## 3 Signal Processing
### Signal Model
\begin{split}& rR_{n,dc}\left(t_{r},f_{a}\right)\\ &=\mathrm{F_{a}}\left[rr_{n,dc}\left(t_{r},t_{a}\right)\right]\\ &=b_{n}\left(f_{a}\right)\sum_{m=0}^{M-1}a_{nm}\text{rect}\left(\frac{t_{r} -\tau_{nm}}{T_{d}}\right)\cdot\text{rect}\left(\frac{t_{r}-\tau_{ref}}{T_{d, ref}}\right)\\ &\quad\cdot\exp\left[j2\pi m\Delta f_{b}\left(t_{r}-\tau_{0}\right)\right] \\ &\quad\cdot\exp\left[-j2\pi k_{r}\left(t_{r}-\tau_{0}\right)\left(\tau_{0}- \tau_{ref}\right)-j\pi k_{r}\left(\tau_{0}-\tau_{ref}\right)^{2}\right]\\ &\quad\cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}-j\frac{\pi\left(\Delta x_ {tx,m}-\Delta x_{rx,n}\right)^{2}}{2\lambda r_{0}}\right]\\ &\quad\cdot\exp\left(j2\pi f_{d}t_{r}\right)\cdot\exp\left(j2\pi f_{c}\tau_ {ref}\right)\\ &\quad\cdot a_{m}\left(f_{a}\right)\cdot R_{az}\left(f_{a}\right)\end{split}  \tag{26}\]  
where \(\mathrm{F_{a}}\left[\cdot\right]\) is the Fourier transform operator with respect to slow time, and \(R_{az}\left(f_{a}\right)\) is calculated as follows:  
\[R_{az}\left(f_{a}\right)=T_{a}\text{sinc}\left(\pi T_{a}\left(f_{a}+k_{a} \Delta t_{a}\right)\right) \tag{27}\]  
which is the Fourier transform of the following azimuth time function, which is given by:  
\[r_{az}\left(t_{a}\right)=\exp\left[-j2\pi k_{a}\Delta t_{a}t_{a}\right] \tag{28}\]  
where \(k_{a}=\frac{2\
u^{2}}{\lambda r_{0}}\) is the chirp rate in azimuth and \(\Delta t_{a}\) is the differential azimuth time between the scatterer and the scene center. Note that \(R_{az}\left(f_{a}\right)\) represents the result of the azimuth dechirping, which is implemented by geometrical steering on the circular path. After substituting Equations (19)-(21) into Equation (26) and changing the notation, we can obtain the dechirped signal at the \(n\)-th Rx channel in the fast time-azimuth angle domain as follows:\[rR_{n,dc}\left(t_{r},\theta\right) \tag{29}\] \[\approx b_{n}\left(\theta\right)\sum_{m=0}^{M-1}a_{nm}\text{rect} \left(\frac{t_{r}-\tau_{nm}}{T_{d}}\right)\cdot\text{rect}\left(\frac{t_{r}- \tau_{ref}}{T_{d,ref}}\right)\] \[\cdot\exp\left[j2\pi m\Delta f_{b}\left(t_{r}-\tau_{0}\right)\right]\] \[\cdot\exp\left[-j2\pi k_{r}\left(t_{r}-\tau_{0}\right)\left(\tau _{0}-\tau_{ref}\right)-j\pi k_{r}\left(\tau_{0}-\tau_{ref}\right)^{2}\right]\] \[\cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp\left(j 2\pi f_{d}t_{r}\right)\cdot\exp\left(j2\pi f_{c}\tau_{ref}\right)\] \[\cdot a_{m}\left(\theta\right)\cdot R_{az}\left(\theta\right)\] \[\equiv b_{n}\left(\theta\right)\sum_{m=0}^{M-1}rR_{m,dc}\left(t_{r },\theta\right)\cdot a_{m}\left(\theta\right)\]  
where \(rR_{m,dc}\left(t_{r},\theta\right)\) is defined as a function of fast time and azimuth angle at the \(m\)-th Tx channel. Therefore, in the fast time-azimuth angle domain, the dechirped signal in the \(N\) Rx channels of the proposed MIMO ViSAR system, described by Equation (29), can be compactly expressed as an \(N\times 1\) vector using Equations (24) and (25) as follows [3]:  
\[\mathbf{r}\mathbf{R}_{R,dc}\left(t_{r},\theta\right)=\mathbf{b}\left(\theta \right)\mathbf{a}\left(\theta\right)^{T}\mathbf{r}\mathbf{R}_{T,dc}\left(t_{r },\theta\right) \tag{30}\]  
where the dechirped signal from the \(M\) Tx channels, described in Equation (30), can be defined in the fast time-azimuth angle domain as an \(M\times 1\) vector:  
\[\mathbf{r}\mathbf{R}_{T,dc}\left(t_{r},\theta\right)=\left[ \begin{array}{c} rR_{0,dc}\left(t_{r},\theta\right)\\ \vdots\\ rR_{M-1,dc}\left(t_{r},\theta\right)\end{array} \right]=\left[ \begin{array}{c} r_{0}\left(t_{r}\right)\\ \vdots\\ r_{M-1}\left(t_{r}\right)\end{array}