# Principle of SAR

SAR is an advanced 2-D high resolution radar, which is able to acquire high imaging resolutions in both the range and azimuth directions. For a real aperture radar, the azimuth resolution relies on both the size of the aperture and the range: bigger aperture and shorter range make higher azimuth resolution. Therefore, in order to improve the azimuth resolution, the aperture size must be enlarged. The innovation of SAR is to simulate an equivalent large azimuth aperture by moving the radar in the azimuth direction with a constant velocity. On the other hand, the high range resolution of SAR is obtained by transmitting and receiving a wide-band linear modulated (LFM) signal.

During the azimuth time *t*_{a}, the platform flies from O to *O _{1}* with a constant velocity V

_{a}.

*R*and

_{0}*R(t*represent the nearest and instantaneous slant ranges between the platform and the target P, respectively. H and X represent the height of the platform and the flat range, respectively. According to Fig. 2.1, R(t

_{a})_{a}) can be expressed as

Expand Eq. 2.2.1 into a Taylor series and keep it to the second-order term of ta, then Eq. 2.2.1 can be approximated as

Suppose the transmitted signal is a LFM signal with a carrier frequency f_{0}, a modulation rate *K _{r}* and a pulse band-width

*T*

_{r}, then the transmitted signal

*s*can be expressed as

_{t}(t_{r})**Fig. 2.1 ****Broadside geometry of a stationary target in airborne SAR**

where rect(-) represents the rectangle window of the transmitted signal, and t_{r }denotes the range time. After the range compression, the echo signal of target P can be expressed as

where *A _{0}* is the complex reflection coefficient of the moving target,

*B*and k are the bandwidth and wavelength of the transmitted signal, respectively, and c is the speed of light.

_{r}From Eq. 2.2.4, the range phase information is contained in the sinc function, which indicates that the range phase is related to the instantaneous slant range R(t_{a}). The azimuth phase information is contained in the exponential terms, which means that it is also related to *R(t*_{a}). Substitute Eq. 2.2.2 into Eq. 2.2.4, it yields

In Eq. 2.2.5, the first exponential term denotes the range location of the target; the second exponential term represents the azimuth phase of the target. It can be noted that the azimuth phase of a point target in stripmap SAR is a second-order function of *t _{a},* which makes it a LFM signal with an azimuth modulation rate

*f*as

_{dr}

Therefore, the azimuth compression can be achieved by utilizing a matched filter with modulation rate *f*_{dr}. If the modulation rate of the matched filter is mismatched, the second-order azimuth phase cannot be accurately compensated, and the azimuth dispersion is induced.

The third exponential term of Eq. 2.2.5 represents the RCM of the target. The RCM contains the range curve migration and the RWM. In broadside stripmap SAR, there is only the range curve migration with the absence of the squint angle. The RCM error will cause an energy dispersion of a target among multiple range cells if not accurately compensated.

According to the analysis above, the core of SAR signal processing is to compensate the phase errors of both the range and azimuth directions. The range and azimuth phase errors are compensated by using the matched filters. Since the transmitted signal is known, the range matched filter is mostly accurate. The azimuth matched filter is calculated and estimated, so it may induce azimuth phase errors if the matched filter is not accurate. The first-order azimuth phase error will cause the dislocation of the target; the second-order azimuth phase error will cause the broadening of the main-lobe; the third-order azimuth phase error will cause the asymmetry of the side-lobes; the fourth-order azimuth phase error will cause the raise of the side-lobes. RCMC is also an important part. Different algorithms have different ways to accomplish the RCMC.

As to a moving target, the signal form is the same as that of a stationary target, while the phase information is different with the existence of the motion. Therefore, the imaging algorithms that suit the stationary target processing are also suitable for the moving targets, if only the imaging parameters are accurately estimated.