# Spectral and Energy Efficiency of DCO-OFDM in Visible Light Communication
Systems with Finite-Alphabet Inputs

Ruixin Yang, Shuai Ma, Zihan Xu, Hang Li, Xiaodong Liu, Xintong Ling, Xiong
Deng, Xun Zhang, Shiyin Li R. Yang, S. Ma and S. Li are with the School of
Information and Control Engineering, China University of Mining and
Technology, Xuzhou, 221116, China. (e-mail: {ray.young, mashuai001,
lishiyin}@cumt.edu.cn).Z. Xu is with Spreadtrum Communications (Shanghai) Co.,
Ltd., Shanghai 201203, China (e-mail: zihan.xu@unisoc.com).H. Li is with the
Shenzhen Research Institute of Big Data, Shenzhen 518172, Guangdong, China.
(email: hangdavidli@163.com).X. Liu is with the School of Information
Engineering, Nanchang University, Nanchang 330031, China (e-mail:
xiaodongliu@ieee.org). X. Ling is with the National Mobile Communications
Research Laboratory, Southeast University, and the Purple Mountain
Laboratories, Nanjing, China. (e-mail: xtling@seu.edu.cn). X. Deng is with the
Center for Information Photonics and Communications, Southwest Jiaotong
University, Chengdu 610031, China.(email: xiongdeng@swjtu.edu.cn).X. Zhang is
with Institut Suprieur dElectronique de Paris, ISEP Paris, France. (e-mail:
xun.zhang@isep.fr).

###### Abstract

The bound of the information transmission rate of direct current biased
optical orthogonal frequency division multiplexing (DCO-OFDM) for visible
light communication (VLC) with finite-alphabet inputs is yet unknown, where
the corresponding spectral efficiency (SE) and energy efficiency (EE) stems
out as the open research problems. In this paper, we derive the exact
achievable rate of the DCO-OFDM system with finite-alphabet inputs for the
first time. Furthermore, we investigate SE maximization problems of the DCO-
OFDM system subject to both electrical and optical power constraints. By
exploiting the relationship between the mutual information and the minimum
mean-squared error, we propose a multi-level mercury-water-filling power
allocation scheme to achieve the maximum SE. Moreover, the EE maximization
problems of the DCO-OFDM system are studied, and the Dinkelbach-type power
allocation scheme is developed for the maximum EE. Numerical results verify
the effectiveness of the proposed theories and power allocation schemes.

###### Index Terms:

Visible light communications, DCO-OFDM, finite-alphabet input, spectral
efficiency, energy efficiency.

##  I Introduction

###  I-A Motivation and Contributions

As the number of Internet of Things (IoT) devices increases tremendously, the
radio frequency (RF) wireless networks are facing an ever-growing bandwidth
burden to support huge and high-speed data transfer [1, 2, 3]. It is reported
that more than $80\%$ of wireless data is generated in the indoor environment
[4, 5]. As a result, visible light communication (VLC)[6] has emerged as a
promising technology to provide high-speed data transmission and illumination
service simultaneously due to the huge unlicensed visible light spectrum for
future IoT applications [7, 8]. By using the ordinary light emitting diodes
(LEDs) at the transmitter side and the simple intensity modulation and direct
detection (IM/DD) at the receiver side, VLC can simultaneously support high-
speed communication and illumination without electromagnetic interference to
the conventional RF networks [9, 10, 11, 12].

From the wireless communication point of view, the VLC system, similar to the
RF system, still faces the issue of inter-symbol interference (ISI) resulting
from limited modulation bandwidth of LEDs [13] and multipath distortion [14].
Thus, the orthogonal frequency division multiplexing (OFDM) based techniques
are introduced into the VLC system against the ISI and enhance the
communication capacity. There are two typical OFDM-based transmission schemes:
direct current (DC) biased optical OFDM (DCO-OFDM) and asymmetrically clipped
optical OFDM (ACO-OFDM). Specifically, to guarantee that the transmitted
signals are positive, DCO-OFDM adds a DC-bias to the time-domain signals and
clips the remaining negative signals to zero, while ACO-OFDM only transmits
the positive parts of the OFDM waveform. Note that if the DC-bias power is
neglected, DCO-OFDM can achieve the Shannon capacity while ACO-OFDM has a 3-dB
penalty [15]. In practice, the DCO-OFDM system usually operates with finite-
alphabet inputs such as pulse amplitude modulation (PAM) and quadrature
amplitude modulation (QAM).

Most of the existing literature on the DCO-OFDM system adopted the assumption
that if the number of subcarriers is large, the time domain signal after the
inverse fast Fourier transform (IFFT) is approximately Gaussian distributed
[16, 17, 18, 19]. However, according to the central limit theorem, only if the
number of subcarriers tends to infinity, and the frequency domain symbols are
independent, the time domain signal equivalently follows the Gaussian
distribution, whatever the alphabet set is a continuous set or a finite
constellation set. Therefore, some approximation errors exist to some extent.
Meanwhile, when the random process is unbounded, the clipping operator cannot
be avoided, and the clipping noise may cause the inevitable information
loss[20, 18]. Under such an assumption, existing models cannot accurately
depict the achievable rate of the DCO-OFDM system with finite-alphabet inputs,
and the corresponding bound of the information transmission rate is yet
unknown. Thus, the spectrum efficiency (SE) and energy efficiency (EE) of the
DCO-OFDM system still need further investigation.

Besides, unlike the conventional RF communication systems, where only the
electrical power constraint is considered in most scenarios, the average
optical power constraint plays an important role in the illumination
requirement [9, 11] and should also be considered in the discussions of SE and
EE.

With the aforementioned issues, in this paper, we consider such a typical DCO-
OFDM-based VLC system and investigate the optimal power allocation over
subcarriers to maximize the SE and EE. Our main contributions of this paper
are summarized as follows:

  * •

To our best knowledge, the theoretical bound of the information transmission
rate of the DCO-OFDM system with finite-alphabet inputs remains unknown. In
this work, we derive the information rate of the DCO-OFDM system with finite-
alphabet inputs for the first time. Specifically, we derive the exact
achievable rate expression of the DCO-OFDM system without the information
loss. Moreover, we derive the closed-form lower bound for the derived
achievable rate. The obtained expressions can be used as the performance
metrics, and we also apply them in the transmission design.

  * •

Based on the closed-form lower bound of the achievable rate, we jointly
optimize DC-bias, and power allocation of subcarriers to maximize the SE of
the DCO-OFDM system under both average optical power and total electrical
transmitted power constraints. We find that the optimal DC-bias without
information loss can be presented in a closed-form expression. Then, by
restricting both optical and electrical power constraints, the optimal power
allocation can be obtained by employing the interior-point algorithm.

  * •

Next, we further study the SE maximization problem based on the exact
achievable rate expression under the same constraints above. By exploiting the
Karush-Kuhn-Tucker (KKT) conditions and the relationship between the mutual
information and the minimum mean-squared error
(MMSE)111$\frac{\partial}{{\partial{\rm{SNR}}}}I\left({{\rm{SNR}}}\right)={\rm{MMSE}}\left({{\rm{SNR}}}\right)$,
where SNR denotes signal-to-noise ratio.[21], we propose a multi-level
mercury-water-filling power allocation scheme to achieve the maximum of SE
[22, 23]. It is shown that the power allocation behaves differently with
respect to the channel gain in the low and high power domains.

  * •

Finally, we investigate the power allocation to maximize the EE of the DCO-
OFDM system derived two achievable rate metrics, respectively. We employ the
Dinkelbach-type algorithm to convert the concave-linear fractional problem
into a sequence of convex sub-problems, and then obtain the optimal power
allocation by the interior-point algorithm. In addition, we reveal that the
optimal power allocation of the EE maximization problem with finite-alphabet
inputs is related to the SE requirement. For the low SE requirement, the
allocated power of each subcarrier is proportional to the channel gain. While
for the high SE requirement, the allocated power of each subcarrier is
inversely proportional to the channel gain.

###  I-B Related Works and Organization

Most of the existing literature on SE and EE of the DCO-OFDM system study that
the time domain signal output by IFFT is approximately Gaussian distributed
when the number of subcarriers is large [16, 17, 18, 19], which causes signal
approximation errors, as well as clipping noise and information loss with the
clipping process [20, 18].

It should be pointed out that the SE of the DCO-OFDM system has been
extensively studied for different input constraints. For example, under the
optical power constraint and a target bit error rate (BER), the adaptive
modulation scheme was employed in [16] to maximize the SE, and it was shown
that the SE of ACO-OFDM is higher than that of DCO-OFDM in the low-SNR region
while it has a $50\%$ reduction compared to DCO-OFDM at high-SNR region. For
uniform power within an optimized band, the authors in [24] proposed a
discrete bit loading algorithm to maximize the achievable rate of the DCO-OFDM
system. For the single-LED case, the signal-to-noise-plus-distortion ratio
(SNDR) of DCO-OFDM was maximized by jointly optimizing both the DC-bias and
the information-carrying power under both the optical and electrical power
constraints [18]. In the case with multiple LEDs, the SNDR maximization
problem in the DCO-OFDM system was studied in [19] by properly designing the
biased beamforming with the optical power constraint. By utilizing generalized
mutual information, the lower bound of information rate was derived in [25]
for the DCO-OFDM system under the average optical power constraint.

Recently, due to the increasing number of IoT devices, the research on EE has
attracted great attention to reduce consumption and prolong the lifetime [26,
27, 28]. It was reported in [17] that both the energy and spectrum efficiency
achieved with DCO-OFDM is higher than that obtained by ACO-OFDM in the case
that a constant DC-bias power is given for the illumination requirement. By
replacing the negative parts of signals with their absolute values, a power-
efficient symbol recovery scheme for the DCO-OFDM system was proposed in [29],
which can improve the symbol error rate (SER) performance for a given DC-bias.
Under the constraints of the BER and the total transmitted power, the authors
in [30] investigated the achievable rate maximization problem for the DCO-OFDM
system, and further optimized transmitted power to achieve the tradeoff
between SE and EE.

It was reported in [31] that the mutual information maximizing design and
classic power allocation scheme based on Gaussian distributed assumption will
lead to a significant loss. However, to the best of our knowledge, only a few
works have investigated the SE and EE of the DCO-OFDM with finite-alphabet
inputs. The most relevant work is [32], where the EE maximization problem of
DCO-OFDM was studied by designing the optimal solution of the modulation order
and power allocation under the minimum SE requirement and a total transmitted
power constraint. However, since the mutual information does not have the
closed-form expression, the authors in [32] adopted the closed-form lower
bound and upper bound to approximate the exact mutual information to solve the
optimal power allocation problem. In our study, we exploit the relationship
between the mutual information and MMSE to deal with the non-closed-form
expression. Moreover, we study the bound of the information transmission rate
of the DCO-OFDM system based on finite-alphabet inputs and the optimal DC-bias
such that the clipping can be avoided, which is different from [18] and [19]
with clipping noise, and propose the optimal power allocation scheme to
maximize SE and EE of the DCO-OFDM system under the constraints of average
optical power and total electrical transmitted power, and further consider the
minimum SE requirement when maximizing EE.

The rest of this paper is organized as follows. The system model of the
considered DCO-OFDM system is presented in Section II. The achievable rate
expressions of DCO-OFDM are derived in Section III. The SE and EE maximization
problems of the DCO-OFDM system are respectively studied in Section IV and
Section V. The simulation results are presented in Section VI. Finally, the
conclusions are drawn in Section VII.

_Notations_ : Expected value of a random variable $z$ is denoted by
${\mathbb{E}}\left\\{z\right\\}$. ${\left(\cdot\right)^{*}}$ represents
conjugate transformation. ${\left[x\right]^{+}}$ denotes
$\max\left\\{{x,0}\right\\}$. ${\mathop{\rm Re}\nolimits}\left(\cdot\right)$
and ${\mathop{\rm Im}\nolimits}\left(\cdot\right)$ denote the real and
imaginary parts of their argument, respectively. ${{\partial
f\left(\cdot\right)}\mathord{\left/{\vphantom{{\partial
f\left(\cdot\right)}{\partial x}}}\right.\kern-1.2pt}{\partial x}}$ represents
the partial derivative of function ${f\left(\cdot\right)}$. Given a variable
$y$, ${\mathbb{E}}\left\\{{z\left|y\right.}\right\\}$ represents the
conditional expectation of $z$ for given $y$. $I\left({X;Y}\right)$ represents
the mutual information of $X$ and $Y$.

##  II System Model

Figure 1:  The schematic diagram of a DCO-OFDM VLC system.

In this paper, we consider both the SE and EE optimization problems for the
DCO-OFDM system with finite-alphabet inputs shown in Fig. 1. At the
transmitter side, the input bit streams are modulated by an $M$-ary QAM after
serial-to-parallel (S/P) conversion. Due to the IM/DD, the transmit signal is
required not only to be non-negative but also to be real-valued. In order to
ensure that the output signal of the IFFT is a real-valued VLC signal, the
IFFT input symbols of $2N$ subcarriers after Hermitian symmetry should satisfy

$\displaystyle\sqrt{{p_{2N-i}}}{X_{2N-i}}=\sqrt{p_{i}}X_{i}^{*},\quad
i=1,...,N-1,$ (1)

where ${X_{i}}$ is the signal of the $i$th subcarrier, which should be
${X_{0}}$ = ${X_{N}}$ = 0 to block the DC component. Without loss of
generality, we can assume
$\mathbb{E}\left\\{{{{\left|{{X_{i}}}\right|}^{2}}}\right\\}=1$. Besides, let
$p_{i}$ denote the allocated power for the $i$th subcarrier, and follow
$p_{0}=p_{N}=0$, and $p_{2N-i}=p_{i},i=1,\dots,N-1$. Then, the IFFT output
signal ${x_{k}}$ in the time domanoin is given by

$\displaystyle{x_{k}}=$
$\displaystyle\frac{1}{{\sqrt{2N}}}\sum\limits_{i=0}^{2N-1}{\sqrt{{p_{i}}}{X_{i}}\exp\left({j\frac{{\pi
ki}}{N}}\right)}$ (2a) $\displaystyle=$
$\displaystyle\frac{1}{{\sqrt{2N}}}\left(\sum\limits_{i=1}^{N-1}\sqrt{{p_{i}}}{X_{i}}\exp\left({j\frac{{\pi
ki}}{N}}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\sum\limits_{i=1}^{N-1}{\left({\sqrt{{p_{i}}}{X_{i}}\exp\left({j\frac{{\pi
ki}}{N}}\right)}\right)}^{*}\right)$ (2b) $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{N}}\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}{\rm{Re}}\left({{X_{i}}\exp\left({j\frac{{\pi
ki}}{N}}\right)}\right)},$ (2c)

where $k=0,...,2N-1$. It is easy to find that
$\mathbb{E}\left\\{{x_{k}}\right\\}=0$.

To guarantee the VLC signal non-negative, the clipping operator is applied.
Specifically, the time domain signal ${x_{k}}$ is converted to clipped signal
${x_{{\rm{clip}},k}}$ by clipping at the level of $-{I_{{\rm{dc}}}}$, where
${I_{{\rm{dc}}}}$ is the DC-bias and the clipping operator is defined as

$\displaystyle{x_{{\rm{clip}},k}}=\left\\{{\begin{array}[]{*{20}{c}}x_{k}\\\
-{I_{{\rm{dc}}}}\\\
\end{array}}\right.\begin{array}[]{*{20}{c}}{x_{k}}\geq-{I_{{\rm{dc}}}},\\\
{x_{k}}\leq-{I_{{\rm{dc}}}}.\\\ \end{array}$ (7)

Then ${x_{{\rm{clip}},k}}$ is added with $I_{\rm{dc}}$ which only affects the
$0$th subcarrier in the frequency domain. Thus, we obtain the non-negative
signal

$\displaystyle{x_{{\rm{dc}},k}}={x_{{\rm{clip}},k}}+{I_{{\rm{dc}}}}.$ (8)

If we want to avoid the information loss brought by the clipping noise, a
feasible way is to avoid the clipping operation, i.e., the amplitude of the
time domain signal $x_{k}$ should be bounded, and an appropriate DC-bias
$I_{\mathrm{dc}}$ should satisfy

$\displaystyle{x_{k}}+{I_{{\rm{dc}}}}\geq\min\left\\{x_{k}\right\\}+{I_{{\rm{dc}}}}\geq
0.$ (9)

Moreover, the digital signal ${x_{{\rm{dc}},k}}$ is converted to the analog
signal via digital-to-analog convertor (DAC) and then transmitted by LED. To
satisfy the illumination and human eye safety requirements, the average
optical power is restricted, i.e.,

$\displaystyle\mathbb{E}\left\\{{x_{{\rm{dc}},k}}\right\\}\leq{P_{o}},$ (10)

where ${P_{o}}$ represents the maximum average optical power budget.

Besides, for the practical electrical circuits consideration, the total
electrical transmitted power of the VLC system should also be constrained,
i.e.,

$\displaystyle\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}\leq{P_{e}}.$
(11)

where $P_{e}$ is the maximum total electrical transmitted power budget.

Note that the channel of the VLC system is generally lowpass [33, 18], and the
channel gain difference between each subcarrier can be utilized via allocating
proper power to each subcarrier to improve the performance of the VLC system.
Indoor VLC channel includes two components: the line-of-sight (LOS) link
between the transmitter and the receiver; and the diffuse link that is the
superposition of all non-LOS components caused by one or more reflections on
the surface of the room.

Let ${H_{{\rm{L}},i}}={\eta_{\rm{L}}}{e^{-j2\pi{f_{i}}{\tau_{\rm{L}}}}}$
denote the channel gain of the LOS link of the $i$th subcarrier, where
${\eta_{\rm{L}}}$ is the generalized Lambertian radiator [8, 33] expressed as

$\displaystyle{\eta_{\rm{L}}}=\frac{{\left({m+1}\right){A_{\rm{r}}}\cos\left(\varphi\right)}}{{2\pi{d^{2}}}}{\cos^{m}}\left(\theta\right)T\left(\varphi\right)G\left(\varphi\right){\rm{rect}}\left({{\varphi\mathord{\left/{\vphantom{\varphi\Psi}}\right.\kern-1.2pt}\Psi}}\right).$
(12)

In (12), $m=-\ln 2/\ln\left({\cos{\Phi_{1/2}}}\right)$ denotes the order of
Lambertian emission, and ${{\Phi_{1/2}}}$ is the semi-angle at half power.
${A_{\rm{r}}}$ denotes the effective detector area of the photodetector (PD),
$\varphi$ and $\theta$ are the incidence and irradiance angles from the LED to
the PD, respectively, ${T\left(\varphi\right)}$ and ${G\left(\varphi\right)}$
are the optical filter gain and the concentrator gain of the receiver,
respectively, $\Psi$ represents the field-of-view (FOV) of the receiver, the
rectangular function ${\rm{rect}}\left(x\right)$ takes 1 whenever
$\left|x\right|\leq 1$, and is 0 otherwise, $f_{i}$ denotes the frequency of
the $i$th subcarrier, $\tau_{\rm{L}}=d/c$ is the signal propagation delay of
the LOS link between the LED to the PD, ${{d}}$ is the distance between the
LED to the PD, and $c$ stands for the speed of light.

Let
${H_{{\rm{D}},i}}=\frac{{{\eta_{\rm{D}}}}}{{1+j2\pi{\tau_{\rm{D}}}{f_{i}}}}$
denote the channel gain of the diffuse link of the $i$th subcarrier [34],
where
${\eta_{\rm{D}}}=\frac{{{A_{\rm{r}}}}}{{{A_{{\rm{room}}}}}}\frac{\rho}{{1-\rho}}$
is the diffuse channel gain factor. Here, ${{A_{{\rm{room}}}}}$ is the surface
of the room and $\rho$ is the average value of the room reflectivity factor.
Moreover,
${\tau_{\rm{D}}}=-\frac{1}{{\ln\rho}}\frac{{4{V_{{\rm{room}}}}}}{{{A_{{\rm{room}}}}c}}$
is the time constant, and ${{V_{{\rm{room}}}}}$ is the volumn of the room.
Then, the total channel gain ${H_{i}}$ of the $i$th subcarrier can be
expressed as

$\displaystyle{H_{i}}={H_{{\rm{L}},i}}+{H_{{\rm{D}},i}},\quad i=0,...,2N-1.$
(13)

At the receiver, the received optical signal is converted to analog electrical
signal by the PD. Then, the digital signal is obtained by an analog-to-digital
convertor (ADC). Finally, the bit stream is recovered through the fast Fourier
transform (FFT) and demodulation operations.

Specifically, the frequency domain expression of the received signal ${Y_{i}}$
can be written as

$\displaystyle{Y_{i}}={H_{i}}\sqrt{{p_{i}}}{X_{i}}+{Z_{i}},\quad i=1,...,N-1,$
(14)

where $Y_{i}$ and $H_{i}$ represent the received signal and the total channel
gain at the $i$th subcarrier, respectively. ${Z_{i}}$ is the additive white
Gaussian noise (AWGN) with zero-mean, i.e.,
${Z_{i}}\sim\mathcal{CN}\left({0,{\sigma^{2}}{W}}\right)$, where
${\sigma^{2}}$ represents the noise power spectral density (PSD), and the
bandwidth of each subcarrier is $W$.

$\displaystyle{R_{{\rm{L}},i}}\left({{p_{i}}}\right)=W\left({{{\log}_{2}}M+1-\frac{1}{{\ln
2}}}\right)-\sum\limits_{n=1}^{M}{\frac{{W}}{M}{{\log}_{2}}\sum\limits_{k=1}^{M}{\exp\left({-\frac{{{p_{i}}{{\left|{{H_{i}}}\right|}^{2}}{{\left|{{X_{i,n}}-{X_{i,k}}}\right|}^{2}}}}{{2{\sigma^{2}}W}}}\right)}}.$
(14)

##  III Achievable Rate of the DCO-OFDM System

Recall that in most existing works, the achievable rates of the DCO-OFDM
system are derived based on the assumption that the time domain signal
obtained after the IFFT is approximately Gaussian distributed, and the exact
achievable rate without the information loss of the DCO-OFDM system with
finite-alphabet inputs is still unknown.

Here, we consider a system that the discrete constellation points are
equiprobably drawn from discrete constellations set
$\left\\{{{X_{i,k}}}\right\\}_{k=1}^{M}$ with cardinality $M$, where $X_{i,k}$
is the $k$th constellation point of the $i$th subcarrier. Thus, the achievable
rate ${R_{{\rm{F}},i}}\left({{p_{i}}}\right)$ of the $i$th subcarrier can be
expressed as

$\displaystyle{R_{{\rm{F}},i}}\left({{p_{i}}}\right)={{I}_{i}}\left({X_{i}};{Y_{i}}\right)$
(15a) $\displaystyle=W\left({{{\log}_{2}}M-\frac{1}{{\ln
2}}}\right)-\sum\limits_{n=1}^{M}{\frac{{W}}{M}{\mathbb{E}_{Z_{i}}}\left\\{{{{\log}_{2}}\sum\limits_{k=1}^{M}{\exp\left({-{d_{n,k}}}\right)}}\right\\}},$
(15b)

where
${d_{n,k}}\buildrel\Delta\over{=}\frac{{{{\left|{{H_{i}}\sqrt{{p_{i}}}\left({{X_{i,n}}-{X_{i,k}}}\right)+{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}$
is a measure of the difference between discrete constellation points $X_{i,n}$
and $X_{i,k}$, and ${\mathbb{E}_{Z_{i}}}\left\\{\cdot\right\\}$ is the
expectation of the noise ${Z_{i}}$. The detailed derivation of (15b) is given
in Appendix A. Moreover, it is easy to find that
${R_{{\rm{F}},i}}\left({{p_{i}}}\right)$ is a concave function with respect to
the power allocation ${p_{i}}$ [31, 35]. Thus, the total achievable rate of
the DCO-OFDM system with finite-alphabet inputs is given by

$\displaystyle{R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\sum\limits_{i=1}^{N-1}{{R_{{\rm{F}},i}}\left({{p_{i}}}\right)}.$
(16)

Note that, for $M\geq 2$, the expectation term in the achievable rate (15b) is
a non-integrable function and lacks of closed-form expression, making it
challenging for resource allocation. To address this challenge, we further
derive the lower bound of (15b) with closed-form expression. Since
${\log_{2}}\left(\cdot\right)$ is a concave function, the upper bound of the
expectation term in (15b) is given by

$\displaystyle{{\mathbb{E}}_{Z_{i}}}\left\\{{\log_{2}}\sum\limits_{k=1}^{M}{\exp\left({-{d_{n,k}}}\right)}\right\\}\leq{\log_{2}}\sum\limits_{k=1}^{M}{{{\mathbb{E}}_{Z_{i}}}\left\\{\exp\left({-{d_{n,k}}}\right)\right\\}}$
(17a)
$\displaystyle={\rm{lo}}{{\rm{g}}_{\rm{2}}}\sum\limits_{k=1}^{M}{\int_{{Z_{i}}}{\exp\left({-{d_{n,k}}}\right)\frac{1}{{\pi{\sigma^{2}}W}}\exp\left({-\frac{{{{\left|{{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}d{Z_{i}}}$
(17b)
$\displaystyle=-1+{\rm{lo}}{{\rm{g}}_{\rm{2}}}\sum\limits_{k=1}^{M}{\exp\left({-\frac{{{p_{i}}{{\left|{{H_{i}}}\right|}^{2}}{{\left|{{X_{i,k}}-{X_{i,m}}}\right|}^{2}}}}{{2{\sigma^{2}}W}}}\right)},$
(17c)

where (17a) is due to the Jensen’s inequality [36].

Thus, let ${R_{{\rm{L}},i}}\left({{p_{i}}}\right)$ represent the lower bound
of mutual information of the $i$th subcarrier with finite-alphabet inputs and
is given by (14).

Similar to ${R_{{\rm{F}},i}}\left({{p_{i}}}\right)$,
${R_{{\rm{L}},i}}\left({{p_{i}}}\right)$ is also a concave function with
respect to the power allocation ${p_{i}}$. The corresponding lower bound of
total achievable rate of the DCO-OFDM system is given by

$\displaystyle{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\sum\limits_{i=1}^{N-1}{{R_{{\rm{L}},i}}\left({{p_{i}}}\right)}.$
(15)

As same as [36], there is a constant gap $C\triangleq\frac{1}{\ln 2}-1$
between $R_{\mathrm{F}}\left(p_{i}\right)$ and
$R_{\mathrm{L}}\left(p_{i}\right)$ when $p_{i}\to\infty$ or $p_{i}\to 0$, and
$R_{\mathrm{A}}\left(p_{i}\right)\triangleq
R_{\mathrm{L}}\left(p_{i}\right)+C$ can also be as a low-complexity
approximation of $R_{\mathrm{F}}\left(p_{i}\right)$, which always converges to
$R_{\mathrm{F}}\left(p_{i}\right)$ with $p_{i}\to\infty$ or $p_{i}\to 0$,
while exceeds $R_{\mathrm{F}}\left(p_{i}\right)$ in the medium region. Then,
the corresponding total achievable rate is given by
${R_{{\rm{A,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\sum\limits_{i=1}^{N-1}{{R_{{\rm{A}},i}}\left({{p_{i}}}\right)}$,
which is still concave with respect to the power allocation $p_{i}$.

##  IV Spectral Efficiency maximization of the DCO-OFDM system

In this section, we investigate the SE maximization problem of the DCO-OFDM
system subject to the maximum average optical power and the maximum total
electrical transmitted power constraints. Specifically, we start from a simple
case of ${{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$ in
(15) with the closed-form expression and then the more challenging case of
${R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$ in (16)
without a closed-form expression.

###  IV-A SE Maximization based on
${{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$

With the closed-form expression of lower bound of the achievable rate (15),
the SE can be expressed as

$\displaystyle{\rm{S}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\frac{{{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}}{{2NW}}.$
(16)

Then, we aim to maximize the SE of the DCO-OFDM system under amplitude
constraint, average optical power constraint and total electrical transmitted
power constraint, which can be mathematically formulated as

$\displaystyle\mathop{{\text{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\},{I_{\rm{dc}}}}$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$
(17a) $\displaystyle{\rm{s.t}}.$
$\displaystyle\quad{x_{k}}+{I_{{\rm{dc}}}}\geq 0,$ (17b)
$\displaystyle\quad\mathbb{E}\left\\{{x_{{\rm{dc}},k}}\right\\}\leq{P_{o}},$
(17c)
$\displaystyle\quad\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}\leq{P_{e}},$
(17d) $\displaystyle\quad{p_{i}}\geq 0,\quad i=1,...,N-1.$ (17e)

Although the joint design of variables DC-bias ${I_{\rm{dc}}}$ and power
allocation ${p_{i}}$ complicates the optimization problem, it can be seen from
problem (17) that the objective function is only related to the power
allocation ${p_{i}}$, and it is monotonically increasing as ${p_{i}}$
increases. Moreover, based on (8) and (17b), the total electrical transmitted
power in (17d) can be expressed as

$\displaystyle\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}$
$\displaystyle=\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{k}^{2}}\right\\}}+\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{I_{{\rm{dc}}}^{2}}\right\\}}$
(18a)
$\displaystyle=2\sum\limits_{i=1}^{N-1}{{p_{i}}}+2N\mathbb{E}\left\\{{I_{{\rm{dc}}}^{2}}\right\\},$
(18b)

where (18a) is true since the time domain signal ${x_{k}}$ is not clipped,
i.e., ${x_{{\rm{clip}},k}}={x_{k}}$, (18b) follows from the Parseval’s
theorem, and the total electrical transmitted power of time domain signal
$x_{k}$ is
$\sum\nolimits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{k}^{2}}\right\\}}=\sum\nolimits_{i=0}^{2N-1}{{p_{i}}}=2\sum\nolimits_{i=1}^{N-1}{{p_{i}}}$.
Thus, combining with the total electrical transmitted power constraint, we
find that more power should be allocated for the information-carrying
${p_{i}}$ and less power should be allocated for the DC-bias ${I_{\rm{dc}}}$
to maximize the achievable rate
${R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$. Specifically,
the optimal value of $I_{{\rm{dc}}}^{{\rm{opt}}}$ should be the minimum DC-
bias without clipping the signal ${x_{k}}$.

Furthermore, from (2c), we have

$\displaystyle{x_{k}}\geq-\sqrt{\frac{2}{N}}\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\left|{{X_{i}}}\right|}.$
(19)

Then, $I_{{\rm{dc}}}^{{\rm{non-clipping}}}$ which minimizes the power of the
DC-bias without clipping the signal ${x_{k}}$ can be written as

$\displaystyle{I_{{\rm{dc}}}^{{\rm{non-
clipping}}}}=\sqrt{\frac{2}{N}}\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\left|{{X_{i}}}\right|}.$
(20)

As a result, combining (8), (17b) and (20), the average optical power can be
rewritten as

$\displaystyle\mathbb{E}\left\\{{x_{{\rm{dc}},k}}\right\\}$
$\displaystyle=\mathbb{E}\left\\{{x_{{\rm{clip}},k}}+{I_{{\rm{dc}}}^{{\rm{non-
clipping}}}}\right\\}$ (21a)
$\displaystyle=\mathbb{E}\left\\{{{x_{k}}}\right\\}+\mathbb{E}\left\\{{{I_{{\rm{dc}}}^{{\rm{non-
clipping}}}}}\right\\}$ (21b)
$\displaystyle=\sqrt{\frac{2}{N}}\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\mathbb{E}\left\\{{\left|{{X_{i}}}\right|}\right\\}}.$
(21c)

According to Cauchy-Schwarz inequality

$\displaystyle{\left({\sum\limits_{i=1}^{n}{{a_{i}}}}\right)^{2}}\leq
n\sum\limits_{i=1}^{n}{a_{i}^{2}},\quad{a_{i}}\geq 0,$ (22)

the upper bound of the square of the average optical power is given by

$\displaystyle{\left({\sqrt{\frac{2}{N}}\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\mathbb{E}\left\\{{\left|{{X_{i}}}\right|}\right\\}}}\right)^{2}}\leq\frac{{2\left({N-1}\right)}}{N}\sum\limits_{i=1}^{N-1}{{p_{i}}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}.$
(23)

Thus, the average optical power constraint (17c) can be restricted to

$\displaystyle\sum\limits_{i=1}^{N-1}{{p_{i}}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}\leq\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}}.$
(24)

Meanwhile, by substituting ${I_{{\rm{dc}}}^{{\rm{non-clipping}}}}$ in (20)
into (18b), the total electrical transmitted power can be written as

$\displaystyle\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}=2\sum\limits_{i=1}^{N-1}{{p_{i}}}+4\mathbb{E}\left\\{{{{\left({\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\left|{{X_{i}}}\right|}}\right)}^{2}}}\right\\}.$
(25)

Applying the inequality (22), the upper bound of the total electrical
transmitted power (25) is given as

$\displaystyle\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}\leq
2\sum\limits_{i=1}^{N-1}{{p_{i}}}+4\mathbb{E}\left\\{{\left({N-1}\right)\sum\limits_{i=1}^{N-1}{{p_{i}}{{\left|{{X_{i}}}\right|}^{2}}}}\right\\}.$
(26)

Thus, the constraint (17d) can be reformulated as

$\displaystyle\sum\limits_{i=1}^{N-1}{{p_{i}}}\leq\frac{{{P_{e}}}}{{4N-2}}.$
(27)

Therefore, the SE maximization problem (17) can be transformed into

$\displaystyle\mathop{{\text{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\}}$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$
(28a) $\displaystyle{\rm{s.t.}}$
$\displaystyle\quad\sum\limits_{i=1}^{N-1}{{p_{i}}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}\leq\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}},$
(28b)
$\displaystyle\quad\sum\limits_{i=1}^{N-1}{{p_{i}}}\leq\frac{{{P_{e}}}}{{4N-2}},$
(28c) $\displaystyle\quad p_{i}\geq 0{\rm{}},\quad i=1,...,N-1.$ (28d)

Note that the optimization problem (28) has a strictly concave objective
function over its input power ${p_{i}}$ and linear constraints, and thus can
be efficiently solved by the interior-point algorithm. Such as the barrier
method, it transforms the convex optimization problems into a sequence of
equality constrained problems and applies Newton’s method to them, or such as
the primal-dual interior-point method, it modifies the corresponding KKT
conditions and solves them by Newton’s method[37, 38]. Besides, It has been
implemented by standard convex optimization solvers such as CVX [39].

Besides, the SE based on
$R_{{\rm{L,total}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$ is denoted as
$\mathrm{SE}_{\mathrm{A}}$, and the corresponding SE maximization power
allocation problem is equivalent to the problem (17) without the optimality
loss, wich can be solved by the similar technique.

###  IV-B SE Maximization based on
${{R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$

In this subsection, we investigate the SE maximization problem of the DCO-OFDM
system based on the exact mutual information (16). The SE achieved with exact
mutual information is given by

$\displaystyle{\rm{S}}{{\rm{E}}_{\rm{F}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\frac{{R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}{{2NW}}.$
(29)

Similarly, the value of the DC-bias ${I_{{\rm{dc}}}}$ is the same as (20) to
ensure that the transmitted DC-bias power is minimized while the signal
${x_{k}}$ is not clipped. Thus, by considering the same constraints in problem
(28), the SE maximization problem with finite-alphabet inputs can be
reformulated as

$\displaystyle\mathop{{\text{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\}}$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{F}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$
(30) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$
$\displaystyle\quad\eqref{power_con1},\eqref{power_con2},\eqref{power_con3}.$

Note that the objective function (30) is concave over ${p_{i}}$, and
constraints (28b) and (28c) are affine functions over ${p_{i}}$. This type of
optimization problem can be efficiently solved based on the KKT conditions. To
this end, we first derive the Lagrangian function of problem (30), which is
given by

$\displaystyle{\mathcal{L}_{\rm{F}}}=-\sum\limits_{i=1}^{N-1}{{R_{{\rm{F}},i}}\left({{p_{i}}}\right)}$
$\displaystyle+{\lambda_{1}}\left({\sum\limits_{i=1}^{N-1}{{p_{i}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}}-\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}}}\right){\rm{}}$
$\displaystyle+{\lambda_{2}}\left({\sum\limits_{i=1}^{N-1}{{p_{i}}}-\frac{{{P_{e}}}}{{4N-2}}}\right),$
(31)

where ${\lambda_{1}}\geq 0$, ${\lambda_{2}}\geq 0$ are the Lagrange
multipliers corresponding to constraint (28b) and (28c) respectively. Then,
the KKT conditions of problem (30) are given as

$\displaystyle\quad\frac{{\partial{\mathcal{L}_{\rm{F}}}}}{{\partial{p_{i}}}}=-\frac{{\partial{R_{{\rm{F}},i}}\left({{p_{i}}}\right)}}{{\partial{p_{i}}}}+{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}=0,$
(32a)
$\displaystyle\quad{\lambda_{1}}\left({\sum\limits_{i=1}^{N-1}{{p_{i}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}}-\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}}}\right)=0,$
(32b)
$\displaystyle\quad{\lambda_{2}}\left({\sum\limits_{i=1}^{N-1}{{p_{i}}}-\frac{{{P_{e}}}}{{4N-2}}}\right)=0,$
(32c)
$\displaystyle\quad\sum\limits_{i=1}^{N-1}{{p_{i}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}}-{\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}}}\leq
0,$ (32d)
$\displaystyle\quad\sum\limits_{i=1}^{N-1}{{p_{i}}}-\frac{{{P_{e}}}}{{4N-2}}\leq
0,$ (32e) $\displaystyle\quad{\lambda_{1}}\geq 0,\quad{\lambda_{2}}\geq
0,\quad p_{i}\geq 0,\quad i=1,...,N-1.$ (32f)

However, it is challenging to directly calculate the partial derivative in
(32a) due to the lack of the closed-form expressions for the achievable rate
${R_{{\rm{F}},i}}\left({{p_{i}}}\right)$. Therefore, we aim to address this
difficulty by exploiting the relationship between the mutual information and
MMSE. Specifically, the MMSE of ${X_{i}}$ is given as

$\displaystyle{\rm{MMS}}{{\rm{E}}_{i}}\left({{\rm{SN}}{{\rm{R}}_{i}}}\right)=\mathbb{E}\left\\{{{{\left|{{X_{i}}-{{\hat{X}}_{i}}}\right|}^{2}}}\right\\},$
(33)

where
${\rm{SN}}{{\rm{R}}_{i}}=\frac{{{{\left|{{H_{i}}}\right|}^{2}}{p_{i}}}}{{{\sigma^{2}}W}}$
is the signal-to-noise ratio of the $i$th subcarrier, and ${{\hat{X}}_{i}}$ is
conditional expectation of ${X_{i}}$, i.e.,
${{\hat{X}}_{i}}=\mathbb{E}\left\\{{{X_{i}}\left|{{Y_{i}}={H_{i}}\sqrt{{p_{i}}}{X_{i}}+{Z_{i}}}\right.}\right\\}$.
According to _Theorem 1_ in [21], the relationship between the mutual
information (15a) and the MMSE (33) is given by

$\displaystyle\frac{\partial}{{\partial{\rm{SN}}{{\rm{R}}_{i}}}}{I_{i}}\left({{X_{i}};{Y_{i}}}\right)={\rm{MMS}}{{\rm{E}}_{i}}\left({{\rm{SN}}{{\rm{R}}_{i}}}\right).$
(34)

Combining (15b) and (34), partial derivative of function
${R_{{\rm{F}},i}}\left({{p_{i}}}\right)$ can be written as

$\displaystyle\frac{{\partial{R_{{\rm{F}},i}}\left({{p_{i}}}\right)}}{{\partial{p_{i}}}}=\frac{{{{\left|{{H_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}{\rm{MMS}}{{\rm{E}}_{i}}\left({\frac{{{{\left|{{H_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}{p_{i}}}\right).$
(35)

By substituting (35) into (32a), we have

$\displaystyle\frac{{{{\left|{{H_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}{\rm{MMS}}{{\rm{E}}_{i}}\left({\frac{{{{\left|{{H_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}{p_{i}}}\right)={\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}.$
(36)

Then, according to (36), the power allocation $p_{i}$ can be obtained as

$\displaystyle{p_{i}}=\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right],$
(37)

where ${\rm{MMSE}}_{i}^{-1}\left(\cdot\right)$ is the inverse function of
${\rm{MMS}}{{\rm{E}}_{i}}\left(\cdot\right)$ with domain in
$\left[{0,1}\right]$ and ${\rm{MMSE}}_{i}^{-1}\left(1\right)=0$ [23].
Therefore, for the SE maximization problem (30), the optimal allocation power
of the $i$th subcarrier is given by (40).

$\displaystyle
p_{i}^{{\rm{opt}}}=\left\\{{\begin{array}[]{*{20}{c}}{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right],}&{0<{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}\leq\frac{{{{\left|{{H_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}},}\\\
{0,}&{{\rm{otherwise}}.}\end{array}}\right.$ (40)

$\displaystyle
G_{i}\left(\lambda_{1},\lambda_{2}\right)\triangleq\begin{cases}\frac{\left|H_{i}\right|^{2}}{\sigma^{2}W\left(\lambda_{1}\mathbb{E}^{2}\left\\{\left|X_{i}\right|\right\\}+\lambda_{2}\right)}-\mathrm{MMSE}^{-1}\left[\frac{\sigma^{2}W}{\left|H_{i}\right|^{2}}\left(\lambda_{1}\mathbb{E}^{2}\left\\{\left|X_{i}\right|\right\\}+\lambda_{2}\right)\right],&0<\lambda_{1}\mathbb{E}^{2}\left\\{\left|X_{i}\right|\right\\}+\lambda_{2}\leq\frac{\sigma^{2}W}{\left|H_{i}\right|^{2}};\\\
1,&\mathrm{otherwise}.\end{cases}$ (40)

Meanwhile, according to the definition of MMSE, $\lambda_{1}$ and
$\lambda_{2}$ should be greater than $0$, otherwise the required power would
be infinity. Therefore, substituting (40) into the complementary slackness
conditions (32b) and (32c), the dual variables $\lambda_{1}$ and $\lambda_{2}$
is the solution of the following equation:

$\displaystyle\sum\limits_{i=1}^{N-1}{{{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right]}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}},$
(39a)
$\displaystyle{\sum\limits_{i=1}^{N-1}{{{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right]}}}=\frac{{{P_{e}}}}{{4N-2}}},$
(39b)

which can obtained by the multi-level mercury-water-filling power allocation
scheme as listed in Algorithm 1 [22, 23]. Moreover, to facilitate the
explanation of the power allocation scheme, an auxiliary function
$G_{i}\left(\lambda_{1},\lambda_{2}\right)$ is defined as follows (40). Then,
the allocated power $p_{i}$ can be represented as

$\displaystyle
p_{i}=\frac{1}{\lambda_{1}\mathbb{E}^{2}\left\\{\left|X_{i}\right|\right\\}+\lambda_{2}}-\frac{\left|H_{i}\right|^{2}}{\sigma^{2}W}G_{i}\left(\lambda_{1},\lambda_{2}\right).$
(41)

Algorithm 1 Multi-level Mercury-water-filling Power Allocation Scheme

1:Given ${\lambda_{1}}\in\left[{0,{{\hat{\lambda}}_{1}}}\right]$,
${\delta_{1}}>0$, where ${{{\hat{\lambda}}_{1}}}$ is the upper bound of
${\lambda_{1}}$ and ${\delta_{1}}$ is a small positive constant that controls
the algorithm accuracy. Initialize
${\lambda_{\min}}=0,{\lambda_{\max}}={{\hat{\lambda}}_{1}}$;

2:while ${\lambda_{\max}}-{\lambda_{\min}}\geq{\delta_{1}}$ do

3: Set
${\lambda_{1}}={{\left({{\lambda_{\min}}+{\lambda_{\max}}}\right)}\mathord{\left/{\vphantom{{\left({{\lambda_{\min}}+{\lambda_{\max}}}\right)}2}}\right.\kern-1.2pt}2}$;

4: Find the minimum ${\lambda_{2}}\geq 0$, with which

5:
$\sum\limits_{i=1}^{N-1}{{{\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right]}\right]}^{+}}}\leq\frac{{{P_{e}}}}{{4N-2}}$;

6: If
${\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}\leq{\left|{{H_{i}}}\right|^{2}}/{\sigma^{2}}W$,
substitute ${\lambda_{1}},{\lambda_{2}}$ to obtain

7:
$p_{i}^{{\rm{opt}}}=\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}{\rm{MMSE}}_{i}^{-1}\left[{\frac{{{\sigma^{2}}W}}{{{{\left|{{H_{i}}}\right|}^{2}}}}\left({{\lambda_{1}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}+{\lambda_{2}}}\right)}\right]$;
otherwise $p_{i}^{{\rm{opt}}}=0$;

8: If
$\sum\limits_{i=1}^{N-1}{{p_{i}^{{\rm{opt}}}}{\mathbb{E}^{2}}\left\\{{\left|{{X_{i}}}\right|}\right\\}}\leq{\frac{{NP_{o}^{2}}}{{2\left({N-1}\right)}}}$,
set ${\lambda_{\max}}\leftarrow{\lambda_{1}}$; otherwise
${\lambda_{\min}}\leftarrow{\lambda_{1}}$;

9:end while

10:$p_{i}^{{\rm{opt}}}$;

##  V Energy Efficiency maximization of the DCO-OFDM system

In this section, we propose the optimal power allocation schemes to maximize
the EE of the DCO-OFDM system subject to the minimum SE threshold, the maximum
average optical power and the maximum total electrical transmitted power
constraints. We first investigate the EE maximization problem based on the
lower bound ${{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$
in (15) with the closed-form expression, and then study the case based on the
exact achievable rate
${R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$ in (16)
without a closed-form expression.

###  V-A EE Maximization based on
${{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$

Based on the rate expression (15), the EE is given by

$\displaystyle{\rm{E}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\frac{{{R_{{\rm{L}},{\rm{total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}}{{2\sum\limits_{i=1}^{N-1}{{p_{i}}}+{P_{{\rm{dc}}}}+{P_{c}}}},$
(42)

where
${P_{{\rm{dc}}}}=\sum\nolimits_{k=0}^{2N-1}{\mathbb{E}\left\\{{{{\left({{I_{{\rm{dc}}}}}\right)}^{2}}}\right\\}}$
denotes the DC-bias power and ${P_{c}}$ is the constant total circuit
consumption of the whole system.

Therefore, the corresponding EE maximization problem can be formulated as

$\displaystyle\mathop{{\rm{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\},{I_{{\rm{dc}}}}}$
$\displaystyle\quad{\rm{E}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$
(43a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$
$\displaystyle\quad\eqref{x_k_idc},\eqref{dimming_aco},\eqref{electrical_aco},\eqref{PA_con3},$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)\geq\gamma,$
(43b)

where $\gamma=\frac{{\bar{R}}}{{2NW}}$ is the minimum SE requirement of the
DCO-OFDM system, and ${\bar{R}}$ is the corresponding minimum threshold of the
total achievable rate.

To ensure that there is no loss of information during the clipping operation,
the value of ${I_{{\rm{dc}}}}$ is given in (20), and ${P_{{\rm{dc}}}}$ is
given as

$\displaystyle{P_{{\rm{dc}}}}=\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{{{\left({{I_{{\rm{dc}}}^{{\rm{non-
clipping}}}}}\right)}^{2}}}\right\\}}=4\mathbb{E}\left\\{{{{\left({\sum\limits_{i=1}^{N-1}{\sqrt{{p_{i}}}\left|{{X_{i}}}\right|}}\right)}^{2}}}\right\\}.$
(44)

Thus, the original joint optimization problem (43) is converted into a
concave-concave fractional problem, which is still complex and hard to solve.
To overcome this challenge, ${P_{{\rm{dc}}}}$ in (44) is reformulated based on
the Cauchy-Schwarz inequality (22) and is given by

$\displaystyle{P_{{\rm{dc}}}}\leq
4\mathbb{E}\left\\{{\left({N-1}\right)\sum\limits_{i=1}^{N-1}{{p_{i}}{{\left|{{X_{i}}}\right|}^{2}}}}\right\\}=4\left({N-1}\right)\sum\limits_{i=1}^{N-1}{{p_{i}}}.$
(45)

Based on (45), the denominator of (42) can be treated as an affine function of
${p_{i}}$. Thus, problem (43) can be transformed into a concave-linear
fractional problem with variable ${p_{i}}$.

Similar to the SE maximization problem, the average optical power constraint
(17c) and total electrical transmitted power constraint (17d) can be
reformulated as (28b) and (28c), respectively. Therefore, the EE maximization
problem (43) can be reformulated as

$\displaystyle\mathop{{\rm{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\}}$
$\displaystyle\quad\frac{{{R_{{\rm{L}},{\rm{total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}}{{\left({4N-2}\right)\sum\limits_{i=1}^{N-1}{{p_{i}}}+{P_{c}}}}$
(46a) $\displaystyle{\rm{s.t.}}$
$\displaystyle\quad\eqref{power_con1},\eqref{power_con2},\eqref{power_con3},$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{{p_{i}}}\right\\}}\right)\geq\gamma.$
(46b)

Note that those constraints of problem (46) form a convex feasible solution
set. It can be seen from the objective function (46a) that the numerator
${R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$ is concave
over its input power and the denominator is an affine function of ${p_{i}}$.
In the following, we employ Dinkelbach-type iterative algorithm [40, 41] to
handle this concave-linear fractional problem by converting problem (46) into
a sequence of convex sub-problems. In particular, by iteratively solving these
convex subproblems, the globally optimal solution of problem (46) can be
obtained eventually [42].

Let us define a new function $f\left({\left\\{{{p_{i}}}\right\\},q}\right)$ as
follow

$\displaystyle
f\left({\left\\{{{p_{i}}}\right\\},q}\right){\rm{}}\buildrel\Delta\over{=}{R_{{\rm{L}},{\rm{total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)-q\left[{\left({4N-2}\right)\sum\limits_{i=1}^{N-1}{{p_{i}}}+{P_{c}}}\right],$
(47)

where $q$ is a real parameter to be found iteratively. When $q$ is as large as
possible, the optimal solution of problem (46) can be obtained by calculating
the roots of the equation $f\left({\left\\{{{p_{i}}}\right\\},q}\right)=0$ in
the feasible constraint set [42].

For a given $q$ in each iteration, the convex subproblem over ${{p_{i}}}$ can
be expressed as

$\displaystyle\mathop{{\text{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\}}$
$\displaystyle\quad f\left({\left\\{{{p_{i}}}\right\\},q}\right)$ (48)
$\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$
$\displaystyle\quad\eqref{power_con1},\eqref{power_con2},\eqref{power_con3},\eqref{EE_lowerbound_ACO_con2}.$

Since the optimization problem (48) has a concave objective function over its
input power and linear constrains, the optimal power $p_{i}^{{\rm{opt}}}$ of
problem (48) can be obtained by the interior-point algorithm. Such as the
barrier method, it transforms the convex optimization problems into a sequence
of equality constrained problems and applies Newton’s method to them, or such
as the primal-dual interior-point method, it modifies the corresponding KKT
conditions and solves them by Newton’s method[37, 38]. Besides, It has been
implemented by standard convex optimization solvers such as CVX [39].

Finally, the EE maximization problem for the lower bound of achievable rate
can be solved by the Dinkelbach-type algorithm. The Dinkelbach-type algorithm
is guaranteed to converge to the optimal solution of problem (46) with a
finite number of iterations [40, 42, 41]. The details of implementation are
shown in algorithm 2.

Similarly, the $\mathrm{EE}_{\mathrm{A}}$ can be defined with
$R_{{\rm{L,total}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$, and the
corresponding EE maximization power allocation problem can be built and solved
as similar to the problem (43).

Algorithm 2 Dinkelbach-type Power Allocation Scheme

1:Given $\delta\to 0,n=0,p_{i}^{{\rm{opt}}}>0,{q^{\left(n\right)}}=0$;

2:while
$\left|{{q^{\left(n\right)}}-{q^{\left({n+1}\right)}}}\right|\leq\delta$ do

3: Compute the optimal solution $p_{i}^{{\rm{opt}}}$ in (48);

4: Calculating the value of function
$f\left({\left\\{{p_{i}^{{\rm{opt}}}}\right\\},{q^{\left(n\right)}}}\right)$;

5:
${q^{\left({n+1}\right)}}={\rm{E}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{p_{i}^{{\rm{opt}}}}\right\\}}\right)$;

6: $n=n+1$;

7:end while

8:${\rm{E}}{{\rm{E}}_{\rm{L}}}\left({\left\\{{p_{i}^{{\rm{opt}}}}\right\\}}\right)$.

###  V-B EE Maximization based on
${{R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$

By applying the exact achievable rate expression given by (16), the EE of
finite-alphabet inputs can be expressed as

$\displaystyle{\rm{E}}{{\rm{E}}_{\rm{F}}}\left({\left\\{{{p_{i}}}\right\\}}\right)=\frac{{{R_{{\rm{F}},{\rm{total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}}{{2\sum\limits_{i=1}^{N-1}{{p_{i}}}+{P_{{\rm{dc}}}}+{P_{c}}}}.$
(49)

Then, the EE maximization problem with finite-alphabet inputs can be
reformulated as

$\displaystyle\mathop{{\text{maximize}}}\limits_{\left\\{{{p_{i}}}\right\\}}$
$\displaystyle\quad{\rm{E}}{{\rm{E}}_{\rm{F}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$
(50a) $\displaystyle{\rm{s}}{\rm{.t}}{\rm{.}}$
$\displaystyle\quad\eqref{power_con1},\eqref{power_con2},\eqref{power_con3},$
$\displaystyle\quad{\rm{S}}{{\rm{E}}_{\rm{F}}}\left({\left\\{{{p_{i}}}\right\\}}\right)\geq\gamma.$
(50b)

Here, ${P_{{\rm{dc}}}}$ in the denominator of (50a) takes the same value and
constraint as (44) and (45). Due to the concave numerator
${R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)$ in (50a),
problem (50) is also a concave-linear fractional problem, which can also be
solved by Dinkelbach-type algorithms.

##  VI Simulation Results and Discussion

In this section, we present numerical results to illustrate the proposed power
allocation schemes for the SE and EE maximization problems of the DCO-OFDM VLC
system. We consider the above DCO-OFDM VLC system is in a $\left({5\times
5\times 3}\right){{\rm{m}}^{3}}$ room equipped with four LED lights, and the
origin $\left({0,0,0}\right)$ of the three-dimensional Cartesian coordinate
system $\left({X,Y,Z}\right)$ is located at one corner on the floor of the
square room. The receiver is located at $\left({0.5,1,0}\right)$m, and the
four LEDs are respectively located at $\left({1.5,1.5,3}\right)$m,
$\left({1.5,3.5,3}\right)$m, $\left({3.5,1.5,3}\right)$m, and
$\left({3.5,3.5,3}\right)$m. The other basic parameters of the system are
listed in Table I.

TABLE I: Simulation Parameters of the DCO-OFDM VLC System. Definition | Value
---|---
Half the number of subcarriers, $N$ | $16$
Room size | $\left({5\times 5\times 3}\right){{\rm{m}}^{3}}$
FOV, $\Psi$ | ${90^{\circ}}$
Lambertian emission order, $m$ | 1
Half power angle, ${\Phi_{1/2}}$ | ${60^{\circ}}$
PD collection area, ${A_{\rm{r}}}$ | $1~{}{\rm{c}}{{\rm{m}}^{2}}$
Reflectivity factor, $\rho$ | 0.8
Circuit power consumption, $P_{c}$ | $0.1~{}{\rm{W}}$
Optical filter gain of receiver, $T\left(\varphi\right)$ | $0~{}{\rm{dB}}$
Concentrator gain of receiver, $G\left(\varphi\right)$ | $0~{}{\rm{dB}}$
Noise PSD, ${\sigma^{2}}$ | $10^{-18}~{}\rm{A^{2}/Hz}$
Modulation scheme | 4-QAM
Bandwidth of each subcarrier, ${W}$ | $1~{}{\rm{MHz}}$
TABLE II: The tightness of the original constraints (17b), (17c), (17d) and (17e) for the solution of problem (17). Definition Value Scenarios |  |  ${P_{o}}=0.5$ W,
---
${P_{e}}=20$ W
|  ${P_{o}}=0.8$ W,
---
${P_{e}}=10$ W
${I_{{\rm{dc}}}^{{\rm{non-clipping}}}}$ | 0.4991 | 0.5491
$\min\left({{x_{k}}}\right)$ |  $-$0.4052 |  $-$0.4175
$\mathbb{E}\left\\{{{x_{{\rm{dc}},k}}}\right\\}$ | 0.4991 W | 0.5491 W
$\sum_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}$ | 8.2369 W | 9.9698 W

In order to solve problem (17), we use inequality to deal with constraints
(17b), (17c), (17d) and (17e) into (28b), (28c) and (28d). To verify the
validity of the inequality treated constraints, we substitute the obtained
solution of problem (28) into the original constraints (17b), (17c) and (17d).
The tightness of the original constraints (17b), (17c), (17d) and (17e) for
the solutions of problem (17) is shown in Table II. It can be seen that in the
scenario of ${P_{o}}=0.5$ W and ${P_{e}}=20$ W,
$\mathbb{E}\left\\{{{x_{{\rm{dc}},k}}}\right\\}=0.4991$ W, thus constraint
(17c), i.e., $\mathbb{E}\left\\{{x_{{\rm{dc}},k}}\right\\}\leq{P_{o}}$, is
almost tight, while in the scenario of ${P_{o}}=0.8$ W and ${P_{e}}=10$ W,
$\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}=9.9698$
W, thus constraint (17d), i.e.,
$\sum\limits_{k=0}^{2N-1}{\mathbb{E}\left\\{{x_{{\rm{dc}},k}^{2}}\right\\}}\leq{P_{e}}$,
is almost tight. Therefore, our treatment of constraints is valid and the
obtained optimal solutions are also high-quality solutions for original
problems.

###  VI-A Simulation Results of SE Maximization Problems

In this subsection, we present the results of the proposed power allocation
schemes for maximizing the SE for finite-alphabet inputs and lower bound of
the mutual information.

In order to illustrate the effect of the difference between channel gain of
each subcarrier on power allocation, the channel gain $H_{i}$, i.e., equation
(13), of half subcarriers is shown in Fig. 2 (a). It can be seen from Fig. 2
(a) that the channel model in our DCO-OFDM system also has low-pass
characteristics. As the subcarrier index $i$ increases, the corresponding
channel gain $H_{i}$ decreases. The reason is that the channel gain $H_{i}$
varies over the subcarrier in the dispersive channel model and high-frequency
subcarriers correspond to higher subcarrier index $i$.

(a)

(b)

(c)

Figure 2: (a) Channel gain $H_{i}$ of subcarrier $i$;(b) Achievable rate
${R_{{\rm{F}},i}}$ versus the allocated power ${p_{i}}$; (c) Gradient function
$\frac{{\partial{R_{{\rm{F}},i}}}}{{\partial{p_{i}}}}$ versus the allocated
power ${p_{i}}$.

Fig. 2 (b) shows the achievable rate ${R_{{\rm{F}},i}}$, i.e., equation (15b),
versus the allocated power ${p_{i}}$, in the case of $i=1$ and $i=15$. It can
be seen that as the allocated power ${p_{i}}$ increases, the achievable rate
${R_{{\rm{F}},1}}$ and ${R_{{\rm{F}},15}}$ first increase fast and then
increase slowly, and ${R_{{\rm{F}},1}}$ gradually approaches ${\log_{2}}M$
($M=4$). This is because the mutual information of $M$-ary discrete
constellation modulation can not exceed ${\log_{2}}M$. Moreover,
${R_{{\rm{F}},1}}$ is higher than ${R_{{\rm{F}},15}}$ since
${H_{1}}>{H_{15}}$.

Fig. 2 (c) shows the gradient function
$\frac{{\partial{R_{{\rm{F}},i}}}}{{\partial{p_{i}}}}$, i.e., equation (35),
versus the allocated power ${p_{i}}$, in the case of $i=1$ and $i=15$. It can
be seen that both $\frac{{\partial{R_{{\rm{F}},1}}}}{{\partial{p_{1}}}}$ and
$\frac{{\partial{R_{{\rm{F}},15}}}}{{\partial{p_{15}}}}$ decrease with the
increase of ${p_{i}}$, and eventually tend to $0$. When ${p_{i}}\leq 5.134$
mW,
$\frac{{\partial{R_{{\rm{F}},1}}}}{{\partial{p_{1}}}}\geq\frac{{\partial{R_{{\rm{F}},15}}}}{{\partial{p_{15}}}}$;
when ${p_{i}}>5.134$ mW,
$\frac{{\partial{R_{{\rm{F}},1}}}}{{\partial{p_{1}}}}<\frac{{\partial{R_{{\rm{F}},15}}}}{{\partial{p_{15}}}}$.
Fig. 2 (b) illustrates that, compared with subcarriers with small channel
gains, when the allocated power is small, the subcarriers with large channel
gains have larger gradient of rate. When the allocated power is large, the
subcarriers with large channel gains have smaller gradient of rate.

(a)

(b)

(c)

Figure 3: Allocated power $p_{i}$ of subcarrier $i$ of $\text{SE}_{\text{F}}$
with ${P_{o}}=10$ W (a) ${P_{e}}=2$ W; (b) ${P_{e}}=10$ W; (c) ${P_{e}}=50$ W.

In the following, three different power allocation scenarios for the SE
maximization problem (30) are compared in the case of low, medium, and high
total transmitted power. Specially, Fig. 3 (a), (b) and (c) are achieved with
the same average optical power ${P_{o}}=10$ W, and different total electrical
transmitted power ${P_{e}}=2$ W, ${P_{e}}=10$ W and ${P_{e}}=50$ W
respectively.

It can be seen from Fig. 3 (a) that when the total allocated power is low
(${P_{e}}=2$ W), more power is allocated to subcarriers with larger channel
gains, which is similar to the classical water-filling solution with the
Gaussian distribution inputs. Combining Fig. 3 (a) and Fig. 2, it can be seen
that this is because when the allocated power is small, the rate of the
subcarriers with larger channel gains increases faster than the subcarriers
with smaller channel gains, thus more power should be allocated to the
subcarriers with larger gradient of rate to maximize the
$\text{SE}_{\text{F}}$ of the system.

Fig. 3 (b) shows that for the case of medium total allocated power
(${P_{e}}=10$ W) , more power is allocated to the subcarriers with moderate
channel gains. Combining Fig. 3 (b) and Fig. 2, it can be seen that this is
because as the allocated power increases, the rate of subcarriers with smaller
channel gains increases faster than the rate of subcarriers with larger
channel gains at this time. Therefore, more power is allocated to the
subcarriers with smaller channel gains to maximize the $\text{SE}_{\text{F}}$
of the system, i.e., power is always preferentially allocated to the
subcarrier with the largest gradient of rate.

It can be seen from Fig. 3 (c) that when the total allocated power is high
(${P_{e}}=50$ W), more power is allocated to subcarriers with smaller channel
gains, which is exactly opposite with the water-filling policy. Combining Fig.
3 (c) and Fig. 2, it can be seen that in this case, the rate of subcarriers
with larger channel gains tends to be saturated (${\log_{2}}M$), thus there is
little incentive to allocate further power to such subcarriers. Rather, to
maximize the $\text{SE}_{\text{F}}$ of the system, the additional power is
better allocated to subcarriers with smaller channel gains whose rate is still
far from saturation.

(a)

(b)

Figure 4: (a) $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ versus electrical power budget $P_{\mathrm{e}}$ with
two different optical power budget $P_{\mathrm{o}}=0.8$ W and
$P_{\mathrm{o}}=\infty$; (b) $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$
and $\text{SE}_{\text{L}}$ versus optical power budget ${P_{o}}$ with two
different electrical power budget ${P_{e}}=20$ W and ${P_{e}}=\infty$.

Fig. 4 (a) illustrates $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ versus total electrical transmitted power budget
$P_{\mathrm{e}}$ with two different optical power budget $P_{\mathrm{o}}=0.8$
W and $P_{\mathrm{o}}=\infty$ (without optical power constraint),
respectively. For the case of $P_{\mathrm{o}}=0.8$ W, as $P_{\mathrm{e}}$
increases, $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ first increase and then get restricted into a constant.
The reason is that the total allocated power is restricted by the optical
power $P_{\mathrm{o}}=0.8$ W. For the case of $P_{\mathrm{o}}=\infty$, as
$P_{\mathrm{e}}$ increases, $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ first increase and then remain constant. This is
because the mutual information of $M$-ary discrete constellation modulation
can not exceed ${\log_{2}}M$. Moreover, similar to $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$, when $P_{\mathrm{e}}>20$ W, $\text{SE}_{\text{F}}$
with $P_{\mathrm{o}}=\infty$ is higher than $\text{SE}_{\text{F}}$ with
$P_{\mathrm{o}}=0.8$ W. This is because the total allocated power is
restricted when $P_{\mathrm{o}}$ takes the value of $0.8$ W. It can be seen
that, using the optimal power allocation policy proposed in this paper,
$\text{SE}_{\text{F}}$ is significantly higher than $\text{SE}_{\text{L}}$
since ${{R_{{\rm{L,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$ is the
lower bound of
${{R_{{\rm{F,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$. Meanwhile,
$\text{SE}_{\text{F}}$ is slightly lower than $\text{SE}_{\text{A}}$ in the
medium $P_{\mathrm{e}}$, however, they will tend to coincide when
$P_{\mathrm{e}}\to\infty$ and $P_{\mathrm{e}}\to 0$, which consistent with the
feature of ${{R_{{\rm{A,total}}}}\left({\left\\{{{p_{i}}}\right\\}}\right)}$
[36].

Fig. 4 (b) depicts $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ versus average optical power budget ${P_{o}}$ with two
different electrical power budgets ${P_{e}}=20$ W and ${P_{e}}=\infty$
(without electrical power constraint), respectively. For the case of
${P_{e}}=20$ W, as ${P_{o}}$ increases, $\text{SE}_{\text{F}}$,
$\text{SE}_{\text{A}}$ and $\text{SE}_{\text{L}}$ first increase and then get
restricted as a constant. This is because the total allocated power is limited
by the electrical power budget ${P_{e}}=20$ W. When ${P_{e}}=\infty$, as
${P_{o}}$ increases, $\text{SE}_{\text{F}}$, $\text{SE}_{\text{A}}$ and
$\text{SE}_{\text{L}}$ first increase and then remain constant. In addition,
similar to $\text{SE}_{\text{A}}$ and $\text{SE}_{\text{L}}$, when
${P_{o}}>0.6$ W, $\text{SE}_{\text{F}}$ with ${P_{e}}=\infty$ is higher than
$\text{SE}_{\text{F}}$ with ${P_{e}}=20$ W. This is because the total
allocated power is restricted when ${P_{e}}$ takes the value of $20$ W.
Besides, we can observe that $\text{SE}_{\text{F}}$ is significantly higher
than $\text{SE}_{\text{L}}$, and $\text{SE}_{\text{A}}$ is slightly highter
than $\text{SE}_{\text{A}}$ but converges if $P_{\mathrm{e}}\to\infty$ and
$P_{\mathrm{e}}\to 0$.

###  VI-B Simulation Results of EE Maximization Problems

In this subsection, we present the simulation results of the proposed power
allocation schemes for maximizing the EE for finite-alphabet inputs and lower
bound of the mutual information.

(a)

(b)

(c)

Figure 5: Allocated power $p_{i}$ of subcarrier $i$ of $\text{EE}_{\text{F}}$
with ${P_{o}}=1$ W, ${P_{e}}=22$ W, (a) $\gamma=0.156$ Mbits/sec/Hz
($\bar{R}=5$ Mbits/sec); (b) $\gamma=0.469$ Mbits/sec/Hz ($\bar{R}=15$
Mbits/sec); (c) $\gamma=0.781$ Mbits/sec/Hz ($\bar{R}=25$ Mbits/sec).

To illustrate the three scenarios when the SE threshold $\gamma$ is small,
medium and large, the allocated power $p_{i}$ of subcarrier $i$ of
$\text{EE}_{\text{F}}$ with $\gamma=0.156$ Mbits/sec/Hz ($\bar{R}=5$
Mbits/sec), $\gamma=0.469$ Mbits/sec/Hz ($\bar{R}=15$ Mbits/sec) and
$\gamma=0.781$ Mbits/sec/Hz ($\bar{R}=25$ Mbits/sec), where ${P_{o}}=1$ W,
${P_{e}}=22$ W, are shown in Fig. 5 (a), (b) and (c) respectively.

It can be seen from Fig. 5 (a) that when the SE threshold $\gamma$ is small
($\gamma=0.156$ Mbits/sec/Hz), more power is allocated to subcarriers with
larger channel gains, and the required power increases with the improvement of
the SE threshold $\gamma$. Besides, Fig. 2 (c) shows that the gradient of the
rate increases synchronously with the channel gain. Thus, more power would be
allocated to the subcarriers with a larger gradient of the rate to maximize
the $\text{EE}_{\text{F}}$ of the system.

Fig. 5 (b) shows that for the case of medium SE threshold $\gamma$
($\gamma=0.469$ Mbits/sec/Hz), more power is allocated to the subcarriers with
moderate channel gains, and as the SE threshold $\gamma$ increases, the power
that needs to be allocated becomes larger. Combining with Fig. 2 (c), it can
be seen that to maximize the $\text{EE}_{\text{F}}$ of the system, the power
is always preferentially allocated to the subcarrier with the largest gradient
of the rate.

It can be seen from Fig. 5 (c) that when the SE threshold $\gamma$ is large
($\gamma=0.781$ Mbits/sec/Hz), more power is allocated to subcarriers with
smaller channel gains. Combining Fig. 5 (c) and Fig. 2 (c), it can be seen
that when the allocated power $p_{i}$ is large, to achieve the same gradient
of the rate, the power allocated to subcarriers with larger channel gains is
much greater than the power allocated to subcarriers with smaller channel
gains. Thus, the additional power is preferentially allocated to subcarriers
with smaller channel gains to maximize the $\text{EE}_{\text{F}}$ of the
system.

(a)

(b)

Figure 6: (a) ${\text{EE}}_{\text{F}}$ and ${\text{EE}_{\text{A}}}$ versus
electrical power budget ${P_{e}}$ with SE threshold $\gamma=0.009$
Mbits/sec/Hz ($\bar{R}=0.3$ Mbits/sec) and two different optical power budget
$P_{\mathrm{o}}=0.06$ W and $P_{\mathrm{o}}=\infty$; (b)
${\text{EE}}_{\text{F}}$, ${\text{EE}_{\text{A}}}$ and
${\text{EE}_{\text{L}}}$ versus SE threshold $\gamma$ with electrical power
budget ${P_{\mathrm{e}}}=5$ W and optical power budget $P_{\mathrm{o}}=1$ W.

Fig. 6 (a) illustrates $\text{EE}_{\text{F}}$ and $\text{EE}_{\text{A}}$
versus electrical power budget $P_{\mathrm{e}}$ with SE constraint
$\gamma=0.009$ Mbits/sec/Hz ($\bar{R}=0.3$ Mbits/sec) and two different
optical power budget $P_{\mathrm{o}}=0.06$ W and $P_{\mathrm{o}}=\infty$
(without optical power constraint), respectively. For the case of
$P_{\mathrm{o}}=0.06$ W, as ${P_{\mathrm{e}}}$ increases,
$\text{EE}_{\text{F}}$ and $\text{EE}_{\text{A}}$ first increase and then get
restricted into a constant. The reason is that the total allocated power is
restricted by the optical power $P_{\mathrm{o}}=0.06$ W. For the case of
$P_{\mathrm{o}}=\infty$, as $P_{\mathrm{e}}$ increases, $\text{EE}_{\text{F}}$
and $\text{EE}_{\text{A}}$ first increase and then remain constant. This is
because $\text{EE}_{\text{F}}$ and $\text{EE}_{\text{A}}$ remain constant when
they reach their maximum values respectively. Moreover, similar to
$\text{EE}_{\text{A}}$, for the large $P_{\mathrm{e}}$, $\text{EE}_{\text{F}}$
with $P_{\mathrm{o}}=\infty$ is higher than $\text{SE}_{\text{F}}$ with
$P_{\mathrm{o}}=0.06$ W since $\text{EE}_{\text{L}}$ and
$\text{EE}_{\text{F}}$ are restricted when $P_{\mathrm{o}}$ takes the value of
$0.06$ W. It can be seen that $\text{EE}_{\text{F}}$ is lower than
$\text{EE}_{\text{A}}$.

Fig. 6 (b) depicts ${\text{EE}}_{\text{F}}$ and $\text{EE}_{\text{A}}$ versus
SE threshold $\gamma$ with electrical power budget $P_{\mathrm{e}}=5$ W and
optical power budget $P_{\mathrm{o}}=1$ W. We observe that
${\text{EE}}_{\text{F}}$ and $\text{EE}_{\text{A}}$ both first keep stable and
then decrease as SE threshold $\gamma$ increases. This is because when the SE
threshold $\gamma$ is small, the performed power allocation can easily satisfy
the SE requirement and thus EE keeps as a constant. When the SE threshold
$\gamma$ becomes larger, more power needs to be consumed to satisfy the rate
constraint, therefore the EE decreases.

_C. Simulation Results of Computational Complexity_

In this subsection, we present the average executing time versus half of
subcarrier number $N$ to evaluate the computational complexity of the proposed
power allocation schemes for SE- and EE-maximization problem. For given $N$
and other fixed parameters, the average executing time is calculated according
on $10000$ repeated simulations, which are performed by MATLAB (2020b) with
Intel(R) Core(TM) i9-10900K 3.70 GHz CPU and 32 GB RAM.

TABLE III: The average executing time (ms) of different scheme versus half of subcarrier number $N$. $N$ Time Scheme | $\mathrm{SE_{F}}$ | $\mathrm{SE_{A}}$ | $\mathrm{EE_{F}}$ | $\mathrm{EE_{A}}$
---|---|---|---|---
4 | $8.34$ | $2.91$ | $18220$ | $1030$
8 | $48.77$ | $2.83$ | $27630$ | $1040$
16 | $126.51$ | $2.99$ | $72500$ | $1040$

For the SE-maximization problem, through the mercury-water-filling method,
Table III depicts the comparing of the scheme base on
$R_{\mathrm{F,total}}\left(\left\\{p_{i}\right\\}\right)$ and based on
$R_{\mathrm{A,total}}\left(\left\\{p_{i}\right\\}\right)$ with electrical
power budget ${P_{e}}=20$ W and optical power budget ${P_{o}}=5$ W. As shows,
the CPU time of the proposed schemes based on
$R_{\mathrm{F,total}}\left(\left\\{p_{i}\right\\}\right)$ increases as $N$
increases, however, for the proposed schemes based on
$R_{\mathrm{A,total}}\left(\left\\{p_{i}\right\\}\right)$, it almost is a
constant. Meanwhile, it is obvious that the scheme based on
$R_{\mathrm{A,total}}\left(\left\\{p_{i}\right\\}\right)$ is much faster than
the scheme based on $R_{\mathrm{F,total}}\left(\left\\{p_{i}\right\\}\right)$.

For the EE-maximization problem, Table III presents the difference of the
scheme based on $R_{\mathrm{F,total}}\left(\left\\{p_{i}\right\\}\right)$ and
$R_{\mathrm{A,total}}\left(\left\\{p_{i}\right\\}\right)$ with electrical
power budget ${P_{e}}=20$ W, optical power budget ${P_{o}}=5$ W, and SE
threshold $\gamma=0.009$ Mbits/sec/Hz,and we can draw a conclusion as same as
the comparing between $\mathrm{SE_{F}}$ and $\mathrm{SE_{L}}$.

##  VII Conclusion

In this work, we investigated the bound of the information transmission rate
of the DCO-OFDM system with finite-alphabet inputs, and proposed the optimal
power allocation schemes to achieve maximum SE and maximum EE of the DCO-OFDM
system, respectively. We first derived the exact achievable rate expression
without information loss. Then, we developed the power allocation to maximize
SE based on the lower bound of achievable rate. Furthermore, we exploited the
KKT conditions and the relationship between the mutual information and MMSE,
and derived the multi-level mercury-water-filling power allocation scheme for
SE maximization. Moreover, we proposed the Dinkelbach-type power allocation
scheme and obtained the optimal power allocation for EE maximization. Finally,
numerical results revealed that the proposed multi-level mercury-water-filling
scheme of SE maximization based on the exact mutual information depends on
both channel gain of each subcarrier and the total transmitted power
constraint. When the total transmitted power is high, the power allocation of
each subcarrier is inversely proportional to the channel gain, which is
different from that of the classical water-filling method. Besides, we
revealed that for the EE maximization problem with finite-alphabet inputs, the
power allocation of each subcarrier is proportional to the channel gain for
the low SE requirement. While for the high SE requirement, the power
allocation of each subcarrier is inversely proportional to the channel gain.
Besides, under the same constraints, the value of SE and EE based on the exact
mutual information is always higher than that based on the lower bound.

##  Appendix A proof of (15b)

According to equation (14), the conditional probability density function
$p\left({{Y_{i}}|{X_{i}}}\right)$ and the probability density function
$p\left({{Y_{i}}}\right)$ corresponding to the channel output ${Y_{i}}$ are
respectively expressed as

$\displaystyle
p\left({{Y_{i}}|{X_{i}}}\right)=\frac{1}{{\pi{\sigma^{2}}W}}\exp\left({-\frac{{{{\left|{{Y_{i}}-{H_{i}}\sqrt{{p_{i}}}{X_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right),$
(51a) $\displaystyle
p\left({{Y_{i}}}\right)={\mathbb{E}_{{X_{i}}}}\left\\{{p\left({{Y_{i}}|{X_{i}}}\right)}\right\\}=\frac{1}{M}\sum\limits_{k=1}^{M}{p\left({{Y_{i}}|{X_{i,k}}}\right)}.$
(51b)

Then, the mutual information between the channel input ${X_{i}}$ and channel
output ${Y_{i}}$ can be obtained as

$\displaystyle
I\left({{X_{i}};{Y_{i}}}\right)=W\sum\limits_{{X_{i}}}{\int\limits_{{Y_{i}}}{p\left({{X_{i}},{Y_{i}}}\right){{\log}_{2}}\frac{{p\left({{X_{i}},{Y_{i}}}\right)}}{{p\left({{X_{i}}}\right)p\left({{Y_{i}}}\right)}}\,\mathrm{d}Y_{i}}}$
(52a)
$\displaystyle=W\sum\limits_{n=1}^{M}{\int\limits_{{Y_{i}}}{\frac{1}{M}p\left({{Y_{i}}|{X_{i,n}}}\right)}{{\log}_{2}}\frac{{p\left({{Y_{i}}|{X_{i,n}}}\right)}}{{p\left({{Y_{i}}}\right)}}\,\mathrm{d}Y_{i}}$
(52b)
$\displaystyle=\frac{W}{M}\sum\limits_{n=1}^{M}{\int\limits_{{Y_{i}}}{p\left({{Y_{i}}|{X_{i,n}}}\right)}{{\log}_{2}}\frac{{p\left({{Y_{i}}|{X_{i,n}}}\right)}}{{\sum\limits_{k=1}^{M}{\frac{1}{M}p\left({{Y_{i}}|{X_{i,k}}}\right)}}}\,\mathrm{d}Y_{i}}$
(52c)
$\displaystyle=\frac{W}{M}\sum\limits_{n=1}^{M}{\int\limits_{{Y_{i}}}{p\left({{Y_{i}}|{X_{i,n}}}\right)}{{\log}_{2}}\frac{{\exp\left({-\frac{{{{\left|{{Y_{i}}-{H_{i}}\sqrt{{p_{i}}}{X_{i,n}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}}{{\sum\limits_{k=1}^{M}{\frac{1}{M}\exp\left({-\frac{{{{\left|{{Y_{i}}-{H_{i}}\sqrt{{p_{i}}}{X_{i,k}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}}}\,\mathrm{d}Y_{i}}$
(52d)
$\displaystyle=-\frac{W}{M}\sum\limits_{n=1}^{M}{\int\limits_{{Z_{i}}}{p\left({{Z_{i}}}\right)}{{\log}_{2}}\sum\limits_{k=1}^{M}{\frac{1}{M}\exp\left({-{d_{n,k}}+\frac{{{{\left|{{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}\,\mathrm{d}Z_{i}}$
(52e)
$\displaystyle=-\frac{W}{M}\sum\limits_{n=1}^{M}{{\mathbb{E}_{{Z_{i}}}}\left\\{{{{\log}_{2}}\sum\limits_{k=1}^{M}{\frac{1}{M}\exp\left({-{d_{n,k}}+\frac{{{{\left|{{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}}\right\\}}$
(52f)
$\displaystyle=-\frac{W}{M}\sum\limits_{n=1}^{M}{{\mathbb{E}_{{Z_{i}}}}\left\\{{{{\log}_{2}}\frac{1}{M}\exp\left({\frac{{{{\left|{{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}}\right)}\right\\}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{W}{M}\sum\limits_{n=1}^{M}{{\mathbb{E}_{{Z_{i}}}}\left\\{{{{\log}_{2}}\sum\limits_{k=1}^{M}{\exp\left({-{d_{n,k}}}\right)}}\right\\}}$
(52g)
$\displaystyle=-\frac{W}{M}\sum\limits_{n=1}^{M}\mathbb{E}_{Z_{i}}\left\\{\log_{2}\left(\frac{\exp\left(\frac{\left|Z_{i}\right|^{2}}{\sigma^{2}W}\right)\sum\limits_{k=1}^{M}\exp\left(-d_{n,k}\right)}{M}\right)\right\\}$
(52h) $\displaystyle=W\left({{{\log}_{2}}M-\frac{1}{{\ln 2}}}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\sum\limits_{n=1}^{M}\frac{W}{M}{\mathbb{E}_{{Z_{i}}}}\left\\{{{{\log}_{2}}\sum\limits_{k=1}^{M}{\exp\left({-{d_{n,k}}}\right)}}\right\\},$
(52i)

where (52c) is due to (51b), (52d) is due to (51a),
${d_{n,k}}\buildrel\Delta\over{=}\frac{{{{\left|{{H_{i}}\sqrt{{p_{i}}}\left({{X_{i,n}}-{X_{i,k}}}\right)+{Z_{i}}}\right|}^{2}}}}{{{\sigma^{2}}W}}$
and the first term in (52i) is based on
${\mathbb{E}_{{Z_{i}}}}\left\\{{{{\left|{{Z_{i}}}\right|}^{2}}}\right\\}={\sigma^{2}}W$.

## References

  * [1] M. A. Razzaque, M. Milojevic-Jevric, A. Palade, and S. Clarke,  “Middleware for internet of things: A survey,”  IEEE Internet Things J., vol. 3, no. 1, pp. 70–95, Nov. 2016.
  * [2] A. Zanella, N. Bui, A. Castellani, L. Vangelista, and M. Zorzi,  “Internet of things for smart cities,”  IEEE Internet Things J., vol. 1, no. 1, pp. 22–32, Feb. 2014.
  * [3] V. W. S. Wong, R. Schober, D. W. K. Ng, and L. Wang, Eds.,  Key Technologies for 5G Wireless Systems,  Cambridge, U.K.: Cambridge Univ. Press, Apr. 2017.
  * [4] L. Feng, H. Yang, R. Q. Hu, and J. Wang,  “MmWave and VLC-based indoor channel models in 5G wireless networks,”  IEEE Wireless Commun., vol. 25, no. 5, pp. 70–77, Oct. 2018.
  * [5] P. D. Diamantoulakis, G. K. Karagiannidis, and Z. Ding,  “Simultaneous lightwave information and power transfer (SLIPT),”  IEEE Trans. Green Commun. Netw., vol. 2, no. 3, pp. 764–773, Sep. 2018.
  * [6] IEEE standard for local and metropolitan area networks-part 15.7: short-range wireless optical communication using visible light,  IEEE Std 802.15.7-2011, Sep. 2011.
  * [7] H. Haas, L. Yin, Y. Wang, and C. Chen,  “What is LiFi?”  J. Lightw. Technol., vol. 34, no. 6, pp. 1533–1544, Dec. 2016.
  * [8] X. Liu, Y. Wang, F. Zhou, S. Ma, R. Q. Hu, and D. W. K. Ng,  “Beamforming design for secure MISO visible light communication networks with SLIPT,”  IEEE Trans. Commun., vol. 68, no. 12, pp. 7795–7809, Dec. 2020\.
  * [9] A. Jovicic, J. Li, and T. Richardson,  “Visible light communication: Opportunities, challenges and the path to market,”  IEEE Commun. Mag., vol. 51, no. 12, pp. 26–32, Dec. 2013.
  * [10] R. Zhang, J. Wang, Z. Wang, Z. Xu, C. Zhao, and L. Hanzo,  “Visible light communications in heterogeneous networks: paving the way for user-centric design,”  IEEE Wireless Commun., vol. 22, no. 2, pp. 8–16, Apr. 2015.
  * [11] P. H. Pathak, X. Feng, P. Hu, and P. Mohapatra,  “Visible light communication, networking, and sensing: A survey, potential and challenges,”  IEEE Commun. Surveys Tuts., vol. 17, no. 4, pp. 2047–2077, 4th Quart. 2015.
  * [12] H. Elgala, R. Mesleh, and H. Haas,  “Indoor optical wireless communication: Potential and state-of-the-art,”  IEEE Commun. Mag., vol. 49, no. 9, pp. 56–62, Sep. 2011.
  * [13] H. Le Minh, D. O’Brien, G. Faulkner, L. Zeng, K. Lee, D. Jung, and Y. Oh,  “High-speed visible light communications using multiple-resonant equalization,”  IEEE Photon. Technol. Lett., vol. 20, no. 14, pp. 1243–1245, Jul. 2008.
  * [14] A. T. Hussein and J. M. H. Elmirghani,  “Mobile multi-gigabit visible light communication system in realistic indoor environment,”  J. Lightw. Technol., vol. 33, no. 15, pp. 3293–3307, Jun. 2015\.
  * [15] S. Dimitrov and H. Haas,  “Information rate of OFDM-based optical wireless communication systems with nonlinear distortion,”  J. Lightw. Technol., vol. 31, no. 6, pp. 918–929, Mar. 2013.
  * [16] L. Wu, Z. Zhang, J. Dang, and H. Liu,  “Adaptive modulation schemes for visible light communications,”  J. Lightw. Technol., vol. 33, no. 1, pp. 117–125, Jan. 2015.
  * [17] X. Deng, S. Mardanikorani, G. Zhou, and J. M. G. Linnartz,  “DC-bias for optical OFDM in visible light communications,”  IEEE Access, vol. 7, pp. 98319–98330, Jul. 2019.
  * [18] X. Ling, J. Wang, X. Liang, Z. Ding, and C. Zhao,  “Offset and power optimization for DCO-OFDM in visible light communication systems,”  IEEE Trans. Signal Process., vol. 64, no. 2, pp. 349–363, Jan. 2016\.
  * [19] X. Ling, J. Wang, X. Liang, Z. Ding, C. Zhao, and X. Gao,  “Biased multi-LED beamforming for multicarrier visible light communications,”  IEEE J. Sel. Areas Commun., vol. 36, no. 1, pp. 106–120, Jan. 2018\.
  * [20] J. Armstrong and B. J. C. Schmidt,  “Comparison of asymmetrically clipped optical OFDM and DC-biased optical OFDM in AWGN,”  IEEE Commun. Lett., vol. 12, no. 5, pp. 343–345, May 2008.
  * [21] D. Guo, S. Shamai(Shitz), and S. Verdu,  “Mutual information and minimum mean-square error in Gaussian channels,”  IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1261–1282, Apr. 2005\.
  * [22] R. Zhang and Y.-C. Liang,  “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,”  IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp. 88–102, 2008.
  * [23] A. Lozano, A. M. Tulino, and S. Verdu,  “Optimum power allocation for parallel Gaussian channels with arbitrary input distributions,”  IEEE Trans. Inf. Theory, vol. 52, no. 7, pp. 3033–3051, Jul. 2006\.
  * [24] S. Mardanikorani, X. Deng, and J. M. G. Linnartz,  “Sub-carrier loading strategies for DCO-OFDM LED communication,”  IEEE Trans. Commun., vol. 68, no. 2, pp. 1101–1117, Feb. 2020.
  * [25] J. Zhou and W. Zhang,  “A comparative study of unipolar OFDM schemes in Gaussian optical intensity channel,”  IEEE Trans. Commun., vol. 66, no. 4, pp. 1549–1564, Apr. 2018.
  * [26] M. Kashef, M. Ismail, M. Abdallah, K. A. Qaraqe, and E. Serpedin,  “Energy efficient resource allocation for mixed RF/VLC heterogeneous wireless networks,”  IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 883–893, Mar. 2016\.
  * [27] W. Guo, H. Zhang, and C. Huang,  “Energy efficiency of two-way communications under various duplex modes,”  IEEE Internet Things J., vol. 8, no. 3, pp. 1921–1933, Feb. 2021\.
  * [28] C. Xiong, G. Y. Li, S. Zhang, Y. Chen, and S. Xu,  “Energy-and spectral-efficiency tradeoff in downlink OFDMA networks,”  IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3874–3886, Dec. 2011.
  * [29] A. Weiss, A. Yeredor, and M. Shtaif,  “Iterative symbol recovery for power-efficient DC-biased optical OFDM systems,”  J. Lightw. Technol., vol. 34, no. 9, pp. 2331–2338, May 2016.
  * [30] Y. Hei, Y. Kou, G. Shi, W. Li, and H. Gu,  “Energy-spectral efficiency tradeoff in DCO-OFDM visible light communication system,”  IEEE Trans. Veh. Technol., vol. 68, no. 10, pp. 9872–9882, Oct. 2019.
  * [31] C. Xiao, Y. R. Zheng, and Z. Ding,  “Globally optimal linear precoders for finite alphabet signals over complex vector Gaussian channels,”  IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3301–3314, Jul. 2011.
  * [32] P. Ge, X. Liang, J. Wang, and C. Zhao,  “Modulation order selection and power allocation for energy efficient VLC-OFDM systems,”  in 2017 9th International Conference on Wireless Communications and Signal Processing (WCSP), pp. 1–6, Oct. 2017.
  * [33] J. M. Kahn and J. R. Barry,  “Wireless infrared communications,”  Proceedings of the IEEE, vol. 85, no. 2, pp. 265–298, Feb. 1997\.
  * [34] H. Schulze,  “Frequency-domain simulation of the indoor wireless optical communication channel,”  IEEE Trans. Commun., vol. 64, no. 6, pp. 2551–2562, Jun. 2016.
  * [35] R. Rajashekar, M. Di Renzo, L. Yang, K. V. S. Hari, and L. Hanzo,  “A finite input alphabet perspective on the rate-energy tradeoff in SWIPT over parallel Gaussian channels,”  IEEE J. Sel. Areas Commun., vol. 37, no. 1, pp. 48–60, Jan. 2019\.
  * [36] W. Zeng, C. Xiao, and J. Lu,  “A low-complexity design of linear precoding for MIMO channels with finite-alphabet inputs,”  IEEE Wireless Commun. Lett., vol. 1, no. 1, pp. 38–41, Feb. 2012\.
  * [37] S. Boyd and L. Vandenberghe,  Convex Optimization,  Cambridge, U.K.: Cambridge Univ. Press, 2004.
  * [38] Joseph-Frédéric Bonnans, Jean Charles Gilbert, Claude Lemarechal, and Claudia A. Sagastizábal,  Numerical Optimization: Theoretical and Practical Aspects,  Universitext. Springer, Berlin ; New York, 2nd ed edition, 2006.
  * [39] M. Grant and S. Boyd,  “CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.
  * [40] W. Dinkelbach,  “On nonlinear fractional programming,”  Manage. Sci., vol. 13, no. 7, pp. 492–498, Mar. 1967.
  * [41] J. P. G. Crouzeix and J. A. Ferland,  “Algorithms for generalized fractional programming,”  Math. Program., vol. 52, pp. 191–207, May 1991.
  * [42] A. Zappone and E. Jorswieck,  “Energy efficiency in wireless networks via fractional programming theory,”  Found. Trends Commun. Inf. Theory, vol. 11, no. 3-4, pp. 185–396, Jun. 2015.