The parametrization of wireless channels by so-called "beyond-diagonal reconfigurable intelligent surfaces" (BD-RIS) is mathematically characterized by a matrix whose off-diagonal entries are partially or fully populated. Physically, this corresponds to tunable coupling mechanisms between the RIS elements that originate from the RIS control circuit. Here, we derive a physics-compliant diagonal representation for BD-RIS-parametrized channels. Recognizing that the RIS control circuit, irrespective of its detailed architecture, can always be represented as a multi-port network with auxiliary ports terminated by tunable individual loads, we physics-compliantly express the BD-RIS-parametrized channel as a multi-port chain cascade of i) radio environment, ii) static parts of the control circuit, and iii) individually tunable loads. Thus, the cascade of the former two systems is terminated by a system that is mathematically always characterized by a diagonal matrix. This physics-compliant diagonal representation implies that existing algorithms for channel estimation and optimization for conventional ("diagonal") RIS can be readily applied to BD-RIS scenarios. We demonstrate this in an experimentally grounded case study. Importantly, we highlight that, operationally, an ambiguous characterization of the cascade of radio environment and the static parts of the control circuit is required, but not the breakdown into the characteristics of its two constituent systems nor the lifting of the ambiguities. Nonetheless, we demonstrate how to derive or estimate the characteristics of the static parts of the control circuit for pedagogical purposes. The diagonal representation of BD-RIS-parametrized channels also enables their treatment with coupled-dipole-based models. We furthermore derive the assumptions under which the physics-compliant BD-RIS model simplifies to the widespread linear cascaded model.

翻译：所谓“超越对角可重构智能表面”（BD-RIS）对无线信道的参数化，在数学上由一个非对角元部分或完全填充的矩阵表征。物理上，这对应于由RIS控制电路产生的RIS单元间的可调耦合机制。本文推导了BD-RIS参数化信道的物理合规对角表示。认识到无论其具体架构如何，RIS控制电路总可表示为一个多端口网络，其辅助端口由可调独立负载端接，我们以物理合规的方式将BD-RIS参数化信道表达为以下三者的多端口链式级联：i) 无线环境，ii) 控制电路的静态部分，以及 iii) 独立可调负载。因此，前两个系统的级联由一个在数学上始终由对角矩阵表征的系统端接。这一物理合规对角表示意味着，现有针对传统（“对角”）RIS的信道估计与优化算法可直接应用于BD-RIS场景。我们通过一个基于实验的案例研究对此进行了验证。重要的是，我们指出，在操作上，仅需对无线环境与控制电路静态部分的级联进行模糊表征，而无需分解其两个组成系统的特性或消除模糊性。尽管如此，为教学目的，我们展示了如何推导或估计控制电路静态部分的特性。BD-RIS参数化信道的对角表示也使其能够用基于耦合偶极子的模型进行处理。此外，我们推导了物理合规的BD-RIS模型简化为广泛使用的线性级联模型所需满足的假设条件。