distributed power control and spectrum sharing in wireless networks
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Distributed Power Control and Spectrum Sharing in Wireless Networks. ECE559VV – Fall07 Course Project Presented by Guanfeng Liang. Outline. Background Power control Spectrum sharing Conclusion. Background. Interference is the key factor that limits the performance of wireless networks - PowerPoint PPT PresentationTRANSCRIPT
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ECE559VV – Fall07 Course Project
Presented by Guanfeng Liang
Distributed Power Control and Spectrum Sharing in Wireless
Networks
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OutlineBackgroundPower controlSpectrum sharingConclusion
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BackgroundInterference is the key factor that limits the
performance of wireless networksTo handle interference, can optimize by
means of Frequency allocation:
Power control:
Or, jointly - spectrum sharing:
f
f
f
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Power ControlN users, M base stations, single channel,
uplinkPj - transmit power of user j
hkj - gain from user j to BS k
zk – variance of independent noise at BS k)()(SIR pp kjj
jikiki
kjjkj up
zph
hp
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General Interference ConstraintsFixed Assignment: BS aj is assigned to user
j
Minimum Power Assignment: each user is assigned to the BS that maximizes its SIR
Limited Diversity: BS’s in Kj are assigned to user j
)()()(
ppp
ja
jFAjjjjaj
j
j uIpup
)(min)()(max
,, p
ppjk
j
k
MPAjjjjkj
k uIpup
)( ,)( , )(
)()(p
p ppp
j
j
Kk jk
jLDjjjKk jkj uIpup
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Standard Interference functionDefinition: Interference function I(p) is
standard if for all p≥0, the following properties are satisfied.Positivity - I(p) ≥0Monotonicity - If p ≥ p’, then I(p) ≥ I(p’).Scalability – For all a>1, aI(p)>I(ap).
IFA, IMPA, ILD are standard.For standard interference functions,
minimized total power can be achieved by updating p(t+1)=I(p(t)) in a distributed fashion, asynchronously. (Yates’95)
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Spectrum Sharing• Power is uniformly allocated across bandwidth
W• Transmission rate is not considered
• What should we do if power is allowed to be allocated unevenly?
• Can “rate” optimality be achieved in a distributed manner?
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SettingsM fixed 1-to-1 user-BS assignmentsNoise profile at each BS: Ni(f)Random Gaussian codebooks – interference
looks like Gaussian noise
i
W
i
W
ij jiji
iiii
Pdffp
dffphfN
fphR
0
0,
,
)( subject to
)()(
)(1log
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Rate RegionRate Region
Pareto Optimal Point
MifpP(f)dfp
dffphfN
fphR
ii
W
i
W
ij jiji
iiii
,...,1for 0)( with and
)()(
)(1log:
0
0,
,
R
MiR,RR
RRRRRR
Mii
iiiM
,...,1for ),,...,~
,...,(~
:),...,(
1
111*
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Optimization ProblemGlobal utility optimization maximization
U(R1,…,RM) reflects the fairness issueSum rate: Usum (R1,…,RM) = R1+…+RM
Proportional fairness: UPF (R1,…,RM) = log(R1)+…+log(RM)
In general, U is component-wise monotonically increasing => optimal allocation must occur on the boundary R*
),...,(subject to
),...,(max
1
1
M
M
RR
RRU
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Examples
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Infinite DimensionTheorem 1:
Any point in the achievable rate region R can be obtained with M power allocations that are piecewise constant in the intervals [0,w1), [w1,w2),…,[w2M-1,W], for some choice of {wi}i=1.
2M-1.
Theorem 2:Let (R1,…,RM) be a Pareto efficient rate vector achieved with power allocations {pi(f)}i=1,…,M. If hi,jhj,i>hi,ihj,j then pi(f)pj(f)=0 for all f [0,W].
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Non-Cooperative ScenariosNon-convex capacity expression -> rate
region not easy to compute
Another approach: view the interference channel as a non-cooperative game among the competing users-> competitive optimal
Assumptions:Selfish usersuser i tries to maximize Ui(Ri) -> maximize Ri
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Gaussian Interference Game(GIG)Each user tries to maximize its own rate,
assuming other users’ power allocation are
known.
Well-known Water-filling power allocation
i
W
i
W
ij jiji
iiii
Pdffp
dffphfN
fphR
0
0 *,
,
)( subject to
)()(
)(1log maximize
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Iterative Water-filling (Yu’02)
)(
)(
1,1
1
fh
fN
)(
)(
2,2
2
fh
fN
22P 11P
1,11,222,22,11 /,/ hhhh
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EquilibriumTheorem 3:
Under a mild condition, the GIG has a competitive equilibrium. The equilibrium is unique, and it can be reached by iterative water-filling.
Nash Equilibrium
MiSs
sssssRssR
ii
MiiiiMi
,...,1, allfor
),...,,,...,(),...,( **1
*1
*1
**1
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Is the Equilibrium Optimal?NO!Example:
h1,1=h2,2=1, h1,2=h2,1=1/4, W=1, N1=N2=1, P1=P2=P
Water-filling -> flat power allocation:
Orthogonal power allocation
PP/PRR as )5log()]41/(1log[21
PPRR as ]21log[)2/1(21
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Repeated GameUtility of user i :
Decision made based on complete history
Advantage: much richer set of N.E., hence have more flexibility in obtaining a fair and efficient resource allocation
)1,0(,)()1(0
t
it
i tRU
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Equilibriums of a Repeated GameFact: frequency-flat power allocations is a N.E. of
the repeated game with AWGN.
Theorem 4:The rate Ri
FS achieved by frequency-flat power spread is the reservation utility of player i in the GIG.
Result: If the desired operating point (R1,…,RM) is component-wise greater than (R1
FS,…,RMFS), there
is no performance loss due to lack of cooperation. (Tse’07)
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Results
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SummaryPerformance optimization of wireless
networks1-D: power = power control
Distributed power control with constant power allocation
2-D: power + frequency = spectrum sharingOne shot GIG – iterative water-fillingRepeated game
3-D: power + frequency + timeCognitive radio
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Thank you and Questions?