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Indoor visible light communications: performance evaluation and optimization
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 228 (2018)
Abstract
For indoor visible light communication (VLC), much work has been done when the noise is independent of the input signal. However, less effort is made when the VLC system suffers from the inputdependent noise. Considering the inputdependent noise, this paper focuses on the performance analysis and optimization for indoor VLC system. The Lambertian emissionbased channel model and onoff keying modulation are employed. According to the system model, the bit error rate (BER) with a closedform expression is derived. To enhance the system performance, an optimization problem that minimizes the BER by tilting the receiver plane is formulated. By solving the problem, the optimal tilting angle of the receiver is obtained. Simulation results verify the derived expression of BER. It is also shown that the BER is strongly affected by the inputdependent noise. Moreover, the optimal tilting angles for the receiver at any position are obtained, which can provide some insights for practical system design.
Introduction
In the past 10 years, the visible light communications (VLC) has gained substantial attention [1]. In VLC, the lightemitting diodes (LEDs) are often employed as the transmitter, while the photodiodes (PDs) are used as the receiver. The illumination and communication in VLC can be implemented at the same time. Moreover, VLC has many advantages, e.g., unlicensed spectrum, immunity to radio frequency interference, and high transmission rate. Consequently, VLC has become one of the most promising candidates for indoor wireless access in the forthcoming fifth generation (5G) communications [2].
The VLC can be implemented in both outdoor and indoor scenarios. For the outdoor scenario, the VLC is mainly used for the intelligent transportation system [3, 4]. However, much researchers concentrate on the indoor VLC. For the indoor scenario, a fundamental analysis for VLC is provided in [5]. In [6], the channel capacity bounds of VLC are derived by considering the constraints of the input signal. Moreover, ref. [7] derives a much tighter upper bound on the channel capacity for indoor VLC. By using pulse amplitude modulation and the inverse source coding, the capacity of VLC is further analyzed in [8]. In [9] and [10], the asymmetrically clipped opticalorthogonal frequency division multiplexing (ACOOFDM) and the direct current opticalOFDM (DCOOFDM) are discussed for VLC, respectively. By employing the color shift keying (CSK) and the generalized space shift keying (GSSK), the error performance of VLC is analyzed in [11] and [12], respectively. In [13] and [14], an optimal spread code and an adaptive equalizer for multipath dispersion in VLC are proposed. It should be emphasized that the noise and the input signal in [5]–[14] are assumed to be independent of each other. However, owing to the stochastic behavior of photon emission in LED, the noise in VLC depends on the input signal [15]. Note that the channel capacity of VLC with the inputdependent noise is investigated in [16], and the mutual information for optical spatial modulation with the inputdependent noise is discussed in [17]. However, the link reliability of the VLC (such as bit error rate (BER)) with the inputdependent noise has not been studied.
In previous work, the normal vectors of the transceiver planes are often supposed to be perpendicular to the ceiling [18–20]. By using this setup, the performance analysis for VLC is more convenient. However, the system performance also degrades dramatically when the distance between the transceiver is large [21]. In other words, when the transceiver distance becomes large, the received signaltonoise ratio (SNR) at the PD will be very small, and thus the BER will be very large. When the distance of the transceiver is large, how to improve the system performance becomes an important problem to be solved. As we know, if the PD inclines its orientation toward the LED, the received SNR will be enhanced observably. Therefore, to obtain better system performance, the optimal tilting angle of the PD should also be investigated.
Methods
In this paper, an indoor VLC system with a single transmitter and a single receiver is considered. The Lambertian emissionbased channel model and the onoff keying (OOK) are employed. In the system, the noises include two parts: inputdependent noise and inputindependent noise. In the presence of two kinds of noises, the BER of the system is analyzed. As a special case, the BER with only the inputindependent noise is also shown. By minimizing the BER, an optimization problem is formulated to improve the system performance. Then, the optimization problem is solved by using the principle of the convex optimization. After solving the problem, the optimal tilting angle of the receiver is obtained. The derived theoretical results of BER are all confirmed by using the MonteCarlo simulations. Moreover, the proposed performance enhancement scheme will expedite the analysis and help gain deeper insights for VLC.
System model
Consider a classic indoor VLC system with a single LED and a single PD, as depicted in Fig. 1. The room size is set to be J×K×L. The LED is employed as the transmitter, which is fixed on the center of the ceiling. The position of the LED is supposed to be [u,v,w]. The PD as the receiver is deployed on a horizontal plane, whose position is assumed to be [a,b,c]. The PD can move to any place of the receiver plane and can change its direction by tilting a proper angle.
Transmitter
At the transmitter, the emitted instantaneous optical intensity signal by the LED is denoted as X, and the average optical intensity of the LED is denoted as P. Without loss of generality, an intensity modulation scheme, OOK modulation, is employed. We assume that X is equal probably generated from the OOK modulation constellation, and thus X belongs to the following set
Channel model
For indoor VLC, the channel gain can be modeled by the Lambertian emission [22]
where m=− ln2/ ln(cosΦ_{1/2}) is order of the Lambertian emission, and Φ_{1/2} is the semiangle at half power of the LED. A is the physical area of the receiver. d is the transceiver distance. α is the irradiance angle of the LED, while β is the incidence angle of the PD. Note that both the optical filter gain and the concentrator gain of the PD are set to be one, and thus they are omitted in (2). Moreover, h in VLC is nonnegative and real, and thus α,β∈[0,π/2].
According to Fig. 1, the geometrical relationship of the transceiver is given by
Define V_{rs} as the vector pointing from the receiver to the transmitter, and V_{nor} as the unit normal vector of the receiver plane. Consequently, we have
where 〈·,·〉 represents the inner product operator, and ∥·∥ represents the norm operator.
In this paper, assume that the position of the PD is not changed by tilting the receiver plane. Therefore, V_{rs}=[u−a,v−b,w−c], V_{nor}=[ cosφ sinθ, sinφ sinθ, cosθ], where θ and φ are the tilting angle of the PD and the azimuth angle. The tilting angle and the azimuth angle are shown in Figs. 1 and 2, respectively. Assume that the projection of the LED onto the XY plane is located in the ith quadrant, and φ can be expressed as [21]
Furthermore, (4) can be written as
Substituting (3) and (6) into (2), the channel gain can be further expressed as
where \(d=\sqrt {(ua)^{2}+(vb)^{2}+(wc)^{2}}\). As can be seen in (7), when the positions of the LED and PD are fixed, the parameters a,b,c,u,v and w are constants. Moreover, according to (5), the azimuth angle φ is also a constant. This indicates that given the positions of the LED and PD, the channel gain h is a function of the tilting angle θ.
Receiver
At the receiver, the corrupted noises include thermal noise, amplifier noise, and shot noise. All of them can be modeled by Gaussian distributions [23]. However, the first two noises are independent of the input signal, while the shot noise depends on the input signal [16, 17]. So, the received electrical signal Y at the PD is given by
where r is the optoelectronic conversion factor of the PD. Z_{0}∼(0,σ^{2}) and Z_{1}∼(0,ς^{2}σ^{2}) are the inputindependent noise, and the inputdependent noise, respectively. ς^{2}≥0 is the scaling factor.
BER analysis
In wireless communications, BER is one of the commonly used indicators to reflect the system performance. Here, the BER and the optimal detection threshold for the considered VLC system will be investigated. Moreover, the behavior of the derived expressions will also be analyzed.
Derivations of BER and optimal detection threshold
From (7), it can be seen that h is fixed when the coordinates of LED and PD are pregiven. When OOK is employed, the BER for VLC can be expressed as [24]
where Pr(off)=Pr(on)=1/2. In the following, the conditional error probabilities Pr(onoff) and Pr(offon) will be analyzed, respectively.
According to (8), conditioned on X, Y is a Gaussian distribution. Therefore, the conditional probability density function (PDF) f_{YX}(yx) is obtained as
In this case, the detection threshold at the receiver is assumed to be η, which can be an arbitrary real number. Let χ=2rhP, and then Pr(offon) can be derived as
where \({\mathcal {Q}}(x) = \int _{x}^{\infty } {{{{e^{ {t^{2}}/2}}} \left /\right. {\sqrt {2\pi } }}} {\mathrm {d}}t\).
Similarly, Pr(onoff) can be derived as
Because Pr(on)=Pr(off)=0.5, (9) can be written as
In (13), the BER is a function of η. Figure 3 shows the value of BER versus η with different ς when r=0.9, (a,b,c)=(2 m,2 m,1.5 m), (u,v,w)=(2.5 m,2.5 m,3 m), P=55 dBm and σ=1. It can be observed that for a given ς, the BER curve has a peak value and a valley value. As it is well known, the optimum detection threshold η^{∗} corresponds to the valley value of BER. By analyzing (13), the following theorem is obtained.
Theorem 1
For the system model in (8), the optimal detection threshold η^{∗} when ς>0 is given by
Proof
See Appendix 1. □
Substituting (14) into (13), the BER when ς^{2}>0 can be finally written as
Behavior analysis
By analyzing Theorem 1, the changing trend of η^{∗} with ς can be obtained in the following theorem.
Theorem 2
When χ is very small, the value of η^{∗} in (14) increases with the increase of ς. However, when χ is very large, the value of η^{∗} in (14) decreases with the increase of ς.
Proof
See Appendix 2. □
In Theorem 1, it can be seen that η^{∗} is a function of ς. To show the relationship between η^{∗} and ς, Fig. 4 shows the value of η^{∗} versus ς with different P when r=0.9, h=8.335×10^{−7}, and σ=1. As is observed, when P is small, the value of η^{∗} increases with the increase of ς. However, when P is large, the value of η^{∗} decreases with ς. This result also verifies the correctness of Theorem 2.
By analyzing (15), the asymptotic behavior of the BER is derived in the following theorem.
Theorem 3
The asymptotic behavior of the BER in (15) is given by
Proof
See Appendix 3. □
It can be known from Theorem 3 that when ς→0, the VLC system with inputdependent noise achieves the same BER performance as the VLC system with only inputindependent noise (i.e., ς=0). In this paper, Eq. (16) can be used as a benchmark.
Problem formulation and solving
In the above section, the BER has been analyzed for the indoor VLC. In this section, an optimization problem to minimize the BER will be investigated. By using the principle of convex optimization, the optimization problem is effectively solved. Finally, the implementation of the optimal tilting angle is discussed.
Problem formulation
The objective of the paper is to minimize the BER of the VLC system. According to the relative positions of LED and PD, the tilting angle θ can not exceed π/2, i.e., θ∈[0,π/2]. Therefore, the optimization problem is given by
Problem solving
When ς^{2}>0, the BER is given by (15); when ς^{2}=0, the BER is given by (16). By analyzing (15) and (16), the following theorem can be derived.
Theorem 4
For all fixed ς^{2}≥0, the BER for the VLC system is a monotonously decreasing function with h.
Proof
See Appendix 4. □
By using Theorem 4, problem (17) can be transformed to
where the channel gain h is given by (7). By analyzing (7), the following theorem can be obtained.
Theorem 5
The objection function h in problem (18) is a concave function of θ.
Proof
See Appendix 5. □
By using Theorem 5, the maximum value of h is achieved by letting the firstorder partial derivative of h with θ to be zero, i.e.,
Therefore, the optimal tilting angle θ^{∗} is obtained as
Implementation of the tilting angle
From (20), it can be found that the optimal tilting angle is a function of the positions of the LED and the PD. In this paper, the position of the LED (i.e., [u,v,w]) is fixed, but the position of the PD (i.e., [a,b,c]) is variable. Therefore, to realize the angle tilting of the receiver plane, the first step is to determine the position of the PD. With an arbitrary tilting angle larger than zero, the tilted receiver can obtain its position by using the threedimensional positioning method proposed in our previous work [25]. After obtaining the position of the PD, the optimal tilting angle for the receiver can be calculated by using (20). Finally, according to the optimal tilting angle, the receiver can realize the angle tilting by employing an electrical machinery. The electrical machinery first makes the tilting angle to be zero and then increases the tilting angle until the optical tilting angle is achieved.
Numerical results and discussion
In this section, a practical VLC system in a room is considered as the test system. The derived theoretical expression of the BER will be confirmed by using the MonteCarlo simulations in MATLAB. Note that the simulation results presented in this section are the average of N=10^{6} independent trials. The detailed simulation process is provided in Algorithm 1. Moreover, the main simulation parameters are illustrated in Table 1.
Figure 5 shows the BER versus ς for different transmit optical powers P. In the simulation, [u,v,w]=[2.5 m, 2.5 m, 3 m] and [a,b,c]=[2 m, 2 m, 1.5 m]. The tilting angle of the PD is set to be zero. The order of the Lambertian emission m=5. It can be seen from the figure that, when P is small, the BER performance improves with the increase of ς. However, when P is large, the value of the BER increases as ς. This indicates that the BER performance is strongly effected by the value of ς. Moreover, when ς tends to a large value, the value of the BER trends to a stable value (i.e., 0.25). From Fig. 5, for a fixed ς, the BER performance improves with the increase of P. As is known, large optical power will generate a high SNR at receiver, and thus it will result in good BER performance. Furthermore, the theoretical results match exactly with the simulation results. This indicates that the derived BER expression can be used to evaluate the system performance without timeintensive simulations.
Figure 6 shows the BER versus the channel gain h with different ς. In this simulation, the transmit optical power of the LED is P=50 dBm. As can be observed, with the increase of h, the BER performance improves accordingly, which coincides with Theorem 4. Moreover, for small value of h as shown in Fig. 6a, the BER performance improves with the increase of ς. However, for large value of h as shown in Fig. 6b‘, the BER performance degrades with the increase of the ς. This conclusion is similar to that in Fig. 5. Once again, it should be noted that the theoretical results match well to the simulation results.
Figure 7 shows the channel gain h versus the semiangle at half power of the LED Φ_{1/2} before and after tilting the receiver plane. In the simulation, the average optical power P is set to be 50 dBm, and the coordinates of LED and PD are set to be [2.5 m, 2.5 m, 3 m] and [0 m, 0 m, 0 m], respectively. It can be observed that, as the increase of Φ_{1/2}, both the two curves increase first and then decrease a little. When Φ_{1/2}=55^{0}, the channel gains of the two curves achieve the maximum values. In addition, by tilting the receiver plane, the channel gain performance improves dramatically. This means that it is necessary to tilt the receiver plane properly to enhance system performance.
Figure 8 plots the distributions of the BER before and after tilting the receiver plane in the room. It can be observed that the worst BER achieves when the PD is located in the corner, while the best BER obtains when the PD is located in the center of the floor. This is n because large transmission distance will result in the system performance degradation. Moreover, the BER performance improves in the whole room after tilting the receiver plane. Specifically, when the PD is located in the corner, the value of the BER reduces from 0.3487 to 0.2270 after tilting the receiver plane. This indicates that it is necessary to tilt the receiver plane properly in VLC system.
Figure 9 shows the distributions of tilting angles in the whole receiver plane after tilting the receiver plane. In this simulation, the LED is fixed on the center of the ceiling, while the PD can move to any place of the receiver plane. In Fig. 9, when the PD is located in the center of the receiver plane, the distance between the transceiver achieves the minimum value, and the tilting angle of the PD in this case is zero. When the PD moves from the center of the receiver plane to other place, the tilting angle of the PD increases gradually. When the PD is located in the corner of the receiver plane, the tilting angle of the PD gets its maximum value, i.e., 1.8032 rad. Therefore, the tilting angle provided in this figure can provide some insights for practical system design.
Conclusions
This paper focuses on an VLC system with inputdependent noise. The main conclusions of this paper are listed as follows:

For the VLC system with inputdependent noise, the optimal detection threshold is obtained. Also, the theoretical expression of the BER is derived, which is quite accurate to evaluate the system performance.

The system performance is strongly affected by the inputdependent noise. For small h or P, the BER decreases with the increase of ς. However, for large h or P, the trend of the BER curve changes. Moreover, the larger the value of the channel gain is, the better the BER performance becomes.

When the transceiver distance is large, the BER performance can be dramatically enhanced by tilting the receiver plane. In practice, the suggested tilting angle of the receiver place is shown in Fig. 9.
Appendix 1: Proof of Theorem 1
To obtain the optimum detection threshold, taking the firstorder partial derivative of \(\phantom {\dot {i}\!}{BER}_{{\varsigma ^{2}} > 0}\) with respect to η and letting it to be zero, we have
i.e.,
By solving the quadratic Eq. (22), we have
According to the curve of BER in Fig. 3, the optimum detection threshold (14) is obtained.
Appendix 2: Proof of Theorem 2
Case (a): When χ is very small, we have
Taking the firstorder derivative of η^{∗} with respect to ς, we have
This indicates that when χ is small, the value of η^{∗} rises with the increase of ς.
Case (b): When χ is very large, we have
Taking the firstorder derivative of η^{∗} with respect to ς, we have
This indicates that when χ is large, the value of η^{∗} decreases with the increase of ς.
Appendix 3: Proof of Theorem 3
Case (a): When ς^{2}→0, using the L’Hospital’s rule, we have
and
Submitting (28) and (29) into (15), the asymptotic BER is given by
Case (b): When ς^{2}=0, the VLC system only includes the inputindependent noise. Under this circumstance, (8) reduces to Y=rhX+Z_{0}. According to Theorem 1, the optimum detection threshold in this case can be easily derived as η^{∗}=χ/2. Moreover, the conditional PDF f_{YX}(yx) becomes
Therefore, the BER becomes
Appendix 4: Proof of Theorem 4
Define I_{1} and I_{2} as
In I_{1}, taking the firstorder partial derivative of I_{4} with h, we can get
Then, taking the firstorder partial derivative of I_{5} with h, we can get
which indicates that I_{5} is a monotonously decreasing function of h. When h→0, I_{5}→0. Since h>0, I_{5}<0 always holds. Moreover, according to (34), we have ∂I_{4}/∂h≤0, which means that I_{4} is a monotonously decreasing function of h. Moreover, I_{3} is a monotonously increasing function of h. Given the above, I_{1} is a monotonously increasing function of h.
Furthermore, taking the firstorder partial derivative of I_{2} with h, we have
where the inequality follows because x> ln(1+x) always holds for all x>0. Therefore, I_{2} is also a monotonously increasing function of h. Moreover, Q(·) is a monotonously decreasing function. Therefore, \(\phantom {\dot {i}\!}\mathrm {BE{R_{{\varsigma ^{2}} > 0}}}\) is a monotonously decreasing function of h, i.e.,
Similarly, when ς^{2}=0, taking the firstorder partial derivative of \(\phantom {\dot {i}\!}\mathrm {BE{R_{{\varsigma ^{2}} = 0}}}\) with respect to h in (16), we have
Appendix 5: Proof of Theorem 5
By taking the secondorder partial derivative of h with θ, we get
In VLC, h is a nonnegative number, and thus ∂^{2}h/∂θ^{2} < 0. Therefore, Theorem 5 always holds.
Abbreviations
 ACO:

Asymmetrically clipped optical
 BER:

Bit error rate
 CSK:

Color shift keying
 DCO:

Direct current optical
 GSSK:

Generalized space shift keying
 LEDs:

Lightemitting diodes
 OFDM:

Orthogonal frequency division multiplexing
 OOK:

Onoff keying
 PD:

Photodiodes
 SNR:

Signaltonoise ratio
 VLC:

Visible light communication
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Acknowledgements
The research presented in this paper was supported by Huaiyin Institute of Technology, China.
Funding
This paper is supported by National Natural Science Foundation of China (61701254), Natural Science Foundation of Jiangsu Province (BK20170901), Key International Cooperation Research Project (61720106003), the open fund for Jiangsu key laboratory of traffic and transportation security, Huaiyin Institute of Technology (TTS201703), the Open Research Fund of the Key Laboratory of Broadband Wireless Communication and Sensor Network Technology, Nanjing University of Posts and Telecommunications, the Ministry of Education (JZNY201706), NUPTSF (NY216009), the Open Research Fund of the Key Laboratory of Intelligent Computing & Signal Processing, Anhui University, and the Open Research Subject of the Key Laboratory (Research Base) of Signal and Information Processing, Xihua University (szjj2017047).
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SHL is the main contributor of the paper. She proposes the main idea, derives the expression of BER, and solves the optimization problem. CL helps to finish the simulation results. XB and JYW help to polish the content of the paper. All authors read and approve the final manuscript.
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Correspondence to JinYuan Wang.
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ShengHong Lin is working toward her PhD degree at Nanjing University of Posts and Telecommunications, Nanjing, China. Her research interest is visible light communications.
Cheng Liu is currently studying for the M.S. degree from the National Mobile Communications Research Laboratory, Southeast University, Nanjing, China. His research interest is physicallayer security in visible light communications.
Xu Bao is currently a professor at Jiangsu Key Laboratory of Traffic and Transportation Security, Huaiyin Institute of Technology, Huai’an, China. His research interest is visible light communications.
JinYuan Wang is currently a lecturer at College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, China. His current research interest is visible light communications.
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Lin, S., Liu, C., Bao, X. et al. Indoor visible light communications: performance evaluation and optimization. J Wireless Com Network 2018, 228 (2018). https://doi.org/10.1186/s136380181243x
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Keywords
 Visible light communications
 BER
 Inputdependent noise
 Tilting receiver plane