## 1 Introduction
The FDTE based on the EP was applied for multiuser detection with the known channel to reduce the complexity [13]. To the best of our knowledge, no detailed investigation was conducted on turbo equalization based on the EP in UWACs. Iterative channel estimation can effectively shorten the necessary length of the training sequence for convergence because of the band-limited nature of the UWAC channel, this process improves the efficiency of the spectrum utilization [14; 15; 16; 17]. However, computational complexity inevitably increases due to the reuse of soft information. Adaptive channel estimation was widely used in UWAC channel estimation due to its simple structure and minimal calculation. However, its accuracy was slightly worse than those of compressed sensing (CS) [18]. The CS-based channel estimation updated tap coefficients through heuristic search and inverse operations, resulting in high computational complexity. The combination of norm constraint and adaptive algorithm can achieve rapid convergence with minimal computation [19]. A SZA normal least mean square (NLMS) (SZA-NLMS) algorithm, belonging to the \(l_{1}\)-norm constrained adaptive algorithm, directly set the tap coefficients below the threshold to zero. Under other similar \(l_{1}\)-norm constrained channel estimations, it had faster convergence and better signal recovery performance than that of the standard NLMS equalization [20; 21]. Recently, to effectively combat the selective fading of underwater acoustic channels for single-carrier deep-sea vertical acoustic communications, an improved proportionate normalized minimum-SER (IPNMSER) algorithm was proposed for adaptive turbo equalization by utilizing the minimum-SER (MSER) criterion to minimize the system's SER directly [22]. Also, a hybrid frequency-time domain turbo equalizer (FTD-TEQ) was proposed to benefit from two turbo equalizers to solve the slow-convergence problem at different iterative stages for UWACs with comprehensive experimental investigations [23]. Furthermore, the block implementation of least mean square based frequency domain direct adaptive turbo equalization was proposed for use in UWACs [24] and it can be combined with excellent Shannon capacity, approaching channel codes for the better performance and security of turbo equalization. Then, security-oriented Polar coding can be adopted based on channel-gain-mapped frozen bits [25]. And decoding algorithms for low-density parity-check (LDPC) codes were proposed [26] and they can be used for such applications in turbo equalization. According to the aforementioned analyses in the literature, efficient turbo equalization with both good performance and low complexity is the exact challenge in UWACs due to the fast time variance fading in UWACs. The limitations of existing equalization schemes means they usually do not make full use of signals, especially the more accurate equalization metrics of the a posteriori probability of transmitted symbols. In addition, the EP interference cancellation is not employed for better equalization performance. Then, the above two aspects motivate the need for an improved approach for further improvement in turbo equalization. In this paper, an improved FDTE (IFDTE) with the EP interference cancellation for iterative channel estimation is proposed to promote the performance of signal recovery with low complexity in UWACs. This scheme is studied in a single-input multiple-output (SIMO) system. Its performance improvement and complexity reduction are analyzed and verified through numerical simulations. Finally, the main contributions are summarized as follows. * A precise sparse adaptive channel estimating using the selective zero attracting penalty term. A SZA-IPNLMS is extended to estimate the channel state information (CSI) in terms of training sequences. Compared with the past sparse adaptive channel estimation, the SZA-IPNLMS gives different constraints to update channel coefficients in accordance with the ratio of channel coefficients to the maximum one.