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 Seminar at Alcatel-Lucent, Stuttgart, 2013-02-06 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH1

2. 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. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH2 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

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 -Beamforming Parametrization Extensions to Practical Conditions -Handling Non-Idealities in Practical Systems 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH3

4. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH4 Introduction

5. 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 Björnson, Post-Doc at SUPELEC and KTH5

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

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

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 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH8

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

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

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

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

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

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

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

16. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH16 Problem Formulation

17. 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 Björnson, Post-Doc at SUPELEC and KTH17

18. Definition: Performance Region R -Contains All Feasible Performance Region 2013-02-06Emil Björnson, 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

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

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

21. Performance Region (4) Which Pareto Optimal Point to Choose? -Tradeoff: Aggregate Performance vs. Fairness 2013-02-06Emil Björnson, 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!

22. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH22 Subjective Resource Allocation

23. 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 Björnson, Post-Doc at SUPELEC and KTH23

24. 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 Björnson, Post-Doc at SUPELEC and KTH24 Pragmatic Approach Try to Select Utility Function to Enable Efficient Optimization

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

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

27. The “Easy” Problem 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH27 Total Power Constraints Per-Antenna Constraints General Constraints, Robustness

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

29. Subjective Approach: Max-Min Fairness (2) 2013-02-06Emil Björnson, 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)

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

31. Subjective Approach: Sum Performance 2013-02-06Emil Björnson, 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!

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

33. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH33 Subjective Approach: Sum Performance (3)

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

35. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH35 Structural Insights

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

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

38. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH38 Extensions to Practical Conditions

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

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

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

42. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH42 Summary

43. 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 Björnson, Post-Doc at SUPELEC and KTH43

44. 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 Björnson, Post-Doc at SUPELEC and KTH44 Promotion Code: EBMC-01069

45. 2013-02-0645Emil Björnson, Post-Doc at SUPELEC and KTH Thank You for Listening! Questions? All Papers Available: http://flexible-radio.com/emil-bjornson

46. 2013-02-06Emil Björnson, Post-Doc at SUPELEC and KTH46 Additional Slides

47. 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 Björnson, Post-Doc at SUPELEC and KTH47

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

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

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

51. 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 Björnson, Post-Doc at SUPELEC and KTH51 Monotonic Optimization