# Performance Analysis of ICA in Sensor Array
## 4 Simulations
In Figure 10e, since \(\beta cos\theta_{1}cos\phi_{1}\
otin\mathbb{Z}\), \(D\) is set to be greater than \((\beta/\beta cos\theta_{1}cos\phi_{1},\)-\([\)\(\beta cos\theta_{1}cos\phi_{1}])=4\); in Figure 10f, \(\beta cos\theta_{1}cos\phi_{1}=1\), thus \(D>\beta>2\) should be enough for the prevention of singular cases and we set \(D=3\); and it should be noticed herein that the mixture would still become singular for \(cos\theta_{2}cos\phi_{2}=0\), which is straightforward from Equation (22). To conclude, the experiment has validated the conclusions obtained in Section 3.2 about the influence of fundamental practical factors on the separation performance, especially those concerned the cases of singular mixtures. Experiment 3: Effectiveness of the proposed countermeasure against singular mixtures. In this experiment, the carrier frequencies of the QPSK and BPSK sources are set at 0.4 MHz and 0.2 MHz respectively, thus \(D=2\). The DOA of source 1 is (\(56^{\circ}\), \(63^{\circ}\)). The sampling frequency is 1.2 MHz and the length of observation is 5120. The input SNR is 30 dB. After numerous experiments, the threshold of the correlation coefficient for judging the singular cases is chosen as 0.99. In Figure 11, the correlation coefficient between source 1 and its corresponding restitution component is adopted as other measurement of performance. In Figure 11a-c, \(\beta=0.75\); and in Figure 11d-f, \(\beta=4.5\). Figure 11a,d demonstrate the separation performance, without the proposed countermeasure. It can be seen that the restitution quality is unsatisfactory while source 2 is from certain directions (the distribution of such DOAs of source 2 can be seen clearer in the subplot northeast of Figure 11a,d). In Figure 11b,e, the proposed countermeasure has been adopted and \(k=1\). Thus, the extra phase delay introduced for these singular cases is \(\Psi_{1}=2\pi kT_{s}(f_{2}-f_{1})=\pi/3\); and since \(k=2\) in Figure 11c,f, the extra delay therein is \(\Psi_{2}=2\pi/3\). It can be seen that restitution qualities in the originally singular points have been dramatically improved, and the failure of separation has been effectively avoided. Thus, the effectiveness of the proposed countermeasure for the singular mixtures has been validated. In Figure 12, the waves of the sources and their estimations are demonstrated to show the improvement of the restitution quality in a more explicit way. Figure 12a presents the original sources in Figure 11a-c. In Figure 12b, the recovered components for DOA of source 2 being (\(-11.5^{\circ}\), \(61.2^{\circ}\)) (the corresponding condition number is 46.7) have been presented, and we can see that the distortion in the wave forms is unbearable. In Figure 12c,d, the proposed scheme is applied for the same situation,  
Figure 11: Effectiveness of the proposed countermeasures. (**a**) \(\beta=0.75\), without extra phase delay (**b**) \(\beta=0.75\), \(\mathrm{k}=1\) (**c**) \(\beta=0.75\), \(\mathrm{k}=2\) (**d**) \(\beta=4.5\), without extra phase delay (**e**) \(\beta=4.5\), \(\mathrm{k}=1\) (**f**) \(\beta=4.5\), \(\mathrm{k}=2\). and it is obvious that the restitution quality has been improved significantly. Similar analysis can be done for Figure 12e-h, which correspond to Figure 11d-f.  
Figure 12: _Cont._  
Figure 12: Restitution quality with the proposed scheme. (**a**) Original Sources (**b**) Recovered components, without extra phase delay (**c**) Recovered components, k = 1 (**d**) Recovered components, k = 2 (**e**) Original Sources (**f**) Recovered components, without extra phase delay (**g**) Recovered components, k = 1 (**h**) Recovered components, k = 2.