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. 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

Emil Björnson Education -2007: Master in Engineering Mathematics, Lund University (fall): PhD in Telecommunications, KTH Research: Wireless Communication -Estimation of channel information -Quantization and limited feedback -Multicell transmission optimization Homepage: Emil Björnson, KTH Royal Institute of Technology4

Background Wireless Communication -One or multiple transmitting base stations -Multiple receiving users – one stream each -Narrowband Uncoordinated or Coordinated Downlink Transmission Emil Björnson, KTH Royal Institute of Technology5 Uncoordinated CellsCoordinated 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 Emil Björnson, KTH Royal Institute of Technology6

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 Technology7

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? Emil Björnson, KTH Royal Institute of Technology8 All improves with SNR Signal Power Noise 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 Emil Björnson, KTH Royal Institute of Technology9 Signal Power Interference + Noise Power

Multi-user Performance Region Emil Björnson, KTH Royal Institute of Technology10 Performance user 1 Performance user 2 Achievable Performance Region Part of interest: Outer boundary Care about user 2 Care about user 1 Balance between 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 Emil Björnson, KTH Royal Institute of Technology11 ConvexConcaveNon-convex Non-concave

Multi-user Performance Region (3) Some Operating Points – Game Theory Names Emil Björnson, KTH Royal Institute of Technology12 Performance user 1 Performance user 2 Achievable Performance Region Utilitarian point (Max sum performance) Egalitarian point (Max fairness) Single user point Which point to 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 w 1 ·R 1 + w 2 ·R 2 + … (w 1 + w 2 +… = 1) -Maximize performance with fairness profile: maximize R tot subject to R 1 =a 1 ·R tot, R 2 =a 2 ·R tot, …(a 1 + a 2 +… = 1) Non-convex problems -Generally hard to solve numerically Emil Björnson, KTH Royal Institute of Technology13 R 1,R 2,… R tot Starts from Performance Starts from Fairness

The “Easy” Problem Given Point (R 1,R 2,…) -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, 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, 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 Emil Björnson, KTH Royal Institute of Technology14 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 Emil Björnson, KTH Royal Institute of Technology15 Rate: user 3 Rate: user 1 Rate: user 2 Interference Channel 3 transmitters w. 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 Emil Björnson, KTH Royal Institute of Technology16

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 Technology17

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, 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: v1. 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 Emil Björnson, KTH Royal Institute of Technology18

Approach 1: Generate Region (3) Emil Björnson, KTH Royal Institute of Technology19 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 R tot subject to R 1 =a 1 ·R tot, R 2 =a 2 ·R tot, …(a 1 + a 2 +… = 1) Geometric Interpretation -Search on ray in direction (a 1,a 2,…) from origin Emil Björnson, KTH Royal Institute of Technology20 (a 1,a 2,…) ·R tot =( a 1 ·R tot,a 2 ·R tot,…) R tot

Approach 2: Geometric Interpretation (2) Emil Björnson, KTH Royal Institute of Technology21 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 optimization Bisection: Linear convergence Good scaling with #users

Approach 2: Geometric Interpretation (3) Maximize weighted sum performance: maximize w 1 ·R 1 + w 2 ·R 2 + … (w 1 + w 2 +… = 1) Geometric interpretation -Search on line w 1 ·R 1 + w 2 ·R 2 = max-value Emil Björnson, KTH Royal Institute of Technology22 But max-value is unknown -Distance from origin unknown -Harder than fairness-profile problem! -Line  hyperplane (dim: #user – 1) -Iterative search algorithm? R 1,R 2,…

Approach 2: Geometric Interpretation (4) Algorithm: Outer Polyblock Approximation Emil Björnson, KTH Royal Institute of Technology23 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 convergence Bad: 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, 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 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, J. Brehmer and W. Utschick, “Utility Maximization in the Multi-User MISO Downlink with Linear Precoding”, Proc. IEEE ICC’09, E. Jorswieck and E. Larsson, “Monotonic Optimization Framework for the Two-User MISO Interference Channel,” IEEE Transactions on Communications, vol. 58, no. 7, pp , Emil Björnson, KTH Royal Institute of Technology24

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 Technology25

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 Emil Björnson, KTH Royal Institute of Technology26 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 Emil Björnson, KTH Royal Institute of Technology27 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) Emil Björnson, KTH Royal Institute of Technology28

Emil Björnson, KTH Royal Institute of Technology Thank You for Listening! Questions? Papers and Presentations Available: