# Experimental Validation of GNSS Interferometric Radio Occultation
## Appendix A Local Oscillator Offset  
In this section we present the equations that show the effect of the local oscillator frequency offset on the Doppler estimates in the interferometric radio-occultations technique. We use the following definitions using boldface letters to design complex signals  
\[\mathbf{x}\left(t\right) = \mathbf{A}\left(t\right)e^{i2\pi f_{\mathrm{c}}t} \tag{1}\] \[\mathbf{y}\left(t\right) = \mathbf{A}\left(t-u\left(t\right)\right)e^{i2\pi f_{\mathrm{c}} \cdot\left[t-u\left(t\right)\right]}\] (2) \[\mathbf{z}_{ref}\left(t\right) = \mathbf{y}\left(t\right)e^{-j2\pi f_{\mathrm{lo}}t} \tag{3}\]  
where \(\mathbf{x}\left(t\right)\) is the transmitted signal, where \(\mathbf{A}\left(t\right)\) is the complex envelope of the signal and \(f_{\mathrm{c}}\) the carrier frequency, \(\mathbf{y}\left(t\right)\) is the signal that suffered a variable delay \(u\left(t\right)\) received at the antenna, and \(\mathbf{z}_{ref}\left(t\right)\) is the down-converted signal where \(f_{\mathrm{lo}}\) is the local oscillator frequency. \(\mathbf{z}_{ref}\left(t\right)\) is our reference signal that we have collected when then transmitting satellite was close to Zenith. The portion of the signal when the satellite was occulting is given by  
\[\mathbf{z}\left(t+\tau\right)=\mathbf{y}\left(t+\tau\right)e^{-j2\pi\left(f_{ \mathrm{lo}}+f_{\mathrm{eff}}\right)\left[t+\tau\right]} \tag{4}\]  
where the local oscillator has now an offset at its frequency given by \(f_{\mathrm{off}}\). Thus, the down-converted signals can be expressed as  
\[\mathbf{z}_{ref}\left(t\right) = \mathbf{A}\left(t-u\left(t\right)\right)e^{j2\pi f_{\mathrm{c}} \left[t-u\left(t\right)\right]}\cdot e^{-j2\pi f_{\mathrm{lo}}t} \tag{5}\] \[\mathbf{z}\left(t+\tau\right) = \mathbf{A}\left(t+\tau-u\left(t+\tau\right)\right)e^{j2\pi f_{ \mathrm{c}}\left[t+\tau\right]}\cdot e^{-j2\pi f_{\mathrm{c}}u\left(t+\tau \right)}\cdot e^{-j2\pi\left(f_{\mathrm{lo}}+f_{\mathrm{eff}}\right)\left[t+ \tau\right]} \tag{6}\]  
The variable delay \(u\left(t\right)\) can be expressed with a sum of two terms, for which the first term \(u_{\mathrm{m}}\left(t\right)\) can be modeled (geometry, standard atmosphere), and the second term cannot be modeled \(u_{\mathrm{x}}\left(t\right)\) and carries the physical information we want to measure. \[u\left(t\right)=u_{\mathrm{m}}\left(t\right)+u_{\mathrm{x}}\left(t\right) \tag{7}\]  
The cross-correlation between the occulting signal \(\mathbf{z}\left(t+\tau\right)\) and the reference signal \(\mathbf{z}_{ref}\left(t\right)\) is given by  
\[ \begin{array}{rcl}\mathbf{R}\left(\tau\right)&=&\int_{0}^{T}\mathbf{z}\left( t+\tau\right)\mathbf{z}_{ref}^{\ast}\left(t\right)e^{-j\theta\left(t\right)} \mathrm{d}t\\ &=&\int_{0}^{T}\mathbf{A}\left(t+\tau-u\left(t+\tau\right)\right)\mathbf{A}^{ \ast}\left(t-u\left(t\right)\right)\\ &&\cdot e^{j2\pi\left(f_{\mathrm{c}}-f_{\mathrm{lo}}\right)\tau}\cdot e^{-j2 \pi f_{\mathrm{eff}}\tau}\\ &&\cdot e^{j2\pi f_{\mathrm{c}}\left[u_{\mathrm{x}}\left(t\right)-u_{\mathrm{ x}}\left(t+\tau\right)\right]}e^{j2\pi f_{\mathrm{c}}\left[u_{\mathrm{m}} \left(t\right)-u_{\mathrm{m}}\left(t+\tau\right)\right]}\\ &&\cdot e^{j2\pi f_{\mathrm{eff}}t}e^{-j\theta\left(t\right)}\mathrm{d}t\end{array}  \tag{8}\]  
The phase term \(\theta\left(t\right)\) is used to compensate all known phase terms so that the remaining phase terms in the integral can be considered constant during the integration time \(T\). In particular it compensates the local oscillator frequency, expressed as the difference between the carrier frequency and the local oscillator, and the variable-delay terms that can be modeled. \[\theta\left(t\right)=2\pi\left(f_{\mathrm{c}}-f_{\mathrm{lo}}\right)\tau-2 \pi f_{\mathrm{c}}\left[u_{\mathrm{m}}\left(t\right)-u_{\mathrm{m}}\left(t+ \tau\right)\right] \tag{9}\]In this case, the correlation function yields  
\[