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?