## 5 Simulations Results and Error Analysis
### Error Analysis  
In this paper, the main errors come from the following aspects: approximation of the slant range \(R(t_{\rm a})\), approximation of the 2-D frequency spectrum derivation, the Taylor expansion of the 2-D frequency spectrum, and approximation of the compensation function. Here, we analyze the error to prove the effectiveness of this algorithm.  
Firstly, the equivalence slant range is kept up to the fourth-order Taylor series expansion of \(R(t_{\rm a})\), where the higher-order terms are ignored. The ideal one is shown in Equation (2) and the error can be expressed as:  
\[\Delta R(t_{\rm a})=R_{\rm r}(t_{\rm a})-\left[R_{\rm r0}+(k_{\rm lc}+k_{1})t_ {\rm a}+k_{2}t_{\rm a}^{2}+k_{3}t_{\rm a}^{3}+k_{4}t_{\rm a}^{4}\right] \tag{25}\]  
Taking the scene center point as the reference point, the parameters are listed in Table 1, and the results are shown in Figure 6.  
The maximum range approximation error is \(3\times 10^{-7}\)m in Figure 6, which contributes to the phase error in Equation (5) as:  
\[\Delta\psi(f_{r},t_{a})=-2\pi\frac{(f_{\rm c}+f_{\rm r})}{c}\Delta R(t_{\rm a}) \tag{26}\]  
Using the simulation parameters into Equation (1), the phase error absolute value of our approximation is about \(6.3\times 10^{-5}\) rad. It meets the demand of \(\Delta\psi\leq\frac{\pi}{4}\).  
Secondly, the approximation of the 2-D frequency spectrum by the method of POSP and series reversion introduces the error. This error partly depends on the range approximation, namely, it is related to the first error. Another aspect is the remaining Taylor expansion orders of \(f_{\rm r}\). In this algorithm, the 2-D frequency spectrum expansion of \(f_{\rm r}\) ignores the terms higher than fourth-order  
Figure 5: Imaging result after geometry correction.  
Figure 6: Slant range approximation error of high orders.  
To analyze this error, the method of numerical calculation is considered as a comparison, which can obtain the accurate 2-D frequency spectrum. Figure 7 shows the imaging results of point targets using the approximation of 2-D frequency spectrum derived in this paper and the method of numerical calculation, respectively.  
The last one, also the greatest cause of algorithm accuracy, is the approximation of k-coefficients which uses \(y_{\text{pc}}\) instead of \(y_{\text{p}}\). Thus, we compare subimages of the center point T5 and edge points T1 and T9, which can be shown in Figure 8.  
From Figure 8, we can find that the focusing performance of the center point is better than the edge point. However, the edge point can still focus well with our algorithm. To make a clearer performance comparison between the proposed algorithm and the traditional RDA, the imaging result obtained by traditional RDA is shown Figure 9.  
Figure 8: Focusing performance of proposed improved RDA: (**a**) Target 1; (**b**) Target 5; and (**c**) Target 9.  
Figure 7: Focusing results of the point target using different 2-D spectra. (**a**) Series reversion; and (**b**) Numerical calculation.  
Figure 9: Focusing performance of traditional RDA: (**a**) Target 1; (**b**) Target 5; (**c**) Target 9.  
Comparing the results, it is easy to find that the proposed algorithm generally offers better focusing capabilities than the traditional RDA. Analyzing the data in Figures 8 and 9, the resolution \(\rho\), peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) are used to evaluate the focusing performance. T1 and T5 are chosen for analysis. The results are listed in Table 2. We can find that higher resolution and the theoretical PSLR and ISLR are obtained by the proposed algorithm.