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Framework for Link-Level Energy Efficiency Optimization with Informed Transmitter

Christian Isheden, Zhijiat Chong, Eduard Jorswieck, Gerhard Fettweis

arXiv:1110.1990v5cs.IT

TL;DR

Rising network infrastructure increases energy consumption, motivating energy-efficiency optimization. The paper presents a fractional-programming framework with efficient solution methods and applies it across several channel settings, including time-varying parallel channels.

  • Problem

    Rapidly increasing energy consumption from network infrastructure motivates energy-efficiency optimization.

  • Method

    The paper coherently presents a framework, carefully described power models, and solution methods for energy-efficiency optimization problems.

  • Results

    The framework applies to time-invariant parallel, time-varying flat-fading, and time-varying parallel channels, with the examples yielding water-filling solutions.

  • Takeaways & Limitations

    The framework illustrates the usefulness of energy-efficiency optimization methods across various channel settings.

Abstract

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The dramatic increase of network infrastructure comes at the cost of rapidly increasing energy consumption, which makes optimization of energy efficiency (EE) an important topic. Since EE is often modeled as the ratio of rate to power, we present a mathematical framework called fractional programming that provides insight into this class of optimization problems, as well as algorithms for computing the solution. The main idea is that the objective function is transformed to a weighted sum of rate and power. A generic problem formulation for systems dissipating transmit-independent circuit power in addition to transmit-dependent power is presented. We show that a broad class of EE maximization problems can be solved efficiently, provided the rate is a concave function of the transmit power. We elaborate examples of various system models including time-varying parallel channels. Rate functions with an arbitrary discrete modulation scheme are also treated. The examples considered lead to water-filling solutions, but these are different from the dual problems of power minimization under rate constraints and rate maximization under power constraints, respectively, because the constraints need not be active. We also demonstrate that if the solution to a rate maximization problem is known, it can be utilized to reduce the EE problem into a one-dimensional convex problem.

I. INTRODUCTION

The paper frames link-level energy-efficiency optimization as a fractional-programming problem for systems with transmit-dependent and transmit-independent power, and develops a broadly applicable solution framework.

  • Motivation: Rising network traffic and infrastructure expansion make reducing energy consumption and improving energy efficiency important tasks in wireless communications.
  • Problem formulation: The paper studies the trade-off between spectral efficiency and energy efficiency in practical systems that dissipate both circuit and transmit-dependent power.
  • Problem formulation: Energy-efficiency maximization differs from rate maximization under power constraints and power minimization under rate constraints because its constraints need not be active.
  • Framework: The proposed framework summarizes fractional-programming results, connects solution approaches through scalarized bi-criterion optimization, and provides an efficient algorithm.
  • Applications: Applications cover time-invariant and time-varying flat-fading channels, parallel channels, empirical power models, and discrete modulation schemes.
  • Applications: The resulting algorithms have low complexity because they use water-filling power allocation, with the water level adjusted to fulfill a maximum-energy-efficiency criterion.

III. FRACTIONAL PROGRAMMING

Fractional programming converts a ratio objective into a parameterized optimization problem whose root characterizes the optimum. Under concavity and convexity conditions, this structure supports convex subproblems and efficient iterative solution methods.

  • Formulation: A fractional program maximizes a ratio of two real-valued functions, with a positive denominator; the concave-convex case additionally uses a concave numerator, convex denominator, and convex feasible set.
  • Properties: For differentiable concave-convex fractional programs, the objective is pseudoconcave, so any stationary point is a global maximum and KKT conditions are sufficient when constraint qualification holds.
  • Algorithms: F(λ) is convex, continuous, and strictly decreasing, enabling root-finding methods such as bisection and the Dinkelbach method.
  • Parametric convex program: The fractional objective can be transformed into the parametric function F(λ) = max x∈S f1(x) −λf2(x), which scalarizes maximizing f1 against minimizing f2.
  • Optimality condition: The optimal ratio q* is characterized by F(λ) = 0, so solving the fractional program is equivalent to finding a root of F(λ).
  • Algorithms: Newton-based updates converge superlinearly, while box constraints can be handled by checking whether the unconstrained optimum lies within the permitted λ interval.

B. Parameter-free convex program

The parameter-free reformulation converts the fractional objective into an equivalent convex problem in transformed variables, preserving optimal solutions and enabling dual-based analysis.

  • Parameter-free reformulation: The feasible set S is formed by imposing convex inequality constraints on a nonempty, convex domain S0.The formulation assumes f2(x) > 0 over the feasible set.
  • Parameter-free reformulation: The parameter-free problem is equivalent to the original fractional problem through a perspective transformation relating x to y/t.The transformed problem is convex because taking a function perspective preserves convexity.
  • Optimality conditions: The Lagrangian introduces λ for the transformed fractional constraint and u for the convex inequality constraints.Stationarity and complementary slackness characterize the optimum.
  • Optimality conditions: The resulting stationarity condition is equivalent to the optimality condition of the parametric convex formulation.Thus, both formulations identify the same optimal solution through their respective optimality systems.

C. Dual program

The dual program provides an alternative characterization of fractional optimization through the optimal ratio parameter λ, while highlighting convexity conditions and scope limitations.

  • Dual characterization: The dual of the equivalent convex program coincides with the dual of the parametric convex program.The same dual problem is obtained when λ is treated as a parameter.
  • Scope and limitations: The dual problem is not generally convex because its equality constraint is typically non-affine.This limits direct convexity guarantees for the dual formulation.
  • Optimality: At optimality, the dual value λ* equals the fractional objective value q(x*).Complementary slackness makes the inequality constraint tight at the optimum.
  • Dual characterization: The dual formulation identifies x* and λ* through a maximization of f1(x) − λ*f2(x) over the feasible set S.The first dual equation is the condition for this parameterized maximization.
  • Convex fractional program: For convex fractional programs, fixing the ratio parameter produces a convex feasibility problem that can be solved by minimizing f2(x) − λf1(x).Bisection can search for the optimal parameter, and the constraint is active at the optimum.
  • EE application: The generic EE problem maximizes achievable rate divided by dissipated power, including transmit-independent power through the offset μ.Increasing μ decreases the optimal objective value and shifts the optimum toward higher sum power.
  • EE application: The framework transforms literature power models into the generic form and permits arbitrary convex transmit-power functions, not only linear models.This extends the treatment beyond the linear power models considered in the examples.

A. Generic base station power model

The paper maps diverse base-station power models and channel settings to a generic energy-efficiency problem, yielding tractable power-allocation solutions under concave-rate conditions.

  • Generic base-station model: A base-station power model accounts for equal transmit-power allocation across antennas, RF-chain hardware, and power-amplifier dissipation.The power amplifier output power is modeled as Pt/(naηPA).
  • Generic base-station model: Static baseband and battery consumption, power-supply efficiency, and cooling-system losses contribute to total power dissipation.These terms enter the generic offset and scaling parameters of the EE metric.
  • Generic base-station model: The EE metric is expressed in bit/J using bandwidth B and parameters p, μ, and C derived from the base-station power model.μ captures transmit-independent components relative to bandwidth, while C captures efficiency losses.
  • Applications: Flat- and frequency-selective fading, static and time-varying channels, SISO and MIMO antennas, and Gaussian or quadratic M-QAM inputs fit the framework when their rate functions are concave.These channel, antenna, and constellation models are treated within the same mathematical framework.
  • Optimization setup: With box constraints 0 ≤ p_i ≤ p_max, the feasible set is compact and convex, allowing parametric and parameter-free solution methods.The paper applies both approaches to the resulting optimization problem.
  • Power allocation: The optimal allocation incorporates box constraints and has a water-filling form with a cutoff CNR below which a subcarrier is unused.The explicit allocation is used in root-finding methods such as Dinkelbach’s algorithm.
  • Power allocation: The EE optimum occurs where the rate–power trade-off tangent passes through the origin; increasing the offset lowers optimal EE and raises sum power.The figure uses sum rate vertically and sum power plus offset horizontally.

2) Parameter-free convex problem:

The EE problem is reformulated as a parameterized convex optimization, with the parameter representing inverse total power dissipation. Dinkelbach iterations and KKT conditions yield the optimal allocation, including idle transmission below a CNR cutoff.

  • The feasible domain requires positive logarithm arguments and a strictly positive denominator.
  • The transformed formulation is a convex problem in y and t, where t corresponds to inverse total power dissipation.
  • Dinkelbach iterations solve a parametric optimization whose stationarity conditions determine the optimal power allocation.
  • For flat fading, the optimal dual variable can be obtained in closed form using the Lambert W function.
  • If the instantaneous CNR is below the cutoff level, the transmitter optimally remains idle.

1) Adding constraints:

Additional rate and power constraints can be represented as bounds on the dual variable, while time-varying parallel channels are optimized over power-allocation functions. The resulting KKT solution separates across subchannels, and numerical integration may become demanding as channel count grows.

  • 1) Adding constraints:: A minimum rate constraint imposes an upper bound λmax, whereas a maximum power constraint imposes a lower bound λmin.
  • Time-varying parallel channels: For time-varying parallel channels, power allocation is optimized as a vector function of each channel realization.
  • Time-varying parallel channels: KKT conditions are sufficient because the parametric objective is a concave functional, and each optimal component depends explicitly only on its own CNR.
  • Time-varying parallel channels: For K > 3, evaluating the required integrals may be demanding, although independence can reduce computation by factorizing the joint PDF.

D. Gap to capacity

The framework incorporates constant capacity gaps and arbitrary modulation through concave mutual-information rate functions. For discrete constellations, MMSE-based conditions produce a mercury/water-filling-like allocation, while mutual information may lack a closed form.

  • Gap to capacity: A constant gap Γ modifies the rate function while remaining independent of the subcarrier CNR.
  • Arbitrary modulation: Arbitrary modulation schemes are handled by defining rate as mutual information for inputs drawn from discrete or continuous modulation sets.
  • Arbitrary modulation: The mutual information rate is strictly concave in transmit power, enabling KKT-based optimization.
  • Implementation: Mutual information is difficult to express in closed form for general constellations, but MMSE and rate values can be tabulated for implementation.
  • Arbitrary modulation: The optimal powers are expressed using the inverse MMSE, with ζi = λ/γi, yielding a graphical interpretation analogous to conventional water-filling.

E. Nested convex problem

Known rate-maximization solutions can be reused to solve EE maximization through a one-dimensional convex problem. The EE allocation has the same functional form as rate-maximizing allocation, but its dual parameter is selected for energy efficiency rather than a power constraint.

  • Rate maximization: Rate maximization over parallel channels yields water-filling solutions whose water level depends on the dual variable.
  • Nested convex problem: The EE problem can be transformed into a one-dimensional convex problem, allowing known rate-maximization results to be reused.
  • Nested convex problem: EE and rate maximization have functionally identical power allocations, but their dual parameters are chosen for different objectives.
  • Nested convex problem: The nesting approach requires no inner optimization in the direct dual-variable formulation, whereas nesting remains attractive when pre-analysis is unavailable or computation time is unimportant.

VI. SIMULATION

Simulations examine energy efficiency across antenna configurations, circuit-power levels, and modulation schemes. They show that antenna activation and modulation order should be selected according to circuit-power and operating-point conditions.

  • Time-varying channel with varying number of antennas: For equal transmit and receive antenna counts, more antennas are more energy-efficient, while optimal EE decreases monotonically with circuit power Pc.When Pc = 0, EE* increases linearly with the common antenna count n.
  • Time-varying channel with varying number of antennas: With one receive antenna, the highest transmit-antenna count is efficient only when circuit power Pc is small.As Pc increases, the EE loss from activating more transmit antennas increases because rate scales sublinearly while power scales linearly with nT.
  • Time-varying channel with varying number of antennas: Additional diversity gain does not always justify activating more antennas, whereas additional multiplexing gain can motivate activation.The simulations therefore recommend activating extra antennas and their RF chains only when their operating-point benefit exceeds their energy cost.
  • Discrete modulation: For discrete modulation, higher-order schemes have higher optimal transmit power p*, and their EE curves are approximately equal under Gaussian noise before diverging at larger µ.If µ is modulation-independent, higher modulation order is always beneficial because no cost is associated with using it.
  • Discrete modulation: A higher modulation order can increase offset power, making a lower modulation order optimal in some cases.The preferred modulation scheme therefore depends on whether the higher order carries an additional power cost.

VII. DISCUSSION

The discussion interprets the fractional-programming parameter λ and summarizes the framework’s applicability and boundaries. It emphasizes that EE water-filling differs from constrained rate or power optimization because constraints need not be active.

  • Interpretation of λ: λ is the denominator’s relative weight in scalarized optimization and the slope of the trade-off curve between sum rate and sum power.At the optimum, λ* equals maximum EE after an appropriate system-dependent adjustment.
  • Interpretation of λ: In the water-filling examples, λ is a cut-off: a channel receives transmission power only when its SNR γ exceeds λ.This interpretation distinguishes the EE solution from standard constrained rate-maximization or power-minimization water-filling.
  • Framework and applicability: The framework unifies fractional-programming results for rate-to-power maximization with carefully specified power models and solution methods.Applications cover time-invariant parallel, time-varying flat-fading, and time-varying parallel channels.
  • Framework and applicability: The framework’s examples include static and time-varying channels, practical modulation, and MIMO channels decomposed into parallel channels.The supplied figures illustrate EE behavior under constraints, antenna-dependent circuit power, and Gaussian versus m-QAM signaling.
  • Scope and extensions: The current framework assumes concave numerator functions; non-concave functions and discrete optimization variables require extensions and additional optimization methods.Discrete variables are relevant to systems with on-off power modes, such as turning off parts of a base station during off-peak hours.
  • Scope and extensions: Additional constraints can penalize EE outside the interval where the optimized parameter remains between its limiting values.For small µ the problem reduces to power minimization, whereas for large µ it reduces to rate maximization.
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