On the Convergence of the Inverse of Toeplitz Matrices and its Applications
Many issues in signal processing involve the inverses of Toeplitz matrices. One widely used technique is to replace Toeplitz matrices with their associated circulant matrices, based on the well-known fact that Toeplitz matrices asymptotically converge to their associated circulant matrices in the weak sense. This often leads to considerable simplification. However, it is well known that such a weak convergence cannot be strengthened into strong convergence. It is this fact that severely limits the usefulness of the close relation between Toeplitz matrices and circulant matrices. Observing that communication receiver design often needs to seek optimality in regard to a data sequence transmitted within finite duration, we define the finite-term strong convergence regarding two families of matrices. We present a condition under which the inverses of a Toeplitz matrix converges in the strong sense to a circulant matrix for finite-term quadratic forms. This builds a critical link in the application of the convergence theorems for the inverses of Toeplitz matrices since the weak convergence generally finds its usefulness in issues associated with minimum mean squared error and the finite-term strong convergence is useful in issues associated with the maximum-likelihood or maximum a posteriori principles.