PaperlayerView exact revision on arXiv

Source-linked AI summary

Large System Achievable Rate Analysis of RIS-Assisted MIMO Wireless Communication with Statistical CSIT

Jun Zhang, Jie Liu, Shaodan Ma, Chao-Kai Wen, Shi Jin

arXiv:2103.09161v1cs.ITmath.IT

TL;DR

The paper studies how to maximize ergodic rate in RIS-assisted MIMO under statistical channel information at the base station. It derives a large-system approximation and uses it to optimize transmission and RIS phases; simulations report agreement with Monte Carlo results, even for small antenna systems.

  • Problem

    The study asks how to choose the base-station transmit covariance and RIS phase shifts to maximize achievable ergodic rate using statistical CSIT.

    Source: §I · p. 4

  • Method

    The paper derives a replica-method large-system approximation of achievable ergodic rate and uses alternating optimization to design the transmit covariance and RIS phase-shifting matrices.

    Source: §I · p. 4

  • Results

    The large-system approximation agrees with Monte Carlo results even for small antenna systems, and the proposed scheme achieves superior performance to the compared schemes.

    Source: §IV · p. 19, §V · p. 21

  • Takeaways & Limitations

    The simulations indicate that placing the RIS near the BS or user yields the maximum achievable ergodic rate, while placing it midway yields the minimum, providing a deployment reference.

    Source: §IV · p. 21

    The RIS phase-shifting matrix solution is only locally optimal, not globally optimal, because of its non-convex constraint.

    Source: D. Proposed Algorithm · p. 16

Abstract

from arXiv · show

Reconfigurable intelligent surface (RIS) is an emerging technology to enhance wireless communication in terms of energy cost and system performance by equipping a considerable quantity of nearly passive reflecting elements. This study focuses on a downlink RIS-assisted multiple-input multiple-output (MIMO) wireless communication system that comprises three communication links of Rician channel, including base station (BS) to RIS, RIS to user, and BS to user. The objective is to design an optimal transmit covariance matrix at BS and diagonal phase-shifting matrix at RIS to maximize the achievable ergodic rate by exploiting the statistical channel state information at BS. Therefore, a large-system approximation of the achievable ergodic rate is derived using the replica method in large dimension random matrix theory. This large-system approximation enables the identification of asymptotic-optimal transmit covariance and diagonal phase-shifting matrices using an alternating optimization algorithm. Simulation results show that the large-system results are consistent with the achievable ergodic rate calculated by Monte Carlo averaging. The results verify that the proposed algorithm can significantly enhance the RIS-assisted MIMO system performance.

I. INTRODUCTION · II. SYSTEM MODEL AND PROBLEM FORMULATION · A. System Model

The paper studies downlink RIS-assisted MIMO with statistical CSIT, deriving a large-system ergodic-rate approximation and using it to design the transmit covariance and RIS phase shifts. Its system model includes direct and RIS-reflected Rician links.

I. INTRODUCTION

  • The introduction motivates RIS as a low-energy, low-cost means of improving propagation, as rising user numbers and application demands strain energy and hardware costs of other wireless technologies.RIS elements can adjust reflection phases or amplitudes, and the surface can be flexibly installed.Source: §I · pp. 2–3
  • Earlier RIS-assisted MISO and MIMO studies considered objectives including energy efficiency, received signal power, sum rate, and joint reflection and transmit-covariance design.The introduction contrasts these works, described as relying on instantaneous CSIT, with the statistical-CSIT setting considered here.Source: §I · p. 3
  • The paper assumes only statistical CSIT at the BS, citing difficult RIS channel estimation as a practical reason; prior statistical-CSIT studies cited address RIS-assisted MISO or MIMO settings.The cited MIMO study excludes a direct BS–user link and assumes Rayleigh fading.Source: §I · pp. 3–4, A. System Model · p. 7

A. System Model

  • The system has a multi-antenna BS and user, a RIS with nearly passive elements, and direct BS–user, BS–RIS, and RIS–user links with LoS and NLoS components.The effective channel is H = H0 + H2ΘH1.Source: §I · p. 4, A. System Model · pp. 5, 7

I. INTRODUCTION

  • The replica-method approximation covers effective channels formed by sums and products of Rician matrices, and simulations report accuracy even for small antenna systems.The paper identifies processing the three Rician matrices together in a single transmission slot as a challenge distinguishing this setting from the cited relay-channel analysis.Source: §I · p. 4
  • The proposed alternating optimization designs the transmit covariance using iterative water-filling and the RIS phase-shifting matrix using projected gradient ascent, based on statistical CSIT.The stated objective is to maximize the downlink RIS-assisted MIMO system's achievable ergodic rate.Source: §I · pp. 4–5

A. System Model

  • The transmitted signal is zero-mean Gaussian with covariance Q, while RIS phases form a diagonal matrix Θ; the model sets each element's reflection amplitude to one and assumes independent Gaussian noise.The text notes that setting all reflection amplitudes to zero represents the case without RIS.Source: A. System Model · p. 6

B. Problem Formulation · III. TRANSMIT COVARIANCE MATRIX AND PHASE-SHIFTING MATRIX OPTIMIZATION · A. Large System Analysis

The paper formulates joint transmit-covariance and RIS phase-shift optimization for ergodic rate, then derives a large-system rate approximation that supports the optimization. The approximation is reported to estimate rates well even with finitely many antennas.

B. Problem Formulation

  • The objective is to maximize achievable ergodic rate by choosing the BS transmit covariance Q and diagonal RIS phase matrix Θ, subject to the BS power constraint and unit-modulus phases.Source: B. Problem Formulation · p. 7
  • The optimization is nonconvex because Θ has unit-modulus diagonal entries, while Monte Carlo channel averaging makes direct rate evaluation computationally prohibitive.Source: B. Problem Formulation · pp. 7–8

A. Large System Analysis

  • The ergodic rate is asymptotically approximated in a regime where N, L, and K grow with L/N and K/L fixed, using six unique equation solutions.Source: A. Large System Analysis · pp. 8–9
  • With no BS-to-user direct link, a corollary gives an asymptotic ergodic-rate approximation for two-hop MIMO Rician product channels.Source: A. Large System Analysis · p. 10
  • The analysis also derives a useful Stieltjes-transform result for Rician random-matrix products using the relationship between Shannon and Stieltjes transforms.Source: A. Large System Analysis · p. 11
  • For Rayleigh channels, another corollary provides an asymptotic approximation of the achievable ergodic rate.Source: A. Large System Analysis · p. 11
  • Substituting Q and Θ into the approximation yields a rate expression whose accuracy remains very good for finite antenna counts.Source: A. Large System Analysis · p. 12
  • The authors propose an alternating method to solve the optimization problem using the large-system approximation.Source: A. Large System Analysis · pp. 12–13

B. Transmit Covariance Matrix Optimization

For fixed RIS phases, the paper derives an asymptotically optimal transmit covariance from the statistical-CSIT-dependent matrix F. The solution uses waterfilling and is computed iteratively because the quantities defining F also depend on Q.

B. Transmit Covariance Matrix Optimization

C. Diagonal Phase-Shifting Matrix Optimization

With fixed Q, the RIS phase-shifting matrix is optimized under unit-modulus diagonal constraints. Because this problem is generally nonconcave, the proposed suboptimal method uses projected gradient ascent to increase the large-system rate while preserving those constraints.

C. Diagonal Phase-Shifting Matrix Optimization

  • The phase-shifting matrix is diagonal, with entries e^{jθ_l} and phases θ_l in [0, 2π).Source: C. Diagonal Phase-Shifting Matrix… · p. 15
  • The optimization is generally nonconcave because each diagonal phase-shifting entry has unit norm, so projected gradient ascent provides a suboptimal search that monotonically increases the rate.The gradient is taken with respect to each phase variable ϑ_l; its expression is given in Appendix B.Source: C. Diagonal Phase-Shifting Matrix… · p. 15
  • Each iteration updates the phase vector using the computed ascent direction, preserves unit magnitude, and forms the next diagonal matrix Θ, as specified in Algorithm 2.Source: C. Diagonal Phase-Shifting Matrix… · p. 15

D. Proposed Algorithm

The proposed algorithm alternates between optimizing the RIS phase-shifting matrix and the BS transmit covariance matrix until the achievable-rate change falls below a tolerance; it is guaranteed to converge despite the phase update being only locally optimal.

D. Proposed Algorithm

  • Starting from Q(0) = I_N and independently uniform RIS phases in [0, 2π), the algorithm alternates Θ updates via Algorithm 1 and Q updates via Algorithm 2 until the rate change is below ε.It returns Qopt and Θopt when |R̄(Q(n+1), Θ(n+1)) − R̄(Q(n), Θ(n))| < ε.Source: D. Proposed Algorithm · p. 16
  • For fixed Θ, Q is optimal by strict concavity, while the nonconvex phase constraint makes Θ only locally optimal; its gradient search increases the rate monotonically, guaranteeing convergence.The existence and uniqueness proofs for the six implicit parameters are omitted; existence is obtained by a subsequence approach, and uniqueness by reduction to absurdity.Source: D. Proposed Algorithm · pp. 16–17

IV. SIMULATION RESULTS · V. CONCLUSION

Simulations show that the large-system approximation predicts achievable ergodic rate reliably and that the proposed statistical-CSIT algorithm improves RIS-assisted MIMO performance. The conclusion reiterates the approximation’s predictive reliability and the algorithm’s effectiveness.

IV. SIMULATION RESULTS

  • The simulations compare the large-system approximation with Monte Carlo achievable-rate results using a setup with P = 10dBm, B = 10MHz, noise power −94dBm, and κ_i = 1.The BS, RIS, and user coordinates are (0; 10m), (0; d(m)), and (80m; 10m), respectively.Source: §IV · p. 17
  • Analytical rates agree with Monte Carlo results; the proposed statistical-CSIT scheme nearly matches perfect CSIT, outperforms no CSIT, and RIS-assisted MIMO outperforms MIMO without RIS.The comparison is shown for N = L = K = 8 and d = 40m.Source: Figure 3 · p. 18, §IV · p. 19
  • The proposed algorithm converges within 25 iterations for P = 10dBm, N = L = K = 8, and d = 40m.The convergence plot tracks the large-system approximation of achievable ergodic rate against iteration count.Source: Figure 4 · p. 18, §IV · p. 19
  • The AF relay system performs almost identically to the system without RIS because its relay-link gain is lower than the direct-link gain, leaving all power allocated to the BS.The AF benchmark uses L = 8 relay antennas and numerically optimized power allocation subject to Ps + Pr = P.Source: §IV · p. 19
  • Achievable ergodic rate increases as the number of RIS elements L grows, and RIS-assisted MIMO outperforms MIMO without RIS.The comparison uses N = K = 10 and d = 40m.Source: §IV · p. 19, Figure 5 · p. 20
  • With optimal Q and Θ, achievable ergodic rate is highest when the RIS is near the BS or user and lowest when it is midway between them.The study compares N = K = 8 with L = 32 or 80.Source: Figure 6 · p. 20, §IV · p. 21

V. CONCLUSION

  • The conclusion states that the replica-method large-system approximation supports designing the BS transmit covariance and RIS phase-shifting matrices to maximize achievable ergodic rate, with numerical results validating prediction reliability and algorithm effectiveness.The investigated downlink system has Rician BS-to-RIS, RIS-to-user, and BS-to-user links.Source: §V · p. 21

APPENDIX · A. Proof of Proposition 1 · 1) Mutual Information:

The appendix derives the mutual-information expression for the MIMO RIS channel using Bayes’ rule and a replica-based representation. It then applies a large-system Gaussian approximation when averaging over H2 and rewrites terms in the derivation through matrix operations and a rate measure.

1) Mutual Information:

  • For fixed input distribution p(s) and channel H, the appendix formulates the conditional mutual information and then expresses the mutual information.The derivation first uses Bayes’ rule to obtain the posterior distribution.Source: 1) Mutual Information · pp. 21–22
  • The derivation introduces replica signal vectors s(0), …, s(r), each drawn under p(s), and collects them into S.This representation is used to rewrite equation (35).Source: 1) Mutual Information · p. 22
  • As N → ∞, V2 converges to a Gaussian random matrix with a specified mean and covariance C2 ⊗ R2.The mean is formed from the replica signals and the channel terms H0, H1, and H2.Source: 1) Mutual Information · pp. 22–23
  • The cited expression contains −log det (I + C2Σ ⊗R2) and −K log (1 + r), including a log-determinant term involving C2Σ and R2.Source: 1) Mutual Information · p. 23
  • The derivation also rewrites equation (43) using the rate measure of μ₂⁽ʳ⁾(C2).The passage introduces this rate measure as the next step in the calculation.Source: 1) Mutual Information · p. 24

1 T2H1S)oo , (48) … 0 B2 ¯H2 ¯H1SA2SH + ¯HH 1 ¯HH 2 B2H0SA2SH

The derivation evaluates channel expectations successively, using Gaussian large-system limits and saddle-point integrations over auxiliary matrices to obtain the rate expression.

1 T2H1S)oo , (48)

  • The integration over C₂ is performed by the saddle-point method.The passage presents the resulting expression after this integration.Source: §1 · p. 24

3) Taking Expectation over H1:

  • As N → ∞, V₀ and V₁ converge to Gaussian random matrices with respective means V̄₀ and V̄₁ and covariances C₀ ⊗ R₀ and C₁ ⊗ R₁.C₀ and C₁ are defined by their respective matrix entries in the passage.Source: 3) Taking Expectation over H1 · p. 24
  • Taking expectation over H₁ rewrites the exponent terms of (44) and (48).The passage states this step is similar to (41).Source: 3) Taking Expectation over H1 · p. 24

0 B2 ¯H2 ¯H1SA2SH + ¯HH 1 ¯HH 2 B2H0SA2SH

  • The expectation over H₀ produces the displayed expression involving C₀ and C₁.The resulting rate measure is μ₀⁽ʳ⁾(C₀, C₁).Source: §0 · pp. 25–26
  • As N → ∞, integrating over C0 and C1 by the saddle-point method yields the stated rate expression.The rate measure is identified as μ₀⁽ʳ⁾(C₀, C₁).Source: §0 · p. 26
  • The final derivation interchanges the N → ∞ and r → 0 limits and combines the preceding expressions.The passage names (35), (38), (49), and the earlier saddle-point expressions as inputs to this step.Source: §0 · pp. 26–27

5) Replica Symmetry:

The section applies replica symmetry to simplify the saddle-point search, then uses the resulting eigenvalue forms to derive saddle-point expressions and the proposition’s parameter definitions.

Replica Symmetry

  • Replica symmetry makes the saddle points independent of replica-index dependence, avoiding a search over all possible C_i and ˜C_i for i = 0, 1, 2.The section adopts this assumption to obtain the saddle points.Source: 5) Replica Symmetry · p. 27
  • Under replica symmetry, the eigenvalues of C_i and ˜C_i are expressed using a_i, b_i, ˜a_i, and ˜b_i, with one distinguished eigenvalue and r repeated eigenvalues.For C_i these are a_i + rb_i and a_i − b_i; for ˜C_i they are ˜a_i + r˜b_i and ˜a_i − ˜b_i.Source: 5) Replica Symmetry · p. 27
  • Substituting the recast terms into the saddle-point expression and setting derivatives with respect to ˜b_i to zero yields expressions for a_i, b_i, and ˜b_i; ˜a_i = 0 when r = 0.These conditions apply for i = 0, 1, 2.Source: 5) Replica Symmetry · p. 29
  • Defining ˜e_i = ˜b_i and e_i = a_i − b_i for i = 0, 1, 2, and substituting the saddle-point expressions into (34), yields the proposition.Source: 5) Replica Symmetry · p. 29

B. Taking the derivative of ¯R (Q, Θ) with respect to ϑl = ejθl

This subsection develops the derivative of ¯R(Q, Θ) with respect to the RIS phase variable ϑ_l = e^{jθ_l} and specifies auxiliary matrices used in its expression.

B. Taking the derivative of ¯R (Q, Θ) with respect to ϑl = ejθl

C. Mathematical Tools

The appendix provides mathematical tools used in the paper, including a Gaussian integral and Hubbard–Stratonovich transformation.

C. Mathematical Tools

  • Lemma 1, [40, Lemma 1], presents a Gaussian integral and Hubbard–Stratonovich transformation for N-dimensional real vectors z and b and a positive definite N × N matrix A.Source: C. Mathematical Tools · p. 31

Related papers

Loading 2103.09161v1…