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Sum Rate Maximization for IRS-assisted Uplink NOMA
M. Zeng, X. Li, G. Li, W. Hao, O. A. Dobre
TL;DR
The paper addresses sum-rate maximization in an IRS-assisted uplink NOMA system under individual user power constraints, where power control and IRS beamforming are jointly optimized. It applies semidefinite relaxation to the nonconvex problem, and numerical results show near-optimal performance and higher sum rates than OMA, with rates increasing linearly with log(N).
Problem
The paper studies how to maximize the sum rate of users in an IRS-assisted uplink NOMA system under individual power constraints.
Method
The proposed approach jointly optimizes user powers and IRS phase shifts using semidefinite relaxation, then extracts a feasible rank-one beamforming solution when needed.
Results
The proposed NOMA solution performs near-optimally, outperforms OMA in sum rate, and yields sum rates that grow linearly with log(N).
Takeaways & Limitations
IRS-assisted uplink NOMA benefits from increasing the number of reflecting elements, while NOMA provides higher sum rate than OMA in the reported evaluations.
Abstract
from arXiv · showhide
An intelligent reflecting surface (IRS) consists of a large number of low-cost reflecting elements, which can steer the incident signal collaboratively by passive beamforming. This way, IRS reconfigures the wireless environment to boost the system performance. In this paper, we consider an IRS-assisted uplink non-orthogonal multiple access (NOMA) system. The objective is to maximize the sum rate of all users under individual power constraint. The considered problem requires a joint power control at the users and beamforming design at the IRS, and is nonconvex. To handle it, semidefinite relaxation is employed, which provides a near-optimal solution. Presented numerical results show that the proposed NOMA-based scheme achieves a larger sum rate than orthogonal multiple access (OMA)-based one. Moreover, the impact of the number of reflecting elements on the sum rate is revealed.
I. INTRODUCTION
The paper extends IRS-assisted NOMA research from downlink to uplink, targeting sum-rate maximization through joint user power control and IRS phase-shift design. Because the resulting problem is nonconvex, it proposes an SDR-based solution and reports higher sum rate than OMA.
- Prior IRS studies [2]– considered OMA-based systems and objectives including weighted sum rate, minimum rate, power minimization, and energy efficiency.
- This paper considers IRS-assisted uplink NOMA and aims to maximize all users’ sum rate under individual power constraints.
- The optimization jointly designs users’ transmit powers and IRS phase shifts, with nonconvexity arising particularly from the IRS element-wise constant-modulus constraint.
- An SDR-based suboptimal solution is proposed, with numerical results indicating near-optimal performance and higher sum rate than the OMA-based counterpart.
A. System Model
The modeled system has single-antenna users and base station communicating only through an IRS. Uplink NOMA uses SIC, but IRS-dependent effective channels complicate decoding-order selection.
- The uplink contains K single-antenna users, a single-antenna BS, and an IRS with N reflecting elements; the direct user–BS link is blocked.
- The received signal comprises users’ transmitted signals, with each sk having unit power and each transmit power Pk constrained by its maximum value.
- The user–IRS channels are hk, the IRS–BS channel is hBS, and Φ is a diagonal matrix of IRS phase shifts satisfying |φi| = 1.
- NOMA mitigates multiuser interference through BS-side successive interference cancellation, typically decoding users with better channel conditions earlier.
- Because the effective channel hBS^HΦhk depends on unknown Φ, users are ordered instead using the simplified effective channel hBS^Hhk.
- Under the NOMA protocol, user k’s SINR determines its achievable data rate through the stated rate expression.
B. Problem Formulation
The paper formulates sum-rate maximization over user transmit powers and IRS passive beamforming. The resulting problem is nonconvex because of its objective and IRS-related constraint.
- The objective is to maximize the users’ sum rate through joint IRS passive beamforming and user power control.
- The transmit-power vector is P = [P1, · · ·, PK], collecting the users’ individual transmit powers.
III. PROPOSED SOLUTION
The proposed solution first fixes users’ transmit powers at their individual maxima, then optimizes IRS phase shifts through reformulation and semidefinite relaxation. The relaxed problem yields an upper bound and a feasible near-optimal beamforming solution.
- The sum rate is independent of user decoding order because its expression reduces to a product of telescoping terms.
- Each user should transmit at full power because the sum rate increases monotonically with each user’s power under any IRS phase-shift matrix.
- The joint optimization is reduced to IRS phase-shift design with unit-modulus constraints after applying the full-power result.
- The IRS phase vector and auxiliary Hadamard-product vector reformulate the effective channel and preserve the unit-modulus constraints.
- Relaxing the element-wise unit-modulus constraint to a sphere constraint produces an upper bound λN^2, attained using the normalized principal eigenvector of H.
- The relaxed upper bound indicates that the sum-rate SINR can potentially grow with the square of the IRS element count.
- Semidefinite relaxation drops the rank-one constraint, then extracts a principal-eigenvector solution and normalizes its elements when necessary to recover feasible unit-modulus phases.
IV. BASELINE SCHEME: OMA
The OMA baseline equally divides time or frequency resources among users and designs IRS beamforming and user power separately for each user. Its per-user optimization is then expressed under unit-modulus IRS constraints.
- OMA equally divides time or frequency degrees of freedom among the K users.
- Under OMA, IRS passive beamforming and user power control can be performed separately for each user.
- The OMA per-user problem uses a unit-modulus constraint for every IRS phase-vector element.
- The optimal OMA beamforming satisfies an element-wise phase condition for the IRS vector.
V. SIMULATION RESULTS
Numerical simulations compare NOMA and OMA under varying reflecting-element and user counts. NOMA-prop remains close to optimal and outperforms OMA when multiple users are served.
- Simulation setup: The simulations average results over 10^3 random trials with three users and 16 IRS reflecting elements as default settings.Distances, propagation models, fading, bandwidth, and noise parameters follow the stated simulation configuration.
- Reflecting elements: Sum rates for all considered schemes grow linearly with log(N) as the number of IRS reflecting elements increases.This trend is reported for NOMA-SDR, NOMA-prop, NOMA-up, and OMA.
- Reflecting elements: NOMA-prop dominates OMA in sum rate for every given value of N.This comparison verifies the reported superiority of NOMA over OMA across reflecting-element counts.
- Number of users: Sum rates increase with the number of users K, but the increase declines at larger K because of the concavity of the logarithm.For K = 1, NOMA-prop matches OMA; for K > 1, it outperforms OMA and the gap increases with K.
VI. CONCLUSION
The paper formulates IRS-assisted uplink NOMA sum-rate maximization as a joint user-power and IRS-beamforming problem. SDR addresses the non-convex formulation, and simulations report NOMA gains over OMA and logarithmic scaling with IRS size.
- VI. CONCLUSION: The considered problem jointly optimizes user transmit powers and passive IRS beamforming for uplink NOMA sum-rate maximization.The problem is non-convex and includes the IRS element-wise constant-modulus constraint.
- VI. CONCLUSION: Semidefinite relaxation transforms the non-convex problem into an SDR that is solved to obtain the proposed solution.The paper compares the resulting NOMA scheme with an OMA solution derived in closed form.
- VI. CONCLUSION: The proposed NOMA scheme achieves higher sum rates than OMA, while both schemes grow linearly with log(N).The reported scaling illustrates the effectiveness of using more reflecting elements at the IRS.