Advances and Trends in Channel Codes for Underwater Acoustic Communications

Abstract

Due to the simultaneous existence of severe difficulties caused by multi-path, Doppler, and environmental noise caused by underwater acoustic channels, designing a stable and reliable underwater acoustic communication system (UWACS) is a challenging task. In addition to channel estimation and equalization, the technique of channel codes, which is capable of approaching Shannon’s capacity at a low encoding and decoding complexity, offers an efficient and reliable solution to improve the performance of the UWACS and thus draws extensive attention from the research community. By seeking a balance between communication efficiency and channel tolerance via multiple coding rate tuning, it has been recognized that the rate-compatible codes can achieve smooth rate-adaptive transmission under highly diverse underwater acoustic channel states. While there has been a substantial number of investigations on rate-compatible channel codes for the UWACS, in this paper, the research of rate-compatible channel codes has been summarized, reviewed, and compared, mainly focusing on the low-density parity-check (LDPC) and polar codes, which render flexibility in terms of code length and rate. Furthermore, typical research was evaluated and compared from the practical viewpoint of performance under real noise in a shallow water environment. Finally, problems encountered by the LDPC and polar codes for the UWACS were highlighted, and several future research issues were discussed.

1. Introduction

The oceans cover 71% of the Earth’s surface and contain valuable resources that support human progress and survival, which is of strategic importance and economic value. Acquiring and communicating marine information has become an essential and valuable aspect of maritime activities due to the rapid development of marine resource exploration and exploitation, fisheries, environmental monitoring, marine scientific research, marine engineering, and the maintenance of maritime rights and interests [1,2,3,4]. Acquiring and communicating marine information in complex marine environments has long been a barrier to progress in marine science and technology in many nations. Acoustic waves are the best choice for long-range underwater wireless communication, as they have low attenuation in seawater [5]. The wide range of oceanic activities necessitates using acoustic communication technology for high-rate, stable, and reliable data transmission, which is imperative and pragmatic.

Marine environmental factors, such as wind speed, temperature, tide level, wind waves, water velocity, internal waves, and topographic structure, can cause varying degrees of change in the underwater acoustic (UWA) channel structure, which results in limited underwater acoustic communication (UWAC) bandwidth, frequency-selective fading, short coherence times, and severe inter-symbol interference (ISI) [6]. The complex characteristics of the UWA channel make developing environmentally robust, reliable, and stable communication systems challenging. In addition to channel estimation and equalization, channel coding is an essential method of combating ISI due to variations in the UWA channel structure, and it is an essential part of the design of the UWACS. Specifically, channel coding offers varying levels of protection by adding redundant bits to the transmitted data and implementing interleaving techniques for correcting random burst errors induced by intricate channels. With the boom in information theory, channel- coding approaches have reached the Shannon limit in recent years. Therefore, the fusion of UWAC technology with advanced channel coding techniques to achieve high-quality transmission of marine data over various channel structures and signal-to-noise ratios (SNR) has become a notable area of research for the pioneering advancement of UWAC technology [7]. Currently, widely employed channel coding schemes in the field of UWAC include repeat-accumulate (RA) codes [8], Bose–Chaudhuri–Hocquenghem (BCH) codes [9], Reed–Solomon (RS) codes [10], convolutional codes [11], fountain codes [12], differential pattern time delay shift coding [13], the extensively researched turbo codes [14], and LDPC [15], whose performance nearly approaches Shannon’s limit. Polar codes can theoretically achieve Shannon’s limit in binary-input discrete memoryless channels (B-DMCs) [16]. Among them, it is worth noting that turbo, LDPC, and polar codes show better performance for longer code lengths. However, the spatial-temporal frequency variable UWA channel requires shorter block data lengths whenever possible. Therefore, research combining the UWA channel characteristics with finite code length coding schemes is imperative for further exploration in practical applications.

Moreover, the complexity and diversity of the marine environment lead to dynamic changes in the multi-path structure of the UWA channel and its temporal variation, posing a significant challenge to the stability of data transmission in the UWACS. UWAC research has recently aimed to resolve concerns about dependability, latency, and bandwidth efficiency, emphasizing adjusting to changing propagation conditions [3]. Rate-compatible coding schemes are commonly employed in wireless communications to flexibly regulate the code lengths and rates of different codes through puncturing. This approach allows for efficient transmission over channels with varying bit error rates. Additionally, this technique can enhance the overall transmission reliability, which is crucial for many communication applications [17,18,19,20]. If a rate-compatible channel coding scheme can be adopted based on the spatiotemporal frequency variation of the UWA channel and based on channel estimation, flexible channel coding rates and lengths can be adjusted according to the different characteristics of the oceanic channel. For example, in harsh communication environments with a low SNR, a low bit rate channel coding can be used to achieve specific performance targets; when the channel has a high SNR, higher code rates and lengths are employed.

The timely use of favorable channel conditions can fully utilize the capacity and ensure stable and low error rate data transmission. Rate-compatible coding schemes have great potential for UWAC and form the basis of adaptive coded modulation [21], an essential technology for designing environmentally adaptive UWACSs.

Based on the open literature, this paper aimed to sort out the current research status and development trend of rate-compatible channel coding schemes in UWAC. This paper is structured as follows: Section 2 outlines the challenges for UWAC, including those arising from UWA channel characteristics, mission characteristics, and the need for rate-compatible UWA channel coding schemes. Section 3 provides a chronological overview of mainstream UWA channel coding developments. Section 4 compares the performance and complexity of various coding schemes while recommending research into LDPC and polar channel coding schemes that meet UWAC requirements. Section 5 presents the conclusions of this paper.

2. UWA Channel Model and Characteristics

The underwater acoustic fluctuation equation provides the theoretical basis for a mathematical model of underwater acoustic propagation. According to different methods of solving the fluctuation equations, acoustic propagation models can be classified into the ray, normal wave mode wave, multi-path expansion, fast field, and parabolic equation models [22]. Each of these underwater acoustic propagation models has its purpose. Considering the time-space-varying nature of changes in the marine environment, the modeling of underwater acoustic channels can quickly evaluate the performance of underwater acoustic devices in different channel environments and facilitate the design of UAC receivers to enhance the performance of UWACSs. However, the complexity and variability of the underwater acoustic channel lead to the lack of generality of these models, and there has yet to be a universally accepted standard underwater acoustic channel model so far.

The actual underwater acoustic channel is affected by wave undulations on the sea surface, turbulent motions in the seawater, topographic structure of the seabed, irregular variations in the temperature gradient in the seawater, and variations in the transmitter and receiver of the communication system, which makes it difficult to perform an accurate characterization. Sound waves are widely reflected, refracted, and scattered in the ocean, propagating through multiple paths to reach the receivers. The underwater acoustic channel has a significant sparsity property, i.e., the channel energy is concentrated on such a small number of paths so that the entire multi-path extension can be used to represent the time-domain impulse response of the entire channel with only a few taps that are much less than the channel length. For this reason, the underwater acoustic channel is usually modeled as a finite multi-path time-delay extended time-varying channel, assuming the amplitude attenuation, time delay, and Doppler effect are the main processes that occur during the propagation of acoustic waves in the ocean. Thus, the UWA channel model is expressed as follows [6]:

$$h(t,\tau )={\displaystyle \sum _{p=1}^{Np}{A}_p(t)\delta (t-({\tau }_p(t)-{\alpha }_p(t)t))}$$

(1)

where $Np$ is the number of multi-paths, and $A_p(t)$ and ${\tau }_p(t)$ are the gain and time delay of $pth$ path, respectively. ${\alpha }_p$ is the doppler factor, and ${\alpha }_p=v_p*\cos ({\theta }_p)/c$, where $c$ denotes the sound speed, and $v_p$ and $\cos ({\theta }_p)$ denote the velocity and direction of motion on the path, respectively. Assuming the symbols to transmit are $s={[s_1\cdots s_N]}^T$, following the application of a double-selective channel, $h={[h_0\cdots h_{P-1}]}^T$, with channel multi-path number $p$, the received symbols are $r={[r_1\cdots r_{N+P-1}]}^T$. Then, the relationship between the received signal and the transmitted signal can be expressed as:

$$r(t)=h(t,\tau )*s(t)+w(t)$$

(2)

The process is expressed as:

$$\begin{array}{c}\left[\begin{array}{c}{r}_1\\ ⋮\\ r_{N+P-1}\end{array}\right]\\ r(n)\end{array}=\begin{array}{c}\left[\begin{array}{ccc}{h}_0& & \\ ⋮& \ddots & \\ h_{P-1}& & h_0\\ & \ddots & ⋮\\ & & h_{P-1}\end{array}\right]\\ H(n)\end{array}\begin{array}{c}\left[\begin{array}{c}{s}_1\\ ⋮\\ s_N\end{array}\right]\\ s(n)\end{array}+\begin{array}{c}\left[\begin{array}{c}{w}_1\\ ⋮\\ w_{N+P-1}\end{array}\right]\\ w(n)\end{array}$$

(3)

$$r(n)={\displaystyle \sum _{i=0}^{P-1}s(i)h(n-i)+w(n)}$$

(4)

where $w(n)$ denotes noise. The received signal is a superposition of various multi-path signals with different time delays, Doppler shifts, and amplitudes.

2.1. Multi-Paths

The UWA channel is an extremely complex stochastic time-space-frequency variable parametric channel with multi-path, fading, high noise, limited frequency band, and other unwanted influences. As shown in Equation (1), the acoustic wave propagates along $Np$ paths underwater, causing signal aliasing at the receiver and degrading the communication quality. It is formed using two primary sources: first, by reflections from objects on the sea surface, sea floor, or in the water, and second, by scattering effects due to inhomogeneities in the water column.

The multi-path in shallow water originates from the reflection of sound waves on the surface and bottom of the water, thus generating the same signal. The phenomenon of arriving through multiple paths is a fast time-varying, different time-delay spread multi-path. In deep water communication, the signal is less affected by the seabed and sea surface, and the multi-path effect in the channel mainly comes from the variation of the velocity of sound waves due to the influence of space during the propagation process, leading to the refraction of sound waves and causing a large time-delay spread with severe multi-path effects. In order to demonstrate the multi-path structure of the real shallow and deep water, Figure 1a,b display the underwater acoustic channels in Xiamen Harbor and the South China Sea obtained via orthogonal matching pursuit [23].

Figure 2 shows a schematic diagram of multi-path propagation. The main constraints on the performance of UWAC are ISI and frequency-selective fading of the received signal due to multi-path effects [24].

2.2. Time-Vary Multi-Path Channel and Doppler

The complex and variable characteristics of the propagation medium, seawater, and the relative motion between the transmitter and the receiver are the leading causes of the time-varying nature of the UWA channel. The changing characteristics of seawater include different scales of time-varying factors, such as seasonal ocean temperature distribution, climate change, ocean circulation, tidal changes, and other large-scale time changes, which exert less impact on the performance of UWAC, and several of these factors have periodical influence; small-scale time changes, such as waves, bubbles, sea surface wind speed, etc., will cause rapid changes in the relative position of the acoustic wave reflection point at the sea surface and, meanwhile, will cause signal scattering and Doppler spreading due to changes in the acoustic propagation distance between communication equipment carrier buoys, wave gliders (WGs), underwater equipment, and mother ship and network nodes, etc. Assuming that the transmitter carrier frequency is $f_0$, the receiver frequency is $f$, and the speed of sound is $c$:

$$f=\frac{c+v_r}{c+v_t}{f}_0$$

(5)

where $v_r$ and $v_t$ denote the speed of the receiver relative to the acoustic propagator; when the receiver travels in the opposite direction to the transmitter, $v_r$ is positive, $v_t$ is negative, and vice versa. As the speed of acoustic waves is much lower than that of electromagnetic waves, the Doppler effect due to relative motion between UWACSs severely impacts communication quality more than in land-based wireless communications. The Doppler effect has two main effects on communications: a frequency shift in the received signal and a compression or expansion of the symbol time domain. A severe Doppler effect can lead to rapid communication performance degradation for both coherent and incoherent demodulation [25].

It can be seen from Figure 3 that the underwater acoustic channel multi-path structure exhibits pronounced time-varying characteristics and multi-paths with long delays, moreover, there are differences in the time-varying characteristics and degree of fading between each data frame [26]. In addition, as shown in Figure 4, the Doppler effect of the multi-path structure with different time delays is extremely time varying and unevenly distributed, which makes it challenging to obtain a unified and concise model for an accurate parameter description of the UWA channel [27]. With the increase in maritime activities in recent years, the kinematic state of the communication equipment carriers has become more complex and diverse, making it more challenging to reduce the Doppler effect, especially for multi-carrier UWAC. Therefore, reducing the ISI caused by the Doppler effect and time vary channel in combination with channel coding is a crucial part of the design of UWACSs.

2.3. Noise and Channel Capacity

Marine environmental noise is one of the main factors limiting the performance of UWACSs. It consists of various noise sources superimposed on the integrated effect, including natural background noise, such as air bubbles, earthquakes, wind and waves, marine organisms, etc., and manufactured noise, such as underwater vehicles, the shore-based industry, etc. As described in the literature [5], SNR are calculated from signal attenuation and the noise power spectrum, as shown in Equation (6), where $P_t$ is the signal transmission power; $\Delta f$ is the noise reception bandwidth; $A(l,f)$ reflects that the SNR is closely related to carrier frequency ($f$) and transmission distance ($l$); and $N(f)$ is the power spectral density of the ambient noise. Hence, the actual UWAC is bandwidth- and power-limited in most scenarios.

$$SNR(l,f)=\frac{P_t}{A(l,f)N(f)\Delta f}$$

(6)

Shannon’s theorem is the cornerstone of modern communication systems. It provides a theoretical formula for channel capacity [28]:

$$C=B{\log }_2(1+\frac{P_r}{B{N}_0})$$

(7)

where $B$ is the bandwidth, $P_r$ is the average power of the received signal, and $N_0$ is the spectral density of the noise power, in which the received SNR is given as:

$$SNR=\frac{P_r}{B{N}_0}$$

(8)

In summary, the SNR severely influences channel capacity, which is the focus of research on channel coding adaptation and innovation in UWA channels and is also an essential consideration in carrier frequency selection, filter design, and receiver design for communication systems.

2.4. Requirements for Channel Coding

The complexity and variability of the UWA channel, multi-path, time variability, Doppler, noise, etc., are challenges to designing an UWACS. As a crucial component of the UWACS for ISI mitigation, the theoretical and applied study of channel coding in UWA channels is paramount, and requirements for various coding schemes have been requested during research. Firstly, reliability, defined as the success probability of transmitting data packets at a certain channel quality [29], is the most crucial indicator of the performance of channel coding and decoding algorithms. For a given data rate, the communication process needs to maximize the reliability of each packet in order to minimize the error rate and thus reduce the number of retransmissions and improve the efficiency of the communication. Secondly, latency is defined as the time to successfully transmit a data block from the transmitter to the receiver via the UWA channel. The running time of the channel coding and decoding algorithms should be as short as possible to ensure real-time data transmission. Finally, regarding complexity and flexibility, UWACSs are mostly battery-powered with limited energy consumption, and optimizing energy consumption is a research hotspot in UWA communication [30]. Low-complexity algorithms form the basis for low energy consumption; moreover, flexible code rates and lengths are crucial to designing UWACSs with the environmental adaptive capability to ensure stable data transmission. Figure 5 illustrates that introducing channel coding schemes to design and optimize UWACSs with excellent performance should consider the trade-offs between reliability, latency, complexity, and flexibility.

3. Channel Codes for UWAS

Channel coding offers protection by adding redundancy to the transmitted information bits through constraint relations to reduce the bit error rate (BER) when transmitted in an interference channel. It is an effective method to combat ISI in UWA communications. This section describes the mainstream coding schemes used in developing and researching UWA communication technologies, including fixed-rate and rate-compatible codes.

3.1. BCH and RS Codes

BCH codes, discovered by Bose, Ray-Chaudhuri, and Hocquenghem et al. [31], are a class of cyclic codes constructed from polynomials in a finite field. The information bits of length $K$ are encoded into a code word of length $N$ with a polynomial in the Galois field. The specific relationship can be expressed as:

$$\begin{array}{l}\mathrm{Code} \mathrm{length}: N=2^m-1\\ \mathrm{Length} \mathrm{of} \mathrm{check} \mathrm{bits}: N-K\le m\times t\\ \mathrm{Hamming} \mathrm{distance}: d\ge 2t-1\end{array}$$

where the positive integers $m\ge 3$, $t<2^{m-1}$, and $t$ denote the number of codewords that can be corrected. It generally employs the bounded distance Berlekamp decoding algorithm. Due to the coding and decoding algorithm’s low complexity and the maximum–minimum distance property to avoid its error leveling problem at low error bit rates, it can be employed in systems with limited power consumption. In 1993, Widmer et al. [32] used a BCH-coded UWACS with 28 check bits added to the message bits to offer extremely reliable communications with an error bit rate of 0.013% in deep water, shallow water, convergence zones, and surface pipeline environments. This guided further development of the system’s error correction capabilities based on BCH codes. In 1999, LR LeBlanc [33] proposed a BCH-coded chirp frequency-shift keying (CFSK) modem for shallow sea UWAC, with BERs of $7\times {10}^{-3}$ and $2\times {10}^{-2}$ at about a 90 m communication distance and SNR of 46 dB and 44 dB, respectively, the feasibility of the BCH code in implementing the UWAC hardware system was verified. In 2001, Stojanovic M et al. [34] employed BCH (63, 10) as an inner code for concatenated RS-BCH-coded multiuser UWAC, as illustrated in Figure 6. The excellent performance of the four mobile users transmitting in severely reverberant environments with a bandwidth expansion factor of seven was verified.

In 2008, Guo et al. [35]. proposed a combination of spread spectrum technology and a BCH-coded code division multiple access (CDMA) UWACS for reliable communication between multiple robots. In 2010, Pompili D et al. introduced a BCH code to an UWAC network and investigated its energy consumption, throughput capacity, and cross-layer optimization [36]. In 2016, Huang et al., for the security of UWAC networks, proposed a protocol that can generate secret keys dynamically based on the channel frequency response (CFR) in orthogonal frequency-division multiplexing systems; BCH codes are used for information reconciliation when the multibit quantization is carried out on the amplitude of each tone [37]. In 2017, Barreto G. et al. improved the energy-efficient communication performance of UWAC networks through the combination and parameter optimization of the BCH and fountain codes. They found that the optimal parameter selection for channel coding is a function of the link distance and the received SNR [38]. In 2021, Niemann E et al. employed the BCH-coded OFDM wideband underwater data communication system in a test tank, achieving zero-error data transmission with a transmission bandwidth of about 280 kHz and a rate of over 200 kbps, and successfully transmitted data over a distance of 200 m in shallow water [39]. Murad M et al. combined BCH codes and XOR ciphering for PAPR reduction of OFDM signals and secure communication, as shown in Figure 7 [10]. They also employed BCH (31, 6)-coded GFDM for UWA communication with a low BER.

It is worth mentioning that Reed–Solomon (RS) codes allow non-binary coding belonging to the BCH family class, which is very effective in dealing with ISI caused by burst errors. Its main attraction is its efficiency increases with the code length [40]. RS codes $(n,k,t)$ are cyclic codes, built from $n$ symbols with a maximum of $n=q-1$, where $q$ is the number of elements in the Galois field ($G{F}^q$ and $q=2^n$). $t$ is the symbol for power-correcting code, so the number of control symbols is $2t$, and the number of information symbols that can be transmitted is $k=n-2t$; the difference with BCH codes is the Hamming distance: $d\ge m_1\times 2t-1$, where $m_1$ is the number of bits carried by each coded symbol. The soft decoding process also uses the Berlekamp and Chien algorithm to correct received symbols [40]. In 2008, ref. [41] developed the UWAC platform TRIDENT (TRansmission d’Images et de Donnes EN Temps réel) to transmit text, images, etc., and the system was tested for its performance using RS codes. As shown in Figure 8, in 2010, the team improved the channel decoding efficiency by combining RS and turbo codes. It tested 300 m, 1000 m, and 2500 transmission distances, three carrier frequencies, and different bit rates, significantly improving system performance compared to RS codes alone [42].

In 2012, Nie X et al. proposed an UWA communication scheme that combines time-reversal processing techniques with RS codes. It effectively reduces ISI and burst errors caused by multi-path effects with reasonable complexity [43]. In 2013, Esmaiel H et al. proposed a high-speed, low-BER underwater image transmission based on RS codes, HQAM modulation, and a ZP-OFDM system without equalizer over a 5 km transmission distance, 40 m water depth, and 8 m/s relative motion speed between the transmitter and the receiver in a simulated environment [44]. In 2014, Diamant R et al. proposed to combine rate-compatible RS codes and the incremental redundancy hybrid automatic repeat request (IR-HARQ) protocol in Haifa port to achieve UWACS performance in terms of throughput and transmission power consumption that is superior to fixed-rate coding schemes [45]. In 2018, Khanai R et al. compared the performance of IDMA-OFDM-MIMO with RS and turbo coding and found that RS codes have higher BERs than turbo codes but consume less power [46]. In 2021, Sklivanitis G et al. found that a suitable combination of channel characteristics and RS codes can securely generate 256-bit encryption keys. The effectiveness of RS codes for UWA physical layer secure applications was verified [47].

In summary, this section reviewed the research of BCH and RS codes applied in UWACSs in chronological order. They both have the advantage of low complexity and can be considered when designing real-time UWACS hardware. BCH codes are effective in reducing the PAPR of OFDM, and RS is very effective in reducing the burst error, which can further improve the performance of UWACSs when combined with different symbol mapping techniques, equalization algorithms, and feedback strategies. Moreover, they have also played a role in the security of the physical layer of UWAC. However, their shortcomings are their flexible code rates and lengths and poor performance compared to the codes in recent years.

3.2. Convolutional Codes

The convolutional codes proposed by Elias in 1955 are described by $(n,m,k)$, where $k$ is the information bits, $m+1$ is the constraint length, $n$ is the code length after coding, and the code rate is $R=k/m$ [48]. The error-correcting ability of convolutional codes increases with $m$ and $k$, but the complexity of the commonly employed decoding Viterbi and BCJR algorithms increases with $m$ [49]. Its application in UWACSs can be traced back to 1989, when Catipovic J et al. developed an acoustic telemetry system, a hardware system using convolutional codes and the MFSK modulation, which operated reliably with a power efficiency of 0.01 Joule/bit in deep and shallow water over a transmission distance of 5 km [50]. In 1990, Catipovic J A et al. further demonstrated that the sequential decoding of long-constrained convolutional codes allows for reliable data telemetry in UWA channels, with an SNR of 14 dB [51]. In 1997, Lam W K et al. employed a convolutional code with a code rate of 1/2 and a constraint length of 7 to combine with the OFDM modulation. Their simulation results demonstrated the excellent BER performance of the coded OFDM with high selective time-frequency and a very low SNR of −9 dB under frequency domain equalization, verifying the advantages of convolutional coding followed by OFDM in UWA communication [52]. In 1998, Stojanovic M et al. combined convolutional codes, interleaving, and spread spectrum techniques to improve UWACS performance [53]. In the same year, Wei-Qing Z et al. used convolutional codes, MFSK, and Viterbi decoding algorithms on AUVs to achieve BERs between ${10}^{-4}$ and ${10}^{-5}$ at transmission rates of 2.5 ks/s and 5.0 ks/s under QPSK modulation in an UWAC test area with a transmission distance of 100–4000 m [54].

In 2001, Blackmon F et al. fed the soft output of a convolutional code back into the DFE to achieve iterative equalization, which improved receiver performance [55]. In 2013, Park J et al. evaluated the effectiveness of code-rate 1/2 convolutional codes in UWA channels. They demonstrated that CC leads to the same Eb/No gain as the theoretical value if the frequency selectivity index is less than about 2.0 [56]. In 2015, de Souza F et al. studied FSK modulation and convolutional codes for UWA sensing networks. They showed that the code rate significantly impacts the overall energy consumption and lifetime of UWAC, suggesting the importance of rate-compatible convolutional codes [57]. In 2016, de Souza F A et al. investigated the impact of using different convolutional codes with rates ranging from $r=$ 0.125 to 0.917 for encoding the multi-hop UWAC, verifying the importance of wide code rates in optimizing the system’s energy consumption [58]. In 2018, Behgam M et al. investigated the effectiveness of the full tail bite (FTB) convolutional code as applied to an UWACS with data blocks of 12, 25, 32, 64, and 512 short packet lengths [59]. As shown in Figure 9, in 2019, Lee H S et al. investigated convolutional coding with code rates of 1/2 and 1/3 and different code lengths in combination with turbo equalization to improve the spectral efficiency and environmental adaptability of UWACSs [60].

Figure 10 depicts a channel-concatenated coding scheme, which was developed in 2022. Channel-concatenated coding was employed to reduce multi-path interference and was designed as iterative joint decoding. The channel-concatenated coding consists of a Hadamard code and a convolutional code. Accordingly, the iterative joint decoding uses a joint Hadamard–Viterbi soft decoding framework and a newly designed branching metric using the Hadamard structure [61].

In 2023, Wang J et al. [62] found that to improve the security of an underwater network, the hash function must be used to authenticate the identity of the legitimate node, and after successful authentication, the measurements of the two communicating parties to be authenticated are obtained via convolutional coding, and the interleaving technique is employed to eliminate the redundancy of the measurements, which effectively reduces the BER of the key. Convolutional codes have been shown to encompass significant advantages in terms of key security and robustness.

In summary, the application of convolutional codes in UWACSs accompanies the development of UWAC technology innovation from the initial fixed code rate hard decoding to soft decoding combined with DFE, turbo equalization, and then the design of the rate-compatible code UWACS to adapt to the environment, combined with other codes to improve the performance of coding and decoding on short packets, and verified the effectiveness in the security of underwater acoustic networks.

3.3. Turbo Codes

In 1993, turbo codes were proposed by C. Berrou et al. by introducing stochastic interweavers into the encoder, parallel cascaded component-wise convolutional encoders, and iterative maximum likelihood decoders. For large blocks, turbo codes are capable of performing within a few tenths of dB from Shannon’s limit [63]. It is revolutionary in coding theory and promotes the development of iterative signal processing. It is worth noting that the resulting turbo equalization further improves the anti-interference capability of UWAC receivers, which is a milestone in their development [64,65]. The structure of a typical turbo encoding and decoding process is shown in Figure 11.

The encoder consists of two parallel encoders, usually recursive systematic convolutional codes with feedback paths (RSC); the interleaver is pseudo-random, whose purpose is to modify the distribution of the code weight and to reduce the correlation of the sequence of information before and after the comparison. It is related to the performance of the whole code. The puncturing process changes the code rate by periodically deleting a number of parity bits. The turbo decoder must interpolate the punctured check bits. It is then fed into a series of convolutional decoders, which iteratively replace the soft decision bits of the decoded outputs to reduce uncertainty and improve overall decoding accuracy as the number of iterations increases. Finally, the decoder produces a soft decision to each message bit in logarithmic form known as a log likelihood ratio (LLR). Typical decoders include SOVA, MAP, Log-MAP, Max-log-MAP, and so on [66]:

$$L_a({\tilde{u}}_k)=\ln \frac{p({\tilde{u}}_k=+1)}{p({\tilde{u}}_k=-1)}$$

(9)

where ${\tilde{u}}_k$ is the decoded bit, and $L_a({\tilde{u}}_k)$ is its log likelihood ratio; the soft decision decoded output is the basis of the iterative equalization for the UWAC receiver. In 2001, Proakis J G. et al. proposed a turbo-coded DFE modem. They demonstrated that the joint decision feedback equalization and turbo decoding are more robust than the turbo equalizer in an UWA sparse channel scenario, thus verifying the feasibility of the turbo code in an UWACS [67]. In 2005, Huang J et al. employed a coded OFDM system with a 1/2 code rate turbo code, which achieved data rates of 9 kbps and 2.8 kbps with BERs lower than ${10}^{-4}$ at distances of 5 km and 10 km, respectively [68]. As shown in Figure 12, in 2007, Roy S et al. proposed layered space–time coding with a wide range code rates of turbo codes, achieving very high spectral efficiencies in a system with many transmit antennas [69].

In 2008, Qiao G et al. compared turbo and convolutional codes with different code rates under different modulations and SNRs via simulation and demonstrated the superior performance of turbo codes under the UWA channel [14]. Since then, Labrador Y et al. investigated the performance of turbo codes combined with higher-order modulation (8 PSK and 16 QAM) under Rician multi-path fading channels. This system’s performance was further improved by combining it with a DFE equalizer [70]. As shown in Figure 13, in 2015, Iruthayanathan N et al. introduced the turbo code in MIMO-OFDM systems, which combines iterative decoding with Log-MAP and a non-linear detector with MMSE-OSIC to mitigate the effects of ISI and acoustic interference effectively and to achieve higher data rates [71].

In 2018, Bocus M J et al. [72] employed 1/3 code rate turbo-coded NOMA-OFDM and NOMA-FBMC systems for a two-user scenario, where both users use the same frequency bandwidth to achieve reliable real-time video transmission between ROVs. Later, the team employed 1/2 code rate turbo-coded OTFS to achieve up to 198.7 kbps between every ROV in the same bandwidth under time-varying UAC over a 1 km vertical UWA channel [73]. In 2022, Minaeva O N et al. proposed a product turbo encoder; the information is encoded line by line using the first encoder, and after interleaving, column by column using the second encoder. Combined with OCDM modulation, its performance outperformed conventional convolutional coding structures [74]. Yang Y et al. [75] designed a high-speed OFDM UWACS based on turbo codes, implemented a fully parallel turbo decoder on an FPGA, and detailed the hardware implementation of the decoder, the design of the algorithmic processing unit, and the interleaving module. In the same year, Jeong H W et al. [76] proposed non-coherent turbo-coded FSK for an efficient receiver structure in combination with spread spectrum technology for high-performance underwater covert communications.

In summary, the research of turbo codes in UWAC ranges from combining single-carrier to multi-carrier communication, optimizing and modifying the coding structure, designing rate-compatible codes to improve coding efficiency, and trying to realize real-time data transmission on the hardware system. More importantly, turbo equalization has significantly contributed to the UWA reducing ISI while combined with DFE, which has always been a research hotspot in designing a robust receiver.

3.4. LDPC Codes

LDPC codes were initially introduced by Gallager in 1962. It was rediscovered in the 1990s, when researchers began to investigate code-on-graph-based and iterative decoding [77]. The performance of non-regular LDPC codes with iterative belief propagation (BP) decoding of a long code length is very close to the Shannon limit. In addition, it can be implemented in hardware for parallel decoding that is rate compatible and has been adopted for data channels in 5G communications, in which two base graphs are designed for the LDPC coding scheme. BG 1 is employed for high data rates and long code lengths; BG 2 is employed for low and short code lengths [78]. LDPC is represented by $(N,K)$, where $K$ is the information bits, and $N$ is the coded code length, which is encoded by a generator matrix ($G$) and a sparse parity check matrix ($H$). Assuming that the parity check matrix $H$ $(3,6)$ is as in Equation (10):

$$H=\left[\begin{array}{cccccc}1& 0& 0& 1& 1& 0\\ 0& 1& 1& 0& 0& 0\\ 1& 0& 0& 0& 1& 1\end{array}\right]$$

(10)

Gaussian cancellation of $H$:$H=[I|P^T]$; then, the generator matrix is $G=[P^T|I]$. The encoded bits are: $C=uG$ and $C{H}^T=0$; $u$ denotes the information bits. However, since $G$ is a non-sparse matrix, this encoding method has high complexity. Therefore, geometric construction methods [79], stochastic construction methods [80], and QC-LDPC [81] have been developed. It is worth mentioning that the Tanner graph, to represent $H$, is the basis for the rediscovery of LDPC codes and an important method for the optimal design of iterative decoding schemes [82]. For example, the Tanner graph representation of Equation (10) is as follows:

The solid circles in Figure 14 are the parity check nodes, and those in the hollow circles are the variable nodes. The variable nodes connected to a parity check node satisfy the relation, e.g., $c_1=v_1+v_4+v_5$. In the design of LDPC codes, the existence of short cycles should be avoided as much as possible so that the minimum length of the cycles is greater than four. The decoding algorithms for LDPC codes are based on Tanner graphs. The most classical one is the BP decoding algorithm, which is an iterative algorithm that uses the log likelihood probability as a message to pass between the nodes until $H{\tilde{u}}_k=0$ or it reaches a pre-set number of decoding iterations, and ${\tilde{u}}_k$ is the decoded output. It has superior performance but high complexity [83]. In 2007, Li, Baosheng et al. introduced a regular LDPC cycle code over GF (64) with a 1/2 code rate into a MIMO-OFDM UWACS; with a 12 kHz bandwidth, the overall data rate was 12.18 kbps, and its performance was found to be much better than that of a convolutional code with the same code rate [84]. Notably, as shown in Figure 15, in 2008, Huang J et al. developed a new method for constructing non-binary regular and irregular LDPC codes, which solved two major problems in OFDM: (1) the poor performance of regular (or uncoded) OFDM in fading channels; and (2) the high peak-to-average power ratio (PAPR) for OFDM transmissions, confirming that the proposed non-binary LDPC codes have the multicarrier UWACS [15].

In 2009, Chen Y et al. derived the probability density function (PDF) of the initial decoded message of the BP algorithm for LDPC codes based on the shallow sea channel model. The effects of multi-path, channel fading, and interleaver on the decoding performance were simulated and analyzed, and their performance was found to be superior to turbo codes [85].

Kang T et al. proposed an iterative LDPC-coded OFDM receiver with CFO and channel estimation, using EXIT charts to optimize the iterative processing between the pilots and the data, and simulations and shallow sea experiments proved the effectiveness of the LDPC code combined with an iterative receiver [86]. In 2012, Qi X K et al. proposed a construction scheme for reversible quasi-cyclic-low-density parity-check (QC-LDPC) codes, which solves the problems of singular parity-check matrices and the high coding complexity in the traditional QC-LDPC codes. The proposed construction outperformed random LDPC in UWAC when the code length was short, and simulations showed that the reversible QC-LDPC code dynamically improved the system’s robustness [87]. In 2013, Tao J et al. introduced LDPC and turbo equalization into a MIMO-OFDM UWACS. They experimentally investigated the selection of the number of subcarriers, iterations of turbo equalization, and LDPC decoding to achieve a trade-off between complexity and performance [88]. In 2014, Chen Y et al. proposed a dynamic coded cooperation (DCC) scheme based on non-binary rate-compatible LDPC-coded OFDM modulation. Simulations and sea tests were carried out to demonstrate the real-time operation of OFDM-DCC in a three-node network [89]. In 2016, Song A et al. proposed the iterative exchange of soft information between a LDPC decoder and a DFE. Using differential evolutionary techniques, they developed an EXIT-assisted method for optimizing LDPC codes for UWA channels [90]. In 2017, Li D et al. proposed a concatenated code based on non-binary LDPC codes and Hadamard code for a non-coherent UWACS, and simulation and experimental studies found that there was a 0.4 dB gain after the concatenation of the two codes, which verifies its advantages in practical applications [91]. In 2020, Padala S K et al. investigated the performance of a spatially coupled (SC) LDPC code based on the protograph for shallow water UWA communication with a communication distance of 1000 m and a channel bandwidth of 10 KHz via simulation. The results showed that the SC-LDPC code improved the performance by 1 dB over the LDPC code with the same delay constraints and a BER of ${10}^{-3}$ [92]. In 2021, Wei X F et al. employed QC-LDPC, combined with spread-spectrum and higher-order modulation techniques, and decoded using log likelihood ratio belief propagation (LLR-BP) to ensure that the system operated reliably under low SNR conditions [93]. In 2023, Zhao Z et al. addressed the problem that the decoding time of LDPC codes needs to be shorter, leading to inefficient communication and being unsuitable for short-block data transmission. As shown in Figure 16, combining the short-block LDPC code with the direct sequence spread spectrum and soft spread spectrum in the UWACS solves the problem that the traditional LDPC code is not applicable, and the simulation tests and pool experiments were verified [94].

In summary, this section reviewed the research of LDPC codes in UWA communication from the initial application research and validation of regular LDPC codes combined with different modulations. Then, introducing non-binary LDPC codes into a MIMO-OFDM UWACS was able to improve system performance and reduce OFDM’s PAPR. The research on the rate-compatible scheme of the LDPC code, which develops the design of the QC-LDPC code with low coding and decoding complexity, flexible code rates and lengths, and its effectiveness in the UWACS, was fully verified; in addition, the concatenation of LDPC codes and other codes can further improve the system’s BER performance. Due to the demand for UWACSs for short-block data transmission and the high complexity of LDPC decoding, the current research mainly considers reducing the decoding complexity and improving the performance on short-block data.

3.5. Polar Codes

In 2009, Arikan E introduced a low-complexity coding method based on channel polarization, which employed successive cancellation (SC) decoding with complexity $O(N\log N)$ and infinite code length, which was mathematically proven in the binary-input discrete memoryless symmetric channel (B-DMC) to be able to reach the Shannon capacity limit [96]. The polar code is constructed using channel polarization by making $N=2^n$ independent copies of a given channel ($W$). After channel merging and splitting, the capacity of $W$ converges to one and zero, as shown in Figure 17. The information bits are then transmitted on the sub-channels with a channel capacity of one, and the bits (freeze bits) that are known to both the transmitter and the receiver are placed on the sub-channels with a channel capacity of zero.

The polar code is denoted by $(N,K)$, where $N$ is the coded code length, $K$ is the information bits, and $N-K$ is the freeze bits. The fundamental element of the code construction is shown in Figure 18.

The polar code can be mathematically expressed as:

$$\left[\begin{array}{c}{c}_0\\ c_1\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{c}{u}_0\\ u_1\end{array}\right]$$

(11)

The construction of the polar code of code length $N=2^n$ can be obtained using $n$ Kronecker products of the basic element as follows:

$$\left[\begin{array}{c}{c}_0\\ c_1\\ ⋮\\ c_N\end{array}\right]=\left[\begin{array}{c}{u}_0\\ u_1\\ ⋮\\ u_N\end{array}\right]{\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]}^{\otimes n}B$$

(12)

where $B$ is the bit reversal permutation matrix. After constructing code $C$, it is necessary to determine the index of the information and frozen bits. Usually, there are Bhattacharyya parameter, Gaussian approximation, density evolution, Monte Carlo [97], polarization weight (PW), etc., methods available. It is worth mentioning that Huawei proposed the PW, a sub-channel reliability assessment method independent of channel characteristics, which is suitable for constructing polar codes for dynamic UWA channels [95]. In polar code decoding with successive cancellation (SC) decoding, which requires sufficiently large code lengths, the block error probability decays exponentially in the square root of the code length. The recursive nature of the SC decoding may impose a large latency depending on the implementation. A major improvement in decoding performance can be achieved using successive cancellation list (SCL) decoding, which keeps a list of most likely decoding paths at all times, unlike the SC decoder, which only keeps one decoding path; that is, it performs a symbol-wise hard decision at each decoding stage. A significant improvement to SCL is the cyclic redundancy check (CRC)-aided SCL (CA-SCL), where the message is encoded with a CRC and then polar coded [98]. So far, CRC-SCL is currently the outstanding decoding method for polar codes.

In 2019, Qiao G et al. employed a rate-compatible polar code to create a nested code structure, which was divided into several subsets (codebook). Each user assigns an independent codebook without successive interference cancellation (SIC) to improve the utilization efficiency of OFDM subcarriers. Simulations and tank experiments verified the effectiveness of the polar codes for OFDM multiuser UWAC [99]. As shown in Figure 19, in 2020, Zhai Y et al. investigated the performance of polarization coding in an OFDM-UWACS both theoretically and simulatively. The simulation results revealed that the BER of polar codes with a code rate of 1/2 can reach ${10}^{-4}~{10}^{-5}$ in underwater time-varying channels with a SNR of 4 dB, which is about 0.5–1 dB higher than that of LDPC codes and turbo codes. In addition, the BER performance of polar codes was compared with different UWA channel models and parameters, code lengths, and code rates, and the validity of the polar codes for the UWA channel was verified [100].

Meanwhile, Falk M et al. compared the performance of mainstream code words in short-block data for UWAC in the competition “Wanted: Best channel codes for short underwater messages”. In addition, non-binary polar codes were studied and found to have the best performance, effectively promoting polar codes’ application in UWACSs [101]. In 2022, Zhou C et al. proposed a joint source and channel coding method based on polar codes, which improves the transmission efficiency by adopting coding with strong error correction capability for important streams and poor or no error correction capability for non-important streams, according to the different importance of the source output streams [102]. Yushuang Z et al. proposed a polar coding construction algorithm for the UWA channel based on the Monte Carlo method; they then optimized Monte Carlo statistical parameters based on the dynamical cognitive of the channel, jointing decoding, and decision feedback channel estimation. It achieved error-free transmission at a signal-to-noise ratio of about 14 dB and a communication distance of about 1 km. Also, it exhibited better error correction than the LDPC-coded modulation system under the same circumstances [103]. Also, Chen R et al. constructed several optimal polar codes with different code lengths and rates based on the optimized Monte Carlo (MC) method for UWA channels, and their performance was found to be better than that of the LDPC-coded OFDM communication system in shallow sea experiments [104]. As shown in Figure 20, in 2023, Liu F et al. proposed a polar coding scheme with less feedback and low complexity based on adaptive equalization. A hybrid automatic repeat request (HARQ) mechanism was provided to overcome the influence of estimation errors. Simulation results demonstrated better BER performance and lower decoding computational complexity than turbo equalization [105].

Yang X et al. proposed a real-time polar-coded orthogonal signal-division multiplexing modem for UWAC, including hardware and software. It achieves error-free accuracy with a data rate of 4.35 kb/s using the minimum mean square error (MMSE) equalizer and polar decoding under slow, time-varying channels, and the communication range reaches more than 5 km [106].

In summary, this section provided an overview of the research on polar codes in UWAC. Polar codes are widely applied in wireless communication due to its low complexity and superior performance. In UWAC, the feasibility is first verified, and the adaptability is studied; then, it is employed with channel equalization and source coding to improve the system’s performance. Meanwhile, it has also been implemented in a hardware real-time transmission system. Current research focuses on the construction of polar codes in UWA channels.

4. Discussion and Research Directions

Most underwater acoustic communication algorithms come from wireless communication systems, and the theoretical development of wireless communication has been extremely mature. Its new frontiers in mobile, ad hoc, and wireless computing can be used for reference to design UWA algorithms with low latency, low energy consumption, etc. [107], in which channel coding is the most critical part of the design process of UWACSs and is employed to reduce ISI. The mainstream algorithms BCH codes, RS codes, convolutional codes, turbo codes, LDPC codes, and polar codes have accompanied the development of UWA communication technology from the initial practical verification of the algorithms to the applicability research. Afterward, their innovation and development were combined with other modules according to the characteristics of the algorithms and communication tasks, and many outstanding works appeared. Moreover, other coding algorithms also play essential roles, such as packet coding and pattern delay difference coding. UWA channels have severe time-varying characteristics, making them unsuitable for transmitting long-block data. It is necessary to compare the performance of different codes to transmit short-block data in UWA channels.

The above review work in previous sections, including Figure 21 and Figure 22, has shown that LDPC and polar codes outperform the other codes in error performance, complexity, and latency on short-block data. Consequently, a profound study of them based on the characteristics of UWA channels is meaningful and promising for improving UWACS performance.

However, the noise introduced in the above simulation work was Gaussian white noise or t-distribution noise [108]. To further verify the effectiveness of LDPC and polar codes in UWACSs, the BER performance of the coded BPSK modulation system under real noise was verified. To collect real noise, as depicted in Figure 23, the field experiments were carried out in the sea area of Xiamen Harbor, Fujian Province, under an average water depth of about 12 m. During the trial, the transmitter and receiver were 3 m in the water, and the communication distance was 1.1 km. The UWA communication system has a sampling frequency of 75 kHz. It transmits signals with a carrier frequency and bandwidth of 15.5 kHz and 5 kHz, respectively.

The code length of the LDPC code was 1052, and its code rates were 1/2, 2/3, 3/4, and 5/6. The codelength of the polar code was 1024, and its code rates were 1/2, 2/3, 3/4, and 7/8. The specific design of two codes will be introduced in our future work. As shown in Figure 24, under the real noise from the shallow sea with BPSK modulation, the polar codes have better performance.

In summary, research on applying LDPC and polar codes in UACS based on underwater acoustic channels is significant. Therefore, a number of potential research directions are introduced.

4.1. Low-Complexity Channel Coding

UWAC modems are primarily integrated into mobile AUVs, ROVs, observation network buoys, subsea network nodes, etc., which place demands on real-time hardware overhead and energy efficiency of UWACSs. Therefore, introducing LDPC and polar codes in the design process of UWACSs requires considering coding and decoding algorithm complexity and latency.

4.2. Joint Channel Coding and Other Modules

Most of the algorithms of each module in the design process of UWACSs originated from wireless communication. Wireless channels have deterministic and unified channel models, a crucial guide for designing and optimizing communication algorithms. The UWA channel exhibits characteristics of random complexity and dynamic time-varying nature, and despite extensive research having been conducted, a unified channel model has yet to be established. Without a unified channel model, each module’s algorithm design and parameter optimization are based on experience, resulting in sub-optimal performance. Moreover, the algorithms of each module are designed and optimized independently and separately in UWACSs traditionally, resulting in the poor performance of the whole communication system. Relevant research has found that both LDPC and polar codes show that the joint optimization of each module can improve the system’s overall performance. From the requirements of UWA communication on the algorithm’s reliability, the joint optimization of the coding and decoding algorithm with other modules is an effective method. Exploiting UWA channel sparsity, relevant schemes can be researched in the future, such as joint channel coding and modulation, joint channel estimation and decoding, and joint source and channel coding. The deep unfolding (driven by joint data and models) method for optimizing joint modules is also worth mentioning.

4.3. Rate-Compatible Codes for Short-Block Data

The dynamic time-varying characteristics of the marine environment make UWACSs unsuitable for long-block data transmission and require environmental adaptability, especially in shallow seas. Therefore, UWACSs require low latency and flexibility in channel coding schemes. In addition, code rate-compatible codes are necessary to fully incorporate the channel capacity, improve transmission efficiency, and reduce the number of data transmission failures. For short-block data, the polar codes were found to perform better than other codes. However, it possesses the disadvantage of having a fixed code length of $2^N$, so further research is needed on the code rate-compatible scheme and how it can be combined with the HARQ scheme to reduce the number of failed retransmissions and improve the environmental adaptability of the system. In addition, the construction of excellent codes based on the characteristics of the UWA channel also needs to be studied in depth; the research of non-binary polar codes also further deserves attention to improve the performance of the codes.

5. Conclusions

This paper introduced the brief principles of mainstream channel coding schemes and reviewed their research in the field of UWA communication. Previous investigations have revealed that compared with BCH codes, RS codes, convolutional codes, turbo codes, LDPC codes, and polar codes generally exhibit superior performance and lower computational complexity. Different types of codes have made good progress according to their own characteristics and communication task requirements.

Overcoming ISI in underwater acoustic channels and achieving efficient and reliable information transmission has always been a critical research hotspot in the field of UAC. The following trends in underwater acoustic communication channel coding technology with rate-compatible and low complexity need to be considered to meet the requirements for the development of powerful three-dimensional marine information systems:

• Under the uncertain marine environment, the fixed data rates of the underwater acoustic communication system make it difficult to ensure stable and efficient data transmission. Design low latency, little feedback, or no feedback modulation transmission strategies based on wide rate polar codes to fully utilize channel capacity, improve underwater acoustic transmission efficiency, and adapt to the environment and tasks.
• Considering the UACS’s energy resource constraints, the complexity of the coding algorithms needs to be optimized and combined with other modules to achieve an overall optimization of the system with less hardware and software overhead.
• Non-binary LDPC codes have been successfully applied to design underwater acoustic communication systems [15]. Research on the excellent characteristics of polar codes, as well as the non-binary polar codes, is noteworthy. Moreover, it is necessary to fully optimize encoding and decoding structures by studying the characteristics of underground acoustic channels in both deep and shallow water.

In summary, the research on underwater acoustic channel coding algorithms must start with their structural characteristics, combine with the characteristics of underwater acoustic channels, and design systems with low energy consumption, high performance, and environmental adaptability.