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Spectral and Energy Efﬁciency of ACOOFDM in
Visible Light Communication Systems
Shuai Ma, Ruixin Yang, Xiong Deng, Member, IEEE, Xintong Ling, Member, IEEE, Xun Zhang, Fuhui Zhou,
Shiyin Li, and Derrick Wing Kwan Ng, Fellow, IEEE
Abstract—In this paper, we study the spectral efﬁciency (SE)
and energy efﬁciency (EE) of asymmetrically clipped optical
orthogonal frequency division multiplexing (ACOOFDM) for
visible light communication (VLC). Firstly, we derive the achiev
able rates for Gaussian distributions inputs and practical ﬁnite
alphabet inputs. Then, we investigate the SE maximization
problems subject to both the total transmit power constraint and
the average optical power constraint with the above two inputs,
respectively. By exploiting the relationship between the mutual
information and the minimum meansquared error, an optimal
power allocation scheme is proposed to maximize the SE with
ﬁnitealphabet inputs. To reduce the computational complexity
of the power allocation scheme, we derive a closedform lower
bound of the SE. Also, considering the quality of service, we
further tackle the nonconvex EE maximization problems of
ACOOFDM with the two inputs, respectively. The problems are
solved by the proposed Dinkelbachtype iterative algorithm. In
each iteration, the interior point algorithm is applied to obtain
the optimal power allocation.The performance of the proposed
power allocation schemes for the SE and EE maximization are
validated through numerical analysis.
Index Terms—ACOOFDM, energy efﬁciency, spectral efﬁ
ciency, visible light communications.
I. INTRODUCTION
Traditional radio frequency (RF) communications are facing
the problem of spectrum crunch because of the exponential
increase in the demand for wireless data trafﬁc [1], [2].
Besides, the tremendous wireless devices consume more than
3% of the global energy [3], [4], and lead to about 5% of the
total CO2emissions worldwide by 2020 [5]–[7]. Therefore,
both spectral and energy resources are severely limited for
next generation wireless communications. Facilitated by the
lowcost and widely installed lighting infrastructure with light
S. Ma, R. Yang and S. Li are with the School of Information and
Control Engineering, China University of Mining and Technology, Xuzhou,
221116, China. (email:mashuai001@cumt.edu.cn; ray.young@cumt.edu.cn;
lishiyin@cumt.edu.cn).
X. Deng is with the Department of Electrical Engineering, Eindhoven
University of Technology (TU/e), Eindhoven, NL. (email: X.Deng@tue.nl).
X. Ling is with the National Mobile Communications Research Laboratory,
Southeast University, and the Purple Mountain Laboratories, Nanjing, China.
(email: xtling@seu.edu.cn).
X. Zhang is with Institut Suprieur dElectronique de Paris, ISEP Paris,
France. (email: xun.zhang@isep.fr).
F. Zhou is with the College of Electronic and Information Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing, 210000, China.
He is also with Key Laboratory of Dynamic Cognitive System of Electromag
netic Spectrum Space, Nanjing University of Aeronautics and Astronautics.
(email: zhoufuhui@ieee.org).
D. W. K. Ng is with the School of Electrical Engineering and Telecom
munications, University of New SouthWales, Sydney, NSW 2052, Australia
(email:w.k.ng@unsw.edu.au).
emitting diodes (LEDs), visible light communication (VLC)
has emerged as a promising green indoor communication
solution enabling simultaneous illumination and wireless data
transmission. Owing to its inherent advantages, such as abun
dant licensefree spectrum, high security, and no interference
to existing RFbased systems, VLC systems are a compelling
supplementary to RF systems for realizing highspeed wireless
data transmissions.
Despite the promising gains brought VLC technologies,
serious intersymbol interference (ISI) creates a system per
formance bottleneck due to the existence of multipath in high
data rate VLC systems. In practice, orthogonal frequency
division multiplexing (OFDM) [8] is an effective solution to
trackle ISI in RFbased systems. However, as VLC systems
exploit intensity modulation and direct detection (IM/DD)
schemes for communication, information of VLC is repre
sented by light intensity and thus transmitted signals should
be realvalued and nonnegative. Thus, conventional RFbased
OFDM techniques cannot directly apply to VLC systems. To
mitigate the ISI issue, asymmetrically clipped optical OFDM
(ACOOFDM) [9], [10], direct current biased optical OFDM
(DCOOFDM) [11]–[13], and Unipolar OFDM (UOFDM)
[14] have been proposed for VLC systems. To generate
nonnegative transmitted signals, ACOOFDM eliminates the
negative component of signals, while DCOOFDM adds a
direct current (DC) bias and then clips the negative parts of
signals by setting them to zero [15]. Moreover, ACOOFDM
transmits data symbols only via odd indexed subcarriers,
whereas DCOOFDM transmits data symbols exploiting all
the subcarriers. Compared with DCOOFDM, ACOOFDM
can generally achieve a lower biterrorrate (BER) for identical
QAM modulation orders such as in [16], [17]. However, due to
only half of the subcarriers to carry information, the spectral
efﬁciency (SE) of ACOOFDM is generally lower than that
of DCOOFDM from moderate to high signaltonoise ratio
(SNR) [18], and the SE of UOFDM is similar to that of ACO
OFDM [19].
Recently, various power allocation schemes have been pro
posed to improve the SE of ACOOFDM VLC systems. For
example, under average optical power constraint, the conven
tional waterﬁlling power allocation scheme can improve the
information rate of ACOOFDM substantially [10]. In [20],
the achievable rates of ACOOFDM and ﬁltered ACOOFDM
(FACOOFDM) were analyzed with both optical power and
bandwidth constraints. Besides, by taking into account both
average optical power and dynamic optical power constraints,
both the error vector magnitude (EVM) and achievable data
arXiv:2108.13906v1 [cs.IT] 31 Aug 2021
2
rates of the DCOOFDM and ACOOFDM systems were
analyzed in [21] showing that ACOOFDM can achieve the
lower bound of the EVM. In [22], two upper bounds of
channel capacity for the intensity modulated direct detection
(IM/DD) optical communication systems were derived based
on an exponential input distribution and clipped Gaussian input
distribution respectively, and a closedformed channel capacity
of ACOOFDM. However, all of them only considered the
average optical power constraint. Then, a more detailed de
scription of the problem was given in [10] with an electrical
power limit or both optical power limit and input power
constraint. For the latter scenario, [10] proved that if the real
and imaginary components of each odd frequency IFFT input
are independent random variables with circular symmetry,
they must follow a zero mean Gaussian distribution and the
outputs of the IFFT are strict sense stationary. Then, a closed
form information rate was derived to satisfy above conditions.
However, the more general question of the information rate
of an ACOOFDM system only with limited average optical
power remains an intractable problem. Also, subject to a given
a target BER requirement, adaptive modulation schemes were
investigated in [23] to maximize the SE of DCOOFDM,
ACOOFDM, and single carrier frequencydomain equaliza
tion (SCFDE) systems, respectively. In spite of the fruitful
research in the literature, the aforementioned studies were
based on the assumption that the input signal follows Gaussian
distribution. Although Gaussian distribution inputs can achieve
the channel capacity under average electrical power con
straints, the optimal distribution with optical power constraints
is still unknown. In fact, practical input signals are often based
on discrete constellation schemes, such as pulse amplitude
modulation (PAM), quadrature amplitude modulation (QAM),
and phase shift keying (PSK). Applying power allocation
schemes based on Gaussian distribution inputs to signals with
practical ﬁnitealphabet inputs may cause serious performance
loss [24]. So far, the SE of ACOOFDM with ﬁnitealphabet
inputs has been rarely considered in the literature. Therefore,
it is necessary to design an optimal power allocation scheme
to unlock the potential of ACOOFDM systems.
In addition to improving the SE, achieving high energy
efﬁciency (EE) is also critical for ACOOFDM VLC systems,
which is usually deﬁned as a ratio of the achievable rate
to the total power consumption [25]. In fact, the improved
SE does not come for free. In particular, the improvement
is always achieved at the expense of increased energy cost.
Unfortunately, most of the aforementioned VLC research [26]
aimed at improving SE, but omitted the EE of ACOOFDM
systems. Recently, there are some works started focusing on
the EE issue in VLC systems. By the joint design of the cell
structure and the system level power allocation, an amorphous
structure of ACOOFDM VLC systems can achieve a higher
EE than that of the conventional cell structures [27]. To ensure
the quality of service (QoS) with affordable energy, the EE of
the conventional and hybrid OFDMbased VLC modulation
schemes was investigated in [28]. However, existing studies
of ACOOFDM’s EE [27], [28] are based on Gaussian distri
bution inputs. As previously mentioned, Gaussian distribution
inputs are difﬁcult to generate in practice. Indeed, practical
inputs are always ﬁnitealphabet inputs, which has been less
commonly studied in literature. Thus, there is an emerge need
for the study of EE of ACOOFDM VLC systems with ﬁnite
alphabet inputs.
In this study, we propose the optimal power allocation
scheme to maximize the SE and the EE of ACOOFDM VLC
systems with Gaussian distribution inputs and ﬁnitealphabet
inputs, respectively. The main contributions of this paper are
summarized as follows:
•We systematically analyze the signal processing module
of a typical ACOOFDM VLC system. Based on the
frequency domain analysis, we ﬁrst derive achievable
rates of the considered system admitting ﬁnitealphabet
inputs from the perspective of practical modulation. Addi
tionally, for both cases of the Gaussian distribution inputs
and ﬁnitealphabet inputs, we develop the corresponding
optical power constraints for ACOOFDM VLC systems.
•Under both the total transmit power constraint and the
optical power constraint, two optimal power allocation
schemes are proposed to maximize the SE of the ACO
OFDM system with Gaussian distribution inputs and
ﬁnitealphabet inputs, respectively. Speciﬁcally, for Gaus
sian distribution inputs, we show that the waterﬁlling
based power allocation scheme can maximize the SE.
Similarly, for ﬁnitealphabet inputs, we derive an opti
mal power allocation scheme to achieve the maximum
SE by exploiting the Lagrangian method, KarushKuhn
Tucker (KKT) conditions, and the relationship between
the mutual information and the minimum meansquared
error (MMSE) [29].
•The optimal power allocation scheme for ﬁnitealphabet
inputs lacks closedform expressions and involves com
plicated computations of MMSE. To reduce the compu
tational complexity, we ﬁrst derive a closedform lower
bound for the achievable rate. Then, based on the pro
posed lower bound, we develop a suboptimal power
allocation scheme to maximize the SE under both the
total transmit power constraint and the average optical
power constraint.
•We propose an explicit EE expression with Gaussian
distribution inputs and ﬁnitealphabet inputs, respectively.
Moreover, under the constraint of maximum transmis
sion power and the minimum data rate requirement, the
nonconvex problem of maximizing EE is investigated.
This problem is solved by applying the Dinkelbachtype
algorithm and the interior point algorithm. Finally, the
relationship between the SE and the EE of the ACO
OFDM system is unveiled.
The rest of this paper is organized as follows. The system
model of ACOOFDM is presented in Section II. The SE of
ACOOFDM system is shown in Section III. The EE of ACO
OFDM system is studied in Section IV and the simulation
results are presented in Section V. Finally, the conclusions are
drawn in Section VI.
Notations: Boldfaced lowercase and uppercase letters rep
resent vectors and matrices, respectively. Expected value of
a random variable zis denoted by E{z}.(·)∗represents
3
conjugate transformation. [x]+denotes max {x, 0}.Re (·)
denotes the real part of its argument. ∂f (·)
∂x represents the
partial derivative operation on xof function f(·). Given a
variable y,E{zy}represents the conditional mean of zfor
given y.min {x, y}represents the minimum value between x
and y.I(X;Y)represents the mutual information of Xand Y.
A complexvalued circularly symmetric Gaussian distribution
with mean µand variance σ2is denoted by CN µ, σ2. A
realvalued Gaussian distribution with mean µand variance
σ2is denoted by Nµ, σ2.
II. SY ST EM MO DE L
Power
Alloca
tion
Herm
itian
Symm
etry
Zero
Clipp
ing
M
QAM D/A LED
IFFT
and
P/S
Optical
Channel
PDA/D
S/P
FFT
MQAM
Demo
dulator
...
...
0
X
2 1N
X
k
x
ˆk
x
...
...
...
P/S
1
Y
2 1i
Y
S/P
0 0
p X
2 1 2 1
N N
p X
 
Fig. 1: A block diagram of an ACOOFDM VLC system.
Consider an ACOOFDM VLC system with total 2Nsub
carriers, as shown in Fig. 1 [17], where the signals are only
transmitted via the odd indexed subcarriers. The information
bit stream is ﬁrst converted to parallel substreams by a
serialtoparallel (S/P) converter. Then, they are modulated by
an MQAM scheme. After applying the inverse fast Fourier
transform (IFFT) and zero clipping on the modulated symbols,
signal ˆxkis nonnegative. Then, the signal passes through
an digitaltoanalog converter (D/A), where the digital signal
ˆxkis converted to an analog signal. After that, the analog
signal is emitted through visible light by an LED. In particular,
by exploiting the IM/DD scheme, the transmitted information
of the VLC system is represented by the signal intensity,
which is real and nonnegative. At the receiver, the received
visible light is transformed into an analog electrical signal
by a photo detector (PD) and then converted to a digital
signal by an analogtodigital converter (A/D). After applying
the fast Fourier transform (FFT) on the digitalized signal,
demodulation is performed at a demodulator to convert the
received MQAM symbols to bit streams.
A. Signal Model
In this section, we discuss the mathematical details of the
considered system. At the transmitter, the raw data bit stream is
going through the modulation. Let Xkdenotes the modulated
signal on the kth subcarrier, and pkdenotes the allocated
power on the kth subcarrier, k= 0, ..., 2N−1. Note that
to ensure real output values at the IFFT, the input of the IFFT
module should satisfy Hermitian symmetry, i.e.,
(X2i=X2(N−i)−2= 0, i = 0, . . . , N/2−1,
X2i−1=X∗
2(N−i)+1, i = 1, . . . , N/2,(1)
where X2i−1is the normalized unitpower input, i.e.,
EnX2i−12o= 1. According to (1), the power allocation
of subcarriers should satisfy
(p2i=p2(N−i)−2= 0, i = 0, . . . , N/2−1,
p2i−1=p∗
2(N−i)+1 ≥0, i = 1, . . . , N/2,(2)
After the IFFT operation, the time domain signal xkis given
as
xk=IFFT n{√p`X`}2N−1
`=0 o(3a)
=1
√2N
2N−1
X
`=0
√p`X`exp jπk`
N(3b)
=r2
N
N/2
X
i=1
√p2i−1Re X2i−1exp jπk (2i−1)
N,
k= 0, ..., 2N−1.(3c)
According to (3c), the obtained time domain signal satisﬁes
antisymmetry as follows
x`=−x`+N, ` = 0, ..., N −1.(4)
Since the transmitted signal should be nonnegative, the nega
tive signals are removed by the clipping process such that
ˆxk=xk
0
xk≥0;
otherwise.(5)
Due to the requirement of the practical system circuit
design, the total electrical transmit power should be lim
ited [10]. Let Pdenote the total electrical transmit power,
i.e., P2N−1
k=0 Eˆx2
k≤P. Combining the clipping process
and Parseval’s theorem [11], [30], we have P2N−1
k=0 pk=
P2N−1
k=0 Ex2
k= 2 P2N−1
k=0 Eˆx2
k. Based on (2), the
electrical transmit power constraint can be rewritten as
PN/2
i=1 p2i−1≤P. For the consideration of human eye safety,
the optical power of VLC signals is generally restricted [31]–
[35]. Let Porepresent the maximum optical power threshold.
The average optical power should satisfy1
E{ˆxk} ≤ Po.(6)
According to the deﬁnition of variance, it is easy to verify that
the sum of average electrical power Pis larger than P2
o.
B. Channel Model
Generally, the VLC channel is characterized by a lineof
sight (LOS) link along with multiple reﬂections of the light
from surrounding objects, such as walls, ﬂoor, and windows.
In this study, we adopt the commonly used frequencydomain
VLC channel model [36], which is not restricted to a ﬁnite
order of reﬂections.
Let Hkdenotes the channel gain of the kth subcarrier, which
includes both the LOS link and the diffuse links as follows
Hk=HL,k +HD,k,(7)
1Due to different distributions, the signals with the same optical power may
have different electrical powers.
4
where HL,k is the gain of the LOS link and HD,i is the gain
of the diffuse links, k= 0, ..., 2N−1.
The LOS link HL,k is expressed as
HL,k =gLe−j2πfkτ,(8)
where gLis the generalized Lambertian radiator [37], fk
denotes the frequency of the ith subcarrier, τis the signal
propagation delay between the transmitter and receiver with
τ=d/c,dis the distance between the transmitter and receiver,
and cis the speed of light, k= 0, ..., 2N−1. The generalized
Lambertian radiator gLcan be expressed as
gL=(m+1)Arcos(ϕ)
2πd2cosm(θ)T(ϕ)G(ϕ)
0
0≤ϕ≤Ψ,
otherwise,
(9)
where mis the order of Lambertian emission, i.e., m=
−ln 2/ln cos Φ1/2,Φ1/2is the half power angle; Aris
the effective detector area of the PD receiver; ϕand θare,
respectively, the incidence and irradiance angle from the LED
to the PD; T(ϕ)and G(ϕ)are the optical ﬁlter gain and the
concentrator gain of the receiver, respectively; Ψrepresents
the ﬁeldofview (FOV) of the receiver.
On the other hand, the gain of the diffuse links HD,k is
given by [38]
HD,k =ηD
1 + j2πτ fk
,(10)
where ηDis the power efﬁciency of the diffuse signal and τis
the exponential decay time. The timedomian diffuse channel
gain hDis given as hD(t) = ηD
τe−t/τ ε(t), where ε(t)is the
unit step function.
Thus, the time domain channel response can be given as
h(t) = gLδ(t) + hD(t−∆T), where δ(t)is the Dirac
function, and ∆Tdescribes the delay between the LOS
signal and the diffuse signal. Besides, the relationship between
the time domain channel response and the corresponding
subcarriers channel gains can be described as H(fk) =
R∞
−∞ h(t)e−j2πfktdt.
C. Performance Metrics
In practice, the signals are transmitted from an LED through
an optical channel. At the receiver, it performs FFT to obtain
the frequencydomain modulated information. However, due
to the zero clipping, the amplitude of the frequency domain
signal at the receiver is half of that at the transmitter [39]. Let
Y2i−1denotes the signals received in the frequencydomain at
the (2i−1)th subcarrier, which is given by
Y2i−1=1
2H2i−1√p2i−1X2i−1+Z2i−1,(11)
where the coefﬁcient 1
2exists since only half subcarriers
are adopted to transmit information, Z2i−1is the additive
white Gaussian noise (AWGN) with zeromean, i.e., Z2i−1∼
CN 0, W σ2,i= 1, ..., N/2, and σ2represents the noise
power spectral density, Wrepresents the bandwidth of each
subcarrier.
Let R2i−1{p2i−1}N/2
i=1 and RACO denote the rate of the
(2i−1)th subcarrier and the total rate of the ACOOFDM
system, respectively, which are given by
R2i−1{p2i−1}N/2
i=1 =I(X2i−1;Y2i−1),(12a)
RACO =
N/2
X
i=1
R2i−1{p2i−1}N/2
i=1 ,(12b)
respectively. Then, the SE of the ACOOFDM VLC system
is deﬁned as the ratio of achievable data rate to the total
bandwidth, which can be expressed as
SE {p2i−1}N/2
i=1 =
N/2
P
i=1
R2i−1{p2i−1}N/2
i=1
2NW ,(13)
where Wdenotes the bandwidth of each subcarrier. At the
same time, the EE of the ACOOFDM VLC system is deﬁned
as the ratio of the capacity to the total power consumption,
which can be expressed as
EE {p2i−1}N/2
i=1 =
N/2
P
i=1
R2i−1{p2i−1}N/2
i=1
2
N/2
P
i=1
p2i−1+Pc
,(14)
where 2PN/2
i=1 p2i−1represents the total electrical power con
sumption of all the subcarriers and Pcdenotes the total circuit
power consumption of the whole system.
III. SPE CT RA L EFFI CI ENCY OF ACOOFDM
In this section, we aim to maximize the SE of the ACO
OFDM system under the electrical transmit power constraint
and taking into account a practical average optical power
constraint. The considered problem can be mathematically
formulated as follows:
maximize
{p2i−1}N/2
i=1
N/2
P
i=1
R2i−1{p2i−1}N/2
i=1
2NW (15a)
s.t.E{ˆxk} ≤ Po,(15b)
N/2
X
i=1
p2i−1≤P, (15c)
p2i−1≥0, i = 1, ..., N/2.(15d)
In the following, we will investigate the SE maximization
problem (15) for the considered ACOOFDM system with
Gaussian distribution inputs and ﬁnitealphabet inputs, respec
tively.
A. Gaussian Distribution Inputs
Assume that the input X2i−1follows independent complex
Gaussian distribution, i.e., X2i−1∼ CN (0,1). According to
the IFFT operation (3c), the time domain signal xkalso fol
lows Gaussian distribution, i.e., xk∼ N 0,2
NPN/2
i=1 p2i−1.
5
Furthermore, based on the relationship in (5), the average
optical power is given by [40], [41]
E{ˆxk}=1
2E{xk} =1
2Z∞
0
xk
1
√2πσs
e−x2
k
2σ2
sdxk
=v
u
u
t1
πN
N/2
X
i=1
p2i−1,(16)
where σ2
s=2
NPN/2
i=1 p2i−1. Additionally, by substituting
(16) into (15b), the average optical power constraint can be
reformulated as
N/2
X
i=1
p2i−1≤N πP 2
o.(17)
According to the Shannon theorem [42], the achievable rate
of Gaussian distribution inputs RG(p2i−1)is given by
RG(p2i−1) = Wlog2 1 + p2i−1H2i−12
4σ2W!.(18)
Then, the SE of the Gaussian distribution inputs
SEG{p2i−1}N/2
i=1 can be expressed as
SEG{p2i−1}N/2
i=1 =
N/2
P
i=1
log21 + p2i−1H2i−12
4σ2W
2N.(19)
Thus, the SE maximization problem with the Gaussian
distribution inputs can be rewritten as
maximize
{p2i−1}N/2
i=1
SEG{p2i−1}N/2
i=1 (20a)
s.t.
N/2
X
i=1
p2i−1≤min P, N πP 2
o,(20b)
p2i−1≥0, i = 1, ..., N/2.(20c)
Problem (20) is a convex optimization problem and satisﬁes
the Slater’s constraint qualiﬁcation [43]. The problem in
(20) can be solved by applying classical convex optimization
approaches. To this end, we ﬁrst need the Lagrangian function
of (20), which is given as
LG=1
2N
N/2
X
i=1
log2 1 + p2i−1H2i−12
4σ2W!
−µ
N/2
X
i=1
p2i−1−min P, N πP 2
o
,(21)
where µ≥0is the Lagrange multiplier associated with
constraint (20b). By setting the differential function to 0, i.e.,
∂LG
∂p2i−1= 0, the optimal p2i−1is given by
p2i−1="1
2N µ ln 2 −4σ2W
H2i−12#+
.(22)
In fact, (22) is known as the classical waterﬁlling solution
and the optimal µcan be found by the conventional gradient
method or the epsilon method [43], [44].
B. Finitealphabet Inputs
In practice, typical inputs are always based on discrete sig
naling constellations, such as MPSK or MQAM, rather than
the ideal Gaussian signals. In this section, we assume that the
inputs are drawn from discrete constellations set {X2i−1,k}M
k=1
with cardinality M, where X2i−1,k is a constellation point of
the (2i−1)th subcarrier. The achievable rate RF(p2i−1)is
given by [24]
RF(p2i−1) = I2i−1(X2i−1;Y2i−1)(23)
=Wlog2M−1
ln 2
−
M
X
n=1
W
MEZ(log2
M
X
k=1
exp (−dn,k)),(24)
where I2i−1(X2i−1;Y2i−1)is the achievable
mutual information over the (2i−1)th channel,
dn,k =1
σ2W1
2H2i−1√p2i−1(X2i−1,n −X2i−1,k) + Z2i−12
is a measure of the difference between input constellation
points X2i−1,n and X2i−1,k,EZ{·} is the expectation of the
noise Z2i−1. Note that RF(p2i−1)is a concave function with
respect to the power allocation p2i−1[24], [45].
According to (5), the average optical power of transmitted
signals is given as [40], [41]
E{ˆxk}=1
2E{xk}
≤1
2√2N
2N−1
X
i=0
E√piXiexp jπki
N
=1
2√2N
2N−1
X
i=0
√piE{Xi},(25)
where the inequality holds due to Piai ≤ Piai, the
value of E{Xi} depends on the speciﬁc modulation schemes,
i.e., (24) and (25) can be apply on another distribution
known discrete signaling constellations, such as OOK, DPSK,
higher order PSK and QAM, and nonuniform discrete inputs.
Furthermore, substituting (25) into (6), the average optical
power constraint is given as
1
√2N
N/2
X
i=1
√p2i−1E{X2i−1} ≤ Po.(26)
Based on the inequality (Pn
i=1 ai)2≤nPn
i=1 a2
i[46],
where ai≥0, the average optical power constraint (26) can
be restricted as
N/2
X
i=1
p2i−1≤4P2
o
E2{X2i−1}.(27)
In other words, (27) is known as a safe approximation of
(26) because the left side of (26) is replaced by its upper
bound. After adopting the optimization algorithm, the optical
power consumption would not exceed Po. Then, the SE of
6
ﬁnitealphabet inputs SEF{p2i−1}N/2
i=1 can be expressed as
SEF{p2i−1}N/2
i=1 =
N/2
P
i=1
RF(p2i−1)
2NW .(28)
Thus, the optimal power allocation problem (15) can be re
formulated as the constellationconstrained mutual information
maximization problem which can be expressed as follows
maximize
{p2i−1}N/2
i=1
SEF{p2i−1}N/2
i=1 (29a)
s.t.
N/2
X
i=1
p2i−1≤ε, (29b)
p2i−1≥0, i = 1, ..., N/2,(29c)
where ε∆
= min nP, 4P2
o
E2{X2i−1}o.
The lack of a closedform expression for the objective func
tion (29a) complicates its solution development. To address
this difﬁculty, we aim to derive the optimal power allocation
scheme for problem (15) by exploiting the relationship be
tween the mutual information and MMSE [47].
To this end, we ﬁrst derive the equivalent Lagrangian
function of problem (29) which is given by
LF=−
N/2
X
i=1
RF(p2i−1) + λ
N/2
X
i=1
p2i−1−ε
,(30)
where λ≥0is the Lagrange multiplier corresponding to
constraint (29b).
Furthermore, the KKT conditions of problem (29) can be
expressed as
−∂RF(p2i−1)
∂p2i−1
+λ= 0,(31a)
λ
N/2
X
i=1
p2i−1−ε
= 0,(31b)
N/2
X
i=1
p2i−1−ε≤0,(31c)
λ≥0, p2i−1≥0, i = 1, ..., N/2.(31d)
According to [47], the relationship between the mutual
information and the MMSE of the (2i−1)th subcarrier is
given by
∂
∂SNRI2i−1(X2i−1;Y2i−1) = MMSE2i−1(SNR) ,(32)
where MMSE2i−1(SNR) = EX2i−1−ˆ
X2i−1
2is the
MMSE of X2i−1, and ˆ
X2i−1is conditional expectation of
X2i−1, i.e.,
ˆ
X2i−1=EX2i−1Y2i−1=1
2H2i−1√p2i−1X2i−1+Z2i−1.
(33)
Combining (23) and (32), the differential function of
RF(p2i−1)can be written as
∂RF(p2i−1)
∂p2i−1
=H2i−12
4σ2WMMSE2i−1 H2i−12
4σ2Wp2i−1!.
(34)
By substituting (34) into (31a), we have
H2i−12
4σ2WMMSE2i−1 H2i−12
4σ2Wp2i−1!=λ. (35)
Then, solving (35) for the power allocation p2i−1yields
p2i−1=4σ2W
H2i−12MMSE−1
2i−1 4σ2W
H2i−12λ!,(36)
where MMSE−1
2i−1(·)is the inverse function of MMSE2i−1(·)
with the domain [0,1] and MMSE−1
2i−1(1) = 0 [29].
Therefore, for the considered ACOOFDM system, the
optimal power allocation scheme of (29) is given by
p∗
2i−1=(4σ2W
H2i−12MMSE−1
2i−14σ2W
H2i−12λ,0< λ ≤H2i−12
4σ2W;
0,otherwise.
(37)
The dual variable λin (37) is the solution of the following
equation
N/2
X
i=1
4σ2W
H2i−12MMSE−1
2i−1 4σ2W
H2i−12λ!=ε, (38)
which can be obtained by a simple bisection method as listed
in Algorithm 1.
Algorithm 1 Bisection Method
Input: Given λ∈h0,ˆ
λi,δ > 0, and initialize λmin =
0, λmax =ˆ
λ, where δdenotes termination parameter, and
ˆ
λrepresents an upper bound of λ;
1: while λmax −λmin ≥δdo
2: Set λ= (1/2) (λmin +λmax);
3: If λ < H2i−12/4σ2W, substitute λto obtain p∗
2i−1=
4σ2W
H2i−12MMSE−1
2i−14σ2W
H2i−12λ; otherwise p∗
2i−1= 0;
4: If
N/2
P
i=1
p∗
2i−1≤ε, set λmax ←λ; otherwise λmin ←λ;
5: end while
Output: p∗
2i−1;
In the following, we explain the differences between the
waterﬁlling power allocation (22) with Gaussian distribution
inputs and the power allocation (37) with ﬁnitealphabet in
puts. To facilitate the presentation, let us introduce a function,
G2i−1(λ), as follows
G2i−1(λ) =
H2i−12
4σ2W λ −MMSE−1
2i−14σ2W λ
H2i−12,
0< λ ≤H2i−12
4σ2W;
1,otherwise.
(39)
7
Then, the power allocation p2i−1can be represented as
p2i−1=1
λ−4σ2W
H2i−12G2i−1(λ).(40)
Note that if G2i−1(λ) = 1, the power allocation (40)
is equivalent to the waterﬁlling power allocation (22) with
the Gaussian distribution inputs; otherwise, the power allo
cation (40) is the power allocation (37) with ﬁnitealphabet
inputs2. Furthermore, based on the function G2i−1(λ), the
power allocation (40) can be interpreted as the mercurywater
ﬁlling scheme [29]. More speciﬁcally, the power allocation
(40) is illustrated in Fig. 2. For the (2i−1)th subcarrier
with noise level 4σ2W
H2i−12, we ﬁrst ﬁll mercury to the height
4σ2W
H2i−12G2i−1(λ), then, pouring water (power) to the height
1
λ. Note that both the noise level and the mercury in the
mercurywaterﬁlling scheme form the bottom level of the
waterﬁlling scheme.
1
l
2 1i
p
Water
Mercury
Noise level
( )
2
2 1
2
2 1
4
i
i
WG
H
sl


2
2
2 1
4
i
W
H
s

Fig. 2: Mercury waterﬁlling scheme.
C. Lower Bound of Mutual Information
For ﬁnitealphabet inputs, the optimal power allocation
scheme (37) involves the calculation of integrals of the MMSE
ranged from −∞ to +∞, which can only be obtained by
Monte Carlo method and numerical integral methods at the
expense of high computational complexity [24], [29]. To strike
a balance between complexity and performance, we further
develop a low complexity power allocation scheme.
As mentioned above, the expression of mutual information
of ACOOFDM is given in (24). The upper bound of the
2The ACOOFDM with ﬁnitealphabet inputs converges weakly to that with
Gaussian distribution inputs as the number of subcarriers Nis sufﬁciently
large [48].
expectation term in (24) is given as [49]
EZ(log2
M
X
k=1
exp (−dnk))≤log2
M
X
k=1
EZ2i−1{exp (−dnk)}
(41a)
= log2
M
X
k=1 ZZ2i−1
exp (−dnk)
πσ2Wexp −Z2i−12
σ2W!dZ2i−1
(41b)
= log2
M
X
k=1
1
2exp −Re2{C2i−1}+ Im2{C2i−1}
2σ2W(41c)
=−1 + log2
M
X
k=1
exp −p2i−1H2i−12X2i−1,n −X2i−1,k2
8σ2W!,
(41d)
where C2i−1
∆
=1
2H2i−1√p2i−1(X2i−1,n −X2i−1,k), in
equality (41a) is based on Jensen’s inequality and (41b) is
the expectation over Z.Besides, the above derivation can
also be used for another distributionknown discrete signaling
constellations, such as OOK, DPSK, and nonuniform discrete
inputs. Note that the upper bound in (41d) is a detertimintic
value without involving integration which can be adopted in
the following for the development of computationally efﬁcient
resource allocation algorithm. Let RL(p2i−1)represents the
lower bound of mutual information in the (2i−1)th subcar
rier. Thus, the lower bound of achievable rate of the ACO
OFDM VLC system with ﬁnitealphabet inputs is given by
RL(p2i−1) = Wlog2M+ 1 −1
ln 2 −
M
X
n=1
W
Mlog2
M
X
k=1
exp −p2i−1H2i−12X2i−1,n −X2i−1,k2
8σ2W!.
(42)
Then, the SE of the lower bound of mutual information
SEL{p2i−1}N/2
i=1 can be expressed as
SEL{p2i−1}N/2
i=1 =
N/2
P
i=1
RL(p2i−1)
2NW .(43)
By substituting the derived lower bound (42) into the
objective of problem of (43), the SE maximization problem
(15) can be reformulated as follows
maximize
{p2i−1}N/2
i=1
SEL{p2i−1}N/2
i=1 (44a)
s.t.
N/2
X
i=1
p2i−1≤ε, (44b)
p2i−1≥0, i = 1, ..., N/2,(44c)
where ε∆
= min nP, 4P2
o
E2{X2i−1}o. This optimization problem
(44) is a standard convex problem, which can be efﬁciently
solved by the interiorpoint algorithm [43], [50]. It can also
be solved by the mercurywaterﬁlling scheme similar to the
one adopted in Section IIIB.
8
IV. ENERGY EFFIC IE NC Y OF ACOOFDM
In this section, we investigate the design of optimal power
allocation scheme to maximize the EE of the ACOOFDM
VLC system which is subject to the QoS requirement,and both
the electrical and optical power constraints. Mathematically,
we can formulate the EE maximization problem of the ACO
OFDM VLC system as
maximize
{p2i−1}N/2
i=1
N/2
P
i=1
R2i−1{p2i−1}N/2
i=1
2
N/2
P
i=1
p2i−1+Pc
(45a)
s.t.E{ˆxk} ≤ Po,(45b)
N/2
X
i=1
p2i−1≤P, (45c)
N/2
X
i=1
R2i−1{p2i−1}N/2
i=1 ≥r, (45d)
p2i−1≥0, i = 1, ..., N/2,(45e)
where ris the minimum achievable rate requirement.
Problem (45) is a general formulation of the EE in ACO
OFDM VLC systems. In the following, we will present the
optimal EE scheme of problem (45) with Gaussian distribution
inputs and ﬁnitealphabet inputs respectively. Note that both
objective function (45a) and the constraint (45d) of the EE
maximization problem (45) are different from that of the SE
maximization problem (15).
A. Gaussian Distribution Inputs
With Gaussian distribution inputs and the achievable
rate expression (18), the EE of Gaussian distribution
EEG{p2i−1}N/2
i=1 can be expressed as
EEG{p2i−1}N/2
i=1 =
N/2
P
i=1
Wlog21 + p2i−1H2i−12
4σ2W
2
N/2
P
i=1
p2i−1+Pc
.
(46)
The EE maximization problem with Gaussian distribution
inputs of ACOOFDM systems can be formulated as
maximize
{p2i−1}N/2
i=1
EEG{p2i−1}N/2
i=1 (47a)
s.t.
N/2
X
i=1
p2i−1≤min P, N πP 2
o,(47b)
N/2
X
i=1
Wlog2 1 + p2i−1H2i−12
4σ2W!≥r, (47c)
p2i−1≥0, i = 1, ..., N/2.(47d)
For the objective function (47a), the numerator is a dif
ferentiable concave function of the variable p2i−1, while the
denominator is an afﬁne function of p2i−1. Thus, the objective
function (47a) is a quasiconcave function of p2i−1. With
a convex constraint set, problem (47) is a typical fractional
problem [51], which is generally nonconvex. To circumvent
the nonconvexity, Dinkelbachtype iterative algorithm [52]–
[54] can be adopted to trackle the problem by converting
the problem (47) into a sequence of convex subproblems. In
particular, solving these convex subproblems iteratively can
eventually obtain the globally optimal solution of problem
(47). Speciﬁcally, let Υdenote the feasible set deﬁned con
straints of problem (47) as follows
Υ = {p2i−1(47b),(47c),(47d), i = 1, ..., N/2}.(48)
Moreover, we introduce a new function f{p2i−1}N/2
i=1 as
follows
f{p2i−1}N/2
i=1 ,
N/2
X
i=1
Wlog2 1 + p2i−1H2i−12
4σ2W!−q
2
N/2
X
i=1
p2i−1+Pc
,
(49)
where qis a given nonnegative parameter to be found
iteratively. Then, by calculating the roots of the equation
f{p2i−1}N/2
i=1 = 0 in the set Υ, the optimal solution of
problem (47) can be obtained.
For a given qin each iteration, the convex subproblem over
p2i−1can be expressed as
maximize
{p2i−1}N/2
i=1
f{p2i−1}N/2
i=1 (50a)
s.t.∀p2i−1∈Υ, i = 1, ..., N/2.(50b)
Since the transformed problem is convex and satisﬁes
the Slater’s constraint qualiﬁcation, we apply the conven
tional optimization techniques by taking the partial deriva
tive of function f{p2i−1}N/2
i=1 and setting it to zero, i.e.
∂f {p2i−1}N/2
i=1
∂p2i−1= 0,which yeilds
ep2i−1="W
2qln 2 −4σ2W
H2i−12#+
.(51)
Then, by projecting ep2i−1into the feasible region Υ, we
obtain the optimal power p∗
2i−1of problem (50) as follows
p∗
2i−1N/2
i=1 = ProjΥ{ep2i−1}N/2
i=1
= arg min
{p2i−1}N/2
i=1
N/2
X
i=1 kep2i−1−p2i−1k2(52)
where ProjΥ{ep2i−1}N/2
i=1 denotes the projection of
{ep2i−1}N/2
i=1 into the subspace Υ.
Note that if {ep2i−1}N/2
i=1 ∈Υ, the power allocation (52)
for the EE maximization is a generalization of the power
allocation (22) for the SE maximization. In particular, for
a small electrical transmit power budget, both SE and EE
maximization show the same waterﬁlling solution. However,
when the budget is sufﬁciently large, once the maximum EE
is achieved, the optimal EE power algorithm would clip the
9
transmit power level, as can be observed in (52).
Finally, the EE maximization problem of Gaussian distribu
tion inputs can be solved by the Dinkelbachtype algorithm.
Under a ﬁnite number of iterations, the Dinkelbachtype
algorithm is guaranteed to converge to the optimal solution
of problem (47), e.g. [52]–[54]. Algorithm 2 shows the detail
of implementation.
Algorithm 2 Dinkelbachtype Algorithm
Input: Given δ→0, n = 0, p∗
2i−1>0, q(n)= 0;
1: while q(n)−q(n+1) ≤δdo
2: Compute the optimal solution p∗
2i−1N/2
i=1 ;
3: Calculating the value of function fp∗
2i−1N/2
i=1 ;
4: q(n+1) = EE p∗
2i−1N/2
i=1 ;
5: n=n+ 1;
6: end while
Output: EE p∗
2i−1N/2
i=1 ;
B. Finitealphabet Inputs
For the ﬁnitealphabet inputs, the achievable rate expres
sion is given by (23), thus, the EE of ﬁnitealphabet inputs
EEF{p2i−1}N/2
i=1 is given by (53).
Furthermore, the average optical power constraint (45b)
can be restricted to constraint (27). Thus, the optimal EE
maximization problem with ﬁnitealphabet inputs under the
electrical power constraint, the average optical power con
straint, and the minimum rate constraint can be expressed as
maximize
{p2i−1}N/2
i=1
EEF{p2i−1}N/2
i=1 (54a)
s.t.
N/2
X
i=1
p2i−1≤min P, 4P2
o
E2{X2i−1},(54b)
N/2
X
i=1
RF(p2i−1)≥r, (54c)
p2i−1≥0, i = 1, ..., N/2.(54d)
Note that there is no closedform expression for the achiev
able rate in (54a), although RF({p2i−1})is strictly concave
over its input power. On the other hand, constraints (54b)
(54d) form a convex feasible solution set. Thus, problem (54)
is a concavelinear fractional problem that can be solved by
Dinkelbachtype algorithms. The details are omitted as it is
similar to the case adopting Gaussian distribution inputs as
discussed in Section IVA.
C. Lower Bound of Mutual Information
Note that in the objective function (54a), complicated inte
grals need to be solved with high computational complexity.
For the same reason as in Section IIIC, we exploit the lower
bound of achievable rate (42) to reduce the complexity, and the
corresponding EE function of the ACOOFDM VLC system
is given by (55).
Furthermore, the EE problem (45) can be reformulated as
maximize
{p2i−1}N/2
i=1
EEL{p2i−1}N/2
i=1 (56a)
s.t.
N/2
X
i=1
p2i−1≤min P, 4P2
o
E2{X2i−1},(56b)
N/2
X
i=1
RL(p2i−1)≥r, (56c)
p2i−1≥0, i = 1, ..., N/2.(56d)
As the lower bound of the achievable rate (42) is concave
and differentiable, problem (56) is also a concavelinear frac
tional problem. At the same time, problem (56) can also be
solved by using Dinkelbachtype algorithms.
V. SIMULATION RESULTS A ND DISCUSSION
This section presents numerical results to evaluate the
proposed power allocation schemes for SE maximization and
EE maximization problems of ACOOFDM VLC systems.
Consider an indoor ACOOFDM VLC system installed with
four LEDs, where a corner of a square room denotes the
origin (0,0,0) of a threedimensional Cartesian coordinate
system (X, Y, Z). The location of receiver is (0.5,1,0)m,
the locations of four LEDs are (1.5,1.5,3)m, (1.5,3.5,3)m,
(3.5,1.5,3)m, and (3.5,3.5,3)m, respectively, and the reﬂec
tion point is (1.5,0,1.5). The basic parameters of the VLC
system are listed in Table I. The channel gain is generated
EEF{p2i−1}N/2
i=1 =
NW
2log2M−1
ln 2 −
N/2
P
i=1
M
P
n=1
W
MEZlog2
M
P
k=1
exp −1
2H2i−1√p2i−1(X2i−1,n−X2i−1,k )+Z2i−12
σ2W
2
N/2
P
i=1
p2i−1+Pc
.
(53)
EEL{p2i−1}N/2
i=1 =
NW
2log2M+ 1 −1
ln 2 −
N/2
P
i=1
M
P
n=1
W
Mlog2
M
P
k=1
exp −1
2H2i−1√p2i−1(X2i−1,n−X2i−1,k )2
2σ2W
2
N/2
P
i=1
p2i−1+Pc
.(55)
10
based on the channel model (7), (8), and (10) [36]–[38].
Besides, the RL(p2i−1)has been added a nonnegative real
constant as same as [49] in all simulations without the opti
mality loss of SE and EEmaximization power allocation.
TABLE I: Simulation Parameters of the ACOOFDM VLC System.
Deﬁnition Value
Number of subcarriers, N64
Transmit angle, θ60◦
FOV, Ψ 90◦
Lambertian emission order, m1
Half power angle, Φ1/260◦
PD collection area, Ar1 cm2
Circuit power consumption, Pc0.2 W
Angle of arrival/departure, ϕ45◦
Optical ﬁlter gain of receiver, T(ϕ) 0 dB
Concentrator gain of receiver, G(ϕ) 0 dB
Noise PSD, σ210−18 A2/Hz
Modulation order, M4QAM
Bandwidth of each subcarrier, W1 MHz
A. Simulation Results of SE Maximization Problem
In this subsection, we present the results of the proposed
three power allocation schemes for maximizing the SE for
Gaussian distribution inputs, ﬁnitealphabet inputs, and lower
bound of the mutual information.
Fig. 3 illustrates allocated power piversus channel gain
Hiof subcarrier iof SEG, SEF, and SEL, where P= 20
(W), Po= 0.25 (W). As can be observed from Fig. 3, the
value of the allocated power piof the SEGis proportional to
the corresponding noise level 4σ2W
H2i−12, which is due to the
waterﬁlling solution for the maximization of the SE with
Gaussian distribution inputs. While for the SEFcase, the
allocated power of pinot only depends on the noise level,
but also depends on the mercury level 4σ2W
H2i−12G2i−1(λ),
which is due to the mercurywaterﬁlling method for the
maximization of the SE with ﬁnitealphabet inputs. For the
case of SEL, the allocated power of each subcarrier is based
on PM
n=1 log2PM
k=1 exp −p2i−1H2i−12X2i−1,n−X2i−1,k 2
8σ2W.
Since Nπ ≥4
E2{X2i−1}, we have N πP 2
o≥4P2
o
E2{X2i−1}.
Therefore, when P≥2N πP 2
o, the allocated power for Gaus
sian distribution inputs is more than that for ﬁnitealphabet
inputs. Moreover, due to the negligible small channel gain,
the allocated power from the 43th subcarrier to the 63th
subcarrier of the three cases are zero, which are not shown
in Fig. 3 for brevity.
Fig. 4(a) illustrates SEG, SEF, and SELversus the electrical
power threshold Pwith an optical power constraint Po= 0.25
(W) and Po=∞(without optical power constraint), respec
tively. As shown in Fig. 4(a), for Po=∞case, as Pincreases,
SEGkeeps increasing, while SEFand SELﬁrst increase and
then remain constant. Moreover, when Pis large, SEGis
higher than both SEFand SEL. This is because the Gaussian
distribution can be regarded as a very highorder constellation
modulation input, which is more suitable for high SNRs. While
1 7 15 23 31 39
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
106
Fig. 3: Allocated power piversus channel gain Hiof subcarrier iof SEG,
SEF, and SELwith P= 20 W, Po= 0.25 W.
0 5 10 15 20 25 30 35 40 45
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(a)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b)
Fig. 4: (a) SEG, SEF, and SELversus electrical power threshold P; (b) SEG,
SEF, and SELversus optical power threshold Po.
11
for Po= 0.25 (W) case, as Pincreases, SEG, SEF, and SEL
ﬁrst increase and then remain constant. This is because the
allocated power is limited by the optical power threshold, i.e.,
Po= 0.25 (W). Moreover, it can be observed from Fig. 4(a)
that the SEFwith ﬁnitealphabet inputs reaches the maximum
point faster than the SEGwith Gaussian distribution inputs.
The reason is that the coefﬁcient of Poof Gaussian distribution
inputs 2Nπ in (20b) is larger than that of ﬁnitealphabet inputs
4
E2{X2i−1} in (29b). Besides, the gap between SEFand SELis
small, which means SELcould approximate SEFwell enough.
Fig. 4(b) illustrates SEG, SEF, and SELversus optical power
threshold Powith an electrical power constraint P= 10 (W)
and P=∞(without electrical power constraint), respec
tively.3.
It can be seen in Fig. 4(b), for P=∞case, as Poincreases,
SEGkeep increasing, while SEFand SELﬁrst increase and then
remain constant. Besides, when Pois large, SEGis higher than
both SEFand SEL. The reason is that the Gaussian distribution
is more suitable for high SNRs, which is similar as that in Fig.
4(a). While for P= 10 (W) case, SEG, SEF, and SELﬁrst
increase and remain as a constant. In fact, the allocated power
is restricted by the electrical power constraint, i.e., P= 10
(W). When Pois small, SEGis higher than SEFand SEL.
As Poincreases, SEGﬁrst reaches its maximum point, while
SEFand SELreach to the corresponding saturation points
later. The reason is that the coefﬁcient of Poof Gaussian
distribution inputs 2Nπ in (20b) is larger than that of ﬁnite
alphabet inputs 4
E2{X2i−1} in (29b). Thus, with the same
optical power threshold Po, the Gaussian distribution inputs
can allocate more power than that of ﬁnitealphabet inputs,
which was also veriﬁed in Fig. 3.
B. Simulation Results of EE Maximization Problems
In this subsection, we present the simulation results for
the evaluation of the EE performance of Gaussian distribution
inputs, ﬁnitealphabet inputs case, and lower bound of mutual
information case for ACOOFDM VLC systems.
Fig. 5 illustrates the different allocated power piof EEG,
EEF, and EELversus channel gain Hiof subcarrier irespec
tively, where P= 20 (W), Po= 0.25 (W), and r= 1
(bits/sec/Hz). From Fig. 5, we can see that the allocated power
of subcarrier iof EEGis proportional to its channel gain,
which is due to the power allocation strategy in (51) and
(52). While the allocated power of subcarrier iof both EEF
and EELdepend on both channel gains and MMSE functions.
Compared with SE maximization problems, both the objective
function and rate constraints of the EE maximization problems
are different, which accounts for the different power allocation
in Fig. 5 compared with that in Fig. 3.
Fig. 6(a) depicts EEG, EEF, and EELversus electrical power
threshold Pwith an optical power threshold Po= 0.03 (W)
and Po=∞(without optical power constraint) respectively,
where the rate constraint r= 0.1(bits/sec/Hz).We see from
Fig. 6(a) that for Po=∞case, as Pincreases, EEG, EEF, and
EELﬁrst increase and then remain constant. This is because
3There might be intersections among SEL,SEF, and SEGas similar as
the Fig. 1, Fig. 2, and Fig. 3 in [49].
1 7 15 23 31 39 47 55 63
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
106
Fig. 5: Allocated power piversus channel gain Hiof subcarrier iof EEG,
EEF, and EELwith P= 20 W, Po= 0.25 W.
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
(a)
0 5 10 15 20
3.3
3.35
3.4
3.45
3.5
3.55
3.6
3.65
3.7
(b)
Fig. 6: (a) EEG, EEF, and EELversus electrical power threshold Pwith rate
constraint r= 0.1bits/sec/Hz and two different optical power thresholds
Po= 0.03 W, and Po=∞; (b) EEG, EEF, and EELversus rate threshold r
with power threshold P= 20 W and optical power threshold Po= 1 W.
12
the optimal EE remains a constant when it has reached the
maximum value. Moreover, the value of EEFapproaches to
that of EEG, which are higher than the value of EEL. While
for Po= 0.03 case, as Pincreases, the EEG, EEF, and EEL
ﬁrst increase and then remain constant. The reason is that EEG,
EEF, and EELis limited by the optical power constraint Po=
0.03 (W). Besides, for the large P, the value of EEGare the
highest of the three power allocation schemes, while EELis
the lowest.
Fig. 6(b) depicts EEG, EEF, and EELversus rate threshold
rwith power threshold P= 20 (W) and optical power
threshold Po= 1 (W). As shown in Fig. 6(b), EEGis higher
than EEFand EEL, and the gap between EEGand EEFis
small. Moreover, the EE of three cases ﬁrst remains constant
and then decreases, as the rate threshold rincreases. Indeed,
when the value of the rate threshold ris small, the performed
power allocation can easier in satisfying the rate requirement
and thus the EE does not change. While for a high rate
threshold r, the resource allocation in the system becomes less
feasible in allocating power as it is forced to consume more
power to satisfy the stringant rate constraint, and therefore the
optimal EE decreases. Moreover, the gap between EEFand
EELincreases as rate threshold rincreases4.
C. Relationship Between SE and EE
To guarantee the QoS to users with affordable energy,
EE and SE in a speciﬁc system are used to evaluate the
performances of energy and spectral usage. Especially, for
achieving a good balance performance of VLC equipments, the
tradeoff between SE and EE should be delicately considered.
Based on (19) and (46), the relationship between SE and EE
of Gaussian distribution for ACOOFDM can be discussed as
EEG{p2i−1}N/2
i=1 =2NW
2
N/2
P
i=1
p2i−1+Pc
SEG{p2i−1}N/2
i=1 .
(57)
Similarly, the relationship between EE and SE of ﬁnite
alphabet inputs and the lower bound of mutual information for
ACOOFDM are similar to that of the Gaussian distribution.
Fig. 7(a) shows the EEG, EEF, and EELversus the SEG, SEF
and SELwith optical power threshold Po= 0.03 (W). It can
be seen from Fig. 7 that there is a nontrivial tradeoff between
the system SE and EE. In practice, as SE increases, the EE
increases at ﬁrst and then decreases. In particular, there exists
an optimal SE to maximize EE. Moreover, the peak of EEG
is the highest while the maximum EEFis higher than that of
EEL, and this phenomenon was also veriﬁed in Fig. 6 (a) and
(b).
Fig. 7(b) depicts the tradeoff between the SE and the EE
with the optical power threshold Po=∞. It can be observed
that, at the low SE region, EEGis close to EEF, and EEFis
close to EELat the high SE region.
4There might be intersections among EEL,EEF, and EEGas similar as
Fig. 1, Fig. 2, and Fig. 3 in [49].
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
(a)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
(b)
Fig. 7: (a) EEG, EEF, and EELversus the SEG, SEF, and SELwith optical
power threshold Po= 0.03 W; (b) EEG, EEF, and EELversus the SEG, SEF,
and SELwith optical power threshold Po=∞.
VI. CONCLUSION
In this study, we addressed the problem of designing optimal
power allocation schemes to maximize the SE and the EE of
ACOOFDM in VLC systems with Gaussian distributions and
ﬁnitealphabet inputs. We ﬁrst derived the achievable rates
and the average optical power constraint for ACOOFDM
VLC systems with the above two mentioned inputs. Then, we
derived the optimal power allocation schemes to maximize
the SE of ACOOFDM systems. Speciﬁcally, for Gaussian
distribution inputs, the waterﬁllingbased power allocation
scheme was presented to maximize the SE. By exploiting the
relationship between the mutual information and MMSE, the
optimal power allocation scheme was derived to maximum the
SE system with ﬁnitealphabet inputs. We further developed
the optimal power allocation scheme to maximize the EE of
ACOOFDM VLC systems with Gaussian distributions and
ﬁnitealphabet inputs. By adopting Dinkelbachtype algorithm,
the EE maximization problems were transformed into convex
problems and the interior point algorithm was exploited to
obtain the optimal solution. Besides, to reduce the computa
tional complexity of ﬁnitealphabet inputs cases, we derived
the closedform lower bounds of mutual information for both
the SE and the EE maximization problems. Finally, we showed
13
the relationship between the SE and the EE of ACOOFDM
VLC systems.
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