## 5 Results
### Smoothing Algorithm  
#### 5.1.1 Smoothing Algorithm Applied with Pulsed Sinusoidal and Chirp RFI  
The use of the FFT implies that the Smoothing Algorithm is implicitly searching for sinusoidal interferences [5; 6; 7], therefore, in order to determine the correct performance of this algorithm, it has been tested with a set of linear chirp and sinusoidal interfering signals as defined in Equation (11). \[RFI_{\text{sin}}[k]=\sum_{p=1}^{p}A_{p}^{\frac{-(k-r_{p})^{2}}{d_{r}^{2}}}\cos \left(\left(2\pi f_{p}+\pi\beta_{p}T_{k}\right)T_{k}k+\varphi_{p}\right)\ +\ \sum_{r=1}^{R}A_{r}\cos \left(2\pi f_{r}T_{k}k+\varphi_{r}\right)\text{rect}\left(\frac{k-\frac{\
u_{r }+\eta_{r}}{2}}{\eta_{r}-\
u_{r}}\right)k=1,...,N \tag{11}\]  
where \(P\) is the number of chirp signals that conform the chirped RFI, \(A_{p}\), \(d_{p}\), \(t_{p}\), \(f_{p}\), \(\varphi_{p}\), \(\beta_{p}\) are the amplitude, effective duration, central time, initial frequency, initial phase, and chirp rate of the \(p^{\text{th}}\) chirp signal respectively; \(R\) is the number of sinusoidal RFI signals, \(A_{r}\), \(f_{r}\), \(\varphi_{r}\), \(\
u_{r}\), \(\eta_{r}\) are the amplitude, frequency, phase, initial time and final time of the \(r^{\text{th}}\) sinusoidal signal respectively; rect() is the rectangular function, \(T_{s}\) is the sampling period, and \(N\) is the number of samples of the RFI signal. The retrieved error in \(T_{A}\) as a function of the threshold's \(P_{\text{fa}}\), the size of the smoothing filter and the power of the interfering signal is presented in Figures 7-9. To test the performance of the Smoothing Algorithm, a 15 \(\times\)15 Hann window is selected. Retrieved \(T_{A}\) and error in the retrieved \(T_{A}\) as a function of the threshold's \(P_{\text{fa}}\) are represented in Figures 8 and 9, respectively. Figure 7 shows the retrieved \(T_{A}\) [K], for different interfering signal power (and even with no interference, black dotted line), with a fixed smoothing filter size (in this case a 15 \(\times\) 15 Hann window), as a function of the \(P_{\text{fa}}\) of the threshold used in our algorithm. The \(T_{A}\) value is obtained by means of 1024 Monte Carlo simulations on a Pentium D at 3.4 GHz with 2 GB of RAM. Simulation time (16 \(P_{\text{fa}}\) levels and 9 INR levels) was 3 h and 5 min. The actual \(T_{A}\) value is 300 K which is asymptotically achieved by the black dotted line (in abscense of RFI) as \(P_{\text{fa}}\) decreases. As it can be seen, the selection of the \(P_{\text{fa}}\) threshold is crucial, as a low \(P_{\text{fa}}\) values decreases the probability of detection of RFI-contaminated pixels, leading to a retrieved temperature higher than the real antenna temperature. On the other hand, a high \(P_{\text{fa}}\) value will produce a high number of false alarms which will produce a clipping in the probability distribution function of the power spectrogram pixels, leading to a retrieved \(T_{\text{A}}\) lower than the real antenna temperature. and not too low to leave a high rate of undetected RFI contaminated pixels (\(P_{\rm fa}<7.24\)\(10^{-4}\) in Figure 8, INR = \(-15\) dB). Figure 8: Error obtained by the Smoothing Algorithm in the retrieved \(T_{\rm A}\) using a \(15\)\(\times\)\(15\) Hann window as smoothing filter. Error is represented in Kelvin (\(y\)-axis), threshold value in \(P_{\rm fa}\) (\(x\)-axis). Colored lines represent RFI contaminated radiometric signals with INR value determined by its label. Black dotted line represents a radiometric signal in abscense of RFI. Figure 7: Retrieved \(T_{A}\) using the Smoothing Algorithm with a \(15\)\(\times\)\(15\) Hann window as smoothing filter, \(T_{\rm A}\)\({}^{\rm ideal}\) = 300 K (gray line). \(T_{\rm A}\) is represented in Kelvin (\(y\)-axis), threshold value in \(P_{\rm fa}\) (\(x\)-axis). Colored lines represent RFI contaminated radiometric signals with INR value determined by its label. Black dotted line represents a radiometric signal in abscense of RFI.