A Gamma-Mixture Solution for Definite Noncentral Complex Gaussian Quadratic Forms

Abstract

Finding exact solutions for the statistics of definite noncentral complex Gaussian quadratic forms (CGQFs) has long been a challenging problem. Among existing exact approaches, Raphaeli's formulation [1] remains the most computationally efficient. However, its analytical tractability is hindered by the dependence of its series coefficients on the evaluation variable, which confines their statistics to the probability density function (PDF) and cumulative distribution function (CDF). This paper overcomes this limitation by deriving novel Gamma-mixture representations for the PDF and CDF of positive- and negative-definite noncentral CGQFs. The resulting structure broadens Raphaeli's statistical framework beyond the PDF and CDF to a wider class of CGQF functionals and a substantial range of practical applications. In particular, expectations of integrable functions of definite noncentral CGQFs are expressed as weighted sums of the corresponding expectations over Gamma random variables, enabling the direct derivation of moments, transforms, and performance measures. Moreover, the proposed formulation reduces the single-point complexity from \(O\!\left(n^2K+nK^2\right)\) for Raphaeli's method to \(O\!\left(nK+K^2\right)\), where \(n\) denotes the quadratic-form dimension and \(K\) the truncation order. The proposed Gamma-mixture structure also enables closed-form approximations that, by construction, remain valid probability distributions for any truncation order. In addition, uniform PDF and CDF truncation-error bounds are derived, allowing prescribed-accuracy truncation. As a sample application, we derive the first reported exact average bit error rate expression for maximal-ratio combining receivers with an arbitrary number of branches under an arbitrary positive-definite Rician correlation structure.