## 1 Introduction  
This paper deals with the blind modulation identification of signals transmitting through underwater acoustic multipath channel. We assume that the received signal is Multiple Phase Shift Keying (MPSK), i.e., BPSK, QPSK, and 8 PSK, but do not assume any prior knowledge of the modulation order. Underwater acoustic MPSK is widely used in information transmission; it is mainly used in high-speed communication scenarios [1; 2]. MPSK can also be used as a part of Direct Sequence Spread Spectrum (DSSS) to achieve long range data transmission [3; 4], and has been abundantly used for underwater Orthogonal Frequency Division Multiplexing (OFDM) subcarrier modulation [5; 6; 7]. With the development of underwater acoustic communication technology and the increasing demand in the field of underwater acoustic signal detection, modulation identification of non-cooperative MPSK has become a research hotspot. MPSK modulation identification is an important basis for parameter estimation, scrambling, and blind demodulation [8; 9; 10], particularly in military applications. Various authors have discussed the MPSK modulation identification in the literature and have proposed many schemes; however, most of the application scenarios are focused on the radio. These methods can be divided into two different groups: likelihood function-based and cyclic cumulant-based methods. Regarding the MPSK modulation identification, various methods have been proposed based on higher-order cumulants and cyclic cumulants [11; 12; 13; 14; 15; 16]. These methods assumed that the signals were in baseband.But in practical application, it is necessary to estimate the parameters of communication system in order to achieve baseband processing. Because the estimated parameters are susceptible to the channel, the higher-order cumulants and cyclic cumulants based method is not effective in practical application. A different class of approaches to PSK modulation identification that employed the likelihood-based method. In [17; 18], the method indicated that the system needed high SNR in order to obtain accurate identification results and this method is only suitable for noise channel. A similar likelihood-based method for identifying MPSK signals was described in [19]. However, this method needs to know the parameters of MPSK modulation. In [20], the authors introduced a likelihood-based method for BPSK, QPSK, and 8 PSK in multipath channels. This method does not take into account that the original constellation is unknown and the frequency-selective channel was slow-varying. In [21], a new MPSK modulation identification method based on likelihood function is proposed. This method identify modulation without knowledge of the symbol rate, carrier frequency and noise power in the received signals. However, this method is suitable for noise channel, its performance is poor in multipath channels. In non-cooperative environment, it is difficult to estimate the channel impulse response (CIR) of multipath channel. Therefore, the likelihood-based modulation identification method is not suitable for multipath channels generally. In [19], the authors concluded that the likelihood-based method performs well in modulation identification when the CIR and noise power are known. Therefore, the likelihood based method draws researchers' attention when the CIR and noise power are unknown in the multipath fading channel. Recently, lots of scholars used the method of expectation maximization (EM) to estimate the CIR and noise power in flat fading channel in view of MPSK and MQAM signals and then they used QHIRT to identify modulation [22; 23; 24].Through these analysis and studies, we found EM-QHIRT can increase the identification rate effectively and make us come up with some new ideas about modulation identification in fading channel. However, EM-QHIRT is only suitable for flat fading channels, but not for underwater acoustic multipath channels. Consequently, the likelihood based methods need concern how to reduce the impact of multipath fading channels.