## 4 Simulation Analysis
### Single-Component Vibration Error Compensation  
In this sub-section, simulations are given to evaluate the compensation performance of the proposed vibration error-compensation method for THz-SAR imaging. The THz-SAR system parameters are shown in Table 1. When the echo signal of multi-point targets is generated, single-component vibration error is added to it. To estimate and compensate the vibration error, the dominant scatter point is firstly selected according to the energies of range bins from the unfocused image. Then, after the target is moved to the image center by a circular shift, the SoI in range-compressed phase history domain is achieved through inverse FFT. To verify the outstanding performance of the proposed vibration error compensation method, the results obtained by the STFT-based rough estimation method and the STFT-based rough estimation method with the optimal window width are also given here, respectively. First, the IF of the SoI is estimated by the STFT and the vibration frequency is obtained from the amplitude spectrum of the IF. On the basis of the estimated vibration frequency, we can obtain the vibration amplitude and initial phase. Table 2 lists the estimated parameters and MLF values. Figure 5 shows the true vibration error and the estimated vibration error. We can see from the Table 2 that the estimated values by the STFT with \(h=20\) have a large deviation from the true values. Hence, the red line in Figure 5 shows that the estimated vibration displacement also deviates from the true one, especially at extreme points of vibration. Accordingly, the MLF value is much smaller than the true MLF value, also indicating that the performance of the method is poor. Although the performance of the STFT with the optimal window width \(h_{op}=26\) is improved, the estimates result still diverges from the true values, due to the singular IF estimates. Fortunately, however, Table 2 shows that the parameter-estimate values using the improved STFT-based fine estimation method are consistent with the true ones. It means that the improved method greatly suppresses both the effects of window width and the singular values of IF estimation. It is also verified by the MLF value in Table 2 and the estimated vibration displacement in the green line of Figure 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline carrier frequency & 220 GHz \\ signal bandwidth & 3 GHz \\ pulse width & 1.5 μs \\ platform velocity & 50 m/s \\ slant range & 2296 m \\ sample frequency & 2.8 GHz \\ pulse repetition frequency & 1050 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\mathbf{a_{1}}\) **(mm)** & \(\mathbf{f_{1}}\) **(Hz)** & \(\mathbf{\phi_{1}}\) **(rad)** & **G** \\ \hline true values & 2.00 & 1.00 & 1.05 & 511.39 \\ \(h=20\) & 2.30 & 9.96 & 0.85 & 119.31 \\ \(h_{op}=26\) & 2.12 & 9.98 & 1.07 & 430.24 \\ \(h_{op}=26\) with RANSAC & 2.02 & 9.98 & 1.05 & 495.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated parameter values. The vibration error is compensated for in the imaging process, and the performance of different estimation methods are then analyzed and compared. Figure 6a shows the imaging results of the RD algorithm without considering the vibration error. It is obvious that the image quality is seriously degraded due to the paired echoes caused by uncompensated vibration error. After the vibration error is estimated and compensated based on the above methods, Figure 6b-d gives the imaging results. The phase difference between the true vibration phase and the estimated vibration phase is given in Figure 7. We can see from Figure 6a,b that the image quality, after vibration error compensation, has improved, but it is still affected by paired echoes.