# LDPC Coded Massive MIMO Systems
## 3 Joint Detection and Decoding for LDPC Coded Massive MIMO System
### Analysis of Joint Detection and Decoding  
We analyze the behavior of JDD in the receiver of coded massive MIMO system in terms of mutual information transfer characteristics, so-called EXIT characteristics, in component units. We focus on the mutual information between coded bits generating transmit symbols and corresponding message variables. In fact, \(\alpha_{ik}^{(l)}(s)\) and \(\beta_{ki}^{(l)}(s)\) at the observation node \(o_{i}^{(l)}\) contain information regarding the transmit symbol \(\bar{x}_{k}^{(l)}\). Thus, for the bit-level EXIT analysis mentioned above, we define new LLR messages of coded bits at observation nodes. Let us consider the \(t\)th bit of a coded bit-stream mapped to \(\bar{x}_{k}^{(l)}\) and define two LLR messages of this bit at the observation node \(o_{i}^{(l)}\). The first LLR is \(L_{ki}^{(l)}(t)=\log\frac{\sum_{s\in\bar{S}_{t}^{-}}\bar{\beta}_{ki}^{(l)}(s)} {\sum_{s\in\bar{S}_{t}^{+}}\bar{\beta}_{ki}^{(l)}(s)}\) sent from \(m_{k}^{(l)}\) to \(o_{i}^{(l)}\), and the second LLR is \(L_{ik}^{(l)}(t)=\log\frac{\sum_{s\in\bar{S}_{t}^{-}}a_{ki}^{(l)}(s)}{\sum_{s \in\bar{S}_{t}^{+}}a_{ki}^{(l)}(s)}\) sent from \(o_{i}^{(l)}\) to \(m_{k}^{(l)}\), where \(k=1,\cdots,2n_{T}\). We let \(L_{o}^{in}\) and \(L_{o}^{out}\) denote random variables representing \(L_{ki}^{(l)}(t)\) and \(L_{ik}^{(l)}(t)\), respectively, for all \(k,i,t\), and let \(U\) denote the corresponding coded bit. We suppose all LLR messages are independent and normally distributed. For each observation node, we define \(I_{OA}=I(U;L_{o}^{in})\) and \(I_{QE}=I(U;L_{o}^{out})\). We define \(I_{VA}=I(U;L_{cv})\) and \(I_{VE}=I(U;L_{vc})\) at variable nodes, where \(L_{cv}\) and \(L_{vc}\) are incoming and outgoing messages at variable nodes, respectively. We also define \(I_{CA}=I(U;L_{vc})\) and \(I_{CE}=I(U;L_{cv})\) at check nodes, where \(L_{vc}\) and \(L_{cv}\) are incoming and outgoing messages at check nodes, respectively. Allowing slight abuse of notation, we use \(I_{VA}(d_{c})\) and \(I_{CE}(d_{c})\) to denote the mutual information between \(U\) and \(L_{cv}\) delivered from degree-\(d_{c}\) check nodes to a variable node. We also use \(I_{VE}(d_{v})\) and \(I_{CA}(d_{v})\) to denote the mutual information between \(U\) and \(L_{vc}\) delivered from degree-\(d_{v}\) variable nodes to a check node. We depict the resultant iterative JDD process represented by transfer blocks of mutual information as in Figure 3.  
Consider a degree-\(d_{v}\) variable node that is connected to \(d_{v}\) check nodes and \(2n_{R}\) observation nodes via a corresponding middle node. The variable node sums up all incoming messages except one from a target node and sends the result to the target node. Thus, \(L_{o}^{in}\) is obtained by summing up \(2n_{R}-1\) copies of \(L_{o}^{out}\) and \(d_{v}\) copies of \(L_{cv}\). It follows that the variance of \(L_{o}^{in}\) is obtained by adding the variance of \(L_{o}^{out}\) multiplied by \(2n_{R}-1\) and the variance of \(L_{cv}\) multiplied by \(d_{v}\). By defining \(J(\sigma_{X})\) as [33]  
\[J(\sigma_{X})=1-\int_{-\infty}^{\infty}\frac{e^{-(\bar{z}-\sigma_{X}^{2}/2)^{ 2}/2\sigma_{X}^{2}}}{\sqrt{2\pi\sigma_{X}^{2}}}\cdot\log_{2}[1+e^{-\bar{z}}]d\xi, \tag{15}\]  
Figure 3: The JDD process in terms of EXIT characteristics. we obtain \(I(U;X)=J(\sigma_{X})\), where \(\sigma_{X}^{2}\) is the variance of a normally distributed random variable \(X\). Then, \(I_{OA}\) is obtained as a function of \(d_{v}\) as  
\[I_{OA}(d_{v})=J\left(\sqrt{\left(2n_{R}-1\right)\cdot\left[J^{-1}\left(I_{OE}( d_{v})\right)\right]^{2}+d_{v}\cdot\left[J^{-1}\left(I_{VA}\right)\right]^{2}} \right), \tag{16}\]  
where \(I_{VA}=\sum_{d_{v}=2}^{d_{v,\max}}\rho_{d_{c}}\cdot I_{VA}(d_{c})\) is the average of \(I_{VA}(d_{c})\) over \(d_{c},\rho_{d_{c}}\) denotes the fraction of edges that are connected to check nodes of degree \(d_{c}\), and \(d_{c,\max}\) denotes the maximum degree of check node.