Modeling and Optimization of Line-Driver Power Consumption in xDSL Systems (länk till annan webbplats)

Modeling and Optimization of Line-Driver Power Consumption in xDSL Systems (länk till annan webbplats)
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This is the published version of a paper published in EURASIP Journal on Advances in Signal Processing.
Citation for the original published paper (version of record):
Wolkerstorfer, M., Trautmann, S., Nordström, T., Putra, B. (2012)
Modeling and Optimization of Line-Driver Power Consumption in xDSL Systems.
EURASIP Journal on Advances in Signal Processing, 2012(Article number 226): 1-17
http://dx.doi.org/10.1186/1687-6180-2012-226
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Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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RESEARCH
Open Access
Modeling and optimization of the line-driver
power consumption in xDSL systems
Martin Wolkerstorfer1* , Steffen Trautmann2 , Tomas Nordström1,3 and Bakti D Putra1
Abstract
Optimization of the power spectrum alleviates the crosstalk noise in digital subscriber lines (DSL) and thereby reduces
their power consumption at present. In order to truly assess the DSL system power consumption, this article presents
realistic line driver (LD) power consumption models. These are applicable to any DSL system and extend previous
models by parameterizing various circuit-level non-idealities. Based on the model of a class-AB LD we analyze the
multi-user power spectrum optimization problem and propose novel algorithms for its global or approximate
solution. The thereby obtained simulation results support our claim that this problem can be simplified with
negligible performance loss by neglecting the LD model. This motivates the usage of established spectral
optimization algorithms, which are shown to significantly reduce the LD power consumption compared to static
spectrum management.
Keywords: Digital subscriber lines, Energy-efficient, Line driver, Optimization
Introduction
This article analyzes the modeling and optimization
of the power consumption in multi-carrier digital subscriber line (DSL) transceivers. The line-driver (LD)
power consumption accounts for the largest part in the
DSL power budget and scales with the transmit power
(TP) [1-3]. With few exceptions [2,4,5], previous study
has therefore focussed on minimizing the transmit sumpower [3,6-8] through power spectral optimization,
also known as dynamic spectrum management (DSM)
[9]. A key feature of this objective is its separability by
subcarriers, which is a prerequisite for the Lagrange
decomposition [10] of the DSM problem. This decomposition results in low-complexity and even distributed
DSM implementations [11-13].
We hypothesize that although TP minimization does
not assume knowledge of the underlying LD power consumption, it achieves energy-efficiency at a negligible performance loss compared to a TP optimization taking the
LD explicitly into account. In order to support this claim
and to realistically assess energy savings by DSM it is
indispensable to have an accurate model of the LD power
*Correspondence: wolkerstorfer@ftw.at
1 FTW Telecommunications Research Center Vienna, Donau-City-Straße1,
A-1220 Vienna, Austria
Full list of author information is available at the end of the article
consumption as a function of the TP. Hence, after providing more background information in Section ‘Background
information’, we begin in Section ‘Line driver models’ by
deriving accurate such models, which are applicable for
any DSL technology and different LD classes. While we
deem a proof of our hypothesis intractable, we exemplarily provide analytical and numerical evidence supporting
our hypothesis based on the proposed enhanced class-AB
LD model in Sections ‘Optimization models and analysis’
and ‘Empirical optimization study’, respectively. For that
purpose we propose two novel numerical approaches for
LD power optimization which are based on successive
geometric programming (GP) [14,15] and difference-ofconvex-functions programming (DCP) [16], respectively.
These techniques help us to motivate the selected scenario
for simulation of a DSL network with realistic parameters
under the two DSM heuristics in [2,3], cf. the introduction in Section ‘Empirical optimization study’ for a more
concise overview of our contributions. The results are
rounded off in Section ‘Average performance evaluation’
by simulations demonstrating the LD power saving potential by energy-efficient multi-user DSM compared to static
spectrum management and rate-maximizing DSM. Our
conclusions are provided in Section ‘Conclusions’.
© 2012 Wolkerstorfer et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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Background information
Energy-efficiency in DSL
In the last ten years, the power consumption of information and communication technology (ICT) has become
an issue on top of our agendas, reflecting our concern on
global warming, CO2 emissions and energy sustainability.
The telecommunication sector is responsible for 25% of
the ICT’s energy consumption [17] and therefore energy
efficiency has naturally become an issue for industry, standardization, as well as governmental bodies. For example,
the share of the fixed broadband access in the telco’s
energy consumption for 2020 is estimated at around 14%
[17]. A related initiative by the European commission aims
at a power reduction of 50% in broadband equipment by
2015 [18].
The power consumption of a DSL transceiver can be
divided according to its three major parts: the digital
front-end (the modem’s digital signal processing); the analog front-end (responsible for the conversion between the
analog and the digital domain, including filters); and the
line driver (the power amplifier driving the line). Depending on the used transmission profile (e.g., bandwidth) the
LD power consumption can be somewhere between 30%
and 60% of the modem’s total power consumption [1-3].
The main focus for energy saving in DSL therefore lies on
the LD power consumption [1,4]. Approaches for reducing the power consumption in DSL can be classified into
three categories [19]: the optimization of hardware components; dynamic rate adaptation (e.g., by spectral optimization); and low-power modes. Our focus is on the first
two approaches, as we a) model the power consumption of
an energy-efficient LD type, and b) study energy-efficient
DSM based on derived LD power consumption models,
leading to lowered transmit rates. We refer to [20,21] for
an introduction to LD design for DSL and to [1,22-25] for
an overview of various energy saving techniques for DSL.
Page 2 of 17
of a class-G LD can be differentiated by whether multiple supplies or internal charge pumps are used to provide
the multiple supply voltages. In the former design the second supply voltage is typically not directly available on a
DSL line card. An additional, costly DC-DC converter is
required which must be included in the LD efficiency calculation. A class-H LD can be seen as a class-G LD with
an infinite number of supply rails, consequently leading
to a higher efficiency at the cost of a more complicated
supply design. Altogether we consider the class-G design
based on internal charge pumps as the most promising
compromise between efficiency and complexity.
As motivated in Section ‘Introduction’, for the evaluation and optimization of the LD power consumption
a realistic functional model is needed which maps the
modem’s TP to its LD power consumption. An empirical
model based on power measurements of a class-AB LD
in ADSL2+ was presented in [2]. However, this model is
not applicable to other DSL technologies or systems with
different physical parameters. A circuit-level model for an
LD of class-AB and G with two supplies has been presented in [4], based on the models in [26]. However, these
models do not precisely account for the non-idealities
of the voltage supply chain [27] (e.g., transformer loss,
impedance synthesize factor, etc.) and the power loss in
the hybrid circuit. Therefore, in Section ‘Class-AB linedriver power model’, we derive an enhanced class-AB
LD power consumption model based on [26] that can
be applied to any DSL profile, and in Section ‘Class-G
line-driver power model’, we propose a novel model for a
class-G LD with charge pumps.
Line driver models
Class-AB line-driver power model
In this section, we enhance the functional class-AB LD
model in [26] based on a circuit analysis, cf. Figure 1. The
total power consumed by a class-AB LD is given as
Line driver modeling
Current DSL systems rely on so called class-AB LDs
as these provide a high degree of linearity over a large
signal bandwidth. The main drawback of this type of
amplifier however lies in its relatively low efficiency. Furthermore, the typical DSL signal exhibits a high crest
factor (CF) with high peak values in comparison to its
root-mean-square (rms) value. Even though those peak
values occur with very low probability, the fixed supply
voltage of a Class-AB LD must be sufficiently high to
provide distortion-free amplification of the highest signal
peaks. This implies that significant power savings could
be obtained by modulating the supply voltage to follow
the envelope of the amplified signal, as done in so-called
class-H LDs. Class-G LDs [20] are class-AB LDs where
the supply rail is switched, e.g., between a lower and a
higher voltage level VL and VH , respectively. The design
PLD(AB) = Pu + Pdiss + PHybrid ,
Figure 1 Class AB Line Driver followed by the hybrid and the
transformer.
(1)
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where Pu is the output power measured at the LD output
of a line indexed by u ∈ U , Pdiss is the total power dissipated inside the LD, and PHybrid is the power consumed
by the hybrid circuit. The level of PHybrid strongly depends
on the hybrid implementation and topology and ranges
from a few mW to several tens of mW. By the reformulation detailed in Appendix 1 the LD power consumption in
(1) can be equivalently written as
2 Pu
+ PHybrid ,
(2)
PLD(AB) = Vs · IQ +
π Rline
where Vs is the supply voltage of the LD, IQ is the qui
escent current, and Rline as defined in Appendix 1 is the
transformed resistance of the line, cf. Figure 1. In the ideal
case the supply voltage Vs can be designed to cover the
output voltage swing CF · Vrms,ideal described by the signal crest factor CF and the ideal rms LD output voltage
Vrms,ideal [26].
However, a more realistic representation of Vs should
include several impairments that will generally be present
in real implementations and significantly influence the LD
efficiency:
1) The achievable signal swing at the LD output is
reduced from its theoretical maximum value Vs by a
voltage drop Vdrop . Its value is typically in the range
between 2 and 4 V, and determined by the design and
the underlying technology of the LD output stage.
2) The resistances of the copper coils and other
non-idealities cause an additional voltage drop over
the transformer. This loss in effective signal power
on the line is called transformer loss TL and can
reach 0.2 to 0.5 dB for EP5 and EP7 transformers as
used in xDSL central office (CO) applications.
3) Another voltage drop occurs in the termination
circuitry. Impedance synthesis is a commonly used
concept in LD system integration [28] to reduce this
Page 3 of 17
loss. More precisely, only a small part of the effective
receive signal termination is provided by an external
resistor, while the main part is actively generated by
the LD itself. The impedance synthesis factor m - that
is the ratio between the external resistor value and
the over-all termination resistance - also determines
the receive signal attenuation and cannot be made
arbitrarily small. Therefore, a voltage drop by a factor
m/(m + 1) must be included in the calculation of the
required LD supply voltage. While for VDSL2 systems
a reasonable choice of m lies in the range from 3 to 6,
for pure ADSL/ADSL2+ systems a more aggressive
choice of m in the range from 6 to 20 is possible.
Using Vrms,ideal = Pu · Rline these additional factors
can be accommodated in the form
Vs = CF ·
m+1
P̂u · Rline · TL ·
+ VDrop ,
m
where P̂u is the maximum transmitted power. Figure 2
depicts an exemplary measurement of a real ADSL2+ LD’s
power consumption, as well as the class-AB LD power
consumption modela in (2), using (a) the mentioned ideal
relation Vs = CF · Vrms,ideal , (b) the relation Vs = CF ·
Vrms,ideal + Vdrop with headroom Vdrop as used in [4], and
(c) the relation derived in (3). From this plot it is visible
that there is a considerable amount of LD power consumption that has not been taken into account by previous
models.
Based on the wide deployment of class-AB LDs and
the simple functional shape of our model we will focus
on this LD type when analyzing the effect of the LD on
energy-efficient DSM in Sections ‘Optimization models
and analysis’, ‘Empirical optimization study’, and ‘Average
performance evaluation’. Another energy-saving approach
mentioned in Section ‘Energy-efficiency in DSL’ is the
0.7
PLD(AB) [W]
0.6
0.5
0.4
0.3
ADSL2+
0.2
VDSL
0.1
0
0
10
20
(3)
Measurement
With Nonidealities
With Headroom Only
Ideal
30
40
50
60
Transmit Power Pu [mW]
70
80
90
Figure 2 Comparison of class-AB LD power models. The figure shows the influence of additional non-idealities considered in our LD model.
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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deployment of more energy-efficient LDs, as analyzed in
the following section.
Class-G line-driver power model
Based on our discussion in Section ‘Line driver modeling’,
we study in this section class-G LDs with a set of internal charge pumps. We refer to Appendix 2 for a model
of an LD with two supply voltages that includes the nonidealities discussed in Section ‘Class-AB line-driver power
model’ into the model in [4,26]. The basic principle of a
charge pump is exemplified in Figure 3. A pair of such
charge pumps is used to generate the high class-G supply voltage VH from a single LD supply voltage which at
the same time serves as the low class-G supply voltage
VL . Under ideal conditions, a maximum voltage ratio of
VH /VL = 3 can be achieved. However, taking technological limitations and various internal voltage drops into
account, assuming a ratio of VH /VL ≈ 2 is more realistic. The total power consumption of a class-G LD with
internal charge pumps is defined as
PLD(G-CP) = PLow,CP + PHigh,CP + PQ,CP + PHybrid , (4)
where PLow,CP is the LD power consumption value of an
equivalent class-AB LD running continuously at the low
voltage supply, and PHigh,CP refers to the additional power
consumption when the LD is switching to the high voltage supply. PQ,CP is the quiescent power dissipation. The
voltage level VH is thought of as the summation of VL and
VH − VL , with the latter being generated by the charge
pumps when needed. The consumed LD power at the low
voltage VL is in analogy to (2) defined as
2 Pu
.
(5)
PLow,CP = VL ·
π Rline
Extending the ideal rms voltage Vrms,ideal = Pu · Rline
with the non-idealities of Section ‘Class-AB line-driver
power model’ we obtain the rms LD output voltage (that
is, before impedance synthesis and transformer) as
m+1
.
(6)
Vrms = Pu Rline · TL ·
m
In analogy to the ideal class-G case [26] the mean
average deviation (MAD) of the LD output voltage for
the cases when the Gaussian distributed output signal is
below and above the threshold Vth = (VL − Vdrop ) is given
as
⎛
⎞
V2
− th
2
2
(7)
Vrms ⎝1 − e 2Vrms ⎠ ,
VMAD,Low =
π
and
VMAD,High =
Off
L
On
Load
Figure 3 Class-G LD: Basic principle behind an internal charge
pump.
V2
− th
2
2
Vrms e 2Vrms ,
π
(8)
respectively. Note that the fraction of time μcp (Pu ) ∈[ 0, 1]
the charge pump is used is higher than the time the output signal exceeds the threshold Vth , the reason being the
additional ramp-up / ramp-down phases between the low
and the high supply. Therefore the output signal’s MAD
during charge pump usage is a combination of that when
the signal is below and above Vth , respectively, weighted
by the corresponding probabilities. The output signal’s
MAD under the assumption of operating below and above
th
)) and
the threshold is given by VMAD,Low /(1 − 2Q( VVrms
th
VMAD,High /(2Q( VVrms
), respectively. Correspondingly we
define the dynamic power PHigh,CP as
PHigh,CP =
(9)
VH − V L
μA · VMAD,Low + μB · VMAD,High · ρ,
Rline TL m+1
m
where
μA =
V
Page 4 of 17
th
μcp (Pu ) − 2Q( VVrms
)
th
1 − 2Q( VVrms
)
,
(10)
μB = 1, Q(·) is the Q-function, ρ is the recharge loss, and
the term Rline TL m+1
m represents the total resistance at the
LD output. Comparing the total dynamic power (the sum
of (5) and (9)) to that under a class-G design with two supplies (the sum of (30a) and (31) in Appendix 2) we find that
the latter one is obtained by setting μA = 0 and ρ = 1.
We emphasize that μcp (Pu ) depends not only on the output power, but, for example, also on the transformer ratio,
the DSL profile, or the way in which the charge pump is
loaded. The quiescent power consists, differently to that
in class-AB LDs, of three main components, given as
PQ,CP = PQ,Low + PQ,High + PQ,classG .
(11)
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The term PQ,Low = VL · IQ is the quiescent power dissipation of an equivalent class-AB LD continuously working
at VL . The additional quiescent power dissipation of the
LD when working at the high voltage supply is defined as
PQ,High = (VH − VL ) · IQ · μcp (Pu ) · ρ.
(12)
The third term in (11) splits into
(13)
PQ,classG =
VL + (VH − VL ) · μcp (Pu ) · ρ · IQ,classG + LclassG ,
where IQ,classG is the additional quiescent current in classG mode and LclassG [W] refers to further fixed losses in the
class-G circuitry.
In Figure 4, we compare the power consumption data
provided for a real class-G LD with charge pumps in [29]
to the three discussed LD models: our model of a class-G
LD with charge pumps in this section, the model of a classG LD with two power supplies in Appendix 2, and the
model of a class-AB LD modeled by Equations (2) and (3),
respectivelyb . In Figure 4a we see that under an ADSL2+
profile the power consumption predicted by our model of
a class-G LD with charge pumps lies between that calcu-
a
LD Power [W]
0.8
0.6
Model: Class AB LD
Model: Class G LD (with charge pump)
Model: Class G LD (two supplies)
Datasheet: Class G LD (with charge pump)
0.4
ADSL2+
0.2
0
5
0
5
10
Transmit Power Pu [dBm]
15
20
lated by the models of a class-AB LD and a class-G LD
with two supplies. Figure 4b shows that the class-G LD
models lead to similar power estimates for transmit powers below 14.5dBm. This is explicable by the fact that a
low transmit power leads to low probabilities μcp (Pu ) and
μ2S (Pu ) of using the high supply in the class-G LD with
charge pumps and with two supplies (see Appendix 2),
respectively. Correspondingly we can approximate the
power consumption of a class-G LD for low transmit
power values by that of a class-AB LD in (5) that operates
at the low supply voltage. However, near the maximum
output power the novel class-G LD model with charge
pumps significantly deviates from the class-G LD model
with two supplies, similarly as the consumption of the real
LD described in [29]. For example, at the maximum transmit power of 20.5dBm the two-supply class-G LD model
underestimates the power consumption of the LD in [29]
by as much as 44mW for ADSL2+ or 83mW for VDSL2 8b.
Regarding for instance the curves for VDSL2 30a we find
that the real consumption values are partially below the
class-G LD models for higher transmit powers. This can
be explained by the deviation of the quiescent current into
the load [28] which is circuit and transmit power dependent. Differently, all the presented LD models (as well as
those in [4,26]) assume a constant quiescent current IQ
that is independent of the transmit power.
In summary, the class-G design with charge pumps
yields substantial energy savings compared to a classAB LD while sparing us the DC-DC conversion needed
for class-G LDs with two supplies. Furthermore, the presented LD power models have a qualitatively similar functional shape for transmit power values below 14.5dBm
as they are all based on the elementary class-AB power
relation in (2). In the following sections, we focus on
the class-AB LD and analyze our hypothesis of Section
‘Introduction’ on the difference between LD power and
TP optimization.
Optimization models and analysis
b
Line Driver Power [W]
Page 5 of 17
0.6
VDSL2 30a
VDSL2 17a
0.4
VDSL2 8b
0.2
0
5
0
5
10
Transmit Power [dBm]
15
20
Figure 4 Comparison of class-G LD power models to a real
class-G LD with charge pumps. Our model of a class-G LD with
charge pumps shows a higher (lower) power consumption compared
to an ideal class-G (class-AB) design. It accounts for additional power
losses at high transmit powers that are also observable in a real
class-G implementation. However, all three LD power models show a
similar shape as a function of transmit power Pu below 14.5dBm.
In this section, we want to formally develop some insight
into when a difference between LD power and TP optimization in terms of the achieved class-AB LD power
might occur, how large it is, and whether this difference
truly occurs under realistic network conditions.
DSL system model and notation
Current DSL systems employ frequency-division duplexing (FDD) and discrete multi-tone (DMT) modulation
which splits the available frequency bandwidth into
C orthogonal subchannels (subcarriers). Our system
model consists of U subscriber lines sharing a single
cable binder. Electromagnetic coupling between the users’
twisted pair wires leads to crosstalk noise at the receivers,
which is the reason for performing the power allocation
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of all users jointly. The achievable rate per DMT-symbol
rcu (pc ) for user u ∈ U = {1, . . . , U} on subcarrier c ∈
C = {1, . . . , C} as a function of the signal to interference and noise ratio (SINR) is modeled by the common
gap-approximation [30]
⎞
⎛
⎜
⎜
rcu (pc ) = log2 ⎜
⎜1 +
⎝
Hcuu puc
Hcui pic
i∈U \u
⎟
⎟
⎟
⎟,
u ⎠
(14)
+ Nc
T
u
where pc =[ p1c , . . . , pU
c ] , pc is the power assigned to
subcarrier c of user u, and the terms Hcuu and Hcui are
the direct channel transfer coefficient of user u and the
cross-channel transfer coefficient from user i to user u
on subcarrier c, respectively. DSM implementations in
standard-compliant DSL systems, including crosstalk estimation functionality, have been reported in [31,32]. The
term indicates the SNR-gap to capacity depending on
the modulation scheme, the targeted bit-error rate, the
coding gain, and the noise margins, while Ncu represents
the total received background noise power on subcarrier
c of user u, including white thermal noise, alien-crosstalk,
and radio-frequency interference.
Based on Section ‘Line driver models’, the LD power
consumption
of a class-AB LD as a function of the total
TP Pu = c∈C puc of user u is given in the form
fuLD (Pu ) = ŵu Pu + w̆u ,
(15)
where the parameters ŵu ∈ R+ and w̆u ∈ R+ are dependent on the hardware and system modelc . In the following
optimization study, we will use two key features of this
function: a) it is monotonously increasing, and b) concave
in Pu (or puc , c ∈ C , respectively).
Optimization problems
Similarly as in previous DSL studies [11,33] we mathematically formulate the problem of minimizing the transmit
sum-power in DSL in the form
J TP = minimize
u
pc ,u∈U ,c∈C
subject to
puc
where p̂cu , c ∈ C , u ∈ U , denotes the PSD mask, B̂
the maximum number of bits that can be allocated to
a single subcarrier, and Ru and P̂u the target-rate per
DMT-symbol and the maximum sum-power of user u,
respectively. In practice numerous other objectives may be
targeted besides energy consumption, including for example sum-rate [33], fairness [34], service coverage [35], the
energy-per-bit [8], or weighted combinations thereof [6].
However, our choice of focusing on energy-minimization
subject to rate-constraints will allow us to study various
defined rate combinations. Similarly to (16), based on the
model in (15) the problem of minimizing the total LD
power consumption in DSL can be stated as
J
LD
= minimize
u
pc ,u∈U ,c∈C
u∈U
puc
(17a)
c∈C
subject to Constraints (16b)–(16e),
(17b)
where for simplicity of exposition we assume identical LD
models for all users. This allows us to omit the added constant w̆u and the factor ŵu , u ∈ U , as they have no influence on the optimal solution. Note that the latter factors
can easily be reintroduced under the numerical optimization approaches in Section ‘Empirical optimization study ’.
For instance, heterogeneous LD models are considered for
the simulations in Section ‘An experiment in real-sized
DSM problems using heuristics’. For brevity we will denote
the optimal per-user sum-power values for the problems
LD ∈ RU , respectively.
in (16) and (17) by PTP ∈ RU
+ and P
+
Analysis of the optimization problems in (16) and (17)
Before turning to the numerical optimization of the problems in (16) and (17) we analyze their solutions and
the difference between their solutions in terms of LD
power independently of their exact solution value. To
begin with we define the set of possible solutions (the
“power-region”).
Definition 1 (Power-region). The power-region associated with the problems in (16) and (17) is defined as the set
(16a)
u∈U c∈C
rcu (pc ) Ru ,
∀u ∈ U ,
(16b)
c∈C
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puc P̂u ,
∀u ∈ U ,
(16c)
c∈C
0 ≤ puc ≤ p̂uc ,
rcu (pc ) ≤ B̂,
∀c ∈ C , ∀u ∈ U ,
∀c ∈ C , ∀u ∈ U ,
(16d)
(16e)
P = {P ∈ RU | ∃puc , c ∈ C , u ∈ U , feasible for
puc }.
constraints (16b)–(16e), Pu =
(18)
c∈C
Proposition 1. The sum-power vectors PTP and PLD
achieved at a solution of the power minimization problems
in (16) and (17), respectively, both lie on the boundary of
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the power-region P as defined in (18), i.e., P ∈ P , P =
PTP , P PTP and P ∈ P , P = PLD , P PLD .
Proof. The proof simply follows from the monotonicity
of the objectives in (16a) and (17a), respectively.
Proposition 1 also suggests a practical heuristic
approach for LD power optimization, namely through
a sequence of weighted sum-power minimizations with
weights based on the projected gradients of the objective
functions fuLD (Pu ), cf. [36] where a similar idea was applied
for a rate-utility maximization problem. However, while in
[36] a non-concave maximization was performed over the
rate-region, here we face a concave minimization problem
over the power-region.
The following proposition identifies the smallest problem instances where a difference between the two problems in terms of LD power may occur, and which we will
further study in Section ‘Empirical optimization study’.
Proposition 2. Differences between the optimal solutions of the problems in (16) and (17) in terms of LD power
can only occur for U ≥ 2 and C ≥ 2.
See Appendix 3 for a proof.
Next, we define the relative gain by LD power optimization in (17) compared to TP minimization in (16)
as
PuTP − PuLD
ŵ u∈U
,
(19)
ξ=
ŵ u∈U PuTP + U · w̆
where in the nominator we have the difference in LD
power between the two optimization approaches we are
interested in, and in the denominator the LD power under
the TP minimization. In other words, the relative gain in
(19) tells us how much more energy-efficient LD power
optimization is compared to classical TP minimization. In
the following we derive a bound on ξ for any number of
users U and subcarriers
C with powers puc summing to the
total power Pu = c∈C puc . More precisely, we have
u∈U
PuTP ≤ J LD ≤
PuTP ,
Page 7 of 17
which is only dependent on the solution of the problem
in (16) and illustrated in Figure 5. Expanding our intuition from Proposition 2, we see that this simple bound
does not allow for any LD power reduction by direct LD
power optimization in (17) compared to TP minimization
in (16) when all but one user transmit with very low power
(e.g., below -20 dBm). Note however that Figure 5 does
not allow us to make any conclusions on possible differences when all lines operate in a high-power regime. For
example, if the solution to the TP minimization problem
in (16) demands all users to use maximum sum-power, by
sum-power optimality in (16) the same must hold in the
LD power minimization problem in (17) and so the difference between the two must actually vanish, differently to
what the bound in (21) indicates. Using Jensen’s inequality we can even bound (21) independently of the solution
PTP , giving
⎞−1
⎛
√
w̆ U
1
⎟
⎜
(22)
ξ ≤ 1− √
⎠ .
⎝1 + U
ŵ
P̂
u
u∈U
The gain√ξ for U = 2 users is for instance bounded
by 1 − 1/ U (≈ 30%). The bounds in (21) and (22) are
identical when PuTP = P̂u = P, ∀u ∈ U .
In this section, we have located the solutions of our two
optimization problems on the boundary of a power-region
and identified potentially insightful problem instances. In
the following section, we will use this information to study
the real gain ξ by directly optimizing the LD power model
through numerical methods.
Empirical optimization study
We will use three approaches to obtain insights into
the differences between TP and LD power minimization in terms of the LD power consumption founded on
the functional model in (15): The first one is based on
an efficient but possibly suboptimal successive geometric
(20)
u∈U
where the first inequality holds due to the monotonicity
and
of the model in (15), and the optimality of
concavity
TP in (16), and the second inequality holds due
P
u∈U u
to feasibility of a solution to the problem in (16) for the
problem in (17). Using (20) in (19) we obtain the bound
TP
TP
ŵ
u∈U Pu −
u∈U Pu
,
(21)
ξ≤
ŵ u∈U PuTP + U · w̆
Figure 5 Bound on the LD power difference between two
optimization objectives. Upper-Bound in (21) on the LD power
difference between the solutions of the problems in (16) and (17) for
U = 2 users.
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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DSM based on successive SINR-approximation and
geometric programming
Geometric programs (GP) are a class of problems which
is not convex but can easily be converted into a convex form by logarithmic transformations [14]. This optimization model was applied to power control in [15,39],
where also successive GP approximations were proposed
for non-convex problems based on monomial [14] or
SINR approximations [11]. For a short introduction to
GP and the corresponding problem transformation of
the LD power optimization problem in (17) we refer to
Appendix 4.
As mentioned above our motivation for applying successive GP is to solve numerous small problem instances
(U = C = 2) in order to identify problem parameters which lead to a substantial gain ξ by LD power
optimization compared to TP optimization. We generated numerous problem instances of (16) and (17) by
setting Hc21 and Hc12 to all combinations out of the set
{−90, −67.5, −45, −22.5, 0}dB, and for each of these combinations forming all target-rate combinations sampling
the users’ possible rates at 20 equi-distant rate-levels
from 0 to the maximum achievable rate (i.e., 400 ratecombinations)d . After running successive GP for the problems in (16) and (17) we re-initialize the algorithm with
the obtained result for the respective other problem and
keep the best solution found for each probleme . Also,
we multiply the per-user sum-powers by a factor of 500
before applying the LD power model in order to obtain
a more realistic estimate of the LD power savingsf . The
result of this experiment can be summarized as follows:
Significant values of ξ occurred under unsymmetric settings of target-rates and crosstalk coefficients, especially
so when the stronger disturber is the one having the
larger target-rate, cf. Figure 6. Intuitively this kind of setup
results in one user operating with low sum-power (where
the derivative of the LD power model in (15) is high) while
the user with the larger target-rate operates with higher
sum-power (corresponding to a lower derivative of the LD
power model in (15)). From a sum-power perspective it
may make sense to allow the strong disturber to interfere with the weak disturber due to his higher target-rates.
However, from an LD power perspective the user with
the low target-rates is worth protecting more due to the
larger derivative of the LD power model at low sum-power
values, cf. the LD power model in Figure 2.
In the following section, we select a specific scenario
based on these insights for further investigation.
Global solutions of non-convex LD power optimization
problems using difference-of-convex-functions
programming (DCP)
Difference-of-convex-functions programming (DCP) [40]
is a widely applicable approach in global optimization
where non-convex objective and constraint functions are
reformulated as the difference of convex functions, cf.
[37,38] for recent applications in power control. Similarly to the reformulation shown in [37,38] for a ratemaximization problem, the rate-constraints in (16b) can
be equivalently written as
ruc pc + Ru = gu (p) − hu (p) ≤ 0, u ∈ U , (23)
−
c∈C
Rate of the Weak Disturber [bits/frame]
programming (GP) approximation used in order to identify problem parameters under which differences between
the optimal solutions of the two optimization problems
occur. While it is known [15] that the TP optimization
problem can be approached by GP, our contribution is to
recognize this fact for the LD power optimization problem. The second approach is based on the globally optimal
solution of both problems in (16) and (17). Global optimality is a necessary property to study the power-region
in Definition 1 and the location of the solutions to the
problems in (16) and (17) in this region. Furthermore, it
allows us to provide an exemplary scenario where provably a difference between the solutions of the two optimization problems occurs. For solving these non-convex
and rate-constrained LD power and TP optimization
problems we found it necessary to develop a problemspecific algorithm. It deviates in various aspects from
the approaches proposed for related rate-maximization
problems in [37,38], e.g., it allows for an optimization
over all subcarriers including a non-convex constraint
set, and uses improved branching and bounding techniques. The third approach is by the heuristic successive convex approximation algorithms proposed in [2,11],
respectively. These two algorithms allow to study problem
instances of realistic size and channel parameters.
Page 8 of 17
H12
= −45 dB, H21
= 0 dB
c
c
3
20
0
2.5
= −67.5 dB, H21
= 0 dB
H12
c
c
20
0
1.5
H12
= −67.5 dB, H21
= −22.5 dB
c
c
20
0
0
2
1
0.5
20
40
Rate of the Strong Disturber [bits/frame]
Figure 6 Gain by direct LD power optimization. Percent gain ξ by
direct LD power optimization compared to TP optimization obtained
through the suboptimal successive GP algorithm.
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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Page 9 of 17
and geometric programming’g . In Figure 7, we show
⎞
⎛
the power-regions and the solutions of the problems
in (16) and (17) for varying crosstalk parameter Hc21 .
c c
c c
log ⎝Huu
pu +
Huj
pj + Nuc ⎠+Ru ,
gu (p) = −
First, we see that both solutions PTP and PLD lie on the
c∈C
j∈U \{u}
power-region, as predicted by Proposition 1. However,
(24) the solutions lie on different contour lines of the function
√
u∈U Pu , meaning that they provably differ in terms of
LD power consumption. While the TP solution minimizes
⎞
⎛
[ 0.5, 0.5] ·P over the power-region, the LD power opti
c c
log ⎝
Huj
pj + Nuc ⎠ , u ∈ U ,
hu (p) = −
mal solution minimizes [ 0.17, 0.83] ·P. In other words,
c∈C
j∈U \{u}
the LD power optimum is attainable by a weighted sum(25) power optimization with specific weights. Searching for
these weights is in fact the idea behind the projected
are convex functions. Writing the objective in (17a) gradient heuristic indicated in Section ‘Analysis of the
formally
as0 − h0 (p) with convex function h0 (p) = optimization problems in (16) and (17)’. With a decreas
u
− u∈U
c∈C pc we can write the problem in (17) as
ing parameter H121 the needed sum-powers for constant
the following DCP problem [40],
target-rates decrease, leading to a decrease of the achievable gain ξ by LD power optimization compared to TP
−
h
(p)
(26a)
minimize
0
puc ,u∈U ,c∈C
minimization, cf. Figure 7.
subject to gu (p) − hu (p) ≤ 0, u ∈ U
(26b)
where
Constraints (16c)–(16e).
An experiment in real-sized DSM problems using heuristics
(26c)
In this section, we compare solutions obtained by two
DSM heuristics and static spectrum management (SSM)
in terms of their LD power: (a) the successive convex
approximation algorithm [3] for the problem in (16) which
is based on the convex approximation r̃cu (pc ) of the ratefunction rcu (pc ) as given in Appendix 4 and introduced in
[11] for a rate-maximization problem in DSL; (b) the successive LP approximation algorithm in [2] for the problem
in (17) which mainly differs from the above approximation heuristic in that the approximation is linear and
the approximated problems are not solved iteratively but
jointly for all users, and (c) single-user water-filling considering a static background noise including the highest
possible crosstalk noise based on the other systems transmitting at PSD mask. A novelty we introduce for the
comparison of suboptimal DSM algorithms is that after
Transmit−Power User 2 [mW]
While in previous applications of DCP in the area of
power control [37,38] the problem was in fact solved
as a concave minimization problem over a convex constraint set, we have additionally complicating DCP constraints in (26b). Correspondingly we developed a more
general solution approach, namely a box-based branchand-reduce algorithm initialized by a successive GP [15]
solution, cf. Appendix 5 for details. Note that this DCP
algorithm can similarly be applied to (optimally) solve the
TP problem in (16).
We use the developed global optimal algorithm to investigate the power-region as given in Definition 1. For
reasons of tractability we restrict ourselves to a specific
scenario (U = C = 2) identified using the heuristic in
Section ‘DSM based on successive SINR-approximation
2
1.5
Contours of the sum−of−square−roots
Power Region
TP solution
LD power solution
1
1
0.5
0
0
H1
21
1
21
1
H21 ... −22.5 dB
ξ ... 1.58 %
T
H21 ... −27.5 dB
H
ξ ... 0.88 %
ξ ... 1.21 %
... −25 dB
... −32.5 dB
[0.5,0.5]
[0.17,0.83]T
ξ ... 0.43 %
2
4
6
8
Transmit−Power User 1 [mW]
10
12
Figure 7 Location of optimization solutions in the power-region. Locations of the solutions to the studied problems in (16) and (17) and the
gain ξ by the latter in terms of the LD power consumption.
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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obtaining the result of a DSM scheme we initialize the
respective other DSM algorithm with this result and keep
the best solutions in terms of LD power and TP objective, respectively. The purpose of this strategy is to avoid
the dependency of the comparison on the initialization
which might have been chosen in favor of one of the
algorithmsh . The difference to the initialization approach
in Section ‘DSM based on successive SINR-approximation
and geometric programming’ is that we cross-initialize
two heuristics, while in Section ‘DSM based on successive
SINR-approximation and geometric programming’ we
applied a single heuristic to two different problems.
Based on the insights of the two previous sections we
design a network scenario with realistic parameters where
we would expect a difference in LD power between the
two considered optimization approaches. This is with
respect to the selected channel model (a 99% worst-case
model [30]), the network topology (a near-far scenario
with one CO deployed line and 7 cabinet deployed disturbers), the bandplan (showing strong crosstalk with the
CO deployed line, see below), the target-rates (low rates
for the CO deployed victim line and high rates for the cabinet lines), and the selected DSL systems (the LD power
model for the VDSL cabinet lines has a lower slope than
that for the ADSL2+ CO line, cf. Figure 2). More precisely,
we consider the near-far downstream scenario shown in
Figure 8 with 8 lines deployed in the same cable bundle,
where 7 VDSL lines are deployed from a cabinet and one
ADSL2+ line is deployed from the CO. We set the parameters of the ADSL2+ line in accordance with the standard
in [41] (using the non-overlapping bandplan with ISDN in
Annex A) and of the VDSL lines according to [42] with
a total SNR gap of = 12.3 dB in both systemsi . The
assigned target-rates are 1, 2, or 3 Mbps and 10, 13, 16, or
19 Mbps for the CO and cabinet deployed lines, respectively, and we investigate all 12 combinations of these
target-ratesj .
We observed that due to the heuristic nature of both
algorithms the LD power optimization did not always give
a better total LD power than the TP optimization (corresponding to a negative gain ξ in (19)). In summary, the
gain ξ in (19) was in the studied 12 scenarios between
−0.01% and +0.01%. DSM gives a more substantial LD
1800 m
Central Office
Cabinet
1500 m
1500 m
Figure 8 Constructed network example with 7 cabinet-deployed
lines disturbing a single CO-deployed line.
Page 10 of 17
power reduction compared to SSM between 20% and 40%.
While this result is no definite answer to whether or not
LD power optimization makes a difference compared to
TP optimization, it is another indication that in practice
the difference may be assumed negligible, which motivates
the simplification of the optimization in this direction.
However, multi-user DSM bares a substantial potential
for energy-reduction compared to SSM, as we shall study
further in a larger set of scenarios in the following section.
Average performance evaluation
Differently to the previous section we will next study
the possible LD power reduction by TP optimization
(DSM) compared to SSM in 300 randomly generated network topologies with simulation parameters as specified
in Section ‘An experiment in real-sized DSM problems
using heuristics’. More precisely, we study two deployment scenarios, where the first one consists of 15 ADSL2+
lines with loop-lengths uniformly sampled between 800 m
and 1600 m. The second type of scenarios consists of 15
VDSL cabinet-deployed lines with loop-lengths between
300 and 800 mk . We compare the TP optimization algorithm in [3] and the SSM algorithm as described in
the previous section. Target-rates are set by multiplying
the (scenario dependent) maximum achievable per-user
rates as achieved by the heuristic in [2] by factors of
{0.2, 0.4, 0.6, 0, 8}. Differently to above, the crosstalk channel model is based on measurements in [43], where we
perform a random cable selection for each network sample. Summarizing, the simulation setup does not exaggerate the inter-user crosstalk (e.g., by near-far scenarios
or worst-case crosstalk couplings) and therefore provides
a realistic evaluation of the energy savings by multi-user
DSM compared to SSM.
Next, we present the average LD power consumption
results together with 99% confidence intervals according
to a student t-test. The average LD power consumption in
the ADSL2+ scenarios obtained by the sum-rate maximizing DSM algorithm in [2] leads already to an LD power
reduction compared to (spectral mask and sum-power
constrained) full-power transmission of 38.70% (±0.97%),
which has to be compared to the maximum possible savings by TP reduction (which is obtained by reducing the
TP to zero) of 85.69%. Hence, even rate-maximizing DSM
can be regarded as an energy saving technology, as already
argued in [44]. In the VDSL scenarios the sum-rate maximization leads to an LD power reduction compared to
full-power transmission of 9.10% (±0.46%). The maximum possible savings are now only 32.14%, due to the
lower sum-power constraint as enforced by the spectral
mask, cf. the LD model for VDSL in Figure 2.
The additional savings by energy-efficient (EE) DSM
compared to rate-maximizing DSM are shown in
Figures 9 and 10. In Figure 9, we see that in the ADSL2+
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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Page 11 of 17
LD Power Reduction [%]
80
60
40
Zero TP vs. Max−Rate DSM
EE−DSM vs. Max−Rate DSM
EE−SSM vs. Max−Rate DSM
EE−DSM vs. EE−SSM
20
0
20
30
40
50
60
Fraction of the Max−Rate [%]
70
80
Figure 9 LD power savings achieved by various TP optimization strategies in ADSL2+.
scenarios multi-user DSM gives (on average) more than
70% LD power reduction at 80% of the maximum rates
compared to sum-rate maximizing DSM, whereas SSM
only results in less than 11% LD power reduction. Hence,
DSM gives substantial improvements compared to SSM,
most noticeable at higher rates. In the VDSL scenarios the
conclusions are qualitatively similar. However, as shown
in Figure 10, the LD power reduction at 80% of the maximum rates is now only 23%, whereas SSM results in less
than 7% LD power reduction.
Multi-user DSM was seen to give substantial energy savings compared to static spectrum management in a large
set of DSL scenarios. Furthermore, through an empirical
simulation study we were able to identify small DSM problem instances where the TP and the LD power optima
provably differ in terms of LD power consumption. However, we were not able to reproduce this difference in simulations for systems of practical size, which suggests that
the multi-user DSM problem can be simplified by optimizing TP instead of LD power at negligible performance
loss.
Conclusions
We derive novel realistic models of the line-driver (LD)
power consumption in class-AB and G LDs as a function of the transmit power (TP) in digital subscriber lines
(DSL). These models include non-idealities of the power
supply and therefore result in more accurate, higher
figures of LD power consumption. Based on the functional
shape of the class-AB LD model we exemplarily study its
optimization by dynamic spectrum management (DSM).
Appendix 1
Derivation of the class-AB LD model
In this appendix, we detail the derivation of (2) based on
(1), adapted from [26]. The output power in (1) is defined
as
Pu =
2
Vrms,ideal
Rline
,
(27)
LD Power Reduction [%]
30
25
20
15
10
5
0
20
30
40
50
60
Fraction of the Max−Rate [%]
Figure 10 LD power savings achieved by various TP optimization strategies in VDSL.
70
80
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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where Vrms,ideal =
2
E{|VO |2 }, VO ∼ N (0, Vrms,ideal
) is the
normal distributed output voltage (cf. Figure 1), Rline =
Rline /n2 is the transformed resistance of the line, and n is
the transformer ratio. The average dissipated power Pdiss
can be decomposed into the quiescent power PQ and the
dissipated power associated with the voltage drop in the
class-AB design [46], according to
|VO |
+ Pu (28a)
Pdiss + Pu = PQ + E (Vs − |VO |) Rline
Vs
(28b)
= PQ + E {|VO |}
Rline
Vs
2
(28c)
Vrms,ideal ,
= V s · IQ + π
Rline
where Vs is the supply voltage and in (28a) we use (27), cf.
[26] for details. Equation (2) derives by (1) and using (27)
in (28c).
Appendix 2
Model of a class G LD with two supplies
The power consumption of a class-G LD with two supply
voltages is given as
PLD(G−2S) = PLow,2S + PHigh,2S + PQ,2S + PHybrid , (29)
where PLow,2S and PHigh,2S are the consumed powers when
the supply voltage is VL and VH , respectively, PQ,2S =
(VL (1−μ2S (Pu ))+VH μ2S (Pu ))·IQ is the quiescent power,
and μ2S (Pu ) ∈[ 0, 1] is the fraction of time the high supply
voltage is active. Assuming a threshold Vth = (VL −Vdrop )
for switching between the two supplies, where Vdrop is
the voltage drop in the class-AB design, and that the LD’s
output voltage VO is Gaussian distributed [26] with zero
2 , we have μ (P ) = 2Q( Vth ),
mean and variance Vrms
2S u
Vrms
Q(·) denoting the Q-function. Furthermore, PLow,2S is
computable as (see [26] for a similar derivation)
Vth
2
− x2
2VL
x
2Vrms dx
e
PLow,2S = √
Rline TL m+1
2πVrms
0
m
= VL ·
⎛
2 Pu ⎝
· 1−e
π Rline
−(VL −Vdrop )2
2
2Pu R
(TL m+1
m )
line
⎞
⎠,
(30a)
(30b)
where the term Rline TL m+1
m in (30a) accounts for the total
LD output resistance, and in (30b) we use the definition
of Vrms in (6). Similarly, the power consumption when the
supply with the higher voltage level VH is active is derived
as
−(VL −Vdrop )2
2 Pu
m+1 2
· e 2Pu Rline (TL m ) .
(31)
PHigh,2S = VH ·
π Rline
Page 12 of 17
These formulas are equivalent to those shown in [4,26],
with the exception of the quiescent power calculation and
the consideration of the resistance Rline at the primary
transformer side, the voltage drop Vdrop , the transformer
loss TL, and the synthesis factor m in the computation of
the voltage-level probabilities. Not included in (29) are the
extra power losses due to the necessary DC-DC conversion, cf. the discussion in Section ‘Line driver modeling’.
We note that the dynamic power (the sum of (30a) and
(31)) can also be written as the sum of the power consumed by a supply always working at VL , and that of a
supply delivering (VH − VL ) during a fraction μ2S (Pu ) of
the time, cf. the class-G LD model with charge pump in
Section ‘Class-G line-driver power model’ that is based on
this interpretation.
Appendix 3
Proof of Proposition 2
Proof. For U = 1 and arbitrary C the objective in
(17a) is simply a single non-linear, monotonously increasing function (a square-root) of the user’s sum-power, and
omitting this function does therefore not change the optimum of the problem in (17) [47], yielding an identical
formulation as of the transmit power minimization problem in (16). In the case of C = 1 and arbitrary U the
target-rates in (16b) uniquely define the minimal peruser transmit powers necessary to support the target-rates
[48]. However, as the LD power model in (15) as a function of the per-user transmit sum-power is monotonously
increasing, any other power allocation feasible in (17b)
than this minimal one would have a higher LD power
consumption, and the minimum TP solution for the problem in (16) is therefore also optimal in the LD power
minimization problem in (17).
Appendix 4
A geometric programming (GP) approach for LD power
optimization
GPs consist of posynomial objective and inequality constraints, as well as monomial
equality constraints. Posynomial functions are sums K
k=1 fk (p) of monomial funcαk
αk
2
1
tions fk (p) : RCU
+ → R of the form fk (p) = ck · p1 · p2 ·
αk
CU
, where ck ≥ 0 and αik ∈ R, 1 ≤ i ≤ CU. We refer
. . . · pUC
to [14] for a more detailed introduction to GPs. Introducing
variables tu , u ∈ U , for the sum-power terms
auxiliary
u in (17a) we obtain the equivalent formulation
p
c∈C c
√
tu
(32a)
puc ≤ 1, ∀u ∈ U ,
(32b)
Constraints (16b)-(16e).
(32c)
minimize
u
pc ,tu ,u∈U ,c∈C
subject to tu−1 ·
u∈U
c∈C
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
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According to the definitions above, the objective in (32a)
is a posynomial function and the auxiliary constraints in
(32b) have posynomial form [14]. As noted in [15] the
constraints in (16b) can also be written as posynomial
constraints when using for instance the SINR approximation [11] rcu (pc ) ≈ r̃cu (pc ) = αcu log2 (SINRuc (p̃c )) +
βcu , c ∈ C , u ∈ U , where SINRuc is the SINR in (14)
and p̃uc , c ∈ C , u ∈ U , is the approximation point. To
see this, one needs to introduce additional
t̃cu ,
variables
ui
i
c ∈ C , u ∈ U , replacing the total noise ( i∈U \u Hc pc +
c. The thereby creNcu ) user u receives on subcarrier
ated additional constraints t̃cu ≥ ( i∈U \u Hcui pic + Ncu ),
c ∈ C , u ∈ U , are posynomial expressions. Under these
additional variables the constraints in (16c) and (16d)(16e) can be seen to be already given in posynomial and
monomial form, respectively. Hence, we have that the
problem in (32) can be approximated as a GP which is
efficiently and optimally solvable by convex optimization
software [49].
Page 13 of 17
value based on a fixed ratio to the value of the maximal
element in splitting dimension.
Another technique integrated in Algorithm 1 is that of
range reduction [51,52]. Briefly speaking, bounds of constraints in the LP used to compute lower bounds can
be tightened based on the obtained optimal dual variables associated with these constraints and the current
incumbent solution, cf. [51,52] for details. Note that we
omitted any local search step for improving the incumbent
solution as is typically done in continuous BnB methods
[52]. We believe the incumbent initialization in Line 1
by the successive geometric programming described in
Appendix 4 is tight enough for the considered applications
to make such a local search in the BnB process redundant. We refer to [50] for a detailed description of a basic
simplicial branch-and-bound algorithm applied to a general DCP problem, and to [51] for an introduction to the
range-reduction technique, as well as to [53] for an application of range reduction in a specific DCP problem with
DCP functions in the objective only.
Appendix 5
A box-based branch-and-reduce algorithm
Algorithm 1 schematically describes the proposed scheme
for global optimization of the DCP problem in (26). The
idea behind the method is to first enclose the set defined
by the mask-constraints in (26c) by a box, cf. Line 2, and
to successively split this set (“branching”) into smaller
boxes, cf. Line 4. We observed that box-based branching repeatedly outperforms simplicial branching [50]. We
believe this is due to the conservative initial search space
in simplicial branching,
which is a simplex with cor
ner points 0, ( u∈U ,c∈C p̂uc )eu , u ∈ U , where eu is the
u’th unit vector. Lower bounds on the objective value in
any box are computed by linear programming (LP) after
linearly approximating (underestimating) all convex functions gu (p) and all concave functions −hu (p) , u ∈ U ,
cf. Line 5. The fact that such a linear underestimation of
convex and concave functions can easily be found [50] is
the key advantage of the DCP formulation in (26). Differently to [50] we propose to apply linear approximations of
all convex functions gu (p) , u ∈ U , not only on a single
point but on various points in the considered box, e.g., in
regular intervals between the center point and each corner point. Based on the lower-bounds and the best feasible
solution found so far (the “incumbent”) the created boxes
are either further split or discarded if the lower-bound
lies above the upper bound, cf. Line 8. More precisely, in
[37] a transformation of variables into dB-scale was proposed. Similarly we perform the branching (bisection) in
dB-scale, which has the advantage that we still consider
the full search-space beginning at a power allocation of
zero. More precisely, in Line 4, we subdivide a box along
its longest edge in dB-scale. In case the value of the minimal element in splitting dimension is zero we use a lower
Algorithm 1 Box-based Branch-and-Reduce Algorithm
1: Initialize the incumbent using a heuristic solution
based on successive geometric programming, cf.
Section ‘DSM based on successive
SINR-approximation and geometric programming’.
2: Initialize the first open, currently active box with
minimal and maximal corner-points 0 ∈ RUC and
p̂ ∈ RUC .
3: while {Any box is open} do
4:
Branching: Generate two new open boxes by
splitting the currently active box in half in
dB-scale in the dimension of its longest edge.
5:
Bounding: Compute objective lower bounds for
both new boxes using an underestimating LP [50]
to the DCP problem in (26) with reduced
variable ranges [51].
6:
Reduction: Try a range-reduction based on the
current incumbent solution [51], and repeat the
lower-bound LP if a range-reduction was
achieved.
7:
Incumbent Update: Update the incumbent
by testing the 2CU−1 new corner points created
through branching and the LP solutions for
feasibility in (26).
8:
Pruning: Close all boxes with a lower bound above
the incumbent solution.
9:
Selection: Choose the open box with the lowest
lower bound as the new active box.
Endnotes
parameters chosen for ADSL2+ are Rline = 100,
n = 1.25, CF = 5.3, P̂u = 19.5dBm, TL = 0.5dB, m = 5,
IQ = 5mA, Vdrop = 4V, and PHybrid = 0. The parameters
a The
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
http://asp.eurasipjournals.com/content/2012/1/226
for VDSL deviating from these values are IQ = 11.1mA
and P̂u = 11.5dBm.
b The selected profiles correspond to downstream ADSL2+
(Annex A) [41] and VDSL2 [45] profiles 8b (Annex A),
17a (Annex B), and 30a. The chosen parameters common to all LD models are Rline = 100, n = 1.4 (as in
[29]), P̂u = 20.5dBm (14.5dBm) for ADSL2+ and VDSL2
profile 8b (VDSL2 profile 17a and 30a), TL = 0.5dB,
m = 5, Vdrop = 5V, and PHybrid = 0. For the class-AB
model we assume CF = 5.3. The quiescent currents
IQ ∈ 0.95 ∗ {7.6, 9.8, 12, 18}mA for the four profiles were
selected according to the values suggested in [29] and
scaled by a factor of 0.95 that accounts for the diversion of
quiescent current to the load [28]. While for the class-AB
LD the optimal supply voltage in (3) is assumed, for the
class-G LD with two supplies we consider VH = 24V, and
for the LD with charge pumps we set VH = 24V +Vdrop,cp ,
IQ,classG = 0.3mA, LclassG = 0mW, and ρ = 1.5dB, where
Vdrop,cp = 2V represents an additional voltage drop due
to the charging circuitry and a margin necessary due to
the permanent discharging of the charge pump capacitors. For both class-G LD types we set VL = 12V and
assume a threshold for switching between high and low
supply of Vth = VL − Vdrop . The usage probability μcp (Pu )
is obtained through simulations for different values of Pu .
The charge pump is assumed to be active for a time-frame
of 0.11μs (ADSL2+ and VDSL2 8b), 0.04μs (VDSL2 17a)
or 0.05μs (VDSL2 30a) when Vrms exceeds Vth . Additionally it is assumed to be active for 0.35μs and 0.5μs
before and after this time-frame, which accounts for the
charging and discharging of the charge pump capacitors,
respectively.
c The specific parameters assumed throughout the rest
of the article are those mentioned in Section ‘Class-AB
line-driver power model’ with the exception of n = 1.2,
CF = 5, and the power limit P̂u = 19.9dBm used for
ADSL2+ lines.
d The remaining relevant parameters are H uu = 1, =
c
12.3dB, = 4.3125 · 103 [Hz], Ncu = 10−140/10 · [mW],
p̂uc = 10−40/10 · [mW], u ∈ U , c ∈ C , B̂ = ∞.
e This sequential re-initialization process is stopped in
case the best solution found for both problems does not
improve for more than three consecutive iterations.
f By multiplication with 500 we heuristically scale the
transmit sum-power values to that of a system with 1000
subcarriers in order to obtain LD power values through
our LD power model which are somewhat comparable
to those under more realistic system parameters in the
following sections.
g The relevant selected parameters are those of
Section ‘DSM based on successive SINR-approximation
and geometric programming’ with the exception
of R1 = 41.36[bits/frame], R2 = 5.9[bits/frame],
Page 14 of 17
Hc12 = −67.5dB and the initial value Hc21 = −22.5dB,
c ∈ C = {1, 2}.
h The sequential re-initialization process is stopped if no
improvement of the best solution found by any of the
algorithms was detected for two consecutive iterations.
The PSD for the TP optimization and its first approximation was initialized at a low level of −120dBm per
subcarrier and user. The trust-region used in the LD
power optimization scheme in [2] is set to −70dBm per
subcarrier and user after being initialized with the solution of the sequential TP minimization algorithm in [3].
i We consider the bandplan setting for fiber-to-theexchange, mask variant B, and un-notched mask M2,
which would not be used in practice in this form due
to the high ingress noise into ADSL lines but serves
our purpose to imitate the insightful scenarios found in
Section ‘DSM based on successive SINR-approximation
and geometric programming’.
j The maximum rate for the VDSL lines in the considered
scenario as found by the LD power optimization algorithm [2] is approximately 19.9Mbps.
k Simulation parameters for both DSL technologies are as
specified in Section ‘An experiment in real-sized DSM
problems using heuristics’, except that for VDSL we use
the bandplan specified in [42] for fiber-to-the-cabinet,
mask variant A-M1.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This work has been funded by BMVIT/FFG under the program FIT-IT. The
Competence Center FTW Forschungszentrum Telekommunikation Wien
GmbH was funded within the program COMET—Competence Centers for
Excellent Technologies by BMVIT, BMWA, and the City of Vienna. The COMET
program was managed by the FFG.
Author details
1 FTW Telecommunications Research Center Vienna, Donau-City-Straße1,
A-1220 Vienna, Austria. 2 Lantiq A GmbH, Siemensstraße 4, A-9500 Villach,
Austria. 3 Centre for Research on Embedded Systems, Halmstad University, Box
823, SE-30118 Halmstad, Sweden.
Received: 16 February 2012 Accepted: 30 August 2012
Published: 25 October 2012
References
1. K Hooghe, M Guenach, in IEEE Global Communications Conference 2011
(Globecom’11). Towards energy-efficient packet processing in access
nodes (Houston, Texas, USA, 2011), pp. 1–6
2. M Guenach, C Nuzman, J Maes, M Peeters, in International Workshop on
Green Communications, IEEE GLOBECOM, vol. 2009. Trading Off Rate and
Power Consumption in DSL Systems (Honolulu, Hawaii, 2009), pp. 1–5
3. M Wolkerstorfer, D Statovci, T Nordström, in IEEE International Conference
on Communications Systems 2008 (ICCS’08). Dynamic Spectrum
Management for Energy-Efficient Transmission in DSL (Guangzhou,
China, 2008), pp. 1015–1020
4. M Guenach, C Nuzman, K Hooghe, J Maes, M Peeters, in International
Workshop on Green Communications, IEEE GLOBECOM 2010. Reduced
dimensional power optimization using class AB and G line drivers in DSL
(USA, Miami, 2010), pp. 1443–1447
Wolkerstorfer et al. EURASIP Journal on Advances in Signal Processing 2012, 2012:226
http://asp.eurasipjournals.com/content/2012/1/226
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
M Guenach, C Nuzman, K Hooghe, J Maes, M Peeters, in On power-efficient
usage of line drivers in copper-based access networks, in IEEE International
Energy Conference and Exhibition (EnergyCon’10) (Manama, Bahrain, 2010),
pp. 131–136
P Tsiaflakis, Y Yi, M Chiang, M Moonen, DSL Green, in IEEE International
Conference on Communications 2009 (ICC’09). Energy-Efficient DSM
(Dresden, Germany, 2009), pp. 1–5
M Guenach, C Nuzman, J Maes, M Peeters, in IEEE International Conference
on Communications Workshops (ICC’09). On Power Optimization in DSL
Systems (Dresden, Germany, 2009), pp. 1–5
M Wolkerstorfer, Energy-efficient resource allocation in multi-carrier
digital subscriber lines. PhD thesis. Vienna University of Technology,
Vienna, Austria, 2012
KB Song, ST Chung, G Ginis, J Cioffi, Dynamic spectrum management for
next-generation DSL systems. IEEE Commun. Mag. 40(10), 101–109
(2002)
B Johansson, P Soldati, M Johansson, Mathematical decomposition
techniques for distributed cross-layer optimization of data networks. IEEE
J. Sel. Areas Commun. 24(8), 1535–1547 (2006)
J Papandriopoulos, J Evans, S C A L E: A low-complexity distributed
protocol for spectrum balancing in multiuser DSL networks. IEEE Trans.
Inf. Theory. 55(8), 3711–3724 (2009)
P Tsiaflakis, M Diehl, M Moonen, Distributed spectrum management
algorithms for multiuser DSL networks. IEEE Trans. Signal Process. 56(10),
4825–4843 (2008)
M Wolkerstorfer, J Jaldén, T Nordström, Column generation for discreterate multi-user and multi-carrier power control. IEEE Trans. Commun.
60(9), 2712–2722 (2012). doi:10.1109/TCOMM.2012.070912.110444
S Boyd, SJ Kim, L Vandenberghe, A Hassibi, A tutorial on geometric
programming. Optimiz Eng. 8, 67–127 (2007)
M Chiang, C Tan, D Palomar, D O’Neill, D Julian, Power control by
geometric programming. IEEE Trans. Wirel. Commun. 6(7), 2640–2651
(2007)
R Horst, P Pardalos, N Thoai, Introduction to Global Optimization, Volume 48
of Nonconvex Optimization and its Applications. (Kluwer Academic
Publishers, Dordrecht, 2000), pp. 1–353
The Climate Group: SMART 2020, Enabling the low carbon economy in
the information age (2008)
EC Directorate-General JRC Joint Research Centre, Institute for Energy,
Renewable Energy Unit: Code of Conduct on Energy Consumption of
Broadband Equipment. Version 4 (2011)
R Bolla, F Davoli, R Bruschi, K Christensen, F Cucchietti, S Singh, The
potential impact of green technologies in next-generation wireline
networks: Is there room for energy saving optimization. IEEE Commun.
Mag. 49(8), 80–86 (2011)
T Piessens, M Steyaert, Design and analysis of high efficiency line drivers for
xDSL. (Kluwer Academic Publishers, Dordrecht, 2004), pp. 1–235
B Serneels, M Steyaert, Design of High Voltage xDSL Line Drivers in Standard
CMOS. (Springer, Dordrecht, 2008), pp. 1–185
M Guenach, C Nuzman, J Maes, M Peeters, Y Li, D Van Bruyssel, F Defoort,
Power efficient copper access. Bell Labs Tech J. 15(2), 117–129 (2010)
M Wolkerstorfer, D Statovci, T Nordström, Energy-Saving by Low-Power
Modes in ADSL2. Elsevier Comput. Netws Special Issue on Green
Communication Networks. 56(10), 2468–2480 (2012)
I Kamitsos, P Tsiaflakis, S Ha, M Chiang, in IEEE Global Communications
Conference 2011 (Globecom’11). Stable Sleeping in DSL Broadband Access:
Feasibility and Tradeoffs (Houston, Texas, USA, 2011), pp. 1–6
K Hooghe, M Guenach, in IEEE Symposium on Communications and
Vehicular Technology in the Benelux 2010 (SCVT’10). Impact of FTTN
architecture on access node energy efficiency (Enschede, The
Netherlands, 2010), pp. 1–5
J Pierdomenico, S Wurcer, B Day, A 684-mW adaptive supply full-rate
ADSL CO driver. IEEE J. Solid-State Circ. 37(12), 1831–1838 (2002)
S Wurcer, Carefully evaluate your ADSL line-driver efficiency. Commun.
Syst. Design. 2003, 21–23 (2003)
R Stephens, Active output impedance for ADSL line drivers. Texas
Instrum. Analog Appl. J. Q4, 24–31 (2002)
TI, H Gated-Class, Dual-Port VDSL2 Line Driver. Tech. Rep. THS 6226, Texas
Instruments (2011)
P Golden, H Dedieu, K Jacobsen, Fundamentals of DSL Technology.
(Auerbach Publications, Boca Raton, 2006), pp. 1–472
Page 15 of 17
31. R Cendrillon, F Liming, J Chou, G Long, C Hung, D Wei, in IEEE Global
Communications Conference 2008 (GLOBECOM’08). D S M from theory to
practice (New Orleans, LA, USA, 2008), pp. 1–4
32. E Medeiros, N Lindqvist, M Monteiro, H Abrahim, F Lindqvist, B Dortschy,
A Klautau, in European Signal Processing Conference 2009 (Eusipco’09). DSM
performance on practical DSL systems based on estimated crosstalk
channel information (Glasgow, Scotland, 2009), pp. 2092–2096
33. R Cendrillon, W Yu, M Moonen, J Verlinden, T Bostoen, Optimal multiuser
spectrum balancing for digital subscriber lines. IEEE Trans. Commun.
54(5), 922–933 (2006)
34. P Tsiaflakis, Y Yi, M Chiang, M Moonen, Fair greening for DSL broadband
access. ACM SIGMETRICS Perf. Eval Rev. 37(4), 74–78 (2010)
35. M Wolkerstorfer, T Nordström, Coverage optimization in DSL networks by
low-complexity discrete spectrum balancing. (Houston, Texas, USA, 2011),
pp. 1–6
36. J Brehmer, W Utschick, Nonconcave utility maximisation in the MIMO
broadcast channel. EURASIP J. Adv. Signal Process , 1–13 (2009). Article ID
645041
37. Y Xu, T Le-Ngoc, S Panigrahi, Global concave minimization for optimal
spectrum balancing in multi-user DSL networks. IEEE Trans. Signal
Process. 56(7), 2875–2885 (2008)
38. K Eriksson, S Shi, N Vucic, M Schubert, E Larsson, in IEEE Global
Telecommunications Conference (GLOBECOM’10). Globally optimal
resource allocation for achieving maximum weighted sum rate (USA,
Miami, 2010), pp. 1–6
39. M Charafeddine, A Paulraj, in Conference on Information Sciences and
Systems (CISS’07). Sequential geometric programming for 2×2
interference channel power control (Baltimore, MD, USA, 2007),
pp. 185–189
40. R Horst, P Pardalos, Handbook of Global Optimization, Volume 2 of
Nonconvex Optimization and its Applications. (Kluwer Academic Publishers,
Dordrecht, 1995), pp. 1–575
41. ITU-T, Asymmetric digital subscriber line (ADSL) transceivers - Extended
bandwidth ADSL2 (ADSL2+). G.992.5 (2005)
42. ETSI, Transmission and Multiplexing (TM); Access transmission systems on
metallic access cables; Very high speed Digital Subscriber Line (VDSL);
Part 1 Functional requirements. TM6 TS 101 270-1, Version 1.3.1 (2003)
43. D Statovci, T Nordström, in European Signal Processing Conference 2004
(Eusipco’04). Adaptive resource allocation in multiuser FDD-DMT systems
(Vienna, Austria, 2004), pp. 1213–1216
44. J Cioffi, H Zou, A Chowdhery, W Lee, S Jagannathan, in IEEE Global
Telecommunications Conference 2008 (GLOBECOM). Greener copper with
dynamic spectrum management (New Orleans, LA, USA, 2008), pp. 1–5
45. ITU-T, Very high speed digital subscriber line transceivers 2 (VDSL2).
G.993.2 (2006)
46. R van der Zee, High Efficiency Audio Power Amplifiers Design and
practical use. PhD thesis, Twente University (1999)
47. S Boyd, L Vandenberghe, Convex Optimization. (Cambridge University
Press, Cambridge, 2004), pp. 1–716
48. R Yates, A framework for uplink power control in cellular radio systems.
IEEE J. Sel. Areas Commun. 13(7), 1341–1347 (1995)
49. ApS Mosek, The MOSEK optimization toolbox for MATLAB manual. Version
6.0 (Rev 106), Copenhagen, Denmark, (2011)
50. R Horst, N Thoai, DC Programming: Overview. J. Optimiz. Theory Appl.
103, 1–43 (1999)
51. H Ryoo, N Sahinidis, A branch-and-reduce approach to global
optimization. J. Global Optimiz. 8(2), 107–138 (1996)
52. N Sahinidis, BARON - Branch and Reduce Optimization Navigator. Version
4.0, University of Illinois at Urbana-Champaign (2000)
53. R Cambini, F Salvi, A branch and reduce approach for solving a class of
low rank d.c. programs. J. Comput. Appl. Math. 233, 492–501 (2009)
doi:10.1186/1687-6180-2012-226
Cite this article as: Wolkerstorfer et al.: Modeling and optimization of
the line-driver power consumption in xDSL systems. EURASIP Journal on
Advances in Signal Processing 2012 2012:226.
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