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Optimizing Performance

Optimizing Performance. In 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.

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Optimizing Performance

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  1. Optimizing Performance In Multiuser Downlink Communication Emil BjörnsonKTH Royal Institute of Technology Invited Seminar, University of Luxembourg

  2. 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. Emil Björnson, KTH Royal Institute of Technology

  3. 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 Emil Björnson, KTH Royal Institute of Technology

  4. Emil Björnson • Education • 2007: Master in EngineeringMathematics, Lund University • 2011 (fall): PhD in Telecommunications, KTH • Research: Wireless Communication • Estimationofchannel information • Quantization and limited feedback • Multicell transmission optimization • Homepage: • http://www.ee.kth.se/~emilbjo Emil Björnson, KTH Royal Institute of Technology

  5. 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 Emil Björnson, KTH Royal Institute of Technology

  6. Background (2) • Downlink Transmission • Multiple transmit antennas • Spatial beamforming • Multiuser communication – co-user interference • System Model • Focus on performance optimization concepts • No mathematical details Emil Björnson, KTH Royal Institute of Technology

  7. 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 Emil Björnson, KTH Royal Institute of Technology

  8. 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 Power • Noise Power • Optimize SNR instead! Emil Björnson, KTH Royal Institute of Technology

  9. 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 Power • Interference + Noise Power Emil Björnson, KTH Royal Institute of Technology

  10. Multi-user Performance Region • Achievable Performance Region – 2 users - Under power budget • Performance user 2 • Care aboutuser 2 • Balancebetweenusers • Part of interest: • Outer boundary • AchievablePerformanceRegion • Care aboutuser 1 • Performance user 1 Emil Björnson, KTH Royal Institute of Technology

  11. 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 Emil Björnson, KTH Royal Institute of Technology

  12. Multi-user Performance Region (3) • Some Operating Points – Game Theory Names • Performance user 2 • Utilitarian point(Max sum performance) Which pointto choose? Optimize: Performance?Fairness? • Egalitarian point(Max fairness) • Single user point • AchievablePerformanceRegion • Single user point • Performance user 1 Emil Björnson, KTH Royal Institute of Technology

  13. 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 • Starts fromPerformance • Starts fromFairness R1,R2,… Rtot Emil Björnson, KTH Royal Institute of Technology

  14. 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) Emil Björnson, KTH Royal Institute of Technology

  15. 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 Interference Channel 3 transmittersw. 4 antennas 3 users • Rate: user 1 • Rate: user 3 • Rate: user 2 Emil Björnson, KTH Royal Institute of Technology

  16. 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 Emil Björnson, KTH Royal Institute of Technology

  17. 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! Emil Björnson, KTH Royal Institute of Technology

  18. 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. Emil Björnson, KTH Royal Institute of Technology

  19. Approach 1: Generate Region (3) • Number of Parameters • Large difference for large problems • High Accuracy Means High Complexity • Heuristic parameters  Often good performance • Number of Transmitters/Receivers Emil Björnson, KTH Royal Institute of Technology

  20. 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 Rtot (a1,a2,…)·Rtot =(a1·Rtot,a2·Rtot,…) Emil Björnson, KTH Royal Institute of Technology

  21. Approach 2: Geometric Interpretation (2) • Simple algorithm: Bisection • Non-convex  Iterative convex • Find start interval • Solve the “easy” problem at midpoint • If feasible: • Remove lower half • Else: Remove upper half • Iterate • Subproblem: Convex optimization • Bisection: Linear convergence • Good scaling with #users Emil Björnson, KTH Royal Institute of Technology

  22. Approach 2: Geometric Interpretation (3) • Maximize weighted sum performance: maximize w1·R1 + w2·R2 + … (w1 + w2+… = 1) • Geometric interpretation • Search on line w1·R1 + w2·R2 = max-value R1,R2,… • But max-value is unknown • Distance from origin unknown • Harder than fairness-profile problem! • Line  hyperplane (dim: #user – 1) • Iterative search algorithm? Emil Björnson, KTH Royal Institute of Technology

  23. Approach 2: Geometric Interpretation (4) • Algorithm: Outer PolyblockApproximation • Find block containing region • Check performance in corners • Select best corner: • Draw line from origin • Search line for boundary point(bisection + “easy” problem) • Remove outer part of block • Iterate • Iterative fairness profile opt. • Good: Global convergence • Bad: No guaranteed speed Emil Björnson, KTH Royal Institute of Technology

  24. 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. Emil Björnson, KTH Royal Institute of Technology

  25. 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 Emil Björnson, KTH Royal Institute of Technology

  26. 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 • Point 1 • Time sharing • Vary between point 1 and point 2 • Achieve everything something in between • This part cannot be reached • Point 2 Emil Björnson, KTH Royal Institute of Technology

  27. 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 Emil Björnson, KTH Royal Institute of Technology

  28. 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) Emil Björnson, KTH Royal Institute of Technology

  29. Thank You for Listening! • Questions? • Papers and Presentations Available: • http://www.ee.kth.se/~emilbjo Emil Björnson, KTH Royal Institute of Technology

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