# Development of the First Portuguese Radar Tracking Sensor for Space Debris
## 3 System Expected Capabilities
### Minimum Detectable Target Size  
The radar range equation represents the physical dependencies of the transmitted power and is used to obtain the power in the receiving antenna [33]. Using the radar \begin{table}
\begin{tabular}{c c} \hline \hline
**Waveform resolution** & 10 ns \\
**Pulse repetition frequency (\(f_{p}\))** & 10 MHz (max) \\
**Duty Cycle** & 10\% (max) \\
**Number of pulses for integration** & Variable \\
**Temperature Monitoring** & Tx, Rx, LNC, Driver, PA, Coupler \\
**System Monitoring** & Pulse shape file name, Tx/Rx gain, \\  & No. Pulses, PRF, Pulse length... \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Controller board parameters. equation, we can obtain the dependence of the signal to noise ratio (SNR) with the radar specifications:  
\[\text{SNR}=\frac{P_{av}G^{2}\lambda^{2}\sigma ne(n)F^{4}}{4\pi^{3}\tau f_{p}R^{4}N _{F}kTBL_{s}} \tag{1}\]  
where:  
* \(P_{av}\) is the average transmitted power (W)
* \(G\) is the antenna gain
* \(\lambda\) is the operating wavelength (m)
* \(\sigma\) is the target radar cross section (RCS) (m\({}^{2}\))
* \(n\) is the number of integrated pulses
* \(e(n)\) is the integration efficiency
* \(F\) accounts for all the propagation effects
* \(\tau\) is the pulse width (s)
* \(f_{p}\) is the pulse repetition frequency (Hz)
* \(R\) is the distance to the object (m)
* \(N_{F}\) is the noise factor of the receiver
* \(k\) is the Boltzmann constant (\(\frac{1}{R}\))
* \(T\) is the receiver temperature (K)
* \(B\) is the bandwidth of the receiver (Hz)
* \(L_{*}\) accounts for all system losses  
Some of the radar specifications are fixed and have already been shown during this article. Regarding propagation losses, it is known that, for frequencies below 10 GHz, losses due to atmospheric absorption may be neglected [34]; for that reason, we neglected for now the \(F\) factor. In future work, we will include a detailed modeling of the atmospheric propagation effects including tropospheric absorption. Accounting for all the losses on the waveguide transition from the antenna to the polarizer/orthomode transducer (OMT)/switch and the coaxial transition from the transmitter/receiver to the switch, we obtained \(L_{s}=0.6\) dB. By changing the pulse length, pulse repetition frequency and the number of integrated pulses, it is possible to establish a desired SNR for an established target RCS. By doing a statistical analysis on the mean elevation speed of the debris obtained from the Space-Track catalogue (explained in Section 3.3) and considering our antenna beamwidth, we can safely define 20 pulses for integration. Since the radar is fully coherent with integration in the digital domain, we can assume a coherent integration with full efficiency (\(e(n)=1\)). In order to obtain the desired SNR, we defined \(\tau=3.3\) ms and \(f_{p}=30\) Hz, obtaining the following performance results:  
* Maximum Unambiguous Range: 5000 km
* Maximum Unambiguous Velocity: 0.4 m/s
* SNR for a 1 m\({}^{2}\) RCS at 10\({}^{3}\) km: 39.55 dB
* SNR for a 10 cm\({}^{2}\) RCS at 10\({}^{3}\) km: 9.55 dB  
We can conclude that we can measure targets up to 1000 km in distance with a minimum RCS of 10 cm\({}^{2}\). Since debris objects in LEO have velocities of around 7.8 km/s, obtaining a maximum unambiguous velocity of 0.4 m/s results in inability to measure the Doppler shift. This is due to the fact that we need a low pulse repetition frequency (PRF) for maintaining range unambiguity. It is still possible to measure velocities without Doppler shift by taking several range measurements and inferring velocity from the measurements, which in fact is what is done in an SST system, by processing several individual tracks of the same object. In order to take advantage of the long idle times between the transmitted and received pulses in a low PRF system, we can use a process called pulse interleaving. Pulse Interleaving is a technique that inserts carefully crafted pulses in the idle time of other pulses.