Optimal Resource Allocation in Coordinated Multi-Cell Systems Emil Björnson Post-Doc Alcatel-Lucent Chair on Flexible Radio, Supélec & Signal Processing Lab, KTH Royal Institute of Technology Seminar Emil Björnson, Post-Doc at SUPELEC and KTH1

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 M. Debbah at Supélec for next 2 years. Optimal Resource Allocation in Coordinated Multi-Cell Systems Monograph Tutorial by E. Björnson and E. Jorswieck Submitted to FnT Communications and Information Theory Emil Björnson, Post-Doc at SUPELEC and KTH2

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 Some Practical Extensions -Handling Non-Idealities in Practical Systems Emil Björnson, Post-Doc at SUPELEC and KTH3

Emil Björnson, Post-Doc at SUPELEC and KTH4 Introduction

Problem Formulation (vaguely): -Transfer Information Wirelessly 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 KTH5

Introduction: Basic Multi-Cell Structure One Base Station per Cell -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 KTH6

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

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 KTH8

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

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

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 (where needed) -Flat-fading Channels (e.g., using OFDM) -Perfect Channel State Information -Ideal Transceiver Hardware -Centralized Optimization Emil Björnson, Post-Doc at SUPELEC and KTH11 Will be relaxed

Introduction: Multi-Cell System Model Emil Björnson, Post-Doc at SUPELEC and KTH12

Introduction: Power Constraints Emil Björnson, Post-Doc at SUPELEC and KTH13 Weight Matrix (Positive semi-definite) Limit (Positive scalar)

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? Emil Björnson, Post-Doc at SUPELEC and KTH14 All improves with SINR: Signal Interf + Noise

Introduction: User Performance Measure Emil Björnson, Post-Doc at SUPELEC and KTH15

Emil Björnson, Post-Doc at SUPELEC and KTH16 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 KTH17

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

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

Performance Region (3) Some Possible Shapes Emil Björnson, Post-Doc at SUPELEC and KTH20 User-Coupling Weak: Convex Strong: Concave Shape is Unknown

Performance Region (4) Which Pareto Optimal Point to Choose? -Tradeoff: Aggregate Performance vs. Fairness Emil 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!

Emil Björnson, Post-Doc at SUPELEC and KTH22 Subjective Resource Allocation

Subjective Approach System Designer Selects Utility Function f : R → R -Describing Subjective Preference -Increasing Function Examples: Sum Performance: Proportional Fairness: Harmonic Mean: Max-Min Fairness: Emil Björnson, Post-Doc at SUPELEC and KTH23

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

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

Subjective Approach (4) When is the Problem Convex? -Most Problems are Non-Convex -Necessary: Search Space must be Particularly Limited We will Classify Three Important Problems -The “Easy” Problem -Weighted Max-Min Fairness -Weighted Sum Performance Emil Björnson, Post-Doc at SUPELEC and KTH26

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

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

Subjective Approach: Max-Min Fairness Emil 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)

Subjective Approach: Max-Min Fairness 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 KTH30

Subjective Approach: Sum Performance Emil 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!

Subjective Approach: Sum Performance 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 Emil Björnson, Post-Doc at SUPELEC and KTH32

Emil Björnson, Post-Doc at SUPELEC and KTH33 Subjective Approach: Sum Performance

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 Emil Björnson, Post-Doc at SUPELEC and KTH34 Pragmatic Resource Allocation Weighted Max-Min Fairness: Weights Enhance Throughput

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 KTH35

Emil Björnson, Post-Doc at SUPELEC and KTH36 Structural Insights

Parametrization of Optimal Beamforming Emil Björnson, Post-Doc at SUPELEC and KTH37

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 KTH38

Emil Björnson, Post-Doc at SUPELEC and KTH39 Some Practical Extensions

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 -More Variables – Same Classifications -Definition: Emil Björnson, Post-Doc at SUPELEC and KTH40

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 KTH41

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 KTH42

Emil Björnson, Post-Doc at SUPELEC and KTH43 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 KTH44

Main Reference 250-page tutorial monograph – currently under review Many more details -Other convex problems and general algorithms -More parametrizations and structural insights -Extensions: multi-cast, multi-carrier, multi-antenna users, etc Emil Björnson, Post-Doc at SUPELEC and KTH45

Emil Björnson, Post-Doc at SUPELEC and KTH Thank You for Listening! Questions? All Papers Available:

Emil Björnson, Post-Doc at SUPELEC and KTH47 Backup Slides

Problem Classifications GeneralZero ForcingSingle Antenna Sum PerformanceNP-hardConvexNP-hard Proportional FairnessNP-hardConvex Harmonic MeanNP-hardConvex Max-Min FairnessQuasi-Convex QoS/Easy ProblemConvex Linear Emil Björnson, Post-Doc at SUPELEC and KTH48

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 Emil Björnson, Post-Doc at SUPELEC and KTH49 Monotonic Optimization

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 KTH50