## 5 Simulation Results and Analyses  
To verify the effectiveness of the proposed scheme, numerical simulations of a LDPC coded BPSK system in an AWGN channel are carried out in this section. The LDPC code (1944, 972) [19] is adopted in the simulations with a maximum of 20 decoding iterations. Generally, the performance is often weighed by bit-error-rate (BER). So Monte Carlo simulations of LDPC decoding will run until either a specified number (e.g., 10 in high SNR and 50 in low SNR) of error frames occur or a total 1 million trials have been run. We choose the above simulation parameters of iteration stop judgment for LDPC decoding by compromising the result reliability and calculation efficiency and also the more practical experiments in some powerful work stations. The SRRC shaped and matched filters both have a roll-off factor of 0.3 and 13 taps. Several timing offset estimation iterations are used. The timing phase and frequency offsets are set over \(\pm 0.5\) T (e.g., 0.3 T and \(T\) is the sampling period) and \(\pm 500\) ppm (e.g., 300 ppm). The parameter \(\sigma/T\) of the random walk is 0.5%, which is similar to that used in [5]. The step size parameters \(\lambda_{1}\) and \(\lambda_{2}\) in Equations (36) and (37) are chosen as 0.2 and 0.5, respectively. Finally, the parameter analyses and the BER performances by the proposed iterative timing recovery are simulated and shown as follows. The partial curves of \(\Psi(\tau,v)\) to \((\tau,v)\), and the \(\Psi(\tau,0)\) to \(\tau\) and \(\Psi(0,v)\) to \(v\), with two, four, and six iterations at \(E_{b}/N_{0}\) of 1.5 dB are shown in Figures 4 and 5, respectively. The timing phase offsets within 0.1 T occur near the optimal position (Figure 4).The closeness to the optimal position is mainly due to the property of the raised cosine waveform by the SRRC-shaped and matched filters, where \(\tau\) within 0.1 T slightly reduces the effective amplitude of the received signals. Furthermore, the influence of \(v\) on \(\Psi(\tau,v)\) changes dramatically (Figure 5). The largest frequency error discrimination occurs within 20 of the optimal point. Thus, approximately 6 iterations are enough for the timing recovery to distinguish the ideal sampling point. Further from this simulation, the effective search ranges for the timing phase and frequency offsets are chosen as \((-0.5\) T, 0.5 T) and (\(-500\) ppm, \(500\) ppm). This may be the deficiency of the proposed timing recovery, where traditional timing recovery should be performed at first to guarantee the residual timing offsets fall into the range of this requirement. However, the proposed timing recovery can obtain much better estimation precision. Here, we define the normalized mean square error (NMSE) simulations for \((\tau,\upsilon)\) as \(NMSE_{\tau}\) and \(NMSE_{\upsilon}\) in Equations (38) and (39) as  
Figure 4: Partial curves of \(\Psi(\tau,0)\) to \(\tau\) with two, four, and six iterations. Figure 5: Partial curves of \(\Psi(0,\upsilon)\) to \(\upsilon\) with two, four, and six iterations. \[NMSE_{\tau} =[(\tau-\bar{\tau})/\tau]^{2}, \tag{38}\] \[NMSE_{v} =[(v-\bar{v})/v]^{2}, \tag{39}\]  
where \((\tau,v)\) are real timing phases and frequency offsets, whereas \((\bar{\tau},\bar{v})\) are their estimates. Moreover, the estimated timing error parameter pair \((\bar{\tau},\bar{v})\) involves an average of 50 times calculation. Meanwhile, the true timing error parameter pair \((\tau,v)\) for test cannot be the vector with any element zero to make sense of Equations (38) and (39) avoiding from divided by zero. Then, the \(NMSE_{\tau}\) and the \(NMSE_{v}\) with four, six, and eight iterations at different SNRs are simulated and shown in Figure 6 as follows. The \(NMSE_{v}\) in Figure 6 is smaller than \(NMSE_{\tau}\), because the influence of \(v\) on object function \(\Psi(\cdot)\) exerts more control than \(\tau\) (Figure 6). Therefore, a total of six iterations are enough for timing recovery (Figure 6).