# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 1 Introduction
The superiority of the proposed scheme has been validated in terms of computational complexity and estimation accuracy. In [14], the authors have presented a new scheme, called sG-MUSIC, which is superior to the previous G-MUSIC algorithm in the viewpoint of a bias of the estimate. The authors have shown that the scheme is quite effective when the number of snapshots is small. In [15], the authors have proposed how the dimension of DOA estimation can be reduced when only a few snapshots are available. The scheme can be applied to 2D-DOA estimation as well as 1-D DOA estimation. In [16], the authors have presented the performance analysis of G-MUSIC algorithm. The authors have shown in what situation G-MUSIC is superior to the original MUSIC algorithm. The authors have also presented asymptotic variances of G-MUSIC. There has been a great deal of research on the determination of emitter location. Localization algorithm consists of two parts: measuring localization parameters between nodes, and the use of these parameters to estimate location. The localization parameters can be angle-of-arrival (AOA), time-difference-of-arrival (TDOA), frequency-difference-of-arrival (FDOA), and received signal strength (RSS). In this paper, we consider performance analysis of the MUSIC algorithm, which is one of the DOA estimation algorithms. The DOA estimation is the main application of array signal processing. Radar and sonar are also good examples of the application of array signal processing. The authors have presented performance analysis for use with amplitude comparison monopulse DOA algorithm under measurement uncertainty. The derived results can be used for determining how the performance of the amplitude comparison monopulse algorithm degrades due to an additive measurement noise without computationally intensive Monte-Carlo simulation. The difference between the scheme in [17] and the proposed scheme in this paper is as follows: the scheme in [17] deals with the performance analysis of monopulse algorithm, which is usually used in tracking radar in the sense that an initial estimate of the DOA should be given in advance. This paper deals with the performance analysis of the MUSIC algorithm, where an initial estimate is not required. In monopulse algorithm, the tracking radar transmits a radar signal and uses a reflected signal from a radar target to update the DOA. On the other hand, in MUSIC algorithm, direction-finding (DF) station does not transmit a signal to estimate the DOA of an emitter: the DF station just receives a signal of an active emitter, and uses a received signal to estimate the DOA of the emitter. Therefore, the monopulse algorithm is an active DOA estimation algorithm, and the MUSIC algorithm is a passive DOA estimation algorithm. In [18], the authors proposed how to suppress motion-induced phase in frequency modulated continuous wave (FMCW) radar for automotive application. The motion compensation method is not computationally intensive. Conventional beamforming algorithm is used for DOA estimation. Although the authors illustrate that the motion compensated algorithm results in smaller estimation error than the uncompensated algorithm, they do not derive the closed-form expression of the MSE of the azimuth estimate: the authors illustrate that the peak of the DOA spectrum for the motion-compensated conventional beamforming algorithm is closer to true incident angle than the peak of the DOA spectrum for the original conventional beamforming algorithm. On the other hand, in this paper, the closed-form expression of the MSE of the azimuth estimate for the MUSIC DOA algorithm is presented. If the MUSIC algorithm, not the conventional beamforming algorithm, is employed for DOA estimation of FMCW radar, the approach leading to Equation (40) of this paper can be applied to get the closed-form expression MSEs of the azimuth estimate both for uncompensated MUSIC algorithm and for compensated MUSIC algorithm.