1. Wireless Networking and Communications Group Resource Allocation in Downlink Multiuser Multicarrier Wireless 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. Wireless Networking and Communications Group November 6, 2007 -2- Embedded Signal Processing Laboratory 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 Sys.Subsys.TheoryAlg.ReleaseDesignEmbed.Release ADSLequalizerYYMatlabYHW/SWDSP/C OFDMres. alloc.YYLabVIEWYSWDSP/C XceiverRFI mitig.YMatlabY DisplayhalftoningYYMatlab/CY EDAfix. pt. con.YMatlabYHW Founded 1996

3. Wireless Networking and Communications Group November 6, 2007 -3- Yousof Mortazavi Aditya Chopra Kapil Gulati Marcel Nassar Today’s ESPL Grad Students 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 Rabih Saliba Greg Allen Hamood Rehman Marcus DeYoung

4. Wireless Networking and Communications Group November 6, 2007 -4- Introduction Weighted-Sum Rate with Perfect Channel State Information Weighted-Sum Rate with Partial Channel State Information Conclusion Outline Dr. Ian Wong

5. Wireless Networking and Communications Group November 6, 2007 -5- Orthogonal Freq. Division Multiplexing subcarrier frequency magnitude channel Bandwidth OFDM Baseband Spectrum 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/g Digital Video Broadcast T/H

6. Wireless Networking and Communications Group November 6, 2007 -6- 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 Orthogonal Frequency Division Multiple Access (OFDMA)... User 1 frequency Base station (Subcarrier and power allocation) User M

7. Wireless Networking and Communications Group November 6, 2007 -7- OFDMA Resource Allocation In downlink direction, OFDMA base station simultaneously transmits data to different users on different subcarriers How do we allocate K data subcarriers 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 M K subcarrier allocations and determine power allocation for each

8. Wireless Networking and Communications Group November 6, 2007 -8- 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 Rate [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 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. Wireless Networking and Communications Group November 6, 2007 -9- 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 Allocation 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 with O(MK 2 ) complexity Dual optimization Multiple small optimization problems with closed-form solutions Practically optimal with O(MK) complexity Adaptive algorithms 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

10. Wireless Networking and Communications Group November 6, 2007 -10- 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 Noise vector Diagonal gain allocation matrix Diagonal channel matrix

11. Wireless Networking and Communications Group November 6, 2007 -11- Frequency-domain channel Stationary and ergodic Complex normal with correlated channel gains for subcarriers Statistical Wireless Channel Model 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. Wireless Networking and Communications Group November 6, 2007 -12- 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 Conclusion Outline

13. Wireless Networking and Communications Group November 6, 2007 -13- Ergodic Continuous Rate Maximization: Perfect CSI and CDI [Wong & Evans, accepted] Powers to determine Average power constraint Subcarrier capacity: Space of feasible power allocation functions: Channel-to-noise ratio (CNR) Constant weights Constant user weights: Perfect channel distribution info (CDI) of  user vector

14. Wireless Networking and Communications Group November 6, 2007 -14- Dual Optimization Framework “Max-dual user selection” Dual problem (convex in ): “Multi-level waterfilling” Duality gap Cut-off CNR is  0,m ( )

15. Wireless Networking and Communications Group November 6, 2007 -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. Wireless Networking and Communications Group November 6, 2007 -16- Optimal Resource Allocation – Ergodic Capacity with Perfect CSI PDF of CNR O (INM) Initialization CNR Realization O (MK) O (K) Runtime I – No. of line-search iterations K – No. of subcarriers M – No. of users N – No. of function evaluations for integration

17. Wireless Networking and Communications Group November 6, 2007 -17- Ergodic Discrete Rate Maximization: Perfect CSI and CDI [Wong & Evans, accepted] Discrete Rate Function: Uncoded BER = 10 -3

18. Wireless Networking and Communications Group November 6, 2007 -18- Dual Optimization Framework “Multi-level fading inversion” w m =1, =1 “Slope-interval selection”

19. Wireless Networking and Communications Group November 6, 2007 -19- 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 I – No. of line-search iterations K – No. of subcarriers L – No. of rate levels M – No. of users N – No. of function evaluations for integration

20. Wireless Networking and Communications Group November 6, 2007 -20- Simulation Results OFDMA Parameters (3GPP-LTE) Channel Simulation

21. Wireless Networking and Communications Group November 6, 2007 -21- Two-User Continuous Rate Region SNR Erg. Rates Algorithm Inst. Rates Algorithm Average number of function evaluations ( N ) 5 dB47.912- 10 dB50.091- 15 dB53.732- Average number of line search iterations ( I ) 5 dB8.0918.344 10 dB7.7278.333 15 dB7.9368.539 Average relative duality / optimality gap (  10 -6 ) 5 dB7.936.0251 10 dB5.462.0226 15 dB5.444.0159 76 used subcarriers

22. Wireless Networking and Communications Group November 6, 2007 -22- Two-User Discrete Rate Region SNR Erg. Rates Algorithm Inst. Rates Algorithm Average number of function evaluations ( N ) 5 dB62.09- 10 dB91.55- 15 dB133.02- Average number of line search iterations ( I ) 5 dB9.81817.241 10 dB10.55017.200 15 dB9.90917.304 Average relative duality/ optimality gap (  10 -4 ) 5 dB0.8713.602 10 dB0.9511.038 15 dB0.5320.340 76 used subcarriers

23. Wireless Networking and Communications Group November 6, 2007 -23- Sum Rate Versus Number of Users Continuous Rate Discrete Rate 76 used subcarriers

24. Wireless Networking and Communications Group November 6, 2007 -24- 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 Conclusion Outline

25. Wireless Networking and Communications Group November 6, 2007 -25- 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:

26. Wireless Networking and Communications Group November 6, 2007 -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. Wireless Networking and Communications Group November 6, 2007 -27- “Multi-level waterfilling on conditional expected CNR” Dual Optimization Framework 1-D Integral (> 50 iterations) 1-D Root-finding (<10 iterations) Computational bottleneck

28. Wireless Networking and Communications Group November 6, 2007 -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. Wireless Networking and Communications Group November 6, 2007 -29- 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

30. Wireless Networking and Communications Group November 6, 2007 -30- Discrete Rate Maximization: Partial CSI with Perfect CDI [Wong & Evans, submitted] Rate levels: Feasible set: Power allocation function given partial CSI: Average rate function given partial CSI: Nonlinear integer stochastic program Derived closed-form expressions

31. Wireless Networking and Communications Group November 6, 2007 -31- Power Allocation Functions Multilevel Fading Inversion Predicted CNR: Optimal Power Allocation

32. Wireless Networking and Communications Group November 6, 2007 -32- Dual Optimization Framework Bottleneck: computing rate/power functions Rate/power functions independent of multiplier Can be computed and stored before running search

33. Wireless Networking and Communications Group November 6, 2007 -33- 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

34. Wireless Networking and Communications Group November 6, 2007 -34- Simulation Parameters (3GPP-LTE) Channel Snapshot

35. Wireless Networking and Communications Group November 6, 2007 -35- Two-User Continuous Rate Region Average number of line search iterations ( I ) 5 dB8.599 10 dB8.501 15 dB8.686 Average relative duality/ optimality gap (  10 -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

36. Wireless Networking and Communications Group November 6, 2007 -36- Two-User Discrete Rate Region Average number of line search iterations ( I ) 5 dB21.33 10 dB21.12 15 dB21.15 Average relative duality/ optimality gap (  10 -4 ) 5 dB71.48 10 dB7.71 15 dB5.66 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

37. Wireless Networking and Communications Group November 6, 2007 -37- 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

38. Wireless Networking and Communications Group November 6, 2007 -38- Comparison with Previous Work Method Criteria Pro- portional [Wong,Shen, Andrews& Evans,‘04] Max Utility [Song&Li ‘05] Weighted Sum Rate [Seong,Mehsini & Cioffi,’06] [Yu,Wang& Giannakis] Weighted Dis-Rate Per-CSI Weight Dis-Rate Imp-CSI Weighted Dis-Rate Imp-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 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. Wireless Networking and Communications Group November 6, 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/bit error rate minimization

40. Wireless Networking and Communications Group November 6, 2007 -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. Wireless Networking and Communications Group November 6, 2007 -41- Questions? 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.

42. Wireless Networking and Communications Group November 6, 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 November 6, 2007 -43- Notation Glossary

44. Wireless Networking and Communications Group November 6, 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 November 6, 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 Two-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 November 6, 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 November 6, 2007 -47- 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)

48. Wireless Networking and Communications Group November 6, 2007 -48- 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)

49. Wireless Networking and Communications Group November 6, 2007 -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. Wireless Networking and Communications Group November 6, 2007 -50- PDF of Discrete Rate Dual Derive the pdf of

51. Wireless Networking and Communications Group November 6, 2007 -51- Performance Assessment - Duality Gap

52. Wireless Networking and Communications Group November 6, 2007 -52- Duality Gap Illustration M=2 K=4

53. Wireless Networking and Communications Group November 6, 2007 -53- Sum Power Discontinuity M=2 K=4

54. Wireless Networking and Communications Group November 6, 2007 -54- 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

55. Wireless Networking and Communications Group November 6, 2007 -55- Closed-form Average Rate and Power Power allocation function: Average rate function: Marcum-Q function

56. Wireless Networking and Communications Group November 6, 2007 -56- 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

57. Wireless Networking and Communications Group November 6, 2007 -57- 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”

58. Wireless Networking and Communications Group November 6, 2007 -58- 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:

59. Wireless Networking and Communications Group November 6, 2007 -59- 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

60. Wireless Networking and Communications Group November 6, 2007 -60- 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

61. Wireless Networking and Communications Group November 6, 2007 -61- 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

62. Wireless Networking and Communications Group November 6, 2007 -62- Subgradient Approximates “Instantaneous multi-level waterfilling with max-dual user selection”

63. Wireless Networking and Communications Group November 6, 2007 -63- Optimal Resource Allocation- Ergodic Proportional Rate without CDI Weighted-sum, Discrete Rate and Partial CSI are special cases of this algorithm

64. Wireless Networking and Communications Group November 6, 2007 -64- Two-User Capacity Region OFDMA Parameters (3GPP-LTE)  1 = 0.1-0.9 (0.1 increments)  2 = 1-  1

65. Wireless Networking and Communications Group November 6, 2007 -65- Evolution of the Iterates for  1 =0.1 and  2 = 0.9 User Rates Rate constraint Multipliers  Power Power constraint Multipliers

66. Wireless Networking and Communications Group November 6, 2007 -66- 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

67. Wireless Networking and Communications Group November 6, 2007 -67- 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