# Performance Analysis of ICA in Sensor Array
## 3 Performance of ICA _vs._ Practical Factors
### Countermeasure for Singular Mixtures  
For most applications, the occurrence of (nearly) singular mixtures would be disastrous, and the intricate actual scenarios have made it somehow inevitable. And here, we attempt to propose certain countermeasure against such situations to avoid extremely poor separation performance. According to Equation (20), reducing the power of noise could be helpful, possibly via means of band pass filtering. Hence, more effective actions may be taken in terms of changing the condition number. One straightforward idea is that: when the mixing process is (nearly) singular, certain extra phase delay \(\Psi\) could be introduced into the observations. This could be effective since when the mixtures are (nearly) singular, \(2\pi(\beta/D)\). \((cos\theta_{2}cos\phi_{2}-Dcos\theta_{1}cos\phi_{1})\) equals or is quite close to \(2m\pi\), \(m\)Z, and the value of the condition number after the introduction of the extra phase delay would be:  
\[cond\left(\mathbf{A}\right)=\frac{4}{\sqrt{2-2 cos[2\pi\left(\beta/D\right) \cdot\left(cos\theta_{2}cos\phi_{2}-Dcos\theta_{1}cos\phi_{1}\right)+\psi]}} \approx\frac{4}{\sqrt{2-2 cos\psi}} \tag{36}\]  
Thus, by setting the proper value of \(\Psi\) the revised condition number would be more beneficial for the separation, and the ideal value of \(\Psi\) would be \((2m+1)\pi\), \(m\)\(\in\)Z. Notice that this could not be realized via directly shifting the phase of the mixtures, since it can be deduced from Equations (8) and (10) that this actually would not change the condition number. In practice, \(\Psi\) could be introduced through shifting the mixtures along the axis of the observation time. More specifically, consider the following scheme depicted by Figure 7. The idea here is that the introduction of extra phase delay in Equation (22) is achieved with one of the observations undergoing an artificial time-lag. For instance, if the observation from the second sensor is shifted backward by \(k\) points, then the equivalent phase delay introduced is:  
\[\psi=2\pi kT_{s}(f_{2}-f_{1}) \tag{37}\]  
in which \(T_{s}\) is the sample interval. It should be stated that the proposed scheme is only designed to combat the situations of (nearly) singular mixtures. In fact, only when originally \(2\pi(\beta/D)\cdot(cos\theta_{2}cos\phi_{2}-Dcos\theta_{1}cos\phi_{1})\) is close to \(2m\pi,m\varepsilon Z\) can it be ensured that the extra phase delay is constructive rather than destructive. One of the actual problems is judging the singularity of the current observations and deciding whether the proposed procedure is needed. Since the condition number of the mixing matrix and the SINR at the output is generally hard to obtain, the correlation coefficient \(\rho\) can act as one practical criterion [34]. For random variables \(x(\xi)\) and \(y(\xi)\), their correlation coefficient is defined as:  
\[\rho_{xy}=\frac{c_{xy}}{\sigma_{x}\sigma_{y}} \tag{38}\]  
in which \(c_{xy}\) is the cross covariance of \(x(\xi)\) and \(y(\xi)\), \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) are the variances of them. The idea is that when the original mixing matrix is (nearly) singular, it can be seen from Equation (8) that the two observations would be (approximately) coherent, and the module value of the correlation coefficient between the two observations would approach 1. More specifically, since the sources are presumed to be zero-mean and independent, it can be calculated from Equations (8) and (38) that the module value of the correlation coefficient between the two observations (denoted as \(\rho_{x1x2}\)) equals:  
\[\left|\rho_{x_{1}x_{2}}\right|=\frac{1}{\sigma  
In practice, a threshold associated with the noise level and the number of samples can be set based on experience, and when the module value of the correlation coefficient between the observations exceeds the threshold, the mixtures can be regarded as (nearly) singular. And then the proposed scheme should be adopted.