# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 1 Introduction
Due to search-free characteristics of evaluating (38) and (39), single evaluation of (38) and (39) once is not computationally intensive at all. On the other hand, many evaluations of (38) and (39) can be computationally intensive, especially for large number of repetitions. Note that (38) and (39) should be evaluated many times to empirically quantify the accuracy of \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) and that the number of repetitions should be large enough to make the estimated accuracy more reliable. To obviate this problem, it is shown in this paper that the MSE of \(\delta\theta_{n}^{(3)}\) can be analytically given by (40). Empirical MSE from many evaluations of (39) can be replaced with analytic MSE given by (40): single evaluation of (40) is much less computationally intensive than many evaluations of (39). It should also be noted that, unlike \(\delta\theta_{n}^{(3)}\), it is impossible to get an analytic expression of the MSE of \(\delta\theta_{n}^{(2)}\). Therefore, the MSE of \(\theta_{n}^{(2)}\) should be evaluated empirically from many evaluations of (38). Different realization of Gaussian-distributed noise should be used for each evaluation of (38). Figure 2: Difference between the existing method and the proposed scheme. In summary, the performance of the MUSIC algorithm can be evaluated very efficiently via (40) in comparison with the Monte-Carlo simulation-based performance analysis. The rest of this paper is structured as follows. In Section 2, we explain how the notation can be used to describe the proposed method. In Section 3, we address the conventional MUSIC algorithm. In Section 4, the explicit expression of the difference between the true covariance matrix and the sample covariance matrix has been derived. In Section 5, we have derived an expression of difference between the signal eigenvector of the true covariance matrix and that of the sample covariance matrix in terms of the difference between the true covariance matrix and the sample covariance matrix. In Section 6, a few closed-form expressions of an estimation error, depending on whether specific approximation is applied or not, are derived. In Section 7, for one of the many closed-form expressions of an estimation error, we have derived an explicit representation of the estimate accuracy in terms of the MSE. In Section 8, we address performance analysis of the MUSIC algorithm for multiple incident signals. The simulation results are shown to validate the closed-form estimation error of the proposed method in comparison with an estimation error of the conventional MUSIC method. In [11], an explicit closed-form expression of the MSE of the direction-of-arrival (DOA) estimation algorithm based on maximum-likelihood (ML) criterion is presented. On the other hand, in this paper, the performance analysis of the MUSIC DOA estimation algorithm for the uniform linear array is proposed. In [11], the derivation is based on the Taylor series expansion of the sample covariance matrix itself since the cost function of the ML DOA estimation algorithm can be explicitly written in terms of the sample covariance matrix. In this paper, we adopt the Taylor series expansion of the eigenvectors of the sample covariance matrix, not the Taylor series expansion of the sample covariance matrix itself, since the cost function of the MUSIC algorithm can be written in terms of the eigenvectors of the sample covariance matrix, not in terms of the sample covariance matrix. In [12], the authors have presented the statistical analysis of MUSIC algorithm for general sparse linear arrays in the presence of sensor location uncertainty. Both cases of deterministic sensor location error and stochastic sensor location error have been considered. In [13], the authors have considered basis mismatch problem in DOA estimation. The authors have indicated why dense sampling grids cannot solve basis mismatch problem completely, and they proposed new scheme which is superior to the previous method for handling basis mismatch problem.