*Ian C. Wong and Brian L. Evans The University of Texas at Austin IEEE Globecom 2007

1. Optimal Downlink OFDMA Subcarrier, Rate, and Power Allocation with Linear Complexity to Maximize Ergodic Weighted-Sum Rates *Ian C. Wong and Brian L. Evans The University of Texas at Austin IEEE Globecom 2007 Washington, D.C. *Dr. Wong is now with Freescale Semiconductor, Austin, TX

2. 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 . . .

3. 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

4. Summary of Contributions

5. Ergodic Discrete Rate Maximization:Perfect CSI and CDI Anticipative and infinite dimensional stochastic program Discrete Rate Function: Uncoded BER = 10-3

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

7. PDF of Discrete Rate Dual • Derive the pdf of via order statistics

8. PDF of CNR Initialization O(INML) Runtime CNR Realization O(MKlog(L)) O(MK) O(K) Optimal Resource Allocation – Ergodic Discrete Rate with Perfect CSI 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

9. Performance Assessment - Duality Gap

10. Duality Gap Illustration M=2 K=4

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

12. Two-User Discrete Rate Region

13. Sum Rate Versus Number of Users Continuous Rate (ICASSP 2007) Discrete Rate

14. Conclusion • Developed a framework for OFDMA downlink resource allocation • Based on dual optimization techniques • Negligible duality gaps with linear complexity • Ergodic discrete rates with perfect CSI • Related work • Continuous rates (capacity-based formulation) • Imperfect CSI • No CDI assumptions

15. Relevant Journal Publications [J1] I. C. Wong and B. L. Evans, "Optimal OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Rates," IEEE Trans. on Wireless Communications, accepted for publication [J2] 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.. 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, ``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?

16. 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

17. 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:

18. Notation Glossary

19. 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]

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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:

26. Diagonal gain matrix Diagonal channel matrix Noise vector OFDMA Signal Model • Downlink OFDMA with K subcarriers and M users • Perfect time and frequency synchronization • Free of inter-symbol and inter-carrier interference • Received K-length vector for mth user at nth symbol

27. 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

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

29. Performance Assessment - Duality Gap

30. Duality Gap Illustration M=2 K=4

31. Sum Power Discontinuity M=2 K=4

32. BER/Power/Rate Functions • Impractical to impose instantaneous BER constraint when only partial CSI is available • Find power allocation function that fulfills the average BER constraint for each discrete rate level • Given the power allocation function for each rate level, the average rate can be computed • Derived closed-form expressions for average BER, power, and average rate functions

33. Average rate function: Closed-form Average Rate and Power Power allocation function: Marcum-Q function

34. Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints Developed adaptive algorithm without CDI Ergodic Sum Capacity • Allows more definitive prioritization among users • Traces boundary of capacity region with specified ratio Average Power Constraint Proportionality Constants Ergodic Rate for User m

35. Dual Optimization Framework • Reformulated as weighted-sum rate problem with properly chosen weights Multiplier for rate constraint Multiplier for power constraint “Multi-level waterfilling with max-dual user selection”

36. Projected Subgradient Search Power constraint multiplier search Multiplier iterates Step sizes Derived pdfs for efficient 1-D Integrals Subgradients Projection Rate constraint multiplier vector search Per-user ergodic rate:

37. Optimal Resource Allocation – Ergodic Proportional Rate with Perfect CSI Initialization PDF of CNR O(INM2) Runtime CNR Realization O(MK) O(MK) M – No. of users K – No. of subcarriers I– No. of subgradient search iterations N – No. of function evaluations for integration O(K)

38. Adaptive Algorithms for Rate Maximization Without Channel Distribution Information (CDI) • Previous algorithms assumed perfect CDI • Distribution identification and parameter estimation required in practice • More suitable for offline processing • Adaptive algorithms without CDI • Low complexity and suitable for online processing • Based on stochastic approximation methods

39. Averaging time constant Subgradient approximates Solving the Dual Problem Using Stochastic Approximation Projected subgradient iterations across time with subgradient averaging - Proved convergence to optimal multipliers with probability one Power constraint multiplier search Multiplier iterates Subgradient Averaging Step sizes Subgradients Projection Rate constraint multiplier vector search

40. Subgradient Approximates “Instantaneous multi-level waterfilling with max-dual user selection”

41. Optimal Resource Allocation- Ergodic Proportional Rate without CDI Weighted-sum, Discrete Rate and Partial CSI are special cases of this algorithm

42. Two-User Capacity Region OFDMA Parameters (3GPP-LTE) 1 = 0.1-0.9 (0.1 increments) 2 = 1-1

43. Evolution of the Iterates for 1=0.1 and 2 = 0.9 User Rates Rate constraint Multipliers  Power Power constraint Multipliers l

44. Summary of the Resource Allocation Algorithms