A New Approach to the Statistical Analysis of Non-Central Complex Gaussian Quadratic Forms with Applications

TL;DR

This paper introduces a new method for analyzing non-central complex Gaussian quadratic forms using an auxiliary random variable, simplifying the derivation of their distribution functions and enabling practical applications in communication systems.

Contribution

The paper presents a novel approach that generates an auxiliary RV converging to the CGQF, providing simple, elementary expressions for PDFs and CDFs applicable to both definite and indefinite forms.

Findings

Derived closed-form expressions for PDFs and CDFs of CGQFs.

Applied the method to Rician channels to compute outage probability.

Demonstrated improved tractability over previous methods.

Abstract

This paper proposes a novel approach to the statistical characterization of non-central complex Gaussian quadratic forms (CGQFs). Its key strategy is the generation of an auxiliary random variable (RV) that converges in distribution to the original CGQF. Since the mean squared error between both is given in a simple closed-form formulation, the auxiliary RV can be particularized to achieve the required accuracy. The technique is valid for both definite and indefinite CGQFs and yields simple expressions of the probability density function (PDF) and the cumulative distribution function (CDF) that involve only elementary functions. This overcomes a major limitation of previous approaches, in which the complexity of the resulting PDF and CDF prevents from using them for subsequent calculations. To illustrate this end, the proposed method is applied to maximal ratio combining systems over correlated Rician channels, for which the outage probability and the average bit error rate are derived.