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Optimal Resource Allocation in Coordinated Multi-Cell Systems Emil Bjrnson Post-Doc Alcatel-Lucent Chair on Flexible Radio, Suplec, France & Signal Processing Lab, KTH Royal Institute of Technology, Sweden Seminar at Alcatel-Lucent, Stuttgart, 2013-02-06 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH1

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Biography: Emil Bjrnson 1983: Born in Malm, 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. Mrouane Debbah at Suplec for 2 years. 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH2 Optimal Resource Allocation in Coordinated Multi-Cell Systems Research book by E. Bjrnson and E. Jorswieck Foundations and Trends in Communications and Information Theory, Vol. 9, No. 2-3, pp. 113-381, 2013

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH3

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH4 Introduction

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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) 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH5

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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) 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH6

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Example: Ideal Joint Transmission All Base Stations Serve All Users Jointly 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH7

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Example: Wyner Model Abstraction: User receives signals from own and neighboring base stations One or Two Dimensional Versions Joint Transmission or Coordination between Cells 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH8

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Example: Coordinated Beamforming One base station serves each user Interference coordination across cells 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH9

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Example: Cognitive Radio Secondary System Borrows Spectrum of Primary System Underlay: Interference Limits for Primary Users 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH10 Other Examples Spectrum Sharing between Operators Physical Layer Security

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH11 Will be relaxed

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Introduction: Multi-Cell System Model 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH12

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Introduction: Power Constraints 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH13 Weighting Matrix (Positive semi-definite) Limit (Positive scalar)

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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 -Still Closest to Reality? 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH14 All improves with SINR: Signal Interf + Noise

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Introduction: User Performance Measure 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH15

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH16 Problem Formulation

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General Formulation of Resource Allocation: Multi-Objective Optimization Problem -Generally Impossible to Maximize For All Users! -Must Divide Power and Cause Inter-User Interference 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH17

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Definition: Performance Region R -Contains All Feasible Performance Region 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH18 2-User Performance Region Care about user 2 Care about user 1 Balance between users Part of interest: Pareto boundary Pareto Boundary Cannot Improve for any user without degrading for other users

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Performance Region (2) Can it have any shape? No! Can prove that: -Compact set -Simply connected (No holes) -Nice upper boundary 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH19 Normal set Upper corner in region, everything inside region

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Performance Region (3) Some Possible Shapes 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH20 User-Coupling Weak: Convex Strong: Concave Shape is Unknown Scheduling Time-sharing between strongly coupled users

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Performance Region (4) Which Pareto Optimal Point to Choose? -Tradeoff: Aggregate Performance vs. Fairness 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH21 Performance Region Utilitarian point (Max sum performance) Egalitarian point (Max fairness) Single user point No Objective Answer Only subjective answers exist!

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH22 Subjective Resource Allocation

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Subjective Approach System Designer Selects Utility Function f : R R -Describes Subjective Preference -Increasing and Continuous Function Examples: Sum Performance: Proportional Fairness: Harmonic Mean: Max-Min Fairness: 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH23

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Subjective Approach (2) Gives Single-Objective Optimization Problem: This is the Starting Point of Many Researchers -Although Selection of f is Inherently Subjective Affects the Solvability 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH24 Pragmatic Approach Try to Select Utility Function to Enable Efficient Optimization

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Subjective Approach (3) Characterization of Optimization Problems Main Categories of Resource Allocation -Convex: Polynomial time solution -Monotonic: Exponential time solution 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH25 Approx. Needed Practically Solvable

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH26

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The Easy Problem 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH27 Total Power Constraints Per-Antenna Constraints General Constraints, Robustness

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Subjective Approach: Max-Min Fairness 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH28 Solution is on this line

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Subjective Approach: Max-Min Fairness (2) 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH29 Simple Line-Search: Bisection -Iteratively Solving Convex Problems (i.e., quasi-convex) 1.Find start interval 2.Solve the easy problem at midpoint 3.If feasible: Remove lower half Else: Remove upper half 4.Iterate Subproblem: Convex optimization Line-search: Linear convergence One dimension (independ. #users)

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH30

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Subjective Approach: Sum Performance 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH31 Opt-value is unknown! -Distance from origin is unknown -Line Hyperplane (dim: #user 1) -Harder than max-min fairness -Provably NP-hard!

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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 Algorithm for Monotonic Optimization -Improve Lower/Upper Bounds on Optimum: -Continue Until -Subproblem: Essentially weighted max-min fairness 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH32

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH33 Subjective Approach: Sum Performance (3)

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH34 Pragmatic Resource Allocation Weighted Max-Min Fairness (select weights to enhance throughput)

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH35 Structural Insights

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Parametrization of Optimal Beamforming 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH36

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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. 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH37

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH38 Extensions to Practical Conditions

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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: 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH39

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH40

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH41

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH42 Summary

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH43

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Main Reference 270 Page Tutorial, Published in Jan 2013 -Other Convex Problems and General Algorithms -More Parametrizations and Structural Insights -Guidelines for Scheduling and Forming Dynamic Clusters -Extensions: multi-cast, multi-carrier, multi-antenna users, etc. Matlab Code Available Online 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH44 Promotion Code: EBMC-01069

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2013-02-0645Emil Bjrnson, Post-Doc at SUPELEC and KTH Thank You for Listening! Questions? All Papers Available: http://flexible-radio.com/emil-bjornson

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2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH46 Additional Slides

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Problem Classifications GeneralZero ForcingSingle Antenna Sum PerformanceNP-hardConvexNP-hard Proportional FairnessNP-hardConvex Harmonic MeanNP-hardConvex Max-Min FairnessQuasi-Convex QoS/Easy ProblemConvex Linear 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH47

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH48

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Further Geometric Interpretations 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH49 Utilities has different shapes Same point in symmetric regions Generally large differences

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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 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH50

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Branch-Reduce-Bound (BRB) Algorithm 1.Cover Region with a Box 2.Divide the Box into Two Sub-Boxes 3.Remove Parts with No Solutions in 4.Search for Solutions to Improve Bounds (Based on Fairness-profile problem) 5.Continue with Sub-Box with Largest Value 2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH51 Monotonic Optimization