1. OFDMA Downlink Resource Allocation for Ergodic Capacity Maximization with Imperfect Channel Knowledge
*Ian C. Wong and Brian L. Evans
The University of Texas at Austin
IEEE Globecom 2007
Washington, D.C.
*Dr. Wong is now with Freescale Semiconductor, Austin, TX

2. Orthogonal Frequency Division Multiple Access (OFDMA)
Used in IEEE d/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
User 1
frequency
Base Station
(Subcarrier and power allocation)
. . .
User M
April 30, 2007

3. 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 e: 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

4. Related Work No Yes No* Yes** Yes*** April 30, 2007 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 No
Yes
No*
Discrete Rates
User prioritization
Solution (algorithm)
Practically optimal
Yes**
Linear complexity
Yes***
Assumption (channel knowledge)
Imperfect CSI
Do not require CDI
* 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

5. Summary of Contributions
Previous Research
Our Contributions
Formulation
Instantaneous rate
Unable to exploit time-varying wireless channels
Ergodic capacity
Exploits time-varying nature of the wireless channel
Previous Research
Our 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(MK2) complexity)
Dual optimization
Multiple small optimization problems w/closed-form solutions
Practically optimal with O(MK) complexity
Previous Research
Our 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(MK2) 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
April 30, 2007

6. OFDMA Signal Model Downlink OFDMA with K subcarriers and M users
Perfect time and frequency synchronization
Free of inter-symbol and inter-carrier interference
Received K-length vector for mth user at nth symbol
Diagonal gain matrix
Diagonal channel matrix
Noise vector
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7. Statistical Wireless Channel Model
Time-domain channel
Stationary and ergodic
Complex normal and independent across taps i and users m
Frequency-domain channel
Stationary and ergodic
Complex normal with correlated channel gains across subcarriers
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8. Partial Channel State Information Model
Stationary and ergodic channel gains
MMSE channel prediction
MMSE Channel Prediction
Ehat
Conditional PDF of channel-to-noise ratio (CNR) – Non-central Chi-squared
Predicted CNR:
Normalized error variance:
April 30, 2007

9. Continuous Rate Maximization: Partial CSI with Perfect CDI
Nonlinear integer stochastic program
Maximize conditional expectation given the estimated CNR
Power allocation a function of predicted CNR
Parametric analysis is not required, thus
April 30, 2007

10. Dual Optimization Framework
“Multi-level waterfilling on conditional expected CNR”
Computational
bottleneck
1-D Integral (> 50 iterations)
1-D Root-finding (<10 iterations)
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11. 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

12. Optimal Resource Allocation – Ergodic Capacity given Partial CSI
Predicted CNR
Runtime
O(MKI (Ip+Ic))
Conditional PDF
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)
April 30, 2007

13. Simulation Parameters (3GPP-LTE)
Channel Snapshot
April 30, 2007

14. Two-User Capacity Region
No. of line search
iterations (I)
5 dB
8.599
10 dB
8.501
15 dB
8.686
Relative
Gap
(x10-4)
0.084
0.057
0.041
Complexity
O(MKI(Ip+Ic))
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
April 30, 2007

15. 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 C-Rate I-CSI
Formulation
Ergodic Rates No
No*
Yes
Discrete Rates
User prioritization
Solution (algorithm)
Practically optimal
Linear complexity
Yes**
Assumption (channel knowledge)
Imperfect CSI
Do not require CDI
* 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

16. Conclusion
Developed a framework for OFDMA downlink resource allocation
Based on dual optimization techniques
Negligible duality gaps with linear complexity
Ergodic capacity with imperfect CSI
Related work
Discrete rate
No CDI assumptions
April 30, 2007

17. Questions? Relevant Journal 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, accepted for publication.
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
April 30, 2007

18. 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

19. Notation Glossary
April 30, 2007

20. 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

21. 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
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
April 30, 2007

22. 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
April 30, 2007

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)
April 30, 2007

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:
April 30, 2007

25. PDF of Discrete Rate Dual
Derive the pdf of
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26. Performance Assessment - Duality Gap
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27. Duality Gap Illustration
M=2
K=4
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28. Sum Power Discontinuity
K=4
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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
April 30, 2007

30. Closed-form Average Rate and Power
Power allocation function:
Average rate function:
Marcum-Q function
April 30, 2007

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

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

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

34. Optimal Resource Allocation – Ergodic Proportional Rate with Perfect CSI
Initialization
PDF of CNR
O(INM2)
Runtime
CNR Realization
O(MK)
O(MK)
M – No. of users
K – No. of subcarriers
I– No. of subgradient search iterations
N – No. of function evaluations
for integration
O(K)
April 30, 2007

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
April 30, 2007

36. Solving the Dual Problem Using Stochastic Approximation
Power
constraint
multiplier
search
Subgradient
approximates
Averaging
time constant
Multiplier
iterates
Step
sizes
Subgradient
Averaging
Projection
Subgradients
Rate
constraint
multiplier
vector
search
Projected subgradient iterations across time with subgradient averaging
- Proved convergence to optimal multipliers with probability one
April 30, 2007

37. Subgradient Approximates
“Instantaneous multi-level waterfilling with max-dual user selection”
April 30, 2007

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

39. Two-User Capacity Region
OFDMA Parameters (3GPP-LTE)
1 = (0.1 increments)
2 = 1-1
April 30, 2007

40. Evolution of the Iterates for 1=0.1 and 2 = 0.9
User Rates
Rate constraint
Multipliers 
Power
Power constraint
Multipliers l
April 30, 2007

41. Summary of the Resource Allocation Algorithms
Initialization Complexity
Per-symbol Complexity
Relative Gap Order of Magnitude
Sum-Rate at w=[.5,.5], SNR=5 dB
WS Cont. Rates Perfect CSI – Ergodic
O(INM)
O(MK)
10-6
2.40
WS Cont. Rates Perfect CSI – Inst. -
O(IMK)
10-8
2.39
WS Disc. Rates Perfect CSI – Ergodic
O(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 (Ip+Ic))
2.37
WS Disc. Rates Partial CSI
O(MK(I+L))
1.09
Prop. Cont. Rates Perfect CSI with CDI - Ergodic
O(INM2)
Prop. Cont. Rates Perfect CSI without CDI - Ergodic
April 30, 2007