Article

# Performance Analysis of a Two–Tile Reconfigurable Intelligent Surface Assisted 2×2 MIMO System

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## Abstract

We consider a two–tile reconfigurable intelligent surface (RIS) assisted wireless network with a two-antenna transmitter and receiver over Rayleigh fading. We show that the average received signal-to-noise-ratio (SNR) optimal combining and transmission vectors are given by the left and right singular spaces of the RIS-receiver and transmit-RIS channel matrices, respectively. Moreover, the optimal phases at the two tiles of the RIS are determined by the phases of the elements of the latter spaces. To further study the effect of phase compensation, we statistically characterize the average SNR of all possible combinations of transmission and combining directions pertaining to the latter singular spaces by deriving novel expressions for the outage probability and throughput of each of those modes. Furthermore, for comparison, we derive the corresponding expressions in the absence of RIS. Our results show an approximate SNR improvement of 2 dB due to the phase compensation at the RIS.

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... Analogous to [22], the outage probability of RIS-aided multiple-input single-output (MISO) systems was approximated with a Gamma distribution in [23]. Furthermore, the outage performance of RIS assisted 2×2 MIMO system was scrutinized by considering perfect CSI at the RIS in [6]. Nonetheless, the above-mentioned literature assumed perfect CSI at the RIS. ...
... By combining (5), (6) and the Mellin transform φ(x) in (20) or (21), the outage probability can thus be expressed as ...
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We thoroughly investigate the outage performance of reconfigurable intelligent surface (RIS) aided multi-input multi-output (MIMO) communications by exploiting only statistical channel state information (CSI). Kornecker channel model is adopted to characterize the impact of spatial correlations among MIMO antennas and reconfigurable reflectors. Mellin transform and random matrix theory are then utilized to derive the exact outage probability in closed-form. With the result, we further conduct the asymptotic outage analysis to obtain insightful findings. In particular, the asymptotic analysis reveals that the number of reflecting elements at the RIS should not be smaller than the total number of MIMO transmit and receive antennas to get rid of the rank deficiency of the cascaded MIMO channels. Moreover, we prove that the transmission rate is a monotonically increasing and convex function of the asymptotic outage probability. The numerical outcomes not only corroborate our analytical results, but also show the negative impact of the spatial correlation and the benefit of increasing the number of Zheng Shi and Guanghua Yang are with the). 2 reconfigurable reflectors. Finally, we apply the asymptotic results to optimally devise the phase shifts with a low computational complexity. Index Terms Diversity order, Kronecker model, MIMO, outage probability, reconfigurable intelligent surface (RIS), spatial correlation.
... Doing so, the corresponding reflected signals can be added either constructively or destructively with other signals to enhance the signal-to-noise ratio (SNR) and/or to suppress the co-channel interference at the receiver. Most of the current art is focused on the incident signals' phase shift optimization and/or knowledge acquisition of the channel state information (CSI) at RIS [5]. Nevertheless, such a condition reflects on a considerably high computational complexity and power consumption. ...
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A multiuser multiple-input multiple-output wireless communication system is analytically studied, which operates with the aid of a reconfigurable intelligent surface (RIS). The intermediate RIS is equipped with multiple elements and operates via random phase rotations to simultaneously serve multiple users. Independent Rayleigh fading conditions are assumed among the included channels. The system performance is analytically studied when the linear yet efficient zero-forcing detection is implemented at the receiver. In particular, the outage performance is derived in closed-form expression for different system configuration setups with regards to the available channel state information at the receiver. Further, a joint coherent/noncoherent linear detection is analytically presented. Finally, some new engineering insights are provided, such as how the channel state information and/or the volume of antenna/RIS arrays impact on the overall system performance as well as the arising efficiency on the performance/complexity tradeoff by utilizing the joint coherent/noncoherent scheme.
... Most previous work on performance analysis considers different wireless networks with single and fixed RIS setup, e.g., [6]- [12] and references therein. For a given set of locations of multiple feedback location information from a part of superior RISs and then selects a RIS among these RISs feeding back. ...
Preprint
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The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the asymptotic expansion of $_0F_0$ in Section 12. Section 13 has some formulae for moments.
Physics-based modeling and scalable optimization of large intelligent reflecting surfaces
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