## 3 Simulated Micro-Doppler Signatures  
### Theoretical Considerations  
The equation for Doppler frequency \(f_{d}\) for a rotating target is given by  
\[f_{d}=2v_{r}/\lambda \tag{1}\]  
where \(v_{r}\) is the radial velocity of the target, and \(\lambda\) is the radar wavelength. The corresponding \(\lambda\) for a 10-GHz carrier frequency is 3 cm. A rotor's rotational velocity \(\varpi\) in revolutions per second is related to the maximum \(v_{r}\) expected of the blade tip through  
\[\varpi=v_{r}/2\pi l \tag{2}\]  
where \(l\) is the blade length. In the following simulations, \(\varpi\) is modeled at 56 revolutions per second, which corresponds to a \(v_{r}\) value of 35.2 m/s for an \(l\) value of 10 cm. This maps to an expected maximum Doppler shift around 2.35 kHz occurring at the blade tips. It is assumed that the blade is along the LOS of the radar. Most UAVs have two or three blades associated with each rotor. Figure 1 shows how the radial component of the rotating blade changes over time. The radar system transmits a continuous signal at a single frequency. These signals bounce off the rotating propellers, and the radar receives these reflected signals. The radial velocity \(v_{r}\) of the spinning blade tips does not remain constant, but the periodicity can be modeled with the use of a velocity-time plot and a spectrogram. In addition, a simple Fast Fourier Transform (FFT) of the raw data provides a power spectral density plot that denotes the peaks of frequencies, corresponding to the radial velocities observed within the recording. An FFT takes a window size of samples and computes the power spectral density of that instance. The spectrogram is known as the Short-Time Fourier Transform (STFT), which computes multiple FFTs and displays them as frequency or velocity over  
Figure 1: Radar signal interaction with a single quadcopter rotor. time. In Figure 2, the power spectral density and the spectrogram of a single propeller are shown. As the blade moves horizontally along the radar's LOS, \(v_{r}\) becomes 0. When the blade begins to move directly to or away from the radar, the maximum \(v_{r}\) values are detected. A target moving towards the radar results in a positive (+) Doppler shift, whereas a target moving away results in a negative (\(-\)) Doppler shift. The maximum peaks in the power spectrum are around 2.5 kHz. In the resulting spectrogram, the maximum velocity is confirmed to be roughly 37.5 m/s which is per Equation (1). Additional Doppler frequencies can be caused by the main body movement of the drone. For a moving drone, the maximum Doppler shift is given by  
\[f_{d}=f_{d\_{\text{body}}}+f_{d\_{\text{micro-Doppler}}} \tag{3}\]  
where \(f_{d\_{\text{body}}}\) is the Doppler shift induced by the main body movement of the drone and \(f_{d\_{\text{micro-Doppler}}}\) is the micro-Doppler shift due to the rotating blades. Furthermore, because the radar system measures radial velocity, the look angle from the radar to the target plays a vital role in the displayed Doppler frequency. Equation (3) can be rewritten as follows when considering the look angle dependence  
\[f_{d}(\theta)=f_{d}\cos\theta \tag{4}\]  
where \(\theta\) is the angle between the drone velocity vector and the radar LOS. The micro-Doppler characteristics of flying objects vary from target to target. For instance, a quadcopter signature will be different from a fixed plane or a helicopter. In addition, many surveillance systems confuse drones with birds due to their similar sizes and speeds. Conveniently, the micro-Doppler phenomenon is a great discriminator between the two classes. In birds, the micro-Doppler of the flapping pattern produces a periodic shape but is not as strong and quick as a drone. The rotation speed for a drone's blades can exceed 6000 RPM, but the wing flaps of a bird are much slower. Through the implementation of spectrograms, the micro-Doppler signatures of different flying targets can be classified. The mechanics of drones in flight are unique.