To mitigate the error propagation of single-carrier time-domain equalization (SC-TDE) with insufficient observation data, this article proposes a low-complexity message-passing-based SC-TDE algorithm. First, the temporal correlation (TC) between subblocks is exploited to improve the performance of conventional message-passing-based sparse Bayesian learning (SBL) when the data are insufficient. The proposed algorithm then models the channel estimation process as a hidden Markov model. It captures the TC property by utilizing a first-order autoregressive model, thus supporting the current subblock with a priori information from the previous subblock. By using belief propagation (BP), the TC and BP-based SBL algorithm (TC-BP-SBL) is derived, which is then approximated to obtain the TC and approximation-message-passing-based SBL (TC-AMP-SBL) with lower computational complexity. Finally, taking advantage of AMP and expectation propagation (EP), a two-layer iterative equalization algorithm is introduced for joint message passing. The inner iteration uses AMP for symbol estimation, and the outer iteration improves the equalization performance by EP based on deterministic approximate variational inference. The proposed algorithm is validated using data collected during the 11th Chinese Arctic Scientific Expedition. The results show that the proposed algorithm can significantly reduce the computational complexity of SC-TDE and effectively mitigate error propagation when the observation data are insufficient.

Underwater signal processing is primarily based on sound waves because of the unique properties of water. However, the slow speed and limited bandwidth of sound introduce numerous challenges, including pronounced time-varying characteristics and significant multipath effects. This paper explores a channel estimation method utilizing superimposed training sequences. Compared with conventional schemes, this method offers higher spectral efficiency and better adaptability to time-varying channels owing to its temporal traversal. To ensure success in this scheme, it is crucial to obtain time-varying channel estimation and data detection at low SNRs given that superimposed training sequences consume power resources. To achieve this goal, we initially employ coarse channel estimation utilizing superimposed training sequences. Subsequently, we employ approximate message passing algorithms based on the estimated channels for data detection, followed by iterative channel estimation and equalization based on estimated symbols. We devise an approximate message passing channel estimation method grounded on a Gaussian mixture model and refine its hyperparameters through the expectation maximization algorithm. Then, we refine the channel information based on time correlation by employing an autoregressive hidden Markov model. Lastly, we perform numerical simulations of communication systems by utilizing a time-varying channel toolbox to simulate time-varying channels, and we validate the feasibility of the proposed communication system using experimental field data.