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Energy-Efficient Transmission Design in Non-Orthogonal Multiple Access
Yi Zhang, Hui-Ming Wang, Tong-Xing Zheng, Qian Yang
TL;DR
The paper asks whether NOMA can improve energy efficiency in multi-user downlink transmission while meeting minimum data-rate requirements. It establishes feasible transmitting power, develops an EE-optimal allocation for the resulting non-convex fractional problem, and reports superior EE for NOMA over conventional OMA.
Problem
Existing NOMA research largely emphasized spectral efficiency, while EE studies were limited, including a fading MIMO case fixed at two users.
Method
The paper determines the minimum power supporting all users’ rate requirements and optimizes EE through decoupled power-allocation subproblems solved using closed form and bisection.
Results
NOMA has superior EE performance compared with conventional OMA, with larger gains as the number of simultaneously served users increases.
Takeaways & Limitations
Simultaneous power-domain service enables more efficient energy use in the studied multi-user SISO downlink setting.
Abstract
from arXiv · showhide
Non-orthogonal multiple access (NOMA) is considered as a promising technology for improving the spectral efficiency (SE) in 5G. In this correspondence, we study the benefit of NOMA in enhancing energy efficiency (EE) for a multi-user downlink transmission, where the EE is defined as the ratio of the achievable sum rate of the users to the total power consumption. Our goal is to maximize the EE subject to a minimum required data rate for each user, which leads to a non-convex fractional programming problem. To solve it, we first establish the feasible range of the transmitting power that is able to support each user's data rate requirement. Then, we propose an EE-optimal power allocation strategy that maximizes the EE. Our numerical results show that NOMA has superior EE performance in comparison with conventional orthogonal multiple access (OMA).
I. INTRODUCTION
The paper examines NOMA’s energy-efficiency benefits in multi-user downlink transmission, extending prior work beyond spectral efficiency and two-user fading MIMO settings. It formulates an EE optimization with per-user QoS requirements and proposes an EE-optimal power allocation strategy.
- NOMA serves multiple users simultaneously through power-domain division, unlike conventional OMA such as TDMA.
- Prior NOMA research primarily focused on spectral efficiency, including ergodic sum rate, outage performance, and user pairing.
- ICT energy consumption motivates studying energy efficiency, while existing NOMA EE work considered only two fixed users in a fading MIMO system.
- The paper studies EE optimization in a multi-user downlink SISO NOMA system with minimum data-rate requirements for every user.
- The proposed strategy first determines feasible transmitting power, then solves the non-convex fractional EE problem through closed-form and bisection-based subproblems.
II. SYSTEM MODEL
The system is a single-antenna base station serving K single-antenna users simultaneously over ordered fading channels. NOMA uses superposition transmission and SIC, with weaker users’ signals decoded before stronger users’ signals.
- A single-antenna base station simultaneously serves K single-antenna users over channels modeled with fading, distance, and path-loss effects.
- Users are ordered by ascending channel gain, with 0 < |h1|^2 ≤ |h2|^2 ... ≤ |hK|^2.
- The base station broadcasts a superposition of K signals, while ak denotes user k’s power fraction of total power P.
- Under SIC, user k decodes and removes users i < k sequentially, while treating users i > k as noise.
- The achievable individual and sum rates are defined from the received signal powers and additive-noise power σ^2.
III. PROBLEM FORMULATION
The paper maximizes energy efficiency, defined as achievable system sum rate divided by total power consumption, while guaranteeing each user’s minimum data rate. Feasibility depends on having sufficient transmitting power.
- Energy efficiency is defined as the ratio of the system’s achievable sum rate to its total power consumption.
- The consumed transmit power is the allocated user-power sum, while Pc represents constant circuit power consumption.
- Each user is assigned a minimum required data rate to provide a quality-of-service guarantee.
- The problem can be infeasible when available power is too small, so a minimum transmitting power PMin must satisfy all users’ rate requirements.
A. Minimum Required Transmitting Power PMin
The minimum feasible transmitting power is obtained by solving a power-allocation problem induced by the users’ minimum-rate constraints. Convexity and active constraints yield a sequential closed-form calculation of the required powers.
- PMin is formulated by denoting Pk as the power allocated to user k’s message and minimizing the power needed to meet all rate requirements.
- Problem (6) is convex, making its KKT conditions necessary and sufficient for the optimal solution.
- All minimum-rate constraints are active at the optimum because their associated Lagrange multipliers are positive.
- The minimum powers are calculated in closed form sequentially from user K down to user 1, and PMin equals their sum.
- The resulting threshold verifies whether the available total power P is large enough to satisfy each user’s data-rate constraint.
IV. ENERGY EFFICIENCY MAXIMIZATION
The EE maximization is reformulated by separating power-allocation coefficients from the actually consumed power ratio θ. The resulting inner problem yields closed-form allocation coefficients, while the outer problem optimizes θ.
- Problem reformulation: The original EE problem is solved under P ≥ PMin, ensuring that all users’ minimum data-rate requirements are feasible.The feasible-power condition is established before optimizing energy efficiency.
- Problem reformulation: θ is defined as the ratio of actually consumed transmitting power Pt to the total available BS power P, and θ may be below one at the EE optimum.The optimization therefore need not use all available transmitting power.
- Problem reformulation: The reformulated problem is decoupled into an inner optimization over power-allocation coefficients and an outer optimization over θ.The two subproblems are solved sequentially.
- Inner optimization: For fixed θ, the proposed power-allocation strategy obtains closed-form expressions for the optimal coefficients {a∗k}.The inner optimization is parameterized by θ.
- Outer optimization: The outer optimization is strict pseudo-concave in its unique argument θ, so bisection finds the EE-maximizing θ∗.This produces the optimal actually consumed transmitting power θ∗P.
A. Optimal Power Allocation Strategy
The inner power-allocation problem is solved by exploiting its shared non-convex subfunction structure. The optimal strategy activates the minimum-rate constraints for users 1 through K−1 and assigns extra power to user K.
- Optimization structure: The inner problem maximizes a sum of K−1 non-convex subfunctions sharing similar forms.The algorithm solves each subfunction and then identifies a common maximizer.
- Closed-form allocation: The component optimization problems have a unique common solution, which is therefore the optimal solution to the inner problem.The solution sets become smaller as more coefficients are determined sequentially.
- Optimization structure: Each subfunction is monotonically increasing in its optimization variable, so maximizing the subfunction is equivalent to maximizing that variable.This reduces the component problems to a uniform formulation.
- Closed-form allocation: The constraints in the component problems are active at the optimum, yielding closed-form expressions for the corresponding power-allocation coefficients.Proposition 1 establishes this active-constraint condition.
- Allocation strategy: The optimal allocation makes the minimum data-rate constraints active for users 1 through K−1 and uses extra power θP − PMin only to increase user K’s data rate.User K has the largest channel gain and can use additional power more efficiently than the other users.
B. Optimal Transmitting Power θ∗P for Maximizing the EE
After substituting the closed-form inner solution, EE becomes a univariate function of θ over the feasible interval. Its strict pseudo-concavity guarantees a unique maximizing transmit-power ratio.
- Outer problem: The closed-form inner solution transforms the outer problem into a univariate optimization with respect to θ.The feasible interval is specified by the constraint PMin ≤ θP ≤ P.
- Outer problem: The feasible range of θ is determined by PMin ≤ θP ≤ P.The lower bound guarantees the users’ rate requirements, while the upper bound reflects the available BS power.
- Optimality: EE(θ) is a strict pseudo-concave function of θ.The proof uses a strict concave numerator and an affine denominator.
- Optimality: Strict pseudo-concavity implies that EE(θ) has a unique maximizer, characterized as the unique root of dEE(θ).The paper applies bisection to locate θ∗ with polynomial complexity.
V. SIMULATION RESULTS
The simulations evaluate EEPA against MaxSE and TDMA under varying power, rate requirements, and user locations. NOMA generally outperforms OMA, but its EE becomes less robust as minimum rate requirements rise.
- Simulation setup: The simulations compare EEPA, MaxSE, and TDMA using 10,000 random channel realizations with specified fading, path-loss, noise, and circuit-power parameters.When available power cannot satisfy all users’ minimum rates, the BS remains silent and EE is set to zero.
- Power variation: NOMA achieves higher EE than OMA, with larger gains as the number of simultaneously served users increases.The paper attributes this to higher diversity gains and spectral efficiency from serving more users simultaneously.
- Power variation: At the Green Point, EE is maximized by both EEPA and MaxSE; beyond it, using full available power is not EE-optimal.Below the Green Point, increasing spectral efficiency also increases EE.
- Rate requirements: As the common minimum required data rate increases, achieving high EE becomes more difficult because more power must support users with worse channel conditions.At very large rate requirements, NOMA’s EE approaches zero faster because available power cannot satisfy the constraints.
- User locations: EE is lowest when all users are far from the BS, while equal average user distance can yield different EE depending on the closest user’s location.The results indicate that the closest user largely determines EE because it is most likely to have the largest channel gain.
VI. CONCLUSION
The paper studies EE optimization for a multi-user SISO NOMA downlink with individual data-rate requirements and proposes an energy-efficient power allocation strategy. Numerical results show superior EE for NOMA over conventional OMA.
- The study optimizes EE in a SISO NOMA system where multiple users have individual data-rate requirements.
- The proposed energy-efficient power allocation strategy maximizes EE for the constrained multi-user downlink.
- NOMA shows superior EE performance compared with conventional OMA because power-domain multiplexing serves multiple users simultaneously.
APPENDIX A PROOF OF PROPOSITION 1
The appendix proves optimality conditions for the convex inner problem by showing that relevant Lagrange multipliers are positive and associated constraints are active. This yields closed-form, sequentially determined coefficients.
- Because problem (19) is convex, its KKT conditions are necessary and sufficient for optimality.
- Positive multipliers imply that constraints (19c) are active for every 1 ≤ k ≤ K0.
- The proof establishes λ = µ1 > 0 by contradiction, then derives µk > 0 for 2 ≤ k ≤ K0.
- Setting those constraints active produces closed-form expressions for the coefficients, which are calculated sequentially in k = 1, 2, ..., K0.
APPENDIX B PROOF OF COROLLARY 1
The appendix uses positivity conditions and mathematical induction to establish the stated corollary, with the relevant quantities calculated sequentially across user indices.
- The proof derives a positive derivative condition from 0 ≤ Dk < 1 for 1 ≤ k ≤ K.
- Mathematical induction verifies that the target relation holds at k = 1 and at k = N + 1 using the induction hypothesis at k = N.
- The quantities are calculated sequentially in the order k = 1, 2, ..., K.