## 3 Simulated Micro-Doppler Signatures
### Simulation Setup and Results  
Before the data were collected experimentally, simulations were created to replicate the micro-Doppler detection scenario. These simulations were created in the MATLAB (R2021a) environment using the Radar Toolbox. The Radar Toolbox expediates and supports the simulation of radar systems through design, deployment, and field data analysis. First and foremost, constants were set for the initial simulation parameters. The simulation parameters are listed in Table 2, denoting the setup for drone behavior throughout the detection process. Using the phased toolkit, a rectangular waveform was used to simulate the CW behavior of the 10-GHz signal. The sampling rate and the Pulse Repetition Frequency (PRF) were at the same value of 44.1 kHz, the default sampling rate of a laptop's audio card. By setting the sampling rate and the PRF to the same value with the pulse width equal to the inverse of the PRF, a 100% duty cycle was simulated. Two antenna elements were used: one for transmission and one for reception. These antenna elements were spaced at half-wavelength (i.e., \(\lambda/2\)) distances from each other. The transmit power used was 70.79 mW with a gain of 22 dB. The receiver preamplifier was set to a gain of 19 dB and a noise figure of 1.7 dB. The environment was free space, where the propagation speed was 3 \(\times\) 10\({}^{8}\) m/s. The drone body was positioned at (10, 0, 10), where the z-axis is the height. The target velocity was 0 for all axes, meaning the drone was hovering in place. The drone behavior was simulated by consisting of the main body RCS of 0.3 m\({}^{2}\) with four accompanying rotors, each evenly positioned 15 cm away from the body. A for loop was created to simulate each 'pulse', sample. With each sample, the drone and each blade position were updated. The sample was transmitted to the target and reflected on the radar. The value was stored in an array until each sample was transmitted and received, summing up to 0.5 s.  
The first plot generated was a time plot shown in Figure 3. This represents the raw values collected by the radar system. Subsequently, the Fast Fourier Transform (FFT) operation was performed on the time signal. Since the FFT is a cyclical process, the frequency axis was split into negative and positive frequencies. The Doppler frequency \(f_{d}\) axis was converted to velocity \(v\) using \(v=\lambda f_{d}/2\). The frequency response is shown in Figure 4. The maximum detectable velocity was around \(\pm 30.24\) m/s but the strongest return was at \(\pm 23.52\) m/s. Finally, spectrograms were generated with a window size of 128 segments with 120 overlapping segments. The time axis, originally in seconds, was converted to milliseconds. The frequency axis was converted to velocity. The spectrogram in Figure 5 accurately displays the micro-Doppler signature of the hovering quadcopter. Periodic maximum velocity values were observed, confirming the cyclical nature of the rotating blades. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Environment & Free space \\ Range & 10 m \\ Dwell time & 0.5 s \\ Sampling rate & 12 kpsps \\ Number of rotors on drone & 4 \\ Blades per rotor & 2 \\ Average blade rotation rate (\(\varpi\)) & 56 rps \\ Blade length & 0.1 m \\ Radar cross section & 0.3 m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation setup parameters. Figure 6 shows the simulated spectrograms for the different window sizes. By adjusting the window size \(N\) of the STFT, a tradeoff between the time resolution and Doppler resolution was observed. When the window size was set to 64 samples, the time resolution was very fine, emphasizing the micro-Doppler periodicity of each blade tip. However, the velocity readings spanned a range of values, resulting in ambiguity regarding the true Doppler velocity. On the other hand, with a window size of 512 segments, the time resolution became ambiguous, but the velocity measurements were finer and more precise.