## 2 Methodology
### Radio Interferometer Phase Detection
The downconversion process removes the carrier exponential term \(2\pi f_{c}t+\phi_{0}\) from Equation (5), yielding the following band-pass-equivalent expression:  
\[v_{BB}(t)=A_{r}A\cdot s(t-t_{d})\cdot e^{i(2\pi(f_{c}-f_{r})t-\alpha_{d}+\phi_{ 0}+\phi_{r}(t))}+A_{r}\omega_{r}(t) \tag{6}\]  
where \(A_{r}\) is the amplitude attenuation due to the receiver gain, \(f_{r}\) is the carrier frequency estimation, and \(\phi_{r}(t)\) is the phase introduced by each receiver downconverter whose average value is constant, thus calibrated, since the receivers local oscillators are synchronized with Phase-Locked-Loop (PLL) subsystems with the interferometer's main clock. The fluctuations on \(\phi_{r}(t)\) are due to the phase noise of the system, which is designed to be smaller than the phase changes induced by atmospheric turbulence. Then, the interferometric signals are obtained from the correlation of the received signal from two different receivers during a limited integration period. The cross-correlation of two power signals, \(v(t)\) and \(w(t)\), is defined as:  
\[R_{vw}(\tau)=\langle v_{BB}(t),w_{BB}^{*}(t-\tau)\rangle \tag{7}\]  
This is a scalar product with the second signal delayed by \(\tau\) relative to the first. For each baseline, the correlator performs a sweep of the delay (\(\tau\)) to find the value (\(\tau_{0}\)) that aligns the two received signals, maximizing the magnitude of the correlation and obtaining the phase differences. The magnitude of the correlation is defined as:  
\[\|R_{vw}(\tau)\|=\frac{R_{vw}(\tau)}{R_{vw}(0)} \tag{8}\]  
where \(R_{vw}(0)\) corresponds to an ideal correlation where the signals are identical and perfectly aligned. It is defined by the maximum value of the integrated samples along the integration time. When normalized, \(\|R_{vw}(0)\|=1\). In our case, the noise of the received signals, atmospheric turbulence and other factors limit the correlation magnitude to 0.8 under experimental conditions. The result of developing Equation (7) yields the correlator's output (\(R_{u}\)):  
\[R_{u}(\tau_{0})=A_{r}^{2}A_{v}A_{w}R_{s}(\tau_{0})\cdot e^{\alpha_{v}-\alpha_{ w}-\alpha_{v_{0}}}+\eta \tag{9}\]  
where \(A_{v}\) and \(A_{w}\) correspond to the amplitude of the received signals at each antenna, \(v\) and \(w\), respectively, and \(\alpha_{v}\) and \(\alpha_{w}\) correspond to the phase shift of each signal along the electric path between the satellite and the receiver. The first term contains the autocorrelation of the received baseband signal (\(R_{s}\)), and \(\alpha_{v_{0}}\) is a residual phase introduced by the discrepancy between the average \(\phi_{r}\) value of each receiver. This is caused by cabling length differences and the phase differences between the local oscillators, which keep the system coherent since they are synchronized with a stable reference main system clock. The \(\alpha_{v_{0}}\) term is determined by means of the interferometric system calibration. The second term (\(\eta\)) groups the noise terms with a zero average value resulting from the cross-products between the signals and noises of the two channels. Note that only a single-lag component of the correlation is required to retrieve the phase difference from a point source. Both received signals are considered properly aligned if the alignment error is smaller than the correlation time of the received signal (\(\Delta\tau\)), which for a DVB-S signal is:  
\[x(t)=s(t)e^{i(2\pi f_{c}t+\phi_{0})} \tag{2}\]0  
Finally, the interferometric phase, \(\alpha\), is obtained as the angle of the interference signal phasor. \[x(t)=s(t)e^{i(2\pi f_{c}t+\phi_{0})} \tag{2}\]1This interferometric phase is used as the orbit observable in order to retrieve the satellite trajectory. Mathematically, the interferometric phase obtained by each pair of antennas in Equation (11) can only be retrieved in the interval \((0,2\pi]\).