## 1. Introduction

As one of the emerging optical wireless communication techniques, the visible light communication (VLC) has drawn considerable attention recently from both the academy and industry [1–3]. Compared to the traditional radio frequency (RF) wireless communication, VLC has many advantages, such as freedom from hazardous electromagnetic radiation, no licensing requirements, low‐cost frontends, large spectrum bandwidth (as shown in **Figure 1**), large channel capacity, and so on. In VLC, both illumination and communication are simultaneously implemented. Moreover, the transmitted optical signal is non‐negative. Therefore, the developed theory and analysis results in traditional RF wireless communication are not directly applicable to VLC.

Up to now, the research on VLC can be divided into two categories: the demo system design and theoretical analysis. As research continues, a variety of demo platforms arise. **Table 1** shows the development of the VLC demo systems. As can be seen in **Table 1**, the transmit rate of the VLC system increases from several Mbps to several Gbps in the last decade, which indicates that the VLC has attractive prospects of development. Specifically, the transmit rates of the early demo systems are low, but the transmit distances are long and the data are processed in real time. With the development of communication techniques, more and more VLC testbeds with high transmit rates are developed successfully, but the real‐time processing becomes very hard. Therefore, more advanced processing techniques are needed for VLC.

In the aspect of theoretical analysis, much work has been done on VLC. In Ref. [4], the channel capacity for VLC using inverse source coding is investigated. However, the theoretical expression of the capacity is not presented. Under the non‐negative and average optical intensity constraints, the closed expression of capacity bounds is derived in Ref. [5]. Based on Ref. [5], a tight upper bound on the capacity is derived in Ref. [6]. By adding a peak optical intensity constraint, tight capacity bounds are further derived in Ref. [7]. In Ref. [8], the capacity bounds for multiple‐input‐multiple‐output VLC are derived. In Ref. [9], the capacity and outage probability for the parallel optical wireless channels are analysed. Furthermore, low signal‐to‐noise ratio (SNR) capacity for the parallel optical wireless channels is obtained in Ref. [10]. It should be noted that the noises in Refs. [4–10] are all assumed to be independent with the input signal. This assumption is reasonable if the ambient light is strong or if the receiver suffers from intensive thermal noise. However, in practical VLC systems, typical illumination scenarios offer very high SNR [11, 12]. For high power, this assumption neglects a fundamental issue of VLC: due to the random nature of photon emission in the light emitting diode (LED), the strength of noise depends on the signal itself [13]. Up to now, the performance of the VLC with input‐dependent noise has not been discussed completely.

In this chapter, we consider a VLC system with input‐dependent Gaussian noise and investigate the fundamental performance of the VLC system. The main contributions of this chapter are given as follows:

A channel model with input‐dependent Gaussian noise for the VLC is considered. In existing literature, the noise is generally assumed to be independent of the signal. However, this assumption is not applicable to the VLC system in some cases. In this chapter, a more general channel model is established which is corrupted by an additive Gaussian noise, however, with noise variance depending on the signal itself.

The mutual information of the VLC system is analysed. Based on the channel model, the exact expression of the mutual information is derived. However, the exact expression of the mutual information is not in a closed form. After that, a closed‐form expression of the lower bound on the mutual information is derived.

The bit error rate (BER) of the VLC system is obtained. By employing the on‐off keying (OOK), the theoretical expression of the BER for the VLC system is derived. Moreover, some asymptotic behaviour for the BER is also presented.

To show the accuracy of the derived theoretical expressions, the theoretical results are thoroughly confirmed by Monte‐Carlo simulations.

The remainder of this chapter is organized as follows. The system model is described in Section 2. Section 3 presents the exact expression and the lower bound of the mutual information. In Section 4, the theoretical expression of the BER is derived. Numerical results are given in Section 5 before conclusions are drawn in Section 6.

## 2. System model

Consider a point‐to‐point VLC system, as shown in **Figure 2**. At the transmitter, an LED is employed as the lighting source, which performs the electrical‐to‐optical conversion. Then, the optical signal is propagated through the VLC channel. At the receiver, a PIN photodiode (PD) is used to perform the optical‐to‐electrical conversion. To amplify the derived electrical signal, a high impedance amplifier is employed. In this chapter, the main noise sources include thermal noise, shot noise and amplifier noise. The thermal noise and the amplifier noise are independent of the signal, and each of the two noise sources can be well modelled by Gaussian distribution [14]. Although its distribution can also be assumed to be Gaussian, the strength of the shot noise depends on the signal itself. Mathematically, the received electrical signal

where

In Eq. (1),

where

Note that the channel gain in Eq. (2) is a constant, where the positions of the LED and the PD are given. Moreover,

In VLC, information is transmitted by modulating the instantaneous optical intensity [17], and thus,

Due to the eye and skin safety regulations, the peak optical intensity of the LED is limited [17], that is,

where

Considering the illumination requirement in VLC, the average optical intensity cannot be changed but can be adjusted according to the users’ requirement (dimming target) [18]. Therefore, the average optical intensity constraint is given by

where

## 3. Mutual information analysis

Mutual information is an important performance indicator for wireless communication systems. In this section, the exact expression of the mutual information and the closed‐form expression of the lower bound on the mutual information for the VLC will be derived, respectively.

### 3.1. Exact expression of mutual information

Assume that

According to Eq. (3), the conditional probability density function (PDF) of

Furthermore, the PDF of

The mutual information between

(10) |

where

From Eq. (3), we have

(11) |

Therefore, Eq. (10) can be further written as

** Remark 1**: Let the average SNR be

which indicates that the maximum value of

Moreover, we have

** Remark 2**: When

### 3.2. Lower bound on mutual information

It should be noted that it is very hard to derive a closed‐form expression of Eq. (12). In this subsection, a lower bound on the mutual information will be derived.

To facilitate the description, Eq. (12) can be further expressed as

(16) |

For

(17) |

Using the Jensen’s inequality for concave function, an upper bound of

(18) |

Substituting Eqs. (17) and (18) into Eq. (16), a lower bound of

** Remark 3**: Obviously,

which indicates that the maximum value of

Moreover, we have

** Remark 4**: According to Eqs. (13) and (20), we have

Similarly, from Eqs. (14) and (21), we have

From Eqs. (22) and (23), it can be concluded that a constant performance gap

** Remark 5**: When

## 4. BER analysis

In this section, the BER of the VLC with input‐dependent noise is analysed. To facilitate the analysis, OOK is employed as the modulation scheme. Suppose that the transmitted optical signal is drawn equiprobably from the OOK constellation and

Therefore, the BER for the VLC with OOK can be written as

where

According to Eq. (8),

where

Moreover,

Therefore, the BER can be finally written as

** Remark 6**: Let the average SNR be

This indicates that the minimum BER and the maximum BER are 0 and 0.5, respectively.

** Remark 7**: When

## 5. Numerical results

In this section, some classical numerical results will be presented. The derived theoretical expressions of the mutual information, the lower bound of mutual information and the BER will be verified.

### 5.1. Results of mutual information

**Figure 3** shows the mutual information (i.e., **Figure 3**, it can be seen that *Remark 1*. Furthermore, the gap between *Remark 4*.

**Figure 4** shows the mutual information (i.e.,

### 5.2. Results of BER

**Figure 5** shows BER versus

**Figure 6** shows the BER versus the SNR with different

## 6. Conclusions

This chapter investigates the performance of the VLC with input‐dependent noise. The theoretical expression of the mutual information is derived, which is not in a closed form. Moreover, the closed‐form expression of the lower bound on the mutual information is obtained. Furthermore, by employing the OOK, the theoretical expression of the BER for the VLC is derived. Numerical results show that the derived theoretical expressions in this chapter are quite accurate to evaluate the system performance without time‐intensive simulations.