Intelligent Reflecting Surface Aided Multiple Access: Capacity Region and Deployment Strategy

I. INTRODUCTION

• IRS uses controllable passive phase shifts to alter wireless channels and improve achievable rates, but multi-user capacity-region characterization remains open.Prior work characterized the fundamental capacity limit only for single-user IRS-aided channels.Source: §I · p. 1
• Distributed deployment places separate IRSs near users, whereas centralized deployment places all M reflecting elements near the AP.The distributed strategy forms two IRSs, each serving one nearby user; the centralized strategy consolidates all elements into one IRS.Source: §I · p. 1, §II · pp. 1–2
• Distributed deployment provides each user only its nearby IRS gain, while centralized deployment offers a larger shared gain that must be divided among users.Signals reflected by far-apart IRSs are weakened by high path loss, whereas a centralized IRS can enhance both users’ channels.Source: Figure 1 · p. 1, §I · p. 1

II. SYSTEM MODEL

• The paper analyzes a two-user MAC in which independent single-antenna users transmit to a single-antenna AP using M passive reflecting elements.The reflecting elements are partitioned across two user-near IRSs or consolidated near the AP, depending on the deployment strategy.Source: §I · p. 1, §II · pp. 1–2

I. INTRODUCTION

• The study compares the capacity regions of both deployments to determine which placement is preferable in multi-user systems.The central comparison concerns the distinct user-AP effective channels induced by the two IRS layouts.Source: §I · p. 1

A. Distributed IRS Deployment

• Each IRS reflection matrix is diagonal, with every reflecting element constrained to unit-modulus phase control.The reflection coefficients satisfy |φ^D_km| = 1 for every element m of IRS k.Source: A. Distributed IRS Deployment · p. 2
• The model neglects signals reflected by the other user’s IRS because sufficiently large user separation makes their path loss prohibitively high.Thus, each user’s effective channel depends on its direct path and serving-IRS reflection rather than the remote IRS.Source: A. Distributed IRS Deployment · p. 2
• Users transmit independent unit-variance CSCG symbols with powers no greater than their respective maximum limits, and the AP receives their superposition plus CSCG noise.The achievable rates are measured in bits per second per Hertz under this signal and noise model.Source: A. Distributed IRS Deployment · p. 2

B. Centralized IRS Deployment

• In centralized deployment, all M reflecting elements form one IRS located near the AP.The centralized IRS uses a single diagonal reflection matrix whose entries have unit modulus.Source: B. Centralized IRS Deployment · p. 2
• Both users’ effective channels depend on all centralized-IRS reflection coefficients, unlike distributed channels that depend only on each user’s serving IRS.This shared coefficient dependence couples the two users’ channel design.Source: B. Centralized IRS Deployment · p. 2
• The centralized model uses the same transmit-signal and receiver-noise assumptions as distributed deployment, with achievable rates defined in bps/Hz.The AP again receives the two users’ superposed signals through their effective channels.Source: B. Centralized IRS Deployment · p. 2
• The paper compares distributed rate pairs (R^D_1, R^D_2) with centralized rate pairs (R^C_1, R^C_2) to study the preferable IRS deployment.The objective is to compare the two capacity regions and derive deployment insights.Source: B. Centralized IRS Deployment · p. 2

III. CAPACITY REGION OF DISTRIBUTED DEPLOYMENT

• Each user’s effective channel gain is upper-bounded by an l1-norm expression, with equality achieved by appropriately aligning its serving-IRS reflection coefficients.This per-user maximization supplies the channel values used in the closed-form region.Source: §III · p. 2
• For fixed reflection coefficients, the achievable rate pairs obey the standard two-user MAC individual-rate and sum-rate constraints.The distributed capacity region is formed from these constraints over feasible IRS designs.Source: §III · p. 2
• Time sharing across feasible reflection designs yields the convex hull of their achievable rate pairs as the distributed capacity region.The region is defined by taking the convex hull of the union over feasible reflection matrices.Source: §III · p. 2
• Theorem 1 gives a closed-form characterization of the two-user MAC capacity region under distributed IRS deployment.The result follows from the special effective-channel structure created by separate user-near IRSs.Source: §III · p. 2
• At the maximizing reflection design, the resulting region also provides a convex-shaped outer bound without requiring an additional convex-hull operation.The cited result identifies the optimized reflection coefficients as sufficient for the stated outer-bound construction.Source: §III · p. 2

• Centralized capacity uses the convex hull of rate pairs over feasible reflection coefficients and time sharing among configurations.Source: §IV · pp. 2–3
• All M centralized reflection coefficients jointly determine both users’ effective channels, coupling the achievable rates.Source: §IV · p. 3
• Unlike distributed deployment, different Pareto-boundary portions generally require different centralized reflection-coefficient configurations.Source: §IV · p. 3
• Exhaustive characterization is prohibitive because uniformly sampling each phase variable requires complexity exponential in M.Source: §IV · p. 3

• The outer region is formed by individual-rate upper bounds and a sum-rate upper bound, yielding a superset of the centralized capacity region.Source: A. Capacity Region Outer Bound · p. 3
• The individual effective-channel bounds become equalities only when all reflection coefficients maximize the corresponding user’s channel gain.Source: A. Capacity Region Outer Bound · p. 3
• The sum-rate bound is difficult because one centralized phase configuration jointly changes both users’ effective channels.Source: A. Capacity Region Outer Bound · p. 3
• Semidefinite relaxation rewrites the quadratic phase-design objective using W = ww^H while relaxing the rank-one constraint.Source: A. Capacity Region Outer Bound · p. 3
• The relaxed semidefinite program is efficiently solvable by an interior-point method with complexity O(M^4.5), and its optimum upper-bounds the original problem.Source: A. Capacity Region Outer Bound · p. 3

• The method solves a series of sum-rate maximization problems using rate profiles to construct an achievable rate region.Source: B. Capacity Region Inner Bound: A… · p. 3
• An alternating optimization algorithm is proposed to obtain high-quality solutions to the rate-profile problems efficiently.Source: B. Capacity Region Inner Bound: A… · p. 3

1) Rate-Profile based Problem Formulation:

• Successive interference cancellation attains the relevant Pareto-boundary rate pairs for each centralized reflection configuration, except pairs requiring time sharing or rate splitting.Source: 1) Rate-Profile based Problem… · p. 3
• The optimization jointly selects centralized reflection coefficients and user transmit powers for a specified rate profile and decoding order.Source: 1) Rate-Profile based Problem… · p. 3
• The two decoding orders are compared for each rate profile, and the order producing the larger optimized sum rate determines the Pareto-optimal pair.Source: 1) Rate-Profile based Problem… · p. 4

Non-convexity

• The remaining optimization is non-convex because unimodular reflection constraints and coupled quadratic channel terms make direct optimization difficult.These constraints motivate convex relaxation and alternating optimization.Source: 2) Proposed Solution to (P1) · p. 4

Problem reformulation

• The centralized capacity-region problem is reformulated through an equivalent optimization problem and rate-profile parameterization over decoding orders.The formulation uses an auxiliary variable and rate ratio to optimize achievable sum rates for each profile.Source: 2) Proposed Solution to (P1) · p. 4

Convex relaxation

• Relaxing each unimodular constraint to a convex magnitude constraint is tight, so the resulting convex problem can be solved efficiently by an interior-point method.The relaxation preserves the optimum because any solution with magnitude below one can be scaled to the boundary.Source: 2) Proposed Solution to (P1) · p. 4

Alternating optimization

• Alternating optimization iteratively solves the convex subproblems, producing a feasible generally suboptimal solution with non-decreasing objective values and monotonic convergence.Solutions for both decoding orders are compared at each rate profile, retaining the one with larger sum rate.Source: 2) Proposed Solution to (P1) · p. 4

Achievable region and complexity

• Time sharing among the optimized rate pairs yields an inner bound, while sampling L rate profiles gives overall complexity O(2MIL), polynomial in M and lower than exhaustive search.The achievable region is constructed from the rate-profile solutions and their convex combinations.Source: 2) Proposed Solution to (P1) · p. 4

Deployment comparison

• Under negligible direct user-AP channels and twin-channel assumptions, the centralized capacity region contains the distributed region, CD ⊆ CC.The comparison is established analytically by coordinating the centralized IRS sub-surfaces through common phase rotations.Source: 2) Proposed Solution to (P1) · pp. 4–5

VI. NUMERICAL EXAMPLE

• The experiment uses M = 30 elements, 15 per distributed IRS, equal user powers of 30 dBm, and noise power σ2 = −90 dBm under Rayleigh fading.The AP and users are placed 500 m apart, with the centralized IRS near the AP and distributed IRSs near the users.Source: §VI · p. 5
• The centralized inner capacity-region bound contains the distributed capacity region, which contains the traditional MAC region without IRS.The centralized comparison uses L = 100 sampled rate profiles.Source: §VI · p. 5
• Centralized deployment outperforms distributed deployment most clearly for asymmetric user-rate requirements because its larger passive beamforming gain benefits the higher-rate user.The figure also compares a heuristic centralized scheme without additional sub-surface phase rotations, whose region is significantly smaller than the proposed method’s region.Source: Figure 2 · p. 5

VII. CONCLUSION

• The distributed deployment capacity region is characterized in closed form, while centralized deployment uses efficient algorithms for inner and outer capacity-region bounds.The centralized approach combines computationally efficient optimization with achievable-region construction.Source: §VII · p. 5
• Centralized deployment outperforms distributed deployment under the practical channel setup, with the largest capacity gain when user rates are asymmetric.This conclusion summarizes the paper’s deployment comparison.Source: §VII · p. 5