## 2 First Step of One-Bounce Scattering Signal Identification Algorithm
### Identification under Condition of Two BSs  
First, consider a simple scenario with only two BSs as depicted in Figure 1. The coordinates of BS A and BS B are \((x_{A},y_{A})\) and \((x_{B},y_{B})\), respectively. BS A detects two one-bounce scattering signals from S1 and S2, path 2 and path 1, their AOA is \(\alpha_{1}\) and \(\alpha_{2}\), respectively. Corresponding to the two one-bounce scattering signals, the distance of TOA from MS to BS A is \(d_{TS_{1}A}\) and \(d_{TS_{2}A}\), respectively. Similarly, BS B detects two one-bounce scattering signals from S1 and S3, path 3 and path 4, their AOA is \(\beta_{1}\) and \(\beta_{2}\), the distance of TOA from MS to BS B is \(d_{TS_{1}B}\) and \(d_{TS_{3}B}\), respectively. For every pair scattering signal combination whose scattering signals are received by two different BSs, for instance, BS A and BS B, the two scattering signal lines intersect at a point, this point may be a scatterer. Suppose that the coordinate of the intersect point is \((x_{s},y_{s})\), based on AOA, coordinates of BS and intersect point, we can obtain a linear equation for every scattering signal line, accordingly, the two scattering signal lines satisfy linear equations as Equation (1):  
\[ \begin{cases}y_{s}-y_{A}=tan\alpha_{i}(x_{s}-x_{A})\\ y_{s}-y_{B}=tan\beta_{i}(x_{s}-x_{B})\end{cases} .\qquad i=1,2 \tag{1}\]  
As depicted in Figure 1, for the four scattering signals that BS A or BS B receive, there are four pairs of scattering signal combination, and four intersection points accordingly, that is, S1, S13, S12 and S23, which are all potential scatterers from which emit two scattering signals arriving BS A and BS B, respectively, the four intersection points are all denoted as S.  
It can be observed from Figure 1 that the intersection points of two scattering signal lines, that is, potential scatterers, can be divided into two categories: the real scatterer that emits two scattering signals arriving BS A and BS B, such as S1; the other just intersection point of two scattering signal lines, but not real scatterers, such as S13, S12 and S23. Obviously, for the identification of a one-bounce scattering signal from the same scatterer, the latter should be identified as a false scatterer, and the corresponding two scattering signals are not scattering signals from the same scatterer. Based on Equation (1), we can get the coordinates of potential scatterers S, that is, \((x_{s},y_{s})\), so the distances between potential scatterer S and BS A, denoted by \(d_{SA}\), can be computed. The distance between potential scatterers S and BS B, denoted by \(d_{SB}\), is similar. As depicted in Figure 1, if the potential scatterer S is a real scatterer between MS and BS A, the TOA distance between MS and BS A, which is denoted by \(d_{TSA}\), is the sum of the distance from MS to scatterer denoted by \(d_{TS}\) and distance from scatterer to BS A denoted by \(d_{SA}\), so we can have  
\[d_{TSA}=d_{TS}+d_{SA} \tag{2}\]  
Similarly, for BS B, we have  
\[d_{TSB}=d_{TS}+d_{SB.} \tag{3}\]  
Hence,  
\[d_{TSA}-d_{SA}=d_{TSB}-d_{SB} \tag{4}\]  
Nevertheless, Equation (4) does not hold for a false scatterer. Therefore, the real scatterer can be distinguished from the false scatterer based on Equation (4). Considering the measurement error, the identification formula is expressed as:  
\[||d_{TSA}-d_{SA}|-|d_{TSB}-d_{SB}||<D, \tag{5}\]  
where \(D\) is the identification threshold. Clearly, with increasing threshold value, a false scatterer is more likely to be identified as a real scatterer, however, a real scatterer is increasingly impossible to be identified as a false scatterer. We define false alarm probability as the probability that a false scatterer is identified as a real scatterer, and alarm dismissal probability is the probability that a real scatterer is identified as a false scatterer. Withincreasing threshold value, the false alarm probability will increase and the alarm dismissal probability will decrease.