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Abstract
Herein, we propose a novel entity extraction neural network (EXNN) with a newly designed sampling convolution kernel approach and a deep learning-based structure to differentiate noise in visible light communication (VLC) systems. In particular, EXNN is used to extract linear and nonlinear distortion in a received signal as an entity and compensate for the signal by removing it. First, we designed a deep learning structure tailored for VLC systems, used experimentation to validate our algorithm’s usability, and determined an appropriate range for the hyper-parameters that govern the PAM-8 system. Second, we compared our approach with existing finite impulse response (FIR) linear and Volterra nonlinear compensation algorithms via experiments. Below the hard-decision forward error correction (HD-FEC) threshold limit of 3.8×10−3, experimental results show that the use of the EXNN increased the operating range of the direct current (DC) bias and the voltage by ∼33.3% and by ∼50% under optimal conditions, respectively. Furthermore, under corresponding optimal power conditions, the proposed approach improved the Q factor of the VLC system by 0.36 and 1.57 dB compared with the aforementioned linear and nonlinear equalizers, respectively. To the best of our knowledge, this is the first time that a deep learning operator has been custom-designed for the VLC system and we have named the completely redesigned network with this sampling convolution kernel operator as EXNN.
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1. Introduction
Visible light communication (VLC) is a crucial component of future sixth-generation blueprints for achieving full spectral coverage. It uses visible light, which does not require a licensed frequency band and provides high potential transmission rates and low transmission latency. Owing to these properties, VLC is suitable for indoor positioning, vehicle networking, deep space communication, and underwater communication [1]. Though possessing all the aforementioned advantages, VLC faces some critical technical challenges that must be resolved to achieve its full standardization, such as noise [2] and nonlinear effects of the VLC system [3,4]. One of the most important challenges is that the performance of a VLC system would greatly deteriorate as the bit error rate (BER) surges when linear and nonlinear distortion increases.
The mechanism of distortion in VLC systems and how to compensate for it have always been a research focus of VLC research. To address the distortion problem, plenty of DSP-based compensation methods are proposed, including FIR-based equalization technology to compensate for linear distortion [5], and the Volterra-based equalization technology to compensate for nonlinear distortion in the system [6]. Though Volterra-based equalizers could deal with nonlinear distortion, due to the complicated expression of the Volterra series, the computational complexity of Volterra-based equalizers is expected to be much higher than FIR-based equalizers given the same number of taps. To lower the computational complexity of Volterra equalizers, the number of taps and the order of the Volterra series are usually restricted, which leads to the degradation of its compensation accuracy.
Machine learning (ML) is introduced as an effective solution to the distortion problem due to its good performance on nonlinear regression and data categorizing. Both traditional ML algorithms and neural networks have been proven effective in compensating for the nonlinear distortion of VLC systems. For example, the Support Vector Machine (SVM) algorithm is used to address phase impairments in VLC systems [7]. For the nonlinearity caused by the electric amplifier and optoelectronic device in the Nyquist PAM-4 VLC system, the damaged signal is compensated based on K-means clustering algorithm [8]. The probabilistic Bayesian algorithm is used to estimate the nonlinear power-current curve of the LED device to compensate the nonlinear distortion of the VLC system [9]. Moreover, many network structures that are well-designed for other fields are transplanted to VLC systems to solve nonlinear distortion. For example, deep neural network (DNN) based equalizer can effectively compensate the high nonlinear distortion of the underwater PAM8 visible light communication channel [10]. In [11], the demodulator based on convolutional neural network compensates the linear and nonlinear distortion in NOMA-VLC system. A post-equalizer based on long short-term memory (LSTM) neural network is used to compensate linear and nonlinear impairments in pulse amplitude modulation visible light communication systems [12]. Neural networks have shown high potential in dealing with nonlinear distortion, but without special changes to the networks’ structures, these networks which are directly transplanted into VLC systems could not realize their real potential.
In this paper, we propose a novel network structure and design a specific sampling convolution kernel to compensate for the damaged signals in VLC systems. In particular, we design an entity extraction neural network (EXNN) with a deep learning-based structure. EXNN uses the sampling convolution kernel to extract linear and nonlinear distortion in the received signal as an entity and then subtracts the extracted distortion entity from the received signal in the time domain to compensate for it. In our experiment, under optimal operating conditions, the proposed EXNN improves the Q factor by at least 0.53 dB compared with the FIR algorithm. Under optimal current and different voltage conditions, the EXNN improved the Q factor by 1.57 and 1.14 dB compared with the FIR equalizer and Volterra equalizers, respectively.
2. Principle
Additive white Gaussian noise (AWGN) and multiple impairments exist in VLC systems, including inter-symbol interference (ISI) due to the limited modulation bandwidth of LEDs, and nonlinear distortion from LED, PIN photodetectors and electronic amplifiers (EA). Our approach is to use the derivation capabilities of neural networks to extract linear and nonlinear distortion generated in the system using a non-classification method. Further, we design special network structures and kernels to compensate for the damaged signals by considering the effect of distortion on the entire system, in place of high-order and high-tap mapping functions. In particular, these kernel function layers provide inference and transformation mechanisms to extract linear and nonlinear distortion from the received signal as an entity.
2.1 EXNN architecture
Our EXNN model combines statistical principles with deep learning techniques to extract both linear and nonlinear system distortions from the VLC system, thereby affording a technique for compensating for the damaged signals. Figure 1 presents the design of our model and the employed signal recovery principle. The EXNN is used to quantify the linear and nonlinear system distortions present in the received signal, and then subtract the network quantized distortion signal from the received corrupted signal to get a compensated signal, while the unresolvable background noise remains intact within the compensation signal.
The symbols and network parameters used in this study are defined in Tables 1 and 2.
Table 1. General symbol definition
Table 2. Parameter settings of the designed EXNN
We subdivide the damage into quantized system distortion and background white noises; then, the received signal in the time domain can be expressed as follows:
where $Tx$ denotes the undamaged signal at the transmitter, ${D_s}$ denotes the system distortion, and ${N_w}$ represents the white noise. The $\tilde{D}$ denotes distortion extracted by the EXNN through a distortion mapping on the received signal $Rx$:Then, $\tilde{D}$ is subtracted from $Rx$ to obtain an equalized signal $Y^{\prime}$, which is expressed as follows:
Here, when the calculated $\tilde{D}$ is equal to ${D_s}$, the final network output constitutes the system transmitter signal and the white noise signal, i.e., $Y^{\prime} = Tx + {N_w}$. The EXNN eliminates the linear and nonlinear distortion in the signal, thus realizing the maximum signal compensation.
2.2 Distortion extraction structure
The EXNN structure eliminates the ${D_s}$ presents in the received signal and enables signal compensation. Figure 2 shows the EXNN structure.
The distortion mapping structure of the received signal $Rx$ comprises four layers. This structure extracts the mapping of the nonlinear and linear relations present in the signal using sampling convolution kernel sets within each layer.
The mapping relation, which is shown in Eq. (2), from the received signal to the system distortion is realized by four layers in EXNN through its sampling convolution kernel sets within each layer. The specific steps to realize distortion mapping are presented below:
Figure 3 shows a diagram of the extraction function structure. The distortion in the signal is extracted using the sampling convolution kernel. In the extraction function, the kernel matrix comprises the Dr, Fs, and Fn. We express this convolution of the signal as
2.3 Extraction function structure diagram
Considering the temporal correlation between long-range signal points and the signal feature reduction caused by the constructed deep network, we designed a sampling convolutional kernel to improve the extraction of the temporal relation between signals and decrease the disappearance of signal features. To remove noise from an image, deep learning techniques extract the noise present in the image using a conventional two-dimensional convolution kernel [13]. We compare the signal extraction structure using conventional convolution kernel and sampling convolution kernel (Fig. 4). The designed sampling convolution kernel is different from the conventional convolution kernel used in CNNs. Figure 4(a) presents the specific structure of the three-layer sampling convolutional layer in the EXNN. Figure 4(b) shows the conventional one-dimensional convolution kernel used in CNNs.
Fig. 4. Signal extraction using (a) a sampling convolutional kernel and (b) a conventional convolutional kernel.
As an example, we use a filter size of 3 for the convolution kernel (Fig. 4). Figure 4(a) presents the signal processing performed using the sampling convolutional layer with different Dr values, where the extraction range of the signal in the previous layer gradually increases. Note that the signal in the previous layer can be fully extracted, ensuring that the next sampling layer contains additional signal information. In Fig. 4(b), the signal is processed using a conventional one-dimensional convolutional layer, in which the Dr value is kept constant at 1 and each layer has a limited extraction range for the signal in the previous layer. Considering this, we can express the signal coverage for each layer as
where $Fr$ denotes the coverage of the signal points of the current layer and k denotes the layer number. By comparing these two kernels, we found that under the same computational complexity the receptive field of the sampling convolutional kernel is wider than the conventional one-dimensional convolution kernel. Thus, it is more suitable for feature extraction in long-distance signals.2.4 Training and updating the parameters
The training process of EXNN is shown in Fig. 5. The compensation performance varies with parameters of EXNN, the parameters are updated to realize the maximum signal compensation during training.
The input is the received signal $Rx$, and $Y_\omega ^{\prime}$ is the compensated signal which is the output of the EXNN when its parameters are $\omega $. The mean square error (MSE) function is used to calculate the loss value of the transmitted signal $Tx$ and ${Y_\omega }^{\prime}$, and the Adam optimizer is used to minimize the loss value to complete the update of the EXNN’s parameters. The updated parameters include the weight and bias parameters of the sampling convolution kernel sets of each layer. The formula to calculate the loss value is as follows:
where the output signal $Y_\omega ^{\prime}$ of the network and the signal $Tx$ of the transmitter are used to calculate the loss value by employing the MSE. The MSE function is defined as where L represents the length of the signal, and $||\, ||_2^2$ denotes the L2 norm of a vector. We use the Adam optimizer to minimize the loss of $Y_\omega ^{\prime}$ and $Tx$, thereby updating the $\omega $ of the network [14], which can be expressed asAfter these parameters are updated using Eq. (11), the parameters $\omega $ of the EXNN are updated by calculating the loss value using the MSE. Here, the input of the MSE function includes the $Y_{{\omega _{t - 1}}}^{\prime}$ and the $Tx$.We reuse the Adam optimizer, this time with enhanced convergence in deep learning, to dynamically adjust the learning rate in updating the weight parameters $\omega $ of the network as the loss value is reduced. When the weight parameter $\omega $ of the network reaches the optimum value, the MSE between $Y_{{\omega _{t - 1}}}^{\prime}$ and $Tx$ reaches the minimum value, and the neural network equalizer with a well-trained weight parameter $\omega $ is obtained.
3. Experimental setup
Figure 6 shows the experimental setup of the PAM-modulated VLC system employing the EXNN. At the transmitting end, the data are converted into PAM-8 symbols, which are up-sampled before up-conversion. The up-conversion function avoids low-frequency noise in VLC systems [15]. Then, the signal is generated using an arbitrary waveform generator (AWG) and passed through a basic hardware equalizer. The LED is driven by current, and to allow the LED to reach the switching current threshold, the signal is passed through the EA. Then, the direct current (DC) is added through the bias tee to drive the LED for electro-optical conversion. A blue LED is used to transmit the light signal through the free-space channel. Some physical parameters used in the experiment are shown in Table 3.
Table 3. System and physical parameters in the experiment
At the receiving end, we use PIN photodiodes to detect light signals. Before employing PIN, a lens is used to focus light. The PIN output is amplified using the EA and then sampled and recorded using an oscilloscope at a sampling rate of 1GSample/s. Finally, the signal is stored in a standard laptop for offline digital signal processing via a network cable/GBIP interface. In offline digital signal processing, down-sampling and down-conversion are performed after synchronization, and then the signal is used as the input of the EXNN equalizer, and the output through the EXNN equalizer is the compensated signal.
4. Experimental results
The experimental results are presented in three sections. In Section 4.1, in this study, the existence of nonlinearity in the system is verified by intuitive Rx-Tx back-to-back data in the experiment and basic polynomial fit. At the same time, the intuitive results of mitigating nonlinear and linear distortion in the signal with EXNN are also shown in this way. Section 4.2 was the parameter definition experiment of EXNN. Compared with neural networks that have been applied in other fields, the parameters of the newly designed EXNN have no reference range, so, the structure and hyperparameters of the EXNN used in the PAM-8 system are optimized by experimental, such as the number of layers, the size of the sampling convolutional kernel (including Fn, Fs, Dr), and the training parameters, i.e., epochs. In Section 4.3, performance comparison experiments are performed. The sampling convolution kernel is compared with the conventional convolution kernel, and the EXNN is compared with the classic digital signal processing algorithms, such as scalar modified cascaded multi-modulus algorithm (S-MCMMA) and Volterra algorithm. It should be noted that, in Section 4, the Q factor and the BER are used interchangeably. In other words, the BER and Q factor values can be converted to each other. At the same time, in the whole experiment, we use different sequences of data as training dataset and test dataset respectively, in order to prevent the overfitting of deep learning.
4.1 Correctness experiment for nonlinear distortion elimination
To obtain direct statistics on the distribution density of the signal, we performed direct polynomial fitting on the received signal to determine the presence of linear and nonlinear distortion in the VLC system (Fig. 7(a)). Figure 7(b) shows the compensation results for the damaged signal using our EXNN algorithm. The nonlinearity problem in the system could be eliminated, and the ISI between the signal level was improved; however, the EXNN could not eliminate some AWGN from the system.
Fig. 7. (a) Receiving signal polynomial fitting curve and (b) signal polynomial fitting curve after EXNN compensation.
4.2 Optimization experiments
In addition to the network structure design (described in Section 2), hyperparameters affect the fitting performance of the EXNN with respect to the signal. To avoid underfitting and overfitting owing to extremely complex or extremely simple networks and thus achieve an optimal EXNN that can compensate the damaged signal, we conducted hyperparameter selection experiments to tune the hyperparameters of the EXNN. The hyperparameters adjusted in our experiments were the Fs, Fn, the number of network layers, and epochs, each of which is described in further detail below.
The Fs determines the signal sampling range of the network. Based on our experiment, we found that when the Fs of the designed EXNN is 3, distortion extraction using the EXNN is optimal. We designed an experiment in which the Fs is varied from 1 to 15 in the EXNN for the same set of received signals. The training dataset of the EXNN contains 15000 data, and the test dataset contain 15000 data. The EXNN achieves the best compensation performance for the signal when the Fs is 3 (Fig. 8(a)). Moreover, the compensation performance gradually decreases as the Fs is increased from 3 to 15.
The Fn determines the extraction strength of the features of the hidden layer in the signal. When the Fn of the EXNN is 64, the optimal compensation performance of the EXNN is realized. If an excessively small Fn is selected, insufficient feature extraction in the signal is observed, resulting in the loss of signal features. Alternatively, if the Fn is excessively large, the network will have a memory of signal characteristics, enabling the network to remember pseudorandom sequences instead of solving nonlinear problems. Memory refers to a unique phenomenon in deep learning; if the pseudorandom number sequence is used in the experiment, the algorithm will remember the training symbol/bit order in some cases, thus providing the illusion of higher performance in experiments than in real applications. Moreover, we use different random numbers $Tx$ to avoid this situation in the experiment. In our experiments, we constructed EXNNs with different Fn values for the same set of signals; in particular, the Fn takes the values [16, 32, 64, 128, 256]. Figure 8(b) shows our experimental results. The maximum compensation performance of the network is achieved when the Fn is 64, and both nonlinear and linear distortions present in the signal are effectively extracted.
To compensate damaged signals, the EXNN structure must be designed for different modulation orders. When passing features through a network structure with a very large depth, feature loss may occur, yielding an incomplete signal in the deep network. Alternatively, a network structure with a very small depth may not learn the features effectively. The structure of the designed EXNN in the PAM-8 system is a three-layer sampling convolution layer that can fully extract the features from the damaged signal. In the experimental environment, the voltage of the system is set to 0.8 V, the DC bias current is adjusted in the range of (5-140 mA), and the data under different currents are collected at the receiving end for experiments. The experimental results are shown in Fig. 9. When the current is 5 mA, the optical power of the system is too low, resulting in a very low Q gain of the system. When the optical power is enhanced to pursue high SNR, the nonlinear effect of the system is enhanced. When the current reaches 140 mA, the nonlinear distortion in the system cannot be compensated. At the same time, we found that in the selection of the number of EXNN’s layers, when the number of network layers is less than 3, the compensation performance of EXNN is enhanced with the increase of optical power. The memory of the training dataset is caused, and the performance of the system is degraded when experiments are performed using the test dataset. According to the above experimental results, we choose the optimal number of network layers to be 3.
In the EXNN structure, high Dr values are set as the number of network layers increases. This can ensure that the high-level network layer contains more extracted features than the low-level network layer. In Section 2.3, the Dr of the sampling convolution kernel represents the interval between the sampling points of the signal. When the Dr of the three-layer structure of the EXNN is set to [1,2,4], we ensure that the deep network can fully learn the low-level signal features. In the experiments, two sets of data were used as training dataset and test dataset for multiple experiments, and the overall performance results of the Dr parameters were represented by boxplots as shown in Fig. 10. The experimental results show that the Dr parameters of the EXNN are selected as [1,2,4] and its experimental performance is the best. In the selection of the Dr parameters of the first layer, it can be observed that the subsequent higher Dr will make the experimental results diverge, and the performance is still lower than the performance when Dr is set to 1. From the analysis of the logical reason, it is because the Dr set too high in the first layer will lead to the loss of the correlation in the continuous sequence signal, and the performance is consistent with the theory. Similarly, in the selection of Dr parameters for the second and third layers, it is found that the higher the Dr parameters are set, the lower the performance. According to the experimental results, the Dr parameters of the second and third layers are set to 2 and 3 respectively.
In the experiment, we determined the number of signals extracted and the number of EXNN parameters updated based on the number of epochs. If the number of epochs is very low, the network could not extract the signals sufficiently, resulting in an inability to fit the network parameters that match the current signal. Alternatively, if the number of epochs is very high, the system tends to memorize the signal, leading to overfitting problems. In our experiments, we used different epochs for the same set of data. The eye diagram performance of the signal is good when the number of epochs is set to 200; however, no significant improvement is observed in the compensation performance when the number of epochs is increased beyond 200. When the number of epochs is set to only 50, the eye opening of the corresponding signal is not good (Fig. 11(a)). When the number of epochs is set to 200, the eye diagram of the signal is clearly opened (Fig. 11(b)). When the number of epochs exceeds 400, the eye diagram in Fig. 11(c) did not show significant improvements relative to that presented in Fig. 11(b). Therefore, the number of epochs for our EXNN is set to 200.
4.3 Comparison experiments
We experimentally verify that the EXNN with sampling convolution kernels (blue curve with circle) proposed in Section 2.3 can enhance damaged signal feature extraction and achieve better equalization performance than conventional convolution kernels (orange curve with square), Additional, the green curve in Fig. 12 shows the BER of the received signals. Since linear distortion predominates in the system, the BER of all received signals before compensation is basically the same though nonlinearity increases as current increases, which is shown in Figs. 12(a)-(c) whose red lines are polynomial fitting on the received signals. Furthermore, two EXNNs with the same structure are constructed—one using a conventional convolutional kernel for distortion extraction and the other using the proposed sampling convolution kernel for experiments. The experimental results are shown in Fig. 12. The EXNN constructed using the sampling convolutional kernel achieves a lower BER than that constructed using the conventional convolutional layer, with the best BER reduction of 42.8% and the average BER reduction of 26.8%. The sampling convolution kernel can greatly improve the relationship between long-distance signals. Compared with the ordinary convolution kernel, its complexity is only different from the constant term. Combined with the special structure of the EXNN, the BER of the signal can be reduced.
We compared the compensation performance of the proposed EXNN and an FIR equalizer on a damaged signal. In our experiment, we selected the S-MCMMA FIR equalizer that achieves optimal signal compensation performance. We examined the distribution of the signal using a histogram; the histogram of the signal after applying various equalization algorithms is shown in Fig. 13. Figure 13(a) shows the histogram of the received signal after passing through the VLC system; here, we observed both linear and nonlinear distortion in the received signal as well as the AWGN signals, whose overall distribution conforms to a Gaussian distribution.
Fig. 13. (a) Received signal histogram. (b) Signal histogram after applying the S-MCMMA algorithm. (c) Signal histogram after applying the EXNN approach. (d) Assessed constellation points of the received signal. (e) Assessed signal constellation points after applying the S-MCMMA algorithm. (f) Constellation points of the verdict signal after applying the EXNN approach.
Under the superimposed interference of noisy signals, the BER is shown in Fig. 13(d). Here, the BER of the constellation points is very high; furthermore, the constellation points intersect with one another, rendering the distinction of the constellation points between the signals impossible. Figure 13(b) presents the histogram of the signal after passing through the S-MCMMA equalizer. Compared with the histogram of the received signal, gaps are detected between the signals after passing through the S-MCMMA equalizer, and eight distinct sets of signal distributions are observed between the signals. Figure 13(e) presents the constellation point assessment after employing the S-MCMMA equalizer. Compared with the corresponding received signal constellation point assessment, obvious gaps are observed between the signal constellation points.
Figure 13(c) shows the histogram of the signal after applying the EXNN equalization. The distribution density of different signal levels is more dispersed; the results of the corresponding constellation point assessment are shown in Fig. 13(f). Overall, the EXNN algorithm afforded a clearer signal constellation point distribution and a considerably lower BER than the S-MCMMA algorithm; in particular, the BER is decreased by 60.8%.
To compare the performance of the EXNN with those of S-MCMMA and Volterra, we designed experiments to obtain signals under specific current and voltage conditions of the VLC system. The results show that the signal compensation performance of the EXNN increases the range of optimal operating conditions under the limit of the hard decision forward error correction (HD-FEC) threshold. Figure 14(a) shows the experiments conducted under different current ranges and the optimal voltage. The distance between the receiver and transmitter is fixed at 1.2 m, and the voltage is fixed at 0.8 V. We collected signals in the current range of 5–140 mA. When the current range is 80–100 mA, the S-MCMMA and Volterra algorithms could not equalize the damaged signal, whereas the EXNN equalizer could effectively equalize the damaged signal and improve the operating range of the DC bias under the HD-FEC threshold by 20 mA.
Fig. 14. Comparison of BERs of S-MCMMA, Volterra, and EXNN algorithms in terms of (a) current and (b) voltage.
Figure 14(b) shows the experimental results at different voltage ranges; here, the distance between the receiver and transmitter is set to 1.2 m, and the current is constant at 100 mA. The signals are collected in the voltage range of 0.2–1.8 V. In the voltage interval of 0.2–0.3 V, the optical signal power is insufficient, the SNR is too low, and all equalization algorithms show poor performance. Alternatively, the voltage range of 0.4–0.6 V covers the linear operating area of the blue light LED used in our experiment. Thus, the EXNN, S-MCMMA, and Volterra algorithms could equalize the linear damage in the signal. In the voltage range of 0.6–0.9 V, the nonlinearity of the system increases when an increase in the voltage and total power of the optical signal; moreover, the signal compensation performance of the Volterra equalization algorithm decreases; however, the EXNN algorithm could still compensate for the nonlinearity of the system. In voltage interval of 0.9–1.8 V, the background noise and nonlinear distortion in the system are in the saturation zone, and the Volterra algorithm led to overfitting for the nonlinearity in this interval; however, the EXNN algorithm has a better compensation effect than the Volterra algorithm, as shown in the figure.
In additional, our proposed EXNN can be generalized in VLC systems. For advanced modulation schemes (such as CAP and QAM-OFDM), EXNN has two compensation schemes.
- 1. The distortion from the signals is removed on the in-phase and quadrature component, respectively, and then the signals obtained after distortion removal are combined and recovered.
- 2. For complex symbols yielded by CAP and QAM-OFDM, the same structure of the EXNN can be employed; however, a higher-dimensional convolution kernel is needed to extract the distortion from the complex symbols and finally complete the signal equalization.
Furthermore, because the PAM system uses the same principle in the EXNN compensator such as PAM-16 or PAM-32, the received signal can be directly provided as input to the EXNN to obtain the signal after distortion removal.
5. Conclusions
In this study, we proposed EXNN with sampling convolution kernels which is a new signal processing method that uses deep learning techniques to compensate for linear and nonlinear distortions in VLC systems. For the damage signal compensation performance from the EXNN in this paper, the proposed sampling convolution kernel reduces the BER by 42.8% compared with the conventional convolution kernel. In our experiments, we adjusted the hyperparameters—the Fs, Fn, Dr, and the number of epochs—to achieve the optimal signal compensation performance of the EXNN. We conducted experiments under different operating conditions to compare the signal compensation performances of the proposed EXNN, a linear FIR equalizer, and a nonlinear equalizer Volterra. Under different DC bias operating conditions, the EXNN increases the operating range of the DC bias by 40 and 20 mA compared with the linear FIR and Volterra equalizers, respectively, and the Q factor by 0.53 and 0.36 dB, respectively. Under different voltage conditions, the EXNN increases the Q factor by 1.57 and 1.14 dB compared with the FIR and Volterra equalizers, respectively. The nonlinear equalizer Volterra induced overfitting problems in the saturated zone of the background noise and nonlinear distortion in the voltage environment, causing equalization failure; however, the EXNN still showed partial compensation performance in highly nonlinear cases.
Funding
Science and Technology Research Program of Chongqing Municipal Education Commission (KJQN202000644); National Natural Science Foundation of China (62006032, 62072066, 61772098).
Acknowledgments
A portion of this work was performed in the Key Laboratory for Information Science of Electromagnetic Waves, Fudan University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this study may be obtained from the authors on reasonable request.
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