Results of the Simulation and Real Data Processing
In this section, we simulate a singleantenna SAR system (see Table 3.1) to validate the proposed algorithm. One stationary target, labeled by T0, and three moving targets, labeled by T1, T2, and T3, are set in the scene. The motion parameters of the moving targets are listed in Table 3.3. According to Table 3.3, T1 and T3 represent the moving targets with slow and fast crosstrack velocities, respectively,
Table 3.3 Target parameters of the simulation
Target 
Motion parameters 

Nearest slant range (m) 
Azimuth location shift (m) 
Crosstrack velocity (m/s) 
Alongtrack velocity (m/s) 
Crosstrack acceleration (m/s^{2}) 

T0 
1100 
10 
0 
0 
0 
T1 
800 
0 
10 
10 
0 
T2 
900 
10 
10 
0 
0 
T3 
1000 
0 
50 
10 
1 
and T2 represents the moving target that is not located at the center of the scene. To better testify the improvement of the proposed algorithm, the conventional algorithm in [13] is operated as a comparison.
Figure 3.14a, b show the processing results of the conventional and the proposed algorithm, respectively. For the slow moving target T1, it is focused and relocated
Fig. 3.14 Simulation results of multiple moving target imaging. a Processing result of the conventional algorithm. b Processing result of the proposed algorithm. c Range compression result comparison of T1. d Azimuth compression result comparison of T1. e Range compression result comparison of T3. f Azimuth compression result comparison of T3
in both Fig. 3.14a, b. However, the image of T3 in Fig. 3.14a is severely smeared since the range walk is not completely corrected by Keystone transform and the third order phase error is not compensated. On the contrary, T3 is wellfocused in Fig. 3.14b. From the comparison, we can conclude that the proposed algorithm is effective in focusing both the slow and fast moving targets. The relocation of T2 in Fig. 3.14a is inaccurate since the energy balance filter method cannot provide the accurate location if the target is not located at the center of the azimuth aperture. However, this problem is avoided by Hough transform, as shown in Fig. 3.14b.
Figure 3.14c, d further compare the range and azimuth compression results of T1. The sidelobe asymmetry is not negligible in Fig. 3.14d although V_{r} of T1 is slow. The range and azimuth compression results of T3 are compared in Fig. 3.14e, f. It can be noted that the range and azimuth resolutions of T3 by the conventional algorithm are both deteriorated by the range walk and the thirdorder phase error.
Motion parameter estimations are listed in Table 3.4. The estimations of Vr and fdc of T2 are incorrect by the conventional algorithm since the azimuth location affects the accuracy of estimation. The estimations of V_{r} and f_{dc} of T3 by the conventional algorithm depart from the actual values because of the Doppler ambiguity, and the estimation of V_{y} of T3 by the conventional algorithm is incorrect since the moving target is assumed with constant velocities in [13]. On the contrary, the proposed algorithm provides accurate motion parameter estimations of all the targets.
Table 3.4 Comparison of the motion parameter estimations
Target 
Parameters 

Doppler centroid (Hz) 
Crosstrack velocity (m/s) 
Alongtrack velocity (m/s) 
Crosstrack acceleration (m/s^{2}) 

T1 
Actual value 
133.33 
10 
10 
0 
Traditional algorithm 
133.00 
9.9750 
10.0333 
0 

Proposed algorithm 
133.30 
9.9975 
10.0667 
0 

T2 
Actual value 
133.33 
10 
0 
0 
Traditional algorithm 
119.1667 
8.9375 
0 
0 

Proposed algorithm 
133.30 
9.9975 
0 
0 

T3 
Actual value 
666.67 
50 
10 
1 
Traditional algorithm 
266.00 
19.9500 
15.5703 
0 

Proposed algorithm 
666.20 
49.9650 
9.7261 
1.0210 
Fig. 3.15 Spectrum analysis of moving target in real data. a Doppler spectrum of the data after range compression. b Azimuth phase angle of the moving target
To further verify our algorithm, we use real Kuband airborne SAR/GMTI data in Sect. 3.4.2 to further demonstrate the robustness of the proposed algorithm. The selected data contains the echo of a vehicle T2 moving along a highway in the scene. T2 is not a controlled target so that the parameter estimation accuracy cannot be compared in this section.
Figure 3.15a shows the Doppler spectrum of the selected data after range compression. From Fig. 3.15a, we can see that the trajectory of moving target, which is tagged by an ellipse, locates at the highband of PRF due to the Doppler centroid shift, and the Doppler spectrum of the moving target exceeds the limit of PRF. Figure 3.15b shows the unwrapped azimuth phase angle of the target. The azimuth phase angle of the moving target is not a standard quadratic curve, i.e., there are higher order phase errors. Therefore, the necessity to compensate for the thirdorder phase error in real data imaging is confirmed.
Figure 3.16a shows the stationary image of the scene without the processing of the moving target. As tagged by the ellipses, the moving target is smeared and dislocated from the highway. Moreover, the image of the moving target splits into two parts because its spectrum expands neighboring PRF. Figure 3.16b shows that with the proposed algorithm, the moving target is focused and relocated on the highway. Therefore, the effectiveness of the proposed algorithm in real airborne SAR data is demonstrated.
Fig. 3.16 Imaging results of the data without and with the proposed algorithm. a Stationary image of the scene without moving target imaging. b Imaging result of the moving target by the proposed algorithm