# LDPC Coded Massive MIMO Systems
## 4 Design of LDPC Coded Massive MIMO System
This verifies the importance of the JDD strategy represented by \(N_{det}\) and \(N_{dec}\) to operate the JDD efficiently. Let us focus on the number of multiplications for a simple analysis of computational complexity. It is clear from Table 2 that the number of multiplications required for detection is much higher than that for decoding, especially with large \(n_{T}\) and \(n_{R}\). Thus, increasing \(N_{det}\) results in much higher computational complexity than increasing \(N_{dec}\). It follows that increasing \(N_{det}\) results in the significant decrease of \(N_{g}\) to maintain the total amount of computational complexity. With a small \(N_{g}\), the JDD does not converge sufficiently so that a low value of threshold is not attained. As a result, we fix \(N_{det}=1\) and adjust \(N_{dec}\) to design the JDD strategy, where \(N_{g}\) is determined by \(N_{dec}\) with other parameters given. We call the JDD with \(N_{det}=N_{dec}=1\) the global iteration only (GIO) JDD. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation** & \multicolumn{2}{c}{**Total Approximate Number of Operations**} \\ \hline \multirow{2}{*}{Addition} & \(N_{g}\left[N_{ch}N_{det}\Theta_{a}^{A}+\Theta_{v}^{A}+N_{dec}(\Theta_{vc}^{A}+ \Theta_{cv}^{A})\right]+(N_{g}-1)N_{ch}\Theta_{s}^{A}\) \\  & \(\approx\bar{d}_{v}/(n_{T}\log_{2}M_{o})N_{g}N^{2}+\{(8n_{R}\sqrt{M_{o}}/\log_{ 2}M_{o})N_{det}+2(\bar{d}_{v}+\bar{d}_{c})N_{dec}\}N_{g}N\) \\ \hline \multirow{2}{*}{Multiplication} & \(N_{g}\left[N_{ch}\left\{N_{det}\Theta_{a}^{M}+(N_{det}-1)\Theta_{g}^{M}\right\} +\Theta_{v}^{M}+N_{dec}\Theta_{v}^{M}\right]+(N_{g}-1)N_{ch}\left(\Theta_{s}^{M} +\Theta_{\beta}^{M}\right)\) \\  & \(\approx\left(30n_{R}\sqrt{M_{o}}N_{det}+(3\bar{d}_{c}(1-R)\log_{2}M_{o})N_{dec} \right)N_{g}N\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Total approximate number of operations required for overall JDD. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Messages** & \multicolumn{2}{c}{**Approximate Number of Operations**} \\ \cline{2-3}  & **Addition** & **Multiplication** \\ \hline \(\alpha_{ik}(s)\) & \(4n_{T}n_{R}(2\sqrt{M_{o}}+1)\triangleq\Theta_{a}^{A}\) & \(2n_{T}n_{R}(11\sqrt{M_{o}}+4)\triangleq\Theta_{a}^{M}\) \\ \hline \(\beta_{ik}(s)\) & \(0\) & \(8n_{T}n_{R}\sqrt{M_{o}}\triangleq\Theta_{\beta}^{M}\) \\ \hline \(L_{v}\) & \((\sqrt{M_{o}}-2)N\triangleq\Theta_{v}^{A}\) & \(4n_{T}n_{R}\sqrt{M_{o}}+N\triangleq\Theta_{v}^{M}\) \\ \hline \(\Pr\{\bar{\chi}_{j}=s\}\) & \(n_{T}\sqrt{M_{o}}\log_{2}M_{o}+\sigma N\triangleq\Theta_{s}^{A}\) & \(2n_{T}\sqrt{M_{o}}(\log_{2}M_{o}-1)\triangleq\Theta_{s}^{M}\) \\ \hline \(L_{vc}\) & \((2\bar{d}_{v}+1)N\triangleq\Theta_{vc}^{A}\) & \(0\) \\ \hline \(L_{cv}\) & \(2\bar{d}_{c}(N-K)\triangleq\Theta_{cv}^{A}\) & \(3\bar{d}_{c}(N-K)\triangleq\Theta_{cv}^{M}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexities for computing messages in the detector and the decoder, where \(\bar{d}_{v}\) and \(\bar{d}_{c}\) denote the average degree of variable nodes and check nodes, respectively. Figure 5: JDD trajectories of LDPC coded massive MIMO system with \(R=1/2\), \(n_{T}=n_{R}=16\) and \(E_{b}/N_{0}=1.50\) [dB]. Consider a LDPC coded massive MIMO system using GIO JDD with \(N_{g}=N_{g}^{GIO}\) as a reference system. The degree distributions of LDPC codes and the JDD strategy in a proposed coded massive MIMO system are determined in the following manner. First, we choose candidate values of \(N_{dec}\) resulting in the number of multiplications equivalent to that of GIO JDD. For each \(E_{b}/N_{0}\), we perform the following optimization for all candidate values of \(N_{dec}\):  
\[\max_{\lambda,\boldsymbol{\rho}} R(\lambda,\boldsymbol{\rho})\] s.t.