# ATLAS: Latest Advancements and First Observations
## 2 Materials and Methods
### Signal Processor  
ATLAS uses an AM Chirp waveform with a bandwidth of 2 MHz and emits bursts of 100 pulses at a frequency of 33 Hz. This configuration results in a range resolution of 75 m after pulse compression and a Doppler resolution of 0.009 m/s. The maximum unambiguous range is 4545 km, and the maximum unambiguous velocity is 0.44 m/s. The signal processor is depicted in Figure 6. The raw data acquired by the ADCs for each pulse are stacked in a range-pulse complex matrix [34] (Chapter 3). The complex data are processed by a matched filter [34] (Chapter 4) and then split into two branches. The first one computes the sample power and passes the signal through a cell-averaging constant false-alarm ratio (CA-CFAR) [34] (Chapter 6) detector with a probability of false alarm of \(10^{-9}\). At the end of this branch, a list of range detections is obtained. The second branch compensates for range migration and performs pulse-Doppler processing. Figure 4: NORAD ID: 25544, ISS. A streak is formed by the echoes received from the sattelite. Figure 5: NORAD ID: 54216, CSS (MENGTIAN). A streak is formed by the echoes received from the sattelite. Range migration compensation is achieved using the Radon transform (RT) for line detection [35] and envelope cross-correlation for range alignment [36; 37]. The RT of a function \(f(x,y)\) is defined as [38]  
\[R(p,\tau)=\int_{-\infty}^{+\infty}f(x,px+\tau)dx, \tag{9}\]  
where \(p\) is the slope of the line and \(\tau\) is its intercept with the \(y\)-axis. So, \(R(p,\tau)\) is the line integral of \(f(x,y)\) over the line defined by the parameters \(p\) and \(\tau\). By applying the RT across a range of slopes and intercepts with the \(y\)-axis, we obtain the sinogram of \(f(x,y)\), which contains maximums for the pairs of \((p,\tau)\) that define existing lines in the \((x,y)\) domain. By combining (7) and (9), it is possible to obtain the RT of the received signal in terms of the range \(r_{0}\) and the radial velocity \(v_{r}\)  
\[R(r_{0},v_{r})=\int_{0}^{CPI}u_{pc}(r_{0}+v_{r}\frac{\Delta t_{s}}{\Delta r_{i }}t_{s},t_{s})dt_{s},r_{0}\in[r_{min},r_{max}], \tag{10}\]  
where \(\Delta t_{s}\) is the pulse repetition interval, and \(\Delta r_{i}\) is the range resolution. The radial velocity can also be defined as the anticlockwise angle from the slow time axis to the range walk line  
\[v_{r}=\tan{(\theta)}\frac{\Delta r_{i}}{\Delta t_{s}},\theta\in[0,\pi]. \tag{11}\]  
If there is one line in \(R(r_{0},v_{r})\), it can be detected using  
\[r_{RT},v_{RT}=\max{(R(r_{0},v_{r}(\theta)),r_{0}\in[r_{min},r_{max}],\theta \in[0,\pi]}, \tag{12}\]  
where \(r_{RT}\) and \(v_{RT}\) are estimates for the initial range and radial velocity computed with the RT. After detecting the line, we can filter out the rest of the components in the sinogram (by  
Figure 6: ATLAS signal processor. The output of the signal processor consists of a list of detections containing range and velocity information. setting them to 0) and apply the inverse Radon transform (IRT), resulting in the detected line being isolated, denoted as \(u_{clean}\)  
\[u_{clean}=\text{IRT}\Bigg{(} \begin{cases}\text{R}(r_{0},v_{r})&\text{if }(r_{0},v_{r})=(r_{RT},v_{RT})\\ 0&\text{else}\end{cases} \Bigg{)}. \tag{13}\]  
Figure 7 showcases the line detection technique applied to the observation shown in Figure 5.  
Applying the envelope cross-correlation to \(u_{clean}\) estimates the range cell shift between the peaks in the line, which are then used to align the predominant scatterer within the same range bin, as depicted in Figure 8.  
Now that the range alignment is completed, we need to correct for the Doppler term (Equation (8)). To do this, we can implement a filterbank, with each filter tuned to a different Doppler frequency. This can be obtained by applying the Fourier transform along the slow time, generating a range-Doppler map [34] (Chapter 5). Due to the usage of a low PRF, the Doppler velocity obtained is highly ambiguous. Unambiguous velocity estimation is obtained from the line parameters found with the RT.