1. Signal Processing for Optimal Radio Resource Management: Fundamentals and Recent Multi-Cell Advances
Emil Björnson
Dept. of Signal Processing, KTH Royal Institute of Technology, Stockholm, Sweden
Alcatel-Lucent Chair on Flexible Radio, Supélec, Gif-sur-Yvette, France
20 March 2014

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 Advisors: Björn Ottersten, Mats Bengtsson
: International Postdoc Grant from Sweden. Work with Prof. Mérouane Debbah at Supélec on “Optimization of Green Small-Cell Networks”
2014: Assistant Professor in Communication Systems, Linköping University, Sweden
Signal processing for optimal radio resource management
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3. Optimal Resource Allocation in Coordinated Multi-Cell Systems
Book Reference
Tutorial is Partially based on a Book:
270 pages
E-book for free (from our homepages)
Printed book: Special price $35, use link:
Matlab code is available online
Visit:
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
Signal processing for optimal radio resource management
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4. Covered by in second 90 min
General Outline
Introduction
Problem formulation and general system model
Optimal Single-Objective Resource Management
Which problems are practically solvable and why?
Multi-Objective Network Optimization
Definitions and potential applications
Generalizations and Specific Properties
How does optimal solutions look like?
Handling imperfect channel knowledge
Distributed resource management
Handling hardware impairments
Fundamentals
Covered by in first 90 min
Recent advances
Covered by in second 90 min
Signal processing for optimal radio resource management
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5. Part 1
Fundamentals
Signal processing for optimal radio resource management
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6. Outline: Part 1 – Fundamentals
Introduction
Multi-cell structure, system model, performance measure
Problem Formulation
Resource management: Multi-objective optimization problem
Subjective Resource Management
Utility functions, different computational complexity
Multi-Objective Network Optimization
What are the future potentials of this theory?
Signal processing for optimal radio resource management
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7. Section
Introduction
Signal processing for optimal radio resource management
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8. Introduction Wireless Connectivity Rapid Network Traffic Growth
A natural part of our lives
Text messages
Voice call
Social networks
Video streaming
Web browsing
Gaming
Rapid Network Traffic Growth
61% annual data traffic growth
Exponential increase!
Extrapolation: 20x until x until x until 2030
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9. Introduction: Evolving Networks for Higher Traffic
Increasing Network Capacity [bit/s/Area]
Consider a given area
How to handle 1000x more data traffic?
Formula for Network Capacity:
Capacity bit/s in area = Spectral efficiency bit/s/Hz/Cell ∙ Cell density Cell/Area ∙ Available spectrum in Hz
Spectral efficiency
Cell density
Spectrum
Total
History ( )
25x
1600x x
Spectral efficiency
Cell density
Spectrum
Total
History ( )
25x
1600x x
Future: Nokia
10x
1 000x
Future: SK Telecom 6x
56x 3x
Considered in this lecture
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10. Introduction: Preliminary Definitions
Problem Formulation (vaguely):
Transfer information wirelessly to users
Divide radio resources among users (time, frequency, space)
Downlink Cellular System
Many transmitting base stations (BSs)
Many receiving users
Multiple-input multiple-output (MIMO)
Sharing a Frequency Band
All signals reach everyone!
Limiting Factor
Inter-user interference
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11. Introduction: Multi-Antenna Transmission
Traditional Ways to Manage Interference
Avoid and suppress in time and frequency domain
Results in orthogonal access techniques: TDMA, OFDMA, etc.
Multi-Antenna Transmission
Beamforming: Spatially directed signals
Adaptive control of interference
Serve multiple users: Space-division multiple access (SDMA)
Main difference from classical resource management!
Signal processing for optimal radio resource management
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12. Introduction: Multi-Antenna Transmission (2)
With channel knowledge: beamforming or precoding
Direct signal towards intended receiver – some interference leaks!
Beamforming Design
Easy for one user
Difficult for multiple users, due to interference
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13. Introduction: From Single-Cell to Multi-Cell
Naïve Multi-Cell Extension
Divide BS into disjoint clusters
SDMA within each cluster
Avoid inter-cluster interference
Fractional frequency-reuse
Coordinated Multi-Cell Transmission
SDMA in multi-cell: All BSs collaborate
Frequency-reuse one: Interference managed by beamforming
Many names: co-processing, coordinated multi-point (CoMP), network MIMO, multi-cell processing
Almost as One Super-Cell
But: Different data knowledge, channel knowledge, power constraints!
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14. Basic Multi-Cell Coordination Structure
General Multi-Cell Coordination
Adjacent base stations coordinate interference
Some users served by multiple base stations
Dynamic Cooperation Clusters
Inner Circle : Serve users with data
Outer Circle : Suppress interference
Outside Circles:
Negligible impact Impractical to acquire information Difficult to coordinate decisions
E. Björnson, N. Jaldén, M. Bengtsson, B. Ottersten, “Optimality Properties, Distributed Strategies, and Measurement-Based Evaluation of Coordinated Multicell OFDMA Transmission,” IEEE Trans. on Signal Processing, 2011.
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15. Example: Ideal Joint Transmission
All Base Stations Serve All Users Jointly = One Super Cell
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16. 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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17. Example: Coordinated Beamforming
One Base Station Serves Each User
Interference Coordination Across Cells
Special Case
Interference channel
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18. Example: Soft-Cell Coordination
Heterogeneous Deployment
Conventional macro BS overlaid by short-distance small BSs
Interference coordination and joint transmission between layers
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19. Example: Cognitive Radio
Secondary System Borrows Spectrum of Primary System
Underlay: Interference limits for primary users
Other Examples
Spectrum sharing between operators
Physical layer security
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20. Optimizing Resource Management: First Definition
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 knowledge
Centralized optimization
Ideal transceiver hardware
Relaxed in Part 2
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21. Multi-Cell System Model
𝐾 𝑟 Users: Channel vector to User 𝑘 from all BSs
𝑁 𝑗 Antennas at 𝑗th BS (dimension of h 𝑗𝑘 ) – small or large
𝑁= 𝑗 𝑁 𝑗 Antennas in Total (dimension of h 𝑘 )
One System Model for All Multi-Cell Scenarios!
Signal processing for optimal radio resource management
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22. Multi-Cell System Model: Dynamic Cooperation Clusters
How are D 𝑘 and C 𝑘 Defined?
Consider User 𝑘:
Interpretation:
Block-diagonal matrices
D 𝑘 has identity matrices for BSs that send data
C 𝑘 has identity matrices for BSs that can/should coordinate interference
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23. Multi-Cell System Model: Dynamic Cooperation Clusters (2)
Example: Coordinated Beamforming
This is User 𝑘
Beamforming: D 𝑘 v 𝑘 Data only from BS1:
Effective channel: C 𝑘 h 𝑘 Interference from all BSs:
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24. Multi-Cell System Model: Power Constraints
Need for Power Constraints
Limit radiated power according to regulations
Protect dynamic range of amplifiers
Manage cost of energy expenditure
Control interference to certain users
𝐿 General Power Constraints:
All at the same time!
Weighting matrix
(Positive semi-definite)
Limit
(Positive scalar)
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25. Multi-Cell System Model: Power Constraints (2)
Recall:
Example 1, Total Power Constraint:
Example 2, Per-Antenna Constraints:
Example 3, Control Interference to User 𝑖
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26. Introduction: How to Measure User Performance?
Mean Square Error (MSE)
Difference: transmitted and received signal
Easy to Analyze
Far from User Perspective?
Bit/Symbol Error Probability (BEP/SEP)
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
Interference + Noise
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27. Introduction: Generic Measure User Performance
Generic Model
Any function of signal-to-interference-and-noise ratio (SINR):
Increasing and continuous function
For simplicity: 𝑔 𝑘 0 =0
Simple Examples:
Information rate:
MSE:
Complicated Function
Depends on all beamforming vectors v 1 ,…, v 𝐾 𝑟
for User 𝑘
User Specific
Measure user’s satisfaction
Signal processing for optimal radio resource management
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28. Section: Introduction
Questions?
Signal processing for optimal radio resource management
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29. Problem Formulation Section
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30. Problem Formulation
General Formulation of Optimal Resource Management:
Multi-Objective Optimization Problem
Generally impossible to maximize for all users!
Must divide power and cause inter-user interference
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31. Performance Region Definition: Achievable Performance Region
Contains all feasible combinations
Feasible = Achieved by some {v 1 ,…, v 𝐾 𝑟 } under power constraints
Care about user 2
Pareto Boundary
Cannot improve for any user without degrading for other users
Balance between users
Part of interest:
Pareto boundary
Other Names
Rate Region Capacity Region MSE Region, etc.
2-User
Performance Region
Care about user 1
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32. Performance Region (2) Definitions of Pareto Boundary
Strong Pareto point: Improve for any user  Degrade for other user
Weak Pareto point: Cannot simultaneously improve for all users
Weak Definition is More Convenient
Boundary is compact and simply-connected
Optimality Condition 1
Sending one stream per user is sufficient (assumed earlier)
Optimality Condition 2
At least one power constraint is active (=holds with equality)
X. Shang, B. Chen, H. V. Poor, “Multiuser MISO Interference Channels With Single-User Detection: Optimality of Beamforming and the Achievable Rate Region,” IEEE Trans. on Information Theory, 2011.
R. Mochaourab and E. Jorswieck, “Optimal Beamforming in Interference Networks with Perfect Local Channel Information,” IEEE Trans. on Signal Processing, 2011.
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33. Upper corner in region, everything inside region
Performance Region (3)
Can the region have any shape?
No! Can prove that:
Compact set
Normal set
Upper corner in region, everything inside region
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34. Performance Region (4) Some Possible Shapes User-Coupling Weak: Convex
Strong: Concave
Scheduling
Time-sharing for strongly coupled users
Select multiple points Hard: Unknown region
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35. Performance Region (5) Which Pareto Optimal Point to Choose?
Tradeoff: Aggregate performance vs. fairness
Utilitarian point (Max sum performance)
Utopia point (Combine user points)
No Objective Answer
Utopia point outside of region
Only subjective answers exist!
Single user point
Egalitarian point (Max fairness)
Performance Region
Single user point
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36. Section: Problem Formulation
Questions?
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37. Subjective Resource Management
Section
Subjective Resource Management
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38. Known as A Priori Approach
Subjective Approach
System Designer Selects Utility Function
Describes subjective preference
Increasing and continuous function
Examples:
Sum performance:
Proportional fairness:
Harmonic mean:
Max-min fairness:
Put different weights to move between
extremes
Known as A Priori Approach
Select utility function before optimization
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39. Subjective Approach (2)
Utilities Functions Has Different Shapes
Curve: 𝑓 g =constant
Optimal constant: Curve intersects optimum
Symmetric region: Same point
Asymmetric region: Different points
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40. Subjective Approach (3)
Utility Function gives Single-Objective Optimization Problem:
This is the Starting Point of Many Researchers
Although selection of is Inherently subjective Affects the solvability
Should always have a motivation in mind!
Pragmatic Approach
Try to Select Utility Function to Enable Efficient Optimization
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41. Complexity of Single-Objective Optimization Problems
Classes of Optimization Problems
Different scaling with number of parameters and constraints
Main Classes
Convex: Polynomial time solution
Monotonic: Exponential time solution
Arbitrary: More than exponential time
Practically solvable
Approximations needed
Hard to even approximate
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42. Complexity of Resource Management Problems
What is a Convex Problem?
Recall definitions:
Convex Function
For any two points on the graph of the function,
the line between the points is above the graph
Examples:
Convex Problem
Convex if objective 𝑓 0 and constraints 𝑓 1 ,…, 𝑓 𝑀 are convex functions
Signal processing for optimal radio resource management
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43. Complexity of Resource Management Problems (2)
When is the Resource Management a Convex Problem?
Original problem:
Rewritten problem (replace SINR 𝑘 with variable ):
Can be selected to be convex
SINR constraints: Main complication!
Convex power constraints
Signal processing for optimal radio resource management
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44. Classification of Resource Management Problems
Classification of Three Important Problems
The “Easy” problem
Weighted max-min fairness
Weighted sum performance
We will see: These have Different Complexities
Difficulty: Too many spatial degrees of freedom
Convex problem only if search space is particularly limited
Monotonic problem in general
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45. Complexity Example 1: The “Easy” Problem
Given Any Point ( 𝑔 1 ,…, 𝑔 𝐾 𝑟 ) or SINRs ( 𝛾 1 ,…, 𝛾 𝐾 𝑟 )
Find beamforming v 1 ,…, v 𝐾 𝑟 that attains this point
Fixed SINRs make the constraints convex:
Global solution in polynomial time – use CVX, Yalmip
Second order cone: Convex
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
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46. Complexity Example 2: 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 )
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47. Complexity Example 2: 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)
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48. Complexity Example 2: Max-Min Fairness (3)
Classification of Weighted Max-Min Fairness:
Quasi-convex problem (belongs to convex class)
Polynomial complexity in #users, #antennas, #constraints
Might be feasible complexity in practice
T.-L. Tung and K. Yao, “Optimal downlink power-control design methodology for a mobile radio DS-CDMA system,” in IEEE Workshop SIPS, 2002.
M. Mohseni, R. Zhang, and J. Cioffi, “Optimized transmission for fading multiple- access and broadcast channels with multiple antennas,” IEEE Journal on Sel. Areas in Communications, 2006.
A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. on Signal Processing, 2006.
E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Trans. on Signal Processing, 2012.
Early work
Main references
Channel uncertainty
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49. Complexity Example 3: Weighted 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
Non-convex problem
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50. Complexity Example 3: Weighted Sum Performance (2)
Classification of Weighted Sum Performance:
Non-convex problem
Power constraints: Convex
Utility: Monotonically increasing or decreasing in beamforming vectors
Therefore: Monotonic problem
Can There Be a Magic Algorithm?
No, provably NP-hard (Non-deterministic Polynomial-time hard)
Exponential complexity but in which parameters? (#users, #antennas, #constraints)
Z.-Q. Luo and S. Zhang, “Dynamic spectrum management: Complexity and duality,” IEEE Journal of Sel. Topics in Signal Processing, 2008.
Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, “Coordinated beamforming for MISO interference channel: Complexity analysis and efficient algorithms,” IEEE Trans. on Signal Processing, 2011.
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51. Complexity Example 3: Weighted Sum Performance (3)
Are Monotonic Problems Impossible to Solve?
No, not for small problems!
Monotonic Optimization Algorithms
Improve Lower/upper bounds on optimum:
Continue until
Subproblem: Essentially weighted max-min fairness problem
Monotonic optimization
Early works
Polyblock
algorithm
BRB algorithm
H. Tuy, “Monotonic optimization: Problems and solution approaches,” SIAM Journal of Optimization, 2000.
L. Qian, Y. Zhang, and J. Huang, “MAPEL: Achieving global optimality for a non- convex wireless power control problem,” IEEE Trans. on Wireless Commun., 2009.
E. Jorswieck, E. Larsson, “Monotonic Optimization Framework for the MISO Interference Channel,” IEEE Trans. on Communications, 2010.
W. Utschick and J. Brehmer, “Monotonic optimization framework for coordinated beamforming in multicell networks,” IEEE Trans. on Signal Processing, 2012.
E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Trans. on Signal Processing, 2012.
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52. Complexity Example 3: Weighted Sum Performance (4)
Branch-Reduce-and-Bound (BRB) Algorithm
Cover performance region with a box
Divide the box into two sub-boxes
Remove parts with no solutions in
Search for solutions to improve bounds
Continue with sub-box with largest value
End when bounds
are tight enough:
Accuracy
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53. Complexity Example 3: Weighted Sum Performance (5)
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)
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54. Complexity Example 3: Weighted Sum Performance (6)
Maximizing Sum Performance has High Complexity
Under linear beamforming
Other Shortcomings
Not all Pareto points are attainable
Weights have no clear interpretation
Not robust to perturbations
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55. Summary: Complexity of Resource Management Problems
General
Sum Performance
NP-hard
Max-Min Fairness
Quasi-Convex
“Easy” Problem
Convex
General
Zero Forcing
Single Antenna
Sum Performance
NP-hard
Convex
Max-Min Fairness
Quasi-Convex
“Easy” Problem
Linear
Proportional Fairness
Harmonic Mean
What about other utility functions?
Recall: The SINR constraints are complicating factor
Three conditions that simplify:
Fixed SINRs (“easy” problem)
Allow no interference (called: zero-forcing)
Multiplication  Addition (change of variable, single antenna BSs)
Signal
SINR
Interference
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56. Summary: Complexity of Resource Management (2)
Recall: All Utility Functions are Subjective
Pragmatic approach: Select to enable efficient optimization
Good Choice: Any Problem with Polynomial complexity
Example: Weighted max-min fairness
Use weights to adapt to other system needs
Bad Choice: Weighted Sum Performance
Generally NP-hard: Exponential complexity (in #users)
Should be avoided – Sometimes needed (virtual queuing techniques)
General
Zero Forcing
Single Antenna
Sum Performance
Convex
Max-Min Fairness
Quasi-Convex
“Easy” Problem
Linear
Proportional Fairness
Harmonic Mean
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57. Summary: Complexity of Resource Management (3)
Complexity Analysis for Any Dynamic Cooperation Clusters
Same optimization algorithms!
Extra characteristics can sometime simplify
Multi-antenna transmission: Higher complexity, higher performance
Ideal Joint Transmission
Coordinated Beamforming
Underlay Cognitive Radio
Signal processing for optimal radio resource management
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58. Section: Subjective Resource Management
Questions?
Signal processing for optimal radio resource management
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59. Multi-Objective Network Optimization
Section
Multi-Objective Network Optimization
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60. Many Network Performance Metrics
How to Improve Network Performance?
Higher user rates
Balance user satisfaction/fairness
Other possible metrics:
Higher total area rate
More active users (per area)
Higher energy efficiency
Considered so far
Can be any performance metrics
Multi-Objective Optimization
Study conflict/alignment of any metrics
Common tool in engineering/economics
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61. Basic Properties Multiple Resources Multiple Objectives
Resource bundle
Feasible resource allocation:
Multiple Objectives
Performance metrics:
Subjective utility: 𝑓(∙)
Performance Region
Compact set
Can be non-normal
Can be non-convex
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62. Visualization Performance Region is Generally Unknown
Sample points can be computed numerically
Helps to visualize tradeoffs/possibilities
Compare 2-3 metrics
Algorithm
Maximize sequence of 𝑓()
Points out different directions from the origin
A Posteriori Approach
Look at region at select operating point
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63. Example: Optimization of Future Networks
Design Cellular Network
Symmetric system
Select: 𝑁 = # BS antennas 𝐾 = # users 𝑃 = power/antenna
Resource bundle:
500
20 W
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64. Example: Optimization of Future Networks (2)
Coordinated beamforming
Each BS serves only its own 𝐾 users
Channels fixed for 𝑇 channel uses
BS knows channels within the cell (cost: 𝐾/𝑇)
Zero-forcing beamforming: no intra-cell interference
Interference leaks between cells
Average User Rate
Array gain
Power/user
Bandwidth
(10 MHz)
CSI estimation
overhead (𝑇=1000)
Noise ·
pathloss
(1.72∙ 10 −4 )
Relative intercell
interference
(0.54)
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65. Example: Optimization of Future Networks (3)
What Consumes Power?
Transmit power (+ losses in amplifiers)
Circuits attached to each antenna
Baseband computations
Coding/decoding
Backhaul/control signaling, cooling etc.
Total Power Consumption
Fixed power
(10 W)
Amplifier
efficiency
(0.31)
Circuit power
per antenna
(1 W)
Circuit power
per user
(0.3 W)
Computing zero-forcing
(2.3∙ 10 −6 ∙𝑁 𝐾 2 )
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66. Example: Results Average user rate Total area rate 3 Objectives
Energy efficiency
3 Objectives
Low user rates,
High area rates
Observations
Area and user rates are conflicting objectives
Different number of users
High user rates,
Low area rates
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67. Example: Results (2) Energy Efficiency vs. User Rates
Utility functions normalized by utopia point
Observations
Aligned for small user rates
Conflicting for high user rates
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68. Summary Multi-Objective Optimization Network Design and Optimization
Rigorous way to study problems with multiple performance metrics
Metrics are generally conflicting, but can be aligned
Network Design and Optimization
Metrics can be: User rates, total area rates, energy-efficiency, etc.
A posteriori approach: Visualize tradeoffs  Make decision
Open problems: How to use framework for design of future networks?
L. Zadeh, “Optimality and non-scalar-valued performance criteria,” IEEE Trans. on Autom. Control, 1963.
R.T. Marler and J.S. Arora, “Survey of multi-objective optimization methods for engineering,” Structural and Multidisciplinary Optimization, 2004.
E. Björnson, L. Sanguinetti, J. Hoydis, and M. Debbah, “Designing multi-user MIMO for energy efficiency: When is massive MIMO the answer?,” in Proc. IEEE WCNC, 2014.
E. Björnson, E. Jorswieck, M. Debbah, and Björn Ottersten, “Multi-Objective Signal Processing Optimization: The Way to Balance Conflicting Metrics In 5G Systems,” IEEE Signal Processing Magazine, to appear.
Signal processing for optimal radio resource management
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69. Section: Multi-Objective Network Optimization
Questions?
Signal processing for optimal radio resource management
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70. Summary: Part 1
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71. Summary: Part 1
Resource Management in Multi-Antenna Multi-Cell Systems
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
Multi-antenna transmission: More possibilities – higher complexity
Pragmatic approach: Select to enable efficient optimization
Polynomial complexity: Weighted max-min fairness etc.
Not solvable in practice: Weighted sum performance etc.
Multi-Objective Network Optimization
Practical optimization problems may have more than one metric
General framework: Study tradeoff between metrics
A posterior approach: Visualize the tradeoff and make decisions
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72. Coffee Break Thank you for listening!
Questions?
After the Break: Part 2 Recent Advances
Imperfect channel knowledge
Structure of optimal beamforming
Distributed resource management
Handling hardware impairments
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73. Part 2
Recent Advances
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74. Outline: Part 2 – Recent Advances
Imperfect Channel Knowledge
Resource management with channel uncertainty
Structure of Optimal Beamforming
Beamforming parametrization and its applications
Distributed Resource Management
Decompose system-wide problem into user-specific subproblems
Handling Hardware Impairments
Can we model and adapt to practical hardware?
Signal processing for optimal radio resource management
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75. Imperfect Channel Knowledge
Section
Imperfect Channel Knowledge
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76. Robustness to Imperfect Channel Knowledge
Perfect versus Imperfect Channel State Information (CSI)
Resource management framework developed with perfect CSI
Practical systems: Uncertainty of h 𝑘
Causes: estimation errors, feedback quantization, delays, etc.
What is the Expected Impact?
Small loss in signal power =200, =190
Larger interference leakage 100−100=0, 98−92=6
Goal: Robustness to Uncertainty
How to define it?
Perfect CSI
Imperfect CSI
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77. Modeling Imperfect CSI
Shape of Channel Estimation Errors
Send: Known “pilot” information
Apply: Optimal channel estimator
Obtain:
Unlimited support: Cannot handle all errors!
Ellipsoidal Uncertainty Sets at BSs
Cut out part of estimation errors
Definition:
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78. Robust Resource Management
Robust User Performance
User k achieves an SINR 𝛾 𝑘 using v 1 ,…, v 𝐾 𝑟 if
where 𝐒 𝑖 = 𝐯 𝑖 𝐯 𝑖 𝐻 is outer product of beamforming vectors
Robust Performance Region
Feasible points { 𝛾 1 ,…, 𝛾 𝐾 𝑟 }
Achieved by v 1 ,…, v 𝐾 𝑟 under power constraints
Good points: On upper boundary
Different system utilities = different points
Unknown shape, but compact and normal
2-User
Performance Region
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79. Robust Resource Management (2)
Formulated as search in robust performance region:
Subjective Utility Functions:
Sum performance:
Proportional fairness:
Harmonic mean:
Max-min fairness:
Recall
We can solve any such problem if the “easy” problem is solvable
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80. Robust Resource Management (3)
Robust “Easy” Problem (with 𝐒 𝑘 = 𝐯 𝑘 𝐯 𝑘 𝐻 ):
Infinitely many constraints!
Should have contained: rank 𝐒 𝑘 =1
Lemma
The solution to the robust “easy” has always rank one.
U. L. Wijewardhana, M. Codreanu, M.Latva-aho, and A. Ephremides, “A Robust Beamformer Design for Underlay Cognitive Radio Networks Using Worst Case Optimization,” EURASIP J. Wireless Communications and Networking, 2014.
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81. S-Procedure Application Lemma: S-Procedure Uncertainty set
SINR ≥ Value
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Application
Replace each uncertainty set with one linear matrix inequality
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82. Solving Robust “Easy” Problem
Proof: Based on S-procedure.
Intuition: Uncertainty sets replaced by new parameters λ 1 ,…, λ 𝐾 𝑟
Theorem
A point is in the robust region if and only if
the following convex feasibility problem is feasible:
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83. Simulation Examples Robust Performance Region
Radius of uncertainty region: ξ
Region shrinks with ξ : Particularly when both users are served
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84. Simulation Examples (2)
Robust Performance Region
Radius of uncertainty region: ξ
Region shrinks with ξ : Particularly when both users are served
Can even turn convex  non-convex
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85. Summary: Robust Resource Management
Channel Uncertainty in Practical Systems
Robust to compact uncertainty sets
Shrinks the performance region - less beneficial to share resources
Robust Resource Management
“Easy” problem is still convex
Consequence: Same problem classifications as with perfect CSI
More optimization parameters  higher complexity
G. Zheng, K.-K.Wong, and T.-S. Ng, “Robust linear MIMO in the downlink: A worst-case optimization with ellipsoidal uncertainty regions,” EURASIP J. on Adv. in Signal Process., 2008.
E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Trans. on Signal Processing, 2012.
C. Shen, T.-H. Chang, K.-Y. Wang, Z. Qiu, and C.-Y. Chi, “Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach,” IEEE Trans. on Signal Processing, 2012.
U. L. Wijewardhana, M. Codreanu, M.Latva-aho, and A. Ephremides, “A Robust Beamformer Design for Underlay Cognitive Radio Networks Using Worst Case Optimization,” EURASIP J. Wireless Communications and Networking, 2014.
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86. Section: Imperfect Channel Knowledge
Questions?
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87. Structure of Optimal Beamforming
Section
Structure of Optimal Beamforming
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88. Line-of-Sight and Non-Line-of-Sight Transmission
Adapt signal phases at antennas
Steer beam towards receiving user
Not a perfect beam: interference
Non-Line-of-Sight
Multipath propagation
Add components coherently
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89. Structure of Optimal Beamforming
Goal: Derive Simple Beamforming Parametrization
For any resource management problem
Step 1: Consider “Easy” Problem ( C 𝑘 = D 𝑘 = 𝐈 𝑁 , 𝐾 users for brevity):
Lagrangian satisfies Karush-Kuhn-Tucker (KKT) conditions:
Convex problem!
Lagrange multiplier
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90. Structure of Optimal Beamforming (2)
Stationarity KKT Condition:
Beamforming direction parameterized by λ 1 ,…, λ 𝐾
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91. Structure of Optimal Beamforming (3)
Power Allocation
Based on SINR constraints
Beamforming Parametrization
In terms of Lagrange multipliers λ 1 ,…, λ 𝐾
Optimal parameters:
(Fixed-point iteration)
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92. Structure of Optimal Beamforming: Summary
Optimal Beamforming for “Easy” Problem:
with
Consider Any Resource Management Problem:
Has 𝐾𝑁 complex optimization variables
Note: Solution coincides with “Easy” problem for some 𝛾 1 ,…, 𝛾 𝐾
Parameterization from above: 𝐾 –1 positive parameters
Equal to 𝑃
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93. Asymptotic Behavior: Signal-to-noise ratio (SNR)
Recall:
Low SNR: 𝜎 2 →∞
Inverse  Identity: Transmit in channel direction
Name: Maximum ratio transmission (MRT)
Maximizes signal gain, ignores interference
Complexity: Prop. to 𝑀𝐾
High SNR: 𝜎 2 →0
Inverse  Project orthogonal to co-users
Name: Zero-forcing beamforming (ZFBF)
Minimizes interference
Under this condition: Signal as large as possible
Complexity: Prop. to 𝑀 𝐾 2
Intended user
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94. Geometric Interpretation
Special Case: 𝐾=2
Beamforming: Linear combination of channel and zero-forcing direction
Tradeoff
Maximize signal vs. minimize interference
Hard to find optimal tradeoff
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95. Heuristic Beamforming at any SNR
Select parameters heuristically
Interpretation: λ 1 ,…, λ 𝐾 are user priorities
Large: High performance requirement or weak channel gain
Small: Low performance requirement or strong channel gain
One Common Heuristic Approach
Set all parameters equal: λ 𝑘 = 𝑃 𝐾
Proposed many times (since 1995):
Transmit Wiener/MMSE filter, Regularized zero-forcing, Signal-to-leakage beamforming, Virtual SINR beamforming, Virtual-uplink MVDR beamforming, etc.
Does it Perform Well?
Symmetric scenarios: Yes, no reason to give different priorities!
Asymmetric scenarios: Okay, but can be improved
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96. Heuristic Beamforming at any SNR (2)
Observations
MRT good at low SNR
ZFBF good at high SNR
Transmit MMSE always good
Example:
One BS with 4 antennas
4 single-antenna users
Maximize sum information rate
Beamforming Schemes
Optimal beamforming (BRB algorithm)
Transmit MMSE filter (regularized ZFBF)
ZFBF
MRT
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97. Asymptotic Behavior: Massive MIMO
Many Antennas: 𝑁→∞ (𝐾 fixed)
Behavior depends on channel model
Suppose: h 1 ,…, h 𝐾 ~ 𝐶𝑁(𝟎, 𝐈 𝑁 )
Asymptotics: 𝐸{ 𝐡 𝑘 2 }=𝑁 and 𝐸{ |𝐡 𝑖 𝐻 𝐡 𝑘 |}≤ 𝑁
Inverse  Project orthogonal to co-users
Once again: Zero-forcing beamforming (ZFBF)
Note
User channels become orthogonal
Consequence: Interference vanish as 𝑁→∞ also with MRT
But: MRT is not asymptotically optimal!
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98. Asymptotic Behavior: Massive MIMO (2)
Example:
10 users, SNR is 15 dB
Beamforming: Transmit MMSE (regularized ZF), ZFBF, and MRT
Multi-cell case: Pilot contamination is 30 dB weaker
Large difference between ZF/MMSE and MRT!
Single-cell case
Multi-cell case
Constant gap
4-5x
Vanishing gap
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99. Extension: Any Dynamic Cooperation Clusters
Any Resource Management Problem is Solved by
Parameters
Priority of User 𝑘: 𝜆 𝑘
Impact of Constraint 𝑙: 𝜇 𝑙
Total: 𝐾 𝑟 +𝐿–2 positive parameters (due to sum constraints)
Lagrange multipliers of “Easy” problem
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100. Summary Structure of Optimal Beamforming Heuristic Beamforming
Simple parametrization – balance signal and interference
Described by 𝐾 𝑟 +𝐿 –2 positive parameters
Will not make it easier to solve an NP-hard problem!
Heuristic Beamforming
MRT: Good at low SNR and lower complexity
ZFBF: Good at high SNR and many antennas
Transmit MMSE/regularized ZFBF: Good at all SNRs/antennas
Good for analysis!
Best in practice!
Early work
State-of- the-art
Tutorial
E. A. Jorswieck, E. G. Larsson, D. Danev, “Complete characterization of the Pareto boundary for the MISO interference channel,” IEEE Trans. on Signal Processing, 2008.
E. Björnson, M. Bengtsson, B. Ottersten, “Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers,” IEEE Trans. on Signal Processing, 2012.
E. Björnson, M. Bengtsson, B. Ottersten, “Optimal Multi-User Transmit Beamforming: Difficult Problem with a Simple Solution Structure,” IEEE Signal Proces. Mag., to appear.
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101. Section: Structure of Optimal Beamforming
Questions?
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102. Distributed Resource Management
Section
Distributed Resource Management
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103. Centralized vs. Distributed Deployments
Centralized Deployment
Assumed in Part 1
Example: Cloud-RAN (China Mobile)
Pro: Highest performance?
Con: Expensive infrastructure
Distributed Deployment
BSs exchange information
Pro: Inexpensive approach (exploit local functionality)
Con: Loss in performance?
Exchange what information?
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104. Distributed Optimization
Coupling Between Users
Users “compete” for resources
Cause inter-user interference
Distributed Optimization Approach:
Represent coupling by coupling variables (CVs)
Make local copies of CVs
Local optimization of resources for fixed CVs
Update CVs iteratively
Popular Techniques
Classical: Dual decomposition
Robustification: Alternating direction method of multipliers (ADMM)
Very essence of radio resource management!
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105. Example: Dual Decomposition
Consider “Easy” Problem
Achieve points { 𝛾 1 ,…, 𝛾 𝐾 𝑟 }
Centralized problem:
Add Auxiliary Variables
Interference from User 𝑖 to User 𝑘:
Believed:
True:
Power for User 𝑘:
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106. Example: Dual Decomposition (2)
How to remove decouple?
Partial Lagrangian for coupling constraints
Lagrange multipliers
Interference: 𝑦 𝑖𝑘
Power: 𝑧 𝑙
Change objective
Remove coupling constraints
Problem is decoupled!
One problem per user (for fixed Lagrange multipliers)
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107. Example: Dual Decomposition (3)
Subproblem: User 𝑘
Convex problem
Requires:
Local CSI
Allowed interference leakage
Master problem
Update Lagrange multipliers
Complicated structure
Subgradient update
Solutions to subproblems
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108. Example: Max-Min Fairness
Iterative Algorithm
Solve subproblems
Update multipliers (step size 𝜉)
Update best achieved point
Iterate
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109. Summary: Distributed Resource Management
Distributed Optimization
Global problem can be decomposed
Proof-of-concept: Dual decomposition
Exchange control variables (interference level, power usage) – no CSI!
Complexity: Can actually be greatly reduced!
Will it Converge to Optimum?
At least for convex problems!
D. Palomar and M. Chiang, “A tutorial on decomposition methods for network utility maximization,” IEEE J. Selected Areas in Communications, 2006.
A. Tölli, H. Pennanen, and P. Komulainen, “Decentralized minimum power multi-cell beamforming with limited backhaul signaling,” IEEE Trans. Wireless Communications, 2011.
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations and Trends in Machine Learning, 2011.
C. Shen, T.-H. Chang, K.-Y. Wang, Z. Qiu, and C.-Y. Chi, “Distributed robust multicell coordinated beamforming with imperfect CSI: An ADMM approach,” IEEE Trans. on Signal Processing, 2012.
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110. Section: Distributed Resource Management
Questions?
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111. Handling Hardware Impairments
Section
Handling Hardware Impairments
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112. Adapting to Transceiver Hardware Impairments
Physical Hardware is Non-Ideal
Phase noise, I/Q-imbalance, non-linearities, etc.
Expensive vs. cheap hardware
Impact reduced by calibration/compensation (not fully removed!)
Exact Characterization is Very Complicated
Many types of impairments and mitigation algorithms
Depends on signal waveform (OFDM, single-carrier, etc.)
Only the combined impact is needed!
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113. Non-Linear Systems Generalized System Model: Bussgang theorem:
Assume: Gaussian input signal (codebook and/or OFDM-signal)
Bussgang theorem:
OFDM systems: 𝑐 equal over subcarriers (can be tracked) 𝑉 is proportional to signal (inter-carrier interference)
W. Zhang, “A general framework for transmission with transceiver distortion and some applications,” IEEE Trans. Commun., 2012.
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114. Orthogonal frequency-division multiplexing (OFDM)
Source: Wikipedia
Transmitter
Receiver
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115. Definition and Interpretation
Transmitter distortion
Receiver distortion
Received Signal at User 𝑘
Gaussian Distortion Noise
Independent between antennas
Depends on beamforming
Error Vector Magnitude (EVM)
Proportional to signal power
Quality of transceivers:
LTE requirements: 0≤EVM≤0.17 (smaller  higher rates)
𝑘 𝑟 at receiver
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116. Resource Management with Hardware Impairments
Generalization of “Easy” Problem:
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117. Resource Management with Hardware Impairments
Generalization of “Easy” Problem:
Picks out signal at 𝑛th antenna
Auxiliary variables for distortion magnitudes
EVM constraints
Lemma
This is a convex problem if 𝑘 𝑡 and 𝑘 𝑟 are constant.
E. Björnson, P. Zetterberg, and M. Bengtsson, “Optimal coordinated beamforming in the multicell downlink with transceiver impairments,” in IEEE GLOBECOM, 2012.
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118. Resource Management with Hardware Impairments (2)
Recall
We can solve any resource management problem if the “easy” problem is solvable
Performance region
Reduces with impairments
Strong users more sensitive
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119. Resource Management with Hardware Impairments (3)
Performance Degradation as Function of SNR
Distortion noise power is proportional to signal power
High SNR  Distortions dominate over “regular” noise
Upper bound on achievable user rates
Simulation
User rate as function of transmit power
Impact increases with power and imperfections
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120. Summary Transceiver Hardware Impairments
Can be reduced but not avoided
Negligible impact at low SNR
Limits performance at high SNR
What is the Point of This Analysis?
Model and adapt beamforming to impairments
Achieve higher performance with same hardware
Achieve same performance with less expensive hardware
T. Schenk, “RF Imperfections in High-Rate Wireless Systems: Impact and Digital Compensation”. Springer, 2008
M. Wenk, “MIMO-OFDM Testbed: Challenges, Implementations, and Measurement Results”. Hartung-Gorre, 2010
E. Björnson, P. Zetterberg, M. Bengtsson, and B. Ottersten, “Capacity limits and multiplexing gains of MIMO channels with transceiver impairments,” IEEE Commun. Letters, 2013.
Books on modeling
Fundamental Capacity Behaviors
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121. Section: Handling Hardware Impairments
Questions?
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122. Summary: Part 2
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123. Summary Framework 1 can be Extended to Practical Conditions
Imperfect channel knowledge
Distributed resource management
Handling hardware impairments
Optimal Beamforming
Balance between signal gain and low interference
Simple beamforming structure
MRT and ZFBF are good for analysis
Transmit MMSE (regularized ZF) has better performance
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124. Many Open Problems Extension to Multi-Hop Networks
Including AF/DF relaying and/or physical layer network coding
Distributed Resource Management using Game Theory
Reach “good” operating points non-cooperatively or cooperatively
Use of More Detailed System Models
Consider imperfect CSI, impairments, etc. at the same time
For amplifiers, hardware imperfections, energy efficiency, etc.
Impact on resource management (e.g., beamforming, power allocation)
Context-Aware Resource Management
Exploit time dimension (e.g., delay constraints, predictable behaviors)
Define and optimize quality-of-experience
Interference and Load Management in Coordinated Multi-Tier Networks
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125. Thank You! Thank You For Listening! More Details in Our book
More parametrizations and structural insights
Guidelines for scheduling and forming clusters
Multi-casting, Multi-carrier Multi-antenna users, etc.
Cognitive radio and physical layer security
300 references to articles in these areas
E-book and code available for free download!
All papers are available:
Signal processing for optimal radio resource management
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