# Experimental Validation of GNSS Interferometric Radio Occultation
## 4 Data Processing, Analysis and Discussion
### Interferometric RO Processing
#### 4.3.3 SNR Analysis  
The precision of Doppler estimation relies on the SNR of the interferometric waveform. In Figure 9, the evolution of the SNR is presented together with the elevation and the iRO Doppler estimation. The SNR for the clean-replica processing has been obtained with an integration time of 1 ms, while for the iRO processing the measurements have been further integrated to 1 s. Thus the vertical scale on the plots shows such a big difference for both types of measurements. It is worth to mention that the SNR of the received signal shows strong dispersion between SoD48000 and SoD50600 with the elevation between 17\({}^{\circ}\) and 0\({}^{\circ}\). That could be explained by the interference from the sea surface reflected signal which causes fading of received signal. At grazing geometries the polarization of the reflected signal is mainly at the same RHCP polarization as the direct signal. \begin{table}
\begin{tabular}{c c c} \hline \hline
**SoD** & **\textless{}48,000** & **48,000–50,600** \\ \hline iRO Raw & 79.0 Hz & 30.5 Hz \\ iRO with Fixed Correction & 19.9 Hz & 10.4 Hz \\ iRO with P2P Correction & 22.4 Hz & 11.1 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard deviation of the estimated Doppler respect to the predicted one in two different time periods. Figure 8: iRO Doppler estimation for PRN24 with removed effect of PRN123. From [14], the SNR of the interferometric waveform \(SNR_{INT}\) can be expressed as  
\[SNR_{INT}=\frac{T_{\mathrm{C}}^{INT}}{T_{\mathrm{C}}^{\mathrm{CR}}}\frac{SNR_{CR}} {1+\frac{1+SNR_{PRE}}{SNR_{PRE}}}=\frac{T_{\mathrm{C}}^{INT}}{T_{\mathrm{C}}^{ \mathrm{CR}}}\frac{SNR_{CR}}{2+\frac{1}{SNR_{PRE}}} \tag{20}\]  
Figure 9: Evolution of SNR for both the clean replica processing and the iRO processing. (**a**) SNR using the clean-replica method. (**b**) SNR using the iRO method. (**c**) iRO Doppler estimation with fixed correction. where \(SNR_{CR}\) is the SNR of the clean-replica open loop waveform, \(SNR_{PRE}\) is the SNR at the input of the correlator, \(T_{\mathrm{C}}^{CR}\) is the coherent integration time of the clean replica waveform and \(T_{\mathrm{C}}^{INT}\) is the coherent integration time of the interferometric waveform. The quotient of the different integration times serves to take into account the different integration times for the clean replica and interferometric observables. Given the bandwidth of the received signal \(B\), we have  
\[SNR_{CR}=B\cdot T_{\mathrm{C}}^{CR}\cdot SNR_{PRE} \tag{21}\]  
With Equations (20) and (21), the relationship between the clean replica and interferometric SNRs can be derived as  
\[SNR_{INT}=\frac{T_{\mathrm{C}}^{INT}}{T_{\mathrm{C}}^{CR}}\frac{SNR_{CR}}{2+ \frac{B\cdot T_{\mathrm{C}}^{CR}}{SNR_{CR}}} \tag{22}\]  
Given the parameters in this signal processing: \(B=2\) MHz, \(T_{\mathrm{C}}^{INT}=1\) s and \(T_{\mathrm{C}}^{CR}=1\) ms, the theoretical relationship between the clean replica SNR and the interferometric SNR is presented in Figure 10 with a continuous line. It can be seen that the relationship between \(SNR_{CR}\) and \(SNR_{INT}\) from the measured data is consistent with that derived theoretically. In addition, the fact that the data points follow the shape of the line is indicating that the SNR variations in Figure 9 are the same for the clean replica and interferometric methods, and are thus not random noise, but correspond to actual fading which is consistently observed with both methods. The actual SNR data plotted in Figure 10 is 2.78 dB (mean value) below the theoretical model, indicating that \(SNR_{INT}\) is lower than expected. A possible explanation to this could be the fact that the raw data has been acquired with 1-bit quantization, which induces a degradation of SNR. However, such degradation has not been taken into account in Equation (22). According to [15] the computation of the cross-correlation of one bit quantized Gaussian noise suffers a SNR loss of \(2/\pi=0.637\), which corresponds to 2 dB.