A Power-Weighted Noncentral Complex Gaussian Distribution

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.

Figures (7)

• Figure 1: Effect of the shape parameter $\alpha$ on the proposed power-weighted noncentral complex Gaussian distribution. From left to right in the top row: $\alpha=0.5, \, 0.75, \, 0.9, \, 1$; from left to right in the bottom row: $\alpha=2, \, 3, \, 4, \, 5$. The other parameters are fixed as $\sigma^2=1$ and $\mu=\frac{1}{2}{\rm e}^{\rm i \frac{\pi}{4}}$. The horizontal and vertical axes represent the real and imaginary axes, respectively, and lower and higher brightness indicate lower and higher probability density.
• Figure 2: Effect of the amplitude of the mean parameter $|\mu|$ on the probability density of the proposed distribution. Top row: $\alpha=0.5$ with $|\mu|=1, \, 2, \, 2.5, \, 3$ (from left to right). Bottom row: $\alpha=5$ with $|\mu|=0.01, \, 0.1, \, 0.5, \, 1$ (from left to right). The other parameters are fixed as $\sigma^2=1$ and $\angle \mu=\frac{\pi}{4}$. The horizontal and vertical axes represent the real and imaginary axes, respectively, and lower and higher brightness indicate lower and higher probability density.
• Figure 3: Examples of the Poisson-type distribution $p(n;\lambda,\alpha)$ distorted by the parameter $\alpha$. The blue curve corresponds to the standard Poisson distribution ($\alpha=1$), while the red and yellow curves represent the cases $\alpha<1$ and $\alpha>1$, respectively.
• Figure 4: Proposed power distribution (solid lines) for different values of $\alpha$. The other parameters are fixed at $\beta=1$ and $\lambda=2$. The dashed lines represent the noncentral gamma distribution with the same parameters.
• Figure 5: Proposed power distribution (solid lines) for different values of $\beta$. The other parameters are fixed at $\alpha=2$ and $\lambda=1$. The dashed lines represent the noncentral gamma distribution with the same parameters.

Theorems & Definitions (2)

• proof
• proof