Title :
Asymptotically Equivalent Sequences of Matrices and Hermitian Block Toeplitz Matrices With Continuous Symbols: Applications to MIMO Systems
Author :
Gutiérrez-Gutiérrez, Jesús ; Crespo, Pedro M.
Author_Institution :
Electron. & Commun. Dept., R&D Center CEIT, San Sebastian
Abstract :
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.
Keywords :
Gaussian channels; Hermitian matrices; MIMO communication; Toeplitz matrices; covariance matrices; decision feedback equalisers; graph theory; intersymbol interference; least mean squares methods; Gray tutorial monograph; Hermitian block Toeplitz matrix; MIMO system; Szego theory; asymptotically equivalent sequence; circulant matrix; fixed input covariance matrix; infinite-length decision feedback equalizer; infinite-length multivariate linear predictor; intersymbol interference; minimum mean square error method; multiple-input-multiple-output channel; time-invariant MIMO Gaussian channel; Algebra; Covariance matrix; Decision feedback equalizers; Eigenvalues and eigenfunctions; Gaussian channels; H infinity control; Intersymbol interference; MIMO; Maintenance engineering; Mean square error methods; (MMSE) linear predictor; Block circulant matrices; Frobenius norm; Hermitian matrices; block Toeplitz (BT) matrices; channel capacity; functions of matrices; multiple-input–multiple-output (MIMO) minimum mean square error decision feedback equalizer (MMSE-DFE); multivariate stationary processes; spectral norm;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2008.2006401