Joint Beamforming and Power Allocation in Downlink NOMA Multiuser MIMO Networks

A. Related Studies and Motivations

• Prior MIMO-NOMA work focused on single-cell settings, while this work considers multicell cooperation to address inter-cell interference.CoMP coordinates base stations; the paper focuses on coordinated beamforming rather than joint processing.Source: A. Related Studies and Motivations · pp. 1–2

B. Contributions

• The design maximizes Group 1 sum-rate while ensuring Group 2 target rates through coordinated joint beamforming and power allocation.The users are grouped by QoS requirements rather than channel qualities.Source: B. Contributions · p. 2
• The iterative SCA algorithm coordinates base stations for joint beamforming and power allocation, and the authors report it converges in no more than 10 iterations on average.The SDR is reported as tight, making the SDP-relaxed problem equivalent to the original joint design problem.Source: B. Contributions · p. 2

II. SYSTEM MODEL

• The network pairs one user from each QoS group in a cluster, with both users sharing the same resource, and assumes each base station has global CSI.Each cell has K users in each group and a base station with M antennas, where M ≥ K.Source: §II · p. 3

A. Zero Forcing Transmission at BSs

• Zero-forcing beamforming suppresses intra-cell interference at Group 1 users, while the beamformer is designed to support Group 2 users’ target rates.The paper states that this zero-forcing design requires only local CSI at the serving base station, reducing cooperation signaling overhead.Source: A. Zero Forcing Transmission at BSs · p. 3

B. SIC at Users in Group 1

• The received Group 1 signal includes the desired signal, intra-cluster interference from the paired user, and inter-cell interference from other base stations.The cited passage identifies these as the three received-signal terms.Source: B. SIC at Users in Group 1 · p. 4
• Group 1 users first decode the paired Group 2 signal to remove its interference through successive interference cancellation.The paper motivates this decoding order by Group 1’s best-effort service and Group 2’s low target-rate requirement.Source: B. SIC at Users in Group 1 · p. 4
• The target rate R0 for Group 2 must be met for successful SIC decoding at the Group 1 user.The paper assumes Group 2 users have the same required target rate R0.Source: B. SIC at Users in Group 1 · p. 4

C. Direct Decoding at Users in Group 2

• The Group 2 receiver’s channel is represented by h, and its received signal includes additive white Gaussian noise.The supplied passage identifies h as the channel vector between the serving base station and the Group 2 user.Source: C. Direct Decoding at Users in Group 2 · p. 4
• The Group 2 received signal contains desired signal power and interference from the paired user, other users in the cell, and other cells.The paper labels these contributions intra-cluster, intra-cell, and inter-cell interference.Source: C. Direct Decoding at Users in Group 2 · p. 4
• Group 2 users’ target-rate QoS must be guaranteed through careful resource allocation.The target condition is expressed as a rate of at least R0.Source: C. Direct Decoding at Users in Group 2 · p. 4

D. Problem Formulation

• The optimization maximizes Group 1 users’ sum data rates while guaranteeing Group 2 target rates and respecting base-station power budgets.The original design jointly chooses beamforming vectors and power-allocation coefficients.Source: D. Problem Formulation · pp. 4–5
• The reformulation’s objective and QoS constraints are non-convex in the power and beamforming variables because those variables are coupled.The paper identifies this coupling as the source of non-convexity in the objective and constraints.Source: D. Problem Formulation · p. 5

A. Problem Reformulation

• The original problem is non-convex and NP-hard, motivating transformation into a rank-constrained semidefinite program.The paper notes that brute force is generally needed for a globally optimal solution.Source: A. Problem Reformulation · p. 5
• The reformulation replaces each beamforming vector with a matrix Q that must be rank-one positive semidefinite to remain equivalent to the original problem.When the rank-one constraint holds, the vector solution can be recovered from the matrix solution.Source: A. Problem Reformulation · p. 5
• The matrix and power-allocation variables remain coupled bilinearly, so dropping the rank constraint alone does not make the reformulated problem tractable.The objective and constraints also have non-convex dependence on the coupled variables.Source: A. Problem Reformulation · p. 5
• Introducing an auxiliary variable ρ bounds the Group 1 SINR from below, and monotonicity of the logarithm makes maximizing the transformed objective equivalent to the original objective.The bound on ρ follows from constraint (13b).Source: A. Problem Reformulation · p. 5

• Bilinear coupling between beamforming matrices and power coefficients makes key constraints nonconvex and obstructs efficient resource allocation.Source: B. Successive Convex Approximation · p. 6
• SCA lower-bounds the difference-of-convex term t_kn^2/w_kn with a first-order approximation and updates its fixed points iteratively.Source: B. Successive Convex Approximation · p. 6
• AGM-based convex upper bounds handle bilinear terms in constraints (11b) and (11c), while the Schur complement reformulates the bilinear constraint in (13b).Source: B. Successive Convex Approximation · p. 6
• The optimal solution to relaxed P3 is always rank one, despite dropping the rank constraint.Source: B. Successive Convex Approximation · p. 6
• The algorithm iteratively tightens upper bounds and generates feasible solutions to convex P3; for fixed auxiliary points, P3 is convex and can be solved efficiently.Source: B. Successive Convex Approximation · pp. 6–7

• Algorithm 1 converges to a stationary point satisfying P1's Karush-Kuhn-Tucker conditions, giving a suboptimal solution with polynomial-time computational complexity.Source: C. Algorithm Convergence Analysis · p. 7

V. NUMERICAL RESULTS

• The simulations use Rayleigh fading and path loss, randomly place users uniformly within 500 m cell discs, and set the iteration error tolerance to 0.001.Source: §V · p. 7
• The study assumes identical BS power budgets and noise powers, defines average transmit SNR as ρ = P/σ^2, and assigns zero Group 1 sum-rate when the optimization is infeasible.Source: §V · p. 7

A. Convergence

• Convergence is faster for M = 4, K = 3 than for M = 4, K = 2 or M = 6, K = 4, as the optimized beamforming solution space is smaller.Source: Figure 2 · p. 7, A. Convergence · p. 8
• After at most 15 iterations on average, the algorithm converges to a stationary point; higher γ lowers Group 1 sum-rate, while target rates have almost negligible impact on convergence rate.Source: A. Convergence · p. 8, Figure 3 · p. 8

C. Sum-rate of Users

• NOMA-CoMP outperforms fixed-power NOMA-CoMP at all tested SNRs, beats uncoordinated NOMA-CoMP at medium-to-high SNR, and slightly underperforms it at low SNR.It also beats OMA-CoMP at medium-to-high SNR but slightly underperforms it at low SNR.Source: Figure 4 · p. 8, C. Sum-rate of Users · pp. 8–9

B. Rank Results

• The minimum ratio of the largest to second-largest beamforming-matrix eigenvalues is sufficiently large across tested M and K, indicating rank-one solutions.Tables I and II report the ratios for different transmit-antenna counts, with K = 2 and K = 3, respectively.Source: B. Rank Results · p. 8, Table II · p. 8

C. Sum-rate of Users

• The proposed algorithm achieves almost the same sum-rate as brute-force optimal search in a two-cell, single-antenna scenario, with much lower computational complexity.The brute-force search cost grows exponentially with problem size.Source: Figure 5 · p. 9, C. Sum-rate of Users · p. 9
• As the number of cells increases, NOMA-CoMP outperforms fixed-power NOMA-CoMP with a widening gap, while its gains diminish as inter-cell interference grows.NOMA-CoMP also outperforms OMA-CoMP; OMA-CoMP's sum-rate falls to zero when the number of cells is large because Group 2 target rates cannot be guaranteed.Source: Figure 6 · p. 9, C. Sum-rate of Users · pp. 9–10
• With increasing path-loss exponent α, Group 1 sum-rate first rises and then falls; the proposed scheme's gain over NOMA without CoMP grows as α decreases.The passage attributes the initial rise to interference attenuating more than desired signals, and the later decline to the shift toward a noise-limited regime.Source: Figure 7 · p. 10, C. Sum-rate of Users · p. 10
• Group 1 sum-rate first increases and then decreases as K grows, while it increases with the number of BS antennas M; proposed NOMA-CoMP always beats fixed-power NOMA-CoMP.The text links the K trade-off to multiuser diversity at small K and increasingly stringent Group 2 rate constraints at larger K.Source: Figure 8 · p. 10, C. Sum-rate of Users · p. 10

VI. CONCLUSION

• The paper formulates QoS-grouped downlink MIMO-NOMA resource allocation as a non-convex problem and proposes an iterative SCA algorithm using SDR at each iteration.The conclusion reports superior sum-rate over existing schemes and fast convergence to a stationary point.Source: §VI · p. 10
• The beamforming design assumes perfect global CSI; the authors identify imperfect-CSI robust optimization and user pairing as future directions.They describe the reported performance as an upper bound for NOMA with imperfect CSI.Source: §VI · p. 11

APPENDIX A OPTIMAL ckn AND dkn

• The AGM inequality identifies an optimal c*_kn that makes F equal to zero.Source: Appendix A · p. 11
• Because F is concave in c_kn, its maximizing value is found by solving ∂F/∂c_kn = 0; d_kn is derived similarly.Source: Appendix A · p. 11

APPENDIX B PROOF OF LEMMA 1

• The proof establishes that the optimal Q_kn of P3 is rank one, a property the authors say is not straightforward to prove using existing SDR results.Source: Appendix B · p. 11
• Because P3 and P4 have the same solutions, the proof studies P4's dual problem and KKT conditions instead.P4 uses the optimal beamforming matrices' total transmit power at each BS.Source: Appendix B · p. 11
• The optimal value of λ5_kn has rank M − K, and λ5_kn Q*_kn = 0 implies that the columns of Q*_kn lie in the null space of λ5_kn.Source: Appendix B · p. 11

APPENDIX C PROOF OF LEMMA 1

• Each iteration replaces G with a differentiable convex function, and the proof checks the surrogate and fixed-point conditions.Source: Appendix C · p. 12
• The proposed algorithm converges to a KKT point of P2a after the third SCA convergence condition is verified.Source: Appendix C · p. 12
• Since P2 is equivalent to P1 when the rank-one property holds, the proof concludes that the algorithm reaches a stationary KKT point of P1.Source: Appendix C · pp. 12–13

APPENDIX D EQUAL POWER CONSTRAINT FOR NOMA WITHOUT COMP

• For NOMA without CoMP, the optimal solution must use the full power budget at each BS.The proof assumes an unsaturated solution, scales the beamforming matrix while retaining feasibility, and obtains a higher objective, contradicting optimality.Source: Appendix D · p. 13
• With CoMP, raising one BS's power can improve its cell while worsening others through increased inter-cell interference.Source: Appendix D · p. 13
• Consequently, CoMP's network sum-rate may not increase with added transmit power, so its power constraint need not be active.Source: Appendix D · p. 13