# Principle of GMTIm Algorithms

After GMTI, the echoes of moving targets are separated from that of the clutter. GMTIm algorithms are used to focus and relocated the moving target by estimating its Doppler parameters. The main operations of GMTIm algorithms contain RCMC and Doppler parameter estimation.

## RCMC of Moving Targets

The RCM of the moving target must be accurately corrected so as to get a focused image of the target. However, the RCM of the moving target is different from that of the clutter, and is unknown without the information of motion parameters. In most articles, the RCMC of moving targets is performed by using the Keystone transform [16].

Before the range compression, the echo of a point target in a broadside airborne SAR can be expressed as

where T_{r} denotes the pulse duration, *f _{c}* denotes the carrier frequency. Transform Eq. 2.5.1 into range-frequency domain, it yields

where f is the range frequency. In Eq. 2.5.2, the first exponential term can be compensated by a range matched filter or the deramping operation so as to accomplish the range compression. The RCM and azimuth phase information are hidden in the second exponential term. Suppose it is represented by *U(t _{a},f*

_{r}), substitute Eq. 2.3.4 into Eq. 2.5.2, it yields

In Eq. 2.5.3, the second exponential term contains the Doppler centroid shift and the RWM; the third exponential term contains the second-order azimuth phase error and the range curve migration. It is noted that the coupling of the range-frequency and azimuth-time denotes the RWM, and the coupling of the range-frequency and the second-order azimuth-time denotes the range curve migration. Transform the azimuth time *t _{a}* by

Substitute it into Eq. 2.5.3, it yields

It is noted that the range-frequency and azimuth-time is decoupled after the transform, and in turn the RWM is corrected. Equation 2.5.4 is the definition of Keystone transform [17-19]. By using the Keystone transform, the RWM can be corrected without the estimation of the cross-track velocity V_{r}.

It is also noted from Eq. 2.5.5 that the range curve migration is not corrected by using the Keystone transform. Thus, researchers proposed the second-order Keystone transform [20], which is defined as

Substitute Eq. 2.5.6 into Eq. 2.5.3, it yields

It can be noted that the range curve migration is corrected after the second-order Keystone transform, and half of the RWM is removed. In the moving target imaging of an airborne SAR system, the impact of the RWM on the imaging quality of a moving target is far severer than the range curve migration [21], so the first-order Keystone transform is the most widely-used algorithm in RCMC of moving targets, and the second-order Keystone transform is only suited for the imaging of along-track moving targets.

Since the azimuth time is a discrete value, the Keystone transform is accomplished by sinc interpolation. The precision of sinc interpolation is highly related to the length of the interpolation core, and long interpolation core requires heavy calculation burden. Therefore, Keystone transform is not suitable for real-time processing.

**Fig. 2.9 ****Flowchart of GMTI and GMTIm in airborne SAR**

Another disadvantage of Keystone transform is that it is only effective without the existence of Doppler centroid ambiguity. If there is Doppler centroid ambiguity in the echo, the Doppler centroid ambiguity number must be estimated before Keystone transform.