DESIGN OF SPECTRUM SENSING AND MAC IN COGNITIVE RADIO NETWORKS ZHENG SHOUKANG NATIONAL UNIVERSITY OF SINGAPORE 2012 DESIGN OF SPECTRUM SENSING AND MAC IN COGNITIVE RADIO NETWORKS ZHENG SHOUKANG (M Eng., National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 To my family i ii Acknowledgements I would like to take this opportunity to express my warmest thanks to many people who have helped and contributed in one way or another to the production of this thesis Without their support, the thesis would not have been written Among the wonderful people, the following individuals deserve special mention I thank Professor Kam Pooi Yuen, my main supervisor, and Dr Liang Ying-Chang, my co-supervisor, for their enlightened supervision, advice, time and tremendous efforts spent in teaching me, discussing with me, refining all my writings and most of all, their overall genuine concern for me I want to thank the Department of Electrical and Computer Engineering, National University of Singapore I also thank the Institute for Infocomm Research for providing me with all the necessary computing and communications facilities, and a congenial environment to carry out my research I must also thank my family for their constant love, care, concern and support during my embarkment and completion of my pursuit of a doctoral degree Last but not least, I would like to express my heartfelt gratitude to my colleagues and friends at the Institute for Infocomm Research for providing hearty help and happy hours together In particular, thanks to my buddies, Dr David Wong and Dr Zeng Yonghong, who have helped to proofread the thesis for me iii Contents Acknowledgements ii Contents iii Summary viii List of Tables xi List of Figures xii Abbreviations xxv Notations xxviii Chapter Introduction 1.1 Background 1.2 Motivation 1.3 Contributions of Thesis 1.3.1 Bayesian Detector for MPSK Modulated Signals 1.3.2 Cross-layered Design of Spectrum Sensing and MAC Protocol 1.3.3 MAC Protocol Design for Cooperative Spectrum Sensing Thesis Organization 1.4 Chapter Literature Review: Spectrum Sensing and Cognitive Radio MAC 11 2.1 Spectrum Sensing 11 2.1.1 Binary Hypothesis Testing 13 2.1.2 Detection Performance and Threshold 15 2.1.3 LRT-based Detection Sensing 16 2.1.4 Energy Detection Based Sensing 17 Contents iv 2.1.5 18 2.1.6 Matched-filter Based Sensing 20 2.1.7 Covariance-based Sensing 21 2.1.8 Wavelet-based Sensing 25 2.1.9 Cooperative Sensing 27 Cognitive Radio MAC 28 2.2.1 MAC for Centralized CRNs 28 2.2.2 MAC for Ad Hoc CRNs 30 2.2.3 2.2 Cyclostationary Detection Based Sensing Joint Design of Spectrum Sensing and MAC 38 Chapter Bayesian Spectrum Sensing for BPSK Modulated Primary Signals 39 3.1 Introduction 39 3.2 System Model and Optimal Detector Structure 40 3.3 Suboptimal Detector Structure 43 3.3.1 Approximation in the Low SNR Regime 44 3.3.2 Approximation in the High SNR Regime 45 Detection and False Alarm Probabilities 45 3.4.1 Detection Probability 46 3.4.2 False Alarm Probability 48 3.4.3 Analysis of ABD1 in the low SNR Regime 48 3.4.4 Analysis of ABD2 in the Low SNR Regime 51 3.4.5 Analysis in the High SNR Regime 52 3.5 Detection Threshold and Number of Samples 53 3.6 Simulation and Numerical Results 55 3.6.1 Low SNR Regime 56 3.6.2 High SNR Regime 61 3.6.3 Impact of Incorrect Prior 64 Summary 66 3.4 3.7 Chapter Bayesian Detector for MPSK Modulated Primary Signals 67 4.1 System Model and Optimal Detector Structure 67 4.2 Suboptimal Detector Structure 71 Contents v 4.2.1 Approximation in the Low SNR Regime 71 4.2.2 Approximation in the High SNR Regime 74 4.3 Detection Threshold and Number of Samples 75 4.4 Simulation and Numerical Results 76 4.4.1 AWGN Channels 77 4.4.2 Rayleigh Fading Channels 81 4.4.3 AWGN Channels versus Rayleigh Fading Channels 90 4.4.4 Performance Comparison of ABDs for B/Q/8PSK Signals 97 4.5 Summary 102 Chapter Bayesian Detector for Unknown Order MPSK Modulated Primary Signals 106 5.1 System Model and Optimal Detector Structure 106 5.2 Suboptimal Detector Structure 108 5.2.1 5.2.2 5.3 Low SNR Regime 108 High SNR Regime 109 Simulation and Numerical Results 109 5.3.1 AWGN Channels 110 5.3.2 Rayleigh Fading Channels 115 5.3.3 AWGN Channels versus Rayleigh Fading Channels 122 5.3.4 Performance Comparison of ABDs for Unknown and Known Order PSK Signals 128 5.4 Summary 133 Chapter Joint Design of Spectrum Sensing and MAC Protocol for Opportunistic Spectrum Access 136 6.1 Introduction 136 6.2 System Model 138 6.2.1 Spectrum Sensing 139 6.2.2 MAC Random Access 6.2.3 Constrained Optimization 143 140 6.3 Cross-layered and Layered Design Approaches 144 6.4 Numerical Results 150 6.4.1 Simulation Model 150 Contents vi 6.4.2 6.4.3 PU Idle Time 154 6.4.4 6.5 Mean Number of Backlogged SUs 150 Interference Constraint 157 Summary 158 Chapter Design of MAC Protocol for Cooperative Spectrum Sensing in Ad Hoc Cognitive Radio Networks 161 7.1 Introduction 161 7.2 Random Access for Cooperative Sensing 163 7.2.1 7.2.2 Cooperative Spectrum Sensing 165 7.2.3 Sequential Detection 167 7.2.4 7.3 System Model 163 Random Access in Control Channel 167 Upper Bound for Overall Throughput 171 7.3.1 7.3.2 Saturation Problem 176 7.3.3 Upper Bound for Overall Throughput 176 7.3.4 7.4 Average Service Time for Sensing Decision 171 Cooperative Sensing-Throughput Tradeoff 177 Sequential Detection with Prioritized Reporting 178 7.4.1 7.4.2 AND-rule Decision Fusion 180 7.4.3 7.5 OR-rule Decision Fusion 180 MAJORITY-rule Decision Fusion 181 Numerical Results 182 7.5.1 7.5.2 Impact of SNR 187 7.5.3 Impact of Sequential Detection 187 7.5.4 7.6 Impact of Frame Size 184 Comparison among Different Decision Rules 189 Summary 190 Chapter Conclusions and Future Work 194 Bibliography 198 Appendix A Computation of µY (k) and σY (k) in (3.50) and (3.52) 215 Appendix B Computation of µY (k) and σY (k) in (4.17) and (4.18) 219 Contents B.1 Computation of µY (k) and σY (k) under H0 B.2 Computation of µY (k) and σY (k) under H1 vii 219 222 Appendix C Proof of Proposition 229 Appendix D Proof in Chapter 230 D.1 Proposition 230 D.2 Proof of Theorem 231 D.3 Proof of Theorem 231 List of Publications 232 A Computation of µY (k) and σY (k) in (3.50) and (3.52) 218 Under H1 , x(k) − 2γ cos(φ(k)) ∼ N (0, 2γ) Since E[x2 (k)] = 2γ + 4γ , E[x4 (k)] = 12γ + 48γ + 16γ , E[x6 (k)] = 120γ + 720γ + 480γ + 64γ , E[x8 (k)] = 1680γ + 13440γ + 13440γ + 3584γ + 256γ , E[x10 (k)] = 30240γ + 302400γ + 403200γ + 161280γ +23040γ + 1024γ 10 , E[x12 (k)] = 665280γ + 7983360γ + 13305600γ +7096320γ + 1520640γ 10 + 135168γ 11 + 4096γ 12 , we obtain µY (k) = 48γ − 48γ + 120γ + γ (96 − 192γ + 720γ ) +γ (−64 + 480γ) + 64γ , σY (k) = (6912γ − 23040γ + 107520γ − 241920γ + 665280γ ) +γ (27648γ − 138240γ + 860160γ − 2419200γ + 7983360γ ) +γ (36864 − 92160γ + 860160γ − 3225600γ + 13305600γ ) +γ (−12288 + 229376γ − 1290240γ + 7096320γ ) +γ (16384 − 184320γ + 1520640γ ) +γ 10 (−8192 + 135168γ) + 4096γ 12 Therefore we can obtain (3.50) and (3.52) (A.9) 219 Appendix B Computation of µY (k) and σY (k) in (4.17) and (4.18) B.1 Computation of µY (k) and σY (k) under H0 Let xn = (k), which is Gaussian distributed, i.e xn ∼ N (µn , σn ), rewriting xn as follows: xn = µn + γ nc (k) cos(φn (k)) − ns (k) sin(φn (k)) N0 Under H0 and H1 , we can get E[x2 x2 ] = E[(xn − µn )2 (xm − µm )2 ] + 4µn µm E[xn xm ] n m 2 +µ2 σm + µ2 σn − 3µ2 µ2 n m n m (B.1) and E[xn xm ] = E[(xn − µn )(xm − µm )] + µn µm , E[x2 ] = σn + µ2 n n (B.2) (B.3) B.1 Computation of µY (k) and σY (k) under H0 220 Then E[vn1 (k)vn2 (k)] = 4γ E[n2 (k) cos(φn1 (k)) cos(φn2 (k))) + n2 (k) sin(φn1 (k)) sin(φn2 (k)))] + µn1 µn2 c s N0 = 2γ cos(φn1 (k) − φn2 (k)) + µn1 µn2 (B.4) Using (A.2), we have E[(vn1 (k) − µn1 )2 (vn2 (k) − µn2 )2 ] = 16γ E N0 −nc (k)ns (k) sin 2(φn1 (k)) + n2 (k) cos2 (φn1 (k)) + n2 (k) sin2 (φn1 (k)) c s × n2 (k) cos2 (φn2 (k)) + n2 (k) sin2 (φn2 (k)) − nc (k)ns (k) sin 2(φn2 (k)) c s N0 16γ = N0 N0 + 2 (cos2 (φn1 (k)) cos2 (φn2 (k)) + sin2 (φn1 (k)) sin2 (φn2 (k))) (cos2 (φn1 (k)) sin2 (φn2 (k)) + sin2 (φn1 (k)) cos2 (φn2 (k)) + sin 2(φn1 (k)) sin 2(φn2 (k))) = 4γ (1 + cos2 (φn1 (k) − φn2 (k))) (B.5) Under H0 , µn = 0, σn = 2γ Therefore E[vn (k)] = 0, (B.6) E[vn (k)] = 2γ (B.7) From (B.5), we obtain 2 E[vn1 (k)vn2 (k)] = 4γ (1 + cos2 (φn1 (k) − φn2 (k))) (B.8) With the fact that n cos2 (kx) = (csc(x) sin(2nx + x) + 2n + 3), k=0 (B.9) B.1 Computation of µY (k) and σY (k) under H0 221 for a given φn2 (k), where n2 ∈ {0, · · · , M/2 − 1},since φn1 (k) = 2n1 π , M where n1 ∈ {0, · · · , M/2 − 1}, we obtain M/2−1 n1 =0 cos2 (φn1 (k) − φn2 (k)) 2π M 2π M csc sin − +1 +2 − +3 M M M = , = (B.10) where M = 2, and for M = 2, M/2−1 n1 =0 cos2 (φn1 (k) − φn2 (k)) = 1, (B.11) Thus by (B.8) and (B.10), M/2−1 2 (k) E n=0 M/2−1 M/2−1 2 E[vn1 (k)vn2 (k)] = n =0 = n =0 2   2M γ , M = 2,   12γ , We can obtain (B.12) M = M/2−1 (k) µY (k) = E n=0 = Mγ, (B.13) M/2−1 σY (k) 2 (k) = E n=0 M/2−1 = E = − µY (k) 2 (k) n=0   2  M γ , M = 2,   8γ , M = Obviously, under H0 , we can get (4.17) − µ2 (k) Y (B.14) B.2 Computation of µY (k) and σY (k) under H1 B.2 222 Computation of µY (k) and σY (k) under H1 Given the transmitted PU signal (k) = √ Es ejφi (k) , γγ cos(φi (k) − φn (k) + θ) γ nc (k) cos(φn (k)) − ns (k) sin(φn (k)) N0 +2 (B.15) Let xn = (k), which is Gaussian distributed, i.e xn ∼ N (µn , σn ) for a given i and θ Under H1 , σn = 2γ With the fact that n cos(kx) = k=1 sin(n + 1)x + sin nx − sin x , sin x (B.16) we obtain M −1 cos k=0 = 1+ 2kπ M sin 2π + sin(M − 1) 2π − sin 2π M M sin 2π M = (B.17) Since n sin(kx) = csc k=0 x nx (n + 1)x sin sin , 2 (B.18) we have M −1 sin k=0 2kπ = M (B.19) It is easy to know that n sin2 (kx) = (− csc(x) sin(1 + 2n)x + 2n + 1) k=0 Thus for x = 2π , M (B.20) M = 2, M −1 k=0 sin2 M 2kπ = M (B.21) B.2 Computation of µY (k) and σY (k) under H1 223 Note that when M = 2, M −1 sin2 k=0 2kπ = M (B.22) Furthermore, we can obtain M −1 cos k=0 M −1 k=0 cos 4kπ +α M    = M , M = 2, (B.23)   cos2 α, M = 2, 4kπ +α M =    0, M = 2, (B.24)   cos α, M = By (B.16), for n1 = 0, 1, · · · , M/2 − and n2 = 0, 1, · · · , M/2 − 1, M/2−1 n1 =0 cos(φn1 (k) − φn2 (k)) M/2−1 = cos k=0 2kπ M = (B.25) Since cos(α) cos(β) = with equiprobability of φi (k) = (cos(α + β) + cos(α − β)), 2iπ ,i M (B.26) = 0, 1, , M − 1, for the case of fading channels with a random θ, we can obtain E[cos(φi (k) − φn1 (k) + θ) cos(φi (k) − φn2 (k) + θ)] E[cos(2φi (k) − (φn1 (k) + φn2 (k)) + 2θ) + cos(φn1 (k) − φn2 (k))] M −1 1 Eθ cos(2φi (k) − (φn1 (k) + φn2 (k)) + 2θ) + cos(φn1 (k) − φn2 (k)) = M i=0 = = cos(φn1 (k) − φn2 (k)) (B.27) B.2 Computation of µY (k) and σY (k) under H1 224 Note that by (B.24), for α = −(φn1 (k) + φn2 (k)) + 2θ and φi (k) = Eθ M −1 i=0 = 2iπ , M we have cos(2φi (k) − (φn1 (k) + φn2 (k)) + 2θ)    0, M = 2,   2Eθ [cos(−(φn1 (k) + φn2 (k)) + 2θ)], M = 2, = (B.28) Specially when n1 = n2 , we have E[cos2 (φi (k) − φn1 (k) + θ)] = (B.29) Note that for M = and constant θ = for AWGN channels, E[cos2 (φi (k) − φn1 (k) + θ)] = (B.30) From the above, for fading channels with a uniformly distributed random variable θ, we can obtain conveniently E[vn1 (k)vn2 (k)] (B.31) = 4γγE[cos(φi (k) − φn1 (k) + θ) cos(φi (k) − φn2 (k) + θ)] + 4γ E[n2 (k) cos(φn1 (k)) cos(φn2 (k))) c N0 +n2 (k) sin(φn1 (k)) sin(φn2 (k)))] s = 2γ cos(φn1 (k) − φn2 (k))(γ + 1) Specially when n1 = n2 , we have E[vn (k)] = 2γ(γ + 1) (B.32) However, for BPSK signals (M = 2) over AWGN channels with constant θ = and γ = γ, we have E[vn (k)] = 2γ(2γ + 1) (B.33) B.2 Computation of µY (k) and σY (k) under H1 225 It is not difficult to derive that E[(nc (k) cos(φn1 (k)) − ns (k) sin(φn1 (k))) × (nc (k) cos(φn2 (k)) − ns (k) sin(φn2 (k)))] N0 cos(φn1 (k) − φn2 (k)), = (B.34) E[(nc (k) cos(φn1 (k)) − ns (k) sin(φn1 (k)))2 ] N0 , = (B.35) E[(nc (k) cos(φn1 (k)) − ns (k) sin(φn1 (k)))2 × (nc (k) cos(φn2 (k)) − ns (k) sin(φn2 (k)))] = 0, (B.36) E[(nc (k) cos(φn1 (k)) − ns (k) sin(φn1 (k)))2 × (nc (k) cos(φn2 (k)) − ns (k) sin(φn2 (k)))2 ] = E −nc (k)ns (k) sin 2(φn1 (k)) + n2 (k) cos2 (φn1 (k)) + c n2 (k) sin2 (φn1 (k)) × n2 (k) cos2 (φn2 (k)) + s c n2 (k) sin2 (φn2 (k)) − nc (k)ns (k) sin 2(φn2 (k)) s N0 = + N0 2 (cos2 (φn1 (k)) cos2 (φn2 (k)) + sin2 (φn1 (k)) sin2 (φn2 (k))) cos2 (φn1 (k)) sin2 (φn2 (k)) + sin2 (φn1 (k)) cos2 (φn2 (k)) + sin 2(φn1 (k)) sin 2(φn2 (k)) = N0 (1 + cos2 (φn1 (k) − φn2 (k))) Note that by (B.23), for φi (k) = 2iπ , M (B.37) α = −(φn1 (k) + φn2 (k)) + 2θ), where θ B.2 Computation of µY (k) and σY (k) under H1 226 is random, it is easy to obtain = = E[cos2 (2φi (k) − (φn1 (k) + φn2 (k)) + 2θ)]    , M =2   2Eθ [cos2 (−(φn1 (k) + φn2 (k)) + 2θ)], M = 2 (B.38) For AWGN channels and M = 2, since n1 = n2 , we have E[cos2 (2φi (k) − (φn1 (k) + φn2 (k)) + 2θ)] = (B.39) By (B.26), (B.28) and (B.38), for random θ, we have E[cos2 (φi (k) − φn1 (k) + θ) cos2 (φi (k) − φn2 (k) + θ)] E cos(2φi (k) − (φn1 (k) + φn2 (k)) + 2θ) + cos(φn1 (k) − φn2 (k)) = = E cos2 (2φi (k) − (φn1 (k) + φn2 (k)) + 2θ) + cos2 (φn1 (k) − φn2 (k)) +2 cos(2φi (k) − (φn1 (k) + φn2 (k)) + 2θ) cos(φn1 (k) − φn2 (k)) = 1 + cos2 (φn1 (k) − φn2 (k)) (B.40) For AWGN channels with constant θ = and M = 2, E[cos2 (φi (k) − φn1 (k) + θ) cos2 (φi (k) − φn2 (k) + θ)] = (B.41) B.2 Computation of µY (k) and σY (k) under H1 227 Thus with (B.29) and (B.27) and (B.34)-(B.40), we can obtain 2 E[vn1 (k)vn2 (k)] = E 4γγ cos2 (φi (k) − φn1 (k) + θ) + −ns (k) sin(φn1 (k)))2 + 8γ 4γ (nc (k) cos(φn1 (k)) N0 γ cos(φi (k) − φn1 (k) + θ) × N0 (nc (k) cos(φn1 (k)) − ns (k) sin(φn1 (k))) × 4γγ cos2 (φi (k) − φn2 (k) + θ) + −ns (k) sin(φn2 (k)))2 + 8γ 4γ (nc (k) cos(φn2 (k)) N0 γ cos(φi (k) − φn2 (k) + θ) × N0 (nc (k) cos(φn2 (k)) − ns (k) sin(φn2 (k))) = 16γ γ E[cos2 (φi (k) − φn1 (k) + θ) cos2 (φi (k) − φn2 (k) + θ)] 16γ γ N0 16γ γ N0 E[cos2 (φi (k) − φn1 (k) + θ)] + + E[cos2 (φi (k) − φn2 (k) + θ)] N0 N0 16γ N + (1 + cos2 (φn1 (k) − φn2 (k))) + + + N0 64γ γ N0 cos(φn1 (k) − φn2 (k))E[cos(φi (k) − φn1 (k) + θ) cos(φi (k) − φn2 (k) + θ)] + N0 + (B.42) For fading channels with a random θ, 2 E[vn1 (k)vn2 (k)] 16γ γ N0 16γ γ N0 1 + cos2 (φn1 (k) − φn2 (k)) + +0+ N0 2 N0 2 2 16γ N + (1 + cos2 (φn1 (k) − φn2 (k))) + + + N0 N0 64γ γ cos(φn1 (k) − φn2 (k)) cos(φn1 (k) − φn2 (k)) + N0 2 = 16γ γ = (1 + cos2 (φn1 (k) − φn2 (k)))2γ (2 + 4γ + γ ) (B.43) B.2 Computation of µY (k) and σY (k) under H1 228 For BPSK signals (M = 2) over AWGN channels with constant θ = and γ = γ, 2 E[vn1 (k)vn2 (k)] = 16γ γ + 16γ γ N0 16γ γ N0 +0+ N0 N0 2 16γ N0 (1 + cos2 (φn1 (k) − φn2 (k))) + + + N0 64γ γ N0 cos(φn1 (k) − φn2 (k)), + N0 + = 4γ (3 + 12γ + 4γ ) (B.44) Applying (B.10) and (B.11) for (B.43) and (B.44), we can obtain M/2−1 2 (k) E n=0 M/2−1 M/2−1 2 E[vn1 (k)vn2 (k)] = n2 =0 n1 =0   M γ (2 + 4γ + γ ), M = 2,     2 = M = 2, fading,  6γ (2 + 4γ + γ ),     4γ (3 + 12γ + 4γ ), M = 2, AWGN (B.45) Hence by (B.32) and (B.45) M/2−1 µY (k) = E = (k)  n=0   Mγ(1 + γ), fading or M = for AWGN, M/2−1 σY (k) (B.46)   2γ(1 + 2γ), M = for AWGN, 2 (k) = E n=0 − µ2 (k) Y   M γ (1 + 2γ),  M = 2,    2 =  4γ (2 + 4γ + 0.5γ ), M = for fading,     4γ (2 + 8γ), M = for AWGN Obviously, under H1 , we can get (4.18) (B.47) 229 Appendix C Proof of Proposition Proof We can obtain e−ax df (x) = (−ax(x + y) + y) dx (x + y)2 Let df (x) dx (C.1) = 0, we get the maximum point x∗ as follows: a(x∗ )2 + ayx∗ − y = or x∗ = y + 4y/a − y (C.2) With the above, we get d2 f (x) dx2 e−ax − (x + y)2 = x=x∗ ′ (ax2 + ayx − y ) − =0 e−ax (2ax + ay) (x + y)2 < x=x∗ (C.3) ∗ Therefore x = √ y +4y/a−y is the maximum point of f (x), which is concave in x We can rewrite the maximum value of f (x) in (C.2) as g(y) = where c = c − y −0.5a(c−y) e , c+y (C.4) y + 4y/a, which is a function of y It is easy to derive dg(y) 2e−0.5a(c−y) (y − c) = dy (c + y)2 Since c > y, dg(y) dy (C.5) < Thus g(y) is a monotonically decreasing function of y so that the maximum value of f (x) is c−yl −0.5a(c−yl ) e , c+yl if the lower bound of y is yl 230 Appendix D Proof in Chapter D.1 Proposition Proposition tr,i (n) and tr (n) is a monotonically decreasing function of n (the number of SUs transmitting sensing reports or decision fusion) for 802.11 DCF and 802.11e EDCA based random access on the control channel Proof Let a = σ − Tc,i and b = (Ts,i −Tc,i )β 1−β have that tr,i (n) = Differentiating tr,i with respect to n gives tr,i (n) ′ = (1−β)n β(1−p) Tc,i +(1−β)n (a+nb) β(1−p) b+ n(a+nb) 1−β After simple manipulation, we Notice that a < 0, b < and ≤ β ≤ and ≤ p ≤ It is clear that tr,i (n) ′ < so tr,i (n) is a monotonically decreasing function of n Similarly, letting a = σ − Tc and b = (Ts −Tc )β , 1−β we can prove tr (n) is a monotonically decreasing function of n Proposition tr,i (n) is an increasing function of i (i is the priority category of AIFS) Proof 10 With the same n, Ptr , Ps , β and p are all the same by 802.11 DCF and 802.11e EDCA model From (7.24), we know the difference among tr,i (n) is in Ts,i and Tc,i , which is an increasing function of AIF Si Notice that AIF Si is D.2 Proof of Theorem 231 an increasing function of i according to (7.12) so tr,i (n) is an increasing function of i D.2 Proof of Theorem Proof 11 According to Proposition 4, the first term at R.H.S of (7.26) is not more than that of (7.27) and (7.28) Furthermore, if DIF S > AIF S1 > AIF S3 , then Ts,3 < Ts Noticing that Tc,i = Tc , we obtain tr,3 (n) ≤ tr (n) Thus we know that tr,3 (M) < tr (M) i.e the first term at R.H.S of (7.26) is less than that of (7.29) If DIF S > AIF S1 , tr,1 (M) < tr (M) Thus the second term at R.H.S of (7.26) is less than that of (7.29) Also, the third term at R.H.S of (7.26) is less than that of (7.29) as it is obvious that Ts,0 < Ts when DIF S > AIF S1 > (i) AIF S0 So Scheme A has a shorter tr for channel i than Scheme B/C/D/E If DIF S > AIF S1 , we can find that Scheme A < D < C < B < E in the increasing order of average service time D.3 Proof of Theorem (i) Proof 12 Since tr of Scheme A is shortest among the four schemes (Scheme A, B, C and D), and if DIF S > AIF S1 , the average service time of Scheme A is also shorter than that of Scheme E, it is straightforward that according to (7.30), Scheme A is better than Scheme B/C/D/E in terms of the overall throughput for (i) the same set of parameters of τ (i) and tb as well as channel selection I (i) 232 List of Publications Conference Papers S Zheng, Y.-C Liang, P Y Kam, and A T Hoang, “Cross-layered design of spectrum sensing and MAC for opportunistic spectrum access,” in Proc IEEE Wireless Communications and Networking Conf WCNC 2009, 2009, pp 1-6 S Zheng, Y.-C Liang, C K Tham, and P Y Kam, “Design of MAC with Cooperative Spectrum Sensing in Ad Hoc Cognitive Radio Networks,” in Proc IEEE 20th Int Personal, Indoor and Mobile Radio Communications Symp, 2009, pp 2581-2585 S Zheng, P.-Y Kam, Y.-C Liang, and Y Zeng, “Bayesian Spectrum Sensing for Digitally Modulated Primary Signals in Cognitive Radio,” in Proc IEEE 73rd Vehicular Technology Conf (VTC Spring), 2011, pp 1-5 (Best Paper Award) Journal Papers S Zheng, P.-Y Kam, Y.-C Liang, and Y Zeng, “Spectrum Sensing for Digital Primary Signals in Cognitive Radio: A Bayesian Approach for Maximizing Spectrum Utilization,” Under preparation ... the design of cognitive radio MAC is no longer an independent task for MAC layer, the researchers have been learning and investigating the approach on joint design of spectrum sensing and MAC. .. spectrum sensing and cognitive radio MAC protocols The fundamentals of various spectrum sensing methods and cognitive radio MAC protocols are described and compared in the literature review on existing.. .DESIGN OF SPECTRUM SENSING AND MAC IN COGNITIVE RADIO NETWORKS ZHENG SHOUKANG (M Eng., National University of Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF