Optimal Resource Allocation in Coordinated Multi-Cell Systems

1. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

2. Biography: Emil Björnson • 1983: Born in Malmö, Sweden • 2007: Master of Science inEngineering Mathematics,Lund University, Sweden • 2011: PhD in Telecommunications,KTH, Stockholm, Sweden • 2012: Recipient of International Postdoc Grant from Sweden. Work with Prof. MérouaneDebbah 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 Emil Björnson, Post-Doc at SUPELEC and KTH

3. 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 • BeamformingParametrization • Extensions to Practical Conditions • Handling Non-Idealities in Practical Systems Emil Björnson, Post-Doc at SUPELEC and KTH

4. Introduction Emil Björnson, Post-Doc at SUPELEC and KTH

5. 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) Emil Björnson, Post-Doc at SUPELEC and KTH

6. 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) Emil Björnson, Post-Doc at SUPELEC and KTH

7. Example: Ideal Joint Transmission • All Base Stations Serve All Users Jointly Emil Björnson, Post-Doc at SUPELEC and KTH

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

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

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

11. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

12. Introduction: Multi-Cell System Model • Users: Channel Vector to User from All BSs • Antennas at th BS • Antennas in Total (dimension of ) Emil Björnson, Post-Doc at SUPELEC and KTH

13. Introduction: Power Constraints • 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) Emil Björnson, Post-Doc at SUPELEC and KTH

14. 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 improveswith SINR: • Signal • Interf + Noise Emil Björnson, Post-Doc at SUPELEC and KTH

15. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

16. Problem Formulation Emil Björnson, Post-Doc at SUPELEC and KTH

17. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

18. Performance Region • Definition: Performance Region R • Contains All Feasible • Care aboutuser 2 Pareto Boundary Cannot Improve for any user without degrading for other users • Balancebetweenusers • Part of interest: • Pareto boundary • 2-User • PerformanceRegion • Care aboutuser 1 Emil Björnson, Post-Doc at SUPELEC and KTH

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

20. Performance Region (3) • Some Possible Shapes User-Coupling Weak: Convex Strong: Concave Scheduling Time-sharingfor strongly coupled users Select multiple pointsHard: Unknown region Emil Björnson, Post-Doc at SUPELEC and KTH

21. 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 • PerformanceRegion • Single user point Emil Björnson, Post-Doc at SUPELEC and KTH

22. Subjective Resource Allocation Emil Björnson, Post-Doc at SUPELEC and KTH

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

24. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

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

26. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

27. The “Easy” Problem • Given Any Point • Find Beamformingthat 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 SemidefiniteOptimization,” 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 PowerConstraints Per-Antenna Constraints General Constraints,Robustness Emil Björnson, Post-Doc at SUPELEC and KTH

28. Subjective Approach: Max-Min Fairness • How to Classify Weighted Max-Min Fairness? • Property: Solution makes the same for all Solution is on this line Line in direction () Emil Björnson, Post-Doc at SUPELEC and KTH

29. 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) Emil Björnson, Post-Doc at SUPELEC and KTH

30. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

31. Subjective Approach: Sum Performance • How to Classify Weighted Sum Performance? • Geometrically: = 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! Emil Björnson, Post-Doc at SUPELEC and KTH

32. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

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

34. 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) Emil Björnson, Post-Doc at SUPELEC and KTH

35. Structural Insights Emil Björnson, Post-Doc at SUPELEC and KTH

36. Parametrization of Optimal Beamforming • Beamforming Vectors: Complex Parameters • Can be Reduced to Positive Parameters • Any Resource Allocation Problem Solved by • Priority of User : • Impact of Constraint : • Tradeoff: Maximize Signal vs. Minimize Interference • Hard to Find the Best Tradeoff Emil Björnson, Post-Doc at SUPELEC and KTH

37. 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. Emil Björnson, Post-Doc at SUPELEC and KTH

38. Extensions to Practical Conditions Emil Björnson, Post-Doc at SUPELEC and KTH

39. 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: Emil Björnson, Post-Doc at SUPELEC and KTH

40. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

41. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

42. Summary Emil Björnson, Post-Doc at SUPELEC and KTH

43. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

44. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

45. Main Reference (2) • Thorough Framework • 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. Emil Björnson, Post-Doc at SUPELEC and KTH

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

47. Additional Slides Emil Björnson, Post-Doc at SUPELEC and KTH

48. 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 Emil Björnson, Post-Doc at SUPELEC and KTH

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

50. 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 Emil Björnson, Post-Doc at SUPELEC and KTH