# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 3 Reconstruction of the Scene
### Recombination  
Reconstruction of the unfolded spectrum of width \(R\cdot PRF\) is achieved by minimizing the distance between the true \(\mathbf{D}\) and the reconstructed \(\mathbf{\hat{D}}\) signal, in some norm. In case of white noise added, the \(\ell_{2}\)-minimization (MMSE in [14]) is performed that corresponds to the pseudo-inverse of \(\mathbf{H}\) in case of diagonal covariance matrix of data:  
\[\mathbf{\hat{D}}=\mathbf{H}^{\dagger}\mathbf{S}, \tag{17}\]  
where \(\mathbf{H}^{\dagger}=(\mathbf{H}^{*}\mathbf{H})^{-1}\mathbf{H}^{*}\) is the Moore-Penrose inverse of matrix \(\mathbf{H}\) and the apex \({}^{*}\) means Hermitian (i.e., conjugate of the transpose). Pseudo-inversion may increase the level of noise, especially if the final sampling step is consistently lower than the target resolution since in that case part of the frequency set \([-R\cdot PRF/2,R\cdot PRF/2]\) has just noise. For this reason, or when correlated noise is present, the general inversion (also called Wiener inversion) that accounts for data and noise covariance matrices is preferred:  
\[\mathbf{\hat{D}}=\mathbf{H}_{\mathbf{s}}^{*}\mathbf{S}=\left[\left(\mathbf{H }\mathbf{C}_{\mathbf{D}}\mathbf{H}^{*}+\mathbf{C}_{\mathbf{N}}\right)^{-1} \mathbf{H}\mathbf{C}_{\mathbf{D}}\right]^{*}\mathbf{S}, \tag{18}\]  
Figure 2: End-to-end model of the acquisition (left part) and focusing (right part). Bistatic transfer functions \(H_{i}(k_{x},\omega)\) are specific of each tx-rx couple; the monostatic transfer function \(H_{m}(k_{x},\omega)\) is common. The \(\mathbf{S}_{\mathbf{i}}\) are the different raw received signals. After range compression, focusing consists of: (i) inversion of the recombination matrix (see Section 3.2), (ii) traditional SAR focusing at full azimuth sampling step (see Section 3.3).  
with \(\mathbf{H_{s}}=\left(\mathbf{H}\mathbf{C_{D}}\mathbf{H}^{*}+\sigma_{n}^{2}\cdot \mathbf{I}\right)^{-1}\mathbf{H}\mathbf{C_{D}}\) and \(\mathbf{C_{D}}\) and \(\mathbf{C_{N}}\) the covariance matrix of the acquisition and of noise, respectively.  
Estimated inversion, \(\mathbf{\hat{D}}\) of size \([RM\times P]\) is, for each range bin \(p\), the value of the spectrum in the \(R\) frequency sets each of width \(PRF\) put in positions \([-R,...,R]\cdot PRF\). Inversion is performed bin-wise for each \(k_{x}\) and for a total of \(M\) times, to fill the spectrum in the unfolded frequency interval.  
The gain of the multistatic formation can be evaluated by estimating the increase of SNR due to recombination. In case of pseudo-inversion, the noise covariance matrix after inversion is:  
\[E\left[\mathbf{H}^{*}\mathbf{N}\mathbf{N}^{*}(\mathbf{H}^{ \dagger})^{*}\right]= \tag{19}\] \[(\mathbf{H}^{*}\mathbf{H})^{-1}\mathbf{C_{N}}=\sigma_{n}^{2}\cdot (\mathbf{H}^{*}\mathbf{H})^{-1}.\]  
In the last step, we supposed uncorrelated noise and diagonal covariance matrix with power \(\sigma_{n}^{2}\). The signal to noise ratio is then:  
\[SNR=\frac{\mathbf{tr}\left\{\mathbf{C_{D}}\right\}}{\sigma_{n}^{2}\cdot \mathbf{tr}\left\{(\mathbf{H}^{*}\mathbf{H})^{-1}\right\}}. \tag{20}\]  
After some algebra, we achieve the SNR in the general case:  
\[SNR=\frac{\mathbf{tr}\left\{\mathbf{H_{s}}^{*}\mathbf{C_{D}}\mathbf{H_{s}} \right\}}{\sigma_{n}^{2}\cdot\mathbf{tr}\left\{(\mathbf{H_{s}}^{*}\mathbf{H_{ s}})\right\}}, \tag{21}\]  
which of course collapses into Equation (20) in case of the diagonal covariance matrix of the acquired signal \(\mathbf{C_{D}}\). Compared to the SNR of the monostatic single sensor, a gain of:  
\[G=\frac{1}{\mathbf{tr}\left\{(\mathbf{H}^{*}\mathbf{H})^{-1}\right\}} \tag{22}\]  
is achieved in an ideal condition and the diagonal covariance matrix of the signal. In case of perfect inversion, the gain is \(N\), which is the upper bound for the recombination gain in power and corresponds to what is expected by theory (the signal sums in coherent way, gaining \(N^{2}\), the noise in incoherent way, gaining only \(N\)) [7].