Generalized Proper Complex Gaussian Ratio Distribution and Its Application to Statistical Inference for Frequency Response Functions

Abstract

The frequency response function governs many important processes. Defined as the quotients of the fast Fourier transform coefficients, frequency response functions can be modeled as ratio random variables in the complex domain. The circularly symmetric complex normal ratio distribution proposed previously is restricted to quantifying the uncertainties for the quotients of complex Gaussian random variables with zero mean. Such limitation motivates research to enlarge the group of probability distributions, allowing a wider scope of applicability. This study provides a theoretical proof for the closed-form of a generalized proper complex Gaussian ratio distribution using the principle of probability density transformation in tandem with an advanced integral technique. The equivalence between the distribution properties of complex ratio random variables and their counterparts in the real-valued domain is also proven. The generalized proper complex Gaussian ratio distribution is then used to infer the statistics of frequency response functions. Stochastic simulation and experimental study are used to demonstrate the goodness and efficiency of the proposed probabilistic model.