  "Toeplitzification" or "redundancy (spatial) averaging", the well-known
routine for deriving the Toeplitz covariance matrix estimate from the standard
sample covariance matrix, recently regained new attention due to the important
Random Matrix Theory (RMT) findings. The asymptotic consistency in the spectral
norm was proven for the Kolmogorov's asymptotics when the matrix dimension N
and independent identically distributed (i.i.d.) sample volume T both tended to
infinity (N->inf, T->inf, T/N->c > 0). These novel RMT results encouraged us to
reassess the well-known drawback of the redundancy averaging methodology, which
is the generation of the negative minimal eigenvalues for covariance matrices
with big eigenvalues spread, typical for most covariance matrices of interest.
We demonstrate that for this type of Toeplitz covariance matrices, convergence
in the spectral norm does not prevent the generation of negative eigenvalues,
even for the sample volume T that significantly exceeds the covariance matrix
dimension (T >> N). We demonstrate that the ad-hoc attempts to remove the
negative eigenvalues by the proper diagonal loading result in solutions with
the very low likelihood. We demonstrate that attempts to exploit Newton's type
iterative algorithms, designed to produce a Hermitian Toeplitz matrix with the
given eigenvalues lead to the very poor likelihood of the very slowly
converging solution to the desired eigenvalues. Finally, we demonstrate that
the proposed algorithm for restoration of a positive definite (p.d.) Hermitian
Toeplitz matrix with the specified Maximum Entropy spectrum, allows for the
transformation of the (unstructured) Hermitian maximum likelihood (ML) sample
matrix estimate in a p.d. Toeplitz matrix with sufficiently high likelihood.
