Performance Predictions of Toeplitz Rectified MUSIC

Multiple Signal Classification (MUSIC) is a high-resolution Direction-Of-Arrival (DoA) estimator that uses the eigendecomposition of the Sample Covariance Matrix (SCM). Random matrix theory (RMT) describes the sample eigenvalues and eigenvectors that determine MUSIC's performance, particularly predicting the phase transition when sample eigenvectors align with the true signal subspace. The phase transition depends on the number of sensors and snapshots used to estimate the SCM, and signal-to-noise ratio. For Uniform Linear Arrays (ULAs) in stationary environments, the ensemble covariance is Toeplitz, but the SCM is not. Enforcing a Toeplitz structure can improve MUSIC's performance for low-SNR sources, especially in snapshot-limited scenarios. Prior work shows that in the presence of loud sources, Toeplitz-Rectified (TR) MUSIC misses weaker sources due to a subspace swap. This paper applies perturbation theory to derive the phase transition for TR-SCM and predict when Toeplitz rectification outperforms conventional MUSIC. Results are validated using Monte Carlo simulations.