Spectral density estimation is a core problem of system identification, which is an important research area of system control and signal processing. There have been numerous results on the design of spectral density estimators. However to our best knowledge, quantitative error analyses of the spectral density estimation have not been proposed yet. In real practice, there are two main factors which induce errors in the spectral density estimation, including the external additive noise and the limited number of samples. In this paper, which is a very preliminary version, we first consider a univariate spectral density estimator using covariance lags. The estimation task is performed by a convex optimization scheme, and the covariance lags of the estimated spectral density are exactly as desired, which makes it possible for quantitative error analyses such as to derive tight error upper bounds. We analyze the errors induced by the two factors and propose upper and lower bounds for the errors. Then the results of the univariate spectral estimator are generalized to the multivariate one.

翻译：谱密度估计是系统辨识的核心问题，也是系统控制与信号处理领域的重要研究方向。关于谱密度估计器设计已有大量成果，但据我们所知，谱密度估计的定量误差分析尚未见诸文献。在实际应用中，谱密度估计的误差主要源于两个因素：外部加性噪声与有限样本数量。本文作为初步研究的初版，首先考虑使用协方差滞后的单变量谱密度估计器。该估计任务通过凸优化方案完成，且估计所得谱密度的协方差滞后与期望值精确匹配，这使得定量误差分析（例如推导严格的误差上界）成为可能。我们分析了两种因素导致的误差，并提出了误差的上界与下界。随后，将单变量谱估计器的结果推广至多变量情形。