Jesús Gutiérrez-Gutiérrez and Pedro M. Crespo (2012), "Block Toeplitz Matrices: Asymptotic Results and Applications", Foundations and Trends® in Communications and Information Theory: Vol. 8: No. 3, pp 179-257. http://dx.doi.org/10.1561/0100000066

© 2012 J. Gutiérrez-Gutiérrez and P. M. Crespo

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**In this article:**

1. Introduction

2. Mathematical Preliminaries

3. Asymptotically Equivalent Sequences of Matrices

4. Block Toeplitz Matrices

5. Block Circulant Matrices

6. Asymptotic Behaviour of Block Toeplitz Matrices

7. Applications to Vector Random Processes

A. Asymptotically 2-Equivalent Sequences of Matrices

B. The Wiener Class

Acknowledgments

References

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.

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1. Introduction

2. Mathematical preliminaries

3. Asymptotically equivalent sequences of matrices

4. Block Toeplitz matrices

5. Block circulant matrices

6. Asymptotic behaviour of block Toeplitz matrices

7. Applications to vector random processes

A. Asymptotically 2-equivalent sequences of matrices

B. The Wiener class

Acknowledgments

References

*Block Toeplitz Matrices: Asymptotic Results and Applications* provides a tutorial introduction and in-depth exposé of
this important mathematical technique used in Communications, Information Theory, and Signal Processing. The matrix representations of
discrete-time causal finite impulse response (FIR), multiple-input multiple-output (MIMO) filters, and correlation matrices of vector wide
sense stationary (WSS) processes are all block Toeplitz.

The monograph deals with the asymptotic behaviour of eigenvalues, products and inverses of block Toeplitz matrices, with the concept of asymptotically equivalent sequences of matrices being the key concept. However, since the blocks of block Toeplitz matrices are, in general, not square, the definition of asymptotically equivalent sequences of matrices are extended to sequences of non-square matrices. Furthermore, the monograph covers any function of block Toeplitz matrices and considers block Toeplitz matrices generated by the Fourier coefficients of any continuous matrix-valued function.

*Block Toeplitz Matrices* is an advanced level tutorial on a mathematical technique with applications in many engineering
disciplines.

**Companion**

*
Toeplitz and Circulant Matrices: A Review
*, Foundations and Trends® in Communications and Information Theory, Volume 2, Issue 3 10.1561/0100000006