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Energy Efficiency and Spectral Efficiency Tradeoff in Device-to-Device (D2D) Communications
Zhenyu Zhou, Mianxiong Dong, Kaoru Ota, Jun Wu, Takuro Sato
TL;DR
The paper addresses the EE–SE tradeoff in D2D communications underlaying cellular uplinks, where channel reuse creates interference and prior SE-focused allocation can overlook UE energy consumption. It models allocation as a noncooperative game and uses nonlinear fractional programming to develop a distributed EE algorithm under SE and power constraints. Simulations show that the proposed algorithm achieves higher normalized average EE than spectral-efficient and random power allocation, while excess transmission power yields little SE improvement but significant EE loss.
Problem
D2D channel reuse creates interference, while prior SE-oriented allocation studies often ignore UE energy consumption even though EE and SE may conflict.
Method
The paper models resource allocation as a noncooperative game and applies nonlinear fractional programming to develop a distributed EE algorithm subject to SE and transmission-power constraints.
Results
0.429 normalized average EE is achieved by the energy-efficient algorithm, compared with 0.124 for random and 0.064 for spectral-efficient allocation.
Takeaways & Limitations
Increasing transmission power beyond the optimum-EE level brings little SE improvement but significant EE loss, so the proposed algorithm delivers significant EE improvement with little SE loss.
Abstract
from arXiv · showhide
In this letter, we investigate the tradeoff between energy efficiency (EE) and spectral efficiency (SE) in device-to-device (D2D) communications underlaying cellular networks with uplink channel reuse. The resource allocation problem is modeled as a noncooperative game, in which each user equipment (UE) is self-interested and wants to maximize its own EE. Given the SE requirement and maximum transmission power constraints, a distributed energy-efficient resource allocation algorithm is proposed by exploiting the properties of the nonlinear fractional programming. The relationships between the EE and SE tradeoff of the proposed algorithm and system parameters are analyzed and verified through computer simulations.
I. Introduction
D2D channel reuse improves network efficiency but creates interference and battery-life challenges, while prior work often optimized SE without accounting for UE energy consumption. The paper therefore studies the EE–SE tradeoff and proposes a distributed EE-oriented resource allocation approach.
- I. Introduction: D2D communications provide proximity, reuse, and hop gains but introduce co-channel interference and battery-life challenges.These challenges arise from spectrum reuse and limited UE battery capacity.
- I. Introduction: Most prior resource-allocation studies optimize SE in interference-limited environments while ignoring UE energy consumption.The paper motivates energy-aware design because handheld UEs can quickly exhaust their batteries when energy consumption is neglected.
- I. Introduction: EE and SE cannot always be achieved simultaneously and may sometimes conflict, leaving their tradeoff in D2D cellular networks insufficiently investigated.The paper identifies this tradeoff as the central research problem.
- I. Introduction: The paper models resource allocation as a noncooperative game and proposes a distributed algorithm that maximizes each UE’s EE under SE and transmission-power constraints.The algorithm is designed for self-interested UEs and addresses the stated tradeoff.
II. System Model
The system models uplink D2D links reusing cellular channels in a single cellular network, with interference coupling the UEs’ energy-efficiency strategies. Each UE’s optimization includes spectral-efficiency, power, and circuit-power considerations.
- II. System Model: The system considers a single cellular uplink in which D2D pairs reuse channels allocated to cellular UEs to improve SE.Cellular UEs receive orthogonal links, while D2D pairs reuse those channels.
- II. System Model: Resource allocation is modeled as a noncooperative game whose players’ EE depends on both their own and other UEs’ strategies.Interdependent strategies capture the interference created by channel reuse.
- II. System Model: The model accounts for transmission powers, channel gains, interference from cellular and D2D transmitters, and receiver noise.These quantities determine the links’ achievable SE and EE.
- II. System Model: UE total power consumption includes transmission power and circuit power, with D2D links including transmitter and receiver circuit power.The PA efficiency is denoted by η, with 0 < η < 1.
- II. System Model: Each D2D and cellular UE maximizes its own EE subject to minimum-SE and non-negative-power constraints.The constraints C1 and C3 impose minimum SE requirements, while C2 and C4 enforce non-negative power allocations.
A. The Objective Function Transformation
The paper transforms the nonconvex fractional EE objectives into concave subtractive-form objectives using nonlinear fractional programming. The transformed problems preserve the original optimum solutions and support best-response optimization.
- A. The Objective Function Transformation: The D2D and cellular EE objectives are nonconvex fractional functions that can be transformed into concave functions using nonlinear fractional programming.This transformation supplies the basis for tractable optimization of both link types.
- A. The Objective Function Transformation: The maximum EE of each D2D transmitter is characterized as its best response given the strategies of the other UEs.The corresponding cellular formulation likewise defines the cellular UE’s maximum EE and best response.
- A. The Objective Function Transformation: Theorem 1 states that the transformed subtractive-form D2D problem is equivalent to the original fractional problem and yields the same optimum solution.The equivalence connects the transformed optimization to the original D2D EE objective.
B. The Iterative Optimization Algorithm
The proposed iterative algorithm repeatedly solves transformed convex EE problems using dual decomposition and multiplier updates. It converges toward optimum EE, while sequential strategy updates support convergence to a Nash equilibrium.
- B. The Iterative Optimization Algorithm: Algorithm 1 iteratively solves the transformed convex problem until the EE change falls within a maximum tolerance.The iteration limit is set to 10, although simulations show convergence in 5 iterations.
- B. The Iterative Optimization Algorithm: The algorithm generates an increasing EE sequence that converges to optimum EE at a superlinear convergence rate.This convergence property is stated for the iterative fractional-programming procedure.
- B. The Iterative Optimization Algorithm: The D2D subproblem uses Lagrange dual decomposition, KKT conditions, and gradient updates for its dual variables.The same transformed-optimization framework is applied to cellular UEs with their corresponding constraints and multipliers.
- B. The Iterative Optimization Algorithm: The resulting transmission-power solution has a waterfilling form in which interference from other UEs lowers the water level.This links the distributed power update directly to measured interference conditions.
C. Complexity Analysis
The iterative algorithm solves a convex problem at each iteration, producing values that converge to the optimum EE. Its complexity is governed mainly by the dual iterations and loop iterations, with a dominant O(I_i,loopK) term when K is large.
- The iterative algorithm solves the convex problem at each iteration and produces an increasing sequence of q_d values that converges to the optimum EE.
- The algorithm’s dominant complexity is O(I_i,dual I_i,loop K) when K is large.I_i,dual is the number of dual iterations required for convergence, while I_i,loop is the number of iterations for solving the dual problem.
- Each D2D pair’s inner maximization allocates power, while the outer minimization finds the corresponding Lagrange multipliers.
- The per-pair implementation requires real additions, multiplications, and comparisons across the inner and outer optimization loops.The supplied complexity expressions account for K, N, I_i,loop, and the associated operation counts.
D. Distributed Implementation
The distributed implementation reduces each D2D pair’s information requirement to interference estimates rather than the specific strategies of other UEs. Sequential strategy updates converge to a Nash equilibrium.
- Each D2D transmitter’s best response depends on the strategies of all other UEs in the formulated EE maximization problem.
- D2D pairs need only estimate interference on available channels instead of knowing the specific strategies of other UEs.The base station estimates D2D interference on each cellular channel and feeds this information back to the corresponding cellular UE.
- Sequential UE strategy updates eventually converge to a Nash equilibrium, whose existence is established in Theorem 3.
- Algorithm 1 iteratively solves the D2D and cellular optimization problems for given EE parameters and stops when the convergence condition is met.
IV. Energy Efficiency and Spectral Efficiency Tradeoff
The paper characterizes how EE changes with SE, transmit power, and interference coupling for D2D links. It also establishes concavity, Nash-equilibrium existence, and convergence of the proposed optimization procedure.
- For D2D links, SE increases with transmit power, while EE first increases and then decreases, making EE quasiconcave in transmit power.
- The transformed subtractive objective function is concave, enabling the nonlinear fractional EE problem to be analyzed through a concave formulation.
- The proposed algorithm converges to the optimum EE, and its resulting optimum strategy set is a Nash equilibrium.
- EE can increase by trading off SE or by simultaneously increasing SE, with the maximum EE increase given by ΔEE = q_d*.
- For fixed transmit power, D2D EE decreases monotonically as the network coupling factor I increases; for finite positive I, EE first increases and then decreases with SE.
- The tradeoff analysis depends on the specific channel realization, and the special-case analysis assumes equal signal and interference gains while neglecting noise relative to interference.
- The paper states that analogous conclusions hold for cellular links but omits them because of space limitations.
V. Simulation Results
Simulations evaluate the proposed EE-maximizing allocation across D2D and cellular links, varying SE requirements, power limits, and interference. The results show convergence, EE–SE tradeoffs, and strong effects from interference, circuit power, channel access, and transmission-power constraints.
- Convergence and algorithm comparison: The proposed energy-efficient algorithm converges to normalized average EE 0.429, versus 0.124 for random allocation and 0.064 for spectral-efficient allocation.Results average 1000 simulations and normalize by the maximum value.
- D2D tradeoff: D2D SE increases monotonically with transmission power, while D2D EE first increases and then decreases as transmission power or SE increases.Beyond the power yielding optimum EE, additional transmission power provides little SE improvement but causes significant EE loss.
- D2D tradeoff: D2D maximum achievable EE and SE decrease monotonically as interference increases; at I = −15 dB, the calculated SE is only 8.6182 bits/s/Hz, making SE requirements of at least 9 bits/s/Hz infeasible.The interference-level simulations use I = −20, −15, and −10 dB.
- Cellular tradeoff: Cellular-link maximum EE is much lower than D2D-link maximum EE because longer transmission distances reduce signal-channel gain, while maximum power limits EE in low- and high-SE regimes.Cellular-link tradeoffs are evaluated with SE requirements from 0 to 10 bits/s/Hz over 500 simulations.
- Cellular tradeoff: For cellular links, increasing interference decreases both maximum achievable EE and SE, and both maxima are lower than for D2D links because cellular links use one channel whereas D2D pairs use K channels.The interference-level results agree with Corollary 3.
VI. Conclusion
The paper proposes a distributed energy-efficient resource allocation algorithm for D2D communications and analyzes the EE–SE tradeoff through nonlinear fractional programming and simulations. Results indicate that operating above the power for optimum EE yields little SE improvement but significant EE loss, while the algorithm achieves significant EE improvement with little SE loss.
- The proposed algorithm uses nonlinear fractional programming to perform distributed energy-efficient resource allocation for D2D communications.
- The EE–SE tradeoff is analyzed and verified through computer simulations.
- Increasing transmission power beyond the power for optimum EE brings little SE improvement but significant EE loss.
- The proposed energy-efficient algorithm can bring significant EE improvement subject to little SE loss.
Appendix B Proof of the Lemma 1
The appendix establishes structural properties of the EE and SE objectives and uses them to support existence, optimality, and convergence results for the noncooperative resource-allocation game. It also shows how EE changes with transmission power and how the proposed iterations approach the optimum.
- SE increases monotonically with transmission power, whereas EE first increases and then decreases.
- The EE objective is quasiconcave because its numerator is concave and its denominator is affine in transmission power.
- The transformed subtractive objective is concave because it combines sums of concave functions with sums of affine functions.
- The strategy sets satisfy the conditions used to establish existence of a Nash equilibrium in the noncooperative game.
- The resource allocation obtained by Algorithm 1 is proved to be the Nash equilibrium.
- The EE increases in each iteration and approaches the optimum EE when the iteration limit is sufficiently large.