Root-MUSIC is a spectral estimation algorithm that approximates the unknown signal frequencies by constructing a high-degree polynomial and finding a subset of roots which are closest to the complex unit circle. Previous works found asymptotic expectation formulas for the performance of Root-MUSIC under the implicit assumption that the aforementioned root selection criterion does not select extraneous roots -- those which are unrelated to the correct parameters. This paper removes the need for this assumption by showing all extraneous roots lie outside an annulus of a certain thickness and therefore are not selected by the algorithm. This paper also provides sharp, non-asymptotic, and explicit error bounds for the correct roots in terms of fundamental model parameters. All results hold under a natural separation condition on the correct signal frequencies and are applicable in both the single- and multi-snapshot models. More specifically, in the multi-snapshot model, we prove that Root-MUSIC estimates the frequencies with error at most $O(σ/(m \sqrt n))$, where $σ^2$ is the noise variance, $m$ is the number of sensors, and $n$ is the number of snapshots. A novelty of this non-asymptotic bound is the explicit $1/m$ decay, which indicates that there is a significant advantage in utilizing additional sensors. Numerical simulations confirm our theory. The main mathematical insight of this paper is a geometric property of the Root-MUSIC polynomial: its correct roots are highly stable to noise while its extraneous roots must lie outside of an annulus.

翻译：Root-MUSIC是一种谱估计算法，通过构造高次多项式并寻找最接近复单位圆的一组根来近似未知信号频率。以往研究在隐含假设——即前述根选择准则不会选取与正确参数无关的额外根——的前提下，推导了Root-MUSIC性能的渐近期望公式。本文通过证明所有额外根均位于某一特定厚度环形区域之外且因此不会被算法选取，消除了这一假设的必要性。本文还针对正确根给出了基于基础模型参数的尖锐、非渐近且显式的误差界。所有结果均在信号频率满足自然分离条件时成立，并适用于单快拍和多快拍模型。具体而言，在多快拍模型中，我们证明Root-MUSIC对频率的估计误差不超过$O(σ/(m \sqrt n))$，其中$σ^2$为噪声方差，$m$为传感器数量，$n$为快拍数。这一非渐近界的创新之处在于显式的$1/m$衰减，表明增加传感器数量具有显著优势。数值仿真验证了我们的理论。本文的主要数学洞见在于Root-MUSIC多项式的一个几何性质：其正确根对噪声高度稳定，而额外根必须位于环形区域之外。