Resource Allocation in Cellular Systems Emil Björnson PhD in Telecommunications Signal Processing Lab KTH Royal Institute of Technology Supélec, Optimization Concepts for

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 1983: Born in Malmö, Sweden 2007: Master of Science in Engineering Mathematics, Lund University, Sweden : Defended doctoral thesis 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 Technology4

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 Technology5

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 Technology6

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 Technology7

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 Technology8

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 Technology9

Emil Björnson, KTH Royal Institute of Technology10 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 Technology11

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? Emil Björnson, KTH Royal Institute of Technology12 All improves with SNR: Signal Power Noise Power Optimize SNR instead!

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 Emil Björnson, KTH Royal Institute of Technology13 Signal Power Interference + Noise Power

Emil Björnson, KTH Royal Institute of Technology14 Multi-user Performance Region

Emil Björnson, KTH Royal Institute of Technology15 Performance User 1 Performance User 2 Performance Region Care about user 2 Care about user 1 Balance between users Achievable Performance Region – 2 users - Under power budget Part of interest: Upper boundary

Multi-user Performance Region (3) Can it have any shape? No! Can prove that: -Compact set -Simply connected (No holes) -Nice upper boundary Emil Björnson, KTH Royal Institute of Technology16 Normal set Upper corner in region, everything inside region

Multi-user Performance Region (3) Possible Shapes of Region -Convex, concave, or neither -In general: Non-convex -In any case: Region is unknown Emil Björnson, KTH Royal Institute of Technology17 ConvexConcaveNon-convex Non-concave

Multi-user Performance Region (3) Some Operating Points – Game Theory Names Emil Björnson, KTH Royal Institute of Technology18 Performance User 1 Performance User 2 Performance Region Utilitarian point (Max sum performance) Egalitarian point (Max fairness) Single user point Which point to choose? Optimize: Sum Performance? Fairness?

Emil Björnson, KTH Royal Institute of Technology19 Performance Optimization

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 w 1 ·R 1 + w 2 ·R 2 + … (w 1 + w 2 +… = 1) -Maximize performance with fairness-profile: maximize R sum subject to R 1 =a 1 ·R sum, R 2 =a 2 ·R sum, …(a 1 + a 2 +… = 1) Non-Convex Optimization Problems -Generally hard to solve numerically Emil Björnson, KTH Royal Institute of Technology20 R 1,R 2,… R sum 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 -Second-order cone or semi-definite program -Global solution in polynomial time – use CVX, Yalmip M. Bengtsson, B. Ottersten, “Optimal Downlink Beamforming Using Semidefinite Optimization,” Proc. Allerton, A. Wiesel, Y. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” IEEE Trans. Signal Processing, W. Yu and T. Lan, “Transmitter optimization for the multi-antenna downlink with per-antenna power constraints,” IEEE Trans. Signal Process., E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Transactions on Signal Processing, To appear Emil Björnson, KTH Royal Institute of Technology21 Single-cell (total power) Single-cell (per ant. power) Multi-cell (general power, robustness)

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 Emil Björnson, KTH Royal Institute of Technology22 Rate: user 3 Rate: user 1 Rate: user 2 Interference Channel 3 transmitters w. 4 antennas 3 users Main part of resource allocation

Geometric Optimization Interpretations Maximize Performance with Fairness Profile: maximize R sum subject to R 1 =a 1 ·R sum, R 2 =a 2 ·R sum, …(a 1 + a 2 +… = 1) Geometric Interpretation -Search on line in direction (a 1,a 2,…) from origin Emil Björnson, KTH Royal Institute of Technology23 (a 1,a 2,…) ·R sum =( a 1 ·R sum,a 2 ·R sum,…) R sum

Geometric Optimization Interpretations (2) Emil Björnson, KTH Royal Institute of Technology24 Simple line-search algorithm: Bisection -Non-convex  Iterative convex (Quasi-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 Line-search: Linear convergence One dimension (independ. #users)

Geometric Optimization Interpretations (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 Technology25 Max-value is unknown! -Distance from origin unknown -Line  hyperplane (dim: #user – 1) -Harder than fairness-profile problem! -Iterative search algorithm? R 1,R 2,…

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 Technology26

Branch-Reduce-Bound (BRB) Algorithm 1.Cover region with a box 2.Divide the box into two sub-boxes 3.Remove parts with no solutions in 4.Search for solutions to improve bounds (Based on Fairness-profile problem) 5.Continue with sub-box with largest value Emil Björnson, KTH Royal Institute of Technology27 Geometric Optimization Interpretations (5)

Emil Björnson, KTH Royal Institute of Technology28 Properties -Global Convergence -Accuracy ε>0 in finitely many iterations -Exponential complexity only in #users -Polynomial complexity in other parameters (#antennas/constraints) Geometric Optimization Interpretations (6)

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 Technology29

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, 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, 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, E. Björnson, G. Zheng, M. Bengtsson, B. Ottersten, “Robust Monotonic Optimization Framework for Multicell MISO Systems,” IEEE Transactions on Signal Processing, To appear Emil Björnson, KTH Royal Institute of Technology30

Emil Björnson, KTH Royal Institute of Technology31 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 1.Select parameters in [0,1] 2.Get an strategy explicitly -Can achieve any point on upper boundary -Only necessary condition Emil Björnson, KTH Royal Institute of Technology32

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, R. Mochaourab, E. Jorswieck, “Optimal Beamforming in Interference Networks with Perfect Local Channel Information,” IEEE Trans. Signal Processing, 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, Emil Björnson, KTH Royal Institute of Technology33

Low-complexity Strategies (3) Emil Björnson, KTH Royal Institute of Technology34 Number of Parameters -Large difference for large problems Number of Transmitters/Receivers

Low-complexity Strategies (4) Emil Björnson, KTH Royal Institute of Technology35

Emil Björnson, KTH Royal Institute of Technology36 Example

Example – Multicell Scenario Emil Björnson, KTH Royal Institute of Technology37 Maximize Weighted Sum Rate -Two base stations: 20 dBm output power -Full, Partial, or No coordination BRB algorithm Heuristic Parameters (=1)

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 Technology38

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