## 1 Introduction
Thus, small channel coefficients are preserved well, and precise CSI is obtained. * Computational complexity reduction with minimal performance loss using the selective update strategy. Compared with traditional iterative channel estimation in UWACs, a threshold is set to selectively update the coefficients with a large offset in terms of the estimated noise variance using the SZA-IPNLMS. Thus, it effectively reduces the unnecessary updating of the channel coefficients. A SZA-DIPNLMS is adopted to replace the SZA-IPNLMS for reducing the counts of update operations for the proportionate step matrix through a fixed update period. The step size is dynamically set in accordance with the noise variance and offset of the estimation to maintain a good performance. In this way, the computational complexity of the iterative channel estimation is effectively reduced with a small bit error rate (BER) loss. * High-quality recovery of UWAC signals by utilizing the IFDTE and bidirectional structure of the equalization. The IFDTE is applied to UWACs combined with iterative channel estimation. Different from the traditional equalization used in UWACs, the IFDTE obtains a precise a posteriori probability of the transmitted symbols by estimating them iteratively based on the EP. In this way, the IFDTE achieves good performance of the UWAC signal recovery with trade-off complexity compared with those of traditional FDTEs. A bidirectional structure of equalization is utilized to acquire the bidirectional gain to promote the performance of the IFDTE. Thus, the bidirectional IFDTE (Bi-IFDTE) achieves a better performance than the IFDTE. Because of the reliable - a posteriori - estimation of symbols obtained by the EP, symbols mapped based on an - a posteriori - estimation served as the training sequence to improve the performance of the channel estimation. Through the high precise channel and symbols, the estimation improves the performance of the interference cancellation in the equalization to achieve a high-quality recovery of UWAC signals. The organization of this paper is briefly introduced as follows. Section 2 presents a SIMO model of UWACs. Section 3 introduces sparse adaptive channel estimation with a SZA term. A low complexity iterative channel estimation with a selective update of channel coefficients is proposed. Different from the traditional turbo equalization in UWACs, the IFDTE based on the EP is introduced in detail to estimate the actual - a posteriori - distribution precisely. A bidirectional structure of equalization is proposed to further enhance the performance of the IFDTE. The complexity of the IFDTE is analyzed in this part. The computational complexity of the IFDTE is slightly higher than the traditional FDTE but lower than the traditional TDTE. Section 4 analyses the MSE, BER performance, and computational complexity of the proposed channel estimation scheme. Extrinsic information transfer (EXIT) charts and BER curves are applied jointly for numerical simulations to verify the good performance of convergence and signal recovery. The results show that the proposed scheme outperforms other existing equalization schemes. Section 5 concludes the whole paper. The list of abbreviations used in this paper is shown in the abbreviations for clarity.