In single-carrier underwater acoustic (UWA) communication systems, sparse multipath channels and long delay spreads pose significant challenges to adaptive equalization, often leading to limited steady-state accuracy and degraded detection performance. To address this issue, this paper proposes a data reuse-based ${\ell }_1$-regularized proportionate recursive least-squares algorithm (DR-${\ell }_1$-PRLS) for sparse adaptive equalization. The proposed method incorporates a data reuse (DR) mechanism into the ${\ell }_1$-PRLS framework, enabling multiple equivalent uses of each received–reference sample pair without increasing pilot overhead. Meanwhile, by combining the proportionate update strategy with the ${\ell }_1$ sparsity constraint, the structural information of sparse channels can be more fully exploited, thereby improving parameter estimation accuracy. Numerical simulations are conducted to evaluate the proposed method in terms of convergence behavior, tracking capability, computational complexity, and bit error rate (BER), and comparisons are made with LMS, RLS, PRLS, ${\ell }_1$-PRLS, and DR-PRLS algorithms. Simulation results show that, under sparse underwater acoustic channel conditions, DR-${\ell }_1$-PRLS achieves lower steady-state error and better BER performance while maintaining good tracking capability, thereby demonstrating its effectiveness and robustness for sparse adaptive equalization in single-carrier underwater acoustic communications.

Accurate estimation of sparse underwater acoustic channels is challenging because of multipath delay spread, correlated inputs, and impulsive non-Gaussian noise. Existing KRSL-based algorithms still suffer from limited convergence speed and tracking capability in time-varying sparse scenarios. This paper proposes a VFF-PRKRSL algorithm, which jointly introduces an error-driven variable forgetting factor and a proportionate gain matrix into the recursive KRSL framework to achieve adaptive historical-information weighting and enhanced updating of dominant taps. Simulation results show that for Bellhop-generated underwater acoustic channels with sparsity levels of 0.125 and 0.4688, the proposed algorithm achieves NMSD values of $-38.4763$ dB and $-37.9417$ dB at the 2000th iteration, improving upon PRKRSL by approximately 5.31 dB and 5.29 dB, respectively. Under Cauchy noise, it reaches an NMSD of $-46.3042$ dB, about 5.95 dB better than PRKRSL. Ablation results indicate that the variable forgetting factor is the main source of the performance gain and is complementary to the proportionate update mechanism. These results demonstrate that VFF-PRKRSL outperforms existing methods in convergence speed, steady-state accuracy, and robustness against impulsive noise.

Aiming at the problem that sparse channel estimation in underwater acoustic communication is susceptible to complex multipath propagation, non-Gaussian impulsive noise, and channel time variations, this paper proposes a bidirectional proportionate recursive maximum correntropy criterion algorithm, referred to as Bi-PRMCC. By introducing a bidirectional filtering structure into the proportionate recursive maximum correntropy criterion (PRMCC) framework, the proposed algorithm jointly exploits the information from forward and backward data sequences, thereby improving the estimation accuracy and block-based channel variation tracking capability for sparse underwater acoustic channels. Meanwhile, the maximum correntropy criterion enhances the robustness of the algorithm against non-Gaussian impulsive noise and outlier error samples, while the proportionate update mechanism improves its identification capability for dominant taps in sparse channels. To verify the effectiveness of the proposed algorithm, short-range sparse underwater acoustic channels and long-range complex multipath underwater acoustic channels are constructed based on the Bellhop ray-tracing model. Simulation experiments are then conducted under three typical non-Gaussian noise environments, namely Cauchy noise, $\alpha$-stable distribution noise, and Middleton noise. The experimental results show that, compared with recursive least squares (RLS), bidirectional recursive least squares (Bi-RLS), proportionate recursive least squares (PRLS), recursive maximum correntropy criterion (RMCC), and PRMCC, Bi-PRMCC achieves a lower steady-state normalized mean square deviation (NMSD) under different non-Gaussian noise conditions, indicating stronger robustness against impulsive noise. Under different signal-to-noise ratio conditions, the proposed algorithm still maintains superior steady-state estimation performance. In addition, in the channel abrupt-change tracking experiment, Bi-PRMCC can rapidly reconverge after channel variations occur, demonstrating favorable reconvergence capability under abrupt channel variations. The ablation study further verifies the stable performance gain brought by the bidirectional structure to PRMCC. Overall, the proposed Bi-PRMCC algorithm exhibits high estimation accuracy, robustness, and reconvergence capability under complex non-Gaussian noise and abrupt channel variation conditions.