## 4 Simulation Results
### Point Target Simulation  
In this sub-section, a single point target simulation, developed in MATLAB with the system parameters listed in Table 1, is used to verify the performance of the proposed MIMO ViSAR system under ideal sensor motion conditions. Figure 11 illustrates the collection geometry, including the target location, for a circular path simulation. Ideal patterns are assumed for all antennas. The transmitted waveforms are generated using Equation (9) and (10), with \(\Delta f_{b}\) = 2 MHz, \(B_{r}\) = 1 GHz, and \(T_{d}\) = 1 ms. The reference signal for dechirping in fast time was simulated as the delayed version of the transmitted signal at Tx channel 1 using Equation (13). \(\tau_{ref}\) was set to 6.67 us in this simulation, which corresponds to the time delay to the scene center. The signal processing procedures presented in Section 3.3 are simulated using the scene geometry and the parameters of the MIMO ViSAR system listed in Table 1.  
RVP terms can be ignored for the parameters used in this simulation. The continuous antenna motion effect for FMCW is compensated using Equation (35). Figure 11: Collection geometry. Figure 10: System block diagram for MIMO video SAR with beat frequency division FMCW waveform in the case of two Tx channels and two Rx channels (\(M=2\), \(N=2\)). DDS: direct digital synthesis; LO: local oscillator; IF: intermediate frequency (IF); Tx: transmit; ADC: analog-to-digital conversion. BFD FMCW demodulation is shown in Figure 12. Figure 12a,b shows the signal in range-frequency domain before and after BFD FMCW demodulation, respectively. Note that in Figure 12a, the responses to different transmitters are separated by the beat frequency division offset, 2 MHz. Due to this spectral separation, transmitted waveforms from each Tx antenna can be distinguished. BFD FMCW demodulation is implemented using the frequency shift compensation detailed by Equation (36), and the phase compensation detailed by Equation (37), as described in Section 3.3. Since two Tx and two Rx channels are used, there are four channels in the virtual array formed by MIMO technology after demodulation, as shown in Figure 12. The distance between the phase centers in the virtual array is 0.02 m. The distance between Tx antennas is 0.04 m, and the distance between Rx antennas is 0.02 m. In this case, uniform spatial sampling is applied according to Equation (8). Azimuth spectrum reconstruction is shown in Figures 13 and 14. Figure 13a,b show the azimuth aliasing spectrum of the first channel (\(l=0\)) in the proposed MIMO ViSAR system before the reconstruction filters. The azimuth spectrum is completely reconstructed, after applying the reconstruction filters of the MCRA, \(\mu_{0}\left(f_{a}\right)\), \(\mu_{1}\left(f_{a}\right)\), \(\mu_{2}\left(f_{a}\right)\) and \(\mu_{3}\left(f_{a}\right)\), as shown in Equation (50), to the data received by the four MIMO channels, respectively, and stitching all four output signals together, as shown in Figure 14. Note that the azimuth frequency axis is expanded to four times its magnitude before applying the MCRA. The MCRA coherently combines the four channel signals in the azimuth and reconstructs a 4 kHz Doppler spectrum, which is four times wider than the single channel signal. Figure 15 plots the simulated 2D SAR image (video frame) of the impulse response function of a point target. Figure 15b shows the contour plot at \(-3\), \(-6\), \(-9\), \(-30\) dB from the peak level. The results of impulse response analysis in the down-range and cross-range are shown in Figure 16a,b respectively. The resolutions at \(-3.9\) dB from the peak level are 0.149 m in the down-range, and 0.081 m in the cross-range. The PSLRs are \(-13.42\) dB in the down-range, and  
Figure 14: Azimuth spectrum after MCRA: (**a**) three-dimensional plot; (**b**) two-dimensional plot at target. Figure 15: SAR image: (**a**) two-dimensional plot; (**b**) contour plot (zoomed). \(-13.41\) dB in the cross-range. No windows are applied for sidelobe reduction, which means \(K_{a}=1\) at \(-3.9\) dB.