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 2012-10-11
2012-10-11
Emil Björnson, Post-Doc at SUPELEC and KTH
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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
2012-10-11
Emil Björnson, Post-Doc at SUPELEC and KTH
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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
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Emil Björnson, Post-Doc at SUPELEC and KTH
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Introduction
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Emil Björnson, Post-Doc at SUPELEC and KTH
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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)
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Emil Björnson, Post-Doc at SUPELEC and KTH
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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)
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Emil Björnson, Post-Doc at SUPELEC and KTH
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Example: Ideal Joint Transmission
• All Base Stations Serve All Users Jointly
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Example: Wyner Model
• Abstraction: User receives signals from own and
neighboring base stations
• One or two dimensional versions
• Joint transmission or coordination between cells
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Emil Björnson, Post-Doc at SUPELEC and KTH
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Example: Coordinated Beamforming
• One base station serves each user
• Interference coordination across cells
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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
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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
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Emil Björnson, Post-Doc at SUPELEC and KTH
Will be relaxed
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Introduction: Multi-Cell System Model
• 𝐾𝑟 Users: Channel Vector h𝑘 to User 𝑘 from All BSs
• 𝑁𝑗 Antennas at 𝑗th BS
• 𝑁=
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𝑗 𝑁𝑗
Antennas in Total (dimension of h𝑘 )
Emil Björnson, Post-Doc at SUPELEC and KTH
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Introduction: Power Constraints
• 𝐿 General Power Constraints:
Weight Matrix
(Positive semi-definite)
Limit
(Positive scalar)
• Many Interpretations
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Protect Dynamic Range of Amplifiers
Limit Radiated Power According to Regulations
Control Interference to Certain Users
Manage Energy Expenditure
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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
All improves
with SINR:
Signal
Interf + Noise
• Information Rate
- Bits per ”channel use”
- Mutual information: perfect and long coding
- Still closest to reality?
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Introduction: User Performance Measure
• Generic Model
- Any Function of Signal-to-Interference-and-Noise Ratio (SINR)
- Continuous and Increasing Function
- Can be MSE, BER/SER, Information Rate, etc.
• Complicated Function
- Depends on All Beamforming Vectors v1 , … , v𝐾𝑟
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Problem Formulation
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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
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Performance Region
• Definition: Performance Region R
- Contains All Feasible
Care about
user 1
Pareto Boundary
Balance
between
users
Part of interest:
Pareto boundary
2 User
Performance
Region
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Cannot Improve
for any user
without degrading
for other users
Care about
user 2
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Performance Region (2)
• Can it have any shape?
• No! Can prove that:
- Compact set
- Simply connected (No holes)
- Nice upper boundary
Normal set
Upper corner in region,
everything inside region
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Performance Region (3)
• Some Possible Shapes
Shape is
Unknown
User-Coupling
Weak: Convex
Strong: Concave
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Performance Region (4)
• Which Pareto Optimal Point to Choose?
- Tradeoff: Aggregate Performance vs. Fairness
Utilitarian point
(Max sum performance)
Egalitarian point
(Max fairness)
Single
user
point
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Performance
Region
No Objective
Answer
Only Subjective
Answers Exist!
Single user point
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Subjective Resource Allocation
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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:
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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
Pragmatic Approach
Try to Select Utility Function to Enable Efficient Optimization
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Subjective Approach (3)
• Characterization of Optimization Problems
• Main Categories of Resource Allocation
- Convex: Polynomial Time Solution
- Monotonic: Exponential Time Solution
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Emil Björnson, Post-Doc at SUPELEC and KTH
Practically Solvable
Approx. Needed
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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
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The “Easy” Problem
• Given Any Point (𝑔1 , … , 𝑔𝐾𝑟 )
- Find Beamforming v1 , … , v𝐾𝑟 that Attains this Point
- Minimize Power
• Convex Problem
- Second-Order Cone or Semi-Definite Program
- Global Solution in Polynomial Time – use CVX, Yalmip
Total Power
Constraints
• 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. Signal Processing, 2006.
Per-Antenna
Constraints
• W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink
with per-antenna power constraints,” IEEE Trans. Signal Process., 2007.
General
Constraints,
Robustness
• E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic
Optimization Framework for Multicell MISO Systems,” IEEE Transactions on
Signal Processing, IEEE Trans. Signal Process., 2012.
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Subjective Approach: Max-Min Fairness
• How to Classify Weighted Max-Min Fairness?
- Property: Solution makes 𝑤𝑘 𝑔𝑘 the same for all 𝑘
Solution is on this line
Line in direction (𝑤1 , … , 𝑤𝐾𝑟 )
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Subjective Approach: Max-Min Fairness
• 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)
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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
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Subjective Approach: Sum Performance
• How to Classify Weighted Sum Performance?
- Geometrically: 𝑤1 𝑔1 + 𝑤2 𝑔2 = opt-value is a line
Opt-value is unknown!
-
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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
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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
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Subjective Approach: Sum Performance
Branch-Reduce-Bound
(BRB) Algorithm
- Global Convergence
- Accuracy ε>0 in finitely
many iterations
- Exponential complexity
only in #users (𝐾𝑟 )
- Polynomial complexity
in other parameters
(#antennas/constraints)
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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: Weights Enhance Throughput
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Why is Weighted Sum Performance bad?
• Some shortcomings
-
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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
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Structural Insights
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Parametrization of Optimal Beamforming
• Any Resource Allocation Problem Solved by
- Original 𝐾𝑟 𝑁 Complex → 𝐾𝑟 + 𝐿– 2 Positive Parameters
- Priority of User 𝑘: 𝜆𝑘
- Impact of Constraint 𝑙: 𝜇𝑙
• Tradeoff: Maximize Signal - Minimize Interference
- Not easy to find the best tradeoff
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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.
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Some Practical Extensions
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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:
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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
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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
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Summary
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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
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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.
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Thank You for Listening!
Questions?
All Papers Available:
http://flexible-radio.com/emil-bjornson
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Backup Slides
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Problem Classifications
General
Zero Forcing Single Antenna
Sum Performance
NP-hard
Convex
NP-hard
Proportional Fairness
NP-hard
Convex
Convex
Harmonic Mean
NP-hard
Convex
Convex
Max-Min Fairness
Quasi-Convex
Quasi-Convex
Quasi-Convex
QoS/Easy Problem
Convex
Convex
Linear
• General: Any Utility and Constraints
• Zero Forcing: Concave Utilities and ZF Constraints
• Single Antenna: 𝑁𝑗 = 1 and one BS Serves Each User
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Monotonic Optimization
• Branch-Reduce-Bound (BRB) Algorithm
1.
2.
3.
4.
Cover region with a box
Divide the box into two sub-boxes
Remove parts with no solutions in
Search for solutions to improve bounds
(Based on Fairness-profile problem)
5. Continue with sub-box with largest value
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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
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