Resource Allocation for OFDMA Systems [1st ed.] 978-3-030-19391-1;978-3-030-19392-8

Table of contents :
Front Matter ....Pages i-xi
Introduction (Chen Chen, Xiang Cheng)....Pages 1-5
Overview of OFDMA and MIMO Systems (Chen Chen, Xiang Cheng)....Pages 7-22
Remarks on Resource Allocation (Chen Chen, Xiang Cheng)....Pages 23-42
Resource Allocation for OFDMA Systems (Chen Chen, Xiang Cheng)....Pages 43-81
Dealing with Imperfect CSI (Chen Chen, Xiang Cheng)....Pages 83-129
Back Matter ....Pages 131-132

Citation preview

Wireless Networks

Chen Chen Xiang Cheng

Resource Allocation for OFDMA Systems

Wireless Networks Series editor Xuemin Sherman Shen University of Waterloo, Waterloo, ON, Canada

More information about this series at http://www.springer.com/series/14180

Chen Chen • Xiang Cheng

Resource Allocation for OFDMA Systems

Chen Chen School of Electronics Engineering & Comp Peking University Beijing, China

Xiang Cheng School of Electronics Engineering Peking University Beijing, China

ISSN 2366-1186 ISSN 2366-1445 (electronic) Wireless Networks ISBN 978-3-030-19391-1 ISBN 978-3-030-19392-8 (eBook) https://doi.org/10.1007/978-3-030-19392-8 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book investigates and surveys resource allocation techniques for MIMOOFDMA systems. The radio resources of traditional communication systems are defined and distributed in a hierarchical structure. However, due to the space, time, and frequency variations and randomness of next-generation wireless communication systems, the traditional hierarchical allocation methods cannot optimize them. Therefore, people pay more attention to cross-layer wireless resource allocation technologies today, so as to achieve joint scheduling and optimization of system space, time, and frequency resources. As several important wireless resources, the rational use and distribution of users, power, and spectrum resources play an extremely crucial role in ensuring communication quality and improving communication performance. This book first introduces the sources and historic collection campaigns of resource allocation in wireless communication systems. Secondly, the unique characteristics of MIMO-OFDMA systems will be thoroughly studied and summarized. Thirdly, remarks on resource allocation and spectrum sharing will be presented, which demonstrate the great value of resource allocation techniques but also introduce distinct challenges of resource allocation in MIMO-OFDMA systems. Fourthly, novel resource allocation techniques for OFDMA Systems will be surveyed from various applications (e.g., for unicast, or multicast with Guaranteed BER and Rate, subcarrier and power allocation with various detectors, low-complexity energy-efficient resource allocation, etc.). Due to the high mobility and low latency requirements of 5G wireless communications, this book will discuss how to deal with the imperfect CSI, e.g., throughput maximization, outage probabilities maximization and guarantee, energy efficiency, physical-layer security issues with feedback channel capacity constraints, in order to characterize and understand the applications of practical scenes. Finally, the challenges and open opportunities of

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resource allocation in MIMO-OFDMA will be broadly investigated and concluded in terms of communications, networking, and energy efficiency, as well as the security issues of private-sensitive scenes. Beijing, China

Chen Chen Xiang Cheng

Acknowledgements

This work was supported in part by the Ministry National Key Research and Development Project under Grant 2017YFE0121400, the National Science and Technology Major Project under Grant 2018ZX03001031, and the Major Project from Beijing Municipal Science and Technology Commission under Grant Z181100003218007.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 OFDM/OFDMA Technology . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Radio Resource Allocation Technology . . . . . . . . . . . . . . . . . . 1.3 The Organizational Structure . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

1 1 2 4 5

2

Overview of OFDMA and MIMO Systems . . . . . . . . . . . . . . . . . . . 2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Channel Characteristics of Mobile Communication . . . . . . . . . . 2.2.1 Large-Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Mesoscale Decline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Small-Scale Decline . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Channel Interference . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiuser MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Signal Detection Method . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Single-User Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Multiuser Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 OFDMA System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 OFDM Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Unicast and Multicast OFDMA Systems . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

7 7 8 8 9 9 10 11 13 14 15 17 17 19 21

3

Remarks on Resource Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dynamic Spectrum Sharing Model and Method Overview . . . . . . 3.2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Dynamic Spectrum Sharing Method and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Current Research Hotspot . . . . . . . . . . . . . . . . . . . . . . . .

23 23 24 24 26 30

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3.3

The Theory of OFDMA System Resource Allocation . . . . . . . . 3.3.1 Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Optimal Solution Algorithm . . . . . . . . . . . . . . . . . . . . . 3.3.3 Suboptimal Solution Algorithm . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

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31 32 34 39 42

Resource Allocation for OFDMA Systems . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Resource Allocation for Multicast OFDMA Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Resource Allocation for MIMO-OFDMA Systems . . . . . . 4.1.3 Resource Allocation for Energy Efficiency in OFDMA Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with Guaranteed BER and Rate . . . . . . . . . . . . . . . . . . 4.2.1 System Model and Problem Formulation . . . . . . . . . . . . . 4.2.2 Optimal Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Suboptimal Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with Various Detectors . . . . . . . . . . . . . 4.3.1 System Model and Problem Formulation . . . . . . . . . . . . . 4.3.2 Subcarrier and Power Allocation . . . . . . . . . . . . . . . . . . . 4.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems . . . . . . . . . . . . . . . . . . 4.4.1 System Model and Problem Formulation . . . . . . . . . . . . . 4.4.2 Energy-Efficient Subcarrier Assignment . . . . . . . . . . . . . 4.4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43

Dealing with Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Channel Model with Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Nosie and Estimation Error . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 CSI Feedback Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Finite-Rate Feedback of Downlink CSI . . . . . . . . . . . . . . 5.3 Downlink Ergodic Throughput Maximization for OFDMA Systems with Feedback Channel Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 44 45 46 46 48 50 55 58 58 58 60 65 69 70 70 71 78 79 79 80 83 83 85 85 86 88

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5.3.2 Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Suboptimal Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Resource Allocation for Maximizing Outage Throughput in OFDMA Systems with Finite-Rate Feedback . . . . . . . . . . . . . 5.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Upper Bound of the Optimal Solution . . . . . . . . . . . . . . . 5.4.3 Suboptimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities . . . . . . . . . . . . . . . . . . . . . 5.5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Suboptimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Energy Efficiency Maximization for Downlink OFDMA Systems with Feedback Channel Capacity Constraints . . . . . . . . . 5.6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Energy-Efficient Resource Allocation with Quantized CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink with Imperfect CSI . . . . . . . . . . . . . . . . . 5.7.1 System Model and Problem Formulation . . . . . . . . . . . . . 5.7.2 Optimal Power Allocation Algorithm When Subcarrier Assignment Is Fixed . . . . . . . . . . . . . . . 5.7.3 Greedy Subcarrier Allocation Algorithm . . . . . . . . . . . . . 5.7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Chapter 1

Introduction

Resource allocation for orthogonal frequency division multiple access (OFDMA) systems which can effectively solve the signal distortion caused by the frequencyselective fading of the wireless channel and provide a high data transmission rate is a challenging issue for next-generation wireless communications. OFDMA is a multiuser OFDM system. It can adaptively allocate subcarriers based on different users’ channel conditions on different subcarriers and adjust power and number of bits on different subcarriers. In an OFDMA system, a proper resource allocation strategy can be used to optimize system performance within a limited spectrum while ensuring service QoS requirements. Therefore, designing an efficient resource allocation strategy for OFDMA systems is an important issue that is attracting attention in current mobile communications.

1.1

OFDM/OFDMA Technology

Looking at the development of wireless communication technology, supporting highspeed broadband data transmission is the main goal of mobile communication development. OFDM technology is considered to be an important technology for 4G wireless communication because of its highly scalable structure, good antimultipath interference capability, and efficient spectrum utilization. However, OFDM-based OFDMA technology can effectively use multiuser diversity. The gain further increases the utilization of spectrum resources. Therefore, OFDM/ OFDMA technology has received extensive attention in recent years. At present, OFDM/OFDMA technology has spread throughout the various fields of wireless communications, leading the future development of mobile communications. OFDM/OFDMA technology has the following advantages [1]: 1. Strong resistance to frequency-selective fading. OFDM divides a frequencyselective fading channel into N parallel-independent flat-fading channels and © Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8_1

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1 Introduction

transmits data independently on each subcarrier at the original rate of 1/N. Therefore, the symbol time on each subcarrier is N times that of a single carrier system. This makes OFDM more resistant to impulse noise and fast channel fading. In addition, OFDM also introduces protection time slots and CPs and makes the guard interval length greater than the maximum delay spread in the wireless channel, which can effectively eliminate ISI during transmission [2–4]. 2. High spectrum utilization. OFDM allows overlapping orthogonal subcarriers to be used as sub-channels instead of using traditional FDM to separate the subcarriers using guard bands so that the bandwidth can be used to transmit data. Since there are no spectral components of other subcarriers at the center frequency point of each subcarrier, it is possible to ensure that the subcarriers are orthogonal in the spectrum overlap and eliminate the inter-subcarrier ICI. 3. OFDM systems have strong bandwidth scalability. OFDM can easily implement data transmission under various bandwidths by using FFT. Because of the development of large-scale integrated circuits, it has become easier to implement FFT [5], which has made OFDM more robust in supporting broadband communications. 4. More flexible resource allocation strategies can be used. In an OFDM system, each subcarrier can independently select an appropriate modulation method, transmission rate, and transmission power. In an OFDMA system, different users can be further assigned different subcarriers. In this way, the base station can dynamically adjust the resource allocation strategy on each subcarrier according to the objective of system optimization, so that the characteristics of frequency diversity and user diversity gain can be utilized to improve the performance of the system [6, 7]. OFDM/OFDMA technology also has disadvantages such as sensitivity to frequency offset and phase noise [8–10], the existence of peak-to-average ratios [11–13], and synchronization [14, 15] and other issues. However, with the in-depth development of wireless communication technology and the solution of many key technical problems, OFDM/OFDMA will play an increasingly important role in the future of wireless communications.

1.2

Radio Resource Allocation Technology

With the popularization of mobile communication technologies, mobile communication technologies are faced with a dramatic increase in the number of users and the continuous improvement of the quality of service (QoS) requirements of users. Wireless resource management has become a key technology to improve the performance of mobile communication. Technology. Wireless resource allocation is responsible for the utilization of air interface resources. Its goal is to provide QoS guarantees for users within the network under limited resources and improve resource utilization, network capacity, and coverage [16].

1.2 Radio Resource Allocation Technology

3

Wireless system resources are the resources that are used when information is transmitted and processed wirelessly. Generally, they can be divided into two categories: 1. Transmission resources: The resources of the channel occupied by signals in the transmission process include: • Frequency resources: The bandwidth and frequency band occupied by the channel. • Time resource: The time slot occupied by the transmission signal. • Code resources: Generally used in CDMA systems, including channelization codes (differentiated channels) and scrambling codes (differing users in uplink and distinguishing cells in downlink). • Space resources: Occupancy of the transmitted signal to the antenna. • Geographical resources: The coverage and access of the coverage area and the community. 2. Node resources: In the process of signal transmission, the occupation of transmission node resources includes: • Power resource: The power consumed by a node when sending a signal. • Time resource: The time consumed by the processor when the node processes signals and allocates resources. • Space resources: Occupation of memory when nodes handle signals and allocate resources. Therefore, from the perspective of transmission resources, the allocation of wireless resources is to use limited resources such as spectrum, time slots, codes, and antennas, so as to effectively increase the capacity of the system and ensure the QoS performance of the service. From the perspective of node resources, the algorithm for wireless resource allocation should make reasonable use of the limited transmit power of the nodes and design suitable algorithms with low time and space complexity to achieve a balance between the complexity of the algorithm and the performance of the system. Resources allocation is to complete the allocation of wireless resources to the service, including: 1. Power control: Different nodes adjust the transmit power on the allocated channel according to the current link status. On the one hand, the power control reduces the interference of the same system and adjacent channels of the entire system by ensuring the QoS requirements of the service. On the other hand, it is also necessary to increase the capacity of the system through power adjustment and improve the utilization of spectrum resources. In addition, for the user, it is also necessary to extend the standby time of the wireless terminal through appropriate power control. In Chap. 2, we give an overview of OMA system. 2. Adaptive rate control/modulation: Rate control is to adjust the channel/source coding rate of information transmission according to the quality of the link under the constraint of the node power limitation. On the one hand, it is necessary to

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1 Introduction

guarantee the QoS of the service, and on the other hand, the limited spectrum resources need to be utilized as much as possible to increase the capacity of the system. 3. Channel allocation: According to the difference among multiple access methods, resources such as transmission frequency, time slot, spreading code, and antenna are allocated for different services. In a multi-cell environment, suitable cell transmission data is also selected and allocated for the wireless service. When the wireless service switches between different cells, an appropriate soft handover or hard handover policy is also selected, so that the service transmission can be continued. Studying resource optimization strategies for different systems has important implications.

1.3

The Organizational Structure

This book mainly studies resource allocation for OMA communication systems to optimize the performance of OMA communication systems under the wireless channel. In Chap. 2, we give an overview of OMA systems, including the related work of OFDMA systems and MIMO systems, the channel characteristics of mobile communication, and the basic theories of OFDMA system and MIMO system. In Chap. 3, we give an initial introduction to three key technologies for radio resource allocation, namely, the multiuser MIMO user selection, OFDMA power allocation, and dynamic spectrum allocation based on cognitive radio and their application in next-generation communication systems. Then we introduce the resource allocation theory of OFDMA system in detail. In Chap. 4, we discuss about some resource allocation scenario in OFDMA systems. We analyze the multicast OFDMA systems, BER performance in MIMOOFDM systems, and the energy efficiency in OFDMA systems. Since the optimal problem is NP-hard which needs very high computation complexity to obtain the optimal solution, we propose the suboptimal algorithm with acceptable complexity for each scenario. Simulation results show that the performance of the suboptimal algorithm is near to the optimal one. In Chap. 5, we discuss the resource allocation problems when the BS can only obtain imperfect CSI. We discuss the ergodic throughput, outage throughput, outage probability, energy efficiency, and secure capacity, respectively. Since the optimal solution has very high computation complexity, we propose the suboptimal algorithms with lower complexity. The simulation results show that the performance of the suboptimal algorithm is very close to the optimal ones.

References

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References 1. A.R.S. Bahai, B.R. Saltzberg, M. Ergen, Multi-Carrier Digital Communications: Theory and Applications of OFDM[M], 2nd (Springer Verlag, 2004) 2. M. Sandell, J.-J. van de Beek, P.O. Börjesson, Timing and Frequency Synchronization in OFDM Systems Using the Cyclic Prefix[C]. Proc. Int. Symp. Synchronization (Essen, 1995), pp. 16–19 3. A. Scaglione, G.B. Giannakis, S. Barbarossa, Redundant filterbank precoders and equalizers. i. Unification and optimal designs[J]. IEEE Trans Signal Process. 47(7), 1988–2006 (1999) 4. M.R.B. Shankar, K.V.S. Hari, Reduced complexity equalization schemes for zero padded OFDM systems[J]. IEEE Signal Process Lett. 11(9), 752–755 (2004) 5. E.O. Brigham, The Fast Fourier Transform and its Applications[M] (Prentice Hall, Englewood Cliffs, 1988) 6. Y.W. Cheong, R.S. Cheng, K.B. Lataief, R.D. Murch, Multiuser OFDM with adaptive subcarrier, bit, and power allocation[J]. IEEE J Sel Areas Commun. 17(10), 1747–1758 (1999) 7. I. Kim, I.-S. Park, Y.H. Lee, Use of linear programming for dynamic subcarrier and bit allocation in multiuser OFDM[J]. IEEE Trans Veh Technol. 55(4), 1195–1207 (2006) 8. T. Pollet, M. van Bladel, M. Moeneclaey, BER sensitivity of OFDM systems to carrier frequency offset and wiener phase noise[J]. IEEE Trans Commun. 43(234), 191–193 (1995) 9. K. Sathananthan, C. Tellambura, Probability of error calculation of OFDM systems with frequency offset[J]. IEEE Trans Commun. 49(11), 1884–1888 (2001) 10. L. Rugini, P. Banelli, BER of OFDM systems impaired by carrier frequency offset in multipath fading channels[J]. IEEE Trans Wireless Commun. 4(5), 2279–2288 (2005) 11. S.H. Muller, J.B. Huber, OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences[J]. Electron Lett 33(5), 368–369 (1997) 12. R.W. Bauml, R.F.H. Fischer, J.B. Huber, Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping[J]. Electron Lett 32(22), 2056–2057 (1996) 13. T. Jiang, Y. Wu, An overview: peak-to-average power ratio reduction techniques for OFDM Signals[J]. IEEE Trans Broadcast. 54(2), 257–268 (2008) 14. M. Gudmundson, P.O. Anderson, Adjacent Channel Interference in an OFDM System[C]. Proc. IEEE VTC’96, vol 2 (Atlanta, 1996), pp. 918–922 15. Y. Mostofi, D.C. Cox, Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system[J]. IEEE Trans Commun. 54(2), 226–230 (2006) 16. J. Zander, S.L. Kim, M. Almgren, O. Queseth, Radio Resource Management for Wireless Networks[M] (Artech House, Boston, 2001)

Chapter 2

Overview of OFDMA and MIMO Systems

In this chapter, we will give an overview of OMA system, including the characteristics of the mobile communication channel, the basic theories of the OFDM system, and the basic theories of the MIMIO system.

2.1

Review

MIMO-OFDM is a key technology for next-generation cellular communications as well as wireless LAN. The basic idea of OFDM is that the transmitting end divides the multipath fading channels into mutually orthogonal sub-channels, converts the high-speed serial data signals into parallel low-speed data streams, and modulates the transmission to each subcarrier. The receiver uses correlation demodulation techniques to separate the orthogonal signals so as to reduce the intercarrier interference (ICI). In an OFDM system, when the bandwidth of each subcarrier is smaller than the relevant bandwidth of the channel, the channel of each subcarrier can be seen as a flat-fading channel, so that intersymbol interference (ISI) can be eliminated, which also facilitates channel equalization. Due to the high-frequency spectrum utilization of OFDM and the strong capability of anti-multipath interference, OFDM technology has also gradually attracted vast research attention from the industrial and academic circles of mobile communications in the past decade. Multiple-input multiple-output (MIMO) is a method for multiplying the capacity of a radio link using multiple antennas in the transmitter and/or receiver to exploit multipath propagation. In MIMO wireless systems, the considerable capacity and increased resilience to fading can be achieved through using the spatial dimension of the channel. H. Sari et al. [1] analyze the performance of OFDMA for the return channel on CATV networks and compare it to the well-known time-division multiple access (TDMA), indicating that OFDMA can offer increased robustness to narrowband interference, impulse noise, and other signal impairments which characterize typical © Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8_2

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CATV networks. S. Barbarossa et al. [2] develop synchronization algorithms for both the downlink and the uplink of quasi-synchronous and asynchronous orthogonal frequency-division multiple access systems. In [3], a coverage and interference analysis based on a realistic OFDMA macro
/femtocell scenario is provided. In [4], a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. In [5], an approach to the design of adaptive observers is presented. It is conceptually simple and computationally efficient, and its global exponential convergence is established for noise-free systems. In [6], a comparison between MIMO wireless communication and MIMO radar is made. Examples are given showing that many traditional radar approaches can be interpreted within a MIMO context.

2.2

Channel Characteristics of Mobile Communication

The channel of mobile communication is different from the fixed cable channel; its performance is affected by the environment of the receiver and transmitter. From the transmitting antenna to the receiving antenna, the signal will pass through a variety of complex propagation paths, including direct, reflection, diffraction, diffraction, and other paths. In addition, the transmitter and the receiver itself will be in motion, so the channel of the mobile communication has the characteristic of time-varying. The total loss of the communication channel is determined by the large-scale path loss (thousands of wavelengths), the mesoscale shadow effect (hundreds of wavelengths), and the small-scale selective fading (tens of wavelengths below the magnitude of the wavelength), respectively. Besides the loss of the wireless multipath channel, the communication system will also be interfered by all kinds of noise and multiple access.

2.2.1

Large-Scale Fading

Large-scale fading, also known as path propagation loss, refers to the loss of radio wave propagation in space. In order to make a quantitative study of large-scale fading, it is necessary to establish a propagation model to predict the average signal intensity within the range of any given transmitter and receiver. Generally speaking, the large-scale propagation model is derived from theoretical deduction and practical measurement. In the mobile communication channel, both theoretical and practical measurements show that the received signal strength decays logarithmically with the distance between the transmit and receive antennas. Therefore, large-scale loss PL can be expressed as a function of distance d.

2.2 Channel Characteristics of Mobile Communication

9

Table 2.1 Path loss parameters Environment Free space Urban honeycomb Building block

Loss n 2 2.6–3.5 4–6

Environment Building line-of-sight propagation Urban honeycomb shadow Factory shielding

d PLðd Þ ¼ PLðd 0 Þ þ 10nlog d0

Loss n 1.6–1.8 3–5 2–3

ð2:1Þ

where n is the path loss parameter and its value depends on the environment of transmission [7], seen in Table 2.1; d0 is a reference distance, which is usually 1 km in a large covered cellular system and 100 m or even smaller in a microcellular system. PLðd 0 Þ can be measured near the far field of the transmitter and can also be calculated according to the assumption of free space at d0. In addition to formula (2.1), which can reflect large-scale decline, there are some specific empirical formulas and models under different environmental conditions, such as Okumura-Hata model [2, 3] and Walfisch-Ikegami model [10].

2.2.2

Mesoscale Decline

When electromagnetic waves are blocked by terrain and tall buildings in space propagation, behind these obstacles, there will be shadows of electromagnetic fields, resulting in changes in the median value of the field strength, known as mesoscale decline, also known as shadow decline. The meso-fading SLs conforms to the lognormal distribution [11] and can be expressed as a function of time t expressed as SLs ðt Þ ¼ xσ ðt Þ where xσ (t) obeys a zero-mean Gaussian distribution of random variables with a standard deviation of σ. Here, the size of σ depends on the frequency of the signal and the characteristics of the obstacle. The high-frequency signal passes through obstacles more easily than the low-frequency signal, and the low-frequency signal has stronger diffraction ability than the high-frequency signal.

2.2.3

Small-Scale Decline

Small-scale fading refers to the tendency of fluctuations in the average level of the reception level in the microscopic range between the transmitter and the receiver. The rate of change of the amplitude of the level of small-scale fading is faster than the shadow fading, and its distribution can be modeled as Rayleigh distribution, Rice

10

2 Overview of OFDMA and MIMO Systems

distribution, and Nakagami distribution [12]. Small-scale decline caused by the following factors: • Multipath effect: When multiple copies of the same signal arrive at the receiver through different paths, the arrival time of each path, the strength of the signal, and the phase of the carrier are all different, which will cause mutual interference. Reflected in the frequency domain, the signal has different fading characteristics at different frequency bands. An important parameter related to the multipath effect is the coherence bandwidth, Bc, of the channel, which reflects the fading correlation of different frequency components of the signal. When the bandwidth B of the information is greater than Bc, the transmitted data can be considered to be subject to frequency selective fading; when the bandwidth B of the information is much smaller than Bc, the transmitted data can be considered to be subject to flat fading. • Doppler shift: Due to the motion of objects around the receiver or the receiver, when the receiver and the transmitter move toward each other, the frequency of the received signal will be higher than the transmission frequency f0; if the two sides do reverse movement, the frequency of the received signal will be lower than the transmission frequency. The Doppler frequency shift causes frequency broadening, which is different in time fading characteristics reflected in the time domain. The parameter associated with the Doppler shift is the coherence time Tc of the channel, which reflects the fading correlation experienced by the signal at different time components. If the period T of a single symbol of the signal is much smaller than the coherence time Tc of the channel, the channel can be considered as a slow fading channel; if the single symbol period is much larger than the correlation time Tc of the channel, the channel can be considered as a fast fading channel. • Spatial selectivity: In a multi-antenna system, signals received at different spatial positions and angles are different, thereby exhibiting spatially selective fading characteristics of the signal. The important parameter for measuring spatial selectivity is the coherence distance, Dc, of the channel, which reflects the correlation of the fading of different spatial components of the signal. When the distance between two antennas is greater than Dc, it can be considered that the fading of the signals received by these two antennas is independent of each other.

2.2.4

Channel Interference

In addition to the fading of the wireless channel itself, the data of the mobile communication will also be subject to a variety of interferences from the outside world and the communication node. It can be roughly divided into the following categories. • White noise interference [13]: It refers to noise with a uniform distribution of power spectral density throughout the frequency domain. White noise originates

2.3 Multiuser MIMO

11

mainly from two sources: (1) passive components, such as the noise caused by the molecular Brownian motion of capacitive, inductive, and other circuit devices (Brownian motion is wide enough in the frequency domain, so its spectral characteristics are flat), and (2) active device, from various types of large-scale integrated circuits, caused by the emission of large amounts of electrons under certain excitation conditions. • Partial band interference [14]: The interference bandwidth only exists in one part of the transmission band, including single-tone interference, multi-tone interference, narrowband interference, etc., where single-tone interference and multitone interference can be modeled as the sum of one or more sinusoidal signals, respectively, and narrowband interference can be modeled as a random process obtained by white noise passing through a bandpass filter. Part of the frequency band interference comes from the interference of other communication systems with the same frequency received by the communication system or from human interference. • Partial time interference (pulse interference) [9]: It refers to the presence of interference at part of the time, which may be either a wideband or narrowband spectrum in the spectrum. It can be described by the interference power and the duty cycle of the interference pulse. • Multiple access interference [15]: When multiple users’ signals cannot be isolated from each other in an orthogonal manner, mutual interference between users may be caused, that is, multiple access interference. In TDMA and FDMA systems, different users can be assigned different time slots and frequencies to distinguish users; however, in CDMA systems, users can be assigned orthogonal spreading codes to distinguish users, but in multipath and time-varying channels, signals often cause distortion during the transmission process, thus undermining the orthogonality of the codes. Therefore, in a CDMA code division system, it is extremely important to suppress multiple access interference between multiple users. Therefore, when signals are transmitted in a mobile broadband communication system, it is necessary to overcome the fading and interference caused by bad radio channels and effectively increase the utilization of spectrum in a limited bandwidth.

2.3

Multiuser MIMO

In a MIMO system, each receiving antenna can receive combined signals from all transmitting antennas. Therefore, signal detection in MIMO systems is an extension of the dimension and alphabet range of signal detection techniques for single-input single output (SISO) systems. However, since the size of the extended alphabet increases exponentially with the number of transmit antennas, the vast majority of exhaustive optimal MIMO detection algorithms have high computational complexity and lack practical value such as maximum likelihood (ML) and maximum

12

2 Overview of OFDMA and MIMO Systems

posterior probability (MAP) detection methods. On the other hand, although the traditional linear detection method has low complexity, but because its receive diversity gain is not high, the performance of the optimal detection method drops significantly when the channel matrix autocorrelation is large. Since the constellation of MIMO signals can be considered as a subset of the lattice, the detection problem of MIMO signals can be transformed into the optimization problem of lattice. The lattice reduction (LR) technique can achieve quasi-orthogonal transformation of the channel matrix in polynomial time. The linear detection method based on this method has the same diversity gain as the optimal detection method, and its performance can approach the optimal detection method. As a key technology for MIMO signal detection, the LR method can achieve a balance between performance and complexity and therefore has important practical value. In a multiuser wireless communication system using MIMO technology, since the user’s geographical location and channel gain are all different, it is possible to obtain an additional diversity gain during user access, that is, multiuser diversity gain. Therefore, in a multiuser MIMO system, selecting an appropriate user to access the system under suitable channel conditions is very important for increasing the system throughput, which directly affects the transmission quality of the channel and the quality of service that the access user can obtain. The literature [16] studies the multiuser diversity gain brought about by multiuser access; literature [17] extends multiuser access to MIMO systems; literature [18] analyzes the relationship between user access strategies and system throughput in a MIMO system. In addition, if the user access problem is regarded as an antenna selection problem, the antenna selection technique in [19–21] can be extended to a multiuser MIMO system. For example, considering the time-varying channel of a mobile cellular system, system throughput may be improved by selecting one or more user accesses with the greatest channel gain within a unit of time. As shown in Fig. 2.1, we can select user 1 at time A and users 2 and 3 at time B to improve system performance. Since the multiuser access problem is naturally an NP-hard problem, such a problem can use an exhaustive search method to obtain an optimal solution. However, considering the high complexity of this method in large-scale systems, it is necessary to design some suboptimal solutions with low complexity. On the other hand, multiuser selection and access criterion for the traditional MIMO system mostly assume that receivers adopt an unrealistic maximum likelihood or linear detection method, leading to many problems in their engineering applications. Therefore, recent researchers have begun research on multiuser access strategies for low-complexity, high-performance detection methods. For example, the literature [22] proposes a multiuser selection strategy based on LR signal detection to minimize the BER; literature [23] designs a low-complexity greedy user selection scheme for LR joint signal detection technology. This section will review the MIMO signal detection method first and then introduce the single-user and multiuser selection strategies based on different detection methods.

2.3 Multiuser MIMO

13

Fig. 2.1 Changes in channel gain at different times

2.3.1

Signal Detection Method

In the signal detection of a MIMO system, the maximum likelihood criterion can be used for joint detection of the received combined signal to achieve the best performance. However, this method has high computational complexity for large-scale MIMO systems, so people have considered a variety of suboptimal detection methods with low complexity. However, although the classic detection methods based on linear and serial interference cancellation (SIC) techniques have lower complexity, they cannot obtain a complete receive diversity gain, and the performance deteriorates significantly with increasing signal-to-noise ratio (compared to ML). In a MIMO system, let the transmit signal vector be s, the MIMO channel matrix be H, and the additive white Gaussian noise vector be n, and then the receive signal vector is y ¼ Hs þ n

ð2:2Þ

The LR process can be expressed as G ¼ HU
1 c ¼ Us

ð2:3Þ

14

2 Overview of OFDMA and MIMO Systems

The matrix U is an integer modulo matrix, and the matrix G is a quasi-orthogonal matrix obtained by subtracting the lattice base. So formula (2.2) can be written as y ¼ Gc þ n

ð2:4Þ

The key problem of joint signal detection based on LR is to complete signal detection and decision in the transformed signal space. Since the coefficients of the lattice are continuous integers, the detection based on LR generally depends on the rounding operation, so that the decision has lower complexity than the conventional detection method. Since the LR detection method has the same diversity order as the maximum likelihood detection method, this method is considered as a key technology for achieving high-performance and low-complexity MIMO detection.

2.3.2

Single-User Selection

When there are multiple users in the MIMO system, the user access system with the best channel condition can be selected in one time slot to obtain additional multiuser diversity gain. For a specific MIMO detector, how to select the right user plays an important role in enhancing the multiuser diversity gain. Due to the need to consider the impact of different detection methods on system performance, the actual transmission rate of the system does not have to be close to the channel capacity. Therefore, the system throughput can be used as a more realistic parameter to measure system performance. The user’s transmission rate and error probability are denoted by Rk and Pe,(k), respectively. If the user is selected to access the system, the system throughput can be written as
 T k ¼ Rk 1
Pe, ðkÞ

ð2:5Þ

From the formula (2.5), it can be seen that under the premise of a certain transmission rate, the throughput of the system can be improved by selecting the user with the smallest error probability in one time slot. In addition, considering that the different detection methods of the receiver have different error probabilities, it is necessary to design the corresponding user selection criteria based on the error probability of the actual detection method so as to optimize the system performance. When the receiver uses the maximum likelihood detection method, the estimated value of the transmitted signal can be expressed as ^s ml ¼ argmin ky
Hsk2

ð2:6Þ

s2S

where x represents the set of all transmitted symbol vectors. Pairing error probability can be expressed as

2.3 Multiuser MIMO

15

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
 1 H sð1Þ
sð2Þ 2
 A P sð1Þ ! sð2Þ ¼ Q@ 2N 0 where QðxÞ ¼

R1 x

pffiffiffiffi 1 2 2π e
z =2

ð2:7Þ

dz, d ¼ s(1)
s(2), N0 represents noise power. Then the

user selection criteria under the corresponding maximum likelihood detection can be further derived from this error probability. When the receiver uses a linear detection method, the estimated value of the transmitted signal can be expressed as ^s ¼ Wy

ð2:8Þ

where W represents a linear filter and the pair error probability is 0

1


 kd k B C P sð1Þ ! sð2Þ ¼ Q@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H  ffiA H 2N 0 d H H d 2

ð2:9Þ

The user selection criteria under the corresponding linear detection method can be further derived from this error probability. When the receiver uses an LR-based detection method, the corresponding pair error probability is 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

 λmin GH G kuk2 A P sð1Þ ! sð2Þ  Q@ 2N 0

ð2:10Þ

where u ¼ c(1)
c(2) ¼ U(s(1)
s(2)), λmin(A) represents the smallest eigenvalue of the matrix A. The user selection criteria under the corresponding LR-based detection method can be further derived from this error probability. In the next chapter, we will introduce these selective strategies in more detail.

2.3.3

Multiuser Selection

Since the user selection problem is a permutation and combination problem in a multiuser environment, if the strategy of single-user selection (selecting one user at a time) is directly extended to multiuser selection (selecting multiple users at a time), it will lead to higher computational complexity in large multiuser MIMO systems. For example, when 10 users need to be selected from 100 users (taking into account user 19 100! ordering), it is necessary to calculate up to ð100
10 joint channel Þ! ¼ 6:2816  10 matrices consisting of candidate user sub-channel matrices, and their computational complexity will increase rapidly with the number of candidate users and the number of selected users. In order to reduce the computational complexity, we can use

16

2 Overview of OFDMA and MIMO Systems

greedy algorithm to achieve multiuser selection [24]. The literature [8, 9] studies the greedy selection method that can maximize the throughput; the literature [14] proposes the greedy choice scheme that minimizes the error rate for the detection method. Greedy algorithm is distributed local optimization solution. For example, if 10 user problems are selected from 100 candidate users as described above, if a greedy algorithm is used, the best user can be selected and removed from the candidate users during the first selection, in the second choice we choose from the remaining 99 alternative users, the third time we choose from the remaining 98 alternative users, and so on. In this way, the number of system channel matrices that we need to calculate is 100 + 99 + 98 +    + 91 ¼ 955. Compared with the previous exhaustive method, the computational complexity is greatly reduced. Considering the greedy selection method in multiuser MIMO systems using LR detection, if the LR operation is re-performed for all user sub-channels including the previously selected user each time a new user is added, the corresponding computational complexity will be very high. As shown in the right-hand side of Fig. 2.2, the newly added fourth user needs to re-perform the LR calculations together with the original channel bases of the three previously selected users, resulting in unnecessary computational load. In order to reduce the complexity, it may be considered that the quasi-orthogonal base information of the selected user is retained after each LR, when the user is newly added, the base of the selected user LR and the original base newly added to the user are used for the local LR. Since the purpose of the LR is to quasiorthogonalize the base, using the base information of the LR to perform local LR can effectively reduce the complexity without losing performance. As shown in the lefthand method of Fig. 2.2, the newly added fourth user can perform local LR calculations together with the channel bases of the three previously selected user LRs.

Fig. 2.2 Comparison of multiple-user selection process updating methods in MIMO system based on LR detection

2.4 OFDMA System

17

Another key technology of space-time wireless resource allocation is OFDMA power distribution technology. In the next section, we will briefly introduce OFDMA systems and their working principles.

2.4

OFDMA System

With the increasing demand for wireless broadband services, improving spectrum utilization in limited bandwidth has become a hot topic in the field of communications in recent years. Orthogonal frequency-division multiplexing (OFDM) is recognized as the core technology of next-generation mobile communications. The basic idea of OFDM is that the transmitting end divides the multipath fading channels into mutually orthogonal sub-channels, converts the high-speed serial data signals into parallel low-speed data streams, and modulates the transmission to each subcarrier; the receiving end uses correlation demodulation techniques that separate mutually orthogonal signals, thereby reducing intercarrier interference (ICI) between subcarriers. In an OFDM system, when the bandwidth of each subcarrier is smaller than the relevant bandwidth of the channel, the channel of each subcarrier can be seen as a flat-fading channel, so that intersymbol interference (ISI) can be eliminated. Each subcarrier is a flat-fading channel, which is advantageous for channel equalization. Due to the high-frequency spectrum utilization of OFDM and strong resistance to multipath interference, OFDM technology has gradually taken the attention of the industrial and academic circles of mobile communications in the past decade. Orthogonal frequency-division multiple access (OFDMA) is an OFDM-based multiuser multiple access technology [25]. In OFDMA systems, different users can use different subcarriers, and because the user’s channel status can be seen as independent, the probability that different users are in deep decline on the same subcarrier is very low. In this way, the base station can adopt a flexible radio resource allocation (RRA) policy to allocate suitable subcarriers, power, and rates for different users, thereby further improving the system’s transmission rate, spectrum utilization, and power utilization. Currently, OFDMA and OFDM transmission technologies have been widely used in next-generation mobile broadband communication systems, such as IEEE 802.16 m, 3GPP LTE, and LTE Advanced systems. In OFDMA system, sharing the subcarriers and transmit power effectively and reasonably according to the channel state information (CSI) has important significance for improving the performance of the OFDMA system.

2.4.1

OFDM Principle

The basic principle of OFDM is to transfer high-speed data streams through serialto-parallel conversion and distribute them to several sub-channels (subcarriers) with

18

2 Overview of OFDMA and MIMO Systems

Fig. 2.3 Baseband diagram of OFDM system: (a) analog and (b) digital

lower transmission rates. Because OFDM widens the symbol period on each subcarrier, it can overcome the delay spread caused by the multipath effect of the channel and thus can reduce the interference between symbols. OFDM system block diagram is shown in Fig. 2.3 [25]. Figure 2.3 shows a block diagram of the analog modulation and digital modulation of the OFDM system. In these two kinds of system structures, the input signal is encoded, converted into N ways through serial and parallel, and then separately modulated. For analog-modulated systems, in one OFDM symbol period, the output modulated signal x(t) can be represented as xð t Þ ¼

N
1 X

X ðnÞexpðj2πf n t Þ,

0  t  Ts

ð2:11Þ

n¼0

where X(n) is the symbol of the nth channel after the input signal has undergone a serial-to-parallel change and fn is the center frequency of subcarrier n, which is 1 Y ðnÞ ¼ Ts

Z

Ts

yðt Þexpð
j2πf n t Þdt

ð2:12Þ

0

where y(t) is the received signal. In analog OFDM systems, the subcarrier signals of the transmitter and receiver are generated by a sinusoidal signal generator, and the subcarriers are accurately synchronized when received. For example, when the number of system sub-channels is large, the system becomes very complicated and expensive. In a digital modulation system, the modulated signal output during one OFDM symbol period is equivalent to sampling analog output signal x(t) at the rate of Ts ¼ n, where Ts is the OFDM symbol periods, i.e.,

2.4 OFDMA System

19

  N
1 X mf T s X ðnÞexp j2π n N n¼0   N
1 X 2π ¼ X ðnÞexp j nm , 0  m  N
1 N n¼0

xðmÞ ¼

ð2:13Þ

In the above equation, x(m) is the level of the OFDM symbol at [mTs/N, (m + 1)Ts/ N]. Therefore, the output signal is equivalent to the inverse fast Fourier transform (IFFT) of the input signal. At the receiving end, the signal on the nth subcarrier can be recovered and obtained through a fast Fourier transform (FFT), which is   N
1 1 X 2π Y ð nÞ ¼ yðmÞexp
j nm N s m¼0 N

ð2:14Þ

The digitally modulated OFDM system can be implemented by integrated circuit IFFT/FFT devices, which greatly simplify the signal modulation and demodulation operation process. Therefore, OFDM has been widely used in the field of mobile communications.

2.4.2

Unicast and Multicast OFDMA Systems

OFDMA refers to a multiuser OFDM system. A common method is to allocate one or a group of subcarriers for each user to implement multiuser access. In an OFDMA system, if the frequency response of a channel remains unchanged within one OFDM symbol, the signal Yk,n of the subcarrier n received by the user k after the FFT (or correlation demodulation) at the receiving end can be expressed as the following equation: Y k, n ¼ H k, n X n þ Z k, n

ð2:15Þ

where Xn is the symbol transmitted on the nth subcarrier and Hk,n and Zk,n are the user’s channel gain and noise on nth subcarrier, respectively. In a multipath channel, user k has different gains Hk,n on different subcarriers n, and the channel gains of different users k on the same subcarrier n can be considered as mutually independent, so the probability that all user k is in a deep fade on subcarrier n is very low. In order to improve the performance of the OFDM system as much as possible, the transmitter can select the appropriate subcarrier allocation, power, and rate allocation for the user according to the channel conditions of different users, so as to optimize the utilization of system resources under the conditions that can meet the service QoS.

20

2 Overview of OFDMA and MIMO Systems

Fig. 2.4 OFDMA system: (a) unicast and (b) multicast

In the OFDMA system, the transmission service can be divided into a unicast OFDMA service and a multicast OFDMA service according to the number of users receiving each service transmission, as shown in Fig. 2.4. In the unicast OFDMA service, each subcarrier is allocated to only one user. Therefore, the unicast OFDMA service is applicable to one-to-one services such as customized services. In the multicast OFDMA service, each subcarrier is used to multicast data to multiple users. If a group of users wants to receive the same data, the multicast information is transmitted only once on each link, and only when the link is bifurcated, the information will be copied. Therefore, the multicast method can achieve efficient use of wireless resources. In addition, different resource allocation methods need to be considered for unicast OFDMA service transmission and multicast OFDMA service transmission. At present, the contradiction between the rapid growth of wireless services and the supply and demand of spectrum resources is becoming increasingly prominent. This increasingly escalating contradiction is caused on the one hand by the scarcity characteristics of spectrum resources, but its most fundamental reason is that the current worldwide static spectrum management model and allocation strategy lead to the unbalanced distribution of available spectrum resources and low spectrum utilization. Academic research and application practice show that the dynamic spectrum sharing method based on cognitive radio technology can greatly change the restraint of static resource management and distribution mode on the development of wireless networks and has become a research hotspot in related fields of production, education, and research. Next, let us understand the so-called dynamic spectrum sharing technology based on cognitive radio technology.

References

21

References 1. H. Sari, Y. Levy, G. Karam, An analysis of orthogonal frequency-division multiple access// Global Telecommunications Conference, 1997. GLOBECOM'97., IEEE. IEEE, 1997, 3, pp. 1635–1639 2. S. Barbarossa, M. Pompili, G.B. Giannakis, Channel-independent synchronization of orthogonal frequency division multiple access systems. IEEE J. Select. Areas Commun 20(2), 474–486 (2002) 3. M. Morelli, C.C.J. Kuo, M.O. Pun, Synchronization techniques for orthogonal frequency division multiple access (OFDMA): a tutorial review. Proc. IEEE 95(7), 1394–1427 (2007) 4. D.J. Love, R.W. Heath, T. Strohmer, Grassmannian beamforming for multiple-input multipleoutput wireless systems. IEEE Trans. Inf. Theory 49(10), 2735–2747 (2003) 5. Q. Zhang, Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems. IEEE Trans. Autom. Control 47(3), 525–529 (2002) 6. D.W. Bliss, K.W. Forsythe, Multiple-input multiple-output (MIMO) radar and imaging: degrees of freedom and resolution//Signals, Systems and Computers, 2004. Conference Record of the Thirty-Seventh Asilomar Conference on. IEEE, 2003, 1, pp. 54–59 7. T. Rappaport, Wireless Communications: Principles and Practice, 2nd edn. (Prentice Hall, New Jersey, 2001) 8. Y. Okumura, E. Ohmori, T. Kawano, K. Fukuda, Field strength and its variability in VHF and UHF land-mobile radio service. Rev. Elec. Commun. Lab 16(9–10), 825–873 (1968) 9. M. Hata, Empirical formula for propagation loss in land mobile radio services. IEEE Trans. Veh. Technol. 29(3), 317–325 (1980) 10. J. Walfisch, H.L. Bertoni, A theoretical model of UHF propagation in urban environments. IEEE Trans. Antennas Propag. 36(12), 1788–1796 (1988) 11. W.C.Y. Lee, Mobile Communications Engineering: Theory and Applications (McGraw-Hill, Inc, New York, 1997) 12. J.G. Proakis, Digital Communication, 4th edn. (McGrawHill, New York, 2001) 13. P. Bergmans, A simple converse for broadcast channels with additive white Gaussian noise (corresp.). IEEE Trans. Inf. Theory 20(2), 279–280 (1974) 14. G.J. Saulnier, Suppression of narrowband jammers in a spread-spectrum receiver using transform-domain adaptive filtering. IEEE J. Select. Areas Commun 10(4), 742–749 (1992) 15. S. Verdu, Minimum probability of error for asynchronous Gaussian multiple-access channels. IEEE Trans. Inf. Theory 32(1), 85–96 (1986) 16. R. Knopp, P. Humblet, Information capacity and power control in single-cell multiuser communications. In Proceedings of IEEE International Conference on Communications, Seattle, WA, 1995, pp. 331–335 17. M. Bengtsson, From single-link MIMO to multiuser MIMO. In Proceedings of IEEE Iternational Conference on Acoustics, Speech, and Signal Processing. 2004: 697–700 18. G. Dimic, N. Sidiropoulos, On downlink beamforming with greedy user selection: Performance analysis and a simple new algorithm. IEEE Trans. Signal Process. 53(10), 3857–3868 (2005) 19. R. Nabar, D. Gore, A. Paulraj, Optimal selection and use of transmit antennas in wireless systems In proceedings of International Conference on Telecommunication, Acapulco, Mexico, May 2000 20. R.W. Heath Jr., S. Sandhu, A. Paulraj, Antenna selection for spatial multiplexing systems with linear receivers. IEEE Commun. Lett 5(4), 142–144 (2001) 21. I. Berenguer, X. Wang, MIMO antenna selection with lattice reduction-aided linear receivers. IEEE Trans. Veh. Technol. 53(5), 1289–1302 (2004) 22. T. Yoo, N. Jindal, A. Goldsmith, Multiantenna broadcast channels with limited feedback and user selection. IEEE J. Select. Areas Commun 25(7), 1478–1491 (2007) 23. Z. Shen, R. Chen, J.G. Andrews, R.W. Heath, B.L. Evans, Low complexity user selection algorithms for multiuser MIMO systems with block diagonalization. IEEE Trans. Signal Process. 54(9), 3658–3663 (2006)

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24. A. Gorokhov, D.A. Gore, A.J. Paulraj, Receive antenna selection for MIMO flat-fading channels: theory and algorithms. IEEE Trans. Inf. Theory 49(10), 2687–2696 (2003) 25. 1900.4, IEEE standard for architectural building blocks enabling network-device distributed decision making for optimized radio resource usage in heterogeneous wireless access networks. 2009:C1–119

Chapter 3

Remarks on Resource Allocation

The radio resources of traditional communication systems are defined and distributed in a hierarchical structure. However, due to the space, time, and frequency variations and randomness of next-generation wireless communication systems, the traditional hierarchical allocation methods cannot optimize them. Therefore, people pay more attention to cross-layer wireless resource allocation technologies today, so as to achieve joint scheduling and optimization of system space, time, and frequency resources. As several important wireless resources, the rational use and distribution of users, power, and spectrum resources play an extremely crucial role in ensuring communication quality and improving communication performance. On the other hand, due to the multiple-input multiple-output (MIMO), orthogonal frequency division multiple access (OFDMA), and cognitive radio technologies in cross-layer design research, these technologies will certainly play an increasingly important role in the future of communications systems. In this chapter, we will firstly introduce the dynamic spectrum allocation based on cognitive radio and their application in next-generation communication systems. Then for OFDMA system, we will introduce its resource allocation theory in detail.

3.1

Review

The management of wireless resources plays a crucial role in improving the performance of mobile systems. It is also important to study resource optimization strategies for different systems. Adaptive resource allocation has been regarded as a key technology for providing efficient adoption of the limited power and spectrum in future wireless systems. In [1], K.B. Letaief et al. provide an overview of recent research on dynamic resource allocation, especially for MIMO and OFDM systems. Ying Jun Zhang et al. [2] propose an adaptive resource allocation methodology with low complexity for cellular orthogonal frequency division multiplexing systems in a multiuser environment. For multiuser multiple-input multiple-output © Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8_3

23

24

3 Remarks on Resource Allocation

(MIMO)/orthogonal frequency-division multiplexing systems, an adaptive resourceallocation approach, which jointly adapts subcarrier allocation, power distribution, and bit distribution according to instantaneous channel conditions, is introduced in [3]. Li Ping Qian et al. [4] propose a real-time pricing scheme that reduces the peakto-average load ratio through demand response management in smart grid systems. In [5], an algorithm, MAPEL, which globally converges to a global optimal solution of the WTM problem in the general SINR regime, is presented.

3.2 3.2.1

Dynamic Spectrum Sharing Model and Method Overview Concept

Defined by the IEEE 1900.4 standard of the Institute of Electrical and Electronics Engineers [6], the dynamic spectrum sharing method is a spectrum access process and mechanism that study how to use a negotiated or non-negotiated method to formulate real-time adjustable spectrum usage rules to avoid harmful mutual interference when different radio access networks and radio terminals dynamically access a certain band of the same frequency band. Generally speaking, the frequency bands allocated to the fixed radio access network are determined, but some of them (frequency points) can be shared by different radio networks and radio terminals, such as the TV blank spectrum and opportunistic spectrum holes that are widely concerned in the ISM band and cognitive radio fields, and so on. At present, the study of blank spectrum dynamic sharing method is a hot academic direction in the field of cognitive radio. As shown in Fig. 3.1, through dynamically sharing a new type of spectrum access process and mechanism [7], it can effectively improve the spectrum utilization rate in the current wireless environment and at the same time improve the service quality of radio access network and cognitive terminals. The IEEE 1900.4 standard states that the implementation mechanism of the dynamic spectrum sharing method includes the following four steps [1], as shown in Fig. 3.2: • • • •

Collect scene information. Evaluate network-side spectrum access performance. Develop a wireless resource selection strategy. Evaluate terminal side parameter reconfiguration performance.

To better understand the implementation mechanism of the dynamic spectrum sharing model, Fig. 3.3 describes a general dynamic spectrum sharing model application case [6] and takes this general model as an example to describe the entire mechanism implementation process in detail: first, network reconfigure managers (NRMs) and terminal reconfigure managers (TRMs) in different carrier networks will collect radio access networks (RANs) and the scene information of the terminal, respectively. At the same time, the collected information is exchanged

3.2 Dynamic Spectrum Sharing Model and Method Overview

25

Fig. 3.1 Spectrum hole and dynamic access schematic

Fig. 3.2 Dynamic spectrum sharing mechanism implementation process

between the RANs and the terminal. By analyzing the scenario information, NRM detects that operator A’s frequency band F1 is overused and operator B’s frequency band F2 is not utilized. Therefore, NRM will make new spectrum sharing decisions: part of the base stations in operator A will share the spectrum resources in operator B. Then, the NRM will initiate a request and control the RAN1 and the RAN2 to perform corresponding reconfiguration operations. At the same time, the NRM will formulate a radio resource selection policy and send it to the corresponding terminal reconfiguration manager to guide the terminal to dynamically access the frequency band of the operator B. Through analyzing the received radio resource selection policy and current scene information, TRM will formulate terminal specific spectrum access decisions and perform corresponding reconfiguration operations for the terminal. Finally, NRM and TRM exchange the current spectrum access decision information and corresponding scene information again to better provide reference for the next spectrum access decision in the time-varying scenario.

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3 Remarks on Resource Allocation

Fig. 3.3 Dynamic spectrum sharing common model

Fig. 3.4 Dynamic spectrum sharing method research content

3.2.2

Dynamic Spectrum Sharing Method and Classification

In the wireless communication process, the time-varying characteristic of the wireless channel requires real-time negotiation between cognitive radio users. From this perspective, the dynamic spectrum sharing method needs to include more features similar to the MAC protocol. At the same time, problems such as coexistence between cognitive users and primary users and joint optimization of multidimensional resources also make the cognitive radio network dynamic spectrum sharing method face many challenges. At present, as shown in Fig. 3.4, the research on the dynamic spectrum sharing method of cognitive radio networks mainly focuses on the following three directions [3]: dynamic spectrum sharing architecture, dynamic spectrum allocation behavior, and dynamic spectrum access technology. Next, we will analyze from the perspective of different technical routes. Research on dynamic spectrum sharing architecture can be divided into centralized and distributed sharing architecture. • Centralized shared architecture: In the centralized framework, the spectrum allocation and access procedures are unified by the central entity. Generally, after the demand and network scene distributed perception, the central control entity will obtain the spectrum allocation requirements of each cognitive radio

3.2 Dynamic Spectrum Sharing Model and Method Overview

27

user and the current network state. Then, according to a specific optimization goal, the central entity will construct the spectrum allocation figure in the current scenario and send it to each demand user for access operation. • Distributed shared architecture: In a distributed framework, each radio user performs distributed spectrum allocation and access procedures in a distributed manner based on their perceived (acquired) scenarios and policy information. In general, distributed approaches emphasize the flexibility and simplicity of sharing methods. Compared to distributed architecture, this method does not require complex central control entities, but it imposes higher requirements on cognitive users’ individual perception capabilities and the flexibility of spectrum sharing methods. Research on dynamic spectrum allocation behavior includes cooperative and noncooperative spectrum sharing methods. • Cooperative spectrum sharing method: In this method, the interference measurement information of each node is shared among all nodes to guide the subsequent dynamic spectrum allocation process. At present, more research is done by clustering and sharing local interference information. This method can effectively balance the performance requirements of the control center’s processing capabilities and the cognitive users’ communication overhead, interference coordination capabilities, and other performance parameters under fully centralized and fully distributed mechanisms. • Noncooperative spectrum sharing method: Compared to collaborative approaches, non-collaborative solutions only consider shared solutions from the perspective of a single node. Because there is no comprehensive consideration of the interference to other nodes in the network, this “selfish” sharing scheme reduces the communication overhead of the nodes in practical applications to some extent, but it also reduces the utilization of the spectrum. Dynamic access technologies are classified into overlay spectrum sharing and underlay spectrum sharing technologies. • Filled spectrum sharing technology: In the filled spectrum sharing mode, the spectrum occupied by the primary user and the secondary user does not overlap, and the secondary user communicates by acquiring the spectrum hole of the primary user. Spectrum hole is a resource block composed of three-dimensional parameters such as time, frequency, and space. Usually, spectrum holes can be managed through a centralized architecture, for example, the concept of spectrum pool proposed by Mitola, the spectrum hole database proposed in the IEEE 802.22 TV band, and many more. Of course, the spectrum holes can also be obtained through a mode of local cognitive user cooperative perception and perform dynamic access through a distributed manner. Here, the filled spectrum sharing technology does not limit the transmission power of the secondary cognitive user, and the secondary cognitive user only pays attention to the usage time and location parameters of the spectrum hole.

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3 Remarks on Resource Allocation

• Underlay spectrum sharing technology: The underlay-type spectrum sharing method is applicable to situations in which a primary user and a secondary user share a certain frequency spectrum at the same time. By adopting complex spread spectrum communication technologies, such as ultra-wideband (UWB) technology, and through power control methods, the secondary user’s transmission interference can be maintained at the authorized user’s normal communication level. Compared to the filler-based spectrum sharing method, secondary users can obtain greater spectrum bandwidth. Cognitive radio dynamic spectrum sharing technology is essentially a multiobjective optimization problem, that is, in different application scenarios, according to the dynamic information requirements of the primary and secondary users, how to dynamically optimize the set goals on multidimensional variables such as time, frequency, space, and power. As cognitive radio networks, all participants (including primary and secondary users) have different goals and interests, and their decisionmaking behaviors also affect each other, and there are competition and collaboration, so how to design spectrum using rules and related access mechanisms, coordinating the behavior of all participants and meeting the different interest needs of all participants, becomes a key issue for spectrum sharing. In general, the current main research direction can be classified from the perspective of application scenarios and application objects of cognitive radio dynamic spectrum sharing, as shown in Fig. 3.5. Among them, considering from the perspective of the scenario, according to whether the primary and secondary user networks belong to the same network, they can be divided into intra-network dynamic spectrum sharing method and internetwork dynamic spectrum sharing method. And according to the user’s position in the cognitive network, they can be divided into horizontal dynamic sharing methods and vertical dynamic sharing methods.

Fig. 3.5 Dynamic spectrum sharing method classification and induction

3.2 Dynamic Spectrum Sharing Model and Method Overview

29

Fig. 3.6 Cognitive radio’s intra-network and inter-network dynamic spectrum sharing scenarios

From the application scenario perspective, the scenarios of cognitive radio’s intra-network and inter-network dynamic spectrum sharing are shown in Fig. 3.6. The intra-network dynamic spectrum sharing scenario refers to spectrum allocation among secondary users in a cognitive radio network. Each secondary user accesses the primary user network without causing interference to the normal communication of the primary user. In fact, the primary user-secondary user sharing scenario is a standard spectrum sharing scenario in the cognitive radio spectrum sharing method. The inter-network dynamic spectrum sharing scenario refers to that when the cognitive radio architecture causes multiple systems to have the same coverage area and use partially overlapping frequency bands when laying the network, the spectrum resources between different systems can be scheduled to use each other. The mutual scheduling of such resources includes the sharing of resources between primary users and primary users and also includes the cross-network resource scheduling of secondary users and secondary users in different cognitive systems. From the perspective of the application object, horizontal sharing refers to the spectrum sharing method between non-authorized users. In the network, all users have the same status, and they will have the same spectrum access rights and interests. Usually this part of the sharing method is also called open spectrum sharing or spectrum commons. This method enjoys spectrum resources in a certain spectrum area through public sharing and is therefore particularly applicable to the unlicensed ISM band. At present, many centralized and distributed spectrum sharing strategies have begun to be explored on this model to deal with the spectrum management challenges under this model. Correspondingly, vertical sharing refers to the sharing method of unlicensed users using authorized user spectrum. This is also a key

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3 Remarks on Resource Allocation

technology that is currently focused on in the field of cognitive radio dynamic spectrum management. Generally, in a cognitive radio spectrum sharing model consisting of a primary user and a secondary user, only the primary user can use the licensed spectrum, and the secondary user needs to use the spectrum opportunistically without affecting the performance of the primary user. It is also called an opportunistic spectrum hole. In this hierarchically shared model, two types of more efficient spectrum sharing methods have been proposed, namely, the filled and underlay spectrum sharing techniques introduced in the previous section.

3.2.3

Current Research Hotspot

Since Mitola proposed the concept of cognitive radio in 1999, scholars have tried to use this innovative theory to solve the problem of dynamic management of wireless resources. At present, the academic community has a certain understanding and application of the dynamic spectrum sharing network architecture, dynamic sharing of basic technical routes, methods, and other research content. However, with the deepening of research on cognitive radio’s dynamic spectrum sharing methods, scholars have begun to pay attention to the distance between current research results and practical applications and look forward to closer links between production, learning, and research. In the current dynamic spectrum sharing architecture, whether it is centralized or distributed, many assumptions have been made on the excavation and acquisition of dynamic spectrum. It is assumed that the spectrum hole or white space has been obtained in a possible representation. In fact, spectrum holes with three-dimensional features of time, frequency, and space are closely related to the rules of the primary user’s activities, and the activities of the primary users need to be established based on statistical analysis of large amounts of data. This brings difficulty to real-time capture of spectrum holes. On the other hand, the proposed multi-user spectrum hole joint sensing method has a certain guiding significance for real-time access to available resources, but this method imposes excessive demands on the perceived ability (sensitivity) of cognitive users and the coordination ability (intelligence) among users. From the perspective of practical application, some new topics have received heated attention from the majority of scholars in recent years. For example, the issue concerns the rights and interests of the primary and secondary users before the use of spectrum resources. In particular, the spectrum resources are regarded as a special kind of market commodity, and the corresponding market mechanism is studied. In addition, the topic focuses on the application of artificial intelligence methods in the process of spectrum access, such as improving resource utilization, reducing interference to primary users, and improving secondary user network performance. • The establishment and regulation of market mechanisms in the spectrum sharing policy domain. Establishing a fair and effective spectrum sharing market is an innovative method for cognitive radio resource management. Since most of the

3.3 The Theory of OFDMA System Resource Allocation

31

available spectrum resources have already been allocated to specific application scenarios, different interest groups have been created. This method provides a clear implementation idea for how the wireless resource static management mode can effectively evolve smoothly to a dynamic management mode. The short-term payment (rent) method of resources based on the market mechanism allows the proponents of licensed bands to have certain economic incentives. At the same time, through the market’s rights and interest norms, it also effectively protects the secondary users’ spectrum use rights and regulates their rational frequency behavior. However, how to maximize the social benefits of the owner of the spectrum resources, how to protect the normal communication quality of the primary user, and how to improve the network performance indicators of the secondary users after accessing the network are problems that need to be reconsidered. • The application of artificial intelligence in dynamic spectrum sharing methods. Active perception, autonomous decision-making, and dynamic reconstruction are the main characteristics of cognitive radio technology. The main reason for possessing these characteristics is the application of artificial intelligence in cognitive radio technology [8]. From the implementation process of the dynamic spectrum sharing method, it can also be seen that learning will play a key role in the entire mechanism implementation step. Regardless of spectrum allocation behavior or spectrum access technology, artificial intelligence methods can greatly improve the cognitive ability of secondary users. However, at present, there are few methodological studies in this area. At the same time, it is limited to the application of less than several artificial intelligence methods in the dynamic spectrum sharing method, and it is necessary to find more suitable targeted application methods to improve the spectrum sharing capability of the network.

3.3

The Theory of OFDMA System Resource Allocation

To study OFDMA resource allocation, the mathematical modeling method is mainly used here to abstract the problem domain into mathematical problems and then design the corresponding algorithm to optimize the OFDMA system. The mathematical modeling process for resource allocation to OFDMA systems includes the following: 1. Identify the problem domain for the study: According to the OFDMA system to be optimized, make reasonable and necessary simplifying assumptions, and then determine the scope of the problem to be studied, including resource allocation parameters, resource constraints, service QoS requirements, and system the goal of optimization. 2. Mathematical model construction: In the problem domain of the OFDMA resource allocation to be studied, use mathematical language and symbols to

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3 Remarks on Resource Allocation

describe the internal laws of objects, and establish mathematical models of variables, constants, etc., such as optimized models, differential equation models, graph models, and game theory models. 3. Solving the problem: Solving the problem through the derivation of analytical expressions, numerical iterations, statistical analysis, and other mathematical methods, thus achieving the goal of reducing the space and time complexity required by the algorithm on the basis of ensuring the performance of the algorithm. 4. Algorithm verification: Return to the actual OFDMA resource allocation scenario and use simulation and actual measurement methods to test the rationality and applicability of the algorithm. In the process of mathematical modeling of OFDMA, a variety of mathematical tools can be comprehensively used. The following first describes the optimization theory and related algorithms involved in OFDMA systems.

3.3.1

Optimization Theory

The resource allocation problem of an OFDMA system can be modeled as an optimization problem of (3.1): min f 0 ðxÞ x




subject to

f i ðxÞ  0, i ¼ 1, . . . , m hi ðxÞ ¼ 0, i ¼ 1, . . . , p

ð3:1Þ

where x ¼ (x1, x2, . . . . xL)T is a parameter that needs to be adjusted in the resource allocation. Parameters that can be adjusted in the OFDMA system include subcarrier allocation, subcarrier power, and rate and other resources; f0(x) is an objective function to be optimized by the system. In the OFDMA system, the target of optimization may be set as the user’s maximum rate, minimum service delay, minimum transmission power, etc. fi(x) and hi(x) are constraints of x and form the constraint domain of problem (3.1). In OFDMA resource allocation, some constraints are used to represent system resource constraints, including subcarrier constraints, transmit power constraints, etc. The other part shows the QoS requirements of different services, which can reflect the error rate, delay, rate and other constraints of the service. Equal constraints. The following describes several special optimization problems.

Convex Optimization Problems First, we give the definition of convex set and convex functions [9]:

3.3 The Theory of OFDMA System Resource Allocation

33

Definition 3.1 For set C, if there are any two points x1 , x2 2 C in set A and any given 0  θ  1, there is. θx1 þ ð1  θÞx2 2 C Then C is a convex set. From a geometric point of view, to verify whether a given set is a convex set, you can take two points from this set and check whether all the points on the line segment that end with these two points belong to this set. The convex and concave functions can be further defined by the definition of convex sets: Definition 3.2 Let f(.) be a function defined on convex set C; for any x1 , x2 2 C and 0  θ  1, there is. f ðθx1 þ ð1  θÞx2 Þ  θf ðx1 Þ þ ð1  θÞf ðx2 Þ Then f(.) is a convex function, and f(.) is a concave function. From a geometric point of view, to verify whether a function is a convex function, you can draw a line segment between any two points on the function. If all points of the line segment are above this function, you can think of the function as a convex one. For the optimization problem of formula (3.1), there is the following definition [9]: Definition 3.3 If the constraint fi(x), i ¼ 0, 1, . . . , m is a convex function with respect to x and hi(x), i ¼ 1, 2, . . . , p is a linear function with respect to x, then the problem (3.1) is a convex optimization problem with respect to x. In addition, there is another equivalent definition of the convex optimization problem [9]: Definition 3.4 If f0(x) is a convex function with respect to x, the constraints formed by constraints hi(x) ¼ 0, i ¼ 1, 2, . . . , p and fi(x), i ¼ 0, 1, . . . , m are a convex set, and then the problem (3.1) is a convex optimization problem with respect to x. The convex optimization problem is a problem of polynomial time complexity. Researchers have conducted in-depth research in the field of convex optimization and developed a general solution. Therefore, whether a problem is a convex optimization problem is a watershed problem to solve the problem [9].

Linear Programming Problems The linear programming problem is a special kind of convex optimization problem [9]. It is required that the objective function f0 and the constraints fi and hi are all linear functions related to x, so that the problem (3.1) can be expressed as:

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3 Remarks on Resource Allocation

min cT x þ d x
Gx  h subject to Ax ¼ b

ð3:2Þ

The constraint domain of linear programming forms a polygon, and it can be shown that one vertex of the polygon must correspond to the optimal solution of problem (3.2) [10]; see Fig. 3.7. Therefore, in addition to solving the linear programming problem according to the standard algorithm of the convex optimization problem, the simplex method can also be used to iterate each vertex of the polygon one by one to find the optimal solution [10].

Integer Planning Problems In problem (3.1), if it is specified that every component of x must be an integer, then problem (3.1) is an integer programming problem [11]. Integer programming problems are mostly NP-hard (non-deterministic polynomial hard) problems [11], that is, an exponential-level complexity is needed to obtain the optimal solution to the problem. However, the literature [12] points out that for the integer programming problem that satisfies the polymatroid structure, the greedy algorithm can be used to obtain the optimal solution. The following problem (3.3) is to satisfy the optimization problem of polymatroid structure: min x1 , ..., xL

L X

f 0, i ðxi Þ i¼1
X

subject to

 V ðS Þ 8S 2 A x1 , . . . , xL 2 N x i2S i

ð3:3Þ

where A is the power set of set {1, 2, . . . , L}, f0,i() is a convex function, N is a set of all nonnegative integers, and V() is a function defined in a subset of set {1, 2, . . . , L} and satisfies the following properties: 1. Normalized: V(ϕ) ¼ 0. 2. Non-decreasing: If S  T  {1, 2, . . . , L}, then V(S)  V(T ). 3. Submodular: For all S, T  {1, 2, . . . , L}, there is V(S) + V (T )  (S [ T) + (S \ T ).

3.3.2

Optimal Solution Algorithm

After modeling the resource allocation problem as a problem (3.1), some algorithms can be used to obtain the optimal solution. In OFDMA resource allocation,

3.3 The Theory of OFDMA System Resource Allocation

35

Fig. 3.7 The optimal solution of the linear programming problem

commonly used algorithms for solving optimal solutions are gradient method, interior point method, exhaustive method, and branch-and-bound method. The first two algorithms are mostly used to find the optimal solution of the convex optimization problem, while the latter two algorithms can solve the optimal solution of the partial non-convex optimization problem.

The KKT (Karush-Kuhn-Tucker) Point for Optimization Problems The KKT point gives the conditions for the optimality of the problem (3.1). Let L be the Lagrangian function of problem (3.1), i.e., Lðx; λ; μÞ ¼ f 0 ðxÞ þ

m X i¼1

λ i f i ð xÞ þ

p X

μi hi ðxÞ

ð3:4Þ

i¼1

where λ ¼ (λ1, λ2, . . . , λm)T, μ ¼ (μ1, μ2, .. . ., μp)T, λi are the Lagrange multipliers of the i-th inequality constraint, μj is the Lagrange multiplier of the j-th equality constraint, and then the KKT point can be defined as. Definition 3.5 If x ¼ (x1, x2, . . . , xL)T, λ ¼ (λ1, λ2, . . . , λm)T, μ ¼ (μ1, μ2, .. . ., μp)T meet the following four conditions: 1. x is a feasible solution to problem (3.1), that is to satisfy fi(x)  0, i ¼ 1, 2, . . . , m and hj(x) ¼ 0, j ¼ 1, 2, . . . , p. 2. The Lagrange multiplier of inequality is nonnegative λi  0. 3. Complementary relaxation conditions, λifi(x) ¼ 0 8 i. 4. The partial derivative of the Lagrange function is 0, ∇xL ¼ 0.

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3 Remarks on Resource Allocation

Then (xT, λT, μT) is called the KKT point of problem (3.1). For the convex optimization problem, the KKT point is the optimal solution to the problem, which exists and is unique [18]. Therefore, the optimal solution to the convex optimization problem comes down to the solution of its KKT point. For non-convex optimization problems, it is possible that the system of equations in Definition 3.5 has no solution. Even if there is a solution, it is mostly a suboptimal solution to problem (3.1). The KKT point for solving the convex optimization problem can be directly solved by the equations satisfied by the KKT point, or it can be solved by a numerical iterative algorithm. The commonly used algorithms include the gradient method and the interior point method.

Gradient Method Gradient method is a numerical iterative algorithm; each step iteration updates the value of the parameter in the direction of the negative gradient of the target function, that is [13]:    xðkþ1Þ ¼ P xðkÞ  αk ∇x f 0 xðkÞ where x(k) is the value of the k-th iteration of the variable x, αk is the iteration step of the k-th step, and PðyÞ is the projection of y within the constraint domain. The gradient method is generally slower than the interior point method, but the algorithm is robust and can resist noise interference during iteration without the need for constraints and the second-order derivative of the objective function.

Interior Point Method The basic idea of the interior point method is to do a second-order approximation of the point where the iteration occurs, find the point where the minimum value of the quadratic function is, and then proceed to the next iteration [18]. The interior point method first uses the inequality constraint as a penalty function, adds it to the objective function, and converts the problem that is required to be solved to a problem that only contains equality constraints, i.e., min ϕðxÞ ¼ f 0 ðxÞ  x

m 1X logðf i ðxÞÞ t i¼1

ð3:5Þ

subject to Ax¼b Among them, because the equality constraint of the convex optimization problem is a linear constraint, here hi(x) ¼ 0, i ¼ 1, 2, . . . , p is expressed as a linear constraint Ax ¼ b; t is a penalty factor, which represents the proportion of the penalty term.

3.3 The Theory of OFDMA System Resource Allocation

37

When t ! 1, it is equivalent to solving the problem (3.1) without considering inequality constraints, and when t > 0, you can limit the solution of problem (3.5) to the constraint domain of problem (3.1). When solving the problem (3.5), we should first perform a second-order approximation on the point x where the current k-th iteration is located and then find the minimum.    T   1 Δx ¼ arg min ϕ xðkÞ þ v þ ∇ϕ xðkÞ v þ vT ∇2 ϕ xðkÞ v V 2  subject to A xðkÞ þ v ¼ b

ð3:6Þ

It can be shown that Δx satisfies the following equations: 

  ∇2 ϕ xðkÞ A

AT 0



Δx w



 ¼

  ∇ϕ xðkÞ 0

ð3:7Þ

In each iteration, the variable for the next iteration can be calculated as xðkþ1Þ ¼ xðkÞ þ αΔx

ð3:8Þ

where α is the search step in the direction of the iteration. In this way, given each t, the solution to (3.5) can be obtained iteratively. If we increase the value of t gradually, we can make the optimal solution of the problem (3.5) and the optimal solution of the original problem (3.1) coincide. The algorithm of the interior point method is shown in Table 3.1: The overall efficiency of the interior point method is higher than that of the gradient method, but since the interior point method needs to invert the matrix when iterating, it is greatly affected by the iteration calculation error, and is not as stable as the gradient method.

Enumeration Algorithm When the value of the variable x is a countable finite set, enumerating all possible values of the variable x and verifying the value of the objective function f0(x) one by one, and then the minimum value of the problem (3.1) is obtained in all possible Table 3.1 Interior point algorithm 1. Given the initial value of x and t and the termination condition of the iteration ε m 2. while do t < ε 3. Solving the problem (3.5) iteratively according to Eqs. (3.7) and (3.8) until convergence 4. Increase the penalty factor t ¼ ηt, where η > 1 is a constant 5. End while 6. Return x

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3 Remarks on Resource Allocation

Table 3.2 Branch-and-bound method 1. Repeat 2. Divide the constrained domain of x into multiple subsets to estimate the upper and lower bounds of the optimal solution for the problem (3.1) in each subset 3. According to the upper and lower bounds of the optimal solution of the problem in each subset, the upper bound fand lower bound f of the optimal solution of the problem (3.1) are determined in the entire constraint domain 4. Until f  f < ε 5. Return the x that makes the upper and lower bounds of the optimal solution close

cases. The enumeration algorithm is suitable for cases where the range of x values is small. However, in general, the range of x values increases exponentially with the increase of the x dimension, so the time complexity of the enumeration algorithm is exponential.

Branch-and-Bound Method The idea of the branch-and-bound method is to divide the set of x satisfying the constraint conditions into multiple subsets, estimate the upper and lower bounds of the problem in each subset, gradually narrow the search range, and determine the final solution of x. The algorithm is shown in Table 3.2. The key to the branch-and-bound method is to estimate the upper and lower bounds of the optimization problem. It is easy to know that for the problem (3.1), it has the following two properties: 1. For any x satisfying the constraints fi(x)  0, i ¼ 1, 2, . . . , m and hi(x) ¼ 0, i ¼ 1, 2, . . . , p, f(x) is an upper bound of the optimal solution (3.1) to the original problem. 2. Let the constraint domain of the problem (3.1) be C; if there is a set C that satisfies  the solution of the following problem can give a lower bound of the C  C, optimal solution of the problem (3.1): min f 0 ðxÞ x

subject to x 2 C

ð3:9Þ

So there are ways to estimate the upper and lower bounds of an optimization problem: 1. Upper Bound Estimation: Finding any point x within the set that satisfies the constraint condition, f0(x) is the upper bound of the optimization problem; x can also be obtained by the suboptimal algorithm given above. 2. Lower Bound Estimation: Relax the constraint of x and solve the optimization problem after relaxation constraint, that is, the lower bound of the problem can be

3.3 The Theory of OFDMA System Resource Allocation

39

obtained; at the same time, the lower bound of the solution can also be obtained by solving the dual problem of the original problem. In the worst case, the branch-and-bound algorithm degenerates to search for all possible values. In this case, the branch-and-bound method has the same time complexity as the enumeration algorithm. However, in many cases, the branchand-bound method still has much less time complexity than the enumeration algorithm.

3.3.3

Suboptimal Solution Algorithm

The resource allocation problems of OFDMA systems are mostly non-convex optimization problems, and solving the optimal solution requires higher time complexity. Therefore, some suboptimal algorithms need to be considered in order to obtain a suboptimal solution that is close to the optimal solution. The following introduces several suboptimal solution algorithms commonly used in OFDMA systems.

Relaxation Constraint Algorithm The algorithm relaxes the constraints of the original problem (3.1) [14]. In some cases, the complex non-convex optimization problem can be converted into a simple convex optimization problem [14]. In this way, the algorithm of the convex optimization problem can be used to solve the new problem after the relaxation constraint, and then the final optimal solution is projected into the constraint domain of the original problem, so that a suboptimal solution is obtained. According to the analysis of the branch-and-bound method, the optimal solution obtained after the variable is relaxed gives a lower bound on the original problem solution, and the solution obtained is an upper bound of the original problem solution. Therefore, this algorithm can be used to measure the distance between the suboptimal solution and the optimal solution.

Duality Algorithm The dual problem of the original question (3.1) can be defined as [18]: max gðλ; μÞ λ, μ subject to λi  0 8i

ð3:10Þ

The dual function is obtained by finding the extremum of the Lagrangian function for the optimization variable, which is

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3 Remarks on Resource Allocation

gðλ; μÞ ¼ inf Lðx; λ; μÞ x

ð3:11Þ

The problem (3.10) has the following properties [18]: Proposition 3.1 The optimal solution of problem (3.10) is the lower bound of the original problem (3.1). And when the original problem (3.1) is a convex optimization problem, there is an x that satisfies the following conditions: f i ðxÞ < 0 i ¼ 1,2, . . . , m hi ðxÞ ¼ 0 i ¼ 1,2, . . . , p

ð3:12Þ

Then the optimal solution of the original problem (3.1) and its dual problem (3.11) is the same. For the convex optimization problem, the condition of Eq. (3.12) is easily satisfied, so in addition to using the optimal method described above to obtain the optimal solution, the solution to its dual problem can also be used to obtain the optimal solution. For non-convex optimization problems, although the complexity of the original problem is high, the dual problem can be solved with a lower complexity algorithm. This is due to the dual nature of the dual problem (3.10) [18]. Proposition 3.2 Problem (3.10) is a convex optimization problem. However, according to the proposition 3.1, in the non-convex optimization problem, the solution of the dual problem can only give the lower bound of the original problem (3.1), i.e., the x obtained by solving (3.10) and (3.11) may not satisfy the problem (3.1). Therefore, the x obtained by solving the dual problem needs to be projected into the constraint domain of problem (3.1), and a suboptimal solution to the original problem is obtained. In summary, the algorithm for the dual problem is shown in Table 3.3: Similar to the relaxation constraint algorithm, the lower bound of the optimal solution obtained by the dual algorithm can be used to measure the distance between the obtained suboptimal solution and the optimal solution.

Alternate Algorithm The basic idea of the alternative algorithm is to consider the optimization variable x  T as two sets of variables x¼ x1T ; x2T , alternately fixing x1 and x2 to solve the Table 3.3 Dual algorithm 1. Convert the original question (3.1) to its dual question (3.10) 2. Solve the problem (3.10) and the corresponding optimization variables (3.11) 3. The x obtained by solving the dual problem is projected into the constraint domain of the original problem (3.1), resulting in x 4. Return x

3.3 The Theory of OFDMA System Resource Allocation

41

Table 3.4 Alternate algorithm 1. Initialize x1 , valLast ¼  1 2. Repeat 3. Record the value of the last iteration, valLast ¼ valCurrent, x1, last ¼ x1 4. Fix x1 ¼ x1, last, solving optimization problem: x2 ¼ arg min f 0 ðx1, last ; x2 Þ x
2 f i ðx1, last ; x2 Þ  0, i ¼ 1,2, . . . , m subject to hi ðx1, last ; x2 Þ ¼ 0, i ¼ 1,2, . . . , p 5. Fix x2 ¼ x2 , solving:   x1 ¼ arg min f 0 x1, x2 x
1   f i x1, x2   0, i ¼ 1,2, . . . , m subject to  hi x1, x2 ¼ 0, i ¼ 1,2, . . . , p   and update the solution valCurrent ¼ f 0 x1 ; x2 6. Until |valCurrent    valLast | < ε x1 7. Return x ¼ x2

optimization problems on x1 and x2 [15]. Alternative algorithms can simplify the optimization problem of high dimensionality and high complexity into the optimization problem of low dimension and low complexity and even the convex optimization problem, so that the algorithm with low time complexity can be used to solve the problem. The alternate algorithm is shown in Table 3.4: In OFDMA systems, the variable x can be used to optimize power, rate, modulation order, and subcarrier allocation. In addition, in the OFDMA resource allocation, since most of the subcarriers are assumed to have equal power and equal bit allocation during initialization, the algorithm in Table 3.4 requires only one round of iteration to obtain a suboptimal solution closer to the optimal solution [16]. Therefore, the alternative algorithm is also called two-step algorithm in OFDMA system [16].

Greedy Algorithm The main idea of the greedy algorithm is to divide the solution of the problem (3.1) into multiple steps. At each step, it is the best choice at present. The greedy algorithm does not take into account the overall optimality. What is obtained is only a partial optimal solution in a certain sense. The literature [12] proves that if the integer programming problem satisfies the structure of polymatroid, this problem can be solved by greedy algorithm. The literature [17] proves, for a single-user OFDMA system, under the constraint of total power, the adaptive modulation algorithm that maximizes the total rate of subcarriers can use the greedy algorithm to obtain the optimal solution. However, for most problems in OFDMA systems, greedy algorithms can only obtain suboptimal solutions, but since greedy algorithms can often simplify the problem (3.1), they are also often used in OFDMA resource allocation.

42

3 Remarks on Resource Allocation

References 1. K.B. Letaief, Y.J. Zhang, Dynamic multiuser resource allocation and adaptation for wireless systems. IEEE Wirel. Commun. 13(4), 38–47 (2006) 2. C.Y. Wong, R.S. Cheng, K.B. Lataief, et al., Multiuser OFDM with adaptive subcarrier, bit, and power allocation. IEEE J. Sel. Areas Commun 17(10), 1747–1758 (1999) 3. Y.J. Zhang, K.B. Letaief, An efficient resource-allocation scheme for spatial multiuser access in MIMO/OFDM systems. IEEE Trans. Commun. 53(1), 107–116 (2005) 4. L.P. Qian, Y.J.A. Zhang, J. Huang, et al., Demand response management via real-time electricity price control in smart grids. IEEE J. Sel. Areas Commun 31(7), 1268–1280 (2013) 5. L.P. Qian, Y.J. Zhang, J. Huang, MAPEL: achieving global optimality for a non-convex wireless power control problem. IEEE Trans. Wirel. Commun. 8(3), 1553–1563 (2009) 6. IEEE standard for architectural building blocks enabling network-device distributed decision making for optimized radio resource usage in heterogeneous wireless access networks, IEEE Standard 1900.4-2009, 2009 7. I.F. Akyildiz, W.-Y. Lee, M.C. Vuran, S. Mohanty, Next generation/dynamic spectrum access/ cognitive radio wireless networks: a survey. Comput. Netw. 50, 2127–2159 (2006) 8. F. Carolina, M. Mihael, Trends in the development of communication networks: cognitive networks. Comput. Netw. (Elsevier) 53, 1354–1376 (2009) 9. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004) 10. E. Lawler, Combinatorial Optimization: Networks and Matroids (Dover Publications, New York, 2001) 11. C.H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity (Dover Publications, Mineola, 1998) 12. A. Federgruen, H. Groenevelt, The greedy procedure for resource allocation problems: necessary and sufficient conditions for optimality. Oper. Res. 34(6), 909–918 (1986) 13. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Programming (Cambridge University Press, Cambridge, 1992) 14. L.M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967) 15. Qifa Ke, T. Kanade, Robust L1 norm factorization in the presence of outliers and missing data by alternative convex programming. Proc. IEEE CVPR ‘05. San Diego, CA, USA, 2005, vol. 1, pp. 739–746 16. I. Kim, I.-S. Park, Y.H. Lee, Use of linear programming for dynamic subcarrier and bit allocation in multiuser OFDM. IEEE Trans. Veh. Technol. 55(4), 1195–1207 (2006) 17. J. Campello, Optimal discrete bit loading for multicarrier modulation systems. Proc. IEEE ISIT ‘98. Cambridge, MA, USA, 1998, p. 193

Chapter 4

Resource Allocation for OFDMA Systems

4.1 4.1.1

Introduction Resource Allocation for Multicast OFDMA Systems

Resource allocation for orthogonal frequency-division multiple access (OFDMA) systems is a challenging issue for next-generation wireless communications. In OFDMA systems, adaptive resource allocations can considerably improve the system performance [1]. Previous research works mainly focused on OFDMA systems for unicast transmissions, wherein each subcarrier is assigned to one user exclusively [1–4]. However, many emerging wireless applications, such as mobile TV and video conference, take the form of multicast transmissions. As shown in Fig. 4.1, in OFDMA systems with multicast services, each subcarrier is assigned to a group of users. Thus, resource allocation for the multicast OFDMA system follows a different approach compared with that for the unicast case. Analyzing OFDMA systems to support multicast services turns to be critical. In recent years, resource allocation for multicast OFDMA systems has been studied under various contexts. The authors in [5] proposed resource allocation algorithms for OFDMA systems with multiple multicast services. Power loading strategies for single-user multicast OFDM systems in the presence of relay nodes were investigated in [6]. Resource allocation algorithms with fair considerations were developed by Ngo et al. [7]. However, the works in [5–7] assumed that the transmission on each subcarrier is based on the worst-channel user. Although this approach enables the QoS requirement of all users to be satisfied, it causes inefficient use of radio spectrums. One approach to this problem is to employ multiple-layer coding. Controlling the number of layers can provide the degree of freedom to satisfy different users [8, 9]. Another approach is to use erasure coding for downlink transmissions. With this technology, the users only need to receive data from subcarriers with good channel conditions [10]. However, both approaches require source coding, which still places the computation burden at the transmitter. © Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8_4

43

44

4 Resource Allocation for OFDMA Systems

User k1,1

Subcarriers User k1,2

User k1,g

User k2,1

User k2,2

BS

User k2,L

User k3,1

User k3,2

User k3,Q

Fig. 4.1 Multicast orthogonal frequency division multiple access transmissions

Moreover, the aforementioned works assume that users decode OFDM symbols independently from different subcarriers. However, if the transmitter can perform error control schemes across the subcarriers, the system performance can be further improved.

4.1.2

Resource Allocation for MIMO-OFDMA Systems

Multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) has been adopted as the technology for several broadband wireless standards, including the IEEE 802.11 ac, IEEE 802.16e, and third-generation partnership project (3GPP) long-term evolution (LTE). Moreover, MIMO-OFDM technology plays a vital role in the fourth-generation wireless communication systems. It has been suggested that multiuser OFDM systems employ adaptive subcarrier allocation to effectively utilize a given bandwidth. For multiuser OFDM systems, two nonlinear optimization problems are extensively investigated: the maximization of the data rate/capacity and the minimization of the overall transmission power. Similarly, many dynamic resource allocation algorithms for multiuser MIMOOFDM systems are used to find the solution to maximize the spectral efficiency [11, 12] or to minimize the overall transmit power [13, 14]. The max-min singular values criterion is proposed to allocate subcarriers [11] by optimizing the subcarrier allocation and antenna selection. In [13], the user with the max-max eigenvalue is selected to minimize the overall transmit power while maintaining a target bit error

4.1 Introduction

45

rate (BER). In optimizing radio resources, although it might be desirable to use the achievable rate as an objective function, there are a few issues in practice. In downlink transmission, receivers at users may not be ideal, and their performance strongly depends on detectors that are actually employed. Thus, we may consider to use the BER as a performance criterion in optimizing radio resources for a given detector. The BER-optimized power allocation problem is addressed for nonMIMO-OFDM systems in [15–17]. In [15], a power allocation scheme for OFDM systems is proposed to minimize the aggregate BER. In [16], an approximate minimum BER power allocation algorithm is proposed for MIMO spatial multiplexing systems. However, to the best of our knowledge, the BER-optimized resource allocation problem for MIMO-OFDM systems has not been studied yet in the literature. As stated above, the BER is depending on the detection methods in MIMOOFDM systems. In this paper, we investigate the subcarrier and power allocation for multiuser MIMO-OFDM systems when a receiver constraint is imposed. For various MIMO detectors, we propose different allocation algorithms to minimize the average BER. The subcarrier and power allocation problem is formulated as a combinatorial optimization problem. We also propose two-step suboptimal algorithms which separate the subcarrier and power allocation. Once the subcarrier allocation is determined, the optimal power allocation can be solved via the method of Lagrange multipliers. For the case of limited computing power, suboptimal power allocation algorithms are also proposed.

4.1.3

Resource Allocation for Energy Efficiency in OFDMA Systems

With the exponential growth of reliable traffic rate in modern wireless communications, energy consumption and environmental problems have attracted more attention recently. Hence, energy-efficient wireless communications are becoming an inevitable trend [18]. Since OFDMA has been adopted in mobile broadband systems as its high efficiency and stable performance against broadband channel fading [19], energyefficient design for OFDMA networks is very important. Previous research on energy-efficient OFDMA networks has mainly focused on downlink scenarios [20–23]. However, it is also urgent to achieve high EE on uplink scenarios due to the limited battery capacity of user equipment (UE). In [24], a suboptimal uplink resource allocation algorithm was proposed to maximize the minimum EE among all users. In [25], the authors developed suboptimal energy-efficient schemes for uplink OFDMA system by considering time-averaged bits-per-Joule metrics. In [26], two suboptimal algorithms were proposed to maximize system EE under quality of service (QoS) constraints. Similarly, in [27], the authors considered the scenario in which the access point serves a subset of UEs when available resources cannot support all of the UEs’ QoS requirements. In [28], the energy-efficient resource

46

4 Resource Allocation for OFDMA Systems

allocation scheme was based on proportion fairness in uplink OFDMA systems. In general, uplink energy-efficient resource allocation was considered less tractable since its individual UEs’ power and QoS constraints. The works in [24] demonstrated the existence of a global-energy efficient solution with a given subcarrier assignment and developed an optimal power allocation algorithm. Therefore, the key issue of the energy-efficient resource allocation lies in subcarrier assignment. However, most existing subcarrier assignment algorithms had high computational complexity, and there was only limited work on subcarrier assignment in uplink OFDMA systems.

4.2

Adaptive Resource Allocation for Multicast OFDMA Systems with Guaranteed BER and Rate

4.2.1

System Model and Problem Formulation

Consider an OFDMA downlink system with a multicast session. The system structure is depicted in Fig. 4.2. In this system, the BS transmits each OFDM symbol to K users over N subcarriers. The OFDM symbol is assumed to have a fixed length equal to C bits, and the instantaneous channel gains are assumed to be known at the BS. According to the corresponding channel information, the BS decides the number of bits to carry on each subcarrier n, cn, and the target SER of each user k on each subcarrier n, SERk,n. Depending on cn and SERk,n, the BS will apply a corresponding modulation scheme and transmit power level. We assume that the multilevel modulation, such as M-ary quadrature amplitude modulation or M-ary phase shift keying, with Gray-mapping rule is employed. Note that traditionally, multicast OFDM provides the same modulation on all subcarriers. However, in this paper, Transmitted symbol, C bits Subcarrier 1 Subcarrier N ...

BS

Error control (bit interleaving)

OFDM modulation

S/P

User k OFDM demodulation

Decoding (bit deinterleaving)

P/S

Received symbol, C bits Subcarrier 1 SERk,1

Fig. 4.2 System model

...

Subcarrier N SERk,N

4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with. . .

47

we assume that modulations on subcarriers are different. Although this requires additional complexity at the receiver as well as additional overhead to communicate the modulation configuration from the transmitter to the receivers, this scheme enables the transmitter to reduce transmit power under a given target average BER. Each user experiences a different channel condition in the frequency-selective channel. Denote by Hk,n the instantaneous channel gain of the user on the k‐th subcarrier. The transmit power must satisfy pn


f ðcn ; SERk, n Þ jH k, n j2

8k, n

ð4:1Þ

where pn is the transmit power on the n‐th subcarrier and f(cn, SERk,n) is the required received power for reliable reception of cn bits given a target SER of user k, SERk,n. The equation in (4.1) implies that the minimum transmit power on subcarrier n is pn ¼ max

f ðcn ; SERk, n Þ

k

jH k, n j2

8n

ð4:2Þ

In the case of Gray mapping, two adjacent symbols in the constellation graph differ by only 1 bit. Because the most probable error is the erroneous selection of an adjacent symbol, the average BER at the k‐th user, BERk, can be approximated by BERk 

N 1 X SERk, n C n¼1

ð4:3Þ

Here, the equation in (4.3) holds only if all SER are small. Consider the case where the error control scheme across subcarriers, such as bit interleaving, is performed at the BS. With this technology, each user k may have different SER on different subcarriers with average BER requirement given by BERk  BER

8k

ð4:4Þ

With the previous notations, to minimize the total transmit power, we formulate the optimization problem as N X f ðcn ; SERk, n Þ min max SERk, n , cn n¼1 k jH k, n j2 8 1 XN > SERk, n  BER 8k > >C < P N n¼1 subject to n¼1 cn ¼ C > > c > n : 2 f0; 1; 2; . . .g SERk, n
0 8k, n

pT ¼

ð4:5Þ

48

4 Resource Allocation for OFDMA Systems

where pT denotes the optimal transmit power and the first and second constraints ensure users’ BER and data rate requirement to be satisfied, respectively. Here, we assume that f(cn, SERk,n) is a monotonically decreasing convex function of SERk,n. For example, in the case of M-ary quadrature amplitude modulation, f(cn, SERk,n) is represented by [11].   2 2σ 2 1 1 SERk, n f ðcn ; SERk, n Þ ¼ Q ð 2c n  1Þ 4 3

ð4:6Þ

where σ 2 is the variance of the noise and Q1(x) is the inverse function of Q(x), 1 QðxÞ ¼ pffiffiffiffiffi 2π

Z

1

et

2

=2

dt

x

It can be seen that f(cn, SERk,n) given by (4.6) satisfies the aforementioned assumption.

4.2.2

Optimal Algorithm

In this section, we propose a full searching algorithm to obtain an optimal solution to the problem in (4.5). Note that given the number of bits cn, the problem in (4.5) becomes N X f ðcn ; SERk, n Þ pT ðc1 ; . . . ; cN Þ ¼ min max k SER jH k, n j2 ( X k, n n¼1 1 N SERk, n ¼ BER 8k subject to C n¼1 SERk, n
0 8k, n

ð4:7Þ

where pT(c1, . . . , cN) denotes the optimal transmit power under the bit c1, . . . , cN. Theorem 4.1 Suppose that f(cn, SERk,n) is a monotonically decreasing convex function of SERk,n.The problem in (4.7) is a convex optimization problem. Proof The convex optimization problem is defined as minimizing a convex objective function subject to a convex set [12]. Because the constraints in (4.7) are linear in variable SERk,n, the feasible set is convex. It remains to prove that the objective function in (4.7) is convex. First, because the function f(cn, SERk,n) is a convex function of SERk,n, f(cn, SERk,n)/|Hk,n|2 is a convex function of SERk,n because the nonnegative scaling does not change the convexity property [12]. Second, because the pointwise supremum preserves the convexity [12], maxkf(cn, SERk,n)/|Hk,n|2 is convex in SERk,n. Because the objective function in (4.7) is a summation of a set of convex functions, the problem in (4.7) is a convex optimization problem.

4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with. . .

49

Therefore, the problem in (4.7) can be optimally solved in polynomial time using standard convex optimization methods. The algorithm proposed in Sect. 4.2.3 can also be employed to solve the problem in (4.7). It turns out that pT(c1, . . . , cN) gives an upper bound on the optimal transmit power pT . To obtain the optimal transmit power pT , we have to search all possible values of cn that satisfy ∑ncn ¼ C. If we solve the problem in (4.6) for each given {cn}, we must solve (C + N  1) ! /(N  1) ! /C! convex optimization problems. To reduce the complexity, we can first estimate the lower bound on pT ðc1 ; . . . ; cN Þ. If we find this lower bound larger than the known upper bound on pT , we do not need to continue to solve the problem in (4.6), because the solution cannot improve the upper bound on the original problem. From the first constraint in (4.5), we have SERk, n  C  BER

ð4:8Þ

Because f(cn, SERk,n) is monotonically decreasing in SERk,n, we have f ðcn ; SERk, n Þ jH k, n j2




  f cn ; C  BER jH k, n j2

ð4:9Þ

Thus, pT ðc1 ; . . . ; cN Þ≜ min SERk, n

N X n¼1

max k

N X

f ðcn ; SERk, n Þ jH k, n j2   f cn ; C  BER


min max SERk, n n¼1 k jH k , n j2 N  X  1 ¼ f cn ; C  BER max k jH k , n j2 n¼1   N X f cn ; C  BER ¼ αn n¼1 ¼ pT ð c 1 ; . . . ; c N Þ

ð4:10Þ

where αn ¼ mink|Hk,n|2 is the lowest channel gain among the K users on the n‐th subcarrier and   N X f cn ; C  BER pT ðc1 ; . . . ; cN Þ ¼ αn n¼1

ð4:11Þ

Here, it can be noted that pT ðc1 ; . . . ; cN Þ gives a lower bound on pT(c1, . . . , cN). Denote by pT the upper bound on the original problem in (4.5). We describe the optimal algorithm in Table 4.1.

50

4 Resource Allocation for OFDMA Systems

Table 4.1 System specification

Parameter Mean channel gain Bandwidth Carrier frequency Vehicular speed

Value 1 1.92 MHz 2.6GHz 50 km/h

Algorithm 4.1 Optimal Algorithm 1: Initialize pT ; 2: for all c1,. . .,cN such that ∑ncn ¼ C and cn 2 {0, 1, 2, . . .} do 3: Calculate pT ðc1 ; . . . ; cN Þ according to (4.11) 4: 5: 6: 7: 8: 9: 10: 11:

if pT ðc1 ; . . . ; cN Þ  pT then Solve the optimal problem in (4.7) to obtain pT(c1,. . ., cN) if pT ðc! ; . . . ; cN Þ  pT then Update pT ¼ pT ðc1 ; . . . ; cN Þ end if end if end for return pT

Complexity In the worst case, the optimal algorithm has to solve the optimization problem in (4.7) for each c1, . . . , cN that satisfies ∑ncn ¼ C. In Subsection 4.2.3, we will show that the problem in (4.7) requires O(N(K + min {K, N})η) for its solution, where η denotes the number of iteration steps for solving (4.7). Thus, the worst-case complexity of the optimal algorithm is O(N(K + min {K, N})η(C + N  1) ! / (N  1) ! /C!).

4.2.3

Suboptimal Algorithm

The algorithm proposed in Subsection 4.2.2 has a high complexity in the worst case. Thus, in this section, we further propose a lower-complexity suboptimal algorithm. In the following, it can be shown that, given the values of SER, SERk,n, the original problem in (4.5) is reduced to a combinatorial problem with variable cn, which can be optimally solved with a greedy algorithm; on the other hand, if the values of bits cn are fixed, the problem in (4.5) becomes a convex optimization problem that can be solved in polynomial time. Thus, this structure enables us to employ the two-step approach to obtain a suboptimal solution to the problem in (4.5): in the first step, bits on each subcarrier are loaded under the assumption that each user has identical target SER over all subcarriers; in the second step, users’ SERk,n on each subcarrier are adjusted depending on the number of bits loaded in the first step.

4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with. . .

51

Bit Loading Assuming that each user has identical target SER on each subcarrier, we will determine the number of bits cn. From the first constraint in (4.7), we have N SERk, n  BER C

ð4:12Þ

Then, the optimization problem reduces to   C  BER f c ; n N X N min cn n¼1 αn PN n¼1 cn ¼ C subject to cn 2 f0; 1; 2; . . .g 8n

ð4:13Þ

The problem in (4.13) can be optimally solved by the Levin-Campello algorithm [13]. This algorithm follows a greedy approach: each time, 1 bit is assigned to a subcarrier, and in each assignment, the subcarrier that requires the least additional power is selected. Denote by

ΔpnðcÞ ¼



 CBER f c þ 1; CBER  f c; N N αn

ð4:14Þ

the additional power required for transmission of one additional bit on the nth subcarrier. The algorithm is described in Algorithm 4.2. Algorithm 4.2 Suboptimal Algorithm: Bit Loading Set cn ¼ 0, and evaluate Δpn(0) for all n; While ∑ncn  C do Find n  ¼ arg minnΔpn(cn); Update cn ¼ cn + 1; Calculate Δpn(cn); end while return cn;

Target Symbol Error Rate Adjusting In this algorithm, the target SER is adjusted according to the number of bits assigned in the previous step. It has been proven in Subsection 4.2.2 that the original problem in (4.5) is reduced to a convex optimization problem in (4.7). Generally, the convex optimization problem can be optimally solved by the interior point method [30]. However, we will not directly employ this method for

52

4 Resource Allocation for OFDMA Systems

the problem in (4.7). First, without exploiting the structure of the problem in (4.6), the interior point method still has a considerably high complexity [30]. Second, the values of SERk,n are usually small; thus, the roundoff error in the iteration cannot be neglected. Because of the second-order property of the interior point method, this roundoff error may cause instability in the calculation, resulting in more iterations. For this reason, we employ the projected subgradient method: variables are updated along the negative subgradient direction of the objective function, and the new variables are then projected back to the feasible set. The projected subgradient method has been demonstrated to be robust against error noise [31] and convergent to the optimal solution [32]. In the l‐th iteration step, the subgradient of the objective function in (4.7) is given by

gk , n ð t Þ ¼

8
0, the solution to the problem in (4.18) is given by the following: Proof First, for the problem in (4.18), we remove the constraint SERk,n(t + 1)
0 and derive the corresponding solution. For any n, according to the first constraint in (4.18), we have SERk, n ðt þ 1Þ ¼ C  BER 

N X

SERk, m ðt þ 1Þ 8n:

ð4:19Þ

m¼1, m6¼n

Substituting (4.19) into the objective function of (4.18) and calculating the derivative, we can obtain that the optimal SERk, m(t + 1) should satisfy ∂d ∂d ∂SERk, n ðt þ 1Þ þ ¼ 0, 8m 6¼ n ∂SERk, m ðt þ 1Þ ∂SERk, n ðt þ 1Þ ∂SERk, m ðt þ 1Þ

ð4:20Þ

where N X ∂d dk, n ðt Þ ¼ 2 C  BER  SERk, m ðt þ 1Þ  SER ∂SERk, m m¼1, m6¼n

 dk, m ðt Þ þ 2 SERk, m ðt þ 1Þ  SER

! ð4:21Þ

Substituting (4.22) into (4.21) gives C  BER 

N X

dk, n ðt Þ ¼ SERk, m ðt Þ  SER dk, m ðt Þ ð4:22Þ SERk, n ðt þ 1Þ  SER

m¼1, m6¼n

Combining with (4.19), we have

 dk, n ðt Þ þ SERk, m ðt þ 1Þ  SER dk, m ðt Þ SERk, n ðt þ 1Þ ¼ SER

ð4:23Þ

Taking the sum operation Σm on the both sides, we have

 P dk, n ðt Þ þ N d SER SER ð t þ 1 Þ  SER ð t Þ k, m k, m m¼1

P m¼1  N d d ¼ N  SERk, n ðt Þ  m¼1 SERk , m ðt Þ  C  BER

N  SERk, n ðt þ 1Þ ¼

PN

ð4:24Þ By dividing by N on both sides, we obtain (4.19). It remains to prove that the SERk,n(t + 1) P given by (4.24) satisfies the constraint for efficiently small α(t). Recalling n SERk, n ¼ C  BER and substituting (4.17) into (4.24), we have

54

4 Resource Allocation for OFDMA Systems

N  SERk, n ðt þ 1Þ ¼ N  SERk, n ðt Þ  αðt Þ Ngk, n ðt Þ 

N X

! gk , m ð t Þ

ð4:25Þ

m¼1

Thus, if Ngk,n(t)  ∑ngk,n(t), then SERk,n(t + 1)
0 is always satisfied because α(t)
0 and SERk,n(t)
0; on the other hand, if Ngk,n(t) > ∑ngk,n(t), then the step size must be αðt Þ 

N  SERk, n ðt Þ PN Ngk, n ðt Þ  m¼1 gk , m ð t Þ

ð4:26Þ

It follows that, for sufficiently small α(t), SERk,n(t + 1)
0. Denote by pT ðt Þ ¼

N X n¼1

max k

f ðcn ; SERk, n ðt ÞÞ jH k , n j2

ð4:27Þ

the transmit power in the t‐th iteration, by Tmax the maximum number of iterations, and by ε the convergence tolerance. The algorithm is described in Algorithm 4.3. Algorithm 4.3 Suboptimal Algorithm: Target SER Adjusting Set SERk, n ðt Þ ¼ C  BER=N, for all k, n, and set pT(0) ¼ 0; Evaluate pT(1) ¼ 0 While t  Tmax and |pT(t)  pT(t  1)|/pT(t)
ε do for n ¼ 1 : N do Determine the worst user kn(t) using (4.16) Calculate the subgradient of user kn(t), gkn ðtÞ ðt Þ using (4.15) ,n

d Calculate the target SER of user kn(t), SER k n ðt Þ, n ðt Þ using (4.17) end for for all worst-channel users k 2 {k1(t), . . . , kN(t)}, do Update SERk, n(t + 1)for all n using (4.19) end for for all non-worst-channel users k, k 2 = {k1(t), . . . , kN(t)}do Update SERk, n(t + 1) ¼ SERk, n(t) for all n end for Evaluate pT(t + 1); end while Find the minimum transmit power over all steps, t  ¼ arg mintpT(t) return SERk, n(t) Assuming that user k is the worst user on the n‐th subcarrier and the non-worst user on the other subcarriers in the t‐th iteration step, the equation in (4.15) implies that gk,n(t) < 0 and gk,m(t) ¼ 0 for m 6¼ n. Also, it follows from (4.25) that SERk, n(t + 1) > SERk,n(t) and SERk,m(t + 1) < SERk,m(t) for all m 6¼ n. Thus, it can be found that on each subcarrier, this algorithm increases the target SER of the worst

4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with. . .

55

user such that the transmit power can be brought down. However, for each worst user, the increment of the target SER on a subcarrier results in the decrement of the target SER on other subcarriers, which may cause this user to become a worst user on some other subcarriers. Thus, this algorithm requires multiple iteration steps until the stopping criterion is met. Complexity For bit loading, it can be noted that the computation is dominated by the calculation of Δpn(c). In step 1, N values of Δpn(0) are evaluated; then, the bit loading iterates C times, and in each iteration from step 4 to step 6, Δpn(c) is evaluated for some subcarrier n. Thus, the number of computations for Δpn(c) is C + N. Thus, the algorithm for bit loading has a complexity of O(C + N ). In the algorithm for target SER adjusting, on each subcarrier n, finding the worst user kn d has a complexity of O(K ); gkn , n ðt Þ and SER kn ðt Þ, n ðt Þ of user kn are evaluated. Thus, the algorithm from step 4 to step 8 has a complexity of O(KN). Then, from step 9 to step 14, N min {K, N} values of SERk,n are updated. Thus, the algorithm for adjusting SERk,n has a complexity of O(N(K + min {K, N})η), where η denotes the number of iteration steps. The overall complexity of the suboptimal algorithm is O(C + N (K + min {K, N})η).

4.2.4

Simulation Results

In this subsection, we numerically evaluate the performance of our proposed transmission scheme. We assume that all users are uniformly distributed in a unit square, where the transmitter is located in the center. We simulate the frequency-selective Rayleigh fading channel by using the COST 259 channel model for a typical urban environment [33] and assume that each multipath tap follows Jakes’ model [34]. The parameters are summarized in Table 4.1. First, in Fig. 4.3, we compare the performance of the proposed suboptimal algorithm with that of the optimal one. The number of users and the number of subcarriers are fixed as K ¼ 9 and N ¼ 6, respectively. Figure 4.3a plots the input SNR ( pT/N(1/2)N0) required to achieve BER under different C. In Fig. 4.3a, the performance gap between the suboptimum and optimum increases when BER becomes larger. However, even at BER ¼ 102 and C ¼ 12 bits, the suboptimal algorithm is only 0.2 dB worse than the optimal one. Figure 4.3b shows the running time of the suboptimal algorithm and the optimal algorithm. We have shown that the optimal algorithm has a complexity that is exponential in the worst case, whereas the suboptimal algorithm has a quadratic complexity in terms of K and N. However, when BER is small (BER < 105 ), the running time of the optimal algorithm is still close to that of the suboptimal one, because the lower bound given in (4.11) can effectively reduce the computation burden for computing SERk,n. However, the running time of the optimal algorithm increases with BER ¼ 102 and C ¼ 12 bits; the optimal algorithm is about 700 times slower than the suboptimal one. The reason is that the bound in (4.11) becomes looser as BER increases, which requires more computations of optimal solutions.

56

4 Resource Allocation for OFDMA Systems

Fig. 4.3 Performance of the suboptimal algorithm and the optimal algorithm: (a) average bit error rate (BER) versus input signal-to-noise ratio (SNR) and (b) average BER versus running time

Next, in Fig. 4.4, we compare the proposed suboptimal algorithm with the conventional multicast approach. Here, in the conventional multicast OFDMA approach, all users have identical target SER over all subcarriers. In Fig. 4.4a, as the number of user K increases from 2 to 64, the SNR difference between the proposed algorithm and the conventional multicast approach increases from 0.4 to 1.6 dB. Thus, it implies that the proposed algorithm can take more advantage of multiuser diversity than the conventional one. Figure 4.4b shows that the proposed algorithm has more SNR gain as the requirement of the average BER is relaxed. As BER increases from 106 to 102, the SNR gain increases from 0.6 to 2.6 dB. The reason for this phenomenon is that, as the BER requirement is relaxed, the BS has more freedom in adjusting users’ target BER across subcarriers. In Fig. 4.4c, the performance gain of the suboptimal algorithm over the conventional one is almost invariant with the number of subcarriers N (between 1.5 and 1.8 dB). Finally, in Fig. 4.5, we compare the proposed suboptimal algorithm with the multilayer multicast approach in [8]. The authors in [8] applied multiple description coding for delivery of multimedia streams and optimized transmission rate under the assumption that the target SER of users on an assigned subcarrier is constant. We use our proposed algorithm to derive input SNR for each given C and then use the algorithm in [8] to maximize the minimum user rate with the input SNR constraint. In Fig. 4.5, we can see that when input SNR ( pT/N(1/2)N0) < 30dB, the minimum rate achieved by our proposed algorithm is worse than that of the multilayer approach; however, as the input SNR increases, the proposed algorithm outperforms the multilayer approach. Thus, by allowing users to have different SNR across subcarriers, our proposed algorithm can improve the performance of multicast OFDMA systems at high SNR.

4.2 Adaptive Resource Allocation for Multicast OFDMA Systems with. . . Fig. 4.4 Performance of the suboptimal algorithm and the conventional approach: (a) number of subcarriers versus input signal-to-noise ratio (SNR) (N ¼ 12, BER ¼ 103 ), (b) average bit error rate (BER) versus input SNR (C ¼ 128 bits), and (c) number of users versus input SNR. (K ¼ 64, BER ¼ 103 )

57

58

4 Resource Allocation for OFDMA Systems

Fig. 4.5 Performance of the suboptimal algorithm and the multilayer approach: input signal-to-noise ratio (SNR) versus minimum rate (N ¼ 32, K ¼ 8, BER ¼ 104 )

4.2.5

Conclusion

In this section, we proposed algorithms for minimizing transmitting power for multicast OFDMA systems with guaranteed average BER and data rate requirement. We first presented the multicast OFDMA transmission scheme. Under the assumption that users can have different target SER on different subcarriers, the required transmit power of the worst user on each subcarrier can be reduced. We then formulated the problem and proposed an algorithm achieving the optimal solution. To reduce the complexity, we further proposed a two-step suboptimal algorithm by separating the procedures of bit loading and target SER adjustment. In the two-step suboptimal algorithm, both subproblems can be solved in low complexity. Finally, we compared our proposed algorithms with the conventional multicast OFDMA approach. We demonstrated that the proposed algorithms can effectively reduce the transmit power and that the performance of the suboptimal algorithm is close to that of the optimal algorithm.

4.3 4.3.1

Subcarrier and Power Allocation for Multiuser MIMOOFDM Systems with Various Detectors System Model and Problem Formulation

Consider a multiuser MIMO-OFDM system with K users in downlink channels, where each user is equipped with Q receive antennas, and the base station (BS) is equipped with M transmit antennas, where Q
M. Figure 4.6 depicts the structure of

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

59

Fig. 4.6 Structure of the downlink multiuser MIMO-OFDM system

a downlink multiuser MIMO-OFDM system. For simplicity, we assume that the channel is invariant during each transmission block [35]. Furthermore, the cyclic prefix is assumed to be sufficient long to avoid inter-block interference caused by the multipath propagation. The MIMO frequency-domain channel on subcarrier n between the base station and user k can be characterized by a Q  M matrix Hk,n. In this paper, we assume that the channel state information (CSI) is perfectly known by the BS. At each transmission block, the BS performs the subcarrier and power allocation for the downlink. Then, the result is sent to all users via a separate control channel. For simplicity, we do not allow more than one user to share a subcarrier. The allocation of the k-th user to subcarrier n is indicated by ρk,n. That is, if the k-th user is allocated to subcarrier n, ρk,n ¼ 1; otherwise, ρk,n ¼ 0. With the subcarrier allocation {ρk,n} is known, the user kn , which subcarrier n is allocated to, is determined. The transmitted signal for user kn can be expressed as Xn ¼ (Xn(1), . . . , Xn(M))T. Denote by pn the transmit power of subcarrier n, and assume the power is equally allocated on each antenna; then the baseband inputoutput relationship is represented as

60

4 Resource Allocation for OFDMA Systems

yk , n

rffiffiffiffiffi pn ¼ Hk , n Xn þ Vk, n , 8k, 8n, M

ð4:28Þ

where yk, n is the Q  1 received signal vector for user k on the n‐th subcarrier and vk, n CN ð0; σ 2 IQ Þ is the noise vector. The notation n CN ðm; RÞ stands for a circularly symmetric complex Gaussian (CSCG) vector with mean m and covariance matrix R. Our objective is to find the optimal subcarrier allocation {ρk,n} and power allocation {ρn} that minimize the overall average BER subject to a total power constraint. The optimization problem is given as follows: 1 XK XN min ρ BERk, n k¼1 n¼1 k , n fρk , n g, fpn g N PN s:t: n¼1 pn ¼ PT , PK ρ k¼1 k , n ¼ 1, 8n ,

ð4:29Þ

where BERk,n and PT represent the BER for user k on the subcarrier n and the total power, respectively.

4.3.2

Subcarrier and Power Allocation

In this subsection, we investigate the optimal algorithms and propose the corresponding suboptimal algorithms of lower complexity depending on the type of actually employed MIMO detector. For simplicity, only binary phase shift keying (BPSK) is considered in this paper.

ML Detector With the ML detector, the estimate of the transmitted signal is given by x~n ¼ arg qffiffiffiffi 2 min yk, n  Hk, n pMn x where χ is a set of the candidate vector signals. A pairwise x2X

error probability (PEP) can be derived as in [28]: 

Pk, n xn ! x~n



0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 pn kHk, n dn k2 A , ¼ Q@ 2Mσ 2

ð4:30Þ

R 1 t2 =2 where dn ¼ xn  x~n and QðxÞ ¼ p1ffiffiffiffi e dt denote the Gaussian-Q function. As 2π x obtaining an explicit expression for the BER of the ML detector for the MIMO

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

61

system is well known to be difficult, we replace the BERk,n in problem (4.29) with an upper bound which can be expressed as BERk, n 

X

X xn 2χ

1 2M

x~n 2 X x~n 6¼ xn

    N e xn ! x~n Pk, n xn ! x~n , M

ð4:31Þ

  where N e xn ! x~n denotes the number of the error bits when the detected signal is not xn but x~n . 1. Optimal Algorithm: Clearly, (4.29) is an NP-hard problem, and an exhaustive searching algorithm can be used. With the subcarrier allocation being known (i.e., the kn ’s are given), the remaining problem to allocate power can be written as 1 XN BERkn , n min n¼1 fpn g N PN s:t: n¼1 pn ¼ PT :

ð4:32Þ

As problem (4.32) is a convex optimization problem, we can solve it by the method of Lagrange multipliers with the following Lagrangian of (4.32): L¼

X N  1 XN  BER þ θ p  P , T k , n n n n¼1 n¼1 N

ð4:33Þ

where θ is the Lagrangian multiplier. Let the derivation of (4.33) with respect to pn be zero, and we have ∂L 1 ∂BERkn , n ¼ þ θ ¼ 0, 8n: ∂pn N ∂pn

ð4:34Þ

From (4.34), we have the following result X xn 2χ

X x~n 2 χ x~n 6¼ xn



N e xn ! x~n



2 !sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hk , n dn 2 pn Hkn , n n  exp  ¼ ~θ, 8n, 4Mσ 2 pn ð4:35Þ

pffiffiffiffiffiffiffi where ~θ ¼ 4NM Mπ 2M σθ is chosen to satisfy the power constraint in (4.29). Since the left-hand side of (4.35) is a continuous and strictly decreasing function of pn, the solution is unique due to one-to-one relation between pn and ~θ. Obviously, the solution of (4.35) must be obtained numerically.

62

4 Resource Allocation for OFDMA Systems

As stated above, obtaining the optimal power allocation needs a numerical search, and the computational complexity for the subcarrier allocation is OðK N Þ. So it is necessary to propose a suboptimal algorithm of lower complexity. 2. Suboptimal Algorithm: We focus on a low-complexity and suboptimal solution  of (4.29) for the ML detector here. The PEP Pk, n xn ! x~n has the following upper bound: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 pn kHk, n dn k A pn kH k , n d n k2 A Q  max Q@ , 2 d2D 2Mσ 2Mσ 2

ð4:36Þ

where D ¼ fd ¼ x  x~jx; x~ 2 X ; x 6¼ x~g. Then, (4.31) can be upper bounded as

BERk, n

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 kHk, n dn k2  M  B pn min C d2D  2  1 Q@ A: 2Mσ 2

ð4:37Þ

To minimize the upper bound (4.37), the subcarrier allocation scheme becomes ρkn , n ¼ 1,ρk0 , n ¼ 0, 8k0 6¼ k n , 8n, where [37]. kn



¼ arg max min kHk, n dk : k

ð4:38Þ

d2D

Once the subcarrier allocation {ρk,n} is known, we could allocate the power via the optimal power allocation algorithm as stated above. However, its complexity to find the optimal solution can be still high as the parameter ~θ must be numerically obtained. Replacing the BERk,n with the upper bound in (4.37), we can use the quasioptimal power loading algorithm [15] to allocate power. The power allocation for the ML detector is given as follows: ζj PT  ζ n X N pn ¼ 2 j¼1 1 þ ζn 1 þ ζ 2j

!1 ,

8n,

ð4:39Þ

2 where ζ n ¼ PT =ð2Mσ 2 Þ min Hkn , n d . d2D

Linear Detector With a linear detector, the estimate of the transmitted signal xn is given by x~n ¼ Gyk, n , where G is a linear filter. For the zero forcing (ZF) detector, we have G ¼ H{k, n with the notation (){ denoting the matrix Moore-Penrose pseudoinverse.

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

63

Then, the post-processing signal-to-noise ratio (SNR) of the signal transmitted on the antenna m is given by SNRkZF , n, m ¼ h

where 1= HkH, n Hk, n

1 i m, m

Mσ 2

h

pn 1 i H Hk, n Hk, n

def

¼

pn γ k, n, m , Mσ 2

ð4:40Þ

m, m

is denoted by γ k,n,m The notation [X]i,j denotes the (i, j)

th element of the matrix X. Thus, the average BER of the k‐th user on subcarrier n can be found as [36]. BERk, n

1 XM ¼ Q m¼1 M

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2pn γ k, n, m : Mσ 2

ð4:41Þ

The analysis of BER is not straightforward for the minimum mean square error (MMSE) detector. Nevertheless, the MMSE detector becomes the ZF detector as the SNR increases. Therefore, in this section we only consider the ZF detector. 1. Optimal Algorithm: We still use a full search algorithm to obtain the optimal solution to problem (4.29). With the given {ρk,n} (i.e., kn ’s are determined), the optimal solution to problem (4.32) can be found by using the standard Lagrange multiplier technique. The Lagrangian of this problem is N N 1 X 1 X Q L¼ N n¼1 M n¼1

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! N X 2pn γkn , n, m pn  PT , þμ Mσ 2 n¼1

where μ is the Lagrange multiplier. Then, the solution can be obtained by solving ∂L/∂pn ¼ 0. Thus, we obtain the following power allocation: X M rffiffiffiffiffiffiffiffiffiffiffiffiffi γ kn n, m m¼1

p γ  n k n n, m  exp  ¼ μ, Mσ 2 pn Mσ 2

ð4:42Þ

pffiffiffi where μ ¼ 2μMN π is chosen to satisfy the total power constraint. The complexity of the optimal algorithm is also prohibitively high. To reduce the complexity, we propose a simple suboptimal algorithm in the next section. 2. Suboptimal Algorithm: As shown in Subsection 4.3.2.1 when considering the ML detector, subcarrier n should be assigned to the user with the minimum BERk,n. Based on the Rayleigh-Ritz Theorem [38], we get h

HkH, n Hk, n

1 i m, m

 ¼

emM

H h

HkH, n Hk, n  H emM emM

1 i M em

  1  λmin HkH, n Hk, n ,

ð4:43Þ

64

4 Resource Allocation for OFDMA Systems

where eiN stands for the i‐th column of IN. λmin(A) denotes the minimum eigenvalue of matrix A. From (4.41) and (4.43), we can obtain the following upper bound:

BERk, n

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi1 2pn λmin HkH, n Hk, n A:  Q@ Mσ 2

ð4:44Þ

This shows that the subcarrier can be allocated to the user with the maximum of the minimum eigenvalues of HkH, n Hk, n . In other words, for the n‐th subcarrier, we have   kn ¼ arg max λmin HkH, n Hk, n :

ð4:45Þ

k

Then, based on the upper bound

in (4.44),  the power allocation is described in H 2  (4.39) with ζ n ¼ 2PT =ðMσ Þλmin Hkn , n Hkn , n : Besides (4.44), we can also get another upper bound from (4.41) as follows: 0

BERk, n

1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2p B C h n  Q @u t 1 i A: H 2 Mσ max Hk, n Hk, n m

ð4:46Þ

m, m

Based on the upper bound in (4.46), the user with the minimum of the maximum  1 diagonal elements of HkH, n Hk, n is assigned to subcarrier $n$ as follows: h 1 i kn ¼ arg min max HkH, n Hk, n k

m

m, m

:

ð4:47Þ

The power allocation is the same as that expressed in (4.39) where ζ n ¼ 2PT =ðMσ 2 Þ min γ kn , n, m . m

The above two algorithms will be referred to as M_eig and M_diag, respectively.

SIC Detector We only consider the ZF-SIC detector in this section. The matrix Hk,n is QR factorized as Hk,n ¼ Qk,nRk,n, where Qk,n is unitary qffiffiffiand ffi Rk,n is upper triangular. Premultiplying QkH, n to yk,n, we have QkH, n yk, n ¼ Rk, n

pn M xn

þ uk, n , where uk,n has the

same statistical properties of vk,n. The error propagation is assumed to be ignored at moderate-to-high SNR, so the SNR of the m‐th signal stream is

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

SNRkSIC , n, m

¼

 ½Rk, n

m, m Mσ 2

2  pn

def

¼

pn~γ k, n, m : Mσ 2

65

ð4:48Þ

Note that (4.48) has the same form as the SNR for the ZF detector in (4.40). Thus, the same procedure is easily adopted to derive the optimal subcarrier and power allocation algorithm for the ZF-SIC detector. From (4.41) and (4.48), we have the following upper bound:

BERk, n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0v  ffi1 u u2p min ½R 2 k , n m, m  C Bt n m C:  QB @ A Mσ 2

ð4:49Þ

Based on the upper bound in (4.49), we get the following suboptimal subcarrier allocation: n o kn ¼ arg max min j½Rk, n m, m j , 8n: k

The suboptimal power ζ n ¼ 2PT =ðMσ 2 Þ min ~γ kn , n, m .

ð4:50Þ

m

allocation

is

given

in

(4.39)

with

m

4.3.3

Simulation Results

This subsection presents simulation results for various detectors. We investigate a MIMO-OFDM system with M ¼ 2 transmit and Q ¼ 2 receive antennas. For simplicity, we assume that the elements of the channel matrix Hk, n are independent zero-mean CSCG random variables with variance 1. The average SNR is defined as PT/(N/Mσ 2). For comparison, two conventional suboptimal algorithms are also considered. One is to allocate power uniformly and then use the mean BER to choose the optimal subcarrier allocation. The other is to allocate each subcarrier to the user with the highest eigenvalue product [39] which is to maximize the capacity and then allocate power with the corresponding optimal approach. These two suboptimal algorithms will be referred to as Opt_u and Prod_opt in Figs. 4.7, 4.8 and 4.9. Due to the high computational complexity of the optimal algorithms due to the exhaustive search, we only consider N ¼ 8 subcarriers and K ¼ 4 users when comparing the optimal algorithm with the suboptimal algorithms. For a large system, we assume N ¼ 64, K 2 {8, 16} and compare the proposed suboptimal algorithm(s) and the two conventional algorithms.

66

4 Resource Allocation for OFDMA Systems 10-1 Optimal Suboptimal Prod_opt Opt_u

BER

10-2

10-3

10-4

10-5 2

3

6 5 7 Average SNR (dB)

4

8

9

10

(a) N=8 10-2 Suboptimal Prod_opt Opt_u

10-3

BER

K=8 K=16

10-4

10-5

10-6 2

3

4

5 6 Average SNR (dB)

7

8

(b) N=64 Fig. 4.7 Average BER performance for the MIMO-OFDM system with the ML detector compared

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

67

10-1 Optimal M_diag M_eig Prod_opt Opt_u

10-2

BER

10-3

10-4

10-5

10-6 4

8

6

10 12 Average SNR (dB)

14

18

16

(a) N=8 10-2 M_diag M_eig Prod_opt Opt_u

K=8

10-3

BER

K=16 10-4

10-5

10-6

3

4

5

7 6 Average SNR (dB)

8

9

10

(b) N=64 Fig. 4.8 Average BER performance for the MIMO-OFDM system with the ZF detector

68

4 Resource Allocation for OFDMA Systems 10-1 Optimal Suboptimal Prod_opt Opt_u

10-2

BER

10-3

10-4

10-5

10-6 2

4

6

8

10

12

14

16

Average SNR (dB)

(a) N=8 10-2 Suboptimal Prod_opt Opt_u

K=8

BER

10-3

K=16

10-4

10-5

10-6 3

4

5

6 7 Average SNR (dB)

8

9

10

(b) N=64 Fig. 4.9 Average BER performance for the MIMO-OFDM system with the SIC detector

4.3 Subcarrier and Power Allocation for Multiuser MIMO-OFDM Systems with. . .

69

Figure 4.7 demonstrates the average BER of the ML detector. It is clearly shown that the performance of the proposed suboptimal scheme appears to be quite close to that of the optimal algorithm as shown in Fig. 4.7 (a), especially in the high SNR region. It is noticed that the proposed suboptimal algorithm can achieve a better performance than Prod_opt and Opt_u algorithms. As shown in Fig. 4.7 (b), at a BER of 106, the suboptimal algorithm has a gain of about 1 dB as compared to Opt_u and Prod_opt algorithms. It is observed that the BER decreases with the number of users, which can be explained by the multiuser diversity effect. As the number of users increases, the probability that a user of high SNR is assigned to a given subcarrier increases as well, resulting in more efficient use of power and better performance. Figure 4.8 shows the simulation results for the ZF detector. It is observed that M_eig and M_diag algorithms appear to be quite close to the optimal algorithm from Fig. 4.8a. When both figures in Fig. 4.8 are compared, it is shown that M_diag algorithm always outperforms M_eig algorithm. This is due to the fact that the upper bound in (4.46) is tighter than the upper bound in (4.44). From Fig. 4.8(a), M_diag algorithm has a gain of about 0.7 dB as compared to Prod_opt algorithm and about 1.6 dB as compared to Opt_u algorithm at a BER of 106 when there are K ¼ 16 users. Figure 4.9 presents the BER performance with different resource allocation algorithms for the ZF-SIC detector. From Fig. 4.9a, we can see that the suboptimal algorithm has a BER performance close to the ideal one of the optimal algorithm. In Fig. 4.9b, it is observed that the suboptimal algorithm outperforms Prod_opt and Opt_u algorithms in terms of BER.

4.3.4

Conclusion

In this section, we investigated the average BER minimization for a multiuser MIMO-OFDM system with a total power constraint. We considered various detectors, such as the ML, linear, and SIC detectors, and derived the optimal subcarrier and power allocation algorithms that are based on the exhaustive search and numerical methods. As the optimal algorithms have a prohibitive high complexity, we also proposed the low-complexity suboptimal algorithms that separately allocate the subcarriers and power. Through the simulations, we demonstrated that the performance achieved by the suboptimal algorithms can be close to that with the corresponding optimal algorithms. We also observed that the suboptimal algorithms outperform two conventional algorithms.

70

4.4

4 Resource Allocation for OFDMA Systems

Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems

In this section, we address energy-efficient subcarrier assignment under QoS constraints in uplink OFDMA networks with frequency-selective fading channels. To make the problem clear, we first consider individual EE. By theoretically analyzing the inherent relation between energy efficiency and the UE’s number of subcarriers, we indicate how to assign a suitable subcarrier to the UE and obtain closed-form expressions to calculate the EE gain of getting or losing a subcarrier based on continuous relaxation. Accordingly, a low-complexity subcarrier assignment algorithm is provided to improve system EE. The algorithm iteratively adjusts the assignment among all UEs according to the EE gain with the change of subcarriers.

4.4.1

System Model and Problem Formulation

We consider a signal-cell uplink OFDMA network with the subcarrier set N ¼ f1; . . . ; n; . . . ; N g and the active user set K ¼ f1; . . . ; k; . . . ; K g. The total bandwidth B is equally divided into N subcarriers, each with a bandwidth of W ¼ B/N. The channel is assumed to be frequency-selective Rayleigh fading. Perfect channel state information is available to both transmitters and receivers. Denote pk,n as the transmission power allocated on subcarrier n and gk,n ≜ |hk, n|2/N0W as the channelto-noise ratio (CNR), where N0 is the single-sided noise power spectral density, and hk,n is the channel frequency response modeled as a zero-mean complex Gaussian random variable. Then, we can express the maximum achievable data rate of the kth UE on the nth subcarrier as   rk, n ¼ W log2 1 þ pk, n gk, n :

ð4:51Þ

We let ρk,n 2 {1, 0} indicate whether the nth subcarrier is assigned to the kth UE. ρ ¼ [ρk,n]K  N and P ¼ [pk,n]K  N are denoted as subcarrier allocation matrix and power allocation matrix, respectively. Accordingly, the data rate of the k th UE is Rk ðρ; PÞ ¼

X

ρ r : n2N k , n k , n

ð4:52Þ

P The overall transmit power of the kth UE is Pk ¼ n2N ρk, n pk, n . In addition to the transmit power, we need to consider the circuit power during transmissions as well. Hence, the total power consumption of the kth UE is Pktot ¼ ζ k Pk þ PkC ,

ð4:53Þ

4.4 Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems

71

where ζ k and PkC represent the reciprocal of drain efficiency of power amplifier and circuit power of the kth UE, respectively. We assume that both ζ k and PkC are constant and same for all UEs here. Furthermore, the individual EE of the kth UE is commonly defined as ηkEE ðρ; PÞ ¼

Rk ðρ; PÞ , Pktot

ð4:54Þ

which means the delivered bits per unit energy (bits/Joule). In this paper, we focus on maximizing the EE of the whole uplink OFDMA network. Therefore, the optimization problem can be formulated as X EE EE max ηsys ðρ; PÞ ¼ ηsys ðρ; PÞ ρ, P k2K subject X to ρk, n r k, n
Rkmin , 8k 2 K, n2N X ρk, n pk, n  Pkmax , 8k 2 K, n2N X ρk, n  1, 8n 2 N ,

ð4:55Þ

k2K

ρk, n 2 f1; 0g, 8n 2 N , 8k 2 K, ρk, n
0, 8n 2 N , 8k 2 K, where Rkmin and Pkmax are the required minimum transmit rate and the maximum transmit power of the kth UE, respectively. The second constraint guarantees the QoS requirements of each UE. The third constraint limits the peak transmit power for each UE. The last two constraints indicate that each subcarrier is exclusively assigned to at most one UE each time to avoid co-channel interference.

4.4.2

Energy-Efficient Subcarrier Assignment

In this section, we first investigate the inherent relation between energy efficiency and the number of subcarriers in a signal UE scenario. Then, based on this, we propose a low-complexity algorithm to solve problem (4.55).

Analysis of Energy Efficiency with Subcarriers Problem (4.55) is generally NP-hard with nonlinear constraints and integer assignment variables ρk,n. To make it more tractable, we divided (4.55) into two subproblems, i.e., power allocation and subcarrier assignment. In this subsection, a signal UE scenario is considered. Therefore, we omit the user index k in the

72

4 Resource Allocation for OFDMA Systems

subsequent discussing. When the users subcarrier assignment is fixed, there exists a global optimal solution maximizing the energy efficiency which has the following properties [24]: 8 ^ ðPÞ > 1 dR > > > 0, if η^EE ðPÞ < > > ζ dP > > ^ ðPÞ η^EE ðPÞ < 1 dR EE d ¼ 0, if η^ ðPÞ ¼ dP > ζ dP > > > ^ ðPÞ > 1 dR > > , : < 0, if η^EE ðPÞ > ζ dP

ð4:56Þ

^ ðPÞ R ^ ðPÞ≜maxR where η^EE ðPÞ≜ ζPþP is the maximum energy efficiency in P and R C  fix  ^ ðPÞ satisfies ρ ; P can be obtained by water-filling method. The derivative of R

^ ðPÞ W log2 e dR  , dP ζμ

ð4:57Þ

where μ is the water-falling level. Authors in [24] solve the power allocation problem with given subcarrier assignments and propose the BPA algorithm to get the optimal power allocation. In the following analysis, we focus on what happens to the EE once the subcarrier assignment is changed. Based on (4.56) and BPA, we let ηN , PN , and μN denote the optimal EE, optimal transmit power, and optimal water-filling level of the UE with N subcarriers. Note that they can all be obtained after performing BPA algorithm. We first address whether the EE increases when the UE obtains a new subcarrier, i.e., whether ηNþ1 > ηN or not. Works in [29] indicate energy efficiency increases with increasing number of subchannels when the channel is flat-fading. Here, in frequency-selective channels, we have the following conclusion: Theorem 4.3 When a UE with N subcarriers obtains a new subcarrier gnew, the energy efficiency will increase if and only if gnew > μ1 ; otherwise the energy N

efficiency remains unchanged. Proof As (4.56) in [24], there are three cases of the optimal EE solution, which are d η^ dPðPÞ > 0, d η^ dPðPÞ ¼ 0, and d η^ dPðPÞ < 0. Here we only consider the second case and the other two cases are similar. We first consider gnew > μ1 . We keep the transmit EE

EE

EE

N

power unchanged and do power allocating using water-filling. With the nature of ^ Nþ1 ðPN Þ > RN ðPN Þ. water-filling, it is easy to prove that the new transmit rate R 1 Hence, the energy efficiency increases. Then, as gnew  μ , if we still keep the N

transmit power unchanged, the new subcarrier cannot be allocated any power. Therefore, the new transmit rate and energy efficiency are also unchanged and still satisfy (4.56), i.e., they are still the optimal solution of the N + 1 subcarriers. This completes the proof of Theorem 4.3.

4.4 Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems

73

Theorem 4.3 provides a criterion of subcarrier assignment for a UE. We name a subcarrier which can lead to the EE increase an efficient subcarrier. Accordingly, only the efficient subcarriers of a UE will be considered during the subcarrier assignment, and the computational complexity will be reduced. Now we define Δη≜ηNþ1  ηN as the specific increase when a UE with N subcarriers gets a new efficient subcarrier. Obviously, BPA can be used to obtain the new optimal energy efficiency ηNþ1 and then Δη. However, this will cause more computational complexity. We need a simple and intuitive mathematical expressions calculating Δη. To make the problem easy to deal with mathematically, we relax the subcarrier index n and the number of subcarriers N from positive integers to positive real numbers n~ and N~ , respectively. Then, the relaxations of PN and RN are defined as   P~ N~ ¼

Z

N~

"

0

  R~ N~ ¼ W

Z

#   1 ~ μ N    d~ n, g n~

ð4:58Þ

     log2 μ N~ g n~ d~ n,

ð4:59Þ



N~ 0

    where μ N~ and g n~ are redefinitions of μN and PN according to the relaxation. Furthermore, we can get the relaxation of ηN   ~η N~ ¼ 

  R~ N~   : ζ P~ N~ þ PC

ð4:60Þ

Since η^EE ðPÞ is strictly quasi-concave in P [24], the following three cases are addressed.   1. PN 2 Pmin ; Pmax : The extreme point of η^EE ðPÞ is included in (Pmin, Pmax) where Pmin is the minimum power for the required minimum transmit rate Rmin, i.e.,  min    ~ ^ ¼ Rmin . The first derivative of ~η N is R P        d~η  N~ W ¼ log2 g N~ þ log2 μ N~    2 dN~ ζ P~ N~ þ PC# "         d~η  N~ d~μ  N~ þ ¼ ζ P~ N~ þ PC  ζ N~ dN~ d N~ ) #Z ~ N         1 þ μ N~    log2 μ N~ g n~ d~ n : g N~ 0   Based on (4.56) and (4.57), when ~η  N~ is the extreme point, we have

ð4:61Þ

74

4 Resource Allocation for OFDMA Systems

  W ~ ¼ η~ N

R N~



0

       ~ g ~n d ~n log2 μ N Wlog e   ¼  2  :  ~ ~ ~ ζ P N þ PC ζμ N

ð4:62Þ

  Then, we simplify the first derivative of ~η  N~ and get   d~η  N~ dN~

        1   ~ ~   log2 μ N g N  log2 e 1   μ N~ g N~   ¼W ζ P~ N~ þ PC

ð4:63Þ

We let ΔN~ ¼ 1 after a new subcarrier joined and get   d~η  N~  ~ Δη  ΔN  dN~ 

¼W

   log2 μN gnew  log2 e 1  μ g1 N new

ζPN þ PC

N~ ¼N

:

ð4:64Þ

2. PN ¼ Pmax : [Pmin, Pmax] is the monotonically increasing interval of η^EE ðPÞ. In this case, PN remains unchanged after getting a new subcarrier. Hence, we can have       dP~ N~ d~μ  N~ 1  ~ ~ ¼μ N þN    ¼ 0: ~ ~ dN dN g N~

ð4:65Þ

  The first derivative of ~η  N~ is   d~η N~ ¼W dN~ 

  ~ e  2 log2 μ N~ þ Nlog 

μ

N~ max

ζP

 

d~μ  N~ dN~

  : þ log2 g N~ ð4:66Þ

þ PC

Then, from the above two equations, we can have   d~η  N~ dN~

¼W

        1   ~ ~   log2 μ N g N  log2 e 1   μ

ζP

max

N~ g N~

þ PC

:

ð4:67Þ

Then we can get the expression of Δη like Case 1

Δη  W

   log2 μN gnew  log2 e 1  μ g1 N new

ζPmax þ PC

:

Note that the two expressions of the above two cases are the same.

ð4:68Þ

4.4 Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems

75

3. PN ¼ Pmin . [Pmin, Pmax] is the monotonically decreasing interval of η^EE ðPÞ. In this case, optimal EE is obtained at Rmin. After getting a new subcarrier, the new optimal EE may be still obtained at Rmin or not, i.e., the UEs condition may change from Case 3 to Case 1. Hence, to guarantee the precision, we still choice BPA to calculate the EE gain. We summarize the aforementioned three cases and give the conclusion that if a UE does not obtain its optimal EE at Rmin, the increase of its energy efficiency after getting a new efficient subcarrier is

Δη  W

   log2 μN gnew  log2 e 1  μ g1 ζPN þ PC

N new

:

ð4:69Þ

Note that we can use (4.69) in the scenario of getting a subcarrier as well as losing a subcarrier by replace gnew with the CNR of the lost subcarrier. Figure 4.10 illustrates the exact energy efficiency by BPA and the approximate energy efficiency by (4.69). As is clear shown, the approximate solution is extremely precise. Accordingly, we can conveniently evaluate the influence of a subcarrier on each UE with different subcarrier sets. It plays an important role in subcarrier assignment, and we will have a detailed interpretation in the following discussion.

Fig. 4.10 Comparison of the exact EE and the approximate EE (CNR ¼ 10 dB)

76

4 Resource Allocation for OFDMA Systems

Energy-Efficient Subcarrier Assignment Algorithm In uplink OFDMA systems, when the subcarrier allocation matrix ρ is given, the optimal power allocation matrix P can be obtained by BPA under the individual power and QoS constraints. Hence, we reformulate problem (4.55) and focus on subcarrier assignment in our algorithm. EE max ηsys ð ρÞ ¼ ρ

subject X to

X

ηkEE ðρÞ

k2K

ρk, n  1, 8n 2 N ,

ð4:70Þ

k2K

ρk, n 2 f1; 0g, 8n 2 N , 8k 2 K, Now, we separate the above subproblem from problem (4.55). In [26], the authors propose MEEI algorithm to assignment subcarriers. The basic idea of MEEI is to assign a subcarrier according to the subcarrier’s EE gain Δηk on different UEs. The UE with the highest EE gain will get the subcarrier in each iteration. In MEEI, the authors use BPA algorithm to calculate Δηk. Here we use (4.69) instead of BPA calculating Δηk to reduce the computation complexity. Based on the inherent relation between energy efficiency and the UE’s number of subcarriers we discussed in last subsection, we propose an efficient subcarrier assignment algorithm. The algorithm is named iteration-adjusting subcarrier adjustment (IASA) and sketched in Algorithm 4.4. Algorithm 4.4 Iteration-Adjusting Subcarrier Assignment Algorithm 1: Initialize the assigned subcarrier sot of each UE S k , minimum EE of each UE ηk, min   ¼ Rkmin = ζPkmax þ PkC ; 2: Step 1: Minimum data rate achievement 3: repeat 4: for each UE unsatisfying QoS do 5: Assign the subcarrier n^k ¼ arg maxn2N r gk, n ; 6: Update N r ¼ N r =fn^k g and S k ¼ S k [ fn^k g; 7: end for 8: until all UEs satisfy QoS 9: for each UE do 10: Obtain ηk, N k , Pk, N k and μk, N k by BPA based on S k ; 11: end for 12: Step 2: Residual subcarriers assignment 13: for each subcarrier n 2 N r do 14: Calculate Δηk of each UE using (4.69); 15: The UE k^ ¼ arg maxk2K Δηk obtains subcarrier n; 10: Update S k^ ¼ S k^ [ fng

4.4 Low-Complexity Energy-Efficient Subcarrier Assignment in Uplink OFDMA Systems

17:

77

Update ηk^, N , Pk^, N and μk^, N by BPA based on S k^ ; k^

k^

k^

18: 19: 20: 21:

end for Step 3: Iterative assignment and adjustment for each iteration do for each subcarrier n 2 N r do    ^ 22: Calculate Δη k n 2 S k and others Δηk k 2 K=k using (??), find k^ ^

^

¼ arg maxk2K= k Δηk ; ^

23:

if Δηk^ > Δη k and η ^

k

24: 25: 20: 27: 28:

 Δη k
η k , min then ^

^

k,N ^

S k ¼ S k =fng, S k^ ¼ S k^ [ fng; ^

^

^

^

Update ηk, N k , Pk, N k and μk, N ^ for UE k^ and k by BPA; k end if end for end forreturn Assigned subcarrier set of each UE S k

1. QoS Achievement: In this step, we aim at meeting the minimum data rate of each UE. In every scheduling, each UE unsatisfying its QoS is assigned the highest gain subcarrier from the residual subcarrier set. 2. Residual Subcarriers Assignment: In this step, we greedily assign each residual subcarrier to the UE who has the most EE gain on it. To reduce the computational complexity, if the subcarrier is efficient based on Theorem 4.3 to UE k, we can use (4.69) to calculate the EE gain. Otherwise, the EE gain is zero. 3. Iteration Adjustment: The above two steps are a complete suboptimal subcarrier assignment. According to (4.69), Δηk of a subcarrier is decided by its channel gain as well as Pk , ηk , and μk of the kth UE. Therefore, when we finish subcarrier assignment, Δηk of a subcarrier will be changed and may be not highest on its owner anymore. Hence, we need to adjust the subcarrier assignment. We denote a subcarrier as an inappropriate subcarrier if the subcarrier is changed from its original owner to another UE, the system EE will increase. In each iteration, we use (4.69) to evaluate each subcarrier. Then we find the inappropriate subcarriers and change their owners. Considering computational complexity, we only adopt BPA when a subcarrier assignment is determined or changed and use (4.69) instead of BPA evaluating a subcarrier. The overall complexity of IASA is OðKN þ NN BPA þ N OL KN Þ, where NBPA is the number of iterations of BPA algorithm and NOL is the number of iteration adjustment in IASA. Ignoring the subcarrier adjustment, the complexity will be OðKN þ NN BPA Þ. For RESP+MEEI in [26], the complexity is OðKNN BPA Þ, and for max-min in [38], the complexity is OðNN BPA Þ. The complexity of our proposed algorithm is between RESP+MEEI and max-min. However, in the following simulation, we can see our proposed algorithm has the best performance.

78

4.4.3

4 Resource Allocation for OFDMA Systems

Simulation Results

In this section, we present simulation results to show the performance of our proposed subcarrier assignment algorithm. We consider an uplink OFDMA network with 72 orthogonal subcarriers. The total bandwidth is 1.08 MHz and each subcarrier’s bandwidth is 15 kHz. For all UEs, each subcarrier is modeled as identical Rayleigh fading independently. The maximum transmit power Pkmax is 300 mW and the circuit power PkC is 80 mW. The reciprocal of drain efficiency of power amplifier ζ k is 50%. For simplicity, we assume that Pkmax , PkC , and ζ k are the same for all UEs. In Fig. 4.11, we compare the RESP-MEEI and IASA. We assume that the minimum data rate requirement is 300 kbps for all UEs. We set the iteration number of IASA zero. From Fig. 4.11, we can see that IASA without Step 3 can achieve the same EE of RESP+MEEI while it has a low complexity. In Fig. 4.12, we present the results of IASA algorithm and choice RESP+MEEI and max-min [24] to be compared. We assume that the minimum data rate requirement is 300 kbps, the CNR is randomly selected in the set {0, 5, 10, 15} dB, and the number of iterations is 2. The EE of all the algorithms are first increase and then decrease. That is because when the number of UEs increase, the subcarriers will be insufficient. We can see in each number of UEs, our algorithm has the best performance of the three algorithms.

Fig. 4.11 Performance of different complexities

4.5 Summary

79

Fig. 4.12 Performance of different algorithms

4.4.4

Conclusion

In this section, we investigated the energy-efficient subcarrier assignment in uplink OFDMA networks. We addressed the relation between energy efficiency and the UE’s number of subcarriers and gave the closed-form expression to approximately calculate the increment of energy efficiency. By this conclusion, we proposed a low-complexity subcarrier assignment algorithm. In this algorithm, we improved the performance by iteratively adjusting the suboptimal subcarrier assignment. Simulation results showed that the proposed algorithm improved the energy efficiency in a low complexity compared with the exist iterative algorithm.

4.5

Summary

In this chapter, we discuss about some resource allocation scenario in OFDMA systems with perfect CSI constraint. We analyze the system capacity in multicast OFDMA systems, BER performance in MIMO-OFDM systems, and the energy efficiency in unicast SISO-OFDMA systems. Since it is NP-hard which need very high computation complexity to obtain the optimal solution, we propose the suboptimal algorithm with acceptable complexity for each scenario. Simulation results show that the performance of the suboptimal algorithm is near to the optimal one.

80

4 Resource Allocation for OFDMA Systems

References 1. Y.W. Cheong, R.S. Cheng, K.B. Lataief, R.D. Murch, Multiuser OFDM with adaptive subcarrier, bit, and power allocation. IEEE J. Select. Areas Commun 17(10), 1747–1758 (1999) 2. I. Kim, I.S. Park, Y.H. Lee, Use of linear programming for dynamic subcarrier and bit allocation in multiuser OFDM. IEEE Trans. Veh. Technol. 55(4), 1195–1207 (2006) 3. Z. Mao, X. Wang, Efficient optimal and suboptimal radio resource allocation in OFDMA system. IEEE Trans. Wirel. Commun. 7(2), 440–445 (2008) 4. I.C. Wong, B. Evans, Optimal resource allocation in the OFDMA downlink with imperfect channel knowledge. IEEE Trans. Commun. 57(1), 232–241 (2009) 5. J. Liu, W. Chen, Z. Cao, K.B. Letaief, Dynamic power and sub-carrier allocation for OFDMAbased wireless multicast systems. Proceedings of IEEE International Conference on Communications 2008, Beijing, China, 2008, pp 2607–2611 6. A.Y. Panah, R.W. Heath, Single-user and multicast OFDM power loading with nonregenerative relaying. IEEE Trans. Veh. Technol. 58(9), 4890–4902 (2009) 7. D.T. Ngo, C. Tellambura, H.H. Nguyen, Efficient resource allocation for OFDMA multicast systems with fairness consideration. Proceedings of IEEE Radio and Wireless Symposium 2009, 2009, pp. 392–395 8. C. Suh, J. Mo, Resource allocation for multicast services in multicarrier wireless communications. IEEE Trans. Wirel. Commun. 7(1), 27–31 (2008) 9. H. Kwon, B.G. Lee, Cooperative power allocation for broadcast/multicast services in cellular OFDM systems. IEEE Trans. Commun. 57(10), 3092–3102 (2009) 10. B. Özbek, R.D. Le, H. Khanfir, Performance evaluation of multicast MISO-OFDM systems. Ann. Telecommun. 63(5), 295–305 (2008) 11. M. Senel, V. Kapnadak, D. Love, Spatial multiplexing with opportunistic scheduling for multiuser MIMO-OFDM systems, in Global Telecommunications Conference, 2006. GLOBECOM ‘06. IEEE, Nov 2006, pp. 1–5 12. W.-C. Pao, W.-T. Lou, Y.-F. Chen, D.-C. Chang, Resource allocation for multiple input multiple output-orthogonal frequency division multiplexing-based space division multiple access systems. IET Commun. 8(18), 3424–3434 (2014) 13. Z. Hu, G. Zhu, Y. Xia, G. Liu, Multiuser subcarrier and bit allocation for mimo-ofdm systems with perfect and partial channel information, in Wireless Communications and Networking Conference, 2004. WCNC. 2004 IEEE, vol. 2, March 2004, pp. 1188–1193 14. M. Moretti, A. Perez-Neira, Efficient margin adaptive scheduling for MIMO-OFDMA systems. IEEE Trans. Wirel. Commun. 12(1), 278–287 (2013) 15. L. Goldfeld, V. Lyandres, D. Wulich, Minimum ber power loading for ofdm in fading channel. IEEE Trans. Commun. 50(11), 1729–1733 (2002) 16. N. Wang, S. Blostein, Approximate minimum ber power allocation for mimo spatial multiplexing systems. IEEE Trans. Commun. 55(1), 180–187 (2007) 17. L. Song, Relay selection for two-way relaying with amplify and forward protocols. IEEE Trans. Veh. Technol. 60(4), 1954–1959 (2011) 18. D. Feng, C. Jiang, G. Lim, L.J. Cimini, G. Feng, G. Li, A survey of energy-efficient wireless communications. IEEE Commun. Surv. Tutor 15(1), 167–178 (2012)., First Quarter 2013 19. Evolved Universal Terrestrial Radio Access (E-UTRA); LTE physical layer; General description, 3GPP, TS36.201, Mar. 2015 20. C. Xiong, G. Li, Y. Liu, Y. Chen, S. Xu, Energy-efficient design for downlink ofdma with delay-sensitive traffic. IEEE Trans. Wirel. Commun 12(6), 3085–3095 (2013) 21. Z. Ren, S. Chen, B. Hu, W. Ma, Energy-efficient resource allocation in downlink ofdm wireless systems with proportional rate constraints. IEEE Trans. Veh. Technol 63(5), 2139–2150 (2014) 22. C. Xiong, G. Li, S. Zhang, Y. Chen, S. Xu, Energy- and spectral-efficiency tradeoff in downlink ofdma networks. IEEE Trans. Wirel. Commun 10(11), 3874–3886 (2011)

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Chapter 5

Dealing with Imperfect CSI

5.1

Introduction

Orthogonal frequency-division multiple access (OFDMA) is a promising multiple-access technique that provides high spectral efficiency for next-generation broadband wireless systems. In the downlink of a cellular OFDMA system, the base station (BS) communicates with users over a set of subcarriers. For systems that employ frequency-division duplexing (FDD), the BS obtains the downlink channel state information (CSI) from users through feedback channels. If perfect CSI is available at the BS, flexible resource allocation schemes can considerably improve system performance. However, conveying perfect CSI requires infinite CSI feedback rate. In practical communication systems, since the capacity of the feedback channel is limited, only quantized feedback CSI can be fed back to the BS. As a result, the performance of resource allocation schemes is degraded due to imperfect CSI. Analyzing the effect of finite feedback rate in OFDMA systems turns to be a crucial problem. The impact of imperfect CSI for OFDM systems has been an active research area in recent years. The effect of feedback delay was addressed in [1]. The author considered a minimum square error channel prediction scheme to overcome the detrimental effect of feedback delay and proposed resource allocation algorithms to maximize the downlink throughput. The works in [2–4] focused on the imperfect CSI resulting from channel estimation error and proposed power loading algorithms for the single-user OFDM system. Resource allocation with quantized CSI was investigated in [5–7]. The authors in [5] assumed uniform power distribution over subcarriers and derived closed-form expressions for the downlink throughput. In [4, 5], the design parameters related to imperfect CSI, such as quantization levels and the feedback period, were optimized to reduce the feedback overhead with a guaranteed system performance for OFDMA systems. However, most previous research works, such as [3–5], were based on suboptimal quantization method.

© Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8_5

83

84

5 Dealing with Imperfect CSI

Most previous research works on OFDMA systems, such as [1, 5–7], take the ergodic throughput as the performance measure. The ergodic throughput is defined as a long-term achievable data rate averaged over all fading states [8]. For applications insensitive to delay, the ergodic throughput is a suitable performance measure. However, for real-time applications that cannot tolerate long delays, it is more appropriate to consider the data rate that can be maintained in all fading states. This gives rise to the concept of the outage throughput, the throughput versus outage probability [9]. The outage probability for a given rate is the probability that the instantaneous channel capacity falls below that rate, and the outage throughput is the expected information delivered to users in non-outage fading states. However, to best of our knowledge, no work has considered the optimization of the outage throughput in OFDMA systems yet. The concept of green communication is promoted due to large energy consumption brought by the high volumes of data traffic in recent years. As OFDMA has been widely adopted by all major wireless communication systems, energy-efficient OFDMA systems play a crucial role in realizing green communication and networks. Most existing works, such as [10–12], have typically assumed that the perfect CSI is known. Unfortunately, this assumption is quite unrealistic for a practical system. The authors in [13] proposed an energy-efficient design scheme considering imperfect CSI by using its probability distribution. However, for systems that employ frequency-division duplexing (FDD), the base station (BS) can obtain quantized CSI in addition to the probability distribution from the feedback channel with limited capacity. Hence, it is critical to analyze the effect of quantized CSI. However, no work to the best of our knowledge has considered the energy efficiency with quantized CSI in OFDMA systems. Secrecy or private message exchanges between mobile users and the base station (BS) are generally needed in present and future wireless systems. Hence, it is essential to integrate physical layer security into the resource allocation problem in multiuser OFDMA systems. In [14], Li et al. investigated independent parallel channels and proved that the secrecy rate of the system is the summation of the secrecy capacity achieved on each independent channel. In [15], Jorswieck and Wolf tried to find the power and subcarrier allocation in an OFDM-based broadband system with the objective of maximizing the sum secrecy rate. Wang et al. [16] considered the problem of secure communications in OFDMA networks in which there are two groups of users: secure users and ordinary users. Their objective is to maximize the ordinary users’ data rate under the individual secrecy rate constraint for secure users and the total transmit power of the BS. In [17], the authors’ objective is to assign subcarriers and allocate the transmit power to optimize the maximumminimum (max-min) secrecy rate among all legitimate users. In most of previous works, it is assumed that the CSI of both the legitimate links and the eavesdropper link are perfectly known at the BS. This assumption is quite unrealistic due to channel estimation errors and, more importantly, channel feedback delay and limited feedback rate. Motivated by the aforementioned observations, we focus on the case where only imperfect (partial) CSI is available in this paper. We consider the resource allocation problem for optimizing the max-min criterion over

5.2 Channel Model with Imperfect CSI

85

Fig. 5.1 Resource allocation in OFDMA system

the user’s secrecy rate under an average total transmitted power constraint with imperfect CSI. To the best of our knowledge, no work has considered the max-min security rate with imperfect CSI of both legitimate users and eavesdropper in OFDMA systems.

5.2

Channel Model with Imperfect CSI

Figure 5.1 shows the resource allocation of the base station based on CSI in OFDMA systems. We assume that there are K users and N subcarriers in the system. The base station obtains the downlink CSI from the uplink feedback and then determines the resource allocation strategy according to the downlink CSI to optimize the system performance. In the TDD system, the base station can directly estimate the CSI of the downlink according to the state of the uplink due to channel heterogeneity. In the FDD system, the user needs to encode the estimated downlink CSI and feed it back to the base station through the uplink. In the partial CSI system, the base station can optimize system performance in a statistically significant sense given the known CSI and the conditional probability distribution of the real CSI relative to the known CSI. For this reason, this section first presents the conditional probability distribution function of the partial CSI obtained by the base station with respect to real CSI in three imperfect CSI scenarios.

5.2.1

Nosie and Estimation Error

In this subsection, we consider the CSI errors caused by noise and estimation error which can be modeled
as a zero-mean complex Gaussian vector with independent entries εk, n
CN 0; σ 2k, n , where σ 2k, n is the mean variance of channel estimation error

86

5 Dealing with Imperfect CSI

^ k , n þ εk , n , H k, n ¼ H

ð5:1Þ

^ k, n denotes the CSI obtained by the BS. Thus, we can have the conditional where H ^ k, n for a given H ^ k, n is also a zero-mean probability density function (pdf) of H 
^ k, n
CN H ^ k, n ; σ 2k, n . Thus, the actual complex Gaussian distribution H k, n j H   ^ k, n 2 is a noncentral power gain αk, n ¼ |Hk, n|2 conditioned on the estimated α^ k, n ¼ H Chi-squared (NCχ 2) distributed random variable with two degrees of freedom with pdf of




f αk, n j^ α k, n ¼

1 σ 2k, n

e

þα α^  k, n2 k, n σ

k, n

I0

! 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α^ k, n αk, n , 2

σ k, n

ð5:2Þ

where I0(.) is the 0th-order modified Bessel function of the first kind.

5.2.2

CSI Feedback Delay

In this subsection, we consider the feedback delays and channel prediction errors of the downlink CSI. Each user estimates the CSI of the downlink channel and sends its channel estimation to the BS. Based on the finite number of past observation of feedback estimates, the BS performs an δmk-step ahead LMMSE (linear minimum mean squared error) prediction method for the CSI of user k, where δmk is equal to the feedback delay of user k. Assuming that the channel gain during the time interval of η OFDM symbols is constant, we denote the index m by the signal sampled at the time mηTs, where Ts is the OFDM symbol period. The baseband channel gain at the k-th user on the n-th subcarrier at the time index m is H k , n ðm Þ ¼

Lk X l¼1



   Nþ1 ak, l ðmÞexp j2πτk, l n  Δf , 2

ð5:3Þ

where Lk is the number of multipath taps, Δf is the subcarrier spacing, and ak, l and τk, l denote the attenuation factor and the propagation delay of the l-th multipath tap at the k-th user’s channel, respectively. The multipath channel taps at the k-th user (ak, l, . . . , ak, L)T can be modeled as a zero-mean circularly symmetric  complex 2 Gaussian (ZMCSCG) vector with independent entries ak, l
CN 0; σ ak:l . Then, we can note that Hk ¼ (Hk,1, . . . , Hk,N)T is also WSS ZMCSCG process H k
CN ð0; ΣHk Þ with the entries of the covariance matrix: ½RHk n1 , n2 ¼ Fk Dak FkH ¼

Lk X l¼1

σ 2ak, l expðj2πτk, l ðn1  n2 ÞΔf Þ,

ð5:4Þ

5.2 Channel Model with Imperfect CSI

87

where Fk is an NhLk matrix i with hentries [F i k]n, l ¼ exp (j2π(n  (N + 1)/2)τk, lΔf ) and Dak ¼ diag E jαk,1 j2 ; . . . ; E jαk, Lk j2 . The identical normalized temporal autocorrelation function of the attenuation factors satisfies Jakes’ model:

   E ak, l ðm þ t Þak, l ðmÞ f h i ¼ J 0 2πvk 0 T s t , ϕk ðt Þ ¼ c0 E jak, l j2

ð5:5Þ

where J0(x) is the 0th-order of the Bessel function of the first kind, vk is the velocity of the user k, f0 is the carrier frequency, and c0 is the speed of the light in vacuum. 
~ k ðmÞ ¼ H ~ k ,1 ; . . . ; H ~ k, N T of user We assume that the channel estimation H ~ k ðmÞ ¼ H k ðmÞ þ εk ðmÞ, where εk ¼ (εk, 1, . . . , εk, N)T denotes the k satisfies H
 estimation error of user k, and it satisfies εk
CN 0; σ 2ε I N . Assume the CSI feedback delay is δmk, the BS predicts the channel gain of user k through a linear
T  ~ ðmÞ; . . . ; H ~ T ðm  Q þ 1Þ T , and ~ Q ðm Þ ¼ H regressive approach with order Q: H k k k suppose that the BS predicts the CSI using LMMSE method ~ Q ðmÞ, ^ k ðm þ δmk Þ ¼ Ak H H k

ð5:6Þ


T  ~ Q ðm Þ ¼ H ~ ðmÞ; . . . ; H ~ T ðm  Q þ 1Þ T is the past feedback of CSI and where H k k k Ak is the optimal predictor in the LMMSE sense, which can be calculated by h
Q  i h Q
Q  i 1 ~ ðm  δmk Þ H E H ~ ðm  δmk Þ H ~ ðm  δmk Þ H A k ¼ E H k ðm Þ H k k k
1 2 ¼ ðΦk  RHk Þ Γ k  RHk þ σ ε INQ , ð5:7Þ where Γ k is an N  N matrix with [Γ k]i, j ¼ ϕk( j  i), Φk ¼ (ϕk(Dk), . . . , ϕk(Dk + Q  1)), and  denotes the Kronecker product. Defining ^ k, n ðmÞ and vk ¼ (vk, 1, . . . , vk, N)T, from the prediction error vk, n ðmÞ ¼ H k, n ðmÞ  H (5.4) and (5.6), we have

 ^ k ðmÞ E ½vk ðmÞ ¼ E ½H k ðmÞ  E H

Q  ~ ðm  δmk Þ ¼ 0 ¼ E ½H k ðmÞ  Ak E H k

ð5:8Þ

and


1 Rvk ≜ E vk vkH ¼ RHk  ðΦk  RHk Þ Γ k  RHk þ σ 2ε INQ ðΦk  RHk ÞH : ð5:9Þ ^ k are WSS ZMCSCG vector processes, the estimation error vk is also Since Hk和H a WSS ZMCSCG vector process. Thus, we can have the prediction error vk is ^ k , and they are independent with each other: orthogonal with H

88

5 Dealing with Imperfect CSI



^ H ¼ 0: E vk H k

ð5:10Þ

^ Then we can have the probability (pdf)
of Hk, n, for  density function   a given 2H k, n 2   ^ ^ ^ is Gaussian distributed with H k, n H k ¼ H k, n H k, n
CN H k, n ; σ k, n , where σ k, n is the n‐th diagonal component of Rvk . It follows that the pdf of the actual power gain   ^ k, n 2 is noncentral αk,n ¼ |Hk,n|2 conditioned on predicted power gain α^ k, n ¼ H Chi-squared (NCχ 2) distributed random variable with two degrees of freedom with pdf of




f αk, n j^ α k, n ¼

5.2.3

1 σ 2k, n

e

α^ þα  k, n2 k, n σ

k, n

I0

! 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α^ k, n αk, n : 2

σ k, n

ð5:11Þ

Finite-Rate Feedback of Downlink CSI

Now we consider the quantization of downlink CSI and determine the capacity of feedback channel required to deliver the quantized CSI using the rate-distortion theory. From this, we can characterize the minimum distortion of the quantized CSI for a given capacity of the feedback channel. his/her knowledge of downlink CSI Hk by an index I k 2 I k The user k describes

¼ 1; 2; . . . ; 2Rk and sends the index Ik to the BS. The BS then reproduces the

^ k from the index Ik, where H ^k ¼ H ^ k ,1 ; . . . ; H ^ k, N T is the quantized channel gain H description of Hk. We introduce the distortion measure function with the following criterion: N  
 X H k , n  H ^k ¼ ^ k , n 2 : d Hk ; H

ð5:12Þ

n¼1

Then, we can define the information RDF of Hk as
 ^k , Rk ðDk Þ ¼ inf I H k ; H

ð5:13Þ


^ k denotes where Dk denotes an upper bound on the quantization error and I H k ; H ^ k. the mutual information between Hk and H By the source-channel separation theorem [18], the quantization error Dk is achievable if and only if the feedback channel’s capacity of the user k satisfies Ck > Rk(Dk). Thus, to characterize the feedback channel’s capacity Ck, we need to find the RDF for Hk. We have the following result.

5.2 Channel Model with Imperfect CSI

89

Theorem 5.1 Suppose that we are given a ZMCSCG vector Hk with the autocorrelation given in (5.4), e.g., H k
CN ð0; RHk Þ. Let the eigenvalue decomposition of RHk be RHk ¼ Uk Λk U H ,

ð5:14Þ

where Uk is the N  N matrix with orthogonal column vectors and Λk is the N  N diagonal matrix with [Λk]n,n ¼ λk,n. Then, 1. The RDF of Hk is given by

Rk ðDk Þ ¼

N X n¼1

 log max

 λk , n ;1 , θk

ð5:15Þ

where Dk ¼

N X

minfθk ; λk, n g:

ð5:16Þ

n¼1

Here, θk is the Lagrangian multiplier which can be decided for given Dk to satisfy (5.16). 2. The test channel that achieves the RDF is given by ^ k þ U k Zk , Hk ¼ H ^ H k ¼ U k Y^ k ,

ð5:17Þ

T
where Y^ k ¼ Y^ k,1 ; . . . ; Y^ k, N and Zk ¼ (Zk,1, . . . , Zk,N)T are mutually independent ZMCSCG vectors with uncorrelated components and Y^ k, n
CN ð0; maxfλk, n  θk ; 0gÞ and Z k, n
CN ð0; minfλk, n ; θk gÞ. Proof We prove the theorem by transforming the correlated ZMCSCG vector Hk to an uncorrelated ZMCSCG vector Yk ¼ (Yk, 1, . . .Yk, N)T by setting Yk ¼ UkH Hk . ^ k . Since Uk is unitary and From (5.14), we have Yk
CN ð0N ; λk Þ. Let Y^ k ¼ U kH H 



^ k and d Y k , Y^ k ¼ d H k ; H ^ k . Now the invertible, we have I Y k ;Y^ k ¼ I H k ; H problem reduces to finding the RDF for the uncorrelated ZMCSCG vector Yk. Following the similar approach which derives the RDF of a parallel Gaussian source in [18, Theorem 10.3.3], we can obtain the result in Theorem 5.1. ^k By the second part of Theorem 5.1, the probability distributions of UkZk and H are Uk Zk
CN ð0N ; Σk Þ,

^ k
CN ð0N ; ΣHk  Σk Þ, H

ð5:18Þ

90

5 Dealing with Imperfect CSI

where Σk ¼ Uk Σzk UkH and Σzk ¼ diagðminfλk,1 ; θk g; . . . ; minfλk, N ; θ; k gÞ. The n‐th σ 2k, n , can be regarded as the diagonal element of Σk, denoted by variance of quantization error for Hk,n From  (5.18), the conditional pdf
(5.17)2 and ^ k, n is H k, n j H ^ k, n
CN H ^ k, n ; σ k, n . Thus, the actual power gain of Hk,n for a given H   ^ k, n 2 is αk,n ¼ |Hk,n|2 conditioned on α^ k, n ¼ H


f αk, n j^ α k, n



1  ¼ 2 e αk , n

α^ k, n þαk , n σ2 k, n

I0

! 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α^ k, n αk, n , 2

σ k, n

ð5:19Þ

where I0() is the 0‐th order modified Bessel function of the first kind. Note that although we only consider the distortion due to the quantization in this paper, it is straightforward to take into account feedback delay through the prediction error as shown in [19, Eq. (7)]. In the above three cases of imperfect CSI conditions, the conditional pdf of Hk for
 ^ k are in the same form H k, n j H ^ k, n
CN H ^ k, n ; σ 2k, n . And the actual power a given H   ^ k, n 2 satisfies the noncentral Chi-squared gain αk,n ¼ |Hk,n|2 conditioned on α^ k, n ¼ H (NCχ 2) distribution with two degrees of freedom in the same form (5.2), (5.11), and (5.19), where σ 2k, n can be regarded as the CSI error caused by feedback delay and feedback limitations. Thus, we can optimize thedifferent performances in OFDMA
system based on the conditional pdf f αk, n j^ α k, n .

5.3

5.3.1

Downlink Ergodic Throughput Maximization for OFDMA Systems with Feedback Channel Capacity Constraints Problem Formulation

The ergodic throughput is defined as the average data rate over all possible fading states. For a given quantized power gain, α^ k, n , the ergodic throughput of the n‐th subcarrier, provided that this subcarrier is assigned to the k‐th user, is expressed as
 T ke, n pn ; α^ k, n ¼ Eαk, n j^α k, n ½log2 ð1 þ αk, n pn Þj^ α k, n 

ð5:20Þ

where pn denotes the input signal-to-noise ratio (SNR) of the n‐th subcarrier, which is proportional to the transmit power on the n‐th subcarrier. In the following, the terms “input SNR” and “transmit power” are used without distinction. Thus, according to the T ke, n , we can define the overall weighted sum of ergodic throughput conditioned on the quantized CSI, α^ k, n ,

5.3 Downlink Ergodic Throughput Maximization for OFDMA Systems with. . .

Te ¼

K X N X


 wk ρk, n T ke, n pn ; α^ k, n ,

91

ð5:21Þ

k¼1 n¼1

where ρk,n denotes the subcarrier indicator: if the n‐th subcarrier is assigned to the k‐ th user, then ρk,n ¼ 1; otherwise ρk,n ¼ 0. wk is the positive constant such that ∑kwk ¼ 1. In this problem, we assume that wk are given by users’ priorities. Now we can formulate the downlink throughput maximization problem under the overall power constraint as follows: max ρk , n , pn

K X N X


 wk ρk, n T ke, n pn ; α^ k, n

k¼1 n¼1

8X ρ ¼ 1 8n > > k k, n > > < ρ 2 f0; 1g 8k, n k, n , subject to X > > p p > n T > n : pn 0 8n

ð5:22Þ

where the first two constraints ensure that each subcarrier is assigned to one user exclusively and the third and fourth constraint is for total transmit power, denoted by pT, and the power allocation should be positive, respectively.

5.3.2

Optimal Solution

The problem (5.22) is a mixed integer programming problem which is NP-hard, and the optimal solution method has higher complexity. In order to get the optimal solution to the problem (5.22), all possible subcarrier allocations should be enumerated. For a given subcarrier allocation, solving the power allocation problem can be achieved by converting it to a convex optimization problem as we discussed in Sect. 5.3.3.2. The optimal solution algorithm is summarized in Algorithm 5.1. Algorithm 5.1 Optimal Algorithm for Ergodic Throughput Maximization 1: Initialize R ¼ 0; 2: for all possible subcarrier assignment ρk,n, where ∑kρk, n ¼ 1 do 3: Calculate the power allocation pn
according  to Sect. 4.2.1; PP 4: Calculate R ¼ k n wk ρk, n T ke, n pn ; α^k , n 5: if R > R then 6: R ¼ R; 7: Save the corresponding ρk,n and pn; 8: end if 9: end for 10: return R, and the corresponding ρk,n and pn.

92

5 Dealing with Imperfect CSI

Since each subcarrier can only be assigned to one user, the complexity of exhausted search method to find the optimal subcarrier assignment is O(KN). As we discussed in Sect. 5.3.3.2, for a given subcarrier assignment, the complexity of power allocation is O(N ). Thus, the complexity of the optimal solution is O(KNN).

5.3.3

Suboptimal Algorithm

In this section, we propose a suboptimal algorithm with an acceptable complexity by breaking the original problem into two steps: subcarrier assignment and power allocation. In the first step, we propose the subcarrier allocation algorithm under the assumption that the power is equally allocated to each subcarrier. Then in the second step, we will calculate the optimal power allocation to each subcarrier according to subcarrier in the last step.

Subcarrier Allocation When the transmit power pn ¼ pT/N is fixed, the original problem (5.22) is decomposed into N independent subproblems: X


 wk ρk, n T ke, n pT =N; α^ k, n (X ρ ¼ 1, 8n k k, n subject to ρk, n 2 f0; 1g, 8k, n

max ρk , n

k

ð5:23Þ

Thus, the optimal subcarrier allocation is given by  ρk , n ¼


 1 k ¼ arg maxk wk T ke, n pT =N; α^ k, n 0 else

ð5:24Þ

Power Allocation When the subcarrier assignment ρk,n is fixed, the problem in (5.22) becomes X


 e ^ w T ; α p k k , n n n n , n k n n X p pT n n subject to pn 0 8n

max pn

ð5:25Þ

where kn ¼ arg maxk ρk,n denotes the assigned user on the n‐th subcarrier. We solve the power allocation problem through the dual approach. The dual of (5.25) is

5.3 Downlink Ergodic Throughput Maximization for OFDMA Systems with. . .

93

min gðμÞ, μ 0

where gðμÞ ¼

max Lðμ; p1 ; . . . ; pN Þ p1 ,...,pN 0 X






X

!

max pn  pT pn ; α^ kn , n  μ p1 ,...,pN 0 n n X ¼ max ðw E½log2 ð1 þ αkn , n pn Þj^ α kn , n   μpn Þ þ μpT , pn 0 kn n ¼

wkn T ken , n

where μ is the Lagrangian multiplier of the first constraint in (5.25). Given μ, the optimal power allocation on the n‐th subcarrier is  pn ðμÞ ¼

p~n 0



wkn E αkn , n j^ α kn , n  μ ln 2 wkn E αkn , n j^ α kn , n μ ln 2

ð5:26Þ

where p~n satisfies 

 αkn , n j^ α k , n ¼ μ ln 2 wkn E 1 þ αkn , n p~n n

ð5:27Þ

According to (5.26) and (5.27), pn is a monotonically decreasing function of μ. Thus, we can use bisection method to search the solution of (5.26). For
each subcarrier n, we need to find the user which has the max  wk T ke, n pT =N; α^ k, n among K users. Thus, the complexity of subcarrier allocation is O(KN). In power allocation, we need to define pn according to (5.26) and (5.27) with given μ. Assuming that there is Iμ searches for the optimal. Thus, the complexity of power allocation is O(IμN ). If we ignore the constant Iμ, the whole complexity of the algorithm is O(KN).

5.3.4

Simulation Results

In Fig. 5.2, we simulate the system performance in the rate-distortion limit under different frequency-selective channels. We also consider the case in which the BS has perfect CSI. In this simulation, we assume that there are N ¼ 32 subcarriers with transmit power pT/N ¼ 20 dB, and K ¼ 8 users with equal weights wk ¼ 1/K. We can see in Fig. 5.2 that ergodic throughput increases as the subcarrier spacing decreases for a given feedback channel’s capacity. At Δf ¼ 15 kHz, the ergodic throughput can achieve 99% of that with perfect CSI when Ck/N ¼ 0.43 b/s/Hz. Thus, we can see that although the feedback channel’s capacity is finite, a near ideal performance can be achieved.

94

5 Dealing with Imperfect CSI

Fig. 5.2 Ergodic throughput versus capacity of feedback channel

5.3.5

Conclusion

In this section, we investigated the downlink ergodic throughput maximization for an OFDMA system with finite feedback rate. First, assuming that the ZMCSCG channel information is fed back to the BS, we derived the RDF for the CSI. According to the rate-distortion theory, the RDF can give a lower bound on the capacity of feedback channel. We also derived the test channel that achieves this RDF. This derived test channel enables us to formulate the resource allocation problems that maximize the ergodic throughput with a rate constraint on feedback channel. Then, we proposed an iterative method to solve the throughput maximization problem and showed that the proposed method achieves the optimum for ergodic throughput. Through numerical results, we found that by exploiting the correlations between subcarriers, the ergodic throughput with a limited feedback rate can approach that with perfect CSI.

5.4

5.4.1

Resource Allocation for Maximizing Outage Throughput in OFDMA Systems with Finite-Rate Feedback Problem Formulation

In this section, we define the outage throughput. Given α^ k, n , the outage probability on the n‐th subcarrier to the k‐th user is ξk, n ¼ Pr ðlog2 ð1 þ αk, n pn Þ < Rj^ α k, n Þ,

ð5:28Þ

5.4 Resource Allocation for Maximizing Outage Throughput in OFDMA Systems. . .

95

where R is the transmission rate. It can be shown that the maximum transmission rate R that can maintain the outage probability ξk, n is 


 R pn ; α^ k, n ; ξk, n ¼ max 1  ξk, n log2 1 þ pn F 1 α k , n ξk , n αk, n j^ ξk , n

ð5:29Þ



  α k, n where F 1 is the inverse function of F 1 αk, n j^ αk, n j^ α k , n ξk , n α k, n ðxÞ ¼ Pr αk , n < xj^ which denotes the cumulative distribution function (cdf) of αk,n conditioned on α^ k, n . Here, we define the outage throughput as the maximum expected information successfully delivered to users,


 T ko, n pn ; α^ k, n ¼ max 1  ξk, n R pn ; α^ k, n ; ξk, n ξk , n 

 ¼ max 1  ξk, n log2 1 þ pn F 1 : α k , n ξk , n αk, n j^ ξk , n

ð5:30Þ


 Setting ~α k, n ¼ F 1 αk, n j^ α k, n ξk , n and substituting (5.19) into (5.30) yield

 
 α k, n log2 1 þ ~α k, n pn T ko, n pn ; α^ k, n ¼ max Pr αk, n ~α k, n j^ ~α k, n sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffi!
 2^ α k, n 2~α k, n log2 1 þ ~α k, n pn ; ¼ max Q 2 2 σ σ ~α k, n k, n
k, n  ¼ max T~ ko, n pn ; α^ k, n ; ~α k, n ~α k, n where


 T~ ko, n pn ; α^ k, n ; ~α k, n ¼ Q

R þ1

ð5:31Þ

rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi
 2^ α k, n 2~ α ; σ 2k, n log2 1 þ ~α k, n pn σ2 k, n

and

k, n

2 2 Qða; bÞ ¼ b
xeðx þa Þ=2 I 0 ðaxÞdx. The following theorem shows that the outage throughput T ko, n pn ; α^ k, n is well defined.

Theorem 5.2 Given pn 0 and α^ k, n 0, there exists a unique globally optimal
 ~α k, n that maximizes T~ ko, n pn ; α^ k, n ; ~α k, n .
pffiffiffi pffiffiffi Proof It has been proved in [19, Theorem 2.7] that ln Q a; b is concave in b for

 b > 0, a 0. Thus, it is easy to see that ln T~ ko, n pn ; α^ k, n ; ~α k, n ¼ ln log2 1 þ ~α k, n pn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ln Q 2^ α k, n =σ 2k, n ; 2~α k, n =σ 2k, n is concave in ~α k, n . In addition, we have
 lim T~ ko, n pn ; α^ k, n ; ~α k, n

~α k, n !0



 ¼ lim log2 1 þ ~α k, n pn Pr αk, n ~α k, n j^ α k, n ~α k, n !0 
¼ log2 ð1 þ 0pn ÞPr αk, n ~α k, n j^ α k, n ¼ 0 ð5:32Þ

and

96

5 Dealing with Imperfect CSI


 T~ ko, n pn ; α^ k, n ; ~α k, n ~α k, n !þ1
 log2 1 þ ~α k, n pn  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2  ~α k, n !þ1 exp
2~α =σ 2  2^ α k, n =σ 2k, n =2 k, n k, n

0

13:46a

¼

lim

1 p  n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , lim ln 2 ~α k, n !þ1
1 þ ~α k, n p exp
1  α^ k, n =~α k, n =σ 2 n k, n 1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  ¼ 0 
exp α k, n =σ 2k, n =2 2~α k, n =σ 2k, n  2^

ð5:33Þ

where we have used L’Hospital’s rule and the upper bound Q(a, b) exp ((b 
a)2 ¼ 2). Thus,  there exists a unique and globally optimal ~α k, n that maximizes T~ ko, n pn ; α^ k, n ; ~α k, n . According 5.2, given pn and α^ k, n , we can use bisection method to
to Theorem  obtain T ko, n pn ; α^ k, n . Thus, the outage throughput of the system with given α^ k, n is To ¼

K X N X


 wk ρk, n T ko, n pn ; α^ k, n

ð5:34Þ

k¼1 n¼1

where ρk, n is subcarrier allocation indicator: if the n‐th subcarrier is assigned to the k‐ th user, ρk,n ¼ 1; otherwise, ρk,n ¼ 0. Now we can formulate the downlink throughput maximization problem as follows: K X N X


 wk ρk, n T ko, n pn ; α^ k, n k¼1 n¼1 8X ρ ¼ 1 8n > > k k, n < 2 f0; 1g 8k, n ρX k, n subject to > p pT > : n n pn 0 8n max ρk , n , pn

ð5:35Þ

where wk is the positive constraint such that ∑kwk ¼ 1. Here, the first two constraints ensure that each subcarrier is assigned to one user exclusively, and the third constraint is for total transmit power, denoted by pT.

5.4.2

Upper Bound of the Optimal Solution

Problem (5.35) is mixed integer programming problem of which optimal algorithm is with high complexity. In this section, we propose the solution to obtain the upper bound of the optimal. Firstly, we can search for the all possible subcarrier allocation ρk,n. In Subsection 5.4.3, we will explain that the power allocation pn with given

5.4 Resource Allocation for Maximizing Outage Throughput in OFDMA Systems. . .

97

subcarrier allocation ρk,n is non-convex optimization problems, but we can use dual method to obtain the upper bound of the outage throughput maximization when subcarrier allocation is given. Thus, we can obtain the upper bound of problem (5.35) by choosing the maximum of all possible subcarrier assignments. The whole algorithm is shown as Algorithm 5.2. Algorithm 5.2 Optimal Algorithm for Outage Throughput Maximization Initialize R ¼ 0; for all possible subcarrier assignment ρk,n, where ∑kρk, n ¼ 1 do Calculate the power allocation pn
(μ) according  to Sect. 4.2.1; PP Calculate R ¼ k n wk ρk, n T ko, n pn ðμ Þ; α^k , n if R > R then R ¼ R; Save the corresponding ρk,n and pn; end if end for  and the corresponding ρk,n and pn. return R, Since each subcarrier can only be assigned to one user, the complexity of exhausted search method to find the optimal subcarrier assignment is O(KN). As we discussed in Subsection 5.4.3, for a given subcarrier assignment, the complexity of power allocation is O(N ). Thus, the complexity of determining the optimal solution upper bound is O(KNN ).

5.4.3

Suboptimal Solution

In order to reduce the computation complexity, we can break the original problem into two subproblems to obtain the suboptimal solution of the problem in (5.35). Similar with Sect. 5.3.3, in the first step, we propose the subcarrier allocation method under the assumption that the power is equally allocated to each subcarrier. And then, we define the power allocation for each subcarrier with the subcarrier allocation in the first step.

Subcarrier Algorithm Under the assumption of pn ¼ pT/N, the optimization problem in (5.35) reduces to X


 wk ρk, n T ko, n pT =N; α^ k, n (X ρ ¼ 1, 8n k k, n : subject to ρk, n 2 f0; 1g, 8k, n

max ρk , n

k

ð5:36Þ

It implies that the subcarriers should be assigned based on the following criterion:

98

5 Dealing with Imperfect CSI

 ρk , n ¼


 1 if k ¼ arg maxk wk T ko, n pT =N; α^ k, n 0 otherwise

Note that if σ 2k, n and wk are identical for all k in (5.34), the subcarrier allocation is simplified as  ρk , n ¼

1 if k ¼ arg maxk α^ k, n 0 otherwise

When a tie occurs, we select the user in random fashion.

Power Allocation Given the subcarrier allocation, the problem in (5.35) becomes X


 wkn T kon , n pn ; α^ kn , n X p pT n n subject to pn 0 8n

max pn

n

ð5:37Þ


 where kn ¼ arg maxk ρk, n. Since T kon , n pn ; α^ kn , n is non-concave function of pn, the optimal solution to the problem (5.37) cannot be obtained using the standard algorithm of convex optimization. We can obtain a suboptimal solution through the dual approach. The dual of the problem in (5.37) is min gðμÞ

ð5:38Þ

μ 0

where gð μ Þ

X






X

!

pn  pT max pn ; α^ kn , n  μ p1 ,...,pN 0 n  X
 ¼ max wkn T kon , n pn ; α^ kn , n  μpn þ μpT pn 0 n

¼

w To n kn kn , n

where μ is the Lagrangian multiplier of the first constraint in (5.37). Given μ, the optimal power allocation on the n‐th subcarrier is
 pn ðμÞ ¼ arg max wkn T kon , n p; α^ kn , n  μp, p

ð5:39Þ

where the optimal μ can be obtained using the bisection method. Note that it is possible that the final power allocation pn(μ) may not satisfy ∑npn(μ) pT. To

5.4 Resource Allocation for Maximizing Outage Throughput in OFDMA Systems. . .

99

project the power allocation to the feasible region, we multiply the final power allocation on each subcarrier pT pn ðμÞ 8n n¼1 pn ðμÞ

pn ¼ P N

ð5:40Þ

In the suboptimal
algorithm, for each subcarrier n, we need to find the user which has the max wk T ko, n pT =N; α^ k, n among K users; thus the complexity of subcarrier allocation is O(KN). At the same time, assuming that the one-dimensional search algorithm for determining the power allocation pn converges in Ip steps and obtaining the optimal μ requires Iμ times iterations, the complexity of power allocation is O(IμIPN ). Since Ip and Iμ are both constants, the whole complexity of the suboptimal algorithm is O(KN).

5.4.4

Simulation Results

We present numerical results to demonstrate the performance of our proposed algorithms in the rate-distortion limit. Our simulation is based on the COST259 channel model in a typical urban environment [20]. First, in Fig. 5.3, we show the 2-user rate region achieved by our proposed suboptimal algorithms for the problem in (5.35) by changing the values of wk. We also plot the rate region achieved by an exhaustive search algorithm. This algorithm searches the maximum outage throughput among all possible subcarrier allocations, and for each subcarrier allocation, it assigns the transmit power using the algorithm given in Subsection 5.4.3.2 without projecting the final power allocation back to the feasible region. This approach can give an upper bound on the optimum throughput, which is the solution of the problem in (5.35) [21]. Here, we assume that the channel gain is independent identically distributed (iid) with N ¼ 8 subcarriers shared by K ¼ 2 users. In addition, the downlink signal noise ratio (SNR) is pT/N ¼ 30 dB, and the capacity of feedback channel is Ck ¼ 24 bps/Hz. In Fig. 5.3, we see that the proposed suboptimal algorithm can achieve near-optimal performance with loss of less than 4%, respectively. Next, in Fig. 5.4, we simulate the system performance under different frequencyselective channels assuming N ¼ 32 subcarriers, K ¼ 8 users, and wk ¼ 1/K. We also consider the case in which the BS has perfect CSI, where the user of the highest power gain is chosen on each subcarrier and the transmit power is assigned by the water-filling method [22]. Figure 5.4 shows that the outage throughput increases as the subcarrier spacing decreases. The reason is that by exploiting the correlation between subcarriers, the quantization distortion of the downlink CSI can be reduced, which can, in turn, boost the performance of the resource allocation at the BS. In the case of Δf ¼ 15 kHz, the outage throughput can achieve 99% of that with perfect CSI when the feedback capacity is Ck ¼ 58 bps/Hz (1.81 bits per subcarrier). On the other

100

5 Dealing with Imperfect CSI

Fig. 5.3 Comparisons of rate region

Fig. 5.4 Performance comparison under different frequency-selective fadings

hand, if no CSI is available, the BS tends to assign the subcarriers randomly to users and allocate equal transmit power
pn on each subcarrier. As a result, the outage 
throughput becomes Nmax~α k, n log2 1 þ ~α k, n pT =N Pr α > ~α k, n =K. This explains why the outage throughput converges to the same value as the feedback channel’s capacity goes to zero for a given SNR.

5.5 Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities

5.4.5

101

Conclusion

In this section, we investigated outage throughput maximization for an OFDMA system with finite feedback rate. First, assuming that the ZMCSCG channel information is fed back to the BS, we derived the RDF for the CSI. According to the ratedistortion theory, the RDF gives a lower bound on the capacity of feedback channel. We also derived the test channel that achieves this RDF. This derived test channel enables us to formulate the resource allocation problems that maximize the outage throughput with a rate constraint on feedback channel. We showed that the optimal algorithm has an exponential complexity, and then, we proposed a low-complexity suboptimal algorithm that divides the problem into two subproblems, namely, subcarrier and power allocation problems. Through numerical results, we found that the performance of the proposed suboptimal algorithms is close to the optimum, and by exploiting the correlations between subcarriers, the outage throughput with a limited feedback rate can approach that with perfect CSI. We also observed that the system performance can be improved as the subcarrier correlation increases.

5.5 5.5.1

Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities Problem Formulation

Based on the predicted CSI α^ k, n , the resource allocation can be carried out for a given performance measure. From this, we can formulate the resource allocation with predicted CSI. Toward this end, we introduce users’ minimum expected rate as the performance measure. For a given outage probability ξk,n, the outage capacity on the n‐th subcarrier of the k‐th user is defined as C~ k, n ¼ max C C     pn subject to Pr log2 1 þ αk, n 2 C ξk, n σr

ð5:41Þ

where pn is the transmit power on the n‐th subcarrier and σ 2r is the noise power. The expected rate that successfully delivered to the user in non-outage stage is represented as
 r k, n ¼ 1  ξk, n C~ k, n :

ð5:42Þ

Using the above notations, we can formulate the resource allocation problem as

102

5 Dealing with Imperfect CSI

max R ρk , n , pn , R 8P N > < Pn¼1 ρk, n r k, n R, 8k K subject to k¼1 ρk , n ¼ 1, ρk , n 2 f0; 1g, 8k, n > : PN n¼1 pn pT ,pn 0, 8n

ð5:43Þ

where pT is total transmit power, and ρk,n is the subcarrier allocation indicator: if the n‐th subcarrier is assigned to the k‐th user, ρk,n ¼ 1; otherwise, ρk,n ¼ 0. Here, the first constraint requires that the rate of each user k should be larger than R, the second constraint ensures that each subcarrier is assigned to one user exclusively, and the third constraint is the total transmit power constraint.

5.5.2

Optimal Solution

Problem (5.43) is a mixed integer programming problem, and the optimal solution has a higher complexity. In order to get the optimal solution to the problem (5.43), all possible subcarrier allocations should be searched. Since each subcarrier is assigned to only one user, there is KN groups of subcarrier allocations. From the analysis in Subsection 5.5.3, we can see that when the subcarrier allocation ρk,n is given, the power allocation problem can be solved by converting it into a convex optimization problem to obtain the optimal solution. Since the complexity of solving power allocation is O(N ), the total complexity of the optimal solution algorithm is O(NKN).

5.5.3

Suboptimal Solution

To make the problem in (5.43) tractable, we first re-express the outage probability and the corresponding throughput. From (5.41), we can show that  
 pn 1 ~ C k, n ¼ log2 1 þ 2 F αk, n j^α k, n ξk, n , σr

ð5:44Þ


α k, n denotes the conditional cumulative distribuwhere F αk, n j^α k, n ðxÞ ¼ Pr αk, n xj^ tion function (cdf) of the actual αk,n conditional on the estimated CSI α^ k, n . Setting
 ~α k, n ¼ F 1 α k, n ξk , n , αk, n j^

ð5:45Þ

we can rewrite (5.42) as




r k, n ¼ 1  ξk, n log2

 1 þ ~α k, n

 pn : σ 2r

ð5:46Þ

5.5 Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities

103

Here, in (5.46), rk, n is determined by the transmit power pn and the outage
probability ξk, n and ~α k, n . To emphasize this, we denote r k, n pn ; ξk, n ; ~α k, n as the rate rk, n as a function of pn, ξk, n and ~α k, n . From (5.46), ~α k, n in (5.45) can be regarded as an equivalent channel gain, which is determined by the predicted CSI, α^ k, n , and the outage probability ξk,n . Given α^ k, n , ~α k, n increases with ξk,n , since higher outage probability allows higher transmission rate. Meanwhile, given ξk,n , ~α k, n increases with α^ k, n , since higher predicted CSI also indicates better channel condition. With the equivalent channel gain ~α k, n , we can now derive the resource allocation algorithm that solves the problem in (5.43). In the following, we propose a suboptimal algorithm by breaking the problem into two steps: subcarrier allocation and power allocation.

Subcarrier Allocation Assuming the transmit power is identical over all subcarriers, pn ¼ pT/N, the original problem in (5.43) becomes max ρk , n , R

R   8 < P N ρ
1  ξ log 1 þ ~α pT R, 8k k, n k, n 2 n¼1 k , n Nσ 2r subject to : : PK k¼1 ρk , n ¼ 1, ρk , n 2 f0; 1g, 8k, n

ð5:47Þ

The problem in (5.47) is an integer programming problem. Thus, finding the optimal solution to (5.47) still has a high complexity. We propose a greedy algorithm that finds a suboptimal solution to (5.47). In each iteration, this algorithm assigns the best subcarrier to the user with the minimum rate. Denote u by the set of subcarriers that have not been assigned to users and Rk by the rate of user k. The algorithm is described as follows: Algorithm 5.3 Greedy Subcarrier Assignment 1: Set U ¼ 1,2, . . . , N, R1 ¼ R2 ¼ . . . ¼ RK ¼ 0 and ρk, n ¼ 0, 8 k, 8 n; 2: while U ¼ 0= do 3: Determine the set of users with the minimum S ¼ fk : Rk Rk0 ; 8k 0 6¼ kg 4: Find the best user-subcarrier pair in the set S  U
 ðk ; n Þ ¼ arg max r n, k pT =N; αn, k ; ξk, n k2U , n2S 5: 6: 7: 8:

Assign the subcarrier n to the user k, ρn , k ¼ 1 Update U ¼ U\ fn g
 Update Rk ¼ Rk þ r n , k pT =N; αn, k ; ξk, n end while

rate,

ð5:48Þ

104

5 Dealing with Imperfect CSI

Note that in step 4, if the outage requirements of all users are equal, ξk,n ¼ ξ, then from (5.46), the expression in (5.48) becomes ðk; nÞ ¼ arg max ~α k, n : k2S , n2U

ð5:49Þ

That is, we choose the best user-subcarrier pair according to the equivalent channel gain ~α k, n .

Power Allocation Given the subcarrier allocation {ρk, n}, the problem in (5.43) is reduced to max R pn , R

  8 < P N ρ
1  ξ log 1 þ ~α pn R, 8k k, n 2 k, n 2 n¼1 k , n subject to : σr : PN p p ,p

0, 8n T n n¼1 n

ð5:50Þ

We can observe that rn,k( pT/N, αn,k, ξk,n) is concave in pn and the constraint is linear in R and pn, respectively. Thus, the problem in (5.50) is a convex optimization problem. We can obtain its optimal solution to (5.50) through a dual approach. The dual problem for (5.50) is min gðλ1 ; . . . ; λK ; μÞ,

λk 0,μ 0

where gðλ1 ; . . . ; λK ; μÞ

¼ max Lðλ1 ; . . . ; λK ; μ; p1 ; . . . ; pN ; RÞ pn 0, R ! !   K N N X X X
 pn ¼ max R þ λk ρk, n 1  ξk, n log2 1 þ ~α k, n 2  R  μ pn  pT pn 0, R σr n¼1 n¼1 k¼1 !   K N X K N X X X
 p ¼ max 1  λk R þ λk ρk, n 1  ξk, n log2 1 þ ~α k, n n2  μpn þ μpT , pn 0, R σ r n¼1 k¼1 n¼1 k¼1 !     K N X X
 p ¼ max 1  λk R þ λkn 1  ξk, n log2 1 þ ~α kn , n n2  μpn þ μpT , pn 0, R σ r n¼1 k¼1

where λk is the Lagrangian multiplier of the first constraint in (5.50) of user k, μ is the Lagrangian multiplier of the second constraint in (5.50), and kn is the index of user assigned to the subcarrier n, i.e., kn ¼ arg maxk ρk,n. Given λk and μ, we can find the optimal pn by differentiating L with respect to pn,
 λk n 1  ξk n , n ~α kn , n ∂L ¼  μ ¼ 0: ∂pn ln 2 σ 2r þ pn ~α kn , n Since pn 0, we have

5.5 Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities

 pn ¼


 þ λkn 1  ξkn , n σ2  r ~α kn , n μ ln 2

105

ð5:51Þ

where (x)+ ¼ max {x, 0}. From the problem (5.50), the rate R is thus given by R ¼ mink

XN n¼1


 ρk, n r n, k pT =N; αn, k ; ξk, n :

To find the optimal λk and μ, we can use subgradient method. In the t‐th iteration, λk and μ are updated by !!þ   pn ð t Þ λk ðt þ 1Þ ¼ λk ðt Þ  βðt Þ ρk, n 1  ξk, n log2 1 þ ~α k, n 2 R σr n¼1 !!þ N X μðt þ 1Þ ¼ μðt Þ  βðt Þ pT  pn ð t Þ N X






n¼1

ð5:52Þ where t is the index of iteration number and β(t) is the t‐th iterative step size. Complexity In the subcarrier allocation method we have just discussed above, we need to determine the user with the smallest current rate every time when a subcarrier is allocated; therefore the complexity is O(K ). Assuming that the number of users with the lowest rate is Umax, we need to compare NUmax values when assigning the best subcarrier to the user with the minimum rate, where the subcarrier number is N. So the complexity of the subcarrier allocation algorithm is O(KN + N2Umax). Since the user’s gain in the subcarriers is a continuous random variable that is independent of each other, the user with the lowest rate is unique, i.e., Umax ¼ 1; the complexity can be expressed as O((K + N )N ). On the other hand, in the power allocation, the power of the N subcarriers needs to be calculated in each iteration step, and K + 1 Lagrange multipliers are updated. Let the number of iteration steps be Iu; the complexity of power allocation algorithm is O(Iu(K + 1 + N )). If we ignore the constant Iu, the total complexity of the suboptimal algorithm can be written as O((K + N )N ). Finally, we can see that this algorithm has a polynomial time complexity.

5.5.4

Simulation Results

We present several simulation results to demonstrate the performance of the proposed algorithm in OFDMA systems with parameters summarized h i in Table 5.1. Here, large estimation errors at users are assumed, σ 2ε ¼ E jH k, n j2 . We simulate the frequency-selective Rayleigh fading channel using the COST 259 channel model for

106 Table 5.1 Simulation parameter

5 Dealing with Imperfect CSI Parameter Mean channel gain (E(|hk, n|2)) Carrier frequency ( f0) Estimation interval (η) Speed (vk)
 Estimation error of users σ 2E Subcarrier spacing (Δf ) Prediction order (Q) Outage probability (ξk, n)

Value 1 2.6 GHz 4 50 km/h 1 45 kHz 5 0.1

Fig. 5.5 Comparison of the proposed suboptimal algorithm and optimum

a typical urban environment [20] and assume that each multipath tap follows Jakes’ model. The feedback delays of all users are equal, δmk ¼ δm. Figure 5.5 shows the comparison of the suboptimum and optimum to the problem in (5.48), where the number of users is K ¼ 8 and the number of subcarriers is N ¼ 2. Here, the suboptimal solution is calculated by the proposed algorithm in Sect. 5.5.3, and the optimal solution is obtained through the full search algorithm: we consider all possible subcarrier allocations, where for each allocation, we assign the power on subcarriers using the algorithm given in Subsection 5.5.3.2. It can be seen that the complexity of the full search algorithm is high. This Fig. 5.5 shows that as the signal error ratio (SNR) increases, the performance gap between the suboptimal solution and the optimal solution is within 5%.

5.5 Resource Allocation for OFDMA Systems with Guaranteed Outage Probabilities

107

Fig. 5.6 Comparison of achieved R under different scenarios

Then, we compare the performance of our proposed resource allocation scheme under different feedback delays, i.e., δm ¼ 1, 8, 15. We also consider another two cases: (1) the BS has perfect knowledge of downlink CSI (Perfect CSI), where we choose ξk, n ¼ 0 and ~α k, n ¼ αk, n; (2) instead of evaluating the equivalent channel gain given by (5.45), the BS directly uses the predicted channel gain as the resource allocation parameter (use predicts), and in this case, we set ξk, n ¼ 0 and ~α k, n ¼ α^ k, n . In all of the aforementioned cases, we choose the number of subcarriers as N ¼ 32 and the number of users as K ¼ 8 and use the two-step suboptimal algorithm discussed in Subsection 5.5.3 to determine subcarrier allocation ρk, n and transmit power pn. Figure 5.6 plots the achieved minimum rate R versus SNR. The achieved minimum rate decreases as the feedback delay increases. When the SNR is larger than 10 dB, the performance loss of the proposed scheme in the case of δm ¼ 1 is within 20% compared to the perfect CSI case. However, if the BS only uses the predicted channel gain α^ k, n to allocate resources, the achieved minimum rate is only 45% of the perfect CSI one and is lower than the proposed scheme when δm ¼ 15. It implies that using the equivalent channel gain ~α k, n as the resource allocation parameter improves robustness of the OFDMA system. Figure 5.7 shows the achieved outage probability under different scenarios. It can be noted that the proposed resource allocation scheme can satisfy users’ outage probability requirement, ξk, n ¼ 0.1. However, if the predicted channel gain is used to assign resources, the outage probability can reach up to 0.5. Figures 5.6 and 5.7 also imply that to achieve the target outage probability requirement, the proposed scheme assigns resources more conservatively under larger delays.

108

5 Dealing with Imperfect CSI

Fig. 5.7 Comparison of achieved outage probability under different scenarios

5.5.5

Conclusion

In this section, we propose a resource allocation scheme for OFDMA systems in the presence of channel estimation errors and feedback delays of downlink CSI. This resource allocation scheme aims at maximizing users’ minimum rates with a guaranteed outage probability. Toward this end, assuming that the MMSE channel prediction scheme is applied, we introduce the parameter for resource allocation, the equivalent channel gain. This parameter is determined by the required outage probability and the statistics of the channel estimates at the BS. With this parameter, we propose a two-step approach to properly assign subcarriers to users and transmit power on each subcarrier. Simulation results show that the proposed scheme can still achieve a moderately high rate while satisfying users’ requirement of outage probabilities under large channel estimation errors and feedback delays and, thereby, improves the robustness of the OFDMA systems with imperfect CSI.

5.6 5.6.1

Energy Efficiency Maximization for Downlink OFDMA Systems with Feedback Channel Capacity Constraints Problem Formulation

Since we have established the relation between feedback channel capacity and quantization errors, we obtained the conditional pdf of γ k, n on ^γ k, n . With the

5.6 Energy Efficiency Maximization for Downlink OFDMA Systems with. . .

109

knowledge of the quantized CSI, ergodic capacity is adopted as the effective capacity. The ergodic capacity for user k on subcarrier n is given by



Rk, n pk, n ; ^γ k, n ¼ γk, n log2 1 þ pk, n γ k, n j^γ k, n

ð5:53Þ

where pk,n is the transmit power of user k on subcarrier n and X fg is the statistical expectation with respect to X. We let P ¼ [pk. n]K  N denote the power allocation matrix and ρ ¼ [ρk, n]K  N denote the subcarrier allocation matrix where the binary variable ρk, n ¼ 1 indicates subcarrier n is assigned to user k and ρk, n ¼ 0 otherwise. Accordingly, the downlink system capacity and the overall transmit power are given by RðP; ρÞ ¼

K X N X
 ρk, n Rk, n pk, n ; ^γ k, n ,

ð5:54Þ

k¼1 n¼1 K X N X

PðP; ρÞ ¼ ζ

ρk , n pk , n þ P 0 ,

k¼1 n¼1

where P0 and ζ represent the circuit power and the reciprocal of drain efficiency of power amplifier, respectively. It is well accepted that the downlink system energy efficiency can be defined as RðP; ρÞ ¼ η¼ PðP; ρÞ

PN

PN

k¼1

ζ

PK

n¼1


 ρk, n Rk, n pk, n ; ^γ k, n

PN

k¼1

n¼1

ρk, n pk, n þ P0

ð5:55Þ

Therefore, the energy efficiency optimization problem can be formulated as RðP; ρÞ max η ¼ P, ρ PðP; ρÞ N X
 s:t: ρk, n Rk, n pk, n ; ^γ k, n Rkmin , 8k; n¼1 K X N X

ρk, n pk, n Pmax ;

ð5:56Þ

k¼1 n¼1 K X ρk, n 1, 8n; k¼1

ρk, n 2 f1; 0g, 8k, n; pk, n 0, 8k, n: where Rkmin and Pmax are the required minimum transmit rate of user k and the maximum transmit power of the downlink system, respectively. The first constraint in (5.56) guarantees the minimum rate requirements of each user. The second

110

5 Dealing with Imperfect CSI

constraint in (5.56) limits the peak transmit power of the system. The third and fourth constraints indicate that each subcarrier is exclusively assigned to at most one user each time to avoid co-channel interference.

5.6.2

Energy-Efficient Resource Allocation with Quantized CSI

Problem (5.56) is generally hard to solve with fractional objective, nonlinear constraints, and integer assignment variables ρk,n. In this section, we give an effective method to solve it with an acceptable complexity. We first use fractional programming theory to transform the fractional problem to an equivalent parametric problem and then solve the equivalent problem by the Lagrange dual decomposition method. As the fractional programming theory, the fractional objective function (5.56) can be associated with the following parametric problem max RðP; ρÞ  ηPðP; ρÞ P, ρ

ð5:57Þ

Let η be the optimal value of original problem (5.56), and then the following statement is equivalent: max RðP; ρÞ  ηPðP; ρÞ ¼ 0 , η ¼ η P, ρ Hence, solving the primal problem (5.56) is equivalent to find η which makes maxP,ρ R(P, ρ)  ηP(P, ρ) ¼ 0. One-dimensional iterative searching method can be useful as long as the optimal value of (5.57) can be obtained for a given η. Therefore, we need to solve new optimization problem as follows: max RðP; ρÞ  ηPðP; ρÞs:t: P, ρ

N X
 ρk, n Rk, n pk, n ; ^γ k, n Rkmin , 8k; n¼1

K X N K X X ρk, n pk, n Pmax ; ρk, n 1, 8n;ρk, n 2 f1; 0g, 8k, n; k¼1 n¼1

k¼1

pk, n 0, 8k, n: The Lagrangian multiplexer function for (5.58) is given by

ð5:58Þ

5.6 Energy Efficiency Maximization for Downlink OFDMA Systems with. . .

111

! K X N K X N X X
 LðP;ρ; λ;μÞ ¼ ρk, n Rk, n pk, n ;^γ k, n  η ζ ρk, n pk, n þ P0 n¼1 k¼1 n¼1 k¼1 ! ! K X N K N X X X
 min þ μ Pmax  ρk , n pk , n þ λk ρk, n Rk, n pk, n ;^γ k, n  Rk k¼1 n¼1

k¼1

n¼1

K N X X


¼ ð 1 þ λk Þ  ρk, n γk, n log2 1 þ ρk, n γ k, n j^γ k, n n¼1

k¼1

 ðμ þ ηζ Þ

K X N X

ρk, n pk, n  ηP0 þ μPmax 

k¼1 n¼1

k X

λk Rkmin

k¼1

where λ ¼ (λ1, λ2, . . . , λK) and μ are the Lagrange multipliers corresponding to (5.58), respectively. The Lagrange dual function is then given as gðλ; μÞ ¼ max LðP; ρ; λ; μÞ K X s:t: ρk, n 1, 8n;

ð5:59Þ

k¼1

ρk, n 2 f1; 0g, 8k, n; pk, n 0, 8k, n: Now, the problem turns to maximize L(P, ρ, λ, μ) from (5.59) with a given λ and μ. We separate the problem into two subproblems: 1. Power Allocation with a given subcarrier assignment When the subcarrier assignment is fixed, the power allocation of user k on subcarrier n can be independently solved. When ρk, n ¼ 0, it is obvious that pk,n should be zero. When ρk,n ¼ 1, taking derivative of L(P, ρ, λ, μ) with respect to pk,n and based on the Karush-Kuhn-Tucker condition, we have  γ k , n

γ k, n j^γ 1 þ pk , n γ k , n k , n







μ þ ηζ ; γk, n γ k, n j^γ k, n ¼ min ð1 þ λk Þlog2 e  μ þ ηζ ; αk, n j^γ k, n : ¼ min ð1 þ λk Þlog2 e

 ð5:60Þ

To further reduce the computational complexity, we use a Gamma distribution to approximate the NC X 2 distribution of (5.19):
  ba a1
f γ k, n j^γ k, n  γ k, n exp bγ k, n ΓðaÞ

ð5:61Þ

112

5 Dealing with Imperfect CSI

2 where a ¼ (κk,n + 1)  is the Gamma pdf shape parameter with κk, n ¼ ^γ k, n
/(2κk,n + 1) =αk, n and b ¼ a= ^γ k, n þ αk, n is the Gamma pdf rate parameter. Using this pdf instead of (5.19), we can derive the closed-form expression of (5.60) according to

 γ k , n

 γ k, n j^γ 1 þ pk , n γ k , n k , n     a b a pkb, n b  e Γ a; pk , n pk , n pk , n

ð5:62Þ

where Γ(a, x) is the incomplete Gamma function. This approximation has been shown to be fully accurate in [19]. Substituting (5.62) into (5.60) and using the bisection method, we can solve the power allocation problem. 2. Subcarrier assignment The problem of (5.59) is equivalent as N X

¼

max

n¼1 N X n¼1

K X


  ρk, n ð1 þ λk ÞRk, n pk, n ; ^γ k, n  ðμ þ ηζ Þpk, n k¼1


  max ð1 þ λk ÞRk, n pk, n ; ^γ k, n  ðμ þ ηζÞpk, n

ð5:63Þ

k

where (5.63) follows from the constraints in (5.58) and pk,n can be decided by solving subproblem 1 with ρk,n ¼ 1. Thus, for each subcarrier, the optimal assignment is  ρk , n ¼

1 if k ¼ arg maxk βk, n 0 otherwise


 where βk, n ¼ ð1 þ λk ÞRk, n pk, n ; ^γ k, n  ðμ þ ηζ Þpk, n from (5.63). When a tie occurs, we can select the user who has the smallest number of subcarriers. After solving the above two subproblems, we can obtain the value of the Lagrange dual function (5.59) for a given λ and μ. Now, we need to find λ and μ which optimize the dual problem as follows: min gðλ; μÞ s:t:λ 0; μ 0:

ð5:64Þ

Subgradient method is an effective method to solve this problem. The idea of the subgradient update method is to design a step size sequence to update the Lagrange multipliers in the subgradient direction. It is performed as follows:

5.6 Energy Efficiency Maximization for Downlink OFDMA Systems with. . .

þ λlþ1 ¼ λkl  sl Δλkl k

 þ μlþ1 ¼ μl  sl Δμl

113

ð5:65Þ

where l is the iteration number, sl is the step size of the lth iteration which can be decided by backtracking line search method, and []+ is defined as []+ ¼ max (, 0). The subgradients of the Lagrange multipliers λ and μ are given by Δλk ¼

N X
 ρk, n Rk, n pk, n ; ^γ k, n  Rkmin n¼1

K X N X Δμ ¼ Pmax  ρk , n pk , n

ð5:66Þ

k¼1 n¼1

It is noteworthy that the dual decomposition method may not obtain the optimal solution because of the non-convex and the integer constraints of problem (5.58). Hence, all minimum rate constraints may not be completely met by the solution of problem (5.58). When this happens, we select the subcarrier assignment obtained by dual method as the optimal subcarrier assignment and then solve the following power allocation problem:

 max R P; ρ fix  ηP P; ρ fix P

s:t:

N X


 ρkfix γ k, n Rkmin , 8k; , n R k , n pk , n ; ^

n¼1 K X N X ρkfix , n pk , n Pmax ; k¼1 n¼1

pk, n 0, 8k, n:

ð5:67Þ

We can see that when the subcarrier assignment matrix ρ is fixed, the problem of (5.58) turns to a convex optimization problem, and then we can use dual method to obtain the optimal power allocation. The whole procedure to solve the primal energy efficiency maximization problem (5.56) is sketched in Algorithm 5.4. Although the solution by using dual decomposition cannot guarantee to be optimal, it is shown in [23, 24] that with practical number of subcarriers, the duality gap is virtually zero and the optimal solution can be efficiently obtained. We will also test the optimality in the next section. Algorithm 5.4 Energy-Efficient Resource Allocation Algorithm 1: Set initial η, iterative accuracy E and the maximum iteration number Nmax; 2: repeat 3: For the given η, use dual method to solve problem (5.46) and obtain ρ and P

114

5 Dealing with Imperfect CSI

4: if Not all rate constraints of (5.46) arc satisfied then Set ρ as the fixed subcarrier assignment and solve the problem (5.5, 5.15) to get the new power allocation P; 5: end if   Þ 6: if jR(P, ρ)  ηP(P, ρ) j > E then Set η ¼ PRððPP ;ρ ;ρ Þ and goto (3); 7: else Subearrier Assignment ρ and power allocation P 8: end if 9: until Iteration number exceeds Nmax Now we consider the computational complexity of our algorithm. Solving problem (5.59) requires OðKN Þ operations. We define NIL and NOL as the number of iterations of the inner-layer subgradient method and the outer-layer iterative method. Thus, the overall complexity of our algorithm is OðN OL N IL KN Þ. If we use exhaustive search method to find the optimal subcarrier assignment, we need KN searches. Hence, the complexity will be OðN OL K N Þ, which is exponentially complex. Therefore, our algorithm has polynomial complexity and is more feasible for practical implementation.

5.6.3

Simulation Results

We present numerical results to evaluate the energy efficiency performance of OFDMA systems under rate-distortion limit using our proposed algorithm. The channel is modeled as frequency-selective Rayleigh fading channel with E[|hk,n|2] ¼ 1. To simulate imperfect CSI, we generate independent ^ k and ek and then generate Hk under rate-distortion limit. Simulation realizations of H parameters are given in Table 5.2. In Fig. 5.8, we test the optimality of our proposed algorithm. We compare our algorithm with the upper bound on the optimum in different minimum rate requirements. To obtain the optimal upper bound, we use exhaustive search method to check all possible subcarrier assignments and allocate the power of each subcarrier using dual method (5.67). We assume that there are N ¼ 8 subcarriers with independent and identical distribution, i.e., ΣHk ¼ IN and K ¼ 2 users. Let the feedback channel’s capacity of each user be 20 kbps. The performance of our Table 5.2 Simulation parameters

Parameter Sub carriers (N ) Users (K ) Subcarrier spacing (Δ f ) Max transmit power (P) Circuit power (P0) Drain efficiency (ζ) Channel model

Simulation 1 8 2 15 kHz 10 W 5W 0.38 i.i.d.

Simulation 2 32 8 15 kHz 20 W 10 W 0.38 COST259

5.6 Energy Efficiency Maximization for Downlink OFDMA Systems with. . .

Fig. 5.8 Performance of different algorithms against average CNR

Fig. 5.9 Energy efficiency against feedback channel’s capacity

115

116

5 Dealing with Imperfect CSI

algorithm is nearly the same of the optimal bound. We can also see that when the average CNR is low, high rate requirements make the energy efficiency decrease. However, when the average CNR is high, this energy efficiency reduction is much slighter. The reason is that when the average CNR is high, the power satisfying the minimum rate requirements becomes low and the minimum rate constraints is not primary of this optimization problem. In Fig. 5.9, we show the energy efficiency of different algorithms against feedback channel capacity. The case in which the BS has perfect CSI is also considered. The channel model is based on COST259 channel model. We assume that there are N ¼ 32 subcarriers and K ¼ 8 users, and the minimum rate requirement is 30 kbps. We can see that all the energy efficiency with imperfect CSI increase as the feedback channel capacity increases and our proposed algorithm is more energy-efficient than the algorithms that optimize the spectral efficiency. We can also see that the energy efficiency of our proposed algorithm can nearly achieve the one with perfect CSI when the feedback channel’s capacity is high.

5.6.4

Conclusion

In this section, we studied the energy-efficient design of downlink FDD-OFDMA systems with feedback channel capacity constraints. By exploiting the relation between the quantized CSI and the finite-rate feedback using the rate-distortion theory, we formulated the energy efficiency problem with finite-rate feedback channel. We showed that the optimal algorithm has an exponential complexity. Thus, an effective algorithm was proposed with acceptable complexity that is more feasible. In our algorithm, we first used generalized fractional programming theory to find the equivalent problem and then solved it by the Lagrange dual decomposition method. Numerical results showed the significant energy efficiency improvement of the proposed algorithm. The energy efficiency with a limited feedback rate can be very close to that with perfect CSI as the feedback rate goes higher.

5.7 5.7.1

Resource Allocation for Physical-layer Security in OFDMA Downlink with Imperfect CSI System Model and Problem Formulation

The system setup is shown in Fig. 5.10. We consider an OFDMA broadband system that consists of K active users and N used subcarriers indexed by the sets K ¼ f1; . . . ; K g and N ¼ f1; . . . ; N g. An eavesdropper who is passive aims to wiretap

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

117

Fig. 5.10 System model

the transmitted signal within each data-bearing subcarrier. We assume that this system employs FDD, the BS obtains the downlink CSI from users’ feedback. The eavesdropper is an honest, but curious, legitimate user who illegally starts wiretapping the messages of the authorized and serviced users. Thus, the BS anticipates the existence of the eavesdropper and can obtain its partial CSI just as it does with other legitimate users. The channel response between user k 2 K and the BS in subcarrier n 2 N is denoted as Hn,k, while the channel response between the ^ n, k and eavesdropper and the BS in subcarrier n 2 N is denoted as Hn,e. We use H ^ n, e denoting the priori information of Hn,k and Hn,e that the BS obtained through H users’ feedback. We have obtained the conditional pdf of both the CNR of the kth user γ n, k ¼ 2 2 2 2 j H n, k j jH^ n, k j jH n, e j jH^ n, e j on ^ γ ¼ and the CNR of eavesdropper γ ¼ on γ ^ ¼ 2 2 2 2 n , k n , e n , e σv σv σv σv . With the knowledge of the feedback CSI, the ergodic capacity for user k on subcarrier n is given by

 Rn, k pn, k ; ^γ n, k ¼ γn, k log2 1 þ pn, k γ n, k ^γ n, k  Z þ1 γ n, k ^ γ n, k 
 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi  γ n, k ^γ n, k dγ n, k , log 1 þ pn, k γ n, k  e α n, k I 0 ¼ αn, k αn, k 0 ð5:68Þ where pn,k is the transmit power of user k on subcarrier n. We assume an exclusive subcarrier assignment, in which each subcarrier is assigned exactly to one user. Let the binary variables ρn, k , n 2 N , k 2 K denote the subcarrier assignment, with

118

5 Dealing with Imperfect CSI

ρn, k ¼ 1 if subcarrier n is assigned to user k. Moreover, let P ¼ [pn, k]N  K denote the power allocation matrix, PT > 0 denote the available amount of power at the BS, and σ 2v denote the ambient noise variance. Regarding (5.68), the following proposition can be stated:
 Proposition 5.1 Under the condition of given αn,k, Rn, k pn, k ; ^γ n, k is an increasing function of ^γ n, k . Proof According to the conditional pdf obtained in (5.19), we can rewrite (5.68) as follows:
 γn, k log2 1 þ pn, k γ n, k ^γ n, k Z þ1
pffiffiffiffi ¼ log2 ð1 þ axÞeðtþxÞ I 0 2 tx dx

ð5:69Þ

0

¼ F ðt Þ, γ

γ^

where x ¼ αnn,,kk , t ¼ αnn,, kk , a ¼ pn, k αn, k . We rewrite the I0() in the form of Taylor series I 0 ðzÞ ¼

þ1 X 1  z 2i : i!i! 2 i¼0

ð5:70Þ

Thus, we have F ðt Þ ¼ et

þ1 i Z þ1 X t log2 ð1 þ axÞex xi dx: i!i! 0 i¼0

ð5:71Þ

The derivative of F(t) is Z þ1 ti log2 ð1 þ axÞ: i!ði þ 1Þ! 0 i¼0 ex ½xiþ1  ði þ 1Þxi dx þ1 t X e ti Gi , ¼ ln 2 i¼0 i!ði þ 1Þ!

F 0 ðt Þ ¼ et

þ1 X

R þ1 where Gi ¼ 0 ln ð1 þ axÞex ½xiþ1  ði þ 1Þxi dx. In addition,

ð5:72Þ

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

Z

119

þ1

ln ð1 þ axÞex xiþ1 dx Z þ1 ln ð1 þ axÞex xi dx  ð i þ 1Þ 0 Z þ1 ¼ ln ð1 þ axÞxiþ1 dex 0 Z þ1  ð i þ 1Þ ln ð1 þ axÞex xi dx 0   x iþ1  x ¼ þ1 ¼  ln ð1 þ axÞe x  x¼0 Z þ1

 þ ex d ln ð1 þ axÞxiþ1 0 Z þ1  ð i þ 1Þ ln ð1 þ axÞex xi dx 0   Z þ1 a ¼ xiþ1 þ ði þ 1Þ ln ð1 þ axÞxi dx ex þ ax 0 Z1 þ1 ln ð1 þ axÞex xi dx  ð i þ 1Þ 0 Z þ1 a xiþ1 dx > 0: ex ¼ 1 þ ax 0

Gi ¼

0

Thus F0(t) > 0, F(t) being an increasing function of t; we obtain the result of Proposition 5.1. Accordingly, the ergodic secrecy rate of user k within n is given by
 n


  C n, k pn, k ¼ γn, k log2 1 þ pn, k γ n, k ^γ n, k


 þ γn, e log2 1 þ pn, k γ n, e ^γ n, e ,

ð5:73Þ

where [x]+ ¼ max {0, x}. With the conclusion we have obtained in Proposition 5.1 and the assumption αn,k ¼ αn, e where αn, e is the eavesdropper’s ratio of quantization error variance to the noise variance, we have

C n, k


  8 ^γ n, k < γ n, k log2
1 þ pn, k γ n, k   ¼ γn, e log2 1 þ pn, k γ n, e ^γ n, e if ^γ n, k > ^γ n, e : 0 if ^γ n, k < ^γ n, e :

ð5:74Þ

Therefore, the problem of maximizing the minimum secrecy rate among all users under total transmitted power and exclusive subcarrier assignment constraints is formulated as

120

5 Dealing with Imperfect CSI

X
 max min ρn, k C n, k pn, k ρn , k , pn , k k n2N X ρn, k pn, k PT , s:t: k2K, n2N X ρn, k ¼ 1, n 2 N ,

ð5:75Þ

k2K

ρn, k 2 f0; 1g, n 2 N , k 2 K, 0 pn, k PT , n 2 N , k 2 K: From a practical point of view, the problem in (5.75) is important mainly for two reasons. First, compared with the max-sum criterion over the total secrecy rate, the max-min criterion can guarantee fairness in the sense that it provides a secrecy rate balancing across the different users for it does not allow to one or more privileged users (i.e., users with much stronger channel than the eavesdropper’s channel) to monopolize the available resources. Second, the capacity of the feedback channel is limited in practical communication systems, so the BS can only obtain quantized CSI. As a result, the performance of resource allocation schemes is degraded due to imperfect CSI. Analyzing the effect of finite feedback rate in OFDMA systems turns out to be a crucial problem. The formulation in (5.75) is a mixed integer nonlinear programming problem due to the existence of the binary variables ρn,k and the integration of logarithm functions Cn,k( pn,k). Hence, we can find that the problem in (5.75) is NP-hard. In the following section, we propose a suboptimal algorithm with an acceptable complexity by breaking the problem into two steps: subcarrier allocation and power allocation.

5.7.2

Optimal Power Allocation Algorithm When Subcarrier Assignment Is Fixed

In this subsection, we aim to specify the optimal power allocation when the subcarrier assignment is given. Let N k denote the set of subcarriers that have been assigned to user k 2 K. In most cases, we assume that Cn,k( pn,k) > 0, n 2 N k , k 2 K, as it is not beneficial to assign channels with Cn,k( pn,k) ¼ 0 to user k. The original problem can be rewritten as max r pn , k , r X
 s:t:0 r Cn, k pn, k , k 2 K, X n2N k pn, k PT : k2K, n2N k For the problem in (5.76), we can have the following result:

ð5:76Þ

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

121

Proposition 5.2 The problem in (5.76) is convex. Proof The objective in (5.76) is linear with respect to the secrecy rate r. Moreover, the constrains in (5.76) are linear functions of pn,k. Thus, we focus on the secrecy constraint in (5.76). The second derivative of Cn,k() with the respect to pn,k is given as ( " # 2 γ 2n, k ∂ C n, k 1  j^γ n, k γ n, k
¼ ln 2 ∂p2n, k 1 þ pn , k γ n , k " #) γ 2n, e  γn, e
2 j^γ n, e : 1 þ pn , k γ n , e  When αn,k is fixed, γ n, k 2

Therefore,

∂ C n, k ∂p2n, k



γ 2n, k

ð1þpn, k γn, k Þ

ð5:77Þ

2

j^γ n, k

is a decreasing function of ^γ n, k .

is always negative for ^γ n, k > ^γ n, e . Thus, Cn,k( pn,k) is a concave

function of pn,k. Therefore, problem (5.76) is a convex problem; we can use any standard convex optimization tool to obtain the unique optimal solution. In what follows, we use a bisection method to derive the optimal solution in a semiclosed form.
 P Let’s assume a fixed value rf such that 0 < r f maxk n2N k RAn, k pn, k , where RAn,k( pn,k) denotes the secrecy rate under the assumption that the transmit power is monopolized by exactly one user and distributed equally to its selected subcarriers. Clearly, the secrecy rate rf can be guaranteed to any user k 2 K if there is a sufficient (possible higher than PT) amount of power at the BS. Nevertheless, we need to solve the new optimization problem as follows to obtain the optimal power allocation that guarantees secrecy rate rf across all the users: min pn, k

X k2K, n2N X

s:t:r f

pn , k
 Cn, k pn, k , k 2 K,

ð5:78Þ

n2N k pn, k 0, n 2 N , k 2 K:

The Lagrangian multiplexer function for (5.78) is given by LðP; λ; μÞ ! XX X X
 ¼ pn , k þ λk r f  C n , k pn , k k2K k2K n2K n2N XX μn, k pn, k , 

ð5:79Þ

k2K n2N

where λ ¼ (λ1, . . . , λK) are the Lagrangian multipliers. Based on the KKT condition, we have

122

5 Dealing with Imperfect CSI

8 ∂LðP; λ; μÞ    > > j pn, k, λk ,μn:k ¼ 0, n 2 N , k 2 K > > > > 2∂pn, k 3 > > < X
 λk 4r f  C n, k pn, k 5 ¼ 0, k 2 K > > > n2N k > > > > μn, k pn, k ¼ 0 > :  pn, k 0,λk 0,μn, k 0,

ð5:80Þ

where pn, k is the optimal power allocation and λk and μn, k are the optimal Lagrange multipliers. Note that μn, k acts as a slack variable which can be eliminated. The optimal power allocation on the n‐th subcarrier which is assigned to user k at the given λk is

pn, k ¼

8 > > < p~n, k > > :0

n



o ln 2 if E γn, k γ n, k j^γ n, k  E γn, e γ n, e j^γ n, e > λk n



o ln 2 :, if E γn, k γ n, k j^γ n, k  E γn, e γ n, e j^γ n, e > λk

ð5:81Þ

where p~n, k satisfies  E γ n, k

   γ n, k γ n, e ln 2 j^γ n, k  E γn, e j^γ n, e ¼ : λk 1 þ pn, k γ n, k 1 þ pn, k γ n, e

ð5:82Þ

We use a Gamma distribution to approximate the NC X 2 distribution of (5.19) to reduce the computational complexity:  
βα α1
γ k, n exp βγ k, n , f γ k, n j^γ k, n  ΓðαÞ

ð5:83Þ

where α
¼ (κn,k + 1)2/(2κ n,k + 1) is the Gamma pdf shape parameter with κ k,n/αk,n and β ¼ α= ^ γ k, n þ αk, n is the Gamma pdf rate parameter. Using this pdf, we can derive the closed-form expression of (5.82) according to  E γ n, k

 γ n, k j^γ 1 þ pn, k γ n, k n, k  α β   α β β  epn, k Γ α; , pn, k pn, k pn , k

ð5:84Þ

where Γ(a, x) is the incomplete Gamma function. This approximation has been shown to be fully accurate in [19]. Substituting (5.84) into (5.82), a bisection method is used to calculate the power allocation problem for
each given λk. To find the P optimal multiplier λk that can satisfy r f ¼ n2N k Cn, k pn, k under the given rf, we can use the bisection method as well. Clearly, when the available power at the BS is

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

123

P PT, the secrecy rate rf can be supported by the system only when n, k pn, k PT . To arrive to a final solution in (5.78), we can use a bisection method to find the optimal rf. For each iteration of the bisection method, the problem in (5.77) is solved, and the total power consumption is calculated. When the total power consumption is higher (lower) than PT, rf is decreased (increased), and the process is repeated. Such a decrease (increase) in rf is interpreted as a decrease (increase) in the multipliers λk, and, thus, more (less) power is allocated to each user. By appropriately setting the lower and upper values for the bisection over r, rmin, and rmax respectively, the process is repeated until the total power consumption to PT is achieved. The whole procedure to solve the power allocation problem for fixed subcarrier assignment is described in Algorithm 5.5. Algorithm 5.5 Optimal Power Allocation
for Fixed Subcarrier Assignment  P 1: Set r min ¼ 0, r max ¼ maxk n2N k RAn, k pn, k 2: repeat 3: Set rf ¼ (rmin + rmax)/2 4: Solve the problem in (5.78) using (5.81). the power allocation is denoted as pn, k , k 2 K, N 2 N P 5: The total power consumption is calculated as Pc ¼ n, k pn, k 6: if Pc < PT then Set rmin ¼ rf and pn, k ¼ pn, k , k 2 K, n 2 N k 7: else Set rmax ¼ rf 8: end if 9: until |rmax – rmin| E 10: Output pn, k , k 2 K, n 2 N

5.7.3

Greedy Subcarrier Allocation Algorithm

Assuming the transmit power is equally split across all subcarriers, pn,k ¼ PT/N, the original problem in (5.75) becomes max R ρn , k , R X
 s:t: ρn, k Cn, k pn, k R, 8k, n2N X ρn, k pn, k PT : k2K, n2N

ð5:85Þ

The problem in (5.85) is an integer programming problem. Thus, finding the optimal solution to (5.85) is still very complex. We propose a greedy subcarrier assignment algorithm that finds a suboptimal solution to (5.85). In each iteration, the user with the currently lowest secrecy rate is enforced to occupy one more subcarrier from the available ones. The subcarrier that is assigned to it is the one that can give

124

5 Dealing with Imperfect CSI

the user the maximum secrecy rate. Denote S as the set of subcarriers that have not been assigned to users and Ck as the rate of user k. The resulting algorithm is described in Algorithm 5.6. Algorithm 5.6 Greedy Subcarrier Assignment 1: Set S denotes the set of available subcarriers. Initially, S ¼ f1; 2; . . . ; N g, C1 ¼ C2 ¼    ¼ CK ¼ 0 and ρn, k ¼ 0, 8 k, 8 n 2: while S 6¼ 0= do 3: Determine the set of users with the minimum secrecy rate, U ¼ fk : Ck C k0 ; 8k0 6¼ kg 4: Find the best user-subcarrier pair in the set S  U
 ðk  ; n Þ ¼ arg max Cn, k pT =N; ^γ n, k k2U , n2S 5: 6: 7: 8:

ð5:86Þ

Assign the subcarrier n to the user k ,ρn , k ¼ 1 Remove n from the set S,
S ¼ S\ fng Update C k ¼ Ck þ C n , k pT =N; ^γ n, k end while

Note that in Step 4, according to the assumption αn,k ¼ αn,e, the expression in (5.86) becomes
 ðk ; n Þ ¼ arg max γ^n, k  ^γ n, e : k2U , n2S

ð5:87Þ

The subcarrier that is assigned to the user with the minimum secrecy rate is the one that has the highest difference in the estimation channel gain with respect to the eavesdropper. If the OFDMA system employs time-division duplex (TDD), the BS estimates the uplink channel and gets the downlink CSI according to the channel reciprocity. If the baseband channel gain from the BS to the kth user on the nth subcarrier satisfies ^ n, k þ En, k , where H ^ n, k denotes the estimation of Hn,k and En,k denotes the H n, k ¼ H
 estimation error that satisfies En, k
CN 0; σ 2n, k , and σ 2n, k is the mean variance of channel estimation error, our proposed algorithm will still be valid. Complexity The complexity of a bisection method is approximately log1E, where the E is the required accuracy. In Algorithm 5.6, an external bisection is required to adjust the value of r, while we use the bisection method per user to search the Lagrange multiplier λk, k ¼ 1, . . . , K and calculate the optimal power policy per user across the selected subcarrier for a given λk. Therefore, the complexity of Algorithm
 5.5 is O KNlog3 ð1=EÞ . In Algorithm 5.6, two maximum-minimum search operations are performed in each subcarrier assignment iteration. Thus, the complexity of Algorithm 5.6 is OðN ðK þ N ÞÞ, and the overall complexity of our proposed algo
 rithm is O KN 2 log3 ð1=EÞK þ N . If we use exhaustive search method to find the optimal subcarrier assignment, we need KN searches. Hence, the complexity will be

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

125

Fig. 5.11 Comparison of the proposed suboptimal algorithm and the optimal algorithm


 O KNlog3 ð1=EÞK N , which is quite high. Therefore, our algorithm has polynomial complexity and is more feasible for practical implementation.

5.7.4

Simulation Results

We present several simulation results to demonstrate the security rate performance of the OFDMA system under different imperfect CSI conditions using our proposed algorithm. We assume that the transmit power PT ¼ 20W, and the channel is modeled as a frequency-selective Rayleigh fading channel with E[|hn, k|2] ¼ E[|hn, e|2] ¼ 1. To simulate imperfect CSI, we generate independent ^ e, and ek, ee according to different imperfect CSI assumption and ^ k, H realization of H then generate Hk which is equal to the sum of them. In Fig. 5.11, we test the optimality of our proposed algorithm. We assume that there are N ¼ 8, 10 subcarriers with independent and identical distribution and K ¼ 2 legitimate users. We use the imperfect CSI assumption discussed in Sect. 5.2.1 with simulation parameters summarized in Table 5.3, and we assume that the feedback delays of all users are equal, δmk ¼ 1. We compare our algorithm with the upper bound on the optimum. To obtain the optimal upper bound, we use exhaustive search method to search the maximum secrecy rate among all possible subcarrier allocations. For each subcarrier allocation, we assign the transmit power using the algorithm given in Subsection 5.7.2. This figure shows that the performance of our algorithm is nearly the same as the optimal bound. The gap between the proposed algorithm and the optimum does not change as the number of subcarriers becomes larger.

126 Table 5.3 Simulation parameters

5 Dealing with Imperfect CSI Parameter Mean channel gain (E(|hk, n|2)) Carrier frequency ( f0) Max transmit power (PT) Speed (vk)
 Estimation error of users σ 2E Subcarrier spacing (Δf ) Prediction order (Q) Channel model

Value 1 2.6 GHz 20 W 50 km/h 1 45 kHz 5 i.i.d

Fig. 5.12 Comparison of achieved secrecy rate under different feedback delays

Then, we compare the performance of our proposed resource allocation algorithm under different feedback delays, i.e., δme ¼ δmk ¼ 1, 5, 8. We use the imperfect CSI assumption discussed in Subsection 5.2.2 with simulation parameters summarized in Table 5.3. We also consider the case in which the CSI is perfect known at the BS. In all of the aforementioned cases, we assume that there are N ¼ 32 subcarriers with independent and identical distribution and K ¼ 4 legitimate users. Figure 5.12 plots the achieved minimum secrecy rate versus the CNR. The simulation results show that the achieved minimum secrecy rate decreases as the feedback delay δmk increases. The performance gap between the proposed algorithm and the perfect CSI case increases as the CNR increases.

5.7 Resource Allocation for Physical-layer Security in OFDMA Downlink. . .

127

Fig. 5.13 Feedback channel’s capcity per user (bps/Hz)

In Fig. 5.13, we show the secrecy rate per user of different algorithm against feedback channel capacity under the imperfect CSI assumption discussed in Subsection 5.2.3. We also consider the case in which the BS has perfect CSI, where we choose the user of the highest power gain gap between the legitimate user and the eavesdropper to each subcarrier, and then use the power allocation algorithm given in Subsection 5.7.2. We assume that there are N ¼ 16 subcarriers with independent and identical distribution and K ¼ 4 users. We can see that the secrecy rate per user with imperfect CSI increases as the feedback channel capacity increases and our proposed algorithm can obtain higher secrecy rate than the algorithms in which the power is distributed equally to each subcarrier. We can also see that the secrecy rate can nearly achieve the rate with perfect CSI when the feedback channel’s capacity is high.

5.7.5

Conclusion

In this section we investigated the problem of resource allocation in a security-aware FDD-OFDMA system with partial CSI constraints. We discussed three kinds of imperfect CSI and established a unified mathematical model of imperfect CSI for the OFDMA system. We formulated the max-min secrecy rate problem under the

128

5 Dealing with Imperfect CSI

condition that the CSI of both legitimate users and the eavesdropper are partially obtained at the BS. Since the optimal algorithm has exponential complexity, an effective algorithm was proposed with acceptable complexity that is more feasible. We proposed a low-complexity suboptimal algorithm that breaks the problem into two steps: subcarrier and power allocation problems. First we solved the power allocation problem for fixed subcarrier assignment by using a bisection method; then, we obtained a suboptimal subcarrier allocation solution through the greedy algorithm. Numerical results showed that the performance of the proposed algorithm is close to the optimum. The secrecy rate per user with a limited feedback rate can be very close to rate with perfect CSI as the feedback rate increases.

5.8

Summary

In this chapter, we discussed the resource allocation problems when the BS can only obtain imperfect CSI. We used the ergodic throughput, outage throughput, outage probability, energy efficiency, and secure capacity as the optimization objective, respectively. Since the optimal solutions have very high computation complexity, we proposed the suboptimal algorithms with lower complexity. The simulation results showed that the performance of the suboptimal algorithm is very close to the optimal ones.

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Summary and Outlook

Summary In this book, we introduce the theory of resource allocation in OFDMA systems in detail. In the first three chapters, we mainly introduced the OFDMA system, MIMO system, and some theories about resource allocation. In Chap. 4, we discuss about some resource allocation scenario in OFDMA systems with perfect CSI constraint and propose the suboptimal algorithm with acceptable complexity for each scenario. In Chap. 5, we discuss the resource allocation problems when the BS can only obtain imperfect CSI and propose the suboptimal algorithms with lower complexity. The main contributions of this book are as follows. We analyze the system capacity in multicast OFDMA systems, BER performance in MIMO-OFDM systems, and the energy efficiency in unicast SISO-OFDMA systems. Since it is NP-hard which need very high computation complexity to obtain the optimal solution, we propose the suboptimal algorithm with acceptable complexity for each scenario. Simulation results show that the performance of the suboptimal algorithm is near to the optimal one. We focus on the case where only imperfect (partial) CSI is available in this book, considering the resource allocation problem for optimizing the max-min criterion over the user’s secrecy rate under an average total transmitted power constraint with imperfect CSI. To the best of our knowledge, no work has considered the max-min security rate with imperfect CSI of both legitimate users and eavesdropper in OFDMA systems.

© Springer Nature Switzerland AG 2020 C. Chen, X. Cheng, Resource Allocation for OFDMA Systems, Wireless Networks, https://doi.org/10.1007/978-3-030-19392-8

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Summary and Outlook

Outlook In this book, we study the adaptive resource allocation of multicast OFDMA systems, MIMO-OFDMA systems, uplink OFDMA systems, and the case when the BS can only obtain imperfect CSI. Based on the research of this book, we can further consider the following areas. 1. Cross-layer joint resource allocation. The performance of the throughput, outage probability, and bit error rate obtained by the OFDMA system resource allocation will affect the upper layer transmission protocol; the upper layer protocol also affects the performance indicators of the physical layer resource allocation. 2. Resource allocation for systems that combine OFDMA and new technologies. When the OFDMA system is applied to a multi-cell, distributed wireless network, it can be combined with new technologies such as network coding and cooperative communication. 3. Resource allocation for OFDMA systems that consider more QoS requirements. In the existing mobile communication systems, different types of services may have different priorities, and different priority services have different weights when allocating resources; in addition, the service also has parameters such as delay and delay jitter, and in order to satisfy these parameter requirements, the base station should also consider the optimization of service queues when allocating resources. 4. Improvement of anti-interference performance of ETF-OFDMA system. The actual interference also includes some time interference such as short-term pulse interference. How to improve the structure of the system for these interferences and optimize the spreading code of the system is also worth studying.