Chiara Masiero: Circulant Rational Covariance Extension as a Tool for Multivariate Spectral Estimation

Rational covariance extension is a classic issue in control theory and system identification. Indeed, under mild assumptions, it allows to model a stationary process as the output of a linear shaping filter fed by white noise. This result paves the way to filtering, estimation and prediction.

In this talk we focus on rational covariance extension for stationary periodic processes.

These can be defined on a finite interval and then are naturally extended by means of modular arithmetic. Stationarity and periodicity imply that their covariance matrix has circulant structure and that their spectral density is defined on the discrete unit circle. Thus, as opposite to regular covariance extension, circulant rational covariance extension leads to the completion of finite (block) circulant Toeplitz matrices, and the resulting ARMA models are bilateral.

Circulant rational covariance extension can be recast in the framework of optimization-based theory of moment problems with rational measures. As a consequence, we can provide a complete parameterization of all bilateral ARMA realizations.

In addition, an algorithm which hinges on Fast Fourier Transform allows to compute the solution efficiently.

The talk illustrates a first step in generalizing this theory to the multivariate case. We also show that circulant rational covariance extension provides a fast approximation procedure for solving the regular rational covariance extension problem for multivariate processes.