## 5 Results
### Smoothing Algorithm
#### 5.1.2 Smoothing Algorithm Applied with Broadband RFI  
As broadband communication systems are increasing exponentially, it is likely to find broadband RFI added to the measured signal. Therefore, the Smoothing Algorithm has been tested with Pseudo-Random Noise (PRN) and Orthogonal Frequency Division Multiplex (OFDM) RFI signals defined in Equations (15) and (17). \[RFI_{{}_{PRN}}\left[k\right]=A_{{}_{PRN}}\cdot PRN\Bigg{(}\frac{T_{{}_{s_{PRN}}} }{T_{{}_{s_{PRN}}}}k\left|\cos\left(2\pi\big{|}f_{{}_{PRN}}-f_{{}_{Rad}}\big{|} T_{{}_{s}}k+\varphi_{{}_{PRN}}\right)\right.\right. \tag{15}\]  
where \(A_{{}_{PRN}}\), \(T_{{}_{s_{PRN}}}\), \(f_{{}_{PRN}}\), \(\varphi_{{}_{PRN}}\), are the amplitude, sampling frequency of the PRN signal, frequency, phase of the OFDM signal respectively, \(f_{Rad}\) is the frequency where the radiometer is measuring and \(PRN[k]\) is an ASK modulated signal composed by the repetition of a the sequence of the firsts 10,230 output bits of a maximum length sequence generator (MLSG) of 14 stages (16). \[MLSG_{{}_{PRN}}=X^{14}+X^{8}+X^{7}+X^{4}+X^{3}+X^{2}+1 \tag{16}\]  
\[RFI_{{}_{OFDM}}\left[k\right]=A_{OFDM}\cdot\sum_{m=0}^{M_{c}-1}S_{m}\cos\left( 2\pi f_{m}k\right)\right. \tag{17}\]  
where \(A_{OFDM}\), is the amplitude of the OFDM signal, \(M_{c}\) is the number of independent carriers (\(i.e.\), data streams), \(S_{m}\) is the \(M_{c}\) parallel modulated source symbols, and \(f_{m}\) is the frequency of the \(m^{\text{th}}\) subcarrier of the OFDM signal, defined as:  
Figure 10: Radiometric sensitivity degradation due to pixel elimination as a function of the filter size. Eight RFI signals have been used, with total INR value labeled for each case. Thresholds used are the most suitable ones for each window size (Table 2). Radiometric sensitivity is normalized by the radiometric sensitivity value obtained when the Smoothing Algorithm is not applied; _i.e._, without the elimination of any pixel of the time-frequency image. Figure 12: _Cont._  
Figure 13: Error obtained by the Smoothing Algorithm in the retrieved \(T_{\text{A}}\) when the RFI is a broadband OFDM signal, using a Hann window as smoothing filter. (**a**) 35 \(\times\)35 Hann window; (**b**) 25 \(\times\) 25 Hann window; (**c**) 5 \(\times\) 5 Hann window; (**d**) without using the Smoothing Algorithm. In Figure 11, the error introduced by the Smoothing Algorithm in the retrieved antenna temperature \(\varepsilon_{T_{\rm A}}\) [K] is represented for different interfering signal powers, with a fixed smoothing filter size: (15 \(\times\)15 Hann window) for two different broadband interfering signal. It can be observed in Figure 11a that a very high \(P_{\rm fa}\) must be used to detect a PRN interfering signal. In this case, the compromise of the \(P_{\rm fa}\) threshold selection leads to a minimum error of retrieved antenna temperature of 14.39 K, (\(P_{\rm fa}=0.0384\)) for an INR value of 5 dB. On the other hand, Figure 11b shows that an OFDM broadband signal can be detected with a threshold with a lower \(P_{\rm fa}\), thus leading to a minimum error of retrieved antenna temperature of 9.07 K, (\(P_{\rm fa}=0.02\)). Following the RFI study in the same way as in the previous section, three additional Monte-Carlo simulations have been performed for different Hann window smoothing filter sizes (35 \(\times\) 35, 25 \(\times\) 25, and 5 \(\times\) 5) for both broadband RFI cases; which are presented in Figures 12-13. Results of Figure 12 shows that it does not exist a \(P_{\rm fa}\)\(|_{\rm Opt}\) for all INR parameter values as in Figure 8 due to the fact that an increase of the INR parameter leads to an increase of the error in the retrieved \(T_{\rm A}\); therefore higher INR parameter values leads to higher error in the retrieved \(T_{\rm A}\) values. In contrast, for the OFDM RFI case, there exists a \(P_{\rm fa}\)\(|_{\rm Opt}\) for all INR parameter values as in the sinusoidal RFI case (Figure 13).