MIMO Communications with 1-bit RIS: Asymptotic Analysis and Over-the-Air Channel Diagonalization

TL;DR

This work analyzes 1-bit RIS-aided MIMO under Ricean fading in the regime where the RIS size dominates the transceiver dimensions. It shows that the dominant SVD components converge to deterministic LoS directions, enabling a closed-form Sign Alignment RIS design based on LoS phase signs and, when the RIS is sufficiently large, OTA diagonalization of the end-to-end channel for interference-free spatial multiplexing without transmitter CSI. The authors introduce a waterfilling-inspired SA (W-SA) RIS element allocation framework that distributes RIS elements across spatial streams based on asymptotic singular values, coupled with a capacity surrogate that leverages diagonalization. Simulations demonstrate that these low-complexity schemes achieve performance close to RMO-based methods while delivering orders of magnitude faster runtimes, and the capacity surrogate provides accurate predictions under large RIS regimes.

Abstract

This paper presents an asymptotic analysis of Multiple-Input Multiple-Output (MIMO) systems assisted by a 1-bit Reconfigurable Intelligent Surface (RIS) under Ricean fading conditions. Using random matrix theory, we show that, in the asymptotic regime, the dominant singular values and vectors of the transmitter-RIS and RIS-receiver channels converge to their deterministic Line-of-Sight (LoS) components, almost irrespective of the Ricean factors. This enables RIS phase configuration using only LoS information through a closed-form Sign Alignment (SA) rule that maximizes the channel gain. Furthermore, when the RIS is asymptotically larger than the transceiver arrays, proper RIS configuration can render the end-to-end MIMO channel in the capacity formula asymptotically diagonal, thereby eliminating inter-stream interference and enabling Over-The-Air (OTA) spatial multiplexing without channel knowledge at the transmitter. Building on this result, a waterfilling-inspired SA algorithm that allocates RIS elements to spatial streams, based on the asymptotic singular values and statistical channel parameters, is proposed. Simulation results validate the theoretical analyses, demonstrating that the proposed schemes achieve performance comparable to conventional Riemannian manifold optimization, but with orders of magnitude lower runtime.

Figures (2)

• Figure 1: Convergence properties of a $K$-factor Ricean fading channel $\mathbf{H} \in \mathbb{C}^{N_{\rm S} \times N_{\rm T}}$. Unless otherwise stated, the parameters $N_{\rm S} = 10^4$, $N_{\rm T} = 10^2$, and $K = 10\,\mathrm{dB}$ were used. Figure \ref{['fig: eigenvalues of single instance']} shows the eigenvalues of $\mathbf{HH}^{\rm H}$ for a single channel instance which are compared with their respective asymptotics computed via \ref{['eq: convergence of all eigenvalues']}; the Normalized Mean Squared Error (NMSE) of this asymptotic approximation is illustrated in Fig. \ref{['fig: eigenvalues versus N']} for different sizes of RISs. In particular, the NMSE is computed for each eigenvalue and the aggregate error is depicted, revealing that asymptotic hardening takes place for increasing $N_{\rm S}$. Figure \ref{['fig: eigenvalues versus K']} depicts the NMSE between the principal eigenvalue of $\mathbf{HH}^{\rm H}$ and $K/(K+1)N_{\rm S}N_{\rm T}$, with the latter being the principal eigenvalue in the asymptotic regime according to Theorem \ref{['theorem: eigenvalue hardening']}. The NMSE is plotted versus $K$ starting with the value $-20$ dB where $1/N_{\rm T}$ equals $K$; recall that, at this point, convergence is lost, since it should hold that $K/N_{\rm T}\to 0$. It can, however, be seen that, if $K\geq 10N_{\rm T}^{-1}$, the NMSE drops below $10^{-2}$ and converges to $0$ for increasing $K$.
• Figure 2: The left subfigure (Fig. \ref{['fig: Capacity Approximation']}) validates the asymptotic diagonalization property established in Theorem \ref{['theorem: channel diagonalization']}, illustrating the NMSE between the actual $\mathcal{C}(\boldsymbol{\phi})$ and the approximated $\hat{\mathcal{C}}(\boldsymbol{\phi})$ capacities, with the latter assuming a diagonal channel; simulation parameters used: $N_{\min}=N_{\rm R}=N_{\rm T}=16$, $K_{\rm T}=K_{\rm R}=0$ dB, and $N_{\rm S}=1\!:\!10\!:\!100\times N_{\max}^3$. Element allocation per stream was random with uniform distribution and weighted to meet $\mathcal{OP}_2$'s constraint. The middle subfigure (Fig. \ref{['fig: Channel Magnitude Optimization']}) illustrates the optimized channel magnitude $||\mathbf{\tilde{H}}||_{\rm F}^2$, considering $N_{\rm T}=N_{\rm R}=100$ and $K_{\rm T}=K_{\rm R}=\{-10,0,10\}$ dB. In particular, the performance of SA is compared with RMO and the asymptotic Lower Bound (LB) derived in Section \ref{['sec: Channel Gain Maximization']}: $0.25\,\frac{K_{\rm T}K_{\rm R}}{(1+K_{\rm T})(1+K_{\rm R})}N_{\rm S}^2N_{\rm T}N_{\rm R}$. The right subfigure (Fig. \ref{['fig: capacity optimisation']}) depicts the achievable rate with the proposed W-SA design against two benchmarks: RMO applied to the original capacity expression \ref{['eq: capacity formula']} and RMO applied to the surrogate formulation \ref{['eq: capacity with diagonalization']}; simulation parameters used: $N_{\rm T}=N_{\rm R}=10$ and $K_{\rm T}=K_{\rm R}=0$ dB. It is shown that, although the RMO variants achieve up to $30\%$ higher capacity than W-SA, the latter requires up to $10^3$ times less computation time (see Table \ref{['tab: running-time capacity']}).

Theorems & Definitions (5)

• proof
• proof
• proof
• Remark 1
• Remark 2