# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 2 SIMO SAR Formation
### Design Bounds  
System requirements to the formation bring to bound some design parameters.  
Figure 1: (**a**) the concept scheme. \(r_{i,k}\) and \(r_{j,k}\) are achieved in Equation (10) by simple geometric considerations; (**b**) a simplified view of the acquisition on a common area. In orange, the antenna pattern; in blue the common area imaged by all the sensors (details are in Section 2.1).  
For a larger swath to be imaged, the Pulse Repetition Frequency (PRF) must be properly lowered to allow the received echoes not to conflict with the transmitted ones. On the other hand, if \(\delta\) is the output (wanted) azimuth resolution, the Doppler bandwidth must be set accordingly:  
\[B_{dop}=0.886\frac{v}{\delta}, \tag{3}\]  
where no distinction has been done between orbital and ground speed \(v\), to avoid confusion (The equations are strictly valid for a rectilinear orbit and flat Earth, but can easily be extended to curvilinear orbit and spherical Earth). The correction factor 0.886 comes from the \(-3\) dB width of a sinc-shaped IRF.  
The well-known conflicting requirement for High Resolution and Wide Swath (HRWS) SAR systems comes just from the previous two conditions since a high Doppler bandwidth is possible, without aliasing, only in case of high PRF, which is not the case. However, using the generalized version of the sampling theorem, an aliased Doppler band can still be reconstructed correctly if we have a sufficient number of acquisitions of the same area--in detail, to assure the correct reconstruction, it is necessary that  
\[B_{dop}\leq R\cdot PRF, \tag{4}\]  
with \(R\) an integer factor describing the ambiguities that must be canceled in the reconstruction, plus one (the ambiguity that is indexed with 0 in the paper is the part of the spectrum to save). Of course, \(R\leq N\), since no more than \(N\) ambiguities can be recombined by using \(N\) acquisitions.  
The PRF is limited below by the achievable azimuth resolution of the monostatic system. In detail, said \(L_{a}\) the antenna length, the monostatic azimuth resolution \(\delta_{m}\geq L_{a}/2\)[19]. After reconstruction, we want to improve the azimuth resolution of a factor up to \(N\). For this reason \(\delta=\delta_{m}/N\). Combining Equation (3) with previous conditions we have:  
\[PRF\geq\frac{2v}{N\cdot L_{a}} \tag{5}\]  
For the current study this may become a tight inferior bound for PRF since micro- or nano-satellites are not expected to mount long antennas. As an example, in case of micro-satellite of mass 100 kg, a deployable antenna array of \(L_{a}=3.5\) m is permitted with the current technology, so that with \(N=5\) satellites and, with \(v=7500\) m/s, we have \(PRF\geq 850\) Hz.  
Finally, an upper bound for the along-track formation extent exists to assure the needed azimuth resolution. The synthetic aperture related to the output resolution \(\delta\) is:  
\[F_{SA}=v\frac{B_{dop}}{|FM|}=\frac{0.886}{2}\cdot\frac{\lambda h}{\delta\cos \theta}, \tag{6}\]  
where \(|FM|=2v^{2}/\lambda r_{0}\) is the Doppler rate, \(\lambda\) is the wavelength, \(h\) is the formation altitude, \(r_{0}\) a reference slant range, and \(\theta\) is the incidence angle at \(r_{0}\). For a given antenna size, the physical footprint on ground is:  
\[F=\frac{\lambda h}{L_{a}\cos\theta}. \tag{7}\]  
The maximum extension of the formation is then limited by the condition for all the satellites to look at a common area, at the same time, large at least \(F_{SA}\) (see Figure 1b). This means that the inter-satellite distance, in case of equally spaced formation is limited by:  
\[dx\leq\frac{F-F_{SA}}{N-1} \tag{8}\]  
This bound limits the formation extension and must be evaluated in the orbit control of the system. As an example, for an X-band formation of \(N=5\) satellites at \(h=500\) km of altitude, with an incidence angle of \(30^{\circ}\) and \(L_{a}=2\) m, if a final resolution of 1 m is required, satellites must be closer than 250 m, which is a safe requirement.