1. Introduction
Underwater acoustic (UWA) communication systems have become a critical component to meet the rising demand for marine exploration and commercial activities. These systems facilitate a diverse set of applications, including oceanographic research, offshore oil and gas exploration, and the Internet of Underwater Things [
1,
2,
3]. The UWA wave is often regarded as one of the most demanding communication mediums as it is characterized by limited bandwidth, significant multipath spread, fast fading, and intricate oceanic noise [
4,
5].
Orthogonal frequency division multiplexing (OFDM) for UWA communication has gained substantial attention due to its high spectral efficiency and resistance to long multipath spread in UWA channels [
6]. The basic idea behind OFDM is to split the channel bandwidth into evenly spaced subchannels in the frequency domain. Given the challenges posed by frequency-selective fading and substantial noise fluctuations in certain UWA environments, accurate channel estimation is crucial for a UWA OFDM communication system. Traditional channel estimation methods, such as the least squares (LS) estimation algorithm, suffer from poor performance in low signal-to-noise ratio (SNR) conditions, as the estimated mean square error (MSE) is inversely proportional to the SNR. In comparison to LS estimation, minimum mean square error (MMSE) estimation utilizing second-order statistics of the channel can achieve higher estimation accuracy. However, it requires prior knowledge of noise and channel statistics, and its complexity is much higher than that of LS estimation.
In recent years, machine learning, especially deep learning (DL), has rapidly progressed and been widely applied in various fields, offering new solutions to the challenges in UWA communication [
7,
8,
9]. Zhang et al. [
10] developed a UWA OFDM communication receiver system based on a five-layer fully connected deep neural network (DNN) for UWA channel estimation and equalization. The model demonstrated advantages over traditional algorithms, especially in scenarios with limited pilot subcarriers in OFDM communication. In a subsequent work, Zhang et al. [
11] introduced a UWA OFDM communication receiver system utilizing a combination of a convolutional neural network (CNN) for feature extraction and multilayer perceptron (MLP) for data symbol recovery. This model outperforms traditional methods and fully connected DNN-based UWA OFDM frameworks. Liu et al. [
12] proposed a CNN-based UWA OFDM receiver system that effectively integrates channel estimation and equalization. The design features an encoder and decoder structure with convolutional layers for feature extraction and signal reconstruction, which reduces network complexity. Qiao et al. [
13] introduced CsiPreNet, a learning model comprising a one-dimensional CNN and long short-term memory (LSTM) network. This model captures the temporal and spectral characteristics of UWA channel state information (CSI) and outperforms existing recursive least square (RLS) predictors. Ouyang et al. [
14] modified a super-resolution neural network to address the channel estimation problem, resulting in the channel super-resolution network (CSRNet). Simulation results showed superior performance compared to LS. Liu et al. [
15] introduced a method for UWA channel estimation based on a denoising sparsity-aware DNN (DeSA-DNN). Their approach uses DNN to simulate the iterative process of classical sparse reconstruction algorithms, leveraging the sparsity of UWA channels. It incorporates an effective denoising module using CNN to mitigate the impact of interference on channel estimation.
In summary, CNN-based methods offer simplicity and flexibility in adapting to the characteristics of UWA channels. However, existing CNN-based channel estimation approaches often focus on single data-block estimations and lack consideration of the temporal correlation within the channel. Moreover, these methods are typically trained within specific SNR ranges, while UWA channels often exhibit significant SNR fluctuations. This can potentially lead to overfitting within the training SNR range and a decrease in performance outside of it.
In this paper, a robust underwater acoustic channel estimation method based on a bias-free CNN is introduced. The main contributions of this research can be summarized as follows:
We incorporate the “bias-free” concept [
16] into denoising convolutional neural network (DnCNN) enhances the stability of the model performance and aims to overcome overfitting the training SNR conditions. And through theoretical justification and framework customization, we develop a specialized neural network for channel estimation known as bias-free complex DnCNN (BF-CDN).
Utilizing the temporal correlation of the channel over a certain time period, the input to the model consists of the coarse channel estimation results of data blocks received within a certain time segment. This results in further improvement and robustness in estimation performance.
Simulations and real sea experimental data results confirm the robustness of the method under different noise conditions and highlight its potential to improve the accuracy and reliability of channel estimation.
The rest of this paper is structured as follows:
Section 2 provides a brief overview of the UWA-OFDM system model.
Section 3 introduces the proposed BF-CDN model for channel estimation and covers the problem transformation, theoretical explanation, and model architecture design.
Section 4 presents simulation and experimental results and provides an analysis of these findings. Finally, in
Section 5, we conclude the paper.
2. UWA-OFDM System Model
Assume that
represents the UWA channel
The UWA channel is assumed to be approximated by N dominant discrete paths, where each path is associated with a complex gain and time delay at the i-th discrete sample time. The formation of multipath in UWA channels is a result of various factors, including reflection on the water’s surface, seafloor, and object surfaces as well as refraction in water. These factors collectively contribute to the time-varying nature of UWA channels.
We consider a cyclic prefix (CP) OFDM baseband system in this context. The block diagram of the system is shown in
Figure 1. After passing through the channel, the signal obtained by the receiver can be expressed as
where ⊗ represents the circular convolution, and
and
denote the transmitted signal and additive noise, respectively. After removing the CP, the received signal in the frequency domain can be obtained by DFT transformation:
where
is the diagonal matrix of the transmitted symbols,
is the corresponding Fourier transform matrix,
is the time domain channel, and
and
denote the frequency domains of
and
, respectively.
5. Conclusions
In this paper, a robust UWA channel estimation method based on the bias-free CNN is introduced. Initially, the LS channel estimation results are employed as a channel affected by noise, and a denoising neural network is applied to achieve accurate channel estimation. Subsequently, the bias-free concept is introduced, and its necessity is theoretically explained and validated through simulations. Then, modifications are made to the model to adapt it for channel estimation. Compared to the DnCNN network, the proposed method exhibits superior robustness for noise fluctuations: even those not encountered during training. Furthermore, simulations confirm that joint estimation further improves the performance and robustness compared to individual estimation by leveraging the temporal correlation of the channel.
Finally, a performance comparison is conducted between the proposed method and classical methods such as LS, OMP, and SOMP. The simulation results show that the method outperforms these classical methods across different SNR levels. At an SNR of 15 dB, the MSE performance improvement is 13.8 dB, 7.9 dB, and 7.3 dB, respectively. Real sea trail data processing further validates the superior performance of the proposed method. In future work, exploration will be conducted on the impact of implementing the bias-free technique within various CNN architectures on channel estimation. The objective is to design and refine models for enhanced performance.