## 4 Proposed Iterative Timing Recovery
So the approximate derivation of \(\Psi(\tau,v)\) to \((\tau,v)\) can be represented as  
\[\frac{\partial\Psi(\tau,v)}{\partial\tau}=2\sum_{i=0}^{K}\left[\Lambda_{i}+E( x_{i}|x_{j,j\
eq i},\tau,v)\right]\frac{\partial y_{i}}{\partial\tau}\cdot \frac{1}{2\sigma^{2}}, \tag{29}\]  
\[\frac{\partial\Psi(\tau,v)}{\partial v}=2\sum_{i=0}^{K}\left[\Lambda_{i}+E(x_{ i}|x_{j,j\
eq i},\tau,v)\right]\frac{\partial y_{i}}{\partial v}\cdot\frac{1}{2 \sigma^{2}}. \tag{30}\]  
By similar techniques in [13], given normalized parameters, the received signal \(y_{i}\) can be approximately modeled as follows:  
\[y_{i}=1-c_{1}\text{sgn}(\tau)\tau-c_{2}i\text{sgn}(v)v+n(t_{i}), \tag{31}\]  
where the amplitudes of the transmitted signals are normalized as \"1\", \(n(t_{i})\) is the channel noise at that time instance, and \"sgn\" is the sign function in mathematics. Here, the approximate transition trend of \(\Psi(\tau,v)\) to \((\tau,v)\) is used to incorporate a more accurate value into positive constant \(c_{1}\) and \(c_{2}\), which can be determined in simulations. Equations (29) and (30) is then approximately calculated as  
\[\frac{\partial\Psi(\tau,v)}{\partial\tau}\approx 2(\frac{2}{\sigma_{ch}^{2}})( -c_{1})\sum_{i=0}^{K}\left[\Lambda_{i}+E(x_{i}|x_{j,j\
eq i},\tau,v)\right] \text{sgn}(\tau), \tag{32}\]  
\[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]0Equations (32) and (33) can be further simplified by uniting all fixed variables or constants into the negative value \(C_{1}\) and \(C_{2}\), respectively, as  
\[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]1  
\[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]2  
Thus, the calculation complexity can be reduced by Equations (34) and (35), because we use the adaptive parameter update algorithm, which just needs the approximate trend of the gradient rather than the more accurate numerical results. So the adaptive timing parameter correction along with the iterative LDPC decoding jointly contribute the good features of both good performance and low complexity. Finally, according to the principles of the adaptive algorithms, \((\tau,v)\) can be estimated and updated with the following Equations (36) and (37) as  
\[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]3 \[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]4  
where \(\lambda_{1}\) and \(\lambda_{2}\) are step parameters for iteratively updating \((\tau,v)\), respectively, \(y_{i}\) in Equation (31) decodes LDPC codes, and the superscript \((n)\) demotes the \((n)\)-th iteration. The partial derivation function \(\partial\Psi(\tau,v)/\partial\tau\) and \(\partial\Psi(\tau,v)/\partial v\) are computed in Equations (34) and (35). The initial parameter pair \((\tau^{(0)},v^{(0)})\) can be set as the zero vector \((0,0)\). The other constants can be verified in the simulations. In addition, a conventional first-order PLL-based structure with a decision-directed Mueller-Muller timing error detector (M&M TED) [6] can also be used as a supplement like [4] to implement the iterative timing tracking. At each iteration, the M&M TED is executed by provided with more accurate symbols decoded by the LDPC decoder and the re-sampled received signals, which are compensated with the optimal estimate of timing phase and frequency offset by our proposed algorithm. Finally, according to the above analyses, the whole timing recovery is designed in \[\Lambda(x_{i}|\mathbf{r})=\log\frac{p(x_{i}=1|\mathbf{r})}{p(x_{i}=-1| \mathbf{r})}\mathbf{=}\frac{2y_{i}}{c_{ch}^{2}}, \tag{24}\]5  
In the whole timing recovery system shown in Figure 3, there are two loops to execute the timing recovery, i.e., Loop1 and Loop2.