Performance Analysis of Real-Valued DOA Estimators via Auxiliary Signal Expansion

This paper presents a theoretical analysis of the real-valued root-MUSIC (RV-root-MUSIC) and RV-MUSIC algorithms. These direction-of-arrival (DOA) estimation methods reduce computational complexity by utilizing only the real part of the covariance matrix, making them attractive for low-complexity implementations. However, RV-based estimators inherently produce a pair of estimates: the true DOA and a virtual mirrored counterpart, which exhibit different estimation biases that have not been fully characterized in the literature. To address this gap, we derive analytical expressions for the biases associated with both the true and mirrored DOA estimates. By introducing an auxiliary signal through conjugate expansion, the bias in the noise subspace of the RV covariance matrix is explicitly characterized, enabling the derivation of theoretical mean-square error (MSE) expressions. The analysis reveals distinct bias behaviors for the true and mirrored DOAs. Compared with existing studies, the proposed framework provides a more explicit characterization of the noise subspace bias and yields more accurate MSE predictions for RV-based algorithms. Simulation results demonstrate close agreement between the theoretical MSEs and empirical results over a wide range of snapshot numbers ($\mathbf{2^{4}}$ to $\mathbf{2^{12}}$) under sufficiently high signal-to-noise ratio ($\mathbf{-3}$ dB to 21 dB) conditions, confirming the validity of the proposed analysis as a reliable performance benchmark for RV-root-MUSIC and RV-MUSIC.