1. Introduction

With the advent of the 5G era, indoor mobile traffic will show explosive growth. VLC is a promising wireless communication technology to solve this problem [1]. VLC uses a light-emitting diode (LED) as the transmitter and PIN or avalanche photon diode (APD) as the receiver to implement data transmission [2,3]. Its many advantages, such as no spectrum limitation, energy and cost-efficient, non-line of sight (NLOS) encryption invalid, and the combination with lighting make it a hotspot in the field of optical wireless research [4,5].

However, the 3dB bandwidth of LED is just 10∼20 MHz, which is a big obstacle to realize a high-speed VLC system [6,7]. In recent years, MIMO VLC has become a hotspot because it can effectively increase the capacity of the system at a low cost. Furthermore, the concept of MIMO accommodates well with the concept of massive access, uplinks, and downlinks in 5G. [8,9]. MIMO-VLC can be divided into two fields: imaging and non-imaging [10]. The imaging MIMO can provide a full-rank channel matrix. However, the requirement for the alignment between LED and detector is strict, which restricts the practical utilization of imaging MIMO. Therefore, most of the current MIMO-VLC research focuses on non-imaging MIMO. However, the correlation of the VLC signal is strong, and the decorrelation may produce an ill-conditioned channel matrix [11]. Traditional MIMO algorithms, such as singular value decomposition (SVD), orthogonal circulant matrix transform (OCT) require a perfect channel matrix [12,13]. Therefore, not all data streams can be decoded in the MIMO-VLC system. Interestingly, the superposition coding modulation (SCM) strategy provides us with another possibility, and it has been proved to be feasible in the MIMO-VLC system. Furthermore, in some scenarios such as the internet of vehicles, the car lights operate at high power. The nonlinear effect of the LED makes the transmitter unable to support high order modulation format. By utilizing two LEDs and SCM, we can reduce the order of modulation format on each transmitter without sacrificing the overall spectrum efficiency. In [14], the author superposes two non-equal probability PAM4 signal to generate an equiprobable PAM7 signal, whose data rate is much higher than the space-time block coding (STBC) PAM4 signal. However, the PAM4 signal is superposed in one dimension, and if the source entropy continues to increase, the system performance will deteriorate dramatically. In [15], the author used QPSK and 16QAM to realize a 64QAM SCM constellation and achieved 1.5 Gbps data transmission with the aid of the look-up table and successive interference cancellation. In [16], the author used a geometrical shaping 8QAM square constellation point combined with 4QAM to implement 32QAM superposition coded modulation, which showed its superiority to the other 32QAM SCM constellation points. However, QAM-based superposition modulation requires both receivers to adopt a differentiated design. For high-order 64QAM, it can only work at a power difference of more than four times. At the same time, big power diversity will introduce power competition between two signals, and the signal-to-noise ratio (SNR) of the signal with lower power will further deteriorate. If the SCM is conducted by two PAM8 streams, we can get a 64QAM constellation at the receiver side as well, and the optimal power ratio is 1 to 1, which will not introduce additional power competition. However, the research of PAM-based SCM signals in MIMO-VLC systems is still lacking.

On the other hand, the nonlinear effect in the VLC system will distort the signal severely [17]. For the SCM signal, it is more susceptible to the nonlinear distortion of the VLC channel. However, the transfer function of the VLC channel is complicated, and it is difficult to be expressed by analytical equations [18]. At this time, with the help of the powerful nonlinear mapping ability of the neural network, the performance of the received signal will be improved to a big degree. In [19], 1.5 Gbps PAM8 transmission over 1.2 m underwater VLC channel is implemented by utilizing a SISO deep neural network (SISO-DNN) as a post equalizer. In [20], the author designed a two tributary heterogeneous neural network (TTHNET) to emulate the VLC channel. The parameters of TTHNET are only 0.8% of the regular multilayer perceptron (MLP). Therefore, it is foreseeable that after the distortion of the MIMO VLC channel, the received SCM signal could be compensated by the neural network. Unfortunately, there is a lack of research on the neural network as a post-equalizer in the MIMO VLC system.

In this paper, we proposed a novel MIMO multi-branch hybrid neural network (MIMO-MBNN) post equalizer and implemented it in a 64QAM SCM and carrier-less amplitude-phase (CAP) modulation single-receiver MIMO (SR-MIMO) VLC system. The superposed 64QAM constellation at the receiver consists of two PAM8 signals from two transmitters. The BER performance is continuous in all the operation range, which is different from that in [15]. Furthermore, the proposed MIMO-MBNN post equalizer shows great superiority to traditional SISO least mean square with Volterra series (SISO-LMS) and SISO-DNN post equalizers when the power of two transmitters are unbalanced. The innovation points are listed as follow:

2. Principles and experimental setup

2.1 Principles of superposed 64QAM constellation

The principles of SCM 64QAM constellations are shown in Fig. 1. In this paper, we define the signal peak to peak voltage of LED1 and LED2 as V_pp1 and V_pp2, respectively. The power ratio ${\mathrm{\beta}} \textrm{ = }{V_{pp2}}/{V_{pp1}}$. The PAM8 SCM scheme is shown in Fig. 1(a). The in-phase part and quadrature part of SCM 64QAM constellation at the receiver side is transmitted by LED1 and LED2, respectively. The original bitstreams of LED1 and LED2 can be modulated into PAM8 symbols, which can be expressed as:

Here ${T_1}(t)$ and ${T_2}(t)$ are the PAM8 symbol column vectors of LED1, and LED2 respectively. $t = 1,2,\ldots ,N$ and N is the total number of transmitted symbols. As the VLC system is intensity modulation and direct detection (IM/DD) system and the signal are modulated by carrier-less amplitude magnitude (CAP) modulation format, the PAM8 signal in LED1 and LED2 are multiplied by cosine subcarrier and sine subcarrier respectively to make them orthogonal. Therefore, the superposed signal at the receiver is a QAM constellation. If we consider no distortion and just additive white gaussian noise in the channel, the received constellation for each transmitted symbol after post equalization in Fig. 1(a) can be expressed as:

 

Fig. 1. Principles of SCM 64QAM schemes. (a) PAM8. (b) QPSK and 16QAM in [15].

Download Full Size | PDF

As a comparison, the QPSK and 16QAM based 64QAM SCM scheme in [15] is shown in Fig. 1(b). From Fig. 1, we can conclude the differences between these two schemes as follow: (i) The optimal power ratio between LED1 and LED2 of the PAM8 SCM 64QAM scheme in Fig. 1(a) is 1, while that in Fig. 1(b) is 4. This indicates that the system in Fig. 1(a) is more conducive to practical application, as the two transmitters do not require a differentiated design. (ii) The valid theoretical value of ${\mathrm{\beta}} $ in Fig. 1(b) is below 0.5 or above 3. When the power ratio is not in the valid operation range, the signal cannot be demodulated. While that in Fig. 1(a) is not limited. Furthermore, the power competition between QPSK and 16QAM will further deteriorate the overall system performance. Therefore, we believe that the PAM8 SCM 64QAM scheme is a better choice for the SR-MIMO VLC system.

Furthermore, we choose the CAP instead of the OFDM as the modulation format because the CAP has a relatively lower peak to average power ratio (PAPR) than the OFDM. As two signals will superpose at the receiver, higher PAPR will clip the signal dramatically and deteriorate the system performance. Therefore, the CAP is superior to the OFDM in the SCM VLC system. Additionally, by utilizing a square-root raised cosine low-pass filter, the CAP signal is more adapted to the high-frequency fading VLC channel than the conventional OFDM signal. Therefore, we choose the CAP as the modulation format in this paper.

2.2 Data flow and experimental setup of SR-MIMO VLC system

The data flow and experimental setup of the SR-MIMO VLC system are shown in Fig. 2. To simplify the expression, we abbreviate Transmitter 1 as Tx1, Transmitter 2 as Tx2, and receiver as Rx. The transmitted signals before up-conversion can be expressed as:

 

Fig. 2. Data flow and experimental setup of the SR-MIMO CAP VLC system. AWG: arbitrary waveform generator; Eq: hardware equalizer; EA: electrical amplifier; OSC: oscilloscope.

Download Full Size | PDF

The received signal after the VLC channel can be expressed as:

The typical value for ${a_1}$, ${a_2}$, ${a_3}$, and b are 1.259, 0.01373,-0.4576 and 0.8165 respectively, which are measured at the 1.4V signal peak to peak voltage (V_pp). After power normalization, ${H_{11}} = 1$ and ${H_{22}}\textrm{ = }{\mathrm{\beta}} $. ${N_1}(t)$ and ${N_2}(t)$ denote the system noise. To reduce the complexity of the analysis, we think that the mutual influence of LED1 and LED2 at the receiver is 0. Therefore, Eq. (5) can be rewritten as:

Here $\tan {{\mathrm{\alpha}} _1} ={-} \frac{{{S_1}(t)}}{{{\mathrm{\beta}} {S_2}(t)}},\tan {{\mathrm{\alpha}} _2} = \frac{{{\mathrm{\beta}} {S_2}(t)}}{{{S_1}(t)}}$. The first two terms of Eq. (10) and Eq. (11) are linear in-band frequency response, which can be treated as the convolution of origin signal ${S_1}(t),{S_2}(t)$ and channel response. As we usually consider the VLC system as a non-time-varying system, the channel response can be treated as a finite impulse response (FIR) filter. Therefore, the first term can be equalized by a linear equalizer. Additionally, for ${Y_1}(t)$, the second term contains ${S_2}(t)$ can be considered as noise and vice versa. Therefore, the value of ${\mathrm{\beta}} $ will significantly affect the SNR of ${S_1}(t)$ and ${S_2}(t)$. The bigger the ${\mathrm{\beta}} $ is, the higher the SNR of ${S_2}(t)$ is. At the same time, the SNR of ${S_1}(t)$ deteriorates. For ${S_1}(t)$, the condition is the contrary. The last two terms are high-frequency components, which can be removed by the square-root raised cosine low-pass filter. According to Eqs. (6), (10) and (11), the expression of ${Y_1}(t)$ and ${Y_2}(t)$ can be easily derived by replacing ${S_1}(t)$ with ${a_2}{S_1}^2(t)/b$ or ${a_3}{S_1}^3(t)/b$ when the system operates at high V_pp. Here we define Eq. (10) and Eq. (11) as:

The expression of ${Y_2}(t)$ is similar to that of Eq. (14). From the above equation, we can see that the received signal after multiplying by $\cos ({{\mathrm{\omega}} _0}t)$ and $\textrm{ - }\sin ({{\mathrm{\omega}} _0}t)$ can be divided into three parts: linear distortion, linear crosstalk, nonlinear distortion and crosstalk. The linear distortion can be successfully compensated by a linear equalizer. For the mitigating of the nonlinear distortion and crosstalk, an additional effective nonlinear equalizer must be considered. Our experimental setup consists of two independent input signals. Each signal should be compensated by two neural networks, one is applied for the linear distortion, the other is for nonlinear distortion. In this way, a total 4 tributaries NN should be used for the signal compensation. Since two input signals are combined together at the single receiver after 1.2m free-space transmission, we assume using only one neural network to compensate the nonlinear distortion for the two signals simultaneously, thus a total triple branches NN can be implemented for post-equalization. Our following experiment results evidently prove the viability of our assumption.

After the processing of three equalizers, the two data streams are loaded by the match filter pairs ${m_1}(t)$ and ${m_2}(t)$ in the CAP modulation. Then the two signals are equalized by two LMS equalizers. Different from the SISO-LMS in case 1, these additional equalizers don’t have the Volterra series, which cannot compensate for the nonlinear distortion. They are utilized to mitigate residual noise. The input taps of the additional two LMS equalizers are 33, and the step is set to 0.007. This way of compensating the received signal with two-stage equalizers has been proven effective in the VLC system [22,23]. Finally, the signals are demodulated and recovered to bitstreams again. The details about the components in Fig. 2 are listed in Table 1.

Table 1. Details of components in Fig. 2.

View Table | View all tables in this article

2.3. Structure of MIMO-MBNN

Figure 3 is the forward propagation of MIMO-MBNN in the SR-MIMO VLC system. The symbols ${Y_i}(t)$, ${S_i}(t)$ and ${P_i}(t)$ are consistent with that in Fig. 2 (i=1,2). For the robustness of the neural network, we should normalize the signals at first. In this paper, we utilize the ‘max absolute value’ normalization function, which can be written as:

 

Fig. 3. Structure of MIMO-MBNN

Download Full Size | PDF

To simplify the principle chart of MIMO-MBNN in Fig. 3, we set the input length of each data stream to 3. In the experiment, the optimal input data length is 53. Compared to the input length of 191 in [20], the input length in this paper is acceptable. Here we define the input vector as $\overrightarrow {Y(t)} = {[{Y_1}(t - 1),Y{(t)_1},{Y_1}(t + 1),{Y_2}(t - 1),{Y_2}(t),{Y_2}(t + 1)]^T}$ and $t = 2,3,\ldots N - 1$, where N is the length of the training data set. Afterward, the data streams are concatenated and loaded by the 1^st, 2^nd linear branch, and the 3^rd nonlinear branch. Then the linear and nonlinear output of 1^st, 2^nd, and 3^rd branches can be expressed as:

3. Results and discussion

Firstly, to show the nonlinear response of the system and the interaction between the two signals, we measured the AM response between the transmitted signal and the received signal. For each point in Fig. 4, the X-axis value is the amplitude of normalized ${X_1}(t)$ in Fig. 2, while the Y-axis value is that of normalized $Y(t)$ in Fig. 2. The normalization way is shown in Eq. (15). Figure 4(a) is the AM response without the interference from Tx2. From Fig. 4(a), we can see that at the lower V_pp1 of 300 mV, the AM response is almost linear. However, with the increase of V_pp1, the AM response at high signal voltage shows apparent nonlinear response between ${X_1}(t)$ and $Y(t)$. Figure 4(b) shows the AM response between Tx1 and Rx at different values of Tx1: Tx2 when V_pp1 is 900 mV. We can easily recognize that with the increase of power ratio of Tx1 to Tx2, the noise power brought by Tx2 is continuously decreasing. This indicates that the signal from Tx1 will inevitably be distorted by that from Tx2 and vice versa.

 

Fig. 4. amplitude magnitude response. (a) Tx1 and Rx without Tx2 at different Vpp. (b) Tx1 and Rx with varying values of Tx1: Tx2 (V_pp1=900 mV)

Download Full Size | PDF

Afterward, the bit error rate (BER) performance versus hidden node numbers and layers in MIMO-MBNN are shown in Fig. 5. We measured the BER performance of MIMO-MBNN with different activation functions and different hidden layers. The MIMO-MBNN model at each point is not overfitting, and ‘Checkpoint’ is utilized to ensure optimal performance of MIMO-MBNN. Additionally, the loss function we utilized is the mean square error (MSE). Figure 5(a) is the BER performance with just 1 hidden layer. The best BER performance is achieved by ‘relu’, and the 1^st node number is 136. Therefore, we set the node number and activation function of 1^st layer to 136 and ‘relu’, respectively. Figure 5(b) shows the BER performance with two hidden layers under different activation function combinations. The 1^st layer node number is fixed to 136. We can conclude from Fig. 5(b) that the optimal choice of the activation function is ‘relu’ + ‘relu’, and the optimal node number of the 2^nd hidden layer is 104. We have verified that further increasing the number of nodes and layers will not significantly improve the system performance. Therefore, the layer number, 1^st hidden layer node number, 2^nd hidden layer node number, and activation function are fixed to 2, 136, 104, and ‘relu’. The parameter details are shown in Table 1. The epoch, batch size, training set ratio, and total sample number are 40, 512, 50%, and 73728, respectively. Therefore, the training set length and testing set length for SISO-LMS, SISO-DNN, and MIMO-MBNN are all 36864. After training, the parameters of MIMO-MBNN, SISO-LMS, and SISO-DNN are fixed. Then we utilize a new set of data to measure the BER or Q performance of the above equalizers and the length is 73728. The computation complexity in terms of the total parameters can be found in Table 2, where N is the input taps of SISO-LMS and SISO-DNN, ${N_v}$ is the input taps of the Volterra series, ${H_1}$, ${H_2}$, and ${H_3}$ are the node number of hidden layers. Clearly, the value of the total parameters depends on the input taps and the node number of hidden layers. In the SISO-LMS and SISO-DNN cases, the optimum input taps is 53, while the optimum input taps for the Volterra series is obtained at 9. Hence the total parameters of SISO-LMS, SISO-DNN, and MIMO-MBNN are 689, 46257, and 29224 respectively. It should be noted that although LMS boasted the lower complexity, the achieved BER performance is above the FEC threshold, which will result in failure detection. The complexity of MBNN is merely 63% of SISO-DNN, clearly validating the feasibility of the proposed scheme.

 

Fig. 5. BER performance versus different node numbers and activation function of MIMO-MBNN. (a) BER performance of 1 hidden layer. (b) BER performance of 2 hidden layers

Download Full Size | PDF

Table 2. Detailed parameters of SISO-LMS, SISO-DNN, and MIMO-MBNN.

View Table | View all tables in this article

Then we measured the frequency response and spectrum mismatch of $Y(t)$, SISO-LMS, SISO-DNN, and MIMO-MBNN, which are shown in Figs. 6(a) and 6(b) respectively. The spectrum mismatch $M({\mathrm{\omega}} )$ is depicted as:

 

Fig. 6. Frequency and spectrum mismatch of the received signal, SISO-LMS, SISO-DNN, and MIMO-MBNN. (a) Frequency response. (b) spectrum mismatch

Download Full Size | PDF

The Q performance versus data rate is shown in Fig. 7. The Q in dB can be expressed as:

 

Fig. 7. Q factor versus data rate

Download Full Size | PDF

Afterward, we measured the system $- {\log _{10}}(BER)$ performance at different V_pp1 and V_pp2. The results are shown in the contours in Fig. 8. The data rate is 2.1 Gbps. We can easily recognize that the system performance after equalization of MIMO-MBNN is better than that of SISO-DNN and SISO-LMS. When V_pp1 is between 250 mV and 350 mV and V_pp2 is between 350 mV and 450 mV, the system performance equalized by MIMO-MBNN owns an operation range above the 3.8×10⁻³ HD-FEC, while SISO-LMS and SISO-DNN own no operation range. Besides, the operation range above the 2×10⁻² HD-FEC threshold of MIMO-MBNN is much bigger than that of SISO-LMS and SISO-DNN as well.

 

Fig. 8. Contour of system performance SISO-LMS, SISO-DNN, and MIMO-MBNN (from 1 to 3 in each row). (a1)-(a3). Average BER performance. (b1)-(b3). Tx1 BER performance. (c1)-(c3). Tx2 BER performance.

Download Full Size | PDF

To further investigate the system performance at different operation points, we measured the system Q performance of LED1, LED2, and the average system Q performance of LED1 and LED2 versus ${\mathrm{\beta}} $ when V_pp1=250 mV, 350 mV, and 600 mV. The results are shown in Fig. 9. When V_pp1=250 mV as Fig. 9(i)–9(iii) show, the operation range above the 2×10⁻² HD-FEC threshold of MIMO-MBNN is from 0.88 to 1.68, which is 8 times and 2.76 times of SISO-LMS and SISO-DNN respectively. In Figs. 9(iv) to 9(vi), when V_pp1 raises to 350 mV, the operation range of MIMO-MBNN is from 0.70 to 1.63, which is 2.74 times and 2.33 times of SISO-LMS and SISO-DNN respectively. Besides, MIMO-MBNN owns the operation range from 0.98 to 1.08 above the 3.8×10⁻³ HD-FEC threshold, while SISO-LMS and SISO-DNN don’t have. When V_pp1 raises to 600 mV as Figs. 9(vii)–9(ix) depict, the system is distorted by the nonlinear AM response, and the operation range of MIMO-MBNN decreases from 0.78 to 0.92, while SISO-LMS and SISO-DNN own no operation range. Additionally, the average Q performance of SISO-LMS, SISO-DNN, and MIMO-MBNN at (c1), (c2), and (c3) in Figs. 9(iv)–9(vi) are 0.94, 2.77, and 6.12 dB respectively. Therefore, MIMO-MBNN shows a Q improvement of 3.35 dB compared with SISO-LMS and SISO-DNN.

 

Fig. 9. Q performance of SISO-LMS (left column), SISO-DNN (middle column), and MIMO-MBNN (right column) at different ${\mathrm{\beta}} $. (i)-(iii) V_pp1=250 mV. (iv)-(vi) V_pp1=350 mV. (vi)-(ix) V_pp1=600 mV.

Download Full Size | PDF

At last, we demonstrate the SCM 64QAM constellations and eye-diagrams of Tx1 and Tx2 at the receiver after post equalization of SISO-LMS, SISO-DNN, and MIMO-MBNN. The marks in Fig. 10 correspond to that in Fig. 9. In Figs. 10(a1)–10(a3), the ${\mathrm{\beta}} = 0.39$, which means the power of LED2 at Rx is about 0.4 times of LED1. Therefore, the SNR of Tx1 is bigger than that of Tx2, which could be easily recognized by the eye diagrams and constellations. When the power of Tx1 and Tx2 are almost equal from Figs. 10(b1) to 10(b3), the SCM 64QAM constellations at the receiver shows the shape of the standard 64QAM, and the eye diagrams of LED1 and LED2 behave similarly to each other. As Figs. 10(c1) to 10(c3) shows, increasing ${\mathrm{\beta}} $ to 1.76 will make Tx2 outperform Tx1, which is contrary to Figs. 10(a1) to 10(a3) and accommodates the analysis in section 2. Besides, the eye diagrams and constellations demonstrate that MIMO-MBNN is superior to SISO-LMS and SISO-DNN.

 

Fig. 10. Constellation and eye diagrams of (a1)-(c3) in Fig. 9.

Download Full Size | PDF

4. Conclusion

In this paper, we proposed for the first time a new structure of neural network post equalizer—MIMO-MBNN in the SR-MIMO VLC system, which modulated by SCM 64QAM constellation and CAP. Experimental results show that the operation range of MIMO-MBNN is much larger than that of SISO-DNN and SISO-LMS above the 2×10⁻² HD-FEC threshold. Additionally, MIMO-MBNN can improve the Q performance by at most 3.35 dB compared with SISO-DNN and SISO-LMS. The data rate of 2.1 Gbps above the 3.8×10⁻³ HD-FEC threshold is achieved by MIMO-MBNN. To the best of our knowledge, this is the highest data rate in the SR-MIMO VLC system, and it is 40% higher than the original record. Further research will focus on the simplification of MIMO-MBNN.