# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 1 Introduction  
In the last few years, the advent of commercial spaceflight companies has created new ways to build satellites to observe the Earth [1]. One of the notable innovations has been the introduction of cheaper systems with the intention to shift the complexity from the space segment to software. The availability of lower cost COTS (Commercial Off-the-Shelf) components or micro-electromechanical systems (MEMS) pushed in this direction too [2]. In this regard, constellations and formations of small satellites fly together with a moderate degree of coordination to provide equivalent or superior performances compared to the current state-of-the-art systems with a minor budget and higher robustness towards failure, at the cost of a major complexity in the control of the formation and ground processing. Along-track formations of small satellites mounting SAR are an interesting case of such a systems since they can solve the well-known contradiction between the increasing requirement of coverage and azimuth resolution [3]. Feasibility of multiple receivers has been widely investigated in the case of multichannel antennas mounted on a single platform (monostatic SAR) [4; 5; 6; 7; 8]. For these systems, the generalized sampling theorem allows the reduction of the Pulse Repetition Frequency (PRF) by a factor N since a proper algorithm (the displaced phase center technique) provides the reconstruction of the full spectrum without ambiguities. In this paper, we focus on multistatic small SAR systems for which higher advantages can be achieved in terms of robustness, flexibility, and reduced costs [8; 9], keeping the advantage of thecoherent formation in fractionating the power with \(N^{2}\) gain. Small satellites are a novel category of systems developed to get faster and cheaper access to space by shifting the complexity from the space to the ground segment. Their weight/size range from the 100-500 kg for small satellites to the 10 kg of the nanosatellites named CubeSAT of size 1-10 U (1 U = 1 liter) ([[https://www.nasa.gov/mission_pages/cubesats/overview](https://www.nasa.gov/mission_pages/cubesats/overview)]([https://www.nasa.gov/mission_pages/cubesats/overview](https://www.nasa.gov/mission_pages/cubesats/overview))). One of the key points in the design of such systems is the matrix inversion implied by the Digital Beam Forming reconstruction [6; 7]. A perfect inversion is achieved if the Phase Centers (PC) are spaced by a distance that is exactly \(1/N\) of the Pulse Repetition Interval (PRI = \(1/PRF\)) in transmission:  
\[dx=\frac{2v\cdot PRI}{N}, \tag{1}\]  
with \(v\) the orbital speed and \(dx\) the phase center distance (each phase center is mid-way between the transmitter and each receiver). For multichannel monostatic systems, \(dx\) and \(v\) are under design control and fixed and the PRF can be programmed to achieve very good performances in terms of ambiguity suppression and recombination gain, i.e., the gain in SNR with respect to a single platform that is due to the coherent summation of the signal and incoherent summation of the thermal noise. For a multistatic SAR formation, the previous equation, even in the ideal case, must be properly changed to account for many \(2v/PRF\) space intervals:  
\[dx_{i}=2v\cdot PRI(\frac{i-1}{N}+k_{i}), \tag{2}\]  
with \(k_{i}\) integers that account for the higher distance between the PC. Discussion in literature on formations of SAR satellites has been faced with different purposes such as geolocation, moving target indication, and single-pass interferometric SAR [10]. Among the different examples of use, anyway, Aguttes [9] seems the most similar to the concept proposed. In [9], the along-track formations of flying satellites are claimed to gain a factor \(N\) on both SNR and ambiguity suppression compared to the single sensor, if properly displaced along orbit.