1   Introduction

Underwater acoustic (UWA) communication is a relatively common communication scenario, which is often affected by the Doppler effect and multipath propagation in UWA channels (Ge et al., 2024; Sun et al., 2023; Liu et al., 2023; Yin et al., 2024). Multipath propagation can lead to intersymbol interference (ISI) (Naman and Abdelkareem, 2023). Conventional turbo equalizers use iterative soft-input soft-output (SISO) equalization and decoding to mitigate ISI under the assumption of time-invariant channel and perfect knowledge of channel state information (CSI) (Tüchler et al., 2002; Zheng et al., 2015). The Doppler effect can lead to inaccuracies in channel estimation, thereby causing errors in the calculation of equalization coefficients (Abdelkareem et al., 2011; Abdelkareem et al., 2016). To compensate for the Doppler effect caused by time-varying UWA channels, utilizing estimated symbols for channel tracking is a common method during turbo iteration. Time-varying UWA channels were estimated by Zhang et al. (2018) by solving the recursive least squares (LS) normal equation, which is established by the symbol a priori information from the turbo decoder. Assuming that the UWA channel follows a first-order autoregressive (AR(1)) process is also a common modeling approach, where the current channel state is influenced by the previous channel state and estimation noise. A joint bidirectional channel estimation (BCE) and turbo equalization algorithm was proposed by Yang et al. (2021), which assumed that the UWA channels satisfied the AR(1) process at different moments and used the superimposed pilot symbols from the turbo decoder to re-estimate the UWA channel. The AR(1) model was combined with the Bayesian framework for channel estimation, which was applied to turbo equalization (Liang et al., 2023). A turbo equalization method was proposed with the joint symbol, channel, and impulse noise estimation based on the AR(1) model (Zhang et al., 2024). These turbo equalizers all assume that the CSI is perfectly known to the receiver. However, this assumption does not hold in practical UWA communication, leading to channel estimation error (CEE), which degrades the bit error rate (BER) performance of the equalizer.

LS estimation (LSE) is widely used in UWA channel estimation because it is simple and does not require channel statistics (Zhang et al., 2022). The combination of LSE and superimposed training by Yang et al. (2022) can track time-varying channels. LSE was utilized by Jiang and Diamant (2023) as a coarse estimate to distinguish the sparsity of UWA channels. However, LSE ignores the sparsity of UWA channels, resulting in considerable CEE compared to compressive sensing methods (Khan et al., 2020; Guo et al., 2021). In our simulation, the BER performance of LSE is considerably inferior to that of sparse Bayesian learning (SBL) channel estimation.

Imperfect CSI refers to a situation where receivers do not perfectly know the CSI and can only obtain the channel estimate, which will be impacted by CEE. There has been extensive research on imperfect CSI in terrestrial radio frequency (RF) communication systems. Li et al. (2016) and Jiang et al. (2018) demonstrated the impact of CEE caused by LSE and proposed turbo equalizers with CEE compensation, resulting in remarkable ISI mitigation, particularly under low signal-to-noise ratio (SNR) conditions. Robust turbo equalizers were proposed by Zhu et al. (2016) and Zhe et al. (2018), which utilized the statistical properties of CEE. To compensate for the CEE due to time-varying channels, closely relevant studies (Fodor et al., 2021; Fodor et al., 2023) considered the memory characteristics of Rayleigh channels and proposed a linear receiver with imperfect CSI that minimizes the mean squared error (MSE). The core idea of utilizing imperfect CSI is to use the channel a priori distribution and statistical properties of the coarse channel estimation to form a more accurate channel a posteriori distribution, which can bring considerable performance gains.

Compared to those in RF wireless channels, the multipath propagation and Doppler effect in UWA channels are more severe because the speed of sound is much lower than the speed of light (Yin et al., 2019). The multipath structure of UWA channels is more complex, and the channel structure varies within tens or hundreds of symbol durations (Roudsari et al., 2017; Zhang et al., 2019). In this scenario, compensating for CEEs caused by LSE is necessary, and the linear receiver proposed by Fodor et al. (2021) may not be optimal. However, to our knowledge, there is little literature utilizing imperfect CSI in UWA communication. Therefore, inspired by the time-varying channel model proposed by Fodor et al. (2021), we aim to use estimated channels at different moments to obtain a channel a posteriori distribution and incorporate it into the design of the turbo equalizer. The contributions of this article can be summarized as follows.

• We give a new derivation for turbo equalization based on the joint Gaussian (JG) criterion, which is equivalent to the turbo equalizer proposed by Yuan et al. (2008) and Guo et al. (2009).

• On the basis of this derivation, we derive the channel a posteriori distribution in fast time-varying scenarios utilizing the AR(1) process and imperfect CSI, which is incorporated into the design of the turbo equalizer. Simulation results show that the proposed algorithm provides a better BER performance than other methods.

This paper is organized as follows. Section 2 provides a new derivation for turbo equalization and describes the UWA channel a posteriori distribution. Section 3 details the proposed turbo equalization for time-varying UWA channels with imperfect CSI. Section 4 shows the simulation results. Section 5 analyzes the computational complexity of the conventional and proposed algorithms. Finally, Section 6 concludes this paper.

Notations: The superscripts (·)^*, (·)^T, (·)^H, and (·)^† represent conjugate, transpose, conjugate transpose, and pseudoinverse, respectively. $\boldsymbol{I}_N$ denotes the N × N identity matrix. $\mathbb{E}_{\boldsymbol{x}}(\boldsymbol{y})$ and $\operatorname{Cov}_{\boldsymbol{x}}(\boldsymbol{y}, \boldsymbol{y})$ represent the expectation and covariance matrix of a random variable y with respect to x, respectively.

2   Turbo equalization and channel estimation

2.1   New derivation for turbo equalization

First, we introduce a new derivation for turbo equalization based on the JG criterion. We consider a widely used UWA communication system that uses single-carrier (SC) block transmission with a cyclic prefix (CP). The insertion of CP can simplify the derivation as the channel matrix is a square matrix. The derivation is performed in the time domain without loss of generality. SC modulation has a lower peak-to-average power ratio and is less sensitive to carrier frequency offset than orthogonal frequency-division multiplexing modulation (Tao, 2015). The signal block structure is shown in Figure 1, with pilot and data (including the CP) lengths of P and N, respectively. The pilot sequences at the beginning and end are used for channel estimations at two moments (denoted t − 1 and t). We assume that the UWA channel is time-invariant within time slots t and (t − 1) rather than within a symbol duration. The transmitted symbols $\left\{x_n\right\}_{n=0}^{N-1}$ modulated by M-ary phase-shift keying (PSK) correspond to the encoded and interleaved bits $\left\{c_{n, j}\right\}_{j=0}^{M-1}$.

Figure  1  Signal block structure

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The bits are mapped to $\left\{x_n\right\}$ according to the M-ary PSK–modulated symbols $\left\{\chi_i\right\}_{i=0}^{2^M-1}$, where $\chi_i$ corresponds to the bits $\left\{q_{i, j}\right\}_{j=0}^{M-1}$. At moment t, assuming that the length of the UWA channel $\boldsymbol{h}(t)$ is L, the received signal $\boldsymbol{y}$ from the hydrophone is

where $\boldsymbol{H}(t) \in \mathbb{C}^{N \times N}$ is a circulant matrix with the first column being $\left[\boldsymbol{h}(t)^{\mathrm{T}}, 0, \cdots, 0\right]^{\mathrm{T}}$ and $\boldsymbol{w}(t)$ is additive white Gaussian noise with zero mean and covariance $\sigma_o^2 \boldsymbol{I}_N$. The detailed modeling of UWA channels can be found in the study by Naman and Abdelkareem (2023).

According to Eq. (1), the a posteriori mean and covariance of y with the JG distribution conditioned on bit $c_{n, j}$ are

where $\chi_{q_{i, j}=c_{n, j}}$ is the mapped symbol $\chi_{i}$ that satisfies $q_{i, j}= c_{n, j}$, and $\boldsymbol{h}_{n}(t)$ is the nth row of $\boldsymbol{H}(t)$. $\mathbb{E}\left(x_{n}\right)$ and v are the a priori mean and variance of $x_{n}$, respectively. The variance v can be approximated from $\operatorname{Cov}(\boldsymbol{x}, \boldsymbol{x}) \approx v \boldsymbol{I}_{N}$ (Guo et al., 2009).

The output log-likelihood ratio (LLR) of the SISO equalizer is

The exponential term of $p\left(\boldsymbol{y} \mid c_{n, j}\right)$ can be simplified as

where $\left.\boldsymbol{A}=\boldsymbol{y}-\boldsymbol{H}(t) \mathbb{E}(\boldsymbol{x})+\boldsymbol{h}_{n}(t) \mathbb{E}(x_{n}\right)$. By substituting Eqs. (2), (3), and (5) into Eq. (4), we have

where $\gamma_{n, j}$ is the a priori LLR of $c_{n, j}$ from the SISO decoder and $\tilde{q}_{i, j}$ satisfies

We set $\beta=\boldsymbol{h}_{n}(t)^{\mathrm{H}} \boldsymbol{\varSigma}^{-1} \boldsymbol{A}$. Specifically, for quadrature PSK (QPSK)-modulated symbols

When $j=0$, Eq. (4) can be written as

where

Similarly, $\lambda_{n, 1}$ can be derived as

The turbo equalizer obtained from Eq. (6) is equivalent to the turbo equalizer proposed by Yuan et al. (2008) and Guo et al. (2009). For binary PSK (BPSK) modulation, because the mapping between symbols and bits is surjective, Eq. (6) can be written as

which is consistent with Eq. (2) in Yuan et al. (2008). For QPSK modulation, Eqs. (9) and (11) are the real and imaginary parts of Eq. (8a) in Guo et al. (2009), and they differ by $\sqrt{2}$ times. This is because it converts the transmitted and received complex vectors into real vectors, which can be regarded as BPSK-modulated symbols, and computes the LLRs using Eq. (12).

2.2   Channel estimation

Let the pilot sequence be $\boldsymbol{s}=\left[s_{0}, s_{1}, \cdots, s_{P-1}\right]^{\mathrm{T}}$. The pilot signal $\boldsymbol{y}_{\text {pilot }}(t)$ received from the hydrophone is

where the dictionary matrix $\boldsymbol{S}$ is a Toeplitz matrix, which is composed as

The LSE of the UWA channel at the t moment can be written as

We assume that the a priori distribution of the UWA channel follows $\boldsymbol{h}(t) \sim \mathcal{C N}\left(\mathbf{0}, \boldsymbol{I}_{L}\right)$. Therefore, the distribution of the estimated channel $\hat{\boldsymbol{h}}(t)$ satisfies

As the pilot sequences are the same at two moments, $\hat{\boldsymbol{h}}(t-1)$ also satisfies Eq. (16).

Given the LS-estimated channels $\hat{\boldsymbol{h}}(t)$ and $\hat{\boldsymbol{h}}(t-1)$, the a posteriori distribution of $\boldsymbol{h}(t)$ satisfies

where

and $\boldsymbol{h}(t) \sim \mathcal{C N}\left(\mathbf{0}, \boldsymbol{I}_{L}\right)$, with $\boldsymbol{R}=\boldsymbol{I}_{L}+\sigma_{\omega}^{2}\left(\boldsymbol{S}^{\mathrm{H}} \boldsymbol{S}\right)^{-1}$ being the covariance of $\hat{\boldsymbol{h}}(t)$ and $\alpha$ being the channel correlation coefficient in the AR(1) process. The proof is in Appendix Ⅱ in Fodor et al. (2021).

The SISO equalization coefficients can be obtained according to the channel a posteriori distribution. It is noted that we utilize LS channel estimation in this article, while other methods, such as MMSE and Kalman channel estimation, can also be used to compute the distribution as above (Zhe et al., 2018; Fodor et al., 2021).

3   Proposed turbo equalization for timevarying UWA channels

This section derives a turbo equalizer for time-varying UWA channels based on the JG criterion. The structure of the turbo equalizer shown in Figure 2 is mainly divided into three parts.

Figure  2  Structure of the proposed turbo equalizer. Π and Π⁻¹ denote interleaving and deinterleaving, respectively

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• SISO equalization. On the basis of the channel a posteriori distribution obtained from the observations at two moments, the SISO equalization coefficients are computed.

• Data-aided channel re-estimation. The channels at two moments are re-estimated using soft decision symbols. The re-estimated channels are then combined with the previously estimated channels to obtain the channel estimation with minimum Bayesian MSE (BMSE).

• A posteriori distribution update. The channel a posteriori distribution is updated according to the re-estimated channels, and the SISO equalization coefficients are recomputed based on the updated distribution.

3.1   SISO equalization conditioned on channel estimation

We assume that the UWA channel, the transmitted acousic signal, and the noise are uncorrelated. Considering the impact of imperfect CSI on the received signal, the mean of $\boldsymbol{y}$ conditioned on the channel estimates $\hat{\boldsymbol{h}}^{(i)}(t)$ and $\hat{\boldsymbol{h}}^{(i)}(t-1)$ in the ith iteration is

where $\tilde{\boldsymbol{H}} \in \mathbb{C}^{N \times N}$ is a circulant matrix with the first column being $\left[\left(\boldsymbol{E}^{(i)} \boldsymbol{\zeta}^{(i)}(t)\right)^{\mathrm{T}}, 0, \cdots, 0\right]^{\mathrm{T}}$ and $\tilde{\boldsymbol{h}}_{n}$ is the $n$th column of $\tilde{\boldsymbol{H}}$.

Because $\boldsymbol{H}(t) \boldsymbol{H}(t)^{\mathrm{H}}=\sum \limits_{k=0}^{N-1} \boldsymbol{h}_{k}(t) \boldsymbol{h}_{k}(t)^{\mathrm{H}}$, the covariance of $\boldsymbol{y}$ conditioned on the estimated channels is

where $\tilde{\boldsymbol{Z}}_{k}=\operatorname{Cov}\left(\boldsymbol{h}_{k}(t), \boldsymbol{h}_{k}(t) \mid \hat{\boldsymbol{h}}^{(i)}(t), \hat{\boldsymbol{h}}^{(i)}(t-1)\right)$.

According to Eq. (7), the LLR $\lambda_{n}$ of the QPSK-modulated symbol from the SISO equalizer can be written as

Using the Woodbury matrix identity, we have

with $\boldsymbol{G} \triangleq v \tilde{\boldsymbol{H}} \tilde{\boldsymbol{H}}^{\mathrm{H}}+\left(v \sum \limits_{k=0, k \neq n}^{N-1} \boldsymbol{Z}_{k}+\sigma_{\omega}^{2} \boldsymbol{I}_{N}\right)$. The LLRs $\lambda_{n, 0}$ and $\lambda_{n, 1}$ are the real and imaginary parts of $\lambda_{n}$, respectively.

3.2   Data-aided channel re-estimation

The LLR $\lambda_{n, j}$ is submitted into the SISO decoder to obtain the soft decision symbols $\mathbb{E}^{(i)}(\boldsymbol{x})$, which can be used to re-estimate the UWA channels $\boldsymbol{h}(t-1)$ and $\boldsymbol{h}(t)$. We have

Similar to Zhu et al. (2016), we aim to enhance channel estimation by utilizing the CSI from the previous iteration. Then, we define

and the re-estimated channels satisfy the distributions of $\hat{\boldsymbol{h}}^{(i)}(t) \sim \mathcal{C N}\left(\mathbf{0}, \boldsymbol{R}^{(i)}(t)\right)$ and $\hat{\boldsymbol{h}}^{(i-1)}(t) \sim \mathcal{C N}\left(\mathbf{0}, \boldsymbol{R}^{(i-1)}(t)\right)$. We combine the channels $\hat{\boldsymbol{h}}^{(i)}(t)$ and $\hat{\boldsymbol{h}}^{(i-1)}(t)$ to obtain a channel estimate $\hat{\boldsymbol{h}}^{B}(t)$ with minimum BMSE (Kay, 1993). It can be shown as

where

Similarly, $\hat{\boldsymbol{h}}^{B}(t-1)$ is the combination of $\hat{\boldsymbol{h}}^{(i)}(t-1)$ and $\hat{\boldsymbol{h}}^{(i-1)}(t-1)$.

3.3   A posteriori distribution update for the re-estimated channels

Autocorrection of the re-estimated channel $\hat{\boldsymbol{h}}^{B}(t)$ is

The cross-correlation of $\hat{\boldsymbol{h}}^{B}(t)$ and $\boldsymbol{h}(t)$ is given by

According to Eqs. (32) and (33), we can update the a posteriori distribution in Eq. (17) as

where

The equalization coefficients in the next iteration can be updated by substituting Eq. (34) into Eq. (24). The values of $\boldsymbol{E}^{(0)}, \boldsymbol{\zeta}^{(0)}(t)$, and $\boldsymbol{Z}^{(0)}$ are obtained from Eq. (17), and $\mathbb{E}^{(0)}(\boldsymbol{x})$ is set to $\mathbf{0}$.

4   Simulation results

We validated the proposed algorithm through a Monte Carlo simulation, utilizing the UWA channels and passband signal measured from a hydrophone mounted on the bow of an autonomous underwater vehicle (AUV) during the experiment in Songhua Lake. The depth of the transmitting transducer was 15 m, and the receiving depth of the AUV was 3 m. The AUV maintained a speed of 3 kn and navigated a course that was perpendicular to the axis of the transmitting transducer. The transmitted frame consisted of five blocks, each with a data length of N = 512. Channel estimations were performed by the pilot sequences with a length of P = 192. The simulation used parameters including a bandwidth of B = 4 kHz, a center frequency of f_c = 12 kHz, and a sampling frequency of f_s = 48 kHz. The transmitted signal was encoded using a rate-1/2 convolutional code and modulated by QPSK. The duration of each frame was 3 s. Figures 3(a) and 3(b) show the pseudocolor map and coherence function of the UWA channels, respectively. We used root-meansquare delay (Bharadwaj and Koul, 2018) to measure the multipath propagation and channel coherence time (Tong et al., 2023) to measure the Doppler effect, which were approximately 20 ms and 0.7 s, respectively.

Figure  3  Measured UWA channels from the AUV experiment in Songhua Lake

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We used the turbo equalizer for channel equalization due to the complex multipath structure of the channel. We considered the following five schemes for comparison, which all used data-aided channel re-estimation to track the UWA channels.

• Case Ⅰ: Conventional turbo equalization (Zheng et al., 2015) with LS channel estimation. We consider Case Ⅰ as the benchmark.

• Case Ⅱ: Conventional turbo equalization (Zheng et al., 2015) with SBL channel estimation.

• Case Ⅲ: Turbo equalization with LS channel estimation and CEE compensation (Li et al., 2016).

• Case Ⅳ: Turbo equalization with BCE (Yang et al., 2021). The AR(1) factor is α = 1.

• Case Ⅴ: Proposed algorithm. The AR(1) factor is α = 1.

Figure 4 compares the BER performance of the proposed algorithm (Case Ⅴ) against those of the other four schemes with iterations of 0, 1, and 5. The performance of Case Ⅰ is the worst in any iteration due to the CEEs caused by LSE and neglect of the sparsity of the UWA channel. Despite LS channel estimation, Case Ⅴ still outperforms Case Ⅱ. This superior performance can be attributed to the fact that the channel a posteriori distribution realized by the AR process and the imperfect CSI provide a higher channel estimation accuracy than SBL channel estimation. The reason for the BER performance gap between Case Ⅲ and Case Ⅴ is that Case Ⅲ does not consider the channel h(t − 1) at the previous moment. Due to the time-varying nature of the channel, the algorithms that consider the AR(1) process, namely Case Ⅳ and Case Ⅴ, perform better than the algorithms that assume that the channel is time-invariant within a block. Although Case Ⅳ uses forward and backward AR processes, its BER is higher than that of Case Ⅴ. For example, at an SNR of 8 dB, the BER of Case Ⅴ is 6.6 × 10⁻⁴ in the fifth iteration, while that of Case Ⅳ is 4.3 × 10⁻³. This is because Case Ⅴ considers the imperfect CSI at moments t and (t − 1), resulting in a more accurate channel a posteriori distribution.

Figure  4  BER performance of the five schemes with different iterations

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Figure 5 shows the BER performance of BCE-Turbo (Case Ⅳ) and the proposed algorithm under different AR(1) factors α in the fifth iteration. The proposed algorithm outperforms BCE-Turbo under any value of α. Moreover, the performance of the proposed algorithm is relatively robust as its BER performance exhibits a minimal variation with changes in α. For example, at a BER of 10⁻³, the maximum gain difference for the proposed algorithm is 0.014 dB, while that for BCE-Turbo is 0.346 dB. Therefore, the proposed algorithm is not sensitive to the selection of α.

Figure  5  BER performance under different α values in the fifth iteration

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5   Computational complexity analysis

We use the method of calculating the floating-point operations (flops) per iteration to assess the computational complexity (Yin et al., 2024). Each flop represents a real addition or multiplication, among other operations. Specifically, a complex addition involves two flops, while a complex multiplication requires six flops. We focus on the computational complexity of the algorithms with similar structures, such as Case Ⅰ, Case Ⅲ, and Case Ⅴ.

5.1   SISO equalization

For Case Ⅰ, the computational complexity primarily arises from calculating $\boldsymbol{\varSigma}$ and its inverse matrix during SISO equalization. The complexity of computing $\boldsymbol{\varSigma}$ mainly exists in calculating $\boldsymbol{H} \boldsymbol{H}^{\mathrm{H}}$. For the nondiagonal elements of $\boldsymbol{H} \boldsymbol{H}^{\mathrm{H}}$, it requires $N^2(N-1)$ complex additions and multiplications, thus needing $N^2(N-1)$ flops. In contrast, for its diagonal elements, because the multiplied elements are conjugates, only 4N² flops are needed. Computing $\boldsymbol{\varSigma}^{-1}$ requires $4 N^3$ flops.

The difference in SISO equalization between Case Ⅰ and Case Ⅲ mainly exists in the composition of the channel matrix $\boldsymbol{H}$, but the overall computational complexity remains the same (Li et al., 2016). Case Ⅴ involves additional computations of $v \sum \limits_{k=0, k \neq n}^{N-1} \tilde{\boldsymbol{Z}}_k$ compared to Case Ⅰ, which requires N² flops because $\tilde{\boldsymbol{Z}}_k$ is a real matrix.

5.2   Initial channel estimation

In the initial iteration of Case Ⅰ, the computational complexity of the LS channel estimation primarily lies in calculating the pseudoinverse of the matrix $\boldsymbol{S}$, requiring $16 L^2 P- 4 P L-12 L^3+20 L^2-4 L$ flops. On the basis of LSE, Case Ⅲ also utilizes the estimated channel to compute the channel a posteriori distribution $p(\boldsymbol{h}(t) \mid \hat{\boldsymbol{h}}(t))$, where the main computational complexity exists in calculating $\boldsymbol{R}$, needing $8 P L^2+12 L^2-4 P L-3 L$ flops. Furthermore, computing the channel a posteriori distribution $p(\boldsymbol{h}(t) \mid \hat{\boldsymbol{h}}(t), \hat{\boldsymbol{h}}(t-1))$ requires $8 P L^2+40 L^3+20 L^2-4 P L-3 L$ flops, as Case Ⅴ considers the channel from the previous moment.

5.3   Channel update

When using the soft information output from the SISO decoder for channel updating, Cases Ⅰ and Ⅲ require only one LSE, and the order of magnitude of flops for calculating $\left(\boldsymbol{S}^{(i)}(t)\right)^{\dagger}$ changes to $\mathcal{O}\left(N L^2+P L^3+L^3\right)$, while Case Ⅴ requires two LSEs. Additionally, in Case Ⅴ, the complexity of calculatin $\hat{\boldsymbol{h}}^B(t)$ primarily resides in computing $\boldsymbol{R}^{(i)}(t)$, with the order of magnitude of flops also being $\mathcal{O}\left(N L^2+\right. \left.P L^3+L^3\right)$. When updating the channel a posteriori distribution, computing $\boldsymbol{R}^B(t)$ requires $12 L^3$ flops, and the number of flops for calculating the channel a posteriori mean and variance remains the same as in the initial channel estimation.

Table 1 shows the final computational complexity of Cases Ⅰ, Ⅲ, and Ⅴ. Figure 6 illustrates the relationship between the flops, block length N, pilot sequence length P, and channel length L. The black circles in the diagram indicate the number of flops required under the parameter settings in Section 4. From the diagram, it is evident that the performance improvement of Case Ⅴ over Case Ⅰ and Case Ⅲ comes at the cost of increased computational complexity. Nevertheless, the computational complexity of the three schemes is of the same order of magnitude. Therefore, the increase in computational complexity caused by the proposed algorithm is acceptable.

Figure  6  Relationship between the flops, block length, pilot sequence length, and channel length

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6   Conclusions

Aturbo equalization algorithm for time-varying UWA channels was proposed. First, we derived the turbo equalizer based on the JG criterion. Subsequently, by leveraging imperfect CSI combined with an AR(1) model, a more precise channel a posteriori distribution is obtained. The distribution is subsequently utilized in the design of the turbo equalizer, performing a more effective mitigation of the multipath propagation and Doppler effect in UWA channels. Simulation results show that: 1) the BER of the proposed algorithm is at least one order of magnitude lower than those of other methods when the SNR exceeds 8 dB after five iterations during the AUV mobile communication. 2) the proposed algorithm has a robust performance with changes in the AR(1) factor, while the performance of the comparison method using AR process will significantly change. 3) The proposed algorithm has increased the acceptable computational complexity compared to other methods.