Asymptotic Eigenvalue Density for the Quotient Ensemble of Wishart Matrices

摘要

The matrix model involving the quotient of two Wishart matrices appears naturally in the investigation of multiple-input multiple-output (MIMO) multiple-access and relay channels. The available exact result for the eigenvalue density for this matrix model involves the determinant of a matrix whose dimension is related to the number of antennas in the channel. Consequently, the exact result becomes impractical while dealing with the case of large number of antennas. In this letter, we derive a novel expression for the asymptotic eigenvalue density of the quotient matrix model. This result is analogous to the Marčenko–Pastur density and can be conveniently applied to deal with the case of large matrix dimensions. Remarkably, satisfactorily results are obtained even for small matrix dimensions. As an application of our asymptotic result, we obtain an upper bound for the ergodic capacity in a full-duplex multi-hop decode-and-forward MIMO relay network.