## Abstract

Direct-current-biased optical orthogonal frequency-division multiplexing (DCO-OFDM) is widely used in high-speed visible light communication (VLC). Due to the limited dynamic range of light-emitting diode (LED) and the unipolarity for the intensity modulation (IM), double-sided clipping is inevitably imposed on the time-domain signal in VLC OFDM systems. Consequently, it calls for proper DCO-OFDM signal shaping by selecting an appropriate bias and time-domain signal power to reduce the clipping distortion and achieve a higher transmission rate. In this paper, we deep dive into the signal shaping design problem for double-sided clipping DCO-OFDM over both flat and dispersive channels. We derive the optimal bias for flat and dispersive channels, and explain its optimality from the perspectives of effective signal-to-noise ratio (SNR) and information theory. We then analytically characterize the optimal power for flat channels and propose a useful algorithm for dispersive channels enlightened by the optimal solution to the flat case. Furthermore, we uncover an inherent relationship between the considered double-sided clipping and the downside-clipping only DCO-OFDM regarding signal shaping optimization, and develop an in-depth understanding of the impact of top clipping based on the established connection. Practical simulations are provided to validate the superiority of our proposed signal shaping over the existing shaping schemes.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Visible light communication (VLC) has emerged as a prospective transmission candidate for the sixth generation (6G) [1]. Thanks to its low cost, high security, simple implementation, and license-free spectrum, VLC is deemed to be a critical enabler for Tbps data transmissions for diverse indoor applications and a valuable complement for conventional radio frequency (RF) communication technologies [2,3]. Usually, VLC employs intensity modulation with direct detection (IM/DD) to convey information. The light-emitting diode (LED) modulates the transmitted bitstream on the emission intensity at the transmitter, and the photodiode (PD) directly detects the luminous intensity at the receiver to recover the emitted waveform [4].

To fulfill very high-speed transmissions, optical orthogonal frequency-division multiplexing (O-OFDM) has been widely applied in broadband VLC systems, since it can effectively eliminate inter-symbol interference (ISI) caused by the channel dispersion [5]. Because of the IM process, the information carried by the intensity waveform has to be real-valued and non-negative. Therefore, various O-OFDM schemes, such as direct-current-biased optical OFDM (DCO-OFDM) [5,6], asymmetrically clipped optical OFDM (ACO-OFDM) [7], multi-band OFDM [8], serial-complex-valued OFDM (SCV-OFDM) [9,10], etc, were proposed to convert the bipolar signals into the unipolar ones. Among them, DCO-OFDM is simple to implement and can fully utilize (almost) all available subcarriers, and thus has been widely adopted for lab-built demonstrations and practical VLC systems [11,12].

The unipolarity constraint of intensity modulation and the limited dynamic range of LED are the two primary reasons leading to the double-sided clipping distortion on the transmitted signal [13,14]. In DCO-OFDM, we often add an extra bias into the transmitted OFDM signal. Owing to the high peak-to-average power ratio (PAPR) of OFDM signals [15], the biased signal of DCO-OFDM is downside clipped to guarantee the waveform nonnegativity. Meanwhile, it is constrained by the limited dynamic range of LED with signal peaks being clipped. Therefore, the signal shaping design on how to set bias and power arises in DCO-OFDM VLC systems. The benefits from proper biasing and clipping for VLC OFDM have been firmly supported by experimental results [16,17].

In principle, a high bias can reduce the downside clipping but is power inefficient since it carries no information. Meanwhile, an excessive bias may result in severe upside clipping due to the saturation characteristics of LED. Also, the clipping distortion is highly related to the signal power, or equivalently the scaling gain. An oversized signal power may increase the distortion, whereas an undersized signal power reduces the nonlinear clipping but may degrade the transmission quality. Consequently, in the signal shaping design, we shall consider not only the bias level but also the subcarrier power allocation in accord with the VLC channel nonlinearity and quality.

#### 1.1 Related works

So far, a lot of works have contributed to the signal shaping problem for DCO-OFDM regarding bias and power. The authors in [18] modeled the clipping process from a statistical perspective, and then did a series of works on the signal shaping design based on the clipper model through numerical approaches [19–21]. The authors in [22] claimed the signal shaping optimization to be nonconvex and derived the optimal bias via minimizing the mean square error (MSE) caused by clipping. In [23], the authors obtained an approximated (but not optimal as claimed) solution under the insignificant clipping assumption. The authors in [24] proposed a low-complexity adaptive scheme to dynamically update bias and power in accord with OFDM symbols. In [25], the authors fixed the bias and only optimized the time-domain signal power. The LDPC-coded DCO-OFDM was investigated in [26], and a feed-forward neural network was proposed to assist in decoding in [27,28]. More recently, the signal shaping has also been considered for DCO-OFDM variants, such as absolute valued DCO-OFDM (AV-DCO-OFDM) [29], non-orthogonal multiple access with DCO-OFDM (NOMA-DCO-OFDM) [30], asymmetrically clipped DCO-OFDM (ADO-OFDM) [31].

As another line of research, studies [32–34] ignored the upside clipping and focused on the downside distortion. The authors in [34] maximized the effective signal-to-noise ratio (SNR) for the optimal bias via the golden section search. In [32,33], the authors provided a globally optimal solution in a closed form for flat channels and proposed effective algorithms for dispersive cases. As a framework, the optimal signal shaping was further expanded into the orthogonal frequency division multiplexing access (OFDMA) [35] and multi-LED systems [36]. Note that these signal shaping schemes are only suitable for the single-sided clipping case, while it is questionable to ignore the upside clipping directly in practical system design.

In summary, we can see the signal shaping problem has been studied by many works and thus is vital to VLC OFDM. However, there still exist several important but unsolved problems. First, although the optimal signal shaping under single-sided clipping was fully characterized, the double-sided clipping case remains open, to the best of our knowledge. Second, most existing works often introduced approximations and heavily relied on numerical methods, heuristic approach, or even simulations because of the nonconvex nature, and thus the optimality of these schemes can hardly be guaranteed. As a direct result, there is a lack of insights and theoretical supports to guide practical system designs. Last but not least, the majority of literature only considered the simplest flat channel case. However, the channel dispersion should be taken into account in practice, since VLC channels often exhibit a static low-pass property [37,38].

#### 1.2 Our contributions

In this paper, we study the signal shaping design for double-sided clipping DCO-OFDM to overcome the barriers mentioned above. Due to the existence of upside clipping, the double-sided clipping signal shaping optimization is much more challenging than the single-sided clipping one. Nevertheless, we will present the optimal signal shaping strategy for both flat and dispersive channels, fully supported by theoretical analysis. Through the derivations, we will provide design principles that are helpful in practice, and further evaluate the impact of upside clipping. The main contributions of this paper are listed as follows.

- • We give the optimal bias for both flat and dispersive channels, and explain the optimality from several different points of view.
- • We analytically characterize the globally optimal subcarrier power over flat channels.
- • We propose an efficient algorithm for the subcarrier power optimization over dispersive channels with guaranteed convergence.
- • We establish an equivalent relationship between the double-sided clipping and single-sided clipping DCO-OFDM signal shaping optimization, and further assess the impact of upside clipping via such an inner connection.
- • The effectiveness and superiority of our proposed signal shaping scheme are validated by simulation results.

## 2. Double-sided clipping DCO-OFDM

#### 2.1 System model

A typical DCO-OFDM system diagram with $2N$ subcarriers is depicted in Fig. 1. At the transmitter, the bitstream is modulated into complex-valued symbols, denoted by $S_{k}$, for subcarrier $k=1,\ldots ,N-1$. Define the power of subcarrier $k$ as $p_{k}\triangleq E\left [|S_{k}|^{2}\right ]$, where $E\left [\cdot \right ]$ represents the expectation operator. To generate the real-valued time-domain signal, the frequency-domain symbols $\left \{ S_{k}\right \} _{k=1}^{N-1}$ are expanded into $\left \{ S_{k}\right \} _{k=0}^{2N-1}$ according to the Hermitian symmetry $S_{k}=S_{2N-k}^{*}$ for $k=1,2,\ldots ,N-1$, and the subcarriers 0 and $N$ are set to $S_{0}=S_{N}=0$. The time-domain signal $s_{n}$ is then obtained from the inverse fast Fourier transform (IFFT) of $S_{k}$. In most practical OFDM systems (as long as $2N\geq 64$ [39]), $s_{n}$ can be approximated by a zero-mean Gaussian distribution with variance $p_{s}$ according to the central limit theorem (CLT). According to Parseval’s theorem [40], we have $\sum _{k=1}^{N-1}p_{k}/N=p_{s}$.

Because of the IM in VLC, the intensity signal is required to be nonnegative. Hence, we need to add a proper bias $b$ into the time-domain signal $s_{n}$ and clip the negative peaks to obtain unipolar signals and drive the LED. (In practice, the input signal should be larger than a positive “turn-on” value $A_{tov}$.) Meanwhile, the time-domain signal $s_{n}$ is also upper bounded by a saturation point $A_{sat}$. Therefore, after proper pre-distortion, we can use a double-sided clipper to model the nonlinear impact in the time domain:

Through the double-sided clipper, the probability density function (PDF) of the clipped signal $s_{clip,n}$ can be expressed as

At the receiver, the PD captures the intensity waveform and converts it into the electrical signal $y_{n}$. Usually, the PD operates in a narrow range so that the distortion is not as significant as the LED [13]. The frequency-domain symbols $Y_{k}$ can be obtained from $y_{n}$ through fast Fourier transform (FFT). In the frequency domain, the VLC channel can be modeled as

where $H_{k}$ and $S_{clip,k}$ are the channel frequency response and the transmitted signal at subcarrier $k$, and the noise $N_{k}$ obeys a complex white Gaussian with zero mean and variance $p_{n}$ [4]. The received data are then recovered by demodulating $Y_{k}$.#### 2.2 Double-sided clipping model

Apparently, the nonlinear clipping distortion is a substantial obstacle in the VLC system design and makes the signal shaping problem intractable to analyze. Before investigating the signal shaping design, we would like to introduce the double-sided clipping model as follows.

**Lemma 1.** *[18] The double-sided clipping process can be modeled in the frequency domain by*

*where*$a$

*is the attenuation factor given by*$a\left (b,p_{s}\right )\triangleq Q\left (-\frac {b}{\sqrt {p_{s}}}\right )-Q\left (\frac {A-b}{\sqrt {p_{s}}}\right )$

*,*$N_{clip,k}$

*is the clipping noise following complex Gaussian distribution with zero mean and variance*$p_{c}\left (b,p_{s}\right )$

*, expressed as*

#### 2.3 Signal shaping design

Our work aims to investigate the DCO-OFDM signal shaping under double-sided clipping by finding an appropriate bias $b$ and subcarrier power allocation $\left \{ p_{k}\right \} _{k=1}^{N-1}$. We adopt the achievable rate $r\left (b,\left \{ p_{k}\right \} \right )$ of all available subcarriers as a key criterion to assess the system performance, which can be expressed in terms of bias $b$ and subcarrier power $\left \{ p_{k}\right \} _{k=1}^{N-1}$ from the signal model Eq. (6):

## 3. Bias optimization

We start from the optimization of the bias $b$. Since the bias is not included in the constraint, we can focus on the objective in $\mathbb {P}$. Observe that, for any fixed power allocation $\left \{ p_{k}\right \}$, there exists a symmetric relation in the objective with respect to $b$, given in Proposition 1.

**Proposition 1.** $r\left (b,\left \{ p_{k}\right \} \right )=r\left (A-b,\left \{ p_{k}\right \} \right )$ *always holds for any given* $\left \{ p_{k}\right \}$ and $\left \{ H_{k}\right \}$.

*Proof*. Observing the expressions of $a\left (b,p_{s}\right )$ and $p_{c}\left (b,p_{s}\right )$, we have $a\left (b,p_{s}\right )=a\left (A-b,p_{s}\right )$ and $p_{c}\left (b,p_{s}\right )=p_{c}\left (A-b,p_{s}\right )$. Based on Eq. (8), we can obtain $\textrm {SNDR}_{k}\left (b,p_{s}\right )=\textrm {SNDR}_{k}\left (A-b,p_{s}\right )$ for each subcarrier, and therefore, the relationship $r\left (b,\left \{ p_{k}\right \} \right )=r\left (A-b,\left \{ p_{k}\right \} \right )$ holds for any given $\left \{ p_{k}\right \}$ and $\left \{ H_{k}\right \}$.

Proposition 1 indicates that both $\textrm {SNDR}_{k}\left (b,p_{s}\right )$ and $r\left (b,\left \{ p_{k}\right \} \right )$ are symmetric about $b=A/2$ for both flat and dispersive channels. The symmetric property inspires us to investigate the value of the objective $r\left (b,\left \{ p_{k}\right \} \right )$ at $b=A/2$. The following theorem reveals that the objective $r\left (b,\left \{ p_{k}\right \} \right )$ is maximized at $b=A/2$ within the dynamic range $\left [0,A\right ]$.

**Theorem 1.** *For any given* $\left \{ p_{k}\right \}$*,* $b^{\star }=A/2$ *is the unique maximizer to the objective* $r\left (b,\left \{ p_{k}\right \} \right )$ *within the dynamic range* $\left [0,A\right ]$.

*Proof*. Now consider two DCO-OFDM systems with bias $A/2-\varepsilon$ and $A/2+\varepsilon$, respectively. As a direct result of $r\left (b,\left \{ p_{k}\right \} \right )=r\left (A-b,\left \{ p_{k}\right \} \right )$ from Proposition 1, these two systems have the same rate. Hence, we have that $\frac {\partial }{\partial b}r\left (b,\left \{ p_{k}\right \} \right )|_{b=\frac {A}{2}}=0$ by letting $\varepsilon \rightarrow 0$, i.e., $b^{\star }=A/2$ is a stationary point of $r\left (b,\left \{ p_{k}\right \} \right )$. By checking the partial derivative of $a\left (b,p_{s}\right )$ with respect to $b$, we can obtain that

Theorem 1 suggests that $b^{\star }=A/2$, the midpoint of the dynamic range, can maximize the achievable rate of double-sided clipping DCO-OFDM. The optimality of $b^{\star }=A/2$ proved by Theorem 1 does not depend on the channel type (flat or dispersive), the channel quality, or even the subcarrier power allocation $\left \{ p_{k}\right \}$. The optimal bias $b^{\star }=A/2$ also minimizes the clipping probability $Q\left (\frac {A-b}{\sqrt {p_{s}}}\right )+Q\left (\frac {b}{\sqrt {p_{s}}}\right )$ of $s_{n}$. Therefore, it does accord with our intuition that $b^{\star }=A/2$ can fully utilize the dynamic range $\left [0,A\right ]$ and reduce the nonlinear distortion at the same time. Furthermore, let us look at the optimal bias from the perspective of information theory.

**Proposition 2.** *Define* ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$ *as the mutual information of a double-sided clipping DCO-OFDM with bias* $b$ *and subcarrier power* $\left \{ p_{k}\right \}$*. Given a VLC channel, we always have*

*and*$b^{\star }=A/2$

*is a local maximum of the mutual information*${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$.

*Proof*. Consider two double-sided clipping DCO-OFDM systems with the dynamic range $[0,A]$. The same time-domain signal $s_{n}$ is biased by $b$ and $A-b$, respectively, and clipped in two approaches, given by

Furthermore, similar to the proof of Theorem 1, we can obtain $\frac {\partial }{\partial b}{\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )|_{b=\frac {A}{2}}=0$ from the symmetric relation, and hence $b^{\star }=A/2$ is a stationary point. By checking the second partial derivative, we further have $\frac {\partial ^{2}}{\partial b^{2}}{\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )|_{b=\frac {A}{2}}<0$, and thus $b^{\star }=A/2$ is also a local maximum point.

From Proposition 2, the relationship ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )={\cal I}\left (y_{n}^{\prime };s_{clip,n}^{\prime }|A-b,\left \{ p_{k}\right \} \right )$ indicates that the mutual information in terms of bias is symmetric about $b=A/2$. We should not take such a property for granted, since a linear transformation usually affects the value of the mutual information in the presence of nonlinear distortions. The numerical results in Fig. 2(b) are in agreement with Theorem 1 and Proposition 2. From the figure, we would like to conjecture that $b^{\star }=A/2$ is not only a local maximum but a global maximizer of ${\cal I}\left (y_{n};s_{clip,n}|b,\left \{ p_{k}\right \} \right )$.

The midpoint $A/2$ has been adopted as the optimal bias in several existing works [23,25,28]. Usually, it is used directly without sufficient theoretical support. In our work, we explain the optimality of the midpoint from the perspective of not only SNDR and also mutual information. Therefore, $b^{\star }=A/2$ is also suitable for DCO-OFDM with channel coding, which can serve as a theoretical foundation for the existing channel coding works [26,28].

According to the PDF of $s_{clip,n}$ in Eq. (2), we have the emitted optical power $E\left [s_{clip,n}\right ]=A/2$, as long as the optimal bias $b^{\star }=A/2$ is adopted. As a result, the average optical power is constant with the optimal signal shaping, even if the VLC channel quality is fluctuating. Therefore, the optimal signal shaping can still guarantee a constant illumination intensity for dual-functional VLC applications.

## 4. Subcarrier power optimization

#### 4.1 Flat channel

Now we consider the subcarrier power allocation $\left \{ p_{k}\right \}$ for the double-sided clipping DCO-OFDM, which is more difficult than the bias optimization and has never been well characterized before. Let us start from the flat channel case, where the channel frequency response satisfies $|H_{k}|=|H|$. Obviously, the uniform subcarrier power allocation $p_{k}=\frac {N}{N-1}p_{s}$ is optimal over flat channels. Now we would like to select the optimal signal power $p_{s}$ to maximize the achievable rate. We define the optical SNR as $\gamma \triangleq \frac {|H|^{2}E^{2}\left [s_{clip,n}\right ]}{p_{n}}$ to evaluate the VLC channel quality. Since the optimal bias is $b^{\star }=A/2$, we thus have the optical SNR $\gamma =\frac {|H|^{2}A^{2}}{4p_{n}}$. Now we characterize the optimal power $p_{s}^{\star }$ in the following theorem.

**Theorem 2.** *Problem* $\mathbb {P}$ *admits a unique optimal power* $p_{s}^{\star }\left (\gamma \right )$*, given by the unique root of an increasing function* $f\left (z;\gamma \right )\triangleq \phi \left (\frac {A}{2\sqrt {z}}\right )-\frac {A}{2\sqrt {z}}Q\left (\frac {A}{2\sqrt {z}}\right )-\frac {A}{4\gamma \sqrt {z}}$.

*Proof*. Taking the derivative of $\textrm {SNDR}_{k}\left (A/2,p_{s}\right )$ with respect to $p_{s}$ yields

Theorem 1 reveals the quasi-concavity of the objective $r\left (A/2,\left \{ p_{k}\right \} \right )$ in $\mathbb {P}$, and $p_{s}^{\star }\left (\gamma \right )$ is the unique maximizer of $r\left (A/2,\left \{ p_{k}\right \} \right )$, given by the root of the increasing function $f\left (z;\gamma \right )$. In principle, we can convey more information by using a larger signal power $p_{s}$ to combat the background noise. However, an excessive signal power $p_{s}$ results in more severe clipping distortion. Therefore, there exists a trade-off between clipping distortion and background noise. $\left \{ p_{k}^{\star }\right \} =\frac {N}{N-1}p_{s}^{\star }\left (\gamma \right )$ pointed out by Theorem 2 is the globally optimal subcarrier power to balance the trade-off and maximize the achievable rate of double-sided clipping DCO-OFDM. Since $f\left (z;\gamma \right )$ is an increasing function in $z$, the optimal power $p_{s}^{\star }\left (\gamma \right )$ can be found via efficient algorithms such as Newton method and bisection method [41]. Now we will reveal more properties of $p_{s}^{\star }\left (\gamma \right )$ in Proposition 3.

**Proposition 3.** *The optimal power* $p_{s}^{\star }\left (\gamma \right )$ *is monotonically decreasing in* $\gamma$.

*Proof*. From the relationship $f\left (p_{s}^{\star }\left (\gamma \right );\gamma \right )=0$, we have $\frac {dp_{s}^{\star }\left (\gamma \right )}{d\gamma }=-\frac {2p_{s}^{\star }\left (\gamma \right )}{2\gamma ^{2}Q\left (\frac {A}{2\sqrt {p_{s}^{\star }\left (\gamma \right )}}\right )+\gamma }<0$, and thus the monotonic relationship is proved.

Proposition 3 yields a fundamental principle for signal shaping design, in accord with intuition. At high optical SNR $\gamma$, we shall set a smaller $p_{s}$ to reduce the clipping distortion. At low optical SNR $\gamma$, the background noise becomes the dominant factor, and thus we should use a larger signal power $p_{s}$. Hence, $p_{s}^{\star }\left (\gamma \right )$ turns out to be the optimal point to balance the impact (or the ratio) of clipping distortion and background noise.

From Theorems 1 and 2, we have obtained and proved $\left (b^{\star }=A/2,\left \{ p_{k}^{\star }\right \} =\frac {N}{N-1}p_{s}^{\star }\left (\gamma \right )\right )$ to be the optimal signal shaping for flat channels, where the optimal bias is constant, and the optimal power can be easily obtained based on the monotonicity of $f\left (z;\gamma \right )$. Therefore, the proposed signal shaping scheme $\left (b^{\star },\left \{ p_{k}^{\star }\right \} \right )$ is simple to implement and can provide useful insights for the practical DCO-OFDM design.

#### 4.2 Dispersive channel

The dispersive channel case, where the channel frequency response $H_{k}$ is not the same for different subcarriers, is more complicated and problematic. That is one possible reason that most existing works simply ignored this vital case. Above all, the uniform power allocation is no longer optimal in the dispersive case. To tackle this more general case, we borrow the principle from the block successive upper-bound minimization (BSUM) algorithm [42] by constructing a sequence of tractable functions serving as lower bounds of the objective, and further iteratively optimize these bounds enlightened by the solution in the flat case.

First of all, from Theorem 1, the midpoint of the limited dynamic range $b^{\star }=A/2$ is always optimal, even for dispersive channels. Define $\gamma _{k}=\frac {|H_{k}|^{2}A^{2}}{4p_{n}}$ as the optical SNR at subcarrier $k$. We further introduce $u_{k}\triangleq p_{k}/p_{s}$ as the normalized power at subcarrier $k$ such that the subcarrier power $\left \{ p_{k}\right \}$ can be decomposed into the normalized subcarrier power allocation $\left \{ u_{k}\right \}$ and the signal power $p_{s}$. From the relationship $\left \{ p_{k}\right \} =p_{s}\left \{ u_{k}\right \}$, the achievable rate $r\left (b^{\star },\left \{ p_{k}\right \} \right )$ can be rewritten as

Now we construct $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ as a lower bound of $r\left (b^{\star },p_{s},\left \{ u_{k}\right \} \right )$, given by

**Theorem 3.** *For given* $p_{s}^{\prime }$, $\left \{ u_{k}\right \}$*, and* $b^{\star }=A/2$, $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ *is a quasi-concave function of* $p_{s}$ *admitting a unique maximizer* $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$*, given by the unique root of an increasing function* ${ f_{d}\left (z;\left \{ \gamma _{k}\right \} \right )}=\sum _{k=1}^{N-1}\frac {u_{k}\textrm {SNDR}_{k}^{2}\left (b^{\star },p_{s}^{\prime }\right )}{1+u_{k}\textrm {SNDR}_{k}\left (b^{\star },p_{s}^{\prime }\right )}f\left (z;\gamma _{k}\right )$.

*Proof*. Taking the derivative of $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ with respect to $p_{s}$ yields

From Theorem 3, the lower bound $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ is also a quasi-concave function with the unique maximizer of $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$, given by the root of the increasing function $f_{d}\left (z;\left \{ \gamma _{k}\right \} \right )$. Observe that $f_{d}\left (z;\left \{ \gamma _{k}\right \} \right )$ is the weighted sum of $f\left (z;\gamma _{k}\right )$ for all the available subcarriers. Since $f\left (z;\gamma _{k}\right )$ determines the optimal power $p_{s}^{\star }\left (\gamma _{k}\right )$ for a flat channel at optical SNR $\gamma _{k}$, Theorem 3 hence provides an insight that, $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ as the maximizer of $r_{lb}\left (p_{s};b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ for a dispersive channel, is the optimal solution for a flat channel whose optical SNR is a specific weighted sum of each subcarrier SNR $\left \{ \gamma _{k}\right \}$. In this sense, the process to obtain $p_{s}^{*}\left (b^{\star },p_{s}^{\prime },\left \{ u_{k}\right \} \right )$ is equivalent to deriving the optimal power by flattening the dispersive channel properly.

Decoupling $\left \{ p_{k}\right \}$ into $\left \{ u_{k}\right \}$ and $p_{s}$, we can optimize the normalized subcarrier power allocation $\left \{ u_{k}\right \}$ for a fixed $p_{s}$ via the well-known waterfilling algorithm. Given $p_{s}^{\prime }$, the expression of $\left \{ u_{k}^{*}\left (p_{s}^{\prime }\right )\right \}$ can be written as

Now we can optimize $\left \{ u_{k}\right \}$ and $p_{s}$ alternatively to obtain a suitable signal shaping for a dispersive channel. The procedure is summarized in Algorithm 1. In the $i$th iteration, given $\left \{ u_{k}^{i}\right \}$ and $p_{s}^{i}$, we can maximize the lower bound $r_{lb}\left (p_{s};b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$ to get $p_{s}^{i+1}\left (b^{\star },p_{s}^{i},\left \{ u_{k}^{i}\right \} \right )$ based on Theorem 3, and then update $\left \{ u_{k}^{i+1}\left (p_{s}^{i+1}\right )\right \}$ via waterfilling in Eq. (11). Hence, we can obtain a sequence of nondecreasing objective values:

## 5. Impact of upside clipping

If we ignore the upside clipping and only consider the downside clipping, the optimal signal shaping has already been well characterized in [33], summarized in Lemma 2.

**Lemma 2.** *[33] For a single-sided clipping DCO-OFDM at optical SNR* $\gamma \triangleq \frac {|H|^{2}E^{2}\left [s_{clip,n}\right ]}{p_{n}}=\frac {|H|^{2}I^{2}}{p_{n}}$ *with the optical power budget* $E\left [s_{clip,n}\right ]=I$*, the optimal bias is* $b^{\star }=I(1-\gamma ^{-1})$ *and the optimal subcarrier power is* $p_{k}^{\star }=\frac {N}{N-1}p_{s}^{\star }=\frac {N}{N-1}\left (\frac {I}{\phi (x^{\star })-x^{\star }Q(x^{\star })}\right )^{2}$ *for* $k=1,\ldots ,N-1$*, where* $x^{\star }$ *is the unique solution to the equation* $\left (1-\gamma ^{-1}\right )\left (\phi (x)-xQ(x)\right )+x=0$.

Since we have developed the optimal shaping scheme for double-sided clipping DCO-OFDM, now we can establish an equivalent relationship between double-sided and single-sided clipping DCO-OFDM, in the sense that they have the same optimal shaping parameters $\left (b^{\star },\left \{ p_{k}^{\star }\right \} \right )$.

**Theorem 4.** *A double-sided clipping DCO-OFDM at optical SNR* $\gamma$ *with the top clipping position* $A$ *is equivalent to a single-sided clipping DCO-OFDM at optical SNR* $2\gamma +1$ *constrained by optical power budget* $\frac {A}{2}\left (1+\frac {1}{2\gamma }\right )$ *in the sense that they have the identical optimal bias and subcarrier power* $\left (b^{\star },\left \{ p_{k}^{\star }\right \} \right )$.

*Proof*. We first prove they have the same optimal bias $b^{\star }$. According to Lemma 2, the optimal bias in the single-sided clipping DCO-OFDM is given by $b_{sgl}^{\star }=\frac {A}{2}\left (1+\frac {1}{2\gamma }\right )\left (1-\frac {1}{2\gamma +1}\right )=\frac {A}{2}$. According to Theorem 1, the optimal bias in the double-sided clipping DCO-OFDM is given by $b_{dbl}^{\star }=\frac {A}{2}$. Therefore, they have the exactly same optimal bias $b^{\star }$ under the given conditions.

We further prove they have identical optimal subcarrier power $\left \{ p_{k}^{\star }\right \}$. Both adopt uniform power allocation $p_{k}^{\star }=\frac {N}{N-1}p_{s}^{\star }$ for flat channels, and thus we focus on $p_{s}^{\star }$. According to Lemma 2, the optimal power in the single-sided clipping DCO-OFDM is given by $p_{s,sgl}^{\star }=\left (-\frac {A}{2x_{sgl}^{\star }}\left (1+\frac {1}{2\gamma }\right )\left (1-\frac {1}{2\gamma +1}\right )\right )^{2}=\left (\frac {A}{2x_{sgl}^{\star }}\right )^{2}$, where $x_{sgl}^{\star }$ is the unique solution to the equation $\left (1-\frac {1}{2\gamma +1}\right )\left (\phi (x)-xQ(x)\right )+x=0$. Recall that in Section 4.1, the optimal power in the double-sided clipping DCO-OFDM is given by $p_{s,dbl}^{\star }=\left (\frac {A}{2x_{dbl}^{\star }}\right )^{2}$, where $x_{dbl}^{\star }$ is also the unique solution to the equation $\left (1-\frac {1}{2\gamma +1}\right )\left (\phi (x)-xQ(x)\right )+x=0$. Therefore, they have the same optimal $p_{s}^{\star }$ due to $x_{sgl}^{\star }=x_{dbl}^{\star }$ and also the identical optimal subcarrier power $\left \{ p_{k}^{\star }\right \}$.

Theorem 4 indicates that a double-sided clipping DCO-OFDM is equivalent to a single-sided clipping DCO-OFDM under some specific conditions. From the inner connection established by Theorem 4, we can assess the impact of upside clipping on DCO-OFDM. First, in a single-sided clipping DCO-OFDM with no power constraint, the optimal bias should be high enough to avoid any possible downside clipping, leading to an infinite emitted optical power. Meanwhile, this case will never happen for a double-sided clipping DCO-OFDM, because the upside clipping limits the maximum amplitude of the time-domain signal and thus constrains the average optical power to be less than $A$ as an implicit average optical power constraint. In some sense, the upside clipping plays a partial role of the optical power constraint.

Second, when the two considered systems are equivalent to each other with the same optimal shaping scheme, the double-sided clipping system requires almost half the optical SNR of the single-sided clipping one ($\gamma$ versus $2\gamma +1$). At the same time, the extra upside clipping makes the double-sided clipping DCO-OFDM suffer (roughly) twice nonlinear distortion than the single-sided clipping system. Therefore, these two systems have very close or even the same ratios between clipping distortion and background noise, which may explain why the same optimal bias and power should be adopted (see the remarks in Section 4.1).

Third, the emitted optical power of the double-sided clipping DCO-OFDM is always $\frac {A}{2}$, whereas that of a single-sided one is $\frac {A}{2}\left (1+\frac {1}{2\gamma }\right )$. Even though they have the same shaping scheme, the double-sided clipping DCO-OFDM always has a lower illumination level, because the time-domain signal is clipped at the top.

Technically, Theorem 4 also provides a new approach to address a double-sided clipping DCO-OFDM by transforming it into a single-sided clipping system. Hence, the methods and insights in the single-sided clipping DCO-OFDM are also available to address this new problem. By establishing such an equivalent relationship, Theorem 4 gives us a new point of view to consider the signal shaping design and evaluate the impact of upside clipping.

## 6. Performance evaluation

#### 6.1 System setup and benchmarks

In this section, we present simulation results to assess the performance of our proposed signal shaping schemes. In the simulation, we use the single-color LED (Osram LE UW S2LN) as the source. According to the datasheet [43], the maximum and minimum forward currents are $A_{sat}=$800mA and $A_{tov}=$100mA, respectively, and thus the dynamic range $A$ is 700mA. The VLC channel bandwidth is set to 100 MHz with $2N=512$ subcarriers. We adopt the same setup as Configuration D in [37], with detailed parameters summarized in Table 1.

We adopt the practical VLC channel model introduced in [37,38] for the simulations. According to the Lambertian radiant model [38], the LOS impulse response $h^{(0)}(t;{\cal S},{\cal R})$ can be expressed as

For dispersive channels, we should take non-LOS paths into account. Given a particular source ${\cal S}$ and receiver ${\cal R}$ in a room with reflectors, the light finally arrives at the receiver after several reflections. Let $h^{(k)}(t;{\cal S},{\cal R})$ represent the response of the light undergoing $k$ reflections. Denote

Furthermore, we also consider the spectral response of the LED itself in the dispersive channel model. We adopt the LED impulse response modeled in [44], given by

where $f_{b}$ is the 3-dB modulation bandwidth and is set to 5 MHz [45] in our simulations.In related existing works, several efficient signal shaping schemes have already been proposed. For comparison, we provide them as benchmarks and the proposed scheme in the simulations, listed as follows:

- • Optimal: the optimal bias $b^{\star }$ and subcarrier power $\left \{ p_{k}^{\star }\right \}$ proposed in this work are adopted.
- • Uniform: the optimal $b^{\star }$ and $p_{s}^{\star }$ are adopted, but the subcarrier power is uniformly distributed instead of the optimal waterfilling, which is equivalent to the “Optimal” scheme for flat channels.
- • Single-sided: the optimal shaping scheme for single-sided clipping DCO-OFDM with the optical power constraint $A/2$ is obtained through Algorithms 1 and 3 in [33] for flat and dispersive channels, respectively.
- • Approximated: the signal shaping scheme is adopted according to [23] (flat channels only).
- • Fixed: biasing and powering are fixed to be given values.

#### 6.2 Simulations

Figure 3(a) demonstrates the performance of the optimal scheme and the benchmarks over flat channels. First of all, one can see that the proposed optimal scheme outperforms the other benchmarks. Both the “Approximated” and “Single-sided” schemes are suboptimal essentially and slightly worse than the optimal scheme. The signal shaping parameters $\left (b,\sqrt {p_{s}}\right )$ of these three schemes versus the optical SNR are plotted in Fig. 3(b). The proposed optimal scheme has the same bias and power as the results obtained from the exhaustive search. At low optical SNR, the two benchmarks have different shaping parameters from the optimal one. However, as the SNR increases, we surprisingly find that the bias and power of both the “Approximated” and “Single-sided” schemes approach to the optimal one. That explains why these benchmarks have a very close performance gap to the optimal one, although they are not exactly optimal.

Let us back to Fig. 3(a). All the fixed strategies exhibit relatively poor system performance. Among them, the performance of $\left (b=A/2,\sqrt {p_{s}}=A/2\right )$ is close to the proposed optimal scheme at low SNR $\gamma$; however, it saturates at high SNR $\gamma$, since the fixed $p_{s}$ is oversized and results in significant clipping distortion. It is consistent with Proposition 3 that a smaller $p_{s}^{\star }\left (\gamma \right )$ should be used at higher SNR $\gamma$. The scheme $\left (b=A/4,\sqrt {p_{s}}=A/8\right )$ is consistently worse than the fixed scheme $\left (b=A/2,\sqrt {p_{s}}=A/8\right )$ with the same power, implying that the clipping distortion becomes more serious if the bias diverges from the optimal bias $b^{\star }=A/2$.

Figure 4(a) shows the achievable rate of different shaping schemes over dispersive channels. The proposed optimal shaping scheme is still effective and superior to the other benchmarks. The channel dispersion widens the performance gap between the “Single-sided” scheme and the optimal one, especially at high optical SNR $\gamma$. Similarly, the “Uniform” scheme shows a larger achievable rate gap to the optimal one, and the gap is reduced with the increase of the optical SNR $\gamma$ since the waterfilling tends to the uniform power allocation at high SNR. The “Fixed” strategies have constant bias and power independent of the channel quality, and still suffer severe performance degradation.

The optimal subcarrier power allocation for dispersive channels is obtained through an iterative process of Algorithm 1, as depicted in Fig. 4(b). In the figure, the maximum achievable rate and the optimal subcarrier power $\left \{ p_{k}^{\star }\right \}$ are obtained from the exhaustive search, where the 2-norm $||\cdot ||_{2}$ represents the distance between the output $\left \{ p_{k}^{i}\right \}$ and the optimal solution $\left \{ p_{k}^{\star }\right \}$. One can see that Algorithm 1 is valid for the signal shaping design over dispersive channels and has excellent convergence properties even the LED response is considered. The shaping schemes corresponding to different initial points have different transmission rates. However, after only a few iterations (less than 5), Algorithm 1 closes the performance gap to the exhaustive search. At the same time, the output of Algorithm 1 approaches to the optimal solution $\left \{ p_{k}^{\star }\right \}$ consistently and stably, even if starting from different initial points, which suggests that Algorithm 1 is insensitive to the initial point.

Figures 5(a) and 5(b) show the inner connection between the single-sided and double-sided clipping DCO-OFDM in terms of the optimal bias $b^{\star }$ and power $p_{s}^{\star }$, respectively. In the figures, we consider not only the single-color LED Osram LE UW S2LN but also another commercial RGBA LED (LZ4-00MA00), where the dynamic range $A$ of this LED is 900mA according to its datasheet [46]. The single-sided clipping DCO-OFDM is constrained by the average optical power $I=E\left [s_{clip,n}\right ]=$ 400mA, 450mA, and 500mA, respectively. We plot the optimal bias $b^{\star }$ and power $p_{s}^{\star }$ for the double-sided clipping DCO-OFDM based on the x-axis $\gamma _{dbl}$ on the bottom, and for the single-sided clipping DCO-OFDM based on the second x-axis $\gamma _{sgl}$ on the top, where $\gamma _{sgl}=2\gamma _{dbl}+1$. We note that the optimal bias of the double-sided clipping DCO-OFDM always equals to $b_{bdl}^{\star }=A/2$, and thus it is a straight line in Fig. 5(a). The double-sided and single-sided clipping DCO-OFDM are equivalent to each other at the intersection points. For example, the red star in Figs. 5(a) and 5(b) represents a double-sided clipping DCO-OFDM with dynamic range $A=700$mA at optical SNR $\gamma =1.15$ and a single-sided clipping DCO-OFDM under the average optical power constraint $I=500$mA at optical SNR $\gamma =3.3$ have the identical optimal bias $b^{\star }=$350mA and signal power $\sqrt {p_{s}^{\star }}=$735.5mA. Note that all these equivalent points do not mean that these two systems have the same achievable rate.

#### 6.3 Coded system

To be more practical, we illustrate the BER performance of the uncoded and coded DCO-OFDM systems in Fig. 6. The convolutional code (CC) and the Bose, Ray-Chaudhuri, Hocquenghem (BCH) code are used as the channel coding in Figs. 6(b), and 6(c), respectively. For the convolutional code, the constraint length is set to 7 with a $1/2$ code rate, and the Viterbi algorithm is adopted at the receiver. For the BCH code, the code length is set to 63 with the code rate $10/21$. The results in Fig. 6 are more convincing since they do not rely on the clipper model in Lemma 1, not to mention that the channel coding is considered. For an uncoded system with a given modulation, the BER minimization is equivalent to the rate maximization, and thus the proposed shaping schemes are still optimal. Furthermore, if we describe the BER of a coded system by $\textrm {BER}_{\textrm {coded}}=C\left (\textrm {BER}_{\textrm {uncoded}}\right )$, the optimality of the proposed shaping scheme can also be guaranteed, as long as $C(\cdot )$ is an increasing function in the considered SNR range. (Note that $C\left (\cdot \right )$ is just a simplified description of the impact of the channel coding.) Therefore, we can observe from all the figures that the optimal scheme outperforms the other benchmarks in both the uncoded and coded systems, consistent with the achievable rate in Fig. 3(a). The “Approximated” and “Single-sided” schemes have slight degradation regarding BER. We also consider the “Adaptive” scheme here, which is worse than the above three schemes but better than the fixed schemes. Not surprisingly, the fixed schemes have the worst performance. Therefore, we can conclude that the proposed optimal shaping does work in practical coded systems.

To be more intuitive, we visualize the constellation diagrams of different signal shaping schemes for uncoded DCO-OFDM in Fig. 7. From the figure, the “Optimal”, “Single-side”, and “Approximated” schemes have similar patterns on the diagrams, which explains why, in Fig. 6(a), their BERs are quite close. The received signal quality of the “Adaptive” scheme is slightly worse than the above three schemes. In the “Fixed” schemes, the received signals are blurred by the clipping distortion and deviate from the constellation points, resulting in the severe symbol errors.

## 7. Conclusions

In this paper, we investigated the optimum signal shaping for double-sided clipping DCO-OFDM in VLC. We formulated the signal shaping design as an optimization problem and decomposed it into two subproblems in terms of bias and subcarrier power. For bias optimization, we illustrated that the midpoint of the limited dynamic range of LED is globally optimal for both flat and dispersive channels, and explained its optimality from the perspective of both SNDR and mutual information. For power optimization, we gave a closed-form solution of the optimal subcarrier power in the flat case, and further provided an efficient algorithm for dispersive channels enlightened by the flat case solution. Based on the optimal solution, we further established an inherent relationship between the double-sided clipping and the single-sided clipping DCO-OFDM, and revealed the impact of upside clipping via the inner connection. At last, we validated our proposed signal shaping scheme via practical simulations.

## Funding

National Key Research and Development Program of China (2018YFB1801103); National Natural Science Foundation of China (61901111, 61971130, 61901110, 61720106003); Natural Science Foundation of Jiangsu Province (BK20190331, BK20160069); Jiangsu Province Basic Research Project (BK20192002); National Mobile Communications Research Laboratory, Southeast University (2019B02); Huawei Cooperation Project; Fundamental Research Funds for the Central Universities.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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