# Performance Analysis of ICA in Sensor Array
## 4 Simulations  
In this section, experiments are carried out using Intel(R) Core (TM) 2 Duo CPU E6550 @ 2.33 GHz and MATLAB 7.12.0(R2011a). Experiment 1: Separation performance of ICA algorithms under varied noise levels and the singularity of the mixing matrix. In this experiment, quadrature phase shift keying (QPSK) and binary phase shift keying (BPSK) signals are adopted as sources with carrier frequencies 0.38, 0.127 MHZ, and symbol rates 2, 1 Kbps, respectively. The sample rate is set as 1 MHz, and the observation length is 1024. The receiving array consists of two sensors with the element spacing being two times the wavelength of source 1. The DOA of source 1 is (25.1\({}^{\circ}\), 40.5\({}^{\circ}\)), while the elevation angle of source 2 is 27.9\({}^{\circ}\). The SNR at the input of the separator and the SINRM adopted here as the measurement for separation performance are defined as in [35]. Typical ICA algorithms tested include: FastICA [22], EASI [23] and Infomax [24]. The initialization of the demixing matrix is generated randomly. The non-linear function selected for FastICA is \(G(y)=\log(0.1+y)\), with more options listed in [22]; the adaptation step for EASI is 0.01, and the nonlinearity is chosen as \(g_{i}(\mathbf{y})=\left\lvert\,y_{i}\,\right\rvert^{2}\!y_{i}\) for \(1\leqslant i\leqslant P\); the updating step for Infomax is 0.1, and the nonlinearity is selected in the form of \(g_{i}(y)=-\left\lvert\,y_{i}\,\right\rvert^{\alpha_{i}-1}\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\  
Experiment 2: Separation performance and practical factors. In this experiment, the same sources have been adopted, except that the carrier frequency of source 2 is set according to different values of \(D\) tested. The DOA of source 1 is (30\({}^{\circ}\), 30\({}^{\circ}\)) in Figure 10a,c,e, and (\(-\)45\({}^{\circ}\), 45\({}^{\circ}\)) in Figure 10b,d,f. The performances have been demonstrated with representative values of \(\beta\) and \(D\), as well as for source 2 from all possible directions. The input SNR is set at 30 dB. Since it has been demonstrated in Figures 8 and 9 that performances of typical ICA algorithms are all influenced by the condition number in ways alike, only results from the FastICA algorithm as described in Experiment 1 are presented, while similar results have been obtained with other ICA algorithms mentioned in Experiment 1 during our simulations. Figure 10 demonstrates the SINRM\({}_{1}\) under varied practical factors simulating possible actual scenarios. It can be seen that in Figure 10a,b, the separation performance endures dramatic deterioration (at least 15 dB) for source 2 from certain groups of directions. This is because in Figure 10a, [\(-\)\(\beta cos\theta_{1}cos\phi_{1}\), (\(\beta/D)-\beta cos\theta_{1}cos\phi_{1}\)] = [-1.5, -0.83], thus \(m=-1\) is valid as analyzed in Section 3.2.2, and the deterioration of performance occurs for \(cos\theta_{2}cos\phi_{2}\) = (\(Dcos\theta_{1}cos\phi_{1}-D/\beta\)); for Figure 10b, valid \(m=\) [-1, 0], and thus corresponding \(cos\theta_{2}cos\phi_{2}\) = [0, 0.75] (notice that groups of (\(\theta_{2}\),\(\phi_{2}\)) correspond to  
Figure 8: Separation performance under varied input SNRs. (**a**) FastICA; (**b**) EASI; (**c**) Informax; (**d**) Equation (20). Figure 9: Influence of the condition number on the separation performance. (**a**) FastICA; (**b**) EASI; (**c**) Informax; (**d**) Equation (20). \(cos\theta_{2}cos\phi_{2}=0\) distribute on the edge of Figure 10b). In Figure 10c,d, \(\beta\) is reduced to be smaller than 1. For Figure 10c, since \(Dcos\theta_{1}cos\phi_{1}>1\), the singular cases have been completely avoided for source 2 from all possible directions; while in Figure 10d, \(Dcos\theta_{1}cos\phi_{1}<1\), thus the mixture would turn singular for \(cos\theta_{2}cos\phi_{2}=Dcos\theta_{1}cos\phi_{1}\), though with \(\beta<1\). In Figure 10e,f, the value of \(D\) is set according to Equation (34).