On slow-fading non-separable correlation MIMO systems

In a frequency selective slow-fading channel in a MIMO system, the channel matrix is of the form of a block matrix. We propose a method to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. We will also calculate the asymptotic eigenvalue distribution of $HH^*$, where the entries of $H$ are jointly Gaussian, with a correlation of the form $E[h_{pj}\bar h_{qk}]= \sum_{s=1}^t \Psi^{(s)}_{jk}\hat\Psi^{(s)}_{pq}$ (where $t$ is fixed and does not increase with the size of the matrix). We will use an operator-valued free probability approach to achieve this goal. Using this method, we derive a system of equations, which can be solved numerically to compute the desired eigenvalue distribution.Comment: 24 pages and 3 figure

Applications of Large Random Matrices in Communications Engineering

This work gives an overview of analytic tools for the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools from probability theory, operator algebra, and statistical physics are reviewed. The survey of analytical tools is complemented by examples of applications in communications engineering. Asymptotic eigenvalue distributions of many classes of random matrices are given. The treatment includes the problem of moments and the introduction of the Stieltjes transform. Free probability theory, which evolved from non-commutative operator algebras, is explained from a probabilistic point of view in order to better fit the engineering community. For that purpose freeness is defined without reference to non-commutative algebras. The treatment includes additive and multiplicative free convolution, the R-transform, the S-transform, and the free central limit theorem. The replica method developed in statistical physics for the purpose of analyzing spin glasses is reviewed from the viewpoint of its applications in communications engineering. Correspondences between free energy and mutual information as well as energy functions and detector metrics are established. These analytic tools are applied to the design and the analysis of linear multiuser detectors, the modelling of scattering in communication channels with dual antennas arrays, and the analysis of optimal detection for communication via code-division multiple-access and/or dual antenna array channels.Comment: arXiv admin note: text overlap with arXiv:0706.1169 by other author