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Intelligent Reflecting Surface Aided Multi-User Communication: Capacity Region and Deployment Strategy

Shuowen Zhang, Rui Zhang

arXiv:2009.02324v2cs.ITeess.SP

TL;DR

The paper asks how IRS deployment affects the capacity of a two-user communication network, comparing distributed IRSs near users with centralized IRS elements near the AP. It characterizes uplink MAC and downlink BC regions using optimization and MAC-BC duality, finding centralized deployment superior under the studied symmetric setup and in numerical comparisons.

  • Problem

    IRS deployment remains insufficiently understood because different placements of a fixed number of elements create different effective channels and can significantly affect multi-user capacity and rates.

  • Method

    The paper characterizes capacity and TDMA/FDMA rate regions for distributed and centralized two-user IRS deployments, using rate-profile optimization, alternating optimization, SDR, and MAC-BC duality.

  • Results

    Under the stated symmetric channel setup, centralized deployment contains the corresponding distributed capacity and achievable-rate regions, with the advantage also holding for the downlink BC.

  • Takeaways & Limitations

    Centralized IRS deployment can provide stronger multi-user rate performance and is especially advantageous for asymmetric user rates and near-far conditions.

  • Takeaways & Limitations

    The paper’s conclusions are established for the studied two-user MAC and BC setting, while extensions to more general network models such as multi-cell multi-user systems remain future work.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is a new promising technology that is able to reconfigure the wireless propagation channel via smart and passive signal reflection. In this paper, we investigate the capacity region of a two-user communication network with one access point (AP) aided by $M$ IRS elements for enhancing the user-AP channels, where the IRS incurs negligible delay, thus the user-AP channels via the IRS follow the classic discrete memoryless channel model. In particular, we consider two practical IRS deployment strategies that lead to different effective channels between the users and AP, namely, the distributed deployment where the $M$ elements form two IRSs, each deployed in the vicinity of one user, versus the centralized deployment where all the $M$ elements are deployed in the vicinity of the AP. First, we consider the uplink multiple-access channel (MAC) and derive the capacity/achievable rate regions for both deployment strategies under different multiple access schemes. It is shown that the centralized deployment generally outperforms the distributed deployment under symmetric channel setups in terms of achievable user rates. Next, we extend the results to the downlink broadcast channel (BC) by leveraging the celebrated uplink-downlink (or MAC-BC) duality framework, and show that the superior rate performance of centralized over distributed deployment also holds. Numerical results are presented that validate our analysis, and reveal new and useful insights for optimal IRS deployment in wireless networks.

I. INTRODUCTION

The paper studies capacity limits and IRS deployment for a two-user AP network, comparing distributed and centralized placements across uplink MAC and downlink BC settings. It develops analytical and computational characterizations, establishes centralized-deployment advantages under symmetric channels, and validates the findings numerically.

  • Motivation: IRS elements passively and independently control phase shifts to reconfigure propagation channels, improving desired signal strength and mitigating interference.The elements collaboratively alter the wireless environment without transmit RF chains.
  • Motivation: The paper addresses the limited understanding of IRS deployment, since distributing a fixed number of elements across locations can change effective channels and achievable rates.Prior work largely assumed IRS locations rather than exploiting deployment flexibility.
  • Contributions: For the uplink MAC, the paper characterizes capacity and TDMA/FDMA rate regions for distributed deployment and uses rate-profile optimization, alternating optimization, and SDR bounds for centralized deployment.The centralized method jointly optimizes IRS reflections and transmit powers across user rate ratios.
  • Contributions: Under a practical symmetric channel setup, centralized deployment contains the corresponding capacity, TDMA, and FDMA regions of distributed deployment.The paper also proves that FDMA contains TDMA for both deployment strategies.
  • Contributions: The uplink results extend to the downlink BC through MAC-BC duality, preserving centralized deployment’s performance advantage over distributed deployment.The paper proposes computationally efficient BC region-characterization methods based on the dual MAC.
  • Numerical insights: Numerical results validate the analysis and show that centralized deployment’s capacity gain is most prominent for asymmetric user rates and helps alleviate near-far effects.These comparisons provide deployment insights for practical systems.

III. DISTRIBUTED IRS DEPLOYMENT

For distributed IRS deployment, each of two nearby IRSs serves its corresponding user, enabling closed-form characterization of the IRS-aided MAC capacity region. The optimal reflections maximize each user’s effective channel independently, and the resulting region has a standard two-user MAC form.

  • Capacity characterization: The distributed deployment characterizes the MAC capacity and TDMA/FDMA achievable regions using the effective channels created by the two user-specific IRSs.The analysis assumes Gaussian user inputs and permits time sharing across reflection configurations.
  • Capacity characterization: Each distributed IRS reflection is designed from its nearby user’s channel, maximizing that user’s effective channel gain through phase alignment.The resulting reflection design simultaneously maximizes both users’ effective gains because the IRSs operate independently.
  • Capacity characterization: The distributed-deployment capacity region is the set of rate pairs satisfying individual-user bounds and a joint sum-rate bound.The paper identifies this region in closed form through Theorem 1.
  • Capacity characterization: The resulting capacity region is convex and also serves as an outer bound for achievable regions generated by other distributed reflection configurations.Consequently, the convex-hull operation is unnecessary for the optimal reflection choice.
  • Decoding: Achieving the distributed MAC capacity region generally requires successive interference cancellation or joint decoding at the AP.With SIC, the AP decodes one user while treating the other as noise, then cancels and decodes the remaining message.

B. Achievable Rate Region with TDMA

Under distributed deployment, TDMA assigns separate time slots to the users, so each user’s rate depends on its own nearby IRS reflection. The resulting TDMA region uses independently maximized effective channels and is contained by the corresponding FDMA region.

  • TDMA formulation: TDMA assigns user 1 a fraction ρT of the time and user 2 the remaining fraction, with each user transmitting in an orthogonal slot.Each user’s achievable rate depends only on its own IRS reflection during its assigned slot.
  • TDMA region: The TDMA rate region is obtained by maximizing each user’s effective channel through the corresponding nearby IRS reflection.The same phase-aligned reflections that maximize the distributed MAC channels are optimal for TDMA.
  • TDMA region: The optimized TDMA region is convex and can therefore be characterized using the independently optimized user channels.The paper establishes this result analogously to the distributed capacity-region derivation.
  • FDMA comparison: FDMA uses orthogonal frequency bands with bandwidth fraction ρF assigned to user 1, while IRS reflections affect the bands identically because the IRS lacks frequency selectivity.The absence of RF chains and baseband processing produces identical reflection coefficients across bands.
  • FDMA comparison: FDMA’s achievable rate region contains TDMA’s for distributed deployment because both schemes use the same optimal reflections and effective user-AP channels.The comparison follows from the corresponding region expressions.

IV. CENTRALIZED IRS DEPLOYMENT

Under centralized IRS deployment, both users’ effective channels depend on all IRS reflection coefficients, making capacity-region characterization more challenging than in distributed deployment. The paper uses rate profiles and SIC-achievable boundary points to characterize the region, while exhaustive search is computationally prohibitive.

  • The centralized MAC capacity region with fixed reflection coefficients is defined by individual-rate and sum-rate constraints, then enlarged through reflection optimization and time sharing.
  • All M centralized reflection coefficients jointly couple the two users’ effective channels, so different Pareto-boundary portions generally require different reflection configurations.
  • A. Rate-Profile based Capacity Region Characterization: Direct exhaustive search over uniformly sampled reflection configurations has complexity exponential in M, motivating alternative inner- and outer-bound methods.
  • A. Rate-Profile based Capacity Region Characterization: SIC achieves the Pareto-boundary rate pairs for each reflection configuration except those requiring time sharing or rate splitting; rate profiles then organize the boundary search.
  • A. Rate-Profile based Capacity Region Characterization: The rate-profile optimization jointly selects IRS reflection coefficients and user powers for prescribed rate ratios and decoding orders, with optimal solutions characterizing the centralized capacity region.
  • A. Rate-Profile based Capacity Region Characterization: The resulting capacity characterization is difficult because the optimization is non-convex under uni-modular constraints and coupled power-reflection variables.

B. Capacity Region Inner Bound

The paper derives a centralized-deployment capacity-region inner bound by transforming the rate-profile problem and solving it with alternating optimization. Convex relaxation of each uni-modular subproblem is tight, yielding an efficient polynomial-complexity approximation.

  • B. Capacity Region Inner Bound: The proposed alternating-optimization algorithm finds a high-quality suboptimal solution to the centralized rate-profile sum-rate problem and therefore an inner bound.
  • B. Capacity Region Inner Bound: The rate-profile problem is transformed into a more tractable formulation before optimizing reflection coefficients and auxiliary variables.
  • B. Capacity Region Inner Bound: Each reflection-coefficient subproblem’s convex relaxation is tight because an optimal relaxed solution satisfies the original unit-modulus constraint.
  • B. Capacity Region Inner Bound: Alternating optimization monotonically improves the objective and converges because the sum rate is bounded by finite transmit powers.
  • B. Capacity Region Inner Bound: The overall proposed complexity is O(2^M(Q + I)L + L log L), which is polynomial over M and lower than exhaustive search.

C. Capacity Region Outer Bound

The centralized-deployment MAC capacity region is bounded by separately optimized individual-user and sum-rate constraints. A semidefinite relaxation provides a tractable outer-bound computation despite the original non-convex unit-modulus optimization.

  • The individual-user effective channel gains are bounded by optimizing each user’s centralized IRS reflection coefficients separately.
  • The centralized sum-rate bound is difficult because one reflection vector jointly changes both users’ effective channels.
  • The resulting quadratic optimization is non-convex because of the unit-modulus reflection constraints.
  • Semidefinite relaxation introduces W = wwH, removes the rank-one constraint, and yields an SDP solvable by an interior-point method with complexity O(M 6.5).
  • The achievable centralized MAC rate regions with TDMA and FDMA are characterized after the outer-bound construction.

D. Achievable Rate Region with TDMA

For centralized IRS deployment, TDMA uses user-specific reflection configurations across time slots, while FDMA is characterized for fixed configurations and then enlarged through time sharing.

  • D. Achievable Rate Region with TDMA: Centralized TDMA assigns each user a time slot with a reflection configuration tailored to that user.
  • D. Achievable Rate Region with TDMA: Choosing each slot’s coefficients to maximize its assigned user’s effective channel gain gives the centralized TDMA achievable region.
  • D. Achievable Rate Region with TDMA: The centralized FDMA region is first defined for any fixed IRS reflection configuration and then combined through time sharing.
  • D. Achievable Rate Region with TDMA: Time sharing among FDMA regions using user-specific maximizing configurations contains the TDMA achievable region.
  • D. Achievable Rate Region with TDMA: Although fixed-configuration FDMA need not contain TDMA, time sharing over different reflection configurations enables FDMA to outperform TDMA in achievable rate region.
  • D. Achievable Rate Region with TDMA: The comparison assumes negligible direct user-AP channels and twin channels linking the two deployment strategies.

A. Capacity Region Comparison

Under negligible direct channels and the twin-channel assumption, centralized deployment contains distributed deployment in capacity-region performance. The proof constructs centralized reflection phases that preserve or improve both users’ effective gains.

  • A. Capacity Region Comparison: Under the stated assumptions, the centralized capacity region contains the distributed capacity region: CD ⊆ CC.
  • A. Capacity Region Comparison: The construction embeds the two distributed IRS sub-surfaces into one centralized IRS and permits separate common phase rotations.
  • A. Capacity Region Comparison: A phase choice exists that preserves or increases both users’ effective channel magnitudes relative to distributed deployment.
  • A. Capacity Region Comparison: The larger centralized passive beamforming gain can benefit both users simultaneously, producing a larger capacity region than two user-specific distributed IRSs.

B. Achievable Rate Region Comparison with TDMA and FDMA

Under the twin-channel condition with negligible direct channels, centralized deployment also contains distributed deployment for TDMA and FDMA achievable regions. The uplink results extend to the BC through MAC-BC duality.

  • B. Achievable Rate Region Comparison with TDMA and FDMA: The centralized FDMA achievable rate region likewise contains the distributed region under the twin-channel assumptions.
  • B. Achievable Rate Region Comparison with TDMA and FDMA: These comparisons require negligible direct channels and the specified twin-channel condition; differing channel distributions make analytical comparison more difficult.
  • B. Achievable Rate Region Comparison with TDMA and FDMA: The centralized TDMA achievable rate region contains the distributed region because its larger IRS provides each user a larger maximum effective channel gain.
  • B. Achievable Rate Region Comparison with TDMA and FDMA: The paper summarizes centralized and distributed IRS capacity and practical multiple-access rate-region results for the two-user MAC.
  • B. Achievable Rate Region Comparison with TDMA and FDMA: For the downlink BC, MAC-BC duality obtains the BC capacity region from dual MAC regions whose user powers sum to the AP power constraint.
  • B. Achievable Rate Region Comparison with TDMA and FDMA: The BC capacity region is computed by searching over power ratios and taking the convex hull of the resulting union set.

B. Centralized IRS Deployment

The centralized IRS broadcast-channel capacity region is derived through MAC-BC duality, with achievable inner bounds obtained by rate-profile optimization and alternating optimization.

  • MAC-BC duality gives the centralized-deployment BC capacity region from the corresponding dual MAC capacity region.
  • Rate-profile optimization over decoding orders produces directly achievable Pareto-optimal BC rate pairs.
  • Alternating optimization iteratively updates auxiliary variables, transmit powers, and IRS phases to obtain a suboptimal solution with monotonic convergence.
  • Time sharing among rate-profile solutions yields an inner bound for the two-user BC capacity region.
  • The resulting algorithm has complexity O(2^M(Q_BC + I_BC)L + L log L).Q_BC is the number of initialization realizations, I_BC the number of outer iterations, and L the number of rate-ratio samples.

2) Capacity Region Outer Bound:

The paper constructs BC outer bounds and practical TDMA/FDMA rate regions by transferring MAC characterizations through MAC-BC duality.

  • BC outer bounds are obtained from MAC outer bounds across power splits satisfying P1 + P2 = P, followed by convexification over rate profiles.
  • Centralized BC TDMA regions use the closed-form centralized MAC TDMA characterization, while FDMA regions use rate-profile optimization with power allocation.
  • The centralized BC FDMA inner bound is computed by alternating optimization over rate, power, time-sharing, and IRS-phase variables.
  • The BC outer bound remains an outer bound for practical TDMA and FDMA achievable regions.

A. IRS-Aided Two-User MAC

Numerical results compare distributed and centralized IRS deployment for two-user MAC and BC settings. Centralized deployment generally provides larger capacity or achievable-rate regions, especially with asymmetric channels or rate requirements.

  • MAC rate regions: Centralized deployment contains distributed deployment in the heterogeneous MAC rate-region comparison, while distributed deployment contains the no-IRS capacity region.
  • MAC common rate: Centralized deployment outperforms distributed deployment in maximum common rate for both NOMA and practical OMA schemes.
  • MAC rate regions: Centralized deployment more strongly benefits the farther user and alleviates the near-far problem under heterogeneous distances.
  • MAC common rate: For distributed deployment, more IRS elements should generally be allocated to a farther-away user, while centralized deployment avoids this allocation issue.
  • BC comparisons: In the BC comparisons, centralized achievable regions contain distributed capacity regions, and centralized gains are more pronounced for the farther user.
  • BC comparisons: TDMA and FDMA approach BC capacity under homogeneous distances but are strictly suboptimal under heterogeneous distances.
  • Conclusions: The paper concludes that centralized regions contain distributed regions for the MAC under the assumed twin-channel setup and that numerical results validate this superiority.

APPENDIX

The appendix supplies proofs supporting the MAC capacity characterization, IRS-phase relaxation, and alternating-optimization procedures.

  • The MAC capacity characterization is proved by establishing achievability and a converse for the proposed region.
  • For the centralized IRS phase design, unit-modulus phases can replace relaxed phases without decreasing the optimization objective.
  • The required phase rotation is selected by cases summarized in Table III, using monotonicity with respect to β.
  • The FDMA subproblem becomes convex after relaxing IRS-phase magnitudes, enabling efficient interior-point solutions within alternating optimization.
  • A phase adjustment can ensure |a_k + b_k e^{jθ}| ≥ |a_k| for both users.
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