# Signal Model and Analysis

The geometry of stationary targets in FMCW SAR is the same as a pulse SAR, so the geometry is the same as Fig. 2.1. However, as analyzed in Sect. 5.2.1, the stop-and-go approximation is not valid in FMCW SAR, so the instantaneous slant range between stationary target P and the platform at azimuth time *t*_{a}** **can be expressed as

It can be noted that the instantaneous slant range is relative to the range time. After Taylor series expansion, Eq. (5.2.2) can be approximated as

where *R*_{A} *=* V**R**T+**Vjtf**, and is approximated as *R*_{A}** **~ **R _{0}**. Compare Eq. 5.2.3 with Eq. 2.2.2, a range-azimuth coupling term is added in FMCW SAR. Since the azimuth information is hidden in the slant range, the echo phase of FMCW SAR is different from that of pulse SAR.

Neglect the impacts of signal amplitude and window function, the transmitted signal of FMCW SAR can be expressed as

And the received signal can be expressed as a copy of the transmitted signal with a time-delay:

Before the received signal is sampled by A/D, the received signal must be deramped by the reference signal. The reference signal is the echo from a known range *R** _{re}f*, which is represented as

Send Eqs. 5.2.5 and 5.2.6 into a conjugate multiplier, set *R _{re}f =* 0, and the beat signal can be presented as

The last term in Eq. 5.2.7 is the RVP term, and must be corrected in the imaging process. After the RVP correction, substitute Eq. 5.2.3 into Eq. 5.2.7 and neglect the higher terms of *t*_{r}, it yields

The first term of Eq. 5.2.8 is a constant term. The second term is the first-order term of range time, which indicates that the range compression can be accomplished by FFT. The third and fourth terms are the azimuth phase and the range curvature. These four terms are all caused by the movement of the radar platform, which are the same as pulse SAR.

The last term of Eq. 5.2.8 is the RWM induced by the movement of the radar platform within a PRI. This term is caused by the invalidation of the stop-and-go approximation, and is ignored in the pulse SAR. Therefore, the imaging algorithms in pulse SAR is not valid in FMCW SAR anymore, and the additional range walk must be corrected.