Resource Allocation in Downlink Multiuser Multicarrier Wireless Systems

1. Resource Allocation inDownlink Multiuser MulticarrierWireless Systems Prof. Brian L. Evans Dept. of Electrical and Computer Eng. The University of Texas at Austin November 6, 2007 Featuring work by my former PhD student Ian Wong

2. Embedded Signal Processing Laboratory Founded 1996 • Signal processing for communication systems • Image acquisition, analysis, and display • Electronic design automation (EDA) tools and methods • 16 PhD, 7 MS, 100 BS alumni + 9 PhD, 2 BS students now

3. Today’s ESPL Grad Students Greg Allen Hamood Rehman • Image processing systems • Communication systems Mitigation of radio freq. interference in laptop embedded transceivers Real-time wired multi-input multi-output (MIMO) multicarrier testbed Wireless multicarrier channel estimation and prediction algorithms Resource allocation algorithms for multiuser multicarrier wireless sys. Wael Barakat Marcus DeYoung Aditya Chopra Yousof Mortazavi Marcel Nassar Rabih Saliba Kapil Gulati

4. Dr. Ian Wong Outline • Introduction • Weighted-Sum Rate withPerfect Channel State Information • Weighted-Sum Rate withPartial Channel State Information • Conclusion

5. Orthogonal Freq. Division Multiplexing • Divides broadband channel into narrowband subchannels Multipath resistant Uses fast Fourier transform “Simpler” channel equalization • Uses static time or frequency division multiple access Digital Audio Broadcast (1996)IEEE 802.11a/gDigital Video Broadcast T/H OFDM Baseband Spectrum channel magnitude subcarrier frequency Bandwidth

6. User 1 frequency Base station User M (Subcarrier and power allocation) Orthogonal Frequency Division Multiple Access (OFDMA) • IEEE 802.16e-2005 (now) and 3GPP-LTE (2009 rollout?) • Multiple users assigned different subcarriers Inherits advantages of OFDM Granular exploitation of diversity among users through channel state information (CSI) feedback . . .

7. OFDMA Resource Allocation • In downlink direction, OFDMA base station simultaneously transmits data to different users on different subcarriers • How do we allocate K datasubcarriers and total power P to M users to optimize some performance metric? E.g. IEEE 802.16e-2005: K = 1536, M40 / sector • Very active research area NP-complete optimization problem [Song & Li, 2005] Brute force optimal solution would search through MK subcarrier allocations and determine power allocation for each

8. Related Work * Considered form of temporal diversity by maximizing an exponentially windowed running average of rate ** Independently developed a similar instantaneous continuous rate maximization algorithm *** Only for instantaneous continuous rate case, and linear complexity not explicitly shown in their papers

9. Summary of Contributions

10. Diagonal gain allocation 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 Single-cell base station (inter-cell interference ignored) • Received K-length vector for mth user at nth symbol Diagonal channel matrix

11. Statistical Wireless Channel Model • Frequency-domain channel Stationary and ergodic Complex normal with correlated channel gains for subcarriers • Time-domain channel Stationary and ergodic Complex normal and independent across tap i and user m Example distribution for fading channel for illustration purposes

12. Outline • Background • Weighted-Sum Rate withPerfect Channel State Information Continuous Rate Case Discrete Rate Case Numerical Results • Weighted-Sum Rate withPartial Channel State Information • Conclusion

13. 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, accepted] Perfect channel distribution info (CDI) of  user vector Space of feasible power allocation functions: Subcarrier capacity:

14. “Max-dual user selection” “Multi-level waterfilling” Duality gap Dual Optimization Framework Dual problem (convex in ): Cut-off CNR is0,m()

15. *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

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

17. Ergodic Discrete Rate Maximization:Perfect CSI and CDI [Wong & Evans, accepted] Discrete Rate Function: Uncoded BER = 10-3

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

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

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

21. Two-User Continuous Rate Region 76 used subcarriers

22. Two-User Discrete Rate Region 76 used subcarriers

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

24. Outline • Background • Weighted-Sum Rate withPerfect Channel State Information • Weighted-Sum Rate withPartial Channel State Information Continuous Rate Case Discrete Rate Case Numerical Results • Conclusion

25. 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

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

27. 1-D Integral (> 50 iterations) Computational bottleneck 1-D Root-finding (<10 iterations) Dual Optimization Framework “Multi-level waterfilling on conditional expected CNR”

28. 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)

29. 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)

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

31. Power Allocation Functions Optimal Power Allocation Multilevel Fading Inversion Predicted CNR:

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

33. 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

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

35. 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

36. 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

37. 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

38. Comparison with Previous Work * Considered form of temporal diversity by maximizing an exponentially windowed running average of 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/bit error rate minimization

40. Future Work • Multi-cell OFDMA and Single Carrier-FDMA Distributed algorithms that allow minimal base-station coordination to mitigate inter-cell interference • Multi-Input Multi-Output (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 for the OFDMA Downlink with Imperfect Channel Knowledge,“ IEEE Trans. on Comm., under review. [J2] I. C. Wong and B. L. Evans, "Optimal Downlink OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Rates," IEEE Trans. Wireless Comm., accepted. 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, Signal Proc., April 16-20, 2007, vol. II, pp. 89-92 [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, Signal Proc., April 16-20, 2007, vol. II, pp. 601-604. [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, to appear. [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, to appear. 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 • Two-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. 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)

48. 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

49. 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:

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