## 2 First Step of One-Bounce Scattering Signal Identification Algorithm
### Identification under Condition of More than Two BSs
To simplify the analysis, we take the deletion of false scattering signal combinations of one scattering signal combination set as an example of derivation in the following analysis, and the corresponding scatterer is referred to as target scatterer S.  
Assuming that there are \(R\) pairs scattering signal combination in the scattering signal combination set corresponding to target scatterer S, the intersection points of \(R\) pairs scattering signal combination are all potential positions of scatterer S. Because of corresponding to the same scatterer S, the \(R\) potential positions of scatterer S should overlap with each other, but the result of estimating does not comply with this constraint for the reason of measurement error. Assuming that the coordinate estimate of potential scatterer S corresponding to scattering signal combination \(j\) is \((\hat{x}_{s_{j}},\hat{y}_{s_{j}}),j=1,2,\ldots,R\), the mean of \(R\) coordinate estimate of scatterer S is  
\[\overline{x}_{s}=\frac{1}{R}\sum_{j=1}^{R}\hat{x}_{s_{j}},\qquad\quad\overline {y}_{s}=\frac{1}{R}\sum_{j=1}^{R}\hat{y}_{s_{j}} \tag{6}\]  
Thus, we can obtain the distance between \((\hat{x}_{s_{j}},\hat{y}_{s_{j}})\) and \((\overline{x}_{s},\overline{y}_{s})\), which is denoted by \(l_{j}\). The mean of \(l_{j}\) for all potential positions \((\hat{x}_{s_{j}},\hat{y}_{s_{j}})\) is denoted by \(l_{mean}\). The scattering signal combinations corresponding to the potential position \((\hat{x}_{s_{j}},\hat{y}_{s_{j}})\), whose \(l_{j}\) is less than \(l_{mean}\), are kept for the second step of the identification algorithm. Obviously, the number of scattering signal combinations \(R\) satisfies the following inequality  
\[R\leq C_{M}^{2}, \tag{7}\]  
and \(R=C_{M}^{2}\) holds only when \(M\) BSs all receive scattering signals from target scatterer S.  
Up to now, we can get multiple scattering signal combination sets corresponding to multiple scatterers. The above procedure is the first step of the identification algorithm, which is also referred to as a one-step identification algorithm for short in the simulation analysis. The aforementioned first step of the identification algorithm can be summarized as Algorithm 2. ```
Input:\(N\) scattering signals from multiple scatterers to \(M\) BSs. Output: multiple scattering signal combination sets corresponding to multiple scatterers. Construct \(P\) pairs scattering signal combinations related to localization of scatterers from \(N\) scattering signals. for each scattering signal combination of \(P\) pairs scattering signal combination do  Compute Inequality (5). if Inequality (5) holds  Corresponding scattering signal combination is kept. else  Corresponding scattering signal combination is discarded. endfor  Divide the rest \(Q\) pairs scattering signal combination according to source scatterer. for each scattering signal combination set corresponding to a source scatterer do  Compute \((\overline{x}_{s},\overline{y}_{s}))\). for each scattering signal combination in set do  Compute \(l_{j}\). endfor  Compute \(l_{mean}\). for each scattering signal combination in set do if\(l_{j}<l_{mean}\)  Corresponding scattering signal combination is kept. else  Corresponding scattering signal combination is discarded. endfor
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**Algorithm 2** First step of the identification algorithm