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Resource Allocation in Intelligent Reflecting Surface Assisted NOMA Systems
Jiakuo Zuo, Yuanwei Liu, Zhijin Qin, Naofal Al-Dhahir
TL;DR
The paper addresses throughput maximization in IRS-assisted NOMA with jointly optimized assignment, decoding order, power, and reflection coefficients. It decomposes the NP-hard formulation into three algorithmic stages and reports improved throughput for IRS-NOMA, including gains over conventional NOMA and IRS-OMA, with placement effects.
Problem
Joint optimization of channel assignment, decoding order, power allocation, and reflection coefficients in IRS-NOMA is NP-hard and non-trivial to solve.
Method
The paper decomposes the problem into many-to-one channel matching, low-complexity decoding-order optimization, and alternating optimization of power allocation and reflection coefficients.
Results
The proposed IRS-NOMA system improves system throughput, outperforms traditional NOMA, and achieves higher throughput than IRS-OMA; gains can increase when the IRS is near receivers.
Takeaways & Limitations
IRS-NOMA resource allocation can improve throughput, while matching-based assignment and low-complexity decoding-order methods achieve near-optimal or comparable performance.
Abstract
from arXiv · showhide
This paper investigates the downlink communications of intelligent reflecting surface (IRS) assisted non-orthogonal multiple access (NOMA) systems. To maximize the system throughput, we formulate a joint optimization problem over the channel assignment, decoding order of NOMA users, power allocation, and reflection coefficients. The formulated problem is proved to be NP-hard. To tackle this problem, a three-step novel resource allocation algorithm is proposed. Firstly, the channel assignment problem is solved by a many-to-one matching algorithm. Secondly, by considering the IRS reflection coefficients design, a low-complexity decoding order optimization algorithm is proposed. Thirdly, given a channel assignment and decoding order, a joint optimization algorithm is proposed for solving the joint power allocation and reflection coefficient design problem. Numerical results illustrate that: i) with the aid of IRS, the proposed IRS-NOMA system outperforms the conventional NOMA system without the IRS in terms of system throughput; ii) the proposed IRS-NOMA system achieves higher system throughput than the IRS assisted orthogonal multiple access (IRS-OMA) systems; iii) simulation results show that the performance gains of the IRS-NOMA and the IRS-OMA systems can be enhanced via carefully choosing the location of the IRS.
I. INTRODUCTION
The paper combines NOMA’s spectrum-efficiency potential with IRS-controlled channel enhancement, while addressing resource-allocation challenges not covered by prior IRS-NOMA studies.
- NOMA background: NOMA allows multiple users to share one orthogonal resource block and uses SIC to manage intra-channel interference.Users with better channel conditions can remove interference from users with poorer channel conditions.
- IRS background: IRS uses passive elements to adjust phase shifts and amplitudes, boosting received signal power with lower energy use than active relaying.The IRS reflects signals without signal-processing capability, distinguishing it from amplify-and-forward relays and active intelligent surfaces.
- Motivation: IRS-assisted NOMA is attractive because IRS control can create additional paths and a stronger combined channel gain.The IRS can modify channel conditions through element-wise phase-shift and amplitude control.
- Research gap: The paper targets joint resource allocation across channel assignment, decoding order, power allocation, and IRS reflection coefficients.Earlier work also considered channel assignment and power allocation in NOMA systems using matching, exhaustive search, convex approximation, Lagrangian, and monotonic optimization approaches.
- Related work: Prior work studies separate IRS-NOMA design components, including power allocation, phase shifts, decoding order, beamforming, and quasi-degraded channels.Existing approaches include alternating optimization, semidefinite relaxation, SOCP-ADMM, zero-forcing, exhaustive search, and alternating DC methods.
B. Motivation and Challenges
The IRS-NOMA system model allows multiple users to share channels, with decoding order and combined channel gains shaped by IRS reflection coefficients.
- System model: The considered system contains one base station, one IRS, and K users communicating over multiple downlink channels.The total bandwidth is divided equally among the channels, and each user is assigned to at most one channel.
- Multiple access: NOMA channels can serve multiple users, whereas the model reduces to IRS-OMA when each channel serves one user.Channel assignment is represented by binary indicators, with a maximum number of users per channel.
- IRS model: Each IRS element controls a phase shift and amplitude, enabling adjustment of the reflected signal component.The phase shift lies in [0, 2π] and the amplitude lies in [0, 1].
- Channel model: The effective channel combines the direct BS-user link with the BS-IRS-user reflected link.The model defines separate channel gains for the IRS-user, BS-IRS, and BS-user links.
- Decoding order: SIC decoding order depends on combined channel gains that can be modified by tuning IRS reflection coefficients.This makes decoding order an optimization variable rather than a fixed ordering based only on static channel gains.
B. Problem Formulation for the IRS-NOMA System
The paper formulates throughput maximization as a joint optimization over assignment, decoding order, transmit power, and IRS reflection coefficients, but the problem is computationally difficult.
- Problem formulation: The objective is to maximize system throughput by jointly optimizing channel allocation, decoding order, power allocation, and reflection coefficients.The optimization includes assignment, power, and decoding-order vectors together with IRS coefficient constraints.
- Problem formulation: The formulation enforces SIC success, minimum user capacity, total transmit-power, reflection-coefficient, channel-capacity, and user-assignment constraints.The constraints require valid decoding, per-user rate guarantees, a power budget, bounded IRS coefficients, and channel-assignment limits.
- Computational complexity: The formulated problem is NP-hard even when only the channel-assignment component is considered.This hardness follows from the binary channel-assignment constraint.
- Challenges: Decoding-order optimization is difficult because IRS reflection coefficients control the channel conditions that determine SIC order.The order cannot be optimized independently of the IRS design.
- Challenges: Power allocation and reflection coefficients are highly coupled, making the joint problem more challenging.This coupling contributes to the difficulty of obtaining a direct solution.
III. THE PROPOSED ALGORITHMS FOR IRS-NOMA SYSTEMS
The proposed solution decomposes the hard resource-allocation problem into channel assignment, decoding-order, and joint power/reflection-coefficient stages, with iterative convex methods for the final stage.
- Channel assignment: The first stage uses a low-complexity many-to-one matching algorithm for channel assignment.The paper reports near-optimal performance compared with exhaustive search.
- Joint design: The third stage alternates power allocation and reflection-coefficient design after channel assignment and decoding order are fixed.The power and reflection variables are separated into subproblems because they are coupled and non-convex jointly.
- Decoding-order optimization: The second stage uses a low-complexity decoding-order optimization algorithm that accounts for IRS reflection coefficients.Its performance is reported as comparable to exhaustive decoding-order search.
- Power allocation: Successive convex approximation transforms the power-allocation subproblem into iterative convex problems solvable by standard methods such as CVX.The iterative algorithm updates feasible variables until the objective converges.
- Convergence: The iterative power-allocation algorithm is guaranteed to converge because its objective sequence is non-decreasing and throughput is finitely bounded.This convergence statement is given in the paper’s remark on Algorithm 1.
- Initialization: A feasibility-search procedure introduces an infeasibility indicator to obtain feasible initial points for the power-allocation algorithm.When the indicator reaches zero, the resulting solutions are feasible for the subsequent convex subproblem.
2) Proposed Algorithm to Solve Subproblem (P2.2):
Subproblem (P2.2) remains non-convex because of constraints (22b) and (22c), so the paper applies successive convex approximation (SCA) iteratively. The resulting reflection-coefficient design procedure is summarized in Algorithm 3.
- Non-convexity handling: SCA approximates the non-convex constraints (22b) and (22c) at each iteration.The approximations transform subproblem (P2.2) into iteratively solvable problems.
- Reflection coefficients design: The proposed iterative reflection coefficients design algorithm solves subproblem (P2.2).The procedure is summarized in Algorithm 3.
- Iterative updates: The algorithm updates eκn,kn and eξn,kn, then solves (P6) to obtain eθ, κn,kn, and ξn,kn.Problem (P6) is convex and can be solved efficiently using CVX.
- Output: The procedure outputs the optimal eθ after the iterative updates.
B. Channel Assignment Algorithm based on Many-to-One Matching
The channel assignment procedure uses many-to-one matching, utility-aware swaps, and channel-gain-based proposals to assign users across channels. Its swapping process terminates when no swap-blocking pair remains, and the algorithm converges to a two-sided stable matching.
- Matching model: Many-to-one matching assigns each user to one channel while limiting the number of users allocated to each channel.A match is mutual between a user and its assigned channel.
- Swapping process: A swap-blocking pair permits two users to exchange channels when no involved utility decreases and at least one utility increases.User and channel utilities are defined within the matching state.
- Initialization process: Unmatched users propose to their best unrejected channel based on equivalent channel gain, while channels accept the highest-gain proposals.The initialization process repeats until users are matched or no further proposals remain.
- Swapping process: The swapping process continues until no swap-blocking pair exists.This refines the channel assignment obtained during initialization.
- Convergence: The proposed channel assignment algorithm converges to a two-sided stable matching within a limited number of iterations.
- Decoding-order optimization: The decoding-order search avoids evaluating all Kn! orders by maximizing the sum of combined channel gains with one optimization problem.The combined gain includes direct and IRS reflection links.
D. Proposed Three-Step Resource Allocation Algorithm for IRS-NOMA Systems
The proposed resource allocation method proceeds in three stages: channel assignment, SIC decoding-order optimization, and joint power allocation with reflection-coefficient design. The final stage uses the results of the first two stages and iterates until convergence.
- Step 1: Channel assignment: Step 1 obtains user index sets through the many-to-one channel assignment algorithm.
- Step 2: SIC decoding order optimization: Step 2 obtains SIC decoding orders for NOMA users in each channel using the low-complexity optimization algorithm.
- Step 3: Joint design: Step 3 jointly designs power allocation and reflection coefficients using the channel assignment and decoding-order results.Feasible initial points are found before alternating power allocation and reflection-coefficient updates.
E. Complexity and Convergence of the Proposed Three-Step Resource Allocation Algorithm
The three-step algorithm combines iterative power, initialization, and reflection-coefficient procedures with matching and decoding-order optimization. Its convergence follows from bounded throughput and a monotonically non-decreasing objective, while the SDP step has complexity on the order of (M+1)^6.
- Algorithm structure: The three-step procedure includes channel assignment, SIC decoding-order optimization, and joint power allocation with reflection-coefficient design.The joint-design stage initializes reflection coefficients and feasible power-related points before iterating until the objective converges.
- Complexity: The maximum number of swap operations in the channel-assignment process is K^2.
- Complexity: The SDP-solving complexity in the decoding-order algorithm is on the order of (M+1)^6.
- Convergence: The proposed algorithm is guaranteed to converge because system throughput is upper bounded and the objective value is monotonically non-decreasing.The monotonicity is established for the objective of (P1).
IV. NUMERICAL RESULTS
Numerical evaluations show that the proposed IRS-NOMA and IRS-OMA resource-allocation algorithms achieve near-exhaustive-search performance with lower complexity, while IRS assistance and more reflecting elements improve throughput.
- Compared algorithms: The proposed algorithms are ThreeStep-IRS-NOMA and TwoStep-IRS-OMA, while exhaustive-search algorithms provide comparison benchmarks.The IRS-OMA benchmark decomposes channel assignment from joint power-allocation and reflection-coefficient design; Exhaust-IRS-NOMA uses exhaustive search for channel assignment and decoding order.
- Channel assignment: 96% and 97.3%: ThreeStep-IRS-NOMA and TwoStep-IRS-OMA achieve these fractions of the utility attained by their exhaustive-search counterparts when N = 4 and Pmax = 15dBm.Exhaustive-search algorithms consistently outperform non-exhaustive methods, but the proposed algorithms remain close at lower complexity.
- Throughput versus reflecting elements: IRS-aided algorithms improve throughput as the number of reflecting elements increases, outperform algorithms without IRS, and IRS-NOMA outperforms IRS-OMA.The paper attributes higher gains with larger M to more passive elements reflecting more BS signal power.
- Throughput versus reflecting elements: 0.38bit/s/Hz and 0.49bit/s/Hz: at M = 20, the proposed IRS-NOMA and IRS-OMA algorithms gain these amounts over NOMA-noIRS and OMA-noIRS, respectively.At M = 140, the gains increase to 1.49bit/s/Hz and 1.86bit/s/Hz, respectively; at M = 80, ThreeStep-IRS-NOMA reaches around 97.6% of Exhaust-IRS-NOMA throughput.
2) System throughput versus the transmit power budget:
The simulations examine how transmit power, decoding order, IRS placement, and the proposed resource-allocation procedures affect IRS-NOMA throughput. IRS-assisted and NOMA-based algorithms outperform their respective baselines, while the proposed low-complexity methods approach exhaustive-search performance.
- System throughput increases as the total transmit power budget Pmax increases for all considered algorithms.
- IRS-assisted algorithms significantly outperform algorithms without the IRS, confirming the throughput advantage of introducing IRS.
- NOMA-based algorithms achieve higher throughput than OMA-based algorithms because NOMA allows users to access the same channel.
- ThreeStep-IRS-NOMA and TwoStep-IRS-OMA achieve near-optimal performance relative to their exhaustive-search counterparts.
- Impact of decoding order: The proposed ThreeStep-IRS-NOMA algorithm performs similarly to Exhaust-IRS-NOMA, outperforms Random-IRS-NOMA, and determines decoding order with lower complexity.
- Impact of the IRS location: ThreeStep-IRS-NOMA throughput gains over NOMA-noIRS are 4.1bit/s/Hz at xIRS = 10m and 6.1bit/s/Hz at xIRS = 45m.
APPENDIX A: PROOF OF THEOREM 1
The appendix proves NP-hardness by reducing a special case of the resource-allocation problem to three-dimensional matching. It also establishes convergence of the matching algorithm through bounded utility improvement and the absence of swap operations.
- The constructed special case of problem (P1) is a three-dimensional matching problem, so (P1) is NP-hard.
- The reduction uses assignments of two users to each channel, represented as triplets in a three-dimensional matching instance.
- A swap operation increases at least one channel utility, while total utility remains upper bounded by the transmit power budget.
- Because users and channels are finite and utility-improving swap-blocking pairs are limited, Algorithm 4 converges when no swap remains.