# Performance Analysis of ICA in Sensor Array
## 3 Performance of ICA _vs._ Practical Factors
### Condition Number vs. Practical Factors
#### 3.2.3. Locations of the Sources  
First, it has been revealed by Equation (22) one of the edges of separation via ICA algorithms over spatial filtering (beamforming), that is: When sources arrive from identical (or very close) directions, the separation based on spatial filtering will become invalid; while it is still possible for ICA algorithms to work, so long as there is difference in carriers of the NB sources (\(D\
eq 1\)). Second, it can be seen that sources from varied locations may correspond to identical condition numbers, so long as their DOAs result in the same value of \(cos\theta_{1}cos\phi_{1}\) (\(i\) = 1,2). More specifically, for fixed \(\beta\), \(D\) and (\(\theta_{1}\),\(\phi_{1}\)), the mixing matrix would be singular so long as:  
\[cos\theta_{2}cos\varphi_{2}=\frac{mD}{\beta}+Dcos\theta_{1}cos\varphi_{1},\ m \in\mathbb{Z} \tag{35}\]  
Since \(cos\theta_{2}cos\phi_{2}\in[0,\ 1]\), the value of possible \(m\) should be within \([-\beta cos\theta_{1}cos\phi_{1},\ (\beta/D)-\beta cos\theta_{1}cos\phi_{1}]\). Suppose that the set of valid \(m\) is \(\{m_{k}\), \(k\) = 1,2,... \(\}\), then for each of its element there are correspondingly a group of (\(\theta_{2k}\),\(\phi_{2k}\)) satisfying \(cos\theta_{2k}cos\phi_{2k}=(Dcos\theta_{1}cos\phi_{1}+m_{k}D/\beta)\), and this is further depicted by Figure 5.  
To avoid the situation where sources from with certain groups of DOAs become inseparable, controls are necessary on the interval \([-\beta cos\theta_{1}cos\phi_{1},\ (\beta/D)-\beta cos\theta_{1}cos\phi_{1}]\), to reduce the number of integers it includes. In terms of the element spacing, if \(\beta<1\), then \(-\beta cos\theta_{1}cos\phi_{1}\in(-1,\ 0]\) and \((\beta/D)-\beta cos\theta_{1}cos\phi_{1}\)\(\in(-1,1)\), thus the only possible \(m_{k}=0\) and the corresponding \(cos\theta_{2k}cos\phi_{2k}=Dcos\theta_{1}cos\phi_{1}\), while \(Dcos\theta_{1}cos\phi_{1}\in[0,1]\). On the other hand \(D>\beta\) for \(\beta cos\theta_{1}cos\phi_{1}\in Z\), then the valid \(m_{k}=-\beta cos\theta_{1}cos\phi_{1}\) and the mixture may turn singular for \(cos\theta_{2}cos\phi_{2}=0\) only; if \(D>\frac{\beta}{\beta cos\theta_{1}cos\phi_{1}-[\beta cos\theta_{1}cos\phi_{1}]}\) for \(\beta cos\theta_{1}cos\phi_{1}\
otin Z\), then since \((\beta/D)-\beta cos\theta_{1}cos\phi_{1}<[\beta cos\theta_{1}cos\phi_{1}]\), the valid \(m_{k}\) does not exist. It can be seen that the analysis here is consistent with the conclusions presented in Sections 3.2.1 and 3.2.2. For situations where values of \(\beta\) and \(D\) can be roughly determined beforehand, Equation (35) might provide helpful guidance in the arrangement of the sensors (such as the microphone) in terms of directivities and locations, to possibly avoid singular mixtures. In Figure 6, condition numbers under varied DOAs have been shown while representative values of \(\beta\) and \(D\) are set for the sake of illustrating conclusions obtained in this section. In Figure 6a, the condition number becomes extremely large for a group of (\(\theta_{2}\),\(\phi_{2}\)) distributed in a circle centered on the origin. Since \([-\beta cos\theta_{1}cos\phi_{1},\ (\beta/D)-\beta cos\theta_{1}cos\phi_{1}]=[-1.5,\ -0.83]\) therein and valid \(m=-1\). Figure 5: Distribution of locations of the source correspond to singular mixtures. It is similar for Figure 6b where [-\(\beta cos\theta_{1}cos\phi_{1}\), (\(\beta/D\)) - \(\beta cos\theta_{1}cos\phi_{1}\)] = [-1.48, -0.48]. Notice that in Figure 6g,h, there exist more circles consisting of singular points, since the set of valid \(m\) are {-3,-2,-1} and {-2,-1,0}, respectively. In Figure 6c,d, \(\beta\) is set to be smaller than 1, and \(Dcos\theta_{1}cos\phi_{1}>1\), thus it can be seen that singular cases have been effectively avoided. In e and f, the value of \(D\) is increased remarkably, and hence, original singular cases are successfully removed.