This paper studies the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. Our most general result is a perturbation bound of the signal space in terms of the minimum singular value of Fourier matrices. When the point sources are located in several separated clumps, we provide an explicit upper bound of the noise-space correlation perturbation error in MUSIC and the support error in ESPRIT in terms of a Super-Resolution Factor (SRF). The upper bound for ESPRIT is then compared with a new Cram\'er-Rao lower bound for the clumps model. As a result, we show that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems. Our theory is validated by numerical experiments.
翻译:本文研究估计固定点源位置的光谱估计问题,这是从一组传感器收集的多张Fourier测量图的多片中估算固定点源的位置。我们证明MUSIC和ESPRIT具有新的稳定性,这是噪音标准偏差、快照数量、源振幅和辅助作用的函数。我们最普遍的结果是信号空间以Fourier矩阵最小单值构成的扰动。当点源位于几个分离的块状中时,我们提供了MUSIC的噪音-空间相关扰动误差和ESPRIT的超分辨率参数支持误的明显上限。然后,ESPRIT的上限值与新的Cram\'er-Rao更低的悬浮模型的下限相比。结果,我们表明,ESPRIT在对噪音、光片和SRF。作为我们分析的副产品,我们发现了一些基本差异,即我们通过验证的单项实验和多项实验,我们发现一些基本差异。