## 1 Introduction  
There has been a lot of research on the direction-of-arrival (DOA) estimation [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. Our interest in this paper is the performance analysis of the maximum likelihood (ML)-based DOA estimation algorithm. In [8], a performance analysis of the ML DOA estimation algorithm for low SNR and small number of snapshots is considered. A threshold effect in the ML DOA algorithm is exploited, and the authors derive approximations to the mean square error and probability of outlier. If the noise variance at each sensor in the array antenna system is equal, the noise covariance matrix is considered to be multiples of an identity matrix. In [9], nonuniform white noise, whose covariance matrix can be expressed as an arbitrary diagonal covariance, is considered, and the new ML DOA algorithm for nonuniform noise is proposed, and the performance analysis of the proposed algorithm is also presented. In [10], the authors addressed the DOA estimation using sparse sensor arrays, where the sensor noises can be uncorrelated between different subarrays due to large intersubarray spacings. The authors proposed a new maximum-likelihood estimator, which can be extended to the uncalibrated arrays with sensor gain and phase mismatch. A new computationally efficient ML DOA algorithm exploiting spatial aliasing is proposed in [11]. Generally, spatial aliasing is undesirable since it degrades the DOA estimation accuracy. In [11], the authors adopted a nested array structure with a doubly spaced aperture. The computational burden of the ML DOA estimation algorithm is reduced by the highly compressed search range and the small number of candidate angles to be searched. The authors also presented Monte Carlo simulation based mean square error (MSE). However, analytic performance analysis of the proposed scheme is not presented in [11]. A new ML DOA algorithm for use with a uniform linear array is proposed [12]. The scheme is superior to the conventional ML algorithm when the true DOAs of two incident signals are very close to each other. The formulation for the new DOA algorithm is based on an asymptotic approximation of the unconditional maximum likelihood (UML) procedure when two closely space signals are incident on the ULA. Taylor approximation is also adopted for derivation of the new algorithm. Empirical MSE is illustrated to validate the proposed scheme. However, the authors do not present analytic performance analysis of the new DOA algorithm. To overcome the problem of large computational complexity for implementation of the ML DOA estimation algorithm, based on a spatially overcomplete array output formulation, an efficient ML DOA estimator is proposed in [13]. Empirical performance from Monte Carlo simulation is present to illustrate the superiority of the proposed scheme over the other DOA algorithms. Analytic performance analysis is not given. Although the ML estimator is known to be optimal in DOA estimation, its computational cost can be quite prohibitive, especially for a large number of incident signals. To solve this problem, in [14], three kinds of natural computing algorithms, differential evolution, clonal selection algorithm, and the particle swarm optimization, are applied for implementation of the multivariable nonlinear optimization of the cost function of the ML DOA estimation algorithm. It turns out that all three natural computing algorithms are capable of optimizing the ML DOA cost function, irrespective of the number of incident signals and their nature. In addition, the number of points evaluated by natural computing algorithms is much smaller than that associated with exhaustive grid search-based algorithms, justifying the application of these natural computing algorithms to the optimization of the cost function of the ML DOA estimation algorithm. In [15], a new implementation of ML DOA estimation, which outperforms the other DOA algorithms for closely spaced incident signals, is proposed.