# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 1 Introduction
Note that, to obtain the closed-form expression of the MSE of the motion-compensated MUSIC algorithm in [18], the derivation leading to (40) in this paper is modified so that the motion-compensation proposed in [18] can be taken into account in the MUSIC DOA estimation. In [19], the authors proposed how to improve performance of the minimum variance distortionless response (MVDR) beamforming algorithm by exploiting signal self-cancellation phenomenon in DOA estimation of underwater acoustic sources. The proposed algorithm is called self-signal cancellation MVDR (SCC-MVDR). In Figures 4 and 5 of [19], they illustrated the root mean square errors (RMSEs) of the MVDR, the MUSIC, the root-MUSIC, TLS-ESPRIT, the propagator method and the SCC-MVDR. The RMSEs are obtained from Equation (17) of [19]: the RMSEs are obtained empirically, not analytically, from the Monte Carlo simulation with the number repetitions of 100. That is, 100 repetitions of the MUSIC algorithm should be performed to get the RMSEs for the MUSIC algorithm. On the other hand, the RMSEs for the MUSIC algorithm can be obtained analytically, without time-consuming repetitive execution of the MUSIC algorithm, by taking the square root of the MSE value in (40) of this paper. In Figure 4 of [19], the RMSEs with respect to SNRs for fixed number of snapshots, 100, are given. In Figure 5 of [19], the RMSEs with respect to the number of snapshots for fixed SNR are presented. Note that, in this paper, the number of snapshots and the SNRs are explicitly taken into account in deriving the MSE in (40), implying that the empirical RMSEs for the MUSIC algorithm in Figures 4 and 5 of [19] can be analytically by taking the square root of the MSE value given by (40). Note that analytical approach is much less computationally intensive than Monte-Carlo simulation-based empirical approach. In [20], a new DOA estimation algorithm for use with direction estimation of sound source. The preprocessing of the scheme is based on the intuition that isolation of a target speaker can be achieved by selecting dominant time-frequency bin of the time-frequency distribution. Furthermore, neural network approach is employed to finally estimate the DOA of sound source. The performance of the proposed DOA estimation algorithm is given by the mean absolute error (MAE), not the MSE. The MAE is empirically estimated from the Monte-Carlo simulation with the number of repetitions of 200. In summary, they proposed a new DOA estimation algorithm, and the performance of the proposed algorithm is given by the empirically obtained MAE. If the MUSIC algorithm is used for the DOA estimation in the sound source experiment described in [20], the expression in Equation (40) of this paper can be employed to yield the performance of the MUSIC algorithm in terms of the MSE analytically, not empirically. In [21], the authors are concerned with how to enhance target sound sound when multiple sound sources are present. Preprocessing for the proposed enhancement scheme is to estimate the DOA of the target sound source. That is, the DOA of the sound source should be available for implementation of the proposed target enhancement scheme. In experiment 3 using simulated data and in experiment 5 using recorded data in an anechoic chamber, the authors show how performance of the source enhancement depends on the DOA mismatch, which is the difference between the true DOA and the estimated DOA. If the MUSIC algorithm is employed for preprocessing DOA estimation, the expression in (38) and (39) can be used for calculation of the DOA mismatch. In the presence of sensor measurement noise, DOA estimate is a random variable. Therefore, it is better to use the MSE of the DOA than to use the DOA mismatch as a measure of DOA estimation accuracy. In that case, the expression in (40) is used to quantify how accurate the DOA estimate for use with the sound source enhancement in [21] is. In [22], a survey on RSS and AOA-based localization scheme is presented.