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Wireless Powered Communications with Non-Orthogonal Multiple Access
Panagiotis D. Diamantoulakis, Koralia N. Pappi, Zhiguo Ding, George K. Karagiannidis
TL;DR
Wireless-powered uplink devices face finite battery lifetimes, motivating energy harvesting, while NOMA is studied to improve individual data rates and fairness. The paper optimizes fixed and time-sharing decoding strategies, solves the formulations with linear programming or convex optimization, and proposes a greedy time-sharing algorithm; simulations show improvements over TDMA in throughput and fairness, alongside relationships among throughput, fairness, harvested energy, and energy efficiency.
Problem
Wireless communication devices operate for finite durations limited by battery lifetimes, motivating energy harvesting and NOMA-based optimization of individual data rates and fairness.
Method
The paper analyzes fixed and time-sharing decoding orders, solves the optimization problems using linear programming or convex optimization, and proposes a greedy algorithm for time-sharing.
Results
The proposed NOMA scheme outperforms baseline TDMA in throughput and fairness, with simulations also showing improved minimum individual data rates and relationships involving harvested energy and energy efficiency.
Takeaways & Limitations
The optimization methods support practical NOMA implementation, while time-sharing exposes trade-offs among rates, energy harvesting, throughput, fairness, and energy efficiency.
Abstract
from arXiv · showhide
We study a wireless-powered uplink communication system with non-orthogonal multiple access (NOMA), consisting of one base station and multiple energy harvesting users. More specifically, we focus on the individual data rate optimization and fairness improvement and we show that the formulated problems can be optimally and efficiently solved by either linear programming or convex optimization. In the provided analysis, two types of decoding order strategies are considered, namely fixed decoding order and time- sharing. Furthermore, we propose an efficient greedy algorithm, which is suitable for the practical implementation of the time-sharing strategy. Simulation results illustrate that the proposed scheme outperforms the baseline orthogonal multiple access scheme. More specifically, it is shown that NOMA offers a considerable improvement in throughput, fairness, and energy efficiency. Also, the dependence among system throughput, minimum individual data rate, and harvested energy is revealed, as well as an interesting trade-off between rates and energy efficiency. Finally, the convergence speed of the proposed greedy algorithm is evaluated, and it is shown that the required number of iterations is linear with respect to the number of users.
I. INTRODUCTION
The paper addresses wireless-powered uplink NOMA to improve individual data rates and fairness, contrasting fixed decoding orders and time-sharing with TDMA. It formulates tractable optimization methods and reports gains in throughput, fairness, and energy efficiency.
- I. INTRODUCTION: Wireless-powered communication addresses finite battery lifetimes by harvesting environmental or wireless energy for sustainable operation.Wireless power transfer can support both battery charging and information transmission.
- I. INTRODUCTION: Wireless-powered nodes can suffer lower individual data rates, making TDMA potentially unsuitable for efficient multiuser transmission.The paper motivates NOMA because it exploits the power domain and uses successive interference cancellation.
- I. INTRODUCTION: Wireless-powered uplink NOMA is studied for increasing individual data rates and user fairness in a network with one BS and multiple energy-harvesting users.The objectives include sum-throughput maximization with minimum-rate improvement and equal individual data-rate maximization.
- I. INTRODUCTION: The study optimizes energy-harvesting time and SIC-related time-sharing variables, while reformulating equal-rate optimization into a tractable problem.Fixed decoding order and time-sharing are the two decoding strategies considered.
- I. INTRODUCTION: All formulated problems can be solved optimally using linear programming or convex optimization, and a greedy algorithm is proposed for efficient time-sharing.The algorithm dynamically handles decoding-order variables and is evaluated for performance and convergence speed.
II. SYSTEM MODEL
The system uses harvest-then-transmit wireless-powered uplink NOMA, with optimization objectives covering asymmetric system throughput and symmetric individual rates. Fixed decoding order and time-sharing yield four optimization schemes, with time-sharing trading higher QoS for computational complexity.
- II. SYSTEM MODEL: The network contains N single-antenna users and one single-antenna BS, with quasi-static perfectly estimated channels during unit-duration frames.The BS-to-user path loss is L_0n and the channel coefficient is h_0n ~ CN(0,1).
- II. SYSTEM MODEL: Harvest-then-transmit assigns 1−T to BS wireless-energy broadcasting and T to simultaneous uplink information transmission using harvested energy.Each user's transmit energy is limited by the energy harvested during the first phase.
- II. SYSTEM MODEL: The asymmetric objective maximizes system sum-capacity while subsequently maximizing the weakest user's individual data rate for fairness.The resulting system throughput is denoted R_tot.
- II. SYSTEM MODEL: The symmetric objective maximizes the common individual rate, which equals the minimum achievable user throughput without necessarily maximizing system throughput.For equal rates, the sum-rate is R_sum = N R_eq.
- II. SYSTEM MODEL: Combining the two objectives with fixed decoding order or time-sharing produces four schemes, while time-sharing improves QoS at higher computational complexity.Its complexity depends on the number of user decoding-order permutations, requiring a balance between optimality and efficiency.
C. Achievable User Throughput in the Case of Fixed Decoding Order
With fixed decoding order, user throughput depends on the sequential interference structure, while time-sharing averages rates across decoding permutations. The achievable NOMA system throughput is independent of decoding order.
- Fixed decoding order: Fixed-order decoding gives earlier-decoded users interference from users decoded later, while the final user experiences no such interference.The n-th user's interference includes users with later decoding positions; the last user's throughput is defined separately.
- Time-sharing: Time-sharing changes decoding order across fractions of the transmission duration, represented by permutation-specific weights τm whose sum is one.The method allows different decoding orders during specific portions of the information-transmission interval.
- Time-sharing: A permutation matrix records which user is decoded at each order, with rows identifying permutations and columns identifying decoding positions.For example, A(2, 4) = 3 means user 3 is decoded fourth under permutation 2.
- Time-sharing: The time-sharing throughput of each user is obtained by incorporating the selected permutation configuration into the achievable-throughput expression.The resulting quantity is denoted by ˜Rn for time-sharing.
- System throughput: NOMA uplink system throughput does not depend on the decoding order, so any arbitrary order can represent the system sum-throughput.Decoding order can therefore affect individual rates without changing achievable system throughput.
III. SYSTEM THROUGHPUT MAXIMIZATION AND MINIMUM THROUGHPUT IMPROVEMENT
The paper maximizes system throughput over the energy-transfer duration and then develops a minimum-throughput improvement procedure. Strict concavity makes the optimal duration unique and expressible using the principal Lambert W function.
- Optimization procedure: The system-throughput optimization first formulates and solves achievable throughput maximization, then improves the minimum individual data rate.The section also introduces a greedy algorithm to reduce the complexity of optimizing time-sharing.
- Throughput maximization: Throughput is zero at T = 0 or T = 1 because one endpoint leaves no time or no energy for user transmission.The optimization therefore considers 0 < T < 1.
- Throughput maximization: Rtot is strictly concave in T over (0, 1), so its maximizing value T* is unique.The unique optimum is obtained from the stated concavity-based optimization.
- Closed-form solution: The optimal duration is expressed in closed form using the principal branch of the Lambert W function.Lambert W is also called the omega function or product logarithm and satisfies x = W(x)e^W(x).
B. Minimum Achievable Throughput Improvement with Descending Decoding Order
Minimum-throughput improvement uses descending channel-gain order for fixed decoding and optimized time-sharing for broader fairness gains. The time-sharing problem is linear but full permutation search becomes inefficient as users increase.
- Descending decoding order: Descending channel-gain order decodes the weakest user’s message without interference and improves fairness and minimum throughput over ascending order.Users are indexed so g1 ≥ ... ≥ gN.
- Time-sharing optimization: Time-sharing sets T = T* to maximize system throughput while optimizing permutation fractions to improve the minimum user throughput.Proper selection of τ can achieve any point of the capacity region, supporting fairness improvement.
- Time-sharing optimization: The time-sharing optimization is a linear program that can be solved efficiently with simplex or interior-point methods.Its worst-case complexity is exponential in the problem dimensions, specified as (N + 1)M.
- Full-space search: Full-space search is optimal because it includes all N! decoding permutations, but it becomes inefficient as the number of users grows.For N = 5, the search must include 120 permutations.
- Greedy algorithm: The greedy algorithm dynamically constructs the permutation set and jointly optimizes the associated time-sharing variables while excluding unnecessary permutations.Only new permutations are inserted, and iterations stop at a maximum K or when a duplicate permutation appears.
- Greedy algorithm: The greedy procedure initializes descending channel-gain order, computes user rates, creates new orders based on current rates, and repeatedly solves the linear program.Users with smaller current throughput receive opportunities to improve while the minimum throughput is not reduced.
E. Examples
The examples show how channel conditions and time-sharing shape achievable throughput regions, revealing a trade-off between system throughput and minimum user throughput.
- Similar-distance users achieve a capacity region in which optimized time-sharing dominates other T selections in both system and minimum throughput.The optimal value is T = 0.7958.
- For similar channels, fixed descending decoding yields R1 = 10.8823 bps/Hz and R2 = 0.7251 bps/Hz.This point is a corner of region D1.
- In the contrasting case, choosing T = 0.54 gives R1 = 7.1242 bps/Hz and R2 = 1.4223 bps/Hz but does not maximize system throughput.
- The examples establish a trade-off between minimum user throughput and system throughput, motivating equal individual data-rate maximization.
IV. EQUAL INDIVIDUAL DATA RATE MAXIMIZATION WITH DESCENDING DECODING ORDER
The fixed descending-order formulation maximizes equal individual data rates under user-throughput constraints. Its concavity and linear components make the problem convex and efficiently solvable through dual decomposition.
- Equal individual data-rate maximization seeks the largest Req subject to Rn ≥ Req for every user and 0 < T < 1.
- The first N constraints are strictly concave, while the objective and final constraint are linear, making the optimization problem convex.
- Dual decomposition solves the convex problem efficiently by calculating optimal Req and T for fixed Lagrange multipliers and iteratively updating them.
- The iterations converge to the optimal solution when the gradient-method step sizes satisfy the stated infinite travel condition.
- A larger multiplier identifies a user constraint with greater influence on Req and prioritizes that user’s throughput in energy-transfer-time optimization.
V. EQUAL INDIVIDUAL DATA RATE MAXIMIZATION WITH TIME-SHARING
With time-sharing, equal individual data-rate maximization jointly optimizes the energy-transfer time and time-sharing configuration, without necessarily maximizing system throughput.
- The time-sharing formulation optimizes both T and the time-sharing configuration to maximize equal individual data rates.
- Its solution does not necessarily maximize system throughput, distinguishing equal-rate optimization from throughput maximization.
A. Problem Formulation and Solution
The time-sharing problem is reformulated by deriving rate constraints that eliminate the time-sharing variables from the equal-rate bound. This reduces the original coupled optimization to tractable subproblems.
- The formulation orders users by channel gains while allowing decoding order to depend on time-sharing.
- For fixed T, the achievable rate region is characterized by inequalities over user sum sets and weakest-link throughput conditions.
- Applying the weakest-user argument successively yields inequalities bounding Req without τ.
- Problem 1 is jointly concave in T and Req, while the subsequent fixed-Req problem is linear and solvable by linear programming or Algorithm 1.
B. Solution of Problem 1
Problem 1 is solved through Lagrange dual decomposition, yielding iterative updates for the equal individual data rate and energy-transfer time. The associated Lagrange multipliers are updated using positive step sizes.
- The Lagrangian is formed after replacing the initial objective with ln(Req).
- The resulting dual problem maximizes the Lagrangian over T and Req for the multiplier vector µ.
- Each iteration determines optimal values of Req and T for the current multipliers.
- The Lagrange multipliers are iteratively updated using positive step sizes.
VI. SIMULATION RESULTS AND DISCUSSIONS
The simulations compare the proposed NOMA optimization schemes with TDMA across throughput, charging time, fairness, and convergence. NOMA with optimized time-sharing improves minimum and equal individual rates, particularly at higher transmit powers, while the greedy algorithm converges efficiently.
- Simulation setup: The simulations average results over 10^5 random channel realizations and include all N! decoding-order permutations for time-sharing optimization.Users are uniformly distributed between radii 5 m and 20 m around the base station.
- Simulation setup: The evaluation compares proposed schemes (a)–(d) against TDMA in system throughput, user throughput, energy-transfer time, energy efficiency, and fairness.The TDMA cases include system-throughput maximization and equal individual data-rate maximization.
- Throughput comparison: NOMA and TDMA achieve the same normalized system throughput for N = 3, while NOMA notably increases minimum user throughput across the full P0 range.This improvement occurs even without time-sharing.
- Throughput comparison: Optimized NOMA time-sharing achieves higher minimum throughput than TDMA at medium and high P0, and higher equal rates across the full transmit-power range.Fixed decoding order reaches only capacity-region corner points, whereas time-sharing can achieve any point of the capacity region.
- Throughput comparison: As the number of users increases, NOMA’s equal individual rate and minimum throughput decrease, while its equal rate remains above TDMA’s.The gap between equal rate and minimum throughput also increases with the number of users.
B. Trade-off Between Energy Harvesting and Information Transmission
The paper identifies trade-offs among energy-transfer time, throughput, energy efficiency, and fairness. NOMA improves fairness and energy efficiency relative to TDMA, while the greedy time-sharing algorithm reaches the optimum in a number of iterations linear in the user count.
- Trade-off Between Energy Harvesting and Information Transmission: When maximizing system throughput, increasing the user count reduces the portion of time dedicated to energy transfer.Users with good channels increasingly dominate system throughput and require less charging time.
- Trade-off Between Energy Harvesting and Information Transmission: When maximizing equal individual rates, increasing the user count increases energy-transfer time so the weakest user can obtain sufficient energy.NOMA uses slightly more harvesting time than TDMA but exploits information-transmission time more efficiently.
- Energy Efficiency: NOMA achieves higher energy efficiency than TDMA across the full P0 range despite transmitting more energy for harvesting.Increasing P0 considerably decreases energy efficiency, revealing a trade-off between equal individual rate and energy efficiency.
- Fairness Comparison: NOMA provides greater Jain’s fairness than TDMA across the full P0 range while the compared schemes achieve the same system throughput.Jain’s fairness index equals 1 when users have equal rates.
- Greedy algorithm convergence: The greedy time-sharing algorithm converges to the optimal minimum throughput within N + 1 iterations for N = 3, 4, 5, and 6.This reduces the search burden relative to considering all N! permutations.
- Conclusions: The study concludes that NOMA outperforms the TDMA baseline in throughput and fairness, with dependence among system throughput, minimum rate, and harvested energy.