Optimal Resource Allocation in Coordinated Multi-Cell Systems Emil Björnson Post-Doc Alcatel-Lucent Chair on Flexible Radio, Supélec, France Signal Processing Lab, KTH Royal Institute of Technology, Sweden Keynote Speech Signal Processing and Optimization for Wireless Communications: In Memory of Are Hjørungnes, Trondheim 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Biography: Emil Björnson 1983: Born in Malmö, Sweden 2007: Master of Science in Engineering Mathematics, Lund University, Sweden 2011: PhD in Telecommunications, KTH, Stockholm, Sweden 2012: Recipient of International Postdoc Grant from Sweden. Work with Prof. Mérouane Debbah at Supélec for 2 years. Optimal Resource Allocation in Coordinated Multi-Cell Systems Research book by E. Björnson and E. Jorswieck Foundations and Trends in Communications and Information Theory, Vol. 9, No. 2-3, pp. 113-381, 2013 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Outline Introduction Multi-Cell Structure, System Model, Performance Measure Problem Formulation Resource Allocation: Multi-Objective Optimization Problem Subjective Resource Allocation Utility Functions, Different Computational Complexity Structural Insights Beamforming Parametrization Extensions to Practical Conditions Handling Non-Idealities in Practical Systems 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Introduction 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Introduction Problem Formulation (vaguely): Transfer Information Wirelessly to Devices Downlink Coordinated Multi-Cell System Many Transmitting Base Stations (BSs) Many Receiving Users Sharing a Common Frequency Band Limiting Factor: Inter-User Interference Multi-Antenna Transmission Beamforming: Spatially Directed Signals Can Serve Multiple Users (Simultaneously) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: Basic Multi-Cell Structure Multiple Cells with Base Stations Adjacent Base Stations Coordinate Interference Some Users Served by Multiple Base Stations Dynamic Cooperation Clusters Inner Circle: Serve users with data Outer Circle: Avoid interference Outside Circles: Negligible impact (impractical to coordinate) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Example: Ideal Joint Transmission All Base Stations Serve All Users Jointly 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Example: Wyner Model Abstraction: User receives signals from own and neighboring base stations One or Two Dimensional Versions Joint Transmission or Coordination between Cells 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Example: Coordinated Beamforming One Base Station Serves Each User Interference Coordination Across Cells 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Example: Cognitive Radio Other Examples Spectrum sharing between operators Physical layer security Secondary System Borrows Spectrum of Primary System Underlay: Interference limits for primary users 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: Resource Allocation Problem Formulation (imprecise): Select Beamforming to Maximize System Utility Means: Allocate Power to Users and in Spatial Dimensions Satisfy: Physical, Regulatory & Economic Constraints Some Assumptions: Linear Transmission and Reception Perfect Synchronization (whenever needed) Flat-fading Channels (e.g., using OFDM) Perfect Channel State Information Ideal Transceiver Hardware Centralized Optimization Will be relaxed 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: Multi-Cell System Model 𝐾 𝑟 Users: Channel Vector h 𝑘 to User 𝑘 from All BSs 𝑁 𝑗 Antennas at 𝑗th BS 𝑁= 𝑗 𝑁 𝑗 Antennas in Total (dimension of h 𝑘 ) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: Power Constraints 𝐿 General Power Constraints: Many Interpretations Protect Dynamic Range of Amplifiers Limit Radiated Power According to Regulations Control Interference to Certain Users Manage Energy Expenditure Weighting Matrix (Positive semi-definite) Limit (Positive scalar) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: User Performance Measure Mean Square Error (MSE) Difference: transmitted and received signal Easy to Analyze Far from User Perspective? Bit/Symbol Error Rate (BER/SER) Probability of Error (for given data rate) Intuitive Interpretation Complicated & Ignores Channel Coding Information Rate Bits per ”Channel Use” Mutual Information: perfect and long coding Anyway Closest to Reality? All improves with SINR: Signal Interf + Noise 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Introduction: User Performance Measure Generic Model Any Function of Signal-to-Interference-and-Noise Ratio (SINR) Increasing and Continuous Function Can be MSE, BER/SER, Information Rate, etc. Complicated Function Depends on All Beamforming Vectors v 1 ,…, v 𝐾 𝑟 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Problem Formulation 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Problem Formulation General Formulation of Resource Allocation: Multi-Objective Optimization Problem Generally Impossible to Maximize For All Users! Must Divide Power and Cause Inter-User Interference 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Performance Region Definition: Performance Region R Contains All Feasible Care about user 2 Pareto Boundary Cannot Improve for any user without degrading for other users Balance between users Part of interest: Pareto boundary 2-User Performance Region Care about user 1 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Performance Region (2) Can it have any shape? No! Can prove that: Compact set Normal set Upper corner in region, everything inside region 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Performance Region (3) Some Possible Shapes User-Coupling Weak: Convex Strong: Concave Scheduling Time-sharing for strongly coupled users Select multiple points Hard: Unknown region 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Performance Region (4) Which Pareto Optimal Point to Choose? Tradeoff: Aggregate Performance vs. Fairness Utilitarian point (Max sum performance) No Objective Answer Only subjective answers exist! Egalitarian point (Max fairness) Single user point Performance Region Single user point 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Resource Allocation 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Subjective Approach System Designer Selects Utility Function Describes Subjective Preference Increasing and Continuous Function Examples: Sum Performance: Proportional Fairness: Harmonic Mean: Max-Min Fairness: 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach (2) Gives Single-Objective Optimization Problem: This is the Starting Point of Many Researchers Although Selection of is Inherently Subjective Affects the Solvability Pragmatic Approach Try to Select Utility Function to Enable Efficient Optimization 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach (3) Classes of Optimization Problems Main Classes in Resource Allocation Convex: Polynomial time solution Monotonic: Exponential time solution Practically Solvable Approx. Needed 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach (4) When is the Problem Convex? Most Problems are Non-Convex Necessary: Search space must be particularly limited Classification of Three Important Problems The “Easy” Problem Weighted Max-Min Fairness Weighted Sum Performance 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH The “Easy” Problem Given Any Point ( 𝑔 1 ,…, 𝑔 𝐾 𝑟 ) Find Beamforming v 1 ,…, v 𝐾 𝑟 that Attains this Point Minimize Emitted Power Convex Problem Second-Order Cone or Semi-Definite Program Global Solution in Polynomial Time – use CVX, Yalmip M. Bengtsson, B. Ottersten, “Optimal Downlink Beamforming Using Semidefinite Optimization,” Proc. Allerton, 1999. A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. on Signal Processing, 2006. W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. on Signal Processing, 2007. E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Trans. on Signal Processing, 2012. Total Power Constraints Per-Antenna Constraints General Constraints, Robustness 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach: Max-Min Fairness How to Classify Weighted Max-Min Fairness? Property: Solution makes 𝑤 𝑘 𝑔 𝑘 the same for all 𝑘 Solution is on this line Line in direction ( 𝑤 1 ,…, 𝑤 𝐾 𝑟 ) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach: Max-Min Fairness (2) Simple Line-Search: Bisection Iteratively Solving Convex Problems (i.e., quasi-convex) Find start interval Solve the “easy” problem at midpoint If feasible: Remove lower half Else: Remove upper half Iterate Subproblem: Convex optimization Line-search: Linear convergence One dimension (independ. #users) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach: Max-Min Fairness (3) Classification of Weighted Max-Min Fairness: Quasi-Convex Problem (belongs to convex class) If Subjective Preference is Formulated in this Way Resource Allocation Solvable in Polynomial Time 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach: Sum Performance How to Classify Weighted Sum Performance? Geometrically: 𝑤 1 𝑔 1 + 𝑤 2 𝑔 2 = opt-value is a line Opt-value is unknown! Distance from origin is unknown Line  Hyperplane (dim: #user – 1) Harder than max-min fairness Provably NP-hard! 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Subjective Approach: Sum Performance (2) Classification of Weighted Sum Performance: Monotonic Problem If Subjective Preference is Formulated in this Way Resource Allocation Solvable in Exponential Time Still Solvable: Monotonic Optimization Algorithms Improve Lower/Upper Bounds on Optimum: Continue Until Subproblem: Essentially weighted max-min fairness 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Branch-Reduce-Bound (BRB) Algorithm Subjective Approach: Sum Performance (3) Branch-Reduce-Bound (BRB) Algorithm Global convergence Accuracy ε>0 in finitely many iterations Exponential complexity only in #users ( 𝐾 𝑟 ) Polynomial complexity in other parameters (#antennas/constraints) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Pragmatic Resource Allocation Recall: All Utility Functions are Subjective Pragmatic Approach: Select to enable efficient optimization Bad Choice: Weighted Sum Performance NP-hard: Exponential complexity (in #users) Good Choice: Weighted Max-Min Fairness Quasi-Convex: Polynomial complexity Pragmatic Resource Allocation Weighted Max-Min Fairness (select weights to enhance throughput) 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Structural Insights 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Parametrization of Optimal Beamforming Beamforming Vectors: 𝐾 𝑟 𝑁 Complex Parameters Can be Reduced to 𝐾 𝑟 +𝐿–2 Positive Parameters Any Resource Allocation Problem Solved by Priority of User 𝑘: 𝜆 𝑘 Impact of Constraint 𝑙: 𝜇 𝑙 Tradeoff: Maximize Signal vs. Minimize Interference Hard to Find the Best Tradeoff 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Parametrization of Optimal Beamforming Geometric Interpretation: Heuristic Parameter Selection Known to Work Remarkably Well Many Examples (since 1995): Transmit Wiener/MMSE filter, Regularized Zero-forcing, Signal-to-leakage beamforming, virtual SINR/MVDR beamforming, etc. 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Extensions to Practical Conditions 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Robustness to Channel Uncertainty Practical Systems Operate under Uncertainty Due to Estimation, Feedback, Delays, etc. Robustness to Uncertainty Maximize Worst-Case Performance Cannot be Robust to Any Error Ellipsoidal Uncertainty Sets Easily Incorporated in the Model Same Classifications – More Variables Definition: 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Distributed Resource Allocation Information and Functionality is Distributed Local Channel Knowledge and Computational Resources Only Limited Backhaul for Coordination Distributed Approach Decompose Optimization Exchange Control Signals Iterate Subproblems Convergence to Optimal Solution? At Least for Convex Problems 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Adapting to Transceiver Impairments Physical Hardware is Non-Ideal Phase Noise, IQ-imbalance, Non-Linearities, etc. Non-Negligible Performance Degradation at High SNR Model of Transmitter Distortion: Additive Noise Variance Scales with Signal Power Same Classifications Hold under this Model Enables Adaptation: Much larger tolerance for impairments 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Summary 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Summary Resource Allocation Divide Power between Users and Spatial Directions Solve a Multi-Objective Optimization Problem Pareto Boundary: Set of efficient solutions Subjective Utility Function Selection has Fundamental Impact on Solvability Pragmatic Approach: Select to enable efficient optimization Weighted Sum Performance: Not solvable in practice Weighted Max-Min Fairness: Polynomial complexity Parametrization of Optimal Beamforming Extensions: Channel Uncertainty, Distributed Optimization, Transceiver Impairments 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Main Reference 270 Page Tutorial Published in Jan 2013 Fundamentals and Recent Advances E-book for free. Printed book €25 (Promo Code: EBMC-01069) Matlab Code Available Online 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Main Reference (2) Thorough Framework Further Extensions: Other Convex Problems and Optimization Algorithms More Parametrizations and Structural Insights Guidelines for Scheduling and Forming Dynamic Clusters Further Extensions: Multi-cast, Multi-carrier, Multi-antenna users, etc. General Zero Forcing Single Antenna Sum Performance NP-hard Convex Max-Min Fairness Quasi-Convex “Easy” Problem Linear Proportional Fairness Harmonic Mean General Sum Performance NP-hard Max-Min Fairness Quasi-Convex “Easy” Problem Convex 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Thank You for Listening! Questions? All Papers Available: http://flexible-radio.com/emil-bjornson 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Emil Björnson, Post-Doc at SUPELEC and KTH Additional Slides 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Why is Weighted Sum Performance Bad? Some Shortcomings Law of Diminishing Marginal Utility not Satisfied Not all Pareto Points are Attainable Weights have no Clear Interpretation Not Robust to Perturbations 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Further Geometric Interpretations Utilities has different shapes Same point in symmetric regions Generally large differences 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH

Computation of Performance Regions Performance Region is Generally Unknown Compact and Normal - Perhaps Non-Convex Generate 1: Vary parameters in parametrization Generate 2: Maximize sequence of utilities f 2013-05-24 Emil Björnson, Post-Doc at SUPELEC and KTH