Channel Estimation with Reconfigurable Intelligent Surfaces -- A General Framework (2110.00553v1)

Abstract: Optimally extracting the advantages available from reconfigurable intelligent surfaces (RISs) in wireless communications systems requires estimation of the channels to and from the RIS. The process of determining these channels is complicated by the fact that the RIS is typically composed of passive elements without any data processing capabilities, and thus the channels must be estimated indirectly by a non-colocated device, typically a controlling base station. In this article, we examine channel estimation for RIS-based systems from a fundamental viewpoint. We study various possible channel models and the identifiability of the models as a function of the available pilot data and behavior of the RIS during training. In particular, we consider situations with and without line-of-sight propagation, single- and multiple-antenna configurations for the users and base station, correlated and sparse channel models, single-carrier and wideband OFDM scenarios, availability of direct links between the users and base station, exploitation of prior information, as well as a number of other special cases. We further conduct numerical comparisons of achievable performance for various channel models using the relevant Cramer-Rao bounds.

• The paper develops a comprehensive theoretical framework and derives the Cramér-Rao Bound (CRB) to analyze the precision of channel estimation in RIS-assisted communication systems.
• It employs parametric channel representations and a Jacobian transformation to derive the CRB for composite channel parameters, addressing estimation ambiguities with imposed constraints.
• The derived CRB provides a foundational measure for evaluating estimator performance and guiding the design of RIS-based communication strategies by quantifying achievable estimation precision.

Analysis of Channel Estimation via CRB Derivation for RIS-Assisted Communication Systems

The paper presents a comprehensive theoretical framework for analyzing the channel estimation in reconfigurable intelligent surface (RIS)-assisted communication systems. The paper focuses on deriving the Cramér-Rao Bound (CRB) as a measure of the precision obtainable in the estimation of the composite channel matrix, leveraging the parametric representations of the channel components and their respective steering vectors.

System Model

The paper assumes a multi-terminal communication scenario where signals transmitted from user equipment (UE) are received at a base station (BS) with the aid of a RIS. Specifically, the system model comprises an $M$-element BS, an $N$-element RIS characterized by a rectangular array structure, and $L$ UE terminals each equipped with $K$ antennas. These elements facilitate the formation of distinct channel paths, namely UE-to-BS $\Fbf_\ell$, UE-to-RIS $\Gbf_\ell$, and RIS-to-BS $\Hbf$, with corresponding complex path gains and angle-of-arrival (AoA) and angle-of-departure (AoD) frequencies.

Parametric Channel Representation

For channel estimation, a parametric model is assumed for the channel components based on uniform linear arrays (ULA) and uniform planar arrays (UPA). The authors meticulously describe the steering vector formulations for the BS, UE, and RIS elements, integrating spatial frequency parameters determined by inter-element spacing and signal angles ($\theta_b$ and $\theta_u$). The reflection coefficients at the RIS are adjustable, synchronizing with UE uplink transmissions, lending an extra degree of freedom in channel manipulation.

CRB Derivation and Ambiguity Resolution

The derivation of CRB involves a meticulous transformation of the composite channel parameters, denoted as $\etabf$, which encapsulates spatial frequencies and path gains for various channel components. The composite channel $\Cbf_\ell(\etabf_\ell)$ integrates these parameters to account for RIS phase modulation effects. To resolve ambiguities inherent in parameter estimation, certain constraints are imposed, such as fixing specific path gains and AoD/AoA frequencies.

Computational Implementation

The paper outlines a MATLAB program design, specifying inputs required for CRB computation. This encompasses channel configuration parameters ($\etabf, M, N, K, L$), spatial frequencies, noise power, transmit powers $P_\ell$, training sample count $T$, and phase shift matrices $\Psibf$. The emphasis on utilizing Jacobians for transforming CRB from $\etabf$ to the channel matrix underscores the intricate mathematical nature of the problem.

Implications and Future Directions

The theoretical insights and numerical results from this research provide significant guidance for the development of RIS-based communication strategies. By determining the achievable bounds in channel estimation, this paper offers a tangible metric for evaluating estimator performance and system design efficacy. The derived CRB aids in quantifying the expected precision of estimated channel parameters, thus playing a crucial role in optimizing resource allocation and improving data throughput.

Future extensions could delve into advanced RIS configurations, optimizing the RIS reflection coefficients dynamically to enhance channel capacity. Additionally, real-time channel estimation algorithms could exploit the derived CRB to facilitate adaptive modulation in rapidly varying communication scenarios, paving the way for more efficient utilization of RIS technology in heterogeneous network architectures.

Conclusion

This rigorous examination of the channel estimation process in RIS-assisted communications provides a foundational measure—the CRB—that effectively represents the estimation accuracy limits. The integration of advanced mathematical constructs with systemic components marks a meaningful contribution to the strategic optimization of next-generation wireless networks. The paper stands as a vital resource for researchers aiming to refine communication systems leveraging RISs and enhance channel estimation methodologies both theoretically and practically.