A Unified Framework for Optimal Resource Allocation in Multiuser Multicarrier Wireless Systems

1. A Unified Framework for Optimal Resource Allocation in Multiuser Multicarrier Wireless Systems Ian C. Wong Supervisor: Prof. Brian L. Evans Committee: Prof. Jeffrey G. Andrews Prof. Gustavo de Veciana Prof. Robert W. Heath, Jr. Prof. David P. Morton Prof. Edward J. Powers, Jr.

2. Outline • Background • Weighted-Sum Rate with Perfect Channel State Information • Weighted-Sum Rate with Partial Channel State Information • Rate Maximization with Proportional Rate Constraints • Conclusion • Background • OFDMA Resource Allocation • Related Work • Summary of Contributions • System Model • Weighted-Sum Rate with Perfect Channel State Information • Weighted-Sum Rate with Partial Channel State Information • Rate Maximization with Proportional Rate Constraints • Conclusion

3. User 1 frequency Base Station User M (Subcarrier and power allocation) Orthogonal Frequency Division Multiple Access (OFDMA) • Used in IEEE 802.16d/e (now) and 3GPP-LTE (2009) • Multiple users assigned different subcarriers • Inherits advantages of OFDM • Granular exploitation of diversity among users through channel state information (CSI) feedback . . .

4. OFDMA Resource Allocation • How do we allocate K datasubcarriers and total power P to M users to optimize some performance metric? • E.g. IEEE 802.16e: K = 1536, M¼40 / sector • Very active research area • Difficult discrete optimization problem (NP-complete [Song & Li, 2005]) • Brute force optimal solution: Search through MK subcarrier allocations and determine power allocation for each

5. Related Work * Considered some form of temporal diversity by maximizing an exponentially windowed running average of the rate ** Independently developed a similar instantaneous continuous rate maximization algorithm *** Only for instantaneous continuous rate case, but was not shown in their papers

6. Summary of Contributions

7. Diagonal gain matrix Diagonal channel matrix Noise vector OFDMA Signal Model • Downlink OFDMA with K subcarriers and M users • Perfect time and frequency synchronization • Delay spread less than guard interval • Received K-length vector for mth user at nth symbol

8. Statistical Wireless Channel Model • Frequency-domain channel • Stationary and ergodic • Complex normal with correlated channel gains across subcarriers • Time-domain channel • Stationary and ergodic • Complex normal and independent across taps i and users m

9. Outline • Background • Weighted-Sum Rate with Perfect Channel State Information • Continuous Rate Case • Discrete Rate Case • Numerical Results • Weighted-Sum Rate with Partial Channel State Information • Rate Maximization with Proportional Rate Constraints • Conclusion

10. Powers to determine Channel-to-noise ratio (CNR) Constant weights Average power constraint Constant user weights: Ergodic Continuous Rate Maximization:Perfect CSI and CDI [Wong & Evans, 2007a] Anticipative and infinite dimensional stochastic program Space of feasible power allocation functions: Subcarrier capacity:

11. “Max-dual user selection” Dual problem: “Multi-level waterfilling” Duality gap Dual Optimization Framework

12. *Optimal Subcarrier and Power Allocation “Multi-level waterfilling” “Max-dual user selection” Marginal dual Power *Independently discovered by [Yu, Wang, & Giannakis, submitted] and [Seong, Mehsini, & Cioffi, 2006] for instantaneous rate case

13. Computing the Expected Dual • Dual objective requires an M-dimensional integral • Numerical quadrature feasible only for M=2 or 3 • O(NM) complexity (N- number of function evaluations) • For M>3, Monte Carlo methods are feasible, but are overly complex and converge slowly • Derive the pdf of • Maximal order statistic of INID random variables • Requires only a 1-D integral (O(NM) complexity)

14. Initialization PDF of CNR O(INM) Runtime CNR Realization O(MK) O(MK) M – No. of users K – No. of subcarriers I – No. of line-search iterations N – No. of function evaluations for integration O(K) Optimal Resource Allocation – Ergodic Capacity with Perfect CSI

15. Ergodic Discrete Rate Maximization:Perfect CSI and CDI [Wong & Evans, submitted] Anticipative and infinite dimensional stochastic program Discrete Rate Function: Uncoded BER = 10-3

16. “Slope-interval selection” “Multi-level fading inversion” wm=1,=1 Dual Optimization Framework

17. PDF of CNR Initialization O(INML) Runtime CNR Realization O(MKlog(L)) O(MK) O(K) M – No. of users; K – No. of subcarriers; L – No. of rate levels; I – No. of line-search iterations; N – No. of function evaluations for integration Optimal Resource Allocation – Ergodic Discrete Rate with Perfect CSI

18. Simulation Results Channel Simulation OFDMA Parameters (3GPP-LTE)

19. Two-User Continuous Rate Region 76 used subcarriers

20. Two-User Discrete Rate Region 76 used subcarriers

21. Sum Rate Versus Number of Users 76 used subcarriers Continuous Rate Discrete Rate

22. Outline • Background • Weighted-Sum Rate with Perfect Channel State Information • Weighted-Sum Rate with Partial Channel State Information • Continuous Rate Case • Discrete Rate Case • Numerical Results • Rate Maximization with Proportional Rate Constraints • Conclusion

23. MMSE Channel Prediction Conditional PDF of channel-to-noise ratio (CNR) – Non-central Chi-squared CNR: Normalized error variance: Partial Channel State Information Model • Stationary and ergodic channel gains • MMSE channel prediction

24. Continuous Rate Maximization:Partial CSI with Perfect CDI [Wong & Evans, submitted] Nonlinear integer stochastic program • Maximize conditional expectation given the estimated CNR • Power allocation a function of predicted CNR • Instantaneous power constraint • Parametric analysis is not required • a

25. 1-D Integral (> 50 iterations) Computational bottleneck 1-D Root-finding (<10 iterations) Dual Optimization Framework “Multi-level waterfilling on conditional expected CNR”

26. Power Allocation Function Approximation • Use Gamma distribution to approximate the Non-central Chi-squared distribution [Stüber, 2002] • Approximately 300 times faster than numerical quadrature (tic-toc in Matlab)

27. Optimal Resource Allocation – Ergodic Capacity given Partial CSI Conditional PDF Runtime O(MKI (Ip+Ic)) Predicted CNR O(1) O(MK) M – No. of users K – No. of subcarriers I – No. of line-search iterations Ip – No. of zero-finding iterations for power allocation function Ic – No. of function evaluations for numerical integration of expected capacity O(K)

28. Derived closed-form expressions Discrete Rate Maximization:Partial CSI with Perfect CDI [Wong & Evans, 2007b] Nonlinear integer stochastic program Average rate function given partial CSI: Rate levels: Feasible set: Power allocation function given partial CSI:

29. Power Allocation Functions Optimal Power Allocation: Multilevel Fading Inversion (MFI): Predicted CNR:

30. Dual Optimization Framework • Bottleneck: computing rate/power functions • Rate/power functions independent of multiplier • Can be computed and stored before running search

31. Conditional PDF Runtime O(MK(I+L)) Predicted CNR O(1) O(1) O(K) M – No. of users K – No. of subcarriers L – No. of rate levels I – No. of line-search iterations Optimal Resource Allocation – Ergodic Discrete Rate given Partial CSI

32. Simulation Parameters (3GPP-LTE) Channel Snapshot

33. Two-User Continuous Rate Region M – No. of users; K – No. of subcarriers I – No. of line-search iterations Ip – No. of zero-finding iterations for power allocation function Ic – No. of function evaluations for numerical integration of expected capacity

34. Two-User Discrete Rate Region No. of rate levels (L) = 4 BER constraint = 10-3 M – No. of users K – No. of subcarriers; I– No. of line search iterations L – No. of discrete rate levels

35. Average BER Comparison No. of rate levels (L) = 4 BER constraint = 10-3 Per-subcarrier Average BER Per-subcarrier Prediction Error Variance BER Subcarrier Index

36. Outline • Background • Weighted-Sum Rate with Perfect Channel State Information • Weighted-Sum Rate with Partial Channel State Information • Rate Maximization with Proportional Rate Constraints • Conclusion

37. Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints Ergodic Sum Capacity • Allows definitive prioritization among users [Shen, Andrews, & Evans, 2005] • Equivalent to weighted-sum rate with optimally chosen weights • Developed adaptive algorithms using stochastic approximation • Convergence w.p.1 without channel distribution information Average Power Constraint Proportional Rate Constraints

38. Comparison with Previous Work * Considered some form of temporal diversity by maximizing an exponentially windowed running average of the rate ** Only for instantaneous continuous rate case, but was not shown in their papers

39. Conclusion • Developed a unified algorithmic framework for optimal OFDMA downlink resource allocation • Based on dual optimization techniques • Practically optimal with linear complexity • Applicable to a broad class of problem formulations • Natural Extensions • Uplink OFDMA • OFDMA with minimum rate constraints • Power/BER minimization

40. Future Work • Multi-cell OFDMA and Single Carrier-FDMA • Distributed algorithms that allow minimal base-station coordination to mitigate inter-cell interference • MIMO-OFDMA • Capacity-based analysis • Other MIMO transmission schemes • Multi-hop OFDMA • Hop-selection

41. Relevant Jounal Publications [J1] I. C. Wong and B. L. Evans, "Optimal Resource Allocation in OFDMA Systems with Imperfect Channel Knowledge,“ IEEE Trans. on Communications., submitted Oct. 1, 2006, resubmitted Feb. 13, 2007. [J2] I. C. Wong and B. L. Evans, "Optimal OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Rates," IEEE Trans. on Wireless Communications, submitted Sept. 17, 2006, and resubmitted on Feb. 3, 2007. Relevant Conference Publications [C1] I. C. Wong and B. L. Evans, ``Optimal OFDMA Subcarrier, Rate, and Power Allocation for Ergodic Rates Maximization with Imperfect Channel Knowledge'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., April 16-20, 2007, Honolulu, HI USA. [C2] I. C. Wong and B. L. Evans, ``Optimal OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Weighted Sum Capacity'', Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Proc., April 16-20, 2007, Honolulu, HI USA. [C3] I. C. Wong and B. L. Evans, ``Optimal Downlink OFDMA Subcarrier, Rate, and Power Allocation with Linear Complexity to Maximize Ergodic Weighted-Sum Rates'', Proc. IEEE Int. Global Communications Conf., November 26-30, 2007 Washington, DC USA, submitted. [C4] I. C. Wong and B. L. Evans, ``OFDMA Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge'', Proc. IEEE Int. Global Communications Conf., November 26-30, 2007 Washington, DC USA, submitted. Questions?

42. Backup Slides • Notation • Related Work • Stoch. Prog. Models • C-Rate,P-CSI Dual objective • Instantaneous Rate • D-Rate,P-CSI Dual Objective • PDF of D-Rate Dual • Duality Gap • D-Rate,I-CSI Rate/power functions • Proportional Rates • Proportional Rates - adaptive • Summary of algorithms

43. Notation Glossary

44. Related Work • OFDMA resource allocation with perfect CSI • Ergodic sum rate maximizatoin [Jang, Lee, & Lee, 2002] • Weighted-sum rate maximization [Hoo, Halder, Tellado, & Cioffi, 2004] [Seong, Mohseni, & Cioffi, 2006] [Yu, Wang, & Giannakis, submitted] • Minimum rate maximization [Rhee & Cioffi, 2000] • Sum rate maximization with proportional rate constraints [Wong, Shen, Andrews, & Evans, 2004] [Shen, Andrews, & Evans, 2005] • Rate utility maximization [Song & Li, 2005] • Single-user systems with imperfect CSI • Single-carrier adaptive modulation [Goeckel, 1999] [Falahati, Svensson, Ekman, & Sternad, 2004] • Adaptive OFDM [Souryal & Pickholtz, 2001][Ye, Blum, & Cimini 2002][Yao & Giannakis, 2004] [Xia, Zhou, & Giannakis, 2004]

45. Stochastic Programming Models [Ermoliev & Wets, 1988] • Non-anticipative • Decisions are made based only on the distribution of the random quantities • Also known as non-adaptive models • Anticipative • Decisions are made based on the distribution and the actual realization of the random quantities • Also known as adaptive models • 2-Stage recourse models • Non-anticipative decision for the 1st stage • Recourse actions for the second stage based on the realization of the random quantities

46. C-Rate P-CSI Dual Objective Derivation Lagrangian: Dual objective Linearity of E[¢] Separability of objective Power a function of RV realization Exclusive subcarrier assignment m,k not independent but identically distributed across k

47. Runtime CNR Realization O(IMK) O(1) O(1) M – No. of users K – No. of subcarriers I – No. of line-search iterations N – No. of function evaluations for integration O(K) Optimal Resource Allocation – Instantaneous Capacity with Perfect CSI

48. Discrete Rate Perfect CSI Dual Optimization • Discrete rate function is discontinuous • Simple differentiation not feasible • Given , for all , we have • L candidate power allocation values • Optimal power allocation:

49. PDF of Discrete Rate Dual • Derive the pdf of

50. Performance Assessment - Duality Gap