# Optimal Power Allocation for Integrated Visible Light Positioning and
Communication System with a Single LED-Lamp

Shuai Ma, Ruixin Yang, Bing Li, Yongyan Chen, Hang Li, Youlong Wu, Majid
Safari, Shiyin Li, and Naofal Al-Dhahir S. Ma, R. Yang, B. Li, Y. Chen and S.
Li are with the School of Information and Control Engineering, China
University of Mining and Technology, Xuzhou, 221116, China. (e-mail:
mashuai001@cumt.edu.cn).

###### Abstract

In this paper, we investigate an integrated visible light positioning and
communication (VLPC) system with a single LED-lamp. First, by leveraging the
fact that the VLC channel model is a function of the receiver’s location, we
propose a system model that estimates the channel state information (CSI)
based on the positioning information without transmitting pilot sequences.
Second, we derive the Cramer-Rao lower bound (CRLB) on the positioning error
variance and a lower bound on the achievable rate with on-off keying
modulation. Third, based on the derived performance metrics, we optimize the
power allocation to minimize the CRLB, while satisfying the rate outage
probability constraint. To tackle this non-convex optimization problem, we
apply the worst-case distribution of the Conditional Value-at-Risk (CVaR) and
the block coordinate descent (BCD) methods to obtain the feasible solutions.
Finally, the effects of critical system parameters, such as outage
probability, rate threshold, total power threshold, are revealed by numerical
results.

###### Index Terms:

Visible light communication, Visible light positioning, Power allocation,
Cramer-Rao lower bound.

##  I Introduction

With the explosively increasing number of Internet of Things (IoT) devices in
beyond fifth generation (B5G) networks, the crisis of radio frequency (RF)
spectrum shortage becomes increasingly challenging, which makes it more
difficult for the RF wireless systems to meet the high speed data transmission
and high accuracy positioning demands simultaneously[1]. It is worth noting
that more than 50$\%$ of voice traffic and 70$\%$ of wireless data traffic
occur in indoor environments [2]. Since the indoor activity requires both
illumination and network access, visible light communication (VLC) [3] and
visible light positioning (VLP)[4], which apply the ubiquitous light emitting
diodes (LEDs) as access points (APs) and anchor nodes, are promising
technologies for indoor IoT applications. Comparing with the conventional
radio frequency (RF) wireless technologies, the distinct advantages of VLC and
VLP are multifold [4] [5], including no electromagnetic interference, high
energy efficiency, high security, and low-cost.

VLC utilizes the simple intensity modulation and direct detection (IM/DD)
mechanism for information transmission, and has attracted significant research
interests as a breakthrough technology for B5G networks [6]. Extensive studies
have been reported to improve VLC networks performance. For example, by
adopting the alternating direction method of multipliers (ADMM), a distributed
coordinated interference management scheme is proposed in [7] for VLC
networks. To balance energy and bandwidth efficiency, both power allocation
and rate splitting are optimized in [8] for DC-biased optical orthogonal
frequency division multiplexing (DCO-OFDM). In [9], in order to jointly
optimize the post-equalizer, the precoder and the DC offset, a gradient
projection-based procedure is presented to minimize the sum mean squared error
(MSE) of the received symbols. Furthermore, VLC has been commercialized in
industry. Some startup companies and existing industry giants, such as
pureLiFi, Philips, and Oledcomm, are providing VLC commercial solutions in
home and business buildings.

Owning to its short wavelength and low multipath interference, VLP can achieve
high indoor positioning accuracy, which can facilitate various applications,
such as indoor navigation, location aware services, logistic management, and
assets tracking, to name few. By exploiting different visible light
characteristics, existing VLP schemes can apply time of arrival (TOA) [10]
[11], time difference of arrival (TDOA)[12], angle of arrival (AOA)[13] and
received signal strength (RSS)[14] techniques for positioning. Among the above
VLP schemes, the RSS based scheme is widely adopted due to its simplicity and
ubiquity, where the distance between the lamp base station (BS) and the device
is calculated based on the channel model. For example, by using the weighted
k-nearest-neighbor (K-NN), a multi-LEDs positioning system is designed in [15]
based on sparse fingerprints. In [16], an artificial neural network
(ANN)-based position estimator is proposed for 3D RSS-based VLP systems.

Most of the existing works only focus on VLC or VLP individually. In practical
indoor applications, an integrated system with both the communication and
positioning functions is highly desirable. So far, only few works considered
the integration of VLC and VLP. Specifically, VLC systems based on orthogonal
frequency division multiplexing access (OFDMA) [17, 18] were proposed to
estimate the receiver position. An integrated visible light positioning and
communication (VLPC) system was designed in [19] by using filter bank
multicarrier-based subcarrier multiplexing (FBMC-SCM). Such FBMC-SCM-assisted
VLP with its high signal processing complexity was proposed to reduce the out-
of-band interference (OOBI). Towards an OFDMA VLPC network, the authors in
[20] jointly optimized the AP selection, bandwidth allocation, adaptive
modulation, and power allocation to maximize the data rate while satisfying
positioning accuracy constraints. In [21], a modified experience replay actor-
critic (MERAC) reinforcement learning (RL) approach was presented to maximize
the sum rate under the users’ minimum data rates and positioning accuracy
requirements. In [22], the authors proposed a coordinated resource allocation
approach to maximize the sum rate while satisfying the minimum data rates and
positioning accuracy requirements of devices.

In the above considered VLPC systems, it is required that at least two lamp
signals are captured at the receiver simultaneously for effective positioning.
Unfortunately, such multi-lamp setup may not fit in many practical scenarios,
such as in a tunnel, corridor, and staircase, where the lamps are sparsely
installed. In these scenarios, the multi-lamp based method will not be as
efficient as in a large and flat room. In terms of system design, most of the
existing VLPC literatures [4, 23] mainly focus on optimizing the resource
allocation in different frequency bands to guarantee quality of service (QoS)
of communication and positioning. However, some fundamental issues have not
been well investigated. Particularly, does the positioning benefit or
compromise the communication? How are the two performances related? Given the
limited power consumption, how to balance the two performances while taking
the positioning error into account?

In this paper, we aim to address the above mentioned fundamental issues, as
well as to provide a robust beamforming and power allocation scheme. The main
contributions of this paper are summarized as follows:

  * •

We establish a VLPC system model with a single LED-lamp and a mobile user with
multiple photoelectric detectors (PD). By leveraging the fact that the VLC
channel model is a function of the receiver’s location, the lamp estimate the
channel state information (CSI) based on the positioning results, instead of
transmitting pilot sequences for CSI estimation, which can significantly
reduce the system overhead.

  * •

We derive the Cramer-Rao lower bound (CRLB) on the positioning error variance,
which is used as the VLP performance metric. In addition, we derive the
achievable rate expression for on-off keying (OOK) modulation, and its closed-
form lower bound. Furthermore, by exploiting CRLB and achievable rate
expressions, we reveal the inner relationship between VLP and VLC for the
first time, i.e., derive the distribution of the CSI error of VLC based on the
positioning error of VLP, and obtain a rate outage probability of VLC.

  * •

Based on the derived model and metrics, we further investigate a joint
positioning and communication power allocation and beamforming problem to
minimize the CRLB subject to rate outage constraints and power constraints.
The outage probability constraint makes the optimization problem non-convex.
Then, we apply the Conditional Value-at-Risk (CVaR) and the block coordinate
descent (BCD) method techniques to convert the original problem into two
convex VLP and VLC sub-problems. Finally, we develop a BCD algorithm for
robust VLPC design, in which the positioning and communication power
allocation are iteratively optimized until convergence.

The rest of this paper is organized as follows. We present the VLPC system
model in Section II. The key performance metrics for the VLPC system are
derived in Section III. In Section IV, we investigate the chance constrained
robust design. Extensive simulation results are presented in Section V.
Section VI concludes the paper. Moreover, Table I and II present the means of
the key notations and the main acronyms of this paper, respectively.

TABLE I: Summary of Key Notations Notation | Description
---|---
${P_{\text{p}}}$ |  | Allocated positioning power
---
${P_{\text{c}}}$ |  | Allocated communication power
---
${{\bf{u}}}$ |  | Location vector of MU
---
${{\bf{e}}_{\rm{p}}}$ |  | Positioning error vector
---
${\bf{\hat{h}}}$ |  | Estimated CSI vector
---
$\Delta{\bf{h}}$ |  | CSI estimation error vector
---
${\bf{I}}$ |  | Identity matrix
---
${{\bf{J}}_{\rm{p}}}$ |  | Fisher information matrix
---
$R_{\rm{c}}^{\rm{L}}$ |  | Lower bound on the the achievable rate
---
$\mathcal{P}$ |  | The set of distributions for ${\Delta{\bf{h}}}$
---
$\bar{r}$ |  | Minimum rate requirement
---
${P_{{\rm{out}}}}$ |  | Maximum tolerable outage probability
---
TABLE II: Summary of Main Acronyms Notation | Description
---|---
VLP |  | Visible light positioning
---
VLC |  | Visible light communication
---
VLPC |  | Visible light positioning and communication
---
MU |  | Mobile user
---
CRLB |  | Cramer-Rao lower bound
---
CSI |  | Channel state information
---
BCD |  | Block coordinate descent
---
CVaR |  | Conditional Value-at-Risk
---

Notations: Boldfaced lowercase and uppercase letters represent vectors and
matrices, respectively. $\mathcal{M}\triangleq\\{1,...,M\\}$. The transpose
and trace of a matrix are denoted as ${\left(\cdot\right)^{\text{T}}}$ and
${\text{Tr}}\left(\cdot\right)$, respectively. ${\left\|\cdot\right\|_{2}}$
denotes 2-norm. $\mathcal{N}$ denotes the Gaussian distribution. $\bf{0}$
denotes a column vector where all elements are $0$. $\mathbb{R}^{n}$
represents the space of $n$-dimensional real matrices. $\mathbb{S}^{n}$
represents the space of $n$-dimensional real symmetric matrices.

##  II System Model

Figure 1:  System model illustration.

Considering a VLPC system, as shown in Fig. 1, where the lamp is equipped with
a single LED that points straight downward, and a mobile user (MU) has a
receiver with $M$ PDs ($M\geq 3$)111For 3D positioning, the number of PDs is
at least 3.. Let
${\bf{l}}={\left[{{{x_{l}}},{y_{l}},{z_{l}}}\right]^{\text{T}}}$,
${{\bf{u}}}={\left[{{x_{u}},{y_{u}},{z_{u}}}\right]^{\text{T}}}$ and
${{\bf{u}}_{i}}={\left[{{x_{i}},{y_{i}},{z_{i}}}\right]^{\text{T}}}$ denote
the locations of the LED, the MU and the $i$th PD, respectively, where
$i\in\mathcal{M}$. Moreover, let
${{\mathbf{v}}_{i}}={\left[{{v_{x,i}},{v_{y,i}},{v_{z,i}}}\right]^{\text{T}}}$
denote the offset of the $i$th PD to the MU, i.e.,
${{\mathbf{u}}_{i}}={\mathbf{u}}+{{\mathbf{v}}_{i}}$.

The wireless channel of the VLPC system has two types of links, i.e. the line-
of-sight (LOS) link and the non line-of-sight (NLOS) link. Generally, the
influence of the LOS link is much stronger than that of the NLOS link [24]. In
order to facilitate the theoretical analysis, the design of VLPC system is
based only on the LOS link, but both the LOS link and NLOS link are considered
in simulation verification. According to the Lambert radiation model [25], the
LOS path gain between the LED and the $i$th PD within field-of-view (FoV) can
be expressed as

$\displaystyle{h_{i}}=$
$\displaystyle\frac{{\left({m+1}\right){A_{{\rm{PD}}}}}}{{2\pi
d_{i}^{2}}}{\cos^{m}}\left({{\phi_{i}}}\right)\cos\left({{\varphi_{i}}}\right)g{T_{f}}.$
(1)

Here, $m$ is the order of Lambertian emission and $m=-\frac{{\ln
2}}{{\ln\left({\cos{\theta_{1/2}}}\right)}}$, where ${{\theta_{1/2}}}$ is the
semi-angle at half power. Other parameters are defined as follows:
${A_{\text{PD}}}$ denotes the PD area; $d_{i}$ is the distance between the LED
and the $i$th PD; ${\phi_{i}}$ and ${\varphi_{i}}$ are the radiance and
incidence angles, respectively; $g$ denotes the gain of the optical
concentrator, and is given by
$g=\frac{{n_{r}^{2}}}{{{{\sin}^{2}}\left({{\psi_{\text{FoV}}}}\right)}}$,
where $n_{r}$ denotes the refractive index, and ${\psi_{\text{FoV}}}$
represents the FoV of receiver; and ${T_{f}}$ denotes the gain of the optical
filter.

Without loss of generality, assume that the PDs are pointing straight upward
[26]. Based on the geometric relationship, the LOS path gain (1) parameters
can be specified as

$\displaystyle{d_{i}}={\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|_{2}},$ (2a)
$\displaystyle\cos\left({{\phi_{i}}}\right)=\frac{{{{\left({{{\bf{u}}_{i}}-{\bf{l}}}\right)}^{\text{T}}}{{\bf{n}}_{{\rm{LED}}}}}}{{{{\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|}_{2}}}}=\frac{{{z_{l}}-{z_{i}}}}{{{{\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|}_{2}}}},$
(2b)
$\displaystyle\cos\left({{\varphi_{i}}}\right)=\frac{{{{\left({{\bf{l}}-{{\bf{u}}_{i}}}\right)}^{\text{T}}}{{\bf{n}}_{i}}}}{{{{\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|}_{2}}}}=\frac{{{z_{l}}-{z_{i}}}}{{{{\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|}_{2}}}},$
(2c)

where ${{\bf{n}}_{{\rm{LED}}}}={\left[{0,0,-1}\right]^{\rm{T}}}$ and
${{\bf{n}}_{i}}={\left[{0,0,1}\right]^{\rm{T}}}$ are unit direction vectors of
the LED and the $i$th PD, respectively. After substituting the above equations
into (1), the LOS path gain can be expressed as

$\displaystyle{h_{i}}=\frac{{\alpha{{\left({{z_{l}}-{z_{i}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{{\bf{u}}_{i}}}\right\|_{2}^{m{\rm{+3}}}}},$
(3)

where $\alpha=\frac{{\left({m+1}\right){A_{{\rm{PD}}}}g{T_{f}}}}{{2\pi}}$.

Figure 2: The frame structure of the considered VLPC system.

As shown in Fig. 2, the operational frame consists of three subframes:
positioning subframe (downlink), feedback subframe (uplink) and data
transmission subframe (downlink). The corresponding durations are
${T_{\rm{p}}}$, ${T_{\rm{u}}}$ and ${T_{\rm{c}}}$, respectively. More
specific, during the positioning subframe, the LED lamp transmits the
positioning symbols to the MU, which estimates the PDs’ locations based on the
RSS, and the corresponding positioning information will be used for channel
estimation in the next sub-frame. Then, during the feedback subframe, the MU
sends feedback signals of the PDs’ locations to the lamp which estimates the
CSI between the LED and the PDs. Finally, during the data transmission
subframe, the lamp transmits data symbols to the MU according to the estimated
CSI. This model can be extented into the multi-user system though proper
multiple access methods. Due to the positioning theory, the positioning signal
and subframe can be shared directly by all users without the multiple access.
However, the multiple acess method, such as OFDMA, time division multiple
address (TDMA), is necessary for the multi-user uplink feedback and downlink
data transmission. Thus, The problem and solution also should refer to the
classical theory of the multi-user networks.

###  II-A Positioning Signal Model and Measurements

In the following, we will specify the signal model in order to analyze the
operation in each subframe. Let ${s_{{\rm{p}}}}\left(t\right)$ denote the
positioning symbol generated at the lamp at time t, and
$\left|{{s_{\rm{p}}}}\left(t\right)\right|\leq A$,
${\mathbb{E}}\left\\{{{s_{\rm{p}}}}\left(t\right)\right\\}=0$,
${\mathbb{E}}\left\\{{s_{\rm{p}}^{2}}\left(t\right)\right\\}=\varepsilon$,
where $A>0$ is the peak amplitude.

For $t\in\left[{0,{T_{\rm{p}}}}\right]$, the transmitted positioning signal
$x_{\text{p}}\left(t\right)$ of the LED is given as

$\displaystyle{x_{\text{p}}}\left(t\right)=\sqrt{{P_{\text{p}}}}{s_{{\rm{p}}}}\left(t\right)+{I_{{\text{DC}}}},$
(4)

where ${P_{\text{p}}}$ indicates the allocated transmission power to the
positioning symbol, and ${I_{{\text{DC}}}}>0$ denotes the direct current (DC)
bias.

To guarantee that the transmitted signal is non-negative, the power
$P_{\rm{p}}$ should satisfy

$\displaystyle\sqrt{{P_{\rm{p}}}}A\leq{I_{{\rm{DC}}}}.$ (5)

Given the human eye safety requirement, the LED optical power is limited,
i.e.$\sqrt{{P_{\rm{p}}}}A+{I_{{\rm{DC}}}}\leq P_{\mathrm{o}}^{\max}$, where
$P_{\mathrm{o}}^{\max}$ denotes the maximum optical power. Thus, the power
level $P_{\rm{p}}$ should also satisfy

$\displaystyle\sqrt{{P_{\rm{p}}}}\leq\frac{{P_{\mathrm{o}}^{\max}-{I_{{\rm{DC}}}}}}{A}.$
(6)

Besides, due to practical circuit limitations, the electrical power of the
transmitted signal is constrained as
$\mathbb{E}\left\\{{x_{\text{p}}^{2}}\left(t\right)\right\\}\leq
P_{\mathrm{e}}^{\max}$, i.e.,

$\displaystyle{P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\leq
P_{\mathrm{e}}^{\max},$ (7)

where $P_{\mathrm{e}}^{\max}$ denotes the maximum LED electrical power.

Based on (5), (6) and (7), the constraint of the power $P_{\rm{p}}$ can be
written as

$\displaystyle
0\leq{P_{\text{p}}}\leq\min\left\\{\frac{{I_{{\rm{DC}}}^{2}}}{{{A^{2}}}},\frac{{{{\left({P_{\mathrm{o}}^{\max}-{I_{{\rm{DC}}}}}\right)}^{2}}}}{{{A^{2}}}},\frac{P_{\mathrm{e}}^{\max}-I_{{\rm{DC}}}^{2}}{\varepsilon}\right\\}.$
(8)

Then, the received positioning signal at the $i$th PD can be expressed as

$\displaystyle{y_{{\rm{p}},i}}\left(t\right)={h_{i}}{x_{\rm{p}}}\left(t\right)+{n_{{\rm{p}},i}}\left(t\right),$
(9)

where $n_{{\text{p}},i}$ denotes the received additive white Gaussian noise
(AWGN), which includes shot noise and thermal noise [27], and
${n_{{\rm{p}},i}}\sim{\cal N}\left({{{0}},\sigma_{\rm{p}}^{2}}\right)$.

Theoretically, the electrical power of the received positioning signal is
given by

$\displaystyle{P_{{\rm{r}},i}}={\mathbb{E}}\left\\{{y_{{\text{p}},i}^{2}}\left(t\right)\right\\}=\left({{P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}}\right){h_{i}^{2}}+\sigma_{{\text{p}},i}^{2},$
(10)

where $\sigma_{{\text{p}},i}^{2}$ denotes noise power. Combining (3) and (10),
we have the following $M$ equations

$\displaystyle\left\\{\begin{array}[]{l}\frac{{{{\left({{z_{l}}-{z_{u}}-{v_{z,1}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{1}}}\right\|_{2}^{m{\rm{+3}}}}}=\frac{1}{\alpha}{\left({\frac{{{P_{{\rm{r}},1}}-\sigma_{{\rm{p}},1}^{2}}}{{{P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}}}}\right)^{\frac{1}{2}}},\\\
{\rm{}}\vdots\\\
\frac{{{{\left({{z_{l}}-{z_{u}}-{v_{z,M}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{M}}}\right\|_{2}^{m{\rm{+3}}}}}=\frac{1}{\alpha}{\left({\frac{{{P_{{\rm{r}},M}}-\sigma_{{\rm{p}},M}^{2}}}{{{P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}}}}\right)^{\frac{1}{2}}}.\end{array}\right.$
(14)

To transform the Equations in (14) into a concise form, we define the
auxiliary variable

$\displaystyle{\eta_{i}}\left({\bf{u}}\right)=\frac{{{{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|_{2}^{m{\rm{+3}}}}}-\frac{1}{\alpha}{\left({\frac{{{P_{{\rm{r}},i}}-\sigma_{{\rm{p}},i}^{2}}}{{{P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}}}}\right)^{\frac{1}{2}}}.$
(15)

Thus, Equation (14) can be equivalently reformulated as follows

$\displaystyle{\eta_{i}}\left({\bf{u}}\right)=0,i\in\mathcal{M}.$ (16)

Here Equation (16) can be solved by using off-the-shelf optimization solvers,
such as FSOLVE in MATLAB[28].

In general, the positioning error is inevitable. Let ${{{\mathbf{\hat{u}}}}}$
and ${{\bf{e}}_{\rm{p}}}$ denote the estimated MU location and the
corresponding positioning error, where
${{\bf{e}}_{\rm{p}}}={\left[{{e_{x}},{e_{y}},{e_{z}}}\right]^{\text{T}}}$.
Their relationship can be written as

$\displaystyle{{\mathbf{e}}_{\text{p}}}={{\mathbf{u}}}-{{{\mathbf{\hat{u}}}}}.$
(17)

Generally, the positioning error ${{\mathbf{e}}_{\mathrm{p}}}$ can be assumed
to follow the Gaussian distribution [29, 30, 31], and then CRLB can be
achieved by the maximum-likelihood (ML) estimator [32, 33]. We use
$f_{\mathbf{e}_{\mathrm{p}}}\left(\mathbf{e}_{\rm{p}}\right)$ to denote the
probability density distribution of ${{\bf{e}}_{\rm{p}}}$, which follows a
Gaussian distribution with mean ${\bf{0}}$ and covariance matrix
${{\bf{E}}_{\rm{p}}}$, i.e.,
${{\bf{e}}_{\rm{p}}}\sim{\mathcal{N}}\left({{\bf{0}},{{\bf{E}}_{\rm{p}}}}\right)$.

###  II-B MU Feedback and Channel Estimation

When $t\in\left[{{T_{\rm{p}}},{T_{\rm{p}}}+{T_{\rm{u}}}}\right]$, the
estimated location of the MU ${{{\mathbf{\hat{u}}}}}$ will be sent to the
lamp, which also serves as an anchor node. Based on ${{{\mathbf{\hat{u}}}}}$,
the lamp can estimate the CSI between the LED and the PDs. Specifically,
${\bf{\hat{h}}}=\left[{{{\hat{h}}_{1}},...,{{\hat{h}}_{M}}}\right]^{\mathrm{T}}\in{\mathbb{R}^{M\times
1}}$ denotes the estimated CSI vector;
$\Delta{\bf{h}}=\left[{\Delta{h_{1}},...,\Delta{h_{M}}}\right]^{\mathrm{T}}\in{\mathbb{R}^{M\times
1}}$ denotes the CSI estimation error vector.

Let ${h_{i}}$ and ${\hat{h}_{i}}$ denote the perfect and estimated CSI between
the LED and the $i$th PD, and $\Delta{h_{i}}$ denote the estimated CSI error,
i.e., ${h_{i}}={\hat{h}_{i}}+\Delta{h_{i}}$. According to (3), the estimated
CSI ${\hat{h}_{i}}$ is a function of the estimated UE’s location
${{{\mathbf{\hat{u}}}}}$ given by

$\displaystyle{\hat{h}_{i}}=\frac{{\alpha{{\left({{z_{l}}-{{\hat{z}}_{u}}-{v_{z,i}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{\hat{u}}}-{{\bf{v}}_{i}}}\right\|_{2}^{m+3}}}.$
(18)

Based on (3) and (18), the estimated CSI error $\Delta{h_{i}}$ is given as

$\displaystyle\Delta{h_{i}}{\rm{}}=\frac{{\alpha{{\left({{z_{l}}-{{\hat{z}}_{u}}-{v_{z,i}}-{e_{z}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{\hat{u}}}-{{\bf{v}}_{i}}-{{\bf{e}}_{\rm{p}}}}\right\|_{2}^{m+3}}}-\frac{{\alpha{{\left({{z_{l}}-{{\hat{z}}_{u}}-{v_{z,i}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{\hat{u}}}-{{\bf{v}}_{i}}}\right\|_{2}^{m+3}}}.$
(19)

###  II-C Data Transmission

Let ${s_{{\rm{OOK}}}}\left(t\right)$ denote the data symbol transmitted from
the LED, and ${s_{{\rm{OOK}}}}\left(t\right)$ takes value $0$ or $A$ with
equal probability, i.e.,
$\Pr\left\\{{{s_{{\rm{OOK}}}}\left(t\right)=0}\right\\}=\frac{1}{2}$, and
${\Pr\left\\{{{s_{{\rm{OOK}}}}\left(t\right)=A}\right\\}=\frac{1}{2}}$, where
$A$ is the peak amplitude of the symbol. Due to $s_{\mathrm{OOK}}\geq 0$,
$I_{\mathrm{DC}}$ can be $0$ in the data transmission.

For
$t\in\left[{{T_{\rm{p}}}+{T_{\rm{u}}},{T_{\rm{p}}}+{T_{\rm{u}}}+{T_{\rm{c}}}}\right]$,
the LED transmitted data signal $x_{\rm{c}}\left(t\right)$ can be expressed as

$\displaystyle{x_{\text{c}}}\left(t\right)=\sqrt{{P_{\text{c}}}}{s_{{\text{OOK}}}}\left(t\right),$
(20)

where ${{P_{\text{c}}}}$ indicates the allocated communication power of the
LED.

Similarly, the communication power ${{P_{\text{c}}}}$ should also meet the eye
safety constraint, i.e. $\sqrt{{P_{\rm{c}}}}A\leq P_{\mathrm{o}}^{\max}$,
where $P_{\mathrm{o}}^{\max}$ denotes the maximum optical power. Thus, the
communication power $P_{\rm{c}}$ should satisfy

$\displaystyle\sqrt{{P_{\rm{c}}}}\leq\frac{{P_{\mathrm{o}}^{\max}}}{A}.$ (21)

Under practical circuit limitations, the electrical power of the transmitted
signal is constrained as
$\mathbb{E}\left\\{{x_{\text{c}}^{2}\left(t\right)}\right\\}\leq
P_{\mathrm{e}}^{\max}$, i.e.,

$\displaystyle{P_{\rm{c}}}\mathbb{E}\left\\{{s_{{\rm{OOK}}}^{2}}\left(t\right)\right\\}=\frac{P_{\mathrm{c}}A^{2}}{2}\leq
P_{\mathrm{e}}^{\max},$ (22)

where $P_{\mathrm{e}}^{\max}$ denotes the maximum LED electrical power.

Based on (21) and (22), the power $P_{\rm{c}}$ should satisfy

$\displaystyle
0\leq{P_{\text{c}}}\leq\min\left\\{{\frac{{{{\left({P_{\mathrm{o}}^{\max}}\right)}^{2}}}}{{{A^{2}}}},\frac{{2P_{\mathrm{e}}^{\max}}}{{{A^{2}}}}}\right\\}.$
(23)

At the receiver, let
${\mathbf{v}}=\left[{{v_{1}},...,{v_{M}}}\right]^{\text{T}}\in{\mathbb{R}^{M\times
1}}$ denote the receive beamforming vector of the MU, and
$\left\|{\bf{v}}\right\|=1$. Therefore, the received data signal at MU can be
expressed as

$\displaystyle{y_{\text{c}}}\left(t\right)={{\mathbf{v}}^{\text{T}}}{\left({{\bf{\hat{h}}}+\Delta{\bf{h}}}\right)}{x_{\text{c}}}\left(t\right)+z_{\text{c}},$
(24)

where
$z_{\text{c}}\buildrel\Delta\over{=}{{\mathbf{v}}^{\text{T}}}{{\mathbf{n}}_{\text{c}}}$,
and ${{\mathbf{n}}_{\text{c}}}\in{\mathbb{R}^{M\times 1}}$ denotes the
receiver Gaussian noise vector, i.e., ${{\bf{n}}_{\bf{c}}}\sim{\cal
N}\left({{\bf{0}},\sigma_{\rm{c}}^{2}{\bf{I}}}\right)$.

##  III Performance Metrics

###  III-A Cramer-Rao Lower Bound

The CRLB represents a lower bound on the variance of the positioning
estimation error. Hence, we adopt CRLB as the performance metric for the
positioning accuracy in this paper. Specifically, we consider three-
dimensional MU location estimation. Considering the received signal model in
(9), the likelihood function of ${{y_{{\rm{p}},i}}}\left(t\right)$ can be
written as

$\displaystyle
f\left({{y_{{\text{p}},i}}\left(t\right);{{\mathbf{u}}}}\right)=\frac{1}{{\sqrt{2\pi}{\sigma_{\rm{p}}}}}{e^{-\frac{{{{\left({{y_{{\rm{p}},i}}\left(t\right)-{h_{i}}{x_{\rm{p}}}\left(t\right)}\right)}^{2}}}}{{2\sigma_{\rm{p}}^{2}}}}}.$
(25)

Therefore, the log-likelihood function of the received signal
$\left\\{{{y_{{\rm{p}},i}}}\left(t\right)\right\\}_{i}^{M}$ is obtained as
follows [34]

$\displaystyle\Lambda\left({{{\mathbf{u}}}}\right)$
$\displaystyle=\ln\left(\mathop{\prod}\limits_{i=1}^{M}f\left({{y_{{\text{p}},i}}\left(t\right);{{\mathbf{u}}}}\right)\right)$
$\displaystyle=\ln\kappa-\frac{1}{{2{\sigma_{\rm{p}}^{2}}}}\sum\limits_{i=1}^{M}{\int_{0}^{T_{\rm{p}}}{{{\left({{y_{{\text{p}},i}}\left(t\right)-{h_{i}}{x_{\text{p}}}\left(t\right)}\right)}^{2}}\,\mathrm{d}t}},$
(26)

where $\kappa$ is a constant that does not depend on the unknown parameters.
Recalling the definition given in (17), and denoting by ${{\bf{E}}_{\rm{p}}}$
the covariance matrix of the positioning error ${{{\bf{e}}_{\rm{p}}}}$. Then,
according to the definition of the CRLB on the variance of any unbiased
estimator [35], a lower limit on the variance of the $i$th element in an
unbiased estimate vector ${\mathbf{\hat{u}}}$ is given by

$\displaystyle\sum_{i}{\left[{{{\bf{E}}_{\rm{p}}}}\right]_{ii}}\geq\sum_{i}{\left[{{\bf{J}}_{\rm{p}}^{-1}}\right]_{ii}},$
(27)

where ${\left[\cdot\right]_{ii}}$ denotes the diagonal element of a matrix,
and ${\bf{J}}_{\rm{p}}$ denotes the Fisher Information matrix (FIM), which is
defined as

$\displaystyle{\left[{{{\mathbf{J}}_{\text{p}}}}\right]_{ij}}=-{\mathbb{E}}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{{\mathbf{u}}_{i}}\partial{{\mathbf{u}}_{j}}}}}\right\\},$
(28)

for $i\in\left\\{1,2,3\right\\}$, $j\in\left\\{1,2,3\right\\}$. Likewise,
${\left[\cdot\right]_{ij}}$ denotes the element on the $i$th row and $j$th
column of a matrix. We show in Appendix A that the FIM for (III-A) is given by

$\displaystyle{\bf{J}}_{\rm{p}}$
$\displaystyle=\frac{{{T_{\rm{p}}}{\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}}{{\sigma_{\rm{p}}^{2}}}{\bf{Q}},$
(29)

where

$\displaystyle{\bf{Q}}=\left[{\begin{array}[]{*{20}{c}}{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}}\\\
{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}}\\\
{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}}&{\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}}\end{array}}\right],$
(30d)
$\displaystyle\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}=\frac{{-\alpha\left({m+3}\right){{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+1}}\left({{x_{u}}+{v_{x,i}}-{x_{l}}}\right)}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}},$
(30e)
$\displaystyle\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}=\frac{{-\alpha\left({m+3}\right){{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+1}}\left({{y_{u}}+{v_{y,i}}-{y_{l}}}\right)}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}},$
(30f)
$\displaystyle\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}=\frac{{-\left({m+1}\right)\alpha{{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m}}}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+3}}}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\frac{{\left({m+3}\right)\alpha{{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+2}}}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}}.$
(30g)

Moreover, let $B$ (Hz) denote the bandwidth of VLC link. Combine the bandwidth
of VLC link, the variance of positioning error ${{\bf{e}}_{\rm{p}}}$ is lower
bounded by

$\displaystyle{\text{Tr}}\left({{{\mathbf{E}}_{\text{p}}}}\right)\geq{\text{Tr}}\left({{\mathbf{J}}_{\text{p}}^{-1}}\right)=\frac{{B\sigma_{\text{p}}^{2}{\text{Tr}}\left({{{\mathbf{Q}}^{-1}}}\right)}}{{{T_{\text{p}}}\left({{P_{\text{p}}}\varepsilon+I_{{\text{DC}}}^{2}}\right)}}.$
(31)

In the following, we will use (31) as the positioning performance metric which
is to be minimized.

###  III-B Achievable Rate for OOK

When considering OOK modulation, the input signal no longer follows the
Gaussian distribution, and thus the Shannon capacity formula based on the
Gaussian assumption cannot be directly applied. In Appendix B, we show that
the mutual information is given by

$\displaystyle
I\left({{x_{\text{c}}};{y_{{\text{c}}}}}\right)=-\frac{1}{2}{\mathbb{E}_{{z_{\rm{c}}}}}\left\\{{{{\log}_{2}}\left({\frac{{{e^{-\frac{{z_{\rm{c}}^{2}}}{{2\sigma_{\rm{c}}^{2}}}}}+{e^{-\frac{{{{\left({{{\bf{v}}^{T}}{\bf{h}}\sqrt{{P_{\rm{c}}}}A+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}}{2}}\right)}\right\\}$
$\displaystyle-\frac{1}{2}{\mathbb{E}_{{z_{\rm{c}}}}}\left\\{{{{\log}_{2}}\left({\frac{{{e^{-\frac{{z_{\rm{c}}^{2}}}{{2\sigma_{\rm{c}}^{2}}}}}+{e^{-\frac{{{{\left({-{{\bf{v}}^{T}}{\bf{h}}\sqrt{{P_{\rm{c}}}}A+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}}{2}}\right)}\right\\}-\frac{1}{{2\ln
2}}.$ (32)

Due to the expectation operation, the expression (III-B) is not analytically
tractable, and can only be calculated numerically at the expense of high
computational complexity. To strike a balance between complexity and
analytical tractability, we derive a closed-form lower bound on the mutual
information (III-B).

Let $R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)$ denote a lower bound on
the achievable rate. Using Jensen’s Inequality and combining
${\bf{h}}={\bf{\hat{h}}}+\Delta{\bf{h}}$, we show in Appendix C that
$R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)$ with the bandwidth $B$ is
given by

$\displaystyle R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)=3B$
$\displaystyle-\frac{B}{{\ln 2}}$
$\displaystyle-2B{\log_{2}}\left({1+{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}\left({{\bf{\hat{h}}}+\Delta{\bf{h}}}\right)}\right)}^{2}}{P_{\rm{c}}}{A^{2}}}}{{4B\sigma_{\rm{c}}^{2}}}}}}\right).$
(33)

The CSI error $\Delta{\bf{h}}$ affects the achievable rate, and
${\Delta{\bf{h}}}$ is a function of the positioning error
${{{\mathbf{e}}_{\text{p}}}}$ as shown in Equation (19), which also depends on
the positioning signal power ${P_{\text{p}}}$. Therefore, both the positioning
signal power ${P_{\text{p}}}$ and communication signal power ${P_{\text{c}}}$
affect the achievable rate, and their allocations need to be carefully
optimized.

##  IV Robust Power Allocation for VLPC Design

In this section, we investigate the positioning error variance minimization
problem via a robust power allocation design by considering the rate outage
probability constraint. Different from existing works, this paper considers
the application of the VLPC system in the 3D case, and establishes the
connection between the rate outage probability and the positioning error for
the first time.

###  IV-A Problem Formulation

We derive the rate outage probability by investigating the relationship
between the positioning error and CSI error. Based on (19), the CSI error
${\Delta{\bf{h}}}$ is a function of the positioning error
${{{\mathbf{e}}_{\text{p}}}}$. Thus, let
$\Delta{h_{i}}\buildrel\Delta\over{=}g_{i}\left({{\bf{e}}_{\rm{p}}}\right)$
denote the CSI error function of the $i$th PD, i.e.,

$\displaystyle\Delta{h_{i}}={g_{i}}\left({{{\bf{e}}_{\rm{p}}}}\right)=$
$\displaystyle\frac{{\alpha{{\left({{z_{l}}-{{\hat{z}}_{u}}-{v_{z,i}}-{e_{z}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{\hat{u}}}-{{\bf{v}}_{i}}-{{\bf{e}}_{\rm{p}}}}\right\|_{2}^{m+3}}}$
$\displaystyle\quad\quad\quad\quad-\frac{{\alpha{{\left({{z_{l}}-{{\hat{z}}_{u}}-{v_{z,i}}}\right)}^{m+1}}}}{{\left\|{{\bf{l}}-{\bf{\hat{u}}}-{{\bf{v}}_{i}}}\right\|_{2}^{m+3}}},$
(34)

where $i\in\mathcal{M}$.

Then, we can write
${{\bf{e}}_{\rm{p}}}=g_{i}^{-1}\left({\Delta{h_{i}}}\right)$, and the
probability density function ${f_{{h_{i}}}}\left({\Delta{h_{i}}}\right)$ is
given by

$\displaystyle{f_{{h_{i}}}}\left({\Delta{h_{i}}}\right)={f_{\bf{e}_{\mathrm{p}}}}\left({g_{i}^{-1}\left({\Delta{h_{i}}}\right)}\right)\left|{\frac{{\partial
g_{i}^{-1}\left({\Delta{h_{i}}}\right)}}{{\partial\Delta{h_{i}}}}}\right|.$
(35)

Unfortunately, an explicit expression of the function
${{\bf{e}}_{\rm{p}}}=g_{i}^{-1}\left({\Delta{h_{i}}}\right)$ is difficult to
derive.

Nonetheless, we can numerically calculate both the mean and covariance matrix
of the CSI error vector ${\Delta{\bf{h}}}$. Specifically, let
${\bm{\mu}}=\mathbb{E}\left\\{{\Delta{\mathbf{h}}}\right\\}\in{\mathbb{R}^{M\times
1}}$ denote the mean vector of the estimated CSI error ${\Delta{\bf{h}}}$,
which is given as

$\displaystyle\mathbb{E}\left\\{{\Delta{{h_{i}}}}\right\\}=\int{{g_{i}}\left({{{\bf{e}}_{\rm{p}}}}\right){{f_{{{\bf{e}_{\mathrm{p}}}}}}\left({{\bf{e}}_{\rm{p}}}\right)}\,\mathrm{d}{{\bf{e}}_{\rm{p}}}}.$
(36)

Furthermore, let
${\bf{D}}=\mathbb{E}\left\\{{\left({\Delta{\bf{h}}-{\bm{\mu}}}\right){{\left({\Delta{\bf{h}}-{\bm{\mu}}}\right)}^{T}}}\right\\}\in{\mathbb{R}^{M\times
M}}$ denote the covariance matrix of the estimated CSI error vector
${\Delta{\bf{h}}}$. Then, the element on the $i$th row and $j$th column of
${\bf{D}}$ is given by

$\displaystyle{\left[{\mathbf{D}}\right]_{ij}}=\mathbb{E}\left\\{\left({\Delta{h_{i}}-\mathbb{E}\left\\{{\Delta{h_{i}}}\right\\}}\right)\left({\Delta{h_{j}}-\mathbb{E}\left\\{{\Delta{h_{j}}}\right\\}}\right)\right\\},$
(37)

where $i\in\mathcal{M}$, and $j\in\mathcal{M}$.

The exact distribution of CSI errors ${\Delta{\bf{h}}}$ is unknown except for
its first and second-order moments. Then, we may define a set $\mathcal{P}$ of
distributions for ${\Delta{\bf{h}}}$ as follows

$\displaystyle\mathcal{P}=\left\\{{\mathbb{P}:{\mathbb{E}_{\mathbb{P}}}\left\\{{\Delta{\bf{h}}}\right\\}={\bm{\mu}},{\rm{Var}_{\mathbb{P}}}\left\\{{\Delta{\bf{h}}}\right\\}={\bf{D}}}\right\\},$
(38)

where $\mathbb{P}$ denotes an arbitrary distribution with the mean
${\bm{\mu}}$ and covariance matrix ${\bf{D}}$. The set $\mathcal{P}$ in (38)
determines the CSI error variation, and the rate outage probability.

Now, we can formulate the positioning error variance minimization through
robust power allocation problem as follows

$\displaystyle\mathop{\min}\limits_{{P_{\rm{p}}},{P_{\rm{c}}},{\bf{v}}}$
$\displaystyle{\rm{Tr}}\left({{\mathbf{J}}_{\text{p}}^{-1}}\right)$ (39a)
$\displaystyle{\rm{s}}.{\rm{t}}.$
$\displaystyle\mathop{\sup}\limits_{\Delta{\bf{h}}\sim{\mathbb{P}},~{}\mathbb{P}\in{\cal
P}}\Pr\left\\{{R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)\leq\bar{r}}\right\\}\leq{P_{{\rm{out}}}},$
(39b) $\displaystyle{P_{\rm{p}}}+{P_{\rm{c}}}\leq{{P_{\rm{T}}}},$ (39c)
$\displaystyle 0\leq{P_{\text{p}}}\leq P_{\rm{p}}^{\max},$ (39d)
$\displaystyle 0\leq{P_{\text{c}}}\leq P_{\rm{c}}^{\max},$ (39e)
$\displaystyle\left\|{\bf{v}}\right\|^{2}=1,$ (39f)

where $\bar{r}$ denotes the minimum rate requirement, ${P_{{\rm{out}}}}$
denotes the maximum tolerable outage probability, ${{P_{\rm{T}}}}$ denotes the
total power,
$P_{\rm{p}}^{\max}\buildrel\Delta\over{=}\min\left\\{\frac{{I_{{\rm{DC}}}^{2}}}{{{A^{2}}}},\frac{{{{\left({P_{\mathrm{o}}^{\max}-{I_{{\rm{DC}}}}}\right)}^{2}}}}{{{A^{2}}}},\frac{P_{\mathrm{e}}^{\max}-I_{{\rm{DC}}}^{2}}{\varepsilon}\right\\}$,
and
$P_{\rm{c}}^{\max}\buildrel\Delta\over{=}\min\left\\{{\frac{{{{\left({P_{\mathrm{o}}^{\max}}\right)}^{2}}}}{{{A^{2}}}},\frac{{2P_{\mathrm{e}}^{\max}}}{{{A^{2}}}}}\right\\}$.

###  IV-B Proposed Robust VLPC Method

The main challenge of problem (39) lies in the chance constraint (39b), which
does not have a closed-form expression. Hence, we will reformulate constraint
(39b). Combined with the lower bound on the achievable rate in (III-B), the
inequality ${R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)\leq\bar{r}}$ can
be equivalently rewritten as

$\displaystyle{\left({{{\bf{v}}^{\text{T}}}\left({{\bf{\hat{h}}}+\Delta{\bf{h}}}\right)}\right)^{2}}\leq\frac{\delta}{{{P_{\rm{c}}}}},$
(40)

where
$\delta\buildrel\Delta\over{=}-\frac{{4B\sigma_{\rm{c}}^{2}}}{{{A^{2}}}}\ln\left({{2^{\frac{3}{2}-\frac{1}{{2\ln
2}}-\frac{{\bar{r}}}{{2B}}}}-1}\right)$. By using the following equivalence
relationship

$\displaystyle{\bf{V}}={\bf{v}}{{\bf{v}}^{\text{T}}}\Leftrightarrow{\bf{V}}\succeq{\bf{0}},{\rm{rank}}\left({\bf{V}}\right)=1,$
(41)

and neglecting the non-convex rank constraint
${\rm{rank}}\left({\bf{V}}\right)=1$, constraints (40) and (39f) can be
respectively relaxed as

$\displaystyle\Delta{{\bf{h}}^{\text{T}}}{\bf{V}}\Delta{\bf{h}}+2{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V}}\Delta{\bf{h}}+{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V\hat{h}}}\leq\frac{\delta}{{{P_{\rm{c}}}}},$
(42a)
$\displaystyle{\rm{Tr}}\left({\bf{V}}\right)=1,~{}{\bf{V}}\succeq{\bf{0}},$
(42b)

In words, we exploit the semidefinite relaxation (SDR) technique to relax (40)
to a semidefinite program (SDP). Then, the outage constraint (39b) can be
recast as

$\displaystyle\Pr\left\\{{\Delta{{\bf{h}}^{\text{T}}}{\bf{V}}\Delta{\bf{h}}+2{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V}}\Delta{\bf{h}}+{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V\hat{h}}}-\frac{\delta}{{{P_{\rm{c}}}}}\leq
0}\right\\}\leq{P_{{\rm{out}}}}.$ (43)

An effective approach to proceed is to transform (43) into a distributionally
robust chance constraint. Then, we can find the worst-case distribution among
all the possible distributions from the ambiguity set, i.e.,

$\displaystyle{\inf_{\mathbb{P}\in\mathcal{P}}}{\mathrm{Pr}_{\mathbb{P}}}$
$\displaystyle\left\\{\Delta\mathbf{h}^{\mathrm{T}}(-\mathbf{V})\Delta\mathbf{h}+2\Delta\mathbf{h}^{\mathrm{T}}(-\mathbf{V})\hat{\mathbf{h}}+\hat{\mathbf{h}}^{\mathrm{T}}(-\mathbf{V})\hat{\mathbf{h}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\left.+\frac{\delta}{P_{\mathrm{c}}}\leq
0\right\\}\geq 1-{P_{{\rm{out}}}},$ (44)

where
$\mathop{\inf}\limits_{\mathbb{P}\in\mathcal{P}}{\Pr_{\mathbb{P}}}\left\\{\cdot\right\\}$
denotes the distribution that can achieve the minimum value of the
probability.

To further deal with the intractability of (IV-B), we introduce a CVaR-based
method [36], which is known as a good convex approximation of the worst-case
chance constraint.

$\textbf{Lemma}\ 1$ (CVaR-Based Method): For a constraint function $L$ that is
concave or quadratic in ${\bm{\xi}}$, the distributionally robust chance
constraint is equivalent to the worst-case constraint, given by [37]

$\displaystyle\mathop{\inf}\limits_{\mathbb{P}\in\mathcal{P}}{{\Pr}_{\mathbb{P}}}\left\\{{L\left({\bm{\xi}}\right)\leq
0}\right\\}$ $\displaystyle\geq 1-\rho$ $\displaystyle\Leftrightarrow$
$\displaystyle\mathop{\sup}\limits_{\mathbb{P}\in\mathcal{P}}\left\\{\mathbb{P}-{{\text{CVaR}}_{\rho}}\left\\{{L\left({\bm{\xi}}\right)}\right\\}\right\\}\leq
0,$ (45)

where the expression
$\mathbb{P}-{{\text{CVaR}}_{\rho}}\left\\{{L\left({\bm{\xi}}\right)}\right\\}$
denotes the CVaR of function $L\left({\bm{\xi}}\right)$ at threshold $\rho$
under distribution $\mathbb{P}$, which is defined as

$\displaystyle\mathbb{P}-{\text{CVa}}{{\text{R}}_{\rho}}\left\\{{L\left({\bm{\xi}}\right)}\right\\}=\mathop{\inf}\limits_{\beta\in\mathbb{R}}\left\\{{\beta+\frac{1}{\rho}{\mathbb{E}_{\mathbb{P}}}\left[{{{\left({L\left({\bm{\xi}}\right)-\beta}\right)}^{+}}}\right]}\right\\}.$
(46)

Here, $\mathbb{R}$ is the set of real numbers,
${\left(z\right)^{+}}=\max\left\\{{0,z}\right\\}$, and $\beta\in\mathbb{R}$ is
an auxiliary variable introduced by CVaR. The worst-case CVaR on the right
hand side of (IV-B) can be converted into a group of SDPs, which will be shown
in the following lemma.

$\textbf{Lemma}\ 2$: Let
$L\left({\bm{\xi}}\right)={{\bm{\xi}}^{\text{T}}}{\mathbf{Q}}{\bm{\xi}}+{{\mathbf{q}}^{\text{T}}}{\bm{\xi}}+{{\mathbf{q}}^{0}}$
denote a quadratic function of ${\bm{\xi}}$,
$\forall{\bm{\xi}}\in{\mathbb{R}^{n}}$. The worst-case CVaR can be computed as
[37]

$\displaystyle\mathop{\sup}\limits_{\mathbb{P}\in\mathcal{P}}$
$\displaystyle\left\\{\mathbb{P}-{{\text{CVaR}}_{\rho}}\left\\{{L\left({\bm{\xi}}\right)}\right\\}\right\\}=\mathop{\min}\limits_{\beta,{\mathbf{M}}}\left\\{\beta+\frac{1}{\rho}{\text{Tr}}\left({{\mathbf{\Omega
M}}}\right)\right\\}$ (47a) $\displaystyle{\text{s}}.{\text{t}}.$
$\displaystyle{\mathbf{M}}\succeq{\mathbf{0}},{\mathbf{M}}\in{\mathbb{S}^{n+1}},$
(47b)
$\displaystyle{\mathbf{M}}-\left[{\begin{array}[]{*{20}{c}}{\mathbf{Q}}&{\frac{1}{2}{\mathbf{q}}}\\\
{\frac{1}{2}{\mathbf{q}}^{\mathrm{T}}}&{{{\mathbf{q}}^{0}}-\beta}\end{array}}\right]\succeq{\mathbf{0}},$
(47e)

where ${\mathbf{M}}$ is an auxiliary matrix variable, and ${\mathbf{\Omega}}$
is a matrix defined as

$\displaystyle{\mathbf{\Omega}}=\left[{\begin{array}[]{*{20}{c}}{{\mathbf{\Sigma}}+{\bm{\mu}}{{\bm{\mu}}^{\text{T}}}}&{\bm{\mu}}\\\
{{{\bm{\mu}}^{\text{T}}}}&{\mathbf{1}}\end{array}}\right],$ (50)

where ${\bm{\mu}}\in{\mathbb{R}^{n}}$ and
${\mathbf{\Sigma}}\in{\mathbb{S}^{n}}$ are the mean vector and covariance
matrix of random vector ${\bm{\xi}}$, respectively.

Let define the continuous quadratic function
$L\left({\Delta{\bf{h}}}\right)=\Delta\mathbf{h}^{\mathrm{T}}(-\mathbf{V})\Delta\mathbf{h}+2\Delta\mathbf{h}^{\mathrm{T}}(-\mathbf{V})\hat{\mathbf{h}}+\hat{\mathbf{h}}^{\mathrm{T}}(-\mathbf{V})\hat{\mathbf{h}}+\frac{\delta}{P_{\mathrm{c}}}$.
By Lemma $2$, the worst-case chance constraint in (IV-B) can be computed by
the optimization problem as similarly as the problem (47). Then, according to
the Lemma $1$, the problem can be equivalent to the following CVaR
constraints:

$\displaystyle\beta+\frac{1}{{{P_{{\rm{out}}}}}}{\rm{Tr}}\left({{\bf{\Omega
M}}}\right)\leq 0,$ (51a)
$\displaystyle{\bf{M}}-\left[{\begin{array}[]{*{20}{c}}-{\bf{V}}&-{{{\bf{V}}^{\text{T}}}{\bf{\hat{h}}}}\\\
-{{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V}}}&-{{{{\bf{\hat{h}}}}^{\text{T}}}{\bf{V\hat{h}}}+\frac{\delta}{{{P_{\rm{c}}}}}-\beta}\end{array}}\right]\succeq{\bf{0}},$
(51d) $\displaystyle{\bf{M}}\succeq{\bf{0}},{\bf{M}}\in{\mathbb{S}}{{}^{4}},$
(51e)

where ${\bf{M}}$ and $\beta$ are two auxiliary variables, and

${\bf{\Omega}}=\left[{\begin{array}[]{*{20}{c}}{{\bf{D}}+{\bm{\mu}}{{\bm{\mu}}^{\text{T}}}}&{\bm{\mu}}\\\
{\bm{\mu}}^{\text{T}}&{\bf{1}}\end{array}}\right].$

Therefore, the original distributionally chance-constrained problem (39) can
be reformulated as follows

$\displaystyle\mathop{\min}\limits_{{P_{\rm{p}}},{P_{\rm{c}}},{\bf{V}},{\bf{M}},\beta}\;~{}$
$\displaystyle{\rm{Tr}}\left({{\bf{J}}_{\rm{p}}^{-1}}\right)$ (52a)
$\displaystyle{\rm{s.t.}}~{}$
$\displaystyle\eqref{robust_cons_Ptotal},\eqref{constraint_Pp},\eqref{constraint_Pc},\eqref{semidefinite
V2},\eqref{CRaV_constraints1},\eqref{CRaV_constraints2},\eqref{CRaV_constraints3}.$

Note that, problem (52) is still non-convex given that the optimization
variables $P_{\rm{p}}$ and ${{\bf{V}}}$ are coupled together in constraint
(51a). However, the problem (52) can be decomposed into two convex subproblems
with two decoupling variables blocks:
$\left\\{{{P_{\rm{c}}},{\bf{V}},{\bf{M}},\beta}\right\\}$ and ${P_{\rm{p}}}$,
respectively. It means that when one of blocks is fixed, the problem becomes
convex in the remaining block of variables, which is called the multi-convex
problem[38]. To solve this kind of multi-convex problem, we propose an
efficient BCD algorithm [39] for robust VLPC design with variables coupling,
which can guarantee to globally converge to the stationary point [40, 41].
Then, at every iteration, the two convex subproblems, i.e., VLP subproblems
and VLC subproblems, are alternatively optimized with respect to one block
variable while the remaining blocks are held fixed. More specifically, for the
$k$th iteration, the VLP and VLC subproblems are optimized follows.

####  IV-B1 VLP subproblem

For fixing variables $P_{\mathrm{c}}^{\left({k-1}\right)}$, the positioning
power $P_{\rm{p}}^{\left({k}\right)}$ is updated via solving the following
convex VLP subproblem

$\displaystyle\mathop{\min}\limits_{{P_{\rm{p}}}}$
$\displaystyle~{}{\rm{Tr}}\left({{\bf{J}}_{\rm{p}}^{-1}}\right)$ (53)
$\displaystyle{\rm{s}}.{\rm{t}}.$
$\displaystyle~{}\eqref{robust_cons_Ptotal},~{}\eqref{constraint_Pp},$

which can be solved using the interior point methods, such as CVX [42].

####  IV-B2 VLC subproblem

With given positioning power $P_{\rm{p}}^{\left({k}\right)}$, the block
variables
$\left\\{P_{\mathrm{c}}^{\left({k}\right)},{\bf{V}}^{\left({k}\right)},\mathbf{M}^{\left(k\right)},\beta^{\left(k\right)}\right\\}$
are updated by solving the following convex VLC subproblem

$\displaystyle\mathop{\min}\limits_{{P_{\rm{c}}},{\bf{V}},{\bf{M}},\beta}$
$\displaystyle~{}{P_{\rm{c}}}$ (54) $\displaystyle{\rm{s}}.{\rm{t}}.$
$\displaystyle~{}\eqref{constraint_Pc},~{}\eqref{semidefinite
V2},~{}\eqref{CRaV_constraints1},~{}\eqref{CRaV_constraints2},~{}\eqref{CRaV_constraints3}.$

In summary, the overall BCD algorithm for robust VLPC design is listed in
Algorithm 1. The solution of the BCD Algorithm 1 is a stationary point of the
joint optimization problem (39) [43, 44]. Note that, due to the SDR, the rank
of ${{{\bf{V}}^{{\left(k\right)}}}}$ may not be $1$. For
${\rm{rank}}\left({{{\bf{V}}^{{\left(k\right)}}}}\right)=1$, the optimal
beamformer $\bf{v}$ can be calculated by eigenvalue decomposition. When
${\rm{rank}}\left({{{\bf{V}}^{{\left(k\right)}}}}\right)>1$, we can calculate
a high-quality feasible solution $\bf{v}$ of problem (54) based on the
Gaussian randomization procedure [45]. Meanwhile, the two SDP problem can be
efficiently solved with a worst case complexity
$\mathcal{O}\left(\max\left\\{m,n\right\\}^{4}n^{0.5}\log\delta^{-1}\right)$,
where $n$ is the problem size $n$, $m$ denotes the number of constraints $m$,
and $\delta$ represents the accuracy of SDP [45]. And the proposed BCD
algorithm has a sub-linear convergence rate,
$\mathcal{O}\left(\frac{1}{k}\right)$, where $k$ is the index of iteration
[46].

Algorithm 1 Block Coordinate Descent Algorithm for Robust VLPC Design

1:Initialize ${\bar{r}}$, ${P_{{\rm{out}}}}$, $P_{\text{p}}^{\left(0\right)}$,
$k=0$ and set the tolerance of accuracy $\epsilon>0$;

2:repeat

3: $k\leftarrow k+1$;

4: Update
$\left\\{P_{c}^{\left({k}\right)},{{\bf{V}}^{\left({k}\right)}},\mathbf{M}^{\left(k\right)},\beta^{\left(k\right)}\right\\}$
by solving VLC subproblem (54) with fixed $P_{\rm{p}}^{\left({k-1}\right)}$;

5: Update $P_{\rm{p}}^{\left({k}\right)}$ by solving VLP subproblem (53) with
given
$\left\\{P_{c}^{\left({k}\right)},{{\bf{V}}^{\left({k}\right)}},\mathbf{M}^{\left(k\right)},\beta^{\left(k\right)}\right\\}$;

6:until
$\left|{{\rm{Tr}}{{\left({{\bf{J}}_{\rm{p}}^{-1}}\right)}^{\left(k\right)}}-{\rm{Tr}}{{\left({{\bf{J}}_{\rm{p}}^{-1}}\right)}^{\left({k-1}\right)}}}\right|\leq\epsilon$;

7:$P_{\rm{p}}^{\left({k}\right)}$, $P_{\rm{c}}^{\left({k}\right)}$ and
${{{\bf{V}}^{{\left(k\right)}}}}$.

##  V  Simulation results

In this section, we present simulation results to evaluate the effectiveness
of the proposed VLPC system design. Consider a VLPC system in a room with size
$\left({5\times 5\times 3{m^{3}}}\right)$, where one corner of the room is the
origin $\left({0,0,0}\right)$ of the Cartesian coordinate system
$\left({X,Y,Z}\right)$. Assume that the LED location is
$\left({2.5,2.5,3}\right)$ and the MU is equipped with $M=3$ PDs.

Moreover, as shown in Fig. 3, we verify the performance of the proposed
optimization method for four different horizontal locations of the MU, i.e.,
${U_{1}}\left({1,1,z_{u}}\right)$, ${U_{2}}\left({1.5,1.5,z_{u}}\right)$,
${U_{3}}\left({2,2,z_{u}}\right)$ and ${U_{4}}\left({2.5,2.5,z_{u}}\right)$,
where the PDs are arranged according to an equilateral triangle with side
length $L$. The other simulation parameters are summarized in Table III.

Figure 3: Locations of MU and LED. TABLE III: Basic Simulation Parameters Parameters | Value
---|---
FoV, ${\psi_{{\rm{FoV}}}}$ | ${90^{\circ}}$
Detector area of PD, $A_{\rm{PD}}$ | $1{\rm{c}}{{\rm{m}}^{2}}$
Half power angle, ${\theta_{1/2}}$ | ${60^{\circ}}$
Gain of an optical filter, $T_{f}$ | 1
Gain of an optical concentrator, $g$ | 1
DC bias, $I_{\rm{DC}}$ | 2
Peak amplitude, $A$ | 0.007
Maximum optical power, $P_{\mathrm{o}}^{\max}$ | 8W
Maximum electrical power, $P_{\mathrm{e}}^{\max}$ | 16W
Bandwidth, $B$ |  $20$MHz
Noise PSD of positioning signal, $\sigma_{\rm{p}}^{2}$ | ${{{{10}^{-21}}{\rm{Watts}}}\mathord{\left/{\vphantom{{{{10}^{-21}}{\rm{Watts}}}{{\rm{Hz}}}}}\right.\kern-1.2pt}{{\rm{Hz}}}}$
Noise PSD of data signal, $\sigma_{\rm{c}}^{2}$ | ${{{{10}^{-21}}{\rm{Watts}}}\mathord{\left/{\vphantom{{{{10}^{-21}}{\rm{Watts}}}{{\rm{Hz}}}}}\right.\kern-1.2pt}{{\rm{Hz}}}}$

###  V-A Positioning Performance

First of all, it should be noted that the positioning error in this section is
the root mean square error (RMSE), which is the error between the average
estimated value obtained from multiple measurements and the true value. The
positioning symbol with normalized power is considered, i.e., $\varepsilon=1$.
The received SNR is defined as the received SNR at the $1$th PD, i.e,
${\rm{SNR=10lg}}\frac{{{P_{\rm{p}}}h_{1}^{2}}}{{B\sigma_{\rm{p}}^{2}}}$.

(a)

(b)

Figure 4:  (a) Positioning error versus SNR when the test point is chosen at
different locations, where $L=0.1m$; (b) Positioning error versus SNR when the
transceiver height difference $\Delta z={z_{l}}-{z_{u}}$ is different, where
$L=0.1m$.

Fig. 4 (a) illustrates the positioning error at the four test points versus
received SNR, where $z_{u}=1m$. We may observe that the positioning error
decreases rapidly at first and then slowly, and finally converge to a constant
as SNR increases. This is because as SNR increases, the influence of the noise
decreases. For high SNR, the localization performance is negligibly affected
by the noise, but still affected by the NLOS link. In addition, it can be seen
from the Fig. 4 (a) that the positioning error at test points $U_{1}$,
$U_{2}$, $U_{3}$ and $U_{4}$ gradually decreases at the same SNR. This is
because $U_{4}$ is the closest to the lamp, while $U_{1}$ is the farthest.
Therefore, when the total transmit power is constant, the SNR at $U_{4}$,
$U_{3}$, $U_{2}$ and $U_{1}$ decreases.

In Fig. 4 (b), we plot the positioning error at test point $U_{1}$ versus SNR,
for different MU heights. The shapes of the curves are similar to Fig. 4 (a),
and the reasons are also similar.

Figure 5: Positioning error versus $L$(cm), where $\Delta z=2m$.

Fig. 5 shows the positioning error versus side length $L$ at high and low SNR.
As we can see, with the increase of relative distance $L$, the positioning
error first decreases rapidly and then slowly when the SNR=5dB, while the
position error first decreases slowly and then remains unchanged when the
SNR=15dB. This is because the larger the relative distance $L$ is, the greater
the difference in signal intensity received by each PD will be. Thus, the
solution of the nonlinear equations in (16) will be more accurate, especially
at low SNR, and the impact of the signal strength difference on accuracy is
more obvious. At the same time, when the relative distance $L$ reaches a
certain value, additional increases will not help. In addition, when $L$ is
constant, the positioning error at $U_{1}$, $U_{2}$ and $U_{3}$ still
decreases sequentially, and the gap between the positioning errors at $U_{2}$
and $U_{3}$ becomes smaller with increasing $L$.

###  V-B Communication Performance

To evaluate the communication performance, we first introduce the non-robust
VLPC design scheme, which ignores the CSI uncertainty $\Delta{\bf{h}}$, and
the estimated CSI $\widehat{\bf{h}}$ is viewed as the perfect CSI ${\bf{h}}$.
Moreover, in our simulations, we choose the following basic parameters: length
of positioning subframe $T_{\mathrm{p}}=0.12\,\mathrm{sec}$ side length
$L=0.1\,\mathrm{m}$ and the test point $U_{3}$ with $z_{u}=1.5\,\mathrm{m}$.

(a)

(b)

Figure 6: In the non-robust VLPC design, the robust VLPC design with outage
probabilities ${P_{{\rm{out}}}}=1\%$ and $5\%$, and the equal power allocation
design: (a) CDF of achievable rate; (b) CDF of CRLB.

Fig. 6 (a) depicts the CDF of achievable rate of the non-robust VLPC design,
the robust VLPC design with the maximum tolerated outage probabilities
${P_{{\rm{out}}}}=1\%$ and $5\%$, and the equal power allocation design
(${P_{\rm{p}}}={P_{\rm{c}}}$), where the total power is ${P_{{\rm{T}}}}=10W$.
It can be seen that the outage probability of the non-robust VLPC design is
$50\%$, which significantly exceeds the maximum tolerated outage probability
requirement. On the other hand, the outage probability of the proposed robust
VLPC design is lower than $5\%$, which meets the outage probability
requirement. Fig. 6 (b) depicts the CDF of CRLB with the same parameters as
Fig. 6 (a). As can be seen from Fig. 6 (b), the positioning error of the
robust VLPC design with ${P_{{\rm{out}}}}=1\%$ is lager than that of the
design with ${P_{{\rm{out}}}}=5\%$. Combined with Fig. 6 (a), the robust VLPC
design allocates more power to communication than non-robust design under the
premise of minimizing positioning accuracy in order to meet the minimum rate
requirements of the system. Therefore, when the total power is limited, the
positioning power decreases correspondingly, resulting in the increase of
positioning error. Compared with the equal power allocation scheme, the robust
VLPC scheme allocates less power to communication under the condition of
satisfying the rate requirement, resulting in less positioning error. Thus,
Fig. 6 demonstrates the effectiveness of our proposed robust VLPC design.

Figure 7: (a) Power allocation of robust VLPC design versus outage probability
${P_{{\rm{out}}}}$; (b) CRLB and average communication rate
${{\tilde{R}}_{\rm{c}}}$ of robust VLPC design versus outage probability
${P_{{\rm{out}}}}$.

Fig. 7 (a) shows the positioning power ${P_{\rm{p}}}$ and the communication
power ${P_{\rm{c}}}$ of the robust VLPC design versus the outage probability
${P_{{\rm{out}}}}$ with $\bar{r}=10{\rm{Mbit}}/\mathrm{sec}$ and
${P_{{\rm{T}}}}=10W$. From Fig. 7 (a), with increasing outage probability
${P_{{\rm{out}}}}$, the positioning power ${P_{\rm{p}}}$ increases, while the
communication power ${P_{\rm{c}}}$ decreases. This is because as the outage
probability ${P_{{\rm{out}}}}$ decreases, the probability that the
communication rate is the threshold $\bar{r}$ decreases, and the robust design
becomes more conservative. Moreover, under the same simulation conditions as
Fig. 7 (a), Fig. 7 (b) depicts the CRLB and average communication rate
${{\tilde{R}}_{\rm{c}}}$ of the robust VLPC design versus the outage
probability ${P_{{\rm{out}}}}$. We observe that, as the outage probability
${P_{{\rm{out}}}}$ increases, the CRLB decreases, and the average
communication rate ${{\tilde{R}}_{\rm{c}}}$ also decreases. This is because
for a given total power, the communication power ${P_{{\rm{c}}}}$ and the
positioning power ${P_{{\rm{p}}}}$ are both related to the communication rate,
and there exists a tradeoff between them.

Figure 8: (a) Power allocation of robust VLPC design versus total power
${P_{{\rm{T}}}}$; (b) CRLB and average rate ${{\tilde{R}}_{\rm{c}}}$ of robust
VLPC design versus total power ${P_{{\rm{T}}}}$.

Fig. 8 show the influence of the total power ${P_{{\rm{T}}}}$ on the robust
VLPC design. Fig. 8 (a) shows the optimized power allocation versus
${P_{{\rm{T}}}}$ with ${P_{{\rm{out}}}}=5\%$ and
$\bar{r}=5{\rm{Mbit}}/\mathrm{sec}$. From Fig. 8 (a), we can observe that as
the total power ${P_{{\rm{T}}}}$ increases, both the positioning power
${P_{\rm{p}}}$ and the communication power ${P_{\rm{c}}}$ increase because
more power can be allocated for both positioning and communication power to
meet the positioning performance requirements and rate constraints. In
addition, Fig. 8 (b) shows the CRLB and average rate ${{\tilde{R}}_{\rm{c}}}$
versus ${P_{{\rm{T}}}}$ with the same parameters as Fig. 8 (a). From Fig. 8
(b), we can observe that as the total power ${P_{{\rm{T}}}}$ increases, the
CRLB decreases and the average communication rate ${{\tilde{R}}_{\rm{c}}}$
increases. This is intuitive since higher total available power improves both
the positioning and communication.

Fig. 9 show the influence of the rate thresholds $\bar{r}$ on the robust VLPC
design, where ${P_{{\rm{T}}}}=8W$ and ${P_{{\rm{out}}}}=5\%$. Fig. 9 (a) shows
the optimized power allocation versus different rate thresholds $\bar{r}$. We
can see that the allocated positioning power ${P_{\rm{p}}}$ decreases while
${P_{\rm{c}}}$ increases as the rate threshold $\bar{r}$ increases. This is
because the robust VLPC system needs more communication power to meet the rate
threshold $\bar{r}$. Moreover, Fig. 9 (b) shows the CRLB and average rate
${{\tilde{R}}_{\rm{c}}}$ versus $\bar{r}$. It can be seen that as the rate
threshold $\bar{r}$ increases, the CRLB increases because the positioning
power ${P_{\rm{p}}}$ decreases as $\bar{r}$ increases. In addition, the
average rate increases because the communication power ${P_{\rm{c}}}$
increases as $\bar{r}$ increases.

Figure 9: (a) Power allocation of robust VLPC design versus rate threshold
$\bar{r}$; (b) CRLB and average rate ${{\tilde{R}}_{\rm{c}}}$ of robust VLPC
design versus rate threshold $\bar{r}$.

##  VI Conclusion

In this paper, we reveal the intrinsic relationship between VLP and VLC based
on the relationship between CSI and location, i.e., the positioning
information can be used to estimate the CSI. Then, both the CRLB for VLP and
the achievable rate of VLC are derived. Furthermore, a robust power allocation
scheme is proposed under practical optical constraints, and QoS requirements.
To tackle the rate outage constraints, the worst-case distribution of the CVaR
is conservatively approximated to a more tractable form. Then, we propose a
BCD Algorithm for robust VLPC design, in which the VLP and VLC sub-problems
are iteratively optimized. Finally, our simulation results demonstrate the
effectiveness of the proposed VLPC scheme for both localization and
communications.

##  Appendix A Derivation of Equation (29)

The Fisher Information matrix (FIM) is given by

$\displaystyle{\mathbf{J}}_{\rm{p}}=\left[{\begin{array}[]{*{20}{c}}{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial
x_{u}^{2}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{x_{u}}\partial{y_{u}}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{x_{u}}\partial{z_{u}}}}}\right\\}}\\\
{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{y_{u}}\partial{x_{u}}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial
y_{u}^{2}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{y_{u}}\partial{z_{u}}}}}\right\\}}\\\
{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{z_{u}}\partial{x_{u}}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial{z_{u}}\partial{y_{u}}}}}\right\\}}&{-\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({\mathbf{u}}\right)}}{{\partial
z_{u}^{2}}}}\right\\}}\end{array}}\right].$ (58)

Then, based on the first partial derivatives of the likelihood function in
(III-A), we have

$\displaystyle\frac{{\partial\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{x_{u}}}}=-\frac{1}{{{\sigma_{\rm{p}}^{2}}}}\sum\limits_{i=1}^{M}{\int_{0}^{T_{\rm{p}}}{\left({{h_{i}}x_{\text{p}}^{2}\left(t\right)-{y_{{\text{p}},i}}\left(t\right){x_{\text{p}}}\left(t\right)}\right)\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\,\mathrm{d}t}},$
(59a)
$\displaystyle\frac{{\partial\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{y_{u}}}}=-\frac{1}{{{\sigma_{\rm{p}}^{2}}}}\sum\limits_{i=1}^{M}{\int_{0}^{T_{\rm{p}}}{\left({{h_{i}}x_{\text{p}}^{2}\left(t\right)-{y_{{\text{p}},i}}\left(t\right){x_{\text{p}}}\left(t\right)}\right)\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\,\mathrm{d}t}},$
(59b)
$\displaystyle\frac{{\partial\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{z_{u}}}}=-\frac{1}{{{\sigma^{2}}}}\sum\limits_{i=1}^{M}{\int_{0}^{T_{\rm{p}}}{\left({{h_{i}}x_{\text{p}}^{2}\left(t\right)-{y_{{\text{p}},i}}\left(t\right){x_{\text{p}}}\left(t\right)}\right)\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}\,\mathrm{d}t}}.$
(59c)

Furthermore, according to the second partial derivatives of the likelihood
function, we obtain

$\displaystyle\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial
x_{u}^{2}}}$
$\displaystyle=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}x_{\rm{p}}^{2}\left(t\right)+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial
x_{u}^{2}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.{-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial
x_{u}^{2}}}}\right)\,\mathrm{d}t,$ (60a)
$\displaystyle\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial
y_{u}^{2}}}$
$\displaystyle=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}x_{\rm{p}}^{2}\left(t\right)+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial
y_{u}^{2}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.{-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial
y_{u}^{2}}}}\right)\,\mathrm{d}t,$ (60b)
$\displaystyle\frac{{{\partial^{2}}\Lambda\left({\bf{u}}\right)}}{{\partial
z_{u}^{2}}}$
$\displaystyle=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}x_{\rm{p}}^{2}\left(t\right)+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial
z_{u}^{2}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.{-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial
z_{u}^{2}}}}\right)\,\mathrm{d}t,$ (60c)
$\displaystyle\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{x_{u}}\partial{y_{u}}}}$
$\displaystyle=\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{y_{u}}\partial{x_{u}}}}=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\left.{+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial{x_{u}}\partial{y_{u}}}}x_{\rm{p}}^{2}\left(t\right)-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial{x_{u}}\partial{y_{u}}}}}\right)\,\mathrm{d}t,$
(60d)
$\displaystyle\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{x_{u}}\partial{z_{u}}}}$
$\displaystyle=\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{z_{u}}\partial{x_{u}}}}=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\left.{+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial{x_{u}}\partial{z_{u}}}}x_{\rm{p}}^{2}\left(t\right)-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial{x_{u}}\partial{z_{u}}}}}\right)\,\mathrm{d}t,$
(60e)
$\displaystyle\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{y_{u}}\partial{z_{u}}}}$
$\displaystyle=\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{z_{u}}\partial{y_{u}}}}=-\frac{1}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\int_{0}^{{T_{\rm{p}}}}{\left({\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}x_{\rm{p}}^{2}\left(t\right)}\right.}}$
$\displaystyle\left.{+{h_{i}}\frac{{{\partial^{2}}{h_{i}}}}{{\partial{y_{u}}\partial{z_{u}}}}x_{\rm{p}}^{2}\left(t\right)-{y_{{\rm{p}},i}}\left(t\right){x_{\rm{p}}}\left(t\right)\frac{{{\partial^{2}}{h_{i}}}}{{\partial{y_{u}}\partial{z_{u}}}}}\right)\,\mathrm{d}t.$
(60f)

Since $\mathbb{E}\left\\{{{s_{\text{p}}}}\right\\}=0$,
$\mathbb{E}\left\\{{s_{\text{p}}^{2}}\right\\}=\varepsilon$, the expectation
of terms in (60) can be simplified as

$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial
x_{u}^{2}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}{\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}},$
(61a)
$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial
y_{u}^{2}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}{\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}},$
(61b)
$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial
z_{u}^{2}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}{\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}},$
(61c)
$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{x_{u}}\partial{y_{u}}}}}\right\\}$
$\displaystyle=\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{y_{u}}\partial{x_{u}}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}{\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}},$
(61d)
$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{x_{u}}\partial{z_{u}}}}}\right\\}$
$\displaystyle=\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{z_{u}}\partial{x_{u}}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}},$
(61e)
$\displaystyle\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{y_{u}}\partial{z_{u}}}}}\right\\}$
$\displaystyle=\mathbb{E}\left\\{{\frac{{{\partial^{2}}\Lambda\left({{{\mathbf{u}}}}\right)}}{{\partial{z_{u}}\partial{y_{u}}}}}\right\\}$
$\displaystyle=-\frac{{{T_{\rm{p}}}\left({P_{\rm{p}}}\varepsilon+I_{{\rm{DC}}}^{2}\right)}}{{\sigma_{\rm{p}}^{2}}}\sum\limits_{i=1}^{M}{\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}},$
(61f)

where ${\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}}$, and
${\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}}$ and
${\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}}$ can be, respectively,
expressed as

$\displaystyle\frac{{\partial{h_{i}}}}{{\partial{x_{u}}}}$
$\displaystyle=\frac{{-\alpha\left({m+3}\right){{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+1}}\left({{x_{u}}+{v_{x,i}}-{x_{l}}}\right)}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}},$
(62a) $\displaystyle\frac{{\partial{h_{i}}}}{{\partial{y_{u}}}}$
$\displaystyle=\frac{{-\alpha\left({m+3}\right){{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+1}}\left({{y_{u}}+{v_{y,i}}-{y_{l}}}\right)}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}},$
(62b) $\displaystyle\frac{{\partial{h_{i}}}}{{\partial{z_{u}}}}$
$\displaystyle=\frac{{-\left({m+1}\right)\alpha{{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m}}}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+3}}}}$
$\displaystyle\quad\quad\quad\quad\quad\quad+\frac{{\left({m+3}\right)\alpha{{\left({{z_{l}}-{z_{u}}-{v_{z,i}}}\right)}^{m+2}}}}{{{{\left\|{{\bf{l}}-{\bf{u}}-{{\bf{v}}_{i}}}\right\|}^{m+5}}}}.$
(62c)

##  Appendix B Derivation of Equation (III-B)

For brevity, we drop the time index $t$ throughout this appendix. Let $s_{1}$
and $s_{2}$ denote values $A$ and $0$, respectively. According to (24), the
PDF of ${{y_{{\text{c}}}}}$ can be written as

$\displaystyle
f\left({{y_{\rm{c}}}}\right)=\frac{1}{{2\sqrt{2\pi}{\sigma_{\rm{c}}}}}\sum\limits_{k=1}^{2}{{e^{-\frac{{{{\left({{y_{\rm{c}}}-{{\bf{v}}^{\rm{T}}}{\bf{h}}\left({\sqrt{{P_{\rm{c}}}}{s_{k}}+{I_{{\rm{DC}}}}}\right)}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}.$
(63)

Then the mutual information of the receiver is derived as

$\displaystyle
I\left({{x_{\text{c}}};{y_{{\text{c}}}}}\right)=h\left({{y_{{\text{c}}}}}\right)-h\left({{y_{{\text{c}}}}\left|{{x_{\text{c}}}}\right.}\right)$
(64a)
$\displaystyle=-\int_{-\infty}^{\infty}{f\left({{y_{\text{c}}}}\right){{\log}_{2}}f\left({{y_{\text{c}}}}\right)\,\mathrm{d}{y_{\text{c}}}}-\frac{1}{2}{\log_{2}}2\pi
e\operatorname{var}\left({z_{\text{c}}}\right)$ (64b)
$\displaystyle=-\int_{-\infty}^{\infty}{\frac{{\sum\limits_{k=1}^{2}{{e^{-\frac{{z_{\rm{c}}^{2}}}{{2\sigma_{\rm{c}}^{2}}}}}}}}{{2\sqrt{2\pi}{\sigma_{\rm{c}}}}}{{\log}_{2}}\frac{{\sum\limits_{j=1}^{2}{{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}}}{{2\sqrt{2\pi}{\sigma_{\rm{c}}}}}\,\mathrm{d}}{z_{\rm{c}}}$
$\displaystyle\quad\quad\quad\quad-\frac{1}{2}{\log_{2}}2\pi
e\sigma_{\rm{c}}^{2}$ (64c)
$\displaystyle=-\frac{1}{2}\int_{-\infty}^{\infty}{\sum\limits_{k=1}^{2}{{f_{{z_{\rm{c}}}}}\left({{z_{\rm{c}}}}\right)}{{\log}_{2}}\frac{{\sum\limits_{j=1}^{2}{{e^{-\frac{{{{\left({{{\bf{v}}^{T}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}}}{{2\sqrt{2\pi}{\sigma_{\rm{c}}}}}\,\mathrm{d}{z_{\rm{c}}}}$
$\displaystyle\quad\quad\quad\quad-\frac{1}{2}{\log_{2}}2\pi
e\sigma_{\rm{c}}^{2}$ (64d)
$\displaystyle=-\frac{1}{2}\sum\limits_{k=1}^{2}\mathbb{E}{{{}_{{z_{\rm{c}}}}}\left\\{{{{\log}_{2}}\sum\limits_{j=1}^{2}{\frac{{{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}}{2}}}\right\\}}$
$\displaystyle\quad\quad\quad\quad-\frac{1}{{2\ln 2}}.$ (64e)

##  Appendix C Derivation of Equation (III-B)

According to Jensen’s Inequality[47], if $f\left(x\right)$ is a convex
function, then we have the inequality
$f\left[\mathbb{E}{\left(x\right)}\right]\geq\mathbb{E}\left[{f\left(x\right)}\right]$.
Since ${\log_{2}}\left(x\right)$ is a concave function with respect to $x$,
according to (64e), a lower bound on the mutual information is derived as

$\displaystyle
I\left({{x_{\rm{c}}};{y_{\rm{c}}}}\right)\geq-\frac{1}{2}\sum\limits_{k=1}^{2}{{{\log}_{2}}\sum\limits_{j=1}^{2}{\frac{1}{2}}\mathbb{E}{{}_{{z_{\rm{c}}}}}\left\\{{{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)+{z_{\rm{c}}}}\right)}^{2}}}}{{2\sigma_{\rm{c}}^{2}}}}}}\right\\}}$
$\displaystyle\quad\quad\quad\quad-\frac{1}{{2\ln 2}}$ (65a)
$\displaystyle=-\frac{1}{2}\sum\limits_{k=1}^{2}{{{\log}_{2}}\sum\limits_{j=1}^{2}{\int_{-\infty}^{\infty}{\frac{{{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)+{z_{\rm{c}}}}\right)}^{2}}+z_{\rm{c}}^{2}}}{{2\sigma_{\rm{c}}^{2}}}}}}}{2{\sqrt{2\pi}{\sigma_{\rm{c}}}}}\,\mathrm{d}{z_{\rm{c}}}}}}$
$\displaystyle\quad\quad\quad\quad-\frac{1}{{2\ln 2}}$ (65b)
$\displaystyle=-\frac{1}{2}\sum\limits_{k=1}^{2}{{{\log}_{2}}\sum\limits_{j=1}^{2}{\frac{1}{{2\sqrt{2}}}}{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}\sqrt{{P_{\rm{c}}}}\left({{s_{k}}-{s_{j}}}\right)}\right)}^{2}}}}{{4\sigma_{\rm{c}}^{2}}}}}}-\frac{1}{{2\ln
2}}$ (65c)
$\displaystyle=-{\log_{2}}\left({1+{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}{\bf{h}}}\right)}^{2}}{P_{\rm{c}}}{A^{2}}}}{{4\sigma_{\rm{c}}^{2}}}}}}\right)-\frac{1}{{2\ln
2}}+\frac{3}{2}.$ (65d)

Supposed that the bandwidth of the VLC link is $B$ (Hz). Then, both the input
and the output of the VLC link can be represented by samples taken
$\frac{1}{{2B}}$ seconds apart. Since the power spectral density of noise is
$\frac{{\sigma_{\rm{c}}^{2}}}{2}$ Watts/Hertz, the noise power is
${B\sigma_{\rm{c}}^{2}}$. For the time interval $\left[{0,T_{\rm{c}}}\right]$,
there are $2BT_{\rm{c}}$ noise samples, and the variance of each sample is
$\frac{{B\sigma_{\rm{p}}^{2}T_{\rm{p}}}}{{2BT_{\rm{c}}}}=\frac{{\sigma_{\rm{c}}^{2}}}{2}$.
Moreover, if the power of signal is $P$, the signal energy per sample is
$\frac{{PT_{\rm{c}}}}{{2BT_{\rm{c}}}}=\frac{P}{{2B}}$.

Therefore, with bandwidth $B$ and ${\bf{h}}={\bf{\hat{h}}}+\Delta{\bf{h}}$,
the achievable rate is given by

$\displaystyle R_{\rm{c}}^{\rm{L}}\left({\Delta{\bf{h}}}\right)$
$\displaystyle=3B-\frac{B}{{\ln
2}}-2B{\log_{2}}\left({1+{e^{-\frac{{{{\left({{{\bf{v}}^{\rm{T}}}\left({{\bf{\hat{h}}}+\Delta{\bf{h}}}\right)}\right)}^{2}}{P_{\rm{c}}}{A^{2}}}}{{4B\sigma_{\rm{c}}^{2}}}}}}\right).$
(66)

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