  A positive semidefinite Toeplitz matrix, which often arises as the finite
covariance matrix of a stationary random process, can be decomposed as the sum
of a nonnegative multiple of the identity corresponding to a white noise, and a
singular term corresponding to a purely deterministic process. Moreover, the
singular nonnegative Toeplitz matrix admits a unique characterization in terms
of spectral lines which are associated to an oscillatory signal. This is the
content of the famous Carath\'{e}odory-Fej\'{e}r theorem. Its importance lies
in the practice of extracting the signal component from noise, providing
insights in modeling, filtering, and estimation. The multivariate counterpart
of the theorem concerning block-Toeplitz matrices is less well understood, and
in this paper, we aim to partially address this issue. To this end, we first
establish an existence result of the line spectrum representation for a finite
covariance multisequence of some underlying random vector field. Then, we give
a sufficient condition for the uniqueness of the representation, which indeed
holds true in the special case of bivariate time series. Equivalently, we
obtain the Vandermonde decomposition for positive semidefinite block-Toeplitz
matrices with $2\times 2$ blocks. The theory is applied to the problem of
frequency estimation with two measurement channels within the recently
developed framework of atomic norm minimization. It is shown that exact
frequency recovery can be guaranteed in the noiseless case under suitable
conditions, while in the noisy case, extensive numerical simulations are
performed showing that the method performs well in a wide range of
signal-to-noise ratios.
