# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 1 Introduction
Based on this intuition, by comparing \(\theta_{n}^{(1)},\theta_{n}^{(2)}\) and \(\theta_{n}^{(3)}\), we can easily determine which approximation results in the dominant approximation error. This inspection cannot be obtained from the scheme presented in the previous study [2]. As far as the authors know, no previous study on the performance analysis of the MUSIC algorithm presented explicit expressions of each estimate in successive approximations [4; 5; 6; 7; 8; 9; 10]. Depending on how many approximations out of all the approximations are applied, the estimates for all the intermediate stages are rigorously derived in this paper. In this paper, we consider the case in which multiple signals are incident on the uniform linear array (ULA). To quantify the difference between the signal subspace of the true covariance matrix and that of the sample covariance matrix, the Taylor series expansion is adopted to get an analytic expression of the MSE. For getting \(\theta_{n}^{(1)}\), for \(n=1,\ldots,d\), in the original MUSIC algorithm, the MUSIC spectrum has to be evaluated as the azimuth angle takes discrete search angles. Note that \(d\) denotes the number of incident signals. To estimate both azimuth and elevation, the MUSIC spectrum has to evaluated as two variables of azimuth and elevation take discrete search angles, which can be computationally intensive, especially for small increments of the azimuth and the elevation. Note that the estimate is obtained from the search angle which optimizes the MUSIC cost function. Therefore, in the original MUSIC algorithm, the final estimate depends on how the search range and the search step are chosen. Improperly chosen search angles in the MUSIC algorithm implementation result in an inaccurate estimate, which degrades the performance of the original MUSIC algorithm. In short, there are two problems in getting \(\theta_{n}^{(1)}\): Firstly, it can be computationally intensive for simultaneous estimation of azimuth and elevation. Secondly, the estimate is highly dependent on the selection of the search range and the search step. It is described how the first problem with \(\theta_{n}^{(1)}\) can be overcome by adopting \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) proposed in this paper. \(\delta\theta_{n}^{(2)}\) is given by (38), and \(\delta\theta_{n}^{(3)}\) is given by (39). Since (38) and (39) are closed-form expressions, evaluation of \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) does not require computationally intensive exhaustive search over search angles. Therefore, getting \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) is much less computationally intensive than getting \(\theta_{n}^{(1)}\). Figure 1: Summary of performance analysis of the multiple signal classification (MUSIC) algorithm. The second problem with \(\theta_{n}^{(1)}\) can also be circumvented. Since \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) are analytically given by (38) and (39), they do not depend on the choice of search parameters such as the search step and the search range. To get \(\delta\theta_{n}^{(2)}\) analytically from (38), the first approximation has to be applied to the cost function of the original MUSIC algorithm. On the other hand, in getting \(\theta_{n}^{(1)}\), no approximation is applied to the cost function of the original MUSIC algorithm. Therefore, \(\delta\theta_{n}^{(2)}\) can be regarded as an estimation error associated with the approximated cost function. Similarly, to get \(\delta\theta_{n}^{(3)}\) analytically from (39), the first approximation and the second approximation should be applied to the cost function of the original MUSIC algorithm. Although \(\theta_{n}^{(1)}\) is obtained from an exhaustive search of the cost function of the original MUSIC algorithm, no approximation is applied to get \(\theta_{n}^{(1)}\). \(\delta\theta_{n}^{(3)}\) can also be regarded as an estimation error associated with the approximated cost function.