## 4 Signal Processing
### Independent Component Analysis
#### 4.4.2 Process of ICA  
The aim of this step is to reduce the dimension of data and separate respiratory signal using blind source separation. In ICA model, the observed signals are described as the linear combination of source signals, including respiratory signal and platform motion signal. Therefore, the observation of each extracted range bin can be written as [29]:  
\[ \begin{array}{l}y_{1}(t)=a_{11}s_{1}(t)+a_{12}s_{2}(t)+\cdots+a_{1n}s_{n}(t) \\ y_{2}(t)=a_{21}s_{1}(t)+a_{22}s_{2}(t)+\cdots+a_{2n}s_{n}(t)\\ \vdots\\ y_{a}(t)=a_{n1}s_{1}(t)+a_{n2}s_{2}(t)+\cdots+a_{m}s_{n}(t)\end{array}  \tag{17}\]  
where \(y_{1}(t),\;y_{2}(t),\cdots,y_{a}(t)\) are the mixed observed signals, \(s_{1}(t),\;s_{2}(t),\cdots,s_{n}(t)\) are the unknown source signals (i.e., independent components) and \(a_{ij}\) is the weight of each source signal. Matrix representation of equation (17) is given as [30]:  
\[\mathbf{Y}=\mathbf{AS}+\mathbf{E} \tag{18}\]where \(\mathbf{A}\in R^{a\times n}\) is the mixing matrix, \(\mathbf{Y}\in R^{a\times m}\) is the matrix of the observed signals, \(\mathbf{S}\in R^{n\times m}\) is the independent component matrix and \(\mathbf{E}\) is the residual matrix, and these matrixes can be expressed as:  
\[\mathbf{A}=\left[ \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{12}\\ a_{21}&a_{12}&\cdots&a_{12}\\ \cdots&\cdots&\cdots&\cdots\\ a_{11}&a_{12}&\cdots&a_{12}\end{array} \right],\ \mathbf{Y}=\left[ \begin{array}{c}y_{1}(t)\\ y_{2}(t)\\ \cdots\\ y_{a}(t)\end{array} \right],\ \mathbf{S}=\left[ \begin{array}{c}s_{1}(t)\\ s_{2}(t)\\ \cdots\\ s_{n}(t)\end{array} \right] \tag{19}\]  
Since the matrixes \(\mathbf{A}\) and \(\mathbf{S}\) are unknown, and the matrix \(\mathbf{Y}\) can be observed, we need to use \(\mathbf{Y}\) to estimate \(\mathbf{A}\) and \(\mathbf{S}\). The fundamental of ICA is to estimate mixing matrix \(\mathbf{A}\) so that the reconstructed components are as independent as possible. The inverse matrix of \(\mathbf{A}\) is denoted by \(\mathbf{W}\), then, the independent components can be calculated as  
\[\mathbf{S}=\mathbf{W}\mathbf{Y} \tag{20}\]  
Here are several existing ICA algorithms to estimate the de-mixing matrix \(\mathbf{W}\). In this paper, we use FastICA to apply blind signal separation on account of its very fast convergence and ease of operation. FastICA is based on a fixed-point algorithm and the first step of pre-processing is to centre the input data, which could simplify the ICA algorithm. This process is achieved by subtracting its mean vector [28]. \[\overset{\boldsymbol{\wedge}}{\mathbf{Y}}=\mathbf{Y}-E(\mathbf{Y}) \tag{21}\]  
The second step of pre-processing is to whiten the input data matrix \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\) and obtain a new white matrix \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\), which aims at eliminating the correlation among the observed signals. Therefore, \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\) satisfies \(E\bigg{(}\overset{\boldsymbol{\wedge}}{\mathbf{Y}}^{\mathsf{T}}\bigg{)}= \mathbf{I}\). Here, the eigen-value decomposition (EVD) is used to whiten data. After EVD, the covariance matrix of \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\) is expressed by  
\[E\bigg{(}\overset{\boldsymbol{\wedge}}{\mathbf{Y}}^{\mathsf{T}}\bigg{)}= \mathbf{U}\boldsymbol{\Lambda}\mathbf{U}^{T} \tag{22}\]  
where \(\mathbf{U}\) and \(\boldsymbol{\Lambda}\) are the eigenvector matrix and diagonal matrix composed of eigenvalues. The detailed process is summarized in Algorithm 1, where the function \(g(\cdot)\) used in the algorithm is [31]  
\[g(u)=u\exp\Bigl{(}-u^{2}/2\Bigr{)} \tag{23}\]  
```
1:Input the observed signals. 2:Centre the data to give \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\). 3:Whiten the data to give \(\overset{\boldsymbol{\wedge}}{\mathbf{Y}}\). 4:Choose the number of independent components \(m\). 5:For\(p=1\) to \(m\)
6:Initialize the weight vector \(\mathrm{w}_{p}\).