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Spectrum sensing and throughput maximization in cognitive radio networks Moghimi, Farzad 2012

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SPECTRUM SENSING AND THROUGHPUT MAXIMIZATION IN COGNITIVE RADIO NETWORKS by Farzad Moghimi B.Sc., Ferdowsi University of Mashhad, Iran, 2006 M.Sc., University of Tehran, Iran, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT 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) July 2012 c Farzad Moghimi, 2012  Abstract In cognitive radio (CR) systems, reliable spectrum sensing techniques are required in order to avoid interference to the primary users (PUs) of the spectrum. In this dissertation two spectrum sensing techniques are developed, and sensing time and power allocation are optimized in multi-input multi-output (MIMO) CR systems. The motivation of the first proposed spectrum sensing technique is that, in practice, CRs also have to cope with various types of non-Gaussian noise such as manmade impulsive noise, and co-channel interference. However, most of the existing literature on spectrum sensing only considers impairment by additive white Gaussian noise (AWGN). To address this issue, we propose an Lp –norm detector which has tunable parameters that can be adjusted for the underlying type of noise. We also propose an adaptive algorithm for optimization of the Lp –norm parameters which does not require any a priori knowledge of the noise statistics. The motivation for the second proposed spectrum sensing technique is that the signals transmitted by PUs often also contain known pilot symbols for synchronization and channel estimation purposes. Coherent correlation based spectrum sensing techniques can exploit these known symbols but waste the energy contained in the data symbols. Hence, while considering AWGN impairment, we propose a hybrid coherent/energy detection scheme which exploits both the pilot and the data symbols transmitted by the PU. Since the complexity of the globally optimal hybrid detection metric is very high, we develop a simple locally optimal hybrid metric, which turns out to be a linear ii  Abstract combination of an energy detection metric and a correlation metric. While the proposed methods improve the accuracy of spectrum sensing, there exists a tradeoff between sensing time and transmission time for CR networks. In this thesis, we investigate this issue for conventional energy detection in MIMO CR networks. Specifically, we optimize the sensing threshold, sensing time, and transmit power of both single-band and multi-band MIMO CR systems for maximization of the opportunistic throughput under transmit power, probability of false alarm, and probability of detection constraints. We also develop efficient iterative algorithms for solving these non-convex optimization problems based on the concept of alternating optimization.  iii  Preface Hereby, I declare that I am the first author of this thesis. Chapters 2-4 are based on work that has been published or submitted for publication. The related publications were co-authored by Professor Robert Schober. The work in Chapter 2 was done in collaboration with Dr. Amir Nasri and the work in Chapters 3 and 4 were done in collaboration with Professor Ranjan K. Mallik from the Indian Institute of Technology, New Delhi. For all publications, I conducted the paper survey on related topics, performed the analysis, and carried out the simulations of the considered communication systems. I also wrote all paper drafts. The following publications are accomplished through this research. Journal Papers • Farzad Moghimi, Amir Nasri, and Robert Schober, “Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks,” IEEE Trans. Commun., vol. 59, no. 7, pp. 1934 - 1945, Jul. 2011. • Farzad Moghimi,  Robert Schober,  and Ranjan K. Mallik,  “Hybrid  Coherent/Energy Detection for Cognitive Radio Networks,” IEEE Trans. Wireless Commun., vol. 10, no. 5, pp. 1594 - 1605, May 2011. • Farzad Moghimi, Ranjan K. Mallik, and Robert Schober, “Sensing Time and Power Allocation in MIMO Cognitive Radio Networks,” submitted. iv  Preface Conference Papers • Farzad Moghimi, Amir Nasri, and Robert Schober, “Lp –Norm Spectrum Sensing for Cognitive Radio Networks Impaired by Non-Gaussian Noise,” in Proc. of IEEE Global Telecommunications Conference (Globecom), Honolulu, HI, December 2009. • Farzad Moghimi,  Robert Schober,  and Ranjan K. Mallik,  “Hybrid  Coherent/Energy Detection for Cognitive Radio Networks,” in Proc. of IEEE Global Telecommunications Conference (Globecom), Miami, FL, December 2010. • Farzad Moghimi, Ranjan K. Mallik, and Robert Schober, “Sensing Time and Power Allocation in MIMO Cognitive Radio Networks,” to be submitted.  v  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  x  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xi  List of Abbreviations and Symbols . . . . . . . . . . . . . . . . . . . . .  xv  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix 1 Introduction 1.1  1.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Channel Access Schemes  1  . . . . . . . . . . . . . . . . . . . . . . . .  3  1.1.1  Underlay Scheme . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.1.2  Overlay Scheme  . . . . . . . . . . . . . . . . . . . . . . . . .  4  1.1.3  Interweave Scheme . . . . . . . . . . . . . . . . . . . . . . . .  4  Spectrum Sensing  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  6  1.2.1  Energy Detection . . . . . . . . . . . . . . . . . . . . . . . . .  7  1.2.2  Coherent Detection  . . . . . . . . . . . . . . . . . . . . . . .  8  1.2.3  Feature-Based Detection . . . . . . . . . . . . . . . . . . . . .  9  vi  Table of Contents 1.2.4  Other Sensing Methods  . . . . . . . . . . . . . . . . . . . . .  9  1.3  Sensing-Throughput Tradeoff . . . . . . . . . . . . . . . . . . . . . .  10  1.4  Contributions and Results  . . . . . . . . . . . . . . . . . . . . . . .  12  1.5  Thesis Organization  . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  2 Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 16 2.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  16  2.2  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19  2.2.1  System Model  . . . . . . . . . . . . . . . . . . . . . . . . . .  19  2.2.2  Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . .  20  Lp –Norm Spectrum Sensing . . . . . . . . . . . . . . . . . . . . . . .  22  2.3.1  Optimal Detection  . . . . . . . . . . . . . . . . . . . . . . .  22  2.3.2  Lp –Norm Detection  . . . . . . . . . . . . . . . . . . . . . . .  24  2.3.3  Optimization of Lp –Norm Decision Statistic  2.3  2.4  2.5  . . . . . . . . .  25  . . . . . . . . . . . . . . . . . . . . . . . . . .  26  2.4.1  Probabilities of False Alarm and Missed Detection for Given h  27  2.4.2  Average Probabilities of False Alarm and Missed Detection  .  29  2.4.3  Asymptotic Analysis of Pf and Pm for N → ∞ . . . . . . . .  32  Performance Analysis  Optimization of Decision Statistic  . . . . . . . . . . . . . . . . . . .  35  2.5.1  Deflection Coefficient  . . . . . . . . . . . . . . . . . . . . . .  35  2.5.2  Relation Between Neyman-Pearson Optimization and dL (p) .  36  2.5.3  Adaptive Algorithm  . . . . . . . . . . . . . . . . . . . . . . .  37  2.6  Numerical Results and Discussions . . . . . . . . . . . . . . . . . . .  39  2.7  Conclusion  49  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3 Hybrid Coherent/Energy Detection for Cognitive Radio Networks 50 3.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50 vii  Table of Contents 3.2  System Model  3.3  Hybrid Coherent/Energy Detection  3.4  3.5  3.6  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  54  . . . . . . . . . . . . . . . . . . . . . . .  54  3.3.1  Optimal Detection  3.3.2  Locally Optimal Detection  Performance Analysis  52  . . . . . . . . . . . . . . . . . . .  56  . . . . . . . . . . . . . . . . . . . . . . . . . .  60  3.4.1  Probability of False Alarm  . . . . . . . . . . . . . . . . . . .  60  3.4.2  Probability of Missed Detection . . . . . . . . . . . . . . . . .  64  3.4.3  Neyman-Pearson Optimization  67  3.4.4  Asymptotic Performance of Neyman-Pearson Optimization  . . . . . . . . . . . . . . . . . .  69  . . . . . . . . . . . . . . . . . . . . . . . . .  72  3.5.1  Known Pilot Positions . . . . . . . . . . . . . . . . . . . . . .  73  3.5.2  Estimated Pilot Positions . . . . . . . . . . . . . . . . . . . .  79  Results and Discussions  Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  84  4 Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85  4.1  Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  85  4.2  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87  4.2.1  System Model and Sensing/Transmission Protocol  . . . . . .  87  4.2.2  CR Pre- and Post-processing  . . . . . . . . . . . . . . . . . .  89  4.2.3  Spectrum Sensing  . . . . . . . . . . . . . . . . . . . . . . . .  91  4.3  4.4  Problem Formulation and Optimization 4.3.1  Problem Formulation  4.3.2  Iterative Algorithm  . . . . . . . . . . . . . . . .  93  . . . . . . . . . . . . . . . . . . . . . .  93  . . . . . . . . . . . . . . . . . . . . . . .  99  Extensions to Multi-Band CR Systems . . . . . . . . . . . . . . . . . 101 4.4.1  Problem Formulation  . . . . . . . . . . . . . . . . . . . . . . 101  viii  Table of Contents 4.4.2  Iterative Algorithm  4.5  Results and Discussions  4.6  Conclusions  . . . . . . . . . . . . . . . . . . . . . . . 104  . . . . . . . . . . . . . . . . . . . . . . . . . 105  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112  5 Summary of Thesis and Topics for Future Research 5.1  Summary of Contributions  5.2  Suggestions for Future Work  . . . . . . . . 114  . . . . . . . . . . . . . . . . . . . . . . . 114 . . . . . . . . . . . . . . . . . . . . . . 116  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118  Appendices A Pm for I.N.D. Case  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128  B Asymptomatic Analysis of Pm C Proof of Theorem 4.1  . . . . . . . . . . . . . . . . . . . . . . 129  . . . . . . . . . . . . . . . . . . . . . . . . . . . 131  ix  List of Tables 2.1  Noise moments for AWGN, GMN, and GGN with variance σz2 , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  4.1  29  Sensing time, power allocation, and opportunistic throughput for 2 × 2 CR MIMO system. γ1 = −12 dB, γ2 = −15 dB, λ1 = 2, λ2 = 0.5, and P = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108  4.2  Sensing time, power allocation, and opportunistic throughput for 2 × 3 CR MIMO system. γ1 = −10 dB, γ2 = −12 dB, γ3 = −15 dB, λ1 = 2, λ2 = 0.5, and P = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . 108  4.3  Sensing time, power allocation, and opportunistic throughput for 2 × 2 multi-band MIMO CR system with K = 4 bands. . . . . . . . . . . . 111  x  List of Figures 1.1  Cognitive radio network. . . . . . . . . . . . . . . . . . . . . . . . . .  2  1.2  Hidden terminal problem. . . . . . . . . . . . . . . . . . . . . . . . .  5  1.3  CR frame structure with frame length T , and sensing time τ . . . . . .  11  2.1  Pf vs. p for Lp –norm detection. L = 1, SNR = −15 dB, and P¯m = 0.1.  39  2.2  Pm vs. p for Lp –norm detection. L = 1, SNR = −15 dB, and P¯f = 0.1.  40  2.3  Deflection coefficient d(p) vs. p for Lp –norm detection. L = 1 and SNR = −15 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.4  41  Squared deflection coefficient dˆ2L (p) vs. iteration number k for adaptive FDSA algorithm with unconstrained p (i.e., p1 = p2 possible) and constrained p (i.e., p1 = p2 = p). For comparison, the deflection coefficient for energy detection (p1 = p2 = 2) is also shown. L = 2 and SNR = −15 dB. Noise I: AWGN in both branches. Noise II: ǫ–mixture noise (κ = 10, ǫ = 0.01) in branch 1 and GGN (β = 3) in branch 2. Noise III: ǫ–mixture noise (κ = 10, ǫ = 0.01) in branch 1 and AWGN in branch 2.  2.5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  Pf vs. N for energy detection (p = 2), Lp –norm detection (p = popt ), and LO detection. L = 1, SNR = −15 dB, and P¯m = 0.1. . . . . . . .  43  xi  List of Figures 2.6  Pf vs. N for Lp –norm detection with optimal p for different numbers of i.i.d. diversity branches L. SNR = −15 dB and P¯m = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3).  2.7  44  Pm vs. N for Lp –norm detection with optimal p for different numbers of i.i.d. diversity branches L. SNR = −15 dB and P¯f = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3).  2.8  . . . . . . . .  45  Pf vs. N for energy detection (p1 = p2 = p3 = 2), Lp –norm detection (constrained and unconstrained p), and LO detection. L = 3 and P¯m = 0.1. Branch 1: SNR = −15 dB, ǫ–mixture noise (κ = 10, ǫ = 0.01). Branch 2: SNR = −18 dB, AWGN. Branch 3: SNR = −21 dB, GGN (β = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.9  46  ROC for energy detection (p = 2) and Lp –norm detection with optimal p for L = 3 diversity branches and N = 20000. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3).  . . . . . . . . . . . . . .  47  2.10 Pm vs. SNR for energy detection (p = 2) and Lp –norm detection with optimal p. L = 1, N = 10000, and P¯f = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3). 3.1  . . . . . . . . . . . . . .  48  Example for two different pilot placement patterns with N = 9 samples, Np = 3 pilots, and pilot-to-received samples ratio β  Np /N =  1/3: (a) Uniform pilot placement with P = {1, 4, 7} and D = {2, 3, 5, 6, 8, 9}. (b) Concentration of pilots at the beginning of frame with P = {1, 2, 3} and D = {4, 5, 6, 7, 8, 9}. . . . . . . . . . . . . . . . . . . . . . . . . . 3.2  53  Pf of hybrid detection and energy detection vs. N . Positions of the pilots are known, β = 0.05, SNR = −10 dB, P¯m = 0.1, and number of diversity branches are L = 1, 2, and 3. . . . . . . . . . . . . . . . . .  74  xii  List of Figures 3.3  Pm of hybrid detection and energy detection vs. N . Positions of the pilots are known, β = 0.05, SNR = −10 dB, P¯f = 0.1, and number of diversity branches are L = 1, 2, and 3. . . . . . . . . . . . . . . . . .  3.4  75  Pf of coherent detection and hybrid detection with and without channel estimation (CE) vs. 100 × β %. Positions of the pilots are known, N = 2000, SNR = −10 dB, and P¯m = 0.1. . . . . . . . . . . . . . . .  3.5  76  Pm of coherent detection and hybrid detection with and without channel estimation (CE) vs. 100 × β %. Positions of the pilots are known, N = 2000, SNR = −10 dB, and P¯f = 0.1. . . . . . . . . . . . . . . . .  3.6  77  ROC of hybrid detection with and without channel estimation (CE), coherent detection, and energy detection for SNR = −5 dB, −10 dB, and −15 dB. Positions of the pilots are known, N = 2000, L = 3, and β = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  3.7  79  Pf of hybrid detection, coherent detection, and energy detection vs. N. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, β = 0.05, SNR = −10 dB, and P¯m = 0.1. . . . . . . . . . .  3.8  81  Pm of hybrid detection, coherent detection, and energy detection vs. N. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, β = 0.05, SNR = −10 dB, and P¯f = 0.1. . . . . . . . . . . .  3.9  82  Pf of hybrid detection, coherent detection, and energy detection vs. 100× β %. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, N = 2000, SNR = −10 dB, and P¯m = 0.1. . . .  4.1  83  Block diagram of the system model. The CR system transmits Mt independent data streams and performs sensing at the receiver in all Mr subchannels. The PU transmits MP independent data streams. .  88  xiii  List of Figures 4.2  Proposed frame structure for CR system with Mt = 3 transmit and Mr = 4 receive antennas. The first Mt = 3 subchannels are used for both sensing and data transmission. The last Mr − Mt = 1 subchannel is only used for sensing.  4.3  89  Opportunistic throughput in single-band case vs. number of convex/waterfilling problems solved. γ  4.4  . . . . . . . . . . . . . . . . . . . . . . . . .  [γ1 . . . γMr ] and λ  [λ1 . . . λMt ]. . . . . . . . . 106  Opportunistic throughput in multi-band case vs. number of convex/waterfilling problems solved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107  4.5  Opportunistic throughput of a single-band MIMO CR system vs. λ1 /λ2 . γ1 = −12 dB, γ2 = −15 dB, and P = 10 dB. . . . . . . . . . . . . . . 110  4.6  Opportunistic throughput of a multi-band MIMO CR system vs. P for K = 2 and K = 4 bands. . . . . . . . . . . . . . . . . . . . . . . . . . 112  xiv  List of Abbreviations and Symbols Acronyms AO  Alternating Optimization  AWGN  Additive White Gaussian Noise  CCI  Co-Channel Interference  CDF  Cumulative Distribution Function  CE  Channel Estimation  CR  Cognitive Radio  CRN  Cognitive Radio Network  CSI  Channel State Information  CSIT  Channel State Information at Transmitter  ED  Energy Detection  ETSI  European Telecommunications Standard Institute  FCC  Federal Communications Commission  FDSA  Finite Difference Stochastic Approximation  GGN  Generalized Gaussian Noise  GMN  Gaussian Mixture Noise  GO  Globally Optimal  IEEE  Institute of Electrical and Electronics Engineers  I.i.d.  Independent and Identically Distributed xv  List of Abbreviations and Symbols I.n.d.  Independent Non-Identically Distributed  INR  Interference-to-Noise Ratio  KKT  Karush Kuhn Tucker  LO  Locally Optimal  LR  Likelihood Ratio  MGF  Moment Generating Function  MIMO  Multiple-Input Multiple-Output  MRC  Maximum Ratio Combining  NTIA  National Telecommunications and Information Administration  Pdf  Probability density function  PSK  Phase-Shift Keying  PU  Primary User  QPSK  Quaternary Phase-Shift Keying  ROC  Receiver Operating Characteristic  SINR  Signal-to-Interference-plus-Noise Ratio  SIR  Signal-to-Interference Ratio  SNR  Signal-to-Noise Ratio  SU  Secondary User  SVD  Singular Value Decomposition  UWB  Ultra-Wideband  Notations and Operators Bold upper case and lower case letters denote matrices and vectors, respectively. The remaining notation and operators used in this thesis are listed below:  xvi  List of Abbreviations and Symbols (·)T  Transpose  (·)H  Hermitian transpose  (·)∗  Complex conjugate  tr(·)  Trace of a matrix  rank(·)  Rank of a matrix  IN ×N  N × N identity matrix  diag(x)  Diagonal matrix with elements of vector x in its main diagonal  0  All-zero column vector  1  All-ones column vector  1X  The X-dimensional all-ones column vector  Ex {·}  Statistical expectation with respect to x  px (·)  Pdf of x  px|y (·|·)  Pdf of x conditioned on y  Pr{A}  Probability of event A  Q(·)  Gaussian Q-function  Γ(·)  Gamma function  Γ(·, ·)  Upper incomplete Gamma function  γ(·, ·)  Lower incomplete Gamma function  sgn(·)  Sign function  N(µ, σ 2)  Gaussian distribution with mean µ and variance σ 2  Γ(α, β)  Gamma distribution with shape parameter α and scale parameter β  A≡B  A is equivalent to B  erf(·)  Error function  det(·)  Matrix determinant  ·  +  max{0, ·} xvii  List of Abbreviations and Symbols ℜ{·}  Real part of a complex number  ℑ{·}  Imaginary part of a complex number  CN ×M  Space of all N × M matrices with complex entries  |·|  Absolute value of a complex number  ·  2  Euclidean norm  xviii  Acknowledgments First and foremost, I would like to express my utmost gratitude to my supervisor, Professor Robert Schober, for his invaluable assistance, support, and guidance throughout the course of this work. This thesis was simply impossible without him. His constant encouragement and support helped me overcome the challenges that I faced while pursuing my PhD. I would also like to thank Dr. Amir Nasri and Dr. Ranjan K. Mallik for their valuable comments and assistance. I am also thankful to my best friends Mahmood, Mahdi, Neda, Nazanin, Hamid, and Mahdi Tayarani. I am grateful to all my friends in communication lab, in particular, Ehsan Bayaki, Mohammad Mohammadnia, Zahra, and Pedram, I really enjoyed working in such an inspirational environment. Many special thanks to all the friends I made along the way in Vancouver, especially Azadeh, Pooya, Pirooz, Nasim, Negar, Nader, Maryam, Behnam, and Sahar. I would like to thank my parents, my brother, Ali, and my sister, Atoosa, for believing in me and encouraging me throughout my life. I am indebted to my father, and my mother for their endless love, care, and support. Last but not least, I would like to thank my lovely wife, Nina. We met at UBC during early stages of PhD and I experienced the best years of my life with her.  xix  Chapter 1 Introduction In traditional spectrum allocation, frequency bands are assigned to specific licensed users and cannot be used by others even if the licensed users are idle. The spectrum allocation is regulated by government agencies such as the Federal Communications Commission (FCC) in the United States and the European Telecommunications Standard Institute (ETSI) in Europe. Recently, the spectrum demand has increased significantly due to new wireless technologies and their diverse applications. However, the inflexible approach to spectrum allocation has led to a perceived scarcity of the frequency spectrum. This fact can be seen from the United States frequency allocation chart1 provided by the National Telecommunications and Information Administration (NTIA) which shows that most of the frequency spectrum has already been assigned and there is almost no room for future wireless technologies and products. However, according to a 2002 FCC report, the licensed spectrum usage is less than 15% on average, while the peak usage is close to 85% [1]. Several other spectrum measurements in different locations also confirmed the underutilization of the spectrum [2]–[5]. Cognitive Radio (CR) has recently emerged as a technology to overcome this problem by allowing unlicensed (secondary) users (SUs) to intelligently access the bands assigned to licensed (primary) users (PUs) without degrading their performance [6, 7, 8]. 1  Available at http://www.ntia.doc.gov/osmhome/allochrt.pdf for the range of 3 kHz to 30 GHz.  1  Chapter 1. Introduction Unlicensed users  Licensed Band-I  Licensed Band-II Spectrum band  Figure 1.1: Cognitive radio network. An example for a CR network is shown in Fig. 1.1. In this figure, three unlicensed users (CRs) are communicating with each other over two licensed frequency bands. CRs should meet several criteria to be able to perform such task. CRs must be intelligent, aware of their environments and the activities of PUs, and be capable of transmitting and receiving on different frequency bands [9]. Thus, by observing the environment, learning from it, and planning accordingly, CRs adjust their transmission strategies such as transmission time, power, and frequency spectrum, such that no harm comes to the PUs [6]. The remainder of this chapter is organized as follows. In Section 1.1, we discuss different channel access schemes in CR networks. In Sections 1.2 and 1.3, different spectrum sensing techniques and the sensing-throughput tradeoff in CR networks are discussed, respectively. Thesis contributions are summarized in Section 1.4, and the thesis organization is provided in Section 1.5.  2  Chapter 1. Introduction  1.1  Channel Access Schemes  Based on the PU network constraints and the awareness of the CRs of their environments, such as channel conditions and codebooks of PUs, three different channel access scenarios for CR networks have been considered: underlay, overlay, and interweave [10]. In the following, we discuss these different schemes.  1.1.1  Underlay Scheme  In this method, there is a constraint on the amount of interference that the PU receivers can tolerate. The CR user needs to have knowledge of the channel between its transmitter and PU receivers in order to estimate the amount of interference caused to the PU. This technique leads to simultaneous transmission of CRs and PUs as long as the amount of interference is kept below a certain threshold. There is a large body of literature in this area, e.g. [11]–[21]. Beamforming is used in [11]–[17] to maximize the capacity of the CR network while the interference to the PU is kept below the threshold. Specifically, in [13], multi-antenna CR and PU networks are considered and the beamforming vector design is formulated as a convex optimization problem and closed-form solutions are derived for some special cases. This approach was later extended to the case where only partial channel state information is available [15]. On the other hand, in [18]–[20] the concept of opportunistic interference alignment was introduced. In this technique, by assuming that the PU is using waterfilling (WF) to maximize its rate, the CR aligns its interference in the direction of unused PU channels to which zero power is assigned by the WF solution. The major drawback of the underlay scheme is the required channel knowledge from the CR transmitter to the PU receivers. In practice, this is very difficult to obtain, especially when no cooperation between PUs and CRs is assumed and the 3  Chapter 1. Introduction PU receivers do not transmit data or pilots.  1.1.2  Overlay Scheme  In this channel access scheme, the CR knows the codebook and the message of the PU and helps the PU by relaying its message. In this scheme, the CR and the PU can simultaneously exploit the spectrum. The CR uses some of its available power for transmitting its own message and uses the rest for relaying the PU message. The PU benefits from CR relaying and thus allows the CR to transmit its own message. As pointed out in [10], it is possible to adjust the power ratio of the CR message and the relayed message such that the signal-to-noise ratio (SNR) of the PU remains constant, i.e., the increase in SNR because of relaying and the decrease in SNR due to the interference cancel each other out. References [22], [23], and [24] provide examples of overlay transmission in CR networks. The major drawback of this scheme is the need for sophisticated coding schemes and the availability of the PU’s message and codebook at the CR transmitter [10].  1.1.3  Interweave Scheme  This is the most common channel access scheme for CR networks. In this scheme, no simultaneous transmission with PUs is allowed and CRs must perform spectrum sensing prior to transmission to ensure that the spectrum is vacant. The unused frequency bands are also referred to as spectrum holes, which vary with time and location. If a spectrum hole is detected, the CR can begin transmission. However, it must periodically perform spectrum sensing to ensure that the PU has not initiated any new transmission. If such transmission is detected, the CR must remain silent until the spectrum becomes vacant or it can switch to another spectrum hole in a 4  Chapter 1. Introduction Licensed Users  Unlicensed users  Interference Zone  Figure 1.2: Hidden terminal problem. different frequency band. However, there exist some challenges such as fading, shadowing, and the hidden terminal problem that affect the sensing performance. In Fig. 1.2, the hidden terminal problem in CR systems is illustrated. In this case, there is an ongoing transmission between the PU transmitter and receivers. However, the CR is out of the PU transmission range and is not capable of sensing it correctly. Hence, it may initiate transmission which interferes with the PU receiver [9]. Fading and shadowing are the main reasons for the hidden terminal problem. Due to the mentioned issues, the PU signal may be very weak at the CR receiver [26]. Thus, CRs are faced with the challenging task of detecting the presence of a very weak signal, with possibly unknown characteristics, in noise. In the IEEE 802.22 standard, it has been suggested that CRs should be able to sense PU receive power levels as low as -116 dBm [27], [28]. The focus of this thesis is on interweave CR systems. One of our research goals is  5  Chapter 1. Introduction to develop low-complexity spectrum sensing techniques capable of performing well for low sensing SNRs. Hence, in the following section, we will discuss different spectrum sensing methods in more detail.  1.2  Spectrum Sensing  In this section, we first define the probabilities of false alarm and missed detection. The probabilities of false alarm and missed detection are widely used as performance criteria for different sensing schemes. The probability of false alarm is the probability that the PU signal is absent but is sensed as present by the CR. This type of error causes no interference to the PU system but it wastes the opportunity of the CR to exploit the spectrum hole. Thus, the effect is the loss of a transmission opportunity of CR system. Another important type of error is the probability of missed detection, which is the probability that the PU is present but is sensed as absent by the CR. This is the main source of interference to PU systems in interweave access scheme. It should be noted that PUs are the owners of the spectrum and are usually not capable of mitigating the interference caused by unlicensed users. To avoid such interference, a certain maximum tolerable probability of missed detection may be imposed by the PU systems [29]. Comparisons between different spectrum sensing techniques are usually done based on these two types of error. A survey on different spectrum sensing algorithms is provided in [25]. In the following, we present several common spectrum sensing techniques used in CR systems. The most popular techniques used for spectrum sensing in CR networks are energy detection, coherent detection, and feature-based detection. These schemes differ in the required amount of a priori knowledge about the PU signal. Here, we briefly discuss each category with its advantages and disadvantages. 6  Chapter 1. Introduction  1.2.1  Energy Detection  Energy detection, which is also known as radiometry, is the most popular technique for spectrum sensing, where the amount of received energy is the criterion for making a decision regarding the presence of a PU [30]–[33]. The performance of energy detection over different fading channels (Rayleigh, Nakagami, and Ricean), and also for different combining methods, such as equal gain combining and selection combining was analyzed in [31]. In [34], multi-band joint energy detection is proposed which enables simultaneous spectrum sensing in multiple frequency bands and expedites channel selection and switching for CRs. While energy detection is easy to implement, it suffers from a relatively poor performance especially for low SNR [35]. Another drawback of energy detection is its poor performance under noise variance uncertainty. In [36], it was shown that in the presence of noise variance uncertainty, for SNRs below a certain threshold, reliable sensing is not possible even with infinite sensing time. This SNR threshold is referred to as SNR wall in the literature. To address the inefficiency in low SNR, and the issue of noise variance uncertainty, as well as the hidden terminal problem, several cooperative spectrum sensing techniques have been proposed in the literature [32], [37]–[43]. In cooperative spectrum sensing each CR performs sensing locally and the result is usually sent to another CR node known as fusion center (FC) which is in charge of making the final decision. A recent survey on different cooperative spectrum sensing techniques and challenges can be found in [44]. Energy detection is the likelihood-ratio optimal detector when both signal and noise are Gaussian random processes. However, a major drawback of energy detection is that, it does not perform well in non-Gaussian noise and interference [7, 26, 36]. In CR systems, non-Gaussian impairments can be due to co-channel interference from  7  Chapter 1. Introduction other CRs, or out-of-band spectral leakage from PUs operating in adjacent frequency bands. Also, man-made impulsive noise [45] and interference from ultra-wideband (UWB) systems, are examples of common non-Gaussian impairments. This issue motivates us to develop a simple spectrum sensing metric similar to energy detection with better performance for different types of non-Gaussian noise. The proposed detector should also be capable of performing well for low sensing SNRs.  1.2.2  Coherent Detection  Coherent detection can be employed for scenarios where some part of the PU transmit signal is known to the SUs. In the extreme case, where the transmitted signal is completely known at the SUs, matched filtering can be employed as the likelihood-ratio optimal sensing method under additive white Gaussian noise (AWGN) impairment [29]. One practical approach to coherent spectrum sensing is to exploit the PU pilot patterns [35], [36], [47]–[50]. However, while pilot symbols are typically emitted by PUs for synchronization and channel estimation purposes, the exact positions of these pilot symbols relative to the data symbols may not be known and have to be estimated by the CR receiver. In [35], coherent pilot-based detection and energy detection were compared and the superiority of coherent detection, especially for low SNRs, was confirmed. In [36] and [47], it was shown that in the low SNR regime, coherent pilot-based detection can overcome possible uncertainties regarding interference and noise distributions, whereas energy detection suffers a significant performance loss in this case. In [48], an experimental comparison of coherent pilot-based sensing and energy detection was performed and it was shown that residual frequency offsets cause an SNR wall in coherent detection. Pilot-based sensing techniques for IEEE 802.22 were proposed  8  Chapter 1. Introduction in [49] and [50], respectively. Although coherent detection outperforms energy detection if the pilot positions are known at the CR receiver, it wastes the energy contained in the PU’s data symbols. In addition, the performance degrades in the presence of pilot position estimation errors. This issue motivates us to develop a simple and efficient spectrum sensing method capable of exploiting both data and pilot symbols which also performs satisfactory under pilot position estimation errors.  1.2.3  Feature-Based Detection  This category includes a wide range of spectrum sensing techniques. It assumes that specific statistical properties of the transmitted PU signal are available a priori at the SUs, and the detector tries to exploit these features. These methods lead to significantly better performance in comparison with energy detection, especially at low SNRs. Cyclostationary-based detection belongs to this category, where the SUs try to sense the channel for a periodicity in the mean and the autocorrelation of the received signal which is usually generated when the modulated signals are coupled with sine wave carriers, pulse trains, hopping sequences or cyclic prefixes [51]–[56]. The required knowledge of the mentioned signal characteristics and a higher complexity than conventional energy detection are the main disadvantages of this category of detectors.  1.2.4  Other Sensing Methods  In addition to the aforementioned categories, here we briefly review some other popular spectrum sensing methods. Wavelet spectrum sensing is an approach that is especially useful for wideband channels. In this approach the frequency band is di9  Chapter 1. Introduction vided into several sub-bands and the spectrum holes can be detected by applying the wavelet transform to the power spectral density of the received signal [57, 58]. Another approach that uses a multitaper spectrum estimation method is proposed in [7]. This algorithm uses tapering to reduce the sidelobe leakage problem and is shown to be nearly optimal for wideband signals in the sense that it almost achieves the Cramer-Rao bound for a spectral estimator. Another option for improving spectrum sensing is to learn the spectrum usage of the PU by tracking the variations in the spectrum and using this information for predicting the future usage of spectrum. As an example, in [59] and [60] semi-Markov models are employed for spectrum usage prediction. Sequential sensing is another spectrum sensing technique. In this framework, the goal is to minimize the sensing time subject to some constraints on detection performance. Thus, instead of a fixed sensing time the samples are taken sequentially and the decision statistic is tested with the desired thresholds with every received sample and the sensing stops if the result is reliable [29]. References [61], [62], and [63] provide examples of sequential sensing methods for CR networks. Compressed sensing can also be applied to CR spectrum sensing [58, 64, 65]. In this technique, a sub-Nyquist sampling rate is used to recover a sparse received signal. This technique is very useful for wideband spectrum sensing where a high sampling rate is required.  1.3  Sensing-Throughput Tradeoff  As mentioned before spectrum sensing must be performed periodically to ensure that the PU is sufficiently protected. A sample frame structure of a CR network is shown in Fig. 1.3. The CR frame comprises T symbol intervals, and the spectrum sensing is 10  Chapter 1. Introduction  T  N  Frame i  T-N  Frame i+1  N  Frame i+2  Figure 1.3: CR frame structure with frame length T , and sensing time τ . performed in the first N symbol intervals of each frame. If the PU is sensed absent, the CR begins transmission in the remaining T − N symbol intervals, but if the PU is present, no transmission occurs. Hence, it is evident that a longer sensing time improves the sensing performance and PU protection, however, it also reduces the CR transmission time. On the other hand, a shorter sensing time reduces the PU protection but increases the transmission opportunity of the CR. This fundamental tradeoff is known as sensing-throughput tradeoff [66]. The sensing-throughput tradeoff has been investigated in several papers for singleantenna CR systems [66]–[72]. In [66], the sensing time was optimized for maximization of the opportunistic throughput of a single-band CR system under a probability of missed detection constraint. In [67, 68, 69], multi-band CR systems were considered and the sensing times and transmit powers were optimized for maximization of the throughput under the assumption that the sensing times of all frequency bands were identical. The resulting nonlinear optimization problems were only convex over a subset of the optimization variables and were solved iteratively by using convex optimization algorithms for these variables and an exhaustive search over the remaining variables. In [70], different frequency bands were allowed to employ different sensing times and the optimization was performed for energy constrained CR systems. The joint optimization of the sensing time and the sensing threshold was considered in 11  Chapter 1. Introduction [71, 72]. The above mentioned papers only consider single-antenna CR networks. However, multi-antenna systems can provide many benefits for CR systems such as diversity and multiplexing gain. These benefits can be exploited to improve the sensing and transmission capabilities of CR systems. The effect of sensing-throughput tradeoff in multi-antenna CR systems is investigated in this thesis. Specifically, our goal is to develop efficient algorithms for optimizing sensing time and transmit power to maximize the throughput of multi-antenna CR systems.  1.4  Contributions and Results  In this thesis, we develop two spectrum sensing techniques. These new detectors overcome some of the disadvantages of the existing detectors in their respective categories. We also propose efficient algorithms for sensing-throughput maximization in multi-antenna CR networks. The main contributions of this thesis are listed in the following. • In Chapter 2, we introduce Lp –norm detection for spectrum sensing in nonGaussian noise environments. This simple detector has tunable parameters that allow it to adapt to the underlying type of noise without requiring knowledge of the full noise distribution. We analyze the probabilities of false alarm and missed detection of the Lp –norm detector in the low SNR regime and investigate their asymptotic behavior for large sample sizes. By employing the Neyman-Pearson optimization framework, we show that while the probability of false alarm decreases exponentially with the sample size for a given probability of missed detection, the probability of missed detection decreases only polynomially with the sample size for a given probability of false alarm. We also propose an adaptive algorithm for real-time estimation of the Lp –norm parame12  Chapter 1. Introduction ters based on the finite difference stochastic approximation (FDSA) framework and the deflection coefficient of the detector. Our simulation results show that the proposed Lp –norm detector yields significant performance gains over the conventional energy detector in various practical environments. • In Chapter 3, we propose a novel hybrid detector for spectrum sensing which exploits both the pilot and the data symbols emitted by a PU. This spectrum sensing scheme is motivated by the fact that the signals transmitted by primary users often contain known pilot symbols for synchronization and channel estimation purposes. Since the optimal detector entails high complexity and is not suitable for practical systems, we derive a locally optimal detector which performs similar to the optimal detector in the low SNR regime but requires a much lower complexity. Interestingly, the locally optimal hybrid detection metric is a linear combination of an energy detection and a coherent correlation metric. Analytical results for the probabilities of false alarm and missed detection are derived. In addition, to gain some insight about the impact of different system parameters on the performance such as the number of diversity branches, percentage of pilots, and SNR, we provide an asymptotic analysis for low SNRs and large sample sizes. The performance gains achieved over energy detection and coherent correlation detection are analytically characterized. Simulation results confirm the superiority of hybrid detection even if the pilot positions are not known a priori. • In Chapter 4, efficient algorithms for optimization of the sensing-throughput tradeoff in multi-antenna CR systems are developed. In particular, the sensing times, sensing thresholds, and transmit power are optimized for single-band and multi-band CR networks. We propose a new sensing/transmission protocol 13  Chapter 1. Introduction for multi-antenna CR systems which allows simultaneous sensing and transmission in different spatial subchannels. An optimization problem is formulated to maximize the opportunistic throughput subject to constraints on transmit power, probability of false alarm, and probability of missed detection. This optimization problem is non-convex in general. However, by using the concept of alternating optimization, we develop efficient algorithms which only solve convex subproblems in each iteration. Thereby, the complexity of algorithms are significantly reduced, and we analytically prove their convergence to a fixed point. Simulation results show that the gain achieved by the proposed algorithms approaches the global optimal solution and is significantly better compared to the conventional methods with equal sensing time or equal power allocation.  1.5  Thesis Organization  The abbreviations and notations used in Chapters 2, 3, and 4 are provided at the beginning of the thesis. In the following, we provide a brief overview of the remainder of the thesis. In Chapter 2, adaptive Lp –norm spectrum sensing for non-Gaussian noise environments is proposed. This spectrum sensing technique includes energy detection as a special case, but has a better performance than energy detection in non-Gaussian noise environments. Also, by employing the proposed adaptive algorithm no a priori knowledge about the noise statistics is required. In Chapter 3, hybrid coherent/energy detection for CR networks is proposed. This technique can exploit both the pilot and the data symbols transmitted by the PU. Exact analytical results are provided for probabilities of false alarm and missed 14  Chapter 1. Introduction detection, and the asymptotic behavior of the CR system for large sample size is investigated. In Chapter 4, optimal sensing time and power allocation in multi-input multioutput (MIMO) CR systems are investigated for both single-band and multi-band systems. Efficient optimization algorithms for maximizing the throughput under transmit power, probability of false alarm, and probability of detection constraints are developed. In Chapter 5, the thesis is concluded and possible directions for future research are introduced.  15  Chapter 2 Adaptive Lp–Norm Spectrum Sensing for Cognitive Radio Networks 2.1  Introduction  As mentioned in the Chapter 1, the most popular techniques for spectrum sensing are coherent, energy, and feature-based detection. Coherent and energy detectors are typically optimized for AWGN impairment. However, in practice, the received signal at the CR may also be impaired by non-Gaussian noise and interference as has been pointed out in [26, 36, 73].2 Examples for non-Gaussian impairments include man-made impulsive noise [45], emissions from microwave ovens [46], co-channel interference from other CRs, out-of-band spectral leakage, and interference from UWB systems. Unfortunately, detectors designed for AWGN do not perform well in nonGaussian noise. Robust nonparametric cyclostationarity-based spectrum sensing in non-Gaussian noise was considered in [51] but requires statistical information about the PU which is assumed to be cylcostationary. However, in practice, simple energy detection [31] is often preferred over more complex coherent and cyclostationarity2  To simplify our notation, in the following, “noise” refers to any additive impairment of the received signal, i.e., our definition of noise also includes what is commonly referred to as “interference”.  16  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks based detection techniques. This motivates the consideration of energy-detection type of detectors for non-Gaussian noise environments. In this chapter, we first investigate globally and locally optimal detectors [74]– [77] for non-Gaussian noise channels. Thereby, we assume that the PU signal is received over multiple diversity branches, which may be accomplished in practice by multiple-antenna CRs or collaborating single-antenna CRs [32]. The optimal detectors require knowledge of the exact noise distribution which may be difficult to estimate and change with time in practice. Therefore, motivated by the structure of the locally optimal detector, we propose suboptimal Lp –norm detectors [78, 79, 80] which include the conventional energy detector as a special case and do not require knowledge of the noise distribution. The proposed Lp –norm detectors have tunable parameters whose optimization requires only knowledge of certain noise moments. We analyze the probabilities of false alarm and missed detection in Rayleigh fading in the low SNR regime and investigate their dependence on the number of samples N used for detection. Interestingly, we can show that for large sample sizes, for a fixed probability of missed detection the probability of false alarm decreases exponentially in N, while for a fixed probability of false alarm the probability of missed detection decreases only polynomially in N. Our analysis is valid for arbitrary Lp –norm metric parameters and all types of circularly symmetric noise with finite moments. The class of circularly symmetric noises with finite moments includes most commonly used noise models such as AWGN, Gaussian mixture noise, Middleton’s Class A noise, ǫ–mixture noise, generalized Gaussian noise, and co-channel interference. The only major exception is α–stable noise which is sometimes used to model impulsive noise but does not have finite moments [81]. Furthermore, using a Neyman-Pearson approach, spectrum sensing may be opti-  17  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks mized either for minimization of the probability of false alarm for a given probability of missed detection [66] or for minimization of the probability of missed detection for a given probability of false alarm [39]. We show that for large enough N both criteria are equivalent to maximizing the deflection coefficient [82, 83] as far as the optimization of the Lp –norm parameters is concerned. The deflection coefficient is defined as the normalized expected decision statistic difference between the cases when the PU is present and absent, respectively, and has the advantage of being independent of the target probabilities of false alarm and missed detection. Based on the deflection coefficient and the finite difference stochastic approximation framework [84], we derive an adaptive algorithm for optimization of the Lp –norm parameters, which does not require any a priori knowledge about the underlying noise statistics and is capable of tracking changes in the noise distribution. Simulation results confirm the excellent performance of Lp –norm detection and show that large performance gains compared to energy detection are possible in non-Gaussian noise. We note that it was also observed in [78, 79, 80] that Lp –norm detection may lead to performance gains compared to energy detection in non-Gaussian noise. However, in [78, 79, 80], the signal model did not include the effects of fading and diversity, a general analysis of the probabilities of false alarm and missed detection was not given, and an adaptive algorithm for optimization of the Lp –norm parameters was not provided. The remainder of this chapter is organized as follows. In Section 2.2, the system model is presented. The proposed Lp –norm detector is derived and analyzed in Sections 2.3 and 2.4, respectively. The optimization of the Lp –norm parameters is discussed in Section 2.5. Simulation results are presented in Section 2.6, and conclusions are drawn in Section 2.7.  18  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  2.2  Preliminaries  In this section, we introduce the system model and noise models used in the remaining of the chapter.  2.2.1  System Model  Spectrum sensing in CR networks can be formulated as a binary hypothesis-testing problem [29], where hypotheses H0 and H1 correspond to the cases when the PU is absent and present, respectively. We consider L diversity branches and assume that the received continuous-time signal in each of the L branches is down-converted into baseband (with respect to the carrier frequency of the PU of the spectrum), filtered with a suitable square-root Nyquist filter (e.g. a low-pass filter eliminating signal components not originating from the PU), and sampled at the symbol rate of the PU. With these assumptions the received signal samples for the two hypotheses may be modeled in equivalent complex baseband representation as  H0 : yl (n) = zl (n),  1 ≤ l ≤ L,  (2.1)  H1 : yl (n) = hl s(n) + zl (n),  1 ≤ l ≤ L,  (2.2)  where yl (n), hl , and zl (n) denote the received signal, the Rayleigh fading channel gain, and the noise samples in the lth diversity branch, respectively, and s(n) is the PU signal. The L diversity branches can be created either by implementing L antennas at the CR or by allowing L single-antenna CRs to cooperate [32]. The channel gains are statistically independent, in general, non-identically distributed (i.n.d.), have variances σh2l  E{|hl |2 }, and are assumed to be constant for the duration of the  spectrum sensing. The PU signal has zero-mean, variance σs2  E{|s(n)|2}, and its 19  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks real and imaginary parts are statistically independent and have both variance σs2 /2. The zero-mean noise samples are, in general, i.n.d. across the diversity branches but are assumed to have identical variance σz2  E{|zl (n)|2 } in different diversity branches  without loss of generality. Both the PU signal and the noise samples are assumed to be temporally independent, identically distributed (i.i.d.). Furthermore, we assume that the noise samples, the fading gains, and the PU signal are mutually independent.  2.2.2  Noise Models  As already mentioned in the introduction, we assume that the underlying noise has finite moments and that its probability density function (pdf) is circularly symmetric [85]. In the following, we present three non-Gaussian noise models meeting these conditions that are relevant in the context of CR and are used in the simulations in Section 2.6. We note that, in general, different diversity branches may follow different noise distributions, especially if collaborative spectrum sensing is used. Gaussian Mixture Noise (GMN) GMN is often used to model man-made noise, impulsive phenomena [45], and certain types of UWB interference [86]. Assuming the lth diversity branch is impaired by GMN, its pdf is given by Il  pzl (zl ) = i=1  where ci,l > 0,  Il i=1 ci,l  ci,l |zl |2 exp − 2 2 π˜ σi,l σ ˜i,l  ,  2 = 1, and σ ˜i,l , 1 ≤ i ≤ Il , are constants and σz2 =  (2.3)  Il 2 ˜i,l . i=1 ci,l σ  One important special case of GMN is ǫ–mixture noise. For ǫ–mixture noise, we have 2 2 2 Il = 2, c1,l = 1 −ǫl , c2,l = ǫl , σ ˜1,l = σz2 /(1 −ǫl + κl ǫl ), and σ ˜2,l = κl σ˜1,l with parameters  20  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0 < ǫl < 1 and κl > 1. Here, ǫl is the probability that the impulsive phenomenon is present, which also follows a Gaussian pdf whose variance is κl times larger than that of the Gaussian background noise. Generalized Gaussian Noise (GGN) GGN is another popular model for non-Gaussian noise [75]. The pdf of GGN is given by 1 βl Γ(4/βl ) exp − pzl (zl ) = 2 cl 2πσz2 (Γ(2/βl )) where cl  |zl | σz  βl  ,  (2.4)  (Γ(2/βl )/Γ(4/βl ))βl /2 and βl , 0 < βl < ∞, denotes the shape parameter  of the lth branch, which can be used to model both heavy-tailed (0 < βl < 2) and short-tailed (βl > 2) noise. GGN contains Laplacian (βl = 1) and Gaussian (βl = 2) noise as special cases. Co-channel Interference (CCI) CCI from other CRs may also impair spectral sensing. Here, we assume the lth diversity branch is impaired by Il synchronous interferers. The corresponding received signal can be modeled as Il  ejθi,l (n) hi,l si (n) + n ˜ l (n),  zl (n) =  (2.5)  i=1  where hi,l , θi,l (n), and si (n) denote the Ricean fading gain, the random, temporally i.i.d. phase uniformly distributed in [−π, π) caused by the lack of phase and frequency synchronization between the interferers and the CR receiver3 , and the signal of the 3  We note that a constant frequency offset causes a deterministic phase change from one symbol interval to the next. However, we assume that the CR receiver is not aware of this frequency offset and model the phase as uniformly distributed and temporally i.i.d. for receiver design.  21  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks ith interferer, respectively. Furthermore, n ˜ l (n) is the Gaussian background noise and the ratio of the interference power and the noise power is denoted by INR. We note that because of the random phase, the pdf of the CCI signal in (2.5) is circularly symmetric even if the interference signals si (n) are improper (e.g. a binary phase shift keying signal) [85].  2.3  Lp–Norm Spectrum Sensing  In this section, we first consider globally and locally optimal detection, and subsequently we introduce the proposed Lp –norm detector and discuss its optimization.  2.3.1  Optimal Detection  Based on the assumptions regarding the signal and noise models introduced in Section 2.2.1, the globally optimal (GO) decision statistic can be expressed as [87]  ΛGO  py|H (y|H1 ) = log = log Es,h py|H (y|H0 )  L  N  pyl |H (yl (n)|H1 , hl , s(n)) pyl |H (yl (n)|H0 ) l=1 n=1  ,  (2.6)  where N is the number of samples used for spectrum sensing in each diversity branch, h  [h1 . . . hL ]T , and vector y contains all NL samples yl (n), 1 ≤ n ≤ N, 1 ≤ l ≤ L.  It is important to note that it is not possible to move the expectation in (2.6) inside the products since hl is constant with respect to n and s(n) is constant with respect to l. Exploiting (2.1) and (2.2), (2.6) can be rewritten as L  ΛGO = log Es,h  N  pzl (yl (n) − hl s(n)) pzl (yl (n)) l=1 n=1  .  (2.7)  22  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks Eq. (2.7) is quite cumbersome to evaluate and requires knowledge of the distribution of the unknown PU signal. However, the pdf of the PU signal may not be known at the CR [26]. Fortunately, (2.7) may be simplified by invoking the low SNR assumption (i.e., σl2 /σz2 → 0, 1 ≤ l ≤ L, where σl2  σs2 σh2l ), which is well justified for typical  CR operating conditions [27, 28]. In particular, introducing the definitions ul (n) hl s(n), uRl (n)  ℜ{ul (n)}, uIl (n)  ℑ{ul (n)}, yRl (n)  ℜ{yl (n)}, and yIl (n)  ℑ{yl (n)}, the Taylor series of the joint pdf of the real and imaginary parts of zl , pzl (yl (n) − ul (n))  p(yRl (n) − uRl (n), yIl (n) − uIl (n)), can be expressed as [82]  ∂ ∂ pzl (y − u) ∼ p(yR , yI ) − uI p(yR , yI ) = p(yR , yI ) − uR ∂yR ∂yI 1 ∂2 ∂2 ∂2 1 p(yR , yI ), (2.8) + u2R 2 p(yR , yI ) + u2I 2 p(yR , yI ) + uR uI 2 ∂yR 2 ∂yI ∂yR ∂yI where we have dropped sample index n and subscript l for conciseness and included only terms up to second order. Before proceeding further, we note that ul (n) = hl s(n) has zero mean, is uncorrelated with respect to both l and n, and has uncorrelated real and imaginary parts. Furthermore, for a given noise variance σz2 the low SNR assumption implies that ul (n) → 0 almost surely as σl2 → 0. Taking this into account, M i=1 (1  we can exploit the approximation  + xi ) ∼ = 1+  M i=1  xi for x1 , . . . , xM → 0  after substituting (2.8) into (2.7), which leads to the locally optimum (LO) decision statistic  ΛLO     ∼ = log  1 + L  ∼ =  l=1  σl2 4  L  N  l=1 n=1  N  n=1  ∂2 2 ∂yR  σl2 4 +  l  ∂2 2 ∂yR  ∂2 ∂yI2  + l    ∂2 ∂yI2  pzl (yl (n))    pzl (yl (n)) l  pzl (yl (n))  l  pzl (yl (n))  ,  (2.9)  23  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks where we exploited again σl2 → 0, 1 ≤ l ≤ L. For low SNR, i.e., σl2 /σz2 → 0, 1 ≤ l ≤ L, the LO detector is equivalent to the GO detector. However, in contrast to the GO detector, the LO detector in (2.9) does not require knowledge of the distribution of the PU signal s(n). For the special case of AWGN, it can be shown that (2.9) can be simplified to ΛLO ≡ ΛED  1 = NL  N  L  σl2 l=1  n=1  |yl (n)|2 ,  (2.10)  where we exploited the fact that the decision statistic is not affected by multiplication of (2.9) with a constant factor. Clearly, (2.10) is the conventional energy detector. Unfortunately, the LO decision statistic in (2.9) still requires knowledge of the noise pdfs pzl (zl ), 1 ≤ l ≤ L, which are difficult to obtain in practice. Therefore, we introduce a suboptimal detector in the next subsection which overcomes this shortcoming.  2.3.2  Lp–Norm Detection  From (2.9) we observe that the LO decision statistic can be equivalently expressed as ΛLO where gl (yRl (n), yIl (n))  1 = NL  L  N  σl2 l=1  gl (yRl (n), yIl (n)),  (2.11)  n=1  NL((∂ 2 /∂yR2 l + ∂ 2 /∂yI2l )pzl (yl (n))/(4pzl (yl (n))) is a bi-  variate non-linear function which depends on the noise pdf. The basic idea behind the proposed suboptimal detector is to use the same structure as the LO detector in (2.11) but to replace the non-linearity by a simpler one that does not involve the noise pdfs but achieves a high performance for a large class of different types of noise. Candidates for such non-linearities include robust metrics such as Huber’s M–metric [88] and Meridian and Myriad metrics [89]. For this work, we adopt the so-called  24  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks Lp –norm metric and set gl (yRl (n), yIl (n))  |yl (n)|pl , where pl , 1 ≤ l ≤ L, are tunable  parameters, which give the resulting decision statistic some flexibility. The main motivation for using the Lp –norm metric is its simplicity and its ability to perform well in both heavy-tailed and short-tailed noise [80]. The proposed suboptimal decision statistic is given by ΛSO  1 = NL  N  L  σh2l l=1  n=1  |yl (n)|pl ,  (2.12)  which also includes the energy detector in (2.10) as a special case for pl = 2, 1 ≤ l ≤ L. The performance of the Lp –norm detector crucially depends on the choice of the metric parameters p  [p1 p2 . . . pL ]T . The optimization of the metric parameters  will be discussed in the next section.  2.3.3  Optimization of Lp –Norm Decision Statistic  We first note that the Lp –norm detector decides in favor of H0 if ΛSO < τ and for H1 otherwise, where τ is a decision threshold, which is adjusted to achieve a desired trade-off between the probabilities of false alarm Pf (τ, p) missed detection Pm (τ, p)  Pr{ΛSO > τ |H0 } and  Pr{ΛSO < τ |H1 }. Depending on the requirements of the  primary and secondary users, the joint optimization of τ and p can be formulated as two different Neyman-Pearson problems [87]. 1) Fixed Probability of Missed Detection: To protect the PU, a certain maximum tolerable probability of missed detection P¯m may be imposed [66]. The SUs are then faced with the problem of minimizing their probability of false alarm for the given P¯m in order to maximize their transmission opportunity. This leads to the following  25  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks optimization problem  min Pf (τ, p)  (2.13a)  subject to Pm (τ, p) ≤ P¯m  (2.13b)  τ,p  2) Fixed Probability of False Alarm: An alternative option is to minimize the probability of missed detection for a prescribed probability of false alarm P¯f [39]. This scenario may be applicable if the SUs pay revenues to the PUs in return for some guaranteed access probability (given by 1 − P¯f ) if the PU is absent. The corresponding optimization problem is given by  min Pm (τ, p)  (2.14a)  subject to Pf (τ, p) ≤ P¯f  (2.14b)  τ,p  Before the optimal decision threshold τ and the optimal Lp –norm parameters p solving problems (2.13) and (2.14) can be found, analytical expressions for the probabilities of false alarm and missed detection have to be obtained. The derivation of these expressions will be tackled in the next section.  2.4  Performance Analysis  In this section, analytical expressions for the probabilities of false alarm Pf (τ, p) and missed detection Pm (τ, p) of the proposed Lp –norm detector are provided. For convenience, we assume first a fixed channel realization h and average subsequently over the pdf of h. Furthermore, to gain more insight, we derive asymptotic expressions for Pf (τ, p) and Pm (τ, p) for large N. 26  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  2.4.1  Probabilities of False Alarm and Missed Detection for Given h  To make the problem analytically tractable, we assume that the number of samples N in any given sensing interval is large enough to invoke the central limit theorem, i.e., we model the decision statistic ΛSO conditioned on the hypothesis under test as a Gaussian random variable. Note that for sufficiently large N the pdf of ΛSO will approach a Gaussian distribution even if the noise zl (n) is non-Gaussian but has finite moments. For a given channel realization h we have ΛSO |H0 , h ∼ N µ0 , σ ¯02  (2.15)  ΛSO |H1 , h ∼ N µ1 , σ ¯12 .  (2.16)  Consequently, dropping the explicit dependence on τ and p for brevity, the corresponding channel-dependent probabilities of false alarm Pf (h) and missed detection Pm (h) are given by  Pf (h)  Pr{ΛSO > τ |H0 , h} = Q  Pm (h)  Pr{ΛSO < τ |H1 , h} = Q  τ − µ0 σ ¯0 µ1 − τ σ ¯1  (2.17) .  (2.18)  In the following, we derive expressions for the means µ0 and µ1 and the variances σ ¯02 and σ ¯12 . We define the pth moment of complex random variable x as Mx (p)  E{|x|p }.  To make the problem tractable, we assume again that the receiver operates in the  27  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks low SNR regime, i.e., σl2 /σz2 → 0, 1 ≤ l ≤ L. For the mean µ0 , we obtain Ez {ΛSO |H0 , h} =  µ0 =  1 L  1 NL  L  N  σh2l l=1  n=1  Ezl {|zl (n)|pl }  L  σh2l Mzl (pl ),  (2.19)  l=1  where we used the fact that zl (n) is temporally i.i.d. For µ1 , we exploit the fact that |hl s(n)| ≪ |zl | almost surely because of the low SNR assumption and use a similar Taylor series approximation as in (2.8) to approximate |hl s(n) + zl (n)|pl around zl (n). This leads to  µ1  Ez,s{ΛSO |H1 , h} 1 = L 1 ∼ = L  L  l=1 L  σh2l Ezl ,s {|hl s(n) + zl (n)|pl } σh2l Mzl (pl ) + p2l  l=1  σs2 |hl |2 Mzl (pl − 2) , 4  (2.20)  where we used the identity E {(∂ 2 /∂zR2 + ∂ 2 /∂zI2 ) |zl |pl } = p2l E{|zl |pl −2 } = p2l Mzl (pl − 2). Note that µ1 does not depend on the pdf of the PU signal s(n) but only on its variance σs2 . The variance σ¯02 can be calculated as σ ¯02  Ez (ΛSO − µ0 )2 |H0 , h 1 = NL2  L  l=1  σh4l Mzl (2pl ) − Mz2l (pl ) .  (2.21)  28  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks Table 2.1: Noise moments for AWGN, GMN, and GGN with variance σz2 , respectively. Noise Model  Moments Mz (p)  AWGN GMN  Γ Γ 1 Γ(2/β)  GGN  p 2  p 2  + 1 σzp I  +1  Γ(2/β) Γ(4/β)  i=1 p 2  Γ  ci σ ˜ip p+2 β  σzp  Furthermore, using a similar approach as for calculation of µ1 , we obtain for σ ¯12 the expression σ ¯12  Ez,s (ΛSO − µ1 )2 |H1 , h 1 = NL2  L  l=1  σh4l Mzl (2pl ) − Mz2l (pl ) − p2l  σs2 |hl |2 2  × [Mzl (pl )Mzl (pl − 2) − 2Mzl (2pl − 2)] .  (2.22)  For a given channel realization h, the probabilities of false alarm and missed detection can be computed from (2.17)–(2.22). The noise moments required in (2.19)–(2.22) are provided in Table 2.1 for AWGN, GMN, and GGN. For CCI, as specified in (2.5), a closed-form expression for the moments does not seem to exist in general. However, in this case, the moments can be easily obtained by Monte-Carlo simulation.  2.4.2  Average Probabilities of False Alarm and Missed Detection  Since µ0 and σ ¯02 are both independent of the channel h, the average probability of false alarm Pf is identical to Pf (h) given in (2.17), i.e., Pf = Pf (h). In contrast,  29  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks both µ1 and σ ¯12 depend on the channel h and can be expressed as L  µ1 = µ0 + l=1  where α ¯l  2 pl 2 σs2 σh  l  4L  α ¯ l |hl |2  Mzl (pl −2) and χ¯l  and σ ¯12 = σ ¯02 +  4 pl 2 σs2 σh  L2  l  1 N  L  l=1  χ¯l |hl |2 ,  (2.23)  (Mzl (2pl − 2) − Mzl (pl )Mzl (pl − 2)/2)  are independent of h. To make the problem tractable, we consider again the low SNR case (σs2 → 0) where χ¯l → 0 and σ ¯12 ∼ ¯02 . With this approximation and (2.18), we =σ can express the average probability of missed detection as L  Pm = E h where ξ  Q l=1  (τ − µ0 )/¯ σ0 , αl  αl |hl |2 − ξ  = Eγ {Q (γ − ξ)} , L l=1  α ¯ l /¯ σ0 , and γ  (2.24)  αl |hl |2 . For calculation of (2.24)  the exact pdf pγ (γ) of γ is needed. Here, we consider two special cases for this pdf. In particular, we consider the i.n.d. case where αi σh2i = αj σh2j , ∀i = j, i, j ∈ {1, . . . , L}, and the i.i.d. case where αi σh2i = ασh2 , ∀i, i ∈ {1, . . . , L}. The corresponding pdfs are given by [90]  pγ (γ) =        L l=1  wl 2 αl σh  l  exp −  γ L−1 2L (L−1)!αL σh  exp  γ 2 αl σh  ,  i.n.d. case, (2.25)  l  − ασγ 2 h  , i.i.d. case,  30  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks L j=1,j=l  where wl =  αl σh2l / αl σh2l − αj σh2j  and γ ≥ 0. It is shown in Appendix A  that for the i.n.d. case, (2.24) can be rewritten as  Pm =     1  √  2π  L l=1     Iind +  wl 2 αl σh  −  e  ξ αl σ 2 hl  0 −  l  √1 2π  L l=1  wl 2 αl σh  π/2  e  | sin φ|e ξ αl σ 2 hl  sin2 φ 2α2 σ 4 l hl  π/2  Jind (φ) − 2| sin φ|e  0  l  | sin φ| √ 2 2αl σh  1 − erf  −  l sin2 φ 2α2 σ 4 l hl  √  ξ 2| sin φ|  erf  dφ,  | sin φ| √ 2 2αl σh  ξ<0 dφ, ξ ≥ 0  l  (2.26)  where Iind and Jind (φ) are given by ξ  L  wl  pγ (γ)dγ =  Iind  l=0  0  | sin φ|e  Jind (φ)  sin2 φ 2α2 σ 4 l hl  1 + erf  −  1−e  ξ αl σ 2 hl  (2.27)  | sin φ| ξ √ −√ 2 2αl σhl 2| sin φ|  .  (2.28)  We note that (2.26) contains only one-dimensional integrals over a finite support which can be easily evaluated numerically. For the i.i.d. case, a similar approach as for the i.n.d. case in Appendix A along with a binomial expansion leads to  L−1    C     k=0  L−1 k  π/2 0  ξ−  sin2 φ 2 ασh  k  √  2| sin φ|  L−k  e  sin2 φ 2α2 σ 4 h  Γ  L−k 2 , Jiid (φ) 2  π/2 sin2 φ L−1 k √ L−k 2α2 σ 4 L−1 sin2 φ Pm = I + C h ξ − ασ2 2| sin φ| e iid  k  h  k=0 0     ×Γ L−k − sgn (J (φ))L−k Γ L−k , J 2 (φ) + 2Γ iid iid 2 2  dφ,  ξ<0  sgn (Jiid (φ))L−k − 1 L−k sin2 φ , 2α2 σ4 2 h  dφ, ξ ≥ 0 (2.29)  31  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks where we have used the definitions C  −  e  ξ ασ 2 h  /(2π (L − 1)!αL σh2L ) and  ξ  pγ (γ) dγ =  Iiid 0  Jiid(φ)  γ L, ασξ 2  h  (L − 1)!  | sin φ| ξ √ . −√ 2 2ασh 2| sin φ|  (2.30) (2.31)  In the above equations upper and lower incomplete gamma functions are defined as Γ(s, x) =  ∞ s−1 −t t e x  dt and γ(s, x) =  x s−1 −t t e 0  dt, respectively. Although both  (2.26) and (2.29) are convenient for numerical evaluation, they both do not offer insight into the impact of design parameters such as L and p on the performance of Lp –norm spectrum sensing. In order to facilitate this insight, we perform an asymptotic analysis of Pm and Pf for large N in the next section.  2.4.3  Asymptotic Analysis of Pf and Pm for N → ∞  In this subsection, we study the asymptotic behavior of Pf when Pm is fixed and the asymptotic behavior of Pm when Pf is fixed, cf. Section 2.3.3. For convenience, we focus on the case where both the fading and the noise are i.i.d, i.e., σh2l = σh2 , αl = α, pl = p, Mzl (p) = Mz (p), 1 ≤ l ≤ L. However, qualitatively similar results can be obtained for the i.n.d. case. Fixed Probability of Missed Detection Pm In this case, the average probability of missed detection is limited to Pm = P¯m , cf. (2.13). On the other hand, it is shown in Appendix B that for asymptotically  32  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks large sample size, i.e., N → ∞, Pm can be approximated as ξ  Pm ∼ =  pγ (γ) dγ = 0  γ L, ασξ 2  h  (L − 1)!  .  √ √ Solving (2.32) for ξ and substituting α = σs2 p2 Mz (p−2) N /[4 L  (2.32)  Mz (2p) − Mz2 (p)],  we observe that for the probability of missed detection to be fixed to Pm = P¯m , threshold ξ has to follow γL−1 P¯m (L − 1)! p2 σs2 σh2 Mz (p − 2) √ ∼ √ N, ξ= 4 L Mz (2p) − Mz2 (p)  (2.33)  where γs−1 (x) is the inverse of γ(s, x), which is a monotone function in x. Based on the asymptotic expression for ξ in (2.33) we can directly obtain an asymptotic expression for Pf = Q(ξ). In particular, using the common approximation Q(x) ∼ = 1/2 exp(−x2 /2), we obtain with the help of (2.33) −1  [γL log(Pf ) ∼ =−  P¯m (L − 1)! ]2 p4 σs4 σh4 Mz2 (p − 2) N − log 2. 32L[Mz (2p) − Mz2 (p)]  (2.34)  Eq. (2.34) nicely illustrates how the various system parameters affect the overall performance for large N if Pm is fixed. In particular, Pf decreases exponentially with N. In other words, for large N, log(Pf ) decreases linearly with N, where the slope is affected by the number of branches L via [γL−1 P¯m (L − 1)! ]2 /L and by the type of noise and the Lp –norm parameter p via p4 Mz2 (p − 2)/[Mz (2p) − Mz2 (p)]. Fixed Probability of False Alarm Pf In this case, the average probability of false alarm is limited to P¯f . Hence, ξ = Q−1 (P¯f ) is constant. For practical scenarios where P¯f ≤ 1/2, we have ξ ≥ 0. Hence, 33  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks only the second part of (2.29) has to be considered. Since α is proportional to √ N and ξ is constant, (2.29) can be considerably simplified by exploiting the ap√ proximations sin2 φ/(ασh2 ) → 0, sin2 φ/(2α2σh4 ) → 0, Jiid → −ξ/( 2| sin φ|), and Iiid = ξ L /(L!αL σh2L ), where we have used γ(s, x) → xs /s for x → 0 [91]. Consequently, for N → ∞, we obtain from (2.29) Pm ∼ =  G(L, P¯f ) ξL + L!αL σh2L 2π(L − 1)!αL σh2L  (2.35)  with L−1  G(L, P¯f ) l=0  π/2  L−1 l  ξl  √  2| sin φ|  L−l  0  ×  1 + (−1)L−l Γ  L−l 2  − (−1)L−l Γ  L−l ξ2 , 2 2 sin2 φ  dφ, (2.36)  which depends on P¯f (via ξ) and L but is independent of N. Substituting α in (2.35), we obtain LG(L, P¯f ) + 2πξ L LL/2 Pm ∼ = 2πL!σh2L A(L,P¯f )  σs2 4  p2 Mz (p − 2)  Mz (2p) −  −L  Mz2 (p)  N −L/2 ,  (2.37)  B(p)  where A(L, P¯f ) is a function of P¯f and L, and B(p) depends on the Lp –norm parameter and the type of noise. To gain more insight, we rewrite (2.37) as L log(Pm ) ∼ = log(A(L, P¯f )) − L log(B(p)) − log N. 2  (2.38)  34  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks From (2.38) we can now readily see the impact of the various system parameters on performance for large N. In particular, on a double-logarithmic scale the probability of missed detection decreases linearly with the number of samples, where the slope is given by −L/2. Note that the dependence of Pm on N for fixed Pf is markedly different from that of Pf for fixed Pm discussed in Section 2.4.3. Furthermore, the type of noise and the Lp –norm parameter influence only B(p). Thus, for large N, on a double-logarithmic scale the Pm –curves for a given L but different types of noise are shifted vertically with respect to each other. The relative shift between different curves can be influenced by optimizing p. This aspect will be investigated more in detail in the next section.  2.5  Optimization of Decision Statistic  Ideally, we want to optimize p based on the Neyman-Pearson problems formulated in Section 2.3.3. However, this approach is hampered by (a) the involved mathematical expressions for Pm (τ, p) in (2.26) and (2.29) and (b) the expected dependence of the optimal p on P¯m and P¯f . To avoid these problems, we first consider the optimization of p based on another commonly adopted approach. In particular, we introduce the deflection coefficient as optimality criterion [82, 83]. Subsequently, we show that the deflection coefficient is closely related to the asymptotic expressions derived for the probabilities of missed detection and false alarm in Section 2.4.3. Finally, we propose an adaptive algorithm for online optimization of p based on the deflection coefficient.  2.5.1  Deflection Coefficient  Deflection is commonly used as performance measure for binary hypothesis testing problems [82, 83]. For the problem at hand, the average deflection coefficient can be 35  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks obtained as  dL (p)  Eh  µ1 − µ0 σ ¯0  =  √  N σs2 4  L l=1 L l=1  σh4l p2l Mzl (pl − 2)  ,  (2.39)  σh4l Mzl (2pl ) − Mz2l (pl )  where (2.19)–(2.21) have been used. The intuition behind (2.39) is that the normalized distance of the means of the pdfs of the two hypotheses is a good measure for the performance of the detector. As can be observed from (2.39), dL (p) only depends on p and the optimal p maximizing dL (p) can be found by an L–dimensional line search. If an L–dimensional line search is computationally too costly for implementation, we may simplify (2.39) by letting p1 = p2 = . . . = pL = p. The corresponding deflection coefficient is given by d(p)  dL (p 1L ). In general, constraining all Lp –norm param-  eters to be equal will entail some loss in performance, except for the special case when both the fading and the noise are i.i.d. Before, we use (2.39) for derivation of an adaptive algorithm for online estimation of p, we investigate the relation between the deflection coefficient and the asymptotic expressions for Pf and Pm in (2.34) and (2.38), respectively.  2.5.2  Relation Between Neyman-Pearson Optimization and dL(p)  In this subsection, we assume i.i.d. fading and i.i.d. noise, i.e., dL (p) = d(p) holds. Exploiting (2.39) for the case when Pm is fixed, we can rewrite (2.34) as −1  [γL log(Pf ) ∼ =−  P¯m (L − 1)! ]2 2 d (p) − log 2. 2L2  (2.40)  36  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks On the other hand, for the case when Pf is fixed, we obtain from (2.38) √ log(Pm ) ∼ = log(A(L, P¯f )) + L log(σh2 L) − L log d(p).  (2.41)  Eqs. (2.40) and (2.41) reveal that, for large N, maximizing the deflection coefficient d(p) is equivalent to minimizing Pf and Pm (for fixed Pm and Pf ), respectively. Furthermore, these equations show that for large sample sizes N, the optimal Lp – norm parameter p does not depend on P¯m and P¯f or, equivalently, not on threshold τ in (2.13) and (2.14). These observations have important practical implications as they justify the adoption of the relatively simple deflection coefficient as optimality criterion for optimization of the Lp –norm parameters. Although, strictly speaking, we have established the relationship between the Neyman-Pearson optimization problem and deflection only for the i.i.d. case, we expect a qualitatively similar relationship to hold for the i.n.d. case. Thus, in the next section, we derive an adaptive algorithm for optimization of p based on the deflection coefficient dL (p).  2.5.3  Adaptive Algorithm  In practice, the statistical properties of the noise impairing a wireless communication system are often not known a priori and an online estimation of the optimal p is desirable. Since dL (p) is always positive, maximizing dL (p) is equivalent to maximizing d2L (p). The adaptive algorithm proposed in this section is based on the finite difference stochastic approximation (FDSA) framework with Kiefer-Wolfowitz gradient estimation [84, 92]. Kiefer-Wolfowitz gradient estimation is particularly well suited for the problem at hand since it avoids the differentiation of d2L (p) which is quite cumbersome. In the kth iteration, the FDSA algorithm improves the parameter 37  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks ˆ (pk ) vector estimate at time k, pk , in the direction of the gradient vector estimate g to obtain a new estimate pk+1 [84]: pk+1 = pk + δk gˆ (pk )  dˆ2L (pk +ζk e1 )−dˆ2L (pk −ζk e1 ) 2ζk   .. ˆ (pk ) =  g .   ˆ2 dL (pk +ζk eL )−dˆ2L (pk −ζk eL ) 2ζk  (2.42)     ,    (2.43)  where δk > 0 and ζk > 0 are the gain sequences of the FDSA algorithm, dˆ2L (p) √ (4/( N σs2 ))2 d2L (p), and el denotes a column vector with a 1 in position l and zeros in all other positions. Since the noise moments Mzl (p), Mzl (p −2), and Mzl (2p) required for computation of dˆ2L (p) are not known a priori in general, they have to be replaced by their respective estimates which can be obtained from the received signal as ˆ z ,k (pl,k ) = (1 − ρk )|yl (k)|pl,k + ρk M ˆ z ,k−1 (pl,k−1), M l l  (2.44)  where ρk , 0 < ρk < 1, is the forgetting factor, which is usually chosen close to one, and pl,k , 1 ≤ l ≤ L, denotes the elements of vector pk . We note that for low SNRs (σl2 ≪ σz2 ), hl s(k) + zl (k) has practically the same moments as zl (k). Therefore, the (approximate) moments of zl (k) can be estimated based on (2.44) without knowing whether or not the PU is present. For moderate to high SNRs, the moment estimation has to be performed when the PU is absent. The convergence theory of the FDSA algorithm [84] states that if yl (k) is stationary and the gain sequences fulfill δk → 0, ζk → 0,  ∞ k=0 δk  = ∞, and  ∞ 2 2 k=0 δk /ζk  < ∞, under some mild conditions on the cost  function, the algorithm is guaranteed to converge to a local maximum. However, since in practice yl (k) is not stationary, we may set δk = δ and ζk = ζ, ∀k, where δ 38  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  Analysis 0.85  Simulation  0.8 0.75  ǫ–mixture (ǫ = 0.01, κ = 10)  0.7  Pf  0.65  AWGN  0.6 0.55  GGN (β = 3)  0.5  CCI (INR=5 dB)  0.45 0.4 0.35  0  1  2  3  4  p  5  6  7  8  Figure 2.1: Pf vs. p for Lp –norm detection. L = 1, SNR = −15 dB, and P¯m = 0.1. and ζ are small positive constants to give the algorithm some tracking capability. We note that a simplified version of the above FDSA algorithm can be obtained by constraining all pl , 1 ≤ l ≤ L, to be equal and by replacing dL (p) by d(p). In this case, the algorithm has to estimate only one Lp –norm parameter and the optimal p is independent of L.  2.6  Numerical Results and Discussions  In this section, we present numerical and simulation results for Lp –norm spectrum sensing. Throughout this section we assume SNR = −15 dB, P¯m = 0.1, P¯f = 0.1, i.i.d. fading, and i.i.d. noise, unless specified otherwise. The simulation results are averaged over 500,000 independent realizations of the Rayleigh fading channel and  39  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  Analysis Simulation  0.8  0.7  ǫ–mixture (ǫ = 0.01, κ = 10) Pm  0.6  0.5  AWGN 0.4  CCI (INR=5 dB) GGN (β = 3)  0.3  0.2  0  1  2  3  4  p  5  6  7  8  Figure 2.2: Pm vs. p for Lp –norm detection. L = 1, SNR = −15 dB, and P¯f = 0.1. compared with analytical results obtained by evaluating (2.17), (2.26), and (2.29). In Figs. 2.1 and 2.2, we show, respectively, the average probability of false alarm Pf for P¯m = 0.1 and the average probability of missed detection Pm for P¯f = 0.1 as functions of p for L = 1, N = 10000, and different types of noise. In particular, AWGN, ǫ–mixture noise, GGN, and CCI are considered. For CCI we assume I = 2 unfaded, equal-power interferers with INR = 5 dB. For each value of p, threshold τ was adjusted to guarantee the prescribed values of Pm = P¯m and Pf = P¯f , respectively. Thereby, Pm and Pf were obtained by simulation and by evaluating (2.29) and (2.17), respectively. Figs. 2.1 and 2.2 show that the theoretical results agree well with the simulations. Furthermore, both figures reveal that whereas for AWGN energy detection (p = 2) achieves the best performance, for the heavy-tailed ǫ–mixture noise and the short-tailed GGN and CCI the optimal values for p are smaller and 40  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 5 4.5  Deflection Coefficient  4  AWGN ǫ-mixture (κ=10, ǫ=0.01)  3.5 3  CCI (INR=5 dB) GGN (β = 3)  2.5 2 1.5 1 0.5 0  1  2  3  4  p  5  6  7  8  Figure 2.3: Deflection coefficient d(p) vs. p for Lp –norm detection. L = 1 and SNR = −15 dB. larger than two, respectively. In Fig. 2.3, we depict the deflection coefficient d(p) (calculated from (2.39)) for the same channel and noise parameters that were considered in Figs. 2.1 and 2.2. A comparison of Fig. 2.3 with the previous two figures shows that the value of p that maximizes the deflection coefficient achieves a close-to-optimal probability of false alarm in Fig. 2.1 and a close-to-optimal probability of missed detection in Fig. 2.2, confirming the conclusions drawn from the asymptotic analysis in Section 2.5.2. Thus, since the deflection coefficient is independent of P¯m and P¯f , it provides a convenient cost function for optimization of the Lp –norm parameters. In Fig. 2.4, we investigate the performance of the proposed FDSA algorithm (ρk = ρ = 0.995, δk = δ = 5 · 10−5 , ζk = ζ = 10−3 , ∀k) for online adaptation of the  41  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 45  Unconstrained p Constrained p  40  p1 = p2 = 2 35  dˆ2L (p)  30  25  20  Noise I Noise II  15  Noise III 10  0  1  2  3  k  4  5  6 4  x 10  Figure 2.4: Squared deflection coefficient dˆ2L (p) vs. iteration number k for adaptive FDSA algorithm with unconstrained p (i.e., p1 = p2 possible) and constrained p (i.e., p1 = p2 = p). For comparison, the deflection coefficient for energy detection (p1 = p2 = 2) is also shown. L = 2 and SNR = −15 dB. Noise I: AWGN in both branches. Noise II: ǫ–mixture noise (κ = 10, ǫ = 0.01) in branch 1 and GGN (β = 3) in branch 2. Noise III: ǫ–mixture noise (κ = 10, ǫ = 0.01) in branch 1 and AWGN in branch 2. Lp –norm parameters p. Assuming L = 2, we consider the cases of unconstrained p (i.e., p1 and p2 may be different) and constrained p (i.e., p1 = p2 ) and compare them with the performance of energy detection (i.e., p1 = p2 = 2). The estimated squared deflection coefficient dˆ2L (p) for these three cases is shown in Fig. 2.4, where we switch abruptly between three different types of noise, which are specified in the caption of the figure. For Noise I (AWGN in both branches), energy detection is optimal and the FDSA algorithms (with and without constraint on p) quickly converge to the same deflection coefficient as energy detection. For Noise II (ǫ–mixture noise  42  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0.6 0.55  ǫ–mixture (ǫ=0.01, κ=10)  0.5  p = 2 (Analysis) popt (Analysis) p = 2 (Simulation) popt (Simulation) LO (Simulation)  AWGN  0.45  Pf  0.4 0.35 0.3 0.25  CCI (INR=5 dB)  0.2 0.15 0.1  GGN (β = 3) 1  1.5  2  2.5  3  N  3.5  4  4.5  5 4  x 10  Figure 2.5: Pf vs. N for energy detection (p = 2), Lp –norm detection (p = popt ), and LO detection. L = 1, SNR = −15 dB, and P¯m = 0.1. and GGN), the FDSA algorithms outperform energy detection. Thereby, the FDSA algorithm without constraint on p achieves a superior performance than the FDSA algorithm with p1 = p2 = p at the expense of a higher complexity. For Noise III the noise in the branch impaired by GGN for Noise II changes abruptly to AWGN. In this case, in steady state, the FDSA algorithm with unconstrained p achieves again a better performance than the constrained algorithm. However, the speed of convergence of the FDSA algorithm with unconstrained p is considerably slower in this case than that for the algorithm with p1 = p2 = p. This is due to the fact that for the constrained case, the optimal p for Noise II is similar to that for Noise III, whereas for the unconstrained case the optimal p’s for both types of noise are considerably different.  43  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0  10  −1  10  L=1 −2  10  Pf  L=3  L=2  −3  10  AWGN (Analysis) ǫ-mixture (Analysis)  −4  10  GGN (Analysis) AWGN (Simulation) −5  10  ǫ-mixture (Simulation) GGN (Simulation)  −6  10  1  1.5  2  2.5  3  3.5  N  4  4.5  5 4  x 10  Figure 2.6: Pf vs. N for Lp –norm detection with optimal p for different numbers of i.i.d. diversity branches L. SNR = −15 dB and P¯m = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3). Fig. 2.5 compares the probabilities of false alarm as functions of the number of samples N for Lp –norm detection (with popt optimized for maximum d(p)) and energy detection (p = 2). The same types of noise as in Fig. 2.1 and L = 1 are considered. For comparison, for ǫ–mixture noise and GGN also simulation results for LO detection are shown. For Lp –norm detection and energy detection the simulation results confirm the theoretical results for all considered types of noise. For the nonGaussian noises, the Lp –norm detector with popt outperforms energy detection for the entire considered range of N and closes the gap to LO detection for both ǫ–mixture noise and GGN. For AWGN there is only one curve since the Lp –norm detector with popt = 2 is equivalent to the LO and the energy detector. In Fig. 2.6, we investigate the influence of the number of i.i.d. diversity branches L 44  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  −1  L=1  Pm  10  AWGN (Analysis)  L=2  ǫ-mixture (Analysis) GGN (Analysis) −2  10  AWGN (Simulation)  L=3  ǫ-mixture (Simulation) GGN (Simulation) 4  10  2×104  N  3×104  4  4×10  5×104  Figure 2.7: Pm vs. N for Lp –norm detection with optimal p for different numbers of i.i.d. diversity branches L. SNR = −15 dB and P¯f = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3). on the probability of false alarm of Lp –norm detection with optimal p for Pm = P¯m = 0.1 and different types of i.i.d. noise. Note that only a single Lp –norm parameter p has to be optimized since both the fading and the noise are i.i.d. In Fig. 2.6, we use a logarithmic scale for Pf but a linear scale for N. As a result, as expected from our asymptotic analysis in Section 2.4.3, the Pf vs. N curves are straight lines whose slopes are not only determined by the number of diversity branches L but also by the type of noise. In particular, short-tailed GGN causes a larger slope than AWGN, whereas heavy tailed ǫ–mixture noise causes a smaller slope than AWGN. Adopting the same channel and noise parameters as in Fig. 2.6, we show in Fig. 2.7 the probability of missed detection of Lp –norm detection with optimal p for Pf = P¯f = 0.1. However, in this case, we use a logarithmic scale for both Pm and N. As 45  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0  10  Analysis Simulation (p 1 = p 2 = p 3 = 2) Simulation (p 1 = p 2 = p 3 = 0.9) Simulation (p 1 = 0.9, p 2 = 1.2, p 3 = 1.9)  Pf  LO Detector −1  10  −2  10  1  1.5  2  2.5  3  N  3.5  4  4.5  5 4  x 10  Figure 2.8: Pf vs. N for energy detection (p1 = p2 = p3 = 2), Lp –norm detection (constrained and unconstrained p), and LO detection. L = 3 and P¯m = 0.1. Branch 1: SNR = −15 dB, ǫ–mixture noise (κ = 10, ǫ = 0.01). Branch 2: SNR = −18 dB, AWGN. Branch 3: SNR = −21 dB, GGN (β = 3). expected from the analysis in Section 2.4.3, the Pm –curves are straight lines and their slope is −L/2 independent of the type of noise. The Pm –curves for different types of noise have a fixed offset which is independent of N. Thereby, the short-tailed GGN has a significantly better performance than AWGN, while the heavy-tailed ǫ–mixture noise has a worse performance. In Fig. 2.8, we investigate the impact of i.n.d. fading and i.n.d. noise on Lp –norm spectrum sensing with L = 3 diversity branches. Thereby, the SNR of branch 1, branch 2, and branch 3 was -15 dB, -18 dB, and -21 dB, respectively. Branch 1 is impaired by ǫ–mixture noise, branch 2 by AWGN, and branch 3 by GGN. Fig. 2.8 shows the probability of false alarm of energy detection, Lp –norm detection with  46  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0  10  SNR=-15 dB −1  10  GGN  AWGN  Pm  ǫ–mixture −2  10  p = 2 (Analysis) −3  10  popt (Analysis) p = 2 (Simulation)  SNR=-10 dB  popt (Simulation) −4  10  −6  10  −5  10  −4  10  −3  10  −2  10  −1  10  0  10  Pf  Figure 2.9: ROC for energy detection (p = 2) and Lp –norm detection with optimal p for L = 3 diversity branches and N = 20000. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3). unconstrained and constrained p, and LO detection for Pm = P¯m = 0.1. For both considered Lp –norm detection schemes the metric parameters were optimized for maximization of the deflection coefficient. Fig. 2.8 shows that although Lp –norm detection yields significant gains compared to energy detection even if the elements of p are all equal (for example, with N = 50000, Pf approximately decreases from 0.1 to 0.04), in case of i.n.d. channels substantial additional gains are possible if the elements of p are not constrained (in this case, with N = 50000, Pf approximately decreases to 0.02), and a performance close to that of the LO detector can be achieved. In Fig. 2.9, we show the receiver operating characteristics (ROCs) of energy detection and Lp –norm detection with optimal p for L = 3 i.i.d. diversity branches, two different SNRs, and three different types of i.i.d. noise. For both ǫ–mixture noise and 47  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks 0  10  ǫ–mixture  Pm  AWGN −1  10  GGN  p = 2 (Analysis) p opt (Analysis) p = 2 (Simulation) p opt (Simulation) −2  10 −20  −18  −16  −14  −12  −10  −8  −6  −4  −2  0  SNR (dB)  Figure 2.10: Pm vs. SNR for energy detection (p = 2) and Lp –norm detection with optimal p. L = 1, N = 10000, and P¯f = 0.1. I.i.d. AWGN, ǫ–mixture noise (κ = 10, ǫ = 0.01), and GGN (β = 3). GGN, Lp –norm detection outperforms energy detection for the considered range of (Pm , Pf )–pairs. For AWGN the Lp –norm detector is identical to the energy detector. The probability of missed detection as a function of the SNR is shown in Fig. 2.10 for energy detection and Lp –norm detection with optimal p for L = 1, P¯f = 0.1, AWGN, ǫ–mixture noise, and GGN. For AWGN only one curve is shown since the Lp –norm detector is identical to the energy detector, i.e., popt = 2. The agreement between simulation results and analysis is excellent for all considered SNR values. Since we use a double-logarithmic scale, for high enough SNR, Pm decreases linearly with the SNR with a slope of −1 for all types of noise and all values of p, cf. (2.37).  48  Chapter 2. Adaptive Lp –Norm Spectrum Sensing for Cognitive Radio Networks  2.7  Conclusion  The conventional energy detection method is the likelihood-ratio optimal detector when both signal and noise are Gaussian processes. However, in practice, the CR received signal is also impaired by non-Gaussian noise and interference. In this chapter, it was shown that energy detection has a poor performance in non-Gaussian noise when the PU received power level is low. Hence, based on the structure of the locally optimal detector, we have introduced an Lp –norm detector for spectrum sensing in non-Gaussian noise. The Lp –norm detector does not require knowledge of the full noise distribution, but has tunable parameters that allow the detector to adapt to the underlying type of noise. Our simulation and analytical results showed that the proposed Lp –norm detector yields large performance gains over the conventional energy detector in both short-tailed and heavy-tailed noise and greatly benefits from diversity. Our asymptotic analysis has shown that while the probability of false alarm decreases exponentially with N for a given probability of missed detection, the probability of missed detection decreases only polynomially in N for a given probability of false alarm. Furthermore, we proposed an adaptive algorithm for online estimation of the Lp –norm parameters that was based on the deflection coefficient of the detector. The proposed adaptive algorithm was capable of tracking changes in the noise without requiring any a priori knowledge about the noise statistics.  49  Chapter 3 Hybrid Coherent/Energy Detection for Cognitive Radio Networks 3.1  Introduction  As mentioned in Chapter 1, the most popular technique for spectrum sensing is energy detection, where the amount of received energy is the criterion for making a decision regarding the presence of a PU [31]–[33], [42]. While this method is easy to implement, it suffers from a relatively poor performance especially for low SNR. A significantly better performance can be achieved by coherent spectrum sensing exploiting known pilot symbols emitted by the PU for synchronization and channel estimation purposes [35, 36], [47]–[50]4 . However, coherent detection techniques fail to utilize the energy contained in the PU’s data symbols. Also, in practice, the exact position of the pilot symbols relative to the data symbols may not be known and has to be estimated by the CR receiver. In this chapter, we propose a novel hybrid detector for spectrum sensing which exploits both the pilot and the data symbols emitted by the PU. We consider the general case where the SU receives the PU signal over multiple diversity branches, 4  For the discussion of these papers please refer to Section 1.2  50  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks which may be accomplished in practice by multiple-antenna CR receivers or collaborating single-antenna CRs [32]. We first show that the optimal likelihood-ratio based detector is difficult to implement and requires knowledge of the data symbol alphabet of the PU, which is not desirable in practice. Subsequently, we derive a locally optimal hybrid coherent/energy detection scheme which is equivalent to the optimal detector in the low SNR regime and is also applicable as a suboptimal detector for moderate and high SNRs. Interestingly, the locally optimal hybrid detection metric is simply a linear combination of an energy detection metric and a coherent correlation metric. Furthermore, we derive analytical expressions for the probabilities of false alarm and missed detection of the proposed hybrid detector. For optimization of the CR system, we adopt a Neyman-Pearson approach, where the probability of false alarm is minimized for a given probability of missed detection or the probability of missed detection is minimized for a given probability of false alarm. For Neyman-Pearson optimization and the asymptotic but practical regime of low SNR and large sample size, we develop simple and elegant expressions for the probabilities of false alarm and missed detection offering significant insight into the impact of system parameters such as the sample size, the number of diversity branches, the percentage of pilot symbols, and the SNR on performance. In particular, our asymptotic analytical results show that the probabilities of false alarm and missed detection of the proposed hybrid detector decay at practically the same rate with increasing sample size as those of the coherent detector and significantly faster than those of energy detection. Moreover, hybrid detection outperforms coherent detection and the gap between both schemes increases with decreasing percentage of pilot symbols. Simulation results confirm the superiority of hybrid detection even if the pilot positions are not known a priori. We note that, in [93, 94] detection of composite signals consisting of one random  51  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks and one deterministic signal in noise have been investigated. However, the signal model did not include the effects of collaborative sensing and diversity on the performance, and a general analysis of the probabilities of false alarm and missed detection was not given. The remainder of this chapter is organized as follows. In Section 3.2, the notations and the system model are presented. The decision metric for the proposed hybrid coherent/energy detection scheme is derived in Section 3.3 and analyzed in Section 3.4. Simulation results are presented in Section 3.5, and conclusions are drawn in Section 3.6.  3.2  System Model  Spectrum sensing in CR networks can be formulated as a binary hypothesis-testing problem [29], where hypotheses H0 and H1 correspond to the cases when the PU is absent and present, respectively. Assuming L diversity branches and sensing at times n ∈ N, N  {1, 2, . . . , N}, the received signal samples for the two hypotheses may  be modeled as  H0 : yl (n) = zl (n),  1 ≤ l ≤ L,  (3.1)  H1 : yl (n) = hl s(n) + zl (n),  1 ≤ l ≤ L,  (3.2)  where yl (n), hl , and zl (n) denote the received signal samples, the channel gain, and zero-mean complex AWGN in the lth diversity branch, respectively, and s(n) is the PU signal. The L diversity branches can be created either by implementing L antennas at the CR receiver or by allowing L single-antenna CRs to cooperate [32]. The channel gains are assumed to be i.n.d. complex Gaussian random variables (i.e., 52  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks  Pilot Symbols s(1) s(2) s(3) s(4) s(5) s(6) s(7) s(8) s(9)  Data Symbols (a) Pilot Symbols s(1) s(2) s(3) s(4) s(5) s(6) s(7) s(8) s(9)  Data Symbols (b) Figure 3.1: Example for two different pilot placement patterns with N = 9 samples, Np = 3 pilots, and pilot-to-received samples ratio β Np /N = 1/3: (a) Uniform pilot placement with P = {1, 4, 7} and D = {2, 3, 5, 6, 8, 9}. (b) Concentration of pilots at the beginning of frame with P = {1, 2, 3} and D = {4, 5, 6, 7, 8, 9}. |hl | is Rayleigh distributed), with variances σh2l  E{|hl |2 }, and are constant for the  duration of spectrum sensing. The AWGN has variance σz2  E{|zl (n)|2 }, 1 ≤ l ≤ L,  and we assume that the noise samples, the fading gains, and the PU signal are mutually independent. Both the PU signal and the noise samples are assumed to be temporally i.i.d. random variables. We assume the PU signal contains pilots for the purpose of synchronization and channel estimation at the PU receiver, cf. Fig. 3.1. The positions of the pilot symbols are contained in set P ⊆ N and there are Np  |P| pilots having power Ep  |s(n)|2 ,  n ∈ P. In practice, the pilot pattern (e.g. every (N/Np )th symbol is a pilot) may be known at the CR receiver but the position of the pilot pattern (e.g. position of first  53  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks pilot symbol in received sequence yl (n), n ∈ N ) relative to the data symbols has to be estimated. In Sections 3.3 and 3.4, we assume that this estimation is perfect and the CR receiver knows P. The effect of pilot position estimation will be studied in Section 3.5. The positions of the data symbols are contained in set D ⊆ N , where D ∩ P = ∅ and P ∪ D = N . The data symbols s(n), n ∈ D, are assumed to have zero mean and variance σs2  E{|s(n)|2}, n ∈ D, and their real and imaginary parts are statistically  independent with identical variance σs2 /2.  3.3  Hybrid Coherent/Energy Detection  In this section, we derive the optimal and the locally optimal detectors for the considered system model. For both schemes, we assume that the set of pilot symbol positions P is perfectly known at the receiver.  3.3.1  Optimal Detection  The optimal detector is based on the likelihood ratio (LR) metric  Λ = Es {Eh {Λ(s, h)}} , where Λ(s, h) is the LR given the PU transmit signal, s the vector of fading gains, h  (3.3)  [s(1) . . . s(N)]T , and  [h1 . . . hL ]T . Based on the signal and noise models  introduced in Section 3.2, Λ(s, h) can be expressed as [87] L  N  pyl |H (yl (n)|H1 , hl , s(n)) = Λ(s, h) = p yl |H (yl (n)|H0 ) n=1 l=1  L  N  pzl (yl (n) − hl s(n)) . p zl (yl (n)) n=1 l=1  (3.4)  54  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks Since we assume impairment by AWGN with pdf pzl (z) = 1/(πσz2 ) exp (−|z|2 /σz2 ), (3.4) can be rewritten as L  N  Λ(s, h) =  −  exp l=1 n=1  2ℜ{yl (n)s∗ (n)h∗l } |s(n)|2 2 |h | + l σz2 σz2  .  (3.5)  B(l,n)  For computation of the optimal LR-based decision metric, Λ(s, h) has to be averaged over both the data symbols s(n), n ∈ D, and the vector of fading gains h. Since the pdfs of the fading gains are given by phl (hl ) = 1/ πσh2l exp −|hl |2 /σh2l , Λ(s) = Eh {Λ(s, h)} can be computed in closed form by exploiting √ ( π/|p|) exp(q 2 /(4p2 )) [95, Eq. 3.323.2]. This leads to L  Λ(s) = l=1  σz2 N n=1  σh2l  |s(n)|2 + σz2   σ2  exp   hl σz2  |  σh2l  ∞ −∞  exp(−p2 x2 ±qx) dx =    N ∗ 2 n=1 yl (n)s (n)| . N 2 + σ2 |s(n)| z n=1  (3.6)  Furthermore, assuming that all data symbols s(n), n ∈ D, have identical magnitude, we have  N n=1  |s(n)|2 = σs2 (N − Np ) + Ep Np , and (3.6) can be further simplified. In  particular, by defining N  ξl2  σz2 =  |s(n)|2 + σz4 /σh2l  n=1 2 σz (σs2 (N  − Np ) + Ep Np ) + σz4 /σh2l ,  (3.7)  the conditional LR in (3.6) can be expressed as L  Λ(s) =  σz4 σ2 ξ 2 l=1 hl l    exp   1 ξl2  N  n=1  2    yl (n)s∗ (n)  .  (3.8)  55  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks To proceed further, we have to average (3.8) with respect to the PU transmit signal. Hence, the optimal LR metric can be written as L  Λ=  σz4 σ2 ξ 2 l=1 hl l  Es        exp   L  l=1  1 ξl2  N  n=1    ∗  . yl (n)s (n)  2  (3.9)  Note that the set of pilots, P, is available at the receiver and the expectation in (3.9) is taken only with respect to the data symbols s(n), n ∈ D. However, it does not seem possible to simplify the optimal metric in (3.9) any further. Thus, the optimal metric has the following short-comings: (a) Computing the expectation in (3.9) involves a sum with M N −Np terms where M is the size of the data symbol alphabet of the PU. This entails a very high complexity even for moderate observation window sizes, e.g. for M = 4, N = 100, and Np = 10, we have M N −Np ≈ 1.5×1054 . (b) The optimal metric requires exact knowledge of the data symbol alphabet of the PU, which may not be known at the SU if the PU employs adaptive modulation. Owing to the drawbacks of the optimal hybrid metric, we develop in the following a locally optimal metric, which is amenable to practical implementation.  3.3.2  Locally Optimal Detection  LO detectors are derived under the assumption of low received SNR and are typically easier to implement than (globally) optimal detectors [82, 96]. Thus, for development of the LO hybrid metric, we assume that the PU signal is very weak when arriving at the CR receiver, which is often a valid assumption in practice. For convenience, (l)  we introduce the data SNRs, γd  (l)  σs2 σh2l /σz2 , and the pilot SNRs, γp (l)  Ep σh2l /σz2 ,  (l)  1 ≤ l ≤ L. Throughout this section we assume γd ≪ 1 and γp ≪ 1, ∀l. For derivation of the LO LR-based detector, we first decompose (3.5) into a pilot  56  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks part, ΛP (s, h), and a data part, ΛD (s, h), i.e., L  L  Λ(s, h) =  exp (B(l, n)) l=1 n∈P  exp (B(l, n)) .  (3.10)  l=1 n∈D ΛP (s,h)  (l)  ΛD (s,h)  (l)  For the low SNR regime (i.e., γd , γp ≪ 1), B(l, n) ≪ 1 is valid. Thus, we can use the second order Taylor series of the exponential function to rewrite the data part of the LR metric as L  ΛD (s, h) =  exp (B(l, n)) l=1 n∈D L  ∼ =  Using the approximation  1 + B(l, n) + l=1 n∈D  N i=1 (1 + xi )  ∼ = 1+  N i=1  B 2 (l, n) 2  .  (3.11)  xi , ∀xi ≪ 1, to simplify (3.11), we  can express ΛD (h) = Es {ΛD (s, h)} as L  ΛD (h) ∼ = 1+  l=1  1 Es {B(l, n)} + 2 n∈D  L  l=1 n∈D  Es B 2 (l, n) .  (3.12)  Next, we substitute B(l, n) from (3.5) into (3.12) and perform the statistical expectation over the data symbols. Doing so and ignoring all terms that depend on (σs2 /σz2 )ν , ν > 1, because of the low SNR assumption, we obtain L  ΛD (h) ∼ =1+  l=1  σs2 σz2  n∈D  |yl (n)|2 −1 σz2  |hl |2 .  (3.13)  For the pilot part of the metric, we first note that the pilot symbols, s(n), n ∈ P, are assumed to be known to the CR receiver. Hence, we can formally write ΛP (h) =  57  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks ΛP (h, s). Thus, we obtain from (3.10) L  ΛP (h) =  l=1  where we exploited  exp −  n∈P  2 Np Ep 2 |h | + l σz2 σz2  n∈P  ℜ{yl (n)s∗ (n)h∗l } ,  (3.14)  |s(n)|2 = Np Ep . Based on (3.13) and (3.14), we obtain the  LO LR metric conditioned on the channel, Λ(h) = ΛP (h) × ΛD (h), which can be expressed as L  Λ(h) ∼ =  σs2 σz2  1+ l=1 L  ×  l=1  exp − +  ℜ{hl } and hI,l  where hR,l  n∈D  |yl (n)|2 −1 σz2  2 Np Ep |hl |2 + 2 ℜ 2 σz σz 2 ℑ σz2  |hl |2 yl (n)s∗ (n) hR,l n∈P  yl (n)s∗ (n) hI,l  ,  (3.15)  n∈P  ℑ{hl }. Since we assume i.n.d. Rayleigh fading,  hR,l and hI,l are real-valued Gaussian random variables with variance σh2l /2, re∞  spectively. Thus, based on (3.15) and the well-known integrals −∞ exp(−p2 x2 ± √ ∞ qx) dx = ( π/|p|) exp(q 2 /(4p2 )) [95, Eq. 3.323.2] and −∞ x2 exp(−p2 x2 ± qx) dx = √ ( π/(2|p|3))(1 + q 2 /(2p2 )) exp(q 2 /(4p2 )) [95, Eq. 3.462.8], the LO LR metric, Λ = E{Λ(h)}, can be obtained as Λ∼ = where Al n∈D (|yl (n)|  L |Bl |2 l=1 Al  exp L l=1  σh2l Al  Np Ep /σz2 + 1/σh2l , |Bl |2 2  L  1+ l=1  |  n∈P  Cl Al  1+  |Bl |2 Al  ,  yl (n)s∗ (n)|2 /σz4 , and Cl  (3.16)  (σs2 /σz2 )  /σz2 − 1). Taking the logarithm of both sides of (3.16) and ignoring  58  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks irrelevant terms, we obtain L  log Λ ∼ = ∼ =  l=1 L  l=1  L  |Bl |2 + log 1 + Al 2  |Bl | + Al  L  l=1  Cl Al  l=1  1+  Cl Al  1+  |Bl |2 Al  |Bl |2 Al  .  (3.17)  In (3.17), we used the approximation log(1 + x) ∼ = x, x ≪ 1, which is applicable  since for low SNR, we have Cl /Al ∼ = (l)  Np γp  (l) 1+Np γp  (l) |hl |2 2 Np γp σh l  (l)  |hl |2 (N −Np )(γd )2 2 (1+N γ (l) ) σh p p l  ≪ 1 and similarly |Bl |2 /Al ∼ =  + 1 ≪ 1, ∀l. By rearranging the terms in (3.17) and exploiting  Cl /Al ≪ 1 once again, we arrive at L  log Λ ∼ =  l=1 L  ∼ =  l=1  |Bl |2 Al  Cl 1+ Al  |Bl |2 + Al  L  l=1  L  + l=1  Cl Al  Cl . Al  (3.18)  Finally, by using Al ∼ = 1/σh2l and the definitions of |Bl |2 and Cl in (3.18) and ignoring irrelevant terms, the LO hybrid coherent/energy detection metric is obtained as L  ΛLO = l=1  σh2l σz2  2  L  ∗  yl (n)s (n) + n∈P ΛP LO  l=1  σs2 σh2l σz2  n∈D  |yl (n)|2 .  (3.19)  ΛD LO  Eq. (3.19) reveals that the LO hybrid coherent/energy detection metric is a simple linear combination of a coherent metric, ΛPLO , which involves the correlation of the symbols received at the pilot positions and the pilots, and an energy detection metric, P D ΛD LO . We note that while both ΛLO and ΛLO have been used for spectrum sensing  before, cf. e.g. [31, 35], the combination of both has not been considered yet. Also, our derivations in this section reveal that linearly combining ΛPLO and ΛD LO is only 59  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks (l)  (l)  optimal when γd → 0 and γp → 0, ∀l. Nevertheless, (3.19) can also be used as a suboptimal metric if these conditions are not strictly fulfilled. In this context, we emphasize again that the implementation of the globally optimal metric in (3.9) is not feasible. It is interesting to note that for both the coherent and the energy detection parts of ΛLO the contributions of the different diversity branches are weighted with the respective variances of the fading gains, σh2l , i.e., the contributions of strong branches are amplified while those of weak branches are attenuated as is also done in maximal ratio combining for data detection [90].  3.4  Performance Analysis  In this section, we derive the probabilities of false alarm and missed detection of the LO hybrid coherent/energy detection metric developed in the previous section. Furthermore, considering a Neyman-Pearson based optimization of spectrum sensing, we derive closed-form expressions for the probabilities of false alarm and missed (l)  (l)  detection for the practical case of low SNR (i.e., γd , γp ≪ 1, ∀l) and large sample size (i.e., N, Np ≫ 1). Throughout this section, we assume that the PU employs a constant envelope modulation format such as phase-shift keying (PSK), where |s(n)|2 = σs2 , n ∈ D. Also, unless stated otherwise, for simplicity of presentation we assume i.i.d. Rayleigh (l)  (l)  fading across the diversity branches, i.e., σh2l = σh2 , γd = γd , and γp = γp , 1 ≤ l ≤ L.  3.4.1  Probability of False Alarm  The proposed detector decides in favor of H0 if ΛLO < τ , and for H1 otherwise, where τ is the decision threshold.  The probability of false alarm is defined as 60  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks Pf (τ )  Pr {ΛLO > τ |H0 }. In the following, we first consider the general case before  we specialize our results to low SNR and large sample size. 1) General Case: In general, the conditional pdf pΛLO (u|H0 ) does not seem tractable. Thus, it is convenient to express the pdf in terms of its moment generE e−sΛLO |H0  ating function (MGF) ΦΛLO (s|H0 ) 1 2πj  c+j∞ c−j∞  via the relation pΛLO (u|H0 ) =  ΦΛLO (s|H0 )eus ds, where c is a small positive constant in the region of con-  vergence of the integral [97]. Consequently, the probability of false alarm can be expressed as −τ  Pf (τ ) = Pr {−ΛLO < −τ |H0 } = c+j∞  =  1 2πj  p−ΛLO (u|H0 )du −∞  ΦΛLO (−s|H0 )e−τ s  ds . s  (3.20)  c−j∞  For calculation of ΦΛLO (s|H0 ), we consider metric ΛLO under H0 . From (3.19), we obtain L  ΛLO |H0 =  l=1  σh σz  L  2  zl (n)s∗ (n)  +  √  l=1 n∈D  n∈P  2  γd zl (n) ,  (3.21)  T2  T1  where T1 is a sum of independent Gaussian noise samples multiplied by known pilot symbols. Consequently, T1 is also Gaussian distributed, T1 ∼ N (0, Np γp σz2 ). Similarly, T2 follows a Gaussian distribution, T2 ∼ N (0, γd σz2 ), too. Hence, we obtain for the pilot part ΛPLO and the data part ΛD LO of hybrid metric ΛLO , L  ΛPLO |H0  l=1  |T1 |2 ∼ Γ L, σz2 Np γp  (3.22)  61  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks and L  ΛD LO |H0  l=1 n∈D  |T2 |2 ∼ Γ L(N − Np ), γd σz2 ,  (3.23)  respectively. ΛLO |H0 is a sum of two independent Gamma distributed random variables with different scale and shape parameters, and as a result, its pdf is not tractable in general. However, due to the statistical independence of the pilot part and the data part of the hybrid metric, the MGF of ΛLO |H0 is simply the product of the individual MGFs. Using the fact that the MGF of a Gamma distributed random variable X ∼ Γ(a, b) is given by ΦX (s) = (1 + bs)−a , we obtain for the MGF of ΛLO |H0 the expression ΦΛLO (s|H0 ) =  1 (1 + γd σz2 s)  L(N −Np )  (1 + σz2 Np γp s)L  .  (3.24)  We point out that the Gamma distribution approaches a Gaussian distribution only if the shape parameter is large enough. However, while the data part ΛD LO |H0 of the hybrid metric has a large shape parameter, L(N − Np ), for large sample size N and can be approximated as Gaussian, this is not true for the pilot part ΛPLO |H0 , whose shape parameter is L. Adopting a similar approach as for the i.i.d. case, (3.24) can be easily generalized to i.n.d. Rayleigh fading, where we obtain  ΦΛLO (s|H0 ) =  1 L  1+ l=1  (l) γd σz2 s  N −Np  .  L  1+  (3.25)  σz2 Np γp(l) s  l=1  The probability of false alarm can be obtained by substituting (3.24) [(3.25)] into (3.20) and evaluating the remaining integral numerically using the well-known Gauss-Chebyshev quadrature rule [98]. We note that the probabilities of false alarm 62  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks of coherent detection and energy detection can be obtained as special cases of (3.20). Namely, for coherent detection we formally let γd = 0 and for energy detection we let γp = 0 and Np = 0 in (3.24) [(3.25)]. 2) Low SNR and Large Sample Size: Although (3.20) can be easily evaluated numerically, it does not offer much insight into how different parameters such as the percentage of pilot symbols, the diversity order, and the SNR affect the overall performance. To gain this insight, in this subsection, we assume low SNR (i.e. γd , γp ≪ 1) and large sample size (i.e. N, Np ≫ 1). Note that, since we are interested in the low SNR regime, large sample sizes are usually required to achieve a reasonable performance. Hence, our assumptions here lead to practically relevant results. Assuming i.i.d. fading, based on (3.20) and (3.24) the probability of false alarm can be expressed as [97]  Pf (τ ) = L−1  1 s (1 − γd σz2 s)  L(N −Np )  (1 − σz2 Np γp s)L  ,  (3.26)  t=−τ  where L−1 {·}t=t0 denotes the inverse Laplace transform at t = t0 . The fraction in (3.26) contains poles at s1 =  1 σz2 γd  and s2 =  1 . σz2 Np γp  Since we assume Np ≫ 1, for  γd ≈ γp , we have s1 ≫ s2 and we can ignore the effect of the poles at s1 . Hence, (3.26) simplifies to Pf (τ ) ∼ = L−1  1 s (1 −  σz2 Np γp s)L  = t=−τ  1 τ Γ L, 2 (L − 1)! σz Np γp  .  (3.27)  Surprisingly, (3.27) reveals that for low SNR and a large number of pilot symbols, the probability of false alarm does not depend on the data part of the hybrid metric. Thus, in this case, for a given τ the proposed hybrid metric ΛLO does not improve the probability of false alarm compared to the coherent metric ΛPLO . We will see in 63  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks the next section that the situation is different for the probability of missed detection.  3.4.2  Probability of Missed Detection  Similar to Section 3.4.1, it is convenient to first discuss the general case before specializing our results to the case of low SNR and large sample size. 1) General Case: The probability of missed detection is defined as Pm (τ ) Pr {ΛLO < τ |H1 }. Using the same approach as for the probability of false alarm, we can express the probability of missed detection as c+j∞  τ  Pm (τ ) =  pΛLO (u|H1 )du =  ΦΛLO (s|H1 )eτ s ds . 2πj s  (3.28)  c−j∞  −∞  For derivation of the MGF ΦΛLO (s|H1 ), we first assume that the fading gains hl , 1 ≤ l ≤ L, and the data symbols s(n), n ∈ D, are given. Consequently, from (3.19) the hybrid metric under hypothesis H1 is given by L  ΛLO |H1 , h, s =  σh σz  l=1  2  n∈P  hl |s(n)|2 + zl (n)s∗ (n) T1,l  L  + l=1 n∈D  √  2  γd (hl s(n) + zl (n)) .  (3.29)  T2,l  The distributions of T1,l and T2,l are the same as those of T1 and T2 in Section 3.4.1, respectively, but with a shifted mean, i.e., T1,l ∼ N (hl Np γp σz /σh , Np γp σz2 ) and T2,l ∼ √ L 2 D N hl s(n) γd , γd σz2 . Consequently, ΛPLO |H1 , h, s l=1 |T1,l | and ΛLO |H1 , h, s L l=1  n∈D  |T2,l |2 follow non-central chi-square distributions, with MGFs given by  64  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks [90],  ΦΛPLO |H1 ,h,s (s) =  ΦΛDLO |H1 ,h,s (s) =  1 (1 + Np γp σz2 s)L  1 (1 + γd σz2 s)L(N −Np )  exp −  exp −  Np 2 γp 2 σz2 Ll=1 |hl |2 s σh2 (1 + Np γp σz2 s)  γd  n∈D  ,  |s(n)|2 Ll=1 |hl |2 s 1 + γd σz2 s  (3.30)  . (3.31)  Hence, the conditional MGF of the hybrid metric is given by ΦΛLO |H1 ,h,s (s) = ΦΛPLO |H1 ,h,s (s) × ΦΛDLO |H1 ,h,s (s). Since we assume constant envelope modulation, we can substitute  n∈D  |s(n)|2 = (N − Np )σs2 in (3.31). For the averaging over the L 2 l=1 |hl |  fading, we define η  1 L−1 2L η (L−1)!σh  is given by pη (η) =  and note that for i.i.d. fading gains, the pdf of η exp − ση2 , η ≥ 0 [90]. Hence, the unconditional h  MGF can be expressed as  ΦΛLO |H1 (s) = ∞  ×  η  L−1  1 (L − 1)!σh2L (1 + Np γp σz2 s)L (1 + γd σz2 s)L(N −Np ) −  exp  0  Exploiting the integral  ∞ 0  Np2 γp2 σz2 s γd σs2 (N − Np )s 1 + + 2 η dη. (3.32) 2 2 2 σh (1 + Np γp σz s) 1 + γd σz s σh  xn exp(−βx)dx =  n! β n+1  [95, Eq. 3.351.3], (3.32) can be  simplified to ΦΛLO (s|H1 ) =  1 (1 + γd σz2 s)  L(N −Np −1)  DL  ,  (3.33)  where  D  1 + σz2 γd + Np γp + Np2 γp2 + γd2 (N − Np ) s + σz4 Np γp γd + Np2 γp2 γd + γd2 γp Np (N − Np ) s2 .  (3.34)  65  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks For i.n.d. Rayleigh fading, a similar approach as for the i.i.d. case leads to L  ΦΛLO (s|H1 ) =  l=1 L  χ(l) 1 + Np γp(l) σz2 s D (l)  (l)  1 + γd σz2 s  (3.35)  N −Np  l=1  l=1  L j=1,j=l  ,  L  1 + Np γp(l) σz2 s  where χ(l) =  (l)  1 + γd σz2 s  (l)  (l)  α(l) σh2l / α(l) σh2l − α(j) σh2j , α(l) = γd (N−Np )σs2 s/(1+γd σz2 s)+ (l)  (l)  Np2 (γp )2 σz2 s/(σh2l (1 + Np γp σz2 s)), and (l)  (l)  D (l) = 1 + σz2 γd + Np γp(l) + Np2 (γp(l) )2 + (γd )2 (N − Np ) s (l)  (l)  (l)  +σz4 Np γp(l) γd + Np2 (γp(l) )2 γd + (γd )2 γp(l) Np (N − Np ) s2 . (3.36) Substituting (3.33) [(3.35)] into (3.28) yields the probability of missed detection. The corresponding integral can again be evaluated numerically using the GaussChebyshev quadrature rule [98] and the result is also valid for the special cases of coherent and energy detection, respectively. 2) Low SNR and Large Sample Size: Similar to the probability of false alarm, it is again insightful to consider the special case of low SNR (i.e., γd , γp ≪ 1) and large sample size (i.e. N, Np ≫ 1). Assuming i.i.d. fading and using (3.28) and (3.33), we obtain  Pm (τ ) = L−1  1 s (1 + γd σz2 s)  L(N −Np −1)  (1 + α1 s + α2 s2 )L  ,  (3.37)  t=τ  where α1 = σz2 γd + Np γp + Np2 γp2 + γd2 (N − Np ) ∼ = σz2 Np γp + Np2 γp2 + γd2 (N − Np ) and α2 = σz4 Np γp γd + Np2 γp2 γd + γd2 Np γp (N − Np ) . The fraction in (3.37) has poles  66  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks s1 = − σ21γd and z  s2,3 = ∼ = where we used  √  −α1 ±  α1 (−1 ± 1 − α12 − 4α2 = 2α2 2α2  α1 −1 ± 1 −  2α2 α21  4α2 ) α21  ,  2α2  (3.38)  1−x∼ = 1 − x2 , ∀x ≪ 1, which is applicable since  α2 α21  ≪ 1 holds in the  considered regime. Consequently, poles s2 and s3 are approximately at s2 = − αα12 + α11 and s3 = − α11 . Hence, for low SNR, it can be seen that |s3 | ≪ |s2 | ≪ |s1 | and poles s1 and s2 can be ignored for the probability of missed detection. Thus, (3.37) simplifies to Pm (τ ) ∼ = L−1  1 s (1 + α1 s)  =  L t=τ  1 τ γ L, (L − 1)! α1  .  (3.39)  We note that, in contrast to the asymptotic expression for the probability of false alarm in (3.27), the probability of missed detection in (3.39) depends on both the pilot part, ΛPLO , and the data part, ΛD LO , of hybrid metric ΛLO .  3.4.3  Neyman-Pearson Optimization  In practice, the decision threshold τ has to be adjusted to achieve a desired tradeoff between the probabilities of false alarm and missed detection. For the analysis, we choose the Neyman-Pearson approach for optimization of the CR system where the probability of missed detection (false alarm) is fixed and the probability of false alarm (missed detection) is minimized. This leads to two different optimization problems. Problem 1: In order to protect the PU from unwanted interference, a certain maximum probability of missed detection P¯m may be imposed on the SU [66]. Consequently, the SU tries to maximize its transmission opportunity by minimizing the  67  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks probability of false alarm for the given P¯m . Hence, this optimization problem can be formulated as  min Pf (τ ) τ  subject to Pm (τ ) ≤ P¯m .  (3.40a) (3.40b)  The constraint in optimization problem (3.40) has to be active for the optimal Pf (τ ), which means that the constraint is satisfied with equality. Using (3.28) we can determine the optimal threshold τ ⋆ which satisfies Pm (τ ⋆ ) = P¯m using a simple onedimensional line search. The found τ ⋆ is then substituted back into (3.20) to obtain the minimum probability of false alarm. Problem 2: Another option for formulating the optimization problem is to minimize the probability of missed detection for a given probability of false alarm [39]. An application of this scenario is when the SU pays revenues to the PU in return for some guaranteed access probability (1 − P¯f ) when the PU is absent. The corresponding Neyman-Pearson problem is given by  min Pm (τ )  (3.41a)  subject to Pf (τ ) ≤ P¯f .  (3.41b)  τ  Similar to Problem 1, the constraint in (3.41b) is satisfied with equality at optimality. Thus, the optimal threshold τ ⋆ and the minimum probability of missed detection can be obtained with a similar procedure as for Problem 1. To gain more insight into the achievable performance of the solutions to Problems 1 and 2, in the following, we will perform an asymptotic analysis assuming low SNR and large sample size. 68  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks  3.4.4  Asymptotic Performance of Neyman-Pearson Optimization  In this section, we assume low SNR (i.e. γd , γp ≪ 1), large sample size (i.e. N, Np ≫ 1), and high accumulated SNR (i.e., Np γp , (N − Np )γd ≫ 1).5 1) Solution of Problem 1 (Fixed Probability of Missed Detection): Exploiting (3.39) for the considered asymptotic scenario, we obtain from Pm (τ ⋆ ) = P¯m threshold τ ⋆ = α1 γL−1 (L − 1)!P¯m , where γs−1 (x) is the inverse of γ(s, x) which is a monotone function of x. Now, by substituting the derived τ ⋆ back into (3.27), we have Np γp + Np2 γp2 + γd2 (N − Np ) 1 Γ L, γL−1 (L − 1)!P¯m (L − 1)! Np γp 1 1 − β γd2 = Γ L, γL−1 (L − 1)!P¯m 1 + + γp βN , (L − 1)! β γp  Pf =  with pilot-to-received-samples ratio β  (3.42)  Np /N. Since we assume that Np γp =  βNγp ≫ 1, the second argument inside the incomplete Gamma function in (3.42) is large. Hence, we exploit the approximation Γ(s, x) → xs−1 e−x , x → ∞, which leads to log Pf ∼ = − log((L − 1)!) + (L − 1) log γL−1 (L − 1)!P¯m − γL−1 (L − 1)!P¯m  1+  1+  1 − β γd2 + γp βN β γp  1 − β γd2 + γp βN β γp  .  (3.43)  Several interesting observations can be made from (3.43). 5  We note that our results in Figs. 3.2 and 3.3 suggest that the asymptotic results derived in this section constitute good approximations even for moderate Np γp (e.g. Np γp ≈ 5).  69  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks a) Asymptotic behavior: For large N, the third term in (3.43) dominates its logarithm in the second term. Consequently, on a logarithmic scale, the probability of false alarm decreases linearly with N with a slope of −γL−1 (L − 1)!P¯m γp β. Hence, the pilot SNR, γp , and the pilot-to-received-samples ratio, β, have a linear effect on the slope. The effect of the diversity order is less obvious and is given by the term γL−1 (L − 1)!P¯m , which is a convex and increasing function in L. To provide more insight, we can further simplify γL−1 (L − 1)!P¯m by considering the special case  P¯m → 0. Exploiting γ(s, x) → xs /s, x → 0, leads to γL−1 (L − 1)!P¯m ∼ =  L  L! P¯m for  P¯m → 0, which implies a sub-linear growth of the slope in L. b) Comparison with coherent detection: By formally setting γd = 0 in (3.43) the corresponding probability of false alarm of coherent detection can be obtained. By doing so, we observe that the resulting asymptotic slope of the probability of false alarm is identical to that for the proposed hybrid detector and also given by −γL−1 (L − 1)!P¯m γp β. Hence, to facilitate the comparison between hybrid and coherent detection, we define GPf = log Pf |γd =0 − log Pf , which represents the gain of hybrid detection compared to coherent detection for a given number of samples N. For large N, GPf becomes a constant implying a vertical offset of the probability of false alarm curves for hybrid and coherent detection  GPf =  γL−1  1 − β γd2 γd2 (1 − β) ¯ (L − 1)!Pm − (L − 1) log 1 + β γp βγp + β 2 γp2 N  1 − β γd2 ∼ . = γL−1 (L − 1)!P¯m β γp  (3.44)  As expected, the gain achievable with hybrid detection over coherent detection increases with increasing data SNR γd and decreasing pilot-to-received-samples ratio β.  70  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks c) Comparison with energy detection: For pure energy detection the pilots are simply also treated as data symbols, i.e., β = 0. However, in this case, (3.43) is not applicable since it was derived under the assumption βNγp ≫ 1. Thus, we borrow a result from Chapter 2, where it was shown that for low SNR and large N, log Pf decreases linearly in N with slope − γL−1 (L − 1)!P¯m  2  γd2 /(2L). Thus, it is  convenient to consider the case γd = γp and to introduce the ratio of the slopes of hybrid detection and energy detection as  ζ Pf =  γd γL−1  2βL . (L − 1)!P¯m  (3.45)  Eq. (3.45) shows that the ratio of the slopes increases linearly with increasing β and decreases with increasing γd . Also, the impact of the diversity order on ζPf is captured in the term L/γL−1 (L − 1)!P¯m , which decreases with increasing L. We note, however, that for typical values of β, γd , and L, ζPf > 1 is valid, i.e., the gap in the probabilities of false alarm of the proposed hybrid detector and the energy detector increases with increasing N. 2) Solution of Problem 2 (Fixed Probability of False Alarm): In this (L − 1)!P¯f , where Γ−1 case, we obtain the optimal threshold τ ⋆ = σz2 Np γp Γ−1 s (x) is L the inverse of Γ(s, x) which is a monotone function of x, by letting Pf (τ ) = P¯f and using (3.27). Threshold τ ⋆ is substituted back into (3.39) to obtain  Pm =    Γ−1 (L − 1)!P¯f L  1 γ L, (L − 1)! 1+  2 1−β γd β γp  + γp βN    .  (3.46)  Considering that γp βN ≫ 1, we can exploit γ(s, x) → xs /s, x → 0, in (3.46) to  71  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks obtain 2 ¯f − L log 1 + 1 − β γd + γp βN (L − 1)! P log Pm ∼ = − log L! + L log Γ−1 L β γp  . (3.47)  Several interesting observations can be made from (3.47). a) Asymptotic behavior: Eq. (3.47) reveals that for the asymptotic case of N → ∞, the probability of missed detection decreases linearly with N on a log-log scale, and the slope is given by −L. b) Comparison with coherent detection: The gain of hybrid detection compared to coherent detection may be defined as GPm = log Pm |γd =0 − log Pm . Thus, based on (3.47) we obtain  GPm = L log 1 +  γd2 (1 − β) βγp + β 2 γp2 N  .  (3.48)  Consequently, for asymptotically large sample size, the gain GPm approaches zero. Nevertheless, we note that for moderate N hybrid detection does achieve a lower probability of missed detection than coherent detection, cf. Section 3.5. c) Comparison with energy detection: For energy detection, it was shown in Chapter 2, that for low SNR, large N, and fixed probability of false alarm, log Pm decreases linearly with log N with slope −L/2. Thus, on a log-log scale, the difference in slope between hybrid detection and energy detection is ζPm = 2.  3.5  Results and Discussions  In this section, we present analytical and simulation results for the proposed hybrid coherent/energy detection scheme. Throughout this section we assume that the PU  72  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks employs quaternary PSK (QPSK) modulation, i.i.d. Rayleigh fading across the diversity branches, SNR γd = γp = −10 dB, and Ep = σs2 unless stated otherwise. The simulation results were averaged over 5,000 independent realizations of the Rayleigh fading channels and for each channel the results were averaged over at least 5,000 independent realizations of the PU transmit signal and the noise. The exact analytical results shown in the following were obtained by evaluating (3.20) and (3.28). In addition, we also show results for the low SNR/large sample size approximation in (3.43) and (3.47).  3.5.1  Known Pilot Positions  Similar to Sections 3.3 and 3.4, we assume in this sub-section that the pilot positions are perfectly known at the CR receiver. In Fig. 3.2, we show the average probability of false alarm, Pf , of the proposed hybrid coherent/energy detector using the metric in (3.19) vs. sample size N for a fixed probability of missed detection of P¯m = 0.1 and different numbers of diversity branches L. We assume β = 0.05, i.e., 5 % of all transmitted symbols are pilot symbols. For comparison, we also show the performance of a pure energy detector, which treats the pilot symbols as additional data symbols. For both considered detectors the agreement between analytical and simulation results is excellent. In addition, the analytical approximation (dashed lines) obtained by evaluating (3.43) approaches the exact results as the sample size N increases. Since we use a log-scale for Pf , for large N all Pf –curves become straight lines. Since, as expected from Section 3.4.4, the curves for hybrid detection have a steeper slope than those for energy detection, hybrid detection yields large performance gains, especially for large N. For both detectors increasing L increases the slope of the curves, and thus, leads  73  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  L=1  −1  10  −2  L=3  L=2  Pf  10  −3  10  Energy Detection (Analysis) Hybrid Detection (Analysis) −4  10  Energy Detection (Simulation) Hybrid Detection (Simulation) Hybrid Detection (Approx. for Large N) 500  1000  1500  2000  2500  N  Figure 3.2: Pf of hybrid detection and energy detection vs. N . Positions of the pilots are known, β = 0.05, SNR = −10 dB, P¯m = 0.1, and number of diversity branches are L = 1, 2, and 3. to significant performance gains. In Fig. 3.3, we investigate the probability of missed detection, Pm , of the proposed hybrid coherent/energy detector for a fixed probability of false alarm of P¯f = 0.1 and different numbers of diversity branches L. Again, we assume β = 0.05 and compare with energy detection. The exact analytical results (from (3.20) and (3.28)) are in excellent agreement with the simulation results. The analytical approximation (3.47) for large N nicely reflects the logarithmic decrease of Pm as a function of N but deviates somewhat from the exact analytical result. This deviation diminishes for larger but less practical values of N (not shown in figure). Fig. 3.3 confirms that hybrid detection also yields large performance gains compared to energy detection when the probability of missed detection is considered as performance metric. 74  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks  L=1  L=2 −1  Pm  10  L=3 Energy Detection (Analysis) −2  10  Hybrid Detection (Analysis) Energy Detection (Simulation) Hybrid Detection (Simulation) Hybrid Detection (Approx. for Large N)  500  1000  1500  2000  2500  N  Figure 3.3: Pm of hybrid detection and energy detection vs. N . Positions of the pilots are known, β = 0.05, SNR = −10 dB, P¯f = 0.1, and number of diversity branches are L = 1, 2, and 3. In Figs. 3.4 and 3.5, we show the probabilities of false alarm and missed detection of the proposed hybrid detector as a function of the pilot-to-received-samples ratio β for fixed probabilities of missed detection (P¯m = 0.1) and false alarm (P¯f = 0.1), respectively. We assume N = 2000 received samples and also show results for coherent detection based on ΛPLO for comparison. For β → 0 the hybrid detector (without channel estimation) becomes a pure energy detector and thus, achieves a significantly better performance than the coherent detector which only relies on (in this case very few) pilot symbols. As β increases, the gap between the hybrid detector and the coherent detector decreases as expected from (3.44) and (3.48), respectively. Nevertheless, for the cases considered in Figs. 3.4 and 3.5 the gain is significant for β ≤ 0.05, especially for L = 3. For example, for β = 0.05 and L = 3, from Figs. 3.4 and 3.5 75  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  Hybrid Detection (Analysis) Coherent Detection (Analysis) −1  Hybrid Detection with CE  L=1  10  Hybrid Detection (Simulation) Coherent Detection (Simulation) −2  Pf  10  L=2 −3  10  L=3  −4  10  0  2  4  6  8  10  12  Percentage of Pilot Symbols  Figure 3.4: Pf of coherent detection and hybrid detection with and without channel estimation (CE) vs. 100 × β %. Positions of the pilots are known, N = 2000, SNR = −10 dB, and P¯m = 0.1. we obtain gains GPf = 1.8 and GPm = 0.43, respectively. These gains are in excellent agreement with the gains predicted by (3.44) [first line of equation: GPf = 1.8; second line of equation: GPf = 2.1] and (3.48) [GPm = 0.48]. We note that although the PU signal may include more than 5 % pilot symbols in practice, the SU may not know all of them. For example, in cellular systems, the SU may know those pilot symbols that are broadcast to all users in the cell but not those that are user specific. We also show in Figs. 3.4 and 3.5 simulation results for a hybrid detector which first n∈P  computes  the  maximum-likelihood  (ML)  channel  estimates  ˆl h  =  yl (n)s∗ (n)/(Np Ep ), 1 ≤ l ≤ L, and subsequently uses a simplified version  76  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  Hybrid Detection (Analysis) Coherent Detection (Analysis)  L=1  Hybrid Detection with CE Hybrid Detection (Simulation) Coherent Detection (Simulation)  −1  10  Pm  L=2  L=3 −2  10  0  2  4  6  8  10  12  Percentage of Pilot Symbols  Figure 3.5: Pm of coherent detection and hybrid detection with and without channel estimation (CE) vs. 100 × β %. Positions of the pilots are known, N = 2000, SNR = −10 dB, and P¯f = 0.1. of metric (3.15), L  ΛCE =  ˆ∗ yl (n)s∗ (n)h l  2ℜ n∈P  l=1 L  + l=1  ˆ l |2 |h  σs2 σz2  n∈D  |yl (n)|2 − (σs2 (N − Np ) + Np Ep ) ,  (3.49)  ˆ l , 1 ≤ l ≤ L, it can be for spectrum sensing. After inserting the channel estimates h observed that the pilot part of (3.49),  L l=1  |  n∈P  yl (n)s∗ (n)|2 /(Np Ep ), is similar to  the pilot part of the hybrid metric without channel estimation in (3.19). The main difference is that the contributions of the different diversity branches are weighted by the channel branch variance σh2l in (3.19) but not in (3.49). Such weighting is benefi77  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks cial in i.n.d. fading. Fig. 3.4 reveals that even for i.i.d. fading hybrid detection with channel estimation is outperformed by hybrid detection without channel estimation for small values of β (e.g. β ≤ 0.02 for L = 3) since in this case, the quality of the channel estimates is very poor. For moderate to large values of β, hybrid detection with channel estimation yields a better performance than hybrid detection without channel estimation at the expense of an increased receiver complexity due to the channel estimation. While these gains are negligible when considering the probability of missed detection (cf. Fig. 3.5), they can be substantial when considering the probability of false alarm (cf. Fig. 3.4). Thus, a further investigation and analysis of metric (3.49) is an interesting topic for future work. In Fig. 3.6, we show the receiver operating curve (ROC) of the proposed hybrid detector, the coherent detector, and the energy detector for three different SNR values. We assume N = 2000 received samples, L = 3 diversity branches, and β = 0.05. Since the accuracy of the derived analytical expressions for the probabilities of false alarm and missed detection was confirmed by simulation in the previous figures, we show only analytical results in Fig. 3.6, except for hybrid detection with channel estimation where simulation results are shown. As can be observed, hybrid detection without channel estimation outperforms coherent and energy detection for all pairs of Pf and Pm and all considered SNR values. The gap between the hybrid detector and the coherent detector increases with increasing SNR, and the gap between the coherent detector and the energy detector is approximately independent of the SNR (on the considered log-log scale). For the considered value of β, hybrid detection with channel estimation performs better than hybrid detection without channel estimation. The gap between both schemes increases with decreasing probability of false alarm and increasing SNR.  78  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  SNR=-15 dB  −1  10  −2  Pf  10  SNR=-5 dB  −3  10  SNR=-10 dB  Energy Detection −4  10  Coherent Detection Hybrid Detection Hybrid Detection with CE −4  10  −3  10  −2  10  −1  10  0  10  Pm  Figure 3.6: ROC of hybrid detection with and without channel estimation (CE), coherent detection, and energy detection for SNR = −5 dB, −10 dB, and −15 dB. Positions of the pilots are known, N = 2000, L = 3, and β = 0.05.  3.5.2  Estimated Pilot Positions  In this sub-section, we investigate the impact of a practical pilot position estimation scheme on the performance of the proposed hybrid detection scheme. We first note that the signal model in (3.2) implies perfect time and frequency synchronization. Blind time and frequency synchronization schemes, which do not require knowledge of the pilot symbols, are available in the literature [99] and we do not investigate the impact of imperfect synchronization here.6 Furthermore, we assume that all the pilot symbols are equally spaced, cf. Fig. 3.1(a). The CR receiver knows that every 6  We note that a constant phase offset does not influence the performance of the proposed hybrid detector. However, phase changes (e.g. induced by a frequency offset) have a negative impact on the coherent part of the hybrid detection metric and should be minimized.  79  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks (N/Np )th symbol is a pilot symbol and also knows the value of the pilot symbol, Ep , n ∈ P, i.e., the pilot pattern is known. However, the CR receiver does  s(n) =  not know where the first pilot symbol is located in the observed signal yl (n), n ∈ N . For estimation of the position of the first pilot symbol, we adopt the following simple correlation metric7 [99]  kˆp = argmax kp ,kp ∈Kp  where Kp      L  Np −1  l=1 ν=−(ω−1)Np  yl  Nν + k p s∗ Np  Nν +1 Np  2    ,  (3.50)    {1, 2, . . . , N/Np }, kˆp is the estimated position of the first pilot symbol,  yl (n), −ωNp ≤ n < 0, are symbols received in previous sensing intervals, and ω denotes the number of sensing intervals employed for pilot position estimation. We assume that the channel is constant for all ω sensing intervals. ω = 1 implies that the pilot positions are determined based on the current sensing interval only, whereas for ω > 1 more averaging is performed for pilot position estimation than for sensing. Using ω > 1 seems reasonable since the pilot positions are not expected to change from sensing interval to sensing interval. We note that if pilots are not present, i.e., H = H0 , the pilot position estimation will fail, of course. However, this does not affect the performance of the hybrid detector for spectrum sensing. Since it does not seem possible to include the effect of pilot position estimation in the analysis of the probabilities of false alarm and missed detection, we resort to simulations in this sub-section. In Fig. 3.7, we compare the probabilities of false alarm of the proposed hybrid detection, coherent detection, and energy detection for the cases of known and esti7  For (3.50) we have assumed that all diversity branches have the same delay which is appropriate if one CR with L receive antennas is employed. For the case, where L single-antenna CRs are used, each diversity branch may have a different delay and the delay of each branch has to be estimated separately.  80  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  Estimated pilot position ω=1 −1  10  Estimated pilot position ω=5 −2  Pf  10  −3  10  −4  10  Known pilot position  Energy Detection Coherent Detection Hybrid Detection  500  1000  1500  2000  2500  N  Figure 3.7: Pf of hybrid detection, coherent detection, and energy detection vs. N. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, β = 0.05, SNR = −10 dB, and P¯m = 0.1. mated pilot positions, respectively. We assume β = 0.05, i.e., every 20th transmitted symbol is a pilot, L = 3, and P¯m = 0.1. For pilot position estimation, we consider ω = 1 and ω = 5, respectively. For energy detection, which does not use pilots, there is only one curve in Fig. 3.7. Note that in case of imperfect pilot position estimation decision threshold τ was adjusted to guarantee P¯m = 0.1 on average even in the presence of pilot position estimation errors. Both hybrid and coherent detection suffer from a loss in performance if the position of the pilot symbols is not known perfectly. However, for ω = 1 and ω = 5 the hybrid detector outperforms energy detection for N > 1500 and N > 500, respectively. Furthermore, for a given ω, the hybrid detector always outperforms the coherent detector. For large enough N, the slopes of the Pf –curves of hybrid and coherent detection do not seem to be affected 81  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks  Estimated pilot position ω=1  Energy Detection Coherent Detection Hybrid Detection  −1  Pm  10  Known pilot position −2  10  Estimated pilot position ω=5 500  1000  1500  2000  2500  N  Figure 3.8: Pm of hybrid detection, coherent detection, and energy detection vs. N. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, β = 0.05, SNR = −10 dB, and P¯f = 0.1. by the imperfect pilot position estimation. Similar observations as for the probability of false alarm in Fig. 3.7 can be made in Fig. 3.8 for the probability of missed detection. For large enough N, the Pm –curves for hybrid and coherent detection with imperfect pilot position estimation are simply shifted versions of the corresponding curves with known pilot positions. As can be observed, increasing ω improves the performance of hybrid and coherent detection with estimated pilot positions and decreases the gap to the case where the pilot symbol positions are known a priori. In Fig. 3.9, we investigate the impact of the pilot-to-received-samples ratio β on the probability of false alarm in the presence of imperfect pilot position estimation. Interestingly, in case of estimated pilot positions, Pf first increases with increasing 82  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks 0  10  Estimated pilot position ω=1 −1  10  Estimated pilot position ω=5  −2  Pf  10  −3  10  −4  10  Energy Detection  −5  10  Coherent Detection  Known pilot position  Hybrid Detection −6  10  0  1  2  3  4  5  6  7  8  Percentage of Pilot Symbols  Figure 3.9: Pf of hybrid detection, coherent detection, and energy detection vs. 100× β %. For known pilot position, and estimated pilot position with ω = 1, and 5. L = 3, N = 2000, SNR = −10 dB, and P¯m = 0.1. β before it decreases again. The reason for the initial increase is the fact that for very small β there are only few pilots and an accurate pilot position estimation is not possible, which impairs the performance of the hybrid detector. However, for ω = 1 and ω = 5 hybrid detection outperforms energy detection for β > 0.042 and β > 0.03, respectively. For pilot-to-received samples ratios of β > 0.05, hybrid detection achieves substantial performance gains compared to energy detection and coherent detection for both ω = 1 and ω = 5.  83  Chapter 3. Hybrid Coherent/Energy Detection for Cognitive Radio Networks  3.6  Conclusion  Coherent detection can be used for spectrum sensing when the pilot symbols transmitted by the PU are known to the CR. As shown in this chapter, coherent detection performs better than energy detection for moderate to high percentage of pilot symbols but it wastes the energy contained in the data symbols. However, when the percentage of known pilot symbols is low, energy detection performs better than coherent detection. To address this issue, in this chapter, we proposed a novel hybrid coherent/energy detection scheme for spectrum sensing and developed a corresponding low-complexity locally optimal decision metric. This hybrid metric is a linear combination of a coherent and an energy detection metric and combines the advantages of these individual metrics as it exploits both the pilot and the data symbols emitted by the PU. We have derived exact analytical expressions for the probabilities of false alarm and missed detection of the hybrid metric which can be efficiently evaluated numerically. Furthermore, assuming a Neyman-Pearson framework for optimization of the CR network, we have derived asymptotic closed-form expressions for the probabilities of false alarm and missed detection valid for low SNR and large sample size which provide significant insight into the impact of various system and channel parameters, such as the percentage of pilot symbols, the sample size, the SNR, and the number of diversity branches on performance. Simulation and analytical results revealed that the proposed hybrid detector achieves considerable performance gains compared to coherent and energy detection, respectively, even if the positions of the pilot symbols are not known a priori.  84  Chapter 4 Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT 4.1  Introduction  As pointed out in Chapter 1, for a fixed frame length in the CR network, longer sensing times improve the PU protection but reduce the time available for CR data transmission. This fundamental tradeoff has been investigated in several papers for single-antenna CR systems [66]–[72]8 . In this chapter, we investigate this issue for multi-input multi-output (MIMO) CR systems. Recently multi-input multi-output (MIMO) CR systems have gained much attention due to benefits such as diversity and multiplexing gains. In [13], spectrum sharing in MIMO CR systems was considered, where the CR transmit power and beamforming matrices were optimized for maximization of the throughput of the CR system under a constraint on the interference caused to the PU receiver. This approach requires knowledge of the channel matrix between the CR transmitter and the PU receiver. However, this knowledge may be difficult to obtain in practice, especially if the PU receiver does not transmit data. Furthermore, in [100], spectrum 8  For the discussion of these papers please refer to Section 1.3.  85  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT sensing methods exploiting multiple antennas at the CR receiver, such as generalized likelihood ratio and energy detection methods, were proposed. In this work, we propose a new protocol for simultaneous data transmission and spectrum sensing in MIMO CR systems. Employing an appropriate pre- and postprocessing at the CR transmitter and receiver, respectively, the MIMO channel is decomposed into independent spatial subchannels. Thus, some subchannels can be used for data transmission while other subchannels are used for spectrum sensing. The transmit power, sensing time, and sensing threshold in each spatial subchannel of the MIMO CR system are optimized for maximization of the throughput under constraints on the maximum transmit power and the probabilities of false alarm and detection. Thereby, we consider both single-band and multi-band MIMO CR systems and employ a modified energy detection scheme for spectrum sensing. The resulting optimization problems are non-convex and finding the global optimal solution entails a high complexity. Thus, we develop two efficient iterative algorithms that are based on the concept of alternating optimization [101]. These algorithms solve only convex problems in each iteration and we prove their convergence to fixed point analytically. Numerical results show that the developed algorithms closely approach the global optimal solution and achieve significant performance gains compared to baseline schemes that enforce equal power allocation or equal sensing times accross the spatial subchannels. The remainder of this chapter is organized as follows. In Section 4.2, the proposed MIMO processing, the sensing/transmission protocol, and the spectrum sensing method are introduced. In Sections 4.3 and 4.4, iterative algorithms for optimization of the transmit powers, sensing times, and sensing thresholds of single-band and multi-band MIMO CR systems are developed, respectively. Simulation results are  86  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT presented in Section 4.5, and some conclusions are drawn in Section 4.6.  4.2  Preliminaries  In this section, we present the considered system model and the adopted spectrum sensing method.  4.2.1  System Model and Sensing/Transmission Protocol  In this and the next section, we concentrate on the case where transmission and sensing are performed in a single frequency band. The extension to the multi-band case will be discussed in Section 4.4. We consider a CR system consisting of a transmitter with Mt antennas and a receiver with Mr antennas, where Mr ≥ Mt , and a primary transmitter with MP antennas. The MIMO channel matrix between the CR transmitter and the CR receiver is denoted by H ∈ CMr ×Mt , and the MIMO channel between the primary transmitter and the CR receiver is denoted by G ∈ CMr ×MP , cf. Fig. 4.1. The elements of H and G are modeled as independent, circularly symmetric, complex Gaussian random variables with zero mean and unit variance. Hence, rank(H) = min(Mt , Mr ) = Mt with probability 1. We assume the CR user has T symbol intervals (which comprise one frame) to sense whether or not a PU is present and to transmit its own data. For convenience and without loss of generality, the duration of the symbol intervals is normalized to 1. Clearly, there is a tradeoff between the throughput of the CR system and the protection of the PU [66]. Long sensing times improve the protection of the PU against interference from the CR user but short sensing times allow the CR user to transmit more data. Using a suitable pre- and post-processing at the CR transmitter and receiver, respectively, Mt spatial subchannels are available for data transmission 87  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT CR transmitter  Mt data streams  V  Mt transmit antennas  Mr receive antennas  CR receiver  Mt data streams  H  U  H  Mr-Mt sensing streams  G MP transmit antennas  VP PU transmitter  Figure 4.1: Block diagram of the system model. The CR system transmits Mt independent data streams and performs sensing at the receiver in all Mr subchannels. The PU transmits MP independent data streams. and Mr spatial subchannels are available for spectrum sensing at the CR receiver, cf. Fig. 4.1 and Section 4.2.2. In this work, we adopt the following sensing/transmission protocol: In the first frame (or in a portion of a frame), only spectrum sensing is performed. If it is decided that the PU is absent, in all following frames, the CR transmitter sends data over the first Mt spatial subchannels in the first T − Ni , 1 ≤ i ≤ Mt , symbol intervals of each frame, and the CR receiver performs spectrum sensing in all Mr subchannels during the remaining Ni , 1 ≤ i ≤ Mr , symbol intervals of the frame. The CR transmitter causes interference to the PU if the PU starts transmitting while the CR also transmits. Since the CR user and the PU are usually not synchronized with each other, this interference is unavoidable regardless of which parts of a frame are allocated to sensing and data transmission, respectively. The problem can be resolved if, prior to data transmission, the PU transmits a pilot sequence with a duration of one frame length of the CR system. For convenience and without loss of generality, 88  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT  T  N1  N2  N3  N4  Frame i  Frame i+1  Frame i+2  Figure 4.2: Proposed frame structure for CR system with Mt = 3 transmit and Mr = 4 receive antennas. The first Mt = 3 subchannels are used for both sensing and data transmission. The last Mr − Mt = 1 subchannel is only used for sensing. in the following, we assume that the first T −Ni , 1 ≤ i ≤ Mt , symbol intervals of each CR frame are used for data transmission and the remaining Ni , 1 ≤ i ≤ Mr , symbol intervals are used for spectrum sensing, cf. Fig. 4.2. We note that in general data reception and sensing are performed concurrently in different spatial subchannels at the CR receiver. If at the end of a frame it is decided that the PU is present, data transmission is stopped and the following frames are used for sensing only until the PU is sensed absent again.  4.2.2  CR Pre- and Post-processing  We assume that the MIMO channel matrix H is known at the CR transmitter and receiver. The CR user transmits Mt independent data streams using a pre-coding matrix V ∈ CMt ×Mt at the transmitter and a post-processing matrix U ∈ CMr ×Mr at the receiver. To maximize the throughput, the CR system employs unitary matrices V and U obtained from the singular value decomposition (SVD) of the channel 89  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT matrix, H = UΛVH , where Λ ∈ CMr ×Mt is a matrix with min(Mr , Mt ) = Mt nonzero main diagonal entries given by λ1 ≥ λ2 ≥ · · · ≥ λMt and zero off-diagonal entries [102], cf. Fig. 4.1. Hence, the received CR signal after post-processing is given by y(n) = Λx(n) + UH GVP s(n) + UH z(n),  where x(n)  [x1 (n), . . . xMt (n)]T ∈ CMt ×1 and s(n)  (4.1)  [s1 (n), . . . sMP (n)]T ∈ CMP ×1  denote the CR and the PU transmit signals, respectively, z(n) ∈ CMr ×1 is the AWGN vector at the CR receiver with distribution z ∼ N (0, σn2 IMr ×Mr ), and VP ∈ CMP ×MP UH z(n) ∈ CMr ×1  is the pre-coding matrix employed by the PU. Noise vector z˜(n)  follows the same distribution as z(n), i.e., z˜ ∼ N (0, σn2 IMr ×Mr ). The power allocated to the CR user’s ith transmitted data stream is denoted by pi total transmit power is given by Pt  E{|xi (n)|2 }, and the  tr E{Vx(n)(Vx(n))H } =  Mt i=1  pi [102]. The  received signal in the ith spatial subchannel of the CR system can be expressed as  yi (n) =     λi xi (n) +    MP j=1  MP j=1  ˜ ij sj (n) + z˜i (n), G  ˜ ij sj (n) + z˜i (n), G  1 ≤ i ≤ Mt Mt + 1 ≤ i ≤ Mr  ˜ ij denotes the element in row i and column j of matrix G ˜ where G CMr ×MP and z˜i (n)  (4.2)  UH GVP ∈  uH i z(n) with ui denoting the ith column of matrix U. We  assume that the PU transmit signals are temporally and spatially i.i.d. with zero mean and variance σs2  E{|si (n)|2 }, 1 ≤ i ≤ MP . If different PU data streams  ˜ ij . Since the CR user have different powers, a corresponding factor is absorbed into G transmits data in subchannel i only during the first T − Ni symbol intervals, we have xi (n) = 0, T − Ni + 1 ≤ n ≤ T , 1 ≤ i ≤ Mt . We note that we assume that the CR transmitter does not know the channel  90  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT matrix between the CR transmitter and the PU receiver as this would require the transmission of pilot symbols by the PU receiver. Therefore, approaches where the CR user avoids interference to the PU via beamforming are not applicable [13]. Instead, similar to other works [66, 72], we assume that the CR receiver knows the MP j=1  compound channel gains, |˜ gi |2  ˜ ij |2 , 1 ≤ i ≤ Mr , between the PU trans|G  mitter and the CR receiver. These gains can be estimated blindly while the PU is known (or estimated) to be present.  4.2.3  Spectrum Sensing  The CR receiver performs spectrum sensing while the subchannels are not used for data transmission. Thus, the two hypotheses to be considered for spectrum sensing are  H0 : yi (n) = z˜i (n),  T − Ni + 1 ≤ n ≤ T, 1 ≤ i ≤ Mr , (4.3)  MP  H1 : yi (n) =  ˜ ij sj (n) + z˜i (n), G  T − Ni + 1 ≤ n ≤ T, 1 ≤ i ≤ Mr . (4.4)  j=1  The CR receiver uses a modified version of energy detection, where the branches of the different subchannels are weighted by the compound interference subchannel gains |˜ gi |2  MP ˜ 2 j=1 |Gij | .  This weighting is akin to maximum ratio combining (MRC)  in data detection and yields a better performance than pure energy detection. The corresponding decision statistic is given by Mr  D=  i=1  T  |˜ gi |  2 n=T −Ni +1  |yi (n)|2 .  (4.5)  We note that even if we used conventional energy detection, knowledge of the compound interference channel gains |˜ gi |2  MP j=1  ˜ ij |2 would be required for the opti|G 91  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT mization problems formulated in Sections 4.3 and 4.4. Therefore, using the improved scheme in (4.5) does not entail any additional overhead. Assuming the number of terms in (4.5) is sufficiently large the central limit theorem (CLT) can be invoked to argue that the pdf of D approaches a Gaussian distribution [66]–[72]. Hence, for a ˜ we have given effective PU channel matrix G, ˜ ∼ N µ0 , σ 2 D|H0, G 0  ˜ ∼ N µ1 , σ 2 , and D|H1, G 1  (4.6)  where the means and variances of the distributions in (4.6) can be expressed as Mr  µ0 µ1  ˜ = σ2 E D|H0 , G n ˜ = σ2 E D|H1 , G n  i=1 Mr  i=1  |˜ gi |2 Ni  (4.7) Mr  |˜ gi |2 Ni + σs2  i=1  |˜ gi |4 Ni  (4.8)  Mr  σ02 σ12  ˜ = σ4 E (D − µ0 )2 |H0 , G n ˜ = σ4 E (D − µ1 ) |H1 , G n  i=1 Mr  2  |˜ gi |4 Ni Mr 4  i=1  (4.9)  |˜ gi | Ni +  2σn2 σs2 i=1  |˜ gi |6 Ni .  (4.10)  To arrive at (4.7)–(4.10) we assumed low SNR (i.e., σs2 /σn2 ≪ 1) and exploited the second order Taylor series approximation of D, similar to Chapter 2. The low SNR assumption is meaningful since CR systems like IEEE 802.22 have to be able to detect PU signal as low as -116 dBm [28]. Hence, in this work, we perform the system optimization and analysis for the worst case, i.e., low SNR. Based on the above model, the probabilities of false alarm, Pf , and detection, Pd ,  92  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT are given by  Pf (n, τ )  ˜ =Q Pr{D > τ |H0 , G}  Pd (n, τ )  ˜ =Q Pr{D > τ |H1 , G}  where τ is the sensing threshold and n  τ − µ0 σ0 τ − µ1 σ1  (4.11) ,  (4.12)  [N1 , . . . , NMr ] is the sensing time vector. In  (4.11) and (4.12), Pf and Pd are written as functions of n and τ to emphasize their dependencies on these parameters.  4.3  Problem Formulation and Optimization  In this section, we formulate a non-convex optimization problem for maximization of the throughput of the CR user under performance guarantees for the PU. Subsequently, we simplify the non-convex problem in several steps and propose an efficient algorithm for its solution.  4.3.1  Problem Formulation  For calculation of the throughput of the CR system, we neglect the achievable throughput of the CR system if the PU is present but not detected. Instead, we only consider the throughput of the CR system for the case when the PU is absent and also sensed as absent by the CR receiver. In general, we expect that the latter throughput will be much larger than the former. Thus, considering the frame structure in Fig. 4.2, the opportunistic throughput of the CR system is obtained as Mt  R (n, τ, p) = (1 − Pf (n, τ ))  i=1  T − Ni pi λ 2 log2 1 + 2 i T σn  ,  (4.13)  93  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT where p  [p1 , . . . , pMt ] is the CR transmit power vector and  T −Ni T  log2 1 +  pi λ2i 2 σn  is the throughput of the ith spatial subchannel. The objective is to maximize the opportunistic throughput of the CR system subject to constraints on the CR transmit power, the probability of false alarm, and the probability of detection. The parameters that can be optimized are the sensing time n, sensing threshold τ , and power allocation p. Accordingly, the maximal throughput is obtained by solving the following optimization problem  max n,τ,p  R (n, τ, p)  (P1)  s.t. C1 : Pd (n, τ ) ≥ P¯d C2 : Pf (n, τ ) ≤ P¯f Mt  C3 : i=1  pi ≤ P  C4 : Ni ∈ {0, 1, . . . , T }, i = 1, . . . , Mr , where P¯d , P¯f , and P are the minimum allowed probability of detection, the maximum probability of false alarm, and the maximum total transmit power of the CR system. We first note that P1 is different from the problems considered in existing work on single-antenna multi-band transmission, e.g. [66]–[72]. In this existing work, there is a separate Pf -constraint for each frequency band which only depends on the SNR and the sensing time of that band. In contrast, for the considered problem, there is only one constraint on the probability of false alarm and Pf is a function of all |˜ gi |2 , 1 ≤ i ≤ Mr , cf. (4.7), (4.9), (4.11) or equivalently of all sensing SNRs γi  σs2 |˜ gi |2 /σn2 ,  and all sensing times, Ni , 1 ≤ i ≤ Mr , which makes the optimization problem more difficult to solve. Furthermore, for the special case of single-antenna CR and PU systems, i.e., Mt = Mr = MP = 1, problems similar to P1 were considered in  94  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT [67]–[71] and shown to be non-convex. Thus, it is clear that P1 is also non-convex for the multi-antenna case and finding the globally optimal solution entails a high complexity. To overcome this problem, in the following, we will transform and relax P1 step-by-step, which will allow us to provide efficient numerical algorithms for finding a near-optimal solution. We first note that the numbers of symbol durations used for sensing, Ni , 1 ≤ i ≤ Mr , are integers, which further complicates P1. To overcome this problem, we relax P1 and assume in the following that the Ni , 1 ≤ i ≤ Mr , are real-valued variables. The optimal sensing times are then rounded to the closest integer value. Next, we introduce the following lemma to simplify problem P1 further. Lemma 4.1: For the optimal solution of P1, constraint C1 has to hold with equality. Proof: From (4.11) and (4.12), we observe that Pf (n, τ ) and Pd (n, τ ) are decreasing functions of τ . In addition, for given n and p the objective function (4.13), is an increasing function of τ . Consequently, for any τ0 in the feasible set of P1 that does not meet C1 with equality, we can always find an τ1 > τ0 in the feasible set that meets C1 with equality and increases the opportunistic throughput. Hence, we have Pd (n, τ ) = P¯d at the optimal point. By using Pd (n, τ ) = P¯d from Lemma 4.1 in (4.12) and substituting τ into (4.11), we have Pf (n, P¯d ) = Q  µ1 −µ0 σ0  +  σ1 −1 ¯ Q (Pd ) σ0  . Substituting (4.7)–(4.10) into this  equation makes the optimization problem untractable. However, from the definition of the variances in (4.9) and (4.10), we observe that σ12 = σ02 + 2σs2 σn2  Mr i=1  |˜ gi |6 Ni .  Thus, for low sensing SNR (σs2 ≪ 1), we have σ12 ∼ = = σ02 . Hence, Pf (n, P¯d ) ∼  95  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT Q  µ1 −µ0 σ0  + Q−1 (P¯d ) holds. Now, by defining  P˜f (n, P¯d )  Q  µ1 − µ0 + Q−1 (P¯d ) σ0    = Q  Mr  i=1    Ni γi2 + Q−1 (P¯d )  (4.14)  with sensing SNR γi = σs2 |˜ gi |2 /σn2 , and using Lemma 4.1, a new optimization problem P2 can be formulated as Mt  max R (n, p) = 1 − P˜f (n, P¯d ) n,p  i=1  pi λ 2 T − Ni log2 1 + 2 i T σn  (P2)  s.t. C1 : P˜f (n, P¯d ) ≤ P¯f Mt  C2 : i=1  pi ≤ P  C3 : 0 ≤ Ni ≤ T, i = 1, . . . , Mr . For low sensing SNR, problems P2 and P1 are equivalent except for the relaxation of Ni , 1 ≤ i ≤ Mr . In the following theorem, we prove that P˜f (n, P¯d ) is a convex, decreasing function of n, for P¯f ≤ 1/2. Theorem 4.1: P˜f (n, P¯d ) is a convex and decreasing function of n if P¯f ≤ 1/2. Proof: Please refer to Appendix C. Based on Theorem 4.1, we obtain the following lemma, which further simplifies problem P2. Lemma 4.2: The maximum of the objective function of problem P2 can be achieved by sensing on the spatial subchannels that are not used for data transmission during the entire frame, i.e., Ni = T , Mt + 1 ≤ i ≤ Mr . Proof: We first note that Ni , Mt + 1 ≤ i ≤ Mr , affects only constraint C1 and the objective function of problem P2 via P˜f (n, P¯d ). Let the optimal sensing time 96  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT vector of problem P2 be n∗  ∗ ], and define a related vector n′ [N0∗ , . . . , NM r  ∗ , T, . . . T ], which differs from n∗ only in the last Mr −Mt entries. Accord[N0∗ , . . . , NM t  ing to Theorem 4.1, we have P˜f (n0 , P¯d ) ≥ P˜f (n′ , P¯d ) since Ni∗ ≤ T , Mt + 1 ≤ i ≤ Mr . Thus, if n∗ is a feasible solution of problem P2, n′ will be feasible as well. Furthermore, replacing n∗ with n′ will also increase the objective function. Thus, the maximum of the objective function of problem P2 can be achieved by setting Ni = T , Mt + 1 ≤ i ≤ Mr . Based on Lemma 4.2, without loss of optimality we set Ni = T , Mt + 1 ≤ i ≤ Mr , in the following and optimize only Ni , 1 ≤ i ≤ Mt . Furthermore, assuming a solution with R(n, p) > 0 exists, we consider the new objective function Mt  log (R(n, p)) = log 1 − P˜f (n, P¯d ) + log  i=1  pi λ 2 T − Ni log2 1 + 2 i T σn  , (4.15)  which leads to the transformed optimization problem  max log(R (n, p)) n,p  (P3)  s.t. C1 : P˜f (n, P¯d ) ≤ P¯f Mt  C2 : i=1  pi ≤ P  C3 : 0 ≤ Ni ≤ T, i = 1, . . . , Mt . Based on Lemma 4.2 it is clear that as long as a solution (n2 , p2 ) with R(n2 , p2 ) > 0 exists for problem P2, the solution to problem P3, (n3 , p3 ), will achieve the same objective value, i.e., R(n3 , p3 ) = R(n2 , p2 ). Achieving R(n2 , p2 ) > 0 is always possible as long as the frame length T is sufficiently long. Unfortunately, problem P3 is still not convex in (n, p). Nevertheless, in the following lemma, we establish two useful facts regarding the convexity of problem 97  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT P3. Lemma 4.3: For a given p, problem P3 is convex in n. Furthermore, for given n, problem P3 is convex in p. Proof: Since all constraints of P3 are convex with respect to n and p, we can concentrate on the concavity of the objective function log (R(n, p)). Using Theorem 4.1, we observe that the first term on the right hand side of (4.15), which only depends on n is concave since it is the logarithm of a concave function. For given p, the second term is also concave in n, since it is the logarithm of an affine function in n. Thus, for given p, we conclude that log(R(n, p)) is a concave function in n. On the other hand, for given n, log(R(n, p)) is a concave function of p, since log2 (1 + pi λ2i /σn2 ), i = 1, . . . , Mt , is concave in p and the logarithm of a sum of concave functions is also concave. Based on the Lagrangian of problem P3 and applying the Karush-Kuhn-Tucker (KKT), for a given n the optimal p is given by  pi =  σ2 T − Ni ν − n2 T λi  where the water-level ν is chosen such that  Mt i=1  +  ,  (4.16)  pi = P holds, and (·)+ = max{·, 0}.  The global optimal solution to problem P3 can be found by computing the optimal p (4.16) for all (T + 1)Mr possible choices for n and picking the p and n that yield the maximum rate R(n, p). Clearly, the complexity of finding the global optimal solution based on this exhaustive search approach is very high. Thus, in the next section, we will exploit Lemma 4.3 to derive a suboptimal but efficient algorithm to solve problem P3.  98  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT Algorithm 1 : Algorithm for solving problem P3. 1: Initialize j ← 0, p(j) , and n(j) . 2: Repeat 3: j ← j + 1. 4: Obtain n(j) by solving P3 for p = p(j−1) using the interior point method. 5: Calculate p(j) for n = n(j) using (4.16). 6: Until p(j) − p(j−1) ≤ ρ and n(j) − n(j−1) ≤ ρ, where ρ is a small constant number.  4.3.2  Iterative Algorithm  In this section, we propose an iterative algorithm for solving P3, which is based on the alternating optimization (AO) concept [101]. In AO, the optimization problem is solved iteratively. For the problem at hand, in each iteration, first the optimal n is found for the p obtained in the previous iteration, and subsequently, the optimal p is found for the previously computed n. Since, according to Lemma 4.3, P3 is convex with respect to n and p, respectively, the solutions to the two optimization problems that have to be solved in each iteration are convex and can be found with low complexity. In fact, for a given n the optimal p can be found from (4.16), and for a given p, the optimal n can be obtained using efficient numerical techniques such as the interior point method [103]. The proposed numerical procedure for solving P3 is summarized in Algorithm 1. In Step 1, the power and sensing times are initialized with feasible vectors p(0) and n(0) . In Step 4, the interior point method is used to solve P3 for a given power allocation and in Step 5 the closed-from waterfilling solution in (4.16) is used. The difference between successive optimal points is checked in Step 6 and if it is less than a given constant, the algorithm stops. In the following lemma, we formally prove that the proposed algorithm converges to a fixed point. Lemma 4.4: For any choice of system parameters and any feasible initial point  99  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT of P3, we have: a) R(n(j) , p(j) ) is upper bounded, i.e., R(n(j) , p(j) ) ≤ log  Mt i=1  log 1 +  P λ2i 2 σn  .  b) R(n(j) , p(j) ) is non-decreasing in j, i.e., R(n(j) , p(j) ) ≤ R(n(j+1) , p(j) ) ≤ R(n(j+1) , p(j+1) ). c) Algorithm 1 converges to some R∗ = limj→∞ R(n(j) , p(j) ). Proof: Part a) can be easily shown by omitting the first term in the right hand side of (4.15), which is negative, and using Ni = 0 and pi = P , 1 ≤ i ≤ Mt , to maximize the second term. We note that this upper bound is not achievable but can be used to show the convergence of the algorithm to a fixed point. Now, we prove part b) by contradiction. We first assume R(n(j) , p(j) ) > R(n(j+1) , p(j) ). This means that in Step 4 of Algorithm 1, the objective function decreases. However, this is not possible since the subproblem is convex and n(j+1) is a feasible point and cannot have a smaller objective function than n(j) . Thus, we have R(n(j) , p(j) ) ≤ R(n(j+1) , p(j) ). Using the same argument, it can also be shown that R(n(j+1) , p(j) ) ≤ R(n(j+1) , p(j+1) ). Consequently, the objective function is non-decreasing in j, i.e., R(n(j) , p(j) ) ≤ R(n(j+1) , p(j) ) ≤ R(n(j+1) , p(j+1) ). The third part, follows directly from part a) and b), since any upper bounded, non-decreasing function converges to a fixed value. Consequently, the convergence of Algorithm 1 is guaranteed. If we assume that the loop in Step 2 of Algorithm 1 needs to be repeated ξ times for convergence, then 2ξ subproblems need to be solved. Extensive simulations (see Section 4.6 for examples) suggest that ξ ≤ 5 is usually sufficient to closely approach the global optimal solution. Thus, because of the polynomial complexity of the interior point method [103], the proposed algorithm has a much lower complexity than the exhaustive search required to find global optimal solution.  100  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT  4.4  Extensions to Multi-Band CR Systems  In this section, we investigate the extension of the proposed MIMO CR system design to a multi-band scenario. We assume there are Q frequency bands and, in each time slot, PUs may randomly choose one or more of these bands for transmission. Throughout this section we add a subscript k to variables that have been previously defined in Section 4.2 and 4.3 to indicated the variable in the kth frequency band, e.g. Ni,k is the sensing time of ith spatial subchannel in the kth frequency band.  4.4.1  Problem Formulation  As customary in the literature, we assume that the CR is capable of sensing the Q bands concurrently, with a different threshold τk for each band, k = 1, . . . , Q, e.g. [34]. Thus, the decision statistic for band k, Dk , is given by Mr  Dk =  i=1  T  |˜ gi,k |2  n=T −Ni,k +1  |yi,k (n)|2 ,  k = 1, . . . , Q.  (4.17)  The corresponding probabilities of false alarm and missed detection are given by  Pf,k (nk , τk )  Q  Pd,k (nk , τk )  Q  τk − µ0,k σ0,k τk − µ1,k σ1,k  ,  k = 1, . . . , Q,  (4.18)  ,  k = 1, . . . , Q.  (4.19)  Among these Q bands, the K ≤ Q bands that were sensed as being not occupied by a PU in the previous frame are used for joint CR data transmission and sensing in the current frame. In the remaining bands only sensing is performed and these bands are not considered for optimization in the current frame. The opportunistic  101  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT throughput in the kth band, k = 1, . . . , K, is given by Mt  Rk (nk , τk , pk ) = (1 − Pf,k (nk , τk ))  i=1  pi,k λ2i,k T − Ni,k log2 1 + T σn2  .  (4.20)  Since the CR user can use all K bands for transmission, our objective is to maximize the sum of the opportunistic throughputs of all bands by optimizing the power, sensing time, and sensing thresholds in all frequency bands. Note that in the multiband scenario, the probabilities of false alarm and missed detection requirements in different bands may be different but there is a joint transmit power constraint for all antennas and frequency bands. Thus, the resulting optimization problem is formulated as K  max  nk ,τk ,pk ,k=1,...,K  R=  R (nk , τk , pk )  (P4)  k=1  s.t. C1 : Pd,k (nk , τk ) ≥ P¯d,k ,  k = 1, . . . , K  C2 : Pf,k (nk , τk ) ≤ P¯f,k ,  k = 1, . . . , K  C3 :  1 K  K  Mt  k=1 i=1  pi,k ≤ P,  C4 : Ni,k ∈ {0, 1, . . . , T },  i = 1, . . . , Mr , k = 1, . . . , K.  For simplification of problem P4, we first extend Lemma 4.1 to the multi-band case. Pf,k and Pd,k are decreasing functions of τk , and R is an increasing function of τk . Hence, C1 in P4 is met with equality at the optimal point for k = 1, . . . , K. Consequently, the optimal τk can be derived from C1 and substituted in C2. By using a low sensing SNR approximation, similar to Subsection 4.3.1, the approximate probability of false alarm in the kth frequency band can be written as P˜f,k (nk , P¯d,k ) = Q  Mr l=1  2 + Q−1 (P¯d,k ) . Consequently, the overall opportunistic throughNl,k γl,k  102  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT put can be expressed as K  R=  k=1      1 − Q     Mr  l=1  2 + Q−1 (P¯d,k ) Nl,k γl,k  Mt  i=1  pi,k λ2i,k T − Ni,k log2 1 + T σn2  .  (4.21)  According to Theorem 4.1, P˜f,k (nk , P¯d,k ) is a convex, decreasing function of nk if P¯f,k ≤ 1/2. Also, by extending Lemma 4.2 to the multi-band case, we observe that without loss of optimality we can set Ni,k = T , for i = Mt + 1, . . . , Mr , k = 1, . . . , K. Thus, we only have to optimize Ni,k , for i = 1, . . . , Mt , k = 1, . . . , K. Based on the above considerations, problem P4 can be reformulated as  max  nk ,pk ,k=1,...,K  R  (P5)  s.t. C1 : P˜f,k (nk , P¯d,k ) ≤ P¯f,k , C2 :  1 K  K  k = 1, . . . , K  Mt  k=1 i=1  pi,k ≤ P  C3 : 0 ≤ Ni,k ≤ T,  i = 1, . . . , Mt , k = 1, . . . , K.  Problem P5 is a non-convex optimization problem. However, for a given nk , k = 1, . . . , K, problem P5 is convex and the solution can be found from the KKT conditions as pi,k =  σ2 T − Ni,k ν − 2n (1 − P˜f,k (nk , P¯d,k )) T λi,k  where variable ν is adjusted to guarantee  K k=1  Mt i=1  +  ,  (4.22)  pi,k = KP . Thus, the global  optimal solution can be found by computing pk , k = 1, . . . , K, based on (4.22) for all (T + 1)KMr possible choices of nk , k = 1, . . . , K. In the next section, we will propose a suboptimal iterative algorithm for solving problem P5 with a much lower complexity.  103  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT Algorithm 2 : Algorithm for solving problem P5. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:  (j)  (j)  Initialize j ← 0, pk = 0, and nk = T · 1, k = 1, · · · , K. Repeat j ← j + 1. For k = 1 to K (j−1) If pk = 0 then (j) nk ← T · 1. Go to Step 4. End if (j) (j−1) Obtain nk by solving P3 for given p = pk using the interior point method. End for (j) (j) Calculate pk , k = 1, . . . , K, for given nk , k = 1, . . . , K, using (4.22). (j) (j−1) (j) (j−1) Until pk − pk ≤ ρ and nk − nk ≤ ρ, k = 1, · · · , K, where ρ is a small constant number.  4.4.2  Iterative Algorithm  We first note that, unlike for the single-band case discussed in Section 4.3.1, applying a logarithm to (4.21) does not make the objective function of problem P5 concave in nk , k = 1, . . . , K, for a given power allocation, since the additional summation over the frequency bands prevents the logarithm from separating the multiplication of (1− P˜f,k (nk , P¯d,k )) and  Mt T −Ni,k i=1 T  log2 1 +  pi,k λ2i,k 2 σn  . Nevertheless, for a given nk ,  k = 1, . . . , K, the optimal transmit powers pk , k = 1, . . . , K, can be obtained from (4.22). Furthermore, for a given power allocation, pk , k = 1, . . . , K, problem P5 can be decoupled into K independent subproblems – one for each frequency band. These subproblems are identical in structure to P2 and can be transformed into convex subproblems by applying the log(·)–function provided that the corresponding achievable rate is positive. Consequently, we obtain K subproblems that are identical to P3. By solving these subproblems we obtain the optimal sensing times in each of the K considered frequency bands for the given power allocation. Based on the above considerations, we propose Algorithm 2 for solving problem 104  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT P5. This algorithm is also based on the concept of alternating optimization. In Step 1 of Algorithm 2, we choose a random feasible starting point which has a positive (0)  opportunistic throughput in each band. Since, pk = 0, k = 1, · · · , K, in the first iteration, all rates are positive and the optimal sensing times for each band are computed in Step 9. In Step 11, for the given sensing times, the optimal powers are found using waterfilling (4.22). Unlike the sensing time optimization, the power allocation has to be performed jointly for all frequency bands. In fact, considering (4.22) it is possible that zero power is allocated to frequency bands with weak CR channels. Hence, for iteration j > 1, Step 9 is skipped for frequency band k if (j−1)  pk  = 0 and the entire frame of this band is used for sensing.  The convergence of Algorithm 2 to a fixed point can be proved using a similar approach as in Lemma 4.4. Assuming the algorithm converges in ξ iterations, the number of convex subproblems that need to be solved is (K + 1)ξ. Because of the polynomial complexity of interior point method, the overall complexity of Algorithm 2 is much lower than that of the exhaustive search required to find the global optimal solution.  4.5  Results and Discussions  In this section, numerical results are presented for the proposed single-band and multiband MIMO CR system with sensing time and power optimization. We assume a frame length of 50 ms for the CR system and a sampling rate of fs = 100 kHz, which corresponds to T = 5000 symbol intervals per frame. We focus on the practical case of low sensing SNRs. The target probabilities of detection and false alarm for the considered single-band system are given by P¯d = 0.9 and P¯f = 0.1, respectively, and the noise power is σn2 = 1. We compare the proposed Algorithms 1 and 2 105  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT 5  Opportunistic Throughput  4.5  4  3.5  3  Global Optimal (Exhaustive Search) 2×2 MIMO, γ = [−12, −15] dB, λ = [0.5, 1] 2.5  3×3 MIMO, γ = [−12, −15, −15] dB, λ = [0.5, 1, 0.6] 3×3 MIMO, γ = [−12, −15, −10] dB, λ = [0.5, 1, 0.8] 4×4 MIMO, γ = [−12, −15, −10, −12] dB, λ = [0.5, 1, 0.8, 0.6]  2  0  1  2  3  4  5  6  7  8  9  10  Number of Convex/Waterfilling Problems Solved  Figure 4.3: Opportunistic throughput in single-band case vs. number of convex/waterfilling problems solved. γ [γ1 . . . γMr ] and λ [λ1 . . . λMt ]. which perform joint power and sensing time optimization with two simpler baseline algorithms. The first baseline algorithm forces all sensing times to be equal and then optimizes both power and sensing time (i.e., the case that the CR receiver receives data on some subchannels while performing sensing on other subchannels is excluded). The second baseline algorithm allocates equal powers to all subchannels and optimizes the sensing times. Unless specified otherwise, we initialize Algorithms 1 and 2 with equal powers and sensing times of T /2 in all subchannels, and the maximum number of iterations is set to ξ = 10. In Figs. 4.3 and 4.4, we investigate the convergence of Algorithms 1 and 2, respectively. In both figures, the horizontal axis shows the number of convex/waterfilling problems solved, which corresponds to Steps 4 and 5 of Algorithm 1 and Steps 9 and 11 in Algorithm 2. For Fig. 4.3, P¯d = 0.9, P¯f = 0.1, and P = 10 dB are valid.  106  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT 10  Opportunistic Throughput  9  8  7  6  Global Optimal (Exhaustive Search)  5  K = 2 Bands K = 3 Bands K = 4 Bands 4  0  5  10  15  20  25  30  Number of Convex/Waterfilling Problems Solved  Figure 4.4: Opportunistic throughput in multi-band case vs. number of convex/waterfilling problems solved. In Fig. 4.4, for K = 2, 3, and 4, we adopt the γ1,k , γ2,k , λ1,k , λ2,k , P¯d,k , and P¯f,k given in the first K columns of Table 4.3 and P = 10 dB. Algorithm 2 was initialized with equal powers and sensing times of T /2 in all subchannels except for band 4, where higher initial sensing times are needed for feasibility. The throughput for the initial values is given by the zeroth iteration in Figs. 4.3 and 4.4. The global optimal throughput obtained through an exhaustive search is also shown for comparison (horizontal lines). As expected from Lemma 4.4, the throughput achieved with both algorithms increases in each step and converges to a fixed point. In all considered cases, convergence is achieved in 10 steps or less, except for the the case of K = 3 in Fig. 4.4, where more iterations are needed for full convergence but a close-to-optimal performance is achieve after 4 iterations. We also note that in all cases, the global optimal performance is approached after a sufficiently large number of iterations.  107  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT Table 4.1: Sensing time, power allocation, and opportunistic throughput for 2 × 2 CR MIMO system. γ1 = −12 dB, γ2 = −15 dB, λ1 = 2, λ2 = 0.5, and P = 10 dB. Algorithm  N1  N2  p1  p2  R(n, p)  Algorithm 1  394  5000  10  0  4.44  Equal Power Allocation  1650  0  5  5  3.70  Equal Sensing Time  1319 1319 6.87 3.13  3.75  Table 4.2: Sensing time, power allocation, and opportunistic throughput for 2 × 3 CR MIMO system. γ1 = −10 dB, γ2 = −12 dB, γ3 = −15 dB, λ1 = 2, λ2 = 0.5, and P = 10 dB. Algorithm  N1  N2  Algorithm 1  0  1912 5000 8.56 1.44  5.35  Equal Power Allocation  0  1250 5000  5.11  Equal Sensing Time  609  609  N3  609  p1  5  p2  5  6.87 3.13  R(n, p)  4.77  In Tables 4.1 and 4.2, we show the sensing times, power allocations, and throughputs obtained with Algorithm 1 and the two baselines schemes for 2 × 2 and 2 × 3 MIMO systems, respectively. For both tables the CR channel had singular values λ1 = 2 and λ2 = 0.5, and the maximum transmit power was P = 10 dB. The sensing SNRs were γ1 = −12 dB, γ2 = −15 dB and γ1 = −10 dB, γ2 = −12 dB, γ3 = −15 dB for Tables 4.1 and 4.2, respectively. For the scenario considered in Table 4.1, the first spatial subchannel, has both a higher sensing SNR and a larger effective channel gain (given by the singular value) than the second subchannel. For Algorithm 1, 108  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT the best strategy is to sense for the entire frame on the second subchannel and to allocate the entire power for transmission to the first subchannel. In contrast, if the power is equally distributed among both subchannels, sensing is performed only in the first subchannel and the second subchannel is used only for data transmission. Since for equal power allocation the properties of the stronger subchannel cannot be fully exploited for data transmission, the baseline scheme seeks to exploit at least the higher sensing SNR of the first subchannel. If the sensing time is constrained to be equal for both subchannels, more power is allocated to the first subchannel but the throughput is considerably lower than that achieved with Algorithm 1. As expected from Lemma 4.2, Table 4.2 shows that the third subchannel is used exclusively for sensing if Algorithm 1 is used. Furthermore, the baseline scheme with equal sensing times suffers a large loss in throughput compared to Algorithm 1 since it only uses a fraction of the third subchannel for sensing, which has a negative effect on the amount of time that subchannels 1 and 2 can be used for data transmission. In Fig. 4.5, we investigate the impact of singular value ratio λ1 /λ2 on the throughput of 2 × 2 MIMO CR systems with λ2 = 0.5 and λ2 = 1, respectively. For both systems, the sensing SNRs are γ1 = −12 dB and γ2 = −15 dB, and the maximum transmit power is P = 10 dB. For low λ1 /λ2 ratios, the throughput achieved with Algorithm 1 becomes constant since if the transmission quality of subchannel 1 is worse than that of subchannel 2, subchannel 1 is used only for sensing and subchannel 2 is used only for data transmission since subchannel 2 also has a worse sensing SNR than subchannel 1. On the other hand, for high λ1 /λ2 , the throughput achieved with Algorithm 1 improves as λ1 /λ2 increases since now subchannel 1 has both better sensing SNR and better transmission quality than subchannel 2, and consequently, it is exploited for both sensing and data transmission. For the entire range of con-  109  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT 5  Global Optimal (Exhaustive Search) Algorithm 1 4.5  Sensing Time Opt. (Equal Power Allocation) Sensing Time and Power Opt. (Equal Sensing Times)  Opportunistic Throughput  4  3.5  λ2 = 1 3  2.5  λ2 = 0.5 2  1.5  1 0.5  0.6  0.7  0.8  0.9  1  1.1  1.2  1.3  1.4  λ1 /λ2  Figure 4.5: Opportunistic throughput of a single-band MIMO CR system vs. λ1 /λ2 . γ1 = −12 dB, γ2 = −15 dB, and P = 10 dB. sidered λ1 /λ2 ratios, Algorithm 1 achieves practically the same throughput as an exhaustive search, which yields the global optimum. We also note that the difference between Algorithm 1 and the baseline scheme with equal power allocation decreases as the λ1 /λ2 ratio increases. This effect is expected as the benefits of optimal power allocation decrease with improving transmission channel quality. In Table 4.3, we show the sensing times, power allocations, and throughput obtained with Algorithm 2 for a 2 × 2 multi-band MIMO CR system with K = 4 bands and P = 10 dB. The sensing SNRs, transmission channel qualities, and probability of false alarm and missed detection requirements are different for each band and are also provided in the table. Since the objective is to maximize the sum throughput, the amount of data transmitted in each band strongly depends its channel conditions and requirements. For example, the fourth band is not used at all for data transmis-  110  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT Table 4.3: Sensing time, power allocation, and opportunistic throughput for 2 × 2 multi-band MIMO CR system with K = 4 bands. k  1  2  3  4  P¯d,k  0.95  0.9  0.93  0.95  P¯f,k  0.05  0.1  0.07  0.05  γ1,k , γ2,k (dB)  -12, -12  -15, -10  -10, -18  -15, -14  λ1,k , λ2,k  0.6, 0.8  0.8, 1  0.5, 0.2  0.2, 0.5  6.31, 0  0, 0  p1,k , p2,k  2.59, 10.91 11.05, 9.13  N1,k , N2,k  2849, 0  0, 984  Rk  3.26  5.52  757, 5000 5000, 5000 1.08  0  sion and zero power is allocated to it because of its weak transmission channel. In contrast, a large portion of the available power is allocated to the second band which enjoys a good transmission channel quality and comparatively high sensing SNRs. In Fig. 4.6, we investigate the opportunistic throughput of two multi-band MIMO CR systems with K = 2 and K = 4 bands vs. transmit power P . The sensing SNRs, transmission channel qualities, and probability of false alarm and missed detection requirements are adopted from the first K columns of Table 4.3. In this figure, for the baseline algorithm with equal sensing times, we keep Ni,k , i = 1, · · · , Mr equal for each k, but different bands may have different sensing times. Restricting the sensing times to be equal in all bands would have a very negative effect on performance as the fourth band has very weak sensing SNRs and a stringent probability of false alarm requirement. From Fig. 4.6, we can observe that, as expected, the performance of 111  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT 25  Algorithm 2 Sensing Time Opt. (Equal Power Allocation) Sensing Time and Power Opt. (Equal Sensing Times) 20  Opportunistic Throughput  K=4  15  10  K=2  5  0  0  2  4  6  8  10  12  14  16  18  20  P in dB  Figure 4.6: Opportunistic throughput of a multi-band MIMO CR system vs. P for K = 2 and K = 4 bands. all schemes improves with increasing transmit power. Algorithm 2 yields significant performance gains compared to the baseline algorithms, especially for K = 4.  4.6  Conclusions  In this chapter, we have investigated the sensing-throughput tradeoff of MIMO CR systems. It was shown that by employing a suitable sensing and transmission protocol different sensing times can be used for different subchannels which greatly improves the system performance. Using this sensing/transmission protocol we formulated non-convex problems for optimization of the transmit powers, sensing times, and sensing thresholds in single-band and multi-band systems. For both cases, efficient iterative algorithms were developed for solving these problems. These algorithms  112  Chapter 4. Sensing Time and Power Optimization in MIMO Cognitive Radio Networks with CSIT are based on the concept of alternating optimization and entail a low complexity as they have to solve only convex problems in each iteration. The convergence of these algorithms to a fixed point was proved analytically. Simulation results showed that the proposed algorithms closely approach the global optimal solution and achieve significant performance gains compared to baseline schemes with equal power or equal transmission time allocation.  113  Chapter 5 Summary of Thesis and Topics for Future Research In this chapter, we first summarize the contributions of the thesis. Then, we propose possible future research directions.  5.1  Summary of Contributions  This thesis focused on CR systems employing interweave channel access scheme. Two spectrum sensing techniques were proposed and efficient algorithms for optimization of the opportunistic throughput in MIMO CR systems were developed. In the following, we summarize the main contributions of the thesis. • In Chapter 2, we have introduced adaptive Lp –norm spectrum sensing for cognitive radio networks. The Lp –norm detector has tunable parameters that can adapt to the underlying circularly symmetric noise with finite moments, without requiring knowledge of the full noise distribution. The probabilities of false alarm and missed detection were analyzed for Rayleigh fading in the low sensing SNR regime. We studied the asymptotic behavior of the probabilities of false alarm and missed detection for large sample size and we observed that while the probability of false alarm decreases exponentially with the sample size for a given probability of missed detection, the probability of missed detection de114  Chapter 5. Summary of Thesis and Topics for Future Research creases only polynomially with the sample size for a given probability of false alarm. Based on the deflection coefficient of the detector we proposed an adaptive algorithm for online estimation of the Lp –norm parameters. Simulation results revealed the significant gain that can be achieved with the proposed detector over the conventional energy detector for various practical noise environments. • In Chapter 3, we developed a locally optimal hybrid coherent/energy detection scheme for CR networks. The decision metric employed by the hybrid scheme is a linear combination of the energy detection metric and the coherent correlation metric. We analyzed the probabilities of false alarm and missed detection for Rayleigh fading for low sensing SNR. Using the Neyman-Pearson optimization framework and the asymptotic analysis for large samples size, we gained significant insight about the effects of different system parameters such as the number of diversity branches, percentage of pilots, and SNRs on the performance. Simulation results confirmed the analytical derivations and revealed that significant performance gains can be achieved in comparison with coherent detection and energy detection, respectively, even if the positions of the pilot symbols are not known a priori. • In Chapter 4, we proposed efficient optimization algorithms for single-band and multi-band MIMO CR networks. By applying suitable pre- and post-processing at transmitter and receiver sides, respectively, we were able to simultaneously perform sensing and transmission for different MIMO subchannels. Thereby, the optimization problems for maximizing the throughput subject to transmit power and probability of false alarm and detection constraints were formulated. These non-convex problems were relaxed and simplified step-by-step and effi115  Chapter 5. Summary of Thesis and Topics for Future Research cient algorithms for optimizing the sensing times, sensing thresholds, and transmit power were developed which only solve convex subproblems in each iteration and entail low complexity. The convergence of the proposed algorithms to a fixed point was proved analytically and simulation results revealed that the performance approaches to the global optimal solution and that significant gain can be achieved with these algorithms compared to baseline algorithms with equal sensing times or equal power.  5.2  Suggestions for Future Work  In the following, we propose several interesting future research directions that are based on the work in this thesis. 1. The performance loss suffered by the hybrid detector in the presence of frequency offset is less than the performance loss of coherent detector due to the additional energy detection metric involved. However, better performance is still desirable. For frequency offset, we have a phase term that increases with the number of samples and prevents the coherent terms in the decision metric to add up constructively. This frequency offset can be caused by a mismatch between transmitter and receiver oscillators and also the Doppler effect. One approach to eliminate the frequency offset is to evaluate the correlation metric over smaller windows, where the phase changes caused by the frequency offset are negligible, and then combine the coherent terms. Hence, providing and analyzing a metric robust to frequency offset, which can perform better than energy detection and coherent detection in pilot-based spectrum sensing, is an interesting topic for future work.  116  Chapter 5. Summary of Thesis and Topics for Future Research 2. While in Chapter 4, modified energy detection is the metric of choice for throughput maximization in MIMO CR networks, it is interesting to consider the optimization of CR systems for other types of detectors. 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As an example, we evaluate the first term (F.T.) π/2  1 F.T. = π √  L  1 =√ 2π  wl − αl σh2 l e αl σh2l  wl − αl σh2 l e αl σh2l  L  l=1  −  e  γ2 − γ2 2 sin2 φ αl σ hl  ξ  | sin φ|e  ∞  sin2 φ 2α2 σ 4 l hl  0  wl − αl σh2 l e αl σh2l  dγ dφ  0 π/2  ξ  l=1  ∞  ξ  l=1  0  2 = π  L  2  e−γ dγ dφ | sin φ| √ 2αl σ 2 hl  π/2  0  | sin φ|e  sin2 φ 2α2 σ 4 l hl  1 − erf  | sin φ| √ 2αl σh2l  dφ.  (A.2)  Performing similar manipulations for the remaining two terms in (A.1) leads to (2.26).  128  Appendix B Asymptomatic Analysis of Pm √ We first note that both ξ = (τ − µ0 )/¯ σ0 and α = α/¯ ¯ σ0 are proportional to N √ since σ¯0 is proportional to 1/ N, cf. (2.21). Hence, for N → ∞, we have ξ > 0, sin2 φ/(ασh2 ) → 0 and Jiid(φ) → −∞. Exploiting furthermore the relations Γ(·, 0) = Γ(·) and Γ(·, ∞) = 0 [91], (2.29) simplifies to the following equation −  ξ 2  e ασh Pm ∼ T1 (L, ξ) +Iiid, = 2π(L − 1)!αL σh2L  (B.1)  T2 (L,ξ)  where, T1 (L, ξ) =  L−1 l=0  L−1 l  π/2  √ ξ l ( 2| sin φ|)L−l 1 + (−1)L−l Γ  0  L−l 2  dφ. Next,  we show that the ratio of the first term and the second term in right hand side of (B.1) goes to zero for N → ∞, since −  ξ 2  T2 (L, ξ) e ασh T1 (L, ξ) = Iiid 2παL σh2L γ (L, ξ/ασh2 ) T1 (L, ξ) = ξl 2παL σh2L Γ(L) ∞ l=L αl σ2l l! h  ≤ = ≃  T1 (L, ξ) 2π Γ(L) ξL L! AL−2 ξ L−2 + AL−3 ξ L−3 + · · · + A0 2π Γ(L) ξL L!  AL−2 ξ L−2 ξL 2π Γ(L) L!  → 0,  (B.2)  129  Appendix B. Asymptomatic Analysis of Pm where coefficients Al  L−1 l  π/2 √ ( 2| sin φ|)L−l 0  1 + (−1)L−l Γ  L−l 2  dφ, 0 ≤ l ≤  L − 2, are independent of ξ and we used the series representation of the lower incomx plete Gamma function γ(s, x) = Γ(s)e−x ∞ k=s k! [91]. The last line in (B.2) is due √ to the fact that ξ is proportional to N. The above considerations show that for k  N → ∞, the behavior of Pm is dominated by Iiid and (2.32) holds.  130  Appendix C Proof of Theorem 4.1 The partial derivative of P˜f (n, P¯d ) with respect to Nj is obtained as     2 ˜ −γj −1  −1 ¯ ∂ Pf exp Q (Pd ) + = √  ∂Nj 2 Mr 2  2 2π i=1 Ni γi  Mr  i=1  2    2 . Ni γi    (C.1)  From (C.1), we observe that P˜f (n, P¯d ) is indeed a decreasing function of Ni , ∀i. The second order partial derivative of P˜f (n, P¯d ) is given by γj2 γk2 ∂ 2 P˜f × = √ r 2 ∂Nj ∂Nk 4 2π M N γ i i i=1 Mr  Q−1 (P¯d ) +  Mr  Ni γi2  Ni γi2 + i=1  − 21  i=1  Q−1 (P˜f )      1 exp − Q−1 (P¯d ) +   2  Mr  i=1  2    2 . Ni γi   (C.2)  Hence, the Hessian of P˜f (n, P¯d ) with respect to n is obtained as Q (P˜f ) + 2 ˜ √ ∇ Pf = 4 2π −1  ˜ where γ  Mr i=1 Mr i=1  Ni γi2  Ni γi2  − 12  exp −  1 Q−1 (P˜f ) 2  2  ˜T , γ˜ γ  (C.3)  2 T ] . We observe that if Q−1 (P˜f ) ≥ 0 or equivalently P¯f ≤ 1/2, [γ12 , . . . , γM r  the Hessian is positive semi-definite, and thus, P˜f (n, P¯d ) is convex with respect to n.  131  

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