# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## Appendix A Probabilistic Approach to Matrix Inversion  
In Section 4, it has been shown that a SAR formation with a random distribution of sensors position may yield insufficient results. For this reason, a different definition of quality can be proposed, dependent on the application. Optical sensors are limited by night and clouds, and are accepted with their drawbacks. HRWS SAR formations are closer to optical systems, in some way. The additional costs of the methodologies that have been proposed to compensate for the uneven distribution of the sensors, i.e., adaptive PRF and/or twin antenna, may or may not balance the improvements in the probability of correct reception, depending on the available technology. The calculation of the probabilities of failure becomes then a necessary step. Let us rewrite the recombination matrix \(\mathbf{H}\) in Equation (14) normalizing the midpoint corresponding to the equivalent monostatic position between the transmitter and a receiver \(i\), i.e., \((dx_{i}+dx_{j})/2\), by relating it to the ground resolution of the current PRF, \(\delta_{m}\):  
\[(dx_{i}+dx_{j})/2=\mu_{i}\cdot\delta_{m}.\]  
Since the on board antenna is shorter than \(2\delta_{m}\), \(R-1\) ambiguities will appear in the focused image, positioned at locations indexed by \(r\):  
\[\xi_{r}=\xi+r\frac{\lambda r_{0}}{2\delta_{m}}.\]  
The corresponding backscattered radiation, within the monostatic approximation, will rotate by \(r\) (the ambiguity index) cycles in the interval \(\delta_{m}\). Previously, \(\xi\) is the position of the illuminated target and \(r_{0}\) its slant range. The signal backscattered from the position \(\xi\) has phase:  
\[\phi=\xi\frac{4\pi}{\lambda}\frac{\delta_{m}\mu_{i}}{2r_{0}}=\frac{\pi\xi_{i} \mu_{i}}{F}, \tag{19}\]  
with \(F\) given by Equation (7). The \(r\)-th aliased phase will be then:  
\[\phi_{r}=(\xi+rF)\,\frac{4\pi\delta_{m}\mu_{i}}{2\lambda R_{0}}=\left(\frac{ \xi}{F}+r\right)\pi\mu_{i}. \tag{20}\]Using the simplified form in Equation (A2) and posing \(\xi/F=q\), the recombination matrix is a Vandermonde matrix, similarly to what already achieved in [15]:  
\[\mathbf{H}=\left[ \begin{array}{ccc}\exp j\mu_{1}q&..&\exp j\pi\mu_{1}\left(R-1+q \right)\\..&..&..\\ \exp j\pi\mu_{N}q&..&\exp j\pi\mu_{N}\left(R-1+q\right)\end{array} \right].\] (A3)  
For a purely random position of the receivers, this matrix assumes the form of a total random complex matrix whose behavior in terms of inversion, can be described by the Marchenko-Pastur theory [30]. There, the probability density function of the distribution of the eigenvalues of \(\mathbf{H}^{*}\mathbf{H}\) is limited in an interval that is a function of the ratio between the number of rows divided by the number of columns. In Figure A1, there is an example of such distribution for different ratios. As shown in the figure, to limit the condition number of \(\mathbf{H}^{*}\mathbf{H}\) to 10 dB (a reasonable value for the matrix conditioning), it is necessary this ratio to be at least 3. In a contrary case, the probability that the inversion is bad-conditioned becomes significant. This fact leads the project towards a number of satellites \(N\) (the number of matrix rows) considerably higher than the number of ambiguities to remove, \(R\) (the number of the matrix columns). In the example of Section 3.3, in case of purely random position, to get a reconstruction of similar quality of Figure 3 with \(R=4\), it would have been necessary to increase the number of satellites to three times as least, i.e., N = 12. The mitigation methods proposed in Section 4 mitigate the performance degradation saving at the same time resources, since displacing the phase center by PRF tuning or by twin antenna may be as effective as increasing the number of satellites. In practical formations, it is possible to visualize the direct effect of a bad conditioning by looking at the increment of the ambiguity level.