Covariance matrices with a circulant structure arise in the context of discrete-time periodic processes and their significance stems also partly from the fact that they can be diagonalized via a Fourier transformation. This note deals with the problem of completion of partially specified circulant covariance matrices. The particular completion that has maximal determinant (i.e., the so-called maximum entropy completion) was considered in Carli et al.  where it was shown that if a single band is unspecified and to be completed, the algebraic restriction that enforces the circulant structure is automatically satisfied and that the inverse of the maximizer has a band of zero values that corresponds to the unspecified band in the data, i.e., it has the Dempster property. The purpose of the present note is to develop an independent proof of this result which in fact extends naturally to any number of missing bands as well as arbitrary missing elements. More specifically, we show that this general fact is a direct consequence of the invariance of the determinant under the group of transformations that leave circulant matrices invariant. A description of the complete set of all positive extensions of partially specified circulant matrices is also given and certain connections between such sets and the factorization of certain polynomials in many variables, facilitated by the circulant structure, is highlighted.
Bibliographical noteFunding Information:
Manuscript received December 22, 2009; revised August 19, 2010; accepted November 24, 2010. Date of publication January 10, 2011; date of current version April 06, 2011. This work was supported by the National Science Foundation and by the Air Force Office of Scientific Research. Recommended by Associate Editor C.-H. Chen. F. P. Carli is with the University of Padova, Padova 35131, Italy (e-mail: firstname.lastname@example.org). T. T. Georgiou is with the University of Minnesota, Minneapolois, MN 55455 USA (e-mail: email@example.com). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2105314
- Circulant matrices
- maximum entropy
- maximum likelihood
- periodic processes