A Well-Posed Multidimensional Rational Covariance and Generalized Cepstral Extension Problem

TL;DR

This paper develops a well-posed method for estimating multidimensional power spectral densities from covariance and cepstral data, enabling the construction of corresponding shaping filters even in high-dimensional settings.

Contribution

It introduces a new multidimensional moment problem incorporating generalized cepstral moments and entropy, providing a unique spectral estimator for any finite dimension.

Findings

Existence of a rational spectral density matching covariances exactly and cepstral coefficients approximately.

Construction of a shaping filter via spectral factorization from the estimated spectral density.

Extension of the theory to dimensions greater than two, filling a significant gap in the field.

Abstract

In the present paper we consider the problem of estimating the multidimensional power spectral density which describes a second-order stationary random field from a finite number of covariance and generalized cepstral coefficients. The latter can be framed as an optimization problem subject to multidimensional moment constraints, i.e., to search a spectral density maximizing an entropic index and matching the moments. In connection with systems and control, such a problem can also be posed as finding a multidimensional shaping filter (i.e., a linear time-invariant system) which can output a random field that has identical moments with the given data when fed with a white noise, a fundamental problem in system identification. In particular, we consider the case where the dimension of the random field is greater than two for which a satisfying theory is still missing. We propose a multidimensional moment problem which takes into account a generalized definition of the cepstral moments, together with a consistent definition of the entropy. We show that it is always possible to find a rational power spectral density matching exactly the covariances and approximately the generalized cepstral coefficients, from which a shaping filter can be constructed via spectral factorization. In plain words, our theory allows to construct a well-posed spectral estimator for any finite dimension.