Block Toeplitz Matrices: Asymptotic Results and Applications

The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block Toeplitz matrices generated by the Fourier coefficients of a continuous matrix-valued function. This study is based on the concept of asymptotically equivalent sequences of non-square matrices. The asymptotic results on block Toeplitz matrices obtained are applied to vector asymptotically wide sense stationary processes. Therefore, this monograph is a generalization to block Toeplitz matrices of the Gray monograph entitled “Toeplitz and circulant matrices: A review”, which was published in the second volume of Foundations and Trends in Communications and Information Theory, and which is the simplest and most famous introduction to the asymptotic theory on Toeplitz matrices.

[1]  Pedro M. Crespo,et al.  Mass concentration in quasicommutators of Toeplitz matrices , 2007 .

[2]  Pedro M. Crespo,et al.  Asymptotically Equivalent Sequences of Matrices and Multivariate ARMA Processes , 2011, IEEE Transactions on Information Theory.

[3]  W. Rudin.  Principles of mathematical analysis , 1964 .

[4]  Jesús Gutiérrez-Gutiérrez,et al.  Geometric MMSE for one-sided and two-sided vector linear predictors: From the finite-length case to the infinite-length case , 2011, Signal Process..

[5]  Jean Pierre Delmas,et al.  Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel identification , 2001, IEEE Trans. Inf. Theory.

[6]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[7]  Robert M. Gray,et al.  Toeplitz and Circulant Matrices: A Review , 2005, Found. Trends Commun. Inf. Theory.

[8]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[9]  Robert M. Gray,et al.  Toeplitz And Circulant Matrices: A Review (Foundations and Trends(R) in Communications and Information Theory) , 2006 .

[10]  James L. Massey,et al.  Proper complex random processes with applications to information theory , 1993, IEEE Trans. Inf. Theory.

[11]  Paolo Tilli,et al.  A note on the spectral distribution of toeplitz matrices , 1998 .

[12]  Jesús Gutiérrez-Gutiérrez.  Powers of tridiagonal matrices with constant diagonals , 2008, Appl. Math. Comput..

[13]  A. Böttcher,et al.  Functions of banded Hermitian block Toeplitz matrices in signal processing , 2007 .

[14]  Pedro M. Crespo,et al.  Asymptotically Equivalent Sequences of Matrices and Hermitian Block Toeplitz Matrices With Continuous Symbols: Applications to MIMO Systems , 2008, IEEE Transactions on Information Theory.

[15]  T. Apostol.  Mathematical Analysis , 1957 .

[16]  William F. Trench,et al.  On the eigenvalue problem for Toeplitz band matrices , 1985 .

[17]  Louis L. Scharf,et al.  Detection of Spatially Correlated Gaussian Time Series , 2010, IEEE Transactions on Signal Processing.

[18]  Pedro M. Crespo,et al.  On the Elementwise Convergence of Continuous Functions of Hermitian Banded Toeplitz Matrices , 2007, IEEE Transactions on Information Theory.

[19]  Robert M. Gray,et al.  On the asymptotic eigenvalue distribution of Toeplitz matrices , 1972, IEEE Trans. Inf. Theory.

[20]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[21]  Sergio Verdú,et al.  The capacity region of the symbol-asynchronous Gaussian multiple-access channel , 1989, IEEE Trans. Inf. Theory.

[22]  Jesús Gutiérrez-Gutiérrez.  Positive integer powers of certain tridiagonal matrices , 2008, Appl. Math. Comput..

[23]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .