## 3 Model Solution
### Selection of Regularization Parameters and Simulation  
Considering that it is difficult to obtain prior knowledge about \(\tilde{n}\) in practical non-uniform sampling problems, we adopt the L-curve criterion to select \(\alpha\). The L-curve is used to describe the relationship between the residual term \(\left\|AS_{n}^{\alpha}-s_{\delta}\right\|\) and the regularized solution \(\left\|S_{n}^{\alpha}\right\|\) under different \(\alpha\). it can effectively utilize the information of \(\left\|AS_{n}^{\alpha}-s_{\delta}\right\|\) and \(\left\|S_{n}^{\alpha}\right\|\), seeking a balance that ensures both values are relatively small [28]. In order to avoid an excessively large regularized solution, we impose a constraint on the norm of \(S_{n}^{\alpha}\) within a certain range. In this case, we select the corner point of the L-curve, which corresponds to the regularization parameter value that maximizes the curvature of the L-curve on a log-log scale. Let \(\rho=\log\left\|AS_{i}^{\alpha}-s_{\delta}\right\|\) and \(\theta=\log\left\|S_{i}^{\alpha}\right\|\), then the chosen \(\alpha\) is determined to maximize the following curvature function \(c(\alpha)\) within a specified range [29]. \[c(\alpha)=\frac{\rho^{\prime}\theta^{\prime\prime}-\rho^{\prime\prime}\theta^{ \prime}}{\left((\rho^{\prime})^{2}+(\theta^{\prime\prime})^{2}\right)^{\frac{ 3}{2}}} \tag{18}\]  
where \({}^{\prime}\) represents the derivative with respect to \(\alpha\). Given the known singular value decomposition of \(A\), it is straightforward to numerically solve for the regularization parameter that maximizes the curvature function \(c(\alpha)\). The solving results of \(s_{1}(t_{i}^{(1)})\) and \(s_{2}(t_{i}^{(2)})\) using the Tikhonov regularization method based on L-curve criterion are shown in Figure 3. In Figure 3a,c, the numbers marked with 'x' represent the values of regularization parameter at those points. The positions marked by the red dashed lines on L-curve indicate the points with maximum curvature. The selected regularization parameters are \(3.5933\times 10^{-12}\) for \(s_{1}(t_{i}^{(1)})\) and \(5.3568\times 10^{-7}\) for \(s_{2}(t_{i}^{(2)})\). It can be observed from Section 3.1 that the regularization parameter selected for \(s_{2}(t_{i}^{(2)})\)  
Figure 2: The solving error of Tikhonov regularization for \(s_{2}(t_{i}^{(2)})\) under different \(\alpha\). based on L-curve criterion is in the same order of magnitude as the one which minimizes the difference from true solution, and there is a very small error present. Compared to \(s_{1}(t_{i}^{(1)})\), \(s_{2}(t_{i}^{(2)})\) is more affected by errors. Therefore, the L-curve exhibits a noticeable corner in Figure 3c and the spectrum of \(s_{1}(t_{i}^{(1)})\) can be perfectly reconstructed in Figure 3b. Moreover, by comparing the results in Figure 3d, we can find that even under conditions where the influence of errors is significant, the proposed algorithm is still able to achieve satisfactory spectral reconstruction results. To further verify the effectiveness of the proposed algorithm, we transform the reconstructed signal's spectrum to the time domain using discrete inverse Fourier transform and compare it with the results obtained through cubic spline interpolation. The cubic spline interpolation method is commonly used due to its simplicity and excellent stability, and it is also frequently employed in addressing the azimuth non-uniform sampling problem in SAR. Therefore, we compare the proposed method with the cubic spline interpolation. Figure 4a-d shows the results of reconstructing \(s_{1}(t_{i}^{(1)})\) and \(s_{2}(t_{i}^{(2)})\) using both methods and comparing them with the original uniformly sampled signals. Figure 4e,f illustrates the comparison results of solving errors between the two methods. This indicate that the proposed method significantly outperforms cubic spline interpolation in terms of reconstructing signals \(s_{1}(t_{i}^{(1)})\) and \(s_{2}(t_{i}^{(2)})\).