Negative Log-Shifted Gamma: A Generalized Pointing Error Model for Directional Wideband Systems

Abstract

Pointing error is critical in highly directional communication systems, where even slight misalignment leads to severe received power reduction. Existing pointing error models fail to cover diverse practical scenarios due to restrictive distributional and design limitations. This paper proposes a novel pointing error model incorporating jitters, AoD/AoA estimation errors, and comprehensive design impacts, including phase shifter quantization, antenna patterns, array design, beamforming architectures, steering capabilities, and bandwidth. To accomplish this, antenna patterns are modeled for both high- and low-gain configurations. Then, the array factor's main lobe is approximated as a Gaussian beam for subarrays and superarrays, accommodating analog and digital beamforming stages in wideband systems under beam squint presence or compensation. Jitters and estimation errors are incorporated via 3D rotations, while phase shift quantization is captured through the phase excitations of the arrays. It is shown that the resulting pointing error follows a novel distribution: the negative log-shifted Gamma (NLSG). Fully characterized by only shape, scale, and location parameters, the NLSG features closed-form statistical expressions. This formulation ensures tractable mathematical manipulation, making it a practical, scalable tool for system design and analysis. Validation via normalized moment error, Jensen-Shannon divergence, and Kolmogorov-Smirnov statistics confirms exceptional accuracy, yielding values primarily below 1%, 0.02, and 0.05, respectively. Furthermore, capacity analysis has demonstrated that specific design choices are paramount for achieving optimal capacity. Finally, important guidelines for designing highly directional wideband communication systems are provided to mitigate the detrimental effects of pointing errors, thereby paving the way for robust next-generation networks.