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Abstract
In this paper, we propose a technique called joint power allocation and orientation of light-emitting diodes (LEDs) to improve the illuminance quality of an indoor visible light communication (VLC) system. The optimization problem for LEDs, considering power allocation coefficients and normal vectors, is solved through an iterative algorithm. The simulation results with various LED geometries show that the proposed scheme can achieve a lower received power variance and a higher quality factor compared to previous power allocation schemes.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recently, visible light communication (VLC) with the widespread deployment of light-emitting diodes (LEDs) has attracted a lot of attention around the world. The optical intensity is utilized to modulate the information bits, which can help alleviate the shortage of available radio frequencies. Moreover, VLC has various applications that cover a wide area and have a high demand value. Furthermore, multiple-input multiple-output (MIMO) techniques can improve the inherently limited modulation bandwidth of VLC via its spectral efficiency [1,2]. Consequently, multiple LEDs and photodiodes (PDs) have been widely employed in MIMO communication systems, thus being an attractive research topic in VLC [3,4].
On the other hand, a uniform illuminance condition is desirable in every VLC system [1]. Firstly, the inherent purpose of any indoor lighting system is to provide illuminance quality as evenly as possible while providing a dimming function with a wide range of brightness. Secondly, the uniformity of light signals around the room is also an important feature in indoor optical wireless communication scenarios since it helps ensure full coverage and improves system performance [5]. However, since it is difficult to get a line-of-sight (LOS) signal in many locations, the illuminance distribution and the uniform illumination condition is generally not taken into account in VLC research [6].
Recently, to achieve uniform irradiance for VLC systems, various techniques have been proposed [7–11]. One approach to realize a uniform illuminance condition is to specifically design the locations of the LEDs in the array. In [11], uniform illumination is achieved by using an ab initio design of an LED array. Moreover, in [8], a numerical optimization method was proposed based on a local search algorithm to design an LED array for a highly uniform illumination distribution around a room. However, specific configurations of the algorithm may lead to local extremum wherein the algorithm cannot obtain a global solution for the design problem, thus reducing the effectiveness of the proposed method. Another approach is the allocation of power for a specific LED geometry. In [7], the uniform irradiance on the receiver plane is achieved by using a simple heuristic power allocation (PA) scheme for a random LED array. Specifically, by modeling the LEDs locations using stochastic geometry with random parameters, the proposed algorithm helps to properly allocate the power for all LEDs to generate average uniform illumination. Recently, in [9], an optimal power allocation scheme was proposed. By considering the variance of the received power on the receiver plane as a metric and reforming the objective function as a convex optimization problem, uniform illuminance can be accomplished. Moreover, various LED array geometries have been considered such as square, circle-square, binomial point process (BPP) [7], and Matern type II hard-core point process (HCPP) [9] to achieve better lighting quality.
In this paper, we aim to further improve the uniform illuminance feature of a VLC system by introducing an additional degree of optimization. In particular, we form and effectively solve a convex optimization problem considering normal vectors and power allocation coefficients of LEDs. By altering the orientation of LEDs in conjunction with optimal power allocation, the variance of the received signals can be further decreased. Moreover, the proposed method can increase the quality factor of the system compared with previous scheme [9]. On the other hand, to facilitate the dimming control capacity in the system, the simulation result with various thresholds on the maximum total power of the system is also considered.
2. System model and the LEDs normal vectors
In Fig. 1, an indoor VLC system with $N$ LEDs is considered. A set of $K$ light received PDs is considered at the receiver plane. Moreover, all the parameters and properties of the system are well defined in [7,9,12]. Here, we only consider a VLC link with LOS characteristics. The channel gain between the $j$-th LED at the $i$-th location, can be simplified as [6]
In the other hand, we have
3. Problem formulation
The channel coefficient with the help of (5) can be expressed as
In the other hand, considering power allocation for the LEDs with the power allocation vector $\mathbf {w}=\left [ w_1\cdots w_N \right ] ^T$ of size $N\times 1$, the received power at the $i$-th PD can be expressed as
4. Proposed solution
It can be seen that the constraints of problem (P1) are separable. Therefore, an iterative algorithm can be used to address problem (P1) with the block variables $\mathbf {t}$ and $\mathbf {w}$ [15]. Specifically, each time we optimize one variable while the other remains fixed [16]. This leads to the following two sub-problems in each iteration.
1 The subproblem w.r.t. $\mathbf {t}$
With fixed power allocation vector $\mathbf {w}$, the variable $\mathbf {t}$ is updated by solving the following optimization problem
where the formation of (P2) can be referred to Appendix A. Moreover, by relaxing and linearizing constraint (17b) to $2\left ( \mathbf {t}_{j}^{\left ( k-1 \right )} \right ) ^T\mathbf {t}_j-\left ( \mathbf {t}_{j}^{\left ( k-1 \right )} \right ) ^T\mathbf {t}_{j}^{\left ( k-1 \right )}\geqslant 1$, the optimization problem (P2) can be solved by2 The subproblem w.r.t. $\mathbf {w}$
With fixed LED normal vectors $\mathbf {t}$, we can reform the object function in problem (P1) according to the variable $\mathbf {w}$. The variable $\mathbf {w}$ is updated by solving the following optimization problem
where the formation of (P4) can be referred to Appendix B. For clarification, the optimization algorithm is summarized in Algorithm 1, which is a two-stage iterative algorithm. The first stage is employed to solve problem (P3) to find the optimal LED normal vectors, while the second stage is employed to update the optimal LED power allocation coefficients. Each iteration of the first stage mainly consists of three steps. In the first step, where the input is the previous value of $\mathbf {w}$, the coefficient matrix $\mathbf {\Pi }$ is constructed and the optimization problem is formed and solved where the output is the vector $\mathbf {t}$. Similarly, in the second stage, the same steps are employed in solving the optimization problem (P4), where the input is the previous vector $\mathbf {t}$. The iteration is terminated when the convergence threshold is met. |
As mentioned in [9], since the variances of received power $\mathbf {t}^T\mathbf {\Pi t}$ and $\mathbf {w}^{\boldsymbol {T}}\mathbf {\Delta w}$ are always positive, $\mathbf {\Pi }$ and $\mathbf {\Delta }$ are positive-definite matrices. Moreover, since all the constraints are linear functions and hence, are convex. Thus the optimization problem (P3) and (P4) with quadratic objective functions are convex optimization problem [15]. Consequently, the optimization problem (P3) and (P4) can be efficiently solved using the CVX solver [17]. Using the solution obtained from previous iterations as the new center point, a new approximated problem can be solved. This procedure is convergent because the sequential iteration is guaranteed to reduce the objective value after each iteration. Moreover, the length of normal vectors in the relaxed constraint (18b) usually around one. Therefore, after each iteration, the normalization step would ensure $\lVert \mathbf {t}_{j}^{\left ( k \right )} \rVert =1$ without significant degradation in the quality of the solution. However, due to the relaxing and linearizing from constraint (17b) to (18b), the joint algorithm in general can only find the local solution of the problem. Therefore, the initialization of input values in the Algorithm 1 needs to be carefully considered. Moreover, through our simulation, we observe that a wise choice of initial vector $\mathbf {t}^{\left ( k \right )}$ can give a favorable solution and usually, with $\mathbf {t}_{j}^{\left ( k \right )}=\left [ \textrm {0} \quad 0 \quad -1 \right ] ^T$, the proposed method gives better performance in comparison with the PA in [9]. In the case of $\gamma \ne 1$, the simplified equation from (5) to (7) should be modified by employing the first order Taylor expansion of the channel coefficient $h_{ij}$ around $\mathbf {t}^{(k)}_j$ and expressing the received power variance in a similar manner. The modified optimization problem can be easily solved for specific $\gamma$ values. Therefore, in this paper, we consider the general case of $\gamma = 1$ which has been employed in various results [7–9,13].
5. Numerical results
In this section, the simulation results are provided for various LED geometries. The system is located inside a room with dimensions of $5\times 5\times 3$m. The receiver plane is 0.85m above the floor. The number of LEDs in each case is 16 with a semiangle of $60^\circ$. The total power of all LEDs is set to $P_{\textrm {total}}=1.1$W. Moreover, for fair comparison, a number of $K=40401$ PDs are uniformly distributed around the room as a square array of $201\times 201$ with each side is composed of 201 PDs. The distance between PDs is $0.025$m, the field of view of PD is $60^o$, and the physical area of the receiver PD is $1\textrm {cm}^2$. For comparison, the LED arrays of square, circle-square, BPP, and HCPP are considered with uniform power allocation and the power allocation scheme in [9]. In Fig. 3, the SNR profiles of different geometries for equal power allocation, the power allocation method in [9], and the proposed method are shown. It can be observed that the joint power allocation and LED orientation can achieve more uniform light since it provides an additional degree of optimization.
The advantage of the joint optimization solution can be seen in Table 1 and Table 2, where the variance of the received power and the quality factor of the schemes with various geometries are presented. The proposed scheme achieves excellent performance in terms of the variance of received power. Meanwhile, the quality factors of the proposed scheme are higher compared to the previous schemes. Moreover, as mentioned in [9], the circle-square geometry and HCPP yield the best quality factors, while the BPP geometry yields the lowest quality factor in both methods. On the other hand, the dimming capability is a very important feature of any VLC system. In the different conditions of lighting intensity required, the total power of the system $P_{\textrm {total}}$ is variously adjusted, and is generally many times higher than the upper threshold of individual LED power. In this section, we take into consideration the maximum value of individual LED power. By adding the constraint $\mathbf {w}\leqslant P_{\textrm {LED}}$ into (P4), we have the modified optimization problem
Table 1. Variance comparison of the received power of the proposed scheme
Table 2. Quality factor comparison of the proposed scheme
Consequently, the Algorithm 1 iteratively solves (P3) and (P4*) to obtain the solution for $\mathbf {w}$ and $\mathbf {t}$. By setting $P_{\textrm {LED}}=\textrm {0.125W}$, where the total power of the LEDs is $P_{\textrm {total}}=1.1$W, the SNR profile results are shown in Fig. 4. On the other hand, the variance of received power and the quality factor are shown in Table 3 and Table 4, respectively. It can be seen that the maximum power threshold has impact on both the power allocation scheme in [9] and the proposed scheme. This results in a significant increase in the variance and a decrease in the quality factor but the proposed scheme still has more advantage in general. Moreover, the joint optimization scheme is less sensitive to the maximum power threshold and consequently, results in a smaller change in the results. The restriction on the individual LED power also has greatly impact on the performance of HCPP geometry. Moreover, to clearly show the advantage of the proposed scheme when the dimming control is realized, in Fig. 5, we present the simulation results with various maximum total powers for the circle-square geometry. The maximum power of individual LEDs is set to $0.125$W, where the total power of the system $P_{\textrm {total}}$ is considered to be $0.03$W to $2$W since the maximum total power of all 16 LEDs is equal to $16\times 0.125$W or $2$W. Overall, the proposed scheme still provides lower variance and a higher quality factor with various values of $P_{\textrm {total}}$. Moreover, with the maximum value at less than $1.15$W, significant gains are observed in the quality factor for both the power allocation and the proposed method. On the other hand, a higher value of the maximum power threshold $P_{\textrm {total}}$ results in the convergence of the quality factor of the three schemes. Therefore, in many scenarios, the maximum total power of the system and the maximum power of each LED should be carefully considered to achieve optimal uniform illumination with high quality factor. Finally, we investigate the advantage of proposed scheme in term of achievable communication performance; more specifically, the instantaneous channel capacity. Unlike the conventional bit error rate evaluation, the capacity analysis is not limited to any specific channel coding or modulation schemes, and can be used to analyze the performance of VLC systems. With the power allocation vector $\mathbf {w}$, the instantaneous unconstrained channel capacity $C_i$ of the $i$-th PD can be expressed as [18]
Table 3. Variance of received power of the proposed scheme with the threshold
Table 4. Quality Factor of the proposed scheme with the threshold
6. Conclusion
In this paper, the joint optimization of LED power allocation and orientation is proposed for uniform illumination in VLC systems. By optimizing the power allocation coefficients with normal vectors for LEDs and solving the problem through an iterative algorithm, the optimal uniform illumination condition can be achieved. Moreover, the optimization with a maximum power threshold for individual LEDs is also considered. The simulation results show that the proposed scheme can achieve lower received power variance and higher quality factor values compared to previous schemes.
Appendix A. Formulation of optimization problem (P2)
We have
Appendix B. Formulation of optimization problem (P4)
We have
Funding
National Research Foundation of Korea (NRF-2016R1D1A1B03934653, NRF-2019R1A2C1005920).
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