# Posterior Probability-Based Symbol Detection Algorithm for CPM in Underwater Acoustic Channels
## 1 Introduction  
The demand for high-speed underwater acoustic (UWA) communication has progressively intensified in tandem with advancements in marine engineering and technology [1, 2, 3]. The conventional phase-shift keying (PSK) modulation scheme faces significant limitations due to the constrained bandwidth of UWA channels and the restricted power output of UWA transducers. Continuous phase modulation (CPM) [4, 5], which has found widespread application in aviation telemetry and satellite communications [6, 7, 8], offers a compelling alternative. CPM enhances the bandwidth and power efficiency of UWA communication by virtue of its constant envelope, minimal out-of-band radiation, inherent robustness, and intrinsic coding gain. Among the various wireless communication channels, UWA channels are among the most intricate, leading to substantial signal losses [9, 10]. The continuous phase and the nonlinear nature inherent to CPM signals further complicate the design of signal receivers. Ideally, a CPM-modulated signal is detected using the Viterbi algorithm for maximum likelihood sequence estimation, involving a search within a singular super grid [11]. However, the practical realization of this approach is impeded by the algorithm's complexity,which escalates exponentially with increasing signal length and the varying characteristics of the communication channel. [PERSON] and [PERSON] [12] mitigated this complexity by employing a delayed decision-feedback sequence estimation method. Nevertheless, significant challenges remain, particularly when addressing long-delay spread channel responses with relatively high complexity. Numerous studies have sought to simplify the receiver algorithm by decoupling equalization and detection, as proposed by [PERSON] [13] and [PERSON] and [PERSON] [14], through CPM decomposition. Additionally, minimum mean square error (MMSE)-based suboptimal linear equalization has been applied to counter inter-symbol interference. [PERSON] and [PERSON] [15] employed Laurent and orthogonal decompositions to achieve frequency domain equalization (FDE) of CPM signals in frequency-selective fading channels. A straightforward differential detection scheme for CPM signals with a modulation index of 1/2 was introduced, significantly reducing the complexity of symbol detection. The effectiveness of the algorithm was validated using minimum shift keying (MSK) and Gaussian MSK (GMSK) signals. [PERSON] and [PERSON] [16] further advanced the development of minimum mean square error (MMSE) equalization for CPM signals in the frequency domain by implementing linear, decision feedback, and turbo equalization techniques. Additionally, leveraging the inherent characteristics of CPM, they proposed an iterative symbol detection algorithm without channel coding, which was integrated with decision feedback equalization. The binary 3-RC format was adopted for performance verification. Several research efforts have focused on simplifying the process for CPM signals. [PERSON] et al. [17] proposed a simplified decision-feedback equalization scheme, which effectively mitigates the computational complexity associated with the Fourier transform and the Viterbi decoder. [PERSON] et al. [18] approximated the autocorrelation matrix of a CPM signal as a diagonal block matrix, thereby simplifying the MMSE equalization into a low-complexity zero-forcing equalizer, significantly reducing the computational burden of FDE. Building on the insights from these studies, [PERSON] and [PERSON] [19] further reduced complexity by simplifying low-energy pulses derived from Laurent decomposition. [PERSON] et al. [20] leveraged the fractional interval representation of CPM to formulate MMSE-based FDE, which approximates the channel matrix and the pseudo-symbol correlation matrix of the CPM signal. Meanwhile, through the use of shaped-offset quadrature PSK, [PERSON] and [PERSON] [21] demonstrated that the FDE techniques proposed by [PERSON] and [PERSON], [PERSON] and [PERSON], and [PERSON] et al.