# LDPC Coded Massive MIMO Systems
## 3 Joint Detection and Decoding for LDPC Coded Massive MIMO System  
### Operation of Joint Detection and Decoding  
We consider an iterative JDD process employing a low-complexity detection algorithm based on FG-GAI BP [22] and a sum-product decoding algorithm. One JDD iteration is composed of \(N_{det}\) detection iterations followed by \(N_{dec}\) decoding iterations, where we call a JDD iteration as a global iteration. Let us consider the \(l\)th channel use. Then, Equation (2) can be written as  
Figure 1: Receiver structure of LDPC coded massive MIMO system. \[\bar{y}_{i}^{(l)}=\sum_{j=1}^{2n_{T}}\bar{h}_{ij}^{(l)}\bar{x}_{j}^{(l)}+\bar{w}_{ i}^{(l)},\qquad i=1,\cdots,2n_{R}. \tag{3}\]  
Each observation node \(o_{i}^{(l)}\) obtains the information of \(\bar{x}_{k}^{(l)}\), \(k=1,\cdots,2n_{T}\), from \(\bar{y}_{j}^{(l)}\) by regarding terms associated with \(\bar{x}_{k}^{(l)}\), \(j\
eq k\), as interferences. For this purpose, we define \(z_{ik}^{(l)}\triangleq\sum_{j=1,j\
eq k}^{2n_{T}}\bar{h}_{ij}^{(l)}\bar{x}_{j }^{(l)}+\bar{w}_{i}^{(l)}\) as the interference plus noise when detecting the symbol \(\bar{x}_{k}^{(l)}\) and rewrite Equation (3) as  
\[\bar{y}_{i}^{(l)}=\bar{h}_{ik}^{(l)}\bar{x}_{k}^{(l)}+z_{ik}^{(l)}. \tag{4}\]  
In the case of using a massive number of transmit antennas, we can approximate \(z_{ik}^{(l)}\) as a Gaussian random variable [22] with the mean \(\mu_{z_{ik}^{(l)}}\) and the variance \(\sigma_{z_{ik}^{(l)}}^{2}\), where  
\[\mu_{z_{ik}^{(l)}}=\mathrm{E}\{z_{ik}^{(l)}\}=\sum_{j=1,j\
eq k}^{2n_{T}}\bar{ h}_{ij}^{(l)}\mathrm{E}\{\bar{x}_{j}^{(l)}\} \tag{5}\]  
and  
\[\sigma_{z_{ik}^{(l)}}^{2}=\mathrm{Var}\{z_{ik}^{(l)}\}=\sum_{j=1,j\
eq k}^{2n_ {T}}\left(\bar{h}_{ij}^{(l)}\right)^{2}\mathrm{Var}\{\bar{x}_{j}^{(l)}\}+ \frac{\sigma^{2}}{2}. \tag{6}\]  
The likelihood of \(\bar{x}_{k}^{(l)}\) at each observation node \(o_{i}^{(l)}\) is approximately obtained by using the Gaussian approximation of \(z_{ik}^{(l)}\) as  
\[\Pr\{\bar{y}_{i}^{(l)}|\mathbf{H}^{(l)},\bar{x}_{k}^{(l)}=s\}\approx\frac{1}{ \sqrt{2\pi\sigma_{z_{ik}^{(l)}}^{2}}}\exp\left(-\frac{(\bar{g}_{i}^{(l)}-\bar{h }_{ik}^{(l)}s-\mu_{z_{ik}^{(l)}})^{2}}{2\sigma_{z_{ik}^{(l)}}^{2}}\right), \tag{7}\]  
where \(s\in\bar{\mathbb{A}}\). Note that \(\mu_{z_{ik}^{(l)}}\) and \(\sigma_{z_{ik}^{(l)}}^{2}\) are computed as  
\[\mu_{z_{ik}^{(l)}}=\sum_{j=1,j\
eq k}^{2n_{T}}\bar{h}_{ij}^{(l)}\left(\sum_{s \in\bar{\mathbb{A}}}s\cdot\Pr^{(i)}\{\bar{x}_{j}^{(l)}=s\}\right) \tag{8}\]  
and  
\[\sigma_{z_{ik}^{(l)}}^{2}=\sum_{j=1,j\
eq k}^{2n_{T}}\left(\bar{h}_{ij}^{(l)} \right)^{2}\bigg{\{}\sum_{s\in\bar{\mathbb{A}}}s^{2}\cdot\Pr^{(i)}\{\bar{x}_{ j}^{(l)}=s\}-\left(\sum_{s\in\bar{\mathbb{A}}}s\cdot\Pr^{(i)}\{\bar{x}_{j}^{(l)}=s \}\right)^{2}\bigg{\}}+\frac{\sigma^{2}}{2}, \tag{9}\]  
where \(\Pr^{(i)}\{\bar{x}_{j}^{(l)}=s\}\) denotes a priori probability of \(\bar{x}_{j}^{(l)}\) at the observation node \(o_{i}^{(l)}\). The extrinsic probability of \(\bar{x}_{k}^{(l)}\) at each observation node \(o_{i}^{(l)}\) is obtained as [22]  
\[\Pr\{\bar{x}_{k}^{(l)}=s|\bar{\mathbf{H}}^{(l)},\bar{\mathbf{y}}_{\setminus i}^ {(l)}\}=\kappa\prod_{j=1,j\
eq i}^{2n_{R}}\Pr\{\bar{y}_{j}^{(l)}|\bar{\mathbf{H }}^{(l)},\bar{x}_{k}^{(l)}=s\}\cdot\Pr\{\bar{x}_{k}^{(l)}=s\}, \tag{10}\]  
where \(\kappa\) is a constant. As simple notations, we let \(\alpha_{ik}^{(l)}(s)\) and \(\beta_{ki}^{(l)}(s)\) denote the likelihood and the extrinsic probability, respectively, of \(\bar{x}_{k}^{(l)}=s\) at the observation node \(o_{i}^{(l)}\), i.e., \(\alpha_{ik}^{(l)}(s)=\Pr\{\bar{y}_{i}^{(l)}|\bar{\mathbf{H}}^{(l)},\bar{x}_{k}^ {(l)}=s\}\) and \(\beta_{ki}^{(l)}(s)=\Pr\{\bar{x}_{k}^{(l)}=s|\bar{\mathbf{H}}^{(l)},\bar{ \mathbf{y}}_{\setminus i}^{(l)}\}\). In the iterative process, the extrinsic probability replaces the role of a priori probability. In other words, \(\Pr^{(i)}\{\bar{x}_{j}^{(l)}=s\}\) in Equations (8) and (9) are replaced by \(\beta_{ji}^{(l)}(s)\).