## 3 Motion Compensation Method
### Proposed Motion Compensation Algorithm  
A motion compensation algorithm based on an improved MIMO sparse array, in which the reference channel is removed, is proposed in this paper. The improved MIMO array is shown in Figure 5a; in addition, the switching sequence of the transmitting and receiving antennas is marked. Figure 5b gives the equivalent array of the improved MIMO array, in which four equivalent array elements overlap (channels 1, 10, 19 and 28), and are located in the center of the array. These overlapping equivalent array elements can be used in motion compensation. Figure 4: The MIMO radar working mode after adding the reference channel. Figure 3: The RCMCA-based array. (**a**) MIMO array; (**b**) equivalent array. The simulation results of the resolution before and after the array improvement are shown in Figure 6. It can be seen that the resolution of the improved array is basically the same as that of the original array. The height sidelobe of the array after improvement deteriorated a little, but remained below \(-14\) dB, which would not affect the subsequent 3D TWI. The single-frame data of the MIMO array contain 32 channels; the switching sequence of the channels is shown in Figure 7, where the blue triangles represent equivalent channels with overlapping positions. The echo time delay of moving targets measured by all channels is \(\tau_{j}=[\tau_{\rm C1}\ \tau_{\rm C2}\ \dots\ \tau_{\rm C32}]\), which can be expressed as:  
\[ \begin{array}{c}\tau_{j}=[\tau_{\rm C1}\ \tau_{\rm C1}+\Delta\tau_{\rm 1}\ \tau_{\rm C1}+2\Delta\tau_{\rm 1 }\ \dots\ \tau_{\rm C1}+8\Delta\tau_{\rm 1}\\ \tau_{\rm C10}\ \tau_{\rm C10}+\Delta\tau_{\rm 2}\ \tau_{\rm C10}+2\Delta\tau_{\rm 2 }\ \dots\ \tau_{\rm C10}+8\Delta\tau_{\rm 2}\\ \tau_{\rm C19}\ \tau_{\rm C19}+\Delta\tau_{\rm 3}\ \tau_{\rm C19}+2\Delta\tau_{\rm 3 }\ \dots\ \tau_{\rm C19}+8\Delta\tau_{\rm 3}\\ \tau_{\rm C28}\ \tau_{\rm C28}+\Delta\tau_{\rm 4}\ \tau_{\rm C28}+2\Delta\tau_{\rm 4 }\ \dots\ \tau_{\rm C28}+4\Delta\tau_{\rm 4}]\end{array}  \tag{3}\]  
where \(\tau_{j}\) represents the echo delay of the target when the \(j\)th channel is working. \(\tau_{\rm C1}\) is the echo delay of the target when channel 1 receives data. The target can be regarded as moving at a constant speed during the time when channel 1 is switched to channel 10. Then, the echo delay of the target increases uniformly, and the increase rate of the delay is \(\Delta\tau_{\rm 1}\). The change in the delay within a single frame time can be divided into four sections, according to the overlapping channels of the equivalent array, channel 1-channel 10, channel 10-channel 19, channel 19-channel 28, and channel 28-channel 32. The increase rates of the delays are \(\Delta\tau_{\rm 1}\), \(\Delta\tau_{\rm 2}\), \(\Delta\tau_{\rm 3}\), and \(\Delta\tau_{\rm 4}\), respectively. It is worth noting that \(\Delta\tau_{\rm 4}\) is  
Figure 5: The improved array. (**a**) MIMO array; (**b**) equivalent array. Figure 6: Resolution comparison. (**a**) Azimuth resolution; (**b**) height resolution. calculated by combining channel 28 of the current frame and channel 1 of the next frame (denoted as channel 33). Channel 1 and channel 10 are in the same position in the first section of the delay change, and the time delay change of the target during this period can be measured. Then, the increase rate of the delay \(\Delta\tau_{1}\) can be calculated as:  
\[\Delta\tau_{1}=[S_{1}(t)\otimes S_{10}(t)]/9 \tag{4}\]  
where \"\(\otimes\)\" represents the convolution operation, which can be used to calculate the time difference of the two signals [10].