Channel Estimation and Iterative Decoding for Underwater Acoustic OTFS Communication Systems

Abstract

Orthogonal Time–Frequency Space (OTFS) is an innovative modulation method that ensures efficient and secure communication over a time-varying channel. This characteristic inspired us to integrate OTFS technology with underwater acoustic (UWA) communications to counteract the time-varying and overspread characteristics of UWA channels. However, implementing OTFS in UWA communications presents challenges related to overspread channels. To handle these challenges, we introduce a specialized OTFS system and offer frame design recommendations for UWA communications in this article. We propose a Doppler compensation method and a dual-domain joint channel estimation method to address the issues caused by severe Doppler effects in UWA communication. Additionally, we propose an OTFS system detection approach. This approach incorporates an iterative detection process which facilitates soft information exchange between a message passing (MP) detector and a low-density parity check (LDPC) decoder. By conducting simulations, we demonstrate that the proposed UWA OTFS system significantly outperforms Orthogonal Frequency-Division Multiplexing (OFDM), Initial Estimate Iterative Decoding Feedback (IE-IDF-MRC), and two-dimensional Passive Time Reversal Decision Feedback Equalization (2D-PTR-DFE) in UWA channels.

1. Introduction

Integrated sensing technology for space–air–ground–ocean (SAGO) contexts is a key research focus for next-generation mobile communications [1]. Underwater communication technology plays a crucial role in SAGO integrated sensing and information transmission systems. Acoustic waves are the sole effective medium for long-range underwater information transmission. However, the underwater acoustic (UWA) channel has numerous drawbacks, including limited bandwidths, significant delay spreads, frequency-selective fading, complex time variability, and susceptibility to the Doppler effect [2,3].

Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier modulation technology in which the frequencies of multiplexed subcarriers are orthogonal to each other. This orthogonality ensures that the peak value of each subcarrier aligns with the zero value of other subcarriers. Consequently, even if subcarriers overlap in the spectrum, they do not interfere with each other at the peak of the desired subcarrier. Moreover, OFDM systems avoid bandwidth efficiency loss, which is common in systems using non-orthogonal carrier sets. Compared to earlier systems, OFDM significantly enhances the spectral efficiency of UWA communication systems. Its low implementation complexity in terms of modulation and demodulation has led to its widespread adoption in high-speed UWA communication and network systems [4,5,6,7,8,9,10,11,12].

However, in high-speed motion scenarios, rapid movement result in Doppler frequency shifts and expansions within the communication system [13]. Additionally, the carrier frequency offset (CFO) present in received signals can impair the orthogonality among subcarriers in OFDM systems, resulting in inter-carrier interference (ICI) which severely compromises system performance [14]. Existing approaches aim to mitigate this phenomenon through shortening the OFDM symbol duration to reduce channel variations within each symbol. However, this method’s primary drawback is reduced spectral efficiency due to symbol duration shortening [15]. Consequently, OFDM modulation is particularly vulnerable to Doppler spread when employed in UWA communication networks, resulting in severe system performance degradation. Addressing efficient data transmission in dynamic and mobile UWA communication scenarios is thus a critical challenge. Enhancing the spectral efficiency in UWA communication networks—particularly in high-Doppler and high-delay UWA channels—has emerged as a primary research focus for future advancements in UWA communication technology.

In 2017, Hadani et al. introduced a novel two-dimensional modulation technology termed Orthogonal Time–Frequency Space (OTFS) [16]. OTFS modulation presents a significant advancement in addressing the challenges posed by severe Doppler effects and channel impairments in high-speed mobile communication systems. Unlike traditional OFDM modulation in the time–frequency (TF) domain, OTFS operates in the delay-Doppler (DD) domain for information modulation. The DD domain exhibits robust tolerance to both time delays and Doppler shifts, offering the potential for full diversity—a key factor supporting reliable communication. Furthermore, OTFS modulation effectively transforms a time-varying fading channel into a stationary and non-fading channel in the DD domain, thereby maintaining nearly constant channel gains across all modulation symbols [16].

OTFS can be conceptualized either as an alternative multi-carrier modulation scheme or as a two-dimensional spread spectrum technique applied to QAM symbols, which can subsequently be used with OFDM or other multi-carrier modulation schemes such as Generalized Frequency Division Multiplexing (GFDM) for modulation. The integration of OTFS technology in UWA communication systems addresses the performance degradation observed in OFDM technology when operating in complex and rapidly changing UWA channels. However, as most existing UWA communication systems are designed for low mobility and low carrier frequency scenarios, OTFS technology introduces new challenges when designing the architectures of and algorithms for transmitters and receivers.

To fully leverage the potential of OTFS technology, several critical issues need to be addressed, including waveform design, channel estimation, detection techniques, and multi-antenna and multi-user designs. Solving these challenges is imperative for realizing the benefits of OTFS technology in UWA communication systems.

To date, research on OTFS modulation technology has primarily been concentrated in the domain of wireless communication, and it remains at an early stage of development. Moreover, there is a paucity of studies exploring the application of OTFS in the field of UWA communication [17,18,19,20,21,22,23]. Nevertheless, some scholars have begun investigating its applicability in UWA communication through leveraging insights from wireless communication research, yielding promising preliminary results. These efforts have primarily been centered on reducing receiver complexity, enhancing channel estimation, and improving signal detection; the research has largely been confined to simulation analyses and field experiments, with no significant progress towards engineering implementation or real-world application.

As an emerging modulation technique, OTFS technology has demonstrated significant potential, warranting further investigation to enhance the performance of UWA communication systems. Although some studies have explored the application of OTFS in UWA communication, the body of literature remains limited and lacks diversity in research directions. The challenge of effectively integrating the performance advantages of OTFS technology in the context of wireless communication into the UWA communication domain remains an area requiring more comprehensive research. This study aims to further combine the OTFS method with UWA communication in order to design OTFS systems based on the unique characteristics of UWA channels. The main contributions of this study are as follows:

•

For the OTFS system transmitter, frame structures tailored to the characteristics of UWA channels are designed by selecting a zero-padding (ZP) frame structure design method. Additionally, the size of the DD domain grid and the length of the guard interval are optimized to enhance the communication efficiency while maintaining performance. The proposed OTFS frame design can significantly reduce the bit error rate (BER) while maintaining the same data rate through careful frame optimization.

•

For the OTFS system receiver, we propose a Doppler compensation method and a dual-domain joint channel estimation method to tackle the issues caused by the strong Doppler effect in UWA communication. Additionally, we propose an OTFS system detection approach based on an iterative decoding feedback-message passing detector (IDF-MP). Based on careful consideration of the characteristics of UWA channels, the proposed OTFS system receiver is shown to provide improvements, when compared with the existing methods of OFDM, Initial Estimate Iterative Decoding Feedback (IE-IDF-MRC) and two-dimensional Passive Time Reversal Decision Feedback Equalization (2D-PTR-DFE), in UWA channels.

The remainder of this paper is organized as follows: Section 2 introduces the related work. Section 3 describes the UWA OTFS system, including OTFS modulation and the UWA OTFS channel model. In Section 4, we propose a comprehensive OTFS system design scheme based on the characteristics of UWA channels. In Section 5, we analyze the communication performance of the proposed UWA OTFS system using simulated UWA channels. Finally, Section 6 presents the concluding remarks and discusses directions for further research.

2. Related Work

To date, research on OTFS modulation technology has primarily focused on the realm of radio communication, and it remains in a nascent stage at present [15,16,17,24,25,26,27,28]. In recent years, numerous enhancements to the original OTFS modulation scheme have been proposed by researchers. For example, Raviteja et al. [15,16,24,25] established the mathematical framework of OTFS signals and derived the modulation and demodulation input–output equations in the DD domain. In [16], an OTFS modulation model for multi-channel dual dispersion channels was established, accounting for time delays within the maximum delay and limited Doppler shift range. However, the original OTFS input–output relationship assumes ideal pulse shaping waveforms to be biorthogonal in both time and frequency, which is unattainable due to the Heisenberg uncertainty principle [16]. To address this, the authors of [24] derived the input–output relationship for actual pulse-shaping waveforms based on the DD domain, extending it to any waveform. Moreover, in [15], it was demonstrated that using the message passing (MP) algorithm for symbol detection in OTFS results in a lower symbol error probability (SEP) compared to OFDM. Building on this, the authors of [24] proposed a scheme that adds a cyclic prefix (CP) only at the end of the OTFS frame, significantly reducing the overhead caused by the loss of biorthogonality of pulse shaping waveforms. Additionally, in [25], a channel estimation method utilizing embedded pilot symbols was introduced, showing that OTFS with non-ideal channel estimation outperforms OFDM with ideal channel estimation in terms of error performance. Further research has delved into the sparsity of the DD domain in OTFS systems [26,27,28]. For example, Tharaj Thaj et al. proposed a low-complexity iterative decision feedback equalizer (DFE) based on maximum ratio combining (MRC), demonstrating its favorable performance and complexity compared to state-of-the-art message passing detectors [17]. These advancements represent significant progress in enhancing OTFS technology, paving the way for its broader application in communication systems.

Compared with the field of terrestrial wireless communication, research on OTFS modulation technology in the field of UWA communication remains relatively limited [17,18,19,20,21,22,23]. Some scholars have explored its applicability in UWA communication through leveraging research findings from wireless communication and have achieved notable results, primarily focusing on receiver complexity reduction, channel estimation, and signal detection. Researchers at Xiamen University have proposed a single-input single-output (SISO) UWA communication scheme based on OTFS [19]. Simulation results comparing OTFS and OFDM in UWA channel environments in terms of bit error rate (BER), spectral efficiency, and peak-to-average power ratio (PAPR) indicated the superior performance of the OTFS-based UWA communication scheme in time-varying multipath UWA channels, compared to OFDM and discrete Fourier transform spread (DFTS) OFDM schemes. In 2019, Bocus et al. from the University of Bristol, U.K., introduced an OTFS system based on large-scale multiple-input multiple-output (MIMO) OFDM to investigate multi-user UWA communication schemes [20]. In the studies [19,20], OTFS-UWA communications in SISO and MIMO scenarios were explored, respectively, with simulation results demonstrating that OTFS outperforms OFDM. However, these studies assume that accurate channel information is known to the receiver. Researchers from Dalian University of Technology have proposed a time-reversal OTFS receiver with adaptive channel estimation [21] and evaluated its system performance in both analog and measured channels. Their results indicated that the proposed two-dimensional passive time reversal (PTR) receiver achieves improved system performance with low complexity. Su Hang et al. have designed a practical UWA OTFS system, incorporating Doppler compensation and channel estimation in DD domain [22]. Additionally, Wei Li et al. have discussed input–output relationships for OTFS under overspread channels while providing a frame design method tailored to the characteristics of UWA channels [23].

In summary, an increasing number of scholars have studied the performance of OTFS in UWA communication. However, most of these studies concentrate on theoretical applications in radio communications, often assuming that the channel is underspread, which is typically not the case for UWA channels that are frequently overspread. Moreover, most UWA OTFS system designs assume perfect channel estimation or only consider the influence of integer Doppler factors. While theoretical investigations into fractional Doppler shifts have been conducted, practical implementation remains challenging. Neglecting fractional Doppler shifts in UWA channels can lead to non-negligible inter-Doppler interference (IDI), severely compromising performance, particularly in UWA communication systems characterized by significant Doppler effects. Therefore, developing a practical OTFS design that is suitable for real-world UWA communication scenarios is imperative. Filling this gap was a key motivation of this work.

3. Underwater Acoustic OTFS System Model

3.1. OTFS Modulation

We assume that one UWA OTFS frame has M subcarriers and N symbols. Then, the bandwidth, signal duration, and symbol number are $M\Delta f$, $NT$, and $MN$, respectively, where $\Delta f$ is the subcarrier spacing and $T=1/\Delta f$ is the symbol duration.

We place OTFS symbols $x(q,l)$ into DD grids (size: $M\times N$) with delay resolution $1/(M\Delta f)$ and Doppler resolution $1/\left(NT\right)$, where $q=0,1,\dots ,N-1$ and $l=0,1,\dots ,M-1$. After inverse symplectic finite Fourier transform (ISFFT), OTFS symbols $x(q,l)$ in the DD domain are transformed into $X(n,m)$ in the time–frequency domain:

$$X(n,m)={\displaystyle \frac{1}{\sqrt{MN}}}\sum _{q=0}^{N-1}\sum _{l=0}^{M-1}x(q,l)e^{j2\pi ({\displaystyle \frac{nq}{N}}-{\displaystyle \frac{ml}{M}})}.$$

(1)

Then, $X(n,m)$ is transformed into the passband time domain signal $\tilde{s}\left(t\right)$ after Heisenberg transformation and up-conversion:

$$\tilde{s}\left(t\right)=s\left(t\right)e^{j2\pi f_{c}t}={\displaystyle \frac{1}{\sqrt{M}}}\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}X(n,m)g(t-nT)e^{j2\pi m\Delta f(t-nT)}{e}^{j2\pi f_{c}t},$$

(2)

where $g\left(t\right)$ is the rectangular window function and $f_c$ is the carrier frequency.

3.2. UWA OTFS Channel Model

In the DD domain, an UAW channel can be modeled as follows:

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

(3)

where $h_i$, ${\tau }_i$, and ${\upsilon }_i$ are the path gain, the path delay, and the Doppler shift of the ith path, respectively.

Then, we can obtain the delay shift and Doppler shift in the DD grid for the ith path:

$${\tau }_i={\displaystyle \frac{l_i}{M\Delta f}},$$

(4)

$${\upsilon }_i={\displaystyle \frac{q_i}{NT}}={\displaystyle \frac{q_{{\upsilon }_i}+{\tilde{q}}_{{\upsilon }_i}}{NT}},$$

(5)

where $l_i$ is an integer delay tap when M is large enough and $q_{{\upsilon }_i}$ and ${\tilde{q}}_{{\upsilon }_i}$ are the integer part and the fractional part of Doppler shift, respectively, with ${\tilde{q}}_{{\upsilon }_i}\in (-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}]$.

Based on the resolution in the DD domain, we define ${\tau }_{max}<{\displaystyle \frac{1}{\Delta f}}$ and ${\upsilon }_{max}<{\displaystyle \frac{1}{T}}$ as the maximum delay and Doppler spread, respectively. Most OTFS system designs suppose that the channel is underspread [22,23]:

$${\tau }_{max}{\upsilon }_{max}<<1.$$

(6)

Based on the underspread assumption, $l_i<M$ and $-{\displaystyle \frac{N}{2}}<q_i<{\displaystyle \frac{N}{2}}$ must be met. The passband OTFS signal after passing through the UWA channel can be represented as follows:

$$\begin{array}{c}\tilde{r}\left(t\right)=\int \int h(\tau ,\upsilon )\tilde{s}(t-\tau )e^{j2\pi \upsilon (t-\tau )}d\tau d\upsilon +\tilde{\omega }\left(t\right)\hfill \\ ={\displaystyle \sum _i^p}{h}_{i}s(t-{\tau }_i)e^{j2\pi f_c(t-{\tau }_i)}{e}^{j2\pi {\upsilon }_i(t-{\tau }_i)}+\tilde{\omega }\left(t\right),\hfill \end{array}$$

(7)

where $\tilde{\omega }\left(t\right)$ denotes white Gaussian noise.

After down-conversion to baseband $r\left(t\right)=\tilde{r}\left(t\right)e^{-j2\pi {f^{\prime }}_{c}t}$, we obtain the following:

$$r\left(t\right)=\sum _i^p{h^{\prime }}_i{e}^{j2\pi \sigma f_{c}t}s(t-{\tau }_i)e^{j2\pi {\upsilon }_i(t-{\tau }_i)}+\omega \left(t\right),$$

(8)

where $h_i^{\prime }=h_i{e}^{-j2\pi f_c\tau }$ is the complex path gain and the exponential term $e^{j2\pi \sigma f_{c}t}$ is caused by residual carrier frequency offset (CFO). After Wigner transform and SFFT, we can obtain the received symbols $y(q,l)$ in the DD domain.

4. Underwater Acoustic OTFS System Design

This part introduces the design of the proposed OTFS system, as shown in Figure 1. It includes the design of the UWA OTFS frame (blue dashed line) and the UWA OTFS receiver (red and green dashed lines).

4.1. Frame Design

In order to avoid inter-block interference, we consider a zero-padded OTFS system design for UWA communications, as shown in Figure 1 [17]. The blue dashed line represents the design of ZP-OTFS frames. Setting the last $2l_g+1$ symbol vectors (rows) in the DD domain grid to zeros, we obtain an interleaved ZP guard band in the time domain. Pilot signals for channel estimation are inserted into the ZP section without additional data overhead. The ZP-OTFS frame in DD domain grid is defined as follows:

$$x(q,l)=\left\{\begin{array}{c}{x}_d(q,l),0\le l<M-(2l_g+1),0\le q\le N-1\\ 0,M-(2l_g+1)\le l\le M-1,0\le q\le N-1\\ x_p(q,l),l=m_p=M-(l_g+1),q=n_p\in [0,N-1]\end{array}\right.,$$

(9)

where $x_d(q,l)$ represents the OTFS data symbol at the lth row and qth column in the Doppler grid, $x_p(q,l)$ is the pilot symbol at the $m_p$th row and $n_p$th column, and 0 signifies the guard symbol outside of $x_d(q,l)$ and $x_p(q,l)$ in the Doppler grid. Assuming that $l_g$ is sufficiently long, channel effects on $x_d(q,l)$ and $x_p(q,l)$ will only extend into the ZP section, thereby effectively avoiding inter-block interference.

4.1.1. Design of M and N

In this study, the channel DD response is assumed to be constant, that is, the parameters $h_i$, ${\tau }_i$, and ${\upsilon }_i$ in Equation (3) are constant within one frame period [23]. Based on this assumption, the OTFS frame length $T_f$ cannot exceed the channel coherence time, which is typically short for a UWA channel. Consequently, N and M should not be excessively large simultaneously. Thus, we must carefully select N and M to ensure the robustness of UWA communication.

Typically, in conventional OTFS systems, a large M is chosen to meet the delay resolution requirements, resulting in a smaller N with bandwidth $B=M\Delta f$ and frame length $T_f=NT$. This often leads to $M>>N$. However, in UWA communication, the system’s bandwidth is limited and the Doppler spread ${\upsilon }_{max}$ is often significant. Consequently, a small N cannot satisfy the Doppler resolution requirement for an underspread condition [23].

Accordingly, we must set a suitable value of M value to avoid significant fractional delays, and appropriately increase N to enhance the system’s Doppler tolerance. Simultaneously, we must ensure that the frame duration $T_f$ remains within the channel coherence time to maintain reliable communication.

4.1.2. Design of $l_g$

Zero-padding can completely avoid inter-block interference when $l_g>l_{max}$ but comes at the cost of a very low communication rate. Thus, the means by which to balance the communication rate and robustness need to be carefully considered. As shown in Figure 2, which displays an example of the UWA channel power response, we can define a maximum additional delay ${\tau }_{\beta }$ as the time at which the multi-path power decays from its initial value by a factor $\beta$ dB, defined as follows:

$$10lg{\displaystyle \frac{P_{max}}{P_{{\tau }_{\beta dB}}}}=\beta ,$$

(10)

where $l_{\beta \mathrm{dB}}$ is the delay tap based on ${\tau }_{\beta \mathrm{dB}}$. Later lower-power paths can be ignored, due to their minimal impact.

In brief, we can set $l_g=l_{\beta \mathrm{dB}}$ to enhance the UWA communication rate and ensure communication robustness.

4.2. Synchronization and Doppler Compensation

The proposed UWA OTFS system employs a linear frequency modulation (LFM) signal for preamble detection and synchronization. However, relying solely on this preamble is inadequate for a UWA OTFS system designed for long delay spreads and significant Doppler effects, due to the requirement that the channel be underspread for OTFS to operate effectively. Therefore, a Doppler compensation method is proposed [29,30].

First, we resample the received signal as $y\left(t\right)=r\left({\displaystyle \frac{t}{1+\widehat{a}}}\right)$, where $\widehat{a}$ is the general Doppler scaler:

$$y\left(t\right)=\sum _i^p{h^{\prime }}_i{e}^{j2\pi {\displaystyle \frac{\sigma f_c}{1+\widehat{a}}}t}s({\displaystyle \frac{t-{{\tau }^{\prime }}_i}{1+\widehat{a}}})e^{j2\pi {{\upsilon }^{\prime }}_i(t-{{\tau }^{\prime }}_i)}+\omega ({\displaystyle \frac{t}{1+\widehat{a}}}),$$

(11)

where ${\upsilon }_i^{\prime }={\displaystyle \frac{{\upsilon }_i}{1+\widehat{a}}}$ and ${\tau }_i^{\prime }={\tau }_i(1+\widehat{a})$.

Second, we multiply $y\left(t\right)$ by $e^{-j2\pi \widehat{\epsilon }t}$ to remove the CFO, where $\widehat{\epsilon }$ is the residual Doppler shift.

We estimate $\widehat{a}$ and $\widehat{\epsilon }$ according to [31]. Through Doppler compensation, the influence of residual Doppler is significantly reduced, allowing the underspread channel assumption of Equation (6) to be satisfied.

4.3. Dual-Domain Joint Channel Estimation

Here, we propose a dual-domain joint channel estimation method, as illustrated in Figure 3. This method first estimates the path delay in the DD domain, then estimates the complete channel impulse response (CIR) in the DT domain based on the path delay estimation results.

4.3.1. Path Delay Estimation in DD Domain

As fractional Doppler shifts do not impact the estimation of delay taps, we can ignore the IDI interference caused by fractional Doppler shifts. The relationship between $y(q,l)$ and $x(q,l)$ with integer Doppler shifts was derived in [15], as follows:

$$y(q,l)=\sum _{q^{\prime }=-q_{max}}^q\sum _{l^{\prime }=0}^{l_g}[d(q^{\prime },l^{\prime }){\xi }_{q^{\prime },l^{\prime }}x((q-q^{\prime })modN,(l-l^{\prime })modM)]+w(q,l),$$

(12)

where $q_{max}$ is the range of fractional Doppler coefficients after Doppler compensation, ${\xi }_{q^{\prime },l^{\prime }}=h^{\prime }(q^{\prime },l^{\prime })\eta (q^{\prime },l^{\prime })$. $d(q^{\prime },l^{\prime })\in \{0,1\}$ is the complex channel gain with the phase $\eta (q^{\prime },l^{\prime })$ and the amplitude $h^{\prime }(q^{\prime },l^{\prime })=h(q^{\prime },l^{\prime })e^{-j2\pi {\displaystyle \frac{q^{\prime }{l}^{\prime }}{NM}}}$, and $d(q^{\prime },l^{\prime })\in \{0,1\}$ is the path indicator; for example, $d(q^{\prime },l^{\prime })=1$ indicates that there is a path with Doppler tap $q^{\prime }$ and delay tap $l^{\prime }$ with corresponding path magnitude ${\xi }_{q^{\prime },l^{\prime }}$. Furthermore, $w(q,l)$ denotes Gaussian noise and ‘(.)mod’ represents the modulus.

The relationship in Equation (12) can be transformed into vector form as follows:

$${\mathbf{Y}}_p={\mathbf{X}}_p\mathbf{h}+\mathbf{w},$$

(13)

in which:

$$\begin{array}{c}\mathbf{h}=[{\xi }_{-q_{max},0},{\xi }_{-q_{max},1},\dots ,{\xi }_{-q_{max},l_g},\hfill \\ {\xi }_{-q_{max}+1,0},{\xi }_{-q_{max}+1,1},\dots ,{\xi }_{q_{max},l_g}{]}^T,\hfill \end{array}$$

(14)

$$\begin{array}{c}{\mathbf{Y}}_p=[y(0,m_p),y(0,m_p+1),\dots ,y(0,m_p+l_g),\hfill \\ y(1,m_p),y(1,m_p+1),\dots ,y(N-1,m_p+l_g){]}^T,\hfill \end{array}$$

(15)

$$\begin{array}{c}{\mathbf{X}}_p=[{\mathbf{x}}_{-q_{max},0},{\mathbf{x}}_{-q_{max},1},\dots ,{\mathbf{x}}_{-q_{max},l_g},\hfill \\ {\mathbf{x}}_{-q_{max}+1,0},{\mathbf{x}}_{-q_{max}+1,1},\dots ,{\mathbf{x}}_{q_{max},l_g}{]}^T,\hfill \end{array}$$

(16)

and

$${\mathbf{x}}_{q^{\prime },l^{\prime }}=\left[\begin{array}{c}x((0-q^{\prime })modN,(m_p-l^{\prime })modM)\hfill \\ x((0-q^{\prime })modN,(m_p+1-l^{\prime })modM)\hfill \\ \dots \hfill \\ x((0-q^{\prime })modN,(m_p+l_g-l^{\prime })modM)\hfill \\ x((1-q^{\prime })modN,(m_p-l^{\prime })modM)\hfill \\ \dots \hfill \\ x((1-q^{\prime })modN,(m_p+l_g-l^{\prime })modM)\hfill \\ \dots \hfill \\ x((N-1-q^{\prime })modN,(m_p+l_g-l^{\prime })modM)\hfill \end{array}\right],$$

(17)

where $\mathbf{h}$ is the equivalent channel with sparsity, ${\mathbf{Y}}_p$ is the received pilot signal, and ${\mathbf{X}}_p$ is the sensing matrix, for $l^{\prime }\in [0,l_g]$ and $q^{\prime }\in [-q_{max},q_{max}]$.

Next, we can estimate the equivalent channel $\mathbf{h}$ using a compressive sensing algorithm, such as the orthogonal matching pursuit (OMP) method [29,32]. Then, the set of path delays $\{\widehat{l}\}$ is obtained according to the index of $\mathbf{h}$.

4.3.2. CIR Estimation in DT Domain

According to [17], the channel impulse response in the DD domain in Equation (3) can be rewritten as follows:

$$h(\tau ,\upsilon )=\sum _{l\in L}\sum _{q\in Q_l}h(l,q)\delta (\tau -lT/M)\delta (\upsilon -q\Delta f/N),$$

(18)

where $L=\left\{l_i\right\}$ is the set of distinct normalized delay shifts in the DD domain, $Q_l=\left\{q_i\right|l=l_i\}$ is the set of normalized Doppler shifts for each path with normalized delay shift $l_i$, and the lth delay tap Doppler response is

$$h(l,q)=\left\{\begin{array}{c}{h}_i,\mathrm{if}l=l_{i}andq=q_i\\ 0,\mathrm{otherwise}\end{array}\right.$$

(19)

Then, the UWA CIR is as follows:

$$\tilde{h}(\tau ,\upsilon )={\int }_{\upsilon }h(\tau ,\upsilon )e^{j2\pi \upsilon (t-\tau )}d\upsilon =\sum _{q\in Q_l}h(l,q)e^{j2\pi q{\displaystyle \frac{\Delta f}{N}}(t-\tau )}.$$

(20)

Discretizing the channel after sampling at $t=aT/M,0\le a\le NM-1$:

$$\tilde{h}(l,a)=\sum _{q\in Q_l}h(l,q)e^{{\displaystyle \frac{j2\pi q(a-l)}{MN}}}.$$

(21)

Setting $a=m+nM$, according to [17], the input–output relationship with the channel in the DT domain can be defined as follows:

$$\mathbf{y}(m,n)=\sum _{l\in L}\tilde{h}(l,m+nM)\mathbf{x}(m-l,n),$$

(22)

where $\mathbf{x}(m,n)$ and $\mathbf{y}(m,n)$ are the transmitted and received symbols in the mth row, respectively, with size $N\times 1$, as shown in Figure 1.

Given an obtained path delay $\widehat{l}$, let $m=m_p+\widehat{l},\widehat{l}\in L$, the input–output relationship of pilot signal can be expressed as follows:

$$\mathbf{y}(m_p+\widehat{l},n)=\sum _{l\in L}\tilde{h}(l,m_p+\widehat{l}+nM)\mathbf{x}(m_p+\widehat{l}-l,n).$$

(23)

Based on Equation (9), $\mathbf{x}(m_p+\widehat{l}-l,n)=0\mathrm{when}0<|l-l^{\prime }|\le l_g$ (i.e., $0<|l-\widehat{l}|\le l_g$), due to the zero-padding. Thus, the CIR can be obtained as follows:

$$\widehat{h}(l,m_p+\widehat{l}+nM)={\displaystyle \frac{\mathbf{y}(m_p+\widehat{l},n)}{\mathbf{x}(m_p,n)}},for\widehat{l}\in L,$$

(24)

where $\mathbf{x}(m_p,n)$ is the transmitted pilot signal. The UWA CIR in the DT domain can be represented using spline interpolation [22]:

$$\widehat{h}(l,a)=spline\_interpolate(\left\{\widehat{h}(l,m_p+\widehat{l}+nM)\right|n\in [0,N-1]\},[m_p+\widehat{l}:M:NM],[0:NM-1]).$$

(25)

Finally, based on $\widehat{h}(l,a)$, the DD domain channel matrix $\mathbf{H}$ can be obtained according to [17].

4.4. Iterative Decoding Feedback-Based MP Detector

The method proposed in this study introduces an OTFS system detection approach based on an iterative decoding feedback message passing detector (IDF-MP). This method enhances the detection performance through an iterative process that enables soft information exchange between the MP detector and a low-density parity check (LDPC) decoder. It utilizes symbol-level soft information output by the detector and bit-level soft information output by the LDPC decoder to improve the information transmission efficiency.

As shown in Figure 4, the process commences with the utilization of the channel equivalent matrix obtained from the preceding channel estimation results. Subsequently, the OTFS DD domain signal and the channel equivalent matrix are fed into the MP detector. The MP algorithm utilized in this study, which has been previously detailed in [15], is specifically designed to leverage the sparse characteristics of the channel matrix $\mathbf{H}$ in the DD domain. We do not further elaborate on this algorithm in this paper.

Figure 4 illustrates the interconnection between the MP detector and the LDPC decoder within the iterative structure. The conversion of symbol-level soft information and bit-level soft information between the MP detector and LDPC decoder is facilitated by log-likelihood ratio (LLR) converters, as well as interleaving and de-interleaving sections.

In the iterative structure of the IDF-MP algorithm, there are three forms of loop counts:

•

The outer-loop iterations $n_{out}$.

•

The inner-loop iterations of the MP detector $n_{in,MP}$.

•

The inner-loop iterations of the LDPC decoder $n_{in,LDPC}$.

The specific information transfer process between the MP detector and LDPC decoder is described in the following.

4.4.1. From the MP Detector to the LDPC Decoder

Within the $n_{out}$ outer loop, when the inner-loop iterations of the MP detector reach the maximum iterations $N_{in,MP}$, the symbol-level outer information output by the detector is as follows:

$$M_{e,MP}^{n_{out}}(x_i=a_j)=p_i^{N_{in,MP}}(x_i=a_j),$$

(26)

where $x_i$ represents the ith symbol after OTFS modulation ($i=1,2,\dots ,NM$), $\mathrm{A}$ represents the set of modulation symbols, $a_j\in \mathrm{A}, j=1,2,\dots ,Q$, and Q is the size of the modulation symbol set $\mathrm{A}$ (i.e., $Q=16$ when the 16 QAM is used for modulation). Furthermore, $p_i^{N_{in,MP}}(x_i=a_j)$ denotes that when the inner-loop iterations of the MP detector reaches the maximum number of iterations $N_{in,MP}$, the probability of the symbol $a_j$ in the quadrature amplitude modulation (QAM) alphabet is calculated within the MP detector. This probability is the symbol-level output of the MP detector.

The symbol-level soft information output by the MP detector $M_{e,MP}^{n_{out}}(x_i=a_i)$ passes through the LLR converter to obtain the bit-level soft information.

$$L_{e,MP}^{n_{out}}(c_{i,k}=a_{j,k})=ln{\displaystyle \frac{{\displaystyle \sum _{x_i\in {\mathrm{A}}_k^1}}{M}_{e,MP}^{n_{out}}(x_i=a_j)}{{\displaystyle \sum _{x_i\in {\mathrm{A}}_k^0}}{M}_{e,MP}^{n_{out}}(x_i=a_j)}},$$

(27)

where $a_{j,k}$ represents the bit value (0 or 1) of the kth bit in the jth element of the modulation symbol set.

Each modulation symbol $a_j\in \mathrm{A}$ corresponds to a binary sequence with a length of $lo{g}_2\left|Q\right|$. Therefore, $c_{i,k}$ represents the bit value (0 or 1) on the kth subcarrier of the ith symbol, where $k=0,1,\dots ,$$lo{g}_2\left|Q\right|$. ${\mathrm{A}}_k^1$ (${\mathrm{A}}_k^0$) represents the modulation symbol set in which the kth subcarrier in the modulation symbol set $\mathrm{A}$ is $1\left(0\right)$.

The bit-level soft information $L_{e,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ output from the LLR in the $n_{out}$ outer loop is de-interleaved to obtain the input information $L_{pr,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ of the LDPC decoding part. $L_{pr,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ serves as the prior information of the LDPC decoder in the $n_{out}$ outer loop.

4.4.2. From the LDPC Decoder to the MP Detector

Within the $n_{out}$ outer loop, the bit-level soft information output from the LDPC decoder is transmitted when the outer-loop iterations of the LDPC decoder reach the maximum number of iterations $N_{in,LDPC}$.

$$L_{e,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})=L_{po,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})-L_{pr,LDPC}^{n_{out}}(c_{i,k}=a_{j,k}),$$

(28)

where $L_{po,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ is the cumulative probability value. Then, after passing through the interleaver, the bit-level soft information $L_{e,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ is transformed into the input information $L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ of the detector. The prior probability value is calculated using Equations (29) and (30):

$$p(c_{i,k}=0)={\displaystyle \frac{1}{1+exp\left(L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})\right)}},$$

(29)

$$p(c_{i,k}=1)={\displaystyle \frac{1}{1+exp(-L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k}))}}.$$

(30)

We introduce the variable ${\lambda }_{i,k}$ and represent Equations (29) and (30) uniformly as shown in Equation (31). When $c_{i,k}=0$, ${\lambda }_{i,k}=-1$; meanwhile, when $c_{i,k}=1$, ${\lambda }_{i,k}=1$.

$$p(c_{i,k}=a_{j,k})={\displaystyle \frac{1}{1+exp(-{\lambda }_{i,k}{L}_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k}))}}={\displaystyle \frac{exp(\frac{{\lambda }_{i,k}{L}_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})}{exp(\frac{L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})+exp(-\frac{L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})}}.$$

(31)

After passing through the LLR converter, the bit-level probability value $L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ is converted into the probability value $M_{pr,MP}^{n_{out}}(x_i=a_j)$ for each modulation symbol. This probability value is then utilized as prior probability information input to the detector, thereby completing an iteration.

$$\begin{array}{c}{M}_{pr,MP}^{n_{out}}(x_i=a_j)=p(x_i=a_j)=\underset{k=1}{\overset{{log}_2\left|Q\right|}{\mathrm{\Pi }}}p(c_{i,k}=a_{j,k})\hfill \\ =\underset{k=1}{\overset{{log}_2\left|Q\right|}{\mathrm{\Pi }}}{\displaystyle \frac{exp(\frac{{\lambda }_{m,k}{L}_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})}{exp(\frac{L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})+exp(-\frac{L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})}{2})}}\hfill \end{array}.$$

(32)

Finally, we repeat the iteration process until the maximum number of iterations $N_{out}$ is reached. Then, the LDPC decoder outputs bit-level LLR information $L_{e,LDPC}^{N_{out}}\left(c_{i,k}\right)$ to make a decision and outputs bit information. The LDPC decoding algorithm in this article adopts the LLR-MP decoding algorithm. The computational complexity of the LDPC decoder is $O\left({log}_2\left(\right|Q\left|\right)NM\right)$, and the computational complexity of MP algorithm is $O\left(NMSQ\right)$, where S represents the number of non-zero elements present in each row or column of the DD domain channel matrix $\mathbf{H}$. Thus, the overall computational complexity of the IDF-MP algorithm is $N_{out}(N_{in,MP}O\left({log}_2\left(\right|Q\left|\right)NM\right)+N_{in,LDPC}O\left(NMSQ\right))$. Algorithm 1 shows the process of IDF-MP detection algorithm based on LDPC.

Algorithm 1 Process of IDF-MP detection algorithm based on LDPC

Require: Time DD domain channel matrix $\mathbf{H}$, received signal $\mathbf{y}$, maximum iteration times $N_{out}$, $N_{in,MP}$, and $N_{in,LDPC}$; Set the outer-loop iteration times $n_{out}=0$.

Ensure: Bit information;

1: When $n_{out}$ is less than $N_{out}$, start MP detection until the number of internal loops in the detector reaches $N_{in,MP}$; 2: Calculate the symbol-level outer information $M_{e,MP}^{n_{out}}(x_i=a_j)$ of the detector output using Equation (26); 3: Convert $M_{e,MP}^{n_{out}}(x_i=a_j)$ to bit-level information $L_{e,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ using Equation (27); 4: After de-interleaving, the $L_{pr,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ is obtained as the prior information for the LDPC decode; 5: Start LDPC decoding until the number of internal loops in the decoder reaches $N_{in,LDPC}$; 6: Calculate the bit-level LLR information $L_{e,LDPC}^{n_{out}}(c_{i,k}=a_{j,k})$ output from the LDPC decoder using formula Equation (28); 7: Obtain the bit-level input information $L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ of the MP detector through an interleaver; 8: Using Equation (32), convert the bit level information $L_{pr,MP}^{n_{out}}(c_{i,k}=a_{j,k})$ into the probability values of each modulation symbol, as the prior probability information $M_{pr,MP}^{n_{out}}(x_i=a_j)$ input to the MP detector; 9: $n_{out}=n_{out}+1$, return to step 2 if $n_{out}$ is less than $N_{out}$, until $n_{out}$ reaches $N_{out}$; 10: return Make a decision to output bit information based on $L_{e,LDPC}^{N_{out}}(c_{i,k}=a_{j,k})$;

5. Performance Evaluation

In this section, the performance of our approach is evaluated.

Table 1. Simulation parameters.

Parameters	Value
Modulation mode	QPSK
Channel coding	1/2 LDPC
Carrier frequency	10 kHz
Bandwidth	4 kHz
Max path delay ${\tau }_{max}$	35 ms
Relative speed of transmitter and receiver v	5 kn/10 kn
Number of subcarriers M	384/512
Number of OTFS symbols N	24/16
Sampling frequency	48 kHz

shows the parameters used in the simulation experiments. The UWA channels were modeled following the methodology described in [33]. The water depth was set to 100 m, with the transmitter positioned at a depth of 30 m and the receiver at 50 m. The horizontal distance between the transmitter and receiver was 2 km. The sound speed profile utilized for channel simulation is illustrated in Figure 5.

Figure 6a illustrates the CIR without Doppler compensation with relative movement speed v = 10 kn. It is evident that the maximum channel delay spread exceeds 15 ms. The relative motion between the receiver and transmitter results in strong time variation in the UWA channel. Figure 6b depicts the Doppler dispersion in the DD domain distributed from 2 Hz to 3 Hz, indicating an overspread UWA channel. After Doppler compensation, Figure 6c shows that the CIR in the DD domain becomes sparser compared to the DT domain, confirming that the modeled channel becomes underspread.

Figure 7 compares the BER performance with different values of N and M under varying v. The frame lengths were identical, and the communication rates were set to the same value of 2.333 kbps. The relative speed between the transmitter and receiver was specified as v = 5 kn and v = 10 kn. In the v = 5 kn or v = 10 kn cases, OTFS frames with larger values of N exhibited higher performance due to their higher Doppler tolerance, provided M is large enough to avoid significant fractional delay. As the Doppler spread increases, the system with larger N shows substantial superiority.

Based on these analyses, setting $M=384$ and $N=24$ proved to be effective, allowing the system to better handle large Doppler spreads and obtain enhanced performance while maintaining a reasonable frame size.

Figure 8 compares the obtained performance based on different ZP lengths when setting the value of $L_g$ to 3 dB, 7 dB, or 10 dB.

The communication rate with maximum ZP length $L_{max}$, maximum additional delay $L_{10dB}$, maximum additional delay $L_{7dB}$, and maximum additional delay $L_{3dB}$ were 2.3333 kbps, 2.5197 kbps, 2.8329 kbps, and 2.9579 kbps, respectively. We can see that setting $L_g$ to the maximum additional delay resulted in a higher communication rate than when using the maximum delay. As $\beta$ in Equation (10) decreases, the communication rate improves. However, when $\beta$ dB decreases, the ZP length may not sufficiently prevent inter-block interference, thereby diminishing its effectiveness and increasing the BER.

In this simulation, setting $L_g=L_{10dB}$ improved the communication rate while ensuring communication reliability. Therefore, subsequent simulations used the parameters $M=384$, $N=24$, and $L_g=L_{10dB}$.

Figure 9 shows the BER performance of the IDF-MP algorithm with different numbers of outer-loop iterations. The outer-loop iteration number $n_{out}$ was set to 0, 1, 2, or 3, and the relative speed v was set to 5 kn or 10 kn. It can be observed that, regardless of whether v = 5 kn or 10 kn, the BER decreased as the number of outer-loop iterations increased. There was a minimal difference in BER performance between three iterations and two iterations. Therefore, subsequent simulations employed two outer-loop iterations.

To comprehensively analyze the IDF-MP algorithm proposed in this study, we compared its BER performance with OFDM, an initial estimate iterative decoding feedback (IE-IDF-MRC) algorithm [17], and a two-dimensional passive time reversal decision feedback equalization (2D-PTR-DFE) algorithm [21]. The IDF-MP algorithm is built upon the proposed dual-domain joint channel estimation technique, distinguishing it from other algorithms that rely on the OMP method for channel estimation. Both the IE-IDF-MRC and IDF-MP algorithms were configured with an outer-loop iteration count of 2, and LDPC codes were employed for channel coding across all algorithms, achieving a communication rate of 2.5 kbps.

As shown in Figure 10, the results demonstrated that the OTFS communication system outperformed the OFDM system, regardless of whether v = 5 kn or 10 kn, with a more pronounced performance difference observed under severe channel time variations. Specifically, the IDF-MP algorithm proposed in this study exhibited the best performance. This superiority can be attributed to the IDF-MP algorithm’s utilization of the soft information output from LDPC decoding as prior information for the MP detector, facilitating iterative information exchange that minimizes information loss and enhances the system’s detection and decoding performance.

6. Conclusions

In this study, we explored the application of OTFS modulation in UWA communications, particularly for overspread UWA channels. At the transmitter of the proposed UWA OTFS system, we introduced a frame structure design which is tailored with respect to the characteristics of UWA channels, in order to avoid inter-block interference and mitigate the impact of overspread channels. At the receiver, we introduced a dual-domain joint channel estimation method and an iterative decoding feedback-based detector to enhance the robustness of UWA OTFS communication systems.

Our simulation results demonstrated the feasibility of the proposed OTFS system in overspread UWA channels. Through meticulous frame design, the OTFS system exhibits significant performance improvements in the case of strong overspread. Moreover, leveraging the proposed channel estimation method and detector, our OTFS system achieved enhanced communication performance when compared to conventional systems such as the OFDM, IE-IDF-MRC, and 2D-PTR-DEF methods.

Our future work will involve deploying the proposed OTFS communication system in real-world UWA environments in order to further analyze its operational performance and effectiveness.