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Achievable Rate Optimization for MIMO Systems with Reconfigurable Intelligent Surfaces

Nemanja Stefan Perović, Le-Nam Tran, Marco Di Renzo, Mark F. Flanagan

arXiv:2008.09563v3cs.IT

TL;DR

The paper addresses achievable-rate optimization for RIS-aided multi-stream MIMO systems, where joint design is challenging and relatively underexplored. It formulates a joint covariance-and-phase optimization and solves it with a projected-gradient method enhanced by convergence controls, while introducing an FSPL-ratio applicability measure. Simulations report AO-level achievable rates with fewer iterations and lower computational complexity, especially demonstrating indoor gains with few RIS elements.

  • Problem

    Achievable-rate optimization for RIS-aided MIMO systems is relatively unknown and more challenging than conventional fixed-channel MIMO optimization.

  • Method

    The paper jointly optimizes the transmit covariance matrix and RIS phase shifts using a projected-gradient method with a Lipschitz-based step size, data scaling, and backtracking line search.

  • Results

    The proposed PGM achieves the same achievable rate as AO with significantly fewer iterations and substantially lower computational complexity.

  • Takeaways & Limitations

    The total FSPL ratio provides a first-order RIS applicability measure, and indoor deployments can obtain higher indirect-link achievable rates with a small number of RIS elements.

Abstract

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Reconfigurable intelligent surfaces (RISs) represent a radical new technology that can shape the radio wave propagation in wireless communication systems and offers a great variety of possible performance and implementation gains. Motivated by this, in this paper we study the achievable rate optimization for a multi-stream multiple-input multiple-output (MIMO) system equipped with an RIS, and formulate a joint optimization problem of the covariance matrix of the transmitted signal and the RIS elements. To solve this problem, we propose an iterative optimization algorithm that is based on the projected gradient method (PGM). We derive the step size that guarantees the convergence of the proposed algorithm and we define a backtracking line search to improve its convergence rate. Furthermore, we introduce the total free space path loss (FSPL) ratio of the indirect and direct links as a first-order measure of the applicability of an RIS in the considered communication system. Simulation results show that the proposed PGM achieves the same achievable rate as a state-of-the-art benchmark scheme, but with a significantly lower computational complexity. In addition, it is demonstrated that the RIS application is particularly suitable to increase the achievable rate in an indoor environment, as in this case even a small number of RIS elements is sufficient to provide a substantial achievable rate gain.

I. INTRODUCTION

RISs address the limits of conventional rate-enhancing technologies by reshaping propagation, while this paper targets the relatively underexplored achievable-rate optimization of RIS-aided multi-stream MIMO systems. It proposes a convergent projected-gradient approach, an FSPL-based applicability measure, and simulations demonstrating comparable rates with lower complexity.

  • Motivation: RISs use reconfigurable passive elements with adjustable phase shifts to modify radio-wave interactions and propagation.Wavefront optimization controls scattering and reflection characteristics.
  • Research gap: Achievable-rate optimization is more challenging with an RIS than without one, and remains relatively unknown for RIS-aided MIMO systems.Existing methods largely focus on downlink systems with single-antenna receivers.
  • Proposed approach: The paper jointly optimizes the transmit covariance matrix and RIS phase shifts for a multi-stream MIMO system using an iterative projected gradient method.The method provides closed-form gradient and projection expressions and converges to a critical point of the nonconvex problem.
  • Proposed approach: A Lipschitz-derived step size, data scaling, and backtracking line search guarantee convergence and significantly improve convergence rate over existing alternating optimization.The line search is reported to outperform the existing AO approach in convergence rate.
  • RIS applicability: The total FSPL ratio between indirect and direct links is introduced as a first-order measure of RIS applicability.Because indirect-link total FSPL is intractable for MIMO, the paper derives it for an SISO system.
  • Results: Simulations show that PGM reaches AO's achievable rate with fewer iterations and substantially lower computational complexity, while indoor systems can benefit from few RIS elements.The indirect link can exceed the direct link's achievable rate with a small number of RIS elements; RIS position primarily determines indirect-link total FSPL.

C. Problem Formulation

The paper formulates achievable-rate maximization for an RIS-assisted MIMO system with a channel that depends on the RIS phase shifts.

  • The achievable rate is optimized for an RIS-assisted wireless communication system using Gaussian signaling and input covariance matrix Q.
  • The optimization jointly accounts for the transmitted-signal covariance matrix and the RIS configuration through the channel matrix H.
  • The RIS-dependent constraints include unit-modulus phase-shift variables for the RIS elements.

III. SOLUTION APPROACH VIA PROJECTED GRADIENT METHOD

The proposed solution applies projected gradient updates to the nonconvex joint optimization, using efficient projections for the RIS phases and covariance matrix. All optimization variables are updated simultaneously, contrasting with alternating optimization.

  • The joint achievable-rate problem is nonconvex, with an objective that is neither convex nor concave in the optimization variables.
  • The achievable rate considered is not the channel capacity because the RIS phase shifts do not encode transmitted data.
  • The projected gradient method solves the problem by projecting gradient-based updates onto the feasible sets of the RIS phases and covariance matrix.
  • The algorithm updates all optimization variables simultaneously in each iteration, whereas alternating optimization updates one variable at a time.
  • The gradient construction uses K(θ, Q) = (I + Z(θ)QZ^H(θ))^-1 and gradients with respect to θ* and Q*.
  • The method is motivated by efficient projection onto the feasible set, including unit-circle projection for RIS elements and water-filling projection for the covariance matrix.

D. Improved Convergence Rate by Data Scaling

The paper improves projected-gradient convergence by scaling variables to balance differently sized gradients, then establishes smoothness and a step-size condition for convergence to a critical point.

  • Unbalanced indirect and direct-link data can make Algorithm 1 converge slowly.
  • A change of variables scales the covariance matrix and RIS phase variables to increase convergence speed.
  • The scaled objective is shown to have an L-Lipschitz continuous gradient, providing the basis for a convergence-guaranteeing step size.
  • The scaling factor k emulates separate step sizes for gradients of different sizes, but its value is selected heuristically through numerical experiments.
  • With step size μ < 1/L, the iterates are bounded and every accumulation point is a critical point of the optimization problem.
  • Because the optimization problem is nonconvex, convergence to a critical point does not generally guarantee global optimality.

B. Complexity Analysis

The complexity analysis approximates per-iteration cost by counting complex multiplications, focusing on the dominant channel, gradient, and linear-system computations.

  • The analysis measures Algorithm 1 complexity using the number of complex multiplications required per iteration.
  • Computing the effective channel Z(θ) is dominated by H2F(θ)H̄1, requiring NrNris + NrNtNris complex multiplications.
  • The matrix A = K(θ, Q)Z(θ) is computed by solving a linear system rather than explicitly forming a matrix inverse.
  • Cholesky decomposition is used to solve the linear system through forward and backward substitution.
  • The resulting computation of A has complexity O(Nr^2Nt + Nr^3) multiplications.

2 AQ ¯HH

The PGM’s computational cost is analyzed per iteration and compared with AO, while backtracking improves convergence despite adding per-iteration work.

  • Complexity analysis: The PGM per-iteration complexity is derived from gradient, update, projection, eigenvalue-decomposition, and water-filling operations.The analysis counts complex multiplications and matrix operations for Algorithm 1.
  • PGM versus AO: Table I compares the PGM and AO computational complexity required to reach 95% of the average achievable rate at the 500th iteration.
  • Complexity analysis: When N_ris is much larger than N_t and N_r, the dominant complexity is approximated by O(N_risN_tN_r).
  • PGM versus AO: The total PGM complexity depends on the number of iterations, whereas AO complexity is expressed through outer iterations containing N_ris + 1 conventional iterations.
  • Backtracking line search: The backtracking line search can select larger step sizes than the conservative convergence-guaranteeing step size.It is based on the Armijo–Goldstein condition and is designed to accelerate convergence.
  • Backtracking line search: Although line search increases per-iteration complexity, its cost can be offset by substantially reducing the required number of iterations and overall runtime.

V. TOTAL FPSL RATIO - A METRIC OF RIS APPLICABILITY

The total FSPL ratio is introduced as a first-order measure of RIS applicability by comparing indirect and direct-link losses. Simulations examine how RIS position and link geometry affect this metric and achievable rate.

  • Metric definition: The indirect-link total FSPL must account for RIS phase shifts, unlike the conventional indirect-link loss definition.
  • Metric definition: Computing indirect-link total FSPL is intractable for MIMO because the optimal RIS phase shifts are initially unknown, so the paper derives a SISO-based measure.
  • Metric interpretation: For T > 1, the direct link is expected to remain stronger and RIS gains are limited; for T < 1, the optimized indirect link is stronger and gains are usually more substantial.
  • RIS position: With l_t = 20 m and l_r = 100 m, T is highest near the RIS’s center position and lowest when the RIS is near the transmitter or receiver.
  • RIS position: The RIS is placed near the transmitter or receiver in subsequent simulations because these positions provide the lowest indirect-link signal attenuation.
  • Simulation results: The PGM converges faster than AO, while both can reach the optimum achievable-rate value; with a blocked direct link, PGM needs only a few iterations.
  • Metric limitations: The SISO-derived T is not entirely trustworthy for predicting MIMO achievable rate, although lower indirect-link FSPL can accompany greater RIS gains.
  • Symmetric geometry: For l_t = l_r = 50 m, the total FSPL ratio is perfectly symmetric with respect to RIS position.

2) Direct link blocked:

With the direct link blocked, the simulations compare PGM-based optimization with AO and examine indirect-link path loss, RIS size, and frequency effects. The results show rapid PGM convergence and increasing achievable rate with more RIS elements, although gains diminish.

  • Direct link blocked:: The PGM reaches the optimum achievable rate in only a few iterations, whereas AO requires approximately Nris + 1 iterations.This behavior is observed when communication occurs solely through the RIS-induced indirect link.
  • Direct link blocked:: The upper-bound expression in (38) closely matches its two sides in SISO simulations, validating (39) as an approximation of the indirect-link total FSPL.The comparison uses Nt = Nr = 1 and averages results over different RIS positions.
  • Direct link blocked:: Doubling Nris increases achievable rate, but the rate curve’s slope gradually decreases as Nris grows.The simulations use square RIS configurations with Nris equal to 49, 196, and 784 rather than exact doublings.
  • Direct link blocked:: At frequencies above 5 GHz, achievable rate remains almost constant whether the direct link is present or blocked because the direct link becomes practically useless.The indirect link maintains approximately the same achievable rate across the considered higher-frequency range.

C. Achievable Rate in Indoor Environments

Indoor simulations evaluate achievable rate, computational complexity, initialization, and realistic RIS imperfections. They show rapid PGM convergence, strong indirect-link performance, and limited rate loss from low-resolution phase shifts or imperfect CSI.

  • C. Achievable Rate in Indoor Environments: In the indoor environment, the PGM converges to the optimal achievable rate with fewer iterations than AO, while achievable rate is almost entirely determined by indirect-link transmission.The smaller indoor distances reduce total FSPL, making the indirect link especially influential.
  • D. Computational Complexity and Run Time Results: The PGM achieves significantly lower computational complexity than AO while requiring a small number of iterations to reach the target achievable rate.With a present direct link, PGM iteration count decreases as Nris increases; with a blocked direct link, it remains constant.
  • D. Computational Complexity and Run Time Results: The PGM converges in extremely low run time, while AO still does not reach the same achievable rate after 500 ms.Approximately the same time is needed for AO merely to select its optimal initial point.
  • C. Achievable Rate in Indoor Environments: Different initial values of θ and Q affect mainly the first few iterations, with all tested cases attaining similar later achievable rates.Fixed initial θ and Q produce a slightly higher rate during the earliest iterations.
  • C. Achievable Rate in Indoor Environments: 1-bit and 2-bit RIS phase shifts reduce the optimal achievable rate by approximately 1.1 bit/s/Hz and 0.2 bit/s/Hz, respectively.The results characterize discretization loss relative to continuous phase shifts.
  • C. Achievable Rate in Indoor Environments: Imperfect CSI with σ2 = 0.2 decreases the optimal achievable rate by approximately 1 bit/s/Hz.The channel estimate is modeled as the true channel plus an i.i.d. Gaussian estimation-error matrix.

APPENDIX A COMPLEX-VALUED GRADIENT OF f(θ, Q)

Appendix A derives the complex-valued gradient of the achievable-rate objective with respect to RIS phase variables and the transmit covariance matrix. The derivation uses matrix differentials, vectorization, and norm inequalities.

  • APPENDIX A COMPLEX-VALUED GRADIENT OF f(θ, Q): The derivation starts from the complex differential of f(θ, Q) with respect to F(θ) = diag(θ) and F*(θ).The appendix follows a general procedure for computing complex-valued gradients.
  • APPENDIX A COMPLEX-VALUED GRADIENT OF f(θ, Q): The appendix obtains the gradient with respect to θ and states that ∇Qf(θ, Q) follows by an analogous derivation.The details of the covariance-gradient proof are omitted because of the page limit.
  • APPENDIX A COMPLEX-VALUED GRADIENT OF f(θ, Q): Trace-vectorization identities and the diagonal-selection matrix Ld rewrite the differential in vectorized form.Ld places the diagonal elements of a square matrix into vec(A) through vec(A) = Ld vecd(A).
  • APPENDIX A COMPLEX-VALUED GRADIENT OF f(θ, Q): The Lipschitz-bound derivation upper-bounds matrix-norm differences using singular-value and product-norm inequalities.Intermediate bounds are substituted into the preceding expressions to obtain the stated result (27).

B. Proof of (28)

The proof of (28) applies the same norm-bounding strategy used for (27). Its details are omitted after establishing that the relevant right-hand-side norms can be upper-bounded.

  • B. Proof of (28): The proof begins from the relation in (17b) and uses it directly in the bounding argument.This provides the starting inequality for deriving (28).
  • B. Proof of (28): The two right-hand-side norms are further upper-bounded by repeating the steps used to prove (27).The appendix omits the detailed algebra because of the page limit.
  • B. Proof of (28): The derivation of (28) is therefore parallel to the proof of (27), rather than introducing a separate bounding technique.The passage explicitly identifies the earlier proof procedure as the template.

APPENDIX C PROOF OF THEOREM 2

The proof establishes convergence-related properties of the iterative procedure by combining smoothness inequalities, boundedness of the feasible iterates, and optimality conditions. It concludes that an accumulation point is a critical point of (11).

  • The proof invokes an inequality for an L-smooth function f.
  • When ¯θ = ¯θ_n, the objective in the considered problem equals 0.
  • The feasible set is closed and bounded, so the iterate sequence (¯θ_n, ¯Q_n) is bounded and has accumulation points.
  • Because f is nondecreasing, all accumulation points share the same objective value f ∗.
  • The proof concludes that an accumulation point is a critical point of (11).

APPENDIX D COMPUTATIONAL COMPLEXITY FOR ALTERNATING OPTIMIZATION (AO)

This appendix derives the computational complexity of the alternating optimization method by accounting for channel construction, covariance optimization, and RIS-element updates. The resulting cost is expressed in terms of the number of outer iterations and system dimensions.

  • The channel matrix is modeled as ˜H = H + RφT, combining direct transmission with the RIS-assisted link.H is the direct channel, T connects the transmitter to the RIS, R connects the RIS to the receiver, and φ models the RIS response.
  • The initial channel-matrix calculations for L_AO RIS realizations require (L_AO + 1)N_rN_tN_ris multiplications.
  • Computing the L_AO covariance matrices Q requires O(L_AO(D^3 + 1/2N_t^2D)) multiplications, with D = min(N_t, N_r).The covariance computation includes a truncated decomposition, water-filling, and matrix multiplication.
  • The AO method alternately adjusts each RIS coefficient α_m and the covariance matrix Q within successive conventional iterations.N_ris + 1 conventional iterations form one outer iteration, and the method stops when its convergence criterion is met.
  • Updating all RIS elements requires auxiliary-matrix computations with O(N_r^2N_t) and O(2N_r^3) costs per element.The subsequent coefficient and matrix updates are described as negligible in complexity.
  • The AO algorithm’s total computational complexity is given by an expression scaled by the number of outer iterations I_OI.The appendix identifies this total cost after combining channel, RIS-element, and covariance-update computations.
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