Statistical analysis of the capacity of MIMO frequency selective Rayleigh fading channels with arbitrary number of inputs and outputs

The classic problem of maximizing the information rate over parallel Gaussian independent sub-channels with a limit on the total power leads to the elegant closed form water-filling solution. In the case of a multi-input multi-output (MIMO) frequency selective channel the solution requires the derivation of the eigenvalue decomposition of the MIMO frequency response which, for every frequency bin, have a generalized Wishart distribution. This paper shows the methodology used to derive the statistics of eigenvalues and eigenvectors and applies this methodology to the derivation of the average channel capacity and of its characteristic function. The rules of exterior differential calculus are used to compute the Jacobian of matrix decompositions and perform integration over matrix groups. Simple expressions are derived for the case of uncorrelated Rayleigh fading and an arbitrary finite number of transmit and receive antennas.