## 4 Simulation and Analysis
### MSE and BER Performances Comparison  
We next discuss the MSEs of the proposed LPA-based methods. The timing error is estimated by Equation (36) with \(BT=0.001\). We use two reduced-complexity detectors. The first detector has a 16-state trellis, as we implemented in Section 3, with TPT, FPT (\(L^{\prime}=2\)), and SSP. The second one only uses TPT and FPT (\(L^{\prime}=2\)) to furhter approach the optimal detector with a 64-state trellis. The results are plotted in Figure 5 and compared with the modified Cramer-Rao bound (MCRB) [26], given by  
\[\text{MCRB}(\tau)=\frac{T^{2}}{8\pi^{2}\overline{h^{2}}C_{a}C_{f}L_{0}}\times \frac{1}{2E_{b}/N_{0}}, \tag{44}\]  
where \(\overline{h^{2}}=1/2\sum_{i=0}^{1}h_{i}^{2}\), \(C_{a}=(M^{2}-1)/3\) for \(M\)-ary symbols, \(C_{f}=3/(8L)\), and \(L_{0}=1/(2 BT)\). Our proposed estimators have ignorable MSE loss compared with the EL-based estimator used in [26; 32], since the same reduced-complexity methods are used to simplify the trellis. When the 64-state trellis detector is used, the proposed SLPA-based estimator achieves results close to the performance of MCRB MSE, except for the larger values of \(E_{b}/N_{0}\).  
In Figure 6, we compare the BERs for \(h=\{\frac{4}{16},\frac{5}{16}\}\) between the proposed methods and EL-based method used in [26; 32] under the same trellis (using TPT, FPT with \(L^{\prime}=2\), and SSP with 4-value phase state partition). This shows that the LPA-based and EL-based methods have comparable BER performances. However, the LPA-based method has much lower complexity than the conventional EL-based method, which is discussed later. The complexity of the SLPA-based method is lower than that of the LPA-based method, and they have almost the same BER performance. The BERs of the SLPA-based method in floating-point simulation and fixed-point hardware tests are also plotted, and both of them  
Figure 4: Power spectrum density comparison.  
are close to the BER curve of ideal synchronization (without timing and phase error). The theoretical BER bound for the multi-\(h\) CPM with \(h=\{\frac{4}{16},\frac{5}{16}\}\) is calculated by  
\[p_{b}=\frac{7}{4^{6}}Q\left(\sqrt{1.288\frac{E_{b}}{N_{0}}}\right)+\frac{(4)(648 )}{4^{6}}Q\left(\sqrt{1.656\frac{E_{b}}{N_{0}}}\right), \tag{45}\]  
where  
\[Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-u^{2}/2}du.\]  
Note that the BER of ideal synchronization has about 0.8 dB degradation to the theoretical BER bound of MLSD at a BER of \(10^{-5}\), which is consistent with the conclusion in [16].  
Next, we show the onboard BER performances of the mentioned three multi-\(h\) CPM schemes. The LPA-based synchronizer is adopted to reduce the implementation complexity. The system is implemented according to Figure 3. The BERs of the three CPM schemes are plotted in Figure 7. It can be seen that all three CPM schemes have BER performances close to the ideal detection without phase and timing errors. Note that the multi-\(h\) CPM scheme is able to achieve the BER performance close to the BER curve of ideal synchronization (without timing and phase error). The theoretical BER bound for the multi-\(h\) CPM with \(h=\{\frac{4}{16},\  
with higher modulation indices has better BER performance. This is because the higher modulation index brings in a larger minimum squared distance, which promises better BER performance [37]. However, the CPM with lower modulation indices has a higher spectral efficiency, as shown in Figure 4. Thus, among the three tested multi-\(h\) CPM schemes, the CPM with intermediate-value modulation indices \(h=\{\frac{5}{16},\frac{6}{16}\}\) brings better tradeoff between the BER and the spectral efficiency than the other two CPM schemes.