## 3 Influence of Timing Errors on LDPC Decoding
\(\mathbf{C}=[c_{1},\cdots,c_{N}]^{T}\) is the codeword vector and the estimate of this vector is \(\mathbf{\hat{C}}=[\hat{c}_{1},\cdots,\hat{c}_{N}]^{T}\). In this step, if \(\mathbf{H}\cdot\mathbf{\hat{C}}=\mathbf{0}(\mathbf{Mod2})\), \(\mathbf{\hat{C}}\) is the decoding result. If \(\mathbf{\hat{C}}\) is the result of the iteration exceeds the maximum times, the iteration is finished. Otherwise, go to next step (2.2). - (2.2) Update the check nodes and then go to step (2.1) -  The decoding metrics updated in the check nodes can be represented as \[L(v_{ji})=2\tanh^{-1}\{\prod_{i^{\prime}\in K(j)\backslash i}\tanh[L(u_{j^{ \prime}j})/2]\},\] (11) and then go to step (2.1) for subsequent processing. In Equation (11), the set of the \(j\)-th variable node connected to the \(i\)-th check node is \(K(j)=\{i:\mathbf{H}_{i,j}=1\}\). Then, the messages from the \(j\)-th variable node and the \(i\)-th check node in LDPC decoding (given as \(u-ji\) and \(v_{ij}\), respectively) are symmetrical with \(\sigma_{m}^{2}=2m\), where \(\sigma_{m}^{2}\) and \(m\) are the variance and mean value of the messages, respectively [17]. The mean value of all messages for variables and check nodes ( \(m_{u}\) and \(m_{v}\) ) are updated in the iterations. The mean values represent the probability distribution of the messages. The mean value of the messages \(m_{v}^{(k)}\) from the variable nodes to the check nodes in \(k\)-th iterations is presented as  
\[m_{v}^{(k)}=m_{u0}+(d_{v}-1)m_{u}^{(k-1)}, \tag{12}\]  
where \(m_{u}^{(k)}\) is the mean value of messages from all check nodes and variable nodes \(v\) after \(k-1\) iterations, \(m_{u0}\) is the mean value of the initial message \(L(c_{i})\), and \(d_{v}\) is the variable node degree. The indexes of \(u_{ji}\) and \(v_{ij}\) can be omitted for simplification, because they are independent and identically distributed (_i.i.d._). Moreover, \(m_{v}^{(k-1)}\) is assigned to a zero mean value, because initial messages from all check nodes are zero. The updated messages coming from check nodes to variable nodes are presented similar to [6] as  
\[F[\tanh(u^{(k)}/2)]=F[\tanh(v^{(k)}/2)]^{(d_{c}-1)}, \tag{13}\]  
where the indexes \(u_{ji}\) and \(v_{ij}\) are omitted because they are _i.i.d._, and \(d_{c}\) is the check node degree. \(F[\tanh(u/2)]\) is presented as  
\[F[\tanh(u/2)]=\frac{1}{\sqrt{4\pi m_{u}}}\int_{R}\tanh(u/2)e^{-(u-m_{u})^{2}/4 m_{u}}du, \tag{14}\]  
where \"\(R\)\" represents that the integral interval falls in the entire field of real number. Moreover, a new function \(\phi(x)\) is defined in [17] as  
\[\phi(x)=\left\{ \begin{array}{cc}1-\frac{1}{\sqrt{4\pi x}}\int_{R}\tanh(u/2) e^{-(u-x)^{2}/4x}du,x>0\\ 1\hskip 14.226378 pt,x=0\end{array} \right.. \tag{15}\]  
Equation (15) is continuous and monotonically decreases on \([0,+\infty)\) with (\(\phi(0)=1\), \(\phi(0)=1\)). The mean value update of all messages from the check nodes to the variable nodes is given by  
\[m_{u}^{(k)}=\phi^{-1}\{1-[1-\phi(m_{u0}+(d_{v}-1)m_{u}^{(k-1)})]^{d_{c}-1}\}, \tag{16}\]  
where \(m_{u}^{(0)}\) is zero, which is the initial value of \(m_{u}\)[18]. The output messages from the variable nodes \(m_{u}^{(k)}\) after \(k\) iterations are represented as  
\[m_{o}^{(k)}=m_{u0}+d_{v}m_{u}^{(k-1)}. \tag{17}\]From Equation (17), \(m_{o}^{(k)}\) is mainly decided by initial message \(m_{u0}\) and message \(m_{u}^{(k-1)}\) after \(k\) iterations. In addition, \(m_{u}^{(k)}\) can be recursively calculated by Equation (16) and \(m_{u}^{(k-1)}\) with an initial value of \(m_{u}^{(0)}=0\). Thus, \(m_{o}^{(k)}\) relies on the mean value of initial message \(m_{u0}\). The properties of \(\phi(x)\) indicate that a larger \(m_{u0}\) leads to similarly larger values of \(m_{u}^{(k)}\) and \(m_{o}^{(k)}\) in Equations (16) and (17), respectively; thus, \(m_{o}^{(k)}\) is mainly decided by \(m_{u0}\). In addition, a larger \(m_{o}\) leads to a larger \(m_{o}^{(k)}\) and sum of all message squares. It can be explained as follows.