# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 5 System Limitations  
### Across-Track Baseline Effects  
For an along-track formation concept, a non zero across-track baseline is an impairment, since even a slight off-track of a sensor provides a received contribution changed by the interferometric phase due to the baseline between the actual and the ideal tracks. This interferometric phase results in a spectral shift and a volumetric decorrelation. The case of a sensor shifted off-track with respect to the formation, assumed in straight orbit, is shown in the geometry of Figure 8a in the along the track, range reference and in Figure 8b in the reference orthogonal to the orbit. This second figure shows how two or more targets, located in the same iso-range with respect to the non-shifted track (the shaded satellite), contribute with different distances, and then different phases to the shifted sensor imagery. This case resembles an interferometric SAR [24], with the major difference that, in the unfocused images, the number of targets that contribute to the same raw data pixel are spread over a wide azimuth aperture. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Optimizer** & **PRF [Hz]** & **Cond Numb** & **Gain [dB]** & **Peak Power [dB]** & **Is Ambig [dB]** & **Az Resol [m]** & **PSLR [dB]** & **2D ISLR [dB]** \\ \hline None & 650 & 77.94 & 10.63 & 16.05 & 1.30 & 1.882 & \(-\)17.81 & \(-\)9.47 \\ _PRF opt_ & 1337.62 & 49.46 & 11.40 & 14.48 & \(-\)58.33 & 1.321 & \(-\)36.21 & \(-\)10.02 \\ _Twin opt_ & 650 & 1.93 & 14.24 & 15.57 & \(-\)27.61 & 1.778 & \(-\)16.46 & \(-\)9.03 \\ _Both opt_ & 1337.62 & 4.19 & 15.02 & 13.44 & \(-\)69.52 & 1.118 & \(-\)30.42 & \(-\)10.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of IRF performance parameters after recombination using different mitigation schemes. Figure 7: The probability that \(10\log 10|\mathbf{H^{*}H}|\) is lower than the ordinate is shown for N = 4, and R = 2 (**left**), R = 3 (**center**), R = 4 (**right**). The color of the dots indicates the number of antenna pairs: blue (none), green (one), red (two), cyan (three). The two graphs of the same color correspond to the two different PRF intervals considered, equivalent to adding one or three sample intervals in the full length of the formation. This fact is better shown in Figure 9, where it is shown in a rough topography that the effective baseline created by the shift causes a fringe pattern that impacts as a multiplicative phase screen in the raw data. The compensation that requires a precise knowledge of the Digital Elevation Model (DEM) is not easy and may be even not possible. In the case of a constant sloped terrain, the impact of an off-track shift of the sensor is a spectral shift [25] whose amount depends on the normal baseline, \(B_{n}\), the incidence angle, \(\theta\), and the local slope, \(\alpha\):  
\[\Delta f=f_{0}\frac{B_{n}}{r_{0}\cdot\tan(\theta-\alpha)}. \tag{26}\]  
The impact of the shift can be mitigated by filtering the common spectral contributions: the ones that would be imaged in both on-track and off-track position. The filtering removes any uncorrelated spectral contributions that would act like clutter noise but reduces the total bandwidth by the same amount, \(\Delta f\). In a general case, where each sensor has its own shift, the normal baseline \(B_{n}\) should be replaced by the total tube, normal to range, in which the formation is kept. For a rough estimation, let us assume the C-band, an incidence angle \(\theta=30^{\circ}\), flat terrain \(\alpha=0\) and range \(r_{0}=600\) km (reasonable for an altitude of 500 km). If we assume 100 MHz bandwidth, and tolerate a total spectral shift that is 10%--leading to a proportional range resolution loss, we would get an orbital tube:  
\[B_{nT}\leq\frac{\Delta f}{f_{0}}\cdot r_{0}\cdot\tan(\theta-\alpha) \tag{27}\]  
Figure 8: Geometry illustrating the off-track shift of a sensor in the formation in the along the track, range (**a**) and elevation, range (**b**) references.