dsl spectrum management - princeton universitychiangm/ele539l12.pdf · wireline communications...

DSL Spectrum Management

Dr. Jianwei Huang

Department of Electrical EngineeringPrinceton University

Guest Lecture of ELE539AMarch 2007

Jianwei Huang (Princeton) DSL Spectrum Management March 2007 1 / 26

Acknowledgements

Collaborations: Raphael Cendrillon, Mung Chiang, Marc Moonen

Sponsorships: Alcatel, NSF


Digitial Subscriber Line (DSL) Networks

Wireline communications networks based telephone copper lines

Cost-effective broadband access network

More than 160 million users world-wide

Speed is the bottleneck

crosstalk

TX

TX RX

RXCO

RT

(Remote Terminal)

(Central Office) Customer

Customer


How DSL Works?

Copper line can support signal transmissions over a large bandwidth

Voice transmission: up to 3.4 KHz

DSL transmissions: up to 30 MHzI Multi-carrier transmissions: Discrete Multitone Modulation

Frequency (KHz)0 3.4

Voice DSL


Network and Channel Model

crosstalk

TX

TX RX

RXCO

RT

(Remote Terminal)


Customer


Mathematical model: multi-user multi-carrier interference channel

Each telephone line is a user (transmitter-receiver pair)

Generate mutual crosstalks over multiple frequency tones


crosstalk

TX

TX RX

RXCO

RT

(Remote Terminal)


Customer


Physical model: mixed CO/RT case

Channel attenuates with distance

Central Office (CO) connect customers who are reasonably close

Remote Terminal (RT) connect customers who are farther away


crosstalk

TX

TX RX

RXCO

RT

(Remote Terminal)


Customer


Frequency-Dependent Channel

Direct channel gain decreases with frequency

Crosstalk channel gain increases with frequency

Lead to near-far problemI RT generates strong crosstalk to CO line, especially in high tonesI CO generates little crosstalk to RT in all tones

Crosstalk System Model

N users (lines) and K tones (frequency bands)

User n’s achievable rate on tone k is

bkn = log(1 + SINRkn

)where

SINRkn =pkn∑

m 6=n αkn,mp

km + σ

kn

Total data rate of user n

Rn =∑k

bkn


Network Objective: Maximize Rate Region


Rate Region: set of all achievable rate vectors

1

R

Rate Region

2

R



Problem A: (Find One Point On the Rate Region Boundary)

maximize{pn∈Pn}n

∑n

wnRn

User n chooses a power vector pn ∈ Pn ={∑

k pkn ≤ Pmaxn , pkn ≥ 0

}.

Changing different weights trace the entire rate region boundary

A suboptimal algorithm leads to a reduced rate region


1

R

Rate Region

2

R



Problem A: (Find One Point On the Rate Region Boundary)

maximize{pn∈Pn}n

∑n

wnRn

User n chooses a power vector pn ∈ Pn ={∑

k pkn ≤ Pmaxn , pkn ≥ 0

}.

Changing different weights trace the entire rate region boundary

A suboptimal algorithm leads to a reduced rate region


R

Rate Region

2

R1

Difficulties of Solving Problem A

Non-convexity: total weighted rate not concave in power.

Physically distributed: local channel information

Performance coupling: across users (interferences) and tones (powerconstraint)


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]

IW

2

R1

R

Algorithm Operation Complexity PerformanceIW Autonomous O (KN) Suboptimal

OSB Centralized O(KeN

)Optimal

ISB Centralized O(KN2

)Near Optimal

ASB Autonomous O (KN) Near Optimal



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]

OSB/ISB

IW

2

R1

R

Algorithm Operation Complexity PerformanceIW Autonomous O (KN) SuboptimalOSB Centralized O

(KeN

)Optimal


)Near Optimal




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]

/ASBOSB/ISB

IW

R

1R

2

Algorithm Operation Complexity PerformanceIW Autonomous O (KN) SuboptimalOSB Centralized O

(KeN

)Optimal


)Near Optimal



Optimal Spectrum Balancing

Global optimization based on dual decomposition

Key: the duality gap is asymptotically zero under frequency-sharingproperty

5

R2

1R

1Rtarget

A

C

B

EL − l

l

D

w = 0

w = 1

w = γ − �

w = γ + �

X

Y

X ∩ Y

Fig. 2. Operating points inX∩Y can be found through a weighted rate-sumoptimization

Theorem 2:For any rate regionX, defineX as the boundaryof X, Y as the convex hull ofX, and Y as the boundaryof Y. Consider any operating pointC , (Rc1, Rc2) which isachievableC ∈ X and on the boundary of the convex hullof the rate regionC ∈ Y as depicted in Fig. 2. There existssomew such that the PSDs at pointC can be found througha weighted rate-sum maximization.

Proof: C is on the boundary of the convex setY. Sothere exists no pointD , (Rd1, Rd2) ∈ Y such thatRd1 > Rc1andRd2 > R

c2. This implies that for somew

wRc1 + (1− w)Rc2 ≥ wRd1 + (1− w)Rd2, ∀ (Rd1, Rd2) ∈ Y.Now sinceX ⊂ YwRc1 + (1− w)Rc2 ≥ wRd1 + (1− w)Rd2, ∀ (Rd1, Rd2) ∈ X.

So the pointC gives the maximum weighted rate-sum of allachievable points within the rate regionX for some particularweightw. Hence the pointC is optimal in the weighted rate-sum (11) for thatw and can be found through a weightedrate-sum maximization.

Corollary 1: For any convex rate-region, all optimal oper-ating points on the boundary of the rate region can be foundthrough a weighted rate-sum optimization.

Proof: In a convex rate region, the boundary of theconvex hull Y, contains the entire boundary of the rateregion andX = Y. All optimal operating points in termsof the original spectrum management problem (3) lie on theboundary of the rate region. Hence Theorem 2 implies thatall optimal operating points can be found through a weightedrate-sum optimization.Theorem 2 implies that any achievable operating point on theboundary of the convex hull of the rate region can be foundthrough a weighted rate-sum optimization. If the rate regionis close to being convex, then the majority of the optimaloperating points can be found. Thankfully this is the case inDSL channels as is now explained.

In the wireline medium there is some correlation betweenthe channels on neighbouring tones. If the channel is sampled

finely enough then neighbouring tones will see almost thesame channels (both direct and crosstalk).

Imagine that the tone spacing is fine enough such thathn,mk ' hn,mk+l , 0 ≤ l ≤ L − 1. Consider two points inthe rate region,A = (Ra1 , R

a2) andB = (R

b1, R

b2) and their

corresponding PSDs(s1,ak , s2,ak ) and(s

1,bk , s

2,bk ). It is possible

to operate at a pointE = ( lLRa1 +

L−lL R

b1,

lLR

a2 +

L−lL R

b2) for

any0 ≤ l ≤ L−1 as depicted in Fig. 2. This is done by settingthe PSDs to(s1,ak , s

2,ak ) on tonesk ∈ {pL+ 1, . . . pL+ l} for

all integer values ofp, and to(s1,bk , s2,bk ) on all other tones.

For example, to operate at a point 2/3 betweenA andB (on the side closer toA), it is required thatl = 2and L = 3. Thus the PSDs are set to(s1,ak , s

2,ak ) on tones

k ∈ {1, 2, 4, 5, 7, 8, . . . ,K − 1} and to (s1,bk , s2,bk ) on tonesk ∈ {3, 6, 9, . . . ,K}. For this to work the tone spacing mustbe small enough such that the channel is approximately flatover L = 3 neighbouring tones. That is, it is necessary thathn,mk ' hn,mk+1 ' hn,mk+2, ∀ k ∈ {1, 4, . . . ,K − 2}.

For largeL (small tone spacing), practically any operatingpoint betweenA andB can be achieved. Thus for any twopoints in the rate region, any point between them is also withinthe rate region. This is the definition of a convex set. As suchthe rate region is approximately convex in DMT systems withsmall tone spacings. This approximation becomes exact asthe tone-spacing approaches zero. For the remainder of thispaper, we assume that the DMT tone spacing is small suchthat the rate region is convex. This is justified for practicalDSL systems for which∆f is 4.3125 kHz.

Note that one should not confuse convexity of the rate-region with convexity of the objective function (11). In practicethe rate regions are seen to be nearly-convex, however theoptimisation problem is highly non-convex, exhibiting manylocal maxima. For this reason conventional convex optimisa-tion techniques cannot be applied and an exhaustive search isrequired on each tone.

B. Dual Decomposition

In the previous section it was shown that the spectrummanagement problem (3) can be solved through a weightedrate-sum optimization (11). It was also shown that in DSLthe rate region is approximately convex, allowing almost alloptimal operating points to be found. This section will showhow the weighted rate-sum optimization can be solved in acomputationally tractable way.

The total power constraints (4) can be incorporated into theoptimization problem by defining the Lagrangian

L , wR1 + (1− w)R2 − λ1∑

k

s1k − λ2∑

k

s2k. (12)

Hereλn denotes the Lagrangian multiplier for usern and ischosen such that either the power constraint on usern is tight∑k s

nk = Pn or λn = 0. The constrained optimization (11)

can now be solved via the unconstrained optimization

maxs1,s2

L(w, λ1, λ2, s1k, s2k). (13)

c©Cendrillon et. al., ICC, 2004



Partial Lagrangian:

L (p1, ...,pN) =∑n

wn∑k

log(1 + SINRkn

)−∑n

λn

(∑k

pkn − Pmaxn

)

Decompose K nonconvex subproblems, one for each tone k:

maximize{pkn}∀n≥0

∑n

wn log(1 + SINRkn

)−∑n

λnpkn

I Joint exhaustive search of optimal transmission power of all users

Optimal values of λ1, ..., λN can be found using bisection orsubgradient search



ProsI Solve a long-standing open problemI Find the global optimal solution (asymptotically)I Linear complexity in K

ConsI Centralized algorithmI Exponential complexity in N


Iterative Water-FillingGame-theoretic model based on selfish optimizationsEach user wants to maximize payoff: total achievable rate

Sn(pn,p−n

)= Rn

(pn,p−n

)

Best Response: the power vector that maximizes payoff

Bn(p−n) , arg maxpn∈Pn

Sn(pn,p−n

)I Convex optimizationI Coupled across tones by total power constraintI Can be solved by dual decompositionI Solution: water-fillingYU et al.: DISTRIBUTED MULTIUSER POWER CONTROL FOR DIGITAL SUBSCRIBER LINES 1109

that no interference subtraction is performed regardless ofinterference strength, the data rates are

(6)

(7)

Comparing the above expression with (2), it is easy to identify

(8)

(9)

and similarly for and . Thus, the simplified modelincurs no loss of generality. The interference channel game con-sidered here is not a zero-sum game, i.e., one player’s loss is notequal to the other player’s gain.

The main objective here is to characterize all pure-strategyNash equilibria in an interference channel game. At a Nash equi-librium, each user’s strategy is the optimal response to the otherplayer’s strategy. So fixing , the optimal must bethe solution to the following optimization problem:

s.t.

(10)

The solution to this problem is the well-known water-fillingpower allocation. More precisely, let

. Then, the water-filling power allocation is

if

if(11)

where is a constant chosen so that the power constraint ismet. Likewise, fixing , the optimal should also bea water-filling power allocation against the combined interfer-ence from and the noise. Thus, a Nash equilibrium isreached if and only if the water-filling condition is simulta-neously achieved for both users. The characterization of Nashequilibria is therefore equivalent to a characterization of “simul-taneous water-filling” points. The idea of simultaneous water-filling is illustrated in Fig. 4. The following theorem offers sev-eral sufficient conditions for the existence and uniqueness of theNash equilibrium in the two-user case.

Theorem 1: Suppose that , ,then at least one pure strategy Nash equilibrium in thetwo-user Gaussian interference game exists. Further, let

, ,, and

. If any of the followingconditions, , , or is satisfied,then the Nash equilibrium is unique and is stable.

The proof of Theorem 1 is lengthy and it is included in theAppendix. The basic idea is that under suitable conditions, the

Fig. 4. Simultaneous water-filling.

Nash equilibrium can be reached by an iterative water-fillingprocedure, where each user successively optimizes his powerspectrum while regarding other users’ interference as noise. Themain purpose of Theorem 1 is to characterize conditions underwhich such an iterative water-filling procedure converges. Thefollowing corollary is a direct consequence of the proof.

Corollary 1: If the condition for existence and uniquenessof the Nash Equilibrium is satisfied, then the iterative water-filling algorithm for the two-user Gaussian interference game,where in every step, each modem updates its PSD regarding allinterference as noise, converges, and it converges to the uniqueNash equilibrium from any starting point.

The condition of Theorem 1 is not a mere technicality.The following simple example illustrates a case wherethe Nash equilibrium is not unique. Consider a two-usercase where there are only two frequencies of concern. Let

. Let powerconstraints and background noise all be 1. The set of powerallocations andis one Nash equilibrium, and the set of power allocations

and is a differentNash equilibrium.

IV. DISTRIBUTED POWER CONTROL

Because of the frequency-selective nature of the DSL channel,power control algorithms for DSL applications need to allocatepower optimally not only among different users, but also inthe frequency domain. This requirement brings in many extra

YU et al.: DISTRIBUTED MULTIUSER POWER CONTROL FOR DIGITAL SUBSCRIBER LINES 1109


(6)

(7)


(8)

(9)



s.t.

(10)



if

if(11)



, ,, and











c©Yu, Ginnis and Cioffi, JSAC, 2002



Sn(pn,p−n

)= Rn

(pn,p−n

)Best Response: the power vector that maximizes payoff


Sn(pn,p−n

)I Convex optimizationI Coupled across tones by total power constraintI Can be solved by dual decomposition

I Solution: water-fillingYU et al.: DISTRIBUTED MULTIUSER POWER CONTROL FOR DIGITAL SUBSCRIBER LINES 1109


(6)

(7)


(8)

(9)



s.t.

(10)



if

if(11)



, ,, and













(6)

(7)


(8)

(9)



s.t.

(10)



if

if(11)



, ,, and











c©Yu, Ginnis and Cioffi, JSAC, 2002



Sn(pn,p−n

)= Rn

(pn,p−n

)Best Response: the power vector that maximizes payoff


Sn(pn,p−n

)I Convex optimizationI Coupled across tones by total power constraintI Can be solved by dual decompositionI Solution: water-fillingYU et al.: DISTRIBUTED MULTIUSER POWER CONTROL FOR DIGITAL SUBSCRIBER LINES 1109


(6)

(7)


(8)

(9)



s.t.

(10)



if

if(11)



, ,, and













(6)

(7)


(8)

(9)



s.t.

(10)



if

if(11)



, ,, and











c©Yu, Ginnis and Cioffi, JSAC, 2002Jianwei Huang (Princeton) DSL Spectrum Management March 2007 13 / 26

Iterative Water-filling

ProsI Autonomous: no explicit communication among users (interference

plus noise can be locally measured)I Low computational complexity of O(KN): separable across users and

tonesI Achieve better performance than the current practice

ConsI Selfish optimizationI No consideration for damages to other usersI Highly suboptimal in the mixed CO/RT case


Autonomous Spectrum Balancing

Key idea: reference line - static pricing for static channel

I A virtual line representative of the typical victim in the networkI Good choice: the longest CO lineI Parameters (power, noise, crosstalk) are publicly known

Each user will choose its transmit power to protect the reference line


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


Reference Line’s Rate

User n’s obtains the reference line parameters locally

Length & LocationReference Crosstalk:

Reference Noise:

Reference Power:OperatorReference Line

Database

pk,ref

σk,ref

αk,refn

The reference line rate

R refn =∑k

log

(1 +

pk,ref

αk,refn pkn + σk,ref

)

I Interference only depends on user n’s transmit power pknI Locally computable without explicit message passing


Frequency Selective Water-filling

Under high SNR approximation of the reference line

Bkn(p−n

)=

wnλn + α

k,refn /σk,ref · 1{pk,ref>0}

−∑m 6=n

αkn,mpkm − σkn

+

I Reference line is not active in high frequency tones

Special case: traditional water-filling (ignore αk,refn /σk,ref)




Bkn(p−n

)=

wnλn + α


−∑m 6=n

αkn,mpkm − σkn

+




Power

Traditional Water−Filling

Frequency

Interference & Noise



Bkn(p−n

)=

wnλn + α


−∑m 6=n

αkn,mpkm − σkn

+




Power

Active Reference Line

Frequency−Selective Water−Filling

Frequency

Interference & Noise

Convergence of ASB Algorithm

ASB Algorithm: users update their individual power allocationaccording to best responses either sequentially or in parallel

Theorem

ASB algorithm globally and geometrically converges to the unique N.E. ifthe crosstalk channel is small, i.e.,

maxn,m,k

αkn,m <1

N − 1.

Independent of the reference line parameters.

Recover the convergence of iterative water-filling as a special case.

Proof: contraction mapping


Proof Outline

1 Key Lemma: min-max of an increasing function and an decreasingfunction is achieved at the intersection.

2 Construct two such functions based on the ASB algorithm.

3 Show the maximum difference between the PSD during adjacentiterations is decreasing.

maxn

max

{∑k

[pk,t+1n − pk,tn

]+,∑k

[pk,t+1n − pk,tn

]−}

≤maxn

max

{∑k

[pk,tn − pk,t−1n

]+,∑k

[pk,tn − pk,t−1n

]−}

I Sequential updates: bound the maximum eigenvalue of the mappingmatrix.

I Parallel updates: more realistic with cleaner proof.


ASB Performance

4 ADSL lines.

Mixed CO/RT deployment.

Practical channel and background noise models.

Both users 2 and 3 acheive fixed rates 2Mbps.

Examine the rate region in terms of users 1 and 4’s rates.

User 4

CP

RT CP

CP

CO CP5Km

4Km

3.5Km

3Km

2Km

3Km

4Km

RT

RT

User 1

User 2

User 3


Achievable Rate Regions of Different Algorithms

12/7/05 Multi-user DSL 60

60Raphael Cendrillon

12/7/05University of Queensland

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 4’s Rate (Mbps)

Use

r 1’s

Rat

e (M

bps)

Optimal (OSB)

Best Available Today (IW)

ASB

R. Cendrillon, M. Moonen, “Iterative Spectrum Balancing for Digital Subscriber Lines”, ICC 2005.


Power Allocation

Power Allocation under ASB Power Allocation under Iterative Waterfilling

R1 = 1 Mbps, R2 = 2 Mbps, R3 = 2 Mbps

R4= 7.3 Mbps under ASB, and 3 Mbps under Iterative Waterfilling.I Around 150% rate increase for user 4


Robustness of Reference Line Choice

downstream transmissions

��

��

��

��

��

��

5km

CO

3km

RT

4km

crosstalk

��

��

Two-line Topology

0 2 4 6 80

0.5

1

1.5

2

RT Rate (Mbps)

CO

Rat

e (M

bps)

4010 m4020 m4050 m4100 m5000 m6000 m

Rate Region w/ various Reference Line Choice

Performance is robust to reference line choices.


Summary

Topic: spectrum management in DSL multiuser interference channels

Key idea: static pricing using reference line

Algorithm: ASB: autonomous, low complexity, and robust

Performance: close to optimal, provable convergence

Practice: achieve significantly larger rate region compared with thestate-of-the-art distributed algorithm

Main contribution: static pricing for static coupling


Background Reading

IW: W. Yu, G. Ginis, and J. Cioffi, “Distributed multiuser powercontrol for digital subscriber lines,” IEEE Journal on Selected Areas inCommunication, June 2002

OSB: R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, andT. Bostoen, “Optimal multi-user spectrum balancing for digitalsubscriber lines,” IEEE Transactions on Communications, May 2006

ASB: R. Cendrillon, J. Huang, M. Chiang, and M. Moonen,“Autonomous spectrum balancing for digital subscriber lines,” toappear in IEEE Transactions on Signal Processing, 2007


Introduction

dsl spectrum management - princeton universitychiangm/ele539l12.pdf · wireline communications...

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