## 4 Numerical Results
### Performance of Sbl-Em-Qflkt  
Two problems need to be considered in the likelihood method. The one is that the original constellation is unknown, and the other one is that the constellation will rotate when obvious frequency offset exist in the underwater acoustic and down sampling will also make the constellation rotate even the original constellation can be known. Obviously, the likelihood method cannot be used to identify modulation directly. Therefore, we use EM to solve these problems. EM sets the original constellation firstly, and then it estimates the amplitude attenuation and phase offset of the channel under all possible modulation modes.  
The simulation process is as follows. QPSK modulation is adopted in simulation and the constellation is [0.707 + 0.707], 0.707 \(-\) 0.707], \(-\)0.707 \(-\) 0.707], \(-\)0.707 + 0.707]], as it is shown in Figure 6a-c show the constellation of the received signal and the constellation after SBL channel estimation and equalization, respectively. By comparison original constellation and the constellation shown in Figure 6c, we can find that constellation will rotate after SBL channel estimation and equalization. In EM, the constellation of MPSK is arbitrarily determined, we set [1, 1], \(-\)1, \(-\)1]] as original constellation in QPSK. We also need to set original constellation in BPSK and 8 PSK. Figure 6d-f show the result of constellation compensated by EM under BPSK, QPSK and 8 PSK. It can be found that constellation in Figure 6c has converged to constellation in Figure 6e that the influence of phase shift has been eliminated. Furthermore, the influence of the channel is further eliminated, which makes the constellation morepolymerized. Now, these two problems have been solved and the likelihood-based method can be used to identify MPSK modulation successfully. Next, we illustrate the influence of parameters of EM algorithm on its performance.  
The influence of different iteration times on the performance of EM method is analysed by this simulation. We set the number of symbols of EM to 300 and the iteration times of EM to 4, 6, 8, and 10 respectively  
Figure 6: Simulation results based on sparse Bayesian learning-expectation maximization (SBL-EM): (**a**) Original constellation in QPSK modulation, (**b**) Constellation of received signal, (**c**) Constellation after SBL, (**d**) Constellation after SBL-EM under BPSK, (**e**) Constellation after SBL-EM under QPSK, (**f**) Constellation after SBL-EM under 8 PSK.  
In the end, we compared the identification rate by using SBL-QHLRT with frequency offset and without frequency offset, also by using EM-QHLRT without SBL and SBL-EM-QHLRT. As shown in the Figure 9 (here we have set EM iteration times to 8 and the number of symbols to 300). SBL-QHLRT with frequency offset cannot identify MPSK modulation successfully because the constellation rotation will make the QHLRT failed. EM-QHLRT also failed to identify MPSK modulation because EM is only suit for flat fading channel. SBL-QHLRT method with known original constellation and have no frequency offset can be useful, but it will be affected by channel estimation error. We can apparently find that SBL-EM-QHLRT can overcome the problem of constellation rotation and reduce the influence of the constellation rotation.  
Figure 8: The influence of the number of symbols on EM.  
Figure 7: The influence of iteration times on EM.  
underwater acoustic fading channel. Figure 10 shows the identification rate of BPSK, QPSK, and 8 PSK under SBL-EM-QHIRT, of which BPSK has the highest identification rate and 8 PSK has the lowest.  
The simulation results show that EM can effectively improve the identification rate of underwater acoustic MPSK. When the number of iterations and the number of symbols of EM meet the requirements, EM can further eliminate the influence of channel and solve the problem of unknown original constellation. Therefore, the likelihood-based identification method can successfully identify MPSK modulation without any prior information.