In this prelinimary version of paper, we propose to give a complete solution to the Truncated Multidimensional Trigonometric Moment Problem (TMTMP) from a system and signal processing perspective. In mathematical TMTMPs, people care about whether a solution exists for a given sequence of multidimensional trigonometric moments. The solution can have the form of an atomic measure. However, for the TMTMPs in system and signal processing, a solution as an analytic rational function, of which the numerator and the denominator are positive polynomials, is desired for the ARMA modelling of a stochastic process, which is the so-called Multidimensional Rational Covariance Extension problem (RCEP) . In the literature, the feasible domain of the TMTMPs, where the spectral density is positive, is difficult to obtain given a specific choice of basis functions, which causes severe problems in the Multidimensional RCEP. In this paper, we propose a choice of basis functions, and a corresponding estimation scheme by convex optimization, for the TMTMPs, with which the trigonometric moments of the spectral estimate are exactly the sample moments. We propose an explicit condition for the convex optimization problem for guaranteeing the positiveness of the spectral estimation. The map from the parameters of the estimate to the trigonometric moments is proved to be a diffeomorphism, which ensures the existence and uniqueness of solution. The statistical properties of the proposed spectral density estimation scheme are comprehensively proved, including the consistency, (asymptotical) unbiasedness, convergence rate and efficiency under a mild assumption. This well-posed treatment is then applied to a system identification task, and the simulation results validate our proposed treatment for the TMTMP in system and signal processing.

翻译：在本论文的初步版本中，我们旨在从系统和信号处理的角度为截断多维三角矩问题提供一个完整的解决方案。在数学意义上的TMTMP中，研究者关注的是对于给定的多维三角矩序列是否存在解，该解可表现为原子测度的形式。然而，在系统与信号处理领域的TMTMP中，为随机过程的ARMA建模，需要获得解析有理函数形式的解（其分子与分母均为正多项式），即所谓多维有理协方差延拓问题。现有文献表明，在给定特定基函数选择的情况下，TMTMP的可行域（即谱密度为正的区域）难以确定，这给多维RCEP带来了严重困难。本文针对TMTMP提出一种基函数选择方案及相应的凸优化估计框架，使得谱估计的三角矩严格等于样本矩。我们给出了保证谱估计正定性的凸优化问题的显式条件，并证明了从估计参数到三角矩的映射是微分同胚，从而确保了解的存在性与唯一性。在温和假设下，我们完整证明了所提谱密度估计方案的统计性质，包括一致性、（渐近）无偏性、收敛速率及有效性。这一适定化处理方法被应用于系统辨识任务，仿真结果验证了我们对系统与信号处理中TMTMP所提解决方案的有效性。