Degrees of Freedom and Beamforming for Large Intelligent Surfaces

Spatial degrees of freedom (DoF), sampling, and beamforming are fundamental to multi-user large intelligent surfaces (LISs), where electromagnetic fields must be shaped, resolved, and focused at multiple near-field locations. This work estimates the number of DoF using closed-form expressions derived from the mutual shadow area for representative LIS configurations. The resulting DoF predictions are validated through numerical singular-value spectra, whose spectral knee points closely match the theoretical estimates. For line-source configurations, an analytic sampling scheme is developed by partitioning the source or observation line into unit-DoF intervals, enabling the selection of spatial samples. Beamforming results using maximum-ratio transmission and zero-forcing demonstrate that approximately the number of DoF independent beams can be formed. Attempting to exceed this limit results in increased interference and degraded performance. For surface-based LIS configurations, sampling points are instead determined numerically using the discrete empirical interpolation method. The corresponding beamforming results further confirm that the target region can support approximately as many independent beams as predicted by the DoF analysis. Finally, a polarization-aware study reveals that the electric-field components contribute unequally to the DoF and that the total-field DoF is twice that of a single polarization component.

Properties of Near Field Focusing for Three-Dimensional Large Intelligent Surface

This work investigates near-field focusing using a three-dimensional (3D) large intelligent surface (LIS) across frequencies and polarizations. Specifically, the LIS elements are distributed in 3D space within a long corridor, rather than being confined to a single planar aperture, and the focal point is located at a prescribed position in the radiating near field. By formulating optimization problems under both local and global power constraints, we obtain the corresponding optima. For continuous apertures, the optimal current magnitude distribution matches time-reversal (TR) solution under the global constraint and conjugate-phase (CP) solution when the local constraint dominates. When both constraints are active, the solution assigns larger excitation magnitudes to elements closer to the illumination field. This behavior remains invariant with respect to frequency and polarization for a fixed-size LIS. These findings are consistent to the more practical case of using discretized apertures in the form of Hertzian dipole arrays, studied using both analytical results and full-wave simulation. In addition, with the CP method, specific polarizations lead to identical transverse and longitudinal resolution, in contrast, under the TR method, these quantities can differ across polarizations.