On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels

The linear vector-valued channel x |→ Π_n M_nx + z with z and M_n denoting additive white Gaussian noise and independent random matrices, respectively, is analyzed in the asymptotic regime as the dimensions of the matrices and vectors involved become large. The asymptotic eigenvalue distribution of the channel´s covariance matrix is given in terms of an implicit equation for its Stieltjes transform as well as an explicit expression for its moments. Additionally, almost all eigenvalues are shown to converge toward zero as the number of factors grows over all bounds. This effect cumulates the total energy in a vanishing number of dimensions. The channel model addressed generalizes the model introduced Muller (see IEEE Trans. Inform. Theory) for communication via large antenna arrays to N-fold scattering per propagation path. As a byproduct, the multiplicative free convolution is shown to extend to a certain class of asymptotically large non-Gaussian random covariance matrices