# Reconfigurable Intelligent Surface Assisted Target Three-Dimensional Localization with 2-D Radar
## 2 System Model  
The target three-dimensional localization with the RIS-assisted 2-D radar is considered. As shown in Figure 1, a single RIS-assisted 2-D radar is given as an example, which is built with a single-station 2-D radar and an RIS. The 2-D radar can only measure the distance from the target to the radar and the azimuth of the target relative to the radar, but it cannot measure the elevation of the target. The detection beam of 2-D radar is usually narrow in the azimuth dimension to achieve high azimuth resolution but wide in elevation dimension to receive the target echoes from different elevations. Taking advantage of the wide beam in the elevation dimension, several UAVs carrying RISs are set in the receiving beam to form multiple auxiliary measurement channels. Hence, an RIS-assisted 2-D radar is constructed to realize target three-dimensional localization. The RISs are mounted on the UAVs and stay in the radar detection beam when the radar is scanning the beam. Different from the conventional radar beam scanning for target detection, the spatial scanning in the RIS-assisted 2-D radar system is accomplished by the joint scanning of the radar beam and the RIS receiving beam. When the radar scans the area of interest, the receiving beam of the RIS should scan all the area covered by the radar beam, and the transmitting beam of the RIS should be always pointed at the radar. Therefore, the radar can receive the target direct echo and the target echo signal reflected by the RISs. The radar transmitting beam points to the detection target, and the received target echo signal includes two parts. The first one is the radar direct receiving channel--the echo from the target directly to the radar, which is called the \"radar direct channel\". The other one is the RIS-assisted receiving channel--the echo signal from the target to the RIS and then reflected from the RIS to the radar receiving channel, which is called the \"RIS auxiliary channel\". In the RIS-assisted 2-D radar, the received and reflected beams of the RIS should be controlled by adjusting the phases of the RIS reflectors. The received beam of the RIS should point to the detected target, and the reflected beam of the RIS should point to the radar. In addition, to ensure enough observation angle difference in different measurement channels, the distribution of RISs is relatively dispersed, and the observation angle is different from that of the radar direct channel. Suppose that there are \(K\) targets in the detection beam of 2-D radar. The locations of the radar are \([x_{R},y_{R},z_{R}]\), and the coordinate of the \(k\)-th target is \([x_{k},y_{k},z_{k}]\). \(M\) auxiliary RISs carried by the UAVs follow the scanning beam of the 2-D radar. The locations of the \(m\)-th RIS are \([x_{r,m},y_{r,m},z_{r,m}]\), \(m=1,\ldots,M\). Since the RISs need to communicate with the 2-D radar, they are usually much closer to the radar than the target. The arrival time of one target in different channels is close on the time axis, forming a time series. In the scenario of sparse multi-targets, it is assumed that the time series of different targets do not overlap on the time axis. The target echo \(r(t)\) received by the radar is divided into the direct channel echo \(r_{k}(t)\) and the auxiliary channel echo \(r_{aux}(t)\):  
\[r(t)=r_{k}(t)+r_{aux}(t) \tag{1}\]  
In the radar direct channel, the target echo \(r_{k}(t)\) can be written as  
\[r_{k}(t)=\sum_{k=1}^{K}\alpha_{k}s(t-t_{k})+w(t) \tag{2}\]  
where \(\alpha_{k}\) is the amplitude of the \(k\)-th target. \(s(t)\) is the radar transmitted signal. \(t_{k}=2R_{T-R}^{k}/c\) refers to the time delay. \(R_{T-R}^{k}\) represents the distance from the \(k\)-th tar  
Figure 1: The proposed RIS-assisted 2-D radar. get to the radar. c is the speed of light, and c \(=3\times 10^{8}\) m/s. \(w(t)\) is the channel noise following the Gaussian distribution.