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Abstract
Considering large dynamic optical intensity range in a water-to-air (W2A) channel, we propose two promising channel coding schemes, namely the concatenated Reed Solomon-Low Density Parity Check (RS-LDPC) code and Raptor code, for W2A visible light communication (VLC). We establish a W2A-VLC link to verify the performance under different wavy water environments and different water depths with a green light emitting diode (LED). A wave generator is adopted to emulate the wavy water surface with wave height up to 0.6 m. The receiver is fixed 3.2 m above the water, and the transmitter varies from 2.5 m to 4.0 m under the water through a up-down-moveable platform. We test the coding schemes with different code lengths and code rates under 5 MSym/s air-interface symbol rate. Experimental results show that both schemes can reduce the bit error ratio (BER) and frame error rate (FER) of a W2A-VLC system, and thus can improve the reliability. Via comparing the two codes with the same overhead and approximately the same code length, it is demonstrated that Raptor code can generally outperform the concatenated RS-LDPC code. Our research provides promising channel coding methods without feedback for a W2A-VLC system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the growing interest in ocean explorations, underwater wireless communication has attracted extensive research interests from both academia and industry [1–3]. Communication through a water-to-air (W2A) interface is also of significant research interest [4–6]. For efficient and secure data transmission in ocean explorations, it is necessary to establish a reliable communication link across the W2A interface.
Three typical wireless carriers can be employed for communication in the air and water, including radio frequency (RF) wave, acoustic wave and optical wave. RF wave can travel a long distance (up to tens of kilometers) and achieve a high transmission rate (up to hundreds of Mbps) in the air, but it can only support short-distance (a few meters) communication at extra-low frequencies (30-300 Hz) due to its high attenuation in the water [7]. Acoustic wave can provide a long link range up to several tens of kilometers in the water. However, the transmission rate of acoustic communication is relatively low (typically in the order of kbps), which cannot satisfy the requirements of real-time large volume data exchange [8]. Additionally, acoustic wave suffers severe reflection at the W2A interface and high attenuation in the air. As a result, neither RF wave and acoustic wave can be applied exclusively for communication across the water surface.
Optical wireless communication (OWC) is a more favorable solution to communication across the W2A interface since it can achieve moderate to high transmission rate in both air and water. It is demonstrated that water exhibits an acceptable attenuation for light in the blue-green window (with wavelength from 450 nm to 550 nm), which implies its feasibility for underwater communication [9]. Compared with acoustic wave and RF wave, optical wave can achieve the highest transmission data rate, up to the order of Gbps [10].
Although OWC enjoys plenty of advantages over RF wave and acoustic wave, realizing W2A-OWC still remains a challenge. In a W2A-OWC channel, due to the effect of water flow, underwater communication nodes inevitably suffer dynamic drift and rotation [11]. Moreover, the water surface shows non-negligible dynamic fluctuation due to both water flow and air flow. As a result, such dynamic factors lead to random spatial modulation to the light [12]. Thus, it is essential to develop OWC techniques to achieve reliable W2A transmission. In this work, assuming no feedback link from the air back to the water, forward error correction (FEC) coding is employed for W2A-OWC systems.
In recent years, some OWC systems across the water surface have been demonstrated [13–15]. In [16], a 2.2 Gb/s OWC system over a 3.6 m underwater and an 8 m air link was established to support the real-time transmission. A low-complexity transmitter-side digital signal processing (DSP) scheme was adopted to improve the optical power sensitivity. However, the above works do not consider the influence of wavy water surface. Another wireless laser communication system was verified in [17] for direct communications across the W2A interface. Experiments were lunched in a canal of the Red Sea based on a pre-aligned link. The system was tested in a diving pool under a mobile signal-searching mode. It achieved a real-time data rate of 87 Mb/s through a tiny wavy channel. Most OWC systems mitigate the wave-induced effects by beam steering techniques [18] and spatial diversity [19]. In [20], a Reed Solomon (RS) code was used for a laser communication system between airborne and underwater platforms, where the airborne platform is 2000 meters high with underwater platform at 80 meters depth. A data rate of 2.5 kbps can be achieved with bit error ratio (BER) lower than $1 \times 10^{-5}$. Although FEC was adopted, the coding structure can be further improved to fit the W2A channel with both noise and deep intensity attenuation.
We propose two coding schemes for W2A visible light communication (VLC) systems. Specifically, we propose the concatenated Reed Solomon-Low Density Parity Check (RS-LDPC) code, which adopts LDPC code to correct random error and RS code to correct the erasure caused by the LDPC decoding failure. Moreover, we propose Raptor code and adopt an interleaver to improve the coding performance. We experimentally demonstrate and compare the coding performance of these two coding schemes. By wave making equipment in a 4-meter deep experimental environment, two types of regular waves are generated with same frequency and different amplitudes. We test the coding schemes under different wave environments and different water depths with a green light emitting diode (LED). Experimental results show that these two coding schemes are both effective in reducing BER and frame error rate (FER) for W2A-VLC channels. Comparing the two codes with same overhead and approximately the same code length, it is demonstrated that Raptor code can generally outperform the concatenated RS-LDPC code. Our research provides promising FEC coding methods without feedback for a W2A-VLC system. Note that in our conference version [21], we have verified that the concatenated RS-LDPC code can improve the reliability of the W2A-VLC system in a laboratory environment with transmitter depth less than $20$ centimeters and wave height lower than $5$ centimeters. In addition to the concatenated RS-LDPC code, we also propose Raptor code for the W2A channel with deep signal intensity attenuation and additive noise in this work. Moreover, we evaluate the performance of the two coding approaches in a more realistic W2A environment. The environment is equipped with a water tank of size 25 $\times$ 2.8 square meters and can generate wavy water surface up to 0.6 meters.
The remainder of this paper is organized as follows. The system model is elaborated in Section 2. Design of the W2A-VLC system is presented in Section 3. The experimental results are presented and analyzed in Section 4. Finally, Section 5 concludes this paper.
2. System model
Consider a W2A-VLC scenario where one underwater transmitter (e.g., unmanned underwater vehicle) sends information to one receiver (e.g., unmanned aerial vehicle) in the air, as shown in Fig. 1. The transmitter is equipped with one green/blue LED at depth $d_w$ under the water surface, and the receiver is equipped with one avalanche photodiode (APD) at height $d_a$ above the water surface. The LED with beam angle $\theta$ transmits the modulated optical signal to the APD. The light first propagates in the water and then is refracted at the wavy water surface. After propagating in the W2A link, the light is detected by the APD for further signal processing.
The block diagram of a W2A-VLC system is shown in Fig. 2. At the transmitter, the encoded signal is modulated and transmitted via LED. At the receiver, the signal received by APD is demodulated and then decoded based on the channel estimate. However, underwater nodes suffer dynamic position and orientation caused by water flow. When the light propagates across the W2A surface, wavy water leads to drift of light spot coverage area. As a result, the received signal strength at a fixed receiver position fluctuates, implying the change of link quality over time. More seriously, sometimes the received signal is extremely weak, which leads to serious packet loss and bit errors. To improve the reliability of W2A-VLC link under wavy water environments, it is essential to find effective coding methods for wavy W2A-VLC system.
3. Design of the W2A-VLC system
Based on Fig. 2, we outline the main blocks for W2A-VLC system, including modulation and demodulation, channel coding design, channel estimation and log-likelihood ratio (LLR) computation.
3.1 OOK modulation
We adopt on-off keying (OOK) modulation for W2A-VLC system due to the weak received signal characteristics. In the OOK transmission, optical signal “on" represents data bit “1", and optical signal “off" represents data bit “0". It is difficult to adopt a lens to concentrate the received optical signal, since the wavy surface causes random drift of the light spot. Thus, the received signals intensity will be significantly attenuated, where low-order OOK modulation is adopted.
3.2 Channel coding design
The dynamic communication environment across W2A interface will cause weak received signal intensity in a certain duration, which can be modeled as channel erasure. Other parts of the received signals can be modeled as being contaminated by additive noise. Based on the noisy and erasure characteristics of W2A-VLC channel, we propose two channel coding schemes, namely the concatenated RS-LDPC code and Raptor code, to improve the communication reliability.
3.2.1 Concatenated RS-LDPC code
Based on the characteristics of W2A-VLC channel, we consider a concatenated coding scheme which combines error-correcting code with erasure-correcting code. Specifically, we adopt Quasi-Cyclic LDPC (QC-LDPC) code for error correction and RS code for erasure correction.
We adopt QC-LDPC code based on circulant permutation matrices [22]. The parity-check matrix of QC-LDPC code is determined by ${\textbf H}_b$ and $L$, where ${\textbf H}_b$ of size $m \times n$ represents the basis matrix and $L$ represents the lifting factor. The parity-check matrix $\textbf H$ of a QC-LDPC code can be represented by
RS code is a non-binary block code, which is defined over the Galois field $GF(2^m)$. An $(n,k)$ RS code over $GF(2^m)$ maps $k$ $m$-bit information symbols into $n$ $m$-bit coded symbols. When the W2A-VLC channel suffers serious interference, LDPC decoding failure will occur at the receiver side, causing the whole frame loss and erasure. For an $(n,k)$ RS code, if the number of correctly received symbols is no less than $k$, the receiver can reconstruct the original data based on these correct data. A systematic RS code is adopted in this work, due to the reason given in section 4.1.
The structure of the concatenated RS-LDPC code [23,24] is shown in Fig. 3. Before encoding, we arrange information bits into a matrix of $P$ rows and $Q$ columns. Firstly, we encode each column of the information matrix based on RS code. Secondly, each RS code symbol is expanded to $m$ bits by row, obtaining a matrix of $M$ rows and $Q$ columns. Finally, we encode each row of the $M$-by-$Q$ matrix via QC-LDPC code, obtaining an encoding matrix of $M$ rows and $N$ columns. The following constraints are satisfied:
where $R_r$ and $R_q$ represent the rates of RS and QC-LDPC codes, respectively. The overhead of the concatenated RS-LDPC code, denoted as $\epsilon _1$, is determined by the rates of both RS code and QC-LDPC code asAt the transmitter side, we send the coded bits in the encoding matrix by rows, which can be seen as multiple continuous LDPC frames. The received LDPC frames are first decoded at the receiver side. For each LDPC frame, we retain the LDPC decoding results if the check bits pass, otherwise we discard the whole frame and set the whole frame as erasure. As a result, the received LDPC frames can be divided into two types, namely correct frames and erasure frames. Then, we arrange the decoded LDPC frames by rows, and decode each column based on RS code. As long as the number of correct frames after LDPC decoding is no less than the number of RS code information symbols, the original data can be reconstructed.
3.2.2 Raptor code
Raptor code is a type of digital fountain code, which combines LT (Luby Transform) code as the inner code with a linear block code as the outer code [25,26]. Raptor code can combat both channel noise and erasure. W2A-VLC channel suffers serious erasure and noise issues under wavy water surface, where the transmission reliability can be improved via adopting Raptor code.
LT code is a typical random code. The encoded data stream of LT code satisfies degree distribution function $\Omega (x)=\sum _{i=1}^{n}\Omega _{i}x^{i}$, where $\Omega _{i}$ represents the probability of degree $i$ coded symbol [27,28]. For LT encoding, we first generate a random degree $d$ based on $\Omega (x)$. Then, we randomly select $d$ bits from the input sequence, and encode these $d$ bits via XOR operation. We adopt the following degree distribution:
To lower the high error floor of the inner LT code, we adopt a high-rate LDPC code as the outer code for Raptor precoding. The information bits are encoded sequentially by LDPC code and LT code. The overhead $\epsilon _2$ of Raptor code is determined by the rates of both the LDPC precode and LT code as
where $R_{lt}$ and $R_l$ represent the code rates of the LT and LDPC codes, respectively.In the W2A-VLC channel, wavy water surface leads to long continuous signal erasure. Based on the joint anti-noise and anti-erasure properties of Raptor code, we model extremely weak received signals as erasure, and other parts of the received signals as noisy signals. In order to avoid one Raptor code frame completely falling into severe weak signal intensity regime, interleaving is adopted to break long continuous erasure into scattered pieces. Before encoding, the information bits are first interleaved at the transmitter side. We arrange information bits into a matrix of $N$ rows and $M$ columns, and then encode each column based on Raptor code. We send the encoding matrix row by row to realize interleaving. At the receiver side, the received signal is firstly deinterleaved, where the long continuous erasure bits are broken down into multiple Raptor sequences. Finally, the Raptor decoding is performed. The structure of the proposed Raptor code scheme is shown in Fig. 4.
In this work, we adopt LLR-Belief Propagation (LLR-BP) algorithm for the decoding of LDPC code and LT code.
3.3 Channel estimation and LLR computation
To calculate soft information for channel decoding, channel estimation and LLR computation are performed. Consider a period of $N$ received signals within channel coherence time. After removing the direct current (DC) component, the received signal in the $n$-th time slot can be expressed as
where $x_n \in \{-1, 1\}$ denotes the transmitted signal; $h$ denotes the W2A-VLC channel gain, which is approximately constant during the channel coherence interval; $z_n \sim N(0,\sigma ^2)$ denotes the independent additive Gaussian noise; $N$ is the length of the observation window. Assuming that the input signal $x_n$ satisfies an equiprobable distribution, the LLR can be expressed asTo obtain the LLR in Eq. (7), it requires accurate estimation of parameters $h$ and $\sigma ^2$. Maximum likelihood (ML) criterion is adopted to estimate the channel gain $h$ and noise variance $\sigma ^2$. Define vectors $\boldsymbol {x} = [x_1,\ldots,x_N]$, $\boldsymbol {y} = [y_1,\ldots,y_N]$, and $\boldsymbol {z} = [z_1,\ldots,z_N]$. Assuming independent symbols $\{x_n\}_{n=1}^N$ and noise components $\{z_n\}_{n=1}^N$, we have
For simplicity, we consider the logarithmic form $\ln p(\boldsymbol {y}|h, \boldsymbol {x}, \sigma ^2)$ of Eq. (8), and solve the following optimization problem
Given non-negative $h$ and $\sigma ^2$, the solution $\hat {x}_n\in \{-1,1\}$ to problem (9) is
Using Eq. (10), problem (9) can be equivalently rewritten as
Then, given $\sigma ^2$, the solution $\hat {h}$ to problem (11) is
Substituting Eq. (12) into Eq. (11) yields the following noise variance estimate
Finally, substituting Eq. (12) and Eq. (13) into Eq. (7), we obtain
For the decoding of LDPC code and LT code, we initialize the LLR according to Eq. (14). Since the link gain under wavy water surface greatly changes over time, the length $N$ of the observation window needs to be shorter than the coherence time, which is in the order of several milliseconds [5].
4. Experimental setup and results
We carry out experiments to evaluate and compare the performance of the two coding schemes for W2A-VLC link. Figure 5 illustrates the experimental W2A-VLC system. We adopt a green LED (Cree XBDGRN, 520 nm $\sim$ 535 nm, 3-dB bandwidth: 5 MHz) as the transmitter, which is sealed in a waterproof container under the water. The depth of the container varies from 2.5 m to 4.0 m through a up-down-moveable platform. The LED is driven by a bias-tee circuit combining the DC bias (DC = 7 V) from a DC power supply (Rigol DP832A) with encoded OOK signals (Vpp = 2 V) from an arbitrary wave generator (AWG, Rigol DG5252). An adjustable lens is placed in front of the LED to concentrate the light, of which the beam angle is approximately $30^{\circ }$.
We use a DC-blocking APD (Hamamatsu C12702-11, bandwidth: 50 kHz - 40 MHz) with an amplification circuit at the receiver, which is 3.2 m above the water surface. The light emitted from the LED reaches the APD after propagating through the W2A link. The APD converts the received optical signal into electrical signal, which is sampled by a data collector (ART Technology PCIe8584, 100 MSa/s) and downloaded to a computer for offline analysis.
We use a wave generator to emulate the wavy water surface in real sea environments. Figure 6 shows the structure of the wave generator equipped with a push board. Figure 6(b) corresponds to the right part of Fig. 6(a). We can generate waves of different amplitudes and frequencies by changing the strength and frequency of the push board, respectively. Although such approach can generate regular wave theoretically, the real generated wave is a mixture of large regular waves and small broken waves due to the collision between wave and environmental substances such as glass wall. The experimental wave parameters are shown in Table 1, where the amplitude is half the wave height and the wavelength is tens of meters.
Table 1. Experimental wave parameters
We generate the pseudorandom coded sequences in an offline manner, and feed them into the AWG. The symbol rate is set to be 5 MSym/s, with sampling rate 50 MSa/s at the receiver. The depth of LED transmitter under the water varies from 2.5 m to 4.0 m with step 0.5 m. At each depth, we conduct the experiments under two different wave conditions, namely Wave 1 and Wave 2 in Table 1.
Take water depth $d_w=2.5$ m for example. We transmit a continuous 5-second-long OOK signal, and the Vpp of the OOK signal is 2 V. The received waveforms are shown in Fig. 7(a). For Wave 1 (above) and Wave 2 (below), we display a continuous 5-second-long signal. Affected by wavy water surface, it is obvious that the received signal strength varies significantly during a short time period. The probability density functions (PDFs) of the received signal strengths under Wave 1 and Wave 2 are given in Fig. 7(b). It is seen that the signal mean under Wave 1 environment is larger than that under Wave 2 environment (Wave 1: 1.22, Wave 2: 1.09). Moreover, Wave 2 environment yields larger signal range and variance than Wave 1 environment (Wave 1: 0.32, Wave 2: 0.41), which implies that larger wave intensity will cause larger signal variation.
Fig. 7. The received signals at depth 2.5 m. (a) The received waveforms. (b) The PDFs of the received signal strengths under Wave 1 and Wave 2 environments.
Remark 1: Note that the experimental condition’s limited space may lead to a reflection of the echo wave superposed to the direct waves. As one common case in real scenarios, a reflection of the echo wave may also exist, e.g., near the ship in the ocean or near the shore. The experimental results in the presence of echo wave provide significant guidance for W2A-VLC systems design and related performance analysis in complicated environments where echo wave cannot be suppressed. It still requires further investigation on the performance comparison in the cases with and without echo wave.
Remark 2: Except the frequency and amplitude of waves, other water parameters (e.g., the salty and temperature) also have an impact on communication quality. These parameters mainly affect attenuation and scattering coefficient of water. Attenuation will only lead to differences on link gain, but does not change W2A-VLC channel variation characteristics. Scattering will weaken the influence of wave water surface to a certain extent, changing W2A-VLC channel variation characteristics. However, the depth of LED transmitter under the water varies from 2.5 m to 4.0 m in our experiment, and scattering has little effect at this depth. In conclusion, the impact of other water parameters is mainly reflected in the difference of link gain but not the fluctuations. As a result, when considering the imapct of other water parameters (e.g., the salty and temperature), we can increase optical power, and the proposed coding schemes in this work are also applicable.
4.1 Performance of the concatenated RS-LDPC code
For the concatenated RS-LDPC code, each data block is composed of $M$ short LDPC frames, and all the $M$ LDPC frames in one data block satisfy RS encoding constrains. We first test two RS-LDPC schemes with the same code length and different code rates. The corresponding encoding parameters are shown in Table 2.
Table 2. Encoding parameters of two concatenated RS-LDPC schemes
We process a continuous 5-second-long signal and calculate the correct frame ratio of each data block after LDPC decoding, as shown in Fig. 8. Subfigures 8(a)–8(d) demonstrate the correct LDPC frame ratios at depth 2.5 m, 3.0 m, 3.5 m, and 4.0 m, respectively. In each subfigure, the left plot and right plot represent the results under Wave 1 and Wave 2, respectively. The RS code rates are given in blue line (code rate 0.75) and red line (code rate 0.50). Scatters above the blue/red line mean that these blocks can be decoded correctly in the corresponding RS code rate. On the contrary, scatters below the blue/red line indicate the decoding failure, which means that a lower code rate is needed. As the water depth increases, the correct frame ratio generally decreases. At water depth 3.5 m, RS code rate 0.50 is more appropriate than RS code rate 0.75. At the same depth, Wave 2 environment yields lower correct frame ratio than Wave 1 environment. The reason is that, as the fluctuation amplitude increases, the W2A-VLC channel suffers greater fading, leading to larger weak link gain regime and thus higher LDPC frame decoding failure ratio. As the water depth increases from 3.5 m to 4.0 m, the SNR decreases such that the decoding failure rate of LDPC codes increases significantly, leading to severe decrease in the correct frame ratios of RS code.
Fig. 8. Correct RS-LDPC frame ratio. (a) Depth 2.5 m. (b) Depth 3.0 m. (c) Depth 3.5 m. (d) Depth 4.0 m.
We demonstrate the BER performance of the concatenated RS-LDPC code. For each data block, if the correct LDPC frame ratio is higher than the corresponding RS code rate, the BER of the code block is zero according to the RS code capability. On the contrary, if the correct LDPC frame ratio is lower than the corresponding RS code rate, the number of errors is beyond the correction capability of RS code. In such case, we adopt a systematic RS code, so that we can extract the original information bits from coding bits to calculate BER. The BER results are shown in Fig. 9. It is seen that Wave 2 environment shows higher BER than Wave 1 environment since the water surface fluctuates more violently. Meanwhile, reducing RS code rate can improve the communication reliability. At depth 3.5 m, the BER falls from the order of $10^{-2}$ to the order of $10^{-3}$ by reducing RS code rate from 0.75 to 0.50.
Moreover, the FER performance of the concatenated RS-LDPC code under different depths is shown in Fig. 10. It is seen that Wave 2 environment yields higher FER than Wave 1 environment. Under water depth 3.5 m and below, the FER falls from the order of $10^{-1}$ to the order of $10^{-2}$ by reducing RS code rate from 0.75 to 0.50. However, for water depth larger than 3.5 m, since the received signal becomes extremely weak, reducing the code rate does not significantly decrease the FER.
Next, we consider RS-LDPC schemes with the same code rate and different code lengths. Since W2A-VLC channel can generate a duration of extremely weak gain due to water wave, increasing the code length can avoid some data blocks completely falling into severe fading. Given water depth 3.5 m and Wave 2 environment, we calculate the correct frame ratios under three different code lengths corresponding to different $m$ in RS code. In Fig. 11, solid lines represent the correct frame ratios under three different RS-LDPC code lengths, where larger parameter $m$ corresponds to longer RS-LDPC code, and dotted lines represent the maximum code rates under different code lengths for zero-BER transmission. It can be seen that, larger $m$ leads to larger code rate for zero-BER transmission. In other words, increasing the code length can achieve reliable transmission at higher code rate. Then, we fix RS code rate as 0.50, and change the code length of RS-LDPC. The BER and FER curves are shown in Fig. 12. It is seen that as the code length increases, the BER and FER decrease under the same code rate.
4.2 Performance of Raptor code
For Raptor code scheme, the interleaving depth is set to be 100. The rate of the outer code is fixed to be 5/6, and the rate of the inner code is adjustable. For a fair comparison between the concatenated RS-LDPC code and Raptor code, the rates of the two codes are set to be the same. The parameters of Raptor code are given in Table 3. The two schemes of Raptor code correspond to the two schemes of concatenated RS-LDPC code with the same code rate.
Table 3. Encoding parameters of Raptor code
The BER and FER of the Raptor code are shown in Fig. 13 and Fig. 14, respectively. As the water depth increases from 2.5 m to 3.5 m, the BER and FER increase gradually. When the depth increases beyond 3.5 m, the BER and FER increase significantly. At the same depth, Wave 2 environment always yields larger BER and FER than those in Wave 1 environment. Reducing Raptor code rate leads to lower BER and FER. When water depth is 3.5 m and below, the BER and FER are zero for Raptor code 2. At depth 4.0 m, although the channel quality is very poor, the BER can still fall one or two orders of magnitude by reducing the entire code rate from 0.375 to 0.25.
Then, we investigate the effect of code length on the BER performance of Raptor code. We fix the code rate (outer code rate: 5/6, inner code rate: 0.45), and change the input data length. We evaluate the BER performance under 3.5 m depth and Wave 2 environment, as shown in Fig. 15(a). As the code length increases, the BER decreases under the same code rate. Next, we study the effect of code rate on the BER performance of Raptor code. We keep the coding length fixed, such that the code blocks with different rates suffer the same fading. We change the inner code rate and calculate the overhead based on Eq. (5). The BER performance under 3.5 m depth and Wave 2 environment is shown in Fig. 15(b), where increasing coding overhead can significantly reduce the BER. When the overhead reaches 2.6, the BER can fall below $10^{-5}$.
Fig. 15. BER of Raptor code with (a) different code lengths and (b) different overheads under 3.5 m depth and Wave 2 environment.
4.3 Comparison between the concatenated RS-LDPC code and Raptor code
Finally, we compare the concatenated RS-LDPC code and Raptor code in terms of the BER and FER. For the concatenated RS-LDPC code, the code length is $N_1=\frac {L_1}{R_r\times R_q}$, where $L_1$ denotes the information bit length of the concatenated RS-LDPC code. For Raptor code, the code length is given by $N_2=\frac {L_2}{R_{lt}\times R_l}$, where $L_2$ denotes the information bit length of Raptor code. For a fair comparison, the same code rate of the two codes are adopted, and the two codes should also suffer the same channel realization time sequence. Since the two codes need to satisfy certain code structure, the code lengths cannot be set to be strictly equal. Thus, in order to show the performance gain of Raptor code, we delicately select the Raptor code parameters, such that the Raptor code length is a bit shorter than the RS-LDPC code length. More specifically, Raptor code 1 and RS-LDPC code 1 have the approximately the same code length and exactly the same code rate (RS-LDPC 1: code length 294336, code rate 0.375; Raptor 1: code length 293000, code rate 0.375), and the same apply to Raptor 2 and RS-LDPC 2 (RS-LDPC 2: code length 294336, code rate 0.250; Raptor 2: code length 291700, code rate 0.250). The BER and FER of the two codes under different depths are shown in Fig. 16. It is seen that at water depth not exceeding 3.5 m, Raptor code yields lower BER than the concatenated RS-LDPC code with a bit shorter code length and the same code rate (as shown in blue dotted bordered rectangle). When the W2A-VLC channel quality is very poor, Raptor code suffers higher BER than the concatenated RS-LDPC code with a bit shorter code length and the same code rate at high code rate (as shown in red dotted bordered rectangle); and Raptor code achieves lower BER than the concatenated RS-LDPC code with a bit shorter code length and the same code rate at low code rate. Moreover, the FER of Raptor code is always lower than that of the concatenated RS-LDPC code with a bit shorter code length and the same code rate.
Fig. 16. The BER and FER comparisons of Raptor code and the concatenated RS-LDPC code under different depths.
Note that even when Raptor code achieves higher BER than the concatenated RS-LDPC code, it still exhibits lower FER. It can be justified by the fact that as the channel error and erasure increase beyond the decoding capability, more errors are generated after BP decoding, resulting in higher residue BER.
Then, we investigate the complexity of the two schemes. For RS-LDPC code, the decoding complexity of an $(n, k)$ erasure RS is in order $\mathcal {O}(n^2)$. The decoding of QC-LDPC code depends on the code length, average degree distribution and the number of iterations. The iterative decoding can be realized in parallel. For Raptor code, the decoding complexity is similar to that QC-LDPC code via parallel realization. For a fair comparison, we test the decoding time of RS-LDPC 1 and Raptor 1 in the MATLAB environment, where the decoding time for RS-LDPC 1 is about 60% times of that for Raptor 1.
Remark 3: As for the experimental results, no matter RS-LDPC code or Raptor code, decreasing of code rate will result in larger number of correct frames. The realization of balancing the code rate and correct frames is to maximize $(1-FER)*CodeRate$ or throughput. Adaptive modulation coded (AMC) technique can be adopted to maximize throughput. The transmitter can obtain channel status information (CSI) through feedback link and adjust code rate in a real-time manner. Specifically, the transmitter can choose high code rate in good channel condition and low code rate in poor channel condition. AMC technique is effective in balancing the code rate and correct frames, and can improve communication efficiency and BER/FER performance simultaneously. The AMC under wavy water surface needs establishing a feedback link from the receiver back to the transmitter with real-time channel state feedback, which needs a large workload and thus remains for future work.
Remark 4: Due to the limitations of experimental conditions, it is infeasible to generate more types of waves. In this work, we compare the coding schemes’ communication performance under two types of waves with different intensities. This work focuses on anti-erasure and error coding methods for W2A-VLC system, where two waveforms can provide us some insights into the wavy impact. In a real sea environment, we can adjust the code rate and code length to adapt to different types of waveforms.
5. Conclusion
In this work, we have adopted the concatenated RS-LDPC code and Raptor code for W2A-VLC systems based on the noisy and erasure characteristics of W2A-VLC channel. To verify the coding performance, we have established a W2A-VLC link with wavy water surface and underwater depth reaching $4$ meters, and tested the coding schemes under different wavy intensities and depths. Experimental results demonstrate the effectiveness of the two coding schemes in improving the BER and FER performance under wavy water environment. Meanwhile, the comparison results show that Raptor code scheme generally outperforms the concatenated RS-LDPC code.
Funding
National Key Research and Development Program of 354 China (2018YFB1801904); National Natural Science Foundation of China (62101526, 62171428); Key Program of National Natural Science Foundation of China (61631018); Key Research Program of Frontier Sciences of CAS (QYZDY-SSW-JSC003); Fundamental Research Funds for the Central Universities (KY2100000118); open research fund of National Mobile Communications Research Laboratory Southeast University (2019D14).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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