MIMO MAC Empowered by Reconfigurable Intelligent Surfaces: Capacity Region and Large System Analysis

TL;DR

This paper calculates the mean of the sum Mutual Information (MI) for the correlated Multiple-Input Multiple-Output (MIMO) Multiple Access Channel (MAC) in the presence of multiple RISs, in the large-antenna number limit, and obtains the capacity region boundaries.

Abstract

Smart wireless environments enabled by multiple distributed Reconfigurable Intelligent Surfaces (RISs) have recently attracted significant research interest as a wireless connectivity paradigm for sixth Generation (6G) networks. In this paper, using random matrix theory methods, we calculate the mean of the sum Mutual Information (MI) for the correlated Multiple-Input Multiple-Output (MIMO) Multiple Access Channel (MAC) in the presence of multiple RISs, in the large-antenna number limit. We thus obtain the capacity region boundaries, after optimizing over the tunable RISs' phase configurations. Furthermore, we obtain a closed-form expression for the variance of the sum-MI metric, which together with the mean provides a tight Gaussian approximation for the outage probability. The derived results become relevant in the presence of fast-fading, when channel estimation is extremely challenging. Our numerical investigations showcased that, when the angle-spread in the neighborhood of each RIS is small, which is expected for higher carrier frequencies, the communication link strongly improves from optimizing the ergodic MI of the multiple RISs.We also found that, increasing the number of transmitting users in such MIMO-MAC-RIS systems results to rapidly diminishing sum-MI gains, hence, providing limits on the number of users that can be efficiently served by a given RIS.

Figures (5)

• Figure 1: The considered MIMO-MAC-RIS communication system comprising ${M}$ TXs each equipped with ${N_t}$ antenna elements, a single ${N_r}$-antenna RX, and $K$ identical RISs. Notations ${\bf G}_{\alpha,\beta}$, detailed in Section \ref{['sec:MIMO channel model']}, represent the channel gain matrices between any pair $\alpha$ and $\beta$ of the latter network nodes.
• Figure 2: The ergodic sum-MI performance in nats per channel use for $M=2$ TXs and for $K=1$ and $2$ RISs versus the angle spread $\sigma$ in degrees. All parameter values of this simulation setup appear in Table \ref{['table1']}, except for the incoming signal azimuth angles from each TX to the RISs, which are set as $\phi_{{\rm in},1}=45^{\circ}$ and $\phi_{{\rm in},2}=-45^{\circ}$, as can be seen in the upper inset. In the case of two RISs, we have kept for simplicity the same parameter values, essentially locating the second RIS at the mirrored location of the first with respect to the plane of the TXs and the RX. The "o" points correspond to the $\hbox{\boldmath$\Phi$}_k$'s obtained using the algorithm of Section \ref{['sec:Analytical_Solution']}, while the solid curves correspond to the optimized $\hbox{\boldmath$\Phi$}_k$'s presented in Section \ref{['sec:Numerical_Solution']}. We see that they practically coincide with each other and with the Monte Carlo simulations. It is also depicted that the benefit of having two RISs (blue curves) serving both TXs is relatively minor, at least as compared with the large performance increase of the presence of one RIS compared to none (dashed curves). We have also added two curves depicting the ergodic sum MI for the case of a single RIS when its phases are $1$-bit quantized to BPSK values (i.e., $e^{i \phi}=\pm 1$) and $2$-bit quantized QPSK values (i.e., $e^{i \phi}=(\pm 1\pm i)/\sqrt{2}$). It is shown that, in both cases, the optimization gain is significant, with the QPSK case being practically identical to the fully optimal case. Similar behavior had been obtained when two RISs are present, but we have not included the curves so as not to clutter the figure. Finally, the lower inset figure depicts the optimal phase distribution on the RIS for the continuous case when the angle spread is equal to $\sigma=5^{\circ}$.
• Figure 3: The ergodic sum-capacity performance in nats per channel use in the presence of $K=1$ RIS for ${M}=2$ (black), $3$ (blue), and $4$ (red) TXs. All incoming and outgoing signals were considered to have angle spread $\sigma=4^{\circ}$. The remaining simulation parameters take values from Table \ref{['table1']}. For the cases where ${M}>1$, the TXs have incoming azimuth angles that are equidistant with maximum angle equal to the one plotted on the $x$-axis of the plot. For concreteness, we have included an inset which shows the azimuth angles for the case of ${M}=3$. The solid lines correspond to the optimized $\hbox{\boldmath$\Phi$}_1$, while the dashed ones correspond to the semi-optimal approach described in Section \ref{['sec:Analytical_Solution']}. The dotted lines correspond to Monte Carlo simulations, while the lower curves depict the sum-MI without any optimization, i.e., for $\hbox{\boldmath$\Phi$}_1={\bf I}_{400}$. It is evident that, for increasing ${M}$, the relative gain is diminishing, Also, for increasing azimuth distance between TXs, it is demonstrated that the capacity gains are decreasing. This happens because the optimum $\hbox{\boldmath$\Phi$}_1$ has difficulties satisfying all TXs effectively.
• Figure 4: Ergodic capacity region for the case of ${M}=2$ TXs and $K=1$ RIS, with incoming signal azimuth angles from each TX set to $\phi_{{\rm in},1}=45^{\circ}$ and $\phi_{{\rm in},2}=-45^{\circ}$, considering the angle spread values $\sigma=4^{\circ}$ (black) and $\sigma=15^{\circ}$ (blue). All other parameter values are included in Table \ref{['table1']}. The upper two curves correspond to the capacity region boundary obtained by optimizing $\hbox{\boldmath$\Phi$}_1$ for $0\leq\mu_1\leq 1$ and $\mu_2=1-\mu_1$ in \ref{['eq:cap_mu12']}. It is clear that (especially for $\sigma=4^{\circ}$) the boundary is not a pentagon, in contrast to the lower two curves for which $\hbox{\boldmath$\Phi$}_1={\bf I}_{400}$. The fact that the lower angle spread has a wider capacity region can be expected from Fig. \ref{['fig:MI_AS_2UE']}. The solid curves correspond to the optimal capacity region boundaries calculated using the full optimization approach of Section \ref{['sec:Numerical_Solution']}. Finally, the rate pairs evaluated from Monte Carlo simulations using the $\hbox{\boldmath$\Phi$}_1$ obtained by numerically optimizing expression \ref{['eq:Lagrangian_mu_cap']} for $\mu_1=0:0.1:1$ and $\mu_2=1-\mu_1$ are illustrated.
• Figure 5: The cumulative distribution distribution of the sum MI in nats with $K=2$ RISs and with ${M}=1$ (black), $2$ (blue), $3$ (red), $4$ (green), and $5$ (magenta) TXs, respectively. We have used ${N_t}=8$ and ${N_r}=4$ antennas and ${N_s}=400$ elements per RIS, and $\rho=10$ dB. The solid curves correspond to a Gaussian approximation using the mean and variance obtained analytically in Proposition \ref{['prop:ergMI']}, and are compared with curves generated via Monte Carlo simulations. For simplicity, we have assumed all channels to be uncorrelated and have set $\hbox{\boldmath$\Phi$}_k={\bf I}_{400}$$\forall$$k$. It can be observed that, for increasing ${M}$, the gain in the median throughput as well as the variance (related to the average slope of the curves) is progressively decreasing. It is also evident that the agreement between the theoretical and numerically generated distributions is particularly good down to $10^{-3}$ outage probability.

Theorems & Definitions (9)

• Proposition 1: MIMO-MAC-RIS Ergodic Capacity Region
• proof
• Proposition 2: Asymptotic Mean and Variance of $I(\{{\bf Q}_m\}_{m=1}^M,\{\hbox{\boldmath$\Phi$}_k\}_{k=1}^K)$
• proof
• Remark 1: Capacity Achieving Covariance Matrices
• Remark 2: Decoupling Effects
• Remark 3: Asymptotic MIMO-MAC-RIS Ergodic Capacity Region
• Remark 4
• Remark 5