Wireless Networking and Communications Group
A Unified Framework for Optimal Resource Allocation
in Multiuser Multicarrier Wireless Systems
Ian C. Wong
Supervisor:
Prof. Brian L. Evans
Committee:
Prof. Jeffrey G. Andrews
Prof. Gustavo de Veciana
Prof. Robert W. Heath, Jr.
Prof. David P. Morton
Prof. Edward J. Powers, Jr.
Wireless Networking and Communications Group
Outline
•
•
•
•
••
Background
– OFDMA Resource
Allocation
Weighted-Sum
Rate
with Perfect Channel State Information
– Related Work
Weighted-Sum
Rate with Partial Channel State Information
– Summary of Contributions
Rate Maximization with Proportional Rate Constraints
– System Model
Conclusion
Weighted-Sum Rate with Perfect Channel State Information
• Weighted-Sum Rate with Partial Channel State Information
• Rate Maximization with Proportional Rate Constraints
• Conclusion
-2April 30, 2007
Wireless Networking and Communications Group
Orthogonal Frequency Division
Multiple Access (OFDMA)
• 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
User 1
...
frequency
Base Station
(Subcarrier and power allocation)
-3April 30, 2007
User M
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
-4April 30, 2007
Wireless Networking and Communications Group
Related Work
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]
Ergodic Rates
No
Yes
No
No*
No
Discrete Rates
No
No
No
Yes
No
User prioritization
No
No
Yes
Yes
Yes
Practically optimal
No
Yes
No
No
Yes**
Linear complexity
No
No
No
No
Yes***
Imperfect CSI
No
No
No
No
No
Do not require CDI
Yes
No
Yes
Yes
Yes
Method
Criteria
Formulation
Solution
(algorithm)
Assumption
(channel
knowledge)
* 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
-5April 30, 2007
Wireless Networking and Communications Group
Summary of Contributions
Previous Research
Our Contributions
Formulation Solution
Instantaneous rate
•Unable to exploit time-varying
wireless channels
Ergodic rate (continuous and discrete)
•Exploits time-varying nature of the wireless
channel
Constraint-relaxation
•One large constrained convex
optimization problem
•Resort to sub-optimal heuristics
((O(MK
O(MK2) complexity)
Dual optimization
•Multiple small optimization problems
w/closed-form solutions
•Practically optimal with O(MK)
O(MK) complexity
•Adaptive algorithms also proposed
Assumption
Perfect channel knowledge
•Unrealistic due to channel
estimation errors and delay
Imperfect channel knowledge
•Allocate based on statistics of channel
estimation/prediction errors
-6April 30, 2007
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
Diagonal gain matrix
Diagonal channel matrix
-7April 30, 2007
Noise vector
Wireless Networking and Communications Group
Statistical Wireless Channel Model
• Time-domain channel
• Frequency-domain channel
– Stationary and ergodic
– Complex normal and independent
across taps i and users m
– Stationary and ergodic
– Complex normal with correlated
channel gains across subcarriers
-8April 30, 2007
Wireless Networking and Communications Group
Outline
• 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
-9April 30, 2007
Wireless Networking and Communications Group
Ergodic Continuous Rate Maximization:
Perfect CSI and CDI [Wong & Evans, 2007a]
Anticipative and infinite
dimensional stochastic program
Constant
weights
Powers to
determine
Channel-to-noise
ratio (CNR)
Average power
constraint
Constant user weights:
Space of feasible power allocation functions:
Subcarrier capacity:
-10April 30, 2007
Wireless Networking and Communications Group
Dual Optimization Framework
“Max-dual user selection”
Dual problem:
“Multi-level waterfilling”
Duality gap
-11April 30, 2007
Wireless Networking and Communications Group
*Optimal Subcarrier and Power Allocation
“Max-dual user selection”
Power
Marginal dual
“Multi-level waterfilling”
*Independently discovered by [Yu, Wang, & Giannakis, submitted] and [Seong, Mehsini, & Cioffi, 2006] for instantaneous rate case
-12April 30, 2007
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)
-13April 30, 2007
Wireless Networking and Communications Group
Optimal Resource Allocation –
Ergodic Capacity with Perfect CSI
Initialization
PDF of CNR
O(INM)
Runtime
CNR Realization
O(MK)
O(MK)
M – No. of users
K – No. of subcarriers
I – No. of line-search iterations
N – No. of function evaluations
for integration
O(K)
-14April 30, 2007
Wireless Networking and Communications Group
Ergodic Discrete Rate Maximization:
Perfect CSI and CDI [Wong & Evans, submitted]
Anticipative
and infinite
dimensional
stochastic
program
Discrete Rate Function:
Uncoded
BER = 10-3
-15April 30, 2007
Wireless Networking and Communications Group
Dual Optimization Framework
“Slope-interval selection”
“Multi-level fading inversion”
wm=1,=1
-16April 30, 2007
Wireless Networking and Communications Group
Optimal Resource Allocation –
Ergodic Discrete Rate with Perfect CSI
PDF of CNR
Initialization
O(INML)
Runtime
CNR Realization
O(MKlog(L))
O(MK)
O(K)
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
-17April 30, 2007
Wireless Networking and Communications Group
Simulation Results
OFDMA Parameters (3GPP-LTE)
-18April 30, 2007
Channel Simulation
Wireless Networking and Communications Group
Two-User Continuous Rate Region
76 used subcarriers
Erg. Rates
Algorithm
Inst. Rates
Algorithm
5 dB
47.91
-
10 dB
50.09
-
15 dB
53.73
-
5 dB
8.091
8.344
10 dB
7.727
8.333
15 dB
7.936
8.539
5 dB
7.936
.0251
10 dB
5.462
.0226
15 dB
5.444
.0159
SNR
No. of
function
evaluations
(N)
No. of
Iterations
(I)
Relative
Gap
(x10-6)
-19April 30, 2007
Wireless Networking and Communications Group
Two-User Discrete Rate Region
76 used subcarriers
SNR
No. of
function
evaluations
(N)
No. of
Iterations
(I)
Relative
Gap
(x10-4)
-20April 30, 2007
Erg. Rates
Algorithm
Inst. Rates
Algorithm
5 dB
47.91
-
10 dB
50.09
-
15 dB
53.73
-
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
Wireless Networking and Communications Group
Sum Rate Versus Number of Users
76 used subcarriers
Continuous Rate
Discrete Rate
-21April 30, 2007
Wireless Networking and Communications Group
Outline
• 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
-22April 30, 2007
Wireless Networking and Communications Group
Partial Channel State Information Model
• Stationary and ergodic channel gains
• MMSE channel prediction
MMSE Channel
Prediction
Conditional PDF of channel-to-noise ratio (CNR) – Non-central Chi-squared
CNR:
Normalized error variance:
-23April 30, 2007
Wireless Networking and Communications Group
Continuous Rate Maximization:
Partial CSI with Perfect CDI [Wong & Evans, submitted]
Nonlinear
integer
stochastic
program
• Maximize conditional expectation given the
estimated CNR
– Power allocation a function of predicted CNR
• Instantaneous power constraint
– Parametric analysis is not required
• a
-24April 30, 2007
Wireless Networking and Communications Group
Dual Optimization Framework
“Multi-level waterfilling on conditional expected CNR”
1-D Integral (> 50 iterations)
Computational
bottleneck
1-D Root-finding (<10 iterations)
-25April 30, 2007
Wireless Networking and Communications Group
Power Allocation Function Approximation
• Use Gamma distribution
to approximate the Noncentral Chi-squared
distribution [Stüber, 2002]
• Approximately 300
times faster than
numerical quadrature
(tic-toc in Matlab)
-26April 30, 2007
Wireless Networking and Communications Group
Optimal Resource Allocation –
Ergodic Capacity given Partial CSI
Conditional PDF
Runtime
O(MKI (Ip+Ic))
Predicted CNR
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
-27April 30, 2007
O(K)
Wireless Networking and Communications Group
Discrete Rate Maximization:
Partial CSI with Perfect CDI [Wong & Evans, 2007b]
Nonlinear
integer
stochastic
program
Rate levels:
Derived
Average rate function given partial CSI: closed-form
expressions
Feasible set:
Power allocation function given partial CSI:
-28April 30, 2007
Wireless Networking and Communications Group
Power Allocation Functions
Optimal Power Allocation:
Multilevel Fading Inversion
(MFI):
Predicted CNR:
-29April 30, 2007
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
-30April 30, 2007
Wireless Networking and Communications Group
Optimal Resource Allocation –
Ergodic Discrete Rate given Partial CSI
Conditional PDF
Runtime
O(MK(I+L))
Predicted CNR
O(1)
O(1)
O(K)
M – No. of users
K – No. of subcarriers
L – No. of rate levels
I – No. of line-search iterations
-31April 30, 2007
Wireless Networking and Communications Group
Simulation Parameters (3GPP-LTE)
Channel Snapshot
20
15
CNR (dB)
10
5
User 1 - Perfect Channel
User 2 - Perfect Channel
User 1 - Predicted Channel
User 2 - Predicted Channel
0
-5
0
10
-32April 30, 2007
20
30
40
50
Subcarrier Index
60
70
80
Wireless Networking and Communications Group
Two-User Continuous Rate Region
No. of line
search
iterations (I)
Relative
Gap
(x10-4)
Complexity
5 dB
8.599
10 dB
8.501
15 dB
8.686
5 dB
0.084
10 dB
0.057
15 dB
0.041
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
-33April 30, 2007
Wireless Networking and Communications Group
Two-User Discrete Rate Region
No. of rate levels (L) = 4
BER constraint = 10-3
No. of line
search
iterations (I)
Relative
Gap
(x10-4)
Complexity
5 dB
21.33
10 dB
21.12
15 dB
21.15
5 dB
71.48
10 dB
7.707
15 dB
5.662
O(MK(I+L))
M – No. of users
K – No. of subcarriers;
I– No. of line search iterations
L – No. of discrete rate levels
-34April 30, 2007
Wireless Networking and Communications Group
No. of rate levels (L) = 4
BER constraint = 10-3
Average BER Comparison
Per-subcarrier Prediction Error Variance
BER
Per-subcarrier Average BER
Subcarrier Index
-35April 30, 2007
Wireless Networking and Communications Group
Outline
•
•
•
•
•
Background
Weighted-Sum Rate with Perfect Channel State Information
Weighted-Sum Rate with Partial Channel State Information
Rate Maximization with Proportional Rate Constraints
Conclusion
-36April 30, 2007
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
-37April 30, 2007
Wireless Networking and Communications Group
Comparison with Previous Work
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
Ergodic Rates
No
No*
No
Yes
Yes
Yes
Discrete Rates
No
Yes
No
Yes
Yes
Yes
User prioritization
Yes
Yes
Yes
Yes
Yes
Yes
Practically optimal
No
No
Yes
Yes
Yes
Yes
Linear complexity
No
No
Yes**
Yes
Yes
Yes
Imperfect CSI
No
No
No
No
Yes
Yes
Do not require CDI
Yes
Yes
Yes
No
No
Yes
Method
Criteria
Formulation
Solution
(algorithm)
Assumption
(channel
knowledge)
* 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
-38April 30, 2007
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
-39April 30, 2007
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
-40April 30, 2007
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 2630, 2007 Washington, DC USA, submitted.
-41April 30, 2007
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
-42April 30, 2007
Wireless Networking and Communications Group
Notation Glossary
-43April 30, 2007
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]
-44April 30, 2007
Wireless Networking and Communications Group
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
-45April 30, 2007
Wireless Networking and Communications Group
C-Rate P-CSI Dual Objective Derivation
Lagrangian:
Dual objective
Linearity of E[¢]
Separability of
objective
Power a function
of RV realization
m,k not independent but identically distributed across k
-46April 30, 2007
Exclusive
subcarrier
assignment
Wireless Networking and Communications Group
Optimal Resource Allocation –
Instantaneous Capacity with Perfect CSI
Runtime
CNR Realization
O(IMK)
O(1)
O(1)
M – No. of users
K – No. of subcarriers
I – No. of line-search iterations
N – No. of function evaluations
for integration
O(K)
-47April 30, 2007
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:
-48April 30, 2007
Wireless Networking and Communications Group
PDF of Discrete Rate Dual
• Derive the pdf of
-49April 30, 2007
Wireless Networking and Communications Group
Performance Assessment - Duality Gap
-50April 30, 2007
Wireless Networking and Communications Group
Duality Gap Illustration
M=2
K=4
-51April 30, 2007
Wireless Networking and Communications Group
Sum Power Discontinuity
M=2
K=4
-52April 30, 2007
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
-53April 30, 2007
Wireless Networking and Communications Group
Closed-form Average Rate and Power
Power allocation function:
Average rate function:
Marcum-Q function
-54April 30, 2007
Wireless Networking and Communications Group
Ergodic Sum Rate Maximization with
Proportional Ergodic Rate Constraints
Ergodic Sum
Capacity
Average Power
Constraint
Ergodic Rate for
User m
Developed
adaptive
algorithm
without CDI
Proportionality
Constants
• Allows more definitive prioritization among users
• Traces boundary of capacity region with specified ratio
-55April 30, 2007
Wireless Networking and Communications Group
Dual Optimization Framework
• Reformulated as weighted-sum rate problem
with properly chosen weights
Multiplier for Multiplier for
rate
power
constraint
constraint
“Multi-level waterfilling with
max-dual user selection”
-56April 30, 2007
Wireless Networking and Communications Group
Projected Subgradient Search
Power
constraint
multiplier
search
Rate
constraint
multiplier
vector
search
Multiplier
Projection
iterates
Step
sizes
Subgradients
Derived pdfs for
efficient 1-D Integrals
Per-user ergodic rate:
-57April 30, 2007
Wireless Networking and Communications Group
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)
-58April 30, 2007
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
-59April 30, 2007
Wireless Networking and Communications Group
Solving the Dual Problem Using
Stochastic Approximation
Power
constraint
multiplier
Multiplier
search
iterates
Rate
constraint
multiplier
vector
search
Projection
Step
sizes
Subgradient
Subgradients
Averaging
Averaging
Subgradient
time constant approximates
Projected subgradient iterations across time with subgradient averaging
- Proved convergence to optimal multipliers with probability one
-60April 30, 2007
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Subgradient Approximates
“Instantaneous multi-level waterfilling with max-dual user selection”
-61April 30, 2007
Wireless Networking and Communications Group
Optimal Resource AllocationErgodic Proportional Rate without CDI
Weighted-sum,
Discrete Rate
and Partial CSI
are special
cases of
this algorithm
-62April 30, 2007
Wireless Networking and Communications Group
Two-User Capacity Region
OFDMA Parameters (3GPP-LTE)
1 = 0.1-0.9 (0.1 increments)
2 = 1-1
-63April 30, 2007
Evolution of the Iterates for 1=0.1 and 2 = 0.9
Power constraint
Multipliers 
Power
Rate constraint
Multipliers  User Rates
Wireless Networking and Communications Group
-64April 30, 2007
Wireless Networking and Communications Group
Summary of the Resource Allocation Algorithms
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)) 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NM2)
O(MK)
10-6
2.40
Prop. Cont. Rates Perfect CSI
without CDI - Ergodic
-
O(MK)
-
2.40
Algorithm
Initialization
Complexity
-65April 30, 2007