## 2 System Model
Since the proposed calibration method is based on receiving GNSS signals during antenna platform motion, Doppler frequency varies due to the changes in the relative velocity between the antenna array and the satellites, which are not negligible. Even in the static case, Doppler frequency is gradually changing due to satellite motion. In the calibration process using a multi-antenna receiver, acquisition, tracking and positioning procedures can be only performed for the received satellite signals at the reference antenna (reference antenna) and the generated local codes can be used to despread the received signals at the other antennas (secondary antennas) as shown in Figure 2.  
During the tracking stage of the signal of each satellite, the phase of the signal at the reference antenna is kept approximately zero and the phase of the signals at the other antennas are measured relative to that of the reference antenna. For simplicity, amplitudes are also normalized with respect to the signal at the reference antenna. This removes the term responsible for Doppler frequency changes from observations as well as phase changes due to the alteration of signal DOA for the master antenna. Therefore, the normalized signal vector that is actually obtained from the multi-antenna receiver, denoted by \(\tilde{\mathbf{y}}\), is related to \(\mathbf{y}\) as:  
\[\tilde{\mathbf{y}}=\frac{\mathbf{y}}{\delta^{T}\mathbf{CPa}e^{\jmath\theta(t)}} \tag{5}\]  
where \(\tilde{\mathbf{s}}_{N\times 1}= \begin{bmatrix}1&0&\cdots&0\end{bmatrix} ^{T}\) (with the first antenna chosen as the reference antenna). In Equation (4), \(\mathbf{P}(\varphi^{B},\lambda^{B})\) is a function of \(\varphi^{B}\) and \(\lambda^{B}\) in the body frame coordinate system. However, a GNSS receiver provides azimuth and elevation angles in the geographic coordinate system such as local coordinate system ENU. Azimuth and elevation angles in the ENU coordinate system, symbolized with \(\varphi^{ENU}\) and \(\lambda^{ENU}\), can be related to the \(\varphi^{B}\) and \(\lambda^{B}\) as:  
\[\mathbf{e}^{B}=\mathbf{R}^{T}\mathbf{e}^{ENU} \tag{6}\]  
in which \(\mathbf{e}^{B}\) and \(\mathbf{e}^{ENU}\) are unit vectors pointing to the satellite in the body frame and ENU coordinate systems, respectively, which can be expressed as:  
\[ \begin{split}\mathbf{e}^{B}=&\left[\sin\left(\varphi^{B }\right)\cos\left(\lambda^{B}\right)\ \ \cos\left(\varphi^{B}\right)\cos\left(\lambda^{B}\right)\ \ \sin\left(\lambda^{B}\right)\right]^{T}\\ \mathbf{e}^{ENU}=&\left[\sin\left(\varphi^{ENU} \right)\cos\left(\lambda^{ENU}\right)\ \ \cos\left(\varphi^{ENU}\right)\cos\left(\lambda^{ENU}\right)\ \ \sin\left(\lambda^{ENU}\right)\right]^{T} \end{split}  \tag{7}\]  
Figure 2: Multi-antenna receiver structure in the calibration mode. In Equation (6), **R** is a transformation matrix from the ENU coordinate to the body frame coordinate defined as:  
\[\textbf{R}= \begin{bmatrix}\cos\big{(}y\big{)}\cos\big{(}r\big{)}-\sin\big{(}r \big{)}\sin\big{(}p\big{)}\sin\big{(}y\big{)}&-\sin\big{(}y\big{)}\cos\big{(}p \big{)}&\cos\big{(}y\big{)}\sin\big{(}r\big{)}+\sin\big{(}y\big{)}\sin\big{(}p \big{)}\cos\big{(}r\big{)}\\ \sin\big{(}y\big{)}\cos\big{(}r\big{)}+\sin\big{(}r\big{)}\sin\big{(}p\big{)} \cos\big{(}y\big{)}&\cos\big{(}y\big{)}\cos\big{(}p\big{)}&\sin\big{(}y\big{)} \sin\big{(}r\big{)}-\cos\big{(}y\big{)}\sin\big{(}p\big{)}\cos\big{(}r\big{)} \\ -\sin\big{(}r\big{)}\cos\big{(}p\big{)}&\sin\big{(}p\big{)}&\cos\big{(}p\big{)} \cos\big{(}r\big{)}\end{bmatrix}  \tag{8}\]  
in which \(r\), \(p\) and \(y\) refer to the roll, pitch and yaw (heading) angles, respectively (shown in Figure 1). In the proposed method, an IMU is providing accurate estimates of these angles.