# LDPC Coded Massive MIMO Systems
## 1 Introduction
#### 2. Modeling of LDPC Coded Massive MIMO System  
Consider a massive MIMO system with \(n_{T}\) transmit antennas and \(n_{R}\) receive antennas. A \(K\)-bit information sequence \(\mathbf{b}\) is encoded to a \(N\)-bit LDPC codeword \(\mathbf{u}\) with the code rate of \(R=K/N\). Then, \(N\) coded bits are modulated as \(M_{o}\)-ary QAM symbols to be transmitted by \(n_{T}\) transmit antennas using a spatial multiplexing over \(N_{ch}\) channel uses. Note that \(n_{T}\) symbols are transmitted at each channel use resulting in \(N_{ch}=\lceil\frac{N}{n_{T}\log_{2}M_{o}}\rceil\). The MIMO channel at the \(l\)th channel use is expressed as  
\[\mathbf{y}^{(l)}=\mathbf{H}^{(l)}\mathbf{x}^{(l)}+\mathbf{w}^{(l)},\quad l=1,2,\cdots,N_{ch}, \tag{1}\]  
where entries of \(\mathbf{w}^{(l)}\) are independent and identically distributed (i.i.d.) zero-mean circular symmetric complex white Gaussian with variance of \(\sigma^{2}\), and entries of \(\mathbf{H}^{(l)}\) are i.i.d. circular symmetric complex Gaussian with zero mean and unit variance. The real-valued representation of Equation (1) is written by  
\[\bar{\mathbf{y}}^{(l)}=\bar{\mathbf{H}}^{(l)}\bar{\mathbf{x}}^{(l)}+\bar{ \mathbf{w}}^{(l)}, \tag{2}\]where  
\[\bar{\mathbf{y}}^{(l)} =\begin{bmatrix}\Re\left\{\mathbf{y}^{(l)}\right\}\\ \Im\left\{\mathbf{y}^{(l)}\right\}\end{bmatrix}\in\mathcal{R}^{2n_{R}\times 1}, \bar{\mathbf{x}}^{(l)} =\begin{bmatrix}\Re\left\{\mathbf{x}^{(l)}\right\}\\ \Im\left\{\mathbf{x}^{(l)}\right\}\end{bmatrix}\in\bar{\mathbb{A}}^{2n_{T} \times 1},\] \[\bar{\mathbf{w}}^{(l)} =\begin{bmatrix}\Re\left\{\mathbf{w}^{(l)}\right\}\\ \Im\left\{\mathbf{w}^{(l)}\right\}\end{bmatrix}\in\mathcal{R}^{2n_{R}\times 1 }, \bar{\mathbf{H}}^{(l)} =\begin{bmatrix}\Re\left\{\mathbf{H}^{(l)}\right\}&-\Im\left\{ \mathbf{H}^{(l)}\right\}\\ \Im\left\{\mathbf{H}^{(l)}\right\}&\Re\left\{\mathbf{H}^{(l)}\right\}\end{bmatrix} \in\mathcal{R}^{2n_{R}\times 2n_{T}}.\]  
The receiver of massive MIMO system with the real-valued representation given in Equation (2) can be expressed by a bipartite graph shown in Figure 1. The receiver consists of a detector and a decoder which exchange messages with each other iteratively by joint detection and decoding. The detector is composed of \(N_{ch}\) detector nodes, \(\text{DET}_{l}\), \(l=1,\cdots,N_{ch}\), while the decoder is composed of \(N\) variable nodes, \(v_{1},\cdots,v_{N}\), and \(N-K\) check nodes, \(c_{1},\cdots,c_{N-K}\). Each detector node \(\text{DET}_{l}\) is composed of \(2n_{R}\) observation nodes, \(o_{1}^{(l)},\cdots,o_{2n_{R}}^{(l)}\), and \(2n_{T}\) middle nodes, \(m_{1}^{(l)},\cdots,m_{2n_{T}}^{(l)}\), connected through edges. Each middle node is connected to \(\log_{2}\sqrt{M_{o}}\) variable nodes, where each real-valued symbol is generated from \(\log_{2}\sqrt{M_{o}}\) bits. We define variable super-nodes, \(\mathbf{v}_{1},\cdots,\mathbf{v}_{N_{ch}}\), each of which is a group of variable nodes associated with symbols transmitted at each channel use. Signals \(\bar{\mathbf{y}}^{(1)},\cdots,\)\(\bar{\mathbf{y}}^{(N_{ch})}\) received over \(N_{ch}\) channel uses are input to detector nodes \(\text{DET}_{1},\cdots,\text{DET}_{N_{ch}}\), respectively.