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 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.
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 , 2013
Emil Björnson, Post-Doc at SUPELEC and KTH

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

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

5. Emil Björnson, Post-Doc at SUPELEC and KTH
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. Emil Björnson, Post-Doc at SUPELEC and KTH
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
Other Examples
Spectrum sharing between operators
Physical layer security
Secondary System Borrows Spectrum of Primary System
Underlay: Interference limits for primary users
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
𝐾 𝑟 Users: Channel Vector h 𝑘 to User 𝑘 from All BSs
𝑁 𝑗 Antennas at 𝑗th BS
𝑁= 𝑗 𝑁 𝑗 Antennas in Total (dimension of h 𝑘 )
Emil Björnson, Post-Doc at SUPELEC and KTH

13. Introduction: Power Constraints
𝐿 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 improves with 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 v 1 ,…, v 𝐾 𝑟
Emil Björnson, Post-Doc at SUPELEC and KTH

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

17. Emil Björnson, Post-Doc at SUPELEC and KTH
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 about user 2
Pareto Boundary
Cannot Improve for any user without degrading for other users
Balance between users
Part of interest:
Pareto boundary
2-User
Performance Region
Care about user 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-sharing for strongly coupled users
Select multiple points Hard: 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
Performance Region
Single user point
Emil Björnson, Post-Doc at SUPELEC and KTH

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

23. Emil Björnson, Post-Doc at SUPELEC and KTH
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. Emil Björnson, Post-Doc at SUPELEC and KTH
The “Easy” Problem
Given Any Point ( 𝑔 1 ,…, 𝑔 𝐾 𝑟 )
Find Beamforming v 1 ,…, v 𝐾 𝑟 that 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 Semidefinite Optimization,” 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 Power Constraints
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 𝑤 𝑘 𝑔 𝑘 the same for all 𝑘
Solution is on this line
Line in direction ( 𝑤 1 ,…, 𝑤 𝐾 𝑟 )
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: 𝑤 1 𝑔 1 + 𝑤 2 𝑔 2 = 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. Branch-Reduce-Bound (BRB) Algorithm
Subjective Approach: Sum Performance (3)
Branch-Reduce-Bound (BRB) Algorithm
Global convergence
Accuracy ε>0 in finitely many iterations
Exponential complexity only in #users ( 𝐾 𝑟 )
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. Emil Björnson, Post-Doc at SUPELEC and KTH
Structural Insights
Emil Björnson, Post-Doc at SUPELEC and KTH

36. Parametrization of Optimal Beamforming
Beamforming Vectors: 𝐾 𝑟 𝑁 Complex Parameters
Can be Reduced to 𝐾 𝑟 +𝐿–2 Positive Parameters
Any Resource Allocation Problem Solved by
Priority of User 𝑘: 𝜆 𝑘
Impact of Constraint 𝑙: 𝜇 𝑙
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. Emil Björnson, Post-Doc at SUPELEC and KTH
Summary
Emil Björnson, Post-Doc at SUPELEC and KTH

43. Emil Björnson, Post-Doc at SUPELEC and KTH
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. Emil Björnson, Post-Doc at SUPELEC and KTH
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 Further Extensions:
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.
General
Zero Forcing
Single Antenna
Sum Performance
NP-hard
Convex
Max-Min Fairness
Quasi-Convex
“Easy” Problem
Linear
Proportional Fairness
Harmonic Mean
General
Sum Performance
NP-hard
Max-Min Fairness
Quasi-Convex
“Easy” Problem
Convex
Emil Björnson, Post-Doc at SUPELEC and KTH

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

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