# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 1 Introduction
For this reason, [PERSON] identified as critical points the width of the tube containing the satellites trajectories (to avoid unknown phase terms in the response) and the control of the sensor position along the track without providing, however, a solution to the problem. Some methods for reconstruction for multichannel SAR have been proposed in literature that could be used, in principle, also for SIMO SAR constellation. In one of these [11; 12], analysis of reconstruction quality with PRF and sampling conditions is provided while in [13] or in [14] the authors propose some reconstruction methods based on pseudo-inversion, maximum likelihood and minimum-mean squared error filters to mitigate the bad conditioning due to non uniform sampling. Irregular sampling position is investigated also in [15] where the authors suggest to adjust with proper weightings the reconstruction filter by looking at the Vandermonde part of the reconstruction matrix, without going, however, into details. In this work, we propose the design of a multistatic along-track formation of small SAR satellites. The novelty of the paper consists in:  
1. A 2D frequency domain spectrum reconstruction method. The effects of noise and bad-conditioned matrix inversion are mitigated by means of a Wiener inversion, similarly to the Minimum Mean Squared Error (MMSE) in [13; 14], and considered an optimum compromise between matched and ML filters. The reconstruction in frequency domain is used to account for range migration, as demonstrated also in [16]. 2. Two practical methods to mitigate the degradation of system performance due to the loss of position control: a dynamic tuning of PRF and the split of the antenna. Both of the methods that can act together aim at modifying the phase centers towards optimal positions. A systematical study of the degradation in phase shift and volume decorrelation, as a consequence of a realistic non zero across-track baseline, leading to the estimation of a critical tube width for the formation to avoid recombination problems. Notably, the recombination is invariant with the along the track baseline for typical formations extents (few hundreds of meters), so there is no need to resort to the analysis of residual phase corrections as in [17]. One of the important results in the discussion is that, for formations of satellites irregularly spaced, even with a significant effort to improve the signals recombination with a tunable PRF or displacing the phase centers using the antenna splitting, optimal results in all situations cannot be achieved. For this reason, the efficiency of the recombination is analyzed by resorting to probabilistic considerations, differently from what proposed was in [14; 15; 18]. The paper is organized as follows: in Section 2, the system is described and the Impulse Response Function (IRF) derived. In Section 3, the recombination algorithm is described. In Section 4, the consequence on system performance of a random positioning of the sensors is analyzed and the methods to mitigate degradation are described and illustrated with simulation results. In Section 5, the effects of a non zero across-track baseline and the possibilities of tomographic formation are discussed. The limits of the system and the conclusions in Section 6 close the paper.