# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 10 Conclusions  
Based on some approximations and the Taylor series expansion, we have derived a few expressions of an approximation of estimate of the MUSIC algorithm. The estimates are obtained by approximating the signal eigenvector and the projection matrix onto the column space of the eigenvector matrix. Two closed-form expressions of the DOA estimate have been derived by applying two approximations. For one of two expressions, closed-form expression of the MSE of the estimate as well as the closed-form of the estimate itself have been derived. The closed-form expression of the MSE has been validated by comparing it with empirically obtained MSE. The derived expression of the MSE of the estimates can be used for the performance analysis of the MUSIC algorithm. From the viewpoint of the computational complexity, an analytic MSE of the estimate can be evaluated much more easily in comparison with an empirical MSE from the time-consuming Monte-Carlo simulation. The proposed scheme can be used for the performance analysis in predicting how accurate the estimate of the MUSIC algorithm is without resorting computationally intensive Monte-Carlo simulation. The performance of the MUSIC algorithm is dependent on various parameters, such as the number snapshots, the number antenna elements in the array, inter-element spacing between adjacent antenna elements and the SNR. Therefore, making Monte-Carlo simulations for different values of the various parameters is computationally very intensive, and the analytic performance analysis proposed in this paper can be employed to predict how well the MUSIC algorithm works, given the specific values of the parameters. [PERSON] mainly made a matlab implementation of the proposed algorithm. [PERSON] formulated the proposed algorithm. [PERSON] and [PERSON] wrote an initial draft. [PERSON] and [PERSON] derived the results in the appendices. [PERSON] helped [PERSON] make a numerical implementation of the proposed algorithm. All authors have read and agreed to the published version of the manuscript. The authors gratefully acknowledge the support from Electronic Warfare Research Center at Gwangju Institute of Science and Technology (GIST), originally funded by Defense Acquisition Program Administration (DAPA) and Agency for Defense Development (ADD). This work was partially supported by Electronics and Telecommunications Research Institute (ETRI) grant funded by the Korean government [20 ZD1110, Development of ICT Convergence Technology for Daegu-GyeongBuk Regional Industry]. The authors declare no conflict of interest. Appendix A Calculation of \(E\left(\left(\delta\mathbf{R}\right)_{kl}^{R}\left(\left(\delta\mathbf{R} \right)_{k^{\prime}l^{\prime}}^{C}\right)\right)^{*}\)  
From (12) and (13), \(E\left(\left(\delta\mathbf{R}\right)_{kl}^{R}\left(\left(\delta\mathbf{R} \right)_{k^{\prime}l^{\prime}}^{C}\right)^{*}\right)\) can be rewritten as  
\[E\left(\left(\delta\mathbf{R}\right)_{kl}^{R}\left(\left(\delta\mathbf{R} \right)_{k^{\prime}l^{\prime}}^{C}\right)^{*}\right)=\left(E\left(\left( \delta\mathbf{R}\right)_{kl}^{R}\right)\right)\left(\left(\delta\mathbf{R} \right)_{k^{\prime}l^{\prime}}^{C}\right)^{*}, \tag{41}\]  
since, from (13), \(\left(\delta\mathbf{R}\right)_{k^{\prime}l^{\prime}}^{C}\) is not stochastic when the noise is zero-mean Gaussian distributed, \(E\left(\left(\delta\mathbf{R}\right)_{kl}^{R}\right)\) is expressed as  
\[E\left(\left(\delta\mathbf{R}\right)_{kl}^{R}\right)=\frac{1}{L}\sum_{i=1}^{L} \sigma^{2}\delta_{kl},\hskip 14.226378 pt\delta_{kl}=\left\{ \begin{array}{ll}1&k=l \\ 0&k\
eq l\end{array}