## 2 Spectrogram Calculation  
In this work, radiometric signals are assumed to be sampled at an adequate sampling rate satisfying the Nyquist criterion and the spectrograms are obtained from these discrete time signals. The spectrogram calculation depends on several parameters which are: the total number of signal samples, the FFT size, the window used for the FFT calculation of the data segments, and the overlapping between these data segments, as shown in Figure 1.  
The spectrogram used in this paper is calculated using discrete samples, each one corresponding to a measurement of the pre-detected voltage signal. The number of samples (N) which forms the spectrogram will determine the radiometric resolution obtained by each spectrogram. The FFT size is the number of samples used to compute each FFT and determines the spectral and temporal resolutions. If the FFT size (L) increases, the spectral resolution increases, while the temporal resolution decreases, since the time lapse between consecutive data segments increases. In contrast, smaller FFT sizes lead to lower frequency resolution, and higher temporal resolution. Therefore, selection of the FFT size must be chosen carefully depending on the RFI present in the radiometric measurements. The FFT calculation requires the use of an analysis window function in order to minimize unwanted \"side lobes\" and ringing in the FFT resulting from abrupt truncations at both ends of the data segment. This situation causes an oscillatory behavior in the FFT called the Gibbs phenomenon [11]. Several functions can be used to avoid this problem, though when the data source is unknown, the Hann window is one of the most common windows used in spectrum analysis because of its excellent roll-off rate at 18 dB/octave [12]. In order to avoid loss of data when using a window in the FFT calculation, some overlapping (O) must exist between consecutive data segments (Figure 1). In addition, overlapping increases the temporal resolution as the beginning of the data segments is reduced. However, increasing the overlap leads to an increase of the total number of samples to be computed and managed, complicating the hardware design. Overlap factor is recommended to be 75% for a Hann window [13,14]; in this work a value of O = 75% is used. The spectrograms calculated in this work have the following parameters:  
* Number of samples: this parameter must be as high as possible to maximize the radiometric resolution, as a higher number of samples means a longer integration time. On the other hand too large values increase the computation time required to perform the Monte-Carlo simulations. The value chosen in this work is N = 2\({}^{18}\) samples. * Overlap: for the case of 2\({}^{18}\) samples, an overlap value of O = 75% is reasonable [13,14] to avoid information loss. * FFT size: The FFT size determines the frequency resolution and the number of FFT segments, determines the temporal resolution. For the simulations performed in this work, and supposing that any information of the RFI is known, the FFT size was chosen to be equal to the number of FFT segments, to have similar spectral and temporal resolutions (both resolutions may not be the optimal ones); thus the FFT size should be the square root of the number of samples (N). Taking into account that there exists a 75% overlap between consecutive Hann windows, the number of FFT segments has to be multiplied by 4. Thus, to equilibrate both resolutions, maintaining the original compromise, the FFTsize has been selected to be \(\mathrm{L}=2^{10}\), and now the total number of samples is \(2^{20}\) (accounting redundant points due to overlapping). As it has been said, the spectrogram operation consists of the power (square of the absolute value) of the STFT magnitude. The spectrogram operation changes the probability density function of the received thermal noise which is a pair of Gaussian distributed random variables (one for the in-phase component and the other one for the quadrature component).