# Performance Analysis of ICA in Sensor Array
## 5 Conclusions  
In this paper, the performance of ICA algorithms in sensor arrays is fully discussed. First, the analytic connection between the environment noise variance, the source variance, the condition number of the mixing matrix and the optimal signal to interference-plus-noise ratio is deduced in Equation (20). It indicates quantitatively how the separation performance would be determined by the environment noise level and the singularity of the mixing process. Next, the relationships between several fundamental practical factors in sensor array receiving and the condition number are analyzed in detail. Factors concerned include the element spacing of the array (depicted by the parameter \(\beta\)), the frequencies (depicted by the parameter \(D\)) and locations of the sources (depicted by the DOAs of sources). Generally speaking, it is found that: (1) the condition number varies with the element spacing in a periodical way, with the period is determined by \(D\) and DOAs; (2) the condition number also varies with the reciprocal of \(D\) periodically, and the period is determined by \(\beta\) and DOAs; (3) as for the locations of sources, we discover that sources from varied locations may lead to identical condition numbers, so long as their DOAs result in the same value of \(cos\theta_{i}cos\phi_{i}\) (\(i\) = 1,2). Considering its significance for applications, the situations where the mixtures become singular have been paid special attention to. Main conclusions concerning the prevention of singular cases include: (1) the cases of singular mixtures can be well prevented for sources with diverse carriers and from all possible directions, so long as the element spacing of the array is smaller than wavelengths of the sources. And the only possible exception exists when \(cos\theta_{2}cos\phi_{2}-Dcos\theta_{1}cos\phi_{1}=0\), then the mixture would be singular regardless of \(\beta\); (2) for certain \(\beta\) and \((\theta_{i}\phi_{i})(i\) = 1,2), the cases of singular mixtures could be avoided by sufficiently large values of \(D\), the reciprocal of which is smaller than \(\min(1/D)_{\max}\) in Equation (34). Moreover, a countermeasure for the singular cases has been proposed, based on the introduction of extra phase delay. And in reality, the scheme is realized with one of the observations undergoing artificial time-shifting. Conclusions and the effectiveness of the proposed countermeasure are validated by succeeding experiments. As a whole, conclusions and results obtained in this paper could be instructive while applying ICA algorithms in sensor arrays or designing the receiving array, especially for the prevention of critical failure in separation. Future works may focus on analyzing the effects of more complicated practical factors such as the multi-path effect or array calibration imperfection, which would involve more complex model of the array output and coupling among different kinds of factors. Also, the effectiveness of the proposed scheme in Section 3.3 could be further promoted, possibly based on the interpolation of the observations to allow better control on the value of the extra phase delay by changing the sample interval. Since the desired phase delay is \((2m+1)\pi\), \(m\in Z\) in theory.  
This research is supported by the National Natural Science Foundation of China (No. 61401490).  
[PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] contributed analysis tools and supervised the experiments; [PERSON] wrote the paper.  
The authors declare no conflict of interest.