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EVM and achievable data rate analysis of clipped OFDM signals in visible light communication
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 321 (2012)
Abstract
Abstract
Orthogonal frequency division multiplexing (OFDM) has been considered for visible light communication (VLC); thanks to its ability to boost data rates as well as its robustness against frequencyselective fading channels. A major disadvantage of OFDM is the large dynamic range of its timedomain waveforms, making OFDM vulnerable to nonlinearity of light emitting diodes. DCbiased optical OFDM (DCOOFDM) and asymmetrically clipped optical OFDM (ACOOFDM) are two popular OFDM techniques developed for the VLC. In this article, we will analyze the performance of the DCOOFDM and ACOOFDM signals in terms of error vector magnitude (EVM), signaltodistortion ratio (SDR), and achievable data rates under both average optical power and dynamic optical power constraints. EVM is a commonly used metric to characterize distortions. We will describe an approach to numerically calculate the EVM for DCOOFDM and ACOOFDM. We will derive the optimum biasing ratio in the sense of minimizing EVM for DCOOFDM. In addition, we will formulate the EVM minimization problem as a convex linear optimization problem and obtain an EVM lower bound against which to compare the DCOOFDM and ACOOFDM techniques. We will prove that the ACOOFDM can achieve the lower bound. Average optical power and dynamic optical power are two main constraints in VLC. We will derive the achievable data rates under these two constraints for both additive white Gaussian noise channel and frequencyselective channel. We will compare the performance of DCOOFDM and ACOOFDM under different power constraint scenarios.
Introduction
With rapidly growing wireless data demand and the saturation of radio frequency (RF) spectrum, visible light communication (VLC) [1–4] has become a promising candidate to complement conventional RF communication, especially for indoor and medium range data transmission. VLC uses white light emitting diodes (LEDs) which already provide illumination and are quickly becoming the dominant lighting source to transmit data. At the receiving end, a photo diode (PD) or an image sensor is used as light detector. VLC has many advantages including lowcost frontends, energyefficient transmission, huge (THz) bandwidth, no electromagnetic interference, no eye safety constraints like infrared, etc. [5]. In VLC, simple and lowcost intensity modulation and direct detection (IM/DD) techniques are employed, which means that only the signal intensity is modulated and there is no phase information. At the transmitter, the white LED converts the amplitude of the electrical signal to the intensity of the optical signal, while at the receiver, the PD or image sensor generates the electrical signal proportional to the intensity of the received optical signal. The IM/DD requires that the electric signal must be realvalued and unipolar (positivevalued).
Recently, orthogonal frequency division multiplexing (OFDM) has been considered for VLC; thanks to its ability to boost data rates and efficiently combat intersymbol interference [5–10]. To ensure that the OFDM timedomina signal is realvalued, Hermitian symmetry condition must be satisfied in the frequencydomain. Three methods have been discussed in the literature for creating realvalued unipolar OFDM signal for VLC.

(1)
DCbiased optical OFDM (DCOOFDM)—adding a DC bias to the original signal [6, 7, 11];

(2)
Asymmetrically clipped optical OFDM (ACOOFDM)—only mapping the data to the odd subcarriers and clipping the negative parts without information loss [8];

(3)
FlipOFDM—transmitting positive and negative parts in two consecutive unipolar symbols [10].
One disadvantage of OFDM is its high peaktoaveragepower ratio (PAPR) due to the summation over a large number of terms [12]. The high PAPR or dynamic range of OFDM makes it very sensitive to nonlinear distortions. In VLC, the LED is the main source of nonlinearity. The nonlinear characteristics of LED can be compensated by digital predistortion (DPD) [13], but the dynamic range of any physical device is still limited. The input signal outside this range will be clipped. A number of papers [14–17] have studied the clipping effects on the RF OFDM signals. However, clipping in the VLC system has two important differences: (i) the RF baseband signal is complexvalued whereas timedomain signals in the VLC system are realvalued; (ii) the main power limitation for VLC is average optical power and dynamic optical power, rather than average electrical power and peak power as in RF communication. Therefore, most of the theory and analyses developed for RF OFDM are not directly applicable to optical OFDM. A number of papers [13, 18–20] have analyzed the LED nonlinearity on DCOOFDM and ACOOFDM and compared their bit error rate, power efficiency, bandwidth efficiency, etc.
In this article, we will investigate the performance of DCOOFDM and ACOOFDM signals in terms of error vector magnitude (EVM), signaltodistortion ratio (SDR), and achievable data rates. EVM is a frequently used performance metric in modern communication standards. In [13, 19], the EVM is measured by simulations for varying power backoff and biasing levels. In this article, we will describe an approach to numerically calculate the EVM for DCOOFDM and ACOOFDM, and derive the optimum biasing ratio for DCOOFDM. We will formulate the EVM minimization problem as a convex linear optimization problem and obtain an EVM lower bound. In contrast to [21] which investigated the achievable data rates for ACOOFDM with only average optical power limitation, we will derive the achievable data rates subject to both the average optical power and dynamic optical power constraints. We will first derive the SDR for a given databearing subcarrier based on the Bussgang’s theory. Upon the SDR analysis, we will derive the achievable data rates for additive white Gaussian noise (AWGN) channel and frequencyselective channel. Finally, we will compare the performance of two optical OFDM techniques.
System model
The system model discussed in this study is depicted in Figure 1. In an OFDM system, a discrete timedomain signal x =[x[0], x[1], …, x[N1]] is generated by applying the inverse DFT (IDFT) operation to a frequencydomain signal X =[X_{0}, X_{1}, …, X_{N1}] as
where $j=\sqrt{1}$ and N are the size of IDFT, assumed to be an even number in this article. In a VLC system using LED, the IM/DD schemes require that the electric signal be realvalued and unipolar (positivevalued). According to the property of IDFT, a realvalued timedomain signal x[n] corresponds to a frequencydomain signal X_{ k } that is Hermitian symmetric, i.e.,
where ∗ denotes complex conjugate.
In Figure 1, y[n] is obtained from x[n] after both a clipping and a biasing operation are implemented. The resulting signal, y[n], is nonnegative (i.e., y[n] ≥ 0) and has a limited dynamic range. In a VLC system, the light emitted is used for illumination and communication simultaneously. The intensity of light emitted by the LED is proportional to y[n], while the electrical power is proportional to y^{2}[n]. In the optical communication literature, the average optical power of the LED input signal y[n] is defined as
where $\mathcal{E}[\xb7]$ denotes statistical expectation. Usually, the VLC system operates under some average optical power constraint P_{ A }, i.e.,
This constraint is in place for two reasons: (i) the system power consumption needs to be kept under a certain limit, (ii) the system should still be able to communicate even under dim illumination conditions. The VLC system is further limited by the dynamic range of the LED. In this article, we assume that the DPD has perfectly linearized the LED between the interval [P_{ L },P_{ H }], where P_{ L } is the turnon voltage (TOV) for the LED. If the TOV is provided by an analog module at the LED and we assume that LED is already turned on, the linear range for the input signal is [0, P_{ H }P_{ L }]. We define the dynamic optical power of y[n] as
G_{ y } should be constrained by P_{ H }P_{ L } as
Moreover, y[n] must be nonnegative, i.e., y[n] ≥ 0.
According to the Central Limit Theorem, x[n] is approximately Gaussian distributed with zero mean and variance σ^{2} with probability density function (pdf):
where $\varphi \left(x\right)=\frac{1}{\sqrt{2\Pi}}{e}^{\frac{1}{2}{x}^{2}}$ is the pdf of the standard Gaussian distribution. As a result, the timedomain OFDM signal x[n] tends to occupy a large dynamic range and is bipolar. In order to fit into the dynamic range of the LED, clipping is often necessary, i.e.,
where c_{ u } denotes the upper clipping level, and c_{ l } denotes the lower clipping level. In order for the LED input y[n] to be nonnegative, we may need to add a DC bias B to the clipped signal $\stackrel{\u0304}{x}\left[n\right]$ to obtain
For y[n] ≥ 0, we need B = c_{ l }.
To facilitate the analysis, we define the clipping ratio γ and the biasing ratio ς as
Thus, the upper and lower clipping levels can be written as
The ratios γ and ς can be adjusted independently causing c_{ u } and c_{ l } to vary.
Clipping in the timedomain gives rise to distortions on all subcarriers in the frequency domain. On the other hand, DCbias only affects the DC component in the frequencydomain. The clipped and DCbiased signal y[n] is then converted into analog signal and subsequently modulate the intensity of the LED. At the receiver, the photodiode, or the image sensor, converts the received optical signal to electrical signal and transforms it to digital form. The received sample can be expressed as
where h[n] is the impulse response of the wireless optical channel, w[n] is AWGN, and ⊗ denotes convolution. By taking the DFT of Equation (14), we can obtain the received data on the k th subcarrier as
where H_{ k } is the channel frequency response on the k th subcarrier.
Based on the subcarrier arrangement, DCbiasing, or transmission scheme, several optical OFDM techniques have been proposed in the literature. In this article, we will focus on the performance analysis of two widely studied optical OFDM techniques, namely, DCOOFDM and ACOOFDM. In the following, we shall use superscripts ^{(D)} and ^{(A)} to indicate DCOOFDM and ACOOFDM, respectively.
In DCOOFDM, subcarriers of the frequencydomain signal X^{(D)} are arranged as
where the 0th and N/2th subcarriers are null (do not carry data). Equation (16) reveals Hermitian symmetry with respect to k = N/2. Let ${\mathcal{K}}_{d}$ denote the set of datacarrying subcarriers with cardinality $\left{\mathcal{K}}_{d}\right$. The set of datacarrying subcarriers for DCOOFDM is ${\mathcal{K}}_{d}^{\left(D\right)}=\{1,2,\dots ,N/21,N/2+1,\dots ,N2,N1\}$ and $\left{\mathcal{K}}_{d}^{\left(D\right)}\right=N2$. The timedomain signal x^{(D)}[n] can be obtained as
which is realvalued. In DCOOFDM, we first obtain a clipped signal ${\stackrel{\u0304}{x}}^{\left(D\right)}\left[n\right]$ similar to the procedure in (8), and then add DCbias B = c_{ l } to obtain the LED input signal
In the frequency domain,
where C_{ k } is clipping noise on the k th subcarrier.
In ACOOFDM, only odd subcarriers of the frequencydomain signal X^{(A)} carry data
and X^{(A)}meets the Hermitian symmetry condition (2). The set of datacarrying subcarriers for ACOOFDM is ${\mathcal{K}}_{d}^{\left(A\right)}=\{1,3,\dots ,N1\}$ and $\left{\mathcal{K}}_{d}^{\left(A\right)}\right=N/2$. Thus, the timedomain signal x^{(A)}[n] can be obtained as
which is realvalued. It follows easily that x^{(A)}[n] satisfies the following negative half symmetry condition:
Denote by z[n] a generic discretetime signal that satisfies z[n + N/2] = z[n], n = 0, 1,…,N/21 and by $\stackrel{\u0304}{z}\left[n\right]$ its clipped version where the negative values are removed, i.e.,
It was proved in [22] that in the frequencydomain,
In ACOOFDM, we obtain the LED input signal y^{(A)}[n] via
Equation (25) can be regarded as a 2step clipping process, whereby we first remove those negative values in x^{(A)}[n], and then replace those x^{(A)}[n] values that exceed c_{ u } by c_{ u }. Since x^{(A)}[n] satisfies (22), we infer based on (24) that
where C_{ k } is clipping noise on the k th subcarrier in the frequencydomain. For ACOOFDM, no DCbiasing is necessary and thus the biasing ratio ς = 0.
As an example, suppose that we need to transmit a sequence of eight quadrature phaseshift keying (QPSK) symbols. Table 1 shows the subcarrier arrangement for DCOOFDM, whereas Table 2 shows the subcarrier arrangement for ACOOFDM. The timemain signals x^{(D)}[n] and x^{(A)}[n] and the corresponding LED input signals y^{(D)}[n] and y^{(A)}[n] are shown in Figure 2. We see that in x^{(A)}[n], the last 16 values are a repetition of the first 16 values but with the opposite sign. It takes ACOOFDM more bandwidth than DCOOFDM to transmit the same message, although ACOOFDM is less demanding in terms of dynamic range requirement of the LED and power consumption.
EVM analysis
EVM is a figureofmerit for distortions. Let ${\mathbf{X}}^{\u2020}=[{X}_{0}^{\u2020},{X}_{1}^{\u2020},\dots ,{X}_{N1}^{\u2020}]$ denote the Nlength DFT of the modified timedomain signal x^{†}. EVM can be defined as
where ${\mathbf{X}}^{\left(r\right)}=[{X}_{0}^{\left(r\right)},{X}_{1}^{\left(r\right)},\dots ,{X}_{N1}^{\left(r\right)}]$ denotes the reference constellation. For DCOOFDM, ${X}_{k}^{\left(r\right)}={X}_{k}^{\left(D\right)}$ for $k\in {\mathcal{K}}_{d}^{\left(D\right)}$. For ACOOFDM, ${X}_{k}^{\left(r\right)}=\frac{1}{2}{X}_{k}^{\left(A\right)}$ for $k\in {\mathcal{K}}_{d}^{\left(A\right)}$.
EVM calculation
In DCOOFDM, clipping in the timedomain generates distortions on all the subcarriers. We denote the clipping error power by ${\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}={\sum}_{k\in {\mathcal{K}}_{d}^{\left(D\right)}}\mathcal{E}\left[\right{X}_{k}^{\left(D\right)}{\stackrel{\u0304}{X}}_{k}^{\left(D\right)}{}^{2}]$. Since the sum distortion power on the 0th and N/2th subcarriers is small relative to the total distortion power of N subcarriers, according to the Parseval’s theorem, we can approximate ${\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}$ as
where $\Phi \left(x\right)=\underset{\infty}{\overset{x}{\int}}\varphi \left(t\right)\mathit{dt}$. Thus, we obtain the EVM for the DCOOFDM scheme as
To find the optimum biasing ratio ς^{⋆}, we take the firstorder partial derivative and the secondorder partial derivative of ${\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}$ with respect to the biasing ratio ς
We can see that if ς=0.5, $\partial {\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}/\mathrm{\partial \varsigma}=0$. The secondorder partial derivative ${\partial}^{2}{\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}/\partial {\varsigma}^{2}>0$ for all ς. Hence, if ς < 0.5, $\partial {\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}/\mathrm{\partial \varsigma}<0$. If ς > 0.5, $\partial {\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}/\mathrm{\partial \varsigma}>0$. Therefore, ς^{⋆} = 0.5 is the optimum biasing ratio which minimizes ${\stackrel{\u0304}{P}}_{\gamma ,\varsigma}^{\left(D\right)}$. By substituting ς^{⋆} into Equation (29) we obtain the EVM for the DCOOFDM scheme at the optimum biasing ratio as
Remark 1

(i)
ς ^{⋆} = 0.5 is the optimum biasing ratio for DCOOFDM, regardless of the clipping ratio. (ii) When ς = 0.5, we infer that c _{ u } = c _{ l }, i.e., when the x ^{(D)}[n] waveform is symmetrically clipped at the negative and positive tails, the clipping error power is always less than that when the two tails are asymmetrically clipped (i.e., when c _{ u } ≠ c _{ l } or when ς ≠ 0.5).
Denote by e[n], n = 0, 1, …, N  1 a generic discretetime signal with DFT E_{ k }, k = 0, 1, …, N  1. When k is odd, E_{ k } can be written as
Let k = 2q + 1, q = 0, 1, …, N / 2  1, Equation (33) can be further written as
Therefore, ${\{{E}_{k}={E}_{2q+1}\}}_{q=0}^{N/21}$ can be viewed as the DFT coefficients of a new discretetime sequence $\underset{n=0}{\overset{N/21}{\{exp\left(j2\Pi \frac{n}{N}\right)\left(e\left[n\right]e[n+N/2]\right)\}}}$. Applying the Parseval’s theorem to $\underset{n=0}{\overset{N/21}{\{exp\left(j2\Pi \frac{n}{N}\right)(e\left[n\right]e[n+N/2]\left)\right\}}}$, we obtain,
In ACOOFDM, we denote the clipping error power by ${\stackrel{\u0304}{P}}_{\gamma}^{\left(A\right)}={\sum}_{k\in {\mathcal{K}}_{d}^{\left(A\right)}}\mathcal{E}\left[{\left{X}_{k}^{\left(A\right)}/2{\stackrel{\u0304}{X}}_{k}^{\left(A\right)}\right}^{2}\right]$. According to (35), we can calculate ${\stackrel{\u0304}{P}}_{\gamma}^{\left(A\right)}$ as
Then we obtain the EVM for the ACOOFDM scheme as
Lower bound on the EVM
Let us consider the setting
where x[n] is the original signal, c[n] is a distortion signal, and the resulting $\widehat{x}\left[n\right]$ is expected to have a limited dynamic range
In (38), all quantities involved are realvalued.
Clipping can produce one such $\widehat{x}\left[n\right]$ signal, but there are other less straightforward algorithms that can generate other $\widehat{x}\left[n\right]$ waveforms that also satisfy (39).
In the frequencydomain,
Since x[n], c[n], and $\widehat{x}\left[n\right]$ are all realvalued, X_{ k }, C_{ k }, and ${\widehat{X}}_{k}$ all should satisfy the Hermitian symmetry condition (2). Therefore, c[n] has the form
We are interested in knowing the lowest possible EVM,
among all such $\widehat{x}\left[n\right]$ waveforms. Afterwards, we can compare the EVM from the DCOOFDM and ACOOFDM methods to get a sense of how far these algorithms are from being optimum (in the EVM sense).
We formulate the following linear optimization problem:
When the distortion of each OFDM symbol is minimized by the above convex optimization approach, the corresponding EVM of $\widehat{x}\left[n\right]$ (which is proportional to $\sqrt{\mathcal{E}\left[{\sum}_{k\in {\mathcal{K}}_{d}}{C}_{k}{}^{2}\right]}$) serves as the lower bound for the given dynamic range 2γσ.
Optimality for ACOOFDM
In this section, we will prove that the ACOOFDM scheme achieves the minimum EVM and thus is optimal in the EVM sense.
For ACOOFDM, let us write
where c[n] is the clipping noise to ensure that $\widehat{x}\left[n\right]$ has a limited dynamic range as described in (39). In the frequencydomain, we have
where the ${X}_{k}^{\left(A\right)}$ subcarriers are laid out as in (20). According to (35), when k is odd, the objective function in (43) can be written as
The dynamic range constraints in problem (43) can be viewed as two constraints put together.
Since x^{(A)}[n] = x^{(A)}[n  N / 2] when N / 2 ≤ n ≤ N  1, Equation (48) can be further written as
From Equations (46), (47), and (49), the problem (43) can be recast as
which is equivalent to
In Appendix, we prove that the solution c^{⋆}[n] to (51) yields
and thus the ACOOFDM scheme is optimum in the EVM sense.
SDR analysis
Based on the Bussgang’s theorem [23], any nonlinear function of x[n] can be decomposed into a scaled version of x[n] plus a distortion term d[n] that is uncorrelated with x[n]. For example, we can write
Let ${R}_{\mathit{\text{xx}}}\left[m\right]=\mathcal{E}\left\{x\right[n\left]x\right[n+m\left]\right\}$ denote the autocorrelation function of x[n], and let ${R}_{\mathit{\text{xy}}}\left[m\right]=\mathcal{E}\left\{x\right[n\left]y\right[n+m\left]\right\}$ denote the crosscorrelation function between x[n] and y[n] at lag m. For any given m, the correlation functions satisfy
Thus, the scaling factor α can be calculated as
Let f(·) denote the function linking the original signal to the clipped signal, it is shown in [24] that the output autocorrelation function ${R}_{\stackrel{\u0304}{x}\stackrel{\u0304}{x}}\left[m\right]$ is related to the input autocorrelation function R_{ xx }m via
where the coefficients
The input autocorrelation function R_{ xx }[m] can be obtained from taking IDFT of the input power spectrum density (PSD)
where ${P}_{X,k}=\mathcal{E}\left[\right{X}_{k}{}^{2}]$ is the expected value of the power on the k th subcarrier before clipping. Then it is straightforward to calculate the output PSD by taking the DFT of the autocorrelation of the output signal:
Taking the DFT of Equation (53), the data at the k th subcarrier are expressed as
Here, we assume that D_{ k } is Gaussian distributed, which is the common assumption when N is large [15]. The SDR at the k th subcarrier is given by
where ${P}_{D,k}=\mathcal{E}\left[\right{D}_{k}{}^{2}]={P}_{\stackrel{\u0304}{X},k}{\alpha}^{2}{P}_{X,k}$ is the average power of the distortion on the k th subcarrier.
According to Equation (56), we can obtain the scaling factor α as a function of the clipping ratio γ and the biasing ratio ς:
Note that in (63) we have used Equations (13) and (12) for c_{ l } and c_{ u }. According to Equation (58), we can obtain the coefficient b_{ ℓ } as a function of the clipping ratio γ and the biasing ratio ς:
where $H{e}_{n}\left(t\right)={(1)}^{\ell}exp\left(\frac{{t}^{2}}{2}\right)\frac{{d}^{\ell}[exp(\frac{{t}^{2}}{2}\left)\right]}{d{t}^{\ell}}$ is the probabilists’ Hermite polynomials [25].
Achievable data rate
In VLC, average optical power and dynamic optical power are two main constraints. Recall from Equation (3), we can obtain the average optical power of y[n] as
Let ${\sigma}_{w}^{2}=\mathcal{E}\left\{{w}^{2}\right[n\left]\right\}$ denote the power of AWGN w[n], we define the optical signaltonoise ratio (OSNR) as
Recall from Equation (5), we can obtain the dynamic optical power of y[n] as
We define the dynamic signaltonoise ratio (DSNR) as
Let η_{OSNR} = P_{ A } / σ_{ w } denote the OSNR constraint and η_{DSNR} = (P_{ H }  P_{ L }) / σ_{ w } denote the DSNR constraint, we have
The maximum σ / σ_{ w } value can be obtained as
by substituting (65) and (67) into the righthand side of (69) and (70), respectively. The ratio η_{DSNR} / η_{OSNR} = (P_{ H }  P_{ L }) / P_{ A } is determined by specific system requirements.
AWGN channel
For AWGN channel, recall from Equation (15), the received data on the k th subcarrier can be expressed as
The signaltonoiseanddistortion ratio (SNDR) for the k th subcarrier is given by
In this article, we assume the power is equally distributed on all datacarrying subcarriers,
then Equation (73) is reduced to
By substituting Equation (71) into (75), we obtain the reciprocal of SNDR at the k th subcarrier:
Therefore, the achievable data rate, as a function of clipping ratio γ, ς, η_{OSNR}, and η_{DSNR}, is given by
Frequencyselective channel
In the presence of frequencyselective channel, the received data on the k th subcarrier obey the following in the frequencydomain:
In this article, we consider the ceiling bounce channel model [26] given by
where H(0) is the gain constant, $a=12\sqrt{11/23}D$ and u(t) is the unit step function. D denotes the rms delay. From Equation (78), the SNDR is given by
With the assumption of equal power distribution, we can obtain the 1 / SNDR as
The achievable data rate, in the presence of frequencyselective channel, is given by
Numerical results
In this section, we show EVM simulation results and achievable data rates of clipped optical OFDM signals under various average optical power and dynamic optical power constraints.
EVM simulation
The EVM analyses for DCOOFDM and ACOOFDM are validated through computer simulations. In the simulations, we chose the number of subcarriers N = 512, and QPSK modulation. One thousand OFDM symbols were generated based on which we calculated the EVM. In order to experimentally determine the optimum biasing ratio for DCOOFDM, we used biasing ratios ranging from 0.3 to 0.7 in step size of 0.02, and clipping ratios ranging from 5 to 9 dB in step size of 1 dB. Their simulated and theoretical EVM curves are plotted in Figure 3. As expected, the minimum EVM was achieved when the biasing ratio was 0.5, regardless of the clipping ratio. This agrees with the analysis in “EVM calculation” section. Next, we compared the EVM for DCOOFDM with biasing ratio 0.5, EVM for ACOOFDM with biasing ratio 0, and their respective lower bounds. To obtain the lower bounds, we used CVX, a package for specifying and solving convex programs [27], to solve Equation (43). The resulting EVM curves for DCOOFDM are plotted in Figure 4. The resulting EVM curves for ACOOFDM are plotted in Figure 5. We see that the EVM for ACOOFDM achieves its lower bound, thus corroborating the discussion in “Optimality for ACOOFDM” section. For DCOOFDM, the gap above the lower bound increases with the clipping ratio (i.e., with increasing dynamic range of the LED). This implies that there exists another (more complicated) way of mapping x^{D}n into a limited dynamic range signal $\widehat{x}\left[n\right]$ that can yield a lower EVM.
Achievable data rates performance
We now show achievable data rates of clipped OFDM signals under various average optical power and dynamic optical power constraints. The number of subcarriers was N = 512. For the frequencyselective channel, we chose the rms delay spread D = 10 ns and sampling frequency 100 MHz. The normalized frequency response for each subcarrier is shown in Figure 6.
As examples, we chose η_{OSNR} = 20 dB, η_{DSNR} = 32 dB, and AWGN channel. Figures 7 and 8 show the achievable data rate as a function of the clipping ratio and the biasing ratio for DCOOFDM and ACOOFDM, respectively. We see that for given η_{OSNR} and η_{DSNR} values, a pair of optimum clipping ratio γ^{†} and optimum biasing ratio ς^{†} exist that maximize the achievable data rate. It is worthwhile to point out that the optimum biasing ratio ς^{†} is different from ς^{⋆} (recall that ς^{⋆} minimizes the EVM). If the system is only subject to the dynamic power constraint, ς^{†} should be equal to ς^{⋆}. If the dominant constraint is the average power, ς^{†} should be less than or equal to ς^{⋆} because reducing the biasing ratio can make the signal average power lower. We can obtain the optimum clipping ratio and biasing ratio for given η_{OSNR}, η_{DSNR} by
Figure 9a shows the optimal clipping ratio as a function of η_{OSNR} for DCOOFDM. Figure 9b shows the optimum biasing ratio as a function of η_{OSNR} for DCOOFDM. Similar plots are shown as Figure 10a,b for ACOOFDM. In all cases, η_{OSNR} varied from 0 to 25 dB in step size of 1 dB, η_{DSNR} / η_{OSNR} = 18 dB, and the channel was AWGN. The main observation is, with a lower average optical power constraint, the clipping ratio and the biasing ratio can be increased to achieve higher data rates. Intuitively, when η_{OSNR} is large, the channel noise has little effect and the nonlinear distortion dominates.
Next, we chose the ratio η_{DSNR}/η_{OSNR} from 6 dB, 12 dB, and no η_{DSNR} constraints. For each pair of η_{OSNR}, η_{DSNR}, AWGN channel, or frequencyselective channel, we can calculate the optimum clipping ratio γ^{†} and biasing ratio ς^{†} according to Equation (84) and the corresponding achievable data rates. Figures 11, 12, and 13 show the achievable data rates with optimal clipping ratio and biasing ratio for the case η_{DSNR} / η_{OSNR} = 6 dB, η_{DSNR} / η_{OSNR} = 12 dB, and no η_{DSNR} constraint, respectively. We observe that the performance of ACOOFDM and DCOOFDM depends on the specific optical power constraints scenario. In general, DCOOFDM outperforms ACOOFDM for all the cases. With the increase of the ratio η_{DSNR} / η_{OSNR}, the average optical power becomes the dominant constraint. The ACOOFDM moves closer to the DCOOFDM.
As seen in Figure 13, when there is no DSNR constraint and the OSNR constraint is large, the DCOOFDM curve closely matches the ACOOFDM curve. This is in contrast to the performance curves in Figures 11 and 12. The reason for the curve coincidence in Figure 13 is twofold. First, we have already discussed that the performance difference between DCOOFDM and ACOOFDM is less when the OSNR constraint dominates, which is the case for Figure 13. Second, the suddenness of the convergence of the two curves can be explained by the fact that with only OSNR constraint, there will be more flexibility in the signal optimization to adjust the clipping ratio and biasing ratio to achieve the best performance. That means that the achievable data rates in the middleOSNR region (5–22 dB) are improved significantly compared with Figures 11 and 12. However, for highOSNR region (greater than 22 dB), since the nonlinear distortion is negligible, the improvement becomes less pronounced compared to Figures 11 and 12. Therefore, the transition from the middleOSNR region to the highOSNR region will become sharper with only an OSNR constraint.
Conclusions
In this article, we analyzed the performance of the DCOOFDM and ACOOFDM systems in terms of EVM, SDR, and achievable data rates under both the average optical power and dynamic optical power constraints. We numerically calculated the EVM and compared with the corresponding lower bound. Both the theory and the simulation results showed that ACOOFDM can achieve the EVM lower bound. We derived the achievable data rates for AWGN channel as well as frequencyselective channel scenarios. We investigated the tradeoff between the optical power constraint and distortion. We analyzed the optimum clipping ratio and biasing ratio and compared the performance of two optical OFDM techniques. Numerical results showed that DCOOFDM outperforms the ACOOFDM for all the optical power constraint scenarios.
Appendix
Proof that ${c}^{\star}\left[n\right]={x}^{\left(A\right)}\left[n\right]{\stackrel{\u0304}{x}}^{\left(A\right)}\left[n\right]$ is optimum for Equation (51)
Denote by u(c[n]) the objective function for the problem in (51):
and denote by g_{ i }(c[n]) the i th constraint function for the problem in (51):
Let μ_{ i } denote the i th Kuhn–Tucker (KT) multiplier. We inter that
Next, we prove that ${c}^{\star}\left[n\right]={x}^{\left(A\right)}\left[n\right]{\stackrel{\u0304}{x}}^{\left(A\right)}\left[n\right]$ satisfies the KT conditions [28].
Stationarity
Primary feasibility
Dual feasibility
Complementary slackness
Substituting c^{⋆}[n] into Equation (89), we obtain,
In order to satisfy all the other conditions (90)–(92), we can choose μ_{ i } as follows

(1)
if ${x}^{\left(A\right)}\left[i\right]={\stackrel{\u0304}{x}}^{\left(A\right)}\left[i\right]$,
$$\phantom{\rule{2em}{0ex}}{\mu}_{i}={\mu}_{i+N/2}=0;$$(94)

(2)
if x ^{(A)}[i]>2γσ and ${\stackrel{\u0304}{x}}^{\left(A\right)}\left[i\right]=2\mathrm{\gamma \sigma}$,
$$\phantom{\rule{2em}{0ex}}{\mu}_{i}=2{\stackrel{\u0304}{x}}^{\left(A\right)}\left[i\right],\phantom{\rule{1em}{0ex}}{\mu}_{i+N/2}=2{x}^{\left(A\right)}\left[i\right];$$(95)

(3)
if x ^{(A)}[i]<0 and ${\stackrel{\u0304}{x}}^{\left(A\right)}\left[i\right]=0$,
$$\phantom{\rule{2em}{0ex}}{\mu}_{i}=2{x}^{\left(A\right)}\left[i\right],\phantom{\rule{1em}{0ex}}{\mu}_{i+N/2}=2{\stackrel{\u0304}{x}}^{\left(A\right)}\left[i\right].$$(96)
Therefore, there exits constants μ_{ i } (i = 0, 1, …, N  1) that make ${c}^{\star}\left[n\right]={x}^{\left(A\right)}\left[n\right]{\stackrel{\u0304}{x}}^{\left(A\right)}\left[n\right]$ satisfy the KT conditions. It was shown in [29] that if the objective function and the constraint functions are continuously differentiable convex functions, KT conditions are sufficient for optimality. It is obvious that u and g are all continuously differentiable convex functions. Therefore, c^{⋆}n is optimal for the minimization problem (51).
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This study was supported in part by the Texas Instruments DSP Leadership University Program.
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Yu, Z., Baxley, R.J. & Zhou, G.T. EVM and achievable data rate analysis of clipped OFDM signals in visible light communication. J Wireless Com Network 2012, 321 (2012). https://doi.org/10.1186/168714992012321
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Keywords
 Orthogonal frequency division multiplexing (OFDM)
 Visible light communications (VLC)
 DCbiased optical OFDM (DCOOFDM)
 Asymmetrically clipped optical OFDM (ACOOFDM)
 Error vector magnitude (EVM)
 Achievable data rate
 Clipping