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Spectral and Energy Efficiency of ACO-OFDM in Visible Light Communication Systems

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.


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 CO 2 emissions 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 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 abundant 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 performance bottleneck due to the existence of multipath in highdata rate VLC systems. In practice, orthogonal frequencydivision 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 represented 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 proposed to improve the SE of ACO-OFDM VLC systems. For example, under average optical power constraint, the conventional 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 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 description 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 closedform 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 equalization (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 constraints, 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 distribution 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 finitealphabet 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. Additionally, 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 Gaussian distribution inputs, we show that the water-fillingbased power allocation scheme can maximize the SE. Similarly, for finite-alphabet inputs, we derive an optimal 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 complicated computations of MMSE. To reduce the computational complexity, we first derive a closed-form lower bound for the achievable rate. Then, based on the proposed 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 transmission 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 represent vectors and matrices, respectively. Expected value of a random variable z is denoted by denotes the real part of its argument. ∂f (·) ∂x represents the partial derivative operation on x of function f (·). Given a variable y, E {z |y } represents the conditional mean of z for given y. min {x, y} represents the minimum value between x and y. I (X; Y ) represents the mutual information of X and Y . A complex-valued circularly symmetric Gaussian distribution with mean µ and variance σ 2 is denoted by CN µ, σ 2 . A real-valued Gaussian distribution with mean µ and variance σ 2 is denoted by N µ, σ 2 . ... Consider an ACO-OFDM VLC system with total 2N subcarriers, 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, signalx k is non-negative. Then, the signal passes through an digital-to-analog converter (D/A), where the digital signal x k is 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 X k denotes the modulated signal on the kth subcarrier, and p k denotes 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., where X 2i−1 is the normalized unit-power input, i.e., E |X 2i−1 | 2 = 1. According to (1), the power allocation of subcarriers should satisfy After the IFFT operation, the time domain signal x k is given as According to (3c), the obtained time domain signal satisfies antisymmetry as follows Since the transmitted signal should be nonnegative, the negative signals are removed by the clipping process such that Due to the requirement of the practical system circuit design, the total electrical transmit power should be limited [10]. Let P denote the total electrical transmit power, i.e., Combining the clipping process and Parseval's theorem [11], [30], we have k . Based on (2), the electrical transmit power constraint can be rewritten as For the consideration of human eye safety, the optical power of VLC signals is generally restricted [31]- [35]. Let P o represent the maximum optical power threshold. The average optical power should satisfy 1 According to the definition of variance, it is easy to verify that the sum of average electrical power P is larger than P 2 o .

B. Channel Model
Generally, the VLC channel is characterized by a line-ofsight (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 H k denotes the channel gain of the kth subcarrier, which includes both the LOS link and the diffuse links as follows where H L,k is the gain of the LOS link and H D,i is the gain of the diffuse links, k = 0, ..., 2N − 1.
The LOS link H L,k is expressed as where g L is the generalized Lambertian radiator [37], f k denotes the frequency of the ith subcarrier, τ is the signal propagation delay between the transmitter and receiver with τ = d/c, d is the distance between the transmitter and receiver, and c is the speed of light, k = 0, ..., 2N − 1. The generalized Lambertian radiator g L can be expressed as where m is the order of Lambertian emission, i.e., m = − ln 2/ ln cos Φ 1/2 , Φ 1/2 is the half power angle; A r is the effective detector area of the PD receiver; ϕ and θ are, 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; Ψ represents the field-of-view (FOV) of the receiver.
On the other hand, the gain of the diffuse links H D,k is given by [38] where η D is the power efficiency of the diffuse signal and τ is the exponential decay time. The time-domian diffuse channel gain h D is given as Thus, the time domain channel response can be given as is the Dirac function, and ∆T describes 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

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 Y 2i−1 denotes the signals received in the frequency-domain at the (2i − 1)th subcarrier, which is given by where the coefficient 1 2 exists since only half subcarriers are adopted to transmit information, Z 2i−1 is the additive white Gaussian noise (AWGN) with zero-mean, i.e., Z 2i−1 ∼ CN 0, W σ 2 , i = 1, ..., N/2, and σ 2 represents the noise power spectral density, W represents the bandwidth of each subcarrier.
and R ACO denote the rate of the (2i − 1)th subcarrier and the total rate of the ACO-OFDM system, respectively, which are given by 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 where W denotes 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 where 2 N/2 i=1 p 2i−1 represents the total electrical power consumption of all the subcarriers and P c denotes the total circuit power consumption of the whole system.

III. SPECTRAL EFFICIENCY 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: 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, respectively.

A. Gaussian Distribution Inputs
Assume that the input X 2i−1 follows independent complex Gaussian distribution, i.e., X 2i−1 ∼ CN (0, 1). According to the IFFT operation (3c), the time domain signal x k also follows Gaussian distribution, i.e., Furthermore, based on the relationship in (5), the average optical power is given by [40], [41] where σ 2 Additionally, by substituting (16) into (15b), the average optical power constraint can be reformulated as According to the Shannon theorem [42], the achievable rate of Gaussian distribution inputs R G (p 2i−1 ) is given by Then, the SE of the Gaussian distribution inputs can be expressed as . (19) Thus, the SE maximization problem with the Gaussian distribution inputs can be rewritten as 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 where µ ≥ 0 is the Lagrange multiplier associated with constraint (20b). By setting the differential function to 0, i.e., ∂LG ∂p2i−1 = 0, the optimal p 2i−1 is given by In fact, (22) is known as the classical water-filling solution and the optimal µ can 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 signaling 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 with cardinality M , where X 2i−1,k is a constellation point of the (2i − 1)th subcarrier. The achievable rate R F (p 2i−1 ) is given by [24] where is a measure of the difference between input constellation points X 2i−1,n and X 2i−1,k , E Z {·} is the expectation of the noise Z 2i−1 . Note that R F (p 2i−1 ) is a concave function with respect to the power allocation p 2i−1 [24], [45].
According to (5), the average optical power of transmitted signals is given as [40], [41] where the inequality holds due to | i a i | ≤ i |a i |, the value of E {|X i |} depends on the specific modulation schemes, i.e., (24) and (25) can be apply on another distributionknown 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 Based on the inequality ( , where a i ≥ 0, the average optical power constraint (26) can be restricted as 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 P o . Then, the SE of can be expressed as Thus, the optimal power allocation problem (15) can be reformulated as the constellation-constrained mutual information maximization problem which can be expressed as follows where ε ∆ = min P, The lack of a closed-form expression for the objective function (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 between the mutual information and MMSE [47].
To this end, we first derive the equivalent Lagrangian function of problem (29) which is given by where λ ≥ 0 is the Lagrange multiplier corresponding to constraint (29b). Furthermore, the KKT conditions of problem (29) can be expressed as According to [47], the relationship between the mutual information and the MMSE of the (2i − 1)th subcarrier is given by where is the MMSE of X 2i−1 , andX 2i−1 is conditional expectation of X 2i−1 , i.e., Combining (23) and (32), the differential function of R F (p 2i−1 ) can be written as By substituting (34) into (31a), we have Then, solving (35) for the power allocation p 2i−1 yields where . Therefore, for the considered ACO-OFDM system, the optimal power allocation scheme of (29) is given by The dual variable λ in (37) is the solution of the following equation which can be obtained by a simple bisection method as listed in Algorithm 1.

Algorithm 1 Bisection Method
Input: Given λ ∈ 0,λ , δ > 0, and initialize λ min = 0, λ max =λ, where δ denotes termination parameter, and λ represents an upper bound of λ; 1: while λ max − λ min ≥ δ do 2: 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 inputs. To facilitate the presentation, let us introduce a function, G 2i−1 (λ), as follows Then, the power allocation p 2i−1 can be represented as Note that if G 2i−1 (λ) = 1, the power allocation (40) is equivalent to the water-filling power allocation (22) with the Gaussian distribution inputs; otherwise, the power allocation (40) is the power allocation (37) with finite-alphabet inputs 2 . Furthermore, based on the function G 2i−1 (λ), the power allocation (40) can be interpreted as the mercury-waterfilling scheme [29]. More specifically, the power allocation (40) is illustrated in Fig. 2. For the (2i − 1)th subcarrier with noise level 4σ 2 W |H2i−1| 2 , we first fill mercury to the height

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 2 The ACO-OFDM with finite-alphabet inputs converges weakly to that with Gaussian distribution inputs as the number of subcarriers N is sufficiently large [48]. expectation term in (24) is given as [49] where C 2i−1 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 R L (p 2i−1 ) represents the lower bound of mutual information in the (2i − 1)th subcarrier. Thus, the lower bound of achievable rate of the ACO-OFDM VLC system with finite-alphabet inputs is given by Then, the SE of the lower bound of mutual information can be expressed as By substituting the derived lower bound (42) into the objective of problem of (43), the SE maximization problem (15) can be reformulated as follows where ε ∆ = min P, 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.

IV. ENERGY EFFICIENCY 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 where r is 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 can be expressed as The EE maximization problem with Gaussian distribution inputs of ACO-OFDM systems can be formulated as For the objective function (47a), the numerator is a differentiable concave function of the variable p 2i−1 , while the denominator is an affine function of p 2i−1 . Thus, the objective function (47a) is a quasi-concave function of p 2i−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 Υ denote the feasible set defined constraints of problem (47) as follows Moreover, we introduce a new function where q is a given non-negative parameter to be found iteratively. Then, by calculating the roots of the equation = 0 in the set Υ, the optimal solution of problem (47) can be obtained.
For a given q in each iteration, the convex subproblem over p 2i−1 can be expressed as Since the transformed problem is convex and satisfies the Slater's constraint qualification, we apply the conventional optimization techniques by taking the partial derivative of function f {p 2i−1 } N/2 i=1 and setting it to zero, i.e.
Then, by projecting p 2i−1 into the feasible region Υ, we obtain the optimal power p * 2i−1 of problem (50) as follows where denotes the projection of 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 transmit power level, as can be observed in (52).
Finally, the EE maximization problem of Gaussian distribution 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.

B. Finite-alphabet Inputs
For the finite-alphabet inputs, the achievable rate expression is given by (23), thus, the EE of finite-alphabet inputs 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 constraint, and the minimum rate constraint can be expressed as Note that there is no closed-form expression for the achievable rate in (54a), although R F ({p 2i−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 integrals 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 As the lower bound of the achievable rate (42) is concave and differentiable, problem (56) is also a concave-linear fractional problem. At the same time, problem (56) can also be solved by using Dinkelbach-type algorithms.

V. SIMULATION RESULTS AND 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 reflection point is (1.5, 0, 1.5). The basic parameters of the VLC system are listed in Table I. The channel gain is generated based on the channel model (7), (8), and (10) [36]- [38]. Besides, the R L (p 2i−1 ) has been added a non-negative real constant as same as [49] in all simulations without the optimality loss of SE-and EE-maximization power allocation.

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 p i versus channel gain H i of subcarrier i of SE G , SE F , and SE L , where P = 20 (W), P o = 0.25 (W). As can be observed from Fig. 3, the value of the allocated power p i of the SE G is proportional to the corresponding noise level 4σ 2 W |H2i−1| 2 , which is due to the water-filling solution for the maximization of the SE with Gaussian distribution inputs. While for the SE F case, the allocated power of p i not only depends on the noise level, but also depends on the mercury level 4σ 2 W |H2i−1| 2 G 2i−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 SE L , the allocated power of each subcarrier is based on o , the allocated power for Gaussian 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 SE G , SE F , and SE L versus the electrical power threshold P with an optical power constraint P o = 0.25 (W) and P o = ∞ (without optical power constraint), respectively. As shown in Fig. 4(a), for P o = ∞ case, as P increases, SE G keeps increasing, while SE F and SE L first increase and then remain constant. Moreover, when P is large, SE G is higher than both SE F and SE L . 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 for P o = 0.25 (W) case, as P increases, SE G , SE F , and SE L first increase and then remain constant. This is because the allocated power is limited by the optical power threshold, i.e., P o = 0.25 (W). Moreover, it can be observed from Fig. 4(a) that the SE F with finite-alphabet inputs reaches the maximum point faster than the SE G with Gaussian distribution inputs. The reason is that the coefficient of P o of Gaussian distribution inputs 2N π in (20b) is larger than that of finite-alphabet inputs 4 E 2 {|X2i−1|} in (29b). Besides, the gap between SE F and SE L is small, which means SE L could approximate SE F well enough. Fig. 4(b) illustrates SE G , SE F , and SE L versus optical power threshold P o with an electrical power constraint P = 10 (W) and P = ∞ (without electrical power constraint), respectively. 3 .
It can be seen in Fig. 4(b), for P = ∞ case, as P o increases, SE G keep increasing, while SE F and SE L first increase and then remain constant. Besides, when P o is large, SE G is higher than both SE F and SE L . 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, SE G , SE F , and SE L first increase and remain as a constant. In fact, the allocated power is restricted by the electrical power constraint, i.e., P = 10 (W). When P o is small, SE G is higher than SE F and SE L . As P o increases, SE G first reaches its maximum point, while SE F and SE L reach to the corresponding saturation points later. The reason is that the coefficient of P o of Gaussian distribution inputs 2N π in (20b) is larger than that of finitealphabet inputs 4 E 2 {|X2i−1|} in (29b). Thus, with the same optical power threshold P o , 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 p i of EE G , EE F , and EE L versus channel gain H i of subcarrier i respectively, where P = 20 (W), P o = 0.25 (W), and r = 1 (bits/sec/Hz). From Fig. 5, we can see that the allocated power of subcarrier i of EE G is proportional to its channel gain, which is due to the power allocation strategy in (51) and (52). While the allocated power of subcarrier i of both EE F and EE L depend 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 EE G , EE F , and EE L versus electrical power threshold P with an optical power threshold P o = 0.03 (W) and P o = ∞ (without optical power constraint) respectively, where the rate constraint r = 0.1 (bits/sec/Hz).We see from Fig. 6(a) that for P o = ∞ case, as P increases, EE G , EE F , and EE L first increase and then remain constant. This is because the optimal EE remains a constant when it has reached the maximum value. Moreover, the value of EE F approaches to that of EE G , which are higher than the value of EE L . While for P o = 0.03 case, as P increases, the EE G , EE F , and EE L first increase and then remain constant. The reason is that EE G , EE F , and EE L is limited by the optical power constraint P o = 0.03 (W). Besides, for the large P , the value of EE G are the highest of the three power allocation schemes, while EE L is the lowest. Fig. 6(b) depicts EE G , EE F , and EE L versus rate threshold r with power threshold P = 20 (W) and optical power threshold P o = 1 (W). As shown in Fig. 6(b), EE G is higher than EE F and EE L , and the gap between EE G and EE F is small. Moreover, the EE of three cases first remains constant and then decreases, as the rate threshold r increases. Indeed, when the value of the rate threshold r is 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 EE F and EE L increases as rate threshold r increases 4 .

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 Similarly, the relationship between EE and SE of finitealphabet inputs and the lower bound of mutual information for ACO-OFDM are similar to that of the Gaussian distribution. Fig. 7(a) shows the EE G , EE F , and EE L versus the SE G , SE F and SE L with optical power threshold P o = 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 EE G is the highest while the maximum EE F is higher than that of EE L , 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 P o = ∞. It can be observed that, at the low SE region, EE G is close to EE F , and EE F is close to EE L at the high SE region. 4 There might be intersections among EE L , EE F , and EE G as similar as Fig. 1, Fig. 2, and Fig. 3 in [49].

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 computational 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 the relationship between the SE and the EE of ACO-OFDM VLC systems.