Spectra of large block matrices

Abstract

In a frequency selective slow-fading channel in a MIMO system, the channel matrix is of the form of a block matrix. This paper proposes a method to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. While it considers random matrices, it takes an operator-valued free probability approach to achieve this goal. Using this method, one derives a system of equations, which can be solved numerically to compute the desired eigenvalue distribution. The paper initially tackles the problem for square block matrices, then extends the solution to rectangular block matrices. Finally, it deals with Wishart type block matrices. For two special cases, the results of our approach are compared with results from simulations. The first scenario investigates the limit eigenvalue distribution of block Toeplitz matrices. The second scenario deals with the distribution of Wishart type block matrices for a frequency selective slow-fading channel in a MIMO system for two different cases of $n_R=n_T$ and $n_R=2n_T$. Using this method, one may calculate the capacity and the Signal-to-Interference-and-Noise Ratio in large MIMO systems.