## 3 Influence of Timing Errors on LDPC Decoding  
The received baseband signals are raised-cosine pulses from the combination of SRRC pulse-shaped filters located at the transmitters and receivers along with the AWGNs. In [6], the sampling positions approached the sampled points from LDPC decoding and corresponding analyses, which led to the increased absolute value of all LDPC decoding metrics, i.e., log-likelihood ratios (\(LLRs\)). A similar effect of the sum of all \(LLR\) squares under these timing errors also manifested. This phenomenon is explained below. Given no noise, a received signal with perfect timing is properly sampled at the center of the symmetry raised cosine signal waveform and it has the most effective amplitude. However, under the overall timing offset as presented in Equation (5), the sampled signal deviates from the optimal center of the waveform with less effective amplitude and it is further deteriorate by the inter-symbol interference (ISI) by adjacent signals. At low SNR, the influence of the ISI can also be approximately modeled as the AWGN [10]. Therefore, the comprehensive effect of the symbol timing error causes a decreased effective SNRfor the coded BPSK modulation system. The equivalent noise variance is the sum of the original noise variance and the new generated equivalent noise variance caused by the ISI. With SNR analyses in [8], more deviations from optimal sampled points cause lower SNRs for the received signals, leading to a decrease of initial messages (i.e., \(LLRs\)) in LDPC decoding. The final output messages in the decoding are dependent on the square of the initial messages, because the messages in the iterations are mainly dependent on the square of previous messages. In addition, the sum of the square of each message increases the deviation effect. Thus, when timing offsets are introduced to reduce the effective SNRs, the amplitude of the initial messages decreases; in turn, this reduces the \(LLRs\) and decreases the sum of the square of all decoding messages. This process can be analyzed with density evolution and Gaussian approximation [17; 18]. Firstly, we give a typical instance of the schematic diagram of the LDPC decoding by the iterative belief propagation (BP) [1; 2] algorithm as shown in Figure 1.  
From Figure 1, at an AWGN channel with zero mean and variance \(\sigma^{2}\), the logarithm version of the LDPC decoding can be expressed as below. * (1) Initialization The initial soft metrics of LDPC decoding, i.e., \(LLRs\), are expresses as \[L(c_{i})=\ln\frac{p(x_{i}=1|y_{i})}{p(x_{i}=-1|y_{i})}=\frac{2y_{i}}{\sigma^{2 }},\] (6) \[L(u_{ij})=L(v_{ji})=0,\] (7) where \(L(\cdot)\) is the logarithm likelihood ratio (\(LLR\)) function and \(L(c_{i})\) is the initial \(LLR\) of the codeword \(c_{i}\). \(x_{i}\) and \(y_{i}\) are the \(i\)-th decided and received signals respectively. \(u_{ij}\) and \(v_{ji}\) are the exchanged messages between the variable node \(j\) and the check node \(i\). * (2) Decoding procedure
* (2.1) Update the variable nodes and perform the final decoding judgment The decoding metrics updated in the variable nodes and final decoding judgment can be expressed as \[L(u_{ij})=L(c_{i})+\sum_{j\in M(i)\backslash j}L(v_{ji}),\] (8) \[L(Q_{i})=L(c_{i})+\sum_{j\in M(i)}L(v_{ji}),\] (9) \[\hat{c}_{i}=\left\{ \begin{array}{ll}0&,&L(Q_{i})\geq 0\\ 1&,&L(Q_{i})<0\end{array} \right.,\] (10) where the set of the \(i\)-th check node connected to the \(j\)-th variable node is expressed as \(M(i)=\{j:\mathbf{H}_{j,i}=1\}\) and \(\mathbf{H}_{i,j}\) is the element of the check matrix \(\mathbf{H}\) with index \((i,j)\). The symbol \"\(\backslash\)\" behind a set means the set with removal of the  
Figure 1: Schematic diagram of the LDPC soft message propagations between the variable nodes and check nodes in BP decoding. element \"\(j\)\". \(Q_{i}\) is the LDPC soft decision metrics of the \(i\)-th variable node of the codeword \(c_{i}\). \(N\) is the code length.