# Joint Iterative Decoding Design of Cooperative Downlink SCMA Systems
## 2 System Model
### The Cooperative SCMA System  
As shown in Figure 2, the two-phase cooperative transmission strategy is deployed. We assume that these \(J\) users are partitioned into two groups, with the first group being located near the BS and the other group being far from the BS and needing a relay to improve system performance. The relay is assumed to work in a half-duplex manner, i.e., it cannot transmit and receive message simultaneously. Specially, this relay is used to decode-and-forward (DF) the message. In the first phase, the BS broadcasts the superimposed signal to all the users and the relay. In the second phase, the relay carries out SCMA detection and then broadcasts the detected messages after SCMA modulation. **Remark 1**. : _The second strategy for the cooperative transmission is to allow the user close to the second group to act as the relay. Specially, in the first phase, the BS broadcasts the superimposed signal to all the users; in the second phase, the above-mentioned user carries out SCMA detection and then broadcasts the detected messages to the users in the second group after SCMA modulation. Since the joint iterative decoding design and the BER analysis of these two transmission strategies are the same, we focus on the first transmission strategy in this work._  
Figure 1: The SCMA factor graph for \(K=4\) and \(J=6\). Figure 2: The system model of a cooperative downlink SCMA system. In what follows, we first formulate the received signal of the \(j\)-th user within the first group and the relay in the first phase, and then the received signal of the \(j\)-th user in the second group is introduced. In the first phase, the received signal \(\mathbf{y}_{j}^{f}=[y_{1,j}^{f},y_{1,j}^{f}\cdots,y_{K,j}^{f}]^{T}\) of the \(j\)-th user can be represented as  
\[\mathbf{y}_{j}^{f}=\mathrm{diag}(\mathbf{h}_{j}^{f})\sum_{i=1}^{J}\mathbf{x}_ {i}^{b}+\mathbf{n}_{j}^{f}, \tag{3}\]  
where \(\mathbf{x}_{i}^{b}=[x_{1,j}^{b},x_{2,j}^{b},\cdots,x_{K,j}^{b}]^{T}\) is the transmitted codeword of the \(i\)-th user, \(i=1,2,\cdots,J\), \(\mathbf{h}_{j}^{f}=[h_{1,j}^{f},h_{2,j}^{f},\ldots,h_{K,j}^{f}]^{T}\) denotes the channel gain vector between the BS and \(j\)-th user, \(\mathrm{diag}(\mathbf{h}_{j}^{f})\) is a diagonal matrix with its \(k\)-th element being \(h_{k,j}^{f}\), and \(\mathbf{n}_{j}^{f}=[n_{1,j}^{f},n_{2,j}^{f},\ldots,n_{k,j}^{f}]^{T}\), in which \(n_{k,j}^{f}\) is the zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma_{f}^{2}\). Similarly, the received signal \(\mathbf{y}_{r}=[y_{1,r},y_{1,r},\cdots,y_{K,r}]^{T}\) of the relay can be written as  
\[\mathbf{y}_{r}=\mathrm{diag}(\mathbf{h}_{r})\sum_{i=1}^{J}\mathbf{x}_{i}^{b}+ \mathbf{n}_{r}, \tag{4}\]  
where \(\mathbf{h}_{r}=[h_{1,r},h_{2,r},\ldots,h_{K,r}]^{T}\) is the channel gain vector between the BS and relay; \(\mathbf{n}_{r}=[h_{1,r},n_{2,r},\ldots,h_{K,r}]^{T}\) is also modeled as the zero mean AWGN with covariance \(E[\mathbf{n}_{r}\mathbf{n}_{r}^{H}]=\sigma_{r}^{2}\mathbf{I}_{K}\). In the second phase, the relay decodes the transmitted signal of all the users, i.e., \(\mathbf{x}_{j}^{r}=[x_{1,j}^{r},x_{2,j}^{r},\cdots,x_{K,j}^{r}]^{T}\), and broadcasts the superimposed signal which is far from the BS to these users in the group. In this way, the received signal \(\mathbf{y}_{j}^{s}=[y_{1,j}^{s},y_{1,j}^{s},\cdots,y_{K,j}^{s}]^{T}\) of the \(j\)-th user in the second group can be denoted as  
\[\mathbf{y}_{j}^{s}=\mathrm{diag}(\mathbf{h}_{j}^{s})\sum_{i=1}^{J}\mathbf{x} _{i}^{r}+\mathbf{n}_{j}^{s}, \tag{5}\]  
where \(\mathbf{h}_{j}^{s}=[h_{1,j}^{s},h_{2,j}^{s},\ldots,h_{K,j}^{s}]^{T}\) is the channel gain vector between the relay and user \(j\) in the second group, \(\mathbf{n}_{j}^{s}=[n_{1,j}^{s},n_{2,j}^{s},\ldots,n_{k,j}^{s}]^{T}\) is the zero-mean AWGN with covariance \(\sigma_{S}^{2}\mathbf{I}_{K}\).