# Cross Sensor Simulation of Tomographic SAR Stacks
## 3 Cross Sensor Simulation
The spectral properties of the target are properly taken into account by the back-projection tomographic algorithm: the reconstructed profile is band-limited and pass-band. However, it lies on the spatial frequencies illuminated by the first system which might be different from the ones required by the second. The reflectivity map returned by the back-projection algorithm when applied to real data will be referred to as \(s_{A}\left(y,z\right)\). This function, if fed directly into Equation (12) would lead to incorrect results the more the simulated system is different from the real one. The convolution expressed in Equation (12) amounts to a product in the wavenumber domain between the Fourier transform of the system filter (whose support corresponds to the spectral enlightenment) and the reflectivity spectrum of the target. Then, for an acquisition to be properly simulated, these two spectral regions must overlap for all the range line, that is, for every look angle. The overlap must occur in the \(ky,kz\) plane so the reflectivity spectrum must match the spectral enlightenment concerning both angle and central frequency. Typically, the angular agreement between the two systems is reached for a narrow interval of distances sampled by the airborne system. It follows that feeding \(s_{A}\left(y,z\right)\) directly into Equation (12) would result in SLCs almost completely empty as a consequence of the different spectral support of \(s_{A}\left(y,z\right)\) and the system filter. Moreover, a large discrepancy in the central frequencies (or a relatively small bandwidth) gets the two spectral regions separated regardless of the look angle. In order to guarantee the correspondence of wavenumbers, the estimated reflectivity profile \(s_{A}\left(y,z\right)\) must be spectrally translated around the wavenumbers illuminated by the simulated system; in the remainder of this section, it is illustrated how this translation can be achieved through a spatial demodulation followed by a modulation. The reflectivity profile at radio-frequency \(s_{A}\left(y,z\right)\) can be taken to baseband (in the \(\left\{ky,kz\right\}\) domain) by compensating the phases associated with the local distances. The spatial map demodulating the reflectivity profile can be expressed as in (13). \[F_{demod}\left(y,z\right)=e^{-j\frac{4\pi}{\lambda_{A}}r_{A}\left(y,z\right)} \tag{13}\]  
where \(r_{A}\left(y,z\right)\) represents the distance of each pixel of \(s_{A}\left(y,z\right)\) from the master sensor and \(\lambda_{A}\) the central wavelength used by the airborne system. The product between \(s_{A}\left(y,z\right)\) and \(F_{demod}\left(y,z\right)\) modifies the spectral support of the reflectivity function and takes it into baseband. Locally, that is for every look angle, it amounts to a translation of the corresponding spectral line along the direction defined by the spectral line itself; however, different look angles are associated with spectral lines with different orientations so that the translation always follows the radial direction. As a consequence, the shape of the spectral support of \(s_{A}\left(y,z\right)\) is modified: from a section of a ring to a \"butterfly\" shaped region as shown in Figure 6. The thickness of the butterfly around the origin in the direction orthogonal to the line-of-sight is connected to the resolution of the reflectivity map along the cross-range direction that the airborne system is able to provide. Real spectra before and after the spatial demodulation are shown in Figure 7. The reflectivity profile can be taken in the surrounding of the wavenumbers illuminated by the spaceborne system by another modulation; the modulating map is shown in Equation (14). \[F_{mod}\left(y,z\right)=e^{j\frac{4\pi}{\lambda_{S}}r_{S}\left(y,z\right)} \tag{14}\]  
where \(r_{S}\left(y,z\right)\) represents the distance of each pixel of \(s_{A}\left(y,z\right)\) from the master sensor of the spaceborne system to be simulated and \(\lambda_{S}\) its central wavelength.