An Upper Bound to the Marginal PDF of the Ordered Eigenvalues of Wishart Matrices

Abstract

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 a Wishart matrix. In order to carry out this calculation, we need the marginal pdf of an arbitrary subset of the ordered eigenvalues. In this letter, 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 be bounded by a multivariate polynomial with a given degree. After a standard bounding procedure in a Pairwise Error Probability (PEP) analysis, by applying the marginal pdf to the calculation of the expectation, the diversity order for a number of MIMO systems can be obtained in a simple manner. Simulation results that support the analysis are presented.