# Adaptive 3D Imaging for Moving Targets Based on a SIMO InISAR Imaging System in 0.2 THz Band
## 3 Adaptive Tracking of the Moving Targets Using Multiple Beams  
### SIMO Signal Model  
The geometry of the tracking and imaging scenario is illustrated in Figure 3. Taking the equivalent phase center of the transmitting antenna as the origin of the radar coordinate system, which is shown as the red dot denoted with \"Tx0\" in the figure. Shown as the green, the phase centers of the four receiving antennas locate symmetrically in the four quadrants of the XOZ plane, with their coordinates denoted as Rx1 : \((\frac{L_{x}}{2},0,\frac{L_{y}}{2})\), Rx2 : \((-\frac{L_{y}}{2},0,\frac{L_{z}}{2})\), Rx3 : \((-\frac{L_{y}}{2},0,-\frac{L_{z}}{2})\) and Rx4 : \((\frac{L_{y}}{2},0,-\frac{L_{z}}{2})\), where \(L_{x}\) and \(L_{z}\) denote the baseline lengths in the horizontal and vertical directions, respectively. Without losing generality, suppose a rigid target which moves in the radar system with a vector velocity \((V_{x},V_{y},V_{z})\), the coordinates of a certain scatterer in the target are denoted as \((x_{p},y_{p},z_{p})\). Then the instantaneous range of the scatterer to the transmitting antenna is expressed as:  
\[ \begin{array}{ll}R_{0P}(t_{m})&=\sqrt{\left(x_{p}+V_{x}t_{m}\right)^{2}+\left( y_{p}+V_{y}t_{m}\right)^{2}+\left(z_{p}+V_{z}t_{m}\right)^{2}}\\ &\approx R_{p}+V_{r}t_{m}+\frac{V_{x}^{2}+V_{y}^{2}+V_{z}^{2}}{2R_{p}}{t_{m}}^{2} \end{array}  \tag{7}\]  
where \(R_{p}\) means the range of the scatterer in the initial position, \(V_{r}\) means the radial velocity synthesized by the velocity vector along the three axes. The instantaneous range to the receiving antenna follows the formula:  
\[ \begin{array}{ll}R_{ciP}(t_{m})&=\sqrt{\left(x_{p}+V_{x}t_{m}-x_{Lci}\right) ^{2}+\left(y_{p}+V_{y}t_{m}\right)^{2}+\left(z_{p}+V_{z}t_{m}-z_{Lci}\right)^ {2}}\\ &\approx R_{0P}(t_{m})+\frac{x_{Lci}^{2}-2x_{Lci}\left(x_{p}+V_{x}t_{m}\right) }{2R_{0P}(t_{m})}+\frac{z_{Lci}^{2}-2z_{Lci}\left(z_{p}+V_{z}t_{m}\right)}{2R _{0P}(t_{m})}\end{array}  \tag{8}\]  
where \(ci=1,2,3,4\) denotes the receiving channel number and \((x_{Lci},0,z_{Lci})\) is the coordinates of corresponding receiving antenna phase center. Then, the received signal after de-chirping processing is expressed as:  
\[s_{ci}(t_{m},\tau)=\sigma e^{-j2\pi\left(f_{c}+\gamma\tau\right)\frac{R_{0P} (t_{m})+R_{ciP}(t_{m})}{c}} \tag{9}\]  
And the corresponding HRRP is:  
\[ \begin{array}{ll}Y_{ci}(t_{m},r)=&\text{sinc}[\frac{2R_{0P}(t_{m})+R_{ciP}( t_{m})}{2})]\\ &e^{-j\frac{2\pi}{\lambda}[R_{0P}(t_{m})+R_{ciP}(t_{m})]}\end{array}  \tag{10}\]  
It is clear that the phase response of a certain scatter in the HRRP is determined by the close wave path from the transmitting antenna to the scatter center and then back to the receiving antenna. It's also implied that the same scatter will respond differently in different receiving channels due to displacement of their antenna phase centers. However, since the baseline length is very short compared with the target range, a certain scatterer will appear in the same range cell on the HRRPs of different channels. Besides, a sum channel signal can be synthesized by accumulating the echoes from the four receiving channels. In this study, the SIMO system can track the moving target continuously, which implies that the target is always located near the antenna axis during observation; thus the instantaneous target range to the transmitting antenna and receiving antennas are approximately identical under the very short baseline configuration. Moreover, in consideration of the symmetrical configuration of the four receiving horns, the phase difference induced by phase center displacement for each channel can be mitigated  
Figure 3: Geometry of the tracking and imaging scenario. when accumulating the signals in the complex domain.