## 1 Introduction  
Underwater acoustic (UWA) communication [1], which has great advantage over the wired communication, is an important technique for ocean research. However, the UWA channel is a complex time-space-frequency variable channel [2]. UWA communication is very challenging due to its characteristic of frequency-selective fading and limited bandwidth [3]. The continuous phase modulation (CPM) [4, 5] technology is characterized by a constant envelope, high spectral efficiency, and noise robustness; it has been widely applied in radio communication [6, 7, 8, 9]. Consequently, the application of CPM to UWA communication may enhance bandwidth and power efficiencies, thereby increasing the performance. Nevertheless, the received signal of a UWA CPM has obvious intersymbol interference (ISI) due to the frequency-selective fading of the UWA channel, so an effective equalization algorithm is required to reduce the ISI. The traditional CPM signal equalization technology is a maximum-likelihood sequence estimation using the Viterbi algorithm [10], which is the optimal detection algorithm. However, the implementation of this algorithm is very challenging due to the fact that the complexity increases exponentially with signal and channel length. Using delayed decision feedback sequence estimation [11], the equalizer can reduce the sequence estimation complexity with a small performance degrade, but it is still complicated for long time delay channels. Consequently, numerous researchers have separated channel equalization and CPM detection, and used frequency-domain equalization (FDE) to simplify the processing. Because CPM is a nonlinear modulation scheme, a linear decomposition is required before the discrete Fourier transform (DFT). [PERSON] et al. [12] deduced single-carrier FDE (SCFDE) coefficients for CPM signals based on Laurent [13] and orthogonal decompositions, and proposed a simple differential detection scheme for CPM signals with the modulation index of \(1/2\). Furthermore, the algorithm's reliability was verified by using both minimum-shift keying (MSK) and Gaussian-filtered MSK (GMSK). Based on the Laurent decomposition, [PERSON] et al. [14] deduced the linear equalization, decision feedback equalization, and turbo equalization techniques of CPM in the frequency domain (FD). Simulation results suggested that the traditional FDE algorithm could provide an acceptable bit error rate (BER) with a reduced amount of computation, whereas the turbo equalization improved performance while also requiring a higher amount of computation. [PERSON] et al. [15] approximated the CPM autocorrelation matrix as a diagonal block matrix, simplifying the algorithm complexity without compromising performance. An analysis of the influence of the sampling interval on FDE performance was conducted, and [PERSON] et al. [16] concluded that the best results can be obtained when three sampling points per bit are obtained. Because the effective channel impulse response is required in FDE, channel estimation is performed and analyzed based on SCFDE [17; 18; 19]. Among them, [PERSON] et al. [17] proposed a method replacing the cyclic prefix (CP) in FDE with the training sequence of the known symbol, which could perform synchronization and channel estimation under the same performance conditions. CPM has not been widely used in UWA communication. [PERSON] discussed the feasibility of blind equalization for MSK UWA communication [20]. [PERSON] [21] and [PERSON] [22] studied MSK and GMSK signals for UWA communication, respectively, and employed the virtual time-reversal mirror to suppress multipath interference. To further enhance the performance of the binary CPM UWA communications, in this study we investigate the FDE technology for a CPM signal based on Laurent decomposition. From the perspective of SCFDE, a prefilter is specified, and a joint detector is consequently designed.