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Capacity Characterization for Intelligent Reflecting Surface Aided MIMO Communication

Shuowen Zhang, Rui Zhang

arXiv:1910.01573v1cs.ITeess.SP

TL;DR

The paper asks how to characterize capacity in point-to-point IRS-aided MIMO systems with multiple antennas at both transmitter and receiver. It jointly optimizes IRS reflection coefficients and transmit covariance matrices using alternating optimization for narrowband and broadband settings. The proposed algorithms improve capacity over traditional MIMO and benchmark schemes while improving channel power, rank, or condition number.

  • Problem

    Capacity characterization for IRS-aided MIMO systems with multiple antennas at both transmitter and receiver remains open because reflection coefficients and transmit covariance must be jointly optimized.

  • Method

    The paper develops alternating optimization methods for narrowband frequency-flat and broadband frequency-selective channels, using convex relaxation for the broadband problem.

  • Results

    The proposed algorithms achieve superior rate performance over benchmark schemes with or without IRS and improve IRS-aided MIMO channel power, rank, or condition number.

  • Takeaways & Limitations

    Jointly designing IRS reflection coefficients and transmit covariance can enhance capacity by improving key properties of the IRS-aided MIMO channel.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is a promising solution to enhance the wireless communication capacity both cost-effectively and energy-efficiently, by properly altering the signal propagation via tuning a large number of passive reflecting units. In this paper, we aim to characterize the fundamental capacity limit of IRS-aided point-to-point multiple-input multiple-output (MIMO) communication systems with multi-antenna transmitter and receiver in general, by jointly optimizing the IRS reflection coefficients and the MIMO transmit covariance matrix. First, we consider narrowband transmission under frequency-flat fading channels, and develop an efficient alternating optimization algorithm to find a locally optimal solution by iteratively optimizing the transmit covariance matrix or one of the reflection coefficients with the others being fixed. Next, we consider capacity maximization for broadband transmission in a general MIMO orthogonal frequency division multiplexing (OFDM) system under frequency-selective fading channels, where transmit covariance matrices can be optimized for different subcarriers while only one common set of IRS reflection coefficients can be designed to cater to all subcarriers. To tackle this more challenging problem, we propose a new alternating optimization algorithm based on convex relaxation to find a high-quality suboptimal solution. Numerical results show that our proposed algorithms achieve substantially increased capacity compared to traditional MIMO channels without the IRS, and also outperform various benchmark schemes. In particular, it is shown that with the proposed algorithms, various key parameters of the IRS-aided MIMO channel such as channel total power, rank, and condition number can be significantly improved for capacity enhancement.

I. INTRODUCTION

The paper addresses the open capacity characterization problem for point-to-point IRS-aided MIMO systems with multiple antennas at both ends. It develops optimization methods for frequency-flat and frequency-selective channels and reports capacity gains associated with improved channel properties.

  • IRS motivation: IRS elements passively alter signal propagation through independently controlled reflection coefficients, enabling constructive signal addition or interference mitigation.This provides an additional degree of freedom for communication design without active transmit RF chains.
  • IRS motivation: IRS deployment is attractive because passive devices can support dense deployment with low cost and low energy consumption compared with more active components.The paper also notes that IRS operation avoids signal amplification, regeneration, and sophisticated self-interference cancellation processing.
  • Research gap: Existing IRS-aided communication studies mainly consider SISO or MISO systems, leaving capacity characterization with multiple antennas at both transmitter and receiver largely open.The open problem requires jointly optimizing IRS reflection coefficients and the MIMO transmit covariance matrix.
  • Paper scope: The paper studies joint IRS reflection-coefficient and transmit-covariance optimization for point-to-point MIMO systems with multiple antennas at both transmitter and receiver.The analysis assumes perfect CSI and fixes all reflection-coefficient amplitudes to one to reduce implementation complexity.
  • Contributions: For narrowband frequency-flat channels, the proposed alternating optimization algorithm iteratively solves transmit-covariance and reflection-coefficient subproblems and is guaranteed to converge to at least a locally optimal solution.The narrowband capacity maximization problem is non-convex.
  • Contributions: For broadband frequency-selective channels, the paper proposes an alternating optimization algorithm based on convex relaxation to obtain a high-quality suboptimal solution.The design accommodates transmit covariance matrices across subcarriers while using common IRS reflection coefficients.
  • Results: Numerical results show higher capacity than traditional MIMO channels without IRS and benchmark schemes, while IRS design improves channel total power, rank, and condition number.These channel-property improvements are reported as supporting capacity enhancement.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system model represents the IRS-assisted link as a direct channel plus an IRS-reflected channel. Capacity is maximized by jointly designing unit-modulus IRS coefficients and the transmit covariance under a power constraint.

  • System model: The model includes a transmitter with Nt antennas, a receiver with Nr antennas, and an IRS with M passive reflecting elements.The paper considers narrowband frequency-flat transmission before extending the framework to broadband frequency-selective channels.
  • System model: The IRS reflection matrix is diagonal, with each coefficient having unit amplitude and a flexibly adjustable phase.Continuous phase adjustment is assumed for characterizing the capacity limit, with extension to discrete phase shifts stated as possible.
  • Effective channel: Neglecting signals reflected more than once, the effective channel is ˜H = H + RφT.The effective channel combines the direct transmitter-to-receiver link with the IRS-mediated link.
  • Signal model: The transmit covariance matrix Q satisfies the sum-power constraint tr(Q) ≤ P, and the received signal is y = (H + RφT)x + z.The receiver noise is modeled as independent circularly symmetric complex Gaussian noise with average power σ2.
  • Capacity objective: Unlike conventional MIMO capacity, IRS-aided capacity depends on both the reflection matrix φ and the resulting optimal transmit covariance matrix Q.The reflection matrix changes the effective channel and thereby affects covariance optimization.
  • Optimization problem: The resulting joint optimization is non-convex because the objective is non-concave in φ, the unit-modulus constraints are non-convex, and Q is coupled with φ.The paper exploits this structure in its proposed solution.

III. PROPOSED SOLUTION TO PROBLEM (P1)

The proposed narrowband solver alternates between transmit-covariance optimization and individual IRS-coefficient optimization. It uses tractable reformulations and closed-form subproblem solutions, with lower-complexity variants for asymptotic SNR regimes and special channel cases.

  • Alternating optimization: The alternating optimization algorithm transforms the capacity objective into an explicit form and iteratively optimizes Q or one reflection coefficient while fixing the other variables.Closed-form solutions for both subproblems enable an efficient locally optimal procedure.
  • Low- and high-SNR methods: The paper derives low-SNR and high-SNR capacity expressions and proposes lower-complexity algorithms specialized to these asymptotic regimes.It also simplifies the capacity algorithms for MISO and SIMO special cases.
  • Channel reformulation: The effective MIMO channel is expressed as the direct channel plus M rank-one reflected components, making the individual coefficient structure explicit.This reformulation supports coordinate-wise optimization of the IRS coefficients.
  • Q optimization: With fixed IRS coefficients, transmit-covariance optimization is convex and the optimal Q follows eigenmode transmission over the effective channel.The solution uses the truncated SVD of ˜H and allocates power across its available data streams.
  • Reflection-coefficient optimization: With Q and the other coefficients fixed, each reflection-coefficient subproblem has a concave objective but a non-convex unit-modulus constraint.The paper derives a closed-form optimal solution by exploiting the subproblem structure.

B. Optimal Solution to Problem (P1-m)

The paper derives closed-form solutions for optimizing one IRS reflection coefficient while the remaining coefficients are fixed. It handles diagonalizable and non-diagonalizable matrix cases separately.

  • The subproblem is reduced to optimizing α_m under the unit-modulus constraint while the other reflection coefficients remain fixed.
  • The derivation exploits the rank-one structure of B_m, determinant identities, and the Sherman–Morrison–Woodbury formula.
  • The resulting optimal values are expressed through log2 det(A_m) and the corresponding scalar correction term.
  • Case I: Diagonalizable A_m^-1B_m: For the diagonalizable case, the optimal coefficient aligns its phase with the relevant eigenvalue, maximizing Re{α_mλ_m}.
  • Case II: Non-Diagonalizable A_m^-1B_m: When A_m^-1B_m is non-diagonalizable with rank zero, any unit-modulus α_m is optimal.

3) Summary of the Optimal Solution to Problem (P1-m):

This section summarizes the optimal solution of the single-reflection-coefficient subproblem and its associated objective value.

  • The optimal solution to (P1-m) is obtained from the diagonalizable and non-diagonalizable cases derived above.
  • These results provide the coefficient update used in the alternating optimization algorithm.
  • The corresponding optimal value of (P1-m) is given by the derived closed-form objective expression.

C. Overall Algorithm

The overall algorithm alternates between optimizing IRS reflection coefficients and the transmit covariance matrix, starting from the best of multiple random initializations. Because each subproblem is solved optimally, the objective converges monotonically to at least a locally optimal solution.

  • Alternating optimization: It alternates between closed-form reflection-coefficient updates and optimal transmit-covariance updates until convergence.
  • Initialization: The algorithm randomly generates L reflection-coefficient realizations and selects the one producing the largest initial objective value.
  • Alternating optimization: Each iteration checks convergence and updates the IRS reflection matrix as φ = diag{α_1, ..., α_M}.
  • Convergence: Because every subproblem is solved optimally, the objective value decreases monotonically over iterations and is bounded by finite channel capacity.
  • Convergence: Every limit point satisfies the KKT condition, and the algorithm is guaranteed to converge to at least a locally optimal solution of (P1).
  • Complexity: The worst-case complexity is polynomial in N_r, N_t, and M.

D. Alternative Solutions to Problem (P1) in Low-/High-SNR Regimes

The paper develops lower-complexity alternatives for low- and high-SNR capacity optimization. Low SNR focuses on the strongest eigenchannel, whereas high SNR approximates capacity maximization by maximizing channel total power.

  • 1) Low-SNR Regime (Strongest Eigenchannel Power Maximization):: At low SNR, capacity maximization reduces to maximizing the strongest singular value of the effective channel.
  • 1) Low-SNR Regime (Strongest Eigenchannel Power Maximization):: The low-SNR solution alternates between IRS reflection coefficients and auxiliary left and right singular vectors.
  • 1) Low-SNR Regime (Strongest Eigenchannel Power Maximization):: The required low-SNR complexity is generally lower because it avoids computing A_m^-1B_m and their eigenvalue decompositions.
  • 2) High-SNR Regime (Channel Total Power Maximization):: At high SNR, the paper proposes maximizing channel total power as an approximate solution to the non-convex capacity problem.
  • 2) High-SNR Regime (Channel Total Power Maximization):: The high-SNR algorithm alternately optimizes one reflection coefficient while fixing the remaining coefficients.
  • 2) High-SNR Regime (Channel Total Power Maximization):: Its complexity is O(N_tN_r(M + N_r)L + N_tN_rMI), generally lower than Algorithm 1.

E. Solution to Problem (P1) with Single-Antenna Transmitter/Receiver

For single-antenna transmitter or receiver cases, the capacity problem simplifies because only one data stream is transmitted. This enables lower-complexity alternating optimization by exploiting explicit transmit covariance and capacity expressions.

  • Special-case simplifications: Single-antenna transmitter or receiver cases reduce the system to one-data-stream transmission with simplified transmit covariance and capacity expressions.The MISO case has Nr = 1, while the SIMO case has Nt = 1.
  • MISO optimization: For the MISO case, optimizing each reflection coefficient with the others fixed converges to at least a locally optimal solution with complexity O(NtML + NtMI).L and I denote the numbers of initializations and outer iterations, respectively.
  • MISO optimization: The MISO algorithm generally has lower complexity than the general alternating algorithm because capacity is an explicit function of the reflection coefficients and avoids iterative Q optimization.This simplification follows from the explicit MISO capacity expression.
  • SIMO optimization: In the SIMO case, the optimal transmit covariance is Q⋆ = P, and capacity can be maximized by maximizing channel total power.The SIMO problem therefore admits an alternating optimization approach analogous to the MISO case.
  • MIMO-OFDM extension: For broadband MIMO-OFDM, transmit covariance matrices are optimized per subcarrier, whereas one common set of IRS reflection coefficients serves all subcarriers.This common-reflection constraint arises because the IRS lacks baseband processing and frequency-selective passive beamforming capability.
  • MIMO-OFDM extension: The MIMO-OFDM problem is harder because its summed log-determinant objective is non-concave and unit-modulus reflection constraints make coefficient optimization non-convex.These properties prevent direct application of the narrowband alternating algorithm.
  • MIMO-OFDM extension: If the relaxed solution has unit-modulus coefficients, it is locally optimal for the original problem; otherwise, coefficient amplitudes are normalized to obtain an approximate solution.With relaxed coefficients, a closed-form optimal covariance solution is generally unknown.

V. NUMERICAL RESULTS

The numerical evaluation examines the proposed capacity-maximization algorithms under specified three-dimensional channel, array, noise, power, and averaging settings. It uses 100 independent channel realizations and compares performance under the paper’s simulation configuration.

  • Simulation setup: The numerical study evaluates the proposed algorithms for IRS-aided MIMO capacity maximization under a three-dimensional Cartesian geometry.The transmitter and receiver use ULAs, while the IRS uses a UPA.
  • Simulation setup: The transmitter, IRS, and receiver are positioned at specified three-dimensional coordinates, with link distances obtained from their geometry.The IRS is placed in the user’s close vicinity to improve performance.
  • Channel and noise assumptions: The channel model uses distance-dependent path loss with β0 = −30 dB at d0 = 1 m and link-specific path loss exponents.The exponents are ᾱD = 3.5, ᾱTI = 2.2, and ᾱIR = 2.8.
  • Channel and noise assumptions: The simulations use a noise power spectral density of −169 dBm/Hz, a 9 dB noise figure, and 10 MHz bandwidth.These settings yield σ² = −90 dBm for narrowband systems.
  • Evaluation protocol: Unless otherwise specified, the downlink transmit power constraint is P = 30 dBm, with 100 random initializations and convergence threshold ϵ = 10^-5.Results are averaged over 100 independent channel realizations.

A. MIMO System under Frequency-Flat Channel

For frequency-flat MIMO channels, the proposed alternating optimization algorithm is evaluated against no-IRS and heuristic benchmarks across SNR regimes, IRS sizes, and channel conditions. The results show gains in rate and favorable channel structure from jointly optimizing the transmit covariance and IRS reflections.

  • Convergence: Algorithm 1 converges monotonically and reaches high precision in about 5 outer iterations for M = 40.Its converged rate is 37.27% higher than at the initial point.
  • Capacity comparisons: Jointly optimizing Q and the IRS reflection coefficients outperforms optimizing reflections with Q fixed from the direct channel.This comparison supports joint optimization rather than direct-channel-only covariance design.
  • Capacity comparisons: Across low- and high-SNR regimes, every IRS-assisted scheme outperforms the no-IRS baseline, with gains increasing as M increases.The proposed algorithm achieves the best performance among the evaluated schemes at all tested M values.
  • Channel structure: Algorithm 1 reshapes the MIMO channel by balancing channel total power maximization against condition-number minimization.The effective channel rank is full for each evaluated scheme.
  • Channel structure: With an IRS, the evaluated schemes provide larger spatial multiplexing gain than the no-IRS scheme in the considered rich-scattering setting.The proposed algorithm’s spatial multiplexing gain is 4 in the reported comparison.

B. MIMO-OFDM System under Frequency-Selective Channel

The MIMO-OFDM study designs a common IRS reflection vector across frequency-selective subcarriers and evaluates its achievable-rate gains against benchmark schemes. The proposed algorithm improves rates over no-IRS transmission, while its gap from an ideal per-subcarrier upper bound grows with system dimensions because the IRS lacks frequency selectivity.

  • System and algorithm: The proposed MIMO-OFDM algorithm optimizes transmit covariance matrices across subcarriers and a common set of IRS reflection coefficients.The frequency-selective setting requires one shared IRS design for all subcarriers; convex relaxation is used in the alternating optimization procedure.
  • Numerical evaluation: The evaluation compares the proposed algorithm with capacity-upper-bound, no-IRS, random-phase, and other IRS benchmark schemes.The achievable rate is examined as a function of M for two parameter sets, including N = 8 and N = 32.
  • Numerical results: 38.82% and 25.07% gains at M = 20 are reported over the no-IRS case for N = 8 and N = 32, respectively.The performance gain increases as M increases, and the proposed algorithm outperforms all IRS benchmarks except the capacity upper bound.
  • Numerical results: The gap from the capacity upper bound generally increases as N and/or M increases.The upper bound permits IRS reflecting elements to be adjusted independently for different subcarriers, unlike the practical common-reflection design.
  • Limitation: The growing upper-bound gap reflects a fundamental limitation caused by the IRS's lack of frequency selectivity.The paper identifies overcoming this limitation as a direction for future research.

VI. CONCLUSIONS

The paper characterizes capacity maximization for IRS-aided point-to-point MIMO in frequency-flat and frequency-selective channels through alternating optimization. Its algorithms achieve superior rates over benchmark schemes, while IRS design improves channel power, rank, or condition number for capacity enhancement.

  • Overall contribution: The paper studies capacity maximization for IRS-aided point-to-point MIMO communication through joint IRS reflection-coefficient and transmit-covariance optimization.This joint design addresses the capacity characterization problem in the considered MIMO settings.
  • Frequency-flat channels: Under frequency-flat channels, alternating optimization finds a locally optimal solution by updating one variable while fixing the others.The variables are the transmit covariance matrix or one IRS reflection coefficient, with optimal updates derived in closed form.
  • Frequency-flat channels: Lower-complexity alternatives are proposed for asymptotically low-SNR and high-SNR regimes as well as MISO/SIMO channels.These alternatives complement the general frequency-flat algorithm.
  • Frequency-selective channels: For frequency-selective MIMO-OFDM, a common IRS reflection-coefficient set is designed across all subcarriers while transmit covariance matrices vary by subcarrier.The proposed alternating optimization uses convex relaxation for this shared-reflection design.
  • Results: The proposed algorithms achieve superior rate performance over benchmark schemes with or without IRS.Extensive numerical results support the reported rate advantage.
  • Results: Judicious IRS reflection-coefficient design significantly improves channel power, rank, or condition number, thereby enhancing capacity.These channel-property changes are identified as mechanisms associated with capacity enhancement.
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