M$^2$-Spectral Estimation: A Relative Entropy Approach

TL;DR

This paper introduces a new spectral estimation method for multivariate signals over multidimensional domains using a relative entropy approach, providing a well-posed, unique, and smoothly tunable solution with promising automotive application results.

Contribution

It proposes a novel optimization technique for the covariance extension problem in multivariate fields using multidimensional Itakura-Saito distance, ensuring existence, uniqueness, and smooth dependence of solutions.

Findings

Solution exists, is unique, and depends smoothly on data.

The method is applicable to periodic random fields.

Simulation shows promising automotive system performance.

Abstract

This paper deals with M$^2$-signals, namely multivariate (or vector-valued) signals defined over a multidimensional domain. In particular, we propose an optimization technique to solve the covariance extension problem for stationary random vector fields. The multidimensional Itakura-Saito distance is employed as an optimization criterion to select the solution among the spectra satisfying a finite number of moment constraints. In order to avoid technicalities that may happen on the boundary of the feasible set, we deal with the discrete version of the problem where the multidimensional integrals are approximated by Riemann sums. The spectrum solution is also discrete, which occurs naturally when the underlying random field is periodic. We show that a solution to the discrete problem exists, is unique and depends smoothly on the problem data. Therefore, we have a well-posed problem whose solution can be tuned in a smooth manner. Finally, we have applied our theory to the target parameter estimation problem in an integrated system of automotive modules. Simulation results show that our spectral estimator has promising performance.