Indoor visible light communications: performance evaluation and optimization

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 input-dependent noise. Considering the input-dependent noise, this paper focuses on the performance analysis and optimization for indoor VLC system. The Lambertian emission-based channel model and on-off keying modulation are employed. According to the system model, the bit error rate (BER) with a closed-form 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 input-dependent noise. Moreover, the optimal tilting angles for the receiver at any position are obtained, which can provide some insights for practical system design.

1 Introduction

In the past 10 years, the visible light communications (VLC) has gained substantial attention [1]. In VLC, the light-emitting 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 optical-orthogonal frequency division multiplexing (ACO-OFDM) and the direct current optical-OFDM (DCO-OFDM) 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 multi-path 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 input-dependent noise is investigated in [16], and the mutual information for optical spatial modulation with the input-dependent noise is discussed in [17]. However, the link reliability of the VLC (such as bit error rate (BER)) with the input-dependent 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 signal-to-noise 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.

2 Methods

In this paper, an indoor VLC system with a single transmitter and a single receiver is considered. The Lambertian emission-based channel model and the on-off keying (OOK) are employed. In the system, the noises include two parts: input-dependent noise and input-independent 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 input-independent 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 Monte-Carlo simulations. Moreover, the proposed performance enhancement scheme will expedite the analysis and help gain deeper insights for VLC.

3 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.

Fig. 1

A classic point-to-point indoor VLC system

3.1 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

$$ X \in \left\{ {0,2P} \right\} $$

(1)

3.2 Channel model

For indoor VLC, the channel gain can be modeled by the Lambertian emission [22]

$$ h = \frac{{(m + 1)A}}{{2\pi {d^{2}}}}{\cos^{m}}(\alpha)\cos (\beta) $$

(2)

where m=− ln2/ ln(cosΦ_1/2) is order of the Lambertian emission, and Φ_1/2 is the semi-angle 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 non-negative and real, and thus α,β∈[0,π/2].

According to Fig. 1, the geometrical relationship of the transceiver is given by

$$ \cos (\alpha) = \frac{{w - c}}{d} $$

(3)

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

$$ \cos (\beta) = \frac{{\left\langle {{\mathbf{V}_{{\text{nor}}}},{\mathbf{V}_{{\text{rs}}}}} \right\rangle }}{{\left\| {{\mathbf{V}_{{\text{nor}}}}} \right\| \cdot \left\| {{\mathbf{V}_{{\text{rs}}}}} \right\|}} $$

(4)

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 i-th quadrant, and φ can be expressed as [21]

$$ \varphi = \left\{ \begin{array}{ll} \arctan \left| {\frac{{v - b}}{{u - a}}} \right|, & i = 1 \\ \pi - \arctan \left| {\frac{{v - b}}{{u - a}}} \right|, & i = 2 \\ \pi + \arctan \left| {\frac{{v - b}}{{u - a}}} \right|, & i = 3 \\ 2\pi - \arctan \left| {\frac{{v - b}}{{u - a}}} \right|,& i = 4 \\ \end{array} \right. $$

(5)

Fig. 2

The azimuth angle when LED’s projection locates in different quadrants of the XY plane. a 1st quadrant. b 2nd quadrant. c 3rd quadrant. d 4th quadrant

Furthermore, (4) can be written as

$$\begin{array}{@{}rcl@{}} \begin{aligned} \cos (\beta) =& [(u - a)\cos\varphi \sin\theta + (v - b)\sin\varphi \sin\theta \\&+ (w - c)\cos\theta ]/d \end{aligned} \end{array} $$

(6)

Substituting (3) and (6) into (2), the channel gain can be further expressed as

$$ \begin{aligned} h\!=&\! \frac{{(m + 1)A{{(w - c)}^{m}}}}{{2\pi {d^{m + 3}}}} [(u \,-\, a)\cos\varphi \sin\theta \!\\ &+\! (v \,-\, b)\sin\varphi \sin\theta \,+\, (w \,-\, c)\cos\theta ] \end{aligned} $$

(7)

where $d=\sqrt {(u-a)^{2}+(v-b)^{2}+(w-c)^{2}}$. 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.

3.3 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

$$ Y = rhX + \sqrt{rhX}{Z_{1}} + {Z_{0}} $$

(8)

where r is the optoelectronic conversion factor of the PD. Z₀∼(0,σ²) and Z₁∼(0,ς²σ²) are the input-independent noise, and the input-dependent noise, respectively. ς²≥0 is the scaling factor.

4 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.

4.1 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 pre-given. When OOK is employed, the BER for VLC can be expressed as [24]

$$ \text{BER} = \text{Pr} (\text{off})\text{Pr} (\text{on}|\text{off}) + \text{Pr} (\text{on})\text{Pr} (\text{off}|\text{on}) $$

(9)

where Pr(off)=Pr(on)=1/2. In the following, the conditional error probabilities Pr(on|off) and Pr(off|on) will be analyzed, respectively.

According to (8), conditioned on X, Y is a Gaussian distribution. Therefore, the conditional probability density function (PDF) f_Y|X(y|x) is obtained as

$$ {f_{Y|X}}(y|x) = \frac{1}{{\sqrt {2\pi (1 + rhx{\varsigma^{2}})} \sigma }}{\exp\left[{ - \frac{{{{(y - rhx)}^{2}}}}{{2(1 + rhx{\varsigma^{2}}){\sigma^{2}}}}}\right]} $$

(10)

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(off|on) can be derived as

$$\begin{array}{@{}rcl@{}} \text{Pr} (\left. {{\text{off}}} \right|\text{on}) &=& \int_{- \infty }^{\eta} {\frac{{e^{- \frac{{{{(y - \chi)}^{2}}}}{{2(1 + \chi{\varsigma^{2}}){\sigma^{2}}}}}}}{{\sqrt {2\pi (1 + \chi{\varsigma^{2}})} \sigma }}} {\mathrm{d}}y \\ &=& {\mathcal{Q}}\left({\frac{{\chi - \eta }}{{\sqrt {1 + \chi{\varsigma^{2}}} \sigma }}} \right) \end{array} $$

(11)

where ${\mathcal {Q}}(x) = \int _{x}^{\infty } {{{{e^{- {t^{2}}/2}}} \left /\right. {\sqrt {2\pi } }}} {\mathrm {d}}t$.

Similarly, Pr(on|off) can be derived as

$$\begin{array}{@{}rcl@{}} \text{Pr} (\left. {\text{on}} \right|{\text{off}}) &=& \int_{\eta}^{\infty} {\frac{1}{{\sqrt {2\pi} \sigma }}{e^{- \frac{{{y^{2}}}}{{2{\sigma^{2}}}}}}} {\mathrm{d}}y \\ &=& {\mathcal{Q}}\left({\frac{\eta }{\sigma }} \right) \end{array} $$

(12)

Because Pr(on)=Pr(off)=0.5, (9) can be written as

$$ \mathrm{BE{R_{{\varsigma^{2}} > 0}}} = \frac{1}{2}\left[ {{\mathcal{Q}}\left({\frac{{\chi - \eta }}{{\sqrt {1 + \chi{\varsigma^{2}}} \sigma }}} \right) + {\mathcal{Q}}\left({\frac{\eta }{\sigma }} \right)} \right] $$

(13)

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.

Fig. 3

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

Theorem 1

For the system model in (8), the optimal detection threshold η^∗ when ς>0 is given by

$$\begin{array}{@{}rcl@{}} \eta^{*} = \frac{{\sqrt {1 + \chi{\varsigma^{2}} + \frac{{{\varsigma^{2}}{\sigma^{2}}(1 + \chi{\varsigma^{2}})\ln(1 + \chi{\varsigma^{2}})}}{{\chi}}} - 1}}{{{\varsigma^{2}}}} \end{array} $$

(14)

Proof

See Appendix 1. □

Substituting (14) into (13), the BER when ς²>0 can be finally written as

$$\begin{array}{@{}rcl@{}} {}\mathrm{BE{R_{{\varsigma^{2}} > 0}}} \!&=&\! \frac{1}{2}\!\!\left[\! {{\mathcal{Q}}\!\!\left(\! {\frac{{\sqrt {1 \,+\, \chi{\varsigma^{2}}} \,-\, \sqrt {1 \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} }}{{{\varsigma^{2}}\sigma }}}\! \right)} \right.\\ &+& \left. {{\mathcal{Q}}\!\left(\!\! {\frac{{ {\sqrt {1 \,+\, \chi{\varsigma^{2}} \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}(1 \,+\, \chi{\varsigma^{2}})\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} \,-\, 1} }}{{{\varsigma^{2}}\sigma }}} \!\!\right)}\!\! \right] \end{array} $$

(15)

4.2 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.

Fig. 4

The value of η^∗ versus ς with different P when r=0.9, h=8.335×10⁻⁷, and σ=1

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

$$ \mathop {\lim }\limits_{\varsigma \to 0} \text{BER}_{{\varsigma^{2}}{> 0}} = \text{BER}_{{\varsigma^{2}} = 0} = {\mathcal{Q}}\left({\frac{{\chi}}{2\sigma }} \right) $$

(16)

Proof

See Appendix 3. □

It can be known from Theorem 3 that when ς→0, the VLC system with input-dependent noise achieves the same BER performance as the VLC system with only input-independent noise (i.e., ς=0). In this paper, Eq. (16) can be used as a benchmark.

5 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.

5.1 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

$$\begin{array}{@{}rcl@{}} &&\;\;\;\; \mathop {\min }\limits_{\theta} \text{BER} \\ \text{s.t.} &&\;\;\;\; 0 \le \theta \le \pi/2 \end{array} $$

(17)

5.2 Problem solving

When ς²>0, the BER is given by (15); when ς²=0, the BER is given by (16). By analyzing (15) and (16), the following theorem can be derived.

Theorem 4

For all fixed ς²≥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

$$\begin{array}{@{}rcl@{}} && \;\;\;\;\mathop {\max }\limits_{\theta} h \\ \text{s.t.} &&\;\;\;\; 0 \le \theta \le \pi/2 \end{array} $$

(18)

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 first-order partial derivative of h with θ to be zero, i.e.,

$$ {{\begin{aligned} \frac{{\partial h}}{{\partial \theta }} \,=\, \frac{{(m \,+\, 1)A{{(w \,-\, c)}^{m}}}}{{2\pi {d^{m + 3}}}} \!\left[\! {\sqrt {{{(u \,-\, a)}^{2}} \,+\, {{(v \,-\, b)}^{2}}} \!\cos \theta \,-\, \left| {w \,-\, c} \right|\sin \theta} \!\right]\! =\! 0 \end{aligned}}} $$

(19)

Therefore, the optimal tilting angle θ^∗ is obtained as

$$ {\theta^{*}} = \arctan \left({\frac{{\sqrt {{{(a - u)}^{2}} + {{(b - v)}^{2}}} }}{{\left| {w -c} \right|}}} \right) $$

(20)

5.3 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 three-dimensional 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.

6 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 Monte-Carlo simulations in MATLAB. Note that the simulation results presented in this section are the average of N=10⁶ independent trials. The detailed simulation process is provided in Algorithm 1. Moreover, the main simulation parameters are illustrated in Table 1.

Table 1 Main simulation parameters

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 time-intensive simulations.

Fig. 5

BER versus ς with different transmit optical powers. a ς∈ (0, 50), b ς∈ (50, 1000)

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.

Fig. 6

BER versus channel gain h with different ς when P=50 dBm. a BER with small h, b BER with large h

Figure 7 shows the channel gain h versus the semi-angle 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=55⁰, 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.

Fig. 7

Channel gain h versus the semi-angle at half power of the LED Φ_1/2

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.

Fig. 8

BER distributions before and after tilting the receiver plane in the whole room

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.

Fig. 9

Tilting angle distributions after tilting the receiver plane in the room

7 Conclusions

This paper focuses on an VLC system with input-dependent noise. The main conclusions of this paper are listed as follows:

• For the VLC system with input-dependent 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 input-dependent 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.

8 Appendix 1: Proof of Theorem 1

To obtain the optimum detection threshold, taking the first-order partial derivative of $\phantom {\dot {i}\!}{BER}_{{\varsigma ^{2}} > 0}$ with respect to η and letting it to be zero, we have

$$\begin{array}{@{}rcl@{}} \frac{{\partial {BER}_{{\varsigma^{2}} > 0}}}{{\partial \eta }} \,=\, \frac{1}{{2\sqrt {2\pi} \sigma }}\!\!\left[\! {\frac{{e^{- \frac{{{{(\chi \,-\, \eta)}^{2}}}}{{2(1 \,+\, \chi{\varsigma^{2}}){\sigma^{2}}}}}}}{{\sqrt {1 \,+\, \chi{\varsigma^{2}}} }} \,-\, {e^{- \frac{{{\eta^{2}}}}{{2{\sigma^{2}}}}}}}\!\! \right]\,=\, 0 \end{array} $$

(21)

i.e.,

$$ \begin{aligned} \frac{{\chi{\varsigma^{2}}}}{{1 + \chi{\varsigma^{2}}}}{\eta^{2}} &+ \frac{{2\chi}}{{1 + \chi{\varsigma^{2}}}}\eta - \frac{{{{\chi}^{2}}}}{{1 + \chi{\varsigma^{2}}}}\\ &- {\sigma^{2}}\ln \left({1 + \chi{\varsigma^{2}}} \right) = 0 \end{aligned} $$

(22)

By solving the quadratic Eq. (22), we have

$$\begin{array}{@{}rcl@{}} \eta \,=\, \frac{{ - 1 \!\pm\! \sqrt {1 \,+\, \chi{\varsigma^{2}} \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}(1 \,+\, \chi{\varsigma^{2}})\ln(1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} }}{{{\varsigma^{2}}}} \end{array} $$

(23)

According to the curve of BER in Fig. 3, the optimum detection threshold (14) is obtained.

9 Appendix 2: Proof of Theorem 2

Case (a): When χ is very small, we have

$$ \eta^{*} \approx \frac{{\sqrt {1 + {\varsigma^{4}}{\sigma^{2}}} - 1}}{{{\varsigma^{2}}}} $$

(24)

Taking the first-order derivative of η^∗ with respect to ς, we have

$$ \frac{{\mathrm{d}}}{{{\mathrm{d}}\varsigma }}\left({\frac{{\sqrt {1 + {\varsigma^{4}}{\sigma^{2}}} - 1}}{{{\varsigma^{2}}}}} \right) = \frac{{2\left({\sqrt {1 + {\varsigma^{4}}{\sigma^{2}}} - 1} \right)}}{{{\varsigma^{3}}\sqrt {1 + {\varsigma^{4}}{\sigma^{2}}} }} > 0 $$

(25)

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

$$ {\eta^{*}} \approx \frac{{\sqrt {\chi + {\varsigma^{2}}{\sigma^{2}}\ln(\chi {\varsigma^{2}})} }}{\varsigma } $$

(26)

Taking the first-order derivative of η^∗ with respect to ς, we have

$$ \frac{{\mathrm{d}}}{{{\mathrm{d}}\varsigma }}\frac{{\sqrt {\chi + {\varsigma^{2}}{\sigma^{2}}\ln \left({\chi {\varsigma^{2}}} \right)} }}{\varsigma }= \frac{{{\varsigma^{2}}{\sigma^{2}} - \chi }}{{{\varsigma^{2}}\sqrt {\chi + {\varsigma^{2}}{\sigma^{2}}\ln \left({\chi {\varsigma^{2}}} \right)} }} < 0 $$

(27)

This indicates that when χ is large, the value of η^∗ decreases with the increase of ς.

10 Appendix 3: Proof of Theorem 3

Case (a): When ς²→0, using the L’Hospital’s rule, we have

$$ {{\begin{aligned} \lim\limits_{\varsigma \to 0} \frac{{\sqrt {1 \,+\, \chi{\varsigma^{2}}} \,-\, \sqrt {1 \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} }}{{{\varsigma^{2}}\sigma }} &=\lim\limits_{\varsigma \to 0} \left[ {\frac{{\chi}}{2{\sigma \sqrt {1 \,+\, \chi{\varsigma^{2}}} }} \,-\, \frac{{\sigma \ln (1 \,+\, \chi{\varsigma^{2}}) \,+\, \frac{{\chi{\varsigma^{2}}\sigma }}{{1 \,+\, \chi{\varsigma^{2}}}}}}{{2\chi\sqrt {1 \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} }}} \right]\\ &= \frac{{\chi}}{2\sigma} \end{aligned}}} $$

(28)

and

$$\begin{array}{@{}rcl@{}} &&\mathop {\lim }\limits_{\varsigma \to 0} \frac{{\sqrt {1 + \chi{\varsigma^{2}} + \frac{{{\varsigma^{2}}{\sigma^{2}}(1 + \chi{\varsigma^{2}})\ln (1 + \chi{\varsigma^{2}})}}{{\chi}}} - 1}}{{{\varsigma^{2}}\sigma }} \\ &&= \mathop {\lim }\limits_{\varsigma \to 0} \frac{{\frac\chi 2 + \frac{{({2}\varsigma {\sigma^{2}} + 4\chi{\varsigma^{3}}{\sigma^{2}})\ln (1 + \chi{\varsigma^{2}})+({\varsigma^{2}}{\sigma^{2}} + \chi{\varsigma^{4}}{\sigma^{2}})\frac{{\chi}}{2{(1 + \chi{\varsigma^{2}}})}}}{{\chi}}}}{{\sigma \sqrt {1 + \chi{\varsigma^{2}} + \frac{{{\varsigma^{2}}{\sigma^{2}}(1 + \chi{\varsigma^{2}})\ln (1 + \chi{\varsigma^{2}})}}{{\chi}}} }} \\ &&= \frac{{\chi}}{2\sigma } \end{array} $$

(29)

Submitting (28) and (29) into (15), the asymptotic BER is given by

$$ \mathop {\lim }\limits_{\varsigma \to 0} {BER}_{{\varsigma^{2}}{> 0}} = {\mathcal{Q}}\left({\frac{{\chi}}{2\sigma }} \right) $$

(30)

Case (b): When ς²=0, the VLC system only includes the input-independent noise. Under this circumstance, (8) reduces to Y=rhX+Z₀. According to Theorem 1, the optimum detection threshold in this case can be easily derived as η^∗=χ/2. Moreover, the conditional PDF f_Y|X(y|x) becomes

$$ {f_{Y|X}}(y|x) = \frac{1}{{\sqrt {2\pi} \sigma }}{\exp\left[{ - \frac{{{{(y - rhx)}^{2}}}}{{2{\sigma^{2}}}}}\right]} $$

(31)

Therefore, the BER becomes

$$ {{\begin{aligned} \mathrm{BE{R_{{\varsigma^{2}} = 0}}} &= \frac 12 \left[\int_{- \infty }^{\chi/2} \!{\frac{1}{{\sqrt {2\pi} \sigma }}{e^{- \frac{{{{(y - \chi)}^{2}}}}{{2{\sigma^{2}}}}}}} {\mathrm{d}}y +\int_{\chi/2}^{\infty}\! {\frac{1}{{\sqrt {2\pi} \sigma }}{e^{- \frac{{{y^{2}}}}{{2{\sigma^{2}}}}}}} {\mathrm{d}}y\! \right] \\ &= {\mathcal{Q}}\left({\frac{{\chi}}{2\sigma }} \right) \end{aligned}}} $$

(32)

According to (30) and (32), Theorem 3 holds.

11 Appendix 4: Proof of Theorem 4

Define I₁ and I₂ as

$$\begin{array}{@{}rcl@{}} \left\{\quad\begin{array}{l}\!\!\!\! {I_{1}} \,=\, \frac{1}{{{\varsigma^{2}}\sigma }}\!\!\left(\!\! {\underbrace {\sqrt {1 \,+\, \chi{\varsigma^{2}}} }_{{I_{3}}} \,-\, \underbrace {\sqrt {1 \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} }_{{I_{4}}}}\! \right)\\\\ \!\!\!\!{I_{2}} \,=\, {\frac{{\sqrt {1\,+\, \chi{\varsigma^{2}} \,+\, \frac{{{\varsigma^{2}}{\sigma^{2}}(1 \,+\, \chi{\varsigma^{2}})\ln (1 \,+\, \chi{\varsigma^{2}})}}{{\chi}}} \,-\, 1}}{{{\varsigma^{2}}\sigma }}} \end{array} \right. \end{array} $$

(33)

In I₁, taking the first-order partial derivative of I₄ with h, we can get

$$ \frac{{\partial {I_{4}}}}{{\partial h}}\! =\! \frac{1}{{2\sqrt {1 \!+ \! \frac{{{\varsigma^{2}}{\sigma^{2}}\ln (1 \!+ \chi{\varsigma^{2}})}}{{\chi}}} }}\frac{{{\varsigma^{2}}{\sigma^{2}}}}{{\chi h}} \underbrace{\left[{\frac{{\chi{\varsigma^{2}}}}{{1 \!+ \!\chi{\varsigma^{2}}}} - \ln (1 \!+ \! \chi{\varsigma^{2}})}\right]}_{I_{5}} $$

(34)

Then, taking the first-order partial derivative of I₅ with h, we can get

$$\begin{array}{@{}rcl@{}} \frac{{\partial {I_{5}}}}{{\partial h}}\!&=&\! \frac{{2rP{\varsigma^{2}}(1 \,+\, \chi{\varsigma^{2}}) \,-\, 2rP{\varsigma^{2}}\chi{\varsigma^{2}}}}{{{{(1 + \chi{\varsigma^{2}})}^{2}}}} \,-\, \frac{{2rP{\varsigma^{2}}}}{{1 \,+\, \chi{\varsigma^{2}}}} \\ &=& \frac{{ -2rP\chi {\varsigma^{4}}}}{{{{(1 + \chi{\varsigma^{2}})}^{2}}}} < 0 \end{array} $$

(35)

which indicates that I₅ is a monotonously decreasing function of h. When h→0, I₅→0. Since h>0, I₅<0 always holds. Moreover, according to (34), we have ∂I₄/∂h≤0, which means that I₄ is a monotonously decreasing function of h. Moreover, I₃ is a monotonously increasing function of h. Given the above, I₁ is a monotonously increasing function of h.

Furthermore, taking the first-order partial derivative of I₂ with h, we have

$$\begin{array}{@{}rcl@{}} \frac{{\partial {I_{2}}}}{{\partial h}}=\frac{1}{{{\varsigma^{2}}\sigma }}\frac{{2rP{\varsigma^{2}}+\frac{{{\varsigma^{2}}{\sigma^{2}}}}{{\chi h}}\left[ {\chi{\varsigma^{2}} - \ln (1 \,+\, \chi{\varsigma^{2}})} \right]}}{{2\sqrt {1 +\chi{\varsigma^{2}} + \frac{{{\varsigma^{2}}{\sigma^{2}}(1 + \chi{\varsigma^{2}})\ln (1 + \chi{\varsigma^{2}})}}{{\chi}}} }} >0 \end{array} $$

(36)

where the inequality follows because x> ln(1+x) always holds for all x>0. Therefore, I₂ 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.,

$$\begin{array}{@{}rcl@{}} \frac{{{\partial}\mathrm{BE{R_{{\varsigma^{2}} > 0}}}}}{{{\partial}h}} < 0 \end{array} $$

(37)

Similarly, when ς²=0, taking the first-order partial derivative of $\phantom {\dot {i}\!}\mathrm {BE{R_{{\varsigma ^{2}} = 0}}}$ with respect to h in (16), we have

$$\begin{array}{@{}rcl@{}} \frac{{{\partial}\mathrm{BE{R_{{\varsigma^{2}} = 0}}}}}{{{\partial}h}} &=& - \frac{{{e^{- \frac{{\chi^{2}}}{{4{\sigma^{2}}}}}}}}{{\sqrt {2\pi} }}\frac{{rP}}{\sigma} < 0 \end{array} $$

(38)

According to (37) and (38), Theorem 4 can be obtained.

12 Appendix 5: Proof of Theorem 5

By taking the second-order partial derivative of h with θ, we get

$$ \begin{aligned} \frac{{{{\partial}^{2}}h}}{{{\partial}{\theta^{2}}}} =& - \frac{{(m + 1)A{{(w - c)}^{m}}}}{{2\pi {d^{m + 3}}}}[(u - a)\cos \varphi \sin \theta \\&+ (v -b)\sin \varphi \sin \theta + (w c)\cos \theta ] = - h \end{aligned} $$

(39)

In VLC, h is a non-negative number, and thus ∂²h/∂θ² < 0. Therefore, Theorem 5 always holds.