Asymptotically Equivalent Sequences of Matrices and Hermitian Block Toeplitz Matrices With Continuous Symbols: Applications to MIMO Systems

For the engineering community, Gray´s tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego theory on large Toeplitz matrices. In this paper, the most important results of the cited monograph are generalized to block Toeplitz (BT) matrices by maintaining the same mathematical tools used by Gray, that is, by using asymptotically equivalent sequences of matrices. As applications of these results, the geometric minimum mean square error (MMSE) for both an infinite-length multivariate linear predictor and an infinite-length decision feedback equalizer (DFE) for multiple-input-multiple-output (MIMO) channels, are obtained as a limit of the corresponding finite-length cases. Similarly, a short derivation of the well-known capacity of a time-invariant MIMO Gaussian channel with intersymbol interference (ISI) and fixed input covariance matrix is also presented.