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Robust game-theoretic
algorithms for distributed
resource allocation in
wireless communications
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• A Doctoral Thesis. Submitted in partial ful llment of the requirements for the award of Doctor of Philosophy of Loughborough University.
Metadata Record: https://dspace.lboro.ac.uk/2134/8619 Publisher: c Amod Jai Ganesh Anandkumar
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Robust Game-Theoretic Algorithms
for Distributed Resource Allocation
in Wireless Communications
by
Amod Jai Ganesh Anandkumar
A Doctoral Thesis
Submitted in partial fulfilment of the requirements for the
award of the degree of Doctor of Philosophy by
Loughborough University
cCERTIFICATE OF ORIGINALITY
This is to certify that I am responsible for the work submitted in this thesis, that the original work is my own except as specified in acknowledgements or in footnotes, and that neither the thesis nor the original work contained therein has been submitted to this or any other institution for a degree.
... (Signed) Amod Jai Ganesh Anandkumar (candidate)
The electronic copy of this thesis differs from the print copy as follows: 1. The citations are listed individually in order to attach hyperlinks to the
corresponding bibliography entries
Abstract
The predominant game-theoretic solutions for distributed rate-maximization algorithms in Gaussian interference channels through optimal power control require perfect channel knowledge, which is not possible in practice due to various reasons, such as estimation errors, feedback quantization and latency between channel estimation and signal transmission. This thesis therefore aims at addressing this issue through the design and analysis of robust game-theoretic algorithms for rate-maximization in Gaussian interference channels in the presence of bounded channel uncertainty.
A robust rate-maximization game is formulated for the single-antenna frequency-selective Gaussian interference channel under bounded channel uncertainty. The robust-optimization equilibrium solution for this game is independent of the probability distribution of the channel uncertainty. The existence and uniqueness of the equilibrium are studied and sufficient condi-tions for the uniqueness of the equilibrium are provided. Distributed algo-rithms to compute the equilibrium solution are presented and shown to have guaranteed asymptotic convergence when the game has a unique equilibrium. The sum-rate and the price of anarchy at the equilibrium of this game are analyzed for the two-user scenario and shown to improve with increase in channel uncertainty under certain conditions. These results indicate that the robust solution moves closer to a frequency division multiple access (FDMA) solution when uncertainty increases. This leads to a higher sum-rate and a lower price of anarchy for systems where FDMA is globally optimal.
A robust rate-maximization game for multi-antenna Gaussian interfer-ence channels in the presinterfer-ence of channel uncertainty is also developed along similar principles. It is shown that this robust game is equivalent to the nominal game with modified channel matrices. The robust-optimization equilibrium for this game and a distributed algorithm for its computation are presented and characterized. Sufficient conditions for the uniqueness of the equilibrium and asymptotic convergence of the algorithm are presented. Numerical simulations are used to confirm the behaviour of these algo-rithms. The analytical and numerical results of this thesis indicate that channel uncertainty is not necessarily detrimental, but can indeed result in improvement of performance of networks in particular situations, where the Nash equilibrium solution is quite inefficient and channel uncertainty leads to reduced greediness of users.
I dedicate this thesis to my early mentors: My grandfather, Y. K. Nanjundaiah S. R. Madhu Rao M. N. Krishna Swamy K. Ramamurthy V. Kesavan
Contents Overview
1 INTRODUCTION 18
2 GAME THEORY — FUNDAMENTALS, NASH
EQUILIBRIUM AND ROBUST GAME THEORY 34
3 ITERATIVE WATERFILLING ALGORITHMS 51
4 ROBUST IWFA FOR SISO FREQUENCY-SELECTIVE
SYSTEMS 92
5 SUM-RATE ANALYSIS IN THE TWO-USER SCENARIO123
6 ROBUST IWFA FOR MIMO SYSTEMS 150
7 SUMMARY, CONCLUSIONS AND FUTURE WORK 167
REFERENCES 174
Statement of Originality
The following aspects of this thesis are believed to be original:• The extension of the MIMO iterative waterfilling algorithm to broad-band Gaussian interference channels and the study of the effect of channel estimation errors on the performance of the MIMO iterative algorithm in Chapter 3.
• The robust rate-maximization game formulation for frequency-selective SISO Gaussian interference channels and the analysis of its equilibrium (Theorem 4.2 on page 105) in Chapter 4.
• The analysis of the effect of channel uncertainty on the sum-rate in the robust SISO rate-maximization game for the two-user case (The-orem 5.1 on page 125 and The(The-orem 5.2 on page 128) in Chapter 5. • The robust rate-maximization game formulation for MIMO Gaussian
interference channels in Chapter 6.
The novelty of this thesis is supported by the following works:
1. Amod J.G. Anandkumar, Animashree Anandkumar, Sangarapillai Lam-botharan, and Jonathon Chambers, “Robust Rate-Maximization Game for MIMO Gaussian Interference Channels Under Bounded Channel Uncertainty”, submitted to IEEE Transactions on Wireless
Commu-nications.
2. Amod J.G. Anandkumar, Animashree Anandkumar, Sangarapillai Lam-botharan, and Jonathon Chambers, “Robust Rate-Maximization Game Under Bounded Channel Uncertainty”, submitted to IEEE
Transac-tions on Vehicular Technology.
3. Amod J.G. Anandkumar, Animashree Anandkumar, Sangarapillai Lam-botharan, and Jonathon Chambers, “Efficiency of Rate-Maximization Game Under Bounded Channel Uncertainty”, 44th Asilomar
Confer-ence on Signals, Systems and Computers, Pacific Grove, CA, Nov. 7
- 10 2010.
4. Amod J.G. Anandkumar, Animashree Anandkumar, Sangarapillai Lam-botharan, and Jonathon Chambers, “Robust Rate-Maximization Game Under Bounded Channel Uncertainty”, 35th IEEE International
Con-ference on Acoustics, Speech, and Signal Processing (ICASSP 2010),
5 Amod J.G. Anandkumar, Sangarapillai Lambotharan, and Jonathon Chambers, “Application of Distributed MIMO Waterfilling Algorithm for Broadband Channels and its Performance in the Presence of CSI Errors”, First UK-India International Workshop on Cognitive
Wire-less Systems (UKIWCWS 2009), New Delhi, India, Dec. 11-12 2009
6 Amod J.G. Anandkumar, Sangarapillai Lambotharan, and Jonathon Chambers, “A Game-Theoretic Approach to Transmitter Covariance Matrix Design for Broadband MIMO Gaussian Interference Chan-nels,” IEEE/SP 15th Workshop on Statistical Signal Processing, 2009
Acknowledgements
I want to first and foremost thank my supervisors Prof. Jonathon Cham-bers and Dr. Sangarapillai Lambotharan for their continuous support and guidance. From my days as an undergraduate summer researcher at Cardiff University in 2007 till the end of my PhD studies, the continued inspiration and advice from Prof. Chambers has been invaluable and have helped shape my outlook towards research. I am also very grateful for the excellent and timely guidance from Dr. Lambotharan, particularly with the more math-ematical aspects of my research. Dr. Lambotharan and Prof. Chambers have been excellent mentors and have provided me with great opportunities to pursue research topics of my interest. I shall treasure their support and words of wisdom and hope to emulate them if I ever mentor any students. I am also thankful for the Engineering and Physical Sciences Research Coun-cil (EPSRC grant EP/F065477/1) for their monetary support which enabled the research presented in this thesis.I am also extremely grateful for the unwavering guidance and support from my sister Dr. Animashree Anandkumar. She has been my greatest supporter and harshest critic, and I am very grateful for both. Her high standards of achievement and success have greatly helped me improve myself. The research conducted in this thesis would not have been possible without her advice on suitably structuring problems to aid in their solution. Her inputs in the art of technical communication have also greatly helped present my work. I would also like to thank the University of California Irvine for hosting me during the winter of 2010.
I am grateful for the opportunity to collaborate with the excellent re-searchers of University of Luxembourg during the summer of 2010 and thank all those who made it possible. I would like to thank Dr. Peter von Wrycza, KTH Royal Institute of Technology and Dr. M.R. Bhavani Shankar and Prof. Bj¨orn Ottersten, University of Luxembourg for valuable discussions and inputs during my visit there. I also thank Dr. Ishai Menache, Microsoft Research for his early input on robust game theory and Dr. Gesualdo Scu-tari, University of Illinois at Urbana-Champaign for initial guidance and advice on waterfilling algorithms.
I am also thankful for the support and friendship of the past and current researchers of the Advanced Signal Processing Group, Loughborough Uni-versity, particularly Ranaji, Vimal, Mohsen, Jo, Georgia and Rahul. I am also extremely grateful for the steadfast friendship and advice of Cumanan throughout my three years at Loughborough.
At this point, I would also like to thank my aunt Dr. Shambhavi Rao, and her family, Dr. Krishna Murthy, Valli and Pramath for their great care and concern and for making me feel at home away from home. Without the efforts and encouragement of my aunt, I would not have visited Cardiff University in 2007, which was a turning point in my life and instrumental in my decision to come to Loughborough in order to work under the guidance of Prof. Chambers and Dr. Lambotharan. I would also like to thank my extended family in the UK for their support and encouragement. I have also received a great deal of support and inspiration from my undergraduate faculty, NITK, particular my dissertation advisor Dr. Sumam David who encouraged my efforts in IEEE student-branch activities which inspired me to join a PhD programme and conduct research myself.
I also wish to thank my grandfather for his years of tutoring in math-ematics and science during my schooling, and my grandmother for always encouraging me to aim high when setting goals for myself. Finally and most importantly, I would like to thank my parents for their invaluable support, motivation and the freedom to pursue my dreams and aspirations. My fa-ther has always inspired me and I owe my passion for technology and its advancement primarily to him. My mother’s steadfast emotional support and encouragement have been the foundation of my achievements to date. I also thank the family concerns M/s. Cam Tools, M/s. Splinewell Engineer-ing and M/s. Turnwell, Mysore for providEngineer-ing me with unique and excellent opportunities to learn about engineering and entrepreneurship over the years.
List of Algorithms
3.1 SISO Iterative Waterfilling Algorithm . . . 72
3.2 MIMO Iterative Waterfilling Algorithm . . . 79
4.1 Robust SISO Iterative Waterfilling Algorithm . . . 106
6.1 Robust MIMO Iterative Waterfilling Algorithm . . . 159
List of Figures
1.1 Growth in mobile broadband internet . . . 19
1.2 Congestion in RF spectrum in USA . . . 21
1.3 Measured spectrum utilization . . . 23
1.4 Concept of a spectrum hole . . . 24
3.1 Waterfilling solution for parallel Gaussian channels . . . 59
3.2 System model for broadband MIMO IWFA simulations . . . 84
3.3 Broadband MIMO IWFA simulation results . . . 86
3.4 Effect of CSI estimation errors on MIMO IWFA . . . 87
4.1 Sum-rate of the system vs. uncertainty δ. . . 109
4.2 Sum-rate of the system vs. number of users, Q. . . 109
4.3 Sum-rate of the system vs. number of frequencies, N . . . 110
4.4 Average number of channels occupied per user vs. uncer-tainty, δ. . . 110
4.5 Average number of channels occupied per user vs. number of users, Q. . . 111
4.6 Average number of iterations vs. uncertainty, δ. . . 111
4.7 Average number of iterations vs. number of users, Q. . . 112
5.1 Two-user two-frequency system model . . . 125
5.2 Equilibrium efficiency – two-user two-frequency case . . . 130
5.3 Equilibrium efficiency – two-user many-frequency case . . . . 132
6.1 Sum-rate vs. channel uncertainty bound, ǫ. . . 163
6.2 Average number of iterations for convergence vs. channel un-certainty bound. . . 163
6.3 Sum-rate vs. number of transmit/receive antennas of each user.164
LIST OF FIGURES 12
6.4 Sum-rate vs. number of users. . . 164
Detailed Contents
1 INTRODUCTION 18
1.1 Radio frequency bands and spectrum management . . . 19
1.2 Emerging wireless paradigms and technologies . . . 22
1.2.1 Cognitive radio and dynamic spectrum access . . . 22
1.2.2 OFDM technology . . . 25
1.2.3 Multi-antenna technology . . . 25
1.3 Motivation . . . 26
1.4 Thesis outline . . . 29
2 GAME THEORY — FUNDAMENTALS, NASH EQUILIBRIUM AND ROBUST GAME THEORY 34 2.1 Introduction to game theory . . . 35
2.1.1 Types of games . . . 38
2.2 Static noncooperative game — Nash equilibrium . . . 40
2.3 Equilibrium efficiency . . . 42
2.3.1 Social welfare . . . 43
2.3.2 Pareto optimality . . . 44
2.3.3 Price of anarchy and price of stability . . . 44
2.4 Moving beyond Nash equilibrium — robust game theory . . . 45
2.5 Robust game model . . . 47
2.6 Summary . . . 49
3 ITERATIVE WATERFILLING ALGORITHMS 51 3.1 Contraction and fixed point theory . . . 52
3.1.1 Existence and uniqueness of a fixed point . . . 52
3.1.2 Convergence of distributed algorithms to a fixed point 54 3.2 Waterfilling: classical results — single-user systems . . . 57 13
Detailed Contents 14
3.2.1 Parallel Gaussian channels . . . 58
3.2.2 MIMO Gaussian channel . . . 60
3.3 Gaussian interference channel — multi-user systems . . . 61
3.3.1 Competitive rate-maximization in the Gaussian inter-ference channel . . . 63
3.4 Iterative waterfilling for frequency-selective GICs . . . 66
3.4.1 System model . . . 66
3.4.2 Rate-maximization game . . . 67
3.4.3 Nash equilibrium . . . 68
3.4.4 Asynchronous iterative waterfilling algorithm . . . 71
3.5 Iterative waterfilling for MIMO GICs . . . 72
3.5.1 System model . . . 72
3.5.2 Rate-maximization game . . . 73
3.5.3 Nash equilibrium . . . 75
3.5.4 MIMO iterative waterfilling algorithm . . . 78
3.6 Iterative waterfilling for broadband MIMO GICs . . . 80
3.6.1 System model . . . 80
3.6.2 Rate-maximization game . . . 83
3.6.3 Numerical results . . . 84
3.7 Effect of CSI estimation errors . . . 85
3.8 Summary . . . 88
3.A KKT Conditions . . . 89
4 ROBUST IWFA FOR SISO FREQUENCY-SELECTIVE SYSTEMS 92 4.1 Related work . . . 93
4.2 System model . . . 95
4.3 Robust rate-maximization game formulation . . . 98
4.4 Robust waterfilling solution . . . 100
4.4.1 Robust waterfilling as a projection operation . . . 101
4.5 Robust-optimization equilibrium . . . 104
4.5.1 Analysis of the equilibrium of game GrobS . . . 104
4.6 Iterative algorithm for robust waterfilling . . . 105
4.7 Simulation results . . . 108
4.8 Summary . . . 114
4.A Proof of Theorem 4.1 . . . 115
4.B Proof of Lemma 4.1 . . . 117
4.C Proof of Theorem 4.2 . . . 121
Detailed Contents 15
5 SUM-RATE ANALYSIS IN THE TWO-USER SCENARIO123
5.1 Two frequency case (N = 2) . . . 124
5.2 Large number of frequencies (N → ∞) . . . 127
5.3 Simulation results . . . 129
5.4 Summary . . . 133
5.A Proof of Theorem 5.1 . . . 133
5.B Proof of Proposition 5.1 . . . 138
5.C Proof of Lemma 5.1 . . . 139
5.D Proof of Theorem 5.2 . . . 149
6 ROBUST IWFA FOR MIMO SYSTEMS 150 6.1 System model . . . 151
6.2 Robust rate-maximization game formulation . . . 153
6.3 Robust-optimization equilibrium . . . 157
6.3.1 Iterative algorithm for robust waterfilling . . . 158
6.4 Simulation results . . . 161
6.5 Summary . . . 166
7 SUMMARY, CONCLUSIONS AND FUTURE WORK 167 7.1 Summary and conclusions . . . 167
7.2 Future work . . . 171
Detailed Contents 16
Abbreviations
CSI Channel state information
DSL Digital subscriber line
FCC Federal Communications Commission
FDMA Frequency division multiple access
GIC Gaussian interference channel
IWFA Iterative waterfilling algorithm
KKT Karush–Kuhn–Tucker
MIMO Multiple-input multiple-output
MUI Multi-user interference
NP Non-deterministic polynomial time
OFDM Orthogonal frequency-division multiplexing
QoS Quality-of-service
RF Radio frequency
SISO Single-input single-output
s. t. subject to
Detailed Contents 17
Notations
a Scalar a a Vector a A Matrix A (·)H Hermitian operator(·)−1 Matrix inverse operator
(·)−H Equivalent to (·)H−1
E{·} Statistical expectation operator
Tr(·) Trace operator
k · k2 Euclidean norm
k · kF Frobenius norm
Diag(·) Diagonal matrix with the arguments as the
diagonal elements
[A]ij (i, j)th element of A
[a]i ith element of a
σmax(A) Largest singular value of A
λi(A) ith eigenvalue of A
λmin(A) Smallest eigenvalue of A
ρ(A) Spectral radius of A
u≥ v [u]i≥ [v]i ∀i
A 0 A is positive semidefinite
Cm×n Set of m × n complex matrices
Rm×n+ Set of m × n matrices with real non-negative
elements
Rm×n
++ Set of m × n matrices with real positive elements
x ∼ N (µ, σ2) Random variable x is drawn from a Gaussian
distribution with mean µ and variance σ2
x ∼ NC(µ, σ2) Random variable x is drawn from a circularly
symmetric complex Gaussian distribution
with mean µ and variance σ2
U (a, b) Uniform probability distribution with interval [a, b]
N (·) Null space operator
PN (A) Orthogonal projection onto the null space of A
(x)+ max(0, x)
[x]b
a Euclidean projection of x onto the interval [a, b]
[X]Q arg min
Chapter 1
INTRODUCTION
Wireless communications technology has become an ubiquitous element of our society, ranging from remote controllers and paging systems to cellular phones and wireless local area networks. With the advent of better battery technology and the relentless progress of Moore’s law, today’s portable de-vices can support a great amount of processing power. This has led to an exponential increase in the usage of portable devices such as cellular phones, tablet computers and laptops for data-intensive broadband internet applica-tions (Figure 1.1).
This huge increase in demand has led to heavy congestion in the radio-frequency (RF) spectrum allocated to these applications. Issues such as call dropping, low download speeds and sporadic availability of network access have become commonplace, particularly in areas with high density of users. This was exemplified at the launch of the popular iPhone 4 cell-phone last year, when Steve Jobs (CEO of Apple Inc.), who is famous for delivering impeccable product-launches, had to briefly suspend the launch midway as the demonstration phone could not access the network. He finally had to 18
Section 1.1. Radio frequency bands and spectrum management 19
Figure 1.1: Projected number of subscriptions (in billions) for mobile broadband and wired broadband internet connections globally [1]. request the 500+ members of the audience to turn off their WiFi devices in order to continue the demonstration [2].
1.1 Radio frequency bands and spectrum management
Wireless communication devices for various applications and services oper-ate in specific pre-determined ranges of the electromagnetic spectrum called
radio frequency (RF) bands. The radio waves used for communication are
transmitted and received through antennas which transform electrical en-ergy to radio waves that propagate through the atmosphere from the source to the destination. Due to varying propagation characteristics of radio waves of different frequencies, specific applications are allocated specific RF bands. For instance, the RF spectrum allocation in the United States of Amer-ica is presented in Figure 1.2. These allocations are determined by various
Section 1.1. Radio frequency bands and spectrum management 20
national and international regulatory bodies (e.g.: the Federal Communi-cations Commission (FCC) in the USA and the Office of CommuniCommuni-cations (Ofcom) in the UK).
When multiple wireless devices operate in the same environment, they often interfere with each other. Spectrum management is needed to control the usage of RF spectrum in order to mitigate interference among wire-less devices and services. The current practice for spectrum allocation by regulatory bodies is known as the command-and-control model [4]. In this approach, the regulators make centralized decisions regarding spectrum al-location and usage, often through an auctioning process commonly referred to as a spectrum auction. After a successful bid, a user/company is awarded the allocation, which is often valid for extended periods of time and over large geographical regions. While the majority of the RF spectrum is man-aged under this scheme, a small region of the RF spectrum, known as the industrial, scientific and medical (ISM) band, is unlicensed and open to any device/application. Some of the technologies using these band are cord-less telephony, bluetooth radio, wirecord-less local area networking and radio-frequency identification (RFID).
The command-and-control model of spectrum management and alloca-tion ensures interference-free operaalloca-tion for the licensed user as it is operating in the band exclusively. Since most of the spectrum is already allocated to various applications (Figure 1.2), emerging wireless applications such as wireless broadband communications face an apparent spectrum scarcity.
Section 1.1. Radio frequency bands and spectrum management 21 U.S . DEP ARTM E NT OF COM MERCE NA TIONAL TELECOMMUN ICATIO NS & IN F OR MA TIONAD MIN ISTR A TI ON MO BIL E ( AER ON AU TIC AL TEL EM ETE RIN G) S) 5.6 8 5.7 3 5.9 0 5.9 5 6.2 6.5 25 6.6 85 6.7 65 7.0 7.1 7.3 7.3 5 8.1 8.1 95 8.8 15 8.9 659.0 40 9.4 9.5 9.9 9.9 95 10 .0 03 10 .0 05.110 10 .1 5 11 .1 75 11 .2 75.411 11 .6 11 .6 5 12 .0 5 12 .1 0 12 .2 3 13 .2 13 .2 6 13 .3 6.413 1 13 .5 7 13 .6 13 .8 13 .8 7.014 14 .2 5 14 .3 5 14 .9 90 15 .0 05.015 10 15 .1 0 15 .615 .8 16 .3 6 17 .4 1 17 .4 8.517 5 17 .917 .9 7 18 .0 3 18 .0 68.118 68 18 .7 8 18 .919 .0 2 19 .6 8.819 0 19 .9 90.919 95 20 .0 05.020 10 21 .0 21 .4 5 21 .8 5 21 .9 24 22 .0 22 .8 55 23 .0 23 .223 .3 5 24 .8 9 24 .9 9 25 .0 05 25 .0 1.025 7 25 .2 1 25 .3 3.525 5 25 .6 7.126 26 .1 75.426 8 26 .9 5.926 6.227 3 27 .4 1.527 4 28 .0 29 .7 29 .829 .8 9 29 .9 1 30 .0
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e) 30 0 32 5 33 5 40 5 41 5 43 5 49 5 50 5 51 0 52 5 53 5 16 05 16 15 17 05 18 00 19 00 20 00 20 65 21 07 21 70 21 73 .59021 .5 21 94 24 95 25 01 25 02 25 05 28 50 30 00 RADIO-LOCATION BRO ADC AST ING FI XE D M OB ILE AM ATE UR RAD IO LO CAT ION MO BIL E FIX ED MAR ITI MEBILMO E MAR ITIME MO BIL E (T ELE PHO NY) MAR ITI MEBILMO E LAN DBILMO E EBILMO EDFIX 30 .0 30 .56 32 .0 33 .0 34 .0 35 .0 36 .0 37 .0 37 .5 38 .0 38 .25 39 .0 40 .0 42 .0 43 .69 46 .647 .0 49 .6 50 .0 54 .0 72 .0 73 .0 74 .6 74 .8 75 .275 .4 76 .0 88 .0 10 8.0 11 7.9 75 121 .93 75.08123 75 123 .58 75 128 .81 25 132 .01 25 13 6.0 13 7.013 7.0 257.113 757.813 258.013 14 4.014 6.014 8.014 9.9 15 0.0 5 15 0.8 15 2.8 55 15 4.0 156 .24 75 157 .03 75.18157 75 15 7.4 51.516 75 16 1.6 25 16 1.7 75 162 .01 25 17 3.2 17 3.4 17 4.0 21 6.0 22 0.0 22 2.022 5.0 23 5.0 30 0 ISM – 6 .78 ± .0 15 MH z IS M – 1 3.56 0 ± .0 07 M Hz IS M – 27. 12 ± .1 63 M H z ISM – 4 0.68 ± .02 MH z IS M – 24. 12 5 ± 0.12 5 GHz 3 0 G H z ISM – 2 45 .0 ± 1GH z IS M – 1 22 .5 ± .5 00 G Hz IS M – 61.25 ± .250 G Hz 30 0.0 32 2.0 32 8.6 33 5.4 39 9.9 40 0.0 5 40 0.1 5 40 1.0 40 2.0 40 3.0 40 6.0 40 6.1 41 0.0 42 0.0 45 0.0 45 4.045 5.0 45 6.0 46 0.0 462 .53 75.73462 75 467 .53 75 467 .73 750.047 51 2.0 60 8.061 4.0 69 8 74 6 76 4 77 6 79 4 80 6 82 182 4 84 9 85 186 6 86 9 89 489 6 90 1 90 1 90 2 92 8 92 9 93 0 93 193 2 93 5 94 0 94 194 4 96 0 12 15 12 40 13 00 13 50 13 9013 92 13 95 20 00 20 20 20 25 21 10 21 55 21 60 21 80 22 00 22 9023 00 23 0523 10 23 20 23 45 23 60 23 85 23 90 24 00 24 17 24 50 24 83 .5 25 0026 55 26 90 27 00 29 00 30 00 14 00 14 2714 29 .5 14 30 14 32 14 35 15 25 15 30 15 35 15 44 15 45 15 49 .5 15 58 .55915 16 10 16 10 .61316 .82616 .5 16 6016 60 .5 16 68 .4 16 70 16 75 17 00 17 10 17 55 18 50 MAR ITIM E M OBILE SAT ELL ITErth)ace to Ea(sp MOB ILE SA TEL LITE (S-E) RAD IO LO CAT IO
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ALATIVIGNA DIORA ON SPA CE RES EAR CH (Pass ive) * E XCEP T AERO M O BIL E (R ) ** E X CEP T AERO M O BILE WAV EL EN GTH BA ND DE SIGNA TIONS ACTIVITIES FREQUENCY 3 x 10 7m 3 x 10 6m 3 x 10 5m 30,0 00 m 3,00 0 m 30 0 m 30 m 3 m 30 cm 3 cm 0.3 cm 0. 03 cm 3 x 10 5Å 3 x 10 4Å 3 x 10 3Å 3 x 10 2Å 3 x 10Å 3Å 3 x 1 0 -1Å 3 x 10 -2Å 3 x 10 -3Å 3 x 10 -4Å 3 x 10 -5Å 3 x 10 -6Å 3 x 10 -7Å 0 10 Hz 10 0 Hz 1 kH z 10 kH z 100 kH z 1 MHz 10 M Hz 100 MHz 1 GHz 10 GHz 100 GHz 1 TH z 10 13 Hz 10 14 Hz 10 15 Hz 10 16 Hz 10 17 Hz 10 18 Hz 10 19 Hz 10 20 Hz 10 21 Hz 10 22 Hz 10 23 Hz 1 0 24 Hz 10 25 Hz TH E RA DI O S PE CT RU M M AG NI FI ED A BO VE 3 kHz 30 0 G Hz VE R Y LO W F R EQ U EN C Y (V LF ) Au di ble R an ge A M B ro ad ca st F M B ro ad ca st R ad ar Su b-M illi m et er Vi sibl e U ltr av io le t G amma-ray Cosmic-ray Infra-sonics So ni cs Ultra-sonics Microwaves Infrared P L S X C R ad ar Ba nd s LF MF HF VHF UHF SHF EHF I N FR AR ED V IS IB LE U LT R AV IO LE T X -R AY G AM M A-R AY C O SM IC -R AY X-ra y
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Section 1.2. Emerging wireless paradigms and technologies 22
However, field studies on actual spectrum usage have shown that the cur-rent spectrum management policy results in highly inefficient spectrum uti-lization (Figure 1.3). This has led to the evolution of new paradigms and technologies for future spectrum management models and wireless commu-nication devices which aim to improve the efficiency of their own spectrum utilization and to exploit the inefficiency of licensed spectrum users.
1.2 Emerging wireless paradigms and technologies
The growing demand for high-throughput wireless communication has led to the emergence of new paradigms and technologies as contenders for next-generation wireless communication networks. In this section, a few of these, namely cognitive radio, dynamic spectrum access, orthogonal frequency-division multiplexing (OFDM) and multi-antenna systems are briefly de-scribed.
1.2.1 Cognitive radio and dynamic spectrum access
Cognitive radio is an emerging paradigm of wireless communication in which an intelligent wireless system utilizes information about the radio environ-ment to adapt its operating characteristics in order to ensure reliable com-munication and efficient spectrum utilization [6]. The main goal of this paradigm is to enable the cognitive radio to exploit the inefficiently utilized licensed spectrum for its own communication needs without significantly af-fecting the licensed user. The band of RF spectrum that is licensed to a user
Section 1.2. Emerging wireless paradigms and technologies 23
Measured Spectrum Occupancy At Seven Locations
0.0% 25.0% 50.0% 75.0% 100.0%
Riverbend Park, Great Falls, VA Tysons Corner, VA NSF Roof, Arlington, VA New York City NRAO, Greenbank, WV SSC Roof, Vienna, VA Chicago, IL Average
Spectrum Occupancy
(a) Overall spectrum occupancy measured at seven locations
Measured Spectrum Occupancy Averaged over Seven Locations
0.0% 25.0% 50.0% 75.0% 100.0%
PLM, Amateur, others: 30-54 MHz TV 2-6, RC: 54-88 MHz Air traffic Control, Aero Nav: 108-138 MHz Fixed Mobile, Amateur, others:138-174 MHz TV 7-13: 174-216 MHz Maritime Mobile, Amateur, others: 216-225 MHz Fixed Mobile, Aero, others: 225-406 MHz Amateur, Fixed, Mobile, Radiolocation, 406-470 MHz TV 14-20: 470-512 MHz TV 21-36: 512-608 MHz TV 37-51: 608-698 MHz TV 52-69: 698-806 MHz Cell phone and SMR: 806-902 MHz Unlicensed: 902-928 MHz Paging, SMS, Fixed, BX Aux, and FMS: 928-906 MHz IFF, TACAN, GPS, others: 960-1240 MHz Amateur: 1240-1300 MHz Aero Radar, Military: 1300-1400 MHz Space/Satellite, Fixed Mobile, Telemetry: 1400-1525 MHz Mobile Satellite, GPS, Meteorologicial: 1525-1710 MHz Fixed, Fixed Mobile: 1710-1850 MHz PCS, Asyn, Iso: 1850-1990 MHz TV Aux: 1990-2110 MHz Common Carriers, Private, MDS: 2110-2200 MHz Space Operation, Fixed: 2200-2300 MHz Amateur, WCS, DARS: 2300-2360 MHz Telemetry: 2360-2390 MHz U-PCS, ISM (Unlicensed): 2390-2500 MHz ITFS, MMDS: 2500-2686 MHz Surveillance Radar: 2686-2900 MHz
Spectrum Occupancy
(b) Measured spectrum occupancy in the 30 MHz-3,000 MHz range averaged over the seven locations
Figure 1.3: Average spectrum occupancy measured at seven locations in the USA in the 30 MHz-3,000 MHz range demonstrating inefficient spectrum utilization [5].
Section 1.2. Emerging wireless paradigms and technologies 24 Time Dynamic spectrum access Frequency Spectrum in use “Spectrum hole” Power
Figure 1.4: Concept of spectrum hole represented in power, time and frequency space wherein there is an opportunity for a cognitive radio to operate [7].
but is not utilized by the licensed user at a particular time and location is called a spectrum hole (Figure 1.4).
One of the key enabling technologies for cognitive radio is dynamic
spec-trum access. The overarching idea behind dynamic specspec-trum access is to
temporarily borrow unused spectrum from licensed users (spectrum holes) without interfering with their operation [8]. Research on various aspects of cognitive radio and dynamic spectrum access has received a great deal of interest in recent times [9, 10, 11, 12, 13] and will help design wireless com-munication systems of the future.
Section 1.2. Emerging wireless paradigms and technologies 25
1.2.2 OFDM technology
Orthogonal frequency-division multiplexing (OFDM) is a frequency-division multiplexing scheme developed to transmit multiple digital signals simulta-neously over a large number of closely-spaced orthogonal sub-carriers [14]. OFDM transmission is used for wideband digital communication, both wired and wireless, and has been adopted for a variety of applications, ranging from digital television to wireless networking.
OFDM technology has many advantages. It has a high spectral effi-ciency and is resilient to interference and multipath effects. It can also easily adapt to severe channel conditions without complicated time-domain equalization and can be efficiently implemented using the fast Fourier trans-form. However, OFDM technology is quite sensitive to Doppler shift and frequency-synchronization issues, despite which it has become the de-facto physical (PHY) layer broadband transmission technology and is particularly advantageous for multiple access systems.
1.2.3 Multi-antenna technology
Multiple antennas at the receiver and/or transmitter of a wireless commu-nication system can be used to improve link performance [15]. The term multiple-input multiple-output (MIMO) is used to describe systems which exploit such antenna diversity. MIMO technology can significantly improve the data throughput and coverage without additional bandwidth or trans-mission power. This is because signals transmitted from multiple antennas
Section 1.3. Motivation 26
experience differing multipath fading and are received by multiple anten-nas. The different multipath signals can be combined coherently to achieve higher data rates and/or lower bit-error rates by using clever signal process-ing techniques. However, this can significantly increase the complexity of the communication system.
1.3 Motivation
The aforementioned technologies and paradigms are integral features of to-day’s high performance networks and/or are strong contenders for wireless networks of tomorrow. OFDM and MIMO technologies are already exten-sively used in wireless networks through standards such as IEEE 802.11n and WiMAX. They are also strong candidate technologies for the next-generation of wireless communication networks [16, 17, 18], which will incorporate prin-ciples of cognitive radio and dynamic spectrum access [19, 20].
A prominent feature of these paradigms is the provision for greater free-dom of action for the users in the network. In such a network, “intelligent” users actively and dynamically manage the resources and characteristics of their transceivers, such as transmit power and bandwidth, in order to op-timize their communication performance in terms of various criteria such as information rate, utilized power and achieved quality-of-service (QoS). With the advent of cognitive radio and dynamic spectrum access, multiple heterogenous wireless technologies and standards of tomorrow are expected to function seamlessly in the same environment and RF spectrum [21]. In
Section 1.3. Motivation 27
such a setting, interference among the users1 becomes a critical issue and
precludes the traditional approach of interference mitigation through prede-termined non-overlapping frequency allocation (FDMA) due to significant demands on coordination among the users. This form of centralized control determining the communication parameters of users leads to heavy signalling overhead as information from all the users needs to be collected, processed and disseminated by the controller.
The issue of maximizing information rates of users in an interference channel by optimizing the power spectral density of the transmitted signal under certain power constraints is of interest in this thesis. The sum-rate maximization problem in a frequency-selective Gaussian interference channel has been proved to be NP-hard [22] and the optimal solution of this problem is of the form of frequency division multiple access (FDMA) [23]. Thus, a centralized solution to this problem not only requires information (in the form of channel state information and noise variances) from all users, but also is computationally unattractive.
The solution to these limitations lies in the framework of distributed
algo-rithms, which enable users of the network to compute their optimal solutions
autonomously with limited (locally available) information. The analysis of such a system of multiple interactive autonomous intelligent users operating in the same environment falls well within the purview of game theory, which
1
Such a system with multi-user interference in a frequency-selective medium (as seen in OFDM transmission) can be modelled as an interference channel, described in Section 3.3 on page 61.
Section 1.3. Motivation 28
was devised to analyze and predict the outcome of situations where multi-ple entities (typically peomulti-ple, companies and nations) interact. In addition, solutions from such a game-theoretic analysis can often be implemented as distributed algorithms. These factors make game theory an attractive tool for the design and analysis of tomorrow’s wireless communication networks. Indeed, it has been shown that the problem of maximizing information rates of users in an interference channel, which is of interest in this thesis, can be modelled as a noncooperative game [24].
However, a vast majority of game-theoretic solutions proposed for wire-less networks in the current literature, including the rate-maximization game, assumes perfect knowledge of channel state information, which is not pos-sible in practice, where such information is estimated with a certain degree of uncertainty. This uncertainty could be introduced though several mecha-nisms, such as estimation errors, feedback quantization and latency between channel estimation and signal transmission. If these solutions are to be implemented in practice, the effect of such uncertainty on the perfor-mance of these game-theoretic solutions needs to be characterized and robust game-theoretic algorithms which perform satisfacto-rily in spite of such uncertainty need to be designed and analyzed. This issue of uncertainty in the parameters of game-theoretic solutions is the central theme of this thesis.
In summary, the design of distributed algorithms based on ideas from game theory for maximizing the information rates of users
Section 1.4. Thesis outline 29
having limited transmit power in single-antenna and multi-antenna interference channels with uncertainty in channel state informa-tion is the focus of this thesis.
1.4 Thesis outline
The aforementioned approaches and issues are addressed over the following chapters of this thesis:
Chapter 2: Game Theory: Fundamentals, Nash Equilibrium and Robust Game Theory
This chapter presents a brief description of the game-theoretic concepts that are of interest in this thesis. The chapter begins with an introduction to game theory and the conditions needed for its application in a given sce-nario, described in Section 2.1 on page 35. This is followed by a discussion on strategic noncooperative games and the concept of Nash equilibrium in Section 2.2 on page 40 and the notion of equilibrium efficiency as a method to quantify the quality of the Nash equilibrium solution in Section 2.3 on page 42.
In Section 2.4 on page 45, a discussion on some of the limitations of the Nash equilibrium concept and the traditional game-theoretic approach to the issue of uncertainty in games is presented. Finally, the robust game model, which is a union of ideas from robust optimization theory and game theory, is introduced in Section 2.5 on page 47 as a suitable candidate for
Section 1.4. Thesis outline 30
resolving the issue of uncertainty in channel state information affecting the performance of game-theoretic solutions.
Chapter:3: Iterative Waterfilling Algorithms
This chapter presents a specific game-theoretic solution and its associated conceptual and mathematical foundations within which the issue of channel uncertainty is investigated in this thesis. This includes results from contrac-tion and fixed point theory, which addresses the formulacontrac-tion and character-ization of distributed algorithms (Section 3.1 on page 52) and information theory, which introduce the waterfilling solution as the optimal solution to the problem of rate-maximization in a single-user context (Section 3.2 on page 57). This is followed by the description of the Gaussian interference channel in the multi-user scenario and a review of the current literature using game theory to address the problem of rate-maximization in this medium in Section 3.3 on page 61.
The predominant game-theoretic solution, namely the iterative water-filling algorithm (IWFA), to the problem of rate-maximization in single-antenna frequency-selective Gaussian interference channels and multi-single-antenna (MIMO) Gaussian interference channels are presented respectively in Sec-tion 3.4 on page 66 and SecSec-tion 3.5 on page 72 and extended to broadband MIMO Gaussian interference channels in Section 3.6 on page 80.
Finally, in Section 3.7 on page 85, the effect of channel state information errors on the performance of the MIMO iterative waterfilling algorithm is
Section 1.4. Thesis outline 31
investigated, which demonstrates the need for robust solutions presented in the subsequent chapters.
Chapter 4: Robust IWFA for SISO Frequency-Selective Systems In this chapter, the analytic framework for a robust formulation of the rate-maximization game for single-input single-output (SISO) frequency-selective Gaussian interference channels is presented. Section 4.1 on page 93 is a re-view of the state-of-the-art in methods that address and investigate the is-sue of channel state information uncertainty in rate-maximization games for Gaussian interference channels. The system under consideration is described in Section 4.2 on page 95. A distribution-free robust rate-maximization game based on the robust game model is formulated in Section 4.3 on page 98. The optimal solution of each user in the form of a robust waterfilling operation is derived and characterized in Section 4.4 on page 100.
In Section 4.5 on page 104, the equilibrium solution of this game, termed the robust-optimization equilibrium, is presented and shown to exist for possi-ble channel values and initializations, and further, to be unique under certain sufficient conditions. A distributed algorithm to compute the equilibrium so-lution iteratively is presented and proved to asymptotically converge when a unique equilibrium is guaranteed in Section 4.6 on page 105. Finally, numer-ical simulations to confirm the behaviour of the algorithm are presented in Section 4.7 on page 108, where an interesting effect of increase in sum-rate with greater channel uncertainty is observed, which merits further analysis.
Section 1.4. Thesis outline 32
Chapter 5: Sum-Rate Analysis in the Two-User Scenario
This chapter is an analytical investigation of the improvement in sum-rate with greater channel uncertainty in the robust SISO rate-maximization game proposed in Chapter 4 for the two-user case. To begin with, the effect of increasing channel uncertainty on the sum-rate and the price of anarchy of a simple two-frequency system are analyzed in Section 5.1 on page 124. Based on these results, conditions for improvement in the sum-rate and the price of anarchy of a system with asymptotically large number of frequencies with an increase in uncertainty are derived in Section 5.2 on page 127. Finally, these results are supported using simulations in Section 5.3 on page 129.
Chapter 6: Robust IWFA for MIMO Systems
In this chapter, a robust rate-maximization game in MIMO Gaussian inter-ference channels in the presence of bounded channel uncertainty is developed. The system model for which the robust game is developed is described in Section 6.1 on page 151. A robust MIMO rate-maximization game for this system is formulated and shown to be a modified MIMO rate-maximization game (Section 3.5) in Section 6.2 on page 153. The robust-optimization equilibrium for this game and an iterative waterfilling algorithm to compute it are presented, along with sufficient conditions for the uniqueness of the equilibrium and asymptotic convergence of the algorithm, in Section 6.3 on page 157. The behaviour of the algorithm under different settings is con-firmed though numerical simulations in Section 6.4 on page 161.
Section 1.4. Thesis outline 33
Chapter 7: Summary, Conclusions and Future Work
The novel results of this thesis and their conclusions are summarized in Section 7.1 on page 167. The work presented in this thesis could be extended in various directions, some of which are described in Section 7.2 on page 171.
Chapter 2
GAME THEORY —
FUNDAMENTALS, NASH
EQUILIBRIUM AND ROBUST
GAME THEORY
The past decade has seen the increasing application of concepts from game theory in wireless communication systems to solve a variety of problems [25, 26, 27, 28, 29, 30, 31]. Game theory has been utilized to solve various resource allocations problems in different scenarios and the problems con-sidered involve bandwidth allocation, power control, medium access control, flow control, routing and pricing issues in wireless networks [32]. Game theory has also been extensively applied in cognitive radio and dynamic spectrum access, particularly to solve issues in spectrum management and spectrum sharing [33, 34, 35, 36, 37].