# An Efficient Quantized Message Passing Receiver Design for SCMA Systems
## 2 System Model
### SCMA System Description  
We consider a typical downlink SCMA system with \(K\) orthogonal resource nodes (RNs), where \(J\) randomly distributed users are served by a base station. In general, the SCMA system works in a overloaded manner to achieve better spectral efficiency, and the overloading factor is defined as \(\lambda=J/K>1\). A unique codebook with cardinality \(M\) is assigned to each user. At the SCMA modulator, the information bits of the \(j\)-th user are sequentially mapped into \(K\)-dimensional complex-valued codewords \(\mathbf{x}_{j}=[x_{1,j},x_{2,j},\cdots,x_{K,j}]^{T}\). Specifically, this codeword \(\mathbf{x}_{j}\) is sparse with \(d_{v}<K\) nonzero elements, which enables MPA detection for SCMA. In the transmitter, the codewords \(\mathbf{x}_{j}\), where \(j=1,2,\cdots,J\), from all the \(J\) users are superimposed and propagate into the air interface. In this way, the received signal \(\mathbf{y}_{j}=[y_{1,j},y_{2,j},\cdots,y_{K,j}]^{T}\) at the \(j\)-th user can be described as [48]  
\[\mathbf{y}_{j}=\text{diag}(\mathbf{h}_{j})\sum_{i=1}^{J}\mathbf{x}_{i}+ \mathbf{n}_{j}, \tag{1}\]  
where \(\mathbf{h}_{j}=[h_{1,j},h_{2,j},\cdots,h_{K,j}]^{T}\) represents the channel gain vector between the base station and the \(j\)-th user, \(\text{diag}(\mathbf{h}_{j})\) refers to the diagonal matrix, with the elements in the first diagonal being the components present in \(\mathbf{h}_{j}\), and \(\mathbf{n}_{j}=[n_{1,j},n_{2,j},\cdots,n_{K,j}]^{T}\) is the Gaussian noise vector with its elements being modeled as independent and identically distributed \(\mathcal{CN}(0,\sigma^{2})\). Thanks to the sparsity of codewords, the sparse structure of SCMA can be represented by a \(K\times J\) binary indicator matrix \(\mathbf{F}\). The element \(f_{k,j}\) corresponds to the \(k\)-th row and \(j\)-th column of \(\mathbf{F}\), and \(f_{k,j}=1\) if and only if the \(k\)-th RN is connected to the \(j\)-th variable node (VN). In the matrix \(\mathbf{F}\), there are \(d_{v}\) and \(d_{c}\) nonzero elements in each row and column, corresponding to the the number of RNs multiplexed by each VN and the number of VNs accessing each RN. In this sense, we define two sets  
\[\mathcal{R}_{j}=\{k|f_{k,j}=1\},j=1,2,\cdots,J, \tag{2}\]  
and  
\[\mathcal{V}_{k}=\{j|f_{k,j}=1\},k=1,2,\cdots,K, \tag{3}\]  
to denote the set of RNs occupied by the \(j\)-th VN and the set of VNs collided over the \(k\)-th RN. In this work, two typical indicator matrices for the SCMA system are considered, which support \(J=6\) and \(J=10\) users, respectively, given by [48]  
\[\mathbf{F}_{4\times 6}=\left[\begin{array}{cccccc}1&1&1&0&0&0\\ 1&0&0&1&1&0\\ 0&1&0&1&0&1\\ 0&0&1&0&1&1\end{array}\right], \tag{4}\]  
and  
\[\mathbf{F}_{5\times 10}=\left[\begin{array}{cccccccccc}1&1&1&1&0&0&0&0&0&0 \\ 1&0&0&0&1&1&1&0&0&0\\ 0&1&0&0&1&0&0&1&1&0\\ 0&0&1&0&0&1&0&1&1\end{array}\right]. \tag{5}\]  
The overloading factor of \(\mathbf{F}_{4\times 6}\) and \(\mathbf{F}_{5\times 10}\) are \(\lambda=150\%\) and \(\lambda=200\%\), respectively. The weights of each column and each row of \(\mathbf{F}_{4\times 6}\) are \(d_{v}=2\) and \(d_{c}=3\), while each VN's degree and each FN's degree are \(d_{v}=2\) and \(d_{c}=4\) for \(\mathbf{F}_{5\times 10}\), respectively. The factor graph to visualize the SCMA configuration in (5) is depicted in Figure 1.