# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 1 Introduction  
The multiple signal classification (MUSIC) algorithm [1] has been one of the most widely used direction-of-arrival (DOA) algorithms [2; 3]. It exploits the orthogonality between the noise eigenvectors of the covariance matrix and the array vectors associated with the true angles of the incident signals. Since the noise subspace is orthogonal to the signal subspace, the array vectors corresponding to the true incident angles should belong to the signal subspace of the covariance matrix. In practice, the covariance matrix itself, which is defined from the ensemble average, is not usually available, and the sample covariance matrix is obtained from time average. In the assumption of the ergodicity, the covariance matrix can be replaced with the sample covariance matrix when the number of snapshots is very large. When the number of snapshots is finite, the time-average-based sample covariance is not equal to the ensemble-average-based true covariance matrix. Since the true covariance matrix is not available, we use the eigenvectors of the sample covariance matrix, not the eigenvectors of the true covariance matrix. When the number of the snapshots is finite, the eigenvectors of the sample covariance matrix are not identical to those of the covariance matrix due to the difference between the true covariance matrix and the sample covariance matrix. Although the array vectors corresponding to the true incident angles belong to the signal subspace of the true covariance matrix, the array vector corresponding to the true incident angles do not belong to the signal subspace of the sample covariance matrix. Note that, in the MUSIC algorithm, anglesspanning the signal subspace of the sample covariance matrix, not the true covariance matrix, are used as estimates of incident signals since true covariance matrix is not available. The mathematical and statistical theories used for derivation of the results are as follows:  
**Statistical description of perturbations**: How various perturbations can be statistically described in terms of probability density function, moments to get a closed-form expression of the MSE of the estimate. **Taylor series**: How Taylor series can be adopted to obtain explicit expression of the MSE of the estimate **Eigenvector perturbation and eigenvalue perturbation**: How the eigenvectors and the eigenvalues of a matrix are perturbed due to the perturbations in the entries of the matrix. In [2], the authors carried out a performance analysis of the MUSIC algorithm and derived an implicit expression of the estimate. On the other hand, in this paper, explicit expressions of the estimate itself and the MSE of the estimate have been derived. In Figure 1, the performance analysis proposed in this paper is summarized. Figure 2 clearly indicates how the proposed scheme in this paper is different from the existing performance analysis of the MUSIC algorithm in [2]. In this paper, we obtain the explicit expressions of \(\theta_{n}^{(1)},\theta_{n}^{(2)}\) and \(\theta_{n}^{(3)}\). The estimate with superscript (1) denotes the estimate of the original MUSIC algorithm. Note that no approximation is used in getting the estimate with superscript (1). The estimate with superscript (2) denotes the estimate based on the first approximation, and that with superscript (3) represents the estimate based on the first approximation and the second approximation. The difference between \(\theta_{n}^{(1)}\) and \(\theta_{n}^{(2)}\) quantifies the error due to the first approximation since the first approximation is applied in getting \(\theta_{n}^{(2)}\). Note that no approximation is applied in getting \(\theta_{n}^{(1)}\). Similarly, the difference between \(\theta_{n}^{(2)}\) and \(\theta_{n}^{(3)}\) quantifies the error due to the second approximation since the first approximation and the second approximations are applied in getting \(\theta_{n}^{(3)}\).