# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 5 Eigenvector Perturbation
The angles, \(\theta=\theta_{1}^{\left(0\right)}+\delta\theta_{1},\cdots\), \(\theta=\theta_{d}^{\left(0\right)}+\delta\theta_{d}\), maximizing \(\hat{F}^{\left(g=1,h=1\right)}\left(\theta\right)\), are given by [28]  
\[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]1 \[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]2  
From (29), (34) is simplified to  
\[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]3Since \(\left(\sum\limits_{l=1}^{M}\sum\limits_{k=1}^{M}\delta(\mathbf{P})_{kl}\delta_{n} \frac{d}{d\theta}f_{kl}(\theta)\Big{|}_{\theta=\theta_{n}^{(0)}}\right)\) is much smaller than \(\left(\sum\limits_{l=1}^{M}\sum\limits_{k=1}^{M}(\mathbf{P})_{kl}\delta_{n} \frac{d}{d\theta}f_{kl}(\theta)\Big{|}_{\theta=\theta_{n}^{(0)}}\right)\) and \(\left(\sum\limits_{l=1}^{M}\sum\limits_{k=1}^{M}\delta(\mathbf{P})_{kl}f_{kl} \left(\theta_{n}^{(0)}\right)\right)\), (35) can be approximated as  
\[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]4  
From (33), \(\frac{df_{kl}(\theta)}{d\theta}\) is explicitly written as  
\[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]5 \[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]6  
**7. Closed-Form Expression of Mean Square Error**  
Let \(\delta\theta_{n}\) in (36) with \(\delta\left(\mathbf{P}\right)_{kl}=\delta\left(\mathbf{P}\right)_{kl}^{(a)}+ \delta\left(\mathbf{P}\right)_{kl}^{(b)}+\delta\left(\mathbf{P}\right)_{kl}^{ (c)}\) be denoted by \(\delta\theta_{n}^{(2)}\):  
\[\delta\theta_{n}^{(2)}=\frac{-\sum\limits  
of \(\delta\theta_{1}^{(2)}\) and \(\delta\theta_{1}^{(3)}\) are close to that of \(\delta\theta_{1}^{(1)}\). In the upper plot of Figure 3, it is clear that the agreement between empirical MSE of \(\delta\theta_{1}^{(3)}\), and analytic MSE of \(\delta\theta_{1}^{(3)}\) improves as the number of snapshots increases, which is quite reasonable. The same is true for the upper plots of the Figures 4 and 5. The above observations for the MSEs are illustrated in the lower plots of Figures 3-5, where the MSEs of the second incident signal are shown. In Figures 3-5, we illustrate how small the MSEs of \(\delta\theta_{n}^{(1)}\), \(\delta\theta_{n}^{(2)}\) and \(\delta\theta_{n}^{(3)}\) are. The results with \"Simulation \(\mathrm{E}\Big{(}\big{(}\delta\theta_{n}^{(1)}\big{)}^{2}\Big{)}\)\", \"Simulation \(\mathrm{E}\Big{(}\big{(}\delta\theta_{n}^{(2)}\big{)}^{2}\Big{)}\)\", and \"Simulation \(\mathrm{E}\Big{(}\big{(}\delta\theta_{n}^{(3)}\big{)}^{2}\Big{)}\)\" are obtained from (48), (49), and (50), respectively. In Figures 3-5, the agreement between the simulation results with superscript (3) and the analytic results improves as the number of snapshots increases, since the analytic results are essentially based on ensemble-average, and the simulation results are based on time-average. Therefore, the agreement between the simulation results and the analytic results should be checked for large number of snapshots. In Figures 3-5, the agreement between the simulation results with superscript (1) and the analytic results is not as good as that between the simulation results with superscript (3) and the analytic results, since the analytic results are associated with the simulation results with superscript (3). Similarly, the agreement between the simulation results with superscript (2) and the analytic results is not as good as that between the simulation results with superscript (3) and the analytic results. \[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\]7 Mean square error of \(\delta\theta\) with respect to the number of snapshots (SNR = \(-10\) dB).