Neural-network-based end-to-end learning for adaptive optimization of two-dimensional signal generation in UVLC systems

Ruizhe Jin; Yuan Wei; Junwen Zhang; Junwen Zhang; Junwen Zhang; Jianyang Shi; Jianyang Shi; Nan Chi; Nan Chi

doi:10.1364/OE.510449

1. Introduction

Visible light communication (VLC), as a short-range optical wireless communication technology, is becoming an increasingly promising candidate in the next generation of communication (6 G) [1–3]. Compared with the existing radio frequency communication system, VLC has the advantages of high data rate, anti-electromagnetic interference, high security, fast switching, etc. These advantages enable its broad applicability in specific communication scenarios, such as inter-satellite communication and underwater wireless communication. In the context of underwater wireless communication, conventional methods typically involve acoustic wave communication and radio frequency communication [4,5]. Among these methods, acoustic wave communication suffers from limited communication rates due to its low bandwidth range (2-5 kHz). On the other hand, radio frequency communication faces challenges as electromagnetic waves experience rapid attenuation in water, restricting its usage to short transmission distances. Therefore, the industry urgently needs a new type of communication method that can achieve a higher communication rate while meeting the requirements of long-distance underwater transmission. UVLC, benefiting from underwater blue-green projection windows, has aroused great interest from industry and research worldwide. [6]. UVLC enables long-distance transmission with minimal loss, further emphasizing its potential for various applications [7].

To achieve stability and wideband characteristics in simulating a practical UVLC channel, it is imperative to accurately emulate the UVLC channel during the system design process. When visible light traverses underwater channels, it is affected by absorption, scattering, and turbulence effects [8,9], resulting in overall energy attenuation and variations in the spatial and temporal distribution of light. The former affects link budget and signal-to-noise ratio calculations, while the latter imposes limitations on the maximum achievable bit rate in underwater communication systems, primarily manifesting as phenomena such as inter-symbol interference [10]. The academic community has extensively researched the modeling of UVLC systems. Techniques including radiative transfer theory, Monte Carlo simulations, and closed-form expressions of double Gamma functions have been applied for underwater channel modeling [11–13]. Xu Ma et al. performed numerical modeling for static UVLC [14], while H.M. Oubei et al. introduced varying temperature differences in underwater environments to simulate weak turbulence scenarios for investigation [15]. Nevertheless, the modeling methods mentioned above solely address the impact on the optical field of visible light after transmission in water, which proves to be inadequate. For a more comprehensive channel modeling, it is imperative to account for factors such as the impulse response of electrical amplifiers, bias current and nonlinear response of LEDs, and modulation formats [16]. This is because alterations in these parameters lead to varying impacts on light signals within different underwater channel conditions. For instance, multi-carrier modulated systems are particularly vulnerable to nonlinear effects induced by weak turbulence resulting from fluctuations in temperature and pressure [17,18]. Neural networks have found extensive application in channel modeling research. While initially successful in modeling indoor wireless communication [19], the performance of neural networks designed for indoor air scenarios is less satisfactory when applied to underwater channel systems [20]. This is attributed to the fact that the channel impulse response of UVLC undergoes changes due to factors such as water impurities, variations in current, and the intensity of background light. These variations are notably different from free-space conditions, making it unsuitable to directly apply neural networks designed for free-space scenarios to underwater channel modeling [21].

UVLC systems can be categorized into two main types: Light emitting diode (LED) systems and laser diode (LD) systems. When comparing UVLC systems, LED-based implementations have notable advantages such as lower cost, enhanced safety, and improved alignment tolerances [22]. Nevertheless, UVLC systems that utilize LEDs encounter a range of challenges. The inherent incoherence of LED systems necessitates the utilization of intensity modulation and direct detection (IMDD) techniques [23]. Moreover, LEDs exhibit an extremely limited modulation bandwidth [4], [22]. When transmitting signals at high baud rates, the high-frequency components are significantly attenuated or suppressed. To combat the challenging underwater channel conditions, UVLC systems often require higher power levels to achieve an improved signal-to-noise ratio (SNR) [24,25]. However, with the increase in power, the UVLC system experiences a substantial rise in nonlinear distortion. These drawbacks collectively impact the overall performance of the UVLC system.

In order to address the signal impairments in UVLC systems, extensive research has been conducted on various compensation methods, focusing on both the transmitter and receiver components. The method of pre-equalization at the transmitter is employed to mitigate the effects of signal attenuation in the high-frequency range. Pre-equalization techniques primarily encompass both hardware pre-equalization and digital pre-equalization methods. Hardware pre-equalization suffers from the challenge of accurately estimating design parameters [26–28]. On the other hand, while digital pre-equalization allows for parameterizing the spectrum of the transmitted signal using the inverse channel response, compensating for severe high-frequency attenuation through this method can result in a significant reduction in overall SNR. At the receiver end, the use of a nonlinear equalizer can help mitigate the detrimental effects of nonlinearity on the signal. Common approaches include employing Volterra technology or utilizing a multi-layer neural network [29,30]. These signal processing blocks are widely adopted in conventional communication systems. While these techniques can achieve locally optimal performance mathematically, they do not guarantee globally optimal performance across the entire communication system.

The literature [31] proposes the replacement of the entire communication system with a neural network and suggests that global optimization of the communication system can be achieved through training the neural network. This work has sparked a multitude of studies on AE techniques in the field of VLC. In the literature [32], a novel scheme is presented for mitigating the Peak-to-Average power ratio (PAPR) in VLC systems. This approach combines the use of a weighted AE and an amplitude clipping method within an end-to-end learning network. The primary goal of this scheme is to overcome the challenges arising from high PAPR and LED nonlinearity in VLC systems. In [33], a designed ANN transmitter is utilized for OOK-modulated VLC, aiming to generate a practical set of OOK pulses. A multi-stage training strategy is implemented to gradually transform the continuous-valued encoder output into an OOK signal. However, it is important to note that the channel model does not take into account the effects of ISI. In Ref. [34], the multicolor VLC scenario is specifically examined, where the entire VLC communication system is represented as an AE model. The end-to-end symbol recovery process encompasses the VLC transceiver pair and the channel layer, which characterizes the optical channel. The aforementioned investigations are solely conducted in a simulation environment, and it is important to note that the channel model used differs from the actual VLC channel. Our previous work, described in literature [35], addressed the issue of inaccurate channel estimation resulting from excessive low frequency noise (LFN) in the VLC system through the implementation of spectrum shifting. However, due to the use of a self-encoder network that attempts to combine various functions such as encoding, modulation, equalization, and the utilization of spectrum shifting during the signal modulation process, the achieved transmission rate is limited to 1.875Gbps. This rate is only marginally better than that of single-carrier pulse amplitude modulation 8 (PAM8) communication system. In our recent advancement within the VLC domain, described in literature [23], we focused on enhancing the equalization function using a neural network while maintaining the conventional coding and modulation schemes of the communication system. The network was trained and tested using discrete multitone (DMT) signals. As a result, the performance of the AE surpassed that of the DMT modulation format by 30.4%. However, it is worth noting that this AE network used in this study primarily focuses on the performance enhancement achieved through equalization and does not take into account the potential improvements stemming from the distribution of constellation points.

In this paper, for the first time, we propose a neural network-based 2D adaptive optimization autoencoder (2D-AOAE) framework in UVLC systems. To the best of our knowledge, this is the first end-to-end learning framework that enables 2D signal generation and restoration while automatically avoiding LFN damage to VLC systems through neural networks. The whole 2D-AOAE includes mapping network and demapping network, modulation network and demodulation network and channel model. Our proposed 2D-AOAE network framework achieves optimal constellation point distribution and globally optimal equalization and filtering functions under different bandwidth and nonlinear conditions. Compared to our previous work using traditional AE as outlined in [35], the 2D-AOAE introduced in this paper achieves an impressive 73% speedup. Furthermore, the framework surpasses the performance of the traditional two-dimensional modulation format 32QAM, increasing the transmission rate by 15.4%. In terms of nonlinear processing capability, the 2D-AOAE solution extends the operating voltage range from 340 mV to 480 mV in a 32QAM communication system using a DNN as a post-equalization network. The experimental results highlight the great potential of the proposed framework in a wide range of multi-scenario applications in the future.

2. Principle

2.1 2D adaptive optimization autoencoder

The schematic diagram of the neural network-based traditional AE model and the 2D-AOAE models is shown in Fig. 1. In the traditional AE scheme, multiple layers of MLP networks are used to generate transmitted signals and recover received signals. However, this single data stream network structure cannot generate a two-dimensional signal like the superposition of two data streams in the I/Q channel of the QAM modulation format. The transmission network can only produce a one-dimensional signal, leading to lower spectral efficiency in the signal. Furthermore, relying solely on neural network training, traditional AE methods cannot address the challenges posed by LFN and high-frequency attenuation in UVLC systems. Unlike other communication systems, the presence of intense LFN in UVLC system results in almost complete signal loss at low frequencies. Hence, in conventional AE, researchers are required to manually adjust the transmitted data generated by the transmission network, shifting the baseband signal up in frequency to avoid the adverse effects of strong LFN before proceeding with transmission. This approach is both intricate and substantially diminishes the effective transmission rate as the additional frequency shifting introduces more symbols into the process. According to Ref. [35], the rate calculation expression of the traditional AE model can be expressed as:

(1)$$\alpha = \frac{{{f_{in - band}}}}{{{f_0}}}$$

(2)$$ R_{A E}=\frac{\alpha \cdot F S}{N_{\text {output }}} \cdot \log _2(M) $$

where $\alpha $ represents the proportion of the in-band spectrum range in the total spectrum. ${R_{AE}}$ represents the effective rate of the AE model. $F{s_{awg}}$ is the sampling rate of AWG. ${N_{output}}$ is the number of the transmit network taps, and M is the order of the modulation format. Therefore, based on the above equation, in the literature [35], when the spectrum shifting ratio is set to 50% and the number of transmit network taps is 4, the effective rate is only half of the conventional communication system with an upsampling rate of 4. This also explains why the traditional AE model, using a 64-QAM modulation scheme, can only be compared with the PAM8 communication system.

Fig. 1. Principle of neural network-based traditional AE model and 2D-AOAE model of UVLC system

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Analysis indicates that the limitations of the traditional AE model are apparent. It requires manual frequency shifting to mitigate the damage caused by LFN and high-frequency attenuation to the signal. Additionally, it is incapable of generating two-dimensional signals with high spectral efficiency. Therefore, we propose the 2D-AOAE model to address the aforementioned shortcomings. We propose the use of a convolutional structure in the transmission network to achieve orthogonal operations on the signal, generating a two-dimensional signal while simultaneously applying pre-equalization. Designing the initial parameters of the convolutional kernels enables the transmitted signal to simultaneously avoid low-frequency noise (LFN) and compensate for high-frequency attenuation, a capability that is challenging for traditional MLP networks to achieve. This eliminates the cumbersome manual frequency shifting operations and significantly enhances the effective transmission rate. Additionally, the 2D-AOAE takes into account the impact of both pre-equalization and post-equalization schemes on transmission performance, while comprehensively considering constellation point distribution. Through global training, the network can simultaneously identify the optimal constellation point distribution and the best equalization scheme for different channels.

In this work, we adopt the one-hot encoding to train our AE network as an ad hoc approach. The conversion of decimal data to one-hot encoding has been demonstrated as an effective approach to minimize symbol error rates [39]. Its straightforward encoding approach ensures that messages maintain a clear and unintended ordering, setting it apart from integer encoding. Although one-hot encoding effectively reduces symbol error rates, it does not achieve optimal performance at the bit level. In our future research, we intend to explore AE network solutions at the bit level, considering the adoption of bit-level labeling and the implementation of bit-level loss functions, among other approaches. The decimal data is first converted into one-hot codes of length M and then enters the mapping model of 2D-AOAE, where it is mapped to constellation point coordinates. The constellation point coordinates are split into I/Q paths, where orthogonalization is achieved through convolutional networks. Subsequently, the results undergo L-fold upsampling and are then added to generate the transmitted data. In the channel model, we have retained the dual-branch network that has been validated to provide sufficiently good performance to simulate the underwater visible light channel. The received data is demodulated by the demodulation model to obtain the constellation point coordinates. After passing through the demapping model, decimal data is obtained, which is then converted into binary codes. The final bit error rate is calculated based on this process. Therefore, in the 2D-AOAE scheme, each symbol can represent $lo{g_2}(M )$ bits. With an upsampling rate of L and a DAC sampling rate of $F{s_{awg}}$, the effective data rate can be expressed as:

(3)$${R_{2D - AE}} = \frac{{F{s_{awg}}}}{L} \cdot {\log _2}(M)$$

The training process is divided into four steps, as shown in Fig. 2. Initially, a sequence of energy-normalized random numbers is generated using an arbitrary waveform generator (AWG) and transmitted through the UVLC system. The transmitted data is subjected to impairments introduced by system components and channels, and the resultant received data is collected at the receiver. This received data is designated as the label for training the channel network model, consequently establishing a digital simulation model for the UVLC system.

Fig. 2. Training and test procedure of 2D-AOAE

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Fig. 3. Schematic of dual-branch neural network channel model

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Subsequently, the trained channel network model remains fixed. The input data undergoes constellation mapping through the mapping model, followed by data stream splitting. The convolution kernel parameters are initialized using a pair of orthogonal shaping filter tap values, known as the Hilbert filter pair. The expressions for these filters are as follows:

(4)$${f_I}(n) = \frac{{\sin [\pi (1 - \alpha )\beta ] + 4\alpha \beta \cos [\pi \beta (1 + \alpha )]}}{{\pi \beta [1 - {{(4\alpha \beta )}^2}]}} \cdot \sin (\pi \beta (1 + \alpha ))$$

(5)$${f_Q}(n) = \frac{{\sin [\pi (1 - \alpha )\beta ] + 4\alpha \beta \cos [\pi \beta (1 + \alpha )]}}{{\pi \beta [1 - {{(4\alpha \beta )}^2}]}} \cdot \cos (\pi \beta (1 + \alpha ))$$

(6)$$\beta = \frac{{n - ({N_{taps}} + 1)}}{N}$$

In which, α represents the roll-off factor, ${N_{taps}}$ is the total tap length, and N is a constant coefficient. In the experiment, we set α=0.205, ${N_{taps}}$ =41, N=8. The low-frequency component of the selected convolutional kernel spectrum is minimal. This characteristic ensures that the signal, after convolution, is scarcely transmitted in the low-frequency range, thereby avoiding the impact caused by LFN. The signal undergoes filtering through the convolutional network, and the shaped signal is then input into the channel network. After being affected by channel impairments, the signal passes through a pair of fixed orthogonal shaping filters for signal recovery. Subsequently, the signal is recombined, and the data is recovered using the demapping model. In this step, only the parameters of the mapping model and demapping model can be trained. This approach enables a faster attainment of a more optimal distribution of constellation points. In the third step, the mapping model and demapping model are fixed, and the stationary filters are replaced with the modulation network. Following data stream separation, the signal traverses two modulation networks individually, facilitating adaptive filtering and pre-equalization functionalities. After passing through the channel network, the impaired signal undergoes post-equalization and recovery using the demodulation network and demapping network at the receiver end. During this step, only the parameters of the modulation network and demodulation network can be trained, expediting the achievement of improved equalization and filtering effects. In the final step, only the parameters of the channel network are fixed, while the other networks undergo global training based on the previously trained models. Different learning rates are applied, and the model is trained with varying learning rates to identify the best-performing model and minimize the risk of getting stuck in a local minimum. This transfer learning approach is adopted because, in our experiments, training the entire 2D-AOAE model from a random state at the beginning resulted in prolonged training times and unstable outcomes due to the substantial number of parameters. Utilizing transfer learning allows us to leverage pre-trained models, leading to more stable and efficient training.

In the testing phase, the data is transmitted through the trained transmit network to generate the transmission signal. After traversing the real UVLC system, the received signal is decoded using the trained receive network, enabling the calculation of the bit error rate (BER). Throughout this process, both the transmitted and received signals can serve as training data for refining the channel neural network. By iteratively executing these operations, a more accurate channel model can be obtained, resulting in enhanced performance. This iterative approach facilitates the continuous refinement of the channel model, thereby improving the overall system performance.

2.2 Dual-branch neural network channel model

In the selection of the channel network model, the double-branch neural network improved for the noise layer in the literature [23] is used. Before deploying the channel model with the noise layer, we conducted performance tests on the channel network without the noise layer. We observed a significant degradation in the AE performance when using a channel model without a noise layer in real-world system environments. This is because a channel network without noise layers essentially represents a deterministic nonlinear system, which deviates significantly from the actual characteristics of a real channel. In the training process of the entire network, a batch size of 256 is employed, and the optimizer is set to Adam. The signal undergoes energy normalization via the energy normalization layer to maintain consistent energy levels in the input data. Subsequently, the data is split into two separate data streams, with each stream directed through specific layers. One stream is directed through the linear branching layer, while the other traverses the nonlinear branching layers. In the linear branch, neither of the signals undergoes an activation function, and the connections between the layers of the neural network are solely linear. After passing through the first hidden layer, the signal is further processed by a noise layer. This noise layer introduces Gaussian white noise to the signal, with the energy size parameter of the noise being updatable during the training process of the neural network. The output of the linear branch can be expressed as

(7)$$Y_{output}^{linear} = w_2^l((w_1^l{X_{input}} + b_1^l) + N(0,{\delta ^{(1)}})) + b_2^l$$

The nonlinear branch maintains a structure identical to the linear branch, incorporating a noise layer positioned between the two hidden layers. Utilizing the Rectified Linear Unit (ReLU) activation function, Gaussian white noise undergoes transformation into nonlinear noise during the signal transmission process. This feature is advantageous for effectively capturing and learning various nonlinear noises inherent in UVLC systems, including signal-signal beat interference (SSBI). The output of the nonlinear branch can be expressed as:

(8)$$Y_{output}^{nonlinear} = relu[w_2^n(relu[w_1^n{X_{input}} + b_1^n] + N(0,{\delta ^{(2)}})) + b_2^n]$$

After the signal goes through the linear branch and the nonlinear branch, it is then passed through another parameter-learnable noise layer following the output layer. The final layer of the noise network is employed to fine-tune the ultimate strength of the Gaussian white noise once again.

Fig. 4. Schematic diagram of the transmission network part in 2D-AOAE.

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Fig. 5. Schematic diagram of the receiving network part in 2D-AOAE.

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2.3 Transmit neural network

Figure 4 shows the network structure of the transmit neural network. The transmission network primarily comprises two components: the mapping network and the modulation network. Initially, decimal data is converted into a one-hot encoding representation. Subsequently, after traversing two linear hidden layers, the network outputs two-dimensional data representing the signal's I/Q coordinates. The mathematical expression for this process is as follows:

(9)$${Y_{I,Q}} = {w_2}(({w_1}{X_{input}} + {b_1})) + {b_2}$$

The two-dimensional data is reshaped into separate dimensions, representing the I-channel and Q-channel data streams. These individual streams are then fed into their respective modulation networks. The data stream is initially upsampled to a higher sampling rate and then passes through a convolutional layer for signal filtering. Subsequently, the output signals from the I-channel and Q-channel modulation networks are combined to obtain the processed transmission signal. Assuming that the two-dimensional data stream obtained after mapping is:

(10)$$Y(t) = {Y_I}(t) + j \cdot {Y_Q}(t)$$

When the upsampling factor is L, the upsampled symbol stream can be expressed as:

(11)$$Y{(t)^{\prime}} = \left\{ {\begin{array}{c} {{Y_{I,Q}}((t - 1)/L + 1),\bmod (t - 1,L) = 0}\\ {0,others} \end{array}} \right.$$

After passing through the modulation network, the transmitted signal obtained can be expressed as:

(12)$$Y = {Y_I}{(t)^{\prime}} \otimes f_T^I + {Y_Q}{(t)^{\prime}} \otimes f_T^Q$$

2.4 Receive neural network

Figure 5 shows the network structure of the receive neural network. The receiving network primarily consists of three components: the post-equalization network, the demodulation network and the demapping network. The received signal undergoes signal equalization by passing through a two-hidden-layer MLP network. The equalized signal is further recovered by two demodulation networks to obtain the final received signal. In the demodulation network, the signal is first filtered through the convolutional layer, and the output signal is:

(13)$${Y_I}{(t)^{\prime}} = Y(t) \otimes f_T^{I^{\prime}},{Y_Q}{(t)^{\prime}} = Y(t) \otimes f_T^{Q^{\prime}}$$

After the signal passes through the convolutional layer, it is down-sampled to achieve signal recovery. This down-sampling operation can be represented as:

(14)$$Y(t) = Y((t - 1)/K + 1),\bmod (t - 1,K) = 0$$

Considering the impact of signal degradation in the UVLC system, and to enhance the network's processing capability, the demapping network employs the ReLU activation function, in contrast to the mapping network, which solely uses linear connections. The data is ultimately transformed into M probability vectors through the SoftMax layer. Each probability vector signifies the likelihood that the input data belongs to a specific category. The training process ensures that the index with the highest probability is designated as the “1” entry in the corresponding one-hot input vector to the transmitter. Throughout the entire 2D-AOAE network training, cross-entropy is employed as the objective function, expressed as:

(15)$$L = \sum\limits_{n = 1}^M - {s_n}\log ({\widehat s_n})$$

3. Experiment setup

The experimental setup of our LED-UVLC system, depicted in Fig. 6, is utilized for our experiments. Once the neural network generates the transmission data, the corresponding transmission file is generated offline using MATLAB. The generated transmission file is then imported into an AWG to generate the actual transmission signal. The sampling rate of the AWG is set within the range of 1.4 GSa/s to 2.4 GSa/s. After passing through an amplifier, the transmission signal is directed through a bias tee to apply a bias voltage. The signal is then emitted via a green LED into a tank approximately 1.2 m long. Due to spatial constraints, we were unable to incorporate multiple water tanks of varying lengths. Presently, we are in the process of constructing additional and longer water tanks to simulate a more diverse range of underwater environments. Additionally, we have plans to conduct experiments in real outdoor underwater settings in the future. The optical signal is converted into an electrical signal using a photodiode. The differential structure is adopted to receive the optical signal, which is beneficial to further reduce the influence of noise on the signal. The amplified signal is passed through a TIA and fed into an oscilloscope for offline processing. The received signal is processed using MATLAB to adjust for data dimensionality, leading to the generation of a received data file. Subsequently, a neural network is employed to recover the data.

Fig. 6. Experimental setup of UVLC system

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Table 1 provides an overview of the parameters utilized in our proposed 2D-AOAE. The mapping network employs only linear layers to accomplish the task of mapping constellation points. Regarding the modulation network, the length of the convolutional kernels will impact the performance of the network. This is because shorter tap lengths may compromise the equalization capability of the filter. Conversely, excessively long tap lengths may lead to overfitting, potentially causing a performance decline in the transmission signal shaping through the actual channel. Therefore, it is crucial to select an appropriate convolutional kernel length to avoid the mentioned issues and ensure optimal performance. In our experiments, we conducted a search for the optimal tap length, and the results are illustrated in Fig. 7(a). Based on the search results, in this experiment, we have chosen $Nc = 41$ as the optimal number of taps for the convolutional layer. We employed grid search to discover the optimal hyperparameter values for the channel network, and the search results are depicted in Fig. 7(b).

Fig. 7. Parameters optimization results of the (a) convolution kernel taps and (b) channel neural network

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Table 1. Structure and parameters of the 2D-AOAE

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Considering a sufficient channel memory length to fully simulate the ISI in the received signal, we select the optimal input and output pair, nearing (40, 20), in the circled-out area of Fig. 7(b). In the end, we chose $Nt = 39$ and $Nr = 21$ as the values for the number of transmit and receive taps, respectively. In the demapping network, the recovered constellation points tend to be scattered. To effectively aggregate and classify these points, we employ the ReLU activation function, which has shown to be effective in capturing non-linear relationships and enhancing the discriminative power of the network.

4. Experiment results and discussions

Figure 8 illustrates the performance obtained with different model frameworks in our UVLC system. We conducted tests to evaluate the performance of the traditional AE model framework compared to a PAM8 communication system. The parameter settings for the traditional AE model was kept consistent with the proposed framework [24]. The transmission network is configured with 4 taps, and the frequency spectrum is reserved at 50%. We opted for 32QAM as the reference modulation format among 16QAM, 32QAM, and 64QAM. This decision is driven by the lower spectral efficiency of 16QAM, limiting its support for higher data rates. Additionally, the performance of 64QAM degrades quickly at higher bit rates due to the system's pronounced high-frequency attenuation. we conducted tests on two 32QAM communication systems, one with neural network pre-equalization and the other without. Notably, the performance curves of these two methods intersected at a data rate of 1.94Gbps. The observed phenomenon can be attributed to the bandwidth limitation of the UVLC system. According to the literature [23], it has been demonstrated that in bandwidth-limited systems, the use of neural network pre-equalization can lead to significant degradation in communication performance due to severe SSBI at the receiving end. Therefore, in comparison to the 32QAM communication system utilizing neural network pre-equalization, the performance of a system that solely applies neural network post-equalization demonstrates better communication performance. The proposed 2D-AOAE in this paper achieves a higher communication rate of 2.85Gbps compared to the 32QAM communication system, which represents an increase of 370Mbps. Furthermore, when compared to the traditional AE scheme, the 2D-AOAE demonstrates a significant improvement with an increase of 1.22Gbps in communication rate. These results indicate that both the traditional 32QAM communication system and the traditional AE autoencoder scheme are inferior by the proposed 2D-AOAE in terms of communication performance.

Fig. 8. BER corresponding to different rates under different algorithms

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Before delving into the evaluation of the 2D-AOAE autoencoder's performance, it is essential to examine the performance of the channel model using the dual-branch neural network. Figure 9 illustrates the disparity between the transmitted signal after undergoing actual underwater channel impairment and the channel model-based impairment. We can observe that within the filtered signal frequency band, the mismatch between the two signals is very small. However, there is a noticeable difference in the low-frequency portion of the signal. This difference arises from the presence of LFN in the UVLC system, which makes it difficult for the neural network to accurately learn the spectral response of the low-frequency components. In the high-frequency region outside the signal frequency band, the signal-to-noise ratio decreases significantly due to high-frequency attenuation and low signal energy. As a result, some deviations can be observed in this region as well. Although there may be discrepancies between the channel model and the real channel outside the signal band, the channel model exhibits consistency with the real channel within the signal band, which is sufficient for experimental purposes. Figure 9(b) depicts the loss curve of the channel model during training, which stabilizes after approximately 15 epochs. Furthermore, Fig. 9(c) and Fig. 9(d) display the spectra within the signal band range for the channel model and the real channel, respectively.

Fig. 9. a) Spectral mismatch between real channel and channel model; (b) Loss varying with epoch in channel model training Spectrum of pre-equalized signal after transmission through (c) real channel and (d) channel model.

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Figure 10 depicts the loss curves during the training process of the second and third steps in the training strategy (Fig. 3). During the training of the mapping network and demapping network, the initial constellation points are randomly distributed, resulting in a relatively high loss. The loss gradually stabilizes after around 20 epochs. After inheriting the parameters of the mapping network and demapping network, the modulation and demodulation network training begins. The convolution kernel parameters are initialized to the Hilbert filter pair tap value. The only network with random initialization is the post-equalization network. Due to less random parameters, the loss convergence speed is faster, and the loss tends to stabilize after approximately 15 epochs, remaining relatively stable thereafter.

Fig. 10. training and validation loss of the 2D-AOAE model

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Figure 11 illustrates the changes of the convolution kernel tap values, convolution kernel spectrum, and constellation point distribution with respect to epoch during the training of the neural network. During the initial stages of training, the convolutional kernel parameters are initialized with the Hilbert filter pair values. As training progresses, it becomes apparent that the convolutional neural network filters and equalizes the signal. This observation highlights the network's ability to learn and execute signal filtering and equalization operations. Through the convolution operation, the neural network selectively reduces the low-frequency component of the signal, mitigating the adverse effects of LFN in the UVLC system and preventing signal degradation. Additionally, it enhances the high-frequency energy of the signal to combat channel fading in the higher frequency range. The distribution of constellation points gradually transitions from the initial random distribution to a mixed constellation point distribution scheme in which the inner circle is hexagonal geometric shaping (GS) distribution and the outer circle is amplitude phase shift keying (APSK) distribution. We opt for the shortest Euclidean distance to quantify the quality of the generated constellation point scheme. During the training process, the shortest Euclidean distance increased from an initial value of 0.10091 to a final value of 0.35763. Nevertheless, it is important to acknowledge that in a strongly non-linear channel, relying solely on the shortest Euclidean distance may not comprehensively capture the quality of the generated constellation point distribution scheme. This limitation arises because each constellation points encounters varying degrees of non-linearity based on its position. As the constellation point distribution extends away from the origin, indicating an increase in energy, the impact of non-linearity intensifies. Consequently, in a non-linear channel, the optimal constellation point distribution is characterized by slightly larger Euclidean distances between points in the outer ring compared to those in the inner ring. This distribution is crucial for ensuring accurate transmission of each individual point. In the constellation point scheme produced by our 2D-AOAE, we observe a distribution that adheres to this theoretical pattern. Remarkably, the neural network autonomously learns and discovers the optimal constellation point distribution based solely on the characteristics of the UVLC underwater channel, without any prior knowledge. During the training process of the kernel taps, it was observed that the orthogonality between the two convolution kernels decreased. The magnitude of this orthogonality is directly related to the pre-equalization effect.

Fig. 11. The convolution kernel spectrum, tap value and constellation point distribution of the received signal after different epoch 2D-AOAE models.

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Figure 12 depicts the high-frequency fading amplitude and the corresponding high-frequency compensation amplitude of the signal after filter equalization within the signal frequency band at various data rates. Under the autoencoder framework, the neural network employs a strategy of allocating a certain proportion of energy to high frequencies during pre-equalization while compensating for high frequency fading. However, this method does not increase the high-frequency energy of the signal too high through pre-equalization like the traditional neural network pre-equalization, but increases the high-frequency energy to an appropriate ratio. The pre-equalization network and post-equalization neural network achieve the best overall equalization performance through the global training of AE. In addition, it can be observed that with the improvement of the pre-equalization effect, the orthogonality of the two convolution kernels will decrease, which is specifically manifested in the increase of the inner product of the two convolution kernels.

Fig. 12. Channel high-frequency attenuation range, model high-frequency boost range and convolution kernel inner product value at different data rates

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Figure 13(a). depicts the performance of different algorithms at 500Mbaud. At this data rate, the system utilizing the neural network for pre-equalization is greatly affected by the presence of SSBI due to the severe bandwidth limitation, resulting in noticeably poor performance. When Vpp is low, the low light intensity at this time results in noise having a dominant impact on the system, leading to a low SNR and resulting in poor system performance. As Vpp gradually increases, the light intensity also increases, and at the same time, the nonlinear effects of the system become stronger. In high Vpp conditions, the nonlinear effects of the system dominate the system's performance. When applying post-equalization algorithms in the 32QAM system, three commonly used approaches are the LMS algorithm, Volterra algorithm, and MLP neural network algorithm. Notably, the LMS algorithm lacks the capability to effectively handle nonlinear compensation, resulting in poorer performance compared to the other algorithms. On the other hand, the Volterra algorithm demonstrates better performance at lower values of Vpp, as it can address certain nonlinear effects. However, as Vpp increases, the level of nonlinearity exceeds the limit that the Volterra algorithm can effectively handle, leading to a rapid deterioration in performance. We also tested the method of using the classic MLP network as a signal post-equalization module, and the parameters of the MLP network used are consistent with the post-equalization network in 2D-AOAE. We choose the BER threshold of 3.8e-3. The received signal can be near-errorless recovered by the HD-FEC with a 7% overhead if the BER before HD-FEC is lower than 3.8E-3 [36]. Ultimately, we achieve a 7% FEC code threshold of 3.8e-3 within a Vpp range of 340 mV. Compared with all the above algorithms, the 2D-AOAE scheme proposed in this paper achieves a Vpp working range of 480 mV, which proves that the scheme has strong robustness in the face of nonlinearity.

Fig. 13. a) Data rate versus Vpp with different digital signal processing algorithms. (b) Data rate versus BER for different digital signal processing algorithms.

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Figure 13(b) displays the graph depicting the relationship between BER and data rate under optimal Vpp conditions. It compares the performance of the Volterra equalization algorithm, MLP neural network post-equalization algorithm, and the 2D-AOAE end-to-end framework in a 32QAM system. When bandwidth is limited, using the neural network for pre-equalization has a detrimental effect on performance due to the presence of SSBI. Solely employing the Volterra algorithm fails to achieve the desired 3.8e-3 threshold due to its inadequate nonlinear equalization capabilities. In contrast, the 2D-AOAE end-to-end learning algorithm outperforms the MLP-based post-equalization algorithm used in the 32QAM system, achieving a rate increase of 0.35Gbps and reaching a rate of 2.85Gbps.

5. Further analysis

We conducted complexity analyses of different algorithms. Table 2 presents the network parameters of the conventional AE that we employed. In the traditional 32QAM algorithm, we employed a NN equalizer with two hidden layers and node counts of (41, 64, 64, 1). We applied a sliding window approach to perform equalization on the received signals. Table 3 lists the detailed computational cost of the E2E framework per symbol in one epoch. With the specific value in our experiment, the computational cost of the E2E framework for traditional AE needs $5.4694 \times {10^{10}}$ multiplications and $5.4677 \times {10^{10}}$ additions in the deployment phase, $2.7824 \times {10^{12}}$ multiplications and $2.7771 \times {10^{12}}$ additions in the training phase. The 2D-AOAE needs $1.1432 \times {10^{10}}$ multiplications and $1.1198 \times {10^{10}}$ additions in the deployment phase, $6.1359 \times {10^{11}}$ multiplications and $6.0774 \times {10^{11}}$ additions in the training phase. The post-equalization neural network needs $4.4460 \times {10^8}$ multiplications and $4.3614 \times {10^8}$ additions in the deployment phase, $2.8940 \times {10^{10}}$ multiplications and $3.5229 \times {10^{10}}$ additions in the training phase. We can observe that the traditional 32QAM scheme exhibits the lowest computational complexity, employing only two layers of hidden neural networks for post-equalization, while the conventional AE approach exhibits the highest computational complexity. As mentioned above in our effective rate calculations, conventional AE requires experimentalists to perform frequency shifting, necessitating a higher number of input taps to accommodate higher data rates. Therefore, with other parameters held constant, the input layer for traditional AE is set to M=64, while for 2D-AOAE, it is M=32. Therefore, although 2D-AOAE has more neural network layers, the computational complexity is lower than that of the traditional AE scheme. This is due to the smaller number of nodes in each layer and the use of convolutional layers instead of fully connected layers. We are also investigating the implementation of novel techniques, such as network pruning and meta-learning-aided online training, to further decrease computational complexity [37,38].

Table 2. Structure and parameters of the traditional AE

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Table 3. The Complexity Analysis of the E2E Framework

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6. Conclusion

In this paper, we propose an end-to-end learning framework 2D-AOAE in UVLC systems. The traditional AE framework suffers from limitations, as it can only generate one-dimensional signals and necessitates spectrum shifting operations to address the impact of LFN. These constraints significantly diminish the potential of traditional AE schemes. To address this issue, we divide the data into two data streams in the 2D-AOAE framework, and complete the generation and recovery of two-dimensional signals through the corresponding modulation and demodulation networks respectively. In a 1.2 m long underwater channel, our proposed 2D-AOAE framework achieves the highest bitrate transmission of 2.85Gbps at 7% HD-FEC threshold. This rate is 0.35Gbps faster than the 32QAM SCM solution using the MLP network as the post-equalization module, and 1.2Gbps faster than the traditional AE solution. These results demonstrate the great potential of 2D-AOAE in realizing underwater high-resolution VLC transmission. Our findings provide valuable insights into the field of end-to-end learning for underwater visible light systems, opening new possibilities for future advancements.

Funding

National Key Research and Development Program of China (2022YFB2802803); National Natural Science Foundation of China (No. 61925104, No. 62031011, No. 62201157).

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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Type of layer	Number of Neurons &. Length of convolutional kernel	Activation
Mapping Model
Input	$M$	$N o n e$
Hidden layer	$4 M$	$N o n e$
Output	$2$	$N o n e$
Modulation Model
Convolutional Layer	$41$	$/$
Channel Model
Input	$L \cdot N t$	$N o n e$
Tributary1 - Hidden Layer1	$L \cdot (N t - 1) / 2$	$N o n e$
Tributary1 – Noise Layer	$L \cdot (N t - 1) / 2$	$/$
Tributary1 - Hidden Layer2	$L \cdot N r$	$N o n e$
Tributary2 - Hidden Layer1	$L \cdot (N t - 1) / 2$	$R e L U$
Tributary2 – Noise Layer	$L \cdot (N t - 1) / 2$	$/$
Tributary2 - Hidden Layer2	$L \cdot N r$	$R e L U$
Output	$L \cdot N r$	$N o n e$
Demodulation Model
Input	$L \cdot N p + 1$	$N o n e$
Hidden layer1	$2 \cdot (L \cdot N p + 1)$	$R e L U$
Hidden layer2	$2 \cdot (L \cdot N p + 1)$	$R e L U$
Output	$1$	$N o n e$
Flatten	/	$/$
Convolutional Layer	41	$/$
Demapping Model
Input	$2$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
Output	$M$	$s o f t m a x$

Type of layer	Number of Neurons	Activation
Transmitter Model
Input	$M$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
output	$L$	$N o n e$
Receiver Model
Input	$L \cdot N r$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
Output	$M$	$s o f t m a x$

Network	Phase	Multiplication	Addition
Channel network	Training	$\begin{matrix} L^{2} N t (N t - 1) + L^{2} (N t - 1) N r \\ + 2 L^{2} N r^{2} \end{matrix}$	$\begin{matrix} (L N t - 1) L (N t - 1) + L N r (L (N t - 1) \\ + 2 L N r - 4) + (L N r - 1) L N r \end{matrix}$
Traditional AE	Training	$\begin{matrix} L N t l o g_{2} (L N t) + N t (20 M^{2} + 4 M L) \\ + L^{2} N t (N t - 1) + L^{2} (N t - 1) N r \\ + 2 L^{2} N r^{2} + 4 M L N r + 20 M^{2} \end{matrix}$	$\begin{matrix} 2 L N t l o g_{2} (L N t) + (L N t - 1) L \\ \cdot (N t - 1) + L N r (L (N t - 1) + 2 L N r \\ - 4 + (L N r - 1) L N r + \\ N t (20 M^{2} - 8 M + 4 M L - L) \\ + 4 M L N r + 20 M^{2} - 9 M \end{matrix}$
Traditional AE	Deployment	$\begin{matrix} N t (20 M^{2} + 4 M L) + 4 M L N r \\ + 20 M^{2} \end{matrix}$	$\begin{matrix} N t (20 M^{2} - 8 M + 4 M L - L) \\ + 4 M L N r + 20 M^{2} - 9 M \end{matrix}$
2D-AOAE	Training	$\begin{matrix} 2 (N t + 10) l o g_{2} (2 (N t + 10)) + \\ (4 M^{2} + 8 M) (N t + 10) + 82 N t L + \\ L^{2} N t (N t - 1) + L^{2} (N t - 1) N r \\ + 2 L^{2} N r^{2} + L (N r - N p) [6 (L N p + 1)^{2} \\ + 2 (L N p + 1) + 82] + 8 M + 20 M^{2} \end{matrix}$	$\begin{matrix} 4 (N t + 10) l o g_{2} (2 (N t + 10)) + \\ (4 M^{2} + 4 M - 2) (N t + 10) + 80 N t L + \\ (L N t - 1) L (N t - 1) + L N r (L (N t - 1) \\ + 2 L N r - 4 + (L N r - 1) L N r + \\ L (N r - N p) [6 (L N p + 1)^{2} - \\ 2 (L N p + 1) + 80] - M + 20 M^{2} \end{matrix}$
2D-AOAE	Deployment	$\begin{matrix} (4 M^{2} + 8 M) (N t + 10) + 82 N t L + \\ L (N r - N p) [6 (L N p + 1)^{2} + \\ 2 (L N p + 1) + 82] + 8 M + 20 M^{2} \end{matrix}$	$\begin{matrix} (4 M^{2} + 4 M - 2) (N t + 10) + 80 N t L + \\ L (N r - N p) [6 (L N p + 1)^{2} - \\ 2 (L N p + 1) + 80] - M + 20 M^{2} \end{matrix}$

Type of layer	Number of Neurons &. Length of convolutional kernel	Activation
Mapping Model
Input	$M$	$N o n e$
Hidden layer	$4 M$	$N o n e$
Output	$2$	$N o n e$
Modulation Model
Convolutional Layer	$41$	$/$
Channel Model
Input	$L \cdot N t$	$N o n e$
Tributary1 - Hidden Layer1	$L \cdot (N t - 1) / 2$	$N o n e$
Tributary1 – Noise Layer	$L \cdot (N t - 1) / 2$	$/$
Tributary1 - Hidden Layer2	$L \cdot N r$	$N o n e$
Tributary2 - Hidden Layer1	$L \cdot (N t - 1) / 2$	$R e L U$
Tributary2 – Noise Layer	$L \cdot (N t - 1) / 2$	$/$
Tributary2 - Hidden Layer2	$L \cdot N r$	$R e L U$
Output	$L \cdot N r$	$N o n e$
Demodulation Model
Input	$L \cdot N p + 1$	$N o n e$
Hidden layer1	$2 \cdot (L \cdot N p + 1)$	$R e L U$
Hidden layer2	$2 \cdot (L \cdot N p + 1)$	$R e L U$
Output	$1$	$N o n e$
Flatten	/	$/$
Convolutional Layer	41	$/$
Demapping Model
Input	$2$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
Output	$M$	$s o f t m a x$

Type of layer	Number of Neurons	Activation
Transmitter Model
Input	$M$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
output	$L$	$N o n e$
Receiver Model
Input	$L \cdot N r$	$N o n e$
Hidden layer1	$4 M$	$R e L U$
Hidden layer2	$4 M$	$R e L U$
Output	$M$	$s o f t m a x$

Neural-network-based end-to-end learning for adaptive optimization of two-dimensional signal generation in UVLC systems

Abstract

1. Introduction

2. Principle

2.1 2D adaptive optimization autoencoder

2.2 Dual-branch neural network channel model

2.3 Transmit neural network

2.4 Receive neural network

3. Experiment setup

4. Experiment results and discussions

5. Further analysis

6. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (3)

Equations (15)

Optics Express