  The rational covariance extension problem to determine a rational spectral
density given a finite number of covariance lags can be seen as a matrix
completion problem to construct an infinite-dimensional positive-definite
Toeplitz matrix the north-west corner of which is given. The circulant rational
covariance extension problem considered in this paper is a modification of this
problem to partial stochastic realization of reciprocal and periodic stationary
process, which are better represented on the discrete unit circle
$\mathbb{Z}_{2N}$ rather than on the discrete real line $\mathbb{Z}$. The
corresponding matrix completion problem then amounts to completing a
finite-dimensional Toeplitz matrix that is circulant. Another important
motivation for this problem is that it provides a natural approximation,
involving only computations based on the fast Fourier transform, for the
ordinary rational covariance extension problem, potentially leading to an
efficient numerical procedure for the latter. The circulant rational covariance
extension problem is an inverse problem with infinitely many solutions in
general, each corresponding to a bilateral ARMA representation of the
underlying periodic (reciprocal) process. In this paper we present a complete
smooth parameterization of all solutions and convex optimization procedures for
determining them. A procedure to determine which solution that best matches
additional data in the form of logarithmic moments is also presented.
