Characterization of Beam-Squint and Beam-Split Effects in RIS-assisted Multi-Frequency Networks

RESUMO

Multi-frequency operation in beyond-5G and 6G systems renders the beam directions of large-aperture arrays frequency-dependent, giving rise to beam misalignment effects critical for reconfigurable intelligent surfaces (RISs). When an RIS phase profile is configured at one frequency but illuminated at another, the reflected field may exhibit main-lobe deviation and additional dominant lobes, degrading beamforming performance. In this paper, we develop a unified analytical framework to characterize these effects for uniform planar array-based RISs under both continuous and finite-resolution phase control. We formalize beam-squint as the deviation of the maximum-gain beam from the desired elevation-azimuth angle pair and beam-split as the emergence of additional maximum-gain beams under frequency mismatch and quantization. For continuous-phase RISs, we derive necessary and sufficient peak conditions, obtain closed-form elevation-azimuth peak families, and establish feasibility conditions as functions of the frequency ratio, incidence and configuration angles, and element spacing. Extending the analysis to b-bit phase quantization, we express the quantized beampattern as a superposition of harmonic array responses via Fourier expansion of the quantization-error phasor, leading to closed-form per-harmonic peak families, dominance ordering, and feasibility conditions. To capture peak displacement induced by harmonic superposition, we further develop a perturbation-based correction leveraging dominant-harmonic curvature and residual-harmonic gradients, improving peak-location accuracy without exhaustive two-dimensional searches. Numerical simulations validate the analysis, quantify peak deviations and beam-split probabilities, and demonstrate sum-rate gains by including analytically predicted squinted/split and harmonic candidates in multi-RIS, multi-user, multi-band networks.