# Cross Sensor Simulation of Tomographic SAR Stacks
## 2 Wavenumbers in SAR Acquisitions
The two-way travel path implies that the wavenumbers of the target responsible for the backscattered echo are twice the ones of the signal impinging on the target itself ([PERSON], 1999). Then, the spectral components giving rise to the radar echo, lie on a line centered on \(k_{0}\) and with length \(\Delta k\) ; these latter quantities are defined by Equations (3) and (4). \[k_{0} =\frac{4\pi}{c}f_{0} \tag{3}\] \[\Delta k =\frac{4\pi}{c}B \tag{4}\]  
The dashed line in Figure 2 is an example of the spectral components illuminated in a highly localized target by a radar in fixed position. The length of this line is connected to the resolution that the radar system can provide along the line-of-sight. The capability of resolving target along the azimuth direction typical of SAR systems can also be interpreted in terms of wavenumbers. A set of acquisitions gathered along the platform trajectory are characterized by spectral lines whose distance from the origin and length are constant (as they depend on radar parameters) but change orientation (being the same as the line-of-sight). The spectral region illuminated by a SAR survey is shown in Figure 2 as a grey surface. The extension of this patch is linked to the resolution that the SAR system can provide; the 2D extent of this surface reflects the lack of resolution of SAR systems along the direction orthogonal to the line-of-sight and to azimuth. That is, standard SAR systems cannot resolve targets lying along the _cross range_ direction \(\zeta\) depicted in Figure 1.  
The grey surface in Figure 2 is the spectral enlightenment associated with a single (or a narrow range) look angle \(\vartheta\); the corresponding surface for a look angle \(\vartheta+\delta\vartheta\) can be obtained by rotating this patch around the \(kx\) axis by an angle \(\delta\vartheta\). The set of all these surfaces defines a volume in the \(\{kx,ky,kz\}\) domain, its edges are shown as thick lines in Figure 2. The spectral spread along the cross range direction should not suggest the possibility of resolving targets in the 3D space though. As a matter of fact, this spectral spread only exists if different parts of the scene are considered together, it disappears for a single target. The very same feature can be appreciated by analyzing the spectral enlightenment along the azimuth direction; in order to do this the constraint of local analysis must be relaxed and the waves must be considered as spreading spherically from the radar rather than being plane. For any fixed azimuth position occupied by the sensor, a large spatial region is reached by the transmitted signal, its extension along \(x\) depending on the antenna beamwidth. It follows that many wavenumbers along \(kx\) are illuminated at once; however, they are associated with different targets on the ground thus providing a narrow band signal (hence no resolution) for each of them. A wideband signal along \(kx\) can only be produced by combining the echoes gathered by the sensor along its path, that is building a synthetic antenna. Analogously, many images of the same target are needed if resolution along the cross range is desired, each one gathered from a different look angle. This issue is discussed more in detail in the following subsection.