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Communication and cooperation in underwater acoustic networks

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Communication and cooperation in underwater acoustic networks

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Content COMMUNICATION AND COOPERATION IN UNDERWATER ACOUSTIC NETWORKS by SRINIV AS YERRAMALLI A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2013 Copyright 2013 SRINIV AS YERRAMALLI Dedication This dissertation is dedicated to my parents and family - without whose support none of this would have been possible. ii Acknowledgements This thesis is the culmination of all the time I have spent at USC and this would not have been possible without the support of my mentor and advisor Prof. Urbashi Mitra. She has been a great advisor, beginning from quickly helping me transfer from the Masters to the Ph D program, helping with my initial research problems, arranging my visit to Dr. Milica Stojanovic at Northeastern University for a year when she was on sabbatical and encouraging me to build a new research direction along with Dr. Rahul Jain resulting in the completion of my thesis. I am extremely thankful for her support and encouragement through all times good and bad. She has been and will always be a role model for me to look up to and learn from and no words will fully express my gratitude for her. I also heartily thank Dr. Milica Stojanovic for hosting me during the year when Dr. Mitra was on sabbatical. She is an amazing person to work with, a true engineer at heart and I have throughly enjoyed my stay at Northeastern. She never allowed me to feel the absence of my advisor and for this I will be ever grateful to her. I also thank Dr. Rahul Jain for working with me and helping me in developing a new direction in my research work and significantly expanding my field of interest. In game theory, I have found a field I truly enjoy and work on this problem would not have been possible without Dr. Jain’s support. I have also greatly benefited from my interactions with Dr. Guiseppe Caire, Dr. Keith Chugg, Dr. Zhen Zhang, Dr. Robert Scholtz and Dr. Alex Dimakis in and outside classes. Finally, I have to mention Dr. Daniel Liu, Dr. Marco Levorato and Dr. Geoff Hollinger for their critical feedback at several stages. iii I cannot begin to describe the hard-work, dedication and support that my parents and family have put into my life. I hope that I turn out to be atleast half the person they deserve. No mention of my life at USC would be complete without a mention of friends. Manish, Gopi, Sanjay, Prashant, Pramod, Karthik, Sundar, Dileep, Harish, Niharika, Nachiketas, Satish, Gautam, Chiru, Daphney, Sunav, Sajjad, Hao, Junyang, Akshara, Nilesh, Dilip, Rameez, Sindhu, Baosheng, Ashish, Parastoo, Sangeeta, Manasa, Salam, Anirudh and Vacha have been the center of my social life during my Ph D years and life as a grad student would have been very difficult without these wonderful people. I am thankful for all the discussions, arguments, fights and fun that we had together. Life at USC was made easy by the expert handling and ever present help provided by Diane Demetras, Mayumi Thrasher, Gerrilyn Ramos and Anita Fung. They were of great help, ever ready and available and I sincerely thank and appreciate their efforts. iv Table of Contents Dedication ii Acknowledgements iii List of Tables ix List of Figures x Abstract xiii Chapter 1: Introduction 1 1.1 Acoustic Channels and Propagation Models . . . . . . . . . . . . . . . 3 1.1.1 Path Loss and Attenuation . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Multipath and Long Delay Spreads . . . . . . . . . . . . . . . 6 1.1.4 Doppler distortion . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.5 Network effects . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Outline of this Document . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2: Partial FFT Demodulation 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Signal Model and Partial FFT Demodulation . . . . . . . . . . . . . . . 18 2.2.1 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Conventional OFDM Demodulation . . . . . . . . . . . . . . . 19 2.2.3 Partial FFT Demodulation . . . . . . . . . . . . . . . . . . . . 20 2.2.4 Combining Partial FFT Outputs . . . . . . . . . . . . . . . . . 22 2.2.5 Partial FFT combining: A window design interpretation . . . . 22 2.3 Computing the Combiner Weights . . . . . . . . . . . . . . . . . . . . 23 2.3.1 Perfect Channel Knowledge Scenario . . . . . . . . . . . . . . 23 2.3.2 No Channel Knowledge Scenario . . . . . . . . . . . . . . . . 25 2.4 Model Based Combining . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.1 OFDM with Carrier Frequency Offset . . . . . . . . . . . . . . 30 2.4.2 OFDM with Time Scaling Distortion . . . . . . . . . . . . . . 31 2.4.3 Model-Based Weight Estimation . . . . . . . . . . . . . . . . . 32 v 2.5 Performance Analysis: Theory . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 Analysis of Model-Based Weight Estimation . . . . . . . . . . 34 2.5.2 Analysis of Recursive Weight Estimation . . . . . . . . . . . . 35 2.5.3 Complexity Requirements . . . . . . . . . . . . . . . . . . . . 39 2.6 Performance Analysis: Simulation . . . . . . . . . . . . . . . . . . . . 42 2.6.1 Underwater Acoustic Communications . . . . . . . . . . . . . 42 2.6.2 Digital Video Broadcasting . . . . . . . . . . . . . . . . . . . . 47 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chapter 3: Optimal Resampling 53 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 Transmit Signal . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.2 Wideband Channel Representation . . . . . . . . . . . . . . . . 58 3.3 Sufficient Statistics and Performance Bounds . . . . . . . . . . . . . . 60 3.3.1 Performance Bounds . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Doppler Compensation using Resampling . . . . . . . . . . . . . . . . 64 3.4.1 Equal Time Scaling on all Paths . . . . . . . . . . . . . . . . . 65 3.4.2 Multi-scale Multi-lag Signals . . . . . . . . . . . . . . . . . . 66 3.4.3 Resampling as approximation with a single scale . . . . . . . . 67 3.5 Analysis of Optimal Resampling . . . . . . . . . . . . . . . . . . . . . 69 3.5.1 One Carrier Scenario . . . . . . . . . . . . . . . . . . . . . . . 70 3.5.1.1 Small scale spread . . . . . . . . . . . . . . . . . . . 71 3.5.1.2 Large scale spread . . . . . . . . . . . . . . . . . . . 71 3.5.2 Simulations and Discussion . . . . . . . . . . . . . . . . . . . 73 3.5.3 Performance Improvement - ML Detection . . . . . . . . . . . 77 3.6 Estimation of the Resampling Parameter from Sam-pled Discrete Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6.1 Blind estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.6.2 Pilot Aided Estimator . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.2.1 Delay parameter estimation . . . . . . . . . . . . . . 83 3.6.3 Packet Length Based Estimation . . . . . . . . . . . . . . . . . 83 3.6.4 Simulation Results - Performance Comparison and Discussion . 85 3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Chapter 4: Channel Estimation 90 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.1 Design of pilot sequences . . . . . . . . . . . . . . . . . . . . 96 4.3.2 Channel estimation . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3.3 Exploiting the relationship between parameters . . . . . . . . . 98 4.3.4 Discussion: Estimation of time scale . . . . . . . . . . . . . . 100 vi 4.4 Pre-processing for optimizing L1 recovery . . . . . . . . . . . . . . . . 101 4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.5.1 Without pre-processing . . . . . . . . . . . . . . . . . . . . . . 104 4.5.2 With pre-processing . . . . . . . . . . . . . . . . . . . . . . . 106 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chapter 5: Transmitter Cooperation for MIMO Multiple Access Channels 108 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2.1 Core and Stability of Cooperation in Partition Form Games . . . 115 5.2.2 Determining nonemptiness of a core . . . . . . . . . . . . . . . 117 5.2.3 Games with empty cores . . . . . . . . . . . . . . . . . . . . . 119 5.3 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3.1 Signal Model with Coalitions and User Cooperation . . . . . . 120 5.4 TX Cooperation Game with Successive Interference Cancelling Receiver 121 5.4.1 Non-cooperative game between TXs . . . . . . . . . . . . . . . 121 5.4.1.1 Uniqueness of NE . . . . . . . . . . . . . . . . . . . 123 5.4.1.2 Relevance of uniqueness of NE utilities . . . . . . . . 123 5.4.1.3 Evaluating the utility function . . . . . . . . . . . . . 127 5.4.2 TX Cooperation - Properties . . . . . . . . . . . . . . . . . . . 129 5.4.3 TX Cooperation - Stability . . . . . . . . . . . . . . . . . . . . 133 5.4.4 SIC receiver with time sharing between decoding orders . . . . 136 5.5 TX Cooperation Game With Single User Decoding Receiver . . . . . . 140 5.5.1 Non cooperative game between TXs . . . . . . . . . . . . . . . 141 5.5.1.1 CFG Model . . . . . . . . . . . . . . . . . . . . . . 141 5.5.1.2 PFG model . . . . . . . . . . . . . . . . . . . . . . . 142 5.5.2 TX cooperation - Properties and Stability . . . . . . . . . . . . 145 5.5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.5.3.1 Which RX is better for enforcing cooperation ? . . . 148 5.5.3.2 Fairness of rate allocation . . . . . . . . . . . . . . . 149 5.5.3.3 Information and computational requirements . . . . . 149 5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Chapter 6: Broadcast Channel Games and Game based MAC-BC Duality 151 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.2.1 Generalized Nash Equilibrium Problems . . . . . . . . . . . . . 154 6.2.2 Characterization of jointly convex GNEPs . . . . . . . . . . . . 157 6.2.3 Uniqueness of NoEs . . . . . . . . . . . . . . . . . . . . . . . 159 6.3 The MIMO BC as a Generalized Nash Equilibrium Problem . . . . . . 163 6.3.1 Uniqueness of NoEs . . . . . . . . . . . . . . . . . . . . . . . 165 6.4 The Sum Power MIMO Multiple Access Channel as a Generalized Nash Equilibrium Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 vii 6.4.1 Signal Model and the Sum Power MAC game . . . . . . . . . . 168 6.4.2 Uniqueness of NoEs . . . . . . . . . . . . . . . . . . . . . . . 169 6.5 Computation of Normalized Equilibrium . . . . . . . . . . . . . . . . . 172 6.5.1 Pareto-Efficiency First Order Conditions . . . . . . . . . . . . . 172 6.5.2 Rosen’s equilibrium first order conditions . . . . . . . . . . . . 173 6.5.3 Relation between Pareto-optimal and NoE solutions . . . . . . 174 6.5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.5.5 Two user Example . . . . . . . . . . . . . . . . . . . . . . . . 177 6.6 Game Theoretic Duality between MIMO Multiple Access and Broadcast Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.6.1 Relationships between NEP and GNEPs . . . . . . . . . . . . . 179 6.6.2 Relationship between two different GNEPs . . . . . . . . . . . 183 6.6.3 Game based MAC-BC duality . . . . . . . . . . . . . . . . . . 185 6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 References 187 viii List of Tables 2.1 Complexity Analysis with Perfect Channel State Information . . . . . . 41 2.2 Complexity Analysis of RW and MW estimators . . . . . . . . . . . . 41 5.1 Table showing the summary of the nonemptiness of the core of the TX cooperation game for various scenarios. SP = sum power constraint and PP = per-antenna power constraint. . . . . . . . . . . . . . . . . . . . . 112 ix List of Figures 1.1 Path Loss of short range UWA channels as a function of distance and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The total signal attenuation as a function of the signaling frequency . . 6 2.1 Symbol Error Rate assuming perfect knowledge of the UWA channel as a function of SNR and model parameters for the partial FFT combiner, banded MMSE and MCMC based MAP SDSC [60]. . . . . . . . . . . 48 2.2 Symbol Error Rate of the RW and MW estimators as a function of SNR and model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 Output SIR as a function of M for an input SNR of 25dB for a UWA OFDM system using MW estimation and partial FFT combining . . . . 50 2.4 Output SIR as a function of M for an input SNR of 25dB for a UWA OFDM system using RW estimation and partial FFT combining . . . . 51 2.5 Symbol Error Rate assuming perfect knowledge of the DVB wireless channel as a function of SNR and model parameters . . . . . . . . . . . 52 3.1 J m (b) derived from the HCRB for the special one carrier scenario show- ing a single peak for small Doppler spread versus multiple distinct peaks when the Doppler spread is large. Channel (smallD s ) = [0.6288, 0.2293, 0.0435], Channel (largeD s ) = [0.4672, 0.5604, 0.5184], Doppler (small D s ) = [0.0025, 0.0020, 0.0028], Doppler (large D s ) = [ -0.004, 0.0005, 0.0045 ],f m = 27kHz,T = 10.7ms. . . . . . . . . . . . . . . . . . . . 74 3.2 Plot of the Hammersley Chapman Robbins Bound as a function of the re- sampling parameter. The minima in the plot occurs at a value very close to the Doppler shift on the strongest tap. Channel = [1.0856, 0.2435, 0.6786, 0.3498], Doppler = [0.0008, 0.0011, 0.0006, 0.0011], K = 4, Δf = 100Hz,f c = 10kHz andD s = 5×10 −4 . . . . . . . . . . . . . . 75 x 3.3 Cramer Rao Bound (CRB) as a function of the resampling parameter for one channel realization showing the performance loss between optimal matched filtering and resampling with various values ofb. The minimum of the CRB (marked in the figure) is very close to the Doppler on the strongest tap and this value minimizes the information loss. Channel magnitude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ] , Doppler = (- 1)* [0.0010, 0.0012 , 0.0005, 0.0004, 0.0011], K = 128, Δf = 39Hz, f c = 15kHz,D s = 8×10 −4 . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Aggregate Mean-Square Error as a function of the resampling parame- ter for one channel realization showing the channel approximation er- ror. The AMSE minimizing value (marked in the figure) is close to the strongest tap Doppler. Channel magnitude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ], Doppler = [-0.001, -0.0012, -0.0005, -0.0004,-0.0011], Δf = 39Hz,f c = 15kHz,D s = 8×10 −4 . . . . . . . . . . . . . . . . . 76 3.5 Symbol Error Rate for ML detection showing the performance improve- ment obtained using optimal resampling versus strongest tap Doppler resampling,Δf = 62.5Hz,fc = 10kHz,K = 8,D s = 2×10 −3 . . . . . 78 3.6 Estimation Error of Blind and Pilot Aided Resampling as a function of resampling parameter for one channel realization. The minimum of each of the functions is exactly equal to the value predicted by the CRB and AMSE criterion in this case. Channel magnitude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ], Doppler = [-0.0010, -0.0012, -0.0005, -0.0004, -0.0011],Δf = 39Hz,f c = 15kHz,D s = 8×10 −4 . . . . . . . . . . . 88 3.7 Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel for Δf = 39Hz,f c = 15kHz,K = 128, Power Profile = [0−3−5− 7−10]dB,B = 5KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccuracies. . . . . . . . . . . . . . . . . 88 3.8 Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel forΔf = 19.5Hz,f c = 15kHz,K = 128, Power Profile = [0−3−5− 7− 10]dB,B = 2.5KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccuracies. . . . . . . . . . . . . . 89 3.9 Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel forΔf = 9.75Hz,f c = 15kHz,K = 128, Power Profile = [0−3−5− 7−10]dB,B = 1.25KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccuracies. . . . . . . . . . . . . . 89 xi 4.1 Performance comparison of channel parameter estimation when using layers 1 and 2 individually and using both layers jointly for uniformly spaced taps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Performance comparison of channel parameter estimation when using layers 1 and 2 individually and using both layers jointly for randomly spaced taps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3 Performance comparison of channel parameter estimation when using layers 1 and 2 with processing using Elad’s and Julio’s method. . . . . . 107 5.1 Cooperative signaling between base station and femto cell network for joint transmission to UE . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Transmitters cooperate to form coalitions. All transmitters in a coalition fully cooperate with each other . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Example of nonempty r-core (shaded region) and s-core (inside dotted region) for a symmetric scenario. . . . . . . . . . . . . . . . . . . . . . 118 5.4 Decoding in an SIC receiver . . . . . . . . . . . . . . . . . . . . . . . 122 5.5 Figure showing the NE rate points for an SIC single antenna receiver for different decoding orders. . . . . . . . . . . . . . . . . . . . . . . . . . 129 5.6 Plot showing the boundary of the region between empty and nonempty core as a function of SNR and the number of players for a symmetric scenario and a single antenna SIC receiver with a fixed decoding order. . 137 5.7 Ratio of approximated to actual utility as a function of SNR for a 3- user MAC with single antenna TXs, unit channel gain and unit power constraint for a SIC RX. Note that the ratio approaches 1 for very high SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.8 Plot showing the r-core of the TX cooperation game for an SNR of 3dB for a 3-user symmetric MAC with single antenna TXs, unit channel gain and unit power constraint for an SIC RX with equal probability of time sharing between decoding orders using the exact utility function. The r-core is highlighted in red. . . . . . . . . . . . . . . . . . . . . . . . . 140 5.9 Figure showing the NE rate point for an SUD single antenna receiver. The equilibrium rate point is in the interior of the capacity region as the receiver treats interference as noise. . . . . . . . . . . . . . . . . . . . 145 6.1 Plot showing the Pareto optimal frontier for a 2-user ADBC and several normalized equilibrium points . . . . . . . . . . . . . . . . . . . . . . 177 xii Abstract In this thesis, we present a study of several problems related to underwater point to point communications and network formation. We explore techniques to improve the achiev- able data rate on a point to point link using better physical layer techniques and then study sensor cooperation which improves the throughput and reliability in an underwater network. Robust point-to-point communications in underwater networks has become increas- ingly critical in several military and civilian applications related to underwater commu- nications. We present several physical layer signaling and detection techniques tailored to the underwater channel model to improve the reliability of data detection. First, a simplified underwater channel model in which the time scale distortion on each path is assumed to be the same (single scale channel model in contrast to a more general multi scale model). A novel technique, which exploits the nature of OFDM signaling and the time scale distortion, called Partial FFT Demodulation is derived. It is observed that this new technique has some unique interference suppression properties and performs better than traditional equalizers in several scenarios of interest. Next, we consider the multi scale model for the underwater channel and assume that single scale processing is per- formed at the receiver. We then derive optimized front end pre-processing techniques to reduce the interference caused during single scale processing of signals transmitted on a multi-scale channel. We then propose an improvised channel estimation technique using dictionary optimization methods for compressive sensing and show that significant performance gains can be obtained using this technique. xiii In the next part of this thesis, we consider the problem of sensor node cooperation among rational nodes whose objective is to improve their individual data rates. We first consider the problem of transmitter cooperation in a multiple access channel and investi- gate the stability of the grand coalition of transmitters using tools from cooperative game theory and show that the grand coalition in both the asymptotic regimes of high and low SNR. Towards studying the problem of receiver cooperation for a broadcast channel, we propose a game theoretic model for the broadcast channel and then derive a game theo- retic duality between the multiple access and the broadcast channel and show that how the equilibria of the broadcast channel are related to the multiple access channel and vice versa. xiv Chapter 1 Introduction The worlds’ oceans and inland waters are of vital importance to resource retrieval and transportation, climate regulation, agriculture and nutrient production, yet they are sig- nificantly under-explored. The design and development of underwater acoustic digital communication systems have received a significant increase in attention over the last few years driven by deep ocean oil and gas exploration, oceanographic data collection, offshore exploration, commercial fishing, coastal and harbor surveillance and assisted navigation. In addition, systems are being deployed for command and control links to submarines, search and rescue operations and detection of underwater events such as tsunamis and underwater earthquakes. Multiple coordinated, unmanned and autonomous underwater vehicles, equipped with sensors, will enable the exploration of natural under- sea resources and gathering of scientific data in collaborative monitoring missions. To meet the communication demands of these applications, there is a need to enable high speed and reliable communications among underwater devices. Underwater sensor nodes and autonomous vehicles must possess self-configuration capabilities, i.e., they must be able to coordinate their operation by exchanging configuration, location and movement information and be able to relay monitored data to an offshore station. The traditional approach for operating underwater wireless networks is to deploy sensors 1 recording data and then recover the instruments to collect the stored information. This approach has several disadvantages: • No real time monitoring: The data is recorded at each node and cannot be accessed until the instruments are recovered. There might be a significant gap between data recording and retrieval and the value of collected data reduces over time as it gets outdated. In applications such as surveillance and monitoring, getting access to real time data might be critical. • No on-line system reconfiguration: Interaction between surface control systems and the monitoring instruments is not possible, which impedes any adaptive tuning and reconfiguration after deployment. • No failure detection: If failures or misconfigurations occur, it may not be possible to detect and correct them until the instruments are recovered which can lead to failed deployments. • Limited storage capacity: The amount of data that can be recorded during the monitoring mission by each sensor is limited by its storage capacity. • No network resource optimization: An online network can optimize resources to maximize the lifetime of every node in the network. Hence, there is a need to deploy underwater networks that will enable real-time moni- toring of the ocean, remote configuration and interaction with onshore human operators. This can be obtained by connecting underwater instruments by means of reliable wire- less links based on acoustic communications. In this report, we investigate the design of systems for high speed underwater acoustic communications. 2 1.1 Acoustic Channels and Propagation Models Underwater acoustic (UWA) communications are mainly influenced by path loss, noise, multi-path, Doppler spread and large propagation delays. The acute path loss severely constraints the bandwidth that is available for signaling and the Doppler spreading caus- ing significant channel variation increasing the complexity of data detection techniques. We first present a detailed characterization of the parameters of the UWA channel rele- vant for horizontal communication in shallow water. 1.1.1 Path Loss and Attenuation One of the key properties that distinguish wireless electromagnetic and acoustic signaling is the fact that the path loss of an acoustic signal depends on the signal frequency. This dependence is a consequence of absorption i.e. the transfer of acoustic energy into heat. In addition to the absorption loss, the signal experiences a spreading loss which increases with distance. The overall path loss can be expressed as A(l,f) =A 0 (l/l r ) k a(f) l−lr , (1.1) wheref is the signal frequency andl is the transmission distance, taken in reference to some l r . The path loss exponent k models the spreading loss and its usual values are between 1 and 2 for cylindrical and spherical spreading respectively. The absorption coefficient a(f) is an increasing function of frequency, and can be represented by an empirical formula given by Thorps’ equation 10loga(f) = 0.11 f 2 1+f 2 +44 f 2 4100+f 2 +2.75×10 −4 f 2 +0.003. (1.2) 3 c Figure 1.1: Path Loss of short range UWA channels as a function of distance and fre- quency This formula is generally valid for frequencies above a few hundred Hz. For very low frequencies, a modified empirical formula may be used: 10loga(f) = 0.11 f 2 1+f 2 +0.002+0.011f 2 . (1.3) Fig. 1.1 shows the attenuation (in dB) as a function of distance and frequency of signaling. Observe that the absorption coefficient increases rapidly with frequency, and is the major factor that limits the maximum usable frequency for an acoustic link for a given distance. The path loss describes the attenuation on a single, unobstructed propagation path. If a tone of frequencyf and power P is transmitted over this path, then the received signal power isP/A(l,f). 4 1.1.2 Noise The ambient noise in the ocean can be modeled using four primary sources: turbulence, shipping, waves and thermal noise. Most of the ambient noise sources can be described by gaussian statistics and a continuous power spectral density (PSD). The following em- perical formulae give the PSD of the four noise components. 10logN t (f) = 17−30logf 10logN s (f) = 40+20(s−0.5)+26logf−60log(f +0.03) 10logN w (f) = 50+7.5w 1/2 +20logf−40log(f +0.4) 10logN th (f) = −15+20logf. (1.4) where, N t (f) is the noise due to the turbulence and influences only the very low fre- quency region, f ≤ 10Hz. Noise caused by distant shipping, denoted by N s (f) is dominant in the frequency region 10 Hz - 100 Hz, and it is modeled through the ship- ping activity factor s, whose value ranges between 0 and 1 for low and high activity, respectively. Surface motion, caused by wind-driven waves (w is the windspeed in m/s) is the majority of the acoustic systems). Finally, thermal noise becomes domi- nant forf > 100 KHz. The overall PSD of the ambient noise is then given byN(f) = N t (f)+N s (f)+N w (f)+N th (f). If we define a narrow band of frequencies of widthdf around some frequencyf, the signal-to-noise ratio (SNR) in this band can be expressed as SNR(l,f) = S l (f) A(l,f)N(f) , (1.5) whereS l (f) is the PSD of the transmitted signal, whose power may be adjusted accord- ing to the distance. For any given distance, the narrow band SNR is thus a function of frequency, as shown in Fig. 1.2. From this figure, it is clear that the acoustic bandwidth 5 c Figure 1.2: The total signal attenuation as a function of the signaling frequency depends on the transmission distance. The bandwidth is severely limited at longer dis- tances: at 100km, only about a kHz is available. At shorter distances, the bandwidth increases, but it will ultimately be limited, thus strongly indicating the need for band- width efficient modulation schemes. Another important observation to be made is that the acoustic bandwidth is centered at low frequencies. In fact, the bandwidth is often of on the order of its center fre- quency f c which makes an acoustic communication system inherently wideband. Thus fact in turn bears significant implications on the design of signal processing methods, as it prevents one from making the typically used narrowband assumption on which radio communication schemes are typically designed. 1.1.3 Multipath and Long Delay Spreads Multipath formation in the ocean is governed by two effects: sound reflection at the surface, bottom and any objects, and sound refraction in the water. The latter is a con- sequence of sound speed variation with depth, which is mostly evident in deep water channels. The impulse response of an acoustic channel is influenced by the geometry of 6 the channel and its reflection properties, which determine the number of significant prop- agation paths, their relative strengths and delays. Strictly speaking, there are infinitely many signal echoes, but those that have undergone multiple reflections and lost much of the energy can be discarded, leaving only a finite number of significant paths. To put a channel model in perspective, let us denote byl p the length of thep th path, with p = 0 corresponding to the first arrival. In shallow water, which is of interest to our work and where the speed of sound can taken as a constant c, path lengths can be calculated using plain geometry, and path delays can be obtained as l p /c. The relative path delays are then given asτ p = l p /c−l 0 /c. However, as done in radio channels, it is not accurate to express the channel impulse response as h(t) = X p h p δ(t−τ p ), (1.6) as the path gain is a function of frequency and distance traveled. For a broadband signal, the frequency response of thep th path is H p (f) = Γ p p A(l p ,f) . (1.7) Hence, each path of the channel acts as a low pass filter, introducing its own dispersion. Path dispersion is much less than the combined multipath spreads of all paths, but must be accounted for nonetheless in an accurate model. The overall channel frequency response in the frequency domain is then H(f) = X p H p (f)e −j2πfτp , (1.8) and the corresponding impulse response can be expressed ash(t) = P p h p (t−τ p ) where h p (t) is the inverse Fourier transform of H p (f). The total multipath spread for typical 7 shallow water channels is of the range of a few milliseconds. 1.1.4 Doppler distortion Relative motion between the transmitter and the receiver or a moving environment causes Doppler distortion, resulting in a time-varying channel impulse response. For wideband underwater acoustic channels, the Doppler effect causes a time scaling of the received signal. The magnitude of the Doppler effect is proportional to the ratio a = v/c of the relative transmitter/receiver velocity to the speed of sound. Because the speed of sound is very low as compared to the speed of electromagnetic waves, motion-induced Doppler distortion of an acoustic signal can be extreme. Autonomous underwater vehi- cles (AUVs) move at speeds that are on the order of few m/s, but even without inten- tional motion, underwater sensors are subject to drifting with waves, currents and tides, which may occur at comparable velocities. In other words, there is always some motion present in the system and a communication system has to be designed taking this fact into account. The major implication of the motion-induced distortion is on the design of receivers for compensating Doppler distortion. As the transmitter and the receiver move relative to each other, the distance between them changes and so does the signal delay. As a consequence, the leading edge of the transmitted signal may experience one delay, while the trailing edge will experience a different delay. Consider a signals(t) being sent over the underwater channel between a transmitter and receiver in relative motion. Assuming a constant velocityv, and ignoring the path gain, the received signal is a time scaled and delayed version of the transmitted signal and can be expressed as r(t) =s(a(t−τ)), (1.9) 8 wherea =v/c+1 is the time scale coefficient. This illustrates that the signal is distorted in two (equivalent) ways: first the signal is time scaled by a factora, so that a transmitted signal of durationT is observed at the receiver as having the durationT/a. Equivalently, its bandwidth B is scaled as aB. If the transmitted signal is centered at a frequency f c Hz, then with respect to the center frequency, we observe that the received signal is experiences an offset (a− 1)f c Hz. The first type of distortion accounts for motion- induced Doppler spreading, while the second accounts for Doppler shifting. For a wireless radio system, if the relative velocity between the transmitter and the receiver is 100mph, then the value of the time scale parameter is given as a = 1+ ≈ 1.5×10 −7 . This value is very small and can be neglected for most narrowband wireless systems. However for an autonomous underwater vehicle (AUV) moving at 1.5 m/sec, the time scale parameter isa = 1+10 −3 , and cannot be ignored in the design of receiver algorithms. The effect of time scale is particularly severe for multi-carrier communica- tion systems like Orthogonal Frequency Division Multiplexing (OFDM) for which the time scale distortion results in a different frequency shift for each subcarrier. This effect of Doppler distortion is exacerbated when the time scale parameters of each path of the channel impulse response is different causing a given subcarrier to experience a different frequency shift on each path. This multi-scale phenomenon, the design of pre-processing, channel estimation and equalization algorithms will be explored in this thesis. 1.1.5 Network effects Due to the small bandwidth available for communication in an underwater network, the achievable data rates on a point to point link are usually of the order of several thousand bits per second. In general, there would be a necessity for several parallel transmissions in an underwater network resulting in interference and severely degrading the achievable 9 data rates on such links. One possible approach to overcome interference is to operate the network in a time-shared manner. However, we know from our understanding of wireless networks that time-sharing is very inefficient in utilizing the available capacity of a net- work. A second strategy to improve data rates in the presence of interference is to allow nodes to cooperate to transmit their data. Such cooperation improves the rates achievable on each link and well as its robustness by reducing the effective amount of interference seen by the network. However, determining which nodes in an underwater network could easily cooperate and how the benefits of this cooperation should be divided among all the cooperating nodes is unknown. In fact, determining node cooperation is an wide open problem in over-the-air wireless communications as well. Exploiting tools from game theory, this thesis explores the problem of understanding cooperation in general wireless networks by considering networks such as the multiple access and broadcast channel and determining the stability of feasibility and stability of cooperation in several scenarios of interest. 1.2 Outline of this Document This thesis is organized as follows. We first consider the problem of communication over a UWA point to point link. Chapter 2 considers the problem of data detection for UWA OFDM signaling under the simplifying assumption that all the channel paths have the same time scale. We propose a new detection schemes called partial FFT demodulation in which the received signal is first partitioned into several intervals with local processing in each interval. Next the information from these smaller intervals is then combined in an appropriate fashion to eliminate time scale distortion for such OFDM systems. Several optimality properties are demonstrated for these proposed techniques and an asymtotic analysis characterizes the optimal number of points into which the interval 10 must be partitioned into. Numerical results demonstrating the efficient performance of this technique are then demonstrated for both UWA channels. Finally, the performance of the proposed method is demonstrated for highly time-varying wireless channels in which the relative velocity between the transmitter and receiver is of the order of hundreds of kilometers per second. In Chapter 3, we consider a UWA channel in which each path has a different time scale also called the multi-scale multi-lag (MSML) channel. Resampling was proposed as the front-end processing scheme for compensating for the Doppler distortion (time scaling) and has been shown to be optimal when all the channel paths have the same time scale parameter. However, resampling is sub-optimal front end processing for a MSML channel. We then discuss the problem of optimizing the resampling parameter to minimize the loss from resampling using the multi-parametric Hammersley Chapman Robbins Bound (HCRB) to compute a lower bound on the error variance for detecting the transmitted sequence. We then analyze the HCRB in some special cases to gain insight into the behavior of the optimal resampling parameter. Our findings suggest that the optimal resampling parameter is nearly equal to the channel gain weighted combination of the time scaling factors on each path when the scale spread is small and is close to the scaling factor on the strongest tap for the scale parameters for large scale spreads. Finally, estimators for evaluating the optimal resampling parameter are derived and their performance analyzed in a range of scenarios. In Chapter 4, we consider the problem of channel estimation for the MSML channel using a recently proposed parametric representation for signaling over the MSML chan- nel. Using multiple co-located frequency bands for layered data transmission, as pro- posed by the parametric representation, allows for multiple observations of the channel impulse response each at a different sampling rate. Directly combining the measurements from the different frequency bands degrades the channel estimation accuracy, due to the 11 relationship between the measurements and the use of sparse recovery techniques with a mismatched basis. By pre-processing the data to optimize the sensing matrix for this problem, we show that the channel estimation error can be reduced significantly when compared to recovery techniques without pre-processing. Next, in Chapter 5 we consider the problem of determining cooperation in a wireless network and in particular underwater networks. We address the stability of the grand coalition of transmitters signaling over a multiple access channel using the framework of cooperative game theory. Single user decoding and successive interference cancelling strategies are examined at the receiver. In the absence of coordination costs, we show that the grand coalition is sum-rate optimal for both strategies. Transmitter cooperation is stable, if and only if the core of the game (the set of all divisions of grand coalition util- ity such that no coalition deviates) is nonempty. Determining the stability of cooperation is a co-NP-complete problem in general. For a single user decoding receiver, transmitter cooperation is shown to be stable at both high and low SNRs, while for an interference cancelling receiver with a fixed decoding order, cooperation is stable only at low SNRs and unstable at high SNR. When time sharing is allowed between decoding orders, it is shown using an approximate lower bound to the utility function that TX cooperation is also stable at high SNRs. We thus demonstrate that ideal zero cost transmitter coopera- tion over a MAC is stable and improves achievable rates for each individual user. Finally, in Chapter 6, we lay the foundation for studying the problem of determining receiver cooperation in a wireless network. To this end, we investigate the behavior of rational receivers over a Gaussian broadcast channel (BC) and a multiple access channel (MAC) with sum power constraints (sum power MAC) using non-cooperative theory. By modeling the scenarios as a generalized Nash equilibrium problem (GNEP) with jointly convex and coupled constraints, we characterizes the existence and uniqueness of equilibrium points for both the Gaussian BC and the sum power MAC. We derive a 12 relationship between Pareto-optimal boundary points and equilibrium points is derived and show that every point on the boundary is a generalized Nash equilibrium. A game- theoretic duality between Gaussian MAC and Gaussian BCs is then derived by showing the achievable equilibria for the MAC can be expressed as achievable equilibria for the BC and vice versa. Using duality, many results known for the MAC can be extended to the BC as well. 13 Chapter 2 Partial FFT Demodulation 2.1 Introduction Orthogonal Frequency Division Multiplexing (OFDM) is now the primary signaling scheme for several wireless communication systems such as Long Term Evolution (LTE), WiMAX, Digital Video Broadcasting (DVB) etc [2, 17, 88] and is also under consid- eration for underwater acoustic (UWA) communications [83, 100, 48, 30]. Interest in OFDM stems from the fact that it decomposes a static frequency selective channel into a number of flat channels, enabling low complexity, single-tap equalization and symbol- by-symbol detection at the receiver. A considerable amount of research on OFDM re- ceivers for highly time-varying scenarios has been conducted. Such channels typically result when either there is high mobility or in wideband signaling such as UWA com- munications where the transmission bandwidth is large relative to the carrier frequency. The Doppler in such highly mobile environments destroys the orthogonality of subcar- riers, that results in inter-carrier interference (ICI), and significantly increases detection complexity required to mitigate the induced interference. The problem of low complexity detection of OFDM signals over time-varying chan- nels has been extensively studied in the literature. Conventional minimum mean-squared 14 error (MMSE) based block equalizers have a complexity that grows cubically in block length and hence challenge practical implementation [73]. By exploiting the banded na- ture of the frequency domain channel matrix, several detection techniques whose com- plexity grows linearly in the number of sub-carriers have been designed. The perfor- mance of these receivers has been further enhanced through iterative detection and inter- ference cancelation algorithms (see [73, 22, 68, 72, 4, 30, 60] and references therein for a detailed overview of time-varying channel equalization for OFDM). In general, lower complexity channel estimation and data detection algorithms are the key to implementing next generation OFDM systems with a large number of subcarriers. In addition to the terrestrial wireless systems noted above, UWA communications also experience highly time-varying channels. The very low speed of sound in water (1500m/s) coupled with mobility results in Doppler distortion, i.e., time scaling of the transmitted signal [48, 72, 12, 30]. The Doppler scaling in UWA communications is similar to that of OFDM radar systems tracking a single target [75]. The time scaling of the signal causes different subcarriers to be shifted by slightly different frequencies, resulting in significant ICI. Distortion compensation and equalization of OFDM signals for UWA channels have been extensively investigated in [86, 48, 12, 78, 99, 30]. In this sequel, we revisit the problem of data detection over highly time-varying chan- nels and propose a new demodulation technique called partial FFT demodulation. The received OFDM symbol is first partitioned into several intervals using non-overlapping rectangular windows and a discrete Fourier transform (DFT) 1 is performed on each win- dowed segment of the received signal. The segments are then weighted and combined. If no weighting is applied, i.e., if the partial FFT outputs for each segment are directly added, the result is equivalent to performing conventional, full FFT demodulation, which 1 The DFT is efficiently implemented using the fast Fourier transform (FFT) and the DFT operation will be referred to as the FFT in the rest of the sequel. 15 results in significant ICI due to uncompensated Doppler distortion. In contrast, by ju- dicious weighted combining of the partial FFT outputs, we show that the ICI can be significantly reduced, improving the detection performance at a complexity that is com- parable to that of typical ICI equalization. The key to improved performance is that the partial FFT outputs contain less mixing of contributions from different symbols and al- low for more effective compensation of ICI versus compensating after matched filtering. It should be noted that while we focus on the interference mitigation using partial FFT processing, further equalization is possible as well. However, for many practical ap- plications, the benefits of improved front-end processing are sufficiently large that they obviate the need for subsequent equalization. In [105], a windowing technique similar to partial FFT technique has been proposed which post-processes and equalizes the received signal by approximating the frequency domain spectrum of a rectangular window. In comparison, our work approximates the time-variation of the channel using a time-domain windowed version of the received sig- nal and develops equalization techniques well suited for UWA communications. While [105] does not analyze the proposed scheme, this sequel presents a detailed theoretical analysis for the proposed equalizers. In [55], an orthogonal chirp type signal basis is used to approximate the ideal signal basis functions. The chirp type functions corre- spond to the signal basis of the fractional Fourier transform (FRFT) in comparison to the sinusoidal basis of the regular Fourier transform. In effect, a multicarrier system similar to OFDM is designed using the FRFT instead of a conventional FFT. A generalization of [55], using the affine Fourier transform as an alternative to the FRFT was proposed in [20] and was shown to have more desirable properties than FRFT based multicarrier systems. Several researchers (see [51] and references therein) have also explored the design of orthogonal pulse shapes using a short-time Fourier (STF) basis for multicar- rier communications. We distinguish our work from [55, 51, 20] by pointing out that 16 we neither design new pulse shapes nor perform alternative transforms to the FFT at the receiver. The proposed method in this sequel is a technique which exploits the structure in the time-variation and pre-processes the input signal by exploiting FFT processing to reduce time-variation. We focus primarily on two scenarios in this sequel: (1) UWA channels with time scal- ing distortion and (2) fast varying terrestrial radio channels experienced by users moving at a very high speed. We first illustrate the partial FFT combining technique and then derive the optimal weighting coefficients for combining the outputs for a generic model of the time-varying channel. The optimal combiner weights are shown to depend only on the channel frequency response at the midpoint of each interval. When the knowledge of the channel impulse response is not available at the receiver, we present a recursive weight estimation algorithm to compute the combiner weights. In UWA channels where the effect of Doppler distortion (time scaling) can be parameterized, the signal structure can be exploited to derive a model-based weight estimation algorithm for the combiner. We present an approximate analysis of the recursive and the model based estimators to determine the optimal number of partial FFTs to be employed in practice. Numerical simulations are presented to illustrate the performance of the proposed algorithms. We show that for signaling over the UWA channel with time scale distortion, the partial FFT method significantly outperforms banded MMSE equalizers and gives comparable performance to algorithms such as MCMC based MAP-SDSC in [60] at a much lower complexity. For scenarios with no channel knowledge or with just knowledge of the distortion process, we show partial FFT processing performs significantly better than algorithms utilizing similar knowledge of the channel at a lower complexity. Finally, for the DVB scenario we show that the proposed method outperforms the conventional banded equalizers in most regimes of interest. The chapter is organized as follows. Section 2.2 presents the OFDM signal model and 17 illustrates the concept of partial FFT demodulation. The optimal combiner weights, and the recursive weight estimation algorithm for scenarios with no prior channel knowledge are derived in Section 2.3, while Section 2.4 illustrates the model-based weight estima- tor for the UWA channel with Doppler distortion. Section 2.5 presents an analysis of the recursive and model-based estimation algorithms, and Section 2.6 shows numerical re- sults that demonstrate the effectiveness of this method in various scenarios. This chapter concludes with remarks in Section 2.7. 2.2 Signal Model and Partial FFT Demodulation 2.2.1 Signal Model Let us consider an OFDM system withK subcarriers. The vectord = [d 1 ,d 2 ,··· ,d K ] of information symbols is modulated onto the K OFDM subcarriers. The transmitted symbols are assumed to be drawn from a finite constellation, such as 4-PSK, which we consider for illustration in this sequel. LetT ,T g andΔf = 1 T denote the duration of the OFDM symbol, duration of the cyclic prefix, and subcarrier spacing respectively. Thek th subcarrier frequency isf k = f 0 +(k−1)Δf, and the total OFDM signaling bandwidth isB =KΔf. The transmitted OFDM signal in passband can be expressed as s(t) = r 1 T Re ( K X l=1 d l e j2πf l t ) ,t∈ [−T g ,T]. (2.1) The passband time-varying channel is modeled as h(τ,t) = P X p=1 h p (t)δ(τ−τ p (t)), (2.2) 18 where h p (t) and τ p (t) are the time-varying gain and delay of the p th path, respectively, andP is the total number of arriving paths. The received signal in passband can then be expressed as r(t) = r 1 T Re ( P X p=1 K X l=1 h p (t)d l e j2πf l (t−τp(t)) ) +z(t), t∈ [−T g ,T], (2.3) wherez(t) is additive white Gaussian noise (AWGN). 2.2.2 Conventional OFDM Demodulation In a conventional receiver, after time and frequency synchronization, the cyclic prefix is discarded and the received signal is transformed into the frequency domain: 2 r k = 1 √ T Z T 0 r(t)e −j2πf k t dt = 1 T K X l=1 d l Z T 0 " P X p=1 h p (t)e −j2πf l τp(t) # | {z } H l (t) e j2π(f l −f k )t dt+z k . (2.4) Here,H l (t) represents the time-varying frequency response of thel th subcarrier, andz k is the AWGN with zero mean and varianceN 0 . From (2.4), we clearly observe how the time-varying channel causes ICI. When H l (t) is time-invariant, the received signal on each subcarrier reduces to r k = d k H k +z k , enabling one-tap equalization and making symbol-by-symbol detection optimal. The time variation in H l (t) destroys this orthog- onality, and necessitates ICI equalization to compensate for the time-varying channel [22, 73]. 2 We use the continuous time Fourier transform for simplicity. In practice as well as in our numerical simulations, FFT is used. 19 2.2.3 Partial FFT Demodulation In partial FFT demodulation, the useful OFDM symbol duration[0,T] is divided intoM non-overlapping intervals (equivalent to multiplying the signal with several rectangular windows) and a Fourier transform is performed on each windowed segment of the signal. The output of the Fourier transform for thek th subcarrier and them th windowed block, henceforth called the partial FFT output, can be expressed as y k (m) = r 1 T Z mT M (m−1)T M r(t)e −j2πf k t dt = 1 T X l d l Z mT M (m−1)T M H l (t)e j2π(f l −f k )t dt+z k (m), m = 1,2,··· ,M. (2.5) In general, it is assumed that the channel parameters vary slowly in comparison to the OFDM symbol duration. We exploit this assumption by approximating the time-varying frequency response in each interval T/M by the midpoint value of the function. The received signal (2.5) can now be simplified as y k (m) ≈ K X l=1 d l H l (m)I l−k (m)+z k (m), H l (m) = P X p=1 h p (m)e −j2πf l τp(m) . (2.6) where H l (m), h p (m) and τ p (m) are the midpoint values of the frequency response, the channel impulse response and path delays, respectively, on the interval[ (m−1)T M , mT M ]. The functionI i (m) captures the effect of partial integration over them th interval, and can be evaluated as I i (m) = 1 T Z mT M (m−1)T M e j2πiΔft dt = 1 M e j2πi(2m−1)/(2M) sinc πi M , i =−(K−1),...(K−1), (2.7) 20 where sinc(x) . = sin(x) x . This function has the property that I 0 (m) = 1 M , m = 1,2,...,M and M X m=1 I i (m) = 0∀i6= 0. We will exploit this property later when we design the combiner. The noise is characterized by the covariances E z k 1 (m 1 )z ∗ k 2 (m 2 ) = ( N 0 M e − jπ(k 1 −k 2 )(2m−1) M sinc π(k 1 −k 2 ) 2M ifm 1 =m 2 =m 0 m 1 6=m 2 . (2.8) The noise components are thus Gaussian and uncorrelated and hence independent across partial FFT outputs for a given subcarrier, but are correlated for a fixedm across subcarri- ers. For thek th subcarrier, let us now definev k = [I k (1),I k (2),··· ,I k (M)] T as the vec- tor containing the partial interval integration coefficients,H k = diag[H k (1),...,H k (M)] as the diagonal matrix containing the channel frequency response in each interval, and y k = [y k (1),y k (2),··· ,y k (M)] T as the vector of partial FFT outputs. 3 Expressing (2.6) in vector form, we have y k = K X l=1 d l H l v l−k +z k . (2.9) Note thatv 0 = 1 M [1,1,··· ,1] T andv H k v 0 = 0∀k 6= 0 which compactly expresses the fact that the OFDM subcarriers are orthogonal to each other. 3 Notation: a T denotes the transpose ofa, anda H denotes its conjugate transpose. 21 2.2.4 Combining Partial FFT Outputs Let us definew k = [w k (1),w k (2),··· ,w k (M)] T as the vector of the combiner weights for thek th subcarrier. The combining then yields x k =w H k y k = K X l=1 d l w H k H l v l−k +w H k z k . (2.10) An appropriate choice of the vectorsw k allows one to compensate for the time-variation of the channel to some degree, thus reducing the ICI but not completely eliminating it. Even if one were to implement optimal front-end filtering, (i.e. a matched filter for each subcarrier), the resulting output would contain ICI. Partial FFT demodulation followed by optimized combining mimics the operation of optimal front-end filtering and cannot completely eliminate the ICI. However, it can significantly reduce its effect possibly even eliminating the need for post-FFT ICI equalization. Clearly, partial FFT demodulation can also be combined with ICI equalization; however, our goal in this sequel is to inves- tigate its performance when used as a stand-alone alternative to equalization of full FFT outputs. 2.2.5 Partial FFT combining: A window design interpretation The proposed technique can be alternatively interpreted by recasting the problem as de- signing a receiver window for each subcarrier. For ease of illustration, we briefly consider a discrete time equivalent model of the received signal. Let us suppose that r(k) is the sampled version of the signalr(t) andr = [r(1),r(2),...,r(K)]. The partial FFT outputs can be expressed in terms ofr as y k (m) =e T k Fdiag{β m }r, (2.11) 22 wheree k is a unit vector whose k th entry is 1, β m is the rectangular window used for computing the m th partial FFT output, and F is the K × K DFT matrix. Now, the combined partial FFT outputs in (2.10) can be written as x k = w H k y k = M X m=1 w ∗ k (m)y k (m) = e T k F " M X m=1 w ∗ k (m)diag{β m } # | {z } ¯ w k r =e T k F[¯ w k ]r. (2.12) Clearly, each subcarrier has its own window, in contrast to the approach in [73, 68], where one window is used for all the subcarriers. In addition, the window ¯ w k is step-wise, while the windows in [73, 68] are in general smooth. Thus, the partial FFT technique can be considered as a generalization of the windowing method adopted in OFDM systems. Though [105] also proposes a similar windowing technique, it significantly differs from this work in the way the combiner coefficientsw k (m) are determined. 2.3 Computing the Combiner Weights 2.3.1 Perfect Channel Knowledge Scenario To derive the MMSE minimizing combiner weightsw opt k , we consider the partial FFT outputs for the k th subcarrier to determine the symbol transmitted on this subcarrier. We emphasize that the only the k th partial FFT outputs are considered for computing w opt k . Assuming that all the channel parameters are deterministic and known, the MMSE combiner weights are the solution to the optimization problem: w opt k = min w k E n d k −w H k y k 2 o ⇒w opt k = (E y k y H k ) −1 E{y k d ∗ k }. (2.13) 23 Substituting fory k from (2.9) and evaluating the cross-correlation and autocorrelation matrices, we obtain R y k d k =E{y k d ∗ k } =H k v 0 , R y k =E y k y H k = K X l=1 H l v l−k v H l−k H H l + N 0 M I M . (2.14) The MMSE optimal combiner weights for the k th subcarrier can then be evaluated as w opt k =R −1 y k H k v 0 . Example - Time Invariant Channel To gain some intuition, let us evaluate the combiner weights for the simple scenario of a linear time-invariant channel. The subcarrier frequency response coefficients H l (m) given by (2.6) are then equal in all theM intervals. The elements of the auto-correlation matrix of the outputs of thek th subcarrier are then R y k (m 1 ,m 2 ) = 1 M 2 K X l=1 |H l | 2 e j2π(l−k) m 1 −m 2 M sinc 2 π(l−k) M + N 0 M δ(m 1 −m 2 ). (2.15) Defining⊕ as addition modulo M, we see thatR y k is a circulant matrix asR y k (m 1 ⊕ M,m 2 ⊕ M) = R y k (m 1 ,m 2 ). Exploiting the fact that any circulant matrix can be diagonalized by a discrete Fourier transform (DFT) matrix, we get R y k = FΛ k F H , whereF is theM×M unitary DFT matrix andΛ k is a diagonal matrix of the eigen-values of R y k . The optimal combiner coefficients now reduce to w opt k = H k M FΛ −1 k F H v 0 = H k λ k (1) v 0 . The closed form solution is obtained by noting that the first column vector of the DFT matrix isv 0 andλ k (1) is the first diagonal element ofΛ k . The eigenvalueλ k (1), corresponding to the first eigenvectorv 0 , can be evaluated from the first row ofR y k as 24 λ k (1) = P M i=1 R y k (1,i) = 1 M (|H k | 2 +N 0 ) and the MMSE optimal combiner weights are w opt k = H k |H k | 2 +N 0 Mv 0 = H k |H k | 2 +N 0 [1,1,...,1] T . (2.16) In other words, optimal processing for a time-invariant channel amounts to adding all the partial FFT outputs for a given subcarrier and then performing single-tap equalization – as expected, this is identical to conventional OFDM processing. 2.3.2 No Channel Knowledge Scenario The MMSE estimate for the combiner weights in (2.13) relies on the knowledge of the channel frequency response at the midpoint of each of the M partial intervals. Since estimating the time-varying channel impulse response may not always be practical, we propose a recursive weight (RW) estimation algorithm that does not utilize any knowl- edge of the time-varying channel. We derive the estimator based on the assumption that the frequency response of the channel changes slowly across subcarriers, and thus the combiner weights w opt k also change slowly as a function of the subcarrier index. This assumption is valid for OFDM systems whose coherence bandwidth is much greater than the subcarrier spacing. For example, this assumption has been exploited in [86] to design low complexity detection algorithms for underwater MIMO-OFDM systems. The RW estimation algorithm is based on the idea that partial FFT combining elimi- nates most of the ICI and that the resulting output after combining fork th subcarrier can be modeled as x k ≈H k d k +w k , (2.17) whereH k is the channel frequency response andw k contains the noise and residual ICI. 25 Algorithm 1 Recursive Weight Estimation 1: INITIALIZATION: 2: Weights:w 1 = [1,1,··· ,1] T M×1 3: Covariance matrix: δ << 1,G 1 =δ −1 I M×M 4: Control parameters: λ = 0.99,α = 0.2 5: Channel estimates: ˆ H 0 = 1 6: for k = 1 toK do 7: COMPUTE SIGNALS: 8: y k = [y 1 (k),··· ,y M (k)] T 9: x k =w H k y k 10: ˆ d k =x k / ˆ H k−1 11: DATA DETECTION/PILOTS: 12: if k∈{k 1 ,k 2 ,··· ,k P } then 13: ˜ d k =d k 14: else 15: ˜ d k = dec( ˆ d k ), dec(·) maps the point to the nearest constellation symbol. 16: end if 17: UPDATE THE CHANNEL: 18: ˆ H k =α ˆ H k−1 +(1−α)x k / ˜ d k 19: UPDATE THE COMBINER (RLS ALGORITHM): 20: e k = ˆ H k ˜ d k −x k 21: g k = G k y k λ+y H k G k y k 22: w k+1 =w k +e ∗ k g k 23: G k+1 = 1 λ (G k −g k y H k G k ). 24: end for 25: RE-COMPUTE FREQUENCY RESPONSE: 26: x p = [x k 1 ,x k 2 ,··· ,x k P ] T 27: S p = diag[d k 1 ,d k 2 ,··· ,d k P ] 28: ˆ h =F † (S p ) −1 x p , F m,l =e j2πkml/K . 29: ˇ H k = P L l=1 ˆ h l e −j2πkl/K 30: DATA DETECTION: 31: ˜ d k = dec(x k / ˇ H k ) 26 Using the estimated values ofw H k , the partial FFT outputsy H k are combined to yield x k = w H k y k . Using an estimate of the frequency response on the previous subcarrier, ˆ H k−1 , the received signal is equalized to form a preliminary estimate of the data symbol, and make the corresponding decision: ˆ d k = x k ˆ H k−1 ; ˜ d k = dec( ˆ d k ). (2.18) Now, using this coarse estimate of the transmitted symbols, the channel frequency re- sponse for the subcarrier of interest is updated as ˆ H k =α ˆ H k−1 +(1−α) x k ˜ d k . (2.19) Assuming correct symbol decisions, or using pilots when available, the error at the com- biner output is evaluated as e k = ˆ H k ˜ d k −x k . (2.20) This error is used to drive an adaptive algorithm for the combiner weights, for example the recursive least squares (RLS) algorithm: w k+1 =w k + RLS[y k ,e k ]. (2.21) The detection method is completely summarized in Algorithm 1. The parameter α ∈ [0,1] controls the update of the channel frequency response and is dependent on the coherence bandwidth of the channel. For channels with small delay spreads, the coher- ence bandwidth is larger and the frequency response changes slowly across subcarriers. However for channels with large delay spreads, the coherence bandwidth is smaller and the frequency gains on subcarriers change rapidly from one subcarrier to another and a 27 smaller value ofα is suited for such scenarios. The parameterλ is the forgetting factor of the RLS algorithm and is chosen to balance between channel dynamics and estimation noise and is typically very close to one. The algorithm is initialized by choosing the channel frequency response to be used for the estimating data on the first subcarrier to be H 0 = 1. The initializing weight vectors are chosen to be [1,1...,1] T as equal gain combining of the partial FFT outputs is optimal when the channel is time-invariant. The parameter δ which determines the initial covariance matrix G 1 is chosen so that the all the variables are independent and have a large variance (i.e.,δ is very small) to simulate the fact that no prior information is available about the statistics of the partial FFT outputs. The first 2M subcarriers are assigned to be pilots to operate the algorithm in pilot-assisted mode which allows better estimation of the RLS errore k to train the system parameters. The algorithm then switches to decision directed model wherein the estimated data symbols are used for computing the error and further updating the weight vector. Note that the inclusion of pilot symbols in the first few subcarriers reduces the rate of data transmission. It is shown in Section 2.5 and Section 2.6 that a small value ofM is optimal and as2M << K, the rate reduction is negligible. Insertion of pilots beyond those necessary for initial RLS convergence is required for channels that exhibit spectral nulls. On such channels, a subcarrier experiencing a deep fade may cause a symbol error, which will then propagate across subcarriers unless corrected. To overcome the loss of detection performance caused by error propagation, pilot symbols are inserted periodically throughout the OFDM symbol and knowledge of the pilot symbol is used in place of estimated data symbols to update the channel frequency response. Once the partial FFT outputs are combined using the RLS to drive the weight estima- tion, the receiver uses the distortion corrected outputsx k and the knowledge of the pilot 28 symbols to compute an estimate of the channel impulse response ˆ h as illustrated in Algo- rithm 1. The channel frequency response, computed from ˆ h as ˇ H k = P L l=1 ˆ h l e −j2πkl/K , can now be used to perform single-tap equalization on x k to recover estimates of the transmitted data symbols. 2.4 Model Based Combining Previously, we have computed the combiner weights when either complete or no knowl- edge of the channel was available at the receiver. In this section, we investigate scenarios where partial knowledge of the distortion process, such as a parametric representation, is available to the receiver. In general the combiner weights do not have any structure and each weight has to be estimated independently. However, in some scenarios of in- terest such as UWA channels, we can obtain a parametric representation of the combiner weights. This parametrization considerably reduces the number of parameters to be esti- mated and enables simpler equalization. To illustrate this method, let us consider OFDM signaling over an UWA channel with Doppler distortion wherein the received signal on each path is a time scaled version of the transmitted signal. Assuming that the channel coefficients are constant for the duration of one OFDM symbol, the time variation of the signal is captured by the time-varying path delaysτ p (t) =τ p −at, wherea =v/c is the ratio of the relative transmitter/receiver velocity to the speed of sound (see [48, 72, 12] and the references therein for the UWA channel model). From (2.3), the received signal in such a scenario can be expressed as r(t) = 1 √ T Re " K X l=1 d l H l e j2πf l t(1+a) # +z(t). (2.22) We observe that for a time-scaled OFDM signal, each subcarrier at frequencyf k is shifted 29 by an offset af k . This model can also be considered as a generalization of the OFDM system with carrier frequency offset (CFO) [50] where all the subcarriers are shifted by the frequencyf d Hz. The partial FFT outputs for the time scale scenario can be expressed as y k = K X l=1 d l H l D ǫ l v l−k +n k , ǫ l =af l T D ǫ l = diag h e j2πǫ l (2m−1) 2M i ,m = 1,2,··· ,M. (2.23) For OFDM with CFO, the partial FFT outputs are obtained by substituting ǫ l = ǫ = f d T, ∀l. Before illustrating the model based weight estimator, we first compute the optimal combiner weights for the two scenarios. 2.4.1 OFDM with Carrier Frequency Offset Evaluating the correlation and covariance matrices needed for the weights in (2.13) using (2.28) withǫ l =f d T, ∀l, we obtain R y k = D ǫ " K X l=1 |H l | 2 v l−k v H l−k + N 0 M I M # D H ǫ , R y k d k =D ǫ H k v 0 . (2.24) Noting thatD ǫ D H ǫ =I M and simplifying the resulting expression similarly as in the case of a time-invariant channel in Section 2.2, the optimal combiner weights are found to be w opt k =D ǫ H k |H k | 2 +N 0 Mv 0 . (2.25) The structure of the optimal combiner suggests that the receiver first compensates for the phase shifte j2πǫ(2m−1)/(2M) at them th partial FFT output and then combines them with equal weights before channel equalization. Note that this method does not completely 30 eliminate ICI as the compensated phase shift does not completely eliminate the CFO. 2.4.2 OFDM with Time Scaling Distortion Evaluating the auto-correlation and the cross-correlation terms as in the previous case, we obtain R y k = K X l=1 |H l | 2 D ǫ l v l−k v H l−k D H ǫ l + N 0 M I M , R y k d k =D ǫ k H k v 0 . (2.26) Expanding the individual entries of the above matrix, we get R y k (m 1 ,m 2 ) = 1 M 2 K X l=1 |H l | 2 e j2π(l−k+ǫ l ) (m 1 −m 2 ) M sinc 2 π(l−k) M + N 0 M δ(m 1 −m 2 ). (2.27) We observe that the autocorrelation matrix is not a circulant matrix and a simple closed form solution for the optimal weights w opt k may be intractable. However, R y k is a Toeplitz Hermitian matrix for each k. Consequently, we define the matrixC y k = circ [R y k (1,m 2 )], m 2 = 1,2,...,M as the related circulant matrix ofR y k . In the limit of large values of M, assuming that the strong norms of the inverses ofR y k andC y k are bounded, we can approximate the Toeplitz matrix by its equivalent circulant matrix [25], and the optimal combiner coefficients in such a scenario are approximately w opt k ≈D ǫ k H k λ k (1) Mv 0 ≈D ǫ k H k |H k | 2 +N 0 Mv 0 ., (2.28) where λ k (1) = P M i=1 C y k (1,i) = P M i=1 R y k (1,i) is the first eigenvalue ofC y k . Note that computingλ k (1) requires the knowledge of the channel frequency response for each subcarrier and the time scale parametera. Clearly, the strategy of first compensating for the phase distortion and then equalizing the received signal is asymptotically optimal. 31 Even though this strategy is not optimal, it will be later shown through simulations that the combiner weights computed using (2.28) result in significant ICI reduction even for small values ofM. We now propose an estimator which exploits the parametric model for the combiner weights in (2.28) to detect data transmitted using OFDM signals over a UWA channel. As the estimator relies on a model of the combiner weights, this estimator is called a model-based weight (MW) estimator. 2.4.3 Model-Based Weight Estimation Let us define {k 1 ,k 2 ,··· ,k P } as the set of subcarriers carrying pilot symbols. Us- ing the model for the received signal in (2.23), the combiner weights are designed to compensate for the phase rotation due to the time scaling. Assuming ˜ a to be a can- didate value of the time scaling factor, let us define the diagonal matrix D ˜ a (m) = diag h e j2πf k 1 ˜ a 2m−1 2M T ,··· ,e j2πf k P ˜ a 2m−1 2M T i to model the phase rotation for the m th partial FFT interval. The partial FFT outputs for the pilot subcarriers for this candidate value of ˜ a are combined as x p ˜ a = M X m=1 D H ˜ a (m)y p (m), (2.29) where the vectory p (m) = y p k 1 (m),··· ,y p k P (m) contains the elements of the partial FFT outputs for the pilot subcarriers in the m th interval. Assuming that this candidate value is the correct value of the time scale factor and thatx p ˜ a is nearly free of distortion, the combined outputs for the pilot subcarriers can be expressed as x p ˜ a =S p Fh+w p , F m,l =e j2πkml/K , (2.30) 32 where S p = diag[d k 1 ,d k 2 ,··· ,d k P ]. The least squares estimate of the channel im- pulse response corresponding to a candidate value of the time scale is now obtained as ˆ h ˜ a =F † (S p ) −1 x p ˜ a . The desired time scale parametera ∗ can then be finally determined using the maximum likelihood estimator [72] by searching over all candidate values and choosing the one which minimizes the metric: a ∗ = argmin ˜ a x p ˜ a −S p F ˆ h ˜ a 2 = argmin ˜ a x p ˜ a −S p FF † (S p ) −1 x p ˜ a 2 . (2.31) This minimization problem can be solved efficiently as it involves projecting a vector over a pre-determined subspace and then using a simple line search for evaluating a ∗ . Using the estimated time scale parameter, the channel impulse response is given as ˇ h = F † (S p ) −1 x p a ∗. The transmitted symbols are now estimated by first computing the channel frequency response across all the subcarriers and then performing one-tap frequency domain equalization: ˇ d k = dec(x k / ˇ H k ), ˇ H k = L X l=1 ˇ h l e −j2πkl/K . (2.32) We conclude this section by emphasizing that the MW estimator is a good choice for scenarios in which a parametric representation of the combiner weights can be derived and the model parameters easily estimated. 2.5 Performance Analysis: Theory In this section, we present an approximate theoretical analysis to characterize the per- formance of the proposed RW and MW estimators. The objective of this analysis is to provide intuition for choosing a practical value of M. We consider a CFO distorted 33 OFDM system to highlight several key properties of partial FFT combining and the pro- posed estimators. 2.5.1 Analysis of Model-Based Weight Estimation For the scenario of OFDM with CFO, we begin by assuming that the parameters of the optimal combiner are estimated correctly. Correcting for the phase distortion at the outputs of the partial FFT as in (2.25), we getx k = P M m=1 e −j2πǫ 2m−1 2M y k (m). Substituting fory k (m) from (2.5), the signal power at the output of thek th subcarrier, S k (M), can be evaluated as S k (M) =|H k | 2 sinc 2 πǫ M . (2.33) Similarly, the total noise-plus-interference power at thek th subcarrier,N k (M), is N k (M) = N 0 + K X l=1,l6=k |H l | 2 sinc 2 π(l−k+ǫ) M × M X m=1 1 M e j2π(l−k) (2m−1) 2M 2 . (2.34) We observe that the interference term P M m=1 1 M e j2π(l−k) (2m−1) 2M 2 is non-zero only for those subcarriers for which (l− k) is a multiple of M. Thus, for an OFDM system with frequency offset on a time-invariant channel, the partial FFT technique completely eliminates interference from all the subcarriers that are not at multiples ofM away from the subcarrier of interest. This is in contrast to a banded equalizer which only eliminates interference from only2M adjacent subcarriers, and thus significantly reduces detection error probability. The SIR of thek th subcarrier for a fixed value ofM is given as SIR ∗ k (M) = S k (M) N k (M) = |H k | 2 sinc 2 πǫ M N 0 + P K l=1,l6=k,l−k=pM |H l | 2 sinc 2 π(pM+ǫ) M (2.35) 34 For a given channel frequency response, assuming that πǫ M ≈ πǫ M+1 ,SIR ∗ k (M) is a mono- tonically increasing function of M. Numerically it is always observed that SIR ∗ k (M) monotonically increases withM and thus choosing the largest value ofM gives the low- est symbol error probability. The optimal M is then only limited by the computational available computational resources. For OFDM signaling over time scale distorted UWA channels, even though the fre- quency offset is slightly different for each subcarrier, it can be treated as a constant in the neighborhood of each subcarrier. This suggests that significant improvement can be obtained for data detection over UWA channels in comparison to banded MMSE equal- ization. 2.5.2 Analysis of Recursive Weight Estimation The RW estimation algorithm operates sequentially over subcarriers and its performance is a function of the rate of channel variation and the parameter M. Intuitively, as M increases from a very small value (M = 1 is the minimum), the performance of the RW estimation improves as the distortion process is better modeled with increasing M. However, with a further increase in M, the number of weights to be estimated in- creases for a fixed amount of information available at the receiver, thus reducing the accuracy of estimating the combiner weights. Hence, we expect a performance degra- dation for larger values of M. To capture the tradeoff between modeling accuracy and over-parametrization, we present a convergence analysis of the RLS under limiting con- ditions. As the number of subcarriers is finite and the channel is frequency selective, the data covariance matrix varies as a function of the subcarrier index, making a general analysis highly intractable. We consider several simplifying assumptions to facilitate our analysis. 35 As typical RLS convergence analysis assumes that the data covariance matrix is fixed [15], we make assumptions to approximate this as closely as possible. We begin by assuming that the channel is frequency flat with gain h 0 . At the receiver, the signal experiences a frequency offsetf d in addition to the AWGN at the input. The partial FFT outputs for the received signal can then be simplified from (2.9) as y k =D ǫ K X l=1 d l h 0 v l−k +n k =D ǫ K−k X l=1−k d k+l h 0 v l +n k , ǫ =f d T. (2.36) The data covariance matrix can now be expressed as R y k =|h 0 | 2 D ǫ " K−k X l=1−k v l v H l # D H ǫ + N 0 M I M . (2.37) Substituting forv l in the expression, we notice that the terms around l = 0 contribute significantly to the covariance matrix while the magnitude of the contribution decreases as we move away from the subcarrier of interest. Using this observation allows us to treat the covariance matrix as being independent of k for most of the subcarriers in the OFDM symbol. Deviations from this assumption are significant at the edge subcarriers, but are ignored for our analysis. Separating the signal and interference terms, we obtain R =|h 0 | 2 D ǫ v 0 v H 0 D H ǫ +R I , whereR I is the covariance matrix of the interfering terms and is given as R I =|h 0 | 2 D ǫ " K X l=1,l6=k v l−k v H l−k # D H ǫ + N 0 M I M . (2.38) 36 Using the matrix inversion lemma, 4 it can be shown that the data and interference covari- ance matrix are related as 1 v H 0 R −1 v 0 =|h 0 | 2 + 1 v H 0 R −1 I v 0 . (2.39) For the RW estimation algorithm using exponential decaying RLS with parameterλ, the optimization function for computing the combiner weights minimizes n X i=1 λ n−i h 0 d i −w H (n)y i 2 . For this scenario, the RLS adaptation rule is given by e k = h 0 d k −w H k−1 y k , w k =w k−1 +e k g k , (2.40) where e k is the RLS prediction error at subcarrier k, and g k is the gain vector from the RLS update. Using the results from RLS convergence theory [19], we know that in the limit of large k, the mean weight vectorw converges to the optimal linear MMSE solution: w ∗ = |h 0 | 2 R −1 D ǫ v 0 . The SIR at the output of the optimal receiver can be computed as in [63]: SIR ∗ (M) = E 2 {w H ∗ y} var{w H ∗ y} =|h 0 | 2 v H 0 D H ǫ R −1 D ǫ v 0 . (2.41) We note that SIR ∗ (M) is derived by using the mid-point approximation for the received signal (see equation (2.6) in Section 2.2), while the SIR evaluated in (2.35) uses the actual SIR by precisely evaluating all the integrals in (2.5). To characterize the performance of the RLS algorithm, we compute the steady-state 4 The matrix inversion identity for appropriately sized matrices A,B,C,D is given as(A+BCD) −1 = A −1 −A −1 B(C −1 +DA −1 B) −1 DA −1 . 37 output error and the steady state SIR at the output of the RLS [63]. The steady state error is the sum of two terms: the optimal MMSE error and the steady-state excess error[19, 63]. In the limit as the number of iterations tends to infinity, the steady state error at the output of the RLS in (2.40) converges to|e k | 2 → e ∗ +e ex (∞), where e ∗ is the mean-square error obtained by optimal MMSE filtering usingw ∗ [63]: e ∗ = E n h 0 d k −w H ∗ y k 2 o =|h 0 | 2 (1−|h 0 | 2 v H 0 D ǫ R −1 D ǫ v 0 ) = |h 0 | 2 1+|h 0 | 2 v H 0 D ǫ R −1 I D ǫ v 0 = |h 0 | 2 1+ SIR ∗ , (2.42) and the steady-state excess mean-square error is given by e ex (∞) = 1−λ 1+λ Me ∗ = ηe ∗ [63]. Now, under the assumption of known data symbols, the steady-state output SIR is defined as SIR ∞ = lim k→∞ E w H k−1 y k 2 var w H k−1 y k . (2.43) The mean output value is E w H k−1 y k = E w H k−1 E{y k }→d k h 0 w H ∗ D H ǫ v 0 =h 0 d k SIR ∗ 1+SIR ∗ , (2.44) where the first equality follows from the independence ofw H k−1 andy k . To compute the steady state output variance we first compute the second moment. E |e k | 2 =E{|d k h 0 −w H k−1 y k |} =|h 0 | 2 +E |w H k−1 y k | 2 −2h 0 d k E w H k−1 y k (2.45) ⇒E |w H k−1 y k | 2 =E |e k | 2 +2h 0 d k E w H k−1 y k −|h 0 | 2 =e ∗ +e ex (∞)−|h 0 | 2 (1−2w H ∗ v 0 ). 38 Using the mean and the second moment of the steady state output, the variance can be obtained as var w H k−1 y k → (1+η)SIR ∗ +η (1+SIR ∗ ) 2 . (2.46) The steady-state output SIR is then SIR ∞ = lim n→∞ E w H k−1 y k 2 var w H k−1 y k = SIR ∗ (1+η)+η/(SIR ∗ ) , (2.47) whereη = 1−λ 1+λ M andSIR ∗ (M) is a function ofM as given by (2.35). We observe that both the numerator and denominator ofSIR ∞ increase asM increases. For smallerM, the parameter η is very small and hence SIR ∞ increases rapidly with M. For larger M, the numerator saturates and the denominator increases and hence SIR ∞ reduces with increasing M. Hence there exists an optimal finite value of M for which the RW estimator gives the best detection performance (see Section 2.6 to see the optimalM as a function of distortion parameters). 2.5.3 Complexity Requirements The computational complexity of partial FFT combining is primarily determined by the algorithm used to compute the combiner weights, the number of subcarriers K and the number of intervalsM. To compute the partial FFT outputs, every algorithm begins by performingM K-point FFTs on a windowed version of the received signal thus having a maximum complexity of MKlogK complex multiplications (CMs) (The actual com- plexity is much lesser than MKlogK as (M − 1)K/M inputs to each FFT block are zeros). When perfect channel knowledge is assumed at the receiver, the optimal com- biner weights are determined using (2.13) which involves solving a system of equations 39 or computing the inverse of a matrix and approximately needs∼O(M 3 ) CMs. Comput- ing the data covariance matrix in (2.13) using (2.14) requires the evaluation of a series withK terms. The major contribution toR y k is by the subcarriers in the neighborhood of the k th subcarrier and thus R y k can be computed by a truncated version of the se- ries in (2.14). The total complexity of computing the optimal combiner weights is then O(M 3 )+O(SM 2 ), whereS is the number of terms used in computingR y k . For the special case of OFDM with time scale distortion or OFDM with CFO, the combiner weights on each subcarrier can be computed quickly using (2.28) or (2.25) respectively with a complexity of M CMs and an additional M CMs to perform the combining. Thus, by exploiting the special structure of the data covariance matrices, the partial FFT method enables computation of combiner weights and equalization inO(M) CMs. A complete summary of the computational complexity of partial FFT combining and a comparison of the computational complexity of several other algorithms proposed in literature is summarized in Table 2.1. The parameter D in Table 2.1 is the number of sub/super diagonals considered for designing the banded equalizer andN d is the total number of iterations required to implement Gibbs sampling in [60]. Clearly for a general scenario such as DVB, the complexity of the partial FFT scheme is larger when com- pared to similar schemes in literature, but for the UWA scenario in which partial FFT combining exploits the structure of the time-varying channels, we clearly observe from Table 2.1 that the complexity of receiver processing is very small and scales linearly with M. In contrast, the MCMC based MAP-SDSC in [60] scales as O(N d D 2 ) while the serial MMSE scales asO(D 3 ) for each subcarrier. Finally, though the PSE algorithm in [4] scales with O(D), it is shown in [3] that the partial FFT technique significantly outperforms the PSE algorithm in [4]. The RW and MW estimators inherently combine channel estimation and computing combiner weights and hence the computational complexity of these estimators includes 40 Table 2.1: Complexity Analysis with Perfect Channel State Information Algorithm Order of Complex Multiplications Partial FFT general scenario O(MKlogK)+O(KM 3 )+O(KM 2 S) Partial FFT UWA scenario O(MKlogK)+O(KM) Serial MMSE [73] O(KlogK)+O(KD 3 ) MCMC based MAP-SDSC [60] O(KlogK)+O(KN d D 2 ) PSE [4] O(Klog(K/M))+O(KD) Table 2.2: Complexity Analysis of RW and MW estimators Algorithm Number of Complex Multiplications RW estimator (M +1)KlogK +K(4M 2 +3M +6)+PL MW estimator (M +1)KlogK +PN s (M +2L+1) the cost of evaluating both sets of parameters. A summary of the total complexity of the RW and the MW estimator is given in Table 2.2, assumingP is the number of pilot symbols used,L is the number of channel taps estimated andN s is the number of search points in the MW estimator. For the RW estimator, the computational complexity is dom- inated by the RLS algorithm which grows asO(M 2 ). However, as shown in the analysis previously in this section, there exists an optimal value ofM for the RW estimator and increasingM beyond this value does not improve the performance of the RW estimator. We will show in Section 2.6 that the optimal value of M is very small compared to K (The maximum value of M is K) and hence the RW estimator has moderate complex- ity. For the MW estimator, the complexity is dominated by the number of points used in the search for the optimal time-scale parametera ∗ (see (2.31)). Once the value ofa ∗ is determined the computation of the channel frequency response and then subsequently combiner weights can be computed usingO(M) operations. 41 2.6 Performance Analysis: Simulation In this section, we evaluate the performance of the proposed partial FFT combining tech- nique using numerical simulations. We primarily consider two scenarios (1) OFDM signaling over time-scale distorted UWA channel [48, 99, 30] and (2) DVB signaling over highly time-varying channels [17, 4, 60]. The simulation setup and the performance results for each of the scenarios is illustrated separately below. 2.6.1 Underwater Acoustic Communications For UWA signaling, we consider an OFDM system withK = 1024 subcarriers operating in a bandwidthB = 12kHz operating at a center frequency off c = 30kHz. The subcar- rier spacing is Δf = 12Hz and the OFDM symbol duration isT = 1/Δf = 83.3ms. A cyclic prefix of lengthT g = T/8 = 10.41ms is added to the OFDM symbol to eliminate inter-symbol interference from the previous symbol. The normalized relative velocity between the transmitter and the receiver is represented bya = v/c, wherev is the rela- tive velocity between the communicating nodes andc = 1500m/sec is the speed of sound in water. The value ofa can reach values in excess ofa = 10 −3 for mobile underwater nodes [72, 30, 48]. As a first step to removing the time-scale distortion, the received signal is resampled using a coarse estimate of v/c. However, even a small error in es- timating a results in residual time scaling on the order of a = 10 −4 . We assume that a is a uniformly distributed random variable on the support [−1.5× 10 −4 ,1.5× 10 −4 ] for our simulations. As the time-scale distortion results in a different frequency shift for each subcarrier, the normalized Doppler for UWA OFDM is a function of the subcarrier frequency. For the system under consideration, the normalized Doppler can be computed asaf k T which is 0.2 for the first subcarrier to about 0.3 for the last subcarrier assuming a = 10 −4 . A normalized Doppler of about0.2 causes strong ICI which spans over several 42 tens of subcarriers. To simulate the underwater channel, we consider a 6-tap sparse chan- nel with power profile[0,−0.9,−4.9,−8.0,−7.8,−23.7]dB and a delay profile with the first tap arriving at0ms and last tap arriving at5ms. Fig. 2.1 shows the Symbol Error Rate (SER) for the partial FFT combiner, banded MMSE equalizer and the MCMC based MAP sequence detection with successive cance- lation (SDSC) scheme in [60] as a function of SNR and the model parameters. Assum- ing perfect knowledge of the time-varying channel, the partial FFT combiner weights have been computed using (2.28). The combiner weights can be computed in O(M) time, however they are not optimal due to the circulant approximation of the Toeplitz- Hermitian matrix (see Section 2.4 for a detailed discussion). For the MCMC based MAP- SDSC in [60], perfect channel state information is available, the interference from all the previous symbols is subtracted out and only the interference from the future symbols is accounted into the receiver design. Gibbs sampling with N d = 30 iterations is used to compute the MAP estimates of symbols with the first 10 iterations being considered as the burn-in period. The complexity of computing the combiner weights is O(N d D 2 ). From Fig. 2.1, it is observed that the partial FFT combiner performs significantly better than the standard banded MMSE equalizer whose complexity isO(D 3 ) per symbol while approaching the performance of the MCMC based MAP SDSC in [60] at moderate to high SNR. Thus, we observe that by exploiting the structure of the UWA time-variation, partial FFT combining can eliminate significant amount of interference and attain a per- formance close to higher complexity and interference canceling data detectors. We also note that the partial FFT method is a pre-processing technique that can be used in com- bination with the MCMC based MAP SDSC and other post-FFT processing techniques as well. Fig. 2.2 shows the SER obtained when performing data detection using the RW and MW estimator. The RW estimator does not assume prior knowledge of the time-varying 43 channel while the MW estimator only assumes knowledge of the distortion process. The RW estimator uses the initialization parameters as shown in Algorithm 1 (see Section 2.3). The parameters λ and α which control the RLS has been numerically optimized to give the best SER. Pilots have been placed uniformly throughout the OFDM symbol with 7 data symbols between each pilot. We note that while some techniques such as [48, 30] use zero symbols surrounding the pilots to reduce ICI experienced by the pilots, our method does not place zero symbols around pilots. However, to train the adaptive equalizer, we assume that the first2M subcarriers of each OFDM symbol are pilots. For the RW estimator in Fig. 2.2(a), we observe that at moderate SNR, the SER improves with increasing M as the time-varying channel is better compensated for thus resulting in reduced ICI. AsM increases further, the number of weights to be estimated increases causing over-parametrization and reduced accuracy of estimating the desired parameters. For example, we see from Fig. 2.2(a) that the performance is the best whenM = 8 and a degradation is observed forM = 2 andM = 32. Observe that this behavior has been predicted by the analysis of OFDM over flat fading channels with CFO in Section 2.5 (see (2.47)). As the RW estimator does not completely eliminate ICI, an error floor is observed at high SNR due to the residual ICI. Fig. 2.2(a) also shows a comparison between the RW estimator and a modified ver- sion of [86]. Both methods use adaptive algorithms to estimate parameters relevant to ICI compensation and neither assumes any knowledge of the channel and both methods use the same number of pilots for a fair comparison. The channel estimates are obtained adaptively (LMS is used in [86]) using pilots and tentative decisions (single tap equaliza- tion and detection ignoring ICI). The estimated ICI is then subtracted from the received data for each subcarrier, and equalization is performed to get new data symbol estimates. In our comparison, we select the best value ofD (one which gives lowest SER) based on 44 numerical simulations. It is clearly observed that the RW method significantly outper- forms the method in [86] under the current simulation scenario. For the MW estimator in Fig. 2.2(b), we assume that K/8 = 128 pilots are placed uniformly throughout the OFDM symbol, the channel length L conservatively set at L = 80 taps and the number of search points in (2.31) for determining the optimal time-scale parameter chosen to be N s = 60. From Fig. 2.2(b), we see that for the MW estimator the SER improves significantly with increasing M. As M grows larger, the performance improvement saturates and results in a tradeoff due to the increasing complexity. Fig. 2.2(b) also shows a comparison between the MW estimator (which es- timates the channel and distortion parameters) and the banded MMSE equalizers (which assume perfect channel knowledge). We clearly observe that even when there is a dif- ference in channel knowledge, the MW estimator significantly outperforms the conven- tional banded MMSE equalizer at moderate to high SNRs at a much lower complexity compared to the banded MMSE equalizer. Comparison of simulation and analytical results The analysis in Section 2.5 assumes a frequency offset scenario, which is a simplification of the more general time scale distortion scenario. To assess the predictive value of these analytical results, we examine the two scenarios: OFDM with CFO and time-scale distortion. Fig. 2.3(a) shows the optimal SIR as a function of the number of OFDM sub-intervals M for various values of the normalized CFO and an input SNR of25dB. The optimal SIR is the output SIR of the MW estimator which has perfect knowledge of the channel and the frequency offset. As pointed out in Section 2.5, the optimal SIR is a monotonically increasing function of M, since the mid-point approximation becomes more accurate with increasing M. We observe that the theoretical output SIR reaches its steady state 45 value even for small values ofM. Fig. 2.3(b) shows the pre-detection SIR for the UWA channel with MW estimation for various values of M and time-scale distortion. All the required channel and time-scale parameters are estimated during the course of the simulation. We observe that except for a small loss in the maximum output SIR (from 25dB to around 21dB) due to errors in channel and time scale estimation, the plot of the output SIR is as predicted in Fig. 2.3(a) thus validating our approximate analysis. Fig. 2.4(a) shows the theoretically expected SIR output for OFDM signaling over a flat fading channel with CFO for various values of normalized CFO and M when all the system and channel parameters are known perfectly. While the SIR ∗ in (2.47) in- creases with M due to more accurate modeling, it saturates at moderate values of M. However, the parameter η that characterizes the adaptive algorithm, increases with M, slowly reducing the steady-state SIR. The SIR curve thus exhibits a maximum, which indicates the optimal number of partial intervals to be used. We also observe from Fig. 2.4(a) that the slope of the SIR curve is different on both sides of the optimal M, and hence, erring on the side of largerM is better than choosing a smaller value. Fig. 2.3(b) show the pre-detection SIR for an UWA channel with RW estimation for frequency se- lective fading channel. We observe that the trend of the SIR curve is as predicted by the theoretical analysis. However, as the curve in Fig. 2.4(a) is for a simplified OFDM system with flat fading and CFO with known parameters, and the curve in Fig. 2.4(b) shows the measured output for a time scaled OFDM system with estimated parameters, there is a mismatch between the two figures which can be attributed to the time-varying nature of the data covariance matrix. However, the purpose of the analysis is to provide intuition for choosing an optimal value ofM and we note from Fig. 2.4(b) that choosing M between 8 and 16 is optimal depending on the level of expected Doppler distortion and agrees well with the results in Fig. 2.2(a). 46 2.6.2 Digital Video Broadcasting For DVB signaling between mobile devices, we consider an OFDM signal with K = 2048 subcarriers in a bandwidth ofB = 2MHz centered atf c = 650MHz. The subcarrier spacing is Δf = 976.5Hz and the OFDM symbol duration is T = 1ms. The cyclic prefix isT g = T/8 = 125µs . The wireless channel between the mobile antenna and the receiver are modeled based on a realistic channel model determined by the COST-207 project and the Typical Urban TU-6 channel model is considered for simulation [60]. The TU-6 channel model has 6 taps and the multipath gains on each tap are modeled with the Jakes model. We consider velocities of 180kmph and 240kmph which result in a normalized Doppler off d T = 0.11 and0.15 approximately. Fig. 2.5(a) and Fig. 2.5(b) show the SER as a function of various system parame- ters for the partial FFT based combiner, banded MMSE equalizer and the MAP based SDSC algorithm in [60] as a function of SNR for velocities 180kmph and 240kmph re- spectively. We observe that in general the partial FFT method performs better than the banded MMSE equalizer of equivalent complexity but there exists a performance gap when compared to the MAP based SDSC. This can be attributed to the fact that the MAP based SDSC is an interference canceling equalizer and thus reduces the effective amount of interference seen by each data symbol. 2.7 Conclusions In this sequel, we considered the problem of OFDM data detection over a highly time- varying channel. A detection technique called partial FFT demodulation was proposed, wherein each OFDM symbol is divided into several smaller intervals and FFT is per- formed on each. Weighted combining of the partial FFT outputs results in a significant 47 0 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 SNR Symbol Error Rate MCMC D=3 MCMC D=7 MCMC D = 31 Partial FFT M = 2 Partial FFT M = 8 Partial FFT M = 32 Banded MMSE D = 3 Banded MMSE D = 7 Banded MMSE D = 31 Figure 2.1: Symbol Error Rate assuming perfect knowledge of the UWA channel as a function of SNR and model parameters for the partial FFT combiner, banded MMSE and MCMC based MAP SDSC [60]. reduction in ICI compared to banded equalization and permits low complexity symbol- by-symbol detection. While banded equalization typically accounts for ICI only from a few adjacent subcarriers, partial FFT demodulation can take into account ICI from many more subcarriers without increasing the computational complexity. Three scenarios were considered in which either full, partial or no knowledge of the channel is available, and techniques for deriving the optimum combiner weights in each of these scenarios have been presented. An approximate theoretical analysis of the proposed weight estimators was presented, showing interesting properties of the estimators that explain the trends observed in numerical simulations. Numerical results for time-varying channels corre- sponding underwater acoustic communications and digital video broadcasting demon- strate the significant improvements that can be obtained using the proposed technique. 48 0 5 10 15 20 25 30 10 −3 10 −2 10 −1 10 0 SNR Symbol Error Rate Modified FD−MMSE Partial FFT M=2 Partial FFT M=8 Partial FFT M=32 (a) RW estimator 0 5 10 15 20 25 30 10 −3 10 −2 10 −1 10 0 SNR Symbol Error Rate Partial FFT M=4 Partial FFT M=8 Partial FFT M=32 Banded MMSE D=7 Banded MMSE D=15 Banded MMSE D= 31 (b) MW estimator Figure 2.2: Symbol Error Rate of the RW and MW estimators as a function of SNR and model parameters 49 10 0 10 1 10 2 8 10 12 14 16 18 20 22 24 26 Number of partial intervals Optimal MMSE Output SIR (dB) Normalized CFO increasing from 0% to 20% in steps of 4% (a) Flat fading channel with CFO 10 0 10 1 10 2 8 10 12 14 16 18 20 22 24 26 M − Number of partial FFT intervals Output SIR a = 0 a = 0.5 x 10 −4 a = 10 −4 a = 1.5 x 10 −4 (b) Frequency selective channel with time scale distor- tion Figure 2.3: Output SIR as a function ofM for an input SNR of25dB for a UWA OFDM system using MW estimation and partial FFT combining 50 10 0 10 1 10 2 8 10 12 14 16 18 20 22 24 26 Number of partial intervals Steady State Output SIR (dB) Normalized CFO increasing from 0% to 20% in steps of 4% (a) Flat fading channel with CFO 10 0 10 1 10 2 8 10 12 14 16 18 20 22 24 26 M − Number of partial FFT intervals Output SIR a = 0 a = 0.5 x 10 −4 a = 10 −4 a = 1.5 x 10 −4 (b) Frequency selective channel with time scale distor- tion Figure 2.4: Output SIR as a function ofM for an input SNR of25dB for a UWA OFDM system using RW estimation and partial FFT combining 51 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 SNR Symbol Error Rate MCMC D=3 MCMC D=5 MCMC D=7 Partial FFT M=2 Partial FFT M=4 Partial FFT M=8 Banded MMSE D=3 Banded MMSE D=5 Banded MMSE D=7 (a) Velocity = 180kmph, Normalized Doppler = 0.11 0 5 10 15 20 25 10 −4 10 −3 10 −2 10 −1 10 0 SNR Symbol Error Rate MCMC D=3 MCMC D=5 MCMC D=7 Partial FFT M=2 Partial FFT M=4 Partial FFT M=8 Banded MMSE D=3 Banded MMSE D=5 Banded MMSE D=7 (b) Velocity = 240kmph, Normalized Doppler = 0.15 Figure 2.5: Symbol Error Rate assuming perfect knowledge of the DVB wireless channel as a function of SNR and model parameters 52 Chapter 3 Optimal Resampling 3.1 Introduction High speed communications in underwater acoustic channels (UAC) has been considered challenging due to the low speed of propagation of sound, fundamental limitations on the available bandwidth due to frequency dependent attenuation, highly time-varying frequency selective channels and colored ambient noise. [82, 64, 80, 81]. The low speed of sound and propagation over multiple paths results in large delay spreads, while the Doppler due to a non-stationary medium and mobile transceivers contributes significantly to the time-varying nature of the channel. Acoustic propagation is best supported at lower frequencies where the available bandwidth is comparable to the center frequency and most acoustic systems are wideband. Orthogonal Frequency Division Multiplexing (OFDM) is now emerging as an attrac- tive signaling scheme for UAC’s [49, 31, 79, 56, 103, 47]. Interest in OFDM stems from the fact that it can decompose a static frequency selective channel into a set of indepen- dent and inter-symbol interference (ISI) free sub-channels which lead to low complexity receivers. However, carrier orthogonality in OFDM is highly sensitive to time-varying multipath and motion induced Doppler distortion. In a narrowband OFDM system, the 53 Doppler distortion is either modeled as a frequency offset [106] or time-varying multi- path coefficients (e.g. Jakes model) [69]. However, in wideband systems such as under- water communications, Doppler distortion on each of the channel paths results in time scaling (dilation or compression) of the signal. The time scaling induces frequency de- pendent offsets for each of the OFDM subcarriers and hence the narrowband model for Doppler distortion cannot be applied. The narrowband model breaks down even further when each channel path has a distinct time-scaling factor as observed in recent experi- ments [49, 56, 8, 11, 103, 87]. In much prior work on UAC’s, all the channel paths were modeled with a common time scaling factor [49, 77, 39, 79]. In [79], the distortion due to time scaling is modeled as a time-varying phase offset and an adaptive decision directed phase compensation algorithm has been proposed. The case of small time scaling values is considered in [36], wherein the Doppler distortion is modeled as a frequency offset and iterative receivers were proposed. The work in [31] is one of the few to provide a high performance solution for highly time-varying channels at the expense of receiver complexity. In [72], the combined effect of time scaling and frequency offsets were discussed and estimation algorithms were derived for the parameters of interest. A [77, 39, 49]. It is straight forward to show that resampling is the optimal pre-processing a scenario with equal time scales on all the paths; this essentially converts a wideband signal into an effective narrowband signal [97] and also allows symbol by symbol detection. However, as mentioned earlier, recent experimental results suggest that in UAC’s - the focus of this work - the effect of mobility in the multipath environment is best described by time domain scale spreading of the transmitted signal. This means that several copies of the transmitted signal combine at the receiver, each with a different time scaling. In addition, each copy of the signal is attenuated and delayed by a different amount [53, 11, 49, 56, 8]. The effect of time scale spreading (referred to as scale spreading 54 henceforth) on OFDM based wideband systems and the trade off between cyclic prefix duration and subcarrier spacing is analyzed in [11]. Several works in literature have also examined the characterization of wideband time-varying systems using scale-lag models; and RAKE-like receivers have been derived to exploit the diversity offered by these systems [54, 33] (and references therein). In this sequel, we investigate the effect of using a single resampling operation (opti- mal processing assuming zero scale spreading) when channel paths have different time scaling values. We presented preliminary results to the question “What is the optimal resampling rate for multipath channels, each with their unique scaling parameter?” in [97]. The primary purpose of this chapter is to provide a comprehensive analysis on resampling OFDM signals and to investigate the performance of various resampling pa- rameter estimators over multi-scale multi-lag channels. Assuming complete knowledge of all the channel parameters, we propose two differ- ent criteria to evaluate the optimal resampling parameter over such channels. First, we examine the multi-parametric Hammersley Chapman Robbins Bound (HCRB[28, 14, 24, 89]) to compute a lower bound on the error covariance for detecting the transmitted se- quence. As the constrained parameter equivalent of the Cramer Rao Bound (CRB), the HCRB is a lower bound on the error covariance for the estimation of constrained param- eters (e.g., BPSK symbols are constrained to take discrete values and the HCRB then lower bounds the error covariance for detecting the discrete parameters). The HCRB is a function of the resampling parameter used at the receiver and the optimal resam- pling parameter is the one that minimizes the HCRB. We also optimize the CRB for soft estimation of transmitted parameters. Next, we show that the sufficient statistics for data detection after resampling can be derived by matched filtering the received signal with an approximated channel having a common time scaling parameter on all the paths. The channel approximation interpretation of resampling suggests a second criterion for 55 estimating the best resampling parameter; one that minimizes the aggregate error be- tween the multi-scale channel and its approximated single scale version. Numerically, it is observed that the aggregate channel error minimizing resampling factor is nearly indistinguishable from the one obtained by using the CRB. We then analyze the HCRB in some special cases to gain insight into the behavior of the optimal resampling parameter. a Next, blind and pilot-aided estimators for the optimal resampling parameter are derived and compared to the classical packet length based scale estimator in [77, 49]. We show the relationship between our estimators and those presented in [72, 62]. Numerical results show that while the pilot-aided estima- tor uniformly outperforms the packet length based estimator by a significant margin, the blind estimator outperforms the packet length based estimator in most scenarios; espe- cially for small scale spreads. It is also shown that the pilot aided estimator is robust to inaccuracies in estimating the path delays. The remainder of this chapter is organized as follows. The system model is described in Section 3.2 and computation of the sufficient statistics for a general case are described in Section 3.3. The two proposed optimization criteria for determining the optimal re- sampling parameter are derived in Section 3.4 and special cases are investigated in Sec- tion 3.5. The derivation of practically implementable estimators is illustrated in Section 3.6 and numerical results are presented. The chapter concludes with remarks in Section 3.7. 56 3.2 System Model 3.2.1 Transmit Signal The transmitted OFDM signal in passband can be compactly written as 1 . s(t) = K−1 X k=0 s k e j2πf k t u(t) u(t) = ( q 1 T ift∈ [−T cp ,T +T ca ] 0 elsewhere (3.1) where s k are the data symbols modulated onto the OFDM subcarriers and let us define s = [s 0 ,s 1 ,...,s K−1 ] as the vector containing all the data modulated onto the subcarri- ers. The frequency of thek th subcarrier isf k = f c −B/2+kΔf, k = 0,1,...,K−1 where f c is the carrier frequency, Δf is the OFDM subcarrier spacing and B = KΔf is the bandwidth of the system. The useful duration of the OFDM symbol is denoted as T = 1/Δf and the cyclic prefix and postfix are given as T cp and T ca respectively. The rectangular pulse u(t) is scaled to have a unit energy in the interval [0,T] and the total duration of an OFDM block isT +T cp +T ca sec. In this work, we consider an isolated OFDM symbol being transmitted over the chan- nel. A packet with several OFDM symbols results in inter-symbol interference (ISI) from adjacent OFDM symbols and is beyond the scope of current work. The cyclic prefix is assumed to be longer than the delay spread and the cyclic postfix is of enough duration to ensure signal continuity in the observation interval. Though this chapter discusses the scenario when a cyclic extension is used, the analysis can be directly applied to Zero Padding OFDM (ZP-OFDM) with minor modifications. 1 A few carriers are unmodulated in a practical system to ensure identifiability of parameters (see Sec- tion 3.6) 57 3.2.2 Wideband Channel Representation As described in Section 3.1, the signal at the receiver is a sum of several scaled copies of the transmitted signal, each of them distinctly dilated and attenuated. This is illustrated by a simple example. Consider a receiver moving with a velocityv. A direct path from the transmitter to the receiver will experience a time scaling of a = v/c, while a path arriving perpendicular to the direction of motion experiences zero scaling. In the extreme case, a path reflected off an obstacle in front of the receiver can experience a negative scaling ofa =−v/c. In general, the scale parameter on a path can take any value from −v/c tov/c. The time-varying channel model is then h(τ,t) = X l h l δ(τ−τ l (t)), τ l (t) =τ l −a l t, (3.2) where h l , τ l and a l are the channel gain, path delay and time scaling on the l th path respectively and thus we have a multi-scale multi-lag channel model (see [53, 66, 33] for a more detailed discussion of the development and analysis of such channel models). The time scale parameters are related to the radial velocity of arrival of the l th path at the receiver v l , by a l = v l /c, where c is the velocity of propagation of the signal in the medium. In general, the coefficients a l are distinct and all the channel parameters are modeled as constant, and deterministic over the duration of one OFDM symbol. We define the scale spread of the channel as D s = max i6=j |a i −a j |. Typical values of the scale parameter depend on the relative velocity between the transmitter and the receiver. The time-varying channel frequency response of the channel, obtained by taking the Fourier transform of (3.2) with respect to the delay variable, is given as H(f,t) = X l h l e −j2πfτ l e j2π(fa l )t . (3.3) 58 From (3.3), a complex sinusoid of frequencyf transmitted over this channel experiences a phase shift ofe −j2πfτ l and a frequency shiftfa l on thel th path. The different scaling across each path causes the complex sinusoid to be shifted by different frequencies on each path. As the transmitted OFDM symbol is a sum of several orthogonal complex sinusoidal functions (see (3.1)), the received signal can be expressed as r(t) = X l h l s(t(1+a l )−τ l )+n(t) = X l h l X k s k e j2πf k (t(1+a l )−τ l ) u l (t)+n(t), (3.4) whereu l (t) = u(t(1+a l )−τ l ) is a time scaled and delayed version of the rectangular pulse u(t) and n(t) is complex additive white Gaussian noise (AWGN) at the receiver front end with zero mean and power spectral density N 0 . Adopting a compact form, using (3.3), we get r(t) = X k s k h k (t)e j2πf k t +n(t), h k (t) = X l h l e −j2πf k τ l e j2π(f k a l )t u l (t). (3.5) In (3.5), h k (t) is the time-varying channel frequency response as seen by the k th sub- carrier. Thek th subcarrier experiences a frequency shift off k a l on thel th path and this frequency shift scales linearly with increasing subcarrier frequency. For example, the fre- quency shift experienced by an OFDM signal occupying the band between 15−30kHz over a channel with scaling parametera = 3×10 −3 is45Hz and90Hz, respectively, for the subcarriers with the least and largest frequencies. 59 3.3 Sufficient Statistics and Performance Bounds In this section, we define the processing to derive a discrete time equivalent model from the continuous time received signal. The information content of the signal is quantified by analyzing a set of sufficient statistics for the detection of the transmitted parameters. Noting that any good receiver design operates on a set of sufficient statistics for the parameters to be determined, we use the Fisher-Neyman factorization theorem to com- pute a set of sufficient statistics for the data symbols [38]. We restrict our observation interval to [0,T]. The choice of interval eliminates the cyclic extension as in a conven- tional OFDM system. As the received signal is a continuous time signal, we examine the likelihood functional of the signal: Λ(r(t)|s,a,h,τ) = βexp − 1 N 0 Z T 0 r(t)− X k s k h k (t)e j2πf k t 2 dt ! , (3.6) wherea = [a 1 ,a 2 ,...,a L ],h = [h 1 ,h 2 ,...,h L ],τ = [τ 1 ,τ 2 ,...,τ L ] andβ is a constant. For simplicity of exposition, we assume that transmitted symbols are modulated onto all the carriers. This assumption is later relaxed when deriving practically implementable algorithms. Expanding the exponent of the right hand side of (3.6), we get three terms. Two of the terms correspond to the energy in the received signal and its noiseless version and do not contribute to the sufficient statistics [38]. The cross term depends on both the received signalr(t) and the data vectors and the desired set of statistics can be derived from this term. The logarithm of the cross term is ℜ " X m s ∗ m Z T 0 r(t)h ∗ m (t)e −j2πfmt dt # , (3.7) 60 and hence one set of sufficient statistics are given by the vectorΦ(s) = [Φ(s 0 ),Φ(s 1 ), ...,Φ(s K−1 )], where Φ(s m ) = Z T 0 r(t)h ∗ m (t)e −j2πfmt dt, m = 0,1,··· ,K−1. (3.8) Substituting for the channelh m (t) in (3.8), each componentΦ(s m ) can be viewed as the sum of outputs ofL filters, each matched to the delay and time scale of one path. Φ(s m ) = X l h ∗ l e j2πfmτ l Z T 0 r(t)e −j2πfm(1+a l )t u l (t)dt, u l (t) = r 1 T ,t∈ " − T cp +τ l 1+a l , T +T ca +τ l 1+a l # . (3.9) The cyclic postfix T ca is chosen such that T+Tca 1+amax ≥ T . This ensures that u l (t) is a constant fort∈ [0,T],∀l, and further reference to the termu l (t) is dropped in the rest of this chapter. The derived set of sufficient statistics is a combination of the outputs of KL filters. While each component Φ(s m ) could somehow be viewed as a statistic for detection of the individual symbols m , we note (by substituting forr(t) in (3.9)) that this is true only when given perfect knowledge of all other transmitted symbols. In general, this implies that the entire collection of sufficient statisticsΦ(s) is required for the detection of each of the transmitted symbols and symbol-by-symbol processing of the received signal is not optimal. This is the effect of inter-carrier interference (ICI) due to the time-varying 61 channel. By substituting forr(t) in (3.8), the vector matrix equation for the discrete time matched filtered equivalent signal is given as Φ = Ps+n where, P mk = Z T 0 h k (t)h ∗ m (t)e j2π(f k −fm)t dt and n m = Z T 0 n(t)h ∗ m (t)e −j2πfmt dt. (3.10) The noise vectorn is colored and its covariance matrix isE[nn H ] =N 0 P. Conditioned on the data sequences and the channel parameters, (3.10) describes a deterministic signal in Gaussian noise and the likelihood function for the sufficient statistics is f(Φ|s) = 1 π K |det(P)| e − 1 N 0 (Φ−Ps) H P −1 (Φ−Ps) . (3.11) 3.3.1 Performance Bounds The Hammersley Chapman Robbins Bound (HCRB) [28, 14, 24, 89] is a lower bound on the estimation error covariance when the parameters of interest are constrained to lie in a subset of the parameter space. As the transmitted data symbols are drawn from a finite constellation, they can be treated as constrained parameters and the HCRB can then be used to lower bound the performance of detectors for these symbols. To determine the HCRB for the detection of the transmitted vector s, we first de- scribe the general form of the multi-parametric HCRB [24, 89]. Consider Θ, a vector of unknown parameters contained in the parameter spaceΘ C with meanm Θ , covariance matrixΣ Θ and density function f Θ . For vectorsv i ∈ Θ C ,i = 1,2,...,M and scalars Δ i ,i = 1,2,...,M, let us define the scalar and vector differencesδ i f Θ andδ i m Θ of the 62 density function and of the mean vector for Θ, respectively, in the direction along the vectorv i as δ i f Θ = f Θ+Δ i v i −f Θ Δ i kv i k , δ i m Θ = m Θ+Δ i v i −m Θ Δ i kv i k , and let us define δf Θ ≡ [δ 1 f Θ ,··· ,δ M f Θ ], δm Θ ≡ [δ 1 m Θ ,··· ,δ M m Θ ]. For any unbiased estimator ˆ Θ, the error covariance matrix satisfies the inequality Σ Θ ≥ sup [δm Θ ] E Θ    " δf Θ f Θ # H " δf Θ f Θ #    ! † [δm Θ ] H , (3.12) where the supremum is over the set of all possible combinations ofv i ,i = 1,2,...,M which are admissible in Θ C and span the parameter space Θ C andE Θ is the expectation operation over the parametersΘ. For the scenario under consideration in (3.11), assuming that the transmitted symbols are BPSK vectors, the parameter space is set of all vectors in{−1,1} K , δ i f Θ = f(Φ|s i )−f(Φ|s) ks i −sk , δ i m Θ = s i −s ks i −sk . (3.13) Substituting the required parameters and computing the expectation in 3.12, the HCRB for the detection ofs is given as Σ s ≥ sup X H G † mf X X = s 1 −s ks 1 −sk ,..., s K −s ks K −sk G mf (i,j) = R C K f(Φ|s i )f(Φ|s j ) f(Φ|s) dΦ−1 ks i −skks j −sk i,j = 1,...,K. (3.14) 63 The notationA ≥ B for matricesA,B implies thatA−B is a positive semi-definite matrix and the subscript ’mf’ denotes matched filtering. The supremum is over the set of all the admissible combinations of vectorss i ∈ {−1,1} N which span the space Θ C . Substituting for the density function in (3.14), we get G mf (i,j) = 1 ks−s i kks−s j k e (s−s i ) H P(s−s j ) N 0 −1 . (3.15) Noting that the use of soft-estimates of discrete parameters is common is modern communications, the Cramer Rao Bound (CRB) is used to lower bound the performance of unbiased estimators. Σ s ≥N 0 P −1 , (3.16) where the matrixP is defined as in (3.10). The HCRB in (3.14) and the CRB in (3.16) are a lower bound on the achievable detection (estimation) performance of any unbiased detector (estimator) and provide a baseline comparison to evaluate the performance of the receiver designs in Section 3.4. 3.4 Doppler Compensation using Resampling In this section, we characterize the effect of resampling on multi-scale multi-lag signals. Resampling of multi-scale signals could be considered as reduced-rank signal processing and may result in information loss. To provide some intuition to the problem at hand, we temporarily focus on the scenario where all paths have the time scale. 64 3.4.1 Equal Time Scaling on all Paths The time varying channel on each sub-carrier can be written as h m (t) = r 1 T " X l h l e −j2πfmτ l # | {z } Hm e j2πfmat , ∀m (3.17) whereH m is the channel gain of them th subcarrier in the absence of time scaling. In this special case, the time varying frequency response observed on them th subcarrier is the product of the gainH m and an exponential function representing the time variation due to Doppler scaling. This representation is true only for channels with a common Doppler shift and is not valid in general. The discrete time equivalent signal after resampling is then r m = 1 T Z T 0 r t 1+a e −j2πfmt dt =H m s m +v m , (3.18) wherev m ∀m is a complex Gaussian random variable with varianceN 0 . After resampling the received signal to eliminate time scaling, the sufficient statistics for data detection derived by examining the likelihood functional are Φ(s m ) =|H m | 2 s m +H ∗ m v m . (3.19) From (3.19), the mixing matrixP = diag[|H 1 | 2 ,|H 2 | 2 ,...,|H K | 2 ] is clearly diagonal clearly showing that resampling completely eliminates ICI in this scenario and symbol by symbol processing is optimal. This proof of optimality is alternate to that provided in [97]. 65 3.4.2 Multi-scale Multi-lag Signals When channel paths have different time scales, the signal is resampled with a parameter b and the discrete time signal model is then r m = 1 T Z T 0 r t 1+b e −j2πfmt dt = 1 T X l,k s k h l e −j2πf k τ l Z T 0 e j2π(f k ( 1+a l 1+b )−fm)t dt+v m . (3.20) Expressed in vector form, we have r =Q(b)s+v, v∼N(0,N 0 I), (3.21) where Q mk (b) = X l h l e −j2πf k τ l 1 T Z T 0 e j2π(f k ( 1+a l 1+b )−fm)t dt. Under the assumption of resampling as front-end receiver processing, the vectorr is a sufficient statistic for the detection of transmitted symbols. However, as noted previously, resampling can be information lossy in the case of multiple, distinct time scales. As in Section 3.3, the statistics are used to derive a lower bound on the detection performance of any unbiased estimator of data fromr. The HCRB is then: Σ s (b) ≥ sup X H G † b X X b (i,j) = 1 ks−s i kks−s j k e (s−s i ) H Q H (b)Q(b)(s−s j ) N 0 −1 , (3.22) 66 whereS and the set over which the supremum is taken are as defined in Section 3.3. The optimal resampling parameter is the one which minimizes the trace of the HCRB matrix: b ∗ HCRB = argmin b Tr[Σ s (b)]. (3.23) When soft estimates of symbols are desired, the CRB gives the lower bound on the estimation error: Σ s (b)≥N 0 (Q H (b)Q(b)) −1 , b ∗ CRB = argmin b Tr[Σ s (b)]. (3.24) The optimal resampling parameterb ∗ CRB then minimizes the trace of the CRB. 3.4.3 Resampling as approximation with a single scale The sufficient statistics in (3.21) allow an alternative interpretation to resampling. Con- sider ˆ h m (t) an approximation to the channel on the m th subcarrier h m (t). The approx- imated channel has the same channel gains and delays as the original channel but a common time scaling factor on each path. ˆ h m (t) = r 1 T X l h l e −j2πfmτ l e j2πfmbt . (3.25) Further processing of the received signal is done assuming that the channel on the sub- carriers is its approximated version. The signal is passed through a set of filters matched 67 to the Doppler shift on the approximated channel. The output of these filters, denoted by ˆ Φ(s m ) are given as ˆ Φ(s m ) = 1 T Z T (1+b) 0 r(t) ˆ h ∗ m (t)e −j2πfmt dt = H ∗ m Z T (1+b) 0 r(t)e −j2πfm(1+b)t dt =H ∗ m r m . (3.26) The upper limit of integration is chosen as T/(1 + b), as the scaled version of all the OFDM sub-carriers e j2πfm(1+b)t complete an integer number of cycles in the interval [0,T/(1+b)] and hence are still orthogonal to each other. In vector form, the output of the matched filters are given as ˆ Φ(s) = diag[H ∗ 1 ,··· ,H ∗ K ]r (3.27) The vector ˆ Φ(s) is a scaled version of ther and thus a set of sufficient statistics them- selves for each value of the resampling parameterb. A set of sufficient statistics can also be generated as outputs of a set of filters matched to a particular time scaling parame- ter. This interpretation of resampling paves the way to a second criterion to compute the optimal resampling factor. Let us define MSE m (b) to be the error between the actual and the approximated channels for them th subcarrier. MSE m (b) = Z T 1+b 0 |h m (t)− ˆ h m (t)| 2 dt, (3.28) 68 The aggregate mean-squared error (AMSE) between channel and its approximated ver- sion for the OFDM symbol is then AMSE(b) = P m MSE m (b). Substituting for the original and approximated channels, we get AMSE(b) = X m X k,l h k h ∗ l e −j2πfm(τ k −τ l ) Z T 1+b 0 (e j2πf k a k t −e j2πfcbt )(e −j2πf k a l t −e −j2πf k bt )dt. (3.29) The optimal resampling parameter then minimizes the aggregate mean-squared error be- tween the actual and the approximated channels. b ∗ AMSE = argmin b AMSE(b). (3.30) Thus optimal resampling can be interpreted as maximizing the projection of the received signal onto a subspace indexed by the resampling parameter. 3.5 Analysis of Optimal Resampling Obtaining a closed form expression for the optimal resampling parameter seems to be an intractable problem in general. We consider some special cases for which we derive approximate solutions to the optimization problem and infer the nature of the optimal resampling parameter. 69 3.5.1 One Carrier Scenario The HCRB expression in (3.22) involves a multi-dimensional numerical optimization whose complexity grows exponentially [24, 89] in the number of parameters. We an- alyze a scenario when only one OFDM subcarrier has data modulated on it. Though this assumption only considers subcarrier self interference and is highly simplifying, it provides intuition into the nature of the optimal solution. Consider a scenario when data is transmitted only on the m th subcarrier. Using the received value on the m th carrier after resampling, the HCRB for data detection can be derived as σ 2 sm ≥ min b sup sm6=s ′ m ks m −s ′ m k 2 e Jm(b)ksm−s ′ m k 2 /(N 0 ) −1 , (3.31) where, J m (b) E s = X l h l e −j2πfmτ l e jπfmT a l −b 1+b sinc πf m a l −b 1+b T 2 . (3.32) For a fixed constellation, we observe from (3.31) that maximizingJ m (b) minimizes the HCRB. The CRB for the soft estimates of the data symbol also yield the same optimiza- tion function. This is however only valid in the single carrier scenario and not true in general when data is transmitted on multiple carriers. For typical scenarios under con- sideration,a max ∼ 10 −3 and hence for all practical values of the resampling parameter, (1 +b) ≈ 1. Applying this approximation, the term (a l −b)/(1 +b) in J m (b) can be approximated with(a l −b). Let us defineg l =h l e −j2πfmτ l e j2πfma l T as the phase rotated channel gain. The optimization function in (3.32) can then be written as J m (b)≈ X l g l sinc(πf m (a l −b)T) 2 . (3.33) Thus, J m (b) is the sum of several overlapping sinc functions and the extent of overlap of the main lobes depends on the scale spreading due to the channel. We analyze the 70 solution to the optimal resampling problem in two extreme cases. 3.5.1.1 Small scale spread When the scale parameters on all channel paths are close to each other, we use a Taylor’s series expansion of the sinc functions around the optimal value b ∗ . Ignoring the higher order terms in this series expansion, the sinc function can be approximated as sinc(x 1 )≈ 1− x 1 2 3 and sinc(x 1 )sinc(x 2 )≈ 1− x 1 2 3 − x 2 2 3 . Substituting in (3.33) and differentiating J m (b) to determine the maximizing value we get, b ∗ ≈ P j (|g j | 2 + P j6=k ℜ(g ∗ j g k ))a j P j (|g j | 2 + P j6=k ℜ(g ∗ j g k )) . (3.34) WhenD s is small, the optimal value of the resampling parameter is a weighted average of the time scaling parameters on each path. Note that when the channel gain on one of the paths dominates, the optimal resampling parameter is nearly equal to the time scale on that path. 3.5.1.2 Large scale spread When the time scales on the paths differ significantly, the elements of{a 1 ,a 2 ,··· ,a L } are not clustered together, but separated. The main lobes of the sinc function in (3.33) have minimal overlap with each other and thus the optimal resampling parameter b ∗ ∈ {a 1 ,a 2 ,··· ,a L }. We now derive a bound on the ratios of channel magnitudes when the time scaling on the strongest path is the resampling parameter. For simplicity of exposition, let us assume that max i |h i | = |h 1 |. Noting that|g l | = |h l |, the sufficient 71 condition for b ∗ = a 1 can be derived as a set of inequalities to be satisfied. Evaluating the functionJ m (b) atb =a 1 , we get: J m (a 1 ) = X l g l sinc(πf m (a l −a 1 )T) 2 . (3.35) Similarly, the value ofJ m (b) at otherb∈{a 2 ,··· ,a L } can be evaluated. Using the fact that sinc(0) = 1 and simple properties of modulus of a number, we derive an upper and lower bound forJ m (b). For example whenb =a 1 , we get J m (a 1 ) ≤ |h 1 |+ X l6=1 |h l sinc(πf m (a l −a 1 )T)| 2 (3.36) J m (a 1 ) ≥ |h 1 |− X l6=1 |h l sinc(πf m (a l −a 1 )T)| 2 . (3.37) Such bounds can be written for all values of b and for b ∗ = a 1 , the sufficient condition would then be |h 1 |− X l6=1 |h l sinc(πf m (a l −a 1 )T)| 2 ≥ |h n |+ X l6=n |h l sinc(πf m (a l −a n )T)| 2 ,2≤n≤L. 72 Simplifying, we get |h 1 |− X j6=1 |h j ||sinc(πf m T(a j −a 1 ))| ≥ |h 2 |+ X j6=2 |h j ||sinc(πf m T(a j −a 2 ))| and ≥ |h 3 |+ X j6=3 |h j ||sinc(πf m T(a j −a 3 ))| and . . . ≥ |h L |+ X j6=L |h j ||sinc(πf m T(a j −a L ))|. (3.38) TheL−1 inequalities in (3.38) define a bound on the channel gains such thatmax|h i | = |h 1 |⇒b ∗ =a 1 . For the purpose of illustration, let us consider a two tap scenario. In this case, the bound in (3.38) can be reduced to |h 1 | |h 2 | > 1+|sinc(πf m T(a 2 −a 1 ))| 1−|sinc(πf m T(a 2 −a 1 ))| . (3.39) When evaluated for a typical underwater system with f m = 27kHz (carrier at center frequency), T = 10ms, |a 1 − a 2 | = 2× 10 −3 , the bound evaluates to |h 1 | |h 2 | > 1.13. This shows that for a two tap scenario with a large scale spread, the optimal resampling parameter is the time scale on the strongest tap even when the channel magnitude on this path is slightly larger than the second. 3.5.2 Simulations and Discussion We first evaluate the cost functionJ m (b) for small and large values of scale spreadD s to illustrate the nature of the optimal resampling parameter. Recall that Equation (3.33) is the function to be optimized for the single carrier scenario. For the one carrier scenario, we note that J m (b) does not depend on the noise variance. We consider a channel with 73 −5 0 5 x 10 −3 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Resampling factor Optimization function Similar Doppler Different Doppler −0.004 0.0005 0.0025 0.0045 Figure 3.1: J m (b) derived from the HCRB for the special one carrier scenario showing a single peak for small Doppler spread versus multiple distinct peaks when the Doppler spread is large. Channel (small D s ) = [0.6288, 0.2293, 0.0435], Channel (large D s ) = [0.4672, 0.5604, 0.5184], Doppler (smallD s ) = [0.0025, 0.0020, 0.0028], Doppler (large D s ) = [ -0.004, 0.0005, 0.0045 ],f m = 27kHz,T = 10.7ms. three paths and D s = 8× 10 −4 and D s = 8.5∗ 10 −3 respectively for the two special cases. Fig. 3.1 shows a plot of J m (b) and illustrates the key points in the analysis for the two scenarios. We observe that when D s is small, J m (b) has a single peak at the weighted value given by (3.34), and for largeD s , the optimization function peaks at the time scaling factor on each path. The optimal resampling parameter is then the scaling factor at whichJ m (b) is maximum, i.e., the time scaling parameter on the strongest path. Next, we consider the scenario where data symbols are modulated on several carriers. Assuming that the data are drawn from a finite constellation, we minimize the HCRB to compute the optimal resampling parameter. In practice, computing the HCRB is com- putationally intensive operation as it involves a multi-dimensional search over parameter space [24] and can be evaluated in a short time only for very small values ofK. For illus- tration, we consider a systemK = 4,Δf = 100Hz andf c = 10kHz and the channel has four paths. Fig. 3.2 shows a logarithmic plot of the HCRB for one particular realization of the channel shown below the figure. It is observed that the minimum of the HCRB is 74 Figure 3.2: Plot of the Hammersley Chapman Robbins Bound as a function of the resam- pling parameter. The minima in the plot occurs at a value very close to the Doppler shift on the strongest tap. Channel = [1.0856, 0.2435, 0.6786, 0.3498], Doppler = [0.0008, 0.0011, 0.0006, 0.0011],K = 4,Δf = 100Hz,f c = 10kHz andD s = 5×10 −4 . very close to the time scale on the strongest tap as expected from our previous analysis (see Section 3.5.1). For largerK, we consider the soft-estimates of discrete parameters and minimize the CRB and the AMSE (see (3.29) and (3.30)) to compute the optimal resampling parame- ter. This optimal resampling parameter is computed based on perfect knowledge of the channel realization and provides a benchmark for evaluating the performance of practical resampling parameter estimation algorithms.Figures 3.3 and 3.4 show a plot of the CRB and the AMSE for one channel realization respectively. The optimal resampling param- eter (as marked in both the figures) is approximately the time scale on the strongest tap. For this particular channel realization, the scale spread is small and hence the optimal re- sampling parameter is nearly equal to the weighted average of the time scales. It has been observed through numerical simulation that the optimal resampling parameter obtained from both the CRB and the AMSE differ negligibly. 75 −3 −2 −1 0 1 2 3 x 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 Resampling factor Cramer Rao Bound Optimal CRB Resampled CRB −0.001 Figure 3.3: Cramer Rao Bound (CRB) as a function of the resampling parameter for one channel realization showing the performance loss between optimal matched filtering and resampling with various values ofb. The minimum of the CRB (marked in the figure) is very close to the Doppler on the strongest tap and this value minimizes the information loss. Channel magnitude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ] , Doppler = (- 1)* [0.0010, 0.0012 , 0.0005, 0.0004, 0.0011], K = 128, Δf = 39Hz, f c = 15kHz, D s = 8×10 −4 . −3 −2 −1 0 1 2 3 x 10 −3 0 50 100 150 200 250 300 350 Resampling Factor Aggregate Channel Error −0.001 Figure 3.4: Aggregate Mean-Square Error as a function of the resampling parameter for one channel realization showing the channel approximation error. The AMSE minimiz- ing value (marked in the figure) is close to the strongest tap Doppler. Channel magni- tude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ], Doppler = [-0.001, -0.0012, -0.0005, -0.0004,-0.0011],Δf = 39Hz,f c = 15kHz,D s = 8×10 −4 . 76 3.5.3 Performance Improvement - ML Detection To illustrate the performance improvement that can be obtained by using optimal re- sampling in comparison to strongest tap resampling in [77, 49], we consider a scenario using K = 8 carriers, a three tap channel with relative power profile of [0,−2,−4]dB andD s = 2×10 −3 . The power profile ensures that most channel realizations have taps of comparable magnitude, while the moderate scale spread ensures that scale parame- ters are not fully clustered together (small D s scenario) or widely separated (large D s scenario). From our previous analysis, this is the regime where the optimal resampling parameter is not necessarily close to the scale parameter on the strongest tap. We as- sume perfect knowledge of all the channel parameters, and as a result, know exactly the time scale associated with the strongest path as well as the optimal resampling factor, and perform Maximum Likelihood (ML) detection assuming optimal and strongest path resampling (We consider a brute force ML search decoding as data detection algorithms which combat residual ICI in this scenario are beyond the scope of this work). Fig. 3.5 shows the symbol error rate (SER) for data detection for optimal and strongest path re- sampling schemes. It is observed that the performance gained by optimal resampling is nearly 1.5dB at an SNR of 15dB and improves with SNR as the effect of mismatch in resampling factor dominates noise. 3.6 Estimation of the Resampling Parameter from Sam- pled Discrete Time Signals In this section, we derive two algorithms for estimating the optimal resampling parameter from sampled versions of the received signal assuming minimal knowledge of channel parameters. The first is blind to the transmitted symbols and the second assumes that 77 5 10 15 20 10 −3 10 −2 10 −1 SNR Symbol Error Rate Optimal Resampling Strongest Tap Resampling Figure 3.5: Symbol Error Rate for ML detection showing the performance improve- ment obtained using optimal resampling versus strongest tap Doppler resampling,Δf = 62.5Hz,fc = 10kHz,K = 8,D s = 2×10 −3 . the receiver has complete knowledge of the symbols through a pilot sequence. We then compare through simulation our algorithms and a packet-length based estimator. Recall that the received signal (from (3.4)) is given as r(t) = X k,l h l s k e −j2πf k τ l e j2πf k (1+a l )t +n(t), (3.40) The samples are obtained by down-convertingr(t) and then sampling the signal at every T/K seconds: The sampled signal in vector form is given as r = X l h l D l A l ΓB l s+n, (3.41) 78 where, for thel th channel tapD l andB l are diagonal matrices,A l is a DFT-like Vander- monde matrix: D l (m) = e j2πfca l mT K , m = 0,1,··· ,K−1 B l (m) = e −j2πfmτ l , m∈K a [A l ] mk = e j2πmk(1+a l ) K , m,k = 0,1,··· ,K−1, (3.42) andΓ is a selection matrix such thatΓ mk = 1 if thek th data symbol is modulated onto them th carrier and thus models the null subcarriers. The matricesD l andA l model the frequency offset and distortion due to time scaling on the l th path andn ∼ N(0,N 0 I). The signal model in (3.41) can be viewed as that for a multi-user data detection problem with each user transmitting the same symbols. Data sequence detection for this signal model, in general, is non-trivial. However, by considering a reduced rank approach, it is possible to derive low complexity signal processing techniques. The first step in data detection is to derive an estimator for the resampling factor using the discrete time model. Applying the channel approximation result in Section 3.4, the received signal is approximated by assuming a common time scaling factorb on each path. The approximated signal is then ˆ r = X l h l D b A b ΓB l s =D b ˜ A b Hs, (3.43) where ˜ A b =A b Γ andD b , H are diagonal matrices given by D b (m) = e j2πfcb mT K , m = 0,1,··· ,K−1 [A b ] mk = e j2πmk(1+b) K , m,k = 0,1,··· ,K−1, H k = X l h l e −j2πf k τ l . (3.44) 79 The optimal resampling parameterb is the one which minimizes the error in approximat- ing the channel (and thus the signal). b ∗ = argmin b |r−ˆ r| 2 = argmin b r−D b ˜ A b Hs 2 . (3.45) Note that since the time scale on each path is approximated with a single common factor b, the approximation is more accurate for small D s . As the scale spread increases, the accuracy of the approximation decreases and may result in degraded estimation accuracy. 3.6.1 Blind estimator Assuming that the receiver does not know the transmitted symbols and frequency re- sponses, we combine them to reduce the number of unknown parameters. The approxi- mated received vector is then ˆ r =D b ˜ A b x, (3.46) where x k = H k s k . In all, we have K equations and |K a | unknowns. The unknown parameter vectorx is uniquely identifiable only whenK > |K a |. This suggests that all subcarriers cannot be loaded with data and at least one row of the matrixΓ is all zero. Let us define the objective functionE(x,b) as the error between the actual and approximated vector, E(x,b) =|r−ˆ r| 2 = r−D b ˜ A b x 2 . (3.47) We utilize the key steps in the deterministic likelihood (DML) method to eliminate the nuisance parameters x in our optimization [72] and estimate the optimum resampling parameterb. For a candidate parameterb, the estimate for the nuisance parameters are ˆ x = ˜ A † b D H b r. (3.48) 80 Substituting the estimate of the nuisance parameters in (3.47), the objective function is now exclusively a function of the resampling parameter. E(b) = r−D b ˜ A b ˜ A † b D H b r 2 . (3.49) IfΓ = I K , then then ˜ A b ˜ A † b = I andE(b) = 0∀b. A detailed discussion on the rank and identifiability is provided in [72]. Simplifying, we have b ∗ = argmin b r−D b ˜ A b ˜ A † b D H b r 2 = argmax b r H D b ˜ A b ˜ A † b D H b r = argmax b Tr n ˜ A b ˜ A † b D H b rr H D b o . (3.50) The optimal value of the resampling parameter maximizes the projection of phase rotated data onto the subspace spanned by ˜ A b ˜ A † b . We observe that the structure ofb ∗ is similar to the estimator structure in [72] as both of them are derived using the deterministic ML method. However, the key difference between the two is the multi-scale signal and the parameter of interest. While our estimator finds the optimal resampling parameter b ∗ , the value which best approximates the multi-scale multi-lag channel with a single time scaling coefficient, the estimator in [72] determines the common time scaling factor on all the channel paths. It can be easily shown that for when the time scaling factor on all the paths are the same, (3.50) reduces to the one in [72]. 3.6.2 Pilot Aided Estimator Using the knowledge of the transmitted symbol, the procedure can be modified to yield a better estimate of the resampling parameter. We assume that a pilot OFDM symbol is 81 transmitted and hence the only unknown parameters are the channel gains. The approxi- mated version of the received vector can be written as ˆ r =D b ˜ A b x, x =SFh, (3.51) whereS is the data vectors expressed as a diagonal matrix. The matrixF relates the subcarrier frequency response in the absence of time scaling, to the channel gain on each path and is given as F =          e −j2πf 1 τ 1 e −j2πf 1 τ 2 ··· e −j2πf 1 τ L e −j2πf 2 τ 1 e −j2πf 2 τ 2 ··· e −j2πf 2 τ L . . . . . . . . . e −j2πf Ka τ 1 e −j2πf Ka τ 2 ··· e −j2πf Ka τ L          . (3.52) and h = [h 1 ,h 2 ,··· ,h L ] is the vector of channel gain on each path. In practice, the delays τ l can be estimated from the transmitted preamble in front of each packet. By approximating the multiple time scales with a single common scaling parameter, and following the procedure earlier described, we get b ∗ = argmin b E(b), E(b) = r−D b ˜ A b SFF † S H ( ˜ A b ) † D H b r 2 . (3.53) We note that the proposed estimator shares the same structure as the high SNR version of [62]. However, there are two key differences between the derivation of the estimators, in addition to the multi-scale channel model considered in this work. The first is that the first stage of the estimator in [62] considers time scale estimation as CFO estimation versus true scale estimation. Secondly, while we treat all parameters as deterministic, [62] assumes that the distributions of the nuisance parameters are known a priori. Thus, 82 while the two schemes have similar structures, their derivations are fundamentally dif- ferent. Similarly the overall performance characterization would be different for both the estimators. Fig. 3.6 shows a plot of the error function for the blind and pilot aided estimators respectively. The channel gain and the scaling factors are identical to the ones used in Figures 3.3 and 3.4 respectively and provide a illustration of the error function for each of the estimators. The minimum of both the error functions (marked in the figure) are as predicted nearly equal to the time scale on the strongest tap. 3.6.2.1 Delay parameter estimation The pilot aided estimator in (3.53) assumes perfect knowledge of the path delays to per- form the optimization. However, we show through simulation, by rounding{τ 1 ,...,τ L } to the nearest sample time in (3.53), that estimation of the optimal resampling parameter is insensitive to accurate knowledge of path delays. The robustness of the estimator to imperfect delays allows the use a simple matched filter correlation based timing estimator in practical systems, without any loss in performance. 3.6.3 Packet Length Based Estimation In [77, 49], a packet length based estimator which utilizes the change in packet length due to compression/dilation is used to estimate the optimal resampling parameter. This estimation algorithm is based on the Doppler tolerant properties of LFM sequences and is as follows. 1. Generate a LFM sequence of duration 100ms, Start Frequency = f c −B/2, End Frequency =f c +B/2. 83 2. Generate a packet with several OFDM symbols. Place the LFM sequence followed by a silence period of 100ms at the beginning of the packet and append a silence period of100ms followed by the LFM sequence at the end of the packet. 3. Simulate the channel by rescaling the packet at a different rate on each path and combining them. 4. Correlate the received packet with the LFM sequence and find the peak at the preamble and postamble. This is similar to performing coarse timing estimation, i.e., finding the location of the path with the strongest correlation peak. 5. Measure the time interval between the two peaks and determine the time scaling using the original packet length. The accuracy of time scale estimation depends on the duration of the packet, larger dura- tion and bandwidth leading to better accuracy. Though this algorithm is computationally inexpensive, it has a few drawbacks because of certain implicit assumptions. First, it is assumed that the strongest path during the preamble and the postamble are the same physical path, which is increasingly violated as the packet duration increases. Also, the time scale resolution and hence the estimation accuracy may be inadequate for packets with very short durations such as control packets. Secondly, the delay spread influences the estimation accuracy as well. For small delay spreads, it is observed that the error increases as each path does not necessarily produce a peak when the LFM sequence is correlated with received packet. Finally, the algorithm implicitly assumes that each path has the same time scaling and the effect of different scales on every path is unknown. In our simulation, we assume that the channel gain remains unchanged through the packet duration and a packet length of1.5secs is used. 84 3.6.4 Simulation Results - Performance Comparison and Discussion We compare the performance of blind, pilot aided and packet length based resampling factor estimators for multi-scale multi-lag signals. We consider an underwater channel with 5 paths. The channel power profile is chosen to simulate a scenario where the energy is spread among several taps and the relative tap strengths are given as [0, -3, -5, -7, -10]dB. When most of the energy is concentrated in one tap, our analysis shows that the time scale on this path is the optimal resampling parameter. However, a scenario where channel energy is spread among several taps allows us to investigate the worst case performance of the derived algorithms. The maximum delay spread is assumed to be 5ms and for each channel realization, the path delays are chosen randomly from [0,5]ms. The time scales are randomly generated and processed to get the desired scale spreading. The performance of these algorithms depends on the subcarrier spacing Δf, as this parameter influences the amount of ICI, the primary cause of detection errors. We consider three different values of Δf in our simulations and the search resolution for the scaling parameter is 10 −4 . For the pilot aided estimator, we consider the case of known and approximate delays (delays rounded to the nearest sample time to illustrate robustness, when path delays are not exactly known). Figures 3.7, 3.8 and 3.9 show the standard deviation of the error in resampling factor estimation for the blind and pilot aided estimators in comparison with the classical packet length based estimator withb ∗ MSE as the reference, for Δf = 39Hz,19.5Hz and 9.75Hz respectively. They also show the comparison between the pilot aided estimator with known and approximate delays. We observe that the pilot aided estimator significantly outperforms the blind and packet length based estimation algorithms for all values of Δf. It is also observed that the increase in standard deviation of the error for pilot aided estimator with approximate delays is much smaller than the search resolution for the 85 optimum resampling parameter; thus illustrating that inaccuracies in estimating delays do not impact the performance of the estimator. For small scale spreads, it is observed that the blind estimator outperforms the packet length based estimator while for larger spreads the relative performance depends on the bandwidth B of the signal. For large values ofB, the packet length estimator performs better than the blind estimator and for smaller B the blind estimator performs better. This is due to the fact that bandwidth of the LFM sequence (same as OFDM signal bandwidth) impacts estimation performance and the resolution accuracy decreases for LFM sequences with smaller bandwidths. Computation cost: The blind and pilot aided estimators project the data onto a sub- space characterized by the resampling parameter and the optimum value is the one which minimizes the error in projection. This involves a search over all possible values of the resampling parameter and the search range is bounded by the maximum relative veloc- ity between the transmitter and the receiver. For the blind estimator, for each candidate parameterb, all the projection matrices can be computed and stored a priori and a vector matrix multiplication and a vector norm computation are the only operations involved. The complexity of the pilot aided estimator is higher as it depends on the path delays, but the matrices need to be computed only once for each packet. Overall, though both the estimators are somewhat memory intensive, they are computationally simple and easily parallelizable. 3.7 Conclusions In this work, we considered multi-scale multi-lag signals occurring in underwater acous- tic communications. Using OFDM signaling, resampling for time scaling compensation was investigated and shown to be a reduced rank signal processing strategy. Two metrics to compute the optimal resampling parameter based on bounds for data detection error 86 and channel approximation were derived assuming all the channel parameters are per- fectly known. An approximate solution for the optimal value of resampling parameter is derived for two special cases. For small scale spreads, the resampling factor is shown to be approximately the weighted average of the time scales on each path and for large spreads, the scaling factor on the strongest tap is the optimal value. In general, it is ob- served that the optimal resampling parameter is very close to the time scale on the path with largest signal energy. However, when the received signal has comparable energy on several paths and moderate scale spreads, optimal resampling results in a significant performance gain even for moderate SNR’s where the effect of ICI dominates over noise. We observe that resampling using the Doppler shift of the dominant path is essentially the classical approach to resampling [49]; herein we have shown under what conditions such an approach is optimal. Blind and pilot aided estimators for the optimal resampling parameter are derived and simulation results show that the proposed algorithms outper- form significantly the commonly used packet length based estimator at the expense of a moderate increase in complexity. 87 −3 −2 −1 0 1 2 3 x 10 −3 10 0 10 1 10 2 10 3 Resampling Factor Estimation Error Blind Estimation Error Pilot Aided Estimation Error −0.001 Figure 3.6: Estimation Error of Blind and Pilot Aided Resampling as a function of re- sampling parameter for one channel realization. The minimum of each of the functions is exactly equal to the value predicted by the CRB and AMSE criterion in this case. Chan- nel magnitude = [0.8571, 0.5549, 0.3825, 0.1578, 0.1971 ], Doppler = [-0.0010, -0.0012, -0.0005, -0.0004, -0.0011],Δf = 39Hz,f c = 15kHz,D s = 8×10 −4 . 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 0 1 2 3 4 5 6 7 8 x 10 −4 Doppler Spread Standard Deviation of Estimation Error Blind Pilot Aided − Known Delays Pilot Aided − Approx. Delays Pkt. Length Based Figure 3.7: Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel for Δf = 39Hz,f c = 15kHz,K = 128, Power Profile = [0− 3− 5− 7− 10]dB,B = 5KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccu- racies. 88 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 0 1 2 3 4 5 6 x 10 −4 Doppler Spread Standard Deviation of Estimation Error Blind Pilot Aided − Known Delays Pilot Aided − Approx. Delays Pkt. Length Based Figure 3.8: Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel for Δf = 19.5Hz,f c = 15kHz,K = 128, Power Profile = [0−3−5−7−10]dB,B = 2.5KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccuracies. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10 −3 0 1 2 3 4 5 6 7 8 x 10 −4 Doppler Spread Standard Deviation of Estimation Error Blind Pilot Aided − Known Delays Pilot Aided − Approx. Delays Pkt. Length Based Figure 3.9: Performance comparison of blind, pilot aided (known and approximated delays) and packet length estimators in a multi-scale multi-lag channel for Δf = 9.75Hz,f c = 15kHz,K = 128, Power Profile = [0−3−5−7−10]dB,B = 1.25KHz. Plot also shows the insensitivity of the pilot aided estimator to delay estimation inaccu- racies. 89 Chapter 4 Channel Estimation 4.1 Introduction Acoustic signaling in underwater communication systems is inherently wideband. Re- cent work models the underwater acoustic (UWA) channel as a multi-scale multi-lag (MSML) channel in which each path experiences a different delay and scale [98, 9]. While over-the-air narrowband communications have significantly benefited by a tapped delay line model for the time-varying channel, such a model has been only recently in- vestigated for wideband channels, and in particular for UWA channels in [32, 94]. While [32] considers the channel modeling in baseband, it employs a wavelet-based pulse hav- ing bandpass characteristics for signaling. In contrast, the model in [94] allows for both parameterization and signaling in baseband. A key appealing feature of [94] is that the received signal can be approximated with the sum of several time-invariant channel each with a different frequency and bandwidth. Additionally, the model proposed in [94] provides a natural framework for parallel transmission over multiple frequency bands (termed multi-layer signaling). In this paper, we consider UWA channel estimation us- ing the proposed signal representation in [94]. The problem of UWA channel estimation has been of recent research interest. In [49], 90 the dominant components the delay-Doppler spread function for time-varying channels are estimated by minimizing the least squares prediction error. Classical techniques such as root-MUSIC and ESPRIT and various pursuit algorithms from sparse approximation theory such as orthogonal matching pursuit (OMP) and basis pursuit (BP) have been proposed in [9] for channel estimation for Orthogonal Frequency Division Multiplexing (OFDM) signaling. In [35], the time-scale characterization proposed in [32] has been exploited for estimating channel parameters and [65] uses homotopy algorithms for esti- mating parameters in sparse underwater channels (sparse approximation theory has been used extensively for channel estimation problems in general and this work does not sur- vey these developments). In this chapter, we consider multi-layer signaling over the UWA channel. The pilots used in each layer can be used to estimate the channel and previously proposed tech- niques such as [9] can be exploited by adapting them to the underlying signaling in each layer. As the underlying channel parameters are common for all the layers, the received samples from each layer can be theoretically combined together to provide better esti- mates of the channel. However, it is observed that directly combining the information from multiple layers leads to degraded channel estimates at the receiver, even for non time-varying channels, due to basis mismatch resulting from quantizing the parameter space for constructing the dictionary. We propose to pre-process the signals using tech- niques from sparse approximation to improve the channel estimates. For this purpose, we utilize several algorithms proposed for designing optimized measurement matrices [18, 13, 93, 104, 102]. Numerical simulations show that the channel estimation error can be significantly reduced by using dictionary modifying pre-processing in all regimes of interest. This chapter is organized as follows. Section 4.2 presents the model for signal param- eterization and briefly describes multi-layer signaling. The channel estimation algorithm 91 is presented in Section 4.3 and Section 4.4 describes methods to improve estimation us- ing dictionary optimizing algorithms. Section 4.5 presents simulation results and Section 4.6 concludes this chapter. 4.2 Signal Model We begin by briefly describing the parameterized baseband data model for representing the received signal for multi-layer transmissions over UWA channels. Letx(t) be the transmitted passband signal over an UWA channel. The signal travels along several paths each characterized by a delayτ and scaleα≈ 1+ 2u c , whereu is the relative velocity between the transmitter and the receiver in the direction of the arriving path and c is the speed of sound in water. The received signal r(t) can be expressed as the superposition of signals arriving from several paths each with a different scale and delay: r(t) = P X p=1 h p √ α p x(α p (t−τ p ))+w(t), (4.1) whereh p ,τ p andα p are the gain, delay and scale parameters of thep th path andw(t) is ad- ditive while Gaussian noise (AWGN). We assume thatα p ∈ [1,α max ] andτ p ∈ [0,τ max ], where α max and τ max are the maximum scale and delay spread respectively. For single layer signalingx(t) =ℜ[¯ x(t)e j2πfct ] with bandwidthW ∗ = 1/T ∗ at the transmitter, the signal at the receiver can be expressed using the parameterization framework in [98] as r SL (t)≈ℜ " R∗ X r=0 e j2πfca r ∗ t L∗(r) X l=0 ¯ h r,l a r/2 ∗ ¯ x(a r ∗ t−lT ∗ ) # +w(t), (4.2) whereR ∗ = ⌈ logαmax loga∗ ⌉ is the maximum number of observed receive layers for a single layer transmission and L ∗ (r) = ⌈a r ∗ τ max /T ∗ ⌉ is the number of uniformly spaced taps 92 at the r th layer. In other words, for a single layer transmission of bandwidth W ∗ and center frequency f c , the received signal can be expressed as a sum of signals on R + 1 layers with bandwidths [W,a ∗ W,...,a R ∗ W] and center frequencies [f c ,a ∗ f c ,...,a R ∗ f c ] respectively each passing through a time-invariant channel. The parametersa ∗ andT ∗ = 1/W ∗ are called the basic scaling factor and the translation spacing and together define the non-uniform grid which represents the signal and ¯ h r,l is the channel gain of the l th delay at ther th layer relative to the parameters of the transmitted layer. In the multi-layer signaling strategy proposed in [94], the overall transmitted signal is the sum of the signals transmitted on K ′ different layers. Let p k (t) = a k/2 ∗ p(a k ∗ t) be a strictly bandlimited pulse with bandwidth a k ∗ W ∗ to be used for signaling in this layer. The baseband signal for thek th layer can be expressed as ¯ x k (t) = X n s k,n p t− n a k ∗ W ∗ = X n s k,n p(t−na k ∗ T ∗ ), (4.3) where s k,n are the data symbols. Now, the passband transmit signal for the k th layer is x k (t) =ℜ[¯ x k (t)e j2πfca k ∗ t ]. We usep(t) = p 1/T sinc(t) as the choice of transmit pulse. This ensures that the signal on any given layer is spectrally non-overlapping with signals on any other layer and hence eliminates crosstalk (see [94] for a detailed explanation). The overall transmitted signal can then be expressed as x(t) = K ′ −1 X k ′ =0 X n ℜ a k ′ /2 s k ′ ,n p(a k ′ t−nT)e j2πfca k ′ t . (4.4) 93 At the receiver, the signal component at the (k +r) th layer (r = 0,1,2...,R) from the signal transmitted on thek th layer can be expressed as r SL k+r←k (t) =ℜ   e j2πfca k+r ∗ t L∗(r) X l=0 ¯ h k r,l a k+r 2 ∗ ¯ x(a k+r ∗ t−lT ∗ )   . (4.5) After down conversion and matched filtering, the discrete time signal at the (k +r) th layer from the signal transmitted at thek th layer can be expressed as ¯ y k+r←k,m = L∗(k+r) X l=0 ¯ h k r,l s k,m−l +w k+r,m , (4.6) where ¯ h k r,l = P X p=1 h p e −j2πfca k+r ∗ τp sinc r− logα p loga ∗ sinc l− a k+r ∗ τ p T ∗ . (4.7) In other words, the samples obtained at thek th layer at the receiver are a combination of signals transmitted from layers(k,k−1,...,k−R) and can be expressed as y k,m = R X r=0 L∗(k+r) X l=0 ¯ h (k−r) r,l s k−r,m−l +w k,m . (4.8) The above expression shows that the signal at each received layer is subject to inter- symbol interference (ISI) from symbols on that layer and inter-layer interference (ILI) from symbols on layers transmitted at lower frequencies. To combat ISI, we consider a block based transmission scheme at each layer with each block containingN +Z data symbols. One such block can be expressed as K ′ −1 X k ′ =0 N+Z−1 X n=0 Re a k ′ /2 s k ′ ,n p(a k ′ t−nT)e j2πfca k ′ t . (4.9) 94 To avoid interference between adjacent blocks, we consider a cyclic prefixed single car- rier transmission, i.e., we introduce a cyclic prefix at the beginning of each block. s k ′ ,n =        b k ′ ,n−Z , forZ ≤n<N +Z b k ′ ,N+n−Z , otherwise, (4.10) whereb k ′ ,n stands for then th information symbol transmitted on thek ′ th layer. Note that interference between adjacent blocks will be completely eliminated when Z is chosen such thatZ ≥⌈ a R+K ′ −1 ∗ τmax T ⌉. At the receiver, after discarding the cyclic prefix on all the blocks, the received signal at each layer can be compactly represented as y k = R X r=0 H (k−r) r b k−r +w k , (4.11) wherey k = [y k,Z ,...,y k,N+Z−1 ] T ,w k = [w k,Z ,...,w k,N+Z−1 ] T ,b k ′ = [b k ′ ,0 ,...,b k ′ ,N−1 ] T andH t r denotes aN×N circulant matrix with first column[h t r,0 ,...,h t r,L∗(k) , 0,...,0 | {z } N−1−L∗(k) ] T . The circulant nature of the channel matrix H t r suggests that further processing could be performed in frequency domain as in a single-carrier frequency domain equalized system. 4.3 Channel estimation We now propose techniques to estimate the channel gain matricesH t r using pilot sym- bols transmitted at each layer. In the first step, we transform the received signal to the frequency domain by multiplying the signal received at each layer with the unitary DFT 95 matrix F n,m = 1 √ N e −j2π mn N . Stacking together the results from each layer, the received signal can be compactly represented as˜ y = ˜ H ˜ b+ ˜ w :          ˜ y 0 ˜ y 1 . . . ˜ y R+K ′ −1          | {z } ˜ y =                 D 0 0 0 . . . . . . . . . D 0 R . . . . . . D K ′ −1 0 . . . . . . 0 D K ′ −1 R                 | {z } ˜ H          ˜ b 0 ˜ b 1 . . . ˜ b K ′ −1          | {z } ˜ b +˜ w (4.12) where D t r = FH t r F H is an N × N diagonal matrix whose n th diagonal element is d t r,n = P L∗(r+k) l=0 h t r,l e −j 2π N nl and ˜ b k =Fb k . Observe that the received samples at thek th layer are dependent on the transmitted symbols from layersk,k−1,...,max(k−R,0) and hence there would be an intermixing of the transmitted symbols from several layers. This also implies that transmitted pilot symbols at a given layer would in general be combined with the pilots in other layers. 4.3.1 Design of pilot sequences In this work, we adopt a pilot design which allows the separation of the effect of symbols from different layers and permits simpler channel estimation at the receiver. From (4.12), we observe that if ˜ b k , ˜ b k+1 ,..., ˜ b k+R have non-overlapping support for any k, then the pilots transmitted from each layer can be separated and used to cleanly estimate the channel matrixD k r between thek th transmit layer andk,k+1,...,k+R received layers. This is feasible as the elements ofD k r are highly correlated: d k r,n = P L∗(r+k) l=0 h t r,l e −j 2π N nl with L ∗ (r +k) << N and hence need very few pilots compared to the length of the 96 blockN. From the literature on OFDM systems, it is well known that placing pilots uniformly in frequency and optimizing the transmitted pilot symbols to follow certain orthogonality constraints can minimize the mean squared estimation error [5]. Using the intuition from [5], we first note that by picking the support of ˜ b k to be uniform in frequency reduces the channel estimation error in general. Let us denote by S k the support of the pilots transmitted on thek th layer in the frequency domain. ClearlyS i ∩S j =∅ for|i−j|≤R. ChooseS k ={k,k +S,k +2S,...,k +⌊ N S ⌋S} (whereS > R) to be the support of the pilots on thek th layer. This choice satisfies the uniform pilot placement criterion and the orthogonality constraint. LetN p be the maximum number of pilots required to estimate the channel from any layerk to layerk+r. We note that this gives a lower bound on the length of the block: N >N p (R+1). For simplicity, we choose all the pilot symbols in frequency domain with non-zero supports to be unit symbols, i.e., ˜ b k S k = 1 for all elements ofS k . The actual transmitted pilot symbols can be obtained from the frequency domain pilot symbols asb k =F H ˜ b k . We note that this choice of transmit symbols is not the pilot sequence that minimizes the channel estimation MSE [5]. Designing optimal pilot sequences for this problem is beyond the scope of this work. 4.3.2 Channel estimation At the receiver, the samples corresponding to the pilot symbols are separated without interference and the received symbols at the(k+r) th layer from transmit symbols in the k th layer can be expressed as ˜ y k+r (S k ) =D k r (S k ,S k ) ˜ b k (S k )+ ˜ w k+r (S k ), (4.13) 97 where the notation˜ y k+r (S k ) represents the components of the vector˜ y k+r correspond- ing to the setS k andD k r (S k ,S k ) is the sub-matrix formed by selecting the corresponding rows and columns ofD k r . Rewriting the above expression in terms of the channel param- eters of interest, we have ˜ y k+r (S k ) = ˜ B(S k ,S k )F(S k ,1 :Z)h k r + ˜ w k+r (S k ), (4.14) whereh k r = [h t r,0 ,h t r,1 ,...,h t r,Z−1 ] T . Substituting for the values of the pilots in the above equation and multiplying with the pseudo-inverse of the partial Fourier matrix, we get ˆ h k r = (F(S k ,1 :Z)) † ˜ y k+r (S k ) =h k r + ˆ w k r , (4.15) where ˆ w k r = (F(S k ,1 : Z)) † ˜ w k+r (S k ). By varying the values of k and r, we note that we can get all the channel gains of interest from (4.15). While this method allows for a simple estimation of all the channel gains, this technique does not exploit the relationship between the various vectorsh k r fork = 0,1,...,K ′ −1 andr = 0,1,...,R given by (4.7). 4.3.3 Exploiting the relationship between parameters Utilizing the underlying model for the channel gains (see (4.7)), the tap strengths can be expressed in terms of the gain, delay and scale on the paths as             h k r,0 . . . h k r,l . . . h k r,Z−1             =       e −j2πfca k+r ∗ τp × sinc r− logαp loga∗ × sinc l− a k+r ∗ τp T∗       | {z } l=0,1,...,Z−1 and p=1,2,...,P          h 1 h 2 . . . h P          + ˆ w k r . (4.16) 98 Clearly, the systems of equations is highly non-linear in the unknown parametersτ p and α p and hence directly exploiting the correlation between the channel gains is computa- tionally expensive. We approach this problem by casting it as selection of columns from an over-complete dictionary, comprised of column vectors parameterized by a selection of possible choices of parameters τ p and α p (see [9] for more details on an application of this technique). Following this approach, the parameterization for the scale and delay values are chosen to be τ ′ = {0, T λ , 2T λ ,...,⌈τ max ⌉} and α ′ ∈ {1,1 + Δα,...,α max }. LetM =|τ ′ |×|α ′ | be the total number of candidate pairs for the delay and scale. Now, the system of equations for the channel gains can be expressed as             h k r,0 . . . h k r,l . . . h k r,Z−1             =       e −j2πfca k+r ∗ τ ′ × sinc r− logα loga∗ × sinc l− a k+r ∗ τ ′ T∗       | {z } l = 0,1,...,Z−1&(τ ′ ,α ′ )          g 1 g 2 . . . g M          + ˆ w k r . (4.17) The desired channel gains from thek th transmit layer to the(k+r) th received layer can be compactly expressed as h k r =A k r g+ ˜ w k r . (4.18) When the actual path delaysτ p and the scalesα p of the channel estimation problem are from the setsτ ′ andα ′ respectively, then there is no basis mismatch and the overcomplete dictionaryA k r can exactly represent the channel gainsh k r,l in the noiseless scenario. For such scenarios, the problem is truly sparse and can be effectively solved using recovery algorithms such as OMP and BP. However, this cannot be expected in a real physical 99 system and in general there exists a mismatch between the columns ofA k r and the ac- tual parameter values. This mismatch results in loss of sparsity in representation and in general degrades the performance of pursuit algorithms such as OMP and BP. Note that (4.18) only considers one of the active links for estimation. In all, we have we have a total of (R + 1)K ′ such available equations for estimating the desired parameters and a subset of such matrix-vector equations can be selected for solving the channel estimation problem.             h 0 0 . . . h 0 R . . . h K ′ −1 R             =             A 0 0 . . . A 0 R . . . A K ′ −1 R             g+             ˜ w 0 0 . . . ˜ w 0 R . . . ˜ w K ′ −1 R             . (4.19) 4.3.4 Discussion: Estimation of time scale In most scenarios of interest, the typical value ofα p is of the order of1+10 −4 to1+10 −3 with α max = 1.01, which corresponds to a maximum velocity of 15m/sec. For these values of scale parameters, we observe that the term sinc r− logα loga∗ in (4.7) is very close to 1 when r = 0 and nearly zero when r 6= 0 for a ∗ = 2 used in this work. As r 6= 0 corresponds to the scenario where the transmitted signal at thek th layer is being observed at the(k+r) th receive layer, it implies that ILI is a atleast an order of magnitude below the noise level whenr 6= 0. This poses significant difficulties in scale estimation when using the parameterized baseband representation framework. When using sparse approximation algorithms, which are approximate algorithms in general, we observe in simulations that determining the scale parameters on each path is not feasible. This difficulty in estimation can be attributed to both the parameterized channel model using 100 a ∗ = 2 as mandated by the choice ofp(t), as well as the sparse approximation framework used to determine the values of the non-linear parameters. However, if desired, all the channel gains can be estimated using (4.15) without exploiting the underlying structure in the model (the estimation accuracy of inter-layer channel gains is poor for this scenario). Designing transmit pulses which have a smallera ∗ and also reduce the error between the received signal r(t) and its parameterized representation r SL (t) is currently a problem under investigation. 4.4 Pre-processing for optimizing L1 recovery From the previous section, we note that using oversampled dictionaries obtained by quan- tizing the space of parameters and applying sparse recovery techniques is one of the tech- niques available for determining the channel parameters which depend non-linearly on the observations. We considerR = 0 which allows the simplification of (4.19). We first note from (4.16) that the columns of the sensing matrix for different layersk 1 andk 2 are very closely related to each other. For example, consider the case when τ p = β p T ∗ for some integer β p . In this case, we see that the magnitude of the columns of the sensing matrix of the higher layer are an upsampled version of the magnitude of the columns of the sensing matrix from the lower layers. Next, we observe that finer quantization (sampling) of the parameter space in (4.16) increases the correlation between the columns ofA thereby increasing the mutual co- herence of the sensing matrix. It is known that high mutual coherence can degrade the performance of sparse recovery algorithms for the perfectly basis matched scenario [18]. However, our simulation results in Section 4.5 show that the degradation caused due to basis mismatch is much higher and having a larger mutual coherence in the mismatched 101 case actually improves recovery. As increasing the number of observations by append- ing observations from more layers typically reduces the mutual coherence, we observe that utilizing all the information available to us actually degrades performance in the mismatched scenario. To remedy this situation, in this section, we use a pre-processing matrix that compresses the number of available observations while simultaneously mod- ifying the coherence properties of the dictionary to enhance the performance of sparse recovery algorithms. We first define some key terms and then briefly describe the tech- niques and metrics proposed in the literature [18, 13, 93, 102] for this purpose. LetP be the pre-processing matrix. Then,PA be the effective dictionary matrix at the receiver andG =A H P H PA be its Gram Matrix. Then, the pre-processed received signal can be expressed as Ph =PAg+P˜ w. (4.20) The mutual coherence of a dictionaryA is the largest absolute value of the normalized inner product between the columns ofA. Formally, µ (A) = max i6=j |a H i a j | ||a i ||·||a j || (4.21) The mutual coherence provides a measure of the worst case similarity between the dic- tionary columns and is closely related to recovery guarantees of pursuit techniques. It has been observed that minimizing a function of the average coherence of the matrixPA would yield better results in general [18, 13]. Define the t-averaged mutual coherence of a dictionary with normalized columnsµ t as: µ t (PA) = P i6=j (|g ij ≥t|)·|g ij | P i6=j (|g ij |≥t) (4.22) 102 and the total coherence of a dictionary as µ t = X i6=j g 2 ij =||G H G−I|| 2 F . (4.23) While the optimization in [18] aims to reduce thet-averaged mutual coherence of the ef- fective dictionaryPA, [13] and [104] provide an approximate and exact solution to mini- mize the total coherence of the dictionary. In this chapter, we investigate the performance of optimizing both the t-averaged mutual coherence [18] (denoted t-averaged MC) and total dictionary coherence [13] for the UWA channel estimation problem. The pre- processing modifies the coherence between the columns and improves the performance of sparse recovery algorithms as illustrated in Section 4.5. Note that pre-processing modifies the noise covariance matrix and may result in noise enhancement. Balancing between optimizing the dictionary coherence and noise enhancement is an open problem and is not considered herein. 4.5 Simulations In this section, we present results from numerical simulations to validate the solution approach presented in this sequel. We consider an underwater acoustic communication system with K ′ = 2 signaling layers. A sinc pulse p(t) with basic scaling factor of a ∗ = 2 is used for data transmission. The first layer has a center frequency off c = 9kHz and a bandwidth ofW ∗ = 2(a ∗ −1)/(a ∗ +1)f c = 2/3f c = 6kHz and the second layer has a center frequency ofa ∗ f c = 18kHz and bandwidth ofa ∗ W ∗ = 12kHz. The translation spacing is given byT ∗ = 1/W ∗ = 0.167ms. The data and pilot symbols on the first layer are modulated on the pulsep(t) = 1 √ T∗ sinc( t T∗ ) and the symbols on the second layer are modulated on a scaled version ofp(t) given by q 2 T∗ sinc( 2t T∗ ). We assume that each block 103 on a transmitted layer carriesN = 256 data symbols. The channel is composed of P = 10 taps distributed over an interval [0,10T ∗ ]. We consider two distributions in this work: (1) a perfectly basis matched scenario in which the path delays are uniform with an inter-tap interval of 1ms, and (2) randomly spaced taps with zero initial tap delay and an exponential inter-arrival time of mean E[τ p+1 − τ p ] = T ∗ . As mentioned in Section 4.3.4, a stationary scenario is assumed for this work and thusα max = 1 andR ∗ = 0. A cyclic prefix ofZ = 32 symbols is affixed in front of each block to prevent inter-block interference in both the layers considered. The support for the pilot symbols the first layer isS 1 ={1,5,9,...,253} and the support on the second layer isS 2 ={2,6,...,256}. Clearly, the supports sets are non-overlapping and allow for channel estimation even for mobile scenarios when α max > 1. An oversampling factor of λ = 4 is used to construct the overcomplete dictionary and the performance of the algorithm is measured by computing the mean-squared error of the channel estimates h k r . 4.5.1 Without pre-processing Fig. 4.1 shows the performance of channel parameter estimation based on the OMP algorithm (10 taps are identified in each case, note that this is equal to the number of arriving paths) for uniformly spaced taps when using the first and second layer individ- ually and both layers jointly. The tap delays are exactly multiples of T λ and hence there is a perfect match with the dictionary constructed by quantizing the parameter space and the columns of A k 0 ,k = 0,1 exactly span the noiseless received signal. The mutual co- herence of the sensing matrix A 0 0 and A 1 0 are 0.91 and 0.67 respectively for λ = 4 and the mutual coherence of the combined matrix [A 0 0 ;A 1 0 ] used for joint estimation exploit- ing information from both the layers is 0.33. From Fig. 4.1, when using only the first 104 0 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 SNR Mean Squared Estimation Error Using Layer 1 Using Layer 2 Using Layers 1 and 2 Figure 4.1: Performance comparison of channel parameter estimation when using layers 1 and 2 individually and using both layers jointly for uniformly spaced taps. layer is better at low SNR but degrades performance significantly at high SNR due to the misidentification of sparsity support at high SNR. When only the second layer is used, the performance significantly improves at high SNR and using both layers jointly yields an additional1dB gain in estimation error at high SNR. Fig. 4.2 shows the performance of channel estimation for taps with exponentially distributed inter-arrival times when using the OMP algorithm (13 taps are identified in each case). For random tap arrivals, which is usually the scenario in a real system, there is a mismatch between the dictionary and the actual parameter values and hence there exists a modeling error. This implies that noiseless received signal in general does not lie in the column span of the sensing matrixA k r . From Fig. 4.2, it is clear that this modeling error causes loss of sparsity and the 13 taps used are no longer sufficient to reduce the error. In this regime, we observe that having a high mutual coherence improves the estimation as basis mismatch has a larger impact on the performance. 105 0 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 SNR Mean Squared Estimation Error Using Layer 1 Using Layer 2 Using Layers 1 and 2 Figure 4.2: Performance comparison of channel parameter estimation when using layers 1 and 2 individually and using both layers jointly for randomly spaced taps. 4.5.2 With pre-processing We now consider a scenario with front-end processing. We focus on the case where both the layers are used for estimation. A total of 16 + 32 = 48 observations are used to determine the channel gains h p and the delays τ p . We now pre-multiply the received signal in (4.19) with a pre-processing matrixP of dimension30×48. This pre-processes compresses the number of measurements and modifies the distribution of the average coherence between columns of the sensing matrix to improve the performance of sparse recovery algorithms. In our simulations, we set the shrinkage factor for the t-averaged MC to be γ = 0.5 and t = 0.2. Fig. 4.3 shows the mean squared channel estimation error using both the pre-processing methods in comparison to a scenario without pre- processing. We observe that the t-averaged MC performs well at high SNR while total coherence method gives a significant performance improvement at all SNRs of interest. Overall, we observe that preprocessing modifies the coherence of the sensing matrix and improves the performance of the proposed channel estimation algorithms. 106 0 5 10 15 20 25 30 10 −3 10 −2 10 −1 10 0 SNR Mean Squared Estimation Error No preprocessing t−averagedMC Total coherence Figure 4.3: Performance comparison of channel parameter estimation when using layers 1 and 2 with processing using Elad’s and Julio’s method. 4.6 Conclusions We considered the problem of channel estimation for underwater acoustic communica- tions using a newly proposed parameterization framework for the received signal which allows multi-layer signaling. We discuss the advantages and drawbacks of the proposed model from the perspective of channel estimation and propose channel estimators by combining the information available from the various signaling layers used. It is ob- served that directly combining information from multiple layers leads to degraded chan- nel estimation performance. We then propose to use pre-processing to improve the per- formance of L1 recovery algorithms and consider two methods for designing optimal pre-processing matrices. Simulations results show that pre-processing the input data be- fore L1 recovery can significantly improve the performance of UWA channel estimation algorithms. 107 Chapter 5 Transmitter Cooperation for MIMO Multiple Access Channels 5.1 Introduction Next generation wireless networks are being designed to operate in a complex and dy- namic environment in which nodes interact and cooperate to improve network throughput (see [71, 76, 45, 1, 46, 70] and the references therein). As nodes signaling via wireless share resources due to the broadcast nature of the medium, cooperation between such nodes has emerged as a key strategy for improving performance [34]. In typical cooper- ative scenarios, it is inherently assumed that all nodes are controlled centrally and hence cooperation can be enforced. However, the emergence of heterogenous networks without a unified central controller challenges this assumption. In such scenarios, it is reasonable to assume that a rational node would willingly cooperate only if cooperation improves its own utility. The problem of determining which nodes in a network would cooperate in a stable fashion and how the benefits of such cooperation would be shared has thus become important, especially for these future heterogenous networks. If the properties of cooperation are well understood in elementary networks such as the multiple access channel (MAC), the broadcast channel (BC), the interference channel (IC) etc., larger 108 Figure 5.1: Cooperative signaling between base station and femto cell network for joint transmission to UE networks can be viewed as a composition of several elementary networks and coopera- tive behavior can then be examined to draw useful insights for system design. Towards this goal, this paper addresses the problem of TX cooperation in a MAC. Consider a example scenario as shown in Fig. 5.1 in which several (service providers) base stations and femtocells (TXs) operating in the same frequency band are in the range of a mobile (RX). The base stations and the femtocells can be connected via a back- bone wired network which can be used to share channel state and codebook information. However, each TX may be owned by a different operator and each operator would like to provide the largest rate to the mobile to increase his revenue. A rational TX with an intent to maximize revenue, could cooperate with other TXs in the range of the mobile by joint encoding and transmission to improve data rates of the mobile (total sum rate) and improve the data rates for each TX. In such a scenario, the TX cooperation game studied in this work, can be used to determine the optimal coalition structure and the individual rates that each TX gets to transmit. Non-cooperative games between TXs in a MAC has been analyzed for various sce- narios such as fading multiple-input single-output (MISO) channels in [44], for fading MIMO channels in [7] and for incomplete channel state information in [29]. To enhance 109 Figure 5.2: Transmitters cooperate to form coalitions. All transmitters in a coalition fully cooperate with each other the sum rate, rational users (TXs here) can form coalitions and cooperate by signaling jointly to the RX (see Fig. 5.2). In the absence of cooperation costs, there is an incentive to form the grand coalition (GC), the coalitions of all users, and signal jointly such that no data streams are decoded in the presence of interference. A larger coalition implies that the benefits of cooperation have to be shared among many users and the GC is sta- ble only when no coalition of users has an incentive to break away and form a smaller coalition. The key question is to determine whether the gains of the GC are sufficient to share the payoff amongst all its members such that no coalition of TXs has an incentive to defect, thus ensuring stable cooperation. This is equivalent to checking the nonempti- ness of the core, the feasible set of a linear program which describes the demands of each coalition. A nonempty core implies that the GC utility can be shared such that no coalition of TXs has an incentive to defect, ensuring its stability. The issue of stability is one of the key questions addressed in this chapter. Several works in the literature have examined the nonemptiness of the core for TX cooperation in a Gaussian MAC under various assumptions, primarily using character- istic form games (CFGs), in which the utility of a coalition is independent of the actions 110 of members outside it. In [42], it is assumed that TXs bargain for higher rates by threat- ening to jam transmissions. A key assumption in [42] is that the jamming users are not interested in transmitting data, in contrast to what we consider herein. In [57], the util- ity of each coalition of TXs is considered to be a minimum assured utility that can be obtained by assuming that all other TXs coordinate to jam the transmissions. This is a very conservative model of utility and it is shown in [57] that the cooperation is unstable and exhibits oscillatory behavior in general. A packetized, slotted, version of the TX cooperation game was considered in [37] and cooperation was shown to be stable for certain scenarios. TX cooperation with non-zero cost of coalition formation in a slotted TDMA system was considered in [70], and [52] discusses the reinterpretation of several information theoretic results, including the MAC, from the point of view of coopera- tive game theory. As the interference generated by external TXs strongly influences the achievable rates in a Gaussian MAC, the CFG model considered in most previous works, does not accurately measure the interference as is needed to analyze TX cooperation in a classical Gaussian MAC. In [71], the need for taking into account the actual interference that affects a coalition using PFGs [85] has been stated and their potential for PFGs to provide a good framework for modeling and analysis of self-organizing next generation communication networks is discussed. The primary contribution of this work is to study the problem of TX cooperation in a MAC under the framework of partition form game theory and show the stability of TX cooperation in several scenarios of interest. Specifically, we consider a MAC with a single user decoding (SUD) RX which treats interference as noise and a successive interference cancellation (SIC) RX in which de- coded signals are canceled out to reduce interference. The TX cooperation game is ana- lyzed in several stages. The existence and uniqueness of a Nash equilibrium (NE) utility (we make the distinction between uniqueness of NE achievable strategies and NE utilities with the later being a weaker notion) for a non-cooperative game between the TXs is first 111 Receiver Regime Power Constraint Core SUD high SNR SP and PP Nonempty low SNR SP Nonempty SIC - fixed decoding order high SNR SP and PP Empty in general low SNR SP Nonempty SIC - time shared decoding high SNR SP and PP Nonempty low SNR SP Nonempty Table 5.1: Table showing the summary of the nonemptiness of the core of the TX co- operation game for various scenarios. SP = sum power constraint and PP = per-antenna power constraint. examined to determine the achievable rates for a given configuration of TXs. Next, TXs are assumed to cooperate to form larger coalitions and the change in utility from cooper- ation is characterized for both the cooperating and the external TXs. Using the utilities determined previously, we consider the various cores of a PFG [27] (PFGs have several different definitions of the core, based on the expected behavior of external coalitions) and examine the the stability of cooperation by investigating their nonemptiness. Table 5.1 shows a summary of results for the various scenarios considered in this work. For an SUD receiver, cooperation is stable at both high and low SNRs, while for a SIC receiver with a given decoding order, cooperation is stable only in the noise-limited regime and may be empty at high SNR. This can be attributed to the asymmetry between the users caused by a fixed decoding order. However, if time sharing between decoding orders is permitted, we show using a high SNR approximation to the utility function that cooper- ation is stable at high SNR as well 1 . Thus, this work demonstrates the role played by the choice of RX and the regime of operation in determining the stability of TX cooperation in a MIMO MAC. The rest of the chapter is organized as follows. Section 5.2 defines several useful 1 Note that while our results are true at high SNR for both sum power and per-antenna power con- straints, the low SNR regime is characterized only for sum power constraints due to the lack of a suitable approximation of the capacity of a MIMO channel with per-antenna power constraints in this regime [91]. 112 concepts related to cooperative game theory and Section 5.3 describes the signal model for a MIMO MAC. The stability of TX cooperation is analyzed for a SIC and SUD RX in Section 5.4 and Section 5.5 respectively. Section 5.6 concludes this chapter. 5.2 Preliminaries We begin by reviewing several game theoretic preliminaries for cooperative games. Let S⊆K,K ={1,2,...,K} denote an arbitrary coalition of TXs. Definition 1. A partitionT ofK is defined as a set of coalitionsS 1 ,S 2 ,...,S N such that S i ∩S j =φ, ∀i,j∈{1,2,...,N}, i6=j and∪ N i=1 S i =K. The set of all partitions ofK is denoted byT . The total number of partitions of a K-element set is called the Bell numberB K and increases exponentially inK. In a cooperative game, players form coalitions and each coalition chooses an action from the set of actions jointly available to it (which may be larger than the set of actions available individually to each of the players). By this choice of actions, each coalition S in partitionT obtains a utility (value) denoted byv(S;T). Games in which the utility of each coalition is dependent on the actions of other coalitions are called partition form games (PFGs) while games in which utility is independent of external actions are called characteristic form games (CFGs), i.e., v(S;T) = v(S) is independent of the specific partitionT . Definition 2. A coalitional game is called a transferable utility (TU) game, if the coop- erative gains achieved by a coalition can be arbitrarily divided among all members of the coalition. In a TU game, that the payoff obtained by cooperation is given to the cooperating coalition to be divided among its members. In contrast, for a non-transferable utility 113 game, cooperation results in payoffs to each individual member of the coalition directly and cannot be redistributed to the other members. In this work, we consider a TU game and denote byx k , the utility allocated to thekth player. Definition 3. A PFG is said to be cohesive if for any partitionT = {S 1 ,S 2 ,...,S N } of K,v(K;K)≥ P N n=1 v(S n ;T). For a cohesive game, the utility obtained by the GC is larger than the sum of utilities of each coalition under any other partition. In other words, the GC maximizes the sum utility among all configurations. Definition 4. A PFG is r-super-additive if for any disjoint coalitions S 1 ,S 2 ,...,S r and any partitionρ ofK\(S 1 ∪S 2 ∪...∪S r ), we havev(S 1 ∪...∪S r ;{S 1 ∪...∪S r }∪ρ)≥ P r t=1 v(S t ;{S 1 ,S 2 ,...,S r }∪ρ). Super-additivity implies that the when coalitions merge to form a larger coalition, the total utility of the larger coalition is greater than the sum of the utilities of its constituent coalitions. In other words, forming larger coalitions improves achievable utility. When r = 2, the above definition reduces to the conventional definition of super-additivity in PFGs [27]. For CFGs, the utility of the coalition is independent of the rest of the partition and hence 2-super-additivity impliesr-super-additivity and cohesiveness. This is however not true for PFGs due to externalities. However, it is clear that if a game is r-super-additive for all feasible values ofr, then the game is cohesive and the GC has the maximum total utility. Definition 5. A PFG is said to have negative externalities if for any mutually disjoint coalitionsS 1 ,S 2 andS 3 and any partitionρ ofK\(S 1 ∪S 2 ∪S 3 ), we havev(S 3 ;{S 3 ,S 1 ∪ S 2 }∪ρ)≤v(S 3 ;{S 1 ,S 2 ,S 3 }∪ρ). 114 A game with positive externalities is defined similarly with the inequality reversed. A game has negative (positive) externalities if a merger between two coalitions does not increase (decrease) the utility of all other coalitions. A game with mixed externalities exhibits both positive and negative externalities for different coalitions or for different realizations of the game parameters. 5.2.1 Core and Stability of Cooperation in Partition Form Games The core of a cooperative game is the set of all divisions of utility such that no user or coalition of users has an incentive to deviate from a given configuration (usually the grand coalition). The core of a cooperative game can be represented as a linear program which describes the demands of each coalition of users, given the behavior of members external to the coalition. If the constructed linear program has a non-empty feasible set, then the core of the game is nonempty and there exists a division of total utility such that no coalition has an incentive to deviate and cooperation is said to be stable. If the feasible set of the linear program describing the core is empty, then the game exhibits oscillatory behavior among several configurations. If the grand coalition (GC) has the highest utility among all possible configurations and the core is nonempty, then the GC is the stable outcome of cooperation. While CFGs have a unique definition of the core, PFGs do not have a unique notion of the core due to the dependence of the behavior of external coalitions. Assuming uniform behavior among external coalitions, several cores have been defined for PFGs to account for different behavior of coalitions [27]. We now state the definitions of a few cores of PFGs suggested in the literature (note that this is not an exhaustive list that are relevant to the work herein). 115 The core of a game with rational expectations, named the r-core, is the feasible region for the set of linear inequalities: X i∈S x i ≥ v ρ ∗ S (S;{S,ρ ∗ S }),∀S⊂K, K X i=1 x i =v(K;K), ρ ∗ S = argmax ρ S X G∈ρ v(G;{S,ρ}). (5.1) The r-core models rational behavior among the remaining players i.e., all the other play- ers excluding the deviating coalition try to maximize the sum utility, assuming that the deviating coalition cannot be changed anymore. The core of the game with merging expectations, named the m-core, is the feasible region for the set of linear inequalities: X i∈S x i ≥ v(S;{S,K\S}),∀S⊂K, K X i=1 x i =v(K;K). (5.2) Each coalitionS evaluates its utility by assuming that all the other players form a coali- tion irrespective of the actual partition of the users external toS. The core of the game with cautious expectations, named the c-core, is the feasible region for the set of linear equalities: X i∈S x i ≥ min ρ v(S;{S,ρ}),∀S⊂K, K X i=1 x i =v(K;K), (5.3) where the minimization is carried out over all partitions ρ of the remaining usersK\ S. Let ρ ∗ S be the minimizing partition. Each coalition S is guaranteed a reward of v(S;{S,ρ ∗ S }), independent of the actual partition of the remaining users and the utility expected by each coalition is a conservative estimate of the actual obtainable utility. 116 Finally, the core with singleton expectations, named the s-core, as the feasible region of the set of inequalities: X i∈S x i ≥ v(S;{S,(K\S)}),∀S⊂K, K X i=1 x i =v(K;K), (5.4) where,(K\S) denotes the partition containing all the singletons. This is in direct contrast to the m-core where the utility of each coalition is computed by assuming that the rest of the users are in a single coalition. Relationship between the various cores: For PFGs with r-super-additivity, the ra- tional behavior of external coalitions is to merge together to form the largest possible coalition and hence the r-core is identical to the m-core in which all the external players are treated as a single entity. For PFGs with negative externalities, the utility of a coali- tion is minimized when all the other coalitions in a partition operate in a unified manner and hence the c-core is identical to the m-core in this scenario. In addition, for games with negative externalities, the constraints that define the s-core are tighter than the con- straints that define the m-core and hence the s-core is a subset of the m-core. Fig. 5.2.1 shows an example of a nonempty r-core and s-core for a symmetric scenario with super- additivity and negative externalities. To summarize, for games with r-super-additivity and negative externalities, we have that s-core⊆ m-core = r-core = c-core and for games withr-super-additivity and mixed externalities, we have that the m-core = r-core. In this work, we primarily focus on the r-core and note that similar results can derived for the other PFG cores. 5.2.2 Determining nonemptiness of a core We now state a necessary and sufficient to determine the nonemptiness of a core. For any S⊆K, denote by1 S ∈R K the characteristic vector ofS given by(1 S ) i = 1 wheni∈S 117 R-core S-core (1,0,0) (0,1,0) (0,0,1) x 1 +x 2 =1 x 1 +x 3 =1 x 2 +x 3 =1 x 2 +x 3 =® Figure 5.3: Example of nonempty r-core (shaded region) and s-core (inside dotted region) for a symmetric scenario. and (1 S ) i = 0 otherwise. The collection (λ S ),S ⊆K of numbers in [0,1] is a balanced collection of weights if for every player k, the sum of (λ S ) over all the coalitions that contain thekth player is 1, i.e., P S⊆K λ S 1 S = 1 K . The Bondareva Shapley theorem [59, p.262] states that a CFG with transferable payoff has a nonempty core if and only if the game is balanced: X S⊆K λ S v(S)≤v(K)∀λ S . (5.5) Though the Bondareva Shapley theorem has been derived in the context of CFGs, it can be applied to PFGs to examine the nonemptiness of the various cores. It is well known that super-additive games may not satisfy the balancedness condition and can have empty cores [16]. Checking whether a game is balanced or not (i.e., verifying the nonemptiness of the core and thus the stability of cooperation) is in general a co-NP- complete problem [26]. In this work, we exploit the structure of the utility function to verify the nonemptiness of the core in several regimes of interest. 118 5.2.3 Games with empty cores In several scenarios, it is feasible for a super-additive game to have an empty core wherein the GC has the highest utility, but cooperation is unstable. Such games ex- hibit oscillatory behavior as described in [57]. Several approaches have been suggested in the literature to enforce stability of cooperation in such scenarios. We consider the m-core for illustration. Theǫ-core of a game is defined as the set of allocations such that X i∈S⊂K x i ≥ min(v(S;{S,K−S})−ǫ,0), K X i=1 x i =v(K;K). (5.6) In effect, the RX penalizes each coalition for leaving the GC. By choosing a large enough value of ǫ, the ǫ-core can always be made nonempty. The least core of the game is defined as theǫ-core for the smallest value ofǫ that makes the core nonempty. The least core can be obtained by solving the optimization problem ǫ ∗ = minǫ subject to (5.6). If cooperation is not stable, the RX can penalize each deviating coalition to the extent determined by ǫ ∗ by refusing to decode signals sent at a higher rate and thus enforcing the stability of the GC. Finally, we note that the amount of penalty imposed on each coalition in (5.6) is an illustration and in general, the penalty can vary depending on the coalition structure and the problem at hand. 5.3 Signal Model Let us consider a MIMO MAC scenario with K users simultaneously transmitting to a M-antenna receiver. Assuming that the kth user has n k TX antennas, the link between 119 the kth TX and the RX is modeled by a deterministic channel matrix G k of dimension M×n k . The signal at the RX can be expressed as Y M×1 = K X k=1 G k X k +Z M×1 , (5.7) where Z ∼ N(0,N 0 I M×M ) is the additive white Gaussian noise and X k is the trans- mitted signal from the kth user. Throughout this work, we assume that the transmitted signals are drawn from a Gaussian codebook with power constraints. 5.3.1 Signal Model with Coalitions and User Cooperation We illustrate the signal model with user cooperation for a partitionT ={S 1 ,...,S N } of users K = {1,2,...,K}. Assume that the cooperating users organize themselves into coalitions, forming a partitionT of the set of users. All the users in a coalitionS n act as a single virtual user and cooperate by jointly encoding the data to be transmitted to the receiver, thus acting as a virtual MIMO system. Note that only the users in each coali- tion cooperate with each other and users across different coalitions do not perform joint encoding. LetH n = [G k 1 ,G k 2 ,...,G k |Sn| ] be the effective channel matrix of dimension M× P |Sn| j=1 n j as seen by thenth coalition. The signal at the RX can then be expressed as Y M×1 = N X n=1 H n X n +Z M×1 , (5.8) where [X n ] P |Sn| j=1 n j ×1 is the transmitted vector for the coalition S n with a covariance matrix given by Q n = E[X n X H n ]. We consider two types of power constraints in this work : (1) a transmit sum power constraint for each coalition, i.e., Tr(Q n )≤ P |Sn| i=1 P Sn(i) whereP Sn(i) is the transmit sum power constraint ofith user in thenth coalition and (2) a per-antenna power constraint for each antenna of each user in a coalition, diag([Q n ])≤ 120 diag([P Sn(1),1 ,P Sn(1),2 ,...]) where P Sn(i),j is the power constraint on the jth antenna of theith user in coalitionS n . . We note that the results in this work rely on key capacity computations that exist in literature for the above signaling models. 5.4 TX Cooperation Game with Successive Interference Cancelling Receiver Consider an arbitrary partition T = {S 1 ,S 2 ,...,S N } of the available TXs transmitting data to a common RX as described in Section 5.3. The RX first announces a decoding orderπ, a permutation of (1,2,...,N), of the coalitions inT . The TX cooperation game is analyzed in several stages: (1) Given T and π, the interaction between the coalitions in T is modeled as a non-cooperative game and the utilities that can be achieved in the configuration examined; (2) next, the properties which influence cooperative behavior among coalitions such as super-additivity, externalities etc. are examined and (3) the stability of cooperation is analyzed by examining the core of PFGs. 5.4.1 Non-cooperative game between TXs The RX signal is the sum of signals arriving from each coalition of TXs and is given by (5.8). For a given partition T and a SIC RX (see Fig. 5.4), the strategy of the kth coalition consists in choosing the vector of transmit covariance matrices Q all k = Q (1) k ,Q (2) k ,...,Q (N!) k each optimized for a given permutation π of {1,2,...,N}. We begin by assuming a fixed decoding order and then extend our results to the case of time-sharing between decoding orders. Without loss of generality, let us assume that the decoding order of the coalitions isπ = (1,2,...,N). The utility obtained by each coali- tion is defined as the maximum achievable rate by all the TXs in the coalition, given the 121 Figure 5.4: Decoding in an SIC receiver transmit covariance matrices of all other coalitions subject to the given power constraint. For a partition T = {S 1 ,...,S N }, under the assumption of AWGN, the obtained utility by a coalitionS n inT adopting a transmit covariance matrixQ n can be expressed as v(S n ;T) = log |N 0 I +H n Q n H H n + P N j=n+1 H j Q ∗ j H H j | |N 0 I + P N j=n+1 H j Q ∗ j H H j | ! . (5.9) Clearly, the utility obtained by each coalitionS n depends on the structure of the par- titionT and the strategiesQ −n adopted by other coalitions inT . Each coalition chooses an action which is the best response to the actions of the users in the other coalitions and hence v(S n ;T) = v n (Q n ,Q −n ) is defined as the NE utility for the nth player and the optimizing Q n is defined as the NE achieving strategy. As the utility achieved by each coalition is dependent on the choices of actions of other players, the cooperative game between the TXs is a PFG. This is in contrast to the CFG model considered previously in [57, 42] where the utility achieved by each coalition is independent of the actions of other players. As stated in Section 5.1, the PFG can capture the exact interference ex- perienced by each coalition and thus is well suited to model scenarios arising in several wireless networks. We now examine the existence of a NE of the non-cooperative game between the coalitions in T which ensures that the NE achievable strategies and thus the NE utility always exist for the coalitions in a partition. The existence of a NE can be proved using 122 the Kakutani fixed point theorem [59, 7]. For illustration, we assume a per-antenna power constraint. Define the setA n ={Q n |diag([Q n ]) diag([P Sn(1),1 ,P Sn(1),2 ,...,])}, i.e.,A n is the set of all covariance matrices which satisfy the per-antenna power constraint for the nth coalition. Clearly,A n is a compact and convex set. The utility functionv(Q n ,Q −n ) is continuous inQ i for alli = 1,2,...,N and is concave inQ n [40, 7]. This satisfies all the conditions of the Kakutani fixed point theorem [59] and ensures the existence of a NE for the non-cooperative game between the coalitions in a partition. 5.4.1.1 Uniqueness of NE We now discuss the uniqueness of the NE achieving strategies and NE utilities. Note that the utility functionv(Q n ,Q −n ) in (5.9) is concave, but not be strictly concave inQ n for a general case. This suggests that there may exist many choices ofQ n which result in the same value of the utility function. In the literature, uniqueness of NE has been used to suggest the uniqueness of NE achieving strategies which in turn implies the uniqueness of NE utilities. However, in this work, we make the distinction between uniqueness of NE achieving strategies and NE utilities due to the non-strict concavity ofv n (Q n ,Q −n ) in (5.9). 5.4.1.2 Relevance of uniqueness of NE utilities As defined in Section 5.2, the core of a PFG is a linear program which describes the demands of each coalition under an assumption on the behavior of external coalitions. The values of the utilities used in defining the PFG core in (5.1), (5.2), (5.3) and (5.4) are the NE utilities which are derived from the non-cooperative game between the coalitions in a given partition. Thus, we see that the uniqueness of the NE utilities allows the core to be well defined. If the NE utilities are not unique, then the core can be written for each combination of possible values of utilities. In the work, we define the core as the union 123 of all the cores that are obtained by choosing from the various possible values of the NE utilities. The uniqueness of NE achieving strategies can be checked by deriving a sufficient condition for uniqueness. In [67, 46, 7] a sufficient condition called diagonally strict concavity (DSC) has been derived to verify the uniqueness of NE achieving strategies. The DSC condition can be interpreted as the case where a player’s utility function is more sensitive to the choice of his own actions as compared to the actions of all the other players. In this work, we generalize the DSC condition proposed in [7]. Though we derive this condition for the scenario with per-antenna power constraints, we note that the condition is applicable to the scenario with sum power constraints as well. Lemma 1. If ˜ Q 1 , ˜ Q 2 ,..., ˜ Q N and ˆ Q 1 , ˆ Q 2 ,..., ˆ Q N be two sets of covariance matrices which are NE to the non-cooperative game between the TXs in (5.9), then C n = Tr h ( ˜ Q n − ˆ Q n )(∇ Qn v n ( ˆ Q n , ˆ Q −n )−∇ Qn v n ( ˜ Q n , ˜ Q −n )) i ≤ 0,∀n = 1,2,...,N. (5.10) Proof. By the definition of an NE, the covariance matrices are the solutions to the op- timization problem in (5.26). The LagrangianL n for the maximization in (5.26) can be written as L n = v n (Q n ,Q −n )+ Tr(L n Q n )− Tr(D n (Q n −R n )), (5.11) whereL n is a positive semi-definite matrix,D n = diag λ Sn(1),1 ,..., is a diagonal ma- trix containing the Lagrange multiplier coefficients and and R n = diag P Sn(1),1 ,..., is another diagonal matrix containing the power constraint values respectively. For non- trivial power constraints, the Slater condition is satisfied and the Karush Kuhn Tucker 124 (KKT) conditions can be written as (a) ˜ Q n 0, ˆ Q n 0. (b) diag([ ˜ Q n ]) diag([P Sn(1),1 ,P Sn(1),2 ,...]). (c) diag([ ˆ Q n ]) diag([P Sn(1),1 ,P Sn(1),2 ,...]). (d) Tr( ˜ L n ˜ Q n ) = 0 and Tr( ˆ L n ˆ Q n ) = 0. (e) Tr( ˜ D n ( ˜ Q n −R n )) = 0 and Tr( ˆ D n ( ˆ Q n −R n )) = 0. (f) ∇ Qn v( ˜ Q n , ˜ Q −n )+ ˜ L n − ˜ D n = 0 and∇ Qn v( ˆ Q n , ˆ Q −n )+ ˆ L n − ˆ D n = 0. Now using the KKT conditions to evaluate and simplifyC n , we get −C n = Tr h ( ˜ Q n − ˆ Q n ) ∇ Qn v( ˜ Q n , ˜ Q −n )−∇ Qn v( ˆ Q n , ˆ Q −n ) i (e) = Tr h ( ˜ Q n − ˆ Q n ) ( ˜ D n − ˜ L n )−( ˆ D n − ˆ L n ) i (c) = Tr h R n ˜ D n − ˜ Q n ˆ D n + ˜ Q n ˆ L n − ˆ Q n ˜ D n + ˆ Q n ˜ L n +R n ˆ D n i (a) ≥ Tr h ˜ D n (R n − ˜ Q n )+ ˆ D n (R n − ˆ Q n ) i (b,d) ≥ 0. The sequence of equalities and inequalities directly follow from the KKT conditions. Lemma 1 shows that if there exist at least two equilibria of the non-cooperative game, thenC n ≤ 0∀n = 1,2,...,N. Remark: From Lemma 1, we can infer that if C n > 0 for at least one value of n, then the NE achieving strategy is unique. This is a refinement of the condition in [67, 7] where one needed to show thatC = P N n=1 C n > 0 for the NE achieving strategy to be unique. The DSC condition derived in [67, 7] holds for problems in which the strategy sets of each player are coupled with each other. In contrast, the refined DSC condition in 125 our work only holds for games where the strategy sets of each player are independent of each other and thus is restricted to the smaller class of NE problems. Lemma 2. For two feasible strategies ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q N ) and ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q N ), we have thatC = P N n=1 C n ≥ 0. Proof. Evaluating C for any two feasible strategies ( ˜ Q 1 ,..., ˜ Q N ) and ( ˆ Q 1 ,..., ˆ Q N ), we get C = N X n=1 Tr h ( ˜ Q n − ˆ Q n )(∇ Qn v n ( ˆ Q n , ˆ Q −n )−∇ Qn v n ( ˜ Q n , ˜ Q −n )) i = N X n=1 Tr " H n ˜ Q n H H n −H n ˆ Q n H H n × (N 0 I + N X j=n H j ˆ Q j H H j ) −1 −(N 0 I + N X j=n H j ˜ Q j H H j ) −1 !# = N X n=1 Tr " (A n −B n ) ( n X j=1 B j ) −1 −( n X j=1 A j ) −1 !# ≥ 0. (5.12) where the matrices A n and B n are defined as A 1 = N 0 I +H N ˜ Q N H H N , B 1 = N 0 I + H N ˆ Q N H H N , A N−n+1 = H n ˜ Q n H H n and B N−n+1 = H n ˆ Q n H H n for n ≥ 2 and the last inequality is true from [7]. Theorem 1. The NE utility of the non-cooperative game between the TXs in a partition T for a given decoding orderπ is unique. Proof. From Lemma 1 and Lemma 2, we infer that given any two NE achieving strategies of the gameC n = 0∀n = 1,2,...,N. Substituting forC n , we get, Tr " H n ˜ Q n H H n −H n ˆ Q n H H n × (N 0 I + N X j=n H j ˆ Q j H H j ) −1 −(N 0 I + N X j=n H j ˜ Q j H H j ) −1 !# = 0. (5.13) 126 Using C N = 0 and the matrix trace inequality in [7], we get that H N ˜ Q N H H N = H N ˆ Q N H H N . Now using C N−1 = 0 and H N ˜ Q N H H N = H N ˆ Q N H H N , we can show that H N−1 ˜ Q N−1 H H N−1 = H N−1 ˆ Q N−1 H H N−1 . Continuing this approach, we can show that H n ˜ Q n H H n = H n ˆ Q n H H n ∀ n = 1,2,...,N. Now, substituting in the utility function in (5.9), it is clear that all NE achieving strategies have the same NE utility and hence the NE utility of the non-cooperative game between the TXs in a partition for a given decoding order is unique. We emphasize that Theorem 1 only shows that the NE utility for all the NE achieving strategies is the same and hence unique. However, there may exist several achievable strategies which would achieve the NE utility. The uniqueness of the NE utility implies that, given the decoding order, each TX can unambiguously evaluate the utility its obtains in a given partition. 5.4.1.3 Evaluating the utility function For a coalitionS n in partitionT , the utility function as defined in (5.9) is the maximum achievable rate over the MIMO channel between the cooperating TXs and the RX, given the interference of all the other coalitions and the decoding order π. For the scenario with sum power constraints, the NE utility in (5.9) can be computed using sequential iterative water filling [101]. On the other hand, for a scenario with per-antenna power constraints, deriving a closed form expression for the capacity remains an open problem and [91] evaluates the capacity in terms of the the variables of the convex dual problem. However, a closed form solution can be derived for both the sum power and per-antenna power constraint scenario and a single antenna RX (M = 1) and we state the NE utility function for this scenario. Let us defineh Sn(i),j as the channel gain from thejth antenna of theith user in the 127 nth coalition to the RX. Using the capacity results in [101] and [91], the NE utility for a coalition of TXs with a sum power constraint can be written as v(S m ;T) = log   N 0 + P N n=1 P |Sn| i=1 P j |h Sn(i),j | 2 P |Sn| i=1 P Sn(i) N 0 + P N n=m+1 P |Sn| i=1 P j |h Sn(i),j | 2 P |Sn i=1 P Sn(i)   , (5.14) and with a per-antenna power constraint can be expressed as v(S m ;T) = log    N 0 + P N n=1 P |Sn| i=1 P j |h Sn(i),j | p P Sn(i),j 2 N 0 + P N n=m+1 P |Sn| i=1 P j |h Sn(i),j | p P Sn(i),j 2    . (5.15) It is clear that beamforming achieves the NE utility for the single antenna RX in both sce- narios. The key difference between the two scenarios is that for the per-antenna power constraint the beam weight has its phase matched to the channel coefficient, but the am- plitude is independent of the channel and fixed based on the power constraint. Computing the NE utility in practice involves full knowledge of the channel gains and the power con- straints of each user at all the players. The evaluated NE utility is used in negotiations to form new coalition structures, in determining the benefits gained by merging with other coalitions and determining the stability of cooperation. The NE utility can be used to evaluate the total utility achievable by the current coalition and is used in negotiations to form new coalition structures, by quantifying the benefits gained by merging with other coalitions and determining the stability of cooperation as demonstrated later in this work. Fig. 5.4.1.3 shows the NE utility for a partition with 2 coalitions and a single antenna receiver for both possible decoding orders. It can be inferred from [101] that the NE utility for the SIC receiver is on the Pareto-optimal boundary of the capacity region of the MAC and thus all the NE of this game are efficient. 128 v 1 v 2 Achievable rates for ¼ = {1,2} Unique NE u!lity for ¼ = {1,2} Unique NE u!lity for ¼ = {2,1} Figure 5.5: Figure showing the NE rate points for an SIC single antenna receiver for different decoding orders. 5.4.2 TX Cooperation - Properties Cooperation between TXs (coalitions of TXs) over the MAC channel with an SIC re- ceiver has two benefits: (1) Cooperating coalitions signal jointly which can result in an improvement in the achievable sum rate; (2) The decoding order of the combined coali- tion improves relatively in comparison to the decoding order of its member coalitions resulting in a further improvement in the achievable rate for the cooperating coalitions. We illustrate this with an example. Consider a MAC scenario with 4 TX coalitions S 1 ,S 2 ,S 3 ,S 4 specified in the order in which they are decoded. Assuming that coalitions S 1 andS 3 cooperate with each other, the receiver first decodes coalitionS 2 . NextS 1 and S 3 , which signal jointly, are decoded followed byS 4 . Clearly,S 1 benefits by moving later into the decoding order and bothS 1 andS 3 benefit by signaling jointly. Joint signaling is achieved in practice by jointly generating the codebook using the knowledge of the joint density function and requires complete channel state information and power constraints of all the users. We formalize this intuition in the following propositions which describe 129 the super-additive property and externalities for coalition formation. Proposition 1. The TX cooperation game with SIC processing at the receiver isr-super- additive and cohesive, i.e., for two given partitionsT 1 = (S 1 ,S 2 ,...,S r ,S r+1 ,...,S N ) with decoding orderπ 1 = (1,2,...,N) andT 2 = (S a 1 ,...,S at ,S b 1 ∪S b 2 ... ∪ S br , S t+r+1 ,...,S N ) with decoding orderπ 2 = (a 1 ,a 2 ,...,a t ,b {12...r} ,t+r+1,...,N). v(S b 1 ∪S b 2 ...∪S br ;T 2 )≥ r X i=1 v(S b i ;T 1 ), (5.16) where all the utilities are as computed in (5.9). Note that (a 1 ,...,a t ,b 1 ,...,b r ) is a per- mutation of (1,2,...,t + r) satisfying a 1 < a 2 < ... < a t and b 1 < b 2 < ... < b r . Proof. Let ˜ Q and ˆ Q be the NE achieving covariance matrix tuples ofT 1 andT 2 respec- tively. Then, v(S b 1 ∪...∪S br ;T 2 ) =I(X b 1 ∪...∪br ;Y|X a 1 ,...,X at )| ˆ Q (a) ≥ I(X b 1 ∪...∪br ;Y|X a 1 ,...,X at )| ˜ Q (b) = I(X b 1 ;Y|X a 1 ,...,X at )| ˜ Q +I(X b 2 ;Y|X a 1 ,...,X at ,X b 1 )| ˜ Q +... +I(X br ;Y|X a 1 ,...,X at ,X b 1 ,X b 2 ,...,X b r−1 )| ˜ Q (c) ≥ r X i=1 I(X b i ;Y|X 1 ,X 2 ,...,X b i −1 )| ˜ Q = r X i=1 v(S b i ;T 1 ), (5.17) where the inequality (a) follows from the assumption of independent signaling among the cooperating coalitions and the definition of the NE, (b) and (c) follow from the chain rule of mutual information and the fact that (a 1 ,...,a t ,b 1 ,...,b r ) is a permutation of (1,2,...,t+r) such that a 1 < a 2 < ... < a t ,b 1 < b 2 < ... < b r . Hence the TX co- operation game with SIC isr-super-additive. Clearly, when all the coalitions cooperate, 130 the TX cooperation game with SIC is cohesive, i.e., the GC has the highest sum utility and hence is the only feasible result of cooperation. Proposition 2. The TX cooperation game with a single antenna SIC receiver has negative externalities. Proof. Using the notation in Prop. 1, we observe thatv(S t+r+i ;T 2 ) = v(S t+r+i ;T 1 ) for all i = 1,2,...,N − t− r as the NE utility of coalition S t+r+i only depends on the undecoded coalitions in their respective partitions, which are identical forT 1 andT 2 (see 5.9). For the other coalitions, assuming a per-antenna power constraint, we have v(S an ;T 2 ) = log N int +α 2 n +( P r i=1 α t+i ) 2 N int +( P r i=1 α t+i ) 2 and v(S an ;T 1 ) = log N int +α 2 n + P r i=1 α 2 t+i N int + P r i=1 α 2 t+i , wheren = 1,...,t, α n = P |Sn| i=1 P j |h Sn(i),j | p P Sn(i),j andN int = N 0 + P t i=n+1 α 2 i + P N i=t+r+1 α 2 i . From the above expressions, it can be clearly seen that v(S an ;T 2 ) ≤ v(S an ;T 1 ) and hence the TX cooperation game has negative externalities for a one an- tenna RX. Note that this result also holds for the sum power constraint scenario as well. Proposition 3. The TX cooperation game with a multiple antenna SIC receiver has mixed externalities. Proof. We show the proposition by constructing examples which have both positive and negative externalities. Consider a 3-user scenario each with a single antenna TX trans- mitting to a 2-antenna RX. Let h k be channel gain vector from the kth TX to the RX and let P k ≤ 1 be the per-antenna power constraint and let T 1 = {{1},{2},{3}} and 131 T 2 = {{1,2},{3}} and let the higher numbered users are decoded first. The utility obtained by user3 under the partitionT 1 andT 2 can be expressed as v(S 3 ;T 1 ) = log(|N 0 I + 3 X i=1 h i h H i |)− log(|N 0 I + 2 X i=1 h i h H i |). (5.18) and v(S 3 ;T 2 ) = log |N 0 I +H 12 Q ∗ 12 H H 12 +h 3 h H 3 | |N 0 I +H 12 Q ∗ 12 H H 12 | , (5.19) respectively with Q ∗ 12 = argmax Q 12 log |N 0 I +H 12 Q 12 H H 12 | |N 0 I| , (5.20) It can be numerically observed that for some realizations of the channel gainsv(S 3 ;T 1 )≤ v(S 3 ;T 2 ) and for other realizationsv(S 3 ;T 1 ) > v(S 3 ;T 2 ). For example, whenN 0 = 1, h 1 = [1.17119,−0.1941], h 2 = [−2.1384,−0.8396] and h 3 = [1.3546,−1.0722], we have thatv(S 3 ;T 1 ) = 0.8580 ≤ v(S 3 ;T 2 ) = 1.0023 while forh 1 = [−1.5771,0.5080], h 2 = [0.2820,0.0335] and h 3 = [−1.3337,1.1275] we have that v(S 3 ;T 1 ) = 0.7593 ≥ v(S 3 ;T 2 ) = 0.7462. Hence the TX cooperation game with a multiple antenna SIC re- ceiver has mixed externalities, in general. To summarize, we have shown that for merging coalitions for TXs, the NE utility achieved by the merging coalitions is at least as large as the sum of the NE utilities achieved by the individual coalitions. r-super-additivity of the PFG implies that the GC, the coalition of all the TXs signaling jointly, has the highest sum utility and hence is the only feasible outcome of cooperation. On the other hand, negative externalities for the single antenna RX imply that as when TXs merge to form larger coalitions, the rate achievable by every other TX reduces. This in turn induces further cooperative behavior as each rational TX tries to improve its utility further (as long as individual allocations 132 increase). For the multi-antenna RX, mixed externalities imply that no general prediction can be made without knowing the specific channels gains and power constraints. 5.4.3 TX Cooperation - Stability Previously, we have discussed the properties of the utility function when several coali- tions merge to form a larger coalition. We now determine the stability of TX cooperation by verifying the nonemptiness of the core. Note that verifying the satisfiability of the Bondareva-Shapley theorem is co-NP-complete even for super-additive games [26] and hence showing the nonemptiness of the core is a difficult problem in general. In this work, we analyze the nonemptiness of the core in the high SNR (low noise) and high SNR (low noise) regime. We first derive an approximation for the capacity in the low SNR regime. Lemma 3. In the low SNR regime, i.e.,N 0 →∞, the capacity achieved by a player (here player 1) under the sum power constraint can be approximated as v(Q 1 ,Q −1 ) = max Tr(Q 1 )≤P 1 log |N 0 I +H 1 Q 1 H H 1 +K intf | |N 0 I +K intf | ≈ σ 2 H 1 P 1 N 0 , (5.21) where σ H 1 is the maximum singular value of H 1 and K intf = P K j=2 H j Q ∗ j H j is the interference from all other users. 133 Proof. We begin by showing that at low SNR, the channel capacity is maximized by allocating all the power to the dominant eigen-mode of the channel. v 1 (Q 1 ,Q −1 ) = max Q 1 log |N 0 I +H 1 Q 1 H H 1 +K intf | |N 0 I +K intf | = max Q 1 log |N 0 I + ˜ H 1 Q 1 ˜ H H 1 | |N 0 I| ! ≈ max Q 1 log |N 0 I +U 1 Σ 1 V H 1 Q 1 V 1 Σ 1 U H 1 | |N 0 I| = max Tr(D 1 )≤P 1 |N 0 I +Σ 1 D 1 Σ 1 | |N 0 I| = max Tr(D 1 )≤P 1 X i log 1+ σ 2 i d i N 0 (a) = log 1+ σ 2 max P 1 N 0 (b) ≈ σ 2 max P 1 N 0 , (5.22) where ˜ H 1 = (I + 1 N 0 K intf ) −1/2 H 1 and ˜ H 1 ≈ H 1 as N 0 → ∞, (a) is true as allocating all the power to the dominant eigen-mode maximizes the expression. Note that the ca- pacity does not depend on any of the interferers in this regime and hence there are no externalities in low SNR regime. Theorem 2. In the low SNR regime, i.e.,N 0 →∞, the TX cooperation game with a sum power constraint has a nonempty core. Proof. From Lemma 3, we know that the capacity at low SNR is effectively independent of the interference experienced by the coalition under consideration. Before showing the nonemptiness of the core, we derive the relation between the joint utilities of cooperating TXs and the utility of each individual TX. Consider two cooperating coalitions with channel gain matricesH 1 andH 2 respectively. The channel gain matrix of the combined coalition can be written asH = [H 1 |H 2 ]. Using the fact thatHH H = H 1 H H 1 +H 2 H H 2 , we get thatσ 2 H i ≤ σ 2 H ≤ σ 2 H 1 +σ 2 H 2 whereσ H ,σ H 1 andσ H 2 are the maximum singular values ofHH H ,H 1 H H 1 andH 2 H H 2 respectively. Now assuming that there are K TXs indexed by K = {1,2,...,K}, the necessary 134 and sufficient condition for the nonemptiness of the core is given by Bondareva-Shapley theorem from (5.5): X S⊂K λ S v S ≤v K ⇒ X S⊂K λ S σ 2 H S P S N 0 ≤ σ 2 H K P K N 0 (5.23) wherev S =σ 2 H S P S /N 0 is the utility of coalitionS from Lemma 3,σ H S is the maximum singular value of the combined channel matrix of cooperating TXs H S , P S = P i∈S P i and λ S is a balanced collection of weights. By substituting the bounds on the singular values of an augmented matrix in (5.23) and comparing coefficients on both sides, it is easy to see that the Bondareva-Shapley theorem holds and hence the core of the TX cooperation game with sum power constraints is nonempty at low SNR. Theorem 3. In the high SNR regime, i.e., N 0 → 0, the TX cooperation game has an empty core for both the sum power and per-antenna power constraints. Proof. We give an example to show that the game has an empty core at high SNR. Con- sider a 4-user MAC (K=4) with one antenna at each TX and the RX. The power constraint on each user isP i ≤ 1 and the noise variance isN 0 = 1 with each user having an identi- cal channel gain ofh i = 1. The receiver performs SIC by decoding the users in the order specified by a permutationπ. Using the above parameters and computing the utilities as in (5.9), it can easily be verified that the r-core of this game is empty. By symmetry, we infer that the r-core is empty for all decoding orders. Discussion: The optimal signaling strategy for each coalition at low SNR is beam- forming along the best eigen-mode of the channel matched to the appropriate power constraint and the core is nonempty in this regime. This implies that the power gain due to beamforming is sufficient to make cooperation stable at low SNR. Note that as the NE utility at low SNR for a sum power constraint does not depend on the utility of other 135 coalitions, the various cores of the PFG are identical in this regime and are all nonempty. For the case with per-antenna power constraints, we note that the nonemptiness of the core cannot be established for the low SNR case due to the lack of suitable approximation of capacity in this regime. In the high SNR regime, it is observed that the r-core (similar statements can be made about other cores) is empty for all decoding orders in general. Fig. 5.4.3 shows a plot of the boundary between the regions of empty and nonempty r- core as a function of the number of players and the SNR (Each player is assumed to have unit per antenna power constraint and unit channel gain and SNR is defined as 1/N 0 , i.e., the scenario where players are identical in all aspects other than the decoding order). Clearly, the core of the game is nonempty at low SNR and is empty at higher SNRs and the minimum SNR at which the core is nonempty reduces with the number of players. The empty core and hence instability of cooperation at high SNR can be attributed to the asymmetry between the TXs caused by a fixed decoding order. In contrast, noise domi- nates the interference in the low SNR regime and the decoding order becomes irrelevant. This removes the asymmetry between the players due to the decoding order and the core is nonempty in this regime. To enforce cooperation in the high SNR regime for a fixed decoding order, the RX can impose penalties on each coalition as described in (5.6). 5.4.4 SIC receiver with time sharing between decoding orders Previously, we have discussed the stability of TX cooperation for an SIC receiver as- suming a fixed decoding order. We now consider the scenario in which time-sharing is permitted between the various decoding orders. From Theorem 2, the core for an SIC re- ceiver is nonempty at low SNRs for a fixed decoding order and hence would be nonempty for a time shared SIC receiver. We now investigate the nonemptiness of the core at high SNRs. 136 4 6 8 10 12 14 16 18 20 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 Number of players SNR Core non−empty Core empty Figure 5.6: Plot showing the boundary of the region between empty and nonempty core as a function of SNR and the number of players for a symmetric scenario and a single antenna SIC receiver with a fixed decoding order. Let Θ be the set of all probability distributions characterizing the time sharing of decoding orders. The setΘ is a convex polyhedron whose corner points are distributions which assign probability 1 to one of the decoding orders. Clearly all the probability distributions in Θ do not contribute to nonempty cores. Determining the subset of Θ for which the core is nonempty seems to be an intractable problem for now. Even the question of determining whether there exists a probability distribution of decoding orders for given channel gain matrices and power constraints appears to be a hard problem as verifying the Bondareva-Shapley theorem becomes highly nontrivial. To simplify the problem at hand, we consider an approximation to the utility function to understand the stability of cooperation. At high SNR, the dominant term can be considered a good approximation to the actual utility as the dominant term increases unboundedly while the other terms have a finite value. We now evaluate the r-core of the game with this approximate utility function. Theorem 4. The core of the TX cooperation game for an SIC receiver with equal prob- ability of time sharing between all decoding orders is nonempty at high SNR to a first 137 order approximation of utility. Proof. For the scenario in which all the decoding orders have equal probability, we first compute the utility function of a given coalition of players. We note that the dominating term in the utility function is the term in which a given coalition S is decoded without any interference and increases unboundedly at high SNR. All the other terms are bounded at high SNR due to the interference present while decoding and can be ignored in this approximate analysis. The utility for coalitionS weighted with equal probability over all decoding orders can then be evaluated as v S = |S| K log |N 0 I +H S Q ∗ S H H S | |N 0 I| , (5.24) where|.| for sets denotes its cardinality. The first order approximation of utility can be interpreted as the scaled (by |S|/K) sum rate obtained when coalition S signals with no interference. This also implies that the approximated utility does not depend on the strategies of other coalitions. Fig. 5.4.4 shows the ratio of the approximated utility to the actual utility for a 3-user symmetric MAC with unit channel gain and power constraints as a function of SNR. We see that in the high SNR regime, the ratio approaches unity, thus showing that the approximation is tight at high SNR. Substituting the utility in the Bondareva Shapley theorem in (5.5), the necessary and sufficient condition to be satisfied can be expressed as X S⊆K λ S |S| K log |N 0 I +H S Q ∗ S H H S | |N 0 I| ≤ log |N 0 I +H K Q ∗ K H H K | |N 0 I| , (5.25) for some balanced set of numbersλ S . The above condition is satisfied as the vector λ S |S| K is a probability distribution and the utility for a coalitionS is always smaller than the utility of the GC. This ensures the core of the TX cooperation game with time sharing 138 −50 0 50 0.4 0.5 0.6 0.7 0.8 0.9 1 SNR(dB) Ratio of approximation to actual utility One user Two user coalition Figure 5.7: Ratio of approximated to actual utility as a function of SNR for a 3-user MAC with single antenna TXs, unit channel gain and unit power constraint for a SIC RX. Note that the ratio approaches1 for very high SNR. SIC receivers is nonempty at high SNR. This result implies that while fixing a decoding order results in an empty core at high SNR, time sharing between the decoding orders can result in an nonempty core (using an approximation to the utility function). Fig. 5.4.4 shows the r-core of the TX cooperation game for a 3-user single antenna MAC with unit channel gain, unit power constraints, N 0 =−3dB (SNR = 3dB) and time sharing with equal probability for all the decoding orders. The exact utility function is used to describe the r-core for this scenario. Clearly, the r-core is nonempty showing the stability of TX cooperation in the high SNR regime as well (Note that the r-core is nonempty for SNRs more than 3dB also and the 3dB has been chosen for illustration in Fig. 5.4.4). From a network design perspective, if a receiver implements SIC with time sharing when receiving data from several TXs, cooperation between the TXs by forming a virtual MIMO system and jointly signaling is stable and each transmitter improves his rate as compared to the non-cooperative scenario. As mentioned previously, accurate character- ization of the nonemptiness of the core for high SNR using the exact utility function is 139 Figure 5.8: Plot showing the r-core of the TX cooperation game for an SNR of 3dB for a 3-user symmetric MAC with single antenna TXs, unit channel gain and unit power constraint for an SIC RX with equal probability of time sharing between decoding orders using the exact utility function. The r-core is highlighted in red. still an open problem. 5.5 TX Cooperation Game With Single User Decoding Receiver In Section 5.4, we considered the problem of TX cooperation with SIC receivers and showed that TX cooperation is super-additive, has mixed externalities in general, coop- eration is stable at low SNR for any decoding order and stable at high SNR when time sharing between decoding orders is permitted. In this section, we consider a RX which performs single user decoding (SUD) of the signals from each coalition. Following the methodology in Section 5.4, we investigate the stability of TX cooperation for such a receiver. 140 5.5.1 Non cooperative game between TXs Consider an SUD receiver which decodes a given coalition’s signal by treating all in- terfering signals from other coalitions as noise. The utility obtained by each coalition is defined as the maximum achievable rate by all the TXs in the coalition, given the transmit covariance matrices of all other coalitions subject to the given power constraint. For a partitionT ={S 1 ,...,S N } of users, under the assumption of AWGN, the obtained utility by a coalitionS n inT adopting a transmit covariance matrixQ n can be expressed as v(S n ;T) =v(Q n ,Q −n ) = max Qn I(X n ;Y)| Q −n = max Qn log |N 0 I +H n Q n H H n + P N j=1,j6=n H j Q j H H j | |N 0 I + P N j=1,j6=n H j Q j H H j | ! . (5.26) By definition, each coalition chooses an action which is the best response to the actions of the users in the other coalitions, hence is the cooperative game between the TXs is a PFG. This model of NE utility is in contrast to the approach adopted in [57] where the SUD RX is designed assuming worst case interference from the other coalitions. We briefly illustrate the problem formulation and state the results from [57] to highlight the key differences between the two approaches (Note that there is no similar CFG model for the TX cooperation game with an SIC RX). 5.5.1.1 CFG Model In the jamming utility model in [57], the utility u(S) of a coalition S is defined as the maximum obtainable rate assuming worst case interference from TXs inS c =K\S. u(S) = min Q S c max Q S I(X S ;Y) = min Q S c max Q S log |N 0 I +H S Q S H H S +H S cQ S cH H S c| |N 0 I +H S cQ S cH H S c| , (5.27) 141 where Q S and Q S c are the transmit covariance matrices of the coalition S and S c re- spectively and are constrained to satisfy the given power constraint. Clearly, u(S) is a very conservative lower bound on the utility that can be obtained in practice as typical TXs will not attempt to jam the transmissions fromS at the cost of their own rate. Un- der the utility model in (5.27), [57] shows that TX cooperation is cohesive. However, using a counter example, [57] demonstrates an empty core in general and conjectures that the core is nonempty when all TXs roughly have similar channel gains. Thus, TX cooperation exhibits oscillatory behavior under this utility model. In contrast, we will demonstrate in this section that under the PFG model for utility, TX cooperation is stable in the high SNR and the low SNR regime. 5.5.1.2 PFG model We now analyze the properties of the non-cooperative game between the coalitions in a partition of TXs to determine the NE utility of a given coalition. As in Section 5.4, the feasible sets of each user are convex and compact and utility functionv n (Q n ,Q −n ) is concave inQ n and continuous inQ i ,i = 1,2,...,N. The existence of an NE then follows from the Kakutani fixed point theorem. We now examine the uniqueness of NE utility for the MIMO SUD receiver. Lemma 4. For two feasible strategies ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q N ) and ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q N ), we get C = P N n=1 C n ≥ 0, whereC n is defined as in Lemma 1. 142 Proof. EvaluatingC for any given feasible strategies, we get C = N X n=1 Tr h ( ˜ Q n − ˆ Q n )(∇ Qn v n ( ˆ Q n , ˆ Q −n )−∇ Qn v n ( ˜ Q n , ˜ Q −n )) i = Tr " N X n=1 H n ( ˜ Q n − ˆ Q n )H H n ! × N 0 I + N X n=1 H n ˆ Q n H H n ! −1 − N 0 I + N X n=1 H n ˜ Q n H H n ! −1 !# ≥ 0. where the inequality follows from that the fact that for positive definite matrices ˜ B = N 0 I + P N n=1 H n ˜ Q n H H n and ˆ B = N 0 I + P N n=1 H n ˆ Q n H H n we have that Tr n ( ˜ B− ˆ B)× ( ˆ B −1 − ˜ B −1 ) o ≥ 0 with equality satisfied only when ˜ B = ˆ B. Proposition 4. From Lemma 1 and Lemma 4, we can infer that if ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q N ) and ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q N ) are two NE achieving strategies of the game between the coalitions of TXs, then P N n=1 H n ˆ Q n H H n = P N n=1 H n ˜ Q n H H n . Proof. Substituting in the definition ofC n , we get C n = Tr h ( ˜ Q n − ˆ Q n )(∇ Qn v n ( ˆ Q n , ˆ Q −n )−∇ Qn v n ( ˜ Q n , ˜ Q −n )) i = Tr " H n ( ˜ Q n − ˆ Q n )H H n × N 0 I + N X n=1 H n ˆ Q n H H n ! −1 − N 0 I + N X n=1 H n ˜ Q n H H n ! −1 !# = 0. ⇔H n ˜ Q n H H n =H n ˆ Q n H H n ∀n = 1,2,...,N(or) N X n=1 H n ˆ Q n H H n = N X n=1 H n ˜ Q n H H n → N X n=1 H n ˆ Q n H H n = N X n=1 H n ˜ Q n H H n . (5.28) This implies that if there exist two distinct Nash equilibrium achieving strategies for the non-cooperative game between the TXs in different coalitions, then the NE achieving 143 strategies satisfy the condition P N n=1 H n ˆ Q n H H n = P N n=1 H n ˜ Q n H H n . Discussion: While the uniqueness of NE utility can be demonstrated for the scenario with an SIC receiver, Proposition 4 shows that this might not be true in general for an SUD receiver. However, numerical simulations suggest that the NE utility for this non- cooperative game is unique in general and a counter-example to this scenario has not been found. In [58], the uniqueness of NE of parallel multiple access channels has been investigated for a SUD receiver and it was shown that the NE achieving strategies and thus the NE utility for this class of channels is unique (almost surely). A special case of our scenario, wherein all the channel matricesH n are diagonal matrices can be modeled as a parallel multiple access channel and we can infer from [58] that the NE utility in this scenario is unique (almost surely). Consider a scenario in which each TX transmits to a common RX over a multipath fading channel. Orthogonal Frequency Division Multiplexing (OFDM) has been one of the schemes used in the literature to overcome multipath effects of the channel. OFDM transforms a multiple access channel into several parallel flat fading channels and the total power available at each user can be distributed among the subcarriers. This signal- ing scenario can be reinterpreted as a parallel MAC with a sum power constraint on the total power for all the subcarriers of a user. Using the result in [58] clearly shows that uniqueness of NE utility for an OFDM signaling scenario when using an SUD receiver, thus demonstrating an important set of channels where the NE is indeed unique. How- ever, the proof technique in [58] does not directly generalize to the case of non-diagonal H n and the general problem of uniqueness for a MIMO SUD receiver remains an open problem. Fig. 5.5.1.2 shows one feasible NE utility for a scenario with 2 users and a single antenna RX. Note that this point is in the interior of the rate region as all the signals are 144 v 1 v 2 MAC rate region NE u!lity for SUD receiver Figure 5.9: Figure showing the NE rate point for an SUD single antenna receiver. The equilibrium rate point is in the interior of the capacity region as the receiver treats inter- ference as noise. decoded with interference. 5.5.2 TX cooperation - Properties and Stability In Section 5.4.2, we demonstrated the properties of TX cooperation for an MIMO MAC with an SIC RX. We now state the properties of TX cooperation for the SUD RX. Proposition 5. The TX cooperation game is r-super-additive and cohesive (see (4)), i.e., for two partitions T 1 = (S 1 ,S 2 ,...,S r ,S r+1 ,...,S N ) and T 2 = (S 1 ∪ S 2 ∪ ...∪ S r ,S r+1 ,S r+2 ,...,S N ) ofK,v(S 1 ∪...∪S r ;T 2 )≥ P r t=1 v(S t ;T 1 ) where all the utilities have been computed as in (5.26). Proposition 6. The TX cooperation game with a single antenna receiver has negative externalities, i.e., for two partitions T 1 = (S 1 ,S 2 ,...,S r ,S r+1 ,...,S N ) and T 2 = (S 1 ∪ S 2 ∪...∪S r ,S r+1 ,S r+2 ,...,S N ) ofK,v(S r+i ,T 2 )≤v(S r+i ,T 1 ) for everyi = 1,...,N−r and∀r. 145 Proposition 7. The TX cooperation game with a multiple-antenna receiver has mixed externalities, i.e., some realizations of the game has positive externalities and other re- alizations have negative externalities. The proof of Propositions 5, 6 and 7 follows exactly along the lines of Propositions 1, 2 and 3 respectively in Section 5.4.2 and are omitted here due to space limitations. We observe that the TX cooperation game for an SUD RX has the same properties as that of an SIC RX. This implies that when coalitions cooperate, the total utility achieved by the combined coalition is larger than the individual utilities and the grand coalition is the only feasible outcome of cooperation. We now investigate the stability of the grand coalition for a SUD RX following the approach in Section 5.4. The key difference between the SUD and the SIC RXs is that the NE utility achieved for an SUD RX is not proven to be unique and hence there may exist multiple feasible NE utilities for a given partition of TXs. To address the issue of uniqueness, we define the core of a game to be union of the cores obtained by choosing all feasible combinations of NE utility. Thus, having a nonempty core from one combination of utilities ensures that the core of the game itself is nonempty ensuring the stability of cooperation. We now determine the stability of TX cooperation by verifying that the Bondareva Shapley theorem in Section 5.2 is satisfied. Theorem 5. In the high SNR regime, i.e. N 0 → 0, the TX cooperation game with a SUD receiver has a nonempty core for both sum power and per-antenna power constraints. Proof. Let us first consider the r-core of the game. The utility of coalitionS in a partition T ={S,S 1 ,...,S N } can be expressed from (5.26) as v(S;T) = log |N 0 I +H S Q ∗ S H H S + P N j=1 H j Q ∗ j H H j | |N 0 I + P N j=1 H j Q ∗ j H H j | ! , (5.29) 146 where(Q ∗ S ,Q ∗ 1 ,...,Q ∗ N ) is an NE achievable strategy. We now show that for any balanced collection of weights λ S , the Bondareva Shapley theorem in Section 5.2 holds at high SNR. Substituting forv(S;T) in (5.5), we get X S⊆N λ S v(S;T) = X S⊆N λ S log |N 0 I +H S Q ∗ S H H S + P N j=1 H j Q ∗ j H H j | |N 0 I + P N j=1 H j Q ∗ j H H j | ! (a) ≤ log |N 0 I +H K Q ∗ K H H K | |N 0 I| =v(K;K), (5.30) where (a) is true for all partitions T as the summation of the LHS has a finite value for large SNR and the RHS terms increases in an unbounded fashion, satisfying the conditions of the Bondareva Shapley theorem. This proves that the r-core is nonempty at high SNR. Using similar arguments, it can be shown that all the various cores of a PFG defined in Section 5.2 are nonempty at high SNR showing that TX cooperation is stable at high SNR for an SUD receiver. In addition, we note that the proof is independent of the nature of the power constraints and hence is valid for scenarios with both sum power and per-antenna power constraints. Theorem 6. In the low SNR regime, i.e. N 0 →∞, the TX cooperation game with a sum power constraint and a SUD receiver has a nonempty core. Proof. We have observed in Theorem 2 that the utility function at low SNR does not depend on the actual interference experienced. Using the same methodology as in The- orem 2, we can show that the TX cooperation game has a nonempty core and hence cooperation is stable at low SNR. As in the previous section, the nonemptiness of the core at low SNR for per-antenna power constraints cannot be determined analytically due to the lack of a suitable approx- imation to the capacity of a MIMO channel at low SNR. However, numerical simulations show that the core is indeed nonempty for this scenario as well. In summary, we note 147 that ideal cooperation is stable for both the high SNR and low SNR regime for a SUD RX. 5.5.3 Discussion 5.5.3.1 Which RX is better for enforcing cooperation ? In a MIMO MAC, this work shows that for the low SNR regime, the GC is always a stable outcome of TX cooperation, under the assumption of no cooperation costs for both the SIC and SUD RXs. In this regime, all the power is allocated to one of the eigen modes of the channel and the power/beamforming gain that is obtained when multiple coalitions merge to form the GC is sufficient to ensure the stability of cooperation. On the other hand, the behavior in the high SNR regime is dominated by the effect of interference. For a SUD RX, any coalition deviating from the GC will experience significant interference and thus looses heavily by deviating from the GC. This ensures the stability of coop- eration at high SNR as shown in Theorem 5. Numerical simulations suggest that this holds for all SNRs of interest in general though it can be analytically shown only for the regimes of extreme SNR. For a SIC RX and a fixed decoding order, we have shown that GC is not stable at high SNR and stable (with an approximation to the utility) for time shared decoding orders. To enforce cooperation at high SNR for a fixed decoding order, we can impose a penalty on any deviating coalitions as indicated in Section 5.2.3. As the utility of the GC is the same irrespective of the RX used to decode the signals (it is the capacity of the virtual MIMO system formed when all the TXs cooperate), our analysis suggests that using a simpler SUD RX is better at enforcing cooperation in comparison to a more capable SIC RX. 148 5.5.3.2 Fairness of rate allocation The core of the TX cooperation game, when nonempty, is a large convex set as it the feasible region of a set of linear inequalities. This implies that, in general, there exist un- countable number of divisions of utility such that no coalition of users has an incentive to deviate from the GC. While the core itself does not take into account fairness of allo- cation, a suitable fairness metric optimized over the core can result in a stable allocation which is fair to the extent allowed by the elements of the core. 5.5.3.3 Information and computational requirements As mentioned previously, computing the NE utility for each configuration and then de- termining the nonemptiness of the core of a game, in general requires complete channel state information at all the players. For a K-player game, we see that O(2 K ) utilities must be evaluated to express the core and evaluating nonemptiness of the core can be evaluated by solving a linear program with exponential (2 K −1) number of constraints (see [26] for further details). While no cooperation costs are assumed in this analysis, sharing the channel state and codebook information in practice incurs costs and may reduce the NE utility that a coalition can achieve. In scenarios such as base station co- operation (see Fig. 5.1), the backbone network connecting the base stations can be used to share the channel state and codebook information. In [70], the cost of cooperation is modeled as a reduction in the power available for data transmission and [57] considers a partial-decode and forward scheme to share the messages to be jointed transmitted. However, accurate modeling of the cost of cooperation is still an open problem in the literature. 149 5.6 Conclusions The question of feasibility of cooperation between rational nodes in a MIMO multiple access channel and whether there exists a fair division of the benefits of cooperation is addressed using partition form cooperative game theory to accurately model the effects of interference. The stability of the grand coalition, the coalition of all transmitters, for SUD and SIC receivers is examined. For an SUD receiver, TX cooperation is shown to be stable at high and low SNRs analytically and at all SNRs numerically, while for an SIC receiver with a fixed decoding order, TX cooperation is only stable at low SNRs where interference is negligible. However, using a high SNR approximation to the utility function, TX cooperation is stable with an SIC receiver implementing equal time sharing between decoding orders. In summary, our work demonstrates that under the assumption of zero costs, voluntary cooperation is feasible and stable between users in a MAC and every user benefits from cooperation. 150 Chapter 6 Broadcast Channel Games and Game based MAC-BC Duality 6.1 Introduction Several wireless and cellular networks currently in operation are centrally controlled by an operator who is assumed to know the state of the network and operate it in an op- timal fashion to maximize throughput and minimize interference in a fair and efficient manner. However, due to the recent emergence of heterogeneous and decentralized wire- less networks in which several nodes are owned by different operators, it can no longer be assumed that there exists a central controller which can optimize the performance of the network as a whole. In recent literature, game theory has been used extensively to model, understand and drive the interactions between nodes in heterogeneous networks whjich act to optimize their own individual objectives. For example, several aspects of the multiple access channel and interference channel have been extensively studied using non-cooperative game theory (see [74, 1] for a detailed survery). In a typical non-cooperative game, the choice of actions by a player affects the util- ity obtained by every player, but does not change the set of available actions for other 151 players. For example, the Gaussian multiple access channel (MAC), the Gaussian inter- ference channel (IC) and its variants belong to this category of games [43, 44, 7, 95]. However, in scenarios, such as the Gaussian broadcast channel (BC), in which the re- ceivers can be considered the players in a game, the choice of actions of all the players is jointly constrained by a transmit power or covariance constraint. Thus, the choice of a transmit covariance matrix for one receiver will constrain the choice of transmit covari- ance matrices for the other receivers. For example, in [61], the problem of maintaining a minimum rate over parallel Gaussian interference channels subject to a sum power con- straint for each user, is considered. The choice of power allocation of a player for a given channel is then influenced by the choice of power allocation of other players on other interference channels to maintain an overall desired rate and hence is also a game with coupled constraints on the strategies adopted by the players. Such joint constraints on the strategies of the players results in the feasible set of each player being a function of the choice of strategies of the other players. Interaction of players at the level of feasible sets makes the analysis of such games much harder than standard non-cooperative games. The problem of determining the equilibrium points of games with coupled constraints is called a generalized Nash equilibrium problem (GNEP) [74, 61, 67] and the points themselves are called generalized Nash equilibria (GNE). From a game theoretic perspective, the broadcast channel (BC) and its related prob- lems have received little attention relative to other channels such as the MAC and the IC. A discrete memoryless BC with 2 users and a resource manager was considered in [84] and the impact of the information available to the resource manager in modifying the utility of each user is studied. A game theoretic model of the BC enables receivers to determine the rate they obtain in a self-enforcing fashion. For example, such a model will pave the way for new cognitive radio scenarios in which the primary and multiple secondary users can operate simultaneously on the same spectrum and in determining 152 the cooperative behavior among receivers in broadcast channels. In this chapter, we first examine a re-interpretation of the GNEP by describing a framework for decoupling the common constraint and treating the resulting problem as a Nash equilibrium problem with penalized utility functions. We then consider the MIMO MAC with sum power constraints (henceforth called the sum power MAC). The sum power MAC is a problem that is closely related to the BC by an information-theoretic duality and [90] shows that the capacity region for the sum power MAC is same as the capacity region for an equivalently defined BC. The results presented herein are a nec- essary precursor towards establishing the notion of duality with respect to MAC and BC games. The sum power MAC is modeled as a GNEP and the existence and uniqueness of equilibrium rates for this problem is investigated. We then illustrate the interplay be- tween uniqueness of equilibrium rates and the choice of decoding order at the receiver for the sum power MAC. By deriving a relation between Pareto-optimal points and equi- librium rates we show that every rate point on the Pareto-optimal boundary of the sum power MAC or the BC is an equilibrium rate point. When paired with a decoupled char- acterization of the GNEP, this relationship enables the enforcement of Pareto-optimal equilibria using decentralized algorithms. We then develop several equivalence relations between equilibria of a class of NEPs and GNEPs and then demonstrate a game-theoretic duality between the MAC and the BC. The game theoretic duality enables us to translate several results from the MAC to the BC and vice versa in an easy fashion and would pave the way for easily analyzing BC games by considering equivalent MAC games and vice versa. The rest of the chapter is organized as follows. Section 6.2 introduces the GNEP and describes a decoupled characterization of some classes of GNEPs. Section 6.3 states the conditions for existence and uniqueness of equilibria of a GNEP and summarizes the results in [96] for the BC while Section 6.4 considers the sum power MAC and shows 153 the existence and uniqueness of equilibria for this problem. The relationship between Pareto-optimal solutions and equilibria is derived in Section 6.5 and the correspondence between the two solutions is discussed with applicability to both the sum power MAC and the BC. Section 6.6 derives several equivalence relations between the MAC and the BC and then shows the existence of a game theoretic duality between the two channels and Section 6.7 concludes this chapter. 6.2 Preliminaries 6.2.1 Generalized Nash Equilibrium Problems Formally, a GNEP consists of K players with each player controlling the variable Q k . We denote byQ, the vector formed by all these decision variablesQ = (Q 1 ,Q 2 ,...,Q K ), and by Q −k the vector formed by the decision variables of all other players except the kth player. Each player has an objective function v k that depends on both his own variables Q k and the controlling variables of all other players Q −k . This func- tion is called the utility function of player k and is formally denoted as v k (Q k ,Q −k ) or v k (Q), Q = (Q 1 ,Q 2 ,...,Q K ) to emphasize the dependence on the controlling variables. Furthermore, each player’s strategy must belong to a setA k (Q −k ) that depends on the rival players’ strategies and that we call the feasible set or strategy space of player k. We emphasize that the setA k (Q −k ) is a function of the strategies of the other players. The aim of thekth player, given the strategies of all the other playersQ −k , is to pick a strategy that solves the maximization problem max Q k v k (Q k ,Q −k ) subject toQ k ∈A k (Q −k ). (6.1) 154 LetΨ k (Q −k ) denote the set of all the solutions to thekth player’s maximization problem givenQ −k . The GNEP is the problem of findingQ ∗ k such that Q ∗ k ∈ Ψ(Q ∗ −k ) for allk = 1,2,...,K. Such as point is called a generalized Nash equilibrium (GNE) or more generally a solu- tion to the GNEP. A pointQ ∗ is therefore an equilibrium, if no player can improve this objective function by changing unilaterally to any other point in his feasible set. If we denote by Ψ(Q) the set Ψ(Q) := × K i=1 Ψ i (Q −i ) we see that Q ∗ is a GNE if and only if Q ∗ ∈ Ψ(Q ∗ ), i.e., if and only if Q ∗ is a fixed point of the mapping Ψ. If the feasible sets do not depend on the rival strategies, then the GNEP reduces to the standard Nash equilibrium problem. In a GNEP, thekth player must know the strategy of the other players to determine his own feasible set, but the other players need to know the strategy of player k to de- termine their own strategy. The key point here is that one cannot imagine a game where the players make their choices simultaneously and it so happens that the constraints are satisfied. However, this view of GNEPs is rather limited [21] and severely undervalues the 1. the explanatory power of the GNEP model; 2. the possibility to use GNEPs to design rules and protocols, set taxes etc and so forth, in order to achieve certain goals. We note that GNEPs have been used in a normative way in the literature. No one is really playing a game; rather a single decision maker establishes that the outcome of a GNEP is desirable and therefore implements it. This perspective is common in many modern 155 engineering applications of GNEPs (see the power allocation problem in a Gaussian interference channel [61]). A GNEP usually has multiple or, in many cases, uncountably many equilibria [67, 21]. While characterizing the GNEs for a general GNEP has proven to be a challenging problem in general, special classes of GNEPs such as jointly convex scenarios have been examined in literature and are briefly illustrated below [67]. Definition 6. A GNEP is said to be concave (convex) if for every playerk and everyQ −k , the utility function v k (Q k ,Q −k ) is concave (convex) and the set Ψ k (Q −k ) is closed and convex. Definition 7. A GNEP is said to be jointly convex if the GNEP is concave (convex) and for some closed setQ and allk = 1,2,...,K, we have Ψ k (Q −k ) ={Q k |(Q k ,Q −k )∈Q}. (6.2) This class of problems has been first studied in detail in a seminal paper by Rosen [67] and have been referred to as GNEPs with coupled constraints or in general jointly convex GNEPs. From [21], we know that if the sets Ψ k (Q −k ) are defined by a system of inequalities, then there exists a functionh(Q) which is component-wise convex with respect to Q k for all k = 1,2,...,K and furthermoreQ = {Q|h(Q)≥ 0}. 1 . In other words, jointly convex GNEPs are characterized by the fact that all the players have the same common constraints. Rosen [67] allows for a discriminatory treatment of players through the introduction of weights r i > 0, i = 1,2,...,K with which the enforcer of the joint constraint can 1 In this work, we will restrict the treatment of GNEPs for a scenario with one joint constraint 156 value each player’s payoff. The main role of the weights in controlling the player’s behavior is that they modify the Karush-Kuhn Tucker multipliers of each player’s utility maximization and entice the players to choose actions that lead to a desirable equilibrium outcome (among the infinitely many possible). 6.2.2 Characterization of jointly convex GNEPs Define the weighted utility function f(B,Q,r) = K X i=1 r i v i (Q 1 ,...,Q i−1 ,B i ,Q i+1 ,...,Q K ), (6.3) for a fixed vector of positive weightsr = (r 1 ,r 2 ,...,r K ). The K-tupleQ ∗ = (Q ∗ 1 ,...,Q ∗ K ) is a normalized equilibrium (NoE) with respect to the weightsr, ifQ ∗ satisfies the fixed point condition Q ∗ = argmax B f(B,Q ∗ ,r), (6.4) Let h(Q 1 ,Q 2 ,...,Q K ) ≥ 0 denote the joint constraint that each player must satisfy in addition to individual constraints on his control variables. Then, Q ∗ is an NoE if and only if it satisfies the following KKT conditions,h(Q ∗ ) = 0,λh(Q ∗ ) = 0 and r k ∇ k v k (Q ∗ k ,Q ∗ −k )+λ k ∇ k h(Q ∗ ) = 0, k = 1,...,K. (6.5) The parameterλ k is called the shadow price of the joint constraint (This is a vector when multiple joint constraints exist). For a general GNEP, the Lagrange multipliers λ k are unrelated to each other in general. However, we consider a special kind of equilibrium which can reflect the different levels of agent responsibility for constraint satisfaction. Definition 8. An equilibrium point is called a Normalized equilibrium point (NoE) if and 157 only ifλ k =λ for allk = 1,2,...,K. The NoEs of a GNEP can be characterized by the fact that the shadow prices of the joint constraints are equal for all the players. The parameter λ r i is the real price that each player pays at NoE for satisfying the joint constraint. Now, we can better characterize the role of the weight vectorr. If the weight of thekth playerr k is greater than that of his competitors, then his real price is reduced and the marginal cost for constraint violation is lower than a player with higher weight. In other words, the choice of the vector r enables the decision maker to decide how the burden of constraint satisfaction is to be divided among all the players in a GNEP. For a GNEP with only one jointly coupled constraint, it can be seen that the weights of each player r k can be modified to r ′ k such that the shadow price of each player is identical. This implies that the NoEs and GNEs of a GNEP with only one jointly coupled constraint are identical. However, we note that this does not generalize to the scenario for a GNEP with multiple jointly coupled constraints. In this work, as we consider BC and sum power MAC problems with only a joint sum power constraint, the set of NoEs and GNEs are identical and the terms will be used interchangeably unless otherwise stated. Once the decision maker of a jointly convex GNEP has established a desired NoE, the equilibrium implementation of the problem is straightforward. The real prices associated with the joint constraints are used as penalty tax rates for constraint violation and the players have to allow for penalties in their payoffs. Define the penalty function: T k (λ ∗ ,r k ,Q 1 ,...,Q K ) = λ ∗ r k max(0,−h(Q 1 ,Q 2 ,...,Q K )), (6.6) whereλ ∗ is the Lagrange multiplier associated with the common constraint,r k the weight of the k th player that defines the responsibility for constraint satisfaction and h(Q) ≥ 0 the common constraint. Clearly, a player with a large weight has a smaller penalty 158 for violating the common constraint while a player with smaller weight suffers a larger penalty. If the constraint is perfectly satisfied then the penalty is zero for all the players. Now, using the penalty function T k , we define a game with modified utility functions which take into account the penalty for common constraint violation. ˜ v k (Q k ,Q −k ) =v k (Q k ,Q −k )−T k (λ ∗ ,r k ,Q 1 ,...,Q K ). (6.7) Each player now needs to satisfy only his decoupled constraints and thus with the mod- ified utility function, the problem becomes a traditional Nash equilibrium problem with decoupled constraints. From [41], we know that the Nash equilibria for the modified utility functions are the same as the NoEs of the original problem with jointly convex constraints. 6.2.3 Uniqueness of NoEs We first define several terms which help us in deriving a sufficient condition for the uniqueness of NoEs for a GNEP. Let σ(Q,r) = K X i=1 r i v i (Q i ,Q −i ), r i > 0, (6.8) to be the weighted sum of the utilities of each player, whereQ are the control variables for all the players andr is a vector containing a set of positive weights. 159 Definition 9. The function g(Q,r) =          r 1 ∇ 1 v 1 (Q 1 ,Q −1 ) r 2 ∇ 2 v 2 (Q 2 ,Q −2 ) . . . r K ∇ K v K (Q k ,Q −k )          , (6.9) where ∇ i is the derivative w.r.t the i th players’control variables is called the pseudo- gradient ofσ(Q,r). Definition 10. The functionσ(Q,r) is called diagonally strictly concave (DSC) in matrix valued strategies for a fixedr > 0, if for every ˜ Q, ˆ Q∈Q, we have that Tr h ( ˜ Q− ˆ Q) T g( ˆ Q,r)+( ˆ Q− ˜ Q) T g( ˜ Q,r) i > 0. (6.10) Proposition 8. If the utility functions of a jointly convex GNEP satisfy the DSC condition for any givenr, then the GNEP has a unique NoE for that given value ofr. The proof of this proposition depends on the nature of the jointly convex constraints in general. We consider a sum power constraint of the form P K i=1 Tr[Q i ] ≤ P tot and note that this proof technique can be directly applied to problems with various kinds of constraints. We begin by assuming that there exists multiple NoEs for the given value of r, and then arrive at a contradiction to the DSC property. 160 Proof. Let us assume that ˜ Q = ˜ Q 1 , ˜ Q 2 ,..., ˜ Q K and ˆ Q = ˆ Q 1 , ˆ Q 2 ,..., ˆ Q K be twoK- tuples of covariance matrices which are NoEs to the game characterized by the weight vectorr. We know from (6.4) that ˜ Q = argmax B f(B, ˜ Q,r) and ˆ Q = argmax B f(B, ˆ Q,r). (6.11) Writing the Lagrangian for the maximization of the weighted utility function, we get L =f(B,Q,r)+λ K X i=1 Tr[Q i ]−P tot ! + K X i=1 Tr[L i Q i ] (6.12) Writing the (KKT) conditions [10] for the two equilibria yields: (a) ˜ Q i , ˆ Q i 0, i = 1,2,...,K (b) P K i=1 Tr[ ˜ Q i ]≤P tot and P K i=1 Tr[ ˆ Q i ]≤P tot . (c) Tr[ ˜ L i ˜ Q i ] = 0 and Tr[ ˆ L i ˆ Q i ] = 0. (d) ˜ λ P K i=1 Tr[ ˜ Q i ]−P tot = 0. (e) ˆ λ P K i=1 Tr[ ˆ Q i ]−P tot = 0. (f) r i ∇ i v i ( ˜ Q)+ ˜ L i − ˜ λI = 0 (g) r i ∇ i v i ( ˆ Q)+ ˆ L i − ˆ λI = 0. where ˜ λ, ˆ λ≥ 0 and ˜ L i , ˆ L i 0 are the Lagrange multipliers associated with sum power 161 constraint and the positive semi-definiteness of the solutions respectively. Now multiply- ing (f) and (g) with ( ˆ Q i − ˜ Q i ) and ( ˜ Q i − ˆ Q i ) respectively, summing oni and taking the trace we get 0 = K X i=1 Tr h ( ˆ Q i − ˜ Q i )(r i ∇ i v i ( ˜ Q)+ ˜ L i − ˜ λI) i + K X i=1 Tr h ( ˜ Q i − ˆ Q i )(r i ∇ i v i ( ˆ Q)+ ˆ L i − ˆ λI) i = K X i=1 Tr h ( ˆ Q i − ˜ Q i )r i ∇ i v i ( ˜ Q)+( ˜ Q i − ˆ Q i )r i ∇ i v i ( ˆ Q) i + K X i=1 Tr h ( ˆ Q i − ˜ Q i )( ˜ L i − ˜ λI)+( ˜ Q i − ˆ Q i )( ˆ L i − ˆ λI) i =α+β (6.13) Re-arranging and evaluating the second term, β = Tr " K X i=1 ( ˜ Q i − ˆ Q i ) n ( ˜ λI− ˜ L i )−( ˆ λI− ˆ L i ) o # (c) = Tr " K X i=1 ( ˜ λ ˜ Q i − ˆ λ ˜ Q i + ˜ Q i ˆ L i − ˜ λ ˆ Q i + ˆ Q i ˜ L i + ˆ λ ˆ Q i ) # (d,e) = ˜ λP tot + ˆ λP tot − Tr " ˆ λ X i ˜ Q i + ˜ λ X i ˆ Q i # + Tr " X i ( ˜ Q i ˆ L i + ˆ Q i ˜ L i ) # (a) ≥ Tr " ˆ λ P tot − X i ˜ Q i !# + Tr " ˜ λ P tot − X i ˆ Q i !# (b) ≥ 0. (6.14) 162 We have shown thatβ≥ 0 and hence forα+β = 0 we need thatα≤ 0. Now α = K X i=1 Tr h ( ˆ Q i − ˜ Q i )r i ∇ i v i ( ˜ Q)+( ˜ Q i − ˆ Q i )r i ∇ i v i ( ˆ Q) i = Tr h ( ˆ Q− ˜ Q) T g( ˜ Q,r)+( ˜ Q i − ˆ Q i )g( ˆ Q,r) i . (6.15) Recognizing thatα > 0 is the DSC condition from Definition 10, we arrive at a contra- diction as we assume that the DSC condition is satisfied for the given value ofr. Hence, when the DSC condition is satisfied, the NoE of the GNEP for a given weight vectorr is unique. Note that though the proof is valid for one jointly convex constraint, it can be ex- tended in a straight forward fashion to a problem to multiple various jointly convex con- straints. Finally, we observe that the DSC condition is one of the sufficient conditions for determining the uniqueness of NoEs and there could exist several other sufficient conditions. 6.3 The MIMO BC as a Generalized Nash Equilibrium Problem Consider a multiple-input multiple-output (MIMO) broadcast channel withK receivers. The transmitted signal, denoted byx nt×1 , wheren t is the number of TX-antennas, is the sum of independent x i , each drawn from a Gaussian codebook and intended for the i th receiver (RX): x = P K i=1 x i , x i ∼ N(0,Q i ). The received signal at the i th RX can be expressed as y i =H i x+n i , z i ∼N(0,N i ), (6.16) 163 where H i is channel gain matrix from the TX to the i th RX and n i is the number of antennas at the i th RX. The signal at each RX is a linear transformation of the sum of the signals intended for all the receivers. Without loss of generality, we assume that TX signaling is constrained by a sum power constraint Tr E[xx T ] = K X i=1 Tr[Q i ]≤P tot , where P tot is the maximum transmit sum power for all the antennas. We begin by assuming that the transmitter performs dirty paper coding with a fixed encoding order π = {K,K−1,...,1}. The rate achieved by thek th receiver, which can be considered as a player in the broadcast channel game, is given as v k (Q k ,Q −k ) = log |N k +H k ( P k i=1 Q k )H H k | |N k +H k ( P k i=1 Q k )H H k | ! . (6.17) In this work, the utility of each player is defined as the achievable rate under the broadcast channel with dirty paper coding and a given encoding order. Note that v k (Q k ,Q −k ) is concave in Q k and continuous in Q i for all i = 1,2,...,K. The controlling variables for the playersQ k are coupled with as sum power constraint which is common to every receiver and hence the broadcast channel game is a GNEP with jointly convex constraints (see Definition 7). We emphasize that the receivers themselves are not playing a game in this scenario. As described in Section 6.2, we assume that the transmitter acts as a single decision maker establishing that the solution of the GNEP is desirable and implements it on behalf of all the receivers. We begin by examining the existence of NoEs for the MIMO BC. From Theorem 3 of Rosen [67], we know that there exists a NoE point for a concave n-person game for every weight vectorr > 0. As the broadcast channel game is a concave game with jointly 164 convex constraints, we immediately conclude that there exists an NoE for every weight vectorr > 0. 6.3.1 Uniqueness of NoEs We begin by considering a special class of broadcast channels called aligned and de- graded broadcast channels (ADBCs). Definition 11. A MIMO BC is aligned and degraded if the BC is aligned, i.e.,n t =n 1 = n 2 =... =n K andH i =I nt×nt and the covariances of the Gaussian noise at the receiver are ordered such that 0≺ N 1 N 2 ... N K , whereA B implies thatB−A is a positive semi-definite matrix [92]. For an ADBC [92], the achievable rate or the utility function of the k th player sim- plifies to v ADBC k (Q k ,Q −k ) = log |N k + P k i=1 Q i | |N k + P k−1 i=1 Q i | ! . (6.18) We know from Section 6.2 that the DSC criterion in Definition 10 is a sufficient con- dition for the uniqueness of NoE for a given positive weight vectorr. We now determine the weight vectors and the decoding orders for which the DSC criterion holds for the ADBC. We first state two trace inequalities that will be used to derive the uniqueness results. Lemma 5. [6] For any positive integer K and a set of positive semi-definite matrices A 1 ,A 2 ,...,A K andB 1 ,B 2 ,...,B K such thatA 1 ≻ 0 andB 1 ≻ 0, we have that Tr    K X k=1 (A k −B k )   k X l=1 B l ! −1 − k X l=1 A l ! −1      ≥ 0. (6.19) 165 Note that this set of inequalities may not be the tightest trace inequalities. For exam- ple, forK = 2 and any positive real numberw, it has been shown in [23] that Tr[(A 1 −B 1 )(B −1 1 −A −1 1 )+4(A 2 −B 2 ) (wB 1 +B 2 ) −1 −(wA 1 +A 2 ) −1 ]≥ 0. (6.20) Theorem 7. For a K-receiver ADBC with dirty paper coding at the TX and encoding order π = {K,K − 1,...,1}, a unique NoE achieving strategy exists for every weight vectorr which satisfiesr 1 ≥r 2 ≥...≥r K > 0. Proof. Let ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q k ) and ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q k ) be any two tuples of covariance matri- ces which satisfy the sum power constraint: P K i=1 Tr[ ˜ Q i ]≤P tot and P K i=1 Tr[ ˆ Q i ]≤P tot . Substituting the utility function for the ADBC from (6.18) in the DSC condition in defi- nition (10), we get Tr " K X k=1 r k ( ˆ Q k − ˜ Q k ) n ∇ k v k ( ˜ Q)−∇ k v k ( ˆ Q) o # (6.21) = Tr " K X k=1 r k ( ˆ Q k − ˜ Q k ) (N k + k X i=1 ˜ Q i ) −1 −(N k + k X i=1 ˆ Q i ) −1 # = K−1 X n=1 (r n −r n+1 )T n +r K T K , (6.22) where the termT n can be expressed as Tr " n X k=1 ( ˆ Q k − ˜ Q k ) (N k + k X i=1 ˜ Q i ) −1 −(N k + k X i=1 ˆ Q i ) −1 # . (6.23) It is now sufficient to show thatT n > 0. Notice that the structure ofT n closely resembles the inequality in (6.19). Choose the quantities A 1 = N 1 + ˜ Q 1 , B 1 = N 1 + ˆ Q 1 , A i = N i −N i−1 + ˜ Q i andB i =N i −N i−1 + ˆ Q i . By definition, sinceN 1 is a positive definite 166 matrix and ˜ Q 1 , ˆ Q 1 are positive semi-definite the matricesA 1 andB 1 are strictly positive definite. From the degradedness of the channel, we get that N i − N i−1 is a positive semi-definite matrix and hence A i and B i are positive semi-definite for i = 2,...,K. Substituting the values of A i and B i in (6.19), it is straight forward to see thatT n ≥ 0. For an ADBC channel having identity channel matrices, we know from [7] that if the NoEs ˜ Q 6= ˆ Q then T n > 0 and hence the NoEs of the ADBC game as unique for r 1 ≥r 2 ...≥r K > 0 and encoding orderπ ={K,K−1,...,1}. It is clear that the weight vectorsr for which uniqueness can be shown is dependent on the tightness of the matrix trace inequalities. ForK = 2, the inequality in (6.19) has been improved to the inequality in (6.20). Thus for the 2-user ADBC, the uniqueness can be derived for a more general set of weight vectors. Theorem 8. For a 2-user ADBC with dirty paper coding at the transmitter and interfer- ence canceling receivers, a unique normalized equilibrium point exists forr 1 ≥ r 2 /4 > 0. Proof. The proof follows exactly on the lines of Theorem 7 with the DSC condition decomposing into two terms given by (r 1 − r 2 4 )T 1 and r 2 4 T 2 . Now using the inequality in (6.20) with w = 1, it is easy to show that there exists a unique NoE for each weight vector which satisfiesr 1 ≥r 2 /4> 0. 6.4 The Sum Power MIMO Multiple Access Channel as a Generalized Nash Equilibrium Problem The capacity regions of the MAC and the BC are closely related by the information- theoretic duality derived in [90]. In particular, it is shown in [90] that the capacity region 167 of the general Gaussian BC is identical to the capacity region of a suitably defined MIMO MAC with a joint sum power constraint across all TXs, in contrast to the individual power constraints for each TX for a regular MAC. In this section, we characterize the NoEs of the sum power MAC which will enable our investigation of duality in Section 6.6. 6.4.1 Signal Model and the Sum Power MAC game Consider a MIMO MAC channel withK TXs (users) with thei th TX hasn i antennas and a common receiver withn r antennas. Each transmitted signal is drawn from a Gaussian codebook x i ∼ N(0,Q i ) and transmitted signals satisfy a sum power constraint given by P K i=1 Tr[Q i ]≤P tot . The received signaly can be written as y = K X i=1 H i x i +z, (6.24) where H i is n r × n i channel matrix from the i th TX to the RX and z ∼ N(0,N 0 I) is the AWGN at the receiver. Each TX is considered as a player in the game with an action space that consists of signaling covariance matrices which satisfy the joint sum power constraint and hence the sum power MAC game can be modeled as a GNEP. The receiver performs successive interference cancellation to decode the signals from each TX and without loss of generality, we assume thatπ = [K,K−1,...,1] is the decoding order. whereH i isn r ×n i channel matrix from thei th TX to the RX andz∼N(0,N 0 I) is the AWGN at the receiver. Each TX is considered as a player in the game with an action space that consists of signaling covariance matrices which satisfy the joint sum power constraint and hence the sum power MAC game can be modeled as a GNEP. The receiver performs successive interference cancellation to decode the signals from each 168 TX and without loss of generality, we assume thatπ = [K,K−1,...,1] is the decoding order. The utility function for each TX is the rate it obtains and can be expressed as v MAC k (Q k ,Q −k ) = log |N 0 I + P k i=1 H i Q i H H i | |N 0 I + P k−1 i=1 H i Q i H H i | ! (6.25) Clearly, the utility function for the k th player (TX) is concave in Q k and a continuous function of all other variables. In addition, as the joint sum power constraint defines a convex set of feasible strategies, the sum power MAC is a concave game and hence from [67] we know that for every weight vector r, there exists at least one NoE achieving strategy. 6.4.2 Uniqueness of NoEs We use the DSC condition to determine the uniqueness of NoE achieving strategies and the uniqueness of the value of the utility function for the sum power MAC. From the derivation of the DSC condition in definition 10, we know that if there exist two NoEs of the game ˜ Q = ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q K ) and ˆ Q = ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q K ) for a given weight vectorr, then K X k=1 r k Tr h ( ˆ Q k − ˜ Q k )∇ k v k ( ˜ Q k )+( ˜ Q k − ˆ Q k )∇ k v k ( ˆ Q k ) i =α≤ 0. (6.26) Theorem 9. For the sum power MAC with a successive interference canceling receiver implementing a decoding order of π = [K,K − 1,...,1], a weight vector r such that 169 r 1 ≥ r 2 ≥ ... ≥ r K > 0 and any two feasible strategies ˜ Q = ( ˜ Q 1 , ˜ Q 2 ,..., ˜ Q K ) and ˆ Q = ( ˆ Q 1 , ˆ Q 2 ,..., ˆ Q K ), we have that K X k=1 r k Tr h ( ˆ Q k − ˜ Q k )∇ k v k ( ˜ Q k )+( ˜ Q k − ˆ Q k )∇ k v k ( ˆ Q k ) i =α≥ 0. (6.27) andH i ˜ Q i H H i = H i ˆ Q i H H i for alli = 1,2,...,K. In other words, the utility achieved by each NoE is unique. Proof. Consider two tuples of covariance matrices in the feasible set ˜ Q = ( ˜ Q 1 ,..., ˜ Q K ) and ˆ Q = ( ˆ Q 1 ,..., ˆ Q K ). Define the matrix valued set of functions φ k (Q) for k = 1,2,...,K as φ k (Q) = ( P k i=1 H i Q i H H i + N 0 I) −1 . By re-arranging terms and sub- stituting the utility function inα, we get K X k=1 r k Tr h ( ˆ Q k − ˜ Q k ) ∇ k v k ( ˜ Q)−∇ k v k ( ˆ Q) i = K X k=1 r k Tr h (H k ˆ Q k H H k −H k ˜ Q k H H k )(φ k ( ˜ Q)−φ k ( ˆ Q)) i = K−1 X n=1 (r n −r n+1 )T n +r K T K , (6.28) whereT n = Tr h P n k=1 ( ˆ Q k − ˜ Q k )(φ k ( ˜ Q)−φ k ( ˆ Q)) i . Choose the quantitiesA 1 =N 0 I+ H 1 ˜ Q 1 H H i , B 1 = N 0 I + H 1 ˆ Q 1 H H 1 , A i = H i ˜ Q i H H i and B i = H i ˆ Q i H H i . By defi- nition, since N 0 I is a positive definite matrix and ˜ Q 1 , ˆ Q 1 are positive semi-definite the matrices A 1 and B 1 are strictly positive definite. In addition, we see that A i and B i are positive semi-definite matrices for i = 2,...,K. Using the trace inequality in [6, 7] which states that P N i=1 Tr h (A i −B i ) n ( P i j=1 B j ) −1 −( P i j=1 A j ) −1 oi ≥ 0, with equality only whenA i =B i , ∀i = 1,2,...,N, we see thatT n ≥ 0 for alln, thus proving 170 the condition in (6.27). From (6.26) and (6.27), we infer that if ˜ Q = ( ˜ Q 1 ,..., ˜ Q K ) and ˆ Q = ( ˆ Q 1 ,..., ˆ Q K ) are two NoEs (and hence are also achievable strategies), thenα = 0. Now from Proposition 9 we know that α = 0⇔A i =B i ∀i⇔H i ˜ Q i H H i =H i ˜ Q i H H i . (6.29) Substituting in the utility function in (6.25), we observe that given a weight vector r such that r 1 ≥ r 2 ≥ ... ≥ r K > 0 and decoding order π = [K,K− 1,...,1], the NoE utility obtained by both the NoE achieving strategies ˜ Q and ˆ Q is identical implying the uniqueness of the NoE utility for a weight vector and its associated decoding order. Note that for a decoding orderπ = [K,K−1,...,1], the NoE exists for every weight vectorr but the NoE utility is unique only for the weightsr 1 ≥ r 2 ≥ ...≥ r K > 0. In addition, as H i ˜ Q i H H i = H i ˆ Q i H H i does not imply that ˜ Q = ˆ Q, the NoE achieving strategies are not unique in general. Similarly, for any weight vectorr and a decoding order in which RXs with the larger weights are decoded later than RXs with smaller weights, we can show that the NoE utility is unique. In other words, for any given weight vectorr, for an appropriate choice of decoding order at the RX, the NoE utility is unique, while for the same r, the NoE utility and hence NoE achieving strategies need not be unique for different decoding orders at the RX. We once again emphasize that the NoE utility is unique only for certain combinations of weight vectors and decoding orders and the uniqueness property does not hold for all weight vectors and decoding orders in general. 171 6.5 Computation of Normalized Equilibrium In this section, we describe the computation of NoEs for the ADBC and the sum power MAC. We have shown previously that both the games are concave and hence for each weight vector r there exists a NoE. Next, we have characterized the weight vectors for which the NoEs are unique. In general, we would like to compute all the NoEs associated with each weight vector r. However, in this work, we focus on Pareto-efficient NoEs, i.e., equilibria which also maximize a weighted sum of utilities. If a rate point is both Pareto-efficient and a NoE, it is a socially optimum solution with self-enforcing prop- erties and enables the implementation of optimal centralized solutions using distributed and decentralized algorithms. To determine Pareto-efficient equilibria, we first derive a relation between the weights γ which characterize a given Pareto-optimal solution and the weightsr for which this rate point is a NoE utility [41]. We generalize the procedure in [41] toK-players and matrix valued strategies to derive the desired relationship. 6.5.1 Pareto-Efficiency First Order Conditions Consider a regulator (for example, a base station or a coordinating entity) who would like to control the rates achievable by each user. The regulator would like choose a feasible strategy to optimize a weighted sum rate of utility functions. LetF denote the set of all K-tuples of covariance matrices which satisfy the sum power constraint. The regulator’s problem can be written as max Q∈F K X i=1 γ i v i (Q i ,Q −i ), (6.30) 172 wherev i is the utility function andγ i is the weight attached to thei th player. We assume that the regulator is interested in solutions which saturate the constraint P K i=1 Tr[Q i ] ≤ P tot . The Lagrangian for the regulator w.r.t the common constraint can be written as L P = K X i=1 γ i v i (Q i ,Q −i )−µ " K X i=1 Tr[Q i ]−P tot # + K X i=1 Tr[L i Q i ], (6.31) whereµ > 0 andL i 0 are the dual variables for the constraints. As the feasible setF is non-empty forP tot > 0, the KKT conditions can be written for allk = 1,2,...,K as: ∂L P ∂Q k = K X i=1 γ i ∂v i (Q i ,Q −i ) ∂Q k −µI +L k = 0 L k 0, Tr[L k Q k ] = 0 µ> 0, K X i=1 Tr[Q i ]−P tot = 0. (6.32) Let us denote the Pareto-optimal solution to beQ ∗ (γ) = (Q 1 (γ),...,Q K (γ)) whereγ = (γ 1 ,...,γ K ) T . Given the concavity of the utility functions in the users’ control variable and convexity of the constraint set, the above conditions are sufficient for the solution of the differential equations to be Pareto-optimal. 6.5.2 Rosen’s equilibrium first order conditions We know from Section 6.2.2 that it is possible to control players, who share a common constraint, to satisfy this constraint by interpreting the GNEP as a modified Nash equi- librium problem with penalties for violating the common constraint. Mathematically, the 173 regulator seeks a solution which is the NE of the modified game. The Lagrangian of the optimization associated with each player can be written as L R k =r k v k (Q k ,Q −k )−λ " K X i=1 Tr[Q i ]−P tot # + Tr[M k Q k ], (6.33) where λ and M k are the dual variables for the constraints. Note that the factor λ is equal for all players as we are interested in NoE where the shadow price ( λ r i is the true price, see Section 6.2) of the common constraint are equal for all players. Now, a set of strategies ˆ Q(r) = (Q 1 (r),...,Q K (r)) is a NoE if it satisfies the first order conditions for allk = 1,2,...,K: ∂L R k ∂Q k =r k ∂v k (Q k ,Q −k ) ∂Q k −λI +M k = 0 M k 0, Tr[M k Q k ] = 0 λ> 0, K X i=1 Tr[Q i ]−P tot = 0. (6.34) 6.5.3 Relation between Pareto-optimal and NoE solutions Now we derive the relation betweenγ andr such that the solutions from obtained from the KKT conditions for the Pareto-optimal problem and the equilibrium problem are identical, i.e.,Q ∗ (γ) = ˆ Q(r). Let us first consider the simple case whenQ ∗ k 6= 0 nt×nt for all k = 1,2,...,K. Defining η = µ λ and manipulating the KKT conditions, we see that for eachk = 1,2,...,K, ηr k Tr ∂v k (Q k ,Q −k ) ∂Q k Q k = K X i=1 γ i Tr ∂v i (Q i ,Q −i ) ∂Q k Q k . (6.35) 174 Let us defineb k =r k h ∂v k (Q k ,Q −k ) ∂Q k Q k i Q ∗ ,b = (b 1 ,b 2 ,...,b k ) T and the elements of matrix A as A ki = Tr h ∂v i (Q i ,Q −i ) ∂Q k Q k i Q ∗ . The above condition can be compactly written as ηb = Aγ. As the weights r i are relative to each other, we assign r K = 1 to constrain our problem. Assuming thatγ andQ ∗ are known, we can solve forr 1 ,r 2 ,...,r K−1 ,η. If there exists an easy method to compute the Pareto-optimal solutions (which is true for the ADBC and the sum power MAC [90]), then using (6.35), the regulator can determine the Lagrange multiplier coefficient λ and the weights r i which enable us to enforce the NoE solution in a distributed fashion. For the ADBC, evaluating the elements of A we get A ki =            Tr[(Q 1 +...+Q k +N k ) −1 Q k ] if k =i Tr[((Q 1 +...+Q i +N i ) −1 −(Q 1 +...+Q i−1 +N i ) −1 )Q k ] if k <i 0 if k >i (6.36) For the regime of γ in which Q ∗ k 6= 0 nt×nt for all k, it is clear from (6.36) that A is an upper-triangular matrix with positive diagonal elements and negative-off diagonal ele- ments. For such scenarios, the matrix A is invertible and its structure ensures that for every weight vector γ and the related dual variable µ , there always exists a vector of positive weightsr and scaling coefficientη. However, the converse is not true and there could be weightsr for which a Pareto-optimal weight vector does not exist. Now, let us consider the scenario in whichQ ∗ k = 0 nt×nt for somek. This implies that no transmission is scheduled for the k th user and this user does not enter the game for the considered value ofγ. In addition, we note that (6.35) reduces to the trivial equation 0 = 0. For such a scenario, we define a new BC withK−1 users by eliminating thek th user. Clearly, the newly defined BC is an ADBC withQ ∗ i 6= 0, ∀i and hence the above procedure can be used to find the weights r i to enforce the Pareto-optimal solution. If 175 all the players except one haveQ ∗ i = 0 nt×nt we have only one player remaining and the problem reduces to a degenerate game and is not considered. We formalize the above discussion in the following theorem. Proposition 9. Every point of the Pareto-optimal boundary of the ADBC is also a NoE of a suitably defined ADBC by eliminating users with zero power allocations. 6.5.4 Discussion • When the control variables of each player are scalars (and not matrices), the con- dition in (6.35) reduces to ηr k ∂v k (Q k ,Q −k ) ∂Q k = K X i=1 γ i ∂v i (Q i ,Q −i ) ∂Q k . (6.37) The above condition has been derived for a 2-player game for scalar control vari- ables in [41]. The derivation in [41] computes the Lagrangian only w.r.t the com- mon constraints and ignores the individual constraints on the control variables. We note that individual constraints must not be ignored in general. • It can easily be seen that for the sum power MAC, for a given decoding order, the matrix A is upper-triangular with non-negative diagonal elements and non- positive off-diagonal elements. Thus, a procedure similar to the above can be used to compute the relationship between the Pareto-optimal points and NoEs. • The similarity of the equilibrium characterization of the ADBC and the sum power MAC suggests that there exists a game-theoretic duality between the MAC and the BC channel. 176 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 User 1 − Achievable Rate User 2 − Achievable Rate (0.4,1.3) (1.5,0.7) (2.4,0) Figure 6.1: Plot showing the Pareto optimal frontier for a 2-user ADBC and several normalized equilibrium points 6.5.5 Two user Example We consider a two-user single antenna ADBC to illustrate the solution and provide some insights into the relation between the weights characterizing equilibria and Pareto- optimal solutions. For a two-user, one-antenna ADBC (n t = n 1 = ... = n K = 1) withQ k representing the power allocated (instead of a covariance matrix) to thek th user. AssumingN 1 < N 2 , the Pareto-optimal utility for the first and second users is given as v 1 (Q 1 ,Q 2 ) = log(Q 1 +N 1 )−log(N 1 ) andv 2 (Q 1 ,Q 2 ) = log(Q 1 +Q 2 +N 2 )−log(Q 1 + N 2 ) respectively withQ 1 +Q 2 =P tot whereP tot is the total power available at the trans- mitter. Substituting the utility functions in (6.35), the relation between the two sets of weights can be derived as: γ 1 γ 2 = r 1 r 2 + (Q 1 +N 1 )(Q 2 ) (Q 1 +Q 2 +N 2 )(Q 1 +N 2 ) . (6.38) Fig. 6.1 shows the Pareto-frontier for the 2-user ADBC withP tot = 10,N 1 = 1 and N 2 = 3 respectively. We first note that each point not on the axes of this curve can be characterized by a unique value ofγ. However several weight vectors result in the same 177 Pareto-optimal boundary when this point is on the axes. For example, allγ 1 ∈ [11/24,1] result in the same Pareto-optimal point(v 1 ,v 2 ) = (2.4,0). In Fig.6.1, the rate tuples for γ 1 = 0.41 and γ 1 = 0.375 are marked and given by (v 1 ,v 2 ) = (1.5,0.7) and (v 1 ,v 2 ) = (0.4,1.3) respectively. Using (6.38), we can show that these correspond to ar 1 /r 2 = 0.35 andr 1 /r 2 = 0.11 respectively. From Proposition 1, we infer that the the equilibrium pointr 1 /r 2 = 0.35 > 0.25 is the unique NoE of the game forγ 1 = 0.41. Thus, a regulator can use these weights to enforce the equilibrium rate (1.5,0.7) using the taxation method described in Section 6.2. In contrast, we see that for the rate tuple corresponding to r 1 /r 2 = 0.11 < 0.25, the corresponding NoE may not be a unique one. Finally, for(v 1 ,v 2 ) = (2.4,0), we note that there is one active player and hence is a degenerate game. However, using (6.38), we see that by choosing any ratio r 1 /r 2 = γ 1 /γ 2 > 11/13 the regulator can try to impose the fact that only the stronger user is allocated all the power for this game. 6.6 Game Theoretic Duality between MIMO Multiple Access and Broadcast Channels In information theory, owing to the structure of the MIMO BC, associated optimization problems such as capacity region computation, and beamforming optimization are typi- cally highly non-convex problems and cannot be solved directly. One feasible approach that has proved effective in the literature is to transform the non-convex BC problems into a convex dual MAC problem, which is easier to deal with and then transform the solution back into the domain of the BC. The conventional information theoretic duality between the BC and the MAC has been established for several scenarios of interest and has been used to solve several problems of interest to the wireless community. In this section, we 178 establish a game-theoretic duality between the MAC and BC by exploiting the properties of the signal transformation developed for establishing the information theoretic duality. We believe that such a transformation can provide new techniques for the computation of GNEs and NoEs for the GNEP. We first show the duality for single-antenna systems. We begin by establishing the relationship between the equilibria of the MAC and the sum power MAC by exploring the relationship between an NEP and a suitably defined GNEP. 6.6.1 Relationships between NEP and GNEPs Let P be the total power available to all the users in a system. Define a NEP with K players, utility functions v k (Q k ,Q −k ) such that v k is concave in Q k and continuous in Q −k , and the feasible strategy of each player as given below. Note that the presentation of this section, while tuned towards showing a duality between MAC and BC, can be extended to constraints and utility functions of a more general nature such as (N1) argmax Q k v k (Q k ,Q −k ) Q k 0, Tr[Q k ]≤P k . (6.39) In addition, we consider NEPs for which the power constraints satisfy the condition P K k=1 P k = P . Note that this constraint is not inherent to the NEP. It just signifies the fact that we are interested in all such NEPs whose individual power constraints for each player sum up toP . Let us now define a GNEP which is very closely related to (NEP1). (G1) argmax Q k v k (Q k ,Q −k ) Q k 0, K X k=1 Tr[Q k ]≤P. (6.40) 179 Observe that (G1) is a GNEP with jointly convex constraints. Clearly, both the games (N1) and(G1) have the same number of players, with each player having the same utility functions. The key difference between the two problems is that while the strategy set of each player is independent of the strategies of other players for(N1), they are dependent on the strategies of other players from (G1). In addition, we note that the feasible set of (N1) is always a subset of the feasible set of (G1). We now discuss the relationship between GNEs of(G1) and the NEs of(N1). Proposition 10. The vector of feasible strategiesQ ∗ is a GNE of (G1) if and only ifQ ∗ is a NE of all problems(N1) for whichQ ∗ is in the feasible set of strategies. In other words, consider all the NEPs for whichQ ∗ is a member of the set of strate- gies, i.e., Tr[Q ∗ k ] ≤ P k with the exogenous constraint that P K k=1 P k = P . ThenQ ∗ is a NE for all such NEPs if and only ifQ ∗ is a GNE of(G1). Proof. Let us begin by assuming thatQ ∗ is a GNE of the jointly convex GNEP defined by(G1). Then, v k (Q ∗ k ,Q ∗ −k )≥v k (Q k ,Q ∗ −k ), (6.41) for allQ k ∈ Ψ k (Q −k ). Now let (P ′ 1 ,P ′ 2 ,...,P ′ K ) be a vector such that 0≤ Tr[Q ∗ k ]≤ P ′ k with P K k=1 P ′ k =P and consider the NEP: (N2) argmax Q k v k (Q k ,Q −k ) Q k 0, Tr[Q k ]≤P ′ k . (6.42) It is straightforward to observe that the feasible set of(N2) is a subset of the feasible set of(G1). Using the fact thatQ ∗ is a GNE of(G1), it is easy to observe that v k (Q ∗ k ,Q ∗ −k )≥v k (Q k ,Q ∗ −k ),∀0≤ Tr[Q k ]≤P ′ k , (6.43) 180 and henceQ ∗ is a NE of (NEP”). As the vector(P ′ 1 ,P ′ 2 ,...,P ′ K ) has been chosen arbitrar- ily, it is clear that ifQ ∗ is a GNE of (G1) it is also a NE for all (N2)s for which it is a member of the feasible set. Next, we assume that Q ∗ is a NE of all the problems of the type (N1) for which it is an element of the feasible set and suppose thatQ ∗ is not a GNE of (G1). Then, there exists am and ˜ Q m such that v m (Q ∗ m ,Q ∗ −m )<v m ( ˜ Q m ,Q ∗ −m ). (6.44) Let Tr[ ˜ Q m ] = ˜ P . Now, consider a vector of the form(P ′ 1 ,...P ′ m−1 , ˜ P,P ′ m+1 ,...,P ′ K ) such that P K k=1,k6=m P ′ k + ˜ P =P and a NEP given by (N3) argmax Q k v k (Q k ,Q −k ) Q k 0, Tr[Q k ]≤P ′ k ,k6=m Q k 0, Tr[Q m ]≤ ˜ P (6.45) As the feasible set of(N3) is a subset of the feasible set of(G1), it is clear that v m (Q ∗ m ,Q ∗ −m )<v m ( ˜ Q m ,Q ∗ −m ). (6.46) for (N3). This contradicts the fact that Q ∗ is a NE of any problem of type (N1) for which it is a member of the feasible set and proves the proposition. 181 Proposition 11. If for every positive weight vector α = (α 1 ,α 2 ,...,α K ), Q ∗ is a GNE for a GNEP of the form, (G2) argmax Q k v k (Q k ,Q −k ) Q k 0, K X k=1 Tr[Q k ] α k ≤ K X k=1 P k α k , (6.47) thenQ ∗ is a NEP of(N1). Proof. Define (G3) which contains the common feasible set of all the GNEPs for every value ofα: (G3) argmax Q k v k (Q k ,Q −k ), Q k 0 [ α>0 ( K X k=1 Tr[Q k ] α k ≤ K X k=1 P k α k ) . (6.48) We first note that the feasible set of(G3) is an intersection of infinitely many convex sets and hence is a convex set itself. Clearly, (G3) is jointly convex. Next, we observe that if Q ∗ is a GNEP of the (G2) for every α, then Q ∗ is also a GNEP for (G3) as both the games have the same utility function. It can be easily shown that the feasible set of(G3) is identical to the feasible set of(N1) and henceQ ∗ is a NE of(N1). We now utilize the relationship between the equilibria of(G1) and(N1) to first derive the relation between the equilibria of the MAC and the sum power MAC. Substituting the utility functionsv k (Q k ,Q −k ) from (6.25) in (N1) and (G1), we infer from Proposition 10 that every GNE of the sum power MAC is also a NE of a MAC with appropriately defined power constraints. From Proposition 11, we then infer that every NE of a MAC is also a GNE of all scaled sum power MACs (where the feasible set scaling is as discussed in Proposition 11). Thus, we have established a relationship between the Nash equilibria 182 of a MAC and the generalized Nash equilibria of a sum power MAC. Note that this result is true for any given decoding order and hence applies to all possible decoding orders. As a next step towards deriving the duality between the MAC and the BC, we now investigate the properties of GNEs when GNEPs are transformed using a special linear transformation. 6.6.2 Relationship between two different GNEPs The information theoretic duality between the MAC and the BC [90] is based on a trans- formation from the variables from the sum power MAC to the BC (and vice versa) such that the utilities are preserved with the transformation. We now explore the relationship between the equilibria of two GNEPs related by a similar linear transformation. Proposition 12. Let us consider two GNEPs (G4) and (G5) defined as follows: (G4) argmax Q k v k (Q k ,Q −k ) Q k 0, K X k=1 Tr[Q k ]≤P, (6.49) (G5) argmax S k u k (S k ,S −k ) S k 0, K X k=1 Tr[S k ]≤P, (6.50) and satisfying the following properties. • v k is concave inQ k and continuous inQ i for alli = 1,...,K. • u k is concave inS k and continuous inS i for alli = 1,...,K. 183 • There exist matrices A k ,B k independent of Q k and S k (but may depend on Q −k and S −k ) such that for the linear transformations S k = A k Q k A H k and Q k = B k S k B H k , we have that • v k (Q k ,Q −k ) =u k (S k ,S −k ) and • P K k=1 Tr(A k Q k A H k )≤P and P K k=1 Tr(B k S k B H k )≤P . Then, Q ∗ is a GNE of (G4) if and only ifS ∗ k = A k Q ∗ k A H k is a GNE of (G5) andS ∗ is a GNE of (G5) if and only ifQ ∗ k = B k S ∗ k B H k is a GNE of (G4). In addition, the utility at equilibriumv k (Q ∗ k ,Q ∗ −k ) =u k (S ∗ k ,S ∗ −k ). This proves the game-theoretic duality between the MAC and BC. Proof. We begin by assuming that Q ∗ is a GNE of (G4). As Q ∗ is a GNE of a jointly convex game with utilityv k (Q k ,Q −k ), we have that Tr ∂v k ∂Q k | Q ∗(Q k −Q ∗ k ) ≥ 0, (6.51) for all k = 1,2,...,K and all feasible Q k . Now, using the fact that v k (Q k ,Q −k ) = u k (S k ,S −k ) we have that ∂v k = ∂u k ∂S k ∂S k = ∂u k ∂S k A k ∂Q k A H k (a) = A H k ∂u k ∂S k A k ∂Q k , (6.52) where (a) is true due to the fact that if a scalar valued function f(X) = AXB, for matricesA,X,B, then∂f(X) =A∂XB =BA∂X. Simplifying, we have that ∂v k ∂Q k =A H k ∂u k ∂S k A k . (6.53) 184 Now, Tr ∂v k ∂Q k | Q ∗(Q k −Q ∗ k ) ≥ 0 ⇒ Tr A H k ∂u k ∂S k | S ∗A k (Q k −Q ∗ k ) ≥ 0 ⇒ Tr ∂u k ∂S k | S ∗(S k −S ∗ k ) ≥ 0. (6.54) This implies that ifQ ∗ is a GNE of (G4), thenS ∗ is a GNE of (G5). The converse can be proved in the same manner, thus completing the proof. Proposition 12 shows that for any two GNEPs which satisfy the given properties, the GNEs and the achieved utilities are identical. In other words, this provides a technique to transform one GNEP to another that might permit simpler analysis. 6.6.3 Game based MAC-BC duality We now present a game-theory based dual relationship between the MAC and the BC. By substituting the utility function for the MAC in Proposition 10, we observe that the equilibria of the MAC and the sum power MAC and related closely to each other. More precisely, every equilibrium point of the MAC with some individual power constraints is an equilibrium point of the sum power MAC with a sum power constraint derived from the individual power constraints. In addition, every equilibrium point of the sum power MAC can be written as an equilibrium for the MAC with appropriately defined power constraints. Proposition 12 provides a way to transform between two different GNEPs of interest. We now use this relationship to show the relationship between the sum power MAC and the BC. Let v k and u k be the utility functions of the sum power MAC and the BC respectively. From [90], we know that there exists linear transformationsS k =A k Q k A H k 185 andQ k = B k S k B H k , withA k andB k independent ofQ −k andS −k , satisfying the power constraints, and which transform the utility function of the sum power MAC (same as the utility function for the MAC) into the utility function of the BC. From Proposition 12, it is clear that for every equilibrium point of the sum power MAC there exists an equilibrium point of the BC and vice versa. Now combining the results from Propositions 10 and 12, we can infer that for every equilibrium point of the MAC there exists an equilibrium point of the BC and vice versa, thus demonstrating a duality between the MAC and the BC. 6.7 Conclusions In this work, we proposed a game theoretic model for the Gaussian broadcast channel and a related problem, the MAC channel with sum power constraints. 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Abstract (if available)

Abstract In this thesis, we present a study of several problems related to underwater point to point communications and network formation. We explore techniques to improve the achievable data rate on a point to point link using better physical layer techniques and then study sensor cooperation which improves the throughput and reliability in an underwater network. ❧ Robust point-to-point communications in underwater networks has become increasingly critical in several military and civilian applications related to underwater communications. We present several physical layer signaling and detection techniques tailored to the underwater channel model to improve the reliability of data detection. First, a simplified underwater channel model in which the time scale distortion on each path is assumed to be the same (single scale channel model in contrast to a more general multi scale model). A novel technique, which exploits the nature of OFDM signaling and the time scale distortion, called Partial FFT Demodulation is derived. It is observed that this new technique has some unique interference suppression properties and performs better than traditional equalizers in several scenarios of interest. Next, we consider the multi scale model for the underwater channel and assume that single scale processing is performed at the receiver. We then derive optimized front end pre-processing techniques to reduce the interference caused during single scale processing of signals transmitted on a multi-scale channel. We then propose an improvised channel estimation technique using dictionary optimization methods for compressive sensing and show that significant performance gains can be obtained using this technique. ❧ In the next part of this thesis, we consider the problem of sensor node cooperation among rational nodes whose objective is to improve their individual data rates. We first consider the problem of transmitter cooperation in a multiple access channel and investigate the stability of the grand coalition of transmitters using tools from cooperative game theory and show that the grand coalition in both the asymptotic regimes of high and low SNR. Towards studying the problem of receiver cooperation for a broadcast channel, we propose a game theoretic model for the broadcast channel and then derive a game theoretic duality between the multiple access and the broadcast channel and show that how the equilibria of the broadcast channel are related to the multiple access channel and vice versa.

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Creator Yerramalli, Srinivas (author)

Core Title Communication and cooperation in underwater acoustic networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 03/01/2013

Defense Date 11/28/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag broadcast channels,channel estimation,Cooperation,Doppler,duality,estimator analysis,game theory,Generalized Nash Equilibrium Problems,interference modeling,multiple access channels,OAI-PMH Harvest,OFDM,partial FFT,partition form games,resampling,time scale,underwater communications,wideband communications,wireless networks

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Mitra, Urbashi (committee chair), Jain, Rahul (committee member), Sukhatme, Gaurav S. (committee member)

Creator Email srinivas.yerramalli@gmail.com,yerramal@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-222584

Unique identifier UC11294886

Identifier usctheses-c3-222584 (legacy record id)

Legacy Identifier etd-Yerramalli-1453.pdf

Dmrecord 222584

Document Type Dissertation

Rights Yerramalli, Srinivas

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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