Resource Allocation in Cellular Systems

1. Supélec, 2012-02-02 Optimization Concepts for Resource Allocation inCellular Systems Emil Björnson PhD in Telecommunications Signal Processing Lab KTH Royal Institute of Technology

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 • 1983: Born in Malmö, Sweden • 2007: Master of Science inEngineering Mathematics,Lund University, Sweden • 2011-11-17:Defended doctoralthesis in telecommunications,KTH, Sweden Multiantenna Cellular Communications Channel Estimation, Feedback, and Resource Allocation • Three Building Blocks of Physical Layer • Mathematical Analysis and Optimization Emil Björnson, KTH Royal Institute of Technology

5. Background • Cellular Communications • Many transmitting multi-antenna base stations • Many receiving single-antenna users • Downlink Transmission • Multiple transmit antennas – exploit spatial dimension • Multiuser transmission Pro: Higher performance, Con: Co-user interference Emil Björnson, KTH Royal Institute of Technology

6. Background (2): Multiple Cells • Uncoordinated Cells: • Simple processing • Simple infrastructure • Uncontrolled interference • Or fractional frequency reuse • Coordinated Cells: • Controlled interference • Backhaul signaling • Computationally complex • Tight synchronization Emil Björnson, KTH Royal Institute of Technology

7. Background (3): Multiple Cells • Dynamic User-Centric Coordination Clusters • Inner Circle (Strong Channels):Consider Transmitting to Users • Outer Circle (Non-negligible Channels):Avoid Interference to Users • Can model any level of coordination Emil Björnson, KTH Royal Institute of Technology

8. Background (4) • Resource Allocation • Select users for transmission • Design beamforming directions • Allocate transmit power • Optimize Resource Allocation • Maximize system performance • Satisfy system constraints (power, interference, fairness) • Any level of coordinated multipoint transmission • Robustness to uncertain channel information • No mathematical details • Focus on performance optimization concepts • Assumption: Linear transmit/receive processing Emil Björnson, KTH Royal Institute of Technology

9. Outline • What is Performance? • Different user performance measures • System Performance vs. user fairness • Multi-user Performance Region • How to interpret? • How to choose operating point? • Performance Optimization • Geometrical interpretation of common formulations • Right problem formulation = Easy to solve • Low-complexity Strategies • Exploit structure from optimal solution Emil Björnson, KTH Royal Institute of Technology

10. What is Performance? Emil Björnson, KTH Royal Institute of Technology

11. What is Performance? • Service Quality • Experienced by users (per-user level) • Can also be measured at system-level • Performance Based on • Average data rate • Latency • Coverage • Battery life • Etc. • Simplified Performance Measures • Necessary for optimization Emil Björnson, KTH Royal Institute of Technology

12. Single-user Performance Measures • 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 • Data Rate • Bits per ”channel use” • Mutual information: 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

13. Multi-user Performance • User Performance Measures • Same measures – but one per user • Performance Limitations • Power Allocation • Co-user interference: SINR= • Why Not Increase Power? • Power = Money & Environmental Impact • Reduce noise  Interference limited • User Fairness • New dimension of difficulty • Heterogeneous user conditions • Depends on performance measure • Signal Power • Interference + Noise Power Emil Björnson, KTH Royal Institute of Technology

14. Multi-user Performance Region Emil Björnson, KTH Royal Institute of Technology

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

16. Multi-user Performance Region (3) • 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 Emil Björnson, KTH Royal Institute of Technology

17. Multi-user Performance Region (3) • Possible Shapes of Region • Convex, concave, or neither • In general: Non-convex • In any case: Region is unknown • Convex • Concave • Non-convexNon-concave Emil Björnson, KTH Royal Institute of Technology

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

19. Performance Optimization Emil Björnson, KTH Royal Institute of Technology

20. System Performance versus Fairness • Always Sacrifice Either • Sum Performance • User 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 Rsum subject to R1=a1·Rsum, R2=a2·Rsum, … (a1 + a2+… = 1) • Non-Convex Optimization Problems • Generally hard to solve numerically • Starts fromPerformance • Starts fromFairness R1,R2,… Rsum Emil Björnson, KTH Royal Institute of Technology

21. The “Easy” Problem • Given Point (R1,R2,…) • Find transmit strategy that attains this point • Minimize power usage • 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 SemidefiniteOptimization,” Proc. Allerton, 1999. • A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Processing, 2006. • W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. Signal Process., 2007. • E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Transactions on Signal Processing, To appear. Single-cell (total power) Single-cell (per ant. power) Multi-cell(general power, robustness) Emil Björnson, KTH Royal Institute of Technology

22. Exploiting the “Easy” Problem • Easy: Achieve a Given Point • Hard: Find a Good Point • Shape of Performance Region • Far from obvious – one dimension per user • Main part of resource allocation Interference Channel 3 transmittersw. 4 antennas 3 users • Rate: user 1 • Rate: user 3 • Rate: user 2 Emil Björnson, KTH Royal Institute of Technology

23. Geometric Optimization Interpretations • Maximize Performance with Fairness Profile: maximize Rsum subject to R1=a1·Rsum, R2=a2·Rsum, … (a1 + a2+… = 1) • Geometric Interpretation • Search on line in direction (a1,a2,…) from origin Rsum (a1,a2,…)·Rsum =(a1·Rsum,a2·Rsum,…) Emil Björnson, KTH Royal Institute of Technology

24. Geometric Optimization Interpretations (2) • Simple line-search algorithm: Bisection • Non-convex  Iterative convex (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, KTH Royal Institute of Technology

25. Geometric Optimization Interpretations (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,… • Max-value is unknown! • Distance from origin unknown • Line  hyperplane (dim: #user – 1) • Harder than fairness-profile problem! • Iterative search algorithm? Emil Björnson, KTH Royal Institute of Technology

26. Geometric Optimization Interpretations (4) • Systematic Search Algorithm • Concentrate on important parts of performance region • Improve lower/upper bounds on optimum: • Continue until • Efficiently Solvable Subproblems • Based on Fairness-profile problem Emil Björnson, KTH Royal Institute of Technology

27. Geometric Optimization Interpretations (5) • Branch-Reduce-Bound (BRB) Algorithm • 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) • Continue with sub-box with largest value Emil Björnson, KTH Royal Institute of Technology

28. Geometric Optimization Interpretations (6) Properties • Global Convergence • Accuracy ε>0 in finitely many iterations • Exponential complexity only in #users • Polynomial complexity in other parameters (#antennas/constraints) Emil Björnson, KTH Royal Institute of Technology

29. Geometric Optimization: Conclusions • Fairness-Profile Approach: Easy • Quasi-Convex: Polynomial complexity • Reason: Only one search dimension • Weighted Sum Performance: Difficult • NP-hard: Exponential complexity (in #users) • 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

30. Geometric Optimization: References • Line-Search Algorithm for Fairness-Profiles • 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,” IEEE Trans. on Signal Processing, Submitted, 2011. • BRB Algorithm • Useful for more than weighted sum performance • E.g. arithmetic, geometric, or harmonic mean performance • H. Tuy, F. Al-Khayyal, and P. Thach, “Monotonic optimization: Branch and cut methods,” Essays and Surveys in Global Optimization, Springer, 2005. • E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic OptimizationFramework for Multicell MISO Systems,” IEEE Transactions on Signal Processing, To appear. Emil Björnson, KTH Royal Institute of Technology

31. Low-complexity Strategies Emil Björnson, KTH Royal Institute of Technology

32. Low-complexity Strategies • Hardware Limitations • Polynomial complexity: Only slowly-varying channels • Exponential complexity: Only suitable for benchmarking • Heuristic Resource Allocation • Find reasonable strategy with little effort • Exploit available insight the optimal structure • Parametrization of Upper Boundary • Select parameters in [0,1] • Get an strategy explicitly • Can achieve any point on upper boundary • Only necessary condition Emil Björnson, KTH Royal Institute of Technology

33. Low-complexity Strategies (2) • Method 1: Interference-temperature Control • Transmitters x (Receivers – 1) parameters • 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, 2011. • R. Mochaourab, E. Jorswieck, “Optimal Beamforming in Interference Networks with Perfect Local Channel Information,” IEEE Trans. Signal Processing, 2011. • Method 2: Exploit Solution Structure of “Easy” Problem • Explicit strategy given by optimal Lagrange multipliers • Always same structure, but different parameters • Take Lagrange multipliers as our parameters! • Transmitters + Receivers – 1 parameters • E. Björnson, M. Bengtsson, and B. Ottersten, “Pareto Characterization of the Multicell MIMO Performance Region With Simple Receivers,” IEEE Trans. Signal Processing, Submitted, 2011. Emil Björnson, KTH Royal Institute of Technology

34. Low-complexity Strategies (3) • Number of Parameters • Large difference for large problems • Number of Transmitters/Receivers Emil Björnson, KTH Royal Institute of Technology

35. Low-complexity Strategies (4) • Parameterized Structure of Resource Allocation • Beamforming directions maximize: • Parameter = Priority of User • Parameter = Based on transmit power of Base station • Heuristic Selection: • All Parameters = 1 • Called: Signal-to-leakage+noise ratio (SLNR) beamforming • Known to work well! Emil Björnson, KTH Royal Institute of Technology

36. Example Emil Björnson, KTH Royal Institute of Technology

37. Example – Multicell Scenario • Maximize Weighted Sum Rate • Two base stations: 20 dBm output power • Full, Partial, or No coordination • BRB algorithm • Heuristic Parameters (=1) Emil Björnson, KTH Royal Institute of Technology

38. Summary • Easy to Measure Single-user Performance • Multi-user Performance Measures • Sum performance vs. user fairness • Performance Region • All combinations of user performance • Upper boundary: All efficient outcomes • Explicit Parametrization: Low-complexity strategies • Two Standard Optimization Strategies • Maximize weighted sum performance • Difficult to solve (optimally – heuristic approx. exists) • Maximize performance with fairness profile • Easy to solve (with line-search algorithm) Emil Björnson, KTH Royal Institute of Technology

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