## 4 Wiener Filtering  
As already discussed, the spectrogram of a noise signal with sinusoidal interference signals can be considered as a noisy image, where the noise is the spectrogram of the radiometric signal (the one we want to measure), and the image to be detected is the spectrogram of the interference (the one to be cancelled). Therefore, designing a filter to eliminate the noise from the image is the way to estimate the RFI, for a later removal of the interference without loss of the radiometric data. The Wiener filter is a well-known adaptive filter used in communications, which provides the best estimation of a signal, equalizes communications channel, and eliminates the noise present in the received signal. In this section the Wiener filter is used to estimate the RFI in the spectrogram image, for a later cancellation and its performance is compared to the smoothing algorithm. Figure 4: Smoothing Algorithm applied to a simulated Radio Frequency Interference (RFI) contaminated radiometric signal. (**a**) RFI contaminated signal’s spectrogram, \(\sigma^{2}_{\text{RFI}}=\sigma^{2}_{n}-5\) dB; (**b**) Convolution of the spectrogram with a N \(\times\) N = 15 \(\times\) 15 Hann window; (**c**) Thresholding for \(P_{\text{fa}}=10^{-4}\): black pixels are considered RFI; (**d**) RFI-mitigated spectrogram using “blanking” of the RFI contaminated pixels. In our case of study, it is not necessary to know the effect of the communications channel in the spectrogram, thus, it will only be necessary to differentiate between the noise and the interfering signal. The way to perform this task is by using a locally adaptive linear minimum mean square error (LLMMSE) [16] to estimate the interfering signal's spectrogram. The LLMMSE algorithm consists of an optimal linear estimator of our interfering signal \(\hat{y}(t_{u},f_{v})\) in combination with additive noise. \[\hat{y}\big{(}t_{u},f_{v}\big{)}=\alpha s\big{(}t_{u},f_{v}\big{)}+\beta \tag{2}\]  
where  
\[s\big{(}t_{u},f_{v}\big{)}=y\big{(}t_{u},f_{v}\big{)}+n\big{(}t_{u},f_{v}\big{)} \tag{3}\]  
and \(y(t_{u},f_{v})\) is the interfering signal spectrogram, \(n(t_{u},f_{v})\) is the spectrogram of the Gaussian noise (independent from the interfering signal spectrogram), \(t_{u}\) is the \(u^{\text{th}}\) time point of the spectrogram, \(f_{v}\) is the \(v^{\text{th}}\) frequency point and \(\alpha\) and \(\beta\) are two parameters chosen to minimize the mean square estimation error criterion \(\varepsilon\) described in Equation (4):  
\[\varepsilon\big{(}\alpha,\beta\big{)}= \tag{4}\]  
The minimum error (\(\varepsilon\)) is found for:  
\[\frac{\partial\varepsilon\big{(}\alpha,\beta\big{)}}{\partial\alpha}= \tag{5.1}\]  
\[\frac{\partial\varepsilon\big{(}\alpha,\beta\big{)}}{\partial\beta}= \tag{5.2}\]  
where \(\partial\) denotes partial derivative. From Equations (4), (5.1) and (5.2), and taking into account that the additive noise power of the spectrogram does not have a zero mean, Equation (6) can be derived:  
\[\hat{y}\big{(}t_{u},f_{v}\big{)}=\big{(}\mu_{s}\big{(}t_{u},f_{v}\big{)}-\mu_ {n}\big{(}t_{u},f_{v}\big{)}\big{)}+\frac{\sigma_{s}^{2}\big{(}t_{u},f_{v} \big{)}-\sigma_{n}^{2}\big{(}t_{u},f_{v}\big{)}}{\sigma_{s}^{2}\big{(}t_{u},f _{v}\big{)}}\big{(}s\big{(}t_{u},f_{v}\big{)}-\mu_{s}\big{(}t_{u},f_{v}\big{)} \big{)} \tag{6}\]  
where \(\mu_{s}(t_{u},f_{v})\) represents the local mean of \(s(t_{u},f_{v})\) around a square of \(L\,\times\,L\) pixels, described in Equation (7), \(\sigma_{s}^{2}(t_{u},f_{v})\) represents the local variance of \(s(t_{u},f_{v})\) around a square window of \(L\,\times\,L\) pixels,  
Figure 5: RFI mitigation technique: an estimation \(\hat{y}(t_{u},f_{v})\) of the RFI signal spectrogram is substracted from the received signal spectrogram. described in Equation (8), \(\mu_{n}\) is the mean of the noise spectrogram, and \({\sigma_{n}}^{2}(t_{u},f_{v})\) is the variance of the noise spectrogram.