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Sum-Rate Optimal Power Policies for Energy Harvesting Transmitters in an Interference Channel
Kaya Tutuncuoglu, Aylin Yener
TL;DR
The paper addresses short-term sum-throughput optimization when two interfering transmitters harvest energy and data may arrive during transmission. It formulates the constrained power-allocation problem, proves convergence of an iterative coordinate-descent approach, and develops directional water-filling, online, and distributed alternatives. The resulting framework yields optimal offline policies in the stated setting and near-optimal policies for online and distributed operation.
Problem
Power allocation in a two-user interference channel must adapt to energy harvesting and, in some settings, intermittent data arrivals under a transmission deadline.
Method
The paper uses iterative coordinate descent, generalized directional water-filling, data-causality penalties, and computationally simpler online or distributed policy designs.
Results
The iterative algorithm converges to the optimal policy, while the paper provides directional water-filling interpretations and near-optimal online and distributed alternatives.
Takeaways & Limitations
Optimal power control must account jointly for harvested energy, interference, and data causality rather than relying solely on conventional directional water-filling.
Abstract
from arXiv · showhide
This paper considers a two-user Gaussian interference channel with energy harvesting transmitters. Different than conventional battery powered wireless nodes, energy harvesting transmitters have to adapt transmission to availability of energy at a particular instant. In this setting, the optimal power allocation problem to maximize the sum throughput with a given deadline is formulated. The convergence of the proposed iterative coordinate descent method for the problem is proved and the short-term throughput maximizing offline power allocation policy is found. Examples for interference regions with known sum capacities are given with directional water-filling interpretations. Next, stochastic data arrivals are addressed. Finally online and/or distributed near-optimal policies are proposed. Performance of the proposed algorithms are demonstrated through simulations.
I. INTRODUCTION
The paper formulates short-term sum-throughput maximization for a two-user Gaussian interference channel whose transmitters harvest energy and may receive data intermittently. It models energy, battery, data-causality, and deadline constraints, then motivates iterative and directional water-filling solutions.
- Motivation: Energy harvesting requires transmission policies to adapt to stochastic, uneven energy availability and limited battery capacity.The paper contrasts these constraints with traditional battery-powered wireless nodes.
- Problem and contribution: The study targets the optimal power schedule maximizing short-term sum throughput before a deadline in a two-user Gaussian interference channel.The transmitters obtain their transmission energy by harvesting from ambient sources.
- Problem and contribution: The interference channel is important because capacity depends strongly on transmitter interaction and receiver interference processing.Known capacity results are available for strong interference and some weak-interference sum-capacity regimes.
- Problem and contribution: The paper extends the analysis from all data available initially to intermittent packet arrivals and studies iterative, online, and near-optimal policy variants.Directional water-filling is adapted to energy arrivals, interference, and data-causality constraints.
- System model: The system uses time slots of length τ, with energy and data arrivals available at each slot’s beginning for immediate use.Energy exceeding battery capacity is lost or truncated, and harvested energy must be stored before consumption.
- System model: Feasible policies obey energy causality, battery capacity, data causality, nonnegative power, and a finite deadline T = N · τ.The optimization represents each user’s transmission powers as a power policy or allocation vector.
III. ITERATIVE SOLUTION
The paper solves the two-user optimization without data causality by alternating optimization over the two users’ power-allocation vectors.
- III. ITERATIVE SOLUTION: The iterative approach partitions the optimization variables into the two users’ power policies, p1 and p2.The method applies cyclic coordinate descent to the two-user problem without the data-causality constraint.
- III. ITERATIVE SOLUTION: Each coordinate update optimizes one user’s power allocation while holding the other user’s allocation fixed.
- III. ITERATIVE SOLUTION: The resulting procedure is designed to solve the short-term throughput problem under independently harvested energy constraints.
A. Iterative Algorithm
The proposed algorithm alternates single-user power-policy maximizations until both users’ policies converge, using independent feasibility constraints and directional water-filling structure.
- A. Iterative Algorithm: The algorithm repeatedly maximizes throughput over one user’s transmission policy while keeping the other policy constant.Iterations begin from an arbitrary initial feasible pair of power policies.
- A. Iterative Algorithm: Independent energy harvesting lets each coordinate subproblem include only the optimized user’s energy constraints.The fixed user’s policy remains feasible because the transmitters consume their own harvested energy.
- A. Iterative Algorithm: The coordinate subproblems are single-user optimizations of concave rate sums over linear constraint sets.
- A. Iterative Algorithm: Directional water-filling is enhanced with directional water flow and taps to represent energy-arrival and interference effects.
B. Convergence
Convergence is obtained by combining concavity and separable convex constraints with a strict-concavity auxiliary penalty when unique coordinate maxima are not otherwise guaranteed.
- B. Convergence: The rate function can be made concave and non-decreasing through time-sharing without performing worse than the original coding scheme.The constructed rate is the maximum of achievable time-shared convex combinations.
- B. Convergence: Theorem 1 states that the iterative algorithm in (6) and (7) converges to the optimal policy.
- B. Convergence: Independent harvesting makes the users’ constraint sets separable, and their linear constraints are convex.These properties provide the Cartesian-product structure required for coordinate-descent convergence.
- B. Convergence: When strict concavity or unique coordinate maxima are unavailable, auxiliary vectors and quadratic penalties make the modified objective strictly concave.The modified objective is g(p1, p2, s1, s2) = r(p1, p2) − ε∥p1 − s1∥2 − ε∥p2 − s2∥2, with ε > 0.
IV. EXAMPLES
The paper interprets single-user subproblems through directional water-filling, deriving the optimal policy with KKT conditions under energy-harvesting constraints.
- IV. EXAMPLES: The directional water-filling algorithm provides insight into the single-user subproblems within the iterative interference-channel solution.The paper first formulates the Lagrangian and then applies KKT stationarity to obtain the optimal power policy.
- IV. EXAMPLES: KKT complementary slackness determines when battery constraints are active and how the water level changes across time slots.Water levels change only when battery-empty or battery-full constraints bind; otherwise, they are equalized subject to directional flow constraints.
- IV. EXAMPLES: The resulting interpretation uses base level 1/h_i, forward-only water flow, and a battery-capacity limit on water transfer between slots.These flow constraints jointly enforce energy causality and finite battery capacity.
B. Asymmetric Interference with ab > 1
For asymmetric interference with ab > 1, the paper alternates directional and generalized directional water-filling to obtain the optimal transmission policy.
- B. Asymmetric Interference with ab > 1: The asymmetric region includes a strong cross channel for one transmitter and a weak cross channel for the other, with the paper assuming a ≤1 and b ≥1.The complementary case is obtained by switching transmitter indices.
- B. Asymmetric Interference with ab > 1: The capacity-achieving scheme treats weaker interference as noise while decoding and removing stronger interference at the receiver.This receiver operation determines the asymmetric-region power-rate structure.
- B. Asymmetric Interference with ab > 1: For T1, fixing T2’s policy converts interference into an effective fading parameter, so its subproblem is solved by directional water-filling.At iteration k, the fading parameter is updated using the previous iteration’s power allocation for T2.
- B. Asymmetric Interference with ab > 1: T2’s rate contains two terms and therefore requires KKT optimality conditions with complementary slackness for its power constraints.The multiplier for the nonnegativity constraint is active when p2 = 0; battery multipliers are active when the battery is empty or full.
- B. Asymmetric Interference with ab > 1: An alternating algorithm using directional water-filling for T1 and generalized directional water-filling for T2 converges to the optimal policy.The convergence follows from the previously established convergence of the iterative algorithm.
C. Asymmetric Interference with ab ≤1
For the complementary asymmetric region with ab ≤1, the paper reduces each fixed-policy subproblem to directional or generalized water-filling and shows that alternating updates converge.
- C. Asymmetric Interference with ab ≤1: This region assumes a ≤1, b ≥1, and ab ≤1, with the complementary transmitter-index case handled symmetrically.The associated power-rate function includes a term involving 1 + b·p1 + p2.
- C. Asymmetric Interference with ab ≤1: The rate is achieved by decoding T1’s interference at R2 while treating interference as noise at R2, with a minimum selecting T1’s limiting receiver rate.Because T1 must be decoded at both receivers, the minimum operation determines which receiver limits its transmission rate.
- C. Asymmetric Interference with ab ≤1: With T2’s policy fixed, the value of p2 determines which term dominates the minimum, independently of the optimization variable p1.This permits T1’s subproblem to be treated as a fading-channel problem with a modified water base level.
- C. Asymmetric Interference with ab ≤1: T2’s single-user subproblem is solved with a generalized water-filling algorithm using an adapted water level.The adapted level accounts for the structure of the asymmetric-region rate expression.
- C. Asymmetric Interference with ab ≤1: The alternating implementation of the two single-user algorithms converges to the short-term throughput-maximizing power policy.The policy combines the T1 directional-water-filling solution with the T2 generalized-water-filling solution.
D. Very Strong Interference
In very strong interference, each interfering signal can be decoded at both receivers, so the sum rate separates into two single-link capacities.
- D. Very Strong Interference: Very strong interference occurs when the cross-channel coefficients are large enough for both receivers to decode the interfering signals.Under these conditions, each user achieves the single-link Gaussian channel capacity.
- D. Very Strong Interference: The iterative algorithm becomes trivial because the two users’ short-term throughput maximization problems are independent.Each user follows a single-link short-term throughput maximization policy, so no further iterations are needed for optimality.
E. Other Regions
For interference regions without simpler single-user rate expressions or known sum-capacity formulas, the paper applies iterative generalized directional water-filling to concave two-user problems. The resulting interpretation retains unidirectional flow and maximum-flow constraints while using an alternative water-level expression.
- Some interference regions lack single-user rate expressions that simplify optimization, or have unknown sum-capacity expressions.
- Any two-user problem with a concave power-rate function can be solved iteratively using generalized directional water-filling for both single-user subproblems.
- The iterative solution is characterized by KKT stationarity and complementary slackness conditions for each user and time slot.
- The optimal power in each time slot is obtained from the solution of the corresponding single-user optimization equation.
- The solution admits a generalized directional water-filling interpretation with an alternative water level based on dr(p_j)/dp_j and unidirectional flow and maximum-flow constraints.
V. EXTENSION TO DATA ARRIVALS
With intermittent data arrivals, data-causality constraints couple users and can destroy convexity, so direct iterative optimization may stall at a non-optimal point. The paper extends directional water-filling with pumps and uses quadratic penalties to improve convergence while accounting for energy, battery, and data constraints.
- Data arrivals: Data arrivals require transmitters to maximize delivered bits when packet availability varies during transmission rather than being known entirely beforehand.
- Data arrivals: Joint data-causality constraints can be non-convex because each user's rate depends on both powers, challenging convergence of iterative algorithms.
- Data arrivals: Figure 3 illustrates direct iteration converging to a non-optimal point when equalizing one user's water levels violates the other user's data-causality constraint.
- Penalty method: The proposed alternative relaxes data-causality handling with a quadratic penalty whose coefficient starts at zero and grows unboundedly across iterations.
- Penalty method: The penalty modifies each user's water level with an offset whenever data causality is violated, with the offset scaled by the iteration penalty parameter.
- Water-filling with pumps: Directional water-filling with pumps handles energy arrivals, battery capacity, and data arrivals through taps and forward or reverse pumps.Battery taps block flow after the capacity limit, while pumps address bit-causality violations and flow direction.
- Water-filling with pumps: When a forward data-causality pump and a battery-capacity tap conflict, the contradictory water is removed without reducing the optimal policy's performance.In the example, energy arriving in the first slot is inevitably lost because the first slot has no data and the second battery is full.
- Special interference cases: For specific asymmetric and very strong interference cases, rate independence removes cross-dependence and permits the pump analogy without backward pumps.
VI. DISTRIBUTED / ONLINE ALGORITHMS
The paper develops lower-information online and distributed power policies inspired by the optimal iterative solution, and evaluates their behavior against optimal and naive strategies. Simulations show that water-filling policies substantially outperform naive transmission, while single-user directional water-filling can closely approach the iterative optimum.
- Distributed / Online Algorithms: The proposed near-optimal algorithms reduce information requirements when transmitters cannot share both users’ energy and data arrivals.The centralized optimal policies require prior knowledge of both transmitters’ arrival settings, whereas the alternatives use less information.
- Distributed / Online Algorithms: With localized power decisions, single-link water-filling uses expected values for unknown parameters and matches the optimal offline policy under very strong interference.For weaker interference, additional iterations provide gradual policy improvements.
- Simulations: Iterative directional water-filling modifies the two users’ allocations through interference-aware interactions rather than independently applying single-user allocations.The simulated optimal policies use a base level for one transmitter and generalized directional water-filling for the other; one transmitter may remain silent while the other uses high power.
- Simulations: In the asymmetric-interference simulation, the optimal policy significantly reduces one transmitter’s power when the other is highly interfering at slots i = {19, 20}.The example illustrates that even two-user interference can materially reshape the power schedules.
- Simulations: Water-filling algorithms provide notable performance gains over naive constant-power transmission, while single-user directional water-filling performs very close to optimal.The comparison includes iterative water-filling, independent water-filling, and naive constant-power nodes.
VIII. CONCLUSION
The paper formulates and solves short-term sum-throughput maximization for a two-user Gaussian interference channel with energy-harvesting nodes. It extends the model to stochastic data arrivals and proposes simpler online and distributed alternatives, whose simulated performance improves notably over naive algorithms.
- Conclusion: The paper solves the short-term sum-throughput maximization problem for a two-user Gaussian interference channel with energy-harvesting transmitters.The formulation targets maximum total transmitted bits under a deadline.
- Conclusion: For asymmetric and very strong interference, generalized iterative water-filling can reduce to modified single-user directional water-filling.The paper also extends the model to stochastic data arrivals using a penalty for data-causality violation.
- Conclusion: The proposed iterative and distributed near-optimal algorithms show a notable performance boost over naive algorithms in simulations.The alternatives are motivated by insight from the optimal solution and target online or distributed operation.
- Conclusion: The results serve as a starting point for energy-harvesting interference networks, with future work extending beyond two users and toward more elaborate multi-hop structures.The conclusion also identifies simpler online algorithms for jointly adapting to energy availability and interference levels as a future direction.