Abstract
The complex Gaussian distribution has been widely used as a fundamental spectral and noise model in signal processing and communication. However, its Gaussian structure often limits its ability to represent the diverse amplitude characteristics observed in individual source signals. On the other hand, many existing non-Gaussian amplitude distributions derived from hyperspherical models achieve good empirical fit due to their power-law structures, while they do not explicitly account for the complex-plane geometry inherent in complex-valued observations. In this paper, we propose a new probabilistic model for complex-valued random variables, which can be interpreted as a power-weighted noncentral complex Gaussian distribution. Unlike conventional hyperspherical amplitude models, the proposed model is formulated directly on the complex plane and preserves the geometric structure of complex-valued observations while retaining a higher-dimensional interpretation. The model introduces a nonlinear phase diffusion through a single shape parameter, enabling continuous control of the distributional geometry from arc-shaped diffusion along the phase direction to concentration of probability mass toward the origin. We formulate the proposed distribution and analyze the statistical properties of the induced amplitude distribution. The derived amplitude and power distributions provide a unified framework encompassing several widely used distributions in signal modeling, including the Rice, Nakagami, and gamma distributions. Experimental results on speech power spectra demonstrate that the proposed model consistently outperforms conventional distributions in terms of log-likelihood.
Abstract (translated)
复高斯分布作为基础频谱与噪声模型在信号处理与通信领域被广泛使用。然而，其高斯结构常限制了对单个源信号中多样化幅度特性的表征能力。另一方面，许多基于超球面模型推导出的非高斯幅度分布虽因幂律结构而具有良好经验拟合效果，却未能明确考虑复值观测数据所固有的复平面几何特性。本文提出一种针对复值随机变量的新概率模型，可视为幂加权非中心复高斯分布。与传统超球面幅度模型不同，该模型直接在复平面上构建，在保留复值观测几何结构的同时维持高维解释性。模型通过单一形状参数引入非线性相位扩散，实现了从相位方向弧状扩散到概率质量向原点聚集的连续几何控制。本文建立了该分布的数学形式并分析了导出幅度分布的统计特性。推导出的幅度与功率分布为信号建模中广泛使用的莱斯分布、中岛分布及伽马分布提供了统一框架。在语音功率谱上的实验结果表明，该模型在对数似然指标上持续优于传统分布。
URL
https://arxiv.org/abs/2603.26344
