A sharp analysis of Root-MUSIC: locations of correct and extraneous roots
Abstract
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 and non-asymptotic 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. 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.
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BibTeX
@misc{huber2026sharp,
title = {{A sharp analysis of Root-MUSIC: locations of correct and extraneous roots}},
author = {Hana Huber and Weilin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.04003},
eprint = {2606.04003},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04003},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A sharp analysis of Root-MUSIC: locations of correct and extraneous roots
AU - Hana Huber
AU - Weilin Li
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.04003
UR - https://arxiv.org/abs/2606.04003
AB - 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 and non-asymptotic 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. 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.
ER -