Ahstract-Partial stochastic realization of periodic processes from finite covariance data leads to the circulant rational covariance extension problem and bilateral ARMA models. In this paper we present a convex optimization-based theory for this problem that extends and modifies previous results by Carli, Ferrante, Pavon and Picci on the AR solution, which have been successfully applied to image processing of textures. We expect that our present results will provide an enhancement of these procedures.
I. INT RODUCTIONThe rational covariance extension problem is an important problem in systems and control with an extensive literature; see, e.g., [2]-[7], [IS], [17], [18], [21], [30] and references therein. Among other things, it is the basic problem in partial stochastic realization theory [3] and Toeplitz ma trix completion problems. Covariance extension for periodic stochastic processes, on the other hand, leads to matrix completion of To eplitz matrices with circulant structure and to partial stochastic realizations in the form of bilateral ARMA models. This connects up to a rich realization theory for reciprocal processes [22]-[2S]. In [12] Carli, Ferrante, Pavon and Picci presented a maximum-entropy approach to this circulant covariance ex tension problem, thereby providing a procedure for deter mining the unique bilateral AR model matching the covari ance sequence. However, recently it was discovered that the circulant covariance extension can be recast in the context of the optimization-based theory of moment problems with rational measures developed in [1], [4], [S], [7]-[9], [11], [19], [20] allowing for a complete parameterization of all bilateral ARMA realizations, and a complete theory for the scalar case was presented in [26]. The present paper provides a first step in generalizing this theory to the multivariable case. The AR theory of [12] has been successfully applied to image processing of textures [14], [31], and we anticipate an enhancement of such methods by allowing for more general ARMA realizations. As pointed out in [26] the circulant rational convariance extension theory provides a fast approximation procedure for solving the regular rational and picci@dei.unipd. it, respectively.covariance extension problem, as it is based on fast Fourier transforms (FFT), and in the present paper we shall provide numerical evidence that this also holds in the multivariable case.The outline of the paper goes as follows. In Section II we review the regular multivariable rational covariance extension problem and harmonic analysis on the discrete unit circle. Then in Section III we present our main result on the mul tivariable circulant rational covariance extension problem, parametrizing the family of solutions, and in Section IV we show how logarithmic moments can be used to determine the best particular solution. Finally, in Section V we provide two numerical examples demonstrating the power of circulant covariance extension as a tool for approximation.
II. PRELIMINARIES
A. The multi variable rational co...