power control for multi-carrier communications

Power Control for Multi-carrier Communications

Jianwei Huang

Princeton University

Joint work with R. Berry, M. Honig, M. Chiang, R. Cendrillon, M. Moonen

Sponsors: NSF, Motorola, Alcatel

Yale Seminar, May 2006

J. Huang (Princeton Univ.) Multi-carrier Power Control May 2006 1 / 48

Multi-carrier Communication Systems

Split transmit bandwidth into many narrow parallel carriers.

Robust to ISI (Inter-Symbol-Interference) due to multi-path fading.

Flexible resource allocation leads to high spectrum efficiency.

Core technology of: Wi-Fi, WiMAX, DAB/DVB, UWB, DSL, etc.


Resource Allocation in Multi-carrier Systems

Resource allocation is challenging.

Simultaneous transmissions over the same carriers lead to interference.

Optimization problem is typically tightly coupled and non-convex.

Objective: design distributed and optimal resource allocationalgorithms to achieve maximum system performance.


Network Models

I: Wireless Ad Hoc Network

CPCPCPCP

COCOCOCO

RT1RT1RT1RT1

RT2RT2RT2RT2

RT3RT3RT3RT3

CPCPCPCP

CPCPCPCP

CPCPCPCP5 km

4 km

3.5 km

3 km

2 km

3 km

4 km

II: Wireline DSL Network


Part I: Wireless Ad Hoc Network

J. Huang, R. Berry and M. Honig, “Distributed InterferenceCompensation for Wireless Networks,” IEEE JSAC, May 2006


Wireless Power Control

Well studied in CDMA cellular systems with fixed SINR targets:[Foschini, Miljanic’93], [Yates’95], etc.

We focus on multi-carrier ad hoc networks and rate adaptive users.I One motivation: dynamic spectrum sharing of multiple radio bands.

Related work: [Chiang’05], [Xi and Yeh’05], etc.I Single carrier network.I Power control with small stepsizes.


Network Model

T1

T2

T3

R1

R2

R3

h1

11 h2

11

h1

12

h2

12

Single-hop transmissions.

Each user is a transmitter/receiver pair.

Transmit over several parallel carriers.I Interference among users in each carrier.

First study the special case of single carrier.

Then consider the generalization for the multi-carrier case.


Single Carrier Model

2

σM

22h

σ212h

11h

σ1

1pTransmitters Receivers

21h

Mp

p

A set of N = {1, ..., n} users.

For each user n ∈ N :I Power constraint: pn ∈ [Pmin

n ,Pmaxn ].

I Received SINR (signal-to-interference plus noise ratio):

γn =hn,npn

σn +∑

m 6=n hn,mpm.

I Utility function Un(γn): increasing, differentiable, strictly concave.


Network Utility Maximization (NUM) Problem

Problem: 1-SC

max{Pmin

n ≤pn≤Pmaxn ,∀n}

∑n

Un(γn).

Technical Challenges:I Coupled across users due to interferences.I Could be non-convex in power.I Want efficient and distributed algorithm, with limited information

exchange and fast convergence.


Benchmark - No Information Exchange

Each user picks power to maximize its own utility, given currentinterference and channel gain.

Results in pn = Pmaxn for all n.

I Can be far from optimal.

We propose algorithm with limited information exchange.I Have nice interpretation as distributed Pigovian taxation.I Analyze its behavior using supermodular game theory.


ADP Algorithm: Asynchronous Distributed Pricing

Price Announcing: user n announces “price” (per unit interference):

πn =

∣∣∣∣∂Un(γn)

∂In

∣∣∣∣ = ∂Un(γn)

∂γn

γ2n

pnhn,n.

Power Updating: user n updates power pn to maximize surplus:

Sn = Un(γn)− pn

∑m 6=n

πmhm,n.

Repeat two phases asynchronously across users.

Scalable and distributed: only need to announce single price, andknow limited channel gains (hm,n).


ADP Algorithm

Interpretation of prices: Pigovian taxationI Tax to correct the negative social side-effects of an activity.I Improve social welfare.

ADP algorithm: distributed discovery of Pigovian taxesI When does it converge?I What does it converge to?I Will it solve Problem 1-SC?I How fast does it converge?


Convergence

Depends on the utility functions.

Coefficient of relative Risk Aversion (CRA) of U(γ):

CRA(γ) = −γU ′′(γ)

U ′(γ).

I larger CRA ⇒ “more concave” U.

Theorem: If for all user n:

(a) Pminn > 0, and

(b) CRA(γn) ∈ [1, 2] for all feasible γn;

then there is a unique optimal solution of Problem 1-SC, and theADP algorithm globally converges to it.

I E.g. condition (b) is always satisfied with log utilities: θn log(γn).I Proof: relating this algorithm to a fictitious supermodular game.


Relationship Summary

Problem 1-SC

ADP AlgorithmFictitious Game(Supermodular)



KKT Points

Fixed PointsNash Equilibria

Problem 1-SC




Price/Power UpdatesBest Response Updates

KKT Points


Problem 1-SC




Multiple KKT Points

Multiple Nash Equilibria / Conditional Convergence Multiple Fixed Points / Conditional Convergence


KKT Points


Problem 1-SC




Unique Nash Equilibrium / Global Convergence Unique Fixed Point / Global Convergence

Unique Optimal Solution

Multiple KKT Points

Multiple Nash Equilibria / Conditional Convergence Multiple Fixed Points / Conditional Convergence


KKT Points


Problem 1-SC



Supermodular Games

A class of games with strategic complementariesI Strategy sets are compact subsets of R; and each player’s pay-off Sn

has increasing differences:

∂2Sn

∂xn∂xm> 0,∀n,m.

Key properties:

(1) A Nash Equilibrium (N.E.) exists.(2) If the N.E. is unique, then the asynchronous best response updates will

globally converge to it.


Convergence Results

Construction of the fictitious game

I Split each user in network into two fictitious players in the game.I Choose players’ payoffs such that best response updates correspond to

the power/price updates in ADP.

Proof: Condition CRA(γn) ∈ [1, 2] guaranteesI The fictitious game is supermodular.I The Problem 1-SC has strictly concave objective function (under log

change of variable [Chiang’05]), with convex feasible set.I Thus there is a unique global optimal solution/fixed point/N.E.


Convergence Speed

0 10 200

0.5

1

Pow

er

ADP Algorithm

200 400 6000

0.5

1

Pow

er

Gradient−based Algorithm

5 10 15 200

20

40

60

80

Iterations

Pric

e

200 400 6000

20

40

60

80

Iterations

Pric

e

Gradient-based algorithm is based on [Chiang’05], 10 users, log utilities


Multi-carrier Model

Problem: 1-MC

max{pn∈Pn,∀n}

∑n

∑k

Ukn (γk

n ).

Assume each user can transmit over K orthogonal channels.

Received SINR in channel k for user n

γkn =

hkn,np

kn

σkn +

∑m 6=n hk

n,mpkm

Can allocate power across channels subject to total power constraint:

Pn :=

{pkn ≥ Pmin

n ,∑k

pkn ≤ Pmax

n

}.


DADP: Dual ADP Algorithm

Solves the dual of Problem 1-MC.

User n relaxes the total power constraint by using a dual price µn.

Primal Updates: solve one subproblem per carrierI Under fixed dual prices.I Each user n announces πk

n as before.I Each user n chooses pk

n to maximize

Skn = Uk

n (γkn )− pk

n

∑m 6=n

hm,nπkm + µn

.

Dual Iterations: optimize the dual function using subgradientinformation.

µn(t) =

[µn(t

−) + κ

(∑k

pkn − Pmax

n

)]+

.


Convergence

Theorem: The DADP algorithm globally and geometrically convergesto the unique optimal solution of Problem 1-MC.

I Under similar restrictions in single carrier case.I With small constant stepsize κ.

Proof:

I Need to show the Lipschitz continuity and strong convexity of thegradient of dual function.

I Separation of time-scales assumption: Primal Updates convergebetween any two adjacent Dual Iterations.


Simulation Results

0 5 10 15 2010

−4

10−3

10−2

10−1

Primal Updates

Max 1 Primal Update / Dual IterationMax 3 Primal Updates / Dual IterationMax 5 Primal Updates / Dual IterationMax 7 Primal Updates / Dual Iteration

(Uto

t*

−U

tot)/

Uto

t*

16 carriers, 50 users, log utilities


Summary

Consider power control in multi-carrier wireless ad hoc networks.

Propose ADP (Asynchronous Distributed Pricing) algorithmswith interpretation of distributed Pigovian taxation.

I Achieve optimal solution with limited information exchange.

Supermodular game theory is the key:I Convergence of fast power updates without small stepsizes.I Analysis of nonconvex problems.

Extend to multi-carrier caseI Primal-dual updates.I Difference kind of pricing.


Part II: Wireline DSL Network

J. Huang, R. Cendrillon, M. Chiang, and M. Moonen, “Autonomousspectrum balancing (ASB) for digital subscriber lines,” to appear inIEEE ISIT, July 2006

R. Cendrillon, J. Huang and M. Chiang, “Autonomous SpectrumBalancing (ASB) for Asynchronous DSL,” submitted to IEEEGlobeCom, 2006


Wireless vs. DSL

Wireless DSL

Time Varying Channel Time-invariant Channel

Mobility Typical Topology

Need Message Passing No Message Passing

Pricing Reference Line


Digital Subscriber Line

Convert telephone lines into broadband communication media.

Core technology: Discrete Multi-Tone (Cioffi, early 90’s).

Utilize the spectrum unused by voice transmissions and divide intolarge number of orthogonal carriers/tones.

Frequency (KHz)0 3.4

Voice DSL


Digital Subscriber Line

Most ubiquitous and cost-effective access network.

Current ADSL standard:I Utilize up to 1MHz bandwidth.I Provide 1.5− 9Mbps download speed over a distance of 2.7− 5.5Km.

Provide a variety of services:I High-speed Internet access.I VoIP.I IPTV (AT&T, Jan. 2006).I Video-on-demand (AT&T, later 2006 summer)

A holistic network optimization may significantly improveperformance.


FAST Copper Project

Joint NSF project among Princeton (Chiang), Stanford (Cioffi) andFraser Research (Fraser).

Collabration with AT&T.

Aim at providing DSL broadband service at 100Mbps by jointoptimization over Frequency, Amplitude, Space and Time.

Today focus on the Frequency aspect: spectrum management.I Performance bottleneck: crosstalks (interferences) among lines.I Current practice: static spectrum management.I Dynamic spectrum management is needed.


Network Model

crosstalk

TX

TX RX

RX Customer

Customer

CO(Central Office)

RT(Remote Terminal)

Each user is a telephone line (transmitter/receiver pair).

Transmits over multiple carriers/tones.

Mixed CO/RT deploymentI Very typical in the United States.I CO (central office) can not reach all customers.I RT (remote terminal) is deployed to increase DSL footprint.


Channel Characteristics

Time-invariant channels.

Topology dependent: channel gain decreases with distance.

Frequency dependent:I Direct channel gain decreases with frequency.I Crosstalk channel gain increases with frequency.The Bandwidth Question

Frequency Response of twisted pairs

No well defined band limits (different from wireless) Bandwidth varies greatly with loop lengthTo use higher BW footprint size must shrinkDirect Channel Attenuation

Comparison of NEXT & FEXT

Crosstalk Channel Attenuation

c© M. Tsatsanis @ Aktino


Mixed CO/RT Case

crosstalk

TX

TX RX

RX Customer

Customer

CO(Central Office)

RT(Remote Terminal)

RT generates strong interference to CO line on high frequencies.

CO generates little interference to RT line on all frequencies.

Major performance bottleneck.

Typical test case for all spectrum management algorithms.


Spectrum Management Problem

Characterize the Pareto optimal boundary of rate region.

Problem: 2A

maximize{pn∈Pn}n

R1

subject to Rn ≥ Rtargetn ,∀n > 1.

I User n’s achievable rate Rn =∑

k log(1 +

pknP

m 6=n αkn,mpk

m+σkn

).

I Total power constraint: Pn ={pk

n ≥ 0,∀k,∑

k pkn ≤ Pmax

n

}.

Technical Difficulty:I Non-convex and tightly coupled problem.I No explicit message passing among users.I We want to design distributed, low complexity algorithm with no

message passing and near optimal performance.


Dynamic Spectrum Management (DSM)

State-of-art DSM algorithms:I IW: Iterative Water-filling [Yu, Ginis, Cioffi’02]I OSB: Optimal Spectrum Balancing [Cendrillon et al.’04]I ISB: Iterative Spectrum Balancing [Liu, Yu’05] [Cendrillon, Moonen’05]I ASB: Autonomous Spectrum Balancing [Huang et al.’06]

Algorithm Operation Complexity Performance

IW Autonomous O (KN) Suboptimal

OSB Centralized O(KeN

)Optimal

ISB Centralized O(KN2

)Near Optimal

ASB Autonomous O (KN) Near Optimal

K : number of carriers N: number of users


Reference Line Concept

Provide partial network information.

Reference Line:I A virtual line representative of typical CO line in the network.I Fixed transmission power and noise on all tones.I Fixed crosstalk channels with actual lines.I All parameters are obtained through measurement and publicly known.

Each user maximizes reference line rate, subject to its rate targetconstraint.

I Protecting the reference line means protecting the worst victim, thuseffectively protecting all other lines.


Reference Line

CP

RT

RT

RT

CP

CO CP

CP


Reference Line

Actual Line

Reference Line

CO

CPCO

RT CP

RT

RT

CP

CP

CP


ASB Algorithm

Under fixed interferences, each user n solves the following problem:

Problem: 2B

maximize{pn∈Pn}

R refn

subject to Rn ≥ Rtargetn

whereR ref

n =∑k

log

(1 +

pk,ref

αk,refn pk

n + σk,ref

)

I Only local information is needed.I The only interference to the reference line is from user n.I User 1’s target rate is set to ∞.

Iterate through users until convergence.


Solve Problem 2B

Still non-convex, but can be solved by dual decompositions.I Duality gap is asymptotically zero with large K [Cendrillon et al.’04].

Relax the rate constraint with wn.

maximize{pn∈Pn}

R refn + wnRn (Step1)

I Adjust wn to achieve target rate constraint R targetn .

Relax the total power constraint with λn.

maximize{pk

n≥0}Rk,ref

n + wnRkn − λnp

kn ,∀k (Step2)

I Adjust λn to achieve∑

k pkn = Pmax

n .I Look at the first order condition and solve a cubic equation.


ASB Algorithm

repeatfor each user n = 1, ...,N

repeatfor each carrier k = 1, ...,K , find

pkn = optimal solution of Step2.

λn =[λn + ελ

(∑k pk

n − Pmaxn

)]+.

wn =[wn − εw

(∑k Rk

n − Rtargetn

)]+.

until convergenceend

until convergence


Simulation Setup

4 ADSL lines.

Mixed CO/RT deployment.

Users 3 and 4 have fixed target rates 2Mbps.

Find the rate region in terms of users 1 and 2’s achievable rates.

COUser 15Km

CP

User 4

User 3

User 24Km

3.5Km

3Km

2Km

3Km

4Km

CP

RT CP

RT

CPRT


Rate Region

0 1 2 3 4 5 6 7 80.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

User 2 Rate (Mbps)

Use

r 1 R

ate

(Mbp

s)

Optimal (OSB)

Best Available Today (IW)

ASB


High SINR Approximation

Assume reference line operates in high SINR regime on active tones.

Rk,refn ≈ log

(pk,ref

σk,ref

)− αk,ref

n

σk,refpkn .

User’s optimal power is frequency-selective water-filling.

pk∗n =

wn

λn + αk,refn /σk,ref

−∑m 6=n

αkn,mpk

m − σkn

+

.


Convergence of ASB

Theorem: ASB algorithm (under high SINR approximation) globallyconverges to the unique fixed point if

maxn,m,k

αkn,m <

1

N − 1.

Independent of the reference line parameters.

Proof: contraction mapping.I Recover the convergence of iterative water-filling as a special case.


Summary

Consider dynamic spectrum management in DSL networks.

Identify the mixed CO/RT case as the major performance bottleneck.

Use reference line to represent partial network information.

Propose ASB (Autonomous Spectrum Balancing) algorithmI Fully autonomous.I Linear complexity in K and N.I Achieves near optimal performance.

Consider high SINR approximation on the reference line.I Close-form optimal solution.I Provable convergence.


Summary

CPCPCPCP

COCOCOCO

RT1RT1RT1RT1

RT2RT2RT2RT2

RT3RT3RT3RT3

CPCPCPCP

CPCPCPCP

CPCPCPCP5 km

4 km

3.5 km

3 km

2 km

3 km

4 km

Part I IIMotivation Ad Hoc DSL

Problem Nonconvex/Convex Nonconvex

Algorithm ADP ASB

Methodology Supermodular Game Theory Reference Line Approximation

Message Passing No Message PassingProperties Optimal Near Optimal

Low Complexity Low ComplexityFast Convergence Provable Convergence


More publications can be found atwww.princeton.edu/∼jianweih


PADP: Primal ADP Algorithm

Solve Problem 1-MC directly.

Price Announcing: user n announces prices πn =(π1

n, ..., πKn

)πk

n =

∣∣∣∣∂Ukn (γk

n )

∂I kn

∣∣∣∣ .Power Updating: user n chooses powers pn =

(p1n, ..., p

Kn

)∈ Pn

to maximize surplus

Sn =∑k

Ukn (γk

n )− pkn

∑m 6=n

hkm,nπ

km

Proof of optimality and convergence:I Supermodular game theory.I Contraction mapping.


Limited Price Information Exchange

ThresholdT1

T2

T3

T4

R1

R2

R3

R4

π1

π1

π2

π3

Previously we assume that each user can decode all the prices.

In practice, users (receivers) may only decode price messages withinthreshold distance.


ADP with Limited Pricing

0 0.2 0.4 0.6 0.8 1 1.2 1.40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Users Per Square Meters

Nor

mal

ized

Util

ity

Maximum Power

Threshold = 0.5m

Threshold = 1m

Threshold = 1.5m

Threshold = 2m

Full Price ADP

utility log(1 + γi ), 10m × 10m area


Comparison of ADP with 802.11 (RTS/CTS)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Users / m2

Nor

mal

ized

Util

ityFull Information ADP

Limited Information ADP

802.11 (RTS/CTS)

Maximum Power

rate utility log(1 + γi ), 10m × 10m area


Comparison of ADP with 802.11 (RTS/CTS)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Users / m2

Nor

mal

ized

Util

ity Full Information ADP

Limited Information ADP

802.11 (RTS/CTS)

Maximum Power

quantized utility log(1 + γi ) ({0, 5, 10, 15, 20}bits/Hz), 10m × 10m area


power control for multi-carrier communications

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