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Modeling and optimization of the linedriver power consumption in xDSL systems
EURASIP Journal on Advances in Signal Processing volume 2012, Article number: 226 (2012)
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 circuitlevel nonidealities. Based on the model of a classAB LD we analyze the multiuser 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.
Introduction
This article analyzes the modeling and optimization of the power consumption in multicarrier digital subscriber line (DSL) transceivers. The linedriver (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 lowcomplexity 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 energyefficiency 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 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 classAB 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 differenceofconvexfunctions 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 energyefficient multiuser DSM compared to static spectrum management and ratemaximizing DSM. Our conclusions are provided in Section ‘Conclusions’.
Background information
Energyefficiency 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 frontend (the modem’s digital signal processing); the analog frontend (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 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 lowpower modes. Our focus is on the first two approaches, as we a) model the power consumption of an energyefficient LD type, and b) study energyefficient 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.
Line driver modeling
Current DSL systems rely on so called classAB 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 rootmeansquare (rms) value. Even though those peak values occur with very low probability, the fixed supply voltage of a ClassAB LD must be sufficiently high to provide distortionfree 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 socalled classH LDs. ClassG LDs[20] are classAB LDs where the supply rail is switched, e.g., between a lower and a higher voltage level V_{ L } and V_{ H }, respectively. The design of a classG 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 DCDC converter is required which must be included in the LD efficiency calculation. A classH LD can be seen as a classG 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 classG 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 classAB LD in ADSL2+ was presented in[2]. However, this model is not applicable to other DSL technologies or systems with different physical parameters. A circuitlevel model for an LD of classAB 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 nonidealities 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 ‘ClassAB linedriver power model’, we derive an enhanced classAB LD power consumption model based on[26] that can be applied to any DSL profile, and in Section ‘ClassG linedriver power model’, we propose a novel model for a classG LD with charge pumps.
Line driver models
ClassAB linedriver power model
In this section, we enhance the functional classAB LD model in[26] based on a circuit analysis, cf. Figure1. The total power consumed by a classAB LD is given as
where P_{ u }is the output power measured at the LD output of a line indexed by\left(\right.separators="">\n \n u\n \u2208\n U\n \n, P_{diss} is the total power dissipated inside the LD, and P_{Hybrid} is the power consumed by the hybrid circuit. The level of P_{Hybrid} 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
where V_{ s } is the supply voltage of the LD, I_{ Q } is the quiescent current, and\left(\right.separators="">\n \n \n \n R\n \n \n line\n \n \n \u2032\n \n \n \n as defined in Appendix 1 is the transformed resistance of the line, cf. Figure1. In the ideal case the supply voltage V_{ s } can be designed to cover the output voltage swing CF·V_{rms,ideal} described by the signal crest factor CF and the ideal rms LD output voltage V_{rms,ideal}[26].
However, a more realistic representation of V_{ s }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 V _{ s }by a voltage drop V _{drop}. 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 nonidealities 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 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 overall 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\left(\right.separators="">\n \n \n \n V\n \n \n rms\n ,\n ideal\n \n \n =\n \n \n \n \n P\n \n \n u\n \n \n \xb7\n \n \n R\n \n \n line\n \n \n \u2032\n \n \n \n \n \n these additional factors can be accommodated in the form
where\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n \n is the maximum transmitted power. Figure2 depicts an exemplary measurement of a real ADSL2+ LD’s power consumption, as well as the classAB LD power consumption model^{a} in (2), using (a) the mentioned ideal relation V_{ s }= CF·V_{rms,ideal}, (b) the relation V_{ s }= CF·V_{rms,ideal} + V_{drop} with headroom V_{drop} 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 classAB LDs and the simple functional shape of our model we will focus on this LD type when analyzing the effect of the LD on energyefficient DSM in Sections ‘Optimization models and analysis’, ‘Empirical optimization study’, and ‘Average performance evaluation’. Another energysaving approach mentioned in Section ‘Energyefficiency in DSL’ is the deployment of more energyefficient LDs, as analyzed in the following section.
ClassG linedriver power model
Based on our discussion in Section ‘Line driver modeling’, we study in this section classG 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 ‘ClassAB linedriver power model’ into the model in[4, 26]. The basic principle of a charge pump is exemplified in Figure3. A pair of such charge pumps is used to generate the high classG supply voltage V_{ H }from a single LD supply voltage which at the same time serves as the low classG supply voltage V_{ L }. Under ideal conditions, a maximum voltage ratio of V_{ H }/V_{ L }= 3 can be achieved. However, taking technological limitations and various internal voltage drops into account, assuming a ratio of V_{ H }/V_{ L }≈ 2 is more realistic. The total power consumption of a classG LD with internal charge pumps is defined as
where P_{Low,CP} is the LD power consumption value of an equivalent classAB LD running continuously at the low voltage supply, and P_{High,CP} refers to the additional power consumption when the LD is switching to the high voltage supply. P_{Q,CP} is the quiescent power dissipation. The voltage level V_{ H } is thought of as the summation of V_{ L }and V_{ H }−V_{ L }, with the latter being generated by the charge pumps when needed. The consumed LD power at the low voltage V_{ L } is in analogy to (2) defined as
Extending the ideal rms voltage\left(\right.separators="">\n \n \n \n V\n \n \n rms\n ,\n ideal\n \n \n =\n \n \n \n \n P\n \n \n u\n \n \n \xb7\n \n \n R\n \n \n line\n \n \n \u2032\n \n \n \n \n \n with the nonidealities of Section ‘ClassAB linedriver power model’ we obtain the rms LD output voltage (that is, before impedance synthesis and transformer) as
In analogy to the ideal classG 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 V_{th} = (V_{ L }−V_{drop}) is given as
and
respectively. Note that the fraction of time μ_{cp}(P_{ u }) ∈ [0,1] the charge pump is used is higher than the time the output signal exceeds the threshold V_{th}, the reason being the additional rampup / rampdown 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 V_{th}, respectively, weighted by the corresponding probabilities. The output signal’s MAD under the assumption of operating below and above the threshold is given by\left(\right.separators="">\n \n \n \n V\n \n \n MAD\n ,\n Low\n \n \n /\n (\n 1\n \u2212\n 2\n Q\n (\n \n \n \n \n V\n \n \n th\n \n \n \n \n \n \n V\n \n \n rms\n \n \n \n \n )\n )\n \n and\left(\right.separators="">\n \n \n \n V\n \n \n MAD\n ,\n High\n \n \n /\n (\n 2\n Q\n (\n \n \n \n \n V\n \n \n th\n \n \n \n \n \n \n V\n \n \n rms\n \n \n \n \n )\n \n, respectively. Correspondingly we define the dynamic power P_{High,CP} as
where
μ_{ B }= 1, Q(·) is the Qfunction, ρ is the recharge loss, and the term\left(\right.separators="">\n \n \n \n R\n \n \n line\n \n \n \u2032\n \n \n TL\n \n \n m\n +\n 1\n \n \n m\n \n \n \n represents the total resistance at the LD output. Comparing the total dynamic power (the sum of (5) and (9)) to that under a classG design with two supplies (the sum of (30a) and (31) in Appendix 2) we find that the latter one is obtained by setting μ_{ A }= 0 and ρ = 1. We emphasize that μ_{cp}(P_{ u }) 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 classAB LDs, of three main components, given as
The term P_{Q,Low}= V_{ L }·I_{ Q }is the quiescent power dissipation of an equivalent classAB LD continuously working at V_{ L }. The additional quiescent power dissipation of the LD when working at the high voltage supply is defined as
The third term in (11) splits into
where I_{Q,classG} is the additional quiescent current in classG mode and L_{classG}[W] refers to further fixed losses in the classG circuitry.
In Figure4, we compare the power consumption data provided for a real classG LD with charge pumps in[29] to the three discussed LD models: our model of a classG 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 classAB LD modeled by Equations (2) and (3), respectively^{b}. In Figure4a we see that under an ADSL2+ profile the power consumption predicted by our model of a classG LD with charge pumps lies between that calculated by the models of a classAB LD and a classG LD with two supplies. Figure4b shows that the classG 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 μ_{cp}(P_{ u }) and μ_{2S}(P_{ u }) of using the high supply in the classG LD with charge pumps and with two supplies (see Appendix 2), respectively. Correspondingly we can approximate the power consumption of a classG LD for low transmit power values by that of a classAB LD in (5) that operates at the low supply voltage. However, near the maximum output power the novel classG LD model with charge pumps significantly deviates from the classG 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 twosupply classG 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 classG 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 I_{ Q } that is independent of the transmit power.
In summary, the classG design with charge pumps yields substantial energy savings compared to a classAB LD while sparing us the DCDC conversion needed for classG 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 classAB power relation in (2). In the following sections, we focus on the classAB LD and analyze our hypothesis of Section ‘Introduction’ on the difference between LD power and TP optimization.
Optimization models and analysis
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 classAB 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 frequencydivision duplexing (FDD) and discrete multitone (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 of all users jointly. The achievable rate per DMTsymbol\left(\right.separators="">\n \n \n \n r\n \n \n c\n \n \n u\n \n \n \n \n \n \n p\n \n \n c\n \n \n \n \n \n for user\left(\right.separators="">\n \n u\n \u2208\n U\n =\n \n \n 1\n ,\n \u2026\n ,\n U\n \n \n \n on subcarrier\left(\right.separators="">\n \n c\n \u2208\n C\n =\n \n \n 1\n ,\n \u2026\n ,\n C\n \n \n \n as a function of the signal to interference and noise ratio (SINR) is modeled by the common gapapproximation[30]
where\left(\right.separators="">\n \n \n \n p\n \n \n c\n \n \n =\n \n \n [\n \n \n p\n \n \n c\n \n \n 1\n \n \n ,\n \u2026\n ,\n \n \n p\n \n \n c\n \n \n U\n \n \n ]\n \n \n T\n \n \n \n,\left(\right.separators="">\n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n is the power assigned to subcarrier c of user u, and the terms\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n uu\n \n \n \n and\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n ui\n \n \n \n are the direct channel transfer coefficient of user u and the crosschannel transfer coefficient from user i to user u on subcarrier c, respectively. DSM implementations in standardcompliant DSL systems, including crosstalk estimation functionality, have been reported in[31, 32]. The term Γ ; indicates the SNRgap to capacity depending on the modulation scheme, the targeted biterror rate, the coding gain, and the noise margins, while\left(\right.separators="">\n \n \n \n N\n \n \n c\n \n \n u\n \n \n \n represents the total received background noise power on subcarrier c of user u, including white thermal noise, aliencrosstalk, and radiofrequency interference.
Based on Section ‘Line driver models’, the LD power consumption of a classAB LD as a function of the total TP\left(\right.separators="">\n \n \n \n P\n \n \n u\n \n \n =\n \n \n \u2211\n \n \n c\n \u2208\n C\n \n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n of user u is given in the form
where the parameters\left(\right.separators="">\n \n \n \n \u0175\n \n \n u\n \n \n \u2208\n \n \n R\n \n \n +\n \n \n \n and\left(\right.separators="">\n \n \n \n \n \n w\n \n \u0306\n \n \n \n u\n \n \n \u2208\n \n \n R\n \n \n +\n \n \n \n are dependent on the hardware and system model^{c}. In the following optimization study, we will use two key features of this function: a) it is monotonously increasing, and b) concave in P_{ u } (or\left(\right.separators="">\n \n \n \n p\n \n \n c\n \n \n u\n \n \n ,\n c\n \u2208\n C\n \n, respectively).
Optimization problems
Similarly as in previous DSL studies[11, 33] we mathematically formulate the problem of minimizing the transmit sumpower in DSL in the form
where\left(\right.separators="">\n \n \n \n \n \n p\n \n \u0302\n \n \n \n u\n \n \n c\n \n \n ,\n c\n \u2208\n C\n ,\n u\n \u2208\n U\n ,\n \n denotes the PSD mask,\left(\right.separators="">\n \n \n \n B\n \n \u0302\n \n \n the maximum number of bits that can be allocated to a single subcarrier, and R_{ u } and\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n \n the targetrate per DMTsymbol and the maximum sumpower of user u, respectively. In practice numerous other objectives may be targeted besides energy consumption, including for example sumrate[33], fairness[34], service coverage[35], the energyperbit[8], or weighted combinations thereof[6]. However, our choice of focusing on energyminimization subject to rateconstraints 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
where for simplicity of exposition we assume identical LD models for all users. This allows us to omit the added constant\left(\right.separators="">\n \n \n \n \n \n w\n \n \u0306\n \n \n \n u\n \n \n \n and the factor\left(\right.separators="">\n \n \n \n \u0175\n \n \n u\n \n \n ,\n u\n \u2208\n U\n ,\n \n 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 realsized DSM problems using heuristics’. For brevity we will denote the optimal peruser sumpower values for the problems in (16) and (17) by\left(\right.separators="">\n \n \n \n P\n \n \n TP\n \n \n \u2208\n \n \n R\n \n \n +\n \n \n U\n \n \n \n and\left(\right.separators="">\n \n \n \n P\n \n \n LD\n \n \n \u2208\n \n \n R\n \n \n +\n \n \n U\n \n \n \n, respectively.
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 “powerregion”).
Definition 1 (Powerregion)
The powerregion associated with the problems in (16) and (17) is defined as the set
Proposition 1
The sumpower vectors P^{TP}and P^{LD}achieved at a solution of the power minimization problems in (16) and (17), respectively, both lie on the boundary of the powerregion\left(\right.separators="">\n \n P\n \n as defined in (18), i.e.,\left(\right.separators="">\n \n \u2204\n P\n \u2208\n P\n ,\n P\n \u2260\n \n \n P\n \n \n TP\n \n \n ,\n P\n \u227c\n \n \n P\n \n \n TP\n \n \n \n and\left(\right.separators="">\n \n \u2204\n P\n \u2208\n P\n ,\n P\n \u2260\n \n \n P\n \n \n LD\n \n \n ,\n P\n \u227c\n \n \n P\n \n \n LD\n \n \n \n.
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 sumpower minimizations with weights based on the projected gradients of the objective functions\left(\right.separators="">\n \n \n \n f\n \n \n u\n \n \n LD\n \n \n (\n \n \n P\n \n \n u\n \n \n )\n \n, cf.[36] where a similar idea was applied for a rateutility maximization problem. However, while in[36] a nonconcave maximization was performed over the rateregion, here we face a concave minimization problem over the powerregion.
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
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 energyefficient 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\left(\right.separators="">\n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n summing to the total power\left(\right.separators="">\n \n \n \n P\n \n \n u\n \n \n =\n \n \n \u2211\n \n \n c\n \u2208\n C\n \n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n. More precisely, we have
where the first inequality holds due to the monotonicity and concavity of the model in (15), and the optimality of\left(\right.separators="">\n \n \n \n \u2211\n \n \n u\n \u2208\n U\n \n \n \n \n P\n \n \n u\n \n \n TP\n \n \n \n in (16), and the second inequality holds due to feasibility of a solution to the problem in (16) for the problem in (17). Using (20) in (19) we obtain the bound
which is only dependent on the solution of the problem in (16) and illustrated in Figure5. 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 Figure5 does not allow us to make any conclusions on possible differences when all lines operate in a highpower regime. For example, if the solution to the TP minimization problem in (16) demands all users to use maximum sumpower, by sumpower 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 P^{TP}, giving
The gain ξ for U = 2 users is for instance bounded by\left(\right.separators="">\n \n 1\n \u2212\n 1\n /\n \n \n U\n \n \n \n (≈ 30%). The bounds in (21) and (22) are identical when\left(\right.separators="">\n \n \n \n P\n \n \n u\n \n \n TP\n \n \n =\n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n =\n P\n ,\n \u2200\n u\n \u2208\n U\n \n.
In this section, we have located the solutions of our two optimization problems on the boundary of a powerregion 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 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 powerregion 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 nonconvex and rateconstrained 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 ratemaximization problems in[37, 38], e.g., it allows for an optimization over all subcarriers including a nonconvex 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.
DSM based on successive SINRapproximation 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 nonconvex 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\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n 21\n \n \n \n and\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n 12\n \n \n \n to all combinations out of the set {−90,−67.5,−45,−22.5,0}dB, and for each of these combinations forming all targetrate combinations sampling the users’ possible rates at 20 equidistant ratelevels from 0 to the maximum achievable rate (i.e., 400 ratecombinations)^{d}. After running successive GP for the problems in (16) and (17) we reinitialize the algorithm with the obtained result for the respective other problem and keep the best solution found for each problem^{e}. Also, we multiply the peruser sumpowers by a factor of 500 before applying the LD power model in order to obtain a more realistic estimate of the LD power savings^{f}. The result of this experiment can be summarized as follows: Significant values of ξ occurred under unsymmetric settings of targetrates and crosstalk coefficients, especially so when the stronger disturber is the one having the larger targetrate, cf. Figure6. Intuitively this kind of setup results in one user operating with low sumpower (where the derivative of the LD power model in (15) is high) while the user with the larger targetrate operates with higher sumpower (corresponding to a lower derivative of the LD power model in (15)). From a sumpower perspective it may make sense to allow the strong disturber to interfere with the weak disturber due to his higher targetrates. However, from an LD power perspective the user with the low targetrates is worth protecting more due to the larger derivative of the LD power model at low sumpower values, cf. the LD power model in Figure2.
In the following section, we select a specific scenario based on these insights for further investigation.
Global solutions of nonconvex LD power optimization problems using differenceofconvexfunctions programming (DCP)
Differenceofconvexfunctions programming (DCP)[40] is a widely applicable approach in global optimization where nonconvex 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 rateconstraints in (16b) can be equivalently written as
where
are convex functions. Writing the objective in (17a) formally as 0−h_{0}(p) with convex function\left(\right.separators="">\n \n \n \n h\n \n \n 0\n \n \n (\n p\n )\n =\n \u2212\n \n \n \u2211\n \n \n u\n \u2208\n U\n \n \n \n \n \n \n \u2211\n \n \n c\n \u2208\n C\n \n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n \n \n we can write the problem in (17) as the following DCP problem[40],
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 boxbased branchandreduce 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 powerregion 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 SINRapproximation and geometric programming’^{g}. In Figure7, we show the powerregions and the solutions of the problems in (16) and (17) for varying crosstalk parameter\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n 21\n \n \n \n. First, we see that both solutions P^{TP}and P^{LD} lie on the powerregion, as predicted by Proposition 1. However, the solutions lie on different contour lines of the function\left(\right.separators="">\n \n \n \n \u2211\n \n \n u\n \u2208\n U\n \n \n \n \n \n \n P\n \n \n u\n \n \n \n \n \n, meaning that they provably differ in terms of LD power consumption. While the TP solution minimizes [0.5,0.5]·P over the powerregion, the LD power optimal solution minimizes [0.17,0.83]·P. In other words, the LD power optimum is attainable by a weighted sumpower optimization with specific weights. Searching for these weights is in fact the idea behind the projected gradient heuristic indicated in Section ‘Analysis of the optimization problems in (16) and (17)’. With a decreasing parameter\left(\right.separators="">\n \n \n \n H\n \n \n 1\n \n \n 21\n \n \n \n the needed sumpowers for constant targetrates decrease, leading to a decrease of the achievable gain ξ by LD power optimization compared to TP minimization, cf. Figure7.
An experiment in realsized DSM problems using heuristics
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\left(\right.separators="">\n \n \n \n \n \n r\n \n ~\n \n \n \n c\n \n \n u\n \n \n (\n \n \n p\n \n \n c\n \n \n )\n \n of the ratefunction\left(\right.separators="">\n \n \n \n r\n \n \n c\n \n \n u\n \n \n (\n \n \n p\n \n \n c\n \n \n )\n \n as given in Appendix 4 and introduced in[11] for a ratemaximization 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) singleuser waterfilling 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 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 algorithms^{h}. The difference to the initialization approach in Section ‘DSM based on successive SINRapproximation and geometric programming’ is that we crossinitialize two heuristics, while in Section ‘DSM based on successive SINRapproximation 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% worstcase model[30]), the network topology (a nearfar scenario with one CO deployed line and 7 cabinet deployed disturbers), the bandplan (showing strong crosstalk with the CO deployed line, see below), the targetrates (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. Figure2). More precisely, we consider the nearfar downstream scenario shown in Figure8 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 nonoverlapping bandplan with ISDN in Annex A) and of the VDSL lines according to[42] with a total SNR gap of Γ = 12.3 dB in both systems^{i}. The assigned targetrates 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 targetrates^{j}.
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 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, multiuser DSM bares a substantial potential for energyreduction 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 realsized DSM problems using heuristics’. More precisely, we study two deployment scenarios, where the first one consists of 15 ADSL2+ lines with looplengths uniformly sampled between 800 m and 1600 m. The second type of scenarios consists of 15 VDSL cabinetdeployed lines with looplengths between 300 and 800 m^{k}. We compare the TP optimization algorithm in[3] and the SSM algorithm as described in the previous section. Targetrates are set by multiplying the (scenario dependent) maximum achievable peruser 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 interuser crosstalk (e.g., by nearfar scenarios or worstcase crosstalk couplings) and therefore provides a realistic evaluation of the energy savings by multiuser DSM compared to SSM.
Next, we present the average LD power consumption results together with 99% confidence intervals according to a student ttest. The average LD power consumption in the ADSL2+ scenarios obtained by the sumrate maximizing DSM algorithm in[2] leads already to an LD power reduction compared to (spectral mask and sumpower constrained) fullpower transmission of 38.70% (±0.97%), which has to to the maximum possible savings by TP reduction (which is obtained by reducing the TP to zero) of 85.69%. Hence, even ratemaximizing DSM can be regarded as an energy saving technology, as already argued in[44]. In the VDSL scenarios the sumrate maximization leads to an LD power reduction compared to fullpower transmission of 9.10% (±0.46%). The maximum possible savings are now only 32.14%, due to the lower sumpower constraint as enforced by the spectral mask, cf. the LD model for VDSL in Figure2.
The additional savings by energyefficient (EE) DSM compared to ratemaximizing DSM are shown in Figures9 and10. In Figure9, we see that in the ADSL2+ scenarios multiuser DSM gives (on average) more than 70% LD power reduction at 80% of the maximum rates compared to sumrate 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 Figure10, the LD power reduction at 80% of the maximum rates is now only 23%, whereas SSM results in less than 7% LD power reduction.
Conclusions
We derive novel realistic models of the linedriver (LD) power consumption in classAB and G LDs as a function of the transmit power (TP) in digital subscriber lines (DSL). These models include nonidealities of the power supply and therefore result in more accurate, higher figures of LD power consumption. Based on the functional shape of the classAB LD model we exemplarily study its optimization by dynamic spectrum management (DSM). Multiuser 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 multiuser DSM problem can be simplified by optimizing TP instead of LD power at negligible performance loss.
Appendix 1
Derivation of the classAB 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
where\left(\right.separators="">\n \n \n \n V\n \n \n rms\n ,\n ideal\n \n \n =\n \n \n E\n {\n \n \n \n V\n \n \n O\n \n \n \n \n \n \n \n 2\n \n \n }\n \n \n \n,\left(\right.separators="">\n \n \n \n V\n \n \n O\n \n \n \u223c\n N\n (\n 0\n ,\n \n \n V\n \n \n rms\n ,\n ideal\n \n \n 2\n \n \n )\n \n is the normal distributed output voltage (cf. Figure1),\left(\right.separators="">\n \n \n \n R\n \n \n line\n \n \n \u2032\n \n \n =\n \n \n R\n \n \n line\n \n \n /\n \n \n n\n \n \n 2\n \n \n \n is the transformed resistance of the line, and n is the transformer ratio. The average dissipated power P_{diss} can be decomposed into the quiescent power P_{ Q }and the dissipated power associated with the voltage drop in the classAB design[46], according to
where V_{ s } 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 classG LD with two supply voltages is given as
where P_{Low,2S} and P_{High,2S} are the consumed powers when the supply voltage is V_{ L }and V_{ H }, respectively, P_{Q,2S}= (V_{ L }(1−μ_{2S}(P_{ u })) + V_{ H }μ_{2S}(P_{ u }))·I_{ Q } is the quiescent power, and μ_{2S}(P_{ u })∈ [0,1] is the fraction of time the high supply voltage is active. Assuming a threshold V_{th} = (V_{ L }−V_{drop}) for switching between the two supplies, where V_{drop} is the voltage drop in the classAB design, and that the LD’s output voltage V_{ O } is Gaussian distributed[26] with zero mean and variance\left(\right.separators="">\n \n \n \n V\n \n \n rms\n \n \n 2\n \n \n \n, we have\left(\right.separators="">\n \n \n \n \mu \n \n \n 2\n S\n \n \n (\n \n \n P\n \n \n u\n \n \n )\n =\n 2\n Q\n (\n \n \n \n \n V\n \n \n th\n \n \n \n \n \n \n V\n \n \n rms\n \n \n \n \n )\n \n, Q(·) denoting the Qfunction. Furthermore, P_{Low,2S} is computable as (see[26] for a similar derivation)
where the term\left(\right.separators="">\n \n \n \n R\n \n \n line\n \n \n \u2032\n \n \n TL\n \n \n m\n +\n 1\n \n \n m\n \n \n \n in (30a) accounts for the total LD output resistance, and in (30b) we use the definition of V_{rms}in (6). Similarly, the power consumption when the supply with the higher voltage level V_{ H } is active is derived as
These formulas are equivalent to those shown in[4, 26], with the exception of the quiescent power calculation and the consideration of the resistance\left(\right.separators="">\n \n \n \n R\n \n \n line\n \n \n \u2032\n \n \n \n at the primary transformer side, the voltage drop V_{drop}, the transformer loss TL, and the synthesis factor m in the computation of the voltagelevel probabilities. Not included in (29) are the extra power losses due to the necessary DCDC 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 V_{ L }, and that of a supply delivering (V_{ H }−V_{ L }) during a fraction μ_{2S}(P_{ u }) of the time, cf. the classG LD model with charge pump in Section ‘ClassG linedriver 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 nonlinear, monotonously increasing function (a squareroot) of the user’s sumpower, 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 targetrates in (16b) uniquely define the minimal peruser transmit powers necessary to support the targetrates[48]. However, as the LD power model in (15) as a function of the peruser transmit sumpower 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\left(\right.separators="">\n \n \n \n \u2211\n \n \n k\n =\n 1\n \n \n K\n \n \n \n \n f\n \n \n k\n \n \n (\n p\n )\n \n of monomial functions\left(\right.separators="">\n \n \n \n f\n \n \n k\n \n \n (\n p\n )\n :\n \n \n R\n \n \n +\n \n \n CU\n \n \n \u2192\n R\n \n of the form\left(\right.separators="">\n \n \n \n f\n \n \n k\n \n \n (\n p\n )\n =\n \n \n c\n \n \n k\n \n \n \xb7\n \n \n p\n \n \n 1\n \n \n \n \n \alpha \n \n \n 1\n \n \n k\n \n \n \n \n \xb7\n \n \n p\n \n \n 2\n \n \n \n \n \alpha \n \n \n 2\n \n \n k\n \n \n \n \n \xb7\n \u2026\n \xb7\n \n \n p\n \n \n UC\n \n \n \n \n \alpha \n \n \n CU\n \n \n k\n \n \n \n \n \n, where c_{ k }≥ 0 and\left(\right.separators="">\n \n \n \n \alpha \n \n \n i\n \n \n k\n \n \n \u2208\n R\n ,\n 1\n \u2264\n i\n \u2264\n CU\n \n. We refer to[14] for a more detailed introduction to GPs. Introducing auxiliary variables t_{ u },\left(\right.separators="">\n \n u\n \u2208\n U\n \n, for the sumpower terms\left(\right.separators="">\n \n \n \n \u2211\n \n \n c\n \u2208\n C\n \n \n \n \n p\n \n \n c\n \n \n u\n \n \n \n in (17a) we obtain the equivalent formulation
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]\left(\right.separators="">\n \n \n \n r\n \n \n c\n \n \n u\n \n \n \n \n \n \n p\n \n \n c\n \n \n \n \n \u2248\n \n \n \n \n r\n \n ~\n \n \n \n c\n \n \n u\n \n \n (\n \n \n p\n \n \n c\n \n \n )\n =\n \n \n \alpha \n \n \n c\n \n \n u\n \n \n \n \n log\n \n \n 2\n \n \n (\n \n \n SINR\n \n \n c\n \n \n u\n \n \n \n \n \n \n \n \n p\n \n ~\n \n \n \n c\n \n \n \n \n )\n +\n \n \n \beta \n \n \n c\n \n \n u\n \n \n ,\n c\n \u2208\n C\n ,\n u\n \u2208\n U\n \n, where\left(\right.separators="">\n \n \n \n SINR\n \n \n c\n \n \n u\n \n \n \n is the SINR in (14) and\left(\right.separators="">\n \n \n \n \n \n p\n \n ~\n \n \n \n c\n \n \n u\n \n \n ,\n c\n \u2208\n C\n ,\n u\n \u2208\n U\n ,\n \n is the approximation point. To see this, one needs to introduce additional variables\left(\right.separators="">\n \n \n \n \n \n t\n \n ~\n \n \n \n c\n \n \n u\n \n \n \n,\left(\right.separators="">\n \n c\n \u2208\n C\n ,\n u\n \u2208\n U\n \n, replacing the total noise\left(\right.separators="">\n \n (\n \n \n \u2211\n \n \n i\n \u2208\n U\n \u2216\n u\n \n \n \n \n H\n \n \n c\n \n \n ui\n \n \n \n \n p\n \n \n c\n \n \n i\n \n \n +\n \n \n N\n \n \n c\n \n \n u\n \n \n )\n \n user u receives on subcarrier c. The thereby created additional constraints\left(\right.separators="">\n \n \n \n \n \n t\n \n ~\n \n \n \n c\n \n \n u\n \n \n \u2265\n (\n \n \n \u2211\n \n \n i\n \u2208\n U\n \u2216\n u\n \n \n \n \n H\n \n \n c\n \n \n ui\n \n \n \n \n p\n \n \n c\n \n \n i\n \n \n +\n \n \n N\n \n \n c\n \n \n u\n \n \n )\n \n,\left(\right.separators="">\n \n c\n \u2208\n C\n ,\n u\n \u2208\n U\n \n, 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].
Appendix 5
A boxbased branchandreduce 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 maskconstraints in (26c) by a box, cf. Line 2, and to successively split this set (“branching”) into smaller boxes, cf. Line 4. We observed that boxbased 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 corner points\left(\right.separators="">\n \n 0\n ,\n (\n \n \n \u2211\n \n \n u\n \u2208\n U\n ,\n c\n \u2208\n C\n \n \n \n \n \n \n p\n \n \u0302\n \n \n \n c\n \n \n u\n \n \n )\n \n \n e\n \n \n u\n \n \n ,\n u\n \u2208\n U\n \n, where e_{ u }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 g_{ u } (p) and all concave functions\left(\right.separators="">\n \n \u2212\n \n \n h\n \n \n u\n \n \n \n \n p\n \n \n ,\n u\n \u2208\n U\n \n, 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\left(\right.separators="">\n \n \n \n g\n \n \n u\n \n \n \n \n p\n \n \n ,\n u\n \u2208\n U\n \n, 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 lowerbounds and the best feasible solution found so far (the “incumbent”) the created boxes are either further split or discarded if the lowerbound lies above the upper bound, cf. Line 8. More precisely, in[37] a transformation of variables into dBscale was proposed. Similarly we perform the branching (bisection) in dBscale, which has the advantage that we still consider the full searchspace beginning at a power allocation of zero. More precisely, in Line 4, we subdivide a box along its longest edge in dBscale. In case the value of the minimal element in splitting dimension is zero we use a lower 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 branchandbound algorithm applied to a general DCP problem, and to[51] for an introduction to the rangereduction technique, as well as to[53] for an application of range reduction in a specific DCP problem with DCP functions in the objective only.
Algorithm 1 Boxbased BranchandReduce Algorithm

1:
Initialize the incumbent using a heuristic solutionbased on successive geometric programming, cf.Section ‘DSM based on successive SINRapproximation and geometric programming’.

2:
Initialize the first open, currently active box withminimal and maximal cornerpoints\left(\right.separators="">\n \n 0\n ?\n \n \n R\n \n \n UC\n \n \n \n and\left(\right.separators="">\n \n \n \n p\n \n ^\n \n ?\n \n \n R\n \n \n UC\n \n \n \n.

3:
while {Any box is open} do

4:
Branching: Generate two new open boxes bysplitting the currently active box in half indBscale in the dimension of its longest edge.

5:
Bounding: Compute objective lower bounds forboth new boxes using an underestimating LP[50]to the DCP problem in (26) with reducedvariable ranges[51].

6:
Reduction: Try a rangereduction based on thecurrent incumbent solution[51], and repeat thelowerbound LP if a rangereduction wasachieved.

7:
Incumbent Update: Update the incumbentby testing the 2^{CU1}new corner points createdthrough branching and the LP solutions forfeasibility in (26).

8:
Pruning: Close all boxes with a lower bound abovethe incumbent solution.

9:
Selection: Choose the open box with the lowestlower bound as the new active box.
Endnotes
^{a}The parameters chosen for ADSL2+ are R_{line} = 100Ω, n = 1.25, CF = 5.3,\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n =\n 19\n .\n 5\n \ndBm, TL = 0.5dB, m = 5, I_{ Q }= 5mA, V_{drop}= 4V, and P_{Hybrid} = 0. The parameters for VDSL deviating from these values are I_{ Q }= 11.1mA and\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n =\n 11\n .\n 5\n \ndBm.
^{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 R_{line} = 100Ω, n = 1.4 (as in[29]),\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n =\n 20\n .\n 5\n \ndBm (14.5dBm) for ADSL2+ and VDSL2 profile 8b (VDSL2 profile 17a and 30a), TL = 0.5dB, m = 5, V_{drop} = 5V, and P_{Hybrid} = 0. For the classAB model we assume CF=5.3. The quiescent currents I_{ Q }∈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 classAB LD the optimal supply voltage in (3) is assumed, for the classG LD with two supplies we consider V_{ H }= 24V, and for the LD with charge pumps we set V_{ H }= 24V + V_{drop,CP}, I_{Q,classG}= 0.3mA, L_{classG} = 0mW, and ρ = 1.5dB, where V_{drop,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 classG LD types we set V_{ L }= 12V and assume a threshold for switching between high and low supply of V_{th} = V_{ L }−V_{drop}. The usage probability μ_{cp}(P_{ u }) is obtained through simulations for different values of P_{ u }. The charge pump is assumed to be active for a timeframe of 0.11μ s (ADSL2+ and VDSL2 8b), 0.04μ s (VDSL2 17a) or 0.05μ s (VDSL2 30a) when V_{rms} exceeds V_{th}. Additionally it is assumed to be active for 0.35μ s and 0.5μ s before and after this timeframe, 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 ‘ClassAB linedriver power model’ with the exception of n = 1.2, CF=5, and the power limit\left(\right.separators="">\n \n \n \n \n \n P\n \n \u0302\n \n \n \n u\n \n \n =\n 19\n .\n 9\n \ndBm used for ADSL2+ lines.
^{d}The remaining relevant parameters are\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n uu\n \n \n =\n 1\n \n, Γ = 12.3dB, Δ = 4.3125·10^{3}[Hz],\left(\right.separators="">\n \n \n \n N\n \n \n c\n \n \n u\n \n \n =\n 1\n \n \n 0\n \n \n \u2212\n 140\n /\n 10\n \n \n \xb7\n \Delta \n \n[mW],\left(\right.separators="">\n \n \n \n \n \n p\n \n \u0302\n \n \n \n c\n \n \n u\n \n \n =\n 1\n \n \n 0\n \n \n \u2212\n 40\n /\n 10\n \n \n \xb7\n \Delta \n \n[mW],\left(\right.separators="">\n \n u\n \u2208\n U\n \n,\left(\right.separators="">\n \n c\n \u2208\n C\n \n,\left(\right.separators="">\n \n \n \n B\n \n \u0302\n \n =\n \u221e\n \n.
^{e}This sequential reinitialization 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 sumpower 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 SINRapproximation and geometric programming’ with the exception of R_{1} = 41.36[bits/frame], R_{2} = 5.9[bits/frame],\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n 12\n \n \n =\n \u2212\n 67\n .\n 5\n \ndB and the initial value\left(\right.separators="">\n \n \n \n H\n \n \n c\n \n \n 21\n \n \n =\n \u2212\n 22\n .\n 5\n \ndB,\left(\right.separators="">\n \n c\n \u2208\n C\n =\n {\n 1\n ,\n 2\n }\n \n.
^{h}The sequential reinitialization 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 trustregion 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 fibertotheexchange, mask variant B, and unnotched 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 SINRapproximation 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 realsized DSM problems using heuristics’, except that for VDSL we use the bandplan specified in[42] for fibertothecabinet, mask variant AM1.
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This work has been funded by BMVIT/FFG under the program FITIT. 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.
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Wolkerstorfer, M., Trautmann, S., Nordström, T. et al. Modeling and optimization of the linedriver power consumption in xDSL systems. EURASIP J. Adv. Signal Process. 2012, 226 (2012). https://doi.org/10.1186/168761802012226
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DOI: https://doi.org/10.1186/168761802012226