1. Introduction

With the progress of worldwide informatization, the conventional radio frequency (RF) communication system is facing great challenge to meet the growing communication requirement [1]. Visible light communication (VLC) may make full use of the ubiquitous illumination systems and the visible light spectrum to relieve the pressure of communication resources, which will be complementary to RF networks [2]. In the era of the Internet of things (IoT) and the 6G standard of wireless systems, VLC is expected to play a more important role in massive intelligent devices, realizing seamless integration for future mobile communication [3,4].

In order to achieve higher-speed wireless communication, the multi-antenna technique is gradually adopted in VLC systems, such as the multiple-input single-output (MISO) [5] and multiple-input multiple-output (MIMO) techniques [6]. Multiple-input VLC systems, which are quite nature in the illumination network, transmit optical signals with multiple light emitting diodes (LEDs) at the same time based on spatial multiplexing. There comes the generalized spatial modulation (GSM) technique that loads additional information with the selection of activated LEDs, improving the channel capacity further with the help of spatial-domain symbols [7,8].

However, the GSM technology for RF systems may not be directly applicable to VLC without modification. On one hand, the transmitted optical signal must be real-valued and non-negative, since it is received by the photodiode (PD) with the method of intensity modulated/direct detection (IM/DD) [9]. On the other hand, the GSM-based VLC channel is different from general RF ones [10], as it is not modeled as a random variable obeying a certain distribution like Rayleigh distribution, but as a function with respect to the random variable of the receiver location, which is called the ergodic VLC channel in this paper.

Therefore, the conventional GSM technologies which are suitable for RF systems should be modified accordingly, taking the specific characteristics of VLC into account. Some researches have achieved improvement of power efficiency in VLC by collaborative encoding based on GSM and MIMO [11], and the MIMO precoder has been verified to be beneficial for the mitigation of multi-user interference [12]. There have also been other researches that focus on the physical-layer security based on spatial modulation (SM) in indoor VLC systems [13]. Nevertheless, the modulations of the GSM systems mentioned above follow conventional constellations like those of quadrature amplitude modulation (QAM), whose performances are not discussed in VLC.

The constellation optimization has attracted the attention of some researchers. There have been researches that build a two-user VLC system with the two-layer 4-QAM constellation, and improve the bit error rate (BER) by optimizing the superposition constellation [14]. The minimum Euclidean distance of symbols is always a key role in the constellation design, which can be the constraint of the constellation optimization for higher power efficiency [15], or the optimization target for a general space-time modulation design [16]. Some other researches also investigate the precompensation of channel distortions like inter subchannel interference while considering the location of constellation symbols [17].

In this work, the constellation optimization is investigated in the GSM-based VLC system, considering both the ergodic VLC channel and the activation choice of LEDs. The asymmetrically clipped optical orthogonal frequency division multiplexing (ACO-OFDM) is taken as an example, while the same principle can be extended to other cases. The distribution of amplitude-phase modulation (APM) symbols is optimized with the algorithm of statistically-convergent gradient descent (SCGD), and the coded modulation average mutual information (CM-AMI) is verified to realize the maximization as the optimization target by the numerical simulation.

2. System model

The most common model of the VLC channel is the Lambertian model [9,10], and only the line-of-sight (LoS) links are taken into consideration in this paper. Figure 1 shows the geometry for the ergodic VLC channel based on GSM, and the network consisting of $N_{\textrm {t}}$ LEDs is arranged on the ceiling. The receiver moves randomly on the ground, while the PD on it can receive optical signals from any LEDs at the same time.

 

Fig. 1. The geometry for the ergodic VLC channel based on GSM.

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The direct distance from the $i$-th LED to the PD with $i$ from 1 to $N_{\textrm {t}}$ is represented by $d_i = (l_i^2 + s_i^2)^{1/2}$ , where $l_i$ and $s_i$ are the vertical and horizontal distances, respectively. $\psi _i$ denotes the incidence angle of the LoS path at the receiver, and $\phi _i$ is the radiance angle of the LED with the maximum value $\Phi _{\textrm {half}}$. On this basis, the direct current (DC) gain of the $i$-th optical path is formulated by

As for the modulation, only $N_{\textrm {a}}$ LEDs are active to transmit the same symbol at the same time based on GSM. There are $N = \tbinom {N_{\textrm {t}}}{N_{\textrm {a}}}$ sets of indexes for activated LEDs, where $\tbinom {\cdot }{\cdot }$ denotes the binomial coefficient. The $n$-th set of activated indexes is represented as $\Omega _n$, and it can be mapped to a spatial-domain symbol $x_n^{\Omega }$. The number of constellation-domain APM symbols is $M$, and the constellation is a set of symbols represented as a vector $\boldsymbol {\xi } = \left [x_1^{\textrm {c}}, x_2^{\textrm {c}}, \ldots , x_M^{\textrm {c}}\right ]\subseteq \mathbb {C}^M$. If the transmitted symbol of all the activated LEDs is $x_m^{\textrm {c}}$ and the set of activated LEDs is $\Omega _n$, the transmitted signal is denoted as a vector, $\textrm {i.e.}$

The ACO-OFDM modulation is adopted in this paper as an example, which ensures that the recovered frequency-domain signal from the clipped time-domain sequence is half of the origin frequency-domain signal with Hermitian symmetry and zero-padded even subcarriers. Consequently, the received signal over the ergodic VLC channel based on GSM is given by

3. GSM-based constellation optimization

The CM-AMI plays a role of optimization target in this paper, which is decided by the probability density function (PDF) of the received signal. Once the ergodic VLC channel $\mathbf {H}$ is given, the PDF of $y$ is described as an explicit analytical formula

The gradient descent (GD) algorithm is adopted to deal with the optimization problem in this paper, and the gradient of the cost function $J(\boldsymbol {\xi })$ is represented as a vector by

The theoretical result of the gradient is a form of mathematical expectation with respect to $\mathbf {H}$, $X^{\Omega }$, and $W$ in Eq. (11), which cannot be calculated in close-form. However, $X^{\Omega }$ and $W$ are random variables, while $\mathbf {H}$ is a matrix function of the receiving location that is also a random variable, so that the mathematical expectation can be calculated based on multiple random experiments according to the Monte Carlo method. The number of experiments in each iteration is represented by $R$, and in the $r$-th experiment, a set of parameters $\left \{\mathbf {H}(r),n(r),w(r)\right \}$ are generated, representing a sampling of the ergodic VLC channel, the index of the selected LED set, and the normalized AWGN, respectively. Thus, the received signal is $y_m(r) = G\mathbf {H}(r)\mathbf {x}_{mn(r)}/2 + w(r)$, and the partial derivative of $J(\boldsymbol {\xi })$ is formulated by the average of all the experimental results as

In the light of the conventional GD principle, the constellation is updated with a step valued $\alpha$ along the gradient calculated based on the current constellation in the $p$-th iteration as following

A larger $R$ can improve the accuracy of the gradient and reduce the vibration amplitude of the convergence curve, while a smaller $R$ can improve the probability of getting out of the local minimum and requires less computation, so it is necessary to set $R$ reasonably according to different demands in the simulations. In addition, the convergence to the local minimum instead of the global minimum is the inherent problem of SCGD. Hence, multiple optimization results from different sets of random initial constellations are obtained, and the best one is selected as the global optimal result in this paper.

4. Simulation results

In ordinary circumstances, the locations of all the LEDs are fixed, and the normal vectors of each light-emitting plane are vertical. Meanwhile, it is assumed that the receiver moves randomly within the intersection of all the LED coverage on the horizontal ground, and the normal vector of PD remains vertical as well. Such an assumption can represent quite a lot of common situations, including the movement of human beings or intelligent robots in the room.

The numerical simulation is adopted for the verification of constellation optimization under the ergodic VLC channel based on GSM. The situation considered in this paper is that there are 4 LEDs on the ceiling of the room with the same height of 5 m, constituting a square with sides that are 5 m in length. Some other key parameters are also listed in Table 1.

Table 1. Simulation Parameters

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The convergence of the proposed SCGD is verified firstly. Figure 2(a) shows an example of the SCGD convergence curve of the cost function with $M = 16$ and $N_{\textrm {a}} = 2$, demonstrating that the cost function vibrates during the iteration process and achieves convergence statistically. Figure 2(b) presents an example of constellation vibration, where the clusters of grey circles are the constellation results of the last 100 iterations in Fig. 2(a). The outline of the optimized constellation can already be figured out from these clusters, and the final statistically-convergent result is the average of each cluster, shown by the black circles in Fig. 2(b).

 

Fig. 2. Convergence verification of the proposed SCGD.

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Figure 3 presents the optimization results of constellations with different numbers of symbols and activated LEDs. The distribution of symbols conforms to the structure of multiple homocentric circles with different numbers of circles and different radii. The symbols are more densely distributed in the center of constellations with $N_{\textrm {a}} = 2$, while the constellations with $N_{\textrm {a}} = 1$ are relatively sparse with larger average Euclidean distance at the center.

 

Fig. 3. The optimal constellations under the ergodic VLC channel based on GSM.

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It can be seen in Eqs. (5) and (6) that CM-AMI will increase with the increase of the SNR, the increase of the average Euclidean distance of constellation symbols, and the variance decrease of the ergodic VLC channel. The ergodic VLC channel with 1 activated LED is proved by Eq. (1) to have smaller average illumination and larger variance with respect to the location of the receiver, so that the increase of the average Euclidean distance is of more necessity for larger CM-AMI.

The CM-AMI comparisons between optimal constellations and QAM ones with different numbers of activated LEDs are shown in Fig. 4, and it is obvious that optimal constellations can provide gains under various SNRs. Specifically, the constellation with 16 symbols and 2 activated LEDs can achieve the SNR gain of around 0.20 dB when reaching the CM-AMI of 3.9 bits, while that with 64 symbols and 1 activated LED obtains about 0.33 dB gain at the CM-AMI of 7.1 bits.

 

Fig. 4. The CM-AMI comparison between optimal constellations and QAM ones.

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5. Conclusions

In this paper, the constellation of APM symbols was optimized with the algorithm of SCGD to realize the maximization of CM-AMI in the GSM-based VLC system, considering both the influence of the ergodic VLC channel and the index selection of activated LEDs. The optimal constellations appear to be of different distributions with different random characteristics of the ergodic VLC channel in the numerical simulation, while they can all lead to higher channel capacity than conventional QAM constellations.