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UBC Theses and Dissertations

Resource allocation for OFDM-based cognitive radio systems Zhang, Yonghong 2008

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RESOURCE ALLOCATION FOR OFDM-BASED COGNITIVE RADIO SYSTEMS by YONGHONG ZHANG B.Eng., Xi’an Jiaotong University, China, 1994 M.A.Sc., University of British Columbia, 2006 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2008 c© Yonghong Zhang, 2008 Abstract Cognitive radio (CR) is a novel wireless communication approach that may alleviate the looming spectrum-shortage crisis. Orthogonal frequency division multiplexing (OFDM) is an attractive modulation candidate for CR systems. In this thesis, we study resource allocation (RA) for OFDM-based CR systems using both aggressive and protective sharing. In aggressive sharing, cognitive radio users (CRUs) can share both non-active and ac- tive primary user (PU) bands. We develop a model that describes aggressive sharing, and formulate a corresponding multidimensional knapsack problem (MDKP). Low-complexity suboptimal RA algorithms are proposed for both single and multiple CRU systems. A sim- plified model is proposed which provides a faster suboptimal solution. Simulation results show that the proposed suboptimal solutions are close to optimal, and that aggressive shar- ing of the whole band can provide a substantial performance improvement over protective sharing, which makes use of only the non-active PU bands. Although aggressive sharing generally yields a higher spectrum-utilization efficiency than protective sharing, aggressive sharing may not be feasible in some situations. In such cases, sharing only non-active PU bands is more appropriate. When there are no fairness or quality of service (QoS) considerations among CRUs, both theoretical analysis and simulation re- sults show that plain equal power allocation (PEPA) yields similar performance as optimal power allocation in a multiuser OFDM-based CR system. We propose a low-complexity discrete bit PEPA algorithm. To improve spectrum-utilization efficiency, while considering the time-varying nature of the available spectrum as well as the fading characteristics of wireless communication channels and providing QoS provisioning and fairness among users, ii this thesis introduces the following novel algorithms: (1) a distributed RA algorithm that provides both fairness and efficient spectrum usage for ad hoc systems; (2) a RA algorithm for non-real-time (NRT) services that maintains average user rates proportionally on the downlink of multiuser OFDM-based CR systems; and (3) cross-layer RA algorithms for the downlink of multiuser OFDM-based CR systems for both real-time (RT) services and mixed (RT and NRT) services. Simulation results show that the proposed algorithms provide sat- isfactory QoS to all supported services and perform better than existing algorithms designed for multiuser OFDM systems. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Co-authorship Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Related Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.1 Resource Allocation Algorithms for OFDM-based CR Systems . . . . 6 1.4.2 Resource Allocation Algorithms for OFDM Systems . . . . . . . . . . 8 1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 2 Subcarrier, Bit and Power Allocation for Multiuser OFDM- based Multi-Cell Cognitive Radio Systems . . . . . . . . . . . . . . . . . 22 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 The Single CRU Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 The Multiple CRU Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iv 2.5.1 Reward/Cost Ratio for Each Constraint . . . . . . . . . . . . . . . . 32 2.5.2 Efficiency Value for Adding One Bit to Subchannel m of CRBS k . . 35 2.5.3 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.1 Single CRU Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6.2 Multiple CRU Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 3 An Efficient Power Loading Scheme for OFDM-based Cognitive Radio Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 An Approximate Solution for OP1 . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 Suboptimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 4 Performance of Equal Power Allocation in Multiuser OFDM- based Cognitive Radio Systems . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Bounds on bit rates for OWF and PEPA . . . . . . . . . . . . . . . . . . . . 71 4.3.1 Upper Bounds on Achievable Bit Rate for OWF . . . . . . . . . . . . 71 4.3.2 Achievable Bit Rate for PEPA . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Rayleigh fading channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1 Opportunistic Subchannel Assignment . . . . . . . . . . . . . . . . . 74 4.4.2 A Fairer Subchannel Assignment Scheme . . . . . . . . . . . . . . . . 74 4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter 5 Subchannel Power Loading Schemes in Multiuser OFDM Sys- tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 Bit rates for OWF and PEPA . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3.1 An Upper Bound on Achievable Bit Rate for OWF . . . . . . . . . . 84 5.3.2 Achievable Bit Rate for Continuous PEPA . . . . . . . . . . . . . . . 86 5.3.3 Achievable Bit Rate for Discrete PEPA . . . . . . . . . . . . . . . . . 86 5.3.4 Achievable Bit Rate for Improved Discrete PEPA . . . . . . . . . . . 87 5.3.5 Improved Discrete PEPA Algorithm . . . . . . . . . . . . . . . . . . . 89 v 5.4 Rayleigh fading channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.4.1 Opportunistic Subchannel Assignment . . . . . . . . . . . . . . . . . 91 5.4.2 A Fairer Subchannel Assignment Scheme . . . . . . . . . . . . . . . . 91 5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Chapter 6 Cross-Layer Resource Allocation for OFDM-based Cognitive Radio Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 Cross-Layer Resource Allocation for RT Services . . . . . . . . . . . . . . . . 111 6.3.1 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3.2 Conversion of MAC Layer Requirements to PHY Layer Requirements 113 6.3.3 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3.4 Estimation of the Number of Available Subchannels in the nextm(t1, t2) slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.4 Cross-Layer Resource Allocation for Mixed Services . . . . . . . . . . . . . . 118 6.4.1 The Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4.2 Conversion of MAC Layer Requirements to PHY Layer Requirements 121 6.4.3 A Goal Programming Approach for Improving Feasibility . . . . . . . 123 6.4.4 The Cross-Layer Resource Allocation Algorithm . . . . . . . . . . . . 125 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.5.1 RT Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.5.2 Mixed Services . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Chapter 7 Resource Allocation for Non-Real-Time Services in OFDM- based Cognitive Radio Systems . . . . . . . . . . . . . . . . . . . . . . . . 151 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.3 Resource Allocation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Chapter 8 A Distributed Algorithm for Resource Allocation in OFDM- based Cognitive Radio Systems . . . . . . . . . . . . . . . . . . . . . . . . 162 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.3 Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.3.1 Determining Achievability of Target Rates . . . . . . . . . . . . . . . 168 8.3.2 Determining HATR in a Resource-limited Situation . . . . . . . . . . 169 8.3.3 Determining HATR in a Resource-abundant Situation . . . . . . . . . 170 vi 8.3.4 The Proposed Distributed Algorithm . . . . . . . . . . . . . . . . . . 171 8.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Chapter 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.1 Contributions and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 182 9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Appendix A Optimal Solutions for Optimization Problems in Chapter 3 189 A.1 Solution for OP1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 A.2 Solution for OP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Appendix B Proofs of Theorems in Chapter 6 . . . . . . . . . . . . . . . . 194 B.1 Proof for Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.2 Proof for Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Appendix C Flowcharts for Algorithm in Section 6.4.4 . . . . . . . . . . . 208 Appendix D Derivation of The Results in (7.8) - (7.10) . . . . . . . . . . 212 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 vii List of Tables 3.1 Actual interference power exceeding I thl , l = 1, 2 by using SUBOPT-APPROX with I th1 = I th 2 = 8× 10−15 W, E{Hl,m} = 10−14, l = 1, 2. . . . . . . . . . . . 62 3.2 Actual interference power exceeding I thl , l = 1, 2 by using SUBOPT-APPROX with I th1 = I th 2 , S = 2.4 W, and E{Hl,m} = 10−14, l = 1, 2. . . . . . . . . . . . 63 6.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2 Dropped packet rates for different values of pn with pn = pa and R RT = 150 kbps.135 6.3 Dropped packet rate with respect to video data rate, RRT with pn = pa = 0.5 136 6.4 Dropped packet rate with respect to video data rate, RRT . pn = 0.5 and pa = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.5 Non-feasible ratio of PHY and HLL. . . . . . . . . . . . . . . . . . . . . . . 142 6.6 Fairness index comparison for three different schedulers. . . . . . . . . . . . . 142 7.1 Fairness index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.1 Subchannel gains (×10−10) from CRP j’s transmitter to CRP i’s receiver. . . 174 8.2 Number of bits per OFDM symbol and fairness index for each of the three algorithms and four different sets of nominal rate requirements. . . . . . . . 178 viii List of Figures 1.1 PU active frequency bands, spectrum holes and CRU OFDM subchannels. . 2 1.2 PU active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels in a protective sharing system. . . . . . . . . . . . . . . . . . . . 4 1.3 Spectrum sharing methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Thesis overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 PU active frequency bands, spectrum holes and CRU OFDM subchannels. . 26 2.2 Average number of bits per OFDM symbol per subchannel as a function of CRU power S, with interference thresholds set to 5× 10−12 W. . . . . . . . . 39 2.3 Average number of bits per OFDM symbol per subchannel as a function of the interference threshold with S = 0.32 W. . . . . . . . . . . . . . . . . . . 40 2.4 Simulation topologies: triangles represent CRBSs and circles represent PU transmitters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 1. . . . . . . . . . . . . 43 2.6 Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 2. . . . . . . . . . . . . . 44 2.7 Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 3. . . . . . . . . . . . . . 45 3.1 PUP active and non-active bands and CRP OFDM subchannels. . . . . . . . 58 3.2 Average number of bits per OFDM symbol (ANB) for each PUP band as a function of E{H1,m} with S = 2.4 W, I th1 = I th2 = 8 × 10−15 W and E{H2,m} = 10−14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Average number of bits per OFDM symbol (ANB) on the whole PUP bands as a function of S with I th1 = I th 2 = 8× 10−15 W, E{Hl,m} = 10−14, l = 1, 2. . 61 3.4 Average number of bits per OFDM symbol (ANB) on the whole PUP bands as a function of I th1 with S = 2.4 W, I th 1 = I th 2 , and E{Hl,m} = 10−14, l = 1, 2. 63 4.1 Markov chain model for the number of available PU bands. . . . . . . . . . . 70 4.2 ABR as a function of number of CRUs for OWF and PEPA. pa = 0.9 . . . . 76 4.3 ABR as a function of pa for OWF and PEPA. K = 6. . . . . . . . . . . . . . 77 5.1 ABR as a function of average SNR γ with K = 12 users for Case A. . . . . . 94 5.2 ABR as a function of average SNR γ with K = 12 users for Case B. . . . . . 95 ix 5.3 ABR difference between OWF and PEPA as a function of average SNR γ for Case A. K = 12 and M = 64. For simulation curves, T = 1 ms and for theoretical curves, T =∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 ABR difference between OWF and PEPA as a function of average SNR γ for Case B. K = 12 and M = 64. For simulation curves, T = 1 ms and for theoretical curves, T =∞. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 ABR as a function of number of subchannels M for Case A. K = 12 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 99 5.6 ABR as a function of number of subchannels M for Case B. K = 12 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞.100 5.7 ABR as a function of number of users K for Case A. M = 64 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. . . . . . 101 5.8 ABR as a function of number of users K for Case B. M = 64 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. . . . . . 102 6.1 Primary users’ active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2 Markov chain model for the number of available PU bands. . . . . . . . . . . 111 6.3 Average CRBS power for U = 1, 2 and 5 slots. . . . . . . . . . . . . . . . . 115 6.4 Transformations and relationships among the optimization prblems. . . . . . 120 6.5 Resource allocation time diagram for CRU 4 with pn = pa = 0.99 and R RT = 150 kbps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.6 Average total power of eight video conference CRUs as a function of pn with pn = pa and R RT = 150 kbps. . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.7 Transmit CRU power with pn = pa = 0.9 and R RT = 150 kbps. . . . . . . . 136 6.8 Average total power for eight video conference CRUs as a function of video data rate, RRT with pn = pa = 0.5. . . . . . . . . . . . . . . . . . . . . . . . 137 6.9 Average total power for eight video conference CRUs as function of video data rate, RRT with pn = 0.5 and pa = 0.1. . . . . . . . . . . . . . . . . . . . . . 138 6.10 Resource allocation time diagram for CRUs 3 and 6. . . . . . . . . . . . . . . 140 6.11 Dropped packet rate of RT CRUs as a function of video bit rate. . . . . . . 143 6.12 Average throughput of NRT CRUs as a function of video bit rate. . . . . . 144 6.13 System throughput as a function of video bit rate. . . . . . . . . . . . . . . 145 7.1 System throughput with respect to number of CRUs with RPRk = 1, k = 1, 2, · · · , K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.1 PU active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.2 Flow chart of the distributed allocation algorithm. . . . . . . . . . . . . . . . 172 8.3 Average number of bits per OFDM symbol duration per CRP as a function of the number of available subchannels with Sk = 10 −3 W, K = 3, RNOM1 = 25, RNOM2 = 30, R NOM 3 = 35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 x 8.4 Average number of bits per OFDM symbol duration per CRP as a function of the number of available subchannels with Sk = 10 −3 W, K = 3, RNOM1 = 20, RNOM2 = 20, R NOM 3 = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 8.5 Average number of bits per OFDM symbol duration per CRP as a function of total power with MCR = 8, K = 3, R NOM 1 = 20, R NOM 2 = 20, R NOM 3 = 20. 176 C.1 Flow chart for the cross layer resource allocation algorithm: Phase 1, the resource-limited phase. Point B refers to the entry point of the resource- abundant phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 C.2 Flow chart for the cross layer resource allocation algorithm: Phase 2, the resource-abundant phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 C.3 Flow chart for the Assignment algorithm used in the cross layer resource allocation algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 xi List of Abbreviations ABR Average Bit Rate. ANB Average Number of Bits per OFDM symbol. AWGN Additive White Gaussian Noise. BER Bit Error Rate. BS Base Station. CDF (cdf) Cumulative Distribution Function. CR Cognitive Radio. CRBS Cognitive Radio Base Station. CRP Cognitive Radio transceiver Pair. CRU Cognitive Radio User. DPR Dropped Packet Rate. DSL Digital Subscriber Line. FCC Federal Communications Commission. FI Fairness Index. HATR Highest Achievable Target Rate. HOL Head of Line. ITWF Iterative Water-Filling. KKT Karush-Kuhn-Tucker. LHS Left Hand Side. MAC Medium Access Control. xii MDKP Multidimensional Knapsack Problem. NABO Non-Active PU Bands Only. NRT Non-Real-Time. OFDM Orthogonal Frequency Division Multiplexing. OWF Optimal Water-Filling. PDF (pdf) Probability Density Function. PEPA Plain Equal Power Allocation. PHY Physical. PR Proportional Rate. PSD Power spectral density. PU Primary User. PUP Primary User transceiver Pair. QoS Quality of Service. RA Resource Allocation. RHS Right Hand Side. RT Real-Time. SINR Signal to Interference plus Noise Ratio. SNR Signal to Noise Ratio. xiii List of Symbols atk,m Subchannel assignment indicator function for subchannel m of CRU k bk,i The length (in bits) of the ith packet in CRU k’s buffer bm The probability of having m available subchannels BOWF Average bit rate for OWF BPEPA Average bit rate for PEPA ck,m(i) (cm(i)) Efficiency capacity of subchannel m of CRU k (the CRU) for constraint i ctk Fraction of service lacking for CRU k at time slot t dk Allowed packet delivery delay after packet’s creation in time slots dmk,l (d m l ) The power gain for subchannel m from PU l to CRU k’s receiver (the CRU’s receiver) fc Carrier frequency fCRk,l,m (f CR l,m ) The interference power introduced by the signal in the mth subchannel of CRU k (the CRU) into PU l’s frequency band fPUk,l,m (f PU l,m ) The interference power generated by PU l to the mth subchannel at CRU k’s receiver (the CRU’s receiver) f t Fairness index at time slot t gm The power gain for subchannel m from the CRP’s transmitter to the CRP’s receiver gmj,i The power gain for subchannelm from CRU i’s transmitter to CRU j’s receiver gtk,m The power gain for subchannel m at time slot t from the CRBS to CRU k’s receiver hml,k (h m l ) The power gain for subchannel m from CRU k (the CRU) to PU l’s receiver xiv htk,m The subchannel assignment function for subchannel m of CRU k in time slot t Hm(f) The OFDM receiver filter frequency response I thl Interference power threshold for PU l Ik,m Interference power from other CRPs on subchannel m of CRP k’s receiver ICRk,m (I CR m ) Interference power from the PU transmitters on subchannel m of CRU k’s receiver (the CRU’s receiver) IPUk,m (I PU m ) Interference power from the other CRUs on subchannel m of CRU k’s receiver (the CRU’s receiver) K Number of CRUs or CRPs KRT Number of RT service CRUs lCR,t The number of available PU bands to a CR system at time slot t L Number of PUs LOWF Water level when using water-filling algorithm mCR,t The number of subchannels available to a CR system at time slot t m(t1, t2) Number of available subchannels from time slot t1 to t2 M Number of subchannels (subbands) M̄CR Expected number of available subchannels Mt (M̄t) The set of available (unavailable) subchannels at time slot t pa The probability of a PU staying in active state pn The probability of a PU staying in inactive state pnm Incremental power required to add the nth bit to subchannel m P Transition probability matrix for the number of available subchannels Q Transition probability matrix for the number of available PU bands rk,m (rm) The number of bits per OFDM symbol that can be supported by subchannel m of CRU k (the CRU) rtk,m The number of bits per OFDM symbol that can be supported by subchannel m of CRU k in time slot t rt,REQk Number of bits that need to be transmitted at time slot t for CRU k xv rMAXk The maximum number of bits that can be transmitted in time slot t for CRU k Rk Total rate over all subchannels of CRU (CRP) k R̄t1,t2k Average data rate of CRU k from time slot t1 to t2 RDATAk The rate at which CRP k can reliably transmit data RMAX The maximum number of bits that can be allocated on any subchannel R̃MAXk Estimated maximum data rate for CRP k RNOMk Nominal rate requirement for CRU (CRP) k RNRTk Rate request of NRT CRU k RPRk The nominal rate requirement of CRU k Rt,REQk Minimum number of bits that needs to be transmitted in time slot 1 to t for CRU k RTARk Target rate of CRP k sk,m (sm) Transmit power for subchannel m of CRU k (the CRU) stk,m Transmit power for subchannel m of CRU k in time slot t Sk (S) Power limit for CRU k (the CRU) tDk,i The delivery time slot of the ith packet in CRU k’s buffer tSk,i The creation time slot of the ith packet in CRU k’s buffer Tstate The number of time slots between possible state transitions for a PU Tsymbol, Ts The duration of an OFDM symbol T thl Interference temperature limit ui the amount of resource i that has already been consumed vk Nominal rate degradation for CRP k wtk Weight factor of CRU k at time slot t Wl Bandwidth of PU l ΦCR(f) Equivalent baseband power spectral density (PSD) of the CRU OFDM signal for a transmit power of 1 W ΦPUl (f) Power spectral density (PSD) of PU l’s signal xvi Γ SNR gap parameter which indicates how far the system is operating from capacity pil The steady-state probability of being in state l Π The steady-state probability vector for the number of available PU bands σ20 Noise power on each subchannel Note: In this thesis, in order to distinguish a random variable from a sample value, the former is denoted by an uppercase letter, whereas the latter is denoted by a lowercase letter. xvii Acknowledgments I would like to offer my enduring gratitude to the faculty, staff and my fellow students at The University of British Columbia (UBC), all of whom have inspired me to continue my work in this field. I owe particular thanks to my supervisor, Prof. Cyril Leung, who has provided insightful guidelines, constructive comments, and invaluable suggestions throughout this study. Without his support, this work would not have been possible. It is my very great privilege to have been one of his students. I would like to express my sincere thanks to my parents for their selfless love and caring. Special thanks are owed to my husband, who has supported me throughout my years of education. My thanks go to the Natural Sciences and Engineering Research Council (NSERC) of Canada for awarding me a Post Graduate Scholarship (PGS) and support under Grant OGP0001731, the Province of British Columbia for awarding me a Pacific Century Graduate Scholarship, UBC for awarding me a University Graduate Fellowship, and the UBC PMC- Sierra Professorship in Networking and Communications for its support. Together, they have provided me with the financial means to engage in and complete this work. xviii To my parents and family. xix Co-authorship Statement Each of Chapters 2 to 8 is based on manuscripts that have been accepted, submitted, or to be submitted for publication in international peer-reviewed journals. The manuscripts are all co-authored by myself as the first author and my supervisor, Dr. Cyril Leung. In all these works, I played the primary role in designing and performing the research, doing data analysis, and preparing manuscripts under the supervision of Dr. Cyril Leung. xx Chapter 1 Introduction 1.1 Background Cognitive radio (CR) is a new technology that has attracted a lot of attention recently. It was first presented by Mitola [1] as a novel wireless communications approach with the ability to sense the external environment, learn from its history, and make intelligent decisions in adjusting its transmission parameters based on the current environment. Haykin [2] defines cognitive radio as follows: “Cognitive radio is an intelligent wireless communication system that is aware of its surrounding environment (i.e., outside world), and uses the methodology of understanding-by-building to learn from the environment and adapt its inter- nal states to statistical variations in the incoming RF stimuli by making corre- sponding changes in certain operating parameters (e.g., transmit-power, carrier- frequency, and modulation strategy) in real-time, with two primary objectives in mind: (1) highly reliable communications whenever and wherever needed; (2) efficient utilization of the radio spectrum.” With the ever increasing demand for mobile and wireless applications, the static assign- ment of radio resources to licensed holders has become a limiting factor in efficient spectrum utilization. In many jurisdictions, there is little spectrum left for exclusive use allocation [3]. However, studies have shown that a large portion of the assigned spectrum is used only sporadically, and that spectrum utilization is generally very low [4]. CR, with its ability 1 to sense the unused bandwidth and adjust its transmission parameters accordingly, is an excellent candidate for improving spectrum utilization. Recognizing this, and to alleviate the looming spectrum-shortage crisis, the FCC [5] has suggested the use of CR technology in order to allow unlicensed users to share radio resources with licensed users while not unduly interfering with them. Orthogonal frequency division multiplexing (OFDM) is a frequency division multiplexing (FDM) scheme that uses a large number of closely spaced orthogonal subcarriers to carry data. It has been considered an appropriate modulation candidate for CR systems [6], not only because of its high spectral efficiency, but also its flexibility in dynamically allocating radio resources to multiple users and its low interference between adjacent subcarriers. 97531 11 13 15 17 19 21 23 25 27 29 31 PU active frequency bands W1 W2 Spectrum hole Spectrum hole Spectrum hole Figure 1.1: PU active frequency bands, spectrum holes and CRU OFDM subchannels. Fig. 1.1 shows the spectrum in a typical OFDM-based CR system. The frequency bands that are currently used by the PUs are the shaded areas marked as W1 and W2. The remaining areas are not occupied by the primary users (PUs) at this time and this geographic location. These vacant frequency bands, termed spectrum holes, can be used by CR users (CRUs). 2 1.2 Scope To implement CR technology, three main tasks are involved [2], namely, radio-scene analysis (radio environment estimation and spectrum hole detection), channel identification (channel- state information estimation and channel capacity prediction), and transmit-power control and dynamic spectrum management. In this thesis, we focus on the last task and aim to design efficient resource allocation (RA) algorithms for OFDM-based CR systems. 1.3 Motivation The introduction of CR technology poses new RA problems that need to be solved. Com- pared to conventional wireless communication systems, two new issues arise, namely, the interference power to the PU bands should be kept below a certain threshold and good qual- ity of service (QoS) should be provided to CRUs in spite of the time-varying nature of the available spectrum. To make unlicensed sharing of the licensed spectrum a reality, PU operation must not be compromised. Thus, CRUs should monitor and keep the generated interference to PU bands to an acceptable level. To this end, the FCC Spectrum Policy Task Force [7] has recommended the use of interference temperature for assessing the level of interference. The specification of an interference temperature limit for a PU corresponds to a maximum allowed level of interference power; CRUs can use PU frequency bands as long as the total generated interference power to the PUs is kept below this limit. In a fading environment, a CRU signal may undergo deep fading and be received with very little power at the PU receiver. As a result, apart from the spectrum holes, CRUs can opportunistically share PU active frequency bands, as long as the total generated interference power at the PU receiver is below the specified interference power threshold. There are two main types of interference generated by CRUs sharing PU bands. One is the co-channel interference generated by CRUs using the PU active frequency bands, and the 3 other is the cross-channel interference from the adjacent channels used by CRUs. Because of orthogonality, inter-carrier interference among CRU subcarriers can be ignored. However, since PUs may not be using OFDM, there could be cross-channel interference [8] generated to the PU bands from adjacent CRU bands and to CRU bands from adjacent PU bands. When a CRU shares spectrum holes as well as PU active-frequency bands, the capacity achievable by the CRU is higher than if PU active-frequency bands are left unused [9]. We refer to this type of sharing as aggressive sharing, since any portion of the spectrum may be utilized at any time. To enable aggressive sharing of the spectrum, new RA algorithms that make efficient use of the radio resource and keep the total generated interference to the PUs below the specified interference power thresholds are necessary. 97531 11 13 15 17 19 21 23 25 27 29 31 PU active frequency bands W1 Guard bands Guard bands W2 Spectrum hole Spectrum hole Spectrum hole Figure 1.2: PU active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels in a protective sharing system. In some practical situations, aggressive sharing may not be possible. This can happen, for example, when the CR system is co-located with a broadcast PU system in which there are so many PU receivers that the probability of keeping the interference power below the specified interference power threshold at all receivers is almost zero. In such situations, PU active frequency bands cannot be shared in order to avoid excessive co-channel interference. To reduce cross-channel interference, appropriate guard bands can be introduced, as indicated in Fig. 1.2. We refer to this type of spectrum sharing, in which interference to PU receivers 4 need not be considered in RA, as protective sharing. Fig. 1.3 shows the spectrum which is shared and the types of interference considered in RA for aggressive sharing and protective sharing. Aggressive sharing - shares both active and non-active PU bands - co-channel and cross-channel interference considered in RA Protective sharing - shares non-active PU bands - CRU interference to PU receivers not considered in RA Figure 1.3: Spectrum sharing methods. Note that RA algorithms designed for aggressive sharing systems can be applied to protective sharing systems by setting the PU interference power threshold at each active PU receiver equals to 0. However, the protective sharing model greatly simplifies RA design because CRU interference to PU receivers does not need to be considered. Since interference to the PUs does not need to be considered in protective sharing systems, it might seem that RA schemes designed for conventional OFDM systems apply directly to OFDM-based CR systems. However, in a CR system, besides the fading characteristics of wireless communication channels, the available transmission spectrum also changes over time. RA algorithms designed for conventional OFDM systems assume that the available spectrum is fixed, which is not the case in CR systems. Thus, new RA algorithms that take into account both the fading characteristics of the transmission channel and the time-varying nature of the available spectrum are needed. 5 1.4 Related Previous Work 1.4.1 Resource Allocation Algorithms for OFDM-based CR Systems Algorithms Dealing with Cross-Channel Interference Cross-channel interference is considered in [10]–[12]. In [10], the bit and power loading problem is studied for the downlink of an OFDM-based CR system, in which the PU channel is located in the middle of a frequency band available to CRUs: an optimal scheme based on a Lagrange formulation and two suboptimal schemes are proposed assuming that there is only one CRU in the system. A similar model is used in [11] and [12] to study subcarrier, power, and bit allocation for multiple CRUs. Greedy algorithms are proposed based on minimum CRU power and minimum PU interference considerations. Algorithms Dealing with Co-Channel Interference The RA problem with co-channel interference has been studied in [13]–[16]. Different opti- mization problems are formulated and solved based on various interference-temperature-limit considerations. In [13], to simplify the problem, this limit is converted to a power constraint on each PU band by defining a protection area for the PUs. The power constraint is calcu- lated based on a path loss factor and the distance between the edge of the protection area and the CRU transmitter. Interference per subchannel is considered in [14] for the single CRU case and in [15] for multiple CRU case. The optimization problems formulated in [16] for a multiple CRU and multiple PU system use two interference temperature models proposed in [17]. The first model, which assumes a unified interference temperature limit on each subchannel, is translated into an average interference power threshold at the measurement point. The second model, which assumes different interference temperature limits on differ- ent PU active frequency bands, is translated into an average interference power threshold at 6 each PU receiver. Instead of the interference temperature limit, some other means of protection for PU signals are considered in [18][19] and [20]. Minimum average rate is guaranteed in [18], by assuming that PUs are willing to be cooperative in RA. PU outage probability is ensured in [19]. In [20], the average PU transmission rate is maintained using CRU cooperation. The above-mentioned algorithms, designed for multiple CRUs, all assume that each sub- channel can only be used by at most one CRU at any given time. In some situations, e.g., in an ad hoc system or a multicell cellular system, allowing multiple CRUs to share each subchannel can result in a higher spectrum utilization. In [21], a two-phase channel and power allocation scheme is proposed for multi-cell CR networks. In the first phase, resource allocation is done for all base stations (BSs) in a way that ensures that the interference power levels at the PU receivers do not exceed the predefined thresholds. In the second phase, the channels are allocated to the CRUs. In [22][23], CR systems with one channel are considered in which all CRUs access the channel at the same time, while keeping the total generated interference below the predefined interference temperature limit at a single measurement point. Two co-located cellular systems, consisting of one PU system and one CRU system, are studied in [24], in which the average generated interference from the CRUs to the PUs is ensured to be below the interference temperature limit. In [25], the generated interference to the PUs is limited by a per channel power mask, which specifies the highest power that can be used by a CRU on each channel. Algorithms Making Use of Spectrum Holes Studies assuming the use of spectrum holes appear in [26]–[28]. A spectrum-selection scheme is proposed in [26] for ad hoc networks, in which each user picks its channels based on a set of rules. The users try to maximize their own performance with minimal regard to overall system performance. In [27],[28], a game theoretic approach is utilized to solve the channel allocation problem based on the observation that users in CR systems may not be willing 7 to cooperate with others but rather may selfishly try to maximize their own performance. A dynamic channel allocation scheme based on a potential game1 approach is proposed for ad hoc networks in [27]. In [28], a non-cooperative game is formulated to model the multi-channel allocation problem. In [30], although PU active-frequency bands are left unused, the subchannels in spectrum holes are shared among CRUs, with the objective of minimizing the total required power consumption while satisfying the CRUs’ data rate and bit error rate (BER) requirements. In [31], cross-layer based medium access control (MAC) protocols are proposed to allow CRUs to share the spectrum holes, which are detected by integrated physical (PHY) layer spectrum-sensing policies. The goal in [32] is to minimize CRU throughput variance in a single-user CR system. In [33], the power allocation problem for a single CRU is cast as a rate-maximization problem that considers the risk of losing a certain subchannel due to PU activity. 1.4.2 Resource Allocation Algorithms for OFDM Systems Centralized Physical (PHY) Layer Approach The bit and power loading problem for single-user OFDM systems can be solved by using the well-known water-filling [34] algorithm if we assume that the number of bits to be loaded is a real number, or implement a greedy approach that assigns one bit at a time to the subcarrier that requires the least additional power for the integer bit case [35]. To reduce computational complexity for the integer bit case, various low complexity algorithms have been proposed, for both optimal (e.g. [36, 37]) and suboptimal solutions (e.g., [38]–[40]). In the case of the downlink transmission of a BS to multiple users, the subchannels need to be assigned to users exclusively [41]. Therefore, RA involves subchannel assignment in addition to power and bit allocation. When the goal is to maximize system throughput, the 1In game theory, a potential game is one in which the incentive of all players to change their strategies can be expressed in one global function, the potential function [29]. 8 problem can be solved in two separate steps [41], namely, assigning each subchannel to the user with the best channel condition, followed by power and bit allocation. When there are QoS or fairness requirements, subchannel, bit, and power allocation be- comes more complicated. Since optimal solutions are generally computationally complex, various sub-optimal solutions have been proposed. In [42]–[45], suboptimal solutions are pro- posed to minimize the total transmit power while satisfying rate and BER requirements for real-time (RT) services. For non-real-time (NRT) services, maximizing system throughput while guaranteeing a certain level of fairness among users is a reasonable goal [46]–[49]. Most of these suboptimal solutions use a divide-and-conquer approach, in which the subcarrier, power, and bit allocation problem is broken down into two steps, i.e., allocate subcarriers to users and load appropriate power and bits to each subcarrier. During the first step, power is often assumed to be the same across all subcarriers so as to simplify the problem. Centralized Medium Access Control (MAC) Layer Approach RA also occurs in the MAC layer, which is responsible for packet scheduling. Almost all existing studies extend opportunistic scheduling [50] strategies for the single carrier case to the multiuser OFDM case with multiple subcarriers. For NRT services, some schemes, e.g., [43][51], extend the proportional fair (PF) rule [52], while others (e.g., [53]) extend the modified-largest weighted delay first (M-LWDF) rule [54] for RT traffic. An urgency and efficiency based packet scheduling (UEPS) algorithm is proposed in [55] for both RT and NRT services using an urgency factor that reflects the urgency of meeting QoS requirements combined with the PF rule to maximize system throughput. The urgency factor approach has previously been used in the single carrier case [56]. Centralized Cross-Layer Approach Some researchers have adopted a cross-layer design approach in allocating system resources. In [57]–[59], sub-optimal algorithms for NRT services are proposed; algorithms for both RT 9 and NRT services are studied in [60] and [61]. In [60], the QoS for RT applications is improved by giving high priority to users whose head-of-line (HOL) packet deadlines are approaching. In [61], the MAC layer QoS requirement for each user is converted to a PHY layer fixed rate requirement based on the average user packet arrival rate and delay constraint. An optimal subchannel and power allocation strategy is proposed that maximizes system throughput subject to a total transmit power limit and user delay requirements. Distributed Approach While centralized RA is suitable for single-cell systems, distributed algorithms may be more appropriate for multi-cell cellular systems or ad hoc systems. Although distributed dynamic channel allocation (DCA) has been studied for multiple cell cellular networks for voice ser- vices, it cannot be easily ported to multiuser OFDM systems. This is because traditional DCA schemes assume homogeneous applications with a pre-determined SINR (signal to noise and interference ratio) threshold, and may not efficiently support services with different re- quirements. To dynamically allocate resources in a multi-cell system or an ad hoc system, subcarriers may be simultaneously shared among served users in order to improve system performance. In this case, co-channel interference has to be considered. In [62], other users’ signals are treated as noise, and the power allocation problem is viewed as a non-cooperative game. A distributed iterative waterfilling (ITWF) algorithm is proposed for digital subscriber line (DSL) systems. To achieve the optimal power allocation solution, the achievable target rates must be known. This is not a big problem for DSL systems, but is unrealistic for time-varying wireless channels. To make ITWF suitable for wireless systems, a scheme is proposed in [63] for multi-cell wireless systems in which a virtual referee is introduced to displace some users out of certain subchannels when necessary, to allow ITWF to converge to good solutions. Power and bit allocation for multiuser OFDM systems with co-channel interference have 10 been formulated as a constrained nonlinear programming problem in [64]. To reduce the complexity of finding a solution, a distributed algorithm is proposed that allocates one bit per iteration. 1.5 Objectives The overall goal of the thesis is to design efficient RA algorithms using both aggressive and protective sharing for OFDM-based CR systems. In the category of aggressive sharing, although cross-channel and co-channel interferences to the PUs have been considered by different researchers separately, they have not been con- sidered jointly. To ensure the PUs’ normal operation, the total generated interference power to the PUs has to be kept below the specified interference power thresholds. Therefore, both cross-channel and co-channel interference have to be taken into account in RA, especially in cases where the PUs do not use OFDM. Our first objective is to Objective 1: Devise efficient RA algorithms to allocate subchannels, powers, and bits in OFDM-based CR systems, which aggressively share both the spec- trum holes and PU active frequency bands while guaranteeing that the total generated interference power due to cross-channel and co-channel interference does not exceed the specified interference power threshold of each PU. In the category of protective sharing, most existing studies focus on dynamic channel allocation, and few consider the influence of the time-varying nature of the available spectrum on QoS and fairness of CRUs. In this thesis, our second objective is to Objective 2: Design efficient RA algorithms in OFDM-based CR systems with QoS provisioning and fairness considerations to operate in a fading environment with time-varying spectrum, and protectively share the spectrum holes without generating undue interference to the PUs. 11 1.6 Thesis Overview This thesis is written in the manuscript-based format according to the guidelines established by The University of British Columbia. Each chapter has its own reference list. The rela- tionships among the chapters are shown in Fig. 1.4 and described below. RA algorithms that aggressively share PU bands are discussed in Chapters 2 and 3, and RA algorithms that protectively share PU bands are studied in Chapters 4 to 8. Subchannnel, power and bit allocation for multiple CRUs in a multi-cell cellular system is studied in Chapter 2. Each cell is treated as a CRU system consisting of one cognitive radio BS (CRBS) and multiple CRUs. Subchannel allocation is performed within each cell. Power and bit allocation is done across all the cells. Considering co-channel interference among multiple CRUs, as well as cross-channel and co-channel interference resulting from CRU sharing of PU bands, the RA problem is formulated as a multidimensional knapsack problem (MDKP). A low-complexity suboptimal solution is proposed for the formulated MDKP problem. In Chapter 3, a simplification of the model proposed in Chapter 2 is formulated which allows for a faster algorithm. The simplification is based on the fact that cross-channel interference from CRUs to PUs is negligible except for a few subchannels adjacent to the PU bands. Assuming that the bandwidth of a PU is much larger than that of a subchannel in an OFDM-based CR system and that there is usually a guard band between two adjacent PU bands, cross-channel interference from any CRU subchannel impacts mostly one PU band, instead of several PU bands as assumed in Chapter 2. In Chapter 4, the performance of the plain equal power allocation (PEPA) algorithm, which allocates the same amount of power to each available subchannel, is studied for the continuous bits case for multiple OFDM-based CR systems. When the goal is to maximize system throughput, the difference between PEPA and the optimal solution is shown to be small. 12 C h a p te r 2 - R A  f o r O F D M -b as ed  m u lt i- ce ll  C R  sy st em s - B o th  c ro ss -c h an n el  a n d  c o -c h an n el  in te rf er en ce  c o n si d er ed  i n  t h e m o d el - A  g en er al iz ed  m u lt id im en si o n al  k n ap sa ck  p ro b le m  f o rm u la ti o n - L o w -c o m p le x it y  s u b o p ti m al  s o lu ti o n  p ro p o se d  u si n g  a  g re ed y  a p p ro ac h C h a p te r 3 - R A  f o r O F D M -b as ed  s in g le -c el l C R  s y st em s - L o w  c o m p le x it y  s u b o p ti m al  so lu ti o n  p ro p o se d C h a p te r 4 - P er fo rm an ce  e v al u at io n  o f P E P A  fo r m u lt iu se r O F D M -b as ed  C R  sy st em s - C o n ti n u o u s b it  c as e - T w o  s u b ch an n el  a ss ig n m en t st ra te g ie s st u d ie d C h a p te r 5 - P er fo rm an ce  e v al u at io n  o f P E P A  fo r m u lt iu se r O F D M  s y st em s,  su it ab le  f o r C R  s y st em s - C o n ti n u o u s an d  d is cr et e b it  c as es - A  s im p le  t o  i m p le m en t d is cr et e b it  P E P A  a lg o ri th m  p ro p o se d C h a p te r 6 - R A  f o r m u lt iu se r O F D M -b as ed  si n g le -c el l C R  s y st em s - D y n am ic  c o n v er si o n  o f C R U  M A C  la y er  Q o S  r eq u ir em en ts  t o  P H Y  la y er  r at e re q u ir em en ts  - P ro b le m  f ea si b il it y  i ss u e so lv ed  u si n g  a  g o al  p ro g ra m m in g  ap p ro ac h R T  S er v ic es - O n -t im e R T  p ac k et  d el iv er y R T  a n d  N R T  S er v ic es - O n -t im e R T  p ac k et  d el iv er y - N R T  u se r n o m in al  r at es C h a p te r 7 - R A  f o r a m u lt iu se r O F D M -b as ed  si n g le -c el l C R  s y st em - U se r p ro p o rt io n al  r at es  m ai n ta in ed C h a p te r 8 - R A  f o r ad  h o c o r m u lt i- ce ll  O F D M -b as ed  C R  s y st em s - D is tr ib u te d  a lg o ri th m  s tu d ie d - S y st em  t h ro u g h p u t m ax im iz ed  w it h  u se r n o m in al  r at es  a ch ie v ed  i f sy st em  r es o u rc e is  p le n ti fu l - F ai r d eg ra d at io n  p ro v id ed  i f sy st em  re so u rc e is  l im it ed A g g re ss iv e sh a ri n g M ai n  f o cu s:  C R U  i n te rf er en ce  p o w er  a t P U  re ce iv er s. A simplified model P ro te ct iv e sh a ri n g M ai n  f o cu s:  t im e- v ar y in g  n at u re  o f th e av ai la b le  s y st em  r es o u rc e,  Q o S  a n d  fa ir n es s am o n g  C R U s. E q u al  s u b ch an n el  p o w er  a ll o ca ti o n C ro ss -l ay er  d es ig n C o n v ex  o p ti m iz at io n G o al  p ro g ra m m in g C en tr al iz ed  a lg o ri th m D is tr ib u te d  a lg o ri th m F ig u re 1. 4: T h es is ov er v ie w . 13 The performance difference between PEPA and the optimal solution for both the con- tinuous and discrete bits case in a multiuser OFDM system is examined in Chapter 5. A low-complexity discrete bit PEPA algorithm is proposed that can also be used in an OFDM- based CR system. In Chapter 6, the subchannel, bit and power allocation problems at the PHY layer and QoS requirements at the MAC layer are considered jointly for RT services on the downlink of a multiuser OFDM-based CR system. The proposed algorithm is designed to provide satisfactory QoS to RT applications is spite of the rapidly changing available resources resulting from PU activities. The RT CRU MAC layer QoS requirements are dynamically converted to PHY layer rate requirements; the conversion depends on the delivery status of queued packets as well as the number of available subchannels. As an extension, the RA problem for a mixture of RT and NRT services is also considered. The time-varying nature of the number of OFDM subchannels available to CRUs gives rise to two resource allocation issues, namely problem feasibility and false urgency. To solve the problem feasibility issue, which arises when resources are insufficient to meet all user QoS requirements, we adopt a goal programming approach. The false urgency issue is effectively avoided by a proposed rate requirement calculation mechanism based on the status of the packets in queue and system resource availability. A optimization problem is formulated and the optimal solution is provided. In Chapter 7, we study the RA problem in a multiuser OFDM-based CR system for NRT applications in which average user data rates are to be maintained proportionally. In contrast to existing algorithms designed for multiuser OFDM systems, which are unable to guarantee users proportional rates when applied to a CR system, we propose an optimal RA algorithm that ensures CR user rates are maintained in proportion to predefined target rates, while at the same time providing an improved system throughput. The protective sharing RA algorithms in Chapters 4 to 7 are designed for systems in which centralized algorithms are appropriate. In Chapter 8, we consider RA in an ad hoc 14 system, in which a distributed algorithm is more practical. In a resource-limited situation under which the nominal rate requirements of users cannot be satisfied, it is desirable to provide fair degradation among users. 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Srivastava, “Subcarrier allocation and bit loading algorithms for OFDMA-based wireless networks,” IEEE Transactions on Mobile Com- puting, vol. 4, no. 6, pp. 652–662, November-December 2005. 21 Chapter 2 Subcarrier, Bit and Power Allocation for Multiuser OFDM-based Multi-Cell Cognitive Radio Systems 2.1 Introduction In many jurisdictions, there is a scarcity of unallocated frequency bands below 6 GHz. At the same time, studies have found that the utilization of a large portion of the allocated (licensed) bands is very low [1]. It has been suggested [2] that one promising approach to solving the spectrum shortage crisis is to use cognitive radio (CR) technology [3][4]. In a CR system, cognitive radio users (CRUs) are allowed to use licensed bands as long as the (licensed) primary users (PUs) are not unduly affected. In order to assess the level of interference, the FCC Spectrum Policy Task Force [5] has recommended the use of the interference temperature. The specification of an interference temperature limit for a PU corresponds to a maximum allowed interference power and makes it possible for CRUs to use both non-active and active PU bands in a controlled fashion. As explained in [6], OFDM is an attractive modulation candidate for CRUs. The subcar- rier, bit and power (resource) allocation optimization problem for OFDM has been studied in the literature (e.g. [7]). However, since a PU may not use OFDM technology, cross-channel interference [8] could be generated by a CRU using a subchannel adjacent to the PU band, and from the PU transmitter to the CRU’s subchannels. Co-channel interference arises when 1The material in this chapter is largely based on the following: (1) Yonghong Zhang and Cyril Leung, “Resource Allocation in an OFDM-based Cognitive Radio System,” accepted for publication in IEEE Transactions on Communications. (2) Yonghong Zhang and Cyril Leung, “Subcarrier, Bit and Power Allocation for Multiuser OFDM-based Multi-Cell Cognitive Radio Systems,” IEEE 68th Vehicular Technology Conference (VTC 2008-fall), Calgary, Canada, Sept. 2008. 22 a CRU uses an active PU band. Thus, for CR systems in which the PUs do not use OFDM, both cross-channel and co-channel interference between the PUs and the CRUs have to be considered. Cross-channel interference is considered in [9][10]. The bit and power loading problem for the downlink of an OFDM-based CR system is studied in [9] using a model in which the active PU’s channel is located in the middle of a frequency band which is available to CRUs. A scheme based on a non-integer Lagrange formulation and two suboptimal schemes are proposed for one CRU. A similar model is used in [10] to study subcarrier, power, and bit allocation for multiple CRUs. A greedy algorithm is proposed based on minimum CRU power and minimum PU interference considerations. The model with a single continuous PU frequency band in [9][10] is somewhat unrealistic in practice. Furthermore, active PU bands can also be utilized as long as such use does not interfere unduly with PU receivers. Some resource allocation (RA) algorithms consider co-channel interference. A single CRU case is studied in [11] and a multiple CRU case is studied in [12], in which each subchannel can only be used by at most one CRU at any given time. Multiple CRUs are allowed to share a subchannel in [13]–[17]. In [13], a two-phase channel and power allocation scheme is proposed for multi-cell CR networks. In the first phase, RA is done for all base stations (BSs), in a way that ensures that the interference power levels at the PU receivers do not exceed the predefined thresholds. In the second phase, the channels are allocated to the CRUs. In [14][15], CR systems with one channel are considered in which all CRUs access the channel at the same time, while keeping the total generated interference below the predefined interference temperature limit at a single measurement point. Two co-located cellular systems, consisting of one PU system and one CRU system, are studied in [16], in which the average generated interference from the CRUs to the PUs is ensured to be below the interference temperature limit. In [17], the generated interference to the PUs is limited by a per channel power mask, which specifies the highest power that can be used by a CRU on each channel. 23 To the best of our knowledge, there is no published work in which both cross-channel and co-channel interference are considered to allow CRUs to use non-active PU bands as well as active PU bands. In this chapter, we study the subcarrier, bit and power allocation problem for multiuser OFDM-based CR systems in which one or more spectrum holes exist between multiple PU frequency bands, and CRUs are able to use any portion of the frequency band as long as this does not interfere unduly with the PUs. We formulate the RA problem as a multidimensional knapsack problem (MDKP) and propose to a low-complexity algorithm to solve it. 2.2 System Model Consider a CR system with total bandwidth W Hz and L PUs. The nominal bandwidth of PU l ranges from fc + F PU l to fc + F PU l +Wl. The equivalent baseband power spectral density (PSD) of PU l’s signal is ΦPUl (f). The maximum interference power that PU l can tolerate is I thl = T th l ∗Wl [5], where T thl is the interference temperature limit of PU l. The frequency bands that are currently carrying PU signals which can be detected by a receiver are referred to as active; non-active bands are also termed spectrum holes. Assume that there are K CR base stations (CRBSs), all of which use OFDM. We are interested in downlink transmissions from the CRBSs to CRUs. The CRBSs may use both active bands and spectrum holes provided that the total interference in PU l’s active band does not exceed I thl . The CR system accomodates M equally spaced OFDM subcarriers; the nominal bandwidth of subband m, m = {1, 2, . . . ,M}, ranges from fc + (m − 1)∆f to fc +m∆f . Each subchannel can be used by a CRBS to transmit to at most one of its CRUs at a time. At each scheduling time, each CRBS chooses the CRU for transmission for each subchannel based on the CRUs’ quality of service (QoS) requirements using some subchannel assignment strategy, e.g., the one described in Section 2.6.2. It is sometimes convenient to refer to 24 CRBS k and its CRUs collectively as cognitive radio pair (CRP) k. The number of CRUs for CRBS k is denoted by nCRU,k. The subchannels are modelled in discrete-time, with the time- varying gain for subchannel m from CRBS i to CRBS j’s CRU (receiver) denoted by √ gmj,i. It is assumed that the power gains { gmj,i } are outcomes of independent random variables (rv’s), and that there is no inter-carrier interference (ICI). Each subchannel is shared by all CRBSs, and the signals from other CRBSs are considered to be interference to any given CRBS’s receiver. The power gains for subchannel m from CRBS k to PU l’s receiver are denoted by hml,k. Each subchannel of a CRP suffers two types of interference, one from the PUs, and the other from other CRBSs which are transmitting on the same subchannel. The signal to interference plus noise ratio (SINR), γk,m, at the output of the OFDM receiver filter for subchannel m of CRP k’s receiver can be written as γk,m = gmk,ksk,m σ20 + I PU k,m + I CR k,m . (2.1) In (2.1), sk,m is the transmit power, σ 2 0 is the additive white Gaussian noise (AWGN) noise power, ICRk,m = ∑ i6=k g m k,isi,m is the interference power from the other CRBSs and I PU k,m =∑L l=1 f PU k,l,m is the interference power from the PU transmitters, where f PU k,l,m can be calculated as fPUk,l,m = ∞∑ n=−∞ dnk,l ∫ (m+n)∆f−(FPUl + 12Wl) (m+n−1)∆f−(FPUl + 12Wl) |Hm(f)|2ΦPUl (f)df . (2.2) In (2.2), Hm(f) , √ 3 2 sinc ( f − (m− 1 2 )∆f + F PUl + 1 2 Wl ) is the OFDM receiver filter fre- quency response, chosen so that the CRBS signal powers at the input and output of the filter are equal. The term dnk,l, n ∈ Z represents the power gain for the subband which ranges from fc + (n− 1)∆f to fc + n∆f . Assuming MQAM modulation and that the total interference can be modelled as AWGN, for a given bit error rate (BER), the number of bits per OFDM symbol, rk,m, which can be 25 supported by subchannel m of CRP k can be approximated in many cases [18, 19] rk,m ≈ log2(1 + γk,m Γ ) (2.3) where Γ is a SNR gap parameter which indicates how far the system is operating from capacity and can be calculated using Γ , − ln(5BER)/1.5. Substituting (2.1) into (2.3), we obtain rk,m = log2 ( 1 + gmk,ksk,m Γ(σ20 + I PU k,m + I CR k,m) ) . (2.4) Spectrum hole 54321 6 7 8 9 10 11 12 13 14 15 16 W2 W3 Spectrum hole Spectrum hole PU active frequency bands f fc+F PU 1 fc+F PU 2 fc+F PU 3 W1 fc Figure 2.1: PU active frequency bands, spectrum holes and CRU OFDM subchannels. The PU active frequency bands, the spectrum holes and the CRU OFDM subbands are shown in Fig. 2.1. 2.3 The Optimization Problem Our objective is to maximize the overall rate achievable by the CRPs, while keeping the interference to the PUs below the specified thresholds I thl , l = 1, 2, . . . , L. The optimization 26 problem can be formulated as max sk,m K∑ k=1 M∑ m=1 rk,m (2.5) subject to M∑ m=1 sk,m ≤ Sk,∀k ∈ 1, 2, . . . , K (2.6) K∑ k=1 M∑ m=1 sk,mf CR k,l,m ≤ I thl ,∀l ∈ 1, 2, . . . , L (2.7) rk,m ∈ {0, 1, 2, . . . , RMAX} . (2.8) In (2.6), Sk is the power limit for CRBS k. In (2.8), R MAX is the maximum number of bits that can be allocated on any subchannel. In (2.7), fCRk,l,m is the interference power introduced by the signal in the mth subchannel of CRBS k into PU l’s frequency band, fCRk,l,m = ∫ FPUl −(m− 12 )∆f+Wl FPUl −(m− 12 )∆f hml,kΦ CR(f)df (2.9) where ΦCR(f) is the equivalent baseband PSD of the OFDM signal in subband m when sk,m = 1 W, i.e. when the transmit power of CRBS k in subchannel m is one watt. 2.4 The Single CRU Case To better understand the problem, we first study a system with only one CRU, i.e., one CR transmitter and one CR receiver. Problem (2.5) can thus be written as max sm M∑ m=1 rm (2.10) 27 subject to M∑ m=1 sm ≤ S (2.11) M∑ m=1 smf CR l,m ≤ I thl ,∀l ∈ 1, 2, . . . , L (2.12) rm ∈ {0, 1, 2, . . . , RMAX}. (2.13) In (2.10)-(2.13), the subscript k is omitted to simplify the notation. In (2.10), since there is no co-channel interference from other CRUs, rk,m in (2.4) can be simplified to rm = log2 ( 1 + gmsm Γ(σ20 + I PU m ) ) (2.14) where gm is the power gain for subchannel m from the CRU transmitter to the CRU receiver. In order to find a low complexity solution, we rewrite the problem as max xnm M∑ m=1 RMAX∑ n=1 xnm (2.15) subject to M∑ m=1 RMAX∑ n=1 pnmx n m ≤ S (2.16) M∑ m=1 RMAX∑ n=1 pnmf CR l,m x n m ≤ I thl ,∀l ∈ 1, 2, . . . , L (2.17) xnm ∈ {0, 1} (2.18) The term pnm ∆ = 2n−1Γ(σ20 + Im)/gm is the incremental power required to add the nth bit to subchannel m and xnm = 1 indicates that the nth bit of subchannel m is allocated. Since pnm increases with n, it follows that x n m = 1 only if x i m = 1 for i = 1, 2, . . . , n− 1. Note that rm = ∑RMAX n=1 x n m and sm = ∑RMAX n=1 p n mx n m. 28 The optimization problem in (2.15) is actually a 0-1 multidimensional knapsack problem (MDKP) of dimension L + 1. Since such problems are NP-hard [20] and our interest is in low-complexity solutions, we focus on greedy-like methods [21]–[23]. Usually, such algorithms make use of an efficiency value for each item which measures the benefit of selecting that item (ı.e. setting xnm to 1). Then at each step, the item with the highest efficiency value is picked until at least one of the constraints no longer holds. For example, (2.15) without constraint (2.17) is a one dimensional 0-1 knapsack problem with the reward value of each item set to 1. The efficiency value of an item can be defined as the ratio of the item’s reward to its cost, i.e. 1/pnm. Then the solution for this problem involves picking the item with the smallest incremental power needed. Since for each sub- channel, the incremental power for adding the nth bit is smaller than that for adding the (n+1)th bit, we need to select the lowest power only from { pr̂m+1m ,m = 1, 2, . . . ,M } , where r̂m is the number of bits allocated on subchannel m. This is just the algorithm proposed in [7] for power allocation for a single user OFDM system. For a MDKP, the challenge of determining an effective efficiency value involves consider- ing multiple costs. Here, we adopt the technique in [23] as it provides a simple to implement solution with low computational complexity. As a first step, we define the efficiency capacity of subchannel m for constraint l as cm(l) =  S−u0 pr̂m+1m , l = 0 Ilth−ul pr̂m+1m h m l f CR l,m , l = 1, 2, . . . , L. (2.19) In the RHS of (2.19), the first line accounts for constraint (2.16) whereas the second line accounts for the set of constraints (2.17). The terms u0 and ul are the costs of resources 29 already allocated, i.e. u0 = M∑ m=1 r̂m∑ n=1 pnm (2.20) ul = M∑ m=1 r̂m∑ n=1 pnmf CR l,m . (2.21) Intuitively, the efficiency capacity, cm(l), in (2.19) represents the maximum number of the current item (i.e. the (r̂m + 1)th bit of subchannel m) that can be accommodated if the entire remaining room for constraint l were to be used for that bit. Second, we define the efficiency value of subchannel m as minl {cm(l)}. We then greedily allocate a bit to the subchannel with the largest efficiency value. This process of allocating one bit at a time is repeated until one of the constraints can no longer hold. Since the next bit is allocated to the subchannel with the maximum efficiency value and the efficiency value is the minimum efficiency capacity, we refer to our algorithm as Max-Min. A pseudo code listing of the Max-Min algorithm is given below. Initialize r̂m = 0, ∀m; ul = 0,∀l while S − u0 > 0 and I lth − ul > 0, l = 1, 2, . . . , L for m = 1 to M calculate cm(l),∀l using (2.19) em = minl {cm(l)} endfor α = argmaxm(em) r̂α = r̂α + 1 update ul,∀l using (2.20) and (2.21) endwhile The Max-Min algorithm has complexity O(RLM), where R is the total number of allo- cated bits. 30 2.5 The Multiple CRU Case In the multiple CRU case, as can be seen from (2.4), a CRU’s transmit power is treated as interference power to other CRUs using the same subchannel. Optimization problem (2.5) can no longer be transformed to a 0-1 MDKP as in the single CRU case in Section 2.4. Nonetheless, optimization problem (2.5) can be viewed as a generalized non-linear MDKP problem with dimension D = K + L. Each dimension corresponds to one of the constraints listed in (2.6) and (2.7). In this section, we extend the Max-Min algorithm to the multiple CRU case. A bit increase in the sum in (2.5) can be viewed as a reward. Because each additional transmitted bit requires a certain amount of CRBS power and generates some interference, the power consumed and the interference generated can be treated as costs. Both CRBS power and interference power are limited system resources and we would like to incur a low cost in adding each bit. If there were only one constraint, we could add the bit that has the largest efficiency value, i.e., ratio of reward to cost. However, for the optimization problem in (2.5), for each bit added, there are altogether K + L reward/cost ratios, one for each constraint. In the process of allocating one additional bit in the system, we can choose any of the M subchannels and any of the K CRBSs. Thus, we select one among K ×M candidate subchannel/CRBS pairs. For each candidate subchannel/CRBS pair, we calculate K + L reward/cost ratios and then its efficiency value. Finally, we add the bit to the subchan- nel/CRBS pair that has the largest efficiency value. In the following, we first discuss how to calculate reward/cost ratio for each constraint when one bit is added to subchannel m of CRBS k. We then define the efficiency value ek,m and describe the multiple CRU Max-Min algorithm. 31 2.5.1 Reward/Cost Ratio for Each Constraint The reward for adding one bit to subchannel m of CRBS k is 1 for any m and k. To calculate the cost to each of the K + L constraints of adding one bit on subchannel m of CRBS k, we need to know the required incremental power which can be calculated from (2.4) as tk,m = 2 rk,mΓ(σ20 + I PU k,m + I CR k,m)/g m k,k. However, we do not know the exact value of I CR k,m since ICRk,m depends on the values of si,m, i ∈ {1, 2, . . . , K \ k}. As we increase CRBS k’s power, the increased power will generate more interference on subchannel m to the receivers of the other CRBSs; to maintain a fixed bit rate, each of the other CRBSs will need to increase its transmitter power, si,m, resulting in a higher value for I CR k,m. As a result, tk,m has to be increased again. The process of increasing powers continues until either an equilibrium point is reached or the powers grow without bound. In order to calculate the reward/cost ratio for each constraint, we next derive an expres- sion for calculating the increase in CRBS powers, Ti,bm = (t i,b 1 , t i,b 2 , . . . , t i,b K ) T if b bits are added to CRBS i on subchannel m. Since we are only concerned with one subchannel in the rest of this subsection, the subscript m is omitted for simplicity of notation. From (2.4), we have K equilibrium equations gk,ksk − (2rk − 1)Γ ∑ l 6=k gk,lsl = (2 rk − 1)Γσ20 . (2.22) In matrix form, the equilibrium powers should satisfy GS = Q (2.23) where S = (s1, s2, ..., sK) T is the kth pair’s transmission power, Q = ((2r1 − 1)ΓN ′1, (2r2 − 32 1)ΓN ′2, · · · , (2rK − 1)ΓN ′k)T , N ′k = σ20 + IPUk , and G is a K ×K matrix that defined by  g1,1 f(1, 2, r1) · · · f(1, K, r1) f(2, 1, r2) g2,2 · · · f(2, K, r2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f(K, 1, rK) f(K, 2, rK) · · · gK,K  where f(i, j, r) = −(2r − 1)Γgi,j. The final equilibrium powers can be obtained as S = G−1Q. (2.24) After the allocation of b more bits to pair i, the equilibrium power vector Si,b (if it exists) can be obtained from Gi,bSi,b = Qi,b (2.25) where Gi,b is the same as G except that the ith row is f(i, 1, ri + b), f(i, 2, ri + b), · · · , gi,i, · · · , f(i,K, ri + b) (2.26) and Qi,b = ((2r1 − 1)ΓN ′1, (2r2 − 1)ΓN ′2, · · · , (2ri+b − 1)ΓN ′i , · · · , (2rK − 1)ΓN ′K)T . (2.27) Subtracting (2.23) from (2.25), the lth equation (l 6= i) is f(l, 1, rl)∆s1 + f(l, 2, rl)∆s2 + . . .+ gl,l∆sl + . . .+ f(l,K, rl)∆sK = 0 (2.28) 33 where ∆sk = s i,b k − sk and si,bk is the kth component of Si,b. The ith equation is f(i, 1, ri + b)s i,b 1 − f(i, 1, ri)s1 + f(i, 2, ri + b)si,b2 − f(i, 2, ri)s2 + . . .+ gi,i(s i,b i − si) + . . .+ f(i,K, ri + b)si,bK − f(i,K, ri)sK = (2ri+b − 1)ΓN ′i − (2ri − 1)ΓN ′i . (2.29) Substituting si,bk , ∀k by ∆sk + sk, we have [f(i, 1, ri + b)− f(i, 1, ri)]s1 + f(i, 1, ri + b)∆s1 + [f(i, 2, ri + b)− f(i, 2, ri)]s2 + f(i, 2, ri + b)∆s2 + . . .+ gi,i∆si + . . . + [f(i,K, ri + b)− f(i,K, ri)]sK + f(i,K, ri + b)∆sK = (2b − 1)2riΓN ′i . (2.30) Moving the terms containing sk from the LHS to the RHS, and noting that f(i, k, ri + b)− f(i, k, ri) = −(2b − 1)2riΓgi,k, we obtain f(i, 1, ri + b)∆s1 + f(i, 2, ri + b)∆s2 + . . .+ gi,i∆si + . . . +f(i,K, ri + b)∆sK = (2 b − 1)2riΓ ( N ′i + ∑ k 6=i skgi,k ) . (2.31) Since ri = log2 ( 1 + sigi,i Γ(N ′i+ ∑ k 6=i skgi,k) ) , we have Γ(N ′i+ ∑ k 6=i skgi,k) = sigi,i 2ri−1 . Equation (2.31) can then be written as f(i, 1, ri + b)∆s1 + f(i, 2, ri + b)∆s2 + . . .+ gi,i∆si + . . . +f(i,K, ri + b)∆sK = (2 b − 1)sigi,i 2 ri 2ri − 1 . (2.32) 34 Combining (2.28) and (2.32), we obtain Gi,bTi,b = P (2.33) where Ti,b ∆ = ( ti,b1 , t i,b 2 , · · · , ti,bK )T = Si,b − S, ti,bk = ∆sk represents the power increase re- sulting from the allocation of b additional bits to pair i, and P is the K × 1 column vector (0, 0, · · · , (2b−1)sigi,i2ri/(2ri−1), · · · , 0)T . The increased power vector Ti,b can be obtained as Ti,b = (Gi,b)−1P. (2.34) Since only the ith component of P is non-zero, Ti,b is dependent only on the ith column of (Gi,b)−1 and ti,bk = (2b − 1)2rigi,isi 2ri − 1 hki (2.35) where hki is the (k, i)th element of (G i,b)−1. We obtain the reward/cost ratio for adding one bit to subchannel m of CRBS i as 1/ti,1k,m for the kth constraint in (2.6) and 1/ ∑K k=1 t i,1 k,mf CR k,l,m for the lth constraint in (2.7), where ti,1k,m is given in (2.35) with b = 1. 2.5.2 Efficiency Value for Adding One Bit to Subchannel m of CRBS k To determine the efficiency value for adding one bit to subchannel m of CRBS k, we need to weigh the relative importance of the K + L constraints. Comparing the reward/cost ratios of the constraints directly does not make a lot of sense because the adding of one bit requires different amount of resources from different constraints and the resources of constraints (2.6) and constraints (2.7) are not comparable. We first define the efficiency 35 capacity of subchannel m of CRP k for constraint i as ck,m(i) =  Si−ui tk,1i,m , i ≤ K Ithi−K−ui∑K j=1 t k,1 j,mf CR j,i−K,m , K < i ≤ K + L. (2.36) In the RHS of (2.36), the first line accounts for the set of constraints (2.6) whereas the second line accounts for the set of constraints (2.7). The terms {ui, i = 1, 2, . . . , K + L} are the costs of resources already allocated, i.e. ui =  ∑M m=1 sk,m i ≤ K∑K k=1 ∑M m=1 sk,mf CR k,i−K,m K < i ≤ K + L . (2.37) Intuitively, the efficiency capacity, ck,m(i), in (2.36) represents the maximum number of the current bit that can be accommodated if the entire remaining room for constraint i were to be used for that bit. The efficiency capacity thus measures how tight or urgent a particular constraint is. The efficiency value of subchannel m of CRBS k is defined as the efficiency capacity of its tightest constraint, i.e. ek,m = mini{ck,m(i)}. 2.5.3 The Proposed Algorithm In the proposed algorithm, we allocate the next bit to the subchannel/CRBS pair which has the largest efficiency value. This process of allocating one bit at a time is repeated until one of the constraints can no longer hold. As in the single CRU case, since the next bit is allocated to the subchannel/CRBS pair with the maximum efficiency value and the efficiency value is the minimum efficiency capacity, we refer to the proposed algorithm as Max-Min. The algorithm allocates the next bit to the subchannel/CRBS pair whose remaining room in its tightest constraint is the largest. A pseudo-code listing of the Max-Min algorithm is given below. The matrix inversion needed in (2.35) requires O(K3) operations. The computational 36 Algorithm 1 Max-Min Algorithm. In the following,k = 1, 2, . . . , K,m = 1, 2, . . . ,M, i = 1, 2, . . . , K + L (1) initialize rk,m = 0, ui = 0 (2) for each subchannel m of each CRP k, do steps (2.1)-(2.2) (2.1) calculate ck,m(i) based on equation (2.36) (2.2) calculate efficiency value ek,m = mini{ck,m(i)} (3) find the subchannel m′ of CRP k′ that has the largest efficiency value as (k′,m′) = argmaxk,m{ek,m}, get power increase tk ′ k,m′ using (2.35) (4) if tk ′ i,m′ + ui ≤ Si, i = 1, 2, . . . , K and∑K j=1 t k′ j,m′f CR j,i−K,m′ + ui ≤ I thi−K , i = K,K + 1, . . . , K + L, do step (4.1); otherwise, stop (4.1) add one bit to subchannel m′ of CRP k′ by adding 1 to rk′,m′ , increase ui by t k′ i,m′ for i = 1, 2, . . . , K and by ∑K j=1 t k′ j,m′f CR j,i−K,m′ for i = K,K + 1, . . . , K + L complexity of the Max-Min algorithm is O(RK4M), where R is the total number of bits loaded. 2.6 Simulation Results To evaluate the proposed Max-Min algorithm, simulations were performed for an OFDM- based CR system for both single CRU and multiple CRUs cases. There are 32 subchan- nels, each with a noise power of 10−16 W. Following [8], the PSDs of the PU signals are assumed to be those of elliptically filtered white noise processes [24], i.e., ΦPUl (f) = [1 + ²2lR 2 n (ξl, f/f0,l)] −1 where n, ²l, ξl, f0,l are the filter parameters and Rn(., .) is the nth- order elliptic rational function; the PSD of the CRU is ΦCR(f) = Ts( sinpifTs pifTs )2, where Ts is the symbol duration. The path loss exponent is 4, the variance of the shadowing effect is 10 dB, and the multipath fading is assumed to be Rayleigh [25]. In (2.4) and (2.8), Γ = 1 and RMAX = 8 respectively. 37 2.6.1 Single CRU Case In the single CRU case, simulations were performed for a system consisting of four PUs and one CRU spread out over a 3 × 3 km area. The transmitters of the PUs and CRU are uniformly distributed within this area. Each PU or CRU receiver is uniformly distributed within a 500-meter circle of its corresponding transmitter. The widths of the PU frequency bands and the spectrum holes are randomly generated with the total PU bandwidth uni- formly distributed between the equivalent of 16 to 26 OFDM subchannels. The transmit power (in W) for PU l is set equal to the number of OFDM subchannels within PU l’s band. In order to evaluate the effectiveness of the proposed Max-Min algorithm, a commercial optimization software package was used to solve (2.10). The software uses an integer nonlin- ear programming optimizer; it indicates whether or not it is able to obtain a global optimal solution for the given problem. For comparison, the performances of the minimum-power (MP) and minimum-interference (MI) algorithms were also evaluated. With MP, each addi- tional bit is allocated to the subchannel with the minimum extra power and with MI, each additional bit is allocated to the subchannel that generates the minimum total interference to all PUs. The bit rate results presented in this section are obtained by averaging over 1,000 different position realizations. Fig. 2.2 shows the average number of bits per OFDM symbol per subchannel as a function of CRU power limit, S. The total interference power that can be tolerated by each PU is 5×10−12 W. It can be seen that the solution obtained using the Max- Min algorithm is very close to the optimal solution given by the commercial optimization software with a difference of less than 0.3%. The largest difference among the 1,000 position realizations is less than 5%. For low average CRU power levels, the system is power-limited: the average bit rate increases with S, the MP algorithm is close to optimal and the MI algorithm performs poorly. For S > 32× 10−2 W, the system becomes interference-limited: the average bit rate 38 00.5 1 1.5 2 2.5 3 3.5 A ve ra ge  n um be r o f b its pe r O FD M  sy m bo l p er  su bc ha nn el   10−7 10−6 10−5 10−4 10−3 10−2 10−1 CRU power limit  S (× 32W) Optimal Max−Min Min−interference Min−power Figure 2.2: Average number of bits per OFDM symbol per subchannel as a function of CRU power S, with interference thresholds set to 5× 10−12 W. increases very little with S and the MP algorithm performs quite poorly compared to the MI and Max-Min algorithms. Fig. 2.3 shows the average number of bits per OFDM symbol per subchannel as a function of the interference power that can be tolerated by each PU. It can again be seen that the Max-Min algorithm is close to optimal with an average difference of less than 0.1%, and is clearly better than the MP and MI algorithms. 39 01 2 3 4 5 6 7 A ve ra ge  n um be r o f b its pe r O FD M  sy m bo l p er  su bc ha nn el   10−14 10−13 10−12 10−11 10−10 Interference threshold  Ithl  (× 5 W) Optimal Max−Min Min−interference Min−power Figure 2.3: Average number of bits per OFDM symbol per subchannel as a function of the interference threshold with S = 0.32 W. 40 2.6.2 Multiple CRU Case To assess the performance of the Max-Min algorithm for multiple CRUs, simulations were performed for the downlink of an OFDM-based CR system consisting of four CRBSs within a 10 km × 10 km area, which is divided into four cells, as shown in Fig. 2.4. A CRBS is located at the center of each cell and its CRUs are uniformly distributed within its cell. The number, nCRU,k, of CRUs for CRBS k is chosen to be 100. A CRBS assigns its subchannel m to the CRU for which the ratio of its current subchannel m gain to its average subchannel m gain is the highest. There are two active PU transmitters and each of the two PU receivers is uniformly distributed within a 500-meter circle of its corresponding transmitter. (a) Scenario 1 (b) Scenario 2 (c) Scenario 3 Figure 2.4: Simulation topologies: triangles represent CRBSs and circles represent PU trans- mitters. Since the PU locations affect the performance of the system, e.g. the system throughput degrades as the PUs get closer to the CRBSs, we consider three scenarios: (1) an optimistic Scenario 1 in which the PU transmitters are located at the corners as shown in Fig. 2.4(a), (2) a pessimistic Scenario 2 in which the PU transmitters are located at the center as shown in Fig. 2.4(b) and (3) an average Scenario 3 in which the PU transmitters are uniformly distributed in the area (Fig. 2.4(c)). The bandwidth for each PU is six subchannels. For simplicity, we assume that all CRBSs have the same total power constraint Sk and all PUs have the same interference threshold 41 I th = 6× 10−15 W. The total power for each PU transmitter is 6 W. To assess the performance of the proposed algorithm, a commercial optimization software package was used to solve the optimization problem in (2.5) without constraint (2.8). Be- cause this software requires several hours to solve the optimization problem with an integer number of bits for each realization (i.e. a randomly generated set of channel gains), the software was used instead to solve the problem assuming the number of bits may assume a real value. The real-valued number of bits results therefore provide upper bounds on the actual performance. The value for each point plotted in the figures is the average over one thousand realizations. To illustrate the benefit of allowing CRPs to share active PU frequency bands, we consider a system in which active PU bands are protected by guard bands on either side of a PU band; these guard bands, as well as the PU bands themselves, may not be used by CRPs. Such a system is hereafter referred to as NABO (non-active PU bands only). For the simulation results, we make the optimistic assumptions that the CRBS OFDM signals have no sidelobes, so that guard band widths are zero and fCRk,l,m = 0, l = 1, 2, . . . , L. The optimization problem is the same as that in (2.5) except that the constraints in (2.7) are always satisfied and the available CRBS bandwidth is only 20 subchannels. The commercial optimization software was used to solve the problem assuming that the number of bits loaded on a subcarrier can be a real value, i.e. constraint (2.8) was ignored; this provides an upperbound on the actual performance. Figures 2.5, 2.6, and 2.7 show the average number of bits per OFDM symbol per sub- channel per CRBS as a function of the CRBS power limit, Sk, for Scenarios 1, 2, and 3 respectively. It can be seen that the solution obtained using the proposed Max-Min al- gorithm is quite close to the optimal upperbound solution obtained using the commercial optimization software. The difference is less than 5% for all scenarios. For Sk = 1 and the Max-Min algorithm, the average number of bits per OFDM symbol per subchannel per CRBS are about 6.1, 5.1 and 5.2 bits for Scenarios 1, 2 and 3 respectively. 42 0.1 0.3 1 3 10 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 CRBS power constraint  Sk (W) A ve ra ge  n um be r o f b its pe r O FD M  sy m bo l p er  su bc ha nn el  p er  C RB S   Optimal (real bits) Max−Min NABO (real bits) Figure 2.5: Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 1. 43 0.1 0.3 1 3 10 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 CRBS power constraint  Sk (W) A ve ra ge  n um be r o f b its pe r O FD M  sy m bo l p er  su bc ha nn el   Optimal (real bits) Max−Min NABO (real bits) Figure 2.6: Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 2. 44 0.1 0.3 1 3 10 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 CRBS power constraint  Sk (W) A ve ra ge  n um be r o f b its pe r O FD M  sy m bo l p er  su bc ha nn el  p er  C RB S   Optimal (real bits) Max−Min NABO (real bits) Figure 2.7: Average number of bits per OFDM symbol per subchannel per CRBS as a function of the CRBS power constraint for Scenario 3. 45 Compared to NABO, the performance improvements for Sk = 1 are 37%, 23%, and 27% for the three scenarios; the performance improvements for Sk = 10 are 45%, 29%, and 32%. These performance improvements are largely due to the fact that NABO can only use 20 subchannels, while the proposed algorithm is able to use all 32 available subchannels. In the simulations, at Sk = 1, the average number of subchannels that are actually used by the proposed algorithm are 32, 31.8, and 31.9 for Scenarios 1, 2, and 3 respectively. 2.7 Conclusions A low-complexity Max-Min algorithm has been proposed for subchannel, bit and power allo- cation in a multiuser OFDM-based multi-cell CR system. The algorithm efficiently utilizes the spectrum holes between PU frequency bands as well as active PU bands while ensur- ing that the generated interference powers do not exceed prescribed thresholds. Simulation results show that the proposed algorithm provides a sum of CRBS bit rates which is close to optimal. The performance improvement is over 25% compared to an algorithm that uses guard bands to protect the active PU bands. 46 References [1] D. Cabric, S. M. Mishra, D. Willkomm, R. 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Honig, “Auction-based spectrum sharing,” Springer Mobile Networks and Applications, vol. 11, no. 3, pp. 405–408, June 2006. [15] Y. Xing, C. N. Mathur, M. A. Haleem, R. Chandramouli, and K. P. Subbalakshmi, “Dynamic spectrum access with QoS and interference temperature constraints,” IEEE Transactions on Mobile Computing, vol. 6, no. 4, pp. 423–433, April 2007. [16] A. Attar, O. Holland, M. R. Nakhai, and A. H. Aghvami, “Interference-limited resource allocation for cognitive radio in orthogonal frequency-division multiplexing networks,” IET Communications, vol. 2, no. 6, pp. 806–814, July 2008. [17] F. Wang, M. Krunz, and S. Cui, “Price-based spectrum management in cognitive radio networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 1, pp. 74–87, February 2008. [18] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 2, pp. 171–178, Febru- ary 2003. [19] A. J. 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Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” IEEE Journal on Selected Areas in Communications, vol. 7, no. 7, pp. 1205–1211, July 1999. 48 Chapter 3 An Efficient Power Loading Scheme for OFDM-based Cognitive Radio Systems 3.1 Introduction Cognitive radio (CR) [1, 2] is a concept which can potentially alleviate the shortage of unlicensed frequency bands. As discussed in [3], orthogonal frequency division multiplexing (OFDM) is an attractive modulation candidate for CR systems. The FCC Spectrum Policy Task Force [4] has recommended the use of the interference temperature for assessing the level of interference. The specification of a primary user pair (PUP) interference temperature limit corresponds to a maximum allowable interference power and makes it possible for CR transceiver pairs (CRPs) to use active PUP bands in a controlled fashion. Two types of interference need to be considered, namely, cross-channel interference and co-channel interference. Since a PUP may not use OFDM, cross-channel interference [5] could be generated by a CRP using an OFDM subchannel close (adjacent) to the PUP band, and from the PUP transmitter to the CRP’s subchannels. Co-channel interference arises when a CRP uses an active PUP band. Only cross-channel interference is treated in [6, 7], since CRPs may only use non-active PUP bands. In [8][9], only co-channel interference is considered since PUPs are assumed to use OFDM. In this chapter, we study the power allocation problem for an OFDM-based CR system in which CRPs may use both non-active and active PUP bands as long as the total cross and co- channel interference powers do not exceed prescribed limits. A mathematical optimization 1A paper based on the material in this chapter has been submitted for publication. Yonghong Zhang and Cyril Leung, “An Efficient Power Loading Scheme for OFDM-based Cognitive Radio Systems”. 49 problem is formulated and the optimal solution as well as a low-complexity suboptimal solution are obtained and compared. 3.2 System Model Since our focus is on power loading, we assume that the assignment of subchannels to CRPs has been made. Thus, without much loss of generality, we consider one CRP in a CR system with access to a band of width W Hz. Non-overlapping portions of this band are licensed to L PUPs, with the nominal bandwidth of PUP l ranging from fc + F PU l to fc + F PU l +Wl, l = 1, 2, · · · , L. The maximum interference power that PUP l can tolerate is I thl = T thl Wl [4], where T thl is the interference temperature limit for PUP l. The CRP can use non-active and active PU bands provided that the total interference in PUP l’s band does not exceed I thl if PUP l is active. The CR system accommodates M equally spaced OFDM subcarriers (subchannels). The nominal bandwidth of subchannel m, m = {1, 2, · · · ,M} ranges from fc + (m − 1)∆f to fc + m∆f . Let Ml be the set of subchannels in PUP l’s band and L be the set of active PUPs. The subchannels are modelled in discrete-time, with the time-varying power gain for subchannel m from the transmitter to the receiver of the CRP denoted by gm. The power gains for subchannel m from the CRP’s transmitter to PUP l’s receiver and from PUP l’s transmitter to the CRP’s receiver are denoted by hl,m and dl,m, respectively. It is assumed that the power gains gm, hl,m, dl,m are outcomes of the random variables (rv’s) Gm, Hl,m, Dl,m, and that there is no inter-carrier interference (ICI). Let eCRl,m be the interference power experienced by PUP l’s receiver due to the CRP signal in subchannel m when hl,m = 1 and sm = 1, where sm is the transmit power. Then, eCRl,m = ∫ FPUl −(m− 12 )∆f+Wl FPUl −(m− 12 )∆f ΦCR(f)df (3.1) 50 where ΦCR(f) is the baseband PSD of the OFDM signal in subchannel m when sm = 1. The interference power generated by subchannel m of the CRP to PUP l’s band is smhl,me CR l,m , which represents cross channel interference when subchannel m is outside PUP l’s band and co-channel interference when subchannel m is within PUP l’s band. 3.3 The Optimization Problem We consider the problem of maximizing the overall rate achievable by the CRP, while keeping the interference power experienced by the PUPs below the specified thresholds I thl , l ∈ L. The optimization problem is formulated as OP1 : max sm M∑ m=1 log2 ( 1 + sm Nm ) (3.2) subject to M∑ m=1 sm ≤ S (3.3) M∑ m=1 smf CR l,m ≤ I thl (3.4) sm ≥ 0,m = 1, 2, · · · ,M . (3.5) In (3.2), log2(1+sm/Nm) is the number of bits per OFDM symbol, which can be supported by subchannel m [10]; Nm = Γ(σ 2 0 + ∑ l∈L f PU l,m )/gm is the equivalent noise power, where Γ is a SNR gap parameter which indicates how far the system is operating from capacity, σ20 is the noise power, fPUl,m is the interference power generated in subchannel m at the CRP receiver by PUP l. In (3.3), S is the CRP power limit. In (3.4), fCRl,m = hl,me CR l,m and f CR l,m = 0 for l ∈ L̄. 51 OP1 is a convex optimization problem. The Lagrangian [11] can be written as F = − M∑ m=1 log2 ( 1 + sm Nm ) +ν ( M∑ m=1 sm − S ) + L∑ l=1 µl ( M∑ m=1 smf CR l,m − I thl ) − M∑ m=1 φmsm (3.6) where ν, µl, l = 1, 2, · · · , L, and φm,m = 1, 2, · · · ,M are Lagrange multipliers. Applying the Karush-Kuhn-Tucker (KKT) conditions [11], we obtain the optimal power allocation as s∗(1)m = [ηm −Nm]+ (3.7) where [x]+ = max(0, x) and ηm = 1∑L l=1 µ ∗ l f CR l,m+ν ∗ with ν ∗ ≥ 0 and µ∗l ≥ 0. Details of the derivation are provided in Appendix A.1. Solving for the L+1 Lagrangian multipliers is computational complex. Also, the interior- point method can be used to solve OP1 with a complexity O(M3) [11]. We next propose an approximate method for solving the power allocation problem with lower complexity. 3.4 An Approximate Solution for OP1 3.4.1 Problem Formulation The cross-channel interference to a PUP band comes mostly from the subchannels imme- diately adjacent to it [5]. We thus view each subchannel as belonging to its closest PUP band and assume that it generates interference only to a single PUP band, namely, its own. Then, OP1 can be simplified to OP2, which is the same as OP1 except that constraint (3.4) is replaced by ∑ m∈Ml smf CR l,m ≤ I thl , l ∈ L. (3.8) Using a similar approach leading to (3.7), we obtain s∗(2)m = [ηm −Nm]+ (3.9) 52 where ηm = 1 µ∗l f CR l,m + ν ∗ . (3.10) In (3.10), ν∗ ≥ 0, µ∗l ≥ 0. For the subchannels in non-active PUP bands, because fCRl,m = 0, the power allocation has a water-filling interpretation with a waterlevel of 1/ν ∗. Note that for a subchannel m in an active PUP band, ηm ≤ 1/ν∗, since both µ∗l and fCRm are non-negative. From (3.9) and (3.10), we have s∗(2)m ≤ [ 1 ν∗ −Nm]+. (3.11) As to be expected, if Nm = N, ∀m, we can see from (3.9) that for a given l, a smaller CRP transmit power will be allocated to subchannels with a higher fCRl,m value. Although solving for the power allocation {s∗(2)m } in (3.9) requires less computation than solving for {s∗(1)m } in (3.7), it may still be computationally impractical in many cases. Thus, faster suboptimal power allocation algorithms are of interest. 3.4.2 Suboptimal Solution In this section, an overview of the procedure for obtaining a suboptimal solution to OP2 is first given. An algorithm for implementing this procedure is then established. If no PUP is active, then constraint (3.8) is satisfied automatically, and the solution to OP2 has the standard water-filling interpretation [12]. In order to design a fast suboptimal solution, we start by assuming that constraint (3.8) is satisfied, and find the resulting wa- terlevel λ and the optimal solution {s∗m = [λ − Nm]+,m = 1, 2, · · · ,M} using water-filling for all subchannels with power limit S. If ∑ m∈Ml s ∗ mf CR l,m ≤ I thl for every l, then the op- timal power allocation has been found. Otherwise, PUP l′ for which ∑ m∈Ml′ s ∗ mf CR l,m /I th l′ is the highest is determined, and the CRP power allocation for its subchannels is adjusted so as to ensure that the total interference power is below I thl′ . From (3.11), in the optimal 53 solution of OP2, the power allocated to the subchannels in Ml′ should be no higher than [1/ν∗−Nm]+. Although we do not know the exact value of 1/ν∗, it can be approximated by λ, i.e., sm ≤ [λ−Nm]+ = s∗m,m ∈Ml′ . The following problem is then solved for PUP l′. OP3 : max sm ∑ m∈M+ l′ log2 ( 1 + sm Nm ) (3.12) subject to ∑ m∈M+ l′ smf CR m ≤ I thl′ (3.13) sm ≥ 0, sm ≤ λ−Nm,m ∈M+l′ (3.14) where fCRm is the simplified notation of f CR l′,m, λ is the waterlevel for subchannels belonging to PUPs whose subchannels has not yet undergone power adjustment andM+l′ ⊆Ml′ contains the subchannels with Nm < λ. For subchannels in Ml′\M+l′ , sm is set to 0. In Section 3.4.2, an algorithm is described which provides a suboptimal solution {s(3)m ,m ∈ M+l′ } for OP3. From (3.14), we have s(3)m ≤ s∗m,m ∈ M+l′ , where s∗m is the power allocation for subchannels belonging to PUPs whose subchannels have not yet undergone power ad- justment. Since sm = 0,m ∈ Ml′\M+l′ , we have ∑ m∈Ml′ s (3) m < ∑ m∈Ml′ s ∗ m. The reduction in CRP transmit power in subchannels m ∈ M+l′ is then redistributed to the subchannels with waterlevel λ. A new waterlevel λ is then calculated. If with the new (higher) water- level, there still exists a PUP l with an interference power higher than I thl , OP3 with l ′ = l is solved again to lower the interference level for PUP l. This process continues until the interference power level for each active PUP l is below I thl . Suboptimal Algorithm for OP2 An implementation of the procedure described above is given in Algorithm 2. 54 Algorithm 2 Power Allocation Algorithm. 1) Initialize M = N = {1, 2, · · · ,M} and p = S. Sort {Nm,m ∈M} in decreasing order with k being the sorted index. Find the waterlevel for subchannels in M using the algorithm proposed in [13] with power limit p as follows: 1.1) Nsum = ∑ m∈N Nm, λ = (Nsum + p)/|N |, n = 1 . 1.2) While Nk(n) > λ, let Nsum = Nsum −Nk(n), N = N\{k(n)}, λ = (Nsum + p)/|N |, n = n+ 1. 1.3) Set sm = [λ−Nm]+,m ∈M. 2) If (3.8) is satisfied for every active PUP l, stop; otherwise, go to Step 3) 3) Determine l′ = argmaxl∈L( ∑ m∈Ml smf CR m Ithl ), let M+l′ =Ml′\{Nm ≥ λ,m ∈Ml′}, and let sm = 0,m ∈Ml′\M+l′ , then execute Algorithm 3 to get s(3)m ,m ∈M+l′ . Set sm = s (3) m ,m ∈M+l′ , p = p− ∑ m∈M+ l′ s (3) m , Nsum = Nsum − ∑ m∈M+ l′ ⋂N Nm, M =M\M+l′ , N = N\M+l′ . 4) Find the waterlevel for subchannels in M with power limit p as follows: Repeat Step 4.1) and Step 4.2) until Nk(n) > λ or n ≤ 0. 4.1) n = n− 1, λ = (p+Nsum +Nk(n))/(|N |+ 1). 4.2) if Nk(n) ≤ λ and k(n) ∈M, then Nsum = Nsum +Nk(n), N = {N , k(n)}. 5) λ = (Nsum + p)/|N |, sm = [λ−Nm]+,m ∈M, go to Step 2). An overall explanation of Algorithm 2 is now given. We initially assume that all sub- channels are in M, the set of subchannels with waterlevel λ, and that all subchannels are also in N , the set of subchannels that have an equivalent noise power lower than λ. To find the first waterlevel, we sort {Nm,m ∈ M} in decreasing order with k(1), k(2), · · · denoting the sorted indices. If Nk(1) is higher than λ = (Nsum + p)/|N |, which is the waterlevel if all subchannels are in set N , then subchannel k(1) should not be in N and removing it from N will result in a lower value of λ. We continue to compare Nk(n) with the updated value of λ until each subchannel m in N has a Nm value that is lower than λ. The resulting λ is the first waterlevel, and n is the index in the sorted {Nm,m ∈ M} list that has the highest equivalent noise power in N . If not every PUP l satisfies (3.8), then in Step 3), the PUP l′ that has the highest interference power to I thl′ ratio is chosen for interference power reduction. After solving OP3 using the algorithm in Section 3.4.2, the reduction in CRP transmit power in subchannels m ∈ M+l′ is redistributed in Step 4) and a new global waterlevel is found. Because the new waterlevel is higher and Nm is sorted in decreasing 55 order, we decrement n in order to find the new waterlevel. Suboptimal Algorithm for OP3 Applying the KKT conditions in OP3, we obtain the following optimal power allocation s∗(3)m =  0, m ∈ S0 β fCRm −Nm, m ∈ S1 λ−Nm, m ∈ S2 (3.15) where β is the Lagrange multiplier with β < Nmf CR m for m ∈ S0, NmfCRm ≤ β ≤ λfCRm for m ∈ S1, and β > λfCRm for m ∈ S2. Details of the derivation are provided in Appendix A.2. Substituting s ∗(3) m into ∑ m∈Ml s ∗(3) m fCRm = I th l , we obtain β = I thl′ − ∑ m∈S2(λ−Nm)fCRm + ∑ m∈S1 Nmf CR m |S1| . (3.16) If the mutually exclusive sets S0, S1, and S2 are known, the optimal power allocation can be found using (3.15). In the first stage of Algorithm 2, which provides a suboptimal solution for OP3, we determine S0. We first set S = ∅, S2 = ∅, S1 = M+l′ , n1 = 1, calculate the initial value of β using (3.16) and sort Nmf CR m in decreasing order with i being the sorted index of the subchannels, i.e., i(1) = argmaxm∈{1,2,··· ,M}Nmf CR m . If β ≥ Ni(1)fCRi(1) , all subchannels satisfy Nmf CR m ≤ β and S0 = ∅. On the other hand, if β < Ni(1)fCRi(1) , subchannel i(1) is added into S0 since from (3.16) removing i(1) from S1 results in a lower value of β, thus ensuring that β < Ni(1)f CR i(1) . We then increment n1 and compare the value of Ni(n1)f CR i(n1) with the value of β, updated using (3.16), until Ni(n1)f CR i(n1) ≤ β. Then S0 = {i(1), i(2), · · · , i(n1)}. In the second stage, we determine S2. We first sort fCRm in decreasing order with j being the sorted index of the subchannels and start with S2 = ∅, n2 = |M+l′ |. If β ≤ λfCRj(|M+ l′ |) , all subchannels satisfy β ≤ λfCRm and S2 = ∅. On the other hand, if λfCRj(|M+ l′ |) < β and 56 j(|M+l′ |) is not in S0, subchannel j(|M+l′ |) is added to S2; from (3.16), removing subchannel j(|M+l′ |) from S1 results in a higher value of β, thus ensuring that λfCRj(|M+ l′ |) < β. We then decrement n2 and compare the value of λf CR j(n2) with the value of β, updated using (3.16), until β ≤ λfCRj(n2). Then S2 = {j(|M+l′ |), j(|M+l′ | − 1), · · · , j(n2)} and S1 =M+l′ \(S0 ⋃S2). Finally, at the end of the second stage, if S1 6= ∅ or S2 = ∅, then S0, S1 and S2 are determined; otherwise, the following adjustment is needed to ensure that (3.8) is satisfied. In this case, s (3) m is either 0 or λ − Nm, one subchannel at a time is transferred from S2 to S0 until (3.8) holds. At each time, the subchannel that has the lowest value of log2(1 + (λ−Nm)/Nm)/((λ−Nm)fCRm ) is selected, because it has the lowest reward/cost ratio, where reward is the bits allocated to the subchannel and cost is the interference power generated on this subchannel. Algorithm 3 Suboptimal Algorithm for solving OP3. 1) Sort Nmf CR m , f CR m in descending order with i, j being the sorted index of the subchannels, respectively. Let S0 = ∅, S1 =M+l′ , S2 = ∅, and calculate β using (3.16). 2) Let n1 = 1. While Ni(n1)f CR i(n1) > β, do S0 = {S0, i(n1)}, S1 = S1\{i(n1)}, β = [β(|S1|+ 1)−Ni(n1)fCRi(n1)]/|S1|, n1 = n1 + 1. 3) Let n2 = |M+l′ |. While λfCRj(n2) < β, do the following: 3.1) if not j(n2) ∈ S0, then set S2 = {S2, j(n2)}, S1 = S1\{j(n2)}, and β = [β(|S1|+ 1)− (λ−Nj(n2))fCRj(n2) −Nj(n2)fCRj(n2)]/|S1|. 3.2) n2 = n2 − 1 4) If S1 6= ∅ or S2 = ∅, then do Step 6); otherwise goto Step 5). 5) Let I = ∑ m∈S2(λ−Nm)fCRm . While I > I thl′ , do m′ = argminm∈S2 log2(1+(λ−Nm)/Nm)(λ−Nm)fCRm , I = I − (λ−Nm′)fCRm′ , and S2 = S2\{m′}. 6) Calculate s (3) m ,m ∈M+l′ using (3.15). The computational complexity of Algorithm 3 is O(|M+l′ | log(|M+l′ |)) and results from the sorting performed in Step 1). The other steps have lower complexity orders. In Algo- rithm 2, the computational complexities in Step 1) and Step 2) are O(M log(M)) and O(L), respectively. For each PUP l with ∑ m∈Ml smf CR m > I th l , Steps 3) to 5) are performed once. So altogether, the complexity of Step 3) is at most ∑L l=1O(|Ml| log(|Ml|)), which is no higher than O(M log(M)). The complexity of Step 4) and Step 5) is O(M) and O(LM), 57 respectively. The overall complexity of Algorithm 2 is O(M log(M)) +O(LM). 3.5 Simulation Results Simulations were performed for an OFDM-based CR system with one CRP and three PUPs. There are M = 24 subchannels with σ20 = 10 −16 and |Ml| = 8, l = 1, 2, 3 as in Fig. 3.1. All links are assumed to undergo Rayleigh fading. PUP 3 is non-active, and the total transmit power for PUP 1 and PUP 2, is 0.8 W each. Following [5], the PSDs of the PUP signals are assumed to be those of elliptically filtered white noise processes [14], i.e. ΦPUl (f) = [1 + ² 2 lR 2 n (ξl, f/f0,l)] −1 where n, ²l, ξl, f0,l are the filter parameters and Rn(., .) is the nth-order elliptic rational function; the PSD of the CRP is ΦCR(f) = Ts( sinpifTs pifTs )2, where Ts = 40 µs is the symbol duration. Other parameters are: E{Dl,m} = 10−15, E{Gm} = 10−14,m = 1, 2, · · · , 24 and OFDM symbol guard interval is 8 µs. 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 PUP 1 band PUP 2 band PUP 3 band Active Non-active Figure 3.1: PUP active and non-active bands and CRP OFDM subchannels. To illustrate the benefit of allowing the CRP to share active PUP frequency bands, we compare the average number of bits per OFDM symbol (ANB) to that of a system in which active PUP bands may not be used by the CRP. In this case, only PUP 3’s band is available to the CRP. The optimal solution for such a system is referred to as NABO (non-active 58 band only). The optimal solution for OP1, OP2, and the suboptimal solution provided by Algorithm 1 are referred to as OPT, OPT-APPROX, and SUBOPT-APPROX, respectively. A commercial optimization software package was used to solve OP1. Actual interference generated by the CRP to the active PUPs, PUP 1 and PUP 2, is calculated using ∑M m=1 smf CR l,m , l = 1, 2 for SUBOPT-APPROX, in order to assess if the approach in Section 3.4 is reasonable. The ANB values (obtained by averaging over ten thousand realizations of the fad- ing gains) for each PUP band for SUBOPT-APPROX and OPT-APPROX as a function of E{H1,m}, is shown in Fig. 3.2, with S = 2.4 W, I th1 = I th2 = 8 × 10−15 W, and E{H2,m} = 10−14. As E{H1,m} decreases, the ANB on PUP 1’s band increases, since the CRP’s interference power to PUP 1’s receiver is reduced. The ANBs on PUP 2’s and PUP 3’s bands decrease with E{H1,m} because more CRP power is being diverted to PUP 1’s band. The ANB difference between OPT-APPROX and SUBOPT-APPROX is less than 2%. Although not shown in Fig. 3.2, the corresponding ANB values for NABO for PUP band 1, 2 and 3 were found to be {0, 0, 34} bits, and the improvement of SUBOPT-APPROX over NABO is 45%, 65%, 80% for E{H1,m} equals 10−11, 10−13, 10−15, respectively. The sum ANB for all subchannels as a function of the total CRP power, S, is shown in Fig. 3.3 with I th1 = I th 2 = 8 × 10−15 W and E{Hl,m} = 10−14, l = 1, 2. As expected, the ANBs of OPT-APPROX, OPT, SUBOPT-APPROX, and NABO increase with S. The ANB difference between OPT-APPROX and OPT is less than 0.2% and the ANB difference between SUBOPT-APPROX and OPT-APPROX is less than 4%. The ANB for SUBOPT- APPROX is over 20% higher than that of NABO at S = 0.024 W and over 60% at S = 2.4 W. To show that when using SUBOPT, the actual generated interference power to active PUPs, PUP 1 and PUP 2, is acceptable, we summarize in Table 3.1 (1) the fraction of realizations for which either PUP 1 or PUP 2 exceeding interference powers in excess of their thresholds I thl , (2) the average % by which the interference threshold limit is exceeded 59  0 5 10 15 20 25 30 35 40 45 50 Average subchannel gain, E{ H1,m} A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l, A N B 10−15 10−13 10−11   PUP 1 band (SUBOPT−APPROX) PUP 1 band (OPT−APPROX) PUP 2 band (SUBOPT−APPROX) PUP 2 band (OPT−APPROX) PUP 3 band (SUBOPT−APPROX) PUP 3 band (OPT−APPROX) Figure 3.2: Average number of bits per OFDM symbol (ANB) for each PUP band as a function of E{H1,m} with S = 2.4 W, I th1 = I th2 = 8× 10−15 W and E{H2,m} = 10−14. 60 010 20 30 40 50 60 70 80 90 100 A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l, A N B   10−3 10−2 10−1 100 CRP power limit,  S (× 24 W) OPT OPT−APPROX SUBOPT−APPROX NABO Figure 3.3: Average number of bits per OFDM symbol (ANB) on the whole PUP bands as a function of S with I th1 = I th 2 = 8× 10−15 W, E{Hl,m} = 10−14, l = 1, 2. 61 considering only those realizations in which the threshold is exceeded, and (3) the % by which the maximum interference power (among the 10, 000 realizations) exceeds the thresholds as a function of S, with I th1 = I th 2 = 8× 10−15 W and E{Hl,m} = 10−14, l = 1, 2. It can be seen that the actual maximum interference power generated by the CRP is less than 1.02I thl . S(×24 W) 10−3 3× 10−3 10−2 3× 10−2 10−1 3× 10−1 100 % realizations exceed I thl 0 0 0.38% 0.38% 1.65% 0.08% 0 average % exceed 0 0 0.51% 0.57% 0.92% 0.72% 0 maximum % exceed 0 0 0.98% 1.23% 1.83% 1.52% 0 Table 3.1: Actual interference power exceeding I thl , l = 1, 2 by using SUBOPT-APPROX with I th1 = I th 2 = 8× 10−15 W, E{Hl,m} = 10−14, l = 1, 2. The sum ANB of all subchannels as a function of the interference threshold, I th1 , is shown in Fig. 3.4 with I th1 = I th 2 , S = 2.4 W, and E{Hl,m} = 10−14, l = 1, 2. The ANBs of SUBOPT-APPROX and OPT-APPROX increase with I th1 as more interference can be tolerated by the active PUPs, PUP 1 and PUP 2. Because NABO only makes use of PUP 3’s band, the ANB of NABO does not change with I th1 . The ANB difference between OPT- APPROX and OPT is less than 0.2% and the ANB difference between SUBOPT-APPROX and OPT-APPROX is less than 4%. The ANB for SUBOPT-APPROX is over 30% higher than that for NABO at I th1 = 8× 10−16 W and over 80% when I th1 ≥ 8× 10−14 W. Table 3.2 lists (1) the fraction of realizations for which either PUP 1 or PUP 2 exceeding interference powers in excess of their thresholds I thl , (2) the average % by which the interfer- ence threshold limit is exceeded considering only those realizations in which the threshold is exceeded, and (3) the % by which the maximum interference power (among the 10, 000 realizations) exceeds the thresholds as a function of I thl , with I th 1 = I th 2 , S = 2.4 W, and E{Hl,m} = 10−14, l = 1, 2. It can be seen that the actual maximum interference power generated by the CRP is less than 1.02I thl . 62  30 35 40 45 50 55 60 65 Interference threshold ,  Ithl  (× 8 W) A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l, A N B   10−16 10−15 10−14 10−13 OPT−APPROX OPT SUBOPT−APPROX NABO Figure 3.4: Average number of bits per OFDM symbol (ANB) on the whole PUP bands as a function of I th1 with S = 2.4 W, I th 1 = I th 2 , and E{Hl,m} = 10−14, l = 1, 2. I thl (×8 W) 10−16 3× 10−16 10−15 3× 10−15 10−14 3× 10−14 10−13 % realizations exceed I thl 0 0.21% 1.46% 0.24% 0.66% 0 0 average % exceed 0 1.48% 0.81% 0.63% 0.6% 0 0 maximum % exceed 0 1.66% 1.92% 1.25% 1.06% 0 0 Table 3.2: Actual interference power exceeding I thl , l = 1, 2 by using SUBOPT-APPROX with I th1 = I th 2 , S = 2.4 W, and E{Hl,m} = 10−14, l = 1, 2. 63 3.6 Conclusions A low complexity, suboptimal solution for the power allocation problem in an OFDM-based CR system, in which the CRP uses both non-active as well as active PUP bands was pro- posed. The proposed algorithm has a lower complexity than the single CRU Max-Min algorithm proposed in Chapter 2. The complexity reduction is achieved by making a reason- able approximation based on: (1) the fact that cross-channel interference from CRUs to PUs is mainly limited to a few subchannels adjacent to the PU bands, and (2) the assumption that the bandwidth of a PU is much larger than that of an OFDM subchannel and that there is usually a guard band between two adjacent PU bands. Simulation results show that the proposed solution is very close to optimal and provides significant improvement over systems that only use non-active PUP bands. It is found that the approximation results in a relatively small performance degradation. 64 References [1] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, February 2005. [2] J. Mitola III and G. Q. Maguire, Jr., “Cognitive radio: making software radios more personal,” IEEE Personal Communications, vol. 6, no. 4, pp. 13–18, August 1999. [3] T. A. 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Leung, “Fair adaptive resource allocation for multiuser OFDM cognitive radio systems,” in Proc. of Second International Conference on Communications and Networking in China (CHINACOM ’07), Shanghai, China, August 2007, pp. 115–119. [8] P. Wang, M. Zhao, L. Xiao, S. Zhou, and J. Wang, “Power allocation in OFDM- based cognitive radio systems,” in Proc. of IEEE Global Telecommunications Conference (GLOBECOM ’07), Washington, DC, USA, November 2007, pp. 4061–4065. [9] P. Cheng, Z. Zhang, H.-H. Chen, and P. Qiu, “Optimal distributed joint frequency, rate and power allocation in cognitive OFDMA systems,” IET Communications, vol. 2, no. 6, pp. 815–826, July 2008. [10] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading chan- nels,” IEEE Transactions on Communications, vol. 45, no. 10, pp. 1218–1230, October 1997. [11] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge Univer- sity Press, 2004. [12] R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley & Sons, 1968. [13] A. Leke and J. M. Cioffi, “A maximum rate loading algorithm for discrete multi- tone modulation systems,” in Proc. of IEEE Global Telecommunications Conference (GLOBECOM ’97), vol. 3, Phoenix, AZ, USA, November 1997, pp. 1514–1518. 65 [14] R. W. Daniels, Approximation Methods for Electronic Filter Design. New York: McGraw-Hill, 1974. 66 Chapter 4 Performance of Equal Power Allocation in Multiuser OFDM-based Cognitive Radio Systems 4.1 Introduction Cognitive radio (CR) [1]-[3] is a concept which can potentially alleviate the pending spectrum shortage crisis. As discussed in [3], orthogonal frequency division multiplexing (OFDM) is an attractive modulation candidate for CR systems. It is well-known that the optimal solution to the problem of determining the capacity of a set of M parallel additive white Gaussian noise (AWGN) subchannels, each of which may have a different noise power level, subject to a total input signal power constraint, has a nice water-filling interpretation [4]. We will refer to this as optimal water-filling (OWF). In OWF, the signal powers allocated to different subchannels are in general different and no power is allocated to “silent” subchannels on which the noise power exceeds a certain threshold (water level). The scheme in which the total available signal power is shared equally among all non-silent subchannels is referred to as constant power water-filling (CPWF) [5] whereas the scheme in which the total available signal power is shared equally among all (silent and non-silent) subchannels is referred to as plain equal power allocation (PEPA). The PEPA scheme is simpler than CPWF since the optimal water level threshold is needed for CPWF but not for PEPA. It is pointed out in [6] that the difference in average achievable bit rates between OWF and CPWF is only about 2% for a Rayleigh fading channel with additive white Gaussian noise. A bound on this difference for the one user case is given in [5]. For the multiple user 1A paper based on the material in this chapter has been accepted for publication. Yonghong Zhang and Cyril Leung, “Performance of Equal Power Allocation in Multiuser OFDM-based Cognitive Radio Systems,” Hindawi Research Letters in Communications. 67 case, simulation results in [7] indicate that the difference between OWF and PEPA is quite small when each subchannel is assigned to the user with the best channel quality for that subchannel in Rayleigh fading channels. Analytical results in [8] show that the performance difference between OWF and PEPA decreases with the number of users and average signal to noise ratio (SNR). In this chapter, we study the performance difference between PEPA and OWF in a multiuser OFDM-based CR system. It is found that PEPA performs almost as well as OWF when there is little variation in CR user (CRU) average subchannel gains or PU activity level is high. 4.2 System Model We consider a CR system with a total bandwidth of W Hz and L PUs; PU l, l = 1, 2, . . . , L has a bandwidth allocation of Wl Hz. Frequency bands carrying PU signals are referred to as active; non-active bands are also termed spectrum holes. In order to reduce the mu- tual interference between secondary CRUs and PUs to acceptable levels, some subchannels adjacent to active PU bands are not used by the CRUs [9]. We are interested in downlink transmissions from one CR base station (CRBS) to K CRUs. It is assumed that the CRBS and the CRUs are able to accurately locate the spec- trum holes. The system bandwidth of W Hz can accommodate M OFDM subbands (or subchannels), each with noise power σ20. Interference among the subchannels is assumed to be negligible. The system is time-slotted with a slot duration equal to an OFDM symbol duration, Ts. The subchannels are modelled in discrete-time, with the gain for subchannel m and time slot t from the CRBS to CRU k denoted by √ gtk,m. For simplicity, it is assumed that for any given value of k, {gk,m,m = 1, 2, · · · ,M} are identically distributed random variables (rv’s) with a common probability function (pdf) and cumulated distribution function (cdf) denoted as fGk(gk) and FGk(gk) respectively. 68 At each time slot t, each subchannel withinMt, the set of available subchannels at time slot t, can be used by the CRBS to transmit to at most one CRU. We use fG(g) and FG(g) to denote the pdf and the cdf of the selected CRUs, respectively. The number, rtm, of bits per OFDM symbol which can be supported by subchannel m in time slot t is given by [10] rtm = log2 ( 1 + gtms t m Γσ20 ) (4.1) where gtm is the subchannel gain of the selected CRU, s t m is the power allocated to subchannel m at time slot t, and Γ is an SNR gap parameter which indicates how far the system is operating from capacity. The available power constraint implies that ∑ m∈Mt stm ≤ S, ∀t (4.2) where S is the total power per time slot. The availability of a PU band is modelled by a two-state Markov chain. During a time slot t, a PU band can be in one of two states: active or inactive [11]. A PU band can change mode once every Tstate slots. At a transition time, the probability of a PU band changing from active to inactive state is 1−pa, and the probability of changing from inactive to active state is 1− pn. The number, lCR,t, of available PU bands at time slots {t, t = 1, 2, . . .} then forms a Markov chain, with a transition probability matrix Q = {qij}, i, j = 0, 1, 2, · · · , L, where state i corresponds to the event that the number of available PU bands at time slot t is equal to i and the probability, qij, of moving from state i to state j is given by qij = L∑ n=0 ( i n ) (1− pn)npi−nn ( L− i n− i+ j ) (1− pa)n−i+jpL−j−na . (4.3) The Markov chain is illustrated in Fig. 4.1. Let Π = (pi0, pi1, · · · , piL)T denote the steady-state probability column vector. Then, we 69 q11 qLLqiiq00 L q01 q0L i q0i qLi q1i q1L qiL qL1qL0 qi1qi0 q10 Figure 4.1: Markov chain model for the number of available PU bands. have [12] Π = Q′Π (4.4) and Π = U−1V (4.5) where U =  1, 1, 1, · · · 1 0, 1− q00 + q10, 1− q00 + q20, · · · 1− q00 + qL0 0, q01 + 1− q11, q01 − q21, · · · q01 − qL1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0, q0L − q1L, q0L − q2L, · · · q0L + 1− qLL  (4.6) and V = (1, 1− q00, q01, · · · , q0L)T . Suppose that each PU band can accommodate the same number of subchannels, i.e. M/L. Then the number of available subchannels at time slot t is mCR,t = lCR,tM/L, and the probability of having m available subchannels is bm =  pil , if m = lM L , l = 0, 1, 2, · · · , L 0 , otherwise (4.7) 70 4.3 Bounds on bit rates for OWF and PEPA Although OWF provides the optimal solution for subchannel power allocation, it is more complex to implement than PEPA. It is therefore useful to compare the bit rates achievable by the two schemes. 4.3.1 Upper Bounds on Achievable Bit Rate for OWF Assuming that OWF is applied to the mCR,t subchannel gains of the selected CRUs at time slot t, we can write the average bit rate (ABR) as BOWF ∆ = limT→∞B (T ) OWF , where [4] B (T ) OWF = 1 TsT T∑ t=1 ∑ m∈Mt:Γσ 2 0 gtm ≤LtOWF log2 ( LtOWFg t m Γσ20 ) . (4.8) In (4.8), Γσ20/g t m can be viewed as the equivalent noise power on subchannel m and L t OWF is the water level at time t. Let Ti be the set of time slots with mCR,t = i, and Ti be the number of elements in set Ti. Grouping the time slots with mCR,t = i, we can re-write (4.8) as B (T ) OWF = 1 TsT M∑ i=1 ∑ ti∈Ti ∑ m∈Mti : Γσ20 g ti m ≤LtiOWF,i log2 ( LtiOWF,ig ti m Γσ20 ) (4.9) where LtiOWF,i is the water level at time slot ti. When mCR,t = i, the ABR if OWF is applied at each time slot ti is smaller than that if OWF is applied in one shot to all the iTi subchannel gains of the selected CRUs over the Ti time slots, i.e. B (T ) OWF ≤ 1 TsT M∑ i=1 ∑ ti∈Ti ∑ m∈Mti : Γσ20 g ti m ≤L(Ti)OWF,i log2 ( L (Ti) OWF,ig ti m Γσ20 ) (4.10) 71 where L (Ti) OWF,i is the global water level for set Ti. Taking the limit as T →∞, we have Ti = biT and BOWF ≤ 1 Ts M∑ i=1 ibi ∫ ∞ Γσ20 L (∞) OWF,i log2 ( L (∞) OWF,ig Γσ20 ) fG(g)dg . (4.11) In (4.10), L (Ti) OWF,i, is lower than the level calculated by including all subchannels at time slots t ∈ Ti because there may exist some subchannels with above water level noises. Therefore, L (Ti) OWF,i ≤ ∑ t∈Ti ∑ m∈Mt Γσ20 gtm iTi + S i . (4.12) Taking the limit as Ti →∞ in (4.12) yields L (∞) OWF,i ≤ E { 1 G } Γσ20 + S i . (4.13) Substituting the two occurrences of L (∞) OWF,i in (4.11) by the RHS of (4.13), we obtain BOWF ≤ 1 Ts M∑ i=1 ibi [( 1− FG ( 1 Li )) log2(Li) + ∫ ∞ 1 Li log2(g)fG(g)dg ] (4.14) where Li = Li and Li , E { 1 G } + S iΓσ20 . The upper bound for L (∞) OWF,i in (4.13) can be quite loose. This is because the RHS of (4.13) includes all subchannels regardless of their equivalent noise powers. As g → 0, the equivalent noise power increases without bound. Consequently, the bound in (4.14) is also loose. To obtain a tighter bound, we turn off any subchannel m ∈ Mt at time slot t ∈ Ti with an equivalent noise power, Γσ20/g t m, greater than the RHS of (4.13), i.e., we consider only the subchannels for which gtm > 1/Li in calculating the water levels. The resulting 72 water level is still higher than L (∞) OWF,i, so that L (∞) OWF,i ≤ Γσ20 ∫∞ 1 Li 1 g fG(g)dg + S i 1− FG ( 1 Li ) . (4.15) Substituting L (∞) OWF,i in (4.11) by the RHS of (4.15), we obtain a tighter bound, namely (4.14) with Li = ∫∞ 1 Li 1 g fG(g)dg + S iΓσ20 1− FG ( 1 Li ) . (4.16) 4.3.2 Achievable Bit Rate for PEPA The ABR for PEPA is BPEPA ∆ = limT→∞B (T ) PEPA where B (T ) PEPA = 1 TsT T∑ t=1 ∑ m∈Mt log2 ( 1 + Sgtm mCR,tΓσ20 ) . (4.17) Grouping the time slots with mCR,t = i, we have B (T ) PEPA = 1 TsT M∑ i=1 ∑ t∈Ti ∑ m∈Mt log2 ( 1 + Sgtm iΓσ20 ) . (4.18) Letting Ti →∞, we obtain BPEPA = 1 Ts M∑ i=1 ibiE { log2 ( 1 + S iΓσ20 G )} . (4.19) 4.4 Rayleigh fading channel In Section 4.3, we studied the ABR for OWF and PEPA for arbitrary pdf’s and cdf’s of the subchannel gains of the selected CRUs. In this section, we obtain the pdf and cdf of the subchannel gains of the selected CRUs for two different subchannel allocation strategies. The subchannel gains of the CRUs are assumed to be Rayleigh distributed, i.e. the power 73 gains are exponentially distributed. 4.4.1 Opportunistic Subchannel Assignment Suppose that at each time t, each of the M subchannels is assigned to the CRU with the highest gain for that subchannel. If the average subchannel power gains for all CRUs are equal, the pdf of the power gain for the CRU assigned to any subchannel is readily obtained using a standard result in order statistics [13], i.e., fG(g) = K [ 1− e− gE{G} ]K−1 e− gE{G} E{G} (4.20) with corresponding cdf FG(g) = [ 1− e− gE{G} ]K . (4.21) 4.4.2 A Fairer Subchannel Assignment Scheme If the average subchannel gains for CRUs are quite different, assigning a subchannel to the CRU with the highest gain may be too unfair to CRUs with poor average subchannel gains. A fairer scheme [14] is to select, for each subchannel, the CRU with the best channel gain relative to its own mean gain, i.e. k∗(t) = argmaxk gk,m(t) E{Gk} . (4.22) The distribution of a CRU’s subchannel gain relative to its own mean is exponential with a mean of 1. Thus the probability of selecting CRU i is 1/K, i.e. P (k∗ = i) = 1/K, i = 74 1, 2, · · · , K. The cdf of the power gain of the selected CRU for a subchannel is FG(g) = P (G ≤ g) = K∑ i=1 P (G ≤ g|k∗ = i)P (k∗ = i) = 1 K K∑ i=1 K∏ j=1 P (Gj ≤ gE{Gj} E{Gi} ) = 1 K K∑ i=1 ( 1− e− gE{Gi} )K . (4.23) The corresponding pdf is fG(g) = K∑ i=1 ( 1− e− gE{Gi} )(K−1) e− gE{Gi} E{Gi} . (4.24) 4.5 Numerical Results To illustrate the ABR difference between OWF and PEPA in a multiuser OFDM-based CR system, OWF, PEPA, and the upper bound for OWF are calculated using (4.8), (4.19), and (4.14) with Li equal to the RHS of (4.16). The two subchannel allocation strategies in Sections 4.4.1 and 4.4.2, hereafter referred to as Case A and Case B respectively, are considered. In Case A, the average subchannel power gain for each CRU is chosen as 2 × 10−13. In Case B, we increase the number of CRUs by six at a time. The average subchannel power gains of the six CRUs are chosen as follows: one with value 10−12, two with value 10−13 and three with value 10−14. The average subchannel gain for the six CRUs is also 2 × 10−13. In our calculations, we also use the following parameter values: Γ = 1, σ20 = 10 −16, S = 0.1 W, W = 2 MHz, Wl = 250 kHz, l ∈ {1, 2, · · · , L}, L = 8, M = 64, pn = 0.9, and Ts = 40 µs. Fig. 4.2 shows the ABR for OWF and PEPA as a function of number, K, of CRUs. For both cases, the ABR of OWF and PEPA increases with K as a result of multiuser diversity. The ABR difference between OWF and PEPA decreases with the number CRUs. The ABR difference between OWF and PEPA in Case A is almost negligible with the improvement of OWF over PEPA being less than 0.05%. In Case B, the improvement of OWF over PEPA 75 6 12 18 24 30 36 42 48 1.5 2 2.5 3 3.5 4  K, number of CRUs A ve ra ge  b it ra te  (M b/s )   ← Case A ← Case B OWF PEPA Bound for OWF Figure 4.2: ABR as a function of number of CRUs for OWF and PEPA. pa = 0.9 76 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5  p a A ve ra ge  b it ra te  (M b/s )   Case B → Case A → OWF PEPA Bound for OWF Figure 4.3: ABR as a function of pa for OWF and PEPA. K = 6. is 9% for K = 6 and 2% for K = 48. For both cases, the proposed upper bound for OWF is very close to the actual OWF curve: the difference is less than 0.001% in Case A and is less than 0.12% for Case B. Fig. 4.3 shows the ABR for OWF and PEPA as a function of pa forK = 6. For both cases, the ABR for OWF and PEPA decreases with pa, due to the reduced number of available subchannels. In Case A, the improvement of OWF over PEPA is 0.1% at pa = 0.1 and 0.05% at pa = 0.9. In Case B, the improvement of OWF over PEPA is 14% at pa = 0.1 and 2% at pa = 0.99. The difference between OWF and PEPA decreases with pa because with a fixed total power, the average SNR for the available subchannels increases. The difference between OWF and PEPA is known to decrease with average SNR [8]. For both cases, the 77 proposed upper bound for OWF is very close to the actual OWF curve and the difference decreases with pa. The results show that the relative performance of PEPA depends on the activity level of the PUs and the variations in average subchannel gains among the CRUs. 4.6 Conclusions The performance difference between the PEPA and OWF subcarrier power allocation schemes in a multiuser OFDM-based CR system was studied. A proposed upper bound for OWF was shown to be tight. When the PU activity is high or the CRU average gains are similar, the simpler PEPA scheme suffers little loss relative to OWF. 78 References [1] S. 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Park, “Wireless packet scheduling algorithm for OFDMA system based on time-utility and channel state,” ETRI Journal, vol. 27, no. 6, pp. 777–787, 2005. 80 Chapter 5 Subchannel Power Loading Schemes in Multiuser OFDM Systems 5.1 Introduction It is well-known that the optimal solution to the problem of determining the capacity of a set of M parallel additive white Gaussian noise (AWGN) subchannels, each of which may have a different noise power level, subject to a total input signal power constraint, has a nice water-filling interpretation [1]. We will refer to this as optimal water-filling (OWF). In OWF, the signal powers allocated to different subchannels are in general different and no power is allocated to “silent” subchannels on which the noise power exceeds a certain threshold (water level). Unfortunately, the implementation and computational complexities associated with OWF are generally high. Thus, lower complexity suboptimal algorithms have been proposed. The scheme in which the total available signal power is shared equally among all non-silent subchannels is referred to as constant power water-filling (CPWF) [2, 3] whereas the scheme in which the total available signal power is shared equally among all (silent and non-silent) subchannels is referred to as plain equal power allocation (PEPA). The PEPA scheme is simpler than CPWF since the optimal water level threshold is needed for CPWF but not for PEPA. The above mentioned algorithms assume that the number of bits that can be loaded on to a subchannel can be a real number although in practice it is integer valued. Both optimal 1Portions of this chapter have been published. Yonghong Zhang and Cyril Leung, “Performance of Equal Power Subchannel Loading in Multiuser OFDM systems,” IEEE Pacific Rim Conference on Communica- tions, Computers and Signal Processing (PacRim 2007), Victoria, Canada, Aug. 2007. The remaining portions have been submitted for publication. 81 solutions (e.g. [4]–[6]) and suboptimal solutions (e.g. [7]–[9]) for the discrete bit loading problem have been proposed for single user systems. As pointed out in [10], continuous bit loading PEPA has a negligible performance loss compared to OWF when the average signal to noise ratio (SNR) is high. In a multiuser OFDM system [11], each subchannel is usually assigned to a user with relatively good channel condition, i.e., high SNR. Consequently, PEPA has been used in many medium access control layer (MAC) resource allocation algorithms, e.g. [12]. Simulation results in [13] show that the difference between OWF and PEPA is quite small when each subchannel is assigned to the user with the best channel quality for that subchannel. In this chapter, we study this difference analytically. Moreover, this best-channel sub- channel assignment strategy may not be fair to all users. We thus examine the difference in performance between OWF and PEPA when a subchannel assignment scheme is used which takes fairness into account. We also study the performance differences between the optimal solution and PEPA for both continuous and discrete bit loading in multiuser OFDM systems. A computationally efficient and simple to implement discrete bit loading algorithm based on PEPA is also proposed. 5.2 System Model Consider the downlink of a multiuser OFDM system with one base station (BS) transmitting to K users sharing M subchannels. The OFDM symbols are assumed to have a duration of one time unit and each subchannel is modelled as a discrete-time channel with AWGN samples n(t), t = 1, 2, . . .. Let σ20 be the noise power (variance) for each subchannel so that the total noise power for the M subchannels is Mσ20. Let the amplitude gain for subchannel m of user k at time t be denoted by the random variable (rv) √ Gk,m(t). It is assumed that the power gain samples for a given value of k {gk,m(t),m = 1, 2, . . . ,M, t = 1, 2, . . . , T} are outcomes of independent, identically distributed (i.i.d.) rv’s with a common probability 82 density function (pdf) fGk(gk) and cumulated distribution function (cdf) FGk(gk), where T is the number of time units considered in the allocation scheme. The average received SNR of all users is γ = ∑K k=1E{Gk}S/(KMσ20), where S is the total available power to be allocated among the M subchannels. Assuming ∑K k=1E{Gk}/K = 1, S is given by S =Mγσ20 . (5.1) At each time t, each subchannel can be used by the BS to transmit to at most one user. We use fG(g) and FG(g) to denote the pdf and cdf of the selected user. Let a t m and s t m denote the user assigned and the power allocated to subchannel m at time t respectively. The number of bits loaded on subchannel m at time t is given by [14] btm = log2 ( 1 + stmg t m Γσ20 ) (5.2) where gtm is a simplified notation for gam,m(t) and Γ is an SNR gap parameter which indi- cates how far the system is operating from capacity. This gap can be estimated using the techniques described in [14]. The available power constraint implies that M∑ m=1 stm ≤ S, ∀t. (5.3) In practical systems, the number of bits loaded on each subchannel at each time is often an integer j, j ∈ {1, 2, · · · , J}, where J is the maximum number of bits that can be loaded on a subchannel. In this case, the number of bits loaded on subchannel m at time t is btm = max (⌊ log2 ( 1 + stmg t m Γσ20 )⌋ , J ) . (5.4) 83 5.3 Bit rates for OWF and PEPA Although OWF provides the optimal solution for subchannel power allocation, its imple- mentation and/or computational complexity is generally high compared to PEPA. Since continuous bit loading PEPA has a performance similar to OWF at high SNRs, PEPA is used in many MAC layer resource allocation schemes. When integer bits loading is used, the floor function in (5.4) results in some power wastage. We now examine the bit rates achievable by OWF and PEPA for both continuous and discrete bit loading. 5.3.1 An Upper Bound on Achievable Bit Rate for OWF Assuming that OWF is applied to the M subchannel gains of the selected users at each time t, t ∈ {1, 2, . . . , T}, we can write the average number of bits per subchannel per OFDM symbol (ABR) as BOWF ∆ = limT→∞B (T ) OWF , where [1] B (T ) OWF = 1 TM ∑ m,t: Γσ20 gtm ≤LtOWF log2 ( LtOWFg t m Γσ20 ) . (5.5) In (5.5), Γσ20/g t m can be viewed as the equivalent noise power on subchannel m and LtOWF is the water level at time t. Taking the limit as M → ∞ and noting that LtOWF = LOWF,M→∞, t = 1, 2, · · · , T , we have B (T ) OWF = ∫ ∞ Γσ20 LOWF,M→∞ log2 ( LOWF,M→∞g Γσ20 ) fG(g)dg . (5.6) In (5.5), the ABR if OWF is applied at each time t is smaller than that if OWF is applied in one shot to all the TM subchannel gains of the selected users over the T time units, i.e. B (T ) OWF ≤ 1 TM ∑ m,t: Γσ20 gtm ≤L(T )OWF log2 ( L (T ) OWFg t m Γσ20 ) . (5.7) 84 Taking the limit as T →∞, we have BOWF ≤ ∫ ∞ Γσ20 L (∞) OWF log2 ( L (∞) OWFg Γσ20 ) fG(g)dg . (5.8) In (5.8), the global water level, L (∞) OWF , is lower than the level calculated by including all subchannels, because there may exist some subchannels with above water level noises. Note that L (∞) OWF = LOWF,M→∞. Therefore, L (T ) OWF ≤ ∑T t=1 ∑M m=1 Γσ20 gtm TM + S M . (5.9) Using (5.1) and taking the limit as T →∞ in (5.9) yields L (∞) OWF ≤ ( E { 1 G } + γ Γ ) Γσ20 . (5.10) Using the RHS of (5.10) to substitute for L (∞) OWF in (5.8), we obtain BOWF ≤ (1− FG( 1 L )) log2 ( E { 1 G } + γ Γ ) + ∫ ∞ 1 L [log2 g] fG(g)dg, (5.11) where L = L and L , E { 1 G } + γ Γ . The upper bound for L (∞) OWF in (5.10) can be quite loose since it includes every subchannel, regardless of its equivalent noise power Γσ20/g t m. As g t m → 0, Γσ20/gtm increases without bound. Consequently, the bound in (5.11) is also loose. We obtain a tighter bound by turning off any subchannel for which Γσ20/g t m is greater than the RHS of (5.10), i.e., we consider only the subchannels for which gtm > 1/L in calculating the water levels. The 85 resulting water level is still higher than L (∞) OWF , so that L (∞) OWF ≤ (∫∞ 1 L 1 g fG(g)dg + γ Γ ) Γσ20 1− FG( 1L) (5.12) Substituting L (∞) OWF in (5.8) by the RHS of (5.12), we obtain a tighter bound, namely (5.11) with L = (∫∞ 1 L 1 g fG(g)dg + γ Γ ) Γσ20 1− FG( 1L) (5.13) 5.3.2 Achievable Bit Rate for Continuous PEPA From (5.2) with stm = S/M , the ABR for PEPA is BPEPA ∆ = limT→∞B (T ) PEPA where B (T ) PEPA = 1 T T∑ t=1 1 M M∑ m=1 log2 ( 1 + Sgtm MΓσ20 ) . (5.14) Using (5.1) in (5.14) and letting T →∞ yields BPEPA = E { log2 ( γ Γ + 1 G )} + E{log2G}. (5.15) 5.3.3 Achievable Bit Rate for Discrete PEPA From (5.4), the ABR for discrete PEPA is BPEPA ∆ = limT→∞B (T ) PEPA where B (T ) PEPA = 1 T T∑ t=1 1 M M∑ m=1 ⌊ log2 ( 1 + Sgtm MΓσ20 )⌋ . (5.16) Using (5.1) in (5.16) and letting T →∞ yields BPEPA = ∫ ∞ 0 ⌊ log2 ( 1 + γ Γ g )⌋ fG(g)dg (5.17) 86 When blog2(1 + gγ/Γ)c = j, j ∈ {1, 2, · · · , J}, we have j ≤ log2 ( 1 + γ Γ g ) < j + 1 (5.18) i.e. (2j − 1)Γ γ ≤ g < (2j+1 − 1)Γ γ (5.19) Using (5.19), we write (5.17) as BPEPA = J−1∑ j=1 j ∫ (2j+1−1)Γ γ (2j−1)Γ γ fG(g)dg + J ∫ ∞ (2J−1)Γ γ fG(g)dg = J−1∑ j=1 j [ FG ( (2j+1 − 1)Γ γ ) − FG ( (2j − 1)Γ γ )] + J [ FG(∞)− FG ( (2J − 1)Γ γ )] = J − J∑ j=1 FG ( (2j − 1)Γ γ ) (5.20) 5.3.4 Achievable Bit Rate for Improved Discrete PEPA Using equal power, S/M , on each subchannel results in some power wastage because the power necessary on subchannelm at time t is (2blog2(1+Sg t m/(MΓσ 2 0))c−1)Γσ20/gtm, which is lower than S/M unless btm = log2(1 + Sg t m/(MΓσ 2 0)). The wasted power can be redistributed to achieve a better ABR. Let δ ∈ (0, 1] and the number of bits loaded on subchannel m at time t be b̄tm = [ log2 ( 1 + Sgtm MΓσ20 )] δ (5.21) where [x]δ =  bxc, if x < bxc+ δbxc+ 1, otherwise. (5.22) Note that [x]1 = bxc. 87 The power necessary on subchannel m is determined by stm = (2 b̄tm − 1)Γσ20/gtm. From (5.3), we have M∑ m=1 (2b̄ t m − 1)Γσ 2 0 gtm ≤ S, ∀t . (5.23) The ABR for the improved discrete PEPA is B̄PEPA ∆ = limT→∞ B̄ (T ) PEPA where B̄ (T ) PEPA = 1 T T∑ t=1 1 M M∑ m=1 [ log2 ( 1 + Sgtm MΓσ20 )] δ . (5.24) Using (5.1) in (5.24) and letting T →∞ yields B̄PEPA = ∫ ∞ 0 [ log2 ( 1 + γ Γ g )] δ fG(g)dg (5.25) From (5.25) and (5.22), it follows that B̄PEPA is a decreasing function of δ. The lowest possible δ that satisfies (5.23) will result in the maximum possible B̄PEPA. In the following, we discuss an algorithm that finds the lowest possible δ, δ∗M→∞, as M →∞. Using (5.1), (5.21) and letting M →∞ in (5.23), we have ∫ ∞ 0 (2[log2(1+ γ Γ g)]δ − 1)1 g fG(g)dg ≤ γ Γ . (5.26) When [log2(1 + γ Γ g)]δ = j, j ∈ {1, 2, · · · , J}, from (5.22), we have j − 1 + δ ≤ log2(1 + γ Γ g) < j + δ (5.27) or equivalently (2j−1+δ − 1)Γ γ ≤ g < (2j+δ − 1)Γ γ . (5.28) Using (5.28), we can rewrite (5.26) as J−1∑ j=1 ∫ (2j+δ−1)Γ γ (2j−1+δ−1)Γ γ (2j − 1)1 g fG(g)dg + ∫ ∞ (2J−1+δ−1)Γ γ (2J − 1)1 g fG(g)dg ≤ γ Γ . (5.29) 88 Letting h(δ) = J−1∑ j=1 ∫ (2j+δ−1)Γ γ (2j−1+δ−1)Γ γ (2j − 1)1 g fG(g)dg + ∫ ∞ (2J−1+δ−1)Γ γ (2J − 1)1 g fG(g)dg − γ Γ (5.30) We can see that δ∗M→∞ is the root of h(δ). Since h(δ) is a monotonically decreasing func- tion, we can find the root using the standard bisection method within log2(1/²) iterations, where ² is the accuracy of the result. The basic idea is as follows. First, we set δ = 0 and δ = 1, representing the lowest and highest possible values of δ∗M→∞. At each iteration, we set δ = (δ+ δ)/2, calculate h(δ) and compare it with 0; if h(δ) > 0, then δ∗M→∞ > δ, and we set δ = δ; otherwise, δ∗M→∞ ≤ δ, and we set δ = δ. This process is repeated until |δ− δ| ≤ ². Then, δ∗M→∞ is set to (δ + δ)/2. Using δ∗M→∞ in (5.25) and following a similar approach as in obtaining (5.20) from (5.17), we have B̄PEPA,M→∞ = J − J∑ j=1 FG ( (2j−1+δ ∗ M→∞ − 1)Γ γ ) (5.31) In (5.31), when δ∗M→∞ = 1, the RHS of (5.31) is equal to BPEPA. As δ ∗ M→∞ decreases, so does FG(·) and B̄PEPA,M→∞ increases. In practice, M is finite and the RHS of (5.31) is an upper bound on B̄PEPA, i.e. B̄PEPA ≤ J − J∑ j=1 FG ( (2j−1+δ ∗ M→∞ − 1)Γ γ ) . (5.32) 5.3.5 Improved Discrete PEPA Algorithm Following the discussion in Section 5.3.4, we can have an ABR higher than BPEPA in (5.20) if (5.21) is used instead of (5.4) and the appropriate δ∗t can be found at time t. In this section, we propose an improved discrete PEPA algorithm that uses (5.21) to perform the bit allocation. The key is to find the lowest value, δ∗t , of δt that satisfies (5.23), which uses the bisection method similar as in finding δ∗M→∞. Once δ ∗ t is obtained, the power on each 89 subchannel can be calculated by using stm = (2 [log2(1+Sg t m/(MΓσ 2 0))]δ∗t − 1)Γσ20/gtm. At time t, btm,m = 1, 2, · · · ,M is first calculated using (5.2), δ is set to 0 and δ is set to 1 representing the lowest and highest possible values for δ∗t , respectively. Then δ ∗ t is found using an iterative method. During each iteration, first, δ is set to δ = (δ + δ)/2, b̄tm,m = 1, 2, · · · ,M is updated by setting b̄tm = [btm]δ and the total required power s is set to the LHS of (5.23); second, if s − S > 0, δ is set to δ because δ∗t > δ, otherwise δ is set to δ. This iteration process continues until |δ − δ| ≤ ², which is the precision of the result. Finally, δ∗t is set to (δ + δ)/2. This algorithm is summarized in Algorithm 4. Algorithm 4 Waterlevel finding algorithm. 1) Calculate btm,m = 1, 2, · · · ,M using (5.2), set δ = 0, δ = 1. 2) If |δ − δ| ≤ ², go to Step 5), otherwise continue. 3) δ = (δ + δ)/2, b̄tm = [b t m]δ,m = 1, 2, · · · ,M . Let s be the value of LHS of (5.23). 4) If s− S > 0, δ = δ; otherwise δ = δ. Go to Step 2). 5) δ∗t = (δ + δ)/2. The computational complexity at each iteration is O(M). Because the range of possible values of δ∗t is halved at each iteration, the number of iterations required is a constant, C = log2(1/²), where ² is the accuracy. So the computational complexity of this algorithm is O(M). 5.4 Rayleigh fading channel In Section 5.3, we studied the ABR for OWF and PEPA for arbitrary pdf’s and cdf’s of the subchannel gains of the selected users. In this section, we obtain the pdf and cdf of the subchannel gains of the selected users for two different subchannel allocation strategies. The subchannel gains of the users are assumed to be Rayleigh distributed, i.e. the power gains are exponentially distributed. 90 5.4.1 Opportunistic Subchannel Assignment Suppose that at each time t, each of the M subchannels is assigned to the user with the highest gain for that subchannel. If the average subchannel power gains for all users are equal, the pdf of the power gain for the user assigned to any subchannel is readily obtained using a standard result in order statistics [15], i.e., fG(g) = K [ 1− e− gE{G} ]K−1 e− gE{G} E{G} (5.33) with corresponding cdf FG(g) = [ 1− e− gE{G} ]K . (5.34) 5.4.2 A Fairer Subchannel Assignment Scheme If the average subchannel gains for users are quite different, e.g., the average subchannel gains of users near the cell edge is much lower than that of the users close to the BS, assigning a subchannel to the user with the highest gain may be too unfair to users with poor average subchannel gains. A fairer scheme [12] is to select, for each subchannel, the user with the best channel gain relative to its own mean gain, i.e. k∗(t) = argmaxk gk,m(t) E{Gk} . (5.35) The distribution of a user’s subchannel gain relative to its own mean is exponential with a mean of 1. Thus the probability of selecting user i is 1/K, i.e. P (k∗ = i) = 1/K, i = 1, 2, · · · , K. The cdf of the power gain of the selected user for a subchannel is FG(g) = P (G ≤ g) = K∑ i=1 P (G ≤ g|k∗ = i)P (k∗ = i) = 1 K K∑ i=1 K∏ j=1 P (Gj ≤ gE{Gj} E{Gi} ) = 1 K K∑ i=1 ( 1− e− gE{Gi} )K . (5.36) 91 The corresponding pdf is fG(g) = K∑ i=1 ( 1− e− gE{Gi} )(K−1) e− gE{Gi} E{Gi} . (5.37) 5.5 Simulation Results To study the performances of continuous and discrete bit loading PEPA and OWF in a multiuser OFDM system, both theoretical and simulation results are provided in this section. For convenience, we will refer to the results from evaluating (5.6) as “OWF”, (5.11) with L equal to (5.13) as “proposed OWF upper bound”, (5.15) as “PEPA”, (5.20) as “D-PEPA”, and (5.32) as “ID-PEPA upper bound”. For comparison, the upper bound in [3] was also evaluated. Since this bound applies to the ABR difference between OWF and CPWF, we added the bound to the ABR of CPWF and refer it to as Adjusted YC upper bound. Simulation results for B (T ) OWF , B (T ) PEPA, B (T ) PEPA, and B̄ (T ) PEPA, obtained using (5.5), (5.14), (5.16), and (5.24) with δ obtained using Algorithm 4, are hereafter referred to as OWF (simu), PEPA (simu), D-PEPA (simu), ID-PEPA (simu). A value ² of 0.01 is used in calculating the ID-PEPA using Algorithm 4. All simulation results presented are based on a time duration of 2.5× 106 OFDM symbols. The two subchannel allocation strategies in Sections 5.4.1 and 5.4.2, hereafter referred to as Case A and Case B respectively, are considered. In Case A, the average subchannel power gain for each user is chosen as 1. In Case B, the number of CRUs was increased in groups of six at a time with the average subchannel gain per group chosen to be 1. The average subchannel power gains of the six CRUs in a group are chosen as follows: one with value 4.85, two with value 0.5 and three with value 0.05. The multipath fading is assumed to be Rayleigh with a maximum Doppler frequency of 20 Hz. An OFDM symbol duration of 40 µs is used and resource allocation is performed once every ms. In our calculations, we also use the following parameter values: Γ = 1, σ20 = 10 −3, and M = 64. 92 Figs. 5.1 and 5.2 show the ABR as a function of average SNR with 12 users for Case A and Case B, respectively. In Case A, the performance of OWF and PEPA are almost the same, with a difference of less than 0.5%. The proposed OWF bound is much tighter than the adjusted YC bound. In Case B, the performance difference between OWF and PEPA is less than 10% when SNR is higher than 5 dB, and the difference decreases with SNR. At low SNR, the difference is much more noticeable, at nearly 20% for SNR = 0 dB. The adjusted YC bound is tighter than the proposed bound for SNR values lower than 2 dB, but is looser at higher SNRs. In both cases, the ABR for ID-PEPA is significantly higher than that for D-PEPA. Figs. 5.3 and 5.4 show the ABR difference of continuous and discrete PEPA compared to continuous OWF as a function of average SNR γ for K = 12 users and M = 64 subchannels for Case A and Case B, respectively. In both Cases, the difference between OWF and PEPA decreases with SNR. In Case A, the differences for D-PEPA and ID-PEPA compared to OWF do not change with SNR, at 0.5 bits and 0.05 bits, respectively. In Case B, the differences for D-PEPA and ID-PEPA compared to OWF decrease with SNR. The difference between D-PEPA and OWF changes from 0.7 bits to 0.5 bits, and the difference between ID-PEPA and OWF decreases from 0.2 bits to 0.03 bits. Specifically, in Case B, when SNR is low, ID-PEPA performs better than PEPA. In both cases, the improvement of ID-PEPA over D-PEPA is significant. Figs. 5.5 and 5.6 show the ABR as a function of number, M , of subchannels for Case A and Case B, respectively. In both cases, the theoretical values of PEPA and D-PEPA do not change with M , and agree closely with the simulation results. OWF and ID-PEPA bound curves are constant since M → ∞ for these curves. As M increases, the simulation values for OWF and ID-PEPA increase and approach their theoretical values. Although OWF increases with M , the increase is very small (less than 0.002% in Case A and less than 2% in Case B), and the performance loss of D-PEPA relative to continuous OWF is about 10% (15%) for Case A (Case B). The performance loss of ID-PEPA relative to OWF decreases 93 0 2 4 6 8 10 1 1.5 2 2.5 3 3.5 4 4.5 5 SNR, γ (dB) A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Adjusted YC upper bound Proposed OWF upper bound OWF PEPA ID−PEPA upper bound D−PEPA Figure 5.1: ABR as a function of average SNR γ with K = 12 users for Case A. 94 0 2 4 6 8 10 0.5 1 1.5 2 2.5 3 3.5 SNR, γ (dB) A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Adjusted YC upper bound Proposed OWF upper bound OWF PEPA ID−PEPA upper bound D−PEPA Figure 5.2: ABR as a function of average SNR γ with K = 12 users for Case B. 95 0 5 10 15 20         SNR, γ (dB) A ve ra ge  b it ra te  d iff er en ce  re la tiv e to  O W F   10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 D−PEPA D−PEPA (simu) ID−PEPA upper bound ID−PEPA (simu) PEPA PEPA (simu) Figure 5.3: ABR difference between OWF and PEPA as a function of average SNR γ for Case A. K = 12 and M = 64. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 96 0 5 10 15 20     SNR, γ (dB) A ve ra ge  b it ra te  d iff er en ce  re la tiv e to  O W F   10−3 10−2 10−1  0 D−PEPA D−PEPA (simu) ID−PEPA upper bound ID−PEPA (simu) PEPA PEPA (simu) Figure 5.4: ABR difference between OWF and PEPA as a function of average SNR γ for Case B. K = 12 and M = 64. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 97 with M , from 10% (15%) at M = 1 to less than 1% (3%) at M = 128 for Case A (Case B). When M = 1, there is no difference in the ABR for D-OWF, D-PEPA, or ID-PEPA. On the other hand, when M = 128, the improvement of using D-OWF or ID-PEPA over D-PEPA is over 10% (16%) for Case A (Case B). This shows that in a system where there is only one channel, e.g., in a narrow band system, D-PEPA is good enough; however, in systems with a large number of subchannels, especially when the average subchannel gains among the users are quite different, the proposed ID-PEPA algorithm provides a noticeable improvement. The performance loss of ID-PEPA relative to D-OWF is small at 0.1% (2%) in Case A (Case B). Figs. 5.7 and 5.8 show the ABR as a function of the number, K, of users for Case A and Case B, respectively. In both cases, the simulation results are very close to theoretical values. The ABR increases with K regardless of the allocation scheme used, due to multiuser diversity. The ABR improvement of ID-PEPA over D-PEPA is around 0.43 bits for the range of K values shown. Since the ABR increases with K, the improvement of ID-PEPA relative to D-PEPA decreases with K, e.g., in Case B, the improvement is 18% at K = 6 and 14% at K = 36. The performance of ID-PEPA is almost the same as D-OWF, with a difference of less than 1% (3%) in Case A (Case B). 5.6 Conclusions A new low computational and implementation complexity discrete bit loading algorithm based on plain equal power allocation for multiuser OFDM systems has been proposed. Bounds for the optimal water-filling algorithm and the proposed bit loading algorithm are derived, and shown to be tight. Simulation results indicate that the proposed bit loading algorithm provides a close to the optimal solution and significant improvement over an equal power allocation scheme in which the number of bits loaded is given by (5.4), especially in a system with widely different average user subchannel gains. 98 1 2 4 8 16 32 64 128 4.4 4.5 4.6 4.7 4.8 4.9 Number of subchannels,  M A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Proposed upper bound OWF OWF (simu) PEPA PEPA (simu) ID−PEPA upper bound D−OWF ID−PEPA (simu) D−PEPA D−PEPA (simu) Figure 5.5: ABR as a function of number of subchannels M for Case A. K = 12 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 99 1 2 4 8 16 32 64 128 2.7 2.8 2.9 3 3.1 3.2 Number of subchannels,  M A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Proposed upper bound OWF OWF (simu) PEPA PEPA (simu) ID−PEPA upper bound D−OWF ID−PEPA (simu) D−PEPA D−PEPA (simu) Figure 5.6: ABR as a function of number of subchannels M for Case B. K = 12 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 100 6 12 18 24 30 36 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 Number of users,  K A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Proposed upper bound OWF OWF (simu) PEPA PEPA (simu) ID−PEPA upper bound D−OWF ID−PEPA (simu) D−PEPA D−PEPA (simu) Figure 5.7: ABR as a function of number of users K for Case A. M = 64 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 101 6 12 18 24 30 36 2.2 2.4 2.6 2.8 3 3.2 3.4 Number of users,  K A ve ra ge  n um be r o f b its  p er  O FD M  sy m bo l pe r s ub ch an ne l, A BR   Proposed upper bound OWF OWF (simu) PEPA PEPA (simu) ID−PEPA upper bound D−OWF ID−PEPA (simu) D−PEPA D−PEPA (simu) Figure 5.8: ABR as a function of number of users K for Case B. M = 64 and γ = 10 dB. For simulation curves, T = 1 ms and for theoretical curves, T =∞. 102 References [1] R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley & Sons, 1968. [2] A. Leke and J. M. Cioffi, “A maximum rate loading algorithm for discrete multi- tone modulation systems,” in Proc. of IEEE Global Telecommunications Conference (GLOBECOM ’97), vol. 3, Phoenix, AZ, USA, November 1997, pp. 1514–1518. [3] W. Yu and J. M. 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Seo, M. Shin, and S. Park, “Wireless packet scheduling algorithm for OFDMA system based on time-utility and channel state,” ETRI Journal, vol. 27, no. 6, pp. 777–787, 2005. 103 [13] J. Jang and K. B. Lee, “Transmit power adaptation for multiuser OFDM systems,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 2, pp. 171–178, Febru- ary 2003. [14] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading chan- nels,” IEEE Transactions on Communications, vol. 45, no. 10, pp. 1218–1230, October 1997. [15] H. A. David and H. N. Nagaraja, Order Statistics. John Wiley & Sons, 2003. 104 Chapter 6 Cross-Layer Resource Allocation for OFDM-based Cognitive Radio Systems 6.1 Introduction Cognitive radio (CR) is a new technology which is attracting a lot of research interest. It was first proposed in [1] as a novel wireless communications approach with the ability to sense and learn its environment, change its transmission and reception parameters, and efficiently utilize the radio spectrum whenever and wherever needed. With the ever-increasing demand for mobile wireless applications, the allocation of spec- trum for the exclusive use of licensed holders is becoming a hurdle to efficient spectrum utilization. There are few unallocated bands below 6 GHz [2]. However, studies have shown that a large portion of the allocated spectrum is used only sporadically, resulting in poor spectrum utilization [3]. CR, with its ability to sense unused bands and adjust transmission parameters, is an excellent candidate for improving spectrum utilization. The FCC [4] is supporting the use of CR technology to allow unlicensed users to share radio resources with licensed users in a way which is transparent to the latter group. As elaborated in [5], or- thogonal frequency division multiplexing (OFDM) is a good modulation candidate for CR systems. However, a number of challenging resource allocation (RA) problems need to be solved in order to realize the full potential of CR. RA in multiuser OFDM systems has been extensively studied at the physical (PHY) layer in terms of subcarrier, bit, and power allocation. In [6]–[9], suboptimal solutions are 1Portions of this chapter have been accepted for publication. Yonghong Zhang and Cyril Leung, “Cross- Layer Resource Allocation for Real-Time Services in OFDM-based Cognitive Radio Systems,” Springer Telecommunication Systems. The remaining portions have been submitted for publication. 105 proposed to minimize the total transmit power while satisfying the fixed rate and bit error rate (BER) requirements for fixed-rate real-time (RT) services. For non-real-time (NRT) services, maximizing system throughput for a given total transmit power while providing a certain level of fairness among users is often considered [10]–[12]. One of the main difficulties in allocating resources in CR systems is that the interference power generated by cognitive radio users (CRUs) at a primary user (PU) receiver should not exceed a predefined threshold. In [13]–[15], the PHY problem of allocating subcarrier, bit and power to maximize system throughput, while guaranteeing that interference power limits are met, is studied for OFDM-based CR systems. In some situations, it may not be practical to determine the interference power levels at all PU receivers and appropriate guard bands are used to protect the PU bands [16]. At first glance, it may appear that the RA algorithms in [6]–[12], designed for conven- tional OFDM systems, can be applied directly to OFDM-based CR systems that use guard bands. However, in contrast to conventional OFDM systems which assume that the available spectrum is fixed, the available transmission spectrum in a CR system changes over time. Existing solutions designed for OFDM systems generally assume a fixed rate requirement for each OFDM symbol duration. This assumption is reasonable in OFDM systems, but may not be feasible when the bandwidth available to the CR system is low. A dynamically adjusted rate requirement is more appropriate. Most existing algorithms designed for mul- tiuser OFDM systems assume that a feasible solution to the RA problem exists. This is generally valid when the available bandwidth is fixed, as in most OFDM systems. However, for CR systems, there may be no feasible solution when the available bandwidth is very low. This problem feasibility issue must be addressed in any CR system in which available resources are changing rapidly. Medium access control (MAC) layer RA algorithms have also been devised to satisfy user quality of service (QoS) requirements. Almost all existing studies (e.g. [7],[17],[18], [19]) extend opportunistic scheduling strategies (which use channel state information) for 106 the single carrier case [20] to the OFDM case with multiple subcarriers. In these algorithms, equal subcarrier powers are generally assumed, which may not be optimal. To improve system throughput, users experiencing poor channel conditions postpone their transmissions until conditions improve. To help RT user packets meet their deadlines, an urgency factor [19] is used to improve transmission opportunities for users whose head-of-line (HOL) packets have imminent deadlines. Such a strategy works well in multiuser OFDM systems. However, this urgency factor ignores the fact that available system resources in a CR system are time- varying, resulting in a false urgency issue. When the available system resources are higher than the assumed fixed system resources, some RT users with bad channel qualities may be given transmission opportunities prematurely. This will result in a lower system throughput. On the other hand, when the available system bandwidth drops below the nominal value, the dropped packet rate (DPR) may become unacceptably high. Some researchers have adopted a cross-layer design approach in allocating system re- sources. In [21]–[23], sub-optimal algorithms for NRT services are proposed; algorithms for both RT and NRT services are studied in [24] and [25]. In [24], the QoS for RT applications is improved by giving high priority to users whose HOL packet deadlines are approaching. Since equal power allocation is assumed in [24], it should be possible to further improve the system performance by relaxing this constraint. Moreover, the high priority strategy is essentially the same as the urgency factor approach and does not resolve the false urgency issue. In [25], the MAC layer QoS requirement is converted to a PHY layer fixed rate re- quirement based on the average user packet arrival rate and delay constraint. An optimal subchannel and power allocation strategy is proposed which maximizes system throughput subject to a total transmit power limit and user delay requirements. The problem feasibility issue is not addressed. To date, there are few studies on QoS provisioning in OFDM-based CR systems in which the available spectrum is time-varying. In [26], cross-layer based MAC protocols are proposed to allow CRUs to share the spectrum holes, which are detected by the integrated 107 PHY layer spectrum-sensing policies. In [27], the average rate of the CRU in a single user case is maintained and the rate variance is minimized, subject to power constraints. The schemes in [26] and [27] are designed for NRT applications. To the best of our knowledge, there is no published work on QoS provisioning for RT applications or a mixture of RT and NRT applications in OFDM-based CR systems. In this chapter, we study RA for both RT and NRT services in a multiuser OFDM- based CR system. Given the time-varying nature of available system resources, we adopt a cross-layer approach which improves system resource utilization, while addressing the problem feasibility and false urgency issues that are not typically encountered in OFDM systems. A system that only supports RT services is first investigated. Since power savings is important for portable wireless devices, an optimization problem which aims to minimize power consumption while guaranteeing QoS for RT services is formulated. In the formulation, the MAC layer packet delay requirements are dynamically converted to PHY layer rate requirements; the conversion is a function of the delivery status of queued packets as well as the number of available subchannels. Then an optimization problem that extends the one for RT services to both RT and NRT services is formulated which dynamically converts RT user delay requirements and NRT user average rate requirements to PHY layer rate requirements. The solution has a multi-level water-filling interpretation with fairness among users achieved by adjusting user water-levels. 6.2 System Model We consider a CR system with a total bandwidth of W Hz and L PUs; PU l, l = 1, 2, . . . , L has a bandwidth allocation of Wl Hz. Frequency bands actually carrying PU signals are referred to as active; non-active bands are also termed spectrum holes. In order to reduce the cross-channel interference between the CRUs and the PUs to acceptable levels, some subchannels (guard bands) adjacent to active PU bands are not used by the CRUs [16]. 108 The active PU bands, PU guard bands, and the spectrum holes in a certain area at time t are shown in Fig. 6.1. For this example, the number, mCR,t, of subchannels available to the CR system (i.e. those within spectrum holes) at time t can range from 0 to 32. Fig. 6.1 depicts a scenario with mCR,t = 15. 97531 11 13 15 17 19 21 23 25 27 29 31 W2 Spectrum hole Spectrum hole PU active frequency bands W1 Guard bands Guard bands fc f Figure 6.1: Primary users’ active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels. Assume that there is one CR base station (CRBS) communicating with K CRUs. We are interested in downlink transmissions from the CRBS to the CRUs. The CRBS and the CRUs are able to accurately locate the spectrum holes. There are M OFDM subbands (or subchannels) in the system, each with noise power σ20. Interference among the subchannels is assumed to be negligible. The nominal bandwidth of subband m, m = {1, 2, . . . ,M} ranges from fc + (m− 1)∆f to fc +m∆f . The system is time-slotted with a slot duration equal to an OFDM symbol duration Tsymbol. The subchannels are modelled in discrete-time, with the gain for subchannel m and time slot t from the CRBS to CRU k denoted by √ gtk,m. For illustrative purposes, it is assumed that the power gains { gtk,m } are outcomes of independent random variables (rv’s), and for any given value of k, { gtk,m, t = 1, 2, · · · , T,m = 1, 2, · · · ,M } are independent identically distributed (i.i.d.) rv’s. The number, rtk,m, of bits per OFDM symbol which can 109 be supported by subchannel m of CRU k in time slot t is given by [28] rtk,m = log2 ( 1 + gtk,ms t k,m Γσ20 ) (6.1) where stk,m is the transmit power and Γ is an SNR gap parameter which indicates how far the system is operating from capacity. For simplicity, we assume continuous bit modulation, i.e. rtk,m can take on real values [28]. The availability of a PU band is modelled by a two-state Markov chain. During a time slot t, a PU band can be in one of two states: active or inactive [26]. A state transition may occur once every Tstate slots. At a transition time, the probability of changing from active to inactive state is 1 − pa, and the probability of changing from inactive to active state is 1 − pn. The number, lCR,t, of available PU bands at time slots {t, t = 1, 2, . . .} then form a Markov chain, with transition probability matrix Q = {qij}, i, j = 0, 1, 2, · · · , L, where state i corresponds to the event that the number of available PU bands is equal to i and the probability, qij, of moving from state i to state j is given by qij = L∑ n=0 ( i n ) (1− pn)npi−nn ( L− i n− i+ j ) (1− pa)n−i+jpL−j−na . (6.2) The Markov chain is illustrated in Fig. 6.2. If each PU band can accommodate an equal number, M/L, of subchannels, then the numberMCR,t of available subchannels at time slots {t = 1, 2, · · · } form a Markov chain with transition probability matrix P = {pmn},m, n = 0, 1, 2, · · · ,M , where state m corresponds to the event that MCR,t = m and the probability, pmn, of moving from state m to state n is given by pmn =  qij, m = iM L , n = jM L 0, otherwise. (6.3) 110 q11 qLLqiiq00 L q01 q0L i q0i qLi q1i q1L qiL qL1qL0 qi1qi0 q10 Figure 6.2: Markov chain model for the number of available PU bands. 6.3 Cross-Layer Resource Allocation for RT Services In this section, we consider a system which supports only RT services. Although only downlink transmissions from the CRBS to the CRUs are studied in this section, the approach proposed is also applicable on the uplink. 6.3.1 The Optimization Problem On arrival, a packet destined for CRU k is placed into CRBS buffer k for transmission in order of packet creation times. The ith packet in buffer k needs to be received by CRU k within dk time slots after packet i’s creation time slot t S k,i; otherwise, the packet becomes stale and will be discarded. Our objective is to minimize the total CRBS transmit power while ensuring that pack- ets are delivered to CRUs within their specified deadlines. The optimization problem is formulated as OP1 : min atk,m,r t k,m T∑ t=1 K∑ k=1 M∑ m=1 atk,m(2 rtk,m − 1) Γσ 2 0 gtk,m (6.4) 111 subject to K∑ k=1 atk,m = 1 (6.5) atk,m ∈  {0, 1} m ∈Mt{0} m ∈ M̄t (6.6) rtk,m ∈ (0, RMAX) (6.7) tSk,i + dk > t D k,i, k ∈ {1, 2, · · · , K} (6.8) atk,m, s t k,m can depend on {guk,m, u ≤ t}, but not on {guk,m, u > t} (6.9) atk,m, s t k,m can depend on the length and creation times of packets which have been placed in buffer k, but not on information about packets yet to arrive. (6.10) In (6.4), T is the number of time slots considered in the allocation scheme, (2r t k,m−1)Γσ20/gtk,m is the power necessary for the OFDM symbol to support a rate of rtk,m as can be seen from (6.1), and atk,m is the subchannel assignment indicator which takes on value 1 if subchannel m is assigned to CRU k at time slot t; otherwise, its value is 0. In (6.6), Mt (M̄t) ⊆ {1, 2, · · · ,M} denotes the set of available (unavailable) subchannels at time t. Constraints (6.5) and (6.6) imply that at most one CRU can use any given subchannel in any given time slot. In (6.7), RMAX is the maximum number of bits that can be allocated on any subchannel. Constraint (6.8) represents the MAC layer QoS requirement that the delivery time tDk,i for packet i of CRU k must be no later than dk time slots after its creation time slot. Conditions (6.9) and (6.10) reflect the causality constraints for the problem. Problem OP1 is difficult to solve as it involves an optimization over multiple time slots t ∈ {1, 2, . . . , T}. We propose to solve a simpler problem, Problem OP3, obtained by first converting the MAC layer constraint (6.8) into (generally different) PHY layer rate requirements for each of the T time slots. As mCR,t → ∞, the optimal solutions for OP1 112 and OP3 will be the same. 6.3.2 Conversion of MAC Layer Requirements to PHY Layer Requirements We now consider the following Problem OP2 OP2 : min atk,m,r t k,m U∑ t=1 K∑ k=1 M∑ m=1 atk,m(2 rtk,m − 1) Γσ 2 0 gtk,m (6.11) subject to K∑ k=1 atk,m = 1 (6.12) atk,m ∈  {0, 1} m ∈Mt{0} m ∈ M̄t (6.13) rtk,m ∈ (0, RMAX) (6.14) U∑ t=1 M∑ m=1 atk,mr t k,m = R U,REQ k (6.15) atk,m, s t k,m can depend on the length and creation times of packets which have been placed in buffer k, but not on information about packets yet to arrive. (6.16) In (6.15), RU,REQk is the minimum number of bits that needs to be transmitted in time slots 1 through U to meet the delay requirements and U ∈ {1, 2, · · · , T}. Note that Prob- lem OP2 is the same as Problem OP1, except that the causality constraint (6.9) is dropped and the MAC layer constraint (6.8) is replaced by the PHY layer constraint (6.15). Theorem 3.1: As mCR,t →∞, t = 1, 2, · · · , U , the optimal solution S∗ = {rt ∗k,m, at ∗k,m} for 113 problem OP2 satisfies M∑ m=1 at ∗k,mr t ∗ k,m = r t,REQ k , t = 1, 2, · · · , U, k = 1, 2, · · · , K, (6.17) where rt,REQk , R U,REQ k mCR,t m(1,U) , and m(1, U) = ∑U t=1mCR,t. Proof: See Appendix B.1. Comment: An intuitive explanation of the theorem is as follows. The optimal solution for Problem OP2 has a water-filling interpretation [29]. When the number of available subchannels in each time slot is large, the local water level at each time slot is nearly equal to the global water level (i.e. the water level if the subchannels in time slots 1, 2, . . . , U are pooled together). The bit rate for CRU k in time slot t is proportional to the fraction of the total number of available subchannels in time slots 1, 2, . . . , U that are available in slot t . Using Theorem 3.1, Problem OP2 which spans the time slots 1, 2, · · · , U can be trans- formed into a series of optimization problems, one for each slot t, t ∈ {1, 2, · · · , U}. The values of at ∗k,m and r t ∗ k,m are obtained as the optimal solution to Problem OP3. OP3 : min atk,m,r t k,m K∑ k=1 M∑ m=1 atk,m(2 rtk,m − 1) Γσ 2 0 gtk,m (6.18) subject to (6.12), (6.13), (6.14) and M∑ m=1 atk,mr t k,m = r t,REQ k . (6.19) In practice, mCR,t is finite. The average CRBS power for Problem OP2 is plotted as a function of the average number, M̄CR,t, of available subchannels in Fig. 6.3 withK = 3 CRUs and different values of U . The number, MCR,t, of available subchannels is assumed to be uniformly distributed between 0.5M̄CR,t and 1.5M̄CR,t and r t,REQ k = 1.5M̄CR,t. In Case A, the average subchannel power gains for the three CRUs are 0.036× 10−9 W, 0.095× 10−9 W and 10−9 W. In Case B, the average subchannel power gains are equal to 10−9 W. In both 114 8 16 32 64 128 256 0 0.02 0.04 0.06 0.08 0.1 0.12 Average number  MCR,t of subchannels A ve ra ge  to ta l p ow er  (W )   − Case A: U=1 Case A: U=2 Case A: U=5 Case B: U=1 Case B: U=2 Case B: U=5 Figure 6.3: Average CRBS power for U = 1, 2 and 5 slots. cases, σ20 = 10 −13 W. It can be seen that as M̄CR,t increases, the difference among the three solutions decreases. The difference increases with variation in the average CRU subchannel power gains. But even in the rather large variation scenario of Case A, the difference is less than 2% when M̄CR,t = 64. Due to running time considerations, each point for U = 1 was obtained by averaging over 1000 random subchannel gain realizations, each point for U = 2 was obtained using 500 realizations, and each point for U = 5 was obtained using 200 realizations. At time slot t, in order to ensure that CRU k’s first packet in queue, i.e., the head of line (HOL) packet, is delivered on time, for the next Tk,1 = t S k,1 + dk − t time slots,∑t+Tk,1−1 u=t ∑M m=1 a u k,mr u k,m should be at least the length bk,1 (in bits) of CRU k’s first packet, 115 i.e. rt,REQk = bk,1 mCR,t m(t,t+Tk,1−1) . Similarly, to ensure that the ith packet in CRBS buffer k is delivered on time, for the next Tk,i = t S k,i+dk− t, ∑t+Tk,i−1 u=t ∑M m=1 a u k,mr u k,m should be at least∑i n=1 bk,n, i.e. r t,REQ k = mCR,t m(t,t+Tk,i−1) ∑i n=1 bk,n. Thus, in order to ensure the on-time delivery of every packet in CRBS buffer k, the CRBS should set rt,REQk to the maximum among the values for all packets in CRU k’s queue at time t, i.e. rt,REQk = maxi [ mCR,t m(t, t+ Tk,i − 1) i∑ n=1 bk,n ] . (6.20) 6.3.3 Proposed Algorithm At time slot t, t = 1, 2, . . . , T , we have an instance of Problem OP3 in which rt,REQk is calculated using (6.20) with mCR,t equal to the number of available subchannels in slot t and m(t, t + Tk,i − 1) can be estimated as discussed in Section 6.3.4. OP3 can be solved by using (B.31) and (B.23) with U = 1. In order to find the set {λ∗k, k = 1, 2, · · · , K} that satisfies ∑M m=1 a t ∗ k,mr t ∗ k,m = r t,REQ k , the initial set of water-levels can be obtained by assuming that each CRU in turn has exclusive use of all subchannels, then iteratively increasing each CRU’s water-level until all CRUs reach their target rt,REQk values. The water-level for a CRU will not change much from slot to slot. So except in cases when new CRUs are joining in, we can use the previous water-levels as the starting point in searching for the optimal solution. The resulting number of bits, rt ∗k,m, loaded at slot t for CRU k at subchannel m are real numbers. For systems that require an integer number of bits allocation, a near-optimal procedure is to set the integer-constrained value, r̃t ∗k,m, to [r t ∗ k,m] where [·] denotes the rounding to the nearest integer operation. Although after rounding off rt ∗k,m, ∑M m=1 a t ∗ k,mr̃ t ∗ k,m may not be equal to rt,REQk , no adjustment is needed unless the corresponding CRU’s HOL packet will miss its deadline. For these CRUs, i.e., for CRU k′ that ∑M m=1 a t ∗ k′,mr t ∗ k′,m < bk′,1 and t = T Sk′,1+d ′ k−1, the following adjustment is necessary: allocate one additional bit at a time to the subchannel that requires the minimum increasing power until ∑M m=1 a t ∗ k′,mr t ∗ k′,m = bk′,1. 116 6.3.4 Estimation of the Number of Available Subchannels in the next m(t1, t2) slots To calculate the appropriate rate requirement for CRU k in time slot t using (6.20), we need to know m(t, t + τ − 1), the total number of available subchannels in the next τ time slots {t, t+ 1, · · · , t+ τ − 1}. At time slot t, the number, mCR,t, of available subchannels is known. Suppose that state transitions for all PUs occur at time slots Tstate, 2Tstate, · · · . Let tn , d tTstate eTstate be the time slot at which the next state transition will take place. Then mCR,t stays unchanged for the next tn − t slots. From time slot tn to tn + Tstate − 1, the probability vector V = (vm0, vm1, · · · , vmn, · · · , vmM) where vmn is the probability of n available subchannels given mCR,t = m is V = XmP. (6.21) In (6.21), Xm = (0, · · · , 1, · · · , 0) is a 1 × (M + 1) vector with all component being zero except for the (mCR,t + 1)th component being 1. Let M̄CR(t1, t2) be the total expected number of available subchannels in time slots t1 through t2. The total expected number of available subchannels in time slots tn through tn + Tstate − 1 is thus M̄CR(tn, tn + Tstate − 1) = TstateVY (6.22) where Y = (0, 1, 2, · · · ,M)T . Similarly, at time slot tn + Tstate to tn + 2Tstate − 1, we get M̄CR(tn + Tstate, tn + 2Tstate − 1) = TstateXmP2Y. (6.23) Let tl , b t+τ−1Tstate cTstate be the time slot at which the last state transition will occur. In 117 time slot tl through t+ τ − 1, we have M̄CR(tl, t+ τ − 1) = (t+ τ − tl)XmP tl−tn Tstate +1 Y . (6.24) Our estimate, m̂(t, t + τ − 1) of the total expected number, m(t, t + τ − 1), of available subchannels in time slots t through t+ τ − 1 is then m̂(t, t+ τ − 1) = min(tn − t, τ)mCR,t + Tstate tl−tn Tstate∑ u=1 XmP uY (t+ τ − tl)XmP tl−tn Tstate +1 Y. (6.25) 6.4 Cross-Layer Resource Allocation for Mixed Services 6.4.1 The Optimization Problem In this section, we consider two types of services, namely, RT services and NRT services. Without loss of generality, we assume that the first KRT CRUs are RT, and the remaining CRUs are NRT. The average data rate for NRT CRU k should be at least RNRTk . Our objective is to maximize system throughput while ensuring that RT CRU packets are delivered within their specified deadlines and that the average data rates for NRT CRUs satisfy their rate requirements RNRTk . The optimization problem can be formulated as OP4 : max atk,m,r t k,m T∑ t=1 K∑ k=1 M∑ m=1 atk,mr t k,m (6.26) 118 subject to K∑ k=1 M∑ m=1 atk,ms t k,m ≤ S (6.27) stk,m = ( 2r t k,m − 1 ) Γσ20 gtk,m (6.28) K∑ k=1 atk,m = 1 (6.29) atk,m ∈  {0, 1} m ∈Mt{0} m ∈ M̄t (6.30) rtk,m ∈ (0, RMAX) (6.31) tSk,i + dk > t D k,i, k ∈ {1, 2, · · · , KRT} (6.32) R̄1,Tk ≥ RNRTk , k ∈ {KRT + 1, KRT + 2, · · · , K} (6.33) atk,m, r t k,m can depend on {guk,m, u ≤ t}, but not on {guk,m, u > t}. (6.34) atk,m, r t k,m can depend on the lengths and creation times of packets already in buffer k, but not on information about packets yet to arrive, k ∈ {1, 2, · · · , KRT}. (6.35) In (6.27), S is the total allowed power per time slot and (6.28) follows from (6.1). Con- straint (6.33) ensures that the average data rate, R̄1,Tk , of NRT CRU k from time slot 1 to time slot T is no smaller than RNRTk . Problem OP4 involves an optimization over the time slots t ∈ {1, 2, . . . , T} and is com- putationally complex. In this section, we formulate a series of simpler related problems, OP5 and OP6. As mCR,t → ∞, the optimal solution for OP6 converges to that for OP4. To deal with the issue of problem feasibility, we adopt a goal programming approach which leads to Problem OP7. The transformations and relationships among these optimization problems are shown in Fig. 6.4. The proposed cross-layer resource allocation algorithm is then described in Section 6.4.4. 119 OP5 - optimization over multiple time slots - problem feasibility issue OP4 - no knowledge of future channel gains - MAC layer QoS requirements - optimization over multiple time slots - problem feasibility issue assume knowledge of future channel gains, transform MAC layer QoS requirements to PHY layer rate requirements d u al to get insights use Theorem 3.1 to transform to optimization over one time slot, knowledge of future channel gains no longer required OP6 - problem feasibility issue use goal programming techniques OP7 Solution transform to convex optimization problem OP2 - optimization over multiple time slots - problem feasibility issue Figure 6.4: Transformations and relationships among the optimization prblems. 120 6.4.2 Conversion of MAC Layer Requirements to PHY Layer Requirements We first transform Problem OP4 into the following Problem OP5 OP5 : max atk,m,r t k,m U∑ t=1 K∑ k=1 M∑ m=1 atk,mr t k,m (6.36) subject to K∑ k=1 M∑ m=1 atk,ms t k,m ≤ S (6.37) stk,m = ( 2r t k,m − 1 ) Γσ20 gtk,m (6.38) K∑ k=1 atk,m = 1 (6.39) atk,m ∈  {0, 1} m ∈Mt{0} m ∈ M̄t (6.40) rtk,m ∈ (0, RMAX) (6.41) U∑ t=1 M∑ m=1 atk,mr t k,m = R U,REQ k (6.42) atk,m, r t k,m can depend on the lengths and creation times of packets already in buffer k, but not on information about packets yet to arrive, k ∈ {1, 2, · · · , KRT}. (6.43) In (6.42), RU,REQk is the minimum number of bits that needs to be transmitted in time slots 1 through U to meet the delay requirements of the RT CRUs and the average data rate requirements of the NRT CRUs, and U ∈ {1, 2, · · · , T}. Note that Problem OP5 is the same as Problem OP4, except that the causality constraint (6.34) is dropped and MAC layer constraints (6.32) and (6.33) are replaced by the PHY layer constraint (6.42). To obtain more insight into Problem OP5, we consider its dual problem [30], which is 121 Problem OP2 that is described in Section 6.3.2. Based on Theorem 3.1, for CRUs who need RU,REQk bits within the next U time slots, requesting RU,REQk mCR,t/m(1, U) bits at time slot t will result in the minimum necessary power. The objective in OP4 is to maximize the system throughput subject to a fixed power constraint. The less is the power used to meet CRU QoS requirements, the more power will be left for system throughput maximization. Using Theorem 3.1, Problem OP5 (which involves a set of time slots 1, 2, · · · , U) can be transformed into a series of optimization problems, one for each time slot t, t ∈ {1, 2, · · · , U}. The values of at ∗k,m and rt ∗k,m are obtained as the optimal solution to Problem OP6, i.e. OP6 : max atk,m,r t k,m K∑ k=1 M∑ m=1 atk,mr t k,m (6.44) subject to (6.37), (6.38), (6.39), (6.40), (6.41) and M∑ m=1 atk,mr t k,m = r t,REQ k . (6.45) We have already obtain the expression for rt,REQk , k ∈ {1, 2, · · · , KRT} in (6.20) for RT. Now we obtain the expression for rt,REQk for NRT CRUs to ensure that their MAC layer QoS requirements are met. Based on Theorem 3.1, the rate requirement for NRT CRU k is set to rt,REQk = R NRT k TsymbolU mCR,t m(1, U) , k ∈ {KRT + 1, KRT + 2, · · · , K} . (6.46) Substituting m(1, U) with M̄CR,tU where M̄CR,t is the average number of available subchan- nels, we obtain rt,REQk = R NRT k Tsymbol mCR,t M̄CR,t , k ∈ {KRT + 1, KRT + 2, · · · , K} . (6.47) 122 6.4.3 A Goal Programming Approach for Improving Feasibility With the fixed system bandwidth in multiuser OFDM systems, an admission control algo- rithm can be used to limit the system load and ensure a feasible solution. However, due to the time-varying nature of the available bandwidth in CR systems, even with a conservative admission control scheme, there will be times when OP6 has no feasible solution. In order to deal with this problem, we adopt a goal programming approach [31]. There are two main steps in applying goal programming. The first step is to introduce slack and surplus variables into the constraints that may not always be satisfied, e.g. (6.45) in OP6. The second step is to rewrite the objective function as a weighted sum of the slack and surplus variables to reflect the goals of the original optimization problem, e.g. (6.44) in OP6. The procedure is now described in more details. First, constraint (6.45) is replaced by M∑ m=1 atk,mr t k,m + n t k − ptk = rt,REQk , ntk ≥ 0, ptk ≥ 0 (6.48) where ntk is the slack variable and p t k is the surplus variable for CRU k. The term n t k represents the achieved rate sum shortfall relative to rt,REQk whereas p t k represents the amount by which rt,REQk is exceeded. Second, we rewrite (6.44) as a weighted sum of the slack and surplus variables by exam- ining our original goals. Because rt,REQk is transformed from CRU MAC layer QoS require- ments, we should reach rt,REQk as close as possible, thus n t k should be minimized. On the other hand, since the bigger the value of ptk, the higher the value of system throughput, p t k should be maximized; and because the value of ptk has the same effect on system throughput, the weight for ptk can be the same for different values of k, thus, we rewrite (6.44) as min atk,m,r t k,m,n t k,p t k K∑ k=1 (wtkn t k − ptk) . (6.49) 123 In (6.49), because achieving QoS requirements of the CRUs is more important than system throughput maximization, wtk should be bigger than 1. To determine the value of w t k, we need to explore a bit more of our goals. When the system resource is not enough for all CRUs to achieve rt,REQk , the actual service each CRU receives may not be the same. In this case, to be fair to the CRUs, we maintain fair service degradation among CRUs, i.e., allocate resource so as to equalize the fractional service degradation, ctk, for all CRUs. c t k represents the fraction of service lacking for CRU k from time slot t−D+ 1 to time slot t, i.e., over a sliding window of size D, and is given by ctk =  Lt−D+1,tk , k ∈ {1, 2, · · · , KRT} 1− R̄ t−D+1,t k RNRTk , R̄t−D+1,tk < R NRT k , k ∈ {KRT + 1, KRT + 2, · · · , K} 0, R̄t−D+1,tk ≥ RNRTk , k ∈ {KRT + 1, KRT + 2, · · · , K} (6.50) where Lt−D+1,tk is the DPR of RT CRU k and R̄ t−D+1,t k is the average data rate for NRT CRU k over the sliding window. The system fairness in allocation of resources to CRUs is measured by the fairness index [32] defined by f t =  ( ∑K k=1 c t k) 2 K ∑K k=1 c t k 2 , ∃k such that ctk > 0 1, ct1 = c t 2 = · · · = ctK = 0 (6.51) f t is maximized at 1 when ct1 = c t 2 = · · · = ctK , i.e., all CRUs have the same service lacking ratio. In order to maximize f t, i.e., maintain equalized ctk for all CRUs, more weights should be given to CRUs with high ctk while less weights should be given to CRUs with low c t k, considering that wtk > 1, we use an exponential function to map the value of c t k to the value of wtk as follows wtk =  α2 β(ctk−c̄t), ctk > c̄ t − 1 β log2(α) 1 + δ, otherwise . (6.52) In (6.52), c̄t = ∑K k=1 c t k/K is the average value of c t k over all CRUs. α is a large positive 124 constant (e.g. MRMAX) so that ntk is given more consideration than p t k. This assigns greater importance to satisfying CRU rate requirements than maximizing system throughput. The parameter β is a positive constant which reflects the relative importance of achieving a high fairness index or high system throughput. The relative importance of fairness increases with β. δ is a small positive number, e.g., 10−6. Using the weights in (6.52), a CRU whose service lacking parameter ctk is higher than the average value c̄t will have a higher value of wtk. As will be shown in Section 6.4.4, a higher value of wtk gives a CRU a better chance to meets its rate requirement which promotes fairness. 6.4.4 The Cross-Layer Resource Allocation Algorithm We now describe the cross-layer RA algorithm. Since the optimization problem depends only on the parameter values in the current time slot, the time index t is dropped to simplify the notation. We thus have OP7 : min ak,m,rk,m,nk,pk K∑ k=1 (wknk − pk) (6.53) subject to (6.27), (6.28), (6.29), (6.30), (6.31), (6.48) and rMAXk − rREQk ≥ pk, k ∈ {1, 2, · · · , KRT} (6.54) In (6.48) and (6.54), rREQk is calculated using (6.20) for RT CRUs and (6.47) for NRT CRUs. In (6.53), wk is obtained using (6.52). In (6.54), r MAX k = ∑Ik i=1 bk,i where Ik is the number of packets in CRU k’s queue. Constraint (6.54) is used to ensure that resources are not wasted in practice if the buffer for RT CRU k is not backlogged, e.g. when channel conditions for CRU k are good. OP7 is a constrained nonlinear programming problem which is, in general, computa- 125 tionally complex. It is shown in Appendix B.2 that Problem OP7 can be converted to a convex problem with the following optimal solution which has a multi-level water-filling interpretation [29]. Theorem 4.1: AsmCR,t →∞, t = 1, 2, · · · , T , the optimal solution S∗ = {a∗k,m, r∗k,m, n∗k, p∗k} for Problem OP7 has the following properties 1. for any given subchannel m, the optimal subchannel assignment strategy is a∗k,m = 1, k = k ∗ and m ∈Mt a∗k,m = 0, otherwise (6.55) where k∗ = argmaxk hk,m (6.56) and hk,m =  0, λk < Γσ20 gk,m λk ln ( λk gk,m Γσ20 ) − ( λk − Γσ 2 0 gk,m ) , Γσ20 gk,m ≤ λk ≤ 2 RMAXΓσ20 gk,m λk ln(2)R MAX − (2RMAX − 1) Γσ20 gk,m , λk > 2R MAX Γσ20 gk,m (6.57) In (6.57), λk is the water level for CRU k. 2. for a given CRU k, for the subchannels with a∗k,m = 0, we have r ∗ k,m = 0; and for the subchannels with a∗k,m = 1, the optimal bit loading strategy is r∗k,m =  0, λk < Γσ20 gk,m log2 ( λk gk,m Γσ20 ) , Γσ20 gk,m ≤ λk ≤ 2 RMAXΓσ20 gk,m RMAX , λk > 2R MAX Γσ20 gk,m . (6.58) 126 3. The water level λk satisfies λk  = wkλB, n ∗ k > 0 ≤ wkλB, n∗k = 0 = λB, p ∗ k > 0, k = KRT + 1, KRT + 2, · · · , K; 0 < p∗k < r MAX k − rREQk , k = 1, 2, · · · , KRT ≥ λB, p∗k = 0 ≤ λB, p∗k = rMAXk − rREQk , k = 1, 2, · · · , KRT . (6.59) where λB ≥ 0 can be treated as the base water level for all CRUs. Proof: See Appendix B.2. Theorem 4.1 shows that if we can find the set of water levels {λk}, then we will get the optimal subchannel and bit allocation from (6.55) and (6.58). From (6.59), we can see that for CRUs with n∗k > 0, i.e., do not achieve their r REQ k , the water levels are proportional to λB. The higher the value of wk, the higher the value of λk, and from (6.55) and (6.58), the higher is the number, ∑M m=1 ak,mrk,m, of allocated bits for CRU k. We also observe that for RT CRUs with 0 < p∗k < r MAX k −rREQk , and for NRT CRUs with p∗k > 0, i.e. are allocated a larger number of bits than r REQ k , the water levels are set at λB. This property makes finding the water levels for these CRUs very simple. For CRUs with n∗k = 0 and p ∗ k = 0, i.e. just meet their r REQ k , the water levels are between λB and wkλB. Moreover, for RT CRUs that have p ∗ k = r MAX k − rREQk , i.e. the allocated number of bits is equal to the maximum number of bits in their buffers, the water levels are lower than λB. Based on the above-mentioned observations, we can devise the following two-phase cross- layer RA algorithm. When some CRUs cannot achieve their rREQk , i.e. ∃k such that n∗k > 0, the system does not have enough resources to satisfy all CRU QoS requirements and we say that the system is resource-limited. On the other hand, if all CRUs can achieve their rREQk , i.e. n∗k = 0,∀k, the system is said to be resource-abundant. 127 In Phase 1, i.e., the resource-limited phase, λB is set to a small number and all CRUs are in set A whose members have a water level of wkλB. A binary search is then used to find the appropriate value of λB that satisfies ∑K k=1 ∑M m=1 ak,msk,m = S. During each iteration of the binary search, λk is set to wkλB for CRUs that are in set A, and the Assignment algorithm performs the subchannel and bit allocation using (6.55) and (6.58). If after the allocation, the total consumed power is less than the total power constraint (i.e. ∑K k=1 ∑M m=1 ak,msk,m < S) and CRU k’s number of allocated bits is higher than its request ( ∑M m=1 ak,mrk,m > r REQ k ), CRU k is removed from set A, indicating that CRU k now has n∗k = 0 and its λk is lower than wkλB. As a result, CRU k’s water level no longer changes with λB but instead its water level changes to ensure that its number of allocated bits is equal to rREQk after the allocation. There are two possible outcomes of Phase 1: Outcome A if we find the optimal solution ( ∑K k=1 ∑M m=1 ak,msk,m = S) and Outcome B if the number of allocated bits for each CRU has reached its rREQk . In the former case, the algorithm terminates; in the latter case, the algorithm proceeds to Phase 2. In Phase 2, i.e., the resource-abundant phase, all CRUs are able to achieve their rREQk and ∑K k=1 ∑M m=1 ak,msk,m < S. Consequently, we have n ∗ k = 0 and p ∗ k ≥ 0, ∀k. In this phase, as a starting point, λB is set to the lowest water level among all CRUs, and set A contains only this CRU. At each iteration, λB is raised to the lowest water level among the CRUs that are not in set A. The Assignment algorithm is then executed: 1) the subchannel and bit allocation using (6.55) and (6.58); 2) for any CRU not in set A, the number of allocated bits is set to rREQk by adjusting λk. After the allocation, if CRU k’s water level is equal to λB, k is added to set A. This process continues until either ∑K k=1 ∑M m=1 ak,msk,m = S, which is the optimal solution, or ∑K k=1 ∑M m=1 ak,msk,m > S. In the latter case, the algorithm uses a binary search to find the appropriate value of λB that satisfies ∑K k=1 ∑M m=1 ak,msk,m = S. A flowchart for the cross-layer RA algorithm is given in Appendix C. 128 6.5 Simulation Results To evaluate the proposed algorithms, simulations were performed for the downlink of a multiuser OFDM-based CR system within a 2.5 km×2.5 km area with the CRBS located at the center. The propagation path loss is calculated using PL = A+ 10γ log10(d/d0) [33], where A = 80 dB is the path loss at the reference point, γ = 4 is the path loss exponent, d is the distance between the transmitter and the receiver and d0 = 100 m is the distance between the transmitter and the reference point. The transmit and receiver antenna gains are 30 dB and 0 dB, respectively. The multipath fading is assumed to be Rayleigh. The parameter Γ is set to 5, obtained using ln(5BER)/(−1.5) [28] with a BER of 10−4. In addition, σ20 = 10 −13 W and RMAX = 8. RT CRUs are assumed to be video conference users, each with an average data rate of RRT = 150 kbps. Each conference call consists of variable-length video packets, generated one every 40 ms, using the algorithm in [34], and dk = 90 ms for all CRUs; the packet lengths are chosen according to a self-similar distribution with an average packet length of 6 kb and a maximum packet length of 25 kb. For RRT 6= 150 kbps, the traffic is obtained by adjusting the packet lengths accordingly. The NRT CRUs are assumed to be always backlogged with 1500 bit long packets and each NRT CRU has a bit rate requirement of RNRT = 150 kbps. There are eight PUs in the system, each with a 250 kHz band (including the guard bands) for a total bandwidth of 2 MHz. For the CR system, the 2 MHz bandwidth is used to support 64 OFDM subchannels. The duration of each OFDM symbol is Tsymbol = 40 µs, the guard interval is 8 µs, and Tstate = 250 time slots. It is assumed that the subchannel gain, g t k,m, is constant over 1 ms periods, i.e. 25 OFDM symbols and independent from period to period. RA is performed once every 1 ms. The simulation parameters are listed in Table 6.1. 129 Cell size 2.5 km ×2.5 km Number of CRUs K = 8 Reference point path loss A = 80 dB Distance of the reference point to the CRBS 100 m Path loss exponent γ = 4 Transmit antenna gain 30 dB Receiver antenna gain 0 dB Multi-path fading Rayleigh Total bandwidth W = 2 MHz Number of PU bands L = 8 Widths of the PU bands W1 = W2 = · · · = W8 = 250 kHz Number of OFDM subchannels M = 64 Duration of OFDM symbol Tsymbol = 40 µs Duration of guard interval 8 µs Noise power, σ20, of each subchannel 10 −13 W Γ 5 RMAX 8 Average video data rate RRT = 150 kbps Packet generation rate 25 packets/second Average packet length 6 kb Maximum packet length 25 kb Packet delay requirement dk = 90 ms Maximum tolerable DPR 0.05% Simulation length for each set of 2, 500, 000 time slots values of pn, pa, and R RT Number of packets generated 2500 Table 6.1: Simulation parameters. 130 6.5.1 RT Services We considered an eight RT CRUs system to assess the performance of the proposed sub- optimal algorithm in Section 6.3.3. The distances of CRU 1, CRU 2,..., and CRU 8 to the CRBS are chosen as 1.2 km, 2.0 km, 2.4 km, 1.8 km, 1.6 km, 0.8 km, 1.4 km, and 1.9 km, respectively. The CRU video conference sessions start at 10 ms, 20 ms, 20 ms, 30 ms, 30 ms, 40 ms, 10 ms, and 20 ms, respectively. To compare the proposed algorithm with existing algorithms designed for OFDM systems, we also simulated the MAC layer solution in [19] and the PHY layer solution proposed in [6] for multiuser OFDM systems. The MAC layer solution uses equal power allocation and treats sets of eight adjacent subchannels as a group, each of which is allocated to the CRU with the best channel gain relative to its own average channel gain to maximize system throughput. An urgency function is introduced to improve on-time delivery of the packets; this function gives higher priority to CRUs as their HOL packets approach their respective deadlines. The power used in the MAC solution was adjusted to achieve a DPR target value of 0.05%. In the PHY layer solution, rREQk is set to 150 kbps × 40 µs = 6 bits during each time slot. To help better understand the proposed algorithm, as an example, we refer to Fig. 6.5 which shows the number of available subchannels, mCR,t, requested rate, and required power for CRU 4 for the proposed, PHY, and MAC algorithms from time 260 ms to 310 ms with pn = pa = 0.99 and R RT = 150 kbps. Since pn = pa, the average number, M̄CR,t, of available subchannels is 32. Fig. 6.5 shows that the requested rate, i.e., rt,REQ4 , is always set to the average require- ment at 150 bits per 1 ms in the PHY solution. In this example, it was always possible to achieve a rate equal to the requested rate for all three algorithms. For the proposed algorithm, rt,REQ4 changes with mCR,t. For example, from time 260 to 270, mCR,t = 40, a relatively high value compared to M̄CR,t and r t,REQ 4 for the proposed algorithm is set to a 131 260 265 270 275 280 285 290 295 300 305 310 16 24 32 40  m CR ,t 260 265 270 275 280 285 290 295 300 305 310 0 100 200 300 400 500 600 700 800 R at es  (b its )   MAC PHY Proposed 260 265 270 275 280 285 290 295 300 305 310 0 1 2 3 4 5 6 7 8 Time (ms) R eq ui re d po w er  (W )   MAC PHY Proposed Figure 6.5: Resource allocation time diagram for CRU 4 with pn = pa = 0.99 and R RT = 150 kbps. 132 value that is higher than 150 bits; from time 285 to 289, mCR,t is only 16 and r t,REQ 4 is set to a lower value. This behaviour follows from (6.20) since rt,REQk is proportional to mCR,t. Theorem 3.1 indicates that setting rt,REQk according to mCR,t minimizes power, and it is thus expected that the proposed algorithm will require less average power than PHY. It might seem surprising that at time 267, PHY requires 0.3 W whereas the proposed algorithm re- quires 0.5 W. But this is because the proposed algorithm has its rt,REQ4 set to a value higher than 150 bits. At time 288, mCR,t is relatively low and PHY requires a power of 1 W to obtain 150 bits, while the proposed algorithm sets rt,REQ4 to a value lower than 150 bits in order to use a power of only 0.51 W, a level which is not very different from that at time 267. For times 267 and 288, the proposed algorithm requires an average power of 0.51 W, while PHY requires 0.65 W. Since the MAC solution assigns eight subchannels as a group to CRUs, CRU 4 is not able to obtain a subchannel assignment at each time; hence when it obtains subchannel assignment, the rate requirement needs to be high in order to achieve an average rate of 150 kbps. As a result, the power required at each time is also higher. This power allocation strategy is far from optimal as can be seen from Fig. 6.5. From time 260 to 310, the average required power per transmitted bit is 0.0037 W, 0.0047 W, and 0.0077 W for the proposed, PHY, and MAC algorithms, respectively. Fig. 6.6 shows the average total power for the eight CRUs as a function of pn with pn = pa. For each value of pn, 2, 500, 000 time slots were simulated. The variance of MCR,t decreases with pn. It can be seen that the proposed algorithm performs much better than the MAC and PHY algorithms. The CRBS transmit power reduction over the PHY solution increases as pn decreases, which shows that the proposed cross-layer algorithm adapts well to the channel variations. The improvement over the PHY solution is over 40% for pn < 0.5, and about 30% for pn = 0.9. The improvement over the MAC solution increases with pn. The reason is that as pn increases, mCR,t stays the same over a longer period of time; when mCR,t is low, to maintain a reasonable DPR value, the required power has to be quite high. 133 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probability of PUs staying in the same state,  p n A ve ra ge  to ta l p ow er  (W )   MAC PHY Proposed Figure 6.6: Average total power of eight video conference CRUs as a function of pn with pn = pa and R RT = 150 kbps. 134 The improvement of the proposed algorithm over the MAC solution is over 65% for pn < 0.5, and nearly 75% for pn = 0.9. The DPR values for the three simulated algorithms for different pn values are listed in Table 6.2. Table 6.2 clearly indicates that DPR is the lowest when using the proposed algorithm, and is the highest when using the MAC layer solution. Note that a DPR value of 0.04% corresponds to one packet dropped among the 2500 packets simulated. pn 0 0.09 0.5 0.9 0.99 0.999 1.00 Proposed 0 0 0 0.04% 0 0 0 PHY 0.04% 0.04% 0.04% 0.04% 0.04% 0.04% 0.44% MAC 0.04% 0.04% 0.04% 0.04% 0.84% 0.92% 0.88% Table 6.2: Dropped packet rates for different values of pn with pn = pa and R RT = 150 kbps. The average power of each CRU is shown in Fig. 6.7 with pn = pa = 0.9 and R RT = 150 kbps. It can be seen that the proposed algorithm requires less average power for all eight CRUs compared to PHY and MAC solutions. CRU 3 requires the highest average power among the 8 CRUs for any of the three algorithms, because it is farthest away from the CRBS and thus has the worst average channel condition. CRU 6 requires the lowest average power because it is closest to the CRBS. The MAC solution assumes equal power on all subchannels, although it takes into account the channel conditions of the CRUs when assigning subchannels. As a result, power is wasted especially for CRUs with very good channel conditions, e.g., for CRUs 6 and 1. Fig. 6.8 and Table 6.3 illustrate the average total power for the eight CRUs and DPR as a function of RRT with pn = pa = 0.5. It can be seen that as the system load (represented by RRT ) increases, the total required CRU power increases. The proposed algorithm performs better than the other two algorithms regardless of the system load. The improvement over the PHY solution is 28% for RRT = 75 kbps, and is about 45% for RRT = 375 kbps. The improvement over the MAC layer solution is over 60% for all values of RRT . Fig. 6.9 and Table 6.4 illustrate the average total power for the eight CRUs and DPR as 135 CRU 1 CRU 2 CRU 3 CRU 4 CRU 5 CRU 6 CRU 7 CRU 8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 A ve ra ge  p ow er  (W )   Proposed PHY MAC Figure 6.7: Transmit CRU power with pn = pa = 0.9 and R RT = 150 kbps. Data rate (kbps) 75 150 225 300 375 Proposed 0.0% 0.0% 0.0% 0.04% 0.04% PHY 0.04% 0.04% 0.04% 0.04% 0.04% MAC 0.04% 0.04% 0.04% 0.04% 0.04% Table 6.3: Dropped packet rate with respect to video data rate, RRT with pn = pa = 0.5 136 75 150 225 300 375 0 0.5 1 1.5 2 2.5 Video data rate (kbps),  RRT A ve ra ge  to ta l p ow er  (W )   MAC PHY Proposed Figure 6.8: Average total power for eight video conference CRUs as a function of video data rate, RRT with pn = pa = 0.5. 137 75 150 225 300 375 0 0.5 1 1.5 2 2.5 Video data rate (kbps),  RRT A ve ra ge  to ta l p ow er  (W )   MAC PHY Proposed Figure 6.9: Average total power for eight video conference CRUs as function of video data rate, RRT with pn = 0.5 and pa = 0.1. a function of RRT with pn = 0.5 and pa = 0.1. As in Fig. 6.8 and Table 6.3, the proposed algorithm performs better than PHY and MAC. The improvement over the PHY solution is over 20% for RRT = 75 kbps and over 30% for RRT = 375 kbps. The improvement over the MAC layer solution is over 60% for all values of RRT . In Fig. 6.8 and Table 6.3, M̄CR,t is 32, whereas M̄CR,t is 41 in Fig. 6.9 and Table 6.4 since the probability, pa, of a PU staying in the active state is reduced from 0.5 to 0.1. As a result, the required power for the proposed, PHY, and MAC solutions are all lower than in Fig. 6.8. The DPRs for the proposed, PHY, and MAC solutions in Fig. 6.9 are similar to those in Fig. 6.8. 138 Data rate (kbps) 75 150 225 300 375 Proposed 0.0% 0.0% 0.0% 0.0% 0.0% PHY 0.04% 0.04% 0.04% 0.04% 0.04% MAC 0.04% 0.04% 0.04% 0.04% 0.04% Table 6.4: Dropped packet rate with respect to video data rate, RRT . pn = 0.5 and pa = 0.1 6.5.2 Mixed Services To evaluate the proposed algorithm for mixed services, we consider a system with four file transfer users representing NRT CRUs, and six video conference users representing RT CRUs. The power limit of the CRBS is S = 12.8 W. The distances of CRUs 1 to 6 (the RT CRUs) to the CRBS are chosen as 1.2 km, 2.0 km, 2.4 km, 1.8 km, 1.6 km, and 0.8 km. The distances of CRUs 7 to 10 (the NRT CRUs) to the CRBS are chosen as 1.4 km, 1.9 km, 1.0 km, and 2.2 km. The RT CRUs video conference sessions start at 10 ms, 37 ms, 3 ms, 25 ms, 30 ms, 18 ms, respectively. It is assumed that the subchannel gain gtk,m is constant during 1 ms periods, i.e. 25 OFDM symbols. Of the 25 OFDM symbols, 15 are used for downlink transmission and 10 are used for uplink transmission. Thus, resource allocation is performed once every 1 ms. In the simulations, OP7 is solved using the algorithm proposed in Section 6.4.4 with α = 512 and β = 20. To illustrate the operation of the algorithm proposed in Section 6.4.4, we plot in Fig. 6.10 the RA time diagram for CRUs 3 and 6 from time 120 ms to 200 ms with pa = pn = 0.9 and RRT = 600 kbps. The number of available subchannels to the CR system is also shown in Fig. 6.10. Recall that CRU 3 has the worst average channel condition among the 10 CRUs. Con- sequently, it receives little extra rate beyond rt,REQ3 . Based on Theorem 3.1, CRU 3’s rate request was dynamically calculated based on (6.20) to use the least possible power so as to preserve power for system throughput maximization. It can be seen that the rt,REQ3 curve follows closely that for mCR,t. The biggest difference is at time 163, when a new packet ar- 139 120 130 140 150 160 170 180 190 200 24 32 40 48 0 150 300 450 600 0 300 600 900 1200 1500 1800 2100 2400 2700 Time  t (ms)  m CR ,t                                                                                  N um be r o f    B its /m s         B its /m s           su bc ha nn el s   fo r C RU  3        fo r C RU  6             Rate request of CRU 6 Rate allocation of CRU 6 Rate request of CRU 3 Rate allocation of CRU 3 Available subchannels Figure 6.10: Resource allocation time diagram for CRUs 3 and 6. rives. Since this new packet is relatively long, using r162,REQ3 cannot ensure on-time delivery of the packet and hence r163,REQk is adjusted to a higher value based on (6.20). Recall that CRU 6 enjoys the best average SNR among the 10 CRUs. Most of the time, it achieves a bit rate that is higher than rt,REQ6 . Consequently, most of its packets are delivered before its next packet arrives. For example, the first packet arrives at time 138 and is delivered at time 160 before the next packet arrives. In order not to waste system resources, rt,REQ6 is set to 0 from time 160 to time 177 since its queue is empty. At time 178, the next packet arrives and r178,REQ6 is set to a non-zero value which can ensure on-time delivery of this packet. To compare the proposed algorithm with existing algorithms designed for OFDM systems, 140 the following algorithms were also simulated. (1) PHY refers to the solution for Problem OP4 with rt,REQk set as follows: r t,REQ k = RRTTsymbol for RT CRUs, and r t,REQ k = R NRTTsymbol for NRT users. (2) MAC refers to the MAC layer solution in [19]. It uses equal power allocation and treats sets of eight adjacent subchannels as a group, each of which is allocated to the CRU with the best channel gain relative to its own average channel gain. An urgency function, which gives higher priority to CRUs as their HOL packets approach their deadlines, is introduced to improve on-time delivery of packets. (3) HLL refers to the cross-layer algorithm in [25] applied to Problem OP4 with both RT CRU delay and NRT CRU average rate requirements converted to fixed rate requirements as follows: rt,REQk = 1.268R RTTsymbol for RT CRUs, and r t,REQ k = R NRTTsymbol for NRT CRUs. (4) JJJ refers to the cross-layer solution in [24] which uses equal power subchannel allo- cation. RT CRU packet delays are improved by giving higher priority to CRUs whose HOL packet’s relative waiting time has exceeded a predefined threshold. The remaining system resources are allocated to CRUs with better average channel conditions among remaining RT CRUs and NRT CRUs. A frame structure is assumed, with each frame lasting 1 ms and containing 15 OFDM symbols. Both PHY and HLL suffer from the problem feasibility issue, although they work quite well for OFDM systems. A commercial optimization software was used to test the feasibility of PHY and HLL solutions. The results, shown in Table 6.5, show that as the system load (represented by RRT ) increases, the feasibility of PHY and HLL solutions decreases. At RRT = 450 kbps, when the overall QoS requirements of the RT and NRT CRUs can be supported by the available system resources1, the non-feasibility ratio, defined as the fraction of RAs with no feasible solution, is 64% for HLL and 35% for PHY. These high non-feasibility ratio values indicate that HLL and PHY are unsuitable for CR systems. 1Fig. 6.11 shows that for RRT = 450 kbps, the DPR for the proposed algorithm, JJJ, and MAC are all below 0.03. 141 RRT (kbps) 75 150 225 300 375 450 525 600 PHY 16% 19% 24% 36% 39% 64% 85% 92% HLL 8% 9% 19% 20% 32% 35% 63% 65% Table 6.5: Non-feasible ratio of PHY and HLL. For the proposed algorithm in Section 6.4.4, MAC, and JJJ, solutions (assuming the number of bits can take on a real value) were obtained. Figs. 6.11, 6.12, 6.13, and Table 6.6 show the DPR, average throughput for NRT CRUs, the system throughput, and fairness index of the three algorithms as a function of video data rate RRT . For each value of RRT , 100, 000 RAs were simulated. RRT (kbps) 75 150 225 300 375 450 525 600 Proposed 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 MAC 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9991 0.9968 JJJ 1.0000 0.9999 0.9686 0.9282 0.9105 0.8795 0.8422 0.8123 Table 6.6: Fairness index comparison for three different schedulers. Figs. 6.11 and 6.12 show that the DPR increases and the average throughput for NRT CRUs decreases with RRT . As the RT CRU rate increases, the system resources needed to satisfy the RT CRU and NRT CRU QoS requirements also increase, leaving less power for the NRT CRUs to increase their throughput. It can be seen that the proposed algorithm performs much better than JJJ or MAC. At RRT = 525 kbps, for RT CRUs, the DPR for the proposed algorithm is less than 1%, but is over 4% for MAC and nearly 6% for JJJ; for NRT CRUs, the average throughput improvement of the proposed algorithm over MAC and JJJ is 57% and 55%, respectively. There are two reasons for the performance improvement achievable with the proposed algorithm compared to MAC and JJJ. First, the proposed algorithm is optimal whereas MAC and JJJ are both suboptimal. Second, both MAC and JJJ suffer from the false urgency issue in CR systems. The urgency mechanism used in MAC and JJJ to improve the on- time delivery of RT CRU packets assumes a fixed amount of available system resources. This 142 75 150 225 300 375 450 525 600 0 0.02 0.04 0.06 0.08 0.1 0.12 Video data rate (kbps),  RRT D ro pp ed  P ac ke t R at e   Proposed JJJ MAC Figure 6.11: Dropped packet rate of RT CRUs as a function of video bit rate. 143 75 150 225 300 375 450 525 600 100 200 300 400 500 600 700 800 900 Video data rate (kbps),  RRT A ve ra ge  th ro ug hp ut  fo r N RT  C RU  (k bp s)   Proposed JJJ MAC Figure 6.12: Average throughput of NRT CRUs as a function of video bit rate. 144 75 150 225 300 375 450 525 600 2.6 2.8 3 3.2 3.4 3.6 3.8 4 Video data rate (kbps),  RRT Th ro ug hp ut  (M bp s)   Proposed JJJ MAC Figure 6.13: System throughput as a function of video bit rate. 145 results in inaccurate information about a packet’s urgency in a CR system with time-varying system resources. At times when there are more available subchannels, some RT CRUs with poor channel qualities might be given unnecessarily frequent transmission opportunities. This lowers the overall system throughput. On the other hand, when the number of available subchannels is low, many packets may fail to meet their deadlines, resulting in a high DPR. Fig. 6.13 shows that the proposed algorithm yields a significantly higher system through- put than JJJ or MAC. The improvement is 52% over MAC and 36% over JJJ at RRT = 75 kbps, and decreases to 15% over both JJJ and MAC at RRT = 600 kbps. The system throughput for the proposed algorithm decreases for RRT > 450 kbps because for high video data rates, system resources may not always be sufficient to ensure on-time delivery for all RT packets. To ensure fair degradation, CRUs that have high DPR values are given high wk values based on (6.52); these correspond to high water levels in Phase 1 of the algorithm based on (6.59) for n∗k > 0. So even though their subchannel conditions are poor, they are able to obtain some subchannels based on (6.55) because of their high water levels. The drawback of using this fair degradation mechanism is a lower system throughput. Table 6.6 shows that the proposed algorithm provides a slightly better fairness index than MAC, and a much higher fairness index than JJJ. JJJ yields a poor fairness index because the NRT CRUs are always given low priority: they are not allocated subchannels unless their average subchannel conditions are relatively high. In the simulations, NRT CRUs 8 and 10 have relatively low average subchannel gains. For RRT = 450 kbps, the average bit rates for the NRT CRUs with JJJ are 502 kbps, 78 kbps, 465 kbps, and 1 kbps, i.e., 335%, 52%, 310%, and 1% of their requested bit rates. Thus, CRUs 8 and 10 are harshly penalized. 6.6 Conclusions Cross-layer RA algorithms combining packet scheduling with subchannel, bit, and power allocation on the downlink of a multiuser OFDM-based CR system were proposed, one for 146 RT applications and one for a mixture of RT and NRT applications. 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Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” IEEE Journal on Selected Areas in Communications, vol. 7, no. 7, pp. 1205–1211, July 1999. [34] M. W. Garrett and W. Willinger, “Analysis, modeling and generation of self-similar VBR video traffic,” ACM SIGCOMM Computer Communication Review, vol. 24, no. 4, pp. 269–280, October 1994. 150 Chapter 7 Resource Allocation for Non-Real-Time Services in OFDM-based Cognitive Radio Systems 7.1 Introduction Cognitive radio (CR) is a concept which may be used to alleviate the looming spectrum shortage crisis [1]. As discussed in [2], orthogonal frequency division multiplexing (OFDM) is an attractive modulation candidate for CR systems. The subcarrier, bit, and power allocation problem for non-real-time (NRT) users in mul- tiuser OFDM systems, subject to total power and user proportional rate (PR) constraints during each OFDM symbol duration, has been studied in [3, 4]. In these papers, suboptimal solutions are proposed that are close to optimal when the number of subcarriers is much greater than the number of users. These algorithms assume that the PR constraint has to be satisfied for each OFDM symbol duration; however, this may not be necessary for most NRT applications because some delay can usually be tolerated. More importantly, in contrast to conventional OFDM systems, in a CR system, the number of available subcarriers is time- varying. When the number of CR users (CRUs) exceeds the number of available subcarriers, and making the usual assumption that each subcarrier is used at any given time by at most one CRU, it may not be possible to maintain PR for all CRUs during each OFDM symbol duration. In this chapter, we take into account the time-varying nature of available spectrum 1A paper based on the material in this chapter has been accepted for publication. Yonghong Zhang and Cyril Leung, “Resource Allocation for Non-Real-Time Services in OFDM-based Cognitive Radio Systems,” IEEE Communications Letters. 151 resources in an OFDM-based CR system, and propose a resource allocation (RA) algorithm for NRT applications which provides statistical PR among CRUs. 7.2 System Model We consider a CR system with a total bandwidth of W Hz and L primary users (PUs). Frequency bands actually carrying PU signals are referred to as active; non-active bands are also termed spectrum holes. It is assumed that some subchannels adjacent to active PU bands are not used in order to reduce the cross-channel interference [5] generated by the CRUs to the PUs to acceptable levels. Thus, in this chapter, we ignore mutual cross-channel interference between PUs and CRUs. Assume that there is one CR base station (CRBS) communicating with K CRUs. We are interested in downlink transmissions from the CRBS to the CRUs. The CRBS and the CRUs are able to accurately locate the spectrum holes. There are M OFDM subbands (or subchannels) in the system, each with noise power σ20. Interference among the subchannels is assumed to be negligible. The system is time-slotted with a slot duration equal to an OFDM symbol Tsymbol. The subchannels are modelled in discrete-time, with the time-varying gain for subchannel m and time slot t from the CRBS to CRU k denoted by √ gtk,m. It is assumed that the power gains {gtk,m} are outcomes of independent random variables (rv’s); furthermore, for any given value of k, the gain rv’s {Gk,m,m = 1, 2, · · · ,M} are identically distributed, with probability density function (PDF) fGk(gk) and cumulative distribution function (CDF) FGk(gk). The number of bits, rtk,m, per OFDM symbol which can be supported by subchannel m of CRU k in time slot t is given by [6] rtk,m = log2 ( 1 + gtk,ms t k,m Γσ20 ) (7.1) 152 where stk,m is the transmit power and Γ is an SNR gap parameter which indicates how far the system is operating from capacity. The availability of a PU band is modelled by a two-state Markov chain. During a time slot t, a PU band can be in one of two modes: active or inactive [7]. A mode transition may occur once every Tstate slots, with a probability 1 − pa of changing from active to inactive mode and a probability 1 − pn of changing from inactive to active mode. The number of available subchannels in time slot t is denoted by mCR,t. The CRUs are NRT users, with CRU k having a nominal rate requirement of RPRk . The rate, R1,Tk , for CRU k within T time slots is required to satisfy R1,T1 RPR1 = R1,T2 RPR2 = · · · = R 1,T K RPRK = d (7.2) where d is the service share for each CRU. In [3, 4], it is implicitly assumed that T = 1. This is overly stringent for most NRT applications which can tolerate some delays. Furthermore, in a CR system in which each subchannel can accomodate at most one CRU at a time, it is not possible to satisfy (7.2) for T = 1 when mCR,t < K. We thus consider the following optimization problem. OP1 : max atk,m,r t k,m T∑ t=1 K∑ k=1 M∑ m=1 atk,mr t k,m (7.3) 153 subject to T∑ t=1 K∑ k=1 M∑ m=1 atk,m ( 2r t k,m − 1 ) Γσ20 gtk,m ≤ TS (7.4) K∑ k=1 atk,m = 1, a t k,m ∈  {0, 1},m ∈Mt{0},m ∈ M̄t (7.5) T∑ t=1 M∑ m=1 atk,mr t k,m = dR PR k (7.6) atk,m, r t k,m can depend on {guk,m, u ≤ t}, but not on {guk,m, u > t}. (7.7) In (7.4), S is the average total allowed power per time slot and (2r t k,m− 1)Γσ20/gtk,m is the power necessary for the OFDM symbol to support a rate of rtk,m as can be seen from (7.1). In (7.5),Mt (M̄t) ⊆ {1, 2, · · · ,M} denotes the set of available (unavailable) subchannels at time t; this constraint also implies that each subchannel can be used by at most one CRU in any given time slot. Constraint (7.6) follows from (7.2) since R1,Tk = ∑T t=1 ∑M m=1 a t k,mr t k,m. Condition (7.7) reflects the causality constraints for the problem. 7.3 Resource Allocation Algorithm Problem OP1 is hard to solve as it involves the causality constraint (7.7). Instead, we consider Problem OP2, which is the same as OP1 except that constraint (7.7) is dropped. The solution to OP2 thus provides an upper bound on that for Problem OP1. OP2 is a constrained nonlinear optimization problem which is still computationally com- plex. We consider the convex optimization problem OP3 obtained by letting atk,m take on a real value in [0, 1] and substituting rtk,m by r̂ t k,m/a t k,m. Using the Karush-Kuhn-Tucker (KKT) conditions [8], it can be shown that as TM →∞ the optimal solution to OP3 has a multi-level water-filling interpretation [9] with the following characteristics. 154 1) For any given subchannel m and time slot t, the optimal subchannel assignment strategy is at ∗k,m =  1, k = k ∗ and m ∈Mt 0, otherwise (7.8) where k∗ = argmaxk h t k,m and htk,m = [ λk ln ( λk gtk,m Γσ20 )]+ − [ λk − Γσ 2 0 gtk,m ]+ . (7.9) In (7.9), [x]+ ∆ = max (0, x) and λk is the global waterlevel over all time slots of subchannels assigned to CRU k. 2) For a given CRU k, for the subchannels with at ∗k,m = 0, we have r t ∗ k,m = 0; when at ∗k,m = 1, the optimal bit loading strategy is rt ∗k,m = [ log2 ( λk gtk,m Γσ20 )]+ . (7.10) The derivation of the above results is provided in Appendix D. Since the variables {at ∗k,m} assume only values 0 or 1, the optimal solution to OP3 is also optimal for OP2. Characteristic 1) shows that subchannel m in time slot t should be allocated to the CRU k with the highest value of htk,m. Characteristic 2) shows that for the subchannels allocated to CRU k, no power is loaded if the equivalent noise, Γσ20/g t k,m, is higher than the waterlevel; otherwise, the power loaded is λk − Γσ20/gtk,m. We note that if we know the waterlevels {λk, k = 1, 2, · · · , K} that satisfy (7.2) and (7.4), then at each time slot t, we can simply use (7.8) and (7.10) to perform the subchannel and bit allocation. However, in Problem OP1, we do not have these waterlevels since we do not know the channel gains {guk,m, u > t} exactly. Nevertheless, as TM → ∞, we can estimate the waterlevels Λ = {λ∞k,m, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} using statistical information about the subchannel gains. In the following, we devise an algorithm for finding the set Λ 155 of waterlevels. When Gk = gk, the probability that CRU k is allocated a given subchannel is pk(gk) = ∏ i 6=k FGi(h −1 i (hk(gk))) (7.11) where h−1i is the inverse of hi and hk is the same as h t k,m in (7.9) with superscript t and sub- script m dropped since for a given CRU k, Gtk,m are independent and identically distributed rv’s. The term FGi(h −1 i (hk(gk))) represents the probability that CRU i has a value of hi that is lower than hk(gk), i.e., the probability that CRU i will not be allocated the subchannel. When there are a total of m subchannels available to all CRUs, the average number of bits that are loaded for CRU k is given by R∞k (m) = m ∫ ∞ Γσ20 λk pk(gk)fGk(gk) log2 ( λkgk Γσ20 ) dgk (7.12) and the average total power used is S∞(m) = m K∑ k=1 ∫ ∞ Γσ20 λk pk(gk)fGk(gk) ( λk − Γσ 2 0 gk ) dgk . (7.13) The basic idea for finding Λ is to iteratively increase the waterlevel for the CRU with the lowest service share by a step size, δ, until S∞(m) is close to S and (7.2) is nearly satisfied. The value of δ is dynamically adjusted to provide rapid convergence to accurate values of {λ∞k,m}. In the first round, δ is set to 1/C, where C > 1 is typically a number between 2 to 4. We then increase λk′ by δλ̃k′ where k ′ = argminkR ∞ k (m)/R PR k and λ̃k′ is the estimated initial waterlevel for CRU k′. This process continues until S∞(m) > S, which indicates that some values of λk are higher than λ ∞ k,m. To address this excess power problem and to ensure that some users do not get unfairly high service shares, we lower each λk to λk − δλ̃k′ and go to the next round with a smaller step size δ = δ/C. We proceed by decreasing the step size at each round until S∞(m) is close to S and (7.2) is nearly satisfied or δ < ², where ² is 156 some small positive number. A pseudo-code listing of the algorithm is shown below. Algorithm 5 Waterlevel finding algorithm. 1) Initialize {λk = ², λ̃k = S/M, k = 1, 2, · · · , K}. 2) For each available subchannel m from M down to 1, do steps 3) and 4). 3) Set δ = 1/C, repeat 3.1) and 3.2) until S∞(m) = S and (7.2) or δ < ². 3.1) Calculate R∞k (m) and S ∞(m) using (7.12) and (7.13). 3.2) If S∞(m) > S, λk = λk − δλ̃k, k = 1, 2, · · · , K, δ = δ/C; otherwise, set λk′ = λk′ + δλ̃k where k ′ = argminkR ∞ k (m)/R PR k . 4) Set λ∞k,m = λk, λ̃k = λk, k = 1, 2, · · · , K. In Step 3), during each round, the number of iterations required is roughly CK, since each CRU needs about C iterations to reach its waterlevel. Since the maximum number of rounds necessary is logC(1/²), the number of iterations required for Step 3) to converge is Nit = CK logC(1/²). Since Steps 3.1) and 3.2) has complexity O(K) and needs CK logC(1/²) iterations to converge, the complexity of Step 3) is O(K2). The complexity of Algorithm 5 is O(K2M) since Step 3) is executed M times. We note that Algorithm 5 does not need to be executed at each time slot t; it is re-run only when the subchannel gain distributions or power limit S change. Once the waterlevels λ∞k,m, k = 1, 2, · · · , K,m = 1, 2, · · · ,M are obtained using Algorithm 5, the subchannel and bit allocation in time slot t proceeds as follows: Algorithm 6 Subchannel and bit allocation algorithm. 1) Set λk = λ ∞ k,mCR,t , k = 1, 2, · · · , K. 2) Allocate subchannels and bits according to (7.8) and (7.10). The computational complexity of Algorithm 6 is O(KM), which is similar to the al- gorithms in [3] and [4]. The proposed algorithm provides a statistical proportional rate solution whereas the algorithms in [3, 4] provide a solution designed to meet power and rate constraints at each time slot. 157 7.4 Simulation Results To assess the performance of the proposed algorithm, we consider the downlink of a multiuser OFDM-based CR system operating within a 2.5 km ×2.5 km cell with the CRBS located in the middle of the cell. The propagation path loss exponent is chosen as 4, the variance of the shadowing effect is 10 dB, and the multipath fading is assumed to be Rayleigh [10]. The number, K, of CRUs in the system varies from 8 to 64 and is increased by adding a group of eight CRUs at a time. The average subchannel gains of the eight CRUs in a group are 0.043× 10−12, 0.062× 10−12, 0.077× 10−12, 0.153× 10−12, 0.260× 10−12, 0.482× 10−12, 1.00 × 10−12 and 2.44 × 10−12. In addition, Γ = 5, σ20 = 10−16 W, S = 1 W, and T = 10, 00 slots (0.4 S). There are eight PUs in the system, each with a 250 kHz band (including the guard bands) for a total bandwidth of 2 MHz. For the CR system, the 2 MHz bandwidth is used to support 64 OFDM subchannels. The duration of each OFDM symbol is Tsymbol = 40 µs, Tstate = 1 slot and pa = pn = 0.5 so that M̄CR = 0.5M . For comparison with existing algorithms (designed for multiuser OFDM systems), we also simulated the algorithms proposed in [3] and [4], hereafter referred to as SAE and MB, respectively. To assess fairness in the service share of CRUs, we use the fairness index in [11] defined as FI = (∑K k=1 R1,Tk RPRk )2 / [ K ∑K k=1 ( R1,Tk RPRk )2] . When all CRU service shares, R1,Tk /R PR k , k = 1, 2, . . . , K, are exactly the same, FI attains its maximum value of 1. K 8 16 24 32 40 48 56 64 Proposed 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 SAE 1.0 1.0 1.0 0.98 0.91 0.80 0.70 0.61 MB 1.0 0.99 0.97 0.91 0.81 0.70 0.61 0.53 Table 7.1: Fairness index. Fig. 7.1 and Table 7.1 show the average throughput and fairness index as a function of the number, K, of CRUs. It can be seen that the proposed algorithm performs better than SAE and MB in terms of both throughput and fairness index. The relative improvement 158 8 16 24 32 40 48 56 64 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8  K, number of CRUs A ve ra ge  th ro ug hp ut   (   ×  M bp s)   Proposed SAE MB Figure 7.1: System throughput with respect to number of CRUs with RPRk = 1, k = 1, 2, · · · , K. 159 increases with K. The throughput improvement of the proposed algorithm over SAE is 8% at K = 8 and 37% at K = 64; the improvement over MB is 13% at K = 8 and 6% at K = 64. The throughputs of both the proposed and MB algorithms increase with K due to the benefit of multiuser diversity. SAE also benefits from multiuser diversity when K << M̄CR (e.g., at K = 8 and K = 16); however, as K increases beyond 16, the throughput decreases with K. This is because SAE is designed for the case K << mCR,t. When this is not the case, the subchannel allocation becomes almost random, resulting in a lower throughput. The FI values for the proposed, SAE and MB algorithms are close to 1 when the number, K, of users is small. As K increases, the FI values for SAE and MB decrease. The reason for this is as follows. Although SAE and MB try to satisfy (7.2) among the CRUs which are allocated at least one subchannel, they cannot provide any rate to CRUs which are not allocated any subchannel. As K increases, so does the number of CRUs with no subchannel allocation and this results in a lower FI value. As an example, for K = 64, the maximum CRU throughputs are 107 kbps (SAE) and 199 kbps (MB), whereas the minimum CRU throughputs are only 245 bits (SAE) and 46 bits (MB). 7.5 Conclusions A resource allocation (RA) algorithm for NRT applications in a multiuser OFDM-based CR system was proposed. Unlike existing RA algorithms designed for multiuser OFDM systems, which are unable to guarantee proportional rate allocation in a CR system when the number of CRUs exceeds the number of available subchannels, the proposed algorithm provides statistically proportional rates as well as an improved system throughput. These benefits are achieved by relaxing the per time-slot power and rate constraints. 160 References [1] Federal Communications Commission, “FCC adopts rule changes for smart radios,” Cognitive Radio Technologies Proceeding (CRTP), ET Docket No. 03-108, 2005. [2] T. A. Weiss and F. K. Jondral, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency,” IEEE Communications Magazine, vol. 42, no. 3, pp. S8–S14, March 2004. [3] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2726–2737, November 2005. [4] C. Mohanram and S. Bhashyam, “A sub-optimal joint subcarrier and power allocation algorithm for multiuser OFDM,” IEEE Communication Letters, vol. 9, no. 8, pp. 685– 687, August 2005. [5] T. Weiss, J. Hillenbrand, A. Krohn, and F. K. Jondral, “Mutual interference in OFDM- based spectrum pooling systems,” in Proc. of IEEE 59th Vehicular Technology Confer- ence (VTC 2004-Spring), vol. 4, Milan, Italy, May 2004, pp. 1873–1877. [6] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading chan- nels,” IEEE Transactions on Communications, vol. 45, no. 10, pp. 1218–1230, October 1997. [7] H. Su and X. Zhang, “Cross-layer based opportunistic MAC protocols for QoS pro- visionings over cognitive radio wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 1, pp. 118–129, January 2008. [8] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge Univer- sity Press, 2004. [9] R. S. Cheng and S. Verdu, “Gaussian multiaccess channels with ISI: capacity region and multiuser water-filling,” IEEE Transactions on Information Theory, vol. 39, no. 3, pp. 773–785, May 1993. [10] V. Erceg, L. J. Greenstein, S. Y. Tjandra, S. R. Parkoff, A. Gupta, B. Kulic, A. A. Julius, and R. Bianchi, “An empirically based path loss model for wireless channels in suburban environments,” IEEE Journal on Selected Areas in Communications, vol. 7, no. 7, pp. 1205–1211, July 1999. [11] R. Jain, D. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimi- nation for resource allocation in shared computer systems,” DEC (Digital Equipment Corporation) Research Report TR-301, September 1984. 161 Chapter 8 A Distributed Algorithm for Resource Allocation in OFDM-based Cognitive Radio Systems 8.1 Introduction It has been suggested [1] that the looming communications spectrum shortage crisis can be alleviated using cognitive radio (CR) technology [2]. It allows unlicensed users to use licensed frequency bands when the licensed users are not active. The highly dynamic nature of the bandwidth available to a CR system makes resource allocation very challenging. Due to its flexibility in dynamically allocating radio resources to multiple users and low interference between adjacent subcarriers, orthogonal frequency division multiplexing (OFDM) is considered an attractive modulation scheme for CR systems [3]. Due to the time-varying resources available in a CR system, it is possible that the nominal bit rates for CR users cannot be provided. In resource-limited situations, it is desirable to provide an equitable level of rate degradation among the CR users. The proportional rate schemes in [4] and [5] provide fair degradation among data users in OFDM systems by guaranteeing specified rate ratios. However, they consider that the rate ratios should be maintained even after user nominal rate requirements have met; this often limits efficient use of resources. The above-mentioned algorithms are centralized. Distributed algorithms may be more suitable in multi-cell systems or ad hoc systems. In such systems, subcarriers are simul- taneously shared among several users and cochannel interference has to be considered in allocating resources. In [6], the cochannel interference in an OFDM-based digital subscriber 1The material in this chapter is largely based on the following paper: Yonghong Zhang and Cyril Leung, “A Distributed Algorithm for Resource Allocation in OFDM-based Cognitive Radio Systems,” IEEE 68th Vehicular Technology Conference (VTC 2008-fall), Calgary, Canada, Sept. 2008. 162 line (DSL) system is treated as noise and the power allocation problem is viewed as a non- cooperative game. A distributed iterative waterfilling (ITWF) algorithm is proposed which requires knowledge of the user highest achievable target rates (HATRs) in order to obtain the optimal power allocation solution. Obtaining this knowledge is difficult for a distributed algorithm in a time-varying wireless channel environment. In [7], a distributed algorithm is designed to maximize system throughput for an OFDM system with no nominal user bit rates. The distributed algorithm for multi-cell systems in [8] can handle nominal user bit rates but does not consider fair user degradation in resource-limited scenarios. In this paper, we consider subcarrier, power, and bit allocation for user applications with nominal bit rates in an OFDM-based CR system. In contrast to conventional OFDM systems, the available resources in a CR system are changing over time. When resources are scarce, the goal is to provide fair degradation among all CR users. When resources are abundant, we try to maximize system throughput while ensuring that the nominal rates for all users are satisfied. 8.2 System Model Consider a CR system, with bandwidth W , in which the (licensed) primary users (PUs) are not active all the time. Assume that there are altogether K CR transceiver pairs (CRPs), all of which use OFDM. The CRPs may use a PU frequency band whenever the PU is sensed not to be active. There are M subbands (subchannels) and the nominal bandwidth of subband m, m = {1, 2, . . . ,M} ranges from fc + (m− 1)∆f to fc +m∆f . The subbands (or subchannels) are modelled in discrete-time, with the time-varying gain for subchannel m from CRP i’s transmitter to CRP j’s receiver denoted by √ gmj,i. Let the power gains{ gmj,i } be outcomes of independent random variables (rv’s), and let { gmi,j,m = 1, 2, · · · ,M } be independent identically distributed (i.i.d.) rv’s for any given values of i and j. It is assumed that there is no inter-carrier interference (ICI). Each subchannel is shared among 163 CRPs, and the signal of any CRP is considered to be interference to other CRPs. We note that in general the interference powers experienced at the two transceivers of a given CRP are not equal. In order to reduce the cross-channel interference between the CRPs and the PUs, the subchannels adjacent to PU bands are not used by CRPs [9]. The PU active frequency bands, PU guard bands, the spectrum holes, and CRU OFDM subchannels are shown in Figure 8.1. Because the active PU bands vary over time, the number, MCR, of subchannels available to CRPs can range from 0 to 32. For the example shown in Figure 8.1, MCR = 15 at this time at this location. 97531 11 13 15 17 19 21 23 25 27 29 31 W2 Spectrum hole Spectrum hole PU active frequency bands W1 Guard bands Guard bands fc f Figure 8.1: PU active frequency bands, guard bands, spectrum holes and CRU OFDM subchannels. The rapidly time-varying nature of available resources complicates resource allocation, especially if CRPs have nominal bit rate targets. Suppose that the nominal rate requirement for CRP k is RNOMk . When the number of available subchannels is low, not every CRP’s nominal rate may be achieved, i.e., for some CRP k, Rk < R NOM k where Rk is the bit rate over all subchannels for CRP k. In this case, we ensure fair degradation among CRPs, i.e. allocate resources so as to equalize the fractional rate degradation, vk = (R NOM k − Rk)/R NOM k , R NOM k > Rk for all CRPs. The system degradation fairness is measured using 164 the fairness index in [10], i.e. FI = ( ∑K k=1 vk) 2 K ∑K k=1 vk 2 . (8.1) When all K CRP rate degradations are the same, i.e. vk = φ,∀k, the fairness index attains its maximum value of 1. The lowest value is FI = 1/K. The FI-value roughly reflects the fraction of CRPs that receive similar service. Under resource-limited conditions, the goal is to maximize some function h of the fairness index, FI, and the sum rate, ∑K k=1Rk, for CRPs. When resources are adequate, i.e., in a resource-abundant situation, the goal is to maximize the throughput of the CR system while ensuring that the nominal user bit rate requirements are satisfied. The optimization problem OP1 can thus be formulated as O1 :max sk,m h ( ( ∑K k=1 vk) 2 K ∑K k=1 vk 2 , K∑ k=1 Rk ) if Rk < R NOM k for some k O2 :max sk,m K∑ k=1 Rk if Rk ≥ RNOMk for all k (8.2) subject to vk =  RNOMk −Rk RNOMk Rk < R NOM k 1 Rk ≥ RNOMk (8.3) Rk = MCR∑ m=1 rk,m (8.4) rk,m = log2 ( 1 + gmk,ksk,m Γ(σ20 + Ik,m) ) (8.5) MCR∑ m=1 sk,m ≤ Sk (8.6) Ik,m = K∑ l=1,l 6=k gmk,lsl,m (8.7) where subscripts k and m refer to CRP k and subchannel m respectively, Sk is the total 165 transmit power constraint for CRP k, Γ is a SNR gap parameter which indicates how far the system is operating from capacity, sk,m is the transmit power, σ 2 0 is the noise power, and Ik,m is the interference power from other CRPs. Problem OP1 is a two-objective non-linear optimization problem, which is generally hard to solve. Here, we study the case in which fair degradation is guaranteed, i.e., FI = 1. In a resource-abundant situation, the condition Rk ≥ RNOMk , ∀k implies that FI = 1 from (8.3). In a resource-limited situation, to ensure that FI = 1, we need Rk RNOMk = 1− φ, 0 ≤ φ ≤ 1, k = 1, 2, · · · , K; if ∃k,Rk < RNOMk . (8.8) As a result, when FI = 1 is guaranteed, Problem OP1 can be transformed to OP2 : max sk,m K∑ k=1 Rk (8.9) subject to constraints (8.4)-(8.7), and (8.8). Because Problem OP2 involves constraint (8.8), which only applies in the resource-limited situations, it is still hard to solve. We can simplify Problem OP2 using a goal programming approach [11]. There are usually two steps involved in goal programming, i.e., (1) intro- duce slack and surplus variables to combine the different requirements in resource-limited situations (require (8.8)) and resource-abundant situations (require Rk ≥ RNOMk ,∀k); (2) rewrite the objective function in (8.9) to reflect the goals of the original problem. Then, the objective of Problem OP2 becomes OP3 : max sk,m K∑ k=1 pk − φ K∑ k=1 RNOMk (8.10) and constraint (8.8) is replaced with Rk + φR NOM k − pk = RNOMk , 0 ≤ φ ≤ 1, pk ≥ 0, φpk = 0 (8.11) 166 where φRNOMk and pk are the slack and surplus variables for CRP k. The slack variable, φRNOMk , is the difference between R NOM k and Rk when Rk < R NOM k . The surplus variable, pk, is the difference between Rk and R NOM k when Rk > R NOM k . Since the slack value for CRP k is proportional to CRP k’s nominal rate requirement, the resulting fairness index will be equal to 1. We will refer to the set of user rates {R∗k, k = 1, 2, . . . , K} which results from solving problem (8.10) as the highest achievable target rate (HATR) set and R∗k as the individual HATR for CRP k. The optimization problem in (8.10) can be solved using a non-linear single objective optimizer. However, the computational complexity for such a centralized approach is generally very high and requires knowledge of the subchannel power gains from each CRP transmitter to each CRP receiver. We next propose a distributed algorithm with lower complexity which is especially suited for ad hoc or multi-cell infrastructure systems because each CRP only needs the subchannel power gains from its own transmitter to its own receiver. 8.3 Distributed Algorithm As mentioned in [6], if each transmitter knows its HATR, then an iterative waterfilling algorithm yields a Nash equilibrium if we view the allocation problem as a non-cooperative game. In the following subsections, we examine the following issues: (1) how can CRPs determine if a set of target rates is achievable, (2) how can a CRP determine its individual HATR when the system is resource-limited, (3) how can a CRP determine its individual HATR when system are abundant, and (4) how to design the distributed algorithm for resource allocation. Since in practical systems, the number of bits has to be an integer, this constraint is taking into account in this section. 167 8.3.1 Determining Achievability of Target Rates Let us first consider the case with only one subchannel. Assume that each receiver of a CRP sends back its current interference level to its corresponding transmitter. Suppose that the target rate for CRP k is rk. Since the subchannel is shared by multiple CRPs, each trying to maintain its target rate, an increase in the power of any CRP transmitter will result in an increase in interference to the other CRP receivers, which would in turn cause an increase in the other CRP transmitter powers. These higher powers will generate more interference to the original CRP’s receiver, causing its transmitter to increase power again. This process of growing CRP transmitter powers continues until either an equilibrium point is attained or some of the CRP transmitters reach their power limits unable to meet their target rates. A CRP thus knows that if it is unable to reach its target rate, given the current interference power level and it has reached its transmit power limit, then that target rate is not achievable. This strategy can be extended to multiple subchannels by treating the RA problem, which minimizes the total transmit power subject to rate requirements {RTARk , k = 1, 2 · · · , K}, as a non-cooperative game [6, 8]. In this game, each CRP competes for data rates with the sole objective of minimizing its own total transmit power while viewing the signals of other CRPs as interference. When the powers of the other CRPs are not changing, the solution to the RA problem for CRP k has a water-filling interpretation [12]. Since a change in one CRP’s power corresponds to changes in the interference power levels of other CRPs, each CRP needs to iteratively use water-filling algorithm based on its current interference power level. If the resulting total rate over all subchannels is still below the CRP’s target rate RTARk when its transmit power limit is reached, then the CRP knows that RTARk is not achievable. The CRP then broadcasts a “Not achievable (NA)” message to inform other CRPs that the current set of target rates is not achievable. A pseudo-code listing for the algorithm is given in Algorithm 7. The parameter trun is the 168 running time of this algorithm. The parameter τ is the time needed to determine whether a given HATR set is achievable or not, which is the time needed for the non-cooperative game to converge. The input parameter RDATAk is the rate at which CRP k can transmit data. Algorithm 7 provides the minimum total transmit power that achieves RTARk for CRP k given that the other CRPs’ power allocation do not change. As a result, the resulting outcome is a Nash equilibrium, i.e., no CRP can improve its total transmit power by only changing its own power allocation. Conditions for the existence and uniqueness of the Nash equilibrium are studied in [6]. If such conditions can be satisfied, then Algorithm 7 converges within τ , which indicates that the rate set {RTARk , k = 1, 2, · · · , K} is achievable; otherwise, the CRP(s) that reach(es) its power limit Sk without achieving R TAR k broadcast(s) message “NA” before time τ expires. Algorithm 7 Achievable Rate Algorithm. input: RTARk , R DATA k output: achievable (1) initialize rm and r DATA m to 0, m = 1, 2, · · · ,MCR (2) if trun ≥ τ , exit; otherwise, do steps (3) to (8) (3) measure current interference power Ik,m for each subchannel (4) get rDATAm and rm by using water-filling (5) if ∑MCR m=1 rm ≥ RTARk , achievable = true; otherwise achievable = false, broadcast message “NA” (6) if receive message “NA”, set achievable = false. (7) load rDATAm bits to subchannel m (8) go to step (2) 8.3.2 Determining HATR in a Resource-limited Situation Initially, each CRP set RNOMk as its target rate. Once it is determined that the target rate set {RNOMk , k = 1, 2, . . . , K} is not achievable, then the goal is to determine the HATR set while ensuring fair degradation. For this purpose, we propose to use a binary search approach. The idea is based on the binary search algorithm [13] for finding a particular value in a sorted list. 169 Each CRP k keeps track of two individual target rates, Rk,l and Rk,h: the value of Rk,l is the current highest ATR plus 1, whereas Rk,h is the current lowest non-achievable TR minus 1. The actual value of HATR is in the set R = {Rk,l − 1, Rk,l, Rk,l + 1, . . . , Rk,h}. Initially, Rk,l is set to 1 and Rk,h is set to R NOM k . Then at each round, the CRP sets its target rate to rk = d(Rk,l+Rk,h)/2e. If the target rate turns out to be achievable, Rk,l is set to rk + 1, otherwise, Rk,h is set to rk − 1. This process continues until the rate Rk,l exceeds Rk,h. At each round, the size of the set R is halved, since either the top half is removed if the target rate is not achievable or the lower half is removed if the target rate is achievable. CRP k can determine its individual HATR in O(log(RNOMk )) rounds. A pseudo-code listing for the binary rate search algorithm is given in Algorithm 8: Rk,min specifies the minimum rate that should be used for its individual HATR. In a resource-limited situation, Rk,min is set to 0. Algorithm 8 Binary Rate Search Algorithm. input Rk,l, Rk,h, Rk,min output RHATRk (1) if Rk,l = Rk,h, then let R HATR k = Rk,l and exit (2) if Rk,l ≤ Rk,h, then do steps (2.1) to (2.3); otherwise, go to (3) (2.1) rk = d(Rk,l +Rk,h)/2e (2.2) call Algorithm 7 with input RTARk = max(rk, Rk,min) RDATAk = Rk,l − 1 (2.3) if the output of Algorithm 7 is achievable, then Rk,l = rk + 1; otherwise, Rk,h = rk − 1. Go to step (2) (3) RHATRk = Rk,l − 1 8.3.3 Determining HATR in a Resource-abundant Situation If the HATR for each CRP k is at least RNOMk , the system is resource-abundant. Based on the current interference level at the rate set {RNOMk , k = 1, 2, · · · , K}, the HATR can be estimated as R̃MAXk using water-filling and assuming that the interference power level will 170 not be raised. Since the level will actually increase if the rate set {R̃MAXk , k = 1, 2, . . . , K} is used, R̃MAXk is an upperbound on the HATR. Since plain equal power allocation, in which power is equally allocated to all subchannels, yields similar results as water-filling [14], we can estimate RMAXk as R̃MAXk = MCR∑ m=1 log2 ( 1 + Skg m k,k MCRΓ (σ20 + Ik,m) ) (8.12) where Ik,m is the interference power that results when all CRP rates are set to their nominal values. The HATR for CRP k can be found using Algorithm 8 with Rk,l = 1 and Rk,h = R̃MAXk . By setting Rk,min to R NOM k , we ensure that the HATR is no less than R NOM k . 8.3.4 The Proposed Distributed Algorithm The proposed distributed algorithm for finding a suboptimal solution to problem (8.10) can be stated as follows. Each CRP k first sets its target rate at RNOMk . Using the procedure in Subsection 8.3.1, it determines if the target rate set {RNOMk , k = 1, 2, · · · , K} is achievable. If the target rate set is not achievable, CRP k then uses Algorithm 8 to find its HATR between 1 and its RNOMk . If the target rate set is achievable, CRP k then uses Algorithm 8 to find its HATR between RNOMk and R̃ MAX k . CRP k then uses its HATR as the target rate and performs iterative waterfilling. Fig- ure 8.2 shows a block diagram of the algorithm. 8.4 Simulation Results To evaluate the effectiveness of the proposed distributed algorithm summarized in Fig. 8.2, simulations were performed on a system covering a 3 km × 3 km area. The transmitters are uniformly distributed within that area and each receiver is uniformly distributed within a 300 meter circle centered on its corresponding transmitter. The propagation path loss is 171 Call Algorithm 1 (input: Rk TAR =Rk NOM, Rk DATA =0) achievable?Y Call Algorithm 2 (input: Rk,l, Rk,h, Rk,min output: Rk BATR) N Iterative water-filling with RBATR Rk,l=1,Rk,h=Rk NOM Rk,min=0 Rk,l=1,Rk,h=R MAX Rk,min=Rk NOM ~ Figure 8.2: Flow chart of the distributed allocation algorithm. calculated using PL = A + 10γ log10(d/d0) [15], where A = 50 dB is the path loss at the reference point, γ = 4 is the path loss exponent, d is the distance between the transmitter and the receiver and d0 = 10 m is the distance between the transmitter and the reference point. The multipath fading is assumed to be Rayleigh. The duration of each OFDM symbol is 40 µs. The CRP bit error rate requirement is 10−4, which corresponds to a Γ value of 5. The number, MCR, of subchannels available to CRPs ranges from 2 to 16 and each subchannel has a noise power σ20 of 10 −15 W. For performance comparison, we obtained the optimal solution for Problem OP3 for each of a number of different realizations of CRP locations using a commercial optimization software package. Similarly, the optimal solution with a fixed fractional rate degradation for all CRPs in resource-limited as well as resource-abundant situations, hereafter referred to as “proportional rate”, was also obtained. The proportional rate constraint is equivalent to R1/R NOM 1 = R2/R NOM 2 = · · · = RK/RNOMK . (8.13) Fig. 8.3 shows the average number of bits per OFDM symbol (ANB) per CRP as a function of the number of available subchannels. The results for the optimal solution is obtained by solving Problem OP3, and the results for the optimal proportional rate solution is obtained by solving Problem OP2 but with constraint (8.8) replaced by (8.13). To keep 172 2 4 6 8 10 12 14 16 10 20 30 40 50 60 70 80 A N B pe r C RP Number of available subchannels,  MCR   Optimal (real bits) Proposed (integer bits) Proportional rate (real bits) Figure 8.3: Average number of bits per OFDM symbol duration per CRP as a function of the number of available subchannels with Sk = 10 −3 W, K = 3, RNOM1 = 25, R NOM 2 = 30, RNOM3 = 35. the running times for obtaining optimal solutions reasonable, the value for each point in Fig. 8.3 is averaged over 1000 realizations of CRP locations. It can be seen from Figure 8.3 that the proposed distributed algorithm provides an ANB (throughput) close to (within 8% of) optimal. The throughput with the proportional rate constraint is considerably lower because it limits efficient use of the extra available system resources. The throughput improvement with the proposed algorithm relative to the proportional rate solution increases with the number of available subchannels from 3% with 4 subchannels to 56% with 16 subchannels. The fairness indices in all three cases are close to 1. 173 The ANB per CRP curves with RNOM1 = R NOM 2 = R NOM 3 = 20 are plotted in Fig. 8.4. The ANB for the proposed distributed algorithm is within 8% of optimal. Compared to Fig. 8.3, the ANB is higher for the optimal and proposed algorithms, especially when MCR is high. The reason is as follows: the nominal CRP rate requirements in Fig. 8.4 are lower than in Fig. 8.3, thus the system is more often in the resource abundant situation resulting in a higher throughput. Since the proportional rate algorithm always maintains CRP rates proportionally, the ANB difference between Fig. 8.3 and Fig. 8.4 is small. The ANB improvement of the proposed algorithm relative to the proportional rate solution increases with MCR, from 4% with 4 subchannels to 80% with 16 subchannels. The fairness indices in all three cases are close to 1. Fig. 8.5 shows the ANB per CRP as a function of the total power constraint, Sk, with MCR = 8. The proposed algorithm provides an ANB which is within 8% of optimal. The ANB of the proposed algorithm is about 50% higher than the proportional rate solution for the range of Sk values plotted. To explain the relative performances of the optimal, proportional rate and proposed algorithms, we consider a system with four subchannels and a randomly selected realization of CRP locations. Table 8.1 lists the subchannel gains from CRP j’s transmitter to CRP i’s receiver. i j sub 1 sub 2 sub 3 sub 4 1 1 919 7723 2058 6910 2 1.77 1.32 4.46 2.19 3 1.23 0.598 0.567 0.324 2 1 0.051 0.07 0.038 0.042 2 140 1696 2227 566 3 0.038 0.052 0.032 0.065 3 1 0.078 0.083 0.052 0.013 2 0.322 0.198 0.009 0.269 3 180 193 135 85 Table 8.1: Subchannel gains (×10−10) from CRP j’s transmitter to CRP i’s receiver. 174 2 4 6 8 10 12 14 16 10 20 30 40 50 60 70 80 90 A N B pe r C RP Number of available subchannels, MCR   Optimal (real bits) Proposed (integer bits) Proportional rate (real bits) Figure 8.4: Average number of bits per OFDM symbol duration per CRP as a function of the number of available subchannels with Sk = 10 −3 W, K = 3, RNOM1 = 20, R NOM 2 = 20, RNOM3 = 20. 175 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 A N B pe r C RP Power limit,  Sk (mW)   Optimal (real bits) Proposed (integer bits) Proportional rate (real bits) Figure 8.5: Average number of bits per OFDM symbol duration per CRP as a function of total power with MCR = 8, K = 3, R NOM 1 = 20, R NOM 2 = 20, R NOM 3 = 20. 176 The three CRPs have equal transmit power limits, namely Sk = 0.1 mW. The total (summed over all four subchannels) number of bits per OFDM symbol duration for each CRP is shown in Table 8.2. Results are given for each of the three algorithms and four different sets of nominal CRP rate requirements. Note that in Table 8.2 the results for the optimal and the proportional rate algorithms have been rounded to the nearest integer. For the first set of nominal CRP rate requirements, i.e. RNOMk = 10 bits per OFDM symbol duration, the system is resource-limited and all three CRPs cannot achieve their nominal rates. The goal is to allocate as many bits as possible while keeping the rate ratios Rk/R NOM k for the three CRPs as close as possible. All three algorithms yield CRP rates of 8 bits per OFDM symbol duration and a fairness value of 1. For the second set of nominal CRP rate requirements, i.e. RNOM1 = 10 bits, R NOM 2 = 15 bits, RNOM3 = 20 bits, the CR system is even more resource-limited. The CRP sum rate and the fairness index for the proposed algorithm is slightly lower than the optimal and proportional rate values. We note that the optimal and proportional rate algorithms produce the same solutions when the system is resource-limited because then the constraint in (8.13) applies to both. For the third and fourth sets of nominal CRP rate requirements, the CR system is resource-abundant. The goal is to maximize the overall system throughput while ensuring that all CRP nominal rates are met. The proposed algorithm gives a fairness value of 1 and sum rates which are within 4% of the optimal values. When the system is resource-abundant, the proportional rate algorithm does not make efficient use of the surplus resources and its sum rates are lower than those of the proposed algorithm by 32% and 74%, respectively. 8.5 Conclusions The subchannel, bit, and power allocation problem for users with nominal bit rate require- ments in an OFDM-based CR system was formulated as a multi-objective non-linear op- 177 RNOM1 = 10, R NOM 2 = 10, R NOM 3 = 10 Algorithm CRP 1 CRP 2 CRP 3 sum rate fair index Optimal 8 8 8 24 1 Proposed 8 8 8 24 1 Proportional rate 8 8 8 24 1 RNOM1 = 10, R NOM 2 = 15, R NOM 3 = 20 Algorithm CRP 1 CRP 2 CRP 3 sum rate fair index Optimal 4 6 8 18 1 Proposed 3 6 8 17 0.997 Proportional rate 4 6 8 18 1 RNOM1 = 6, R NOM 2 = 6, R NOM 3 = 4 Algorithm CRP 1 CRP 2 CRP 3 sum rate fair index Optimal 24 17 8 49 1 Proposed 23 17 7 47 1 Proportional rate 12 12 8 32 1 RNOM1 = 1, R NOM 2 = 2, R NOM 3 = 4 Algorithm CRP 1 CRP 2 CRP 3 sum rate fair index Optimal 24 17 8 49 1 Proposed 23 17 7 47 1 Proportional rate 2 4 8 14 1 Table 8.2: Number of bits per OFDM symbol and fairness index for each of the three algorithms and four different sets of nominal rate requirements. 178 timization problem. The goal is to provide a fair bit rate degradation among users in resource-limited situations and to maximize system throughput while satisfying user nom- inal rate requirements when resources are plentiful. A goal programming approach was used to transform the problem into a single objective non-linear optimization problem. A distributed algorithm was designed to solve the problem. Simulation results were provided which show that the proposed algorithm has a performance which is within 8% of optimal. 179 References [1] Federal Communications Commission, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies,” notice of proposed rulemaking and order, FCC 03-322, 2003. [2] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23, no. 2, pp. 201–220, February 2005. [3] T. A. Weiss and F. K. Jondral, “Spectrum pooling: an innovative strategy for the enhancement of spectrum efficiency,” IEEE Communications Magazine, vol. 42, no. 3, pp. S8–S14, March 2004. [4] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2726–2737, November 2005. [5] G. Yu, Z. Zhang, Y. Chen, P. Cheng, and P. Qiu, “Subcarrier and bit allocation for OFDMA systems with proportional fairness,” in Proc. of IEEE Wireless Communica- tions and Networking Conference (WCNC 2006), vol. 3, Las Vegas, NV, USA, April 2006, pp. 1717–1722. [6] W. Yu, G. Ginis, and J. M. Cioffi, “Distributed multiuser power control for digital subscriber lines,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 5, pp. 1105–1115, June 2002. [7] G. Kulkarni, S. Adlakha, and M. Srivastava, “Subcarrier allocation and bit loading algorithms for OFDMA-based wireless networks,” IEEE Transactions on Mobile Com- puting, vol. 4, no. 6, pp. 652–662, November-December 2005. [8] Z. Han, Z. Ji, and K. J. R. Liu, “Non-cooperative resource competition game by virtual referee in multi-cell OFDMA networks,” IEEE Journal on Selected Areas in Commu- nications, vol. 25, no. 6, pp. 1079–1090, August 2007. [9] T. Weiss, J. Hillenbrand, A. Krohn, and F. K. Jondral, “Mutual interference in OFDM- based spectrum pooling systems,” in Proc. of IEEE 59th Vehicular Technology Confer- ence (VTC 2004-Spring), vol. 4, Milan, Italy, May 2004, pp. 1873–1877. [10] R. Jain, D. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimi- nation for resource allocation in shared computer systems,” DEC (Digital Equipment Corporation) Research Report TR-301, September 1984. [11] M. J. Schniederjans, Goal programming: methodology and applications. Springer, 1995. [12] R. G. Gallager, Information Theory and Reliable Communication. New York: John Wiley & Sons, 1968. [13] D. Knuth, The Art of Computer Programming, Volume 3: Sorting and Searching. Addison-Wesley, 1997. 180 [14] Y. Zhang and C. Leung, “Performance of equal power subchannel loading in multiuser OFDM systems,” in Proc. of IEEE Pacific Rim Conference on Communications, Com- puters and Signal Processing (PacRim 2007), Victoria, BC, Canada, August 2007, pp. 526–529. [15] J. B. Andersen, T. S. Rappaport, and S. Yoshida, “Propagation measurements and mod- els for wireless communications channels,” IEEE Communications Magazine, vol. 33, no. 1, pp. 42–49, January 1995. 181 Chapter 9 Conclusions 9.1 Contributions and Discussions In this thesis, we studied various resource allocation (RA) problems in OFDM-based CR systems. The sharing of PU bands by CRUs is categorized into aggressive or protective sharing. In aggressive sharing, any portion of the spectrum can be shared by CRUs as long as such use does not interfere unduly with PU operation. To successfully share PU bands aggressively, the key is to monitor the generated interference to the PUs by CRUs. Two types of CRU-generated interference must be considered, namely, cross-channel interference which results from sharing of subchannels adjacent to PU active frequency bands and co- channel interference from sharing of PU active frequency bands. In the literature, each type of interference has been considered separately by different researchers, but not jointly. The two types of interference are studied jointly in Chapters 2 and 3. In Chapter 2, a model that takes into account both cross-channel and co-channel inter- ference is presented for a multiuser OFDM-based multi-cell CR system. A suboptimal but low-complexity algorithm, Max-Min, is proposed for the RA problem. Simulation results show that the Max-Min algorithm yields solutions that are close to optimal (within 0.1% for the single-user case and 5% for the multi-user case). The multi-user Max-Min algorithm can be applied on the downlink of a multi-cell cellular system. Although the Max-Min has a much lower computational complexity than algorithms for solving general MDKP problems, e.g. [1], its complexity is still high. For example, for the 182 single-user case, Max-Min has complexity O(LMR), where L is the number of PUs, M is the number of subchannels, and R is the total number of allocated bits. A simplified model of that in Chapter 2 is therefore proposed in Chapter 3, based on: (1) the fact that cross- channel interference from CRUs to PUs is mainly limited to a few subchannels adjacent to the PU bands, and (2) the assumption that the bandwidth of a PU is much larger than that of an OFDM subchannel and that there is usually a guard band between two adjacent PU bands. A power and bit allocation algorithm for a single user is proposed with complexity O(M log(M)) + O(LM), which is much lower than that of Max-Min. Simulation results show that the proposed suboptimal solution is close to optimal, with a difference of less than 4%. Simulation results in Chapters 2 and 3 show that aggressively sharing the whole band can provide a substantial performance improvement over protective sharing. Aggressive sharing is preferable to protective sharing in terms of spectrum efficiency; however, in some practical situations, only protective sharing is possible. This situation can occur in a broadcast PU system, in which PU receivers are densely located so that the probability of a CRU transmission not unduly disturbing any PU receiver is almost zero. Note that RA algorithms proposed in Chapters 2 and 3 designed for aggressive sharing systems can be applied to protective sharing systems by setting the PU interference power threshold at each active PU receiver equals to 0. However, the protective sharing model greatly simplifies RA design because CRU interference to PU receivers does not need to be considered. For example, the single CRU Max-Min algorithm has a complexity O(LMR), whereas similar algorithms assuming protective sharing has a complexity O(M) (e.g. [2]). Although interference to PUs is not a factor in protective sharing, the time-varying nature of the available spectrum due to PU activities is a hurdle to meeting CRU QoS/fairness constraints. Few studies in the literature have considered such QoS provisioning difficulties in resource allocation for CR systems. In Chapters 4 to 8, we presented RA algorithms that provide good QoS to the users to operate in a fading environment with time-varying spectrum. 183 Assuming that the objective is to maximize system throughput, we showed in Chapter 4 and 5 that the performance difference between the PEPA and optimal solutions is quite small. This difference increases with the number of subchannels available to CRUs. A discrete- PEPA algorithm with low-complexity is proposed in Chapter 5 that yields close to optimal solutions. Since both continuous and discrete bits PEPA are very simple to implement, it is preferable in CR systems when simplicity is more important than performance. To improve spectrum utilization efficiency, a cross-layer approach was adopted in Chapter 6 for both real-time (RT) services and mixed services. A MAC-PHY RA scheme combin- ing packet scheduling with subchannel, bit, and power allocation for RT applications was proposed. The MAC layer QoS requirements are dynamically converted to PHY layer rate requirements in a way that takes into account the delivery status of packets waiting in the CRBS buffers as well as the number of available subchannels. Simulation results show that the proposed RA algorithm can provide substantial CRBS transmit power reductions compared to existing PHY layer and MAC layer solutions designed for multiuser OFDM systems. The algorithm was extended to include NRT services, each of which has a nominal rate requirement. Similar to the RT services, the nominal rate requirements in the MAC layer are converted to PHY layer rate requirements. We then formulated a rate maximiza- tion problem in the PHY layer, of which the optimal solution was provided. In contrast to existing algorithms designed for multiuser OFDM systems, the proposed algorithm solves the problem feasibility issue by using the goal programming approach and avoids the false urgency issue caused by variations in available system resources by using a proposed rate requirement calculation mechanism. Simulation results show that the proposed algorithm provides fairness and satisfactory QoS to both RT and NRT CRUs, and performs much better than existing algorithms designed for multiuser OFDM systems. In Chapter 7, we proposed a RA algorithm for a type of NRT service, in which the average user rates are to be maintained proportionally. As an example, this type of NRT service could be used in a wireless router to maintain proportional rates for different flows. 184 In contrast to existing RA algorithms designed for multiuser OFDM systems, which are unable to guarantee proportional rate allocation in a CR system when the number of CRUs exceeds the number of available subchannels, the proposed algorithm provides statistically proportional rates as well as improved system throughput. The protective sharing RA algorithms presented in Chapters 4 to 7 are designed for cel- lular systems, in which centralized algorithms are appropriate. In Chapter 8, we considered RA in an ad hoc system, in which distributed algorithms are more practical. The subchannel, bit, and power allocation problem for CRUs with nominal rate requirements in an OFDM- based CR system was formulated as a multi-objective non-linear optimization problem. The goal is to provide a fair bit rate degradation among users in resource-limited situations and to maximize system throughput while satisfying user nominal rate requirements when re- sources are plentiful. A distributed algorithm was designed to solve the problem. Simulation results show that the proposed distributed algorithm provides good fairness and significantly higher system throughput compared with the proportional rate algorithm when resources are abundant. To summarize, we showed in Chapters 2 and 3 that aggressive sharing provides a sig- nificant performance improvement over protective sharing. Although it is more efficient, aggressive sharing is more difficult to implement and may not be possible in some situations; protective sharing is easier to implement and can be used in most scenarios. RA algorithms using protective sharing are proposed in Chapters 4 to 8. We showed, in Chapters 4 and 5, that PEPA provides a close to optimal performance if the objective is to maximize system throughput. PEPA is therefore preferable in CR systems in which simplicity is more impor- tant than performance. To improve spectrum utilization efficiency while meeting fairness and QoS contraints, efficient RA algorithms designed for RT and NRT services are proposed in Chapters 6 to 8 for CR systems with time-varying available system resources. 185 9.2 Future Work As a new technology, cognitive radio presents new challenges in resource allocation. One of the main difficulties is that the interference power generated by CRUs at a PU receiver has to be kept below a predefined interference power threshold. There are mainly two ways of controlling this interference power: (1) aggressive sharing, which makes use of the whole spectrum with close monitoring of the total generated interference power; and (2) protective sharing, which avoids CRU interference to PU bands by allowing sharing of only the spectrum holes. Because interference avoidance is adopted in protective sharing, the interference-to- PU issue does not arise. In aggressive sharing, both cross-channel interference and co-channel interference to PU bands need to be considered. Another difficulty is caused by the effects of the time-varying nature of the available spectrum on CRUs. Existing RA algorithms (e.g. [3]–[5]) at the PHY layer for multiuser OFDM systems generally assume implicitly that the available spectrum is fixed; although the fading characteristics of wireless communication channels is taken into account in these algorithms, the time-varying nature of the available spectrum has a bigger impact that must be considered in resource allocation in OFDM-based CR systems to provide satisfactory QoS to CRUs. In this thesis, we have studied the interference-to-PU issue in aggressive sharing and have addressed the spectrum time-varying issue in protective sharing in RA for OFDM- based CR systems. However, the spectrum time-varying issue in aggressive sharing was not addressed, although similar approaches to our proposals for protective sharing could be adopted. Furthermore, most of the algorithms (except the one in Chapter 8) proposed in the thesis are centralized; distributed algorithms would be useful as well. These topics are left for further investigation. The following is a list of specific topics for future study: • In aggressive sharing, in addition to the cross-channel and co-channel interference generated by CRUs to PU bands, time-varying nature of the available spectrum should 186 be taken into account in order to provide CRUs with satisfactory QoS. RA algorithms that address both issues together need to be developed. • To operate in fading environment with time-varying available spectrum either using aggressive or protective sharing, appropriate MAC layer RA algorithms would be of interest. • Distributed RA algorithms that allow aggressive sharing of PU bands need to be de- veloped for ad hoc or multi-cell cellular systems that deal with both interference to PU bands and the time-varying nature of the available spectrum. • Distributed cross-layer RA algorithms that protectively share PU bands should be de- signed for ad hoc or multi-cell cellular systems to efficiently use the available spectrum. 187 References [1] B. Gavish and H. Pirkul, “Efficient algorithms for solving multiconstraint zero-one knap- sack problems to optimality,” Springer Mathematical Programming, vol. 31, no. 1, pp. 78–105, January 1985. [2] J. Campello, “Practical bit loading for DMT,” in Proc. of IEEE International Conference on Communications (ICC ’99), vol. 2, Vancouver, BC, Canada, June 1999, pp. 801–805. [3] C. Y. Wong, R. S. Cheng, K. B. Letaief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE Journal on Selected Areas in Com- munications, vol. 17, no. 10, pp. 1747–1758, October 1999. [4] M. Ergen, S. Coleri, and P. Varaiya, “QoS aware adaptive resource allocation techniques for fair scheduling in OFDMA based broadband wireless access systems,” IEEE Trans- actions on Broadcasting, vol. 49, no. 4, pp. 363–370, December 2003. [5] Z. Shen, J. G. Andrews, and B. L. Evans, “Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints,” IEEE Transactions on Wireless Communications, vol. 4, no. 6, pp. 2726–2737, November 2005. 188 Appendix A Optimal Solutions for Optimization Problems in Chapter 3 A.1 Solution for OP1 Let {s∗(1)m , ν∗, µ∗l , φ∗m, l = 1, 2, · · · , L,m = 1, 2, · · · ,M} be the optimal solution; then Karush- Kuhn-Tucker (KKT) conditions [1] state that ν∗ ≥ 0, µ∗l ≥ 0, φ∗m ≥ 0 (A.1) ν∗ ( M∑ m=1 s∗(1)m − S ) = 0 (A.2) µ∗l ( M∑ m=1 s∗(1)m f CR l,m − I thl ) = 0 (A.3) φ∗ms ∗(1) m = 0 (A.4) − 1 Nm + s ∗(1) m + ν∗ + L∑ l=1 µ∗l f CR l,m − φ∗m = 0 (A.5) in addition, the optimal values {s∗(1)m , ν∗, µ∗l , φ∗m, l = 1, 2, · · · , L,m = 1, 2, · · · ,M} need to satisfy (3.3)-(3.5). Equation (A.5) is obtained by setting ∂F ∂sm = 0 where F is the Lagrangian in (3.6). Solving for s ∗(1) m from (A.5), we obtain s∗(1)m = 1∑L l=1 µ ∗ l f CR l,m + ν ∗ − φ∗m −Nm (A.6) 189 Since s ∗(1) m ≥ 0, from (A.6), we have 1∑L l=1 µ ∗ l f CR l,m + ν ∗ − φ∗m ≥ Nm (A.7) When Nm ≤ 1∑L l=1 µ ∗ l f CR l,m+ν ∗ , it follows from (A.7) that φ ∗ m = 0. Thus from (A.6), we have s∗(1)m = 1∑L l=1 µ ∗ l f CR l,m + ν ∗ −Nm (A.8) When Nm > 1∑L l=1 µ ∗ l f CR l,m+ν ∗ , from (A.7), we have 1∑L l=1 µ ∗ l f CR l,m + ν ∗ − φ∗m ≥ Nm > 1∑L l=1 µ ∗ l f CR l,m + ν ∗ (A.9) It follows from (A.9) that φ∗m > 0. Based on (A.4), we must have s ∗(1) m = 0. Together with (A.8), we have (3.7). A.2 Solution for OP3 The Lagrangian [1] can be written as F = − ∑ m∈M+ l′ log2 ( 1 + sm Nm ) +ρ  ∑ m∈M+ l′ smf CR m − I thl − ∑ m∈M+ l′ γmsm+ ∑ m∈M+ l′ αm(sm−λ+Nm) (A.10) where ρ, and γm, αm,m ∈M+l′ are Lagrange multipliers. Let {s∗(3)m , ρ∗, γ∗m, α∗m,m ∈ M+l′ } be the optimal solution; then KKT conditions [1] state 190 that ρ∗ ≥ 0, γ∗m ≥ 0, α∗m ≥ 0 (A.11) ρ∗  ∑ m∈M+ l′ s∗(3)m f CR m − I thl  = 0, l ∈ L (A.12) γ∗ms ∗(3) m = 0 (A.13) α∗m(s ∗(3) m − λ+Nm) = 0 (A.14) − 1 Nm + s ∗(3) m + ρ∗fCRm − γ∗m + α∗m = 0 (A.15) in addition, the optimal values {s∗(3)m , ρ∗, γ∗m, α∗m,m ∈ M+l′ } need to satisfy (3.13)-(3.14). Equation (A.15) is obtained by setting ∂F ∂sm = 0, where F is given in (A.10). Solving for s ∗(3) m from (A.15), we obtain s∗(3)m = 1 ρ∗fCRm − γ∗m + α∗m −Nm (A.16) Now, we consider the cases when 1 ρ∗fCRm < Nm, Nm ≤ 1ρ∗fCRm ≤ λ, and 1 ρ∗fCRm > λ in the following. (1) 1 ρ∗fCRm < Nm. Since s ∗(3) m ≥ 0, from (A.16), we have 1 ρ∗fCRm − γ∗m + α∗m ≥ Nm . (A.17) From (A.17) and the condition that Nm > 1 ρ∗fCRm , we have 1 ρ∗fCRm − γ∗m + α∗m > 1 ρ∗fCRm . (A.18) Thus, we have −γ∗m + α∗m < 0. Together with (A.11), we must have γ∗m > 0. From (A.13), we obtain s ∗(3) m = 0. 191 (2) Nm ≤ 1ρ∗fCRm ≤ λ. Because 0 ≤ s ∗(3) m ≤ λ−Nm, from (A.16), we get 0 ≤ 1 ρ∗fCRm − γ∗m + α∗m −Nm ≤ λ−Nm (A.19) From (A.19) and the condition Nm ≤ 1ρ∗fCRm ≤ λ, we have 1 ρ∗fCRm = 1 ρ∗fCRm − γ∗m + α∗m (A.20) Thus, we have −γ∗m + α∗m = 0. From (A.13) and (A.14), γ∗m and α∗m cannot be positive at the same time. Together with (A.11), we must have γ∗m = 0 and α ∗ m = 0. From (A.16), we obtain s ∗(3) m = 1ρ∗fCRm −Nm. (3) 1 ρ∗fCRm > λ. Because s ∗(3) m < λ−Nm, from (A.16), we get 1 ρ∗fCRm − γ∗m + α∗m −Nm < λ−Nm (A.21) From (A.21) and the condition that λ < 1 ρ∗fCRm , we have 1 ρ∗fCRm − γ∗m + α∗m < 1 ρ∗fCRm (A.22) Thus, we get −γ∗m + α∗m > 0. It follows from (A.11) that α∗m > 0. From (A.14), we must have s ∗(3) m = λ−Nm. Summarize the three cases, we have s∗(3)m =  0 , 1 ρ∗ < Nmf CR m 1 ρ∗fCRm −Nm , NmfCRm ≤ 1ρ∗ ≤ λfCRm λ−Nm , λfCRm < 1ρ∗ (A.23) Substituting 1 ρ∗ with β, we obtain (3.15). 192 References [1] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge University Press, 2004. 193 Appendix B Proofs of Theorems in Chapter 6 B.1 Proof for Theorem 3.1 In Problem OP2, if we relax constraint (6.6) to allow atk,m to take on a real value in [0, 1], and use the transformation r̂tk,m = a t k,mr t k,m, the following convex optimization problem is obtained OP− B1 : min atk,m,r̂ t k,m U∑ t=1 K∑ k=1 M∑ m=1 atk,m(2 r̂tk,m at k,m − 1) Γσ 2 0 gtk,m (B.1) subject to K∑ k=1 atk,m − 1 = 0 (B.2) Rt,REQk − U∑ t=1 M∑ m=1 r̂tk,m = 0 (B.3) r̂tk,m −RMAXatk,m ≤ 0 (B.4) −r̂tk,m ≤ 0 (B.5) atk,m − 1 ≤ 0, m ∈Mt (B.6) −atk,m ≤ 0, m ∈Mt (B.7) atk,m = 0, m ∈ M̄t . (B.8) Functions atk,m(2 r̂tk,m at k,m −1)Γσ20/gtk,m in (B.1) are convex functions in convex set C = {atk,m ∈ [0, 1], r̂tk,m ∈ [0, RMAXatk,m]}, because the Hessian matrix of each function is positive semidef- 194 inite [1] throughout C. The functions in the LHS of (B.2) to (B.8) are affine. The Lagrangian [2, 3] for OP-B1 can be written as L = U∑ t=1 K∑ k=1 M∑ m=1 atk,m(2 r̂tk,m at k,m − 1) Γσ 2 0 gtk,m + U∑ t=1 M∑ m=1 γtm ( K∑ k=1 atk,m − 1 ) + K∑ k=1 λk ( Rt,REQk − U∑ t=1 M∑ m=1 r̂tk,m ) + U∑ t=1 K∑ k=1 M∑ m=1 µtk,m ( r̂tk,m −RMAXatk,m )− U∑ t=1 K∑ k=1 M∑ m=1 ζtk,mr̂ t k,m + U∑ t=1 K∑ k=1 M∑ m=1 φtk,m(a t k,m − 1)− U∑ t=1 K∑ k=1 M∑ m=1 βtk,ma t k,m (B.9) where {γtm, λk, µtk,m, ζtk,m, φtk,m, βtk,m, t = 1, 2, · · · , U, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} are the Lagrange multipliers. Let {at ∗k,m, r̂t ∗k,m, γt ∗m , λ∗k, µt ∗k,m, ζt ∗k,m, φt ∗k,m, βt ∗k,m, t = 1, 2, · · · , U, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} be an optimal solution. Then the Karush-Kuhn-Tucker (KKT) conditions state that [3] γt ∗m ≥ 0, λ∗k ≥ 0, µt ∗k,m ≥ 0, ζt ∗k,m ≥ 0, φt ∗k,m ≥ 0, βt ∗k,m ≥ 0 (B.10) γt ∗m ( K∑ k=1 at ∗k,m − 1) = 0 (B.11) λk(R U,REQ k − U∑ t=1 M∑ m=1 r̂t ∗k,m) = 0 (B.12) µt ∗k,m(r̂ t ∗ k,m −RMAXat ∗k,m) = 0 (B.13) ζt ∗k,mr̂ t ∗ k,m = 0 (B.14) φt ∗k,m(a t ∗ k,m − 1) = 0 (B.15) βt ∗k,ma t ∗ k,m = 0 (B.16) Γσ20 gtk,m 2 r̂t ∗k,m at ∗ k,m ln(2)− λ∗k + µt ∗k,m − ζt ∗k,m = 0 (B.17) Γσ20 gtk,m ( 2 r̂t ∗k,m at ∗ k,m − 1− 2 r̂t ∗k,m at ∗ k,m ln(2) r̂t ∗k,m at ∗k,m ) + γt ∗m −RMAXµt ∗k,m + φt ∗k,m − βt ∗k,m = 0 .(B.18) 195 In addition, the optimal values {at ∗k,m, r̂t ∗k,m, γt ∗m , λ∗k, µt ∗k,m, ζt ∗k,m, φt ∗k,m, βt ∗k,m, t = 1, 2, · · · , U, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} need to satisfy (B.2)-(B.8). Equations (B.17) and (B.18) are obtained by setting ∂L ∂r̂tk,m = 0 and ∂L ∂atk,m = 0, respectively. From (B.17), when at ∗k,m 6= 0, we have r̂t ∗k,m = a t ∗ k,m log2 ( (λ∗k + ζ t ∗ k,m − µt ∗k,m)gtk,m ln(2)Γσ20 ) . (B.19) Now, we consider the cases when λ∗k ln(2) < Γσ20 gtk,m , Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2RMAX Γσ20 gtk,m , and λ∗k ln(2) > 2R MAX Γσ20 gtk,m in the following. (1) Case λ∗k ln(2) < Γσ20 gtk,m : from (B.19) and the fact that r̂t ∗k,m ≥ 0, we obtain (λ∗k+ζ t ∗ k,m−µt ∗k,m)gtk,m ln(2)Γσ20 ≥ 1. This together with the condition λ∗k ln(2) < Γσ20 gtk,m , yields ζt ∗k,m − µt ∗k,m > 0 . (B.20) From (B.10), (B.13) and (B.14), µt ∗k,m and ζ t ∗ k,m are both nonnegative and cannot be both positive; using (B.20), we must have µt ∗k,m = 0 and ζ t ∗ k,m > 0. Then, from (B.14), we must have r̂t ∗k,m = 0. (2) Case Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2RMAX Γσ20 gtk,m : from (B.19) and the fact that 0 ≤ r̂t ∗k,m ≤ RMAXat ∗k,m, we have 1 ≤ (λ ∗ k+ζ t ∗ k,m−µt ∗k,m)gtk,m ln(2)Γσ20 ≤ 2RMAX . This together with the condition Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2R MAX Γσ20 gtk,m , yields ζt ∗k,m − µt ∗k,m = 0 . (B.21) Since µt ∗k,m and ζ t ∗ k,m are both nonnegative and cannot be both positive; using (B.21), we must have µt ∗k,m = 0 and ζ t ∗ k,m = 0. Then it follows from (B.19) that r̂ t ∗ k,m = a t ∗ k,m log2( λ∗kg t k,m ln(2)Γσ20 ). (3) Case λ∗k ln(2) > 2R MAX Γσ20 gtk,m : from (B.19) and the fact that r̂t ∗k,m ≤ RMAXat ∗k,m, we obtain (λ∗k+ζ t ∗ k,m−µt ∗k,m)gtk,m ln(2)Γσ20 ≤ 2RMAX . This together with the condition λ∗k ln(2) > 2R MAX Γσ20 gtk,m , yields ζt ∗k,m − µt ∗k,m < 0 . (B.22) 196 Since µt ∗k,m and ζ t ∗ k,m are both nonnegative and cannot be both positive; using (B.22), we must have µt ∗k,m > 0 and ζ t ∗ k,m = 0. Then, from (B.13), we must have r̂ t ∗ k,m = R MAXat ∗k,m. Combining the three cases, and since rt ∗k,m = r̂ t ∗ k,m/a t ∗ k,m, the optimal bit allocation for CRU k for subchannels with at ∗k,m > 0 is thus given by rt ∗k,m =  0, λ∗k ln(2) < Γσ20 gtk,m log2 ( λ∗kg t k,m ln(2)Γσ20 ) , Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2R MAX Γσ20 gtk,m RMAX , λ∗k ln(2) > 2R MAX Γσ20 gtk,m . (B.23) From (B.23) and (6.1), we obtain the optimal power allocation for CRU k for subchannels with at ∗k,m > 0 as st ∗k,m =  0, λ∗k ln(2) < Γσ20 gtk,m λ∗k ln(2) − Γσ20 gtk,m , Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2R MAX Γσ20 gtk,m (2R MAX−1)Γσ20 gtk,m , λ∗k ln(2) > 2R MAX Γσ20 gtk,m . (B.24) In (B.24), λ∗k/ ln(2) can be interpreted as the water level for CRU k. Substituting (B.23) into (B.18) and noting that λ∗k = Γσ20 gtk,m 2 r̂t ∗k,m at ∗ k,m ln(2) + µt ∗k,m − ζt ∗k,m from (B.17), we have htk,m − γt ∗m − φt ∗k,m + βt ∗k,m = 0 (B.25) where htk,m =  0, λ∗k ln(2) < Γσ20 gtk,m λ∗k log2 ( λ∗kg t k,m ln(2)Γσ20 ) − ( λ∗k ln(2) − Γσ20 gtk,m ) , Γσ20 gtk,m ≤ λ∗k ln(2) ≤ 2R MAX Γσ20 gtk,m λ∗kR MAX − (2RMAX − 1) Γσ20 gtk,m , λ∗k ln(2) > 2R MAX Γσ20 gtk,m (B.26) Suppose that at ∗k,m > 0 for CRUs k1, k2, · · · , kC , C > 1. Constraint (B.2) implies at ∗k,m < 1, k = k1, k2, · · · , kC . From (B.15) and (B.16), we have φt ∗k,m = 0 and βt ∗k,m = 0. Then, from 197 (B.25), we conclude that htk,m = γ t ∗ m , k = k1, k2, · · · , kC , i.e. htk1,m = h t k2,m = · · · = htkC ,m. (B.27) From (B.26), unless λ∗k ln(2) < Γσ20 gtk,m , it is highly unlikely that any of the two htk,m will be equal since htk,m is a function of g t k,m, and {gtk,m, k = k1, k2, · · · , kC} are outcomes of independent, real-valued random variables. We conclude that for any given time slot t and subchannel m ∈ M, there is only one CRU, say CRU k′, that has a non-zero value and at ∗k′,m = 1 from (B.2). In the following, we will show how to find this CRU k′. Since at ∗k′,m = 1, from (B.15) and (B.16), we obtain φ t ∗ k′,m ≥ 0 and βt ∗k′,m = 0, respectively. Then from (B.25), we have γt ∗m = h t k′,m − φt ∗k′,m . (B.28) For CRUs that k 6= k′, since at ∗k,m = 0, from (B.15) and (B.16), we obtain φt ∗k,m = 0 and βt ∗k,m ≥ 0, respectively. Then from (B.25), we have γt ∗m = h t k,m + β t ∗ k,m . (B.29) From (B.28), (B.29), and (B.10), we obtain htk′,m ≥ htk,m . (B.30) As mentioned earlier, it is highly unlikely that any two htk,m will be equal, so we conclude that the subchannel allocation strategy for subchannel m at time slot t should be at ∗k,m = 1, k = k ∗ and m ∈Mt at ∗k,m = 0, otherwise (B.31) 198 where k∗ = argmaxk h t k,m . (B.32) In the unlikely event that for a given t and m, the htk,m values are equal for several users, we arbitrarily set at ∗k,m to be 1 for one of the users and 0 for all the others. We now complete the proof of Theorem 3.1. Suppose that the set of {λk, k = 1, 2, · · · , K} which allows the user rates {RU,REQk , k = 1, 2, · · · , K} to be achieved is {λ∗k, k = 1, 2, · · · , K}. It follows from (6.15) and (B.23) that RU,REQk = U∑ t=1 M∑ m=1  0, λk ln(2) < Γσ20 gtk,m at ∗k,m log2 ( λkg t k,m ln(2)Γσ20 ) , Γσ20 gtk,m ≤ λk ln(2) ≤ 2R MAX Γσ20 gtk,m RMAX , λk ln(2) > 2R MAX Γσ20 gtk,m . (B.33) Let the probability that a given subchannel is assigned to CRU k be pk. Assume that the probability that the power gain of the assigned channel satisfies Γσ20/g t k,m ≤ λk/ ln(2) ≤ 2R MAX Γσ20/g t k,m, i.e. Γσ 2 0 ln(2)/λk ≤ gtk,m ≤ 2RMAXΓσ20 ln(2)/λk, is pak, and let the mean of log2(G t k,m) for such subchannels be Ḡk. Suppose that the probability that the power gain of the assigned channel satisfies λk/ ln(2) > 2 RMAXΓσ20/g t k,m is p b k. As m(1, U)→∞, we have RU,REQk = ∑ Γσ20 ln(2) λk ≤gtk,m≤ 2R MAX Γσ20 ln(2) λk ,at ∗k,m=1 log2 ( λkg t k,m ln(2)Γσ20 ) + ∑ gtk,m> 2R MAX Γσ20 ln(2) λk ,at ∗k,m=1 RMAX = m(1, U)pkp a k log2 ( λk ln(2)Γσ20 ) + ∑ Γσ20 ln(2) λk ≤gtk,m≤ 2R MAX Γσ20 ln(2) λk ,at ∗k,m=1 log2(g t k,m) +m(1, U)pkp b kR MAX = m(1, U) ( pkp a k log2 ( λk ln(2)Γσ20 ) + pkp a kḠk + pkp b kR MAX ) (B.34) 199 Rewriting (B.34), we have RU,REQk m(1, U) = pkp a k log2 ( λk ln(2)Γσ20 ) + pkp a kḠk + pkp b kR MAX (B.35) At time t, the optimal number of bits allocated to CRU k is M∑ m=1 at ∗k,mr t ∗ k,m = M∑ m=1 at ∗k,m log2 ( 1 + gtk,ms t ∗ k,m Γσ20 ) . (B.36) As mCR,t →∞ and using the same subchannel assignment strategy as in (B.31), the prob- ability of a subchannel being assigned to CRU k, the probability that the power gain of the assigned channel satisfies λk/ ln(2) > 2 RMAXΓσ20/g t k,m, the probability that the power gain of the assigned channel satisfies Γσ20/g t k,m ≤ λk/ ln(2) ≤ 2RMAXΓσ20/gtk,m and the mean of log2(g t k,m) of these subchannels are given by pk, p b k, p a k, and Ḡk, respectively. Using a similar procedure to that leading to (B.33),(B.34) and (B.35), we obtain ∑M m=1 a t ∗ k,mr t ∗ k,m mCR,t = pkp a k log2 ( λk ln(2)Γσ20 ) + pkp a kḠk + pkp b kR MAX . (B.37) Finally, from (B.35) and (B.37), we have M∑ m=1 at ∗k,mr t ∗ k,m = R U,REQ k mCR,t m(1, U) . (B.38) B.2 Proof for Theorem 4.1 In Problem OP7, if we relax constraint (6.30) to allow ak,m to be a real value in [0, 1], and replace rk,m by r̂k,m = ak,mrk,m, the following convex optimization problem is obtained OP− B2 : min ak,m,r̂k,m,nk,pk K∑ k=1 (wknk − pk) (B.39) 200 subject to K∑ k=1 M∑ m=1 ak,m ( 2 r̂k,m ak,m − 1 ) Γσ20 gk,m − S ≤ 0 (B.40) ak,m − 1 ≤ 0,−ak,m ≤ 0, K∑ k=1 ak,m − 1 = 0 (B.41) − r̂k,m ak,m ≤ 0, r̂k,m ak,m −RMAX ≤ 0 (B.42) pk − rMAXk + rREQk ≤ 0, k ∈ {1, 2, · · · , KRT} (B.43) rREQk − M∑ m=1 r̂k,m − nk + pk = 0 (B.44) −pk ≤ 0,−nk ≤ 0 . (B.45) The functions {ak,m(2 r̂k,m ak,m − 1)Γσ20/gk,m, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} in (B.40) are convex in convex set C = {ak,m ∈ [0, 1], r̂k,m ∈ [0, RMAXak,m]} since the Hessian matrix of each function is positive semidefinite [1] in C. The functions in (B.39), and the LHS of (B.41) to (B.45) are affine. The Lagrangian [3][2] for OP-B2 can be written as L = K∑ k=1 (wknk − pk) + ν ( K∑ k=1 M∑ m=1 ak,m ( 2 r̂k,m ak,m − 1 ) − S ) + M∑ m=1 ϕm ( K∑ k=1 ak,m − 1 ) + K∑ k=1 M∑ m=1 φk,m(ak,m − 1)− K∑ k=1 M∑ m=1 βk,mak,m + K∑ k=1 M∑ m=1 µk,m ( r̂k,m ak,m −RMAX ) − K∑ k=1 M∑ m=1 ζk,m r̂k,m ak,m + KRT∑ k=1 ηk(pk − rMAXk + rREQk ) + K∑ k=1 γk(r REQ k − M∑ m=1 r̂k,m − nk + pk)− K∑ k=1 ψknk − K∑ k=1 ξkpk (B.46) where {ν, ϕm, φk,m, βk,m, µk,m, ζk,m, γk, ψk, ξk,m = 1, 2, · · · ,M, k = 1, 2, · · · , K}, and {ηk, k = 1, 2, · · · , KRT} are the Lagrange multipliers. Let {a∗k,m, r̂∗k,m, n∗k, p∗k, ν∗, ϕ∗m, φ∗k,m, β∗k,m, µ∗k,m, ζ∗k,m, γ∗k, ψ∗k, ξ∗k,m = 1, 2, · · · ,M, k = 201 1, 2, · · · , K}, and {η∗k, k = 1, 2, · · · , KRT} be an optimal solution; then the Karush-Kuhn- Tucker (KKT) conditions state that [3] ν∗ ≥ 0, ϕ∗m ≥ 0, η∗k ≥ 0, γ∗k ≥ 0, ψ∗k ≥ 0, ξ∗k ≥ 0, φ∗k,m ≥ 0, β∗k,m ≥ 0, µ∗k,m ≥ 0, ζ∗k,m ≥ 0 (B.47) ν∗ ( K∑ k=1 M∑ m=1 a∗k,m ( 2 r̂∗k,m a∗ k,m − 1 ) − S ) = 0 (B.48) ϕ∗m ( K∑ k=1 a∗k,m − 1 ) = 0 (B.49) φ∗k,m(a ∗ k,m − 1) = 0, β∗k,ma∗k,m = 0 (B.50) µ∗k,m ( r̂∗k,m a∗k,m −RMAX ) = 0, ζ∗k,m r̂∗k,m a∗k,m = 0 (B.51) η∗k(p ∗ k − rMAXk + rREQk ) = 0, k = 1, 2, · · · , KRT (B.52) γ∗k(r REQ k − M∑ m=1 r̂∗k,m − n∗k + p∗k) = 0 (B.53) ψ∗kn ∗ k = 0 (B.54) ξ∗kp ∗ k = 0 (B.55) ν∗2 r̂∗k,m a∗ k,m Γσ20 gk,m ln(2) + µ∗k,m a∗k,m − ζ ∗ k,m a∗k,m − γ∗k = 0 (B.56) ν∗ Γσ20 gk,m ( −1 + 2 r̂∗k,m a∗ k,m − r̂ ∗ k,m ln(2) a∗k,m 2 r̂∗k,m a∗ k,m ) + ϕ∗m + φ ∗ k,m − β∗k,m − µ∗k,mr̂ ∗ k,m a∗ 2k,m + ζ∗k,mr̂ ∗ k,m a∗ 2k,m = 0 (B.57) wk − γ∗k − ψ∗k = 0 (B.58) −1 + γ∗k + η∗k − ξ∗k = 0, k = 1, 2, · · · , KRT (B.59) −1 + γ∗k − ξ∗k = 0, k = KRT + 1, KRT + 2, · · · , K . (B.60) In addition, the optimal values ν∗, {a∗k,m, r̂∗k,m, φ∗k,m, β∗k,m, µ∗k,m, ζ∗k,m, m = 1, 2, · · · ,M, k = 1, 2, · · · , K}, {n∗k, p∗k, γ∗k, ψ∗k, ξ∗k, k = 1, 2, · · · , K}, {ϕ∗m,m = 1, 2, · · · ,M}, and {η∗k, k = 1, 2, · · · , KRT} need to satisfy (B.40) to (B.45). 202 Equations (B.56), (B.57), and (B.58) are obtained by setting ∂L r̂k,m = 0, ∂L ak,m = 0, and ∂L nk = 0, respectively. Equations (B.59) and (B.60) are obtained by setting ∂L pk = 0 for RT CRUs and NRT CRUs, respectively. It follows from (B.56) that µ∗k,m − ζ∗k,m ν∗a∗k,m ln(2) = γ∗k ν∗ ln(2) − 2 r̂∗k,m a∗ k,m Γσ20 gk,m . (B.61) From (B.51), we know that µ∗k,m and ζ ∗ k,m cannot be both positive, and from (B.47), we know that both µ∗k,m and ζ ∗ k,m are nonnegative. Hence the values of µ ∗ k,m and ζ ∗ k,m have to be in one of the following three cases µ∗k,m > 0, ζ ∗ k,m = 0 or µ ∗ k,m = 0, ζ ∗ k,m > 0 or µ ∗ k,m = 0, ζ ∗ k,m = 0 (B.62) When γ∗k ν∗ ln(2) < Γσ20 gk,m , it follows from (B.61) that µ∗k,m − ζ∗k,m < 0 and from (B.62), we must have µ∗k,m = 0 and ζ ∗ k,m > 0. From (B.51), we must have r̂ ∗ k,m = 0. When Γσ20 gk,m ≤ γ∗k ν∗ ln(2) ≤ 2R MAX Γσ20 gk,m , it follows from (B.61) and (B.42) that Γσ20 gk,m ≤ γ∗k ν∗ ln(2) − µ∗k,m−ζ∗k,m ν∗a∗k,m ln(2) ≤ 2RMAX Γσ20 gk,m . Thus, we must have µ∗k,m − ζ∗k,m = 0. Solving for r̂∗k,m from (B.61), we obtain r̂∗k,m = a ∗ k,m log2( γ∗kgk,m ν∗ ln(2)Γσ20 ). When γ∗k ν∗ ln(2) > 2 RMAX Γσ 2 0 gk,m , it follows from (B.61) that µ∗k,m−ζ∗k,m ν∗a∗k,m ln(2) = 2R MAX Γσ20 gk,m −2r̂∗k,m/a∗k,m Γσ20 gk,m > 0, so that µ∗k,m− ζ∗k,m > 0. Hence from (B.62), we must have µ∗k,m > 0 and ζ∗k,m = 0. Finally, from (B.51), we must have r̂∗k,m = a ∗ k,mR MAX . To summarize, we have the following optimal bit distribution policy r̂∗k,m =  0, λk < Γσ20 gk,m a∗k,m log2 ( λk gk,m Γσ20 ) , Γσ20 gk,m ≤ λk ≤ 2 RMAXΓσ20 gk,m a∗k,mR MAX , λk > 2R MAX Γσ20 gk,m (B.63) 203 where λk = γ ∗ kλB (B.64) and λB = 1 ν∗ ln(2) . (B.65) The term λk can be viewed as the waterlevel for CRU k. When the equivalent noise Γσ20/gk,m of subchannel m for CRU k is higher than the waterlevel, no bit is loaded onto the channel; when the waterlevel exceeds the threshold, 2R MAX Γσ20/gk,m, needed to achieve a rate of RMAX , a∗k,mR MAX bits are loaded onto the channel; otherwise, s∗k,m = λk − Γσ20/gk,m and r̂∗k,m = a ∗ k,m log2(λkgk,m/(Γσ 2 0)) bits are loaded onto the channel. Substituting r̂∗k,m in (B.57) by (B.63), and using (B.51) and (B.61) we have ν∗hk,m − ϕ∗m − φ∗k,m + β∗k,m = 0 (B.66) where hk,m is defined by hk,m =  0, λk < Γσ20 gk,m λk ln ( λk gk,m Γσ20 ) − ( λk − Γσ 2 0 gk,m ) , Γσ20 gk,m ≤ λk ≤ 2 RMAXΓσ20 gk,m λk ln(2)R MAX − (2RMAX − 1) Γσ20 gk,m , λk > 2R MAX Γσ20 gk,m . (B.67) Suppose that a∗k,m > 0 for CRUs k1, k2, · · · , kC , C > 1. Constraint (B.41) implies that a∗k,m < 1, k = k1, k2, · · · , kC . From (B.50), we have φ∗k,m = 0 and β∗k,m = 0. Then, from (B.66), we conclude that ν∗hk,m = ϕ∗m, k = k1, k2, · · · , kC , i.e. hk1,m = hk2,m = · · · = hkC ,m. (B.68) From (B.67), we note that unless λk < Γσ20 gk,m , it is highly unlikely that any two of {hk,m, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} will be equal since hk,m is a function of gk,m, and {gk,m, k = 204 k1, k2, · · · , kC} are outcomes of independent, real-valued random variables. We conclude that for any given time slot t and subchannel m ∈ M, there is only one CRU, say CRU k′, that has a non-zero value of a∗k′,m and a ∗ k′,m = 1 from (B.41). In the following, we describe how to determine k′. From (B.50), we have φ∗k′,m ≥ 0 and β∗k′,m = 0, and it follows from (B.66) that ϕ∗m = ν ∗hk′,m − φ∗k′,m . (B.69) For a CRU k 6= k′, a∗k,m = 0, and from (B.50), we have φ∗k,m = 0 and β∗k,m ≥ 0. It follows from (B.66) that ϕ∗m = ν ∗hk,m + β∗k,m . (B.70) From (B.69) and (B.70), we have hk′,m ≥ hk,m . (B.71) The subchannel allocation strategy for subchannel m at time slot t is then a∗k,m = 1, k = k ∗ and m ∈Mt a∗k,m = 0, otherwise (B.72) where k∗ = argmaxk hk,m . (B.73) In the unlikely event that for a given t and m, the hk,m values are equal for several users, we arbitrarily set a∗k,m to be 1 for one of the users and 0 for all the others. When n∗k > 0, it follows from (B.54) that ψ ∗ k = 0. Then, from (B.58), we must have γ∗k = wk, and from (B.64), we have λk = wkλB. When n∗k = 0, it follows from (B.54) that ψ ∗ k ≥ 0. Then, from (B.58), we must have 205 γ∗k ≤ wk, and from (B.64), we have λk ≤ λBwk. For NRT CRUs, when p∗k > 0, it follows from (B.55) that ξ ∗ k = 0. Then, from (B.60), γ∗k = 1 and from (B.64) λk = λB. Similarly, when p ∗ k = 0, γ ∗ k = 1 + ξ ∗ k and λk ≥ λB. For RT CRUs, when 0 < p∗k < (r MAX k − rREQk ), it follows from (B.52) and (B.55) that η∗k = 0 and ξ ∗ k = 0. From (B.59) and (B.64), we have γ ∗ k = 1 and λk = λB. When p ∗ k = 0, it follows from (B.52) and (B.55) that η∗k = 0 and ξ ∗ k ≥ 0. Then from (B.59) and (B.64), we have γ∗k = 1+ξ ∗ k and λk ≥ λB. When p∗k = (rMAXk −rREQk ), it follows from (B.52) and (B.55) that ξ∗k = 0 and η ∗ k ≥ 0. Finally, from (B.59) and (B.64), we have γ∗k = 1− η∗k and λk ≤ λB. 206 References [1] D. G. Luenberger, Introduction to Linear and Nonlinear programming. Reading, Mas- sachusetts: Addison-Wesley, 1973. [2] ——, Optimization by Vector Space Methods. New York: Wiley, 1969. [3] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge University Press, 2004. 207 Appendix C Flowcharts for Algorithm in Section 6.4.4 Fig. C.1 and Fig. C.2 show the flow charts for the resource-limited phase and the resource- abundant phase of the cross-layer RA algorithm, respectively. The flow chart for the As- signment algorithm is shown in Fig. C.3 in which fk,m is defined by fk,m =  h −1 k,m(maxk hk,m + δ) ak,m = 0 ∞ otherwise (C.1) where δ is a small positive number. 208 Figure C.1: Flow chart for the cross layer resource allocation algorithm: Phase 1, the resource-limited phase. Point B refers to the entry point of the resource-abundant phase. 209 Figure C.2: Flow chart for the cross layer resource allocation algorithm: Phase 2, the resource-abundant phase. 210 Figure C.3: Flow chart for the Assignment algorithm used in the cross layer resource allo- cation algorithm. 211 Appendix D Derivation of The Results in (7.8) - (7.10) In Problem OP2, if we relax constraint (7.5) to allow atk,m to be a real value in [0, 1] and replace rtk,m by r̂ t k,m = a t k,mr t k,m, the following convex optimization problem is obtained OP3 : max atk,m,r̂ t k,m T∑ t=1 K∑ k=1 M∑ m=1 r̂tk,m (D.1) subject to K∑ k=1 atk,m − 1 = 0 (D.2) T∑ t=1 K∑ k=1 M∑ m=1 atk,m ( 2 r̂tk,m at k,m − 1 ) Γσ20 gtk,m − TS ≤ 0 (D.3) RPRk d− T∑ t=1 M∑ m=1 r̂tk,m = 0 (D.4) −r̂tk,m ≤ 0 (D.5) atk,m − 1 ≤ 0, m ∈Mt (D.6) −atk,m ≤ 0, m ∈Mt (D.7) atk,m = 0, m ∈ M̄t (D.8) −d < 0 (D.9) The functions {atk,m(2 r̂tk,m at k,m−1)Γσ20/gtk,m, t = 1, 2, · · · , T, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} in (D.3) are convex in convex set C = {atk,m ∈ [0, 1], r̂tk,m ∈ [0,∞)}, since the Hessian matrix of each function is positive semidefinite [1] in C. The functions in (D.1), the LHS of (D.2), 212 and (D.4) to (D.9) are affine. The Lagrangian [2, 3] for OP3 can be written as L = − T∑ t=1 K∑ k=1 M∑ m=1 r̂tk,m + T∑ t=1 M∑ m=1 ϕtm ( K∑ k=1 atk,m − 1 ) +ν ( T∑ t=1 K∑ k=1 M∑ m=1 atk,m ( 2 r̂tk,m at k,m − 1 ) Γσ20 gtk,m − TS ) + T∑ t=1 K∑ k=1 M∑ m=1 φtk,m(a t k,m − 1)− T∑ t=1 K∑ k=1 M∑ m=1 βtk,ma t k,m + K∑ k=1 γk(R PR k d− T∑ t=1 M∑ m=1 r̂tk,m)− T∑ t=1 K∑ k=1 M∑ m=1 ζtk,mr̂ t k,m − θd (D.10) where {ϕtm, ν, φtk,m, βtk,m, γk, ζtk,m, θ, t = 1, 2, · · · , T, k = 1, 2, · · · , K,m = 1, 2, · · · ,M} are the Lagrange multipliers. Let {at ∗k,m, r̂t ∗k,m, φt ∗k,m, βt ∗k,m, ζt ∗k,m, t = 1, 2, · · · , T, k = 1, 2, · · · , K, m = 1, 2, · · · ,M}, d∗, ν∗, θ∗, {ϕt ∗m , t = 1, 2, · · · , T,m = 1, 2, · · · ,M}, {γ∗k, k = 1, 2, · · · , K} be an optimal 213 solution; then the Karush-Kuhn-Tucker (KKT) conditions state that [3] ϕt ∗m ≥ 0, ν∗ ≥ 0, φt ∗k,m ≥ 0, βt ∗k,m ≥ 0, γ∗k ≥ 0, ζt ∗k,m, θ∗ ≥ 0 (D.11) ϕt ∗m ( K∑ k=1 at ∗k,m − 1 ) = 0 (D.12) ν∗ ( T∑ t=1 K∑ k=1 M∑ m=1 at ∗k,m ( 2 r̂t ∗k,m at ∗ k,m − 1 ) Γσ20 gtk,m − TS ) = 0 (D.13) φt ∗k,m(a t ∗ k,m − 1) = 0 (D.14) βt ∗k,ma t ∗ k,m = 0 (D.15) ζt ∗k,mr̂ t ∗ k,m = 0 (D.16) γ∗k(R PR k d ∗ − T∑ t=1 M∑ m=1 r̂t ∗k,m) = 0 (D.17) θ∗d∗ = 0 (D.18) ν∗ Γσ20 gtk,m ( 2 r̂t ∗k,m at ∗ k,m − 1− 2 r̂t ∗k,m at ∗ k,m ln(2) r̂t ∗k,m at ∗k,m ) + ϕt ∗m + φ t ∗ k,m − βt ∗k,m = 0 (D.19) −1 + ν∗ ln(2)2 r̂t ∗k,m at ∗ k,m Γσ20 gtk,m − γ∗k − ζt ∗k,m = 0 . (D.20) In addition, the optimal values {at ∗k,m, r̂t ∗k,m, φt ∗k,m, βt ∗k,m, ζt ∗k,m, t = 1, 2, · · · , T, k = 1, 2, · · · , K, m = 1, 2, · · · ,M}, d∗, ν∗, θ∗, {ϕt ∗m , t = 1, 2, · · · , T,m = 1, 2, · · · ,M}, {γ∗k, k = 1, 2, · · · , K} need to satisfy (D.2)-(D.9). Equations (D.19), (D.20) are obtained by setting ∂L ∂atk,m = 0, ∂L ∂r̂tk,m = 0, respectively. Solving for r̂t ∗k,m/a t ∗ k,m from (D.20) and notice that r t ∗ k,m = r̂ t ∗ k,m/a t ∗ k,m ≥ 0, we obtain rt ∗k,m = a t ∗ k,m [ log2 ( λk gtk,m Γσ20 )]+ . (D.21) where λk = γ∗k+1 ν∗ ln(2) can be interpreted as the water level for CRU k. Substituting (D.21) into (D.19), we get ν∗htk,m − ϕt ∗m − φt ∗k,m + βt ∗k,m = 0 . (D.22) 214 where htk,m is calculated using (7.9). Suppose that at ∗k,m > 0 for CRUs k1, k2, · · · , kC , C > 1. Constraint (D.2) implies at ∗k,m < 1, k = k1, k2, · · · , kC . From (D.14) and (D.15), we have φt ∗k,m = 0 and βt ∗k,m = 0. Then, from (D.22), we conclude that ν∗htk,m = ϕ t ∗ m , k = k1, k2, · · · , kC , i.e. htk1,m = h t k2,m = · · · = htkC ,m. (D.23) From (7.9), we note that unless λk < Γσ20 gtk,m , it is highly unlikely that any two of {htk,m, t = 1, 2, · · · , k = 1, 2, · · · , K,m = 1, 2, · · · ,M} will be equal since htk,m is a function of gtk,m, and {gtk,m, k = k1, k2, · · · , kC} are outcomes of independent, real-valued random variables. We conclude that for any given time slot t and subchannel m ∈M, there is only one CRU, say CRU k′, that has a non-zero value and at ∗k′,m = 1 from (D.2). In the following, we describe how to determine k′. Since at ∗k′,m = 1, from (D.14) and (D.15), we obtain φ t ∗ k′,m ≥ 0 and βt ∗k′,m = 0, respectively. Then from (D.22), we have ϕt ∗m = ν ∗htk′,m − φt ∗k′,m . (D.24) For CRUs that k 6= k′, since at ∗k,m = 0, from (D.14) and (D.15), we obtain φt ∗k,m = 0 and βt ∗k,m ≥ 0, respectively. Then from (D.22), we have ϕt ∗m = ν ∗htk,m + β t ∗ k,m . (D.25) From (D.24), (D.25), and (D.11), we obtain htk′,m ≥ htk,m . (D.26) As mentioned earlier, it is highly unlikely that any two htk,m will be equal, so we obtain (7.8). In the unlikely event that for a given t and m, the htk,m values are equal for several users, 215 we arbitrarily set at ∗k,m to be 1 for one of the users and 0 for all the others. From (7.8) and (D.21), we obtain Characteristic 2). 216 References [1] D. G. Luenberger, Introduction to Linear and Nonlinear programming. Reading, Mas- sachusetts: Addison-Wesley, 1973. [2] ——, Optimization by Vector Space Methods. New York: Wiley, 1969. [3] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge: Cambridge University Press, 2004. 217

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