## 6 Conclusions and Summary  
In Figure 6, the proposed algorithm is outlined. Note that Equations (A)-(D) in this section refer to Equations (A)-(D) in Figure 6, respectively. The constraints used for derivation of estimation errors are equations (A) and (B): equation (A) is valid since, in a noiseless environment, no estimation error occurs in the azimuth estimates. The estimation error due to an additive noise in noisy environment is formulated as equation (B). To quantify the estimation error due to an additive noise, equations (A) and (B) and two approximations of \(U\) approximation and \(V\) approximation are used. Note that the covariance matrix in equation (A) and that in equation (B) are associated with noiseless response and noisy response, respectively. Applying the Taylor series approximation in equation (B) and using equation (A) in the approximated expression, the azimuth estimation error in equation (C) can be derived. To get an explicit expression of the MSE of the azimuth estimate, \(V\) approximation is applied, and the closed-form expression of the MSE of the estimate is obtained from equation (D). To the best of our knowledge, no previous study used explicit equations (A) and (B) to derive the azimuth error in equation (C). One of the novelties of this paper is that the derivation is based on the observation that the azimuth error in the subscript in equation (B) can be analytically derived since equation (A) is true for noiseless covariance matrix. In summary, applying \(U\) approximation to equation (B) and using the constraint in equation (A), the estimation error in equation (C) can be obtained. The MSE of the azimuth estimation error in equation (D) is obtained from equation (C) and \(V\) approximation. Note that, to get equation (D) from equation (C), the statistics of an additive noise should also be exploited. Quantitative study on the estimation error for direction-of-arrival estimation in terms of standard deviation of an additive noise has been addressed in this paper. In this paper, in case of estimating azimuths of multiple incident signals, the closed-form expression of the MSE of the DOA estimate for the ML algorithm has been derived by stepwise  
Figure 6: Outline of the proposed performance analysis scheme. approximations. The type of antenna array is assumed as ULA. The closed-form of the MSE has been derived by using the Taylor series approximation which linearizes the nonlinear parts of the array vector and additional approximation based on the assumption that the estimation error is very small at high SNR. The closed-form of the MSE has been verified through numerical results. The closed-form of the MSE has been verified through numerical results. All of the stepwise approximated simulation results and the results obtained from the closed-form of the MSE show good agreement. Although the formulation and the numerical results for two incident signals are presented in this paper, an extension to multiple incident signals is quite intuitively clear and straightforward. In this paper, we rigorously derive how the MSE of the ML algorithm for direction-of-arrival can be expressed in terms of various parameters, which include the number of sensor elements, the number of incident signals, the number of snapshots, and the variance of additive noises on the antenna elements. Although, for convenience, the statistics of additive noise is assumed to be non-zero-mean Gaussian distributed, the derivation in the Appendices can be extended to the case where noise can be modeled as any other random variable as long as the moments of the random variable are analytically available. [PERSON] and [PERSON] made a Matlab implementation of the proposed algorithm. [PERSON] formulated the proposed algorithm. [PERSON], [PERSON], and [PERSON] wrote an initial draft, and [PERSON] and [PERSON] revised the manuscript. [PERSON] and [PERSON] derived the results in the appendices. All authors have read and agreed to the published version of the manuscript.