A sharp analysis of Root-MUSIC: locations of correct and extraneous roots

2026-05-26

AI Summary

This work addresses the susceptibility of the Root-MUSIC algorithm to spurious roots lying outside the unit circle, which can degrade frequency estimation accuracy in spectral analysis. By analyzing the geometric structure of its associated polynomial, the study rigorously establishes—without additional assumptions—that all spurious roots reside strictly outside a fixed-thickness annular region beyond the unit circle, thereby precluding their erroneous selection. Moreover, it derives a non-asymptotic, explicit error bound for the true roots, revealing their high robustness to noise. Under natural frequency separation conditions, the estimation error in the multi-snapshot setting is shown to be at most \(O(\sigma/(m\sqrt{n}))\), with an explicit \(1/m\) decay rate in the number of sensors \(m\). These theoretical guarantees hold for both single- and multi-snapshot scenarios and are corroborated by numerical experiments.

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, 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.

Problem

• Root-MUSIC
• extraneous roots
• spectral estimation
• non-asymptotic analysis
• frequency estimation

Innovation

• Root-MUSIC
• non-asymptotic analysis
• extraneous roots
• spectral estimation
• sensor array