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, cross-track velocity, cross-track acceleration and along-track velocity are the main parameters that affect the moving target imaging. In fact, the impact of the along-track 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 along-track acceleration; secondly, in the real flight of airborne SAR, the platform velocity is not stable, and the along-track 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 along-track 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 third-order 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 third-order Taylor expansion term of a moving target is related to the cross-track velocity and the along-track velocity, which increases accordingly with the cross-track velocity, and he third-order term is no longer negligible if the cross-track 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 cross-track velocity satisfies Eq. 2.3.7, the Doppler ambiguity will be induced. Assuming a Ku-band airborne SAR system with PRF of 1500 Hz, carrier frequency of 0.0194 m, then the Doppler ambiguity will be induced if the cross-track 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 cross-track 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 cross-track 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 third-order 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 third-order phase error increases with the cross-track velocity. The third-order phase error will lead to the asymmetry of the side-lobes, and is not negligible for the fast moving target. Therefore, the third-order phase error must be compensated during the fast moving target imaging.

We simulated a single-antenna 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 cross-track 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 dis-located from their real azimuth location in the image with the existence of the cross-track velocity. The dis-location degree of T2 is larger than that of T1 since T2 has a larger cross-track 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 third-order phase error compensation. In the simulation, assuming that the RCM and azimuth second-order phase error are accurately corrected, it is noted that the impact of third-order phase error is negligible in the case of T1. On the contrary, the third-order phase error induces side-lobe 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 cross-track 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 third-order 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)

Cross-track velocity (m/s)

Along-track velocity (m/s)

Cross-track 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