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OFDMA Downlink Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge

OFDMA Downlink Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge. *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. User 1. frequency.

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OFDMA Downlink Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge

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  1. OFDMA Downlink Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge *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. 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

  5. Summary of Contributions

  6. 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

  7. 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

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

  9. Continuous Rate Maximization:Partial CSI with Perfect CDI Nonlinear integer stochastic program • Maximize conditional expectation given the estimated CNR • Power allocation a function of predicted CNR • Parametric analysis is not required, thus

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

  11. 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)

  12. Conditional PDF Runtime O(MKI (Ip+Ic)) Predicted CNR O(1) O(MK) O(K) Optimal Resource Allocation – Ergodic Capacity given Partial CSI 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

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

  14. Two-User Capacity 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

  15. 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

  16. Conclusion • Developed a framework for OFDMA downlink resource allocation • Based on dual optimization techniques • Negligible duality gaps with linear complexity • Ergodic capacity with imperfect CSI • Related work • Discrete rate • No CDI assumptions

  17. Relevant Journal 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, accepted for publication. 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 Questions?

  18. 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

  19. Notation Glossary

  20. 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]

  21. 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

  22. 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

  23. 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

  24. 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:

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

  26. Performance Assessment - Duality Gap

  27. Duality Gap Illustration M=2 K=4

  28. Sum Power Discontinuity M=2 K=4

  29. 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

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

  31. 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

  32. 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”

  33. 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:

  34. 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)

  35. 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

  36. 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

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

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

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

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

  41. Summary of the Resource Allocation Algorithms

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