ESPRIT和空间平滑ESPRIT的非渐近性能分析
Nonasymptotic performance analysis of ESPRIT and spatial-smoothing ESPRIT
摘要：
本文研究了多快拍数据的频率估计问题。众所周知，ESPRIT（以及存在相干源或给定有限快拍的空间平滑ESPRIT）可以在快拍数或信噪比（SNR）接近无穷大的情况下定位真实频率。本文分析了ESPRIT和空间平滑ESPRIT在有限快拍数和有限信噪比下的非渐近性能。我们证明了ESPRIT的绝对频率估计误差（或空间平滑ESPRIT）以压倒性的概率由$C\frac{\max(σ, σ^2)}{\sqrt{L}}$从上方界定，其中$σ^2$表示高斯噪声方差，$L$是快照的数量，并且$C$是独立于$L$和$σ^2$的系数，当且仅当真实频率可以由ESPRIT（或空间平滑ESPRIT）在没有噪声或具有无限多个快照的情况下定位。我们的结果是通过导出新的矩阵扰动界和推广经典的Schur乘积定理得到的，这可能是独立的兴趣。还对MUSIC和SS-MUSIC进行了扩展。数值结果证实了我们的分析。
英文摘要：
This paper is concerned with the problem of frequency estimation from multiple-snapshot data. It is well-known that ESPRIT (and spatial-smoothing ESPRIT in presence of coherent sources or given limited snapshots) can locate the true frequencies if either the number of snapshots or the signal-to-noise ratio (SNR) approaches infinity. In this paper, we analyze the nonasymptotic performance of ESPRIT and spatial-smoothing ESPRIT with finitely many snapshots and finite SNR. We show that the absolute frequency estimation error of ESPRIT (or spatial-smoothing ESPRIT) is bounded from above by $C\frac{\max(σ, σ^2)}{\sqrt{L}}$ with overwhelming probability, where $σ^2$ denotes the Gaussian noise variance, $L$ is the number of snapshots and $C$ is a coefficient independent of $L$ and $σ^2$, if and only if the true frequencies can be localized by ESPRIT (or spatial-smoothing ESPRIT) without noise or with infinitely many snapshots. Our results are obtained by deriving new matrix perturbation bounds and generalizing the classical Schur product theorem, which may be of independent interest. Extensions to MUSIC and SS-MUSIC are also made. Numerical results are provided corroborating our analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
