## 3 System Model
\begin{split}\mathbf{\tilde{s}}_{k}(t)&=\mathbf{U}_{ k}^{\mathrm{H}}\mathbf{V}_{k}(t)\\ &=\sum_{i=1}^{K}\gamma_{\mu}\mathbf{U}_{k}^{\mathrm{H}}\mathbf{H} _{ik}\mathbf{V}_{k}\mathbf{s}_{i}(t)+\mathbf{U}_{k}^{\mathrm{H}}\mathbf{n}_{k} (t)\\ &=\gamma_{ik}\mathbf{U}_{k}^{\mathrm{H}}\mathbf{H}_{ik}\mathbf{V} _{k}\mathbf{s}_{k}(t)+\sum_{i=1,j\
eq k}^{K}\gamma_{\mu}\mathbf{U}_{k}^{ \mathrm{H}}\mathbf{H}_{ik}\mathbf{V}_{k}\mathbf{s}_{i}(t)+\mathbf{U}_{k}^{ \mathrm{H}}\mathbf{n}_{k}(t)\end{split}  \tag{10}\]  
where:  
\[\mathbf{U}_{k}= [\mathbf{U}_{k,1},\mathbf{U}_{k,2},...,\mathbf{U}_{k,d}] \tag{11}\] \[\mathbf{U}_{k,l}\in\mathbf{C}^{\mathrm{N}\times 1},\,l=1,\,2,\,...,\,d\] (12) \[\parallel\mathbf{U}_{k,l}\parallel= 1 \tag{13}\]  
At the demodulator, the received signal vector \(\mathbf{\tilde{s}}_{k}(t)\) is correlated with the match filter to yield the test statistic vector, _i.e._,  
\[\mathbf{\hat{s}}_{k}= \int_{0}^{T_{k}}\mathbf{\tilde{s}}_{k}(t)g_{k}(t)dt \tag{14}\]  
Finally, the demodulated bit vector \(\mathbf{\hat{m}}_{k}\) is determined according to the real part of the test statistic vector, _i.e._,  
\[\hat{m}_{k,l}= \begin{cases}1,&\text{when}\quad\text{real}(\hat{s}_{k,l})>0\\ -1,&\text{when}\quad\text{real}(\hat{s}_{k,l})<0\end{cases} ,\,l=1,\,2,\,...,\,d \tag{15}\]  
where:  
\[\mathbf{\hat{s}}_{k}= \begin{bmatrix}\hat{s}_{k,1},\hat{s}_{k,2},...,\hat{s}_{k,d} \end{bmatrix} ^{\mathrm{T}},\,\,\mathbf{\hat{m}}_{k}= \begin{bmatrix}\hat{m}_{k,1},\hat{m}_{k,2},...,\hat{m}_{k,d}\end{bmatrix} ^{ \mathrm{T}} \tag{16}\]  
According to [11], the IA feasible condition can be given as follows:  
\[K\leq(M+N)\,/\,d-1 \tag{17}\]  
Under the constraint of Equation (17), the system is likely to be feasible, and perfect IA can be achieved if the following conditions are satisfied:  
\[\mathbf{U}_{k}^{\mathrm{H}}\mathbf{H}_{y}\mathbf{V}_{j}=0;k,j= 1,2,...,K;k\
eq j \tag{18}\] \[\text{rank}(\mathbf{U}_{k}^{\mathrm{H}}\mathbf{H}_{ik}\mathbf{V} _{k})=d \tag{19}\]  
Many works have concentrated the design of the precoding and decoding matrices for IA [23, 24, 25, 26]. One of the most representative methods is the Max-SINR algorithm as shown in Algorithm 1. The Max-SINR algorithm highlights the advantage of good sum rate achievement. In this paper, the Max-SINR algorithm is selected as the strategy for designing the corresponding matrices. Perfect CSI and large-scale channel attenuation \(\gamma_{ki}\) are assumed to be known at each node. Although the situationsof imperfect CSI are interesting and more practical, detailed discussion on CSI feedback is beyond the scope of this paper. The readers who are interested in this area can refer to the \"Related Work\" section and the references herein. Besides, we assume that the nodes in the WSNs can communicate with each other. In the scenario where the distances between some nodes are too large, the relay nodes [36,37] can be added to transmit data between them. ```
1. Set the maximal iterations and the initial precoding matrices as \(\mathbf{V}_{k}\) (\(k\) = 1, 2,..., \(K\)). The columns of \(\mathbf{V}_{k}\) are linearly independent unit vectors. 2. (Forward Iteration) Calculate the interference-plus-noise covariance matrices \(\mathbf{J}_{k,l}\) (\(l\) = 1, 2,..., \(d\)) in the original channel, _i.e._, \[\mathbf{J}_{k,l}=\sum_{j=1,j\
eq k}^{K}\mathbf{H}_{ij}\mathbf{V}_{j}\mathbf{V} _{j}^{\mathrm{H}}\mathbf{H}_{ij}^{\mathrm{H}}+\sum_{r=1,r\
eq l}^{d}\mathbf{H} _{ik}\mathbf{V}_{k,r}\mathbf{V}_{k,r}^{\mathrm{H}}\mathbf{H}_{ik}^{\mathrm{ H}}+\sigma^{2}\mathbf{I}\] (20) Formulate the decoding matrix \(\mathbf{U}_{k}\), the \(l\)-th column of which can be represented as \[\mathbf{U}_{k,l}=\mathbf{J}_{k,l}^{-1}\mathbf{H}_{ik}\mathbf{V}_{k,l}/\parallel \mathbf{J}_{k,l}^{-1}\mathbf{H}_{ik}\mathbf{V}_{k,l}\parallel\] (21)
3.