## 5 Numerical Results  
In Section 3, \(U\) approximation and \(V\) approximation have been described and analytic mean square error(MSE) have been derived in Section 4. Empirical MSEs of the azimuths are defined as  
\[\text{Simulation }\;E\left[\left(\hat{\theta}_{1}-\theta_{1}^{(0)}\right)^{2} \right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{1(w)}-\theta_{1}^{(0)} \right)^{2} \tag{39}\] \[\text{Simulation }\;E\left[\left(\hat{\theta}_{1}^{(u=1)}-\theta_{1}^{(0)} \right)^{2}\right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{1(w)}^{(u=1)} -\theta_{1}^{(0)}\right)^{2}\] (40) \[\text{Simulation }\;E\left[\left(\hat{\theta}_{1}^{(u=1,\
u)}-\theta_{1}^{(0 )}\right)^{2}\right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{1(w)}^{(u=1, \
u)}-\theta_{1}^{(0)}\right)^{2}\] (41) \[\text{Simulation }\;E\left[\left(\hat{\theta}_{2}-\theta_{2}^{(0)} \right)^{2}\right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{2(w)}^{(u=1)} -\theta_{2}^{(0)}\right)^{2}\] (42) \[\text{Simulation }\;E\left[\left(\hat{\theta}_{2}^{(u=1)}-\theta_{2}^{(0 )}\right)^{2}\right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{2(w)}^{(u=1 )}-\theta_{2}^{(0)}\right)^{2}\] (43) \[\text{Simulation }\;E\left[\left(\hat{\theta}_{2}^{(u=1,\
u)}-\theta_{2}^{(0 )}\right)^{2}\right]=\frac{1}{W}\sum_{w=1}^{W}\left(\hat{\theta}_{2(w)}^{(u=1,\
u)}-\theta_{2}^{(0)}\right)^{2} \tag{44}\]  
where \(W\) denotes the number of repetitions. The subscript \((w)\) denotes the estimate associated with the \(w\)-th repetition out of \(W\) repetitions. \(\hat{\theta}_{1}\) and \(\hat{\theta}_{2}\) in (39) and (42) are given by (9). Similarly, \(\hat{\theta}_{1}^{(u=1)}\) and \(\hat{\theta}_{2}^{(u=1)}\) in (40) and (43) are given by (29) and (30). \(\hat{\theta}_{1}^{(u=1,\
u=1)}\) and \(\hat{\theta}_{2}^{(u=1,\
u=1)}\) in (41) and (44) are obtained from (34) and (35). 'Simulation \(E\left[\left(\hat{\theta}_{1}-\theta_{1}^{(0)}\right)^{2}\right]\), 'Simulation \(E\left[\left(\hat{\theta}_{1}^{(u=1)}-\theta_{1}^{(0)}\right)^{2}\right]\), 'Simulation \(E\left[\left(\hat{\theta}_{1}^{(u=1,v)}-\theta_{1}^{(0)}\right)^{2}\right]\), and 'Analytic \(E\left[\left(\hat{\theta}_{1}^{(u=1,v)}-\theta_{1}^{(0)}\right)^{2}\right]\), are obtained from (39), (40), (41), and (36), respectively. Figure 1: Analytic and simulated MSEs (Mean Square Errors) of \(\theta_{1}\) and \(\theta_{2}\) with respect to SNR (Signal to Noise Ratio). Figure 2: Analytic and simulated MSEs \(\theta_{1}\) and \(\theta_{2}\) with respect to the number of snapshots (SNR = 0 dB). Figure 3: Analytic and simulated MSEs \(\theta_{1}\) and \(\theta_{2}\) with respect to the number of snapshots (SNR = 5 dB). Similarly, the results with 'Simulation \(E\left[\left(\hat{\theta}_{2}-\theta_{2}^{(0)}\right)^{2}\right]\)', 'Simulation \(E\left[\left(\hat{\theta}_{2}^{(u=1)}-\theta_{2}^{(0)}\right)^{2}\right]\)', 'Simulation \(E\left[\left(\hat{\theta}_{2}^{(u=1,r)}-\theta_{2}^{(0)}\right)^{2}\right]\)', and 'Analytic \(E\left[\left(\hat{\theta}_{2}^{(u=1,r)}-\theta_{2}^{(0)}\right)^{2}\right]\)' are obtained from (42), (43), (44) and (36), respectively. Figure 4: Analytic and simulated MSEs \(\theta_{1}\) and \(\theta_{2}\) with respect to the number of snapshots (SNR = 10 dB). For all the results in Figures 1-4, the difference between Simulation \(E\left[\left(\hat{\theta}_{1}-\theta_{1}^{(0)}\right)^{2}\right]\), and 'Simulation \(E\left[\left(\hat{\theta}_{1}^{(u=1)}-\theta_{1}^{(0)}\right)^{2}\right]\)', is much larger than that between 'Simulation \(E\left[\left(\hat{\theta}_{1}^{(u=1)}-\theta_{1}^{(0)}\right)^{2}\right]\)', and 'Simulation \(E\left[\left(\hat{\theta}_{1}^{(u=1,v)}-\theta_{1}^{(0)}\right)^{2}\right]\)', which implies that \(U\) approximation results in much greater error than \(V\) approximation. Therefore, to improve DOA estimation performance, second-order Taylor expansion, which corresponds to \(u=2\), can be used.