding assisted source nodes. The set of linear inequality constraints with positive variable in (8.47) is compact and nonempty. Hence the optimization problem set is always feasible. Moreover, since the objective function is an increasing function of power, the ﬁrst inequality constraint should meet with equality at the optimal point. The convex optimization problem can be written in its standard form: minimize
−t
subject to: t − ri ≤ 0 ,
PRS ki ≤ PRmkax ,
i = 1, 2, ..., M, k = 1, 2, ..., N,
S i ∈S(R k )
PRS ki ≥ 0, t ≥ 0,
(8.48)
where t is a slack variable. The solution of this problem gives the optimal power allocation among the relays that maximizes the worstuser rate. In the special
8.3 Multiuser cooperation
223
case, in which all the users share the same set of relays the rates of all users are equal at the optimal point. This is easily understood from the max–min problem of singleuser case where the capacity is maximized with equality at the optimal point.
Weightedsum fairness The problem of the max–min ratebased power allocation is that it degrades the sum capacity (or total network throughput). This is because it tends to improve the performance of the worstuser rate by allocating more power to the poor links. The weightedsum rate maximization can potentially achieve certain fairness for diﬀerent users by allocating large weights to the users in unfavorable channel conditions while maintaining good network performance. Let wi denote the weight allocated to user i, then the weightedsum rate power allocation problem can be formulated as maximize
M
wi ri
i= 1
subject to:
PRS ki ≤ PRmkax ,
k = 1, 2, ..., N,
S i ∈S(R k )
PRS ki ≥ 0.
(8.49)
The solution of this convex optimization problem gives better network throughput, which indicates that the weightedsumratebased power allocation favors users with good channel conditions. However, it does not severely penalize users with bad channel conditions either. In fact, because the objective function (weightedsum rate) is concave and increasing with respect to the allocated powers, the increment in the function value is higher when the power is low. Therefore to maximize the objective function value, it is obvious to allocate more power to ‘bad’ users operating at low SNR.
8.3.3
Distributed power allocation The centralized allocation discussed in the previous subsection may not be very practical for a number of reasons. Firstly, it requires a large signaling overhead since the channel gain estimations between various S–R and R–D have to be transmitted to the base station, which allocates power based on these estimates. Moreover, the wireless channels are time varying and frequency selective. Therefore, the channel gain information needs to be constantly updated, which requires a lot of data to be transmitted to the base station using the control channels. In fact, in mobile communication systems with constant changes in the channels between various devices, such a centralized power allocation scheme could become impractical. Secondly, these schemes are highly complex and would create a large computational load at the base station. Thus, it is very important
224
Radio resource optimization in cooperative cellular wireless networks
to devise distributed schemes in which each relay calculates its own optimal power allocation based on the information available from neighboring nodes and hence reduces the overall signaling overhead as well as the large computational complexity in the base station. Several distributed power allocation schemes for cooperative communication systems have been discussed in the literature. The distributed power allocation scheme for the multiuser AF relay network was developed in [21] using the Lagrange dual decomposition method. For the DF scheme, the distributed power allocation problem with partial CSI was addressed in [23] and computationally eﬃcient optimal schemes were proposed. Both relay selection and power allocation for AF wireless networks with reduced complexity were investigated in [24]. In the following we discuss the distributed power allocation strategy for both AF and DF schemes.
Case 1 AF scheme The centralized optimization problem for power allocation based on weightedsum rate maximization is given in (8.49). Here we solve the same problem in a distributed manner. The main idea is to separate the original problem in (8.49) into independent subproblems using the Lagrange dual decomposition method. The Lagrangian of problem (8.49) is L(PRS ki , λ) =
M
wi ri −
i=1
N
⎛
⎞
λk ⎝
PRS ki − PRmkax ⎠ ,
(8.50)
S i ∈S(R k )
k =1
where λ = [λ1 , λ2 , ..., λN ] represents the Lagrange multipliers corresponding to the N constraints of (8.49). Using (8.42) we can write the following: N
λk
k =1
PRS ki =
S i ∈S(R k )
M
λk PRS ki .
(8.51)
i=1 R k ∈R(S i )
The Lagrangian in (8.50) is rewritten using (8.51) as L(PRS ki , λ) =
M
⎣wi ri −
⎤ λk PRS ki ⎦ +
R k ∈R(S i )
i=1
=
⎡
M
Li (PRS ki , λ) +
i=1
N
N
λk PRmkax
k =1
λk PRmkax ,
(8.52)
k =1
where Li (PRS ki , λ) corresponds to the ith value of the Lagrangian. The corresponding Lagrange dual function is given by g(λ) = max L(PRS ki , λ). S
P R i ≥0 k
(8.53)
8.3 Multiuser cooperation
225
This dual function can be obtained by solving M separate subproblems corresponding to M diﬀerent users as maximize Li (PRS ki , λ) subject to: PRS ki ≥ 0, Rk ∈ R(Si ).
(8.54)
Since the original problem given in (8.49) is convex, strong duality holds and we can get the optimal power allocation by solving the dual problem: minimize
g(λ)
subject to: λk ≥ 0, k = 1, 2, ..., N.
(8.55)
Let the solution of the optimization problem in (8.54) be L∗i (λ), which corresponds to the optimal value of Li (PRS ki , λ). Then the dual problem can be rewritten as minimize
M i=1
L∗i (λ) +
N
λk PRmkax
k =1
subject to: λk ≥ 0, k = 1, ..., N.
(8.56)
The distributed power allocation algorithm aims to solve (8.54) and (8.56) sequentially. Let the Lagrange multiplier λk ≥ 0 denote the price per unit power at relay k. Therefore λk PRS ki represent the total price that user i must pay for using PRS ki power at each relay Rk ∈ R(Si ). Thus each user tries to maximize the weighted rate minus the total price it has to pay given the price coeﬃcients at the relays. Since the dual function g(λ) is diﬀerentiable, the problem can be solved iteratively by the gradient projection method. The receiver of user i ﬁnds its optimal power with given λ from the assisting relays Rk ∈ R(Si ) as ⎧ ⎫ ⎨ ⎬ λk [t]PRS ki (λ[t]) . (8.57) PRS ki (λ[t + 1]) = arg max wi ri − ⎩ ⎭ R k ∈R(S i )
The receiver of user i then sends this information to all the relays allocated to source i. Based on PRS ki values received from destinations, the relays update their dual variable λk as follows: ⎡ ⎛ ⎞ ⎤+ PRS ki (λ[t])⎠⎦ , (8.58) λk [t + 1] = ⎣λk [t] − δ ⎝PRmkax − S i ∈S(R k )
where t is the iteration index, [.]+ denotes the projection onto the feasible set of nonnegative orthants, and δ is the step size parameter. Update equation (8.58) can be interpreted as the price (λk ) change by Rk depending on the requested power levels from its users. The price increases when the power allocated to the relay exceeds its maximum limit. This pricebased distributed algorithm requires
226
Radio resource optimization in cooperative cellular wireless networks
Figure 8.9. Pricebased distributed power allocation algorithm. message passing only between each receiver and its assisting relays. The overall structure of the distributed power allocation is shown in Figure 8.9.
Example 5 In Figure 8.10, we compare the worstuser rate for both the weightedsum and max–min fairness algorithms for a network with number of source nodes, M = 8 and number of relays, N = 4. The equal power allocation among the relays is also included for reference. We see that the worst user obtains the best rate under the max–min fairness scheme and the worst rate under weightedsum scheme with equal weight coeﬃcients. However, Figure 8.11 shows the weightedsum scheme results in maximum capacity or network throughput. The max–min scheme targets improving the performance of the worst user, which results in a signiﬁcant loss in the sum capacity of the whole network.
Case 2 DF scheme Distributed power allocation strategies for the twohop DF scheme were investigated in [23]. Here the relay decides its power so that the destination achieves the target SNR assuming that it is the only relay to the destination. The relay
8.3 Multiuser cooperation
227

Figure 8.10. Comparison of the worstuser rate for the weightedsum rate and max–min rate algorithms.
Figure 8.11. Comparison of the sum capacity for the weightedsum rate and max–min rate algorithms.
228
Radio resource optimization in cooperative cellular wireless networks
decides to forward the received message from the source to the destination if its channel gain hk D satisﬁes hk D 2 > γ,
(8.59)
where γ is a given threshold value. The transmit power of the relay is given by Pk∗ =
(SN Rtar g et − Ps hS k 2 )+ . hk D 2
(8.60)
The distributed power allocation algorithm is formulated as follows: E[Pk ] minimize Ps + k ∈N R (P s )
subject to: Prob (SN RD ≤ SN Rtar g et ) ≤ ρtar g et , Ps hk D 2 ≥ SN Rtar g et ,
k ∈ NR ,
(8.61)
where NR denotes the set of reliable relays which satisfy (8.61). SN RD is the SN R at destination node and ρtar g et is the target outage probability. Note that, even though a relay ultimately determines its own power, it does not take much of the computational load. This scheme, however, requires knowledge of all the channels gains at the receiver. Therefore, several suboptimal schemes should be derived, in which the source can calculate γ and Ps based on the channel characteristics without requiring feedback from the receivers.
8.4
Relay selection Until now we have assumed that the relays are assigned a priori to power allocation. The joint relay selection and power allocation problem is a mixed integer optimization problem and is very diﬃcult to solve. Therefore, in most of the literature, the problem is divided into two subproblems [25]. The relays are chosen assuming ﬁxed powers (e.g., equal power allocation) at the relays. Once the relays are selected, the techniques mentioned in previous sections of this chapter are used to allocate power to the selected relays. In this section, some of the relay selection algorithms are presented. In spite of assuming ﬁxed power at the relays, relay selection is still an NPhard problem and the optimum solution can only be achieved through exhaustive search [22]. Several greedy algorithms for relay selection for DF systems have been proposed in the literature. The general strategy is to select the best relay or a group of relays which satisﬁes a certain performance improvement over the direct S–D link [26]. In [27], a relay is selected based on the amount of time taken to deliver a ﬁxed number of messages. Two optimal algorithms are provided where in the ﬁrst one the relays are adaptively selected and in the second one the optimal number of relays is predetermined without relying on
8.4 Relay selection
229
the speciﬁc network realization. Alternatively, in [24], a relay is selected if it oﬀers an increase in capacity compared with source destination channel. In [28], a distributed algorithm is presented to select the ‘best’ relay that provides the best endtoend path between the source and the destination. Each relay is assigned a timer with initial value inversely proportional to either hk = min(hS k 2 , hk D 2 ) or hk = 2hS k 2 hk D 2 /(hS k 2 + hk D 2 ). The relay with the highest value of hk is considered to be the one with the best endtoend path between source and destination. All relays start their timers simultaneously with diﬀerent initial values which depend upon the channel realizations. The relay whose timer expires ﬁrst (i.e., the one with maximum hk ) starts relaying the message from the source as soon as its time expires. All the relays listen to the other relays and, therefore, know whether some other relay has started forwarding the message or not. When they recognize that a particular relay has started transmission, they do not forward the source message to the destination. Joint relay selection and power allocation algorithms were studied in [29] for multisource AF systems. The following max–min optimization problem formulated: ⎛ maximize min Si
1 ⎜ log2 ⎝1 + 2
subject to:
Rk
⎞ " " " " " S i "2 " D i "2 S i "hR k " "hR k " PR k PS i xSRik ⎟ " "2 " "2 ⎠ " Si " " i " Si Si P x + 1 "hR k " PS i + "hD " Rk Rk Rk
PRS ki ≤ PRmkax ,
Si
xSRik = xSmi ax ,
Rk
PRS ki ≥ 0, xSRik ∈ {0, 1}.
(8.62)
This is a nonconvex optimization problem and hence a unique analytical solution cannot be achieved. Therefore, a suboptimal solution was proposed by relaxing the integrality constraints on xSRik and allowing them to take all values in the interval [0, 1]. The relaxed optimization problem is convex and can be easily solved using convex programming techniques. A rounding technique is then used to obtain a suboptimal solution for the original optimization problem from the solution of the relaxed optimization problem. Joint relay selection and resource allocation is still an open research issue. The challenge is to develop lowcomplexity schemes that treat all users fairly and that can be implemented feasibly without the requirement of signiﬁcant additional system resources.
230
Radio resource optimization in cooperative cellular wireless networks
8.5
Conclusion We have studied the problem of radio resource allocation in cooperative wireless networks. Our objective was to show how the usage of the available radio resources in these networks can be maximized by using diﬀerent optimization techniques. In formulating these optimization models, we have taken into consideration the architecture and various cooperation schemes deployed in these networks along with the constraints on the available radio resources such as bandwidth and transmit power. In a cooperative network, relay nodes play a vital role in achieving the intended beneﬁt of cooperative communication. The performance of the entire network depends on how these relays are selected for individual communications between source and destination nodes and how resource is allocated on the links between the relays and the communicating nodes. Jointly optimizing relay selection and resource allocation is an NPhard problem and is often handled as two separate subproblems. Although the bulk of this chapter has been devoted to analyzing diﬀerent optimizationbased approaches for resource allocation once relays are selected, we have also highlighted the diﬀerent relay selection methods that are proposed in the literature and several suboptimal methods to solve these subproblems jointly. Resource optimization in cooperative wireless networks remains, however, an open research problem and demands indepth future research.
References [1] T. Cover and A. E. Gamal, “Capacity theorems for the relay channels,” IEEE Transactions on Information Theory, vol. 25, no. 5, pp. 572–584, Sep. 1979. [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, part I: System description,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity, part II: Implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, Nov. 2003. [4] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Communications Magazine, vol. 42, no. 10, pp. 74–80, Oct. 2004. [5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [6] Y. W. Hong, W. J. Huang, F. H. Chiu, and C. C. J. Kuo, “Cooperative communications in resourceconstrained wireless networks,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 47–57, May 2007.
References
231
[7] Y. Liang and V. V. Veeravalli, “Gaussian orthogonal relay channel: Optimal resource allocation and capacity,” IEEE Transactions on Information Theory, vol. 51, no. 9, pp. 3284–3289, Sep. 2005. [8] M. Janani, A. Hedayat, T. E. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: Spacetime transmission and iterative decoding,” IEEE Transactions on Signal Processing, vol. 52, no. 2, pp. 362– 371, Feb. 2004. [9] T. E. Hunter and A. Nosratinia, “Outage analysis of coded cooperation,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 375–391, Feb. 2006. [10] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorems for relay networks,” IEEE Transactions on Information Theory, vol. 51, no. 9, pp. 3037–3063, Sep. 2005. [11] A. H. Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 2020–2040, Jun. 2005. [12] L. Le and E. Hossain, “Multihop cellular networks: Potential gains, research challenges and resource allocation framework,” IEEE Communications Magazine, vol. 45, no. 9, pp. 66–73, Sep. 2007. [13] Q. Zhang, J. Zhang, C. Shao, Y. Wang, P. Zhang, and R. Hu, “Power allocation for regenerative relay channel with rayleigh fading,” in Proc. of IEEE Vehicular Technology Conference (VTC), vol. 2, pp. 1167–1171, May 2004. IEEE, 2004. [14] T. Cover and J. Thomas, Elements of Information Theory, Wiley, 1991. [15] J. Zhang, Q. Zhang, C. Shao, Y. Wang, P. Zhang, and Z. Zhang, “Adaptive optimal transmit power allocation for twohop nonregenerative wireless relay system,” in Proc. of IEEE Vehicular Technology Conference (VTC), vol. 2, pp. 1213–1217, May 2004. IEEE, 2004. [16] E. I. Telatar, “Capacity of multiantenna gaussian channels,” European Transactions on Telecommunications, vol. 10, no. 6, pp. 585–595, Nov./Dec. 1999. [17] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [18] I. Maric and R. D. Yates, “Bandwidth and power allocation for cooperative strategies in gaussian relay networks,” in Proc. of 38th Asilomar Conference on Signal, System and Computers, vol. 2, pp. 1907–1911, Nov. 2004. IEEE, 2004. [19] J. N. Laneman and G. W. Wornell, “Distributed spacetimecoded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003. [20] I. Maric and R. D. Yates, “Forwarding strategies for Gaussian parallelrelay networks,” in Proc. of Conference on Information Sciences and Systems (CISS), vol. 2, pp. 591–596, Mar. 2004. Princeton University, 2004.
232
Radio resource optimization in cooperative cellular wireless networks
[21] K. T. Phan, L. B. Le, S. A. Vorobyov, and T. LeNgoc, “Centralized and distributed power allocation in multiuser wireless relay networks,” in Proc. of IEEE International Conference on Communications (ICC), pp. 1–5, Jun. 2009. IEEE, 2009. [22] Y. Zhao, R. S. Adve, and T. J. Lim, “Improving amplifyandforward relay networks: Optimal power allocation versus selection,” IEEE Transactions on Wireless Communications, vol. 6, no. 8, pp. 3114–3123, Aug. 2007. [23] M. Chen, S. Serbetli, and A. Yener, “Distributed power allocation strategies for parallel relay networks,” IEEE Transactions on Wireless Communications, vol. 7, no. 2, pp. 552–561, Feb. 2008. [24] J. Cai, X. Shen, J. W. Mark, and A. S. Alfa, “Semidistributed user relaying algorithm for amplifyandforward wireless relay networks,” IEEE Transactions on Wireless Communications, vol. 7, no. 4, pp. 1348–1357, Apr. 2008. [25] Z. Han, T. Himsoon, W. P. Siriwongpairat, and K. J. R. Liu, “Energyeﬃcient cooperative transmission over multiuser OFDM networks: Who helps whom and how to cooperate,” in Proc. of IEEE Wireless Communications and Networking Conference (WCNC), vol. 2, pp. 1030–1035, Mar. 2005. IEEE, 2005. [26] A. Nosratinia and T. E. Hunter, “Grouping and partner selection in cooperative wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp. 369–378, Feb. 2007. [27] S. Nam, M. Vu, and V. Tarokh, “Relay selection methods for wireless cooperative communications,” in Proc. of Conference on Information Sciences and Systems (CISS), pp. 859–864, Mar. 2008. Princeton University, 2008. [28] A. Bletsas, A. Lippnian, and D. P. Reed, “A simple distributed method for relay selection in cooperative diversity wireless networks, based on reciprocity and channel measurements,” in Proc. of IEEE Vehicular Technology Conference (VTC), vol. 3, pp. 1484–1488, May 2005. IEEE, 2005. [29] K. T. Phan, D. H. N. Nguyen, and T. LeNgoc, “Joint power allocation and relay selection in cooperative networks,” in Proc. of IEEE Global Telecommunications Conference (Globecom), pp. 1–5, Nov. 2009. IEEE, 2009.
9
Adaptive resource allocation in cooperative cellular networks Wei Yu, Taesoo Kwon, and Changyong Shin
9.1
Introduction The cellular structure is a central concept in wireless network deployment. A wireless cellular network comprises base stations geographically located at the centre of each cell serving users within the cell boundary. The assignment of the users to the base stations depends on the relative channel propagation characteristics. As a mobile device can usually receive signals from multiple base stations, the mobile is typically assigned to the base station with the strongest channel gain. Signals from all other base stations are regarded as intercell interference. However, at the cell edge, it is often the case that the propagation pathlosses from two or more base stations are similar. In this case, the signaltonoiseandinterference ratio (SINR) could be close to 0 dB, even if the mobile is assigned to the strongest base station. To avoid excessive intercell interference in these cases, traditional cellular networks employ a ﬁxed frequency reuse pattern so that neighboring base stations do not share the same frequency. In this manner, neighboring cells are separated in frequency so that celledge users do not interfere with each other. The traditional ﬁxed frequency reuse schemes are eﬀective in minimizing intercell interference, but are also resource intensive in the sense that each cell requires a substantial amount of nonoverlapping bandwidth, so that only a fraction of the total bandwidth can be made available for each cell. Consequently, the standardization processes for future wireless systems have increasingly targeted maximal frequency reuse, where all cells use the same frequency everywhere. In these systems, it is crucial to manage intercell interference using dynamic power control, frequency allocation, and rate allocation methods. Wireless channels are fundamentally impaired by fading, propagation loss, and interference. Intensive research has been focused on the mitigation of shortterm fading, where spatial, temporal, and frequency diversity techniques have been devised to combat the shortterm variation of the channel over time. Largescale fading, propagation loss, and intercell interference, however, call for diﬀerent approaches. As largescale channel and interference characteristics can often be Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
234
Adaptive resource allocation in cooperative cellular networks
estimated at the receivers and made available at the transmitter, rather than combating largescale fading the right approach is to adapt to it. To this end, cooperative communication has emerged as a promising future technology for dealing with the largescale channel impairments. This chapter considers two types of cooperative networks that speciﬁcally address the issues of intercell interference and pathloss.
r Base station cooperation This type of cooperative network explores the possibility of coordinating multiple base stations. In a traditional cellular network, each base station operates independently. In particular, each base station adapts to the channel propagation condition within each cell without considering the intercell interference it causes the neighboring cells. The intercell interference is always treated as part of the background noise. A network with base station cooperation is a network in which the transmission strategies among the multiple base stations are designed jointly. In particular, the base stations may cooperate in their power, frequency, and rate allocations in order to jointly mitigate the eﬀect of intercell interference for users at the cell edge. Such a cooperative network can also be thought of as an adaptive frequency reuse scheme where the frequency usage and transmission power spectrum are designed speciﬁcally according to the mobile locations and user traﬃc patterns. r Relay cooperation This type of cooperative network explores the use of relays to aid the direct communication between the base station and the remote subscribers. The pathloss is a fundamental characteristic of the wireless medium. The pathloss exponent, which is determined by the physics of electromagnetic wave propagation environment, typically ranges from 2 to 6. Consequently, propagation distance is a crucial factor that aﬀects the capacity of the wireless channel. The use of cooperative relays in a cellular network can be thought of as a method for reducing the propagation distance. Instead of adding more base stations to the network (which is costly), the idea of a cooperative relay network is to deploy relay stations within each cell so that the mobile users may connect to the nearest relay, rather than the base station which may be far away. Relay deployment substantially improves the areaspectral eﬃciency of the network. In both types of cooperative networks, resource allocation is expected to be a crucial issue. In a network with base station cooperation, base stations must jointly determine their respective power and bandwidth allocation for the purpose of minimizing intercell interference. In a cooperative relay network, power and bandwidth assignments need to be made for each of the basestationtorelay and relaytomobile links. The optimal allocation of these network resources has a signiﬁcant impact on the overall network performance. This chapter provides an optimization framework for power, bandwidth, and rate allocation in cooperative cellular networks. The network is assumed to employ orthogonal frequency division multiple access (OFDMA) which provides
9.2 System model
235
ﬂexibility in power, subchannel, and rate assignment for each link. This chapter covers the theory and practice of cooperative network design, and makes a case that cooperative communication is a key future technology that could signiﬁcantly improve the overall capacity of wireless networks. Throughout this chapter, it is assumed that the network employs an initial channel estimation phase so that the frequency selective channel gain between any arbitrary transmitter and receiver pair can be estimated and made known throughout the network. The assumption of channel knowledge is necessary in order to optimize the allocation of power, bandwidth, and rate in the network. In this chapter it is further assumed that channel estimation is perfect. In practical situations where channel estimation error exists, robust optimization design is needed. The impact of imperfect channel knowledge on the resource allocation of cooperative cellular networks has been dealt with in [1, 2], but is not directly addressed in this chapter. It should be noted that in this chapter we consider cooperative networks in which transmitting nodes cooperate in their transmission strategies only (e.g., power, bandwidth allocation), but not in actual signals. It is possible to envision a networkwide cooperative system where all the antennas from all the base stations are pooled together as a single antenna array at the signal level. Such a network multipleinput multipleoutput (MIMO) system is capable of achieving the ultimate areaspectral eﬃciency limit of the network, but is outside of the scope of this chapter.
9.2
System model In this chapter we consider wireless cellular networks employing an OFDMA scheme, where the total bandwidth is divided into a large number of subchannels, and where arbitrary scheduling, as well as power, frequency, and rate allocation, may be made for any transmitter–receiver pair throughout the network. The ﬂexibility of the OFDMA system in assigning resources throughout the network is one of its key advantages, but it also presents a challenge in resource optimization, as the number of optimization variables is typically quite large in a realistic network. This section presents a system model for the OFDMA network.
9.2.1
Orthogonal frequencydivision multiplexing (OFDM) The OFDM scheme was originally conceived as a way to combat the multipath or frequencyselective nature of the wireless channel. By utilizing an N point inverse fast Fourier transform (IFFT) at the transmitter and an N point fast Fourier transform (FFT) at the receiver, the available frequency band is divided into N orthogonal subchannels on which independent data transmissions take place. The orthogonalization of frequency dimensions relies on the use of a cyclic
Adaptive resource allocation in cooperative cellular networks
Frequency
236
3 2 3
2 4 1 1
3 1 3 1 2 2 1 1 2
3 2 3 2
2 3 2 1 4
1 1 1 1 4
1 2 2 2 2
4 2 4 2 1 1 4 1 2 2 4 4 2 4 3 2 3 Time
Figure 9.1. In an OFDMA system, the time and frequency dimensions are partitioned and can be assigned arbitrarily to multiple users in the cell.
preﬁx, and on the assumption that the channel is stationary within each OFDM symbol, an assumption which is made throughout this chapter. The OFDM system can also be thought of as a multipleaccess scheme, in which multiple users may occupy orthogonal frequency subchannels without interfering with each other. For example, in a cellular network, diﬀerent mobile users may communicate with the base station on nonoverlapping sets of frequency tones, so that diﬀerent users’ signals are separated in frequency. This is known as OFDMA. The idea of OFDMA can also be extended to a cooperative relay network in which diﬀerent transmitter–receiver pairs in the network use nonoverlapping sets of frequency tones. Orthogonalization within each cell is, in general, a good idea, as whenever a receiver is close in range to a nonintended transmitter, orthogonalization is needed to avoid mutual interference. The use of OFDM enables orthogonalization in the frequency domain, which along with scheduling (which is essentially orthogonalization in the time domain) allows an arbitrary division of orthogonal dimensions among users within each cell. The assignment of dimensions can be visualized in a time–frequency map as shown in Figure 9.1. It is implicitly assumed in the preceding discussion that when multiple transmitter–receiver pairs use OFDMA, the FFT at each receiver orthogonalizes not only the intended transmit signal, but also all the interfering signals. For this to happen, the received OFDM symbols from all transmitters must be symbol synchronized, as otherwise a leakage would occur from one tone to its neighboring tones [3]. For the downlink cellular setting, symbol synchronization is automatic. For the uplink, transmit timing oﬀset can be introduced to ensure synchronization at the receiver. A more challenging case is the relay cooperative network, where it is possible to have one relay communicating with a mobile on one set of frequency tones, while another relay communicates with a diﬀerent mobile on adjacent tones. In this case, simultaneous symbol synchronization at
9.2 System model
237
two diﬀerent receivers becomes diﬃcult. It is possible to use advanced techniques (such as a cyclic suﬃx in addition to a cyclic preﬁx [4]) to correct for these eﬀects. For simplicity, in the rest of this chapter it is assumed that leakage of this type is suﬃciently small that its eﬀect can be ignored.
9.2.2
Adaptive power, spectrum, and rate allocation An OFDM system allows arbitrary assignment of power, modulation format, and rates across the frequency domain for each transmitter–receiver pair. Assuming a ﬁxed type of modulation, e.g., quadrature amplitude modulation (QAM), and a ﬁxed target probability of error, the maximum bit rate in each OFDM tone is a function of the SINR on that tone. The overall achievable rate of the link is the sum of the bit rates across the tones, which can be expressed as R=
SINR(n) , log 1 + Γ n =1 N
(9.1)
where SINR(n) is the ratio of the received signal power to the noise and interference power at the receiver in tone n, and Γ is the SNR gap, which is a measure of the eﬃciency of the particular modulation and coding scheme employed. With strong coding, Γ can be made to be close to the informationtheoretical limit of 0 dB. In practical wireless systems, Γ can range from around 6 dB to 12 dB. The exact value of Γ depends on the modulation scheme, coding gain, and the probabilityoferror target. The use of the SNR gap to relate the SINR with the transmission rate is an approximation which is accurate for moderate and high SNRs. The exact relation between the SINR and the rate depends on a detailed probabilityoferror analysis, and would give rise to complex functional forms. The advantage of using the SNR gap approximation is that the resulting functional relation is amenable to analytic optimization, and it closely resembles the Shannon capacity formula for the additive white Gaussian noise channel. Note that because of the presence of intercell and intracell interference within the network, the SINRs of diﬀerent links in a cellular network are interdependent. For this reason, the optimization of achievable rates over all users across the network is, in general, a nontrivial problem.
9.2.3
Cooperative networks In this chapter we consider two types of cooperative cellular networks: networks where multiple base stations from diﬀerent cells may cooperatively set their power allocation across the frequency tones; and networks where relay stations may be deployed to transmit and receive information from the mobile users. It is assumed that an OFDMA scheme is used within each cell, and no two links within each cell can use the same frequency tone at the same time. This
238
Adaptive resource allocation in cooperative cellular networks
eliminates intracell interference. Intercell interference is still present especially for celledge users. Given a ﬁxed frequency and power allocation for all transmitters, the SINR for each link in every frequency tone can be easily computed as a function of the transmit powers and the direct and interfering channel pathlosses. The network optimization problem is that of coordinating the allocation of frequency tones and time slots to diﬀerent links within each cell and the allocation of power across time and frequency subject to total and peak power constraints, so that an overall network objective function is maximized.
9.3
Network optimization
9.3.1
Singleuser waterﬁlling The OFDM transmit power optimization problem for a single link has a classic solution known as waterﬁlling. In this section, we brieﬂy review the optimization principle behind waterﬁlling and set the stage for subsequent development for multiuser network optimization. For the singlelink problem where the noise and interference are assumed to be ﬁxed, the optimization of the achievable rate subject to a total power constraint can be formulated as N
h(n)2 P (n) log 1 + Γσ 2 (n) n =1
maximize
N
subject to :
P (n) ≤ Ptotal ,
n =1
0 ≤ P (n) ≤ Pm ax ,
(9.2)
where the optimization is over P (n), the transmit power on the frequency tone n. The channel pathloss h(n)2 and the combined noise and interference σ 2 (n) are assumed to be ﬁxed. The optimization is subject to a total power constraint Ptotal and a pertone maximum powerspectraldensity (PSD) constraint Pm ax . The waterﬁlling solution arises from solving the above optimization problem via its Lagrangian dual. Let λ be the dual variable associated with the total power constraint, then the Lagrangian of the above optimization problem is N N h(n)2 P (n) −λ (9.3) log 1 + P (n) − Ptotal . L(P (n), λ) = Γσ 2 (n) n =1 n =1 The constrained optimization problem is now reduced to an unconstrained one in which λ can be interpreted as the power price. Optimizing the above Lagrangian subject to peak power constraints by setting its derivative to be zero gives
Γσ 2 (n) 1 − P (n) = λ h(n)2
P m a x , 0
(9.4)
9.3 Network optimization
239
Figure 9.2. Singleuser waterﬁlling solution.
where [·]ba denotes a limiting operation with lower bound a and upper bound b. The optimal λ can then be found based on the total power constraint, either by a bisection or by using algorithms based on the sorting of the subchannels by their eﬀective noises. Equation (9.4) is the celebrated waterﬁlling solution for transmit power optimization over a single link. The name waterﬁlling comes from the interpretation that the eﬀective noise and interference Γσ 2 (n)/h(n)2 can be thought of as the bottom of a bowl, 1/λ can be thought of as the water level, and the power allocation process can be thought of as that of pouring water into the bowl. The optimal power is the diﬀerence between the water level and the bottom of the bowl, as illustrated in Figure 9.2. The fundamental reasons that an (almost) analytic and exact solution exists for this problem are that the objective function of the optimization problem (9.2) is a concave function of the optimization variables and the constraints are linear. Therefore, convex optimization techniques such as Lagrangian dual optimization can be applied. Modern wireless communication systems often implement adaptive power control and adaptive modulation schemes that emulate the optimal waterﬁlling solution. It should be noted that the exact shape of the optimal power allocation is not important. If one approximates the optimal solution by a constant power allocation where all subchannels that would receive positive power in the optimal solution receive equal power in this approximate solution, the value of the objective function would be close to the optimum [5, 6]. This is because the waterﬁlling relation, i.e., (9.4), operates on a linear scale on P (n), while the rate expression, i.e., (9.2), is a logarithmic function, which is not sensitive to the exact value of P (n) when it is large. Thus, in the implementation of waterﬁlling in practice, while it is important to identify the minimum channelgaintonoise ratio beyond which transmission should take place, the exact value of P (n) is not as important.
240
Adaptive resource allocation in cooperative cellular networks
In a cellular setting, whenever a particular cell implements power adaptation, it changes its interference pattern on its neighbors. Thus, when every cell implements waterﬁlling at the same time, the entire network eﬀectively reaches a simultaneous waterﬁlling solution, where the optimal power allocation in each cell is the waterﬁlling solution against the combined noise and interference from all other cells. Such a simultaneous waterﬁlling solution can typically be reached via an iterative waterﬁlling algorithm in a systemlevel simulation where the waterﬁlling operation is performed on a percell basis iteratively [7]. Mathematically, the most general condition for the convergence of such an iterative algorithm is not yet known, but iterative waterﬁlling has been observed to converge in most practical situations.
9.3.2
Network utility maximization In a network with multiple users, the transmit power, bandwidth, and rate allocation problem becomes considerably more complicated, as the achievable rates of various users become interdependent. There are several consequences of such interdependency. First, the improvement in the rate of one user generally comes at the expense of other users in the network. For example, in a multicell OFDMA setup, to improve the rate of one user, one has to increase either its frequency allocation or its transmit power. The former comes at the expense of the bandwidth allocation for other users within the cell. The latter comes at the expense of more interference for users in adjacent cells. In both cases, there is a tradeoﬀ between the rates of various users. The concept of rate region is often used to characterize such a physicallayer tradeoﬀ among the rates of various users as a function of their power and bandwidth allocations. Further, in a realistic network with many users running diﬀerent applications, the same rate improvement often brings a diﬀerent amount of beneﬁt to diﬀerent users depending on the application layer characteristics. For example, a rate improvement could result in higher video quality for one user engaged in a videoondemand service, or a faster ﬁle transfer by a diﬀerent user. The network must decide which of the two alternatives is preferable. Such a choice depends not only on the nature of the application, but also on revenue considerations. Therefore, a tradeoﬀ also exists in the application layer. Network utility maximization (NUM) is an optimization framework that captures both the physicallayer and the applicationlayer tradeoﬀs [8]. In this framework, each user has an associated utility, which is a function of its (windowed) ¯ i ). The utility is an increasing function and is average rate, denoted as Ui (R ¯ i for the assumed to be concave; it captures the desirability of having a rate R user i. The NUM problem is that of maximizing the sum of the utilities over all users in the network, subject to the achievability of these rates in the physical
9.3 Network optimization
241
layer, i.e., maximize
K
¯i ) Ui (R
i=1
subject to:
(R1 , R2 , · · · , RK ) ∈ R,
(9.5)
where the constraint is that each instantaneous ratetuple must be inside R, the achievable rate region at each time instance, which is deﬁned as the convex hull of the union of all achieveable ratetuples. It is implicitly assumed in the above problem formulation that the utility functions for diﬀerent users in the network are independent. This is a realistic assumption for the case where each user runs a separate application. In a specialized network, such a sensor network, where users collaborate in a speciﬁc task, it is conceivable that the utility of the network could depend jointly on all the ¯2 , . . . , R ¯ K ). A generalization of ¯1 , R rates, i.e., the objective is to maximize U (R NUM in this setting has been treated in [9].
9.3.3
Proportional fairness ¯i ) = A common choice of the utility function is the logarithm function, i.e., Ui (R ¯ log(Ri ). The choice of logutility leads to a proportional fair rate allocation, which is described in detail below. The network’s objective is to maximize the sum utility of the average rates of diﬀerent users in the network. The averaging is typically done in a windowed fashion, or more commonly, exponentially weighted as ¯ i + αRi , ¯ i = (1 − α)R R
(9.6)
where 0 < α < 1 is the forgetting factor. Assuming that αRi is small, the new contribution of the instantaneous rate Ri to the overall utility can be approximated as " ∂Ui "" ¯ ¯ (αRi ). (9.7) Ui (1 − α)Ri + αRi ≈ Ui (1 − α)Ri + ∂Ri "R i = R¯ i Under this approximation, the maximization of the sum utility, which is equivalent to the maximization of the incremental utility, becomes the maximization of the weighted sum rate, where the weights are determined by the derivative of the utility function evaluated at the present average rate. When the utility function is the logarithm function, the equivalent maximization problem reduces to maximize
K
wi Ri
i=1
subject to:
(R1 , R2 , · · · , RK ) ∈ R,
(9.8)
242
Adaptive resource allocation in cooperative cellular networks
¯ i . This is a weighted rate sum maximization problem for the where wi = 1/R instantaneous rates with weights equal to the inverse of the average rates. As these weights ﬂuctuate, and as the rate region R changes over time (due to, for example, user mobility and the fading characteristics of the underlying fading channel), the above optimization problem needs to be solved repeatedly. The proportional fair rate allocation was originally devised in the context of user scheduling [10]. The above discussion shows that it is also applicable to the power allocation problem. The use of proportional fairness utility is not the only way to reduce the network utility maximization to a weighted rate maximization problem. Alternatively, one may consider a system in which each user has an associated input queue, and where weights of the weighted rate sum maximization problem are determined as a function of the input queue length of each respective user [11, 12]. The important point is that in either approach, the application layer demand for rates (expressed in either the utility function or the queue length) is decoupled from the physical layer provision of rates. In both cases, the physicallayer problem is reduced to a weighted rate sum maximization problem.
9.3.4
Rate region maximization The reduction of the network utility maximization problem to a weighted rate sum maximization problem is a crucial step in the development of resource allocation algorithms for OFDMA networks. In an OFDMA network, the rate of each link is the sum of bit rates across the frequency tones. The reduction of the maximization of a nonlinear utility function of link rates to a weighted rate sum maximization essentially linearizes the objective function and decouples the objective function on a pertone basis, which simpliﬁes the problem signiﬁcantly. The rate region maximization problem also often has constraints that couple across the frequency tones. For example, each user may have a power constraint across the frequency. In addition, there is typically the constraint that no two users should occupy the same frequency tone within each cell. To solve the rate region maximization problem eﬃciently, it is important to decouple these constraints across the frequency tones as well. A key technique for achieving decoupling is to utilize the Lagrangian duality theory in optimization. For example, consider the case where N
hii (n)2 Pi (n) log 1 + Ri = Γ( j = i hj i (n)2 Pj (n) + σ 2 (n)) n =1
(9.9)
subject to a power constraint N n =1
Pi (n) ≤ Pi,total ,
(9.10)
9.3 Network optimization
243
where Pi (n) denotes the transmit power of user i in tone n, and hij (n) is the complex channel gain from the transmitter of user i to the receiver of the user j. The weighted rate sum maximization problem subject to the power constraint can be solved by dualizing with respect to the power constraint. This results in a dual function g(λi ) deﬁned as N 4 g(λi ) = max wi Ri − λi . (9.11) Pi (n) − Pi,total P i (n )
n =1
The point is that when the objective function is a weighted rate sum and the constraint is linearized via the use of Lagrangian dual variable λi , the above optimization problem reduces to N pertone problems: 4 hii (n)2 Pi (n) − λi Pi (n) . (9.12) max wi log 1 + Γ( j = i hj i (n)2 Pj (n) + σ 2 (n)) P i (n ) Just as in singleuser waterﬁlling, where the solution to a convex optimization problem reduces to solving the problem for each λ, then ﬁnding the optimal λ, similar algorithms based on λ search can be applied here. The reduction to an N pertone optimization problem ensures that the computational complexity for each step of this optimization problem with ﬁxed λi is linear in the number of tones. The theoretical justiﬁcation for the above duality approach is convexity. For convex optimization problems where the feasible set has a nonempty interior (which is almost always true in engineering applications), the maximum value of the original objective is equal to the minimum of the dual optimization problem min g(λi ).
λ i ≥0
(9.13)
The optimum λi can be found using search methods such as the ellipsoid method (which is a generalization of bisection search to higher dimensions) or the subgradient method (see, for example, [13]). An interesting fact is that this duality technique remains applicable even when the functional form of the rate expression is nonconvex as is the case in (9.9), as long as the OFDM system has a large number of dimensions in the frequency domain, which allows an eﬀective convexiﬁcation of the achievable rate region as a function of the power allocation. More rigorously, under general conditions, the duality gap between the original optimization problem and its dual goes to zero as the number of OFDM tones goes to inﬁnity [13, 14]. This fact allows the duality technique to be used in a wide variety of applications. To summarize, under the NUM framework, the network optimization problem for an OFDMA network under peruser power constraints reduces to a pertone weighted sum rate maximization problem with a linear power penalty term. The weights in weighted rate sum maximization are determined by the utility function. The power penalty weighting can be found using a generalization of bisection.
244
Adaptive resource allocation in cooperative cellular networks
Figure 9.3. A multicell network in which the base stations cooperate in user scheduling and power allocation across the frequency spectrum.
The rest of this chapter is devoted to two examples of cooperative OFDMA networks where the adaptive scheduling, and power, frequency, and rate allocation problem can be solved eﬃciently using this methodology.
9.4
Network with base station cooperation
9.4.1
Problem formulation Consider an OFDMAbased cellular network as shown in Figure 9.3 in which the base stations cooperate in setting their downlink transmit power, and the mobiles likewise jointly set their uplink transmit power in order to avoid excessive interference between the neighboring cells. The optimal design of this multicell cooperative network becomes that of designing a joint scheduling and power allocation scheme that decides in each frequency tone:
r Which user should be served in each cell? r What are the appropriate uplink and downlink transmit power levels? Scheduling can be thought of either as the optimal partitioning of the frequency among the users within each cell, or as the optimal assignment of users in each time slot for each frequency tone. Scheduling and power allocation need to be considered jointly in order to reach an optimal solution for the entire network. Assuming a proportional fairness objective function, the network optimization problem for the downlink is a weighted rate sum maximization problem max
K L l=1 k =1
wD ,lk RD ,lk ,
(9.14)
9.4 Network with base station cooperation
245
where RD ,lk is the instantaneous downlink rate of the kth user in the lth cell in a network consisting of L cells with K users per cell. The weights wD ,lk = ¯ D ,lk are the proportional fairness variables determined by the exponentially 1/R weighted average rates. Let k = fD (l, n) be the downlink scheduling function, which assigns a user k to the nth frequency tone in the lth base station. The downlink rate expression RD ,lk is then PDn ,l hnllk 2 , (9.15) log 1 + RD ,lk = Γ(σ 2 + j = l PDn ,j hnjlk 2 ) n ∈Dl k
where the summation is over frequency tones assigned to the kth user in the lth cell, i.e., Dlk = {nk = fD (l, n)}. Here, hnjlk is the channel transfer function from the jth base station to the kth user in the lth cell and in the nth tone, and PDn ,l is the downlink power allocation for the lth base station in the nth tone. The optimization is over PDn ,l . The weighted rate maximization problem is to be solved under the perbasestation power constraint for the downlink N
Pl (n) ≤ Pl,total
(9.16)
n =1
as well as possibly peak power constraints: m ax 0 ≤ PDn ,l ≤ SD .
(9.17)
In addition, there is the OFDMA constraint that no two users should occupy the same frequency tone within each cell. Finally, note that a similar optimization problem with corresponding rate and power expressions can be written for the uplink. The joint scheduling and power allocation problem formulated above has been studied in several works [15–17], in which key ideas such as iterative optimization of scheduling and power allocation and numerical methods for power adaptation have been proposed. The following section outlines the approach based on these works and provides a performance projection for networks with base station power cooperation.
9.4.2
Joint scheduling and power allocation The dual decomposition technique outlined in the previous section can be used to tackle the joint scheduling and power allocation problem above. The key fact is that after dualizing with respect to the total power constraint, the optimization problem decouples on a tonebytone basis: PDn ,l hnllk 2 − λD ,l PDn ,l , wD ,lk log 1 + maximize Γ(σ 2 + j = l PDn ,j hnjlk 2 ) l
subject to:
m ax 0 ≤ PDn ,l ≤ SD
∀l,
(9.18)
246
Adaptive resource allocation in cooperative cellular networks
where the optimization is over both the power variables PDn ,l as well as the scheduling function k = fD (l, n) across the L base stations for a ﬁxed tone n. The duality theory for OFDMA networks states that if the above pertone optimization problem can be solved exactly for each set of ﬁxed λD ,l s, an ellipsoid or subgradient search over λD ,l in an outer loop can be carried out to ﬁnd the optimal λD ,l , which then leads to the global optimum of the overall network optimization problem. The optimization problem (9.18) is a mixed integer programming problem with nonconvex objective function. Finding the global optimum for such an optimization problem is known to be a diﬃcult task. However, many approximation algorithms that can reach at least a local optimum exist. Although strictly speaking, the duality theory for OFDMA networks requires the global optimum solution of the pertone optimization problem, the local optimum solution already works quite well in practice. The rest of this section focuses on solving (9.18) using local optimum approaches. Observe ﬁrst that for the downlink, the scheduling step and the power allocation step can be carried out separately. This is because the scheduling choice at one base station does not aﬀect the amount of intercell interference at the neighboring cells. The intercell interference is a function of the power allocation only. Thus, an iterative algorithm can be devised so that one can ﬁnd the best schedule for a ﬁxed power allocation, then ﬁnd the best power allocation for the ﬁxed schedule [15]. The iteration always increases the objective function monotonically, so it is guaranteed to converge to at least a local optimum of the joint scheduling and power allocation problem. For the downlink, because the intercell interference is independent of the scheduling decisions at each cell, ﬁnding the best schedule for a ﬁxed power allocation is a percell optimization problem. In other words, each base station only needs to ﬁnd the user in each tone that maximizes the weighted rate. This amounts to a simple search among the K users in each cell. For a ﬁxed user schedule, the optimal power allocation problem becomes that of solving (9.18) for the set of scheduled users in each tone. This is a nonconvex problem with potentially multiple local optima. The ﬁrstorder condition for this optimization problem can be found by taking the derivative of the objective function and setting it to be zero:
PDn ,l hnllk 2
wD ,lk hnllk 2 = tnD ,j l + λD ,l , n n 2 2 + Γ(σ + j = l PD ,j hj lk  )
(9.19)
j = l
where k = fD (l, n) for l = 1, . . . , L,
tnD ,j l
Γhnlj k 2 = wD ,j k n PD ,j hnjj k 2
(SINRnD ,j )2 1 + SINRnD ,j
,
(9.20)
9.4 Network with base station cooperation
247
Figure 9.4. Waterﬁlling where the waterﬁlling level is modiﬁed by the tnD ,j l pricing terms. and SINRnD ,j =
PDn ,j hnjj k 2 Γ(σ 2 + i = j PDn ,i hnij k 2 )
(9.21)
with k = fD (j, n). The ﬁrstorder condition gives a waterﬁlling like condition if the terms tnD ,j l are considered to be ﬁxed. In this case, (9.19) suggests that the following power allocation is a local optimum of the pertone optimization problem: 6S Dm a x 5 2 n n 2 + P h  ) Γ(σ w D ,j j lk D ,lk j = l PDn ,l = − , (9.22) n hnllk 2 j = l tD ,j l + λD ,l 0
where k = fD (l, n). Note that this is similar to the singleuser waterﬁlling power allocation (9.4), except that the power is allocated with respect to the combined noise and interference, and that the waterﬁlling level λD ,l is modiﬁed by the additional tnD ,j l terms. This process is called modiﬁed waterﬁlling [18] and is illustrated in Figure 9.4. The tnD ,j l term can be interpreted as a summary of the eﬀect of allocating additional power at the lth base station on the downlink rate at the neighboring jth cell. A larger value of tnD ,j l signals a larger interference eﬀect from the lth cell to the jth cell. The multiuser waterﬁlling condition in (9.22) implies that when interference is present, the waterﬁlling level needs to be modiﬁed. The waterﬁlling level should decrease if the eﬀect of interference is strong, which suggests that the power allocation should be reduced. Note that the waterﬁlling level is also aﬀected by the proportional fairness weights wD ,lk . A larger weight suggests a higher waterﬁlling level. The terms tnD ,j l also have a pricing interpretation [19–22], which comes from the fact that tnD ,j l is the derivative of the jth base station’s data rate with respect to the lth base station’s power, weighted by the proportional fairness variable.
248
Adaptive resource allocation in cooperative cellular networks
A higher value of tnD ,j l suggests that the lth base station must pay a higher price for allocating its power in tone n, which is reﬂected in the modiﬁcation of the waterﬁlling level. The waterﬁlling condition (9.22) suggests that one way to coordinate multiple base stations in a cooperative cellular network is to allow base stations to exchange values of tnD ,j l with their neighbors. Note that the value of tnD ,j l depends on the ratio of the direct and the interfering channel gains, which can be easily estimated using pilot signals. Knowing tnD ,j l , each base station may use (9.22) to update its power allocation. This results in an iterative process. When it converges, it reaches a local optimum of the weighted rate sum maximization problem (9.18). This procedure is known as the modiﬁed iterative waterﬁlling algorithm [18]. In practice, it may be necessary to damp the iteration to ensure convergence [17]. Alternatively, one may resort to a direct numerical optimization of (9.18) [16, 17]. Starting from an initial power allocation, one may compute a gradient or Newton’s increment direction for the optimization objective, then successively improve the objective function until a local optimum is reached. The gradient can be computed locally at each base station based on the pricing terms tnD ,j l . To summarize, the coordination of base stations for the downlink can be eﬃciently implemented using an approach that iterates between coordinated scheduling and coordinated power allocation. The scheduling step for the downlink can be eﬃciently implemented on a percell basis; the power allocation step can be implemented if certain exchange of pricing information is allowed among the base stations. This iterative process, together with an outer loop that ﬁnds the optimal power prices λD ,l , reaches a local optimum of the weighted rate sum maximization problem. Much of the discussion in this section is also applicable to the uplink, except that optimal scheduling is no longer a percell problem. In the uplink, the assignment of users in each cell directly aﬀects the interference in neighboring cells, so an optimal uplink scheduler needs to consider the eﬀect of the interference as well. However, there is evidence suggesting that if one uses identical scheduling for both the uplink and the downlink, the network often already performs very well [17]. This can be justiﬁed in part by the fact that there is a duality between uplink and downlink channels. The capacity regions of the uplink and downlink channels are identical under the same power constraint.
9.4.3
Performance evaluation To illustrate the performance of the proportionally fair joint scheduling and power allocation method described in Section 9.4.2, we present simulation results for a multicell network with base station cooperation. The simulated network consists of 19 cells hexagonally tiled with 40 users per cell, occupying a total bandwidth of 10 MHz partitioned into 256 subchannels using OFDMA. For
249
9.4 Network with base station cooperation
Table 9.1. Uplink (UL) and downlink (DL) sum rates over 19 cells with 40 celledge users per cell with proportional fairness joint scheduling and cooperative power allocation among the base stations [17] 2.8 km
1.4 km
Base to base distance
UL
DL
UL
DL
Fixed power spectrum Adaptive power spectrum Improvement
125 Mbps 185 Mbps 48%
129 Mbps 181 Mbps 40%
137 Mbps 228 Mbps 66%
142 Mbps 227 Mbps 60%
simplicity, a maximum transmit PSD of −27 dBm/Hz is imposed at both the base stations and the remote users, but no total power constraint is imposed. A multipath fading channel model is used with 8 dB of lognormal shadowing. The channel pathloss is modeled as a function of distance d as 128.1 + 37.6 log10 (d) (in decibels). The background noise level is assumed to be −169 dBm/Hz. The joint proportionally fair scheduling and adaptive power allocation is expected to provide the largest improvement in performance for users at the cell edge where intercell interference is dominant. To illustrate the performance gain for celledge users, in the simulation users are placed at the cell edge on purpose. Table 9.1 illustrates a comparison of the achievable sum rates over all users in 19 cells for the adaptive power allocation algorithm vs. the constant transmit PSD scheme with proportionally fair scheduling. These results were originally reported in [17] and are consistent with other studies in this area [16]. It can be seen that depending on the base station to base station distance, a sum rate improvement of 40–60% is possible. The improvement is larger when the base stations are closer, because in this case the intercell interference is also larger. It is worth emphasizing that the sum rate improvement reported in Table 9.1 is for celledge users. If averaged over all users uniformly placed over the cell, the sum rate improvement would have been about 15–20%. Figure 9.5 illustrates the convergence behavior of the joint proportional fair scheduling and power allocation algorithm. Each iteration here consists of either an adaptive power allocation step or a scheduling step. Up to ten subiterations are performed within each power allocation step. The sum rates of each of the 19 cells are plotted. Note that the proportional fairness weights are also updated in each iteration. These weights ensure that the rates are allocated to all users with fairness. The simulation results clearly illustrate the value of coordinating base station PSDs in an interferencelimited multicell environment. The projected performance improvement is obtained by allowing base stations to exchange pricing information with each other, and by iteratively converging to a joint networkwide optimum.
250
Adaptive resource allocation in cooperative cellular networks
45
Downlink sum rate per cell (Mbps)
40 35 30 25 20 15 10 5 0
0
20
40
60
80 Iterations
100
120
140
160
Figure 9.5. Convergence of downlink sum rates in each of the 19 cells using modiﬁed iterative waterﬁlling with proportional fairness joint scheduling and power allocation.
9.5
Cooperative relay network Base station cooperation addresses the intercell interference problem for celledge users in a cellular network, but the celledge users’ performances are also fundamentally aﬀected by pathloss, which is distancedependent, and shadowing. A viable approach for dealing with pathloss is to deploy relay stations throughout the cell, so that a mobile user may connect to the base station via a relay, thereby reducing the eﬀective path distance. This section addresses the network optimization problem for the cooperative relay network. The resource allocation problem for the cooperative relay network has attracted much attention in the wireless cellular communication literature (e.g., [23–26]). There are many diﬀerent ways in which a relay may help the communication between a transmitter and receiver pair (also known as the source and the destination in the relay literature). In a decodeandforward protocol, the relay decodes the message from the source then reencodes and transmits to the destination. Alternatively, a relay may amplify and forward, or quantize its observation and forward it to the destination. In general, decodeandforward is a sensible strategy when the relay is located closer to the source than to the destination, while amplifyandforward and quantizeandforward are more suitable when the relay is closer to the destination. However, the question of which strategy is the most suitable is a complicated one, as it also depends on the power allocation at the source and at the relay, as well as the enduser’s rate requirement or utility function. An
9.5 Cooperative relay network
251
optimization framework for choosing the best cooperation strategy has been dealt with for a singlerelay link in [27], but the general optimization of relay strategies for a cellular network is likely to be quite hard. In the chapter we focus instead on a simpliﬁed model where only the decodeandforward protocol is used. This is done for the following reasons. First, the primary focus of this chapter is the use of relay for enhancing cellular coverage at the cell edge, in which case a sensible relay location within the cell is somewhere close to the halfway point between the base station and the cell edge. In this case, for both uplink and downlink transmissions, the lengths of both the source–relay and the relay–destination paths are about the same, making decodeandforward a suitable strategy. Second, decodeandforward oﬀers a digital approach to relaying. It eliminates the noise enhancement problem inherent in amplifying, or quantizing the relay observation. Third, in this chapter we consider the deployment of ﬁxed infrastructurebased relay stations. These relay stations typically have the computational resources to perform decoding and reencoding. Further, as the primary focus here is the use of relays to combat distancedependent pathloss, in this chapter attention is restricted to a twohop relay strategy, where the direct path from the source to the destination is ignored (as it is typically very weak). In the ﬁrst hop, the source transmits information to the relay. In the second hop, the relay decodes and retransmits the same information to the destination. Under these assumptions, the capacity of a single source–relay–destination link is simply the minimum of the source–relay and the relay–destination link capacities. The characterization of capacity becomes more involved if one considers the possibility that a single relay deployed in a cellular network may help multiple mobiles at the same time. Further, each mobile has a choice of either connecting to the base station directly, or through relays. The mobile may even choose to use diﬀerent relays for diﬀerent frequency tones. These possibilities are coupled with the allocation of power across the frequency tones. In the rest of this section we use the network optimization framework introduced earlier and provide a solution based on the duality theory to solve the bandwidth, rate, and power allocation problem for OFDMA relay networks. The methodology used here is the one proposed in [23] and [26].
9.5.1
Problem formulation Consider a wireless cellular network in which each cell is equipped with M relay stations located somewhere between the base station and the cell edge and at angles 360/M degrees apart from each other. There are K mobiles in each cell. Each mobile may connect either directly to the base station or through one of the relays in each frequency tone, (but the mobile may possibly use diﬀerent relays in diﬀerent tones.) There are a total of M + 1 links emanating from each mobile. These mobileoriginated links, plus the M links connecting the relays
252
Adaptive resource allocation in cooperative cellular networks
h(1,1) h(2,0) h(0,1) h h(3,2) (0,2)
h(0,3) h(5,0)
h(4,2)
h(5,2)
Figure 9.6. A cooperative relay network in which the mobiles may connect directly to the base station or through the relays. to the base station, give a total of K(M + 1) + M links in the entire cellular network. Label the base station as node 0, and the mobiles as nodes k = 1, . . . , K. Label each link by a pair of indices as follows: the basestation–relay links are labeled (0, m), with m = 1, . . . , M ; the mobile–basestation links are labeled (k, 0) and the mobile–relay links are labelled (k, m) with k = 1, . . . , K and m = 1, . . . , M . This labeling convention is illustrated in Figure 9.6. We consider a setup in which each cell employs an OFDMA scheme. Further, it is assumed that in each frequency tone, at most one link may be active at any given time. This assumption allows the intracell interference to be avoided completely, and simpliﬁes the numerical solution considerably. For simplicity, we also assume that scheduling and power allocation are done on a percell basis (i.e., without base station cooperation). The problem formulation presented here can be extended to a more general setting where spatial reuse is enabled within each cell, or where intercell cooperation is enabled across the cells, but the resulting optimization problem would become considerably more complex. Consider now the uplink scenario. Deﬁne the scheduling function as a mapping from the frequency tone to the link index, i.e., fU (n) = (i, j). With the assumptions stated above, the achievable rate for each link (i, j), denoted rU,(i,j ) , can be expressed as n n 2 PU,(i,j ) h(i,j )  , (9.23) log 1 + rU,(i,j ) = Γσn2 n ∈U( i , j )
where the summation is over all frequency tones assigned to that link, i.e., n n 2 U(i,j ) = {n(i, j) = fU (n)}, P(i,j ) denotes the transmit power, h(i,j )  denotes
9.5 Cooperative relay network
253
the channel gain for the link (i, j) at tone n, and σn2 denotes the combined intercell interference and noise. The achievable rate for each user, denoted Rk , is the sum of achievable rates of all links emanating from the mobile, i.e., RU,k =
M
rU,(k ,j ) .
(9.24)
j =0
At each relay, a ﬂow conservation constraint must be satisﬁed so that all the incoming traﬃc can be forwarded to the base station. This results in M constraints as follows: K
rU,(k ,m ) ≤ rU,(0,m ) .
(9.25)
k =1
The above equation is an example of the general ﬂow conservation formulation [23, 26]. It is now straightforward to write down the uplink percell optimization problem for the cooperative relay network, which consists of both the allocation of power and bandwidth for each link, and the routing of the information within each cell. Under the network utility maximization framework, the optimization problem can be reduced to a weighted rate sum maximization problem across ¯ U,k : the K users with weights wU,k = 1/R maximize
K
wU,k
subject to :
rU,(k ,m )
,
m =0
k =1 K
M
rU,(k ,m ) ≤ rU,(0,m )
∀m = 1, · · · , M,
max pnU,(0,m ) ≤ PU,R ,m
∀m,
k =1 N n =1 N M
max pnU,(k ,m ) ≤ PU,M ,k
m =0 n =1 max 0 ≤ pnU,(k ,m ) ≤ SU,(k ,m ) pnU,(k ,m ) pnU,(k ,m ) = 0,
∀k, ∀k, m,
∀n,
∀(k, m) = (k , m )
∀n, (9.26)
where rU,(k ,m ) is as expressed in (9.23), and the optimization is over power max allocations pnU,(k ,m ) , subject to the permobile total power constraint PU,M ,k , max the perrelay total power constraint PU,R ,m , as well as the peak PSD constraints max at both the mobiles and the relays SU,(k ,m ) . The last constraint ensures that no two links share the same frequency tone within each cell. Note that the downlink problem can be formulated in a similar fashion.
254
Adaptive resource allocation in cooperative cellular networks
9.5.2
Joint routing and power allocation The network utility maximization problem for the cooperative relay network is essentially a joint routing and power allocation problem, as each mobile has the option of either transmitting information bits directly to the base station or routing through one or more of the relays. One way to solve this problem is to dualize with respect to the ﬂow conservation constraint (9.25) using dual variables µm , so that the objective function becomes a new weighted rate sum maximization problem over all link rates (rather than the enduser rates in (9.26)). The new objective function is K k =1
wU,k
M
rU,(k ,m )
M
−
m =0
m =1
=
M m =1
µm
K
rU,(k ,m ) − rU,(0,m )
k =1
µm rU,(0,m ) +
M K
(wU,k − µm )rU,(k ,m ) , (9.27)
k =1 m =0
subject to the peak and total power constraints in (9.26). Let g(µ1 , · · · , µM ) denote the maximum value of (9.27) subject to the power constraints for any ﬁxed set of µm . Because of the zerodualitygap property of the OFDMA system, the solution of the original problem then reduces to the maximization of g(µ1 , · · · , µM ) over all µm . The dual variables, µm , enter the new objective function as weights to the weighted rate sum maximization problem over the link rates. Roughly speaking, a higher value for µm indicates congestion in the link between the base station and the mth relay and that more rate should be allocated to that link to release congestion. A lower value of µm indicates the opposite. The weighted link rate sum maximization problem subject to the power constraints can itself be solved by yet another dual decomposition step with respect to the total power constraints, as treated earlier in this chapter. In this case, the optimization problem is again decoupled on a tonebytone basis. Because of the assumption that only one link may be active in any given time slot and frequency tone, the weighted rate sum maximization then reduces to the selection of the best link for each frequency tone, which involves a simple search. Finally, the optimization of g(µ1 , · · · , µM ) over all µm can be handled by either an outer loop using the ellipsoid or the subgradient method. The search over the optimal set of µm balances the incoming and outgoing ﬂows at each relay.
9.5.3
Performance evaluation The optimization framework described above can be used to evaluate the eﬀectiveness of deploying relay stations in a cellular network. To take into account the cost of relay deployment, the performance of a baseline system with a cell diameter of 1.4 km is compared with a relay network in which the cell area is doubled
255
9.5 Cooperative relay network
Table 9.2. The achievable minimum and sum rates for a sevencell network: the baseline vs. relay scenarios (RS) with a varying number of relays per cell and relay locations [26] Scenario Relays per cell Relay distance Mobiles per cell Cell diameter (km) Cell area (km2 ) Mininum rate (Mbps) Sum rate (Mpbs)
Baseline
RS 1
RS 2
RS 3
RS 4
0 n/a 9 1.4 1.54 0.193 96.4
3 r 18 1.98 3.08 0.583 75.4
4 r 20 1.98 3.08 0.972 80.2
5 r 20 1.98 3.08 0.705 72.7
3 r 18 1.98 3.08 0.578 87.2
2 3
2 3
2 3
1 3
(with diameter 1.98 km), but with three, four or ﬁve relays deployed within each cell. The rationale is that if a relay station costs roughly 13 – 15 of a base station, then the deployment cost of both systems would be approximately the same. In both cases, the achievable uplink transmission throughput is computed for a sevencell system with users uniformly placed within the area. The user densities in both cases are the same. When the relays are deployed, they are located between 13 r and 23 r from the base station, symmetrically in the angular direction. Again, standard cellular channel models are used with both distancedependent pathloss and lognormal shadowing components. The simulation setting is as presented in [26]. The simulation methodology presented in the previous section is for a weighted rate–sum maximization problem formulation. However, instead of maximizing a proportionally fair utility function, the numerical results in this section pertain to a maximization of the minimum rate over all users in the system (similar to [23]), which requires a slight modiﬁcation of the optimization problem (9.26). First, the additional constraint that each user must have a rate larger than some minimal rate Rmin , where Rmin is a constant added. Then, the resulting optimization problem is solved with successively larger Rmin until the problem becomes infeasible. The largest such Rmin is the maximum minimal rate. In practice, the maximum Rmin can be found eﬃciently using a bisection. For simplicity, the adaptive scheduling and power and rate allocation here are implemented on a percell basis. For simulation purposes, the network throughput is computed using an iterative approach, in which the intercell interference is updated in each iteration, and the cellular network eventually reaches an equilibrium. Table 9.2 shows the maximum minimal rates for the baseline network without relays and for a number of relay scenarios. These results were ﬁrst presented in [26]. The most interesting feature here is that the addition of relays to the infrastructure improves the minimal rate dramatically; however, it does not make
256
Adaptive resource allocation in cooperative cellular networks
much diﬀerence at all in the sum rate. This illustrates that the beneﬁt of relays is concentrated on users at the celledge. As far as the sum rate is concerned, when the users are uniformly distributed throughout the cell, the sum rate is dominated by the rates of those users closest to the base station, which are not helped by the relays.
9.6
Conclusion In this chapter we have presented a network utility maximization framework for cooperative networks employing OFDMA. It has been shown that the objective of maximizing the sum of utilities of multiple users in a multicell network can be eﬃciently carried out using a number of techniques, including proportional fairness scheduling, dual optimization, the descent method for local optimization, and the network ﬂow conservation principle. A central observation here is that because the OFDM scheme partitions the frequency domain into many parallel subchannels, the NUM problem often decomposes into a tonebytone optimization problem, which is considerably easier to solve. In this chapter we have focused on two types of cooperative networks and formulated the corresponding joint scheduling, power adaptive and rate allocation problem in each case. For networks with base station cooperation, it has been shown that adaptively adjusting power allocation across the base stations has the eﬀect of reducing intercell interference, hence improving the throughput of the celledge users in the network. The celledge performance can also be reduced by deploying relays throughout the cells. The relays have the eﬀect of enhancing the coverage at the cell edge, which improves the minimal service rate within each cell. Base station cooperation and relay deployment are technologies with the potential to signiﬁcantly enhance the performance of the traditional wireless cellular network structure, especially at the cell edge. The beneﬁts brought by these cooperative techniques are particularly valuable to network service providers, because celledge users are the bottleneck in the current generation of wireless networks.
References [1] I. C. Wong and B. L. Evans, “Optimal resource allocation in the OFDMA downlink with imperfect channel knowledge,” IEEE Trans. Commun., vol. 57, no. 1, pp. 232–241, Jan. 2009. [2] Y. Cui, V. K. N. Lau, and R. Wang, “Distributive subband allocation, power and rate control for relayassisted OFDMA cellular system with imperfect system state knowledge,” IEEE Trans. Wireless Commun., vol. 8, no. 10, pp. 5096–5102, Oct. 2009.
References
257
[3] V. M. K. Chan and W. Yu, “Multiuser spectrum optimization for discrete multitone systems with asynchronous crosstalk,” IEEE Trans. Signal Processing, vol. 55, no. 11, pp. 5425–5435, Nov. 2007. ¨ [4] F. Sj¨ oberg, M. Isaksson, R. Nilsson, P. Odling, S. K. Wilson, and P. O. B¨orjesson, “Zipper: A duplex method for VDSL based on DMT,” IEEE Trans. Commun., vol. 47, no. 8, pp. 1245–1252, Aug. 1999. [5] P. S. Chow, Bandwidth optimized digital transmission techniques for spectrally shaped channels with impulse noise, PhD thesis, Stanford University, 1993. [6] W. Yu and J. M. Cioﬃ, “Constant power waterﬁlling: Performance bound and lowcomplexity implementation,” IEEE Trans. Commun., vol. 54, no. 1, pp. 23–28, Jan. 2006. [7] W. Yu, G. Ginis, and J. M. Cioﬃ, “Distributed multiuser power control for digital subscriber lines,” IEEE J. Select. Areas Commun., vol. 20, no. 5, pp. 1105–1115, June 2002. [8] M. Chiang, S. H. Low, A. R. Calderbank, and J. C. Doyle, “Layering as optimization decomposition: A mathematical theory of network architectures,” Proceedings of IEEE, vol. 95, no. 1, pp. 255–312, Jan. 2007. [9] J. Yuan and W. Yu, “Joint source coding, routing and power allocation in wireless sensor networks,” IEEE Trans. Commun., vol. 56, no. 6, pp. 886–898, June 2008. [10] E. F. Chaponniere, P. J. Black, J. M. Holtzman, and D. N. C. Tse, “Transmitter directed, multiple receiver system using path diversity to equitably maximize throughput,” US Patent 6,449,490, ﬁled July 1999. [11] L. Tassiulas and A. Ephremides, “Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE Trans. Automat. Contr., vol. 37, no. 12, pp. 1936– 1949, Dec. 1992. [12] M. J. Neely, E. Modiano, and C. E. Rohrs, “Dynamic power allocation and routing for time varying wireless networks,” IEEE J. Select. Areas Commun., vol. 23, no. 1, pp. 89–103, Jan. 2005. [13] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarrier systems,” IEEE Trans. Commun., vol. 54, no. 6, pp. 1310–1322, Jun. 2006. [14] Z.Q. Luo and S. Zhang, “Dynamic spectrum management: Complexity and duality,” IEEE J. Select. Areas Signal Processing, vol. 2, no. 1, pp. 57–73, Feb. 2008. [15] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling and power allocation in downlink multicell OFDMA networks,” IEEE Trans. Veh. Technol., vol. 6, no. 58, pp. 2835–2848, Jul. 2009. IEEE, 2009. [16] A. L. Stolyar and H. Viswanathan, “Selforganizing dynamic fractional frequency reuse for besteﬀort traﬃc through distributed intercell coordination,” in Proc. of IEEE INFOCOM, Apr. 2009. IEEE, 2009.
258
Adaptive resource allocation in cooperative cellular networks
[17] W. Yu, T. Kwon, and C. Shin, “Joint scheduling and dynamic power spectrum optimization for wireless multicell networks,” in Proc. of Conf. Inf. Sci. Sys. (CISS), Princeton, NJ, USA, Mar. 2010. Princeton University, 2010. [18] W. Yu, “Multiuser waterﬁlling in the presence of crosstalk,” in Proc. of Inf. Theory Appl. (ITA) Workshop, La Jolla, CA, USA, Jan.–Feb. 2007. University of California San Diego, 2007. [19] J. Huang, R. A. Berry, and M. L. Honig, “Distributed interference compensation for wireless networks,” IEEE J. Select. Areas Commun., vol. 24, no. 5, May 2006. [20] C. Shi, R. A. Berry, and M. L. Honig, “Distributed interference pricing for OFDM wireless networks with nonseparable utilities,” in Proc. of Conf. Inf. Sci. Sys. (CISS), Princeton, NJ, USA, Mar. 2008, pp. 755–760. Princeton University, 2008. [21] F. Wang, M. Krunz, and S. Cui, “Pricebased spectrum management in cognitive radio networks,” IEEE J. Sel. Top. Signal Processing, vol. 1, no. 2, pp. 74–87, Feb. 2008. [22] J. Yuan, Optimization techniques for wireless networks, PhD thesis, University of Toronto, 2007. [23] S.J. Kim, X. Wang, and M. Madihian, “Optimal resource allocation in multihop OFDMA wireless networks with cooperative relay,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1833–1838, May 2008. [24] R. Kwak and J. M. Cioﬃ, “The subchannelallocation for OFDMA relaying downlink systems with total power constraint,” in Proc. of IEEE Globecom, Dec. 2008. IEEE, 2008. [25] T. Ji, D. Lin, A. Stamoulis, A. Khandekar, and N. Bhushan, “Relays in heterogeneous networks,” in Proc. of Inf. Theory Appl. (ITA) Workshop, San Diego, CA, USA, 2009. University of California, San Diego, 2009. [26] J. Ji and W. Yu, “Bandwidth and routing optimization in wireless cellular networks with relays,” in Proc. of 5th Workshop on Resource Allocation, Cooperation and Competition in Wireless Networks (RAWNET/WNC3), Seoul, Korea, June 2009. IEEE, 2009. [27] T. C.Y. Ng and W. Yu, “Joint optimization of relay strategies and resource allocations in a cooperative cellular network,” IEEE J. Select. Areas Commun., vol. 25, no. 2, pp. 328–339, Feb. 2007.
10 Crosslayer scheduling design for cooperative wireless twoway relay networks Derrick Wing Kwan Ng and Robert Schober 10.1
Introduction Background. The degrees of freedom introduced by multiple antennas at the transmitters and receivers of wireless communication systems facilitate multiplexing gains and diversity gains [1, 2]. A wireless pointtopoint link with M transmit and N receive antennas constitutes an M byN multipleinput multipleoutput (MIMO) communication system. The ergodic capacity of an M byN MIMO fading channel increases almost linearly with min{M, N } provided that the fading meets certain mild conditions [2, 3]. Hence, it is not surprising that MIMO has attracted a lot of research interest since it enables signiﬁcant performance and throughput gains without requiring extra transmit power and bandwidth. However, limitations on the number of antennas that a wireless device is able to support as well as the signiﬁcant signal processing power and complexity required in MIMO tranceivers limit the gains that can be achieved in practice. To overcome the limitations of traditional MIMO, the concept of cooperative communication has been proposed for wireless networks such as ﬁxed infrastructure cellular networks and wireless adhoc networks [4, 5]. The basic idea of cooperative communication is that the singleantenna terminals of a multiuser system can share their antennas and create a virtual MIMO communication system. Thereby, three diﬀerent types of cooperation may be distinguished, namely, user cooperation, base station (BS) cooperation, and relaying. Theoretically, user cooperation and BS cooperation are able to provide huge performance gains, when compared with noncooperative networks. However, the required information exchange between users and BSs may make these options less attractive in practice. In contrast, cooperative relaying with dedicated relays requires signiﬁcantly less signaling overhead and allows for lowcost implementations while achieving signiﬁcant coverage extensions, diversity gains, and throughput gains compared with noncooperative transmission. Therefore, cooperative relaying has attracted signiﬁcant interest from both academia and industry. In theory, relays can transmit and receive signals at the same time and over the same frequencies; this is known as fullduplex relaying. However, building such relays requires Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
260
Crosslayer scheduling design for cooperative wireless relay networks
Figure 10.1. Oneway halfduplex relaying system.
Figure 10.2. Twoway halfduplex relaying system.
precise and expensive components which is undesirable in practice. Alternatively, relays may operate in a oneway halfduplex mode as shown in Figure 10.1, i.e., relays do not receive and transmit simultaneously at the same time and frequency. These relays are also referred to as cheap relays in the literature [6]. The main disadvantage of oneway halfduplex relaying is a loss in throughput compared with fullduplex relaying. Fortunately, this throughput loss can be recovered by twoway halfduplex relaying [12–15]. Compared with traditional oneway halfduplex relaying [16–18], twoway halfduplex relaying achieves higher power and spectral eﬃciencies, by allowing simultaneous message exchanges between a BS and the users, see Figure 10.2. Both amplifyandforward (AF) and decodeandforward (DF) protocols can be used for twoway halfduplex relaying. The AF protocol may be more appealing in practice because of its simple transceiver design. Crosslayer design. Combining crosslayer design with cooperative relaying has become an active research area [7–11]. Orthogonal frequency division multiple access (OFDMA) is a particularly attractive multiple access technique because of
10.1 Introduction
261
its ﬂexibility in resource allocation and its robustness against multipath fading. Crosslayer design can lead to signiﬁcant improvements in the overall system performance by facilitating the exchange of useful information between diﬀerent layers of the protocol stack. The eﬀect of crosslayer design on quality of service (QoS) provisioning in multiuser systems is an interesting, yet challenging, topic. Intuitively, by allocating resources to users with better channel qualities, the scheduler can maximize the overall system throughput and exploit the socalled multiuser diversity (MUD) gain. However, it may degrade other QoS metrics such as delay and fairness, since users are suspended from transmission when their channels are poor. Next generation wireless communications systems are expected to provide resourcehungry services such as video streaming and realtime video conferencing with certain QoS requirements. The combination of OFDMA/OFDM with twoway halfduplex relaying may be instrumental in meeting these requirements, particularly for users at the cell edge. In [32, 33], besteﬀort resource allocation for twoway halfduplex relayassisted OFDMA and OFDM systems for homogeneous users is studied. In these works, perfect global channel state information (CSI) of each link is assumed to be available at the BS such that optimal resource allocation can be performed. However, in practice, users are heterogeneous with diﬀerent QoS requirements, such as the maximum tolerable outage probability and minimum required data rate, which besteﬀort resource allocation cannot guarantee. Besides, perfect CSI at the transmitter (CSIT) cannot be achieved in practice for the relaytouser links due to the mobility of the users. When the CSIT is imperfect, there is a ﬁnite probability that the scheduled data rate exceeds the instantaneous channel capacity despite the use of strong forward error correction (FEC) codes, causing the transmitted packet to be corrupted which is known as channel outage. The conventional performance measure, ergodic capacity, fails to account for the penalty of channel outage. Furthermore, the asymmetric nature of fading in the BStorelay and relaytouser links in relay networks has often been overlooked in the system modeling, e.g., [12–18, 32, 33]), which may lead to inaccurate results with regard to the performance gain achievable with relays in a practical system. In addition, though the multiuser diversity capacity gain has been known to scale with the number of users K in the order of O(log log K) [26, 34] in Rayleigh faded singlehop systems with perfect CSIT, it is not clear how the system performance scales with the number of users and relays in a twoway halfduplex relayassisted OFDMA system with imperfect CSIT and asymmetric fading links. Furthermore, existing works such as [32–36] focus on centralized resource allocation at the BS. As the numbers of users, relays, and subcarriers in the system increase, the overhead in collecting CSI for scheduling becomes signiﬁcant and the computational complexity increases exponentially at the BS which limits the scalability of the system in practice. Contributions. Motivated by the aforementioned prior works, we propose and analyze in this chapter a novel crosslayer scheduling design for twoway
262
Crosslayer scheduling design for cooperative wireless relay networks
halfduplex relayassisted OFDMA systems. In particular, we focus on the following issues:
r We consider crosslayer scheduling for an AF twoway halfduplex relayassisted OFDMA system when both delaysensitive users and nondelaysensitive users are coexisting in the system. The considered problem is formulated as a mixed integer and nonconvex optimization problem which takes into account imperfect CSIT and the heterogeneous data rate requirements of users. After a proper transformation of the original nonconvex problem into a convex one, dual decomposition is used to derive a novel distributed iterative resource allocation algorithm with closedform solutions for the power, data rate, and subcarrier allocation. The proposed algorithm’s fast convergence and robustness to quantization eﬀects in the information exchanged between the BS and the relays in each iteration make it attractive for implementation. r We investigate the asymptotic behavior of the system goodput of proportional fair (PF) schedulers in a twoway halfduplex relay system with respect to the numbers of nondelaysensitive users and relays by using tools from extreme value theory. Closedform asymptotic order growth expressions are derived which reveal that an extra gain from a large number of users and relays can be achieved only under certain conditions. These expressions also allow us to quantify the penalties on the system performance caused by imperfect CSIT and a lineofsight (LoS) path. r We propose an eﬃcient computational burden reduction scheme which is aimed at alleviating the computational load at each relay. The proposed scheme preserves the essential MUD gain for various scheduling policies, such as maximum system goodput scheduling and PF scheduling. On the other hand, the computational load can be explicitly controlled by adjusting a corresponding threshold. Simulation results demonstrate signiﬁcant computational savings at the cost of a small performance degradation. Organization. The rest of the chapter is organized as follows. In Section 10.2, we discuss some basic concepts in crosslayer scheduler design. In Section 10.3, we outline the model for the considered twoway halfduplex relayassisted OFDMA system, and in Section 10.4, we formulate the crosslayer design for this system as an optimization problem. In Section 10.5, the problem considered is solved by dual decomposition, and in Section 10.6, we analyze the asymptotic order growth of the system goodput for large numbers of users and relays and introduce a computational burden reduction scheme. Simulation results for the distributed algorithm are provided in Section 10.7 and some conclusions are drawn in Section 10.8. Notation. E[·] and (·)∗ denote statistical expectation and complex conjugation, respectively. A complex Gaussian random variable with mean µ and variance σ 2 is denoted by CN (µ, σ 2 ), and ∼ means “distributed as”. 1(·) denotes an indicator function which is 1 when the event is true and 0 otherwise. O(g(x)) denotes an asymptotic upper bound. Speciﬁcally, f (x) = O(g(x)) if limx→∞ f (x)/g(x) ≤ W
10.2 Crosslayer scheduling design – some basic concepts
263
Figure 10.3. Traditional OSI reference model [19]. for W ∈ (0, ∞). Moreover, Q(c, d) is the ﬁrstorder Marcum Qfunction, I0 (·) is the zeroth order modiﬁed Bessel function of the ﬁrst kind, J0 (·) is the zeroth order Bessel function of the ﬁrst kind. u(·) is the unit step function and (x)+ = max{0, x}. Finally, all logarithms, unless further speciﬁed in the subscript, are assumed to have base e.
10.2
Crosslayer scheduling design – some basic concepts Traditional communication systems can be viewed as a hierarchy of layers, which is known as the open system interconnection (OSI) reference model (Figure 10.3). A layer is a collection of conceptually similar system functions which provide services to other layers. For example, the duty of the physical (PHY) layer is to transmit a bit stream over a given physical channel with a target bit error rate (BER), while the media access control (MAC) layer is charged with giving multiple users access to the channel and error checking. The layering concept emphasizes the isolation of the diﬀerent layers which means that each layer is optimized individually without considering the requirements of other layers. In other words, a layer treats the other layers as black boxes when performing its own duty. This isolated approach provides a higher ﬂexibility for the deployment of new protocols and simpliﬁes debugging and standardization compared with a joint optimization of all layers. However, while the rigid separation of layers is well suited for timeinvariant wireline communication channels, it is not
264
Crosslayer scheduling design for cooperative wireless relay networks
Figure 10.4. Timing diagram for crosslayer scheduling.
compatible with the requirements of next generation wireless communications systems. Current and future wireless communication systems not only have to support concurrent transmission of realtime voice and video data, but also nonrealtime services such as email and web surﬁng. In practice, these services require diﬀerent levels and diﬀerent types of QoS. For instance, realtime video conferencing is a delaysensitive application but it is relatively robust with respect to decoding errors. On the other hand, email is a delay nonsensitive application but does not tolerate even a single bit error. Unfortunately, the PHY layer as it is deﬁned in the OSI model does not guarantee any QoS since it does not receive any side information regarding QoS requirements from the upper layers. Therefore, crosslayer design/optimization is essential for a better utilization of the limited system resources, while guaranteeing the required QoS for each application at the same time. We would like to point out that it is a common misconception that crosslayer design is aimed at removing the concept of layering and performing an overall optimization of the communication system. Instead, the crux of crosslayer design is to enhance the system performance by allowing minimal information exchange between the layers.
10.2.1
Utility functionbased crosslayer optimization In each scheduling slot, the scheduler selects the users for the next transmission frame and determines their power and rate allocation based on the information available at the scheduler such as the CSI of all users and the length of their queues (Figure 10.4). Ideally, a crosslayer scheduler should exploit both the
10.2 Crosslayer scheduling design – some basic concepts
265
information from the PHY layer and that from the layers above the MAC layer in order to achieve the best possible performance. In the literature, utilityfunctionbased crosslayer optimization [20, 21] is one of the widely used methods for solving resource allocation problems. There are two main functionalities of the utility function U(·). First, it captures the essential scheduling criteria by using the available crosslayer information building a bridge between diﬀerent layers. Second, it maps the users’ utilization of the system resources into a level of satisfaction providing a tangible performance metric. In the following, we brieﬂy discuss two utility functions commonly used for crosslayer scheduling. Maximum throughput scheduler. In most wireless applications, the aggregate data rate of users is the most important ﬁgure of merit for evaluation of the system performance from the service provider’s point of view. Considering a system with K users, the corresponding utility function can be expressed as 6 5K K (10.1) Rk = E rk , UThp (R1 , . . . , RK ) = k =1
k =1
where Rk = E[rk ] is the average throughput of user k and rk is the instantaneous throughput of user k in each time slot. Schedulers designed to maximize the above utility function achieve the highest average system capacity and are referred to as maximum throughput schedulers. Proportional fair (PF) scheduler. Although the maximum throughput scheduler results in the optimal utilization of the system resources, it does not take into account fairness in the scheduling process. Users with poor channel conditions may suﬀer from starvation since they are rarely selected for transmission which is undesirable from the users’ point of view. Therefore, PF scheduling [1, 2] was proposed to resolve the fairness issue. PF schedulers are popular because they allow a balance to be struck between overall system capacity and fairness among users, and they have been implemented in third generation (3G) cellular systems for delaytolerant applications. The corresponding utility functions are given by UPF,1 (R1 , . . . , RK )
K
˜k ) log2 (R
(10.2)
6 K rk . ≈E ˜k R k =1
(10.3)
≈
k =1
and
5
UPF,2 (R1 , . . . , RK )
˜k The utility functions in both (10.2) and (10.3) achieve proportional fairness. R is the approximation of Rk . In order to implement PF schedulers, the system has ˜ k . For the (n + 1)th scheduling slot, the average to keep track of the value of R
266
Crosslayer scheduling design for cooperative wireless relay networks
˜ k [n + 1] of user k can be updated as follows: data rate R ⎧ 1 ˜ 1 ⎪ ⎪ Rk [n] + rk [n] , k = k∗ , ⎨ 1− tc tc ˜ Rk [n + 1] = 1 ˜ ⎪ ⎪ 1− Rk [n] , k= k∗ , ⎩ tc
(10.4)
where tc is the time constant for the averaging window and k ∗ is the selected user at the (n + 1)th time slot.
10.2.2
Qualityofservice (QoS) measure The increasing demand for highdatarate wireless service networks imposes great challenges on crosslayer optimization since operators are required to satisfy the diverse QoS requirements of heterogeneous user populations. Diﬀerent QoS measures have to be incorporated in the crosslayer optimization in order to overcome these challenges. While many diﬀerent QoS measures have been considered in the literature, we discuss here only the two most important ones. Minimum data rate requirement. In practical systems, users are usually heterogeneous with diﬀerent minimum data rate requirements imposed by the maximum delay constraints for the respective applications they are running. One way to handle these requirements, is to incorporate data rate constraints in the formulation of the crosslayer optimization problem [22]. Intuitively, the scheduler will ﬁrst try to serve the delaysensitive users which have nonzero data rate constraints. Once these users are served, the remaining resources will be allocated to the nondelaysensitive users. However, it is clear that this kind of resource allocation results in a degradation of the overall system performance since the scheduler loses degrees of freedom in the user selection. Frame error rate (FER). Although at the PHY layer the BER is usually considered as a performance measure, at the MAC layer the FER is more relevant. In general, the FER is hard to calculate analytically and typically results in complicated expressions which are not useful for crosslayer scheduling design. However, in slow fading channels, if channel capacity achieving codes are used for error protection (such as, e.g., turbo codes or lowdensity paritycheck codes (LDPCs)), the outage probability is a good approximation for the FER [23, 24]. This connection between the outage probability and FER can be often exploited to arrive at a simple resource allocation algorithm.
10.2.3
Multiuser diversity gain In a multiuser wireless communication system, diﬀerent users experience diﬀerent fading conditions. When the number of users in the system is large, there is a high probability that at least one user has a very good channel at any time. This
10.2 Crosslayer scheduling design – some basic concepts
267
eﬀect is known as multiuser diversity [25]. It can be exploited by scheduling a user for transmission only when his/her channel is in a favorable condition. Therefore, the average system throughput increases with the number of users if only good users are selected for transmission in each scheduling slot. This performance gain plays an important role in the crosslayer scheduling design. In the following, we present a simple example to illustrate the concept of multiuser diversity in a system with scheduling and resource adaptation. Example 10.1 We consider a wireless system with ﬁve users having inﬁnite backlog in their buﬀers as illustrated in Figure 10.5. We assume that only one user is scheduled in each scheduling slot and the channel state of each user is either good or bad with equal probability. If the channel states of the users are perfectly known at the transmitter and a capacity achieving code is applied for error protection, then 6 bps/Hz and 1 bps/Hz can be successfully delivered1 to scheduled users having good and bad channels, respectively. We consider ﬁrst a simple roundrobin (RR) scheduling2 scheme where users take turns to access the channel periodically regardless of their actual channel states. Since all users have the same channel access probability, the average system throughput of this scheduling scheme is 6 × 12 + 1 × 12 = 3.5 bps/Hz. On the other hand, by taking advantage of CSIT, the scheduler can select a user whose channel is in the good state (provided there is such a user) which results in maximum throughput scheduling. The average system throughput of the latter scheduling scheme is 215 × 1 + (1 − 215 ) × 6 = 5.84375 bps/Hz. Maximum throughput scheduling tries to operate the system at the peak transmission rate based on CSIT knowledge and the resulting performance gain is the multiuser diversity gain. In the following, we will quantify the multiuser diversity gain for maximum throughput scheduling for a large number of users based on extreme value theory. Asymptotic multiuser diversity gain. Extreme value theory is a branch of statistics which deals with the extreme value, such as the maximum or the minimum, of a set of random variables. It has been widely used for analyzing the asymptotic performance of maximum throughput scheduling [26, 27]. In the following, we introduce a fundamental theorem from extreme value theory. Theorem 10.1 (Generalized extreme value distribution [28]) Let x1 , x2 , . . . , xZ , be a sequence of Z independent and identically distributed (i.i.d.) random 1 2
Spectral eﬃciencies of 6 bps/Hz and 1 bps/Hz correspond to 64ary quadrature amplitude modulation (64QAM) and binary phase shift keying (BPSK), respectively. RR is the simplest scheduling technique for scheduling users and is widely used in TDMA systems such as GSM.
268
Crosslayer scheduling design for cooperative wireless relay networks
(6 bps/Hz) (1 bps/Hz)
feedback b
Figure 10.5. An example for crosslayer scheduling with two channel states.
variables and let xmax denote the maximum among the Z random variables, i.e., xmax = max(x1 , x2 , . . . , xZ ).
(10.5)
If there exist some constants aZ ∈ R, bZ > 0 and some nondegenerate distribution G(x) such that the distribution of aZ (xmax − bZ ) converges to G(x), then G(x) must be one of the following three standard extreme value distributions: Gumbel distribution, Fr´echet distribution, and Weibull distribution. Next, in the following lemma, we introduce a suﬃcient condition on the distribution of xi such that the limiting maximum distribution is a Gumbel distribution with cumulative distribution function (CDF) G(x) = Pr{X ≤ x} = exp(− exp(−x)),
x ∈ R.
(10.6)
Lemma 10.1 (Suﬃcient conditions for converging to Gumbel distribution [28]) Let {x1 , x2 , . . . , xK } be a sequence of K positive i.i.d. random variables with probability density function (PDF) f (·) and twice diﬀerentiable CDF F (·). Let xmax = max{x1 , x2 , . . . , xK } denote the maximum among the K random variables. If the reciprocal hazard function g(x) satisﬁes lim g(x) = lim
x→∞
x→∞
1 − F (x) = c, f (x)
(10.7)
269
10.2 Crosslayer scheduling design – some basic concepts
6
Average system throughput (bps/Hz)
5
4
3
2 Simulation at SNR = 10 dB Asymptotic trend at SNR = 10 dB Simulation at SNR = 5 dB Asymptotic trend at SNR = 5 dB Simulation at SNR = 0 dB Asymptotic trend at SNR = 0 dB
1
0
0
10
20
30
40
50 60 Number of users
70
80
90
100
Figure 10.6. Average system throughput vs. number of users for a Rayleigh fading channel and diﬀerent received SNRs. where c is a constant, then xmax − lK converges to a Gumbel distribution where lK is given by F (lK ) = 1 − 1/K. This suggests that xmax grows like lK for K → ∞. Now, let us consider K Rayleigh distributed channel gain coeﬃcients, {h1 , h2 , . . . , hK }, i.e., {h1 2 , h2 2 , . . . , hK 2 } are exponentially distributed with unit mean. By evaluating the corresponding reciprocal hazard function, it can be shown that (1 − F (x))/f (x) = 1 and lK = log K. Therefore, by selecting the maximum channel gain among K users, the multiuser gain grows in the order of log K. Figure 10.6 depicts the system goodput as a function of the number of users K assuming that all users have i.i.d. Rayleigh fading channels. Both simulation results and analytical results obtained by exploiting the asymptotic growth of the maximum channel gain are shown. If the maximum throughput scheduler is used for scheduling, it can be observed from Figure 10.6 that the multiuser diversity provides a signiﬁcant throughput gain as the number of users increases. However, as the number of users becomes large the growth rate of the multiuser diversity gain decreases since the average system throughput grows with the number of users in the order of log log(K). Now, that we have established the basic idea behind crosslayer scheduling, we will focus in the remainder of this chapter on a particular application, namely crosslayer scheduling for a twoway halfduplex AF relayassisted OFDMA network.
270
Crosslayer scheduling design for cooperative wireless relay networks
1 2
Figure 10.7. Relayassisted packet transmission system model with K = 9 users, A = 3 sectors, and M = 3 relays.
10.3
Network model for relayassisted OFDMA system In this section, we ﬁrst establish the adopted system and channel models, and subsequently discuss the CSI model assumed for crosslayer scheduling.
10.3.1
System model We consider a singleantenna twoway halfduplex relay OFDMA network which consists of one BS, M relays, and K mobile users which belong to one of two categories, namely, delaysensitive users and nondelaysensitive users. Without loss of generality, we assume that the ﬁrst K1 users are delaysensitive users belonging to set D = {1, 2, . . . , K1 } and the remaining K − K1 users are nondelaysensitive users belonging to set N = {K1 + 1, K1 + 2, . . . , K}. A single cell with two ringshaped boundary regions as shown in Figure 10.7 is studied. The cell coverage is divided into A equal size sectors and each user is assigned to a group of GRj > 0 relays such that M = A j =1 GRj . In this chapter, we focus on the crosslayer scheduling design for relayassisted users and we assume that the resource allocation for nonrelayassisted users (e.g., users close to the BS) is done separately.3 In the model considered, there is no direct link between the BS and the users due to pathloss and heavy blockage. We adopt the frame structure 3
We note that a joint resource allocation for relayassisted and nonrelayassisted users would entail a signiﬁcantly higher computational complexity at the BS than separate resource allocations.
10.3 Network model for relayassisted OFDMA system
271
Figure 10.8. Frame structure and CSI correlation model.
of IEEE 802.16m [30] where a superframe is divided into F frames (Figure 10.8). The channel is assumed to be slowly time varying over a superframe but to remain constant within a frame. In each scheduling slot, at the beginning of each superframe, scheduling and resource allocation are performed. In each frame, the information exchange between the BS and the users via the relays is accomplished in two phases. In the ﬁrst phase, the BS and the users transmit their signals to the relay stations through a multiple access channel. Then, in the second phase, the relay stations amplify the previously received signals and forward them to the corresponding users.
10.3.2
Channel model We consider an OFDMA system with nF subcarriers. The channel impulse response is assumed to be timeinvariant (slow fading) within a frame. In the ﬁrst phase of frame t ∈ {1, . . . , F }, the received symbol in subcarrier i ∈ {1, . . . , nF } at relay m ∈ {1, . . . , M } for user k ∈ {1, . . . , K} is given by @ @ (t,k ) (t,k ) (t) (t) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) PB R m ,i lB R m HB R m ,i Xi + PU R m ,i lU R m HU R m ,i Wi +ZR m ,i , YR m ,i = (10.8) (t,k )
(t,k )
where PB R m ,i and Xi are the transmit power and the transmit symbol for the link between the BS and relay m in subcarrier i in the ﬁrst phase of transmission (t) of frame t, respectively. lB R m represents the pathloss between the BS and relay (t,k )
(t,k )
(t,k )
, and lU R m are deﬁned in a similar manner as the m. Variables PU R m ,i , Wi corresponding variables for the BStorelay links except that the signaling direction is from the users to the relays. ZR m ,i is the additive white Gaussian noise (t) (t,k ) (AWGN) in subcarrier i at relay m, and HB R m ,i and HU R m ,i are, respectively the smallscale fading coeﬃcients between the BS and relay m and between relay m and user k in subcarrier i. In practice, diﬀerent links in a relay network can experience asymmetric fading conditions [37]. For instance, a strong LoS propagation channel is expected between the BS and the relays, since they are placed
272
Crosslayer scheduling design for cooperative wireless relay networks
Phase 1
Phase 2
Broadcasts to the BS and users
Signals from the BS and users
f tr t
Figure 10.9. Subcarrier mapping in a twoway halfduplex relay OFDMA system for users 1, 2, and 3. in relatively high positions in practice and the number of blockages between them (t) is limited. Hence, HB R m ,i is modeled as Rician fading with Rician factor κ, i.e., (t) (t,k ) HB R m ,i ∼ CN ( κ/(1 + κ), 1/(1 + κ)). On the other hand, we model HU R m ,i as (t,k )
Rayleighdistributed, i.e., HU R m ,i ∼ CN (0, 1), since the users are generally surrounded by a large number of scatterers. In order to optimize the system performance [33], the signals received at relay m from the BS and user k in subcarrier i are mapped to subcarrier p ∈ {1, . . . , nF } in the second transmission phase as shown in Figure 10.9. Furthermore, the signal in subcarrier p is forwarded @ to the destination after being ampliﬁed by a gain factor
(t,k )
(t,k )
GR m ,i,p PR m ,p , where
(t,k )
PR m ,p is the transmit power of relay m in subcarrier p for user k and the BS, and (t,k )
GR m ,i,p normalizes the input power of the relay. Since the channel is assumed to be timeinvariant for the two transmission phases, channel reciprocity is preserved. Therefore, the signal received at user k in subcarrier p from relay m in frame t is given by @ (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) Y˜U m ,p = GR m ,i,p PR m ,p lU R m HU R m ,p @ (t,k ) (t) (t) (t,k ) (t,k ) PB R m ,i lB R m HB R m ,i Xi × + Im ,i + ZR m ,i + Zp(k ) , (10.9) (t,k )
where Im ,i =
@
(t,k )
(t,k )
(t,k )
(t,k )
PU R m ,i lU R m HU R m ,i Wi (k ) Zp
represents the selfinterference of
user k in subcarrier i and is the AWGN at user k in subcarrier p. For simplicity and without loss of generality, we assume a normalized noise variance
10.3 Network model for relayassisted OFDMA system
273
of N0 = 1 at all transceivers. By estimating the channel coeﬃcients in each frame and exploiting the channel reciprocity, the selfinterference is perfectly known (t,k ) at user k and can be subtracted from Y˜U m ,p [12]. Therefore, the received signal after selfinterference cancellation can be written as @ (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) YU m ,p = GR m ,i,p PR m ,p lU R m HU R m ,p @ (t,k ) (t) (t) (t,k ) × PB R m ,i lB R m HB R m ,i Xi + ZR m ,i + Zp(k ) . (10.10) Similarly, the received signal at the BS in subcarrier p from user k is given by @ (t,k ) (t,k ) (t,k ) (t) (t) YB m ,p = GR m ,i,p PR m ,p lB R m HB R m ,p @ (t,k ) (t,k ) (t,k ) (t,k ) × PU R m ,i lU R m HU R m ,i Wi + ZR m ,i + Zp , (10.11) where Zp is the AWGN at the BS in subcarrier p. Following [12], the gain is chosen as GR m ,i,p −1 = 1 + PB R m ,i lB R m HB R m ,i 2 + PU R m ,i lU R m HU R m ,i 2 . (t,k )
10.3.3
(t,k )
(t)
(t)
(t,k )
(t,k )
(t,k )
(10.12)
Channel state information (CSI) In the system considered, both the BS and the users exchange information with the help of relays. As mentioned before, we assume the CSI of all links to be constant for the duration of one frame. Thus, the BS, the relays, and all (scheduled) users can accurately estimate the CSI of their links using training symbols in each frame for relaying and signal detection purposes. For scheduling, we assume that the relays perfectly know the pathlosses of their respective BStorelay and relaytouser links due to accurate longterm measurements. Furthermore, to perform scheduling for the next superframe, the relays estimate the smallscale fading coeﬃcients of their respective BStorelay and relaytouser links based on training symbols sent by the BS and all users at the beginning of the scheduling slot (Figure 10.8). Since we assume that both the BS and the relays are static, the associated channel is timeinvariant and is assumed to remain constant for the duration of the entire superframe. In contrast, due to the mobility of the users, the CSI of the relaytouser links changes slowly and the CSIT of these links used for scheduling becomes outdated over the duration of a superframe. To capture this eﬀect, we model the CSI of the relaytouser links using Jake’s model [38]. For simplicity, we assume that each scheduling slot has the length of one frame. As illustrated in Figure 10.8, the correlation between the scheduling slot (frame 0) and frame t in subcarrier i of the link between relay m and user (t,k ) (0,k ) k is given by E[HU R m ,i (HU R m ,i )∗ ] = ρ(t × τ ) with ρ(τ ) = J0 (2πfD τ ), where τ and fD are the time duration of one frame and the maximum Doppler frequency, (t,k ) respectively. Therefore, for scheduling purposes, the actual CSI, HU R m ,i , in frame t ∈ {1, . . . , F } for the link between user k and relay m in subcarrier i can be
274
Crosslayer scheduling design for cooperative wireless relay networks
expressed as (t,k )
HU R m ,i = (t,k )
ˆ (t,k ) + ∆H (t,k ) , 1 − σe2 (t)H U R m ,i U R m ,i
(10.13)
(0,k )
ˆ where H U R m ,i = HU R m ,i is the outdated CSI used for scheduling at the beginning (t,k )
of the superframe. Here, ∆HU R m ,i ∼ CN (0, σe2 (t)) represents the CSIT error with (0,k )
(t,k )
variance σe2 (t) = 1 − ρ2 (t × τ ) and E[HU R m ,i ∆HU R m ,i ] = 0.
10.4
Crosslayer design for twoway relayassisted OFDMA systems In this section, we introduce the system goodput as a performance measure for crosslayer scheduling and formulate the crosslayer optimization problem.
10.4.1
Instantaneous channel capacity and system goodput Given perfect CSI at the receiver (CSIR), the downlink (DL) instantaneous channel capacity between the BS and user k in using subcarrier pair (i, p) through relay m in frame tis given by the mutual information which can be expressed as (t,k ) (t,k ) 1 CD L m ,i,p ≈ 2 log2 1 + ΓDLm ,i,p with equivalent DL SNR (t,k )
ΓDLm ,i,p = (t,k )
(t)
(t)
(t,k ) (t,k )
(t,k )
PB R m ,i lB R m HB R m ,i 2 PR m ,p lU R m HU R m ,p 2 (t,k )
(t)
(t)
(t,k )
(t,k )
(t,k )
(t,k ) (t,k )
(t,k )
PB R m ,i lB R m HB R m ,i 2 + PU R m ,i lU R m HU R m ,i 2 + PR m ,p lU R m HU R m ,p 2
,
(10.14) 1 2
where the approximation and the prelog factor in the channel capacity equation are due to a highSNR assumption and the two channel uses necessary for transmitting one message, respectively. Similarly, the channel capacity for the endtoend uplink (UL) from user k in using subcarrier pair (i, p) through relay (t,k ) (t,k ) m in frame t is given by CU L m ,i,p ≈ 12 log2 (1 + ΓULm ,i,p ) with equivalent UL SNR (t,k )
ΓULm ,i,p = (t,k )
(t,k )
(t,k )
(t,k ) (t)
(t)
PU R m ,i lU R m HU R m ,i 2 PR m ,p lB R m HB R m ,p 2 (t,k )
(t)
(t)
(t,k )
(t,k )
(t,k )
(t,k ) (t)
(t)
PB R m ,i lB R m HB R m ,i 2 + PU R m ,i lU R m HU R m ,i 2 + PR m ,p lB R m HB R m ,p 2
.
(10.15) Now, we are ready to deﬁne the instantaneous goodput (bps/Hz successfully delivered) of DL and UL transmission for user k who is assigned to relay m as (k )
ρD L m (k ) ρU L m
=
nF F nF 1 (t,k ) (t,k ) (t,k ) (t,k ) s r 1(rD L m ,i,p ≤ CD L m ,i,p ), Fn F t=1 i=1 p=1 m ,i,p D L m , i , p
=
nF F nF 1 (t,k ) (t,k ) (t,k ) (t,k ) s r 1(rU L m ,i,p ≤ CU L m ,i,p ), Fn F t=1 i=1 p=1 m ,i,p U L m ,i,p
(10.16)
10.4 Crosslayer design for twoway relayassisted OFDMA systems
275
(t,k )
where Fn F = F × nF , sm ,i,p ∈ {0, 1} is the subcarrier pair allocation indicator, (t,k )
(t,k )
and rD L m , i , p and rU L m , i , p are the transmission data rates for DL and UL, respectively. The average weighted system goodput, which is deﬁned as the total average number of bps/Hz successfully decoded at the BS and the K users via the M relays (averaged over multiple scheduling phases), is given by 5 M 6 (k ) (k ) (k ) w (ρD L m + ρU L m ) , (10.17) Ug oodpu t (P, R, S) = E m =1 k ∈Um
where P, R, and S are the power, rate, and subcarrier allocation policies, respectively. Um is the set of users associated with relay m and w(k ) is a positive constant that can be used to enforce certain notions of fairness such as proportional fairness and max–min fairness [39].
10.4.2
Crosslayer design problem In practice, a channel outage occurs in slow fading channels whenever the data rate exceeds the channel capacity. Furthermore, users are heterogeneous with different data rate requirements. Therefore, a practical scheduler has to be able to fulﬁll the diﬀerent data rate requirements of the users as well as their channel outage probability requirements. This leads to the following optimization problem.
Problem (Crosslayer optimization problem) The optimal power allocation policy, P ∗ , rate allocation policy, R∗ , and subcarrier allocation policy, S ∗ , are given by
subject to:
(P ∗ , R∗ , S ∗ ) = arg max Ug oodpu t (P, R, S), P,R,S 0 / (t,k ) (t,k ) C1: Pr rD L m ,i,p > CD L m ,i,p Ξm ≤ ε, ∀k, t, m, i, p, / 0 (t,k ) (t,k ) C2: Pr rU L m ,i,p > CU L m ,i,p Ξm ≤ ε, ∀k, t, m, i, p, C3: C4:
M m =1 M
(k )
(k )
ρD L m + ρU L m
nF nF F
m =1 k ∈Um t=1
≥ R(k ) , (t,k )
∀k ∈ D ∩ U m ,
(t,k )
(t,k )
(t,k )
sm ,i,p (PU R m ,i + PB R m ,i + PR m ,p ) ≤ PT , i=1 p=1 (t ,k )
Pm , i , p
C5:
nF M
(t,k )
sm ,i,p = 1, ∀p, t,
m =1 k ∈Um i=1
C6:
nF M
(t,k )
sm ,i,p = 1,
m =1 k ∈Um p=1 (t,k ) C7: sm ,i,p ∈ {0, 1}, ∀m, i, p, k, t, (t,k ) (t,k ) (t,k ) C8: PB R m ,i , PU R m ,i , PR m ,p ≥ 0,
∀i, t,
∀m, i, p, k, t.
(10.18)
276
Crosslayer scheduling design for cooperative wireless relay networks
ˆ U R m , Lm ] is the CSI matrix, where H ˆ U R m , HB R m , and Lm Here, Ξm = [HB R m , H (k ) ˆ are vectors which contain the estimated CSIT, HU R m ,j , for all links from relay m to users k ∈ Um , the actual CSIT, HB R m ,i , for the link between the BS and relay m, and the pathloss for all links involving relay m, respectively. D ∩ U m is (t,k ) (t,k ) (t,k ) (t,k ) the intersection of sets Um and D, and Pm ,i,p = PU R m ,i + PB R m ,i + PR m ,p is the power usage for one subcarrier pair. Furthermore, C1 (C2) represents the outage probability requirement of the DL (UL) for user k in each frame and limits the maximum outage probability to ε. This constraint is used as a measure for the QoS with respect to the FER which can be well approximated by the channel outage probability if capacity achieving codes are applied [23, 24]. C3 enforces the minimum required data rate for delaysensitive users which are chosen by the application layer. C4 is the joint power constraint for the BS, relays, and users with total maximum power PT . Although the BS, relays, and users have diﬀerent power supplies in practice, a joint power optimization provides insight into the power usage of a whole communication link (both UL and DL) rather than the perhop required power. Furthermore, the global CSI of the entire system is needed in all devices in order to derive the optimal power allocation if separate power constraints are used [32], which would limit the system’s scalability due to signiﬁcant signalling overheads. Constraints C5, C6, and C7 are imposed to guarantee that each subcarrier pair is only used by one user in each frame.
10.5
Crosslayer optimization solution In this section, the problem considered is solved by dual decomposition and a novel distributed iterative scheduling algorithm is derived to reduce the computational complexity at the BS.
10.5.1
Transformation of the optimization problem The problem considered is a mixed combinatorial and nonconvex optimization problem. The combinatorial nature of the problem is due to the integer constraint for subcarrier allocation while the nonconvexity is caused by the power allocation variables in the objective function. In general, a brute force approach is needed to obtain the globally optimal solution. However, such a method does not provide any system design insights and has limited scalability. In order to obtain an insightful iterative solution, we ﬁrst assume that the ratio between the DL and UL transmitted power is ﬁxed, i.e., (t,k )
PB R m ,i (t,k )
P(B + U ) m ,i
(t,k )
(t,k )
(t,k )
(t,k )
(t,k )
= αm ,i P(B + U ) m ,i , PU R m ,i = (1 − αm ,i )P(B + U ) m ,i , (10.19) (t,k )
(t,k )
= PB R m ,i + PU R m ,i ,
277
10.5 Crosslayer optimization solution
(t,k )
where 0 < αm ,i < 1 controls the transmit power ratio between UL and DL, and (t,k )
P(B + U ) m ,i represents the power consumption on subcarrier i by the BS and user k. We introduce the following useful lemma. Lemma 10.2 (HighSNR power consumption) Assuming a high SNR, the optimal power allocation for the BS, relay m, and user k in using subcarrier pair (i, p) in frame t is given by (t,k )
(t,k )
PR m ,p
(t,k )
=
Pm ,i,p Pm ,i,p (t,k ) , P(B + U ) m ,i = , 2 2
=
(1 − αm ,i )Pm ,i,p αm ,i Pm ,i,p (t,k ) , PB R m ,i = . 2 2
(t,k )
(t,k ) PU R m ,i
(t,k )
(t,k )
(t,k )
(10.20)
Proof. Please refer to the Appendix at the end of this chapter. A key step in solving the optimization problem in (10.18) is to incorporate the outage probability constraints in C1 and C2 into the objective function. In general, a very tedious expression will be obtained if we try to incorporate the inequality in C1 and C2 into the objective function and it is virtually impossible to obtain a tractable resource allocation solution. To obtain ﬁrstorder design insight and a simple resource allocation algorithm, we restrict the problem such that the constraints in C1 and C2 are fulﬁlled with equality for the optimal solution. Simulations (not shown here) suggest that C1 and C2 are always satisﬁed with equality for the low outage probabilities required in practical applications (e.g., ε ≤ 0.1). We are now ready to introduce the following lemma. Lemma 10.3 (Equivalent data rate incorporating outage probability) For a given outage probability ε in C2, the equivalent UL data rate in using subcarrier pair (i, p) is (t,k )
C2 ⇒ rU L m ,i,p =
1 (t,k ) log2 1 + ΛUL m ,i,p , ∀i, p 2
(10.21)
with equivalent UL receive SNR (t,k )
ΛULm ,i,p = (t,k )
(t,k )
(t,k )
−1(t,k )
(t)
(t)
Pm ,i,p (1 − αm ,i )lU R m FU R m ,i (ε)lB R m HB R m ,p 2 (t,k ) (t)
(t)
(t,k )
(t,k )
−1(t,k )
(t)
(t)
2 αm ,i lB R m HB R m ,i 2 + (1 − αm ,i )lU R m FU R m ,i (ε) + lB R m HB R m ,p 2
, (10.22)
−1(t,k )
where FU R m ,i (ε) is the inverse CDF of a noncentral chisquare random variable ˆ (t,k ) 2 /σe2 (t). Here, with two degrees of freedom and noncentrality parameter H U R m ,j 2 σe (t) is the estimation error variance deﬁned in (10.13).
278
Crosslayer scheduling design for cooperative wireless relay networks
On the other hand, for a given outage probability ε in C1, the equivalent DL data rate in using subcarrier pair (i, p) is (t,k ) (t,k ) (10.23) C1 ⇒ rD L m ,i,p = 12 log2 1 + ΛDLm ,i,p (t,k )
with equivalent DL receive SNR ΛDL m ,i,p at user k given by ⎧ (t,k ) (t,k ) (t) (t) (t,k ) −1(t,k ) ⎪ Pm ,i,p αm ,i lB R m HB R m ,i 2 lU R m FU R m ,i (ε) ⎪ ⎪ , if i = p, ⎪ ⎪ (t,k ) (t) (t) ⎪ 2 + (2 − α(t,k ) )l(t,k ) F −1(t,k ) (ε) ⎨ 2 αm l H  ,i B R m m ,i B R m ,i U R m U R m ,i (t,k ) ΛDLm ,i,p = (10.24) (t,k ) (t,k ) (t) (t) −1(t,k ) 2 ⎪ Pm ,i,p αm ,i lB R m HB R m ,i  (FU R m ,i,p (ε))2 ⎪ ⎪ ⎪ , if i = p, ⎪ ⎪ −1(t,k ) ⎩ 2 α(t,k ) l(t) H (t) 2 + (F (ε))2 m ,i
B Rm
B R m ,i
U R m ,i,p
−1(t,k )
where FU R m ,i,p (ε) is the inverse CDF of a random variable which is the ratio of two independent nonidentical Rice distributed random variables (10.42). Proof. Please refer to the appendix. A new objective function which incorporates the outage of both UL and DL can be obtained by substituting (10.21) and (10.23) into the original objective function (10.17). The next step in solving the problem is to handle the combi(t,k ) natorial constraint in C7. We adopt a timesharing approach4 by relaxing sm ,i,p (t,k )
to be a real value between 0 and 1 instead of a Boolean, i.e., 0 ≤ sm ,i,p ≤ 1. Furthermore, since the power constraint is instantaneous, average weighted system goodput maximization is equal to the instantaneous weighted goodput maximization in each scheduling slot. Thus, the crosslayer scheduling optimization problem is transformed into the following convex optimization problem. Problem (Transformed crosslayer optimization problem) arg max
P,R,S
(t,k ) nF nF M F w(k ) sm ,i,p m =1 k ∈Um t=1 i=1 p=1
subject to: C4:
2Fn F
nF nF M F m =1 k ∈Um t=1
(t,k ) (t,k ) P˜(B + U ) m ,i + P˜R m ,p ≤ PT , C3, i=1 p=1
C5, C6,
(t ,k ) P˜m , i , p
(t,k )
C7: 0 ≤ sm ,i,p ≤ 1, C8:
(t,k ) C˜m ,i,p
∀m, i, p, t, k,
(t,k ) (t,k ) (t,k ) P˜B R m ,i , P˜U R m ,i , P˜R m ,p
≥ 0,
∀m, i, p, t, k,
(10.25)
(t,k ) (t,k ) (t,k ) (t,k ) (t,k ) where C˜m ,i,p = log2 (1 + ΛDL m ,i,p /sm ,i,p ) + log2 (1 + ΛUL m ,i,p /sm ,i,p ) is the data (t,k ) rate incorporating the eﬀects of outage and time sharing. P˜(B + U ) m ,i = (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) s P , P˜ =s P , and P˜ =s P are auxiliary m ,i,p
4
(B + U ) m ,i
R m ,p
m ,i,p
R m ,p
m ,i,p
m ,i,p
m ,i,p
In which the authors showed that this relaxation is asymptotically optimal w.r.t. the number of subcarriers.
279
10.5 Crosslayer optimization solution
(t,k ) (t,k ) (t,k ) power variables, and P˜m ,i,p = P˜(B + U ) m ,i + P˜R m ,p is the total power usage of subcarrier (i, p) in frame t for user k. Since the Hessian matrix of the objective function in (10.25) is negative semideﬁnite, the problem is jointly concave w.r.t. the optimization variables, and the duality gap is equal to 0 under some mild conditions [40]. Thus, centralized numerical methods such as the interiorpoint method can be used to solve the above problem. However, for centralized methods, the overhead required for collecting the CSI of all links and the computational complexity at the BS grows exponentially w.r.t. the number of subcarriers and users. Therefore, an optimal distributed iterative solution with reduced signaling overhead in the CSI collection and lower computational complexity will be derived in the next section.
10.5.2
Dual problem formulation In this subsection, the transformed crosslayer scheduling optimization problem is decomposed into one master problem (solved at the BS) and several subproblems (solved at each relay) by using dual decomposition. For this purpose, we ﬁrst need the Lagrangian function of the primal problem. Upon rearranging terms, the Lagrangian is given by L(λ, β, δ, υ, P, R, S) = (t,k ) nF nF M F M sm ,i,p (w(k ) + δk ) (t,k ) C˜m ,i,p − R(k ) δk 2F n F m =1 t=1 i=1 p=1 m =1 k ∈Um
+
nF F
k ∈Um
(t)
(βi
(t)
+ υi ) − λ
nF nF M F
(t,k ) P˜m ,i,p + λPT −
m =1 k ∈Um t=1 i=1 p=1
t=1 i=1 nF nF M F
(t)
(t,k )
(βp(t) + υi )sm ,i,p ,
(10.26)
m =1 k ∈Um t=1 i=1 p=1
where λ is the Lagrange multiplier corresponding to the joint power constraint and δ is a vector of Lagrange multipliers corresponding to the data rate constraint with elements δk , k ∈ {1 . . . K}, with δk = 0 for nondelaysensitive users. Furthermore, Lagrange multiplier vectors β and υ are associated with the sub(t) (t) carrier usage constraints and have elements βp , p ∈ {1, . . . , nF }, and υi , i ∈ {1, . . . , nF }, t ∈ {1, . . . , F }, respectively. Thus, the dual problem is min
max
λ,β,δ,υ≥0 P,R,S
10.5.3
L(λ, β, δ, υ, P, R, S).
(10.27)
Distributed solution – subproblem for each relay station By dual decomposition, relay station m can solve the following subproblem without any assistance from other relays. max
P,R,S
Lm (λ, β, δ, υ, P, R, S),
280
Crosslayer scheduling design for cooperative wireless relay networks
where Lm (λ, β, δ, υ,P, R, S) = (t,k ) nF nF F sm ,i,p (w(k ) + δk ) (t,k ) (t,k ) (t) (t,k ) C˜m ,i,p− λP˜m ,i,p − (βp(t) + υi )sm ,i,p . 2F n F t=1 i=1 p=1 k ∈Um
(10.28) Using standard optimization techniques and the Karush–Kuhn–Tucker (KKT) condition, the optimal power allocation in high SNR for subcarrier pair (i, p) is given by + 1 (w(k ) + δk ) 1 ∗(t,k ) (t,k ) ∗(t,k ) (t,k ) ˜ − Pm ,i,p = sm ,i,p Pm ,i,p = sm ,i,p − , (t,k ) (t,k ) λ 2ΛUL m ,i,p 2ΛDL m ,i,p (10.29) where δk is equal to 0 for nondelaysensitive users, i.e., δk = 0, ∀k ∈ N . Power allocation (10.29) can be interpreted as a multilevel waterﬁlling scheme as the water levels of diﬀerent users can be diﬀerent. Speciﬁcally, the water levels of delaysensitive users, i.e., (w(k ) + δk /λ), are generally higher than those of nondelaysensitive users, in order to satisfy constraint C3 in (10.18). To obtain the optimal subcarrier allocation, we take the derivative of the subproblem with (t,k ) respect to sm ,i,p and substitute the optimal power allocation (10.29) into the derivative. The resulting subcarrier pair selection determined by relay station m is given by ⎧ (t,k ) (t,k ) (t) (t) ΛUL m ,i,p ΛDL m ,i,p ⎪ 2(βp + υi ) ∗(t,k ) ⎨ ˜ + + , 1, if Cm ,i,p ≥ ∗(t,k ) (t,k ) (t,k ) sm ,i,p = w(k ) + δk 1 + ΛUL m ,i,p 1 + ΛDL m ,i,p ⎪ ⎩ 0, otherwise, (10.30) ∗(t,k )
(t,k )
(t)
(t)
where C˜m ,i,p = C˜m ,i,p P˜ ( t , k ) = P˜ ∗( t , k ) . The dual variables βp and υi m ,i,p
can be inter
m ,i,p
preted as the shadow price associated with the usage of subcarrier pair (i, p) in frame t. Dual variable δk forces the scheduler to assign more subcarrier pairs to delaysensitive users by lowering the price for satisfying the data rate requirements. The Lagrange multipliers λ, β, δ, and υ are provided by the BS in each iteration. Finally, by substituting (10.20) and (10.29) into the equivalent packet ∗(t,k ) outage constraints in (10.21) and (10.23), the optimal rate allocation rD L m ,i,p ∗(t,k )
and rU L m ,i,p can be calculated.
10.5.4
Solution of the master problem at the BS To solve the master problem at the BS, each relay calculates the local resource ∗(t,k ) ∗(t,k ) ∗(t,k ) ∗(t,k ) usages and passes this information, i.e., rD L m ,i,p , rU L m ,i,p , sm ,i,p , and Pm ,i,p , to the BS. Since the dual function is diﬀerentiable, the gradient method can be used to solve the minimization in (10.27) at the BS. Thus, the solution is given
281
10.6 Asymptotic performance analysis
by 5 βp(t) (n
+ 1)
βp(t) (n)
=
− ξ1 (n) 1 −
5 (t) υi (n
+ 1)
/ δk (n + 1)
=
=
6+ (t,k ) sm ,i,p
m =1 k ∈Um i=1
− ξ2 (n) 1 −
nF M
∀p, t, (10.31)
, 6+
(t,k ) sm ,i,p
∀i, t, (10.32)
,
m =1 k ∈Um p=1
0+ δk (n) − ξ3 (n) ∆R(k ) 1 ∆R(k ) < 0 , ∀k ∈ D ∩ U m ,
5 λ(n + 1)
(t) υi (n)
=
nF M
λ(n) − ξ4 (n) PT −
M
nF nF F
(10.33)
6+ (t,k )
Pm ,i,p
,
(10.34)
m =1 k ∈Um t=1 i=1 p=1
where ∆R(k ) is the diﬀerence between the scheduled data rate and the target (k ) (k ) (k ) ). data rate for delaysensitive user k, i.e., ∆R(k ) = M m =1 (ρD L m + ρU L m − R Index n is the iteration index, ξ1 (n), ξ2 (n), ξ3 (n), and ξ4 (n) are positive step sizes, and convergence to the optimal solution is guaranteed under some mild conditions on the step sizes [41]. The overall algorithm tries to select the best user who can maximize the outage incorporated data rate for both DL and UL while any selected subcarrier pair is eventually occupied by one user only. In summary, in each iteration, the BS broadcasts the Lagrange multipliers, which indicate the price of global resource usage for all relays. Then each relay solves the subproblem based on its local CSI and passes the solution back to the BS. Finally, the BS updates the Lagrange multipliers according to (10.31)–(10.34) and broadcasts them to all relays in the next iteration. This process is repeated until convergence or the maximum number of iterations are reached.
10.6
Asymptotic performance analysis and computational complexity reduction scheme In this section, we analyze the asymptotic order growth of the average system goodput with respect to the numbers of nondelaysensitive users KN = K − K1 and relays M . In addition, an eﬃcient computational complexity reduction scheme is proposed for the relays in order to achieve the simplicity required for a practical implementation.
10.6.1
Asymptotic analysis of system goodput The schedulers sacriﬁce system performance in order to fulﬁll the data rate requirements of the delaysensitive users, since system resources are allocated to them regardless of their actual channel conditions. In contrast, nondelaysensitive users allow a more ﬂexible resource allocation which beneﬁts the overall
282
Crosslayer scheduling design for cooperative wireless relay networks
system performance. Therefore, in this section, we analyze the asymptotic order growth of the average weighted system goodput w.r.t. the numbers of nondelaysensitive users KN = K − K1 and relays M . In order to obtain a tractable result, we assume that the distances between the BS and each relay are identical and that relays do not perform subcarrier mapping, and we focus on the study of PF schedulers with longterm fairness consideration. PF scheduling provides a good compromise between maximizing system capacity and achieving fairness among users and has been implemented in practical systems such as the highspeed downlink packet access (HSDPA). For longterm fairness, the pathloss of the users is disregarded by the PF scheduler and the user selection is based only on the instantaneous i.i.d. smallscale fading channel gain [29]. In other words, the near–far eﬀect is omitted and each user is served at its own peak channel gain, and therefore each user has the same channel access probability. The analysis is divided into two cases. In case I a large number of nondelaysensitive users KN and a growing number of relays M such that their ratio can be written as limK N ,M →∞ KN /M → ∞ are considered. In case II a large number of relays M and a growing number of nondelaysensitive users KN with ratio limK N ,M →∞ M/KN → ∞ are studied. The results are summarized in the following theorem. Theorem 10.2 (Asymptotic system goodput for PF scheduler in high SNR5 ) log M . (10.35) Case I: Ug oodpu t (P, R, S) = O log2 κ+1 Case II: Ug oodpu t (P, R, S) for 0 ≤ σe2 (·) < 1. = O log2 (1 − σe2 (F × τ )) log KN (10.36) Proof. Please refer to the Appendix. Theorem 10.2 illustrates that the asymptotic growth of the average system goodput is indeed the asymptotic growth of the cutset bound of a twoway relay channel [14] in both cases. If KN and M do not satisfy either limK N ,M →∞ M/KN → ∞ or limK N ,M →∞ KN /M → ∞, the gain achieved by either large KN or large M will be quickly saturated due to noise ampliﬁcation in the AF relays. On the other hand, both Case I and Case II demonstrate the beneﬁts of using twoway halfduplex relays which are able to recover the spectral eﬃciency loss of 12 asymptotically when compared with oneway halfduplex relays. Moreover, the terms κ + 1 and 1 − σe2 (F × τ ) act as penalties on the system performance introduced by the LoS path and the imperfect CSIT, respectively. Thus, we need an exponentially larger number of nondelaysensitive users KN 5
For a better illustration of the eﬀect of imperfect CSIT and LoS path on the system performance, we preserve some terms which do not grow with either K N or M .
10.7 Results and discussions
283
and relays M to compensate the penalties on the system performance caused respectively by an LoS path and imperfect CSIT.
10.6.2
Scheme for reducing computational burden at each relay In this section, a novel computational burden reduction scheme is introduced to reduce the computational load of the relays. In general, the distributed resource allocation and scheduling algorithm alleviates the computational burden at the BS by distributing computation to the relays. However, the relays may not be able to handle the additional computational complexity when the number of users is large, since relays have usually limited computational power. Therefore, a computational burden reduction scheme for the relays is needed. Our proposed scheme oﬀers two advantages. First, it preserves the desirable properties of the scheduler under various scheduling policies. In other words, the essential gain achieved by MUD is preserved as will be demonstrated in the simulation section. Second, the total computational load can be conveniently controlled by adjusting a corresponding threshold. We assume that subcarrier mapping is not performed such that the combination of subcarrier pairs in the view of each relay increases linearly w.r.t. nF instead of n2F . As can be observed from (10.30), the subcarrier selection criterion at the relays is based on the data rate incorporating outage and the global (t) (t) resource usage which is reﬂected in the shadow price, i.e., βp and υi . For a reasonably large number of users K, due to MUD, the probability that a user with ∗(t,k ) a small value of C˜m ,i,p is assigned any subcarrier pairs is low. Hence, computing and allocating any resources to them is wasteful and should be avoided. In the proposed computational burden reduction scheme, the scheduler only considers subcarrier i of user k for scheduling when the following condition is fulﬁlled: 4 Θ th 4nF +4 (t,k ) −1(t,k ) ( w k + δ k ) ( 1 −ε ) 2 , (10.37) lU R m FU R m ,i (ε) ≥ PT where Θth is a threshold. Since only some users are considered for scheduling, the computational burden for the calculation of the resource allocation is reduced. Threshold Θth allows us to trade performance for complexity. The larger Θth , the fewer the subcarriers that are considered, which leads to a lower complexity but may cause some degradation in performance. For the derivation of (10.37), please refer to the Appendix.
10.7
Results and discussions In this section, we evaluate the system performance using simulations. A single cell with two ringshaped boundary regions is considered. The outer boundary and the inner boundary have radii of 1 km and 500 m, respectively. The M relay stations are equally distributed in the area between the inner and the
284
Crosslayer scheduling design for cooperative wireless relay networks
outer boundaries, which is divided into A sectors of equal sizes. Each user is served by GRj = M/A, 1 ≤ j ≤ A, relays. In a superframe, there are N = 5 frames and each frame has a length of 2 ms. The number of subcarriers is nF = 128 with carrier center frequency 2.5 GHz and the 3GPP pathloss model is (t,k ) adopted [42]. αm ,i = 2/3 such that the transmit power of the BS is twice that of the mobile users. The smallscale fading coeﬃcients of the BStorelay links are modeled as i.i.d. Rician random variables with Rician factor κ = 6 dB, while the smallscale fading coeﬃcients of the relaytouser links are i.i.d. Rayleigh random variables. The target packet outage probability is set to ε = 0.01 for illustration. The quantizer used to quantize the information exchanged between the relays and the BS in each iteration of the distributed resource allocation algorithm is designed oﬄine using the Lloyd–Max algorithm. We study the performance of the maximum system goodput and PF schedulers. Maximum goodput scheduling can be obtained by using weights wk = 1, ∀k, while PF scheduling is performed by adapting the weights of each user according to [29]. The average weighted system goodput is obtained by counting the number of packets successfully decoded by all users averaged over both the macroscopic and microscopic fading.
10.7.1
Convergence of the distributed resource allocation algorithm Figures 10.10, 10.11, and 10.12 illustrate the evolution of the Lagrange multi(1) pliers λ, β1 , and δ1 for the distributed maximum goodput and PF scheduling algorithms over time for diﬀerent maximum transmit powers Pt . There are K = 15 users, A = 3 sectors, and M = 3 relays and the packet outage probability was chosen to be ε = 0.01. Each user has mobility 5 km/h and there are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz. The results are averaged over 1000 independent adaptation processes. As can be observed from Figures 10.10–10.12, the proposed distributed algorithm converges fast and typically achieves 90–95% of the optimal value within ten iterations even if the value of the dual variables which are exchanged between the BS and relays are quantized to 3 bits.
10.7.2
Average system goodput vs. transmit power and user mobility Figure 10.13 illustrates the average weighted system goodput vs. the transmit power for M = 3 relays and A = 3 sectors. In the cell, there are K = 15 users with mobilities of 5 km/h. Three users are delaysensitive with data rate requirement R(k ) = 1 bps/Hz, ∀k ∈ D, while the remaining users are nondelaysensitive. We study the performance of maximum system goodput and PF scheduling with the proposed distributed algorithms. For comparison, we also simulate the centralized schedulers associated with the scheduling algorithms considered. The centralized scheduler is implemented at the BS which is assumed to have the global CSI of the entire network and resource allocation is performed by using a brute force search. It can be observed that the performance of the proposed distributed
10.7 Results and discussions
285
PT=38 dBm, PT=38 dBm, PT=38 dBm, PT=35 dBm, PT=35 dBm, PT=35 dBm, PT=35 dBm, PT=38 dBm, PT=38 dBm, PT=35 dBm, PT=35 dBm, PT=35 dBm,
Figure 10.10. Dual variable λ vs. the number of iterations with K = 15 users, A = 3 sectors, M = 3 relays, and packet outage probability ε = 0.01. Each user has mobility 5 km/h and there are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz.
(1)
Figure 10.11. Dual variable β1 vs. the number of iterations with K = 15 users, A = 3 sectors, M = 3 relays, and packet outage probability ε = 0.01. Each user has mobility 5 km/h and there are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz.
286
Crosslayer scheduling design for cooperative wireless relay networks
Figure 10.12. Dual variable δ1 vs. the number of iterations with K = 15 users, A = 3 sectors, M = 3 relays, and packet outage probability ε = 0.01. Each user has mobility 5 km/h and there are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz.
algorithm closely approaches the optimal centralized scheduling algorithm for both maximum goodput and PF scheduling after only ten iterations, which conﬁrms the practicality of the distributed algorithm. On the other hand, although a better performance can be achieved when subcarrier mapping is implemented at the relays, it provides less than 1 dB gain in terms of system goodput at the cost of a higher implementation complexity. Therefore, subcarrier mapping should be avoided if the system is operating in high SNR and the computational complexity is a concern. We also study the impact of the proposed computational burden reduction scheme and quantization of the information exchanged between the BS and the relays in each iteration of the distributed algorithm on the system performance. In particular, three bits are used for quantization and the threshold Θth , deﬁned in Section 10.6.1, is chosen such that the relays only need to process 20% of the CSI coeﬃcients. It can be observed that the quantization and the computational burden reduction scheme cause only a small loss in performance while signiﬁcantly reducing the required signaling overhead and computational complexity. It is not surprising that the maximum goodput scheduler has a better performance than the PF scheduler in terms of the average weighted system goodput. Nevertheless, the maximum goodput scheduler results in a extremely unfair resource allocation, since only users with good channel conditions (users who are located near the relays) are selected for transmission. In contrast, the PF scheduler maintains fairness among users by sacriﬁcing performance. Figure 10.13 also shows the performance of a baseline oneway halfduplex relay [16] in which subcarrier mapping is performed and a brute force approach
287
(bps/Hz)
10.7 Results and discussions
PT
Figure 10.13. Average weighted system goodput vs. the total transmit power for diﬀerent scheduling algorithms with K = 15 users, A = 3 sectors, M = 3 relays, and packet outage probability ε = 0.01. Each user has a mobility of 5 km/h and there are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz.
is used for resource allocation. The proposed scheduler achieves a substantial gain in average weighted system goodput when compared with the baseline scheme, especially in the high transmit power regime. This is because the proposed scheduler acquires a better spectral eﬃciency by utilizing simultaneous message exchanges between the BS and users, while the oneway halfduplex relay uses two phases to transmit only one message.
288
Crosslayer scheduling design for cooperative wireless relay networks
Figure 10.14. Average weighted system goodput vs. the user mobility. A = 3 sectors, M = 3 relays, packet outage probability ε = 0.01, and K = 15 users. There are three delaysensitive users in the system. Each curve corresponds to diﬀerent transmit power levels and diﬀerent data rate requirements for the delaysensitive users. Figure 10.14 illustrates the average system goodput vs. the user mobilities for K = 15 users where three users are delaysensitive with diﬀerent data rate requirements. It can be observed that as the speed of the users increases, the system performance decreases since the corresponding CSI at the proposed schedulers becomes more outdated, and the schedulers have to be conservative in the
10.7 Results and discussions
289
resource allocation in order to satisfy the outage probability requirements of each user. In addition, the PF scheduler is more sensitive to the CSIT quality degradation due to the mobility of the users when compared with the maximum goodput scheduler, since the PF scheduler only considers the smallscale fading in the long run while the maximum goodput scheduler considers both the pathloss and smallscaling fading. It can be observed that the average weighted system goodput diminishes as the data rate requirements become more stringent, since most of the resources are consumed by the delaysensitive users regardless of their actual channel quality. In other words, the degrees of freedom in the resource allocation decrease when the delaysensitive users become more resource hungry, and hence the performance gain achieved by multiuser diversity diminishes. In contrast, the nondelaysensitive users cannot be served even if they have very good channel conditions, because the scheduler needs to fulﬁll the data rate requirements of the delaysensitive users.
10.7.3
Asymptotic system goodput performance of PF scheduling In this section, we focus on the asymptotic performance of PF scheduling with respect to the numbers of users K and relays M for A = 3 sectors. All users in the system have a mobility of 5 km/h. There are three delaysensitive users with data rate requirement R(k ) = 1 bps/Hz, ∀k ∈ D, and KN = K − 3 nondelaysensitive users. The number of iterations for the distributed scheduler is set to 20 and subcarrier mapping is not performed. Figure 10.15 illustrates the average system goodput vs. the number of relays for K = 300 users and diﬀerent transmit powers. As expected, the average system goodput grows with order O (log2 (log M /(κ + 1))) which matches the predicted asymptotic trend closely. Although the asymptotic expressions in Theorem 10.2 are derived for nondelaysensitive users, they can be used to approximate the system performance in the considered case as well. This is because the performance can again be mainly attributed to the nondelaysensitive users since they provide more degrees of freedom for the resource allocation. Figure 10.16 illustrates the average system goodput as a function of the number of users K for M = 300 relays for diﬀerent total transmit powers and each user has a mobility of 5 km/h. As be observed, the average system goodput can follows the order growth of O log2 (1 − σe2 (F × τ )) log KN closely. This result suggests that in order to fully exploit the MUD gain in the considered system, the number of relay stations should grow faster than the number of users to compensate for the noise ampliﬁcation in the AF process at the relay, which corresponds to an impractical scenario. We also study in Figures 10.15 and 10.16 the impact of the proposed computational burden reduction scheme and quantization on the system performance. It can be observed that both techniques do not change the asymptotic trend of the system performance which conﬁrms that the proposed computational burden
290
Crosslayer scheduling design for cooperative wireless relay networks
PT =38 dBm
(bps/Hz)
PT =35 dBm
PT =32 dBm
PT t PT t PT t PT t PT t PT t
M
Figure 10.15. Average weighted system goodput of PF scheduler vs. the number of relays M with K = 300 users. Tthree users are delaysensitive with data rate requirements R(k ) = 1 bps/Hz, ∀k ∈ D. The cell is divided into A = 3 sectors, each user has mobility of 5 km/h, and the packet outage probability requirement is ε = 0.01. reduction scheme is able to preserve the essential multiuser diversity gain even if the scheduler takes into account only 20% of the users for resource allocation.
10.8
Conclusion We have explored some basic ideas in crosslayer scheduling design and technical challenges arising in practical implementations of crosslayer scheduling. In particular, we have studied the crosslayer scheduling design for twoway halfduplex AF relayassisted OFDMA systems. The problem considered has been formulated as a mixed combinatorial and nonconvex optimization problem, in which imperfect CSIT and heterogeneous user QoS requirements have been taken into consideration. After a proper transformation, dual decomposition has been used to derive an optimal, iterative, and distributed resource allocation solution, which requires only local CSI at each relay. Furthermore, an eﬃcient computational burden reduction scheme has been proposed, which reduces the computational
10.8 Conclusion
291
PT =38 dBm
PT =35 dBm
PT =32 dBm
PT PT PT PT PT PT
Figure 10.16. Average weighted system goodput of PF scheduler vs. the number of users K with M = 300 relays. There are three delaysensitive users with data rate requirements R(k ) = 1 bps/Hz, ∀k ∈ D. The cell is divided into A = 3 sectors, each user has mobility of 5 km/h and packet outage probability requirement ε = 0.01. complexity at the relays signiﬁcantly, while preserving the essential gain obtained from multiuser diversity. In addition, the asymptotic order growth of the average system goodput in terms of the number of users and relays has been derived to obtain useful system design insights. Our simulation results have demonstrated the excellent performance of the proposed schedulers which approach the optimal centralized solution within a small number of iterations.
Appendix
Proof of Lemma 10.2 For convenience of notation, we deﬁne H1 2 , H2 2 , H3 2 , and H4 2 to represent (t) (t) (t) (t) (t,k ) (t,k ) (t,k ) (t,k ) lB R m HB R m ,p 2 , lB R m HB R m ,i 2 , lU R m HU R m ,p 2 , and lU R m HU R m ,i 2 , respec2 2 tively. Without loss of generality, we assume that H1  ≥ H2  ≥ H3 2 ≥ H4 2 . (t,k ) (t,k ) Furthermore, for high SNR, log2 (1 + ΓDLm ,i,p ) and log2 (1 + ΓULm ,i,p ) can be (t,k )
(t,k )
approximated as log2 (ΓDLm ,i,p ) and log2 (ΓULm ,i,p ), respectively. By applying (10.20) in (10.14) and (10.15), upon rearranging terms, the channel capacity becomes
(t,k )
(t,k )
CD L m ,i,p + CU L m ,i,p ≈ ; (t,k ) 2(t,k ) (t,k ) (t,k ) 2 2 2 2 1 2 log2 (αm ,i − αm ,i ) + 2 log2 (PR m ,p P(B + U ) m ,i ) + log2 (H1  H2  H3  H4  ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) − log2 αm ,i P(B + U ) m ,i H2 2 + (1 − αm ,i )P(B + U ) m ,i H4 2 + PR m ,p H3 2 < (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) − log2 αm ,i P(B + U ) m ,i H2 2 + (1 − αm ,i )P(B + U ) m ,i H4 2 + PR m ,p H1 2 ⎛ ⎞ (t,k ) (t,k ) (t,k ) 2(t,k ) PR m ,p P(B + U ) m ,i log ((αm ,i − αm ,i )H1 2 H2 2 H3 2 H4 2 ) ⎠+ 2 ≥log2 ⎝ (t,k ) (t,k ) 2 P +P R m ,p
(B + U ) m ,i
− log2 (H1  ). 2
(10.38)
It is interesting to note that on the right hand side of (10.38), (t,k ) (t,k ) (t,k ) (t,k ) log2 PR m ,p P(B + U ) m ,i /(PR m ,p + P(B + U ) m ,i ) is the only term related to the (t,k )
(t,k )
power allocation variables and it is jointly concave w.r.t. PR m ,p and P(B + U ) m ,i . (t,k )
(t,k )
(t,k )
Therefore, for a given power Pm ,i,p = PR m ,p + P(B + U ) m ,i and using standard (t,k )
(t,k )
optimization techniques, the optimal values of PR m ,p and P(B + U ) m ,i can be shown (t,k )
to be identical and are given by Pm ,i,p /2. On the other hand, using an approach similar to that above, we can show that the obtained power allocation also max(t,k ) (t,k ) imizes the cut set bound [14], i.e., log2 (PR m ,p P(B + U ) m ,i ), which is a capacity upper bound for the relay channel.
Appendix
293
Proof of Lemma 10.3 We assume that the subcarrier pair (i, p) is used for transmission in the ﬁrst and second time slots through relay m for user k. Then, the outage probability in C2 (t,k ) for the UL data rate rU L m ,i,p is given by ⎤ ⎡ 1 (t,k ) (t,k ) Pr ⎣rU L m ,i,p > log2 1 + ΛUL m ,i,p Ξm ⎦ 2 6 5 (t) (t) (t) (t) z(lB R m HB R m ,i 2 + lB R m HB R m ,p 2 ) (t,k ) 2 = Pr HU R m ,i  ≤ Ξm (t) (t) (t,k ) (lB R m HB R m ,p 2 − z)lU R m (t) (t) (t) (t) z(lB R m HB R m ,i 2 + lB R m HB R m ,p 2 ) (t,k ) (10.39) = FU R m ,i (t) (t) (t,k ) (lB R m HB R m ,p 2 − z)lU R m (k )
(t,k )
(t,k )
where z = 2(22r m , i , j − 1)/Pm ,i,p , and FU R m ,i (·) denotes the CDF of a noncentral chisquare random variable with two degrees of freedom and noncentrality (k ) 2 ˆ (t,k ) 2 /σ 2 (t). Note that P (t,k ) l(t) H (t) parameter H e U R m ,i R m ,p B R m B R m ,p  > z since rm ,i,j will not exceed the channel capacity of the BStorelay links as the corresponding perfect CSI is available at the scheduler. Using the above result, the target outage probability in constraint C2 in (10.18) is equivalent to (t) (t) (t) (t) z(lB R m HB R m ,i 2 + lB R m HB R m ,p 2 ) (t,k ) = ε (10.40) C2 ⇒ FU R m ,i (t) (t) (t,k ) (lB R m HB R m ,p 2 − z)lU R m (t)
⇒ ⇒
(t)
(t)
(t)
z(lB R m HB R m ,i 2 + lB R m HB R m ,p 2 ) (t) (lB R m
(t) HB R m ,p 2
−
(t,k ) z)lU R m
−1(t,k )
= FU R m ,i (ε)
(t,k ) (t,k ) rU L m ,i,p = log2 1 + ΛUL m ,i,p ,
−1(t,k )
(t,k )
where FU R m ,i (·) represents the inverse function6 of FU R m ,i (·). On the other (t,k )
hand, in order to obtain the DL data rate rD L m ,i,p which incorporates outage, we need ﬁrst to derive the PDF of the outage event. For the case of i = p, since there is only one random variable in the view of the scheduler, we can use the same approach as for the UL to ﬁnd the outage incorporated data rate. (t,k ) (t,k ) For the case of i = p, we ﬁrst deﬁne random variables Xm ,p = HU R m ,p 2 and (t,k )
(t,k )
Ym ,i = HU R m ,i 2 , which are noncentral chisquare distributed with two degrees ˆ (t,k ) 2 and s2y = H ˆ (t,k ) 2 , and of freedom, noncentrality parameters s2x = H U R m ,p
U R m ,i
variances σx2 and σy2 , respectively. Conditional on the power allocation variables, the CSI of the BStorelay link and the imperfect CSI of the relaytouser link, upon rearranging terms, the outage probability of the DL transmission is given 6
The inverse of the noncentral chisquare CDF is commonly implemented as an inbuilt function in software such as MATLAB.
294
Crosslayer scheduling design for cooperative wireless relay networks
by
= ⎛= > (t,k ) (t,k ) ⎞ > (t,k ) (t,k ) (t,k ) > m ,i ηm ,i,p > m ,i lU R Xm ,p m ⎠ < ? (t,k ) Pr ⎝? (t,k ) (t,k ) (t,k ) (t,k ) m ,i + θm ,i Ym ,i m ,i − ηm ,i,p = ⎛ > (t,k ) (t,k ) ⎞ (t,k ) > m ,i ηm ,i,p Am ,i,p ⎠, = Pr ⎝ (t,k ) < ? (t,k ) (10.41) (t,k ) Bm ,i,p m ,i − ηm ,i,p @ @ (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) (t,k ) Am ,i,p = m ,i lU R m Xm ,p , Bm ,i,p = m,i + θm,i Ym,i ,
where (t,k )
(t,k ) (t)
(t)
m ,i = αm ,i lB R m HB R m ,i 2 , (t,k )
(t ,k )
(t,k )
(t,k )
ηm ,i,p = (22r D L m , i , p − 1)2/Pm ,i,p ,
(t,k )
(t,k )
(t,k )
and
(t,k )
θm ,i =
(t,k )
(1 − αm ,i )lU R m . Note that in (10.41) m ,i > ηm ,i,p as rD L m ,i,p will not exceed the channel capacity of the BStorelay links since the corresponding perfect (t,k ) (t,k ) (t,k ) (t,k ) CSI is available at the schedulers. Let ZU R m ,i,p = Am ,i,p /Bm ,i,p , where Am ,i,p (t,k )
and Bm ,i,p are Rician random variables with two degrees of freedom, noncen(t,k ) (t,k )
(t,k )
(t,k )
trality parameters s2a = m ,i lU R m s2x and s2b = m ,i + θm ,i s2y , and variances (t,k ) (t,k )
(t,k )
σa2 = ( m ,i lU R m )2 σx2 and σb2 = (θm ,i )2 σy2 , respectively. Thus, the CDF of (t,k )
ZU R m ,i,p , which is the ratio of two independent nonidentical Rice distributed random variables, is (t,k )
FU R m ,i,p (z) = Q(c, d) − (σa2 c2 /s2b z 2 ) exp (−(c2 + d2 )/2)I0 (cd)
(10.42)
for z ∈ R+ , where c = (s2b z 2 /(σb2 z 2 + σa2 ))1/2 and d = (s2a /(σb2 z 2 + σa2 ))1/2 . Thus, when combining (10.41) and the constraint C1 in (10.18), we can solve for (t,k ) rD L m ,i,p , which yields ⎛= ⎞ > (t,k ) (t,k ) > η m ,i m ,i,p (t,k ) ⎠=ε FU R m ,i,p ⎝? (t,k ) (t,k ) ( m ,i − ηm ,i,p ) (t,k ) (t,k ) −1(t,k ) Pm ,i,p m ,i (FU R m ,i,p (ε))2 1 (t,k ) ⇒ rD L m ,i,p = log2 1 + . (10.43) (t,k ) −1(t,k ) 2 2( m ,i + (FU R m ,i,p (ε))2 ) −1(t,k )
The inverse CDF FU R m ,i,p (ε) can be implemented as a lookup table or by using the bisection method for practical implementation.
Asymptotic analysis Here, we ﬁrst show the asymptotic order growth of Rician and Rayleigh random variables, before we obtain the asymptotic order growth of the average system goodput for the PF scheduler. The order of growth of Rician fading coeﬃcients with perfect CSIT and Rayleigh fading coeﬃcients with imperfect CSIT can be derived by using Lemma 10.1. Suppose {h1 , h2 , . . . , hM } are M i.i.d. Rician random variables with Rician factor κ. Let a = κ/(1 + κ), v = 1/(κ + 1), and xi = hi 2 . Then the set {x1 , x2 , . . . , xM } of the magnitude squared of the Rician
295
Appendix
random variables with PDF f (·) and CDF F (·) is twice diﬀerentiable for all x. Therefore, the limit of the reciprocal hazard function is given by [31] lim g(x) = lim
x→∞
x→∞
1 1 − F (x) ≈ f (x) v
(10.44)
which satisﬁes the suﬃcient condition in Lemma 10.1 and therefore xmax − lM converges to a Gumbel distribution in the limiting case of M → ∞. Then, solving F (lM ) = 1 − 1/M for lM , we obtain √ 2 v log M + a + O(log log M ). (10.45) lM = Therefore, the growth of xmax is given by O((log M )/(κ + 1)) for suﬃciently −1(t,k ) large M . On the other hand, we consider FU R m ,i (ε), k ∈ {1, 2, . . . , KN } for nondelaysensitive users as deﬁned in Section 10.5.1. By adopting a framework similar to that in and applying Lemma 10.1, it can be shown that for CSIT −1(t,k ) error σe2 (t) ∈ [0, 1), the growth of max1≤k ≤K N FU R m ,i (ε) is given by variance O (1 − σe2 (t)) log KN for suﬃciently large KN and the term (1 − σe2 (t)) acts as a penalty on the multiuser diversity gain due to imperfect CSIT for σe2 (t) ∈ [0, 1). Remark 10.1 Since we are interested in the asymptotic performance of the PF scheduler with longterm fairness, the selection of the relaytouser links will be based on the i.i.d. smallscale fading coeﬃcients only [29], and thus, the extreme value theory for i.i.d. random variables is applicable. Therefore, by considering only the ﬁrst order growing terms, the asymptotic order growth of the average weighted system goodput is given by Ugoodput (P, R, S)
=
nF nF ,k ) M F w (k ) s(t m ,i,p 2Fn F m = 1 k ∈N t = 1 i = 1 p = 1 m
⎧ ⎛ ⎛ ⎞⎞ (t , k ) (t , k ) (t , k ) (t ) ⎨ Pm , i , p (1 − αm , i )lU R m (1 − σe2 (t))(log KN )lB R m (log M/(κ + 1)) ⎠⎠ × O⎝log2 ⎝ (t , k ) (t ) (t , k ) (t , k ) ⎩ 2 (1 + αm , i )lB R m (log M/(κ + 1)) + (1 − αm , i )lU R m (1 − σe2 (t)) log KN ⎛ ⎛ ⎞⎞⎫ (t , k ) (t , k ) (t ) (t , k ) ⎬ Pm , i , p αm , i lB R m (log M/(κ + 1))lU R m (1 − σe2 (t)) log KN ⎠⎠ +O ⎝log2 ⎝ (t , k ) (t ) (t , k ) (t , k ) ⎭ 2 α l (log M/(κ + 1)) + (2 − α )l (1 − σ 2 (t)) log K m ,i
B Rm
m ,i
U Rm
e
N
(10.46)
where Nm = Um ∩ N is the intersection of sets Um and N . We are now ready to derive the results of Case I and Case II. Case I (Asymptotic system goodput for a large number of nondelaysensitive users KN and a growing number of relays M ) In this case, we assume that the number of nondelaysensitive users KN is always larger than the number of relays M and KN grows with M such that limK N ,M →∞ KN /M → ∞. By only considering the ﬁrst order growing terms, the growth of the average system
296
Crosslayer scheduling design for cooperative wireless relay networks
goodput is given by Ugoodput (P, R, S) (a)
≈
(t,k ) nF nF M F w(k ) sm ,i,p m =1 k ∈Nm t=1 i=1 p=1
(b)
=
log M , O log2 κ+1
2Fn F
× O 2 log2
(t,k ) (t)
Pm ,i,p lB R m (log M/(κ + 1)) 2
(10.47)
where (a) is due to the assumption that limK ,M →∞ K/M → ∞ and (b) is because the channel coeﬃcients of the BStorelay links are identical distributed since the distances between the BS and relays are assumed to be same. Case II (Asymptotic system goodput for a large number of relays M and a growing number of nondelaysensitive users KN ) In this case, we assume that the number of relays M is always larger than the number of users KN and M grows with KN such that limK N ,M →∞ M/KN → ∞. By replacing the channel gain variables with the associated asymptotic growth expression, we obtain the growth of the average system goodput which is given by Ug oodpu t (P, R, S) (c)
≈
(t,k ) nF nF M F w(k ) sm ,i,p m =1 k ∈Nm t=1 i=1 p=1
(d)
≈
2Fn F
× O 2 log2
(t,k ) (t,k )
Pm ,i,p lU R m ((1 − σe2 (t)) log KN ) 2
O log2 (1 − σe2 (F × τ )) log KN ,
(10.48)
where (c) is due to the assumption that limK N ,M →∞ M/KN → ∞ and (d) is due to the fact that the growth of the system goodput is limited by the largest CSIT estimation error variance. It can be observed that in both Case I and Case II, the twoway relay compensates the loss of spectral eﬃciency due to halfduplexing, i.e., the prelog factor 1 2.
Derivation of feedback condition Let Θth be the threshold representing the global subcarrier usage and replacing (γi + βi ) in (10.30). Furthermore, assume that the SNR is high such (t,k ) (t,k ) (t,k ) (t,k ) that ΛULm ,i,i /(1 + ΛULm ,i,i )+ΛDLm ,i,i /(1 + ΛDLm ,i,p ) ≈ 2 in (10.30). With these (k )
assumptions and sm ,i,j = 1, based on (10.30), we can establish the following (t,k )
upper bound for 0 < αm ,i < 1, which is given by 2 log2
(t,k ) (t,k ) (t,k ) ΛDL m ,i,i ΛUL m ,i,i Pm ,i,p (t,k ) −1(t,k ) (t,k ) lU R m FU R m ,i (ε) − 2 ≥ C˜m ,i,i − − (t,k ) (t,k ) 2 1 + ΛDL m ,i,i 1 + ΛUL m ,i,i ≥
2Θth . (wk + δk )(1 − ε)
(10.49)
References
297
The above upper bound has the same asymptotic order growth as Case II in Theorem 10.2, which suggests that the adopted upper bound is an appropriate choice. Furthermore, the user selection does not depend on the CSI of the BStorelay link. Therefore, based on (10.49), the proposed user feedback criterion can be obtained as (t,k ) 4Θ th 4Θ th Pm ,i,p (a) 4nF +4 +4 (t,k ) −1(t,k ) ( w k + δ k ) ( 1 −ε ) ( w k + δ k ) ( 1 −ε ) 2 ≈ 2 , lU R m FU R m ,i (ε) ≥ 2 PT (10.50) where (a) is due to the fact that the users do not know the ﬁnal power allocation before the schedulers perform the resource allocation, and hence equal power allocation is assumed for user selection.
References [1] A. Goldsmith, Wireless Communications. 1st ed. Cambridge University Press, 2005. [2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. 1st ed. Cambridge University Press, 2005. [3] V. K. N. Lau and Y. K. R. Kwok, ChannelAdaptive Technologies and CrossLayer Designs for Wireless Systems with Multiple Antennas: Theory and Applications. 1st ed. WileyInterscience, 2006. [4] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun Magazine, 42 (2004),74–80. [5] R. Pabst, B. H. Walke, D. C. Schultz, et al., “Relaybased deployment concepts for wireless and mobile broadband radio,” IEEE Commun Magazine, 42 (2004), 80–89. [6] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “On capacity of Gaussian ’cheap’ relay channel,” in Proc. of IEEE Global Telecommun. Conf., 2003, pp. 1776–1780. IEEE, 2003. [7] T. C. Y. Ng and W. Yu, “Joint optimization of relay strategies and resource allocations in cooperative cellular networks,” IEEE J. Select. Areas Commun., 25 (2007), 328–339. [8] P. Liu, Z. Tao, Z. Lin, E. Erkip, and S. Panwar, “Cooperative wireless communications: A crosslayer approach,” IEEE Wireless Commun. Magazine, 13 (2006), 84–92. [9] Y. Yuan, Z. He, and M. Chen, “Virtual MIMObased crosslayer design for wireless sensor networks,” IEEE Trans. Veh. Technology, 55 (2006), 856–864. [10] S. J. Kim, X. Wang, and M. Madihian, “Optimal resource allocation in multihop OFDMA wireless networks with cooperative relay,” IEEE Trans. Wireless Commun., 7 (2008), 1833–1838. [11] J. Wang, Y. Zhao, and T. Korhonen, “Cross layer optimization with complete fairness constraints in OFDMA relay networks,” in Proc. of IEEE Global Telecommun. Conf., 2008. IEEE, 2008.
298
Crosslayer scheduling design for cooperative wireless relay networks
[12] B. Rankov and A. Wittneben, “Spectral eﬃcient protocols for halfduplex fading relay channels,” IEEE J. Select. Areas in Commun., 25 (2007), 379– 389. [13] T. J. Oechtering, C. Schnurr, I. Bjelakovic, and H. Boche, “Broadcast capacity region of twophase bidirectional relaying,” IEEE Trans. Inform. Theory, 54 (2008), 454–458. [14] A. S. Avestimehr, A. Sezgin, and D. N. C. Tse, “Approximate capacity of the twoway relay channel: A deterministic approach,” in Proc. of IEEE 46th Annual Allerton Commun., Control, and Computing, 2008, pp. 1582–1589. IEEE, 2008. [15] B. Rankov and A. Wittneben, “Achievable rate regions for the twoway relay channel,” in Proc. of IEEE Int. Symp. on Inform. Theory, 2006, pp. 1668–1672. IEEE, 2006. [16] D. W. K. Ng and R. Schober, “Crosslayer scheduling for OFDMA amplifyandforward relay networks,” IEEE Trans. Vehi. Technology, to appear. [17] G. D. Yu, Z. Y. Zhang, Y. Chen, S. Chen, and P. L. Qiu, “Power allocation for nonregenerative OFDM relaying channels,” in Proc. of IEEE Int. Conf. on Wireless Commun., Networking and Mobile Computing, 2005, pp. 185– 188. IEEE, 2005. [18] Y. Li, W. Wang, J. Kong, W. Hong, X. Zhang, and M. Peng, “Power allocation and subcarrier pairing in OFDMbased relaying networks,” in Proc. of IEEE Int. Conf. on Commun., 2008, 2602–2606. IEEE, 2008. [19] C. Comaniciu, N. B. Mandayam, and H. V. Poor, Wireless Networks: Multiuser Detection in CrossLayer Design. 1st ed. Springer, 2005. [20] G. Song and Y. Li, “Crosslayer optimization for OFDM wireless networksPart I: theoretical framework,” IEEE Trans. Wireless Communications, 4 (2005), 614–624. [21] D. P. Palomar and M. Chiang, “A tutorial on decomposition methods for network utility maximization,” IEEE J. Select. Areas in Commun., 24 (2006), 1439–1451. [22] M. Tao, Y. C. Liang, and F. Zhang, “Resource allocation for delay diﬀerentiated traﬃc in multiuser OFDM systems,” IEEE Trans. Wireless Commun., 7 (2008), 2190–2201. [23] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inform. Theory, 45 (1999), 1468–1489. [24] R. Narasimhan, “FiniteSNR diversitymultiplexing tradeoﬀ for correlated Rayleigh and Rician MIMO channels,” IEEE Trans. Inform. Theory, 52 (2006), 3965–3979. [25] R. Knopp and P. A. Humblet, “Information capacity and power control in singlecell multiuser communications,” in Proc. of IEEE Int. Conf. on Communications, 1995, pp. 331–335. IEEE, 1995. [26] G. Song and Y. Li, “Asymptotic throughput analysis for channelaware scheduling,” IEEE Trans. Commun., 54 (2006), 1827–1834.
References
299
[27] M. Sharif and D. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inform. Theory, 51 (2005), 506–522. [28] H. A. David Order Statistics. 1st ed. Wiley, 1970. [29] G. Caire, R. R. Muller, and R. Knopp, “Hard fairness versus proportional fairness in wireless communications: The singlecell case,” IEEE Trans. Inform. Theory, 53 (2007), 1366–1385. [30] IEEE 802.16m System Description Document [Draft]; 2009. Available: http://wirelessman.org/tgm/docs/80216m08 003r9a.doc.zip [31] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inform. Theory, 48 (2002), 1277–1294. [32] K. Jitvanichphaibool, R. Zhang, and Y. C. Liang, “Optimal resource allocation for twoway relayassisted OFDMA,” IEEE Trans. Vehi. Technology, 58 (2009), 3311–3321. [33] C. K. Ho, R. Zhang, and Y. C. Liang, “Twoway relaying over OFDM: Optimized tone permutation and power allocation,” in Proc. of IEEE Int. Conf. on Commun., 2008, pp. 3908–3912. IEEE, 2008. [34] S. Sanayei and A. Nosratinia, “Opportunistic downlink transmission with limited feedback,” IEEE Trans. Inform. Theory, 53 (2007), 4363–4372. [35] X. J. Zhang and Y. Gong, “Adaptive power allocation in twoway amplifyandforward relay networks,” in Proc. of IEEE Int. Conf. on Commun., 2009. IEEE, 2009. [36] M. Chen and A. Yener, “Power allocation for multiaccess twoway relaying,” in Proc. of IEEE Int. Conf. on Commun., 2009. IEEE, 2009. [37] H. A. Suraweera, R. H. Y. Louie, Y. Li, G. K. Karagiannidis, and B. Vucetic, “Two hop amplifyandforward transmission in mixed Rayleigh and Rician fading channels,” IEEE Commun. Letters, 13 (2009), 227– 229. [38] J. G. Proakis, Digital Communications. 4th ed. McGraw Hill Higher Education, 2000. [39] G. Song and Y. Li, “Crosslayer optimization for OFDM wireless networkspart II: Algorithm development,” IEEE Trans. Wireless Commun., 4 (2005), 625–634. [40] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [41] S. Boyd, L. Xiao, and A. Mutapcic, Subgradient Methods. Notes for EE392o Stanford University Autumn. 2003–2004. [42] Spatial Channel Model for MultipleInput MultipleOutput (MIMO) Simulations. 3GPP TR 25.996 V7.0.0 (2007–06).
11 Green communications in cellular networks with ﬁxed relay nodes Peter Rost and Gerhard Fettweis
11.1
Introduction Mobile communication systems have to provide exponentially increasing data rates for an increasing number of subscribers using ubiquitous data services. As the capacity per cell is limited by the available bandwidth, the same time– frequency resources must be spatially reused. Hence, the more the user density increases, the higher the spatial reuse must be to satisfy the demand for high data rate services. This chapter discusses relaying as a candidate technology to increase the spatial reuse and therefore to provide the required data rates while reducing energy consumption in mobile communication systems.
11.1.1
Two motivating examples A challenging property as well as an opportunity for exploiting the wireless channel is nonlinear signal attenuation (pathloss), which oﬀers the possibility to concentrate power at certain points in the network and spatially reuse resources within a mobile communication network. Consider an additive white Gaussian noise (AWGN) channel with a pathloss exponent α = 4, receiver noise power N , and transmission power P . Given these qualities and assuming a downlink transmission where a terminal can use the received signals from each radio access point (RAP), the observed signaltonoise ratio (SNR) at a normalized distance d is given by ρ(d) = i P/N · di − d−α , where di is the position of the ith RAP. Figure 11.1 compares the received SNR in two diﬀerent conﬁgurations of a onedimensional wireless network: A: RAPs are deployed PA such that PA /N B: RAPs are deployed PB such that PB /N
at di = [0, 3, 6, . . . , i · 3] and each transmits with power = 10 at di = [0, 2, 4, . . . , i · 2] and each transmits with power = 2.
Conﬁguration B increases the RAP density by a factor of 1.5 compared to conﬁguration A but its RAPs transmit with only oneﬁfth the power PA . Therefore, Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
11.1 Introduction
10
0 (a)
20
ρ(d) (dB)
ρ(d) (dB)
20
301
0
2 4 6 8 10 (b) Normalized distance d
10
0
0
2 4 6 8 Normalized distance d
10
Figure 11.1. Spatial power distribution for two diﬀerent deployment scenarios: (a) conﬁguration A; (b) conﬁguration B.
the total energy consumption in conﬁguration B is less than half of the energy consumption in conﬁguration A, but both achieve the same minimum SNR. In addition to the lower energy consumption, conﬁguration B provides a higher density of RAPs and therefore also a higher spatial reuse, which potentially increases the overall system throughput. Consider a similar setup with two RAPs, which are placed at distance d from each other, and one user terminal (UT), which is placed at distance ∆d, ∆ < 0.5, from its serving RAP. The signaltointerferenceandnoise ratio (SINR) experienced at the UT is given by ρA =
P (∆d)−α . N + P ((1 − ∆)d)−α
(11.1)
Let conﬁguration B again use a higher RAP density, i. e., dA > dB , and let the sum power spent in both conﬁgurations be normalized, i. e., the transmission power per node in conﬁguration B is given by P · dB /dA . The ratio of the SINR received at the user terminal in the conﬁgurations is given by 1−α dA + P /N ((1 − ∆)dA )−α ρA dB = < 1. (11.2) ρB 1 + P /N ((1 − ∆)dA )−α The probability that a UT experiences a certain ∆ is the same for both scenarios, which implies that even though we move RAPs closer to each other and only exploit the signal received from the assigned RAP, we increase the SINR (in this linear model). In next generation cellular systems not only will the distance between two RAPs decrease but also the carrier frequency will increase. Therefore, in the interferencelimited regime (P N ) let dA = dB and αA < αB in
302
Green communications in cellular networks with ﬁxed relay nodes
order to account for the higher pathloss at higher carrier frequencies: ρA = ρB
∆ 1−∆
α B −α A < 1.
(11.3)
Hence, if we use a higher carrier frequency and move RAPs closer to each other we are able to improve the observed SINR and the spatial reuse of resources.
11.1.2
Scope and key problems These examples illustrate that densely deployed networks potentially provide higher data rates while requiring less transmission energy. In addition, future networks face the challenge of connecting RAPs through a backhaul using outofband resources in order to exploit interfering signals. Backhaul requirements such as the availability of a highspeed wired connection or microwave links to a central server diminish the deployment ﬂexibility and raise necessary expenditures. Hence, deployment and transmission strategies in next generation networks should be able to mitigate intercell interference and increased deployment costs due to the higher RAP density. In this chapter, we investigate relaying as an opportunity to increase the RAP density in a next generation mobile communication system operating at a carrier frequency of about 4 GHz. Future wireless networks are likely to use multicell multipleinput multipleoutput (MIMO) where multiple RAPs cooperatively serve UTs. More speciﬁcally, techniques introduced in the context of the MIMO broadcast channel (BC) [1, 2] and MIMO multiple access channel (MAC) [3, 4] are applied to an array of physically separated antennas. Multicell MIMO is able to (partly) cancel interference and to provide an additional array gain. However, in addition to highquality channel state information (CSI), it requires the exchange of user data over a highperformance backhaul link. In this chapter diﬀerent aspects of a mobile communication system, which uses both multicell MIMO for interference cancelation and relaying in order to improve the spatial resource reuse, are analyzed and discussed. Studies have shown that the energy consumption in cellular networks already represents a major cost driver, which emphasizes that decreasing the systemwide energy consumption is not only of ecological but also economical interest. In a mobile communication system, we distinguish between the energy consumed by the RF frontend (transmission power) and the energy required for signal processing as well as encoding and decoding (computation power). It does not suﬃce to solely reduce the transmission power by using (almost) capacityachieving codes and decoding algorithms, which imply that the additional computation power required outweighs the reduced transmission power [5]. Our goal is to employ lowcomplexity approaches, which provide a higher throughput while reducing the overall energy spent in the system. The scope of this chapter is the analysis of the cost–beneﬁt tradeoﬀ in relay networks as well
11.2 System model
303
as the tradeoﬀ between computation power, transmission power, and the system performance.
11.1.3
Outline and contributions We explore system architectures based upon the transmission schemes introduced in [6] and apply these techniques to two major scenarios: the widearea scenario and the Manhattanarea scenario. While previous work considered either downlink or uplink performance, we analyze both jointly. As previously mentioned, the key to green communication is to use lowcomplexity algorithms as well as to reduce the overall consumed transmission energy. In order to show how conventional and relaybased systems perform with less complex encoding and decoding algorithms, we provide results for diﬀerent SNR gaps [7, p. 66]. The SNRgap approximation applies a constant SNR scaling under the assumption that for the considered SNRs the diﬀerence between the employed encoding/decoding scheme and the maximum achievable rate is only an SNR shift. Furthermore, we compare femtocells with high datarate infrastructure connection and relaying where only inband connections are available. Each additional relay node (RN) implies additional costs for the deployed system but also provides a performance gain for its users. This relation is examined using a cost– beneﬁt tradeoﬀ analysis, which compares the system performance under a cost constraint. In Section 11.2, we introduce the considered system model and we review the most important parameters of the physical layer. In Section 11.3 we discuss the system and protocol design, and summarize the approaches for the considered mobile communication system. The main part is the discussion of numerical results in Section 11.4. Section 11.5 concludes the chapter.
11.2
System model
11.2.1
Propagation scenarios Our analysis in Section 11.4 is based on two propagation scenarios, which were deﬁned by the European research project WINNER: a macrocellular deployment (referred to as widearea) and a microcellular deployment (referred to as Manhattanarea) [8]1 . Figure 11.2(a) illustrates the widearea scenario with uniformly deployed sites at an intersite distance dis,r ef = 1000 m. At each site three base stations (each indicated by “BS”) serve three adjacent sectors, each supported by two ﬁxed RNs (indicated by triangles). Each BS uses a directed antenna (main lobe directions are indicated by arrows) with a downtilt of 5 degrees and an antenna attenuation at an enclosed angle of ϕ given 1
All the WINNER documents referred to are publicly available on http://www.istwinner.org.
304
Green communications in cellular networks with ﬁxed relay nodes
BS BS BS
BS BS
BS BS BS
BS BS BS
BS
BS
BS BS BS
BS
BS BS BS
BS BS
30 m
BS BS BS
BS BS
BS
BS BS
BS BS BS
BS
(a)
BS
200 m
BS
(b)
Figure 11.2. The two reference scenarios considered in this chapter: (a) widearea scenario with one tier of interfering sites. Triangles indicate relay nodes and arrows the main lobe direction in each cell; (b) Manhattanarea scenario as deﬁned in [9]. Triangles indicate outdoor relays and squares indoor relays.
by 12 (ϕ/70)2 dB. Throughout our analysis, we consider only the performance results for the central site, which is surrounded by 18 interfering sites. We randomly place users according to a uniform distribution and an average density of 90 users per square kilometer [10]. Users are assigned to RAPs based on their pathloss to the respective base station (BS) or RN. Due to the pathlossbased cell assignment, the actual instantaneous cell layout may diﬀer from the regular layout in Figure 11.2(a). Figure 11.2(b) illustrates the second scenario and its regular street grid as originally deﬁned by the UMTS 30.03 recommendation [9]. In contrast to the widearea scenario where BSs employ directed antennas, omnidirectional antennas are used in the Manhattan scenario. Each BS is supported by four RNs of which two are placed indoors (indicated by a square) and two are placed outdoors (indicated by a triangle). Both outdoor and indoor RNs are able to increase a user’s lineofsight (LoS) probability with its assigned RAP and therefore promise, in particular, to improve data rates indoors, which has been previously demonstrated using ﬁeld trial results in [11]. The numerical results in Section 11.4 only include the inner three BSs, which are surrounded by 44 interfering cells. User terminals are again randomly placed on streets and within buildings according to a uniform distribution with an average density of 250 users per square kilometer [10]. For each diﬀerent link between user terminals and RAPs, an LoS probability, a pathloss model, and a power delay proﬁle are deﬁned in [12] for the smallscale
11.2 System model
305
Table 11.1. Used channel models, which are deﬁned in [12] Link BS to RN BS to indoor UT BS to outdoor UT RN to indoor UT RN to outdoor UT Indoor RN to indoor UT
Manhattanarea
Widearea
B5c B4 B1 B4 B1 A1
B5a C2 C2
fading process, where each tap is assumed to be Gaussian distributed. Table 11.1 lists the channel models, which are used for the numerical evaluation.
11.2.2
Air interface and scheduling Both the widearea and the Manhattanarea scenario use orthogonal frequencydivision multiple access (OFDMA) [13, 14] at a carrier frequency fc = 3.95 GHz. The system resources are divided in resource blocks of 15 OFDM symbols and 8 subcarriers. Uplink and downlink operate in timedivision duplex (TDD) such that the whole bandwidth is alternately occupied by either uplink or downlink (which are separated by a duplex guard of 8.4 µs). Two consecutive resource blocks are assumed to be timeinvariant, i.e., we model no Doppler spread resulting from user mobility. Therefore, both uplink and downlink experience the same channel realizations but use a separate radio resource management (RRM). All individual channel models for each link are listed in Table 11.2 and are described in detail in [8]. In order to use the relevant capacity expressions deﬁned in [6], we assume throughout our analysis Gaussian alphabets, perfect rate adaptation, and very large blocklengths. Nonetheless, in order to achieve realistic results we consider a rate clipping at 8 bits per channel use (bpcu) or (bits/symbol) enforced by a corresponding SNR clipping at 28 − 1. We assume that CSI is perfectly available at the receiver as well as at the BS in the downlink. Furthermore, the system is perfectly synchronized. Each transmitter is assumed to have a full queue, which implies that each node transmits at the maximum rate and exploits all available resources. Both scenarios deploy halfduplex RNs, which are connected to their assigned BS using an inband feederlink operating on the same time–frequency resources as UTs. Furthermore, it might happen that in the downlink an unexpected high throughput between the RN and UTs cannot be supported due to insuﬃciently ﬁlled relay buﬀers (and similarly in the uplink between the RN and BS). Hence, relay buﬀer management is an important part of the system in order to guarantee
306
Green communications in cellular networks with ﬁxed relay nodes
Table 11.2. Parameters of the underlying WINNER system used for the numerical evaluation User density Average no. users/cell Channel models Number of antennas BS/RN/UT BS transmit power RN transmit power RN transmit power UT noise ﬁgure Noise power spectral density FFT size Carrier frequency System bandwidth OFDM symbol duration Superframe duration Guard interval Used subcarriers Channel state information
90 per km2 in widearea, 250 per km2 in Manhattanarea 26 in wide/Manhattanarea As deﬁned in 4/1/1 46 dBm 37 dBm 24 dBm 7 dB −174 dBm/Hz 2048 3.95 GHz 100 MHz 20.48 µs 5.89 ms 2.00 µs [−920; 920] \ {0} Assumed to be perfectly known at transmitter and receiver
that all resources are exploited while keeping the latency as low as possible. Due to the assumption of perfect rate adaptation in our system, we do not consider any automatic repeatrequest (ARQ) protocols.
11.3
System and protocol design Assume that two BSs are connected by a highperformance backhaul that does not limit the data exchange between them. Both BSs can be regarded as one virtual antenna array, which realizes an interpath cooperation based on the same approaches as introduced in the context of the BC. Since RNs are not connected to the backhaul, they implement interpath coordination approaches as introduced in the context of the interference channel (IC) [15, 16]. The relayinterference channel combines both models and has been used in [6] to derive and evaluate strategies for a relayassisted cellular system. In this chapter, we brieﬂy discuss these strategies and apply them to the presented scenarios. However, we do not
11.3 System and protocol design
307
consider intersubsystem cooperation where signals transmitted from the BS and the RN are combined, i.e., cooperative relaying [17].
11.3.1
Nonrelaying protocols Conventional mobile communication systems such as GSM and UMTS do not provide the possibility of interpath cooperation but solely coordinate their resources. However, the spatial resource reuse can be improved if all BSs use the same resources to serve UTs experiencing a high SINR (cellcenter users) while all other UTs are served using orthogonal resources (celledge users) [18]. The actual assignment of users to the individual groups is based upon the expected longterm downlink SINR, which is assumed to be perfectly known. In our analysis, all users with an SINR below 10 dB [19] are assigned to the celledge group and all other users are regarded as cellcenter users. Such a protocol requires intercell coordination because a change of the resource assignment at one BS directly aﬀects the SINR experienced at UTs assigned to other BSs. As the density of nodes increases, this resource coordination becomes even more complicated using the described conventional protocol. One way to improve the SINR for users suﬀering from high intercell interference is to employ multicell MIMO transmission approaches [20, 21], which improve the performance by investing in computation power instead of transmission power. The idea of multicell MIMO transmission is that multiple BSs can build one virtual antenna array and exploit interference using coherent transmission. It was shown in [1, 2] that dirty paper coding (DPC) [22] is the capacity approaching strategy for the downlink scenario. In the uplink, multicell MIMO can be regarded as a MIMOMAC between UTs in diﬀerent cells and the assigned BSs. For this scenario, the capacity is known [3, 4] and is achieved by successive interference cancelation (SIC), which assumes perfect rate adaptation in order to cancel interference. Furthermore, the application of SIC based on the (quantized) received signal of other BSs again requires a high datarate backhaul. If the channel is changing quickly, the CSI feedback delay causes a signiﬁcant performance loss [23] for downlink multicell MIMO transmission. Furthermore, using multicell MIMO requires a veryhighperformance backhaul for both uplink and downlink to exchange parts or complete user data in addition to the CSI [21, 24]. In our analysis, we assume that all BSs are connected by an unconstrained backhaul link, which provides the means to exchange user data, unquantized received signals, and perfect channel state information at the transmitter (CSIT). However, we show how the performance is aﬀected in a scenario without intercell backhaul links where only BSs at the same site cooperate.
11.3.2
Relayonly protocol One key problem of nonrelaying protocols is a strongly nonuniform power distribution. A more uniform power distribution is the requirement to lower the
308
Green communications in cellular networks with ﬁxed relay nodes
systemwide spent transmission energy. Multicell MIMO improves the power distribution by exploiting interfering signals, which otherwise worsen the SINR particularly at the cell edge. An alternative is to increase the RAP density while still keeping the computational overhead reasonably low, for instance using RNs, which aggregate data of multiple users and forward the data to the BSs. If RNs employ lowcomplex algorithms, not only the transmission power but also the computation power can be decreased. Otherwise, the saved transmission power is traded for computation power, which is then spatially distributed among the RNs. Our analysis considers a protocol in which all users are served by RNs, which employ Han–Kobayashi (HK) coding [25] on the RNtoUT links. In a halfduplex relaybased system, the inband feederlink 2 represents the bottleneck, as it reduces the amount of available resources for the R−D links. Based on the results in [6, 26], we assume that RNs are served using multicell MIMO. In comparison to multicell MIMO between BSs and UTs, the signaling overhead is reduced as relays are stationary and can be placed such that they experience a strong LoS towards the base station. This implies a higher coherence bandwidth and coherence time, which allow less complex algorithms. Furthermore, the introduction of RNs might allow a lower density of BSs, which reduces the backhaul requirements, initial deployment costs, and the systemwide spent computation energy.
11.3.3
An integrated approach Users in the cell center of their assigned BS are likely to experience an LoS link towards the BS without strong multipath fading. Hence, for those users the feedback overhead can be reduced by serving them without multicell MIMO or using multicell MIMO involving only BSs at one site. Our analysis considers a simple protocol supporting both multicell MIMO and relaying in order to exploit the beneﬁts of both protocols and to reduce the signaling overhead for multicell MIMO. We constrain the system such that either multicell MIMO or relaying is used on the same time–frequency resource within one cell and the resource assignment to the individual protocols is done adaptively using a fair scheduler. Such an approach is scalable as it exploits relaying for coverage extension and supports multicell MIMO if only intercell interference cancelation is capable of providing the required data rates. Furthermore, since RNs can provide the same wireless interface as BSs, UTs need not implement relayingspeciﬁc functions, which reduces the required complexity at UTs and improves the legacy ability.
2
Note that we diﬀerentiate between the backhaul connecting multiple sites and inband feederlinks, which connect BSs and RNs.
11.4 Numerical analysis
11.3.4
309
Simpliﬁcations In [2], it has been shown that DPC is the capacity achieving approach for the MIMO BC. However, it requires timesharing of multipleuser orderings in which user streams are encoded and it also requires the optimization of the precoding matrix. In the case of perantenna power constraints this is an extensive optimization with nondeterministic complexity [27, 28], which might not be realizable in a realtime system. In this work, we apply zeroforcing DPC (ZFDPC) [29], which uses the LQ decomposition of the channel matrix. By precoding with the Hermitian of Q, the channel becomes a triangular matrix. Afterwards, DPC is applied such that individual interferencefree links result and the power assignment is determined such that the antenna constraints are satisﬁed [30]. Similarly, at the BS the decoding process in the uplink is based on the QR decomposition. By multiplying the received signal with the Hermitian of Q, the resulting channel matrix is again a triangular matrix. Then, beginning with the ﬁrst interferencefree user stream, the decoder applies SIC in order to obtain interferencefree links, which requires perfect rate adaptation in order to avoid error propagation. HK coding requires timesharing of multiple power assignments, which are diﬃcult to determine in real time. Therefore, we apply Etkin–Tse–Wang (ETW) coding [31], which provides a rule for determining the power levels of common and private messages, and guarantees rates within 1 bpcu of capacity. Applying ETW coding based on the fastfading information requires a recomputation of the power levels in every frame as well as a signiﬁcant signaling overhead. Hence, instead of using the fastfading information, we use the longterm SINR statistics to determine the common and private message power level.
11.4
Numerical analysis The previous sections introduced diﬀerent approaches for the integration of multicell MIMO transmission and relaying in a next generation mobile network (NGMN). This section presents numerical results for the widearea and Manhattanarea scenarios in order to demonstrate how both approaches aﬀect the system performance. In particular, we discuss ﬁrst the ability to counteract intercell and intracell interference and how the indoor service quality is aﬀected by relaying. We further evaluate the energy performance and cost–beneﬁt tradeoﬀ of a relayassisted system, where the provided performance gains are compared based on normalized energy consumption and normalized expenditures.
11.4.1
Simulation methodology The previously presented protocols are analyzed using a snapshotbased systemlevel simulation. Instead of explicitly modeling the user mobility, it models the
310
Green communications in cellular networks with ﬁxed relay nodes
Table 11.3. Numerical results for the widearea scenario using inband feederlinks and full intersite cooperation Scenario/protocol
θ (MBit/s)
σ (MBit/s)
θ5% (MBit/s)
Downlink
Conventional Multicell MIMO 2 Relays, relayonly 2 Relays, mixed
1.3 2.4 3.4 4.9
2.4 3.0 3.1 3.9
2.3 · 10−2 0.2 5.9 · 10−2 0.7
Uplink
Conventional Multicell MIMO 2 Relays, relayonly 2 Relays, mixed
0.1 0.2 2.1 3.0
0.5 0.6 2.5 3.0
4.3 · 10−4 3.7 · 10−3 1.8 · 10−2 0.1
eﬀects of user mobility on the experienced channel. For each placement of a group of users and for each frame consisting of 16 OFDM symbols we randomly generate an independent channel realization and perform the resource assignment as well as user scheduling. Blockfading models the situation in which users experience slowly varying channels but the connection spans multiple blocks with independent channel realizations. Due to the fact that we do not explicitly consider user mobility, we also do not have a continuous user context to which the scheduling can be applied. Hence, we partition the overall area into rectangles of equal size, i. e., in the widearea scenario they are of size 30 m × 30 m and in the Manhattanarea scenario they are of size 10 m × 10 m. For each snapshot consisting of 16 frames, i. e., 256 OFDM symbols, users are randomly placed according to a uniform distribution. Then, for each individual user the corresponding spatial block (x, y) is determined. In order to obtain suﬃcient statistics, we keep track of all results for all users, such as the average throughput θ(x, y) = Et {θ(x, y, t)}, which is used by a fair scheduler to determine the number of resources of a user placed in spatial block (x, y). More −1 speciﬁcally, the number of resources is proportional to θ (x, y). Alternatively, we could use the 5% quantile of throughput θ5% , with Pr {θ(x, y, t) ≤ θ5% } ≤ 5%. In this case, a user with a highly varying performance would receive even more resources at the expense of other users’ performance.
11.4.2
Throughput performance in the widearea scenario We analyze the ability of relaying and multicell MIMO to mitigate and cancel intercell interference in the macrocellular widearea scenario. Consider Figure 11.3, which shows the cumulative distribution function (CDF) of θ(x, y, t) over all analyzed user snapshots. In addition, Table 11.3 lists the average
311
+
+ 101
+
+
Pr {θ(·, ·) ≤ θ}
+
+
100
+
11.4 Numerical analysis
102 102
101
(a)
Conventional Virtual MIMO 2 Relays, relayonly 2 Relays, mixed
100 101 Throughput θ in MBit/s
+ 102 103
(b)
102
+
+
101
+
Pr {θ(·, ·) ≤ θ}
+
+
+ + + +
100
102
Conventional Virtual MIMO 2 Relays, relayonly 2 Relays, mixed
101 100 Throughput θ in MBit/s
101
Figure 11.3. Marginal throughput for the widearea scenario using inband feederlinks and full intersite cooperation: (a) downlink; (b) uplink.
throughput θ, the 5% quantile throughput θ5% , and the standard deviation σ = Var{θ(x, y, t)} for both uplink and downlink. Multicell MIMO doubles the average throughput and improves the 5% quantile throughput by a factor of approximately 6. Due to the improved spatial reuse the cell throughput is increased (average throughput), and in addition due to the additional intercell interference cancelation as well as the array gain using coherent transmission the worstuser performance (5% quantile throughput) is signiﬁcantly higher. Using the relayonly approach, the average throughput is further improved (by a factor
312
Green communications in cellular networks with ﬁxed relay nodes
+ + + +
+
+
101
+
+
+
Pr {θ(·, ·) ≤ θ}
+
+
+
+ + + + + + +
100
102 105
104
103
102
101
Conventional Multicell MIMO Relayonly, 2 RNs Mixed, 2 RNs Relayonly, 4 RNs Mixed, 4 RNs
100
101
102
Throughput θ in MBit/s Figure 11.4. Marginal uplink throughput for the Manhattanarea scenario using inband feederlinks. of 3 in the uplink) but the 5% quantile throughput does not improve alike (it even decreases in the downlink). Relay nodes only improve the pathloss and LoS conditions for celledge users and users close to a relay node. Users that are closer to the BS now suﬀer from a high pathloss, which causes a worse service quality for those users. By contrast, the mixed approach signiﬁcantly improves both average and 5% quantile throughput and outperforms all other protocols. Users near a BS are directly served using multicell MIMO and users at the cell edge or close to a RN are served by relay nodes. The improved spatial reuse of resources using concurrently transmitting relays and the interference mitigation are able to outweigh the loss due to the halfduplex constraint. In this chapter we consider downlink and uplink separately using the marginal CDF of both although both directly aﬀect each other. A joint uplink–downlink analysis taking into account how the net and the gross rate of uplink and downlink inﬂuence each other was given in [32].
11.4.3
Throughput performance in the Manhattanarea scenario The previous analysis showed that relaying improves both average throughput and worstuser throughput, in particular for those users at the cell edge. In a microcellular scenario such as the Manhattanarea scenario, we face the problem of providing high data rates for indoor users. Consider Figure 11.4 and Table 11.4, which show the uplink performance for the Manhattanarea scenario (the
11.4 Numerical analysis
313
Table 11.4. Numerical results for the Manhattanarea scenario using inband feederlinks and full intersite cooperation Scenario/protocol
θ (MBit/s)
σ (MBit/s)
θ5% (MBit/s)
Downlink
Conventional Multicell MIMO Relayonly, 2 RNs Relayonly, 4 RNs Mixed, 2 RNs Mixed, 4 RNs
0.8 0.6 0.5 5.4 1.2 5.8
2.1 1.1 1.0 10.6 2.5 13.0
1.6 × 10−4 1.3 × 10−3 4.0 × 10−4 0.2 2.1 × 10−3 0.2
Uplink
Conventional Multicell MIMO Relayonly, 2 RNs Relayonly, 4 RNs Mixed, 2 RNs Mixed, 4 RNs
0.2 0.2 0.4 5.3 0.9 5.7
0.8 0.8 1.0 10.4 2.5 12.0
4.4 × 10−6 3.0 × 10−5 2.0 × 10−5 0.1 7.4 × 10−5 0.2
downlink shows diﬀerent quantitative but the same qualitative results as the uplink). Both conventional and multicell MIMO transmission are unable to provide fair and suﬃcient throughput results, particularly for users that are indoors. In contrast, relaying is able to signiﬁcantly improve the worstuser performance and to deliver a suﬃcient service quality to indoor and outdoor users. Table 11.4 further highlights that the required performance is only provided by four RNs per cell of which two are indoors and two are outdoors. More speciﬁcally, the average rate is improved by a factor of 10 and the 5% quantile throughput by a factor of approximately 103 in the downlink and 104 in the uplink compared with the deployment without indoor relays. There is no signiﬁcant diﬀerence between the performance of the relayonly and mixed approaches, which implies that the system could use only relays in a microcellular scenario and apply multicell MIMO exclusively to the links between BSs and RNs. Since four relays consume less transmission power than one BS (a diﬀerence of 3 dB in our setup), the average downlink transmission power is reduced in the relayonly setup. Moreover, the improved performance oﬀers the possibility to use less complex algorithms, which require less computation power.
11.4.4
Femtocells vs. relaying Femtocells [33] have attracted signiﬁcant attention as they provide an eﬃcient way of improving coverage and high data rates for indoor users. Femtocells are homedeployed by users and exploit the available wired network as the backhaul
314
Green communications in cellular networks with ﬁxed relay nodes
Table 11.5. Numerical results for the Manhattanarea scenario and using femtocells instead of inband feederlinks Scenario/protocol
θ (MBit/s)
σ (MBit/s)
θ5% (MBit/s)
Downlink
Conventional Multicell MIMO Femtoonly, 2 RNs Femtoonly, 4 RNs Mixed, 2 RNs Mixed, 4 RNs
0.8 0.5 0.5 13.4 1.3 13.7
2.4 1.0 1.0 27.5 2.4 31.1
1.2 × 10−4 1.2 × 10−3 5.0 × 10−4 0.7 2.3 × 10−3 0.7
Uplink
Conventional Multicell MIMO Femtoonly, 2 RNs Femtoonly, 4 RNs Mixed, 2 RNs Mixed, 4 RNs
0.2 0.2 0.3 12.1 0.7 12.6
1.3 0.7 0.7 25.3 2.0 29.0
3.3 × 10−6 1.9 × 10−5 2.4 × 10−5 0.7 6.9 × 10−5 0.6
connection. Table 11.5 lists the numerical results for femtocells using the same parameters as for the previous relay setup except for a perfectly operating backhaul that provides very high data rates to each RN and a perfectly operating relay buﬀer. The average throughput for the case of four RNs improves by a factor of approximately 2 due to the increased UT resources and the perfectly operating relay buﬀer. Furthermore, the 5% quantile throughput improves by a factor of 3–5 as a major part of the additional resources are employed for indoor users. The similar uplink and downlink performances suggest that a majority of the SINR values are cut at 8 bpcu. Hence, the higher transmission power in the downlink does not necessarily provide higher data rates, which implies that the relayassisted network allows a lower RAP transmission power. Femtocells do not improve the performance if only two outdoor RNs are deployed, which underlines that the major bottleneck is the high pathloss from outdoor BSs to indoor UTs. This supports the ﬁeld trial results presented in [11], which show signiﬁcant performance gains for a relaybased communication system in an LTE environment. Femtocells provide major performance improvements but the necessity to connect indoor RAPs to the data network makes the deployment less ﬂexible. Furthermore, the deployment of femtocells is more expensive and it is not clear whether the backhaul resources require less computation and transmission power than a direct link between BS and RN. Nonetheless, the results prove the ﬂexibility of the relaying concept as indoor relays could be equipped with both
315
11.4 Numerical analysis
101 Uncoded QAM [7] 0
10
Turbo Codes [7]
+ 2
10
0
2
+
8
10
+
0
+
102
+
101
+
100
103 (b)
4 6 SNRGap Γ
101
+
Average throughput in MBit/s
(a)
Virtual MIMO Conventional 2 Relays, relayonly 2 Relays, mixed
+
103
+
+
+
101
+
+
+
Average throughput in MBit/s
Trellis Codes + THP [35]
4 6 SNRGap Γ
8
10
Virtual MIMO Conventional 2 Relays, relayonly 2 Relays, mixed 2
Figure 11.5. Average uplink throughput performance for varying SNRgap values: (a) Widearea scenario; (b) Manhattanarea scenario. a wireless interface to connect them directly to the BS and an interface to the wired broadband network.
11.4.5
Computationtransmissionpower tradeoﬀ An energyeﬃcient mobile communication system must consider the tradeoﬀ between computation and transmission power, which diﬀerently aﬀect the achieved performance. One way to examine this tradeoﬀ is the SNR gap approximation [7, pp. 66]. This approximation deﬁnes an SNR gap Γ ≥ 1 between
316
Green communications in cellular networks with ﬁxed relay nodes
practically possible modulation and coding schemes (MCS) and the theoretically possible capacity [35, 36]. More formally, given an SNR gap Γ we apply the modiﬁed capacity expression SINR . (11.4) CMCS = log2 1 + Γ Hence, the less eﬃcient the MCS, the higher the SNR gap Γ. In particular, it is a useful means of approximating the eﬀects of channel estimation errors, ﬁnite blocklengths, and dirty RF eﬀects. In Figure 11.5, we see the average uplink throughput for both scenarios and varying SNRgaps. It further shows that for uncoded QAM Γg ap = 8.8 dB for an error probability of 10−6 . More complex MCS such as turbo codes [37] achieve smaller SNR gaps of about Γg ap = 1 dB [7]. In [34], the authors showed that the SNR gap for trellis coding and Tomlinson–Harashima precoding [38, 39], which is a practical implementation of DPC, achieve Γg ap = 1.5 dB and up to about Γg ap = 5 dB. Furthermore, in [40] Kannan derived the SNR gaps for diﬀerent blocklengths and MIMO algorithms such as DBLAST. The performance drop in Γ for both relaying approaches in the widearea scenario has a slightly smaller slope than conventional and multicell MIMO transmission. The reason for this behavior is the very high SINR values in a relaybased system and the resulting rate cut at 8 bpcu. For the same reason the slope in the Manhattanarea scenario is smaller than in the widearea scenario. In order to achieve the same SINR after applying a practical MCS with SNR gap Γ, we had to spend at least additional transmission power Γ. If most SINR values exceed 28 − 1, which occurred for both relaying approaches in the Manhattanarea scenario, we do not need to increase the transmission power by Γ. Hence, the overall systemwide spent computation energy is reduced while providing almost the same performance. However, the optimal operating point is still diﬃcult to determine as the computation power for a speciﬁc approach can only be approximated. Nonetheless, this analysis gives an indication that relaying is able to cope with less powerful MCS while still providing signiﬁcant performance gains over conventional nonrelayingbased approaches using more powerful and more complex MCS. For instance, based on Figure 11.5, relaying with uncoded QAM provides higher data rates than conventional and multicell MIMO transmission without an SNR gap.
11.4.6
Reduced backhaul requirements In the relaybased system, multicell MIMO is particularly important for the area between two cells served at the same site while the celledge area between two diﬀerent sites is covered by RNs. Due to the low intercell interference, the cellcenter area in the main lobe direction of the BS could be served using conventional transmission. Since RAPs are uniformly distributed, the main interferers for the BStoRN links are located at the same site instead of diﬀerent sites. This
11.4 Numerical analysis
317
Table 11.6. Numerical results for the widearea scenario using inband feederlinks and limited intersite cooperation. Full cooperation uses virtually inﬁnite backhaul between multiple sites and limited cooperation only uses cooperation at the same site Scenario/protocol
θ (MBit/s)
σ (MBit/s)
θ5% (MBit/s)
Downlink
Conventional Multicell, full Multicell, limited Mixed, full Mixed, limited
1.3 2.4 1.9 4.9 4.9
2.4 3.0 2.7 3.9 3.9
2.3 × 10−2 0.2 8.6 · 10−2 0.7 0.7
Uplink
Conventional Multicell, full Multicell, limited Mixed, full Mixed, limited
0.1 2.39 1.92 4.95 4.94
0.5 3.03 2.71 3.91 3.94
4.3 × 10−4 0.16 0.09 0.66 0.65
motivates the investigation of a system where only BSs at the same site cooperate. Such a system uses the same hardware for all BSs at one site and therefore does not increase the backhaul requirements compared to a conventional system. In this Table 11.6 system is compared with limited cooperation and a system with virtually inﬁnite intersite backhaul between two physically separated sites (full cooperation). Only multicell MIMO transmission is aﬀected by the limitation of the intersite cooperation. More speciﬁcally, the average throughput is decreased by about 20 % and the 5% quantile throughput by approximately 50%. The decrease in the worstuser performance is mainly caused by the cell edge between two sites where full cooperation enables interference cancelation. In a relaybased system, most of the celledge users between two sites are served by RNs and therefore full and limited cooperation provide the same performance. Such a system using RNs and limited intersite cooperation is more robust against multicell MIMO synchronization errors, low coherence time/frequency, and the quality of the available CSI. In addition, it requires less resourceintensive backhaul network, which contributes a major part to the energy consumption of a mobile network.
11.4.7
Cost–beneﬁt tradeoﬀ So far we have assumed that the additional performance gains from relaying are provided for free, e. g., the increased costs have not been considered and the possible inﬂuence on the computation and transmission power in our network
318
Green communications in cellular networks with ﬁxed relay nodes
Table 11.7. Example for initial and operating costs for a relay node and site with three base stations Item of expenditure
Relay node
Site (with 3 BS)
Hardware Loan Installation Rental costs Energy costs (at 10 cent per kWh) Life span
10 kEUR 3 years at 8 % 5 kEUR 100 EUR per month
80 kEUR 8 years at 8 % 30 kEUR 1000 EUR per month
438 EUR p.a. (500W)
4.4 kEUR p.a. (5kW)
10a
10a
Cost ratio
αRN,
ref
= CRelay /CSite = 0.1
has not yet been evaluated. In order to provide fair results considering among things the energy costs of RAPs, we have the following options: (1) compare the performance for constant energy consumption per area element; (2) compare the required energy and costs for a constant performance measure; and (3) compare the performance for constant costs (including energy costs). In this chapter, we choose the third option and compare the performance for normalized system costs. Let CS ite denote the capital and operational expenditures of a site (consisting of three BSs) including hardware, site rental, average costs for power and backhaul supply. In addition, let CR elay denote the cost for a relay and αR N = CR elay /CS ite the relative costs of an RN compared to a site. Table 11.7 shows an example of the necessary costs of a BS and RN, which imply a relative costs of αR N = 0.1. If we assume a regular grid of sites as shown in Figure 11.2(a) with NR elay = 6 RNs assigned to each site, the intersite distance dis (αR N ) normalizing the system expenditures is given by (dis,r ef is the intersite distance in a system without relays): dis (αR N ) = dis,r ef 1 + αR N NR elay . (11.5) Figure 11.6 presents the 5% quantile uplink throughput depending on the relative relay costs αR N . The mixed approach provides a signiﬁcant performance gain for all plotted αR N and the relayonly approach for αR N < 0.2. Downlink and uplink transmission show the same behavior (downlink performance results are not shown in this section) although the performance beneﬁts of the downlink transmission are not as dominant. Our analysis uses inband feederlinks, which still require signiﬁcant resources for the BStoRN links. In the case of
319
11.4 Numerical analysis
+
+
+
Virtual MIMO Conventional 2 Relays, relayonly 2 Relays, mixed
+
102
+
101
+
5% quantile in MBit/s
+
100
0.10
0.15
0.20
0.25
αRN, ref
103
104
0
0.05
RNtoBS cost ratio αRN Figure 11.6. Cost–beneﬁt tradeoﬀ for uplink transmission and 5% quantile throughput.
femtocells these resources are used by RNs and the shown performance advantage would be even higher. Our cost–beneﬁt analysis included among other things the energy costs, which represent about 13% of the overall expenditure. However, assume that the target is to normalize the system’s energy consumption and let the energy consumption at the relay be ER N = βR N ES ite . Then, Figure 11.6 illustrates the performance gain for a normalized energy consumption depending on the relative energy consumption βR N . Using the particular setup in Table 11.7, an energy normalization implies for βR N < 0.1 that the costs for the resulting relaybased system are also reduced. For βR N = 0.1 and αR N = 0.1 the system is both costnormalized and energynormalized while providing signiﬁcant performance gains over conventional transmission techniques. A provider of mobile communication services is undoubtedly not exclusively interested in the ecological aspects of green communication but also in the economical aspects. Hence, the cost–beneﬁt tradeoﬀ provides an interesting tool to analyze the tradeoﬀ between the system expenditures, the performance, and the energy consumption of the system. In our example we are able to normalize both costs and energy while providing an improved performance in the relayassisted system. In addition, the computation energy can be reduced due to the higher performance in a relaybased system and the much higher SINR values. In this chapter we have not quantitatively considered the fact that relaying allows more deployment ﬂexibilities, very high data rates in otherwise shadowed areas,
320
Green communications in cellular networks with ﬁxed relay nodes
and the reduced backhaul requirements. All three of these contribute to the energy consumption of the system and further emphasize the ability of relaying to improve both the energy consumption in the system and to reduce the expenditures.
11.5
Conclusion We have introduced an approach to integrate relaying and multicell MIMO in next generation mobile communications systems. In particular, we have compared the achievable throughput performance, the energy saving potential, and how relaying might reduce deployment costs. Relays provide a ﬂexible way to improve the spatial reuse, are less complex than BSs and therefore cheaper to deploy, and reduce both the computation and transmission power in the system. Hence, relays are an ecological and economic alternative to systems based on direct transmission. These beneﬁts result from the more homogeneous power distribution due to a denser RAP distribution and the interference mitigation at the relay nodes. Furthermore, the ﬂexible and scalable deployment of RNs allows for a reuse of existing 2G and 3G sites, which further reduces deployment costs. In addition, a mixed approach using both relaying and onsite multicell MIMO signiﬁcantly reduces the backhaul requirements, which otherwise increase the deployment costs and require signiﬁcant energy resources. The probably most challenging problems in a relaybased mobile communication system are the synchronization of all RAPs, the resource assignment and synchronization of multiple cells, and the handover mechanisms. Furthermore, the optimization of the chosen MCS in order to exploit the available resources appears to be challenging. Practical results such as in [11] suggest that the promising results in this chapter are realizable and allow signiﬁcant performance improvements particularly for indoor users. Femtocells already represent a way to implement relaying using an existing infrastructure link, but are less ﬂexible and are not an option for those areas where the mobile communications system is the only choice to provide broadband access.
Acknowledgments Part of this work has been performed in the framework of the IST project IST4027756 WINNER II, which has been partly funded by the European Union, and the Celtic project CP5026 WINNER+. The authors would like to acknowledge the contributions of their colleagues in WINNER II and WINNER+, although the views expressed are those of the authors and do not necessarily represent the project.
References
321
References [1] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sumrate capacity of Gaussian MIMO broadcast channels,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2658–2668, October 2003. [2] H. Weingarten, Y. Steinberg, and S. Shamai, “The capacity region of the Gaussian MIMO broadcast channel,” in Proc. of IEEE International Symposium on Information Theory, Chicago (IL), USA, June 2004, p. 174. IEEE, 2004. [3] R. Ahlswede, “Multiway communication channels,” in Proc. of International Symposium on Information Theory, Tsakkadsor, Armenian SSR, 1971, pp. 23–52. Akad´emiai Kiad´ o, 1971. [4] H. Liao, Multiple access channels, PhD dissertation, University of Hawaii, 1972. [5] A. Sahai and P. Grover, “The price of certainty: Waterslide curves and the gap to capacity,” CoRR, vol. abs/0801.0352, 2008. [6] P. Rost, G. Fettweis, and J. Laneman, “Opportunities, constraints, and beneﬁts of relaying in the presence of interference,” in Proc. of IEEE International Conference on Communications, Dresden, Germany, June 2009. IEEE, 2009. [7] B. Li and G. St¨ uber, Orthogonal Frequency Division Multiplexing for Wireless Communications. Birkh¨ auser, 2006. [8] IST4027756 WINNER II, “D6.13.7 Test scenarios and calibration issue 2,” December 2006. [9] ETSI recommendation TR 30.03, “Selection procedure for the choice of radio transmission technologies of the UMTS,” November 1997. [10] IST4–027756 WINNER II, “D6.11.2 key scenarios and implications for WINNER II,” September 2006. [11] G. Fettweis, J. Holfeld, V. Kotzsch, et al. “Field trial results for LTEadvanced concepts,” in Proc. of IEEE International Conference. on Acoustics, Speech and Signal Processing (ICASSP), Dallas, TX, USA, March 2010. IEEE, 2010. [12] IST4027756 WINNER II, “D1.1.2 WINNER II channel models,” September 2007. [13] S. Weinstein and P. Ebert, “Data transmission by frequencydivision multiplexing using the discrete fourier transform,” IEEE Transactions on Communications, vol. 19, no. 5, pp. 628–634, October 1971. [14] L. Cimini, “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing,” IEEE Transactions on Information Theory, vol. 33, no. 7, pp. 665–675, July 1985. [15] C. Shannon, “Twoway communication channels,” in Proc. of 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 611–644. University of California Press, 1961.
322
Green communications in cellular networks with ﬁxed relay nodes
[16] A. Carleial, “Interference channels,” IEEE Transactions on Information Theory, vol. IT24, pp. 60–70, January 1978. [17] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, December 2004. [18] T. Nielsen, J. Wigard, and P. Mogensen, “On the capacity of a GSM frequency hopping network with intelligent underlayoverlay,” in Proc. of IEEE Vehicular Technology Conference (VTC), vol. 3, Phoenix, (AZ), USA, May 1997, pp. 1867–1871. IEEE, 1997. [19] F. Boye, Analyse und Bewertung der Implikationen von Relaying auf die Systemeigenschaften eines Zellularen 4gmobilfunknetzes mit Mehrantennen¨ ubertragung, Master’s thesis, Technische Universit¨at Dresden, 2007. [20] S. Shamai and B. Zaidel, “Enhancing the cellular downlink capacity via coprocessing at the transmitting end,” in Proc. of IEEE Vehicular Technology Conference (VTC), vol. 3, Rhodes, Greece, May 2001, pp. 1745–1749. IEEE, 2001. [21] P. Marsch and G. Fettweis, “On downlink network MIMO under a constrained backhaul and imperfect channel knowledge,” in Proc. of IEEE Global Communications Conference, Honolulu (HI), USA, December 2009. IEEE, 2009. [22] M. Costa, “Writing on dirty paper,” IEEE Transactions on Information Theory, vol. IT29, no. 3, pp. 439–441, May 1983. [23] A. Lapidoth, S. Shamai, and M. Wigger, “On the capacity of fading MIMO broadcast channels with imperfect transmitter side information,” in Proc. of Allerton Conference on Communications, St. Louis, MO, USA, November 2005. Curran Associates, Inc. 2005. [24] P. Marsch and G. Fettweis, “On uplink network MIMO under a constrained backhaul and imperfect channel knowledge,” in Proc. of IEEE International Conference on Communications, Dresden, Germany, June 2009. IEEE, 2009. [25] T. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Transactions on Information Theory, vol. IT27, no. 1, pp. 49–60, January 1981. [26] P. Rost, Opportunities, beneﬁts, and constraints of relaying in mobile communication systems, PhD dissertation, Technische Universit¨ at Dresden, 2009. [27] W. Yu and T. Lan, “Mimimax duality of Gaussian vector broadcast channels,” in Proc. of IEEE International Symposium on Information Theory, Chicago, IL, USA, June 2004, p. 175. IEEE, 2004. [28] W. Yu and T. Lan, “Transmitter optimization for the multiantenna downlink with perantenna power constraints,” IEEE Transactions on Signal Processing, vol. 55, no. 6, pp. 2646–2660, June 2007.
References
323
[29] G. Caire and S. S. (Shitz), “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Transactions on Information Theory, vol. 49, no. 7, pp. 1691–1706, July 2003. [30] K. Karakayali, G. Foschini, R. Valenzuela, and R. Yates, “On the maximum common rate achievable in a coordinated network,” in Proc. of IEEE International Conference on Communications, vol. 9, Istanbul, Turkey, June 2006, pp. 4333–4338. IEEE, 2006. [31] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Transactions on Information Theory, vol. 54, no. 12, pp. 5534–5562, December 2008. [32] P. Marsch, P. Rost, and G. Fettweis, “Application driven joint uplinkdownlink optimization in wireless communications,” in Proc. of 2010 International ITG/IEEE Workshop on Smart Antennas, Bremen, Germany, February 2010. IEEE, 2010. [33] V. Chandrasekhar, J. Andrews, and A. Gatherer, “Femtocell networks: A survey,” IEEE Communications Magazine, vol. 46, no. 9, pp. 59–67, September 2008. [34] W. Yu, D. Varodayan, and J. Cioﬃ, “Trellis and convolutional precoding for transmitterbased interference presubtraction,” IEEE Transactions on Information Theory, vol. 53, no. 7, pp. 1220–1230, July 2005. [35] C. Shannon, “Mathematical theory of communication,” Bell System Technical Journal, vol. 27, pp. 379–423 and 623–656, July and October 1948. [36] C. Shannon, “Communication in the presence of noise,” Institute of Radio Engineers, vol. 37, no. 1, pp. 10–21, 1949. [37] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit errorcorrecting coding and decoding: Turbocodes,” in Proc. of IEEE International Conference on Communications, vol. 2, May 1993, pp. 1064–1070. [38] M. Tomlinson, “New automatic equaliser employing modulo artihmetic,” IEE Electronics Letters, vol. 7, no. 5, pp. 138–139, March 1971. [39] H. Harashima and H. Mikayawa, “Matchedtransmission technique for channels with intersymbol interference,” IEEE Transactions on Communications, vol. COM20, no. 4, pp. 774–780, August 1972. [40] A. Kannan, Communication strategies for single user and multiuser slow fading channels, PhD dissertation, Georgia Institute of Technology, 2007.
12 Network coding in relaybased networks Hong Xu and Baochun Li
12.1
Introduction Since its inception in information theory, network coding has attracted a signiﬁcant amount of research attention. After theoretical explorations in wired networks, the use of network coding in wireless networks to improve throughput has been widely recognized. In this chapter, we present a survey of advances in relaybased cellular networks with network coding. We begin with an introduction to network coding theory with a focus on wireless networks. We discuss various network coded cooperation schemes that apply network coding on digital bits of packets or channel codes in terms of, for example, outage probability and diversity–multiplexing tradeoﬀ. We also consider physicallayer network coding which operates on the electromagnetic waves and its application in relaybased networks. Then we take a networking perspective, and present in detail some scheduling and resource allocation algorithms to improve throughput using network coding in relaybased networks with a crosslayer perspective. Finally, we conclude the chapter with an outlook into future developments. Network coding was ﬁrst proposed in [1] for noiseless wireline communication networks to achieve the multicast capacity of the underlying network graph. The essential idea of network coding is to allow coding capability at network nodes (routers, relays, etc.) in exchange for capacity gain, i.e., an alternative tradeoﬀ between computation and communication. This can be understood by considering the classic “butterﬂy” network example. In Figure 12.1, suppose the source S wants to multicast two bits a and b to two sinks D1 and D2 simultaneously. Each of the links in the network is assumed to have a unit capacity of 1 bit per time slot (bps). With traditional routing, each relay node between S and the two sinks simply forwards a copy of what it receives. It is then impossible to achieve the theoretical multicast capacity of 2 bps for both sinks, since the thick link in the middle can only transmit either a or b at a time. However, with network coding, the intermediate relay node (shaded in the ﬁgure) can perform coding, in this case a bitwise exclusiveor (XOR) operation, upon the two information bits and generate a + b to multicast towards its two outgoing links. D1 receives Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
12.1 Introduction
S
S
a
b
a
a
b
a
a? b?
b
a b
a
b
D2 (a)
b
a+b a+b
a+b
D1
325
D1
D2 (b)
Figure 12.1. The “butterﬂy” network example of network coding: (a) with traditional routing, the link in the middle can only transmit either a or b at a time; (b) with network coding, the relay node can mix the bits together and transmit a + b to achieve the multicast capacity of 2 bps.
a and a + b, and recovers b as b = a + (a + b). Similarly, D2 receives b and a + b and can recover a. Both sinks are therefore able to receive at 2 bps, achieving the multicast capacity. In the above example, the network coding operation is bitwise XOR, which can be viewed as linear coding over the ﬁnite ﬁeld GF (2). Following the seminal work of [1], Li et al. [2] showed that a linear coding mechanism suﬃces to achieve the multicast capacity. Ho et al. [3, 4] further proposed a distributed random linear network coding approach, in which nodes independently and randomly generate linear coeﬃcients from a ﬁnite ﬁeld to apply over input symbols without a priori knowledge of the network topology. They proved that receivers are able to decode with high probability provided that the ﬁeld size is suﬃciently large. These works lay down a solid foundation for the practical use of network coding in a diverse set of applications. After the initial theoretical studies in wireline networks, the applicability and advantages of network coding in wireless networks were soon identiﬁed and investigated extensively [5]. Though the noiseless assumption no longer holds for wireless communications, the wireless medium does provide a unique characteristic conducive to network coding operations – the inherent broadcasting capability. Again this can be best understood by a classic example of the “Alice and Bob” topology as in Figure 12.2. Assume that Alice and Bob want to exchange their information represented by bits a and b, respectively, and each link has a static capacity of 1 bps. It can readily be seen that under the traditional routing paradigm, four time slots are needed to exchange the bits through relay R, which
326
Network coding in relaybased networks
Figure 12.2. The beneﬁts of network coding in wireless networks: (a) four time slots are needed for Alice and Bob to exchange information bits through relay R by plain routing; (b) with network coding, R can XOR the bits and broadcast the coded bit to both parties simultaneously, reducing it to three time slots. sequentially forwards one bit at a time. In contrast, with network coding, R can XOR the two bits together and transmit the coded bit. Because of the broadcast nature of wireless medium, this transmission can be heard by both Alice and Bob. Alice then receives a + b, and recovers Bob’s bit b as b = a + (a + b). Similarly Bob can recover a. Therefore only three time slots are needed in this case, which represents a 25% throughput improvement for both parties. Inspired by the mathematical simplicity and practical potential of network coding, the communications and networking communities have devoted a signiﬁcant amount of research to utilizing it in a number of wireless applications, ranging from opportunistic routing in mesh networks to distributed storage and link inference in sensor networks. Our focus in this chapter is on relaybased networks, which essentially generalizes the example above in various ways depending on the network model. Here, the relays can refer to dedicated stations solely providing traﬃc relaying for others’ beneﬁts, or to users relaying one another’s signal, i.e., user cooperation. The purpose of relaying can be merely to extend the coverage of a cellular network when the base station is too distant from the mobile station (multihopping), or to combat fading by providing additional cooperative diversity with more advanced receiver hardware. Our discussion will assume that cooperative diversity is exploited whenever possible, i.e., whenever the receiver receives multiple transmissions that contain the same data, and relaying amounts to mere multihopping only when the receiver obtains one copy of the data.
12.2
Network coded cooperation In both wireline and wireless networks, network coding usually operates over bits, or symbols. The general use of this conventional form of network coding in
327
12.2 Network coded cooperation
a
b
a
a b
A
R
a+b
B
(a)
A
R
b
B
(b)
Figure 12.3. (a) Plain relaying without network coding; (b) network coded cooperation with one dedicated relay. cooperative diversity is usually termed network coded cooperation in the literature. As was originally invented for wireline networks, network coding can work at or above the MAC layer over the data bits of packets when applied to relaybased wireless networks, assuming the link layer delivers errorfree data. In other words, it serves as an “addon” component to the existing lower layer techniques. Alternatively, it can also be applied at the link layer, by a more dedicated joint design with channel coding/decoding.
12.2.1
Simple network coded cooperation Consider a simple network model where there are two mobile stations transmitting on the uplink to the base station. The simplest and the most naive way to attain cooperative diversity, shown in Figure. 12.3(a), is to replicate the classic threeterminal model in information theory by assigning a relay to users on orthogonal channels (this can be achieved by time, frequency, or code division). The cooperative transmission progresses in two phases [6]. In the ﬁrst phase, each mobile station transmits its own data on orthogonal channels while its relay receives and decodes the data. In the second phase the relay forwards the data to the base station, again on orthogonal channels. The base station receives both the original and relayed signals which results in a diversity order of 2 for each user. If timedivision is used to orthogonalize channels, a total of four time slots is needed for two mobile stations, 2 for the ﬁrst phase and 2 for the second phase. Chen et al. [7] were among the ﬁrst to point out that the same diversity gain can be achieved with less spectrum cost by network coding as shown in Figure 12.3(b), where the relay assists both users at the same time by transmitting the XORed version of information from both users in the second phase. If
328
Network coding in relaybased networks
any two of the three transmissions succeed, the base station can still recover both a and b, therefore the diversity order is 2 for both users. By a probabilistic analysis, it can be shown [7] that the above network coded cooperation also provides a lower system outage probability at high SNR if the total power consumption for the system is ﬁxed, since only three transmissions are required for a complete round of cooperative transmission. The above network coded cooperation scheme allows the relay node to ﬁrst combine information overheard from multiple sources with linear network coding, and then forward the coded data to the destination. This provides diversity for multiple sources with better spectral eﬃciency because the relaying bandwidth is suppressed. Though the analysis in [7] is fairly elementary, this key idea demonstrates the relevance of network coding in cooperative diversity, and is largely followed in the community. Now consider the more general model of a cellular network with N ≥ 2 users and M ≥ 1 relays communicating with the base station. Following [7], Peng et al. [8, 9] proposed an extended network coded cooperation for multiuser networks. Although their scheme was designed for multiple unicast sessions with N distinct destinations, it can be readily applied to a multiuser cellular network, as outlined above, with one common destination, the base station. In their scheme, each user still transmits its own data in the ﬁrst phase on orthogonal channels achieved by timedivision. Then in the second phase, a single “best” relay is selected from the M candidates that maximizes the worst instantaneous channel conditions of links from users to the relay and from the relay to the base station, and broadcasts the XORed version of data received from each user to the base station. The idea of using a single best relay instead of all available ones is rooted in opportunistic relaying, ﬁrst proposed in [10]. Recall the conventional cooperation protocols mandate that a relay transmission must be coupled with each source transmission, no matter whether the relay transmission uses distributed spacetime coding across multiple relays or opportunistically utilizes a single best relay [10]. Thus with timedivision, a total of 2N time slots is needed with N time slots in the ﬁrst phase and N time slots in the second phase for N users. In contrast, the network coded cooperation in [9] takes only one time slot in the second phase to broadcast the XORed message for all N users, and only N + 1 time slots are required. Therefore, intuitively, the multiplexing gain of the system can be improved by the proposed scheme. The question is, however: can it also maintain the full diversity gain of M + 1 provided by the M + 1 possible paths to the base station as the conventional schemes? A comprehensive diversity–multiplexing tradeoﬀ analysis of the selectionbased network coded cooperation was oﬀered in [9]. Before we present the results, let us recall the deﬁnition of the diversity–multiplexing tradeoﬀ. Deﬁnition 12.1 A scheme is said to achieve spatial multiplexing gain Rnorm and diversity gain D if the data rate R satisﬁes lim R(ρ)/ log ρ = Rnorm
ρ→∞
(12.1)
12.2 Network coded cooperation
329
and the average error probability pe satisﬁes − lim log pe (ρ)/ log ρ = D ρ→∞
(12.2)
where ρ denotes the signaltonoise ratio (SNR). We can see that essentially it is a fundamental tradeoﬀ between error probability and transmission rate, and has been widely recognized and adopted in the literature. Speciﬁcally, the following theorem was proved in [9]. Theorem 12.1 [9] In a singlecell network composed of N users and M relays, the above network coded cooperation scheme achieves a diversity–multiplexing tradeoﬀ given as follows: N +1 N Rnorm , Rnorm ∈ 0, . (12.3) D(Rnorm ) = 2 1 − N N +1 As a comparison, conventional cooperation protocols for multi–user networks (distributed spacetime coding or opportunistic relaying) achieve a diversity multiplexing tradeoﬀ of D(Rn or m ) = (M + 1) (1 − 2Rn or m ) for Rn or m ∈ 0, 12 . Therefore, only a diversity gain of 2 can be achieved at high SNR, while the multiplexing gain can indeed be improved from 12 to N/N + 1. In other words, the proposed network coded cooperation scheme achieves a diﬀerent tradeoﬀ. The reason why XOR network coding does not help in attaining the full diversity gain in this case is that, although the coded message can be potentially helpful for any user, it can only help at most one user provided that all the other N − 1 users’ data are decoded correctly, no matter what the number of relays is. The endtoend performance of one user is bottlenecked by all the other users, attributing to the diversity order1 of 2. Interestingly, as a special case, when M = 1, i.e., only one relay exists in the network, network coded cooperation achieves the full diversity order of 2 and has a strictly better diversity–multiplexing tradeoﬀ than the conventional scheme 2 (1 − 2Rn or m ) for Rn or m ∈ 0, 12 . It entails less loss in spectral eﬃciency to achieve the same diversity gain, and oﬀers larger diversity order at the same spectral eﬃciency in this case. Finally, Figure 12.4 graphically summarizes the above discussion and comparison of the diversity–multiplexing tradeoﬀ for diﬀerent cooperative diversity schemes. 1
In the case of multiple unicast sessions, it can be shown that this network coded cooperation has a diversity–multiplexing tradeoﬀ of D(R n o r m ) = (M + 1) (1 − N + 1/N R n o r m ) , R n o r m ∈ (0, N/N + 1), assuming that each distinct destination dj can reliably overhear messages from other sources sj , j = i. This assumption, however, implicitly requires enormous power consumption during sj s transmission and is strongly weakened when the network scales. It may not be convincing to demonstrate the superiority of networkcoded cooperation.
330
Network coding in relaybased networks
D(Rnorm) M
M
N N
Rnorm
Figure 12.4. Diversity–multiplexing tradeoﬀ comparison.
Another user cooperation protocol, named adaptive network coded cooperation (ANCC), which essentially achieves the same diversity–multiplexing tradeoﬀ as conventional cooperation protocols in the multiuser setting, was proposed in [11]. It also takes 2N time slots to complete one round of cooperation for N users. The basic idea is that in the second phase, each user, acting now as a relay, takes turns to select (randomly) a subset of messages it correctly overheard, and to transmit the binary checksum of them. From a coding perspective, the network coded cooperation can be viewed as matching networkongraph, i.e., instantaneous network topologies described in graphs, with the wellknown class of codeongraph, i.e., lowdensity paritycheck (LDPC) codes, on the ﬂy; hence the term adaptive. Analysis of the achievable rate and outage rate coupled with numerical evaluations shows that ANCC outperforms repetitionbased cooperation and is on a par with spacetime coded cooperation. This scheme, however, does introduce some level of overhead in transmitting a bitmap so the base station knows how the checksums are formed. Further, decoding also requires an adaptive architecture, and may not be practical in reality. From the above discussion, we can see that in a multiuser cellular network, the superiority of network coded cooperation mainly lies in its adaptivity to the lossy nature of the wireless media, and its operational simplicity. The peruser complexity, for both schemes in [9, 11], is invariant as the network grows, compared with spacetime coded cooperation, whose complexity and overhead increases linearly with the network size. We also note that, despite the fact that these proposals are available for combining network coding and cooperative relaying and show the beneﬁts of network coding in some scenarios, further research is required to unleash full potential of network coded cooperation and make it practical in more general scenarios.
12.2 Network coded cooperation
12.2.2
331
Joint network and channel coding/decoding The discussion so far has focused on using network coding on packets produced by channel decoding. Network coding essentially works as an erasure–correction code that operates above the link layer and oﬀers reliable communication through redundant transmission of packets. Channel coding, on the other hand, is an errorcorrection technique in a wireless environment. It is used in the link layer to recover erroneous bits through appending redundant paritycheck bits to a packet. By treating them separately, a certain degree of performance loss is introduced. Erasure–correction decoding cannot utilize the redundant information in those packets that fail the channel decoding, and hence are discarded at the link layer, while errorcorrection decoding cannot take advantage of the additional redundancy provided in the erasure codes. Indeed, in one early work on network coding [12] it was shown that, in general, capacity can only be achieved by a joint treatment of channel and network coding. A number of research eﬀorts have directed at unifying the two techniques in order to obtain further performance improvement. We survey some important ones in the following.
3 1
x1
x3 3
x2
x1 4
x3
x2 2 (a)
(b)
Figure 12.5. (a) A multipleaccess relay channel; (b) a twoway relay channel.
Let us ﬁrst revisit the example of Figure 12.3 where two users share one relay on the uplink to the base station, which resembles the multipleaccess relay channel in coding and information theory as shown in Figure 12.5(a). Hausl et al. [13] ﬁrst considered joint networkchannel decoding in this model, assuming that the relay can reliably decode messages from both users. A distributed regular LDPC code is used for networkchannel coding. Unlike in [7] in which the two messages transmitted from the users with the network coded message transmitted from the relay are separately decoded, they proposed to decode the three messages jointly on a single Tanner graph [14] with the iterative messagepassing algorithm. More precisely, let ui denote the information bits of user i, i ∈ {1, 2}, xi the networkchannel code, and Gij the generator matrix on the link from i to j.
332
Network coding in relaybased networks
Then the network encoder of [14] at the relay produces2 x3 = u1 G31 + u2 G32 .
(12.4)
The encoding operations at the users and the relay can be jointly described as
0 G31 G14 = uG. (12.5) x = [x1 x2 x3 ] = [u1 u2 ] 0 G24 G32 Then the paritycheck matrix H of the networkchannel code has 2N + NR columns and 2(N − K) + NR rows and has to fulﬁll GHT = 0, where N and NR denote the channel code length and network code length, respectively and K denotes the length of information bits u1 , u2 . The decoder at the base station decodes the LDPC code with paritycheck matrix H on the Tanner graph and exploits the diversity provided by network coding. In [13] the diversity and code length gain of networkchannel code was shown by numerically comparing it with two references systems, one where the relay is shared without network coding, and another where no relay is employed. The diversity and code length gain is not diﬃcult to explain intuitively. For the comparison to be fair, the numbers of code bits used in all systems, 2N + NR , have to be equal. Consider the setup with K = 1500 and N = NR = 2000, which corresponds to a channel code rate of 0.75 in the ﬁrst phase of cooperation. Networkchannel code achieves the full diversity order of 2 since x4 contains both u1 and u2 . For the system without network coding, the relay is shared and only transmits 1000 code bits for each user, while there are 1500 information bits. Thus it is impossible to achieve full diversity. Now consider another setup with K = 1500 and N = NR = 4000. Without network coding, the relay transmits 2000 code bits for each user and full diversity order can be achieved, with a channel code rate of 0.75 to the base station in the relaying phase. The networkchannel code, in this case, provides a more robust code rate of 0.375 to the base station for each user. In [13] the LDPCbased networkchannel coding was not compared with separate network and channel coding. In [15] the same idea was followed and turbocodebased networkchannel coding for the multipleaccess relay channel was proposed. Moreover, it was shown that even though full diversity order can be obtained with separate networkchannel coding, joint networkchannel coding based on turbo codes is able to exploit the additional redundancy in the relay transmission and exhibits strictly better bit error rate (BER) and frame error rate (FER) performance. Hausl further extended joint networkchannel coding to a similar model, the twoway relay channel [16] as in Figure 12.5(b). It is essentially an abstraction of the “Alice and Bob” example shown in Figure 12.2. Note that simple network 2
Note that the scheme in [7] does not consider interleaving and produces x 3 = (u 1 + u 2 )G 3 , which is slightly inferior in terms of bit error probability, though a diversity order of 2 is achievable.
12.2 Network coded cooperation
333
coded cooperation proposed for the multipleaccess relay channel as in [7] works without any change for the twoway relay channel, and full diversity order can be exploited. However, joint networkchannel coding does call for a diﬀerent design. The subtle diﬀerence is that in the twoway relay channel, each user exchanges information through a relay in the middle and decodes the other’s message, whereas in the multipleaccess relay channel a common destination decodes both messages from each user. An example of the twoway relay channel is the uplink and downlink transmissions between a mobile station and a base station with a relay. Thus the joint networkchannel decoder at the mobile station takes as inputs the channel code from the base station and the networkchannel code from the relay, and jointly decodes them to obtain the data from the base station. Its own data are utilized during the decoding process, since the networkchannel encoder at the relay is a convolutional encoder coding the interleaved bits from both the mobile station and the base station together. The networkchannel decoder at the base station works in a similar way. Again, joint networkchannel decoding is able to obtain the full diversity order of 2 here with better BER performance. The proposals in [13, 15, 16] rely on the key assumption that the relay is able to reliably decode the messages from both ends, and utilize the simplest XOR network coding. Yang and Koelter [17] and Kang et al. [18] broke this assumption and designed iterative network and channel decoding when the relay cannot perfectly recover packets. Guo et al. [19] took one step further and proposed a practical scheme, called nonbinary joint networkchannel decoding, which couples nonbinary LDPC channel coding and random linear network coding in a highorder Galois ﬁeld. A joint networkchannel coding scheme was also proposed in [20] for user cooperation that endows users with eﬃcient use of resources by transmitting the algebraic superposition of their locally generated information and relayed information that originated at the other user. Finally, we oﬀer an information theoretic perspective regarding joint networkchannel coding on the twoway relay channel in Figure 12.5(b) before we end the discussion. The second phase of transmission involves the relay broadcasting to two receivers at each end, and can be seen as the wellstudied broadcast channel. Joint networkchannel coding, as we discussed above, performs network coding across channel codewords, i.e., x3 = π32 (u1 ) + π31 (u2 ),
(12.6)
where πij (·) denotes the channel encoding function for link ij. In general, the capacityachieving code rate varies for diﬀerent links, and by coding channel codewords the capacity of each link involved in the broadcast is achieved as in the singlelink transmission.3 The only diﬀerence here is that for user i, certain 3
Note that all works we have described, including [13, 15, 17–20] which were not designed for the twoway relay channel, assume that the code rate is the same for all channel codes involved on all links to ease the decoding design. These proposals will work for the more
334
Network coding in relaybased networks
bits in the channel code are ﬂipped with a known pattern π3j (ui ) at the relay as in (12.6), and they are ﬂipped back after demodulation at i. If separate network channel coding is used, x3 = π3 (u1 + u2 ). We can readily see that the rate of π3 has to be conﬁned to the minimum rate of links 31 and 32 for both users to be able to decode, resulting in performance loss in terms of achieving the broadcast channel capacity. This is referred to as the coding to the worst rate problem. It is easy to verify that this problem does not exist in the multipleaccess relay channel.
12.3
Physicallayer network coding The main distinguishing characteristic of wireless communications is its broadcast nature. The radio signal transmitted from one node is often overheard by many neighboring nodes, and causes interference to their reception. Conventional wisdom treats interference as a nuisance and strives to avoid it by making the transmissions orthogonal to each other in time, frequency, or by code, which in eﬀects disguises a wireless link as a wired one. In the preceding discussions of network coded cooperation, network coding takes advantage of the broadcast nature without changing this fundamental interferenceavoiding structure of wireless transmissions. Opposite to this line of thinking, a novel paradigm that embraces interference as a unique capacityboosting advantage was developed in [21] and is gaining momentum. The idea is to encourage interference from concurrent transmissions, and by smart physicallayer techniques transform the superposition of the electromagnetic waves as an equivalent network coding operation that mixes the radio signals in the air. An apparatus of network coding is then created at the physical layer, and works on the electromagnetic waves in the air rather than on the digital bits of data packets or channel codes. Therefore, this is termed physicallayer network coding (PNC), or analog network coding, while the term digital network coding refers to its conventional counterpart. More speciﬁcally, Figure 12.6(a) illustrates the working and power of PNC, on the familiar twoway relay channel. Using PNC, only two time slots are needed for each user to exchange information, as opposed to three using digital network coding and four using direct transmission. In the ﬁrst time slot each user concurrently transmits to the relay. For simplicity, assume that BPSK modulation is used, and symbollevel and carrier phase synchronization and channel preequalization are done perfectly, so that the frames from users arrive at relay with the same amplitude and phase. Then the baseband signal received by the
general case where code rate is diﬀerent, with proper modiﬁcation, mostly of the decoding algorithm.
12.3 Physicallayer network coding
335
Time
–1 2,–2
Time
Figure 12.6. (a) The intuitive beneﬁts of physicallayer network coding. The top graph shows the transmission schedule using digital network coding, and the bottom graph shows that of physicallayer network coding. (b) BPSK modulation/demodulation mapping for physicallayer network coding. relay is r(t) = S1 (t) + S2 (t) = (a1 + a2 ) cos ωt,
(12.7)
where Si (t) is the baseband signal from user i. The relay does not decode both BPSK signals a1 and a2 from r(t). Instead it tries to decode and transmit the signal a1 + a2 . The original BPSK scheme has only two signals, 1 and −1 corresponding to data bits of 1 and 0, respectively. However r(t) has three signals, −2, 0, and 2. A modulation/demodulation mapping, shown in Figure 12.6(b), was developed for BPSK in [21] so that superposition of the analog signals a1 + a2 can be mapped to arithmetic addition of the digital bits they represent in GF (2). In theory, physicallayer network coding has the potential to improve throughput performance greatly (by 100% compared with direct transmissions and by 33.3% compared with digital network coding as in Figure 12.6(a)). This, however, does come with a nonnegligible price – the loss of diversity gain. Recall that the full diversity order of 2 is achievable in a twoway relay channel for conventional and networkcoded cooperation as discussed in Section 12.2.2. Since PNC entails concurrent transmissions in the ﬁrst time slot, both users only receive one transmission in the second slot when the relay broadcasts,4 whereas they receive a direct transmission and a relay transmission in conventional and networkcoded cooperation. Hence, cooperative diversity cannot be exploited and only a diversity order of 1 can be obtained with PNC, and relaying here boils down to simple multihopping. For this reason, to our knowledge almost all subsequent works on the subject compare PNC with multihopping and draw conclusions without considering cooperative diversity, even if it is possible. Hence unless otherwise 4
The standard assumption of halfduplex radio hardware is assumed.
336
Network coding in relaybased networks
speciﬁed, our discussion of PNC hereafter inherits this assumption from the literature, the validity and fairness of which, however, remain disputable and left to the judgment of readers. Given its promising potential, the idea of physicallayer network coding has generated a considerable amount of interest. Hao et al. [22] ﬁrst analyzed the achievable rates of PNC, assuming the additive white Gaussian noise (AWGN) channel model. Through numerical analysis it was shown that PNC outperforms digital network coding and direct transmission signiﬁcantly, and approaches the capacity limit of the twoway relay channel with appropriate modulation schemes. This is not unexpected as illustrated in the aforementioned example in Figure 12.6(a). Several estimation techniques were developed for PNC to deal with noise in the decoding process in [23]. The rigid and unpractical requirements of perfect synchronization and channel preequalization then became the focal point of critics of PNC, as the detrimental eﬀect of imperfect synchronization can be substantial (6 dB loss as shown in [22]). General PNC schemes that relax the synchronization requirement while preserving the performance superiority are actively being pursued. An amplifyandforward scheme, in which the relay directly ampliﬁes and forwards the interfered signal to both users as opposed to the decodeandforward strategy in [21], was proposed and studied in several works [22, 24–27]. It was generally found that amplifyandforward PNC is more robust and oﬀers better performance when synchronization is absent than decodeandforward PNC [22], whereas when perfect synchronization is provided amplifyandforward PNC suﬀers from a loss of optimality in terms of achievable rate [25]. The robustness of amplifyandforward PNC is more pronounced for general scenarios because no channel preequalization is required. In [26, 27] the case in which there are M relays available for the twoway relay channel was considered. To eﬀectively utilize multiuser diversity oﬀered by relays, a distributed relay selection strategy was proposed with selection criteria speciﬁcally designed for amplifyandforward PNC. Two information theoretic metrics, outage and ergodic capacities, closely related to the diversity and multiplexing gains of the system, were analytically and numerically evaluated to conﬁrm its advantage. Indeed, the following theorem can be proved as in [27]. Theorem 12.2 [27] The amplifyandforward PNC scheme in [27] achieves a diversitymultiplexing tradeoﬀ of the following: D(Rnorm ) = M (1 − Rnorm ), Rnorm ∈ (0, 1].
(12.8)
Clearly, multiuser diversity is capitalized as the diversity order is M . Moreover, the same multiplexing gain of 1 as direct transmission can also be attained, echoing our remark on the beneﬁt of network coding in suppressing bandwidth for relaying in Section 12.2.1. On the other hand, decodeandforward PNC is not able to utilize multiuser diversity gain, due to the operational requirement
12.4 Scheduling and resource allocation: crosslayer issues
337
of channel preequalization before decoding in order to make sure both signals are received with equal amplitude. One immediately realizes that, despite the technical diﬃculty, channel preequalization eﬀectively inverses the fading eﬀect of the channel, and results in loss of diversity gain. Joint networkchannel coding/decoding is also explored in the area of PNC. The main objective is to design a good coding/decoding scheme that maximizes the amount of information that can be reliably exchanged through the twoway relay channel. More advanced coding that applies lattice code on the relay was discussed in [28, 29]. We make a few comments about open issues and future directions of physicallayer network coding. First, several theoretic issues remain unsolved or unexplored, as research in this area is still in its nascent stage. Most importantly, only the simplest XOR network coding on GF (2) is realized with signal superposition in the air, while linear network coding can work on a much larger Galois ﬁeld. Note that in wireless communications, pathloss and channel fading eﬀectively couple to the transmitted signal a complex coeﬃcient, which, to some extent, resembles the multiplication operation in a ﬁnite ﬁeld of complex numbers, while signal superposition resembles the addition operation. It is therefore interesting, but certainly nontrivial, to investigate whether the complex channel coeﬃcient can be “transformed” to an equivalent network coding coeﬃcient to realize an even larger improvement, in performance. In addition, all the developed schemes remain fairly theoretical, and very few researchers have attempted to evaluate the true performance of PNC in a real hardware implementation. To the best of our knowledge, the only work in which the practical coding and decoding algorithms for MSK modulation based on amplifyandforward PNC is sketched, and the system using software radios implemental is [24]. It is asserted that PNC is indeed practical, and achieves signiﬁcantly higher throughput than digital network coding. While this conveys an optimistic message, one should realize that physicallayer network coding marks a signiﬁcant departure from the conventional wisdom, and tremendous eﬀorts have to be made across layers of the protocol stack which are designed to avoid interference before it can be implemented in general scenarios.
12.4
Scheduling and resource allocation: crosslayer issues The discussion so far has largely focused on the lower layers of the protocol stack. From communications and information theory point of view, numerous schemes have been developed based on network coding to utilize a given budget of resources to transmit information as eﬃciently as possible. In other words, they answer the question of how to transmit in a given scenario for individual nodes. From a networking point of view, provided with the lowerlayer transmission protocol, upperlayer protocols have to coordinate the transmissions between nodes in the network, and allocate resources networkwise (such as channels) and
338
Network coding in relaybased networks
Figure 12.7. (a) A cellular network with multiple relays. Solid lines denote direct transmissions while dotted ones denote relay transmissions. All transmissions happen on orthogonal OFDM subchannels. Note that one mobile station (MS) can be paired up with multiple relays (RSs), while one relay could help multiple mobile stations, complicating the scheduling and resource allocation. (b) XORassisted cooperative diversity as adapted from [31]. Here diﬀerent line types denote diﬀerent subchannels. (BS – base station).
nodewise (such as power) adequately to these competing sessions so as to optimize some networkwise metrics. In other words, we need to create scenarios amenable for network coding aided relay transmission protocols such that performance is maximized from a network perspective. The use of network coding certainly calls for new designs of crosslayer scheduling and resource allocation protocols on existing cellular network architectures. To this end, there exist only a few publications in the literature. Zhang and Li [30] were arguably the ﬁrst to venture into this area. They considered the use of simple network coded cooperation on the twoway relay channel, with the base station serving as the relay to XOR the incoming packets and broadcast to two users. They assumed an orthogonal frequencydivision multipleaccess (OFDMA) based cellular network model, and developed a coding aware dynamic subcarrier assignment heuristic in a frequencyselective multipath fading environment. The idea is that in such an environment, diﬀerent OFDM subcarriers have independent channel gains to the same mobile station (multichannel diversity), and even the same subcarrier fades diﬀerently on diﬀerent mobile stations (multiuser diversity). By dynamically matching the subcarriers to the best mobile stations for network coded cooperation while taking fairness into account, a substantial throughput improvement was reported in [30]. Another work, that of Xu and Li [31], represents an indepth investigation in this area, which is discussed in detail here. Consider the more general scenario in Figure 12.7(a). Assume multiple relays are available for the OFDMAbased network, and all transmissions happen on orthogonal OFDM subchannels. An XORassisted cooperative diversity scheme, named XORCD, is employed by replicating the twoway relay channel using bidirectional traﬃc of a given mobile
12.4 Scheduling and resource allocation: crosslayer issues
339
station as in Figure 12.7(b). Speciﬁcally, it works as follows. In the ﬁrst phase of cooperation, on the uplink MS sends packet A to the base station and on the downlink base station sends packet B to mobile station, simultaneously on orthogonal subchannels. Transmissions are overheard by the relay. In the second phase, the relay broadcasts A + B using another orthogonal subchannel. We can readily see that cooperative diversity can still be exploited as in conventional schemes, which require four subchannels and more power to complete the same job. Extending to the network scale, while XORCD shows promise, it seems prohibitive in terms of cost and complexity to design good scheduling and resource allocation algorithms for the following reasons. First, one mobile station can be paired up with multiple relays and one relay could help multiple mobile station, as can be illustrated in Figure 12.7(a). Second, for one mobile station, direct transmission, conventional cooperative diversity, and XORCD can be utilized at the same time on diﬀerent subchannels, depending on the dynamic channel conditions and resource availability. Third, there is an intricate interplay between three of the problems, namely relay assignment, relaystrategy selection, and subchannel assignment, further aggravating the problem. The contributions of [31] is twofold. First, a unifying optimization framework is developed that jointly considers relay assignment, relaystrategy selection, and subchannel assignment for both mobile station and relays. Second, the joint optimization problem, referred to as a relay assignment, relaystrategy selection and subchannel assignment (RSS) XOR problem, is shown to be NPhard, and an eﬃcient approximation algorithm is proposed based on set packing with a provable approximation ratio. Speciﬁcally, the following theorem is shown to hold. Theorem 12.3 [31] The RSSXOR problem is equivalent to a maximum weighted threeset packing problem, and is NPhard. Proof. Construct a collection C of sets from a base set ζ ∪ ψ as shown in Figure 12.8, where ζ is the set of data subchannels and ψ the set of relay subchannels. As we can see, there are three kinds of sets, representing three possible transmission modes (ci ), ci ∈ ζ, represents the direct transmission with data subchannel ci : (ci , cr ), ci ∈ ζ, cr ∈ ψ; corresponds to the conventional cooperative diversity with data subchannel ci and relay subchannel cr ; ﬁnally, (ci , cj , cr ), ci , cj ∈ ζ, cr ∈ ψ, corresponds to XORCD with data subchannel pair (ci , cj ) and relay subchannel cr . Sets intersect if they share at least one common channel, and are otherwise said to be disjoint. Each set has a corresponding weight, denoting the maximum marginal utility found across all possible assignments of this set to diﬀerent combinations of relays and links.5 The utility function is deﬁned such that proportional fairness is taken into account. For (ci , cj , cr ), its weight is found 5
One mobile station has two links, namely the uplink and the downlink.
340
Network coding in relaybased networks
Figure 12.8. Set construction and transformation into an intersection graph with two data subchannels and two relay subchannels. Vertices in GC correspond to sets in C. Edges are added between vertices whose corresponding sets intersect. over all possible assignments of this set to combinations of relays and both links of a MS. The RSSXOR problem is to ﬁnd the optimal strategy to choose the transmission mode and assign relays and channels to each link in order to maximize the aggregated throughput. The maximization is done across all links. Equivalently, we can also interpret it as ﬁnding the optimal strategy to select disjoint channel combinations, and assign relays and links to them so as to maximize the objective. In this alternative interpretation, the maximization is done over all possible channel sets by matching them to the best possible links and relays, without violating the obvious constraint that each channel can only be used once. The number of elements in a set is at most three; therefore, this problem is essentially a weighted threeset packing problem [32], which is NPcomplete. To propose an approximation algorithm, we ﬁrst construct an intersection graph GC , such that the a vertex in the vertex set VC corresponds to a set in the set system C, and there is an edge between two vertices in GC if the two corresponding sets intersect, as shown in Figure 12.8. Weighted set packing can then be generalized as a weighted independent set problem, the objective of which is to ﬁnd a maximum weighted subset of mutually nonadjacent vertices in GC [33]. The size of sets is at most three, therefore GC is threeclaw free. Here, a dclaw c is an induced subgraph that consists of an independent set Tc of d nodes. The bestknown approximation algorithm for the weighted independent set problem in a clawfree graph is proposed in [33] and then acknowledged in [32], which is then adopted in [31] as the solution algorithm with an approximation ratio of 23 as proved in [33]. As a side note, it was also shown in [31] that the joint optimization problem that involves only conventional cooperative diversity and direct transmission can be cast as a weighted bipartite matching problem, which admits polynomialtime algorithms to obtain the optimal solution. This demonstrates the interesting point that, although network coding provides signiﬁcant performance gains, it may render the scheduling and resource allocation problems even more involved at least in some cases. Moreover, this also reﬂects the importance of developing
12.5 Conclusion
341
crosslayer protocols to fully realize the beneﬁts of network coding in relaybased networks. While an attempt was made to address these upperlayer issues in [30, 31], the algorithms developed are still impractical for implementation in realworld cellular networks. Full channel state information (CSI) is assumed to be available at the base station, which may not be the case for fast fading environments. The complexity of the algorithms is high for realtime scheduling at the time scale of 5–10 ms, which is the common frame length. Therefore, substantial eﬀorts in design, implementation, and evaluation of practical protocols are imperative to further understand and to conquer the challenges network coding brings to relaybased networks.
12.5
Conclusion Network coding, by allowing network nodes to mix information ﬂows, represents a paradigm shift for communication networks. With the broadcast nature of the wireless medium, it can be naturally applied to wireless communications and brings the promise of performance improvement in relaybased cellular networks. The question is: what is the most eﬃcient way to utilize network coding in relaybased networks, and how practical is it to be adopted in the real world? We have started with networkcoded cooperation that applies network coding on digital bits of information, both without and with a joint consideration of channel codes. We have noted the operational simplicity and multiplexing gain of networkcoded cooperation, and have shown that in general a joint design of network and channel coding/decoding is required to achieve the information theoretic capacity, at a cost of increased decoding complexity. Then we have extended the discussion to physicallayer network coding, a radical yet interesting idea that treats the signal superposition in the air as the XOR networkcoding operation. By embracing interference, physicallayer network coding has the potential dramatically to improve the throughput performance of relaybased networks. From the communications and information theory perspective, the overall message appears to be very optimistic. From a networking perspective, although network coding is mostly applied in the lower layers of the protocol stack, upper layers have to be codingaware and perform scheduling and resource allocation accordingly. In other words, crosslayer eﬀorts are needed to realize network coding in a practical network setting. To this end, we have presented some initial crosslayer studies on cellular relay networks. An important lesson is that network coding may render the scheduling and resource allocation more complicated, at least in some cases, because it mixes information ﬂows from diﬀerent sessions. Despite a signiﬁcant amount of research, the use of network coding in relaybased cellular networks remains largely theoretical. Substantial eﬀorts on implementation and evaluation of networkcodingaided transmissions in largescale
342
Network coding in relaybased networks
relay networks are imperative at this stage. With the proliferation of wireless technologies and devices, and the evergrowing bandwidth demand from dataintensive mobile applications, we envision that network coding could become a practical and important component in future wireless technologies, and play an important role in the quest of high throughput and ubiquitous connectivity in wireless communications.
References [1] R. Ahlswede, N. Cai, S.Y. R. Li, and R. W. Yeung, “Network information ﬂow,” IEEE Trans. Inform. Theory, 46:4 (2000), pp. 1204–1216. [2] S.Y. R. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. Inform. Theory, 49:2 (2003), pp. 371–381. [3] T. Ho, M. M´edard, J. Shi, M. Eﬀros, and D. R. Karger, “On randomized network coding,” in Proc. of 41st Allerton Conference on Communications, Control, and Computing, 2003. University of Illinois at UrbanaChampaign, 2003. [4] T. Ho, M. M´edard, R. Koetter, et al., “A random linear network coding approach to multicast,” IEEE Trans. Inform. Theory, 52:10 (2006), pp. 4413–4430. [5] Y. Wu, P. A. Chou, and S.Y. Kung, “Information exchange in wireless networks with network coding and physicallayer broadcast,” in Proc. of 39th Annual Conference on Information Sciences and Systems (CISS), 2005. Princeton University, 2005. [6] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior,” IEEE Trans. Inform. Theory, 50:12 (2004), pp. 3062–3080. [7] Y. Chen, S. Kishore, and J. Li, “Wireless diversity through network coding,” in Proc. of IEEE Wireless Communications and Networking Conference (WCNC), 2006. IEEE, 2006. [8] C. Peng, Q. Zhang, M. Zhao, and Y. Yao, “On the performance analysis of networkcoded cooperation in wireless networks,” in Proc. of IEEE INFOCOM, 2007. IEEE, 2007. [9] C. Peng, Q. Zhang, M. Zhao, Y. Yao, and W. Jia, “On the performance analysis of networkcoded cooperation in wireless networks,” IEEE Trans. Wireless Commun., 7:8 (2008), pp. 3090–3097. [10] A. Bletsas, A. Khisti, D. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., 24:3 (2006), pp. 659–672. [11] X. Bao and J. Li, “Adaptive network coded cooperation (ANCC) for wireless relay networks: Matching codeongraph with networkongraph,” IEEE Trans. Wireless Commun., 7:2 (2008), pp. 574–583.
References
343
[12] M. Eﬀros, M. M´edard, T. Ho, S. Ray, D. Karger, and R. Koetter, “Linear network codes: A uniﬁed framework for source, channel and network coding,” in Proc. of DIMACS Workshop on Network Information Theory, 2003. American Mathematical Society, 2003. [13] C. Hausl, F. Schreckenbach, I. Oikonomidis, and G. Bauch, “Iterative channel and network decoding on a Tanner graph,” in Proc. of 43rd Allerton Conference on Communications, Control, and Computing, 2005. Curran Associates, Inc., 2006. [14] R. M. Tanner, “A recursive approach to lowcomplexity codes,” IEEE Trans. Inform. Theory, 27:5 (1981), pp. 533–547. [15] C. Hausl and P. Dupraz, “Joint networkchannel coding for the multipleaccess relay channel,” in Proc. of International Workshop on Wireless Adhoc and Sensor Networks (IWWAN), 2006. IEEE, 2006. [16] C. Hausl, and J. Hagenauer, “Iterative network and channel decoding for the twoway relay channel,” in Proc. of IEEE ICC, 2006. IEEE, 2006. [17] S. Yang and R. Koetter, “Network coding over a noisy relay: A belief propagation approach,” in Proc. of IEEE International Symposium on Information Theory (ISIT), 2007. IEEE, 2007. [18] J. Kang, B. Zhou, Z. Ding, and S. Lin, “LDPC coding schemes for error control in a multicast network,” in Proc. of IEEE International Symposium on Information Theory (ISIT), 2008. IEEE, 2008. [19] Z. Guo, J. Huang, B. Wang, J.H. Cui, S. Zhou, and P. Willett, “A practical joint networkchannel coding scheme for reliable communication in wireless networks,” in Proc. of ACM MobiHoc, 2009. ACM, 2009 [20] L. Xiao, T. E. Fuja, J. Kliewer, and D. J. Costello, Jr., “A network coding approach to cooperative diversity,” IEEE Trans. Inform. Theory, 53:10 (2007), pp. 3714–3722. [21] S. Zhang, S. C. Liew, and P. Lam, “Hot topic: Physicallayer network coding,” in Proc. of ACM Mobicom, 2006. ACM, 2006. [22] Y. Hao, D. L. Goecket, Z. Ding, D. Towsley, and K. K. Leung, “Achievable rates of physical layer network coding schemes on the exchange channel,” in Proc. of IEEE Military Communications Conference (MILCOM), 2007. IEEE, 2007. [23] S. Zhang, S. C. Liew, and L. Lu, “Physicallayer network coding schemes over ﬁnite and inﬁnite ﬁelds,” in Proc. of IEEE Global Communications Conference (Globecom), 2008. IEEE, 2008. [24] S. Katti, S. Gollakota, and D. Katabi, “Embracing wireless interference: Analog network coding,” in Proc. of ACM SIGCOMM, 2007. ACM, 2007 [25] P. Popovski and H. Yomo, “Physical network coding in twoway relay channels,” in Proc. of IEEE ICC, 2007. IEEE, 2007. [26] Z. Ding, K. K. Leung, D. Goeckel, and D. Towsley, “On the capacity of network coding with diversity,” in Proc. of First Annual Conference of the International Technology Alliance, 2008. Imperial College London, 2008.
344
Network coding in relaybased networks
[27] Z. Ding, “On the study of network coding with diversity,” IEEE Trans. on Wireless Commun., 8:3 (2009), pp. 1247–1259. [28] K. R. Narayanan, M. P. Wilson, and A. Sprintson, “Joint physical layer coding and network coding for bidirectional relaying,” in Proc. of 45th Annual Allerton Conference on Communications, Control and Computing, 2007. University of Illinois at UrbanaChampaign, 2007. [29] B. Nazer and M. Gastpar, “Computeandforward: Harnessing interference with structured codes,” in Proc. of IEEE Intl. Symposium on Information Theory (ISIT), 2008. IEEE, 2008. [30] X. Zhang and B. Li, “Joint network coding and subcarrier assignment in OFDMAbased wireless networks,” in Proc. of 4th International Workshop on Network Coding, Theory and Applications (NetCod), 2008. The Chinese University of Hong Kong, 2008. [31] H. Xu and B. Li, “XORassisted cooperative diversity in OFDMA wireless networks: Optimization framework and approximation algorithms,” in Proc. of IEEE INFOCOM, 2009. IEEE, 2009. [32] B. Chandra and M. M. Halld´ orsson, “Greedy local improvement and weighted set packing approximation,” J. Algorithms, 29:2 (2001), pp. 223– 240. [33] P. Berman, “A d/2 approximation for maximum weight independent set in dclaw free graphs,” Nordic J. Computing, 7:3 (2000), pp. 178–184.
Part IV Game theoretic models for cooperative cellular wireless networks
13 Coalitional games for cooperative cellular wireless networks Walid Saad, Zhu Han, and Are Hjørungnes
13.1
Introduction Cooperation among wireless nodes has attracted signiﬁcant attention as a novel networking paradigm for future wireless cellular networks. It has been demonstrated that, by using cooperation at diﬀerent layers (physical layer, multiple access channel (MAC) layer, network layer), the performance of wireless systems such as cellular networks can be signiﬁcantly improved. In fact, cooperation can yield signiﬁcant performance improvement in terms of reduced bit error rate (BER), improved throughput, eﬃcient packet forwarding, reduced energy, and so on. In order to reap the beneﬁts of cooperation, eﬃcient and distributed cooperation strategies must be devised in future wireless networks. Designing such cooperation protocols encounters many challenges. On the one hand, any cooperation algorithm must take into account not only the gains but also the costs from cooperation which can both be challenging to model. On the other hand, the wireless network users tend to be selﬁsh in nature and aim at improving their own performance. Therefore, giving incentives for these users to cooperate is another major challenge. Hence, there is a strong need to design cooperative strategies that can be implemented by the wireless nodes, in a distributed manner, while taking into account the selﬁsh goals of each user as well as all the gains and losses from this cooperation. This chapter describes analytical tools from game theory that can be used to model the cooperative behavior in wireless cellular networks. In this context, the key tool that will be explored is the framework of game theory/coalitional game theory. For instance, coalitional game theory is the main branch of game theory/cooperative game theory and it describes the formation of cooperating groups of players, referred to as coalitions [1, 2]. Coalitional games prove to be a very powerful tool for designing fair, robust, practical, and eﬃcient cooperation strategies, which can be used to model cooperative behavior in wireless networks. The main goal in this chapter is to give the reader a good understanding of how cooperative behavior in a wireless system can be modeled through coalitional game theory. In particular, the chapter gives a better understanding Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
348
Coalitional games for cooperative cellular wireless networks
of two main applications of coalitional games in wireless networks: (i) distributed cooperation for virtual multipleinput multipleoutput (MIMO) formation; and (ii) the impact of cooperative transmission and relaying on network formation in a wireless network. To achieve these goals, the chapter is organized as follows. In Section 13.2, we provide a brief introduction to the main ideas of coalitional game theory. Then, in Section 13.3, we present a coalitional game model that allows single antenna users to beneﬁt from the performance advantage of MIMO systems, through cooperation. Further, in Section 13.4, we discuss how coalitional game theory can be used to study the structure and dynamics of the wireless network architecture in the presence of relay station nodes utilizing cooperative transmission. Finally, a chapter summary is provided in Section 13.5.
13.2
A brief introduction to coalitional game theory In general, coalitional game theory involves a set of players that are seeking to form cooperative entities, i.e., coalitions. Let N denote the set of these players. By forming any coalition S ⊆ N , the players can improve the utility they obtain in the game. Thus, another main concept of coalitional game theory is the value or utility which describes the total beneﬁt that the members of a coalition S can obtain when acting cooperatively. The value of a coalitional game can have diﬀerent forms: characteristic form, partition form, or graph form. Brieﬂy, a coalitional game is in a characteristic form if the utility of a coalition S depends solely on the members of that coalition, with no dependence on the players in N \ S. In contrast, a game is in partition form if, for any coalition S ⊆ N , the coalitional value depends on both the members of S as well as the coalitions formed by the members in N \ S. In certain coalitional games, the diﬀerent players are connected to each other through a graph structure. Consequently, for modeling such coalitional games, the value is considered in graph form, i.e., for each graph structure, a diﬀerent utility can be assigned. Independent of the form of the game, every coalitional game is uniquely deﬁned by the pair (N , v), where N is the players’ set and v is the coalitional value. In any coalitional game, it is always important to distinguish between two entities: the value of a coalition and the payoﬀ received by a player. The value of a coalition represents the amount of utility that a coalition, as a whole, can obtain. In contrast, the payoﬀ for a player represents the amount of utility that a player, the member of a certain coalition, will obtain. For instance, depending on how the value is mapped into payoﬀs, the coalitional game can be either with transferable utility (TU) or with nontransferable utility (NTU). A TU game implies that the total utility received by any coalition S ⊆ N can be apportioned in any manner between the members of S. A prominent example of TU type games is when the value represents an amount of money, which can be distributed in any way between coalition members. In contrast, in an NTU,
13.2 A brief introduction to coalitional game theory
349
the individual payoﬀ obtained by any player within a coalition S is restricted by some underlying structure of the game. In a TU environment, the utility obtained by a coalition is characterized by a function over the real line, while in an NTU framework, the utility of a coalition S is a set of payoﬀ vectors of size 1 × S (S represents the cardinality of set S), whereby each element of any vector, represents the payoﬀ that a particular player in S receives. Depending on the metric being used as a utility, the game can be transferable or nontransferable. Consequently, depending on the form, type, and objectives of a coalitional game, three classes of games can be distinguished [3]: canonical coalitional games, coalition formation games, and coalitional graph games. Each class has its own properties, solutions, and challenges. For instance, the canonical coalitional game class can be used to model problems where: (i) the value is in characteristic form (or can be mapped to the characteristic form through some assumptions); (ii) cooperation is always beneﬁcial, i.e., including more players in a coalition does not decrease its value; and (iii) there is a need to study how payoﬀs can be allocated in a fair manner that stabilizes the grand coalition, i.e., coalition of all players. In wireless networks, such problems are of interest to study the limits and fairness of cooperation, when no cost for forming coalitions exists. For example, using canonical coalitional games, the work in [4] focused on devising a cooperative model for rate improvement through ideal receivers cooperation. Using canonical games, the authors of [4] showed that, for the receiver coalition game in a Gaussian interference channel and synchronous CDMA multiple access channel (MAC), a stable grand coalition of all receivers can be formed if no cost for cooperation is taken into account. In addition, using canonical games, the fair allocation of rate for cooperating users in an interference channel was studied in [5] for the transmitters. Under some assumptions on the users’ behavior, the authors showed that a unique rate allocation exists verifying certain welldeﬁned fairness axioms from canonical coalitional games. In canonical games, there is an implicit assumption that forming a coalition is always beneﬁcial. In contrast, coalition formation games consider cooperation problems in the presence of both gains from and costs of cooperation. This is quite a useful class of games since, in several problems, forming a coalition requires a negotiation process or an information exchange process which can incur a cost, thus reducing the gain from forming the coalition. In general, coalition formation games are of two types: static coalition formation games and dynamic coalition formation games [3]. On the one hand, for static coalition formation games, a coalitional structure is imposed on the game through some external factor (e.g., a network owner), and, hence, the goal of static games is to study the already formed structure. On the other hand, in dynamic coalition formation games, the main objectives are to analyze the formation of a coalitional structure, through players’ interaction, as well as to study the properties of this structure and its adaptability to environmental variations or externalities. As a result, a key question that dynamic coalition formation games seek to answer is “which
350
Coalitional games for cooperative cellular wireless networks
coalitions will form in a given game?”.1 For instance, in any problem where the beneﬁt–cost tradeoﬀ from cooperation is of interest, coalition formation games constitute a wellsuited analytical tool since they provide rules, concepts, and algorithms for characterizing the coalitional structure that can emerge for a given coalitional game. A coalition formation game can be in either characteristic or partition form, as the network structure plays a key role in any solution. In wireless systems, coalition formation games are quite a useful tool for studying various problems such as virtual MIMO formation [6], cognitive radio networks [7, 8], or physical layer security [9, 10]. In both canonical and coalition formation games, the utility or value of a given coalition has no dependence on how the players inside (and outside) the coalition communicate. Nonetheless, in certain scenarios, the underlying communication structure, i.e., the graph that represents the connectivity between the players in a coalitional game can have a major impact on the utility and other characteristics of the game [1, 3]. In such scenarios, coalitional graph games constitute a strong tool for studying the graph structures that can form based on the cooperative incentives of the various players. In coalitional graph games, instead of focusing on forming or studying the properties of coalitions, one is interested in studying the properties of various graph structures. In this regard, coalitional graph games can be used in many wireless and communications applications such as routing, network formation, and vehicular networks. In the remainder of this chapter, we will illustrate, in detail, the use of coalition formation games and coalitional graph games in two main applications in cooperative wireless cellular networks. On the one hand, we will discuss how coalition formation games can be used for distributed virtual MIMO formation. On the other hand, we will formulate a coalitional graph game for studying the network architecture that will form in the uplink nextgeneration wireless systems when cooperative transmission is used. Finally, we note that, although canonical coalitional games also admit many interesting applications within cooperative wireless cellular networks (notably when cooperation has no costs), their treatment is outside the scope of the present chapter and the interested reader is referred to [3–5].
13.3
A coalition formation game model for distributed cooperation In this section, we investigate the use of coalition formation games for modeling cooperative behavior among the users of a wireless network. In particular, we emphasize the problem of the formation of multiple antenna systems through cooperation among singleantenna transmitters. First, we motivate the problem
1
In the remainder of this chapter, the term “coalition formation game” refers to a dynamic coalition formation game.
13.3 A coalition formation game model for distributed cooperation
351
and provide a suitable system model. Then, we formulate the problem using coalitional game theory, we describe the properties of the resulting game, and we derive an algorithm for coalition formation.
13.3.1
Motivation and basic problem An important application for cooperation in wireless cellular networks is the formation of virtual MIMO systems through cooperation among singleantenna devices. In this context, a number of singleantenna devices can form virtual multipleantenna transmitters or receivers through cooperation, and consequently, beneﬁt from the advantages of MIMO systems without the extra burden of having multiple antennas physically present on each transmitter or receiver. Thus, the basic idea of virtual MIMO is to rely on cooperation among mobile devices in order to beneﬁt from the widely acclaimed performance gains of MIMO systems. An intensive research eﬀort has been dedicated to information theoretic studies of virtual MIMO systems. For instance, the authors of [11] showed the interesting gains in terms of outage capacity resulting from the cooperation of two singleantenna devices that are transmitting to a far away receiver in a Rayleigh fading channel. Further, the work in [12, 13] considered cooperation among multiple singleantenna transmitters as well as receivers in a broadcast channel. Diﬀerent cooperative scenarios were thus studied and the results showed the beneﬁts of cooperation from a sumrate perspective. It is important also to note that virtual MIMO gains are not only limited to rate gains. For example, forming virtual MIMO clusters in sensor networks can yield gains in terms of energy conservation [14]. Implementing distributed cooperation algorithms that allow the wireless network to reap these capacity or energy beneﬁts in a practical wireless network is challenging and desirable. In this regard, using concepts from coalitional game theory to design such distributed algorithms is quite appealing [3]. Thus, consider the single cell of a wireless cellular network having N singleantenna transmitters (e.g., mobile users) sending data in the uplink to their serving base station (BS) which possesses K > 1 receive antennas. Let N = {1 . . . N } denote the set of all N users in the network. For multiple access, we consider a TDMA transmission in the network,2 thus, in a noncooperative manner, the N users require a time scale of N slots to transmit as every user occupies one time slot. In order to improve their system performance, the users can cooperate. For instance, if the users cooperate, each group, i.e., coalition S ⊆ N , of cooperating singleantenna transmitters can be seen as a singleuser MIMO device (since the system already has multiple antennas at the receiving end, i.e., the BS). These cooperative
2
Note that the subsequent model and formulation can also be tailored for other multiple access schemes such as OFDMA or CDMA.
352
Coalitional games for cooperative cellular wireless networks
Coalition 2 User 4
User 6
User 5
Coalition 3 Base station with K antennas User 2
Time slot 1
User 1
Time slot 2 Time slot 3 Time slot 4
User 2
User 3
User 4
Time slot 5 Time slot 6
User 5
User 6
User 1 User 3
Time slot 1
Coal. 1
Time slot 2 Time slot 3 Time slot 4
Coal. 1
Coal. 1
Coal. 2
Time slot 5 Time slot 6
Coal. 3
Coal. 2
Coalition 1 Figure 13.1. An illustrative example of a single cell with N = 6 users cooperating for virtual MIMO formation.
coalitions subsequently transmit in a TDMA manner, i.e., one coalition per time slot. Hence, during the time scale N , each coalition is able to transmit within all the time slots previously held by its users. By doing so, there is a cooperative scheme within every cell in a wireless cellular network. An illustrative example of this network model is shown in Figure 13.1 for N = 6. The key issues that need to be tackled in this model are twofold: (i) What are the beneﬁts and costs from cooperation? (ii) Given the beneﬁt–cost tradeoﬀ which groups of users should cooperate? Given a coalition S ⊆ N with S users ( ·  represents the cardinality of a set), we consider a block fading K × S channel matrix H S with a pathloss model between @ the users in S and the BS with each element of the matrix hi,k = ej Φ i , k κ/dµi,k with µ the pathloss exponent, κ the pathloss constant, Φi,k the phase of the signal from transmitter i to the BS receiver k, and di,k the distance between transmitter i and the base BS’s receiver k. Further, for the considered TDMA system, we deﬁne a ﬁxed transmit power constraint per time slot, i.e., a power constraint per coalition P˜ . This average power constraint is applied to all the transmitters that are part of the coalition active in the slot. In the noncooperative scenario, this same power constraint per slot is simply the power constraint per individual user active in the slot. In fact, due to ergodicity, for each time slot, the average longterm power constraint per individual user and the power constraint per slot (i.e., constraining all transmitters of a coalition active in a slot) are the same [12, 13]. Subsequently, each coalition transmits in a slot, hence, perceiving no interference from other coalitions during transmission. As a result, in a slot, the sumrate capacity of the virtual MIMO formed by a
13.3 A coalition formation game model for distributed cooperation
353
coalition S, under a power constraint PS with Gaussian signaling is given by [15] CS = max I(xS ; y S ) = max(log det(I K + H S · QS · H †S )) QS
QS
subject to: tr[QS ] ≤ PS ,
(13.1)
where xS and y S are, respectively, the transmitted and received signal vectors of coalition S of size S × 1 and K × 1, QS = E [xS · x†S ] is the covariance of xS and H S is the channel matrix with H †S its conjugate transpose. The considered channel matrix H S is assumed perfectly known at the transmitter and receiver. Thus, the maximizing input signal covariance is given by QS = V S D S V †S [15], where V S is the unitary matrix given by the singular value decomposition of H S = U S ΣS V †S and D S is an S × S diagonal matrix given by D S = diag(D1 , . . . , DF , 0, . . . , 0). Here F ≤ min (K, S) is the number of positive singular values of the channel H S (eigenmodes) and each Di + is given by Di = (µ − λ−1 by waterﬁlling to satisfy i ) , where µ is determined the coalition power constraint tr[QS ] = tr[D S ] = i Di = PS , and λi is the ith eigenvalue of H †S H S . Using [15], the resulting capacity, in a slot, for a coalition S is CS =
K
(log (µλi ))+ .
(13.2)
i=1
To form the considered virtual MIMO coalitions and beneﬁt from the capacity gains, the users need to exchange their data information and their channel (userBS) information. For this purpose, we will consider a cost for information exchange in terms of transmit power. This transmit power cost mainly models the data exchange penalty. As we consider block fading channels with a long coherence time, the additional power penalty for exchanging the userBS channel information can be deemed as negligible relative to the data exchange cost, since the considered channel varies slowly (for example, exchange of the channel information can be done only periodically). Consequently, the cost for information exchange is taken as the sum of the powers required by each user in a coalition S to broadcast to its corresponding farthest user inside S. Due to the broadcast nature of the wireless channel, once a coalition member broadcasts its information to the farthest user, all the other members can also receive this information simultaneously. The power needed for broadcast between a user i ∈ S and its corresponding farthest user ˆi ∈ S is ν0 · σ 2 , P¯i,ˆi = h2i,ˆi
(13.3)
where ν0 is a
[email protected] SNR for information exchange, σ 2 is the noise variance, and hi,ˆi = κ/dµi,ˆi is the channel gain between users i and ˆi with di,ˆi the distance between users i and ˆi. In consequence, the total power cost for a
354
Coalitional games for cooperative cellular wireless networks
coalition S having S users is given by PˆS as follows: PˆS =
S 
P¯i,ˆi .
(13.4)
i=1
It is interesting to note that the deﬁned cost depends on the location of the users and the size of the coalition; hence, a higher power cost is incurred whenever the distance between the users increases or the coalition size increases. Thus, the actual power constraint PS per coalition S with cost is PS = (P˜ − PˆS )+ ,
(13.5)
where P˜ is the average power constraint per coalition (per slot), PˆS is the cooperation power cost given in (13.4), and a+ max (a, 0). In order to achieve the capacity in (13.2), within a slot, each user of a coalition S adjusts its power value based on the waterﬁlling solution, taking into account the available power constraint PS . Note that, since the power constraint P˜ applied over a coalition is the same as the maximum power constraint per individual user in the coalition, the waterﬁlling solution always yields a power value per user that does not violate the user’s available power after deducing the cost for cooperation in (13.3) from its individual longterm power constraint. The considered power cost does not take into account the interference during exchange of information between users and can be considered as a lower bound of the penalty incurred by cooperation. In addition to this power cost, a fraction of time may be required for the data exchange between the users prior to cooperation. However, due to the fact that the power cost given in (13.4) depends on distance and coalition size, the formed coalitions will typically consist of small clusters of nearby users, and thus the users can exchange information at high rates rendering the time for data exchange negligible relative to the transmission time slot (typically, the distance between the users of a coalition and the BS is larger than the distance between the coalition users themselves). Furthermore, in practice, cooperating to form a virtual MIMO formation can require a synchronization at the carrier frequency between the nodes, yielding some costs for practical implementation. In this chapter, we will not account for these carrier synchronization costs; however, this could constitute quite an interesting future direction. The coalition formation results derived in this chapter could also be applied for other cost functions without loss of generality. For example, the cost of power can be replaced by the cost of bandwidth where one could quantify the use of an additional band for information exchange, orthogonal to the band of transmission. Given this beneﬁt–cost tradeoﬀ, the following subsections mainly deal with: (i) formulating a coalitional game among the users by deﬁning an appropriate value function, and (ii) classifying and studying the properties of the game.
13.3 A coalition formation game model for distributed cooperation
13.3.2
355
Distributed virtual MIMO coalition formation game Coalition formation game formulation By investigating the illustrative example in Figure 13.1, one can easily see that to form a virtual MIMO system a coalitional game can be deﬁned with the players set being the set N of all users. The next step is to deﬁne an appropriate value for the game. For instance, based on the capacity beneﬁt and power cost deﬁned, respectively, in (13.2) and (13.5), over the TDMA time scale of N , for every coalition S ⊆ N , the utility (value) function in characteristic form is deﬁned as S · CS , if PS > 0, (13.6) v(S) = 0, otherwise, where PS is given by (13.5), CS is given by (13.2), and S is the number of users in S. This utility represents the total capacity achieved by coalition S during the time scale N while accounting for the cost through the power constraint. A coalition of S users will transmit with capacity CS during S time slots, thus achieving a total sumrate of v(S) during the time scale N (e.g., in Figure 13.1, during N = 6 coalition 2 consisting of two users achieves C2 in slot 4 and C2 in slot 6; thus a total of 2 · C2 during N = 6 slots). The second case in (13.6) implies that if the power for information exchange is larger than (or equal to) the available power constraint the coalition cannot be formed due to a utility of 0. By a close inspection of (13.6), one can immediately make the following remark. Remark 13.1 The virtual MIMO formation game can be formulated as an (N , v) coalitional game in characteristic form with transferable utility (TU). For instance, it is easily seen that the value function in (13.6) depends only on the users inside S and, thus, the game is in characteristic form. Moreover, the utility in (13.6) represents the sumrate achieved by coalition S, therefore, this sumrate can be divided in any manner between the coalition members in order to obtain the individual user payoﬀ achieved. This individual user payoﬀ, denoted φvi , represents the total rate achieved by user i during the transmission time scale N . Due to the TU property of the game, the payoﬀ φvi of each user i in a coalition S can be computed by a fair division of the utility v(S) through various criteria such as proportional fair division, Shapley value (SV) division, egalitarian fair division, or max–min fair division using the nucleolus. In this chapter, we restrict ourselves to the proportional fair and SV divisions deﬁned below (interested readers can see [6] for information on the other rules). Proportional fairness In practice, a network user experiencing a good channel might not be willing to cooperate with a user with bad channel conditions unless the payoﬀ it receives
356
Coalitional games for cooperative cellular wireless networks
takes into account its contribution to the coalition. To account for the channel diﬀerences, one can use the proportional fairness criterion, in which the extra beneﬁt from forming a coalition is divided in weights according to the users’ noncooperative utilities. Thus, ⎛ ⎞ v({j})⎠ + v({i}), (13.7) φvi = wi ⎝v(S) − j ∈S
where i∈S wi = 1 and within the coalition wi /wj = v({i})/v({j}). Thus, within the coalition for proportional fair division, the users with good channel conditions deserve more extra beneﬁts than the users with bad channel conditions. Shapley value (SV) fairness Another measure of fairness for payoﬀ division can be done using the SV [1], deﬁned as follows. Deﬁnition 13.1 An SV φv is a function that assigns to each possible characteristic function v a vector of real numbers, i.e., φv = (φv1 , φv2 , . . . , φvN ), where φvi represents the worth or value of user i in the game (N , v). There are four Shapley axioms that φv must satisfy: (1) eﬃciency axiom: i∈N φvi = v(N ); (2) symmetry axiom: if user i and user j are subject to v(S ∪ {i}) = v(S ∪ {j}) for every coalition S not containing user i and user j, then φvi = φvj ; (3) dummy axiom: if user i is subject to v(S) = v(S ∪ {i}) for every coalition S not containing i, then φvi = 0; (4) additivity axiom: if u and v are characteristic functions, then φu + v = φv + u = φu + φv . It is shown [1] that there exists a unique function φv satisfying the Shapley axioms given by (S)!(N − S − 1)! [v(S ∪ {i}) − v(S)]. (13.8) φvi = N! S ⊆N −{i}
The SV provides a fair division which takes into account the randomly ordered joining of the users in the coalition. Under the assumption of randomly ordered joining, the Shapley function of each user is its expected marginal contribution when it joins the coalition [1]. In general, the SV is used in games where the grand coalition, i.e., the coalition of all users, will form (such as in canonical games). However, whenever independent coalitions can form in the network such as in the virtual MIMO game (as will be seen later in this section), given any coalition S, one can consider the payoﬀ division by SV by applying (13.8) on each restricted game (S, v) in the network. In fact, it is shown in [16] that, in a game where a coalitional structure is
13.3 A coalition formation game model for distributed cooperation
357
present (not just a single grand coalition), the SV of the whole game (N , v) is found by using the SV on the game restricted to each coalition S in the structure. Given the formulated (N , v) TU coalitional game for virtual MIMO formation, we investigate some of the properties of this game in order to map it into a suitable class of coalitional game theory. First, we investigate the property of superadditivity deﬁned as follows: Deﬁnition 13.2 A coalitional game (N , v) with transferable utility is said to be superadditive if for any two disjoint coalitions S1 , S2 ⊂ N , v(S1 ∪ S2 ) ≥ v(S1 ) + v(S2 ). Theorem 13.1 The virtual MIMO formation (N ,v) coalitional game is nonsuperadditive. The proof of this theorem can be found in [6]. Theorem 13.1 implies that, for the virtual MIMO formation game, due to the cost for cooperation implies, whenever the coalition grows, the total utility, i.e., sumrate, that it generates may decrease (depending on the users’ channels). For instance, the cost for information exchange grows with the size of the coalition as well as the distance between these users. Further, we deﬁne the following concepts of coalitional game theory [1]. Deﬁnition 13.3 A payoﬀ vector φv = (φv1 , . . . , φvN ) for dividing the value v of v a coalition is said to be group rational or eﬃcient if N i=1 φi = v(N ). A payoﬀ vector φv is said to be individually rational if the player can obtain a beneﬁt no less than acting alone, i.e., φvi ≥ v({i}), ∀ i. An imputation is a payoﬀ vector satisfying the above two conditions. Deﬁnition 13.4 An imputation φv is said to be unstable through a coalition S if v(S)> i∈S φvi , i.e., the players have an incentive to form coalition S and reject the proposed φv . The set C of stable imputations is called the core, i.e., 4 v v v C= φ : φi = v(N ) and φi ≥ v(S) ∀ S ⊆ N . (13.9) i∈N
i∈S
A nonempty core means that the players have an incentive to form the grand coalition. Remark 13.2 In general, the core of the (N ,v) virtual MIMO formation game is empty. In the discussed model, the costs of cooperation for a coalition S increase as the number of users in a coalition increases as well as when the distances between the users increase, hence aﬀecting the topology. In particular, consider the grand coalition N of all N users in the network. This coalition consists of
358
Coalitional games for cooperative cellular wireless networks
a large number of users who are randomly located at diﬀerent distances. Hence, the grand coalition will often have a value of v(N ) = 0 due to the cooperation costs and several coalitions S ⊂ N have an incentive to deviate from this grand coalition and form independent disjoint coalitions. Consequently, an imputation that lies in the core cannot be found, and, due to cost, the core of the virtual MIMO formation (N ,v) game is generally empty. Hence, as seen in Figure 13.1 and corroborated by the nonsuperadditivy of the game and the emptiness of the core, in general, due to the cost for coalition formation, the grand coalition will not form. Instead, independent and disjoint coalitions appear in the network as a result of the virtual MIMO formation process. In this regard, the game is classiﬁed as a coalition formation game [3], and the objective is to ﬁnd the coalitional structure that will form in the network, instead of ﬁnding only a solution concept, such as the core. Further, note that one implication of the nonsuperadditivity of the game is that the SV might not be individually rational, hence, any coalition formation algorithm that will be constructed must handle this property with an appropriate coalition formation decision. In the next section, we will devise an algorithm for coalition formation that can take into account all these properties of the virtual MIMO coalition formation game and that allows to characterize the network structure in the presence of the users’ cooperative behavior.
Coalition formation algorithm Coalition formation has been a topic of great interest in game theory [2, 3, 17, 18]. By using mathematical tools from coalition formation games, one can build algorithms to form coalitions dynamically among a group of players, and, thus these algorithms can be applied in a distributed manner, notably in a wireless scenario. To devise a suitable algorithm for virtual MIMO formation, several concepts from coalition formation games need to be deﬁned as follows. Deﬁnition 13.5 A collection of coalitions, denoted by S, is deﬁned as the set S = {S1 , . . . , Sl } of mutually disjoint coalitions Si ⊂ N . In other words, a collection is any arbitrary group of disjoint coalitions Si of N , not necessarily spanning A all players of N . If the collection spans all the players of N ; i.e., lj =1 Sj = N , the collection is a partition of N . Deﬁnition 13.6 A preference operator or comparison relation is an order deﬁned for comparing two collections R = {R1 , . . . , Rl } and S = {S1 , . . . , Sp } that are partitions of the same subset A ⊆ N (i.e., same players in R and S). Therefore, R S implies that the way R partitions A is preferred to the way S partitions A. Various wellknown orders can be used as comparison relations in diﬀerent scenarios [3]. For the virtual MIMO coalition formation game, we deﬁne the following order.
13.3 A coalition formation game model for distributed cooperation
359
Deﬁnition 13.7 Consider two collections R = {R1 , . . . , Rl } and S = {S1 , . . . , Sm } that are partitions of the same subset A ⊆ N (same players in R and S). For a collection R = {R1 , . . . , Rl }, let the utility of a player j in a coalition Rj ∈ R be denoted by φvj (R). Then R is preferred over S by Pareto order, written as R S, iﬀ R S ⇐⇒ {φvj (R) ≥ φvj (S) ∀ j ∈ R, S} with at least one strict inequality (>) for a player k.
(13.10)
In other words, a collection is preferred by the players over another collection, if at least one player is able to improve its payoﬀ without hurting the other players. Subsequently, to perform autonomous coalition formation between the users in the wireless network, we construct a distributed algorithm based on two simple rules denoted as “merge” and “split” [17] deﬁned as follows. Deﬁnition 13.8 Merge rule Merge any set of coalitions {S1 , . . . , Sl } whenever the merged form is preferred by the players, i.e., where {
l 3
Sj } {S1 , . . . , Sl }.
j =1
A Therefore, {S1 , . . . , Sl } → { lj =1 Sj }. Deﬁnition 13.9 Split rule Split any coalition is preferred by the players, i.e., where {S1 , . . . , Sl } {
l 3
Al j =1
Sj whenever a split form
Sj }.
j =1
Al
Thus, {
j =1
Sj } → {S1 , . . . , Sl }.
In brief, multiple coalitions will merge (split) if merging (splitting) yields a preferred collection based on the Pareto order. Thus, coalitions will merge only if at least one user can enhance its individual payoﬀ through this merge without decreasing the other users’ payoﬀs. Similarly, a coalition will split only if at least one user in that coalition is able to strictly improve its individual payoﬀ through the split without hurting other users. A decision to merge or split is thus tied to the fact that all users must beneﬁt from the merge or split, therefore any merged (or split) form is reached only if it allows all involved users to maintain their payoﬀs with at least one user improving. In summary, using merge and split one can devise a coalition formation algorithm with partially reversible agreements [2], where the users sign a binding agreement to form a coalition through the merge operation (if all users are able to improve their individual payoﬀs from the previous state) and they can only split this coalition if splitting does not decrease the payoﬀ of any coalition member (partial reversibility).
360
Coalitional games for cooperative cellular wireless networks
For the virtual MIMO formation game, the coalition formation algorithm consists of two phases: adaptive coalition formation and transmission. In the adaptive coalition formation phase, an iteration of sequential mergeandsplit rules is performed until a ﬁnal network partition composed of independent disjoint coalitions is obtained. In the transmission phase, the formed coalitions transmit in their corresponding slots in a TDMA manner. The transmission phase may occur several times prior to the repetition of the coalition formation phase, notably in a lowmobility environment where changes in the coalition structure due to mobility seldomly occur. Although any arbitrary merge process can be used, we consider a distributed costbased merge procedure allowing the coalitions (users) to perform a local search for partners. Consequently, the decision to merge with neighboring coalitions is taken based on the Pareto order proceeding from the partner that provides the lowest cost. In order for coalition S1 to merge with another coalition S2 , the utility of the formed coalition through merge must be positive; i.e., v(S1 ∪ S2 ) > 0 otherwise no beneﬁts exist for the merge. Thus, based on the deﬁned power cost (13.4) and utility (13.6), coalitions can only merge when the cost for cooperation is less than P˜ . Otherwise, when the cost is greater than or equal P˜ , through (13.6) the utility of the merged coalition will be 0 and there is no mutual beneﬁt. Thus, using (13.4) the merge is possible (nonzero utility) for S 1 ∪S 2  ¯ Pi,ˆi < P˜ , which, by (13.3), yields S1 with S2 if PˆS 1 ∪S 2 < P˜ , i.e., i=1 S 1 ∪S 2 
i=1
P˜ 1 . µ < di,ˆi ν0 · σ 2 · κ
(13.11)
Thus, a coalition will only attempt to merge with other coalitions where (13.11) can be veriﬁed. Each stage of the coalition formation algorithm starts from an initial network partition T = {T1 , . . . , Tl } of N . In this partition, any random coalition (user) can start with the merge process. For implementation purposes, assume that the coalition Ti ∈ T which has the highest utility in the initial partition T starts the merge by attempting to cooperate with the coalition yielding the lowest cost. On one hand, if merging occurs, a new coalition T˜i is formed and, in its turn, coalition T˜i will attempt to merge with the lowestcost partner. On the other hand, if Ti were unable to merge with the lowestcost coalition, it would try the next lowestcost partner, proceeding sequentially through the coalitions verifying (13.11). The search ends with a ﬁnal merged coalition Tiﬁnal composed of Ti and one or more of the coalitions in its vicinity (or just Ti , if no merge occurred). The algorithm is repeated for the remaining Ti ∈ T until all the coalitions have made their local merge decisions, resulting in a ﬁnal partition W. The coalitions in the resulting partition W are next subject to split operations, if any are possible. An iteration consisting of multiple successive mergeandsplit operations is repeated. As shown in [17], any arbitrary sequence of mergeandsplit rules will terminate, and, thus, the convergence of the ﬁrst phase of the algorithm is
13.3 A coalition formation game model for distributed cooperation
361
Table 13.1. One stage of the mergeandsplit coalition formation algorithm Initial state The network is partitioned by T = {T1 , . . . , Tk } (at the beginning of all time T = N = {1, . . . , N } with noncooperative users). Coalition formation algorithm Phase I Adaptive coalition formation: Coalition formation using mergeandsplit occurs. repeat (a) Coalitions begin the local search merge operation: W = Merge(T ). (b) Coalitions in W decide to split based on the Pareto order. T = Split(W). until mergeandsplit iteration terminates. Phase II Virtual MIMO transmission: The coalitions transmit during the time scale N with 1 coalition per slot with each coalition occupying all the time slots previously held by its members. The algorithm is repeated periodically, enabling the users to autonomously selforganize and adapt the topology to environmental changes such as mobility.
always guaranteed. Table 13.1 shows a summary of one round of the coalition formation algorithm. For networks where the environment is changing, e.g., due to mobility, the algorithm in Table 13.1 can be repeated periodically in order to adapt the network structure to the environmental changes. The result of the algorithm in Table 13.1 is a network partition composed of disjoint independent coalitions. First and foremost, one would note that, in the ﬁnal network structure, due to convergence, no coalition has an incentive to pursue any further merge or split procedure. Thus, the resulting network structure is mergeandsplit stable in the sense that no coalition has an incentive to deviate from this structure through merge or split rules. This stability of the resulting network partition can be further investigated with respect to the concept of a defection function D [17]. Deﬁnition 13.10 A defection function D is a function which associates with each partition T of N a family (group) of collections in N . A partition T = {T1 , . . . , Tl } of N is Dstable if no group of players is interested in leaving T when the players who wish to leave can only form the collections allowed by D.
362
Coalitional games for cooperative cellular wireless networks
The most important defection function is the Dc (T ) function (denoted Dc ) which associates with each partition T of N the family of all collections in N . This function allows any group of players to leave the partition T of N through any operation and create an arbitrary collection in N . From the Dc function one can deﬁne a strong stability concept. Deﬁnition 13.11 A partition T of the players’ set N is said to be Dc stable, if no players in T are interested in leaving T to form other collections in N . For instance, whenever a Dc stable partition exists for a given coalitional game, this partition possesses the following properties: (1) If it exists, a Dc stable partition is the unique outcome of any arbitrary iteration of merge and split. (2) A Dc stable partition T is a unique maximal partition, i.e., for all partitions T = T of N , T T . In the case where represents the Pareto order, this implies that the Dc stable partition T is the partition that presents a Pareto optimal payoﬀ distribution for all the players. Clearly, a Dc stable partition is an optimal partition that the wireless network can seek as it provides a payoﬀ distribution that is Pareto optimal for all users with respect to any other network partition. In addition, this partition is a unique outcome of any arbitrary iteration of mergeandsplit rules. However, the existence of a Dc stable partition is not always guaranteed [17]. The Dc stable partition T = {T1 , . . . , Tl } of the whole space N exists if a partition of N veriﬁes two necessary and suﬃcient conditions [17]: (1) For each i ∈ {1, . . . , l} and each pair of disjoint coalitions A and B such that {A ∪ B} ⊆ Ti we have {A ∪ B} {A, B} (referred to as cond. (A) hereafter). (2) For the partition T = {T1 , . . . , Tl } a coalition S ⊂ N formed of players belonging to diﬀerent Ti ∈ T is T incompatible if for no i ∈ {1, . . . , l} we have S ⊂ Ti . Dc stability requires that for all T incompatible coalitions {S}[T ] {S}, where {S}[T ] = {S ∩ Ti ∀ i ∈ {1, . . . , l}} is the projection of coalition S in partition T (referred to as cond. (B) hereafter). If no partition of N can satisfy these conditions, then no Dc stable partition of N exists. Since the Dc stable partition is a unique outcome of any arbitrary mergeandsplit iteration, we have Lemma 13.1 For the (N , v) virtual MIMO coalition formation game, the mergeandsplit algorithm of Table 13.1 converges to the optimal Dc stable partition, if such a partition exists. Otherwise, the algorithm yields a ﬁnal network partition that is mergeandsplit proof.
13.3 A coalition formation game model for distributed cooperation
363
In the transmitter cooperation game, the existence of a Dc stable partition depends on various factors. For instance, cond. (A) states that, for a Dc stable partition T , every coalition Ti ∈ T must verify the Pareto order not only at the level of the whole coalition Ti but also at the level of all disjoint coalitions subsets of Ti . Verifying the Pareto order requires that the utility of every union of any two disjoint coalitions subsets of a coalition Ti must yield an extra utility over the disjoint case; i.e., v(A ∪ B) > v(A) + v(B) ∀ A, B ⊂ Ti . In an ideal case with no cost, as the number of transmit antennas is increased for a ﬁxed power constraint, the overall system diversity increases as the data pass through diﬀerent channel values allowing, with adequate coding, the symbols to be recovered without error at a higher rate [19]. In such a case, since A ∪ B has a larger number of antennas than A and B, ∀ A, B ⊂ Ti and for each Ti we have CA ∪B > max (CA , CB ) and thus A ∪ B ·CA ∪B >A · max (CA , CB ) + B · max (CA , CB ), A ∪ B · CA ∪B >A · CA+B · CB ⇔ v(A ∪ B)>v(A)+v(B),
(13.12)
which is suﬃcient to verify cond. (A) for Dc stability when adequate payoﬀ divisions are done. However, due to the cost CA ∪B , CA and CB can have diﬀerent power constraints and (13.12) may not be guaranteed ∀A, B ⊂ Ti . Guaranteeing this condition is directly dependent on the cooperation cost within the coalitions in T and, thus, on the users’ location. In practical networks, verifying cond. (A) for Dc stability depends on the users’ random locations. Cond. (B) for the existence of a Dc stable partition T is that players formed from diﬀerent Ti ∈ T have no incentive to form a coalition S outside of T . In the transmitter cooperation game, cond. (B) is also dependent on the location of the coalitions Ti ∈ T ; speciﬁcally on the distance between the users in diﬀerent coalitions Ti ∈ T . Thus, cond. (B) is veriﬁed whenever two users belonging to diﬀerent coalitions in a partition T are separated by a large distance. A suﬃcient condition for verifying this second requirement can be derived. Theorem 13.2 For a network partition T = {T1 , . . . , Tl } resulting from the coalition formation algorithm, if the distance di,j between any two users i ∈ Ti µ1 and j ∈ Tj with Ti = Tj veriﬁes di,j > κ · P˜ /2 · ν0 · σ 2 = dˆ0 , then the second condition for Dc stability, cond. (B), is veriﬁed. Proof. Since a Dc stable partition is a unique outcome of any mergeandsplit iteration, we will consider the partition T = {T1 , . . . , Tl } resulting from any mergeandsplit iteration in order to show when cond. (B) can be satisﬁed. A T incompatible coalition is a coalition formed from users belonging to different Ti ∈ T . Consider the T incompatible coalition {i, j} that can potentially form between two users i ∈ Ti and j ∈ Tj with Ti = Tj . The total power cost for {i, j} is given by (13.3) as Pˆ{i,j } = P¯i,j + P¯j,i = 2 · P¯i,j . In the case where the total power cost is larger than the constraint, we have Pˆ{i,j } ≥ P˜
364
Coalitional games for cooperative cellular wireless networks
and thus P¯i,j ≥ P˜ /2 which yields the required condition on distance di,j ≥ 1 (κ · P˜ /2 · ν0 · σ 2 ) µ = dˆ0 . We have by (13.6) v({i, j}) = 0, and, thus, φvi ({i, j}) = φvj ({i, j}) = 0. Or we have that {i, j}[T ] = {{i, j} ∩ Tk ∀ k ∈ {1, . . . , l}} = {{i}, {j}}, and, thus, φvi ({i, j}[T ]) = v({i}) > φvi ({i, j}) = 0 and φvj ({i, j}[T ]) = v({j}) > φvj ({i, j}) = 0. Consequently, {i, j}[T ] {i, j} and cond. (B) is veriﬁed for any T incompatible coalition formed of two users. Moreover, when any two users i ∈ Ti and j ∈ Tj with Ti = Tj are separated by dˆ0 , T incompatible coalitions S with S > 2 have a cost PˆS > Pˆ{i,j } ≥ P˜ and thus v(S) = 0; yielding S[T ] S for all T incompatible coalitions S. Hence, when any two users in the network partition T resulting from merge and split are separated by a distance larger than dˆ0 , then cond. (B) for Dc stability existence is veriﬁed. In summary, the existence of the Dc stable partition is closely tied to the users’ location, which is a random parameter in practical networks. However, when such a partition does exist, the network resulting from the coalition formation algorithm will converge to that Dc stable partition as per Lemma 13.1. The algorithm in Table 13.1 can be implemented in a distributed way. As the user can detect the strength of other users’ uplink signals (through techniques similar to those used in the adhoc routing discovery), nearby coalitions can be discovered and the local merge algorithm performed. Each coalition surveys neighboring coalitions satisfying (13.11) and attempts to merge based on the Pareto order. The users in a coalition need only to know the maximum distances with respect to the users in neighboring coalitions. Moreover, each formed coalition internally decides to split if its members ﬁnd a split form by Pareto order. By using a control channel, the distributed users can exchange some channel information and then cooperate using our model (exchange data information if needed, form a coalition then transmit). Signaling for this handshaking can be minimal.
Numerical results To assess the performance of coalition formation for distributed virtual MIMO, we set up a singlecell composed of a single BS with K = 3 equally spaced antennas located at the center of a square of 4 km× 4 km. Without loss of generality, at the receiver, we consider antennas that are separated enough while3 Φi,k = 0 ∀i, k. The propagation loss is set to µ = 3 and the pathloss constant κ = 1. The power constraint per slot is P˜ = 10 mW, the cost SNR target for information exchange is ν0 = 10 dB, and noise level is −90 dBm. First, we set up a network with N = 10 users where the users are located in a way that a Dc stable partition exists. Figure 13.2 shows that the coalition 3
This choice provides a lower bound on the performance gain of the coalition formation algorithm (i.e., the gains mainly stemming from the transmitters cooperation which is the main objective of this section); considering diﬀerent phases certainly yields an additional multiplexing gain and it does not aﬀect the analysis or the results hereafter.
13.3 A coalition formation game model for distributed cooperation
365
Figure 13.2. Convergence of the algorithm to a ﬁnal Dc stable partition. formation algorithm converges to the ﬁnal Dc stable network partition T = {T1 , . . . , T5 } (valid for both proportional fair and SV fairness criteria). Cond. (A) for Dc stability is easily veriﬁed for coalitions consisting of at most two users since such coalitions do not form unless the Pareto order is internally veriﬁed (deﬁnition of the merge rule). For the threeusers coalition T2 = {7, 9, 10} Table 13.2 shows the payoﬀs of the diﬀerent subcoalitions for both fairness types. Table 13.2 shows that the Pareto order is internally veriﬁed for T2 , that is ∀A, B ⊂ T2 ; {A ∪ B} {A, B} for all fairness cases. In addition, by inspecting Figure 13.2 it is clear that any two users belonging to T incompatible coalitions are separated by a distance larger than the maximum distance, which is dˆ0 = 0.793 km, computed using Theorem 13.2 for the simulation parameters. Thus, Theorem 13.2 is satisﬁed and cond. (B) is veriﬁed. For example, for the T incompatible coalition S = {4, 7} equation (13.6) yields v(S) = 0 and thus φv4 (S) = φv7 (S) = 0 due to the distance between users 4 and 7. The projection of S in T is S[T ] = {{4, 7} ∩ T1 , {4, 7} ∩ T2 , {4, 7} ∩ T3 , {4, 7} ∩ T4 , {4, 7} ∩ T5 , {4, 7} ∩ T6 } = {{4}, {7}}. In S[T ], the payoﬀs of users 4 and 7 are respectively φv4 (S[T ]) = v({4}) = 7.6069 and φv7 (S[T ]) = v({7}) = 3.1077, by Pareto order φv4 (S[T ]) > φv4 (S) and φv7 (S[T ]) > φv7 (S), thus, S[T ] S. In Figure 13.3, we show the average total individual user payoﬀ (rate) improvement achieved during the whole transmission time scale as a function of the network size. This result is averaged over the random locations of the users. We compare the performance of the coalition formation algorithm to that of
366
Coalitional games for cooperative cellular wireless networks
Table 13.2. Payoﬀs for coalition T2 = {7, 9, 10} of Figure 13.2 and its subcoalitions Proportional fair User 7 {7} {9} {10} {7, 9} {7, 10} {9, 10}
3.1077
T2 = {7, 9, 10}
4.6431
User 9
User 10
Shapley User 7
User 9
3.1077 2.7173
2.7173 2.7345
4.0416 4.0648
User 10
3.5338 3.6206
3.5779 3.5967
4.0869
4.0598
2.7345 3.9829 4.7863 4.5210
3.5925 3.5996
4.4406 3.6177
4.1558
4.1131
Figure 13.3. Cooperation gains in terms of the average individual user payoﬀ achieved by the mergeandsplit scheme during the whole transmission duration compared with the noncooperative case and the centralized optimal solution for diﬀerent network sizes and diﬀerent fairness criteria. the noncooperative case as well as the optimal partition found by a centralized entity through exhaustive search. For the cooperative case, the average user’s payoﬀ increases with the number of users N since the possibility of ﬁnding cooperating partners increases. In contrast, the noncooperative approach presents an
13.3 A coalition formation game model for distributed cooperation
367
fair SV
Figure 13.4. Frequency of mergeandsplit operations per minute for diﬀerent speeds in a mobile network of Mt = 50 users. almost constant performance with diﬀerent network sizes. Cooperation presents a signiﬁcant advantage over the noncooperative case in terms of average individual utility for all fairness types, and this advantage increases with the network size. The proportional fair division presents the best performance, as it allows an improvement of up to 40.42% over the noncooperative case at N = 100. This result also highlights the tradeoﬀ between fairness and cooperation gains. For instance, while the proportional fairness presents an advantage in terms of payoﬀ gain, since it allows larger coalitions to form (due to it being less strict in fairness than the SV), the SV presents lower gains but more fairness in allocating payoﬀs. Furthermore, compared to the optimal solution, the mergeandsplit algorithm achieves a highly comparable performance with a performance loss not exceeding 1% at N = 20 users. This clearly shows that, by using the distributed mergeandsplit algorithm, the network can achieve a performance that is very close to optimal. Note that, for more than 20 users, ﬁnding the optimal partition by exhaustive search is mathematically and computationally intractable. To show how the coalition formation algorithm can handle mobility, we deploy a network of N = 50 mobile users (random walk mobility model) for a period of 5 minutes. For N = 50, each TDMA transmission requires 50 × θ seconds with θ the slot duration (we let θ = 10 ms). The results in terms of frequency of mergeandsplit operations per minute are shown in Figure 13.4 for various speeds. As the speed increases, for both fairness types, the number of mergeandsplit operations increases due to the changes in the network structure incurred by mobility. Fairness types that potentially yield large coalitions, such as proportional
368
Coalitional games for cooperative cellular wireless networks
fair, incur a higher frequency of merge and split since such coalitions require additional merge operations and are more prone to splitting due to mobility. In contrast, strict fairness criteria, such as the SV, yield relatively lower frequency of merge and split. In a nutshell, Figure 13.4 demonstrates that, by periodic runs of the coalition formation algorithm and through adequate mergeandsplit decisions, the users can selforganize and adapt to the changes in the environment.
13.4
Coalitional graph game among relay stations In this section the challenges of deploying relay stations in next generation wireless systems are studied. For this purpose, we examine the use of coalitional graph games for modeling the interactions among the relay stations seeking to send data in the uplink of a wireless system. First, we motivate the problem and provide a suitable system model. Then, we formulate a network formation game among the relay stations, investigate the resulting graph structure, and study their properties.
13.4.1
Motivation and basic problem In order to mitigate the fading eﬀects of the wireless channel, several nodes or relays can cooperate with a given source node in the transmission of its data to a far away destination, thereby, providing spatial diversity gains for the source node. By doing so, one can signiﬁcantly improve the performance of the source node. This class of cooperation is commonly referred to as cooperative communications [20]. It has been shown that by using one or more relays [19–21] a signiﬁcant improvement can be obtained in terms of BER, throughput, or other quality of service (QoS) parameters. Moreover, diﬀerent aspects of cooperative transmission have been discussed in the literature such as performance analysis and resource allocation [19–21]. Due to this proven advantage of cooperative communications, a key feature in next generation wireless systems, such as 3GPP’s longterm evolution advanced (LTEAdvanced) [22], or the forthcoming IEEE 802.16j WiMAX standard [23], is the introduction of the relay station (RS) nodes which can enable relaying in the network. Deploying RSs in a wireless system faces several challenging issues. For instance, as they are deployed in the network, the RSs need to route, among them, diﬀerent packets received from external mobile stations using advanced techniques such as cooperative transmission. Due to the presence of RSs, the architecture of next generation wireless networks is governed by a multihop network structure formed between the RSs and their serving BS. Consequently, one prominent challenge is the design and study of the network multihop architecture that will connect the BS to the RSs in its coverage area given the cooperative strategies of the RSs and the existing network traﬃc. Thus, consider the uplink of a wireless system with M RSs (ﬁxed, mobile, or nomadic) under the coverage area of a single BS. The RSs transmit their
13.4 Coalitional graph game among relay stations
369
data in the uplink to the BS through multihop links, and, therefore, a tree architecture needs to form, in the uplink, between the RSs and their serving BS. Once the uplink tree is formed, mobile stations (MSs) can hook up to the network by selecting a serving RS (or directly connecting to the BS). Each MS is assigned to the closest RS. We consider that the MSs deposit their packets to their serving RSs using direct transmission. In their turn, the serving RSs act as source nodes transmitting the received MS packets to the BS through one or more hops in the formed tree, using cooperative transmission. This assumption of direct transmission between MS and RS allows the provision of a tree formation algorithm that can be easily incorporated in a new or existing wireless network without relying on external entities such as the MSs. For cooperative transmission between the RSs and the BS, the decoded relaying multihop diversity channel of [21] is considered whereby each intermediate node on the path between a transmitting RS and the BS combines, encodes, and reencodes the received signal from all preceding terminals before relaying (decodeandforward relaying). Formally, each MS k in the network is considered as a source of traﬃc following a Poisson distribution with an average arrival rate λk . With such Poisson streams at the entry points of the network (the MSs), we assume that for every RS incoming packets are stored and transmitted in a ﬁrstin ﬁrstout (FIFO) fashion and that we have Kleinrock’s independence approximation [24, Chap. 3] with each RS being an M/D/1 queueing system. With this approximation, the generation of the total traﬃc that RS i receives from the MS that it is serving is a Poisson process with an average arrival rate i λl , where Li is the number of MSs served by RS i. Moreover, RS i of Λi = Ll=1 also receives packets from RSs that are connected to it with a total average rate ∆i . For these ∆i packets, the sole role of RS i is to relay them to the next hop. In addition, any RS i that has no assigned MSs (Li = 0, Λi = 0, and ∆i = 0), transmits “HELLO” packets, generated with a Poisson arrival rate of η0 in order to keep its link to the BS active during periods of no actual MS traﬃc. An illustrative example of this model is shown in Figure 13.5. The main objective in this section is to provide a distributed algorithm that allows the RSs in a wireless system to autonomously form the uplink tree structure such as in Figure 13.5, adapting it to environmental changes as the network evolves. Further, another key goal is to model the gains in terms of cooperative transmission, while accounting for the costs in terms of the buﬀering and transmission delay incurred by the multihop transmission. In this context, the remainder of this section is dedicated to providing an accurate model for the uplink tree formation problem.
13.4.2
A network formation game among relay stations Game formulation To model the interactions among the RSs seeking to form the uplink network structure, we refer to coalitional graph games [3] which is a class of coalitional
370
Coalitional games for cooperative cellular wireless networks
Figure 13.5. A prototype of the uplink tree model that governs the network architecture of next generation wireless systems. game theory that deals with the formation of network graphs among a number of nodes seeking to cooperate. In particular, for the RSs game, we utilize a subclass of coalitional graph games known as network formation games. For instance, network formation games are a hybrid class of games that combine concepts from both cooperative and noncooperative game theory [3, 18, 25]. In such games, several independent decision makers (players) interact in order to form a network graph among them. Depending on the goals of each player, a ﬁnal network graph G, resulting from individual players’ decisions, forms. We model the uplink tree formation problem among the RSs as a network formation game with the RSs being the players. The result of the interactions among the RSs is a directed graph G(V, E) with V = {1, . . . , M + 1} denoting the set of all vertices (M RSs and the BS) and E denoting the set of all edges (links) between pairs of RSs. Each link between two RSs i and j, denoted (i, j) ∈ E, corresponds to an uplink traﬃc ﬂow between RS i and RS j. First and foremost, we deﬁne the notion of a path: Deﬁnition 13.12 A path between two nodes i and j in the graph G is deﬁned as a sequence of nodes i1 , . . . , iK such that i1 = i, iK = j and each directed link (ik , ik +1 ) ∈ G for each k ∈ {1, . . . , K − 1}. This section considers solely multihop tree (or forest, if some parts of the graph are disconnected) architectures, since such architectures are ubiquitous in next
13.4 Coalitional graph game among relay stations
371
generation networks [22, 23]. In this regard, throughout the remainder of this section we adopt the following convention. Remark 13.3 Each RS i is connected to the BS through at most one path, and, thus, we denote by qi the path between any RS i and the BS whenever this path exists. Further, we delineate the possible actions or strategies that each RS can take in the network formation game. The strategy space of each RS i consists of the RSs (or the BS) to which i wants to connect. Consequently, the strategy of RS i is to select the link that it wants to form from the available strategy space. We note that RS i cannot connect to RS j, which is already connected to i, in the sense that if (j, i) ∈ G, then (i, j) ∈ / G. Formally, for a current network graph G, let Ai = {j ∈ V \ {i}(j, i) ∈ G} be the set of RSs from which RS i accepted a A link (j, i), and Si = {(i, j)j ∈ V \ ({i} Ai )} as the set of links corresponding to the nodes (RSs or the BS) with whom i wants to connect (note that i cannot connect to RSs that are already connected to it, i.e., RSs in Ai .). Consequently, the strategy of RS i is to select the link si ∈ Si that it wants to form, i.e., choose the RS to which it will connect. Based on Remark 13.3, an RS can be connected to at most one other node in our game so choosing to form a link si implicitly implies that RS i will replace its previously connected link (if any) with the new link si . Having modeled the tree formation problem as a network formation game, we introduce a utility function that takes into account the QoS in terms of the packet success rate (PSR) as well as the delay incurred by multihop transmission. Consider an actual network tree graph G where each RS extracts a positive utility from the packets successfully transmitted to the BS out of the packets received from the MSs. Each transmitted packet is subject to a BER due to the transmission over the wireless channel using one or more hops. For example, for a transmission between RS V1 ∈ V and destination Vn +1 (the destinaton is always the BS) going through n − 1 intermediate relays {V2 , . . . , Vn } ⊂ V , let Nr be the set of all receiving terminals, i.e., Nr = {V2 . . . Vn +1 } and Nr (i) be the set of terminals that transmit a signal received by a node Vi . Hence, for a relay Vi on the path from the source V1 to the destination Vn +1 , we have Nr (i) = {V1 , . . . , Vi−1 }. Given this notation, the BER between a source RS V1 ∈ V and the destination Vn +1 = BS is computed along the path qV 1 = {V1 , . . . , BS} using the tight upper bound given in [21, Eq. (10)] for the decoded relaying multihop diversity channel with BPSK modulation and Rayleigh fading as follows: ⎛ ⎤⎞ ⎡ 1⎜ ⎢ ⎥⎟ γk ,i γk ,i ⎜ ⎥⎟ . ⎢ × 1 − PqeV ≤ ⎝ ⎣ 1 2 γk ,i − γj,i γk ,i + 2 ⎦⎠ N i ∈Nr
N k ∈Nr ( i )
N j ∈Nr ( i ) N j = N k
(13.13)
372
Coalitional games for cooperative cellular wireless networks
Here, γi,j = Pi · gi,j /σ 2 is the average received SNR at node j from node i, where Pi is the transmit power of node i, σ 2 the noise variance, and gi,j = 1/dµi,j is the pathloss with di,j the distance between i and j and µ the pathloss exponent. Without loss of generality, we assume that all the RSs transmit with equal power Pi = P˜ , ∀i. Finally, for RS i which is connected to the BS through a direct transmission path qid ∈ Qi with no intermediate hops, the BER is given 1 e by Pq d = 2 1 − γi,B S /(1 + γi,B S ) [21], where γi,B S is the SNR at the BS i for transmission from RS i. Using the BER expression in (13.13) and with no channel coding, the PSR ρi,q i perceived by RS i over a path qi is deﬁned as follows: ρi,q i (G) = (1 − Pqei )B ,
(13.14)
where B is the number of bits per packet. The PSR is a function of the network graph G as the path qi varies depending on how RS i is connected to the BS in the formed network tree structure. Transmitting over multihop links incurs a signiﬁcant delay due to buﬀering and multiple transmissions. For this purpose, we consider the average delay τi,q i along the path qi = {i1 , . . . , ik } from RS i1 = i to the BS. A measure of the average delay over qi in a network with Poisson arrivals at the entry points and considering the Kleinrock approximation as in the previous section (each RS is an M/D/1 queueing system) is given by [24, Chap. 3, eqs. (3.42), (3.45), and (3.93)] τi,q i (G) =
(i k ,i k + 1 )∈q i
Ψi k ,i k + 1 1 + 2µi k ,i k + 1 (µi k ,i k + 1 − Ψi k ,i k + 1 ) µi k ,i k + 1
,
(13.15)
where Ψi k ,i k + 1 = Λi k + ∆i k is the total traﬃc (packets/s) originating from MSs (Λi k ) and from RSs (∆i k ) traversing link (ik , ik +1 ) ∈ qi between RS ik and RS ik +1 . The ratio 1/µi k ,i k + 1 represents the average transmission time (service time) on link (ik , ik +1 ) ∈ qi with µi k ,i k + 1 being the service rate on link (ik , ik +1 ). This service rate is given by µi k ,i k + 1 = Ci k ,i k + 1 /B where Ci k ,i k + 1 = W log (1 + νi k ,i k + 1 )
(13.16)
is the capacity of the direct transmission between RS ik and RS ik +1 , νi k ,i k + 1 = P˜ gi k ,i k + 1 /σ 2 is the received SNR from RS ik at RS ik +1 , and W is the bandwidth available for RS ik which is assumed the same for all RSs in the set of vertices V, without loss of generality. Similarly to the PSR, the delay depends on the paths from the RSs to the BS, hence is a function of the network graph G. For VoIP services, given the delay and the PSR, an appropriate utility function can be deﬁned through the concept of the Rfactor [26]. The Rfactor is an expression that links the delay and packet loss to the voice quality as
13.4 Coalitional graph game among relay stations
373
follows [26]: Ui (G) = Ωa − 1 τi,q i (G) − 2 (τi,q i (G) − 3 )H − υ1 −υ2 ln (1 + 100υ3 (1 − ρi,q i (G))),
(13.17)
where τi,q i is the delay given by (13.15) expressed in milliseconds, 100(1 − ρi,q i ) represents the packetloss percentage (we consider only the packet loss due to errors through ρi,q i in (13.14), ignoring packet loss due to overloaded links). The remaining parameters are constants deﬁned as follows: Ωa = 94.2, 1 = 0.024, 2 = 0.11, 3 = 177.3, H = 0 if τi,q i < 3 , H = 1 otherwise. The parameters υ1 , υ2 , and υ3 depend on the voice speech codec. The relationship between the Rfactor and the VoIP service quality is such that as the Rfactor increases, the voice quality improves. For diﬀerent voice codecs, diﬀerent Rfactor ranges provide an indication of the voice quality varying through poor, low, medium, high to best as the Rfactor increases [26]. For example, for certain speech codecs as the Rfactor increases in steps of 10 from 50 to 100, the voice quality is poor, low, medium, high and best, respectively [26]. Although the RSs are the players of the game, in the ﬁnal results the performance of the MSs must be assessed in terms of the Rfactor achieved (considered as MS utility). To compute the Rfactor of the MSs, the PSR, and the delay for the whole transmission from MS to BS must be considered. For instance, the PSR perceived by each MS i served by RS j is given by ζi,j (G) = ρi,ij · ρj,q j (G),
(13.18)
where ρi,ij is the PSR on the direct transmission between MS i and RS j (independent of network graph G) and ρj,q j (G) is the PSR from RS j to the BS along path qj given by (13.14) (which can be either a multihop transmission or a direct transmission depending on how RS j is connected in the tree graph G). Moreover, the delay perceived by an MS i served by RS j is given by (13.15) by taking into account, in addition to the delay on the path qj , the traﬃc on the link (i, j) between the MS and the RS, i.e., the buﬀering and transmission delay at the MS level. Having the PSR given by (13.18) and the delay, the utilities of the MSs can be computed for performance assessment.
Network formation algorithm Having formulated a network formation game between the RSs, the next step is to deﬁne an algorithm of interaction between the RSs in order to form the desired network graph. For the RSs game, prior to providing a network formation algorithm, we ﬁrst highlight that any such algorithm will result in a connected tree structure, as follows: Proposition 13.1 Any network graph G resulting from a network formation algorithm applied to the RSs network formation game is a connected directed tree structure rooted at the BS.
374
Coalitional games for cooperative cellular wireless networks
The proof of this property can be found in [27]. Brieﬂy, due to the high disconnection cost, if an RS is unable to ﬁnd any partner suitable for forming a link, it will connect to the BS through direct transmission. Thus, the network initially starts with all the RSs connected to the BS (star topology), before engaging in the network formation game. Let Gs i ,s−i denote the graph G formed when RS i plays a strategy si ∈ Si while all other RSs maintain their vector of strategies s−i = [s1 , . . . , si−1 , si+1 , . . . , sM ]. We deﬁne the best response for an RS as follows [25]. Deﬁnition 13.13 A strategy s∗i ∈ Si is a best response for RS i ∈ V if Ui (Gs ∗i ,s−i ) ≥ Ui (Gs i ,s−i ), ∀si ∈ Si . Thus, the best response for RS i is to make the selection of the link that maximizes its utility given that the other RSs maintain their vector of strategies. By using the diﬀerent properties of the RS network formation game, one can build a best responsebased algorithm that allows a distributed formation of the network graph. In this algorithm, the RSs are assumed to be myopic in the sense that the RSs aim at improving their payoﬀ considering only the current state of the network without taking into account the future evolution of the network. Finding an optimal network formation algorithm is a very complex problem, and no strict rules exist for doing so in the literature [18, 25]. Therefore, for each network formation game model, diﬀerent operations must be applied suited to the model considered. Network formation literature encompasses several myopic dynamics for various game models with directed and undirected graphs [18, 25]. Inspired by [18] and [25], an algorithm composed of several rounds is constructed. In this algorithm, each round consists mainly of two phases: a fair prioritization phase and a myopic network formation phase. In the fair prioritization phase, we consider a priority function that assigns a priority to each RS. In the myopic network formation phase, by increasing priority, the RSs are allowed to interact. Therefore, each round of the network formation algorithm begins with the fair prioritization phase where each RS is assigned a priority depending on its actual perceived BER: RSs with a higher BER are assigned a higher priority. The motivation behind this procedure is to allow RSs that are perceiving a bad channel fairly to possess an advantage in selecting their partners for the purpose of improving their BER. Thus, the RSs experiencing a high BER can select their partners out of a larger space of strategies during the dynamics phase. Other priority functions can also be used, and in a general case, a random priority function can be deﬁned. Following prioritization, the RSs start selecting their strategies sequentially in order of priority. During its turn, each RS i chooses to play its best response s∗i ∈ Si in order to maximize its utility at each round given the current network graph resulting from the strategies of the other RSs. The best response of each RS can be seen as a replace operation, whereby the RS will replace its current link to the BS with another link that maximizes its utility (if such a link is available).
13.4 Coalitional graph game among relay stations
375
Table 13.3. One round of the RSs network formation algorithm Initial state All the RSs start by directly connecting to the BS in a star topology. Two phases in each round of network formation Phase 1 Fair prioritization: Prioritize the RSs from the highest to the lowest current BER. Phase 2 Myopic network formation: The RSs take action sequentially by priority. Each RS i plays its best response s∗i , maximizing its utility (Rfactor). The best response s∗i of each RS is a replace link operation through which a RS i splits from its current parent RS and replaces it with a new RS that maximizes its utility. Multiple rounds are run until convergence to the ﬁnal Nash tree G† after which no RS can improve its utility by a unilateral change of strategy.
Multiple rounds consisting of the above two phases are run until convergence to the ﬁnal tree structure G† after which the RSs can no longer improve their utility through best responses. A summary of this algorithm is shown in Table 13.3. The stability of the ﬁnal graph G† is given using the concept of Nash equilibrium applied to network formation games [25]. Deﬁnition 13.14 A network graph G in which no node i can improve its utility by a unilateral change in its strategy si ∈ Si is a Nash network. Therefore, a Nash network is a network where the links chosen by each node are the best responses. Hence, in such a network, the nodes are in a Nash equilibrium with no node able to improve its utility by unilaterally changing its current strategy. When our dynamics converge, and as an immediate consequence of playing a best response dynamics, we have: Lemma 13.2 The ﬁnal tree structure G† resulting from the RSs network formation algorithm is a Nash network.
Numerical results In order to highlight the properties and performance of the network formation game among the RSs, we consider a square area of 4 km × 4 km with the BS at the center. We deploy the RSs and the MSs within this area. Further, we set the
376
Coalitional games for cooperative cellular wireless networks
Figure 13.6. Snapshot of a tree topology formed using the RSs network formation algorithm with 10 RSs before (solid line) and after (dashed line) the random deployment of 50 MSs (MS positions not shown for clarity).
transmit power to P˜ = 50 mW (RSs and MSs), the noise level to −110 dBm, and the bandwidth per RS to W = 100 kHz. For pathloss, we set the propagation loss to µ = 3. For the VoIP traﬃc, we consider a traﬃc of 64 kbps, divided into packets of length B = 200 bits with an arrival rate of 320 packets/s. For the HELLO packets we set η0 = 1 packet/s with the same packet length of B = 200 bits. For voice, we select the G.729 codec, hence υ1 = 12, υ2 = 15 and υ3 = 0.6 [26]. In Figure 13.6 we randomly deploy M = 10 RSs within the BS area. The network formation game starts with the star topology where all RSs are connected directly to the BS. Prior to the presence of the MSs in the network (only HELLO packets present), the RSs interact and converge to a ﬁnal Nash tree structure shown by the solid lines in the ﬁgure. This ﬁgure clearly shows how the RSs connect to their nearby partners, forming the tree structure. Furthermore, we randomly deploy 50 MSs in the area, and show how the RSs autonomously adapt the topology to this incoming traﬃc. The resulting network structure upon deployment of MSs is shown in Figure 13.6 in dashed lines. The RSs autonomously selforganize and adapt to the deployed traﬃc. For instance, RS 3 can no longer accommodate the traﬃc generated by RS 2 as it drastically decreases its utility. As a result, RS 2 takes the decision to disconnect from RS 3 and improve its Rfactor by directly connecting to the BS. Similarly, RS 7 disconnects from RS 10 and connects directly to the BS. Moreover, RS 9 ﬁnds it beneﬁcial to replace its link with RS 5 with a link with the less loaded RS 6.
13.4 Coalitional graph game among relay stations
377
Figure 13.7. Selfadaptation of the network’s tree topology to mobility shown through the variation of the utility of RS 5 as it moves upwards on the yaxis prior to any MS presence. Furthermore, we assess the eﬀect of mobility on the network structure. We consider the network of Figure 13.6 prior to the deployment of the MSs and we assume that RS 5 is moving upwards on the positive yaxis while the other RSs remain ﬁxed. The changes in utility of the concerned RSs during the movement of RS 5 are shown in Figure 13.7. As RS 5 moves upwards, its utility starts by dropping since the distance to its serving RS (RS 6) increases. Upon moving 0.3 km, RS 5 ﬁnds it beneﬁcial to replace its link with RS 6 with a direct link to the BS, adapting the topology. Meanwhile, RS 9 decides to remain connected to RS 5 since it cannot improve its utility elsewhere. However, when RS 5 has moved 0.4 km, RS 9 decides to disconnect from RS 5 and connect directly to RS 6 improving its utility. This decreases the traﬃc on RS 5 which is now closer to RS 3 and can maximize its utility by connecting to RS 3. As RS 5 moves closer to RS 3, its utility continues to improve. After moving 1.2 km, RS 5 approaches RS 8 which ﬁnds it beneﬁcial to disconnect its link with RS 3 and connect to RS 5. At this point the utility of RS 5 drops due to the newly accepted traﬃc; moreover, this utility drops further as RS 5 distances itself from its serving RS (RS 3). Meanwhile, the utility of RS 8 continues to improve as RS 5 moves closer to it. Through these results, we clearly illustrate how the RSs can autonomously selforganize adapting the topology to mobility. Similar results can be shown when new RSs enter the network or RSs leave the network with or without the presence of MSs. In Figure 13.8, we show the average achieved utility per MS as the number of RSs, M , in the network increases. The results are averaged over random positions
378
Coalitional games for cooperative cellular wireless networks
Figure 13.8. Performance assessment of the distributed tree formation algorithm for a network with 40 MSs shown through the average achieved MS utility vs. number of RSs M in the network (average over random positions of MSs and RSs). of the MSs and the RSs in a network having 40 MSs. We compare the performance of the network formation algorithm with the star topology whereby each RS is directly connected to the BS as well as the scenario where there are no RSs in the network. In this ﬁgure, we clearly see that as the number of RSs in the network increases the performance of the network formation algorithm as well as of the star topology improves. However, for the star topology the slope of increase is much lower than that of the network formation algorithm. In addition, the network formation algorithm presents a clear performance advantage, increasing with the number of the RSs, and reaching 40.3% and 42.75% (at M = 25 RSs) relative to the star topology and the no RSs case, respectively.
13.5
Conclusion Coalitional game theory presents a rich framework that can be used to model various aspects of cooperative behavior in wireless cellular networks. On the one hand, for ideal cooperation, one can utilize the various solution concepts of canonical coalitional games to study the stability and fairness of allocating utilities when all the users in the network cooperate. Although in this chapter canonical games have not been explored, their applications are numerous and they provide useful analytical tools for studying the limits of cooperation and the
13.5 Conclusion
379
feasibility of providing incentives for the wireless users to maintain a cooperative behavior, when cooperation is ideal. On the other hand, whenever there is a beneﬁt–cost tradeoﬀ for cooperation, one can revert to a class of coalitional games, known as coalition formation games, to derive models and algorithms that can help in analyzing the cooperating groups that will emerge in a given wireless network. For example, using simple concepts from coalition formation game theory such as the mergeandsplit rules, singleantenna users can cooperate to form virtual MIMO coalitions and, thus, beneﬁt from the advantages of multipleantenna systems. Using merge and split for cooperation is not limited to the single cell of a wireless cellular system nor to the virtual MIMO application. In fact, cooperative approaches based on coalition formation games can extend to multicell systems and can model a variety of scenarios. For example, in a wireless cellular network, one can adopt a mergeandsplit algorithm for cooperation among the BSs in order to beneﬁt from receive diversity, receive or transmit beamforming, interference cancelation, and other advanced communication techniques. In brief, the coalition formation game class constitutes an appropriate framework for modeling the cooperative behavior in wireless systems, and, thus, it will admit numerous applications in the future. Further, for analyzing routing problems, network structure formation, and graph interconnection, coalitional graph games provide several algorithms and solutions, notably through the framework of network formation games. As demonstrated in this chapter, network formation games are quite useful for modeling the interactions among the relay stations that are bound to be deployed in next generation cooperative wireless systems. In this context, one can design eﬃcient and robust strategies for forming the network structure that will govern the architecture of wireless systems. The application of network formation games is not limited to the RSs problem studied in this chapter. In fact, this framework admits numerous uses in wireless systems such as modeling the downlink transmission path in multihop systems, studying network routing in cooperative wireless cellular systems, identifying the hierarchy that can govern the architecture of wireless systems, modeling wireless peertopeer interaction. Also, an interesting aspect of network formation is to study the cooperative behavior of the nodes whenever they are far sighted, i.e., they make decisions based on future rewards. In a nutshell, using coalitional game theory one can design eﬃcient, robust, and fair models for cooperative wireless cellular networks. Consequently, the use of these games and models is bound to be prevalent in the design and analysis of cooperative behavior in future wireless communication networks.
Acknowledgments This research is supported by the Research Council of Norway through projects 183311/510, 176773/510, 18778/VII.
380
Coalitional games for cooperative cellular wireless networks
References [1] R. B. Myerson, Game Theory, Analysis of Conﬂict. Harvard University Press, 1991. [2] D. Ray, A GameTheoretic Perspective on Coalition Formation. Oxford University Press, 2007. [3] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Ba¸sar, “Coalitional game theory for communication networks: A tutorial,” IEEE Signal Processing Mag., Special issue on “Game Theory in Signal Processing and Communications”, vol. 26, no. 5, pp. 77–97, Sep. 2009. [4] S. Mathur, L. Sankaranarayanan, and N. Mandayam, “Coalitions in cooperative wireless networks,” IEEE J. Select. Areas Commun., vol. 26, pp. 1104–1115, Sep. 2008. [5] R. La and V. Anantharam, “A gametheoretic look at the Gaussian multiaccess channel,” DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 66, pp. 87–106, 2003. American Mathematical Society, 2003. [6] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Ba¸sar, “A distributed coalition formation framework for fair user cooperation in wireless networks,” IEEE Trans. Wireless Commun., vol. 8, no. 9, pp 4580–4593, Sep. 2009. [7] W. Saad, Z. Han, M. Debbah, and A. Hjørungnes, “Coalitional games for distributed collaborative spectrum sensing in cognitive radio networks,” in Proc. of IEEE INFOCOM, Rio de Janeiro, Brazil, Apr. 2009. IEEE, 2010. [8] W. Saad, Z. Han, T. Ba¸sar, A. Hjørungnes, and J. B. Song, “Hedonic coalition formation games for secondary base station cooperation in cognitive radio networks,” in Proc. of IEEE Wireless Communications and Networking Conference (WCNC), Networking Symposium, Sydney, Australia, April 2010. IEEE, 2010. [9] W. Saad, Z. Han, T. Ba¸sar, M. Debbah, and A. Hjørungnes, “Physical layer security: Coalitional games for distributed cooperation,” in Proc. of 7th International Symp. Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), Seoul, South Korea, June 2009. ICST, 2009. [10] W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Ba¸sar, “Coalition game theory for communication networks: A tutorial,” in Proc. of 3rd ICST/ACM International Workshop on Game Theory in Communication Networks, Pisa, Italy, Oct. 2009. ICST, 2009. [11] K. Yazdi, H. E. Gamal, and P. Schitner, “On the design of cooperative transmission schemes,” in Proc. of Allerton Conference on Communication, Control, and Computing, Illinois, IL, USA, Oct. 2003. University of Illinois at UrbanaChampaign, 2003. [12] C. Ng and A. Goldsmith, “Transmitter cooperation in adhoc wireless networks: Does dirtypayer coding beat relaying?,” in Proc. of International
References
[13]
[14]
[15] [16] [17]
[18] [19]
[20]
[21]
[22]
[23]
[24] [25]
[26] [27]
381
Symposium on Information Theory, Chicago, IL, USA, June 2004. IEEE, 2004. M. Jindal, U. Mitra, and A. Goldsmith, “Capacity of adhoc networks with node cooperation,” in Proc. of International Symposium on Information Theory, Chicago, IL, USA, June 2004. IEEE, 2004. A. Coso, S. Savazzi, U. Spagnolini, and C. Ibars, “Virtual MIMO channels in cooperative multihop wireless sensor networks,” in Proc. of Conference on Information Sciences and Systems, New Jersey, NY, USA, Mar. 2006, pp. 75–80. Princeton University, 2006. I. E. Telatar, “Capacity of multiantenna Gaussian channels,” European Trans. Telecommun., vol. 10, pp. 585–595, Dec. 1999. R. Aumann and J. Dr`eze, “Cooperative games with coalition structures,” Int. J. Game Theory, vol. 3, pp. 317–237, Dec. 1974. K. Apt and A. Witzel, “A generic approach to coalition formation,” in Proc. of International Workshop on Computational Social Choice (COMSOC), Amsterdam, the Netherlands, Dec. 2006. University of Amsterdam, 2006. G. Demange and M. Wooders, Group Formation in Economics: Networks, Clubs and Coalitions. Cambridge University Press, 2005. E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications. Cambridge University Press, 2007. J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Eﬃcient protocols and outage behavior,” IEEE Trans. Information Theory, vol. 50, pp. 3062–3080, Dec. 2004. J. Boyer, D. Falconer, and H. Yanikomeroglu, “Multihop diversity in wireless relaying channels,” IEEE Trans. Commun., vol. 52, pp. 1820–1830, Oct. 2004. S. W. Peters, A. Panah, K. Truong, and R. W. Heath, “Relay architectures for 3GPP LTEAdvanced,” EURASIP J. Wireless Commun. Networking, vol. 2009, May 2009. The Relay Task Group of IEEE 802.16, “The p802.16j baseline document for draft standard for local and metropolitan area networks,” 802.16j06/026r4, Jun. 2007. D. Bertsekas and R. Gallager, Data Networks. Prentice Hall, Mar. 1992. J. Derks, J. Kuipers, M. Tennekes, and F. Thuijsman, “Local dynamics in network formation,” in Proc. of 3rd World Congress of The Game Theory Society, Chicago, IL, USA, July 2008. Northwestern University, 2008. ITUT Recommendation G.107, “The emodel, a computational model for use in transmission planning,” ITUT, Jun. 2002. W. Saad, Z. Han, M. Debbah, A. Hjørungnes, and T. Ba¸sar, “A gamebased selforganizing uplink tree for VoIP services in IEEE 802.16j networks,” in Proc. of IEEE International Conference on Communications (ICC), Dresden, Germany, June 2009. IEEE, 2009.
14 Modeling malicious behavior in cooperative cellular wireless networks Ninoslav Marina, Walid Saad, Zhu Han, and Are Hjørungnes 14.1
Introduction Future communication systems will be decentralized and adhoc, hence allowing various types of network mobile terminals to join and leave. This makes the whole system vulnerable and susceptible to attacks. Anyone within communication range can listen to and possibly extract information. While these days we have numerous cryptographic methods to ensure highlevel security, there is no system with perfect security on the physical layer. Therefore, the physical layer security is attracting renewed attention. Of special interest is socalled informationtheoretic security since it concerns the ability of the physical layer to provide perfect secrecy of the transmitted data. In this chapter, we present diﬀerent scenarios of a decentralized system that protects the broadcasted data on the physical layer and makes it impossible for the eavesdropper to receive the packets no matter how computationally powerful the eavesdropper is. In approaches where informationtheoretic security is applied, the main objective is to maximize the rate of reliable information from the source to the intended destination, while all malicious nodes are kept ignorant of that information. This maximum reliable rate under which a perfectly secret communication is possible is known as the secrecy capacity. This line of work was pioneered by Aaron Wyner, who deﬁned the wiretap channel and established the possibility of secure communication links without relying on private (secret) keys [1]. Wyner showed that when the eavesdropper channel is a degraded version of the main channel, the source and the destination can exchange perfectly secure messages at a nonzero rate. The main idea proposed by him is to exploit the additive noise impairing the eavesdropper by using a stochastic encoder that maps each message to many codewords according to an appropriate probability distribution. With this scheme, a maximal equivocation (i.e., uncertainty) is induced at the eavesdropper. In other words, a maximal level of secrecy is obtained. By ensuring that the equivocation rate is arbitrarily close to the message rate, one can achieve perfect secrecy in the sense that the eavesdropper is limited to learning almost nothing about Cooperative Cellular Wireless Networks, eds. Ekram Hossain, Dong In Kim, and Vijay K. Bhargava. Published by Cambridge University Press. C Cambridge University Press, 2011.
14.1 Introduction
C
S
383
D
Cm E Figure 14.1. The eavesdropper channel.
the source–destination messages from its observations. Followup work by LeungYanCheong and Hellman characterized the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel [2]. In their landmark paper, Csisz´ ar and K¨ orner generalized Wyner’s approach by considering the transmission of conﬁdential messages over broadcast channels [3]. There have been considerable eﬀorts to generalize these studies to the wireless channel and multiuser scenarios ([2, 4–11]). The basic scenario, called the eavesdropper channel, is shown in Figure 14.1. There is a source S that is transmitting a message to a destination D, while an eavesdropper E is trying to listen to the communication between S and D. In the following text we shall also use the term malicious node for the eavesdropper. A perfect informationtheoretically secret system is a system with positive secrecy capacity. In addition to the secrecy capacity being positive it is beneﬁcial for the source to have it as large as possible to enable higherrate secret communication. For the class of channels of interest here, the secrecy capacity is deﬁned as Cs
=
max(C − Cm , 0),
(14.1)
where C is the capacity of the direct pointtopoint channel between the source and the destination and Cm is the capacity of the channel between the source and the eavesdropper. Having a positive secrecy capacity guarantees perfect secrecy. Note that it is in the interest of the system designer to make Cs as large as possible. To increase Cs , it is obvious from (14.1) that one has to either increase C or decrease Cm or both. In the rest of this chapter we describe three diﬀerent scenarios for how the secrecy capacity could be changed by cooperation. First in Section 14.2 we describe how cooperation with some friendly jamming nodes can decrease Cm , which results in increasing Cs . In order to study what the incentive is for the friendly jammer to help the source, we analyze the system by using the Stockholder game. In Section 14.3 we study how the cooperating nodes can help the source by relaying the useful information to the destination while at the same time jamming the eavesdropper. Here we realize that for such a model
384
Modeling malicious behavior in cooperative cellular wireless networks
there is an area that the eavesdropper must not be allowed to enter in order to ensure a positive secrecy capacity. We call this region the vulnerability region and we study its size as the number of cooperative relaying nodes increases. Finally, in Section 14.4 we warn that as well as the “good guys”, the eavesdroppers can also organize themselves, making the secrecy capacity lower. We study a scenario in which eavesdroppers form coalitions, thereby decreasing the secrecy capacity acted much more than if the eavesdroppers act individually. The cooperative behavior of the eavesdroppers is studied by using coalition games.
14.2
Cooperating jammers In this section, we investigate the interaction between the source and friendly jammers, who help the source by jamming the eavesdropper. Although the friendly jammers help the source by reducing the data rate that is “leaking” from the source to the malicious node, at the same time they also reduce the useful data rate from the source to the destination. Using wellchosen amounts of power from the friendly jammers, the secrecy capacity can be maximized. In the game that we deﬁne here, the source pays jammers to interfere with the malicious eavesdropper and therefore to increase the secrecy capacity. The jammers charge the source a certain price for jamming the eavesdropper. It can be seen that there is a tradeoﬀ when deciding the price: If the price of a certain jammer is too low, its proﬁt is also low; if its price is too high, the source will buy from another jammer. In modeling the outcome of the above games our analysis uses the Stockholder type of game. Initially, the existence of equilibrium will be studied. Then, a distributed algorithm will be proposed and its convergence will be investigated. The outcome of the distributed algorithm will be compared with the centralized genieaided solution. Some implementation concerns are also discussed. From the simulation results, we can see the eﬃciency of friendly jamming and the tradeoﬀ for setting the price; the source prefers buying service from only one jammer, and the centralized scheme and the proposed game scheme have similar performances. Jamming [12–14] has been studied for a long time to analyze the hostile behaviors of malicious nodes. Jamming has been employed as a physical layer security method to reduce the eavesdropper’s ability to decode the source’s information [15]. In other words, the jamming is friendly in this context. Moreover, the friendly helper can assist the secrecy by sending codewords, and bring further gains relative to unstructured Gaussian noise [15–17]. Game theory [18] is a formal framework with a set of mathematical tools to study some complex interactions among interdependent rational players. There has been a surge in research activity that employs game theory to model and analyze modern distributed communication systems. Most of this
14.2 Cooperating jammers
C
S
D
Cm
385
S – Source D – Destination M – Malicious node (Eavesdropper) J1,…,JJ – J Friendly jammers
J1 M
Useful data Interference Payment
J2 JJ
Figure 14.2. System model for the proposed informationtheoretic security game.
research [19–22] concentrated on the distributed resource allocation for wireless networks.
14.2.1
System model We consider a network with a source, a destination, a malicious eavesdropper node, and J friendly jammer nodes as shown in Figure 14.2. The malicious node tries to eavesdrop the transmitted data coming from the source node. When the eavesdropper channel from the source to the malicious node is a degraded version of the main source–destination channel, the source and destination can exchange perfectly secure messages at a nonzero rate. By transmitting a message at a rate higher than the rate of the malicious node, the malicious node can learn almost nothing about the messages from its observations. The maximum rate of secrecy information from the source to its intended destination is deﬁned by the term secrecy capacity. Suppose the source transmits with power P0 . The channel gains from the source to the destination and from the source to the malicious node are Gsd and Gsm , respectively. Each friendly jammer i, i = 1, . . . , J, transmits with power Pi and the channel gains from it to the destination and the malicious node are Gid and Gim , respectively. We denote by J the set of indices {1, 2, . . . , J}. If the pathloss model is used, the channel gain is given by the distance to the negative power of the pathloss coeﬃcient. The thermal noise for each channel is σ 2 and the bandwidth is W . The channel capacity for the source to the destination is P0 Gsd . (14.2) C = W log2 1 + 2 σ + i∈J Pi Gid
386
Modeling malicious behavior in cooperative cellular wireless networks
The channel capacity from the source to the malicious node is P0 Gsm . Cm = W log2 1 + 2 σ + i∈J Pi Gim
(14.3)
As we know from (14.1), the secrecy capacity is Cs = max(C − Cm , 0). Both C and Cm are decreasing and convex functions of the jamming powers Pi . However, Cs = C − Cm is not a monotonous and convex function. This is because the jamming power might decrease C faster than Cm . As a result, Cs might increase for some values of Pi . So, the questions to be considered are whether or not Cs can be increased, and how the jamming power can be controlled in a distributed manner. We will solve the problems in the following text using a game theoretical approach.
14.2.2
The game Next we describe how game theory can be used to analyze informationtheoretic security in a cooperative network. First, we deﬁne the game between the source and the friendly jammers. We optimize the source side and the jammer side, respectively. Then, we prove some properties of the proposed game. Furthermore, a comparison is made with the centralized scheme. Finally, we discuss some implementation concerns. The source can be modeled as a buyer who wants to optimize its secrecy capacity minus cost by modifying the “service” (jamming power Pi ) from the friendly jammer, i.e., source’s game: max Us = max(aCs − M ), subject to : Pi ≤ Pm ax ,
(14.4)
where a is the gain per unit capacity, Pm ax is the maximal power that a jammer can provide, and M is the cost to pay for the other friendly jamming nodes. Here pi P i , (14.5) M= i∈J
where pi is the price per unit power for the friendly jammer, Pi is the friendly jammer’s power, and J is the set of friendly jammers. From (14.4) we note that the source will not participate in the game if C < Cm , or, in other words, the secrecy capacity is zero. For each jammer, Ui (pi , Pi (pi )), is the utility function of the price and power bought by the source. For the jammer’s (seller’s) utility, we deﬁne the following utility Ui = pi Pic i ,
(14.6)
where ci ≥ 1 is a constant to balance the payment pi Pi from the source and the transmission cost Pi . Notice that Pi is also a function of the vector of prices (p1 , . . . pN ), since the power that the source will buy also depends on the price that the friendly jammers ask. Hence, for each friendly jammer, the optimization
387
14.2 Cooperating jammers
problem is friendly jammer’s game: max Ui .
(14.7)
pi
In the following text, we analyze the optimal strategies for the source and friendly jammers to maximize their own utilities. Introducing A = P0 Gsd /σ 2 , B = P0 Gsm /σ 2 , ui = Gid /σ 2 , and vi = Gim /σ 2 , i ∈ J , we have ⎛ ⎛ ⎞ ⎞⎞+ ⎛ ⎜ ⎜ ⎜ Us = aW ⎜ ⎝log ⎝1 +
1+
A
u j Pj
⎟ ⎜ ⎟ − log ⎜1 + ⎠ ⎝
1+
j ∈J
B
vj P j
⎟⎟ ⎟⎟ − pj P j . ⎠⎠ j ∈J
j ∈J
(14.8) For the source (buyer) side, we analyze the case C > Cm . By diﬀerentiating (14.4), we have ∂Us ∂Pi
= − +
(1 + A +
aW Aui / ln 2 j ∈J uj Pj )(1 + j ∈J uj Pj )
(1 + B +
aW Bvi / ln 2 − pi = 0. j ∈J vj Pj )(1 + j ∈J vj Pj )
(14.9)
Rearranging the above equation, we have Pi4 + Fi,3 Pi3 + Fi,2 (pi )Pi2 + Fi,1 (pi )Pi + Fi,0 (pi ) = 0,
(14.10)
where Fi,3 = (2 + 2αi + A)2 + (2 + 2βi + B)2 , B (2 + 2αi + A)(2 + 2βi + B) Li Ki aW A , + 2 + 2 − − Fi,2 (pi ) = u i vi vi ui p i u i vi vi ui Li Ci + Ki Di aW (ADi − BCi ) Fi,1 (pi ) = + , (14.11) u2i vi2 pi u2i vi2 Ki Li aW (Aui Li − Bvi Ki ) Fi,0 (pi ) = 2 2 + , ui vi pi u2i vi2 and αi =
Gj d Pj ,
βi =
j = i
Gj m Pj ,
(14.12)
j = i
Ki = (1 + αi )(1 + αi + A),
(14.13)
Li = (1 + βi )(1 + βi + B),
(14.14)
Ci = ui (2 + 2αi + A),
(14.15)
Di = vi (2 + 2βi + B).
(14.16)
The solutions of the quartic equation (14.10) can be expressed in a closed form but this is not the primary goal here. It is important that the solution we are
388
Modeling malicious behavior in cooperative cellular wireless networks
interested in is given by the following function: Pi∗ = Pi∗ (pi , A, B, {uj }, {vj }, {Pj }j = i ).
(14.17)
Note that 0 ≤ Pi ≤ Pm ax . Since Pi satisﬁes the polynomial function, we can have the optimal strategy as Pi∗ = min[max(Pi , 0), Pm ax ].
(14.18)
Because of the complexity of the closedform solution of the quartic equation in (14.18), we also consider two special cases: the low interference case and the high interference case. (a) Interference at the destination is much smaller than the noise (low interference case) Remember the deﬁnitions: A = P0 Gsd /σ 2 , B = P0 Gsm /σ 2 , ui = Gid /σ 2 , and vi = Gim /σ 2 . Imagine a situation in which all jammers are close to the malicious node and far from the destination node. In that case the interference from the jammer to the destination is very small in comparison with the additive noise and therefore Us ≈ aW
log (1 + A) − log 1 +
1+
+
B
j ∈J
vj P j
−
pj Pj .(14.19)
j ∈J
Then aW Bvi / ln 2 ∂Us − pi = 0. = ∂Pi (1 + B + j ∈J vj Pj )(1 + j ∈J vj Pj )
(14.20)
Rearranging we get Pi2 +
2 + 2βi + B (1 + βi )(1 + B + βi ) aW B = 0. Pi + − vi vi2 pi vi ln 2
(14.21)
Solving the above equation we obtain a closedform solution 7 2 + 2β (1 + βi )(1 + B + βi ) aW B + B (2 + 2βi + B)2 i Pi∗ = − + − + 2vi 4vi2 vi2 pi vi ln 2 zi = qi + wi + , (14.22) pi where 2 + 2βi + B , 2vi (2 + 2βi + B)2 (1 + βi )(1 + B + βi ) − , 4vi2 vi2 aW B . vi ln 2
qi
= −
(14.23)
wi
=
(14.24)
zi
=
(14.25)
14.2 Cooperating jammers
389
Finally, by comparing Pi∗ with the power under the boundary conditions (Pi = 0, Pi = Pm ax , and Cs = 0), the optimal Pi∗ in the lowSNR region can be obtained. (b) One jammer with interference that is much higher than the noise but much smaller than the received power at the destination and the malicious node In this case the interference from the jammer is much higher than the additive noise but much smaller than the power of the received signal at the destination and the malicious node. In other words, 1 1, ηI(p) ≥ I(ηp). In [24], it has been proved that the price will converge to the ﬁxed point (i.e., the Stockholder equilibrium in our case) from any feasible initial price vector. The positivity is very easy to prove. If the price pi goes up, the source will buy less from the ith friendly jammer. As a result, ∂Pi∗ /∂pi in (14.27) is negative, and we prove positivity pi = Ii (p) > 0. We prove the monotonicity and scalability only for the lowinterference case. In this case, from (14.22) it is obvious that 2 wi p2i + zi pi (qi pi + wi p2i + zi pi ) (Pi∗ ) Ii (p) = − = , (14.39) ∂P ∗ ci zi ci i ∂pi which is monotonically increasing in pi . For scalability, we have wi p2i + zi pi /η(qi pi + wi p2i + zi pi /η) Ii (ηp) = 1.
(14.40)
392
Modeling malicious behavior in cooperative cellular wireless networks
For more general cases, the analysis is not tractable. In our simulations, we employ general simulation setups. The simulation results show that the proposed scheme can converge and outperform the nojammer case. Traditionally, the centralized scheme is employed assuming all channel information is known. The objective is to optimize the secrecy capacity under the constraints of maximal jamming power: ⎡ ⎛ ⎞ ⎤ ⎢ ⎜ max Cs = max ⎢ W log2 ⎜ ⎣ ⎝ Pi
1+ 1+
σ2 +
σ2 +
P0 Gs d i ∈J P i G i d
P G 0 sm i∈J Pi Gim
⎟ ⎥ ⎟ , 0⎥ ⎠ ⎦
subject to 0 ≤ Pi ≤ Pm ax , ∀i.
(14.41)
The centralized solution is found by maximizing the secrecy capacity only. If we do not consider the constraint, we have ∂Cs ∂Pi
=
−AW ui (1 + αi + ui Pi )(1 + A + αi + ui Pi ) BW vi = 0. + (1 + βi + ui Pi )(1 + B + βi + ui Pi )
(14.42)
Rearranging we get Au2i (2 + B + 2βi ) − Bvi2 (2 + A + 2αi ) Pi Au3i − Bvi3 Aui (1 + βi )(1 + B + βi ) − Bvi (1 + αi )(1 + A + αi ) + = 0. (14.43) Au3i − Bvi3
Pi2 +
Using the Karush–Kuhn–Tucker (KKT) condition [25], the ﬁnal solution would be obtained by comparing the boundary conditions (i.e., Pi = 0, Pi = Pm ax , and Cs = 0). Notice that our proposed algorithm is distributive, in the sense that only the pricing information needs to be exchanged. In the simulation results, we compare the proposed gametheoretical approach with this centralized scheme. There are several implementation concerns for the proposed scheme. First, the channel information from the source to the malicious eavesdropper might not be accurately known. Under this condition, the secrecy capacity formula should be rewritten considering the uncertainty. If the direction of arrival is known, multiple antenna techniques can be employed such as in [11]. Second, the proposed scheme needs to update the price and power information iteratively. A natural question arises if the distributed scheme has less signalling than the centralized scheme. The comparison is similar to distributed and centralized power control in the literature [23, 24]. Since the channel condition is continuously changing, the distributed solution only needs to update the diﬀerence of the parameters such as power and price to be adaptive, while the centralized scheme requires
14.2 Cooperating jammers
393
1 Jammer location (50,75) Jammer location (10,75)
0.9 0.8
Secrecy capacity Cs
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
0.005
0.01 Jamming power
0.015
0.02
Figure 14.3. Secrecy capacity vs. the power of the single jammer.
all channel information in each time period. As a result, the distributed solution has a clear advantage and dominates current and future wireless network design. For example, in the power control for cellular networks, the openloop power control is done only once during the link initialization, while the closedloop power control (distributed power allocation such as [24]) is performed 1500 times for UMTS and 800 times for CDMA2000. Finally, for the multisource multidestination case, there are two possible choices to solve the problem. First, we can use a clustering method to divide the network into subnetworks, and then employ the singlesource–destination pair and multiplefriendlyjammers solution proposed in this section. If we believe that the jamming power can be useful for multiple eavesdroppers, techniques such as double auction could be investigated.
14.2.3
Simulation results The simulation was set up as follows. The source and friendly jammers have power of 0.02, the bandwidth is 1, the noise level is 10−8 , the propagation loss factor is 3, and an additive white Gaussian noise (AWGN) channel is assumed. The source, destination, and eavesdropper are located at the coordinates (0,0), (100,0), and (50,50), respectively. Here we select a = 2 for the friendly jammer’s utility in (14.6). For the singlefriendlyjammer case, we show the simulation with the friendly jammer at a location of (50,75) and (10,75). In Figure 14.3, we show the secrecy
Modeling malicious behavior in cooperative cellular wireless networks
0.016 Jammer location (50,75) Jammer location (10,75)
0.014 0.012 Amount of power bought
394
0.01 0.008 0.006 0.004 0.002 0
0
50
100 Jammer price
150
200
Figure 14.4. How much power the source buys as a function of the price.
capacity as a function of the jamming power. We can see that with the increase of the jamming power, the secrecy capacity ﬁrst increases and then decreases. This is because the jamming power has diﬀerent eﬀects on C and Cm . There is an optimal point for the jamming power. Also the optimal point depends on the location of the friendly jammer, and a friendly jammer close to the eavesdropper is more eﬀective in improving the secrecy capacity. Moreover, notice that the curve is neither convex nor concave. Figure 14.4 shows how the amount of the power bought by the source from the jammer depends on the requested price. We can see that the power is reduced if the price is high. At some point, the source would stop buying the power. So there is a tradeoﬀ for setting the price, i.e., if the price is too high, the source will buy less power or even stop buying it. For the twojammer case, we set up the following simulations. The malicious node is located at (50,90), the ﬁrst friendly jammer is located at (50,50), and the second friendly jammer is located at (50,75). Figures 14.5, 14.6, and 14.7 show, respectively, the source’s utility Us , the ﬁrst jammer’s utility U1 , and the second jammer’s utility U2 as a function of both users’ price. We can see that the source will buy service from only one of the friendly jammer. If the friendly jammer asks too low a price, the jammer’s utility is very low. On the other hand, if the jammer asks too high a price, it risks the situation in which the source will buy the service from another friendly jammer. There is an optimal price for each friendly jammer to ask, and the source will always select the one that can provide the best performance improvement. Next, we set up a simulation of mobility. The ﬁrst friendly jammer is ﬁxed at (50,50), while the second friendly jammer moves from (−50,75) to (100,75).
14.2 Cooperating jammers
395
1
Source Us
0.8
0.6
0.4
0.2 0 100 User 1 Price p 2
0 50
200 100 150 300
User 2 Price p 1
Figure 14.5. Us vs. the prices of both users. (Source (0,0), destination (100,0) malicious node (50,90), user 1(50,50), user 2(50,75).)
'
Figure 14.6. U1 vs. the prices of both users. (source (0,0), destination (100,0) malicious node (50,90), user 1(50,50), user 2(50,75).)
In Figure 14.8, we show the source utilities for the centralized scheme and the proposed game. The centralized scheme serves as a performance upper bound. We observe that the game result is not far away from the upper bound, while the game solution can be implemented in a distributed manner. The performance diﬀerence is insigniﬁcant when friendly jammer 2 is close to the malicious eavesdropper (e.g., the friendly jammer is at locations from (20, 75) to (70, 75)). In
396
Modeling malicious behavior in cooperative cellular wireless networks
Figure 14.7. U2 vs. the prices of both users. (Source (0,0), destination (100,0) malicious node (50,90), user 1(50,50), user 2(50,75).)
Figure 14.8. Us vs. the location of the second jammer.
Figure 14.9, we show the jammer power as a function of the location of jammer 2. We can see that depending on the jammers’ locations, the source switches between the two jammers for the best performance. Moreover, the source also buys the optimal amount of jamming power: when the jammer is close to the malicious eavesdropper, the source buys less power since the jammer is more
14.2 Cooperating jammers
397
Figure 14.9. Power vs. the location of the second jammer.
utility
utility utility
location
Figure 14.10. Utility vs. the location of the second jammer. eﬀective at improving the secrecy capacity. In Figure 14.10, we show the corresponding friendly jammers’ utilities of the proposed game. Finally, we show the eﬀect of parameter a on the friendly jammer’s utility in (14.6). When a is large, the friendly jammer’s utility reduces quickly if the source does not buy the service. As a result, the friendly jammer would not ask an arbitrary price, and the performance gap to the optimal solution is small. In
398
Modeling malicious behavior in cooperative cellular wireless networks
1 Optimal solution Game result
0.95
Secrecy capacity
0.9 0.85 0.8 0.75 0.7 0.65
2
2.5
3
3.5 Factor a
4
4.5
5
Figure 14.11. Eﬀect of the parameter a on the game.
Figure 14.11, we show the secrecy capacity as a function of a when the second jammer is located at (0,75). We can see that the performance gap shrinks when a increases. To summarize, if the source pays friendly jammers to interfere with the malicious eavesdropper, the secrecy capacity, and therefore the security of the network is increased. The friendly jammers charge the source a price for the jamming. To analyze the game outcome, we investigated the Stockholder game and constructed a distributed algorithm. Some properties such as equilibrium and convergence were analyzed. From the simulation results, we conclude the following. First, there is a tradeoﬀ for the price: if the price is too low, the proﬁt is low; and if the price is too high, the source will not buy or buy from another jammer. Second, for the multiplejammer case, the source will buy service from only one jammer. Third, the centralized scheme and distributed scheme have similar performances, especially when a is suﬃciently large. Overall, the proposed game theoretical scheme can achieve a comparable performance with distributed implementation.
14.3
Cooperating relays In this section, we observe how node cooperation improves the informationtheoretic security of a simple wireless network by reducing the surface of the geographical area in which the malicious nodes can listen to the data transmitted from the source to the destination. Our analysis and simulation
14.3 Cooperating relays
399
Relay 1
Relay 2
Source node
Destination node
Relay 3
Relay M
Figure 14.12. Cooperative network with a source, a destination, M cooperative relays, and a malicious node. results show a dramatic improvement even for cooperation with only one relay node. Adding more cooperating nodes gives greater improvement. We also observe that when cooperating nodes are closer to the line that connects the source and the destination node, the region in which the malicious node can proﬁt from the eavesdropping is smaller.
14.3.1
System model We analyze the Gaussian parallel multiplerelay network. Although the capacity of this network is not known, we use its upper and lower bounds, with a hope that our results will initiate further thought in this interesting and important research area. Observe the network in Figure 14.12. There is a source node that transmits data to a destination node, while a malicious node “listens” to the transmitted information. There are several relay nodes that help the source by relaying the transmitted data. We consider the additive complex white Gaussian noise model. A message at the source node is encoded into a codeword {X[j]}nj=1 of length n. The components X[j] are complex numbers that satisfy the power constraint n−1
n
E[X[j]2 ] ≤ P.
(14.44)
j =1
At time j each of the M relays observes an attenuated and noisy version of the input, i.e., Yi [j] = hi,s X[j] + Zi [j],
i = 1, 2, . . . , M,
400
Modeling malicious behavior in cooperative cellular wireless networks
where Zi [j] is an independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variable of zero mean and (without loss of generality) variance σ 2 . Moreover, Zi and Z are independent for i = . The coeﬃcients hi,s , i = 1, 2, . . . , M , are ﬁxed realvalued constants, assumed to be known throughout the network. For that reason we may assume that all noise processes Zi are of same variance σ 2 without loss of generality. Observing {Yi [j]}nj=1 at each relay i produces a sequence {Xi [j]}nj=1 that must be causal, i.e., Xi [j] = fi,j (Yi [j − 1], Yi [j − 2], . . . , Yi [1]), and satisﬁes the individual power constraint n−1
n
E[Xi [j]2 ] ≤ Pi .
(14.45)
j =1
The received signal at the destination node is Y [j] = hd,s X[j] +
M
hd,i Xi [j] + Zd [j],
i=1
where Zd [j] is a sequence of i.i.d. zeromean circularly symmetric complex Gaussian random variables of variance σ 2 . The coeﬃcients hd,s and hd,i , where i = 1, 2, . . . , M , are ﬁxed and assumed to be known throughout the network. We assume that the coeﬃcient ha,b between two nodes a and b is −β /2
ha,b = da,b ,
(14.46)
where β is the pathloss exponent and da,b is the distance between the nodes a and b. In this case, the distance between the source and the destination is denoted by dd,s , that between the source and relay i, by di,s , i = 1, 2, . . . , M , and that between relay i and the destination, by dd,i , where i = 1, 2, . . . , M .
14.3.2
Secrecy capacity Here we develop expressions for the secrecy capacity of the cooperative system described in Figure 14.12. It is assumed that the malicious node may eavesdrop on the source as well as on the cooperating relay nodes. Note, however, that in order to get full use of the signals transmitted from the source and the relay nodes, the source must be fully synchronized with all of them. To capture the eﬀect of synchronization, we model the received signal at the malicious node as follows: √ hm ,i ki Xi [j] + Zm [j], Ym [j] = hm ,s qX[j] + M
i=1
14.3 Cooperating relays
Zm ∼ CN
0, σ + (1 − 2
q)P d−β m ,s
+
M
401
(1 −
ki )Pi d−β m ,i
,
(14.47)
i=1
where Zm [j] is a sequence of i.i.d. circularly symmetric complex Gaussian ran√ dom variables of zero mean and variance σ 2 + (1 − q)P + M i=1 hm ,i 1 − ki Pi , −β /2 −β /2 hm ,s = dm ,s , and hm ,i = dm ,i , i = 1, 2, . . . , M . How well the malicious node is synchronized with the source is modeled by q ∈ [0, 1], while how well it is synchronized with the relay node i is modeled by ki ∈ [0, 1] for i = 1, 2, . . . , M . More precisely, q is the fraction of the source transmitting power that will be received by the malicious node as useful signal for itself, while (1 − q) is the fraction of the source power that makes interference at the malicious node. If q = 1, then the malicious node is perfectly synchronized with the source node, while if q = 0 there is no synchronization and the malicious node receives only noise from the source. Similarly, ki is the fraction of the transmitting power of relay node i that will be received by the malicious node as useful signal for itself, while (1 − ki ) is the fraction of the power of relay node i that makes interference at the malicious node. The same explanation is valid for the parameters ki . That means, an omnipotent eavesdropper will have q = ki = 1 for all i = 1, 2, . . . , M and a “dummy” eavesdropper will have q = ki = 0 for all i = 1, 2, . . . , M . In the former case, the vulnerability region will be maximal, while in the latter, it vanishes, i.e., we have a perfect secrecy system. In order to make the eavesdropper capabilities closer to reality, we introduce two models to describe the synchronization parameters as a function of the distance between the eavesdropper and the eavesdropped node. Model 1 is the exponential model, deﬁned as q
= e−d m , s ,
ki
= e−d m , i .
(14.48)
Model 2 is the squared exponential model (Gaussian model), deﬁned as q
= e−d m , s ,
ki
= e−d m , i .
2
(14.49)
2
For notational convenience, we use deﬁnitions similar to those in, i.e., aM = d−β d,s +
M
d−β i,s ,
i=1
bM =
M 2 d−β i,s P + σ i=1
dβi,s d−β d,i
dM = d−β d,s +
M i=1
,
d−β d,i .
402
Modeling malicious behavior in cooperative cellular wireless networks
With all these assumptions we will be able to determine the bounds on the secrecy capacity, since the capacity of a general nondegraded relay channel is not known in general. First, the capacity of the main channel (the parallel relay channel between the source and the destination) is upper bounded by min{aM P, (P + M i=1 Pi )dM } . (14.50) C ≤ log2 1 + σ2 The lower bound of the capacity of the main channel is given by ⎞ ⎛ @ −β M ⎟ ⎜ (aM − d−β ⎜ i=1 Pi ⎟ d,s ) + 2 bM dd,s / ⎟. P C ≥ log2 ⎜ 1 + ⎟ ⎜ P + σ2 ) bM (d−β ⎠ ⎝ d,s 2 σ + −β M (aM − dd,s ) i=1 Pi
(14.51)
The capacity of the eavesdropper channel (the maximum rate that can be achieved by the malicious node) is given by ⎛ ⎞ M ki Pi qP + ⎜ ⎟ β ⎜ ⎟ dβm ,s i=1 dm ,i ⎟ Cm = log2 ⎜ 1 + (14.52) ⎜ ⎟ M ⎝ (1 − q)P (1 − ki )Pi 2⎠ + +σ dβm ,s dβm ,i i=1 where Pi is the transmit power of relay i. Note that it might be obtained after some allocation process since we have the total power constraint applied to all relays, given by (14.45). The secrecy capacity is then deﬁned as [2] Cs = max{C − Cm , 0}.
(14.53)
In other words, the secrecy capacity is positive only if C > Cm . In order to analyze how cooperation improves the secrecy, we introduce the following two deﬁnitions. Deﬁnition 14.3 The geometrical area (region) in which the secrecy capacity is positive is called the secrecy region. Deﬁnition 14.4 The geometrical area (region) in which the secrecy capacity vanishes is called the vulnerability region. Obviously, we want to keep all malicious nodes away from the vulnerability region. In other words, the system is more secure if its vulnerability region is minimized, or, equivalently, if its secrecy region is maximized. Note that if there are no cooperating relay nodes (M = 0), it can be shown that the vulnerability region is a circle (disk) since in that case the capacity of the main channel is the capacity of the pointtopoint Gaussian channel between the source and the
14.3 Cooperating relays
destination. More precisely the region is determined from P d−β d,s C = log2 1 + σ2 qP d−β m ,s < log2 1 + = Cm . 2 (1 − q)P d−β m ,s + σ
403
(14.54)
Solving (14.54), we get that the vulnerability region is a disk centered at the source with radius d∗m ,s
= dd,s (max {0, q(1 + γ) − γ})1/β ,
2 where γ = P d−β d,s /σ . In other words, ⎧ ⎨0, d∗m ,s = ⎩dd,s (q(1 + γ) − γ)1/β ,
(14.55)
γ for 0 ≤ q ≤ , 1+γ γ ≤ q ≤ 1. for 1+γ
This means that for a high SNR, it is easier to get a perfectly secure system if q ≤ γ/(1 + γ). If q = 1, d∗s,m
= dd,s ,
while for 0 ≤ q ≤ γ/(1 + γ) the vulnerability region vanishes, which is a desired situation. In order to compare the vulnerability region to some reference region we introduce the normalized vulnerability region. Deﬁnition 14.5 The normalized vulnerability region is the ratio of the vulnerability region to the surface of the disk with radius dd,s . The deﬁnition tells us that if the normalized vulnerability region of a cooperative system is less than 1, we get a smaller vulnerability region than for a noncooperative system, or in other words, cooperation increases network security. In our numerical analysis we observe the following bounds on the secrecy capacity: CsU B
=
max(C U B − Cm , 0),
CsL B
=
max(C L B − Cm , 0).
In a cooperative system, where M ≥ 1, we can determine numerically the lower and the upper bound of the normalized vulnerability region. It is natural that they depend heavily on the position of the cooperating relays, and that the lower bound of the vulnerability region is determined by the upper bound of the capacity of the main channel (14.50), while the upper bound of the vulnerability region is determined by the lower bound on the capacity of the main channel (14.51).
404
Modeling malicious behavior in cooperative cellular wireless networks
LB
UB
LB
UB
LB
UB
Figure 14.13. Examples of the lower bound (LB) and upper bound (UB) on the shape of a typical vulnerability region for M = 1 (upper), M = 3 (middle), and M = 5 (lower).
14.3.3
Simulation results In this subsection, we show several examples of the shape of the secrecy region in a cooperative system. We also observe how the average surface depends on the number of relays. We ﬁx the parameters as follows: σ 2 = 1, P = Pi = 1, i = 1, 2, . . . , M , and β = 2. For these parameters, typical vulnerability regions are described in Figure 14.13 for M = 1, M = 3, and M = 5, using model 2. The source node is represented by a centrally positioned star, the destination node by a square positioned to the right of the source, and the randomly placed relay nodes by diamonds. When the malicious node is positioned in a dark area, perfectly secure communication is possible (i.e., the secrecy capacity is positive). The white areas (islands) represent the vulnerability regions. It is easily noticeable that an increased number of cooperating relays reduces the vulnerability region, hence, increasing the system secrecy.
14.3 Cooperating relays
405
0.6
0.5
Bounds
0.4
0.3
UB
0.2
0.1 LB
(a)
0 1
2
3
4
5
6
4
5
6
M 0.6
0.5
Bounds
0.4
UB
0.3
0.2 LB 0.1
0
(b)
1
2
3 M
Figure 14.14. Lower bound (LB) and upper bound (UB) of the normalized vulnerability region for β = 2 and (a) correlation model 1 and (b) correlation model 2 as a function of the number of cooperating relays M . In order to understand the importance of the cooperation in increasing the secrecy capacity or, equivalently, the secrecy region of a certain network, we characterize the surface of the vulnerability region. To that end, we analyze how the normalized vulnerability region depends on the number of cooperative relays. The dependence of the lower and upper bounds of the normalized vulnerability region on the number of relay nodes, for β = 2, for both correlation models is shown in Figure 14.14. Our simulations indicate that choosing cooperation relays that are closer to the line that connects the source and the destination, minimizes the vulnerability region. In Figure 14.15 we observe t