## 2 First Step of One-Bounce Scattering Signal Identification Algorithm
### Identification under Condition of More than Two BSs  
In practical environments, there are more than two BSs, suppose there are \(M\) BSs, the coordinate of the \(i\)th BS is \((x_{i},y_{i}),i=1,2,\ldots,M\), the coordinate of MS is \((x,y)\) and scatterers distribute around the MS. Moreover, suppose that the \(M\) BSs receive \(N\) one-bounce scattering signals from multiple scatterers. For the \(N\) one-bounce scattering signals, there are \(C_{N}^{2}\) pairs of scattering signal combinations, but only the scattering signal combinations whose two scattering signals arrive at two different BS are related to the localization of scatterers. Assuming that the number of scattering signal combinations related to the localization of scatterers is \(P\), there are \(P\) potential scatterers corresponding to the \(P\) pairs scattering signal combination. Identifying with inequality Equation (5), we assume that the remaining are \(Q\) pairs scattering signal combinations. For urban or indoor environments, there are multiple scatterers usually, the \(Q\) pairs scattering signal combination may come from different scatterers, these scattering signal combinations should be divided into multiple groups according to their source scatterer. For the division of scattering signal combinations according to source scatterer, firstly, we introduce the definition of adjacent combination. **Definition 1**. : _(adjacent combination): if A scattering signal combination and B scattering signal combination have one same scattering signal, then A scattering signal combination is B scattering signal combination's adjacent combination, and the converse is true, that is, A scattering signal combination and B scattering signal combination is the mutual adjacent combination._  
For instance, \(A\) scattering signal combination consists of scattering signal 1 and scattering signal 2, and \(B\) scattering signal combination consists of scattering signal 2 and scattering signal 4, the two combinations have the same scattering signal 2, so they are adjacent combinations. When the number of BS is three, there are three scattering signals from a given scatterer to BSs at most, and the three scattering signals can form three pairs of scattering signal combinations, one is the other's adjacent combination. When the number of BS is four, there are four scattering signals at most from a given scatterer to BSs, and they can form six pairs of scattering signal combinations at most, accordingly. In the six pairs scattering signal combination, any scattering signal combination could find its adjacent combination in the rest other five pairs scattering signal combination. When the number of BS is more than four, among all scattering signal combinations of a given scatterer, any scattering signal combination could find its adjacent combination in the rest other scattering signal combinations, too. Moreover, the number of scattering signals from any scatterer should be less than or equal to the number of BS, therefore, if the number of scattering signals from one scatterer is larger than the number of BS in the result, corresponding data should be excluded as wrong data resulting from measurement error. Based on the aforementioned characterization of the one-bounce scattering signal from the same scatterer, we can get Algorithm 1, which divides scattering signal combinations according to source scatterers, and each scattering signal combination set in Algorithm 1 corresponds to a source scatterer. After dividing scattering signal combinations according to source scatterer, for each scattering signal combination set, we should further delete false scattering signal combinations resulting from measurement error, which may result in large errors in scatterer localization.