## 2 System Model  
Assume an _N_-element antenna array with an arbitrary configuration placed on a platform which is fixed with respect to the body frame coordinate system. In this configuration, one antenna is chosen as a reference antenna. For the sake of simplicity, it is assumed that the origin of the body frame coordinate system is located at the reference antenna location shown in Figure 1.  
Under the assumption of perfect antenna elements (identical isotropic elements) and ignoring the measurement noise, the _N_\(\times\)1 complex baseband signal vector consisting of measured phases and amplitudes at different antennas received from one satellite can be written as:  
\[\textbf{y =Ca}e^{\mathit{\beta(t)}} \tag{1}\]  
where \(\mathit{\beta(t)}\) considers the phase changes due to the Doppler frequency variation, which is a function of time due to both satellite and vehicle motions and **a** is an _N_\(\times\)1 vector representing the steering vector of the satellite signal defined as [16]:  
Figure 1: Antenna array configuration in the body frame coordinate system. \[\mathbf{a}\triangleq \begin{bmatrix}1\\ e^{j\frac{2\pi}{\kappa}\left(\mathbf{e}^{\mu}\right)^{T}\mathbf{r}_{2}}\\ \vdots\\ e^{j\frac{2\pi}{\kappa}\left(\mathbf{e}^{\mu}\right)^{T}\mathbf{r}_{\mu}}\end{bmatrix}  \tag{2}\]  
in which \(\kappa\) is the wavelength of the signal and \(\mathbf{r}_{n}\), \(n=2,\ldots,N\) is a 3\(\times\)1 vector pointing to the \(n^{\text{th}}\) antenna element and \(\mathbf{e}^{n}\) is a 3\(\times\)1 unit vector pointing to the satellite in the body frame coordinate system. Without loss of generality, the first antenna is chosen as the reference antenna where the coordinate system origin is located. In Equation (1), \(\mathbf{C}\) is a \(N\times N\) matrix representing constant uncertainties due to any imperfection from antennas to digitizers. The diagonal elements of \(\mathbf{C}\) model the difference in amplitudes and phases due to unequal cable lengths and other electronic parts and its off-diagonal elements represent the coupling coefficients between antennas. Nevertheless this model is not yet sufficiently accurate since the antenna elements are not isotropic and they have different gain and phase radiation patterns. Moreover, phase center variations and uncertainties of the array configuration should be also taken into the account to increase the accuracy of the calibration process. To include these variant uncertainties, \(\mathbf{y}\) should be re-expressed as:  
\[\mathbf{y}=\mathbf{C}\mathbf{P}(\varphi^{B},\lambda^{B})\mathbf{a}e^{j\theta \left(x\right)} \tag{3}\]  
where \(\mathbf{P}(\varphi^{B},\lambda^{B})\) is a \(N\times N\) matrix defined as:  
\[\mathbf{P}(\varphi^{B},\lambda^{B})\] \[\triangleq \begin{bmatrix}a_{1}(\varphi^{B},\lambda^{B})e^{j\alpha _{1}(\varphi^{B},\lambda^{B})}e^{j\frac{2\pi}{\kappa}\left(\mathbf{e}^{\mu} \right)^{T}\alpha_{2}}\\ 0&a_{2}(\varphi^{B},\lambda^{B})e^{j\alpha_{1}(\varphi^{B},\lambda^{B})}e^{j \frac{2\pi}{\kappa}\left(\mathbf{e}^{\mu}\right)^{T}\alpha_{2}}\\ \vdots&&\ddots&\vdots\\ 0&0&\cdots&a_{N}(\varphi^{B},\lambda^{B})e^{j\alpha_{N}(\varphi^{B},\lambda^ {B})}e^{j\frac{2\pi}{\kappa}\left(\mathbf{e}^{\mu}\right)^{T}\alpha_{N}}\end{bmatrix}  \tag{4}\]  
in which \(\Delta\mathbf{r}_{n}\) is a 3\(\times\)1 vector representing the perturbation of the \(n^{\text{th}}\) antenna element position in the body frame coordinate system, and \(a_{n}(\varphi^{B},\lambda^{B})\) and \(\omega_{n}(\varphi^{B},\lambda^{B})\) are amplitude and phase deviations of the received signal at elevation \(\lambda^{B}\) and azimuth \(\varphi^{B}\) in the body frame coordinate system for the \(n^{\text{th}}\) antenna element. This term considers imperfection of the antenna elements, radiation patterns, phase center variations and any other signal DOA-dependent phenomena.