DSL Spectrum Management - Princeton Universitychiangm/ele539l12.pdf · Wireline communications...
Transcript of DSL Spectrum Management - Princeton Universitychiangm/ele539l12.pdf · Wireline communications...
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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
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Acknowledgements
Collaborations: Raphael Cendrillon, Mung Chiang, Marc Moonen
Sponsorships: Alcatel, NSF
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 2 / 26
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 3 / 26
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 3 / 26
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 4 / 26
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Network and Channel Model
crosstalk
TX
TX RX
RXCO
RT
(Remote Terminal)
(Central Office) Customer
Customer
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 5 / 26
Mathematical model: multi-user multi-carrier interference channel
Each telephone line is a user (transmitter-receiver pair)
Generate mutual crosstalks over multiple frequency tones
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Network and Channel Model
crosstalk
TX
TX RX
RXCO
RT
(Remote Terminal)
(Central Office) Customer
Customer
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 5 / 26
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
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Network and Channel Model
crosstalk
TX
TX RX
RXCO
RT
(Remote Terminal)
(Central Office) Customer
Customer
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 5 / 26
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
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Network and Channel Model
crosstalk
TX
TX RX
RXCO
RT
(Remote Terminal)
(Central Office) Customer
Customer
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 5 / 26
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
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 6 / 26
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Network Objective: Maximize Rate Region
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 7 / 26
Rate Region: set of all achievable rate vectors
1
R
Rate Region
2
R
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Network Objective: Maximize Rate Region
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 7 / 26
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
Rate Region: set of all achievable rate vectors
1
R
Rate Region
2
R
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Network Objective: Maximize Rate Region
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 7 / 26
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
Rate Region: set of all achievable rate vectors
1
R
Rate Region
2
R
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Network Objective: Maximize Rate Region
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 7 / 26
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
Rate Region: set of all achievable rate vectors
R
Rate Region
2
R1
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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)
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 8 / 26
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 9 / 26
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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]
OSB/ISB
IW
2
R1
R
Algorithm Operation Complexity PerformanceIW Autonomous O (KN) SuboptimalOSB Centralized O
(KeN
)Optimal
ISB Centralized O(KN2
)Near Optimal
ASB Autonomous O (KN) Near Optimal
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 9 / 26
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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]
/ASBOSB/ISB
IW
R
1R
2
Algorithm Operation Complexity PerformanceIW Autonomous O (KN) SuboptimalOSB Centralized O
(KeN
)Optimal
ISB Centralized O(KN2
)Near Optimal
ASB Autonomous O (KN) Near Optimal
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 9 / 26
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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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 10 / 26
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Optimal Spectrum Balancing
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 11 / 26
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Optimal Spectrum Balancing
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 12 / 26
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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
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
c©Yu, Ginnis and Cioffi, JSAC, 2002
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 13 / 26
-
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 decomposition
I 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
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
c©Yu, Ginnis and Cioffi, JSAC, 2002
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 13 / 26
-
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
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
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 14 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 15 / 26
-
Reference Line
CP
RT
RT
RT
CP
CO CP
CP
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 16 / 26
-
Reference Line
Actual Line
Reference Line
CO
CPCO
RT CP
RT
RT
CP
CP
CP
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 16 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 17 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 17 / 26
-
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)
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 18 / 26
-
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)
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 18 / 26
Power
Traditional Water−Filling
Frequency
Interference & Noise
-
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)
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 18 / 26
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 19 / 26
-
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.
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 20 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 21 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 21 / 26
-
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.
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 22 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 23 / 26
-
Robustness of Reference Line Choice
downstream transmissions
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5km
CO
3km
RT
4km
crosstalk
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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.
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 24 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 25 / 26
-
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
Jianwei Huang (Princeton) DSL Spectrum Management March 2007 26 / 26
Introduction