Source-linked AI summary
Throughput Maximization for the Gaussian Relay Channel with Energy Harvesting Constraints
Chuan Huang, Rui Zhang, Shuguang Cui
TL;DR
The paper asks how to maximize throughput in a Gaussian relay channel when source and relay transmissions draw on predictable but intermittent harvested energy. It analyzes delay-constrained and no-delay-constrained traffic, develops power-allocation solutions, and shows that flexible delays can exploit energy diversity under stated conditions.
Problem
The paper studies throughput maximization for a three-node Gaussian relay channel whose source and relay operate under deterministic energy-harvesting constraints and different decoding-delay requirements.
Method
It formulates DC and NDC power-allocation problems for decode-and-forward relaying, using joint optimization for DC and an optimal source-relay separation principle for NDC.
Results
The NDC case is no worse than the DC case, and it is strictly better exactly under the conditions characterized by Proposition 5.3.
Takeaways & Limitations
No-delay-constrained transmission can exploit energy diversity created by independent source and relay energy availability over time, even with time-invariant channels.
Takeaways & Limitations
The study assumes deterministic energy arrivals, infinite energy storage, and decode-and-forward relaying in an orthogonal relay channel.
Abstract
from arXiv · showhide
This paper considers the use of energy harvesters, instead of conventional time-invariant energy sources, in wireless cooperative communication. For the purpose of exposition, we study the classic three-node Gaussian relay channel with decode-and-forward (DF) relaying, in which the source and relay nodes transmit with power drawn from energy-harvesting (EH) sources. Assuming a deterministic EH model under which the energy arrival time and the harvested amount are known prior to transmission, the throughput maximization problem over a finite horizon of $N$ transmission blocks is investigated. In particular, two types of data traffic with different delay constraints are considered: delay-constrained (DC) traffic (for which only one-block decoding delay is allowed at the destination) and no-delay-constrained (NDC) traffic (for which arbitrary decoding delay up to $N$ blocks is allowed). For the DC case, we show that the joint source and relay power allocation over time is necessary to achieve the maximum throughput, and propose an efficient algorithm to compute the optimal power profiles. For the NDC case, although the throughput maximization problem is non-convex, we prove the optimality of a separation principle for the source and relay power allocation problems, based upon which a two-stage power allocation algorithm is developed to obtain the optimal source and relay power profiles separately. Furthermore, we compare the DC and NDC cases, and obtain the sufficient and necessary conditions under which the NDC case performs strictly better than the DC case. It is shown that NDC transmission is able to exploit a new form of diversity arising from the independent source and relay energy availability over time in cooperative communication, termed "energy diversity", even with time-invariant channels.
I. INTRODUCTION
The paper studies throughput maximization in an orthogonal Gaussian relay channel powered by deterministic energy-harvesting sources, under strict and flexible decoding delays. It develops separate optimization strategies for delay-constrained and no-delay-constrained traffic while addressing time-varying source and relay energy availability.
- System model: The system uses a half-duplex orthogonal Gaussian relay channel with energy-harvesting source and relay nodes under a deterministic energy model.Energy arrival times and amounts are known before transmission, and harvested energy is subject to causal consumption constraints.
- Traffic models: Delay-constrained traffic requires each source message to be decoded immediately after source reception and next-block relay forwarding.The protocol therefore uses one-block decoding delay at the destination and requires immediate relay forwarding.
- Traffic models: No-delay-constrained traffic permits arbitrary decoding delays up to the end of the N-block transmission, allowing the relay to store and forward multiple decoded messages flexibly.This flexibility is associated with energy diversity from independent source and relay energy arrivals, even over time-invariant channels.
- Contributions: For the DC case, the paper formulates a convex throughput maximization problem and develops a joint source-relay power allocation algorithm using KKT conditions and non-decreasing optimal power profiles.The algorithm searches forward over the two-dimensional harvested energy profiles.
- Contributions: For the NDC case, an optimal separation principle decouples the non-convex problem into source and relay power allocation subproblems solved through a two-stage strategy.The source is optimized first independently of the relay, followed by relay optimization using the resulting source solution.
- System model: The model adopts decode-and-forward relaying with orthogonal relay-destination transmission and equal source-relay and relay-destination bandwidths.Each source block carries a new message, while the relay transmits a binning index in the subsequent block or aggregates indices under NDC traffic.
B. No-Delay-Constrained Case
In the NDC case, relay transmissions may carry binning indices for multiple source messages, producing a non-convex throughput problem whose optimum is obtained through separate source and relay allocation.
- NDC relaying allows each relay transmission to carry binning indices for the current and earlier source messages.
- The NDC throughput maximization problem is non-convex because of its first coupling constraint.
- The NDC problem’s maximum throughput is no smaller than the DC problem’s maximum throughput.
- For optimal profiles, source energy constraints are active when 0 < h0 < 1, while at least one source or relay energy constraint is active when h0 = 0.
IV. OPTIMAL SOLUTION FOR THE DC CASE
For DC traffic, the paper solves a convex joint source–relay power-allocation problem using KKT conditions and monotonicity, then implements the solution with a forward-search algorithm.
- Monotonic Power Allocation: An optimal DC power solution is non-decreasing over transmission blocks.
- The optimal DC source and relay power profiles must be jointly optimized because each message’s achievable rate depends on both powers.
- Optimal Power Structure: KKT conditions show that source power changes only when source energy is exhausted or when it transitions between the two power expressions.
- Optimal Power Structure: Between successive source energy-exhausting blocks, optimal source powers follow one of three scenarios: identical values, or a single transition between two values.
- Algorithm I: Algorithm I identifies energy-exhausting blocks and scenarios, then computes optimal profiles through a forward search from the first to the N-th block.
- Solution Properties: The resulting source power solution is unique for 0 < h0 < 1, whereas the relay solution may be non-unique but achieves minimum relay energy consumption.
C. The Case Without Direct Link
Without a direct source–destination link, the DC allocation simplifies: source and relay powers are constant between energy-exhaustion events, and Algorithm I reduces to Algorithm II.
- For the minimum-energy optimal solution, source and relay power levels are set equal, although optimal profiles need not generally be unique.
- Source and relay power profiles change values only when harvested energy at the source or relay is exhausted.
- Algorithm I can be simplified into Algorithm II to compute the optimal source and relay allocation without a direct link.
- When h0 = 0, the relay-channel model becomes a cascade of two AWGN point-to-point channels.
V. OPTIMAL SOLUTION FOR THE NDC CASE
For NDC traffic, the non-convex throughput problem admits an optimal separation principle: source allocation is solved first, followed by relay allocation. This yields globally optimal profiles efficiently, with source-power uniqueness depending on the direct-link condition.
- A. Optimal Source Power Allocation: The NDC problem is solved by first obtaining source power while ignoring the relay, then optimizing relay power using that source solution.
- A. Optimal Source Power Allocation: The optimal source powers for the relay-ignored problem are non-decreasing over transmission blocks.
- A. Optimal Source Power Allocation: The source power profile from the relay-ignored problem is globally optimal for the full NDC problem.
- A. Optimal Source Power Allocation: Although the NDC throughput problem is non-convex, the separation principle produces its globally optimal solution efficiently.
- A. Optimal Source Power Allocation: For 0 < h0 < 1, the optimal source profile is unique, whereas for h0 = 0 it need not be unique because source energy may remain unused.
B. Optimal Relay Power Allocation
The relay allocation is obtained through a convex reformulation and forward search after fixing the optimal source profile. Rate scheduling then uses surplus rates to satisfy deficient blocks, with backward search producing a feasible schedule when needed.
- B. Optimal Relay Power Allocation: After fixing the optimal source profile, relay allocation is formulated through a rate-variable transformation that makes the problem convex.
- B. Optimal Relay Power Allocation: A forward-search algorithm computes the optimal relay allocation, whose relay powers are non-decreasing over time.
- B. Optimal Relay Power Allocation: The optimal source and relay power solutions can both be chosen non-decreasing across transmission blocks.
- C. Optimal Rate Scheduling: For NDC rate scheduling, surplus rates from some blocks are used to transmit earlier source-message binning indices with deficient rates.
- C. Optimal Rate Scheduling: When the area needing filling exceeds available surplus, the corresponding binning-rate values are not unique.
- C. Optimal Rate Scheduling: A backward-search algorithm initializes binning rates conservatively and adjusts them from block N to block 1 using accumulated surplus.
- C. Optimal Rate Scheduling: When h0 = 0, the source profile may be updated to achieve the same maximum throughput with minimum source energy consumption.
D. Throughput Comparison: DC vs. NDC
The paper compares delay-constrained (DC) and no-delay-constrained (NDC) throughput under deterministic energy harvesting, using numerical results to examine power-allocation schemes and energy diversity. NDC can outperform DC when delayed decoding exploits independently timed source and relay energy availability.
- Throughput comparison: NDC throughput is no smaller than DC throughput, with strict improvement characterized by Proposition 5.3.The condition is necessary and sufficient for strictly larger average throughput.
- Numerical setup: The numerical study uses periodic source and relay energy profiles with B = 100, N = 40, θ = 5π/4, and AS = AR = 200.The proposed algorithms are compared with a greedy strategy using current harvested energy.
- Numerical results: 0.387 bps/Hz is the throughput limit as the direct-link gain h0 increases.This limit is reached by NDC near h0 = 0.05, whereas DC reaches it only when h0 exceeds 0.75.
- Numerical results: NDC outperforms DC only at small direct-link gains, where delayed decoding exploits energy diversity.The greedy strategy can lose substantially, especially at small h0, relative to the proposed DC allocation.
- Scope and extensions: The paper leaves random EH models, finite storage, and relaying methods beyond DF as future extensions.These assumptions limit the present study to predictable energy arrivals, infinite storage, and decode-and-forward relaying.
APPENDIX A PROOF OF PROPOSITION 4.1
The proof establishes structural properties of optimal source and relay powers by showing that alternative consecutive allocations can be modified to satisfy energy constraints while increasing the sum rate. The remaining feasible ordering yields Proposition 4.1.
- Case analysis: The proof considers consecutive source and relay power pairs and analyzes three possible cases.Each case is ruled out through a feasible power redistribution that improves the sum rate.
- Improvement argument: Concavity of the rate function and monotonicity of logarithmic expressions support the improving reallocations.The proof repeatedly uses energy-feasible transfers between adjacent blocks to obtain larger two-block sum rates.
- Case elimination: The first two candidate orderings cannot be optimal because their reallocations preserve energy feasibility and strictly improve the objective.The argument applies source and relay updates while retaining the relevant source powers or adjusting relay powers.
- Conclusion: Therefore, the only remaining power ordering must hold, and Proposition 4.1 follows.The appendix explicitly concludes the proposition after ruling out the alternatives.
APPENDIX B THE OPTIMALITY PROOF OF ALGORITHM I
The appendix proves Algorithm I optimal for the delay-constrained problem by comparing all possible source-power transition scenarios and showing that infeasible or strictly suboptimal alternatives can be excluded. It then establishes optimality of the corresponding relay profile.
- Source-profile optimality: The appendix verifies that the source profiles produced by Algorithm I are optimal for Problem (P1).Cases involving exhausted or non-exhausted source constraints are handled separately.
- Scenario exclusion: Scenario I and Scenario II are excluded when they violate source-energy exhaustion or energy-constraint conditions.The proof uses contradictions involving unavailable exhaustion blocks or violated source constraints.
- Scenario exclusion: Scenario III is either impossible under Algorithm I’s defining conditions or strictly suboptimal after a feasible source-power redistribution.Concavity of the rate function yields a larger sum rate for the modified allocation.
- Relay-profile optimality: After source optimality is established, the relay powers generated by Algorithm I are also optimal for Problem (P1).The proof checks the relay-power cases associated with the identified source scenarios.
- Structural property: For 0 < h0 < 1, optimal source powers in the related problem can be chosen non-decreasing over time.The appendix derives this property from concavity and monotonicity of the relevant rate expression.
APPENDIX D PROOF OF PROPOSITION 5.3
The proof establishes the sufficient and necessary condition for NDC throughput to exceed DC throughput. It does so by constructing improved DC allocations when the condition holds and proving equality otherwise.
- Sufficiency: Source-power redistribution across two blocks can increase the DC rate while preserving the source energy constraint.The rate gain follows from the concavity of the rate function.
- Sufficiency: Relay-power redistribution likewise improves the sum rate when relay energy is exhausted later than the compared block.The construction shifts relay energy and compensates message binning rates across blocks.
- Necessity: If the condition does not hold, the optimal NDC allocation produces DC block rates equal to the NDC rates.Searching the full DC feasible set then shows identical throughput in both cases.