# Experimental Validation of GNSS Interferometric Radio Occultation
## 2 Review of the iRO Processing
If its variation during the integration period is small enough so that it can be considered constant, it can be taken out of the integral. Then, the remaining integral can be approximated by the auto-correlation of the envelope \(\mathbf{A}\left(t\right)\). Note that the smearing of the auto-correlation function caused by the \(u\left(t+t_{1}\right)\) and \(u\left(t+t_{1}-\tau\right)\) terms is minimal, because the temporal duration of the auto-correlation function is much larger than the difference of delays. Note also, that in our case the envelope is real-valued, and any possible smearing of the envelope has no effect on the phase of the resulting auto-correlation. Taking into account only the geometry between transmitter and receiver, the delay \(u\left(t\right)\) is directly proportional to the distance \(r\left(t\right)\) between them  
\[u\left(t\right)=r\left(t\right)/c \tag{7}\]  
Figure 1: Delay of the signal at different instants of time. being \(c\) the speed of light in free space. So, the desired phase term can be written as a function of the distance, speed or Doppler frequency  
\[e^{-j2\pi f_{c}\frac{1}{2}\left[r\left(t+t_{1}\right)-r\left(t+t_{1}-\tau\right) \right]}=e^{-\beta\pi f_{c}\frac{1}{2}\frac{\left[r\left(t+t_{1}\right)-r\left(t +t_{1}-\tau\right)\right]}{\tau}\tau}=e^{-j2\pi f_{c}\frac{1}{2}v\left(t_{1}, \tau\right)\tau}=e^{-j\phi}=e^{-j2\pi f_{d}\tau} \tag{8}\]  
where \(v\left(t_{1},\tau\right)\) is the mean speed in an interval of \(\tau\) length around time instant \(t_{1}\), \(f_{\mathrm{d}}\) is the Doppler frequency, and \(\phi\) is the measured phase. It is necessary to remark that the Doppler frequency estimate in Equation (8) is obtained from a phase measurement, which has a \(k2\pi\) ambiguity. This translates to an ambiguity in the Doppler frequency estimate given by  
\[\hat{f}_{d}=\frac{\phi}{2\pi\tau}+\frac{k}{\tau} \tag{9}\]  
where \(k\) is an integer number. This ambiguity can be solved by using a-priori knowledge of the expected Doppler frequency derived from the transmitter and receiver positions and velocities. We will consider the phase term within the integral Equation (6) to be constant when its variation during the integration interval is smaller than a 100 th of a cycle. This means that  
\[\Delta f_{\mathrm{d}}<\frac{1}{100\tau} \tag{10}\]  
For \(\tau=1\) ms this means that the _change of the Doppler frequency_ during the integration interval \(T_{i}\) must be smaller than 10 Hz. When the Doppler dynamics has larger variations, as it is the case for space-borne RO, the signal processing approach cannot be based on the direct computation of the auto-correlation function of the received signal. A phase compensated auto-correlation function as shown next shall be employed. An open-loop model of the expected phase is used inside the integral to reduce its dynamics to a level, where it can be considered constant within the integration interval as in Equation (6). The phase-compensated auto-correlation is  
\[ \begin{array}{rcl}\mathbf{R}_{z}\left(t_{1},\tau\right)&=&\frac{1}{T_{i}} \int_{-\frac{T_{i}}{2}}^{+\frac{T_{i}}{2}}\mathbf{z}\left(t+t_{1}\right) \mathbf{z}^{*}\left(t+t_{1}-\tau\right)m\left(t_{1},t\right)\mathrm{d}t\\ &=&\frac{1}{T_{i}}\int_{-\frac{T_{i}}{2}}^{+\frac{T_{i}}{2}}\mathbf{z}\left(t +t_{1}\right)\mathbf{z}^{*}\left(t+t_{1}-\tau\right)e^{j\theta\left(t_{1},t \right)}\mathrm{d}t\end{array}  \tag{11}\]  
with  
\[m\left(t_{1},t\right)=e^{i\theta\left(t_{1},t\right)}=e^{j2\pi\hat{f}_{d}\tau} \tag{12}\]  
being a phase model that guarantees that the residual phase term inside the integral is kept almost constant during the integration interval, by using a good prediction (\(\hat{f}_{d}\)) of the Doppler frequency.