Channel Codes for Underwater Acoustic Communications: Comparison
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Please note this is a comparison between Version 2 by Sirius Huang and Version 1 by Yuehai Zhou.

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.

  • underwater acoustic communication systems
  • channel coding
  • polar codes
  • LDPC codes

1. Introduction

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. The following sections describe the mainstream coding schemes used in developing and researching UWA communication technologies, including fixed-rate and rate-compatible codes.

2. BCH and RS Codes

BCH codes, discovered by Bose, Ray-Chaudhuri, and Hocquenghem et al. [31][1], 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:

Code   length :   N = 2 m − 1 Length   of   check   bits :   N − K ≤ m × t Hamming   distance :   d ≥ 2 t − 1

where the positive integers 𝑚≥3, 𝑡<2𝑚−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][2] 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][3] proposed a BCH-coded chirp frequency-shift keying (CFSK) modem for shallow sea UWAC, with BERs of 7×10−3 and 2×10−2 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][4] employed BCH (63, 10) as an inner code for concatenated RS-BCH-coded multiuser UWAC, as illustrated in Figure 61. The excellent performance of the four mobile users transmitting in severely reverberant environments with a bandwidth expansion factor of seven was verified.

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Figure 61.
A concatenated coding method with the RS and BCH codes.
In 2008, Guo et al. [35][5]. 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][6]. 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][7]. 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][8]. 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][9]. 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]2 [10]. They also employed BCH (31, 6)-coded GFDM for UWA communication with a low BER.
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Figure 72.
A BCH-coded OFDM UWACS. (red box represents the channel codes employed by the system).
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][11]. 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 (𝐺𝐹𝑞 and 𝑞=2𝑛). 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≥𝑚1×2𝑡−1, where 𝑚1 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][11]. In 2008, ref. [41][12] 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 83, 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][13].
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Figure 83.
The UWAC platform TRIDENT and the RS coding scheme.
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][14]. 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][15]. 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][16]. 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][17]. 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][18].
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. 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][19]. 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][20]. 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][21]. 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][22]. 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][23]. In 1998, Stojanovic M et al. combined convolutional codes, interleaving, and spread spectrum techniques to improve UWACS performance [53][24]. 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][25].
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][26]. 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][27]. 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][28]. 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][29]. 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][30]. As shown in Figure 94, 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][31].
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Figure 94.
The structure of the convolutional-coded AMC transceiver model.
Figure 105 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][32].
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Figure 105.
A channel-concatenated scheme with convolution and Hadamard coding.
In 2023, Wang J et al. [62][33] 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.

4. 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][34]. 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][35][36]. The structure of a typical turbo encoding and decoding process is shown in Figure 116.
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Figure 116.
Encoder and decoder of a turbo code. (
a
) Encoder. (
b
) Decoder.
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][37]:
 
L a ( u ˜ k ) = ln p ( u ˜ k = + 1 ) p ( u ˜ k = − 1 )

where 𝑢˜𝑘 is the decoded bit, and 𝐿𝑎(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][38]. 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][39]. As shown in Figure 127, 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][40].

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Figure 127.
Layered space–time coding with turbo encoder.
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][41]. 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][42]. As shown in Figure 138, 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][43].
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Figure 138.
The turbo-coded MIMO-OFDM system.
In 2018, Bocus M J et al. [72][44] 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][45]. 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][46]. Yang Y et al. [75][47] 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][48] 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.

5. 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][49]. 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][50]. 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 (3):
L a ( u ˜ k ) = ln p ( u ˜ k = + 1 ) p ( u ˜ k = − 1 )
Gaussian cancellation of 𝐻:𝐻=[𝐼|𝑃𝑇]; then, the generator matrix is 𝐺=[𝑃𝑇|𝐼]. The encoded bits are: 𝐶=𝑢𝐺 and 𝐶𝐻𝑇=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][51], stochastic construction methods [80][52], and QC-LDPC [81][53] 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][54]. For example, the Tanner graph representation of Equation (10) is as follows:
The solid circles in Figure 149 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., 𝑐1=𝑣1+𝑣4+𝑣5. 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 𝐻𝑢˜𝑘=0 or it reaches a pre-set number of decoding iterations, and 𝑢˜𝑘 is the decoded output. It has superior performance but high complexity [83][55]. 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][56]. Notably, as shown in Figure 150, 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][57].
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Figure 149.
Tanner graph of the parity check matrix.
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Figure 150.
Non-binary LDPC-coded OFDM system.
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][58].
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][59]. 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][60]. 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][61]. 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][62]. 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][63]. 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][64]. 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][65]. 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][66]. 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 shCown 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][67].
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.

6. 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(𝑁log𝑁) 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][68]. The polar code is constructed using channel polarization by making 𝑁=2𝑛 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 171. 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.
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Figure 171.
Channel polarization with
N
= 1024.
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 182.
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Figure 182.
Basic element for constructing the polar code.
The polar code can be mathematically expressed as:
 
[ c 0 c 1 ] = [ 1 1 0 1 ] [ u 0 u 1 ]
The construction of the polar code of code length 𝑁=2𝑛

can be obtained using 𝑛 Kronecker products of the basic element as follows:

 
[ c 0 c 1 ⋮ c N ] = [ u 0 u 1 ⋮ u N ] [ 1 1 0 1 ] ⊗ n B

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][69], 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][70]. 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][71]. 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][72]. As shown in Figure 193, 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][73].
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Figure 193.
A polar-coded OFDM-UWACS.
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][74]. 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][75]. 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][76]. 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][77]. As shown in Figure 2014, 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][78].
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Figure 2014.
The polar coding scheme with the HARQ mechanism.
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][79].
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.

References

  1. Gorenstein, D.; Zierler, N. A class of error-correcting codes in pm symbols. J. Soc. Ind. Appl. Math. 1961, 9, 207–214.
  2. Widmer, H.P.; Lev, E.; Costabile, J.J. Demonstration of a Low-Frequency, Long-Range Acoustic Communications System. Johns Hopkins APL Tech. Dig. 1993, 14, 174.
  3. LeBlanc, L.R.; Beaujean, P.P.; Singer, M.; Boubli, C.; Strutt, G.T. Chirp FSK modem for high reliability communication in shallow water. In Proceedings of the Oceans’ 99. MTS/IEEE. Riding the Crest into the 21st Century. Conference and Exhibition. Conference Proceedings (IEEE Cat. No. 99CH37008), Seattle, WA, USA, 13–16 September 1999; Volume 1, pp. 222–227.
  4. Stojanovic, M.; Freitag, L. Multiuser undersea acoustic communications in the presence of multipath propagation. In Proceedings of the MTS/IEEE Oceans 2001. An Ocean Odyssey. Conference Proceedings (IEEE Cat. No. 01CH37295), Honolulu, HI, USA, 5–8 November 2001; Volume 4, pp. 2165–2169.
  5. Guo, S.; Zhao, Z.; Pan, Q. A CDMA acoustic communication system for multiple underwater robots. In Proceedings of the 2008 IEEE International Conference on Robotics and Biomimetics, Bangkok, Thailand, 22–25 February 2009; pp. 1522–1526.
  6. Pompili, D.; Akyildiz, I.F. A multimedia cross-layer protocol for underwater acoustic sensor networks. IEEE Trans. Wirel. Commun. 2010, 9, 2924–2933.
  7. Huang, Y.; Zhou, S.; Shi, Z.; Lai, L. Channel frequency response-based secret key generation in underwater acoustic systems. IEEE Trans. Wirel. Commun. 2016, 15, 5875–5888.
  8. Barreto, G.; Simão, D.H.; Pellenz, M.E.; Souza, R.D.; Jamhour, E.; Penna, M.C.; Brante, G.; Chang, B.S. Energy-efficient channel coding strategy for underwater acoustic networks. Sensors 2017, 17, 728.
  9. Niemann, E.; Nicolaides, K.; Dhuness, K. A prototype broadband acoustic underwater data communication system for high data rates. In Proceedings of the OCEANS 2021, San Diego, CA, USA, 20–23 September 2021; pp. 1–6.
  10. Murad, M.; Tasadduq, I.A.; Otero, P. Ciphered BCH Codes for PAPR Reduction in the OFDM in Underwater Acoustic Channels. J. Mar. Sci. Eng. 2022, 10, 91.
  11. Reed-Solomon Codes and Their Applications; John Wiley & Sons: Hoboken, NJ, USA, 1999.
  12. Trubuil, J.; Goalic, A.; Beuzelin, N. Synchronization and channel coding in shallow water acoustic communication. In Proceedings of the OCEANS 2008, Quebec, QC, Canada, 15–18 September 2008; pp. 1–5.
  13. Goalic, A.; Trubuil, J.; Laot, C.; Beuzelin, N. Underwater acoustic communication using Reed Solomon Block Turbo Codes channel coding to transmit images and speech. In Proceedings of the OCEANS 2010 MTS/IEEE SEATTLE, Seattle, WA, USA, 20–23 September 2010; pp. 1–6.
  14. Nie, X.; Xu, W.; Zheng, J. Time reversal acoustic communication with concatenated reed-solomon code. In Proceedings of the 2012 Oceans-Yeosu, Yeosu, Republic of Korea, 21–24 May 2012; pp. 1–4.
  15. Esmaiel, H.; Jiang, D. Image transmission over underwater acoustic environment using OFDM technique with HQAM mapper. In Proceedings of the 2013 IEEE Third International Conference on Information Science and Technology (ICIST), Yangzhou, China, 23–25 March 2013; pp. 1596–1601.
  16. Diamant, R.; Lampe, L. Adaptive error-correction coding scheme for underwater acoustic communication networks. IEEE J. Ocean. Eng. 2014, 40, 104–114.
  17. Khanai, R.; Shahapur, S.S.; Torse, D. Performance analysis of underwater acoustic communication using IDMA-OFDM-MIMO with Reed Solomon and turbo code. Int. J. Comput. Netw. Inf. Secur. 2018, 10, 41–46.
  18. Sklivanitis, G.; Pelekanakis, K.; Yıldırım, S.A.; Petroccia, R.; Alves, J.; Pados, D.A. Physical layer security against an informed eavesdropper in underwater acoustic channels: Reconciliation and privacy amplification. In Proceedings of the 2021 Fifth Underwater Communications and Networking Conference (UComms), Lerici, Italy, 31 August–2 September 2021; pp. 1–5.
  19. Viterbi, A. Convolutional codes and their performance in communication systems. IEEE Trans. Commun. Technol. 1971, 19, 751–772.
  20. Moon, T.K. Error Correction Coding: Mathematical Methods and Algorithms; John Wiley & Sons: Hoboken, NJ, USA, 2020.
  21. Catipovic, J.; Deffenbaugh, M.; Freitag, L.; Frye, D. An acoustic telemetry system for deep ocean mooring data acquisition and control. In Proceedings of the Oceans, Seattle, WA, USA, 18–21 September 1989; Volume 3, pp. 887–892.
  22. Catipovic, J.A.; Baggeroer, A.B. Performance of sequential decoding of convolutional codes over fully fading ocean acoustic channels. IEEE J. Ocean. Eng. 1990, 15, 1–7.
  23. Lam, W.K.; Ormondroyd, R.F. A coherent COFDM modulation system for a time-varying frequency-selective underwater acoustic channel. In Proceedings of the Seventh International Conference on Electronic Engineering in Oceanography—Technology Transfer from Research to Industry, Southampton, UK, 23–25 June 1997.
  24. Stojanovic, M.; Proarkis, J.G.; Rice, J.A.; Rice, J.A.; Green, M.D. Spread spectrum underwater acoustic telemetry. In Proceedings of the IEEE Oceanic Engineering Society. OCEANS’98. Conference Proceedings (Cat. No. 98CH36259), Nice, France, 28 September–1 October 1998; Volume 2, pp. 650–654.
  25. Zhu, W.-Q.; Wang, C.-H.; Pan, F.; Zhu, M.; Wang, R.; Zhang, X.-J.; Dai, Y.-M. Underwater acoustic communication system of AUV. In Proceedings of the IEEE Oceanic Engineering Society. OCEANS’98. Conference Proceedings (Cat. No. 98CH36259), Nice, France, 28 September–1 October 1998; Volume 1, pp. 477–481.
  26. Blackmon, F.; Sozer, E.; Murandian, M.; Proakis, J.; Salehi, M. Performance comparison of iterative/integral equalizer/decoder structures for underwater acoustic channels. In Proceedings of the MTS/IEEE Oceans 2001. An Ocean Odyssey. Conference Proceedings (IEEE Cat. No. 01CH37295), Honolulu, HI, USA, 5–8 November 2001; Volume 4, pp. 2191–2200.
  27. Park, J.; Seo, C.; Park, K.C.; Yoon, J.R. Effectiveness of convolutional code in multipath underwater acoustic channel. Jpn. J. Appl. Phys. 2013, 52, 07HG01.
  28. de Souza, F.; Souza, R.; Brante, G.; Pellenz, M.; Rosas, F.; Chang, B. Code rate optimization for energy efficient delay constrained underwater acoustic communications. In Proceedings of the OCEANS 2015-Genova, Genova, Italy, 18–21 May 2015; pp. 1–4.
  29. de Souza, F.; Souza, R.; Brante, G.; Souza, R.D.; Pellenz, M.E.; Rosas, F. Optimizing the number of hops and retransmissions for energy efficient multi-hop underwater acoustic communications. IEEE Sens. J. 2016, 16, 3927–3938.
  30. Behgam, M.; Zheng, Y.R.; Liu, Z. Coding for short messages in multipath underwater acoustic communication channels. In Proceedings of the OCEANS 2018 MTS/IEEE Charleston, Charleston, SC, USA, 22–25 October 2018; pp. 1–5.
  31. Lee, H.S.; Jung, J.W.; Baek, C.U.; Lee, A.H.; Kim, W.J. Adaptive modulation and coding for underwater acoustic communication. In Proceedings of the 2019 Eleventh International Conference on Ubiquitous and Future Networks (ICUFN), Zagreb, Croatia, 2–5 July 2019; pp. 54–56.
  32. Wu, F.Y.; Tian, T.; Su, B.X.; Song, Y.C. Hadamard–Viterbi Joint Soft Decoding for MFSK Underwater Acoustic Communications. Remote Sens. 2022, 14, 6038.
  33. Wang, J.; Zhang, S.; Luo, P.; Zhao, X. A Strong Security Key Agreement Scheme for Underwater Acoustic Networks. In Proceedings of the International Conference on Innovative Computing, Singapore, 1–3 February 2023; Springer Nature Singapore: Singapore, 2023; pp. 666–677.
  34. Berrou, C.; Glavieux, A.; Thitimajshima, P. Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1. In Proceedings of the ICC’93-IEEE International Conference on Communications, Geneva, Switzerland, 23–26 May 1993; Volume 2, pp. 1064–1070.
  35. Tuchler, M.; Koetter, R.; Singer, A.C. Turbo equalization: Principles and new results. IEEE Trans. Commun. 2002, 50, 754–767.
  36. Zheng, Y.R.; Wu, J.; Xiao, C. Turbo equalization for single-carrier underwater acoustic communications. IEEE Commun.-Tions Mag. 2015, 53, 79–87.
  37. Jasim, S.; Abbas, A. Performance of turbo code with different parameters. J. Univ. Babylon 2017, 25, 1684–1692.
  38. Sozer, E.M.; Proakis, J.G.; Blackmon, F. Iterative equalization and decoding techniques for shallow water acoustic channels. In Proceedings of the MTS/IEEE Oceans 2001. An Ocean Odyssey. Conference Proceedings (IEEE Cat. No. 01CH37295), Honolulu, HI, USA, 5–8 November 2001; Volume 4, pp. 2201–2208.
  39. Huang, J.; Sun, J.; He, C.; Sheng, X.H.; Zhang, Q.F. High-speed underwater acoustic communication based on OFDM. In Proceedings of the 2005 IEEE International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, Beijing, China, 8–12 August 2005; Volume 2, pp. 1135–1138.
  40. Roy, S.; Duman, T.M.; McDonald, V.; Proakis, J.G. High-rate communication for underwater acoustic channels using multiple transmitters and space–time coding: Receiver structures and experimental results. IEEE J. Ocean. Eng. 2007, 32, 663–688.
  41. Xu, X.; Qiao, G.; Su, J.; Su, J.; Hu, P.; Sang, E. Study on turbo code for multicarrier underwater acoustic communication. In Proceedings of the 2008 4th International Conference on Wireless Communications, Networking and Mobile Computing, Dalian, China, 12–14 October 2008; pp. 1–4.
  42. Labrador, Y.; Karimi, M.; Pan, D.; Miller, J. Modulation and error correction in the underwater acoustic communication channel. Int. J. Comput. Sci. Netw. Secur. 2009, 9, 123–130.
  43. Iruthayanathan, N.; Vishvaksenan, K.S.; Rajendran, V.; Mohankumar, S. Performance analysis of turbo-coded MIMO–OFDM system for underwater communication. Comput. Electr. Eng. 2015, 43, 1–8.
  44. Bocus, M.J.; Agrafiotis, D.; Doufexi, A. Non-orthogonal multiple access (NOMA) for underwater acoustic communication. In Proceedings of the 2018 IEEE 88th Vehicular Technology Conference (VTC-Fall), Chicago, IL, USA, 27–30 August 2018; pp. 1–5.
  45. Bocus, M.J.; Doufexi, A.; Agrafiotis, D. Performance of OFDM-based massive MIMO OTFS systems for underwater acoustic communication. IET Commun. 2020, 14, 588–593.
  46. Minaeva, O.N.; Miroshnikova, N.E.; Sattarova, A.I. Research of the Convolutional and Product Turbo Codes Characteristics in Conditions of Underwater Channel. In Proceedings of the 2022 Systems of Signals Generating and Processing in the Field of on Board Communications, Moscow, Russian, 15–17 March 2022; pp. 1–5.
  47. Yang, Y.; Li, Y. Research and Implementation of Turbo Coding Technology in High-Speed Underwater Acoustic OFDM Communication. J. Robot. 2022, 2022, 2576303.
  48. Jeong, H.W.; Jung, J.W.; Kim, I.S. Non-coherent turbo coded frequency shift keying for reliable covert underwater acoustic communications. J. Acoust. Soc. Am. 2022, 151, A235.
  49. Thorpe, J. Low-density parity-check (LDPC) codes constructed from protographs. IPN Prog. Rep. 2003, 42, 42–154.
  50. Richardson, T.; Kudekar, S. Design of low-density parity check codes for 5G new radio. IEEE Commun. Mag. 2018, 56, 28–34.
  51. Richardson, T.J.; Urbanke, R.L. Efficient encoding of low-density parity-check codes. IEEE Trans. Inf. Theory 2001, 47, 638–656.
  52. Kou, Y.; Lin, S.; Fossorier, M.P.C. Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Trans. Inf. Theory 2001, 47, 2711–2736.
  53. Li, Z.; Chen, L.; Zeng, L.; Fong, W.H. Efficient encoding of quasi-cyclic low-density parity-check codes. IEEE Trans. Commun. 2006, 54, 71–81.
  54. Dehghan, A.; Banihashemi, A.H. On the Tanner graph cycle distribution of random LDPC, random protograph-based LDPC, and random quasi-cyclic LDPC code ensembles. IEEE Trans. Inf. Theory 2018, 64, 4438–4451.
  55. Chen, J.; Fossorier, M.P.C. Density evolution for two improved BP-based decoding algorithms of LDPC codes. IEEE Commun. Lett. 2002, 6, 208–210.
  56. Li, B.; Zhou, S.; Stojanovic, M.; Freitag, L.; Huang, J.; Willett, P. MIMO-OFDM over An Underwater Acoustic Channel. In Proceedings of the OCEANS 2007, Vancouver, BC, Canada, 29 September–4 October 2007; pp. 1–6.
  57. Huang, J.; Zhou, S.; Willett, P. Nonbinary LDPC coding for multicarrier underwater acoustic communication. IEEE J. Sel. Areas Commun. 2008, 26, 1684–1696.
  58. Chen, Y.; Xu, X.; Zhang, L. Performance analysis of LDPC codes over shallow water acoustic channels. In Proceedings of the 2009 5th International Conference on Wireless Communications, Networking and Mobile Computing, Beijing, China, 24–26 September 2009; pp. 1–4.
  59. Kang, T.; Iltis, R.A. Iterative carrier frequency offset and channel estimation for underwater acoustic OFDM systems. IEEE J. Sel. Areas Commun. 2008, 26, 1650–1661.
  60. Qi, X.-K.; Li, Y.; Huang, H.-N. Construction of Reversible QC-LDPC Codes and Its Performance in Underwater Acoustic Communication System. J. Electron. Inf. Technol. 2013, 34, 1986–1992.
  61. Tao, J.; Zheng, Y.R. Turbo detection for MIMO–OFDM underwater acoustic communications. Int. J. Wirel. Inf. Netw. 2013, 20, 27–38.
  62. Chen, Y.; Wang, Z.; Wan, L.; Zhou, S.; Xu, X. OFDM-modulated dynamic coded cooperation in underwater acoustic channels. IEEE J. Ocean. Eng. 2014, 40, 159–168.
  63. Liu, S.; Song, A. Optimization of LDPC codes over the underwater acoustic channel. Int. J. Distrib. Sens. Netw. 2016, 12, 8906985.
  64. Li, D.; Wu, Y.; Zhu, M. Nonbinary LDPC code for noncoherent underwater acoustic communication under non-Gaussian noise. In Proceedings of the 2017 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), Xiamen, China, 22–25 October 2017; pp. 1–6.
  65. Padala, S.K.; D’Souza, J. Performance of spatially coupled LDPC codes over underwater acoustic communication channel. In Proceedings of the 2020 National Conference on Communications (NCC), Kharagpur, India, 21–23 February 2020; pp. 1–5.
  66. Wei, X.F.; Chen, J. Performance Simulation of Underwater Acoustic Spread Spectrum Communication Based on LDPC Code. In Proceedings of the Journal of Physics: Conference Series. IOP Publ. 2021, 1873, 012018.
  67. Zhao, Z.; Sun, Z. Short-Block-Length Low-Density Parity-Check Codes-Based Underwater Acoustic Spread-Spectrum Communication System. Electronics 2023, 12, 3884.
  68. Arikan, E. Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels. IEEE Trans. Inf. Theory 2009, 55, 3051–3073.
  69. Vangala, H.; Viterbo, E.; Hong, Y. A comparative study of polar code constructions for the AWGN channel. arXiv 2015, arXiv:1501.02473.
  70. He, G.; Belfiore, J.C.; Land, I.; Yang, G.; Liu, X.; Chen, Y.; Li, R.; Wang, J.; Ge, Y.; Zhang, R. Beta-expansion: A theoretical framework for fast and recursive construction of polar codes. In Proceedings of the GLOBECOM 2017–2017 IEEE Global Communications Conference, Singapore, 4–8 December 2017; pp. 1–6.
  71. Niu, K.; Chen, K. CRC-aided decoding of polar codes. IEEE Commun. Lett. 2012, 16, 1668–1671.
  72. Qiao, G.; Xing, S.; Zhou, F. A multi-user detection scheme based on polar code construction in downlink underwater acoustic OFDM communication system. IEEE Access 2019, 7, 65973–65981.
  73. Zhai, Y.; Li, J.; Feng, H. Research on Polar coding application for underwater acoustic OFDM communication system. In Proceedings of the 2020 IEEE 6th International Conference on Computer and Communications (ICCC), Chengdu, China, 11–14 December 2020; pp. 322–328.
  74. Falk, M.; Bauch, G.; Nissen, I. On channel codes for short underwater messages. Information 2020, 11, 58.
  75. Zhou, C.; Cao, H.; Li, Q. Joint coding of underwater acoustic communication source channel based on polar code. In Proceedings of the International Conference on Network Communication and Information Security (ICNCIS 2021), Sanya, China, 3–5 December 2021; Volume 12175, pp. 19–23.
  76. Yushuang, Z.; Haihong, F.; Jilong, L.I.; Feng, H. Polar code construction method for the polar coded modulation orthogonal frequency division multiplexing underwater acoustic communication. Tech. Acoust. 2023, 48, 482–495.
  77. Chen, R.; Wu, W.; Zeng, Q.; Liu, S. Construction and application of polar codes in OFDM underwater acoustic communication. Appl. Acoust. 2023, 211, 109473.
  78. Liu, F.; Wu, Q.; Li, C.; Chen, F.; Xu, Y. Polar coding aided by adaptive channel equalization for underwater acoustic communication. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences. 2023, Volume 106, pp. 83–87. Available online: https://search.ieice.org/bin/summary.php?id=e106-a_1_83 (accessed on 26 October 2023).
  79. Yang, X.; Zhou, Y.; Wang, R.; Tong, F. Research and Implementation on a Real-time OSDM MODEM for Underwater Acoustic Communications. IEEE Sens. J. 2023, 23, 18434–18448.
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