On the central Chi-square distribution with even degrees of freedom and correlated multivariate complex components

This paper presents the derivation new expressions for the statistics of a Chi-square distribution with $n$ degrees of freedom and where n is an even number. The complex Gaussian components of the chi-square distribution are modelled with a linear correlated model using different statistics (multi-rate) for each component. We focus on the specific expressions for the probability density function (PDF) and complementary cumulative density function (CCDF). Unlike previous approaches, we use a frequency domain interpretation that allows us to derive a closed form expression for the characteristic function (CF) as an inverse polynomial equation. Using the roots of this polynomial equation, it is possible to decompose the CF as a partial fraction expansion (PFE). This allows us to obtain a simple expression for both the PDF and CCDF by simply using the inverse Fourier transform of PFE decomposition of the CF. The statistics derived here have a much lower complexity than the expressions obtained from conventional non-frequency domain methods at the expense of the complexity of the polynomial root solution scheme. In scenarios where the average statistics of the components do not change over some periods of time, the proposed expressions provide the lowest possible complexity, as the polynomial rooting process needs to be conducted only once and potentially offline.