Hausdorff Moment Transforms and Their Applications to Wireless Networks

In the analysis of wireless networks, the standard signal-to-interference (SIR) distribution does not capture the performance at the individual link level. The meta distribution (MD) of the SIR resolves this problem by separating different sources of randomness, such as fading and point process(es). While it allows for a much sharper performance characterization, it can in most cases only be calculated based on the moments of the underlying conditional distribution, i.e., by solving a Hausdorff moment problem. Several methods to reconstruct MDs from the moments have been proposed but a rigorous analysis, comparison of their performance, and practical implementations are missing.

This dissertation focuses on three parts: the reconstruction of MDs, the fundamental problem behind it, i.e., the truncated Hausdorff moment problem, and the sensitivity issues related to the moments subject to perturbations. As for the truncated Hausdorff moment problem, we establish a method of comparison for the performance of the approximations. Three ways of producing random moment sequences are discussed and applied. Also, some of the approximations have been rewritten as linear transforms and detailed accuracy requirements are analyzed. Our finding shows that the performance of the approximations differ significantly in their convergence properties, accuracy, and numerical complexity, and that the decay type of the moment sequence strongly affects the accuracy requirement. As for the reconstruction of MDs, we introduces a tweaking mapping for adjusting approximations, presents terminology to categorize the quality of approximations, proposes the use of the Fourier-Legendre method, which has not previously been applied to MDs, and provides the achievable lower and upper bounds on the MD given the first $n$ moments. Further, to facilitate the use of MDs, we give comprehensive guidance on the selection of the best method to determine MDs, and we offer ready-to-use implementations of the proposed algorithms. This study fills an important gap in the literature by rigorously analyzing the MDs, comparing the performance of different methods, and offering user-friendly implementations for recovering MDs from moments. As for the sensitivity issues, we explore the reliability and robustness of these techniques. We analyze the sensitivity of commonly used MD reconstruction methods in the presence of perturbations to moments and provide valuable guidelines for the application of these methods. Furthermore, we quantify the impact of inaccurate moments on MD reconstructions, examining the validity of perturbed moment sequences and demonstrating the critical importance of moment accuracy. Our investigation demonstrates the necessity for precise moment computation. Succinctly put, moment quality is preferred over moment quantity.

在无线网络分析领域，传统的信噪比（SIR）分布无法充分反映个体链路层的性能。信噪比元分布（MD）通过分离不同随机源，例如衰落和点过程（们），解决了这一问题。尽管它允许对性能进行更为精确的描述，但在大多数情况下，MD的计算只能基于条件分布的矩，即通过解决Hausdorff矩问题来实现。尽管已提出多种从矩重建MD的方法，但对其性能的严格分析、比较和实际应用尚显不足。本论文聚焦于三个主要部分：MD的重建、其背后的基本问题，即截断Hausdorff矩问题，以及受扰动矩相关的敏感性问题。对于截断Hausdorff矩问题，我们建立了一种比较近似性能的方法，讨论并应用了三种产生随机矩序列的方式。此外，一些近似被重新表述为线性变换，并对其精确度要求进行了详细分析。我们的发现表明，近似性能在收敛性、精确度和数值复杂度方面存在显著差异，且矩序列的衰减类型对精确度要求有重要影响。至于MD的重建，我们引入了一种调整近似的微调映射，提出了用于分类近似质量的专业术语，提出了将傅里叶-勒让德方法应用于MD的建议，该方法之前尚未应用于此领域，并提供了基于前$n$个矩的MD可达到的上下界。此外，为了便于MD的应用，我们提供了关于选择最佳方法以确定MD的全面指导，并提供了现成的算法实现。本研究通过严格分析MD、比较不同方法的性能并提供从矩恢复MD的用户友好实现，填补了文献中的重要空白。至于敏感性问题，我们探讨了这些技术的可靠性和鲁棒性。我们分析了在矩受扰动的情况下常用MD重建方法的敏感性，并提供了这些方法应用的宝贵指南。此外，我们量化了不准确矩对MD重建的影响，检验了扰动矩序列的有效性，并证明了矩精确性的关键重要性。我们的研究证明了精确矩计算的必要性。简而言之，矩的质量优于矩的数量。