# Performance Analysis of ICA in Sensor Array
## 1 Introduction
Several criteria have been put forth based on the matrix theory or the structure characterization theory of random variables [27; 28; 29]. For instance, the concept of the \(m\)-row decomposability is introduced in [27] and it is proved therein that if the mixing matrix is \(m\)-row decomposable, then in principle it is possible to separate the source signals into \(m\) groups. In [28], the authors proved that the signals can be decoupled, or separated, using only the condition that they are statistically independent, and found even weaker sufficient conditions involving their cross-polyspectra. The results of these works are significant, but they lack further discussion on the separation quality when decoupling is possible, which is crucial and of great concern for the applications. The work had been carried on by several researchers. In [30], [PERSON] studied the most common class of orthogonal invariant algorithms for the blind separation of independent sources in terms of performance, quantified by the rejection rate, and a pairwise lower bound on the performance was given. [PERSON] analyzed the theoretical asymptotic performance of a contrast-based BSS scheme in [31]. Finally, in [32], the Cramer-Rao lower bounds for linear ICA as an algorithm independent theoretical limit of the achievable separation quality was deduced. Limitations and performance bounds of certain ICA algorithms have been indicated in these works, but little attention has been paid to the actual performance under different situations depicted by the mixing processes, the sources as well as the environment, while the issue is actually significant for the applications. Analysis of the performance of several ICA algorithms under varied conditions was referred to in the scenario of blind suppression of interfering signals in direct sequence spread spectrum communication systems [33], but the work was mainly based on computer simulations with little in the way of analytical expression and conclusions. Generally speaking, we think it is of great significance to analyze the performance of ICA algorithms under varied conditions depicted by the mixing matrix, the sources as well as the environment, since conclusions (especially quantitative ones) from such analysis may act as guidance for the applications of wide variety. For instance, it should be helpful to learn how to design the receiving array so as to expect better separation performance and more robustness against intricate situations? Why the algorithm works well for some mixtures while being unsatisfactory for the others? Under what kind of scenarios would the ICA algorithms fail? and further, what measures can be taken when faced with such failure? Answers to such questions are interesting and of great importance. However, little attention has been paid to the issue in the current literature, especially analytical form works. In this manuscript, the typical scenario of antenna array receiving is taken as an illustration for the varied applications involving sensor arrays. The problem of concern is the performance of ICA algorithms under varied situations depicted by the mixing matrix, the sources and the environment noise. The influence of several fundamental factors in practice has been analyzed with quantitative results; factors concerned include the environment noise level, the element spacing of the array, the frequencies of the sources as well as their locations. The analytic connection between the noise variance, the source variance, the condition number of the mixing matrix and the optimal signal to interference-plus-noise ratio is given. Also, the question how the mixing matrix is affected by the element spacing of the array, the frequencies and locations of the radiators is considered, with special concern on the prevention of (nearly) singular mixing matrices. What is more, an effective countermeasure against the situations of singular mixtures has been proposed, on the basis of previous analysis. The manuscript is arranged as follows.