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Uncertainty Bounds for Spectral Estimation

机译：光谱估计的不确定性界限

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摘要

The purpose of this paper is to study metrics suitable for assessing uncertainty of power spectra when these are based on finite second-order statistics. The family of power spectra which is consistent with a given range of values for the estimated statistics represents the uncertainty set about the “true” power spectrum. Our aim is to quantify the size of this uncertainty set using suitable notions of distance, and in particular, to compute the diameter of the set since this represents an upper bound on the distance between any choice of a nominal element in the set and the “true” power spectrum. Since the uncertainty set may contain power spectra with lines and discontinuities, it is natural to quantify distances in the weak topology—the topology defined by continuity of moments. We provide examples of such weakly continuous metrics and focus on particular metrics for which we can explicitly quantify spectral uncertainty. We then consider certain high resolution techniques which utilize filter-banks for preprocessing, and compute worst case a priori uncertainty bounds solely on the basis of the filter dynamics. This allows the a priori tuning of the filter-banks for improved resolution over selected frequency bands.
机译：本文的目的是研究适用于评估功率谱不确定性的指标，这些指标基于有限的二阶统计量。与给定统计值的给定值范围一致的功率谱系列表示有关“真实”功率谱的不确定性设置。我们的目标是使用适当的距离概念来量化此不确定性集合的大小，尤其是要计算集合的直径，因为这代表了集合中任何标称元素与“真”功率谱。由于不确定性集合可能包含具有线和不连续性的功率谱，因此很自然地可以量化弱拓扑中的距离，该弱拓扑是由矩连续性定义的拓扑。我们提供了此类弱连续指标的示例，并重点关注可以明确量化频谱不确定性的特定指标。然后，我们考虑某些利用滤波器​​组进行预处理的高分辨率技术，并仅根据滤波器动力学来计算最坏情况的先验不确定性范围。这允许对滤波器组进行先验调谐，以在选定频带上提高分辨率。

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