# Performance Analysis of ICA in Sensor Array
## 3 Performance of ICA _vs._ Practical Factors  
### Environment Noise and the Condition Number  
Since restitution of sources via typical ICA algorithms equals solving the linear equation set in nature, it has been well-known that the performance of ICA estimators in the noisy case depends on the condition number of the mixing matrix and the environment noise level. More specifically, it should be intuitively understood that the ICA algorithms would be more sensitive to the noise item when the singularity of the mixing matrix is stronger. Here, we attempt to push the work further via figuring out the upper bound of the ICA separation performance, under certain noise level and the condition number. The concept of the optimal separator that BSS methods aim at implementing asymptotically is introduced in [35], which indicates the upper limit of the separation performance for all ICA algorithms. In [35], the separation performance is measured by the criterion signal to interference-plus-noise ratio (SINR), and the SINR of source \(k\) at output \(i\) of separator \(\mathbf{B}\) is defined therein as:  
\[\mathrm{SINR}_{k}(\mathbf{b}_{i})\triangleq\sigma_{\widetilde{s}_{k}}^{2} \frac{|\mathbf{b}_{i}\mathbf{a}_{k}|^{2}}{\mathbf{b}_{i}\mathbf{R}_{i\backslash k }\mathbf{b}_{i}^{H}} \tag{14}\]  
where \(\mathbf{b}_{i}\) is the \(i\)-th row of \(\mathbf{B}\), \(\sigma_{\widetilde{s}_{k}}^{2}=E\left\{|\widetilde{s}_{k}(t)|^{2}\right\}\) (\({}^{\prime}E\left\{\cdot\right\}^{\prime}\) stands for mathematical expectation), \(\widetilde{s}_{k}(t)\) is the \(k\)-th equivalent source in Equation (5), \(\mathbf{a}_{k}\) is the steering vector of source \(k\), \(\mathbf{R}_{k}\) is the total noisecorrelation matrix temporal mean for the source \(k\), defined by \(\mathbf{R}_{\backslash k}\triangleq\mathbf{R_{x}}-\sigma_{\tilde{\mathbf{x}}_{k}}^{2 }\mathbf{a_{i}}\mathbf{a_{k}^{H}}\), \(\mathbf{R_{x}}\triangleq E\left\{\mathbf{x}(t)\mathbf{x}(t)^{H}\right\}\) is the auto-correlation of the observation, and the superscript \"\(H\)\" indicates conjugate transpose. The restitution quality of the source \(k\) by the separator \(\mathbf{B}\) can be evaluated by the quantity \(\text{SINRM}_{k}\), which is the maximum value of \(\text{SINR}_{k}(\mathbf{b}_{i})\) among all \(1\leqslant i\leqslant N\). It has been verified in [35] that under this criterion, \(\mathbf{B}\) and \(\mathbf{B}\mathbf{A}\mathbf{\Pi}\) have the same performance, in which \(\mathbf{\Lambda}\) is a diagonal matrix and \(\mathbf{\Pi}\) is a permutation matrix. This indicates that the criterion itself is not susceptible to the intrinsic scale-permutation ambiguity of the ICA. For the case of two sources, the \(\text{SINRM}_{1}\) at the output of the optimal separator is given by:  
\[\text{SINRM}_{1}=\varepsilon_{1}(1-\frac{\varepsilon_{2}}{1+\varepsilon_{2}} \left|\alpha_{12}\right|^{2}) \tag{15}\]  
where \(\varepsilon_{k}\triangleq\sigma_{\tilde{\mathbf{x}}_{k}}^{2}\mathbf{a_{k}^{H }}\mathbf{R_{v}^{-1}}\mathbf{a_{k}}\) denotes the signal-to-noise ratio (SNR) at the input of the separator for source \(k\), \(\mathbf{Rv}\) is the auto-correlation of the noise item. \(\alpha_{12}\) is the spatial correlation coefficient between sources \(1\) and \(2\) in the matrix of \(\mathbf{R_{v}^{-1}}\), defined by:  
\[\alpha_{12}\triangleq\frac{\mathbf{a_{1}^{H}}\mathbf{R_{v}^{-1}}\mathbf{a_{2 }}}{\left(\mathbf{a_{1}^{H}}\mathbf{R_{v}^{-1}}\mathbf{a_{1}}\right)^{1/2} \left(\mathbf{a_{2}^{H}}\mathbf{R_{v}^{-1}}\mathbf{a_{2}}\right)^{1/2}} \tag{16}\]  
One thing to be noticed is that when the noise on the sensors are additive white Gaussian (as presumed in this paper), we have \(\mathbf{R_{v}}=\sigma_{\mathbf{v}}^{2}\mathbf{I}\) (where \(\mathbf{I}\) is the unit matrix) and thus \(\varepsilon_{k}=\left(\sigma_{\tilde{\mathbf{x}}_{k}}^{2}/\sigma_{\mathbf{v}}^ {2}\right)\mathbf{a_{k}^{H}}\mathbf{a_{k}}\).