# Performance Analysis of ICA in Sensor Array
## 3 Performance of ICA _vs._ Practical Factors
### Condition Number vs. Practical Factors
#### 3.2.2 Carrier Frequencies of the NB Sources  
From Equation (22), it can be deduced that the condition number varies with the reciprocal of \(D\) in a periodical way. The period is determined by \(\beta\) and DOAs as:  
\[\Delta\left(\frac{1}{D}\right)=\left(\frac{1}{\beta cos\theta_{2}cos\varphi_{2 }}+\frac{cos\theta_{1}cos\varphi_{1}}{cos\theta_{2}cos\varphi_{2}}\right),\; cos\theta_{2}cos\varphi_{2}\
eq 0 \tag{29}\]  
Take the partial derivative of Equation (29) against \(\beta\), we have:  
\[\frac{\partial\Delta\left(1/D\right)}{\partial\beta}=-\frac{1}{cos\theta_{2} cos\varphi_{2}}.\frac{1}{\beta^{2}},\;cos\theta_{2}cos\varphi_{2}\
eq 0 \tag{30}\]  
Thus, the period decreases along with the increase of \(\beta\). Values of \((1/D)\) correspond to minima (denoted as (\((1/D)_{\mathrm{min}}\))) and maxima (denoted as \((1/D)_{\mathrm{max}}\)))) of the condition number that can be obtained as:  
\[ \begin{split}(1/D)_{\mathrm{min}}&=\frac{(m+1/2)+ \beta\text{cos}\theta_{1}cos\varphi_{1}}{\beta cos\theta_{2}cos\varphi_{2}}\\ (1/D)_{\mathrm{max}}&=\frac{m+\beta cos\theta_{1} cos\varphi_{1}}{\beta cos\theta_{2}cos\varphi_{2}}\end{split} ,\;m\in\mathbb{Z},\;  
Though Equation (31) indicates values of \((1/D)\) correspond to the maxima of the condition number, it should be noticed that a maximal point is valid only when it is within the interval of \((0,1]\). This means that \(m\) which relates to a valid \((1/D)_{\max}\) should satisfy:  
\[-\beta cos\theta_{1}cos\varphi_{1}<m\leq\beta cos\theta_{2}cos\varphi_{2}-\beta cos \theta_{1}cos\varphi_{1}, \tag{32}\]  
\[m\in\mathbb{Z},\ cos\theta_{2}cos\varphi_{2}\
eq 0\]  
If \(\beta>1\), it should be obvious that there must exist \(m\in[[-\beta cos\theta_{1}cos\varphi_{1}]+1\), \(|\beta cos\theta_{2}cos\varphi_{2}-\beta cos\theta_{1}cos\varphi_{1}]]\) for certain groups of \((\theta_{i}\varphi_{i})\) (\(i=1\),2), that would validate Equation (32). If \(\beta=1\), then the valid \(m\) would be 1 for \(cos\theta_{1}cos\phi_{1}=0\), or 0 for \(cos\theta_{1}cos\theta_{1}\
eq 0\). If \(0<\beta<1\), then for \(cos\theta_{1}cos\phi_{1}=0\), there would not be \(m\in\mathbb{Z}\) satisfying Equation (32); for \(cos\theta_{1}cos\phi_{1}\
eq 0\), the only valid \(m\) would be 0, with \((1/D)_{\max}=cos\theta_{1}cos\phi_{1}/cos\theta_{2}cos\varphi_{2}\). Notice that this is consistent with the analysis on the element spacing, where it has been stated that if \(\beta<1\), the cases of singular mixtures would not occur for all \(D\geq 1\) and \(\theta_{i}\in[-90^{\circ},90^{\circ}]\), \(\phi_{i}\in[-90^{\circ},90^{\circ}]\), with the only exception of \(cos\theta_{2}cos\phi_{2}-Dcos\theta_{1}cos\phi_{1}\). From Equation (31), it could also be found that the minimum of \((1/D)_{\max}\) (denoted as \(\min(1/D)_{\max}\)) is:  
\[\left\{ \begin{array}{l}\frac{-|\beta cos\theta_{1}cos\varphi_{1}|+\beta cos \theta_{1}cos\varphi_{1}}{\beta cos\theta_{2}cos\varphi_{2}},\ \beta cos\theta_{1}cos\varphi_{1}\
otin\mathbb{Z}\\ \frac{1}{\beta cos\theta_{2}cos\varphi_{2}},\ \beta cos\theta_{1}cos\varphi_{1} \in\mathbb{Z}\end{array} \right. \tag{33}\]  
where '\([\)x\(]\)' indicates the largest integer no greater than \(x\). This indicates that for certain \(\beta\) and \((\theta_{i}\varphi_{i})\) (\(i=1\),2), the cases of singular mixtures could be avoided by sufficiently large values of \(D\), the reciprocal of which is smaller than \(\min(1/D)_{\max}\). Further, for the adaptation of varied \((\theta_{2}\varphi_{2})\), Equation (33) could be transferred to:  
\[\left\{ \begin{array}{l}\frac{-|\beta cos\theta_{1}cos\varphi_{1}|+\beta cos \theta_{1}cos\varphi_{1}}{\beta},\ \beta cos\theta_{1}cos\varphi_{1}\
otin\mathbb{Z}\\ \frac{1}{\beta},\ \beta cos\theta_{1}cos\varphi_{1}\in\mathbb{Z}\end{array} \right. \tag{34}\]  
Equations (33) and (34) could be meaningful in the situation where one of the sources is fixed in both of the frequency and its location, while the other source is a moving one.