## 5 Results
### Smoothing Algorithm
#### 5.1.1 Smoothing Algorithm Applied with Pulsed Sinusoidal and Chirp RFI
Combining different simulations with RFIs of different powers (INR from 5 dB to \(-30\) dB compared to the noise power) it is observed that the threshold with optimum performance has a \(P_{\text{fa}}\sim 7.24\,\times 10^{-4}\) (\(P_{\text{fa}}|_{\text{Opt}}\) in Figure 8) with a maximum RMS error value of 2.33 K (Max(\(\varepsilon_{T_{\text{A}}}\)) in Figure 8), and RMS error value without RFI of 1.84 K (\(\varepsilon_{T_{\text{A}}}\))\({}_{\text{RFI-0 dB}}\) in Figure 8) which it will be the error obtained in case of RFI free situation; the threshold value associated to \(P_{\text{fa}}|_{\text{Opt}}\) is equal to 1.37 \(\sigma^{2}_{\text{ }n}\). Three additional Monte Carlo sets of simulations have been performed with a size of the Hann window smoothing filter of \(35\,\times 35\), \(25\,\times 25\), and \(5\,\times 5\). Similar results have been obtained (Figure 9a-c respectively) which are summarized in Table 1. In this table, the values of the threshold with the lower \(T_{\text{A}}\) RMS error independently of the RFI power are presented, in addition to the \(P_{\text{fa}}\) associated with this threshold and the \(T_{\text{A}}\) RMS error obtained with this threshold in the absence of RFI. For all cases, it is important to have a low \(T_{\text{A}}\) RMS error value for the most suitable threshold in the black dotted curve, as absence of RFI should be the most probable case. In Table 1 it is observed that, as the filter size used in the Smoothing Algorithm increases, the maximum \(T_{\text{A}}\) RMS error value decreases; in addition this value also decreases in absence of RFI. Figure 9: Error obtained by the Smoothing Algorithm in the retrieved \(T_{\text{A}}\) 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 Smoothing Algorithm. The best performance is obtained with the largest window (35 \(\times\) 35 Hann window). However, large smoothing windows exhibit a poorer radiometric resolution, as explained below. The radiometric resolution of an ideal total power radiometer is inversely proportional to the square root of the product of the noise bandwidth and the integrating time,  
\[\Delta T=\frac{T_{rec}+T_{A}}{\sqrt{B\tau}} \tag{12}\]  
where \(\Delta T\) is the radiometric resolution, \(T_{rec}\) is the receivers temperature, \(B\) is the noise bandwidth and \(\tau\) is the integration time. Spectrogram pixels have a resolution of a determined bandwidth \(\Delta B\) and a determined integration time \(\Delta\tau\), and the sum over all pixels of \(\Delta B\)\(\Delta\tau\) corresponds exactly to the product of \(B\) and \(\tau\). Pixel elimination produced by the Smoothing Algorithm decreases the number of pixels available for radiometric measurements and the radiometric resolution degrades. In Equations (13) and (14), the relationship between the radiometric resolution after and before applying the Smoothing Algorithm is developed. \[\frac{\Delta T\big{|}_{\mathrm{5A}}}{\Delta T}=\frac{\frac{T_{rec}+T_{A}}{ \left(\sqrt{B\tau}\right)|_{\mathrm{5A}}}}{\frac{T_{rec}+T_{A}}{ \sqrt{B\tau}}}=\sqrt{\frac{\Delta B\Delta\tau N}{\Delta B\Delta\tau N\big{|}_{ \mathrm{5A}}}}=\sqrt{\frac{N}{N\big{|}_{\mathrm{5A}}}}=\sqrt{\frac{N}{N-N_{d}}} \tag{13}\]  
\[\Delta T\big{|}_{\mathrm{5A}}=\sqrt{\frac{N}{N-N_{d}}}\Delta T \tag{14}\]  
where \(\Delta T\)/\({}_{\mathrm{5A}}\) is the radiometric resolution after applying the Smoothing Algorithm, \(\Delta B\) and \(\Delta\tau\) are the frequency and time resolutions of the spectrogram pixels, \(N\) is the number of pixels of the spectrogram, \(N\)/\({}_{\mathrm{5A}}\) is the number of pixels of the spectrogram that passes the Smoothing Algorithm filtering process, and \(N_{el}\) is the number of eliminated pixels by the Smoothing Algorithm. Figure 10 shows the degradation of the radiometric resolution due to loss of radiometric data. It is observed that the degradation increases with the window size.