# Cross Sensor Simulation of Tomographic SAR Stacks
## 2 Wavenumbers in SAR Acquisitions  
This section recalls the basic principles of any SAR surveys. The transmitted signal, the scene under observation and their interaction are described in two conjugated domains. The former is the direct domain, that is time or space; the latter is the frequency domain, usually referred to as _wavenumber_ domain when spatial coordinates are transformed. This representation is particularly useful due to the possibility of interpreting SAR acquisitions in terms of enlightenment of the target's components. Also, the simulation procedure, core of this work, can only be described by resorting to this approach. The change of domain is achieved by computing the Fourier transform [19] of the function of interest, that is projecting it on a set of complex exponentials of the form shown by Equations (1) and (2) for signals defined over time and spatial maps respectively. \[Ge^{j\omega t} = e^{j2\pi ft} \tag{1}\] \[e^{jk\cdot r} = e^{j\left(kx\cdot x+ky\cdot y+kz\cdot z\right)} \tag{2}\]The basis function expressed by Equation (2) has unitary magnitude and phase varying along a specific direction only. This direction is defined by the direction of the real vector \(\mathbf{k}\); by moving on the plane perpendicular to \(\mathbf{k}\) no changes occur. For this reason, vector \(\mathbf{k}\) is referred to as _wavevector_, its components \(kx\), \(ky\) and \(kz\) are known as _wavenumbers_. The speed of the phase rotation is determined by the magnitude instead: the longer \(\mathbf{k}\) the faster the oscillations. Due to the orthogonality of two complex exponentials having different wavevectors, the function in Equation (2) defines a point in the \(\{kx,ky,ky\}\) domain, that is the wavenumber or the spatial frequency domain. Signals traveling in both space and time can be described either in the frequency domain or in the wavenumber domain by freezing time or by focusing on a specific spatial coordinate; the relationship between the propagation in time and in space is ruled by the propagation speed, the speed of light in vacuum \(c\) is assumed throughout all this discussion. The transmitted radar pulse can be described in the temporal frequency domain through a function defined on a limited spectral support: it is centered on the carrier frequency \(f_{0}\) and width is given by the bandwidth \(B\). In the far field approximation, this very same signal can be described in the spatial domain as a plane wave. According to this approximation, the direction of propagation, that is the direction of the wavevector \(\mathbf{k}\), is defined by the line joining the sensor and the point (\(P\) in Figure 1) far away where the signal is evaluated. The extended bandwidth implies the existence of many waves of the form expressed by Equation (1), each one oscillating at a different frequency. Then, the corresponding spectral support in the wavenumber domain is a line directed as \(\mathbf{k}\), with length proportional to the system bandwidth and distance from the origin linked to the carrier frequency. It follows that the link between temporal and spatial frequencies depends on the geometry of acquisition, in particular on the _look angle_ (or _off-nadir_ angle) specified by \(\hat{\theta}\) in Figure 1.  
Figure 1: Schematic representation of a zero squint synthetic aperture radar (SAR) acquisition. The off-nadir angle \(\hat{\theta}\) (also known as look angle) defines the line-of-sight and hence the cross range axis \(\zeta\); both of them lying in the zero-Doppler plane, orthogonal to azimuth. The cross range (proportional to the elevation) is a blind direction for single-image SAR systems: sensitivity to the \(\zeta\) direction is provided by interferometric or tomographic SAR systems. The interaction between wave and target generates a set of waves scattered in all directions. In this discussion only the waves traveling back to the radar are considered, that is, only the monostatic case is analyzed.