## 3 Simulation Results  
The simulation is performed to verify the algorithm mentioned above. In our simulation, the cube-shaped target with size 0.5 m \(\times\) 0.5 m \(\times\) 0.5 m is investigated. Firstly, the attitude angle is 45\({}^{\circ}\), and the spin rate of the target is set at 5 Hz. According to the formula \(10\lg(P_{s}/P_{n})\) (\(P_{s}\) is the signal power, \(P_{n}\) is the noise power), different laser emission powers are selected to set three different signal noise ratios (SNRs), i.e., SNR = 10 dB, 2 dB, and \(-\)10 dB, respectively. The estimation of spin rate from the simulation is illustrated in Figure 4. As shown in this figure, the characteristic frequency \(f_{f}\) = 20 Hz can be acquired under different SNRs. However, the peak value of the 2D Fourier-transformed spectrum for the case of large noise is relatively lower. According to Equation (8), the spin rate can be obtained as \(f_{s}=f_{f}/4\) = 5 Hz, which is consistent with the setting value in the simulation. In addition, the attitude angle \(\alpha\) = 15\({}^{\circ}\), 30\({}^{\circ}\), 45\({}^{\circ}\), 60\({}^{\circ}\), 75\({}^{\circ}\), 90\({}^{\circ}\), spin rate \(f_{s}\) = 0.5 Hz, 1 Hz, 2 Hz, 3.5 Hz, and 5 Hz are investigated as well. Our algorithm all gives accurate spin rates. To sum up, simulation results illustrate the validity of the proposed algorithm. It should be noted that large noise affects the peak value of the transformed spectrum. However, when the SNR is larger than \(-\)10 dB, the accurate spin rate can be extracted. The absolute error \(\delta_{\alpha}\) and \(\delta_{f_{s}}\) of the estimation results of the attitude angle in the simulation is shown in Figure 5 for SNR = 10 dB with known target size. With the variation of the LOS angle and spin rate, one can see that the maximum absolute error is less than 2\({}^{\circ}\), and the average absolute error is less than 1.5\({}^{\circ}\). In particular, in a low-noise environment, our proposed approach is capable of estimating the attitude angle effectively. The accuracy control is within 2\({}^{\circ}\), and the absolute error possesses its maximum at \(\alpha\) = 75\({}^{\circ}\) (see Figure 5a). In addition, it is found that the absolute error for \(\alpha\) = 75\({}^{\circ}\) is significantly large, which implies that we should avoid the detection under this LOS angle. Instead, we need to select an LOS angle between 15\({}^{\circ}\) and 60\({}^{\circ}\). This is called the best LOS angle window, in which the absolute error is less than 0.5\({}^{\circ}\). It provides a scientific basis for arranging the optimal measurement strategy for high-precision satellite attitude angle measurement. For an unknown target size, the estimation results for SNR = 10 dB and \(\Delta\) = 15\({}^{\circ}\) are displayed in Figure 6. With the variation of the LOS angle and spin rate, one can see that the maximum absolute error is less than 4.5\({}^{\circ}\), and the average absolute error is less than 3\({}^{\circ}\). The relatively larger error in the case of unknown target size stems from the twice edge extraction of TF spectra. It can be seen from Figure 6a that when the LOS angle is between 15\({}^{\circ}\) and 60\({}^{\circ}\), the mean absolute error is less than 2\({}^{\circ}\). When the spin rate is low (0.5 Hz), the absolute error becomes larger. This is due to the smaller Doppler frequency shift induced by the slow spin. In addition, Figure 7 displays simulation results of the side length estimation for \(L\) = 0.5 m, and SNR = 10 dB. It can be seen from Figure 7 that the maximum absolute error \(\delta_{\alpha}\) of the estimated side length is less than 0.003 m. When the LOS angle is between 15\({}^{\circ}\) and 60\({}^{\circ}\), the absolute error almost vanishes. In addition, the maximum relative error \(\gamma_{\alpha}\) of the estimated side length is no greater than 6% (see the inset of Figure 7). In summary, one can obtain a high-precision estimation of the side length.