## 2 Modeling and the Proposed Method  
As displayed in Figure 1, the radar is stationary and located at the origin \(Q\) of the radar coordinate system (\(U\), \(V\), \(W\)). The target is depicted in a local coordinate system (\(x\), \(y\), \(z\)) attached to it and has translations and rotations with respect to the radar coordinates. The origin \(O\) of the local coordinates is at a distance \(R_{0}\) from the radar. The azimuth and elevation angle of the target in the radar coordinates (\(U\), \(V\), \(W\)) are \(\mu\) and \(\eta\), respectively. In this work, we choose \(\eta=0\) to simplify the calculation, and thereby, the unit vector of the target along the line of sight (LOS) of the radar in the local coordinate system (\(x\), \(y\), \(z\)) is given by \(\mathbf{n}=[0,\,\sin\,\alpha,\,\cos\,\alpha]\), where the attitude angle \(\alpha\) denotes the angle between the LOS of the radar and the \(z\)-axis. In this work, we focus on the spin motion estimation of the target. Any point on the target depicted in the local coordinate system (\(x\), \(y\), \(z\)) will move to a new position due to the rotation of the target. The new position can be obtained from its initial position vector multiplied by an initial rotation matrix determined by Euler angles (\(\phi\), \(\theta\), \(\varphi\)), where the angle \(\phi\) rotates about the \(z\)-axis, the angle \(\theta\) rotates about the \(x\)-axis, and the angle \(\varphi\) rotates about the \(z\)-axis again. Viewed in the target local coordinate system, when a target rotates about \(z\)-axes with an angular velocity \(\Omega\). The rotation matrix is expressed as  
\[\mathbf{R}_{rot}=\left[ \begin{array}{ccc}cos\,\Omega t&-sin\,\Omega t&0\\ sin\,\Omega t&cos\,\Omega t&0\\ 0&0&1\end{array} \right]\]  
The micro-Doppler effect caused by the micro-motion of a target or structures on the target can be derived from the theory of the electromagnetic back-scattering field. It can be mathematically formulated by extending the conventional Doppler effect analysis to utilize micro-motions. In this work, a spinning cube-shaped target is considered. When the radar transmits a sinusoidal waveform with a carrier frequency \(f_{0}\), the baseband of the signal returned from one scattering point P on the cube surface is expressed as [6]. \[s_{m}(t)=\rho exp\left\{i2\pi f_{0}\left[2r_{p}(t)/c\right]\right\} \tag{1}\]  
Here, \(\rho\) is the reflectivity function, \(c\) is the speed of the electromagnetic wave propagation, and \(r_{p}(t)\) the distance between the scattering point of the target and the detector at time \(t\). By taking the time derivative of phase \(\Phi=2\pi f_{0}\) [\(2r_{p}(t)/c\)], the Doppler frequency shift induced by the target's motion is obtained as [23]. \[2\pi f_{D}=d\Phi/dt=4\pi f_{0}dr_{p}/dt/c \tag{2}\]  
In our model, the target consists of \(K\) scattered points. Thereby, the returned superposition signal from the surface scattering of the laser light is  
\[s_{l}(t)=\sum_{k=1}^{K}\rho_{k}exp\{i2\pi f_{0}[2r_{k}(t)/c]\} \tag{3}\]  
\(\rho_{k}\) and \(r_{k}(t)\) represent the reflectivity function and distance of the \(k\)-th scattering point, respectively. The time-frequency (TF) analysis that provides localized time-dependent frequency information is explored to analyze the micro-Doppler signal. In this work, the 2D TF image is obtained by using the short-time Fourier transform (STFT), which includes linear transforms [6; 24]. After taking the logarithm of the intensity and performing Gaussian filtering of the frequency, low-frequency details can be reserved, the noise is effectively suppressed with edge-preserving, and a smooth TF image is eventually acquired. Table 1 displays the simulation parameter selection for the 3 cm wavelength radar system, and Figure 2 shows the corresponding TF images. In contrast, Table 2 illustrates the simulation parameter selection for the 1064 nm wavelength radar system, and Figure 3 shows the corresponding TF images.