*5.2. Real Data Processing Result Analysis*

In this section, two parts of RADARSAT-1 Vancouver scene data [1,26] are utilised to validate the presented algorithm. These real radar data are recorded by a C-band space-borne SAR system. The main radar system parameters are summarised in Table 3. The detailed parameters of these real data are given by [1,26]. The azimuth pulse number of selected data is 1500.

**Table 3.** Main radar parameters for RADARSAR-1.


#### 5.2.1. Processing Result of a Single Target

Figure 17a shows the image of the selected scene, where the interested target is highlighted in the figure. Figure 17b depicts the trajectory for the target of interest after range compression. The trajectory distributes into multiple range sample bins due to the serious RCM, which indicates the typical defocusing. Figure 17c illustrates the result of RCMC in the range–Doppler domain. The target energy is focused in the same range bin after RCMC, but the effect of DFM still remains. As shown in Figure 17d, an evident peak appears in the 1D SCFT domain. According to the peak position in Figure 17d, the target can be focused by using the proposed method, as exhibited in Figure 17e. Therefore, the above-mentioned real data processing results demonstrate the effectiveness of the presented method.

**Figure 17.** Real data processing results of a single target: (**a**) image of the selected scene; (**b**) trajectory of interested target after range compression; (**c**) result after RCMC; (**d**) 1D SCFT result; (**e**) focusing result for a single target of interest.

#### 5.2.2. Processing Result of Two Targets

Figure 18a displays the image of the selected scene, where the two targets of interest, which are denoted by Target 1 and Target 2, are marked in the figure. Figure 18b shows the trajectories of Target 1 and Target 2 after range compression. The profile along the 420th azimuth sample bin of Figure 18b is depicted in Figure 18c,d. Two trajectories span over several range sample bins due to the serious RCM. Figure 18e depicts the result of RCMC in the range–Doppler domain. The profile along the 420th azimuth Doppler sample bin of Figure 18e is shown in Figure 18f,g. The RCM is effectively removed, but the DFM still exists. The trajectory in the middle of trajectories for Target 1 and Target 2 is a potential cross term according to the analysis in Section 4.2. Figure 18h–j illustrate 1D SCFT results of data in the 69th range sample bin (Target1), 79th range sample bin (cross term) and 89th range sample bin (Target2), respectively. Given that the peak positions in Figure 18h–j satisfy the symmetric features described in Figure 7, the peak shown in Figure 18i is preliminarily identified as a potential spurious peak. The spurious peak recognition results are shown in Figure 19a,b. The peak illustrated in Figure 18i is confirmed as the spurious peak because the symmetric peaks with the same amplitudes appear in 10th and −10th range time sample bins of the recognition function. The cross term is removed, and the focused results of Target 1 and Target 2 are depicted in Figure 19c,d, respectively. These real data processing results verify that the proposed method can be used to focus multiple targets and validate the proposed spurious peak recognition procedure.

**Figure 18.** *Cont.*

(**j**)

**Figure 18.** RCMC and 1D SCFT results: (**a**) image of the selected scene; (**b**) range compression result for two targets of interest; (**c**) profile along the 420th azimuth sample bin of Figure 18b; (**d**) dB version of amplitude for Figure 18c; (**e**) result after RCMC; (**f**) profile along the 420th azimuth Doppler sample bin of Figure 18e; (**g**) dB version of amplitude for Figure 18f; (**h**) 1D SCFT result of the 69th range sample bin (Target 1); (**i**) 1D SCFT result of the 79th range sample bin (cross term); (**j**) 1D SCFT result of the 89th range sample bin (Target 2).

**Figure 19.** Potential spurious peak recognition and final focusing results: (**a**) 1D SCFT result for the −10th range time sample bin of the recognition function; (**b**) 1D SCFT result for the 10th range time sample bin of the recognition function; (**c**) focusing result of Target 1; (**d**) focusing result of Target 2.

#### **6. Discussion**

#### *6.1. Computational Complexity*

In this section, we discuss the computational complexity of the proposed method, FOKT-based method [25], stationary phase-based method [28] and SOKT-GHHAF method [32]. Similar to [32], the number of complex multiplications is utilised to indicate the computational complexity. We suppose that *G* represents the number of range bins and *P* denotes the number of pulses. For convenience, we assume that the SRCM correction function in Equation (10) is effective. The computational burden of the proposed algorithm includes a range FFT operation, a *G* × *P*

point matrix complex multiplication, an SRCM correction operation, a 1D SCFT processing, an azimuth FFT operation, a matched filtering processing and a 2D IFFT operation. Notably, the 1D SCFT in Equation (14) can be easily implemented using the NUFFT of low computational burden. The detailed analysis of the NUFFT has been provided in [43,44]. The computational complexity of the NUFFT-based 1D SCFT is obtained using *O*(*Plog*2*P*) [43,44]. Thus, the total computational cost of the proposed method is denoted as *O*(2*PGlog*2*G*) + *O*(2*GPlog*2*P*) + *O*(*Plog*2*P*) + 3*GP*. We assume that the searching times of β<sup>2</sup> and Doppler ambiguity number are represented by *I*<sup>2</sup> and *Id*, respectively. The computational cost of the SOKT-GHHAF method is denoted as *O GP*<sup>2</sup> + *<sup>P</sup>*(*<sup>P</sup>* <sup>−</sup> <sup>1</sup>)*<sup>G</sup>* + *O GP*<sup>2</sup> + *O P*3 + *O I*2*Plog<sup>P</sup>* 2 [32]. The computational burden for stationary phase-based method is obtained using *O*[*IdI*2(*PGlog*2*G* + *GPlog*2*P*)] [28]. The computational complexity of the FOKT-based method is represented as *O*[(*Id* + 1)(*GPlog*2*P* + *PGlog*2*G*)] + *IdGP* [25].

In the case of the SRCM correction function mismatch, the SOKT operation should be added in the proposed method. The chirp-z-based SOKT is applied to compensate the SRCM. The computational cost of chirp-z-based SOKT is represented as *O*(*GPlog*2*P*) [45]. The total computational burden is denoted as *O*(2*PGlog*2*G*) + *O*(3*GPlog*2*P*) + *O*(*Plog*2*P*) + 3*GP*. In this case, the computational complexity of the proposed algorithm is slightly increased. However, the proposed method still has low computational complexity.

Table 4 exhibits the detailed computational costs of the above-mentioned algorithms. The table shows that proposed method<sup>1</sup> denotes the computational complexity of the proposed method in the case of SRCM correction function matching, and proposed method2 represents the computational complexity of the proposed method in the case of SRCM correction function mismatch. In summary, the computational complexity of the proposed algorithm is lower than that of the SOKT-GHHAF, stationary phase-based and FOKT-based methods because it can be implemented using FFT, IFFT and NUFFT. The parameter searching procedure is avoided.



*6.2. Some Remarks*

**Remark 1.** *The di*ff*erent moving targets may have various scattering intensities for multiple target focusing. If the intensities of these targets are significantly di*ff*erent, then the target with higher intensity may submerge the target with a lower value; this condition a*ff*ects the performance of the presented method. In this case, the CLEAN technique in [46,47] can be provided to remove the strong target e*ff*ect. The strong and weak moving targets can be focused iteratively*.

**Remark 2.** *The proposed method has a relatively high demand on the target input SNR*/*signal-to-clutter and noise ratio (SCNR) because its processing procedure contains a nonlinear operation. Therefore, the presented algorithm is suitable for the fast realisation for refocusing of fast-maneuvering targets in the case of relatively high SNR*/*SCNR. However, a slow and weak moving target may be drowned by the strong clutter background. In this case, the performance will degrade. At this time, many excellent clutter rejection methods, such as displaced phase centre antenna [48] and space-time adaptive processing [49,50], can be performed to reject the clutter. After clutter suppression, the proposed method can be used to refocus the moving target given that the target input SCNR is significantly increased. The e*ff*ectiveness of clutter suppression in the moving target refocusing application has been validated in previous studies [16,17,25,27,29,32]. Interested readers may refer to [27,29]*

*for a detailed analysis about clutter rejection in the moving target refocusing applications. The fast realisation for refocusing of moving targets with a low SNR*/*SCNR in strong or extremely heterogeneous background is still a challenging work and will be investigated in the future.*
