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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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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 No Yes No No* No
Discrete Rates No No No Yes No
User 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-
Wireless Networking and Communications Group
Summary of ContributionsPrevious Research Our Contributions
Fo
rmu
latio
n
Instantaneous rate•Unable to exploit time-varying wireless channels
Ergodic rate•Exploits time-varying nature of the wireless channel
So
lutio
n
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
Assu
mp
tion
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
Fo
rmu
latio
n
Instantaneous rate•Unable to exploit time-varying wireless channels
Ergodic rate•Exploits time-varying nature of the wireless channel
So
lutio
n
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
Fo
rmu
latio
n
Instantaneous rate•Unable to exploit time-varying wireless channels
Ergodic rate (continuous and discrete)•Exploits time-varying nature of the wireless channel
April 30, 2007-7-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Dual Optimization Framework“Max-dual user selection”
Dual problem:
“Multi-level waterfilling”
Duality gap
April 30, 2007-12-
Wireless Networking and Communications Group
*Optimal Subcarrier and Power Allocation“Multi-level waterfilling” “Max-dual user selection”
Ma
rgin
al
du
al
Po
we
r
*Independently discovered by [Yu, Wang, & Giannakis, submitted] and [Seong, Mehsini, & Cioffi, 2006] for instantaneous rate case
April 30, 2007-13-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Dual Optimization Framework
“Multi-level fading inversion”
wm=1,=1
“Slope-interval selection”
April 30, 2007-17-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Simulation Results
OFDMA Parameters (3GPP-LTE) Channel Simulation
April 30, 2007-19-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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
Relative
Gap
(x10-4)
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-
Wireless Networking and Communications Group
Sum Rate Versus Number of UsersContinuous Rate Discrete Rate
76 used subcarriers
April 30, 2007-22-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
“Multi-level waterfilling on conditional expected CNR”
Dual Optimization Framework
1-D Integral (> 50 iterations)
1-D Root-finding (<10 iterations)Computationalbottleneck
April 30, 2007-26-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Power Allocation Functions
Multilevel Fading Inversion (MFI):
Predicted CNR:
Optimal Power Allocation:
April 30, 2007-30-
Wireless Networking and Communications Group
Dual Optimization Framework
• Bottleneck: computing rate/power functions• Rate/power functions independent of multiplier
– Can be computed and stored before running search
April 30, 2007-31-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Simulation Parameters (3GPP-LTE)
0 10 20 30 40 50 60 70 80-5
0
5
10
15
20
Subcarrier Index
CN
R (
dB)
User 1 - Perfect Channel
User 2 - Perfect ChannelUser 1 - Predicted Channel
User 2 - Predicted Channel
Channel Snapshot
April 30, 2007-33-
Wireless Networking and Communications Group
Two-User Continuous Rate RegionNo. of line search
iterations (I)
5 dB 8.599
10 dB 8.501
15 dB 8.686
Relative
Gap
(x10-4)
5 dB 0.084
10 dB 0.057
15 dB 0.041
Complexity (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-
Wireless Networking and Communications Group
Two-User Discrete Rate RegionNo. of line search
iterations (I)
5 dB 21.33
10 dB 21.12
15 dB 21.15
Relative
Gap
(x10-4)
5 dB 71.48
10 dB 7.707
15 dB 5.662
Complexity (MK(I+L))M – No. of usersK – No. of subcarriers;I– No. of line search iterations
L – No. of discrete rate levels
No. of rate levels (L) = 4BER constraint = 10-3
April 30, 2007-35-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
• 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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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
FormulationErgodic Rates No No* No Yes Yes Yes
Discrete Rates No Yes No Yes Yes Yes
User prioritization Yes Yes Yes Yes Yes Yes
Solution (algorithm)
Practically optimal No No Yes Yes Yes Yes
Linear complexity No No Yes** Yes Yes Yes
Assumption (channel knowledge)
Imperfect CSI No No No No Yes Yes
Do 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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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.
April 30, 2007-42-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Notation Glossary
April 30, 2007-44-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
PDF of Discrete Rate Dual
• Derive the pdf of
April 30, 2007-50-
Wireless Networking and Communications Group
Performance Assessment - Duality Gap
April 30, 2007-51-
Wireless Networking and Communications Group
Duality Gap Illustration
M=2K=4
April 30, 2007-52-
Wireless Networking and Communications Group
Sum Power Discontinuity
M=2K=4
April 30, 2007-53-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Closed-form Average Rate and Power
Power allocation function:
Average rate function:
Marcum-Q function
April 30, 2007-55-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Dual Optimization Framework
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
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-
Wireless Networking and Communications Group
Subgradient Approximates
“Instantaneous multi-level waterfilling with max-dual user selection”
April 30, 2007-62-
Wireless Networking and Communications Group
Optimal Resource Allocation- Ergodic Proportional Rate without CDI
Weighted-sum,Discrete Rateand Partial CSIare specialcases of this algorithm
April 30, 2007-63-
Wireless Networking and Communications Group
Two-User Capacity Region
OFDMA Parameters (3GPP-LTE)
1 = 0.1-0.9 (0.1 increments)2 = 1-1
April 30, 2007-64-
Wireless Networking and Communications Group
Evolution of the Iterates for 1=0.1 and 2 = 0.9U
ser
Rat
es
Rat
e
cons
trai
ntM
ultip
liers
P
ower
Pow
er
cons
trai
ntM
ultip
liers
April 30, 2007-65-
Wireless Networking and Communications Group
Summary of the Resource Allocation Algorithms
Algorithm 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 (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