# Performance Analysis of ICA in Sensor Array
## 2 Hypotheses, Definitions and Notations
### Sensor Array Output  
In this manuscript, the typical scenario of a receiving antenna array such as in spaceborne non-cooperative communication is taken as an illustration for the varied applications involving sensor arrays, as depicted by Figure 1.  
The array consists of \(P\) narrowband (NB) omnidirectional antennas and we call \(\mathbf{x}(t)\) the vector of complex amplitude of signals at the output. Each antenna is assumed to receive the contribution of \(N\) zero-mean stationary NB sources that are statistically independent. Under these assumptions, we model \(\mathbf{x}(t)\) as:  
\[\mathbf{x}(t)=\sum_{i=1}^{N}\alpha_{i}(t)\cdot e^{j2\pi v_{i}t}\cdot e^{-j2\pi f _{i}\tau_{i}(t)}\cdot s_{i}(t)\cdot\mathbf{a}_{i}(t)+\mathbf{v}(t),\ t\in[0,T] \tag{4}\]  
where \"\(j\)\" is the imaginary unit, \(T\) is the observation length, \(\alpha_{i}(t)\), \(v_{i}\) and \(\tau_{i}(t)\) are respectively the attenuation, the frequency offset and the propagation delay of the channel corresponding to source \(i\) at time \(t\);\(f_{i}\) is the carrier frequency of the NB source \(i\); \(s_{i}(t)\) is the complex amplitude of the corresponding source; \(\mathbf{a}_{i}(t)\) is the equivalent steering vector (or spatial signature) of source \(i\) at time \(t\), which is a \(P\times 1\) vector depending on the array, the direction of arrival (DOA) and is also relevant with carrier frequencies of sources. \(\mathbf{v}(t)\) is the noise vector, assumed to be zero-mean, stationary, circular, Gaussian and spatially white, with variance \(\sigma_{\
u}^{2}\) in the sampling band of frequency. In addition, the array is set as a uniform linear array (ULA), the element spacing of which is \(d\), and sensors are disposed along the (\(Ox\)) axis. Notice that the sources are presumed to be statistically independent, which is generally the case for physically separated sources in reality. Figure 1: Spaceborne non-cooperative communication with antenna array. Regarding the first unit of the array as the datum unit and denoting \(N\) complex equivalent sources arriving at the datum unit as \(\widehat{s}_{i}(t)\), we have:  
\[\widehat{s}_{i}(t)\doteq\alpha_{i}(t)\cdot e^{[2\pi v_{i}t}\cdot e^{-2\pi f_{i} \gamma_{i}(t)}\cdot s_{i}(t) \tag{5}\]  
Then Equation (4) can be rearranged as:  
\[\mathbf{x}(t)=\sum_{i=1}^{N}\widehat{s}_{i}(t)\cdot\mathbf{a}_{i}(t)+\mathbf{v }(t),\ t\in[0,T] \tag{6}\]  
Regarding the steering vector to be constant within the observation window, then \(\mathbf{a}_{i}(t)\) can be explicitly expressed as:  
\[ \begin{split}&\mathbf{a}_{i}(t)=\mathbf{a}_{i}(\theta_{i},\varphi_{ i},\lambda_{i})=[1,\mathbf{a}_{i2}(\theta_{i},\varphi_{i},\lambda_{i}),..., \mathbf{a}_{iP}(\theta_{i},\varphi_{i},\lambda_{i})]^{T}\\ &=\exp\left\{\frac{\left(2\pi[\mathbf{x}\cdot cos(\theta_{i}) \cdot cos(\varphi_{i})+\mathbf{y}\cdot sin(\theta_{i})\cdot cos(\varphi_{i})+ \mathbf{z}\cdot sin(\varphi_{i})]\right)}{\lambda_{i}}\right\}\end{split}  \tag{7}\]  
where \(\theta_{i}\) and \(\phi_{i}\) are the azimuth and elevation incident angles of the source \(i\), \(\theta_{i}\in[-90^{\circ},90^{\circ}]\); \(\varphi_{i}\in[-90^{\circ},90^{\circ}]\); \(\lambda_{i}\) is the wavelength corresponding to the carrier frequency of the NB source \(i\), \(\lambda_{i}=c/f_{i}\) (\(c\) denotes the speed of light); (\(\mathbf{x}\),\(\mathbf{y}\),\(\mathbf{z}\)) are the coordinates of sensors in the array, as depicted in Figure 1; the superscript \"\(T\)\" denotes transpose. Without the loss of generality, we consider the fundamental situation where \(P=N=2\). The coordinate of the second sensor is (\(d\),\(0\),\(0\)) and DOAs of sources are (\(\theta_{1}\),\(\varphi_{1}\)) and (\(\theta_{2}\),\(\varphi_{2}\)) respectively, as depicted by Figure 2. Moreover, the frequency offsets of sources are supposed to be negligible, and the propagation channels are assumed to be time invariant over the observation duration. Hence, the mixing matrix \(\mathbf{A}\) whose columns are vectors \(\mathbf{a}_{i}(\theta_{i}\varphi_{i},\lambda_{i})\) will be:  
\[\mathbf{A}=\left[