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Spectral and Energy Efficiency of ACO-OFDM 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 (ACO-OFDM) 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 mean-squared error, an optimal
power allocation scheme is proposed to maximize the SE with
finite-alphabet inputs. To reduce the computational complexity
of the power allocation scheme, we derive a closed-form lower
bound of the SE. Also, considering the quality of service, we
further tackle the non-convex EE maximization problems of
ACO-OFDM with the two inputs, respectively. The problems are
solved by the proposed Dinkelbach-type 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—ACO-OFDM, 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
low-cost 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. (e-mail: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. (e-mail: X.Deng@tue.nl).
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. Zhang is with Institut Suprieur dElectronique de Paris, ISEP Paris,
France. (e-mail: 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.
(e-mail: 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
(e-mail: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 license-free spectrum, high security, and no interference
to existing RF-based systems, VLC systems are a compelling
supplementary to RF systems for realizing high-speed wireless
data transmissions.
Despite the promising gains brought VLC technologies,
serious inter-symbol 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 RF-based 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 real-valued and nonnegative. Thus, conventional RF-based
OFDM techniques cannot directly apply to VLC systems. To
mitigate the ISI issue, asymmetrically clipped optical OFDM
(ACO-OFDM) [9], [10], direct current biased optical OFDM
(DCO-OFDM) [11]–[13], and Unipolar OFDM (U-OFDM)
[14] have been proposed for VLC systems. To generate
nonnegative transmitted signals, ACO-OFDM eliminates the
negative component of signals, while DCO-OFDM adds a
direct current (DC) bias and then clips the negative parts of
signals by setting them to zero [15]. Moreover, ACO-OFDM
transmits data symbols only via odd indexed subcarriers,
whereas DCO-OFDM transmits data symbols exploiting all
the subcarriers. Compared with DCO-OFDM, ACO-OFDM
can generally achieve a lower bit-error-rate (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 ACO-OFDM is generally lower than that
of DCO-OFDM from moderate to high signal-to-noise ratio
(SNR) [18], and the SE of U-OFDM is similar to that of ACO-
OFDM [19].
Recently, various power allocation schemes have been pro-
posed to improve the SE of ACO-OFDM VLC systems. For
example, under average optical power constraint, the conven-
tional water-filling power allocation scheme can improve the
information rate of ACO-OFDM substantially [10]. In [20],
the achievable rates of ACO-OFDM and filtered ACO-OFDM
(FACO-OFDM) 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 DCO-OFDM and ACO-OFDM systems were
analyzed in [21] showing that ACO-OFDM 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 closed-formed channel capacity
of ACO-OFDM. 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 ACO-OFDM 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 DCO-OFDM,
ACO-OFDM, and single carrier frequency-domain equaliza-
tion (SC-FDE) 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 finite-alphabet inputs may cause serious performance
loss [24]. So far, the SE of ACO-OFDM with finite-alphabet
inputs has been rarely considered in the literature. Therefore,
it is necessary to design an optimal power allocation scheme
to unlock the potential of ACO-OFDM systems.
In addition to improving the SE, achieving high energy
efficiency (EE) is also critical for ACO-OFDM 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 ACO-OFDM
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 ACO-OFDM 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 OFDM-based VLC modulation
schemes was investigated in [28]. However, existing studies
of ACO-OFDM’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 finite-alphabet inputs, which has been less
commonly studied in literature. Thus, there is an emerge need
for the study of EE of ACO-OFDM VLC systems with finite-
alphabet inputs.
In this study, we propose the optimal power allocation
scheme to maximize the SE and the EE of ACO-OFDM VLC
systems with Gaussian distribution inputs and finite-alphabet
inputs, respectively. The main contributions of this paper are
summarized as follows:
•We systematically analyze the signal processing module
of a typical ACO-OFDM VLC system. Based on the
frequency domain analysis, we first derive achievable
rates of the considered system admitting finite-alphabet
inputs from the perspective of practical modulation. Addi-
tionally, for both cases of the Gaussian distribution inputs
and finite-alphabet inputs, we develop the corresponding
optical power constraints for ACO-OFDM 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
finite-alphabet inputs, respectively. Specifically, for Gaus-
sian distribution inputs, we show that the water-filling-
based power allocation scheme can maximize the SE.
Similarly, for finite-alphabet inputs, we derive an opti-
mal power allocation scheme to achieve the maximum
SE by exploiting the Lagrangian method, Karush-Kuhn-
Tucker (KKT) conditions, and the relationship between
the mutual information and the minimum mean-squared
error (MMSE) [29].
•The optimal power allocation scheme for finite-alphabet
inputs lacks closed-form expressions and involves com-
plicated computations of MMSE. To reduce the compu-
tational complexity, we first derive a closed-form 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 finite-alphabet inputs, respectively.
Moreover, under the constraint of maximum transmis-
sion power and the minimum data rate requirement, the
non-convex problem of maximizing EE is investigated.
This problem is solved by applying the Dinkelbach-type
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 ACO-OFDM is presented in Section II. The SE of
ACO-OFDM 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{z|y}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 complex-valued circularly symmetric Gaussian distribution
with mean µand variance σ2is denoted by CN µ, σ2. A
real-valued 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
M-QAM
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 ACO-OFDM VLC system.
Consider an ACO-OFDM 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 sub-streams by a
serial-to-parallel (S/P) converter. Then, they are modulated by
an M-QAM scheme. After applying the inverse fast Fourier
transform (IFFT) and zero clipping on the modulated symbols,
signal ˆxkis non-negative. Then, the signal passes through
an digital-to-analog 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 non-negative. 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 analog-to-digital converter (A/D). After applying
the fast Fourier transform (FFT) on the digitalized signal,
demodulation is performed at a demodulator to convert the
received M-QAM 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 unit-power input, i.e.,
En|X2i−1|2o= 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 line-of-
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 frequency-domain
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 field-of-view (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 time-domian 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 frequency-domain 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 frequency-domain 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 zero-mean, 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 ACO-OFDM
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 ACO-OFDM 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 ACO-OFDM 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 ACO-OFDM
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 ACO-OFDM system with
Gaussian distribution inputs and finite-alphabet 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−1|H2i−1|2
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−1|H2i−1|2
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−1|H2i−1|2
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−1|2#+
.(22)
In fact, (22) is known as the classical water-filling solution
and the optimal µcan be found by the conventional gradient
method or the epsilon method [43], [44].
B. Finite-alphabet Inputs
In practice, typical inputs are always based on discrete sig-
naling constellations, such as M-PSK or M-QAM, 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| ≤ Pi|ai|, 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 non-uniform 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
finite-alphabet 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 constellation-constrained 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 closed-form 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−1|2
4σ2WMMSE2i−1 |H2i−1|2
4σ2Wp2i−1!.
(34)
By substituting (34) into (31a), we have
|H2i−1|2
4σ2WMMSE2i−1 |H2i−1|2
4σ2Wp2i−1!=λ. (35)
Then, solving (35) for the power allocation p2i−1yields
p2i−1=4σ2W
|H2i−1|2MMSE−1
2i−1 4σ2W
|H2i−1|2λ!,(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 ACO-OFDM system, the
optimal power allocation scheme of (29) is given by
p∗
2i−1=(4σ2W
|H2i−1|2MMSE−1
2i−14σ2W
|H2i−1|2λ,0< λ ≤|H2i−1|2
4σ2W;
0,otherwise.
(37)
The dual variable λin (37) is the solution of the following
equation
N/2
X
i=1
4σ2W
|H2i−1|2MMSE−1
2i−1 4σ2W
|H2i−1|2λ!=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−1|2/4σ2W, substitute λto obtain p∗
2i−1=
4σ2W
|H2i−1|2MMSE−1
2i−14σ2W
|H2i−1|2λ; 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
water-filling power allocation (22) with Gaussian distribution
inputs and the power allocation (37) with finite-alphabet in-
puts. To facilitate the presentation, let us introduce a function,
G2i−1(λ), as follows
G2i−1(λ) =
|H2i−1|2
4σ2W λ −MMSE−1
2i−14σ2W λ
|H2i−1|2,
0< λ ≤|H2i−1|2
4σ2W;
1,otherwise.
(39)
7
Then, the power allocation p2i−1can be represented as
p2i−1=1
λ−4σ2W
|H2i−1|2G2i−1(λ).(40)
Note that if G2i−1(λ) = 1, the power allocation (40)
is equivalent to the water-filling power allocation (22) with
the Gaussian distribution inputs; otherwise, the power allo-
cation (40) is the power allocation (37) with finite-alphabet
inputs2. Furthermore, based on the function G2i−1(λ), the
power allocation (40) can be interpreted as the mercury-water-
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−1|2, we first fill mercury to the height
4σ2W
|H2i−1|2G2i−1(λ), then, pouring water (power) to the height
1
λ. Note that both the noise level and the mercury in the
mercury-water-filling scheme form the bottom level of the
water-filling 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 water-filling scheme.
C. Lower Bound of Mutual Information
For finite-alphabet 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 ACO-OFDM is given in (24). The upper bound of the
2The ACO-OFDM with finite-alphabet 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−1|2
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−1|H2i−1|2|X2i−1,n −X2i−1,k|2
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 distribution-known discrete signaling
constellations, such as OOK, DPSK, and non-uniform 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 finite-alphabet inputs is given by
RL(p2i−1) = Wlog2M+ 1 −1
ln 2 −
M
X
n=1
W
Mlog2
M
X
k=1
exp −p2i−1|H2i−1|2|X2i−1,n −X2i−1,k|2
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 interior-point algorithm [43], [50]. It can also
be solved by the mercury-water-filling scheme similar to the
one adopted in Section III-B.
8
IV. ENERGY EFFIC IE NC Y OF ACO-OFDM
In this section, we investigate the design of optimal power
allocation scheme to maximize the EE of the ACO-OFDM
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 finite-alphabet 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−1|H2i−1|2
4σ2W
2
N/2
P
i=1
p2i−1+Pc
.
(46)
The EE maximization problem with Gaussian distribution
inputs of ACO-OFDM 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−1|H2i−1|2
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 quasi-concave function of p2i−1. With
a convex constraint set, problem (47) is a typical fractional
problem [51], which is generally non-convex. To circumvent
the non-convexity, Dinkelbach-type 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−1|H2i−1|2
4σ2W!−q
2
N/2
X
i=1
p2i−1+Pc
,
(49)
where qis a given non-negative 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−1|2#+
.(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 water-filling 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 Dinkelbach-type algorithm.
Under a finite number of iterations, the Dinkelbach-type
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 Dinkelbach-type 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. Finite-alphabet Inputs
For the finite-alphabet inputs, the achievable rate expres-
sion is given by (23), thus, the EE of finite-alphabet 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 finite-alphabet 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 closed-form 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 concave-linear fractional problem that can be solved by
Dinkelbach-type algorithms. The details are omitted as it is
similar to the case adopting Gaussian distribution inputs as
discussed in Section IV-A.
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 III-C, we exploit the lower
bound of achievable rate (42) to reduce the complexity, and the
corresponding EE function of the ACO-OFDM 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 concave-linear frac-
tional problem. At the same time, problem (56) can also be
solved by using Dinkelbach-type 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 ACO-OFDM VLC systems.
Consider an indoor ACO-OFDM VLC system installed with
four LEDs, where a corner of a square room denotes the
origin (0,0,0) of a three-dimensional 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−1|2
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 non-negative real
constant as same as [49] in all simulations without the opti-
mality loss of SE- and EE-maximization power allocation.
TABLE I: Simulation Parameters of the ACO-OFDM 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, M4-QAM
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, finite-alphabet 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−1|2, which is due to the
water-filling 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−1|2G2i−1(λ),
which is due to the mercury-water-filling method for the
maximization of the SE with finite-alphabet inputs. For the
case of SEL, the allocated power of each subcarrier is based
on PM
n=1 log2PM
k=1 exp −p2i−1|H2i−1|2|X2i−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 finite-alphabet
inputs. Moreover, due to the negligible small channel gain,
the allocated power from the 43-th subcarrier to the 63-th
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 high-order 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
10-6
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 finite-alphabet 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 finite-alphabet 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 finite-alphabet 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, finite-alphabet inputs case, and lower bound of mutual
information case for ACO-OFDM 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
10-6
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 ACO-OFDM 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
ACO-OFDM 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 non-trivial 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
ACO-OFDM in VLC systems with Gaussian distributions and
finite-alphabet inputs. We first derived the achievable rates
and the average optical power constraint for ACO-OFDM
VLC systems with the above two mentioned inputs. Then, we
derived the optimal power allocation schemes to maximize
the SE of ACO-OFDM systems. Specifically, for Gaussian
distribution inputs, the water-filling-based 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 finite-alphabet inputs. We further developed
the optimal power allocation scheme to maximize the EE of
ACO-OFDM VLC systems with Gaussian distributions and
finite-alphabet inputs. By adopting Dinkelbach-type 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 finite-alphabet inputs cases, we derived
the closed-form 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 ACO-OFDM
VLC systems.
REFERENCES
[1] H. Zhang, N. Liu, K. Long, J. Cheng, V. C. M. Leung, and L. Hanzo,
“Energy efficient subchannel and power allocation for software-defined
heterogeneous VLC and RF networks,” IEEE J. Sel. Areas Commun.,
vol. 36, no. 3, pp. 658–670, Mar. 2018.
[2] V. W. S. Wong, R. Schober, D. W. K. Ng, and L. Wang, Eds., Key
Technologies for 5G Wireless Systems, Cambridge University Press,
Apr. 2017.
[3] G. P. Fettweis and E. Zimmermann, “ICT energy consumption trends
and challenges,” in Proc. 11th Int. Symp. Wireless Personal Multimedia
Commun., pp. 1–4, Sep. 2008.
[4] G. Y. Li, Z. Xu, C. Xiong, C. Yang, S. Zhang, Y. Chen, and S. Xu,
“Energy-efficient wireless communications: tutorial, survey, and open
issues,” IEEE Wireless Commun., vol. 18, no. 6, pp. 28–35, Dec. 2011.
[5] S. Buzzi, C. I, T. E. Klein, H. V. Poor, C. Yang, and A. Zappone, “A
survey of energy-efficient techniques for 5G networks and challenges
ahead,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 697–709, Apr.
2016.
[6] M. Ismail and W. Zhuang, “Network cooperation for energy saving in
green radio communications,” IEEE Wireless Commun., vol. 18, no. 5,
pp. 76–81, Oct. 2011.
[7] Q. Wu, G. Y. Li, W. Chen, D. W. K. Ng, and R. Schober, “An overview
of sustainable green 5G networks,” IEEE Wireless Commun., vol. 24,
no. 4, pp. 72–80, Aug. 2017.
[8] J. Armstrong, “OFDM for optical communications,” J. Lightw. Technol.,
vol. 27, no. 3, pp. 189–204, Feb. 2009.
[9] J. Armstrong and A. J. Lowery, “Power efficient optical OFDM,”
Electron. Lett., vol. 42, no. 6, pp. 370–372, Mar. 2006.
[10] X. Li, J. Vucic, V. Jungnickel, and J. Armstrong, “On the capacity
of intensity-modulated direct-detection systems and the information rate
of ACO-OFDM for indoor optical wireless applications,” IEEE Trans.
Commun., vol. 60, no. 3, pp. 799–809, Mar. 2012.
[11] 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.
[12] 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.
[13] L. Chen, J. Wang, J. Zhou, D. W. K. Ng, R. Schober, and C. Zhao,
“Distributed user-centric scheduling for visible light communication
networks,” Opt. Express, vol. 24, no. 14, pp. 15570–15589, Jul. 2016.
[14] D. Tsonev, S. Sinanovic, and H. Haas, “Novel unipolar orthogonal
frequency division multiplexing (U-OFDM) for optical wireless,” in
IEEE Veh. Tech. Conf., Yohohama, Japan, pp. 1–5, May 2012.
[15] 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.
[16] S. Dimitrov, S. Sinanovic, and H. Haas, “Clipping noise in OFDM-
based optical wireless communication systems,” IEEE Trans. Commun.,
vol. 60, no. 4, pp. 1072–1081, Apr. 2012.
[17] S. D. Dissanayake and J. Armstrong, “Comparison of ACO-OFDM,
DCO-OFDM and ADO-OFDM in IM/DD systems,” J. Lightw. Technol.,
vol. 31, no. 7, pp. 1063–1072, Apr. 2013.
[18] S. Mazahir, A. Chaaban, H. Elgala, and M. Alouini, “Effective
information rates of single-carrier and multi-carrier modulation schemes
for bandwidth constrained IM/DD systems,” in Proc. IEEE Int. Conf.
Commun., pp. 1–6, May 2017.
[19] Z. Wang, T. Mao, and Q. Wang, “Optical OFDM for visible light
communications,” in Proc. 13th Int. Wireless Commun. Mobile Comput.
Conf. (IWCMC), Valencia, Spain, pp. 1190–1194, Jun. 2017.
[20] S. Mazahir, A. Chaaban, H. Elgala, and M. Alouini, “Achievable rates
of multi-carrier modulation schemes for bandlimited IM/DD systems,”
IEEE Trans. Wireless Commun., vol. 18, no. 3, pp. 1957–1973, Mar.
2019.
[21] Z. Yu, R. J. Baxley, and G. T. Zhou, “EVM and achievable data
rate analysis of clipped OFDM signals in visible light communication,”
EURASIP J. Wireless Commun. Netw., vol. 2012, no. 1, pp. 321, Dec.
2012.
[22] X. Li, R. Mardling, and J. Armstrong, “Channel capacity of IM/DD
optical communication systems and of ACO-OFDM,” in 2007 IEEE
Int. Conf. Commun., pp. 2128–2133, 2007.
[23] 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.
[24] 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.
[25] S. Ma, T. Zhang, S. Lu, H. Li, Z. Wu, and S. Li, “Energy efficiency
of SISO and MISO in visible light communication systems,” J. Lightw.
Technol., vol. 36, no. 12, pp. 2499–2509, Jun. 2018.
[26] A. W. Azim, Y. Le Guennec, and G. Maury, “Spectrally augmented
hartley transform precoded asymmetrically clipped optical OFDM for
VLC,” IEEE Photon. Technol. Lett., vol. 30, no. 23, pp. 2029–2032,
Dec. 2018.
[27] R. Zhang, H. Claussen, H. Haas, and L. Hanzo, “Energy efficient visible
light communications relying on amorphous cells,” IEEE J. Sel. Areas
Commun., vol. 34, no. 4, pp. 894–906, Apr. 2016.
[28] Y. Sun, F. Yang, and L. Cheng, “An overview of OFDM-based visible
light communication systems from the perspective of energy efficiency
versus spectral efficiency,” IEEE Access, vol. 6, pp. 60824–60833, Oct.
2018.
[29] 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.
[30] A. Oppenheim, A. Willsky, and S. Nawab, Signals and Systems,
Prentice-Hall signal processing series. Prentice Hall, 1997.
[31] B. Bai, Z. Xu, and Y. Fan, “Joint LED dimming and high capacity
visible light communication by overlapping PPM,” in The 19th Annual
Wireless and Opt. Commun. Conf. (WOCC 2010), pp. 1–5, May 2010.
[32] K. Ying, H. Qian, R. J. Baxley, and G. T. Zhou, “MIMO transceiver
design in dynamic-range-limited VLC systems,” IEEE Photon. Technol.
Lett., vol. 28, no. 22, pp. 2593–2596, Nov. 2016.
[33] S. Rajagopal, R. D. Roberts, and S. Lim, “IEEE 802.15.7 visible
light communication: modulation schemes and dimming support,” IEEE
Commun. Mag., vol. 50, no. 3, pp. 72–82, Mar. 2012.
[34] Q. Gao, C. Gong, and Z. Xu, “Joint transceiver and offset design for
visible light communications with input-dependent shot noise,” IEEE
Trans. Wireless Commun., vol. 16, no. 5, pp. 2736–2747, May 2017.
[35] J. Wang, Q. Hu, J. Wang, M. Chen, and J. Wang, “Tight bounds on
channel capacity for dimmable visible light communications,” J. Lightw.
Technol., vol. 31, no. 23, pp. 3771–3779, Dec. 2013.
[36] H. Schulze, “Frequency-domain simulation of the indoor wireless
optical communication channel,” IEEE Trans. Commun., vol. 64, no. 6,
pp. 2551–2562, Jun. 2016.
[37] J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc.
IEEE, vol. 85, no. 2, pp. 265–298, Feb. 1997.
[38] V. Jungnickel, V. Pohl, S. Nonnig, and C. von Helmolt, “A physical
model of the wireless infrared communication channel,” IEEE J. Sel.
Areas Commun., vol. 20, no. 3, pp. 631–640, Apr. 2002.
[39] 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.
[40] S. Dimitrov, S. Sinanovic, and H. Haas, “Double-sided signal clipping
in ACO-OFDM wireless communication systems,” in Proc. of IEEE
International Conference on Communications (IEEE ICC 2011), Kyoto,
Japan, 2011.
[41] J. Armstrong and A. J. Lowery, “Power efficient optical OFDM,”
Electronics letters, vol. 42, no. 6, pp. 370–372, Mar. 2006.
[42] T. M. Cover and J. A. Thomas, Elements of Information Theory, New
York: Wiley, Oct. 1990.
[43] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, U.K.:
Cambridge Univ. Press, 2004.
[44] R. W. Hamming, Numerical Methods for Scientists and Engineers,
Dover Publications, Inc., USA, 1986.
[45] 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.
[46] G. Chrystal, Algebra: An Elemenary Text-Book for the Higher Classes
of Secondary Schools and for Colleges, RI: AMS Chelsea, 1999.
[47] 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.
[48] S. Wei, D. L. Goeckel, and P. A. Kelly, “Convergence of the complex
envelope of bandlimited OFDM signals,” IEEE Trans. Inf. Theory,
vol. 56, no. 10, pp. 4893–4904, Oct. 2010.
14
[49] 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.
[50] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex
programming,” http://stanford.edu/boyd/cvx, Jun. 2009.
[51] D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-efficient resource al-
location in multi-cell OFDMA systems with limited backhaul capacity,”
IEEE Trans. Wireless Commun., vol. 11, no. 10, pp. 3618–3631, Oct.
2012.
[52] W. Dinkelbach, “On nonlinear fractional programming,” Manage. Sci.,
vol. 13, no. 7, pp. 492–498, Mar. 1967.
[53] 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.
[54] J. P. G. Crouzeix and J. A. Ferland, “Algorithms for generalized
fractional programming,” Math. Program., vol. 52, pp. 191–207, May
1991.
