Echo Model of Fast Moving Target
The echo model of fast moving target is established in this section. No matter what the motion parameters of the target are, the geometry of the moving target in broadside SAR is unchanged. Therefore, the echo model is established based on Figs. 2.2 and 2.3.
According to Eq. 2.3.4, crosstrack velocity, crosstrack acceleration and alongtrack velocity are the main parameters that affect the moving target imaging. In fact, the impact of the alongtrack acceleration is existed. However, its impact is neglected for the following reasons: first of all, the Doppler centroid and Doppler modulation rate are not related to the alongtrack acceleration; secondly, in the real flight of airborne SAR, the platform velocity is not stable, and the alongtrack acceleration of the moving target is negligible compared with the changes of the platform velocity, and can be compensated during the autofocus process. Thus, the impact of the alongtrack acceleration is neglected, and the instantaneous slant range of the moving target can be expressed as
Expand Eq. 3.2.1 into a Taylor series and keep it to the thirdorder term of t_{a}, then Eq. 3.2.1 can be approximated as
Substitute Eq. 2.3.3 into Eq. 3.2.2, it yields
According to Eq. 3.2.3, the thirdorder Taylor expansion term of a moving target is related to the crosstrack velocity and the alongtrack velocity, which increases accordingly with the crosstrack velocity, and he thirdorder term is no longer negligible if the crosstrack V_{r} is fast. Substitute Eq. 3.2.3 into Eq. 2.2.4, it yields
The second exponential term of Eq. 3.2.4 is the Doppler centroid term of a moving target. If the crosstrack velocity satisfies Eq. 2.3.7, the Doppler ambiguity will be induced. Assuming a Kuband airborne SAR system with PRF of 1500 Hz, carrier frequency of 0.0194 m, then the Doppler ambiguity will be induced if the crosstrack velocity of a moving target is faster than 14.55 m/s. In real applications, both civil and military vehicles can travel above this speed, which indicates that Doppler ambiguity cannot be neglected in the real data moving target processing.
The fourth exponential term is the RWM term of the moving target. It can be noted that the RWM significantly increases in the case of fast moving targets. The RWM and Doppler centroid are both induced by the crosstrack velocity. The Doppler centroid is sampled by PRF, which is aliased with a period of PRF. However, the RWM is not affected by the Doppler ambiguity. Therefore, the fast crosstrack velocity can be extracted from the RWM.
Compare Eq. 3.2.4 with Eq. 2.3.5, the only difference is that there is a thirdorder phase error in Eq. 3.2.4, i.e., the last exponential term in Eq. 3.2.4. It can be represented as
It can be noted that the thirdorder phase error increases with the crosstrack velocity. The thirdorder phase error will lead to the asymmetry of the sidelobes, and is not negligible for the fast moving target. Therefore, the thirdorder phase error must be compensated during the fast moving target imaging.
We simulated a singleantenna SAR system (see Table 3.1) to validate the conclusions above. In the observation area, four targets are set as shown in Table 3.2. It can be noted that T0 is a stationary target, T1 and T2 are slow moving target with a small crosstrack velocity, and T3 is a fast moving target.
The processing results of the four targets are shown in Fig. 3.1. The images after the range compression are shown in Fig. 3.1a. It can be noted that the RWM curve of T0 are paralleled with the azimuth axis, while the RWM curves of T1, T2 and T3 are slant in different scale. The focusing results of these targets are shown in Fig. 3.1b by using the stationary target parameters. It can be noted that all targets are smeared excepted T0. Also, T1 and T2 are dislocated from their real azimuth location in the image with the existence of the crosstrack velocity. The dislocation degree of T2 is larger than that of T1 since T2 has a larger crosstrack velocity. However, T3 appears in the upper half of the image, which is induced by the Doppler ambiguity. This result proves that the Doppler centroid cannot be correctly estimated by using the azimuth location of the fast moving target.
Figure 3.2a, b compare the azimuth compression results of T1 and T3 with and without thirdorder phase error compensation. In the simulation, assuming that the RCM and azimuth secondorder phase error are accurately corrected, it is noted that the impact of thirdorder phase error is negligible in the case of T1. On the contrary, the thirdorder phase error induces sidelobe symmetry of fast moving target T3.
According to the echo model and analysis in this section, three conclusions can be drawn: first of all, the large crosstrack velocity of a fast moving target leads to the Doppler ambiguity; secondly, the Doppler ambiguity leads to the azimuth location aliasing, while the RWM is not affected; thirdly, the thirdorder azimuth phase error must be compensated during the imaging process.
Table 3.1 System parameters of the simulation
System parameter 
Value 
System parameter 
Value 
Central slant range 
1000 m 
Range sampling rate 
60 MHz 
Carrier frequency 
2 GHz 
PRF 
400 Hz 
Pulse time width 
5 us 
Platform velocity 
100 m/s 
Pulse bandwidth 
30 MHz 
Antenna size 
1 m 
Table 3.2 Target parameters of the simulation
Target 
Motion parameter 

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

T0 
800 
0 
0 
0 
T1 
900 
1 
3 
0.1 
T2 
1000 
3 
3 
0.1 
T3 
1200 
18 
3 
0.1 
Fig. 3.1 Simulation results of the targets. a Range compression result. b Azimuth compression result
Fig. 3.2 Azimuth compression results of T1 and T3. a Azimuth compression result of T1. b Azimuth compression result of T3