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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 Presentation
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ECE559VV – Fall07 Course Project
Presented by Guanfeng Liang
Distributed Power Control and Spectrum Sharing in Wireless
Networks
OutlineBackgroundPower controlSpectrum sharingConclusion
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
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
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
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)
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?
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
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*
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
Examples
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].
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
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
Iterative Water-filling (Yu’02)
)(
)(
1,1
1
fh
fN
)(
)(
2,2
2
fh
fN
22P 11P
1,11,222,22,11 /,/ hhhh
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
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
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
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)
Results
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
Thank you and Questions?