# Experimental Validation of GNSS Interferometric Radio Occultation
## 2 Review of the iRO Processing  
As follows, we review the interferometric RO technique that we have used in our validation experiment, and that was first described in [8]. Let \(\mathbf{x}\left(t\right)\) be the signal transmitted by the GNSS satellite  
\[\mathbf{x}\left(t\right)=\mathbf{A}\left(t\right)e^{-\left/2\pi f_{\mathrm{c}} t} \tag{1}\]  
where \(\mathbf{A}\left(t\right)\) is the complex envelope of the signal and \(f_{\mathrm{c}}\) the carrier frequency. In our experiment we have used GPS C/A signals because of the 1 ms periodicity of the PRN sequences. The complex envelope of GNSS signals is given by a sequence of chipping pulses multiplied by a sequence of navigation bits. \[\mathbf{A}\left(t\right)=\sum_{k=0}^{K-1}c_{k}\cdot p\left(t-kT_{p}\right) \cdot\sum_{l=0}^{L-1}b_{l}q\left(t-lT_{l}\right) \tag{2}\]  
where \(p(t)\) is the chipping pulse, \(T_{p}\) the chip duration, \(c=\pm 1\) the chip, \(q(t)\) is the navigation bit pulse, \(T_{l}\) is the navigation bit period and \(b_{l}=\pm 1\) is the navigation bit. The chipping sequence has a length of \(K\) chips, and the temporal duration of the complete code is \(T_{code}=KT_{p}\). For GPS C/A signals \(T_{l}=20T_{code}\). The navigation message bit sequence has length \(L\). The signal \(\mathbf{y}\left(t\right)\) reaching the receiving antenna suffers a non-constant delay \(u\left(t\right)\), that has a monotonous variation. This delay is caused by the fact that transmitter and receiver are in motion, and by the medium through which the signal travels (e.g., atmosphere, ionosphere). The delay \(u\left(t\right)\) can be modeled using a polynomial. \[\mathbf{y}\left(t\right) = \mathbf{x}\left(t-u\left(t\right)\right)=\mathbf{A}\left(t-u \left(t\right)\right)e^{\left/2\pi f_{\mathrm{c}}\left(t-u\left(t\right) \right)} \tag{3}\] \[u\left(t\right) = u_{0}+u_{1}t+u_{2}t^{2}+\ldots=\sum_{s=0}^{S-1}u_{s}t^{s} \tag{4}\]The receiver down-converts the received signal to IF with  
\[ \begin{array}{rcl}\mathbf{z}\left(t\right)&=&\mathbf{y}\left(t\right)e^{-j2\pi f _{\mathrm{lo}}t}\\ &=&\mathbf{A}\left(t-u\left(t\right)\right)e^{-j2\pi f_{\mathrm{lo}}u\left(t \right)}e^{j2\pi f_{\mathrm{if}}t}\end{array}  \tag{5}\]  
where \(f_{\mathrm{lo}}\) is the local oscillator frequency, and \(f_{\mathrm{if}}=f_{\mathrm{c}}-f_{\mathrm{lo}}\) the intermediate frequency. Once the down-converted signal has been recorded, the software post-processor computes the auto-correlation function of the recorded signal at delay \(\tau\), coincident with the code duration. The integration interval is chosen to be the same length as the duration of the code, \(T_{i}=T_{code}\). Finally, the auto-correlation function is computed at a time instant \(t_{1}\). \[ \begin{array}{rcl}\mathbf{R}_{z}\left(t_{1},\tau\right)&=&\frac{1}{T_{i}} \int_{-\frac{T_{i}}{T}}^{+\frac{T_{i}}{T}}\mathbf{z}\left(t+t_{1}\right) \mathbf{z}^{*}\left(t+t_{1}-\tau\right)\mathrm{d}t\\ &=&e^{j2\pi f_{\mathrm{if}}\tau\frac{1}{T_{i}}}\int_{-\frac{T_{i}}{T}}^{+ \frac{T_{i}}{T}}\mathbf{A}\left(t+t_{1}-u\left(t+t_{1}\right)\right)\mathbf{A }^{*}\left(t+t_{1}-\tau-u\left(t+t_{1}-\tau\right)\right)\\ &&e^{-j2\pi f_{\mathrm{c}}\left[u\left(t+t_{1}\right)-u\left(t+t_{1}-\tau \right)\right]}\mathrm{d}t\end{array}  \tag{6}\]  
The complex exponential term outside the integral is dependent on the intermediate frequency \(f_{\mathrm{if}}\) and on the delay \(\tau\) at which the auto-correlation is computed. To be able to compensate this term, precise knowledge of these terms is required. In our experiment \(\tau\) is perfectly known, as it is proportional to the lag spacing, which is the inverse of the sampling frequency \(F_{s}\), and the sampling frequency is measured precisely respect to the GPS time frame. The intermediate frequency \(f_{\mathrm{if}}\) is also known as it is coherently derived from the sampling frequency. The phase term inside the integral is the one we are interested in. It depends on the variation of the delay \(u\left(t\right)\). Figure 1 shows a conceptual plot of this delay as a function of time.