Optimal Resource Allocation in Coordinated Multi-Cell Systems

Optimal Resource Allocation in Coordinated Multi-Cell SystemsEmil Bjrnson

Post-DocAlcatel-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 KTH1Biography: Emil Bjrnson1983: Born in Malm, 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. Mrouane Debbah at Suplec for 2 years.

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH2Optimal Resource Allocation in Coordinated Multi-Cell SystemsResearch book by E. Bjrnson and E. JorswieckFoundations and Trends in Communications and Information Theory, Vol. 9, No. 2-3, pp. 113-381, 2013

OutlineIntroductionMulti-Cell Structure, System Model, Performance Measure

Problem FormulationResource Allocation: Multi-Objective Optimization Problem

Subjective Resource AllocationUtility Functions, Different Computational Complexity

Structural InsightsBeamforming Parametrization

Extensions to Practical ConditionsHandling Non-Idealities in Practical Systems2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH32013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH4IntroductionIntroductionProblem Formulation (vaguely):Transfer Information Wirelessly to Devices

Downlink Coordinated Multi-Cell SystemMany Transmitting Base Stations (BSs)Many Receiving UsersSharing a Common Frequency BandLimiting Factor: Inter-User Interference

Multi-Antenna TransmissionBeamforming:Spatially Directed SignalsCan Serve Multiple Users(Simultaneously)2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH5

Introduction: Basic Multi-Cell StructureMultiple Cells with Base StationsAdjacent Base Stations Coordinate InterferenceSome Users Served by Multiple Base Stations

Dynamic Cooperation Clusters Inner Circle: Serve Users with DataOuter Circle: Avoid InterferenceOutside Circles: Negligible Impact (Impractical to Coordinate)2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH6

Example: Ideal Joint TransmissionAll Base Stations Serve All Users Jointly2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH7

Example: Wyner ModelAbstraction: User receives signals from own and neighboring base stations

One or Two Dimensional VersionsJoint Transmission or Coordination between Cells2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH8

Example: Coordinated BeamformingOne base station serves each userInterference coordination across cells2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH9

Example: Cognitive RadioSecondary System Borrows Spectrum of Primary SystemUnderlay: Interference Limits for Primary Users2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH10

Other Examples

Spectrum Sharing between Operators

Physical Layer SecurityIntroduction: Resource AllocationProblem Formulation (imprecise):Select Beamforming to Maximize System UtilityMeans: Allocate Power to Users and in Spatial DimensionsSatisfy: Physical, Regulatory & Economic Constraints

Some Assumptions:Linear Transmission and ReceptionPerfect Synchronization (whenever needed)Flat-fading Channels (e.g., using OFDM)

Perfect Channel State InformationIdeal Transceiver HardwareCentralized Optimization2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH11Will be relaxedIntroduction: Multi-Cell System Model2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH12

Introduction: Power Constraints2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH13

Weighting Matrix(Positive semi-definite)Limit(Positive scalar)

Introduction: User Performance MeasureMean Square Error (MSE)Difference: transmitted and received signalEasy to AnalyzeFar from User Perspective?

Bit/Symbol Error Rate (BER/SER)Probability of Error (for given data rate)Intuitive InterpretationComplicated & Ignores Channel Coding

Information RateBits per Channel UseMutual Information: perfect and long codingStill Closest to Reality?

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH14All improveswith SINR:

SignalInterf + Noise14Introduction: User Performance Measure2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH15

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH16Problem FormulationProblem FormulationGeneral Formulation of Resource Allocation:

Multi-Objective Optimization ProblemGenerally Impossible to Maximize For All Users!Must Divide Power and Cause Inter-User Interference2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH17

Definition: Performance Region R Contains All Feasible

Performance Region2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH182-UserPerformanceRegionCare aboutuser 2Care aboutuser 1BalancebetweenusersPart 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 setSimply connected (No holes)Nice upper boundary2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH19Normal setUpper corner in region, everything inside regionPerformance Region (3)Some Possible Shapes

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH20

User-Coupling

Weak: ConvexStrong: ConcaveShape is UnknownScheduling

Time-sharingbetween strongly coupled users

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

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH21PerformanceRegionUtilitarian point(Max sum performance)

Egalitarian point(Max fairness)

Single user point

Single user point

No Objective Answer

Only subjective answers exist!2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH22Subjective Resource AllocationSubjective ApproachSystem Designer Selects Utility Function f : R R Describes Subjective PreferenceIncreasing and Continuous Function

Examples:

Sum Performance:Proportional Fairness:Harmonic Mean:Max-Min Fairness:2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH23

Subjective Approach (2)Gives Single-Objective Optimization Problem:

This is the Starting Point of Many ResearchersAlthough Selection of f is Inherently SubjectiveAffects the Solvability2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH24

Pragmatic Approach

Try to Select Utility Function to Enable Efficient OptimizationSubjective Approach (3)Characterization of Optimization Problems

Main Categories of Resource AllocationConvex: Polynomial time solutionMonotonic: Exponential time solution2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH25

Approx. NeededPractically SolvableSubjective Approach (4)When is the Problem Convex?Most Problems are Non-ConvexNecessary: Search Space must be Particularly Limited

Classification of Three Important ProblemsThe Easy ProblemWeighted Max-Min FairnessWeighted Sum Performance

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH26The Easy Problem2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH27Total PowerConstraints

Per-AntennaConstraints

General Constraints,RobustnessSubjective Approach: Max-Min Fairness2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH28

Solution is on this line

Subjective Approach: Max-Min Fairness (2)2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH29Simple Line-Search: BisectionIteratively Solving Convex Problems (i.e., quasi-convex)Find start intervalSolve the easy problem at midpointIf feasible: Remove lower halfElse: Remove upper halfIterate

Subproblem: Convex optimizationLine-search: Linear convergenceOne dimension (independ. #users)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 WayResource Allocation Solvable in Polynomial Time

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

Subjective Approach: Sum Performance2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH31

Opt-value is unknown!Distance from origin is unknownLine Hyperplane (dim: #user 1)Harder than max-min fairnessProvably NP-hard!Subjective Approach: Sum Performance (2)Classification of Weighted Sum Performance:Monotonic Problem

If Subjective Preference is Formulated in this WayResource Allocation Solvable in Exponential Time

Algorithm for Monotonic OptimizationImprove Lower/Upper Bounds on Optimum:

Continue UntilSubproblem: Essentially weighted max-min fairness

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH32

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH33Subjective Approach: Sum Performance (3)

Pragmatic Resource AllocationRecall: All Utility Functions are SubjectivePragmatic Approach: Select to enable efficient optimization

Bad Choice: Weighted Sum PerformanceNP-hard: Exponential complexity (in #users)

Good Choice: Weighted Max-Min FairnessQuasi-Convex: Polynomial complexity2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH34Pragmatic Resource Allocation

Weighted Max-Min Fairness(select weights to enhance throughput)2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH35Structural InsightsParametrization of Optimal Beamforming2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH36

Parametrization of Optimal BeamformingGeometric Interpretation:

Heuristic Parameter SelectionKnown to Work Remarkably WellMany 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

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH38Extensions to Practical ConditionsRobustness to Channel UncertaintyPractical Systems Operate under UncertaintyDue to Estimation, Feedback, Delays, etc.

Robustness to UncertaintyMaximize Worst-Case PerformanceCannot be Robust to Any Error

Ellipsoidal Uncertainty SetsEasily Incorporated in the ModelSame Classifications More VariablesDefinition:

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

Distributed Resource AllocationInformation and Functionality is DistributedLocal Channel Knowledge and Computational ResourcesOnly Limited Backhaul for Coordination

Distributed ApproachDecompose OptimizationExchange Control SignalsIterate Subproblems

Convergence to Optimal Solution?At Least for Convex Problems2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH40

Adapting to Transceiver ImpairmentsPhysical Hardware is Non-IdealPhase Noise, IQ-imbalance, Non-Linearities, etc.Non-Negligible Performance Degradation at High SNR

Model of Transmitter Distortion:Additive NoiseVariance Scales with Signal Power

Same Classifications Hold under this ModelEnables Adaptation: Much larger tolerance for impairments2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH41

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH42SummarySummaryResource AllocationDivide Power between Users and Spatial DirectionsSolve a Multi-Objective Optimization ProblemPareto Boundary: Set of efficient solutions

Subjective Utility FunctionSelection has Fundamental Impact on SolvabilityPragmatic Approach: Select to enable efficient optimizationWeighted Sum Performance: Not solvable in practiceWeighted Max-Min Fairness: Polynomial complexity

Parametrization of Optimal Beamforming

Extensions: Channel Uncertainty, Distributed Optimization, Transceiver Impairments2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH43

Main Reference270 Page Tutorial, Published in Jan 2013Other Convex Problems and General AlgorithmsMore Parametrizations and Structural InsightsGuidelines for Scheduling and Forming Dynamic ClustersExtensions: multi-cast, multi-carrier, multi-antenna users, etc.Matlab Code Available Online

2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH44Promotion Code:EBMC-010692013-02-0645Emil Bjrnson, Post-Doc at SUPELEC and KTHThank You for Listening!

Questions?

All Papers Available:http://flexible-radio.com/emil-bjornson2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH46Additional SlidesProblem ClassificationsGeneralZero ForcingSingle AntennaSum PerformanceNP-hardConvexNP-hardProportional FairnessNP-hardConvexConvexHarmonic MeanNP-hardConvexConvexMax-Min FairnessQuasi-ConvexQuasi-ConvexQuasi-ConvexQoS/Easy ProblemConvexConvexLinear2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH47Why is Weighted Sum Performance Bad?Some ShortcomingsLaw of Diminishing Marginal Utility not SatisfiedNot all Pareto Points are AttainableWeights have no Clear InterpretationNot Robust to Perturbations2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH48

48Further Geometric Interpretations2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH49

Utilities has different shapes

Same point in symmetric regions

Generally large differences

Computation of Performance RegionsPerformance Region is Generally UnknownCompact and Normal - Perhaps Non-Convex

Generate 1: Vary parameters in parametrizationGenerate 2: Maximize sequence of utilities f2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH50Branch-Reduce-Bound (BRB) AlgorithmCover Region with a BoxDivide the Box into Two Sub-BoxesRemove Parts with No Solutions in Search for Solutions to Improve Bounds(Based on Fairness-profile problem)Continue with Sub-Box with Largest Value2013-02-06Emil Bjrnson, Post-Doc at SUPELEC and KTH51

Monotonic Optimization