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Distributed Power Splitting for SWIPT in Relay Interference Channels using Game Theory
He Chen, Yonghui Li, Yunxiang Jiang, Yuanye Ma, Branka Vucetic
TL;DR
The paper addresses distributed power splitting for SWIPT in relay interference channels, where energy-harvesting relays must divide received signals between information processing and forwarding energy. It formulates game-theoretical power-splitting schemes for pure and hybrid AF/DF networks, proves unique equilibria with convergent distributed algorithms, and reports near-optimal average network-wide performance. The implementation discussion assumes simple processing costs are negligible and requires relays to obtain specific channel-related parameters.
Problem
SWIPT power splitting in relay interference channels is difficult because each relay’s ratio affects individual rates and the network-wide sum-rate under interference.
Method
The paper models each source-relay-destination link as a strategic player selecting its relay’s power splitting ratio, analyzes AF, DF, and hybrid games, and develops best-response distributed algorithms.
Results
The formulated games have unique Nash equilibria, and simulations show near-optimal network-wide performance on average, especially under relatively low and moderate interference.
Takeaways & Limitations
Game-theoretical distributed power splitting provides a practical approach for coordinating relay ratios without centralized optimization in the considered SWIPT relay-interference settings.
Takeaways & Limitations
The implementation discussion assumes relay transmit-processing power is negligible and requires each relay to obtain parameters Xi, Yi, Zi, and Wi.
Abstract
from arXiv · showhide
In this paper, we consider simultaneous wireless information and power transfer (SWIPT) in relay interference channels, where multiple source-destination pairs communicate through their dedicated energy harvesting relays. Each relay needs to split its received signal from sources into two streams: one for information forwarding and the other for energy harvesting. We develop a distributed power splitting framework using game theory to derive a profile of power splitting ratios for all relays that can achieve a good network-wide performance. Specifically, non-cooperative games are respectively formulated for pure amplify-and-forward (AF) and decode-and-forward (DF) networks, in which each link is modeled as a strategic player who aims to maximize its own achievable rate. The existence and uniqueness for the Nash equilibriums (NEs) of the formulated games are analyzed and a distributed algorithm with provable convergence to achieve the NEs is also developed. Subsequently, the developed framework is extended to the more general network setting with mixed AF and DF relays. All the theoretical analyses are validated by extensive numerical results. Simulation results show that the proposed game-theoretical approach can achieve a near-optimal network-wide performance on average, especially for the scenarios with relatively low and moderate interference.
I. INTRODUCTION
The paper studies SWIPT in relay interference channels with energy-harvesting relays and develops a distributed game-theoretical approach to power splitting. It addresses pure AF, pure DF, and hybrid AF/DF networks, with simulations indicating near-optimal average network-wide performance.
- Motivation: Relay interference channels support multiple simultaneous source-destination pairs communicating through dedicated relays over shared spectral resources.The relays lack their own power supply and harvest energy from received source signals before forwarding information.
- Motivation: Each relay splits its received signal between information processing and energy harvesting using a power splitting ratio.The harvested energy is used for forwarding in the second time slot.
- Motivation: Choosing all relays’ splitting ratios is difficult because each ratio affects both its own link and the interference-coupled network-wide sum-rate.The paper therefore models links as strategic players that maximize individual achievable rates.
- Contributions: The framework formulates distributed non-cooperative games for pure AF and pure DF networks, with each link selecting its dedicated relay’s splitting ratio.The framework is also extended to hybrid networks containing both AF and DF relays.
- Contributions: The pure-network games have existence and uniqueness results for their Nash equilibria, and distributed algorithms with provable convergence are developed to reach them.The algorithms update relay power splitting ratios from non-equilibrium states.
- Results: The proposed game-theoretical approach achieves near-optimal network-wide performance on average according to extensive numerical simulations.The paper considers AF and DF protocols and validates the analytical results numerically.
A. AF Relaying
For AF relaying, the relay exhausts harvested energy to amplify and forward its information-processing signal, yielding an end-to-end SINR and achievable-rate formulation. The paper uses these expressions to characterize the AF link performance under power splitting.
- AF operation: An AF relay exhausts its harvested energy to amplify and forward the signal received by its information-processing unit.The paper assumes this behavior because each relay seeks to maximize its achievable rate.
- AF signal model: The destination signal model separates the useful forwarded signal from the remaining interference-plus-noise terms.This separation leads to the end-to-end SINR expression for the AF link.
- AF performance: The AF formulation uses channel-dependent quantities whose physical meanings include relay-side SNR and INR, destination-side SNR, and destination-side INR.These quantities describe limiting cases with full information forwarding or full energy harvesting.
- AF performance: The AF link’s achievable rate is expressed from its end-to-end SINR as a function of the relays’ power splitting ratios.The ratio vector is denoted by ρ = [ρ1, ..., ρN]^T.
B. DF Relaying
The DF-relaying section models each link’s performance through relay decoding, destination forwarding, and an end-to-end achievable-rate expression. It then formulates distributed power splitting as a coupled non-cooperative game because interference links the players’ decisions.
- DF relays first decode information from the received information signal before forwarding decoded data to their destinations.
- The destination receives forwarded decoded information in the second time slot using energy harvested during the first time slot.
- The DF destination SINR uses parameters Zi and Wi, and the resulting achievable rate is treated as the link’s end-to-end performance.
- The network-wide objective is the sum-rate of all links, motivating a distributed power-splitting scheme rather than direct centralized optimization.
- The pure-network game has N S-R-D links as players, with each link selecting ρi ∈ Ai to maximize its own achievable rate.
- The centralized optimization is non-convex for AF networks and additionally non-differentiable for DF networks because of the min operator, creating signaling and computational difficulties.
A. Existence of the Nash Equilibrium
The paper establishes existence of Nash equilibria for the AF and DF power-splitting games, then analyzes uniqueness through best-response functions and standard-function properties. A special AF case yields equal splitting when the two-hop SINRs are balanced.
- A Nash equilibrium is a feasible strategy profile where no single player can improve its utility by deviating while others’ strategies remain fixed.
- The existence theorem requires compact, convex action sets and utilities that are continuous and quasi-concave in each player’s action.
- The AF utility is quasi-concave in ρi, and both the AF and DF power-splitting games possess at least one Nash equilibrium.
- The standard-function analysis uses positivity and scalability properties to establish uniqueness for the AF game.
- When (Xi + Yi)(Wi + 1) = Zi, the AF best response is ρi* = 1/2 because ρi(1 − ρi) is maximized at equal splitting.
- The AF game always admits a unique Nash equilibrium after uniqueness is reduced to the fixed point of its best-response function.
C. Uniqueness for the NE of the game GDF
For the DF game, the paper addresses uniqueness despite the utility’s non-differentiability by deriving best-response functions and applying standard-function analysis. It also introduces the distributed algorithm used to reach the equilibrium.
- The DF utility’s min operator makes standard differentiability-based uniqueness methods difficult to apply, so the analysis uses best-response functions and standard functions.
- The DF game always possesses one and only one Nash equilibrium.
- Algorithm 1 initializes each player with a feasible random power-splitting ratio and iteratively updates the ratio using best responses until termination.
- The proposed distributed algorithm is needed to achieve the unique equilibrium from non-equilibrium states, not merely to establish that the equilibrium exists.
1) Algorithm Description:
The paper extends best-response power splitting to hybrid AF/DF relay networks and provides a distributed implementation. The hybrid game has a unique equilibrium and an algorithm obtained by replacing the pure-network update rule.
- 1) Algorithm Description:: Algorithm 1 converges from any initial point to the unique Nash equilibrium of the pure AF and DF games.
- 2) Implementation Discussion:: Each link is modeled as a virtual single player, while a geographically distributed node such as the relay acts as the link coordinator in practice.
- 2) Implementation Discussion:: The relay coordinator needs channel, transmit-power, and received-signal measurements to calculate the parameters required by the best-response function.
- IV. EXTENSION TO HYBRID NETWORK: In the hybrid network, each link still maximizes its achievable rate over its power-splitting ratio, using the AF or DF rate expression associated with its relay protocol.
- IV. EXTENSION TO HYBRID NETWORK: The hybrid game GHD has one and only one Nash equilibrium, and Algorithm 2 uses its hybrid best-response update to achieve that equilibrium.
V. NUMERICAL RESULTS
Numerical experiments validate convergence to the games’ Nash equilibria and compare the proposed distributed power-splitting scheme with random and centralized alternatives across AF, DF, and hybrid relay networks. The game-theoretical approach achieves near-optimal average network performance particularly under relatively low or moderate interference, while interference and network size shape its gains and splitting ratios.
- Convergence validation: The proposed algorithms converge to the corresponding Nash equilibria from different starting points in two-link pure and hybrid games.The two-link experiments show unique equilibrium intersections, while a four-link hybrid example converges to the same values from two initial points.
- Two-link sum-rate comparisons: In two-link AF and DF networks, the game-theoretical scheme outperforms random splitting under sufficiently separated links and approaches the centralized optimum as interlink distance increases.It loses performance relative to the optimum at very small interlink distances, but approaches or coincides with the centralized solution as interference decreases.
- Multi-link effects: As link count increases, average sum-rates first rise from multiplexing gains and then decline as stronger interlink interference dominates.The proposed scheme outperforms random splitting for all tested link counts, with a significantly larger performance gap in DF than AF networks.
- Power-splitting behavior: AF networks maintain higher average power-splitting ratios than DF networks, and both ratios increase with link count as mutual interference grows.The same ratio-increase pattern occurs when all source transmit powers rise.
- Per-link rates: The game-theoretical scheme improves the average rates of both the best and worst links in AF and DF networks, with a larger improvement in DF.This comparison uses the same multi-link setting as the transmit-power experiments.
- Overall numerical conclusion: The framework’s numerical results support near-optimal network-wide performance on average, especially in scenarios with relatively low and moderate interference.The experiments validate the theoretical analyses and distributed algorithms across the studied network settings.
APPENDIX
The appendix proves existence of Nash equilibria for the pure AF and pure DF games by establishing suitable utility-function properties on each relay’s feasible power-splitting domain.
- Pure AF network: The AF utility is quasi-concave because it increases before a unique point ϵ_i and decreases afterward on [0, 1].The sign change follows from κ_i(0)>0 and κ_i(1)<0, ensuring a zero crossing within the feasible domain.
- Pure AF network: The pure AF game has at least one Nash equilibrium because its feasible strategy sets are compact and convex, and its utilities are continuous and quasi-concave.
- Pure DF network: The DF utility is concave, and therefore quasi-concave, because its SINR is concave in ρ_i and log(1+x) is concave and non-decreasing.
- Pure DF network: The pure DF game also admits at least one Nash equilibrium under the established continuity and quasi-concavity conditions.
B. Proof of Lemma 1
The proof derives the AF best response by analyzing a quadratic equation whose valid solution lies within the feasible power-splitting interval [0, 1].
- Best-response derivation: The AF best response is identified with the point ϵ_i where the utility’s derivative changes sign.
- Case analysis: When C_i = 0, the quadratic equation used to determine ϵ_i simplifies before the best-response expression is substituted into the final result.
- Case analysis: When C_i > 0, the valid root ϵ_i,2 lies in (0, 1), while the other root exceeds 1 and is infeasible.
- Case analysis: When C_i < 0, one root is negative and the other lies in (0, 1), so the latter is the valid feasible solution.
C. Proof of Proposition 2
The proof establishes that the AF best-response mapping is a standard function by verifying positivity, monotonicity, and scalability.
- Standard-function properties: The AF best-response mapping is positive for every player and every strategy profile.
- Standard-function properties: The mapping is monotone because increasing the relevant interference-related term W_i increases the best response.
- Case visualization: Figure 9 organizes the possible shapes of κ_i(ρ_i) for the cases C_i > 0 and C_i < 0 considered in the proof.
- Standard-function properties: The mapping satisfies scalability, completing the verification that it is a standard function.
D. Proof of Lemma 2
The proof characterizes the equilibrium power-splitting ratio as the feasible root of a quadratic equality associated with the ith link’s best-response condition.
- Equilibrium condition: At equilibrium, the ith relay’s power-splitting ratio must satisfy the best-response condition for the given strategy profile ρ.
- Quadratic solution: The equilibrium condition is rearranged into a quadratic equality that can be solved for ρ_i*.
- Feasibility selection: The quadratic has one root in (0, 1) and another in (1, +∞), so feasibility selects the smaller root as ρ_i*.
E. Proof of Proposition 3
The proposition is proved by showing that the DF game's best-response function BDF(ρ) is standard, satisfying positivity, monotonicity, and scalability.
- BDF(ρ) is positive because every player's best-response function is larger than 0 for any strategy profile ρ.
- Monotonicity is established by comparing two strategy profiles ρ and ρ′ with ρ ≥ ρ′ and analyzing the corresponding best responses.The proof uses the positivity of Xi/(2YiZi) and the square-bracket term, together with Wi > 0.
- Scalability is proved by defining Fi(α,ρ) = αBDFi(αρ) for α > 1 and showing Fi(α,ρ) > 0.Since Fi(1,ρ) = 0, the argument proceeds by showing that Fi(α,ρ) increases with α through derivative analysis.
- The derivative calculations and positivity conditions complete the proof of the scalability property and hence the proposition.The proof derives first- and second-order derivatives with respect to α and additional derivatives with respect to Wi.