Model-Driven Deep-Learning-Based Underwater Acoustic OTFS Channel Estimation

Abstract

Accurate channel estimation is the fundamental requirement for recovering underwater acoustic orthogonal time–frequency space (OTFS) modulation signals. As the Doppler effect in the underwater acoustic channel is much more severe than that in the radio channel, the channel information usually cannot strictly meet the compressed sensing sparsity assumption in the orthogonal matching pursuit channel estimation algorithm. This deviation ultimately leads to a degradation in system performance. This paper proposes a novel approach for OTFS channel estimation in underwater acoustic communications, utilizing a model-driven deep learning technique. Our method incorporates a residual neural network into the OTFS channel estimation process. Specifically, the orthogonal matching pursuit algorithm and denoising convolutional neural network (DnCNN) collaborate to perform channel estimation. The cascaded DnCNN denoises the preliminary channel estimation results generated by the orthogonal matching pursuit algorithm for more accurate OTFS channel estimation results. The use of a lightweight DnCNN network with a single residual block reduces computational complexity while still preserving the accuracy of the neural network. Through extensive evaluations conducted on simulated and experimental underwater acoustic channels, the outcomes demonstrate that our proposed method outperforms traditional threshold-based and orthogonal matching pursuit channel estimation techniques, achieves superior accuracy in channel estimation, and significantly reduces the system’s bit error rate.

1. Introduction

Underwater acoustic (UWA) communications have been widely applied in various domains, including marine environmental monitoring [1], resource exploration [2], underwater target localization [3], etc. However, its practical implementation is constrained by characteristics of UWA communication channels [4]. The UWA communication channels exhibit multipath delay and Doppler shift characteristics that are significantly greater in magnitude than radio communication channels, leading to severe time-selective fading and frequency-selective fading [5]. In UWA communication, orthogonal frequency division multiplexing (OFDM) has the advantages of high spectrum utilization and robustness against frequency selective fading [6]. However, the severe Doppler shift in the UWA channel will harm the orthogonality of OFDM subcarriers, thereby degrading the communication performance. Different from the time–frequency (TF)-domain-modulated OFDM, orthogonal time–frequency space (OTFS) communication modulates the signals in the delay-Doppler (DD) domain. The DD domain channel that corresponds to the TF dual-selective fading channel does not exhibit significant selective fading characteristics in the DD domain. Therefore, the symbols within a DD domain frame experience nearly time-invariant fading [7].

In OTFS channel estimation, an exploratory work [8] proposes a threshold-based channel estimation method. The pilot symbols and guard symbols are embedded in the DD domain OTFS frame. The receiver employs a threshold criterion to estimate a finite number of channel impulsive responses through the least squares (LS) algorithm. The threshold-based channel estimation method often exhibits limited performance in low SNR channels. To address this limitation, ref. [9] proposes an enhanced OTFS channel estimation method that employs a highly correlated pseudorandom noise sequence as the pilot. And it estimates three parameters, delay offset, Doppler offset, and channel fading coefficient for OTFS channel estimation. An OTFS channel estimation approach is proposed in [10], utilizing the orthogonal matching pursuit (OMP) algorithm with reduced requirements on the two-dimensional correlation of pilot sequences. Moreover, the channel sparsity in the DD domain is exploited within OMP to enhance estimation performance. The OMP channel estimation is further developed into three dimensions [11]. In addition to the DD domain, the angle domain has also been introduced. Leveraging the channel sparsity in this three-dimensional domain, a compressed sensing algorithm has been proposed to effectively address the challenge of recovering sparse signals in OTFS channel estimation based on prior knowledge of their sparsity conditions. The effective design of pilot and guard space for OTFS channel estimation is also studied. A superimposed pilot pattern is proposed [12], which superimposes one pilot symbol with one data symbol. This approach can address the low spectrum efficiency issue in embedded pilot patterns. A multiple superimposed pilots scheme is further proposed [13], which assumes constant delay and Doppler effects across consecutive OTFS frames.

Especially in the UWA channel, the channel information usually cannot strictly meet the sparsity assumptions. The UWA channel exhibits significant multipath propagation and Doppler effects. Upon transforming the channel from the time domain to the DD domain, the delay and Doppler grids of the channel undergo dispersion along their corresponding axes in the DD domain. Consequently, the channel distribution in the DD domain becomes less sparse. In this situation, the performance of typical OMP-based channel estimation methods will degrade. In communication systems, deep learning (DL) has the potential to extract hidden correlation features of signals and fit the complex nonlinear system for performance improvement. The DL-based communication can be mainly categorized into data-driven and model-driven methods.

Data-driven DL-based OFDM channel estimation [14,15] does not require prior information. However, its performance is highly dependent on the quality and quantity of the training data. In actual communication systems, obtaining high-quality datasets is often challenging [16,17]. In OTFS investigation, data-driven DL-based methods have been explored for signal detection [18,19,20]. In the classic block-based OTFS communication system, a two-dimensional convolutional neural network (CNN) is utilized for substituting the signal detection process [18]. In order to construct the dataset, the data augmentation technology is used to enhance the dataset for learning. The use of ResNet, DenseNet, and RDN for OTFS signal detection is proposed and compared in [19], respectively. The three network structures can mitigate the gradient explosion and gradient disappearance problems existing in FC-DNN and CNN, consequently yielding superior performance outcomes. In UWA OTFS, the authors have proposed a cascaded neural network structure for signal detection, consisting of a skip connection (SC)-CNN and bidirectional long short-term memory (BiLSTM) network [20]. Through the evaluation in both simulation and experimental UWA channels, the proposed neural network exhibits better performance compared with the classic linear-, nonlinear-, DNN-, and CNN-based signal detection methods.

Model-driven DL utilizes prior knowledge, which requires smaller datasets than data-driven DL-based methods. In wireless radio communication, model-driven DL-based channel estimation is investigated for MIMO based on a deep residual network [21] and YOLO network [22]. The neural networks can refine the preliminary channel matrix to be more accurate. The thought of image super-resolution can be also used for channel estimation, considering the estimated channel of pilots as a low-resolution image. With the preprocessing of interpolation, the proposed ChannelNet [23] can reconstruct the complete high-resolution channel impulse response (CIR) from the low-resolution CIR of pilots. Further, a enhanced super-resolution CNN-based channel estimation method, named SRDnNet, was introduced in [24]. The enhanced SRDnNet can directly learn from the input to determine the appropriate interpolation relationship, enabling the generation of a raw channel estimation matrix without interpolation. By employing an element-wise subtraction structure, SRDnNet leverages the denoising mechanism to recover the accurate channel coefficients from the raw channel matrix. However, the super-resolution-based methods are not suitable for UWA OTFS channel estimation due to channel sparsity.

With the aim of alleviating the performance degradation of OMP channel estimation in UWA OTFS, this paper proposes a model-driven DL-based UWA OTFS channel estimation method. Our main contributions are as follows:

• We address the design of system parameters for enabling effective UWA OTFS communication, considering the specific characteristics of the UWA channel. We discuss the configuration of subcarrier spacing, the number of subcarriers per symbol, and the number of symbols per frame in the delay-Doppler domain, taking into account the influence of multipath and Doppler effects.
• We propose a model-driven deep learning technique for UWA OTFS channel estimation. Considering the more pronounced Doppler effect in the UWA channel compared with the radio channel, the channel information often deviates from the compressed sensing sparsity assumption typically assumed in the classical OMP estimation algorithm. To address this issue, our method incorporates the OMP algorithm and denoising convolutional neural network (DnCNN) collaboration for channel estimation. The use of a lightweight DnCNN network with a single residual block reduces computational complexity while still preserving the accuracy of the neural network. The proposed method can obtain better channel estimation results by denoising the preliminary channel estimation.

The rest of this paper is organized as follows. Section 2 presents UWA OTFS system and typical channel estimation model. Section 3 proposes DnCNN-enhanced OMP channel estimation for UWA OTFS. Section 4 evaluates the performance of DnCNN-based channel estimation by both simulation and experimental data. Section 5 provides a discussion of the study and results. Section 6 draws a conclusion for our research.

2. UWA OTFS System Model

2.1. UWA OTFS System Model

In comparison with OFDM employing TF domain modulation, OTFS modulates information in the DD domain, which has the advantages of Doppler and delay resilience, reduced signaling latency, and reduced complexity of implementation [25]. Figure 1 illustrates the block diagram of a UWA OTFS system.

The transmitter constructs a DD grid as $\mathrm{\Lambda }=\left\{\right(l\Delta \tau ,k\upsilon ),l=0,...,M-1,k=0,...,N-1\}$, where $\Delta \tau$ is the grid interval in delay dimension, and $\Delta \upsilon$ is the grid interval in Doppler dimension. The number of grids in Doppler dimension is N, and the number of grids in delay dimension is M. The modulation module maps the one-dimensional constellation symbols $x=[x_1,...,x_{NM}]$ into two-dimensional transmission symbols using the specified modulation mode. The two-dimensional symbols are distributed across OTFS DD data grids. A pilot symbol is inserted in the two-dimensional symbols. And in the DD domain, the pilot is placed in the middle of the DD grids, while the guard intervals exist between the pilot symbol and the data frame for interference mitigation. Using inverse symplectic finite Fourier transform (ISFFT), the DD domain symbols can be converted into the TF domain as

$$X[n,m]=\frac{1}{\sqrt{MN}}\sum _{k=0}^{N-1}\sum _{l=0}^{M-1}x[k,l]e^{j2\pi (\frac{nk}{N}-\frac{ml}{M})},$$

(1)

where $m=0,1,2,...,M-1,$ and $n=0,1,2,...,N-1$. Next, a time–frequency modulator converts the samples $X[n,m]$ to a continuous time waveform $s\left(t\right)$ with a shaping waveform $g_{tx}\left(t\right)$ by Heisenberg transform as

$$x\left(t\right)=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}X[n,m]g_{tx}(t-nT)e^{j2\pi m\Delta f(t-\Delta f)},$$

(2)

where $\Delta f$ is the subcarrier spacing, $T=1/\Delta f$ is symbol duration, and $g_{tx}$ represents the transmit pulse shaping filter.

The channel can be represented by CIR in the DD domain, which is expressed as

$$h(\tau ,\upsilon )=\sum _{i=1}^p{h}_i\delta (\tau -{\tau }_i)\delta (\upsilon -{\upsilon }_i),$$

(3)

where $h_i$ is the channel coefficient of path i, ${\upsilon }_i$ is the frequency bias of path i, and ${\tau }_i$ is time delay of path i.

The received signal is represented by the transmitted signal propagating through the UWA channel. The channel is modeled by CIR and additive noise. Therefore, the signal at the receiver can be expressed as

$$r\left(t\right)=\sum _{i=1}^p{h}_i{e}^{j2\pi {\upsilon }_i(t-{\tau }_i)}x(t-{\tau }_i)+w\left(t\right).$$

(4)

At the receiver, the received time-domain signal undergoes a series of steps to be transformed into the delay-Doppler (DD) domain; then, channel estimation and signal detection processes are subsequently executed to recover the transmitted data. At the receiver, the received time-domain signal $r\left(t\right)$ first undergoes a transformation into a TF domain signal, using the Wigner transform as

$$Y(n,m)=\left[\int g_{rx}^{*}(t-\tau )r\left(t\right)e^{-j2\pi f(t-\tau )}dt\right],$$

(5)

where $\tau =nT$, $\upsilon =m\Delta f$. Then, the TF domain signal is transformed into DD domain using the symplectic finite Fourier transform (SFFT) as

$$y[k,l]=\frac{1}{\sqrt{MN}}\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}Y[n,m]e^{-j2\pi (\frac{nk}{N}-\frac{ml}{M})}.$$

(6)

Finally, channel estimation is performed using the pilot data, and signal detection is performed using the channel information to recover the transmitted signal $s[k,l]$.

2.2. Parameter Design of UWA OTFS Communication

In UWA communication, the multipath effect and Doppler shift are more severe than those in terrestrial radio communication.

Table 1. Comparison of terrestrial radio and UWA communications.

Parameters	Wireless Radio Communication	UWA Communication
Propagation speed c	$3\times {10}^8$ m/s	1500 m/s
Typical carrier frequency $f_c$	Several GHz	Several kHz
Subcarrier spacing $\Delta f$	Several KHz	Several Hz
CFO $f_D$, when $v_t=$ 1 m/s	Several Hz	Several Hz
Normalized CFO $\theta$, when $v_t=$ 1 m/s	≈1$\times {10}^{-3}$	≈1

shows a comparison of channel characteristics between terrestrial radio and UWA communications. The propagation speed of UWA waves is significantly slow: five orders of magnitude slower than the speed of radio waves. Moreover, the UWA channel experiences significant distance- and frequency-dependent attenuation, resulting in low frequency and limited bandwidth for long-range communication, typically in the KHz range. As a consequence, in the UWA channel, even minor movements can result in noticeable Doppler shifts.

In the time-varying UWA channel, assume that the maximum multipath delay is represented by ${\tau }_{max}$, and the maximum Doppler shift is denoted by ${\upsilon }_{max}$. The setting of OTFS parameters are influenced by the UWA channel conditions, particularly when considering the characteristics in the DD domain. In the Doppler axis, the maximum supportable Doppler is restricted by the symbol duration, denoted as ${\upsilon }_{max}<1/T$. In the delay axis, the maximum supportable multipath delay is restricted by $1/\Delta f$, as ${\tau }_{max}<1/\Delta f$. In UWA channels, the maximum multipath delay ${\tau }_{max}$ is as large as tens of milliseconds; so, the corresponding designing parameter, $\Delta f$, needs to be set at a small value. And the corresponding T is large as $T=1/\Delta f$.

An UWA OTFS system has a total bandwidth $B=M\Delta f$ and a frame duration $T_f=NT$ with $NM$ subcarriers per frame. Since $\Delta f$ is set to a small value, the setting of a large value of M will benefit broadband and high-data-rate communication. Considering the large value of T and the need of avoiding excessively long frame duration $T_f=NT$ for demodulation latency, the value of N should not be set to be too large. In order to ensure effective UWA OTFS communication, it is advisable to set a small value of N and a large value of M for efficient and reliable communication.

2.3. OMP Channel Estimation Algorithm

Regarding channel sparsity in the DD domain, the OTFS channel estimation problem can be formulated as a reconstruction problem involving sparse signals. In the signal reconstruction problem, each column of the perception matrix represents an atomic signal. The matching pursuit (MP) algorithm employs the linear operations with these atomic vectors to progressively approximate the observed signal. In the MP algorithm, the selected atoms are not always orthogonal to the residuals, which results in a decrease in convergence speed. The OMP algorithm addresses this issue by ensuring the selected atoms and the residual are always orthogonal, and the inner product of the selected atom and the residual in the next cycle is 0. OMP avoids the problem of repeated atom selection, thus guaranteeing the efficiency of the iteration process.

Algorithm 1 depicts the OMP-based OTFS channel estimation algorithm. In OMP-based channel estimation, the atoms are set as the input signal x, each selected atom is put in the matrix D as a reconstructed set, the observation vector is the received signal y, and the residual is r. By the compressed sensing technique, the channel matrix H with sparse characteristics is obtained by OMP algorithm.

Algorithm 1 OMP-Based OTFS Channel Estimation

1: Input: The observation vector y; perception signal x; Signal sparsity P; 2: Output: Channel Matrix H; 3: Initialization: the residuals $r_0=y$, the index set $A\in Ø$ and the reconstructed vector set $D\in Ø$; 4: for each t do 5:    Calculate the inner product of x and r; 6:    Find the optimal reconstruction vector in x with a maximum inner product and put it in the set D. Put the index of the optimal vector into A. 7:    Reconstruct the t-th estimated channel matrix vector $H={\left(D_t^T{D}_t\right)}^{-1}{D}_t^{T}y$ by LS algorithm; 8:    Update the residual $r_t=y-D_{t}H$; 9:    If $t>P$, stop. Else, continue. 10: end for 11: Obtain the final channel estimation result H.

3. DL-Based OTFS Channel Estimation

The OMP channel estimation algorithm relies on a sparse channel matrix. However, in UWA channels, the limited bandwidth and frame duration constraints lead to restricted resolution in the DD grid. Consequently, fractional Doppler shifts or delay shifts may span multiple lattices instead of aligning precisely with a single lattice. This phenomenon introduces channel expansion in the DD domain, making the UWA channel matrix nonsparse. As a result, the performance of the OMP channel estimation algorithm is degraded.

This paper incorporates a residual network [26], referred to as the DnCNN network, into OTFS channel estimation to alleviate the performance degradation of the OMP channel estimation algorithm with nonstrict sparse channel information. DnCNN and ResNet share a common feature of residual learning. However, DnCNN does not incorporate connections between various layers of the neural network. Instead, it directly modifies the output of the network learning process to represent residuals. This unique characteristic enables faster convergence of the entire network and reduces training time compared with ResNet.

Figure 2 illustrates the system structure of the proposed DnCNN-based OTFS channel estimation method. The pilot symbols in the received signal are input into the OMP channel estimation module, which generates raw channel estimation results. Then, the results are input into the DnCNN network for further processing. The DnCNN network for UWA OTFS channel estimation includes the following layers:

• Convolutional (Conv.) layer with Rectified Linear Unit (ReLU) activation: This initial layer is responsible for extracting the channel information features.
• Conv.+batch normalization(BN)+ReLU layer: This combination of layers forms a convolution block that focuses on signal reconstruction. The BN layer enhances the flow of gradients through the network, improving training speed and enhancing the network’s learning capabilities.
• Conv. layer: The final layer solely consists of a convolutional layer, which is utilized for information feature extraction.
• Residual block: After the last convolutional layer, the residual noise n is output. This residual noise is then processed by a residual block, and the channel estimation results ${\widehat{H}}_{DL}$ is obtained by the raw estimation and the residual noise.

The DnCNN network uses residual learning with skip connections to mitigate gradient disappearance and gradient explosion while denoising, thus achieving better performance than traditional CNNs. In DnCNN-based OTFS channel estimation, the DnCNN learns the correlation between the input raw estimated channel information and the residual noise using the residual block. As shown in Figure 2, the output channel estimation information ${\widehat{H}}_{DL}$ obtained after the residual block can be expressed as

$${\widehat{H}}_{DL}={\widehat{H}}_{OMP}-n,$$

(7)

where n is the trained residual noise, n and ${\widehat{H}}_{OMP}$ are the inputs of the residual block, and ${\widehat{H}}_{DL}$ is the output of the residual block. The residual block learns to minimize the difference between the input n and the output ${\widehat{H}}_{DL}$. So, the DnCNN learns more accurate channel information.

The DnCNN-based channel estimation algorithm is initially trained offline and then deployed online. In the offline training phase, the input of the DnCNN training is the raw channel estimation result, the training labels are the actual channel responses, and the objective of the training is to minimize the difference between the network output and the labels. During the DnCNN training process, the mean-square error (MSE) serves as the loss function, which can be denoted as

$$L=MSE(H,{\widehat{H}}_{DL})=\frac{1}{n_B}\sum _1^{n_B}{(H-{\widehat{H}}_{DL})}^2,$$

(8)

where $n_B$ is the batch size.

In the online deployment phase, a test set is generated, and the previously trained network model is loaded. The DnCNN network module is deployed on the receiver to output the estimated channel information ${\widehat{H}}_{DL}$.

The process of DnCNN-based OTFS channel estimation is shown in Algorithm 2.

Algorithm 2 DnCNN-Based OTFS Channel Estimation

1: Training process  2: Input: The observation vector y; perception signal x;  3: Output: ${\widehat{H}}_{DL}$;  4: Initialization: the residuals $r_0=y$, initialize DnCNN weights and bias randomly;  5: for each $Epoch\in \left[1,e\right]$ do  6:    for each $Iterations\in \left[1,i\right]$ do  7:      Using OMP algorithm to obtain raw channel information ${\widehat{H}}_{OMP}={\left(D_t^T{D}_t\right)}^{-1}{D}_t^{T}y$  8:      ${\widehat{H}}_{OMP}$ is input into the DnCNN network for training;  9:      Calculate loss function, and return the gradient; 10:     Update network parameters with certain learning rate and optimizer. 11:    end for 12: end for 13: Save the trained DnCNN with optimal weights and biases. 14: Testing process 15: Load the trained DnCNN; 16: The current estimated channel information ${\widehat{H}}_{OMP}$ is input to the loaded DnCNN model; 17: The channel estimation result is obtained according to Equation (7).

After continuous testing, the DnCNN network for UWA OTFS channel estimation is optimized.

Table 2. Parameters of DnCNN for UWA OTFS channel estimation.

Paraments	Input Size	Output Size
Conv + ReLU	(32, $3\times 3$)	(8, M, N)
Conv + BN + ReLU	(32, $3\times 3$)	(8, M, N)
Conv + BN + ReLU	(32, $3\times 3$)	(8, M, N)
Conv + BN + ReLU	(32, $3\times 3$)	(8, M, N)
Conv	(2, $3\times 3$)	(2, M, N)

lists the parameters of the DnCNN network for UWA OTFS channel estimation.

Both the first layer and the middle layers have 32 convolutional kernels, and the last layer has 2 convolutional kernels. All the convolutional kernel sizes in the DnCNN are set to $3\times 3$. The convolutional kernel size of $3\times 3$ has good performance, which is the size used in the paper that proposed DnCNN [27]. The setting is validated based on [28]. The same padding is used in the online DnCNN to ensure that the size of the convolution remains constant after convolution.

The learning rate is set to 0.001, the batch size is 50, and a total of 12 training rounds are performed. The parameters of the DnCNN are updated using the Root Mean Square propagation (RMSprop), which employs an exponentially weighted moving average to adjust the gradient accumulation, effectively discarding distant historical gradient information [29].

For DnCNN-based channel estimation, $30,000$ samples are generated under the DD channel. The data samples are divided into a training set, validation set, and test set in the ratio of 2:1:1. We can see that the model-driven DL-based method only takes a small number of data samples for efficiency learning.

4. Numerical Results

For the evaluation of UWA OTFS system performance, the OTFS frame size is set to $(N,M)$= (8, 64), signifying that each frame consists of 8 symbols and 64 subcarriers within the TF domain. The carrier frequency is set to 6 kHz. In the sea experiment, the maximum multipath delay is approximately 100 ms, which determines the setting of the subcarrier spacing to be $\Delta f$ = 10 Hz. Binary phase-shift keying (BPSK) is utilized to map symbol constellation.

4.1. System Performance under Simulated UWA Channel

The system performance is analyzed using the simulated UWA channel with the statistical channel model. The parameters of the simulated UWA channel are specified as presented in

Table 3. Parameters for simulation of UWA channel.

Paraments	Value
Number of multipaths	4, 8
Maximum multipath delay/${\tau }_{max}$	100 ms
Channel gain of multipaths/ $h_i$	Rayleigh distribution
Speed of sound/c	1500 m/s
Moving speed/${\upsilon }_{max}$	1, 2, 3 knots
Maximum Doppler spread/$f_D$	2, 4, 6 Hz
Doppler distribution	equal probability

. For the analysis of the Doppler shift’s impact on OTFS channel estimation in UWA channel, the following parameters are employed: The number of multipaths is set at 4 and 8, with a maximum multipath delay of 100 ms. These eight paths are randomly distributed within the maximum delay range, and their channel gains conform to independent Rayleigh distributions. The speed of sound, denoted as c, is fixed at 1500 m/s. To investigate the influence of Doppler shift, node movement speeds in water are set to 1, 2, and 3 knots, corresponding to maximum Doppler expansions of 2, 4, and 6 Hz. The Doppler coefficients for each path are generated uniformly from the range of $[-f_D,f_D]$ with equal probability.

Figure 3 shows the MSE performance under simulated UWA channel with four multipaths. The proposed DnCNN-based channel estimation outperforms the OMP channel estimation at various Doppler shifts. Compared with OMP channel estimation, the proposed DnCNN-based channel estimation has about 0.9 dB gain at 4 Hz Doppler shift and 1.7 dB gain at 6 Hz Doppler shift, at an MSE of $0.8\times {10}^{-3}$. The heuristic nature of OMP makes it difficult to find the optimal solution. The DnCNN-based method can overcome this limitation by data training fitting to obtain the final optimal solution. With the well-trained neural network, the proposed method can address non-Gaussian noise and improve the estimation accuracy by denoising.

Comparing the performance of different scales of Doppler shifts for OMP channel estimation, at an MSE of $0.8\times {10}^{-3}$, the SNR required for a 2 Hz Doppler shift is about 1.4 dB, which is 2.8 dB lower than that for 4 Hz and 6Hz Doppler shift, respectively. At the same MSE, for the proposed DnCNN-based OTFS channel estimation, the required SNR for a 2 Hz Doppler shift is about 1.3 dB, which is 1.9 dB lower than that for 4 Hz and 6 Hz Doppler shift, respectively. It can be seen that the DnCNN-based channel estimation with different Doppler shifts shows a narrower performance gap compared with the OMP channel estimation. In UWA channels, a larger Doppler shift can lead to channel expansion and introduce fractional delay, diminishing the sparsity of channel information in the Doppler domain. This poses a challenge for accurate channel estimation using the OMP algorithm, especially when the Doppler shift is large. The proposed DL-based method has better accuracy and robustness in various Doppler scales under complex UWA channels.

Figure 4 shows the MSE performance under simulated channel with eight multipaths. Similarly, the proposed DnCNN-based channel estimation outperforms the OMP channel estimation at various Doppler shifts. From the figure, we can observe the following: (1) Comparing with OMP channel estimation, the proposed DnCNN-based channel estimation has about 1.1 dB gain at 4 Hz Doppler shift and 3.5 dB gain at 6 Hz Doppler shift, at an MSE of $0.8\times {10}^{-3}$. (2) Comparing the performance of different scales of Doppler shifts for OMP channel estimation, at an MSE of $0.8\times {10}^{-3}$, the SNR required for a 2 Hz Doppler shift is about 1.8 dB, which is 4.8 dB lower than that for 4 Hz and 6 Hz Doppler shift, respectively. Comparing the effect of various Doppler for DnCNN-based channel estimation at the same MSE, the required SNR for a 2 Hz Doppler shift is about 1 dB, which is 1.7 dB lower than that for 4 Hz and 6 Hz. The deterioration of channel sparsity in the Doppler domain poses a challenge for the OMP algorithm to achieve accurate channel estimation, while our proposed method has better accuracy and robustness under complex UWA channels.

It can be seen from Figure 3 and Figure 4 that the channel estimation performance of the system with eight multipaths is worse than that of four multipaths, as the channel structure of eight multipaths is more complex than that of four multipaths. Even in this case, our proposed method shows better performance than the OMP channel estimation. At an MSE of $0.8\times {10}^{-3}$ and Doppler of 6 Hz, our proposed method has a 1.7 dB gain compared with the OMP under the channel of four multipaths. At the same MSE and Doppler, it has a 3.5 dB gain compared with the OMP under the channel of eight multipaths. With the number of multipaths increasing, the SNR gain of the proposed method becomes larger. Even if the channel becomes complex, the cascaded DnCNN in the proposed method can further denoise the preliminary channel estimation of the OMP algorithm and can finally achieve accurate channel estimation results.

4.2. System Performance under Experimental UWA Channel

The system validation utilized the UWA experimental channels from the sea trial dataset named WATERMARK [30], which is a benchmark dataset characterized by time-varying impulse response measurements at sea. The raw CIRs measured at Norway-Oslofjord (NOF) and Kauai 1 (KAU1) were employed.

Table 4. Parameters of channel dataset.

Paraments	NOF	KAU1
Environment	Fjord	Shelf
Range	750 m	1080 m
Water depth	10 m	100 m
Transmitter deployment	Bottom	Towed
Receiver deployment	Bottom	Suspended
Doppler coverage	7.8 Hz	7.8 Hz

lists the parameter settings of the two experimental channels.

Figure 5 and Figure 6 show the CIR of the NOF channel in both the time domain and DD domain. In Figure 5, the time-domain CIR of the NOF channel exhibits clear time-varying multipaths. Correspondingly, Figure 6 presents the DD domain CIR, where the Doppler shift of each path can be observed. The Doppler range spans from −4 to 4.

Figure 7 and Figure 8 show the CIR of the KAU1 channel in the time domain and DD domain. Compared with the NOF channel, the KAU1 channel exhibits a more complex CIR structure and pronounced variation in both the time and DD domain. In the DD domain, the maximum Doppler shift of KAU1 is larger than that of NOF. The larger Doppler shift in the KAU1 channel is attributed to the combined effects of tugboat motion and the movement of seawater.

In the UWA channel, the presence of Doppler effects is noticeable even when the transmitter and receiver are not moving fast. Multiple unique characteristics in the actual UWA environment can cause complex varying Doppler shifts, including seawater motion, distributed bubble layer, and inhomogeneous sea surface and seabed. The complex varying UWA channels are often difficult to model accurately, but the DL-based method has the potential to learn and capture the specific UWA channel characteristics and then fit a more effective UWA channel model.

Figure 9 and Figure 10 show the MSE of the three channel estimation methods applied to the NOF and KAU1 channels, respectively. For threshold-based OTFS channel estimation, the threshold is set to $3\sqrt{{\sigma }_p^2}$ according to the paper that proposed threshold-based OTFS channel estimation [8], where ${\sigma }_p^2$ denotes the effective noise power of the pilot signal. From the two figures, the MSE performance of the proposed DnCNN-based channel estimation method performs better than the OMP-based channel estimation method. And the OMP channel estimation is better than the threshold-based channel estimation method.

In Figure 9, the DnCNN-based channel estimation requires 5 dB lower SNR than the OMP-based channel estimation method at MSE of $0.65\times {10}^{-3}$. This significant improvement in SNR requirements highlights the superior performance of the DnCNN-based method in accurate channel estimation.

Similarly, in Figure 10, focusing on the more complex KAU1 channel, the DnCNN-based channel estimation method demonstrates enhanced performance, with approximately 1 dB lower SNR requirements compared with the OMP channel estimation method at the MSE of ${10}^{-3}$. The DnCNN-based method employs denoising through DnCNN with residual blocks for fine estimation. By leveraging a well-trained model, this method effectively addresses non-Gaussian errors and noise, enabling a closer approximation to the optimal solution and resulting in enhanced accuracy in channel estimation.

Comparing Figure 9 with Figure 10, the MSE performance of the DnCNN-based channel estimation method under the NOF channel is better than that under the KAU1 channel, as the KAU1 channel exhibits more complex CIR structure and significant channel variations than NOF.

Figure 11 and Figure 12 show the bit error rate (BER) of the uncoded OTFS system with various channel estimation and signal detection methods under NOF and KAU1 channels, respectively. The OTFS communication system using the DnCNN-based channel estimation method outperforms the system using the conventional OMP- and threshold-based channel estimation methods, regardless of whether zero-forcing (ZF) or linear minimum mean squared error (LMMSE) signal detection is employed.

As shown in Figure 11, in the NOF channel scenario, when employing the LMMSE signal detection algorithm, the proposed DnCNN-based channel estimation method has 0.5 dB gain over the OMP-based channel estimation method and 0.7 dB SNR gain over the threshold-based channel estimation method at a BER of $3\times {10}^{-3}$. When employing ZF as the signal detection method, the DnCNN-based channel estimation method still has better performance than the conventional methods.

As shown in Figure 12, in the KAU1 channel scenario, when employing LMMSE signal detection, the DnCNN-based channel estimation method has an SNR gain of about 1dB and 0.5 dB compared with the OMP- and threshold-based channel estimation methods at a BER of $3\times {10}^{-3}$.

In the UWA channel, the obvious channel expansion and the limited resolution of the DD grid result in fractional delay and Doppler frequency shift. The conventional OMP channel estimation algorithm, relying on compressed sensing, suffers from inaccurate estimation due to the degradation of channel information sparsity. In contrast, the proposed method incorporates a residual neural network to reconstruct and denoise the channel information, therefore enhancing the accuracy of the channel estimation and system reliability.

5. Discussion

This paper focuses on addressing the challenge of achieving accurate channel estimation in the UWA OTFS system operating in the presence of severe Doppler and multipath effects. To tackle this issue, this paper has developed a DL-based UWA OTFS channel estimation method, taking into account the distinctive attributes of the UWA OTFS channel within the DD domain. The proposed method incorporates a DnCNN-based approach that combines the OMP algorithm with DnCNN to achieve precise channel estimation and reliable communication. The results, findings, and implications of this research are as follows.

To enable reasonable UWA OTFS communication, this paper discusses the design of the system parameters according to the features of the UWA channel. UWA communication has more severe multipath and Doppler effects compared with radio communication. Multipath delays in UWA communication typically range from tens of milliseconds to a hundred of milliseconds. Doppler shifts in UWA communication can reach several Hz, even without any movement of the transmitter. Consequently, it is crucial to configure the parameters of UWA communication in accordance with the specific characteristics of the UWA channel. The parameters are designed considering both limitations imposed by the channel and the expected performance. Due to the large multipath delay and Doppler shift of the UWA channel, $\Delta f$ is only as small as several Hz, and the corresponding symbol duration T is tens of milliseconds. For high-data-rate $M\Delta f$, the value of M should be large. For not too large frame duration $T_f=NT$, N should be set smaller. To facilitate effective OTFS communication in the UWA channel, it is advisable to design the parameters by selecting a larger M value and a smaller N value.

In the UWA channel, the channel sparsity usually cannot strictly meet the sparsity assumption of compressed sensing in OMP channel estimation. To enhance the precision of UWA OTFS channel estimation, this paper proposes a model-driven DL-based UWA OTFS channel estimation method. In the proposed method, the OMP algorithm collaborates with DnCNN to estimate UWA channel, while the concatenated DnCNN denoises the preliminary channel estimation results generated by the OMP algorithm, thereby obtaining more accurate OTFS channel estimation results. The evaluation results using experimental sea channel, as illustrated in Figure 10 and Figure 11, demonstrate the superior performance of our proposed method compared with conventional OMP and threshold-based approaches. The findings reveal that the proposed method achieves higher channel estimation accuracy and significantly reduces the BER of the system.

This paper employs a DnCNN network with a residual block, which is a lightweight solution for OTFS channel estimation. When comparing neural networks with equivalent structure, the DnCNN network employed in this paper achieves lightweighting by reducing both the number of parameters and the number of floating-point operations. The proposed method employs approximately one-thirds fewer parameters and two-thirds fewer floating-point operations compared with the classical ResNet.

6. Conclusions

This paper proposes a DnCNN-based OTFS channel estimation method to address the issue of performance degradation in OMP channel estimation in the UWA channel with severe Doppler shift. The proposed method incorporates a residual network to alleviate the nonstrict sparsity problem of channel information in the OMP algorithm. The effectiveness of the proposed method is evaluated in both simulated and experimental UWA channels. The results demonstrate that the proposed DnCNN-based channel estimation method performs robustly with increasing Doppler shifts. Furthermore, the proposed method demonstrates superior MSE and BER performance compared with the conventional threshold-based and OMP channel estimation methods. These findings highlight the efficacy of the DnCNN-based approach in addressing the challenges associated with UWA channel estimation and improving overall system performance.