## 3 Signal Processing
### Signal Processing Procedure  
Signal processing for the proposed MIMO ViSAR system proceeds in the following parts: BFD FMCW demodulation, including Doppler shift compensation, azimuth ambiguity suppression using the MCRA, and video frame formation, with image rotation to the CDU using the PFA. The RVP terms in the dechirped signal described in Equation (32) can be ignored or compensated using the RVP correction method (\"range deskev\") described in [2] as follows:  
\[ \begin{array}{l}r_{m,DRVP}\left(t_{r}\right)\\ =a_{nm}\text{rect}\left(\frac{t_{r}-\tau_{nm}}{T_{d}}\right)\cdot\text{rect} \left(\frac{t_{r}-\tau_{ref}}{T_{d,ref}}\right)\\ \cdot\exp\left[j2\pi m\Delta f_{b}\left(t_{r}-\tau_{0}\right)\right]\\ \cdot\exp\left[-j2\pi k_{r}\left(t_{r}-\tau_{0}\right)\left(\tau_{0}-\tau_{ref} \right)\right]\\ \cdot\exp\left[-j\frac{4\pi}{\lambda}\tau_{0}\right]\cdot\exp\left(j2\pi f_{d} t_{r}\right)\cdot\exp\left(j2\pi f_{c}\tau_{ref}\right)\end{array}  \tag{34}\]  
where the fourth exponential term represents the Doppler shift due to continuous antenna motion within one sweep, as mentioned earlier. The continuous antenna motion at the \(n\)-th Rx channel is compensated by multiplying Equation (34) with the Doppler frequency correction factor giving the following expression:  
\[ \begin{array}{l}r_{m,DDOP}\left(t_{r}\right)\\ =r_{m,DRVP}\left(t_{r}\right)\cdot\exp\left(-j2\pi f_{d}t_{r}\right)\\ =a_{nm}\text{rect}\left(\frac{t_{r}-\tau_{nm}}{T_{d}}\right)\cdot\text{rect} \left(\frac{t_{r}-\tau_{ref}}{T_{d,ref}}\right)\\ \cdot\exp\left[j2\pi m\Delta f_{b}\left(t_{r}-\tau_{0}\right)\right]\\ \cdot\exp\left[-j2\pi k_{r}\left(t_{r}-\tau_{0}\right)\left(\tau_{0}-\tau_{ref} \right)\right]\\ \cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp\left(j2\pi f_{c} \tau_{ref}\right)\end{array}  \tag{35}\]  
where the first exponential term represents the frequency and phase offset due to BFD FMCW modulation. Next, BFD FMCW demodulation is carried out at the \(n\)-th Rx channel in two processing steps: frequency shift and phase shift compensation. The frequency shift, which results from the beat frequency division offset, is compensated by multiplying Equation (35) with the correction factor, giving the following expression:  
\[ \begin{array}{l}r_{m,BFD1}\left(t_{r}\right)\\ =\exp\left(-j2\pi m\Delta f_{b}t_{r}\right)\cdot r_{m,DDOP}\left(t_{r}\right) \\ =\exp\left(-j2\pi m\Delta f_{b}t_{r}\right)\exp\left[j2\pi\left\{m\Delta f_{b} -k_{r}\left(\tau_{0}-\tau_{ref}\right)\right\}\left(t_{r}-\tau_{0}\right) \right]\\ \cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp\left(j2\pi f_{c} \tau_{ref}\right)\\ \cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp\left(j2\pi f_{c} \tau_{ref}\right)\end{array}  \tag{36}\]  
where the second exponential term represents the phase offset due to BFD FMCW modulation. The phase shift is then compensated by multiplying Equation (36) with the correction factor, giving the following:  
\[ \begin{array}{l}r_{m,BFD2}\left(t_{r}\right)\\ =r_{m,BFD1}\left(t_{r}\right)\cdot\exp\left(j2\pi m\Delta f_{b}\tau_{0}\right) \\ =\exp\left[-j2\pi k_{r}\left(\tau_{0}-\tau_{ref}\right)\left(t_{r}-\tau_{0} \right)\right]\cdot\exp\left[-j\frac{4\pi}{\lambda}r_{0}\right]\cdot\exp\left( j2\pi f_{c}\tau_{ref}\right)\end{array}  \tag{37}\]  
Therefore, using Equation (37), after BFD FMCW waveform demodulation, the signal from Equation (30) can be expressed as:  
\[\mathbf{r}\mathbf{R}_{BFD}\left(t_{r},\theta\right)=\mathbf{b}\left(\theta \right)\mathbf{a}\left(\theta\right)^{T}\mathbf{M}_{BFD}\left(t_{r},\theta\right) \tag{38}\]where the BFD FMCW waveform demodulation matrix can be defined using Equations (27) and (37) as the following \(M\times M\) diagonal matrix:  
\[