Constellation Design for Single Photodetector Based CSK With Probabilistic Shaping and White Color Balance

Research and development of Li-Fi (Light Fidelity) in Internet-of-things (IoT) has created significant interest among researchers. The low-cost version of standard color shift keying (CSK) modulation scheme having single photodetector (PD) at the receiver can become an alternative to on-off keying (OOK) in IoT sensor networks as an $M$ -ary CSK modulation scheme provides $log_{2}M$ times more data rate than that of OOK. This work revolves around optimizing the constellation points of single PD based CSK (CSK-1PD) to achieve a more power-efficient modulation scheme. The optimization has been done with and without average white tone constraint for uniform and non-uniform (exponential, Maxwell-Boltzmann, Pareto) source distributions. The constellation design strategy is based on maximizing the minimum distance among the differently distributed symbols. In the case of non-uniform distributions, the constellations points have been optimized by geometrically shaping the points based on their probabilistic shaping (PS). The optimized constellation points (OCPs) of CSK-1PD without white tone constraint provide higher signal-to-noise ratio (SNR) gain at forward error correction (FEC) limit symbol error rate (SER) as compared with strict white tone constraint. Further, the strict white tone constraint has been relaxed by considering the whole white light region in the CIE 1931 chromaticity diagram as white tone. The OCPs for uniform data with relaxed white tone constraint achieves SNR gain similar to without white tone constraint. However, for probabilistically shaped symbols, the SNR gain achieved from OCPs improves from strict white tone constraint but remains less than the without white tone constraint. A novel method to utilize an additional RGB LED at the transmitter side has been proposed and designed to maintain any white tone light in the white light region without degrading the maximum achieved SNR gain.

communication along with illumination for indoor environments in buildings. CSK maintains a constant instantaneous luminance across each constellation, preventing intensity flicker and sudden flux of currents in LED devices. Thus, CSK reduces the imminent human health complications, such as nausea and epilepsy, related to light intensity fluctuations [5], [6]. The proposed work optimizes constellation points for single photodetector based CSK (referred to as CSK-1PD).
A. BACKGROUND: CSK-1PD Fig.1 shows the block diagram of single photodetector based CSK (CSK-1PD) modulation scheme in VLC [7]. CSK-1PD utilizes the property of photodetector (PD) to generate specific electrical current to an incident optical power of a given wavelength. The electrical current s generated at the PD is linearly related to the transmitted optical LED power and the PD responsivity ρ at each red, green and blue wavelength of the RGB LED as shown in (1). The optical power of red, green, and blue LED lights are denoted by P r , P g , and P b , respectively.
The arithmetic operator · in (1) defines the inner product between the transmitted RGB optical power vector, p = [P r , P g , P b ] T such that P r + P g + P b = P T , satisfying the constant output power criteria. The PD responsivity vector, ρ = [ρ(red), ρ(green), ρ(blue)] where, the vector ρ = [0.42, 0.32, 0.22] has been obtained from the response curve of a practical PD (PDA36A2) from Thorlabs [8] for the central wavelength of red (640 nm), green (530 nm) and blue (465 nm) light respectively. PDA36A2 detects light signals ranging from 350 to 1100 nm and comes with an inbuilt gain amplifier. The power received at the PD is affected by the channel gain (H ) and noise w. The proposed work considers the most accepted and widely used indoor VLC channel gain, i.e., DC optical channel gain with additive white Gaussian noise (AWGN) at the receiver PD [9], [10]. The received electrical current s rec at the output of the PD is represented through equation (2).
The constellation symbols of M -CSK-1PD at the transmitter side can be represented by RGB optical powers whereas at the receiver side it is physically realized with electrical currents and denoted by symbols (s i , i ∈ 1, 2, . . . M ). M is the number of different symbols. The receiver performs symbol detection in M -CSK-1PD by computing the minimum Euclidean distances between the received and the transmitted symbols in the form of electric currents. One of the primary criteria of any VLC modulation schemes is to maintain illumination with white tonality. The average color obtained from the CSK constellation symbols may not always lie inside the white region of the CIE 1931 chromaticity diagram, which generates color unbalances perceived by the human eye [7], [11]. IEEE 802.15.7 standard has already proposed the adoption of a pseudo-random sequence, known as scrambler to overcome the color unbalance problem. The scrambler has been used at the transmitter in [7] to maintain an average white tone output light in CSK-1PD for uniform input data. The actual input data can be obtained at the receiver by a descrambler.

B. RELATED WORK
The growing interests on designing and optimizing the constellation points mostly in terms of reduced received bit/symbol error rate for standard CSK over the past few years have led to the emergence of diverse techniques in the literature. Most of the constellation design problems aim to improve color balancing limitations or data rate transmissions. The authors in [11] proposed a genetic algorithm to design the constellation symbols for multiple LED transmitters at different wavelengths in the VLC system. Drost and Sadler are the first ones to investigate billiards algorithms for the design of CSK constellation points in [12]. Later, in paper [13], they compensated the channel distortions by designing a framework for optimizing the constellation points for VLC systems employing multiple emitters at different wavelengths. In [14], Monteiro et al. designed constellation points for CSK using interior-point methods and extended it to higher-order CSK constellations in [5]. In [15], the authors presented a power-efficient high dimensional constellation design scheme for the standardized TLEDs. To the best of the authors' knowledge, the constellation design of CSK-1PD has been performed solely in [7] till date, which has stated numerical optimization solution for designing CSK-1PD constellations points for uniform input data.

C. MOTIVATION AND PROPOSED WORK
CSK-1PD is a low-cost color space-based modulation scheme [7], which can be exploited in Li-Fi (light fidelity) in IoT indoor scenario. CSK-1PD modulation scheme can be utilized in implementing IoT sensor networks in a smart home or hall scenario for communication as well as illumination, where resources are constrained and thus, low-cost and power-efficient systems are preferred. CSK-1PD constellations can be designed for maximum power efficiency in terms of low signal-to-noise ratio (SNR) required at forward error correction (FEC) limit symbol error rate (SER) within its optimization constraints.
Limited research work has motivated the authors to design optimized constellation points for CSK-1PD modulation scheme. The proposed work consists of designing optimized constellation points for CSK-1PD with uniform input data and then with three different non-uniform probabilistically shaped (exponential, Maxwell-Boltzmann, and Pareto) input data. Additionally, CSK-1PD with the four different input data types are optimized considering white color balance. Finally, a novel method to maintain any desired white tone light as CSK-1PD modulated output light has been proposed.

D. CONTRIBUTION
The key contribution of the proposed work can be enumerated as follows: 1) This article proposes a rigorous framework to design and optimize the constellation points for uniform and non-uniform source distribution in CSK-1PD to reduce the received SER. 2) The constellation points of non-uniformly distributed input symbols in CSK-1PD are optimized by geometrically shaping (GS) the symbols based on their preceding probabilistic shaping (PS). The consecutive PS-GS technique to optimize the constellation points has been scantly investigated hitherto. 3) A relaxed white tone constraint has been formulated by considering the whole white light region in CIE 1931 chromaticity diagram to maintain a white color balance with reduced SER. 4) A novel technique to maintain any desired average white tonality light without degrading the SNR gain by inserting an extra RGB LED at the transmitter.
The remainder of this article is structured as follows; Section II introduces the constellation optimization problem and proposes the objective functions. Section III mentions the constraints of the optimization problem statements with different aspects. The novel solutions to maintain average white tone have been proposed in Section IV. Section V analyzes and discusses the results. Finally, Section VI presents the conclusions.
Notations: The vector and the matrix are denoted as x and X respectively. The rows of matrix X is denoted as X R . The objective function for optimization is denoted by F(X). The set of pairwise Euclidean distance is denoted by d k . The average of a scalar is denoted by x.

A. PROBLEM FORMULATION
In this article, the constellation optimization problem is formulated by adopting the widely accepted constellation design problem. It defines a constrained optimization strategy that maximizes the minimum Euclidean distance among the constellation symbols (s i,j ) while maintaining some of the aspects such as constant output power, average white color balance with and without PS. The constellation points are designed for the minimization of SER at the receiver. The optimization problem, as shown in (3), becomes a max-min problem, and therefore the objective function becomes a non-convex problem [16]. The pairwise Euclidean distance of symbols s i and s j is denoted by |s i − s j |. The objective function is both non-convex and non-differentiable, making it difficult to solve analytically [7], [14]. Further, the minimum function of the objective function is discontinuous, which makes it incompatible with efficient, gradient-based techniques [11]- [14]. The M = 4 constellation symbols (s i,j ) have been investigated in the paper for CSK-1PD which are in the form of electrical currents.

B. OBJECTIVE FUNCTION
To solve this non-convex problem, researchers have suggested many approaches; one of them is the differential approximation which removes the non-differentiable nature by using well known ''log sum-exponentials'' continuous approximation [16]. The approximated objective function in (4) still remains non-convex; however, optimization techniques based on the derivation of the Hessian of the objective function can find the locally optimal symbol sets. For non-convex problems, commercially available optimization algorithms exclusively discover local minima [14]. However, the likelihood of realizing a global optimum is improved by optimizing the objective function multiple times over a small set of random starting points and tuning the β parameter where β determines the accuracy of the approximation between (3) and (4) as discussed in [11]. The PS of input data already provides SNR gain in optical fiber and VLC systems [17]- [19]. The optimized constellation points obtained from (4) for uniform input data can be further optimized for non-uniform input data by adapting the concept of hybrid probabilistic shaping and geometric shaping [18], [21]. The PS CSK-1PD constellation points are designed in the confined region of constellation symbols according to the probability of occurrence of symbols. Therefore, the PS constellation symbols are shifted geometrically according to the known PS of the symbols to get minimum SER. Hence, the objective function in (4) needs to be modified to include the probability of symbols.
The PS in constellation symbols follows optimum probability order obtained in [19], employing the highest probabilities to the minimum error causing pair of symbols. The probability of occurrence of symbols p(s) for non-uniform source distributions has been calculated at unity achievable information rate (AIR) to match it with the uniform source distribution AIR, to retain a consistent AIR in results for a fair comparison [19]. The plot of probability distribution function (PDF) of 4-CSK-1PD for uniform, exponential, Maxwell-Boltzmann, and Pareto source distributions at unity AIR has been presented in Figs. 2 (a), 2 (b), 2 (c), and 2 (d), respectively. The optimum probability order for CSK-1PD in an AWGN channel follows p( When the symbols are non-equiprobable the detection threshold for received signal in an AWGN channel shift from the midway point between two symbols based on the logarithmic ratio of the probability of occurrence of symbols [20]. Henceforth, the pairwise Euclidean distances of non-equiprobable symbols get affected by the logarithmic ratio of the likelihood of occurrence of symbols as shown in (5). The logarithmic ratio alters the pairwise Euclidean distances so that the minimum of log sum-exponentials procures the pairwise Euclidean distance of minimum error causing pair of symbols.

C. OPTIMIZATION ALGORITHM
As stated in section II. B, finding the analytical solution of the non-convex optimization problem becomes a difficult task, so usually, this type of optimization problem is solved with numerical optimization algorithms [22]. The continuous approximated objective functions can be solved using commercially available algorithms such as MATLAB's fmincon, which finds the minimum of a constrained nonlinear multivariable function. The maximum of a function can also be found by fmincon, by taking negative of the functions in (4) and (5). This work utilizes MATLAB's fmincon based on an interior-point algorithm to obtain the optimized constellation points (OCPs) of CSK-1PD with the constraints (explained in the upcoming section III). The numerical optimization algorithm process is summarized in Algorithm 1. The objective function F(X), has been formulated for constellation points in the form of matrix X. The set of pairwise Euclidean distances is denoted by (6), which can be obtained from matrix X and a sparse matrix formed out of ρ.
The tuning parameter β in Algorithm 1 produces poor result for smaller values, whereas, for arbitrarily large values, the large gradients cause the algorithm to converge slowly. Therefore, the objective function applies the interior point method for multiple iterations, starting with a smaller β and increasing β in each iteration [14]. This approach provides a reliable approximation without slow convergence and with each successive β the optimal solution of the previous β acts as the starting point for the next. However, the solution obtained at each successive β may not always be feasible with respect to the linear inequality constraints. Therefore, in Algorithm 1, the basic concept of the genetic algorithm to produce good solutions by avoiding the trajectory towards bad solutions has been used [23]. It helps in avoiding the non-feasible solutions. The minimum Euclidean distance (MED) constraint updates itself for every feasible solution and thus along with genetic algorithm reduces the number of iterations for the optimization process with β. The SER of feasible solutions at design SNR is calculated until the occurrence of the first non-feasible solution to form a potential population set P SNR . The choice of design SNR is medium to high SNRs as the SER expression with union upper bound is fairly tight in those SNRs [24]. Further, the SER is greater than 0.5 in SNRs below 15 dB, optimizing points for lower SNRs may not completely optimize the points for higher SNRs. This work is looking for SNR gain at FEC limit SER. Hence, the design SNR is 15 dB, the average of SNRs at worst, and FEC limit SER. For faster computation the theoretical CSK-1PD SER expression from [19] has been considered in the algorithm, as shown in (7). where, The apriori probability of symbol s i is represented by p(s i ) and N 0 is the noise variance. At the occurrence of non-feasible solution the best fitting solution, i.e., the constellation points providing minimum SNR from the potential population set is chosen as the starting point. The solutions obtained by the numeric algorithm depend on the initial estimate of the constellation points (X0) used to start the optimization process [21]. The random starting points must be a strictly feasible constellation points (as shown in Table. 1), fulfilling the optical power and geometric constraint with strict inequality range constraint (as explained in section III). The constellation symbols are represented by s i , where i = {1, 2, 3, . . . M }. Algorithm 1 can be used for optimizing constellations of any size with arbitrary convex constraints.

III. CONSTRAINTS FORMULATION
The constellation design in CSK-1PD scheme for uniform input data without white color balance requires optical power based constraint, geometric and range constraint. Additionally, it requires MED constraint, which updates with updated starting constellation points X0 in the optimization process in Algorithm 1. For non-uniform input data; optical power, geometric and range constraints remain the same, whereas the PS of symbols modifies the MED constraint as shown in section III-C. Later, to maintain white-tone output light in both uniform and non-uniform source distributions, color constraints have been discussed.

A. OPTICAL POWER CONSTRAINT
The signal envelope in CSK scheme has a non-varying power as it modulates data for constant output power. Constellation points have been designed to keep the desired total optical Algorithm 1 Numerical Optimization Algorithm Initialize: Random initial constellation points, X0, satisfies power, geometric and range constraints.
subject to Constraints. if X obtained satisfies the constraints to become a feasible solution then X0 ← X power, P T , as a positive non-zero constant. The optical powers P r , P g , and P b may take any non-negative value less than its peak values (9). P pr , P pg , and P pb denote the peak optical powers of red, green, and blue LEDs, respectively. The peak powers and P T are considered 1 to work on normalized values. The constant optical power constraint is analogous to the constant energy condition of many optimization processes. The optical power constraint, as shown in (10) is a mandatory constraint for designing constellation points in CSK-1PD.

B. GEOMETRIC AND RANGE CONSTRAINT
The geometric condition for CSK-1PD symbols as shown in (11) is that each vector formed by the difference of any two symbols should not be orthogonal to the responsivity vector ρ (as discussed in section I-A). It is simply to avoid same electric current s at the receiver, from any of the two constellation points so as to have a distinct set of VOLUME 8, 2020 symbols.
The region of CSK-1PD constellation points in terms of normalized electrical current is limited to one dimension (1-D) only as seen in Fig. 3. The 1-D region range depends on the responsivity of respective red, green and blue lights on the photodetector, as shown in (12). The maximum and minimum range of a constellation point (in terms of normalized electrical current) remains the same in an RGB LED as the highest, and lowest responsivity corresponds to red, and blue colour LEDs, respectively. Therefore, the values of all possible constellation points lie between the normalized electrical currents defined by the highest and lowest responsivities. The electrical symbols corresponding to the two extremes of responsivity values provides the maximum Euclidean distance between any two symbols in the constellation set. Hence, the additional range constraint, as shown in (13) has been included in the optimization process for complete range constraint. Constraint in (13) can be further reduced to (14), when the matrix X0 and X have rows always arranged in increasing current fashion from top to bottom as seen in Table 1. The matrix arrangement has been kept the same in all the simulations and results. The constraints have been expressed as a set of linear equalities and inequalities as per fmincon solver.
The vector dot product changes to matrix dot product on considering all the four constellation points in a matrix, i.e., X0. X0 R 1 , and X0 R 4 represent the first, and fourth row of the matrix X0. The results are simulated in matrix notation to obtain all four symbols in one go in the matrix notation X o .

C. MINIMUM EUCLIDEAN DISTANCE CONSTRAINT
The optimization process takes random initial constellation points, as shown in Table 1. The minimum Euclidean distance between any two symbols among all the symbols in the initial random constellation points, X0, is taken as a constraint for next optimization iteration (t). The minimum Euclidean distance (MED) constraint is updated according to the successive optimized constellation points (X) in (t + 1) iteration of β for much faster convergence. The only difference in this constraint for uniform (in (15)) and non-uniform (in (16)) input data is the logarithmic factor.  [25] has been shown in Fig. 4. The color of any visible light wavelength can be represented by the chromaticity coordinates (x, y). CIE 1931 color space typically considers eleven mean white tonalities at different color temperatures for white light evaluation. Table 2, shows the chromaticity coordinates producing white tones. The constellation points consist of RGB LED optical power vector for a given symbol. The (x, y) chromaticity coordinates have been represented by (x, y) coordinates in the rest of the paper. Equation (17) provides a relation between x, y chromaticity coordinates and LED output power for a given wavelength of light. Hence, from X0 one can find the value of (x, y) in the CIE 1931 chromaticity diagram as follows where the coordinates of central wavelength of red LED light is (x r , y r ), green LED light is (x g , y g ) and blue LED light is (x b , y b ) in the CIE chromaticity color space. The central wavelength coordinates are known from the CIE 1931 color space as shown in Table 3. The CSK-1PD constellation symbols on an average should emit white colour light. The average white tone can be obtained in terms of chromaticity coordinates as where p(s i ) is the probability of symbol s i , and (x, y) are the chromaticity coordinates representing the average perceived color of the transmitted symbols. When (x, y) represents a perceived white color from any of the eleven mean white tones (refer Table 2), the constellation symbols are said to have a white color balance. In order to fix the perceived color to white tone, an average white tone constraint is required on the constellation symbols as follows {(0.418, 0.390), . . . , (0.300, 0.305)} forms the set of eleven mean white tones. The optimization is done by considering one out of eleven mean white tones at a time which makes it a strict constraint. The average white tone constraint is given by the affine equality for a color temperature of 3100 K, (x, y) = (0.418, 0.390). Similarly, optimization is done for all other white tones. The average tone of optimized constellation points without color constraint obtained and shown in the results section for different source distributions has been calculated to know the chromaticity coordinates. It is clear from Table 4, that none of them fulfills the white tone balance. Therefore, a white tone constraint is required for maintaining an average white tone in the constellation points. There is a limitation to this  white tone constraint as the OCPs with strict white tone constraint reduces the SNR gain of the system in comparison to the OCPs without white tone constraint. Further, not all the white tones can be achieved by the constellation points optimized for minimum SNR. However, the desired white tones are achieved with reduced SNR performance. Therefore, the authors have explored two alternatives to manage white tones in the optimized CSK constellation points.

IV. SOLUTIONS FOR AVERAGE WHITE LIGHT
The CIE 1931 chromaticity diagram in Fig. 4 shows a wide area for white light. The strict white tone constraint can be extended to the whole white region to achieve SNR gain close to OCPs of without white tone constraint. The white region constraint has been derived in section IV-A, which relaxes the strictness of the previous mean white tone constraint. Another way to obtain white tone output light can be the addition of another RGB LED at the transmitter side. The use of an extra RGB LED helps in maintaining white tone without even disturbing the OCPs obtained without considering the white tone constraint. The extra RGB LED method has been described in section IV-B.

A. RELAXED WHITE TONE CONSTRAINT
The strict white tone constraint considers only eleven mean white tones, so it is not possible to get optimized constellation points for all the eleven white tones. The relaxed white tone constraint regards any (x, y) coordinates falling inside the white region (see Fig. 4) as desired white. The white light region has been extracted out from the CIE 1931 chromaticity diagram in Fig. 5 for deriving the approximate mathematical model of the region. Every (x, y) coordinates in the whole white region does not give pure white light; however, (x, y) coordinates inside the white light region has been considered in designing LEDs in literature [26]- [29]. There is a trade-off between the purity of white tone and SNR performance; the focus of the proposed work is to obtain optimized constellation points which maintain the bare minimum acceptable white light.
The desired (x, y) coordinates of the OCPs lie inside the white light region, which is an irregular shape. The research VOLUME 8, 2020 literature is replete with algorithms to determine whether a point is within a polygon or not. Therefore, the most common approach is to first approximate an irregular shape into a polygon from the set of points, R forming the irregular shape. Once the irregular shape is decomposed to a polygon, deciding whether a point lies inside is the point-in-polygon (PIP) test problem. The irregular shape can be decomposed to triangles using polygon triangulation method [30]. The triangulation method chooses a subset of points, P from the set R to approximate the white region as a mesh of triangles. All the P points form the vertices of the white light region, hence with the triangulation method, the white region can be approximated to an arbitrary polygon with few points. In polygonal representation, the curved shapes are represented using tangential approximations in the form of straight lines, which ultimately depends on the number of vertices considered. Therefore, there is always an approximation error (a e ), which can be realized in terms of percentage of area left uncovered by the polygon approximation. Since the white region is in two dimensions (2-D) only, there are a number of algorithms for the 2-D PIP problem. The authors opted for the efficient standard algorithm in paper [31], which is capable of solving PIP problem for any arbitrary polygon, convex or non-convex. Further, during the optimization process, the algorithm shall be called so often, hence an accelerated algorithm that grows linearly with the number of polygon vertices, v n , is a viable choice. Therefore, the complexity of the PIP algorithm used in this work is O(v n ). The polygonal representation of white light region depends on PIP algorithm as well and thus becomes an optimization problem as follows: where a e decreases with more number of v n while the PIP algorithm complexity increases with more number of v n . The complexity and approximation error values have been normalized to [01]. They have been weighted equally in the minimization problem. The number of vertices which pro-  vides minimum complexity and area error has been obtained and shown in Fig. 6. The white light region (W) in the CIE 1931 chromaticity diagram has been approximated to a polygon in Fig. 7 with vertices, v n = 35. The relaxed white tone constraint ensures (x, y) ∈ W, and has been formulated as follows: where, x P and y P belongs to the x and y coordinates of polygon points P. The PIP algorithm returns logical 1, whenever a point is inside or on the polygon.

B. EXTRA RGB LED
Researchers have used scrambler for maintaining white tone in CSK-1PD [7]. Linear feedback shift register (LFSR) based scrambler is not very expensive; however, its complexity increases with increase in the length, l. The color balance in [7] is achieved for uniform input data at l > 20. The input data scrambled for length l > 20, provides color balance by averaging (P r , P g , P b ) to get values around (0.33, 0.33, 0.33), which corresponds to a single chromaticity coordinates (0.3330, 3718). The scrambler needs to be designed specifically for different PS input data, and a particularly designed scrambler can produce only one type of white tone. The scramblers used for this purpose are usually additive, so it must be reset by the frame synchronization bits otherwise massive error propagation happens [4]. Further, with the non-uniformity in the input data, the length in scrambler may increase, which in turn increases the complexity drastically.
In order to escape the problems mentioned above, the authors in this article have suggested the utilization of an extra RGB LED at the transmitter to maintain any white tone in the white light region without hampering the SNR gain obtained from specific OCPs for source distributions. One can design a constant color light with the extra RGB LED to mix with the RGB modulated symbols to produce the desired average white tone (x d , y d ). It provides flexibility and removes the constraint on the length or number of the input symbols. Further, RGB LEDs being inexpensive does not incur much cost to the system. The chromaticity coordinates (x e , y e ) of extra RGB LED can be obtained as where Y e and Y 1 are the luminance corresponding to extra RGB LED chromaticity coordinates, and average tone coordinates obtained from the specific OCPs for source distributions, respectively. The normalized range of luminance is between 0 to 1 [32]. (x e , y e ) can be expressed in the form of RGB LEDs optical powers by substituting (17) in (22) with the following constraint P r + P g + P b = 1, where, (P r , P g , P b ) ≥ 0 and 1 ≥ Y e > 0. (23) The computational complexity of this method depends on designing the RGB optical powers of extra LED, which forms linear equations in five variables on considering (22) and (23). There are five unknown variables with their respective upper and lower bounds and three linear equations. Therefore the linear equations are solved for different values of Y 1 to obtain (Pr, Pg, Pb) in terms of Y e and then Y e can be obtained from (23). The PD receives the combined light of RGB modulated and extra RGB LED signal. The receiver side knows the excess current generated because of the additional RGB LED. The total current generated shifts the symbols with additional electric current values, hence on subtracting the incremental current yields corresponding current of the modulated signal. It is analogous to removing the constant ambient light noise at the VLC receiver. Henceforth, with linear computational complexity at the transmitter and receiver, the concept of using an extra RGB LED can be a better option for colour rendering and balancing.

V. RESULTS AND DISCUSSION
The SNR performance gain with respect to SER for 4-CSK-1PD with and without white tone for uniform and non-uniform input data has been obtained and reasoned in the following sections. The simulation parameters have been kept same for this work as in [19] for fair comparison. The list of simulation parameters have been shown in Table. 5. The possibility of maintaining any desired white tone has been validated by finding non-negative RGB optical powers for the desired white tone in section V-C.
A. WITHOUT WHITE TONE CONSTRAINT Table. 6 shows the OCPs of all the source distributions considering all constraints as discussed in section III except white tone constraint. The SNR in dB for FEC limit SER can be observed in Fig. 8 for different source distributions utilizing their specific OCPs. It has been observed that the optimized constellation points for uniform input data (OCPu) perform better than the OCP of uniform source distribution from [7] by 0.75 dB. The OCPu results in equidistant normalized electric current symbols, which is equivalent to the analytically known global optimum placement of equiprobable symbols in pulse amplitude modulation (PAM) for minimum SER [20]. The normalized design range for CSK-1PD is just a straight line, so following the equidistant symbol spacing criteria for equiprobable symbols yields the same d i = 0.0667 for all i (check equation (8) for d i ). The OCPu achieve a minimum Euclidean distance of 0.0667 for all d i .
It has been observed that the SNR performance of non-uniform source distributions utilizing specific OCPs   obtained in this work provides improved SNR gain as compared to the work in [19], where OCPs designed in [7] for uniform source distribution was utilized. The respective SNR values have been tabulated for comparison in Table 7. In the case of specific OCPs, the constellations points are further optimized by geometrically shaping the symbols based on their probability of occurrence. Therefore, the improved performance of non-uniform distributions in case of specific OCPs is because of the consecutive PS based GS technique. It is interesting to note that with specific OCPs the performance of non-uniform source distributions approach close to each other which is again due to the specific shifting of symbols according to PS.

B. WITH WHITE TONE CONSTRAINT
The constellation points are optimized so as to get average white tone light as the transmitted output.

1) STRICT CONSTRAINT
The constellation points are obtained considering all constraints in section III along with strict white tone constraint in (19). The OCPs obtained with strict white tone are abbreviated as S-OCPs. The uniform and Pareto (non-uniform) source distribution OCPs are obtained and shown in Table. 8.  Table. 8.
The corresponding mean white tone chromaticity coordinates can be seen in the last column. The SER versus SNR plots in Fig. 9, demonstrates the impediment of considering strict white tone constraint. The SNR for uniform distribution is 31.4 dB and for Pareto is 30.2 dB at 10 −3 SER. It can be observed that the performance of both uniform and Pareto have decreased from OCPu and OCPp, respectively. The chromaticity coordinates of the four OCPs do not lie in the white region except one, and for maintaining average white tone, the optimum position of symbols in terms of SNR is changed and therefore reduces the SNR gain.

2) RELAXED CONSTRAINT
In this case, the whole white region has been considered in the white tone constraint of the optimization process. Table. 9 shows the OCPs of all the source distributions considering all constraints and the relaxed white tone constraint. The respective white tone coordinates can be seen at the last row. The SNR at FEC limit SER can be observed in Fig. 10 for different source distributions considering OCPs for relaxed white tone (R-OCPs). The SNR values for FEC limit SER have been enumerated in Table. 10 for comparison. It has been observed that SNR performance gain of uniform has achieved the optimum SNR gain of OCPu. The equiprobability of symbols in uniform distribution assist it to render ideal white color balance with optimum SNR gain. However, for non-uniform distributions, SNR gain has improved from the strict white tone constraint, whereas it is still less than the without white tone OCPs performance gain. The range TABLE 9. Optimized CSK-1PD constellation points with relaxed white tone constraint for uniform, exponential, Maxwell-Boltzmann, and Pareto distributed input data.   of choice of chromaticity coordinates has increased here tremendously, and hence the performance of the R-OCPs in comparison to S-OCPs. The non-uniformity of symbols with s 1 (p 1 in terms of RGB power) as highest probable symbol sabotage the reach to optimum SNR gain in R-OCPs. Moreover, the average white tone chromaticity coordinates of non-uniform source distributions are precisely the same, but the OCPs are different. The reason behind this comes from Table. 4, where the average tone of non-uniform distributions is close to each other. Hence, during optimization with white tone constraint, the different distributions look for a nearby valid white tone coordinates for optimized SNR gain with distinct OCPs.

C. WITH EXTRA RGB LED
The chromaticity coordinates of extra RGB LED for the vertices of the white light region (see Fig. 7) in Pareto distribution have been obtained and shown in Fig. 11 for different values of luminance Y 1 . The extra RGB LED coordinates gets skewed to remain inside the CIE chromaticity diagram. The non-negative RGB optical powers of LED fulfilling all the linear equalities and inequalities have been obtained. The RGB optical powers obtained for the extra LED can be viewed in Table. 11.

VI. CONCLUSION
This article, for the very first time, tries to optimize the constellation points of CSK-1PD with and without average white tone for uniform and non-uniform (exponential, Maxwell-Boltzmann, and Pareto) source distributions. The specific OCPs without white tone constraint gives improved SNR performance gain in all the source distributions with respect to the OCPs of [7]. However, the OCPs with strict white tone constraint maintains the white tone on an average with SNR gain reduced by 1.4 dB and 3.2 dB from without white tone constraint for uniform and Pareto distribution, respectively. The optimization process for OCPs with relaxed white tone constraint considers the whole white region in white tone constraint to boost the SNR gain. The R-OCPs for uniform input data achieves the same SNR gain (30 dB) as OCPu. However, for non-uniform, the R-OCPs still performs 1.25-1.45 dB lesser than the non-uniform OCPs without white tone constraint. Furthermore, this article proposes a novel solution to use two RGB LEDs at the transmitter, one for modulation and another for maintaining white tone. It has been observed from results that the RGB optical powers of extra RGB LED can be designed for any desired chromaticity coordinates inside the white light region retaining maximum SNR gain with specific OCPs for uniform and non-uniform source distributions.