## 1 Introduction  
Underwater acoustic communications (UWACs) have been widely used as sensing and communication techniques in underwater sound measurement and fishing. They are adequately developed and equipped in the underwater sensors of ships, submarines, autonomous underwater vehicles, and so on. However, UWAC channels are time-varying and multipath-fading. The received signals usually have serious Doppler distortion and inter-symbol interference [1]. Their high-quality channel recovery has attracted wide attention in the literature. Turbo equalization is used to effectively recover original transmission signals by exchanging the extrinsic information between an equalizer and a decoder iteratively [2]. Many schemes of turbo equalization were proposed for UWACs in the past years. The first Turbo equalization applied to UWACs was a joint maximum a posteriori probability (MAP) equalization [3]. Although the pre-survivor processing was only adopted to track the most likely path of trellis states for MAP equalization, the computational complexity remains high. Consequently, the design of low complexity turbo equalization based on a minimum mean square error (MMSE) was proposed. Time-domain turbo equalization (TDTE) in UWACs extending conventional MMSE linear equalization or soft decision feedback equalization (SDFE) was reported in the literature [4; 5]. The receivers exhibit good performance with a computational complexity quadratic scaling at the block length and the length of finite channel impulse response (CIR). The length of the filter for equalization is longer than the UWAC channels, and the lengths of UWAC channels are often longer than 50 [2]. Therefore, the time-domain turbo equalization still experiences huge computational complexity. A linear frequency domain turbo equalization (FDTE) was used in UWACs to initially process the received signals and to reduce the computational complexity of time-domain turbo equalization [6]. In the FDTE, the parallel block equalization filters were implemented using fast Fourier transformation (FFT) with much efficient computation of parallel convolution; it thus improved the computational complexity significantly by reducing many multiplications in the filtering calculations of the TDTE. In addition, some more techniques, such as the selective zero attracting penalty, selective update strategy, bidirectional structure, and so on, can be combined closely in the FDTE to further improve the performance. Good performance was obtained in low-order modulation. However, the performance degraded radically due to severe Doppler shift under high-order modulation. The performance gap between the FDTE and TDTE still exists. Consequently, nonlinear FDTE was applied in UWACs to improve the performance of receivers. An improved FDTE (IFDTE) with interference cancellation and phase-locked loop was applied in UWACs [7]. Although it effectively overcame the phase ambiguity problem and achieved better performance compared with linear FDTE, its limitation was the high complexity of multiple layers for the equalization of symbols. A frequency domain decision feedback FDTE (FDDF-FDTE) scheme with iterative channel estimation was adopted to achieve a good trade-off between performance and complexity; its feasibility was then verified in terms of the SPACE'08 sea trial data [8]. Coarse estimation based on the Bayesian principle was utilized to estimate the a posteriori probability of transmitted symbols and to enhance the performance of the FDDF-FDTE [9]. Therefore, the turbo equalization with the estimation of a posteriori distribution is important in practice. Expectation propagation (EP) was widely used as the machine learning for signal detection, using simple distributions to approximate complex ones. The block-EP [10], EP-Filter [11], and the DFE-IC EP [12] had outperformed traditional algorithms under the MMSE criterion. The complexity is relatively higher than that of the traditional TDTE due to the self-iteration of the EP.