# Identification of C-Band Radio Frequency Interferences from Sentinel-1 Data
## 4 Sensitivity  
The Effective Isotropic Radiated Power (EIRP) of the transmitter causing the RFI contributes to S1 receiver with a power:  
\[P_{R}=\frac{P_{EIRP}\cdot A_{S}\cdot\eta}{4\pi R^{2}}, \tag{4}\]  
where \(As\) is the S1 antenna area, \(\eta\) the total losses and \(R\) the range. The sensitivity is the RFI power that cause the same contribution as noise at S1 receiver:  
\[P_{n}=K_{B}\cdot T_{S}\cdot B_{RFI}, \tag{5}\]  
\(K_{B}\) being Boltzmann constant, \(B_{RFI}\) the RFI bandwidth, \(T_{S}\) is S1 equivalent noise temperature. We can then define the minimal power \(P_{EIRPZ}\) of the RFI that gives the same contribution as thermal noise, by equating (4) and (5):  
\[P_{EIRPZ}=\frac{K_{B}\cdot T_{S}\cdot B_{RFI}\cdot 4\pi R^{2}}{A_{S}\cdot \eta}. \tag{6}\]  
Figure 6: Example of detection of low-power RFI. Top and middle panels on the left: spectrograms for V and H polarizations. Each vertical strip corresponds to the 9 pulses at the beginning of a bursts, all coming from the same sub-swath (IW1). The strongest RFI are encircled. Lower left panel: plot of KL divergence for each burst (\(x\)-axis define the burst number, and it has been aligned with the spectrograms above). Right panels: histograms, in log scale, and fitting normal PDF, for the bursts with K \(>\) 0.025 and K \(<\) 0.025. We can achieve the same result by assuming that S1 is illuminating a homogenous target with backscatter \(\sigma_{NESZ}\) (where NESZ stands for Noise Equivalent Sigma Zero), by a means of a mean power that is the product between S1 peak power and the duty cycle, and contributing to the RFI bandwidth \(B_{RFI}\):  
\[P_{EIRPZ}=P_{S}\cdot dc\cdot\sigma_{NESZ}\cdot\frac{B_{RFI}}{B_{S}} \tag{7}\]  
The evaluation of the sensitivity from (7) is straightforward, given the S1 peak power, \(Ps\) = 5.2 kW, the duty cycle, \(dc\) = 9%, the bandwidth, \(Bs\) = 50-70 MHz (depending on the swath, see Figure 4) and the \(\sigma_{NESZ}\) as from [14; 15]. If we assume S1 requirement \(\sigma_{NESZ}<-22\) dB, and a ratio \(B_{RFI}/Bs\) of 100 looks over the 20,000 spectral samples in range, we get from (7) \(P_{EIRPZ}\) = 15 mW for a single tone RFI. This sensitivity becomes as small as 8 mW if the actual figure for S1 \(\sigma_{NESZ}=-25\) dB is assumed [14; 15]. In that case, the equivalent noise temperature from (6) and (7):  
\[T_{s}=\frac{P_{S}\cdot dc\cdot\sigma_{NESZ}\cdot A_{S}\cdot\eta}{K_{B}\cdot B_ {RF}\cdot 4\pi R^{2}} \tag{8}\]  
results in roughly 800 K, by assuming an equivalent antenna area \(A_{S}\) = 9.6 m\({}^{2}\), total losses \(\eta=-4.5\) dB and \(R\) = 840 km [14; 15]. The sensitivity here computed refers to a single tone RFI, and does not allow to detect such slight RFI in the raw data in time domain. However, for the strongest RFI power exceeds by far that value, say over a factor 20,000/100 = 20, a time domain identification is possible, like the case shown in Figure 7, that refers to the strongest RFI found. The peak power spectrum is 32 dB above the thermal noise level, and the bandwidth is about 5 MHz, that, from (7) evaluated for IW3, gives \(P_{EIRP}\) = 250 W. Such a high power is still in the dynamic range of S1, from the level of \(-3\) dB to \(\sigma_{NESZ}\)+39 dB, say with 10 dB margin. The method here proposed has been used for estimating RFI all over Europe, in different times--by exploiting ascending and descending passes, and different periods: from November to December 2016 and from February to March 2017. A total of 960 products were considered, spanning 32,000 bursts. In the analysis, we aimed to a very high sensitivity, therefore we discarded the first one-two pulses at the beginning of each burst, since there were still affected by some residual ripples, as shown in Figure 5.  
Figure 7: Example of high-power RFI in the spectrogram (**top**); and in the time-along track domain (**bottom**). Color scales are in dB. Horizontal axis is in number of rank pulses that is 10 for IW3.