1. Introduction

With the rapid development of the information age, the conflict between individuals’ demand for high-speed wireless communication and limited communication bandwidth is one of the main contradictions in the current wireless communication field. In the indoor environment, visible light communication (VLC) is an effective means to solve this contradiction [1]. VLC is an optical wireless communication system that uses light-emitting diodes as transmitters and positive-intrinsic-negative (PIN) or avalanche photodiodes (APD) as receivers [2,3]. The higher carrier frequency makes VLC possible to realize a larger bandwidth optical wireless communication system [4]. Combining the characteristics of lighting and communication, the VLC communication system has great application potential in the current environment where LED is the main lighting appliance [5,6]. Providing high-speed wireless communication services while lighting is very energy-saving and cost-efficient. At the same time, since the blue and green light is in the transmission window of the seawater, the VLC system with blue and green LEDs as the transmitter can provide underwater workers with high-speed optical wireless communication services [7–9].

Due to the lower modulation bandwidth (∼15 MHz) of LEDs, the maximum achievable bit rate of the single-input-single-output VLC system is severely restricted [10]. Utilizing the concept of multiple-input-multiple-output (MIMO) that is widely used in 5G, can effectively help the VLC system to realize high-speed optical wireless communication [11]. However, since the VLC system is usually based on intensity modulation and direct detection (IM/DD) technology, the decorrelation of multiple input signals at the receiving end has become a new challenge [12]. Since the MIMO-VLC signal may cause the channel matrix to be ill-conditioned during the decorrelation process, algorithms such as orthogonal circulant matrix transform (OTC) and singular value decomposition (SVD) cannot be applied to the MIMO-VLC system. Imaging receivers have been proved to be a potential way to reduce the correlation of VLC signals to ensure a perfect channel matrix after the decorrelation process. However, the imaging receiver has very demanding requirements for the alignment of the transmitter and the receiver, which limits the application potential of the imaging receiver (especially in dynamic scenarios) [13]. Fortunately, VLC systems based on superimposed code modulation (SCM) do not require imaging receivers, which reduces the production cost and difficulty of the transmitter and receiver. Assuming that the final received signal is 16QAM, SCM can be composed of two kinds of streams, the first is the superposition of two 4QAM streams, and the second is the superposition of two PAM8 signal streams. The difference between these two schemes is that the SCM conducted by two 4QAM streams requires that the ratio of the power of the two signal sources is not equal to 1. When the power ratio is equal to 1, the constellation diagram of the two data streams at the receiving end will be degenerate. To ensure the bit error rate of the received signal when the signal-to-noise ratio is low, the Vpp of one stream is at least 1.5 times that of the other stream. The power mismatch will lead to power competition between the two signals, which will reduce the signal-to-noise ratio (SNR) of the signal with lower power. For the SCM conducted by two 4PAM streams, the best working condition is that the power of the two signal sources is the same. And in the case of any power ratio, the constellation diagram at the receiving end will not be degenerate, which makes the SCM conducted by two 4PAM streams has greater application potential in VLC systems.

In the single-input-single-output (SISO) VLC system, a powerful post-equalization algorithm is generally required to equalize the nonlinear distortion in the system. At present, the discussion and research of post-equalization algorithms based on artificial neural networks (ANNs) are most concerned. In 2014, Paul Haigh proposed an ANN equalizer in the on-off keying modulated VLC system for the first time. He experimentally proved that the ANN-based post-equalization algorithm has significantly better BER performance than adaptive decision feedback (DF) and linear equalizers [14]. A Gaussian kernel-aided deep neural network was reported, which could achieve 1.5Gbps in a PAM8 underwater VLC system [15]. Furthermore, a dual-branch multi-layer perceptron-based post-equalizer (DBMLP) has been reported, which could achieve 3.2Gbps in a CAP64 underwater VLC system [9]. However, there are few studies on ANN as a post-equalizer in MIMO VLC systems. In [13], the author proved that the SISO-DNN equalizer does not perform well in the MIMO-VLC system, and further proposed a multi-branch hybrid neural network (MBNN) with higher space complexity but better BER performance. However, excessive complexity will increase the demand for computing power at the receiving end, and further increase the energy consumption of the receiving end, which will lead to a decrease in the practicality of the algorithm. In addition, because MBNN has two hidden layers, the design of the hidden layer hyperparameters will become very difficult. For instance, the number of nodes in each layer and the choice of activation function both faces a huge number of combinations. However, it is almost impossible to traverse these combinations in the process of network structure design. Therefore, the research of ANN post-equalization algorithm in MIMO VLC system should be further carried out.

In this paper, we propose two adaptive artificial neural network post-equalizer (AANN) algorithms based on power ratio $\alpha $ in 16QAM SCM and carrier-less amplitude-phase (CAP) SR-MIMO VLC systems. By introducing the MIMO-LMS algorithm into the SISO DNN algorithm, the issues that the SISO DNN cannot work effectively in the MIMO VLC system proposed in [13] is solved. Since the ANN algorithms utilized in the proposed AANNs have only one hidden layer, the difficulty of hyperparameter design is reduced. The hyperparameter combination between hidden layers no longer exists, which reduces the design cost of the ANN post-equalization algorithm. Furthermore, by introducing the adaptive strategy, the computational spatial complexity of AANNs is only 10% of MBNN without damaging the BER performance of the SR-MIMO VLC system. The following content of this paper will elaborate on the design ideas and process of AANNs, and experimentally measure its BER performance in the practical MIMO VLC system.

2. Principle

2.1 Principle of the PAM4 SCM 16QAM SR-MIMO VLC system

Figure 1 illustrates the principle of the SR-MIMO VLC system in detail. At the transmitting end, the two original binary data streams are first mapped into two PAM4 data streams respectively. To reduce the inter-signal interference (ISI), the original two PAM4 data streams should be up-sampled to get two up-sampled data streams (${U_1}(t )$ and ${U_2}(t )$). To further reduce ISI, we utilize square-root raised cosine (SRRC) filters ($r(t )$) to complete the pulse shaping of ${U_1}(t )$, and ${U_2}(t )$. U1 and U2 are respectively multiplied by the cosine sub-carrier ($\cos ({2\pi {f_0}t} )$) and the sine sub-carrier ($- \sin ({2\pi {f_0}t} )$) to ensure the orthogonality of the two signal streams while completing the up-conversion process. ${f_0}$ is the center frequency of the up-conversion process. Finally, the signal is converted from digital to analog (DAC) by the AWG and emitted as a light signal by the LEDs. The transmitted signal streams of two transmitters could be expressed as

 

Fig. 1. The block diagram of a SR-MIMO VLC system.

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The received signal streams we get after passing through the VLC channel can be expressed as

At the receiving end, a PIN converts optical signals into electrical signals. An Oscilloscope (OSC) is utilized to complete the analog-to-digital conversion (ADC) process. At this stage, the signal can still be expressed as the function (3). It should be noted that h(t) and g(x) contains the response of the receiver. At this point, we noticed that both $g(x)$ and $h(t)$ act on $I(t)$, which is the sum of the power of the two signal streams. Since the two optical signals are superimposed on the PIN of the receiving end, it is easy to make the optical power on the PIN too high, which will cause the response of the PIN to enter the saturation interval. It further leads to more serious nonlinear distortion. Theoretically, we can model and equalize $g(x)$ and $h(t)$ at this stage. This is also the theoretical basis that SISO ANN can perform nonlinear post-equalization in MIMO VLC systems. As $\alpha $ is hidden in $I(t)$, SISO ANN cannot estimate $\alpha $ at this stage. When the power of the two signals matches, $\alpha \approx 1$, and we no longer need to estimate $\alpha $. Consequently, SISO ANN could effectively equalize the linear and nonlinear distortion in the MIMO VLC system. This inference will be verified in subsequent experiments.

When the power of the two received signals differs greatly, the influence of $\alpha $ on the received signal can no longer be ignored. Therefore, we need to separate two signal streams (${Y_1}(t)$ and ${Y_2}(t)$) from $R(t)$ form the received signal by multiply by $\cos (2\pi {f_0}t)$ and $\sin (2\pi {f_0}t)$. To equalize the influence caused by $\alpha $, both ${Y_1}(t)$ and ${Y_2}(t)$ need to be input to the post-equalizer as features. ${Y_1}(t)$ and ${Y_2}(t)$ could be expressed as follows,

Since the form ${Y_2}(t)$ is similar to that of ${Y_1}(t)$, it will not be repeated here. Since the design of the two transmitters is the same (this is also the lowest cost solution), When the two channels are at the maximum power output, the total power of the receiving end is the largest, and the nonlinear distortion in the VLC system is also highest. Then, $\alpha $ of the received signal is close to 1, and the main factor causing bit errors is the nonlinear distortion caused by the nonlinear response in the VLC system. When the power difference between the two channels is large, the total power of the system is reduced, and the nonlinear distortion is reduced accordingly. The main factor causing the bit error is the constellation point offset caused by the power mismatch between ${X_1}(t)$ and ${X_2}(t)$ . Therefore, we only need an on hidden layer MBNN (OHL MBNN) or a MIMO-LMS to effectively equalize the signal distortion caused by power mismatch. In this state, the use of MBNN will cause a lot of waste of computing resources.

The received signal is equalized by a post-equalizer to compensate for the distortion in the received signal. Then, the equalized signal is filtered by the root cosine filter to filter out-of-band noise and decrease the inter-symbol interference. Then the signal is restored to the state before the up-sampling by down-sampling. After that, the down-sampled signal is linearly equalized by the single-in-single-out LMS. Then, the signal is classified into different categories in the PAM demodulation stage. Finally, the two outputs signal are combined and mapped to binary signal. The BER is obtained by comparing the received binary signal with the transmitted binary signal.

2.2 Strategy and structure of the AANN

In the process of explaining the principle of the SR-MIMO VLC system in the previous section, Function 3 shows that the SISO ANN equalizer can effectively compensate for the linear and nonlinear distortions in the system when $\alpha \approx 1$ . However, SISO ANN cannot compensate for the distortion caused by the power mismatch between the two transmitters. Therefore, when the powers of the two transmitting ends are close, the SISO ANN equalizer can effectively equalize the distortion existing in the SR-MIMO VLC system. When the power difference between the two channels is large, the total power of the system will decrease. At the same time, the nonlinear distortion of the system will be reduced. Instead, the constellation diagram caused by the power mismatch will become the main factor affecting the BER of the SR-MIMO VLC system. In this state, a linear MIMO-LMS equalizer or OHL MBNN equalizer can effectively correct the distorted constellation diagram.

Therefore, we propose three improved ANN post-equalization algorithms. DBMLP-LMS (the combination of DBMLP and MIMO-LMS), OHL MBNN, and L-DBMLP-L (the combination of MIMO-LMS, DBMLP, and MIMO-LMS) are provided in Fig. 2(a), (b) and (c), respectively. Table 1 provides the structure of Fig. 2(a), (b), (c) and (d) in detail.

 

Fig. 2. Diagram of proposed (a) DBMLP-LMS, (b) OHL MBNN, (c) L-DBMLP-L and (d) MIMO-LMS. (e) is the flow chart of the proposed adaptive algorithm.

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Table 1. The structure of MIMO-LMS, DBMLP-LMS, LMS-DBMLP-LMS and OHL MBNN.

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Then, according to Fig. 2(e), an adaptive algorithm is designed and utilized to combine DBMLP and OHL MBNN or L-DBMLP-L, which creates two AANN algorithms (ADP L/DBMLP-L and ADP MIMO ANN). ADP L/DBMLP-L and ADP OHL-MBNN adopt the DBMLP in our previous work and use tanh as the activation functions [9]. AANNs calculate the average power of the synchronized (SYN) data (${S_1}(t)$ and ${S_2}(t)$) of the two signals to obtain $\alpha $ according to the following function,

After that, AANNs will determine whether the value of $\alpha $ is between $1\textrm{ - }{\sigma _1}$ and $1\textrm{ + }{\sigma _2}$ (We learned through follow-up experiments that ${\sigma _1}\textrm{ = }0.35$ and ${\sigma _1}\textrm{ = }0.43$). If yes, input the data to the upper branch of Fig. 2(a) and (b). If not, input the transmission data to the lower branch. The upper branch of the two algorithms is a combination of SISO DBMLP and MIMO LMS. When, the value of $\alpha $ is not located in the range between $1\textrm{ - }{\sigma _1}$ and $1\textrm{ + }{\sigma _2}$, ADP L/DBMLP-L performs MIMO-LMS post-equalization on the signal, and equalizes the linear distortion caused by power mismatch. Then the two equalized signal streams are combined into one signal stream and input into the DBMLP to equalize the non-linear distortion existing in the SR-MIMO-VLC system. Finally, perform MIMO-LMS post-equalization on the signal stream output by DBMLP to compensate for the linear distortion caused by the power mismatch that may exist in the SR-MIMO VLC system. The algorithm of the lower branch can be abbreviated as L-DBMLP-L. The algorithm and structure of MIMO-LMS have been provided in Fig. 3(a). On the counterpart, the lower branch of ADP MIMO ANN is been replaced by a one hidden layer MBNN (OHL MBNN), which could effectively equalize the power mismatch distortion and weak nonlinear distortion (Because the structure of OHL MBNN is simple, there is only one hidden layer, and a small number of nodes, the nonlinear equalization ability is not as good as the complexity MBNN.). The performance of ADP MIMO ANN is better than ADP L/DBMLP-L. However, ADP L/DBMLP-L has lower spatial complexity.

 

Fig. 3. The experimental setup

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In the training process, the weights optimization process is the same as the traditional ANNs. AANNs calculate the MSE after completing the forward propagation process. Then, the gradient of the trainable weights is obtained based on the chain rule. Finally, the trainable weights of AANNs are optimized based on the Adam optimizer [16]. The total size of the data set is 70,000, of which the size of the training set and the test set are both 35,000. In order to make our experimental results comparable with existing research results, we set epoch and batch to 20 and 512, respectively [13].

3. Experimental setup

Figure 3 shows the SR-MIMO VLC experimental platform used in our experiment. The electric signal emitted by the AWG is amplified by the electric amplifier (EA). Then, the signal is coupled with the bias current to drive the LED through the Bias-Tee. Finally, the signal is emitted in the form of a light signal from the LED. The distance between the LED and PIN at receiving end is 1.2 m. The two optical signals are irradiated on a PIN together. A PIN converts the optical signal into an electrical signal. The electrical signal is transmitted to the oscilloscope after being amplified by EA. The detail of devices and work condition has been provided in Table 2.

Table 2. The detail of devices and work conditions in the experimental setup

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In the experiment, we compared the performance of various post-equalization algorithms including, without post-equalization, MIMO-LMS, LMS-DBMLP-LMS, DBMLP, DBMLP-LMS, ADP LMS/DBMLP-LMS, MBNN, OHL MBNN, and ADP MIMO ANN.

4. Experimental results

To visually show the influence of $\alpha $ on the received signal, we measured the receiving constellation diagram under the two values of $\alpha $ under four equalization algorithms. When $\alpha \textrm{ = }0.8$, From the constellation diagram of W/o MIMO LMS, we can see that the receiving constellation point is distorted on the x-axis. If a hard decision algorithm is made based on the standard constellation point, a large number of constellation points will be misjudged. Figure 4((a)(ii)) proves that the MIMO-LMS algorithm can correct this distortion. According to Fig. 4((a)(iii)), DBMLP as a kind of SISO ANN could effectively equalizer the linear and nonlinear distortion exited in the SR-MIMO VLC system to get a more convergent constellation diagram. However, DBMLP could not correct the distortion caused by power mismatch. Therefore, if we combine DBMLP and MIMO LMS we get Fig. 4((a)(iv)). The nonlinear distortion and power mismatch distortion of the received signal are both equalized by DBMLP-LMS, which also shows that the nonlinear and power-mismatch distortion in the SR-MIMO VLC system is relatively independent and can be equalized by DBMLP and MIMO LMS respectively. Additionally, we measure the constellation of received signals at $\alpha \textrm{ = }1.25$ in Fig. 4(b). The conclusions that can be drawn from Fig. 4(b) are the same as those in Fig. 4(a).

 

Fig. 4. Constellation distortion when the power of two transmitters is mismatched. (a) $\alpha \textrm{ = }0.8$; (b) $\alpha \textrm{ = 1}\textrm{.25}$

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Then the contour of system performance is based on different post equalizer algorithms provided in Fig. 5. If we observe Fig. 5 along the n-axis, what we get is the ability of various equalizers to compensate for the nonlinear distortion. If $\alpha $ is close to 1, the performance of DBMLP-LMS is almost the same as that of MBNN, which shows that DBMLP-LMS is close to MBNN in its ability to deal with the nonlinear distortion in the SR-MIMO VLC system. If we observe Fig. 5 along the m-axis, what we get is the ability of various equalizers to compensate for power mismatch distortion. By combining MIMO-LMS and DBMLP, DBMLP gets the ability to equalize the signal distortion caused by the mismatch of the transmission power of the two transmitters. Due to the low complexity of MIMO-LMS and DBMLP, the complexity of the AANN formed by them is much lower than MBNN. From Fig. 5(d), (e) and (c), it can be noticed that with the utilization of MIMO-LMS, the BER performance of the DBMLP equalized signal on the m-axis gradually becomes better. The performance of LMS-DBMLP-LMS and OHL MBNN are similar to MBNN, which shows that LMS-DBMLP-LMS and OHL MBNN have similar performance with MBNN in their ability to deal with the power mismatch distortion in the SR-MIMO VLC system. Therefore, we combine DBMLP-LMS with L-DBMLP-L and OHL MBNN to form ADP L/DBMLP-L and ADP MIMO ANN. By comparing Fig. 5(g), Fig. 5(f), and Fig. 5(i), We can qualitatively conclude that the two AANN algorithms and the MBNN algorithm are the same in terms of equalizing nonlinear distortion and power mismatch distortion.

 

Fig. 5. The contour of system performance based on different post-equalizers.

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Figure 5 qualitatively and intuitively shows the BER performance of two AANNs. And got the conclusion that the performance of AANNs is the same as that of MBNN. Figure 6 shows the relationship between the BER performance of SR-MIMO systems based on different equalizers and $\alpha $ when the Vpp of Tx1 is a certain value. Since the adjustment range of the Vpp of the two transmitters in our experiment is from 0.2 V to 1.2 V, when the Vpp of Tx1 is 0.2 V, the value of $\alpha $ will be always greater than or equal to 1. This makes the BER caused by the power mismatch dominate the BER of the system in Fig. 6(a). Due to the low Vpp of Tx1, the signal-to-noise ratio (SNR) of the received signal is too low, the overall BER of the SR-MIMO VLC system is relatively high, which could hardly be modified by post-equalizations. On this condition, the BER performances of OHL MBNN and ADP MIMO ANN are consistent with that of MBNN. As the power mismatch dominate the BER of the system, The MIMO LMS in ADP L/DBMLP-L could not effectively equalize the power-mismatching distortion. Therefore, the BER performance of ADP L/DBMLP-L is worse than MBNN. According to Fig. 6(b) and Fig. 6(c), as Tx1 Vpp increases, the range of $\alpha $ approaches 1. At the same time, the distortion caused by power mismatch will also be reduced. Therefore, the performance of ADP L/DBMLP-L increases. When $\alpha $ is close to 1, the performance of ADP L/DBMLP-L is similar with ADP MIMO ANN, OHL MBNN, and MBNN. As the power of Tx1 is further improved, the maximum power that the system can reach continues to increase. The nonlinear distortion in the system is constantly getting stronger and dominates the distortion in the system. As the power of Tx1 is further improved, the maximum power that the system can reach continues to increase. The nonlinear distortion in the system is constantly getting stronger and dominates the distortion in the system. Under this condition, the performance of OHL MBNN is not as good as MBNN. Because the insufficient complexity of OHL MBNN is not large enough to equalize the nonlinear distortion in the system.

 

Fig. 6. the BER vs. power ratio $\alpha $ when the Vpp of Tx1 equal to (a) 0.2 V, (b) 0.4 V, (c) 0.6 V, (d) 0.8 V, (e)1 V and (f)1.2 V.

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According to Fig. 6(d), (e), and (f), when $0.65 \le \alpha \le 1.43$, the performance of the DBMLP-LMS is similar to the MBNN. In this state, the algorithm of the ADP MIMO ANN and the ADP L/DBMLP-L is the same as the DBMLP-LMS. Therefore, the performance of AANNs is similar to the MBNN. when $\alpha \textrm{ < }0.65$ or $\alpha \textrm{ > }1.43$, the performance of the OHL MBNN and the L-DBMLP-L are similar to the MBNN. In this state, the algorithms of the ADP MIMO ANN and the ADP L/DBMLP-L are the same as the OHL MBNN and the L-DBMLP-L, respectively. Therefore, the performance of AANNs are similar to the MBNN as well.

In general, to reduce production costs, the design parameters of the two transmitters are completely the same. Therefore, we tested the impact of Vpp and bit rate changes on the system BER under the condition of two channels with the same Vpp ($\alpha \textrm{ = }1$). In this state, both AANNs will choose DBMLP-LMS as the post-equalization algorithm. Therefore, in Fig. 7, the BER performance of ADP L/DBMLP-L and ADP MIMO ANN is consistent with DBMLP-LMS. According to Fig. 7(a), when the Vpp is lower than 0.6 V the performance of the OHL MBNN, MBNN and AANNs are similar. However, as the Vpp increases, the nonlinear distortion in the SR-VLC system increases, and the BER performance of the OHL MBNN declines due to insufficient complexity. Figure 7(b) shows that changes in the bandwidth of the transmitted signal will not cause any difference in the BER performance of OHL MBNN, MBNN, and AANNs. Therefore, we can conclude that, AANNs with lower space complexity has more application potential and research value than MBNN in SR-MIMO systems.

 

Fig. 7. BER performance of different equalizer-based SR-MIMO VLC systems at different (a) Vpp and (b) Bitrate.

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Table 3 shows the detailed hyperparameter settings of equalizers in Fig. 7. We can notice that in a relatively complex state, the spatial complexity of ADP L/DBMLP-L is 6.4% MBNN. The spatial complexity of ADP MIMO ANN is 10% MBNN. In the best condition, the spatial complexity of two AANNs is only 5.7% of MBNN. It can be seen that AANNs have obvious advantages over MBNN in terms of spatial complexity. The following experiment only needs to verify that the BER performance of ANNs is not weaker than that of MBNN, which can prove that the adaptive design strategy can effectively reduce the spatial complexity of the ANN equalizer without reducing the BER performance of the ANN equalizer. Furthermore, the theory proposed in section 2.1 could be verified.

Table 3. Comparison of spatial complexity.

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5. Conclusion

Through the above experiments, we have proved that it is feasible to deal with the distortion caused by nonlinearity and power imbalance existing in the SR-VLC system separately. We verify that the proposed AANNs have lower spatial complexity and at the same time have a BER performance that is not inferior to that of MBNN. This low-complexity AANNs algorithm that can be used in SR-MIMO VLC systems has the characteristics of low spatial computational complexity, low energy consumption, and excellent BER performance, which makes it have greater application potential.