a unified framework for optimal resource allocation in multiuser multicarrier wireless systems

Wireless Networking and Communications Group

A Unified Framework for Optimal Resource Allocation in Multiuser Multicarrier Wireless Systems

Ian C. WongSupervisor:

Prof. Brian L. EvansCommittee:

Prof. Jeffrey G. AndrewsProf. Gustavo de Veciana Prof. Robert W. Heath, Jr.

Prof. David P. MortonProf. Edward J. Powers, Jr.

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

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

frequencyBase Station(Subcarrier and power allocation)

User M

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 MK subcarrier

allocations and determine power allocation for each

April 30, 2007-5-


Related Work

MethodCriteria

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]

FormulationErgodic Rates No Yes No No* NoDiscrete Rates No No No Yes NoUser prioritization No No Yes Yes Yes

Solution (algorithm)

Practically optimal No Yes No No Yes**

Linear complexity No No No No Yes***

Assumption (channel knowledge)

Imperfect CSI No No No No No

Do not require CDI Yes No Yes Yes Yes

* 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

April 30, 2007-6-


Summary of ContributionsPrevious Research Our Contributions

Formulatio

n

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(MK2) complexity)

Dual optimization•Multiple small optimization problems w/closed-form solutions•Practically optimal with O(MK) complexity

Assum

ption

Perfect channel knowledge•Unrealistic due to channel estimation errors and delay

Imperfect channel knowledge•Allocate based on statistics of channel estimation/prediction errors

Previous Research Our Contributions

Formulatio

n


Ergodic rate•Exploits time-varying nature of the wireless channel

Solution

Constraint-relaxation•One large constrained convex optimization problem•Resort to sub-optimal heuristics ((MK2) complexity)

Dual optimization•Multiple small optimization problems w/closed-form solutions•Practically optimal with (MK) complexity•Adaptive algorithms also proposed

Previous Research Our Contributions

Formulatio

n


Ergodic rate (continuous and discrete)•Exploits time-varying nature of the wireless channel

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 vectorDiagonal gain matrix Diagonal channel matrix

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

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

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:

April 30, 2007-11-


Dual Optimization Framework“Max-dual user selection”

Dual problem:

“Multi-level waterfilling”

Duality gap

April 30, 2007-12-


*Optimal Subcarrier and Power Allocation“Multi-level waterfilling” “Max-dual user selection”

Mar

gina

l du

alP

ower

*Independently discovered by [Yu, Wang, & Giannakis, submitted] and [Seong, Mehsini, & Cioffi, 2006] for instantaneous rate case

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(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)

April 30, 2007-14-


Optimal Resource Allocation – Ergodic Capacity with Perfect CSI

PDF of CNR(INM)

Initialization

CNR Realization

(MK)

(MK)

(K)

Runtime

M – No. of usersK – No. of subcarriersI – No. of line-search iterationsN – No. of function evaluations for integration

April 30, 2007-15-


Ergodic Discrete Rate Maximization:Perfect CSI and CDI [Wong & Evans, submitted]

Discrete Rate Function:

UncodedBER = 10-3

Anticipative and infinite dimensional stochastic program

April 30, 2007-16-


Dual Optimization Framework

“Multi-level fading inversion”

wm=1,=1

“Slope-interval selection”

April 30, 2007-17-


Optimal Resource Allocation – Ergodic Discrete Rate with Perfect CSI

PDF of CNR

CNR Realization

(INML)

(MKlog(L))

(MK)

(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

April 30, 2007-18-


Simulation Results

OFDMA Parameters (3GPP-LTE) Channel Simulation

April 30, 2007-19-


Two-User Continuous Rate RegionSNR Erg. Rates

AlgorithmInst. Rates Algorithm

No. of function evaluations(N)

5 dB 47.91 -

10 dB 50.09 -

15 dB 53.73 -

No. of Iterations(I)

5 dB 8.091 8.344

10 dB 7.727 8.333

15 dB 7.936 8.539

Relative Gap (x10-6)

5 dB 7.936 .0251

10 dB 5.462 .0226

15 dB 5.444 .0159

76 used subcarriers

April 30, 2007-20-


Two-User Discrete Rate RegionSNR Erg. Rates

AlgorithmInst. Rates Algorithm

No. of function evaluations(N)

5 dB 47.91 -

10 dB 50.09 -

15 dB 53.73 -

No. of Iterations(I)

5 dB 9.818 17.24

10 dB 10.550 17.20

15 dB 9.909 17.30


5 dB 0.8711 3.602

10 dB 0.9507 1.038

15 dB 0.5322 0.340

76 used subcarriers

April 30, 2007-21-


Sum Rate Versus Number of UsersContinuous Rate Discrete Rate

76 used subcarriers

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

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:

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

April 30, 2007-25-


“Multi-level waterfilling on conditional expected CNR”


1-D Integral (> 50 iterations)

1-D Root-finding (<10 iterations)Computationalbottleneck

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)

April 30, 2007-27-


M – No. of usersK – No. of subcarriersI – No. of line-search iterationsIp – No. of zero-finding iterations for power allocation functionIc – No. of function evaluations for numerical integration of expected capacity

Optimal Resource Allocation – Ergodic Capacity given Partial CSI

Predicted CNR

(1)

(MK)

(K)

Runtime

(MKI (Ip+Ic))

Conditional PDF

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

April 30, 2007-29-


Power Allocation Functions

Multilevel Fading Inversion (MFI):

Predicted CNR:

Optimal Power Allocation:

April 30, 2007-30-



• Bottleneck: computing rate/power functions• Rate/power functions independent of multiplier

– Can be computed and stored before running search

April 30, 2007-31-


Optimal Resource Allocation – Ergodic Discrete Rate given Partial CSI

Predicted CNR

(1)

(1)

(K)

Runtime

M – No. of usersK – No. of subcarriersL – No. of rate levelsI – No. of line-search iterations

(MK(I+L))

Conditional PDF

April 30, 2007-32-


Simulation Parameters (3GPP-LTE)

0 10 20 30 40 50 60 70 80-5

0

5

10

15

20

Subcarrier Index

CN

R (d

B)

User 1 - Perfect ChannelUser 2 - Perfect ChannelUser 1 - Predicted ChannelUser 2 - Predicted Channel

Channel Snapshot

April 30, 2007-33-


Two-User Continuous Rate RegionNo. of line searchiterations (I)

5 dB 8.599

10 dB 8.501

15 dB 8.686


5 dB 0.084

10 dB 0.057

15 dB 0.041Complexity (MKI(Ip+Ic))

M – No. of users; K – No. of subcarriersI – No. of line-search iterationsIp – No. of zero-finding iterations for power allocation functionIc – No. of function evaluations for numerical integration of expected capacity

April 30, 2007-34-


Two-User Discrete Rate RegionNo. of line searchiterations (I)

5 dB 21.33

10 dB 21.12

15 dB 21.15


5 dB 71.48

10 dB 7.707

15 dB 5.662Complexity (MK(I+L))M – No. of users

K – No. of subcarriers;I– No. of line search iterationsL – No. of discrete rate levels

No. of rate levels (L) = 4BER constraint = 10-3

April 30, 2007-35-


Average BER ComparisonPer-subcarrier Average BER Per-subcarrier Prediction Error Variance

Subcarrier Index

BE

R

No. of rate levels (L) = 4BER constraint = 10-3

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

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

April 30, 2007-38-


Comparison with Previous Work

MethodCriteria

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-RateI-CSIAdaptive

FormulationErgodic Rates No No* No Yes Yes YesDiscrete Rates No Yes No Yes Yes YesUser prioritization Yes Yes Yes Yes Yes Yes

Solution (algorithm)

Practically optimal No No Yes Yes Yes YesLinear complexity No No Yes** Yes Yes Yes

Assumption (channel knowledge)

Imperfect CSI No No No No Yes YesDo not require CDI Yes Yes Yes No 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

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

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

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, `Òptimal 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, `Òptimal 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, `Òptimal 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, `ÒFDMA Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge'', Proc. IEEE Int. Global Communications Conf., November 26-30, 2007 Washington, DC USA, submitted.

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

April 30, 2007-43-


Notation Glossary

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]

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 1st stage– Recourse actions for the second stage based on the realization

of the random quantities

[Ermoliev & Wets, 1988]

April 30, 2007-46-


C-Rate P-CSI Dual Objective DerivationLagrangian:

Dual objective

Linearity of E[¢]

Separability of objective

Power a function of RV realization

Exclusive subcarrier assignmentm,k not independent but identically distributed across k

April 30, 2007-47-


Optimal Resource Allocation – Instantaneous Capacity with Perfect CSI

CNR Realization

O(1)

O(1)

O(K)

Runtime

M – No. of usersK – No. of subcarriersI – No. of line-search iterationsN – No. of function evaluations for integration

O(IMK)

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:

April 30, 2007-49-


PDF of Discrete Rate Dual• Derive the pdf of

April 30, 2007-50-


Performance Assessment - Duality Gap

April 30, 2007-51-


Duality Gap Illustration

M=2K=4

April 30, 2007-52-


Sum Power Discontinuity

M=2K=4

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

April 30, 2007-54-


Closed-form Average Rate and PowerPower allocation function:

Average rate function:

Marcum-Q function

April 30, 2007-55-


Ergodic Sum Rate Maximization with Proportional Ergodic Rate Constraints

Ergodic Sum Capacity

Average Power Constraint

Proportionality Constants

Ergodic Rate forUser m

• Allows more definitive prioritization among users• Traces boundary of capacity region with specified ratio

Developed adaptive algorithm without CDI

April 30, 2007-56-



Multiplier forpower constraint

Multiplier forrate constraint

• Reformulated as weighted-sum rate problem with properly chosen weights

“Multi-level waterfilling with max-dual user selection”

April 30, 2007-57-


Projected Subgradient Search

Power constraintmultipliersearch

Rate constraint multipliervectorsearch

Multiplier iterates

Step sizes

SubgradientsProjection Derived pdfs forefficient 1-D Integrals

Per-user ergodic rate:

April 30, 2007-58-


Optimal Resource Allocation – Ergodic Proportional Rate with Perfect CSI

PDF of CNR

CNR Realization

O(INM2)

O(MK)

O(MK)

O(K)

Initialization

Runtime

M – No. of usersK – No. of subcarriersI– No. of subgradient search iterationsN – No. of function evaluations for integration

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

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 constraintmultipliersearch

Rate constraint multipliervectorsearch

Multiplier iterates

Step sizes

SubgradientsProjection Averaging time constant

Subgradient approximates

April 30, 2007-61-


Subgradient Approximates

“Instantaneous multi-level waterfilling with max-dual user selection”

April 30, 2007-62-


Optimal Resource Allocation- Ergodic Proportional Rate without CDI

Weighted-sum,Discrete Rateand Partial CSIare specialcases of this algorithm

April 30, 2007-63-


Two-User Capacity Region

OFDMA Parameters (3GPP-LTE)

1 = 0.1-0.9 (0.1 increments)2 = 1-1

April 30, 2007-64-


Evolution of the Iterates for 1=0.1 and 2 = 0.9U

ser

Rat

es

Rat

e co

nstra

int

Mul

tiplie

rs

Pow

er

Pow

er

cons

train

tM

ultip

liers

April 30, 2007-65-


Summary of the Resource Allocation AlgorithmsAlgorithm Initialization

ComplexityPer-symbol Complexity

Relative Gap Order of Magnitude

Sum-Rate at w=[.5,.5], SNR=5 dB

WS Cont. Rates Perfect CSI – Ergodic (INM) (MK) 10-6 2.40

WS Cont. Rates Perfect CSI – Inst. - (IMK) 10-8 2.39

WS Disc. Rates Perfect CSI – Ergodic (INML) (MKlogL) 10-5 1.20

WS Disc. Rates Perfect CSI – Inst. - (IMKlogL) 10-4 1.10

WS Cont. Rates Partial CSI - (MKI (Ip+Ic)) 10-6 2.37

WS Disc. Rates Partial CSI - (MK(I+L)) 10-4 1.09

Prop. Cont. Rates Perfect CSI with CDI - Ergodic

(INM2) (MK) 10-6 2.40

Prop. Cont. Rates Perfect CSI without CDI - Ergodic

- (MK) - 2.40

a unified framework for optimal resource allocation in multiuser multicarrier wireless systems

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