## 3 Signal Processing
### Signal Model
\(f_{d}\) results from the continuous antenna motion of the platform, since the traditional stop-and-go approximation is not valid for FMCW SAR [11]. In Equation (17), the first exponential term is the frequency and phase shift due to the BFD offset. The second exponential term represents the range signal including the RVP component, which is the second term in the square brackets. The third exponential term is the azimuth phase history of the proposed MIMO ViSAR. The fourth exponential term represents the Doppler shift due to continuous antenna motion within one sweep. The fifth exponential term is the phase delay corresponding to the time delay to the scene center. Therefore, the vector of a dechirped received signal in the proposed MIMO ViSAR system can be expressed as:  
\[\mathbf{r}_{R}\left(t_{r},t_{a}\right)\equiv\left[ \begin{array}{c}rr_{0,dc} \left(t_{r},t_{a}\right)\\ \vdots\\ rr_{n,dc}\left(t_{r},t_{a}\right)\\ \vdots\\ rr_{N-1,dc}\left(t_{r},t_{a}\right)\end{array} \right] \tag{18}\]  
\(\bullet\) Signal model in the fast time-azimuth angle domain. This model is convenient for understanding the subsequent signal processing steps of the MCRA, as it is implemented in the fast time-azimuth frequency domain. The model has a close relationship with the digital beamforming technique mentioned in [7]. The received signal can be expressed using a steering vector in the azimuth angle domain. This signal model explicitly shows the relationship between the digital beamforming techniques used for Tx and Rx in the proposed MIMO ViSAR system. The relationship between the azimuth frequency domain, \(f_{a}\), and the azimuth angle domain, \(\theta\), is as follows [7]:  
\[f_{a}=\frac{2v}{\lambda}\sin\theta \tag{19}\]  
The azimuth time delay term in Equation (17) can be expressed in the azimuth frequency domain, using the Fourier transform property as:  
\[b_{n}\left(f_{a}\right)a_{m}\left(f_{a}\right)=2\pi f_{a}\frac{\Delta x_{tx,m} +\Delta x_{rx,n}}{2v} \tag{20}\]  
which can be written in the azimuth angle domain using Equation (19) as:  
\[ \begin{array}{l}b_{n}\left(\theta\right)a_{m}\left(\theta\right)\\ =2\pi f_{a}\frac{\Delta x_{tx,m}+\Delta x_{rx,n}}{2v}\\ =2\pi\left(\frac{2v}{\lambda}\sin\theta\right)\frac{\Delta x_{tx,m}+\Delta x _{rx,n}}{2v}\\ =\frac{2\pi\left(\Delta x_{tx,m}+\Delta x_{rx,n}\right)}{\lambda}\sin\theta \end{array}  \tag{21}\]  
where \(a_{m}\left(\theta\right)\) are individual elements of the Tx steering vector, \(\mathbf{a}\left(\theta\right)\), and are given by:  
\[a_{m}\left(\theta\right)=\frac{2\pi\Delta x_{tx,m}}{\lambda}\sin\theta \tag{22}\]  
and \(b_{n}\left(\theta\right)\) are individual elements of the Rx steering vector, \(\mathbf{b}\left(\theta\right)\), and are given by:  
\[b_{n}\left(\theta\right)=\frac{2\pi\Delta x_{rx,n}}{\lambda}\sin\theta \tag{23}\]The Tx steering vector, \(\mathbf{a}\left(\theta\right)\), can be expressed as an \(M\times 1\) vector [3]:  
\[\mathbf{a}\left(\theta\right)\equiv\left[ \begin{array}{c}\frac{2\pi}{\lambda} \Delta x_{tx,0}\sin\theta\\ \vdots\\ \frac{2\pi}{\lambda}\Delta x_{tx,m}\sin\theta\\ \vdots\\ \frac{2\pi}{\lambda}\Delta x_{tx,M-1}\sin\theta\end{array} \right] \tag{24}\]  
and the Rx steering vector, \(\mathbf{b}\left(\theta\right)\), can be expressed as an \(N\times 1\) vector:  
\[\mathbf{b}\left(\theta\right)\equiv\left[ \begin{array}{c}\frac{2\pi}{\lambda} \Delta x_{rx,0}\sin\theta\\ \vdots\\ \frac{2\pi}{\lambda}\Delta x_{rx,n}\sin\theta\\ \vdots\\ \frac{2\pi}{\lambda}\Delta x_{rx,N-1}\sin\theta\end{array} \right] \tag{25}\]  
Therefore, after applying the Fourier transform in the azimuth frequency domain to Equation (17), the dechirped signal can be written as:  
\[