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Physics-based Modeling and Scalable Optimization of Large Intelligent Reflecting Surfaces

Marzieh Najafi, Vahid Jamali, Robert Schober, Vincent H. Poor

arXiv:2004.12957v2cs.ITeess.SP

TL;DR

Large IRSs are needed to address far-field path loss, but optimizing their many unit cells is not scalable for online transmission. The paper develops a physics-based tile model and a two-stage optimization framework, showing that large IRSs with thousands of elements can be efficiently configured.

  • Problem

    Optimizing the many unit-cell phases of large IRSs is not scalable, while far-field scenarios require large surfaces to address end-to-end path loss.

  • Method

    The paper partitions unit cells into tiles, derives physics-based tile responses and end-to-end channels, and uses offline transmission-mode design followed by online tile-mode optimization.

  • Results

    The proposed framework efficiently configures large IRSs containing thousands of elements, with increasing tile count beyond a certain number yielding negligible performance improvement.

  • Takeaways & Limitations

    Grouping unit cells into tiles provides a scalable performance–complexity tradeoff for optimizing large IRSs.

Abstract

from arXiv · show

Intelligent reflecting surfaces (IRSs) have the potential to transform wireless communication channels into smart reconfigurable propagation environments. To realize this new paradigm, the passive IRSs have to be large, especially for communication in far-field scenarios, so that they can compensate for the large end-to-end path-loss, which is caused by the multiplication of the individual path-losses of the transmitter-to-IRS and IRS-to-receiver channels. However, optimizing a large number of sub-wavelength IRS elements imposes a significant challenge for online transmission. To address this issue, in this paper, we develop a physics-based model and a scalable optimization framework for large IRSs. The basic idea is to partition the IRS unit cells into several subsets, referred to as tiles, model the impact of each tile on the wireless channel, and then optimize each tile in two stages, namely an offline design stage and an online optimization stage. For physics-based modeling, we borrow concepts from the radar literature, model each tile as an anomalous reflector, and derive its impact on the wireless channel for a given phase shift by solving the corresponding integral equations for the electric and magnetic vector fields. In the offline design stage, the IRS unit cells of each tile are jointly designed for the support of different transmission modes, where each transmission mode effectively corresponds to a given configuration of the phase shifts that the unit cells of the tile apply to an impinging electromagnetic wave. In the online optimization stage, the best transmission mode of each tile is selected such that a desired quality-of-service (QoS) criterion is maximized. We show that the proposed modeling and optimization framework can be used to efficiently optimize large IRSs comprising thousands of unit cells.

I. INTRODUCTION

IRSs use programmable sub-wavelength elements to reshape reflected electromagnetic waves, but far-field path loss and element-level optimization make large-scale deployment and online configuration challenging. The paper addresses this with physics-based modeling and tile-based scalable optimization.

  • Motivation: IRSs comprise programmable sub-wavelength unit cells whose phase distributions can redirect reflected wavefronts.Properly designed phase distributions enable generalized Snell’s law, while passive construction supports large surfaces.
  • Challenge: Directly optimizing each unit-cell phase is non-convex and becomes unmanageable as the number of optimization variables grows.The unit-modulus constraint |e^jβq| = 1 contributes to the non-convexity, while large surfaces can contain thousands of cells.
  • Motivation: Far-field IRS-assisted links suffer significant end-to-end path loss because the transmitter-to-IRS and IRS-to-receiver losses combine.The paper notes that very large IRSs may be needed to overcome this loss in practice.
  • Challenge: Physics-based modeling is needed because tile gain depends on incident angle, reflection angle, polarization, amplitude, and phase.Highly directive tiles can provide negligible power to receivers outside their main radiation lobe, making phase-only combining inefficient.
  • Proposed framework: The proposed framework partitions Q unit cells into N ≪ Q tiles and optimizes tile configurations through offline design and online selection.An offline transmission-mode codebook and online selection of relevant modes reduce the search space from |B|^Q.
  • Proposed framework: The framework models tiles using physically derived response functions and can configure IRSs with thousands of cells using only a few tens of tiles.Discrete tiles with spacing below λ/2 can accurately approximate continuous tiles without significant performance degradation.

II. END-TO-END CHANNEL MODEL FOR IRS-ASSISTED WIRELESS SYSTEMS

The paper builds an end-to-end far-field channel model by deriving tile response functions for continuous and discrete IRS tiles, then incorporating all tile responses and propagation angles into system channels. The resulting path-loss formulation expresses the IRS-assisted link through the transmitter-to-IRS loss, IRS-to-receiver loss, and tile response.

  • B. Tile Response Function: The response function is derived from integral equations for electric and magnetic vector fields, first for continuous tiles and then for discrete tiles.The paper derives g from physical principles rather than using an empirical response model.
  • B. Tile Response Function: For a given transmission mode, the tile response function g(Ψt, Ψr) maps an incident wave and polarization to the reflected field toward a receiver.The unit cells act as secondary sources, and |g(Ψt, Ψr)|^2 is the radar cross section of the tile.
  • C. End-to-End Channel Model: The end-to-end channel model accounts for the impact of all IRS tiles, their transmission modes, and incident, reflection, and polarization angles.This enables modeling the superposition of multiple waves arriving at receivers in the far field.
  • A. IRS Structure: The IRS is partitioned into tiles, each containing programmable sub-wavelength unit cells; continuous and discrete tiles represent idealized and practical implementations.The total number of cells is Q = NQxQy, with tile dimensions and spacings determining Qx and Qy.
  • B. Tile Response Function: The model focuses on far-field propagation, where the incident wave at a tile can be treated as a plane wave.The formulation characterizes incident direction and polarization together with the reflection direction.
  • C. End-to-End Channel Model: The IRS-assisted path loss is expressed through the transmitter-to-IRS path loss, IRS-to-receiver path loss, and the tile response function.Lemma 1 provides the free-space path-loss expression in terms of g, while reflected-wave polarization mismatch can be absorbed into IRS-to-receiver channel gains.

C. Continuous Tiles

The continuous-tile model characterizes how an arbitrarily polarized, incident electromagnetic wave is reflected toward arbitrary observation directions. It derives response functions from electromagnetic field equations and examines beamwidth, reflection behavior, and path-loss implications.

  • Continuous-tile formulation: The incident wave is parameterized by its direction and polarization, while the reflected field is characterized through a tile response function g(Ψt, Ψr).The formulation generalizes electric and magnetic fields to arbitrary incident angles and polarizations.
  • Electromagnetic derivation: The scattered fields are derived by replacing the IRS with an equivalent PMC surface, introducing an electric current, and applying Image Theory in an obstacle-free system.The formulation then obtains reflected electric and magnetic fields from the equivalent current.
  • Continuous-tile formulation: The model uses a prescribed phase-shift profile β(x, y), including a linear profile designed to realize generalized Snell’s law for a desired reflection direction.The resulting response is evaluated for arbitrary incident and observation directions.
  • Path-loss implications: For fixed transmitter, receiver, and direct-link distances, the required continuous-IRS area decreases at higher carrier frequencies and when the IRS is placed near an endpoint.The area is defined as the smallest IRS area yielding the unobstructed direct link’s free-space path-loss.
  • Response characteristics: Increasing tile dimensions narrows the response beam; even a 20λ × 20λ tile has an approximately 6-degree 10-dB beamwidth, potentially causing far-field interference.The response peak and beamwidth for anomalous reflection can also differ from those for specular reflection.

D. Discrete Tiles

The discrete-tile model represents a tile as a superposition of unit-cell responses and analyzes spacing, quantized phase shifts, and the number of cells needed to match direct-link path-loss. It also identifies conditions under which discrete and continuous models coincide.

  • Discrete-tile model: A discrete tile consists of many uniformly spaced sub-wavelength unit cells whose responses are superposed to obtain the tile response function gd(Ψt, Ψr).The unit-cell factor captures radiation dependence on polarization, incident angle, cell size, and observation angle.
  • Discrete-tile model: The phase-shift design imposes constant cell shifts that realize generalized Snell’s law and produce the corresponding discrete-tile response.The resulting amplitude and phase are evaluated for arbitrary incident and reflection directions.
  • Continuous approximation: When unit-cell spacing tends to zero and cell size equals spacing, the discrete response becomes identical to the continuous response: gd(Ψt, Ψr) = gc(Ψt, Ψr).With gaps between cells, the effective tile size decreases; the analysis illustrates this for Luc = 0.8d.
  • Phase quantization: A 3-bit uniform phase quantization produces a response very close to ideal real-valued shifts, whereas 1-bit quantization reduces the peak and alters sidelobes.The 1-bit response retains a similar overall shape despite these deviations.
  • Cell-count requirement: At 5, 10, and 28 GHz, at least 3333, 6666, and 18667 unit cells, respectively, are required for the IRS link to match unobstructed direct-link free-space path-loss.The required minimum number of cells increases with carrier frequency, unlike the required minimum continuous-IRS area.

E. End-to-End System Model

The paper develops a transmission-mode end-to-end model for IRS-assisted systems that captures tile responses, angle-dependent amplitudes, and low-rank propagation. This model addresses dependencies and physical-factor limitations of conventional phase-shift models while enabling reconfigurable channel selection.

  • Transmission-mode model: The transmission-mode model characterizes each tile through a response function whose amplitude depends on the incident and reflected angles.Each tile and mode has an angle-dependent phase-shift matrix and response function.
  • Phase-shift model: The conventional phase-shift model represents IRS unit cells with a diagonal phase-shift matrix and constant unit-cell amplitude factors.The model combines direct, BS-to-IRS, and IRS-to-user channels with per-cell phase shifts.
  • Model limitations: Large-IRS channels exhibit dependencies because their ranks are limited by the number of environmental scatterers, contrary to independent-entry assumptions.The paper identifies this as a drawback of commonly used channel models for large Q.
  • Model limitations: The unit-cell factor depends on unit-cell properties, incident and reflected angles, and wave polarization rather than being universally constant.Many prior works assume the ideal value ¯guc = 1, whereas the proposed model relates it to the physical factor guc.
  • Transmission-mode model: Under the low-rank formulation, tile channels are constructed from steering matrices and scatterer gains for the BS-to-IRS and IRS-to-user links.The resulting tile-mode channel is represented by Hn,m,k = eHr,kGn,m,k eHt.
  • Transmission-mode model: Selecting one transmission mode per tile yields |M|N possible end-to-end channel matrices, enabling a smart reconfigurable wireless environment.The selection variables identify the active mode for each tile within a chosen subset of modes.

III. TWO-STAGE OPTIMIZATION FRAMEWORK FOR IRS-ASSISTED COMMUNICATIONS

The proposed IRS optimization framework separates design into offline and online stages. This two-stage structure is introduced for efficient optimization of large IRSs.

  • Two-stage framework: The framework consists of an offline design stage and an online optimization stage.The framework is developed from the end-to-end channel model.

A. Offline Tile Transmission Mode Codebook Design

The offline stage constructs a finite transmission-mode codebook by discretizing tile reflection parameters and wavefront phase. Reflection slopes control reflected-wave direction, while the affine phase controls wavefront phase.

  • Reflection codebooks: Directly discretizing ¯βx and ¯βy avoids discretizing the expected incident angle and exploits periodic equivalence in the tile response.The paper notes that ¯βi and ¯βi + 1 yield the same response function.
  • Codebook construction: The transmission-mode codebook is formed from discretized reflection parameters ¯βx, ¯βy and wavefront phase ¯β0.Its size is M = |Bx| × |By| × |B0|.
  • Reflection codebooks: The reflection codebooks can be built by uniformly discretizing the supported ranges of ¯βx and ¯βy.All tiles use the same reflection codebook Bi for each spatial direction.
  • Reflection codebooks: For the illustrated angular scenario, choosing |Bx| = |By| = 9 produces reflection codebooks with nine values per spatial direction.The example uses unit-cell spacing dx = dy = λ and uniformly distributed elevation and azimuth angles over specified intervals.
  • Reflection-codebook example: The resulting tile response functions cover the considered reflected-wave elevation range θr ∈[0, π/4].The example evaluates the response for a 10λ × 10λ tile with λ-spaced unit cells.
  • Wavefront phase codebook: The wavefront phase parameter ¯β0 compensates position-dependent phase differences between tiles and controls the phase of the reflected wavefront.Tiles with identical reflection parameters require appropriate ¯β0 values to coherently reflect an incident wave.

B. Mode Pre-selection for Online Optimization

The online stage first pre-selects transmission modes with non-negligible channel gains, reducing the search space before resource allocation. A threshold controls the performance–complexity trade-off.

  • Mode pre-selection: For a given transmitter-receiver pair, only a few transmission modes may have non-negligible tile-channel Frobenius norms.This sparsity motivates pre-selecting a subset of modes before online optimization.
  • Mode pre-selection: The pre-selection set M contains mode indices retained for subsequent online resource allocation, with the exact criterion depending on the application.Alternative criteria can account for interference or information leakage.
  • Performance–complexity trade-off: Choosing a smaller threshold δ selects more modes, potentially improving performance at the cost of higher online complexity.Thus, δ is a design parameter for trading performance against computational complexity.
  • Numerical illustration: The numerical example evaluates one IRS tile in a single-transmitter, two-receiver system with free-space path loss over 1000λ links.The IRS parameters and reflection codebook match those used in the codebook-design example.
  • Numerical illustration: 10 modes out of 81 are retained when δ = −130 dB, reducing the subsequent online optimization complexity.The 81 candidates arise from 9 × 9 reflection-codebook elements, and the threshold is nearly 20 dB below the maximum channel gain.

C. Online Optimization

The paper formulates IRS online configuration as joint binary tile-mode selection and BS precoder optimization under users’ SINR constraints. It develops alternating-optimization and greedy approaches that reduce transmit power while providing scalable subproblems and convergence guarantees.

  • Online Optimization: Both algorithms optimize discrete tile modes rather than individual IRS unit-cell phase shifts, enabling online operation with reduced configuration complexity.The methods assume Nt ≥ K and use the tile channel coefficients and required SINRs as inputs.
  • Problem formulation: The online problem jointly selects binary transmission modes for all tiles and designs the BS precoder to minimize transmit power under minimum user SINR constraints.The resulting problem is non-convex because of mode–precoder products, quadratic SINR terms, and binary variables.
  • Alternating Optimization-Based Solution: The alternating-optimization method decomposes the problem into tile configuration with power minimization and joint power–beamforming design.The first subproblem optimizes a tile mode and transmit power for fixed other tiles and normalized precoder; the second optimizes the precoder for fixed tile selections.
  • Alternating Optimization-Based Solution: Each alternating-optimization subproblem is solved globally, and the resulting sequence of transmit powers is non-increasing and converges to a locally optimal solution.The beamforming subproblem can drop the rank-one constraint while recovering an equivalent rank-one solution.
  • Greedy Iterative Solution: The greedy iterative algorithm configures one tile per iteration, selecting the user with the largest power contribution and the mode that most improves that user’s end-to-end channel.After all N iterations, it solves the beamforming subproblem for the obtained tile configuration.

2) Greedy Iterative Solution:

The greedy method incrementally configures tiles by alternating precoder design, user selection, and mode selection, then performs a final precoder optimization. Its complexity depends on the number of tiles and modes rather than directly on the number of IRS unit cells.

  • Greedy Iterative Solution: The greedy algorithm runs for N iterations and configures one additional tile in each iteration.The first iteration starts with only the direct link represented in the initial tile-selection matrix.
  • Greedy Iterative Solution: At each iteration, the algorithm first designs a precoder for the current configuration, then selects the user contributing most to BS power consumption.It configures the new tile to improve the selected user’s channel.
  • Greedy Iterative Solution: For the selected user, the algorithm chooses the tile-codebook mode that most improves the end-to-end channel gain, then optimizes the final precoder after all tiles are configured.The resulting tile-selection matrix is fixed during this final beamforming step.
  • Complexity Analysis: The stated complexity orders are O(|M|) for Algorithm 1’s first subproblem, O(KN_t^3.5) for its SDP subproblem, and O(N_itr(N|M| + KN_t^3.5)) overall.Algorithm 2 has complexity order O(N(|M| + KN_t^3.5)).
  • Complexity Analysis: The proposed search space is reduced from |B|^Q in the phase-shift model to |M|^N in the transmission-mode model.The mode count |M| and tile count N are design parameters that trade performance for complexity.
  • Complexity Analysis: For Q = 3600 IRS unit cells, the paper uses N = 9 tiles and |M| = 32 modes, values reported as comfortably handled by the proposed algorithms.The proposed online complexities depend on N and |M| rather than directly on Q.

D. Channel Estimation

The channel-acquisition procedure estimates end-to-end channels only for offline-codebook transmission modes, so overhead scales with the number of modes. Simulations show that a small number of tiles can optimize large IRSs effectively, although required tile counts depend on system and channel parameters.

  • Limitations: Channel-estimation errors are unavoidable, and the paper leaves the minimum required estimation quality and robust transmission under such errors for future work.A detailed treatment of channel estimation is explicitly outside the paper’s scope.
  • Channel acquisition: Channel estimation is performed for the transmission modes in the offline codebook, making estimation overhead and complexity scale with the number of modes.The IRS configures each mode, the BS sends a pilot, and the user estimates the resulting end-to-end channel or physical parameters.
  • Channel acquisition: The overhead can be reduced by identifying satisfactory modes early or searching hierarchical multi-resolution codebooks instead of testing every mode.The paper notes that small offline codebooks are important for improving channel-estimation quality.
  • Simulation results: Algorithm 1 converges within 1–3 iterations for the considered initializations and typically within 1–5 iterations in extensive simulations.The convergence behavior depends on initialization because different starting points can lead to different locally optimal solutions.
  • Simulation results: An IRS with nine 10λ × 10λ tiles reduces required transmit power by approximately 30 dBm compared with no IRS for the considered channel realization.The same simulations report less than 3 dB improvement for Algorithm 1 relative to Algorithm 2 in that realization.
  • Simulation results: Using 24 modes causes no degradation for 8×8×4 offline-codebook entries and approximately 2 dB higher transmit power for 16×16×4 entries.Increasing the wavefront phase-codebook size from 4 to 8 provides only a small additional reduction.
  • Simulation results: Increasing tile count beyond N = 9 yields only a small additional power reduction, particularly for large reflection codebooks.The results therefore indicate that thousands of unit cells need not be optimized individually online in the shown 3600-cell example.
  • Simulation results: Required transmit power decreases as channel scatterers increase because additional angles of arrival and departure provide more strong paths to exploit.The number of tiles needed for a target performance depends on transmitters, receivers, and channel scatterers.

V. CONCLUSIONS

The paper presents a physics-based channel model and scalable two-stage optimization framework for large IRSs. Tiles are designed offline for multiple transmission modes and selected online to optimize QoS, enabling efficient configuration of IRSs with thousands of elements.

  • Framework: The framework partitions IRS unit cells into tiles and optimizes them through offline design and online transmission-mode selection.Offline design jointly configures each tile for different transmission modes; online optimization selects each tile’s best mode for a given fading realization and QoS objective.
  • Physics-based modeling: The proposed physics-based model derives tile response functions for continuous and discrete tiles and incorporates them into an end-to-end IRS-assisted channel model.The tile response characterizes how a tile affects the wireless channel for transmitter and receiver directions.
  • Online optimization: Online optimization formulates a mixed-integer problem that selects the best transmission mode for each tile to maximize a desired QoS criterion.For an exemplary downlink, the paper also studies minimizing base-station transmit power subject to users’ QoS constraints and develops alternating-optimization and greedy algorithms.
  • Results: The framework can efficiently configure large IRSs containing thousands of elements.This result is supported by computer simulations of the proposed modeling and optimization approach.
  • Results: Beyond N = 9 tiles, increasing the number of tiles for a given IRS size yields only a negligible performance improvement.The number of tiles needed for a target performance depends on system parameters.

APPENDIX A

Appendix A derives the electromagnetic tile response and connects incident fields, scattered fields, radiation intensity, and received power. The derivation uses far-field approximations and separates elevation and azimuth field components.

  • Field and power relations: The incident radiation power intensity at the IRS is related to the electric and magnetic fields through the time-averaged Poynting vector.The impedance, real-part operator, cross product, and complex conjugate determine the field-based intensity expression.
  • Tile response: The tile response function relates the incident electric field at the IRS to the electric field at the receiver.The resulting receiver power is obtained by multiplying receive-side radiation intensity by the receive-antenna area.
  • Far-field approximation: In the far field, the radial electric-field component becomes negligible because it decays faster than the elevation and azimuth components.The electric and magnetic fields therefore become perpendicular to the propagation direction, simplifying the field expressions.
  • Field derivation: The elevation and azimuth field components are derived from the tile’s generating current components at the observation point.The derivation projects current components into spherical coordinates and evaluates the resulting integrals using an integral identity.
  • Discrete tiles: For discrete tiles, identities applied to the phase shifts simplify the response into expressions involving directional differences and phase terms.The resulting amplitude and phase functions complete the discrete-tile response derivation.

APPENDIX D

Appendix D analyzes a convex optimization problem using duality and KKT conditions. The argument establishes that an optimal solution exists with a rank-one matrix variable.

  • Convexity and duality: The optimization problem is jointly convex in Q_k and satisfies Slater’s constraint qualification.Consequently, strong duality holds, allowing the dual problem to characterize the optimal primal solution.
  • Optimality conditions: The Lagrangian dual function is formed using multipliers associated with the problem’s constraints.The optimal solution is then examined through the corresponding KKT necessary optimality conditions.
  • Eigenvalue analysis: The eigenvalues of Δ_k are ordered descending, with δ_max defined as the largest eigenvalue.The proof considers separately the cases δ_max < 1 and δ_max = 1 while enforcing the stated constraints.
  • Rank structure: The KKT conditions constrain the rank and null-space structure of the optimal matrix variables.In particular, the columns of Q_k must satisfy the relevant null-space condition associated with X_k.
  • Rank-one optimality: There always exists an optimal solution for which rank(Q_k*) = 1.When an optimal solution has higher rank, the proof constructs another solution that preserves optimality while reducing the rank.
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