Optimizing Performance In Multiuser Downlink Communication Emil Björnson KTH Royal Institute of Technology Invited Seminar, University of Luxembourg

Optimizing PerformanceIn Multiuser Downlink Communication

Emil Björnson

KTH Royal Institute of Technology

Invited Seminar, University of Luxembourg

KTH in Stockholm

KTH was founded in 1827 and is the largest of Sweden’s technical universities.

Since 1917, activities have been housed in central Stockholm, in beautiful buildings which today have the status of historical monuments.

KTH is located on five campuses.

22010-11-12 Emil Björnson, KTH Royal Institute of Technology

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A top European grant-earning university

• Europe’s most successful university in terms of earning European Research Council Advanced Grant funding for ”investigator-driven frontier research”

5 research projects awarded in 2008:• Open silicon-based research platform for emerging devices • Astrophysical Dynamos • Atomic-Level Physics of Advanced Materials • Agile MIMO Systems for Communications, Biomedicine, and

Defense • Approximation of NP-hard optimization problems

2010-11-12 Emil Björnson, KTH Royal Institute of Technology

Emil Björnson

• Education- 2007: Master in Engineering Mathematics, Lund University- 2011 (fall): PhD in Telecommunications, KTH

• Research: Wireless Communication- Estimation of channel information- Quantization and limited feedback- Multicell transmission optimization

• Homepage:- http://www.ee.kth.se/~emilbjo

2010-11-12 Emil Björnson, KTH Royal Institute of Technology 4

Background

• Wireless Communication- One or multiple transmitting base stations- Multiple receiving users – one stream each- Narrowband

• Uncoordinated or Coordinated Downlink Transmission


Uncoordinated Cells Coordinated Cells

Background (2)

• Downlink Transmission- Multiple transmit antennas- Spatial beamforming- Multiuser communication – co-user interference

• System Model- Focus on performance optimization concepts- No mathematical details


Outline

• How to Measure Performance?- Different performance measures- Performance vs. user fairness

• Multi-user Performance Region- How to interpret?- How to generate?

• Performance Optimization- Geometrical interpretation of standard strategies- Right problem formulation = Easy to solve


Single-user Performance Measures

• Mean Square Error- Difference: transmitted and received signal- Easy to analyze- Far from reality?

• Bit/Symbol Error Rate (BER/SER)- Probability of error (for given data rate)- Intuitive interpretation- Complicated & ignores channel coding

• Data Rate- Bits per ”channel use”- Ideal capacity: perfect and long coding- Still closest to reality?


All improveswith SNR

Signal PowerNoise Power

Optimize SNR

instead!

Multi-user Performance

• Performance Measures- Same – but one per user

• Performance Limitations- Division of power- Co-user interference: SINR=

• Why Not Increase Power?- Power = Money- Removes noise interference limited

• User Fairness- New dimension of difficulty- Different user conditions- Depends on performance measure


Signal PowerInterference + Noise Power

Multi-user Performance Region


Performance user 1

Performance user 2

AchievablePerformance

Region

Part of interest:Outer boundary

Care aboutuser 2

Care aboutuser 1

Balancebetween

users

• Achievable Performance Region – 2 users - Under power budget

Multi-user Performance Region (2)

• Different Shapes of Region- Convex, concave, or neither- If convex: Simplified optimization- In general: Non-convex- Never any holes


Convex Concave Non-convexNon-concave

Multi-user Performance Region (3)

• Some Operating Points – Game Theory Names


Performance user 1

Performance user 2

AchievablePerformance

Region

Utilitarian point(Max sum performance)

Egalitarian point(Max fairness)

Single user point

Single user point

Which pointto choose?

Optimize:Performance?

Fairness?

Performance versus Fairness

• Always Sacrifice Either- Performance- Fairness- Or both: optimize something in between

• Two Standard Optimization Strategies- Maximize weighted sum performance:

maximize w1·R1 + w2·R2 + … (w1 + w2+… = 1)

- Maximize performance with fairness profile: maximize Rtot

subject to R1=a1·Rtot, R2=a2·Rtot, … (a1 + a2+… = 1)

• Non-convex problems- Generally hard to solve numerically


R1,R2,…

Rtot

Starts fromPerformance

Starts fromFairness

The “Easy” Problem

• Given Point (R1,R2,…)- Find transmit strategy that attains this point- Minimize power usage

• Convex Problem (for single-antenna users, single user detection)

- Second order cone program- Global solution in polynomial time – use CVX

• A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 161–176, 2006.

• W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2646–2660, 2007.

• E. Björnson, N. Jaldén, M. Bengtsson, B. Ottersten, “Optimality Properties, Distributed Strategies, and Measurement-Based Evaluation of Coordinated Multicell OFDMA Transmission,” IEEE Trans. Signal Process., Submitted in July 2010.


Single-cell(total power)

Single-cell(per ant. power)

Multi-cell(general power)

Exploiting the “Easy” Problem

• Easy to Achieve a Given Operating Point- But how to find a good point?

• Shape of Performance Region- Far from obvious – one dimension per user


Rate: user 3

Rate: user 1

Rate: user 2

Interference Channel

3 transmittersw. 4 antennas

3 users

Two Optimization Approaches

• Approach 1: Generate Performance Region- Parametrization – simplifies search- Heuristic solutions

• Approach 2: Geometric Interpretation- Algorithms for non-convex problems – global convergence- Sometimes in polynomial time

• Both Exploit the ”Easy” Problem


Approach 1: Generate Region

• Approach 1:- Generate sample points of performance region- Evaluate performance at all points – select best value

• Searching All Transmit Strategies- One complex variable per link (transmit receive antenna)- Generally infeasible!


Approach 1: Generate Region (2)

• Simplifying Parameterizations- Vary parameters from 0 to 1

• Method 1: Interference-temperature Control- Transmitters x (Receivers – 1) parameters

• E. Jorswieck, E. Larsson, and D. Danev, “Complete characterization of the Pareto boundary for the MISO interference channel,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 5292–5296, 2008.

• X. Shang, B. Chen, and H. V. Poor, “Multi-user MISO interference channels with single-user detection: Optimality of beamforming and the achievable rate region,” IEEE Trans. Inf. Theory, arXiv:0907.0505v1.

• Method 2: Exploit Solution Structure of “Easy” Problem- Transmitters + Receivers parameters

• E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers,” Submitted to ICC 2011.


Approach 1: Generate Region (3)


2 3 4 5 6 7 8 9 100

20

40

60

80

100

Method 1Method 2

• Number of Parameters- Large difference for large problems

• High Accuracy Means High Complexity- Heuristic parameters Often good performance

Number of Transmitters/Receivers

Approach 2: Geometric Interpretation

• Maximize Performance with Fairness Profile: maximize Rtot

subject to R1=a1·Rtot, R2=a2·Rtot, … (a1 + a2+… = 1)

• Geometric Interpretation- Search on ray in direction (a1,a2,…) from origin


(a1,a2,…)·Rtot =(a1·Rtot,a2·Rtot,…)

Rtot

Approach 2: Geometric Interpretation (2)


• Simple algorithm: Bisection- Non-convex Iterative 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 optimizationBisection: Linear convergenceGood scaling with #users


• Maximize weighted sum performance: maximize w1·R1 + w2·R2 + … (w1 + w2+… = 1)

• Geometric interpretation- Search on line w1·R1 + w2·R2 = max-value


But max-value is unknown

- Distance from origin unknown- Harder than fairness-profile problem!- Line hyperplane (dim: #user – 1)- Iterative search algorithm?

R1,R2,…


• Algorithm: Outer Polyblock Approximation


1. Find block containing region

2. Check performance in corners

3. Select best corner:

Draw line from origin

4. Search line for boundary point(bisection + “easy” problem)

5. Remove outer part of block

6. Iterate

Iterative fairness profile opt.Good: Global convergenceBad: No guaranteed speed

Approach 2: References

• Bisection Algorithm for Fairness Profile• M. Mohseni, R. Zhang, and J. Cioffi, “Optimized transmission for fading

multiple-access and broadcast channels with multiple antennas,” IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp. 1627–1639, 2006.

• J. Lee and N. Jindal, “Symmetric capacity of MIMO downlink channels,” in Proc. IEEE ISIT’06, 2006, pp. 1031–1035.

• E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers,” Submitted to ICC 2011.

• Polyblock Algorithm- Useful for more than weighted sum performance

• H. Tuy, “Monotonic optimization: Problems and solution approaches,” SIAM Journal on Optimization, vol. 11, no. 2, pp. 464–494, 2000.

• J. Brehmer and W. Utschick, “Utility Maximization in the Multi-User MISO Downlink with Linear Precoding”, Proc. IEEE ICC’09, 2009.

• E. Jorswieck and E. Larsson, “Monotonic Optimization Framework for the Two-User MISO Interference Channel,” IEEE Transactions on Communications, vol. 58, no. 7, pp. 2159-2169, 2010.


Approach 2: Conclusions

• Fairness Profile: Easy- Linear convergence, Convex subproblems

• Weighted Sum Performance: Difficult- No guaranteed speed, Iterative fairness profiles- Reason: Optimizes both performance and fairness

• Every Weighted Sum = Some Fairness Profile- Easier to solve when posed as fairness profile problem- Parameter relationship non-obvious


Why Weighted Sum Performance?

• Difficult to solve optimally – easier with fairness profile- Heuristic solutions (using Approach 1)

• Better Practical Interpretation?- Fairness part of optimization

• Some boundary points cannot be achieved- Non-convex part of region


This part cannot be reached

Time sharing

Vary between point 1 and point 2

Achieve everything something in

between

Point 1

Point 2

Example – Two Performance Measures


• 3 Transmit Antennas- Per antenna constraints- SNR 10 dB (single user)

• 2 Single-antenna Users

• Performance Region- One i.i.d. realization- Upper: Data rate- Lower: SER

Summary

• Easy to Measure Single-user Performance• Multi-user Performance Measures

- Sum performance vs. user fairness

• Performance Region- Illustrated using parameterizations (new parametrization)- Useful for heuristic solutions- Can generate many points and evaluate performance

• Two Standard Optimization Strategies- Maximize weighted sum performance• Difficult to solve (optimally – heuristic approx. exists)

- Maximize performance with fairness profile• Easy to solve (with bisection algorithm)


2010-11-12 29Emil Björnson, KTH Royal Institute of Technology

Thank You for Listening!

Questions?

Papers and Presentations Available:http://www.ee.kth.se/~emilbjo

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Optimizing Performance In Multiuser Downlink Communication Emil Björnson KTH Royal Institute of Technology Invited Seminar, University of Luxembourg