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Abstract
Linear and nonlinear impairments in underwater wireless optical communication (UWOC) systems caused by the limited bandwidth and nonlinearity of devices severely degrade the system performance. In this paper, we propose a sparse Volterra series model-based nonlinear post equalizer with greedy algorithms to mitigate the nonlinear impairments and the inter-symbol interference (ISI) in a UWOC system. A variable step size generalized orthogonal matching pursuit (VSgOMP) algorithm that combines generalized orthogonal matching pursuit (gOMP) and adaptive step size method is proposed and employed to compress the Volterra equalizer with low computational cost. A maximum data rate of 500 Mbps is realized with the received optical power of −32.5 dBm in a 7-m water tank. In a 50-m swimming pool, a data rate of 500 Mbps over 200-m underwater transmission is achieved with a BER lower than the forward error correction (FEC) threshold of 3.8 × 10−3. The number of kernels of the sparse Volterra equalizer is reduced to 70% of that of the traditional Volterra equalizer without significant BER performance degradation. Compared with orthogonal matching pursuit (OMP) scheme and regularized orthogonal match pursuit (ROMP) scheme, the VSgOMP scheme reduces the running time by 68.6% and 29.2%, respectively. To the best of our knowledge, this is the first time that a sparse Volterra equalizer combined with VSgOMP algorithm is employed for the nonlinear equalization in a long-distance high-speed UWOC system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, ocean exploration has become a topic of concern due to the increasing shortage of land resources. Underwater wireless communication (UWC) plays an indispensable role in the accelerated exploration of ocean resources. The communication distance of radio frequency (RF) is limited by the severe attenuation in seawater [1]. The underwater acoustic communication (UAC) can achieve a long transmission distance, but faces the challenges of high propagation latency and low data rate [2]. In contrast, underwater wireless optical communication (UWOC) has been widely studied for its advantages of high bandwidth and low delay [3].
Blue-green (400-550 nm) light with low underwater attenuation coefficients has been extensively researched [4]. Compared with light-emitting diodes (LEDs), laser diodes (LDs) with smaller divergence angles and higher bandwidth can effectively reduce the geometric loss and achieve high-speed communication systems [5]. Some researches on LD-based UWOC have improved the system performance in terms of data rate and transmission distance. Recent works of long-distance UWOC systems based on LDs with data rates higher than hundreds of Mbps are summarized below. In 2016, a 1.5-Gbps UWOC system with a distance of 20 m was achieved by using non-return-to-zero on-off keying (NRZ-OOK) modulation [6]. In 2017, through a 21-m underwater channel and a 5-m air channel, a 5.5-Gbps cross-media wireless optical communication system utilizing orthogonal frequency division multiplexing (OFDM) and power loading was established [7]. In the same year, a UWOC system with a distance of 34.5 m and a data rate of 2.7 Gbps was reported in [8]. To further increase the system capacity, digital signal processing (DSP) technology has been employed in UWOC. By using a nonlinear post equalizer, a 2.5-Gbps UWOC system was implemented through a 60-m underwater channel [9], and a similar algorithm was employed to realize a 500-Mbps UWOC system at 100-m underwater transmission distance [10]. In addition, a data rate of 12.6 Gbps over a 35-m transmission distance was demonstrated by adopting probabilistic constellation shaping (PCS) technique [11]. In 2020, frequency-domain equalization and time-domain noise prediction technology were applied to achieve a data rate of 3.31 Gbps with an underwater transmission distance of 56 m [12]. In 2021, a 5-Gbps UWOC system using a low-complexity two-level chaotic encryption scheme was reported with a distance of 50 m [13]. By combining partial response shaping, interleaving, precoding, and TCM technology, a UWOC system was successfully demonstrated at a data rate of 500 Mbps and a transmission distance of 150 m in a swimming pool [14].
Comparing with the other aforementioned works, the communication distance was extended to 150 m in [14] by using a photomultiplier tube (PMT) as the detector. Such detectors with high sensitivity have great potentials to further extend communication distances. In [15], an 8.39-Mbps communication data rate was achieved with an attenuation length of 24 using a multi-pixel photon counter (MPPC) as the detector. Nevertheless, the performance of the systems is commonly restricted by the limited bandwidth of high-sensitivity detectors.
To increase the data rate, adaptive equalizers based on least mean squares (LMS) [16] and recursive least square (RLS) [17] were adopted to mitigate the linear distortion. However, the nonlinear distortion of systems, mainly caused by the nonlinearity of LDs [18], electric amplifiers [19], and high-sensitivity detectors [20], is difficult to be compensated by linear equalizers. The simplified Volterra series model-based nonlinear post equalizer was employed in a wireless optical communication system [21]. The nonlinear post equalizers based on the Volterra series, memory polynomial model, and memoryless polynomial were systematically compared in [22]. Due to the universal approximation property of deep neural network (DNN), the DNN-based nonlinear equalizers were used in UWOC systems to improve the performance [23]. To speed up the convergence rate and reduce the space complexity, Gaussian kernel-aided method and partial pruning strategy were proposed and experimentally investigated [24,25].
The Volterra equalizer has been widely employed to mitigate nonlinear distortion [26]. However, the training complexity of high-order Volterra equalizer is unbearable [27]. Even for a simplified third-order Volterra equalizer with a long memory length, the complexity is still high. Compared with linear programming (LP) methods, greedy algorithms have lower computational complexity, which have been widely used in model compression [28]. Due to the sparse property of kernels [29], three different greedy algorithms, including match pursuit (MP), orthogonal match pursuit (OMP), and regularized orthogonal match pursuit (ROMP), were applied to obtain sparse nonlinear equalizers by selecting and retaining significant kernels [30]. Comparing with MP, the OMP algorithm effectively avoids the reselection of kernels by making each latest selected kernel orthogonal to all the previously selected kernels [31]. However, OMP selects only one kernel in each iteration, resulting in long computing time. The generalized orthogonal matching pursuit (gOMP) and improved ROMP algorithms speed up the compression process by selecting multiple kernels in each iteration [30,32]. Furthermore, as the residual between the transmitted signal and the estimated signal decreases, the sparse property of the Volterra equalizer for the proposed UWOC system is not constant. In other words, the degree of sparsity is small at the initial stages. As more kernels are selected, the kernels gradually become sparser. To maintain desirable performance, gOMP and improved ROMP generally preset a small step size and low sparsity level, respectively, which lead to more computational cost. For complex and dynamic underwater environments, high computational cost will limit the application of sparse nonlinear equalizer in real high-speed UWOC systems. Aiming at the image recognition, a modified adaptive orthogonal matching pursuit algorithm was proposed to improve the recognition accuracy and efficiency [33]. However, the distributions of signal and noise in UWOC systems are different from that in the computer vision (CV) field. How to effectively reduce the computational overhead of the Volterra equalizer compression process is still an urgent and unsolved problem.
In this paper, we propose a sparse nonlinear equalizer with variable step size generalized orthogonal matching pursuit (VSgOMP) for a long-distance high-speed UWOC system. The sparse nonlinear equalizer was applied to mitigate linear and nonlinear distortion with low space complexity. VSgOMP was applied to reduce the computational cost of the compression process without significant performance degradation. In 7-m water tank experiments, the minimum required received optical power to achieve a data rate of 500 Mbps is lower than −32.5 dBm. In a 50-m swimming pool, by employing the proposed sparse Volterra equalizer, a data rate of 500 Mbps is realized with an underwater transmission distance of 200 m. The achieved data rate is 14.9% and 4.5% higher than that of the least square equalization scheme and memory polynomial equalization scheme, respectively. The running time of compression process using the proposed VSgOMP scheme is 31.4%, 70.8%, and 74.2% of that of the OMP scheme, ROMP scheme, and gOMP scheme, respectively. To the best of our knowledge, it is the first time that the improved gOMP is applied to compress nonlinear equalizer in a long-distance high-speed UWOC system, which effectively reduces the computational overhead.
The rest of the paper is organized as follows. In Section 2, we introduce the principles of Volterra series model, gOMP, and VSgOMP. Section 3 describes the experimental setup of the proposed UWOC system. The experimental results are presented and analyzed in Section 4. Finally, Section 5 concludes the paper.
2. Principle
In general, the driver circuit of light source, electrical amplifier, and photodetector will bring varying degrees of nonlinearities to UWOC systems [18-20]. Nonlinear post equalization is an effective method to compensate the distortion caused by ISI and nonlinearity.
2.1 Principle of Volterra series model
Volterra series model-based nonlinear equalizer has been employed to improve UWOC system capacity [21,22], which takes both nonlinearity and memory effects into account. The output of the Volterra equalizer can be expressed as
For p training symbols, the (2) can be rewritten in a vector form as
where $\boldsymbol{\hat{d}}$, $\boldsymbol{w}$ and $\boldsymbol{X}$ can be expressed as2.2 gOMP and VSgOMP
As a greedy algorithm, OMP has been widely applied to model compression. The compression method of OMP is to select the most important kernel according to the correlation of signal in each iteration, and then the weights of the selected kernels are updated. After multiple iterations, all the required kernels are obtained. The gOMP algorithm with the similar principle is an extension of the OMP and was elaborated in [32]. For the sparse nonlinear equalizer, the transmitted signal vector $\boldsymbol{d}$ is estimated by linear combination of partial vectors of $\mathbf{X}$. The main four operations of gOMP are kernel identification, kernel index augmentation, weight estimation, and residual update. At the initial stage, $\boldsymbol{d}$ is assigned to the residual $\boldsymbol{r}$, and the correlation between $\boldsymbol{r}$ and each vector of $\mathbf{X}$ is obtained by calculating the normalized inner product. The indices of vectors with the highest correlation are selected and incorporated into the support set. In the weight estimation, the least squares (LS) method is used to calculate the corresponding weight, which effectively avoids kernel reselection. Then, the residual is updated. Repeating the aforementioned operations until the residual is lower than the preset threshold or the number of selected kernels reaches the required number E.
By combining gOMP algorithm and variable step size method, VSgOMP algorithm is proposed and employed to adjust the number of selected kernels adaptively in each iteration to further reduce the computational time of the compression process. The main operations of VSgOMP are similar to gOMP. The difference is that the number of required kernels L in each iteration will be updated according to the residual of last iteration. At the initial stages, a relatively large step size is chosen, which is affected by step size coefficient $\alpha$. As the mean of residual decreases, the step size decreases gradually. Due to the rapid change of the residual at the initial stages, logarithmic function is adopted to keep the change of step size relatively smooth. It is worth noting that in order to adapt to different sparsity degrees, the step size is related to the number of required kernels. The proposed VSgOMP algorithm is summarizes in Algorithm 1.
Figure 1 shows the schematic diagram of the proposed sparse Volterra equalizer. The yellow, green, and blue parts represent the structure of the 1st-order linear kernels, 2nd-order Volterra series kernels, and 3rd-order memory polynomial kernels, respectively. Since the kernels of different orders are correlated, all kernels are compressed together. By employing VSgOMP, the required kernels and corresponding weights are obtained. The gray dashed lines and multipliers represent unselected kernels and omitted calculation processes after the compression process. It is worth noting that the gray parts in Fig. 1 are for description, not the actually ignored parts. Since the kernels with little effect on system performance are eliminated, the complexity of the equalizer is effectively reduced without excessive performance degradation.
2.3 Complexity analysis
In Table 1, the complexity of each operation in the $i\textrm{th}$ iteration is summarized. Here, real-valued multiplications are applied to estimate the computational cost. The specific content of the ROMP algorithm can be found in [30]. Compared with other schemes, the ROMP additionally contains a regularization operation. It can be found that when the number of the selected kernels is small enough, the computational complexity mainly depends on the identification of kernels. As the number of required kernels increases, the operation of weight estimation will be the main computational cost. Therefore, the final computational complexity mainly depends on the number of iterations when the number of total selected kernels N is large. According to Table 1, when the values of N and L are the same, the computational cost for a single iteration of gOMP and VSgOMP is similar, which is higher than that of OMP and lower than that of ROMP. In practice, when the number of required kernels E increases, the iteration step size of VSgOMP will increase, resulting in a computational advantage due to the fewer iterations over OMP and gOMP schemes. The number of iterations and computing time will be further experimentally analyzed in the fourth section.
Table 1. Complexity of the four greedy algorithms (${i}$th iteration)a
3. Experimental setup
The experimental setup of the proposed UWOC system enabled by sparse Volterra equalization is shown in Fig. 2. At the transmitting end, Mersenne Twister (MT) algorithm described in [35] was used to generate a pseudo-random binary sequence with a length of $5 \times {10^5}$. The binary data were mapped into PAM4 symbols, followed by a 2-times up-sampling. A root-raised cosine (RRC) filter with a length of 401 and a roll-off coefficient of 0.01 was further applied for pulse-shaping. Then, the signal was fed into an arbitrary waveform generator (AWG, Tektronix AWG70002A) to generate the electrical signal. After being amplified by a power amplifier (AMP, 100kHz-75MHz) with a gain of 37 dB, the amplitude of the electrical signal was further adjusted by a variable electrical attenuator (VEA, KT2.5-90/1S-2S). A 450-nm blue laser diode was driven by the electrical signal, which was added with direct current (DC) by a bias tee. Before being injected into the underwater channel, the optical signal was collimated by a lens. The LD and lens were fixed in a watertight cabin. In a swimming pool, the light was reflected three times by using three mirrors ($30\textrm{ }\textrm{cm} \times 30\textrm{ }\textrm{cm}$) and the UWOC link was expanded to 200 m ($50\textrm{ }\textrm{m} \times 4$).
Fig. 2. Experimental setup of the proposed UWOC system. Insets: (i) the transmitter cabin, (ii) the receiver cabin, (iii) the mirror, and (iv) the 50-m swimming pool.
At the receiving end, the optical signal was received by a PMT, which was fixed in another watertight cabin. The output signal of the PMT was recorded by a mixed-signal oscilloscope (MSO, Tektronix MSO71254C) with a sampling rate of 3.125 GSamples/s. The electrical signal was resampled and filtered by the RRC filter with the same parameters as the transmitting end. After synchronization and down-sampling, part of the signal was used for training and the VSgOMP was employed to select the significant kernels of the Volterra equalizer by reiteration. Then, the obtained sparse Volterra equalizer was applied to mitigate the ISI and nonlinear impairments of the received signal. Finally, the binary sequence was recovered by demapping, and the BER of the system was calculated. In addition, an optical power meter (Thorlabs PM200) was used to measure the received optical power. In Fig. 3(a), the output characteristics of the 450-nm LD are illustrated. The measured normalized frequency response of the whole system is shown in Fig. 3(b), indicating that the 20-dB bandwidth is approximately 100 MHz. Table 2 lists the main parameters of the experimental system.
Table 2. Parameters of the experimental system
4. Experimental results and discussion
In this section, the performance of the proposed sparse Volterra equalizer is experimentally analyzed in the UWOC system. To analyze the characteristics of the proposed system and the sparse Volterra equalizer, we first carried out experiments in a 7-m water tank filled with tap water.
First, we studied the nonlinearity of the system by transmitting sawtooth wave at a sampling rate of 500 MSamples/s. In Fig. 4, the curve fitted by the received data deviates from the linear curve, which shows the nonlinearity of the system. Then, the effects of system parameters including the memory length, sparsity degree, and step size coefficient $\alpha$ on the BER performance are analyzed.
In Fig. 5, the linear and nonlinear memory lengths of the simplified third-order Volterra equalizer are investigated. Firstly, the linear kernels are considered in Fig. 5(a). At the data rates of 450 Mbps, 500 Mbps, and 550 Mbps, the BER performance tends to be stable when the linear memory lengths ${M_1}$ are 45, 46, and 53, respectively. Since the absolute values of the linear kernels’ weights change irregularly, there is a sudden improvement of performance in the middle of the three curves. Then, the 2nd-order nonlinear kernels are analyzed with 71 linear kernels. Figure 5(b) shows that the BER performance tends to be stable at aforementioned data rates when the nonlinear memory length ${M_2}$ equals to 7, 11, and 15, respectively. In Fig. 5(c), the BER performance shows small fluctuations under different third-order memory lengths with the fixed 2nd-order memory length of 15. It can be found that with the increase of the data rate, the required memory lengths of linear terms and 2nd-order nonlinear terms correspondingly increase. To balance the complexity and performance, the memory lengths of linear kernels, 2nd-order kernels, and 3rd-order kernels used in the following experiments were 71, 15, and 5, respectively.
Fig. 5. BER performance with different memory lengths of (a) linear terms, (b) 2nd-order nonlinear terms, and (c) 3rd-order nonlinear terms at the data rates of 450 Mbps, 500 Mbps, and 550 Mbps.
In addition, to further study the phenomenon of sudden improvement in Fig. 5(a), the weights of the linear kernels are shown in Fig. 6. At the data rates of 450 Mbps, 500 Mbps, and 550 Mbps, there are obvious peaks at the 72nd, 74th, and 28th tap positions, respectively. The corresponding linear memory length can be expressed as ${M_1} = 2|{n_p} - 51|,$ where ${M_\textrm{1}}$ is the corresponding linear memory length, ${n_p}$ is the peak position of tap. By calculation, it can be found that the peak positions at these data rates correspond to the linear memory lengths of 42, 46, and 46, respectively, which are consistent with the positions of sudden improvement in Fig. 5(a).
Fig. 6. Normalized weights of linear kernels at the date rates of (a) 450 Mbps, (b) 500 Mbps, and (c) 550 Mbps.
VSgOMP was employed to compress the aforementioned simplified Volterra equalizer. Here, we define sparsity as the ratio of the number of ignored kernels to the number of total kernels, which represents the degree of sparsity and is used in following analysis of experimental results. Figure 7(a) shows that the BER performance improves with the decrease of the sparsity and step size parameter $\alpha$ of VSgOMP at the data rate of 500 Mbps. High sparsity results in low complexity of the equalizer, but the performance will degrade accordingly. The increase of $\alpha$ will increase the number of selected kernels in each iteration, resulting in less compression time and worse BER performance. In Fig. 7(b), at the data rates of 450 Mbps, 500 Mbps, and 550 Mbps, the BER will not degrade significantly when the sparsity is lower than 0.4, 0.3, and 0.2, respectively with the $\alpha$ of 0.3. To reduce the complexity of sparse equalizer while maintaining the BER performance, the preset sparsity and optimized $\alpha$ in the following experiments were both fixed at 0.3.
Fig. 7. (a) The effect of sparsity and step size parameter $\alpha$ of VSgOMP on the BER performance; (b) Relationship between sparsity and BER performance at the data rates of 450 Mbps, 500 Mbps, and 550 Mbps.
In addition, the performance of different equalization schemes at different bias currents was studied. The LS scheme was used as the linear equalizer. The memory lengths of other schemes are the same with the simplified Volterra equalizers. As shown in Fig. 8(a), with different equalization schemes, the best BER performance appears when the bias current is 0.4 A. Memory polynomial and Volterra equalizers have better performance than linear equalizer based on LS, confirming the existence of nonlinear impairments in the system. In addition, the dynamic ranges of bias current of the nonlinear equalizers are wider than the linear scheme. Compared with memory polynomial equalizer, Volterra equalizer with off-diagonal kernels can further improve performance. The number of kernels of the sparse Volterra equalizer is reduced by 58 compared with the Volterra equalizer. Since the sparse Volterra equalizer retains most of the significant kernels, the BER performance of OMP and VSgOMP schemes are similar to that of the Volterra scheme. The bias current of the laser and the attenuation value of the electric attenuator used in the following experiments were 0.4 A and 6 dB, resulting in an output optical power of 24.6 dBm. To investigate the reliability of the proposed UWOC system over a long-distance, we used optical attenuators and the optical power meter to adjust the received optical power. It can be found in Fig. 8(b) that when the optical power is lower than −27.5 dBm, the BER performance gradually improves with the increase of the optical power. Since PMT starts to get saturated, the BER increases significantly when the optical power is larger than −20 dBm. At a data rate of 500 Mbps, the minimum required optical power for the linear equalizer, memory polynomial equalizer, and sparse Volterra equalizer is about −31.5 dBm, −33 dBm, and −33.5 dBm, respectively.
Fig. 8. (a) Measured BER versus bias direct current at the data rate of 500 Mbps; (b) BER performance of different received optical power at the data rate of 500 Mbps.
Then, to study the effect of sparse Volterra equalizer enabled by VSgOMP on the UWOC system, we experimentally compared the VSgOMP scheme with OMP, ROMP, and gOMP schemes in terms of BER performance and time cost of the compression process with the received optical power of around −32.5 dBm and the data rate of 500 Mbps. After optimization, the sparsity level and regularization parameter of ROMP were set as 4 and 1.4, respectively, and the number of selected kernels in each iteration of gOMP was set as 3. Figure 9 shows that the performance of system with the VSgOMP scheme is almost the same when the sparsity is lower than 0.4, and the performance is similar with that of other schemes when the sparsity is lower than 0.5. Under high sparsity, since OMP selects one kernel in each iteration, the correlation between selected kernels is smaller and the performance of OMP is the best. In addition, VSgOMP has better performance than gOMP scheme owing to its adaptive step size.
In the process of equalizer compression, multiple iterations are required to find the kernels. Due to the calculation of inverse matrix in each iteration, the computational time of each iteration gradually increases with the accumulation of the selected kernels. The mean square error (MSE) between the equalized signal and the transmitted signal can effectively reflect the BER performance. Figure 10(a) describes the MSE performance versus the number of iterations when the sparsity is 0.3. To better compare performance, only part of the curves is shown. The convergence rate of MSE with the OMP scheme is the slowest due to selecting the kernels one by one. Conversely, the VSgOMP scheme with a large step size at the initial stages has the fastest convergence rate of MSE. The required numbers of iterations for OMP, ROMP, gOMP, and VSgOMP schemes are 139, 61, 47, and 24, respectively. To reduce MSE to 5% of the initial value, the corresponding numbers of iterations are 17, 9, 7, and 3, respectively.
Fig. 10. (a) MSE performance versus number of iterations; (b) Running time as a function of the number of required kernels.
Furthermore, we analyze the required running time for obtaining different numbers of required kernels using the four greedy algorithms, as shown in Fig. 10(b), which corresponds to the BER performance in Fig. 9. The running time under different sparsity was measured using Matlab2019b program on a personal computer under Intel Core i7 processor and Microsoft Windows 10 environment. Overall, it can be found that the running time of the OMP scheme is the longest. When the number of required kernels is small, the ROMP, gOMP, and VSgOMP schemes have similar running time. Since the adaptive step size is affected by the number of the required kernels, the initial step size of VSgOMP scheme increases as the number of required kernels increases, which will reduce the required number of iterations. When the number of required kernels is greater than 118, the running time of VSgOMP is significantly shorter than that of the ROMP and gOMP schemes. In addition, the trend is more obvious as the number of required kernels increases. When the number of required kernels is 138, that is, the sparsity is 0.3, the running time of the VSgOMP scheme is around 31.4%, 70.8%, and 74.2% of that of OMP scheme, ROMP scheme, and gOMP scheme, respectively.
To better analyse the sparse property of the Volterra equalizer used in the proposed UWOC system, the heat maps of 2nd-order kernel weights are shown in Fig. 11. The normalized weights of the 2nd-order kernels before sparsity are shown in Fig. 11(a). The values of x axis and y axis represent the memory serial number ${m_{21}}$ and ${m_{22}}$, respectively. The VSgOMP was employed to choose the significant kernels and recalculate the weights of kernels. When the sparsity is 0.3, the weights of kernels are shown in Fig. 11(b). After sparsity process, there are more white areas and around 45.8% of the 2nd-order kernels are deleted, thus the time complexity and space complexity of the equalizers are correspondingly reduced.
After completing the above experiments, we used the same experimental devices to further verify the performance and complexity of the proposed scheme in an indoor standard 50-m swimming pool. To calculate the attenuation coefficient of the underwater channel, we measured the received optical power using an optical power meter at different distances in the air and swimming pool. At the transmission distance of 100 m, 150 m, and 200 m, the received optical power is about −7.88 dBm, −16.04 dBm, and −23.67 dBm, respectively. After removing the measured geometric loss, the values of the received optical power and the fitted curve are shown in Fig. 12, and the estimated attenuation coefficient is 0.141 dB/m (0.0325 m−1).
The measured BER performance under an underwater transmission distance of 200 m is depicted in Fig. 13. In the insert, the red crosses are the misjudged signals and the blue points are the signals which are judged correctly. The sparsity of the sparse Volterra equalizer used in the experiment is 0.3. Under this sparsity, the performance of ROMP and gOMP schemes is similar to the VSgOMP scheme. The performance of the four greedy algorithms under different sparsity is shown in Fig. 9 and analyzed above, thus the results of ROMP and gOMP schemes are not shown in this part. With hard decision forward error correction (HD-FEC) threshold of $3.8 \times {10^{ - 3}}$, the achievable data rate of the least square equalization scheme is lower than 450 Mbps. The achievable capacity of the memory polynomial scheme is about 494.5 Mbps. The BER of the proposed VSgOMP scheme is $2.9 \times {10^{ - 3}}$ at a data rate of 500 Mbps, which is similar to Volterra and OMP schemes. Compared with the Volterra scheme, the number of sparse equalizer kernels is reduced by 30% with sacrificing only $2.5\%\sim 4.5\%$ BER performance. In the compression process, the required iterations of the VSgOMP scheme is 115 fewer than that of the OMP scheme. The required running time of the VSgOMP scheme can be reduced to 31.4% of that of the OMP scheme, verifying the feasibility and advantages of employing VSgOMP to obtain a sparse Volterra equalizer. The frequency spectra of the transmitted signal and received signal without equalizer and with VSgOMP-based sparse nonlinear equalizer at 500 Mbps are shown in Fig. 14(a) and (b), respectively. It can be found that the sparse nonlinear equalizer effectively compensates the impairments of signal in frequency domain.
Fig. 13. Measured BER performance with the data rates of 450 Mbps, 500 Mbps, and 550 Mbps through 200-m underwater transmission. Insert: signal amplitude distributions (i) without equalizer, with (ii) linear equalizer, and (iii) VSgOMP-based sparse nonlinear equalizer at the data rate of 500 Mbps.
Fig. 14. Frequency spectra comparison of the transmitted signal and received signal (a) without equalizer and (b) with VSgOMP-based sparse nonlinear equalizer.
5. Conclusion
In this paper, we propose a sparse Volterra equalizer with VSgOMP in a long-distance high-speed UWOC system. For the first time, the memory lengths of equalizers, step size coefficient of VSgOMP, and sparsity degree were analyzed and determined in experiments. At a data rate of 500 Mbps, the minimal received optical power is −32.5 dBm under the optimal bias current. In a 50-m swimming pool, the proposed UWOC system successfully achieves a data rate of 500 Mbps through 200-m underwater transmission. The system capacity with sparse Volterra equalizer is 14.9% and 4.5% higher than that of the least square equalization scheme and the memory polynomial equalization scheme, respectively. The experimental results prove that compared with the conventional Volterra equalizer, the sparse Volterra equalizer can effectively reduce the number of kernels by 30% with only $2.5\%\sim 4.5\%$ degradation of BER performance. Compared with the OMP and ROMP schemes, the VSgOMP scheme can reduce the running time by 68.6% and 29.2%, respectively. In the future, we will further optimize the nonlinear equalizer to improve UWOC system capacity.
Funding
National Natural Science Foundation of China (61971378); Guangdong Science and Technology Planning Project (2019A050503003); National Key Research and Development Program of China (2016YFC1401202, 2017YFC0306100, 2017YFC0306601); Strategic Priority Research Program of the Chinese Academy of Sciences (XDA22030208); Zhoushan-Zhejiang University Joint Research Project (2019C81081).
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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