Multiuser Full-Duplex Two-Way Communications via Intelligent Reflecting Surface

I. INTRODUCTION

• Prior IRS two-way studies targeted sum rate, while this work addresses fairness in a multiuser full-duplex two-way network.The authors describe this as the first work to consider fairness in that setting with IRS assistance.Source: §I · p. 2
• The proposed design jointly optimizes BS precoding and IRS reflection coefficients to maximize all users’ weighted minimum rate under transmit-power and unit-modulus constraints.The IRS enables simultaneous uplink and downlink transmissions on the same frequency, potentially doubling spectral efficiency.Source: §I · p. 2
• The WMMSE reformulation enables BCD to alternate between precoding and reflection subproblems, formulated respectively as an SOCP and a quasi-SOCP with a non-convex quadratic constraint.The original objective is non-differentiable and the optimization variables are highly coupled.Source: §I · p. 2
• The lower-complexity MM method smooths the non-differentiable subproblem objectives and derives minorizing functions that yield surrogate problems with closed-form solutions.The proposed MM algorithm iteratively solves both approximated subproblems.Source: §I · p. 2
• Simulations verify algorithm convergence and efficiency and highlight a placement trade-off between stronger user-side self-interference elimination and fewer propagation blockages near the BS.The authors also report the feasibility of IRS assistance for full-duplex two-way communication.Source: §I · p. 2

A. Signal Transmission Model

• The system assumes no direct BS-user links because of path loss and blockages, using an IRS to establish non-line-of-sight links for simultaneous full-duplex transmissions.The BS has multiple transmit and receive antennas, while each user has a transmit and receive antenna.Source: A. Signal Transmission Model · p. 3
• The IRS has M passive elements with unit-modulus reflection coefficients, and the BS is assumed to know all quasi-static channel state information perfectly.The model includes BS-IRS, IRS-BS, user-IRS, IRS-user, and loop channels.Source: A. Signal Transmission Model · p. 3
• At users, residual loop and self-interference are modeled with coefficients, while user-to-user interference is treated as additive white Gaussian noise because blockages are expected to make it small.At the BS, loop interference is assumed effectively canceled, and IRS-reflected self-interference is assumed known and mitigated.Source: A. Signal Transmission Model · pp. 3–4

• The objective maximizes weighted minimum rate across users by jointly optimizing the precoding matrix F and reflection vector φ, with weights ω_l,k ≥ 1.Source: B. Problem Formulation · p. 4
• The reflection vector φ is constrained to unit modulus, and the optimization is difficult because F and φ are coupled and this constraint is non-convex.Source: B. Problem Formulation · p. 4
• Weights represent inverse user priority: the solution tends to equalize weighted uplink and downlink rates, while larger ω_l,k lowers that user's rate to accommodate asymmetric rate needs.Source: B. Problem Formulation · p. 4

III. SOCP-BASED BCD METHOD

• The method transforms the original problem into a tractable WMMSE form, then uses BCD to optimize its subproblems.The reformulation uses the equivalence between WMR and WMMSE.Source: §III · p. 4

A. Reformulation of the Original Problem

• Decoding variables and nonnegative auxiliary weights express the downlink and uplink rates through MSE-based WMMSE terms.The resulting rate functions are concave in each variable set when the others and the reflection vector are fixed.Source: A. Reformulation of the Original Problem · pp. 4–5
• The auxiliary weights are obtained from the rate expressions, while the linear receivers are derived by setting first-order derivatives to zero.These updates provide the optimal auxiliary variables used in each BCD iteration.Source: A. Reformulation of the Original Problem · p. 5
• With the auxiliary variables updated, BCD alternates over the precoding matrix F and reflection vector φ as the main optimization tasks.The optimal decoding variables and weights are given at each iteration.Source: A. Reformulation of the Original Problem · p. 5

B. Optimizing the Precoding Matrix F

• For fixed φ, F does not affect the uplink rate, so its subproblem can be simplified to optimize the downlink objective.The subproblem uses the weighted downlink rate functions hD,k(F).Source: B. Optimizing the Precoding Matrix F · p. 5
• A selection vector tk isolates user k when deriving the precoding subproblem from the MSE expression and defining hD,k(F).The derivation also specifies Bk, Ck, and constk for the resulting expressions.Source: B. Optimizing the Precoding Matrix F · pp. 5–6
• Introducing δ for the pointwise minimum reformulates the precoding problem as an SOCP that existing tools such as CVX can solve optimally.The reformulation is stated as Problem (25).Source: B. Optimizing the Precoding Matrix F · p. 6

C. Optimizing the Reflection Coefficient Vector φ

• The preceding reformulations collect diagonal elements of channel-related matrices into vectors g_D,k and g_U,k.The passages place these definitions in reformulations of equations (14) and (13), respectively.Source: C. Optimizing the Reflection… · p. 6
• The derivation defines h_l,k(φ) = ω_l,k r_l,k(φ) and obtains expressions by substituting equations (26) and (27) into equations (15) and (16).The resulting expressions use a_l,k, A_l,k, and const_l,k.Source: C. Optimizing the Reflection… · p. 6
• The reflection-vector subproblem maximizes a common lower bound ε on h_l,k(φ) for every l and k, subject to the reflection constraint set.The formulation introduces ε as an auxiliary variable equivalent to the original subproblem.Source: C. Optimizing the Reflection… · p. 6
• Replacing the unit-modulus set with |φ_m| ≤ 1 for each element transforms the subproblem into an SOCP that can be solved optimally.This optimality applies to the relaxed problem, not necessarily the original one.Source: C. Optimizing the Reflection… · p. 6
• The method obtains an approximate solution to the original problem by applying element-wise exp(·) to the relaxed solution.The passages identify exp(·) as an element-wise operation.Source: C. Optimizing the Reflection… · p. 6
• The projected solution is not guaranteed to be globally optimal at each iteration, so the method adopts a strategy to ensure convergence.The supplied passage does not describe the strategy's steps.Source: C. Optimizing the Reflection… · p. 6

1) SOCP based BCD algorithm:

• The BCD algorithm alternately updates decoding variables, linear receivers, auxiliary variables, the BS precoding matrix, and the IRS reflection coefficients to maximize WMR.Source: 1) SOCP based BCD algorithm · p. 6
• Problem (25) is solved globally at each iteration, but the relaxation in Problem (30) means overall optimality is not guaranteed; simulations report limited loss when the IRS is deployed at the BS side.Source: 1) SOCP based BCD algorithm · pp. 6–7
• Starting from feasible precoding and reflection variables, each iteration updates decoding, receiver, and auxiliary variables before optimizing the precoding matrix and reflection coefficients.Source: 1) SOCP based BCD algorithm · p. 7
• The iterations stop when the objective-function value converges.Source: 1) SOCP based BCD algorithm · p. 7

2) Complexity Analysis:

• The complexity analysis accounts for computing decoding variables, linear receivers, and auxiliary variables before evaluating the two SOCP solves.Source: 2) Complexity Analysis · p. 7
• Problem (25) has K rate constraints and one power constraint, each of dimension KNt, and its complexity order is analyzed using [38].Source: 2) Complexity Analysis · p. 7
• The relaxed Problem (30) has 2K rate constraints of dimension M and M constant-modulus constraints of dimension one.Source: 2) Complexity Analysis · p. 7
• The per-iteration complexity of Algorithm 1 is dominated by solving Problems (25) and (30).Source: 2) Complexity Analysis · p. 7

IV. LOW-COMPLEXITY ALGORITHM DEVELOPMENT

• The proposed low-complexity algorithm replaces the SOCP and quasi-SOCP solved at each BCD iteration with an approach based on smooth approximations and MM.The paper introduces this approach to reduce computational load and obtain closed-form solutions.Source: §IV · p. 7
• Two smoothing functions, f(F) and f(φ), approximate the objective functions of Problems (23) and (29), respectively, and are lower bounds that are concave with respect to F and φ, respectively.Source: §IV · p. 7
• The smoothing parameter μ must be capped and adjusted: large values can hinder early progress, improve later approximation accuracy, and produce extremely small intermediate variables.The paper therefore requires an upper bound μmax.Source: §IV · p. 7

A. Optimizing the Precoding Matrix F

• For precoding, the smoothed objective is continuous and concave but too complex to optimize directly, so MM uses tractable surrogate problems.The resulting sequence is guaranteed to converge to a KKT point of Problem (37), and its objective values are monotonically non-decreasing.Source: A. Optimizing the Precoding Matrix F · pp. 7–8
• A quadratic minorizer of the objective at the current F iterate provides the surrogate problem used to update the precoding matrix.The paper gives the quadratic minorizer in Theorem 1 and forms the surrogate by replacing the objective of Problem (37).Source: A. Optimizing the Precoding Matrix F · p. 8
• The surrogate precoding update has a closed-form solution derived using a Lagrange multiplier and the power constraint.The left-hand side of the power-constraint equation decreases with the multiplier, which is used to obtain the optimal F iterate.Source: A. Optimizing the Precoding Matrix F · p. 8

B. Optimizing the Reflection Coefficient Vector φ

• For the reflection-coefficient update, MM minorization conditions are modified to accommodate the non-convex constraint, including directional-derivative agreement along feasible tangent-cone directions.The resulting surrogate subproblems have closed-form solutions.Source: B. Optimizing the Reflection… · pp. 8–9

C. Algorithm Development

• SQUAREM accelerates the MM algorithm, reducing the MM iterations required at each precoding-matrix or reflection-vector update to 2.The paper introduces acceleration because loose minorizing functions can require many iterations despite low computation per iteration.Source: C. Algorithm Development · p. 9

1) BCD-MM algorithm:

• Algorithm 2 defines fixed-point iteration maps for the precoding matrix and reflection vector and uses an adjustment factor to gradually increase µ to µmax.It also evaluates the original objective at the current precoding matrix and reflection vector.Source: 1) BCD-MM algorithm · p. 9
• The BCD-MM objective is non-decreasing and bounded, guaranteeing convergence; its solution is close to a KKT point of Problem (12) when the smoothing parameter is sufficiently large.Gradually increasing the smoothing parameter reduces the approximation gap and gives the algorithm a relatively high probability of converging to a good locally optimal solution.Source: 1) BCD-MM algorithm · p. 9
• The BCD-MM procedure updates decoding and receiver variables, then alternates the precoding-matrix and reflection-vector updates before checking its stopping condition.It terminates when the objective change is below εe or the iteration count reaches nmax.Source: 1) BCD-MM algorithm · p. 9
• The converged solution satisfies the KKT conditions of the surrogate problems (37) and (46), and therefore of Problem (12) when the stated equivalences hold.The paper notes that the approximations become very tight when µ is sufficiently large.Source: 1) BCD-MM algorithm · p. 9

Complexity Analysis

• Computing UD, UU, WD, and WU has complexity O(M^2 + NtM + NrM).Source: 2) Complexity Analysis · p. 9
• Optimizing F is driven mainly by computing V and α, with their costs depending on gD,k and tp2k, respectively.Source: 2) Complexity Analysis · p. 9
• Each BCD iteration requires only two MM iterations to calculate Fn+1 and φn+1.Source: 2) Complexity Analysis · p. 9
• Algorithm 2 has overall complexity O(K^2NtM + K^2NrM + K^2N^2 + KM^3 + KNtM^2 + K^2M^2), and MM greatly reduces its computational load.Source: 2) Complexity Analysis · p. 10

V. SIMULATION RESULTS

• The simulations assess the proposed multiuser IRS-aided full-duplex two-way communication system’s performance.The section presents extensive simulation results for this purpose.Source: §V · p. 10

A. Simulation Setup

• The simulated network has three randomly placed users, no direct BS-user link, and an IRS at (xIRS, 20), defaulting to xIRS = 10 m.Users occupy a 40 m by 20 m region centered at (120, 0); the BS is at (0, 0).Source: A. Simulation Setup · p. 10
• The IRS links use path-loss exponents αBI = αIU = αIRS = 2.2, while small-scale fading is Rician with factor κ = 3.The reference path loss is -30 dB at 1 m, and the non-line-of-sight component is Rayleigh distributed.Source: A. Simulation Setup · p. 10
• Unless otherwise stated, results average 200 independent channel realizations using 10 MHz bandwidth, 4 BS antennas per side, 16 IRS elements, and 1 W maximum BS power.The reflection phases are initialized uniformly at random; the precoding matrix is initialized from independent Gaussian real and imaginary parts and scaled to satisfy (3).Source: A. Simulation Setup · p. 10

B. Baseline Schemes

• SOCP+MM benchmarks the proposed quasi-SOCP solution by replacing BCD-MM’s steps 4–9 with Algorithm 1’s SOCP step.MOSEK optimally solves Problem (25) and is used for Problem (25) and the relaxed Problem (30).Source: B. Baseline Schemes · p. 10
• BCD-SOCP, Rand and BCD-MM, Rand skip updates to the IRS reflection vector to assess joint-optimization benefits.These schemes optimize only the precoding matrix.Source: B. Baseline Schemes · pp. 10–11
• The practical 2-bit baselines quantize each optimized IRS element to one of four phase shifts, then update the precoding matrix.The resulting variants are BCD-SOCP, 2 bit; BCD-MM, 2 bit; and SOCP+MM, 2 bit.Source: B. Baseline Schemes · p. 11

C. Convergence of Proposed Algorithm; VI. CONCLUSIONS

• BCD alternates optimization of the BS precoder and IRS coefficients, while closed-form MM iterations reduce complexity; the algorithms converge within 80 iterations, and BCD-MM is faster in CPU time.BCD-SOCP and BCD-MM reach basically the same converged WMR, slightly below SOCP+MM. The conclusion reports high convergence speed and communication performance for the proposed algorithm.Source: C. Convergence of Proposed Algorithm · p. 11, §VI · p. 14

D. Impact of the IRS Location

• For all eight schemes, IRS deployments nearer the BS improve WMR; jointly optimized schemes outperform random-phase schemes, while 2 bit phases perform between continuous and random phases.The results identify two WMR peaks, one on the BS side and one on the user side, with path loss contributing to declines at extreme locations and a valley near xIRS ≈ 60.Source: D. Impact of the IRS Location · pp. 11–12, Figure 4 · p. 12

E. Impact of the SI Coefficient

• At xIRS = 120, reducing the SI coefficient ρS from 1 to 0.1 raises BCD-MM and SOCP+MM WMR from 0.85 to 0.95, but this ideal SI-elimination case needs experimental verification.Even with lower SI, IRS deployment near users yields lower WMR than deployment near the BS, because user-received signals also experience co-channel interference, which grows with user count.Source: E. Impact of the SI Coefficient · p. 12

F. Impact of Asymmetric Priority

• In most priority scenarios, IRS deployment is preferable on the BS side; only uplink-first scenarios with arbitrary phase shifts show a slightly different outcome.The WMR depends on the weighting factors and is not equivalent to data rate; the authors conclude that common FD two-way scenarios favor a location between the BS and users, closer to the BS, if reflection links are favorable.Source: Figure 5 · p. 12, F. Impact of Asymmetric Priority · p. 12

G. Impact of the Weights and the Achieved Fairness

• Equal weights produce balanced user rates despite differing path loss, while assigning user 2 greater activity yields the highest rate for user 2 and an essentially constant average rate.The tested unequal-activity setting uses ω2 = 1 and ω1 = ω3 = 2; weights can represent user priority and activity in resource allocation.Source: Figure 6 · p. 13, G. Impact of the Weights and the… · p. 13

H. Impact of the Path Loss Exponent

• For each scenario, WMR increases more than twofold for every 0.2 decrease in αIRS, while performance decays to 0 at high path-loss exponents.The results indicate that IRS gains depend strongly on channel conditions, favoring deployment where there are fewer obstacles.Source: Figure 7 · p. 13, H. Impact of the Path Loss Exponent · p. 13

I. Impact of the Rician Factor

• As the Rician factor κ rises and multipath diversity gain decreases, WMR decreases; at κ = 0, random-phase schemes have significantly lower multipath diversity gain than other methods.The comparison in the rich-scattering Rayleigh environment highlights the benefits of joint optimization.Source: Figure 8 · p. 13, §I · p. 13

J. Impact of the Number of IRS Reflection Elements

• From M = 8 to 64, the diminishing WMR gains from adding IRS elements are not very significant, supporting the cost-effectiveness of deploying more reflection elements.The paper relates the gain to an approximately logarithmic increase in channel capacity as M increases.Source: J. Impact of the Number of IRS… · pp. 13–14, Figure 9 · p. 14

APPENDIX A PROOF OF THEOREM 1

• A quadratic minorizer is proposed for f(F), with D determined by matching directional derivatives under condition (A3).The proof notes that each hD,k(F) is quadratic and uses conditions (A2) and (A3) to determine D and M.Source: Appendix A · p. 14
• For condition (A2), the minorizer must remain a lower bound along every line segment from Fn, with curvature checked over η ∈ [0, 1].The functions agree at η = 0 and have matching first derivatives there; a sufficient condition is that jF’s second derivative is at least JF’s throughout the interval.Source: Appendix A · p. 14
• The curvature matrix can be chosen as M = αI, with α = λmin(Ξ), but computing α directly is costly, so the proof derives a simpler lower bound.This choice satisfies the required matrix condition, and the lower bound is used to reduce the computation needed to obtain the minorizer.Source: Appendix A · p. 15
• Substituting the derived lower bound into the curvature expression yields equation (38), completing the proof.Source: Appendix A · p. 15

APPENDIX B PROOF OF THEOREM 2

• A quadratic function minorizes f(φ), with undetermined parameters N and d chosen to satisfy conditions (B2) and (B3).Conditions (B1) and (B4) are already satisfied.Source: Appendix B · p. 15
• For (B3), the minorizer’s directional derivative at φn is matched to that of f(φ), using gl,k(φn) as defined in (49b).The comparison is made in the direction φt − φn for φt ∈ Sφ.Source: Appendix B · p. 15
• For (B2), a sufficient condition is established along φ = φn + η(φt − φn), η ∈ [0, 1], using matching values and gradients at η = 0 and concavity of Jφ.The proof compares second-order behavior to ensure the required bound holds along the segment.Source: Appendix B · p. 16
• The proof chooses N = βI = λmin(Ω)I, replaces β with its lower bound from (49c), and uses φHφ = (φn)Hφn = M under unit modulus to derive (79).The resulting expression uses v and consφ from (48a) and (48b).Source: Appendix B · pp. 16–17