The distribution of the l-th largest eigenvalue of central Wishart matrices and its application to the performance analysis of MIMO systems

The research of closed form expressions for the pdf of the $ith{ell}$ ordered eigenvalue of a Wishart matrix has received a great attention in the past years owing to its applications in the performance analysis of multiple input multiple output (MIMO) systems in fading environments. Although several closed form expressions for this pdf were obtained in the past years, to the authors' knowledge, no one was very friendly for further analysis. We propose a methodology to obtain the pdf for the $ith{ell}$ largest eigenvalue whose expression is given as a sum of terms $x^{beta} e^{-x delta}$. This expression is easily usable to obtain closed form results for the performance of many MIMO systems, such as, for instance, MIMO beamforming, and MIMO with Singular Value Decomposition (SVD). The methodology is valid for both uncorrelated and correlated central Wishart, allowing the investigation of MIMO systems with uncorrelated and correlated Rayleigh fading.