# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 4 Control of Sensor Position
### Mitigation of System Performance Degradation
#### 4.2.1 Tuning the PRF  
According to Equation (2), the ideal position of sensors even in a not equally spaced formation is given by the choice of proper integers \(k_{i}\), once a PRF is selected. The sensibility of this solution to sensor position is, however, very high: change of the \(i\)-th sensor position of \(\delta x\) implies an adjustment of the corresponding PRF of:  
\[P\hat{R}F\approx PRF-\frac{2v}{dx_{i}^{2}}\delta x. \tag{24}\]  
For an LEO orbit, with \(v=7500\) m/s and a \(dx_{i}=50\) m, 1 m of uncontrolled shift brings to a PRF adjustment of 6 Hz. This means that the degradation of the formation performance is very sensible to little shifts of a sensor's position and that PRF must be accurately tuned accordingly. Of course, it may not exist a single value of PRF able to satisfy Equation (2) with integers \(k_{i},i=1,...,N-1\) so a sub-optimal PRF must be found in practice. For a given formation, the true recombination performance is evaluated through the IRF quality (resolution, PSLR, ISLR) and the reduction of azimuth ambiguities. However, it is very difficult to relate such values to the research of a sub-optimal PRF (there is not a simple closed-form expression relating satellite distance, PRF, and performance). For this reason, we adopted an exhaustive research of the optimal PRF according to the quality of recombination, simply measured through the ratio between the recombination gain, given by Equation (22) and the condition number of \(\mathbf{H}^{*}\mathbf{H}\):  
\[\mathbf{F}=\frac{1}{\text{tr}\left\{(\mathbf{H}^{*}\mathbf{H})^{-1}\right\}} \cdot\frac{\max\text{eig}\left\{\mathbf{H}^{*}\mathbf{H}\right\}}{\min\text{ eig}\left\{\mathbf{H}^{*}\mathbf{H}\right\}}. \tag{25}\]  
The recombination gain is related to the peak gain, the condition number to the quality of the matrix inversion (and so to the reduction of ambiguities). The two quantities are related each other, but they do not express the same thing since the recombination gain is the harmonic mean of the eigenvalues of \(\mathbf{H}^{*}\mathbf{H}\), while the condition number is the ratio of the maximum and minimum eigenvalues and provides a measure of the matrix conditioning. The optimization has been applied in a practical case with \(N=5\), \(v=7500\) m/s, \(L_{a}=3.5\) m, \(\lambda=0.055\) m, sensor altitude 500 km and \(dx=\)[\(-250\) -96.52 20.95 166.55 250]m. The starting value of the figure of performance (the ratio of the reconstruction gain and the condition number), is: 9.2 \(\times 10^{-7}\) (\(G=-17.64\) dB, cond num = 1.87 \(\times 10^{4}\)) for a PRF = 880 Hz, so a bad result is expected for ambiguity ratio (PTAR) and IRF main lobe (see Figure 4a,b). After an exhaustive search of the optimum PRF in the frequency set [880,1500] Hz (geometrical constraints have been ignored here), a solution has been found at 1376.33 Hz, for which the figure of performance increased to 3.39 (\(G=12.55\) dB, cond num = 5.31). The corresponding results, shown in Figure 4c,d, are visually better. The stochastic nature of the mitigation method is better illustrated by evaluating the number of _successes_ achieved through the PRF tuning for many different formation with random sensors' location. In Figure 5a, the histograms of the figure **F** before and after the optimization show the wide beneficial effect even when satellites start from purely random position, but that bad cases may still exist and can not be corrected. Figure 4: Focused IRF of a point target for N = R = 5 and sensors in random position. (**a,c**) Along-track cut of the focused IRF; (**b,d**) oversampling of the 2D focused IRF. Case (**a,b**): figure of performance 9.2 \(\times 10^{-7}\); case; (**c,d**): figure of performance 3.39. Figure 5: Histogram of the logarithm of the figure of performance for different random positioning of the sensors in a formation of 400 m of extent, N = R = 5 and parameters as above. (**Left**) \(\log\)**F** before (blue) and after (red) the PRF tuning.