Effective Electromagnetic Degrees of Freedom in Backscatter MIMO Systems

While the quantification of the multiplexing gain in a static linear multiple-input multiple-output (MIMO) system in terms of its effective electromagnetic degrees of freedom (EEMDOFs) 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 random variable, whose distribution depends on the mutual coupling between the backscatter elements, the considered distribution of load configurations, 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 large reconfigurable intelligent surfaces, including an example in which the mean of the number of BS-EEMDOFs (with optimized coherent illumination) exceeds the number of conventional EEMDOFs. Our work provides an electromagnetically consistent way of quantifying and optimizing the multiplexing gain in BS-MIMO systems.