基于协方差滞后的谱密度估计器的定量误差分析
Quantitative Error Analyses of Spectral Density Estimators Using Covariance Lags
摘要：
谱密度估计是系统辨识的核心问题，也是系统控制和信号处理的重要研究领域。在谱密度估计器的设计上已经有了许多结果。然而，据我们所知，定量的误差分析的频谱密度估计还没有提出。在实际应用中，导致谱密度估计误差的主要因素有两个，即外部加性噪声和有限的样本个数。在本文中，这是一个非常初步的版本，我们首先考虑一个单变量谱密度估计使用协方差滞后。估计任务由凸优化方案执行，并且所估计的谱密度的协方差滞后完全符合期望，这使得可以进行定量误差分析，例如导出紧误差上界。我们分析了这两个因素引起的误差，并提出了误差的上界和下界。然后将一元谱估计的结果推广到多元谱估计。
英文摘要：
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.
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