**7. Conclusions**

The unknown motion parameters of ground fast-maneuvering targets induce high-order RCM and DFM (i.e., CRCM and QDFM), which make the target energy seriously defocused. Fast-maneuvering targets easily exhibit complex Doppler ambiguity due to the limitation of PRF for SAR systems. These factors result in the difficulty in focusing of moving targets. In this work, a new computationally efficient algorithm is proposed to focus fast-maneuvering targets. The characteristics of the presented algorithm are summarised as follows: (1) the presented algorithm can effectively focus fast-maneuvering targets in the range–azimuth time domain because the acceleration and third-order phase are considered; (2) the proposed method is computationally efficient; (3) the proposed algorithm has a wide applicability because two constant factors ε and ϕ are introduced; (4) a new deramp function is proposed to further address the complex Doppler ambiguity including Doppler centre blur and spectrum ambiguity; (5) the cross-term interference for multiple target focusing is analysed, and a corresponding recognition procedure is proposed to identify the spurious peak. The effectiveness of the moving target refocusing algorithm and spurious peak recognition procedure has been confirmed by simulated and real data processing results. However, the proposed method introduces the nonlinear operation due to fast implementation for refocusing of fast-maneuvering targets; this condition weakens the performance in the case of low SNR/SCNR. This problem will be investigated in the future.

**Author Contributions:** Conceptualization, J.W. and L.Z.; Data curation, J.W. and Y.Z.; Formal analysis, J.W. and Z.C.; Funding acquisition, Y.Z. and L.Z.; Investigation, J.W. and H.Y.; Methodology, J.W. and Z.C.; Project administration, Y.Z. and L.Z.; Supervision, L.Z.; Validation, J.W., Y.Z. and Z.C.; writing—original draft, J.W.; writing—review and editing, Y.Z., L.Z., Z.C. and H.Y.

**Funding:** This research was funded by the National Natural Science Foundation of China, Grant Nos. 61871305, 61671361, and 61731023; and the APC was funded by the National Natural Science Foundation of China, Grant No. 61871305.

**Acknowledgments:** The authors would like to thank the anonymous reviewers for their valuable and useful comments and suggestions that helped improve the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

In this appendix, the selection criterion of the constant factor ε for 1D SCFT in our proposed method is discussed. In accordance with the peak in Equation (14), the equation is obtained as follows:

$$\hat{f}\_{\mathbf{f}\_u^2} = -\frac{2\left(\upsilon - \upsilon\_a\right)^2 - 2R\_0 a\_c}{\varepsilon \lambda R\_0}.\tag{A1}$$

To ensure the constant factor ε matching, the following inequality should be satisfied:

$$f\_{t\_n^{2}\max} \ge \left| \frac{2\left(v - v\_{\rm tr}\right)^2 - 2R\_0a\_c}{\varepsilon \lambda R\_0} \right|,\tag{A2}$$

where *f t* 2 *nmax* denotes the maximum value of *f t* 2 *n* .

We assume that the value scopes of target along-track velocity *va* and cross-track acceleration *ac* are [−*vamax*, *vamax*] and [−*acmax*, *acmax*]. Accordingly, the following equation is obtained:

$$\left|\frac{2(\upsilon - \upsilon\_a)^2 - 2R\_0 a\_c}{\varepsilon \lambda R\_0}\right|\_{\text{max}} = \frac{2(\upsilon + \upsilon\_{a\text{min}})^2 + 2R\_0 a\_{c\text{max}}}{\varepsilon \lambda R\_0}.\tag{A3}$$

By substituting Equation (A3) into Equation (A2), we obtain:

$$f\_{t\_n^2 \max} \ge \frac{2\left(v + v\_{a\min}\right)^2 + 2R\_0 a\_{c\max}}{\varepsilon \lambda R\_0}.\tag{A4}$$

In accordance with Equation (A4), the selection scope of constant factor ε is obtained as follows:

$$
\varepsilon \ge \frac{2\left(v + v\_{\rm{amin}}\right)^2 + 2R\_0 a\_{\rm{max}}}{f\_{\rm{t}^2\_n \rm{max}} \lambda R\_0}. \tag{A5}
$$

The constant factor should satisfy the inequality in Equation (A5). According to Equation (15), if a large constant factor is chosen, then the estimation error will be increased. Therefore, a smaller constant factor can be selected to improve estimation accuracy under the condition described in Equation (A5).

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
