## 1 Introduction
The concept of Monte Carlo importance sampling is applied. The superiority of the proposed scheme comes from its better convergence to a global maximum in comparison with other iterative approaches. Although analytical performance analysis of the proposed scheme is not presented, empirical performance of the propose algorithm and the other DOA algorithms is given. Note that Monte Carlo simulation is employed to get empirical performance in terms of the MSE of the DOA estimate. A heuristic optimization algorithm, called gravitational search algorithm, is presented to optimize the cost function of the ML DOA estimation algorithm for a uniform circular array [16]. It is empirically shown that the proposed algorithm is superior to the MUSIC algorithm and particle swarm optimization-based ML algorithm. Analytic performance analysis of the proposed scheme is not presented in [16]. To reduce computational burden of optimizing the cost function of ML DOA estimation algorithm, the artificial bee colony (ABC) algorithm is applied to maximize the cost function of the ML DOA estimation algorithm [17]. It is empirically shown that the proposed scheme is superior to other ML-based DOA estimation methods in the view point of efficiency in computation and statistical performance. Analytic performance analysis of the proposed scheme is not presented in [17]. DOA estimation of narrowband sources in unknown nonuniform white noise is considered in [18]. The stepwise concentration of the log-likelihood function with respect to the signal parameters and noise parameters is obtained by alternating minimization of the Kullback-Leibler divergence. Closed-form expressions for the signal parameters and noise parameters are derived, implying that the proposed scheme results in significant reduction in computational complexity in comparison with exhaustive multidimensional search-based ML DOA algorithms. In [19], a new wideband ML DOA estimation algorithm for an unknown nonuniform sensor noise is proposed to reduce the performance degradation due to nonuniformity of the noise. Two associated implementation schemes are proposed: one is iterative and the other is non-iterative. Simulationresults show that the performance of two processing algorithms is consistent with the Cramer-Rao lower bound. Analytic performance analysis, more specifically the Cramer-Rao lower bound, of the proposed algorithm is presented in [19]. In this paper, we are concerned with quantitative study on how much estimation error is induced due to an additive Gaussian noise on array antennas. More specifically, mean-squared error (MSE) of direction-of-arrival estimation in terms of a standard deviation of an additive noise is derived. In this paper, performance analysis of azimuth estimation using uniform linear array (ULA) is presented. In this paper, the estimate with no superscript denotes the estimate of the original ML algorithm. Note that no approximation is used in getting the estimate with no superscript. In this paper, the estimate with the superscript \((u=1)\) denotes the estimate by using the first approximation, and that with the superscript \((u=1,v)\) represents the estimate by using the first approximation and the second approximation. The difference between the estimate with no superscript and the estimate with the superscript \((u=1)\) quantifies the error due to the first approximation since the first approximation is applied in getting the estimate with the superscript \((u=1)\). Note that no approximation is applied in getting the estimate with no superscript. Similarly, the difference between the estimate with the superscript \((u=1)\) and the estimate with the superscript \((u=1,v)\) quantifies the error due to the second approximation since the first approximation and the second approximations are applied in getting the estimate with the superscript \((u=1,v)\). Based on this intuition, by comparing these three estimates, we can easily determine which approximation results in the dominant approximation error.