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.

本文涉及M${}^2$信号，即在多维上定义的多元（或向量值）信号域。特别是，我们提出了一种优化技术来解决平稳随机矢量场的协方差扩展问题。多维Itakura–Saito距离被用作优化标准，以在满足有限矩约束的光谱中选择解。为了避免在可行集的边界上可能发生的技术性，我们处理离散形式的问题，其中多维积分由黎曼和近似。频谱解决方案也是离散的，当基础随机场是周期性的时，它自然发生。我们表明，存在离散问题的解决方案，它是唯一的，并且平稳地依赖于问题数据。因此，我们有一个问题很明确的问题，它的解决方案可以平稳地进行调整。最后，我们已经将我们的理论应用于汽车模块集成系统中的目标参数估计问题。仿真结果表明，我们的频谱估计器具有良好的性能。