**A. Experimental Conditions**

**1. Experimental Parameters**

This paper uses the linear frequency modulation pulse to simulate radar signal with the following parameters: carrier frequency\({f_c}=4 \times {10^9}Hz\), signal bandwidth\(B=4 \times {10^8}Hz\), pulse duration\({T_1}=6.4 \times {10^{ - 7}}s\), pulse repetition frequency\(PRF=100\), and the number of pulses\(N=128\). As shown in Fig. 5, the target is simulated using 110 scatters. The target parameters are velocity\(v=50m/s\), acceleration\(a=10m/{s^2}\), and the angle between the target’s direction of motion and the radar line of sight\(\alpha ={0^ \circ }\).

**2. Methods Used in The Experiment**

This section will use conventional and improved methods to conduct a comparative study. The differences between the two methods are shown in Table I.

TABLE I

COMPARISON OF TRADITIONAL AND IMPROVED METHODS

| Range Bins Selection | Windowing |
---|

The Conventional Method | Contrast Value-Based Conditional Judgment Method | Rectangular Window |

The Improved Method | Amplitude-Based Traversal Method | Kaiser Window |

**B. Image-Focusing Effect**

To visualize the difference in imaging effect between the improved and conventional methods, we first compare the images obtained by two methods, as shown in Fig. 6. Without loss of generality, we select the images obtained by two methods for a comparative study when the SNR is -5dB, 0dB, and 5dB.

The focusing effect of the images obtained by the improved method is better than those obtained by the conventional method at different SNRs. As shown in Fig. 6, the improved method can better focus on the main scattering points. Meanwhile, the target outline is more evident. However, the images obtained by the conventional method contain clutter components, the target outline of the images are blurred, and the focusing effect is poor. The improved method uses Kaiser windowing process, so the high-frequency phase information in the range bins can be preserved. The conventional windowing process makes it unable to estimate high-frequency phase information. In addition, the amplitude-based traversal method can also improve the phase gradient estimation accuracy to some extent.

Under different SNRs, the improved method is more robust than the conventional method. In Fig. 6, the improved method can obtain better focusing effects under different SNRs. In contrast, the images obtained by the conventional method are highly random. For example, when SNR = 0dB, the conventional method’s result can show the target outline relatively clearly. However, under the condition of higher SNR of SNR = 5dB and lower SNR=-5dB, the images are blurred severely. It indicates that the imaging effect of the conventional method is volatile due to the noise. In the improved method, the Kaiser window keeps the noise and clutter components continuously suppressed as the iterations proceed. Therefore, the improved method has stronger robustness under different SNRs.

To further illustrate that the improved method gives better-focusing results after phase autofocus for different types of range bins than the conventional method, we select the range bins containing one and multiple strong scatters from the images obtained by both methods for comparison. Figure 7 shows the focusing results of the 20th and 30th range bins.

The improved method achieves better focus in different range bins than the conventional method. As can be seen in Fig. 7, although the results obtained by the improved method still have small clutter, the single or multiple strong scatters in the corresponding range bin have achieved better focus. The results obtained by the conventional method, on the other hand, contain a large amount of clutter, although focusing can be achieved with only one strong scatter in the range bin. At the same time, the intensity of the scatter recovered by the conventional method is much smaller than the result obtained by the improved method, and the focusing effect is poor. For the case of multiple strong scatters in the range bin, the results obtained by the conventional method still contain much clutter and are not well focused. Even when SNR=-5dB, all strong scatters cannot be recovered correctly. According to the previous analysis, the phase information retained by the conventional method is minimal because of the rectangular windowing process. When there are multiple strong scatters in the range bin, the conventional method is more likely to miss practical phase information, and its phase estimation ability will be further degraded.

Next, entropy is introduced to quantify the difference between the results obtained by the two methods and the ideal results. The entropy formula is shown in Eq. (15). Due to the randomness of complex Gaussian white noise, here we conduct 100 rounds of Monte Carlo experiments and find the average entropy values of all experimental results. The results are shown in Table II.

TABLE II

ENTROPY STATISTICS RESULTS

| -5dB | 0dB | 5dB | 10dB | 15dB | 20dB |
---|

Ideal Results | 6.889 | 6.766 | 6.723 | 6.708 | 6.703 | 6.701 |

Conventional Method | 8.266 | 8.231 | 8.221 | 8.219 | 8.217 | 8.200 |

Improved Method | 7.584 | 7.471 | 7.426 | 7.411 | 7.409 | 7.411 |

The mean squared error (MSE) between the entropy obtained by the two methods and the entropy of the ideal result is further solved using Eq. (19), as shown in Fig. 8. In Fig. 8, the blue and red curves indicate the results corresponding to the conventional and improved method, respectively.

$$MSE=\frac{1}{K}\sum\limits_{{k=1}}^{K} {{{\left[ {E(k) - \tilde {E}(k)} \right]}^2}}$$

19

In Table II, under different SNRs, the entropy values of the results obtained by the improved method are smaller than those obtained by the conventional method. It further shows that under different SNRs, the focusing effect of the images obtained by the improved method is better than that obtained by the conventional method. In Fig. 8, the curve corresponding to the improved method can maintain a straight line, while the curve corresponding to the conventional method fluctuates and undulates. It shows that the improved method is more robust than the conventional method under different SNRs. In addition, the MSE values obtained by the improved method are consistently around 0.5 under different SNRs, which are much smaller than those obtained by the conventional method. It shows that the improved method can obtain images closer to the ideal results than the conventional method.

**C. Algorithm Computational Efficiency**

We counted the average time required for each iteration of the two methods in 100 Monte Carlo experiments. The experiments were completed using MATLAB 2021 on a device with a processor model of Intel(R) Pentium(R) CPU G3250 3.20GHz and a built-in memory of 4.00GB. The improved method takes an average of 0.174s per iteration, while the conventional method takes 0.233s. Since the time required for each iteration of the two methods is not significantly different, the computational efficiency of the two methods can be compared by comparing the number of iterations required to achieve convergence of the entropy between the two methods.

We count the average entropy change of the results of each iteration in 10 rounds of iterations and the number of iterations required to achieve entropy convergence for both methods. The results are shown in Fig. 9 and Table III, respectively.

TABLE III

THE NUMBER OF ITERATIONS CORRESPONDING TO THE TWO METHODS

| -5dB | 0dB | 5dB | 10dB | 15dB | 20dB |
---|

Conventional Method | 9 | 10 | 10 | 11 | 12 | 13 |

Improved Method | 8 | 8 | 7 | 7 | 6 | 6 |

Compared with the conventional method, the improved method can converge the entropy value to a lower level faster under different SNRs. According to Fig. 9 and Table III, the improved method under different SNRs can achieve the convergence of the entropy value through about seven iterations. In comparison, the conventional method needs about ten iterations. Thanks to the improvements in range bins selection and windowing, the improved method can utilize more useful phase information in the target and range bins for phase estimation and autofocus. Therefore, the improved method converges faster and has a lower convergence entropy.

In Table III, the number of iterations of the improved method increases as the SNR decreases, while the number of iterations of the conventional method decreases. However, this does not mean that the conventional method has the advantage of reducing the number of iterations compared with the improved method at low SNR. To further compare the ability of the conventional and improved methods to improve the focusing effect, the entropy values obtained by the two methods after the first iteration under different SNRs and the entropy values that reach convergence are subtracted here. The entropy values obtained after the first iteration are marked in Fig. 9, the entropy values that reach convergence are shown in Table II and the results are shown in Table IV.

TABLE IV

THE DIFFERENCE BETWEEN THE ENTROPY VALUE OBTAINED BY THE TWO METHODS

| -5dB | 0dB | 5dB | 10dB | 15dB | 20dB |
---|

Conventional Method | 0.134 | 0.138 | 0.138 | 0.138 | 0.139 | 0.143 |

Improved Method | 0.693 | 0.684 | 0.653 | 0.613 | 0.573 | 0.561 |

In Table IV, as the SNR decreases, the entropy value difference obtained by the conventional method continuously decreases, while the improved method is the opposite. In addition, as shown in Fig. 9, as the SNR decreases, although the entropy values of the results obtained by both methods after the first iteration are increasing, the improvement in entropy values by the conventional method in Table IV is decreasing. Therefore, the conventional method requires fewer iterations to achieve entropy convergence. It also explains why the entropy values of the conventional method in Table II are maintained at around 8.2. In contrast, the improvement of the entropy by the improved method increases as the SNR decreases, so more iterations are needed to achieve the entropy convergence. The reduction in the number of iterations at low SNR does not indicate that the conventional method has the advantage of reducing the number of iterations at low SNR. Because this ‘advantage’ is obtained by sacrificing the improvement of the entropy value at low SNR.