## 5 Conclusions  
This paper developed a novel likelihood-based modulation identification algorithm for MPSK signals in underwater acoustic multipath channels. The simulation results showed that the proposed method can identify the modulation mode of MPSK well in underwater acoustic multipath channel.  
Figure 10: Comparison of BPSK, QPSK, and 8 PSK.  
Figure 9: Comparison of identification method.  
In the aspect of blind channel estimation, the NMSE of SBL based blind channel estimation algorithm is below \(-20\) dB when the SNR is higher than \(15\) dB. In addition, EM algorithm makes the received constellation more convergent and solves the problem of unknown original constellation. In view of the simulation results, some discussions for the SBL-EM-QHLRT method are given, as follows:  
In the SBL channel estimation method, the channel 2 and 3 are more accurately estimated than channel 1 due to the setting constraint conditions on channel 1. Adding constraints will lead to gain and delay in the estimated CIR. Channel 2 and channel 3 can be normalized to eliminate the gain, but channel 1 cannot eliminate the gain due to constraints, which results in poor performance of channel 1 as compared with channel 2 and 3. The estimated result of channel 1 does not affect the identification rate, because we only need to take the received signal of one of the paths for processing.  
The SBL blind channel estimation method proposed in this paper can also be applied to other MPSK modulation identification methods. After eliminating the channel interference, the features that are used for identification in these methods become more apparent.  
From the simulation results, when the iteration times and symbol length of EM increases, the identification rate will be better, but the computational efficiency should be considered in practical use.  
Further improvements of the blind likelihood-based modulation identification method for signals in different channel environment, such as Doppler shift and MIMO, is needed in the future.  
[PERSON] conceived the algorithm and designed the simulations; [PERSON] and [PERSON] analyzed the results; [PERSON] and [PERSON] revised the manuscript. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Key R&D Program of China(Grant Nos. 2018 YFC0308500), National Natural Science Foundation of China (Grant Nos. 61771152, 11974090, 11774074 and 11704090), the Natural Science Foundation of Heilongjiang Province of China (Grant No. YQ2019F002). The authors wish to thank all participants of the research.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} EM & Expectation Maximization \\ QHLRT & Quasi-Hybrid Likelihood Ratio Test \\ SBL & Sparse Bayesian Learning \\ MPSK & Multiple Phase Shift Keying \\ MQAM & Multiple Quadrature Amplitude Modulation \\ SIMO & Single Input Multi Output \\ CIR & Channel Impulse Response \\ GMM & Gaussian Mixture Model \\ NMSE & Normalized Mean Square Error \\ SNR & Signal to Noise Ratio \\ DSSS & Direct Sequence Spread Spectrum \\ OFDM & Orthogonal Frequency Division Multiplexing \\ \end{tabular}