Intelligent Reflecting Surface-Aided Joint Processing Coordinated Multipoint Transmission

I. INTRODUCTION

• IRS research addresses energy and hardware costs associated with other capacity-enhancing technologies, while the combination of IRS and JP-CoMP remained an emerging research area.Source: §I · pp. 1–3

A. System Model

• JP-CoMP lets multiple base stations jointly serve cell-edge users, whose received signals combine direct BS-user and IRS-reflected paths; the model assumes known channel state information at the central processor.Source: §I · p. 2, A. System Model · p. 4

I. INTRODUCTION

• The design targets fairness by maximizing the minimum achievable rate across cell-edge users through joint optimization of base-station beamforming and IRS phase shifts.Source: §I · p. 3
• The joint problem is non-convex; an MSE reformulation supports iterative algorithms using dual subgradient and MM methods for one user, and SOCP and SDR for multiple users.Source: §I · p. 3
• Simulations show that IRS-aided JP-CoMP outperforms IRS-aided CS/CB-CoMP in max-min rate and improves throughput over designs without an IRS.Source: §I · p. 3

B. Problem Formulation

• The objective is to maximize the minimum user rate by jointly optimizing downlink beamforming and IRS phase shifts, subject to base-station transmit-power constraints.Source: B. Problem Formulation · p. 4
• Coupled beamforming and IRS-phase variables make the problem difficult to solve optimally, motivating an equivalent MSE-based formulation.Source: B. Problem Formulation · p. 4
• The formulation assumes sufficient fronthaul capacity for data and CSI exchange among base stations and the central processor.Source: B. Problem Formulation · p. 4
• The MSE construction introduces matrices Q_k and U_k, with the MSE matrix forming part of the transformed problem.Source: B. Problem Formulation · p. 5
• The transformed problem is equivalent in the sense that both formulations share the same optimal solution.Source: B. Problem Formulation · p. 5, §III · p. 5

III. SINGLE CELL-EDGE USER SYSTEM

• For the single-user case, K = 1, and the formulation simplifies while retaining coupled optimization variables.Source: §III · p. 5
• The single-user solution alternates between beamforming optimization at fixed IRS phases and phase optimization at fixed beamforming.Source: §III · p. 5, B. Problem Formulation · p. 5

A. Transmit Beamforming Matrix Optimization with Fixed Phase Shift Matrix

• With the IRS phase shifts fixed, the transmit-beamforming subproblem is convex and satisfies Slater’s condition, so strong duality holds.Source: QWH ¯HHU · p. 5

QWH ¯HHU

• For a fixed dual variable, the optimal transmit beamforming matrix has a closed-form solution, enabling subgradient optimization of the unconstrained dual problem.Source: QWH ¯HHU · p. 6
• The method updates the dual variables using positive step sizes until the dual objective’s fractional increase falls below a predefined threshold.Source: QWH ¯HHU · p. 6

B. Phase Shift Matrix Optimization with Fixed Transmit Beamforming

• With transmit beamforming fixed, the phase-shift optimization remains non-convex, and the unit-modulus constraints motivate use of the MM method.Source: B. Phase Shift Matrix Optimization… · p. 6, QWH ¯HHU · p. 6

QWH ¯HHU

• MM constructs convex surrogate functions that upper-bound the objective and match its value and gradient at the current point, yielding at least a locally optimal solution with low computational complexity.Source: QWH ¯HHU · p. 6
• The MM procedure iteratively updates the phase shifts using the surrogate-based solution until the objective’s fractional decrease falls below a threshold.Source: QWH ¯HHU · p. 7

C. Overall Algorithm

• Each iteration updates the variables through the two subproblem solutions, with each optimization performed while the other variables are fixed.Source: C. Overall Algorithm · p. 7
• The overall algorithm is guaranteed to converge because its objective is non-decreasing across iterations and bounded above by finite BS transmit power and IRS element count.Source: C. Overall Algorithm · p. 7

IV. MULTIPLE CELL-EDGE USERS SYSTEM

• In the multiuser setting, constraint (10) prevents applying the single-user dual-subgradient and MM methods; SOCP handles transmit beamforming and SDR handles phase shifts.Source: §IV · p. 8
• The multiuser design decomposes the problem into beamforming optimization with fixed phase shifts and phase-shift optimization with fixed beamforming.Source: §IV · p. 8

A. SOCP for Transmit Beamforming Matrix Optimization

• With IRS phase shifts fixed, the beamforming problem is convex and can be solved optimally as an SOCP, with lower worst-case complexity than the SDP alternative.The equivalent SOCP is solved by an interior-point method.Source: A. SOCP for Transmit Beamforming… · p. 8

B. SDR Technique for Phase Shift Matrix Optimization

• With beamforming fixed, the phase-shift problem is non-convex, so the method relaxes its rank-one constraint using SDR and solves the resulting SDP.Source: B. SDR Technique for Phase Shift… · pp. 8–9
• If the relaxed solution is rank one, singular-value decomposition recovers the phase vector; otherwise, randomization constructs a suboptimal rank-one solution.Source: B. SDR Technique for Phase Shift… · p. 9

C. Overall Algorithm

• Algorithm 4 alternates updates to its optimization variables, including beamforming and the IRS phase-shift matrix.Source: C. Overall Algorithm · p. 9
• Gaussian randomization may fail to improve the objective, so the algorithm retains the previous phase-shift matrix when the candidate objective is lower.The simulations use 1000 random realizations per iteration and select the best candidate.Source: C. Overall Algorithm · p. 9
• The objective is non-decreasing and bounded across iterations, but Algorithm 4 does not guarantee a local or global optimum.The paper states that the algorithm terminates and provides a feasible solution with low complexity.Source: C. Overall Algorithm · p. 9

V. NUMERICAL RESULTS

• The setup places each BS at the center of a hexagonal cell and assumes BS and IRS altitudes of 10 m.Source: §V · p. 10
• The simulations assume LoS BS–IRS and IRS–user links with Rician fading, Rayleigh fading for BS–user links, and average results over 500 channel realizations.The path-loss exponents are 2.2 for BS–IRS and IRS–user links, and 3.6 for BS–user links.Source: §V · p. 10
• The evaluation also considers practical IRS phase shifters with a finite number of discrete values, represented by b bits of resolution.The discrete phases are derived from the continuous phase shifts produced by the single-user or multiuser algorithm.Source: §V · p. 10

A. Single-user System

• The iterative algorithms converge quickly: Algorithm 3 takes about 4 iterations at M = 20 and fewer than 10 at M = 100, while Algorithm 4 takes about 5 and 25 iterations, respectively.Algorithm 3’s rate increases quickly across the tested element counts; Algorithm 4’s average max-min rate also increases quickly with iterations.Source: A. Single-user System · p. 10, Figure 2 · p. 10, Figure 7 · p. 11
• In the single-user case, MM with continuous phases performs nearly as well as continuous-phase SDR while requiring much lower computational complexity; both outperform no-IRS JP.Random-phase IRS also outperforms the no-IRS scheme, while discrete-phase SDR incurs losses that are reduced at higher phase resolution.Source: Figure 3 · p. 10, A. Single-user System · p. 11
• Under Nt = 2, average achievable rate increases with IRS element count: No IRS achieves about 1.29 bps/Hz at M = 50, while SDR, JP, continuous achieves 4.62 there and 7.76 bps/Hz at M = 300.Source: A. Single-user System · p. 11, Figure 4 · p. 11
• Moving the IRS closer to the single user raises average achievable rate, which the authors attribute to reduced reflection path loss.The simulation fixes M = 100 and Nt = 2 and varies the IRS’s horizontal position from −150 m to 150 m.Source: A. Single-user System · p. 11, Figure 5 · p. 11

B. Multiuser System

• As BS antenna count increases, all schemes’ average max-min rates rise; two-bit optimized-phase JP-CoMP loses little versus continuous phases and outperforms counterpart CS/CB-CoMP schemes.The authors relate the JP-CoMP advantage to exploiting the system’s degrees of freedom for resource allocation.Source: B. Multiuser System · p. 12, Figure 8 · p. 12
• At Pmax = 10 dB, optimized-phase JP-CoMP reaches 6.1837 bps/Hz versus 3.7246 bps/Hz for optimized-phase CS/CB-CoMP, a nearly 40% increase.The performance gap becomes more pronounced at larger BS transmit-power budgets.Source: B. Multiuser System · p. 12, Figure 9 · p. 12
• For multiuser transmission, JP-CoMP outperforms CS/CB-CoMP in average max-min rate, with the gap widening as the number of IRS elements increases.The authors attribute the stronger IRS beam to more focused and powerful reflections at larger M.Source: B. Multiuser System · p. 12, Figure 10 · p. 13
• The optimized IRS scheme outperforms half-duplex AF relaying, with a small gap at low M and a significant gap at large M.The comparison ignores the direct BS-to-user link; the paper attributes the relay’s disadvantage to its induced time-slot delay.Source: B. Multiuser System · p. 13, Figure 11 · p. 13

• IRS-aided JP-CoMP significantly outperforms CS/CB-CoMP in max-min rate, while IRS deployment achieves significant throughput over the no-IRS case.Source: §VI · p. 13
• The design maximizes the minimum cell-edge-user rate by jointly optimizing IRS phase shifts and BS transmit beamforming in multicell downlink systems.Source: §VI · p. 13
• For the single-user system, an iterative algorithm combines subgradient-based optimal transmit beamforming for fixed IRS phases with MM-based phase optimization for fixed beamforming.Source: §VI · p. 13
• For the multiuser system, the transmit-beamforming subproblem is solved as an SOCP and the IRS phase-shift subproblem uses semidefinite relaxation in an iterative algorithm.Source: §VI · p. 13
• The study leaves extensions to multiple IRSs, frequency-selective channels, and imperfect CSI for future work.Source: §VI · p. 13

APPENDIX A

• APPENDIX A identifies the Lagrangian for (P1) and shows it also represents (P) when λ_k is the multiplier for constraint (5), completing the equivalence proof.Source: Appendix A · p. 14
• APPENDIX A sets derivatives with respect to Q_k and U_k to zero, then uses the resulting expressions and the Duncan-Guttman formula to establish that (P) and (P1) have the same optimal primal solution.Source: Appendix A · p. 14

APPENDIX B

• APPENDIX B vectorizes the beamforming matrices to rewrite the BS power constraint and substitutes a rewritten E_k expression into constraint (10) to complete Theorem 2’s proof.Source: Appendix B · p. 14