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Abstract
Superposed constellation for spatial multiplexing visible light communication (VLC) systems has recently attracted significant attention as a promising method to alleviate the effect of nonlinearity on light-emitting diodes (LEDs) that has achieved significant performance improvements in VLC systems. A large minimum Euclidean distance (ED) value of the superposed signals can be achieved at the receiver side by employing optimal power ratios transmitted from different LEDs. However, the power allocation strategy using fixed shaped sub-constellations at the LEDs has the inherent limitation of low optimization dimensions. In this paper, we propose optimizing the sub-constellations at individual LEDs instead of just the power allocation coefficients to further increase the minimum ED value. Moreover, intra-ED and inter-ED terms are used to reduce the complexity of the optimization solving process. The simulation results show that the proposed constellation design scheme can improve the symbol error rate performance compared to the conventional one, and the right choice of pre-defined shape of the sub-constellations at the LEDs has an important role in the optimization process.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Visible light communication (VLC) in conjunction with the growing development of light-emitting diode (LED) technology has become a research area of significant interest in short-range wireless broadband communication [1]. This is owing to the current replicable infrastructure, which can integrate both lighting and communication services. In particular, the subsequent distortion of the nonlinearity of LEDs has a significant impact on system transmission performance. It is well known that an LED as a transmitter can only operate in a limited linear voltage zone, while the photodetector (PD) as the receiver works in a larger one [2–4]. Although several modulation schemes can mitigate the nonlinearity of LEDs by sacrificing spectral efficiency [5], such as on-off keying (OOK) modulation [2], pulse width modulation (PWM) [6], and pulse position modulation (PPM) [7], all of these binary signals modulations have very limited spectral efficiency and are not good in high rate transmission systems. On the other hand, various modulation schemes have been proposed for VLC systems with multiple LED transmissions, such as spatial multiplexing (SMP) [8] and repetition coding (RC). Despite its simple design principle, spatial modulation (SM) [9] has been envisioned as one prospective digital modulation approach among the various technologies to achieve high spectral eficiency and energy eficiency. Even so, SM performance is also impaired by the high-level correlation of the optical channel, especially when the channel links are identical, thus making it very difficult to identify the active LEDs in the SM VLC system.
It has been proved that spatial summing is useful in mitigating LED nonlinearity with ideal LED devices when the channel correlation is high. Coincidentally, low-order constellations are more appropriate at the transmitters whereas higher-order constellations are acceptable at the receivers. Therefore, higher-order modulation can be employed without non-linear distortion in VLC systems. The promising approach of an SMP scheme using the principle of superposed signals has recently been suggested for a 2$\times$1 multi-input single-output (MISO) VLC system [10]. In the proposed method, two data streams multiplied by various power ratios are transmitted from two LEDs and then uniquely superposed modulations are recreated in the receiving PD. Consequently, multiplexing gains can be achieved in the highly correlated MISO channels. In [11], the authors suggested a novel 2$\times$2 VLC system with a superposed 32 quadrature amplitude modulation (QAM) constellation scheme with the conclusion that only low-order constellations are required at LEDs to have a reasonable minimum Euclidean distance (ED). Previous studies such as [10–12] mainly focused on an effective algorithm to separate and detect a signal transmitted by individual LEDs while an optimal power allocation coefficient was exhaustively determined to improve the transmission performance. The power allocation is one way of reducing the error probability at the receiver side by increasing the minimum ED between the received superposed constellation points. This has proved to be remarkably effective since different values of coefficients generate different error probabilities. However, this strategy has the inherent limitation of low optimization dimensions and can only improve the transmission performance up to a certain point.
In this paper, instead of the conventional way of optimizing the power allocation for individual LEDs in the superposed constellations of a multi-dimensional VLC system [11,12], we propose designing the sub-constellations at each LED to increase the minimum ED of the superposed constellation at the receiver side, resulting in a significant improvement in the symbol error rate (SER) of the system. More specifically, an optimization problem formed by using the constellations at individual LEDs as unknown parameters is linearized and iteratively solved. The optimal result vector is then used to construct the sub-constellations at each LED so that the minimum ED is as large as possible according to the constraints on total electrical power and maximum transmitted power. Furthermore, we propose the inter-ED and intra-ED terms to reduce the number of constraints in the optimization problem, while some pre-defined shape sub-constellations are utilized to remarkably enhance the minimum ED values and improve the SER performance. The simulation results demonstrate the advantages of the proposed constellation design method. The superposed constellation at the receiver can have different minimum ED values (and thus different error rate probabilities) depending on the pre-defined shape of the sub-constellations. Our approach promises to deliver favorable performance and affordable complexity costs with appropriately chosen parameter values such as constellation size and shape.
2. System model
First, we consider a VLC system consisting of $N$ LEDs as the transmitter and one PD as the receiver. At each period $t$, the input information bits are modulated into the input signal vector that will later be emitted by all LEDs, $\left [ s_1\left ( t \right ) ,s_2\left ( t \right ) ,\ldots ,s_N\left ( t \right ) \right ] ^{^T}$. On the receiver side, after the propagation of the signal through the optical channel, it can be assumed that the signals received at the PD are superpositions of those transmitted from each LED. For example, as in [11,12], the 32-QAM constellation on the receiver can be obtained using 4-QAM and 8-QAM on the two LEDs on the transmitter side. The relationship between the received superposed signals at the PD and the independent transmitted signals from the LEDs can then be defined as
Any particular superposed constellation composition can be defined by the corresponding power ratio of the LEDs. For example, in the case of a VLC system with two LEDs and without the loss of generality, the real and imaginary components of the $\mathrm {LED}_1$ constellation are kept as invariant, with scaling factor $\alpha$ to express the power ratio between the two LEDs. Therefore, $\alpha$ can be multiplied by the real and imaginary components of the $\mathrm {LED}_2$ to adjust its amplitude. Consequently, the optimal constellation with favorable performance can be obtained by exhaustively searching for the optimal value of $\alpha$. On the other hand, specific receivers have been suggested to recognize the original symbols conveyed by LEDs, with the maximum likelihood (ML) or lookup table (LUT) detector being by far the most straightforward and fundamental signal detector. Notably, the ML detector algorithm is used to estimate the received mixture from various transmitted combinations as via the ML algorithm, it tries to find a combination that generates the minimum ED with the obtained signal. The successive interference cancellation-LUT (SIC-LUT) was proposed by [11], while in [12], the least-squares algorithm is utilized in the frequency domain to separate the superposed signals. Moreover, the ED is commonly used as a distance measure in signal processing. In the present study, the ED between any two received constellation points is dened as $d^2\left ( \left ( x^{\left ( i \right )},y^{\left ( i \right )} \right ) ,\left ( x^{\left ( i\prime \right )},y^{\left ( i\prime \right )} \right ) \right ) =\left ( x^{\left ( i \right )}-x^{\left ( i\prime \right )} \right ) ^2+\left ( y^{\left ( i \right )}-y^{\left ( i\prime \right )} \right ) ^2$. The overall pairwise-error probability (PEP) for all available pairs of superposed symbols is generally employed as the upper limit for the system SER performance.
3. Proposed constellation design scheme
Without the loss of generality, the real part of the received signal in (1), $r_x\left ( t \right )$, can be rewritten by using the vector form as
where $\mathbf {h}\left ( t \right ) =\left [ h_1\left ( t \right ) \cdots h_n\left ( t \right ) \cdots h_N\left ( t \right ) \right ]$ and $\mathbf {x}\left ( t \right ) =\left [ x_1\left ( t \right ) \cdots x_n\left ( t \right ) \cdots x_N\left ( t \right ) \right ] ^{\mathrm {T}}$ represent the channel gain and the real parts of the transmitted symbols, respectively. Using the presumption in [11,12], when the propagation distance and range between the positioned LEDs are relatively small, we can presume that in an ideal scenario, all of the channel gains have approximately the same values. Subsequently, signals from all of the LEDs are superposed in the PD, and the system can be simplified as the single-input single-output (SISO) model. Therefore, the difference between the channel coefficients is very small, and the received symbols in Eq. (2) can be expressed as where $\mathbf {x}_{\mathrm {sup}}=\left [ x_{1}^{\left ( 1 \right )}\cdots x_{1}^{\left ( M_1 \right )}x_{n}^{\left ( 1 \right )}\cdots x_{n}^{\left ( M_n \right )}x_{N}^{\left ( 1 \right )}\cdots x_{N}^{\left ( M_N \right )} \right ]$ is the $\sum _{n=1}^N{M_n}$-length vector which composed of all real parts of symbols set $\mathcal {S}$. Besides, $\mathbf {p}\left ( t \right ) =\left [ \begin {matrix}{} \mathbf {p}_1 & \cdots & \mathbf {p}_n & \cdots & \mathbf {p}_N\\\end {matrix} \right ] ^{\mathrm {T}}$ is a vector of the same length whereas each individual $\mathbf {p}_n$ is an $M_n$-length vector in which all of the elements are zero except for the one at the position of the transmitted symbol of the $n$-th LED. For example, if the first symbol of the $\mathrm {LED}_1$ and the second symbol of the $\mathrm {LED}_2$ with 4-QAM constellation are chosen, we have $\mathbf {x}_{\mathrm {sup}}=\left [ x_{1}^{\left ( 1 \right )}x_{1}^{\left ( 2 \right )}x_{1}^{\left ( 3 \right )}x_{1}^{\left ( 4 \right )}x_{2}^{\left ( 1 \right )}x_{2}^{\left ( 2 \right )}x_{2}^{\left ( 3 \right )}x_{2}^{\left ( 4 \right )} \right ]$ and $\mathbf {p}\left ( t \right ) =\left [ 1\quad 0\quad 0\quad 0\quad 0\quad 1\quad 0\quad 0 \right ] ^T$. Notice that we denote the imaginary part of the transmitted symbol $y_n\left ( t \right )$ as $\left ( x_n\left ( t \right ) ,y_n\left ( t \right ) \right ) \in \mathcal {S}_n$. Therefore, with the imaginary part of the transmitted signal, we can represent the superposed constellation $\mathbf {r}\in \mathcal {R}$ at the receiver as$\mathbf {t}=\left [ \begin {matrix} \mathbf {x}_{\mathrm {sup}} & \mathbf {y}_{\mathrm {sup}}\\\end {matrix} \right ]$, $\mathbf {p}^{\begin {array}{c} \left ( i,i' \right )\\\end {array}}=\mathbf {p}^{\left ( \begin {array}{c} i\\\end {array} \right )}-\mathbf {p}^{\left ( \begin {array}{c} i\\\end {array}' \right )}$, and $\mathbf {p}^{\left ( \begin {array}{c} i\\\end {array} \right )}$ is the vector represents the index of the transmitted symbol at the transmitter. Intuitively, at a high signal-to-noise (SNR) level, the minimum ED between any two received constellation points is the predominant factor in the average error probability. To design a constellation that maximizes the minimum of all of the pair-wise distances among the constellation symbol set, optimal vector $\mathbf {t}$ can be determined through an optimization problem $\mathcal {P}_0$ as
4. Low complexity constellation design
In this section, we explain the low complexity constellation design principle with some examples. Moreover, we (1) define the intra-ED and inter-ED to reduce the number of constraints and (2) propose using the pre-defined sub-constellation shapes to reduce the total unknown variables in the optimization problem.
4.1 Principles of intra-ED and inter-ED
Consider a simple case of 32-QAM superposed constellation $\mathcal {S}$ using two LEDs with the constellation $\mathcal {S}_1$ as 4-QAM at $\mathrm {LED}_1$, and $\mathcal {S}_2$ as 8-QAM at $\mathrm {LED}_2$. We can define the intra-ED between two superposed constellation points $\mathbf {r}^{\left ( i \right )}$ and $\mathbf {r}^{\left ( i' \right )}$ which comprise the same signal from one LED (either $\mathrm {LED}_1$ or $\mathrm {LED}_2$). For example, as illustrated in Fig. 1, constellation points $\mathbf {r}^{\left ( i \right )}$ or $\mathbf {r}^{\left ( i' \right )}$ of $\mathcal {S}$ are both superposed from $\left ( x_{1}^{\left ( 4 \right )},y_{1}^{\left ( 4 \right )} \right )$ of $\mathrm {LED}_1$ with either $\left ( x_{2}^{\left ( 7 \right )},y_{2}^{\left ( 7 \right )} \right )$ or $\left ( x_{2}^{\left ( 8 \right )},y_{2}^{\left ( 8 \right )} \right )$ of $\mathrm {LED}_2$, respectively. As a result, the ED between these two constellation points $\mathbf {r}^{\left ( i \right )}, \mathbf {r}^{\left ( i' \right )}\in \mathcal {R}$ can be computed as $d^2\left ( \mathbf {r}^{\left ( i \right )},\mathbf {r}^{\left ( i' \right )} \right ) =\left [ \left ( x_{1}^{\left ( 4 \right )}+x_{2}^{\left ( 7 \right )} \right ) -\left ( x_{1}^{\left ( 4 \right )}+x_{2}^{\left ( 8 \right )} \right ) \right ] ^2+\left [ \left ( y_{1}^{\left ( 4 \right )}+y_{2}^{\left ( 7 \right )} \right ) -\left ( y_{1}^{\left ( 4 \right )}+y_{2}^{\left ( 8 \right )} \right ) \right ] ^2=\left [ x_{2}^{\left ( 7 \right )}-x_{2}^{\left ( 8 \right )} \right ] ^2+\left [ y_{2}^{\left ( 7 \right )}-y_{2}^{\left ( 8 \right )} \right ] ^2=d^2\left ( \mathbf {s}_{2}^{\left ( 7 \right )},\mathbf {s}_{2}^{\left ( 8 \right )} \right )$. Therefore, regardless of the transmitted signal in $\mathrm {LED}_1$, the distance between the two superposed constellation points with the intra-ED property always equals $d^2\left ( \mathbf {s}_{2}^{\left ( 7 \right )},\mathbf {s}_{2}^{\left ( 8 \right )} \right )$, which is equivalent to the ED between the transmitted symbols at $\mathrm {LED}_2$. In this way, all of the intra-ED constraints with the same transmitted symbols at $\mathrm {LED}_1$ can be reduced to just one constraint. On the other hand, the inter-ED constraints that are the ED between two superposed constellation points $\mathbf {r}^{\left ( i \right )}$ and $\mathbf {r}^{\left ( i' \right )}$ belonging to $\mathcal {S}$ composed of different symbols from all of the LEDs can be considered as nontrivial. In this way, the $\sum _n{M_n}\left ( \sum _n{M_n}-1 \right ) /2$ constraints in the optimization problem $\mathcal {P}_2$ can be reduced to $\,\,\sum _{k=0}^{N-2}{\left [ \prod _{l=0}^k{\left ( M_{N-l} \right ) ^2\times M_{N-k-1}}\left ( M_{N-k-1}-1 \right ) \right ]}$ and comprises both the inter-ED and intra-ED constraints.
4.2 Pre-defined shaped sub-constellations
As mentioned before, the unknown vector $\mathbf {t}$ has a total of $2\sum _{{n}=1}^{{N}}{M_{{n}}}$ variables that becomes obstructively large as the number of LEDs and the constellation size increase. For example, for $N = 3$, a 64-point constellation with all of the LEDs using 4-QAM constellation, $\mathbf {t}$ contains 24 variables. Besides, the approximated optimization problem with a large number of variables becomes ineffective since the random initial vectors of $\mathbf {t}$ pose a huge degree of uncertainty in the solving process, thereby leading to a poor quality solution and degradation in the system performance. Therefore, to further reduce the complexity cost on the optimization problem-solving process and improve the optimal solutions, we further modify the optimization problem using pre-defined shaped sub-constellations for each LED. Figure 2 shows examples of various sub-constellation types that can be employed at the LEDs. As illustrated in the simulation results, comparisons are made between the pre-defined shaped sub-constellation QAM combinations, such as square-shaped (sQAM), rectangular-shaped (rQAM), and circular-shaped (cQAM). A comparison with conventional and the aforementioned proposed high complexity method clearly shows that the pre-defined shape constellation of the LED helps to reduce the complexity cost of the optimization problem-solving process while significantly improving the quality of the resulting constellation.
Fig. 2. Sub-constellation examples: (a) square-shaped (4-sQAM), (b) rectangular-shaped (8-rQAM), and (c) circular-shaped (8-cQAM).
4.2.1 Example: N = 2, M = 32 using rQAM constellations
In this scenario, the transmitter is equipped with two LEDs, $\mathrm {LED}_1$ and $\mathrm {LED}_2$, with rQAM constellations, denoted as $M_1$-rQAM and $M_2$-rQAM, respectively. For example, 4-rQAM at $\mathrm {LED}_1$ and 8-rQAM at $\mathrm {LED}_2$ can be employed to obtain a 32-QAM superposed constellation at the receiver. The unknown 4-rQAM constellation at $\mathrm {LED}_1$ can be denoted as $\mathcal {S}_1=\left \{ \left ( x_{1}^{\left ( 1 \right )},y_{1}^{\left ( 1 \right )} \right ) ,\left ( x_{1}^{\left ( 1 \right )},-y_{1}^{\left ( 1 \right )} \right ) ,\left ( -x_{1}^{\left ( 1 \right )},y_{1}^{\left ( 1 \right )} \right ) ,\left ( -x_{1}^{\left ( 1 \right )},-y_{1}^{\left ( 1 \right )} \right ) \right \}$ while 8-rQAM at $\mathrm {LED}_2$ can be denoted as $\mathcal {S}_2=\Big \{\left ( 3x_{2}^{\left ( 1 \right )},y_{2}^{\left ( 1 \right )} \right ) ,\left ( 3x_{2}^{\left ( 1 \right )},-y_{2}^{\left ( 1 \right )} \right ) ,\left ( x_{2}^{\left ( 1 \right )},-y_{2}^{\left ( 1 \right )} \right ) ,\left ( -x_{2}^{\left ( 1 \right )},-y_{2}^{\left ( 1 \right )} \right ) ,\left ( -3x_{2}^{\left ( 1 \right )},-y_{2}^{\left ( 1 \right )} \right )$,
$\left ( -3x_{2}^{\left ( 1 \right )},y_{2}^{\left ( 1 \right )} \right ) ,\left ( -x_{2}^{\left ( 1 \right )},y_{2}^{\left ( 1 \right )} \right ) ,\left ( x_{2}^{\left ( 1 \right )},y_{2}^{\left ( 1 \right )} \right ) \Big \}$. Consequentially, $\mathbf {x}_{\mathrm {sup}}$ can be expressed as $\mathbf {x}_{\mathrm {sup}}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{1}^{\left ( 1 \right )}\,\,-x_{1}^{\left ( 1 \right )}\,\,-x_{1}^{\left ( 1 \right )}\,\,3x_{2}^{\left ( 1 \right )}\,\,3x_{2}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,-x_{2}^{\left ( 1 \right )}\,\,-3x_{2}^{\left ( 1 \right )}\,\, -3x_{2}^{\left ( 1 \right )}\,\,-x_{2}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )} \right ] =\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )} \right ]$ $\left [ \begin {matrix} \mathbf {q}_{x}^{\left ( 1 \right )} & \mathbf {0}_{1\times M_2}\\ \mathbf {0}_{1\times M_1} & \mathbf {q}_{x}^{\left ( 2 \right )}\\ \end {matrix} \right ] , $ where $\mathbf {q}_{x}^{\left ( 2 \right )}=\left [ 3\quad 3\quad 1\quad -1\quad -3\quad -3\quad -1\quad 1 \right ]$ and $\mathbf {q}_{x}^{\left ( 1 \right )}=\left [ 1\quad 1\quad -1\quad -1 \right ]$. Similarly, $\mathbf {y}_{\mathrm {sup}}$ can be expressed using just two unknown variables: $\left [ y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )} \right ]$ with $\mathbf {q}_{y}^{\left ( 1 \right )}=\left [ 1\quad -1\quad 1\quad -1 \right ]$ and $\mathbf {q}_{y}^{\left ( 2 \right )}=\left [ 1\quad -1\quad -1\quad -1\quad -1\quad 1\quad 1\quad 1 \right ]$. Consequently, $\mathbf {t}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )} \right ] \mathbf {U}$, where $\mathbf {U}=\left [ \begin {matrix} \mathbf {q}_{x}^{\left ( 1 \right )} & \mathbf {0}_{1\times M_2} & \mathbf {0}_{1\times M_1} & \mathbf {0}_{1\times M_2}\\ \mathbf {0}_{1\times M_1} & \mathbf {q}_{x}^{\left ( 2 \right )} & \mathbf {0}_{1\times M_1} & \mathbf {0}_{1\times M_2}\\ \mathbf {0}_{1\times M_1} & \mathbf {0}_{1\times M_2} & \mathbf {q}_{y}^{\left ( 1 \right )} & \mathbf {0}_{1\times M_2}\\ \mathbf {0}_{1\times M_1} & \mathbf {0}_{1\times M_2} & \mathbf {0}_{1\times M_1} & \mathbf {q}_{y}^{\left ( 2 \right )}\\\end {matrix} \right ]$. Finally, the constraint (8b) can be expressed as $2\mathbf {\bar {t}}^{\left ( k \right )}\mathbf {\bar {Q}}^{\left ( i,i' \right )}\mathbf {\bar {t}}^{\mathrm {T}}-\mathbf {\bar {t}}^{\left ( k \right )}\mathbf {\bar {Q}}^{\left ( i,i' \right )}\left ( \mathbf {\bar {t}}^{\left ( k \right )} \right ) ^{\mathrm {T}}\ge q, \forall i\ne i',$ where $\mathbf {\bar {Q}}^{i,i'}=\mathbf {UQ}^{i,i'}\mathbf {U}^{\mathrm {T}}$ and $\mathbf {\bar {t}}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )} \right ]$. This obviously helps to alleviate the computational cost as the number of unknown values is just 4 instead of 24.
4.2.2 Example: N = 2, M = 32 using 4-rQAM and 8-cQAM constellations
A 4-rQAM constellation is used at $\mathrm {LED}_1$ and an 8–cQAM constellation is used at $\mathrm {LED}_2$. Similarly, we have $\mathbf {x}_{\mathrm {sup}}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{1}^{\left ( 1 \right )}\,\,-x_{1}^{\left ( 1 \right )}\,\,-x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,0 -x_{2}^{\left ( 1 \right )}\,\,0 x_{2}^{\left ( 2 \right )}\,\,-x_{2}^{\left ( 2 \right )}\,\,-x_{2}^{\left ( 2 \right )}\,\,x_{2}^{\left ( 2 \right )} \right ] =\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 2 \right )} \right ]$ $\left [ \begin {matrix} \mathbf {q}_{1x}^{\left ( 1 \right )} & \mathbf {0}_{1\times 4} & \mathbf {0}_{1\times 4}\\ \mathbf {0}_{1\times 4} & \mathbf {q}_{2x}^{\left ( 1 \right )} & \mathbf {0}_{1\times 4}\\ \mathbf {0}_{1\times 4} & \mathbf {0}_{1\times 4} & \mathbf {q}_{2x}^{\left ( 2 \right )}\\\end {matrix} \right ]$ with $\mathbf {q}_{1x}^{\left ( 1 \right )}=\left [ 1\quad 1\quad -1\quad -1 \right ]$, $\mathbf {q}_{1x}^{\left ( 1 \right )}=\left [ 1\quad 0\quad -1\quad 0 \right ]$, and $\mathbf {q}_{1x}^{\left ( 2 \right )}=\left [ 1\quad -1\quad -1\quad 1 \right ]$. Similarly, $\mathbf {y}_{\mathrm {sup}}$ can be expressed using just three unknown variables: [$y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 2 \right )}$] with $\mathbf {q}_{1y}^{\left ( 1 \right )}=\left [ 1\quad -1\quad 1\quad -1 \right ]$, $\mathbf {q}_{2y}^{\left ( 1 \right )}=\left [ 0\quad 1\quad 0\quad -1 \right ]$ , and $\mathbf {q}_{2y}^{\left ( 2 \right )}=\left [ 1\quad 1\quad -1\quad -1 \right ]$. Consequently, the optimization problem $\mathcal {P}_2$ can be reformed using a similar form where $\mathbf {\bar {t}}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 2 \right )}\,\,y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 2 \right )} \right ]$.
4.2.3 Example:N = 3, M = 64 using 4-rQAM constellations
Similar to the previous scenario, when $N = 3$ and $M = 64$, we assume that all three LEDs use 4-rQAMs in the transmission. For example, with $\mathrm {LED}_1$, the selected constellation is $\mathcal {S}_1=\left \{ \left ( x_{1}^{\left ( 1 \right )},y_{1}^{\left ( 1 \right )} \right ) ,\left ( x_{1}^{\left ( 1 \right )},-y_{1}^{\left ( 1 \right )} \right ) ,\left ( -x_{1}^{\left ( 1 \right )},y_{1}^{\left ( 1 \right )} \right ) ,\left ( -x_{1}^{\left ( 1 \right )},-y_{1}^{\left ( 1 \right )} \right ) \right \}$. Consequently, the unknown vector can be denoted as $\mathbf {\bar {t}}=\left [ x_{1}^{\left ( 1 \right )}\,\,x_{2}^{\left ( 1 \right )}\,\,x_{3}^{\left ( 1 \right )}\,\,y_{1}^{\left ( 1 \right )}\,\,y_{2}^{\left ( 1 \right )}\,\,y_{3}^{\left ( 1 \right )} \right ]$.
5. Numerical results
The performance of the proposed constellation was evaluated via simulation studies. Similar to previous researches [10,11], we assume that the channel coefficients are the same between all of the LEDs and the receiver. A conventional constellation was utilized with optimal power allocation coefficients at the LEDs while the proposed ones were employed without the need for power allocation. More specifically, both the conventional and proposed low complexity constellations were built using the resulting optimal vector $\mathbf {t}$ from the optimization solving process. With each of 50 random initial vectors $\mathbf {t}^{\left ( 1 \right )}$, the solver iteratively solved $\mathcal {P}_2$ within 5 loops. Moreover, $I_{\max }=1$ and $P_{\max }=1$ are used in the constrains of the optimization problems. Last, the optimal $\mathbf {t}$ was the one that gave the largest ED value.
Fig. 3. Superposed 32-point constellation configurations using two LEDs: (a) proposed constellation at two LEDs without pre-defined shaped {($\mathrm{LED}_1$: 4-QAM (red points), $\mathrm{LED}_2$: 8-QAM (green points))}, (b) proposed with pre-defined shaped (4-QAM, 8-QAM), (c) conventional (4-sQAM, 8-sQAM), (d) conventional (4-sQAM, 8-cQAM), (e) proposed with pre-defined shaped (4-rQAM, 8-cQAM), (f) proposed with pre-defined shaped (4-rQAM, 8-rQAM).
The resulting constellations for the case of $N = 2$ and $M = 32$, where the cardinalities of $\mathcal {S}_1$ and $\mathcal {S}_2$ are 4 and 8, respectively, are shown in Fig. 3. With the case of proposed method without pre-defined shaped sub-constellations, the constellations of the individual LEDs are shown in Fig. 3(a), while the resulting superposed constellation at the receiver side is shown in Fig. 3(b). It is clear that with a large number of constraints and without any restraint on the shape of the sub-constellations, it is difficult to find an acceptable constellation that can provide a sufficiently large minimum ED. This issue was confirmed later by the error probability of the constellations using this method being worse than the proposed pre-defined shaped sub-constellations. On the other hand, the conventional superposed constellations of 4-sQAM at $\mathrm {LED}_1$ and 8-sQAM at $\mathrm {LED}_2$ (Fig. 3(c)), and 4-sQAM at $\mathrm {LED}_1$ and 8-cQAM at $\mathrm {LED}_2$ (Fig. 3(d)), had optimal power allocation coefficients at the two LEDs. Finally, the constellations from the proposed low complexity method of 4-rQAM at $\mathrm {LED}_1$ and 8-cQAM at $\mathrm {LED}_2$, and 4-rQAM at $\mathrm {LED}_1$ and 8-rQAM at $\mathrm {LED}_2$, are shown in Figs. 3(e) and 3(f), respectively.
A minimum ED value comparison between the conventional constellations and the proposed ones is summarized in Table 1. It can be seen that the proposed low complexity method with the pre-defined shape of 4-rQAM and 8-cQAM obtained the largest minimum ED values, while those of the proposed method without the pre-defined shapes provided relatively small. Besides, the proposed low complexity method with 4-rQAM and 8-rQAM achieved a similar minimum ED value to the conventional one while the conventional 4-sQAM and 8-rQAM obtained the worst minimum ED value. This means that the choice of the shape of the constellation for the individual LEDs matters as it decides the minimum ED values.
Table 1. Minimum ED comparison of the 32-point superposed constellations.
The SER performances of the various configurations detailed in Table 1 are exhibited in Fig. 4, in which it is evident that the constellation combination with the largest minimum ED value had the best SER performance. While the proposed low complexity constellation with a 4-rQAM at $\mathrm {LED}_1$ and 8-cQAM at $\mathrm {LED}_2$ achieved the best performance, the proposed (4-rQAM, 8-rQAM) and conventional (4-sQAM, 8-sQAM) configurations gave performances with SNR gaps similar to the best one at around 1 dB. Moreover, it is not surprising that the proposed method without a pre-defined shape constellation obtained a poor SER result, although a little better than the conventional (4-sQAM, 8-rQAM) and conventional (4-sQAM, 8-cirQAM) configurations.
The minimum EDs for the scenario of 3 and 4 LEDs at the transmitter are reported in Table 2. The first four entries are for 3-LED systems used to generate a 64-point superposed constellation of at the receiver. Various sub-constellations were used at each LED. For example, the best conventional constellation was 3 4-sQAMs at 3 LEDs while the proposed one can use constellations range from binary phase-shift keying (BPSK) to an 8-QAM to ensure that the resulting constellation would have 64 distinguishable superposed points at the receiver side. In a similar case with a system of 4 LEDs, a 128-point superposed constellation was generated at the receiver. For both the 64-and 128-point constellations, the choice of pre-defined shape constellation at the transmitter dominated the performance of the superposed constellations at the receiver, which was similar to the previous cases with two LEDs. The best constellations were still the low complexity ones involving cQAMs.
Table 2. Minimum ED comparison of 64- and 128-point superposed constellations.
Finally, Figs. 5(a) and 5(b) exhibit the SER performances for the 64-point and 128-point constellations, respectively. Similar to the 32-point constellations, the SER performance with a high SNR level closely followed the minimum ED values: constellations with the largest minimum ED values had the lowest error probability whereas the one with the smallest minimum ED value (the proposed high complexity method) had the worst performance in comparison with others. Moreover, the findings in [12] indicate that square-shaped sub-constellations of LEDs in combination with optimum power allocation coefficients can provide a good superposed constellation. Nevertheless, it is clear that with the help of pre-defined formed sub-constellations, solving the optimization problem can generate constellations with better performance. Hence, the circle-shaped constellation is a potential candidate to construct excellent superposed constellations for many scenarios with various numbers of LEDs and constellation sizes.
Fig. 5. SER comparison of a (a) 3-LED system with 64-point and (b) 4-LED system with 128-point superposed constellations.
6. Conclusion
We proposed a novel method to design sub-constellations for the superposed constellation scheme of a MISO VLC systems that can be regarded as an alternative solution to achieve high-order modulations. We solved the optimization problem to determine the sub-constellations with the largest minimum ED values of the superposed constellation at the receiver under power constraints. From the simulation results, a significantly larger SNR gap could be achieved using pre-defined shaped sub-constellations at the LEDs than with traditional optimal power allocation superposed constellations.
Funding
National Research Foundation of Korea (NRF-2019R1A2C1005920).
Acknowledgments
Both authors discussed the contents of the manuscript and contributed to its presentation. M.L.T. designed and implemented the proposed scheme, analyzed the simulation results, and wrote the paper under the supervision of S.K.
Disclosures
The authors declare no conflicts of interest.
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