Joint Beamforming Design and Power Splitting Optimization in IRS-Assisted SWIPT NOMA Networks

Proposed framework

• The paper proposes an IRS-assisted SWIPT NOMA framework in which users share resource blocks while receiving information and energy.The BS transmit-power minimization jointly optimizes SIC decoding order, beamforming, PS ratio, and IRS phase shift under QoS and harvested-energy constraints.Source: §I · pp. 2, 4, 5

Motivation

• NOMA performance can suffer when users’ channel conditions are similar because practical wireless channels are random and difficult to control.IRS is introduced to adjust channels and increase channel-condition differences among users.Source: §I · pp. 2–3
• SWIPT lets users receive information and energy simultaneously, while power splitting divides received power between information decoding and energy harvesting.This supports large-scale, low-power, energy-constrained IoT devices.Source: §I · p. 3
• Conventional SWIPT suffers sharply reduced WPT efficiency as distance increases because of severe propagation loss.Strengthening channel conditions can improve WPT efficiency and coverage.Source: §I · p. 3

IRS-assisted networking

• IRS uses passive reflecting elements to adjust wireless channels, improving NOMA performance, SWIPT WPT efficiency, and network coverage.Its passive operation avoids additional noise and requires lower hardware cost and power consumption than MIMO.Source: §I · p. 3

Results

• The proposed algorithm reduces BS transmit power relative to baseline algorithms, while more IRS reflecting elements further reduce the required power.The numerical results also report good convergence.Source: §I · pp. 5–6

• The network comprises one BS, one IRS with M reflecting elements, and K single-antenna users served by an N-antenna ULA BS.Source: A. System Model · p. 6
• The IRS reflection uses amplitude coefficients β_m ∈ [0,1] and phase shifts θ_m ∈ [0,2π], with β_m typically set to 1 to maximize reflected signal power.Source: A. System Model · p. 7
• The wireless channel consists of BS–IRS, IRS–user, and direct BS–user links, modeled with Rician fading for links involving the IRS and Rayleigh fading for direct links.Source: A. System Model · pp. 7–8
• Each user receives a dedicated information beam through linear transmit precoding, with independently distributed CSCG data symbols.Source: A. System Model · p. 9
• Each user applies power splitting, directing a ρ_k portion of received power to information decoding and the remaining 1 − ρ_k to energy harvesting.Source: A. System Model · p. 9
• IRS phase shifts alter combined channel gains, making SIC decoding order dependent on the IRS-assisted channel conditions rather than only direct BS–user gains.Source: A. System Model · pp. 9–10

• The objective is to minimize BS transmit power by jointly optimizing SIC decoding order, transmit beamforming, received power-splitting ratios, and IRS phase shifts.Source: B. Problem Formulation for the… · p. 11
• QoS constraints enforce each user’s SINR threshold γ_k, while energy constraints require harvested power to reach threshold e_k.Source: B. Problem Formulation for the… · p. 11
• SIC constraints ensure that users decoding another user’s information achieve the required comparative SINR conditions.Source: B. Problem Formulation for the… · p. 11
• The formulation is non-convex because beamforming, PS ratios, and IRS phases are coupled, while phase shifts and IRS-dependent SIC orders introduce additional nonlinearity.Source: B. Problem Formulation for the… · p. 11

Two-stage optimization

• The algorithm decouples the problem into SIC decoding-order determination followed by alternating optimization of beamforming, PS ratios, and IRS phases.Source: §III · p. 12
• The second stage applies SDR, SCA, and Gaussian randomization to optimize the coupled design variables for a fixed SIC order.Source: §III · p. 12

SIC decoding order determination

• The first stage maximizes the sum of users’ combined channel gains because IRS phase shifts affect all users’ gains and therefore their SIC order.Source: A. SIC Decoding Order Determination… · p. 12

SDR and randomization

• SDR converts the relaxed first-stage problem into a semidefinite program, but the solution is generally high-rank and supplies an upper bound rather than an exact rank-one solution.Source: A. SIC Decoding Order Determination… · p. 13
• Gaussian randomization reconstructs rank-one candidate reflection solutions from the relaxed result and selects the candidate maximizing the combined channel gain.Source: A. SIC Decoding Order Determination… · pp. 13–14

Performance guarantee

• π/4 approximation is guaranteed for the optimal objective value of the first-stage problem, while the complete JDBPR algorithm obtains a sub-optimal solution to problem (P1).Source: A. SIC Decoding Order Determination… · p. 14

• Given the SIC decoding order, PS ratio, and IRS phase shift, the problem is transformed into a BS beamforming feasibility-check problem.Source: B. BS Beamforming Vector Optimization · pp. 14–15
• SDR relaxes the rank-one constraint, while SCA converts the remaining non-convex constraint into a tractable form.Source: B. BS Beamforming Vector Optimization · pp. 14–15
• The relaxed problem is a standard SDP that can be solved using CVX.Source: B. BS Beamforming Vector Optimization · p. 15
• The optimal relaxed solution satisfies Rank(W*_k) = 1, so the SDR for the BS beamforming problem is tight.Source: B. BS Beamforming Vector Optimization · p. 15

• With the BS beamforming vector and IRS phase shift fixed, the original problem becomes a PS-ratio feasibility-check problem.Source: C. PS Ratio Optimization · p. 16
• SCA transforms the non-convex PS-ratio constraint into a tractable approximation.Source: C. PS Ratio Optimization · p. 16
• The resulting PS-ratio problem is a standard convex optimization problem solved using the CVX toolbox.Source: C. PS Ratio Optimization · p. 16

Stage 1: SIC decoding order determination

• Stage 1 optimizes IRS phase shifts to maximize combined channel gain, then determines SIC decoding order from users’ combined channel gains.Source: E. Two-stage Overall Optimization… · p. 18

Stage 2: Joint optimization

• Stage 2 alternately optimizes the BS beamforming vector, PS ratio, and IRS phase shift using SDR, SCA, and Gaussian randomization.Source: E. Two-stage Overall Optimization… · p. 18

Convergence

• Gaussian randomization achieves an approximate value of at least π/4 for the first-stage optimal objective, while the overall JDBPR solution is sub-optimal.Source: E. Two-stage Overall Optimization… · p. 18

Stage 2: Joint optimization

• The algorithm updates beamforming, PS ratio, and IRS phase shift in sequence until the fractional objective decrease falls below ε.Source: E. Two-stage Overall Optimization… · p. 19

Convergence

• The objective function is non-increasing after each second-stage iteration, and finite lower bounds guarantee convergence.Source: 2) Convergence analysis · p. 20

Convergence

• The JDBPR algorithm converges by the 8-th iteration, with BS transmit power gradually decreasing as iterations increase.Source: §IV · pp. 20–21

QoS and harvested-energy thresholds

• BS transmit power increases with users’ QoS threshold under energy harvested thresholds e = 0dBm and e = −10dBm.Source: §IV · p. 22

Baseline comparisons

• TS-JDBPR-opt has transmit-power performance similar to EX-JBPR-opt but substantially lower complexity than exhaustive search.Source: §IV · pp. 22–23
• TS-JDBPR-opt outperforms the other baselines because it uses global optimization, optimized beamforming, and optimized IRS phase shifts.Source: §IV · p. 22

QoS and harvested-energy thresholds

• BS transmit power increases as the users’ energy harvested threshold rises, and higher QoS thresholds require more power at the same harvested threshold.Source: §IV · pp. 22–23

IRS resources

• Increasing the number of IRS reflecting elements lowers BS transmit power and enlarges the performance gap over non-IRS-assisted networks.Source: §IV · p. 23

BS antennas

• Increasing the number of BS antennas decreases BS transmit power, while TS-JDBPR-opt retains a performance advantage over the comparison algorithms.Source: §IV · pp. 24–25

• The paper investigates BS transmit-power minimization in IRS-assisted SWIPT NOMA networks under users’ QoS and energy-harvesting constraints.Source: §V · p. 25
• SIC decoding order, BS beamforming, PS ratio, and IRS phase shift are jointly optimized.Source: §V · p. 25
• A two-stage algorithm determines SIC decoding order first, then alternately optimizes beamforming, PS ratio, and IRS phase shift using SDR, SCA, and Gaussian randomization.Source: §V · p. 25
• The proposed JDBPR algorithm is analyzed for computational complexity and convergence.Source: §V · p. 25
• Numerical results show that the proposed algorithm significantly reduces BS transmit power compared with baseline algorithms.Source: §V · p. 25
• IRS assistance can greatly relieve pressure on the BS with low practical cost.Source: §V · p. 25

• The proof takes the logarithm of the relevant constraint before reformulating it.Source: Appendix A · p. 25
• SCA provides upper bounds for concave terms, yielding the transformed constraint in Eq. (27).Source: Appendix A · p. 26
• The beamforming subproblem is convex, so Slater’s condition holds and the duality gap is zero.Source: Appendix B · p. 26
• The proof formulates the Lagrangian and dual problem, then uses KKT conditions to investigate the optimal solution structure.Source: Appendix B · p. 27
• The analysis investigates whether the optimal beamforming matrix W*_k is rank-one using the structure of the dual solution.Source: Appendix B · p. 27

• Constraint (29c) is rewritten into an equivalent expression before approximation.Source: Appendix C · p. 28
• SCA obtains an upper bound for the concave fourth term on the left-hand side.Source: Appendix C · p. 28
• The proof takes logarithms of both sides of the constraint before deriving the transformed form.Source: Appendix D · pp. 28–29
• SCA upper bounds the first and fourth left-hand-side terms, producing the transformed constraint in Eq. (30).Source: Appendix D · p. 29