  The inverse Toeplitz eigenvalue problem (ToIEP) concerns finding a vector
that specifies the real-valued symmetric Toeplitz matrix with the prescribed
set of eigenvalues. Since phase "calibration" errors in uniform linear antenna
arrays (ULAs) do not change the covariance matrix eigenvalues and the moduli of
the covariance matrix elements, we formulate a number of the new ToIEP problems
of the Hermitian Toeplitz matrix reconstruction, given the moduli of the matrix
elements and the matrix eigenvalues. We demonstrate that for the real-valued
case, only two solutions to this problem exist, with the "non-physical" one
that in most practical cases could be easily disregarded. The computational
algorithm for the real-valued case is quite simple. For the complex-valued
case, we demonstrate that the family of solutions is broader and includes
solutions inappropriate for calibration. For this reason, we modified this
ToIEP problem to match the covariance matrix of the uncalibrated ULA. We
investigate the statistical convergence of the ad-hoc algorithm with the sample
matrices instead of the true ones. The proposed ad-hoc algorithms require the
so-called "strong" or "argumental" convergence, which means a large enough
required sample volume that reduces the errors in the estimated covariance
matrix elements. Along with the ULA arrays, we also considered the fully
augmentable minimum redundancy arrays that generate the same (full) set of
covariance lags as the uniform linear arrays, and we specified the conditions
when the ULA Toeplitz covariance matrix may be reconstructed given the
M-variate MRA covariance matrix.
