Random Matrix Theory for Wireless Communications

This course gives an overview of analytic tools to 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, the free central and free Poisson limit theorem. The replica method developed in statistical physics for the purpose of analyzing spin glasses is reviewed from the view point 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.

Knowledge: The student shall become familiar with random matrix theory and its use to understand and design wireless communication systems with many users. Abilities: (S)he shall be able to calculate asymptotic eigenvalue statistics for various system settings by means of Stieltjes-transforms, R-transform, S-transform and the replica method. General competence: (S)he shall understand that wireless communication systems are a particular case of statistical mechanics and be enabled to use the vast literature in this area to his/her advantage when designing new systems.

Forelesning