# Motion Error Analysis

# Platform Velocity Error

In the stationary SAR imaging, the platform velocity error will lead to the inconstant Doppler modulation rate. In order to solve this problem, the SAR system is changed from the time-constant sampling system to the spatial-constant sampling system. The PRF of the spatial-constant sampling system is instantaneously adjusted to the platform velocity, so as to keep the PRF-to-velocity ratio constant. In such systems, the time-variant PRF can be defined as

In the spatial-constant sampling system, the Doppler bandwidth of the clutter can be expressed as

where *L _{a}* denotes the azimuth size of the antenna. According to Eqs. 6.4.1 and 6.4.2, both

*B*and

_{a}(t_{a})*PRF(t*are linear functions of

_{a})*V*Therefore, the spectrum of the clutter in each azimuth sample can be coherently accumulated during

_{a}(t_{a}).*T*However, the Doppler centroid of the moving target is independent of V

_{a}._{a}(t

_{a}). If V

_{a}(t

_{a}) is inconstant, the location of

*f*in each azimuth sample is different from one another, which will make the spectrum of the moving target spread on multiple Doppler centroids. The impact of the platform velocity error is illustrated in Fig. 6.7.

_{dc}In Fig. 6.7, the red breaking lines represent the Doppler histories of the moving target. In Fig. 6.7a, the Doppler history of the moving target is a rectilinear trajectory. On the contrary, the trajectory is not a straight line in Fig. 6.7b with the PRF variation. In the case of Fig. 6.7b, the energy of the moving target cannot be compressed into one Doppler centroid. In practical airborne SAR applications, especially for unmanned missions, the platform velocity changes severely in the existence of the air turbulence. Therefore, the platform velocity error must be compensated in the moving target imaging.