The dramatic growth in the number of communication devices and users in recent years has posed a significant challenge to traditional radio-frequency (RF) communication systems, where limited spectrum resources have become one of the most critical issues^[1]. Visible light communication (VLC) employs light-emitting diodes (LEDs) as transmitters and has evolved into an alternative to congested RF communication systems. Compared to RF communication, VLC has the advantages of rich and unregulated spectrum resources, low cost, immunity to electromagnetic interference, and high physical layer security. Consequently, VLC has emerged as a potential communication technology for sixth-generation mobile networks and Internet of Things systems^[2,3].

The limited modulation bandwidth of commercial LEDs is one of the main bottlenecks restricting the data rate of VLC systems. Multiple-input multiple-output (MIMO) systems, which can multiply the data rate by simultaneously transmitting independent data streams, are considered a promising solution to compensate for bandwidth limitations^[4,5]. However, high channel correlation often occurs in VLC systems, which hinders the implementation of MIMO^[6]. Once a channel is highly correlated, the receivers cannot separate the multiple data streams. Consequently, superposed constellations have recently been proposed to solve this problem^[7–12]. MIMO based on a superposed constellation can be implemented regardless of the extent of channel correlation because MIMO detection is realized by simple constellation demapping rather than data-stream separation.

The principle of the superposed constellation was first proposed in Ref. [7], in which two signals modulated with 4-quadrature amplitude modulation (4QAM) and 16QAM were sent from two LEDs to superpose a 64QAM signal at the receiver. However, this scheme suffered from performance loss caused by LED nonlinearity and photodiode (PD) power competition, owing to the power imbalance of the transmitted signals. Therefore, a flipped superposition scheme was proposed in Ref. [10]. In addition to achieving equal power superposition of transmitted signals, this approach had the advantages of thorough gray coding and an improved received signal-to-noise ratio (SNR). In Ref. [11], Zou et al. studied a superposed 32QAM constellation with probabilistic shaping to further reduce the nonlinear distortion of VLC systems. Subsequently, a scalar-superposed coded modulation scheme was proposed in Ref. [12], in which two probabilistically shaped (PS) 6-pulse amplitude modulation (PAM6) signals were superposed to form a PS 36QAM signal. In these studies, PS signals were generated using a constant composition distribution matcher (CCDM) with the addition of redundant bits, which reduced the data rate. To deal with this issue, nonlinear coding instead of CCDM was introduced to realize probabilistic shaping in Ref. [13], which was initially proposed for spectral shaping of PAM4 signals in Ref. [14]. In this paper, uniformly distributed PAM4 signals were encoded into PS PAM6 signals using nonlinear coding and superposed as PS 36QAM signals. The study indicated that nonlinear coding generated PS signals by increasing the modulation order rather than by adding redundant bits. Another advantage of nonlinear coding is that it can achieve modulation of any order and does not have to be a power of 2. Hence, the MED does not decrease significantly because of the increase in the modulation order. However, adjacent constellation points are more likely to overlap when the powers of the transmitted signals are not equal owing to the use of scalar superposition with PAM signals. In addition, the data rate was relatively low because only 4-order modulation was employed.

In this Letter, we propose a novel superposed constellation scheme based on nonlinear coding that extends nonlinear coding to the generation of PS QAM signals for the first time, to our knowledge. In this scheme, a nonlinear coding equation is proposed to generate higher-order PS signals, which convert 8-level signals with equal probabilities into PS 9- or 10-level signals. Subsequently, the PS 9- or 10-level signals are mapped to the PS 9QAM or 10QAM signals, where square-shaped 9QAM and trapezoid-shaped 10QAM constellations are introduced to maximize the MED of the superposed constellations. The PS 9QAM or 10QAM signals are then superposed with the uniformly distributed 4QAM signals in a flipped manner to obtain PS 36QAM or 40QM signals at the receiver. Benefiting from probabilistic and geometrical shaping, the proposed superposed 36QAM and 40QAM constellation schemes increase the MED of the superposed constellation, thus obtaining a better bit error rate (BER) performance. In addition, the proposed PS QAM signals are more robust to system nonlinearity. Furthermore, a new detection algorithm is proposed for nonlinearly coded superposed constellation schemes. The Viterbi decoding algorithm is employed to avoid error propagation and improve detection performance because the PS 9QAM and 10QAM signals generated by nonlinear coding are time-dependent. Finally, an experimental demonstration of a $2\times 2$ MIMO VLC system was established to evaluate the system performance. The experimental results verified the superiority of the proposed schemes over the traditional scheme, where a lower BER and a larger dynamic range of driving peak-to-peak voltage (Vpp) were achieved. The superposed 36QAM constellation scheme achieved a better BER performance than that of the 40QAM constellation scheme at a lower driving Vpp owing to its advantage in MED. However, with the driving Vpp increasing, the superposed 40QAM constellation scheme performed better, thus validating its superiority in terms of anti-nonlinearity.

In this Letter, nonlinear coding is introduced to generate PS signals. Rather than adding redundant bits, nonlinear coding generates PS signals by increasing the modulation order. The higher the modulation order, the more constellation points of probability shaping. However, an increase in modulation order may lead to a decrease in MED. Fortunately, nonlinear coding can achieve modulation of any order and does not have to be a power of 2. Therefore, we propose a nonlinear coding equation that converts uniformly distributed 8-level signals to nonuniform 9- or 10-level signals with only one or two increments of modulation order. The proposed nonlinear coding equation is expressed as $$v_n=\{\begin{array}{ll}{u}_n& \text{for}{u}_n\ge \lfloor \alpha v_{n-1}+A\rfloor \\ u_n+8& \text{for}{u}_n<\lfloor \alpha v_{n-1}+A\rfloor \end{array},$$(1)where $u_n$ denotes an original 8-level symbol, $v_n$ denotes the encoded 9- or 10-level symbol, and $n$ denotes the corresponding time index. The operator $\lfloor ·\rfloor$ denotes the floor function. The level number of the encoded symbol $v_n$ is dependent on $\alpha$, which increases with the increase of $\alpha$, while $A$ ensures that the probability values are generated in pairs. Nonlinear coding can be interpreted as follows. The output signal $v_n$ is related to the signal of the previous state $v_{n-1}$ and current input signal $u_n$. PS 9-level signals can be generated according to Eq. (1) when $\alpha$ and $A$ are set to 1/5 and 0, respectively. The trellis diagram for nonlinear coding with $\alpha =1/5$ and $A=0$, representing all the possible state transition paths for the 9-level signals, is shown in Fig. 1. The branches and nodes in the trellis diagram represent the original $u_n$ and encoded $v_n$ symbols, respectively. As shown, the number of possible paths associated with each state is different, leading to a nonuniformly distributed 9-level signal, i.e., the state with more paths corresponds to a higher probability.

According to Eq. (1), the symbol transition is restricted by one-memory nonlinear coding. This Markov chain has a one-step transition probability, which is expressed as $$P_{ij}=P(v_n=j|v_{n-1}=i).$$(2)

Thus, the transition matrix can be expressed as $${\mathbf{P}}_1=\left[\begin{array}{ccccccccc}1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 0\\ 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 0\\ 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 0\\ 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 0\\ 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 0\\ 0& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8\\ 0& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8\\ 0& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8\\ 0& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8& 1/8\end{array}\right].$$(3)

The state probability vector ${\mathbf{w}}_1$ corresponds to the probability of the 9-level symbols and can be derived from the eigenvector of the transition matrix, which is expressed as $${\mathbf{w}}_1=\left[\begin{array}{ccccccccc}\frac{4}{56}& \frac{7}{56}& \frac{7}{56}& \frac{7}{56}& \frac{7}{56}& \frac{7}{56}& \frac{7}{56}& \frac{7}{56}& \frac{3}{56}\end{array}\right].$$(4)

Similarly, the PS 10-level signals can be generated according to Eq. (1) when $\alpha$ and $A$ are equal to 1/4. In this case, the state probability vector ${\mathbf{w}}_2$, i.e., the probability of the 10-level symbols, can be expressed as $${\mathbf{w}}_2=\left[\begin{array}{ccccccccc}\frac{1}{32}& \frac{3}{32}& \frac{4}{32}& \frac{4}{32}& \frac{4}{32}& \frac{4}{32}& \frac{4}{32}& \frac{4}{32}& \frac{3}{32}\end{array}\frac{1}{32}\right].$$(5)

The principle of the proposed superposed 36QAM and 40QAM constellation schemes is shown in Fig. 2. The PS 9-level signals were mapped to the PS 9QAM signals, and a square-shaped 9QAM constellation was designed to improve the MED of the superposed constellation, as shown in Fig. 2(a). The mapping relationship between the 10-level and 10QAM signals is presented in Fig. 2(b), where a trapezoid-shaped 10QAM constellation is introduced. Flipped superposition was selected to further reduce the nonlinear distortion and power competition because the transmitted signals were superposed with equal power^[10]. In addition, this approach benefits from thorough gray coding and the improvement of the received SNR. Based on the concept of flipped superposition, the PS 9QAM or PS 10QAM signal was first optimally biased, power-normalized, and flipped according to the value of the 4QAM signal. The signals were then superposed with the 4QAM signal to obtain the target superposed 36QAM or 40QAM constellation, as shown in Fig. 2. The 36QAM or 40QAM constellation was also probabilistically shaped. Table 1 lists the MED of the superposed constellation when the power of both transmitted signals is equal to one. The superposed 32QAM constellation scheme was introduced for comparison, in which uniformly distributed square-shaped 8QAM and 4QAM signals were superposed in a flipped manner^[8]. Although the modulation order is increased in the proposed schemes, both superposed 36QAM and 40QAM constellations achieve higher MED because of the probabilistic shaping. The MED of the superposed 36QAM constellation is the largest, indicating that it has the strongest resistance to noise.

A novel detection algorithm was further proposed for the nonlinearly coded superposed constellation scheme, in which Viterbi decoding was introduced to decode the PS 9QAM and 10QAM signals. The Viterbi algorithm is well known for the state estimation of Markov chains, which can fully utilize the temporal correlation of signals created by nonlinear coding. The 4QAM and PS 9QAM or 10QAM signals were decoded serially rather than jointly to reduce complexity. The superposed 36QAM constellation scheme was used as an example to introduce the detection algorithm principle.

First, the 4QAM signal was detected by determining the quadrants of the received superposed signal. The received noisy 9QAM signal was calculated using $$X_{9\mathrm{QAM}}=X_{36\mathrm{QAM}}-{\widehat{X}}_{4\mathrm{QAM}},$$(6)where $X_{36\mathrm{QAM}}$ denotes the received superposed 36QAM signal and ${\widehat{X}}_{4\mathrm{QAM}}$ denotes the estimated 4QAM signal.

Second, the Viterbi algorithm was employed to decode the noisy 9QAM signals into original 8-level signals. Compared with the decoding algorithm in Ref. [13] that mainly depends on the concept of decision feedback and hard decision, Viterbi decoding avoids error propagation and improves the BER performance. Furthermore, soft decision can be used for Viterbi decoding, in which the path metric is measured using the Euclidean distance. Only two consecutive moments of the signal values are involved in the measurement of the path metric because the memory length of the proposed nonlinear encoder is one. The cumulative path metric of the $j$th moment is expressed as $${\mathbf{m}}_j^T={\mathbf{m}}_{j-1}^T+{\mathbf{\Delta }}^T.$$(7)

Equation (7) indicates that the cumulative path metric at the $j$th moment is obtained from the value at the ($j-1$)th moment plus the increment. Here, ${\mathbf{m}}_j={[m_{1,j},m_{2,j},\dots ,m_{9,j}]}^T$ denotes the cumulative metric vector of the state, and $\mathbf{\Delta }={[{\mathrm{\Delta }}_1,{\mathrm{\Delta }}_2,\dots ,{\mathrm{\Delta }}_9]}^T$ denotes the increment vector at the current moment, which is calculated using $${\mathrm{\Delta }}_i=\underset{{\mathbf{v}}_{k,i}\in {\mathbf{\Omega }}_i}{\min }({|[X_{9\mathrm{QAM},j-1},X_{9\mathrm{QAM},j}]-{\mathbf{v}}_{k,i}|}^2),$$(8)where $[X_{9\mathrm{QAM},j-1},X_{9\mathrm{QAM},j}]$ is a vector comprising the 9QAM signals at two consecutive moments. ${\mathbf{\Omega }}_i$ denotes the set of possible transition paths for the $i$th state. ${\mathbf{v}}_{k,i}$ represents the $k$th state transition path of ${\mathbf{\Omega }}_i$.

A matrix $\mathbf{R}$ was defined to record the input signal corresponding to the possible path, where the elements of the $i$th row and $j$th column are expressed as $$r_{i,j}=\underset{{\mathbf{v}}_{k,i}\in {\mathbf{\Omega }}_i}{\arg \min }({\mathrm{\Delta }}_i).$$(9)

Equation (9) indicates that $r_{i,j}$ is the input signal that minimizes the increment at the $j$th instant.

Finally, the surviving path is the one with the smallest cumulative path metric. The decoded 8-level signals can be obtained after the surviving path is determined, according to the matrix $\mathbf{R}$.

A $2\times 2$ MIMO VLC system was constructed for an experimental demonstration to further verify the superiority of the superposed PS QAM constellation scheme. The system block diagram and experimental setup are shown in Fig. 3. First, two binary data streams were generated: one was directly modulated with the 4QAM format and the other was nonlinearly encoded and mapped to PS 9QAM or 10QAM signals. Then, discrete Fourier transform (DFT) spread signals were generated using an M-point DFT to reduce the peak-to-average power ratio of the transmitted signals^[10,15]. A training sequence (TS) was inserted for synchronization and channel estimation. After upsampling and Hermitian symmetry, an inverse fast Fourier transform was implemented to obtain the real-valued orthogonal frequency-division multiplexing (OFDM) signals. Finally, a cyclic prefix (CP) was added to resist multipath propagation. The above signal processing was programmed using MATLAB, and offline transmitted signals were generated.

The generated signals were loaded into an arbitrary function generator (AFG, Tektronix AFG3252C). The electrical signals were then amplified using an electrical amplifier (EA, Mini-Circuit ZHL-6A-S+) and direct current (DC) was supplied through a DC bias (Mini-Circuit ZFBT-4R2GWFT+) to ensure that the signals were positive. Subsequently, the mixed signals were used to drive red LEDs (Cree XLamp XP-E), where the signals were transmitted in the form of light. After approximately 1.5 m of free-space transmission, two PDs (Hamamatsu C12702-11) were used to convert the optical signals into electrical signals at the receiver. Each PD simultaneously received light from the two LEDs owing to the light diffusion. The two PDs were placed approximately 1.5 cm apart considering the terminal size and power consumption in practical applications, resulting in a highly correlated MIMO channel. Lenses were placed between the LEDs and PDs to focus the light and improve the received SNR. Finally, the signals were recorded using a high-speed digital oscilloscope (OSC, Tektronix MDO4104C) that was set for offline detection.

In the offline signal processing, frame synchronization was realized by correlation operation with TS to detect the starting position of the data stream. We used two PDs to improve the received SNR, where the two received signals were merged based on the equal-gain combining criterion^[8,16]. The channel gains were estimated, and signal distortion was compensated using channel equalization after CP removal, downsampling, and OFDM demodulation. Subsequently, the inverse DFT converted the signals from the frequency domain to the time domain. Finally, the original binary data streams were recovered using the proposed detection algorithm.

The system parameters in the experiment were set as follows: the number of DFT points was 128, the number of subcarriers in the OFDM modulation was 256, and the CP length was 8. The upsampling rate was 4, and the DC bias current was set to 100 mA. Except for the experiment in Fig. 8, the sampling rate of the other experiments was set to 100 Mbps. The distance between the LED and PD was approximately 1.5 m, and the distances between each pair of LEDs and PDs were 40 and 1.5 cm.

Initially, the BERs of the different superposed constellation schemes under different driving Vpp values of LED2 (Vpp2) were measured, as shown in Fig. 4. The 4QAM signal was sent by LED1, while the PS 9QAM, PS 10QAM, and normal 8QAM signals were sent by LED2, depending on scheme. The driving Vpp of LED1 (Vpp1) was maintained at 500 mV. The traditional superposed 32QAM constellation scheme was used for comparison. The original signals were modulated at orders of 8 and 4 in all schemes. Therefore, the data rates of the schemes were consistent to ensure a fair comparison. As can be seen, all schemes achieved the best BER performance, and the constellation points were uniformly distributed when Vpp1 and Vpp2 were both equal to 500 mV. Increasing or decreasing the Vpp2 caused a reduction in the MED. In addition, nonlinear distortion and power competition also deteriorated the BER performance when Vpp2 was increased. The superposed 32QAM constellation scheme always exhibited the worst BER performance because of the smallest MED. When Vpp2 was lower than 600 mV, the superposed 36QAM constellation scheme obtained the lowest BER owing to its highest MED. However, the BER performance of the superposed 40QAM constellation scheme was close to that of the superposed 36QAM constellation scheme when Vpp2 was higher than 600 mV. This indicated that the PS 10QAM signal was more robust to LED nonlinearity because the probability of the highest power points in the PS 10QAM constellation was smaller than that in the PS 9QAM constellation. The Vpp2 dynamic ranges of the superposed 32QAM, 36QAM, and 40QAM constellation schemes were 240, 360, and 330 mV, respectively, considering the 7% pre-forward error correction (pre-FEC) BER threshold of $3.8\times {10}^{-3}$.

The BER performances of different schemes were compared when the LEDs worked in more severe nonlinear regions to further highlight the advantage of the superposed 40QAM constellation scheme in resisting the nonlinear effects of LEDs. The BERs were measured when Vpp2 was fixed at 900 mV, where severe nonlinearity always occurred in LED2, as shown in Fig. 5. In contrast to the previous experimental results shown in Fig. 4, the superposed 40QAM constellation scheme achieved the best BER performance regardless of the value of Vpp1. This result confirms the superiority of the superposed 40QAM constellation scheme, which was the most robust to nonlinearity. Considering the 7% pre-FEC BER threshold of $3.8\times {10}^{-3}$, the Vpp1 dynamic range of the superposed 40QAM constellation scheme reached a maximum value of 520 mV, which was 79% larger than that of the 32QAM scheme. Concurrently, the dynamic range of the superposed 36QAM constellation scheme was 59% larger than that of the 32QAM scheme.

In Fig. 6, BER performance was evaluated when Vpp1 was always equal to Vpp2. Hence, a superposed constellation with uniformly distributed constellation points can always be obtained for all schemes. The relationship between SNR and Vpp1/Vpp2 is also presented in Fig. 6. Initially, the SNR improved with the increase of Vpp1 and Vpp2. However, it began to decrease when Vpp1 and Vpp2 were greater than 600 mV. The reason is that the appearance of nonlinearity leads to the deterioration of SNR. Therefore, the BER performance also declined. Benefiting from the advantage in MED, the superposed 36QAM constellation scheme still performed best in the initial phase. However, when nonlinearity occurred, the superposed 40QAM constellation scheme had the lowest BER.

In Fig. 7, the BER performance of 9QAM and 10QAM signals with and without Viterbi algorithm was compared. Here, the decoding algorithm in Ref. [13] represents the algorithm without Viterbi decoding. The experimental setup was the same as above. As evident, the BER performance with Viterbi decoding algorithm is much better. This is because the Viterbi decoding algorithm utilizes the temporal correlation of signals created by nonlinear coding, and can avoid error propagation.

In Fig. 8, we measured the data rate versus BER for different schemes. In the experiment, the Vpp1 and Vpp2 were both fixed at 600 mV. The data rate was changed by adjusting the sampling rate of AFG. In the figure, the data rate is the sum rate of the two transmitted signals. Hence, the data rates of two separate data streams can be obtained from two-fifths and three-fifths of the sum rate, respectively. The experimental results showed that the BER increased with the data rate growing. This is because an increase in the sampling rate leads to an increase in the modulated bandwidth. As the frequency increased, the frequency response of the LED decreased exponentially, resulting in a decline in BER performance.

Finally, the dynamic ranges of the driving Vpp were measured for different schemes, as shown in Fig. 9. It was evident that the proposed 36QAM and 40QAM schemes exhibited better BER performance and achieved larger dynamic ranges of driving Vpp than the traditional superposed 32QAM constellation scheme. The proposed scheme took advantage of a larger MED and stronger ability to resist nonlinearity owing to the probabilistic shaping. The dynamic ranges of driving Vpp1 and Vpp2 with a 7% pre-FEC BER of $3.8\times {10}^{-3}$ as the threshold are denoted by the black outlined shapes in Fig. 9. The areas of the driving Vpp1 and Vpp2 for the superposed 36QAM and 40QAM constellation schemes were approximately $3.5\times {10}^5$ and $3.4\times {10}^5{\mathrm{mV}}^2$, which were 52% and 48% larger than that of the superposed 32QAM constellation scheme ($2.3\times {10}^5{\mathrm{mV}}^2$), respectively.

This work proposed a superposed QAM constellation scheme for MIMO VLC systems, where PS 9QAM or 10QAM signals were superposed with 4QAM signals in a flipped manner to obtain PS 36QAM or 40QAM signals. In the scheme, a novel nonlinear coding equation was proposed to generate PS signals without adding redundant bits, which encoded uniformly distributed 8-level signals into PS 9- or 10-level signals. Then the PS 9- or 10-level signals were mapped to the PS 9QAM or 10QAM signals, where square-shaped 9QAM and trapezoid-shaped 10QAM constellations were introduced to optimize the MED of the superposed constellation. The Viterbi decoding algorithm was introduced to achieve better performance, since nonlinear coding created a temporal correlation of the PS 9QAM and 10QAM signals. The proposed scheme was experimentally investigated comprehensively, where the system performance was evaluated under different driving Vpps, data rates, and detection algorithms. Experimental results showed that the superposed 36QAM constellation scheme achieved the best BER performance when Vpp was low, owing to its maximum MED. However, the superposed 40QAM constellation scheme performed best under high driving Vpp and proved to be more robust to nonlinearity. Considering the 7% pre-FEC BER threshold of $3.8\times {10}^{-3}$, the Vpp dynamic ranges of the superposed 36QAM and 40QAM constellation schemes were 52% and 48% larger than that of the superposed 32QAM constellation scheme. This result is encouraging for large-power VLC systems because high power is always necessary to satisfy the requirement of sufficient illumination.