## Calibration Procedures
### Distance Calibration with an Unmanned Aerial Vehicle  
The distance resolution \(\Delta r\) of FMCW radars is calculated using the chirp bandwidth \(B\). An error in the estimation of this term introduces a bias in the range estimation proportional to the target distance. In addition, uncorrected time delays in the microwave components can introduce a constant offset in the beat frequency compared to what is expected from the physical time of flight, and this can produce a constant bias in distance estimation.  
To avoid biases in the distance estimation, we implement an external range calibration experiment based on the method published by [PERSON] et al. [19]. This method is developed for FMCW radars and consists in using external targets placed at known distances from the radar to calibrate the relationship between the IF frequency \(F_{d}\) and range retrievals.  
The target used in this experiment is a commercial Unmanned Aerial Vehicle (UAV) Xiaomi MI. The UAV is programmed to hover at three known heights above the radar (200, 300, and 400 m), using its internal GPS to control its position. Each position is held for 45 s, to guarantee an accurate estimation of the \(F_{d}\) frequency associated with each radar position. The experimental setup is presented in Figure 5.  
Figure 6a shows the raw received power versus distance for the complete experiment and their associated Uncalibrated Doppler Velocities. Valid measurements must have a power value 2 dB higher than the noise floor, sampled using the power at the penultimate range bin where no external signal is  
Figure 5: Image of the distance calibration procedure, which uses a UAV as target and the internal GPS of the UAV to estimate its height. (**a**) The radar setup in the Astronomy Campus at the University of Chile (33\({}^{\circ}\)23\({}^{\circ}\)48\({}^{\prime\prime}\)5, 70\({}^{\circ}\)32\({}^{\prime}\)13.3\({}^{\prime\prime}\)W). (**b**) Picture taken by the UAV during the procedure.  
observed. Uncalibrated Doppler velocity is proportional to physical velocity, and it is positive when the target moves towards the radar.  
Figure 6: Distance calibration using a UAV as the range reference. (**a**) Vertical Raw Power profile for the range calibration experiment. (**b**) Doppler velocity profile. Positive velocities indicate movement towards the radar. Time is in seconds from the begining of the Radar measurements. The UAV hovering height versus time is indicated with the black lines; (**c**) Fitting line between the IF frequency \(F_{d}\) and the UAV position.  
Table 1 summarizes the results from the calibration experiment. To estimate the \(F_{d}\) value associated with each UAV position, we select the profiles within the hovering periods. Then, we retrieve for each profile a single IF frequency associated with the profile maximum power. Finally, we take the median to estimate \(F_{d}\) for the UAV range. Results are shown in Table 1.  
The results from Table 1 are related using a linear fit, as shown in Figure 6b. There we observe that the radar follows a linear relationship between frequency and range, as expected from theory (see Equation (1)). However, the slope of \(1.471\cdot 10^{-2}\) [MHz m\({}^{-1}\)] retrieved experimentally is larger than what is expected from theory (\(\approx\)1.00\({}^{-2}\) [MHz m\({}^{-1}\)]). Additionally, we also observe an offset. With this data, we calculate the coefficients needed to get the range \(r\) for each \(F_{d}\) value [19], obtaining the relationship of Equation (6):  
\[r[m]=\frac{F_{d}[MHz]-0.8824}{1.471\cdot 10^{-2}}. \tag{6}\]  
With this calibration relationship and considering the \(F_{d}\) resolution (300 kHz), we obtain a range resolution (\(\Delta r\)) of 20 m.