# Signal Model of Moving Target with Non-ideal Motion Error

The high order phase error is assumed to be compensated by PGA and INS data. However, the images of moving targets are mostly smeared in real SAR data processing. This indicates that there are other non-ideal phase errors that lead to the de-focusing.

We believe that the first-order azimuth phase error is the cause of the de-focusing. In stationary SAR imaging, the energy of a target is assumed to be scattered from a single Doppler centroid. In practical SAR systems, the LOS of the antenna remains steady, so that the Doppler centroid of the clutter is single. Therefore, the focused stationary SAR image can be obtained after high order phase error is compensated.

However, in the case of moving targets, the single Doppler centroid may not be satisfied. In our research, we found that the Doppler centroid of a moving target is not a constant value in practical SAR data, and it will severely deteriorate the imaging quality of moving targets.

In traditional GMTI and GMTIm algorithms, the platform is modeled as moving with a constant velocity along a straight course, and the moving target is modeled as moving with a constant velocity and acceleration during the synthetic aperture time. The typical synthetic aperture time of an airborne SAR is around 1-2 s, so the assumption is rational in most cases. However, in the existence of non-ideal motion errors, a more precise signal model must be established.

Since the non-ideal trajectory error can be compensated by INS and PGA, the airplane is assumed to move along a straight trajectory. The platform velocity *V _{a}(t*

_{a}) is variant to azimuth time. The geometry of a moving target with non-ideal motion error is illustrated in Fig. 6.6.

In the slant range domain, the cross-track velocity is denoted by V_{r}(t_{a}), the along-track velocity is denoted by *V _{y} (t_{a}).* When the platform flies O from to

*O*the moving target moves from P to

_{1},*P*Suppose the nearest slant range of the moving target is R

_{1}._{0}, the instantaneous slant range

*R(t*can be expressed as

_{a})**Fig. 6.6 ****Geometry of a moving target with non-ideal motion errors**

Expand Eq. 6.3.1 into a Taylor series and keep it to the second-order term of t_{a}, then Eq. 6.3.1 can be approximated as

Substitute Eq. 6.3.2 into Eq. 2.2.4 and expand, it yields

In Eq. 6.3.3, the first two terms are the azimuth phase of the target. The first denotes the Doppler centroid of the target, and the second denotes the second- and higher-order phase error. The third and fourth terms are the RCM of the moving target. In traditional SAR imaging algorithm, the focus of a target mainly relies on the compensation of the second term in Eq. 6.3.3. In the GMTIm, this term can be accurately compensated by the precise motion parameter estimation and PGA.

However, the PGA algorithm cannot correct the error caused by the Doppler centroid. In Eq. 6.3.3, the Doppler centroid is related to *t _{a}* and V

_{r}(t

_{a}). If the Doppler centroid is constant, the energy of the target will be concentrated into a point after the higher order azimuth phase error is corrected. However, since

*t*and V

_{a}_{r}

*(t*are inconstant, the energy of moving target will not be concentrated into a point, which makes the image of the target smear in azimuth.

_{a})Azimuth time *t _{a}* is sampled by PRF, and PRF is determined by the platform velocity

*V*in a space-constant PRF system. Therefore, the Doppler centroid is affected by both the platform and the moving target.

_{a}(t_{a})