## 3 Influence of Timing Errors on LDPC Decoding
For the same LDPC codeword, the main code parameters, i.e., the degree distribution, are the same. So the variable node degree \(d_{v}\) and check node degree \(d_{c}\) are equal for a specific node message processing under different channel state information, or the initial LLR represented in Equation (6). From Equation (15), the function \(\phi(x)\) is continuous and monotonically decrease on its definition domain \([0,+\infty)\) with (\(\phi(0)=1\), \(\phi(+\infty)=0\)). Then, due to the relationship that a function and its inverse function is just mirror symmetrical about the function \(y=f(x)=x\), the inversion function \(\phi^{-1}(y)\) is also decrease on \((0,+1]\) with (\(\phi(0)=+\infty\), \(\phi(1)=0\)). So the first recursive iteration of the messages from the check nodes to the variable nodes in Equation (16) is given as  
\[m_{u}^{(1)}=\phi^{-1}\{1-[1-\phi(m_{u0})]^{d_{c}-1}\}, \tag{18}\]  
where \(m_{u0}\) is the initial message and it is got from Equation (6). Obviously, due to the monotonically decrease property of \(\phi(\cdot)\) and \(\phi^{-1}(\cdot)\), larger \(m_{u0}\) causes the larger \(m_{u}^{(1)}\). For the second recursive iterations, Equation (16) is expressed as  
\[m_{u}^{(2)}=\phi^{-1}\{1-[1-\phi(m_{u0}+(d_{v}-1)m_{u}^{(1)})]^{d_{c}-1}\}. \tag{19}\]  
From Equation (19), other parameters being the same, the larger initial message \(m_{u0}\) along with the corresponding larger previous generated message \(m_{u}^{(1)}\) leads to a much larger \(m_{u}^{(2)}\) than that generated from a smaller initial message. The rest can be done in the same manner and in the higher order of recursive iterations, larger initial message \(m_{u0}\) surely causes larger solution result \(m_{u}^{(k)}\) after \(k\) recursive iterations. Therefore, the conclusion is true for an LDPC-coded system with an all-0-bit input. A symmetry condition (\(f(x)=f(-x)e^{x}\)) is used to preserve the above conclusion for all messages, because initial negative messages can be processed for positive ones as follows [18]. First, the channel symmetries are given by  
\[p(y_{i}=q|x_{i}=1)=p(y_{i}=-q|x_{i}=-1). \tag{20}\]  
The sign factor of the check node message map \(\phi_{c}^{(k)}\) is given by:  
\[\phi_{c}^{(k)}(b_{1}m_{1},\cdots,b_{d_{c}-1}m_{d_{c}-1})=\phi_{c}^{(k)}(m_{1}, \cdots,m_{d_{c}-1})\prod_{i=1}^{d_{c}-1}b_{i}, \tag{21}\]  
where \(b_{i}\) belongs to \(\{\pm 1\}\) sequence. The most important sign inversion invariance property of the variable node message map \(\phi_{v}^{(k)}\) is expressed as:  
\[\phi_{v}^{(k)}(-m_{0},\cdots,-m_{d_{c}-1})=-\phi_{v}^{(k)}(m_{0},\cdots,m_{d_{ c}-1}),k\geq 1. \tag{22}\]  
The final output message \(m_{o}^{(k)}\) is dependent on the square of initial messages based on Equations (12)-(22). The non-0-input messages, which correspond to the minus data in the iterations, are also dependent on the square of message \(m_{i}\) based on the symmetry properties shown in Equations (20)-(22). Thus, larger squares of initial legal LDPC codeword messages lead to larger squares of final output messages. The sum of the above equations amplifies this effect. Initial message amplitudes (i.e., \(LLRs\)) decrease when timing offsets are introduced to reduce the effectiveness of SNR. The decrease reduces the cost function, i.e., the sum of all the square of the \(LLRs\) Thus, a timing offset estimate (including timing phase and frequency offsets) is obtained by searching possible ranges in obtaining the maximum cost function. Moreover, random walk has limited destructive effects, because it is limited by oscillation quality. Therefore, optimal timing offsets are estimated and corrected by maximizing the sum of all the square of the \(LLRs\).