An Upper Bound to the Marginal PDF of the Ordered Eigenvalues of Wishart Matrices and Its Application to MIMO Diversity Analysis

Diversity analysis of a number of multiple-input multiple-output (MIMO) applications requires the calculation of the expectation of a function whose variables are the ordered multiple eigenvalues of the Wishart matrices. To solve this, we need the marginal pdf of an arbitrary subset of the ordered eigenvalues. The marginal pdf shown in the literature is useful in numerical analysis, but not beneficial to diversity analysis. In this paper, we derive an upper bound to the marginal pdf of the eigenvalues. The derivation is based on the multiple integration of the well-known joint pdf, which is very complicated due to the exponential factors of the joint pdf. We suggest an alternative function that provides simpler calculation of the multiple integration. As a result, the marginal pdf is shown to consist of a multivariate polynomial with a given degree. By applying the marginal pdf to the calculation of the expectation, the diversity order for a number of MIMO systems can be calculated. Simulation results that support the analysis are presented.