In a typical airborne SAR system, the synthetic aperture time is several seconds. During the synthetic aperture time, a target is radiated and reflected the microwave continuously. In most articles, the moving target is assumed to be moving with a constant velocity or a constant acceleration during the synthetic aperture time. According to this assumption, A broadside SAR configuration with a moving target in the slant range domain is shown in Fig. 2.2.

Fig. 2.2 Broadside geometry of a moving target in a single-antenna SAR system

The platform parameters are the same as in Fig. 2.1, whereas the target is a moving target with cross-track velocity V_{x}, cross-track acceleration a_{x}, along-track velocity V_{y} and along-track acceleration a_{y}. During the azimuth time t_{a}, the platform flies from O to O_{1} with a constant velocity V_{a}, while the moving target moves from P to P_{1}. Therefore, the instantaneous slant range R(t_{a}) can be expressed as

Expand Eq. 2.3.1 into a Taylor series and keep it to the second-order term of t_{a}, then Eq. 2.3.1 can be approximated as

The relationship of the motion parameters between the slant range domain and range domain can be expressed as

where V_{r} and a_{r} represent cross-track velocity and acceleration in the slant-range domain respectively as shown in Fig. 2.3. Substitute Eq. 2.3.3 into Eq. 2.3.2, it yields

Fig. 2.3 Projection relationship between the range and slant-range domain

Substitute Eq. 2.3.4 into Eq. 2.2.4, the echo signal of the moving target can be expressed as

Compare Eq. 2.3.5 with Eq. 2.2.5, it is noted that the signal model of the moving target is different from that of a stationary target. The first exponential term of Eq. 2.3.5 is the Doppler centroid phase, which results in the azimuth dis-location of the moving target. The second exponential term is the second-order azimuth phase, it is also different from a stationary target, indicating that the Doppler modulation rate of a moving target differs from that of a stationary target. The third and fourth exponential terms are the RCM of the moving target. Compared with a stationary target, an additional RWM is induced by the motions, and the range curve migration is different from a stationary target. The RCM will deteriorate the imaging resolution if not accurately compensated, which is also a reason why a moving target appears smeared in a stationary SAR image. The Doppler parameters of a moving target can be represented as

where f_{dc} represents the Doppler centroid, H_{rwm} and H_{rcm} represent RWM and RCM, respectively.

From Eqs. 2.3.6a, 2.3.6b, 2.3.6c, 2.3.6d it can be noted that Doppler centroid and RWM are impacted by the cross-track velocity of a moving target, whereas the Doppler modulation rate and RCM is affected by both the along-track velocity and the cross-track acceleration.

Therefore, the Doppler parameters of a moving target changes correspondingly with the motion parameters of the moving target. The relationship between the Doppler parameters and the motion parameters is the reason why a moving target appears smeared and dislocation in the image, and also the lead to indicate and focus a moving target. All moving target processing algorithms are based on one or several parameters in Eqs. 2.3.6a, 2.3.6b, 2.3.6c, 2.3.6d, thus the impact of each parameter is analyzed as follows.