# Performance Analysis of ICA in Sensor Array
## 3 Performance of ICA _vs._ Practical Factors
### Condition Number vs. Practical Factors
#### 3.2.1 Element Spacing of the Array  
From Equation (22), it is obvious that for fixed \(D\) and DOAs, the condition number varies with \(\beta\) in a periodical way, with the period being:  
\[\beta=\left|\frac{D}{cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1}} \right|,\ cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1}\
eq 0 \tag{23}\]  
The periodicity indicates that, increasing the element spacing of the array may have different impacts under varied situations depicted by \(D\) and DOAs. Taking the partial derivative of Equation (23) against \(D\), we have:  
\[ \begin{array}{c}\frac{\partial\Delta\beta}{\partial D}=\frac{\text{sign}( cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1})\cdot cos\theta_{2}cos \varphi_{2}}{\left(cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1} \right)^{2}},\\ \\ cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1}\
eq 0\end{array}  \tag{24}\]  
where \(\text{sign}(\cdot)\) is the sign function. Hence, if \(cos\theta_{2}cos\varphi_{2}>cos\theta_{1}cos\phi_{1}\), then the period increases along with the value of \(D\) within the interval \(D\in[1,\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}}),\ cos \theta_{1}cos\varphi_{1}\
eq 0\); and decreases along with it once \(D\) is greater than\(\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}}\). If \(cos\theta_{2}cos\varphi_{2}<cos\theta_{1}cos\phi_{1}\), the period decreases along with the value of \(D\). The influence of the element spacing on the condition number is depicted by Figure 3.  
The periodicity has been clearly demonstrated, with the period determined by varied \(D\) and DOAs. In Figure 3A, \(\theta_{1}=31.3^{\circ}\), \(\phi_{1}=71.5^{\circ}\), \(\theta_{2}=-25.3^{\circ}\), \(\phi_{2}=15.1^{\circ}\), thus \(\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}}=3.2194\). We can see that for (a) and (b), where \(D\in[1,\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}})\), the period is lengthened with greater \(D\) ; while for (c) and (d), where \(D>\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}}\), the period is shortened with increasing \(D\). In Figure 3B, \(\theta_{1}=-29.3^{\circ}\), \(\phi_{1}=43.1^{\circ}\), \(\theta_{2}=-56.3^{\circ}\), \(\phi_{2}=12.1^{\circ}\), and \(\frac{cos\theta_{2}cos\varphi_{2}}{cos\theta_{1}cos\varphi_{1}}=0.8520\). We can see that the value of \(\beta\) keeps decreasing with greater element spacing. The minimal and maximal values of the condition number also arise periodically, and they appear respectively at \(\beta_{\min}\) and \(\beta_{\max}\) that equal:  
\[ \begin{array}{l}\beta_{\min}=\frac{(m+1)D}{cos\theta_{2}cos\varphi_{2}-Dcos \theta_{1}cos\varphi_{1}}\\ \beta_{\max}=\frac{mD}{cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1} }\end{array} ,\;cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1}\
eq 0,m\in\mathbb{Z} \tag{25}\]  
According to previous analysis, the maximal values of the condition number which correspond to the singular mixtures should be avoided in applications, in order to prevent the critical failure of separation. In practice, the expected circumstance is that: with well-designed sensor arrays, the separation via ICA algorithms is satisfactory and can handle with mixtures consisted of sources with diverse carriers (within the band that the array is designed for) and from all possible locations. This is meaningful considering the intricate and unforeseen situations in applications. And here, we attempt to achieve this via designing properly the element spacing of the array. One straightforward idea is that if the actual \(\beta\) is smaller than all possible \(\beta_{\max}\), then the singular case would not occur. While \(D\) and DOAs are fixed, this indicates that:  
\[ \begin{array}{l}\beta<\left|\frac{D}{cos\theta_{2}cos\varphi_{2}-Dcos\theta _{1}cos\varphi_{1}}\right|,\\ cos\theta_{2}cos\varphi_{2}-Dcos\theta_{1}cos\varphi_{1}\
eq 0\end{array}