Wireless Networking and Communications Group Adaptive Downlink OFDMA Resource Allocation Ian Wong* and Brian L. Evans** * Freescale Semiconductor, Austin, Texas **The University of Texas at Austin October 29, 2008 IEEE Asilomar Conference on Signals, Systems and Computers

Wireless Networking and Communications Group -2- 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 a/g Digital Video Broadcast T/H

Wireless Networking and Communications Group -3- IEEE e-2005 (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

Wireless Networking and Communications Group -4- OFDMA Resource Allocation Each subcarrier used by at most one user How do we allocate K data subcarriers and total power P to M users to optimize some performance metric? –For example, in IEEE e-2005, K = 1536, M  40 / 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

Wireless Networking and Communications Group -5- Method Criteria Max-min [Rhee & Cioffi,‘00] Sum Rate [Jang,Lee &Lee,’02] Prop- ortional [Wong,Shen, Andrews& Evans,‘04] Max Utility [Song&Li, ‘05] Weighted Sum [Seong,Mehsi ni & Cioffi,’06] [Yu, Wang & Giannakis] Ergodic Weighted Sum [Wong& Evans, 2007] Formulation Ergodic Rates NoYesNoNo*NoYes Discrete Rates No YesNoYes User prioritization No Yes Solution (algorithm) Practically optimal NoYesNo Yes**Yes Linear complexity No Yes***Yes Assumption (channel knowledge) Imperfect CSI No Yes 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

Wireless Networking and Communications Group -6- 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 p m,k [n] power allocated to user m in subcarrier k at time n Noise vector Diagonal gain allocation matrix Diagonal channel matrix

Wireless Networking and Communications Group -7- Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints Ergodic Sum Capacity Average Power Constraint Proportionality Constants Ergodic Rate for User m Specifies prioritization among users Non-convex constraint space K set of subcarriers M set of users  m,k CNR p m,k power

Wireless Networking and Communications Group -8- Dual Optimization Framework Multiplier for power constraint Multiplier vector for rate constraint Reformulate as weighted-sum rate problem with properly chosen weights Form dual problem using Lagrangian “Multi-level waterfilling with max-dual user selection” K set of subcarriers M set of users  m,k CNR p m,k power

Wireless Networking and Communications Group -9- 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: Need to know channel distribution information

Wireless Networking and Communications Group -10- 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

Wireless Networking and Communications Group -11- 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

Wireless Networking and Communications Group -12- Optimal Resource Allocation- Ergodic Proportional Rate without CDI Weighted-sum, Discrete Rate and Partial CSI are special cases of this algorithm

Wireless Networking and Communications Group -13- Two-User Capacity Region OFDMA Parameters (3GPP-LTE)  1 = (0.1 increments)  2 = 1-  1

Wireless Networking and Communications Group -14- Evolution of Parameters for  1 =0.1 and  2 = 0.9 User Rates Rate constraint Multipliers  Power Power constraint Multipliers

Wireless Networking and Communications Group -15- Conclusion Developed an adaptive framework for OFDMA resource allocation Converges to dual optimal solution with probability one Easily extensible to similar OFDMA allocation formulations

Wireless Networking and Communications Group -16- References I. C. Wong and B. L. Evans, “Optimal Resource Allocation in the OFDMA Downlink with Imperfect Channel Knowledge”, IEEE Transactions on Communications, accepted for publication.Optimal Resource Allocation in the OFDMA Downlink with Imperfect Channel Knowledge” I. C. Wong and B. L. Evans, ”Optimal Downlink OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Rates”, IEEE Transactions on Wireless Communications, vol. 7, no. 3, Mar. 2008, pp ”Optimal Downlink OFDMA Resource Allocation with Linear Complexity to Maximize Ergodic Rates” I. C. Wong and B. L. Evans, Resource Allocation in Multiuser Multicarrier Wireless Systems, Springer, 2007, ISBN Resource Allocation in Multiuser Multicarrier Wireless Systems

Wireless Networking and Communications Group -17- 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

Wireless Networking and Communications Group -18- Notation Glossary

Wireless Networking and Communications Group -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]

Wireless Networking and Communications Group -20- 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]

Wireless Networking and Communications Group -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

Wireless Networking and Communications Group -22- 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)

Wireless Networking and Communications Group -23- 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)

Wireless Networking and Communications Group -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:

Wireless Networking and Communications Group -25- PDF of Discrete Rate Dual Derive the pdf of

Wireless Networking and Communications Group -26- Performance Assessment - Duality Gap

Wireless Networking and Communications Group -27- Duality Gap Illustration M=2 K=4

Wireless Networking and Communications Group -28- Sum Power Discontinuity M=2 K=4

Wireless Networking and Communications Group -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

Wireless Networking and Communications Group -30- Closed-form Average Rate and Power Power allocation function: Average rate function: Marcum-Q function

Wireless Networking and Communications Group -31- 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

Wireless Networking and Communications Group -32- 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”

Wireless Networking and Communications Group -33- 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:

Wireless Networking and Communications Group -34- 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

Wireless Networking and Communications Group -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

Wireless Networking and Communications Group -36- 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

Wireless Networking and Communications Group -37- Subgradient Approximates “Instantaneous multi-level waterfilling with max-dual user selection”

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

Wireless Networking and Communications Group -39- Two-User Capacity Region OFDMA Parameters (3GPP-LTE)  1 = (0.1 increments)  2 = 1-  1

Wireless Networking and Communications Group -40- Evolution of the Iterates for  1 =0.1 and  2 = 0.9 User Rates Rate constraint Multipliers  Power Power constraint Multipliers

Wireless Networking and Communications Group -41- 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) WS Cont. Rates Perfect CSI – Inst. - O (IMK) WS Disc. Rates Perfect CSI – ErgodicO (INML) O (MKlogL) WS Disc. Rates Perfect CSI – Inst. - O (IMKlogL) WS Cont. Rates Partial CSI - O (MKI (I p +I c )) WS Disc. Rates Partial CSI - O (MK(I +L)) Prop. Cont. Rates Perfect CSI with CDI - Ergodic O (I  NM 2 ) O (MK) Prop. Cont. Rates Perfect CSI without CDI - Ergodic - O (MK) -2.40

Wireless Networking and Communications Group -42- 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