1. Wireless Networking and Communications Group 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. Wireless Networking and Communications Group April 30, 2007 -2- 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 Background Weighted-Sum Rate with Perfect Channel State Information Weighted-Sum Rate with Partial Channel State Information Rate Maximization with Proportional Rate Constraints Conclusion Outline

3. Wireless Networking and Communications Group April 30, 2007 -3- 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 Orthogonal Frequency Division Multiple Access (OFDMA)... User 1 frequency Base Station (Subcarrier and power allocation) User M

4. Wireless Networking and Communications Group April 30, 2007 -4- OFDMA Resource Allocation How do we allocate K data subcarriers 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 M K subcarrier allocations and determine power allocation for each

5. Wireless Networking and Communications Group April 30, 2007 -5- Related Work Method Criteria Max-min [Rhee & Cioffi,‘00] Sum Rate [Jang,Lee &Lee,’02] Proportional [Wong,Shen, Andrews& Evans,‘04] Max-utility [Song&Li, ‘05] Weighted-sum [Seong,Mehsini&Cioffi,’06] [Yu,Wang&Giannakis] Formulation Ergodic Rates NoYesNoNo*No Discrete Rates No YesNo User prioritization No Yes Solution (algorithm) Practically optimal NoYesNo Yes** Linear complexity No Yes*** Assumption (channel knowledge) Imperfect CSI No Do not require CDI YesNoYes * 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. Wireless Networking and Communications Group April 30, 2007 -6- Summary of Contributions Previous ResearchOur Contributions Formulation Instantaneous rate Unable to exploit time-varying wireless channels Ergodic rate Exploits time-varying nature of the wireless channel Solution Constraint-relaxation One large constrained convex optimization problem Resort to sub-optimal heuristics (O(MK 2 ) complexity) Dual optimization Multiple small optimization problems w/closed-form solutions Practically optimal with O(MK) complexity Assumption Perfect channel knowledge Unrealistic due to channel estimation errors and delay Imperfect channel knowledge Allocate based on statistics of channel estimation/prediction errors Previous ResearchOur Contributions Formulation Instantaneous rate Unable to exploit time-varying wireless channels Ergodic rate Exploits time-varying nature of the wireless channel Solution Constraint-relaxation One large constrained convex optimization problem Resort to sub-optimal heuristics (O(MK 2 ) complexity) Dual optimization Multiple small optimization problems w/closed-form solutions Practically optimal with O(MK) complexity Adaptive algorithms also proposed Previous ResearchOur Contributions Formulation Instantaneous rate Unable to exploit time-varying wireless channels Ergodic rate (continuous and discrete) Exploits time-varying nature of the wireless channel

7. Wireless Networking and Communications Group April 30, 2007 -7- 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 Noise vector Diagonal gain matrix Diagonal channel matrix

8. Wireless Networking and Communications Group April 30, 2007 -8- Frequency-domain channel –Stationary and ergodic –Complex normal with correlated channel gains across subcarriers Statistical Wireless Channel Model Time-domain channel –Stationary and ergodic –Complex normal and independent across taps i and users m

9. Wireless Networking and Communications Group April 30, 2007 -9- 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 Outline

10. Wireless Networking and Communications Group April 30, 2007 -10- Ergodic Continuous Rate Maximization: Perfect CSI and CDI [Wong & Evans, 2007a] Powers to determine Average power constraint Subcarrier capacity: Space of feasible power allocation functions: Anticipative and infinite dimensional stochastic program Channel-to-noise ratio (CNR) Constant weights Constant user weights:

11. Wireless Networking and Communications Group April 30, 2007 -11- Dual Optimization Framework “Max-dual user selection” Dual problem: “Multi-level waterfilling” Duality gap

12. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -13- Computing the Expected Dual Dual objective requires an M-dimensional integral –Numerical quadrature feasible only for M=2 or 3 O(N M ) 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. Wireless Networking and Communications Group April 30, 2007 -14- Optimal Resource Allocation – Ergodic Capacity with Perfect CSI PDF of CNR O (INM) Initialization CNR Realization O (MK) O (K) Runtime M – No. of users K – No. of subcarriers I – No. of line-search iterations N – No. of function evaluations for integration

15. Wireless Networking and Communications Group April 30, 2007 -15- Ergodic Discrete Rate Maximization: Perfect CSI and CDI [Wong & Evans, submitted] Discrete Rate Function: Uncoded BER = 10 -3 Anticipative and infinite dimensional stochastic program

16. Wireless Networking and Communications Group April 30, 2007 -16- Dual Optimization Framework “Multi-level fading inversion” w m =1, =1 “Slope-interval selection”

17. Wireless Networking and Communications Group April 30, 2007 -17- Optimal Resource Allocation – Ergodic Discrete Rate with Perfect CSI PDF of CNR CNR Realization O (INML) O (MKlog(L)) O (MK) O (K) Initialization Runtime 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

18. Wireless Networking and Communications Group April 30, 2007 -18- Simulation Results OFDMA Parameters (3GPP-LTE) Channel Simulation

19. Wireless Networking and Communications Group April 30, 2007 -19- Two-User Continuous Rate Region SNR Erg. Rates Algorithm Inst. Rates Algorithm No. of function evaluations ( N ) 5 dB47.91- 10 dB50.09- 15 dB53.73- No. of Iterations ( I ) 5 dB8.0918.344 10 dB7.7278.333 15 dB7.9368.539 Relative Gap (x10 -6 ) 5 dB7.936.0251 10 dB5.462.0226 15 dB5.444.0159 76 used subcarriers

20. Wireless Networking and Communications Group April 30, 2007 -20- Two-User Discrete Rate Region SNR Erg. Rates Algorithm Inst. Rates Algorithm No. of function evaluations ( N ) 5 dB47.91- 10 dB50.09- 15 dB53.73- No. of Iterations ( I ) 5 dB9.81817.24 10 dB10.55017.20 15 dB9.90917.30 Relative Gap (x10 -4 ) 5 dB0.87113.602 10 dB0.95071.038 15 dB0.53220.340 76 used subcarriers

21. Wireless Networking and Communications Group April 30, 2007 -21- Sum Rate Versus Number of Users Continuous Rate Discrete Rate 76 used subcarriers

22. Wireless Networking and Communications Group April 30, 2007 -22- 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 Outline

23. Wireless Networking and Communications Group April 30, 2007 -23- Stationary and ergodic channel gains MMSE channel prediction MMSE Channel Prediction Partial Channel State Information Model Conditional PDF of channel-to-noise ratio (CNR) – Non-central Chi-squared CNR:Normalized error variance:

24. Wireless Networking and Communications Group April 30, 2007 -24- 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 a Nonlinear integer stochastic program

25. Wireless Networking and Communications Group April 30, 2007 -25- “Multi-level waterfilling on conditional expected CNR” Dual Optimization Framework 1-D Integral (> 50 iterations) 1-D Root-finding (<10 iterations) Computational bottleneck

26. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -27- M – No. of users K – No. of subcarriers I – No. of line-search iterations I p – No. of zero-finding iterations for power allocation function I c – No. of function evaluations for numerical integration of expected capacity Optimal Resource Allocation – Ergodic Capacity given Partial CSI Predicted CNR O (1) O (MK) O (K) Runtime O (MKI (I p +I c )) Conditional PDF

28. Wireless Networking and Communications Group April 30, 2007 -28- Discrete Rate Maximization: Partial CSI with Perfect CDI [Wong & Evans, 2007b] Rate levels: Feasible set: Power allocation function given partial CSI: Average rate function given partial CSI: Nonlinear integer stochastic program Derived closed-form expressions

29. Wireless Networking and Communications Group April 30, 2007 -29- Power Allocation Functions Multilevel Fading Inversion (MFI): Predicted CNR: Optimal Power Allocation:

30. Wireless Networking and Communications Group April 30, 2007 -30- Dual Optimization Framework Bottleneck: computing rate/power functions Rate/power functions independent of multiplier –Can be computed and stored before running search

31. Wireless Networking and Communications Group April 30, 2007 -31- Optimal Resource Allocation – Ergodic Discrete Rate given Partial CSI Predicted CNR O (1) O (K) Runtime M – No. of users K – No. of subcarriers L – No. of rate levels I – No. of line-search iterations O (MK(I +L)) Conditional PDF

32. Wireless Networking and Communications Group April 30, 2007 -32- Simulation Parameters (3GPP-LTE) Channel Snapshot

33. Wireless Networking and Communications Group April 30, 2007 -33- Two-User Continuous Rate Region No. of line search iterations ( I ) 5 dB8.599 10 dB8.501 15 dB8.686 Relative Gap (x10 -4 ) 5 dB0.084 10 dB0.057 15 dB0.041 Complexity O (MKI (I p +I c )) M – No. of users; K – No. of subcarriers I – No. of line-search iterations I p – No. of zero-finding iterations for power allocation function I c – No. of function evaluations for numerical integration of expected capacity

34. Wireless Networking and Communications Group April 30, 2007 -34- Two-User Discrete Rate Region No. of line search iterations ( I ) 5 dB21.33 10 dB21.12 15 dB21.15 Relative Gap (x10 -4 ) 5 dB71.48 10 dB7.707 15 dB5.662 Complexity O (MK(I +L)) M – No. of users K – No. of subcarriers; I – No. of line search iterations L – No. of discrete rate levels No. of rate levels (L) = 4 BER constraint = 10 -3

35. Wireless Networking and Communications Group April 30, 2007 -35- Average BER Comparison Per-subcarrier Average BER Per-subcarrier Prediction Error Variance Subcarrier Index BER No. of rate levels (L) = 4 BER constraint = 10 -3

36. Wireless Networking and Communications Group April 30, 2007 -36- Background Weighted-Sum Rate with Perfect Channel State Information Weighted-Sum Rate with Partial Channel State Information Rate Maximization with Proportional Rate Constraints Conclusion Outline

37. Wireless Networking and Communications Group April 30, 2007 -37- Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints Ergodic Sum Capacity Average Power Constraint Proportional Rate Constraints 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

38. Wireless Networking and Communications Group April 30, 2007 -38- Comparison with Previous Work Method Criteria Proportional [Wong,Shen, Andrews& Evans,‘04] Max-utility [Song&Li, ‘05] Weighted [Seong,Mehsini &Cioffi,’06] [Yu,Wang& Giannakis] Weighted or Prop. D-Rate P-CSI Weighted or Prop. D-Rate I-CSI Weighted or Prop. D-Rate I-CSI Adaptive Formulation Ergodic Rates NoNo*NoYes Discrete Rates NoYesNoYes User prioritization Yes Solution (algorithm) Practically optimal No Yes Linear complexity No Yes**Yes Assumption (channel knowledge) Imperfect CSI No Yes Do not require CDI Yes No Yes * 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. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -41- Questions? 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.

42. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -43- Notation Glossary

44. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -45- Stochastic Programming Models 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 1 st stage –Recourse actions for the second stage based on the realization of the random quantities [Ermoliev & Wets, 1988]

46. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -47- Optimal Resource Allocation – Instantaneous Capacity with Perfect CSI CNR Realization O(1) O(K) Runtime M – No. of users K – No. of subcarriers I – No. of line-search iterations N – No. of function evaluations for integration O(IMK)

48. Wireless Networking and Communications Group April 30, 2007 -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. Wireless Networking and Communications Group April 30, 2007 -49- PDF of Discrete Rate Dual Derive the pdf of

50. Wireless Networking and Communications Group April 30, 2007 -50- Performance Assessment - Duality Gap

51. Wireless Networking and Communications Group April 30, 2007 -51- Duality Gap Illustration M=2 K=4

52. Wireless Networking and Communications Group April 30, 2007 -52- Sum Power Discontinuity M=2 K=4

53. Wireless Networking and Communications Group April 30, 2007 -53- 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

54. Wireless Networking and Communications Group April 30, 2007 -54- Closed-form Average Rate and Power Power allocation function: Average rate function: Marcum-Q function

55. Wireless Networking and Communications Group April 30, 2007 -55- Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints Ergodic Sum Capacity Average Power Constraint Proportionality Constants Ergodic Rate for User m Allows more definitive prioritization among users Traces boundary of capacity region with specified ratio Developed adaptive algorithm without CDI

56. Wireless Networking and Communications Group April 30, 2007 -56- Dual Optimization Framework Multiplier for power constraint Multiplier for rate constraint Reformulated as weighted-sum rate problem with properly chosen weights “Multi-level waterfilling with max-dual user selection”

57. Wireless Networking and Communications Group April 30, 2007 -57- Projected Subgradient Search Power constraint multiplier search Rate constraint multiplier vector search Multiplier iterates Step sizes Subgradients Projection Derived pdfs for efficient 1-D Integrals Per-user ergodic rate:

58. Wireless Networking and Communications Group April 30, 2007 -58- Optimal Resource Allocation – Ergodic Proportional Rate with Perfect CSI PDF of CNR CNR Realization O(I  NM 2 ) O(MK) O(K) Initialization Runtime M – No. of users K – No. of subcarriers I  – No. of subgradient search iterations N – No. of function evaluations for integration

59. Wireless Networking and Communications Group April 30, 2007 -59- 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

60. Wireless Networking and Communications Group April 30, 2007 -60- Subgradient Averaging 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 Rate constraint multiplier vector search Multiplier iterates Step sizes Subgradients Projection Averaging time constant Subgradient approximates

61. Wireless Networking and Communications Group April 30, 2007 -61- Subgradient Approximates “Instantaneous multi-level waterfilling with max-dual user selection”

62. Wireless Networking and Communications Group April 30, 2007 -62- Optimal Resource Allocation- Ergodic Proportional Rate without CDI Weighted-sum, Discrete Rate and Partial CSI are special cases of this algorithm

63. Wireless Networking and Communications Group April 30, 2007 -63- Two-User Capacity Region OFDMA Parameters (3GPP-LTE)  1 = 0.1-0.9 (0.1 increments)  2 = 1-  1

64. Wireless Networking and Communications Group April 30, 2007 -64- Evolution of the Iterates for  1 =0.1 and  2 = 0.9 User Rates Rate constraint Multipliers  Power Power constraint Multipliers

65. Wireless Networking and Communications Group April 30, 2007 -65- Summary of the Resource Allocation Algorithms AlgorithmInitialization Complexity Per-symbol Complexity Relative Gap Order of Magnitude Sum-Rate at w=[.5,.5], SNR=5 dB WS Cont. Rates Perfect CSI – ErgodicO (INM) O (MK) 10 -6 2.40 WS Cont. Rates Perfect CSI – Inst. - O (IMK) 10 -8 2.39 WS Disc. Rates Perfect CSI – ErgodicO (INML) O (MKlogL) 10 -5 1.20 WS Disc. Rates Perfect CSI – Inst. - O (IMKlogL) 10 -4 1.10 WS Cont. Rates Partial CSI - O (MKI (I p +I c )) 10 -6 2.37 WS Disc. Rates Partial CSI - O (MK(I +L)) 10 -4 1.09 Prop. Cont. Rates Perfect CSI with CDI - Ergodic O (I  NM 2 ) O (MK) 10 -6 2.40 Prop. Cont. Rates Perfect CSI without CDI - Ergodic - O (MK) -2.40