Effective Electromagnetic Degrees of Freedom in Backscatter MIMO Systems

TL;DR

The paper tackles the challenge of defining and quantifying BS-EEMDOFs for backscatter MIMO systems where mutual coupling induces a nonlinear load-to-field mapping. It develops a Jacobian-based framework, deriving a closed-form Jacobian from multiport-network theory and showing that BS-EEMDOFs form a distributed quantity whose value depends on the backscatter loads and the coherent illumination. It demonstrates that BS-EEMDOFs reside in the end-to-end channel's column space, but their number generally differs from conventional EEMDOFs and can be tuned via the illumination, with practical validation through numerical simulations and experiments across diverse radio environments. The results highlight the potential to optimize RIS-based backscatter links by shaping the distribution of BS-EEMDOFs, offering insights for electromagnetic information theory and wave-domain processing.

Abstract

While the definition of the effective electromagnetic degrees of freedom (EEMDOFs) of a static linear multiple-input multiple-output (MIMO) system is well established, the counterpart for a backscatter MIMO (BS-MIMO) system is so far missing. A BS-MIMO system encodes the input information into the loads of backscatter elements. Due to mutual coupling, the mapping from load configuration to observed fields is fundamentally non-linear, which complicates the analysis of BS-EEMDOFs. We introduce a definition of BS-EEMDOFs based on the Jacobian of the observed fields with respect to the load configuration. We derive a closed-form expression from multiport network theory which demonstrates that the number of BS-EEMDOFs is fundamentally a distributed variable, whose distribution depends on the mutual coupling between the backscatter elements and the coherent illumination. The modes associated with BS-EEMDOFs lie in the column space of the end-to-end channel matrix from backscatter array ports to receiver ports, but the number of BS-EEMDOFs is generally different from the number of benchmark EEMDOFs associated with the same array being coherently fed rather than tunably terminated. The dependence on the coherent illumination yields optimized coherent illumination as a control knob for the number of BS-EEMDOFs. We present numerical and experimental results for the evaluation and optimization of the number of BS-EEMDOFs in different radio environments with reconfigurable intelligent surfaces.

Figures (4)

• Figure 1: Taxonomy of considered MIMO systems. (a) In a conventional MIMO system, the input information is encoded into the input wavefront $\mathbf{x}$, and the output information is extracted from the output wavefront $\mathbf{y}$. (b) In an RIS-enhanced conventional MIMO system, the configuration $\mathbf{r}$ of an RIS is optimized to shape the end-to-end MIMO transfer function. (c) In an RIS-based backscatter MIMO (BS-MIMO) system, the input information is encoded into the RIS configuration $\mathbf{r}$ (rather than into the input wavefront $\mathbf{x}$), while the output information is again extracted from the output wavefront $\mathbf{y}$. (d) In a benchmark MIMO system for the BS-MIMO system from (c) we remove the RIS loads and inject a coherent wavefront $\mathbf{a}$ carrying the input information via the RIS array, while extracting the output information again from the output wavefront $\mathbf{y}$. ML denotes matched load.
• Figure 2: Numerical setup from tapie2023systematic used in Sec. \ref{['sec_numerics']}. The setup comprises 7 antennas and 64 RIS elements.
• Figure 3: Probability density functions (PDFs) of the number of BS-EEMDOFs in the numerical setup from Fig. \ref{['Fig2']} for different constraints on the entries of $\mathbf{r}$ (different line colors), different choices of $\mathbf{x}$ (different line thicknesses; MAX vs. MIN distinction by inspection), and different choices of $\mathcal{T}$ (different panels). In addition, the number of conventional EEMDOFs for the same array in the same radio environment is shown as a vertical line. Each displayed PDF is evaluated based on 10,000 samples using (\ref{['eq:JBS_full']}).
• Figure 4: For each of the four considered radio environments, we display a photographic image in the top row and the PDFs of the number of BS-EEMDOFs for three distributions of $\mathbf{x}$ (RAND, MAX, MIN; distinction by inspection) for the PIN constraint on the entries of $\mathbf{r}$ in the bottom row. Each displayed PDF is evaluated based on 10,000 samples using a finite-difference evaluation of the Jacobian with respect to the RIS control variables, leveraging accurate predictions of the end-to-end MIMO channel matrix based on a proxy MNT model.