# Along-Track Multistatic Synthetic Aperture Radar Formations of Miniatellites
## 2 SIMO SAR Formation
### The Model in the Frequency Domain  
As widely shown in literature [21], representation of the transfer function of a bistatic system is a very hard task. Usually, an equivalent monostatic approximation is adopted for a reasonable distance between the transmitter and receiver. This assumption can be considered valid for a formation extent but not for constellation of satellites [17]. At least, some phase correction can be added in the focusing, as demonstrated in [22].  
In the equivalent monostatic approximation and neglecting the antenna terms, the Impulse Response Function (IRF) in Equation (9) can be written as [7]:  
\[\begin{split} h_{b,i}(t,\eta)\approx\exp\left\{-J\frac{4\pi}{ \lambda}r_{0}\right\}.\\ \exp\left\{-J\frac{2\pi}{\lambda}\frac{v^{2}(\eta-(dx_{i}+dx_{j}) /(2v))^{2}}{r_{0}}\right\}.\\ \exp\left\{-J\frac{\pi}{2\lambda}\frac{(dx_{i}-dx_{j})^{2}}{r_{0 }}\right\}.\end{split} \tag{11}\]  
In the monochromatic approximation, i.e., when \(\lambda\approx c/f_{c}\) (valid for relative narrow band), the first term is constant w.r.t. \(\eta\) and a function of the minimum formation-target distance; the second term is the phase of an equivalent monostatic SAR [7] delayed of \((dx_{i}+dx_{j})/2\), i.e., halfway between the transmitter and receiver; the third term is a pure bistatic term and it is constant too.  
The previous approximation can be adopted to write the 2D signal spectrum in a sampled context. Supposing a backscattering field \(\Gamma(x,r)\) acquired by the bistatic system \(ij\) (receiver-transmitter couple). The 2D spectrum of the received signal is:  
\[S_{i}(k_{x},\omega)=\sum_{r}\Gamma(k_{x}+r\cdot k_{x,s},\omega)\cdot H_{b,i}( k_{x}+r\cdot k_{x,s},\omega), \tag{12}\]  
where \(S_{i}(k_{x},\omega)\) is the spectrum of the signal observed by the \(i\)-th receiver, \(k_{x}\) is the azimuth wavenumber, \(\omega\) the range pulse, and the repetition in the wavenumber domain \(k_{x,s}\) is \(2\pi\cdot PRF/v\).  
This signal spectrum is the summation (in principle endless) of the aliased versions of the spectra of the target reflectivity \(\Gamma(k_{x},\omega)\) passed through the (aliased) transfer function of the bistatic SAR system. The transfer function of the bistatic SAR is the 2D-Fourier transform of Equation (11):  
\[\begin{split} H_{b,i}(k_{x},\omega)=\exp\left\{-J\frac{4\pi}{ \lambda}r_{0}\right\}\cdot F\left\{\exp\left[-J\frac{2\pi}{\lambda}\frac{(v \eta)^{2}}{r_{0}}\right]\right\}\cdot\\ \exp\left\{-jk_{x}\frac{dx_{i}+dx_{j}}{2}\right\}\cdot\exp\left\{ -J\pi\frac{(dx_{i}-dx_{j})^{2}}{2\lambda r_{0}}\right\}.\end{split} \tag{13}\]  
In the previous transfer function, the first two terms correspond to the transfer function of the equivalent monostatic SAR:  
\[H_{m}(k_{x},\omega)=\exp\left\{-J\frac{4\pi}{\lambda}r_{0}\right\}\cdot F \left\{\exp\left[-J\frac{2\pi}{\lambda}\frac{(v\eta)^{2}}{r_{0}}\right]\right\}. \tag{14}\]  
The third and fourth terms are specific to the bistatic acquisition:  
\[H_{i}(k_{x},\omega)=\exp\left\{-jk_{x}\frac{dx_{i}+dx_{j}}{2}\right\}\cdot \exp\left\{-J\pi\frac{(dx_{i}-dx_{j})^{2}}{2\lambda r_{0}}\right\}. \tag{15}\]  
A block scheme of the acquisition steps, where bistatic and monostatic acquisition are conceptually divided, and of the corresponding recombination and focusing of the N received signals, is in Figure 2.